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A TEXT-BOOK OF 
 
 PRACTICAL PHYSICS 
 
BY THE SAME AUTHOR 
 
 A TEXT-BOOK OF PHYSICS 
 
 With 579 Illustrations and a Collection &f 
 Examples and Questions with Answers. 
 
 Large crown 8vo, 10s. 6d. 
 
 ELEMENTARY 
 PRACTICAL PHYSICS 
 
 With 120 Illustrations and 193 Exercises. 
 Crown 8vo, 2s. 6d. 
 
 LONGMANS, GREEN, AND CO. 
 
 LONDON, NEW YORK, AND BOMBAY 
 
A TEXT-BOOK OF 
 PRACTICAL PHYSICS 
 
 BY 
 
 WILLIAM WATSON 
 
 M 
 
 A.R.C.S., D.Sc. (LOND.), F.R.S. 
 
 LONGMANS, GREEN, AND CO. 
 39 PATERNOSTER ROW, LONDON 
 
 NEW YORK AND BOMBAY 
 1906 
 
 All rights reserved 
 

 
PREFACE 
 
 THE following pages are intended to serve as a book of refer- 
 ence to the student working in a physical laboratory. The 
 experiments described are not intended for a beginner, but are 
 suited for a student who has already spent a little time in the 
 laboratory and worked through a more elementary course 
 of experiments, such as those described in the author's 
 Elementary Practical Physics. 
 
 It is not intended, or expected, that any one class will 
 work through all the experiments described in this book, but 
 that the teacher will select those which he considers most 
 suitable, being guided by the requirements of his pupils and 
 the resources of his laboratory. It is, however, hoped that 
 teachers and students will find that all the experiments which 
 can be performed with advantage in a laboratory having the 
 ordinary equipment are described. 
 
 In almost every case the descriptions and hints apply to 
 any pattern of apparatus, no attempt being made to give 
 elaborate instructions for working some particular form of 
 instrument. It is hoped, however, that the figures will be 
 found of assistance by teachers when making the apparatus 
 needed to perform many of the experiments. The aim of 
 the book is to draw attention to those points which require 
 care, and to indicate the sources of error which are common 
 to all the instruments which are likely to be employed. It 
 is hoped that in this way the teacher will be relieved from 
 
vi PREFACE 
 
 that constant repetition of warnings and elementary discussion 
 of the errors with which the author has often found his time 
 employed, and will thus have more leisure to investigate the 
 peculiar difficulties of individual students, such as may be 
 typified by the " loose contact " et hoc genus omne. 
 
 It is hoped that the series of tables at the end will be 
 found of service, and every endeavour has been used to 
 eliminate errors. But there, as well as in the body of the 
 book, the author from bitter experience fears that mistakes 
 will have crept in, and he will be much obliged to any readers 
 who discover such if they will send him a note of them. 
 
r THE 
 UNIVERSITY 
 
 OF 
 
 ^*$* 
 
 CONTENTS 
 
 CHAPTER I 
 
 METHODS USED IN THE REDUCTION AND DISCUSSION OF 
 THE RESULTS OF PHYSICAL MEASUREMENTS 
 
 PAGES 
 
 1. Correct apportionment of the accuracy with which the different observations 
 are to be made in an experiment. 2. On the application of corrections to 
 observed quantities. 3. Use of tables and curves of corrections when reducing 
 observations. 4. Representation of a series of results. 5. To find an alge- 
 braical expression to represent a series of observations. 6. Mathematical 
 tables. 1. The slide rule. 8. Arithmometers. 9. Determination of the 
 area of plane figures. 10. The planimeter. 11. Harmonic analysis . . 1-40 
 
 CHAPTER II 
 MEASUREMENTS OF LENGTH 
 
 12. Standards of length. 13. The vernier. 14. The micrometer screw. 15. 
 The spherometer. 16. The micrometer screw gauge. 17. The travelling 
 microscope. 18. Illumination of the divisions of a standard scale. 19. The 
 comparator. 20. Correction of measurements of length to allow for the 
 effects of temperature. 21. The cathetometer. 22. Measurement of a ver- 
 tical length by means of a separate scale and reading telescopes. 23. The 
 optical lever. 24. To measure the radius of curvature of a spherical surface 
 with the optical lever 41-62 
 
 CHAPTER III 
 WEIGHING 
 
 25. The balance. 26. Weighing by the method of oscillation and to compare 
 the lengths of the arms of the balance. 27. Calibration of a set of weights. 
 28. Reductions of weighings to vacuo 63-78 
 
viii CONTENTS 
 
 CHAPTER IV 
 DENSITY 
 
 PAGES 
 
 29. Density. 30. The measurement of density corrections for temperature 
 of water and buoyancy of air. 31. Measurement of the density of a solid 
 heavier than water by the method of Archimedes. 32. Measurement of the 
 density of a solid lighter than water. 33. Measurement of the density of a 
 solid in the form of small pieces the pyknometer. 34. Measurement of the 
 density of a liquid with a sinker. 35. Measurement of the density of a liquid 
 with specific gravity bottle. 36. Hare's apparatus for the comparison of the 
 densities of liquids. 37. The hydrometer. 38. Nicholson's hydrometer. 
 39. Calibration of a burette . 79-< 
 
 CHAPTER V 
 ELASTICITY OF SOLIDS 
 
 40. Young's modulus by stretching a wire. 41. Young's modulus by the 
 flexure of a beam (1). 42. Young's modulus by the flexure of a beam (2). 
 43. Measurement of the period of an oscillating body by the eye and ear 
 method comparison of moment of inertias. 44. To calculate the simple 
 rigidity of a wire from observations of the period of a torsional pendulum. 
 45. Determination of Young's modulus and rigidity of a wire by Searle's 
 method . . . 99-117 
 
 CHAPTER VI 
 
 THE PENDULUM MEASUREMENT OF g" AND RATING 
 A CHRONOMETER 
 
 46. The pendulum. 47. Determination of "g" by means of a Borda pendulum. 
 48. To examine the laws of the compound pendulum. 49. Determination of 
 " g" by Kater's pendulum. 50. Determination of the rate of a clock by the 
 occultation of a star by a terrestrial obstacle. 51. The sextant and its adjust- 
 ment. 52. Determination of the error of a chronometer with the sextant . 118-138 
 
 CHAPTER VII 
 SURFACE TENSION AND VISCOSITY 
 
 53. Measurement of surface tension by the elevation in a capillary tube. 54. 
 Measurement of the surface tension of a soap film. 55. Measurement of the 
 viscosity of a liquid by its rate of flow through a capillary tube. 56. Measure- 
 ment of the logarithmic decrement of a vibrating body. 57. Measurement- of 
 the viscosity of a liquid by the oscillating disc method .... 139-154 
 
CONTENTS ix 
 
 CHAPTER VIII 
 THE BAROMETER 
 
 PAGES 
 
 58. Measurement of the atmospheric pressure ...... 155-161 
 
 CHAPTER IX 
 TH ERMOME TR Y 
 
 59. Calibration of the tube of a mercury thermometer. 60. Determination of 
 the upper fixed points of a mercury thermometer. 61. Determination of 
 the lower fixed point of a mercury thermometer. 62. Application of the 
 correction for fundamental interval. 63. Deternu'nation of the corrections to 
 a thermometer by comparison with a standard. 64. Auxiliary fixed points 
 transition temperatures. 65. Correction of thermometer reading for emergent 
 column. 66. Reduction of the readings of mercury thermometers to the 
 hydrogen or air scales. 67. Changes in zero of mercurial thermometers. 
 68. The movaUe zero method of using a mercury thermometer. 69. Measure- 
 ment of the depression in a mercury thermometer 162-186 
 
 CHAPTER X 
 EXPANSION OF SOLIDS AND LIQUIDS 
 
 70. Linear expansion. 71. Measurement of the linear expansion of a rod. 
 72. Cubical expansion. 73. The weight thermometer or dilatometer. 74. 
 Measurement of the cubical expansion of glass with a simple form of weight 
 dilatometer. 75. Measurement of the coefficient of expansion of a liquid with 
 the volume dilatometer. 76. The hydrostatic balance method of measuring 
 the coefficient of expansion of a liquid. 77. Determination of the coefficient 
 of apparent expansion of mercury in glass by means of a thermometer . 187-202 
 
 CHAPTER XI 
 THERMAL EXPANSION OF GASES 
 
 78. The air thermometer. 79. Measurement of the coefficient of increase of 
 pressure of air with the air thermometer. 80. Determination of the corrections 
 to a mercury thermometer at high temperatures by means of the constant 
 volume air thermometer. 81. The constant pressure air thermometer . 203-212 
 
 CHAPTER XII 
 CA LO RIM E TR Y 
 
 82. Calorimetry the method of mixture. 83. Measurement of the specific 
 heat of glass by the method of mixture. 84. Measurement of the specific heat 
 of a liquid by the method of mixture. 85. Measurement of the specific heat 
 of a liquid by the method of cooling 213-227 
 
x CONTENTS 
 
 CHAPTER XIII 
 CALORIMETRY LATENT HEAT 
 
 PAGES 
 
 86. Measurement of the density of ice by Bunsen's method. 87. Bunsen's ice 
 calorimeter. 88. Measurement of the latent heat of fusion of ice. 89. Measure- 
 ment of the heat of vaporisation of water. 90. Measurement of the heat of 
 vaporisation by Berthelot's method. 91. Joly's steam calorimeter. 92. 
 Measurement of the heat of solution ........ 228-244 
 
 CHAPTER XIV 
 
 VAPOUR PRESSURE 
 
 93. The measurement of vapour pressure. 94. Ramsay and Young's method of 
 measuring vapour pressure. 95. Determination of the dew-point. 96. The 
 wet and dry bulb thermometer. 97. The absorption hygrometer . . 245-252 
 
 CHAPTER XV 
 
 VAPOUR DENSITY FREEZING AND BOILING POINTS 
 OF SOLUTIONS 
 
 98. Measurement of vapour density by Victor Meyer's method. 99. Measure- 
 ment of vapour density by Hofmann's method. 100. Measurement of the 
 freezing point of a solution. 101. Calculation of molecular weight from the 
 depression of the freezing point. 102. Correction of the readings of a thermo- 
 meter in which the quantity of mercury is variable. 103. Measurement of the 
 boiling point of a solution 253-266 
 
 CHAPTER XVI 
 
 MELTING POINT RATIO OF SPECIFIC HEATS- 
 CONDUCTIVITY 
 
 104. Determination of the melting point by the method of cooling. 105. Deter- 
 mination of the ratio of the specific heats for air by Clement and Desorme's 
 method. 106. Measurement of the heat conductivity of copper. 107. 
 Measurement of the heat conductivity of glass 267-274 
 
 CHAPTER XVII 
 SOUND 
 
 108. Determination of the pitch of a tuning fork by means of a clock and a 
 rotating drum. 109. Stroboscopic method of measuring the frequency of a 
 fork. 110. Measurement of the pitch of a fork by means of a string. 111. 
 Comparison of the frequency of two forks. 112. Measurement of the velocity 
 of sound in air by resonance. 113. Measurement of the velocity of sound in 
 a gas by Kundt's method. 114. To calculate the ratio of the specific heats 
 of gases from measurements of the velocity of sound. 115. Measurement of 
 the wave-length of a high note by interference 275-288 
 
CONTENTS xi 
 
 CHAPTER XVIII 
 REFRACTIVE INDEX 
 
 PAGES 
 
 1-16. Adjustment of a telescope for parallel rays. 117. Measurement of the angle 
 of a prism with the spectrometer. 118. Refractive index of a solid or a liquid 
 in the form of a prism. 119. Measurement of the refractive index of a liquid 
 by total reflection, using an air film. 120. The measurement of refractive 
 index by total reflection, using a prism (first method). 121. The measurement 
 of refractive index by total reflection, using a prism (second method) . 289-304 
 
 CHAPTER XIX 
 DISPERSION AND WAVE-LENGTH MEASUREMENTS 
 
 122. The measurement of dispersion sources of light. 123. Light filters. 
 
 124. Calibration of a spectroscope by the use of lines of known wave-lengths. 
 
 125. Calibration of a spectroscope by interference fringes. 126. Measurement 
 
 of wave-length with the diffraction grating 305-315 
 
 CHAPTER XX 
 INTERFERENCE 
 
 127. Measurement of the wave-length of light with Fresnel's biprism. 128: 
 Measurement of the wave-length of light with Fresnel's biprism and a 
 spectrometer. 129. Measurement of the wave-length of light by means of 
 diffraction fringes. 130. On the localisation of interference fringes. 131. 
 Michelson's interferometer. 132. Newton's rings. 133. Measurement of the 
 wave-length of light by Newton's rings. 134. Diffraction through a slit and 
 limit of resolution of a telescope. 135. Resolving power of a spectroscope. 
 136. Measurement of the resolving power of a prism spectroscope. 137. 
 Measurement of the resolving po\\er of a grating. 138. Rayleigh's refracto- 
 meter 316-347 
 
 CHAPTER XXI 
 LENSES AND MIRRORS 
 
 139. Measurement of the focal length of a thin lens. 140. Measurement of the 
 radius of curvature of a concave surface. 141. Measurement of the radius of 
 curvature of a convex surface. 142. To test the flatness of the faces of a piece 
 of glass. 143. Thick lenses and systems of lenses. 144. To find the focal 
 length and the positions of the principal planes of a thick lens or system of 
 lenses. 145. Test of a photographic lens. 146. Measurement of the magni- 
 fying power of a telescope. 147. Measurement of the magnifying power of a 
 microscope . 348-369 
 
xii CONTENTS 
 
 CHAPTER XXII 
 POLARISED LIGHT 
 
 PAGES 
 
 148. Rotation of the plane of polarisation by optically active substances. 149. 
 Methods of measuring the rotation of the plane of polarisation. 150. Measure- 
 ment of the rotatory power of solutions of turpentine. 151. Measurement of 
 the strength of a solution of sugar. 152. Elliptically and circularly polarised 
 light. 153. Analysis of an elliptical vibration ...... 370-381 
 
 CHAPTER XXIII 
 PHOTOMETRY AND COLOUR VISION 
 
 154. Standards of light. 155. Photometers. 156. Determination of the con- 
 nection between the candle-power of an incandescent lamp and the watts 
 consumed for different voltages. 157. The spectro-photometer. 158. Measure- 
 ment of the absorption curve of a solution. 159. Colour mixture with the 
 colour top. 160. Measurement of the luminosity of pigment colours. 161. 
 Test for colour-blindness with Holmgren's wools 382-395 
 
 CHAPTER XXIV 
 MEASUREMENT OF THE EARTH'S MAGNETIC FIELD 
 
 162. Determination of the direction of the magnetic meridian. 163. Determina- 
 tion of the geographical meridian. 164. Measurement of the horizontal 
 component of the earth's field. 165. Comparison of the values of " //" at 
 different parts of a building. 166. Measurement of " H" with the Kew- 
 pattern unifilar magnetometer. 167. Determination of the temperature 
 coefficient of a magnet. 168. Measurement of the dip with the Kew-pattern 
 circle. 169. Comparison of the magnetic moments of two magnets . . 396-419 
 
 CHAPTER XXV 
 ADJUSTMENT AND USE OF GALVANOMETERS 
 
 170. Measurement of angles of rotation by means of a mirror and scale. 171. 
 Adjustment of a telescope and scale to meisure a rotation by means of a 
 mirror. 172. Adjustment of a suspended needle reflecting galvanometer. 
 173. Adjustment of a suspended coil galvanometer. 174. Determination of 
 the figure of merit of a galvanometer. 175. Comparison of electromotive 
 forces with the galvanometer ..,... 420-431 
 
CONTENTS xiii 
 
 CHAPTER XXVI 
 MEASUREMENT OF RESISTANCE 
 
 PAGES 
 
 176. The Wheatstone's network of conductors. 177. The metre form of Wheat- 
 stone's bridge. 178. Calibration of a bridge wire by Carey Foster's method. 
 179. Calibration of a wire by means of a double circuit. 180. Calibration of 
 a wire by Strouhal and Barus's method. 181. Comparison of two nearly 
 equal resistances by Carey Foster's method. 182. To determine the ratio of 
 two nearly equal resistances. 183. The Post Office form of Wheatstone bridge. 
 184. The dial-pattern Wheatstone's bridge. 185. Calibration of a dial-pattern 
 Wheatstone's bridge. 186. The Callendar and Griffiths pattern bridge. 
 187. Calibration of a Callendar-Griffiths bridge. 188. Measurement of 
 high resistances. 189. Measurement of the insulation resistance of a piece 
 of rubber covered cable. 190. Measurement of low resistances Mathissen 
 and Hocking's method of the projection of equal potentials. 191. Measure- 
 ment of low resistance the method of auxiliary conductors, or Kelvin's double 
 bridge. 192. The differential galvanometer. 193. Comparison of two low 
 resistances with the differential galvanometer. 194. The shunt potentiometer 
 method of comparing two low resistances 432-474 
 
 CHAPTER XXVII 
 THE RESISTANCE OF ELECTROLYTES 
 
 195. Mance's method of measuring the resistance of a battery. 196. Beetz's 
 method of measuring the resistance of a battery. 197. Measurement of the 
 resistance of electrolytes. 198. Measurement of electrolytic resistance by the 
 passage of a direct current. 199. Comparison of the specific resistances of 
 electrolytes by the potentiometer method. 200. Measurement of electrolytic 
 resistance by Kohlrausch's method with alternating currents. 201. The 
 preparation of solutions for the determination of resistance . . . 475-486 
 
 CHAPTER XXVIII 
 MEASUREMENT OF ELECTROMOTIVE FORCE 
 
 202. The construction of standard cadmium (Weston) cells. 203. The con- 
 struction of the standard Clark cell. 204. Comparison of E.M.F.'s with the 
 potentionmeter. 205. Potentiometers. 206. The capillary electrometer. 
 207. To examine the relation between the surface tension of mercury and an 
 electrolyte and the E.M.F. acting across the surface 487-500 
 
xiv CONTENTS 
 
 CHAPTER XXIX 
 RESISTANCE THERMOMETERS AND THERMO- J UNCTIONS 
 
 PAGES 
 
 208. Determination of the constants of a platinum thermometer. 209. Measure- 
 ment of the E.M.F. of a thermocouple. 210. Standardisation of a thermo- 
 couple '...... 591-507 
 
 CHAPTER XXX 
 MEASUREMENT OF CURRENT 
 
 211. The tangent galvanometer. 212. Adjustment of a tangent galvanometer. 
 213. The sine galvanometer. 214. Measurement of a current by the deposi- 
 tion of copper. 215. Calibration of an ammeter by copper deposition. 
 216. The silver voltameter. 217. The potentiometer method of measuring a 
 current 508-517 
 
 CHAPTER XXXI 
 
 THE BALLISTIC GALVANOMETER AND MEASUREMENT 
 OF CAPACITY 
 
 218. The ballistic galvanometer. 219. Determination of the constant of a ballistic 
 galvanometer. 220. Standardisation of a ballistic galvanometer by means of 
 a solenoid inductor. 221. To compare the capacities of two condensers with 
 the ballistic galvanometer. 222. Comparison of electromotive forces with 
 the ballistic galvanometer. 223. Measurement of a magnetic field with the 
 earth inductor. 224. Measurement of the strength of a magnetic field with 
 an exploring coil. 225. Comparison of capacities by de Sauty's method. 
 226. Comparison of capacities by the method of mixtures. 227. Measurement 
 of a capacity in electromagnetic units in terms of a resistance and a time . 518-535 
 
 CHAPTER XXXII 
 MEASUREMENT OF SELF AND MUTUAL INDUCTION 
 
 228. Measurement of the coefficient of self-induction of a coil by Rayleigh's 
 method. 229. Use of a rotating commutator when making measurements of 
 induction (Ayrton and Perry's secohmmeter). 230. Measurement of self- 
 induction by comparison with a capacity (Rimington's method). 231. Measure- 
 ment of the self-induction of a coil by comparison with a capacity (Anderson's 
 method). 232. Measurement of self-induction by means of a variable standard 
 of self-induction. 233. To compare the mutual-induction between two coils 
 with the self-induction of one of them. 234. Measurement of the coefficient of 
 mutual-induction between two coils by means of a variable standard of self- 
 induction. 235. The comparison of two mutual- inductances. 236. The 
 measurement of self-induction with alternating currents (Wien-Dolezalek 
 method) .... 536-551 
 
CONTENTS xv 
 
 CHAPTER XXXIII 
 PERMEABILITY 
 
 PAGES 
 
 237. Measurement of the permeability of iron by the magnetometer method. 
 238. Correction on account of the finite length of the specimen. 239. The 
 ballistic method of measuring permeability. 240. To demagnetise a specimen 
 of iron 552-568 
 
 CHAPTER XXXIV 
 THE QUADRANT ELECTROMETER 
 
 241. The quadrant electrometer. 242. Determination of the capacity of a 
 
 quadrant electrometer. 243. Determination of the saturation current . 569-574 
 
 APPENDIX 
 
 1. Glass blowing. 2. Working fused silica manufacture of quartz fibres. 3. 
 Making divided scales. 4. Silvering glass. 5. Mounting cross-wires in 
 telescopes and microscopes. 6. On the use of manganine wire for the con- 
 struction of resistance coils 575-596 
 
 Table I. Reduction of barometer readings to C . . . To face page 596 
 > II. Approximate formulae 597 
 
 III. Reduction of weighings to vacuo 598 
 
 IV. Density of dry air at different temperatures and pressures . 599 
 V. Density of water at various temperatures measured on the 
 
 hydrogen scale 600 
 
 VI. Correction for buoyancy in density measurements . . . 601 
 
 VII. Moments of inertia . . 601 
 
 ,, VIII. Reduction of periodic time to infinitely small arc, and correc- 
 tion on account of chronometer rate 602 
 
 IX. Correction of the sun's or a star's apparent altitude to allow 
 
 for parallax and refraction ...... 602 
 
 X. Reduction of barometer readings to C 603 
 
 ,, XI. Correction to the height of the barometer to allow for the 
 
 effect of capillarity 604 
 
 XII. Reduction of the volume of a gas to standard pressure . . 605 
 ,, XIII. Values of (l + o.^) for reducing the volume of gases to standard 
 
 temperature for values of t from to 40 C. . . 606 
 
 ,, XIV. Tension of water vapour and mass of water in a cubic metre 
 
 of saturated air 607, 608 
 
 XV. Boiling point of water 609 
 
 ,, XVI. Vapour pressure of liquids suitable for use in vapour jackets . 610 
 
 ,, XVII. Vapour pressure of mercury 611 
 
 XVIII. Depression of zero 611 
 
 ,, XIX. Corrections to reduce readings on mercury-in-glass thermo- 
 meters to the hydrogen or air thermometer . . . 612 
 
xvi CONTENTS 
 
 PAGFS 
 
 Table XX. Coefficients of expansion of water and mercury . . . 013 
 
 XXI. Specific heat of water 614 
 
 ,, XXII. Refractive indices for sodium light 614 
 
 XXIII. Wave-lengths . 615 
 
 ,, XXIV. Correction to scale readings when a mirror and scale is used 
 
 to measure a rotation . . . . . . 616 
 
 XXV. Conductivity of electrolytes 617 
 
 XXVI. E.M.F. of Clark and cadmium cells at different temperatures . 618 
 ,, XXVII. Connection between the units in the C.G.S. system and the 
 
 practical system 619 
 
 ,, XXVIII. Density . 620 
 
 INDEX 
 
 . 621-626 
 
F THE 
 
 UNIVERSITY 
 
 or 
 
 A TEXT-BOOK OF 
 
 PRACTICAL PHYSICS 
 
 CHAPTER I 
 
 ERRATA 
 
 Page 168, table, mean value ought to be 9'89. 
 254. table, for ' analine ' read ' aniline.' 
 ,, 267, last line, for ' DesormeV read ' DesormesV 
 , , 360, line 8 , for ' fv = 3/ ' read ' v = 3/. ' 
 ,, 391 and 393, /or ' tip ' read ' top ' in several places. 
 
 409, line 16, for 1+ - read 1+ rj -i*, and make the same alteration 
 r HJM. 
 
 in the following table and in the example given on 
 page 411. 
 ,, 411, in the example the value of tPK ought to be 3'42247, and that 
 
 of log Mil 2-22057. 
 
 413, in the example the values of log |r 3 ought to be 4'13066 and 
 4'50548, and the corresponding corrections, as alsu that for 
 the log of Mil (see above), to be applied all down, when the 
 values obtained will be: log M= 2*96180 and 2 '96185, 
 and hence J/ = 915'80 and 915'90. 
 
 437, line 13, for \ 7 = ?-^? read \q = l^ 
 n 1 n 1 
 
 456, line 7 from end, for ' two hundredths ' read ' one-fifth.' 
 
 ,, 462, ,, 4 ,, for ' Mathissen and Hocking ' read ' Mathiessen 
 
 and Hockin.' 
 474, ,, 4 ,, ,, for ' Mathissen and Hocking ' read ' Mathiessen 
 
 and Hockin.' 
 ,, 550, line 12, for ' inductors ' read ' conductors.' 
 
 597, formula (14), read ' cos (0a) = cos 6 + a sin 6.' 
 ,, 622, line 12, for k Desorme's' read ' DesorrnesV 
 
 624, ,, 15, for 'Mathissen and Hocking' read 'Mathiessen and 
 Hockin.' 
 
xvi CONTENTS 
 
 PAGFS 
 
 Table XX. Coefficients of expansion of water and mercury . . . 613 
 
 XXI. Specific heat of water 614 
 
 ,, XXII. Refractive indices for sodium light 614 
 
 XXIII. Wave-lengths ' . 615 
 
 ,, XXIV. Correction to scale readings when a mirror and scale is used 
 
 to measure a rotation . . . . . . 616 
 
 XXV. Conductivity of electrolytes 617 
 
 XXVI. E.M.F. of Clark and cadmium cells at different temperatures . 618 
 ,, XXVII. Connection between the units in the C.G.S. system and the 
 
 practical system 619 
 
 ,, XXVIII. Density 620 
 
 INDEX . 621-626 
 
F THE 
 
 UNIVERSI 
 
 or 
 
 A TEXT-BOOK OF 
 
 PRACTICAL PHYSICS 
 
 CHAPTER I 
 
 METHODS USED IN THE REDUCTION AND DISCUSSION OF THE 
 RESULTS OF PHYSICAL MEASUREMENTS 
 
 1. Correct Apportionment of the Accuracy with which the different 
 Observations are to be made in an Experiment. When performing 
 an experiment which involves the measurement of a number of 
 different quantities in order to obtain the result which is sought, it is 
 of the utmost importance that before starting on the experiments the 
 student should satisfy himself as to the effect such errors as he is 
 likely to make, in each of the individual measurements, will produce on 
 the result. Having done this he will be able to adjust his observations 
 so that he shall measure each of the quantities with an accuracy such 
 that the possible error in the final result produced by each individual 
 measurement shall be approximately equal. For example, suppose 
 that in a determination of specific heat by the method of mixtures the 
 rise of temperature of the calorimeter is about five degrees, while the 
 thermometer used to measure this rise is only divided into half degrees. 
 With such a thermometer it will be almost impossible to read the rise of 
 temperature nearer than a twentieth of a degree, that is, to 1 per cent. 
 of the total rise. Suppose that the calorimeter is made of copper and 
 weighs about twenty grams, while it holds two hundred grams of water. 
 The specific heat of copper being about O'l, the water value of the calori- 
 meter is only 1 per cent, of the weight of the water. Now it would 
 be quite easy to weigh the calorimeter to within a milligram, that is, to 
 within one part in 20,000. Since, however, we only know the rise of 
 temperature to one part in 100, such refinement would be quite useless, 
 as the accuracy of the value obtained for the specific heat would in no 
 way be increased thereby. It will be quite sufficient if we measure the 
 weight of the calorimeter to within a gram, so that we know its water 
 value to within '1 calory. In the same way the water need not be 
 weighed to nearer than half a gram, that is, to within one part in 400. 
 Not only does a careful consideration of the relative accuracy with 
 
 A 
 
 tL. 
 
REDUCTION OF EXPERIMENTAL RESULTS [ 1 
 
 which the different component measurements can be made tend very 
 materially to increase the accuracy of the results obtained, by directing 
 attention to those measurements which it is necessary to make with 
 particular accuracy, but also considerable time will in the long run 
 be saved. 
 
 We are thus led to examine the extent to which a small error in the 
 different magnitudes which are actually observed will affect the quantity 
 which is being deduced. 
 
 Suppose that the quantity to be measured, y, is connected with the 
 quantities x v x 2 , x s , &c., which are actually observed by the relation 
 
 V=f( x v x v x v &c -)> 
 
 and that the observed quantities receive small changes, so that x l becomes 
 x l + 8x l , x 2 becomes x 2 + 8x 2 , &c., and as a result y becomes 
 Then it is shown in books on the differential calculus l that 
 
 8fo. . Ac.) MC.) 
 
 where ^ & - means the partial differential coefficient of 
 
 /(a?!, x v ajg, &c.) with respect to x v and so on. 
 
 The above expression allows of our calculating what will be the effect 
 on the quantity we are measuring of any given small error in any one of 
 the observed quantities. Thus an error of magnitude 8x l in x 1 will pro- 
 
 8f(x x. x &c ) 
 
 duce an error in y of magnitude - -J* l - 8x r The applica- 
 tion of this method of estimating the effects of errors in the measured 
 quantities on the result can most easily be studied by means of a few 
 examples. 
 
 As a first case, let us suppose that we are measuring the volume of a 
 circular cylinder by determining the length / and diameter d. Here we 
 
 have y = -. d 2 l. Hence 
 
 or dividing each side by y or its equivalent - d 2 l, 
 
 8 I = 8l +() 8d 
 y I " d' 
 
 Let us now suppose that keeping d fixed we change I by 1 per cent. 
 so that 81 is 'Oil. Then 
 
 l-oi, 
 
 1 Lamb. Infinitesimal Calculus, p. 138. 
 
2] ON THE APPLICATION OF CORRECTIONS 3 
 
 i.e. 8y is 1 per cent, of y. Next keeping I fixed, let us change d by 1 
 per cent. Then in the same way 
 
 ^=02, 
 
 y 
 
 i.e. 8y is 2 per cent, of y. This shows that a given percentage error in 
 d will produce twice as great an error in the volume as would an equal 
 percentage error in /. Thus if the cylinder has a length of 10 cm. and 
 a diameter of 1 cm., and that we are only able to read the diameter on 
 our measuring instrument to O'OOOl cm., i.e. one part in 10,000, then 
 we need only measure the length to one part in 5000, that is, to 0*002 cm. 
 As a second example, let us suppose that we require to determine the 
 moment of inertia of the above cylinder about an axis through its centre 
 of gravity at right angles to the axis. In this case 
 
 where M is the mass of the cylinder. 
 
 rpi, a ( P d2 \ * Ml */ Md 
 
 Then 8y = -. + 8M + 81 + 
 
 and 
 
 In the present case, where 1= 10 and </= 1, this equation reduces to 
 
 8y_8M 800 81 6 8d 
 
 ^~ il*4ffl ~l + 403 ~d' 
 or quite nearly enough for our present purpose, 
 
 d 
 
 Here it is at once evident that an error in the diameter produces 
 quite an insignificant effect compared to a proportional error in the 
 length. If we can only measure I to 0'001'cm., i.e. to one part in 10,000, 
 we need not measure the diameter nearer than to about one part in 150, 
 i.e. to 0-07 mm. 
 
 The above two examples illustrate the way in which a preliminary 
 discussion helps us to concentrate our attention on those measurements 
 which require making with especial care, for we see that when it is the 
 volume we require we have to take particular pains over the measure- 
 ment of the diameter, while to obtain the moment of inertia we must 
 measure the length with special care. 
 
 2. On the Application of Corrections to Observed Quantities. It 
 is the exception rather than the rule that the actual measurements we 
 make can be used without change in deducing the desired result. In 
 
4 REDUCTION OF EXPERIMENTAL RESULTS [ 2 
 
 general these measurements will require the application of certain small 
 corrections. Thus where a length is measured by comparison with a 
 standard scale, a correction will generally be necessary to allow for the 
 fact that the distance between the two divisions of the scale is not what 
 it purports to be, (1) on account of errors of graduation, (2) on account 
 of the temperature of the scale at the time of comparison not being the 
 same as it was when the scale was standardised. In the following 
 pages we shall have numerous examples of the application of such 
 corrections. Since, however, a considerable amount of time and labour 
 may often be saved in the application of corrections by adopting certain 
 devices, we shall consider the point at the present time. This saving 
 in time is particularly the case where a number of observations of the 
 same kind have to be reduced, the corrections in the various cases 
 differing slightly. 
 
 When correcting observations, more particularly for temperature, much 
 time will be saved by making use of the fact that when the corrections 
 are small we may apply them by addition in place of multiplication. 
 This can be shown as follows. Suppose the quantity y which has to 
 be determined is obtained from the observed quantities a, b, c by means 
 of the relation 
 
 where D is some constant factor. If, now, the measured quantities a, b, c 
 have all to be reduced to what they would be at some fixed temperature 
 we should have 
 
 (1) 
 
 where t l9 2 , and t z are the amounts by which the temperatures at which 
 a, b, and c were observed exceed the fixed temperatures, while a, /3, and y 
 are the coefficients which express the changes of the quantities a, &, and 
 c with temperature. If, as is almost always the case, t v t y and 3 are 
 not very large, while a, /?, and y are quite small quantities, we have, 
 neglecting the squares and higher powers of the small quantities, 1 
 
 y = D~(l+at l + pt 2 -yt,) ...... (2) 
 
 Hence by calculating the value of the expression within the brackets 
 we can at once, by multiplying the uncorrected value of y, namely, 
 
 D , by this quantity get the corrected value. When calculating the 
 
 
 
 1 In general if 
 
 s=f(x + h, y + k, z + l, &c.), 
 we have by Taylor's theorem (Lamb, Infinitesimal Calculus, p. 595) 
 
 where is the partial differential coefficient of s with respect to x, and so on. 
 
3] USE OF TABLES AND CURVES 5 
 
 value of the terms at v fit. 2 , and yt 3 it will generally be found sufficiently 
 accurate to employ a slide rule, so that the evaluations of the correction 
 factor can be rapidly performed. 
 
 To test whether the above approximation is sufficiently accurate in 
 any given case, it will be as well to employ the formula (1) for one of 
 the observations, and compare the two values obtained for y. If these 
 do not differ by more than, say, a unit in the last significant figure 
 which may fairly be retained in the value for y, then the approximation 
 may be considered sufficiently good. 
 
 A list of approximate formulae which may often be of service when 
 applying corrections will be found in the Appendix (Table 2). 
 
 3. Use of Tables and Curves of Corrections when reducing 
 Observations. When a number of reductions of the same kind have 
 to be performed which involve the application of corrections to the 
 observed quantities, a very considerable amount of time may often 
 be saved by preparing suitable tables or curves. When the range of 
 the quantity the variation of which causes the necessity for the correc- 
 tion in most cases in practice this is the temperature is great it is 
 usually best to employ a curve, as interpolation is so much easier in the 
 case of a curve than in the case of a table, unless the entries in the latter 
 are very numerous. 
 
 When preparing such a curve it is necessary to exercise some 
 discrimination, otherwise in order to obtain a sufficiently open scale 
 the dimensions of the curve will be excessive. This point may best 
 be illustrated by means of some examples. 
 
 When using a platinum thermometer to measure temperatures it is 
 necessary to reduce the observations to the air thermometer scale. If 
 the resistance of a piece of pure platinum wire at C. is 7? , its resist- 
 ance R t at a temperature t on the air thermometer may be expressed by 
 a formula of the type 
 
 B t = R (l+at + W) . (1) 
 
 Hence if R t and R^ are measured, and we know the constants a and 6, 
 it is possible to calculate the value of t. This calculation, however, would 
 obviously be very laborious. Even if we calculated the values of R t for 
 several values of t, and drew a curve through the points obtained, the 
 result would not be satisfactory, as to obtain an accuracy of, say, 0'01 
 it would be necessary to have a curve on such an enormous scale. 
 Thus to be able to read all temperatures between and 100, the curve 
 would have to extend in both directions for about 200 cm. We may, 
 however, make use of the fact that the change in resistance is nearly 
 proportional to the change in temperature, and as a first approximation 
 assume that 
 
 R t = R (l+ct) (2) 
 
 So that if R liw is the resistance at 100, we have, using tp to represent 
 the temperature as deduced on this assumption, 
 
 (3) 
 
6 REDUCTION OF EXPERIMENTAL RESULTS [ 3 
 
 Now Callendar has shown that throughout a very wide range the 
 deviation between the temperature on this platinum scale and the 
 temperature on the air thermometer scale may be represented by 
 
 ^ fM \ 2 M 
 t - tp = 8 U - - V 
 
 ,100/ 100/ 
 
 where 8 is a constant for any particular sample of wire, and generally 
 has a value very near 1-5. The maximum value of t -tp between and 
 100 is less than 0'5, and hence to be able to read the correction to 0'01, 
 a curve 5 cm. high would be ample. That the values obtained by means 
 of equations (2), (3), and (4) are the same as those given by (1) can be 
 shown as follows : 
 
 Substituting the value of tp from (3) in (4), we have 
 
 But from (2) 
 
 c - ^100 ~ ^o 
 
 Hence 
 
 t '-=. - 
 
 ioo72 
 
 t \'_ t \ 
 lOO/ 100/ 
 
 - 
 
 which is the same as (1) if 
 
 / 
 
 a = c ( 
 
 i ^ i. 
 
 = c ( 1 + ^^ ) and b = - 
 
 OO; - loo* 
 
 As another example, suppose that a current has been measured by 
 the potentiometer method, in which the drop of potential produced in 
 a resistance R when traversed by the current is balanced against the 
 E.M.F. E of a cadmium cell. If the temperature changes, then a cor- 
 rection will have to be applied on account of the variation of the 
 resistance and of the E.M.F. of the cell. Suppose that the resistance 
 has its nominal value at 15 C., while the E.M.F. of the cell at a tem- 
 perature t e is given by 
 
 K t = 1 -0184(1 - 0-000037& - 20) - 0'00000064(* e - 20) 2 }. 
 If the resistance is of manganine the temperature coefficient will be about 
 + 0-00003, and for small ranges of temperature the change of resistance 
 with temperature may be taken as linear, so that 
 
 72 t = 72 16 {l + '00003(^-1 5)}, 
 
 where t r is the temperature of the coil. Now if the observations all take 
 place at temperatures ranging from, say, 18 to 22, the term in t 2 in the 
 expression for E t is negligible, and the current is given by 
 
 C= 1 -01847^1 - 0-000037(f, - 20)} {1 + 0'00003(/ r - 15)} 
 = 1-018472 - 1-018472 x 0'000037( e - 20) 
 + 1-018472 x -00003(* r - 15) + . 
 
4] REPRESENTATION OF A SERIES OF RESULTS 7 
 
 Now in this expression the first term on the right is a constant, while the 
 two other terms, which are small, vary with the temperature. Hence 
 if two curves be drawn showing the values of these two terms for 
 different temperatures, the corrections which have to be applied to allow 
 for the fact that the resistance and the cell are not at their standard 
 temperatures can easily be read off. In obtaining the curves only one 
 point need be calculated in each case, say, for 10 in the case of E and 25 
 in the case of R, for by the assumptions we have made the curves will 
 both be straight lines, and one will cut the axis at a temperature of 20 
 and the other at a temperature of 15. 
 
 It will at once be evident that if a number of observations have to be 
 reduced, the above plan will be much quicker than correcting E and R 
 separately for temperature and then finding the product of these two 
 corrected values to give the value of the current. 
 
 In such a case as that just considered, where the temperature 
 coefficients are both very small, a table can easily be constructed which 
 in some cases is more easily used than a curve. Thus suppose that the 
 value of R at 15 is 1 ohm, then the following table gives the corrections 
 over the range of ordinary room temperatures : 
 
 Temperature. 
 
 Correction for Cell. 
 
 Correction for Resistance. 
 
 15 
 
 + 00019 
 
 
 16 
 
 + 00015 
 
 + 00003, 
 
 17 
 
 + 00014 
 
 + '00006 
 
 18 
 
 + 00008 
 
 + 00009 
 
 19 
 
 + 00004 
 
 + 00012 
 
 20 
 
 
 + 00015 
 
 21 
 
 -00004 
 
 + 00018 
 
 22 
 
 - '00008 
 
 + 00021 
 
 Thus if the cell has a temperature of 16 '2 and the resistance coil a tem- 
 perature of 20 '6, the value of the current is 
 
 C= 1-0184 + 0-0001 +0-0002 
 = 1-0187 amperes. 
 
 4. Representation of a Series of Results. Frequently as a result of 
 a physical investigation we obtain a series of values of the quantity which 
 is being investigated for different values of one of the quantities on which 
 it depends, as, for instance, we may have measured the rotation of the 
 plane of polarisation by a magnetic field in water at different tempera- 
 tures. In such a case the question arises as to the best manner of 
 exhibiting the results so as to show the connection between the two. 
 Two methods are commonly employed for this purpose. In one of these 
 a curve is plotted with the two variables as co-ordinates, and in the other 
 an algebraical expression is sought which shall express the relation be- 
 tween them. We will first consider the representation of a series of 
 
8 REDUCTION OF EXPERIMENTAL RESULTS [ 4 
 
 measurements by means of a curve. The first thing to do is to choose a 
 suitable scale on which to plot the curve and the kind of ruling to be 
 used for the paper. For most purposes paper divided up into milli- 
 metre squares will be found most convenient. Such paper can now be 
 obtained in sheets of almost any size and with very regular rulings 
 at quite a moderate price. It is convenient to have the half centimetre 
 lines slightly darker than the general ruling, and the centimetre lines 
 slightly darker still. 
 
 When choosing the scale on which to plot the observations it will not do 
 to use too small a scale, as then the points on the curve cannot be plotted 
 with as great an accuracy as that with which the observations are taken, 
 and thus we shall be failing to make use to the full of the time which has 
 been spent in making the observations. On the other hand, it is possible 
 to use too large a scale, for if such an exaggerated scale is employed then 
 the unavoidable errors of observation will become so prominent that the 
 points which represent the observations may appear so scattered that it 
 will be difficult to draw a curve evenly between them. 
 
 As a general rule it will be found sufficient to so choose the scale that 
 half a millimetre, or at most a millimetre, represents a unit in the last 
 place which, from a discussion of the probable sources of error of our 
 measurements, we consider we are justified in retaining. Occasionally it 
 may happen that a very considerable change of one of the quantities only 
 involves a comparatively small change in the other, in which case it is 
 often sufficient to plot the quantity which changes rapidly on a consider- 
 ably smaller scale than that given by the above rule. Thus suppose 
 that we are plotting the refractive index of water at different tempera- 
 tures. The refractive index at is 1-3341 and that at 100 is 1'3184, 
 that is, for a difference of temperature of 100 there is a change of '0257. 
 We will suppose that measurements of refractive index are such that we 
 are not entitled to retain more than four places of decimals, while the 
 temperature has been read to a tenth of a degree. According to our rule, 
 we should take a millimetre to represent either one or two units in the 
 fourth place of the refractive index and a tenth of a degree in the tem- 
 perature. This scale would, however, unduly extend the temperature 
 scale, for a unit in the last place of the refractive index corresponds to 
 a change of temperature of nearly half a degree. Thus if we take a 
 millimetre to represent a degree, or at most half a degree, we shall obtain 
 a scale quite large enough. 
 
 Having fixed on the scale and written along the bottom and left-hand 
 edge of the paper the values corresponding to the chief division lines, the 
 points representing the observations must be inserted. The point corre- 
 sponding to each observation must be indicated either by a small dot, 
 which for distinctness may be surrounded by a circle of about 2 milli- 
 metres in diameter, or by the intersection of two short lines forming a 
 cross, each arm of the cross being about 2 millimetres long. 
 
 A curve has now to be drawn which shall as nearly as possible pass 
 through all the points, and yet shall not show any sudden twists and 
 
4] REPRESENTATION OF A SERIES OF RESULTS 9 
 
 turns. Owing to the unavoidable errors to which all observations are 
 liable it will probably be impossible to draw a regular curve so as 
 to pass exactly through all the points. It must, however, be so drawn 
 that at every point of the curve the points are distributed fairly evenly 
 on the two sides of the line. The drawing of a curve to represent the 
 results of a series of observations so that it shall give a truthful repre- 
 sentation of the results is an operation of very considerable difficulty, and 
 requires both practice and skill. The first thing to test is whether the 
 points lie on a straight line, or at any rate whether a straight line can be 
 drawn which shall pass evenly between the points. For this purpose a strip 
 of glass or celluloid about 4 centimetres wide, along which a fine line 
 has been ruled with a writing diamond or splinter of carborundum, will 
 be found of great assistance. The glass is placed on the paper with the 
 side on which the line is ruled downwards, and is moved about till the 
 points appear evenly distributed on the two sides throughout the whole 
 length. When this adjustment has been satisfactorily performed the 
 readings where the line cuts the edge of the paper must be noted, and 
 
 FIG. 1. 
 
 then by means of a straight-edge a line must be ruled joining these two 
 points. It will be found very difficult to settle on the position of such a 
 line by means of a straight-edge alone, as the ruler will cover all the points 
 on one side and render it almost impossible to arrange that the points 
 shall be evenly distributed on the two sides of the line. 
 
 If a straight line cannot be drawn to pass evenly through the points, 
 it will be necessary to draw a curved line. The most convenient arrange- 
 ment in such a case is that shown in Fig. 1. A thin flexible wooden lath 
 or spline is bent so that one edge passes evenly between the points, being 
 held in position by a number of lead weights in the manner shown. 
 These weights are supplied with a wooden base, which is cut to a point, 
 and they must, throughout that part of the spline where the curve has to 
 be drawn, be placed on the same side of the spline, otherwise they will 
 prevent a pencil being passed along the edge of the spline when drawing 
 the curve. Since half the points will be more or less hidden by the 
 spline, it is rather difficult to adjust the spline. This difficulty is in a 
 measure removed if the points are represented by a dot, and circles of 
 equal size are drawn round the dots by means of a pair of compasses. 
 Then, although the dot itself may be hidden by the spline, a portion of 
 
10 REDUCTION OF EXPERIMENTAL RESULTS [ 5 
 
 the circle may be visible and allow of the position of the dot being 
 inferred. If an estimate has been made of the error which is possible 
 in the measurement of the quantity being plotted it is a good plan to 
 make the radius of these circles equal to this error, so that if the curve 
 passes everywhere through these circles it will at no point depart from 
 the observed values by more than the possible errors of experiment. 
 Hence if one or more points are found to be so placed that without 
 introducing irregularities in the curve we cannot make the curve pass 
 through the circles which surround them, it will be advisable to check 
 the calculations made in reducing the observations corresponding to these 
 points and to make certain that no error has been made in plotting. 
 Particular care must be taken in adjusting the position of the spline at 
 the ends of the curve, as there is a tendency to give undue weight to the 
 end point. 
 
 Sets of splines are made of varying flexibility, and it is advisable to 
 use a spline of such a thickness that, while it can be bent to follow the 
 curve and be held in place by the weights, yet it is as stiff as possible, for 
 by this means the chances of forcing the spline to take up an irregular 
 form to pass through a point which, owing to errors of experiment, is 
 somewhat displaced will be minimised. For use with curves in which 
 the curvature is considerably greater at one end than at the other splines 
 having a tapering section may be used. 
 
 As a check on the accuracy with which the curve has been drawn the 
 ordinates of the curve corresponding to the abscissae of the observed 
 points should be taken, and a table drawn up showing the differences 
 between these readings and the observed values. If the curve passes 
 evenly between the points, then the sum of the positive differences ought 
 to be numerically equal to the sum of the negative differences. There 
 ought also to be no marked preponderance of differences of one sign at 
 any part of the curve. 
 
 5. To find an Algebraical Expression to represent a Series of 
 Observations. It is often convenient to express the result of a series 
 of measurements of the connection between two physical quantities in 
 the form of an algebraical equation. The first thing to do is to dis- 
 cover the most suitable type of equation to employ a matter often of 
 no little difficulty and then from the observed numbers we have to 
 determine the values of the constants which appear in this equation. 
 In some cases theory suggests the type of equation which must be 
 employed, and in such a case we have only to determine the best values 
 for the constants. For instance, suppose we have measured the distance 
 a body falling freely traverses in different times. In this case we know 
 that the space s varies as the square of the time of fall t, so that 
 
 where a is some constant, the value of which we can deduce from the 
 results of the measurements. If, however, theory does not suggest a 
 suitable type, then the first thing to do is to plot the results on squared 
 
5] REPRESENTATION OF A SERIES OF RESULTS 11 
 
 paper, as described in the preceding section. If the points lie uniformly 
 about a straight line, then the connection between the two quantities can 
 be represented by a linear equation of the form 
 
 y = a + bx ......... (1) 
 
 where a and b are two constants, the values of which have to be deduced 
 from the observations. To determine the values of these constants, 
 read off two pairs of values of x and y from the straight line, the points 
 chosen being near the extremities of the line so as to give as great an 
 interval between the two points as possible, and then substituting first 
 one set of values of x and y in the equation, and then the other, two 
 simultaneous equations are obtained from which the values of a and b 
 can be deduced in the ordinary manner. 
 
 The values of the constants a and b can also be obtained from the 
 curve, for when x = 0, y = a, so that a is the value of y where the curve 
 cuts the T axis. Calling this value y^ we have 
 
 Now in the right-angled triangle, of which the co-ordinates of the angular 
 points are (Oy ), (#y ), and (xy), the side parallel to the X axis has a length 
 , and the side parallel to the Taxis has a length y - y Qt the hypotenuse of 
 the triangle being formed by the curve. Thus the tangent of the angle 
 made by the straight line with the line through the point (Oy ) parallel 
 
 to the axis of X is ^ ~ ^. Hence we see that the constant b is the 
 
 x 
 
 tangent of the angle the straight line through the observed points makes 
 with the axis of X. 
 
 If a straight line cannot be drawn through the points, equations of 
 other types must be tried. A hint as to a suitable form can often be 
 obtained by plotting, not the actual values of the quantities, but their 
 logarithms, or by plotting the value of one quantity against the logarithm 
 of the other. In order to facilitate the plotting of these curves paper 
 can be obtained in which, instead of the lines being equally spaced, they 
 are spaced at distances proportional to the logarithms of the natural 
 numbers to the base ten. A portion of such a sheet of logarithm paper 
 is shown in Fig. 2. 1 The distance from A to B or A to c being taken 
 as the unit, the distance from A to the point D, marked 2, or 2 multi- 
 plied by some power of 10, is equal to log 2, i.e. 0'301 of the distance 
 AB. From A to E is equal to log 3, and so on. It will be noticed that 
 the spacing of the lines is periodically repeated. This is owing to the 
 fact that the logarithm of two numbers, one of which is a power of ten 
 times the other, differs by a whole number. Thus the logarithm of 20 is 
 equal to 1 + log 2, of 200 is 2 + log 2, and so on. 
 
 Now suppose that two quantities x and y are related to one another 
 in such a way that 
 
 y = ax 11 ......... (2) 
 
 i In this figure the lines for 1, 1-5, 2, 2-5, &c., only are shown. 
 
12 REDUCTION OF EXPERIMENTAL RESULTS [ 5 
 
 where a and n are constants. Then taking logarithms, \\e get 
 log y = log a + n log x. 
 
 Hence if instead of plotting x and y we plot log x and log .y, the result- 
 ing curve will be of the form 
 
 y' = c + nx', 
 
 and this represents a straight line. Of course we could plot such a curve 
 on ordinary squared paper by looking out the values of the logarithms 
 in a table and then plotting these values. Such a procedure would, 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 *v 20 000 
 
 
 
 
 
 
 
 
 
 
 / 
 
 (3> 
 
 
 
 
 
 
 
 
 u 
 
 Q. 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 j 
 
 
 
 
 
 
 
 
 i 
 
 n; 
 
 uj 10000 
 
 
 
 
 
 
 
 
 
 
 // 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f j 
 
 
 
 
 
 
 
 
 
 2 
 
 M 
 
 
 
 
 
 
 
 
 
 r 
 
 ig 
 
 
 
 
 
 
 
 
 
 K 
 
 s 
 
 
 
 
 
 
 
 
 j 
 
 
 :?s= 
 
 
 
 
 
 
 
 
 ".'. i 
 
 5 
 
 
 
 
 
 
 
 
 
 . ^ 
 
 
 
 
 
 
 
 
 
 
 >- 4000 
 
 
 
 
 
 
 
 
 7 
 
 > 
 
 
 
 
 
 
 
 
 
 
 I * 
 
 
 
 
 
 
 
 
 !; , 
 
 
 
 
 
 
 
 
 
 
 
 3000 
 
 
 
 
 
 
 
 / 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ) 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 2000 
 
 Bi 
 
 
 
 
 \ 
 
 / 
 
 A 
 
 " / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ; 
 
 
 ) 
 
 A 
 
 i ' 
 
 
 
 
 
 
 
 
 
 
 
 
 A 
 
 
 / 
 
 i 
 
 
 / 
 
 / 
 
 
 
 () 
 
 
 
 
 
 
 
 
 
 2000 3000 4000 sm 6000 10.000 
 
 INDUCTION 
 FIG. 2. 
 
 20,000 
 
 100.000 
 
 however, be somewhat lengthy, while by using logarithmically divided 
 paper we can plot the points without the necessity of looking out the 
 logarithms. As an example of the utility of logarithmic paper, we have 
 plotted on Fig. 2 the results obtained by Ewing for the energy loss per 
 cubic centimetre per cycle of three samples of iron for varying maximum 
 inductions. It will, in the first place, be noticed that in each case a 
 straight line passes very fairly between the various points; further, 
 that all three straight lines are approximately parallel, that is, the con- 
 
5] REPRESENTATION OF A SERIES OF RESULTS 13 
 
 stant n has the same value for each sample of iron, for n is equal 
 to the tangent of the angle which the straight line makes with the 
 axis of X. 
 
 To find the value of a we notice that when a?= 1, log# = 0, and hence 
 
 log y = log a. 
 
 Hence we have to find the value of y' corresponding to x' = 1 . Now in 
 Fig. 2 neither of the curves cuts the ordinate through the abscissae x' 1 
 on the diagram. They do, however, cut the ordinate through the point 
 x = 10000, and if y' is the ordinate at the point of intersection, we have 
 
 or log a = y' - k.n. 
 
 Now from curve (1), when #' = 10000, y= 11800 and ?/ = log 11800. 
 Hence 
 
 log a = log 11800 -4x1 -58 
 = 4-07-6-32 
 = 3-75 
 .-. a = 0-0056. 
 
 Hence the observations corresponding to curve (1) can be represented 
 
 by 
 
 n = -0056 J? 1 ' 58 , 
 
 where 17 is the energy loss due to hysteresis per cubic centimetre per 
 cycle. 
 
 In the same way the other curves are represented by 
 
 Thus by plotting the results of these observations of .Ewing's we are 
 at once led to investigate an expression similar to Steinmetz's expression 
 for the loss due to hysteresis, namely, 
 
 where a is a constant for any one sample of iron, but varies from one 
 sample to another. 
 
 If when the observations are plotted on logarithmic paper the curve 
 drawn through the points does not depart greatly from a straight line for 
 large values of the ordinate or abscissa, while for small values the curvature 
 is considerable, it is advisable to try whether a formula of the type 
 
 will fit the observations ; that is, we have to see whether we can find a 
 constant a or /3 which, added to x or to y respectively, will give a straight 
 line on the logarithmic chart. To illustrate the method of procedure, 
 
14 
 
 REDUCTION OF EXPERIMENTAL RESULTS 
 
 [5 
 
 suppose a series of measurements had given the results shown in the follow- 
 ing table, and which are also shown plotted on the curve AB in Fig. 3 : 
 
 X 
 
 y 
 
 X 
 
 y 
 
 1-00 
 
 3-40 
 
 5-79 
 
 5-87 
 
 1-50 
 
 3-74 
 
 6-61 
 
 6-20 
 
 T90 
 
 4-00 
 
 7-70 
 
 6-60 
 
 2-80 
 
 4-50 
 
 8-80 
 
 6-97 
 
 3-82 
 
 5-00 10-00 
 
 7-35 
 
 4-90 
 
 5-50 
 
 
 
 
 
 
 
 It will be noticed that for the higher values of x the curve is very 
 nearly a straight line, while for low values of x the curve is distinctly 
 
 ! 2 c 
 
 I 4 
 
 \ 567891 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r 
 
 r^ 
 
 x. 
 
 r p 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 x-^ 
 
 * 
 
 ^ 
 
 <" 
 
 
 
 
 
 
 
 
 
 
 
 < ^- 
 
 ^ 
 
 ., 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 x ^ 
 
 
 ^, 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 ^ 
 
 
 r*< 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 ^r 
 
 
 ^ 
 
 ^ 
 
 c 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 ^^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 -"V 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 s^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 2 3 < 
 
 FIG. 3. 
 
 5 6 7 8 9 1C 
 
 concave upwards. We will, therefore, try the effect of the addition of a 
 constant quantity to x. 
 
 Selecting two points, A and E, near the ends of the curve and a point c 
 
5] REPRESENTATION OF A SERIES OF RESULTS 15 
 
 about half-way between, for convenience choosing points where the curve 
 is cut by the thick vertical lines, take a straight-edge, or strip of glass 
 with a line ruled down the centre, and lay it so that it passes through the 
 points A and E. Now moVe the straight-edge till it intersects the 
 horizontal lines through A and E one division to the right of the points 
 A and E respectively, and count how many divisions to the right the 
 straight-edge intersects the horizontal line through the point c. If this 
 were one division, then we should have for the three points ACE that by 
 adding '1 to a? the resulting points lie on a straight line, and we should 
 then add '1 to the x of each of the other points and see if all the fresh 
 points lay on a straight line, or at any rate were evenly distributed 
 about such a line. If the agreement at the three points does not occur, 
 then move each end of the straight-edge through another division to the 
 right and again try. Proceeding in this way in the case of the example, 
 it will be found that when the x has been increased by 1*2 the three 
 points A'C'E' all line on a straight line DF. Adding 1 '2 to the abscissae 
 of each of the other points, it will be found that all the points lie 'on or 
 very near to the straight line DF. 1 We therefore infer that the obser- 
 vations can be represented by an expression of the form 
 
 and it now remains to determine the values of the constants a and n. 
 Writing z for x + 1*2, the expression takes the form 
 
 y = 2 n > 
 
 or taking logarithms, 
 
 log y = log a -f n log z. 
 
 When z is unity, log y = log a, that is, the point where the straight line 
 DF cuts the Y axis, gives the value of the constant a, which in the 
 example is 2 '34. Also, it can be at once shown, as on page 1 1, that n is 
 equal to the tangent of the angle which the line DF makes with the axis 
 of JT, that is 
 
 which in the example gives n = 0*476. 
 
 An approximate value of /? can be calculated from three points on the 
 curve in the following manner. Let a^ y lt x. 2 y^ and x s y s be the co-ordi- 
 nates of the three points so chosen that 
 
 tog 2/2 = \ lo g y l + f log y s , 
 
 or = .......... 4 
 
 1 After plotting the points in this way it may often be seen that by slightly 
 altering the value of /3 a more uniform distribution of the points on the two sides 
 of the straight line can be obtained, and that we can obtain a better value of /3 
 by making successive approximations. 
 
16 
 
 SEDUCTION OF EXPERIMENTAL RESULTS 
 
 [5 
 
 then if the points obtained by adding ft to each of the x's are to be in a 
 straight line we must have 
 
 log (* 2 + f}) = l log (x l +P) + 2 log (x s + P), 
 
 or 
 that is 
 
 (5) 
 
 Thus the procedure is to choose two points x l y l and x 3 y z , find from the 
 curve and point X 2 y 2 such that the condition expressed by (4) is fulfilled, 
 and then substituting the values of x lt x, and x 3 in (5) calculate f3. 
 
 If the connection between the observed quantities jr and y is of the 
 forms 
 
 (6) 
 
 or 
 
 so that log y = log a + ft#, 
 
 or log y = log a + Mnx, 
 
 where M is the modulus to reduce Napierian logarithms to the base e to 
 common logarithms to the base 10 (the value of M is '43429), we may 
 with advantage plot log y against x. In this case we get a straight line 
 of which the equation is of the form 
 
 y' = b + ex, 
 where b and c are constants. 
 
 To illustrate the advantages of this semi-logarithmic plotting we may 
 take the observations made by Weber l on the diffusion of zine sulphate 
 through water given in the following table, where the concentration is 
 measured in arbitrary units : 
 
 
 Difference in Concentration be- 
 
 
 Time in 
 Days. 
 
 tween two given Points. 
 
 Difference. 
 
 
 Observed. 
 
 Calculated. 
 
 
 
 
 871 5 
 
 883-1 
 
 -1-6 
 
 1 
 
 717-8 
 
 720-5 
 
 -2-7 
 
 2 
 
 589-0 
 
 587-8 +2-8 
 
 3 
 
 483-2 
 
 480-0 +3-2 
 
 4 
 
 394-0 
 
 391-2 +2-8 
 
 5 320-4 
 
 319-1 +1-1 
 
 6 260-1 
 
 260-4 - 0-3 
 
 7 212-5 
 
 212-4 +0-1 
 
 8 173-2 
 
 173-3 -0-1 
 
 9 141-2 
 
 141-4 -0-2 
 
 10 115-0 
 
 115-3 
 
 -0-3 
 
 If these numbers are plotted on ordinary paper, they will lie on a 
 1 Wied. Ann. (1879), vi. p. 485. 
 
5] REPRESENTATION OF A SERIES OF RESULTS 17 
 
 curve. If, however, as shown in Fig. 4, we plot them on semi-logarithm 
 paper, i.e. paper divided logarithmically in one direction and in equal 
 parts in the other, all the points lie on a straight line AB. 
 
 'OOC 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 JUU 
 
 s. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 z 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 f- 
 
 
 
 
 1 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ttROO 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .00 
 
 
 
 
 
 
 ^ 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ld 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 O 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 z ioo 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 LJ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 ^v 
 
 
 
 
 
 
 
 
 z 
 
 Ul 
 
 a 
 u 
 
 U_2Oo 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 \ 
 
 
 
 
 
 
 u." 00 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 Sk 
 
 
 
 c 
 
 too 
 
 r\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 s, 
 
 \ 
 
 \ 
 
 
 
 
 
 
 ? 
 
 
 1 
 
 1 
 
 I 
 
 1 
 
 
 f 
 
 5 
 
 
 r 
 
 
 3 
 
 < 
 
 ) 
 
 1 
 
 TIME IN DAYS 
 FIG. 4. 
 
 Calling log?/ y', the equation to the straight line AB is 
 
 t/ = h + nx, 
 
 where b is a constant. Hence if A?/ is the change in y' produced by a 
 change Ax in x, we have 
 
 Taking A^=10, the corresponding Ay/' is represented by AC. Since 
 the logarithm of 10 is unity, the length OE represents unity on the scale 
 on which AC represents Ay. Hence by measuring up AC and OE we 
 have 
 
 OE 
 
 = --884, 
 
 since y f decreases as x increases. 
 
18 KEDUCTION OF EXPERIMENTAL RESULTS [ 5 
 
 Thus = - 
 
 From the original equation 
 
 y 
 
 we see that when x = Q, y = a. Hence a = 883, which is the reading on 
 fche y scale where the straight line cuts the axis of y. 
 
 The equation which represents the observation is therefore 
 
 = 883 10-' 0884 *- 
 or in terms of natural logarithms 
 
 The values calculated by this expression are entered in the table, and 
 it will be observed that they agree very fairly with the observed numbers. 
 
 When there is no reason for using any special form of function 
 to give the relation between the two variables involved in a series of 
 measurements, or where by plotting on ordinary and on logarithmic 
 paper no such form of function has been suggested, it will often be 
 found convenient to express the results of a series of measurements by an 
 expression of the form 
 
 y = a + bx + cx 2 + dx? + &c ....... (7) 
 
 where a, b, c, d, &c., are constants. In practice it is hardly ever necessary 
 to go beyond the term in .T 3 , and in fact very seldom beyond the term in 
 x 2 . We have now to consider how the values of the constants in the 
 above expression can be deduced from the experimental numbers. 
 
 Suppose that x v x. 2 , # 3 , and y lt y z , y% are three corresponding sets of 
 values of x and y, we then have, substituting these values in equation (7), 
 and omitting the term in x 3 , 
 
 y l = a + bx l + cx-f 
 
 y a = a + bx% + cx z 2 . 
 
 By solving these three equations we could calculate the values of the 
 three constants a, b, and c. 
 
 By taking any other three pairs of values of x and y, we could in the 
 same way calculate the values of a, b, and c; we should probably, 
 however, find that the values now obtained do not exactly agree with the 
 former values. This is owing to the fact that all measurements are 
 subject to a certain error. If we only have three observations, then the 
 solution of the equations gives the best values of the constants a, b, and 
 c we can deduce from the measurements. In general, however, there are 
 more than three measurements, and, as we have mentioned above, any 
 three will give a set of values for a, b, and c, but the different sets of 
 three will give different values. We have, therefore, to consider how we 
 may best use our available data to calculate the constants. Perhaps the 
 simplest way to proceed is to plot the observations on squared paper and 
 
5] REPRESENTATION OF A SERIES OF RESULTS 19 
 
 to draw a curve evenly between the points. Then choosing three points, 
 one near either end of the curve and one near the middle, read off the 
 corresponding values of x and ?/, and substitute these values in the three 
 equations, and thence deduce the values of the constants. Having done 
 this, proceed to calculate the values of y according to the expression 
 obtained for each value of x corresponding to an observed point, and 
 draw up a table in which there are four columns, the first column 
 containing the value of x, the second the observed values of y, and the 
 third the calculated values of y. In the fourth column enter the 
 differences between the observed and calculated values of y. 
 
 As an example to illustrate the methods of deducing a formula, we 
 may take the following series of values of the temperature t and of 
 a quantity y which have been read off from a curve drawn through the 
 observed points which themselves were not made at equal intervals 
 of temperature : 
 
 
 
 y 
 
 
 , 
 
 
 < 
 
 from Curve. 
 
 calculated 
 from (A). 
 
 Difference. 
 
 calculated 
 from (B). 
 
 Difference. 
 
 10 
 
 20-8 
 
 21-11 
 
 -31 
 
 20-91 
 
 -11 
 
 20 
 
 34-1 
 
 34-10 
 
 + -00 
 
 33-90 
 
 + 20 
 
 30 
 
 49'2 
 
 49-11 
 
 + 09 
 
 48-91 
 
 + 29 
 
 40 
 
 65-9 
 
 66-14 
 
 -24 
 
 65-94 
 
 -04 
 
 50 
 
 84-8 
 
 85-17 
 
 -37 
 
 84-97 
 
 -17 
 
 60 
 
 106-2 
 
 106-22 
 
 -02 
 
 106-02 
 
 + 18 
 
 70 
 
 129-1 
 
 129-27 
 
 -17 
 
 129-07 
 
 + 03 
 
 80 
 
 1541 
 
 154-34 
 
 -24 
 
 154-14 
 
 -04 
 
 90 
 
 180-8 
 
 181-43 
 
 -73 
 
 181-23 
 
 -53 
 
 100 
 
 210-5 
 
 210-52 
 
 00 
 
 209-32 
 
 + 18 
 
 Mean difference . 
 
 
 -20 
 
 
 00 
 
 Since it very much simplifies the calculations if the difference in i 
 between the first and second points chosen is equal to the difference 
 between the second and third, we will choose the three values corre- 
 sponding to the temperatures of 20, 60, and 100. Substituting these 
 values in the equations, we get 
 
 Subtracting 
 Subtracting 
 
 34-1 =a + 206 + 400c 
 106-2 = a + 606 + 3600e 
 210-5 =a+ 1006 +10000c. 
 
 72-1 = 406 + 3200c 
 104-3 = 40& + 6400c. 
 
 09.9 
 ~- 
 3200 
 
 =-01006. 
 
20 REDUCTION OF EXPERIMENTAL RESULTS [ 5 
 
 Hence 
 
 , 72-1 - 3200 x -01006 
 b = 40 
 
 -998 
 
 and a = 34'1 - 20 x -998 - 400 x -01006 
 -10-12. 
 
 Hence the expression for y is 
 
 y = 10-12 + -998* + -01006i 2 . ...... (A) 
 
 and the table shows the agreement between the numbers given by this 
 expression and those read from the curve. 
 
 The mean of the differences, taking account of sign, is - -20, that is, 
 the calculated values are on the whole too small, though of course 
 they agree with the curve values at the temperatures 20, 60, and 100, 
 which were used in deducing the constants a, b, and c. It would 
 evidently be better, therefore, if we were to use slightly different values 
 for the constants, so that on the whole the differences were evenly dis- 
 tributed both as to magnitude and as to sign. To see which of the 
 constants a, I, or c ought to be altered, the most convenient method is 
 to plot the differences against the values of t on squared paper. If the 
 points thus obtained lie fairly evenly about a straight line parallel to the 
 axis of t, then a change in a is what is required. If the points lie evenly 
 about a straight line inclined to the t axis, but crossing this axis mid- 
 way between the origin and the highest value of t, then a change in ft 
 will be sufficient. If, however, the straight line does not pass through this 
 mid point, then a change in both a and b will be necessary. The amount 
 by which b has to be changed can at once be obtained by dividing the 
 difference between the ordinates at the two ends of the line by the 
 difference in the corresponding abscissae. If the line which lies most 
 evenly between the points is curved, then a change in all three constants 
 may be necessary. The sign of the change in c can at once be seen by 
 observing whether the curve is convex or concave to the axis of /. In 
 most cases, however, it will save time to recalculate the values of a, />, 
 and c- by the method which follows rather than to attempt to adjust the 
 values of all three constants. 
 
 In the case before us the points lie fairly evenly on both sides of a 
 straight line drawn parallel to the axis of t corresponding to a difference 
 of - -20. Thus we have to reduce the value of a by this amount, so that 
 the expression for y becomes 
 
 y = 9-92 + -998 + -01006* 2 ....... (B) 
 
 The values for ?/ calculated from this expression are entered in the fifth 
 
 column of the above table and the differences in the sixth column, and 
 
 it will be noticed how much more evenly the differences are distributed. 
 
 By taking any three other points and solving we should get different 
 
5] REPRESENTATION OF A SERIES OF RESULTS 21 
 
 values for the constants a, b, and c. Thus taking the values of y at 10, 
 50, and 90, the expression 
 
 ?/ = 9 -80 + 1 -000* +'01 OOCtf 2 
 
 is obtained, and this would have to l>o adjusted by considering the 
 differences. As any such adjustment is a matter of judgment, a method 
 of combining all the observations when calculating the values of the con- 
 stants, and in which the differences will necessarily be distributed as we 
 require, is often a desideratum. Such a method we now proceed to describe. 
 Suppose that y v y^ . . . //, and x v # 2 , . . . x n are the observed values 
 of y and x, the calculated value of y corresponding to x lt namely 
 
 a + bx 1 + cx^ 
 
 will differ from the observed value y r Let this difference be called 8 l 
 and so on, then we shall have- 
 
 + car,, 2 ) = S, 
 
 Now 6 p 8.,, etc., will be the differences between the observed and calcu- 
 lated values of y, and we require to so choose the constants a, b, and c 
 that these differences shall be as small as possible, but at the same time 
 there shall be no preponderance of positive or negative values. Now in 
 books on the method of least squares 1 it is shown that these conditions 
 are likely to be fulfilled if we so choose the constants that the sum of the 
 squares of the differences is a minimum. Further, that this sum will be 
 a minimum if we deduce the values of the constants in the following 
 manner : 
 
 Take the observational equations in the form 
 
 a + bx 1 + ex* - v/j = 
 
 = Q) 
 
 and first 2 add all the equations together. Then multiply each equation 
 through by the coefficient of b in that equation and add the equations 
 thus obtained. Finally, multiply each equation through by the coefficient 
 of c in that equation and add the equation thus obtained. In this 
 way the following equations, which are called noimal equations, are 
 obtained : 
 
 2a + 62a; + cZa; 2 -2 = ) 
 
 > ..... (10) 
 
 1 Merriman, A Text-book of Least Squares. 
 
 2 We assume that all the observations are made with the same accuracy. If 
 the different observations have different weight, then the observational equations 
 will require adjustment before deducing the normal equation. 
 
22 
 
 REDUCTION OF EXPERIMENTAL RESULTS 
 
 [5 
 
 the sum of the squares of 
 
 where 2# means the sum of the x's, 
 the x's, and so on. 
 
 These three normal equations being solved will give values for 
 the constants a, b, and c, which will fulfil the condition that the 
 sum of the squares of the differences are a minimum. The calculation 
 of the coefficients in the normal equations, if there are many observational 
 equations, is a matter of very considerable labour. The work may, 
 however, be considerably reduced if we have a series of values of y for 
 values of x which differ by a constant quantity, for we may then take 
 this constant quantity as the unit of x while making the calculations. 
 Further, it is advisable to take the first value of x as the zero of x, and 
 the first value of y as the zero of y, so that in terms of these new co- 
 ordinates, when x = 0, y = 0. In this case 2x is the sum of the first 
 n - 1 natural numbers, 2x 2 the sum of the squares of the first n l 
 natural numbers, and so on. 1 
 
 As an illustration of the method, we proceed to apply it to the 
 numbers used in the preceding pages. 
 
 In the first place, we take 10 as our unit of x, and deduct 1 from 
 each of the resulting numbers ; thus on this new scale the first value for 
 x is zero. In the same way we deduct 20*8 from each of the values 
 of y. We have then to calculate for each pair of values of x and y the 
 quotients xy and x 2 y, and obtain the sums ^xy and 2x 2 y, as shown in 
 the following table : 
 
 X. 
 
 y 
 
 xy 
 
 <*9 
 
 
 
 
 
 
 
 
 
 1 
 
 13'3 
 
 13-3 
 
 13-3 
 
 2 
 
 28'4 
 
 56-8 
 
 113-6 
 
 3 
 
 45-1 
 
 135-3 
 
 405-9 
 
 4 
 
 64-0 
 
 256-0 
 
 1024-0 
 
 5 
 
 85'4 
 
 427-0 
 
 2135-0 
 
 6 
 
 108-3 
 
 649-8 
 
 3898-8 
 
 7 
 
 133-3 
 
 933-1 
 
 65317 ' 
 
 8 
 
 160-0 
 
 1280-0 
 
 10240-0 
 
 9 
 
 189-7 
 
 1707-3 
 
 15365-7 
 
 
 
 5458-7 
 
 39728-0 
 
 1 These quantities can be calculated from the following expressions : 
 1+2+3+ +m : 
 
 1) (3m 8 + 3m-l) 
 30 
 
5] REPRESENTATION OF A SERIES OF RESULTS 23 
 We have also to calculate 
 
 +9 3 ) = 2025 
 +9*) = 15333. 
 
 Since with the present values for the quantities x and y, when x = 0, 
 y = 0, the observational equations are of the form 
 
 y = bx + ex 2 , 
 and the normal equations are 
 
 Hence 
 
 Substituting the values for the various sums, we have 
 
 15333 x 54587 - 2025 x 39728 
 "285 x 15333 -(2025) 2 " 
 
 _ 83698200 - 80449000 
 : 4369900 -4100620" 
 
 26861 
 26928 
 
 = -9975. 
 
 c ~ 26928 
 
 Thu 
 
 If, however, we change back to the original values of x and y we have 
 y - 20-8 = 12-065*--i + -9975 (~~~ 
 
 - 1-2065* - 12-065 + -009975(* 2 - 20* + 100) 
 = - 11-068+ 1-0065*+ -009975^ 
 
 (C) 
 
24 
 
 REDUCTION OF EXPERIMENTAL RESULTS 
 
 The following table exhibits the differences obtained by using this 
 expression for y : 
 
 t 
 
 y 
 
 from Curve. 
 
 a 
 
 calculated from (C). 
 
 Difference. 
 
 10 
 
 20-8 
 
 20-80 
 
 + 
 
 20 34-1 
 
 33-85 
 
 + 25 
 
 30 49-2 
 
 48-91 
 
 + 29- 
 
 40 
 
 65-9 
 
 65-95 
 
 - -05 
 
 50 
 
 84-8 
 
 85-00 
 
 -20- 
 
 60 
 
 106-2 
 
 106-03 
 
 + 17 
 
 70 
 
 129-1 
 
 129-07 
 
 + 03 
 
 80 
 
 154-1 
 
 154-09 
 
 + 01 
 
 90 
 
 180-8 
 
 181-12 
 
 - -32 , 
 
 100 210-5 
 
 210-13 
 
 + 47 
 
 When very large numbers of observational equations have to be dealt 
 with, special schemes for conducting the necessary calculations are best 
 adopted, for which see Kohlrausch's Physical Measurements, p. 21. 
 
 As has already been mentioned, to find an algebraical expression to 
 represent the results of a series of observations, where theory does not 
 indicate what the form of the expression should be, is often a matter of 
 extreme difficulty, and there are no hard-and-fast rules which shall guide 
 us in such a search. What has been said in the preceding pages, and 
 the following list of expressions which may be tried, may be of some 
 assistance : 
 
 + bx ... A straight line on squared paper. 
 ^7 ,2 /A curved line on squared paper, the slope being 
 \ proportional to x. 
 
 (4) y = 
 
 (10) 
 
 ax 
 
 \ Plot y and - on squared paper, when a straight 
 I line will be obtained. 
 
 {Plot and y on squared paper, when a straight 
 line will be obtained. 
 
 (v 
 Plot -^ and y on squared paper, when a straight 
 
 line will be obtained. 
 
 TPlot y and x on semi-logarithm paper, when a 
 \ straight line will be obtained, 
 f Plot on logarithm paper, when a straight line will 
 \ be obtained. 
 
MATHEMATICAL TABLES 
 
 25 
 
 6] 
 
 6. Mathematical Tables. The most generally useful aid to the 
 calculations necessary in the laboratory is a table of the logarithms of 
 the natural numbers and of the circular functions. For almost all the 
 calculations necessary in the laboratory a table giving the logarithms to 
 five places of decimals is sufficient, and it is only in very exceptional 
 cases that it is necessary to employ six or seven figures. For many 
 purposes, notably the calculation of corrections and the like, four figures 
 are sufficient. Since a table of the logarithms of the numbers from 100 
 to 1000 to four places can be printed on a sheet about 9 inches by 6 
 inches, such a table is extremely handy, for no turning over of pages is 
 involved, and thus much time is saved. 1 
 
 The five-figure table 2 ought to contain the logarithms of the numbers 
 from 1 to 10000, and will occupy some twenty to thirty pages. It ought 
 also to contain the logarithms of the sines, tangents, cosines, and co- 
 tangents of each minute of arc between and 90. A table of the 
 natural sines and tangents proceeding by half degrees, and giving to three 
 places of decimals, will also be found of service. 
 
 For such cases as demand more than five places, the book of seven- 
 figure logarithms published by Chambers will be found suitable. 
 
 Much time may sometimes be saved by the use of a table, such as 
 that of Barlow (published by Spon), containing the squares, cubes, 
 square roots, cube roots, and reciprocals of numbers from 1 to 10000. 
 
 7. The Slide Rule. In almost all calculations where an accuracy of 
 about one in two or three hundred is sufficient, the most convenient 
 method is to employ a slide rule. The theory of the slide rule is ex- 
 tremely simple, nevertheless it requires some practice to learn to use it 
 with facility. In the physical laboratory, however, so much time can 
 be saved by the use of the slide rule that all students will be well advised 
 to acquire the necessary skill. 
 
 The most usual form of slide rule is shown in Fig. 5. A convenient 
 size consists of a wooden body about 10 inches long, which carries two 
 
 I 
 
 
 FIG. 5. 
 
 divided scales, A and D. Working in a groove in this base there is a 
 slider which carries two scales, B and c. The scales A and B are the 
 same, while the scale c is the same as the scale D. In the case of the 
 
 1 A four-figure table, printed on thin card, is issued by Messrs. Vincent and 
 Wilson, of Cambridge, price 6d. The Board of Education, South Kensington, 
 also issue such a table as part of a set of examination tables, price one penny. 
 
 2 The most convenient five-figure table with which the author is acquainted 
 is one by Dupnis, and is published by Hachette, Paris and London, price 2s. 2d. 
 
26 REDUCTION OF EXPERIMENTAL RESULTS [ 7 
 
 two upper scales the unitf is equal to half the length of the rule, and the 
 divisions are so spaced and numbered that the distance between the 
 extreme left-hand division, which is marked 1, and any division is equal 
 to the logarithm of the number placed opposite this division. Thus the 
 distance between the division 1 and the division 2 is equal to '301 of 
 half the length of the rule, for *301 is the logarithm of 2 to the base ten. 
 This scale is repeated twice on each of the scales A and B, each half 
 being numbered from 1 to 9. The scales c and D are divided in the 
 same logarithmic manner ; the unit taken, however, is the whole length of 
 the scale ; that is, the unit in the case of these two lower scales is twice 
 as great as the unit in the case of the two upper scales. 
 
 A light metal frame, called a cursor, moves in grooves in the body of 
 the instrument, and is fitted with a plate of glass or celluloid, on which 
 is engraved a fine line perpendicular to the length of the scales. This 
 line serves as a reference mark, and also for obtaining the reading on the 
 scale D corresponding to a reading on the scale A. 
 
 Suppose, as shown in the figure, that the slide is drawn to the right 
 so that the unit on the scale B is opposite 2 on the scale A. Then 
 opposite 3 on the scale B we shall have that division of the scale A of 
 which the distance from the zero is equal to the distance from 1 to 2 
 plus the distance from 1 to 3, that is, the reading will represent the 
 sum of these distances. But the distances represent the logarithms of 
 2 and 3 respectively, and therefore the sum of these distances will re- 
 present the logarithm of the product of 2 and 3 ; that is, the reading 
 on the scale A opposite 3 on the scale B will be 6, the product of the 
 numbers 2 and 3. In the same way, opposite each division on the scale 
 B will be the number corresponding to the product of 2 into the reading 
 corresponding to the division considered on scale B. 
 
 In the same way, opposite any division of scale A will be found the 
 quotient of the number corresponding to this division divided by 2. 
 
 Hence by one setting of the slide we can directly read off" the pro- 
 ducts or quotients of any number of numbers by some constant number. 
 This property of the slide rule will be found of great value, for we often 
 require to multiply or divide a column of figures by some common multi- 
 plier or divisor. 
 
 The manner in which the slide rule is used where there are more than 
 two factors to be multiplied or divided is best explained by means of 
 some examples. 
 
 1. Let us suppose that the fraction to be evaluated is 
 
 980 x 2-43 x 10'3 
 27-5 x 6-31 * 
 
 (1) Set 9'8 on the scale A opposite 2'75 on the scale B. 
 
 (2) Set the black line on the cursor to 243 on B. 
 
 (3) Move the slide, the cursor remaining in place, till 6*31 on B is 
 opposite the line on the cursor. 
 
7] THE SLIDE KULE 27 
 
 (4) Opposite 1*03 on B read off the required result, namely, 1'4 15 
 on scale A. 
 
 We thus obtain the value 1415 for the result, and the position of the 
 decimal place remains to be fixed. Inspection of the fraction shows that 
 its value must lie between 100 and 1000, hence we conclude that the 
 true value is 
 
 141-5. 
 
 This example could also be worked on the two lower scales in exactly 
 the same manner, and owing to the more open nature of the scales the 
 accuracy attainable would be increased. In the following two examples 
 the advantage of having the scale duplicated will appear : 
 
 2. 45-5x31-2 
 
 12-6 
 
 Suppose we use the lower scales, and set 1-26 on scale c opposite 4*55 
 on scale D, then the result will be opposite 3 '12 on scale c. This 
 reading on c is, however, beyond the end of scale D. To obtain the 
 reading we must set the cursor on the left-hand end division of scale c, 
 and then move the slide to the left till the right-hand end division on c 
 coincides with the cursor line. The reading on scale D opposite 3-12 on 
 scale c will give the required result, namely, 1-125, or adjusting the 
 position of the decimal point, 
 
 112-5. 
 
 Here to obtain the required result it has been necessary to move the 
 slider through its whole length, and to do this a setting of the cursor 
 has been made independent of those necessary while manipulating the 
 numbers. If the calculation be made by means of the two upper scales, 
 owing to the repetition of the scales this movement of the slide will no 
 longer be necessary, and hence some time will be saved. As, however, 
 the scales are not so finely divided as the lower scales, the accuracy with 
 which the settings can be made is not so great. 
 
 In certain cases, although it will not be necessary to move the slide, 
 it becomes necessary to change from one half of the scales to the other, 
 that is, to multiply or divide by ten. This is illustrated in the following 
 example : 
 
 3. 68-5x3-45x95-2. 
 
 Putting the left-hand (1) division of scale B opposite 6 -85 on scale A, 
 move the cursor to 3 "45 on scale B. If, now, the first division on the left- 
 hand side of scale B is placed at the line of the cursor it will be found 
 that 9-52 on scale B is beyond the end of scale A. If, however, the 
 middle division (marked 1) on scale B is placed so as to coincide with 
 the mark on the cursor, then opposite the 9'52 division on B will be 
 found the required product, namely, 
 
 2-24, 
 that is, the product is 22400. 
 
 The above examples will give some hints as to the methods which 
 
28 REDUCTION OF EXPERIMENTAL RESULTS [ 7 
 
 have to be adopted to perform simple calculations on the slide rule, and 
 it is only for simple calculations that it will be in the long run found 
 advisable to use the rule. In using the rule it is well to remember that 
 whatever the relative positions of the two scales which are adjacent to 
 one another, if a lt a. 2 are two readings on one scale and b v b< 2 are the 
 corresponding readings on the other, then 
 
 a, a., a-, b-. 
 
 -1 = p or J = r*. 
 
 QI p a a -2 2 
 
 Thus, suppose we require to find the quotient of b/c. Calling the quotient 
 a, we have 
 
 b a 
 
 or 
 
 the first of these expressions corresponds to the method of working 
 described above, namely, b on one scale is made to coincide with c on the 
 other, and the quotient is found opposite 1 on this latter scale. The 
 second expression corresponds to the following method of working, which 
 is often more convenient. Bring c on one scale opposite the unit on the 
 other, and then opposite b on the first scale will be found the desired 
 quotient a on the second scale. 
 
 Squares and square roots may be directly read off by means of the 
 cursor. Thus by setting the cursor to any reading on the D scale, the 
 corresponding reading on the A scale will give the square. Conversely, 
 by setting the cursor to a given reading on the A scale where the 
 division line of the cursor cuts the D scale will give the square root. ~ 
 
 Cubes may be found by obtaining the square and then setting the 
 unit division on scale c opposite the given number on scale D, and then 
 reading off the number on scale A opposite the given number on scale B. 
 
 To obtain cube roots the slide must be taken out and turned upside 
 down, so that scale c is opposite scale A. The unit division on scale c 
 must then be placed opposite the given number on scale A, and scales B 
 and D examined for the point where the readings on the two scales 
 which are opposite one another are the same. The reading correspond- 
 ing to this coincidence is the required cube root. Thus if the cube root 
 of 27 is required, the unit on scale c is put opposite the left hand 2 '7 of 
 scale A, when it is found that 3*0 of scale B coincides with 3*0 of scale 
 D, and hence 3 is the required cube root. If the unit on scale c had 
 been placed opposite the right hand 2' 7 on scale A it would be found 
 that nowhere did the readings on scales B and D coincide. 
 
 The back of the slider of most rules carries three other scales ; these 
 are used for finding logarithms, sines, and tangents. They are seldom 
 required in the laboratory, as it is generally more convenient to use tables. 
 An examination of the rule will, however, at once indicate how they are 
 to be used. 
 
(UNIVERSITY! 
 8] ARITHMOMETERS 29 
 
 8. Arithmometers. There are certain calculations which, when a 
 great mass of figures have to be dealt with, are most expeditiously 
 performed by means of an instrument called an arithmometer. A 
 common form of arithmometer is shown in Fig. 6. It is called 
 the Thomas de Colmar pattern, and is very convenient to use, the only 
 objections being the high price (20 to 50) and somewhat large size. 
 
 Iss^^^ 
 
 FIG. 6. 
 
 The instrument consists of a movable portion AB, which on being raised 
 slightly can be slid along to the right, and on closing always rests so 
 that the upper line of perforations AB lie opposite to the number slides 
 EF on the fixed portion. Through each of the holes in the row AB can 
 be seen one of the figures on a series of discs, each disc carrying the 
 numbers from to 9. There is also a lower series of perforations CD ; 
 each of these also allows one of the numbers on a disc similar to the 
 upper row to be read. The sliders EF can be moved up grooves, and 
 a small pointer attached to each slider moves on a scale which is 
 numbered from to 9. A handle G can be put in two positions, one 
 labelled " addition and multiplication," and the other " subtraction and 
 division." A handle H can be rotated in the clockwise direction, and 
 serves to set the mechanism working. Two levers, A and D, when 
 pulled to the left and right respectively after the slider is raised, serve 
 to set all the discs in the two rows to zero. 
 
 If all the sliders EF are at zero, then turning the handle H produces 
 no result on the upper row of figures If, however, any of the sliders 
 are moved up, then on turning the handle through a complete turn the 
 discs on the upper row AB turn so that the numbers showing through 
 the windows are the same as the numbers to which the sliders which 
 happen to be exactly opposite them are set. At the same time the disc 
 on the lower row CD, which is opposite the handle H, turns so that 
 1 appears, this representing the number of times the handle has been 
 turned. If, now, the handle is given a second turn, then the readings on 
 the upper row of discs will increase by an amount corresponding to the 
 readings of the sliders EF, while the fact that the handle has been 
 
30 REDUCTION OF EXPERIMENTAL RESULTS [ 8 
 
 turned twice will be indicated by 2 appearing on the disc of the lower 
 row immediately opposite the handle. Now the number represented on 
 the sliders having been twice added to the reading (zero) on the upper 
 row of discs, the number which is now shown on these discs will be 
 twice the number shown on the sliders. In the same way, if the handle 
 is turned three times, the number shown on the upper row of discs will 
 be three times the number shown on the sliders, and so on. Let us 
 suppose that the slider being as far to the left as it will go, and all the 
 discs showing zero, we place the number 111111 on the sliders, that is, 
 move each slider up till its pointer points to 1 on the scale. If, then, 
 the handle is turned once the number 111111 will appear on the last 
 six of the discs on the upper row, while 1 will appear on the last disc 
 of the lower row, that is, on the disc opposite the handle. Next move 
 the slider one notch to the right so that the second disc on the lower 
 row is opposite the handle. If, now, the handle is turned once, each of 
 the discs which is now opposite one of the sliders will have its reading 
 increased by unity, that is, by the amount shown on the corresponding 
 slider. The result now shown on the upper row of discs is not twice 
 111111, as it would have been had the slider not been moved, but it is 
 the product of 11 arid 111111, for the machine has performed exactly the 
 same sequence of operations that is used when doing multiplication by 
 the ordinary method, namely, it has multiplied the multiplicand by the 
 unit digit of the multiplier, then multiplied by the ten digit, displacing 
 this quotient one place to the left, and then added the two results 
 together. If the handle is turned twice when in the second position, 
 the product of 111111 by 21 will appear on the upper row of discs, 
 while the lower row will exhibit the multiplier 21. In the same way, 
 by displacing the slider another place to the right and turning the 
 handle once, we shall obtain the product of 111111 by 121, and so on. 
 
 If the lever G is placed opposite the inscription "subtract or divide," 
 then on turning the handle once the number represented on the slides 
 will be subtracted from any number which may be exhibited on the 
 upper row of discs. To perform division, the dividend is placed on the 
 upper row of discs as near the left-hand end of the slider as possible. 
 This can be done either by setting this number on the slides and then 
 turning the handle once while the lever G is at "add," or, what is 
 generally more convenient, by raising the slider and then turning, by 
 means of a small stud which projects immediately below the window, 
 each disc till the required digit appears in the window. The divisor 
 is then placed on the slides, also as far to the left as possible. The 
 slider is moved till the number shown on the upper row of discs is 
 larger than that shown on the slides, the unit digit in the two cases 
 being supposed to be vertically over one another. The lever G being 
 at "subtract," the handle is turned till the number on the discs is 
 smaller than that on the slides. The slider is then moved one place 
 to the left, and the handle again turned till the number on the discs 
 is less than that on the slides, and so on till the required number of 
 
8] ARITHMOMETERS 31 
 
 places is obtained, when the quotient will appear on the lower row of 
 discs. If, when performing this operation, the handle is accidentally 
 turned once too often, this will be indicated by a number of nines 
 appearing to the left of the numbers on the upper row of discs. In 
 such a case the lever must be put at " add " and the handle given one 
 turn, which will exactly annul the superfluous turn which has been 
 given, and then again setting the lever at " subtract," the slider can be 
 moved one step to the left and the division proceeded with. It will at 
 once be evident that the machine proceeds in exactly the same manner 
 as is done when doing long division by the ordinary method. 
 
 The arithmometer does not do division with anything like the rapidity 
 with which it does multiplication. Hence if several numbers have to be 
 divided by the same number, it will save time if the reciprocal of this 
 number is obtained, either by means of the machine or from tables, arid 
 then each of the numbers is multiplied by this reciprocal in the manner 
 described below. 
 
 When a number of numbers which do not differ much have to be 
 multiplied by the same number, an operation which is of frequent 
 occurrence in Physics, after performing the first multiplication it is not 
 necessary to wipe out the numbers which appear on the discs before 
 proceeding to the second multiplication. All that has to be done is by 
 suitably manipulating the slide and the lever G, turning the handle 
 when necessary of course, to change the reading on the lower set of discs 
 till the second number appears there, when the product will be given on 
 the upper row of discs. Thus, suppose the common multiplier is 64799, 
 and that two of the numbers to be multiplied are 17628 and 17635. 
 First set the common multiplier, 64799, on the slides. Then starting 
 with the slider as far to the left as it will go, turn the handle eight 
 times; move the slider one place to the right and turn the handle 
 twice ; move slider another place to right and turn six times, and so on. 
 The product is 
 
 1142276772, 
 
 and as many digits of this as are required must be noted down. Next 
 placing the slider so that the 8 of the number on the lower row of discs 
 is opposite the handle and setting the lever G at " subtract," turn the 
 handle three times, so that the 8 becomes 5. Next move the slider one 
 place to the right, so that the 2 comes opposite the handle. Place the 
 lever G at "add" and turn the handle once, so that the 2 becomes 3. 
 The number now shown on the lower row of discs is 17635, and the 
 product of this number by 64799, namely, 
 
 1142730365, 
 
 is shown on the upper row of discs. Thus we have obtained the second 
 product by only turning the handle four times, while if we had started 
 ab initio the handle would require to. have been turned twenty-two times. 
 The above arithmometer may be used for adding together long columns 
 of figures. For this purpose each number is in turn placed on the slides, 
 
32 REDUCTION OF EXPERIMENTAL RESULTS [ 9 
 
 and the handle given one turn so as to add the number on the slides to 
 the number already shown on the upper row of discs. The method is 
 not a very rapid one, since the slides have to be manipulated each time, 
 and, where much addition has to be performed, an instrument called a 
 comptometer 1 may with advantage be used. In this machine there are 
 a number of keys, something like those of a typewriter, arranged in 
 columns, each column containing keys numbered from to 9, and when- 
 ever a key is depressed the number on it is added to the number which 
 appears at the top of that column on the recording dials. 
 
 9. Determination of the Area of Plane Figures. A problem which 
 occurs fairly frequently in Physics is the measurement of the area of 
 a plane figure. Thus the work done by the gas in the cylinder of a 
 steam or gas engine is obtained by measuring the area of the indicator 
 diagram. This area can either be obtained by measuring a number of 
 ordinates and using one of the rules given below, or an instrument called 
 a planimeter can be employed. 
 
 The most convenient rule to employ when calculating the areas of a 
 figure from the measurement of a number of ordinates is as follows : 
 
 Draw two parallel lines touching the extremities of the figure, and 
 a line at right angles to these two lines. Then divide the portion of 
 this cross line intercepted between the parallel lines into an even number 
 of parts (preferably 20, 30, 40), and through the 1st, 3rd, 5th, &c., 
 of these division points draw lines parallel to the first two lines. In 
 this way half as many of these intermediate ordinates will be obtained 
 as there were parts into which the cross line was divided, the interval 
 between the end ordinates and the lines touching the boundaries of the 
 figure being half the interval between the intermediate ordinates. The 
 length of each of the intermediate ordinates intercepted between the 
 boundaries of the figure is measured, and the mean value of this inter- 
 cept is multiplied by the length of the cross line included between the 
 two tangent lines, and the product is the area of the figure. The appli- 
 cation of the method to an indicator diagram is shown in Fig. 7. Where 
 a number of indicator diagrams have to be measured, it will save much 
 time if a piece of glass be prepared with a scale, as shown in Fig. 7. 
 This can be prepared either by ruling on the glass or by drawing the 
 scale on paper, and then photographing it and printing a positive on a 
 photographic plate. 
 
 Another rule frequently employed, and known as Simpson's rule, is 
 as follows : 
 
 Divide the area into an even number of strips by an odd number of 
 equally spaced ordinates, the first and last touching the boundary of the 
 figure. Measure the lengths of the ordinates intercepted by the figure, 
 and then add together the sum of the first and last ordinates, four times 
 the sum of the even ordinates, and twice the sum of the odd ordinates, 
 
 1 Made by Felt & Tarrant Manufacturing Company, 52 Illinois Street, Chicago, 
 U.S.A. Price, 25. 
 
10] THE PLANIMETER 33 
 
 omitting the first and last. This total must be multiplied by one-third 
 the distance between consecutive ordinates to give the area. 
 
 STROKE 10-16 CM. > 
 
 FIG. 7. 
 
 It will be seen that the first rule is considerably easier to apply, 
 and the result is very little, if at all, less accurate than that given by 
 Simpson's rule. 
 
 10. The Planimeter. The commonest form of planimeter is that 
 invented by Amsler, and shown in Fig. 8. It consists of two pivoted 
 arms AB and CD, with a sharp point at D, which, on being pressed into 
 the paper, serves as a centre round which the instrument can turn. A 
 small weight is generally attached above D to keep this point in the hole 
 
 in the paper. At the end A of the other arm is a tracing-point, which 
 is guided by hand round the perimeter of the figure of which the area 
 is being measured. Attached to the arm AB is a wheel E, which rests on 
 the paper, and can turn about an axis parallel to AB. The edge of this 
 wheel is fairly sharp, and attached to the side of the wheel is a divided 
 drum, by means of which, and a fixed vernier F, the angle through which 
 the wheel has turned can be read. The whole turns of the wheel are 
 registered by means of a divided disc G, which is rotated by a worm 
 attached to the spindle which carries the wheel E. By means of a clamp 
 
 c 
 
34 REDUCTION OF EXPERIMENTAL RESULTS [ 10 
 
 i, and a fine adjustment H, the distance between the point where the 
 arm CD is pivoted and the tracing-point A can be varied. By varying 
 this length the factor by which the change in reading on the wheel has 
 to be multiplied to give the area can be altered. There are generally 
 division lines on the arm AB, so that when these are made to coincide 
 with a fixed mark on the part which carries the pivot for the arm CD the 
 readings on the wheel give the area in square centimetres or square 
 inches, as the case may be. 
 
 The position of the fixed point D having been so chosen that it is 
 possible to trace the whole of the perimeter of the figure with the point 
 A, this pointer is brought to some marked position on the curve, and 
 either the integrating wheel E is adjusted to zero, or the reading is 
 noted. The tracing-point A is then carried round the perimeter of the 
 figure in the clockwise direction and brought back to the starting-point, 
 and the reading of the wheel again noted. The difference in the read- 
 ings of the wheel at the end and the beginning, when multiplied by the 
 appropriate factor, as shown on the side of the arm AB, will give the 
 area of the figure. 
 
 If the figure is so large that the instrument will not stretch far 
 enough to allow of the tracing-point being taken right round, the figure 
 must be divided into parts, and the area of each part be determined 
 separately. 
 
 It is only when the fixed point D is outside the area to be measured 
 that the difference in the readings of the integrating wheel gives directly 
 the area. If the fixed point lies within the area, then on tracing the 
 perimeter of the figure the instrument will make a whole revolution 
 round the fixed point. In such a case the area, as given by the differ- 
 ence in the readings of the wheel, must be added to the area of what is 
 called the datum circle, the area of this circle being marked by the maker 
 on the top of the arm AB. The datum circle is the circle swept out by 
 the pointer when the two arms make such an angle with one another 
 that the plane of the edge of the integrating wheel passes through the 
 fixed point. It is evident if, while the arms make this angle, the instru- 
 ment is rotated round the fixed point; since the wheel will always be 
 moving in a direction parallel to its axis, it will not rotate, and hence 
 will not record the area swept out by the tracing-point. If the area of 
 that portion of the figure which lies outside the datum circle is less than 
 the difference in area between the remainder of the figure and the datum 
 circle, then on tracing out the figure in the clockwise direction the read- 
 ing on the integrating wheel will decrease. This decrease in the reading 
 corresponds to a negative area, and hence to obtain the area of the figure 
 this area, as read from the wheel, has to be subtracted from the area of 
 the datum circle to obtain the area of the figure. 
 
 For the theory of the Amsler planimeter the student must refer to 
 books on the integral calculus. A most interesting description of the 
 theory of planimeters in general, and of Amsler's planimeter in particular, 
 with a discussion of the errors to which it is liable, by Professor O. Hen- 
 
11] HARMONIC ANALYSIS 35 
 
 rici, will be found on p. 496 of the Report of the British Association for 
 the Oxford Meeting of 1894. 
 
 11. Harmonic Analysis. According to Fourier's theory, any finite 
 periodic function can be expressed as a series of terms, each of which is 
 a simple harmonic function of this variable. Thus if F is any periodic 
 function of the time, according to Fourier's theory F can, by suitably 
 choosing the constants A^ A lt A 2 , &c., v B^ &c., be expressed by a 
 series of terms as follows : 
 
 F= AQ + A l sin kt + A 2 sin 2kt + A B sin 3kt + <kc. ) ,^ 
 
 + B l cos kt + J5 2 cos 2kt + B B cos Ut + &c. j ' 
 
 This series, containing, as it does, terms involving both sines and cosines, 
 is sometimes less convenient than the following form : 
 
 F= A + M l sin (kt + ej + If 2 sin (2kt + e. 2 ) + &c., 
 
 where M v M. 2 , M^, &c., and e v e 2 , e B , &c., are constants. 
 
 The connection between the constants which appear in the two series 
 is as follows : 
 
 , &c, 
 
 73 D 
 
 tan e x = -i; tan e 2 = 2 , &c. 
 
 A l A 2 
 
 A problem which occurs in the study of alternating current machinery, 
 in acoustics, and in meteorology, is that, given the form of the curve 
 which expresses the connection between the function F and the vari- 
 able t , to deduce the values of the coefficients of the terms in the Fourier 
 expansion. Although instruments have been devised for mechanically 
 performing the analysis, yet owing to their great price and scarcity it is 
 hardly necessary to describe them here. A large number of methods 
 have been devised for performing the analysis by calculation both with 
 and without the aid of graphical methods. 
 
 The principles on which the methods of determining the values of 
 the coefficients of a Fourier series to represent any given periodic curve 
 are founded are as follows. By an easy application of the integral 
 calculus it can be shown that the following definite integrals, taken over a 
 whole period of the circular function, have the values shown, the constants 
 a and h having any values, and k being 27T/7 7 , where T is the period : 
 
 /T rT 
 
 (a) I sin akt .dt= \ cos akt .dt=Q, 
 Jo Jo 
 
 rT 
 
 (/,) I sin akt . cos bkt .dt = 0, 
 
 J o 
 
 rT 
 
 (e) I sin akt . sin bkt . dt = 0, 
 
 ./o 
 
36 REDUCTION OF EXPERIMENTAL RESULTS [ 11 
 
 cos akt . cos bkt . dt = 0, 
 
 rr CT rp 
 
 (e) I sin 2 akt .dt= \ cos 2 akt .dt=--. 
 Jo Jo A 
 
 Suppose that we multiply both sides of \he expression (1) given on 
 p. 35 for the Fourier series by sin nkt and integrate between and T, 
 that is, over a complete period, we have 
 
 fT CT rT 
 
 \ Fsinnkt.dt=A () l sin nkt . dt + A^ I sin kt . sin nkt . dt + . . . 
 
 Jo Jo Jo 
 
 I'T 
 
 + A n l sin' 2 nkt. dt 
 
 J o 
 
 + B, I cos kt . sin nkt .dt + ... + B n I cos nkt . sin nkt . dt. 
 
 Jo Jo 
 
 Now the left-hand side of this equation does not necessarily vanish 
 for F is a function of t. Of the terms on the right-hand side the onlj 
 one which is not zero, according to the relations given above, is the tern 
 involving A n , and the integral will have the value 7 7 /2. Hence 
 
 A n = -l F sin nkt.dt. 
 
 TJo 
 
 But I F sin nkt . dt 
 
 is the mean value of the ordinates of the original curve each multipliec 
 by the appropriate value of sin nkt. 
 
 Hence the coefficient A n is twice the mean value of the product o: 
 each ordinate corresponding to an epoch kt multiplied by the sine o: 
 n times this epoch. Similarly for the cosine terms we must multiply 
 through by cos nkt and obtain the mean ordinate. Thus to analyse th< 
 curve we have to multiply each ordinate by sin kt, cos kt, sin '2kf 
 cos 2kt, and so on, and obtain the means. To perform this multiplica 
 tion for many terms is' very laborious ; the calculation may, however, b< 
 considerably simplified by making use of the fact that as kt increase: 
 from to 360 the sine and cosine each pass four times through thi 
 same numerical value, twice positive and twice negative. Hence certaii 
 of the ordinates will have to be multiplied by the same value of the sini 
 or cosine as the case may be, and therefore labour will be saved b] 
 adding or subtracting the ordinates first and then performing the multi 
 plication on the sum or difference. 
 
 Various schedules 1 have been prepared which give directions fo 
 
 1 In addition to the methods described below, a series of schedules has beei 
 prepared by General Strachey which are particularly applicable to meteoro 
 logical phenomena of which we have either hourly observations throughout tb 
 day or daily observations throughout the year. These schedules and a numbe 
 of tables to assist in the computations are printed in Part IV. of the Ilourl, 
 Readings, 1884, published by the Meteorological Council (Eyre & Spottiswoode). 
 
11] HARMONIC ANALYSIS 37 
 
 performing the necessary additions, subtractions, &c., and of these those 
 prepared by Professor C. Runge are particularly suited to the wants of 
 the physicist. Runge l gives two schemes, one in which twelve equi- 
 distant ordinates are used, and the other in which there are thirty-six 
 ordinates. Since the scheme for thirty-six ordinates is somewhat com- 
 plicated, and for most purposes sufficient accuracy can be obtained with 
 twelve ordinates, the scheme for this latter will alone be given. 
 
 Let the values of F taken at twelve equidistant points throughout 
 the period be represented by y v y 2 , y^ . . . y l2 . These values must 
 then be arranged in two horizontal lines, as shown in the schedule 
 below, the lower line being written in the reverse direction. The 
 sums j, a 2 , a 3 , &c., and the differences e 2 , e 3 , e 4 , &c., of the various 
 columns having been obtained, these numbers are written down in the 
 order shown in Schedules 2 and 3, and the sums and differences of the 
 columns of these tables obtained. These sums and differences are used to 
 construct Schedules 4 and 5, and in one case the sums, and in the other 
 the differences, are obtained. 
 
 When obtaining the sums and differences great care must be taken 
 that the signs are correct. This is a matter of such vital importance 
 that the author considers it advisable whenever any of the measured 
 values of F are negative to increase each of the values of y by 
 such an amount that all the numbers are positive. The only effect of 
 this increase will be to increase the constant term A by an equal 
 amount. 
 
 The numbers obtained by these various operations must now be 
 entered in the indicated places in the Schedule 6, each number having 
 first been multiplied by the number given at the left-hand end of the 
 horizontal row in which it is to be entered. Thus/j has to be multiplied 
 by 0'5 and then entered at the top left-hand corner of the division 
 which is headed 1 and 5. It will be observed that the numbers in each 
 of these divisions are .not all arranged vertically under one another, but 
 form two columns. The totals obtained by separately adding up the 
 numbers in these columns are to be entered at the bottom of the 
 divisions, the sum corresponding to the first column being placed verti- 
 cally over that obtained from the second column. The sums and 
 differences of these totals are then obtained, and these numbers when 
 divided by 6 (in the case of A Q divided by 12) give the values of the 
 coefficients. 
 
 (1) 
 
 Values of jr. . ^ y, y B y y. ^ 
 Ordinates / . . y 12 y n y 10 y Q y 8 ?/ y 
 
 Sums . . .a, etc, a* a A a, a r a* 
 
 1 J o 4 O i 
 
 Differences . . e z e 3 e 4 e 5 e 6 
 
 1 C. Runge, Zdtschrift fur Mathematik und Physik (1903), 48, 443. 
 
38 REDUCTION OF EXPERIMENTAL RESULTS 
 
 (2) (3) 
 
 [n 
 
 Sums . . . b l 
 Differences . c x 
 
 (*) 
 
 6, 
 
 1 /, 
 
 'i ft 
 
 (5) 
 
 Sums . . d l 
 
 Differences . h^ 
 (6) 
 
 Multipliers, 
 
 Sine Terms. 
 
 Cosine Terms. 
 
 1st. 5th. 
 
 2nd. 4th. 
 
 3rd. 
 
 1st. 5th. 
 
 2nd. 4th. 
 
 3rd. 
 
 0. Oth. 
 
 5 
 
 866 
 1 
 
 /3 
 
 9i 92 
 
 *l 
 
 C 3 
 C-2 
 
 c l 
 
 -63 & 2 
 b, -\ 
 
 h 2 
 
 rfj d 2 
 
 Total 1st column 
 Total 2nd column 
 
 
 
 ... 
 
 ... 
 
 ... 
 
 ... 
 
 
 ... 
 
 Sums 
 
 6^j 
 
 6-4 5 
 
 64 2 
 6^1 4 
 
 6^3 
 
 BB, 
 
 6^5 
 
 65. 2 
 6* 4 
 
 6^3 
 
 12A 
 12t 
 
 Differences .... 
 
 15 
 
 10 
 
 10 
 
 It will be observed that since the multiplication by 0*5 and by 1 can 
 
 be done as the figures are being 
 copied down, there only remain four 
 figures which have to be multiplied 
 by 0-886, that is, sine 60, all the 
 rest being addition and subtraction. 
 The following example taken from 
 Professor Runge's paper will show 
 how the calculation ought to be 
 set out, while in Fig. 9 is given 
 the form of the curve which the 
 numbers represent : 
 
 Values of 
 Ordinates 
 
 14-0 
 
 15-8 12-0 
 -6-8 -9-9 
 
 7-7 
 -9-3 
 
 4-3 1-4 
 -8-2 -6-8 
 
 -4-6 
 
 Sums (a) . .14-0 + 9-0 +2-1 -1-6 - 3'9 - 5'4 
 
 Differences (e) ... +22'6 +21-9 +17'0 +12-5 + 8'2 
 
 14-0 9-0 2-1 -1-6 22-6 21-9 
 
 -4-6 -5-4 -3-9 8-2 12'5 
 
 Sums (b) . 9-4 
 Differences (c) 18 '6 
 
 3-6 -1-8 
 14-4 6-0 
 
 (/)30-8 
 (g) 14-4 
 
 34-4 
 9-4 
 
 -4-6 
 
 17-0 
 
 17-0 
 
11] 
 
 HARMONIC ANALYSIS 
 
 39 
 
 9-4 
 -1-8 
 
 3-6 
 -1-6 
 
 30-8 
 17-0 
 
 18-6 
 6-0 
 
 Sums ((?) 7-6 2-0 Differences (h) 13-8 12'6 
 
 Multipliers. 
 
 Sine Terms. 
 
 Cosine Terms. 
 
 1st. 5th. J2nd. 4th. 
 
 3rd. 
 
 1st. 5th. 
 
 2nd. 4th. 
 
 3rd. 
 
 0. 6th. 
 
 0-5 
 866 
 
 i-o 
 
 15-4 
 
 29-8 
 17-0 
 
 12-5 8-1 
 
 13-8 
 
 3-0 
 12-5 
 18-6 
 
 0-9 1-8 
 9-4 1-6 
 
 12-6 
 
 7'6 2-0 
 
 1st column . 
 2nd column . 
 
 32-4 
 
 29-8 
 
 12-5 
 
 81 
 
 ... 
 
 21-6 
 12-5 
 
 10-3 
 3-4 
 
 ... 
 
 7-6 
 
 2-0 
 
 Sums .... 
 Differences . 
 
 62-2 
 2-6 
 
 20-6 
 4-4 
 
 13-8 
 
 34-1 
 9-1 
 
 13-7 
 6-9 
 
 12'6 
 
 9-6 
 5-6 
 
 
 ^ = 10-37 
 A 5 = 0-43 
 
 4 = 3-43 
 4 = 0-73 
 
 4 3 = 2-30 
 
 ^ = 5-68 
 5 =1-52 
 
 ^2 = 2-28 
 4 =M5 
 
 53=2-10 
 
 A ='80 
 5 fi =-47 
 
 ^'=0-80 + 10-37 sin Atf + 3'43 sin 2kt + 2*30 sin 3kt 
 
 + 0'73 sin &kt + 0'43 sin 5kt 
 + 5-68 cos kt +2-28 cos 2^ + 2-10 cos 3kt 
 
 + 1-15 cos kt + 1'52 cos 5/^ + 0*47 cos Q/cf. 
 
 Where only terms up to the third harmonic are required, Professor 
 Silvanus Thompson has devised the following very simple method of per- 
 forming the calculation. The values of the function are obtained every 
 60, giving six values of ?/, and an additional ordinate must be measured 
 at 90 ; let this ordinate be called y 90 . Then arrange as follows, and take 
 the sums and differences : 
 
 y\ 2/2 2/3 
 
 Sums . . . j a 2 a 3 
 Differences . e l e. 2 e z 
 
 The values of the coefficients are then : 
 
 AQ = (a x + 2 + a 3 ) -r 6 
 A l = (e l + e 2 ) -j- 3*464 
 ^ 2 = ( ai _a 2 )^-3-464 
 A B = A + A l - 2 -y QQ 
 B l = (e l -e 2 -2e 3 )-^Q 
 ^ 2 = (2a 3 -a 1 - 2 ) + 6 
 
 In the case of most of the curves which occur in the study of alter- 
 nating currents, the two halves of the curve are always similar. In such 
 
40 REDUCTION OF EXPERIMENTAL RESULTS [ 11 
 
 a case the Fourier series contains only odd terms, and the analysis can 
 be considerably simplified. Professor S. P. Thompson 1 has modified 
 Runge's method for use when only odd harmonics appear in the following 
 manner : 
 
 Draw the curve with the axis of x halfway between the highest and 
 lowest points, so that in the expression there will be no constant term. 
 Also, take the origin where the curve crosses the axis of x and measure 
 the height of eleven ordinates, so spaced as to divide the half period 
 into twelve equal parts, and call the values of these ordinates y v y 2 , 
 
 y - - 2/11- 
 
 Arrange these values of y as under, and add and subtract : 
 
 y\ 
 
 2/2 3^3 
 
 2/io 2/9 
 
 Sums . . 
 Differences 
 
 2/4 
 
 j/8 
 
 a A 
 
 2/6 
 
 y- 
 -, 
 
 *j o 2 03 04 c< 5 
 
 Group these numbers as follows to obtain the numbers b v 6 2 , /: 
 
 Next insert the numbers obtained in the following schedule, having first 
 multiplied each by the multiplier given at the left-hand end of the line 
 on which the product has to be entered : 
 
 Multipliers. 
 
 Sine Terms. 
 
 Cosine Terms. 
 
 1st. llth. 
 
 3rd. 9th. 
 
 5th. 7th. 
 
 1st. llth. 
 
 3rd. 9th. 
 
 5th. 7th. 
 
 0-259 
 0-500 
 0-707 
 0-866 
 0-966 
 1-000 
 
 "5 
 
 , 
 
 6 
 
 5 
 
 ' 
 
 C 4 
 
 Total 1st column 
 Total 2nd column 
 
 
 ... 
 
 
 ... 
 
 ... 
 
 6^5 
 
 Sums 
 
 ell, 
 
 6^9 
 
 el 5 
 
 at, 
 
 6,69 
 
 Differences .... 
 
 1 Proceedings of the Physical Society of London (1905), xix. p. 443. 
 
CHAPTER II 
 
 MEASUREMENTS OF LENGTH 
 
 12. Standards of Length. Since in almost every case the quantity 
 which is actually observed when making any kind of physical measure- 
 ment is a length, the consideration of this subject is one of particular 
 importance. Thus, although the consideration of many arrangements, 
 which are of special assistance in measuring the lengths which are 
 required in certain measurements, are more conveniently dealt with in 
 connection with the descriptions of these measurements, yet it will be 
 convenient to describe at some length the arrangements more commonly 
 employed at the very commencement. The student, at the same time, 
 will find that the practice gained by performing the experiments given to 
 illustrate the use of the various instruments and their adjustment a most 
 valuable introduction, to those more elaborate measurements which inci- 
 dentally involve measurements of length. 
 
 We have to consider two things when dealing with the measurement 
 of length, namely, the most suitable form to give the standard scale with 
 which the distance to be measured is to be compared, and how it is to be 
 graduated, and, secondly, the manner in which the distance to be measured 
 is to be compared with the standard scale. 
 
 The conditions which a standard scale has to fulfil are as follows : 
 (1) The bar of metal on which the divisions are traced must be so stiff 
 that when a reasonable amount of care is taken in supporting the scale 
 it shall not, owing to its weight, bend appreciably ; (2) the material of 
 which the scale is composed must be such that whenever the temperature 
 has any given value the distance between any two divisions must be the 
 same ; (3) the division lines must be fine and of uniform thickness 
 throughout the scale. 
 
 In addition, it is a great convenience if the coefficient of expansion 
 of the material of which the scale is composed is small, so that the 
 true distance between the graduations does not alter much with the 
 temperature. It is also a great convenience if the graduations are so 
 evenly spaced that, except in the case of measurements of the utmost 
 refinement, we may take them as perfectly equal, and so do not have to 
 apply a calibration correction. Although it is possible to obtain scales 
 which are so accurately divided as to make the application of a calibra- 
 tion correction in most cases unnecessary, this accuracy cannot be taken 
 for granted. 
 
 The cross-section adopted for accurate length standards of 50 cm. 
 
 41 
 
42 MEASUREMENTS OF LENGTH [ 12 
 
 and over is shown in Fig. 10. The divisions of the scale are engraved 
 on the surface AB of the cross-bar of the H. The advantages of this 
 shape are twofold. In the first place, this form of cross-section gives very 
 great stiffness combined with lightness, so that the 
 bar does not bend owing to its own weight. This 
 question of the bending of length standards under 
 their own weight has been investigated by Chree, 1 
 who has shown that a scale having this cross-section 
 and supported at two points only, the points of sup- 
 port being at 0'22 of the length of the bar from 
 the ends, does not bend sufficiently to alter the 
 FIG. 10. distance between any two divisions of the scale by 
 
 an appreciable quantity. Secondly, the graduations 
 being placed where they are, if the bar bends, that portion of the bar 
 which carries the graduations is neither stretched nor compressed owing 
 to the bending. The reason for this is that the scale occupies the 
 position of the neutral fibres, so that, while the metal on one side is 
 stretched, that on the other side is compressed. Incidentally this form of 
 scale serves to protect the graduations from accidental injury. 
 
 Up to the last year or two the material of which almost all standard 
 scales were constructed was brass, the divisions in the finer scales being 
 engraved on a thin strip of silver, which was let infc> the upper surface of 
 the scale. The disadvantage of brass is that its coefficient of expan- 
 sion with rise of temperature is fairly large, namely, about O0000187. 
 Within the last few years Guillaume has carried on an important series 
 of investigations as to the best material to use for the construction of 
 scales, and as a result has discovered an alloy which has a very much 
 smaller coefficie-nt of expansion than brass, and yet seems quite as un- 
 alterable. This alloy consists of about thirty-six parts of nickel to sixty- 
 four parts of iron, and its coefficient of thermal expansion (linear) is 
 0*0000009. This material takes a fine polish, so that the graduations can 
 be made directly on the nickel alloy, and it is not necessary to insert a 
 strip of silver to receive the divisions. It has the further advantage that 
 it does not tarnish or rust when exposed to air. 
 
 Standard scales about a metre long are generally graduated in milli- 
 metres. In the case of very finely divided scales it is usual to subdivide 
 a single millimetre at either end into tenths. It is also of importance to 
 have an additional graduation put on at either end. Thus if the scale is 
 intended to measure up to a metre an additional millimetre division ought 
 to be placed at either end, so that the total length of the divisions is 
 1002 mm. These additional divisions are used when standardising the 
 scale. 
 
 Th"e divisions of a good standard scale ought not to be more than two 
 or three microns 2 wide, and hence they are almost invisible to the naked 
 
 1 Proc. Physical Society of London (1901), xviii. ; and Philosophical Magazine 
 (1901), 2, p. 532. 
 
 2 A micron ^ is a thousandth of a millimetre = '001 mm. 
 
13] THE VERNIER 43 
 
 eye. Some kind of auxiliary apparatus has, therefore, to be employed to 
 read the scale. Again, the form of the scale does not permit of the object 
 which has to be measured being placed alongside the divisions, so that some 
 instrument is required to transfer the length from the object to the scale. 
 Finally, it is generally necessary to make the measurements accurate to 
 within a quantity which is considerably smaller than the smallest sub- 
 division of the scale. We shall first consider the various methods which 
 can be employed for the subdivision of the scale divisions, and in this 
 connection shall for convenience deal not only with devices suitable for 
 use with the form of scale described above, but also with other devices 
 which are employed for this purpose, though they could not be applied to 
 this particular form of scale. 
 
 13. The Vernier. The vernier consists of a small auxiliary scale 
 which can slide alongside the graduations of the main scale. The dis- 
 tance between the graduations of the vernier is either a little smaller or a 
 little larger than the distance between the graduations of the scale. Thus, 
 frequently, ten divisions of the vernier are equal to nine divisions of the 
 scale, so that each division 
 
 10 15 
 
 of the vernier is "1 of a 
 division less than the divi- 
 sions of the scale. Sup- A I I I I | I I I IT 
 pose, as in Fig. 11, the [0 5 10 
 
 zero division of the vernier FlG 
 
 lies between the fourth and 
 fifth divisions of the scale, the reading being 4 + x divisions where x is 
 a fraction of a division. Then, since the length of a vernier division 
 is less than the length of a scale division by a tenth of a scale 
 division, .. the interval between the second division of the vernier and 
 the fifth division of the scale is x- 'I. In the same way the interval 
 between the third division of the vernier and the sixth division of the 
 scale is x - '2, and so on. But in the figure the interval between the 
 seventh division of the vernier and the eleventh division of the scale 
 is zero ; that is, we have 
 
 z-'7 = 0, 
 or #='7. 
 
 Hence by simply noting the division of the vernier which coincides with 
 a division of the scale we get the fraction of a division by which the 
 zero of the vernier is distant from the division of the scale immediately 
 below. This result may be generalised as follows : If m divisions of 
 the vernier are equal in length to m + 1 divisions of the scale, and co- 
 incidence occurs at the Tith division of the vernier, then the reading of 
 the vernier, i.e. the distance between the zero of the vernier and the 
 preceding division on the scale, is n/mfhs of a division of the scale. 
 
 Verniers are sometimes constructed so that m divisions are equal in 
 length to m + l divisions of the scale. In such a case an argument 
 similar to that given above shows that if the vernier is numbered in the 
 
44 MEASUREMENTS OF LENGTH [ 14 
 
 reverse direction to the scale, the fiducial mark of the vernier being still 
 at the end of the vernier nearest the zero of the scale, and coincidence 
 occurs at the nth division, the distance between the fiducial mark of the 
 vernier (i.e. the rath division) and the preceding division of the scale is 
 w/raths of a division. The advantage of this form of vernier is that the 
 distance between the graduations is rather greater, i.e. the scale more 
 open. It is, however, rather more confusing to read, and is seldom used 
 on that account. 
 
 14. The Micrometer Screw. When a screw such as A, Fig. 12, is 
 rotated through a whole turn it advances through the nut B by an 
 amount equal to the distance, measured parallel to the axis, between 
 consecutive threads. Thus, suppose that the screw has twenty threads 
 to the centimetre, when the screw is rotated through a whole turn it 
 advances through half a millimetre. By means of a divided head c 
 attached to the screw the fraction of a turn through which the screw 
 has been rotated can be read. With a screw having a pitch of half a 
 millimetre, which for most purposes is a convenient quantity, it is usual 
 to divide the head into fifty parts, so that one division of the head 
 corresponds to a motion of the end of the screw of 0*01 mm. In order 
 to determine the whole number of turns through which the screw has 
 been turned a scale D is placed alongside the divided head. This scale 
 is divided into half millimetres, and a circle engraved on the divided 
 head or on the edge of the head serves to read the position of the screw 
 to the nearest half millimetre, the fractions being obtained from the 
 reading on the divided head opposite to the edge of the scale D. When 
 using a micrometer screw for any but comparatively rough measurements 
 it will not do to assume that the distance between the threads is exactly 
 what it purports to be, nor that the threads are quite uniform, so that it 
 becomes necessary to calibrate a micrometer screw. Various methods of 
 performing such a calibration will be described. 
 
 In order to be able to adjust a micrometer screw to make a measure- 
 ment, it is essential that it should move fairly easily in its nut. As a 
 result, particularly when a screw has been used for some time, there will 
 be a certain amount of play between the screw and the nut. Hence the 
 position of the screw will vary slightly according as to whether the 
 reading obtained has been approached by turning the screw in one 
 direction or the other. This effect is referred to by the term backlash, 
 and to ensure its not affecting the readings it is essential that the screw 
 should always be brought up to its final position by turning the head in 
 the same direction, so that the threads of the screw shall always bear on the 
 same side of the threads in the nut. If by accident the screw is turned 
 too far, it must not simply be turned back till the reading is correct, but 
 must be moved back about a quarter of a turn beyond the correct position 
 and then gradually turned forward till the adjustment is complete. 
 
 Two simple instruments in which a screw is made use of to measure 
 a length are the spherometer and the micrometer screw gauge, and it will 
 be found advisable to obtain some practice in the use of a micrometer 
 
FIG. 12. 
 
 15] THE SPHEROMETER 45 
 
 screw by making the measurements indicated in the following pages 
 before proceeding to use a more complicated instrument. 
 
 15. The Spherometer. The usual form given to the spherometer 
 is shown in Fig. 12, from which the construction of the instrument will 
 be clear. The screw is generally one having 
 a pitch of half a millimetre, and the divided 
 head c is divided into 500 parts, so that 
 each division represents a movement of the 
 screw parallel to its axis of O'OOl mm. The 
 hundreds divisions on the head are num- 
 bered 0/5, 1/6, 2/7, &c. This means that 
 starting, say, from the zero when the division 
 marked 0/5 is opposite the zero of the fixed 
 scale D, and turning the head till the 
 division 1/6 is reached, we have to take the 
 upper number 1, and the screw has advanced 
 through *01 mm. When the division marked 
 0/5 is reached we have now to take the lower 
 number 5, and when we come to the next 
 division the lower number 6, and so on till we 
 get to the division 0/5, when we start with the upper numbers again, but 
 have a whole millimetre in addition to the fraction shown on the divided 
 head. This whole millimetre will be shown on the scale D. This scale 
 will also serve to show in any case whether we have to read the upper or 
 the lower figure on the divided head, for if the reading on the fixed scale is 
 between a whole millimetre division and the half millimetre division we 
 have to take the upper figure, while if it is between the half millimetre 
 division and the next whole millimetre then we have to take the lower 
 figure. In some instruments the chief divisions on the head are simply 
 numbered 1, 2, 3, <fec , up to 5, and 0'5 has to be added to the reading 
 when the scale D shows that the reading ought to be between 0*5 and 
 1*0 mm. 
 
 The first thing to do when making a measurement with a spherometer 
 is to determine the reading of the micrometer head when the ends of the 
 three fixed legs and of the micrometer screw are in the same plane. To 
 make this test the instrument is generally provided with a disc of glass, 
 one surface of which is ground to a plane surface. In the absence of 
 such a disc a piece of plate glass, so long as it is not cut from near the 
 edge of the original sheet of glass as supplied by the glass works, will do 
 quite well. In fact, by selecting suitable pieces of plate glass by the 
 method given in 140 it is possible to obtain pieces having surfaces 
 much more nearly plane than are the ground surfaces ordinarily supplied 
 with spherometers. 
 
 Place the piece of glass firmly on the table so that it does not rock, 
 and on it stand the spherometer and roughly adjust the micrometer 
 screw so that the four legs are in contact at the same time. The final 
 adjustment can be made in the following manner : Lightly tap the 
 
46 MEASUREMENTS OF LENGTH [ 15 
 
 spherometer in a horizontal direction in such a manner as to tend to 
 make it spin round the middle leg, when it will be found that if only 
 the outside legs are in contact with the glass that the instrument will 
 not turn. If, however, the middle leg is projecting beyond the others, 
 then even a very light tap will be sufficient to make the instrument 
 turn. Hence the adjustment is made by starting with the middle leg 
 too short and then gradually turning the micrometer screw till the 
 instrument just shows an inclination to turn when gently tapped. If 
 by accident the micrometer is turned too far, it must be turned back 
 and then brought up to the reading in the forward direction again. 
 This adjustment must be made three or four times, being careful always 
 to bring the screw up to its final position by turning it in the same 
 direction, and the mean taken to represent the zero reading of the 
 instrument. 
 
 Next use the spherometer to measure the thickness of a microscope 
 cover glass. The upper surface of the glass plate used when determining 
 the zero reading must be cleaned, particular care being taken that no 
 specks of grit are left. The cover glass must then be cleaned in the 
 same way by being drawn between two folds of a soft clean rag held 
 between the thumb and first finger of the right hand. If the cover 
 glass and the support are properly clean, on placing the cover glass 
 on the support and gently pressing it down the cover glass will 
 adhere quite firmly. Next place the spherometer on the plate and 
 adjust the micrometer screw so that it just touches the upper surface 
 of the cover glass, the three fixed legs resting on the glass plate. 
 Repeat this adjustment three times, and then the difference between 
 the mean of the readings and the zero reading will give the thickness 
 of the cover glass. 
 
 The spherometer may be used to determine the curvature of a 
 spherical surface, that is, the radius of the sphere of which the surface 
 forms a part. The radius of the surface of a lens or spherical mirror 
 can be measured in this way, and it is from its use to measure 
 the curvature of such surfaces that the instrument derives its 
 name. 
 
 When the three fixed legs rest on a spherical surface the plane 
 passing through their tips will cut off a spherical cap ABC, Fig.. 13, 
 
 and the radius of the circle bound- 
 B ing this cap will be equal to the 
 
 distance between the outside legs 
 and the middle leg of the instru- 
 
 D x ment. When the micrometer is 
 
 FIG. 13. adjusted so that the central leg 
 
 just touches the surface of the lens 
 
 at the point B, the difference between the reading and the zero read- 
 ing will give the length of the line BD. Calling this length x, the dis- 
 tance between the fixed legs and the central leg r, and the curvature 
 of the surface of the lens, that is, the radius of the circle, R, then since 
 
16] THE MICROMETER SCREW GAUGE 47 
 
 AD = DC = r arid x and 1R - x are the two segments of the diameter of 
 the circle which passes through D, we have 
 
 x 2 + r 2 x 
 * = - = 
 
 In general the quantity x will be very small, so that x/2 will be 
 negligible compared with ?* 2 /2#, in which case we have 
 
 It will be noticed that to employ this method of measuring the curva- 
 ture of a surface it is necessary that the surface should be so large that 
 the three fixed legs of the spherometer can all rest on the lens at the 
 same time. 
 
 Since the distance between the central leg and the outside ones is 
 involved in the expression for the radius of curvature of the spherical 
 surface, and further, since this length occurs to the second power, it is 
 necessary to determine its value with accuracy. The maker generally 
 attempts to make it exactly an inch or 2 centimetres, according as the 
 screw has a pitch measured in English or metric units. A method of 
 determining the distance between the central leg and the outside ones 
 is to lay a sheet of thin and uncrumpled tin-foil on the glass plate used 
 to determine the zero. The instrument is then placed on the tin-foil 
 and lightly pressed down, so that all four legs mark the foil. The 
 distance between the central dot thus obtained and the three outside 
 dots is then measured with a travelling microscope (see 17). The 
 mean of the three distances must be taken as the quantity r in the 
 formula. If the travelling microscope has sufficient overhang, the 
 distances between the legs can be measured directly by placing the 
 spherometer on the base of the microscope with the points of the legs 
 turned upwards. Although it may be impossible to measure the 
 distances from the central leg to the outside ones, the micrometer 
 microscope will generally allow of the distances between the outside 
 legs being measured. If d is the mean of the three distances between 
 the outside legs, then 
 
 16. The Micrometer Screw Gauge. A common form of micrometer 
 screw gauge is shown in Fig. 14. The cap G is attached to the screw, 
 and it carries on its lower edge a series of division marks, by means 
 of which the fractions of a turn of the screw can be read off. The 
 number of whole turns made by the screw can be read off on the 
 scale F, which is engraved on the nut, and is uncovered by the cap o as 
 the screw is turned. The usual pitch for the screw is half a millimetre, 
 and the scale on the edge of the cap is generally divided into fifty parts, 
 
48 
 
 MEASUREMENTS OF LENGTH 
 
 [17 
 
 so that each division represents 0'02 of a complete turn, or a movement 
 of the end of the screw of 001 mm. The divisions on the cap are 
 
 usually numbered 0/5, 1/6, &c., as in 
 the case of the spherometer, and the 
 upper or lower figure has to be taken 
 according as the screw has made an 
 odd or even number of complete turns. 
 If the gauge is in exact adjust- 
 ment, the scales will be at zero when 
 the screw point is in contact with 
 the stud B. In general, however, 
 FIG. 14. this adjustment is not quite correct, 
 
 and the first thing to be done when 
 
 making a measurement is to determine the amount and sign of the zero 
 error. For this purpose, first clean the end of the screw and the stud, 
 and then gently turn the screw till its end touches the stud. When 
 screwing the micrometer screw up to the stud, or when making a 
 measurement of thickness, it is of importance that it^ should be only 
 lightly brought into contact with the object. For this reason the cap 
 by which the screw is turned ought to be grasped very lightly between 
 the thumb and first finger, so that whenever contact is secured the 
 fingers may slip and so not strain the screw. 
 
 Having determined the zero reading and noted whether the quantity 
 has to be added to or subtracted from the subsequent readings, the 
 object to be measured can be placed between the jaws and the reading 
 obtained when contact is secured. This reading, with the zero correction 
 applied, will be the thickness of the object. 
 
 Determine by means of the gauge the thickness of the microscope 
 cover glasses which were measured with the spherometer, and deter- 
 mine whether there seems to be 
 any consistent difference between the 
 values obtained with the two instru- 
 ments. 
 
 Determine with the gauge the 
 diameter of a piece of steel or brass 
 wire. The diameter ought to be 
 measured at several points, while at 
 each point the diameter should be 
 measured in two directions at right 
 angles. In this way ascertain whether 
 the wire has a circular section, and also 
 if its diameter is uniform throughout 
 1 its length. 
 
 17. The Travelling Microscope. 
 An instrument which is of very great 
 
 FIG. 15. 
 
 service in making a large number of measurements of length, such 
 as are constantly required in the physical laboratory, is shown in Fig. 15, 
 
17] THE TRAVELLING MICROSCOPE 49 
 
 and is sometimes called a travelling microscope. There are various forms 
 of the instrument made, but they all consist essentially of a microscope M, 
 Fig. 16, which is mounted on a frame so that it can be moved in a 
 direction perpendicular to its length, and the distance through which 
 it has been moved can be read off by means of a scale and vernier or 
 a micrometer screw A. The microscope can be moved up and down 
 within the upright of the carriage so as to allow of the microscope being 
 focused on the object which is being measured. Two cross-wires are 
 fixed within the microscope, and the eye-piece can slide in and out 
 so that these cross-wires can be focused. The object on which the 
 measurement has to be made must be placed below the microscope, with 
 the line of which the length has to be determined parallel to the direction 
 in which the microscope traverses. To assist in making this adjustment 
 it is useful to have a series of lines ruled on the base parallel to the slide 
 along which the microscope moves. If these lines have not been put on 
 by the maker they can be ruled by attaching a piece of pointed wire to 
 act as a scriber and then traversing the microscope throughout the whole 
 length of its travel, so that a line is ruled on the base by the point 
 of the wire. 
 
 Determine by means of the travelling microscope the errors in the 
 lengths of the divisions of a small scale ruled on glass. For this purpose 
 the first adjustment to be made is to focus the cross-wires and microscope 
 correctly, and as this is an adjustment which has constantly to be made, 
 the following method had better be employed. Remove the microscope 
 from the carriage, and directing it towards a window or some brightly 
 illuminated white surface look into the eye-piece with the right eye. 
 While doing this do not shut the left eye, but keep it fixed and focused 
 on some object, such as a window bar, which is more than two yards 
 away. With a little practice it will be possible to see the bright field of 
 the microscope with the cross-wires, and apparently alongside them the 
 window bar. The eye-piece must then be adjusted so that both the 
 cross- wires and the window bar are sharply in focus at the same time. 
 This method of focusing the eye-piece ensures that when the eye is 
 focused on the cross-wires the accommodation muscles of the eye are not 
 unduly strained. Unless some such device is employed it often happens 
 that the eye-piece is so focused that although it is possible to see the 
 cross-wires quite distinctly, yet to accomplish this one has unconsciously 
 to strain the eye, so that if the instrument is used much in this condition 
 very serious injury to the eyesight may ensue. Having focused the 
 cross-wires in this way the microscope must be replaced in the instru- 
 ment, being careful not to displace the eye-piece, and then by moving 
 the microscope as a whole it must be focused on the divisions of the 
 scale which has to be measured. When making this adjustment it is 
 important that the image of the scale produced by the object-glass of 
 the microscope should be in the same plane as the cross- wires. To see 
 whether this adjustment is complete place the vertical cross-wire so that 
 it appears immediately over one of the divisions of the scale, then 
 
 D 
 
50 MEASUREMENTS OF LENGTH [ 18 
 
 move the eye from side to side as far as the aperture in the eye-piece 
 will allow. If the image of the division on the scale and the cross- wires 
 are in the same plane, they will not appear to alter their relative positions 
 as the eye moves. If, however, the focus is not quite right, so that the 
 image is nearer to the eye or farther away than the cross- wires, then, 
 owing to parallax, there will appear to be relative motion when the eye 
 is moved from side to side. The whole microscope must be moved up 
 and down till there is no parallax between the image and the cross-wires. 
 When this is secured, the focusing is complete. The reason that it is 
 important to make this adjustment with great care is, that if parallax 
 exists the setting of the cross-wires on a division of the scale when 
 making a measurement will depend on the position of the eye with 
 reference to the eye-piece. Hence as we cannot ensure that the eye 
 occupies always exactly the same relative position when making the 
 settings at the two ends of the object, an error of some considerable 
 'magnitude might be produced in this way. 
 
 Having adjusted the focus of the microscope, the next thing to be 
 done is to arrange the scale which has to be measured on the bed of the 
 instrument in such a way that the divided portion is parallel to the slide 
 along which the microscope moves. To do this, move the microscope to 
 the extreme right-hand end of its travel and adjust the scale so that one 
 end of the division lines corresponds to the intersection of the cross- 
 wires. Then move the microscope to the other end of its slide, and again 
 adjust the scale. By repeating this operation a few times it will be 
 possible to arrange the scale so that, as the microscope is moved along, 
 the ends of the division lines of the scale always coincide with the 
 intersection of the cross- wires of the microscope, and hence the scale 
 is parallel to the slide along which the microscope travels. Next turn the 
 microscope, without altering its focus, so that one of the cross-wires is 
 parallel to the division lines of the scale. The instrument is now in 
 adjustment. 
 
 Take a reading of the micrometer screw when the cross-wire of the 
 microscope coincides with each whole centimetre division of the scale, and 
 draw up a table showing the difference between the nominal distance 
 between the division lines of the scale and the distances as obtained with 
 the micrometer microscope. 
 
 18. Illumination of the Divisions of a Standard Scale. An im- 
 portant point when using scales with fine division lines is the illumina- 
 tion. In Figs. 16 and 17 are shown two ways in which the scale D can 
 be illuminated. In Fig. 16 the light from a small incandescent lamp, 
 or the light of a window reflected from a mirror, strikes the lens c ; it is 
 then partly reflected downwards at the surface of a plane piece of un- 
 silvered glass B, and having traversed the lenses of the object-glass A of the 
 microscope, strikes the scale D. The light, after reflection at the surface 
 of the scale, passes through the object-glass and through the glass plate 
 B, and finally reaches the eye-piece of the microscope. In the second 
 arrangement (Fig. 17), it is not necessary to alter the objective by cutting 
 
19] THE COMPARATOR 51 
 
 a hole in the side and by the insertion of a glass plate. To a small brass 
 ring, which fits on the end of the objective, is attached a small ring- 
 shaped metallic mirror B, inclined at 45 to the axis of the microscope. 
 The rays of light from some source pass through the lens c, and are 
 
 FIG. 16. 
 
 FIG. 17. 
 
 reflected by the mirror B on to the scale. After reflection at the scale 
 they pass through the hole in the centre of the mirror B, and enter the 
 microscope. 
 
 In both of the above arrangements the illumination of the gradua- 
 tions of the scale is symmetrical. If a source of light placed on one side 
 of the microscope is used the illumination is asymmetrical, and the 
 graduation being better illuminated on one side than the other an 
 incorrect setting may result. 
 
 19. The Comparator. The travelling microscope is unsuited for 
 measuring lengths greater than about 5 cm. When greater lengths, up 
 . a metre, have to be measured with an accuracy greater than the 
 tenth of a millimetre, the principle of the comparator has to be employed. 
 This instrument l consists essentially of two micrometer microscopes 
 attached to a rigid bed in such a way that the distance between the 
 carriages which carry the microscopes can be varied. The object to be 
 measured is placed on the bed of the instrument with the length to be 
 measured parallel to the line along which the microscope carriages move, 
 and the microscopes are so adjusted that their cross-wires coincide with 
 the marks which define the ends of the line which has to be measured. 
 The t.'bjeet is then removed, care being taken not to disturb the micro- 
 scopes, and a standard scale placed in its place. The cross-wires of the 
 
 1 An interesting account of the comparators in use at the Bureau Interna- 
 tional des Poids et Mesures at Paris wfll be found in a book by Guillaume, called 
 La Convention du Jfttrc, published by Gauthier-Villars, Paris. 
 
MEASUREMENTS OF LENGTH 
 
 [20 
 
 microscopes will not in general coincide with a division of the scale at 
 either end. The divided heads of the micrometers having been read, 
 the microscopes are moved till the cross-wires coincide with the nearest 
 division line of the scale, when the micrometers are again read. The 
 difference between the readings of the micrometers will give the amounts 
 by which the object exceeds or falls short of the length between the two 
 whole divisions of the scale. Unless the accuracy of the micrometer 
 screws has been ascertained, it is as well to take the reading correspond- 
 ing to two divisions of the scale, one on either side of the original position 
 of the microscopes. In this way the amount by which the object exceeds 
 the whole number of divisions of the scale can be obtained without in- 
 volving the pitch of the micrometer screws. The following example will 
 illustrate how the observations ought to be recorded, and the method of 
 obtaining the fractions of a scale division from the micrometer read- 
 ings : 
 
 Left-hand Microscope. 
 
 Right-hand 
 
 Microscope. 
 
 Temperature. 
 
 Object . 
 0-0 of scale 
 O'l of scale 
 
 3-1475 cm. 
 3-1251 
 
 3-2266 
 
 Object . 
 50" 1 of scale 
 50-2 of scale 
 
 0-1 of scale 
 Dif. object 
 50-2 . 
 
 o-os 
 
 . 1-7751 cm. 
 . 1-7036 
 
 . 1-8048 
 
 = 0-1012 cm. 
 to 
 . = 0-0297 
 597 
 
 18-1 
 18-1 
 18-1 
 
 0*1 of scale 
 Dif. 0-0 to object 
 
 = 0*1015 cm. 
 = 0-0224 
 
 0-1015 
 
 = 0-02207,, 
 
 (VK 
 
 _ I UZJoO ,, 
 
 
 Length of object (uncorrected for scale error) = 50 '2 -0*02207 -0-02935 
 
 = 50-1486 cm. at 18-1. 
 
 20. Correction of Measurements of Length to Allow for the Effects 
 of Temperature. When making accurate measurements of length it is 
 of importance to note the temperature of the body being measured, as 
 also of the scale, for, owing to the fact that bodies change their length 
 with change of temperature, the length will differ at different temperatures. 
 It is only at some fixed temperature that the divisions of a standard scale 
 have the length they profess to have. In the case of a scale divided in 
 metric measure the divisions are generally correct at C. Scales divided 
 in English measure are generally correct at 62 F., or 16'67 C. If, 
 when making a measurement, the temperature of the scale is higher 
 than the temperature at which it is correct, each of the divisions will be 
 too long by an amount which depends on the coefficient of expansion of 
 the material of which the scale is composed and the amount by which 
 the temperature exceeds the standard temperature. If the scale is of 
 brass it will, except in cases where the utmost refinement is required, 
 be sufficiently accurate to assume that the coefficient of expansion is 
 
21] THE CATHETOMETER 53 
 
 0-0000187 ; while if the scale is made of steel, the coefficient of expan- 
 sion may be taken as 0*000011. The method of correcting a length 
 measurement for the temperature of the scale is given below, where the 
 measurement given on p. 52 is reduced. 
 
 The excess of the length of the scale over its nominal value at 50 '1 
 cm., as derived from the tabulated set of corrections, is - 0*0028, that 
 is, the true distance between the zero graduation and 50' 1 at a tempera- 
 ture of C. is 50-9972. Hence, applying this correction, the length of 
 the object becomes 50'1458. The temperature coefficient of the scale is 
 0-0000187. Hence the actual distance between the zero and 50*1458 
 at a temperature of 18* 1 is 
 
 50-1458(1 + 0-0000187 . 18-1), 
 or 50-1628. 
 
 The above results would be entered thus 
 
 Uncorrected length at 18' 1 . . . =50*1486 cm. 
 
 Correction for error of scale at . . = - "0028 
 
 Corrected length, scale at 18 -1 . . =50*1458 
 Correction to allow for expansion of scale 
 
 between and 18 *1 . . . = -f* 0170 ,, 
 
 Corrected length of body at 18 -1 . . =50*1628 
 
 If we required the length of the body at any fixed temperature, say, 
 C., then a knowledge of the coefficient of expansion would be neces- 
 sary. The body measured in the above example being made of brass, 
 we may assume its coefficient of expansion as 0*0000187. The length at 
 would then be given by 
 
 50*1628{1 - 0*0000187 x 18*1}. 
 
 When applying corrections for temperature, particularly when they 
 have to be applied to the standard, great care must be taken that the 
 correct sign is used. 
 
 21. The Cathetometer. The comparator is only suitable for measur- 
 ing lengths when the object can be rested in a horizontal position on the 
 base of the instrument. In many cases, however, the length to be measured 
 is vertical, as, for instance, the height of a column of liquid in a tube. 
 In such a case an instrument called a cathetometer is often employed, 
 though of late years the arrangement described on p. 58 is frequently 
 used for this purpose, since it avoids many of the errors and troublesome 
 adjustments which are involved in the use of the cathetometer. 
 
 Cathetometers are of many patterns, though they all consist of an 
 upright slide along which a telescope can be moved, the axis of the tele- 
 scope being horizontal. One form of cathetometer is shown in Fig. 18. 
 An upright metal pillar AB is fixed to a heavy metal stand in such a way that 
 it can be made to rotate about a vertical axis, the rotation being limited 
 when necessary by two adjustable stops. The pillar has a scale engraved 
 
54 MEASUREMENTS OF LENGTH [ 21 
 
 along one face. The telescope T is supported by a carriage which can 
 slide along the pillar, a second carriage D sliding along the same pillar 
 being connected with the carriage which supports the telescope by a 
 micrometer screw E. This carriage D can be clamped to the pillar, and 
 then by turning the screw B the telescope can be moved up or down 
 
 through a small distance and so 
 its position adjusted. The position 
 of the carriage c can be read off 
 on the scale on the pillar by means 
 of a vernier v. A spirit-level L 
 serves to show when the axis of 
 the telescope is horizontal, and a 
 screw F serves to slightly tilt the 
 telescope, and thus permits of its 
 axis being adjusted. 
 
 The cross- wires of the telescope 
 are attached to a ring, which ring 
 is supported by four screws, as 
 shown in Fig. 19. These screws 
 pass through holes in the side 
 of the telescope tube and are 
 threaded into the ring, so that 
 by slackening off one screw and 
 tightening the opposite one the 
 position of the ring, and hence also 
 of the cross- wires, can be altered. 
 
 The line joining the centre of 
 the object-glass to the intersection 
 of the cross-wires is called the 
 optic axis of the telescope, and 
 the object in having the cross- 
 wires adjustable is to allow of the 
 optic axis being made parallel to 
 the axis of the tube of the tele- 
 scope. This adjustment, which is 
 generally referred to as the col- 
 limation adjustment, is performed 
 
 by turning the telescope to some 
 
 fairly distant object having a sharp 
 
 FIG. 18. horizontal edge or line, and bring- 
 
 ing the horizontal wire into coin- 
 cidence with the image of this edge. The telescope is then turned about 
 its own axis through 180, the springs or clamps which hold it to the 
 carriage having been loosened if necessary. If the horizontal wire still 
 coincides with the image of the edge, this wire is correctly adjusted ; if not, 
 the cross-wire must be moved half-way between its position and the image 
 of the edge. It will generally be necessary to repeat the operation two or 
 
FIG. 19. 
 
 21] THE CATHETOMETER 55 
 
 three times before the adjustment is complete. The vertical wire must 
 
 be adjusted in the same manner. When both wires are adjusted, then on 
 
 rotating the telescope the intersection of the wires ought not to move 
 
 relatively to the image formed in 
 
 the telescope. When the collima- 
 
 tion of the telescope is correct we 
 
 are then certain that if we arrange 
 
 that the axis of the tube of the 
 
 telescope is horizontal, then the 
 
 optic axis of the telescope will also 
 
 be horizontal. 
 
 When being used to measure a 
 vertical distance a cathetometer has 
 to fulfil the following conditions : 
 (1) The vertical column which 
 carries the scale, and on which the 
 telescope slides, must be truly 
 vertical; (2) as the telescope is 
 moved up and down the column 
 the axis of the telescope must 
 remain parallel to some fixed direction. The fixed direction chosen 
 is a horizontal plane. 
 
 As an exercise in the 'use of the cathetometer, hang up a standard 
 metre scale or other finely divided scale. By means of a plumb line 
 attached to the stand which carries the scale, adjust the scale so that it 
 is accurately vertical. Find, by moving backwards and forwards a piece 
 of paper on which there is some writing, the shortest distance at which 
 the cathetometer telescope will focus an object. Place the cathetometer 
 at a distance from the suspended scale slightly greater than this distance, 
 and in such a position that the scale on the cathetometer can be con- 
 veniently read. 
 
 The first adjustment to make is to set the axis of the telescope parallel 
 to the axis of the level. To do this adjust the inclination of the telescope, 
 either by means of the levelling screws at the base or by the screw p, till 
 the bubble is at the centre of the scale. Then remove the level, and 
 leaving the telescope untouched turn the level end for end. If the bubble 
 is not now at the centre of the scale bring it half-way back by tilting the 
 telescope by the screw F, and the other half by means of the adjusting 
 screw attached to the level itself. It will generally be found necessary to 
 repeat the reversal two or three times before the adjustment is complete. 
 
 Great care must be taken when reading or adjusting the level not 
 to heat the tube, either by breathing on it or by touching it with the 
 hand. The extent to which a trifling amount of heat can affect the read- 
 ing can be tested by lightly touching the tube near one end of the bubble 
 for a few seconds. The bubble will be found to move towards the 
 heated end, and it will be some little time before it returns to its correct 
 position. 
 
56 
 
 MEASUREMENTS OF LENGTH 
 
 [21 
 
 The next adjustment to be made is to set the column of the catheto- 
 meter vertical. To do this turn the column till the telescope is parallel 
 to the line joining two of the levelling screws of the base, and by turning 
 these screws bring the bubble of the telescope level to the centre of the 
 scale. Next turn the column through 180, and if the bubble is not at 
 the centre of the scale bring it half-way back by turning the levelling 
 screws, and the other half-way by turning the screw F (Fig. 18). Now turn 
 the column so that the telescope is at right angles to the line joining the 
 two levelling screws that have been used up to now, and bring the bubble 
 back to the central position by turning the remaining levelling screw. It 
 will generally be necessary to repeat the adjustment, as moving the third 
 screw will often slightly throw out the adjustment of the others. When 
 the adjustment is complete the level bubble ought to remain at the zero 
 of its scale, whatever the direction in which the telescope is turned. 
 
 The theory of this adjustment is as follows : Suppose AB (Fig. 20) 
 represents the position of the vertical axis, and CD that of the telescope. 
 
 On turning the instrument through 180 
 the vertical axis remains unaltered, but 
 the telescope takes up the position C'D'. 
 Now if a is the angle the vertical axis 
 makes with the vertical, then the angle 
 between C'D' and the horizontal is 2a. 
 Hence if the bubble of the telescope level 
 is brought half-way back by moving the 
 vertical axis through an angle a, the tele- 
 scope would be in the position C"D", 
 making an angle a with the horizontal. 
 Hence when the other half of the move- 
 ment of the bubble is taken out by the 
 screw which tilts the telescope, the axis 
 of this latter takes up the horizontal 
 position C'"D'". 
 
 The eye-piece of the telescope must 
 now be focused on the cross-wires and 
 the telescope on the scale by the method given on p. 49, so that 
 there is no parallax. The adjustments of the instrument being com- 
 plete, set the telescope so that the horizontal cross-wire coincides with 
 the lower division of the suspended scale, and then read the vernier on 
 the cathetometer scale. Next move the telescope up so that the hori- 
 zontal cross-wire coincides with the image of the upper division of the 
 scale. If the level bubble is found to be exactly at the zero of its scale, 
 the vernier can now be read. If the telescope is not exactly level adjust 
 its position by means of the screw F, and then again set the cross-wire to 
 coincidence with the upper division of the scale, and if the telescope has 
 still remained level read the vernier. The difference between the two 
 readings of the vernier wil) give the distance between the two divisions 
 of the scale in terms of the cathetometer scale. As a check on the results 
 
 FIG. 20. 
 
22] MEASUREMENT OF A VERTICAL LENGTH 57 
 
 the adjustment of the cathetometer ought to be thrown out and then 
 made again and the measurement repeated. 
 
 To obtain some idea of the accuracy which a careful observer may 
 expect to get, examine the vernier and find the smallest movement which 
 .could be read. Twice this amount will be the largest error in the dis- 
 tance between the two positions of the telescope Avhich ought to be 
 obtained in any setting. This distance, however, does not necessarily 
 equal the distance between the two divisions of the scale which was 
 measured, for the cathetometer scale may not have been vertical, or, what 
 is of very much greater importance, the axis of the telescope may not 
 have been exactly parallel when making the two readings at the two ends 
 of the object. To ascertain what influence on the final result a possible 
 error in this adjustment may make, we require to know through what 
 angle the telescope has to be tilted to cause the level bubble to move 
 through one division of its scale, and, further, within what fraction of a 
 division of the level we are able to read and adjust the position of the 
 bubble. The angle through which the telescope has to be tilted so that the 
 bubble may move through one division may be determined in the follow- 
 ing manner : By means of the screw F tilt the telescope so that the 
 bubble is near one end of the scale, and then read the division of the 
 suspended scale corresponding to the horizontal cross-wire of the telescope 
 and the position of the bubble. Next tilt the telescope in the opposite 
 direction, and again read the scale and bubble. Let the difference of the 
 readings on the suspended scale be s, and the difference in the positions 
 of the bubble, expressed in divisions of the level scale, be b. Measure 
 the distance between the pivot about which the telescope turns when 
 tilted by the screw F and the suspended scale, and call it D. Then if a 
 is the angle through which the telescope has been tilted we have 
 
 S I D t a 
 or since a is very small 
 
 Hence one division of the level scale corresponds to a tilt of a/b ex- 
 pressed in circular measure, or to 206300 a/b seconds of arc. 
 
 If, now, an examination of the level indicates that we are not able to 
 fix the reading to less than I /nth of a division of the scale, this means 
 that the two positions of the telescope may have been inclined to one 
 another when measuring the scale by I /nth of the angle found for one 
 division. Hence, knowing the distance of the telescope from the scale, 
 the uncertainty of the length of the scale corresponding to this un- 
 certainty of level can immediately be calculated. 
 
 22. Measurement of a Vertical Length by Means of a Separate 
 Scale and Reading Telescopes. The cathetometer is by no means a 
 satisfactory instrument with which to measure a vertical distance. In 
 
58 MEASUREMENTS OF LENGTH [ 22 
 
 the first place, in order to give results which are at all accurate it is 
 necessary that the instrument should be constructed with very great 
 care as to its design and workmanship, which makes the instrument 
 very costly. Secondly, the adjustments which have to be made are 
 troublesome and require repeating at each reading, so that it takes a 
 very considerable time to make a measurement. Thirdly, unless the 
 instrument is supported on a masonry pillar the weight of the observer 
 will bend the floor sufficiently to alter the level of the telescope, and 
 hence influence the readings. Fourthly, the presence of the observer's 
 
 FIG. 21. 
 
 body in the immediate neighbourhood of the divided scale on the catheto- 
 meter column causes a very considerable uncertainty as to what is the 
 actual temperature of this scale, and hence the actual distance between 
 the divisions. These objections may in a great measure be obviated by 
 separating the scale with which the object is to be compared from the 
 telescope used in making the comparison and placing this scale as near 
 to the object as possible, while the telescope simply serves to transfer the 
 length from the object to the scale. The method of measuring thus 
 resembles that used with the comparator, except that in place of moving 
 the object and replacing it by the scale while the microscopes are kept 
 
23] THE OPTICAL LEVER 59 
 
 fixed, the object and scale are here kept fixed, and the vertical column 
 which carries the telescopes is rotated about a vertical axis. 
 
 The method of supporting the scale s, and one form of upright for 
 carrying the telescopes, is shown in Fig. 21. The telescopes are each 
 provided with a micrometer screw MJ and M 2 , by means of which the 
 horizontal cross- wire can be moved in a vertical direction, the amount of 
 the movement being read off on the divided head of the screw. When 
 making an observation the scale must be placed as near to the object as 
 is convenient, and at the same distance from the telescopes as the object. 
 Using a plumb line as a guide, the scale must be made vertical. The 
 axis about which the column AB, which carries the telescopes, turns 
 must be set accurately vertical. This is done in exactly the same way 
 as was described in the case of the cathetometer (p. 56). Next the 
 axes of the telescopes must be made horizontal, which is also done by 
 the method previously described (p. 55), namely, by first turning the 
 telescopes on their v's and adjusting the cross-wires so that the line of 
 collimation is parallel to the tube on which the level rests and which 
 itself rests in the v's, and secondly by reversing the level till the reading 
 of the bubble is the same in either position. 
 
 The telescopes and scale being in adjustment, bring the cross-wires 
 of the telescopes to coincide with the upper and lower marks on the 
 object, and read the two micrometers MJ and M 2 . Now turn the 
 column carrying the telescopes so that the scale is seen in the telescopes, 
 and take the readings of each of the micrometers when the cross-wires 
 coincide in turn with each of the division lines of the scale on either side 
 of the positions they occupied after being adjusted on the object. From 
 these readings the fractions of a scale division can be immediately 
 deduced, exactly as was described in the case of the comparator ( 19). 
 
 In cases where an accuracy of a tenth or twentieth of a millimetre is 
 sufficient, one of the simplest and most satisfactory methods of measuring 
 a length, whether horizontal or vertical, so long as the scale can be 
 brought within about 2 cms. of the object, is to use a scale divided on 
 looking-glass. The glass used must be not less than 7 mm. thick. Such 
 glass, silvered on the back, can be obtained in strips of about 6 or 8 cms. 
 wide, and up to about a metre long, from most makers of pier glasses. 
 These strips are the wastage when the glass is cut to size to fit the frames. 
 The method of dividing such a scale is described in the Appendix. 
 
 As an example of the application of such a mirror scale to the 
 measurement of a length we may take the measure- <\ 
 
 ment of vapour density described in 98. A 
 
 23. The Optical Lever. The optical lever consists 
 of a lozenge-shaped metal plate A (Fig. 22), having 
 four needle-points projecting from the under side, 
 and a mirror M attached to the upper side. The two middle needle- 
 points project a very little farther than the end ones, so that when 
 the lever is placed on a plane surface it can ro^ slightly about the 
 middle points. In addition to the lever, a stand with a piece of good 
 
 
60 
 
 MEASUREMENTS OF LENGTH 
 
 [23 
 
 plate glass cemented to the top will be required, also a reading telescope 
 with a vertical scale. A convenient form of reading telescope is shown 
 in Fig. 23. The telescope can be moved up and down along the vertical 
 
 rod, and clamped by means of the screw A. 
 The telescope can also be moved about a 
 horizontal axis, and clamped by the screw B. 
 A scale s can be placed either vertical when 
 measuring rotations about a horizontal 
 axis, or horizontal when measuring rota- 
 tions about a vertical axis. 
 
 Set up the stand for the lever on a 
 shelf attached to the wall, or on a very 
 firm table, and place the telescope at a 
 distance of about a metre from the stand, 
 and focus on the scale as seen reflected 
 in the mirror of the lever. 
 
 The readings of the scale must now be 
 taken when the lever is tilted, so as to 
 rest first on the front leg and then on the 
 back, and let the difference of the readings 
 be d. We will suppose that the thickness 
 of a piece of microscope cover glass has 
 to be measured. The cover glass having 
 been well cleaned is placed on the stand 
 in such a way that the central legs of the 
 lever rest on the cover glass, but the end legs 
 are clear of the edge and rest on the plate 
 glass. The readings of the scale, as the lever rests first on one end leg 
 and then on the other, must again be taken, the difference now being D. 
 The thickness of the plate placed beneath the central legs is given by 
 
 FIG. 23. 
 
 where I is half the distance between the points of the end legs, and L is 
 the distance between the mirror and the scale. 
 
 Suppose that when the lever is placed on a plane surface the 
 angle through which it rocks is a. This also is the angle through which 
 the mirror turns, and since a beam of light reflected from a mirror turns 
 through twice the angle through which the mirror turns (Watson's 
 Physics, p. 441), it follows that the angle included between the two rays 
 MG and MF (Fig. 24) is 2a. But if a is small, we have 2a = OF/MS = d/L. 
 Now DE is the amount by which the central legs project beyond the end 
 legs. From the triangle ADE we have 
 
 or 
 
 since a is small. 
 
 ) == tan DA K= tan a/2, 
 
 =l tan a/2 =7u/'2, 
 
24] THE OPTICAL LEVER 
 
 Hence, substituting for a the value already obtained, 
 
 61 
 
 When the lever is placed with its central legs on a plate of thickness 
 
 FIG. 24. 
 
 t, and the difference in the readings' on the scale is D, we have in the 
 same way 
 
 Hence 
 
 It will be noticed that in obtaining the above formula we have 
 supposed that the angle through which the lever is tilted is small, hence 
 the central legs must not project more than one or two tenths of a 
 millimetre beyond the end legs, and the lever must only be used for 
 measuring thin plates. 
 
 24. To Measure the Radius of Curvature of a Spherical Surface 
 with the Optical Lever. The optical lever can be employed to measure 
 the radius of a spherical surface, such as a lens or mirror, so long as the 
 radius of curvature is large and the surface of sufficient size to allow all 
 the legs of the lever to rest on the surface. The method of making the 
 measurement is exactly the same as that described in the preceding 
 section, except that the lens will require supporting in such a way that 
 it will not rock as the lever rocks, which may be done by attaching it 
 with some soft wax. 
 
 If 1) is the difference in the scale readings when the lever is placed 
 on the spherical surface, then the radius of curvature R of the surface 
 is given by 
 
 2L 7 
 
62 
 
 MEASUREMENTS OF LENGTH 
 
 [24 
 
 where b is half the distance between the central legs. To deduce this 
 formula, let the end legs touch the lens at the points AC (Fig. 25, a), 
 and the centre legs at the points EE (Fig. 25, b). Through the mid 
 
 w 
 
 FIG. 25. 
 
 point o of AC draw the perpendicular OFG. Then if R is the radius of 
 the sphere of which the surface is a part, that is, the radius of the circle 
 AGC, we have (Euclid, III. 35) 
 
 But if R is large, AO is very nearly equal to the distance between the end 
 and middle legs. 
 
 Hence, neglecting the square of the small distance OG, we have 
 
 2/2 . OG = 2R((JF+ FO) = I 2 . 
 But by tilting the lever we measure the distance FO, so that 
 
 Also from Fig. 25 (ft), we see in the same way that 
 Hence 
 
 or 
 
 2L /2- 
 D-d I 
 
 D-d 
 
CHAPTER III 
 
 WEIGHING 
 
 25. The Balance. The theoretically perfect balance beam would be 
 one in which the three knife-edges were in the same plane, and were 
 parallel to one another. Further, they would remain so whatever the load 
 in the pans. Practically, however, it is impossible to secure the exact 
 fulfilment of the above conditions, although in a well-made balance they 
 are so nearly fulfilled as to produce practically negligible errors. It can 
 easily be shown that if the central knife-edge is below the plane contain- 
 ing the terminal knife-edges then the balance will be more sensitive with 
 a large load than with a small one. In the same way, if the central 
 knife-edge is above the terminal ones the sensitiveness will decrease as 
 the load increases. Thus if the relative position of the knife-edges did 
 not alter as the load was varied we could, by measuring the change of 
 sensitiveness with change of load, determine the relative positions of the 
 knife-edges. Owing, however, to unavoidable flexure of the beam with 
 increasing loads the relative positions of the knife-edges alters as the 
 load is increased. Makers generally arrange that the central knife-edge 
 is a little below the terminal ones when there is no load in the pans. 
 Thus at first the sensitiveness increases as the load increases. When, 
 owing to the flexure of the beam, all three knife-edges are in the same 
 plane, then for small changes of load in this neighbourhood the sensitive- 
 ness will not vary. If the load is increased above this point the terminal 
 knife-edges will be below the central one, and so the sensitiveness will 
 decrease as the load is further increased. 
 
 If the terminal knife-edges are not parallel to the central one, then, if 
 the scale pan stirrup were so constructed that by altering the position of 
 the weight placed in the pan we could cause either one or other end of 
 the terminal knife-edge to carry the weight, we should alter the arm of the 
 lever at which the pan and its contents works, and so change the position 
 of equilibrium of the balance. In order that this may not occur in 
 practice, owing to the slight unavoidable errors in the adjustment of the 
 knife-edges, the pan supports are made with one or more movable links, 
 such as that shown in Fig. 26, so that wherever in the pan the weight is 
 placed the weight may be distributed uniformly along the knife-edge, or 
 at any rate the distribution of the weight along the knife-edge may 
 remain unaltered. 
 
 The accuracy of the adjustment of the terminal knife-edges to 
 parallelism to the central knife-edge may be tested by doing away with 
 
FIG. 26. 
 
 64 WEIGHING [ 25 
 
 the freedom to swing about the point A of the scale pan, for which 
 purpose some balances have a set screw, B, provided. Place a weight of 
 about a quarter the maximum the balance is supposed to carry at the edge 
 of the pan farthest from the front of the balance- case, and counterpoise 
 
 by placing weights in the other pan. Then 
 move the weight in the clamped pan to the 
 edge nearest the front of the case, and see 
 whether the balance is still in equilibrium. 
 In practice, however, the more important 
 thing to test is whether, when the pans are 
 not clamped but are allowed all the freedom 
 that the construction permits, on changing 
 the position of the weights in the pan the 
 equilibrium of the balance is affected. If 
 any measurable difference is found it shows 
 that the knife-edges are out of adjustment, 
 and that the pan supports are not sufficiently 
 free to eliminate the error thus introduced. 
 If cleaning the links, or whatever arrange- 
 ment is used to secure the freedom of the 
 pans, does not remedy the matter the 
 balance must be sent back to the makers to 
 
 have the knife-edges readjusted, since this adjustment is one which no 
 one but a practised balance-maker can perform satisfactorily. When 
 using a balance it is always advisable to be careful to place the weights, 
 &c., in the pans in such a way that the pans hang in about the same 
 position that they occupy when no load is in the pans. 
 
 The weights of the pans and their supports ought to be the same, 
 though their exact equality is not essential. To test whether they are 
 of equal weight, remove the two pans and adjust the vane of the beam 
 so that the pointer is at the zero of the scale. Then attach the pans and 
 read the position of the pointer. Interchange the pans and supports and 
 again read the position of the pointer. The readings corresponding to 
 the two positions of the scale pans and supports ought to be the same. 
 If they are different, they can be adjusted to equality by carefully filing the 
 bottom of the heavier pan or adding a piece of platinum wire to the lighter. 
 The pans having been adjusted to approximate equality, the beam 
 may be brought to the horizontal position, that is, the pointer to the 
 middle of the scale, by adjusting the vane on the beam. As it is not a 
 good thing to be continually readjusting the vane, unless the pointer, 
 when there is no load on the pans, comes to rest at some distance from 
 the centre of the scale, it is better not to alter the vane, but to take the 
 reading corresponding to no load as the zero. 
 
 Balances nowadays are made with very much shorter beams than 
 they used to be, the requisite sensitiveness being obtained by using a 
 long pointer and reading lens or microscope. The advantage of the 
 short beam is that the time of vibration of the balance is considerably 
 
25] 
 
 THE BALANCE 
 
 65 
 
 reduced, while the beam itself can be made much lighter without loss of 
 rigidity. In some balances a mirror is attached to the beam, and the 
 deflections are read either with a telescope and scale or with a lamp and 
 scale, as described in 168. 
 
 On account of the shortness of the beam, if the rider used weighed a 
 centigram, and, in order to obtain the milligrams and fractions of a 
 milligram, were moved from the centre of the beam to the end, the 
 graduations of the beam would need to be very close, rendering it 
 difficult to obtain the exact position of the rider. For this reason in 
 
 FIG. 27. 
 
 short beam balances it is usual to employ a rider which weighs only half 
 a centigram and to attach to the front of the beam a light scale DE, 
 Fig. 27, the graduated portion being of the same length as the distance 
 between the end knife-edges. The rider is then placed on this scale, 
 and the vane, by means of which the beam is balanced, is so adjusted 
 that when the rider is at the extreme left-hand edge (i.e. at the zero of 
 the graduations) the pointer of the balance is at zero. When the rider 
 is at the extreme right-hand end of the scale the effect is as if half a 
 centigram had been moved to the end of an arm twice as long as the 
 balance arms, that is, as if the weights in the right-hand pan had been 
 increased by 1 centigram. Thus the rider scale is twice as open as it 
 
66 WEIGHING [ 26 
 
 would be with the old arrangement. The separate scale to carry the 
 rider has a further great advantage in that it allows of the rider being 
 placed as near as we like to its zero position ; while in the old arrange- 
 ment, owing to the presence of the gravity bob, it was usually impossible 
 to place the rider very near the centre of the beam. 
 
 26. Weighing by the Method of Oscillation and to Compare the 
 Lengths of the Arms of the Balance. If a balance is to be sensitive 
 it is necessary to reduce the friction between the knife-edges and the 
 planes to the minimum, so that a good balance, unless some special 
 device is provided for damping the oscillations, will make a great 
 number of vibrations before the beam comes to rest. Thus if we had 
 to wait in every case till the pointer came to rest before we could take 
 a reading, the operation of weighing would take a very long time. It 
 is, however, unnecessary to wait till the balance comes to rest, for we 
 can, by observing the readings of the scale corresponding to the turning 
 points of the pointer as the balance swings, calculate what would be the 
 reading corresponding to the position of rest. This method of per- 
 forming a weighing, which must always be used when we are attempting 
 to perform the weighing with the utmost accuracy with which the balance 
 is capable, is called the method of oscillation. 
 
 Confusion and possible error will be saved if the scale of a balance 
 with which the method of vibrations is to be used is numbered from the 
 left, the extreme left being called 0, the middle division 50, and the 
 extreme right-hand division 100. 
 
 The first thing to do is to determine the zero. For this purpose gently 
 raise and lower two or three times the beam from off the arresters, to 
 ensure that the planes have settled down into their correct positions. 
 Next raise the beam in such a way that the pointer swings over between 
 five and ten divisions on either side of the zero, and after the beam has 
 made two or three swings to get into a state of regular motion, read the 
 turning points of the pointer on the scale. The 
 turning points must be estimated to within a tenth 
 of a division of the scale, and if the divisions are 
 at all fine a lens fixed to the front of the balance 
 'case will be found of great assistance. Since a 
 lens will often be used for reading a scale, it is 
 convenient to have several having a focal length 
 of about 5 cm. mounted in such a way that they 
 can be placed in any desired position. One of the 
 simplest, and at the same time most convenient, 
 forms of mounting is to have the lens fixed in 
 FIG. 28. a brass hoop, this hoop being soldered to a short 
 
 length of lead wire A (Fig. 28). The lower end of 
 the lead wire is attached to a small base piece made of lead, and by 
 bending the lead wire the lens can be adjusted to any desired position. 
 
 As in the case of most balances the pointer swings in front of the 
 scale, care must be taken to avoid parallax. For this purpose the 
 pointer ought to be bent so that it only just clears the surface of the 
 
26] WEIGHING BY METHOD OF OSCILLATION 67 
 
 scale. If a lens is used, then parallax can be avoided by arranging that 
 the eye always occupies the same position relative to the lens. This 
 can be secured by sticking a piece of paper to the front glass of the 
 balance case with a small hole in such a position that when the eye is 
 placed opposite the hole the scale can be focused. 
 
 When taking the turning points of the pointer of the balance, readings 
 must be taken on both sides of the position of rest of the pointer and an 
 odd number of readings must be made, so that the first and last turning 
 points observed are on the same side of the zero. It will in general be 
 fbund best to take five readings, two on one side and three on the other. 
 Then take the mean of the three readings corresponding to the turning 
 points in one direction and the mean of the two in the other, and the 
 mean of these two means will be the scale reading corresponding to the 
 position of rest. The reason that an odd number of elongations have 
 to be observed is that the arc over which the pointer is swinging is 
 continually growing less, and thus if we only took two observations, say, 
 one to the left and 
 then one to the right, 
 we should obtain a 
 position of rest too far 
 to the left. This will 
 be at once apparent 
 from a study of Fig. 
 
 29. In this figure FIG. 29. 
 
 the abscissae represent 
 
 time, and the ordinates the position of the pointer measured from the 
 position of rest, which is represented by the axis o x. The elongations are 
 represented by the ordinates through the points A, B, c, n, E. The mean 
 of the first two elongations would give the point F as the position of rest, 
 which is considerably above the true position of rest. The mean of any 
 odd number of elongations obtained as above will, so long as the decrease 
 in the successive elongations is not very rapid, represent the position of 
 rest with a degree of accuracy quite sufficient for the purpose in hand. 
 
 The determination of the position of rest ought to be repeated three 
 or four times, the beam being arrested between each observation. If the 
 balance is working satisfactorily, the values obtained for the position of 
 rest ought not to differ by more than a few 7 tenths of a scale division. If 
 larger discrepancies than a tenth of a division occur, it indicates that 
 the balance beam is not swinging freely, or that there are air currents 
 within the balance case which are causing the beam to swing unevenly. 
 To remedy the first of these defects, the pan supports and the beam 
 must be removed, and the knife-edges and planes be gently rubbed with 
 a clean piece of pith or soft cork and then dusted with a camel's hair 
 pencil. If this cleaning does not remove the irregularity, and, further, 
 the rate at which the elongations decrease is small, then probably air 
 currents are the cause of the irregularity. These air currents may be 
 produced by the heat from a neighbouring gas flame, or from the balance 
 case being placed too near a window. 
 
68 WEIGHING [ 26 
 
 Having determined the position of rest when there is no load in the 
 pans, proceed to determine the ratio of the lengths of the arms of the 
 balance. For this purpose place two equal weights in the pans. These 
 weights ought to be about half the maximum load the balance will 
 carry, so that for a balance intended to carry 200 grams a weight 
 of 100 grams ought to be placed in each pan. Adjust the position of 
 the rider so that it is at the nearest milligram division l to the point 
 it would have to occupy to bring the pointer to the zero position, and 
 determine by the method of vibrations the position of rest (A). Then 
 move the rider so as to alter the weight on one side by 1 milligram, 
 and again determine the position of rest (B). The weights in the pans 
 must now be interchanged, and the readings (c and r>) for two positions 
 of the rider determined as before. 
 
 From the two sets of readings before and after changing the weight 
 by 1 milligram calculate the number of scale divisions which cor- 
 respond to 1 milligram, or, what is for most purposes more convenient, 
 the increase of weight which will produce a deflection of one scale 
 division. Let W and W be the true weights of the weights placed in 
 the pans, which true weights may, and probably will, differ from their 
 nominal values. When W is in the right-hand pan and W in the left, 
 calculate the weight w which has been added to the one which appears 
 to be lighter, say, W, to. produce equilibrium. The whole number of 
 milligrams in w will be obtained from the position of the rider on the 
 beam, and the tenths and hundredths will be obtained by multiplying 
 the difference between the scale reading for the zero and the scale reading 
 corresponding to the position of rest (A) above by the weight correspond- 
 ing to a scale division. In the same way calculate the weight w' which 
 must be added to W when W is in the left-hand pan to bring the balance 
 to the zero position. The quantity w may be either positive or negative. 
 
 Let the length of the right-hand arm of the beam be called It, and that 
 of the left L. Then in the first position of the weights we have ( W + w)li 
 = W'L, and in the second position W'R = (W+w')L. Multiplying 
 these two equations together we get W(W+w)l& = W'( W+w')L?. 
 
 Thus 
 
 tff w w 
 
 l + 2W + )( l -2W + 
 
 1 In the case of a very sensitive balance the position of the rider can be 
 adjusted to the nearest division in the scale which carries the rider, while the 
 change in the position of the rider made to determine the sensitiveness will 
 have to be less than 1 milligram. 
 
26] WEIGHING BY METHOD OF OSCILLATION 69 
 
 For as w and w are small compared with W or W, we may neglect the 
 
 w . w 
 terms in squares and higher powers of ^ and -=, 
 
 If the arms are found to be unequal it will be necessary either to 
 correct all weighings by multiplying them by the ratio of the arms 
 determined in this way, a process which is not very satisfactory if great 
 accuracy is desired, since owing to unequal heating of the two arms 
 this ratio may not be constant; or we may so conduct the weighings 
 that the ratio of the lengths of the arms is eliminated from the result. 
 There are two methods by which this may be performed. One method, 
 called the method of double weighing, or Gauss's method, is to weigh 
 the body first when it is placed in the right-hand pan, and then when 
 it is placed in the left-hand pan. Let the apparent weights be w and 
 w ; then if W is the true weight, we have 
 
 WR wL 
 
 or 
 
 W= 
 
 Since w and w' are always very nearly equal, the arithmetical mean 
 ^(w + w') will not differ appreciably from the geometrical mean Jivw' . 
 Thus since the arithmetical mean is so very much easier to calculate, the 
 true weight of the body is taken as the mean of the weights obtained 
 when it is placed in the two pans. The other method is called Borda's 
 method, or the method of substitution. In this method the body, say, is 
 placed in the left-hand pan, and weights are added to the right-hand 
 pan till equilibrium is obtained. Then, leaving these weights in the 
 right-hand pan, the body is removed and weights are added to the left- 
 hand pan till the balance is again in its zero position. The sum of the 
 weights in the left-hand pan will then be equal to the weight of the 
 body, independent of any inequality in the lengths of the balance arms. 
 
 Of these two methods of making the weighing independent of the 
 lengths of the balance arms, the first, or the method of double weighing, 
 is the better, as in addition to being somewhat quicker and involving only 
 the use of a single set of weights, it is slightly more accurate, since two 
 weighings of the body are made, and the probable error of the final value 
 is smaller than in the method of substitution, where the body is weighed 
 and then the weights are weighed. 
 
 When two very nearly equal weights have to be compared, as occurs 
 when determining the value of a weight from that of a standard, or when 
 calibrating a set of weights, the method of double weighing is employed. 
 Since in general the differences of the weights will be very small, instead 
 of adding weights to one side till the balance comes back to zero, the 
 position of rest is noted, first, when the weight W, say, is in the right 
 pan, and, second, when W is in the left pan. If O l and 2 are these 
 
70 WEIGHING [ 26 
 
 readings on the scale, we have, if R and L are the lengths of the balance 
 
 arms, 
 
 MI - (1) 
 
 8<9 2 (2) 
 
 where 8 is a constant depending on the position of the centre of gravity 
 of the beam, weight of the beam, length of pointer, &c. &c. 
 Subtracting (2) from (1) 
 
 W(R + L)- W'(R + L) = 8(0! - 2 ), 
 
 or 
 
 W- W = 
 
 or since R and L are very nearly equal, and the whole difference W - W 
 is small compared to either, we may take 
 
 ~ OD\1 2/" *" \ I 
 
 If, now, the sensitiveness (s) of the balance (that is, the deflection pro- 
 duced by one milligram) is determined, we have 
 
 or 
 
 R 
 
 Hence substituting this value in (3) 
 
 W- W' = 6 -^-^ milligrams. 
 
 The following example will illustrate the way of entering results 
 when using the method of vibrations, also the manner in which the 
 readings are arranged when comparing two nearly equal weights. The 
 numbers were obtained when comparing a 1 gram weight (indicated 
 by 1 .) with a standard 1 gram weight (l,. fc ) which has itself been 
 compared with the standards at the Bureau International at Sevres. 
 
 Left Pan. 
 
 Right Pan. 
 
 Readings of Extremities 
 of Swings. 
 
 Position 
 of Rest. 
 
 Means. 
 
 
 j 
 
 61-0 49-9 
 
 60-3 
 
 50-6 
 
 59-6 
 
 55-27^1 
 
 
 1.-. gram . . 
 
 l s . fc . gram. . I 
 
 50-9 
 50-3 
 
 60-9 
 60-1 
 
 51-1 60-2 
 50-8159-8 
 
 51-6 
 51-2 
 
 55-88 1 
 55-36 | 
 
 55-46 
 
 
 1 
 
 50-8 
 
 59-8 
 
 51-1 
 
 59-2 
 
 51-6 
 
 55-34 J 
 
 
 l.-. + O'l mg. . 
 
 U . . j 
 
 49-1 
 50-2 
 
 40-3 
 40-5 
 
 48-8 
 49-9 
 
 41-0 
 41-0 
 
 48-3 
 49-5 
 
 44-69 { 
 45-31 ] 
 
 45-00 
 
 
 I 
 
 49-0 
 
 40-9 
 
 48-9 
 
 41-4 
 
 48-6 
 
 44-99 \ 
 
 
 u. .... 
 
 
 50-0 
 49-0 
 
 42-3 
 
 42-2 
 
 49-4 
 
 48-7 
 
 42-9 
 
 42-8 
 
 48-8 
 48-2 
 
 46-00 
 45-56 f 
 
 45-76 
 
 
 (|50-4 
 
 42-9 
 
 50-0 
 
 43-0 
 
 49-7 
 
 46-49 J 
 
 
 l..fc .... 
 
 l.-. + O-l mg. < 
 
 49-9 
 50-7 
 
 61-7 
 
 62-4 
 
 50-2 
 51-0 
 
 61-1 
 62-0 
 
 50-6 
 51-4 
 
 55-81 ) 
 56-61 i 
 
 56'21 
 
27] CALIBRATION OF A SET OF WEIGHTS 71 
 
 Deflection for 0*1 mg. = 55 f 46 - 45-00 = 10*46 scale divisions 
 =56-21 -45-76 = 10-45 
 
 Sensitiveness (deflection for 1 mg.) = 104-5 scale divisions. 
 Difference of reading on reversing weights = 2{1 . - l 8-fc- } 
 = 55-46 - 45-76 = 9'70 scale divisions. 
 
 Hence 1 . - I 8k = 4 ' 85 = 0-0464 milligrams. 
 104'5 
 
 But the standard l sk is really 1 gram + 0-083 mg. 
 Hence the true weight of 1.. is 1-000129 grams. 
 
 27. Calibration of a Set of Weights. Although a set of weights by 
 a good maker when new will be wonderfully correct, yet when making 
 really accurate measurements it is necessary to determine carefully the 
 amounts by which the individual weights differ from their nominal values, 
 and, further, if the weights are much used it will be necessary to repeat 
 this calibration from time to time. To obtain the absolute values of the 
 weights it will be necessary to have a standard weight the absolute value 
 of which is known, and in order to obtain accurately the values of the 
 other weights of the set it is essential that this standard weight should 
 have the same nominal value as one of the largest weights of the set 
 which is being tested. Thus if the set to be tested consists of weights 
 from 100 grams downwards the standard ought to be a 100 gram weight, 
 though a 50 gram weight would do fairly well. It would be quite impos- 
 sible to determine the errors of such a set of weights with any reasonable 
 accuracy by comparing them with a standard 1 gram weight. If no 
 standard weight is available then we can still determine the errors in the 
 ratios which the several weights bear to one another, and for many pur- 
 poses this is all that is necessary. Thus when making a chemical 
 analysis, or determining a density, the absolute values of the weights is 
 immaterial ; all that we wish to know is the relation the weights bear one 
 to the others. 
 
 A set of weights from 100 grams downwards generally consists of the 
 following pieces : 100, 50, 20, 10, 10, 5, 2, 1, 1, 1, -5, '2, -1, -1, '05, 
 '02, -01, *01, and a rider weighing -01 gram or "005 gram. For purposes 
 of calibration, however, a more convenient arrangement is the following : 
 -100, 50, 20, 20, 10, 5, 2, 2, 1, 1, -5,, -2, -2, -1, -05, '02, -02, -01, -01, 
 and a centigram or a half centigram rider. We shall describe two 
 methods of conducting the calibration, the first a comparatively short and 
 simple one, applied to the first of the above arrangements of weights, and 
 the second a more elaborate and accurate method, which has been worked 
 out by M. Rene Benoit of the Bureau International, and arranged to suit 
 a set of weights having the second arrangement. 
 
 Unless the weights in the set which have the same denomination 
 have received distinguishing marks it will be necessary to provide such 
 marks. The best way of doing this is to take a sharp-pointed sewing- 
 
72 WEIGHING [ 27 
 
 needle and immediately after the figure denoting the denomination 
 of the weight make a small dot or set of dots by gently pressing the 
 point of the needle vertically down on the weight. Thus the 1 gram 
 weights would be marked 1., !..,!.. . This method of marking 
 the weights is preferable to attempting to scratch letters or any such 
 device on the weights, as being more legible and less likely to remove 
 any of the metal. 
 
 The first operation to be performed is to compare the 50 gram weight 
 with the sum of the 20 gram to 1 gram weights inclusive. This com- 
 parison must be performed by the method of reversal, in which the 50 gram 
 weight is first placed in the left-hand pan and the others in the right, 
 and then the loads in the pans are interchanged, the calculation being 
 performed as in the example given in the last section. If the weights 
 differ much from their nominal values, it will sometimes be necessary to 
 add a small weight to one pan or the other to produce equilibrium. In a 
 balance having each of the arms graduated into a hundred parts and 
 using a centigram rider, this would involve moving the rider from side to 
 side or using two riders. In that form of balance in which the whole 
 beam from one end to the other is divided into a hundred parts, and a 
 half centigram rider is used, it is impossible by means of the rider to add 
 a small weight to the left-hand side. Hence in either case it is advis- 
 able to place a weight of half a centigram in the left-hand pan and keep 
 it there throughout the weighings. By this means the rider, when no 
 other weights are placed in the pans, will be half-way along the scale, 
 and hence by moving it to one side or other of this position we are able 
 to increase the load in either pan. 
 
 Place the 50 gram weight in the left-hand pan and the others in the 
 right, and find the position of the rider which will most nearly bring the 
 pointer to' the zero of the scale. Then determine very carefully by the 
 method of vibrations the scale reading corresponding to the position of 
 rest. Next reverse the weights and repeat. The results should be 
 entered as in the example given in the last section. 
 
 In the same way perform the comparisons indicated in the following 
 table, in which the expressions (20), &c., represent the actual weight of 
 the weight whose nominal weight is 20 grams, and the quantities a, 6, c, 
 &c., are the differences deduced from the weighings, and which may be 
 either positive or negative : 
 
 (50) - 
 
 (2) 
 (3) 
 (l.-.)}^/ .... (4) 
 
 .)} = e . . . . . (5) 
 
 (2) -{(!.) + (!..)}=/ ........ (6) 
 
 (l.)-(l-. ) = g . . ..... '. . (7) 
 
 (1 .>-/,)-* ......... (8) 
 
27] CALIBRATION OF A SET OF WEIGHTS 73 
 
 Consider, first, the first four weighings, and using the symbol (-10) 
 for the true weight of the five smaller weights, we have 
 
 From (4) (10 .) = (210) + d (9) 
 
 From 
 From 
 From 
 
 3) and (9) (10. .) = (210) + d + c (10) 
 
 2) and (10) (20) = 2(21Q) + 2d+<H-fc. . . .(11) 
 
 and (11) (50) = 5(210)-f-4tZ+2c + fr + a . . . (12) 
 (210) = i(50) - i{4d + 2c + b + a}. 
 
 Or if we assume for the present that the 50 gram weight is correct, 
 and write 
 
 (50) = 50-0. 
 
 Next to obtain the errors of the units we have from (5, 6, 7, and 8) 
 above 
 
 adding 2(10) =10(1 /.] 
 
 or since 2(10)= 10-5 
 
 . 
 
 If we call 
 
 (5) - 5 - 5s + 4/t + 2</ +/+ e. 
 
 n this way the errors of the weights taken with respect to the 50 
 gram weight are obtained, and for many purposes this will be sufficient. 
 Occasionally, however, as, for instance, when a current is measured by the 
 weight of metal deposited in a voltameter, it is necessary to know the 
 actual errors of the weights. To obtain these it is necessary to compare 
 the weights with a standard weight, the value of which is known in 
 terms of the standard kilogram kept at Sevres. In order to obtain the 
 errors of the whole set of weights with an accuracy comparable with the 
 accuracy with which two weights can be compared on the balance used, 
 it is necessary that the standard weight should be at least equal to the 
 
74 WEIGHING [ 27 
 
 largest of the weights in the set being tested. If this is not done, and, 
 say, we attempt to standardise a set of weights up to 50 grams by com- 
 parison with a standard 1 gram weight, the error of the standard and 
 of the comparison of the standard with one of the 1 gram weights of the 
 set will be magnified fiftyfold in the case of the 50 gram weight. Thus, 
 in the case before us, it is necessary to have either a 50 or a 100 gram 
 standard. Suppose that we have a standard 100 gram weight, then we 
 compare this with the sum of all the weights from 50 to 1 gram. This 
 observation must be repeated several times, and the mean of the differ- 
 ences taken to represent the quantity x in the expressions below. Then 
 
 (2100) = (100) + x, 
 
 where (100) represents the actual weight of the standard as given on its 
 certificate. 
 
 Let 2y be the algebraical sum of all the errors of the weights from 
 20 to 1 gram obtained when the 50 gram was taken as correct. Then 
 
 from which the error of the 50 gram weight can at once be obtained, and 
 thence the errors of the other weights. 
 
 When comparing the standard weight it is important to ascertain 
 whether its density is the same as that of the weights being tested. If 
 not, a correction for buoyancy will have to be made, as described in 28. 
 The density of the standard weight will be given on its certificate, since 
 it will have been measured before it was standardised. The density of 
 the weights being calibrated will have to be measured by weighing one 
 of them, say, the 50 gram weight, first in air, and then in water (see 
 31). This, however, cannot be done if the weight is not made in a 
 single piece of metal as it ought to be, that is, if it has the knob at the 
 top screwed into the main portion. In such a case, the only thing will 
 be to take the density as being equal to that of the particular metal of 
 which the weights are supposed to be composed, as given in Table 
 28. In any case, it is a good thing to compare the weight of the 
 weight on which the measurement of density has been made both 
 before and after being immersed in water. If the weight has increased, 
 it will indicate that there is some hollow in the weight into which the 
 water has penetrated, and the weight will require readjusting. 
 
 The errors of the fractions of the gram and of the rider can be de- 
 termined in the same manner. To obtain accurate results it will, how- 
 ever, be necessary to use a lighter and more sensitive balance than was 
 used for the larger weights. 
 
 The following account of the method he uses for calibrating a 
 set of weights is due to M. Kene Benoit, Director of the International 
 Bureau at Sevres, and has been communicated to the author in a 
 letter : 
 
 The set of weights is supposed to consist of the following : 100, 50, 
 20, 20 . , 10, 5, 2, 2 . , 1, 1, and so on, and the 100 gram weight is sup- 
 
27] 
 
 CALIBRATION OF A SET OF WEIGHTS 
 
 75 
 
 posed to have been compared with the standards, the error of comparison 
 being negligible. 
 
 The (100) is then compared with the (2100), the operation being 
 performed three or four times, the mean result being that 
 
 (50) + (20) + (20.) + (10) = ^ (1) 
 
 The additional comparisons in the following equations are then made ; 
 in each case the weights to which a + sign are prefixed being compared 
 with those to which a - sign is prefixed : 
 
 + (50) + (20) + (20.) + (10) =M 
 
 + (50) -(20) -(20.) -(10) =! 
 
 + (50) - (20) - (20 .) - (210) = 2 
 
 + (20) - (20 .) + (10) - (210) = a s 
 
 + (20) - (20 .) - (10) + (210) = a 4 
 
 + (20) -(20.) 
 
 + (20) -(10)-(210) = a 6 
 
 + (20.) -(10)-- (210) = a, 
 
 + (10) -(210) = 08 
 
 Here we have nine equations and only five unknowns, namely, the 
 values of the five weights (50), (20), (20.), (10), (210), the sum of the 
 units being considered as a single weight. The equations must, therefore, 
 be reduced to five normal equations. The rule founded on the principle 
 of least square is to multiply each equation through by that coefficient 
 of one of the unknowns which occurs in that equation, and to add all 
 the equations thus obtained to form a new equation, which is called a 
 normal equation. Thus in the case of the (20) we have to multiply the 
 first equation through by + 1, the second and third equations through 
 by - 1, and the next four equations through by + 1, while the last 
 two equations, as they do not contain (20), i.e. we multiply through 
 by 0, vanish. Then adding, we get 
 
 - (50) + 7(20) + (10) = M - a x - 
 
 + a. 
 
 Proceeding in this way the following normal equations are obtained : 
 
 (20.) 
 
 (10) 
 
 3(50)- (20)- 
 -(50) + 7(20) + 
 
 -(50) +7(20.)+ 
 
 + (20)+ (20.) + 7(10)- (SIO) 
 -(50) - (10) + 6(210) 
 
 =M+A. 2 
 
 (3) 
 
 riiere 
 
 - a^ - a 2 + a 3 + ff 4 + a 5 
 
 (1^ +#,! 
 
 1 o 4 
 
 tt.-) vn ~f~ d>* 
 
 + a, 
 
76 WEIGHING [ 28 
 
 Solving the set of simultaneous equations (3) we obtain 
 
 (5) 
 
 (50) = J3f+ V {7A 1 - A+ A 5 } 
 (20) = M+ jfol A^^+ 5A 2 - A 4 } 
 
 - A} 
 
 h 5^ 6 [ 
 (210) =^M + T Jo{7^!+ 3^ 4 + 25^1 5 } 
 
 Calling now the correction to (210) M lt so that 
 
 we may proceed in exactly the same manner to obtain the corrections to 
 these weights, and then having obtained the correction to the 1 gram 
 weight, the corrections to the fractions can be obtained by a similar 
 process. 
 
 M. Benoit has shown that by combining the comparisons in the 
 above manner there is no accumulation of errors, and that any prob- 
 able error made in comparing the (2100) with the standard becomes 
 of less and less influence as the weights of lower denomination are 
 reached. 
 
 In the event of a standard 100 gram weight not being available, the 
 ratios of the weights can be obtained by the above process, for if we put 
 M=0 in equations (5), we shall obtain the corrections to the weights on 
 the supposition that the sum of the weights 50 + 20 + 20 . + 10 is correct. 
 
 28. Reductions of Weighings to Vacuo. The downward force 
 exerted by a body on a scale pan in air is slightly less than the weight 
 of the body owing to the buoyancy of the air, the actual pressure on 
 the scale pan being equal to the weight of the body less the weight of 
 the air displaced by the body. Thus, unless the volume of the body 
 being weighed and that of the weights is the same, the amounts by 
 which their weights appear to be diminished on account of the buoyancy 
 of the air will be different, and hence, although they may be in equi- 
 librium when placed in the opposite pans of a balance, they have not 
 exactly the same weight. If, however, the weighings were made in 
 vacuo, there would be no such difference depending on the difference 
 between the density of the body and of the weights. Since it would be 
 very inconvenient to perform weighings in vacuo, we have to apply a 
 correction to allow for the buoyancy of the air, an operation called 
 reducing the weighings to vacuo. 
 
 Let W be the true weight of the body, i.e. its weight in vacuo, and 
 w the true weight of the weights which counterpoise the body. Then if 
 D is the density of the body, 8 that of the weights, and or that of the air 
 at the time of the weighing, we have that the volume of the body is 
 W/D, and that of the weights is w/8. Hence the weight of the air dis- 
 placed by the body is a- W/D, and of that displaced by the weights is 
 trw/8, and therefore 
 
28] REDUCTIONS OF WEIGHINGS TO VACUO 77 
 
 w w(l-<r/8) 
 Hence ^T^/lP 
 
 or since o- is generally small compared with D or 8, 
 
 From this expression it will be seen that when the density of the weights 
 is greater than that of the body, we have to slightly increase the apparent 
 weight of the body to get the true weight, while if the density of the 
 weights is less than that of the body, we have to reduce the apparent 
 weight. 
 
 The density of the air, <r, varies slightly from day to day, and where 
 the utmost accuracy is desired it must be calculated from the expression l 
 
 0-001293 B - 3/8A 
 1+0-00367* ~ 760 
 
 ~ 
 
 where B is the height of the barometer reduced to 0, h is the tension of 
 aqueous vapour present in the air, and t is the temperature. Where no 
 measurements of the hygrometric state have been made, it will generally 
 be best to assume that the air is half saturated at a temperature equal 
 to that inside the balance case. 2 Of course if the interior of the case is 
 very carefully dried by means of some desiccating agent, h may be taken 
 as equal to zero. For most measurements made in the physical laboratory 
 it is quite accurate enough to take a mean value for the density of air, 
 and the number generally employed is 0'0012. In the same way we 
 may take the density of the weights if of brass as 8 '4, if of platinum as 
 21 '5, if of quartz or aluminium as 2 '65. Assuming the above values for 
 the densities of air and the weights, it is possible to calculate a table to 
 facilitate the correction of weighings for the buoyancy of the air. 
 
 Table 3 has been worked out on these assumptions, and contains 
 in the first column the density of the body being weighed, and in 
 the second, third, and fourth columns the value of the expression 
 
 0012( - - j x 1000, that is, the correction in milligrams to the weight 
 
 of the body for each gram of this weight. For instance, if a body of 
 density 3'0 is found to weigh 12*3456 grams when brass weights are 
 used, then the buoyancy correction is + 12'35 x 0'26 = +3'21 milli- 
 grams, so that the weight of the body in vacuo would be 12 '3488 grams. 
 The correction on account of buoyancy in the case of water when, as 
 is usual, brass weights are used is very appreciable. Thus in the case 
 of 10 grams of water the correction amounts to 10'6 milligrams. Thus 
 when water is used to gauge the volume of a vessel, the correction for 
 "moyancy must be made if results correct to 1 in 1000 are required. 
 
 001^5)3 
 
 1 The values of "'^ for various temperatures are given in Table 4. 
 
 1 4~ O'OUoD ( t 
 
 2 For the tensions of aqueous vapour at different temperatures, see Table 14. 
 
78 WEIGHING [28 
 
 It is important to remember that weighings have to be reduced to 
 vacuo not only when the absolute weight of a body is required, but also 
 in some cases when we have to do with the ratio of the weights of two 
 bodies, as, for instance, in a chemical analysis. Thus if two of the 
 constituents of the body being analysed have very different densities in 
 the form in which they are weighed, the ratio of their apparent weights 
 in air will not be the same as the ratio of their true weights. Thus, 
 suppose that the true weight of one of the constituents is W, its density 
 D, and that its apparent weight is w, the density of the weights being 8, 
 while let the same letters dashed represent the corresponding quantities 
 for the second constituent. Then we have the true ratio of the weights 
 of the bodies given by 
 
 w "fr-Ks-S)} 
 w 
 
 since a- is small compared with D, D', or 8. It will be seen from this 
 expression that we have not to correct for the air displaced by the 
 weights used in performing the weighings if, as we have supposed, 
 weights of the same density are used in the two cases. 
 
 Thus, suppose a mass of sea water (density 1*026) is weighed, the 
 apparent weight being w, and the apparent weight of the silver chloride 
 formed when the chlorides are precipitated by adding silver nitrate is w. 
 Then, since the density of silver chloride is 5 -6, the true ratio of the 
 weight of the silver chloride to the weight of water is 
 
 ^,= w -\ i - -0012 ( * - nL) j -?,{ i + -0011 ) 
 
 W w I \5'6 1'026/J w I j 
 
 so that the correction amounts to about 1 in 1000. 
 
CHAPTER IV 
 
 DENSITY 
 
 29. Density. By the density of a substance is meant the mass of unit 
 volume. Thus in the C.G.S. system the density of a body is the mass 
 in grams of a cubic centimetre. Since, however, the cubic centimetre is 
 denned not as the volume of a centimetre cube, but as the thousandth part 
 of a litre, and a litre as the volume of a kilogram of water at a temper- 
 ature of 4 C., it follows that the density of a body is really the ratio of 
 the mass of any given volume of the substance to that of an equal volume 
 of water taken at a temperature of 4 C. The term specific gravity is 
 sometimes taken to represent the ratio of the mass of any given volume of 
 the substance to the mass or weight of an equal volume of water taken at 
 the same temperature as the substance. Since different bodies expand 
 with heat at different rates, the change in density which takes place as the 
 temperature rises is not the same for all substances, and hence the specific 
 gravity as defined above is not a constant, but depends on the temperature 
 at which the comparison is made. The following ratios are sometimes 
 used to express the density or specific gravity of substances ; their use, 
 however, has nothing to recommend them : 
 
 Mass of substance at 
 Mass of equal volume of water at ' 
 
 Mass of substance at 15 
 
 Mass of equal volume of water at 15 
 
 Throughout this book, when we use the term density of a substance 
 we shall imply the following definition : 
 
 The density of a substance at a temperature t is the ratio of the mass 
 or weight of a given volume of this substance taken at the temperature t 
 to the mass or weight of an equal volume of water taken at 4 C. 
 
 Of course the observations need not necessarily be made in water at 
 4 C., since elaborate experiments have been made to determine the density 
 of water, as above defined, at all temperatures. Table 5 gives the density 
 of water at temperatures between and 35 C. 
 
 30. The Measurement of Density Corrections for Temperature 
 of Water and Buoyancy of Air. The determination of the density of 
 a substance, whether it be a solid, a liquid, or a gas, consists of two 
 parts. In the first place, we determine the weight of a given volume 
 of the substance, and, secondly, we determine, directly or indirectly, the 
 weight of an equal volume of water at a temperature of 4 C. Both the 
 
 79 
 
80 DENSITY [ 30 
 
 weights will require the application of certain corrections, and as these 
 are the same whatever the method employed, it will be convenient to 
 consider these corrections once for all before proceeding to give detailed 
 instructions for performing the measurements. 
 
 Let w be the apparent, i.e. not corrected for buoyancy of the air, 
 weight of the body in air, or, if we are determining the density of a 
 liquid by weighing a solid sinker in it, the apparent loss of weight of 
 the sinker as it is transferred from air to the liquid. Let w be the un- 
 corrected weight of a volume of water at temperature t and density A 
 equal to the volume of the body. Lastly, let o- be the density of the air 
 at the time of the observations, which we may in general take as being- 
 equal to -00 12. 
 
 In the first place, we may note that as we shall throughout be 
 dealing with ratios of weights it will be unnecessary to correct for the 
 air displaced by the weights if we use weights of the same density 
 throughout (see p. 78). 
 
 If the volume of the body is called v, then the weight of air it dis- 
 places is t'o-, and thus its weight in vacuo will be w + vo-. If, however, 
 the substance is a fluid, and we obtain the weight of a given volume v 
 by weighing a sinker of this volume first in air and then in the fluid, 
 the apparent weight of the sinker in air will be the true weight less vo-, 
 and its apparent weight in the fluid will be its true weight less the true 
 weight of the volume v of the fluid. Hence the loss of weight as the 
 sinker is moved from air to the fluid will be the true weight of the 
 volume v of the fluid less the weight of air it displaced ; that is, 
 
 Loss of weight of sinker = true weight of sinker - wr 
 {true weight of sinker - true weight of v c.c. of liquid}. 
 
 Thus the true weight of the volume v of the fluid will be the loss of 
 weight plus wr. Therefore, whatever the method employed for deter- 
 mining the weight of the body being measured, we have that the true 
 weight is w + vo-. 
 
 Next we have to consider the weight of an equal volume of water 
 at 4 C. Suppose, in the first place, that we actually weigh the volume 
 v of water in the air, as when we use a specific gravity bottle. Then the 
 true weight of the water will be w +vo: Next, if we obtain the weight 
 of the water by weighing the body being measured in the water by the 
 method of Archimedes, then, as explained above, the observed loss of 
 weight will be less than the weight of water displaced by the amount vcr. 
 Hence in this case also the weight of the water of density A, which 
 occupies a volume equal to that of the body, is w' + vo-. Thirdly, if we 
 have weighed a sinker, first, in the given fluid, and, secondly, in water, we 
 have as before to add to the loss the quantity vo- to get the true weight 
 of the water which occupies a volume equal to that of the sinker, and 
 hence equal to the volume considered of the given fluid. 
 
 Now, what we require is not the weight of the volume v of water 
 at a temperature t, but the weight of the same volume of water at 4 C. 
 
30] THE MEASUREMENT OF DENSITY 81 
 
 Now at 4 C. the density of the water is 1, while at t its density is A. 
 Hence to obtain the weight of water which will fill any given volume, we 
 have to divide the weight of water at a temperature t, which will fill the 
 same volume, by A. 
 
 Thus whatever the method used for measuring the density, the weight 
 of water at 4 C. which would occupy the same volume as the body is 
 given by (w + w)/A. 
 
 But we have seen that the true weight of the body is w + wr. Hence 
 the density is given by 
 
 W + V<T 
 
 We have now to find an expression for v in terms of the measured 
 quantities. 
 
 The volume v is equal to the weight of water, w' + va; which occupies 
 the volume v, divided by the density of the water A. Hence 
 
 w' + va- 
 
 or 
 
 -. 
 
 A-o- 
 
 Substituting this value for v, we get 
 
 D = 
 
 A-o- 
 
 W<T 
 
 (\ 
 1 - jj- - + terms in higher powers of o-/(A o-) JA. 
 
 w 
 
 Since o- is small compared to A, we may neglect squares and higher 
 
 powers of - - . Hence 
 A -o- 
 
 =,+_ _ + . 
 
 w A - a- w A - o- 
 
 Now the two last terms on the right are both small compared to the 
 first term, so that we may in these small terms neglect o- compared to A, 
 so that the expression reduces to 
 
 w 
 
 If iy is the density, uncorrected for buoyancy, that is, D' = A, 
 
 w 
 
 then since A is always nearly unity, we may substitute D' for wjw in 
 the buoyancy correction term. Hence 
 
 /y) ....... (2) 
 
 F 
 
82 
 
 DENSITY 
 
 [31 
 
 When correcting observations of density by this expression, it will 
 generally be quite sufficient to take the mean value 0'0012 for the density 
 of the air. Thus o- being constant, we may construct a table of the values 
 
 of the correction a-{ 1 - I for different values of wlw' that is, of the 
 I w J 
 
 uncorrected density. Such a table is given in the Appendix (Table 6). 
 It will be seen from the table, as is also at once evident from the formula, 
 that the correction vanishes when the density of the body is unity. For 
 bodies of density less than unity the correction is positive, while for 
 densities greater than unity it is negative. 
 
 31. Measurement of the Density of a Solid heavier than Water 
 by the Method of Archimedes. Suppose we have to measure the 
 density of a solid body which is not soluble in water and has a density 
 
 greater than unity, such as a 
 piece of quartz or glass. The 
 operations to be performed is to 
 determine the weight of the body, 
 first in air and, secondly, when 
 suspended so that it is totally 
 immersed in water at a known 
 temperature. In order to be able 
 to place a vessel .containing the 
 water so that the body when 
 suspended from the arm of the 
 balance may hang in the water, 
 either a short pan support, as 
 shown in Fig. 30 (a), may be used, 
 or we may use a small wooden 
 stand to bridge over the pan, as 
 shown in Fig. 30 (b). Generally 
 this latter arrangement will be 
 found the most convenient. When 
 being weighed in water the body 
 will require to be suspended in 
 some manner. The usual arrange- 
 ment is to hang it by means of 
 FIG. 30. a fine fibre of unspun silk or 
 
 a fine platinum wire. Cotton 
 
 must on no account be used, since the water would be sucked up 
 along the cotton fibres, and would thus make the weight when 
 weighing in water appear greater than it should be. Of course with 
 this arrangement what is really measured is the mean density of the 
 body and a certain amount of the silk fibre, so that it is essential that 
 the fibre should be as fine as possible. When the utmost accuracy is 
 desired, and whenever a large number of measurements of the density of 
 solids have to be made, it is much preferable to use the arrangement shown 
 in Fig. 31. A light metal pan, preferably of platinum, is suspended 
 
31] MEASUREMENT OF DENSITY OF A SOLID 83 
 
 by a fine platinum wire as shown, and throughout the measurements 
 
 remains immersed in the water. When determining the weight in air 
 
 the body is placed in the upper pan, and when measuring the weight in 
 
 water in the lower. Care must be taken always 
 
 to bring the pointer of the balance to the same 
 
 division of the scale and to keep the level of 
 
 the water constant, so that the same amount 
 
 of the suspension wire is immersed in the 
 
 water, when the loss of weight on the body 
 
 being transferred to the lower pan will be due 
 
 alone to the loss of weight in the body when 
 
 transferred from air to water. 
 
 Owing to the effects of capillarity at the 
 point where the suspension wire cuts the 
 surface of the water, the sensitiveness and 
 accuracy of a balance is appreciably less when 
 used to weigh a body in water. Hence it 
 is advisable, whenever possible, to use a fair- 
 sized piece of the material which is being 
 tested. 
 
 Kohlrausch l has shown that the effect FIG. 31. 
 
 of capillarity in producing irregularities, 
 
 where a platinum suspension fibre cuts the surface of the water, can 
 be very much reduced if the wire is treated in the following 
 manner : The wire is used as the negative electrode in an electrolytic 
 cell containing a dilute solution of platinic chloride (1 part platinic 
 chloride, 30 parts water, and O008 part lead acetate, i.e. a trace), and 
 an E.M.F. of about 4 volts is applied. In this way, after washing and 
 drying, a black mat surface is obtained. The wire is then heated to 
 redness in a small Bunsen flame, so that the surface turns grey. Care 
 must be taken not to dirty the portion of the wire which will cut the 
 surface of the water, even by touching it with a clean finger. 
 
 Take a piece of glass rod weighing about 10 grams, and determine its 
 weight in air. Then tie a silk fibre round the middle and suspend it 
 from the support which carries the pan, and again weigh. The differ- 
 ence in the weights will give the weight of the fibre. Next take a beaker 
 and nearly fill it with distilled water, and, having well stirred the water, 
 place the beaker on a stand so that the glass is immersed in the water 
 and does not touch the sides of the beaker. Also place a thermometer 
 in the water. Examine the glass very carefully to see whether there are 
 any air bubbles adhering. If so, these must be removed. This can best 
 be done by means of a small camel's hair pencil. Next determine the 
 weight of the glass as it hangs in the water, and then read the ther- 
 mometer. 
 
 In very accurate measurements it is necessary to take account of 
 the effect of the suspension fibre. To do this, estimate what proportion 
 
 1 Wiedemann Annalen (1895), Ivi. 185. 
 
84 DENSITY [32 
 
 of its total length is immersed in the water, and then from the weight 
 of the whole fibre and its density (platinum is generally used in such 
 delicate measurements, density 21*5) calculate the volume of the fibre 
 which was immersed in the water. The weight of this volume of water 
 must then be subtracted from the total weight of the fibre, and the 
 difference subtracted from the weight of the substance in water. 
 
 If the weight of the body in air is w, and its loss of weight in water 
 is w', the density of the water at the temperature of observation as 
 obtained from Table 5 being A, and the density of the air at the time 
 of observation o-, then the density of the body D is given (see p. 81) by 
 
 w ( w 
 
 32. Measurement of the Density of a Solid lighter than Water. 
 
 If the density of the solid is less than unity, it will float in water. Hence 
 to determine the weight of water it will displace we have to attach a 
 weight to act as a sinker. A piece of lead or copper, the weight of 
 which need only be just sufficient to sink the body, must be weighed in 
 water, and let the weight be s. Next weigh the body in air, and let its 
 weight be w. The sinker must be attached to the body and the weight 
 of the two determined in water, and let the weight be x. Then the 
 weight of the body alone in water will be given by 
 
 x - s. 
 
 Thus the apparent weight w' of the water displaced by the body is 
 given by 
 
 w' = w x + s, 
 
 and this value substituted in the expression given at the end of the last 
 
 section will give the density of the body. 
 
 If a number of determinations of density of bodies lighter than water 
 
 have to be made, it will be worth while to make a small cylindrical cage 
 of wire gauze, as shown in Fig. 32, and to suspend this by 
 a fine wire so that it is immersed in water. The weight s 
 necessary to counterpoise the cage when immersed in water 
 is first determined. Then when measuring the density of 
 the body it is first weighed when placed in the scale pan, 
 the cage being immersed in water, and the difference be- 
 tween the weight and s gives the weight of the body in air. 
 The body is then transferred to the cage and the weight 
 determined, which has to be added to the right-hand pan 
 to produce balance. The difference between the weights 
 32. on the right-hand pan when the body is in the pan and in the 
 
 cage gives the apparent weight of the water displaced, w'. 
 Bodies which are soluble in water, or on which water acts, must be 
 
 weighed in some fluid in which they are unacted upon. The density 
 
 of this fluid must be measured by one of the methods given later if it 
 
33] MEASUREMENT OF DENSITY OF A SOLID 85 
 
 is not already known. The quantity A in the formula in this case must 
 be taken as representing the density of the liquid employed. 
 
 33. Measurement of the Density of a Solid in the form of Small 
 Pieces The Pyknometer. If the solid, the density of which is to be 
 determined, is in the form of small pieces, the previous methods are 
 inapplicable, and we must use a specific gravity bottle or pyknometer. 
 Two forms of specific gravity bottle are shown in Fig. 33. The bottle 
 figured at (a) is fitted with a carefully ground-in stopper, which stopper is 
 pierced with a fine hole. The bottle is always filled to the top, and then 
 the stopper is replaced, and the excess water 
 which escapes through the hole in the stopper 
 is removed by wiping the bottle with a clean 
 duster or filter paper. The bottle shown at 
 (b) has a fine mark etched round the neck, 
 and in use is always filled up to this 
 mark. 
 
 First weigh the bottle empty and carefully 
 dried. Then fill the bottle full of water 
 which has been boiled for a short time to 
 get rid of dissolved air and has then been 
 allowed to cool. The bottle must then be 
 stood in a beaker of water which is at the temperature of the room 
 or a degree or two hotter, and must remain there for at least twenty 
 minutes, the water being kept well stirred the while. A thermometer 
 placed in the water is read just before the bottle is removed, and this 
 will give the temperature of the water. In the case of the bottle with 
 the mark on the neck, the quantity of water in the bottle must be ad- 
 justed to the mark before the bottle is removed from the water. For 
 this purpose a piece of glass tube, the end of which is drawn out into 
 a fine capillary, or a small twist of filter paper, may be employed. After 
 carefully drying the outside, the bottle must be again weighed. If the 
 pattern of bottle with the mark on the neck is employed, it will be found 
 best to fill it at a temperature of C. For this purpose, first fill it 
 with water at the temperature of the room to a little above the mark, 
 and then place the bottle in a mixture of pounded ice and distilled water 
 so that the mark is just flush with the upper surface of the ice, and allow 
 it to stand for about half-an-hour. Having allowed the water in the 
 bottle to acquire the temperature of the ice, adjust the surface of the 
 water to the mark on the neck. 
 
 From the difference between the weighi of the bottle empty and full 
 of water the volume of the bottle can be calculated. 
 
 The bottle having been emptied and dried is now about three parts 
 filled with the solid, say, sand, and then weighed. The difference be- 
 tween this weight and the weight of the bottle empty is the apparent 
 weight of the sand. 
 
 The bottle has now to be filled up to the mark with water. This 
 operation is often one of considerable difficulty, owing to the way in 
 
86 DENSITY [ 33 
 
 which air bubbles cling to a finely divided solid. If an air-pump is avail- 
 able the bottle may be nearly filled up to the mark, or at any rate well 
 above the surface of the sand, and then placed beneath the receiver of 
 the pump. On working the pump the air will be extracted from between 
 the grains of sand. In the absence of a pump the bottle, after partly 
 filling with water, must be heated till the water just boils. In this way 
 the air will be expelled. After the expulsion of the air the bottle must 
 be filled a little above the mark, and then by immersion in water or ice, 
 as the case may be, brought to a constant temperature, which must be 
 noted, and then, after drying outside, be again weighed. 
 
 In the first place, let us suppose that the temperature at which the 
 bottle has been weighed partly filled with the sand and partly filled with 
 the water is the same as the temperature at which the bottle was filled 
 entirely with water, so that as far as the expansion of the bottle is con- 
 cerned the volume was the same in the two cases. If A is the density 
 of water at this temperature and the following are the results of the 
 weighings 
 
 Weight of bottle empty . . . . . B 
 
 Weight of bottle full of water . W 
 
 Weight of bottle and sand . . . . S 
 
 Weight of bottle with the sand and water . . R, 
 
 then the apparent weight of the sand is S B. The apparent weight of 
 the water, which occupies the same volume as the sand, is given by 
 
 w'=W+S-B-R. 
 
 Hence the density, unconnected for the reduction of the weighings to 
 vacuo, is given by 
 
 jy- S ~ B A 
 W+S-B-R 
 
 Correction (1). To reduce the weighings to vacuo we must add the 
 term o-(l - Z>'), so that if D is the corrected density, 
 
 The value of the correction is at once obtained from Table 6. 
 
 Correction (2). If the temperature at which the bottle is filled with 
 water when it contains the sand is not the same as that at which it was 
 filled with water when its volume was being determined, a correction 
 will have to be applied to allow for the fact that, owing to tfre expan- 
 sion of the glass, the volume changes with temperature. To calculate 
 the value of this correction we must know the value of the coefficient of 
 cubical expansion of the glass of which the bottle is made. The manner 
 in which this quantity may be determined is described in 73, but as 
 the measurements of density made with the specific gravity bottle are 
 always made at temperatures not far removed from the temperature at 
 which the bottle was gauged, it is generally sufficient to take the value 
 0'000023 for the coefficient of cubical expansion of glass. 
 
34] MEASUREMENT OF DENSITY WITH A SINKER 87 
 
 Suppose that when the bottle is filled with water the temperature 
 is ^ and that the density of water at this temperature is A , while when 
 the bottle contains water and sand the temperature is ^ and the density of 
 the water is A r The volume of the bottle at ^ is ( W - -B)/A , and hence 
 if y is the coefficient of cubical expansion of the glass, the volume at t 1 is 
 
 Thus the weight of water which would fill the bottle at the temperature 
 $! is given by 
 
 The actual weight of water contained in the bottle, in addition to the 
 sand, is li - S. Hence the weight of the water displaced by sand is 
 
 This expression may be simplified by writing l+(Aj-l) for Aj and 
 1 + ( A - 1 ) for A ? and dividing, when, since \ and A are each very 
 nearly equal to unity, we get 
 
 Also since y and (A x - A ) are each small, the expression reduces to 
 
 The above expression is the value of w' corrected for the temperature, 
 and it must now be substituted in the expression 
 
 0-*=**,, 
 
 w 
 
 To obtain the density reduced to vacuo the correction a( 1 D) must be 
 added. 
 
 34. Measurement of the Density of a Liquid with a Sinker. If the 
 
 liquid is not too volatile and can be procured in sufficient quantity, one of 
 
 the most convenient ways of measuring its density is to 
 
 determine the weight of a suitable sinker when suspended, 
 
 first, in water at a known temperature, and, secondly, in the 
 
 liquid. A convenient form of sinker is shown in Fig. 34, 
 
 and consists of a glass bulb partly filled with shot so that 
 
 it will sink in the densest liquid likely to be measured. It 
 
 is made by blowing a bulb on the end of a piece of glass 
 
 tubing, introducing the shot, and then drawing off the neck 
 
 and at the same time forming a small hook by means of FIG. 34. 
 
 which the bulb is attached to the suspension wire. 
 
 Suspend the bulb from the left-hand arm of a balance by means of a 
 silk fibre or fine wire, and determine the weight of the bulb in air. 
 
88 DENSITY [ 34 
 
 Next weigh the bulb when immersed, first in water and, secondly, in the 
 given liquid. The temperature of the water and of the liquid must be 
 noted in each case, and care must be taken to immerse the same length 
 of the suspension fibre each time. This can be secured if the level of 
 the upper surface of the liquid is always brought to the same height 
 above the surface of the balance case. An allowance will have to be 
 made for the weight of that portion of the suspension fibre above the 
 surface of the liquid. This can most conveniently be done by placing 
 a piece of the same kind of fibre or wire equal in length to that of the 
 portion of the suspension fibre which is above the surface of the liquid 
 in the right-hand pan of the balance and keeping it there throughout the 
 weighings. 
 
 Let the weight of the sinker in air be B, and its weight in water, of 
 which the temperature is and the density A , be W, while its weight in 
 the liquid is S, and first we will suppose that the temperature of the liquid 
 is also t Q . Then the apparent volume of the sinker is (B - TP)/A , and 
 the apparent weight of this volume of the liquid is B - S. Hence the 
 density, without reduction to vacuo, is given by 
 
 Correction (1). Eeduced to vacuo the correction o-(l - D') has to be 
 added to the uncorrected value given above. 
 
 Correction (2). If the temperature of the liquid is not the same as 
 that of the water, but is, say, t lt then a correction must be applied for 
 the expansion of the sinker with increase of temperature. Let y be the 
 coefficient of cubical expansion of the material of which the sinker is 
 made (it will in the case of a glass sinker generally be sufficiently 
 accurate to take the value '000023 for y). Then the volume of water 
 at t Q displaced by the sinker is (B - W}j\. The volume of the sinker 
 at a temperature ^ is therefore 
 
 Hence correcting for change in temperature we get 
 D= _(-Jty 
 
 since y is small. 
 
 The correction to reduce to vacuo is as before, <r(l - D). 
 
 Kohlrausch has pointed out that when using a glass float to deter- 
 mine the density of water solutions it is advantageous to observe at a 
 temperature of between 5 and 7, for over this range of temperature the 
 coefficient of expansion of water is practically the same as that of glass. 
 
35] MEASUREMENT OF DENSITY OF A LIQUID 89 
 
 If a silver lioat is employed, the temperature ought to be somewhere 
 between 7 and 10. 
 
 35. Measurement of the Density of a Liquid with Specific Gravity 
 Bottle. When only a comparatively small quantity of liquid is avail- 
 able, the specific gravity bottle may be used for measuring the specific 
 gravity of a liquid. Either of the forms of bottle shown in Fig. 33 may 
 be used, but on the whole the form with a mark on a narrow neck is to 
 be preferred to that which has a removable stopper. If the liquid is 
 volatile, then it will be necessary to employ some means of closing the 
 top. For this purpose either a glass stopper may be ground into the 
 
 {a) 
 
 FIG. 35. 
 
 top of a bottle of the form shown in Fig. 35 (a), or a bottle of the form 
 shown at (6), with a tap above the mark on the neck, may be employed. 
 As no grease must be used on the stopper or tap, care must be taken 
 not to force these in too tightly or they will stick, and great difficulty 
 may be experienced in removing them. 
 
 The method of measuring the density of a liquid with the specific 
 gravity bottle consists in first measuring the volume of the bottle by 
 weighing the water which it will contain. If, then, the weight of any 
 other liquid which fills it up to the mark is determined, we can 
 immediately calculate the density of the liquid. 
 
 When the liquid to be experimented upon is valuable, as, for instance, 
 a sample of some specially purified chemical compound, the filling of the 
 
90 
 
 DENSITY 
 
 [35 
 
 bottle in such a way that there shall be no chance of the liquid being 
 lost or contaminated is of great importance. A piece of apparatus which 
 may be used in such a case is shown in Fig. 36. It consists of two glass 
 vessels AB, one of which A is fixed to an upright board, and the other 
 B is movable, these being connected by a length of rubber tube c. The 
 vessel B can be placed in two positions, a loop attached to the top being 
 
 FIG. 36. 
 
 hooked on either of the screws K or K'. The vessel A communicates, by 
 means of a three-way tap, with the open air or a side tube E. This side 
 tube is connected by rubber tubing with the side tube of a small glass 
 T-piece F. The lower arm of the T-piece passes through a cork which 
 fits into the neck of the bottle G, in which the liquid to be used is 
 stored. Through the vertical arm of the T-piece passes a capillary tube 
 H bent into a U, a small length of rubber tubing being employed to 
 make an air-tight joint. The other leg of the capillary passes down into 
 
35] MEASUREMENT OF DENSITY OF A LIQUID 91 
 
 the pyknometer i, which is supported on a shelf J, the height of which 
 is adjustable. 
 
 The vessels A and B being partly filled with dry mercury, when the 
 vessel B is hung on the peg K the air in A is compressed, and by 
 suitably turning the tap D the pressure is communicated to the bottle 
 G, forcing the liquid over into the pyknometer. To withdraw the liquid 
 from the pyknometer the vessel A is made to communicate with the 
 open air, B is raised till A is very nearly full of mercury, and then the 
 tap D is closed. B is then lowered, and the tap turned so as to put A into 
 communication with the bottle G. In this way a reduction of pressure 
 is produced in the bottle G, and hence the liquid in the pyknometer is 
 forced over by the atmospheric pressure. 
 
 A form of pyknometer which can easily be filled, and has the further 
 advantage that it can easily be cleaned and dried, is shown at (c) in 
 Fig. 35. The capillary A being dipped below the surface of the liquid, 
 the liquid is drawn up into the bulb by applying a gentle suction to B 
 through a piece of rubber tubing. The bottle is filled till the surface 
 of the liquid is above the mark c. The bottle is then immersed in ice 
 so that the ends of the capillaries project above the ice, and when the 
 temperature of the liquid has fallen to zero the instrument is tilted so 
 that the liquid flows to the end of the capillary A. Then by holding 
 a piece of filter paper against A, so much liquid is removed that the 
 surface of the liquid in the capillary B coincides with the mark c. When 
 a volatile liquid is being used, both capillaries must be closed by caps. 
 
 When great accuracy is desirable, it is almost essential that the bottle 
 should be filled with water and with the liquid at a temperature of C. 
 Even when a comparatively rough result is aimed at, yet, since the 
 temperature at which the filling takes place will have to be determined, 
 it* will generally be found to save time to work at the temperature of 
 melting ice, since this temperature is so easily maintained and reproduced. 
 The bottle must be filled slightly above the mark, and then placed in a 
 mixture of pounded ice and distilled water so that the mark is on a 
 level with the top surface of the ice. After the bottle has been in the 
 ice for between ten and fifteen minutes the surface of the liquid must 
 be adjusted to the mark by means of a pipette, made by drawing a piece 
 of glass tube down to a fine capillary, or with filter paper if the bottle 
 is of the double capillary pattern. The outside having been carefully 
 dried, the bottle is then weighed. 
 
 To remove the water from a bottle with a single neck the instrument 
 shown in Fig. 36 may be employed, or holding the bottle upside down, 
 blow air into it through a capillary glass tube attached to a piece of 
 india-rubber tubing. Having removed as much as possible of the water, 
 the bottle must be rinsed out twice with a small quantity of absolute 
 alcohol and then once with ether. To remove the ether, blow air 
 through a capillary into the bottle by means of a foot blow-pipe. The 
 mouth must on no account be used to blow air through the bottle, since 
 the water vapour in the breath would condense in the bottle. If no 
 
29 DENSITY [ 36 
 
 absolute alcohol is available for washing out the bottle, it may be dried 
 by heating over a flame and blowing a current of air through it with 
 a bellows. This will have to be continued for some time, and the 
 heating must be very thorough, care being taken not to crack the bottle 
 by too sudden heating. 
 
 Next fill the bottle with the liquid to be measured, using the same 
 precautions as in the case of the water, and again weigh. 
 
 If, as directed above, the bottle has been filled in both cases at the 
 same temperature, namely, that of melting ice, its volume will be the 
 same when filled with the liquid and with water, and no correction for 
 the expansion of the glass will be necessary. If the weight of the 
 empty bottle is B, that of the bottle when full of water W, and when 
 full of the liquid S t the value of the density, not reduced to vacuo, is 
 
 where A is the density of water at 0. 
 
 Correction (1). To reduce to vacuo, the correction o-(l - D) must be 
 added. 
 
 Correction, (2). When the temperature at which the bottle is filled 
 with the liquid is different from that at which it was filled with water, 
 a correction has to be applied for the expansion of the bottle with rise 
 of temperature. If t$ and A are the temperature and density of the 
 water, and t : is the temperature of the liquid, while the coefficient of 
 expansion of the glass of which the bottle is made is y, then the volume 
 
 W B 
 
 of the bottle at t is given by -. . 
 
 A o 
 Hence the volume of the bottle at is 
 
 Thus the density of the liquid is given by 
 
 = , 
 
 W-B 
 
 and to allow for buoyancy a further correction of cr(l - D) must be 
 added. 
 
 36. Hare's Apparatus for the Comparison of the Densities of 
 Liquids. A convenient piece of apparatus for comparing the densities 
 of two liquids, of which about 50 c.c. are available, when an accuracy of 
 about 1 in 200 is sufficient, is shown in Fig. 37, and is called Hare's 
 apparatus. It consists of a stand carrying two glass tubes AB, AC, which 
 are joined together at the top and to a third tube at A. A piece of india- 
 
36] 
 
 DENSITY OF LIQUIDS 
 
 93 
 
 rubber tube is attached at A, and either a tap or a pinch-cock allows of 
 this branch being closed at will. The lower ends of the long tubes 
 dip in two small glass vessels, in which are placed the two liquids the 
 densities of which are to be compared. A 
 divided scale is fixed alongside the long 
 tubes, and two metal rods H, H', pointed 
 at the bottom, and having cross pieces, as 
 shown at (a), pass through small blocks of 
 wood, so that the cross-bars rest against the 
 scale. A small plum bob a is suspended 
 alongside the upright of the stand, and the 
 thread by which it is carried passes through 
 a small metal ring i, which is so arranged 
 that when the thread passes centrally then 
 the tubes and scale are vertical. If the 
 density of a liquid is required it is placed 
 in one of the vessels and water in the other, 
 and then by sucking at the end of the 
 rubber tube the liquids are drawn up into 
 the tubes till the level of the surface of the 
 lighter liquid is near the top of the scale. 
 The points of the rods H, H' are then ad- 
 justed so that they just touch the surfaces 
 of the liquids in the vessels, for which 
 purpose the rods can be moved vertically 
 through the wood blocks. When this 
 adjustment is complete, the readings on 
 the scale corresponding to the upper surfaces 
 of the liquids in the tubes are taken, as 
 also those corresponding to the tops of the 
 cross-bars of the rods H and H'. The length 
 of the rods being known they may con- 
 veniently be 10 cm. the heights of the 
 columns of the liquid and of water re- 
 spectively can at once be obtained. Since 
 the pressure at top and bottom of the two 
 
 columns is the same, and the pressure exerted by a column of liquid 
 is directly proportional to the height and the density of the liquid, we 
 have, if h Q is the length of the water column and A x that of the liquid, 
 
 Density of liquid _ h 
 Density of water h^ 
 
 Hence if the temperature of the water, which may be taken as equal to 
 that of the air as given by a thermometer suspended alongside the tubes, 
 is t, and the density of water at this temperature is A, we get 
 
 FIG. 37. 
 
 Density of liquid = r~ 
 
94 DENSITY [ 37 
 
 In order that capillarity may not appreciably affect the results, it is 
 necessary that the tubes AB, AC should have a diameter not less than 
 6 mm. 
 
 37. The Hydrometer. A hydrometer is an instrument for measuring 
 the density of liquids, and consists essentially of a narrow stem attached 
 to a fairly large bulb, a second and smaller bulb which is partly filled 
 with shot or mercury serving to ballast the instrument so that it may 
 float in an upright position. A scale graduated on the stem serves to 
 determine the depth to which the instrument sinks in the liquid. 
 
 t When the hydrometer is floated in a liquid it will always sink to 
 such a depth that the weight of the volume of the liquid displaced by 
 the immersed part will be equal to the weight of the instrument. Sup- 
 pose that the scale is divided into centimetres and millimetres, and that 
 the zero is near the top of the stem, and let Fbe the volume of the 
 instrument up to the zero of the scale, and v the volume of unit length 
 of the stem, the stem being cylindrical. Then if when immersed in a 
 liquid of density D l the instrument sinks to the graduation s v while 
 when placed in a liquid of density D 2 it sinks to * 2 , then we have 
 
 Weight of instrument = (V- vs-^D-^ = (V- vs. 2 )D 2 . 
 
 D, V-vs* 
 Hence =} = = -. 
 
 I j I/ SjtQ 
 
 JJ 2 fWj 
 
 If the zero of the scale is so adjusted that when the instrument is 
 placed in a liquid of density unity the surface of the liquid coincides 
 with the zero, so that when D 2 = 1, 2 = 0, we get 
 
 ''"\-AT 
 
 If when placed in a second liquid the instrument sinks to s , we have 
 
 V^VTC!/: 
 
 Hence s - , = 
 
 From this expression it will be seen that for equal increments of 
 density (D 2 - D-^ the increments in the scale reading (s. 2 - Sj) will not 
 be equal. As D l and D. 2 increase the increase in scale reading ( 2 - s^ 
 corresponding to a given increase in density (D 2 - DJ will get smaller 
 and smaller. Hence if instead of dividing the scale in equal lengths 
 it is divided so that consecutive divisions correspond to equal increments 
 
37] THE HYDROMETER 95 
 
 in density, the interval between the divisions will get less and less as we 
 go down from the top of the stem. 
 
 Since the actual distance between the graduations also depends on 
 the ratio of the total volume up to the zero to the volume of unit length 
 of the stem, a scale to read directly in densities will have to be specially 
 divided for each individual hydrometer. As the cost of preparing such 
 a special scale for every individual hydrometer is considerable, various 
 manners of dividing the scale in equal parts have come into use, the 
 actual density corresponding to any given division being obtained from 
 tables. 
 
 The best known of these hydrometers with scales having divisions of 
 equal length is Baume's. There are two Baume scales, one for liquids 
 heavier than water, and the other for liquids lighter than water. 
 
 In the instrument for use with liquids heavier than water one fixed 
 point on the scale is given by immersing the instrument in distilled 
 water at a temperature of 15 C., which gives the zero of the scale near 
 the top of the stem. The second point is obtained by immersing the 
 instrument in a solution of sodium chloride (common salt) containing 
 15 parts by weight in 100 parts by weight of the solution, the tempera- 
 ture of the salt solution being 15 C. This second point is called 15, 
 and the interval between the zero and this point is divided into 15 equal 
 parts, and the graduations are continued down to the bottom of the 
 stem. 
 
 In the case of the hydrometer for use with liquids of less density than 
 water the two fixed points are, 0, when the instrument is immersed in a 
 solution of sodium chloride containing 10 parts by weight of salt in 
 100 parts of the solution, the temperature being 15. The second point, 
 called 10, is obtained by immersing the instrument in water at a tem- 
 perature of 15 C. The interval is divided into 10 equal parts, and these 
 divisions are continued up to the top of the stem. The density corre- 
 sponding to a reading r on Baume's hydrometer scale is given by the 
 following expressions : 
 
 146-3 
 
 1. For liquids heavier than water D= -. ^/?.o _ 
 
 1 A Q 
 
 2. For liquids lighter than water ^ = i^^To . 
 
 Neglecting the effect of purely graduation errors, the reading of a 
 hydrometer will only give the correct density at one particular tem- 
 perature, since for all temperatures higher than this temperature the 
 immersed portion* up to any given reading will be greater, while the 
 weight of the instrument will be the same as at the standard temperature, 
 so that the instrument will read too high. Similarly, at all temperatures 
 below the standard the instrument will read too low. 
 
 If t is the temperature at which the hydrometer is correct, and D' 
 
96 DENSITY [ 38 
 
 is the reading on the hydrometer when immersed in a liquid at a tem- 
 perature t, the density of the liquid D t at the temperature t is given by 
 
 where y is the coefficient of cubical expansion of the material of which 
 the hydrometer is constructed. 
 
 Although in theory the hydrometer may be made as sensitive as 
 we please by reducing the diameter of the stem and increasing the size of 
 the bulb, yet in practice it is found that the limit of accuracy attainable 
 is soon reached. It has been found that this limitation is caused by the 
 influence of capillarity at the point where the stem meets the surface 
 of the liquid. If the effect of capillarity remained constant, then its 
 influence would not be so injurious. Experiment has, however, shown 
 that impurities derived from the surface of the hydrometer and the 
 jar in which the liquid is contained have such an enormous 
 influence on the surface tension that the effect on the 
 hydrometer is by no means constant. 
 
 38. Nicholson's Hydrometer. Nicholson's hydrometer 
 (Fig. 38) is one which in use is always brought to the 
 same immersion, that is, till a mark on the stem is flush 
 with the surface of the liquid. It can be used for 
 measuring the density of solids as well as of liquids. 
 
 Let the weight of the hydrometer be H, and the weight 
 which has to be added to the upper pan B to sink the 
 instrument to the mark in water of density A ft be W, 
 FIG. 38. while when the instrument is placed in a liquid of density 
 D let the weights required to sink it to the mark be w. 
 Then since the volume of the hydrometer immersed, and therefore the 
 volume of liquid displaced, is the same in the two cases, we have 
 
 H+W H+w 
 
 Volume up to mark = A ^ . 
 
 ZA Q JJ 
 
 H+w . 
 Hence ^ = 1T+1F * 
 
 When using the instrument to measure the density of a solid, first 
 place the solid in the upper pan and determine the weight w } which has 
 to be added to bring the instrument to the mark. Then remove the 
 solid from the upper pan and place it in the lower pan and determine the 
 weight w 2 , which has now to be placed in the upper pan to bring the 
 instrument to the mark. Then the weight of the solid is W-iv^ and 
 the weight of water displaced by the solid is w 2 - w r Hence the 
 density of the solid is given by 
 
 If the experiment is made at temperatures such as ordinarily occur 
 
39] CALIBRATION OF A BURETTE 97 
 
 it will be sufficiently accurate to take A = l, and no correction need be 
 applied for air displacement, since the measurements are not capable of 
 being made with sufficient accuracy to warrant such corrections. The 
 chief cause of this want of accuracy is the effect of capillarity on the 
 stem where it passes through the surface of the liquid. When used in 
 water, this effect may be reduced by placing, just before taking a reading, 
 a drop of methylated spirit at the point where the stem cuts the surface. 
 
 A tall jar is provided for the liquid in which the instrument is 
 immersed. Before placing any weights on the upper pan bend two 
 pieces of wire across the top of the jar, so that while the stem of the 
 hydrometer can pass freely between them the upper pan cannot. In this 
 way, if by accident too heavy a weight is placed in the upper pan the 
 whole instrument will not be able to sink and thus wet, and perhaps 
 damage, the weights. 
 
 39. Calibration of a Burette. The accurate reading of the position 
 of the meniscus in a burette is a somewhat difficult operation, owing to 
 indistinctness of the meniscus and the fact that the scale is at some dis- 
 tance from the portion of the meniscus to which we read. The best 
 position for the burette, at any rate when being used to measure off a 
 transparent liquid, is before a bright window, when the meniscus will 
 appear dark on a bright ground. The lowest part of the curved meniscus 
 is in all cases to be read. In order to avoid errors due to parallax it is 
 advisable, where possible, to choose some distant object at the same level 
 as the eye, and always when making a reading to bring this object and 
 the meniscus into line. Since, if the object is at some distance, 
 lines joining it to the highest and lowest positions of the eye will be 
 sensibly parallel, we thus secure that the line joining the eye to the 
 meniscus is always parallel to the same direction. 
 
 A straightforward, if somewhat lengthy, method of calibrating a 
 burette is to weigh the quantity of water which must be run off to lower 
 the meniscus, say, every 2 c.c. of the scale. The first operation neces- 
 sary is to thoroughly clean the burette, and in particular remove every 
 trace of grease. For this purpose the burette should be filled with strong 
 sulphuric acid in which a little bichromate of potash has been dissolved. 
 Allow this mixture to remain in the burette for some hours, and then 
 thoroughly wash out with tap water and finally with distilled water. 
 Attach a thermometer to the outside of the burette and then fill the 
 burette with distilled water and set up in a clamp before a window, using 
 a plumb line to set the burette vertical. Clean and weigh a small beaker 
 capable of containing as many cubic centimetres as the burette will hold. 
 Into another beaker run off the water from the burette till the meniscus 
 coincides with the first division of the scale. Touch the point of the burette 
 with the side of the beaker to remove any adhering drop, and replace by 
 the weighed beaker. Into this beaker run the water till the meniscus 
 coincides with the 2 c.c. mark. Having removed the drop by touching 
 the point with the side of the beaker, read the thermometer and weigh 
 the beaker and water. Then run in two more cubic centimetres, and again 
 
 G 
 
B 
 
 98 DENSITY [ 39 
 
 weigh. Proceed in this way till the whole scale has been traversed, and 
 then from the density of water at the temperature 
 of the experiment calculate the volumes between 
 the graduations and the zero, and plot a curve 
 showing the true volume between the zero and each 
 of the graduations. 
 
 Another method of calibrating a burette is bj 
 means of a small volume gauge. The arrangement 
 recommended by Ostwald for this purpose is showrj 
 in Fig. 39. The volume of the gauge A between 
 the two marks B and c is obtained by weighing, as 
 in the previous method. The burette having been 
 filled with water, care being taken to get rid oi 
 all air bubbles in the connecting tubes, by opening 
 the clip E the level of the meniscus in the burette 
 is adjusted to the zero of the scale. Then the 
 clip E being closed, the water is run out of the gauge 
 till the surface stands at the mark c. The clip I 
 is then opened and water allowed to flow intc 
 the gauge from the burette till the surface stands 
 at the mark B, when the reading on the burette 
 is noted. The gauge having been emptied to ( 
 is again filled from the burette and the reading 
 taken, and so on. The readings of the burette giv< 
 the positions of the meniscus when the volume 
 from the zero is v, 2v, 3v, &c., where v is the 
 volume of the gauge, which may conveniently b< 
 2 c.c. A 'curve showing the true volumes foi 
 
 the different readings can then be prepared as before. 
 
 FIG. 39. 
 
CHAPTER V 
 
 ELASTICITY OF SOLIDS 
 
 40. Young's Modulus by Stretching a Wire. If a wire of length 
 L and radius r becomes elongated by an amount e when a tension F is 
 applied, then since 
 
 Young's modulus = 
 
 strain 
 
 and the stress is ^/(area of cross-section) and the strain is the elongation 
 per unit length, we have 
 
 A simple piece of apparatus for measuring the elongation of a wire under 
 the influence of a stretching force is shown in Fig. 40. It consists of a scale 
 A divided into half millimetres, which is attached to 
 the lower end of a wire of the material to be tested. 
 A second wire of the same material carries the 
 vernier B, which is capable of sliding along the edge 
 of the scale with very little friction. The upper 
 ends of the two wires are soldered or clamped to 
 the same support immediately alongside one another, 
 so that if, owing to the load applied to one wire, 
 the support bends, the point of support of both wires 
 will drop to exactly the same amount, and hence 
 the relative positions of the vernier and scale will 
 not alter. This method of supporting the scale also 
 eliminates any correction for changes in temperature 
 affecting the lengths of the wire being tested ; for 
 as the wires are of the same material and are sus- 
 pended close together they will have the same 
 temperature, and so whenever one expands the 
 other will expand to an equal extent, so that the reading on the scale is 
 unaltered on this account. 
 
 The wire which carries the scale has a weight attached to its lower 
 end, which serves to keep the wire taut and free from kinks. The other 
 wire carries a scale pan, so that by placing weights in this pan the wire 
 can be stretched. 
 
 First ascertain the; breaking weight of the wire. To obtain this 
 
 FIG. 40. 
 
100 
 
 ELASTICITY OF SOLIDS 
 
 [40 
 
 quantity measure the diameter of the wire and calculate the area of cross- 
 section in square centimetres. Then multiply this area by the breaking 
 stress for the substance given in the following table. This will give the 
 breaking weight with sufficient accuracy for the present purpose. Having 
 ascertained the breaking weight, care must be taken never to load the wire 
 to more than half this amount : 
 
 BREAKING STRESS. 
 
 
 Kilos, per sq. cm. 
 
 Brass .... 
 Copper .... 
 Iron 
 Steel .... 
 
 3400 
 3000 
 6000 
 8000 
 
 Place the maximum load permissible in the scale pan and allow the 
 wire to remain stretched for some minutes, and then remove the greater 
 part of the weight, leaving, however, enough to keep the wire taut and 
 free from kinks. Read the position of the vernier, and then increase 
 the weight in the pan by equal amounts so chosen that ten readings will 
 be obtained before the maximum load is reached. Repeat the readings as 
 the weights are removed. The readings obtained while putting on and 
 taking off the weights ought to agree within a fifth of a millimetre. If 
 they do not, either the wire was slightly kinked, and as the weights were 
 added the kinks straightened out, or the maximum load was too great, so 
 that permanent elongation has been produced. 
 
 The readings must be entered in a table, as in the following example, 
 and the mean elongation worked out by subtracting the first reading from 
 the sixth, the second from the seventh, and so on, and then taking the 
 mean of these differences : 
 
 
 Vernier Reading. 
 
 
 
 Load in Pan. 
 
 
 Mean. 
 
 Elongation 
 for 5 kilos. 
 
 Load Increasing. 
 
 Load Decreasing. 
 
 3 kilos. 
 
 4'65 mm. 
 
 4'70 mm. 
 
 4-67 mm. 
 
 4 
 
 6-00 
 
 6-05 
 
 6-02 
 
 
 5 
 
 7-30 
 
 7-35 
 
 7-32 
 
 6 
 
 8-65 
 
 8-70 
 
 8-67 
 
 7 
 
 10-00 
 
 10-00 
 
 10-00 
 
 8 
 
 11-35 
 
 11-40 
 
 11-37 6'70 mm. 
 
 9 
 
 12-75 
 
 12-75 
 
 12-75 6-73 
 
 10 
 
 14-05 
 
 14-10 
 
 14-07 6-75 
 
 11 
 
 15-40 
 
 15-45 
 
 15-42 
 
 6-75 
 
 12 
 
 16-75 
 
 16-80 
 
 16-77 
 
 6-77 
 
 
 
 
 Mean . 
 
 6'74 mm. 
 
40] YOUNG'S MODULUS" ' 101 
 
 Elongation for 1 kilo (981000 dyries) ~M&J$ Cm.' , ' i '( & ' - 
 Mean diameter = '06224 cm. 
 
 Length ..... = 837'5 cm. 
 
 981000x837-5 
 Youngs modulus- 
 
 
 = 2-003 x 10 12 dynes per sq. cm. 
 
 Measure the diameter of the wire by means of a micrometer screw 
 gauge. The diameter must be measured at a number of points along the 
 length of the wire, and at each point a measurement must be made along 
 two diameters of the wire at right angles to one another. In this way 
 errors due to want of uniformity in the wire may be reduced to a minimum. 
 To obtain the length of the wire a steel tape may be suspended from the 
 support which carries the upper ends of the wires, arid the readings taken 
 corresponding to the point of attachment of the wires and the point where 
 the upper screw which clamps the vernier to the wire touches the wire. 
 If a steel scale is not available the length may be measured by using a 
 long wooden lath on which marks are made corresponding to the upper 
 and lower ends of the wire, which length is then determined by measure- 
 ment with a millimetre scale. 
 
 The accuracy which it is necessary to attain in the measurement of 
 the total length of the wire may be calculated in the following manner : 
 Let L be the length of the wire and r its radius, and the mean elonga- 
 tion as obtained above be e. The expression for calculating Young's 
 modulus is 
 
 FL 
 
 Ti-rV 
 
 and we wish to arrange that each of the factors which enter into this 
 expression, namely, F, L, r 2 , and e, are measured with the same per- 
 centage accuracy. If the vernier reads to 1/20 mm., the percentage 
 accuracy to be expected of a single reading of the elongation is about 
 5/e, since e is supposed to be given in centimetres. If the screw gauge 
 reads to "001 cm., the percentage accuracy of a single setting is about 
 O'l/2r, which corresponds to a percentage accuracy of 0'1/r in r 2 . Since 
 a number of measures of the above two quantities are made, we may in 
 general suppose that the means are correct to about a fifth of the 
 number given above, so in calculating the accuracy with which we have 
 to know L we may suppose that we know e to 'l/e per cent., and r 2 to 
 '02/r per cent. Then if 8L is the magnitude of the error we can allow 
 in measuring L we have to arrange so that 
 
 8 r 
 
 -f- x 100 is not greater than l/e or '02/r. 
 l_j 
 
 In the same way, if SW is the error of the weights W employed, 
 
102 
 
 ELASTICITY OF SOLIDS 
 
 [40 
 
 1003 WjW must' net l^c greater than 'l/e or *02/r. In the example given 
 above, e= 13 cm., r= '031, Z = 837 cm., TF= 1000 grams. Hence 
 
 = ^TO = "76 per cent., 
 
 and 
 
 02 
 
 T 
 
 02 
 031 
 
 = '64 per cent. 
 
 Hence we see that the error to be expected owing to our method of 
 measuring r is slightly greater than that due to e. Further, = 
 
 must not be greater than '6, and therefore 8L must not be greater than 
 5 cm. Hence if we measure the total length of the wire to within a 
 centimetre or two, it will be quite accurate enough. In the same way, 
 if the weights are correct to 2 or 3 grams in the kilogram, they are 
 sufficiently accurate. 
 
 The above investigation illustrates the care which has to be exercised 
 to avoid spending a lot of time and trouble in measuring with great 
 accuracy some one of the factors which enter into the final calculation 
 
 when, owing to unsurmountable 
 difficulties in making some other 
 measurement or want of uniformity 
 in the quantity measured, the 
 accuracy of the final result is almost 
 entirely governed by the accuracy 
 with which we can measure this 
 latter quantity. Thus if we are able 
 to use a wire of the length con- 
 sidered above, a vernier which reads 
 to one-twentieth of a millimetre will 
 measure the elongation with a per- 
 centage accuracy greater than that 
 with which we are able to measure 
 the radius of the wire. If, how- 
 ever, it is for some reason impossible 
 to use such a long wire, or if measure- 
 ments have to be made on wires of 
 such a large diameter that the 
 elongation which can conveniently 
 be obtained with the weights at our 
 disposal is much smaller than that 
 considered above, then some more 
 FlG - 41. refined method of measuring the 
 
 elongation becomes necessary. In 
 
 the case of the short wire, we shall also have to measure the length 
 with greater accuracy. This last measurement is not a matter of very 
 great difficulty, since it is easier to measure shorter lengths than lengths 
 of about 8 metres. 
 
41] YOUNG'S MODULUS 103 
 
 A device for measuring the elongation of a short wire, designed by 
 Mr. G. F. C. Searle, 1 is shown in Fig. 41. The two wires of the same 
 material A, A' are fastened at the top to the same support, and at the 
 bottom to two metal frames CD, C'D'. These frames are connected 
 together by two links K, K', which allow of their free relative movement 
 in a vertical direction. One end of a sensitive level L is pivoted to the 
 frame CD, while the other end rests on the point of a micrometer screw 
 s. From the lower parts of the frames hang a stretching weight M and 
 a scale pan p. When using the instrument, a sufficient weight having 
 been put in the pan to keep the wire straight, the bubble of the level is 
 brought to the centre of the scale by turning the screw s, and the 
 reading on the scale K, and divided head s are recorded. A weight is 
 then added to the pan, and the bubble being brought back to the centre 
 
 H 
 
 (ay 
 
 FIG. 42. 
 
 of the scale, the micrometer is again read. The difference in the 
 micrometer readings of course gives the elongation produced by the 
 additional load. 
 
 41. Young's Modulus by the Flexure of a Beam (1). A convenient 
 arrangement for measuring the flexure of a loaded beam is shown in Fig. 
 42. The beam rests on the upper edges of two cast iron stands A and 
 B, which are filed up to a blunt knife-edge. These uprights also carry 
 a wooden board CD, which supports a microscope M, used to measure 
 the flexure. This board is not permanently attached to the uprights but 
 simply rests on two small projections, so that the distance between the 
 uprights, and hence the length of the beam, can be altered. The micro- 
 scope has a horizontal cross-wire in the eye-piece, and is focused on a 
 small scale s, which rests on the beam. This scale is shown on an en- 
 larged scale at (a). It may conveniently be divided into quarter milli- 
 metres, with every tenth division numbered. The scale pan in which the 
 
 1 Proceedings of the Cambridge Philosophical Society (1900), vol. x. p. 318. 
 
104 ELASTICITY OF SOLIDS [ 42 
 
 weights are placed is attached to a hook E, which rests on the upper 
 surface of the beam at its mid point, passing between the feet of the 
 scale. 
 
 With a pencil make a mark at the mid point of the beam, and then 
 draw transverse lines at either side at distances of, say, 30, 40, and 50 
 centimetres from the middle. Place the uprights at a distance of 
 100 cm. apart and adjust the beam so that the two marks at 50 cm. 
 from the middle are immediately over the knife-edges, and also place the 
 scale and scale pan in position at the middle of the beam, taking care 
 that the scale is vertical. Focus the microscope on the scale, and take 
 the reading corresponding to the cross-wire of the microscope. Then 
 gradually add weights to the scale pan, and take for each load the 
 reading on the scale. Repeat the readings while removing the load, 1 
 and enter the results as in the example given in 40. Repeat 
 the measurements with the supports at 80 and 60 cm., also with 
 the beam turned so that what was the depth becomes the breadth, and 
 vice versa. 
 
 Measure the depth and breadth of the bar at ten points at equal 
 intervals along its length, and take the means. Since the depth occurs 
 to the third power, it is important that this quantity should be measured 
 with care. The measurement is best made with a micrometer screw gauge. 
 If, however, the dimensions of the bar are such that the micrometer will 
 not open sufficiently wide, a pair of vernier callipers must be used. 
 
 As the length also occurs to the third power, this will also require 
 measuring with care. It can be measured by placing a steel metre scale 
 divided into millimetres across the uprights, with the face which carries 
 the divisions vertical, and reading the distance between the knife-edges. 
 
 The method to be adopted in recording and reducing the observations 
 is the same as that used in the preceding section. 
 
 If e is the deflection of the mid point of the beam produced by a load 
 of P dynes, the length, breadth, and depth of the beam being L, b, and d 
 respectively, then Young's modulus T is given by 
 
 From this expression it will be seen that if P is constant, the quotient 
 L 3 /e is a constant for different lengths. Hence use the mean value 
 of LPje for the different lengths of beam when calculating the value of 
 Young's modulus. 
 
 42. Young's Modulus by the Flexure of a Beam (2). In place of 
 measuring the depression at the centre of a beam supported at the ends 
 and loaded in the middle, we may measure the inclination of the ends 
 
 1 If the readings taken when removing the load differ appreciably, it generally 
 indicates that the beam has been loaded beyond its elastic limit, and a smaller 
 maximum load should be tried 
 
42] 
 
 YOUNG'S MODULUS 
 
 105 
 
 of such a beam. If an inclination < is produced by a load P in a beam 
 of length L, depth d, and breadth b, then we have 
 
 0) 
 
 4 d 3 btan 
 
 In order to measure < two mirrors MJ and M 2 are attached to the 
 bar beyond the knife-edges which support the bar in such a way that 
 the scale s (Fig. 43) is seen in the telescope T after reflection in both 
 
 FIG. 43. 
 
 mirrors. Then if the reading on the scale corresponding to the cross- 
 wire of the telescope changes by x cm. when the bar is loaded and the 
 distance between the scale and the mirror M 9 is D, and that between 
 the two mirrors is d, we have to a first approximation, if <j> is the angle 
 through which each of the mirrors turns, 
 
 $ = tan < = 
 
 Hence substituting this value for tan <f> in (1) above 
 
 _3(2D 
 
 (2) 
 
 To obtain the expression for < in terms of x, let A and B (Fig. 44) be 
 the mirrors, T the telescope, and c the reading on the scale before the 
 beam is bent. The path of the axial pencil of rays from c to the 
 telescope is along CBAT. Next suppose the mirror A alone rotated in 
 the clockwise direction through an angle <j>, an incident ray TA would 
 now be reflected along AB', the angle BAB' being 2<. If a is the angle 
 TAB, or CBA, the ray AB will be incident on B at an angle a/2, while the 
 ray AB' will be incident at an angle of a/2 - 2</>. If, now, the mirror B be 
 rotated through an angle < in the anti-clockwise direction, the angle of 
 incidence of the ray AB' will be a/2 - 3</>. Hence if B'C is the reflected 
 
106 
 
 ELASTICITY OF SOLIDS 
 
 [42 
 
 ray, the angle AB'C' is a - 6<. If B'D' is a line through B' parallel to 
 BA, the angle AB'D' is 2<jf>. Hence the angle C'B'D' is a - 4<. 
 
 FIG. 44. 
 
 If, now, the scale cc' is placed so that it is at right angles to the line 
 AB, we have 
 
 CD = D tana 
 
 Hence 
 
 DD' = d tan 2</>. 
 x = CO' = D tan a - D tan (a - 4<) + d tan 2<f>. 
 
 If both a and < are so small that the angles may be put equal to the 
 tangents, we get 
 
 or 
 
 In order to ensure that this approximation is permissible, the scale 
 should be placed at a considerable distance, and the telescope adjusted so 
 that the angle a is as small as is compatible with clearing the tops of 
 the mirrors. 
 
 By replacing the mirrors by two right-angled prisms, as shown in plan 
 in Fig. 45, a can be made rigorously zero. When prisms are used, the 
 
 SI 
 
 / 
 
 
 (i! 4x 
 
 <r 
 
 
 V 
 
 V r j l > 
 
 \ 
 
 
 IP 
 
 1 
 
 
 y 
 
 FIG. 45. 
 
 quantity D in the formula is the distance between the scale and the 
 hypotenuse face of the further prism +8/[i, where \L is the refractive 
 index of the glass, and 3 is the height of the prism measured from the 
 hypotenuse face. In the same way d is the distance between the 
 hypotenuse faces of the two prisms + 28/p. 
 
43] 
 
 COMPARISON OF MOMENT OF INERTIAS 
 
 107 
 
 B 
 
 The advantages of the reflection method are that, owing to the 
 sensitiveness of the method employed for measuring the deformation 
 of the beam, smaller loads can be used, and therefore the chance of 
 passing the elastic limit of the material is reduced. In the second 
 place, the fact that as the load increases the supporting knife-edges 
 slightly sink into the material has no effect on the readings, as is the 
 case when the deflection at the centre is measured. 
 
 43. Measurement of the Period of an Oscillating Body by the Eye 
 and Ear Method Comparison of Moment of Inertias. You are 
 supplied with a light metal cradle A (Fig. 46) suspended by a phosphor 
 bronze wire, and having a mirror M attached, by means of which an image 
 of a scale is reflected into a telescope. The upper end of the suspension 
 wire is soldered to a metal rod, 
 which is itself firmly clamped to 
 a cross-bar B that rests in two 
 v's attached to an upright tube c, 
 which is itself carried by a box 
 with glass sides. This box serves 
 to protect the suspended part of the 
 apparatus from draughts. Metal 
 cylinders of various shapes are 
 provided, which can be placed in 
 the v's of the cradle. The object 
 of the experiment is to determine 
 the period of the cradle alone, and 
 also when it carries each of the 
 metal cylinders in succession. From 
 these periods we shall be able to 
 compare the moments of inertia of 
 the cylinders, and also, if we know 
 the moment of inertia of one of 
 the cylinders, determine the tor- 
 sional rigidity of the suspension 
 wire. 
 
 Adjust the telescope and scale 
 so that when the cradle is empty FIG. 46. 
 
 and at rest the vertical cross-wire 
 
 of the telescope coincides with a whole division near the centre of the 
 scale. It will generally be found advisable to mark this division in some 
 conspicuous manner, so that it catches the eye when the mirror is 
 vibrating. For this purpose a small pointed piece of black paper may 
 be attached to the scale, so that the point coincides with the upper 
 end of the central division line but does not cover the line. 
 
 Next set the cradle vibrating by slightly twisting it, and then releasing 
 it. Care must be taken not to produce pendulum vibrations during this 
 process. If the cradle is set swinging like a pendulum, this motion may 
 be stopped without greatly checking the torsional vibrations by lightly 
 
108 ELASTICITY OF SOLIDS [ 43 
 
 touching the suspension wire just above the cradle with the finger or 
 with a feather or brush. 
 
 Determine the time occupied by fifty complete vibrations by means 
 of a stop-watch, and suppose that the value of the period determined in 
 this way is 2 J . 
 
 In order to determine the period accurately, some more reliable method 
 of measuring the time than the stop-watch will be required. For this 
 purpose a chronometer or good clock, which ticks every half second, will be 
 required. If a clock which ticks every second is employed, a very slight 
 modification in the procedure will be required, which modification will 
 be quite apparent, and in what follows we shall suppose that a chrono- 
 meter beating half seconds is used. 
 
 Since it is impossible to watch the chronometer and look through the 
 telescope at the same time, we have to make use of the ear to follow 
 the chronometer, while the eye is used to follow the movement of the 
 image of the scale as seen through the telescope after reflection in the 
 mirror. There are various methods in use for counting the number 
 of seconds which elapse between the instant when the chronometer is 
 looked at and the instant when the event which is being timed occurs, 
 but the author has found that in his experience the following is least 
 likely to lead to a mistake in the counting : At each tick corresponding 
 to a half second say tick, while at each tick corresponding to a whole 
 second say the number of the second. Thus starting at the whole minute 
 one counts in time with. the beats of the chronometer as follows : tick, 
 one, tick, two, tick, three, &c. Up to the twelfth second this is quite 
 simple, but from thirteen up to sixty the numbers consist of more than 
 one syllable, and the question arises, which of the syllables should be 
 made to coincide with the beat of the chronometer. The author is in 
 the habit of strongly accenting the first syllable of those numbers which 
 have more than one syllable, and making this first syllable coincide with 
 the beat of the chronometer. The syllables after the first are hardly 
 sounded at all, and in the three-syllable numbers the middle syllable is 
 almost entirely slurred. This method of counting may perhaps be indi- 
 cated in the following manner, in which the asterisk indicates the instant 
 
 # * # * * * 
 
 of the chronometer beat : tick, thir-teen, tick, four-teen, . . . tick, twen-ty, 
 
 * * * * 
 
 tick, twen-t'one, tick, twen-t'two, &c. The above may at first seem some- 
 what complicated, but a very little practice will enable the reader to count 
 on for several minutes, if need be, without making any mistake, and 
 practically without mental effort. When a little practice has been gained 
 it will not be necessary to count out loud, though at first this will be 
 found an assistance. It is important to practise taking the time from 
 the chronometer and start counting the seconds at any beat, so as not to 
 have to wait till the end of the minute. Thus if when you look at the 
 chronometer it has just ticked on the eighth second, you must with the 
 next beat say tick, and then go on, nine, tick, ten, tick, &c. 
 
 Returning to the suspended cradle : this, without any metal cylinder, 
 
43] COMPARISON OF MOMENT OF INERTIAS 109 
 
 will generally have a period of about a second, and the following will 
 be found a convenient method of measuring such a period with accuracy : 
 Set the cradle swinging over pretty well the whole range of the scale, 
 then, taking the time from the chronometer, count on while you watch 
 the movements of the scale in the telescope. Go on watching till, exactly 
 on a beat of the chronometer, the zero mark of the scale crosses the 
 cross- wire of the telescope, apparently moving from left to right. Imme- 
 diately write down the time at which this transit occurred, which will 
 be a complete whole or half second. Then take up the time again and 
 watch till the zero mark sweeps across the cross-wire exactly on a beat, 
 but this time moving from right to left. Write this time down in a 
 column alongside the first, as shown in the example given below. 
 Shortly before a minute after the first observation again take up the 
 time, and determine the time when the zero crosses at a beat, moving 
 from left to right. Then make an observation as the scale swings from 
 right to left, and so on, making an observation of transit each way about 
 every minute till the amplitude of the oscillations have so far died 
 down as to make the exact determination of the instant of transit 
 doubtful. 
 
 From the times obtained in this way we are able, by means of the 
 approximate value of the period which we have already determined, to 
 calculate the number of vibrations which have occurred between any two 
 observations of the time of transit. The method employed can most 
 easily be explained by means of an example. The following figures refer 
 to the oscillations of a cradle, such as that shown in Fig. 46 : 
 
 Approximate period : 100 vibrations took 2 minutes 4*5 seconds. 
 Hence period is 1'245 seconds. 
 
 The times of transit given on the next page were determined with 
 a chronometer, and were entered in the two columns A and B as the 
 vibrations proceeded. 
 
 The columns c and D contain the differences between successive 
 transits in the same direction. By dividing these intervals by the 
 approximate time of vibration we obtain the number of vibrations which 
 occur between the transit observations, and these numbers are entered in 
 columns E and F. It will be found very convenient to use a slide rule 
 for dividing the intervals in columns c and D by the approximate 
 period, for if the 1 in the bottom, or D scale, is put opposite the approxi- 
 mate time of oscillation on the third or c scale, then opposite any 
 interval between transits on the c scale will be found the number of 
 vibrations on the D scale and a single setting of the slide will do 
 for all the intervals. 
 
 We now have to calculate the number of vibrations between the first 
 transit taken as and each of the others. This is at once obtained from 
 the numbers in the columns E and F by addition. As, however, we 
 shall only require the first and last five, the sums of the numbers in- 
 cluded by the brackets can be obtained once for all, and used to calculate 
 the last five. 
 
110 
 
 ELASTICITY OF SOLIDS 
 
 A. 
 
 C. 
 
 E. 
 
 G. 
 
 B. 
 
 D. 
 
 F. 
 
 H 
 
 
 a 
 
 A 
 
 ce 
 
 
 d 
 
 k 
 
 A 
 
 
 o 
 
 3 
 
 j3 
 
 
 0) 
 
 2 ^ 
 
 ^ 
 
 Time of 
 
 1-2 
 
 > I J 
 
 
 Time of 
 
 l| 
 
 > 1-2 
 
 > 1 
 
 Transit. 
 
 ,0'M 
 
 "o ~S'i 
 
 t 'w J 
 
 Transit. 
 
 J2 53 
 
 "o"S'| 
 
 "o^ 
 
 > 
 
 || 
 
 ill 
 
 jji^ 
 
 
 Is 
 S-H 
 
 5 M *-i 
 
 I.! 
 
 * 
 
 
 <B 
 
 3. 
 
 S "^ 
 
 
 <D 
 
 3 2 
 
 S "*" 
 
 
 ~ 
 
 
 
 
 
 ~ 
 
 3 '-i 3 
 
 & 
 
 H. M. S. 
 
 Seconds. 
 
 
 
 H. M. S. 
 
 Seconds. 
 
 
 
 3 42 25-0 
 
 
 
 
 
 3 42 37-0 
 
 
 
 
 
 60 
 
 48 
 
 
 
 61 
 
 49 
 
 
 3 43 25-0 
 
 
 
 48 
 
 3 43 38-0 
 
 
 
 
 
 61 
 
 49 
 
 
 
 61 
 
 49 
 
 
 3 44 26-0 
 
 
 
 97 
 
 3 44 39'0 
 
 
 
 
 
 61 
 
 49 
 
 
 
 60 
 
 48 
 
 
 3 45 27-0 
 
 
 
 146 
 
 3 45 39-0 
 
 
 
 1 
 
 
 61 
 
 49 
 
 
 
 61 
 
 49 
 
 
 3 46 28-0 
 
 
 
 195 
 
 3 46 40-0 
 
 
 
 1 
 
 
 60 
 
 48 
 
 
 
 61 
 
 49 
 
 
 3 47 28-0 
 
 
 
 
 3 47 41-0 
 
 
 
 
 
 61 
 
 49 
 
 
 
 61 
 
 49 
 
 
 3 48 290 
 
 
 
 
 3 48 42-0 
 
 
 
 
 
 61 
 
 49 
 
 
 
 60 
 
 48 
 
 
 3 49 30-0 
 
 
 
 
 3 49 42-0 
 
 
 
 
 
 60 
 
 48 
 
 
 
 61 
 
 49 
 
 
 3 50 30-0 
 
 
 
 03* 
 
 3 50 430 
 
 
 
 t/i 
 
 
 61 
 
 49 
 
 o 
 
 
 61 
 
 49 
 
 
 3 51 31-0 
 
 
 
 i 
 
 3 51 44-0 
 
 
 
 2 
 
 
 61 
 
 49 
 
 ^ 
 
 
 60 
 
 48 
 
 o3 
 
 3 52 32-0 
 
 
 
 s 
 
 3 52 44-0 
 
 
 
 .^ 
 
 
 60 
 
 48 
 
 ** 
 
 
 61 
 
 49 
 
 i > 
 
 3 53 32-0 
 
 
 
 CO 
 CO 
 
 3 53 45-0 
 
 
 
 /P 
 
 
 61 
 
 49 
 
 
 
 61 
 
 49 
 
 Li 
 
 3 54 33-0 
 
 
 
 "8 
 
 3 54 46-0 
 
 
 
 o 
 
 
 61 
 
 49 
 
 "3 
 
 
 61 
 
 49 
 
 13 
 
 3 55 34-0 
 
 
 
 1 
 
 3 55 47-0 
 
 
 
 L 
 
 
 61 
 
 49 
 
 
 
 
 61 
 
 49 
 
 C 
 
 3 56 35-0 
 
 
 
 M 
 
 3 56 48-0 
 
 
 
 K- 
 
 
 60 
 
 48 
 
 
 
 60 
 
 48 
 
 
 3 57 35-0 
 
 
 
 
 3 57 48-0 
 
 
 
 
 
 61 
 
 49 
 
 
 
 61 
 
 49 
 
 
 3 58 36-0 
 
 
 
 
 3 58 49-0 
 
 
 
 
 
 61 
 
 49 
 
 
 
 61 
 
 49 
 
 
 3 59 37-0 
 
 
 
 
 3 59 50-0 
 
 
 
 
 
 60 
 
 48 
 
 
 
 60 
 
 48 
 
 
 4 37-0 
 
 
 
 876 
 
 4 50-0 
 
 
 
 
 
 61 
 
 49 
 
 
 
 61 
 
 49 
 
 
 4 1 38t) 
 
 
 
 925 
 
 4 1 51-0 
 
 
 
 
 
 61 
 
 49 
 
 
 
 61 
 
 49 
 
 
 4 2 39-0 
 
 
 
 974 
 
 4 2 52-0 
 
 
 
 
 
 60 
 
 48 
 
 
 
 60 
 
 48 
 
 
 4 3 391) 
 
 
 
 1022 
 
 4 3 52-0 
 
 
 
 1 
 
 
 61 
 
 49 
 
 
 
 61 
 
 49 
 
 
 4 4 40-0 
 
 
 . 
 
 1071 
 
 4 4 53-0 
 
 
 
 1 
 
43] COMPARISON OF MOMENT OF INERTIAS 111 
 
 We can now obtain five intervals corresponding to the scale moving 
 in either direction, the number of vibrations being known, as shown 
 below : 
 
 Time. 
 
 Interval. 
 
 Number of 
 Vibrations. 
 
 Period. 
 
 At Start. 
 
 At End. 
 
 M. S. 
 
 M. S. 
 
 " M. S. 
 
 
 
 g> f42 25-0 
 
 60 37-0 
 
 18 12'0 
 
 876 
 
 1-2466 
 
 >+> 43 25-0 
 
 61 38-0 
 
 18 13-0 
 
 877 
 
 1-2463 
 
 |.2M44 26-0 
 
 62 39-0 
 
 18 13-0 
 
 877 
 
 1-2463 
 
 w 45 27-0 
 
 63 39-0 
 
 18 12-0 
 
 876 
 
 1-2466 
 
 g~ U6 28-0 
 
 64 40-0 
 
 18 12-0 
 
 876 
 
 1-2466 
 
 02 
 
 
 
 
 
 a f42 37-0 
 
 60 50-0 
 
 18 13-0 
 
 877 
 
 1-2463 
 
 14*143 38-0 
 
 61 51-0 
 
 18 13-0 
 
 877 
 
 1-2463 
 
 S ^ 1 44 39-0 
 
 62 52-0 
 
 18 13-0 
 
 877 
 
 1-2463 
 
 3 145 39-0 
 
 63 52-0 
 
 18 13-0 
 
 877 
 
 1-2463 
 
 1 146 40-0 
 
 64 53-0 
 
 18 13-0 
 
 877 
 
 1-2463 
 
 
 
 
 Mean . 
 
 1-2464 
 
 Having determined the period of the cradle alone, that of the cradle 
 loaded with one of the metal cylinders has to be determined. Before 
 placing the cylinder in the cradle, adjust the horizontal cross-wire of the 
 telescope so that it coincides with what appears as the upper edge of the 
 division lines of the scale. Then when placing the cylinder on the cradle, 
 adjust its position till the horizontal cross-wire again coincides with the 
 upper ends of the division lines. In this way you can ensure that the 
 cradle hangs in the same position with reference to the axis of suspension 
 as it did when empty, and hence that its moment of inertia about this 
 axis is unaltered. At the same time it ensures that the centre of 
 gravity of the cylinder is on the axis of rotation, an important matter 
 when we come to calculate its moment of inertia from its dimensions. 
 
 The period with the cylinder will be considerably greater than that 
 of the empty cradle, and a slightly different procedure will be required to 
 measure it. 
 
 Determine the time taken to make about fifty vibrations with a stop- 
 watch, and calculate the approximate period. 
 
 Next set the arrangement swinging over almost the whole of the 
 scale, and taking the time from the chronometer determine the time of 
 transit of the zero across the cross-wire when moving from left to right. 
 By noting the position of the cross-wire on the scale with reference to the 
 zero at the tick immediately preceding and following the transit it will 
 be found possible, after a little practice, to determine the time of transit 
 to within one- or two-tenths of a second. Having written down the time 
 of transit, again take the time from the chronometer and determine in 
 the same way the time of transit when the scale is travelling from right 
 to left. When nearly a minute has elapsed from the first transit again 
 determine the time of transit from left to right, and proceed in this way 
 
112 ELASTICITY OF SOLIDS [43 
 
 till five transits in each direction have been recorded, as in the example 
 below. It will now be sufficient to record a transit in either direction 
 every two or even four minutes till about twenty minutes after the first 
 transit. Then note the times' of five transits in either direction at 
 intervals of a minute, as at the start. 
 
 The reason why observations are only made every two or four minutes 
 during the middle portion is, that these observations are only used to 
 count the number of vibrations made in this interval. The only thing 
 we have to be careful to secure is that the interval during which we allow 
 the instrument to vibrate without making a reading of the time of transit 
 is not of such a length that when we divide by the approximate time of 
 vibration there should be a doubt as to what the number of vibrations 
 ought to be. If the interval is too long then, although if wo knew 
 the correct period this would divide into the interval almost exactly 
 a whole number of times, thus giving the number of vibrations in the 
 interval, yet the approximate period might divide into this interval a 
 different number of times. Thus if the quotient obtained by dividing 
 the interval by the approximate period consists of a whole number, 
 n, plus '4 or over, there will be considerable doubt as to whether 
 we ought to take n or n + 1 as the number of vibrations in the interval. 
 
 Suppose that the true period is T and that the approximate period 
 may differ from this by 8T, then the number of vibrations, n, allowed to 
 elapse between consecutive observations of transit must be such that 
 n .8T is not greater than about T/3. In practice it will generally be found 
 advisable to make the observations of transit at least twice as frequently 
 as indicated by this expression, since then if for any reason one of the 
 transit observations is incorrect, owing to a mistake in taking the time or 
 other cause, it will be possible to neglect this observation and still be 
 able to calculate the number of vibrations in the interval from the 
 transits on either side of the one which has been neglected. When 
 taking the time of vibration with a stop-watch the possible error in the 
 interval taken will be quite 0*5 second, so that if we suppose that no 
 errors have been made in counting we have, in the case of the example 
 given above for the carriage only, that 87 T is '005. Thus the limiting 
 value which it is safe to take for n is given by 
 
 14 
 
 005 x 3 
 
 A short calculation such as the above will always give the safe 
 interval between successive transits, and ought always to be made when 
 using this method of counting the number of vibrations performed by a 
 body of which the period is being determined. The method of redwing 
 the observations of period made with the cylinder in the cradle will be at 
 once evident from the following example : 
 
 Determination of approximate period: 40 vibrations took I .'{0*7 
 seconds. Hence the period is 3*418 seconds. 
 
43] COMPARISON OF MOMENT OF INERTIAS 113 
 
 A. 
 
 C. 
 
 E. 
 
 G. 
 
 B. 
 
 D. 
 
 F. 
 
 H. 
 
 
 8 
 
 <D 
 
 Is 
 
 | 
 
 
 1 
 
 | r 
 
 1 
 
 Time of 
 Transit. 
 
 II 
 
 >! 
 
 8J | 
 
 |L 
 
 Time of 
 Transit. 
 
 || 
 
 
 fl* 
 
 - 
 
 
 
 I'M 
 
 I ^ 
 
 
 |i 
 
 ill 
 
 I 1 
 
 
 
 -2 
 
 S.2 
 
 8 '** 
 
 
 
 s! 
 
 S -^ 
 
 
 t 1 
 
 fc"* 
 
 fc 
 
 
 c 
 
 ~* 
 
 | 
 
 H. M. S. 
 
 
 
 
 H. M. S. 
 
 
 
 
 2 11 11-2 
 
 
 
 
 
 2 11 23-1 
 
 
 
 
 
 
 61-5 
 
 18 
 
 
 
 61-5 
 
 18 
 
 
 2 12 12-7 
 
 
 
 18 
 
 2 12 24-6 
 
 
 
 18 
 
 
 61-4 
 
 18 
 
 
 
 61-5 
 
 18 
 
 
 2 lit 14-1 
 
 
 
 36 
 
 2 13 26-1 
 
 
 
 36 
 
 
 61-5 
 
 18 
 
 
 
 61-4 
 
 18 
 
 
 2 14 15-6 
 
 
 
 54 
 
 2 14 17-5 
 
 
 
 54 
 
 
 61-4 
 
 18 
 
 
 
 61-5 
 
 18 
 
 
 2 15 17-0 
 
 
 
 72 
 
 2 15 29-0 
 
 
 
 72 
 
 
 122-8 
 
 36 
 
 
 
 122-8 
 
 36 
 
 
 2 17 19-8 
 
 
 
 
 2 17 31-8 
 
 
 
 
 
 123-0 
 
 36 
 
 5 
 
 
 122-8 
 
 36 
 
 a 
 
 2 19 22-8 
 
 
 
 o 
 
 2 19 34-6 
 
 
 
 o 
 
 
 122-9 
 
 36 
 
 g 
 
 
 123-0 
 
 36 
 
 1 
 
 2 21 25-7 
 
 
 
 
 
 2 21 37-6 
 
 
 
 
 
 122-9 
 
 36 
 
 > 
 
 
 122-9 
 
 36 
 
 *> 
 
 2 23 28-6 
 
 
 
 00 
 
 2 23 40-5 
 
 
 
 > 
 
 
 122-9 
 
 36 
 
 IM 
 
 
 122-9 
 
 36 
 
 w 
 
 2 LT) 31-5 
 
 
 
 
 
 2 25 43-4 
 
 
 
 o 
 
 
 122-8 
 
 36 
 
 1 
 
 
 122-8 
 
 36 
 
 "3 
 
 L> 27 34-3 
 
 
 
 E 
 
 2 27 46-2 
 
 
 
 C 
 
 
 122-9 
 
 36 
 
 e 
 
 
 122-9 
 
 36 
 
 fl 
 
 L' L".I 37-2 
 
 
 
 M 
 
 2 29 49-1 
 
 
 
 HH 
 
 
 122-8 
 
 36 
 
 ( 
 
 
 122-9 
 
 86 
 
 
 2 31 40-0 
 
 
 
 360 
 
 2 31 52-0 
 
 
 
 360 
 
 
 81*8 
 
 18 
 
 
 
 61-5 
 
 18 
 
 
 2 32 41-fi 
 
 
 
 378 
 
 2 32 53-5 
 
 
 , 
 
 378 
 
 
 61'4 
 
 18 
 
 
 
 61-4 
 
 18 
 
 
 2 33 i:;-o 
 
 
 
 396 
 
 2 33 54-9 
 
 
 
 396 
 
 
 61-4 
 
 18 
 
 
 
 61-5 
 
 18 
 
 
 2 34 44-4 
 
 
 
 414 
 
 2 34 56-4 
 
 
 
 414 
 
 
 61-5 
 
 18 
 
 
 
 61-3 
 
 18 
 
 
 2 35 45-9 
 
 
 
 432 
 
 2 35 57-7 
 
 
 
 432 
 
 From these numbers we obtain the following values for the interval 
 occupied by 360 vibrations : 
 
 Scale moving to Right. Scale moving to Left. 
 
 M. s. M. 8. 
 
 20 28-8 20 28-9 
 
 _'n -J8-9 20 28'9 
 
 20 28-9 20 28-8 
 
 20 28*8 20 28-9 
 
 20 28-9 20 28-7 
 
 Mean 20 minutes 28 '85 seconds. 
 Hence Period = 3 '4 137 seconds. H 
 
114 ELASTICITY OF SOLIDS [43 
 
 If t Q is the time of vibration of the cradle only, and if k is the 
 moment of inertia of the cradle, we have 
 
 
 where c is a constant depending on the length, diameter, and nature 
 of the suspension wire. This constant c represents the couple which, 
 if applied at one end of the suspension wire while the other end is held 
 fixed, would twist the end of the wire through one radian. 
 
 If t l is the time of vibration when a cylinder of which the moment 
 of inertia is K is placed in the cradle we have 
 
 K+k 
 
 ^~T~ 
 
 *t*-& 
 
 Substituting from (1) above 
 
 *i 
 = / 2 _ / 2 
 
 Ej [Q 
 
 47r2A% 2 
 also c = - 1 
 
 Hence if we calculate the value of K from the dimensions of the cylinder, 
 we can from our observations deduce both k and c. 
 
 The moment of inertia of a solid cylinder about an axis passing 
 through its centre of gravity and perpendicular to the axis is given 
 by (see Table 7) 
 
 1-2 ^ r '2 1 
 
 12 4 f ' 
 
 where M is the mass of the cylinder, I its length, and r its radius. 
 
 Measure the length, diameter, and weight of each of the cylinders 
 which have been used in the vibration experiment, and calculate the 
 moment of inertia of each. Then using these calculated moments of 
 inertia and the periods determined, calculate as many values of k and c as 
 there are experiments with cylinders, and enter the results in a table. 
 
 If the shape of a body is such that we are not able to obtain the 
 moment of inertia by calculation, or if the density l of the body is not 
 
 1 All the expressions for the moment of inertia given in Table 7 only hold if 
 the density of the body is quite uniform. It is rather the exception than the 
 rule for a mass of metal to have uniform density throughout, and so in accurate 
 measurements involving the moment of inertia to test for the uniformity of the 
 density is most important. 
 
44] SIMPLE RIGIDITY OF A WIRE 115 
 
 uniform throughout, the moment of inertia K can be obtained by deter- 
 mining the period ^ when it is placed in the cradle, and then measuring 
 the period t> 2 when a body of known moment of inertia K. 2 is placed in 
 the cradle. For we have 
 
 whence 
 
 the second form of the expression being more convenient for use when 
 logarithms are employed. 
 
 44. To Calculate the Simple Rigidity of a Wire from Observations 
 of the Period of a Torsional Pendulum. The quantity c, which is the tor- 
 sional couple called into play when the end of the wire used in the last 
 section is twisted through one radian, depends on the length Z of the wire, 
 its radius r, and on the material of which the wire is composed. It can 
 be shown that 
 
 _ 
 
 '~ 
 
 where n is called the simple rigidity of the material of which the wire 
 is composed. The simple rigidity of a material is a constant for that 
 material. It must be observed, however, that the simple rigidity of a wire 
 depends in a great measure on the state of hardness of the wire. Thus 
 annealing a wire often alters its rigidity very considerably, while stretch- 
 ing a wire so as to give it a permanent set will also alter the rigidity. 
 
 Measure the length and diameter of the wire used to suspend the 
 carriage in the previous exercise, and by means of the values obtained for 
 c calculate the simple rigidity of the wire. Since in the expression for 
 the rigidity the radius of the wire occurs raised to the fourth power, care 
 must be taken to measure r with the utmost possible accuracy. Since a 
 wire is seldom perfectly cylindrical, the diameter must be measured at a 
 number of points along its length, and at each point two measurements 
 must be taken in directions at right angles to one another. 
 
 45. Determination of Young's Modulus and Rigidity of a Wire 
 by Searle's Method. A method of measuring Young's modulus of a wire 
 which only requires a comparatively small length of the wire, and in 
 which the measurement of an elongation or of deflection is replaced by a 
 measurement of periodic time, has been devised by Searle. 1 
 
 1 Phil. Mag. (Feb. 1900), p. 193. 
 
116 
 
 ELASTICITY OF SOLIDS 
 
 [45 
 
 The arrangement used is shown in Fig. 47. It consists of two metal 
 
 bars AB and CD, each about 30 cm. long, and of either square or circular 
 
 cross - section, the area of cross- 
 section being about 1'5 sq. cm. 
 These bars are connected at their 
 mid points by the wire which is 
 under test, the method by which 
 the wire is attached being shown 
 at (a). The wire is soldered 
 into a hole drilled in the head 
 of a screw J, which is held 
 in place by a nut K, this nut 
 being of such a size that it would 
 just fill up the hole left in the 
 bar when the screw is in place. 
 Two light wire hooks are attached 
 to the bars at their mid points, 
 and serve for the attachment of 
 two pieces of thread about a 
 metre long. The upper ends 
 of these threads are fixed to a 
 
 support, so that the threads are parallel and the bars lie in the same 
 
 horizontal plane. 
 
 The bars are caused to vibrate in a horizontal plane, each about its 
 
 mid point, and if ^ is the period, Young's modulus for the wire is given 
 
 by 
 
 FIG. 47. 
 
 Y= 
 
 (1) 
 
 where r and I are the radius and length of the wire EF, and A' is the 
 moment of inertia of one of the bars. 
 
 The bars are then unhooked from the suspension fibres, and while one 
 bar is held in a clamp, the period of the other, acting as a torsion pendu- 
 lum, is determined. If n is rigidity of the material of the wire, we have, 
 as in the last section, 
 
 K having the same value as before, since the bars are either of square or 
 circular cross-section. 
 
 If the mass of the wire is negligible compared to that of the bars, in 
 the first experiment the wire will be bent into a circular arc as the bars 
 vibrate. If p is the radius of the arc into which the wire is bent, 
 
 the bending moment is ~ .* Further, if <j> is the angle either bar 
 
 * See Minchin's Statics, vol. ii. p. 424. 
 
45] DETERMINATION OF YOUNG'S MODULUS 117 
 
 makes with its position of rest p = l/2<j>, so that the bending moment 
 
 Since the bending moment of the wire is equal to the couple acting 
 on either bar tending to bring it back to its position of rest, the equation 
 of motion of either of the bars is 
 
 21 
 
 and hence 
 
 *i~ 2 *\/JPr 
 
 Y -t,S ' ' ! ' V ' ' < 3 ) 
 
 To perform the experiment, the dimensions of the bars are measured and 
 the moment of inertia K is calculated. The bars having been suspended 
 by the threads, a loop of cotton is tied round the ends of the bars, so that 
 they are slightly drawn together. The bars are then started into vibra- 
 tion by burning through the cotton, and the period measured either with 
 a stop-watch or by the eye and ear method, using a chronometer. The 
 determination of the period as a torsion pendulum is performed as in 
 43. 
 
CHAPTER VI 
 
 THE PENDULUM MEASUREMENT OF " g " AND RATING A 
 CHRONOMETER 
 
 46. The Pendulum. It can be shown (see Watson's Physics, p. 123) that 
 so long as the arc over which a simple pendulum swings is very small, 
 the time of vibration i is given by 
 
 ,W 
 
 where I is the length of the pendulum, and g is the acceleration of 
 gravity. Since at any one place g is constant, it 
 follows that t' 2 /l must also be constant. 
 
 If the angle through which the pendulum swings 
 is greater than one or two degrees, then the observed 
 time of vibration will be appreciably greater than 
 what it would be supposing the amplitude were very 
 small, and so a correction will have to be applied on 
 this account. 
 
 Thus let OA (Fig. 48) be the position of rest of the 
 pendulum, and OB the position occupied at a given 
 instant , then the force acting on the bob in the 
 direction of its motion is - mg sin 0, where m is the 
 mass of the bob. The acceleration with which the bob 
 
 is moving is - , where ds is an element of the circular arc in which 
 
 the bob moves. Hence 
 
 d' 2 s 
 Force acting in bob = m - 2 = - mg sin 6. 
 
 Now if I is the length of the pendulum, s = 10. Thus 
 
 dt dt 
 
 i^ 
 a dt* 
 
 Therefore 
 
 If both sides are multiplied by 2d6 and we integrate, then 
 
 l{ - } = 2<7 cos 6 + a constant 
 \dt/ 
 
 (2) 
 
 118 
 
46] THE PENDULUM 119 
 
 If we measure t from the instant of maximum elongation # () , so 
 that when t = 0, # = # , and since at maximum elongation the velocity 
 
 is zero, we have from (2) 
 
 = 2g cos + const. 
 Hence substitute this value for the constant in (2) 
 
 (dO\* 2 
 (dt) =j 
 
 n_ _do^ 
 
 V 2g (cos 0- cos 
 This may be written 
 
 Take a new variable </> such that 
 
 sin 
 Sm ^sin- 
 
 which is always possible, since >0. Differentiating 
 
 /T I _d&_ 
 
 ^ Ig sin (0 /2) (1 - sin' (tf/2)Jsiir (6^2))* 
 
 003 
 
 sn 
 and M 
 
 COS 
 
 2 sin (0 n /2) cos 
 
 Substituting in (3) 
 
 dt= + V-fn: -j-iTJ 
 
 To find the time taken to reach the equilibrium position, i.e. the 
 quarter period, we have to integrate between = and = 0, that is, 
 between < = vr/2 and < = 0. 
 
 Hence 
 
 The integral is a complete elliptic integral of the first kind, and 
 its value can be obtained from tables. We may, however, expand 
 
120 THE PENDULUM [46 
 
 (1-sin 2 (#0/2) sin 2 <)~i by the binomial theorem, and integrate the 
 terms separately, 
 Thus 
 
 sn 
 
 or r=2*r-{l + -sinX0o/2) + sin* (00/2) + } . . . (5) 
 
 If T is the time of vibration of the pendulum when the amplitude is 
 very small, we may write the expression for t as follows : 
 
 or T =T(l-^ 
 
 = T(l- e A 
 
 so long as # is not very great. 
 
 When making an actual determination of period, the amplitude 
 gradually decreases during the experiment, and having observed the 
 initial amplitude and the final amplitude 0', we require to calculate 
 the period with infinitely small arc from the observed period t. When the 
 amplitudes are not very large, we may employ the following approximate 
 formula 1 
 
 fc-rfl- 
 
 00\ 
 16 / 
 
 In Table 8 the values of the correction term for different values of 
 the initial and final amplitudes are given, and this table will be found 
 of use in reducing observations of period, both in the present case and 
 in others which we shall consider later, as, for instance, in determining 
 the period of a magnet oscillating in the earth's field. 
 
 Next suppose in place of an ideal simple pendulum we have a com- 
 pound pendulum, that is, a rigid body of any shape capable of rotation 
 about a horizontal axis. Let r be the distance of a given point P of the 
 body from the axis about which it can rotate (the axis of suspension), 
 and let y be the perpendicular distance of the point from a vertical plane 
 through the axis. Thus if o> is the angular velocity with which the body 
 is moving, the linear velocity of the point P is wr, and the acceleration 
 
 1 Q and 6' are supposed to be measured in radians. 
 
46] THE PENDULUM 121 
 
 is r . If in is the mass of the particle at P, the force acting on 
 
 (Jit 
 
 P is mr - , and the moment of this force about the axis is rar 2 -- 
 
 ir dt 
 
 Hence for the whole body the moment of the force acting about the 
 
 axis is 2(mr 2 ). Now the only force acting on the particle m is the 
 dt 
 
 attraction of gravity, and the moment of this force about the axis is mgy, 
 and for the whole body the moment is ^(mgy) ; but this is equal to the 
 moment of the whole mass M of the body, supposed concentrated at the 
 centre of gravity, about the axis. Hence if h is the distance of the 
 centre of gravity from the axis, and 6 is the angle through which the 
 body has turned from the equilibrium position, we have 
 
 Moment of forces acting on the body = g M h sin 0. 
 Hence equating the two expressions for the moment of the forces, we get 
 
 ~2(mr 2 )= 
 
 dB 
 
 Now (o= . 
 
 dt 
 
 Hence ^2 2 ( mr ') - ~ 9 Mh sin - 
 
 The quantity 2(mr 2 ) is the moment of inertia of the body about the 
 axis. Hence if R is the radius of gyration about a parallel axis through 
 the centre of gravity, 
 
 Thus 
 
 d*e gh 
 
 _ y 
 
 f) 
 
 
 In the case of a simple pendulum we have 
 
 Hence if 
 
 the simple pendulum of length I will have the same period as the com- 
 pound pendulum. 
 
 If the body is now supported by an axis in the same plane as the 
 
122 THE PENDULUM [ 47 
 
 first axis and the centre of gravity, and parallel to the first, and at a 
 distance I from it, we get similarly 
 
 where h' is the distance of the new axis from the centre of gravity. 
 But h' = l-h = IP/h. Hence 
 
 m 
 l' = + h = l ........ (8) 
 
 that is, the period about the new axis will be the same as about the old. 
 The point where the first axis cuts the vertical plane drawn through the 
 centre of gravity perpendicular to the axis being called the centre of 
 suspension, the point where the second axis cuts this plane is called the 
 centre of oscillation. 
 
 In the case of a heavy sphere of radius r suspended by a fine wire of 
 negligible weight, and length A, we have (see Table 8) 
 
 2 
 
 7?2 _ r '2 
 
 ~5* ' 
 
 Hence the length of the equivalent simple pendulum is given by 
 
 2r 2 
 
 Hence the periodic time of such a pendulum is 
 
 .... 
 
 47. Determination of " g " by Means of a Borda Pendulum. The 
 
 expression found in the last section for the period of a heavy bob sus- 
 pended by a fine wire may be employed for measuring g. For (10) may 
 be written 
 
 A piece of apparatus suitable for determining the value of g from the 
 period of such a pendulum is shown in Fig. 49. It consists of a bracket 
 A, to the upper surface of which a piece of plate glass is attached. This 
 glass serves as a plane on which the knife-edge which supports the pen- 
 dulum rests. A slot in the glass allows the suspension wire to pass. 
 The bob consists of a brass sphere having a diameter of about 5 cm., 
 to which the suspension wire is attached by being soldered into a fine 
 hole bored radially into the sphere. The upper end of the wire passes 
 through a knife-edge B, and is held by being clamped by the screws at c. 
 
47] DETERMINATION OF "</" BY A BORDA PENDULUM 123 
 
 A scale D is attached to the wall in such a way that the wire passes just 
 clear of its surface, and is used to measure the initial and final amplitude 
 of the pendulum. A screw E 
 serves, as will be explained 
 later, to assist in measuring the 
 length of the pendulum. 
 
 A good clock beating 
 seconds or a chronometer will \ 
 be required to measure the \ 
 time of oscillation, also a 
 metal rod having a diameter ; 
 of about a centimetre, and : 
 about 105 cm. long, will be : 
 required to assist in measuring ; 
 the length. The ends of this 
 rod must be carefully turned ' 
 so as to be planes perpen- 
 dicular to the axis of the '. 
 rod. If the rod has not been 
 exactly adjusted to 105 cm. 
 long, or if its length is not 
 known, it will require to be 
 measured, which can be done 
 with the comparator ( 19). 
 
 The first operation to be 
 performed is to adjust the 
 knife-edge so that without the 
 suspension wire it will have 
 a period of two seconds. The 
 period can be adjusted by 
 screwing the weight w up or 
 down. Since the length of the 
 pendulum will be adjusted so 
 
 that the period is nearly two seconds, when the knife-edge is adjusted 
 in this way it will have no appreciable influence on the period of the 
 pendulum. Of course it is only when the natural periods of the 
 pendulum and knife-edge are exactly equal that the knife-edge has no 
 influence whatever on the period of the pendulum. Since, however, the 
 knife-edge is purposely made as light as possible, any slight departure 
 from isochronism will not appreciably affect the result. 
 
 Having adjusted the period of the knife-edge, pass the suspension 
 wire through the hole, and adjust the length of the suspension wire so 
 that the distance between the centre of the bob and the knife-edge is 
 about 99 '5 cm. Place the knife-edge on the glass plate, and steady 
 the bob so that the pendulum comes to rest. 
 
 Set up a telescope at a distance of about 2 metres in front of the 
 scale D, and focus it on this scale. The suspension wire will be seen 
 
 FIG. 49. 
 
124 THE PENDULUM [ 47 
 
 crossing the scale, and the reading corresponding to the point where it 
 crosses the scale must be taken. This will give the position of rest of 
 the pendulum, and this point must be marked on the scale by a small 
 blackened pointer. 
 
 The pendulum must now be set in vibration, so that the bob moves 
 through about 8 cm. on each side of the zero. The best way of 
 starting the swings without setting up irregular vibrations is to tie 
 a loop of cotton round the bob, and by means of a piece of cotton 
 attached to this loop to draw the bob aside, and fasten it in this 
 position by attaching the cotton to a stand. Then by burning the 
 cotton the bob will be released and the pendulum started. Having 
 noted the readings on the scale corresponding to the end of the swing 
 on either side of the zero, the period must be determined, and, finally, a 
 second reading made of the scale corresponding to the extreme positions. 
 To determine the period, the method given in 43 is employed, that is, 
 we note the time when the pendulum crosses its position of rest exactly 
 when the clock or chronometer gives a tick. 
 
 Taking the time from the chronometer, or clock, watch the move- 
 ment of the suspension thread through the telescope till it is seen to 
 cross the zero mark, apparently going from left to right, exactly on a 
 tick. Having written this time down, again take up the time, and from 
 the chronometer note when the pendulum crosses from right to left on 
 a tick. Continue observing the coincidences in either direction for at 
 least half-an-hour. The interval between consecutive coincidences will 
 depend on the relative periods of the pendulum and the clock or 
 chronometer ; the nearer these agree in period, the longer the coincidence 
 period. The coincidence period ought not to be less than two minutes. 
 After the first pair of coincidences have been observed it will not be 
 necessary to watch the pendulum, as the time when to expect the next 
 coincidence can be readily calculated and the time can be taken up, and 
 the watching started, about twenty seconds before the expected time. 
 
 A note must be made as to whether the pendulum or the clock is 
 going the faster. This can at once be seen by watching the next few 
 transits after a coincidence. If on the ticks following a coincidence the 
 pendulum appears to lag, so that it does not reach its zero position on 
 the tick, the clock is going faster than the pendulum, and vice versa. 
 
 We have now to calculate how many vibrations have been made by 
 the pendulum between the first and last coincidence in either direction. 
 
 Let us suppose that the pendulum is vibrating quicker than the 
 clock or chronometer. By this we mean that while the pendulum makes 
 one vibration the clock pendulum has not quite completed one vibration, 
 that is, it has not made two ticks, or that the chronometer has not quite 
 made four ticks. The periodic time of the pendulum is very nearly two 
 seconds, so let its period be called 2 - x. Then starting from the 
 instant when there is coincidence, at its next transit the time will be 
 2 - x t at the next transit 2(2 - x\ and so on till after n vibrations of 
 the pendulum the time will be ?i(2 - x) or 2n -nx. Now, as the 
 
47] DETERMINATION OF "#" BY A BORDA PENDULUM 125 
 
 pendulum vibrates it gradually has fallen behind the ticks of the clock, 
 till finally it will have fallen behind the clock by a whole tick when the 
 next coincidence will take place. This occurs when nx is equal to the 
 interval between the ticks. Hence we have 
 
 and 
 
 nx=l for the clock beating seconds, 
 
 '5 for the chronometer beating half seconds. 
 
 nx 
 
 But the coincidence interval is equal to 2n - nx seconds, 
 the coincidence interval c, we have 
 
 Hence calling 
 
 For the clock 
 
 or 
 
 ?i , IP ,1 c = %-* 
 
 j ' I and for the I c + -5 
 
 = C - ' ~~ --, I chronometer 1 or n = . 
 
 If the pendulum is moving slower than the clock, then by an exactly 
 similar argument we can show that 
 
 In the clock 
 
 -{ c - '5 
 
 chronometer or n = , 
 j 12 
 
 Hence, knowing c, we can immediately calculate the number of vibra- 
 tions, w, which the pendulum has made between the coincidences. 
 Proceeding in this way we can obtain the total number of vibrations 
 made by the pendulum during the whole time of vibration, and then 
 from the times of the first and last transits observed (or the first two 
 or three and last two or three) we can calculate the period of the 
 pendulum. The following scheme shows the method of recording such 
 a set of observations : 
 
 Pendulum moving to Right. 
 
 Pendulum moving to Left. 
 
 Coincidence 
 Time of Interval 
 Coincidence. (Pendulum 
 ' gaining). 
 
 Number of 
 Vibrations. 
 
 Time of 
 Coincidence. 
 
 Coincidence 
 Interval 
 (Pendulum 
 gaining). 
 
 Number of 
 Vibrations. 
 
 H. M. S. M. S. 
 
 10 7 23 ; 
 
 
 H. M. S. 
 
 10 8 39 
 
 M. S. 
 
 
 9 54 2 31 
 
 76 
 
 11 12 ! 2 33 
 
 77 
 
 12 27 2 33 77 
 
 13 43 ! 2 31 
 
 76 
 
 To measure the length of the pendulum, the screw E (Fig. 49) is 
 adjusted till it just touches the bottom of the bob. By allowing the 
 bob to swing over a small arc while moving the screw, it will at 
 once be seen .when the point of the screw touches the bob. When 
 
126 THE PENDULUM [ 48 
 
 making this adjustment, it is important to see that the point of the 
 screw is vertically below the point of support of the pendulum. This 
 can be secured by bringing the pendulum to rest, and then moving the 
 knife-edge on the plane till the bob appears to be arranged symmetrically 
 with regard to the screw. 
 
 Having completed the adjustment of the screw, remove the pendulum 
 and stand the metal rod on the point of the screw with its upper end 
 in the slot of the glass plate on which the knife-edge rested. To 
 determine the amount by which the length of the pendulum differs from 
 the length of the rod, determine the zero reading of a spherometer and 
 then the reading when the centre leg touches the end of the rod and the 
 three fixed legs rest on the surface of the glass. The difference between 
 the spherometer readings will give the amount by which the length from 
 the point of support to the bottom of the bob exceeds or falls short of, 
 as the case may be, the length of the rod. To obtain the quantity 7, 
 we have to decrease this length by the radius of the bob. Measure the 
 diameter of the bob with the vernier callipers, and thus obtain the 
 radius. 
 
 The only thing that remains is to calculate the initial and final 
 amplitudes in minutes of arc, so that we may apply the correction to 
 reduce to infinitely small arc. If a and a are the readings on the scale 
 i) at the commencement and end of the observation of period, and D is 
 the distance of the scale from the plane on which the knife-edge rests, 
 we have 
 
 Initial amplitude = tan" 1 a/D = a/D radians, 
 
 since the angle is always very small. But a radian contains 3438 minutes 
 of arc. Hence the initial amplitude in minutes is given by 
 
 ock 
 iard 
 
 <9 = 3438-. 
 
 Similarly, calculate the final amplitude. 
 
 The time of vibration will require correction for the rate of the cl 
 or chronometer. This can be obtained by comparison with a standa 
 clock, of which the rate is known from astronomical observations. In 
 the absence of such a clock, one of the methods described in 50, 52 
 may be employed to rate the clock or chronometer. 
 
 48. To Examine the Laws of the Compound Pendulum. The 
 apparatus used for this experiment consists of a uniform brass bar 
 90 cm. long, the cross-section being a rectangle, of which the sides are 
 2 cm. and 0'5 cm. respectively. Holes 6 mm. in diameter are bored at 
 equal intervals throughout the length of the bar, which allow of its being 
 suspended from a knife-edge which passes through the hole. This knife- 
 edge consists of a piece of hard steel ground to a sharp edge and driven 
 into the edge of a piece of wood. This wooden base has two fixed legs 
 in front, formed by the heads of two round-headed screws, and an 
 adjustable levelling screw behind. The object of the levelling screw is 
 
48] 
 
 LAWS OF THE COMPOUND PENDULUM 
 
 127 
 
 to allow of the knife-edge being made horizontal, so that the bar will 
 swing regularly without twisting as it swings. 
 
 Determine with a stop-watch the period of the bar when suspended 
 from each of the holes in turn, by observing the time taken to make 
 from 50 to 100 vibrations. It will be found, when the centre of the bar 
 is approached, that the period gets very long, and only a few vibrations 
 can be observed. At the centre hole the period will be too long to observe, 
 and the bar will probably refuse to vibrate steadily. If the knife-edge 
 were placed exactly at the mid point of the bar, instead of at a distance 
 from the middle point equal to the radius of the hole, the bar when 
 deflected would not tend to come back into its original position, that is, 
 its period would be infinite. 
 
 Next measure the distance from one end of the bar to the point 
 of support in each case, and plot the results on a piece of squared paper, 
 
 10 20 30 4O 50 6O 7O 50 6 
 
 DISTANCE OF POINT OF SUPPORT FROM ONE END. GM - 
 
 FIG. 50. 
 
 taking the lengths measured from one end of the bar as abscissae, and the 
 times of a single vibration as ordinates. 
 
 The curve drawn through the observed points will be similar to that 
 shown in Fig. 50, and consists of two branches, which ought, as the 
 pendulum is uniform, to be exactly similar. If we draw any horizontal 
 line, such as A BCD, it will cut the curve four times, which means that 
 there are four points of suspension from which the bar would have a 
 period equal to 1'5. 
 
 Now it was shown in 46 that if we can find two points on a body 
 which are not symmetrically placed with reference to the centre of gravity, 
 such that the period when suspended from either of these points is the 
 same, then the distance between these points is equal- to the length of 
 the simple pendulum which would have the same period as has the body. 
 These points are called the centre of suspension and the centre of oscilla- 
 tion or percussion respectively. From a consideration of the curve in 
 
128 THE PENDULUM [49 
 
 Fig. 50, we see that the pairs of points A and c and B and D fulfil the 
 above conditions, so that the length AC or BD will be the length of 
 the simple pendulum, which would have the period 1*5 seconds. 
 
 On your curve draw a horizontal line through the first observed 
 point, and determine the length of the equivalent simple pendulum 
 from the curve. Then fit up a simple pendulum of this length by 
 suspending a small lead bullet by a silk fibre, and prove experi- 
 mentally that it has the same period as the rod when suspended from 
 the end hole. 
 
 Since we can obtain from the curve the length of the equivalent 
 simple pendulum which would have a period equal to the ordinate of 
 the straight line which is drawn parallel to the axis of JT, we can 
 immediately calculate the value of g by means of the formula g 4?r 2 ^ 2 . 
 
 Measure off the lengths of the simple pendulum for three or four 
 lines drawn parallel to the X axis, and for each calculate the value of g. 
 
 49. Determination of "g" by Eater's Pendulum. The method 
 of deducing the length of the equivalent simple pendulum from the 
 distance between the centre of suspension and the centre of oscillation 
 was first used by Kater in 1818, and it remains to this day the most 
 accurate method for measuring g. Various forms of Kater's pendulum 
 have been devised, but they all consist essentially of a metal rod 
 weighted at one end so that the centre of gravity is much nearer one end 
 than the other and having two knife-edges, one near either end. In 
 addition, there is some method of adjusting the periods to equality when 
 the pendulum is suspended from either knife-edge. In one form, while 
 one of the knife-edges is fixed the other can be moved through a small 
 distance along the stem of the pendulum. This form has the dis- 
 advantage that it is difficult to ensure that the movable knife-edge is 
 always parallel to the fixed knife-edge. There is the further disadvan- 
 tage that every time the movable knife-edge is moved the distance 
 between the knife-edges alters and has to be re-measured. In the other 
 form the knife-edges are fixed at almost the correct distance apart, and 
 the final adjustment is performed by sliding a weight along the stem of 
 the pendulum. In this form, since the distance between the knife-edges 
 remains the same, all the measurements made of this distance can be 
 combined together and the mean taken as the length used in reducing 
 the different observations of period which are made. 
 
 Whichever the form of pendulum used, it will be found a very tedious 
 operation to attempt to adjust the periods about the two knife-edges to 
 exact equality. This exact adjustment is also unnecessary, for, as we 
 shall see, if the periods are very nearly the same we can calculate a 
 correction to allow for the want of equality in the periods. Suppose that 
 the periods about the two knife-edges are ^ and 2 , and that the distances 
 of the knife-edges from the centre of gravity of the pendulum are A x and 
 Ji. 2 respectively. Then if R is the radius of gyration of the pendulum 
 about an axis passing through its centre of gravity and parallel to the 
 edges of the knife-edges, and M is the mass of the pendulum, we have 
 
49] DETERMINATION OF 
 
 that the moment of inertia about the knife-edges is 
 M(R 2 + h/). Hence the periods are given by 
 
 BY RATER'S PENDULUM 129 
 
 -f} and 
 
 Squaring and multiplying across we get 
 
 and 
 
 Eliminating R by means of these two equations we obtain 
 
 1 -945 
 
 Since (7? 1 + h 2 ) is the distance between the knife-edges, the first term on 
 
 the right above can be at once calculated. Also, since |the periods about 
 
 the two knife-edges are always 
 
 very nearly equal, so that 
 
 t-f - 2 2 is small, the second 
 
 term is small compared to the 
 
 first. Hence we need only 
 
 determine approximately the 
 
 distances of the knife-edges 
 
 from the centre of gravity. 
 
 The position of the centre of 
 
 gravity can be determined with 
 
 sufficient accuracy by balancing 
 
 the pendulum. 
 
 If the pendulum is one in 
 which the knife-edge is mov- 
 able, then a slightly different 
 procedure to avoid having to ad- 
 just to strict equality of period 
 is possible. Having approxi- 
 mately determined the position 
 of the movable knife-edge for 
 equality, determine the period 
 about both knife-edges ; and suppose the distance slightly smaller than that 
 corresponding to equality of periods. Then move the knife-edge so that it is 
 
 Q 
 
 O , . 940 
 
 s 
 
 O. 
 
 1-935 
 
 93 93-5 
 
 DISTANCE BETWEEN KNIFE EDGES 
 
 FIG. 61. 
 
130 THE PENDULUM [ 49 
 
 approximately as far beyond the correct position as it was short of it when 
 the above measurements were made, and again determine the period about 
 both knife-edges. Next plot the observed times and distances between 
 the knife-edges on squared paper, as is done in Fig. 51, the points A 
 and B corresponding to the heavy bob being uppermost and the points 
 c and D to the bob being below, and connect the points representing 
 the period about each of the knife-edges by straight lines AB and CD. 
 Then the length and period corresponding to the point P, where these 
 straight lines intersect, will give the distance between the knife-edges 
 and the corresponding period which would be found if the knife-edge 
 were adjusted for strict equality in the periods. Thus in the example 
 the length of the equivalent simple pendulum is 93'33 cm., and its period 
 is 1-9378 seconds. 
 
 To determine the period, the method of coincidences must be em- 
 ployed. Since the conduct of this part of the experiment differs in no 
 manner from that described in 47, it will not be again described, 
 but reference must be made to that section. The distance between the 
 knife-edges is a measurement which is of very considerable difficulty. 
 By placing the pendulum on a horizontal table, and supporting a steel 
 scale alongside so that the divided surface of the scale is on a level with 
 the edges of the knife-edges, it is possible with care, and by using a 
 reading microscope, to obtain the distance to within one- or two-tenths of 
 a millimetre. To obtain greater accuracy it will be necessary to employ 
 a vertical comparator, in which the microscopes are first focused 
 on the knife-edges as the pendulum hangs, and then, replacing 
 the pendulum by a standard scale, the distance between the 
 cross-wires of the microscopes is obtained. It will in general 
 be unnecessary in an ordinary physical laboratory experiment 
 to go to this refinement, since to justify it we should have to 
 take many precautions and make corrections which would be 
 outside the range of the means ordinarily at the disposal of the 
 student. Thus the slightest flexure of the stand under the 
 influence of the swinging pendulum would have to be measured 
 and allowed for, and a correction, the magnitude of which can 
 only be obtained by swinging the actual pendulum used in a 
 vacuum, would have to be applied for the effect of the sur- 
 rounding air on the period of the pendulum. Special pre- 
 cautions would have to be taken to allow of the temperature 
 being kept constant throughout the measurements, and so on. 
 
 The effect of air friction may be made the same, whichevei 
 end of the pendulum is uppermost, by fitting a dummy bob ai 
 the end remote from the real bob. A form of pendulum whicl 
 FIG. 52. satisfies this requirement is shown in Fig. 52. The bras 
 cylinder A is filled with lead, while the similar brass cylinde 
 B is empty. The rod connecting these consists of a brass tube, to whicl 
 are fixed the knife-edges D and E. The weight c slides on the tube 
 and can be clamped by means of a set-screw. The rod is graduate* 
 
50] DETERMINATION OF THE RATE OF A CLOCK 131 
 
 iu millimetres to allow of the position of c being identified. The pendulum 
 is supported from a bracket, such as shown at A in Fig. 49. Two small 
 wires F and G serve to read the amplitude of movement on a scale. 
 
 50. Determination of the Eate of a Clock by the Occultation of 
 a Star by a Terrestrial Obstacle. A simple method of rating a clock 
 or chronometer, which is capable of giving results sufficiently accurate 
 for many purposes, and does not involve any elaborate instrument, is as 
 follows : Attach a metal plate, in which there has been bored a small 
 hole (about 3 mm. in diameter), to the top of a wall or the side of a 
 house in such a position that, looking through the hole in a southerly 
 direction, some object, such as a lightning conductor, which has a smooth 
 vertical edge, can be seen at a considerable elevation. A chimney is not 
 a suitable object to select, owing to the heat causing the air in the 
 neighbourhood to shimmer. In order to rate the chronometer look 
 through the hole in the metal plate, and by the eye and ear method 
 determine the time when a fairly bright star vanishes behind the 
 lightning conductor. The instant of occultation can be determined 
 with very considerable accuracy, since the disappearance of the star 
 is most surprisingly sudden. 1 The time at which the same star is 
 occulted on the following night must be determined in the same way. 
 The interval between the two occultations will be equal to one sidereal 
 day, that is, to a mean solar day less 3 minutes 55*91 seconds, and by 
 comparing this interval with the interval shown by the clock or chrono- 
 meter the rate of the latter can be obtained. 
 
 51. The Sextant and its Adjustment. The usual form of sextant 
 is shown in Fig. 53. It consists of a triangular framework ABC, having 
 a graduated circular arc AB. An arm cv, called the index arm, is pivoted 
 at the centre of the arc, and carries a vernier v. This arm is supplied 
 with a clamp screw and tangent screw, by means of which the position 
 can be adjusted. A mirror i, called the index glass, is fixed to the arm 
 cv, the plane of the mirror being perpendicular to the plane of the 
 circular arc, which we may call the plane of the instrument. A second 
 mirror H, called the horizon glass, is fixed with its plane perpendicular 
 to the plane of the instrument. The lower half only of the horizon glass 
 is silvered, the upper half being left clear. A telescope T is mounted so 
 that its optic axis is parallel to the plane of the instrument, and passes 
 through the centre of the horizon glass. Two sets of coloured glasses, 
 E and F, are provided for use when observing the sun, while a lens G is 
 provided for reading the vernier v. 
 
 When using the sextant to measure the angle subtended at the 
 observer by two distant objects, the telescope is directed so that one 
 object is seen through the clear part of the horizon glass. The index 
 arm cv is then moved, the instrument being so held that its plane passes 
 through the two objects till an image of the second object, formed by 
 reflection in the mirrors I and H, appears in the field of view of the tele- 
 scope. The index arm is then clamped and the tangent screw adjusted 
 
 1 It is advisable to note the times of occultation of three or four stars. 
 
 
132 
 
 THE SEXTANT 
 
 [ 51 
 
 till the two images appear to coincide. The angle which the index arm 
 now makes with its position when the two mirrors I and H are parallel, 
 is half the angle subtended by the two objects at the mirror I (see 
 Watson's Physics, 336). Since the arc AB is so graduated that the 
 zero reading corresponds to the position of the index arm when the 
 mirrors I and H are parallel, and each half degree is numbered as a degree, 
 the arc as read off will give directly the angle between the two objects. 
 
 In order that the angle as measured with the sextant should be correct, 
 the following adjustments must be made : 
 
 (1) The plane of the index glass I must be perpendicular to the plane 
 of the instrument. To test whether this adjustment is complete, set the 
 
 FIG. 53. 
 
 radius bar to about the middle of the arc. Then with the eye near the 
 index glass look obliquely into the glass, so as to see at the same time 
 part of the arc direct and part after reflection in the mirror. If the two 
 portions of the arc appear in the same plane, the mirror is in adjustment. 
 If not, then the screws at the back of the mirror must be moved till the 
 adjustment is complete. 
 
 (2) The plane of the horizon glass must be perpendicular to the plane 
 ot the instrument. Direct the instrument so as to view some small well- 
 defined distant object, and move the radius arm till two images of the 
 object appear in the field of view. If by adjusting the radius arm the 
 two images can be made to exactly coincide, then the mirror H is parallel 
 to the mirror i, and hence is perpendicular to the plane of the instrument. 
 If turning the radius arm will not bring the two images into exact 
 
51] THE SEXTANT AND ITS ADJUSTMENT 133 
 
 coincidence, the screw at the back of the mirror H must be adjusted till 
 complete coincidence of the images can be secured. 
 
 (3) The line of collimation of the telescope must be parallel to the 
 plane of the instrument. Turn the eye-piece of the telescope so that 
 one of the wires, or two of the wires if there are four wires, is parallel 
 to the plane of the instrument. 
 
 The images of two stars at a distance of between 90 and 120 are 
 then brought to coincidence on the wire farthest from the plane of the 
 instrument, or, if there is only one wire, near the edge of the field. If 
 on tilting the sextant so that the images are near the other wire, or 
 near the other edge of the field, they remain superposed, then the axis 
 of the telescope is correctly adjusted. If the images separate, then the 
 telescope must be adjusted by the screws which hold the collar. 
 
 Another way of testing this adjustment is to prepare two L-shaped 
 pieces of brass, the upright part being of such a height that when they 
 are placed in the arc of the sextant the upper edges are at the same 
 level as the optical axis of the telescope. The heights of these two pieces 
 of brass must be very carefully adjusted to equality. This may be 
 tested by first carefully levelling a small table by means of a delicate 
 spirit-level, then placing the Us on the table and resting the level on 
 the tops. These L's are then placed on the arc of the sextant as far 
 apart as possible, and, the instrument being placed on a firm table, a 
 sight is taken over the tops of the L's on some distant object. The 
 telescope is then adjusted so that the sighted point on the distant object 
 appears in the centre of the field as defined by the cross-wires. 
 
 Whenever the sextant is used to measure an angle, the images must 
 be brought into contact at the centre of the field. If contact is secured 
 near the edge of the field, then the angle as measured will be too great. 
 
 (4) The zero of the scale is supposed to coincide with the position of 
 the index arm when the two mirrors are parallel. This adjustment is 
 seldom exactly correct, and hence the correction (called the index cor- 
 rection) which has to be applied must be determined. 
 
 If the reading on the vernier when the mirrors are parallel is e, then 
 the index correction is - e, and this quantity has to be added to all the 
 measured angles. The mirrors will be parallel when the images of a 
 very distant object, or a star, coincide. The reason why a very distant 
 object must be chosen is that we suppose that the lines drawn from the 
 object to the mirrors I and H are parallel, and this is only strictly true 
 if the object is at an infinite distance. An easy calculation will show 
 at what distance the object must be so that the error produced may not 
 be appreciable on the scale of the instrument. With sextants of the 
 ordinary size the angle subtended by the distance between the two 
 mirrors at half a mile is about 20 seconds. Hence if the sextant reads 
 to 10 seconds, the object chosen must be more than a mile away. Having 
 brought the images of such a distant object into coincidence, read the 
 vernier, and let the reading be e. If e is positive, that is, if the reading 
 is on the side of the zero towards B (Fig. 53), then the quantity e has to 
 be subtracted from all the readings, and vice versa. 
 
134 
 
 THE SEXTANT 
 
 [51 
 
 FIG. 54. 
 
 The sextant is most often employed to measure the altitude of the 
 sun or stars, and in such a case, on land at any rate, what is called an 
 artificial horizon is necessary. This consists of a brass or gun-metal 
 
 stand A (Fig. 54) fitted with 
 levelling screws c. On the 
 upper surface is a shallow 
 recess B, in which is placed a 
 small quantity of mercury. 
 The bottom of the recess B 
 is amalgamated by filling it 
 with a solution of mercuric 
 nitrate. In this form of 
 artificial horizon accidental tremors die down very rapidly, while in the 
 ordinary form in which a deep pool of mercury is used it is quite impossible 
 to observe in a town, since the mercury surface is never sufficiently still. 
 
 The surface of the mercury will become dull owing to the formation 
 of an amalgam, but this skin may be removed before taking an obser- 
 vation by drawing a piece of paper or card across the surface. The 
 levelling screws must be adjusted so that the film of mercury (about 
 1 mm. thick) extends all over the recess. If observations are to be 
 made in a wind the mercury surface must be protected by a small cover 
 D, which has two windows, one of which is shown at E. These windows 
 may be closed by pieces of " cabinet " mica. 
 If glass is used, it is necessary to use optically 
 worked glass in which the surfaces are strictly 
 parallel. 
 
 When determining the altitude of a 
 heavenly body, say, a star, the telescope of the 
 sextant is turned so as to view directly the 
 image of the star formed by reflection in the 
 mercury. Then the radius bar is turned till 
 the image formed by reflection in the two 
 mirrors, but without reflection in the mercury, 
 coincides with the image seen by reflection in 
 the mercury. When this adjustment is com- 
 plete, the angle read off on the sextant is 
 twice the altitude of the star. For if AB 
 (Fig. 55) is the horizontal surface of the 
 mercury, N being the normal to the surface, 
 
 the angle measured is the angle sio. But the star being at such a great 
 distance, si is parallel to s'o, and therefore if 10 is produced to c, the 
 angle given by the sextant is the angle s'oc. Now, since the angle of 
 incidence is equal to the angle of reflection, the angle S'ON is equal to the 
 angle ION, and hence the angle S'OB is equal to the angle IOA or to the 
 angle BOG. Thus the angle s'oc is equal to twice the angle S'OB, that is, 
 the measured angle is equal to twice the altitude S'OB of the star. 
 
 When using the sextant to measure altitudes with the artificial 
 
52] DETERMINATION OF ERROR OF A CHRONOMETER 135 
 
 horizon it will be found a great assistance to hold the sextant in a 
 clamp attached to a retort stand. 
 
 52. Determination of the Error of a Chronometer with the Sex- 
 tant. The two methods suitable for determining time with a sextant 
 are by (1) single altitude, and (2) equal altitudes of the sun or a star. 
 Since it is in general more convenient to observe during the day, we 
 shall assume, in what follows, that the sun is used. 
 
 Since in both these methods the time is deduced from the altitude, it 
 is of importance to observe when the altitude is changing as rapidly as 
 possible. It can be shown that the altitude is changing most rapidly 
 when the sun or star is crossing the prime vertical, that is, is due east 
 or west of the observer. Further, errors in the assumed values for the 
 latitude will produce the least effect when the star is near the prime 
 vertical. The hour angle, that is, the time which will elapse, or has 
 elapsed, before the sun or star crosses the observer's meridian when the 
 sun or star is crossing the prime vertical, is given in the following table : 
 
 HOUR ANGLE OF THE SUN OR A STAR ON THE PRIME VERTICAL. 
 
 
 Declination (North). 
 
 T 4-"4- A 
 
 
 
 
 
 5 
 
 10 
 
 15 
 
 20 25 
 
 
 
 
 
 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. 
 
 H. M. H. M 
 
 50 
 
 6 
 
 5 43 
 
 5 26 
 
 5 8 
 
 4 49 4 28 
 
 52 
 
 6 
 
 5 44 
 
 5 28 
 
 5 11 
 
 4 54 4 36 
 
 54 
 
 6 
 
 5 45 
 
 5 31 
 
 5 15 
 
 4 59 4 42 
 
 56 
 
 6 
 
 5 46 
 
 5 33 
 
 5 18 
 
 53 4 47 
 
 58 
 
 6 
 
 5 47 
 
 5 35 
 
 5 21 
 
 57 4 52 
 
 60 
 
 6 
 
 5 48 
 
 5 37 
 
 5 24 
 
 5 11 4 57 
 
 It will often be impossible to observe when the sun is crossing the 
 prime vertical, since with the artificial horizon it is impossible to observe 
 altitudes much less than 20. Hence it may be necessary to observe 
 some time after the sun has crossed the prime vertical. Observations 
 ought not, however, to be made within about two hours of noon. Thus 
 in the winter, when in the latitude of Great Britain the altitude of the 
 sun does not become sufficiently high till near noon, it will be necessary 
 to use a star, one with northerly declination being chosen. 
 
 (1) Time by Sinf/le Altitude. In this method the altitude of the sun 
 at a time noted on the chronometer is measured by means of a sextant 
 and artificial horizon. 
 
 The artificial horizon having been set up on a steady support and 
 levelled so that the mercury covers the dish, the sextant is adjusted so that 
 two images of the sun are seen, one after reflection in the mercury, and the 
 dark glasses are so adjusted that the two images appear equally bright. 
 
136 
 
 THE SEXTANT 
 
 [52 
 
 If the observations are being made before noon, move the index arm 
 till the images are separated slightly ; if a reversing telescope is used, 
 the image which moves with the movement of the index arm is to be upper- 
 most. Then set the vernier to the next whole ten minute division, and 
 taking the time from the chronometer, note when the two images just 
 touch. Next move the vernier forward ten minutes, and again note the 
 time of contact. Repeat this observation so as to obtain five readings 
 of the scale and chronometer, that is, of the double altitude of the sun's 
 upper limb at certain observed times. 
 
 Next move the radius arm so that the images of the sun overlap, and 
 note the times when the images separate, obtaining, as before, five obser- 
 vations. These observations give the double altitudes of the sun's lower 
 limb at the observed times. 
 
 If the observations are being made in the afternoon, then the lower 
 limb should be observed first. 
 
 Enter the observations as in the following example, applying the 
 correction for index error. The apparent altitude will require correction 
 for refraction and parallax, the amount of which two corrections can be 
 obtained from Table 9, the correction in every case being subtractive : 
 
 Date, 3 April 1906. 
 
 Latitude of station, 51 29' 48". 
 
 Longitude of station, h. m. 41'5 s. W. 
 
 Approximate error of chronometer (G.M.T.) 10 s. fast. 
 
 
 Altitude. 
 
 Time. 
 
 Upper limb 
 
 D. M. 
 
 68 50 
 
 H. M. S. 
 
 9 34 45 '5 
 
 Lower limb 
 
 69 
 10 
 20 
 30 
 69 
 
 35 30-0 
 36 15-0 
 37 0-5 
 37 45-5 
 40 14-0 
 
 
 10 
 20 
 30 
 40 
 
 40 59-0 
 41 44-5 
 42 30-0 
 43 16-0 
 
 Sums .... 
 
 150 
 
 390 O'O 
 
 Means 
 
 69 15' 0" 
 
 H. M. S. 
 
 9 39 O'O 
 
 Index correction 
 
 + 50 
 
 
 
 
 
 Double apparent altitude .... 
 Apparent altitude . . ... 
 
 69 15 50 
 34 37 55 
 
 
 Refraction and parallax .... 
 
 -1 17 
 
 
 Altitude (a) 
 
 34 36 38 
 
 
 
 
 
52] DETERMINATION OF ERROR OF A CHRONOMETER 137 
 
 Declination. 
 
 Equation of Time. 
 
 At Noon. 
 
 Variation. 
 
 At Noon. 
 
 Variation. 
 
 5 3' 58" '6 N. 
 -2 16-4 
 
 57-55 
 2-37 
 
 + 3 32-8 
 -1-8 
 
 75 
 
 2-4 
 
 115-10 
 17-26 
 4-03 
 
 1-50 
 30 
 
 1-80 
 
 136-39 
 
 5 1' 42"-2 
 
 
 + 3 310 
 
 
 
 84 58' 17"-8 = Polar distance (90 -declination). 
 
 
 
 Altitude =a= 34 3 
 Latitude = 1= 51 2 
 Polar distance =p= 84 5 
 
 6' 38" 
 9 48 log sec 1 = -20582 
 8 18 log cosec p = -00167 
 
 Sum =171 
 iSum =S= 85 3 
 S-a = 50 5 
 
 4 44 
 2 22 log cos S =2-89083 
 5 44 log sin (S - a) = 1 '89006 
 
 Angle to Time. 
 
 Sum = 2'98838 
 
 H. M. S. 
 
 36 = 2 24 
 21' = 1 24 
 44" = 2-9 
 
 Log sin /z//2 ^ sum 1-49419 
 &/2 = 18 10' 52" 
 Hour angle =/i = 36 21' 44" 
 
 H. M. S. 
 
 2 25 26-9 
 
 
 Hour angle .... 
 
 H. M. S. 
 
 . . 2 25 26'9 
 
 Apparent local time . 
 Equation of time 
 
 9 34 33-1 
 + 3 31 '0 
 
 
 
 Mean local time 
 
 9 38 4-1 
 
 Longitude (W.) 
 
 4-41-5 
 
 G.M.T 
 
 Time by chronometer 
 
 9 38 45-6 
 9 39 0-0 
 
 Chronometer error (fa 
 
 it) . . . . 14-4 
 
138 THE SEXTANT [ 52 
 
 The formula employed to deduce the hour angle h is 
 
 COS 
 
 where I is the latitude, p the polar distance of the sun (i.e. 90 - sun's 
 declination), a is the observed altitude, and S is written for 
 
 a + l+p 
 
 2 
 
 The declination and equation of time are obtained from the Nautical 
 Almanac for noon, and the quantities at the actual time of the observa- 
 tions are calculated from the change in one hour given in the almanac. 
 The method of carrying out the calculation is clearly indicated in the 
 example, and it will be found advantageous to set out the different steps 
 of the calculation as shown there. 
 
 (2) In the method of equal altitudes the time the sun or star has 
 the same altitude is observed on the two sides of the meridian. If the 
 declination of the heavenly body observed remained constant, then the 
 mean of the two observed times will give the time of meridian passage 
 on the chronometer, and the G.M.T. of this passage can at once be 
 obtained from the Nautical Almanac if the longitude of the observing 
 station is known. 
 
 In the case of the sun the declination alters appreciably during the 
 time which must elapse between the two altitude observations, and hence 
 a correction, called the equation of equal altitudes, must be applied. 
 For the method of calculating this correction, reference may be made 
 to Martin's Navigation (Longmans), p. 345. When only a few observa- 
 tions have to be reduced, the simplest procedure is to treat separately 
 each of the two sets of altitude observations made before and after noon, 
 as in the case of single altitudes, and then take the mean of the two 
 chronometer corrections deduced. 
 
 The advantage of having the two sets of observations is that errors 
 in the adjustment of the sextant, and in the values assumed for the 
 refraction, latitude, and declination, are in a great measure eliminated. 
 
 When the rate only of the chronometer is desired, then if the time 
 at which the same star has any given altitude on successive nights is 
 observed, the interval observed on the chronometer compared with a 
 sidereal day (23 h. 56 m. 4'1 s.) will give the rate. When making the 
 observations, five or ten altitudes ought to be observed on each night, 
 a star being chosen as near the prime vertical as possible. 
 
 For methods of determining time involving the use of a transit 
 instrument or a theodolite, reference must be made to works on practical 
 astronomy. 
 
CHAPTER VII 
 
 SURFACE TENSION AND VISCOSITY 
 
 53. Measurement of Surface Tension by the Elevation in a Capillary 
 Tube. Take three lengths of thermometer tubing having different bores 
 ranging from 0'5 mm. to 1'5 mm., the lengths of the pieces being about 
 15 cm. The bores of these tubes must be very carefully washed, 1 first 
 with sulphuric acid, then with a strong solution of caustic soda, and 
 finally rinsed out by passing a stream of water from the tap through 
 them for some minutes. It is best to use tap water for this purpose, as 
 distilled water is very apt to have a minute trace of grease, which would 
 be quite fatal to the accuracy of the measurements. If the tap is 
 allowed to run freely for a minute or so before the tubes are placed in 
 the stream, any grease which may be on the tap will be carried away by 
 the surface of the water as it streams out. The fact that the tap water 
 may contain a small quantity of dissolved salts is of less importance, as 
 far as its surface tension is concerned, than the presence of grease or 
 dirt of a greasy nature. When waiting to be used the tubes must be 
 kept immersed in a beaker of tap water drawn off with the precautions 
 observed above, and the top of the beaker must be kept covered with a 
 clock glass. 
 
 A convenient instrument for measuring the elevation of a liquid in a 
 capillary tube is the small cathetometer microscope shown in Fig. 56. 
 When this, or a similar type of instrument, is not available, a scale 
 engraved on glass may be employed. Since it is difficult to focus the 
 microscope on the surface of the liquid contained in the beaker, 2 a needle 
 is fixed to the tube in the manner shown. A convenient form of clip 
 may be made out of a small piece of sheet brass, and an ordinary stout 
 sewing-needle is soldered to the brass, as shown on a large scale at (). 
 An india-rubber ring serves to hold the clip against the tube. The 
 needle cannot be placed close alongside the tube, since, owing to capillarity, 
 the water is raised up where the tube cuts the surface. When making 
 a measurement the needle is adjusted so that its point just touches the 
 surface of the liquid, and then the distance between the top of the 
 
 1 Instead of using old tubes, which require very careful cleaning, new 
 capillaries may be prepared by drawing out pieces of clean glass tube. The 
 ends of these capillaries must be sealed up to exclude dust till the tube is 
 required. 
 
 2 Glass vessels with plate glass sides, through which good definition can 
 be obtained, are supplied by Leybold's Nachfolger, Koln, Germany. 
 
 139 . 
 
140 
 
 SURFACE TENSION AND VISCOSITY 
 
 [53 
 
 needle and the surface of the liquid in the capillary is measured. By 
 adding the length of the needle to this distance the capillary elevation is 
 obtained. 
 
 When a glass scale is employed, the arrangement used by Quinke, 
 shown in Fig. 57, will be found convenient. The bent pointer B is 
 cemented to the glass scale A, to which the capillary tube is also attached 
 
 by two india-rubber bands. The scale is either clamped to the side, if a 
 square vessel is employed to contain the liquid, or is held in the clip of a 
 retort stand. The reading on the scale corresponding to the tip of the 
 pointer B can be obtained by means of a small set-square, and the 
 difference between this reading and that corresponding to the meniscus 
 in the capillary tube gives the capillary elevation. 
 
 To make a measurement of the surface tension of water, clean out 
 
53] 
 
 MEASUREMENT OF SURFACE TENSION 
 
 141 
 
 a small beaker or rectangular glass trough by the method used in 
 cleaning the capillaries, and fill the beaker with water from the tap. 
 Take a capillary tube from the water in which they have been stored, and 
 having attached a short length of rubber tubing 
 to one end, fix it in a clamp c (Fig. 56) so that 
 its lower end dips into the water, and make it 
 vertical by means of a plumb line. Attach a 
 needle as shown, and adjust the needle so that its 
 point just touches the surface of the water. 
 Set up the table cathetometer at such a distance 
 that the capillary tube can be focused, and by 
 means of a striding level on the microscope make 
 the axis about which the microscope turns vertical. 
 To allow of this adjustment the base of the 
 cathetometer is provided with levelling screws. 
 The microscope is first turned parallel to the 
 line joining two of the screws A and B, say, and 
 the level bubble brought to zero. The micro- 
 scope being turned through a right angle, the 
 level is brought to zero by means of the third 
 screw c. After repeating this adjustment once 
 or twice the level will remain at zero in whatever direction the microscope 
 is turned. 
 
 The needle must be placed so that it is at the same distance from the 
 vertical axis of the cathetometer as is the bore of the capillary tube, so 
 that having focused the microscope on the meniscus in the tube, and 
 turned the microscope about the vertical axis so that the needle appears 
 in the field, the image seen may be sharply defined. 
 
 Having first slightly raised the water in the tube by pinching the 
 rubber tube in two places, 1 measure the height of the water surface 2 
 above the top of the needle. Place a mark on the outside of the tube at 
 the point occupied by the surface of the raised liquid column, and then 
 raise the tube through about half a centimetre, and having again adjusted 
 the needle, measure the elevation. Make a series of measurements in 
 this way, raising the tube half a centimetre at a time till observations 
 have been made with the surface at a number of points ranged along 
 about 3 centimetres of the tube, and mark the lowest point occupied 
 by the surface. We have now to measure the radius of the capillary 
 
 1 Never raise the liquid by sucking the tube, as doing so is almost certain to 
 contaminate the surface. 
 
 2 It is best to measure to the lowest part of the concave surface. A correction 
 must be applied to allow for the weight of the liquid above the lowest point of 
 the surface. In fairly narrow tubes we may assume that the surface is a hemi- 
 sphere, so that the volume of liquid in question is the difference between the 
 volume of a cylinder of height r and radius r and a hemisphere of radius r, 
 that is, 7rr 3 -2/37rr 3 or l/3?rr 3 . Hence if we increase the measured height A 
 by an amount r/3, we shall allow for the weight of the liquid above the point to 
 which h was measured. 
 
142 
 
 SURFACE TENSION AND VISCOSITY 
 
 [53 
 
 tube. The elevation of the water depends on the radius at the point 
 occupied by the water surface, and not at all on the radius either above 
 or below this point. The measurement of the diameter at a point can 
 only be made by cutting the tube and using a microscope, and it is 
 difficult to obtain any great accuracy in this way. By determining the 
 weight of a quantity of mercury which occupies a measured length of 
 the tube, we can calculate the mean radius of the tube in the part which 
 the column occupied when we measured its length. If, then, we weigh a 
 column of mercury which occupies approximately the portion of the tube 
 between the two marks made when making the observations of elevation, 
 we can obtain the mean radius of this portion of the tube, and this mean can 
 be used with the mean of the elevations to calculate the surface tension. 
 
 Having carefully dried the tube by drawing hot air through it, care 
 being taken not to displace the marks, introduce a column of mercury of 
 such a length that it will approximately extend from one mark to the 
 other. The mercury may be drawn in by attaching a piece of india- 
 rubber tubing to the tube, and by pinching this tube in two places the 
 mercury may be drawn up. Measure with a vernier microscope the 
 length of the mercury column when it is arranged symmetrically with re- 
 spect to the marks, and then run the mercury into a weighed watch-glass 
 and weigh. 
 
 Then if D is the density of the mercury at the temperature at which 
 the length of the column was measured, I the measured length, 1 and w 
 the weight of the mercury, we have 
 
 -4 
 
 Knowing the elevation h, the radius r, and taking the density p of 
 the liquid at the temperature of the experiment from a table, we can 
 now calculate the surface tension T on the supposition that the liquid 
 wets the walls so that the angle of contact is zero by the expression 2 
 
 where g is the acceleration of gravity. 
 
 Repeat the experiments, using the other tubes, and enter your results 
 as deduced from the various tubes in a table such as that below : 
 
 
 I 
 
 Tube. 
 
 Radius 
 r. 
 
 Elevation 
 h. 
 
 Surface Tension 
 T. 
 
 
 
 
 
 1 A correction must be applied for the hemispherical ends of the mercury 
 column ; see note on previous page. 
 
 2 Watson's Physics, p. 185. 
 
54] 
 
 MEASUREMENT OF SURFACE TENSION 
 
 143 
 
 54. Measurement of the Surface Tension of a Soap Film. The 
 
 pressure inside a soap bubble of radius R exceeds the external pressure 
 by an amount p, which is given by the relation 1 
 
 li 
 
 (1) 
 
 where T is the surface tension of the soap solution. Hence if we 
 measure p and R we can calculate the surface tension. 
 
 A piece of apparatus suitable for performing this experiment is shown 
 in Fig. 58 (a). It consists of a wooden box A, the front and back being 
 
 closed by pieces of plate glass. A glass tube B passes through the side 
 of the box, and is connected by a rubber tube to a manometer. A small 
 watch-glass c is supported by a wire which passes through a cork in the 
 top of the box, and can be moved up and down by the handle D. This 
 watch-glass serves to contain the soap solution, which by raising the 
 handle D and by rotating the tube B can be brought up to the end of 
 the tube. 
 
 1 Watson's Physics, p. 183. 
 
144 SURFACE TENSION AND VISCOSITY [ 54 
 
 Since the pressure inside a bubble of sufficient size to be satis- 
 factorily measured is about equal to the pressure exerted by a column of 
 water of 2 millimetres in height, a special form of manometer will be 
 required. 
 
 A suitable manometer is shown in Fig. 58 (b). It consists of a base- 
 board A, to which a second board B is hinged at H, the other end of 
 this board being supported by a micrometer screw M, the point of 
 which bears on a small glass plate cemented to the base-board. 
 
 An upright piece of wood cc' supports a bent glass tube DBF, the two 
 portions DE and EF being inclined at equal angles to the board B. The 
 tube has a bore of about 3 millimetres, and the turned-up ends expand 
 into small bulbs. One end is attached to a three-way cock, by means of 
 which the air in the tube DF may either be put in communication with 
 the air, or through a rubber tube may be connected to the soap bubble 
 through the tube F (Fig. 58 (a)). In the bent tube DF is a thread of 
 xylene about 10 centimetres long, and a scale attached to the upright 
 c enables the horizontal distance between the ends of the thread to be 
 measured. A lens L attached to a cork, which rests on the upper surface 
 of B, serves to assist in reading the position of the meniscus. 
 
 To obtain a reading of pressure with this manometer, first place the 
 tube DF in communication with the air, and adjust the board B approxi- 
 mately horizontal. Then read and record the positions of the two ends 
 of the liquid thread, and also note the reading on the scale s and the 
 divided head of the micrometer M. 
 
 The tap G having been turned so as to put the instrument in con- 
 nection with the vessel in which the pressure is to be measured, the 
 liquid column will be moved to the left or right according as the pressure 
 to be measured is greater or less than atmospheric pressure. Suppose it 
 is greater, so that the liquid moves to the left. The screw M must be 
 turned so as to raise the end B of the board BB' till the liquid column 
 comes back to the original position. When this adjustment is complete 
 the micrometer is again read, the difference in the readings giving the 
 height H, through which the end of the board B has been raised. 
 
 If I is the distance between the ends of the column of liquid KJ, as 
 measured on the scale Q, p the density of the liquid, and h the vertical 
 height of the meniscus K above the meniscus J, the pressure being 
 measured exceeds the atmospheric pressure by an amount p, which is 
 given by 
 
 But if is the angle through which the board BB' has been tilted, 
 h l sin 6. 
 
 Further, if L is the distance between the micrometer screw and the 
 hinge H, we have 
 
 =tan 6. 
 
54] MEASUREMENT OF SURFACE TENSION 145 
 
 Now in practice is always small, and hence we may take sin 9 = tan 0. 
 
 Thus p = H -^. 
 
 L 
 
 Hence, if we know the density p of the xylene, the measured quantities 
 allow us to calculate the pressure. 
 
 To measure the pressure in a soap bubble some soap solution 1 is 
 placed in the watch-glass c (Fig. 58 (a)), and the tube B is turned so that 
 the open end is downwards. Then by raising D the end of B is made to 
 dip in the solution. The tap G (Fig. 58 (b)) being turned so as to cut off 
 communication with the manometer, a bubble is formed by blowing into 
 the tube E. The glass tube B is then turned into the position shown 
 in the figure, and the bubble put in communication with the manometer, 
 and the pressure is measured as described above. 
 
 To measure the diameter of the bubble, a convex lens having a focal 
 length of about 30 cm. is arranged in a stand, so as to form an image 
 of the bubble on a scale divided on cardboard. A gas flame placed 
 behind the box containing the bubble serves to illuminate the bubble. 
 The distance u from the bubble to the lens, and that, v, from the lens to 
 the scale, must be measured. Then if is the diameter of the image, 
 the radius of the bubble is given by 
 
 The measurement ought to be repeated, using bubbles of varying size, 
 the value of the surface tension being calculated in each case by means 
 of the equation 
 
 55. Measurement of the Viscosity of a Liquid by its Rate of Flow 
 through a Capillary Tube. When a liquid flows through a tube, at any 
 rate so long as the liquid wets the wall of the tube, the layer of liquid in 
 immediate contact with the wall of the tube remains at rest. The speed 
 with which the liquid moves increases as we go from the surface of the 
 tube up to a maximum along the axis of the tube. Hence if we imagine 
 the liquid to consist of a number of hollow cylinders coaxial with the 
 
 1 The soap solution can be prepared as follows : Fill a clean bottle (about a 
 quart) three-quarters full of distilled water, and add one-fortieth of the weight 
 of the water of pure fresh oleate of soda. After the oleate has dissolved fill up 
 the bottle with Price's glycerine, and well shake. Leave the bottle to stand for 
 a week in a dark place, and then by means of a syphon draw off the clear liquid 
 and add one or two drops of strong ammonia to every pint of the liquid. This 
 solution must be preserved in a well-stoppered bottle, preferably in a dark place, 
 and will be found to improve with keeping. When making the solution do not 
 warm the liquid or filter it. Exposure to the air for any length of time will 
 injuriously affect the solution. 
 
 K 
 
146 
 
 SURFACE TENSION AND VISCOSITY 
 
 [55 
 
 tube, the fluid within each of these cylindrical shells will be moving with 
 the same speed, but will be moving more slowly than that in the shell 
 immediately inside, and more quickly than that in the shell immediately 
 outside. This relative motion of adjacent layers of the liquid is resisted 
 by what is called the viscosity of the liquid. The viscosity of a liquid is 
 measured by the tangential force exerted on unit area of either of two 
 planes which are parallel to the direction of motion of the fluid and at 
 unit distance apart, one plane moving in its own plane with a velocity 
 greater than the other by unity, the velocity of the liquid being supposed 
 to change at a uniform rate as we pass from one plane to the other. 
 
 As long as the velocity with which the liquid is forced through the 
 tube does not exceed a certain value, depending on the bore of the tube 
 and the viscosity of the liquid, there is the following relation between 
 the coefficient of viscosity r), the radius of the tube R, its length I, the 
 difference of pressure p acting on the fluid at the two ends of the tube, 
 and the volume V of fluid which passes through in the time t * 
 
 Thus by measuring the volume of liquid which flows through a tube 
 
 of known dimensions under 
 a measured head, we can cal- 
 culate the coefficient of vis- 
 cosity. 
 
 An arrangement suitable 
 for measuring the coefficient 
 of viscosity of a liquid such 
 as water is shown in Fig. 
 59. The capillary tube E is 
 attached by a short piece of 
 thick-walled rubber tube to 
 a glass bulb A, capable of 
 holding about 100 c.c. Two 
 circles are etched, one at B 
 above the bulb and the other 
 at c below, and the volume 
 between these marks is deter- 
 FIG. 59. mined by filling the bulb up 
 
 to the upper mark with water, 
 
 and then running off 7 the water down to the lower mark into a weighed 
 
 beaker and again weighing. 
 
 A tube about 30 cm. long and with a bore of about 2 mm. must be 
 
 selected, and the uniformity of its bore tested by introducing a column 
 
 of mercury about 5 cm. long, and measuring the length occupied by the 
 
 1 See p. 148. 
 
55] MEASUREMENT OF VISCOSITY OF A LIQUID 147 
 
 mercury in different parts of the tube. If the lengths of the column 
 differ by more than about '5 per cent., the bore is too irregular, and 
 another tube must be tried. A suitable tube having been selected, a 
 column of mercury must be introduced, extending as nearly as possible 
 the whole length of the tube, and the length of this column and the 
 weight of the mercury must be measured in the manner described in 
 53. And as described in that section, the length having been 
 corrected for the curvature of the ends of the column, the mean radius 
 must be calculated. The total length of the tube having been measured, 
 it must be fitted up as shown in Fig. 59, being held firmly in position 
 by a clamp. The bulb A having been filled with water, the pinch-cock n, 
 serving to close the lower tube, is then connected to the upper end 
 of the capillary, and a beaker is placed to receive the water as it 
 flows out. 
 
 Having opened the pinch-cock to its full extent, the times are noted 
 when the surface of the liquid passes the marks B and c, the difference 
 giving the interval taken by a volume V of the liquid to flow through' 
 the tube. 
 
 In order to obtain the mean difference of pressure between the ends 
 of the tube, the heights of the two marks B and c and of the free end 
 of the tube above the surface of the table must be measured. Then the 
 mean of the heights of B and c, less the height of the free end, will give the 
 mean head. This multiplied by the density of the fluid and by g will be 
 the difference of pressure. 
 
 The experiment must then be repeated, but the bulb A must be 
 placed with the mark c uppermost, and the mean of the two times and 
 pressures used to calculate the coefficient of viscosity. The reason for 
 this second experiment is that the bulb A may not be quite symmetrical, 
 and hence, say, more than half the liquid may flow out at a pressure 
 higher than the mean if the bulb is pear-shaped with the larger end 
 uppermost. 
 
 Since viscosity changes rapidly with temperature, the liquid should 
 have stood for some time in the room before the experiment so as to 
 acquire the room temperature, and by means of a thermometer placed 
 in the beaker which catches the liquid as it leaves the tube, the temperature 
 should be read and recorded. 
 
 In the case of tubes where the velocity of efflux is considerable, a 
 correction will have to be made to allow for the fact that part of the 
 head of liquid is employed, not in overcoming the viscosity of the fluid, 
 but in imparting to the fluid the kinetic energy with which it escapes 
 from the end of the tube. The amount by which the head is reduced on 
 
 p 
 this account is ' -, where p is the density of the liquid, so that the 
 
 TT-'t~'Jl 
 
 expression for the coefficient of viscosity becomes 
 
 
 irptW pV 
 
 ~8VC ~ 
 
148 SURFACE TENSION AND VISCOSITY [ 55 
 
 In general, if long narrow tubes are used, the second term on the right 
 is negligible. 
 
 The expression for the quantity of liquid which flows through a tube 
 can be obtained in the following manner : Let the speed with which the 
 liquid is moving parallel to the axis at a distance r from the axis be v. 
 Then the stress due to viscosity parallel to the direction of motion acting 
 
 on unit area is given by ^ . Hence if we consider unit length of the 
 dr 
 
 tube, the stress over the cylindrical surface of radius r is 
 
 dv 
 
 . 
 dr 
 
 The stress over a cylindrical surface of radius r + dr will be 
 
 1 ,. 
 d(r + dr) 
 
 Now, if we write /(?) for r-=-, we have by Taylor's theorem 
 
 or, neglecting terms in (dr)* 2 , we have 
 
 / j \ dv dv d / dv\ , 
 
 (r + dr) -- = r + ( r-- )dr + . 
 'd(r-\-dr) dr dr\ drj 
 
 Hence the difference in the tangential forces acting on the two surfaces 
 of the cylindrical shell of thickness dr is 
 
 ~ d 
 
 Now, if p is the difference in pressure between the two ends of the tube of 
 length I, the difference in pressure between the ends of the shell will be 
 p/l, which we will call a. Hence, since the area of the end of the 
 annulus is ^Trrdr, the resultant force acting on the annulus, owing to the 
 difference of the pressures, is 
 
 Zirradr ......... (2) 
 
 As the motion of the liquid is supposed to be steady, this force must 
 be equal and opposite to the resistance due to viscosity, or 
 
 d ( dv 
 
 ) ? ) dr = ZTrradr. 
 dr\ dr 
 
 Hence integrating 
 where G is a constant. 
 
56] LOGARITHMIC DECREMENT 149 
 
 Dividing by r and again integrating, we get 
 
 r 2 a 
 -r)v = -^- + G log r + Cf, 
 
 where G' is a second constant. We have now to evaluate the values of 
 the constants G and G'. In the first place, we see that since log = oo , 
 if the constant G is not zero, then v must be infinite when r = 0, i.e. 
 along the axis of the tube. This of course is impossible, and hence C 
 is zero. To obtain G' we note that experiment has shown that at the 
 surface of the tube v = Q. Hence if R is the radius of the tube we get 
 
 and hence 
 
 ,-%&-*) . . , . ", . . . (3) 
 
 The volume of liquid which in a time t passes through the annulus is 
 
 2irrdr . vt. 
 The total volume which passes across any section of the tube is therefore 
 
 fM 
 
 V = %7rt I vrdr 
 Jo 
 
 or 
 
 If we have to allow for the kinetic energy acquired by the liquid, then 
 it can be shown l that 
 
 pV 
 
 where p is the density of the liquid. 
 
 56. Measurement of the Logarithmic Decrement of a Vibrating 
 Body. On account of the resistance of the air and the viscosity of the 
 suspension fibre, the amplitude of the vibrations of a torsion pendulum 
 gradually decreases. In such a case it can be shown that if the resist- 
 ance experienced is proportional to the velocity with which the vibrating 
 body is moving, then the ratio of the amplitude of any vibration to the 
 
 1 See Couette, Ann. de Chim. Phys. (1890), [6] xxi. 433. 
 
150 SURFACE TENSION AND VISCOSITY [ 56 
 
 amplitude of the succeeding vibration is constant. Now, experiment has 
 shown that in most cases of vibrating bodies this equality of the ratios 
 of the amplitudes of successive vibrations holds, so that to a near 
 approximation we may suppose that the resistance experienced by the 
 vibrating body is proportional to the velocity. 
 
 Suppose that the vibrations of a torsion pendulum are being watched 
 by observing the image of a scale seen reflected in a mirror attached to 
 the pendulum, and that the readings on the scale corresponding to the 
 cross-wire of the telescope at the extremities of the swings are x 11 # 2 , # 3 , 
 a: 4 , &c., where x v x s , x 6 , &c., are the swings to the right, and a? 2 , je 4 , x & 
 &c., those to the left. If, further, the zero of the scale is at one end so 
 that all the readings are positive and the reading corresponding to the 
 position of rest is , the amplitudes of the successive swings are 
 
 Calling these amplitudes a v a 2 , 3 , a 4 , &c., we have 
 
 CL-, rtn o 
 
 = = = = a constant = p. say. 
 a 2 3 4 
 
 Taking logarithms, we get 
 
 log e ^ - log, 2 = log e 2 - log e a g = log e a g - log c a 4 
 = a constant = A., say 
 
 The quantity X, which is the difference between the logarithms of succes- 
 sive amplitudes, or the logarithm of the constant ratio of successive 
 amplitudes, is called the logarithmic decrement. The logarithms con- 
 sidered above are natural or Napierian logarithms. Hence if I is the 
 difference of the logarithms to the base ten of the successive amplitudes 
 
 A = 2-3026 I. 
 
 In order to obtain the quantities a 1; a. 2 , 3 , &c., it is necessary to 
 know the scale reading corresponding to the position of rest. As in 
 practice this is a decided inconvenience, the following change is advisable. 
 Since if 
 
 ^l = ^_% = 4 == 
 
 2 3 4 a b 
 Then aM 
 
 _ 
 
 2 33 4 a 4 + a 5 ~ 
 Hence 
 
 But 
 
 -x 2 = x l - a 2 = A V say, 
 -XQ = x 3 -x 2 = A v say, 
 
56] LOGARITHMIC DECREMENT 151 
 
 Now x l - x. 2 is the distance between the first extreme right-hand elonga- 
 tion and the next extreme left-hand elongation, and so on, and the 
 determinations of these quantities does not involve the necessity of 
 knowing the position of rest. 
 
 Hence for practical purposes A is derived from 
 
 Hence if we were to observe n readings of extremities of swing 1 we 
 should have, adding the above equations together, 
 
 or 
 
 J. 
 
 where as before the logarithms are natural logarithms. 
 
 If, however, we employ this formula the accuracy of the result will 
 depend on the accuracy with which we observe the first and last 
 elongations. Thus if the vibrations are fairly rapid, so that it is difficult 
 to make accurate readings of the scale at the extremities of the swing, 
 considerable error may result. Suppose, however, we start by observing 
 an even number (2r) extremities of swing and obtain the amplitudes 
 
 -"It ^3' -^5 "'jr-li 
 
 then after a time, when the amplitude has fallen to about a third of 
 its initial value, again observe 2r extremities of swing and obtain the 
 amplitudes 
 
 " -"n+2> -"-n+4: -"-n+2r-l* 
 
 Now 
 
 ~J = 3 _ _ 
 
 A A ' ' * 
 
 -^n+2 - 
 
 Hence 
 
 "* " + -"-H.+9 "I" -"- 
 
 Thus 
 
 and we make use of a number of scale readings of extremities of swing at 
 either end of the series of observations. 
 
 If the vibrations die down fairly rapidly, so that only a comparatively 
 small number of elongations can be observed, then it will be advisable to 
 note all the elongations till the amplitude has died down to about a 
 quarter of its initial value. Then divide the observations into two 
 groups and treat them by the method indicated by (3) above ; n being in 
 this case equal to 2r, and one group following immediately after the other. 
 
 In cases where the logarithmic decrement is very small, so that the 
 
 1 That is, n half-vibrations. 
 
152 
 
 SURFACE TENSION AND VISCOSITY 
 
 [56 
 
 amplitude only decreases very slowly, the above method of observation 
 would be very tedious, since in order to get a sufficiently great decrease in 
 elongation to give an accurate value of A would involve making a very 
 large number of readings. In such a case it is best to determine the 
 period T of the vibrations, then observe, say, ten or twenty turning points 
 so as to allow of the calculation of the elongations A lt A B , A 5t &c. Then 
 start a stop-watch, or note the time on a chronometer. After the 
 vibrations have died down to about a third of their initial value 1 stop 
 the watch or note the time, and then again observe ten or twenty turning 
 points. Now from a knowledge of the period and the interval which has 
 elapsed we can at once calculate how many vibrations took place between 
 the two sets of observations of turning points. As it will in general be 
 found inconvenient to start the watch or read the time at the same 
 instant as make the last observation of turning point, it is better to count 
 one or two vibrations after the last reading before starting the watch or 
 taking the time. In the same way, at the end count one or two 
 vibrations after stopping the watch before beginning to read the turning 
 points. Of course the number of swings thus counted will have to be 
 added to the number deduced from the time as indicated by the watch. 
 The method of making and recording the observations will be clear from 
 the following example : 
 
 DETERMINATION OF LOGARITHMIC DECREMENT. 
 
 Turning Points. 
 
 
 Turning Points. 
 
 
 
 Elongation. 
 
 
 Elongation. 
 
 Left. 
 
 Right. 
 
 
 Left. 
 
 Eight. 
 
 
 9-50 
 
 43-80 
 
 34-30 A l 
 
 13-90 (6) 
 
 39-30 
 
 25-40 A m 
 
 9-55 
 
 43-70 
 
 34-15 A 3 
 
 14-00 
 
 39-25 
 
 25-25 A U5 
 
 9-60 
 
 43-65 
 
 34-05 A 5 
 
 14-00 
 
 39-20 
 
 25-20 A U7 
 
 9'70 
 
 43-55 
 
 33-85 A 7 
 
 14-10 
 
 39-15 
 
 25-05 &c. 
 
 975 
 
 43-50 
 
 33-75 A 9 
 
 14-15 
 
 39-10 
 
 24-95 
 
 9-80 
 
 43-40 
 
 33-60 A u 
 
 14-20 
 
 39-05 
 
 24-85 
 
 9-90 
 
 43-35 
 
 33-45 A 13 
 
 14-25 
 
 39-00 
 
 24-75 
 
 10-00 
 
 43-30 
 
 33-30 A 16 
 
 14-30 
 
 38-95 
 
 24-65 
 
 10-05 
 
 43-20 
 
 33-25 A# 
 
 14-35 
 
 38-90 
 
 24-55 
 
 10-10 
 
 43-15 (a) 
 
 33-05 A 19 
 
 14-40 
 
 38-85 
 
 24-45 
 
 Sum . . 336-75 
 
 Sum . . 249-10 
 
 Approximate period = 6 '7 6 seconds. 
 Interval by watch 6 minutes 46 seconds. 
 
 Number of whole vibrations = = = 60. 
 
 6"7o 
 
 1 If the logarithmic decrement is excessively small, then it will generally be 
 unnecessary to wait till the amplitude has decreased so much as this. 
 
57] MEASUREMENT OF VISCOSITY OF A LIQUID 153 
 
 There was one half-vibration between the reading marked (a) and 
 starting the watch, and two half-vibrations after stopping the watch 
 before the reading marked (6). 
 Hence 
 
 71 = 20+120 + 3 = 143. 
 Thus 
 
 = log 10 336-75 -Iog 10 249*10 
 = 2-52731-2-39637 
 = 13094 
 
 and /= = 0-000922 
 
 A = 0-000922 x 2-3026 = 0-002123. 
 
 57. Measurement of the Viscosity of a Liquid by the Oscillating 
 Disc Method. It has been shown by O. E. Meyer 1 that when a thin 
 disc oscillates in its own plane, in a liquid of which the coefficient of 
 viscosity is 77, under the influence of the torsion of the suspending wire, 
 then 
 
 where the symbols have the following meanings : 
 
 M = moment of inertia of the disc and its supports. 
 
 R and 8 = the radius and thickness of the disc. 
 
 T=the time of vibration of the disc when vibrating in air. 
 
 A and X x = the logarithmic decrements of small oscillations of 
 
 the disc in air and in the liquid respectively. 
 p = the density of the liquid. 
 
 The theoretical deduction of the above formula is not above criticism ; 
 it may, however, be regarded as an empirical formula which gives results 
 agreeing very fairly with those obtained by other methods. 
 
 A piece of apparatus suitable for performing this experiment is shown 
 in Fig. 60. It consists of a wooden box, the sides of which can be closed 
 by sheets of glass sliding in grooves, and having an upright brass tube 
 with a V notch at the top. The disc A hangs from a rod B, which carries 
 a mirror M. This rod is soldered to the bottom of a fine steel wire, the 
 top end of the wire being soldered to the rod c, which is clamped to a 
 cross-bar D. This cross-bar rests in the notch at the top of the tube. A 
 window E allows the vibrations of the disc to be watched, a telescope and 
 scale being used in the ordinary manner (see 168). The liquid to be 
 experimented upon is contained in a glass dish F, which must be of such 
 a size as to allow at least 3 cm. clear all round the disc. 
 
 1 Poygendorf Annalen, 113, p. 55. 
 
154 
 
 SURFACE TENSION AND VISCOSITY 
 
 [57 
 
 To perform the experiment, first set the disc vibrating, and having 
 damped out any pendulum vibrations by lightly touching the bottom of 
 
 the suspending fibre, deter- 
 mine the period of the disc 
 in air by the method given 
 in 43. Next determine the 
 period when the disc is loaded 
 with the brass ring G, the 
 outside diameter of which is 
 equal to the diameter of the 
 disc. Having determined the 
 moment of inertia of the ring 
 by calculation from its dimen- 
 sions, calculate the moment 
 of inertia of the disc and the 
 supporting rod and mirror by 
 the method used in 43. 
 
 After removing the ring, 
 determine the logarithmic 
 decrement of the disc by the 
 method described in the last 
 section. 
 
 Having filled the vessel F 
 with water, stir it well, then 
 start the disc vibrating, and 
 FIG. 60. rea( i the temperature as 
 
 shown by the thermometer T. 
 
 Having determined the logarithmic decrement, again read the tempera- 
 ture. The mean of the two temperature readings may be taken as the 
 temperature for which the viscosity has been determined. Since the 
 viscosity changes very rapidly with temperature, it is important either 
 to work at the room temperature, or, if observing at other temperatures, 
 to protect the vessel from loss or gain of heat as much as possible by 
 surrounding it with a layer of felt or other bad conductor of heat. 
 
CHAPTER VIII 
 
 THE BAROMETER 
 
 58. Measurement of the Atmospheric Pressure. In practice the 
 value of the atmospheric pressure is always obtained by reading the 
 height of the barometer, that is, by measuring the height of the column 
 of mercury which will exert a pressure equal to that of the atmosphere. 
 The form of barometer generally employed is that known as Fortin's 
 cistern barometer (see Watson's Physics, 133). 
 
 The chief difficulty with the Fortin form of barometer is setting the 
 surface of the mercury to coincide with the tip of the ivory point which 
 marks the zero of the scale. When the barometer is quite new, and 
 therefore the surface of the mercury in the cistern is perfectly clean, this 
 adjustment can be made with considerable accuracy, particularly if a lens 
 having a focal length of about 5 cm. is used. Griffiths recommends that 
 the sides of the cistern be covered with black paper, small windows being 
 left in front and behind the point. A fine black wire is stretched across 
 the back window, and the position of the reading lens is so adjusted that 
 an image of the line is seen formed by rays reflected in the mercury near 
 the point. Directly the point touches the mercury, the image will be- 
 come distorted. The mercury must always be brought up to the point 
 when making a setting whatever method is employed, since if the sur- 
 face is lowered after having been raised too high it will be found that the 
 mercury is apt to cling to the point, being raised up above the general 
 level. 
 
 After some time it will be found that the surface of the mercury 
 becomes more or less dirty, and then the difficulty of making a setting 
 becomes very great. The condition of the surface can sometimes be 
 temporarily improved by lowering the mercury in the cistern as much 
 as possible and then bringing it back to its proper level. During this 
 process some of the scum on the surface of the mercury is left clinging 
 to the walls of the reservoir, and a fresh and clean surface is produced 
 near the point. 
 
 By replacing the ivory point by one made of platinum iridium and 
 having an insulated platinum wire dipping in the mercury of the cistern, 
 an electrical method of noting when the point touches the mercury can 
 be employed. The case of the barometer and the insulated wire are 
 joined, through a fairly high resistance and a telephone, to a Leclanche 
 cell, when a distinct click will be heard in the telephone when the 
 mercury touches the point. 
 
 155 
 
156 
 
 THE BAEOMETER 
 
 [58 
 
 A form of Fortin barometer which can be made at a small fraction of 
 the cost of the ordinary pattern is shown in Fig. 61. The tube A has a 
 bore of about a centimetre at the top and bottom, and is bent at the 
 bottom as shown, a fine point of black glass 
 being attached at the bend, the extremity of the 
 point lying on the prolongation of the axis of 
 the upper part of the tube. A steel collar c is 
 cemented to the tube just above the bend. A 
 fine screw is cut on this collar, arid a steel nut 
 which engages with this screw is cemented inside 
 a glass thimble E. This thimble has a small 
 hole F in the side, which may be kept lightly 
 plugged with cotton wool in order to exclude 
 dust as much as possible. The surface of the 
 mercury can be brought into coincidence with 
 the point B by turning the thimble. The upper 
 part of the tube is graduated in millimetres, the 
 divisions being numbered as if the scale started 
 at the end of the point B. If it is only required 
 to obtain the height of the barometer to within 
 one- or two-tenths of a millimetre, then the 
 scale may be read by eye. When, however, it is 
 desired to obtain closer readings, the small slider 
 shown at G can be used. There is a spring H 
 in the slider which keeps the v's on one side 
 pressed against the tube. Two openings i, one in 
 front and the other behind, serve to view the 
 mercury column, the upper edges being brought 
 into line with the top of the mercury meniscus. 
 A vernier v engraved on the slider is used to 
 read the scale. This vernier has ten divisions, 
 equal to eleven divisions of the scale, and 
 is numbered downwards (see p. 43). The 
 barometer is mounted on a board as shown, 
 and a thermometer T is attached to the board, 
 the bulb being enclosed in a piece of the same 
 
 tube used to form the barometer. This piece of tube is filled with 
 mercury, so that the thermometer bulb may, as far as possible, be 
 subjected to the same conditions as it would if enclosed within the mer- 
 cury column. 
 
 To take a reading of the height of a Fortin barometer we must 
 proceed in the following manner : 
 
 1. Read the temperature of the barometer as given by the attached 
 thermometer. The temperature as read will generally require correcting 
 for the error of the thermometer. 
 
 The reason for reading the thermometer before the barometer is, that 
 the presence of the observer's body will most likely cause the reading of 
 
 FIG. 61, 
 
58] MEASUREMENT OF ATMOSPHERIC PRESSURE 157 
 
 the thermometer to go up, though the heat will not have time to reach 
 the column of mercury, and so affect the height, before the reading 
 is made. 
 
 2. Having gently tapped the tube of the barometer near the upper 
 surface of the column to' 'prevent sticking of the mercury to the sides of 
 the tube, adjust the level of the mercury in the cistern so that the 
 point, which corresponds to the zero of the scale, just touches the sur- 
 face. The surface is raised till the point and its image, seen by reflection 
 in the mercury surface, just appear to touch, and yet there is no depres- 
 sion of the mercury surface as indicated by the irregularity in the 
 reflected image of some straight object, such as a wire, the reflection 
 taking place alongside the point. 
 
 3. Adjust the vernier so that the bottom appears to be a tangent to 
 the convex surface of the mercury. To make this adjustment satis- 
 factorily a piece of white paper or milk glass placed behind the tube 
 must be brightly illuminated. In order to avoid parallax in making 
 the setting, the movable tube which carries the vernier is trimmed so 
 that the back edge and the front edge, which corresponds to the zero of 
 the vernier, are in the same horizontal plane. Thus if when making 
 the adjustment the eye is placed so that the front and back edges 
 coincide and further appear to touch the top of the mercury meniscus, 
 then the vernier will be set correctly. When the adjustment is correct 
 it should just be possible to see some light at the edges of the tube, but 
 on slightly moving the eye up and down no light should appear at the 
 middle. The vernier attached to the slider must then be read. 
 
 To the reading obtained as described above a number of correc- 
 tions will have to be applied, and we now proceed to consider these in 
 detail. 
 
 1. Correction for Temperature. Since the pressure exerted by a 
 column of liquid of given height depends on the density of the liquid, 
 and that the density of mercury varies with the temperature, a cor- 
 rection will have to be applied on this account. Further, a correction 
 will have to be applied for the expansion with rise of temperature of the 
 scale which has been used to measure the height of the column, for it 
 is only at one temperature that the distance between the divisions of 
 the scale are what they profess to be. Hence we have to reduce the 
 observed height h t to what it would be if the temperature of the mercury 
 had some standard value, for which purpose C. is always taken, and 
 further correct for the expansion of the scale. If a is the coefficient of 
 linear expansion of the material of which the scale is composed, and it 
 is correct at 0, then the actual length at a temperature t corresponding 
 to the observed length 7i t is h t (l + at). Since the height of a column of 
 liquid which will exert a given pressure varies inversely as the density 
 of the liquid. If p t is the density of mercury at # and p the density at 
 0, we have 
 
 Height of mercury column at _ p ( 
 Height of mercury column at t p 
 
158 THE BAROMETER [58 
 
 But if v and v t are the volumes of unit mass of mercury at and t 
 respectively, and 8 is the coefficient of cubical expansion of mercury, 
 then 
 
 Pt _ v _ v 1 
 
 ~P~fy~ v(l + 8t) ~ T+8i' 
 
 Hence the corrected height H of the mercury column is given by 
 
 . I . '; .*-;-& ......... I 
 
 which may be written 
 
 l+St-St + at, 
 l+St "' 
 
 that is, the correction to be applied to the observed height h t is 
 
 3- a , 
 
 In most cases it will be sufficiently accurate l to neglect 67 compared to 
 unity, when the correction takes the simplified form 
 
 - (8 - a)t . *,. 
 
 If the scale instead of being correct at a temperature of is correct 
 at a temperature tf , then the true length corresponding to an observed 
 length h t at a temperature t is h t {l 4- a(t - )}. Hence the reduced height 
 of the barometer in this case is given by 
 
 or approximately, if we neglect 8t compared to unity, 
 
 H={l~(S-a)t-at } ....... (5) 
 
 The quantity 8 is the same for all mercury barometers, and is equal to 
 0*0001819 when the temperature is measured in degrees Centigrade. 
 The value of a, however, depends on the material which has been used 
 for the scale. If the scale is of brass, then we may take a as being 
 
 1 In the case of a brass scale, if h t is 760 mm. and the temperature is 30 C. 
 we have -- a th t = 3'707 mm. and (5-a)tf = 3'728, thus the difference amounts 
 
 1 + St 
 
 to 0'021 mm. Hence if it is required to obtain the height correct to a hundredth 
 of a millimetre, the more accurate expression must be employed. 
 
58] MEASUREMENT OF ATMOSPHERIC PRESSURE 159 
 
 0-0000184. Thus for a brass scale 8 - a is 0-0001635. If the scale is 
 of glass (verre dur) a = 0-0000087 and (8 - a) = '0001732. It is convenient 
 to calculate a table which shall, for different temperatures in the neigh- 
 bourhood of ordinary room temperatures and for different values of 
 h t between about 65 and 79 cm., give the value of the correction to 
 reduce the height of the barometer to C. ; that is, we tabulate for 
 
 different values of t and of h t the values of h t \ ~ a }*. See Table 10. 
 
 1 + ot 
 
 A very handy graphical method of obtaining the barometer correction, 
 which is capable of giving results correct to about '01 mm., has been 
 described by Mehmke. 1 A diagram drawn by his method is given in the 
 Appendix. It ought to be cut out and mounted on card, since it requires 
 to be used quite flat. A straight line from the observed height on the 
 scales to the left or right, according as the barometer has a brass or a 
 glass scale, to the temperature on one of the two middle scales will 
 intersect the corresponding inclined scale at the required correction, 
 which, for all temperatures shown, has to be subtracted from the observed 
 height. The most convenient way of reading off the correction is by 
 means of a straight line ruled on a sheet of glass or of celluloid. 
 
 In the case of some English barometers the scale is divided into 
 inches, and the thermometer reads in degrees Fahrenheit. The scale in 
 such a case is correct, not at the temperature of melting ice (32 F.), but 
 at 62 F. The reduction formula in this case becomes 
 
 H= h t {l - (8' - a')(t F - 32) - a'(62 - 32)} 
 
 = h t {l-(S'-a')t F + 328' -62a'} (6) 
 
 Here of and 8' are the coefficients of linear expansion of the scale and 
 of cubical expansion of mercury respectively in the Fahrenheit scale. 
 For brass a' = 0-00001022, for glass a =0*00000483, and for mercury 
 8' = 0-0001010. 
 
 Hence, substituting these values for a' and 8' in (6) we get 
 
 Brass scale . . . #= ^{1 -002598 - 0'0000908^}. 
 Glass scale . . . J7= ^{1-002933 -0'0000962y. 
 
 It will be observed that at a temperature of 2 8 -6 F. the expression 
 within the brackets is unity in the case of a brass scale. Hence at this 
 temperature the barometer reading is correct. In the case of the glass 
 scale, the temperature at which the barometer reading is correct is 
 30 -5 F. 
 
 2. Correction for Capillarity and Index Error. Owing to the effect 
 of capillarity at the surface of the mercury in the tube the column is 
 shorter than it ought to be. The smaller the bore, the greater is this 
 capillary depression. For any given tube it also depends to a certain 
 extent on whether the mercury at the time of observation is rising or 
 
 1 Wiedemann Annalen (1890), xli. p. 892. 
 
160 THE BAROMETER [ 58 
 
 falling, that is, on the height of the meniscus. Tables have been pre- 
 pared showing the amount of the correction for tubes of different bores 
 with different heights of meniscus. See Table 11. 
 
 Since the value of the capillary depression depends very much on the 
 cleanliness of the mercury and the state of the walls of the tube, it is 
 better, when the height of the barometer is to be measured accurately, to 
 employ a barometer with so wide a tube that the correction is negligible. 
 Such is the case when the diameter of the bore of the tube is greater 
 than 2 '5 cm. In the case of a barometer having a narrower bore, the 
 best way of obtaining the capillary depression is to compare its reading 
 with a standard barometer having a wide bore. 
 
 The index correction is to allow for the point of the index not being 
 exactly at the zero of the scale. This also is best determined by com- 
 paring the reading of the barometer with that of a standard. ,. In this 
 way a single correction is obtained which includes both the capillary 
 correction and the correction for index error. Such a comparison is 
 generally performed at some observatory, and in England accurate 
 barometers are sent to the Kew Observatory, which issues a certificate 
 showing the correction. 
 
 3. Reduction to Sea-level and Latitude 45. By the application of 
 the corrections considered above we obtain the height of the mercury 
 column which at C. would exert a pressure at the place of observation 
 equal to the atmospheric pressure. Since, however, the value of gravity 
 varies from place to place, the same atmospheric pressure would not be 
 represented by columns of mercury of the same height at every station, 
 but only so long as the value of gravity is the same at all stations. 
 Hence to obtain a number which shall represent the pressure and shall 
 not involve in any way any local peculiarity of the station at which 
 the mercury column was measured, we have either to allow for the actual 
 value of gravity at the station and reduce the pressure to dynes per 
 square centimetre, or calculate what would be the height of the mercury 
 column suppose gravity had some particular fixed value. Since the 
 value of gravity varies with the latitude of the place of observation, and 
 also with the height of the station above the sea-level, the reduction 
 consists of two parts to allow for these two effects. The standard 
 value of gravity which is adopted is that at latitude 45 and at the sea- 
 level. 
 
 According to Broch, 1 the value of gravity at a station of which the 
 latitude is < and the height above the sea-level is I metres is given by 
 
 - 0-00259 cos 2<)(1 - 0'000000196 1} 
 
 - 0-00259 cos 2< - 0'000000196 /) very nearly. 
 
 According to Helmert, the value of <7 45 is 980 '5966 cm. /sec 2 . 
 
 Since the height of the mercury column which will exert a given 
 
 1 Traveau et Mtmoires du Bureau International des Poids et Mesures (1881), 
 vol. i. p. 9. 
 
58] MEASUREMENT OF ATMOSPHERIC PRESSURE 161 
 
 pressure is inversely as the value of gravity, if H is the height reduced to 
 C., and H is the height corresponding to the standard value of gravity, 
 we have 
 
 H^ = H{1- 0-00259 cos 2< - 0-000000196 1\. 
 
 The labour involved in the application of this correction can be consider- 
 ably reduced by tabulating the values of the term '00259 cos 2< for 
 different values of <, and of the term 0-000000196 I for different values of 
 I, in each case for different heights of the barometer. Such tables will 
 be found in the Smithsonian Physical Tables, and in Landolt and 
 Bernstein's Tables. Since, however, for any given station both </> and I 
 are constant, it is quite easy for that place to make a table of the correc- 
 tions to be applied to different heights of the barometer, or a curve may 
 be drawn giving the correction for any height. 
 
CHAPTER IX 
 
 THERMOMETRY 
 
 59. Calibration of the Tube of a Mercury Thermometer. A degree 
 Centigrade on the mercury-in-glass thermometer is defined as such that 
 when the temperature of a thermometer which is graduated on this scale 
 is raised through 1 degree, the apparent increase in the volume of the 
 mercury is 1 /100th of the apparent decrease which takes place when the 
 thermometer is cooled from the temperature of the steam given off by water 
 boiling under standard conditions to the temperature of melting ice. If 
 the tube of the thermometer is perfectly cylindrical, and the space 
 between the marks corresponding to and 100 is divided into a 
 hundred equal parts, then the volume of the bore between each of these 
 divisions and the next will be exactly a hundredth of the volume between 
 the and the 100 marks. With a thermometer having such a perfectly 
 cylindrical tube, the temperature could be obtained directly by reading 
 the position of the end of the mercury column. In practice, however, it 
 is found that the bores of the glass tubes of which thermometers are 
 made are never perfect cylinders ; in other words, the bore of all tubes is 
 found to vary from point to point. Thus if the space between the 
 and the 100 marks is divided into a hundred equal parts, it will not 
 follow that when the end of the mercury column advances through one 
 of these divisions that the temperature of the thermometer has been 
 raised through 1 degree, for if the bore between the two divisions 
 considered happens to be wider than the mean bore between the fixed 
 marks, then the volume of the part of the bore intercepted between these 
 divisions will be greater than l/100th of the volume of the bore between 
 the fixed marks. Thus the thermometer will have been heated through 
 more than 1 degree, and to obtain the true temperature from the 
 reading given by such a thermometer we shall have to apply a correction, 
 the magnitude of this correction being different at different parts of the 
 scale. The process of determining the values of the corrections to allow 
 for this irregularity of the bore of the tube is called calibrating the tube. 
 In the case of all thermometers which are required for accurate work the 
 space between the fixed marks is divided into divisions of equal length, 
 and then the corrections which have to be applied are determined. In 
 the case of thermometers which are only required for rough measure- 
 ments, the maker sometimes roughly examines the bore of the tube, and 
 then instead of dividing the stem into divisions of equal length he 
 attempts to so regulate the spacing of the divisions that the volume of 
 
 162 
 
59] CALIBRATION OF TUBE OF A THERMOMETER 163 
 
 the bore included between any two adjacent divisions should be the 
 same. This adjustment of the scale is in general a very rough affair, 
 so that there are residual errors which have to be allowed for which 
 necessitates the preparation of a table of corrections. Further, although 
 the changes in the diameter of the bore are gradual, the maker, instead 
 of adjusting the length of the divisions in the same gradual manner, a 
 process which cannot be easily performed with a dividing engine, is 
 satisfied with dividing the scale into a number of portions, throughout 
 each of which he makes the divisions of equal length, the change from 
 divisions of one length to the next taking place suddenly. 
 
 In order to determine the corrections which have to be applied to the 
 readings of a thermometer to allow for the irregularities of the bore a 
 thread of mercury is broken off the end of the mercury column, and the 
 length of this thread is measured when it occupies different portions of 
 the tube. Since the volume of the thread is constant, it follows that in 
 a perfectly cylindrical tube, or in the case of a thermometer where the 
 maker had perfectly adjusted the divisions, the length of the portion of 
 the thread would be everywhere the same. In general, however, the 
 length will vary as the thread is moved from one part of the tube to the 
 other, and from the variations in the length we can calculate the correc- 
 tions which have to be applied to different parts of the scale. 
 
 We will suppose that it is desired to calibrate the tube of a thermo- 
 meter which reads from to 100, and that it has been decided that 
 it will be sufficient to determine the corrections at every 10 degrees 
 between the two fixed marks. Suppose that a length of the mercury 
 column about equal to 10 degrees has been separated, and that, starting 
 with one end of this column near the zero and the other end at or near 
 the 10 mark, we measure the length of the column when it lies between 
 and 10, between 10 and 20, and so on. Let the lengths thus 
 measured be l lt 1 2 , Z 3 , &c., and let the mean of all the lengths, namely, 
 TO ft + Z a + . . + IIQ), be L. If the bore were perfectly uniform, the 
 length of the column would be everywhere the same and equal to L. Now 
 L represents the number of degrees which the column would occupy if it 
 were placed in a tube having a perfectly uniform bore, and such that a 
 length of it equal to the length of the tube being calibrated between the 
 and 100 marks would have the same volume as the volume of the 
 tube between these marks. If, then, we suppose that we had two tubes 
 the actual one and the corresponding one having a uniform bore and 
 that they contained a thread of the same volume ; when the ends of the 
 two threads are both at the other end of the thread in the uniform 
 tube will be at L, and that of the thread in the actual tube at l v Now 
 the position of the thread in the uniform tube gives the position which 
 ought to be occupied by the end of the thread in the actual tube. 
 Hence the correction which has to be applied at 10 is L - Z 1? say 8 V for 
 if we add 8 l to / x we get*.L, which is what the reading ought to be. 
 Now move both threads so that they have one end at the 10 marks and 
 the other ends at or near the 20 marks. The reading for the end in 
 
164 THERMOMETIIY [ 59 
 
 the case of the uniform tube will be 10 + L, while that for the actual 
 tube will be 10 + L 2 . Thus if we call L-l. 2 S 2 , and suppose for the 
 moment that the 10 mark on the actual tube is correct, the correction 
 near 20 would be 8 2 . But the 10 is not correct, for, as we have already 
 seen, the correction there is S r If, then, the end of the thread instead of 
 being at 10 were placed at the reading which really corresponds to 10, 
 namely, 10 + 8 1 , its other end would also be moved through the distance 
 8 l ; in other words, the reading corresponding to this end would now be 
 lO + L + ^ + Sg, while the reading corresponding to the same end of the 
 thread in the uniform tube would be 1 + Z/. Thus the correction to the 
 tube at 20 is 8 1 + S 2 . In the same way it can be shown that the cor- 
 rection at 30 is 8 l + 8. 2 + 8 3 , and so on. 
 
 Since we have assumed that the scale is correct at 100, it follows 
 that the correction at 100 obtained in this way, namely, 0^ + 6^ + . . . 
 + 8 10 , must be zero, and we shall thus have a test of the accuracy with 
 which we have made our calculations. The fact that this sum is zero is, 
 however, no test of the accuracy of the observations. 
 
 In the process above described it is important to note that since 
 each of the measurements of thread length we make is subject to 
 a possible observational error the results will be affected by 
 these errors, and that the magnitude of the error at any of the 
 points at which we have determined the correction depends on the 
 number of times we have measured the length of the thread between 
 the extreme points, that is, on the number of places along the scale 
 at which we have determined the correction. Thus it is quite 
 possible that by taking a very short thread, and thus determining the 
 corrections at a great number of points of the scale, we may obtain a 
 less accurate correction curve than if we had been content to use a 
 longer thread and to obtain the corrections at the intermediate points by 
 interpolation, say, by plotting the corrections obtained on squared paper 
 and then drawing a curve freehand between these points. The probable 
 error of the corrections is a maximum at the middle point of the scale, 1 
 and hence if the tube is so irregular that a large number of points have 
 to be observed on the correction curve, it is best to determine this curve 
 by two sets of experiments. In the first set a comparatively long thread, 
 say, J, |, or J of the length between the extreme points, is used to 
 determine the correction at 1, 2, or 3 equidistant points between the 
 extremes, and then by means of a short thread the corrections at points 
 intermediate between these principal points are obtained. 
 
 Unless the glass of which the thermometer is made is either " verre 
 dur " or Jena normal glass, any very great refinement in the determina- 
 tion of the calibration curve will be thrown away, owing to the fact that 
 different glasses expand at different rates at the different temperatures. 
 
 1 For a discussion of the probable error and a description of this and other 
 methods of calibrating a tube, see the Keport of the Committee on the 
 Calibration of Mercurial Thermometers, British Association Report for 1882 
 (Southampton Meeting). 
 
59] CALIBRATION OF TUBE OF A THERMOMETER 165 
 
 This fact is illustrated in Fig. 62, which shows the corrections l which 
 have to be applied to the readings of thermometers made of different 
 kinds of glass to reduce the temperatures measured on these thermometers 
 to the hydrogen scale. It must be borne in mind that all these curves 
 refer to thermometers which have been calibrated to allow for irregu- 
 larities in the bore. Practically, unless the glass is undoubtedly either 
 one of the Jena glasses or " verre dur," we shall be uncertain of the 
 temperature on account of the properties of the glass alone to about 0*05 
 
 tO 2O 3O 40 50 6O 70 80 9O 
 TEMPERATUREON HYDROGEN THERMOMETER 
 FIG. 62. 
 
 1OO 
 
 Centigrade, and hence it is unnecessary to determine the calibration 
 corrections to less than this quantity. 
 
 When making a calibration, the first process to be performed is to 
 detach a portion of the mercury column of suitable length from the rest. 
 We will suppose that the corrections have to be determined for every 
 tenth degree of a thermometer which reads from to 100. We have 
 thus to separate a thread of mercury which shall occupy very nearly 10 
 degrees on the scale of the thermometer. The method adopted at the 
 Bureau International at Sevres, and given by Guillaume, is as follows : 
 The thermometer is held in a vertical position, with the bulb uppermost 
 and the other end resting on the tip of the index finger of the right 
 hand. By giving the thermometer slight jars in an up-and-down 
 direction the mercury is caused to run down the tube, and then by a 
 
 1 These curves have been plotted from numbers given by Chree, Phil. Mag., 
 March 1898, p. 218. 
 
166 THE11MOMETRY [ 59 
 
 rapid movement of the right hand the thermometer is reversed, and by 
 striking it transversely against the left hand the mercury column is 
 caused to separate at the junction of the stem and bulb. The thread of 
 mercury left in the tube is then used to separate a thread of the desired 
 length on account of the property of fine mercury columns of separating 
 at the point of the bore where a column, such as has now been separated, 
 is allowed te rejoin the rest of the column. The thermometer is placed 
 in a horizontal position, and the lower part of the detached thread is 
 brought to some noted division on the scale, say, the 15 division, and 
 then by gently heating the bulb in the left hand, or over a very small 
 flame, the column is made to join on to the detached thread. Then the 
 bulb is allowed to cool till the desired length of thread is beyond the 
 division at which the junction took place, that is, in the case we have 
 taken, when the extreme end of the mercury column is at 25. When 
 this point* is reached the thermometer is given a slight jar in the longi- 
 tudinal direction, when the column separates at the point of the tube 
 where the junction took place. The separation is caused by the fact 
 that when the junction took place a very minute bubble of air was left 
 sticking to the side of the tube. The presence of the air is due to the 
 fact that it is never possible to entirely deprive the mercury used to fill 
 a thermometer of air. 
 
 The above method of separating a thread of the mercury is by no 
 means easy, and requires a very considerable amount of skill and practice. 
 The following method is much easier, but involves heating a portion of 
 the tube of the thermometer to the temperature at which mercury boils. 
 A very small gas flame is obtained by using as burner a finely drawn- 
 out piece of glass tube (Fig. 63). The gas must be turned down till 
 
 the flame is about 4 millimetres high and 
 is non-luminous. The thermometer is 
 then held in a horizontal position, and the 
 tube is heated at a point which intercepts 
 a thread of the desired length. The tube 
 must be brought near the flame slowly 
 and kept in continuous rotation, so that 
 it shall be heated uniformly all round. 
 FIG. 63. The heating is continued till the column 
 
 separates. The separated thread is then 
 
 run up to the top of the tube, and the bulb heated so that the mercury 
 column passes over the heated part, and thus collects all the globules of 
 mercury which will have condensed on the sides of the tube. 
 
 If reasonable care is used in the heating, the second method is quite 
 safe. It is not, however, applicable to thermometers of the German 
 pattern, in which the stem of the thermometer consists of a capillary 
 tube placed alongside a milk glass scale, and enclosed together with this 
 scale in an outside glass tube. Thermometers of this form necessitate 
 the use of the first method given above. 
 
 Having separated a thread of the desired length, we have now to 
 
59] CALIBRATION OF TUBE OF A THERMOMETER 167 
 
 measure its length at different points along the tube. The length of 
 the column has to be measured in terms of the scale of the thermometer, 
 and hence it is necessary to measure in each position of the thread the 
 amount by which the ends of the thread project beyond some whole 
 number of divisions. To do this we may either estimate the fractions 
 of a division by eye, aided if necessary by a lens, or we may employ an 
 instrument, such as that shown in Fig. 64, called a vernier microscope. 
 The chief objection to the use of a microscope for the purpose of measur- 
 ing the thread length is that it is impossible to focus on the thread and 
 the graduations of the scale at the same time. Unless the slide in which 
 the microscope moves is very well made, it is not safe to focus first on 
 the thread and then move the microscope so that it is in focus on the 
 graduations. If the microscope is of fairly long focus, it will generally 
 be advisable to so arrange matters that the thread and the divisions of 
 the scale are about equally clearly in focus. In this case there is some 
 danger of errors arising owing to parallax if the eye is not always placed 
 
 FIG. 64. 
 
 in exactly the same position with reference to the eye-piece. This may 
 to a certain extent be obviated in the case where the diameter of the 
 eye-hole is greater than the diameter of the pupil of the eye by fixing a 
 piece of black card pierced with a hole the size of the pupil over the 
 eye-hole of the eye-piece. 
 
 When using the vernier microscope three readings will have to be 
 made at each end of the thread, one reading on the end of the thread, 
 and the others on the division marks between which the end of the 
 thread lies. The difference between these latter readings will give the 
 length of the division in terms of the scale of the microscope, and from 
 this the fraction of a division of the scale of the thermometer by which 
 the thread projects beyond the division can immediately be calculated. 
 
 It will in general be found sufficient if the length of the thread is 
 measured by eye, using a low power lens. In order to avoid parallax 
 the thermometer must be placed on a sheet of mirror glass, and when 
 taking a reading the eye must be so placed that the image of the eye in 
 the mirror is alongside the position of the end of the mercury thread 
 which is being observed, or the instrument shown in Fig. 67 may be 
 
168 
 
 THERMOMETRY 
 
 [59 
 
 employed. It will generally be found advisable to adjust the position 
 of one end of the thread so that it lies exactly on one of the division 
 marks, so that the fraction of a division has only to be estimated at one 
 end. If the division lines are very coarse, it may be better to adjust one 
 end of the thread so that it lies half-way between two divisions. Having 
 noted the length of the thread when the lower end coincides with the 0, 
 10, 20, &c., divisions, the measurements ought to be repeated, starting 
 with the upper end in coincidence with the 100 division, then with the 
 90 division, and so on. The means of the lengths obtained in the two 
 sets of measurements will then have to be used in the calculations. The 
 object in thus beginning first at one end and then at the other is to 
 vary to a certain extent the fractions which have to be estimated, and 
 also to allow for any steady change in temperature which may take place 
 during the measurements. Such a change in temperature will cause a 
 change in the length of the thread. A change of temperature of 1 
 degree Centigrade will cause the thread to expand by O0002 of its length. 
 The numbers obtained are to be entered in a table, as in the following 
 example : 
 
 
 ] 
 
 
 Readings for Ends of 
 Thread. 
 
 Length of Thread. 
 
 Difference from 
 Mean. 
 
 Corrections. 
 
 A. 
 
 B. 
 
 
 Means 
 I, &c. 
 
 L-l, &c. 
 
 
 o-oo 
 
 9'84 - 
 
 9-84- 
 
 9'85 
 
 + 04 
 
 + 04 
 
 10-00 
 
 19-78 
 
 9-78 
 
 9-79 
 
 + 10 
 
 + 14 
 
 20-00 
 
 29-92 
 
 9'92 
 
 9-93 
 
 -04 
 
 + 10 
 
 30-00 
 
 39-98 
 
 9-98 
 
 9'98 
 
 -09 
 
 + 01 
 
 40-00 
 
 49-92 
 
 9-92 
 
 9-92 
 
 -03 
 
 -02 
 
 50-00 
 
 59-86 
 
 9-86 
 
 9-87 
 
 + 02 
 
 + 04 
 
 60-00 
 
 69-84 
 
 9-84 
 
 9-85 
 
 + -04 
 
 + 08 
 
 70-00 
 
 79-90 
 
 9-90 
 
 9-90 
 
 -01 
 
 + 07 
 
 80-00 
 
 89-96 
 
 9-96 ' 
 
 9-95 
 
 -06 
 
 + 01 
 
 90-00 
 
 99-90 
 
 9-90 
 
 9-90 
 
 -01 
 
 00 
 
 90-10 100-00 
 
 9-90 
 
 Si$g94 
 
 
 
 80-06 . 90-00 
 
 9-94 
 
 Mean = 99 
 
 
 
 70-10 80-00 
 
 9-90 
 
 = L 
 
 
 
 60-14 70-00 
 
 9-86 
 
 q c 9-L 
 
 
 
 50-12 60-00 
 
 9'88 
 
 /* I 
 
 
 
 40-08 
 
 50-00 
 
 9'92 
 
 
 
 
 30-02 40-00 
 
 9-98 
 
 
 
 
 20-06 30-00 
 
 9'94 
 
 
 
 
 10-20 
 
 20-00 
 
 9-80 
 
 
 
 
 0-14 
 
 10-00 
 
 9-86 - 
 
 
 
 
 ! 
 
 
 
 
 
 In order to determine the corrections at intermediate points of the 
 scale the observed points must be plotted on squared paper, the abscissa 
 
60] DETERMINATION OF FIXED POINTS 169 
 
 being the readings on the thermometer scale, and the ordiuates the 
 corrections, taken above the zero line if they are positive, and below 
 if they are negative (see Fig. 70). A curve must then be drawn 
 passing as evenly as possible through all the points, remembering that 
 the correction is zero at and 100. A suitable scale for the ordinates 
 will be one on which 2 centimetres represent the smallest division on 
 the scale of the thermometer. Thus if the thermometer is divided in 
 half degrees, 2 centimetres would be taken as representing half a 
 degree, while if the thermometer is divided in tenths of a degree, each 
 centimetre would represent a twentieth of a degree. 
 
 If on plotting the observations it is found that the corrections vary 
 considerably from point to point so that a curve passing through the 
 points would be very wavy, it indicates either that the observations are 
 not sufficiently accurate, the irregularities of the curve being due to the 
 errors of experiment, or that the bore of the tube is very irregular, arid 
 we are not justified in filling in the parts of the curve between the 
 observed points. A consideration of the individual readings will gene- 
 rally indicate which of the above is the true explanation. In particular, 
 if the corrections are calculated independently for the readings made, 
 starting from either end of the scale, the agreement or otherwise between 
 the curves obtained will indicate the divergence to be expected on 
 account of errors of observation. 
 
 60. Determination of the Upper Fixed Points of a Mercury 
 Thermometer. The upper fixed point (100) is defined as the tempera- 
 ture of the steam given off from water boiling under a pressure of a 
 standard atmosphere. A standard atmosphere is the pressure equivalent 
 to the weight of a column of mercury 76 cm. high at the sea-level 
 and at latitude 45, the temperature of the mercury being that of 
 melting ice. 
 
 The reason that in the above definition the temperature of the steam 
 is taken rather than that of the water is, that it is found that the 
 temperature of the water depends to a slight extent on the nature of 
 the vessel in which ebullition takes place. A more important reason, 
 however, is that slight impurities, in the form of dissolved salts, in the 
 water affect the temperature of the water to a considerable extent. The 
 temperature of the steam given off from water containing dissolved salt 
 is, however, the same as that of the steam given off from pure water. 
 Thus by immersing the thermometer in the steam we avoid the 
 necessity of employing perfectly pure water. 
 
 Since the temperature at which water boils depends on the pressure, 
 we have to measure the pressure to which the water is subjected at the 
 time when the reading of the thermometer is made. For this reason 
 the barometric height must be read immediately before the thermometer 
 is read. The barometric height, as given by the reading, will have to 
 be reduced to what it would be if the scale employed were at the 
 temperature at which it is correct and the mercury were at the tempera- 
 ture of melting ice (0). Further, the height will have to be reduced 
 
170 
 
 THERMOMETRY 
 
 [60 
 
 to what it would be at sea-level and latitude 45. The method of 
 making these corrections has already been described in 58. 
 
 Knowing the pressure to which the water was subjected while 
 determining the boiling point, the actual temperature of the steam is 
 obtained from tables giving the boiling point of water for different 
 temperatures. Such a table is given in the Appendix (Table 15). The 
 difference between the temperature corresponding to the barometric 
 
 pressure and the reading of the thermometer 
 will give the correction to the upper fixed 
 point of the scale. 
 
 The thermometer of which the upper 
 fixed point has to be determined is placed 
 within a double walled steam-jacket, such 
 as is shown in Fig. 65. It is held in place 
 by means of a cork E, or is suspended from 
 a retort stand in such a way that the 100 
 mark is just showing above the cork which 
 closes the steam-jacket. The upright tube 
 D serves as a condenser, so that no steam 
 escapes into the air. The escape of steam 
 into the air of a laboratory is often objection- 
 able, and, further, if the steam is not con- 
 densed and the water returned to the boiler, 
 this latter is apt to run dry and be damaged. 
 The tube F by which the condensed water 
 is returned to the boiler must be long 
 enough to dip below the surface of the water 
 in the boiler. The U tube H has a little 
 water at the bend, and serves as a manometer 
 to show whether the pressure inside the 
 steam-jacket is the same as the atmospheric 
 pressure. It is of importance that the 
 pressure inside the apparatus should not 
 exceed the external pressure, for it is this 
 external pressure which is measured by a 
 barometer, and we assume, when calculating 
 the actual temperature at which the water 
 boils, that this external pressure is the 
 pressure to which the boiling water is subjected. If the manometer 
 shows that the pressure inside the apparatus is higher than the external 
 pressure, this shows either that the exit tube D is stopped up, or that 
 the boiler is being heated too much, so that steam is being developed 
 at such a rate that it is unable to escape as quickly as it is being 
 generated. The flame under the boiler ought to be adjusted so that 
 while the lower part of the tube D is quite hot, yet no steam is seen to 
 issue from the top. 
 
 The thermometer ought to be exposed to the steam for about ten 
 
 FIG. 65. 
 
60] 
 
 DETERMINATION OF FIXED POINTS 
 
 171 
 
 FIG. 66. 
 
 minutes, a reading should then be taken, and the height of the barometer 
 noted, together with the temperature of the barometer . The ther- 
 mometer must then be again read, and the reading should 
 be the same as before. The barometer is then again 
 read, including a readjustment of the level of the 
 mercury in the cistern. Finally, the thermometer must 
 be read a third time. The mean of the three readings 
 of the thermometer is taken as the reading for the 
 boiling point, while the mean of the two barometer 
 readings is used in calculating the actual temperature 
 of the steam. 
 
 While the thermometer is in the steam, preparation 
 must be made for the determination of the lower fixed 
 point, so that immediately after the last reading in 
 steam the thermometer may be removed, and after the 
 reading has fallen to about 50 it may be placed in the ice. 
 
 In order to avoid parallax, the thermometer ought to 
 be read by means of a telescope. The telescope should be 
 capable of focusing the scale of the thermometer when 
 placed at a distance of about 40 cm. Care must be 
 taken that the thermometer is held vertical, while by means of a level 
 the axis of the telescope is set horizontal. As these two adjustments 
 are often troublesome where, as here, the greater part of the stem of the 
 thermometer is hidden, the following 
 modification, which may often be 
 used when reading a temperature, 
 may be employed : A small piece of 
 mirror glass M (Fig. 66), to which is 
 attached a metal clip AB, as shown, 
 is fixed to the thermometer, so that 
 the lower edge of the mirror is below 
 the 100 mark. The telescope is then 
 adjusted and focused so that an 
 image of the object glass, seen by 
 reflection in the mirror, occupies the 
 centre of the field. The telescope 
 being firmly clamped in this position, 
 the draw-tube is carefully pulled out 
 till the scale and thread are in focus. 
 By this means we adjust the axis of 
 the telescope normal to the mirror, 
 and since the thermometer is held by 
 the clip against the surface of the mirror, the axis of the telescope 
 must be perpendicular to the stem of the thermometer. 
 
 Another method of avoiding parallax when reading a thermometer 
 is to make use of the small instrument shown in Fig. 67. 1 It consists 
 
 1 These thermometer microscopes may be obtained from Messrs. Griffin, 
 Kings way, London, W.C. 
 
 FIG-. G7. 
 
172 
 
 THERMOMETRY 
 
 [61 
 
 FIG. 68. 
 
 of a short metal tube AB, which has two V notches G and a spring H, by 
 
 means of which the tube is held against 
 the thermometer. A second tube CD 
 slides within the first, and has a plano- 
 convex lens L of about 5 centimetres 
 focus at one end, and an eye-hole E at 
 the other end. Two metal pointers p are 
 fixed on a diameter of the tube at right 
 angles to the two notches G. The 
 pointers are always brought opposite the 
 point at which the reading is to be 
 made, and hence the line of sight, that 
 is, the line joining E to the line 
 joining the pointers, is always perpen- 
 dicular to the thermometer stem. The 
 illumination is somewhat improved by 
 having a hole I cut in the side of the 
 tube AB. 
 
 61. Determination of the Lower Fixed Point of a Mercury 
 Thermometer. The lower fixed point on the Centigrade scale, or C., 
 is the temperature of fusion of pure 
 ice. Pressure has such a small in- 
 fluence on the temperature of fusion 
 of ice that within the range of pressure 
 ordinarily met with no appreciable 
 variation takes place. 
 
 Since the presence of even a small 
 quantity of impurity in the ice will 
 appreciably lower the temperature 
 of fusion it is most important that 
 only pure ice should be employed, and 
 that the vessel in which it is placed 
 should be clean and the distilled .water 
 used free from all dissolved salts. 
 The ice, having been washed in run- 
 ning water to remove the dirt on the 
 outside of the lump, must be either 
 pounded till the individual lumps are 
 not larger than a pea, or, what is 
 more convenient, cut into shavings 
 with the machine shown in Fig. 68. 
 The finely powdered ice is intro- 
 duced into a glass funnel A (Fig. 69) 
 capable of holding from two to three 
 litres, and fitted at the bottom with a length of india-rubber tubing, which 
 can be closed by means of a pinch-cock c. The ice is saturated with dis- 
 tilled water, and by means of a piece of wood the ice is firmly pressed down, 
 
 FIG. 69. 
 
62] CORRECTION FOR FUNDAMENTAL INTERVAL 173 
 
 additional ice being added so as to completely fill the funnel. The water 
 having been run off the funnel is again filled with distilled water, the ice 
 again pressed well down, more ice being added if necessary. A little of 
 the water is then run off till the upper surface of the ice begins to become 
 dry. A small hole is then produced in the ice by means of a glass rod 
 held in the clip B. This hole must not go beyond the centre of the funnel. 
 
 The thermometer having been removed from the steam in which the 
 upper fixed point was determined is allowed to cool till the temperature 
 indicated is between 45 and 50. It is then introduced into the ice, 
 being supported by the clip B, with the zero mark just above the surface 
 of the ice. The zero reading should be taken immediately the mercury 
 ceases to fall, a telescope being with advantage used to read the scale, as 
 in the case of the boiling point. The reason the freezing point is deter- 
 mined as rapidly as possible after the boiling point is that there is a 
 progressive rise in the zero, which sets in as soon as the temperature 
 commences to fall (see 66). This rise is, however, small, and, in the case 
 where we are only attempting to read the temperature to within a tenth 
 of a degree, need hardly be considered. 
 
 62. Application of the Correction for Fundamental Interval. 
 Suppose that by the methods described in the last two sections it has 
 been found that the correction to a thermometer at is A and at 100 
 it is A 100 ; that is, that the reading on the thermometer when placed in 
 melting ice at is - A , while the reading when in steam at the 
 standard pressure is 100 - A 100 . Thus the interval between the fixed 
 points, or the fundamental interval as it is called, instead of being 
 100 degree divisions is (100 - Aj 00 + A ) divisions. In other words, 
 the interval between the fixed points, in place of being divided into 
 100 degrees, is really divided into (100 - A 100 + A ) degrees. Each 
 degree division on the thermometer therefore really corresponds to 
 
 i? 5 degree. Thus when the thermometer reads t degrees the 
 
 true temperature will be obtained by adding to the correction at the 
 zero of the scale, namely, A n , the correction due to the part of the scale 
 between the zero and the division t. Now the correction to be applied 
 on account of the error between the zero of the scale and t consists of 
 two parts. In the first place, there is the calibration correction 8 t to 
 allow for the irregularity in the bore in the tube, which has been obtained 
 on the supposition that the scale was correct at the two fixed points. 
 Secondly, a correction has to be applied on account of the error in the 
 fundamental interval. Since each degree on the scale really corresponds, 
 as we have seen above, to (100 - A 100 + A )/100, the correction on t 
 
 degrees on account of the fundamental interval amounts to L_J^ 
 
 1U 
 Thus the total correction to be applied to the reading t is 
 
174 THERMOMETRY [ 63 
 
 By a slight addition to the curve (Fig. 70) on which the calibration 
 corrections are plotted, we are able to read off directly the whole correc- 
 tion. Thus suppose on the calibration curve we take a point A on the 
 
 o 
 
 QJ 
 C A 
 
 or i 
 
 o 
 
 o 
 
 O 50 100 
 
 READING ON THERMOMETER 
 
 FIG. 70. 
 
 ordinate through the zero such that OA is equal to - A . Also at the 
 100 ordinate take a point B such that its distance from "the axis of 
 temperature is - A 100 . Then if the points A and B are joined by a straight 
 line, the ordinate drawn through any temperature t intercepted by this 
 line will be equal to 
 
 A A io 
 
 oo 
 
 But the ordinate through t of the calibration curve is S,. Hence the 
 difference of the two ordinates will be equal to 
 
 Thus if we take as a new axis the line AB, then the readings on the cali- 
 bration curve measured from this new axis will give the whole correction 
 to be applied. 
 
 Since it is not very convenient to read from such an inclined axis as 
 AB, it will be convenient to construct a new curve, such as that given in 
 Fig. 71, where the ordinates measured from the line ox are equal to the 
 corresponding ordinates measured from the axis AB in Fig. 70. The 
 values of the fixed point corrections for the thermometer for which the 
 calibration curve is drawn were A = + 0'08, A 100 = +0*11. Hence OAis 
 taken downwards, i.e. towards the negative direction, and made equal to 
 - '08 ; O'B in same way is - '1 1. 
 
 63. Determination of the Corrections to a Thermometer by Com- 
 parison with a Standard. In most cases in the laboratory it is most 
 
 
63] 
 
 STANDARDISING A THERMOMETER 
 
 175 
 
 convenient to use a thermometer of comparatively limited range, but 
 which includes the temperatures to be measured. Thus to measure 
 the temperature of, say, a resistance coil, a thermometer reading from 
 to 30 will have ample range, and without being unwieldy it can 
 be given a much more open scale than if it were capable of reading 
 from to 100. It is, however, important that every thermometer 
 should be capable of registering for one or two degrees on both sides 
 of one or other of the fixed points. For simplicity in checking, and 
 to avoid heating the thermometer to a high temperature, it is prefer- 
 able that the point chosen should be the freezing point. If the 
 thermometer is to read, say, between 15 and 25, it would render the 
 stem unduly long to make it read from to 25, so what is usually 
 done is to blow a small auxiliary bulb, so arranged that when the ther- 
 mometer is at zero the end of the mercury column is between the main 
 bulb and this auxiliary bulb, and when the thermometer is at 15 the 
 
 g 
 o 
 
 DC 
 
 O 
 
 O 5O 100 
 
 READING ON THERMOMETER 
 
 FIG. 71. 
 
 end of the column just projects above the auxiliary bulb. The reason 
 for having one of the fixed points on the thermometer is to allow of the 
 secular rise of the zero being determined. 
 
 Such thermometers as the above cannot conveniently have their cor- 
 rections determined by a process of calibration, and so the method 
 employed to determine their errors at different parts of the scale is to 
 compare their readings with those of a thermometer which has itself 
 been calibrated. It will in general be quite sufficient if the thermo- 
 meters are compared at every 5 degrees throughout the scale of the 
 short-range instrument, since modern thermometer makers are able 
 to obtain tube in which sudden irregularities of bore are very seldom 
 met with. 
 
 The thermometer or thermometers to be compared must be placed 
 together with the standard in a position in which the temperature can 
 be maintained constant for some time at all points where the thermo- 
 meters are to be compared. For temperatures between the air tempera- 
 
176 THERMOMETEY [ 63 
 
 ture and about 40 a water bath will be found suitable. A suitable 
 vessel for containing the water is a copper box in the form of a hollow 
 cube open at the top. This box can be placed in a wooden box which 
 is large enough to allow a space of about 5 cm. on all sides, which space 
 can be packed with cotton wool. The most convenient arrangement for 
 stirring consists of a small propellor c (Fig. 72) driven by a cord, which 
 
 is itself driven by a small electric motor. 
 In order to heat the liquid in the bath, 
 by far the most convenient arrange- 
 ment is a small coil of fine manganine 
 wire which has been treated with 
 shellac varnish in the manner described 
 in the Appendix. This coil is placed in 
 series with an adjustable carbon resist- 
 ance and a storage cell. Owing to 
 the passage of the current heat is de- 
 veloped in the coil, and the rate of 
 FIG. 72. heat evolution can be adjusted with 
 
 great ease by adjusting the resistance. 
 
 Another method of heating the water is to use an L-shaped piece of 
 stout copper bar, of which one arm dips in the water and the other arm 
 is horizontal and projects over the edge of the wooden box. By placing 
 a Bunsen burner under the projecting portion of the copper bar heat is 
 conveyed by conduction through the copper to the water, and moving 
 the flame to different distances along the bar allows of the supply of heat 
 being adjusted with very considerable delicacy. 
 
 The thermometers to be compared must be suspended in such a 
 manner that the tops of the mercury threads can be seen over the edge 
 of the box and so that all their bulbs are close together. 1 It is generally 
 best to arrange a separate reading telescope to read each of the thermo- 
 meters, as in this way there is little delay when taking a set of readings 
 between reading the several thermometers. 
 
 The resistance in the heating circuit is so adjusted that the tempera- 
 ture of the bath becomes practically stationary at the different tempera- 
 tures at which a comparison is to be made. When the temperature is 
 stationary, the rate at which heat is developed within the coil is exactly 
 equal to the rate at which the bath is losing heat by radiation, con- 
 duction, and evaporation, &c. The reason the temperature has to be 
 pretty well stationary when a comparison is made, is, that unless the 
 thermometers are all exactly alike the quickness with which the thermo- 
 meters follow the changes in temperature of the bath will be different, 
 and hence their readings will differ from what they would be if the 
 temperature of the bath were stationary by different amounts. If it is 
 found impossible to arrange for the temperature of the bath being 
 stationary, the effects of the different heat capacities of the different 
 bulbs can in a great measure be got over by immersing all the bulbs of 
 
 1 It, may be necessary to apply corrections for emergent column. See f>5. 
 
63] 
 
 CORRECTIONS TO A THERMOMETER 
 
 177 
 
 the thermometers in a small vessel of mercury supported in the middle of 
 the bath. 
 
 For temperatures over 40 a water bath is not suitable, owing to the 
 rapid loss of heat which goes on owing to evaporation, and also to the 
 moisture which condenses on the stems of the thermometers. By 
 replacing the water by glycerine or oil and heating the bath below by 
 means of a flame, the comparison can be 
 continued above 40. It is, however, 
 necessary that the bath should contain a 
 very considerable volume of liquid and be 
 very well stirred. Owing to the difficulty 
 of adjusting the supply of heat so that 
 the temperature may remain stationary 
 while the thermometers are read the 
 experiment is very tedious, and unless 
 a number of thermometers have to be 
 compared with a standard at the same 
 time, the instrument described below will 
 be found much more convenient and 
 accurate. 
 
 The action of the instrument depends 
 on the fact that the temperature of the 
 vapour given off by a pure liquid boiling 
 under a constant pressure is constant, 
 and hence if the thermometers to be 
 compared are surrounded by such a 
 vapour, their temperature must be the 
 same. Further, by merely altering the 
 pressure we can alter the temperature of 
 the vapour. 
 
 A general view of the instrument is 
 shown in Fig. 73. A glass tube AB, 
 closed at the top and open below, about 
 76 cm. long and 2*5 cm. in diameter, 
 passes up inside a second glass tube of 
 about the same length and 4 '7 cm. in 
 diameter. The space between these two 
 tubes is at the bottom closed by an 
 india-rubber cork c (Figs. 73 and 74), 
 
 while at the top the inner tube is kept in place by three small pieces of 
 glass rod fused on and forming a triradiate star. The upper end of the 
 outside tube is fused on to a Liebeg condenser D, the upper end of the 
 condenser being connected by a rubber joint fitted with a glycerine seal to 
 a manometer E, a stoppered funnel F, and a large glass bottle which acts 
 as a reservoir. This bottle is packed round with cotton wool, and is en- 
 closed in a wooden box G. The air can be exhausted from or admitted to 
 this bottle by means of a three-way tap H. Some mercury is placed on the 
 
 FIG. 73. 
 
178 THERMOMETRY [ 64 
 
 top of the cork c (Fig. 74) to prevent the liquid used to form the vapour 
 touching the cork. The liquid is heated by means of a spiral of fine 
 uncovered platinum wire, the terminals being fused into glass tubes 
 which pass down through the mercury and the cork. 
 
 The vapour of the boiling liquid rises between the inside and outside 
 tube, is condensed in the condenser D, and runs down to the bottom again. 
 The thermometers are fastened together, their bulbs resting in a 
 small glass vessel K (Fig. 74), which is filled with mercury. This vessel 
 is supported by a glass rod which rests on the table on 
 which the instrument stands. To remove the thermometers 
 the instrument is drawn to the edge of the table till this 
 rod can be lowered, and with it the thermometers. The 
 scales are read by means of two telescopes, it being quite 
 easy to see through the vapour if suitable liquids are used ; 
 carbon bisulphide (20 to 46), ethyl alcohol (46 to 79), 
 and chlorobenzene (79 to 120) do very well. Water cannot 
 be used, as it forms drops on the inside of the glass so that 
 the thermometers cannot be read. 
 
 In order that the india-rubber cork may form a thoroughly 
 air-tight joint it must be well cleaned with benzene, coated 
 with india-rubber solution, and put in place while the solution 
 is wet. When the solution has got thoroughly dry, which 
 takes four or five days, the joint will be quite air-tight. 
 
 With such an arrangement the thermometers can easily 
 be maintained at a temperature constant to within 0'01 for 
 three or four hours together. The manometer is only used to 
 adjust the temperature to the desired point, a table of vapour- 
 pressures being employed. In about half-an-hour after alter- 
 FIG. 74. ing the pressure, and hence the temperature, the thermometer 
 readings become quite constant. The instrument once 
 started can be left entirely to itself ; the only thing the observer has to 
 do is to read the thermometers, then let a little more air in to get the 
 next higher temperature, and so on. 
 
 64. Auxiliary Fixed Points Transition Temperatures. It is 
 sometimes convenient to have some other fixed temperatures besides 
 and 100, and thus to directly determine the error of a thermometer at 
 these points. For high temperatures the fixed points used are generally 
 the boiling points of sulphur, mercury, and the freezing points of some 
 metals. Since, however, mercury thermometers cannot satisfactorily be 
 used at temperatures much over 200, the description of the methods 
 of using such fixed points will be postponed till the sections dealing with 
 platinum resistance thermometers and thermocouples. 
 
 It is, however, very convenient to have a fixed point somewhere near 
 
 the ordinary air temperature. Richards 1 has shown that such a fixed 
 
 point is supplied by the temperature at which hydrated sodium sulphate 
 
 (Na.,SO 4 , 10H 2 O) melts. There is only one temperature, the pressure 
 
 1 Proc. American Acad. (1902), xxxviii. p. 431. 
 
64] 
 
 TRANSITION TEMPERATURES 
 
 179 
 
 being kept constant, at which anhydrous sodium sulphate (Na 9 SO 4 ), 
 hydrated sodium sulphate (Na^SO^, 10H. 2 O), the saturated solution and 
 the vapour can coexist in a state of equilibrium. This temperature 
 is called the transition temperature. If a mixture of the above sub- 
 stances is placed in an environment at a higher temperature than the 
 transition temperature part of the hydrated salt will go into solution, 
 a process which, owing to the latent heat of the solid, involves an 
 absorption of heat. If the surrounding temperature is below the transi- 
 tion temperature, then some of the saturated solution will form crystals of 
 the hydrated salt and heat will be evolved. In either case, therefore, the 
 temperature of the mixture will remain constant till either all the crystals 
 of the hydrated salt are melted, or till all the solution has crystallised. 
 
 The sodium sulphate crystals must be pure, and they can be prepared 
 by recrystallising two or three times the commercial "pure" salt. The 
 crystallising ought to be continued till the 
 transition temperature remains constant. 
 
 The arrangement used in testing a ther- 
 mometer at the transition temperature of 
 sodium sulphate is shown in Fig. 75. The 
 thermometer T passes through a cork in the 
 mouth of the glass boiling-tube A, the sodium 
 sulphate being contained in this tube. The 
 tube A itself passes through a cork in the outer 
 glass tube B, the interspace forming an air-jacket 
 to prevent the too rapid heating of the sulphate. 
 The outer tube B is placed in a water bath c, 
 this bath being kept well stirred and at a 
 temperature of 33 C. The necessary supply 
 of heat can most easily be provided by a coil 
 of German silver wire immersed in the water, 
 an electric current being passed through the wire 
 and, by means of a variable resistance in the 
 circuit, the current being so adjusted that the 
 temperature remains constant. 
 
 The sodium sulphate crystals are powdered, and if necessary some 
 anhydrous sodium sulphate added. The efflorescence which naturally 
 occurs when the hydrated crystals are kept is in general quite sufficient to 
 provide the necessary quantity of anhydrous salt. The reason for adding 
 the anhydrous salt is to be certain that there is no water present 
 besides the water of crystallisation. The crystals having been introduced 
 the thermometer is put in place, and the tube A is heated till, by the 
 melting of some of the crystals, a pasty mass results. The tube A is then 
 placed within the tube B, and the whole placed in the water bath. The 
 crystals are kept well stirred, and the thermometer read at intervals. 
 When the thermometer reading becomes constant, the temperature is the 
 transition temperature, which Richards has found to be 
 
 32'383 C. 
 
 FIG. 75. 
 
180 THERMOMETBY [ 65 
 
 on the hydrogen scale. By comparing the reading of the thermometer 
 with the above value, the error of the thermometer is obtained. 
 
 If the temperature of the bath is maintained between 32'5 and 33, 
 the temperature of the sodium sulphate will be found to remain constant 
 for two or three hours. The accuracy with which this temperature can 
 be reproduced is about the same as that with which the freezing point 
 can be determined, and is greater than that with which the boiling point 
 can be measured unless very accurate pressure measurements are made. 
 The use of this fixed point has another recommendation in that it avoids 
 the necessity of heating a thermometer, which is ordinarily only used at 
 temperatures near that of the air, to such a high temperature as 100, 
 and hence producing a serious depression of the zero (see 67). 
 
 65. Correction of Thermometer Reading for Emergent Column. 
 In many cases where a mercury thermometer is used, it is not possible 
 to arrange so that the temperature of the thread is the same as that 
 of the bulb. If, as is usually the case, the temperature of the thread 
 is less than that of the bulb the reading will be too low, for the mercury 
 in the cold part of the thread will not have expanded so much as it would, 
 supposing the temperature had been the same throughout. 
 
 By making the assumption that the bulb and stem, up to the division 
 corresponding to the temperature t , is at one temperature, and that the 
 rest of the stem is at a temperature ', then we can calculate the correc- 
 tion which must be applied to the observed reading t to obtain the true 
 temperature to which the bulb and lower part of the scale is subjected. 
 Let v be the volume of 1 degree of the capillary, then the volume of 
 the mercury in the emergent portion of the column is v(t - t Q ), and if 
 this volume of mercury were heated to a temperature t, its apparent 
 increase in volume would be 
 
 (7 -c)v(t-t Q )(t-t'), 
 
 where y is the coefficient of expansion of mercury, and c the coefficient 
 of cubical expansion of the glass. The number of degrees corresponding 
 to this apparent increase of volume is 
 
 ( y - c )(t-t )(t-t f ) > 
 
 and hence this quantity expresses the correction for emergent column. 
 For most kinds of thermometer glass (y - c) is .- 
 
 The correction calculated in this way is not of much value, since the 
 assumptions we have made are certainly erroneous. Thus the transition 
 from the portion of the stem, which is at a temperature t, to that at a 
 temperature t' is gradual, and not sudden, as has been assumed. 
 
 An elaborate series of experiments has been made by Rimbach l to 
 determine experimentally the correction for the emergent cool column, 
 and tables have been drawn up embodying the results. The following 
 table gives the correction in the case of (1) the ordinary solid stem ther- 
 mometers, and (2) the German pattern, in which a fine capillary is enclosed 
 in an outer glass tube. In the table t is the temperature as shown by 
 1 Zeitsch. fur Instrumentcnlcunde (1890), x. 153. 
 
65] 
 
 CORRECTION FOR EMERGENT COLUMN 
 
 181 
 
 the thermometer, and t' the temperature shown by an auxiliary thermo- 
 meter, of which the bulb is on a level with the mid point of the exposed 
 thread, and at a distance of 10 cm. from it. This auxiliary thermometer 
 must be screened from the direct action of any source of heat used to heat 
 the vessel in which the main thermometer is placed. The quantity n is 
 the length of the exposed thread measured in 1 degree divisions of the 
 thermometer. The correction, when the temperature of the thread is 
 below that of the bulb, is additive. The tables are for thermometers of 
 Jena glass having the degree intervals from '9 to I'l mm. long. 
 
 THERMOMETER WITH ENCLOSED SCALE (GERMAN PATTERN). 
 
 
 t-t' 
 
 
 
 70 
 
 80 
 
 90 
 
 100 
 
 110 
 
 120 140 
 
 100 
 
 180 
 
 
 10 
 
 01 
 
 01 
 
 03 
 
 04 
 
 06 
 
 07 '10 
 
 13 
 
 17 
 
 10 
 
 20 
 
 08 
 
 12 
 
 14 
 
 19 
 
 23 
 
 25 ! -28 
 
 32 
 
 40 
 
 20 
 
 30 
 
 25 
 
 28 
 
 32 
 
 36 
 
 39 
 
 42 -48 
 
 54 
 
 66 
 
 30 
 
 40 
 
 30 
 
 35 
 
 41 
 
 48 
 
 54 
 
 60 1 -67 
 
 77 
 
 92 
 
 40 
 
 50 
 
 41 
 
 46 
 
 52 
 
 59 
 
 70 
 
 79 I -89 
 
 98 
 
 1-16 
 
 50 
 
 60 
 
 52 
 
 60 
 
 68 
 
 79 
 
 89 
 
 99 1-11 
 
 1-23 
 
 1-46 
 
 60 
 
 70 
 
 63 
 
 74 
 
 85 
 
 98 
 
 Ml 
 
 1-20 j 1 32 
 
 1-45 
 
 1-70 
 
 70 
 
 80 
 
 75 
 
 87 
 
 1-01 
 
 1-15 
 
 1-28 
 
 1-38 1-53 
 
 1-70 
 
 1-98 
 
 80 
 
 90 
 
 87 
 
 99 
 
 1-13 
 
 1-28 
 
 1-45 
 
 1-62 1-82 
 
 1-94 
 
 2-25 
 
 90 
 
 100 
 
 98 
 
 1-12 
 
 1-29 
 
 1-47 
 
 1-65 
 
 1-82 2-03 
 
 2-20 
 
 2-55 
 
 100 
 
 120 
 
 
 
 
 1-88 
 
 2-10 
 
 2-28 2-49 
 
 2-68 
 
 3-13 
 
 120 
 
 140 
 
 
 
 
 
 
 2'75 2-97 
 
 3-22 
 
 3-75 
 
 140 
 
 160- 
 
 
 ... 
 
 
 
 
 ... | 3-35 
 
 1 
 
 3-80 
 
 4-35 
 
 160 
 
 THERMOMETERS WITH SOLID STEM, SCALE ON STEM. 
 
 n 
 
 t-t' 
 
 n 
 
 
 
 
 
 
 
 
 
 70 
 
 80 90 100 
 
 110 
 
 120 
 
 140 
 
 160 
 
 180 
 
 
 10 
 
 02 
 
 03 -05 -07 
 
 09 
 
 11 
 
 17 
 
 21 
 
 27 
 
 10 
 
 20 
 
 13 
 
 15 -18 -22 
 
 26 
 
 29 
 
 38 
 
 46 
 
 53 
 
 20 
 
 30 
 
 24 
 
 28 1 -33 -39 
 
 44 
 
 48 
 
 59 
 
 70 
 
 78 
 
 30 
 
 40 
 
 35 
 
 41 -48 -56 
 
 62 
 
 68 
 
 82 
 
 94 
 
 1-04 
 
 40 
 
 50 
 
 47 
 
 53 i -62 -72 
 
 81 
 
 88 
 
 1-03 
 
 1-17 
 
 1-31 
 
 50 
 
 60 
 
 57 
 
 66 '77 '89 
 
 1-00 
 
 1-09 
 
 1-25 
 
 1-42 
 
 1-58 
 
 60 
 
 70 -69 
 
 79 -92 1-06 
 
 1-19 
 
 1-30 
 
 1-47 
 
 1-67 
 
 1-86 
 
 70 
 
 80 -80 
 
 91 1-05 1-21 
 
 1-37 
 
 1-52 
 
 1-71 
 
 1-94 
 
 2-15 
 
 80 
 
 90 -91 
 
 104 1-19 1-38 
 
 1-56 
 
 1-73 
 
 1-96 
 
 2-20 
 
 2-42 
 
 90 
 
 100 
 
 1-02 
 
 1-18 1-35 1-56 
 
 1-79 
 
 1-97 
 
 2-18 
 
 2-45 
 
 2-70 
 
 100 
 
 120 
 
 
 1-98 
 
 2-23 
 
 2-43 
 
 2-69 
 
 2-95 
 
 3-26 
 
 120 
 
 140 
 
 
 
 
 2-92 
 
 3-22 
 
 3-47 
 
 3-86 
 
 140 
 
 160 
 
 
 
 ... 
 
 
 
 4-00 
 
 4-46 
 
 160 
 
182 THERMOMETRY [ 67 
 
 66. Reduction of the Readings of Mercury Thermometers to the 
 Hydrogen or Air Scales. As has already been mentioned, the scales 
 of mercury thermometers, made of different kinds of glass, do not agree, 
 the differences amounting, as will be seen from Fig. 62, to nearly a tenth 
 of a degree at 50, even in the case of the standard glasses. Whenever, 
 therefore, an accuracy of a tenth of a degree is aimed at, it is always 
 advisable to express all temperature measurements on the scale of either 
 the hydrogen or the air thermometer. For all temperatures below 100 
 the hydrogen thermometer is taken as the standard, but for temperatures 
 above 100 it is usual to employ either the nitrogen or the air scale. 
 
 In Table 19 are given the corrections which must be applied to 
 the readings of thermometers made of the standard glasses to reduce to 
 either the hydrogen or the air scale. 
 
 67. Changes in Zero of Mercurial Thermometers. After a mercury 
 thermometer has been filled, it is in general kept for some time before 
 being graduated, so that the bulb may in a measure recover from the 
 effects of the extreme heating to which it was subjected in the process of 
 manufacture. It is, however, found that the bulb goes on slowly con- 
 tracting, as indicated by a gradual rise in the zero reading, for many 
 years. This slow rise of the zero is called the secular rise. 
 
 The secular rise is much more rapid at high than at low temperatures, 
 and hence it is advisable to keep a thermometer after filling, and before 
 graduation, at a temperature of about 300 for a day or so, and then 
 to very gradually reduce the temperature. This annealing process has 
 been elaborated at the Jena Glass Works, where in certain cases the glass 
 is cooled at a regular rate from 465 to 370, the whole operation occu- 
 pying four weeks. Glass so treated is found remarkably free from strain, 
 and is said to have been " fine annealed." 
 
 All thermometers ought to be made of one of the standard glasses. 
 These are (1) verre dur, as used by the French thermometer maker 
 Tonnelot ; (2) the normal thermometer glass made at Jena, and designated 
 by the number 16'"; and (3) the Jena borosilicate glass, designated 59'". 
 The Jena glass, 16'", is the most commonly met with, and it may be 
 known by the presence of a thin violet line near the surface. 
 
 In the case of the glasses used more than ten years ago the secular 
 rise was very considerable, and went on for many years. In the case of 
 the French verre dur and the Jena glasses (16"' and 59'") the secular 
 rise is comparatively small, amounting to about 0'04 in four years, after 
 which time the rise is very small. The secular rise can be taken into 
 account in the case of thermometers made of either of the above glasses 
 by making occasional observations of zero, and then assuming that the 
 rise has taken place at a uniform rate between the determinations. 
 
 In addition to the secular rise, which is probably due to the recovery 
 of the glass from a state of strain produced during the process of manu- 
 facture of the thermometer, there are other and more troublesome changes 
 in zero produced when the thermometer is heated to quite moderate 
 temperatures. Thus if after a thermometer has been kept at a tempera- 
 
68] 
 
 MOVABLE ZERO METHOD 
 
 183 
 
 tare of zero for some time its zero is determined, and the thermometer 
 is then heated to 100 for about half-an-hour, and as soon after as 
 possible the zero is again tested, it will be found that the zero is now 
 lower than it was before the heating to 100. The depression obtained 
 in this way by heating a thermometer to 100 and then immediately 
 determining the zero is called the depression constant of the thermometer. 
 The depression constants of the three standard thermometer glasses are 
 shown in the following table : 
 
 Glass. 
 
 Verre dur . 
 Jena glass (16'") 
 Jena glass (59'") 
 
 Depression Constant. 
 
 O'll 
 0-08 
 0'03 
 
 At ordinary room temperatures the depression takes two or three 
 days to die out, at higher temperatures the recovery is more rapid. The 
 time the heating at 100 has to be continued to produce the full depres- 
 sion is only two or three minutes in the case of the standard glasses. 
 With ordinary crystal glass, however, it requires about half-an-hour's 
 heating to produce the full depression, which may amount to half a 
 degree. The fact that the standard glasses reach their full depression 
 rapidly makes their use in hypsometric thermometers advisable, even 
 if there were not other reasons for their employment in this case. 
 
 If a thermometer is maintained for a long time at a high temperature 
 it is found that there is a progressive rise in the zero. Thus in the case 
 of a verre dur thermometer, examined by Guillaume, heating to 200 for 
 about six hours produced a rise of about 0'4, while heating for thirty hours 
 to 370 produced a rise of about 3'l. In the case of the Jena glasses 
 Wiebe found that heating a thermometer of 16'" to 300 for twenty hours 
 produced a rise of about l - 6. In the case of 59'", heating to 300 for 
 thirty hours produced a rise of 3*9. If, however, the thermometer had 
 been subjected to the fine annealing process referred to on p. 182, then 
 the rise was only 0'2. 
 
 The rise produced by prolonged heating to high temperatures appears 
 to be permanent, and further exposure to a slightly lower temperature 
 produces less and less effect as the period of the heating is prolonged. If 
 a thermometer of Jena glass, 1 6"', is heated to 300 for twenty-four hours 
 before it is graduated, then for temperatures up to, say, about 250 the 
 rise produced is for most purposes inappreciable. 
 
 68. The Movable Zero Method of Using a Mercury Thermometer. 
 The phenomenon of the depression of the zero of mercury in glass 
 thermometers indicates that when such a thermometer is heated, even to 
 temperatures below 100, then, in addition to the reversible expansion 
 
184 THERMOMETRY [ 68 
 
 which disappears when the bulb is cooled, there is an irreversible expan- 
 sion which does not at once vanish when the thermometer is cooled, and 
 which produces the depression of the zero. This irreversible effect intro- 
 duces a complication when we come to consider the definition of the scale 
 of the mercury-in-glass thermometer. When the zero of the thermometer 
 is determined immediately after the boiling point, then, since it is found 
 that the irreversible effect persists for some time, the fundamental interval 
 obtained corresponds to the reversible effect only. Suppose, now, the 
 fundamental interval is divided into degrees of equal volume, the scale, 
 therefore, corresponding to the reversible effect only, and that after the 
 thermometer has been preserved for some time at a temperature not much 
 above zero it is heated to 50 as indicated by the scale. Now the tem- 
 perature will not be 50, for, in the first place, if D is the depression 
 constant of the thermometer before the thermometer was heated, the zero 
 will have risen by an amount Z>, so that if there were no irreversible effect 
 produced by heating to 50 the true temperature corresponding to the 
 reading 50 would be 50 - D. There is, however, an irreversible expan- 
 sion produced by the heating to 50, which we may call cZ, and on this 
 account the thermometer will read too low by an amount d, so that the 
 actual temperature corresponding to the reading 50 is 50 - (D - d). 
 Now, if after taking the reading at 50 we had immediately plunged the 
 thermometer into melting ice and determined the zero, the reading would 
 have been D - d. If, then, we apply this reading as a correction to the 
 reading at 50, that is, take the zero as found immediately after the 50 
 reading as the zero corresponding to that reading, we shall have allowed 
 for the irreversible expansion of the glass. This method of referring 
 every reading of a mercury thermometer to a special zero taken immedi- 
 ately after the reading is called the movable zero method of using a 
 thermometer. 
 
 It is quite obvious that in many cases it would be impossible to 
 interrupt a series of temperature measurements to take a zero reading 
 after every reading of the thermometer. With a view of overcoming this 
 difficulty various observers have made determinations of the depressions at 
 different temperatures for thermometers made of the standard glasses, 
 and have drawn up tables giving these depressions. The results ob- 
 tained have not been very concordant, so that in Table 18 the means of 
 the results obtained by the different observers are given and these are 
 probably correct to one or two hundredths of a degree. 
 
 In order to use the tables of depression of zero the thermometer to be 
 used must be kept at zero for some considerable time so that the depres- 
 sion produced by previous heating has time to disappear, which will be 
 indicated by the zero becoming constant. Let the zero reading obtained 
 in the way be D. Then when the reading of the thermometer is , the 
 temperature having been kept at t for such a time that the glass has 
 reached the permanent condition corresponding to this temperature, the 
 true temperature is t - (D - d t ), where d t is the depression corresponding 
 to the temperature t as obtained from the tables. 
 
69] DEPRESSION OF THE ZERO 185 
 
 For example, suppose that the thermometer for which Fig. 70 gives 
 the correction curve after being kept in ice for two or three days reads 
 -f 0*03, and when placed in a given enclosure reads 78'32, to find the 
 temperature of the enclosure. From the correction curve we have that 
 the correction at 78-32 is -f0 0t 13. If the thermometer is of verre dur 
 the depression at 78'32 is 0'08, that is, if the zero had been determined 
 immediately after heating to 78 0- 32 the depression would have been O08, 
 and, therefore, the zero reading + -03 - -OS or - 0'05. Hence the 
 correction for the irreversible effect is + 0'05, and the true tempera- 
 ture is 
 
 78-32 + 0-13 + 0-05, 
 
 or 78 0- 50 on the scale of the verre dur thermometer. To reduce the 
 hydrogen scale a correction - 0'05 (see Table 19) has to be applied, so 
 that in the hydrogen scale the temperature was 78'45. 
 
 For a description of the corrections which must be applied and the 
 precautions to be taken if attempts are made to read temperatures 
 Avith mercury thermometers nearer than 0- 01, reference must be made to 
 Guillaume's Traite de la Thermomctrie de Precision. 
 
 69. Measurement of the Depression in a Mercury Thermometer. 
 Take a thermometer graduated in tenths, or at least fifths, of a degree 
 and keep it in melting ice for two or three days so as to allow any 
 depression existing to disappear. The thermometer used ought not to 
 have been heated above the ordinary room temperature for at least a 
 week before the experiment. 
 
 Determine the zero as described in 61, and then heat the 
 thermometer for ten minutes to a temperature of 50, and again deter- 
 mine the zero as quickly as possible. The difference between the zero 
 now found and that obtained before the heating will give the depression 
 for 50. 
 
 Next place the thermometer in a boiling point apparatus, and read 
 the boiling point every minute for half-an-hour, or till it becomes 
 constant. The barometer must be read between each thermometer 
 reading to make certain that the differences obtained are not due to 
 variations of atmospheric pressure. Having applied, if necessary, a 
 correction for the effects of variation in pressure, draw up a table 
 showing the rapidity with which the thermometer takes up its final 
 condition. 
 
 Remove the thermometer from the steam, and as soon as safe plunge 
 it into ice and take a zero reading. The difference between the first 
 zero reading and the one obtained after heating to 100 will be the 
 depression constant of the thermometer. 
 
 A zero reading should then be taken every day for a few days to 
 determine how long the thermometer takes to recover from its heating 
 to 100. 
 
 To a near approximation the depression <l t at any temperature t, 
 below 100, may be calculated on the supposition that the depression 
 
186 THERMOMETRY [ 69 
 
 is proportional to the temperature to which the thermometer has been 
 raised. Thus 
 
 Dt 
 
 Resistance and thermoelectric methods of measuring temperature are 
 described in the later chapters, where the electrical phenomena on which 
 they depend are considered. 
 
CHAPTER X 
 
 EXPANSION OF SOLIDS AND LIQUIDS 
 
 70. Linear Expansion. If the distance between two marks on a solid 
 body at the temperature is L Q , and when the temperature is raised 
 to t the length between the marks is L t , then, so long as t is not too 
 great, it is in general possible to express L t by an equation of the form 
 
 (1) 
 
 where a and I are constants. If the range of temperature t is small, 
 then the term in t 2 can generally be neglected, so that the increase in 
 length is directly proportional to the increase in temperature, and we 
 have 
 
 L t = L (l+at) (2) 
 
 In these circumstances it is usual to call a the coefficient of linear 
 expansion of the solid. 
 
 If the lengths of the solid at two temperatures ^ and 2 are L^ and 
 //. then we have 
 
 Hence 2 = 
 
 L^ 1 + a^ + U* 
 
 or since both a and b are very small, neglecting terms involving the 
 squares, or higher powers, of a and b, we have 
 
 t^} ..... (3) 
 If the difference of temperature ( 2 - ^) is small, this reduces to 
 
 ....... (4) 
 
 Hence if L 2 and L^ are the lengths measured at the two temperatures 
 t 2 and t v we can from this expression calculate a. 
 
 If the term in t 2 in the expression (1) above is not negligible, then 
 the value obtained for a at different temperatures, i.e. for different 
 
 values of the mean temperature ( 2 -- 1 J, will not be the same, or, in 
 other words, the coefficient of expansion denned by (2) above will 
 
188 EXPANSION OF SOLIDS AND LIQUIDS [ 71 
 
 be a variable. In such a case the coefficient of linear expansion at a 
 temperature t is defined by the expression 
 
 8L 
 
 a < = L8t> 
 
 where L is the length at a temperature t, and L + 8L is the length at a 
 temperature t + St, 8t being small. 
 
 Leaving out of account methods depending on the use of levers, which 
 are not suitable for accurate measurements, methods for measuring the 
 coefficient of linear expansion may be divided into two classes. In one 
 of these a comparatively long piece of the solid (about a metre) is used, 
 and the elongation measured by means of a micrometer screw or a 
 microscope with an eye-piece scale, and in the other a short length of 
 2 or 3 centimetres is used, and the elongation is measured by various 
 devices depending on the interference of light. 
 
 If the length of a solid at a temperature t is L, and when the 
 temperature increases by At the length becomes L + AL, then the 
 coefficient of expansion between these limits of temperature is 
 
 AL 
 L.&i 
 
 This shows that to determine the coefficient of expansion we have to 
 measure the increase of length AL, the rise in temperature Atf, and the 
 original length L. As all these quantities appear in the expression to 
 the first power, we ought to measure them all with the same accuracy. 
 Since AZ, is small, the most difficult part of the measurement consists 
 in determining the increase in length, the measurement of the original 
 length, and of the increase in temperature being in general much less 
 difficult. 
 
 71. Measurement of the Linear Expansion of a Rod. An arrange- 
 ment suitable for measuring the coefficient of thermal expansion of a 
 
 FIG. 7(5. 
 
 metal rod about a metre long is shown in Fig. 76. The expansion is 
 read by means of two reading microscopes which are clamped at a fixed 
 distance apart, the expansion being read off on eye-piece scales or by 
 means of a micrometric eye-piece. In order to obtain an accurate 
 measure of the expansion, it is essential that two very fine lines should 
 
71] LINEAR EXPANSION OF A ROD 189 
 
 be ruled on the metal rod to act as fiducial marks between which the 
 expansion shall be measured. To give a suitable surface for these 
 marks, half the thickness of the rod is filed away for a short distance 
 at either end, as shown. The flat surface must then be very carefully 
 polished, first with fine emery paper, and finally with rouge. While 
 performing the final stages of the polishing, it is essential that the 
 rubbing should take place in a direction parallel to the length of the 
 rod. When the polishing is complete, a line parallel to the axis of the 
 rod must be ruled on the polished surface. This can be done by means 
 of a pair of dividers, one leg being held against the side of the rod. 
 This longitudinal line need not be very fine, but may be made so as 
 to be easily seen by the naked eye. The transverse line, on the other 
 hand, must be so fine as to be practically invisible except with a lens. 
 A very sharp needle is selected by examining the points of a number 
 of needles under a low power microscope, and then a transverse line is 
 ruled with this needle, a very light touch being employed. A small 
 square resting against the side of the rod can be used to guide the 
 needle. It is advisable after ruling the line, and before moving the 
 square, to make with another needle a somewhat coarse mark as a 
 prolongation of either end of -the fine line. This coarse 
 mark will be found of great assistance when adjusting 
 the apparatus. 
 
 Another way of making a suitable mark is to file 
 a notch at each end of the rod, as shown in Fig. 77, and 
 then solder a very fine metallic wire across the notch. 
 The wire ought to have a diameter of about '05 mm., FIG. 77. 
 
 and is preferably of platinum, as it does not then corrode. 
 In the absence of sufficiently fine wire a fine thread of glass may be used, 
 but the attachment presents some difficulty. If the rod is not going 
 to be heated above a temperature of 100, a little melted shellac does 
 fairly well. When making the measurements, one of the edges of the 
 wires is used as the mark. 
 
 The rod is enclosed, except for about 2 centimetres at either end, in 
 a metal tube about 6 centimetres in diameter. At one end A (Fig. 76) 
 the rod passes through a cork which fits tightly; at the other end it 
 passes through a short tube soldered to the disc which closes the end of 
 the jacket. The joint between the tube and the rod is made water-tight 
 by slipping a short length of india-rubber tubing over the rod and the 
 projecting tube. This joint will allow of the free expansion or contraction 
 of the rod as the temperature is changed. 
 
 A thermometer passes through a small hole at either end, and is held 
 in place by a small cork. Two side tubes c and D allow of a stream of 
 steam or cold water being passed through the jacket. A rubber tube 
 from one of these leads to a sink, while the other is connected to a 
 T-piece F. The other arms of this T-piece are connected to the water 
 supply and a can in which water can be boiled respectively. By clamping 
 one or other of these tubes, water or steam may be passed in succession 
 
190 EXPANSION OF SOLIDS AND LIQUIDS [ 71 
 
 through the jacket, and the change from one to the other can be made 
 without disturbing the jacket and the rod. A small vertical tube E 
 having a rubber tube attached which can be closed by means of a pinch- 
 cock is placed at the top of the jacket. This tube allows of the escape 
 of the air when filling the jacket with water. The jacket is supported on 
 two v's at such a height that the microscopes can be focused on the 
 marks on the rod. 
 
 To perform the experiment the two microscopes are clamped firmly 
 down on a steady table, or preferably on a stone or slate shelf, and 
 the rod is adjusted so that the transverse marks are in focus and the 
 longitudinal mark is near the centre of the field. 
 
 The microscopes must be so adjusted that when the rod is cold the 
 transverse marks on the rod are near the ends of the scales nearest to the 
 centre of the rod, otherwise when the rod is heated it may expand so 
 much that the marks would go out of the field of view. Of course if 
 either of the microscopes can be moved parallel to the length of the rod 
 by a sufficiently fine micrometer screw this adjustment is of little or 
 no importance. 
 
 Water is passed through the jacket till the readings of the thermo- 
 meters become constant, when the reading on the two microscope scales 
 are taken, as well as the readings of the thermometers. The water 
 is then run out and steam passed through the jacket till again the 
 temperature is constant. If, owing to the expansion, the mark at the 
 end B is beyond the scale, the jacket and rod must be moved through the 
 v's till the mark comes into view again. The readings of the microscopes 
 and thermometers are then taken. Cold water is again passed through 
 the jacket, and the readings at the low temperature repeated. The two 
 sets of readings at the low temperature ought to agree. If they do not 
 the microscopes have probably been moved, and the whole set of readings 
 must be repeated. 
 
 It is a good thing to take a number of readings of the microscopes at 
 each temperature, and, to prevent being biased by the readings already 
 taken, to move the rod longitudinally in the v's between each reading. 
 In this way the readings of the microscope scales will be different, but if 
 the scales both read in the same direction the sum of the two readings 
 ought to be constant. 
 
 The values of the scale divisions of the micrometer scales of the 
 microscopes must be determined by means of a fine scale ruled on glass, 
 on which each microscope is focused in turn. 
 
 To determine the distance between the two marks on the rod a 
 divided metre scale can be supported in place of the rod, and the readings 
 on the microscope scales obtained between any two of the divisions of the 
 scale. Then from the readings of the microscopes corresponding to the 
 marks on the rod the distance between these marks can be immediately 
 determined. 
 
 If the material of which the coefficient of thermal expansion is to be 
 measured is in the form of a tube, then it will be possible to obtain a very 
 
72] CUBICAL EXPANSION 191 
 
 fairly accurate measure by alternately passing cold water and steam through 
 the tube itself, the outside of the tube, except near the ends where the 
 marks are made, being wrapped round by some non-conductor of heat, 
 such as felt or cotton wool. 
 
 72. Cubical Expansion. Suppose that the volume of a body at 
 C. is F , then the volume V t at a temperature t is given by an 
 expression of the form 
 
 ....... (1) 
 
 where a. and b are constants. Some experimenters have expressed their 
 results by a formula involving the cube of t, such as 
 
 
 but it has been found that a formula involving only the first and second 
 powers of t is sufficient to express the results given by experiment to 
 an accuracy as great as that of the measurements, and hence it is 
 unnecessary to include the term in t 3 . 
 
 If p Q and p t are the density of the substance at and t respectively, 
 
 then Pt /p Q - VJ V t = 1/(1 + at + to 2 ). 
 
 Now in every case where a and b are small we may neglect the squares 
 and higher powers of these quantities as well as their products. Hence 
 1/(1 + at + to 2 ) = 1 -at- M\ so that 
 
 Pt = Po (l-at-M*) ....... (2) 
 
 If we are given the volume V t at a temperature t and require to find the 
 volume at a temperature t' we have 
 
 F,= F O (!+*+ to 2 ) 
 
 V t ,= V Q (\+at' + to' 2 ). 
 Hence 
 
 F _ F 1 + at + to' 2 
 * I+at + bt 2 
 
 (3) 
 
 if as before we suppose that a and b are both small quantities. 
 
 If we define the mean coefficient of expansion between any two 
 temperatures t and t' as the change of volume per degree of unit volume 
 of the body at the lower temperature t, or in symbols as 
 
 V,(t'-t) 
 
192 EXPANSION OF SOLIDS AND LIQUIDS [ 71 
 
 we get from (3) that the mean coefficient of expansion 
 
 = E*rJj z^-j 2 ) 
 
 V t (t' -t)~ t'-t 
 
 = a + b(t' + t) .......... (4) 
 
 if a and b are small. 
 
 If, on the other hand, we define the mean coefficient of expansion 
 between t and t' as the change in volume per degree of unit volume 
 of the body 'at (7., or in symbols as 
 
 we have that the mean coefficient of expansion j8' is given by 
 V v -V t FoMf - /) + 6(*' 2 - * 2 )} 
 
 Hence we see that when a and b are so small that the squares and higher 
 powers of these quantities can be neglected, the value of the mean 
 coefficient of expansion is the same whichever of these definitions we 
 employ. 
 
 The true coefficient of expansion at any given temperature t is the 
 rate at which the volume varies at that temperature per unit volume, or, 
 if we like, is the mean coefficient of expansion between two temperatures 
 taken one on either side of t, and so near t that the expansion does not 
 change appreciably over the range of temperature considered. Thus the 
 true coefficient of expansion at t is obtained by making t' coincide with 
 t in the above expressions for /?, and we thus obtain that the true 
 coefficient of expansion at t is 
 
 a + 2bt ......... (5) 
 
 We may also obtain this expression by noting that by definition the true 
 coefficient of expansion is 
 
 1 dV t 
 
 F dt 
 
 which from (1) is a + 2bt. 
 
 In the case of liquids and solids, a and b are so small that the 
 approximations we have made above are sufficiently close. In the case 
 of gases, however, this is not the case, and therefore to prevent confusion 
 we shall for gases always adopt the definition of the coefficient of 
 expansion in terms of the volume at C., that is, in symbols 
 
73] 
 
 THE WEIGHT THERMOMETER 
 
 193 
 
 In the case of an isotropic solid, if the length at t is L t , and that at 
 L Q , we have 
 
 The determination of the absolute coefficient of cubical expansion of a 
 liquid is a problem which can only be solved by means of elaborate experi- 
 ments, such as those of Regnault, by the method of balancing columns. 
 We can, however, measure the relative expansion of a liquid 
 and a solid containing vessel, and then, if we know the 
 absolute expansion of the solid, we can at once deduce the 
 absolute coefficient of expansion of the liquid. To obtain 
 the coefficient of expansion of the envelope, we may make 
 use of the fact that the expansion of mercury and of water 
 have been determined with great accuracy. Thus an experi- 
 mental determination of the value of the coefficients of 
 apparent expansion of mercury or water when enclosed in 
 this envelope will allow us to deduce the coefficient of 
 expansion of the envelope. 
 
 The methods generally employed for measuring the 
 apparent expansion of a liquid are (1) the weight thermo- 
 meter, (2) the volume dilatometer, and (3) the hydrostatic 
 balance. 
 
 73. The Weight Thermometer or Dilatometer. The 
 simplest form of weight thermometer is shown in Fig. 78. 
 It consists of a glass bulb with a capillary neck, the neck 
 being generally bent round as shown. The dilatometer 
 having been weighed is filled with the liquid at a tem- 
 perature t lt and again weighed. The difference in the weights gives the 
 weight w of the contained liquid. The dilatometer is then heated to 2 , 
 some of the liquid escaping. By again weighing, the weight z0 2 of liquid 
 which fills the dilatometer at t 2 is obtained. 
 
 If v l p l and v 2 p. 2 are the volumes of unit mass and density respec- 
 tively of the liquid at the two temperatures, and a is the true coefficient 
 of expansion of the liquid, we have by definition 
 
 FIG. 78. 
 
 1 ^2z3 = l f%- i \ 
 
 t 2 - ^ V 1 * 2 - ^ If7, 
 
 m 
 
 If V l and F 9 are the volumes of the dilatometer at the temperatures T 
 and t.y and /3 is the coefficient of cubical expansion of the material of 
 which the dilatometer is constructed, then 
 
 Now 
 
 and 
 
194 EXPANSION OF SOLIDS AND LIQUIDS 
 
 Hence _2 = L! = _1 . _2 
 
 [73 
 
 Substituting this value for v 2 /v l in (1), we get 
 
 (2) 
 
 If d is the weight of liquid which escapes at the higher temperature, so 
 that w l - w. 2 = d, then (2) may be written 
 
 o d ( I \ 
 
 a ~ P = - \ . T- + \ 
 
 w l U? - ^ J 
 
 (3) 
 
 a form sometimes convenient. 
 
 Thus if either a or /3 is known, the other can immediately be cal- 
 culated. The coefficients of expansion of water and mercury can be 
 obtained from Table 20, so that for these liquids a is known, and /3 
 can therefore be calculated. 
 
 The coefficients of expansion of the standard thermometer glasses have 
 been measured with considerable care, and thus if a dilatometer is con- 
 structed of either of these glasses the value of ft is known, and the 
 experiment with mercury or water can be omitted. 
 
 The following are the values 1 of the coefficients of cubical expansion, 
 that is, the coefficients ft and /3' in the equation 
 
 
 H 
 
 i<w 
 
 Verre dur . 
 
 2225-2 
 
 1-083 
 
 Jena (16'") .... 
 
 2316-7 
 
 1-071 
 
 Jena (59'") .... 
 
 1703-9 
 
 0-746 
 
 In the expression (2) the factor w L /w 9 is often very nearly unity. If 
 we assume that it is so nearly unity that we may put it equal to 1, then 
 (2) reduces to 
 
 f) 1 W~t % / I \ 
 
 *2-*l W '2 
 
 This simplified formula must, however, never be used unless we have 
 satisfied ourselves by actual calculation that the assumption is justified. 
 
 1 Thiesen, Scheel, and Sell, Zeit. fiir Instrumentenkunde (1896), xvi. p. 54. 
 
74] MEASUREMENT OF EXPANSION OF GLASS 195 
 
 The magnitude of the error which may be caused by this assumption 
 is shown in the following example : 
 
 Weight of dilatometer empty . . . = 3'0510 grams. 
 . full of mercury at 29 C. = 76*0747 
 
 98 C. = 75-2925 
 Then =-0001569 + 1-018 a. 
 
 Taking the coefficient of expansion of mercury between 29 and 98 as 
 0001818 we get 
 
 a = -00002446. 
 
 If, however, we had put the coefficient of a as unity, we should obtain 
 
 a = -0000249, 
 
 and hence have produced an error of about 2 per cent. 
 
 74. Measurement of the Cubical Expansion of Glass with a Simple 
 Form of Weight Dilatometer. Take a piece of glass tube having a bore 
 of about a centimetre and close one end, carefully removing any blob of 
 glass produced, so that the thickness of the glass is the same at the end 
 as at the sides. Then about 5 centimetres from the closed end heat 
 the glass in the blowpipe flame. During the heating 
 keep the glass in continuous rotation, but do not 
 draw the tube out. In this way the glass will become 
 thickened at the point heated. When the bore 
 has been reduced to about half, remove the glass 
 from the flame, and then slowly draw out the heated 
 portion till the capillary has a bore of about a 
 millimetre. In this way the walls of the capillary 
 will be almost as thick as the rest of the bulb, and 
 by gently heating in a small flame the capillary can 
 easily be bent into the form shown in Fig. 78. 
 
 Having carefully cleaned the outside of the bulb, 
 weigh it. 
 
 The bulb has now to be filled with clean mercury. 
 This operation is one of some little difficulty. The 
 bulb is supported, as shown in Fig. 79, being sus- 
 pended from a retort stand by some copper wire, the 
 stand itself being placed in a mercury tray to 
 catch the mercury in case the bulb breaks. The 
 mercury is placed in a porcelain crucible, the end of the capillary 
 dipping below the surface. The bulb and the mercury are gently 
 heated with a Bunsen flame, expelling some of the air. On allow- 
 ing the bulb to cool some of the mercury will be driven back into the 
 bulb. The whole of the dilatometer, together with the mercury in the 
 crucible, must then be heated up to the boiling point of mercury, and 
 nearly the whole of the mercury in the bulb allowed to boil away. On 
 removing the flame the mercury vapour will condense, and the mercury 
 
 FIG. 79. 
 
196 
 
 EXPANSION OF SOLIDS AND LIQUIDS 
 
 [74 
 
 will be driven over into the bulb. It is most important that the 
 mercury in the crucible should be kept hot, otherwise when the mercury 
 commences to pass over into the bulb the glass will be cracked owing to 
 the cooling produced by the cold mercury. When the mercury is quite 
 cold, it will generally be found that there is a small bubble of air left. 
 By shaking the dilatometer this bubble should be made 'to move as far up 
 the capillary as possible. Then placing the end of the capillary below 
 the surface of the mercury, gently heat the bulb so as to drive the bubble 
 out of the capillary. 
 
 Next suspend the dilatometer so that the bulb and as much of the 
 neck as possible is immersed in water at a temperature a degree or so 
 above that of the air of the laboratory, the end of the capillary still 
 dipping below the surface of the mercury in the crucible. An arrange- 
 ment for supporting the dilatometer is shown in 
 Fig. 80. The cork is split, and fits tightly into 
 a hole in the wooden cross-bar B. Keep the 
 water stirred, and allow the dilatometer to remain 
 for at least a quarter of an hour. Then read 
 the temperature of the water, and remove the 
 dilatometer from the water and from the mercury 
 in the crucible. Carefully dry the dilatometer 
 and weigh. While drying the dilatometer and 
 transferring it to the balance case be careful 
 not to warm it, otherwise some of the mercury 
 will be expelled owing to expansion. It was for 
 this reason that the temperature of the water 
 was taken slightly above that of the air. 
 
 The dilatometer must now be suspended in a beaker of boiling water 
 so that it is entirely immersed, capillary and all, and the water kept boil- 
 ing for at least twenty minutes. The temperature having been read, 
 remove the dilatometer, dry, and again weigh. 
 
 As a check on the weighings transfer the mercury which has escaped 
 into the beaker in which the dilatometer has been heated to a watch 
 glass, and weigh. The weight of this mercury ought to be equal to the 
 difference in the weights of the dilatometer. 
 
 If the only delicate balance available will not carry the weight of the 
 full dilatometer, then weigh the dilatometer empty and full at the lower 
 temperature on a less sensitive balance which will carry this load, and 
 thus obtain the quantity w^. Then weigh the globule of mercury which 
 escapes when the dilatometer is heated to 100 on the sensitive balance, 
 and thus obtain d. By this method of procedure the quantities u\ and 
 d, which enter into the expression for calculating the coefficient of expan- 
 sion of the glass, can be determined with about the same percentage 
 accuracy ; while if the less sensitive balance had been used to weigh 
 the dilatometer at both temperatures, and the quantity d deduced 
 from the difference of the weighings, this quantity would not be known 
 with anything like the same percentage accuracy as is w lm 
 
 FIG. 80. 
 
75] MEASUREMENT OF EXPANSION OF A LIQUID 197 
 
 By taking the lower temperature, when part of the capillary has to 
 project outside the bath in which the dilatometer is immersed, nearly 
 equal to the room temperature, the error produced on account of the 
 temperature of the mercury in the capillary not being the same as that 
 of the bulb is reduced to a minimum. It is, however, sometimes neces- 
 sary to work with the lower temperature differing from that of the room. 
 For instance, it may be desired to measure the mean coefficient of cubical 
 expansion of a liquid between and 100. In such a case it is evident 
 that it will be impossible to weigh the bulb when it has been filled at 0, 
 since when it is warmed up in the balance case some of the liquid would 
 escape. It is, therefore, best, having filled the dilatometer, to pack it 
 round with melting ice, leaving as little of the capillary projecting as 
 possible, and allow the end of the capillary to dip below the surface of 
 some of the liquid contained in a small crucible. When the bulb has 
 taken up the temperature, the crucible is removed and weighed. The 
 end of the capillary having been again dipped in the liquid in the weighed 
 crucible, the bulb of the dilatometer is placed in boiling- 
 water. When the temperature has had time to become 
 steady, the crucible is removed and again weighed. The 
 difference in the weights of the crucible gives the quantity d. 
 The dilatometer is then removed from the boiling water, and 
 when it has cooled, the capillary in this case no longer dipping 
 below the surface of the liquid, the dilatometer is weighed, 
 and from this weighing, and that of the dilatometer empty, 
 the quantity w 2 is deduced. Then w = w^ + d. 
 
 Having measured the coefficient of cubical expansion 
 of the glass, the weight dilatometer may be used to measure 
 the true coefficient of expansion of any other liquid. The 
 procedure to be adopted is the same as that described above, 
 and then since we now know ft equations (2) or (3) of the 
 last section will allow of our calculating a. 
 
 75. Measurement of the Coefficient of Expansion of a 
 Liquid with the Volume Dilatometer. A simple form of 
 volume dilatometer is shown in Fig. 81. It consists of a 
 bulb A, with a capillary stem BC, on which a scale has been 
 etched. When using the instrument to measure the expansion 
 of volatile liquids, and also for keeping out dust when the FlGt 
 instrument is not in use, it is a good thing to have a tap D 
 at the top. There also ought to be a small auxiliary bulb at c. This 
 form of dilatometer suffers from the same objections as the form of weight 
 thermometer shown in Fig. 78, in that it is difficult to fill and clean, 
 but to a somewhat smaller degree. Owing to the fact that the capillary 
 is straight, it is possible to fill the bulb by means of a tube drawn 
 down to a fine capillary and the arrangement shown in Fig. 36, also 
 since the dilatometer can be plunged into a water bath till only a few 
 millimetres of the liquid column project above the surface of the water, 
 the correction for cool column can be rendered practically negligible. 
 
198 EXPANSION OF SOLIDS AND LIQUIDS [ 75 
 
 In order to reduce the measurements with the volume dilatometer it is 
 necessary to know the volume of the bulb up to the zero graduation of 
 the scale, and also the volume of the capillary from the zero of the 
 graduations up to any division. These quantities can be determined by 
 means of mercury. First the weight w of the dilatometer when empty is 
 determined, then the weight Wx when filled with mercury up to a point 
 (x) near the top of the scale, the temperature being that of melting ice. 
 Then the volume up to the division x at C. is given by 
 
 Wx-w 
 
 where A d is the density of mercury at 0. 
 
 Next some of the mercury is removed till, when the whole is at a 
 temperature of melting ice, the end of the mercury column is near the 
 zero of the scale, say at the division y, and let the weight be Wy. Then 
 the volume up to the y division is 
 
 Wy - w 
 
 and the volume of one division of the scale is 
 
 Wx-Wy 
 
 (*-y)\ 
 
 Hence the volume up to the zero is 
 
 Wy-w_Wx-Wy , 
 
 AC (*-y)\ y 
 
 We have, in the above, supposed that the bore of the capillary is 
 cylindrical. This must always be tested by measuring the length of a 
 thread of mercury when it occupies different parts of the scale. If the 
 length varies from one part of the scale to another, indicating that the 
 bore is irregular, then the scale will have to be calibrated in the same way 
 as the tube of a thermometer, and a table of corrections prepared, by 
 means of which the volume from the zero up to any division can be 
 obtained. 
 
 To determine the coefficient of expansion of the glass, an experiment 
 must be performed with mercury. 
 
 Let the volume, as read off on the scale, at a temperature f l be V v 
 and the volume at a temperature 2 be F 2 . Then the volume of the 
 liquid at t. 2 is F^l +/3(> - ^)}, where /? is the coefficient of cubical ex- 
 pansion of the liquid. 
 
 The volume occupied in the dilatometer is apparently F|>, but owing 
 to the expansion of the glass this volume is really F 2 {1 + a.(t.> - ^)}, where 
 a is the coefficient of cubical expansion of the glass. 
 
75] MEASUREMENT OF EXPANSION OF A LIQUID 199 
 Hence the increase in volume of the liquid is 
 
 and this must be equal to 
 
 (3) 
 
 Thus knowing /3 } in the case of mercury, we can calculate a, and 
 then in the case of any other liquid we can use this value of a to calculate 
 the coefficient of cubical expansion of the liquid. 
 
 The form of volume dilatometer shown in Fig. 81, but without the tap, 
 can easily be made by the student, the stem being formed by a piece of 
 
 wide bore thermometer tube. In the absence of a tap, the top can be 
 closed by about a centimetre of rubber tube 
 and a small piece of glass rod. The gradua- 
 tions on the stem can be made by the method 
 described in the Appendix. 
 
 If an air-pump is available, then the 
 arrangement shown in Fig. 82 will be found 
 
 to save an enormous amount of time. It 
 consists of a small bottle with a well-fitting 
 india-rubber cork. This cork is pierced by 
 two holes. The dilatometer stem passes 
 through one, and a piece of bent glass tube 
 through the other. The liquid is placed in 
 the bottle, and the bent tube is connected by 
 means of a length of thick-walled rubber 
 tube to the air-pump. To fill the dilato- 
 meter the pump is worked, and the bulb of 
 the dilatometer is heated with a Bunsen 
 name to drive off any moisture. On ad- 
 mitting air from the pump the liquid is driven into the bulb. By 
 admitting the air very slowly, and when enough liquid has entered to 
 fill the dilatometer up to the required height in the capillary, raising the 
 dilatometer through the cork, the advent of any further liquid can be 
 prevented. 
 
 If the dilatometer is fitted with a tap, then the cork must be split on 
 one side. 
 
 Having weighed the dilatometer empty, fill it with mercury up to a 
 little above the top graduation, and place it in melting ice prepared as 
 described in 61. When the temperature has become stationary, read the 
 position of the top of the mercury column, and then again weigh. Next 
 heat the dilatometer so that some of the mercury escapes, and hence when 
 
 f';he instrument is placed in ice the mercury meniscus is near the bottom 
 )f the scale. When the temperature is again 0, read the position of the 
 
 FIG. 82. 
 
200 EXPANSION OF SOLIDS AND LIQUIDS [ 76 
 
 meniscus, and again weigh the dilatometer. From the readings and weigh- 
 ings calculate the volume up to the zero of the scale and the volume of 
 one division of the capillary. 
 
 The dilatometer being placed in steam read the position of the 
 meniscus, also the barometer. Having calculated the temperature of the 
 steam from the barometer reading in the manner described in 60, 
 calculate by (3) the coefficient of cubical expansion of the glass. 
 
 To remove the mercury, replace the dilatometer in the filler (Fig. 82), 
 and produce as good a vacuum as possible. If the mercury does not flow 
 out, gently heat the bulb. 
 
 The procedure when measuring the expansion of a liquid is practically 
 the same as the above, namely, the bulb is filled so that at the 
 meniscus is near the bottom of the scale. The position of the meniscus 
 is then read off when the instrument is maintained at different tempera- 
 tures, and the coefficient of expansion calculated by (3). 
 
 76. The Hydrostatic Balance Method of Measuring the Coefficient 
 of Expansion of a Liquid. This method consists in determining the loss 
 of weight of a piece of glass, or other solid, when immersed in the liquid 
 at different temperatures. It is only applicable when a comparatively 
 large volume of the liquid is available. In order to obtain the absolute 
 expansion of the liquid, we require to know the coefficient of cubical 
 expansion of the solid used. This solid generally consists of a cylindrical 
 piece of glass tube, closed at the ends, with a small hook formed at one 
 end for the attachment of the suspending wire. A little mercury inside 
 the glass serves to ensure its sinking. 
 
 To determine the expansion of the glass, an experiment can be made 
 with a liquid of known coefficient of expansion. Since mercury cannot 
 be used for this purpose, on account of its great density, water must be 
 employed. The expansion of the glass can also be measured by the 
 method of the weight thermometer, after which the neck of the thermo- 
 meter is fused off so that the bulb forms the sinker. 
 
 Let V l be the volume of the sinker at a temperature ^ and V< 2 the 
 volume at t 2 , these volumes being obtained by weighing in water, the 
 formula used being 
 
 where ?0' is the weight of the sinker in air, and w" its weight when 
 immersed in water at a temperature t lt Aj is the density of the water as 
 given in Table 5, and <r is the density of the air (see 30). 
 
 Then F 2 = ^{1 +/?(* -fj)}, 
 
 -'- 1 
 
 where /? is the coefficient of cubical expansion of the glass. 
 
 If the loss of weight of the sinker when immersed in a liquid at 
 
77] EXPANSION OF MERCURY IN GLASS 201 
 
 temperatures t l and t 2 is w l and w 2 respectively, then the coefficient of 
 cubical expansion of the liquid between these temperatures is given by 
 formulae (2) and (3) on p. 194. 
 
 The sinker is suspended by means of a fine platinum wire (diameter 
 about '004 cm.), the upper end of the wire being attached to the pan of 
 a balance, which must be one in which there are no pan arrestors. A hole 
 must be bored through the base of the balance to allow of the passage 
 of the wire, and the wire ought to be at least 40 cm. long, so that the 
 heated liquid is not near the balance. For this reason also it is as 
 well to allow the wire to pass through a small hole in the table which 
 carries the balance, the liquid being supported on a stand under the table. 
 It is not necessary for the whole of the wire to be of platinum. Only 
 the portion below the surface of the liquid and for about 10 cm. above 
 the surface need be of this metal ; the rest can be of fine copper wire. 
 
 In order to reduce the disturbance produced by capillarity where the 
 wire cuts the surface of the liquid, it is almost necessary that the surface 
 of the platinum wire should be treated as described on p. 83. 
 
 Having suspended the sinker, obtain its weight in air. Next im- 
 merse the sinker in the liquid, the vessel containing the liquid being 
 packed round with non-conducting material, so that the rate of loss of 
 heat is small. The liquid must be well stirred just before an observa- 
 tion, and then, when the currents set up by the stirring have subsided, 
 the balance must be read, as well as a thermometer in the liquid. The 
 bulb of this thermometer must be placed as near the sinker as possible. 
 
 When the temperature of the liquid is higher than that of the sur- 
 rounding air, it is as well to adjust the weights so that the pointer is 
 slightly deflected towards the side indicating that the sinker is too 
 heavy. As the liquid cools its density will increase, and the loss of 
 weight of the sinker therefore will also increase, and so the balance will 
 gradually come to its zero position. When the pointer passes the zero, 
 read the thermometer, and take this reading as giving the temperature 
 corresponding to the weights in the balance pan. 
 
 77. Determination of the Coefficient of Apparent Expansion of 
 Mercury in Glass by means of a Thermometer. A method of obtaining 
 the mean coefficient of apparent expansion (e) of mercury in glass when 
 a calibrated thermometer of suitable range is available, has been used at 
 Reichsanstalt. 1 Suppose that we have a thermometer reading from 
 - 100 C. to + 100, the bore having been calibrated, and that the volume 
 of the bulb up to the zero division is equal to u times the volume between 
 two consecutive degree divisions on the stem. If, now, the thermometer is 
 immersed first in steam at 100 C. and then in ice at 0, and the number 
 of degrees of the scale between these two fundamental points is /, we 
 have 
 
 1 Wissen. Abhandlungen, i. 102 ; see also Chree, Philosophical Magazine [5] 
 (1898), xlv. 322. 
 
202 EXPANSION OF SOLIDS AND LIQUIDS [ 77 
 
 Next run some of the mercury into the bulb at the top of the stem, and 
 repeat the observations at and 100. If the mercury run into the top 
 bulb would at fill a degrees, and if the fundamental interval is 
 now /j, we have 
 
 - A 
 
 ~100(n-a) 
 
 Next repeat the observation when a different quantity (b) of mercury is 
 run into the top bulb, so that we have 
 
 e= 
 
 100(-d) 
 
 (3) 
 
 From equations (1) and (2) or (1) and (3) we can deduce both n and e. 
 In this experiment a ought to correspond to about 50 and b to about 
 100 in order to obtain a sufficiently great variation in / to give a 
 satisfactory measure of e. 
 
CHAPTER XI 
 
 THERMAL EXPANSION OF GASES 
 
 78. The Air Thermometer. There are two forms of air thermometer. 
 In one the volume of the gas is supposed to remain constant throughout, 
 and the temperature is deduced from the change in pressure. In the 
 other form the pressure is kept constant, and it is the change in volume 
 of the gas which is used to give the temperature. In practice, however, 
 it is found impossible to keep the volume in the one case and the pres- 
 sure in the other rigorously constant. Hence it is advantageous to 
 obtain expressions for the temperature when both the pressure and 
 volume are supposed simultaneously to vary. These expressions can then 
 be modified to suit the case when either the volume only changes 
 slightly or the pressure only changes slightly, as the case may be. 
 Let 
 V t = volume of bulb, and that part of the capillary which is heated, 
 
 at a temperature t ; 
 
 v t = volume of gas in the remainder of the capillary, the manometer, 
 ifec., i.e. in the dead space when the bulb is at a temperature 
 t, the temperature of this air being t' ; 
 
 p f = the pressure to which the gas is subjected at a temperature t. 
 Also let 
 
 F , u , and ;> be the corresponding quantities at a temperature 
 
 of C. ; and 
 V v v v and^j at the temperature of steam when the barometeric 
 
 pressure is B. 
 
 Then if p is the density of air at and standard pressure, the 
 density of the air at a temperature t and pressure p t is 
 
 
 PoPt 
 
 where a is the coefficient of expansion of air. Hence the mass of air 
 contained in the bulb, of which the volume is V t , is 
 
 PoPt v 
 
 and the mass of air contained in the dead space is 
 
 PoPt _ 
 (l+uf)7G '' 
 
204 THERMAL EXPANSION OF GASES [ 79 
 
 Hence the total mass m of air contained in the thermometer is given by 
 
 or {V'/(l + at) + v t /(l + ait')}p t = = constant . . . (1) 
 
 Po 
 
 This is the fundamental equation of gas thermometry. It may be 
 written in a slightly different form. Thus if 6 is the temperature 
 measured from the absolute zero corresponding to t, we have 
 
 Hence equation (1) can be written 
 
 { 5 + |j^ = constant^/,: (say) ..... (2) 
 
 where & is the absolute temperature corresponding to t'. 
 
 If an observation is made at C. or # on the absolute scale, 
 we have 
 
 Further, if an observation is made in steam at a pressure 13, the 
 temperature of the steam at this pressure being 100 + 1> or # + 100 + b 
 on the absolute scale, we have 
 
 If ft is the coefficient of cubical expansion of the material of which the 
 bulb is constructed, then 
 
 pt)=v {i+p(0-e )} ..... (5) 
 
 These equations will serve to perform all the calculations necessary 
 when using air thermometers, whether of the constant volume or the 
 constant pressure type. In the constant volume type v t remains very 
 nearly constant, while p t varies; while in the constant pressure type 
 v t varies considerably, while p t is kept as nearly as possible constant. 
 
 79. Measurement of the Coefficient of Increase of Pressure of 
 Air with the Air Thermometer. A simple form of constant volume 
 air thermometer is shown in Fig. 83. The bulb A is connected to the 
 manometer by a fine capillary tube, a three-way tap c being included. 
 The glass reservoir K has a black glass point D attached near the top, and 
 the surface of the mercury is always brought to coincide with this point. 
 The object of the reservoir is that when the bulb is heated the air may 
 not be driven over into the manometer tube, even if the pressure is not 
 
79] CONSTANT VOLUME AIR THERMOMETER 205 
 
 increased. The lower part of the reservoir K is connected by means of a 
 three-way cock with the manometer tube EF and a length of thick-walled 
 rubber tubing, which is attached to the mercury reservoir G. This 
 mercury reservoir can be raised or lowered by means of a cord which, 
 after passing over a pulley near the top of the apparatus, is wound on a 
 drum. This drum can be rotated by a tangent screw actuated by a 
 
 FIG. 83. 
 
 handle J. The manometer tube EF is designed to measure the whole 
 pressure to which the gas is subjected, that is, it is not necessary to read 
 an auxiliary barometer. The top of the tube is fused to a well-ground 
 tap F and a small separating funnel. Just below the tap the tube is 
 constricted, as shown at I on the enlarged drawing (a). 
 
 The manometer tube is filled in the following manner : The tap F 
 being open, and the tap E turned so as to cut off connection with K, the 
 reservoir G is raised till the mercury flows into the funnel. The tap F is 
 
206 THERMAL EXPANSION OF GASES [ 79 
 
 then closed, and the reservoir lowered as far as possible. In this way a 
 Torricellian vacuum will be left in the upper part of the tube EF. 
 Owing, however, to moisture condensed on the sides of the tube the 
 vacuum will generally be bad. To get rid of this moisture the tube must 
 be heated, the mercury reservoir remaining at its lowest. Then by 
 raising the reservoir, and when it is at its highest point opening the tap r, 
 the vapour can be driven out. By repeating this operation several times 
 it %ill be possible to get quite a good vacuum. The object of the con- 
 striction i is that owing to capillarity the mercury in the small bulb 
 above the constriction will remain, and if any air leaks in at the tap F 
 this will at once be indicated by the mercury in this bulb being forced 
 down. Thus as long as there is mercury in the bulb we can feel 
 confident that the vacuum remains good. 
 
 When the tap E is turned so as to connect K with the manometer, and 
 the surface of the mercury is adjusted to coincide with the end of the point D, 
 then the height of the mercury meniscus L above the point D will give 
 the pressure to which the gas in the bulb is subjected. The height of 
 this mercury column is read off on a divided scale fastened alongside the 
 tube, the zero of the scale being adjusted to coincide with the point D. 
 
 The bulb A has now to be filled with dry air. To do this the side 
 tube M is connected, through some drying tubes containing phosphorus 
 pentoxide, with an air-pump. The mercury surface in the reservoir K 
 having been lowered as much as possible, the tap E is turned so as to 
 shut off K from the manometer and reservoir G, and then the bulb is 
 exhausted. While the vacuum is kept as good as possible, the bulb A, 
 the connecting tubes, and the vessel K are warmed with a flame so as to 
 drive off all moisture. Air is then admitted through the drying tubes. 
 The operation of exhaustion and admission of air, which latter operation 
 ought to be performed very slowly, is repeated several times. In this 
 way the bulb can be filled with dry air, and the film of moisture which 
 always clings to the surface of glass can in a great measure be removed. 
 
 Having either adjusted and filled the instrument as described above, 
 or ascertained that it has been so filled, place the bulb A inside a funnel 
 which is supported on a retort stand, and pack with pounded ice, treated 
 in the manner described in 61. When the bulb has been in the ice for 
 a few minutes, adjust the surface of the mercury to coincide with the 
 end of the point D, using the reflection of the point in the mercury 
 surface to assist in making the adjustment. If after two or three 
 minutes the adjustment is still complete, read the thermometer T and the 
 position of the mercury meniscus L. 
 
 The bulb must now be placed either in boiling water, the temperature 
 of which is taken by a thermometer of which the boiling point error is 
 known, or in a steam-jacket, such as is shown in the figure. The wooden 
 lid s is split in halves so that it may be fitted round the bulb stem. 
 Steam is passed in at o, and the position of the meniscus roughly ad- 
 justed to coincide with the point D. After the steam has been passing 
 for about ten minutes, the adjustment of the mercury meniscus is com- 
 
79] CONSTANT VOLUME AIR THERMOMETER 207 
 
 pleted, and the thermometer T and the position of the meniscus L are 
 again read. The height of the barometer must also be noted, so as to 
 allow of the temperature of the steam being obtained. 
 
 The height of the mercury column in the manometer will have to be 
 reduced to what it would be at (and in very accurate measurements to 
 standard gravity) in the manner described in 58. The temperature of 
 the steam must also be calculated as in 60. 
 
 The end of the point D may not exactly coincide with the zero of the 
 scale used to read the height of the column. This point may be tested, 
 and the correction, together with that for the effect of capillarity ( 57), 
 can be obtained if a standard barometer is available. The tap c being 
 turned so as to connect the upper part of K with the open air, the 
 mercury meniscus is adjusted to the point D, and the temperature and 
 height of the mercury column are read. The barometer being also read, 
 the difference between the values for the barometric pressure obtained 
 after reducing to will give the correction to be applied to the readings 
 of the manometer. 
 
 It is possible to obtain a formula by means of which the coefficient of 
 increase of pressure a is expressed in terms of the observed quantities. 
 Such an expression, however, is very complicated and quite un suited to 
 computation. It is much easier to perform the reduction by successive 
 steps, using the equations given in the last section. We have from 
 equations (3), (4), and (5), since in this experiment v 1 = v Qt 
 
 IPO/?) 
 
 or fl LM.^f 1 100_ !_ V \ 
 
 \0o <V vj IQ \e~+ 100 + b + e~~+ 100 '+& + #/ * ^) Pl 
 
 Neglecting the effect of the dead space and the expansion of the bulb, 
 this gives as a first approximation for # 
 
 .Po_ Pi 
 
 or 
 
 We may now employ this approximate value of # to calculate the values 
 of OQ and 6^ from the observed temperatures given by the thermometer 
 T. Further, to obtain the values of the correction terms in (6) we 
 require to know the values of /3 and of v IV Q . These quantities do not 
 require to be known with any very great degree of accuracy, since they 
 are only involved in small correction terms. 
 
 (1) If some of the glass of which the bulb A is made is available, 
 then p may be measured by using this glass to make a weight dilato- 
 
208 THERMAL EXPANSION OF GASES [80 
 
 meter, and observing in the manner described in 74. For most purposes 
 it will be sufficiently accurate to assume the value 0'0000232 for p. 
 
 (2) The quantity ^ /F , which is the ratio of the volume V Q of the 
 dead space to the volume V Q of the bulb, may be determined in the 
 following manner : The bulb being immersed in water, which has stood 
 for some time so as to acquire the temperature of the air, the mercury is 
 adjusted to the point D, and the height TT I of the mercury column is read. 
 The reservoir G is then very slowly raised till the mercury has almost 
 reached the bend in the capillary just above the bulb, and the pressure 
 7r is read off. We then have 
 
 or 
 
 The values of the various quantities are then inserted in equation (6), 
 which then becomes a simple equation involving the single unknown # , 
 and from it we therefore deduce this quantity, and then a= 1/0 . 
 
 80. Determination of the Corrections to a Mercury Thermometer 
 at High Temperatures by means of the Constant Volume Air Ther- 
 mometer. Determine the pressure of the air thermometer when the 
 bulb is in melting ice and steam respectively, as in the preceding section. 
 Then place the bulb in a beaker containing glycerine, the top of the 
 beaker being closed by a wooden disc similar to that used to close the 
 steam-jacket in Fig. 83. There must be two additional holes in the 
 wooden disc, one for the passage of a mercury thermometer reading to 
 300, and the other for the handle of a stirrer. The beaker must rest 
 on a sheet of asbestos card, and it is an advantage to wrap the sides 
 of the beaker with asbestos cloth, so as to reduce the loss of heat from 
 the sides. Place a small Bunsen burner below the beaker, and adjust 
 the size of the flame so that the temperature of the glycerine becomes 
 stationary at about 150. Then read the thermometer and the height 
 of the mercury column in the manometer, the meniscus having been 
 adjusted to the standard position. Then increase the size of the flame, 
 and in the same way obtain readings at 200, 250, and 300. 
 
 Having first reduced all the manometer readings to 0, proceed as in 
 the previous section to calculate the value of $ . Then from equations 
 (2), (3), and (5), 78, we have 
 
 in which all the quantities except 6 are known ; for in the term fit we 
 may, without causing appreciable error, take t as the temperature of the 
 glycerine bath as given by the mercury thermometer, while # /F can 
 be obtained as in the last section. Then the temperature t a of the bath 
 on the scale of the air thermometer is given by 
 
81] CONSTANT PRESSURE AIR THERMOMETER 209 
 
 The difference between the reading of the mercury thermometer and 
 t a gives the correction to the thermometer. 
 
 81. The Constant Pressure Air Thermometer. A form of constant 
 pressure air thermometer which is fairly simple is represented in Fig. 84. 
 It is slightly modified from that described by Callendar and Griffiths. 1 
 The thermometer bulb A is connected through a capillary to a second 
 bulb B, and also to an oil manometer c. The lower part of the bulb B 
 has two tubes attached, each of which is furnished with a stop-cock. 
 
 FIG. 84. 
 
 Through E mercury can be passed into B from the reservoir G, and 
 through D mercury can be run out of B into a weighed beaker placed 
 to receive it. The bulb B is surrounded , by water contained in a glass 
 cylinder J, the temperature of the water being given by a finely divided 
 thermometer T. The manometer c is fitted with a three-way cock F, 
 and the limb nearest the bulb is graduated in millimetres. The bulb A 
 may either be connected to the rest of the apparatus by a continuous 
 length of capillary, or, what is generally much more convenient, the 
 bulb B and manometer c may be made in one piece, and this be joined 
 
 1 Philosophical Transactions of the Royal Society (1891), vol. clxxxii. p. 119. 
 
 
 
210 THERMAL EXPANSION OF GASES [ 81 
 
 to the capillary attached to the bulb A by a joint H. This joint is made 
 by drawing down a piece of glass tube till it makes a tight fit on 
 the outside of the capillary, then cutting off about 2 centimetres of 
 this tube, slipping it over the ends of the capillaries, and having well 
 heated the glass, running in sealing-wax between the glass sleeve and 
 the capillaries. A section of the joint is shown 
 ^^^ on a large scale in Fig. 85. 
 
 ' Before the instrument is mounted on the 
 
 stand and the bulb attached it will be necessary 
 to measure the volume of the bulb up to a 
 FIG. 85. point about 2 centimetres beyond the bend 
 
 in the capillary, up to which point the heat- 
 ing will in each case be made. This volume may be determined 
 with sufficient accuracy by weighing the bulb empty, and then when 
 filled up to the given point, which should be indicated by a mark 
 on the capillary, with water at 0. The water can be removed by 
 connecting the end of the capillary to a water-pump, and then heating 
 the bulb to a temperature over 100. The volume of the capillary 
 between the mark and the end where it is to be joined to the rest of 
 the instrument must also be determined. This can be done by drawing 
 up a thread of mercury by first warming and then cooling the bulb, and 
 then weighing the thread. 
 
 Finally, the volume of that part of the capillary between the joint H, 
 the zero of the scale on the manometer, and a mark K on the top of the 
 bulb must be determined, as well as the volume of unit length of the 
 divided portion of the manometer. These volumes can be determined 
 by weighing the mercury which fills the various portions of the tube. 
 
 To fill the bulb with dry air a pump is connected to I, and the tap D 
 is connected to some drying tubes, and the air is alternately exhausted and 
 readmitted through the drying tubes, the whole apparatus being kept 
 warm and the bulb A heated to a fairly high temperature. Dry and 
 clean mercury is then run in through E till the surface reaches almost 
 up to where the manometer joins the other capillary. Some oil, such 
 as is used in the Fleuss pump, is then sucked in through the tap I, the 
 tap being so turned that connection is cut off from the bulbs A and B. 
 The tap F is then turned so that the side tube I is closed, and by slightly 
 lowering the mercury in B the oil is caused to stand at the same level in 
 the two limbs of the manometer. To prevent the oil from being sucked 
 over into the bulbs, owing to the pressure in the bulbs falling below that 
 of the atmosphere, it is necessary to take care that, except when taking 
 a reading, the tap F is turned so as to cut off the two halves of the 
 manometer from one another. 
 
 To perform an experiment, place the bulb A in melting ice, and by 
 raising the reservoir G bring the surface of the mercury to the mark K. 
 When the temperature of the bulb has had time to reach 0, run off 
 mercury by the tap D into a weighed beaker till the manometer shows 
 that the pressure within the instrument is equal to that of the atmosphere. 
 
81] CONSTANT PRESSURE AIR THERMOMETER 211 
 
 Immediately read the temperature t' on the thermometer T and the 
 barometric height. Let the weight of mercury run out be w . 
 
 Next heat the bulb in steam, and determine the weight of mercury 
 w l which has to be run out so as to make the pressure inside equal to 
 the external pressure, and read the temperature t' on the thermometer T 
 and the barometric height. It is not essential that the pressure shall 
 be always adjusted to exactly atmospheric pressure, but if this adjust- 
 ment is not quite exact, then the difference in the levels of the oil in 
 the two limbs of the manometer, as well as the reading on the scale c, 
 must be noted. These readings allow of the pressure to which the gas 
 is subjected being calculated, as well as the volume between the zero of 
 the scale c and the top of the oil column. 
 
 The method of reducing the observations is very much the same as 
 that adopted in the case of the constant volume instrument. The dead 
 space here, however, may be considered as consisting of three portions. 
 The first portion consists of the volume v of the capillary up to the 
 mark K, and the zero of the scale c. This quantity remains constant 
 throughout the observations, and has been determined once for all before 
 the instrument was mounted. Secondly, there is the volume v of the 
 space between the zero of the scale c and the top of the oil column. 
 The value of this volume is obtained from the reading of the scale c 
 and the calibration of the manometer tube made before mounting. 
 Thirdly, there is the volume v" of the space occupied by the air between 
 the top of the mercury in the bulb B and the mark K. The volume of 
 this space is obtained from the weight of mercury run out. Thus 
 if p is the density of mercury at the temperature t' of the water bath J, 
 the value of v" when the bulb was in ice is given by V Q " = w /jo, while 
 when the bulb was in steam we have i\" = (w + MI) /p. If, however, the 
 temperature of the water bath J changes, then a correction will have 
 to be applied to allow for the expansion of the bulb B and the mercury 
 remaining in the bulb. The magnitude of this correction can easily 
 be obtained ; it is, however, simpler if the conditions are such as to 
 cause the temperature of J to vary to fill the jacket with melting ice, 
 so that the temperature of the bulb B is always the same, namely, 0. 
 
 The general equation (2), p. 204, may in this case be written 
 
 V f v v' 
 
 where we assume that the whole dead space has the same temperature 
 B' as the jacket J. 
 
 In the ice experiments we have 
 
 and in steam at the temperature 
 
212 THERMAL EXPANSION OF GASES [ 81 
 
 We may then evaluate the various terms, as was done in the case of 
 the constant volume air thermometer, and deduce a value of # , just as 
 in that case. 
 
 If we assume that we know the value of $ {272*2}, then the observa- 
 tions in ice and steam will allow of F , the volume of the bulb at 0, 
 being calculated. 
 
 The method of using this thermometer to calibrate a mercury 
 thermometer is exactly similar to that described in the last section, 
 and the method of reducing the results will be obvious from what has 
 been said above. 
 

 CHAPTER XII 
 
 CALOK1METRY 
 
 82. Calorimetry The Method of Mixture. In the method of mixtures 
 for determining the specific heat of a body, the body is heated to a 
 known high temperature, and is then introduced into a mass of water 
 contained within a vessel called the calorimeter, and from the rise in 
 temperature of the water the specific heat of the body is calculated. 
 Although the principle of the method is very simple, yet to obtain 
 accurate results it is necessary to take several important precautions 
 and to introduce corrections, some of which, at any rate, are by no 
 means easy to determine. 
 
 F Since the heat given out by the hot body is employed not only in 
 heating the water in the calorimeter but also in heating the calorimeter 
 itself, a correction has to be applied to allow for this heat, and hence 
 it is of importance to make the mass of the calorimeter as small as 
 possible. Further, since to calculate the heat given to the calorimeter 
 we shall have to multiply its mass by its specific heat and by the rise 
 in temperature, it is important that the calorimeter should be at the 
 same temperature throughout. Now, it being impossible to fill the 
 calorimeter quite full of water, unless the material of which the calori- 
 meter is constructed is a good conductor of heat, the portions of the 
 calorimeter above the surface of the water will be at a different tempera- 
 ture to the rest, and hence the change in temperature of the water 
 within the calorimeter will not be the same as the change in temperature 
 of the whole of the calorimeter. For this reason it is important to make 
 the calorimeter of a good conductor of heat, and always to take care 
 that it is nearly filled with water. The materials best suited for the 
 construction of calorimeters are copper and silver, for both these metals 
 are very good conductors of heat. If a liquid has to be used in the 
 calorimeter which would corrode either of these metals a glass calori- 
 meter must be employed, but in this case it is essential that the glass 
 should be very thin and that the liquid should fill the calorimeter 
 almost up to the top, so that the mass of glass above the surface of the 
 liquid should be as small as possible. 
 
 Since if the temperature of the calorimeter is different from that of 
 the surrounding air it will either lose or receive heat at a rate dependent 
 on its surface and the difference in temperature between its surface and 
 the surrounding air, it is of importance to make the surface of the 
 calorimeter (including the free surface of the water) as small as possible, 
 
214 
 
 CALOEIMETRY 
 
 [82 
 
 so that if the calorimeter is cylindrical in form the height ought to be 
 made equal to the diameter of the base, for in this case for a given 
 volume the ratio of the volume to the surface will be greatest. This 
 shape of calorimeter has one disadvantage, namely, that the free surface 
 of the liquid in the calorimeter is considerable, and hence if the liquid 
 being experimented upon is at all volatile, the evaporation may introduce 
 considerable error. This is particularly the case with such a liquid as 
 water, which has a very large latent heat of evaporation. Hence it is 
 generally advisable to make the height about 1'25 times the diameter. 
 The evaporation may be checked by providing the calorimeter with a 
 lid. Such lid ought to be made of the thinnest possible metal, and 
 must be pierced with a slot to allow of its being removed for the 
 introduction of the hot body without disturbing the thermometer. 
 
 An important adjunct to the calorimeter is the stirrer, for it is only 
 by keeping the contents of the calorimeter well stirred that we can 
 ensure that they are at a uniform temperature. The conditions aimed 
 at in the design of a stirrer are that it should be light, and that when 
 in use it should thoroughly mix the liquid without requiring to be 
 worked at a very great rate. Where possible, it is also an advantage 
 to U8e a form of stirrer which, when being worked, does not move in 
 
 such a way that the handle moves in and 
 out of the water, for such a motion of the 
 handle must necessarily encourage the 
 evaporation of the liquid. 
 
 Since it is important to reduce the rate 
 of loss of heat from the outside surface of 
 the calorimeter to a minimum the outside 
 of the calorimeter ought to be polished, 
 for the radiation from a polished surface is 
 less than that from a black surface. Also 
 to prevent loss of heat by conduction the 
 calorimeter ought to be suspended by three 
 fine strings, or supported on three small 
 cones made of cork. For most purposes the 
 cork supports will be found the more con- 
 venient, since with them it is so much easier 
 to remove the calorimeter. The bases of the 
 three cork cones ought to be glued to a disc 
 of brass or copper which will just fit inside 
 the water-jacket referred to below. 
 In order to render the loss of heat from the outside of the calori- 
 meter regular, and' hence allow of a correction being calculated, it is 
 necessary to keep the calorimeter during the observations within a 
 vessel which is itself kept at a constant temperature. The best method 
 of securing such a constant temperature surrounding is to have a double- 
 walled metal vessel, the space between the walls being filled with water. 
 A form of calorimeter which fulfils the conditions mentioned above 
 
 FIG. 86. 
 
82] 
 
 THE METHOD OF MIXTURE 
 
 215 
 
 is shown in Fig. 86. It consists of a cylindrical copper vessel 10 cm. 
 in diameter and 12 cm. high. Two small notches are cut in the top 
 edge at opposite ends of a diameter. The stirrer consists of a thin 
 horizontal copper rod AB, bent as shown in the figure, and resting in the 
 notches. Two thin copper plates CD are soldered to this rod, while a 
 small glass handle is cemented to the outside. By giving this handle 
 a to-and-fro motion the paddles c and D are moved backwards and 
 forwards through the liquid, and thus stir it. The thermometer T is 
 supported in a clip, so that its bulb is near the centre of the calorimeter. 
 A simpler form of stirrer consists of a copper annulus of such a diameter 
 
 
 FIG. 87. 
 
 that it loosely fits inside the calorimeter, and having a short length 
 of copper wire soldered to the upper surface. A piece of glass tube is 
 cemented to this wire, and forms a handle by which the annulus can 
 be moved up and down through the liquid, the bulb of the thermometer 
 passing through the hole in the centre. This form of stirrer suffers 
 from the objection that as it is used part of the stem moves in and 
 out of the liquid, and thus promotes evaporation. 
 
 The general arrangement of the apparatus when measuring the 
 specific heat of a solid by the method of mixtures is shown in Fig. 87. 
 The calorimeter A is placed inside a double-walled copper vessel B, the 
 space between the walls being filled with water. The thermometer T is 
 
216 CALORIMETEY [ 82 
 
 supported by a clip attached to an arm F, which is itself movable on a 
 vertical rod attached to the upright board D. This board serves as a 
 screen to protect the calorimeter from heat radiated from the heater c. 
 The heater consists of a double-walled copper vessel, so that steam can 
 be passed through the space between the walls. 1 The heater is carried 
 by a hinged arm E, and the position of the calorimeter is so adjusted 
 that when the heater is swung round till the arm E touches two stops 
 GG, the heater is vertically over the calorimeter. The substance of 
 which the specific heat is to be determined is suspended within the 
 heater alongside the bulb of the thermometer /, the top and bottom 
 of the inner part of the heater being closed by plugs of cotton wool. 
 By swinging the heater over the calorimeter and releasing the string H 
 the substance can be dropped into the liquid contained in the calori- 
 meter, the heater being immediately swung back into its position behind 
 the screen D. 
 
 The thermometer T used in the calorimeter must be divided into at 
 least tenths of a degree, since, otherwise, the comparatively small rise in 
 temperature which takes place on the introduction of the hot body will 
 not be measured with reasonable accuracy. The range of the thermo- 
 meter need not be very great, say, from 15 to 30. Since it is im- 
 portant to be able to quickly read the thermometer, it is advisable to 
 use a thermometer in which the thread can be distinctly seen. Further, 
 since it is not very convenient to use a reading telescope, it is advisable 
 to have the divisions as near the thread as possible, so as to avoid errors 
 due to parallax. For this reason the German pattern of thermometer 
 has decided advantages, for the capillary having a thin wall, the scale 
 is nearer the mercury thread than in the ordinary English pattern, 
 where the stem has to be made thick to give strength. 
 
 If M is the mass of the body of which the specific heat is being 
 measured, W the mass of the water in the calorimeter, and ^ is the 
 initial temperature of the body, t 2 the initial temperature of the water, 
 and t 3 the final temperature of the water and body, then to a first 
 approximation the specific heat s of the body is given by 
 
 The result obtained in this way will require correction to allow for 
 a number of things. 
 
 1. The heat given out by the body has been expended, not only in 
 raising the temperature of the water in the calorimeter, but also in 
 raising the temperature of the calorimeter itself, of the stirrer, and of 
 the portion of the thermometer which dips in the water. If we assume 
 that the whole of the calorimeter and the copper part of the stirrer are 
 
 1 Another form of beater is shown in Fig. 94. This heater can be easily 
 brought over the calorimeter and the body tipped into the water by inverting 
 the heater. 
 
82] THE METHOD OF MIXTUEE 217 
 
 always at the same temperature as the water, which we may expect if 
 the calorimeter is very nearly filled and only a small part of the stirrer 
 projects beyond the calorimeter, then we can at once calculate the 
 amount of heat spent in heating these. Thus if w is the mass of the 
 calorimeter and stirrer (without the glass handle), and o- is the specific 
 heat of the material of which the calorimeter and stirrer are made, the 
 heat used up in raising their temperature is w<r(t 3 - 1. 2 ). The quantity 
 w<r, which represents the quantity of heat necessary to raise the tempera- 
 ture of calorimeter and stirrer through 1 degree, is called the water 
 value of the calorimeter and stirrer. If w is small compared to the 
 weight of water W, and the calorimeter is made of copper, then it will 
 in general be quite sufficiently accurate to take the value '1 for cr. 
 
 The determination of the water value of the immersed portion of 
 the thermometer is a matter of much greater difficulty. For most 
 purposes it will, however, be sufficiently accurate to proceed in the 
 following manner: The density of mercury being 13*6, and its specific 
 heat O0333, the quantity of heat required to raise the temperature of 
 unit volume of mercury through 1 degree is 13 '6 x 0'0333, or '453. 
 The density of Jena glass 16'" is 2 '58, and its specific heat is 0'199. 
 Hence the quantity of heat necessary to raise the temperature of unit 
 volume of this glass 1 degree is 2'58 x '199, or "513. It will thus 
 be seen that approximately it requires the same quantity of heat to 
 raise the temperature of unit volume of mercury and glass through 
 the same range. Hence if we measure the volume of the portion 
 of the thermometer which is immersed in the calorimeter, and assume 
 that the whole of this volume is filled with mercury, we shall not 
 be much in error. If v is the volume of the immersed part of the 
 thermometer, then the water value of the thermometer is given by 
 
 The quantity v can be measured by immersing the thermometer, as far 
 as it is immersed in the calorimeter, in the water contained in a graduated 
 tube, such as a burette, and reading off the volume corresponding to the 
 rise of the surface of the water. 
 
 2. Some heat will have been lost during the transfer of the body 
 from the heater to the calorimeter. It being practically impossible to 
 estimate the amount of this loss, we are constrained to reduce it as much 
 as possible. It is for this reason that the heater is made to swing over 
 the calorimeter, so that the body may drop straight down into the water. 
 The importance of the error caused by this loss of heat will become less 
 as the mass of the body is increased, so that it is always advisable to 
 use as much of the substance as the heater will conveniently hold. In 
 this case, as in nearly all the errors which are met with in performing 
 a determination of specific heat by the method of mixtures, the error is 
 approximately as the surface of the body or calorimeter, as the case may 
 be, while the quantity affected by the error varies as the volume. Hence 
 the percentage error produced decreases as the volume increases, for the 
 increase in the volume is more rapid than the increase in the surface. 
 
 
218 CALORIMETRY [ 82 
 
 3. Between the time when the hot body is introduced and the instant 
 when the temperature of the water in the calorimeter, as indicated by 
 the reading of the thermometer, reaches its maximum, there will in 
 general be a loss of heat by radiation and convection from the surface 
 of the calorimeter, so that the final temperature attained is not so high 
 as it ought to be. There are various ways of allowing for this loss of 
 heat by radiation (and conduction). 
 
 In Rumford's method the initial temperature of the water is taken 
 below that of the surrounding water-jacket, so that during the first half 
 of the rise the calorimeter will gain heat from the surroundings, while 
 during the second half it will lose heat. If the rate of rise of temperature 
 in the calorimeter were even approximately constant, then a fairly good 
 compensation could be secured by Rumford's method. In practice, 
 however, the temperature rises at first very rapidly, and the maximum 
 is only reached slowly, so that the interval during which the calorimeter 
 is losing heat is greater than that during which it is gaining heat, 
 if the initial cooling is such as to make the mean between the initial 
 and final temperatures equal to the temperature of the surroundings. 
 To allow for this effect, it has been proposed to cool the water, so that 
 its initial temperature is twice as much below that of the surroundings 
 as its final temperature is above. It is, however, much preferable to 
 measure the loss of heat due to radiation and to make a correction on 
 this account, as described below. 
 
 Regnault's method of performing an experiment so as to correct for 
 radiation is as follows : 
 
 The experiment is divided into three periods. During the first period 
 the temperature of the calorimeter is noted at intervals during three or 
 four minutes, the temperature of the jacket being also noted. In the 
 second period the hot body is introduced at a noted time, and the 
 temperature of the calorimeter is read every fifteen or thirty seconds till 
 the maximum reading is obtained. During the third period the tem- 
 perature of the calorimeter is read every minute for about ten minutes. 
 The temperature of the jacket must also be noted. 
 
 By plotting the numbers obtained during the first period against the 
 time, the temperature of the calorimeter at the instant when the hot 
 body was introduced can be obtained by a slight extrapolation. 
 
 From the numbers obtained in the third period, calculate the fall of 
 temperature per second when the calorimeter has its highest temperature. 
 For this purpose it will be best to take the fall of temperature during 
 four or five minutes and divide by the number of seconds in the interval. 
 Then on a piece of squared paper, taking the abscissae as the temperatures 
 of the calorimeter, and as ordinates the fall of temperature per second, 
 plot the point Q (Fig. 88) to correspond to the value of the rate of fall 
 of temperature at the high temperature obtained above. Also take the 
 point P on the axis of temperatures corresponding to the temperature of 
 the water-jacket surrounding the calorimeter, for when the temperature 
 of the calorimeter is equal to that of the surroundings, the rate of fall 
 
82] 
 
 THE METHOD OF MIXTURE 
 
 219 
 
 J 
 
 TEMPERATURE OF CALORIMETER-* 
 FIG. 88. 
 
 of temperature will be zero. If, then, we assume that the rate of fall of 
 temperature is proportional to the excess of the temperature of the calori- 
 meter over the temperature of the 
 jacket, the straight line joining p 
 and Q will give the rate of fall of 
 temperature for all temperatures of 
 the calorimeter. The assumption 
 made above, namely, that Newton's 
 law of cooling holds, is quite 
 justified so long as the excess of 
 temperature of the calorimeter does 
 not exceed a few degrees, which 
 difference in temperature must never 
 be exceeded if an accurate determina- 
 tion of specific heat is being made. 
 
 Next, the numbers obtained in the second period must be plotted, 
 taking the times as abscissae and the temperatures of the calorimeter as 
 ordinates, and in this way a curve ABODE (Fig. 89) is obtained. 
 
 Divide- the curve into segments AB, BC, CD, &c., which are approxi- 
 mately straight. Let the temperatures corresponding to the points A and 
 B be #! and 2 , so that the mean temperature during the interval OM is 
 
 l * 2 . From the cooling curve (Fig. 88) obtain the rate of cooling 
 
 corresponding to this mean temperature, and multiply by the time OM. 
 
 The product, which we may 
 
 call t v will be the amount 
 
 by which the temperature at 
 
 the end of the interval OM 
 
 will be below what it would 
 
 have been suppose there had 
 
 been no loss of heat due to 
 
 radiation, &c. Hence if we 
 
 increase the ordinate MB by 
 
 an amount BB' equal to t v MB' 
 
 will represent what would 
 
 have been the temperature of 
 
 the calorimeter at the end of 
 
 the interval OM if there had 
 
 been no loss of heat. In the 
 
 same way, read off the mean 
 
 M N 
 
 TIME 
 
 FIG. 89. 
 
 temperature for the interval 
 
 MN, and multiply the rate of 
 
 cooling for this temperature by the interval, and obtain the correction 
 
 t. 2 for loss of heat during the second interval. If the ordinate NO 
 
 is then increased by an amount cc' equal to ^ + * 2 NC ' ^^ represent 
 
 what the temperature would be at the end of the second interval 
 
 if there had been no loss of heat, for the loss of heat at the end of the 
 
220 CALOEIMETRY [ 82 
 
 second interval is equal to the sum of the losses during the first and 
 second intervals. Proceeding in this way a new curve AB'C'D'E' is 
 obtained, which represents what would have been the temperature of the 
 calorimeter supposing there had been no loss of heat due to radiation, 
 conduction, and convection. The part D'E' of this curve ought to be 
 horizontal, since we are correcting for the fall of temperature due to loss 
 of heat, and if there is no loss of heat, then the temperature of the calori- 
 meter would remain constant. The final temperature to be used in 
 calculating the specific heat is that corresponding to the horizontal part 
 of the curve, viz. D'E'. 
 
 If the temperature of the jacket surrounding the calorimeter varies 
 during the experiment, then when obtaining the cooling curve and 
 calculating the corrections the difference between the temperature of 
 the calorimeter and that of the jacket must be used when obtaining the 
 cooling curve and when calculating the corrections. 
 
 The importance of the correction for radiation increases as the time 
 which elapses between the introduction of the hot body and the tempera- 
 ture reaching the maximum gets greater. Thus it is more important 
 in the case of a bad conductor of heat, such as glass, which only slowly 
 parts with its heat, than in the case of a good conductor, such as a 
 metal. In order to reduce the time for equilibrium to be set up it is 
 advisable, where possible, to use the solid in the form of small pieces, 
 which may be contained within a cage made of thin copper gauze. 
 The water value of the cage will have to be calculated, and the heat 
 conveyed by it to the calorimeter allowed for in the calculations. When 
 the cage is suspended in the heater, the bulb of the thermometer ought 
 to be surrounded by the pieces of solid contained in the cage. When 
 the cage is in the calorimeter it can be used as the stirrer, being moved 
 round and round by the string by which it was suspended. 
 
 Regnault's method of correcting for the loss of heat due to radiation 
 is unsatisfactory in one particular. It assumes that the temperature of 
 the thermometer follows exactly the temperature of the water in the 
 calorimeter, and, further, that the temperature of the outside of the calori- 
 meter does not lag behind that of the contents. If the calorimeter is 
 very thin and well polished outside, this last assumption is probably 
 justified. The question of the lag of the thermometer is, however, more 
 important. Since the rise to be measured is small the thermometer 
 requires to be very sensitive, and hence the bulb has to be large, and it 
 is quite certain that when the temperature is varying rapidly that the 
 temperature of the mercury in the thermometer will be less than the 
 temperature of the surrounding water. 
 
 The correction for lag of the thermometer can be obtained in the 
 following manner : Having the calorimeter filled with water at the same 
 temperature as the jacket, cool the thermometer to about 6 degrees lower, 
 and then introduce it into the calorimeter, and, keeping the water 
 stirred, take the reading of the thermometer every fifteen seconds till 
 the readings become constant. Then plot the readings of the thermometer 
 
82] 
 
 THE METHOD OF MIXTURE 
 
 221 
 
 against the time and obtain a curve, such as OPQ (Fig. 90). If we draw 
 a tangent at any point P of this curve, and call the thermometer reading 
 r and the corresponding time t, the rate at which the thermometer read- 
 ing is changing is , and if is the angle the tangent makes with the 
 
 ctt 
 
 ,. ,. dr 
 
 axis or time we have - 
 dt 
 
 tan 0. 
 
 Now at the point P the lag of the 
 
 thermometer behind the temperature of the surrounding water is repre- 
 sented by MN, so that when the temperature of the thermometer is chang- 
 ing at a rate represented by tan 6, the lag is MN. 
 
 Suppose, then, we require to find the lag when the reading is changing 
 at the rate of half a degree in twenty seconds. Take a point s such that os 
 represents twenty seconds, and take SR equal to half a degree, and join OR. 
 Then the tangent of the angle EOS is equal to the rate of change of the 
 thermometer reading. Thus if we draw a line AP tangential to the curve 
 and parallel to OR, the lag will be equal to MN. To apply the corrections 
 to the specific heat observa- 
 tions, the lag must be calculated 
 for each reading of the thermo- 
 meter and added to the reading 
 to give the actual temperature 
 of the water in the calorimeter. 
 The readings thus obtained 
 corrected for thermometer lag 
 must then be plotted on a 
 curve, and the radiation cor- 
 rection applied as described 
 above. 
 
 We have hitherto tacitly 
 assumed that the specific heat 
 of water is a constant, so 
 
 that the heat required to heat a gram of water between any two tem- 
 peratures is numerically equal to the difference in the temperatures. 
 This assumption is, however, not correct, for the quantity of heat 
 necessary to heat a gram of water through 1 degree is different at 
 different temperatures. Hence it is necessary in the first place to choose 
 some temperature, and to say that at that temperature the specific heat 
 of water is unity, and then determine the specific heat of water at other 
 temperatures in terms of this unit. Various temperatures have been 
 suggested for the standard temperature. Perhaps the most frequently 
 used is 15 C., a calorie being defined as the heat required to raise the 
 temperature of 1 gram of water from 14*5 to 15'5. Callendar has 
 advocated the employment of a mean calorie taken between 15 and 25, 
 this mean calorie being practically the same as the calorie at 20. 
 
 When we come to consider the variations of the specific heat of 
 water, the results obtained by different observers differ very considerably. 
 
222 CALORIMETRY [ 83 
 
 As far as the present state of our knowledge is concerned, the following 
 formula proposed by Callendar 1 seems as accurate as any, and gives 
 results for the specific heat of water in terms of the calorie at 20 over 
 the range 10 to 60 : 
 
 s = '9982 + p-0000045( - 40) 2 ; 
 for temperatures between and 20 an additional term is required, giving 
 
 s = -9982 + 0'0000045(* - 40) 2 + 0*0000005(20 - t)*. 
 For temperatures between 60 and 100 the formula proposed is 
 s =1 + -0001 4(-60). 
 
 The values of the specific heat of water calculated by these formulae 
 are given in Table 21, from which the mean specific heat over any range 
 can at once .be calculated. It is the mean specific heat of the water over 
 the range used in the experiment which must be employed when calculat- 
 ing out the specific heat of a body, if the accuracy aimed at in the ex- 
 periment is sufficiently great to necessitate a correction on account of the 
 variation of the specific heat of the water. 
 
 83. Measurement of the Specific Heat of Glass by the Method of 
 Mixture. Make a bundle of pieces of fine-bore glass tube, each piece 
 being about 6 centimetres long, using a weighed length of copper wire to 
 fasten the glass together. Suspend the bundle in the steam heater, and 
 fix the bulb of a thermometer reading to 100 alongside. Pass steam 
 from a boiler through the heater, using rubber tube to conduct the 
 steam to the heater and to lead away the excess steam. The glass must 
 be left for at least half-an-hour in the heater, and if convenient an hour 
 is a distinct advantage. 
 
 While the glass is being heated weigh the calorimeter and stirrer, and 
 calculate their water value. Then fill .the calorimeter about two-thirds 
 full of water, and again weigh. Place the calorimeter in the water- 
 jacket, and having adjusted the position of the thermometer, place a card 
 with a slit cut to allow of the passage of the thermometer over the top 
 of the calorimeter. The thermometer used in the calorimeter ought to be 
 one having a very open scale divided into tenths of a degree, each degree 
 corresponding to about a centimetre. With such a thermometer hun- 
 dredths of a degree can be read with a very fair degree of accuracy. 
 
 When the glass has been in the heater long enough to have acquired 
 the temperature of the steam, read the calorimeter thermometer every 
 half minute for a few minutes. Then read the thermometer in the heater, 
 and exactly at a whole minute swing the heater round and lower the 
 glass into the calorimeter. The glass must be lowered quickly, but not 
 in such a way as to risk spilling the water by splashing. Immediately 
 after the introduction of the glass swing the heater back, and keeping 
 the water stirred read the thermometer as quickly as possible, and con- 
 tinue taking a reading every twenty seconds till the maximum has been 
 
 1 Phil. Trans. . S. (1902), clxix. 142. 
 
84] 
 
 SPECIFIC HEAT OF A LIQUID 
 
 223 
 
 passed. Continue reading the thermometer every minute for about five 
 minutes. 
 
 Note how much of the thermometer has been immersed in the calori- 
 meter, and determine the volume in cubic centimetres of this part of the 
 thermometer, and multiply this volume by 0*45 to get the water value of 
 the thermometer (see 82). 
 
 From your numbers calculate the specific heat of the glass by the 
 method described in the previous section. 
 
 84. Measurement of the Specific Heat of a Liquid by the Method 
 of Mixture. The method of mixtures may be employed for measuring 
 the specific heat of a liquid. One method of experiment is to use a 
 solid of which the specific heat has already been measured, preferably a 
 metal having a high thermal conductivity, such as copper. The solid is 
 heated and then transferred to the liquid which replaces the water in the 
 calorimeter, the method of performing the experiment being exactly the 
 same as that described in the preceding section. 
 
 The above method is only applicable when a fairly large volume of 
 the liquid is available. If only a small quantity of the liquid is avail- 
 able, then it is introduced into a weighed glass tube shaped like a weight 
 dilatometer, but with a straight capillary. The liquid must only about 
 three-quarters fill the bulb, otherwise when the temperature is raised 
 there is danger of the bulb being burst. The end of 
 the capillary having been closed in the blow-pipe flame, 
 the tube is bent round to form a hook by which the 
 bulb may be suspended. By again weighing the bulb, 
 the weight of the liquid is obtained. 
 
 If the liquid does not boil below 100 C., then 
 the bulb can be suspended within a steam heater ; 
 otherwise the bulb must be placed, together with a 
 thermometer, in a test-tube, which is itself immersed 
 in water which is maintained at a temperature a 
 little below the boiling point of the liquid. 
 
 The heated bulb is introduced into the water 
 contained in the calorimeter, and the experiment 
 proceeds as in the preceding section, an allowance 
 being of course made for the water value of the glass 
 of the bulb. 
 
 In place of sealing up the liquid in a glass bulb, 
 Berthelot recommends that the liquid of which the 
 specific heat is to be measured be contained in a 
 small bottle made of platinum capable of containing 
 from 50 to 100 cubic centimetres. Since, however, 
 such a platinum bottle is seldom available, we may 
 use a thin-walled glass bottle of the shape shown 
 in Fig. 91, and made by drawing down a thin- walled glass boiling tube. 
 A sensitive thermometer similar to that used in the calorimeter is fixed 
 in the mouth of this bottle by means of a tight-fitting cork. 
 
 FIG. 91. 
 
 
224 CALORIMETRY [ 85 
 
 Berthelot's method of conducting the experiment is as follows : 
 The bottle is weighed and its water value, together with that of the 
 thermometer, is calculated. The water value of the cork can in general be 
 neglected. The bottle is then filled about two-thirds full of the liquid 
 and again weighed, the difference in the two weights giving the weight 
 of the liquid. The bottle is then placed inside a fairly tight-fitting metal 
 cylinder B, the side of which is wrapped round with a layer of asbestos 
 cloth c. The cylinder is rested in a sand-bath or over a sheet of asbestos 
 cardboard, and a small flame is placed beneath. While the liquid is heating 
 the calorimeter is filled with water and weighed, and after placing inside 
 a water-jacket the thermometer is read every minute in the manner 
 described in 82. 
 
 By occasionally shaking the cylinder containing the bottle the heating 
 of the liquid may be kept uniform. When the temperature of the liquid 
 is sufficiently high, remove the cylinder and the contained bottle to the 
 neighbourhood of the calorimeter, and having well shaken, read the 
 thermometer very carefully and immediately remove the bottle from the 
 cylinder, using the thermometer as a handle, and immerse in the 
 calorimeter, being careful to keep the cork about a centimetre above the 
 surface of the water. 
 
 Keeping the water in the calorimeter well stirred by moving the 
 bottle about, using the thermometer as a handle, read both the thermo- 
 meters every half minute. When the reading of the thermometers are 
 within a degree of one another give a final good shake and carefully read 
 both thermometers, noting the time, and then remove the bottle. Next 
 observe the fall of temperature in the calorimeter for five minutes. 
 
 Although the water value of the calorimeter and its contents during 
 the last period is less than it was during the middle period when the 
 bottle was immersed, yet it will in general be found quite sufficiently 
 accurate to assume that the rate of fall of temperature deduced from the 
 observations when the bottle is removed applies to the case when the 
 bottle is in the calorimeter, and then to calculate the correction to 
 the maximum reading of the calorimeter thermometer in the manner 
 described in 82. Knowing the water value of the calorimeter and 
 its contents and the rise in temperature, the amount of heat acquired 
 can at once be calculated. In the same way the heat given out by the 
 bottle and its contents can be calculated in terms of the specific heat of 
 the liquid, and equating the two quantities of heat, the specific heat of the 
 liquid can be deduced. 
 
 85. Measurement of the Specific Heat of a Liquid by the Method 
 of Cooling. The specific heats of two liquids may be compared by 
 noting the times taken by a calorimeter filled, first with the one and then 
 with the other, to cool from a temperature ^ to a temperature t. 2 , the 
 conditions as to the temperature, &c., of the surroundings being kept the 
 same in the two cases. 
 
 Let w be the water value of the calorimeter, thermometer, and stirrer, 
 and W the weight of liquid of specific heat s placed in the calorimeter. 
 
85] SPECIFIC HEAT OF A LIQUID 225 
 
 If by some means or other the calorimeter were maintained at the 
 temperature t v the temperature of the surroundings being # , then during 
 a second the calorimeter and its contents would lose a quantity of heat 
 which would depend (1) on the nature and extent of the surface of the 
 calorimeter, and (2) on the difference between the temperatures ^ and t Q . 
 Since the surface and the surroundings are kept constant throughout, we 
 only have to consider the effect of the difference between the temperature 
 of the calorimeter and of the surroundings on the rate of loss of heat. 
 Thus we may represent the rate of loss of heat at the temperature ^ by 
 /(^), where /(^) is some function of the temperatures ^ and t the 
 value of which we do not require to know. Next suppose that instead 
 of keeping the temperature constant we allow the calorimeter to cool, and 
 we notice the time 8 l T which is taken to cool from the temperature 1 1 to 
 the temperature ^ - 8t, where 8t is very small. Since the change of 
 temperature is very small, we may without appreciable error suppose that 
 the rate of loss of heat remains unchanged. The heat lost during the 
 time the temperature is falling by the amount 8t is therefore 8tf(t l )8 l T. 
 The heat lost by the calorimeter is given by ( W + w)s8t. Hence 
 
 
 Next suppose that the time taken for the temperature to fall from 
 t l - 8f to f^ - 28t to be 8 2 7 7 , we then get in the same way 
 
 
 
 Proceeding in this way till the temperature has fallen to and 
 adding all the expressions similar to (1) and (2), we get 
 
 Bu 
 
 &c. 
 
 ( W+w)s /(^) T /(*, - 8t) T /(*j - 28#) 
 
 t S 1 7' + 8.,7 T +8o7 7 +&c. is the total time T taken for the calorimeter 
 
 to cool from ^ to f . Hence 
 
 
 
 (W+w) 
 
 where the sign 2 indicates the sum of the terms like 
 
 If, now, we repeat the experiment, using a weight W of a liquid of 
 specific heat s in the calorimeter, starting at the same temperature t l and 
 finishing at the same temperature t^ we obtain in the same way as 
 before, if T' is the total time taken to cool from t l to t. 2 , 
 
 T' 1 
 
226 CALORIMETRY [ 85 
 
 Now in the expressions (3) and (4) the parts on the right-hand side 
 are the same, since they only depend on the surface of the calorimeter, the 
 temperature range, and the temperature and nature of the surroundings. 
 Hence equating the two left-hand expressions we get 
 
 * T'(W+w) 
 
 ' 
 
 Hence if one of the liquids is water, or some other liquid of known 
 specific heat, we can immediately obtain the specific heat of the other. 
 
 In the practical application of this method it is important to secure 
 the rigorous fulfilment of the conditions we have postulated above. 
 Thus the outer surface of the calorimeter must remain exactly the same 
 during the two experiments. For this reason it is not safe to heat the 
 liquid up to the initial temperature in the calorimeter itself, as the flame 
 or other source of heat would be likely to alter the surface. The liquid 
 must be heated in a beaker or flask, and then poured into the calorimeter. 
 The weight of liquid used can be found by weighing the calorimeter after 
 the end of the cooling experiment. In order to secure that the sur- 
 rounding conditions remain the same, the calorimeter must be placed 
 inside a hollow double-walled copper vessel, the space between the walls 
 being filled with water or melting ice. 
 
 The calorimeter itself must be made of some good conducting material 
 such as copper or silver, the walls being as thin as possible. The reason 
 for this is, that we assume that the temperature of the outside is the 
 same as the temperature of the liquid. As any evaporation of the liquid 
 during the course of the experiment would vitiate the result, owing to 
 the latent heat of the vapour formed, the calorimeter must be fitted with 
 a lid. This lid must be pierced with holes for the thermometer and 
 stirrer. The stirrer must be a rotary one, for an up-and-down stirrer 
 would cause a portion of the stem of the stirrer to move in and out of the 
 liquid, and thus encourage evaporation. This effect may easily produce 
 a material error, since the stirring has to be vigorous in order to ensure 
 that the temperature remains uniform throughout the liquid. The best 
 shape for the calorimeter is a cylinder, of which the length is about four 
 times the diameter, and the lid must fit tightly, so that the vessel may be 
 filled with liquid almost up to the top without risk of any of the liquid 
 escaping. 
 
 When performing the experiment clean out the calorimeter and black 
 the outside with dead black varnish, formed by mixing vegetable black 
 with methylated spirits and adding a few drops of shellac varnish. Then 
 weigh the calorimeter, stirrer, and thermometer. Heat the liquid to 
 a temperature of about 50, if this can be done without the vapour 
 tension becoming too high. Pour the liquid into the calorimeter and 
 place it in the water-jacket. Stir the liquid well, and read the tem- 
 perature every half minute till the temperature has fallen to a little more 
 than half-way to the temperature of the water-jacket. Then remove the 
 calorimeter and weigh. 
 
85] SPECIFIC HEAT OF A LIQUID 227 
 
 Having cleaned the calorimeter, being careful not to touch the outside 
 more than is absolutely necessary, repeat the experiment using water, 
 starting and finishing at about the same temperatures as before. 
 
 Two curves must now be plotted showing the relation between the 
 temperature and the time for the two experiments. From these curves 
 the times corresponding to equal initial and final temperatures can be 
 obtained. The initial temperature must not be taken till a minute or 
 two after the observations are started, so as to allow time for a regular 
 state of affairs to have set in. The final temperature ought to be about 
 half-way between the initial temperature and the temperature of the 
 surroundings. 
 
 If the temperature of the water-jacket varies between the two 
 experiments, then so long as the variation is not more than a degree 
 or two we may allow for this effect by taking our initial and final 
 temperatures such that there is the same difference between the tempera- 
 ture of the calorimeter and of the jacket in the two experiments. 
 
 It is extremely difficult to obtain accurate measurements by this 
 method. The errors are chiefly caused by change in the condition of the 
 surface of the calorimeter and evaporation of the liquid. The method, 
 however, being simple, it is often convenient where an approximate value 
 of the specific heat of a liquid is required. It is quite inapplicable 
 to solids, even when good conductors, and in the form of powder, the 
 temperature not remaining sufficiently uniform throughout the solid. 
 
CHAPTER XIII 
 
 B 
 
 CALORIMETKY LATENT HEAT 
 
 86. Measurement of the Density of Ice by Bunsen's Method. 
 Take a piece of glass tube a little over a centimetre in diameter and 
 about 30 cm. long, and having well cleaned it, dry it thoroughly. Near 
 one end thicken the glass by rotating it before a small blow-pipe flame, 
 and draw it down so as to make a narrow and 
 short neck as at A (Fig. 92). Near the other end 
 thicken the glass, and then draw out into a 
 narrow capillary. Where this capillary leaves 
 the main tube form a slight neck, as shown at D. 
 Finally heat the middle of the tube in a fairly 
 large flame, and allow the glass to get quite 
 thick. During this process the ends of the 
 tube may be very slightly forced together. 
 When the glass is thoroughly thickened and 
 uniformly hot, withdraw from the flame and then 
 very slowly draw out till a neck about 8 cm. 
 long is formed, and while the glass is still hot 
 bend round so that the two parts of the tube 
 are parallel. Finally, bend the capillary, as 
 
 shown at E in the figure, and make a fine mark on the neck A with a 
 wire file. 
 
 The U-tube thus formed must be filled about half full with pure dry 
 mercury, and then weighed carefully. 
 
 Now fix the tube in a retort clamp, so that the end of the capillary 
 dips into a beaker containing distilled water. Tne beaker must stand 
 on some wire gauze and the water be boiled. After the water has boiled 
 for some minutes tilt the U-tube so that the mercury runs into the arm 
 DC and nearly fills it and drives out the air. On tilting the tube back 
 into the vertical, water will be drawn over into the arm DC. Again tilt 
 the tube, driving out the remaining air and most of the water. Repeat 
 the process of drawing in water and forcing it out again two or three 
 times, the water in the beaker being kept boiling all the time. In this 
 way the upper half of the arm DC will be filled with water which is 
 almost free from dissolved air. Having fixed the U-tube in the vertical 
 position, direct a small flame from a hand blow-pipe against the tube 
 at D, and when the glass becomes soft draw off the capillary, leaving the 
 end of the arm DC sealed. The operation of sealing off a tube filled with 
 
 FIG. 92. 
 
86] MEASUREMENT OF THE DENSITY OF ICE 229 
 
 water in this way is really not a matter of any difficulty. What is 
 chiefly required is boldness. This, together with a little practice on 
 odd pieces of glass, will enable the operation to be performed without 
 cracking the glass. 
 
 Carefully dry the portion of the capillary removed, and weigh the 
 U-tube and this portion of glass. The difference in the two weighings 
 will give the weight of the water which has been introduced. Next fill 
 up the arm AB a little above the mark at A with mercury, and immerse 
 the whole up to A in melting ice, prepared as described in 61. After 
 about an hour the water and mercury will have attained a temperature 
 of zero. The excess of mercury above the mark A must then be removed 
 with a small pipette having a capillary tube. 
 
 Over the closed limb must now be placed a wide glass tube, shown 
 dotted in the figure, and closed below by a cork. This tube is filled with 
 pounded ice, and then a small cone of paper must be placed round the 
 extreme end of the closed limb, and some freezing mixture, consisting 
 of a mixture of pounded ice and crystallised calcium chloride, placed 
 in the cone. In this way the extreme end of the water column will 
 become frozen. When this occurs, pull up the outer glass tube till the 
 cork is just below the frozen part, and fill the tube with the freezing 
 mixture. By slowly moving the outside tube down, the water may then 
 be entirely frozen, the freezing proceeding regularly from top to bottom. 
 The reason for freezing in this way is that if the water at the bottom 
 became frozen first, when the upper portions froze the expansion would 
 burst the tube. When the whole of the water has been frozen, remove 
 the outer tube and carefully wipe the U-tube and introduce it into the 
 pure melting ice again. After it has been in the ice for an hour, 
 carefully remove the mercury which is above the mark A with a pipette, 
 and place this mercury in a weighed watch-glass and weigh. 
 
 Let the weight of the mercury expelled by the freezing of the water 
 be iv, and d the density of mercury at 0. Then the increase in volume 
 of the water in freezing is w/d. If W is the weight of the water, and D 
 is the density of the water at 0, then the volume of the water at is 
 W/D. Hence the volume of the ice at is W/D + w/d. Thus if 8 is 
 the density of ice at 0, we have 
 
 W = W w 
 8 D d' 
 
 . WDd 
 
 Wd + wD 
 
 The value of D is 0'999868, and that of d is 13'5954. 
 
 It has been found that the density of ice varies very considerably 
 according to the method used to freeze the water. The value obtained 
 by this method is, however, probably the one to use when reducing the 
 observations taken by the ice calorimeter. 
 
230 
 
 CALORIMETRY LATENT HEAT 
 
 [87 
 
 87. Bunsen's Ice Calorimeter. In Bunsen's ice calorimeter the heat 
 given out by a body as it cools from a given temperature to C., is 
 measured by obtaining the quantity of ice which will be melted by this 
 heat. In order to measure the quantity of ice melted, use is made of the 
 change in volume which takes place during fusion. 
 
 The increase in volume of 1 grain of ice when it becomes ice at 
 C. is O0907 c.c. Thus the communication of one calorie will cause 
 
 a change of volume of about 
 0-00113 c.c. 
 
 The usual form of calori- 
 meter consists of an inner glass 
 tube A (Fig. 93) closed at the 
 bottom, which serves to receive 
 the body of which the specific 
 heat is being measured. The 
 lower part of this tube is sur- 
 rounded by a glass bulb B, 
 which is connected to a side tube 
 D. A piece of fine-bore thermo- 
 meter tubing F about a metre 
 long is connected to E by being 
 cemented into an ebonite 
 stopper E. An iron screw G 
 passing through the stopper 
 serves to adjust the position 
 of the mercury in the tube F. 
 The upper part of the bulb B 
 is filled with air-free distilled 
 water, while the lower part, 
 the tube D, and part of F are 
 filled with mercury. By a 
 process to be described later, 
 a cap of ice K is formed round 
 the lower part of the tube A, 
 and when a hot body is in- 
 troduced into A its heat melts 
 some of the ice, and the 
 contraction which takes place 
 
 causes the mercury meniscus to travel along the tube F towards the 
 instrument. If we know the cross-section of the tube we can calculate, 
 from the distance traversed by the meniscus, the diminution of volume, 
 and. hence, knowing the densities of water and ice at C., obtain the 
 weight of ice melted, and therefore, knowing the latent heat, the quantity 
 of heat given out by the body placed in the instrument. This method 
 of obtaining the result not only involves a knowledge of the change of 
 volume which takes place when ice melts, a quantity which seems to 
 depend in some measure on the conditions under which the ice was 
 
 FIG. 93. 
 
87] BUNSEN'S ICE CALORIMETER 231 
 
 formed, but also the latent heat of ice. Further, a correction would have 
 to be applied to allow for the 5 contraction of the mercury which flows 
 from the tube F, where it is at the temperature of the room, into the 
 tube D, where it is at C. For these reasons it is usual to calibrate 
 the instrument by introducing a known weight of water at a measured 
 temperature, and to note the range through which the mercury meniscus 
 moves. In this way the quantity of heat which has to be communicated 
 to A to cause the mercury to move through 1 centimetre is obtained, 
 and can be used to reduce the observations when bodies of unknown 
 specific heat are used. 
 
 In the ordinary manner of performing the experiment, the instru- 
 ment, as described above, is placed in a vessel containing pounded ice or 
 snow, the top of the tube A and the tube F alone projecting above the 
 ice. It is, however, found that there is always a small difference in 
 the freezing point of the ice in the instrument and that of the ice 
 outside. If the temperature of the outside ice is the higher, then there 
 will be a slow melting of the ice in the instrument, which will cause a 
 continuous creep of the mercury meniscus towards the instrument. If 
 the freezing point of the ice outside is lower than that of the ice in the 
 instrument, then there will be a slow freezing of the water in the instru- 
 ment, causing the meniscus to creep away from the instrument. This 
 creep generally amounts to 2 or 3 centimetres per hour, and is 
 sufficient to make it very difficult to obtain trustworthy measurements. 
 A slight addition 1 to the instrument will almost eliminate the creep, 
 reducing it to about a tenth. This addition is shown in Fig. 93, and 
 consists in placing the instrument in an empty vessel H, the top of which 
 is closed by a cork through which the tubes A and D pass. This vessel 
 is surrounded by the pounded ice. Owing to air being a bad conductor 
 of heat, heat can now only very slowly pass between the instrument and 
 the surrounding ice. Another method of securing the same conditions 
 has been employed by Callendar, who uses a second glass bulb surrounding 
 the bulb B and produces a vacuum in the space between the two bulbs. 
 The vacuum being a w r orse conductor of heat than air, the passage of 
 heat from the surrounding ice to the instrument is in this way reduced 
 yet more. When this device is used, when originally cooling the 
 instrument down to zero the process will take a considerable time, unless 
 a stream of ice-cool water is passed through the tube A. 
 
 Since when a body has once been placed in the tube A it is very 
 difficult to extract it, it will be found of advantage to place a thin -walled 
 test-tube inside A. This test-tube must be a fairly good fit, and a little 
 glycerine can be placed in A so as to fill the interspace, and thus facilitate 
 the passage of heat from the hot body to the ice. When it is required 
 to remove a body from the calorimeter, the test-tube is removed, and the 
 body can then be tilted out. 
 
 The filling of the Bunsen calorimeter with water and mercury is an 
 operation of some delicacy. The following method is recommended by 
 
 i Boys, Philosophical Magazine (1887), xxiii. 214. 
 
232 CALOKIMETRY LATENT HEAT [ 87 
 
 Wads worth. 1 The required quantity of mercury is first introduced by 
 means of a funnel with the stem drawn out into a fine tube capable 
 of reaching down to the bend of the tube D. The calorimeter is then 
 inverted, the mercury running to the top of the bulb B, and immersed 
 upside down in a beaker full of distilled water. The water must cover 
 the entire calorimeter. The water is then heated to boiling and then 
 cooled. When the bulb B is partly filled with water, the beaker is again 
 heated and the water allowed to boil for some considerable time. In 
 this way the remaining air in the bulb B will be replaced by water 
 vapour, and, on allowing the whole to cool, the water will be forced into 
 the bulb. When the whole is cool enough to handle the end of the 
 tube D is closed with the finger, and the calorimeter removed and turned 
 into the upright position. More mercury must now be introduced by 
 means of the funnel till the side tube is entirely filled with mercury. 
 
 The screw G having been withdrawn from the stopper, the stopper 
 with the tube F attached must be placed in the end of the side tube and 
 cemented in place with a little shellac varnish or india-rubber cement. 
 On replacing the screw, the mercury will be forced up the tube F. 
 
 Before freezing the cap of ice, the whole instrument must be cooled 
 down to a temperature of about 2 C. by immersing it in a beaker full of 
 water containing some ice. To freeze the cap of ice round A a freezing 
 mixture, consisting of finely powdered ice and crystalline calcium chloride, 
 must be prepared, and introduced within the tube A. This freezing 
 mixture will require renewing from time to time till the ice cap formed 
 extends about half-way from the inside tube to the outside bulb. The 
 freezing being complete, the inside of the tube A must be rinsed out 
 with ice-cold water, and dried by means of a roll .of blotting-paper. A 
 thin test-tube which fits fairly well inside A must be weighed and then 
 put in place, and the instrument must then be introduced in the ice-box 
 and allowed to stand for several hours before an experiment is made. 
 The reason is that the temperature of the ice after its formation by the 
 freezing mixture will be below zero, and it must have time to reach 
 this, temperature, also the surrounding water must have time to cool 
 down to zero. 
 
 When the instrument has been in the ice for an hour or two, the 
 mercury meniscus must be adjusted to near the far end of the tube F by 
 means of the screw G. By watching the movement of the end of the 
 column it will then be possible to judge when the temperature of the 
 instrument becomes constant. When the movement becomes small 
 (2 or 3 millimetres per hour), the rate of movement must be carefully 
 observed. 
 
 Some water having been heated to a temperature of about 40, the 
 temperature must be carefully read, and then enough of the hot water 
 poured into the test-tube to fill it about two-thirds of the way up to the 
 top of the ice cap. The movement of the mercury meniscus is then 
 again observed, the observations being continued till the rate of move- 
 
 1 American Journal of Science (1897), iv. 282. 
 
88] 
 
 LATENT HEAT OF FUSION OF ICE 
 
 233 
 
 ment is the same as it was before the introduction of the hot water. The 
 test-tube is then removed and again weighed, and in this way the weight 
 of water is obtained. Knowing this weight and the initial temperature, 
 the heat given out by the water can at once be calculated. 
 
 The movement of the mercury meniscus consists of two parts, that 
 due to the ice melted by the hot water, and that due to the slow creep 
 which occurs when no heat is being supplied to the tube A, due to th*e 
 introduction of a hot body. The time the experiment lasted being ob- 
 served, the amount of the creep can be calculated, and this amount must 
 be added to or subtracted from the observed displacement of the 
 meniscus, according as the creep takes place away from or towards the 
 instrument. The corrected displacement divided by the number of 
 calories given up by the water gives the displace- 
 ment per calorie, and this quantity is used when 
 measuring the specific heat of other bodies. 
 
 The test-tube having been dried, a small pellet 
 of cotton wool is placed at the bottom, and the 
 tube is replaced in the instrument, and after about 
 half-an-hour the rate of movement of the meniscus 
 is again observed. The body of which the specific 
 heat is to be measured is placed in the test-tube 
 A of the heater shown in Fig. 94. This heater 
 consists of an outer tube B, closed at either end by 
 corks. Through the upper cork the test-tube A 
 passes, while through the lower cork there pass the 
 tubes c and D. These tubes are attached to two 
 pieces of rubber tubing about 2 feet long. Steam 
 is passed in through the tube F, and escapes through 
 E. The heater is wrapped round with felt. This 
 felt serves the double purpose of decreasing the 
 loss of heat, and of allowing the heater being taken 
 in the hand. To transfer the heated body to the 
 caloriAeter, the thermometer T is read and then 
 the cork is removed, and, the heater being inclined, the hot body is tilted 
 into the ice calorimeter. 
 
 The movement of the meniscus is observed as in the case of the water, 
 and a correction having been applied for the steady creep, the total dis- 
 placement divided by the displacement per calorie obtained from the 
 water experiment gives the heat given out by the hot body. Hence, 
 knowing the mass and initial temperature of the hot body, the specific 
 heat can at once be calculated. 
 
 When using the ice calorimeter for accurate measurements, it is 
 necessary to calibrate the capillary tube. This can be done by means 
 of a mercury thread in the manner described in 59. 
 
 88. Measurement of the Latent Heat of Fusion of Ice. The 
 chief difficulty to be overcome in determining the latent heat of fusion of 
 ice consists in introducing blocks of ice into the calorimeter which, while 
 
 FIG. 94. 
 
234 CALORIMETRY LATENT HEAT [88 
 
 having a temperature of 0, shall be entirely free from attached water. 
 Further, it is important that during the time necessary for melting that 
 the ice should be kept below the surface of the liquid in the calorimeter, 
 otherwise the gain of heat from surrounding objects will be considerably 
 increased. 
 
 Although by taking the ice in the form of small fragments we can 
 make the time occupied by the melting short, and hence reduce the 
 radiation correction, yet it is hardly advisable to do so on account of the 
 difficulty of removing the adherent water from the large surface of the 
 ice. Better results will be obtained by taking the ice in one piece of 
 such a shape that it can easily be freed from attached water. If the 
 calorimeter has a capacity of about 500 c.c., a cylinder of ice about 
 3 cm. in diameter and 5 cm. long will produce a suitable fall of tempera- 
 ture (about 5). Such cylinders of ice may be prepared by freezing 
 water in a test-tube of the appropriate size by means of a freezing 
 mixture. By slightly warming the test-tube, the block of 
 ice can be removed. 
 
 Until required, the cylinder of ice may be wrapped inside 
 a roll of blotting-paper and placed in a large test-tube, the 
 test-tube being immersed in melting ice. The ice must be 
 left for at least an hour, so as to allow it to acquire the 
 temperature of 0. 
 
 In order to hold the ice below the surface of the water 
 in the calorimeter, the stirrer has the form shown in Fig. 95. 
 It consists of a cup formed of copper gauze soldered to the 
 handle B. The ice having been thrown into the water, the 
 stirrer is raised and the ice imprisoned within the gauze 
 IG. 9o. CU p^ an( j ^ me iting i s facilitated by the motion of the 
 
 stirrer through the water. 
 
 In order to allow for radiation the experiment is divided into three 
 parts, as in the case described on p. 219. In the first part, the calori- 
 meter, having been filled with water arid weighed, is introduced within 
 the water-jacket, and the course of the thermometer watched Huring 
 about five minutes. At a noted time the ice is then introduced, and the 
 thermometer read every half minute till all the ice is melted. Finally, 
 the course of the thermometer is watched during another five minutes. 
 The correction for radiation is calculated exactly as previously described. 
 The weight of ice melted is obtained by weighing the calorimeter and its 
 contents at the end of the experiment. 
 
 By packing the cylinder of ice round with blotting-paper as described, 
 it is generally quite dry ; if not, then it can be dried in the following 
 way : A strip of clean blotting-paper is cut, having a length of about 
 50 cm. and a width slightly greater than the length of the ice cylinder. 
 The cylinder of ice is rolled up in this strip of paper, and by folding in the 
 ends the attached water is sucked up. Then, holding the paper cylinder 
 over the calorimeter between the finger and thumb of the left hand, the 
 ice is forced out by means of a piece of glass rod held in the right hand. 
 
. 
 
 89] HEAT OF VAPORISATION 235 
 
 The magnitude of the radiation correction can be very considerably 
 reduced by using Rumford's device. 1 Having roughly calculated the 
 drop of temperature which may be expected from the size of the ice 
 cylinder and the capacity of the calorimeter, heat the calorimeter to a 
 temperature higher than that of the surrounding water-jacket by half 
 this amount. It will be found impossible to secure exact compensation 
 between the heat lost by radiation during the first half of the experiment 
 and that gained during the second, and the observed drop of tempera- 
 ture will still require a certain amount of correction. By reducing the 
 amount of the correction, however, we minimise the effects of slight 
 errors in determining the rate of cooling. 
 
 Using the following symbols for the quantities entering into the 
 calculations : 
 
 W= weight of water in the calorimeter. 
 w = water value of the calorimeter, thermometer, and stirrer. 
 x the weight of ice melted. 
 #! = the initial temperature of the water. 
 t. 2 = the final temperature of the water. 
 
 8t = the correction to the final temperature to allow for radiation. 
 L = the latent heat of ice. 
 
 Then the following is the thermal equation 
 
 xL + x{t 2 + St] = { W+ ;}{*! - (* 2 + &)}, 
 
 from which the latent heat can be calculated. 
 
 Further correction. The thermometer readings may be corrected for 
 the effect of lag as described on p. 221. The results will require 
 correction to allow for the variation of the specific heat of water. Thus, 
 if s l is the mean specific heat of water between and 2 , as given in 
 Table 21, the heat required to raise the temperature of the water formed 
 by melting the ice is s^t^ + 8t}. In the same way, if s. 2 is the mean 
 specific heat of water between t 2 and t v the heat given out by the water 
 in the calorimeter is Ws^fa - (t z + 8t)}. It will in general be found that 
 the accuracy of the results obtained does not warrant the application of 
 this correction. 
 
 89. Measurement of the Heat of Vaporisation of Water. The 
 determination of the heat of vaporisation of a liquid is a matter of very 
 great difficulty if an accuracy greater than about 1 per cent, is desired. 
 The usual method is to pass the vapour of the liquid into a vessel which 
 is immersed in a calorimeter and is at a temperature lower than the 
 boiling point of the liquid. The weight of vapour condensed is obtained 
 
 1 In this experiment the rate at which the temperature of the calorimeter 
 falls is approximately constant, and so Rumford's method is applicable. See 
 p. 218. 
 
 
236 
 
 CALORIMETRY LATENT HEAT 
 
 [89 
 
 by weighing the vessel before and after the experiment, while the heat 
 liberated by the condensation of the vapour and the cooling of the liquid 
 formed to the temperature of the calorimeter is obtained from the rise of 
 temperature of the calorimeter. 
 
 The chief sources of error when performing an experiment such as 
 outlined above consist of (1) Heat communicated to the calorimeter by 
 conduction through the tube through which the vapour is conducted from 
 
 the boiler into the calorimeter. 
 (2) The vapour is very apt to 
 carry over into the calorimeter 
 small particles of the liquid. 
 The presence of this liquid in 
 the vapour may be due either 
 to spirting, owing to the 
 boiling being too rapid, or to 
 the fact that some of the 
 vapour has been condensed 
 into liquid owing to loss of 
 heat during the passage of the 
 vapour through the tube con- 
 necting the boiler with the 
 calorimeter. It is a very diffi- 
 cult matter to estimate the 
 magnitude of the corrections 
 to be applied to allow for 
 these two errors, or even to 
 ensure that the magnitude of 
 the errors remains the same 
 from one experiment to the 
 next. 
 
 The manner in which the 
 above difficulties can in a 
 measure be overcome, where 
 time is no object, and it is 
 possible to work with large 
 quantities of material, will be 
 
 FIG. 96. found in the account Regnault 
 
 gives of his determination of 
 the latent heat of steam. 1 
 
 An arrangement by means of which the latent heat of steam can be 
 measured to within 1 or 2 per cent, is shown in Fig. 96. The copper 
 drum A has two copper uprights, one of which B serves for the admission 
 of the steam, and the other c communicates with the air. This vessel is 
 attached to a thick disc of varnished wood D, which serves to form a lid 
 
 1 See Regnault, Relation dcs experiences, tfcc., vol. i. p. 635 ; also Cours de 
 Physique, by Jamin and Bouty, vol. ii. p. 159. 
 
90] HEAT OF VAPORISATION 237 
 
 to the calorimeter. The uprights attached to a stirrer E and a ther- 
 mometer also pass through the lid. 
 
 The steam is conducted from the boiler, through a short length of 
 rubber tube, to the side tube G of a small separator, which serves to 
 catch any condensed steam which would otherwise be carried over into 
 the calorimeter. 
 
 To conduct an experiment with this apparatus, first weigh the 
 calorimeter, stirrer, thermometer, the internal vessel A, and the cover. 
 Having filled the calorimeter with water to within about 2 cm. of the 
 top, again weigh. The difference in the weights will give the weight of 
 water in the calorimeter. Place the calorimeter within a double-walled 
 vessel (see p. 214), and watch the changes in temperature during five 
 minutes, keeping the water stirred all the time. Having allowed the 
 steam to pass through the trap F for a few minutes, at a noted time 
 place the end of the glass tube H into the upper end of the upright B, so 
 that the steam passes into the calorimeter. Keep the water in the calori- 
 meter well stirred, and read the thermometer every thirty seconds till the 
 temperature has risen about 10 degrees. At a noted time withdraw the 
 tube H, and continue stirring and reading the thermometer for another 
 five minutes, then weigh the calorimeter and its contents, and thus obtain 
 the weight of steam condensed. 
 
 The rise in temperature of the calorimeter will have to be corrected 
 for loss of heat, due to radiation and conduction, in the manner described 
 on p. 219. It will also be necessary to obtain the water value of the 
 calorimeter, &c., which can be done either by calculation, if the specific 
 heats of the constituents are known, or it may be determined experi- 
 mentally by pouring water at a known temperature (about 100) into A, 
 and proceeding as in the method of mixtures. 
 
 The method of calculating the value of the latent heat is exactly the 
 same as that employed in the previous section. 
 
 90. Measurement of the Heat of Vaporisation by Berthelot's 
 Method. Berthelot has devised a form of apparatus which is designed 
 for the determination of the latent heat of a vapour when only a small 
 quantity of the material is available. His form of the apparatus is 
 described in Watson's Physics, p. 240, and possesses the great defect 
 that it is almost impossible to avoid the vapour becoming superheated, 
 owing to the proximity of the tube leading the vapour from the boiler 
 to tne calorimeter to the flame used to heat the boiler. The presence of 
 this flame also causes a considerable difficulty in screening the calori- 
 meter and thermometer. Finally, the liquid is very apt to boil with 
 bumping. A modification of Berthelot's apparatus, in which the above 
 objections are avoided, has been used by Kahlenberg. 1 In this form, 
 which is shown in Fig. 97, the heating is electrical. The liquid is con- 
 tained in a test-tube shaped tube A, which is closed by a cork D. Two 
 glass tubes E pass through this cork, and have pieces of stout platinum 
 wire fused at their bottom ends. A coil of fairly fine platinum wire F is 
 
 1 Journal of Physical Chemistry (1901), v. 222. 
 
238 
 
 CALORIMETRY LATENT HEAT 
 
 [90 
 
 attached to these stout platinum wires. These tubes are partly filled 
 with mercury, and the wires from the battery dip into the mercury. 
 The vapour produced by the boiling of the liquid passes down an upright 
 tube c into a glass worm in the calorimeter. The worm terminates in 
 
 O' 
 
 FIG. 97. 
 
 an enlargement B, in which the liquid accumulates, and an upright tube 
 G, which is open to the air. The boiler A is not permanently attached 
 to the worm, but the lower part of the tube c fits loosely into the 
 expanded upper part of the worm, the joint being made tight by a ring 
 of cork or of rubber tube. The calorimeter is closed by a thick wooden 
 
91] 
 
 JOLY'S STEAM CALORIMETER 
 
 239 
 
 or ebonite cover, and a thermometer and stirrer pass through this cover. 
 The boiler A is wrapped round with a close fitting cover of felt, so as to 
 check the loss of heat due to radiation and conduction. 
 
 A simpler form of the boiler is shown in Fig. 97 (a). Here the 
 boiler consists of a test-tube A, closed by a cork M, through which pass 
 a tube c, and the wires o attached to the platinum spiral. To introduce 
 the liquid the cork is removed, the required quantity of liquid introduced 
 into the test-tube, and the cork replaced. The whole is then inverted 
 into the position shown in the figure. 
 
 When performing an experiment, the worm is carefully dried and 
 weighed. The calorimeter is weighed first empty, and also full of water. 
 The liquid having been in- 
 troduced into the boiler, 
 the apparatus is put to- 
 gether, and the thermometer 
 read at intervals for five 
 minutes. The current is 
 then switched on and the 
 vapour allowed to pass into 
 the calorimeter till the tem- 
 perature has risen about 5 
 degrees, the thermometer 
 being read every thirty 
 seconds. The current is 
 then stopped, the boiler 
 removed, and the march of 
 the thermometer followed 
 for five minutes. The 
 worm is then removed from 
 the calorimeter, dried out- 
 side, and weighed. If the 
 liquid is at all volatile, corks 
 must be placed in the open 
 ends of the worm directly 
 the boiler is removed. 
 
 The rise of temperature FIG. 98. 
 
 of the calorimeter will re- 
 quire correction for gain or 
 manner. 
 
 91. Joly's Steam Calorimeter. 1 In the steam calorimeter the heat 
 necessary to raise the temperature of a body from the air temperature 
 to 100 is measured by determining the weight of steam which must be 
 condensed into water at 100 to supply this heat. The instrument is 
 somewhat troublesome to use, but in the hands of a careful observer 
 it gives very accurate results.,. 
 
 The body K (Fig. 98) is supported on a light platinum or copper 
 1 Proceedings of the Royal Society (1889), xlvii. 218. 
 
 loss of heat by radiation in the ordinary 
 
240 CALORIMETRY LATENT HEAT [ 91 
 
 dish J, which is suspended by a fine platinum wire G from one of the 
 pans of a balance, which rests on the wooden shelf L. The pan hangs 
 within a vessel A, made of thin brass and covered with felt. The upper 
 part B of this vessel is attached to the stand by means of the tube c, 
 which also serves for the admission of the steam. The lower part can 
 be removed to get at the pan, and is attached to the top by a simple 
 bayonet joint. A drain pipe E allows the superfluous steam to escape. 
 This pipe can be closed when necessary by closing the rubber tube by 
 means of a pinch-cock. The suspension wire passes through a light 
 metal disc H, which is pierced with a hole having a diameter of about 
 two-thirds of a millimetre, which rests lightly on the upper opening of 
 the calorimeter. As the pan swings about after placing the body in it 
 the suspension wire moves this disc, and when the vibrations die out the 
 disc is left in such a position that the wire passes through the hole without 
 touching the edge. To prevent the steam condensing on the suspension 
 wire and on the disc H an electric current is passed through a coil of fine 
 platinum wire I, which it heats to redness. The steam is brought from 
 a boiler through a wide bore piece of india-rubber tube to a short length 
 of brass tube D, which can be fitted into the side tube c. The steam is 
 generated in a boiler, which is shown on a smaller scale at M. In addition 
 to the bent tube to which the rubber tube leading to the calorimeter is 
 attached, there is a short upright tube with a sharp edge. This tube 
 is closed by a light sheet of mica, and acts as a safety-valve to allow the 
 steam to escape when the tube leading to the calorimeter is closed. To 
 prevent the mica being shot off, a fine upright wire passes loosely through 
 a hole in the mica. 
 
 The procedure adopted when making a measurement is as follows : 
 The empty scale pan having been counterpoised, the body is placed on 
 the pan, and the lower part of the calorimeter having been put in place, the 
 weight is determined which has to be added to the other pan of the balance 
 to produce equilibrium. Thus the weight of the body is obtained. The 
 body must be left in the calorimeter till its temperature, as shown by the 
 thermometer T, has become constant, and even in the case of good con- 
 ductors of heat, at least half-an-hour must be allowed. During this 
 time the drain tube at the bottom of the calorimeter must be closed. 
 The tube from the boiler must also be removed, and a cork inserted into 
 the opening to the tube c. While the temperature of the body is becoming 
 constant the steam is raised in the boiler, care being taken to preserve the 
 calorimeter from radiation from the burner or boiler by interposing screens. 
 The steam must be allowed to pass freely through the tube D, so as to 
 heat it. 
 
 The temperature having been read, the thermometer T must be with- 
 drawn, and the hole F closed with a cork. The current having been 
 switched on to the spiral i, the drain tube must be fully opened and the 
 steam pipe connected to the calorimeter, so as to allow the steam to enter 
 the calorimeter suddenly. Some of the steam will be condensed on the 
 body, and will be caught in the pan J. After from one to four minutes 
 
92] HEAT OF SOLUTION 241 
 
 the body will have reached the temperature of the steam, and it will be 
 possible to adjust the balance. Till the body reaches the temperature 
 of the steam the weight will increase steadily, but when the body is 
 completely heated, the weight will only increase very slowly, about half 
 a milligram in five minutes. Finally, the barometric height must be 
 read. 
 
 A similar experiment will have to be made with the empty pan to 
 allow for the increase in weight due to the steam condensed while 
 raising the temperature of the pan and stirrup to the temperature of 
 the steam. 
 
 If t. 2 is the temperature of the steam as deduced from the barometric 
 height, fj the initial temperature of the body, S its specific heat, W its 
 weight, w the weight of steam condensed, and L the latent heat of steam, 
 while w' is the w r eight of steam condensed by the pan alone when its 
 temperature is raised from t^ to /, we have 
 
 A correction will have to be applied to allow for the fact that the 
 density of steam at 100 is about half that of air at ordinary temperatures, 
 and so the buoyancy correction in steam is considerably less than in air 
 at the lower temperature. The deduction from the weight w on this 
 account will be equal to the product of the volume of the body into the 
 difference in density between the air at the lower temperature and that 
 of steam at the higher. Of course the actual density of the steam will 
 vary with its temperature, that is, with the height of the barometer. 
 For most practical purposes, however, it is sufficient to take a mean 
 value for the density of steam, and the following table, which gives the 
 relative density of air at various temperatures to steam at 100, will serve 
 for the calculation of the correction : 
 
 Initial Correction 
 
 Temperature. Factor. 
 
 0-00069 
 
 5 0-00066 
 
 10 0-00064 
 
 15 0-00062 
 
 20 0-00060 
 
 The volume of the body must be multiplied by the most suitable of 
 the above factors, and the product deducted from w to give the actual 
 weight of steam condensed. 
 
 92. Measurement of the Heat of Solution. If two substances 
 which are soluble are taken at the same temperature and mixed, then 
 in general the act of solution is accompanied by a change in temperature. 
 The thermal change may indicate either an absorption or an evolution of 
 
 Q 
 
242 CALORIMETRY LATENT HEAT [ 92 
 
 heat, according to the nature of the substances. The quantity of heat 
 which has to be communicated or abstracted from the substances to 
 maintain the temperature constant during the solution of 1 gram of 
 the solute in a given weight of the solvent is called the heat of solution 
 of the solute for the given concentration. 
 
 If p grams of a salt, of which the molecular weight is m, are 
 dissolved in q grams of a solvent, the heat developed during solution 
 being H, then mUjp is called the molecular heat of solution for the 
 
 solution containing per cent, of the solute. With strong solutions 
 
 the molecular heat of solution in general decreases as p increases. 
 
 In the practical determination of heats of solution it is not possible 
 to keep the temperature constant during solution and to measure the 
 heat which has to be abstracted or communicated. What is done is to 
 prevent as far as possible all exchange of heat with surrounding bodies, 
 and to measure the change in temperature which occurs on solution. 
 From this change in temperature we can calculate the heat of solution 
 if we know the specific heat of the solution. 
 
 Let s be the specific heat of the solution, and w the water value of 
 the calorimeter, thermometer, and stirrer. Then if is the initial 
 temperature of the salt and the solvent, and ^ is the final temperature 
 after solution is complete, the heat of solution is given by 
 
 The heat of solution at any temperature having been measured, we 
 can calculate the heat of solution at any other temperature if we know 
 the specific heats of the salt, of the solution, and of the solvent. For 
 suppose we start with the salt and solvent at the lower temperature t v 
 we may obtain a solution at the higher temperature 9 in two ways. In 
 the first way we dissolve the salt at t v the heat of solution being Q tl . 
 We then heat the solution to t 2 , during which we have to supply an 
 amount of heat 
 
 Next we may heat the salt and solvent separately to t 2 , during which 
 we shall have to supply a quantity of heat which, if s p and x q are the 
 specific heats of the salt and of the solvent respectively, is equal to 
 
 and then dissolve the salt at 2 , the heat of solution being Q tz . 
 
 If we neglect the external work due to the very small changes in 
 
92] 
 
 HEAT OF SOLUTION 
 
 volume which take place, the quantities of heat required in the two 
 cases must be the same. Hence 
 
 (p* 
 
 or 
 
 
 But p - - is the mean specific heat of the constituents, thus if this 
 
 is greater than s, the specific heat of the solution, the heat of solution 
 decreases with rise of temperature, and vice versa. 
 
 A convenient form of calorimeter for use when measuring heats of 
 solution is shown in Fig. 99. A thin beaker B, which would hold about 
 half a litre, is supported on a cardboard cross 
 within a larger beaker A. A wood or ebonite cover 
 c is pierced with three holes, one of which serves 
 to give passage to a thin-walled test-tube D. A 
 delicate thermometer T and a stirrer s pass through 
 the other holes. The stirrer is best made of glass 
 rod, since in many cases the solutions would act 
 upon metal. 
 
 As an example of the measurement of the heat 
 of solution of a salt, we may take the case of sodium 
 chloride in water. The salt must be well dried over 
 a water bath and finely powdered. The beaker B 
 having been weighed, it must be about three-quarters 
 filled with distilled water, and then again weighed. 
 The test-tube D must also be weighed, and suffi- 
 cient of J;he salt put in to form a solution of the required strength. 
 The test-tube containing the salt having been introduced within the 
 calorimeter, it must be left there for at least half-an-hour, to allow 
 the salt to acquire the same temperature as the water. During the 
 last ten minutes the water must be kept stirred, and the temperature 
 read at intervals of a minute or half a minute. The temperature of 
 the water and salt being the same, the bottom of the test-tube must 
 be broken by pushing down a glass rod, and the solution of the salt 
 assisted by vigorous stirring. The thermometer must be read every 
 half minute during the time solution is taking place, and then every 
 minute for five minutes after all the salt has dissolved. 
 
 The observed fall of temperature will have to be corrected for gain 
 of heat by radiation exactly in the manner described in 82. The 
 water value of the beaker, test-tube, thermometer, and stirrer must be 
 obtained either by calculation from their weights and the specific heat 
 of glass, or by experiment. If the specific heat of the solution is not 
 
 FIG. 99. 
 
244 
 
 CALORIMETRY LATENT HEAT 
 
 [92 
 
 known, 1 it will have to be measured by one of the methods previously 
 described ( 84, 85). The heat of solution is then calculated by 
 equation (1) above. 
 
 1 Where no very great accuracy is required, we may assume that the solution 
 requires the same quantity of heat to raise its temperature 1 as would the water 
 it contains alone require. The magnitude of the error produced by this assump- 
 tion will be apparent from the following table, in which a few of Berthelot's 
 results are given : 
 
 Composition of Solution. 
 
 3600 grams H 2 + 
 
 80 grams SO 3 
 63 HNC-3 
 
 36-5 
 40-0 
 58-5 
 74-6 
 53-5 
 
 HC1 . 
 NaHO 
 NaCl . 
 KC1 . 
 NH 4 C1 
 
 Heat in Calories 
 
 required to raise 
 
 Temperature 1 C. 
 
 3595 
 
 3597 
 
 3561 
 
 3578 
 
 3578 
 
 3565 
 
 3588 
 
CHAPTER XIV 
 
 VAPOUR PRESSURE 
 
 93. The Measurement of Vapour Pressure. The methods in use for 
 measuring vapour pressure may be divided into two classes. In the 
 first class, called the statical methods, a limited portion of the liquid 
 to be experimented upon is introduced into a confined space, and evapo- 
 ration is allowed to go on till this space is saturated with the vapour. 
 The temperature and pressure are then measured. In the second class, 
 called the dynamical methods, the liquid is caused to boil under a 
 constant pressure, and the temperature of ebullition gives the tempera- 
 ture at which the vapour pressure is equal to the pressure to which 
 the boiling liquid is subjected. Both of these methods were used by 
 Regnault in his classical researches on the vapour pressure of water. 
 More recent experiments have, however, shown that the statical 
 method is in general unreliable. The reason is that it is practically 
 impossible to obtain absolutely pure liquids. In the statical method 
 any trace of impurity will, owing to the small quantity of liquid 
 evaporated, be liable to produce quite a considerable variation in the 
 pressure exerted by the vapour of the liquid. 
 
 94. Ramsay and Young's Method of Measuring Vapour Pressure. 
 A very convenient and accurate method of measuring the vapour pressure 
 of a liquid by the dynamical method has been worked out by Ramsay 
 and Young. 1 Their arrangement is shown in Fig. 100, and consists of 
 a glass boiling tube A, with a side tube B. A thermometer T and the 
 stem of a tap funnel c pass through a cork, which closes the top of the 
 tube A, the joint being made quite air-tight. The bulb of the thermo- 
 meter is wrapped round with some porous material, such as cotton wool 
 or asbestos, and the drawn-out end of the tap funnel is bent round 
 so as to allow the liquid to drop on the bulb. The side tube B is 
 connected to a small bottle D in the manner shown on the figure, while 
 this bottle is connected through a three-way tap with a large bottle F 
 and a tube H leading to a manometer. A side tube G, which can be 
 closed with a tap, is connected with an air-pump, by means of which 
 the pressure of the air in the apparatus can be altered. All the joints 
 must be made air-tight by being painted, if necessary, with india-rubber 
 cement. The bottle F must be protected from change in temperature 
 by being packed round with cotton wool or immersed in a bucket of 
 water. The function of this bottle is to act as an air reservoir, and so 
 
 1 Journal of the Chemical Society (1885), xlvii. 42. 
 245 
 
246 
 
 VAPOUR PRESSURE 
 
 [94 
 
 reduce the effects of any slight leak, or variation in the rate of formation 
 of vapour, on the pressure. 
 
 When performing the experiment, the liquid to be measured is placed 
 in the funnel c, and the tube A is placed in a bath, which is heated to a 
 temperature slightly higher than the boiling point of the liquid under 
 the pressure at which the experiment is to be made. By means of the 
 air-pump the air is exhausted till this pressure is reached, when the 
 tap G is closed. The bottle D is placed in a mixture of ice and water, 
 or ice and calcium chloride, according to the volatility of the liquid, 
 and serves to catch the vapour as it passes over from the tube A. 
 
 The liquid is allowed to flow freely till the bulb of the thermometer 
 
 FIG. 100. 
 
 is thoroughly moist, and then the tap on the funnel is turned off to 
 such an extent that the supply of liquid is just sufficient to allow for 
 the evaporation which takes place as the vapour is drawn off through 
 the tube B. When the reading of the thermometer becomes constant, 
 say, at t, the pressure as indicated by the manometer is read as well as 
 the height of the barometer. From these readings the pressure at 
 which the vapour was formed can be at once calculated, the usual 
 corrections for the temperature of the mercury columns, &c., being 
 applied. This pressure will be the vapour tension of the liquid at the 
 temperature t. 
 
 By altering the pressure in the apparatus, a series of measurements 
 of vapour tension at different temperatures can be made. 
 
95] THE DEW-POINT 247 
 
 95. Determination of the Dew-Point. One of the most convenient 
 instruments for measuring the dew-point is Regnault's hygrometer (see 
 Watson's Physics, 220). The aspirator should be placed at least 
 2 metres from the instrument, and preferably at a lower level, the 
 connection being conveniently made by a length of lead or " compo " 
 tubing. The silver thimbles must be very well polished by rubbing 
 with a soft leather and polishing powder (jewellers' rouge) till the sur- 
 face looks almost black. As the presence of the observer in the neigh- 
 bourhood of the instrument is certain to alter the hygrometric state of 
 the air it is best to use two telescopes, one to view the silver thimble, 
 and the other to read the thermometer in the cooled thimble. If it is 
 impossible to observe by means of telescopes, then a sheet of glass at 
 least 50 centimetres in each direction must be held between the observer 
 and the instrument. The thimble which is going to be cooled must be 
 so placed that a bright light, such as a window or a gas flame, is seen 
 reflected in the silver. 
 
 To perform an experiment, fill the thimble to which the aspirator 
 is attached with ether well above the thermometer bulb. To assist the 
 other thermometer taking up the temperature of the air it is advisable to 
 place some liquid, such as xylene or toluene, in the other thimble. Water 
 is not a good liquid to use for this purpose, since it has such a high 
 specific heat. Neither is ether, since it is so volatile, and if any evapora- 
 tion takes place it will cause the temperature to fall, owing to the latent 
 heat of the liquid. 
 
 Start the aspirator, and watch the reflection of the light as seen in the 
 silver, and as soon as any dulness appears reduce the current of air 
 through the ether till the dew on the silver disappears, at the same time 
 noting the temperature. Having in this way roughly determined the 
 dew-point, slightly increase the rate of flow of the water, and immediately 
 any sign of dew appears read the thermometer. Decrease the flow of 
 water, and read the air temperature. Then watch the dew, and as soon 
 as it vanishes, again read the thermometer. The mean of the tempera- 
 tures corresponding to the deposition and disappearance of the dew is 
 taken as the dew-point. The observation should be repeated several 
 times. 
 
 From a table of the vapour pressures of water (Table 14) 
 the vapour pressures corresponding to the dew-point and the 
 temperature of the air can be obtained. The ratio of the actual 
 pressure as deduced from the dew-point to the maximum pressure 
 corresponding to the temperature of the air is the hygrometric 
 state. 
 
 If the actual weight of water contained in the unit volume, or what 
 is more usual in a cubic metre, be required, a small correction will have 
 to be applied to the number corresponding to the dew-point obtained 
 from Table 14. 
 
 Let the weight of water which is required to saturate a cubic metre 
 of air at the temperature of the dew-point r be w. Now if we start with 
 
248 VAPOUR PRESSURE [ 96 
 
 a cubic metre, of unsaturated air at the temperature of the air t, and 
 cool it down to the dew-point T, the volume will be 
 
 273 + r ,. 
 - -- cubic metres. 
 273 + if 
 
 But w is the water contained- in a cubic metre at T. Hence a cubic 
 metre of the air at the air temperature t will contain less than w grams 
 of water, the true weight of water being 
 
 273 + r 
 2-73^1" gramS " 
 
 96. The Wet and Dry Bulb Thermometer. A convenient way of 
 obtaining an approximate value for the hygrometric state of the air is 
 to measure the cooling of a thermometer, which is kept moist with water, 
 owing to the evaporation of the water. Two similar thermometers are 
 generally fixed on a stand, the one being used to give the temperature 
 of the air. The bulb of the other is surrounded with muslin, which is 
 kept moist by a piece of lamp wick which dips in a vessel containing 
 water. 
 
 The drier the air, the more rapid the evaporation of the water on the 
 wet bulb, and hence the greater the difference in the readings of the two 
 thermometers. The rate of evaporation of the water, however, does not 
 depend exclusively on the hygrometric state of the air, but also on the 
 nature of the surroundings of the thermometers, and more particularly 
 whether the air in the neighbourhood of the wet bulb is in motion 
 or not. 
 
 If t w = the temperature given by the wet bulb thermometer, 
 
 t d = the temperature of the air as given by the dry bulb thermometer, 
 F w = the maximum tension of water at the temperature t w , 
 B = the height of the barometer in millimetres, 
 
 then the actual tension of the water vapour present in the air is given by 
 
 where c is a constant depending on the nature of the surroundings. The 
 following values of c may be used in the conditions mentioned : 
 
 In the open air with slight wind c = 0*0008. 
 In the open air with no wind c = 0'0009. 
 In a small closed room c = 0*001. 
 
 Owing to the uncertainty as to the correct value for c to use on any 
 given occasion, it is generally quite sufficient to take for B a mean value, 
 such as 750 mm. 
 
 Somewhat more consistent results are obtained if the wet bulb ther- 
 mometer is kept swinging backwards and forwards for a minute or two 
 before taking a reading. In such a case c may be taken as equal to 
 
97] 
 
 THE ABSORPTION HYGROMETER 
 
 249 
 
 FIG. 101. 
 
 O'OOOT. The motion may be communicated by attaching the ther- 
 mometer to a heavy pendulum, which is set swinging. 
 
 97. The Absorption Hygrometer. The most accurate method of 
 determining the quantity of moisture contained in a cubic metre of air 
 is to draw a measured volume of the air through some tubes in which 
 are placed drying materials, such as strong sulphuric acid or phosphorus 
 pentoxide. The difference between the 
 weight of these drying tubes before and 
 after the passage of the air gives the weight 
 of the moisture present in the volume of air 
 passed through. 
 
 A convenient form of absorption tube 
 for use in this experiment is shown in Fig. 
 101, two tubes being generally ample to 
 absorb all the moisture. The test tubes are 
 about 10 cm. long and 2J cm. wide, while 
 the glass tubes through which the air 
 passes have a bore of about 5 mm. These 
 tubes are filled with pumice saturated with 
 strong sulphuric acid. In order to prevent 
 leakage at the corks these are pushed right 
 into the top of the test tubes, and then a layer of melted paraffin wax 
 is poured in. 
 
 The pumice is prepared in the following manner : Having been 
 broken into lumps about the size of a pea, the dust is removed and 
 the remainder is heated to redness in a crucible, and then thrown, while 
 still red-hot, into strong ("pure re- 
 distilled ") sulphuric acid. The super- 
 fluous sulphuric acid having been poured 
 off, the pumice is preserved in a well- 
 stoppered bottle. 
 
 When the drying tubes are not in use, 
 the ends of the tubes must be closed by 
 means of a short length of rubber tubing 
 and a piece of glass rod. These stoppers 
 are weighed with the tubes. During 
 the weighing the tubes may be suspended 
 by means of a loop of wire attached to 
 the bent tube joining the two test-tubes. 
 
 As it is necessary to know the 
 volume of air which passes through 
 
 the drying tubes, some form of aspirator which is capable of giving this 
 volume must be used. The usual form is shown at A (Fig. 102). The 
 vessel is filled with water and the upper tube connected with the drying 
 tubes through a length of rubber tubing. On opening the tap on the 
 bottom tube the water runs out and the air is drawn in to take its place. 
 The volume of the vessel, and hence that of the air, is obtained by 
 
 FIG. 102. 
 
 
250 VAPOUR PRESSURE [ 97 
 
 weighing the vessel empty and full of water. Another form of aspirator, 
 which can readily be set up, is shown at B, and its construction will be 
 obvious from the figure. 
 
 When making an experiment the aspirator must be placed at a 
 distance from the point where the air is drawn into the drying tubes, 
 otherwise the water used in the aspirator will influence the hygrometric 
 state of the air. It is best not to have a long tube between the intake 
 and the drying tubes owing to the errors which might be caused by the 
 evaporation from, or deposition of moisture on, the tube. A long length 
 of tubing, so long as it is air-tight, has, however, no ill effect when placed 
 between the drying tubes and the aspirator. 
 
 It will be advisable to include a third drying tube, which need not be 
 weighed, between the two weighed tubes and the aspirator, to prevent 
 any moisture from the aspirator finding its way to these tubes. 
 
 The tubes having been weighed, connect them to the aspirator and 
 start the current of air. The air should pass at the rate of about a litre 
 per minute. If the current is too rapid, all the aqueous vapour may not 
 be removed in the drying tubes; while if the current is too slow, the 
 experiment will take an unduly long time. A thermometer, hung with 
 its bulb freely exposed to the air in the neighbourhood of the intake, must 
 be read at regular intervals during the course of the experiment. After 
 about 15 to 20 litres of air have been drawn through the apparatus the 
 drying tubes must be again weighed. The increase in weight divided 
 by the volume of air, in cubic metres, which has passed gives the weight 
 of water contained in a cubic metre. The weight of moisture which 
 would be required to saturate the air is obtained from the air tempera- 
 ture by means of Table 14. 
 
 Corrections. The volume F, which is the volume of water which 
 has been run out of the aspirator, will require a slight correction to 
 obtain the exact volume of air drawn into the drying tubes. Thus if 
 t is the temperature of the air, and t f that of the water in the aspirator, 
 f is the tension of the water vapour actually present in the air, and f 
 the maximum tension of water at ', while the barometric height is B. 
 Then since the air in the aspirator becomes saturated owing to its con- 
 tact with the water, the pressure of the dry air would be B -f ; that 
 is, if the water vapour were removed, the volume remaining F, the pres- 
 sure would fall to Bf. If, further, the temperature were reduced to 
 zero, the dry air would have a volume F/(l+u^), the pressure being 
 B-f. 
 
 Now when drawn into the drying tubes the air was at a temperature 
 t, and at this temperature its volume, the pressure being B -/', would be 
 
 "-'. The actual pressure to which the dry air was subjected was, 
 1 + at 
 
 however, B -f. Hence the volume of the air before it entered the 
 drying tubes was 
 
 B-f ' 
 
97] THE ABSOEPTION HYGROMETER 251 
 
 From the weight of moisture w contained in a cubic metre of 
 air we may calculate the tension of the aqueous vapour /. Thus if 
 a is the mass of a cubic metre of air at 760 mm. pressure and C., 
 the mass of a cubic metre of air at a pressure / and a temperature t is 
 
 . If, then, the density of aqueous vapour in terms of that of 
 760(1 + at) 
 
 dry air at the same pressure and temperature is 8, the mass of water 
 vapour contained in a cubic metre of the moist air is given by 
 
 a/8 
 " 760(1 + at)' 
 
 Hence 
 
 ,_ 760(1 +at)w ( . 
 
 7 a8 
 
 Now Regnault has shown that if 8 be taken as a constant and equal 
 to 0'6221, the above expression gives the correct value for /, so that we 
 are justified in using this value for 8. The value of a is 1292*8 grams 
 per cubic metre, and a is '00367. Hence substituting these values 
 we get 
 
 /= 0-9449(1 + 0-00367 *)w ...... (3) 
 
 Thus if W is the total increase in the weight of the drying tubes when 
 V cubic metres of water are run out of the aspirator we have, combining 
 (1) and (3), 
 
 . 0-9449(1 +at)W 
 J~ B-f 
 B-f ' 
 
 f _ 0-9449(1+ 0-00367 JQTF ( } 
 
 B-f~ (B-f)V 
 
 The above method of conducting the experiment takes some consider- 
 able time and requires constant attention, since the aspirator has to 
 be refilled and the temperature of the air read. We cannot make use of 
 a water pump, since we require to know the volume of air which has 
 passed through the apparatus. By arranging the experiment, however, so 
 that we directly determine the weight of water contained in a given 
 volume of air, and also the weight of water contained in the same volume 
 of air saturated at the temperature of the air, then it will be unnecessary 
 to know either the volume of the air or its temperature in order to obtain 
 the hygrometric state. 
 
 The air is first passed through two drying tubes, such as those shown 
 in Fig. 101. It then passes through a series of tubes, which are freely 
 exposed to the air so that they and their contents are at the air tempera- 
 ture. These tubes contain water, and during its passage through the 
 water the air becomes saturated. Next the air passes through a second 
 
252 VAPOUR PRESSURE [ 97 
 
 set of drying tubes. The increase of weight in the first set of drying tubes 
 gives the weight of water actually contained in the volume of air drawn 
 through the apparatus, while the increase of weight of the second set 
 gives the weight of water contained in this volume of air when saturated 
 with water at the air temperature. The hygrometric state is then 
 obtained by dividing the first weight by the second. The tubes used 
 to saturate the air with moisture may consist of three or four large 
 U -tubes packed with glass wool, sufficient water being added to saturate 
 the wool and fill the bend for about 5 centimetres. After passing 
 through these U -tubes the air is passed through a tube lightly packed 
 with cotton wool to prevent any drops of water being mechanically 
 carried into the drying tubes by the passing air. 
 
 The air may be sucked through the apparatus by means of a water 
 pump, and by allowing the pump to work slowly we may continue the 
 experiment for twenty-four hours, and thus obtain the mean hygrometric 
 state of the air during the day. 
 
CHAPTER XV 
 
 VAPOUR DENSITY FREEZING AND BOILING POINTS 
 OF SOLUTIONS 
 
 98. Measurement of Vapour Density by Victor Meyer's Method. 
 
 A convenient method of measuring the vapour density of a body which 
 can be volatilised below a temperature of about 400 C., and where 
 an accuracy of something less than 
 5 per cent, is sufficient, is jthat 
 devised by Victor Meyer. A slightly 
 modified form of the apparatus is 
 shown in Fig. 103. The wide tube 
 A contains at the bottom a liquid 
 which has a higher boiling point 
 than the substance of which the vapour 
 pressure has to be measured. The 
 substance is vaporised in the bulb B, 
 and in doing so displaces its own 
 volume of air, which is driven up the 
 tube c through a length of thick- 
 walled in lii-rubber tubing attached 
 to the side tube D and into a graduated 
 burette E. This burette is supported 
 in the manner shown within a glass 
 cylinder F, the bottom of the cylinder 
 being closed either by being drawn 
 down as shown, or with a cork and 
 a drainage tube attached, which can 
 be closed by a pinch-cock G. The 
 liquid to be experimented upon is 
 contained either in a small bottle 
 with a glass stopper, or in a thin- 
 walled glass bulb, which at the 
 commencement of the experiment is 
 hung from a wire J, which passes 
 through the cork which closes the 
 top of the tube c. The form of this 
 wire is shown on a large scale at I. In the position shown the bottle 
 hangs from the lower end of the wire, a loop of wire being attached to 
 the neck of the bottle for the purpose. On turning the wire through 
 
 253 
 
 FIG. 103. 
 
254 VAPOUR DENSITY [ 98 
 
 two right angles by means of the handle J, the wire loop slips off the 
 hook, and the bottle falls down the tube c. A little asbestos placed 
 in the bulb B serves to prevent the bottle breaking the tube in its fall. 
 
 The following liquids may be used in the heater, and will serve 
 for substances which vaporise at temperatures below those given opposite 
 each liquid : 
 
 Maximum Boiling 
 
 Liquid in Heater. Point of 
 
 Substance. 
 
 Water 80 
 
 Analine 150 
 
 Methyl salicylate . . . 200 
 
 Sulphur 400 
 
 To conduct an experiment, the small bottle or glass bulb which is to 
 contain the liquid is weighed. It is then filled with liquid, and again 
 weighed. If a glass bulb is used, after the introduction of the liquid the 
 ends of the capillaries used in filling the bulb, which may have the 
 shape shown in Fig. 103 at L, must be sealed off. This may readily be 
 done by holding the ends in the edge of a Bunsen flame. 
 
 The flame used to heat the tube A is adjusted so that the vapour of 
 the liquid used extends to within about a third of the length of the 
 tube from the upper end. As it is of importance that the temperature of 
 the various parts of the tube c should remain constant throughout the 
 experiment, it is important to screen the flame and the boiling tube from 
 draughts by means of an asbestos or cardboard tube. After the boiling 
 liquid in A has reached a steady state, the bottle containing the substance 
 is attached to the hook and the cork inserted in c. The level of the water 
 in the tube F is then adjusted so that the surfaces of the water in the 
 burette and in the tube F are exactly at the same level, and the reading 
 of the water surface in the burette is noted. The pin J is then turned, 
 allowing the bottle to fall into the bulb B. Owing to the heat the 
 stopper is forced out of the bottle or the bulb is burst, as the case may be, 
 and the liquid vaporises, driving air over into the burette E. By allowing 
 water to run out, the level of the water inside and out is kept the same, 
 and as soon as the volume becomes constant the reading on the burette 
 is noted. Immediately after the thermometer T is read, and also the 
 height of the barometer. 
 
 After taking the second reading of the burette, the cork closing the 
 tube c should be removed to prevent any chance of the water in the 
 burette being forced over into the bulb B. 
 
 The difference between the two readings of the burette gives the 
 volume which the vapour would occupy at the temperature shown by the 
 thermometer T and at the pressure corresponding to the barometric height, 
 subject to a correction to allow for the volume .occupied by the water 
 vapour In the upper part of the burette. 
 
 Let V be the volume as read off on the burette corrected for any 
 graduation error which may exist, T the temperature, and B the height 
 
99] 
 
 VAPOUR DENSITY 
 
 255 
 
 of the barometer. Then if F is the tension of the water vapour at the 
 temperature T, the actual pressure of air in the burette is H - F. 
 Thus the volume of the vapour at the temperature T and the pressure 
 H-F is F. If the volume at C. and the standard pressure of 
 76 cm. is F () , we have 
 
 v __ V(H-F) 
 
 K "76(l+a7')' 
 
 But the difference between the weight of the bottle full and empty is 
 the weight W of the vapour. Hence the density D is given by 
 
 V(H-F} ' 
 
 To obtain the molecular weight of the vapour we have to calculate 
 the density in terms of that of hydrogen taken as 2. Now 1 cubic 
 centimetre of hydrogen at and 76 cm. 
 pressure weighs '0000900 grains. Hence 
 the molecular weight M is given by 
 
 M 
 
 99. Measurement of Vapour Density 
 by Hofmann's Method. A more accurate 
 method of determining vapour density 
 than that described in the previous section 
 is that devised by Hofmann. A glass 
 tube A (Fig. 104) having a diameter of 
 about 2 cm. and a length of about 90 cm., 
 and closed at the top, is filled with dry 
 mercury and inverted in a trough D contain- 
 ing mercury. This tube is surrounded by 
 a wide glass tube, B, which serves as a 
 vapour jacket for heating the inner tube. 
 This outer tube also dips in the mercury 
 in the trough, resting in a shallow groove 
 cut in the bottom of the trough and held 
 at the top by a loosely fitting clamp H, 
 which is attached to a wooden upright. 
 The upper end of the tube A is held in a central position by a ring of cork 
 E, which is cut away in the manner shown at F to allow of the passage 
 of the vapour used to heat the jacket. A glass tube E, bent in the manner 
 shown, has its upper end ground flat, and is cemented into the trough, 
 so that when the mercury just comes to the top there is a layer of about 
 2 cm. of mercury in the trough. This tube serves as a drain tube to carry 
 
 Fin. 104. 
 
256 VAPOUR DENSITY [ 99 
 
 away the liquid which condenses in the jacket. The heating vapour enters 
 the jacket through a tube c, which passes through the cork which closes the 
 top of the vapour jacket. A thermometer T serves to give the tempera- 
 ture of the vapour. 
 
 When performing an experiment the tube A is thoroughly cleaned 
 and well dried by driving a current of hot air up to the closed end 
 through a glass tube. The tube is then filled with dry mercury, all air 
 bubbles attached to the side being removed by running a bubble of air 
 up the tube. This removal of - the air can further be facilitated by 
 placing the lower end of the tube in the mercury bath, so that a Torri- 
 cellian vacuum is formed. If, then, the tube is slowly inclined till the 
 mercury reaches the top, the air which has collected will appear as a 
 bubble, which may be run up to the open end and removed. The tube 
 filled with mercury is then introduced into the jacket, and the whole 
 inverted over the mercury bath. The liquid to be observed is introduced 
 in a small stoppered bottle or glass bulb, as in the previous method. In 
 order to facilitate the introduction of the liquid, the bottom of the 
 mercury bath has a groove cut below the end of the tube A, as shown in 
 the figure. 
 
 The vapour of the substance which is to be used to heat the jacket is 
 passed in through the tube c. Since if too large a quantity of liquid is 
 not used the vapour will exist under a pressure less than one atmosphere, 
 the jacket need not be heated to a temperature higher than the boiling 
 point of the substance. In fact, by using a wide tube and a small 
 quantity of the substance, so that the pressure within the tube is very 
 small, the temperature of the jacket may well be very considerably lower 
 than that of the boiling point under atmospheric pressure. 
 
 When the vapour used to heat the jacket has been issuing from the 
 tube E for some time, so that the mercury column has had time to reach 
 the temperature of the jacket, the volume of the vapour and the height 
 of the mercury column must be read. In some forms of the apparatus the 
 tube A is divided along its whole length in millimetres. In this case 
 the readings corresponding to the mercury meniscus at L and at M must 
 be made,' and to facilitate this measurement it is well to provide the 
 mercury trough with plate glass at front and back. In the other 
 arrangement a millimetre scale engraved on mirror glass is attached 
 to the wooden stand immediately behind the jacket, and in order to 
 facilitate the reading of the position of the mercury surface in the trough 
 a small metal rod G with a pointed end is attached to the upright. The 
 lower point of this rod is brought into contact with the surface of the 
 mercury, and the position of the upper end is read on the scale. Then by 
 measuring the length of the rod we can immediately obtain the reading 
 corresponding to the surface of the mercury. When the tube is not 
 graduated, it will be necessary not only to take the reading on the scale 
 corresponding to the meniscus L, but also to read the position of the top 
 of the tube K, or of a fine line engraved on the side of the tube. 
 
 To obtain the volume of the .vapour it is next necessary to determine 
 
99] VAPOUR DENSITY 257 
 
 the volume of the tube A from the closed end down to the position 
 occupied by the mercury meniscus when the vapour was present. Fcr 
 this purpose the tube must be cleaned and dried, and fixed in an upright 
 position with the closed end downwards. Mercury must be poured in 
 till the meniscus occupies the position it did in the experiment. If 
 the tube itself is graduated, this position can be identified from the 
 graduations. If the tube is not graduated, then the position must be 
 determined by measuring from the top or the fixed mark, as the 
 case may be. The volume of the mercury is then determined 
 by pouring the mercury into a weighed beaker and weighing. 
 A correction will have to be applied to allow for the 
 fact that the. curvature of the mercury meniscus is in the 
 opposite direction when measuring the volume from what 
 it was when the vapour was present. Thus if the readings 
 have been made, as they should, to the top of the meniscus, 
 the volume obtained from the weight of mercury will have to FIG. 105. 
 be increased by the volume of the shaded part shown in w ^~~ . _ ~ 
 Fig. 105. The amount of the corrections which have to be applied with 
 tubes of different diameters have been determined by Bunsen, and are 
 given below. 
 
 Diameter of tube, 1-4 1-5 1'6 1-7 1'8 1'9 2'0 2-1 cm. 
 Correction . . . 0'175 0-187 0'195 0'198 0'193 0'181 0'163 0'139 c.c. 
 
 The correction in each case is the volume which has to be added to 
 the volume as deduced from the weight of mercury to give the volume of 
 the vapour. 
 
 The volume as determined above will require correction to allow for 
 the expansion of the glass. In calculating this correction it will be 
 sufficiently accurate to take a mean value, 0*0000232, for the coefficient 
 of cubical expansion of glass. If v is the volume of the tube as measured 
 at a temperature t, and T is the temperature at which the vapour was 
 measured, we have that the volume V of the vapour is given by 
 
 V=v{l+ 0-0000232(7'- t)}. 
 
 To find the pressure to which the vapour is subjected, we must deduct 
 from the height of the barometer the pressure due to the column of 
 mercury in the tube A. Let the height of this column be h ; then 
 the height H if the temperature of the mercury were would be 
 given by l 
 
 #=/*{! -0-0001827 7 }. 
 
 Thus the pressure within the tube is equal to B - H, where B is the 
 height of the barometer at the time of the experiment, reduced of course 
 to 0. Theoretically, these heights ought to be reduced to what they 
 would be at the sea-level and at latitude 45, but in practice this is 
 hardly necessary. 
 
 1 See 58. 
 
 
258 
 
 VAPOUR DENSITY 
 
 [100 
 
 The pressure inside the tube is, however, not entirely due to the 
 vapour, for the mercury has a vapour tension, and so part of the pressure 
 is due to the presence of mercury vapour, and hence at temperatures 
 above 100 a correction becomes necessary on this account. If e is the 
 vapour tension of mercury at the temperature T, as given in Table 17, 
 the actual pressure exerted by the vapour is 
 
 Hence the density of the vapour is given by 
 
 where W is the weight of liquid vaporised. 
 
 In accurate determinations an allowance will have to be made for the 
 
 volume of the bottle or bulb in which the liquid was contained. This 
 
 can at once be calculated from the weight, taking the value 2 '5 for 
 
 the density of glass. 
 
 100. Measurement of the Freezing Point of a Solution. A very 
 
 convenient piece of apparatus for determining the lowering in the 
 freezing point produced by the solution of a given 
 substance has been devised by Beckmann, and is 
 shown in Fig. 106. It consists of a glass test-tube 
 A provided with a side tube B. This side tube may 
 in many cases be dispensed with, and a simple boiling 
 tube used. The tube A is fitted, by means of a cork, 
 within a slightly wider tube c, which is itself im- 
 mersed in water or some freezing? mixture contained 
 in a beaker D, the liquid in this beaker being main- 
 tained at a temperature of a few degrees below the 
 freezing point of the solution placed in A. A delicate 
 thermometer T passes through the cork which closes 
 the top of A, as also does a glass rod which carries 
 a loop of stout platinum wire at its lower end and 
 acts as a stirrer. 
 
 In order to allow of its being used with solvents 
 which have freezing points at different temperatures 
 the thermometer is fitted at the top with an auxiliary 
 bulb, shown on a large scale at F. By allowing more 
 or less of the mercury to flow into this bulb, and then 
 detaching it from the rest of the mercury column, we 
 
 can adjust matters so that when the bulb is at the freezing point of the 
 
 solvent being examined, the top of the mercury thread is near the top of 
 
 the thermometer scale. 
 
 The quantity of mercury in the thermometer is adjusted in the 
 
 following manner : The bulb of the thermometer is placed in some 
 
 water heated to" a temperature 1 or 2 degrees above the freezing point 
 
 FIG. 106. 
 
100] FREEZING POINT OF A SOLUTION 259 
 
 of the solvent, and when the thermometer has had time to acquire the 
 temperature of the water, by a sudden jerk the mercury column is caused 
 to break at the top of the bulb F. The thermometer is 'then cooled to 
 the melting point of the solvent, and if the end of the mercury column 
 is within about a degree of the top of the scale, the adjustment is 
 complete. If not, the operation must be repeated. The mercury in the 
 bulb F can be made to join up with the rest of the mercury by heating 
 the bulb till the mercury reaches the top of the bulb F, and then inverting 
 the thermometer so that the mercury at the bottom of the bulb joins on 
 to that at the end of the column. 
 
 Having adjusted the thermometer, the next operation is to determine 
 the freezing point of the pure solvent. The tube A, with the thermo- 
 meter, stirrer, &c., is weighed, and then enough of the solvent is intro- 
 duced to well cover the bulb of the thermometer. By again weighing, 
 the weight of solvent is obtained. About 20 grams of solvent ought 
 to be employed, so that a balance which will weigh to within about 
 0*02 gram will be sufficiently accurate. During the weighing the 
 apparatus can be suspended from the hook which carries the balance pan 
 by means of a wire sling G. 
 
 The tube A, without the air jacket c, is immersed in the outside bath 
 till the solvent commences to freeze. The outside of A having been wiped, 
 it is placed within the air jacket, and the contents are allowed to cool 
 slightly below the freezing point, if this can be done without the whole 
 mass of the solvent becoming solid. By working the stirrer the solidi- 
 fication of a portion of the supercooled solvent is caused, and the reading 
 of the thermometer first rises and then remains constant. This constant 
 temperature is the freezing point of the solvent. If the solvent is 
 not quite pure there may be a slow fall of the thermometer reading while 
 solidification goes on, owing to the change in the composition of the fluid 
 portion of the liquid, due to one constituent of the liquid freezing 
 out more rapidly than the other. 
 
 Having determined the freezing point of the pure solvent, the 
 tube A is removed and a weighed quantity of the substance is introduced 
 through the side tube B. The solidified portion of the solvent is allowed 
 to melt, and the solution of the substance promoted by stirring. When 
 solution is complete the tube A is replaced in the air jacket, and the 
 temperature allowed to fall a little below the freezing point. Freezing is 
 then started in the supercooled liquid by stirring, when the thermo- 
 meter rises slightly, becomes constant, and then slowly falls. If the 
 supercooling has not been too great, the maximum temperature corre- 
 sponds to the freezing point of the solution of the strength used. 
 
 It is sometimes found difficult to start solidification, the solution 
 becoming much supercooled. In these circumstances some of the 
 solvent must be placed in a test-tube immersed in the outside bath 
 and caused to solidify. A glass rod is dipped into this test-tube and 
 allowed to remain till the end has become thoroughly cold. The side 
 tube B is then uncorked and the stirrer E raised and touched with 
 
260 DEPRESSION OF THE FREEZING POINT [ 100 
 
 the end of the glass rod, to which some of the solidified solvent must 
 be attached. On coming in contact with the solidified solvent the 
 solution attached to the rod will solidify, and then on stirring the 
 solution solidification of part of the solvent will be set up. 
 
 Correction 1. Owing to the fact that some of the solid solvent 
 has to separate out from the solution to cause the rise of temperature 
 which occurs when solidification takes place, the strength of solution, 
 for which the maximum temperature shown by the thermometer is the 
 freezing point, is less than that of the original solution. A correction 
 will have to be applied to the results to allow for this effect. Of 
 course the correction becomes less and less as the amount of the 
 supercooling of the solution is decreased. Hence it is advisable to 
 reduce the amount of the supercooling as much as possible, and not 
 to attempt to start solidification by increasing the supercooling, but to 
 use a trace of the solid solvent in the manner described above. If 
 c is the specific heat, w the weight of the solution, L the latent 
 heat of the solvent, and A the amount by which the temperature 
 just before solidification starts is below the maximum reading given 
 by the thermometer, so that A is the supercooling, then the heat 
 necessary to raise the temperature of w grams of the solution A degrees 
 is cwA. This quantity of heat has to be supplied by the latent heat 
 of the solidified solvent. Hence if x is the weight of solvent which 
 solidifies, we have 
 
 xL = cwA, 
 
 ctrA 
 or x=- r . 
 
 Hence when working out the strength of the solution we must deduct 
 the quantity x from the weight of solvent in order to obtain the 
 strength of the solution which has the observed freezing point. 
 
 Correction 2. A further correction has to be applied, if very accurate 
 measurements are to be made, to allow for the temperature of the 
 freezing solution being affected by exchange of heat with the sur- 
 roundings and heat developed by the stirrer. Owing to these causes 
 the liquid in the tube A, supposing no "freezing" take place, will 
 settle down to a temperature differing from that of the external vessel 
 containing the freezing mixture, this temperature being called by 
 Nernst and Abegg the convergence temperature. Thus when the 
 solution has been supercooled and freezing takes place we shall have 
 two effects going on. In the first place, there will be a supply of heat 
 owing to the freezing of the solvent causing the temperature to ap- 
 proach the true freezing point of the solution. At the same time the 
 temperature will, owing to radiation, conduction, and heat development 
 from stirring be approaching the convergence temperature. The actual 
 maximum temperature reached will be intermediate between the true 
 freezing point and the convergence temperature. Hence it is of 
 advantage to make the convergence temperature as nearly equal to the 
 
101] CALCULATION OF MOLECULAR WEIGHT 261 
 
 freezing point as possible. For this reason the temperature of the 
 freezing mixture should not be more than 1 or 2 degrees below 
 the freezing point of the solution. If 7' is the observed freezing 
 point, T the true freezing point, and T c the convergence temperature, 
 then 
 
 where A is a constant the value of which depends on the circumstances 
 of the experiment, and is equal to the change in the observed freezing 
 point produced by a change of 1 in the convergence temperature. For 
 Beckmann's form of apparatus, the value of A is approximately 0*0005 
 in the case of water solutions. 
 
 For a discussion of the corrections to be applied to the readings 
 of a Beckmann thermometer, see 102. 
 
 101. Calculation of Molecular Weight from the Depression of 
 the Freezing Point. If a dilute solution containing s grams of the 
 solute in S grams of the solvent has a freezing point A degrees below 
 that of the pure solvent, then the molecular w r eight of the solute is given 
 
 by 
 
 where K is a constant, the value of which depends on the nature of the 
 solvent. 
 
 The values of K for some solvents are given in the following table : 
 
 Solvent. 
 
 Freezing Point. 
 
 AC 
 
 Water, HgOj .... 
 
 
 
 1850 
 
 Benzene, C (i H 6 .... 
 
 5'5 
 
 5010 
 
 Nitro-benzene, C H C NO 2 . 
 Naphthalene, C 10 H 8 . 
 
 5-3 
 
 80 -1 
 
 7050 
 6850 
 
 Phenol, C f H 5 OH ... 39 
 
 7300 
 
 Acetic acid, CH 3 CO 2 H 
 
 17 
 
 3900 
 
 In the cases of solvents other than water, besides the series of sub- 
 stances which give normal depressions of the freezing point, there is in 
 general a series of other substances which give depressions which are 
 only about half the normal. These abnormally small depressions are 
 supposed to be due to the formation of aggregates of two molecules of 
 the solute, so that in the solution the molecular weight is double the 
 normal value. Such polymerisation appears in the case of solutions of 
 ethyl alcohol, methyl alcohol, phenol, &c., in benzene, and with sulphuric 
 and hydrochloric acid in acetic acid. 
 
262 DEPRESSION OF THE FREEZING POINT [ 102 
 
 In the case of solutions in water, in addition to the normal series it 
 is found that in the case of electrolytes the depression is generally twice 
 as great as the normal. This effect is supposed to be due to dissociation 
 of the solute into simpler molecules. (See Watson's Physics. 8 225, 
 543.) 
 
 102. Correction of the Readings of a Thermometer in which the 
 Quantity of Mercury is Variable. Thermometers of the Beckmann 1 
 type, described in 100, are so very useful wherever small differences of 
 temperature are to be measured, that it becomes necessary to study a 
 correction which must be applied to allow for the fact that the quantity 
 of mercury in the bulb varies according to the temperature at which the 
 instrument has been adjusted. Since practically all the Beckmann 
 thermometers in the market are made of Jena normal glass (16'"), it will 
 simplify the description if we only consider a thermometer made of this 
 glass. The coefficient of apparent expansion of mercury in Jena glass 
 (16"') is 0-000157, or 1/6370. 2 Hence if we have a thermometer 
 such that when placed in melting ice the end of the mercury column 
 is at the division of the scale, and take as our unit of volume 
 the volume of 1 of the capillary, the volume of the mercury will be 
 6370. For if the temperature is raised 1, the apparent increase 
 in volume of the mercury will be 6370 x 1/6370, or 1, i.e. 1 
 degree. 
 
 Next suppose that enough mercury to fill n degrees is passed over 
 into the auxiliary bulb. The mercury will now be at the of the scale 
 at a temperature n. If, now, the temperature is raised to n + 1 degrees C., 
 
 the apparent increase in volume of the mercury will be or 
 
 1 - go-TQ- I n other words, when the temperature is increased 1 the 
 
 reading will only increase 1 - rc/6370 degrees, that is, the value of a 
 degree on the scale will be greater than 1 C. Thus suppose w=10, 
 then each degree on the scale of the thermometer now corresponds to 
 1-S-{1 10/6370}, or 1 0> 0016, quite an appreciable correction, since such 
 thermometers are generally divided into 0'01. 
 
 We have in the above supposed that when the thermometer was 
 placed in melting ice it read exactly 0, and further that when heated to 
 1 it read exactly 1 on the scale, and that the bore of the capillary was 
 cylindrical. These, however, are points which have to be tested experi- 
 mentally. In the first place, the uniformity of the bore has to be tested 
 by a calibration performed as described in 59. When performing this 
 
 1 This form of thermometer was really devised by Walferdin, who called 
 them metastatic thermometers. They are so widely known, however, as Beck- 
 mann thermometers, that it seems hardly possible to designate them by any other 
 name without risking confusion. 
 
 2 The coefficient of apparent expansion of mercury and Jena glass 59'" is 
 1/6080. 
 
[ 102 BECKMANN THERMOMETER 263 
 
 calibration the and, say, the 6 are assumed correct, and the interval 
 is subdivided into portions of equal volume. 
 
 In order to test whether, when all the mercury is in the main bulb, 
 the thermometer reads correctly, resource must be had to a standard 
 thermometer (preferably made of the same kind of glass) of which the 
 errors are known. The Beckmann thermometer, having had its zero 
 determined in the ordinary manner, must be compared with the standard 
 at a temperature near the top of the scale, preferably near the division 
 used as a fixed point when performing the calibration. In this way the 
 error at the upper fixed point of the calibration is determined, and then 
 a curve showing the errors at the intermediate points can be drawn up, 
 as described in 62. When any mercury is placed in the upper bulb, 
 then the difference in the readings, corrected from the curve, and multi 
 
 plied by ^07?) - , will give the difference in temperature on the scale of 
 
 the mercury thermometer. To reduce to the hydrogen thermometer, we 
 must remember that when using a Beckmann thermometer we are only 
 concerned with the accurate measurement of a difference of temperature ; 
 in other words, it is only the difference in the corrections to reduce to 
 the hydrogen scale at the upper and lower points which has to be con- 
 sidered. For example, if the correction to reduce to the hydrogen scale 
 was the same at the upper and the lower temperatures, it is evident that 
 the difference in temperature obtained with a mercury thermometer would 
 be the same as that given by a hydrogen thermometer. If the correction 
 to reduce to the hydrogen thermometer, as given in Table 19, at the 
 lower temperature # , as measured on the mercury scale, is 6 , and that at 
 the higher temperature t l is 8 p then the true difference of temperatures is 
 
 . 
 
 that is, the correction to be applied to the difference of temperature as 
 read on the thermometer is 8 t - S . This correction may become quite 
 appreciable for those temperatures where the curves given in Fig. 62 are 
 steep. Thus suppose t = and t l = 5, so that for Jena glass (16'") <$ = 
 and 8j = - "03, then the true difference of temperature on the hydrogen 
 scale is 5 0> 00-0 0> 03, or 4'97, the correction amounting to nearly 1 
 per cent. 
 
 Another correction which may often be very considerable is that due 
 to emergent column. This point has been examined in the case of Beck- 
 niann's pattern of thermometer by Griitzmacher. 1 His results are given 
 in the following table : 
 
 Zeit.fur Instrumentenlunde (1896), xvi. 202. 
 
264 
 
 BOILING POINT 
 
 [103 
 
 CORRECTIONS FOR EMERGENT COLUMN IN BECKMANN'S THERMOMETER 
 (the whole column is supposed outside the bath, as in Fig. 106) : 
 
 Temperature Interval 
 Measured. 
 
 Mean Temperature of 
 Column. 
 
 Correction per 1 for 
 Effect of Cool Column. 
 
 -35 to -30 
 
 
 
 - 0'005 
 
 to +5 
 
 15 
 
 -0'002 
 
 45 to 50 
 
 26 
 
 + 0'004 
 
 95 to 100 
 
 32 
 
 + 0'011 
 
 145 to 150 
 
 38 
 
 + 0'018 
 
 195 to 200 
 
 44. 
 
 + 0'025 
 
 245 to 250 
 
 50 
 
 + 0'031 
 
 Thus if in the case considered above the room temperature was such 
 that the mean temperature of the column was 15, only the bulb being 
 
 immersed in the liquid, the 
 true difference of temperature 
 would be 4-97 (1 - '002), or 
 4-96. 
 
 103. Measurement of the 
 Boiling Point of a Solution. 
 The accurate determination 
 of the boiling point of a 
 solution is rendered difficult 
 owing to the readiness with 
 which almost all solutions 
 become superheated. The 
 form of apparatus ordinarily 
 adopted is that devised by 
 Beckmann, and shown in Fig. 
 107. The three-necked flask 
 A is filled about a third full 
 of glass 'beads or garnets, 
 while a piece of very thick 
 platinum wire is fused 
 through the bottom. These 
 two devices are used to secure 
 regular ebullition and prevent 
 superheating. A thermo- 
 meter T passes through the 
 ^, lft _ cork which closes the middle 
 
 neck, while a condenser c is 
 
 attached to one of the side necks. The third neck B serves for the 
 addition of the substance, the effect of which on the boiling point of the 
 solvent is being measured. The instrument is stood on a sheet of thick 
 
103] 
 
 BOILING POINT OF A SOLUTION 
 
 265 
 
 asbestos card D, which is pierced with a hole. This card rests on a sheet 
 of wire gauze E. A short glass cylinder F rests on the upper side of the 
 card G, leaving an air space all round the bulb. There is a plate of mica 
 or asbestos G used to close the upper end of this air jacket. The height 
 of the flame is adjusted so that about five drops a minute fall from the 
 end of the condenser. This form of apparatus is by no means free from 
 superheating, and Beckmann has devised a more complicated form, in 
 which the inner bulb containing the solution is jacketed, except at the 
 bottom, with the vapour of the solvent boiling in a separate vessel. 
 
 The only really satisfactory method of avoiding superheating is to 
 use electrical heating, and 
 as no modern physical 
 laboratory is without a 
 supply of accumulators, 
 there is no reason why this 
 method should not be used 
 exclusively. By this means, 
 not only is regular boiling 
 secured, and hence super- 
 heating avoided, but owing 
 to the ease with which the 
 supply of heat can be 
 regulated and kept con- 
 stant the time necessary 
 for the experiment can be 
 materially reduced. 
 
 A convenient form of 
 apparatus, which is a very 
 slight modification of that 
 described by Christensen, 1 
 is shown in Fig. 108. It 
 consists of a glass test-tube 
 A closed by a cork, through 
 which passes the thermo- 
 meter T, two thin glass 
 tubes B, and two other 
 tubes E and F (shown at a). 
 
 The bottom of the tubes B are fused round a short length of stout 
 platinum wire G, while a spiral of thin platinum wire c is attached to 
 these pieces of wire either by autogeneous soldering or with pure gold. 
 The tubes B are partly filled with mercury, and are connected to a battery 
 of three or four accumulators and an adjustable resistance by copper 
 wires, which dip in this mercury. A tube E, which is normally closed 
 with a cork, serves for the introduction of the substance. The end of a 
 small condenser F, or, in the case of a solvent with a high boiling point, 
 simply an air condenser passes through the cork in the top of the tube A. 
 1 J. C. Christensen, Journal of Physical Chemistry (1900), iv. 585. 
 
 G~L~ ^^~Z=- 
 xVr-V-'r-o 
 
 FIG. 108. 
 
266 BOILING POINT [ 103 
 
 The end of the condenser is slightly bent as indicated, and cut off at an 
 angle, so that by holding the tube A slightly inclined the condensed 
 solvent may run down the side of the outer tube, and not come in contact 
 with the thermometer. To prevent any chance of the thermometer being 
 affected by the rising bubbles of superheated vapour a small thimble- 
 shaped glass dish D is supported in the position shown by two pieces of 
 platinum wire, which are fused to the edge of the glass and wound 
 round the tubes B. When making an experiment the tube A is packed 
 round with cotton wool or asbestos cloth, according to the temperature at 
 which the solvent boils. 1 
 
 When making an experiment the tube A, without the cork and the 
 pieces of apparatus attached, is weighed, the solvent introduced, so that 
 it will quite cover the bulb of the thermometer, and the whole again 
 weighed. During this weighing the tube may be suspended from the 
 hook attached to the stirrup of the balance by means of a wire sling H. 
 If the weights are determined to within O'Ol gram, it will be quite 
 sufficient. The thermometer, condenser, &c., are now attached, and the 
 tube packed round with some non-conducting material, and the current 
 started. The resistance in the circuit must be adjusted, so that when 
 boiling has become regular the vapour only extends a very short distance 
 up the condenser tube. 
 
 The boiling must be continued, the thermometer being read every 
 five minutes till the temperature becomes constant. This in general 
 takes about ten minutes. Then the temperature must be read, as also 
 the height of the barometer. A weighed quantity of the substance is 
 then introduced through the tube E, the current having been turned off 
 before removing the cork. The current having been switched on, the 
 reading of the thermometer must be made every five minutes till the 
 temperature becomes constant, when the thermometer and barometer are 
 again read. 
 
 The solution of which the boiling point is actually determined is 
 slightly more concentrated than would appear from the weighings, owing 
 to some of the solvent during the ebullition being absent from the solution, 
 since it is in the form of vapour in the upper portion of the tube, and of 
 liquid in the condenser and on the sides of the tube A. The exact 
 evaluation of the amount which has to be deducted from the weight 
 of solvent on this account is difficult, and so it is important to reduce 
 the correction to a minimum by adjusting the rate of boiling so that the 
 vapour only just reaches the condenser, and using as large a volume of 
 the solvent as the apparatus will allow. 
 
 1 The instrument may be placed inside a Dewar vacuum vessel. 
 
CHAPTER XVI 
 
 MELTING POINT RATIO OF SPECIFIC HEATS CONDUCTIVITY 
 
 104. Determination of the Melting Point by the Method of Cooling. 
 
 Since when a liquid solidifies heat is evolved, the rate of fall in the 
 temperature of a liquid placed in a region at a lower temperature than 
 the melting point will be decreased, or even become zero, during the 
 time when solidification is taking place. Hence by noting the tempera- 
 ture at which this change in the rate of decrease of temperature takes 
 place, the melting point can be determined. 
 
 The substance to be experimented upon is contained in a thin-walled 
 test-tube. This test-tube passes through a cork which is placed in the 
 mouth of a wider test-tube, this outer tube being placed in a beaker 
 of water. In this way the rate of loss of heat is made fairly regular. 
 A thermometer passes through the cork which closes the inner tube. 
 
 Sufficient of the solid substance having been placed in the inner tube 
 to well cover the bulb of the thermometer, the solid is melted and its 
 temperature raised about 10 degrees above the melting point by placing 
 the inner tube in hot water or over a flame. During this operation it is 
 advisable to remove the thermometer. 
 
 The inner tube having been replaced in the outer, and the whole 
 immersed in water at a temperature considerably below the melting point, 
 the reading of the thermometer is taken every half minute till a tempera- 
 ture below the melting point is reached. The temperatures and times 
 are then plotted on squared paper and a curve is drawn through the points. 
 The temperature or temperatures at which the curve becomes distinctly 
 more nearly parallel to the axis of time correspond to the melting points 
 of the various constituents of the substance. 
 
 If the changes in the slope of the cooling curve are very gradual, so 
 that it is difficult to determine the temperature which corresponds to the 
 melting point, it probably means that the outside layers of /the liquid 
 have solidified before the temperature of the inside portions has fallen to 
 the melting point. This effect will be found to be particularly well 
 marked if a somewhat wide test-tube is employed with substances, such 
 as paraffin wax, which are very bad conductors of heat, and have only a 
 small latent heat. In such a case a very much sharper inflection in the 
 curve will be obtained if the liquid is kept well stirred during the time 
 it is cooling. 
 
 105. Determination of the Ratio of the Specific Heats for Air by 
 Clement and Desorme's Method. Suppose that the air within a vessel 
 
 
 267 
 
268 MELTING POINT [ 105 
 
 is suddenly slightly compressed, so that the pressure rises from the 
 atmospheric pressure B to p r Since the compression which takes place 
 in the air within the vessel is rapid, there will be little transference of 
 heat between the air and the walls of the vessel while the compression 
 is going on, so that the compression is adiabatic. Hence if V Q and v l are 
 the initial and final volumes of unit mass of the air within the vessel, 
 and k is the ratio of the specific heats, we have 1 
 
 (~ l } = - (1) 
 
 V'V ih 
 
 After a time the heat which has been caused by the adiabatic com- 
 pression of the air will be dissipated by conduction, &c., to the walls of 
 the vessel, till finally the temperature regains its initial value. Owing 
 to this fall in the temperature of the air within the vessel, the pressure 
 will fall, say, to p 2 . Now, since the total volume of the gas is the same 
 as it was at the end of the adiabatic compression, the volume of unit 
 mass is the same as at the end of the compression, namely, v r Further, 
 since the temperature is now the same as the start, we have, from Boyle's 
 law, 
 
 ..... 
 
 V P-2 
 
 Hence, substituting this value in equation (1) above, we get 
 
 or, taking logarithms of both sides, 
 
 &(log B - log p. 2 ) = log B - log ^ 
 
 _ 
 log /f-logj0 2 
 
 A form of apparatus suitable for applying this method of determining 
 the ratio of the specific heats is shown in Fig. 109. 
 
 It consists of a glass flask A, about 30 centimetres in diameter, which 
 is placed inside a wooden box and packed round with cork dust, cotton 
 wool, or other bad conductor of heat. The neck of the flask is closed by 
 a well fitting rubber cork, which is pierced by three holes through which 
 pass three glass tubes. One of these, E, is closed by a tap, another F 
 is attached to a manometer G, while the third c is of at least 1 centi- 
 metre in bore, and is connected by a short length of wide bore rubber 
 
 1 See Watson's Physics, 259. 
 
105] RATIO OF THE SPECIFIC HEATS FOR AIR 269 
 
 tubing with the mercury reservoir H. This rubber tube can be closed by 
 a pinch-cock i. Two small shelves J and K are attached to the side of 
 the box, and each has a slot 
 of such a size as to allow the 
 flange on the neck of the 
 reservoir H to catch. 
 
 The quantity of mercury 
 in the apparatus is so ad- 
 justed that when the reservoir 
 is on the lower shelf K the 
 surface of the mercury in the 
 flask covers the cork to 
 the depth of about 1 centi- 
 metre. A small quantity of 
 concentrated sulphuric acid is 
 then introduced through the 
 tube E so as to form a layer L 
 on the surface of the mercury. 
 This sulphuric acid serves to 
 keep the air within the flask 
 dry, a matter of great import- 
 ance. Some oil, such as is used 
 in the Fleuss pump, is intro- 
 duced into the manometer 
 tube, and the apparatus is then 
 ready for use. 
 
 To perform an experiment 
 in which the air is suddenly 
 compressed, place the mercury 
 reservoir on the lower shelf, 
 and open the pinch-cock and 
 the tap E. Then close the pinch-cock, and raise the mercury reservoir 
 on to the upper shelf. After allowing the apparatus to stand for a few 
 minutes, so that the temperature of the air in the flask may become 
 uniform, close the tap E. Then open the tap F on the manometer, and 
 fully open the pinch-cock, thus allowing the mercury to flow into the 
 flask and compress the air. Watch the manometer, and as soon as the 
 reading is a maximum close the tap F. By this preliminary experiment 
 the liquid in the manometer has been adjusted almost to the position 
 which it will occupy at the end of the compression. Leaving F closed, 
 lower the mercury reservoir and allow the mercury to run out from the 
 flask, close the pinch-cock, and raise the reservoir on to the upper shelf. 
 Open the tap E and allow the apparatus to stand for a few minutes, then 
 close E. Next open the pinch-cock, and when the mercury has nearly all 
 run into the flask open the tap F, and carefully note the highest reading 
 given by the manometer. If the preliminary adjustment of the mano- 
 meter has been properly performed there will only be a small movement 
 
 FIG. 109. 
 
270 MELTING POINT [ 106 
 
 of the column, and the maximum reading will be attained without 
 oscillations being set up. 
 
 As the heat developed owing to the compression of the air is dissi- 
 pated, the pressure will fall and the reading of the manometer when this 
 fall is complete, which will occur after three or four minutes, must be 
 recorded. 
 
 If Tij and 7i are the differences in the level of the oil in the mano- 
 meter at the two readings, and J is the density of the oil, and B the 
 barometer height, then we have 
 
 and substituting these values in (3) the value of A- is obtained. 
 
 If the compression is small, so that - i is small, then the formula 
 
 13'6 
 
 can be very much simplified, for 
 
 + cVrc. 
 
 13-65 2V13-65, 
 Thus if we may neglect the square and higher powers of - 'j we get 
 
 Similarly log p. 2 = log B + -' ^ 
 
 and hence substituting in (3) 
 
 an expression which does not involve a knowledge of the density of the 
 oil in the manometer. 
 
 Experiments should also be made in which the air in the instrument 
 is first compressed by raising the mercury reservoir to J, then closing the 
 pinch-cock I, and lowering the mercury reservoir to K. On opening the 
 pinch-cock the air expands and becomes cooled. The pressure before 
 expansion, immediately after expansion, and after the air has regained its 
 initial temperature, are determined by means of the manometer in a 
 manner similar to that described above. 
 
 106. Measurement of the Heat Conductivity of Copper. The 
 measurement of the conductivity of a solid is in general rendered difficult 
 owing to the loss of heat from the surface of the rod or lar used, the 
 
106] 
 
 HEAT CONDUCTIVITY OF COPPER 
 
 271 
 
 evaluation of this loss being a matter of some ditliculty. In the case of 
 a good conductor, such as copper, we may, however, so arrange the ex- 
 periment that the proportional loss of heat from the surface may be 
 negligible. A piece of apparatus designed with this view is shown in 
 Fig. 110 The copper rod AB has a total length of 15 centimetres and 
 a diameter of 3 centimetres, and has a copper box c soldered on one 
 end. Steam is passed into this box through the tube p, the excess 
 escaping through the tube E. A second copper box F is soldered to the 
 other end of the rod, and is fitted with an annular partition which com- 
 pels the water that enters at G and leaves at i to flow in the manner 
 indicated by the arrows. Two thermometers T l and T. 2 serve to indicate 
 the temperature of the incoming and escaping water. Both the boxes 
 
 T 
 
 r r 
 
 T 4 
 
 ^\.J 
 
 FIG. 110. 
 
 are carefully lagged with felt, so as to protect them as much as possible 
 from external loss of heat. 
 
 Two narrow holes are drilled almost right through the copper rod at 
 a distance of 8 centimetres apart, and in 'these holes are inserted the 
 bulbs of two small thermometers T 3 and T 4 . The holes ought to be of 
 such a size that the bulbs of the thermometers are a fairly good fit. The 
 surface of the copper rod between the two boxes may with advantage be 
 lightly packed round with cotton wool. 
 
 Water is passed through the instrument by connecting the side tube 
 G to the rubber tube r of a pressure tank A (Fig. 111). This tank con- 
 sists of a small metal vessel having three tubes soldered through the 
 bottom. One of these tubes is connected through the rubber tube B 
 with the water supply. The middle tube extends about two-thirds up 
 
272 MELTING POINT [ 107 
 
 the vessel and serves as an overflow, while the third tube is connected to 
 the conductivity apparatus. The supply tap is turned on so that a little 
 water always escapes through the tube c, and hence 
 the head of water always remains the same, and so 
 the rate of flow through the apparatus is constant. 
 
 Steam being passed into the box c, adjust the 
 flow of water through the box F by means of a 
 pinch-cock on the supply tube, so that the rise of 
 temperature of the water is about 10 degrees, and 
 read the four thermometers at intervals till the 
 readings become constant. When this is the case, 
 read and record the temperatures indicated by the 
 thermometers, and place, at a noted instant, a weighed 
 beaker below the water outlet i. Continue collecting 
 the water for about five minutes, and at a noted 
 instant remove the beaker and again read the thermo- 
 meters. By weighing the beaker determine the 
 quantity of water W which has passed in the 
 time T. 
 
 If the mean temperatures indicated by the thermometers T 1? T. 2? T 3 , 
 and T 4 are t v t. 2 , t 9 and 4 , then the heat which has been conducted along 
 the bar to the water is W(t<> - ^). Hence if I is the distance between the 
 thermometers T 2 and T 3 , and s is the cross-section of the copper rod, the 
 conductivity k is given by 
 
 Correction. Since most of the stem of the thermometers T 3 and T 4 are 
 not heated to the temperature of the bulb, the correction for emergent 
 column may be considerable. The magnitude of the correction may be 
 determined experimentally, or the corrections given in the tables in 65 
 may be used. 
 
 107. Measurement of the Heat Conductivity of Glass. In the 
 case of a bad conductor of heat, such as glass, the following method of 
 measuring the conductivity is capable of giving results which are 
 accurate to within about 5 per cent. : 
 
 The glass is taken in the form of a tube AB (Fig. 112), having a bore 
 of about 1'5 centimetres. The tube is enclosed for the greater part of 
 its length in a larger tube c, which forms a steam jacket, the steam 
 entering at D and leaving at E. Two T-pieces are connected to the ends 
 of the tube AB, and each encloses a thermometer, T X and T 2 , by means of 
 which the initial and final temperature of a stream of water which 
 traverses the tube AB can be determined. 
 
 The T-piece G is connected by the rubber tube H to the constant 
 head reservoir shown in Fig. 111. The tube AB is inclined so that the 
 water flows upwards, and in this way any air evolved from the water 
 owing to the rise in temperature is swept out. 
 
107] 
 
 HEAT CONDUCTIVITY OF GLASS 
 
 273 
 
 Since it is important that the temperature of the water at any cross- 
 section of the tube AB shall be the same, the water is caused to traverse 
 the tube spirally. For this purpose a piece of rubber tube is wound in a 
 very open spiral round a thin brass rod, and the spiral is placed within 
 the tube AB. 
 
 When making an experiment steam is passed through the jacket, and 
 the stream of water adjusted so that the temperature of the water rises 
 about 10. When the thermometer readings have become constant, the 
 water which passes in a noted time is collected in a weighed beaker. If 
 
 FIG. 112. 
 
 W is the weight of water which passes in a time T, and t l and / 2 are 
 the initial and final temperatures, then the quantity of heat which has 
 passed through the glass is W(t^ - ^). 
 
 The temperature 2 of the external surface of the glass can be 
 obtained from the barometric height and Table 15. The temperature, 
 
 f\, of the internal surface may be taken as the mean of t l and # 2 . 
 If I is the length of the glass tube AB exposed to the steam, and 1\ and 
 2 are the radii l of the internal and external surfaces of the tube, then 
 1 If r and r + dr are the radii of the surfaces of a thin cylindrical shell in 
 the glass, and the difference in temperature between these surfaces is dd, since 
 the quantity of heat Q must pass in a time T through this cylinder, and also 
 through every similar cylindrical shell into which we may imagine the glass 
 subdivided, we have 
 
 . T 
 dr ' 
 
 or 
 
 d6= 
 
 Qdr 
 
 Integrating between the inner and outer surfaces of the glass tube 
 
 dr 
 r ' 
 
 ,-,= log 
 
274 CONDUCTIVITY [ 101 
 
 the quantity of heat which passes through the glass in a time T is 
 given by 
 
 Since Q is known, this expression enables us to calculate /:, the con- 
 ductivity of the glass. 
 
 If the glass is thin, so that r. 2 - 1\ is small, we may expand Iog 6 - 2 
 
 '*! 
 
 and replace it by the first term, namely 
 
 '-*+! 
 
 or 2 ? ' 2 -- r i, 
 
 and hence 
 
 a= 
 
 ~ 
 
 Now 27T-1 U is the mean, ^4, of the areas of the external and internal 
 
 surfaces of the glass, and r 2 - 1\ is the thickness, d, of the glass, so that 
 we have 
 
 The external radius is obtained by measuring the* diameter of the 
 glass tube in a number of places with callipers and taking the mean. To 
 obtain the mean internal radius two corks are fitted in-to the ends of the 
 tube, and the whole is weighed. One cork is then removed, and the tube 
 filled with water. After replacing the cork, the distance between the 
 inner surfaces of the corks is measured, and the tube is again weighed. 
 If w is the increase in weight, that is, the weight of the water column, 
 and L is its length, the mean radius, r lt is given by 
 
 r,- " 
 
 where A is the density of the water at the temperature of the experi- 
 ment 
 
CHAPTER XVII 
 
 SOUND 
 
 108. Determination of the Pitch of a Tuning Fork by means of a 
 Clock and a Rotating Drum. One method of measuring the number 
 of vibrations a tuning fork makes in a second is to cause the fork to 
 trace a wavy line on a smoked drum by attaching a fine style to one of 
 the prongs, and at the same time making a trace alongside with a second 
 style attached to a small electro-magnet, the circuit of this electro-magnet 
 being broken once every second by the movement of the pendulum of 
 a clock. 
 
 Since the attachment of a style will slightly lower the pitch of the fork, 
 it is important to use as light a style as possible. The most convenient 
 form of style is a fairly fine pig's bristle, attached to the prong with the 
 
 M 
 
 FIG. 113. 
 
 minimum quantity of soft wax which will make it stick. The drum 
 must be so mounted that as it rotates it is traversed through a distance 
 of between 1 and 2 cm. for each revolution. If the drum is rotated 
 mechanically, and has a diameter of 20 cm., it ought to make a revolution 
 in two seconds, though, of course, the minimum speed to give a legible 
 trace varies with the pitch of the fork. 
 
 The electro-magnetic style (Fig. 113) must be placed so that its tracing 
 point lies alongside the style attached to the fork, so that the two traces are 
 about half a centimetre apart. If the clock is not fitted with an arrange- 
 ment for making or breaking an electric circuit every second, such an 
 arrangement may be prepared as follows : A short and fairly stout piece 
 of platinum wire is fixed to the bottom of the pendulum, and, if the 
 pendulum rod is of wood or other non-conductor, a fine wire must be run 
 
 275 
 
276 
 
 SOUND 
 
 [108 
 
 from this platinum point along the pendulum and attached to the clamp 
 which is fixed to the suspension spring. The platinum point makes 
 
 contact with a thread of mercury, B (Fig. 
 114) contained in a narrow channel cut 
 in a wooden block. This block is shown 
 in plan at (a), and in section at (b). The 
 groove widens out into a circular pool A, 
 which serves to contain a reserve of 
 mercury. Owing to capillarity the mercury 
 in the groove stands up above the wood, 
 and when the platinum point sweeps 
 through the mercury the circuit containing 
 the electro-magnetic style and a battery is 
 completed, one terminal being connected 
 to the metal frame of the clock, and the 
 other to the mercury in A. If it is found 
 that the platinum poini does not make 
 contact, heat the point to redness and 
 plunge it into mercury, thus amalgamating 
 the point. Then heat the point gently so 
 as to drive off the mercury and replace 
 the wire, being careful not to touch the point with the fingers, and 
 so grease the wire. 
 
 The fork must be held firmly in a clamp in such a position that the 
 end of the style just touches the surface of the cylinder. The surface of 
 the cylinder may itself be blacked, or a sheet of glazed paper may first 
 be stretched over the surface, the edge being pasted down. To smoke 
 the surface the cylinder must be slowly revolved over the flame of a 
 paraffin lamp, or over some burning turpentine. A thin but uniform 
 coating of soot is to be aimed at. 
 
 Having bowed the fork, so that it is vibrating strongly, start the drum, 
 and continue the motion as long as the fork gives a trace of sufficient 
 amplitude to be read. Then draw a line parallel to the axis of the 
 cylinder through the point where the first time mark occurs, also through 
 the third or fifth, seventh, &c., according to the length of the trace, and 
 count the number of vibrations which the fork has made between these 
 two marks. The reason for using only every other mark is that the time 
 marks used at the beginning and end should be made by the pendulum 
 when passing through the contact maker in the same direction, for unless 
 the contact is made rigorously at the middle of the swing the intervals 
 between successive passages "will not be equal, though the interval 
 between alternate passages will be so. By dividing the number of 
 vibrations by the time the frequency of the fork with the style attached 
 will be obtained, and it will then be necessary to determine the correction 
 which has to be made to allow for the effect of the style. For this pur- 
 pose obtain a second fork of the same frequency as that being tested, and 
 determine the number of beats made by the two forks in five or ten 
 
I 
 
 109] FREQUENCY OF A FORK 277 
 
 seconds. 1 Also, by slightly loading the auxiliary fork with wax, deter- 
 mine which fork is the sharper. In this way the difference in the pitches 
 of the two forks is obtained. Next remove the style from the fork, and 
 again determine by means of beats the difference in the frequencies. 
 Thus the effect of the style is obtained, and by adding the number of 
 vibrations per second, due to the addition of the style, to that obtained 
 by means of the drum, the frequency of the fork without the style is 
 obtained. 
 
 The correction obtained in this manner for the effect of the style does 
 not take account of any influence which may be exerted on the period of 
 the fork owing to friction of the style on the drum. If care is taken 
 that the style is very pliable, and that it only rests very lightly on the 
 surface, such an influence on the period will be very small. This point 
 may be investigated by determining the number of beats made by the 
 fork when the style rests against the surface of the drum, the drum 
 being rotated as in the actual experiment, and comparing this number 
 with the number obtained when the style does not touch the surface of 
 the drum. 
 
 Since the frequency of a fork depends on the temperature, a note of 
 the temperature, t, must be taken when the determination of frequency 
 is made. The frequency at any other temperature, t', can be obtained 
 by means of the following expression, which holds for forks made of 
 ordinary steel : 
 
 < = n t { l-O'OOO 11 2(f- *)}. 
 
 A correction would have also to be applied for the rate of the clock if it 
 were excessive, but such rates as ordinarily occur will hardly affect the 
 results obtained, unless extreme accuracy is aimed at. Thus a rate of 
 five seconds a day will necessitate a correction amounting to one part in 
 17300. 
 
 A modification of this method which does not necessitate the use of 
 an electro-magnetic style, and a clock with a make-and-break device, has 
 been devised by Koenig. A. second fork is taken, which is also provided 
 with a style, and by loading its prongs it is made to give about four 
 beats per second with the fork to be measured, and the number of beats 
 (a) given per second are accurately determined. The two forks are then 
 caused to record alongside on the rotating drum, and by counting the 
 waves the ratio (b) of the frequencies of the two forks is determined. If, 
 then, T&J is the frequency of one fork, and n. 2 that of the other, we have 
 
 n^ - n. 2 = a, and njn% = b. 
 
 Thus 
 
 al 
 
 b-l 
 
 109. Stroboscopic Method of Measuring the Frequency of a Fork. 
 Two small and light plates, 2 c and D (Fig. 115), are attached to the 
 
 1 See also 111. 
 
 2 These plates may be made of thin card or aluminium foil. 
 
278 SOUND [ 109 
 
 ends of the prongs A and B of the fork. Each of these plates is pierced 
 by a narrow slit, and the plates are so placed that when the fork is at 
 rest the two slits are opposite each other. Thus, when the fork is 
 
 c- A 
 
 LA 
 
 D- B 
 
 B 
 
 FIG. 115. 
 
 sounding, light can only pass through the two slits when the prongs are 
 passing through their position of rest. Hence if n is the frequency of 
 the fork, light will pass through the slits 2n times psr second. 
 
 A disc on which are painted a number of rows of equidistant dots, 
 
 as shown in Fig. 116, is mounted 
 on the axle of a small electric motor, 
 and a worm-wheel attached to the 
 motor axle works against the edge of 
 a wheel which has a hundred teeth 
 cut in its circumference. A pin 
 attached to this wheel completes an 
 electric circuit containing a battery 
 and an electric bell once during 
 each revolution of the wheel, that 
 is, for every 100 revolutions of the 
 disc. 
 
 The disc is placed behind the 
 fork, and the surface is brightly 
 illuminated, and is looked at through 
 the slits attached to the prongs of 
 the fork. An adjustable resistance is placed in the circuit of the motor, 
 so that by altering this resistance the speed of the motor can be varied. 
 
 Start the fork and set the disc in rotation, and looking at the disc 
 through the slits, gradually increase the speed of th.3 motor till the dots in 
 one of the rings on the disc appear stationary. By adjusting the resistance, 
 or lightly touching the axle of the motor with the finger, keep the speed 
 constant, so that the dots appear at rest, and with a stop-watch or a 
 chronometer beating half seconds determine the time the disc takes to 
 make 500 or 1000 revolutions, that is, the interval corresponding to five 
 or ten strokes of the bell, and from the result obtain the number of 
 revolutions JV the disc makes in one second. 
 
 Unless the fork is maintained electrically, it will be necessary to bow 
 it lightly every now and then in order to keep up the vibrations. Koenig 
 has shown that such bowing does not affect the phase of the vibrations. 
 
 FIG. 116. 
 
110] PITCH OF FORK BY MEANS OF A STRING 279 
 
 If there are m dots in the ring which appeared stationary, then in 
 one second Nm dots have passed any given point placed near the edge 
 of the disc. Now the reason the row of dots appears stationary is that 
 each time the disc is seen any given dot has moved on, so that it occupies 
 the place previously occupied by the preceding dot. Thus as a dot has 
 moved on one place each time the disc has been seen, it follows that the 
 disc must have been seen mN times in a second. But if n is the 
 frequency of the fork, the disc is seen 2n times in a second. Hence 
 
 The measurement should be repeated, varying the speed of the motor, 
 so that different rings of dots appear stationary. 
 
 It is important to make certain that the dots when they appear 
 stationary are at the same distance apart as when the disc is stationary, 
 for if the disc is going at half the right speed the dots will appear 
 stationary, but there will seem to be twice the right number of dots. 
 This is due to the fact that, owing to persistence of vision, we are unable 
 to detect that we only see a dot in each position every alternate time we 
 see the disc. By placing a piece of card near the edge of the disc, and 
 making two marks on this card at a distance apart equal to the distance 
 between the dots, it will be at once apparent if the disc is turning too 
 slowly. Similarly, if the disc is turning twice or three times too fast 
 the dots will still appear stationary, for now between each view of the 
 disc each dot will have moved up two, three, &c., places. By watching 
 the disc as the motor gains speed, however, it is easy to make certain 
 of picking out the lowest speed at which the right number of dots 
 is seen. 
 
 A correction will have to be applied to allow for the effect of the 
 metal plates attached to the prongs if the fork is to be used without 
 these plates. The method of determining the correction is similar to 
 that described in the last section. The temperature of the fork must 
 also be noted. 
 
 A modification of the above method, which does not involve attaching 
 anything to the prongs of the fork, consists in replacing the disc with 
 dots by a disc with a number of equally spaced narrow radial slits near 
 the circumference. This disc is placed near the fork, and the extremity 
 of the prong is examined through the slits in the disc, which is rotated 
 by a motor as in the previous method. When the interval between the 
 passage of successive slits past the eye is equal to the periodic time of 
 the fork, the fork, though sounding, will appear to be at rest. Thus by 
 noting the time taken by the disc to make, say, 500 revolutions, the 
 frequency of the fork can at once be deduced. 
 
 110. Measurement of the Pitch of a Fork by means of a String. 
 When a string of length I is stretched by a tension T, the mass of unit 
 
280 SOUND [110 
 
 length of the string being m, the frequency of the note produced when 
 the string is sounding its fundamental is given by 
 
 or 
 
 _ * I T 
 
 ~2/V m' 
 
 I I~T 
 " = 2V J 
 
 (a) 
 
 where M is the mass of a length I of the string. 
 
 When using a string to measure the pitch of a note, it is better to use 
 a monochord which can be hung against the wall than one which is 
 
 placed horizontal and in which the 
 string passes over a pulley, since, owing 
 to the friction of the pulley, the tension 
 in the horizontal portion of the string 
 is never the same as that in the vertical 
 portion. A form of monochord which 
 can be easily made in the laboratory 
 is shown in Fig. 117. The bridge A is 
 v fixed, and the movable bridge B, shown 
 I on a larger scale at (a), is about 2 
 millimetres higher than A. The tops of 
 the bridges are formed of a piece of brass 
 wire formed into a staple and driven 
 into the wood. This wire is then filed 
 to a knife-edge. 
 
 Take a length of steel piano wire 
 having a diameter of about 0*5 mm., and 
 soften about 2 inches at either end by 
 heating in a flame. Turn the softened 
 end into a loop, and attach one loop to the 
 pin of the monochord, and hang a weight 
 of about 10 kilos from the other. 
 Adjust the movable bridge till the note 
 given by the wire is in unison with the 
 fork, and measure with a scale the dis- 
 tance between the two bridges. Repeat 
 the observation several times, and then 
 alter the weight, and again repeat. 
 While the weight is still attached to 
 the wire make two light file marks on 
 the wire at a distance apart about equal 
 to that used when in unison with the 
 fork, and then, having measured the dis- 
 tance between the two marks, cut the wire at these points and weigh, 
 thus obtaining the weight of unit length of the wire. The tension 
 
 FIG. 117. 
 
111] FREQUENCY OF TWO FORKS 281 
 
 of the wire is obtained by multiplying the stretching weight in grams 
 by g (981). Inserting the values for m, I, and T, calculate the frequency 
 of the fork by the formula given above. 
 
 When tuning the string to unison with the fork the length of the 
 wire should be varied till beats are heard, and then the position of the 
 movable bridge adjusted till these beats vanish. The position of unison 
 should be approached both by lengthening the wire and by shortening it. 
 When unison is attained, on striking the fork and bringing the end of 
 the stem in contact with the base of the monochord the wire will be set 
 into violent motion, owing to resonance, and if difficulty is found in 
 finding the position for unison by ear, this resonance effect may be used 
 to detect the point at which unison is attained. 
 
 111. Comparison of the Frequency of Two Forks. The method of 
 comparing the frequency of two forks by tracing curves on a smoked 
 drum has already been described on p. 277. This method is applicable, 
 however large the interval between the forks. When the frequencies of 
 the forks do not differ from each other by more than about four vibrations 
 per second, then this difference may most conveniently be deduced from 
 the number of beats. 
 
 Take two forks which give about four beats per second, and having 
 fixed them firmly in a stand, bow them, and count the number of beats in 
 twenty or thirty seconds, and thus obtain the number of beats per second, 
 that is, the difference in the frequency of the forks. To ascertain which 
 fork has the lower frequency attach a very small pellet of wax to the 
 prong of one fork, and again determine the number of beats per second. 
 If there are more beats than before the attachment of the wax, it follows 
 that the loaded fork was that of lower pitch, since further lowering its 
 pitch by loading has increased the difference between the pitches of the 
 forks. If loading reduces the number of beats, the loaded fork was the 
 higher. 
 
 If the forks to be compared are fitted with mirrors, then Lissajous' 
 figures form a convenient method of comparing the frequency of the 
 forks. This method can be applied when the ratios of the frequencies of 
 the forks do not differ from the ratio of any two simple whole numbers 
 by more than about four vibrations per second. 
 
 The forks are supported in clamps, with their mirrors turned towards 
 each other and their planes of vibration at right angles. A small hole is 
 pierced in a piece of card and placed before a source of light, and the 
 image of this hole, seen after reflection in succession in the two mirrors, 
 is examined by means of a low powered telescope. On setting the forks 
 in motion the spot of light seen in the telescope will trace out the figure 
 corresponding to the ratio of the frequencies of the forks, and if the ratio 
 is not exactly correct, the figure will in turn take all the forms corre- 
 sponding to this ratio, according as the relative phases of the two simple 
 harmonic motions alters. The time taken to go through a complete 
 series of changes is the interval in which the phase of the quicker fork 
 has gained a whole vibration on the lower. Thus suppose that the 
 
282 SOUND [112 
 
 figures traced out are those shown in Fig. 118, that is, that the 
 frequencies of the forks are nearly as 2 to 3. If, then, the whole series 
 of figures is gone through in x seconds, and if n l is the frequency of the 
 higher fork, and n. 2 that of the lower, so that njn^ is very nearly 3/2, 
 
 FIG. 118. 
 
 then during the interval x one fork will make n^x vibrations, and the 
 other n. 2 x vibrations. During this interval the higher fork has either 
 made one more vibration or one less vibration than 3/2 times the number 
 made by the slower. Hence we have 
 
 w i x _l = ? 
 
 o' 
 
 iiijtj & 
 
 or 
 
 n. } 
 
 In order to^ determine whether to take the plus or minus sign, 
 slightly load the higher fork, and thus lower its frequency. If x in- 
 creases, this shows that reducing n^ brings the forks more nearly to the 
 interval 3/2, and hence we must take the plus sign, and vice versa. 
 
 This method is very convenient for adjusting the pitches of two forks 
 to give any desired interval, so long as this interval can be expressed by 
 the ratio of two whole numbers, neither of which differ much from unity. 
 
 If a vibration microscope * is available, then this method can be 
 applied to compare the frequency of the fork of the microscope with that 
 of any other fork, and it will not be necessary to attach a mirror or style 
 to the fork, and thus alter its period. 
 
 112. Measurement of the Velocity of Sound in Air by Resonance. 
 If /\ is the wave-length in air of the sound produced by a fork of fre- 
 quency 7i, then, on bringing the fork near the mouth of a tube closed at 
 the other end, there will be strong resonance if the length of the tube 
 
 is equal to ; ' t &c. ; subject to a small corrction to allow for 
 
 the fact that in such a tube the first loop is not exactly at the open end 
 of the tube. Since, however, if we use tubes of the same diameter, the 
 correction x for the end will be the same for two tubes having lengths 
 
 1 See Watson's Physics, % 304. 
 
112] VELOCITY OF SOUND IN AIR 283 
 
 of A/4 - x and 3 A/4 - x respectively, the difference in the two lengths 
 will be equal to A/2, and the unknown correction for the end will be 
 eliminated. A convenient arrangement for applying this method of 
 measuring the wave-length in air of the note given by a fork is shown in 
 Fig. 119. It consists of an upright 
 metal tube A supported on a firm 
 base, and fitted at its upper end 
 with a glass window B. A narrower 
 brass tube c, which is open at both 
 ends, is supported within the tube 
 A by means of three cords, which, 
 after passing over pulleys D, are 
 attached to a lead counterpoise E. 
 The tube A is filled with water up 
 to about the middle of the window, 
 the water serving to close the lower 
 end of the tube c. By holding the 
 sounding-fork ove'r the open end 
 of c and moving this tube up and 
 down, so as to alter the length of 
 the column of air contained within 
 the tube, the positions at which the 
 resonance is a maximum can be 
 determined. The lengths of the 
 tube corresponding to these posi- 
 tions can be read off on a scale 
 divided in millimetres which is 
 engraved on the outside of the 
 tube c, the top end of this tube F 1Gi 119. 
 
 being at the zero of the scale. By 
 
 deducting the reading of the scale . corresponding to the shortest column 
 which is in resonance with the fork from the readings corresponding to 
 the other positions of resonance, the lengths of A/ 2, A, 3 A/2, ttc., as the 
 case may be, will be obtained, and the mean value of A can be calculated. 
 Knowing the wave-length, the velocity of sound in the air at the tem- 
 perature of the experiment is given by v = X.N. 
 
 t To obtain the velocity in dry air at C., this'result will have to be 
 corrected (1) to allow for the effect of the moisture present on the density, 
 and (2) to reduce to 0. 
 
 To apply the correction for moisture, we require to know the humidity 
 of the air. In the experiment as described above, we may suppose that 
 the air is saturated and at the temperature of the water. 1 If 8 is the 
 density of dry air at 760 mm. pressure and at the temperature t of the 
 experiment (Table 4), and e is the vapour pressure of water at the same 
 
 1 If the air is not saturated, then the quantity c in equation (1) will be the 
 actual vapour pressure of the moisture present in the air. 
 
284 SOUND [113 
 
 temperature (Table 14), then the density of the moist air at t and a 
 pressure of 760 mm. is given by 
 
 j, = 760-0-378e 
 
 But if v is the velocity of sound in dry air at 0, and v t is the velocity in 
 air, of which the density is 8 tt we have (see next section) 
 
 * < S > 
 
 Hence the observed velocity v t can be reduced to 0, and dry air by means 
 of equations (1) and (2). 
 
 113. Measurement of the Velocity of Sound in a Gas by Kundt's 
 Method. The apparatus used in this experiment consists of a glass tube 
 CD (Fig. 120), having a diameter of about 4 centimetres and a length of 
 about 2 metres, closed at one end by a tightly fitting cork D. A glass tube 
 
 FIG. 120. 
 
 about 180 centimetres long passes through this cork, so that it is sup- 
 ported at its mid point. The end A of this tube is furnished with a 
 small card disc, which has a diameter a few millimetres less than the 
 inside of the tube CD. The end c of the large tube can be closed by a 
 piston E, consisting of a cork, round the outside of which a strip of 
 chamois leather is glued ; this piston can be moved in and out by the 
 rod F. 
 
 The tube CD must be thoroughly dried by heating it over a flame and 
 blowing air through with a bellows, and then a narrow streak of well- 
 dried lycopodium powder is poured down the tube. This can be done 
 by inclining the tube and pouring the lycopodium from the blade of a 
 knife, the tube being tapped to cause the powder to run down. The tube 
 is then rested in a horizontal position on two V blocks, and the rod AB is 
 fixed in place. Starting with the piston E at the extremity c of the tube, 
 the rod AB is caused to vibrate longitudinally by stroking with a clamp 
 cloth, while the piston is gradually moved in till strong vibrations are set 
 up in the air inside the tube, which is manifest by the lycopodium being 
 strongly agitated, and the formation of groups of characteristic ridges. 
 When this resonance takes place, the piston E will be at a node and the 
 disc A at a loop of the stationary vibrations set up in the gas in the tube, 
 while the transverse ridges will be at the loops, the distance bi-twci-n 
 jcoj^e^utive_loQps_ being equal to half the wave-length of tlie"note pro- 
 duced by the rod AB in the gas. Measure the distance between the end 
 
113] VELOCITY OF SOUND IN A GAS 285 
 
 groups of ridges, and by dividing by the number of intervals get the 
 mean value for half a wave-length, twice this quantity being the wave- 
 length. The temperature of the gas must also be noted, for which 
 purpose a thermometer is placed alongside the tube. 
 
 To obtain the velocity, we require to know the frequency of the 
 waves as well as the wave-length. To measure the frequency of the note 
 given by the rod, determine the length of the wire of a monochord in unison 
 with the rod, and also that, in unison with a fork of known pitch, and 
 from the relation that the frequencies of two notes given by a stretched 
 string, the tension being constant, are inversely proportional to the lengths, 
 calculate the frequency n of the note given by ( the rod. Then the velocity 
 v of sound in the gas at the temperature of the experiment is given by 
 
 v t = nX, 
 the velocity at being given by 
 
 v = v t (l- 0-00183*). 
 
 When measuring the velocity in gases other than air, the gas can be 
 fed into the apparatus through a tube which acts as the handle for 
 moving the piston E, while an exit tube is fitted through the cork which 
 closes the end D of the outside tube. Unless the gas is quite dry it will 
 be necessary to pass the gas through a series of drying tubes before pass- 
 ing into the instrument, as a minute trace of moisture is sufficient to 
 prevent the lycopodium powder moving freely, and thus indicating the 
 position of the loops. 
 
 If it is only wished to compare the velocity of sound in two gases, 
 then it will not be necessary to measure the frequency of the rod AB, for 
 the velocities will be proportional to the wave-lengths in the two gases. 
 
 By surrounding the tube DC with a larger tube, and passing a current 
 of steam through the space between the two tubes, it is possible to 
 determine the ratio of the velocity of sound at 100 to that at ordinary 
 temperatures, and thus obtain the rate at which the velocity changes 
 with temperature. 
 
 The change of velocity with temperature may, however, be at once 
 calculated from the coefficient of expansion of the gas. If v t is the 
 velocity of sound in a gas at a temperature #, the density at this tempera- 
 ture being p and k is the ratio of the specific heats, we have 
 
 If v n and /) are the values of the velocity and density at 0, then 
 r^M 
 
286 SOUND [ 114 
 
 where a (-00366) is the coefficient of expansion of a gas. Hence 
 
 pk 
 
 p (l - at) <x/l - at 1 
 
 or approximately, 
 
 114. To Calculate the Ratio of the Specific Heats of Gases from 
 Measurements of the Velocity of Sound. If v is the velocity of sound 
 in a gas, the density of which at the temperature of observation and at 
 a pressure p is p, then 
 
 B 
 
 E 
 
 1 
 
 ll 
 CJID 
 
 where 7c is the ratio of the specific heats. 
 
 Having measured the velocity of sound in air, carbon-dioxide, 
 hydrogen, and ammonia by the method described in the last section, 
 calculate for each gas the value of k. 
 
 115. Measurement of the Wave-Length of a High Note by Inter- 
 ference. The most convenient source of sounds of high frequency is a 
 
 kind of bird-call. This consists of 
 two brass tubes, A and B (Fig. 121), 
 which can slide one over the other. 
 The end of one of these tubes is 
 closed by a flat disc of thin copper 
 foil, at the centre of which a hole 
 having a diameter of about 1*5 mm. 
 is bored. It is important that this 
 hole should be quite central, and, 
 further, that the edge of the hole be 
 FIG. 121. sharp and smooth. A triangular 
 
 piece D of the same copper foil is 
 
 soldered in the tube B, and has an equal-sized hole bored at the middle, 
 so as to be exactly opposite the hole in the disc c. By sliding the tube B 
 over the tube A the distance between the copper plates can be varied, 
 and thus to a certain extent the pitch of the note produced altered. Air 
 at a pressure of about 15 centimetres of water is forced into the tube A 
 through a tube E. The air must be contained in a gas bag, and the 
 desired pressure obtained by weighting the pressure boards. As it is 
 essential, if the pitch of the note is to remain constant, that the pressure 
 should be kept the same, it is desirable to attach a water manometer, 
 consisting of a U-tube, to a T-piece forming part of the air supply tube. 
 To detect the vibrations of high pitch produced by the bird-call 
 a sensitive flame is used. Such a flame is obtained by forcing coal-gas 
 
115] WAVE-LENGTH OF A HIGH NOTE 287 
 
 from a gas bag under a pressure of about 25 centimetres of water through 
 a pin-hole burner. This burner consists of a plug of steatite attached to 
 the end of a metal tube, the plug being pierced by a hole a little less 
 than a millimetre in diameter. The gas is supplied from a gas bag, and 
 care must be taken that there are no obstructions in the supply tube 
 near the burner. The quantity of gas must be adjusted by a stop-cock 
 near the bag, and a fair length of rubber tube used to convey the gas to 
 the burner. It is sometimes rather difficult to obtain a thoroughly 
 satisfactory burner, the reason probably being that the holes are not 
 quite smooth and regular. In general, such a burner gives a flame which 
 is much more sensitive to waves coming in one direction than to waves 
 coming in other directions. The sensitive side of the burner ought 
 therefore to be determined, and this side turned towards the direction in 
 which the waves are coming. 
 
 Fix a flat surface, such as a sheet of glass or a drawing-board, perpen- 
 dicular to the table, and in front and at a distance of about 2 metres 
 place the bird-call, the call being about 30 centimetres above the surface 
 of the table. Between the call and the reflecting surface a board is 
 placed on the table. This board has a scale divided into millimetres 
 attached to one edge, and the sensitive burner is attached to a block of 
 wood which rests with one side against the scale. A mark on the base 
 of the burner enables the distance through which the burner is moved to 
 be read off on the scale. 
 
 Light the gas and adjust the pressure so that a flame about 40 centi- 
 metres high is obtained, but the flame does not flare. Then start the 
 bird-call, so adjusting the air pressure that a clear note, as distinct from 
 a hissing noise, is produced. Starting with the burner as near to the 
 reflecting surface as possible, slowly move it towards the call, noting the 
 readings on the scale at the points where the flame does not flare. The 
 points where the flame does not flare are the nodes ; that is, the points of 
 maximum changes in pressure of the stationary waves produced by the 
 direct and reflected systems of waves. Since the distance between con- 
 secutive nodes is equal to half the wave-length, the wave-length is at 
 once obtained from the readings on the scale. 
 
 In place of the sensitive flame the ear may be used to detect the 
 positions of the nodes and loops of the stationary waves. For this pur- 
 pose the pin-hole burner is replaced by a piece of glass tube having a 
 bore of about 5 mm. This tube is held on a block of wood similar to 
 that which carried the burner, with its upper end on the level with the 
 call, while the lower end is connected by means of a length of thin india- 
 rubber tubing with a piece of tube which is fitted into the ear. On 
 moving the stand carrying the glass tube backwards and forwards 
 maxima and minima of sound will be heard, and. the distances between 
 consecutive maxima or minima will be equal to half the wave-length. 
 In this case the minima of sound occur at the loops, while the maxima 
 occur at the nodes, exactly the opposite to what occurred in the case of 
 the flame, The reason is that the flame is sensitive to changes in motion 
 
288 SOUND [ 115 
 
 of the air, which is a maximum at the loops, while the maximum effect 
 as far as producing waves in the tube is concerned occurs when the 
 pressure of the air at the open end of the tube varies most. 
 
 Having measured the wave-length of the sound given by the bird-call, 
 then, from the velocity of sound in air at the temperature of the experi- 
 ment, it is possible to calculate the frequency of the note given by the 
 call. 
 
CHAPTER XVIII 
 
 REFRACTIVE INDEX 
 
 116. Adjustment of a Telescope for Parallel Rays. An adjust- 
 ment which has frequently to be made is to focus a telescope so that a 
 pencil of parallel rays falling on the objective are brought to a focus in 
 the plane of the cross-wires. The first thing to be done is to focus the 
 eye-piece on the cross-wires. The best way of doing this is to turn the 
 telescope towards some fairly brightly lighted object, such as a white 
 wall, which is at a distance of at least 2 metres. Then adjust the object 
 glass so that the wall is not at all in focus, and so that on looking into 
 the telescope a bright field of view is seen, but no image. If, now, one 
 eye is used to look into the eye-piece, and the other is kept open and 
 turned towards the wall, the appearance presented will be that the cross- 
 wires of the telescope seem to cross the wall. The position of the eye- 
 piece is then adjusted till the cross- wires appear quite distinct as seen 
 with one eye, while with the other eye the wall is also seen distinctly. 
 
 In this way we ensure that whenever the cross- wires are seen dis- 
 tinctly the eye is adjusted for seeing a distant object, which for most 
 persons is an unstrained condition. In the case of a short-sighted 
 person the wall may be replaced by the open page of a book, supported 
 at such a distance from the eye-piece that with the unaided eye the print 
 can be seen distinctly, and then proceeding to adjust the eye-piece so that 
 the print can be seen distinctly with the unaided eye at the same time as 
 the cross-wires with the other eye. 
 
 If the telescope is to be continually used by the same observer it will 
 be well, having once fixed the correct position of the eye-piece, to make 
 a light scratch corresponding to the end of the tube in which the eye- 
 piece fits, so that the eye-piece can always be placed in the correct 
 position without having to go through the above described procedure. 
 
 The eye-piece having been focused, the telescope must be supported 
 in a clamp, so that it may be focused on some distant object. The 
 object chosen ought to have a clearly defined outline, and a lightning 
 arrestor or a telegraph wire will be found suitable. If the telescope is 
 placed inside a room the wincTow must be opened, since window glass is 
 so irregular that it will be found impossible to obtain a sharp focus if 
 the rays have passed through the glass. The telescope must then be 
 adjusted till the image formed by the object glass is in the plane of the 
 cross-wires. This will be secured when, on moving the eye from side to 
 
 289 
 
B 
 
 290 REFRACTIVE INDEX [117 
 
 side as far as the occular of the telescope will allow, the cross-wires do 
 not seem to move relatively to the image. 
 
 Another method of focusing a telescope for parallel rays is described 
 on p. 291. 
 
 117. Measurement of the Angle of a Prism with the Spectrometer. 
 When making a measurement of the angle of a prism with the spectro- 
 meter, the following adjustments have to be made : 
 
 1. The optical axes of the telescope and collimator have to be made 
 perpendicular to the axis about which the telescope and table turn. 
 
 2. The telescope focused for parallel rays. 
 
 3. The collimator focused for parallel rays. 
 
 4. The prism adjusted so that the faces which include the angle which 
 is to be measured are both parallel to the axis about which the telescope 
 and table turn. 
 
 1. If the instrument is provided with a Gauss eye-piece, that is, with 
 an eye-piece with a side opening between the eye-piece lenses and a plate 
 
 of unsilvered glass B (Fig. 122) inclined 
 at 45 to the axis, the following method 
 * can be employed : When a light is 
 
 placed opposite the opening the rays are 
 
 reflected by the glass along the axis of 
 the telescope, and if, after they leave the 
 object glass, they are reflected back by,a 
 FIG. 122. plane mirror, an image of the cross- wires 
 
 will be seen alongside the cross- wires 
 
 themselves. A piece of plate glass is mounted on the table with its sides 
 vertical, and adjusted so that the image of the cross-wires formed by re- 
 flection at the surface of the glass coincides with the cross- wires them- 
 selves. The telescope is then turned through 180, so that the light is 
 now reflected back into the telescope from the other surface of the glass. 
 If the image of the cross- wires now coincides with the wires, the axis of 
 the telescope is perpendicular to the axis about which the telescope turns. 
 If the image does not coincide with the wires, it must be brought half-way 
 back by adjusting the telescope, 1 and half-way by adjusting the table on 
 which the plate glass is supported. This adjustment must, if necessary, 
 be repeated till the image in both positions coincides with the wires. 
 The adjustment will not be absolutely correct unless the two sides of 
 the plate glass are exactly parallel. In general, however, the angle 
 included between the two sides is so small as to produce no material 
 error. 
 
 If the telescope is not fitted with a Gauss eye-piece, we may assume 
 that the optic axis of the telescope is parallel to the telescope tube, and 
 
 1 This adjustment is made by tilting the telescope about a horizontal axis 
 with reference to the movable arm which carries the telescope. The arrange- 
 ments for making this adjustment vary in different instruments, but an exami- 
 nation will show the manner in which the adjustment is to be made in the case 
 of any particular instrument. 
 
117] MEASUREMENT OF THE ANGLE OF A PRISM 291 
 
 proceed as follows : A striding level, that is, a level provided with a V at 
 either end, which V's can rest on the upper surface of the telescope tube, 
 is placed on the telescope. The telescope is then turned so that it lies 
 parallel to the line joining two of the levelling screws of the instrument, 
 and by adjusting these screws the bubble is brought to the centre. The 
 telescope is then turned through 180, and if the bubble is no longer at 
 the centre, it is brought half-way back by adjusting the levelling screws, 
 and the other half by adjusting the telescope in its support. It will 
 generally be found necessary to repeat the adjustment two or three 
 times before the bubble remains exactly at the centre as the telescope 
 is turned. 
 
 The principle on which this method of adjusting is made is the same 
 as that described on p. 56 with reference to the cathetometer. 
 
 To adjust the collimator, a thin wire must be stretched across the 
 middle of the slit, and the screws which attach the collimator altered till 
 the image of this cross-wire, as seen in the telescope, coincides with the 
 intersection of the cross-wires in the telescope. 
 
 2. The telescope may be focused for parallel rays in the manner 
 described in the previous section, but a more convenient and accurate 
 way in the case of a spectrometer is as follows : Place a prism on the 
 table, and turn the telescope so as to see the refracted image of the slit, 
 which must be illuminated by monochromatic light. This light is most 
 conveniently produced by holding in the flame of a Bunsen burner a piece 
 of asbestos soaked in a strong solution of sodium chloride. The prism is 
 then rotated till it is in the position of minimum deviation. The tele- 
 scope having then been turned to a position corresponding to a greater 
 deviation than the minimum, it will be found that there are two positions 
 of the prism for which a refracted image of the slit will be seen in the 
 telescope. In one of these positions (a) the angle of incidence of the 
 light on the first face will be greater than for minimum deviation, and in 
 the other position (b) less. Turn the prism into position (a), and focus 
 the telescope till the image of the slit appears sharp. Then turn the 
 prism into the position (b), and focus the collimator till the image is 
 sharp. Turn back into position (a) and again focus the telescope, and so 
 on. After one or two trials the image will remain in good focus for both 
 positions of the prism. When this is the case, both the collimator and 
 the telescope are focused for parallel rays. In practice it is quite 
 unnecessary to try and remember the order in which the telescope and 
 collimator are to be focused, since after one or two alternations, i.e. 
 focusing the telescope in one position of the prism and the collimator 
 in the other position of the prism, the adjustment will either get very 
 much better, indicating that the right order has been adopted, or the 
 adjustment will get worse, when the order must be reversed. 
 
 4. The prism must now be mounted on the table with its centre o 
 (Fig. 123) over the centre of the table, and with one of the faces which 
 includes the angle to be measured, A, perpendicular to a line joining two 
 of the table levelling screws. To assist in placing the prism in this 
 
292 
 
 REFRACTIVE INDEX 
 
 FIG. 123. 
 
 [117 
 
 position, it is an advantage to have a series of lines scribed on the table 
 top, as indicated in the figure. 
 
 The telescope having been turned so that its axis makes an angle 
 of about 90 with the axis of the collimator, the prism is turned till an 
 
 image of the slit is seen produced by light 
 reflected from the face AB, i.e. the face which 
 is perpendicular to the line joining two of 
 the levelling screws, and the levelling screws 
 adjusted so that the image of the wire 
 stretched across the slit coincides with the 
 intersection of the cross-wires. The prism 
 is then turned so that the light reflected 
 from the face AC enters the telescope, and 
 the image of the wire is made to coincide 
 with the intersection of the cross-wires by 
 turning the levelling screw F. Since altering 
 the screw F will move the face AB in its 
 own plane, by proceeding in this way the adjustment of this face will 
 not be thrown out when adjusting the face AC, as it would if the prism 
 had not been placed in this particular position on the table. The 
 adjustment of the prism is now complete, and we may proceed to the 
 measurement of the angle. 
 
 There are two methods which may be used. If, however, the table 
 is not provided with verniers, by means of which its rotation can be 
 measured with as great an accuracy as that with which the rotation of 
 the telescope can be read, then only the second method is available. 
 
 Method 1. Keeping the telescope in the position in which it was 
 placed when adjusting the prism, and the centre of the prism being over 
 the centre of the table, turn the table till the image reflected from the 
 face AB coincides with the intersection of the cross-wires, and read the 
 verniers attached to the table. Then turn the table till the reflection 
 from the face AC coincides with the intersection of the cross-wires, and 
 again read the vernier. The difference between the two readings sub- 
 tracted from 180 will give the angle CAB. 
 
 Method 2. Place the prism on the table so that the angle at A is 
 turned towards the collimator and vertically over the axis of the instru- 
 ment, and obtain the readings on the verniers attached to the telescope 
 when the intersection of the cross-wires coincides with the image of the 
 slit when seen reflected first from the side AB, and then from the side AC, 
 the prism remaining fixed. The difference between these readings will 
 give twice the angle CAB.* 
 
 If the instrument is so constructed that the collimator can be moved 
 to different parts of the divided circle, the measurements ought to be 
 repeated with the collimator making angles of 90, 180, and 270 with 
 its original position, in order to eliminate errors of graduation of the 
 circle as much as possible. 
 
 If the difference between the angle, as measured by these two methods, 
 
117] MEASUREMENT OF THE ANGLE OF A PRISM 293 
 
 is greater than would be expected from the accuracy with which the 
 circle can be read, it most probably is due to the fact that the collirnator 
 has not been correctly adjusted for parallel rays, and this adjustment 
 must be repeated. 
 
 A study of the effect of want of adjustment of the collimator on the 
 observed angles by the two methods is instructive. Consider, in the first 
 place, the first method, in which the prism is turned and the telescope is 
 kept fixed. Let ABC (Fig. 124) be the position of the prism when the image 
 of the slit reflected in the face AB is on the cross-wires of the telescope, 
 and let o be the centre of the divided circle, that is, the centre about which 
 the prism is turned. If the collimator is not correctly adjusted, then the 
 image of the slit it produces will not be at infinity ; thus, suppose that 
 this image is virtual 
 and situated at s, 
 then the image 
 formed by reflection 
 in the face AB will 
 be situated at s lt 
 and' the axis of the 
 telescope will be 
 directed along the 
 straight line Tos r 
 
 Next suppose 
 the prism rotated 
 into the position 
 
 image of the slit re- 
 flected in the side 
 BC is seen on the 
 cross - wire of the 
 telescope, i.e. this 
 image is situated on 
 the axis of the 
 telescope, and there- 
 fore somewhere on the line 
 
 A' 
 
 B 
 
 FIG. 124. 
 
 Now, unless the centre 
 
 !, say, at s. 2 . 
 
 o, about which the prism turns, is at the centre of the triangle ABC, 
 when the prism has been turned so that A'C' is parallel to BA, the perpen- 
 dicular distance of s from A'C'D will be different from the perpendicular 
 distance of s from the line BAD, and hence the images Sj and s. 2 will not 
 coincide, but in the case shown in the figure we must turn the prism 
 through a further angle BDA' or S 1 ss., before the image s., comes 
 on the line TOS r In other words, the angle through which the 
 prism has been turned is 180+ <BDA' - <BAC. Hence to obtain a 
 correct result the angle BDA' or SjSS. 2 must vanish. This occurs (1) 
 when s is at infinity, or (2) when o coincides with the centre of the 
 triangle ABC. Hence when using this method adjust the collimator 
 as well as possible, and then to reduce the effect of any outstretching 
 
294 
 
 REFRACTIVE 
 
 INDEX [117 
 
 the axis about which the 
 
 error set the prism so that its centre is over 
 table turns. 
 
 Next consider the second method, in which the prism is kept fixed and 
 the telescope is rotated. Thus let ABC (Fig. 125) represent a section of 
 the prism, the centre of the divided circle being at o, and the image of 
 the slit formed by the collimator in place of being at infinity being at s. 
 Then after reflection in the face AB the light proceeds as if it came from 
 
 c' 
 
 Fia. 125. 
 
 S 2 , which is the image of s formed by reflection in the face AB. Similarly, 
 Sj is the image of s formed by reflection in the face AC. Then it is at once 
 evident from the similar triangles SAC', SjAc', and SAB', S 2 AB' that the angle 
 SjASa is twice the angle BAC, that is, the angle T/AT/ is twice the angle BAG. 
 Since, however, the telescope turns about the point o as centre, the 
 angle measured is the angle TjOTg or SjOS.,, and this angle differs from 
 twice the angle BAC by the sum of the angles oSjA, os 2 A. Hence the 
 condition that the angle measured by the rotation of the telescope shall 
 be twice the angle of the prism is, that the angles oSjA and os 2 A shall 
 vanish. This condition is fulfilled if either o coincides with A or if s, and 
 
118] REFRACTIVE INDEX OF A SOLID OR LIQUID 295 
 
 s., are at an infinite distance from o, that is, if s is at infinity, and there- 
 fore the collimator is correctly adjusted. Since it is impossible to be 
 absolutely certain that the adjustment of the collimator is quite accurate, 
 the effect of this want of adjustment should be rendered as small as possible 
 by placing the angle A vertically over the centre of the divided circle. 
 
 An objection to the second method is that at most half the objective 
 of the telescope is used for either reading, and it is not the same half 
 which is used in the two cases. Thus errors 
 may be caused by spherical aberration in 
 the lenses of the telescope. In a well con- 
 structed telescope, however, the error from 
 this cause will not be important. 
 
 Both the telescope and the table ought 
 to be provided with two verniers at 180 
 apart, for unless the centre of the divided 
 circle exactly coincides with the axis of 
 rotation of the telescope or table, then the 
 angle read off by means of a single vernier 
 may differ from the true angle. The mean, 
 however, of the angles of rotation as read 
 off on the two verniers is free from error 
 on account of excentricity of the circle. 
 For suppose c (Fig. 126) is the centre of the divided circle, and o is the 
 centre about which the telescope rotates. Then if AB and A'B' are two 
 positions of the verniers, the angle through which the telescope has 
 actually been rotated is AOA' or BOB', while the angles read off on the 
 verniers are ACA' and BOB'. But since the angle subtended by the arc 
 AA' at the centre c is twice the angle it subtends at the circumference of 
 the circle, we have 
 
 also 
 Hence 
 
 ACA' = ZAB'A' = 2{AOA' - B'AB}, 
 BOB' = 2B'AB. 
 
 or 
 
 that is, the mean of the angles read off on the verniers is equal to the 
 true angle. 
 
 118. Refractive Index of a Solid or a Liquid in the form of a 
 Prism. If the angle of the prism is 6, and the angle of minimum 
 deviation 8, then the refractive index //, is given by l 
 
 sin -5 (0 + 8) 
 
 /* = 
 
 . 1 a 
 
 sin U 
 '2 
 
 Hence to determine the refractive index of the material of which a prism 
 1 Watson's Physics, 349. 
 
296 REFRACTIVE INDEX [118 
 
 is composed, we have to measure the refracting angle by one of the 
 methods given in the preceding section, and then determine the angle of 
 minimum deviation for light of some definite wave-length. 
 
 To determine the angle of minimum deviation the prism is placed on 
 the table of the spectrometer with its centre over the centre of the table, 
 and the faces which include the refracting angle which is to be used are 
 adjusted parallel to the vertical axis of the instrument in the manner 
 described on p. 292. Then turn the refracting edge of the prism to 
 the right and adjust the telescope so that the refracted image is seen, 
 and turning the prism in such a way that the deviation decreases, follow 
 the movement of the refracted image with the telescope till the image 
 commences to turn back, and by trial adjust the telescope so that when 
 the prism is turned the image just reaches the intersection of the cross- 
 wires and then moves back. Read the verniers attached to the telescope, 
 and turning the prism so that the refracting edge is turned towards the 
 left, repeat the adjustment for minimum deviation. The difference 
 between the readings for the telescope verniers in the two positions will 
 be twice the angle of minimum deviation. 
 
 Since the refractive index changes with temperature, a note of the 
 temperature must be made. For this purpose a thermometer can be 
 suspended with its bulb near the prism, care being taken when adjusting 
 the prism for minimum deviation not to handle the prism itself, but to 
 turn it by means of the table. 
 
 If the prism is in the form of an equilateral triangle, so that all the 
 angles are very nearly equal to 60, it will be unnecessary to measure 
 the angles if measurements of minimum deviation are made with all the 
 angles in turn, and the mean is taken as the deviation corresponding to 
 a refracting angle of 60 . 1 
 
 In the case of a liquid, a hollow glass prism has to be employed to 
 contain the liquid. The sides of the prism are formed by two plates of 
 glass with optically worked surfaces, so as to be plane and parallel. By 
 taking a little trouble it is possible to select pieces of plate glass which 
 shall be as good as any but the very best, and therefore very expensive, 
 optically worked glass. The manner in which a suitable piece can be 
 selected is described in 142.. These glass sides are either cemented 2 
 to the body of the prism which is the most convenient arrangement 
 when a series of liquids of the same nature have to be examined, so that 
 the same cement is unattacked by all the liquids or the plates are 
 simply held in place by india-rubber bands, the plates having been made 
 such a good fit that the liquid will not escape. In this case care must 
 be taken that the prism is not left on the spectrometer table longer than 
 is necessary to take a reading. To protect the table in case some of the 
 liquid does escape, a shallow glass dish may be cemented to the top of 
 the table, and the prism placed in this dish. Since the refractive index 
 
 1 See Proceedings of the Royal Society, vol. Ixx. p. 330. 
 
 2 For water and solutions in water, marine glue can be used as a cement. 
 For liquids such as alcohol and carbon bisulphide glycerine glue is suitable. 
 
119] 
 
 REFRACTIVE INDEX OF A LIQUID 
 
 297 
 
 of a liquid changes rapidly with temperature, a small thermometer must 
 pass through the cork which closes the opening at the top of the prism, 
 the bulb being immersed in the liquid. This thermometer must be read 
 immediately after each setting for minimum deviation. 
 
 When measuring the angle of the hollow prism, unless the faces of 
 the plates of glass which form the sides are exactly parallel, two images 
 of the slit will be seen in each position, one formed by reflection at the 
 outside surface of the glass plate, and the other at the inside surface. To 
 ascertain which image is formed by the inside surface, fill the prism with 
 liquid, when the brightness of one of the reflected images will be de- 
 creased. The image in which this decrease in brightness takes place is 
 the image formed by light reflected at the inside surface of the face of 
 the prism. When measuring the angle of the prism, the images formed 
 by reflection from the outside surfaces must be used. To allow for the 
 effect of the prismatic form of the faces, two sets of observations of angle 
 and of minimum deviation must be made. In the second set the prism 
 faces must be rotated in their own plane through an angle of 180, that 
 is, the edge of the glass plate which was nearest the refracting angle in 
 the first observation must be turned so as to be nearest the base in the 
 second observation. The mean of the values obtained for the refracting 
 angle and the angle of minimum deviation must be employed when 
 calculating the refractive index. 
 
 119. Measurement of the Refractive Index of a Liquid by Total 
 Reflection, using an Air Film. 1 If two parallel sided plates of glass 
 
 
 FIG. 127. 
 
 parated by an air film are immersed in a liquid contained in 
 sided cell, which is placed between a collimator and a telescope, and the 
 two positions of the glass plates are determined for which the light is just 
 
 Watson's Physics, 360. 
 
298 
 
 REFRACTIVE INDEX 
 
 [119 
 
 cut off owing to total internal reflection at the air film, the refractive 
 index from air to the liquid is given by 
 
 where 2 a is the angle through which the glass plates have to be turned 
 to pass from one position of extinction to the other. 
 
 An arrangement by means of which an ordinary spectrometer can be 
 used to measure the refractive index of a liquid by total reflection is 
 shown in Fig. 127. It consists of an upright A carried by the prism 
 table, which is bent twice at right angles, and supports the two glass 
 plates with the enclosed film of air. A small metal plate c is carried 
 on a bracket D attached to the telescope or collimator, and on this 
 plate is placed a box with plate glass sides, E, in which the liquid is 
 contained. 
 
 When performing an experiment the liquid is placed in the glass 
 trough, and the double plate B is introduced and set approximately per- 
 pendicular to the incident light. The telescope is then turned so that the 
 image of the slit is seen to coincide with the intersection of the cross- 
 wires. The prism table, carrying with it the double plate, is then turned 
 till the image of the slit is extinguished, and the readings of the verniers 
 taken. The prism table is then turned in the opposite direction till 
 
 the image again vanishes, 
 and the verniers are read. 
 The difference between the 
 readings gives twice the 
 critical angle from air to the 
 liquid. A thermometer ought 
 to be placed in the liquid, and 
 the temperature read both 
 before and after the measure- 
 ments. 
 
 This method requires a 
 considerable volume of the 
 liquid, and hence in many 
 cases, where only a small 
 quantity of liquid is avail- 
 able, it is inapplicable. The methods described in the following sections, 
 since only a few drops are required, may be used in all cases. 
 
 Corrections. The formula given above assumes that the sides of the 
 plate of glass which encloses the air film on the side- next the collimator 
 are parallel. If they are not parallel, then a correction must be applied. 
 Suppose the two sides AB, CD (Fig. 128) make an angle 0, and that PA 
 and QB are the directions of the incident rays when the refracted rays in 
 the glass are incident on the surface CD at the critical angle /3. If, then, 
 
 O 
 
 FIG. 128. 
 
120] REFRACTIVE INDEX BY TOTAL REFLECTION 299 
 
 /A is the refractive index from air to the liquid in which the rays PA and 
 QB are incident we have, as in 360 of Watson's Physics, 
 
 _ sin(/3 + 0) _ sin (ft - 0) 
 
 
 sin a sin (3 sin a' sin f? 
 = sin ft cos + cos /? sin sin /:? cos - cos /3 sin 
 
 sn a sn sn a sn 
 
 But since is small, we may put sin 0=0, and cos 0=1. 
 
 Hence . _ sin ft + cos ft = sin ff - cos ff 
 
 sin a sin [3 sin a' sin ft ' 
 
 or /x sin a = 1 + cot /?, 
 
 and /A sin a' = 1 - cot ft. 
 
 Hence adding /x (sin a + sin a/) = 2 
 
 or /x= . 
 
 sin a -f sin a 
 
 1 
 
 sin - (a + a') COS - (a - a') 
 
 Hence the correction to allow for the effect of the glass plate is to 
 multiply by the factor I/cos - (a - a'). To obtain the magnitude of this 
 
 correction it is necessary to measure the difference of the angles 
 a and a'. 
 
 120. The Measurement of Refractive Index by Total Reflec- 
 tion, using a Prism (First Method). A very convenient and accurate 
 method of measuring the refractive index of a liquid or solid which is 
 available if we possess a prism having a higher refractive index than that 
 to be measured is as follows : Suppose that one side BC (Fig. 129) of a 
 prism is unpolished, and that this side is illuminated by monochromatic 
 light We may then consider each point on the side BC as being self- 
 luminous, sending rays into the prism in all directions. Let p be such a 
 point, and PQ be the ray that strikes the surface AB at an angle of inci- 
 dence just greater than the critical angle. This ray will be reflected along 
 QR, and after refraction at the surface AC, will travel along RS, and will 
 be very little diminished in intensity during the process. Similarly, a ray 
 such as PQ 7 will reach s' with little loss. It will, however, be quite 
 otherwise with a ray such as PQ" which strikes the face AB at an angle of 
 incidence less than the critical angle, for part of the light will be refracted 
 along Q"T, and only part will be reflected along Q"R", and so the ray R"S" 
 
300 EEFRAOTIVE INDEX [ 120 
 
 will be much more feeble than the rays us or R'S'. If, then, a telescope is 
 placed so as to look into the face AC, with its axis parallel to SR, the field 
 of view will be seen divided into two parts, one half being much brighter 
 than the other, the line of demarcation corresponding to the direction 
 SR of the ray for P, which has fallen on BA at the critical angle. 
 
 An exactly similar argument would apply to the rays sent out from 
 any other point on the face BC, and the direction of the rays which leave 
 
 B 
 
 p Mat Surface 
 
 FlG. 129. 
 
 the face AC and limit the brighter part of the field would all be parallel to 
 RS. Hence if the telescope is focused for parallel rays, all these rays 
 would be focused along a line parallel to the edge A of the prism. 
 
 Let a be the angle RS makes with the normal RN' to the face CA, 
 9 the angle of the prism, and {$ the angle QRN. Further, for generality, 
 let us suppose that the medium (2) outside the face AB is not the same as 
 the medium (3) (air) outside the face AC. Then using the symbol ^ for 
 the refractive index from the medium (2) to the glass of the prism, ^ 
 for the refractive index from air to the glass of the prism, and 3 /x. 2 for the 
 refractive index from air to the medium (2), we have, since the angle 
 NQR is the critical angle from medium (1) to medium (2), and the angle 
 
 1 
 
 sin ((9-0) 
 
 also 
 
 sin a 
 sin0 
 
 (1) 
 
 (2) 
 
120] REFRACTIVE INDEX BY TOTAL REFLECTION 301 
 
 and 3^ = ^. (See Watson's Physics, 345) (3) 
 
 2/*l 
 
 Hence 3 ju,., = 3^ sin (0 /?) = ^i l {sin 6 cos /3 - cos sin /?}. 
 
 Substituting from (2) we get 
 
 3 /x 2 = sin # VG^i 2 " s i n2 a ) ~ cos $ sin a ..... (4) 
 If the angle is 90, then this formula reduces to 
 
 i 2 - sin2 a ) ...... ( 5 ) 
 
 In the case where medium (2) is air, then 3 p-.,= l, and equation (4) 
 reduces to 
 
 a formula which may be employed to measure the refractive index of the 
 prism. 
 
 The angle of the prism must not be greater than a certain limiting 
 value, or the ray QR will suffer total internal reflection at R. The limiting 
 value for is such that the ray RS is parallel to the face, i.e. that 
 
 sin/8= But from (1) ^ = sin- 1 l/ 2 /*! + P- Hence the limiting con- 
 
 .3/*l 
 
 dition is that 
 
 < sin- 1 -. + sin iJL ....... (7) 
 
 2/*i sP-i 
 
 If the medium (2) is air, this reduces to 
 
 ^<2siri- 1 - ....... (8) 
 
 3/^1 
 
 and it is evident that the smaller the refractive index of the glass, the 
 larger may be the angle. For a glass having a refractive index of 1 '5 the 
 limiting value of 6 is 83 38'. If the refractive index of the glass is 
 1-62, the limiting angle is 76 14'. If with this glass 61 = 90, then the 
 method is not applicable to finding the refractive index of the glass. It 
 may, however, be used to measure the refractive index of a liquid or 
 solid, so long as (1) the refractive index of this substance is less than 1'62 
 and greater than a value which we can at once deduce from equation (7). 
 
 We have, since 2 /Xj = 3 ^i, in the limiting case 
 
 -1-27. 
 Thus the method may be used for substances having refractive indices 
 
302 REFRACTIVE INDEX [ 120 
 
 between 1*3 and 1*6. In the same way the limits may be obtained for a 
 prism of any other angle and refractive index. 
 
 The prism to be used ought to have a high refractive index, since the 
 method presupposes that the refractive index of the prism is higher than 
 that of the medium (2) of which the refractive index is to be measured. 
 
 (a) To measure the refractive index of the prism by this method, the 
 prism is mounted on the table of a spectrometer with its centre over the 
 axis of the instrument, and the faces AB, AC (Fig. 129) are adjusted 
 parallel to the axis by the method described in 117, and the telescope 
 is focused for parallel rays. The collimator is not employed, but the 
 source of monochromatic light, such as a Bunsen flame coloured with 
 salt, is placed opposite the unpolished face BC at a distance of 10 to 
 20 cm. The telescope is then turned to look into the face AC, and 
 adjusted till the line separating the bright from the dark portion of 
 the field is on the cross-wires. The position of the telescope having 
 been read on the divided circle, the telescope is turned to look into the 
 face AB, the prism remaining fixed, and the cross-wires are again adjusted 
 to coincide with the line of separation between the bright and dark 
 portions of the field, this reading corresponding to the direction ED 
 (Fig. 129). The difference, 8, between the readings for the position of 
 the telescope gives the angle EMS. But the angle 
 
 Hence 8 
 
 7T+0-S 
 
 a--^-, 
 
 Hence substituting this value of a in equation (6), and knowing the 
 value (see 117), the refractive index of the prism can be calcu- 
 lated. 
 
 (b) To measure the refractive index of a liquid, which must be less 
 than that of the prism, a drop of the liquid is placed on the face AB, and 
 a microscopic cover glass placed over. The cover glass is either kept in 
 place by capillarity, or a little wax may be used. 
 
 The telescope having been adjusted on the line of separation between 
 the light and dark parts of the field, the verniers are read. In this case 
 it is not convenient to turn the telescope to face the other face of the 
 prism, since this would involve cleaning off the liquid from the face AB 
 and putting it on the face AC, during which operation the prism might 
 accidentally get moved. The best way is to employ a Gauss eye-piece 
 (see Fig. 122), and having illuminated the cross-wires, turn the telescope 
 till the image of the wires, as seen reflected in the face AC, coincides 
 with the wires themselves. The reading of the verniers then gives 
 the direction of the normal RN', and the difference in the vernier 
 readings for the two positions of the telescope gives directly the value of 
 a, and from this, together with the value of the angle of the prism and 
 
121] REFRACTIVE INDEX BY TOTAL REFLECTION 303 
 
 its refractive index, the refractive index of the liquid can be obtained by 
 equation (4) above. 
 
 (c) To measure the refractive index of a solid. If a plate of some 
 solid is placed against the face AB, a layer of a liquid of higher refractive 
 index than the solid being interposed, then we may proceed exactly 
 as described in (6) above and obtain the refractive index of the solid. A 
 suitable liquid to use is mono-bromnapthaline, which has a refractive 
 index of 1*65 for sodium light. 
 
 121. The Measurement of Refractive Index by Total Reflection, 
 using a Prism (Second Method). A modification of the method described 
 in the last section consists in using monochromatic light at grazing 
 incidence on the face AB (Fig. 129), the source of the light being in the 
 plane of the face, and nearer B. In this case the portion of the field of 
 the telescope, directed along SR, on the side next A will appear bright, 
 while the other side of the 
 field will be dark, as is 
 evident from a study of the 
 figure. The formulae (4), 
 (5), and (6) of the last 
 section apply to this case 
 also. The advantage of the 
 method lies in the fact that 
 in place of having to set the 
 cross-wire of the telescope 
 on the line separating two 
 regions, the brightness of 
 which do not differ greatly, 
 in this case one portion of 
 the field is quite dark, and 
 so the line of separation is 
 much more easily seen. FIG. 130. 
 
 When an extended source 
 
 of light, such as a Bunsen flame coloured with some salt, is employed, 
 the source can be placed at a distance of from 10 to 30 cm. from 
 the prism, so that the centre of the flame is in the plane of the face 
 BA. If a small source of light, such as a vacuum tube, is employed, 
 then it is advisable to use a lens of about 10 cm. focal length between 
 the source and the prism, the position of the lens being adjusted so as 
 to give a slightly convergent beam of light. 
 
 When measuring the refractive index of the prism no difficulty will 
 be met with, particularly if small cardboard screens are placed so as to 
 cut off stray light from the telescope. When applying the method to 
 measuring the refractive index of liquids and solids, however, some 
 points have to be attended to, or it will be found impossible to obtain 
 a .sharp line of separation between the two halves of the field. 
 
 Thus when only a drop of the liquid is used, the cover glass em- 
 ployed must have a straight edge, and this edge must be adjusted to 
 
304 REFRACTIVE INDEX* [121 
 
 lie parallel to the edge of the prism where the light enters. If more 
 than a drop of the liquid is available, then the form of prism shown in 
 Fig. 130 may advantageously be employed. The angle B of a prism of 
 dense glass is ground off by a plane perpendicular to the face BA, and 
 then two pieces of glass, BD, DA, are cemented as shown. These pieces 
 of glass, together with a plate of glass cemented to the bottom, enclose 
 a prism-shaped space in which the liquid is contained, the angle at D 
 serving to allow the introduction of the bulb of a thermometer. The 
 light is incident parallel to PQ, and the limiting rays are parallel to QR, RS. 
 If the corner at E is slightly bevelled off as shown, so that none of the 
 cement used to attach the plate DB interferes with the light being 
 incident at 90, the field of view is very sharply divided, and very 
 accurate settings can be made. 
 
 The angle BAG of the prism must be measured before the glass plates 
 are cemented in place. 
 
 The advantages possessed by this arrangement over the hollow prism 
 are as follows : (1) Less liquid is required. (2) The plates of glass 
 used to enclose the liquid need not be optically worked, and the fact 
 that their faces are not parallel produces no effect. (3) The angle of 
 the glass prism has only to be measured, and this can be done once for 
 all, while with the hollow prism the measurement of the angle has to be 
 repeated each time the side plates are removed. 
 
CHAPTER XIX 
 
 DISPERSION AND WAVE-LENGTH MEASUREMENTS 
 
 122. The Measurement of Dispersion Sources of Light. The 
 
 dispersion of a medium can be determined by measuring the refractive 
 index by one of the methods previously described for light of different 
 wave-lengths. For this purpose it is necessary to have sources of light 
 which shall give light of some definite and known wave-length. Since 
 such sources of light of definite wave-length will be required for several 
 other measurements, it will be convenient to consider them here in detail, 
 and under different heads. 
 
 (a) Flames coloured icith metallic salts. A source of light which 
 is often convenient is a Bunsen flame coloured with some metallic salt. 
 Since the brightness of the coloration depends on the temperature of 
 the flame being high, it is of importance that the support used to 
 introduce the salt into the flame should not 
 be such as to cool down the flame more 
 than is absolutely necessary. If the light 
 is only required for a short time, a piece 
 of fine platinum wire dipped in a solution 
 of the salt and held in the outside portion 
 of a Bunsen flame about 2 cm. from the 
 bottom of the flame will be found con- 
 venient. As, however, the wire soon loses 
 all salt, this arrangement is not suitable 
 when the light is required for any time. 
 In such a case, a small scoop of thin platinum 
 foil fused to the end of a short platinum 
 wire will be found convenient for containing 
 the salt. A small shred of asbestos or a 
 piece of wood charcoal which has been 
 dipped into a strong solution of the salt 
 can also be used, being held in the edge 
 of the flame. 
 
 A much brighter light than can be 
 obtained with the Bunsen burner is pro- 
 duced by means of the arrangement shown in Fig. 131. It consists of a 
 strong glass test-tube A, which contains a solution of the salt. A thick 
 copper tube H is connected to a steel bottle of compressed hydrogen gas, 
 such as is used in connection with the lime light. This tube is connected 
 
 305 
 
 FIG. 131. 
 
306 
 
 DISPERSION 
 
 [122 
 
 direct to the bottle without the intervention of any pressure regulator, 
 so that the full pressure of the gas can act. The lower end of the tube is 
 closed by a screw D, while a very fine hole is pierced through the side of 
 the tube at B. The gas escaping through this hole, under great pressure, 
 churns the solution into a fine spray, which is carried up the tube c. A 
 concentric tube E, connected with a side tube o, serves for feeding oxygen 
 gas round the burning hydrogen. The oxygen need not be supplied at 
 any great pressure, so that it is best to use a regulator in connection 
 with the bottle in which this gas is stored. 
 
 If possible, it is always advisable to place the flame under a hood, so 
 that the fumes may not escape into the room. This is particularly 
 necessary in the case of salts of thallium, which are very poisonous. 
 
 The chief difficulty when using this burner is to prevent the fine 
 hole becoming clogged up with dirt. A small plug of cotton wool 
 placed in the tube H will to a certain extent prevent dirt carried over 
 by the gas reaching the nozzle. 
 
 The following table gives the salts which can be used in a Bunsen 
 flame to give light of definite wave-length. 
 
 Salt to be Used. 
 
 How Introduced into 
 Flame. 
 
 Wave-length in 
 lO- 8 cm. 
 
 Colour of 
 Light. 
 
 Potassium nitrate 
 
 On platinum wire 
 
 ( 7668-5 
 I 7701-9 
 
 Red 
 
 (double), i 
 
 Lithium chloride 
 
 On charcoal 
 
 f 6708-2 
 \ 6103-8 
 
 Red. 
 Orange. 
 
 Sodium chloride 
 
 On charcoal 
 
 1 5890-2 
 
 I 5896-2 
 
 Orange 
 (double). 
 
 Thallium chloride 
 
 On platinum wire 
 
 5350-7 
 
 Green. 
 
 Strontium chloride On charcoal 
 
 4607-5 
 
 Blue. 
 
 Calcium chloride On charcoal 
 
 4226-9 
 
 Blue. 
 
 Potassium nitrate On platinum wire 
 
 j 4044-3 
 | 4047-4 
 
 Violet 
 - (double). 
 
 Rubidium chloride On platinum wire 
 
 J 4202-0 
 | 4215-7 
 
 Violet 
 (double). 
 
 (b) A source of light which is in many ways more convenient 
 than a flame coloured by metallic salts, since it does not require the 
 constant attention this latter requires, consists of a tube fitted with 
 electrodes, and filled with certain gases under reduced pressure. On 
 passing an electric discharge through such a tube by means of an 
 induction coil the contained gas will become luminous, the spectrum 
 produced consisting of lines. 
 
 The commonest form of such a vacuum tube is shown in Fig. 132. 
 
 % It consists of two wider portions A and B, connected by a narrow tube c 
 
 having a bore of about 2 mm. The electrodes consist of short aluminium 
 
 wires fused on to short pieces of platinum wire. The platinum passes 
 
 through the glass and terminates outside in a small loop, to which the 
 
122] 
 
 THE MEASUREMENT OF DISPERSION 
 
 307 
 
 B 
 
 wires from the induction coil are attached. On passing a discharge 
 through the tube, the gas within the capillary tube c glows brightly. 
 The disadvantage of this form 
 of tube is that the bright part 
 of the tube can only be used 
 sideways, since the electrodes 
 prevent the light which passes 
 along the axis of the tube 
 being used. A better form 
 of tube is that shown in Fig. 
 133. Here the capillary tube 
 c is so placed that nothing 
 interferes with the light which 
 passes axially along the tube. 
 This is an advantage, since 
 the light given along the axis 
 is very much brighter than 
 that given at right angles to 
 the axis. It is also an ad- 
 vantage to have electrodes of 
 considerable size, and hence the aluminium wire is coiled into a spiral. 
 
 The following table gives a list of the more useful gases for use 
 in vacuum tubes, with the wave-lengths of the principal lines in air at 
 15 and 760 mm. pressure : 
 
 B 
 
 FIG. 132. 
 
 FIG. 133. 
 
 Gas. 
 
 Wave-length in 10~ 8 cm. 
 
 Colour. 
 
 Hydrogen 
 
 6563-0 
 
 Red. 
 
 
 4861-5 
 
 Blue-green. 
 
 
 4340-7 
 
 Violet, 
 
 Helium . . 
 
 7065-2 
 
 Red. 
 
 
 6678-1 
 
 Red. 
 
 
 5875-6 
 
 Yellow. 
 
 
 5015-7 
 
 Green. 
 
 , 
 
 4921-9 
 
 Blue. 
 
 
 4713-2 
 
 Blue. 
 
 
 4471-5 
 
 Violet. 
 
 Vacuum tubes containing gases are found to deteriorate fairly rapidly, 
 and for almost all purposes the light given out by mercury vapour, 
 caused to glow by the passage of an electric discharge, is more con- 
 venient. There are two distinct types of tubes for giving the mercury 
 spectrum. Where a small source of light with a fair amount of 
 brilliancy is sufficient, a tube such as that shown in Fig. 133, and in which 
 a little mercury has been enclosed, can be used with the discharge from 
 In order to fill the tube with mercury vapour, it is 
 
 an induction coil. 
 
308 DISPERSION [ 122 
 
 necessary to heat the tube over a small Bunsen flame. A small shield of 
 asbestos cardboard placed over the tube will be found useful in keeping 
 off draughts and securing uniform heating. 
 
 Where, however, the electric light is available, what is called a 
 mercury, or Cowper-Hewett, lamp is by far the most convenient arrange- 
 ment to use, since it does not require an induction coil, and gives a very 
 brilliant light. A very convenient form is the Bastian lamp, which is 
 shown in Fig. 134. The exhausted tube AB through which the dis- 
 charge passes, is bent into a zig-zag, and there is a small mercury 
 reservoir at the end A. The electrodes are connected, through a 
 resistance adjusted to suit the voltage of the supply, to an ordinary 
 incandescent electric light socket. The lamp with its resistance is 
 
 FIG. 134. 
 
 screwed to a board i), which is pivoted on a horizontal axis between the 
 uprights EF. 
 
 To start the lamp, the board D is turned so that the lamp is facing 
 downwards, and then the tube is tilted so that the mercury flows from 
 the reservoir A and fills the tube, thus completing the electric circuit. On 
 releasing the lamp, the tube tilts so that the mercury flows back into the 
 reservoir A, and in doing so breaks the circuit in the tube, and the small 
 arc formed at the point of the rupture vaporises some of the mercury, 
 and the discharge continues to pass through this mercury vapour. When 
 the lamp is started, the board D can be turned up into a vertical plane, 
 in which positions the vertical portions of the tube form convenient 
 sources of light for use with a spectrometer. 
 
122] 
 
 THE MEASUREMENT OF DISPERSION 
 
 309 
 
 The light given by the mercury lamp contains a number of lines 
 but in the following table only the more conspicuous ones are given : 
 
 WAVE-LENGTHS IN AIR AT 15 AND 760 MM. PRESSURE OF 
 THE CHIEF LINES GIVEN BY THE MERCURY LAMP. 
 
 Colour. 
 
 Wave-length in 10~ 8 cm. 
 
 Remarks. 
 
 Bed 
 
 6232 
 
 Fairlv bright. 
 
 Yellow . . . 
 
 5790-7 
 
 Very bright. 
 
 Yellow . . . 
 
 5769-6 
 
 Very bright. 
 
 Green .... 
 
 5460-7 Very brio-lit. 
 
 Green-blue . . 
 
 4959-7 
 
 Fairly bright. 
 
 Blue . . 
 
 4916-4 
 
 Faint. 
 
 Blue .... 
 
 4358-3 
 
 Bright. 
 
 Violet .... 
 
 4078-1 
 
 Bright. 
 
 Violet. . . . 
 
 4046-8 
 
 Faint. 
 
 The light given by cadmium or zinc, when the vapour of these metals 
 is rendered incandescent by the passage of an electric discharge, is often 
 useful. These metals require to be heated to a considerable temperature 
 before they vaporise, and it is found that the vacuum tubes in which 
 they are used deteriorate rather rapidly. A form of tube which can 
 easily be made by even an inexperienced glassblower has been devised 
 
 o< 
 
 FIG. 135. 
 
 by Lord Rayleigh, and is shown in Fig. 135. The central part of the 
 tube is similar to that shown in Fig. 132. Since, however, the insertion 
 of the electrodes through the glass is a matter of some difficulty, the 
 glass showing a marked inclination to crack at the point where the 
 platinum wire is inserted when the tube is heated, the aluminium wires 
 are inserted through two long tubes CA and BD, the joints at c and D 
 being made air-tight with sealing-wax. The tubes JA, BD are each about 
 20 centimetres long, and have a bore of 1'5 millimetre. A little zinc or 
 cadmium is introduced, and the tube is then connected to the pump at 
 F, and when exhaustion is complete is sealed off at E. When in use the 
 central part AB of the tube is enclosed in a little metal box, in which a 
 hole is pierced so that the light may escape, and this box is heated over 
 a name. The tubes CA and BD project beyond the box, and are so long 
 
310 
 
 WAVE-LENGTH MEASUKEMENTS 
 
 [123 
 
 that the ends do not become sufficiently hot to soften the wax. After 
 the tubes have been in use for a few hours they will require re-exhaustion. 
 The wave-lengths of the chief lines in air at 15 and 760 mm. pres- 
 sure are : 
 
 Metal. 
 
 Wave-length in 10~ 8 cm. 
 
 Colour. 
 
 Cadmium . 
 
 6438-47 
 
 Red. 
 
 
 5085-82 
 
 Green. 
 
 
 4799-91 
 
 Blue. 
 
 Zinc .... 
 
 6362-34 
 
 Red. 
 
 
 4810-53 
 
 Blue. 
 
 
 4722-16 
 
 Blue. 
 
 
 4680-14 
 
 Blue. 
 
 123. Light Filters. When using such a source of light as a mercury 
 lamp, which gives light of several wave-lengths, with a spectrometer, no 
 inconvenience is caused, for the prism itself sorts out the rays of the 
 different colours. For small deviation this is also true when a grating is 
 being used. If, however, a grating of high dispersive power is used, or 
 if spectra of a high order are to be employed, then inconvenience will be 
 caused by the overlapping of the light of the wave-length being investi- 
 gated by light of another colour, and corresponding to a spectrum of a 
 
 FIG. 136. 
 
 different order. In the same way, when using the mercury light with 
 interferometers the interference fringes due to the different wave-lengths 
 would overlap and obscure each other. In such a case we have to use 
 some piece of auxiliary apparatus to sift out the light of the wave-length 
 we are using from the other light. A convenient arrangement for this 
 purpose is shown in Fig. 136. It consists of a mercury lamp H, which 
 is at the principal focus of the collimation lens A, a prism B, which need 
 not be of any very dispersive glass, and a lens c, at the principal focus of 
 
124] CALIBRATION OF A SPECTROSCOPE 311 
 
 which is placed a slit of adjustable width D. Finally, in many cases, 
 it is advisable to use a third lens E, with the slit at its principal focus. 
 With this arrangement a beam of parallel light falls on the prism, and, 
 if the lenses are so arranged that the deviation is a little greater than 
 minimum, any line of the spectrum formed in the plane of the slit by 
 the lens c can be brought on the slit by slightly rotating the prism. In 
 the cases where a beam of parallel monochromatic light is required, this 
 can be obtained by means of the lens E. The light of wave-lengths 
 other than that which is required is cut off by the slit. When used with 
 a grating spectroscope the slit D and lens E are unnecessary, since the 
 collimator of the instrument with its slit perform the functions of this 
 slit and lens. In such a case the spectrum formed by the prism and 
 lens c is projected on the slit of the instrument. 
 
 124. Calibration of a Spectroscope by the Use of Lines of known 
 Wave-lengths. Owing to the fact that the deviation produced by a 
 prism is not proportional to the wave-length, when using a prism 
 spectroscope to measure wave-lengths it is necessary to calibrate the 
 spectroscope, that is, to draw a curve giving the deviation in terms of 
 the wave-length. One way of obtaining such a curve is to measure the 
 deviation for a number of different lines of known wave-length fairly 
 uniformly distributed throughout the spectrum, and then to draw a 
 uniform curve through the points obtained. 
 
 To obtain such a curve, having adjusted the spectroscope in the 
 manner described in 1 1 7, set the prism to minimum deviation for some 
 known line. For this purpose the more refrangible of the two sodium 
 lines will be found convenient. Then keeping the prism fixed, determine 
 the deviation for a number of lines throughout the spectrum, using as 
 source of light one of those described in the previous section. It will 
 be necessary, if great refinement is required, to either always work at 
 the same temperature, or by observing at different temperatures to deter- 
 mine how the deviation for each of the standard lines varies with the 
 temperature. 
 
 Having obtained the readings for the deviations./^fot a curve having 
 the wave-lengths for ordinates and the corresponding deviations for 
 abscissae. When using the spectroscope to nJeasure the wave-length 
 of some unknown line, the prism must be so/f for minimum deviation 
 for sodium light, and the deviation corresponding to the unknown line 
 measured. Then from the curve the wave-length corresponding to this 
 deviation can be immediately read off. 
 
 The curve obtained by plotting wave-lengths as ordinates and devia- 
 tions as abscissae is approximately a rectangular hyperbola, and since the 
 curvature is very considerable, unless there are a large number of observa- 
 tions, it will be found difficult to draw an accurate curve. If, however, 
 in place of plotting wave-lengths we plot frequencies, or what comes 
 to the same thing, the number of waves in 1 centimetre, against the 
 deviations, the curve obtained is much less curved, and hence can be 
 more accurately drawn. A curve which is very nearly straight can be 
 
312 WAVE-LENGTH MEASUREMENTS [ 125 
 
 obtained by plotting I/A 2 against the deviations, and where great accuracy 
 is desired, the extra arithmetical labour is quite worth while. 
 
 Where the wave-length of a single line is desired, it is sometimes 
 convenient to avoid the trouble of making a whole series of observations 
 so as to plot the complete curve. In such a case two lines may be 
 observed, one on either side of the unknown line, and the wave-length of 
 this line obtained by arithmetical interpolation. 
 
 If the two known lines are very near the unknown line, we may 
 interpolate by proportional parts. In general, however, such a pro- 
 cedure would lead to a very erroneous result. Since, however, the 
 deviations are very nearly proportional to the inverse square of the wave- 
 lengths, we may use this property for interpolation without causing any 
 very great error. 
 
 Let Aj and A 2 be the wave-lengths of the two known lines, A the 
 wave-length of the unknown line, and n^ n 2 , and n the corresponding 
 deviations. Then 
 
 A 2 = \~ 2 ' 
 from which y-, and hence A can at once be calculated. 
 
 A. 
 
 125. Calibration of a Spectroscope by Interference Fringes. 1 A 
 
 method of calibrating a spectroscope, which is particularly convenient 
 when the spectrum is photographed, is to make use of interference 
 fringes. If a very thin film of air is enclosed between two optically 
 plane pieces of glass and placed immediately before the slit of a spectro- 
 meter, then on illuminating the slit with a slightly convergent beam of 
 white light the spectrum will be crossed by a number of dark bands. 
 These bands are due to the interference between the light which has 
 passed straight through the air film and that which, owing to internal 
 reflection, has traversed the air film three times. The number and 
 breadth of the bands depends on the thickness of the air film, being 
 more numerous the thicker the film. In order to render the bands quite 
 dark, it is necessary to partly silver the glass surfaces which enclose the 
 film so as to strengthen the portion of the light which undergoes re- 
 flection within the film. The silvering can be performed by the process 
 described in the Appendix, and must be carried on till about three- 
 quarters of the incident light is reflected at the silvered surface. The 
 following is the method recommended by Edser and Butler for preparing 
 
 i Edser and Butler, Phil. Mag. [5] (1898), vol. xlvi. p. 207. Proc. Physical 
 Society of London (1898), xvi. 207. 
 
125] CALIBRATION OF A SPECTROSCOPE 313 
 
 the air film : Having selected two pieces of plate glass which, when 
 tested by the method given in 142, show broad and straight inter- 
 ference fringes, silver them on one side. Place a little soft red wax 
 round the edges of the plates, and press them firmly together while look- 
 ing through them at a bright spot of light. A long train of images will 
 be seen, due to multiple reflections. Squeeze the glass till only a single 
 image is obtained, when on looking through the glasses at a sodium 
 flame interference fringes will be seen. Adjust the glasses till these 
 fringes are as broad as possible. When the adjustment is nearly com- 
 plete it will be necessary to look at the fringes from a little distance, 
 owing to the fact that with a parallel air film viewed normally the bands 
 are formed at an infinite distance in front of the film. 
 
 Having prepared an air film, illuminate the slit of the spectroscope 
 with a slightly convergent beam of light from an arc, and place the air 
 film as near the slit as possible. The spectrum will then be found to 
 consist of bright lines separated by almost dark intervals. 
 
 To take a photograph, half of the slit can be covered while the inter- 
 ference bands are being photographed, while, when the spectrum is being 
 photographed, this half of the slit can be exposed, that part previously 
 uncovered being now covered. In this way a photograph of the interfer- 
 ence bands will appear alongside the photograph of the spectrum. 
 
 If d is the thickness of the air film, the length of the path traversed 
 by the twice reflected part of the light will exceed that of the light which 
 has gone straight through by 2d, and there will be a bright band in 
 the spectrum wherever the wave-length goes exactly an integral number 
 of times into 2d ; that is, if 
 
 are the wave-lengths corresponding to the bright bands, then 
 
 where n is some integer. Suppose that the wave-lengths A and A TO , cor- 
 responding to two of the bands, are known, then n can be determined 
 from the equation 
 
 m\ m 
 
 where A m is the wave-length of the mfh band from that corresponding to 
 A.,,, A,, being the band nearer the red end of the spectrum. The wave- 
 length A,., corresponding to the rth band, counting from A in the same 
 way, is given by 
 
 n + r 
 
 Thus if we know the wave-lengths of any two bands sufficiently far 
 apart so as to be able to calculate the value of n, we can, by simply 
 
314 WAVE-LENGTH MEASUREMENTS [ 126 
 
 counting the number of bands from the first line up to any given line, 
 determine the wave-length corresponding to this line. 
 
 126. Measurement of Wave-length with the Diffraction Grating. 
 If a parallel beam of monochromatic light of wave-length A. falls 
 normally on a grating, either transparent or reflecting, having N lines to 
 the centimetre, then if 6 is the angle between the direct light and the 
 deviated light in the first spectrum on either side, we have (see Watson's 
 Physics, 377) 
 
 A = sin BjN. 
 
 In the second spectrum, in the same way 
 
 and so on for the spectra of higher orders. Hence if we know N and 
 measure 9, we can immediately calculate A. 
 
 When performing the experiment, having adjusted the telescope and 
 collimator for parallel rays, as described in 116, 117, the first thing is 
 to take the reading < corresponding to the direct light. We have then 
 to set up the grating on the spectrometer table so that it is normal 
 to the incident light. Place the grating on the table so that the ruled 
 surface is immediately over the centre of the circle and parallel to the 
 line joining two of the levelling screws (see p. 292), and adjust the 
 plane of the grating parallel to the axis of the circle by the method given 
 on p. 290. Next set the telescope so that the readings on the attached 
 verniers are < + 90, that is, so that the axis of the telescope is at right 
 angles to the direction of the rays of light proceeding from the collimator. 
 Turn the grating till the image, which is simply reflected from the surface 
 of the grating and not diffracted, coincides with the cross-wires of the 
 telescope, and read the verniers attached to the table. The plane of 
 the grating now makes an angle of 45 with the incident light. Add or 
 subtract, as the case may be, 45 to the reading for the table, and set it 
 to this new angle. The plane of the grating will now be perpendicular 
 to the incident light. 
 
 The final adjustment is to set the lines of the grating parallel to the 
 axis about which the telescope turns. To do this, having adjusted a wire 
 across the slit, illuminate the slit with monochromatic light, say, sodium 
 light, and adjiist the table levelling screw which lies in the plane of the 
 grating, till the intersection of the cross- wires coincides with the horizontal 
 wire across the slit in the case of all the diffracted images which can be 
 observed on both sides. When this adjustment is complete, the rulings 
 on the grating are parallel to the axis of the instrument. The slit should 
 then be made very narrow, and, having placed the telescope so as to 
 observe one of the diffracted images, the slit must be turned so that the 
 image is as sharp as possible. In this way the slit is made parallel to 
 the rulings on the grating. 
 
 Having arranged a source of light to give the lines required, deter- 
 mine the readings on the circle corresponding to the diffracted images 
 
126] MEASUREMENT OF WAVE-LENGTH 315 
 
 corresponding to the spectra of each order. The difference between the 
 readings corresponding to the diffracted images on the two sides corre- 
 ponding to any given order will be equal to twice the angle 6. 
 
 If the number of lines to the centimetre is not known, it can at once 
 be determined by making a series of measurements with light of known 
 wave-length. For this purpose sodium light will be found as convenient 
 as any. It is sometimes convenient, when using a reflection grating, to 
 place the grating in such a way that the incidence is not normal. If the 
 angle of incidence of the light on the grating is <, and is the angle 
 between the axis of the telescope and collimator for a line in the spectrum 
 of the nth order, it can at once be shown that 
 
 n\ 
 
 sin (0 - <) - sin < > 
 
CHAPTER XX 
 
 INTERFERENCE 
 
 127. Measurement of the Wave - length of Light with Fresnel's 
 
 Biprism. Fresnel's biprism is a method of obtaining interference fringes 
 between waves of light which, having started from the same point of the 
 source, traverse slightly different paths and then meet. The theory of 
 the biprism follows at once from a consideration of Fig. 137. If pis the 
 source of light, then after passing through the biprism AB the light which 
 has passed through the two halves of the prism will travel as if it came 
 
 B 
 
 FIG. 137. 
 
 from the two virtual images p' and P" respectively. If a screen is placed at 
 DE, then at the point o, which is on the straight line drawn through the 
 source of light and the centre of the biprism, the two trains of waves will 
 always be in the same phase, and will therefore strengthen each other. 
 At a point Q at a distance x from o there will be interference, owing to 
 the two trains of waves reaching Q in opposite phases, if P"Q exceeds P'Q 
 by an odd number of half wave-lengths, that is, if 
 
 Calling the distance PO between the source and the screen D, and the 
 distance P'P" between the images 2d, we have 
 
 and 
 Hence 
 
 316 
 
127] MEASUREMENT OF WAVE-LENGTH OF LIGHT 317 
 
 or since .r + d is always very small compared to D 
 
 In the same way 
 
 Hence 'P'Q 
 
 or if there is interference at Q 
 
 or 
 
 
 Now the distance between consecutive interference bands is obtained by 
 giving to n the values of two consecutive integers, and obtaining the 
 difference in x. Calling the distance between the bands y we get 
 
 y = \D,'2d, 
 x 2d 
 or n^' 
 
 Thus to obtain the wave-length we have to measure the distance between 
 the bands, the distance between the source of light and the screen, and 
 the distance between the virtual images formed by the biprism. To carry 
 out the measurement we shall require a narrow slit to form the source of 
 
 M 
 
 FIG. 138. 
 
 light, an adjustable upright to carry the biprism, and a micrometer eye- 
 piece to measure the distance between the bands. In addition, a convex 
 
318 INTERFERENCE [ 127 
 
 lens will be required to assist in measuring the distance between the 
 images. A form of optical bench suitable for this purpose is shown in 
 Fig. 138. It consists of a strong metal bed AB, along which can slide a 
 number of uprights which are used to carry the slit, biprism, &c. There 
 is a scale attached to this bed, and verniers attached to the uprights serve 
 to read the positions of these uprights. Two of the uprights, c and D, are 
 fitted with metal jaws, which serve to clamp the slit, biprism, &c., while 
 a third upright E carries a micrometer M. This micrometer consists of a 
 vertical cross-wire and a positive or Ramsden's eye-piece. The eye-piece 
 and the cross- wire can be moved in a direction at right angles to the 
 length of the bed by means of a micrometer screw. There is a fourth 
 upright F, which carries the convex lens L. The upright D, which carries 
 the biprism p, has a slide and screw by means of which it can be moved 
 in a direction at right angles to the bed. 
 
 When making a measurement, the following adjustments have to be 
 made : 
 
 1 . Focus the eye-piece on the cross-wire, and turn it till it is vertical. 
 Then place the lens between the slit and the eye-piece, and adjust till an 
 image of the slit is seen in the eye-piece. Turn the slit about a horizontal 
 axis till the slit is parallel to the cross-wire. To allow of this adjustment 
 being made, the jaws which carry the slit (as also those which carry the 
 biprism) can be rotated about a horizontal axis by means of a tangent 
 screw T. 
 
 2. Move the eye-piece up to the slit and adjust them to the same 
 vertical height, at the same time adjusting the micrometer slide perpen- 
 dicular to the length of the bed. 
 
 3. Place the biprism in the jaws of the upright, and adjust so that the 
 prism is at the same height as the slit, and the plane face is at right 
 angles to the length of the bed. 
 
 4. Having illuminated the slit, which should be narrow, with sodium 
 light, on looking through the eye-piece indistinct bands will probably be 
 seen. If no bands are visible, move the upright carrying the biprism at 
 right angles to the bed till the bands appear. Turn the biprism about a 
 horizontal axis by means of the tangent screw till the bands appear as 
 distinct as possible. When this occurs, the refracting edge of the biprism 
 will be parallel to the slit. 
 
 5. Draw the upright which carries the eye-piece away from the 
 biprism, and observe whether the bands appear to travel across the field 
 of view. If they do, adjust the positions of the biprism and the cross- 
 wire in directions at right angles to the bed till such a transverse motion 
 on moving back the eye-piece no longer takes place. When this adjust- 
 ment is complete, the line joining the slit to the refracting edge of the 
 biprism will be parallel to the scale along the bed. 
 
 The preliminary adjustments are now complete, but before proceeding 
 to measure the distance between the bands it will be convenient to deter- 
 mine the distance between the slit and the focal plane of the eye-piece, 
 i.e. the plane of the cross- wire. For this purpose remove the upright 
 
127] MEASUREMENT OF WAVE-LENGTH OF LIGHT 319 
 
 which carries the biprism, taking care not to disturb the adjustments of 
 this latter. Also remove the cap which carries the cross-wire, leaving 
 the eye-piece in place. Then hold one end of a metal rod about 20 cm. 
 long against the middle of the slit, and move the upright which 
 carries the eye-piece till the other end of the rod is in focus when seen 
 through the eye-piece. Measure the length a of the rod, and take the 
 readings on the scale corresponding to the uprights which carry the slit 
 and eye-piece. If the difference between these readings is b, then the 
 correction which has to be added to the distance as given by the readings 
 on the scale to obtain the true distance will be a b. 
 
 Having replaced the upright which carries the biprism and readjusted 
 the cross- wire, it will be necessary to adjust the width of the slit so that 
 the bands may be as distinct as possible. If the slit is too wide, only a 
 few bands of varying width will be seen ; these are diffraction bands, 
 and at present we are not concerned with them. On narrowing the slit 
 the diffraction bands will be less obvious, while the interference bands due 
 to the biprism will become more distinct. If, however, the slit is made 
 too narrow, the light will be so feeble that the bands will be difficult to 
 see. To obtain good measurements it is necessary to use a bright source 
 of light. It will be found a great advantage to mount a convex lens of 
 about 4 cm. diameter and 3 cm. focal length so as to form an image of 
 the flame on the slit. 
 
 Take the reading on the micrometer corresponding to every fifth or 
 tenth bright band right across the field, and by dividing the readings into 
 two groups in the method described on p. 100, calculate the mean 
 distance between consecutive bands. The distance between the bright 
 bands is the same as that between the dark bands, and since it 
 is easier to see the cross-wire when it is at the centre of a bright 
 band, it will be found best to measure the distance between the bright 
 bands. 
 
 To measure the distance between the images place the convex lens l 
 between the biprism and the eye-piece, taking care not to alter the relative 
 positions of the slit and biprism. The eye-piece having been removed to 
 a distance from the slit greater than four times the focal length of the 
 lens there will be found two positions for the lens, in each of which a 
 sharp pair of images of the slit are formed. The distance between the 
 images in either position must be measured with the micrometer. Let 
 these distances be c x and c 2 . Then the distance between the virtual 
 images formed by the biprism is given by 
 
 2d= ^/CjCg. 
 This follows at once, for if in the first position of the lens the distance 
 
 1 It will be found of advantage to cover the lens with a disc of paper having 
 a hole of about 5 mm. diameter pierced in the centre, so that only the central 
 part of the fens is used. In this way, and using monochromatic light, good images 
 will be obtained with a single piece lens. Care must be taken that the axis of 
 the lens is parallel to the bed. 
 
320 INTERFERENCE [ 128 
 
 between the lens and the slit is d^ and the distance between the lens and 
 the cross-wires is d. 2 , we have 
 
 In the second position, in the same way 
 
 2d = d 2 
 C 2 ~d l ' 
 
 2d c 9 
 Hence = <rS. 
 
 C-t JjCu 
 
 or 2d= ^/CjCg. 
 
 128. Measurement of the Wave-length of Light with Fresnel's 
 Biprism and a Spectrometer. In the absence of an optical bench, such 
 as that described in 127, we may employ a spectrometer in its place. 
 Having adjusted the collimator to give parallel light, mount the biprism 
 on the table with the refracting edge perpendicular, and its plane face in 
 the plane which passes through the vertical axis of the instrument and 
 one of the levelling screws of the table. The final adjustment of the edge 
 in the plane of the biprism can be left, and the plane face can be set 
 parallel to the axis of the spectrometer by any of the methods described 
 in 117. The biprism can also be set with its plane face perpendicular 
 to the axis of the collimator by the method employed in the case of the 
 diffraction grating ( 126). 
 
 Turn the telescope so that the vertical cross-wire is half-way between 
 the two images of the slit, and unscrew the object glass. On illuminating 
 the slit with monochromatic light, interference fringes will be seen in the 
 field of view of the eye-piece. Rotate the biprism in its own plane by 
 using the levelling screw of the table which is immediately under the 
 prism, till the fringes are the most distinct, i.e. till the refracting edge of 
 the prism is parallel to the slit. The angle subtended by the interval 
 between two adjacent fringes has now to be determined. This is best 
 obtained by taking the reading corresponding to every tenth fringe right 
 across the system of bands, and then obtaining the mean width of a band 
 in the manner described on p. 100. If the circle of the spectrometer is 
 not very finely divided, say, to read to 10", it will be better to attach a 
 small plane mirror to the telescope, and to read the rotation by means of 
 a scale and telescope, as described in 170. 
 
 Having replaced the object glass of the telescope, measure the angle a 
 subtended by the two images of the slit. The distance d between the 
 focal plane of the eye-piece and the axis of the spectrometer must also be 
 measured. 
 
 The distance y between consecutive fringes is then given by y = d6, 
 
128] 
 
 FRESNEL'S BIPRISM 
 
 321 
 
 where 6 is the angle subtended by consecutive bands measured in 
 circular measure. The wave-length of the light is obtained from 
 
 X = 2y sin a/2. 
 
 This formula can be obtained as follows: Let AB (Fig. 139) be 
 the biprism, and OP the direction of the incident light. Then the wave- 
 front, such as GH, before reaching the prism is at right angles to OP. 
 After passing through the prism let the wave-fronts be co and OD, which 
 
 B D 
 
 FIG. 139. 
 
 from symmetry are equally inclined to the line OP. At p the waves will 
 always be in the same phase, and if QE and QF are drawn from any point 
 Q normal to the wave-fronts, then the waves will reach Q in the same 
 phase if 
 
 But 
 and 
 Hence 
 
 FQ=OQsm QOF, 
 EQ=OQsin QOE. 
 FQ- EQ = OQ, (sin QOF- sin QOE) 
 
 5} 
 
 where a is the inclination of the normals to the two wave-fronts EO 
 and OD. Calling PQ a?, there will be a bright band at Q if 
 
 o a 
 
 2x sin -. 
 
 2 
 
322 INTERFERENCE [ 129 
 
 Hence if y is the distance between consecutive bands, 
 
 A = 2y sin ^ 
 
 The distance y between consecutive bands is given by - cosec 7 , and 
 
 where a is the inclination of the wave-fronts after leaving the prism, and 
 this angle depends on the angle of the biprism 8, and the refractive index 
 of the glass of which it is made. Thus to increase the distance between 
 the bands it is necessary to decrease a by using a biprism having a 
 refracting angle very nearly equal to 180. When used in the optical 
 bench with divergent light the distance between the bands could be' 
 increased by either decreasing the distance between the slit and the 
 biprism, or increasing the distance between the biprism and the eye-piece. 
 For this reason the optical bench is on the whole the more convenient 
 arrangement. By doing away with the collimator lens, so as to use 
 divergent light, and placing the slit fairly near the axis of the spectro- 
 meter, the width between the bands can be increased. The formula to 
 be used in this case can at once be obtained. 
 
 129. Measurement of the Wave-length of Light by means of 
 Diffraction Fringes. The optical bench described in 127 can be 
 used for determining the wave-length of light from measurements made 
 on diffraction fringes. The object, edge, wire or slit, as the case may be, 
 is mounted in place of the biprism, and the distance between the fringes 
 is measured with the micrometer. 
 
 1. Diffraction at a Straight Edge. A metal plate with a carefully 
 worked straight edge is mounted on the upright D (Fig. 138), the edge 
 being vertical, and in such a position that the line joining the slit to the 
 edge is parallel to the bench. The edge is then adjusted parallel to the 
 slit by turning the plate in its own plane till the diffraction bands are 
 as distinct as possible. 
 
 If x is the distance of a band from the geometrical shadow of the 
 edge, then it can be shown 1 that if a is the distance of the edge from the 
 slit, and b the distance of the focal plane of the eye-piece from the edge, 
 then for a bright band 
 
 and for a dark band 
 
 where n is an integer. 
 
 i For an elementary discussion of diffraction bands the student is recom- 
 mended to read Chapter V. of Schuster's Theory of Optics (Arnold). 
 
129] DIFFRACTION BANDS 323 
 
 Taking the case of the bright bands, and giving n in succession the 
 values 0, 1, 2, &c., we get the following values : 
 
 No. of 2 Distance between 
 
 Band. Adjacent Bands. 
 
 -1) or -732 
 
 &c. &c. 
 
 Measure by means of the micrometer the distances between tho 
 different bands, and by means of the above expressions deduce values 
 for the wave-length of the light, and show that within the limits of the 
 accuracy of the experiment the above expressions are correct. 
 
 To obtain from the measurements the best value for the wave-length, 
 measure the distance between the second band and the farthest visible, 
 say, the mih ; if this distance is d, we have 
 
 from which A. can be calculated. 
 
 t2. Diffraction by a Wire. Mount a wire having a diameter of about 
 t mm. in a vertical position, and with its length parallel to the slit, and 
 t a distance of about 6 cm. 
 Three systems of fringes will be seen, two series outside the 
 eometrical shadow, in which the spacing of the bands is unequal, and 
 rfiich are similar to those produced by a single edge considered above, 
 and a third set within the geometrical shadow, which are approximately 
 equally spaced. 1 
 
 If b is the distance between the wire and the focal plane of the eye- 
 piece, and c is the width of the wire, the distance between consecutive 
 bands of the internal set is 
 
 Measure the distance between as many bands of the internal set as 
 possible, and deduce the wave-length of light. 
 
 > ' With a very narrow obstacle these internal bands spread out beyond the 
 limits of the geometrical shadow. 
 
324 INTERFERENCE [ 130 
 
 The inverse operation may, however, often be more convenient ; 
 that is, knowing the wave-length of the light, we may use the bands to 
 measure the diameter of the wire. This is the principle on which 
 Young's Eriometer depends. 
 
 3. Diffraction by a Slit. Using one slit as a source, mount a second 
 slit at a distance of about 60 cm., and examine the fringes with the eye- 
 piece at different distances. When the slit is fairly broad, and the 
 eye-piece is near the slit, a system of unequally spaced bands will be 
 seen within the geometrical image. There is also an external system, so 
 long as the slit is not too broad, in which the bands are equally spaced. 
 If the slit is made very narrow the internal system will disappear, while 
 the external system will be mdre widely separated. 
 
 If c is the width of the slit, and b the distance between the slit and 
 the focal plane of the eye-piece, the distance between consecutive bands is 
 
 Since it is very difficult to accurately measure the width of the slit, 
 these bands can hardly be used to measure the wave-lengths. They do, 
 however, form a means of measuring the width of the slit, which may on 
 occasion be of service. 
 
 A convenient way of observing and measuring the diffraction bands 
 due to a slit is to use a spectrometer. Focus the telescope and collimator 
 for parallel light, and mount an adjustable slit on the table with its 
 length parallel to the slit of the collimator. Using sodium light, 
 measure the angle subtended by five or six of the diffraction bands, and 
 dividing this angle by the number of intervals obtain the angle between 
 two consecutive bright bands or, what is the same thing, between two 
 consecutive dark bands. It can at once be shown that this angle 
 is equal to the angle 6 between the central bright band and the first 
 dark band. Calculate from your result and the wave-length of sodium 
 light the width of the slit, and compare the number obtained with the 
 width as determined by direct measurement with a micrometer micro- 
 scope. Repeat the measurements for different widths of slit. 
 
 130. On the Localisation of Interference Fringes. Since inter- 
 ference fringes are often employed for the purpose of measuring small 
 lengths, it will be worth while considering the position such fringes appear 
 to occupy in some detail ; that is, supposing we have a set of fringes 
 produced by some arrangement, and it is desired to examine the fringes 
 in a telescope, on what point are we to focus the telescope 1 From the 
 point of view of the measurement of length the fringes which are 
 of importance are those which are produced by a thin film, which in 
 practice is always an air film between two glass surfaces. 
 
 The following method of investigating the problem is due to Edser : T 
 
 1 Light for Students, by E. Edser (Macmillan). 
 
130] LOCALISATION OF INTERFERENCE FRINGES 325 
 
 Let AB and CD (Fig. 140) be the surfaces of the air film, these surfaces 
 being inclined to one another at an angle 0. Then if IM is a ray 
 incident on the first surface at an angle a, it will be partly reflected 
 
 FIG. 140. 
 
 along MR and partly refracted along ML, the angle NML being /?. This 
 refracted ray is reflected at L, refracted at M 2 , and finally proceeds along 
 M 2 R. From M 2 draw M 2 F perpendicular to MR, M 2 E perpendicular to 
 ML, and M 2 HG perpendicular to CD, and produce ML to meet this line 
 
326 INTERFERENCE [ 130 
 
 at G. Calling the angle of incidence at L /?', and the refractive index of 
 the material above AB u we have 
 
 MF= J/J/ 2 sin a 
 ME = j/F 2 sin 
 
 and sin /2 = ^ sin a. 
 
 Hence ~ 
 
 and thus ME and MF are equal optical paths. Thus the difference in 
 optical length of the paths MLM 2 R and MR is EL + LM^. But the 
 triangles M 2 LH, GLH being equal, LG is equal LM 2 , and hence the difference 
 in optical length is EG. But EG = M 2 G cos M 2 GE, and the angle M 2 GE is 
 equal to /?', since M 2 G and N'L are each perpendicular to CD. Thus If we 
 call the thickness M 2 H of the film T, we have that the difference in optical 
 length of the two paths which meet at R is 
 
 IT cos ft'. 
 
 Taking account of the fact that there is a 1 change of phase l of A/2 at 
 the reflection at M I} we shall get an interference band at R if 
 
 (1) 
 
 and we may consider that this relation fixes the value of ($' for any 
 given band, and the values of ft' for the different bands are obtained 
 by giving to n the values 1, 2, 3, 4, &c. 
 
 If we now suppose /3' given, we may obtain an expression for the 
 position of R. Let < be the angle MRM 2 , and D the distance M 9 E, then 
 we have from the triangle MM 2 R 
 
 sin (7T/2 - a) cos a 
 and from the triangle MM 2 G 
 
 MM. 2 
 
 M 2 sin (7T/2 - P) _cos j ^ > 
 ~ siiL/^' ~~amfS' 
 
 T^ iif~^n n m cos a sin 13' 
 
 Hence D = M^R = 27' J: . 
 
 sin <p cos p 
 
 1 See Watson's Physics, 378. 
 
131] MICHELSON'S INTERFEROMETER 327 
 
 But in practice the angles a, /?, ^', and < are all small, hence approxi- 
 mately 
 
 But * = N'MR - N'"M 2 R, 
 
 and sin N'"M. 2 R = /* sin JV" Jf 2 L - j* sin (' - 0), 
 
 or since the angles are small 
 
 and hence </> = a - 
 
 or since a = /3//x = (/?' + 0)/p 
 
 Hence D = 1 - .......... (2) 
 
 u 
 
 Remembering that /?' is given by equation (1), equation (2) shows that 
 if T is very small, i.e. the air film is very thin, then D is very small also, 
 unless is exactly zero ; that is, the interference bands apparently 
 coincide with the film, and to see them an observer's eye, or a telescope, 
 should be focused on the film. If 6 = 0, and T is not infinitely small, 
 then D = oo, and the bands appear to be at an infinite distance, and the 
 observing telescope should be focused for parallel rays. If is small, 
 while T is considerable, the value of D will vary rapidly with /?'. Now, 
 if K is the object-glass of the observing telescope, and R' is the conjugate 
 focus of B,, any phase difference which exists at R. will be reproduced at 
 R', and hence if there is interference at R there will be interference at R', 
 and R' will be a point on one of the interference bands seen in the 
 telescope. If the neighbouring band is seen at R", then by drawing a 
 similar figure, in which the difference in phase is now A greater than that 
 at K, we should find that the rays which interfere are incident on AB and 
 CD at a different angle, i.e. ft' would be different. But if /3' is different, 
 D will be different, and hence all the interference bands cannot be in focus 
 at the same time. 
 
 Again, suppose that while 6 remains constant T is varied, as is the 
 case in the Michelson interferometer ( 131), then D will also vary, and, 
 therefore, the focus of the observing telescope will have to be continually 
 altered unless 6 is zero ; that is, the film of air is parallel sided. 
 
 131. Michelson's Interferometer. The principle on which this 
 instrument depends is as follows : A ray of light is incident in the 
 direction 10 (Fig. 141) on a parallel sided plate of glass G 1 . Part of the 
 light passes straight through this plate and through a similar plate G 2 , 
 and is reflected back at a plane mirror M r On again reaching the 
 
328 
 
 INTERFERENCE 
 
 [ 131 
 
 surface of the plate G I part of this light will be reflected in the direction 
 OR r Part of the incident beam, instead of passing through the plate G p 
 will be reflected at its second surface, and will reach the mirror M 2 . 
 Here it will be reflected, and part passing through the plate G l will travel 
 in the direction R 2 . Thus the incident ray of light has been split up 
 into two rays which, having traversed different paths, are finally brought 
 
 FIG. 141. 
 
 back to traverse the same path. In the figure the reflected rays from the 
 mirrors M X and M 2 are shown slightly displaced so as to avoid confusion. 
 In reality the rays RJ and R 3 coincide. The object of the plate G.> is that 
 by its introduction the two rays traverse exactly the same thickness of 
 glass, otherwise the light reflected from the mirror M x would only traverse 
 the glass plate once, while that reflected in the mirror M 2 would do so 
 three times. 
 
 The two systems of waves, which have traversed the two paths, are 
 in such a condition that they can interfere, and thus a system of inter- 
 ference bands will be seen on looking in the direction RjO'. If one of 
 the mirrors, say, M 15 can be moved parallel to itself the bands will appear 
 to move, and it is evident that a movement of the mirror through a 
 distance of A/2 will cause the displacement of each band into the -position 
 previously occupied by the adjacent band. 
 
 A form of Michelson interferometer is shown in Fig. 142. One of 
 the mirrors, M p is carried on a carriage A, which moves along carefully 
 worked ways planed on the bed. This carriage is moved by means of a 
 fine micrometer screw. A divided head B is attached to the screw, and 
 a tangent screw c allows of the micrometer screw being rotated slowly ; 
 at the same time a divided head attached to the tangent screw serves for 
 the determination of the fractions of a division of the scale on B. The 
 second mirror, M 2 , is capable of slight rotation about a vertical and hori- 
 zontal axis by means of the screws F and E. The points of these screws 
 do not press directly against the mirror or its frame, but against two small 
 springs which are attached to the frame. The frame itself is carried by a 
 small horizontal metal pillar, which is screwed to an upright attached to 
 
131] MIOHELSON'S INTERFEROMETER 329 
 
 the bed. The pressure of the screws against the springs cause this pillar 
 to bend slightly, and thus the plane of the mirror is moved. This arange- 
 ment allows of a very delicate adjustment of the plane of the mirror, 
 for, owing to the elasticity of the springs, the frame does not move 
 through anything like the distance through which the points of the 
 screws are moved. The glass plate G 2 is also carried on a pillar attached 
 
 FIG. 142. 
 
 to the bed, and a screw D, with a divided head acting on a spring 
 attached to the support, serves to give this glass plate a small rotation 
 about a vertical axis. 
 
 Adjustment of the Interferometer. By means of a scale or pair of 
 dividers adjust the movable mirror M x so that it is at the same distance 
 from G I as is the fixed mirror M.,. 
 
 Place either a bright sodium flame s or a white light at the principal 
 focus of the lens L, and fix a fine needle point between the lens and G r 
 Four images of the needle point will in general be seen, two formed by 
 reflection in each of the mirrors M X and M 2 . This is owing to the fact 
 that light is reflected from both sides of the glass plate GJ, the way in 
 which the four images are formed being indicated in Fig. 143. Inter- 
 ference can take place when either of the reflected beams (1) and (2) 
 coincide with either (3) or (4). When (2) coincides with (3) the inter- 
 fering beams have traversed equal thicknesses of glass. When, however, 
 (1) coincides with (4), one beam (4) has traversed five times the thick- 
 
330 
 
 INTERFERENCE 
 
 [131 
 
 ness of glass traversed by the other (1). Hence it is (2) and (3) that 
 should be brought together, and the screws at the back of the mirror M 9 
 must be adjusted till these two images are exactly superposed. It is 
 easy by screening either of the mirrors with a card to see which pair 
 belong to the adjustable mirror. When this adjustment has been care- 
 fully made, monochromatic light must be used at s, when the fringes 
 ought to be visible. If they are not visible, and a slight alteration in 
 the adjustment of MJ does not bring them into view, the setting with the 
 needle point must be repeated. 
 
 We may look upon Michelson's instrument as being an arrangement 
 in which interference takes place in the film of air inclosed between the 
 mirror M 1 and the image of the mirror M 2 formed by reflection in the 
 
 FIG. 143. 
 
 58 plate G r Now, when the mirror M 2 is adjusted so that its image 
 is parallel to M I} the interference fringes will be circles, with their 
 centres at the foot of the perpendicular drawn from the observer's 
 eye to the mirror M r The air film having parallel sides, it follows, 
 from what has been said in the preceding section, that the fringes are 
 at infinity, and hence an observing telescope should be adjusted for 
 parallel rays. 
 
 If the image of M 2 is inclined at a finite angle to M I? then the 
 fringes are approximately arcs of circles, having their centres on a line 
 perpendicular to the line of intersection of the image and mirror. The 
 greater the value of #, the narrower will be the bands. In this case the 
 apparent position of the bands will depend on and the thickness of 
 the air film, and will vary as this thickness varies. Hence as M X is 
 
131] MICHELSON'S INTERFEROMETER 331 
 
 moved, the focus of the observing telescope will need to be continually 
 altered if the bands are to appear as sharp as possible. For this reason 
 it is advisable whenever possible to use the circular bands corresponding 
 to = 0. By altering the adjusting screws at the back of the mirror M 2 , 
 any desired form of bands can be obtained. 
 
 Since with white light there are only a few fringes formed, these will 
 only appear when the optical length of the two paths are exactly equal. 
 The most convenient way of finding the position of the movable mirror 
 for which the paths are equal is to adjust M 2 so that the fringes are arcs 
 of very large circles. It will then be found, as the carriage is moved 
 backwards and forwards, that the direction of the curvature of the fringes 
 changes when the carriage passes through a certain position. Note the 
 reading on the tangent screw and on the scale B when the fringes are 
 just perceptibly curved, first in one direction and then in the other. 
 Having placed a source of white light at s, move the carriage very 
 slowly from one of the positions just found to the other, when the inter- 
 ference bands will be seen to sweep across the field. The movement 
 must be very slow, otherwise the bands will pass across so rapidly as 
 not to be noticed. For this reason it is not a bad plan to leave the 
 sodium flame in place, putting the source of white light just behind it. 
 By watching the motion of the sodium fringes, the speed at which the 
 tangent screw is turned can then be regulated, and, what is also of 
 importance, the eye can be kept focused on the bands. When making 
 measurements with the interferometer, using straight fringes, it will be 
 necessary to observe the fringes with a low-power telescope fitted with a 
 cross-wire, so as to provide some fixed point with reference to which the 
 movement of the fringes can be noted. 
 
 (a) To determine the pitch of the screw of the interferometer in 
 terms of the wave-length of sodium light. Adjust the mirror to give 
 circular fringes, and move the carriage towards G l till the fringes begin 
 to get indistinct, and read the position of the scale B and the tangent 
 screw. Then slowly move the carriage away from G 15 counting the 
 number of fringes which vanish at the centre till the screw has made 
 about one whole turn, and again read the scale and the tangent screw. 
 If n fringes have vanished, the distance through which the carriage has 
 moved is n\/2. Hence if the screw has made x turns, as given by the 
 reading at the commencement and end, the pitch of the screw is n/2x 
 times the wave-length of sodium light. 
 
 (b) A stage micrometer divided, say, into tenths and hundredths of 
 a millimetre can be calibrated by use of the interferometer by attaching 
 the micrometer to the carriage and clamping a microscope to the bed so 
 that it can be focused on the scale. Then by counting the number of 
 fringes which vanish while the scale is moved, so that the cross-wire of 
 the microscope coincides in turn with each of the divisions of the scale, 
 the distance between successive divisions in terms of the wave-length of 
 sodium light is at once obtained. 
 
 By using a mercury tube or lamp and the arrangement shown in 
 
 
332 INTERFERENCE [ 132 
 
 Fig. 136, the wave-lengths of the mercury lines can be used in place of 
 the sodium. 
 
 (c) The interferometer may be used to examine whether a given 
 source of light produces strictly monochromatic light, or whether, say, 
 light of two nearly identical wave-lengths, as also to determine in such 
 a case the difference in the wave-lengths of the two lines. Either the 
 yellow lines of mercury or the sodium lines may be employed for this 
 purpose. Suppose that we adjust the instrument to give approximately 
 straight bands, and start with the two paths of the interferometer of 
 equal length, that is, with the central band produced with white light on 
 the cross-wire, and that we use a source of light the spectrum of which 
 consists of two neighbouring lines A and B, having wave-lengths \ a 
 and A 6 . Each of these kinds of light will produce a set of interfer- 
 ence bands, and these two sets will coincide in the position in which 
 the paths are equal. If, now, we increase one of the paths by 
 moving back the movable mirror, since the wave-length of the two lights 
 are different, so that the bands due to the smaller wave-length are nearer 
 together than those due to the other, it follows that the agreement between 
 the two sets of fringes will gradually be lost. Thus, after a certain time, 
 a bright band for one system will coincide with a dark band for the 
 other, and hence, if the two lines are equally bright, one set of fringes 
 will exactly destroy the other, and no fringes will be seen. On further 
 increasing the difference in the two paths, the fringes will gradually 
 reappear and get more and more distinct, till finally there is exactly one 
 whole wave-length more of the system A than of the system B in the 
 distance by which the one path has been increased. Hence if the 
 movable mirror has been moved through a distance d between two 
 maxima of distinctness of the bands the increase of path is 2cZ, and the 
 number of waves of the system A in this distance is 2d/X a , and the 
 number of waves in the system B is 2d/\b. 
 
 Hence as there is one more wave in the system A than in the system 
 B we have 
 
 2^_2d_ 
 ~~~ 
 
 A* ~2d 
 
 Examine by this method the yellow light given by a mercury tube, and 
 determine the distance through which the carriage has to be moved 
 between consecutive maxima of the bands, and taking the value of 
 X a from the table on p. 309, calculate X. b . Repeat the experiment, 
 using sodium light. 
 
 132. Newton's Rings. If a convex lens, having a radius of curva- 
 ture of R, is placed in contact with a plane surface of glass, and the point 
 of contact examined, there will be seen a system of rings both when the 
 transmitted light and the reflected light is examined. If white light is 
 
132] NEWTON'S KINGS 333 
 
 used, only a few rings, and these coloured, will be visible. With mono- 
 chromatic light, however, a large number of rings are visible. These 
 rings are due to the interference produced in the air film between the 
 two glass surfaces. The theory of these rings is given in books on light 
 (see Watson's Physics, 379), and it is shown that in the case of the 
 transmitted light, which falls normally on the plane surface, the radii r 
 of the successive dark rings are given by 
 
 where A' is the wave-length of the light in the medium between the 
 surfaces, and n is any whole number. If A is the wave-length in air, and 
 the medium between the surfaces has a refractive index /x, then 
 
 Hence for the dark rings we have 
 
 Similarly the radii of the bright rings, are given by 
 
 In general the medium between the surfaces is air, so that /x = 1 . Hence 
 knowing R we can determine the wave-length of the light by measuring 
 the diameters of the rings, or conversely, knowing A we can obtain R. 
 In the case of reflected light, the angle of incidence being ft, we have 
 
 f j i / 9 n\R 
 
 tor dark rings r* = - ^ : 
 
 cos/3 
 
 for bright rings r' n 2 = < 2 * + - 
 
 2 cos p 
 
 where the symbols have the same meaning as before. 
 
 If in place of using a plane surface a concave surface of radius R' 
 greater than R is employed, then the radii of the dark rings are given by 
 
 Transmitted light r* = A { 
 
 2 { 
 
 Reflected light /* - - wX | 
 
 cos f3 I 
 
 The formulae given above are not in general applicable, since it is impos- 
 sible to be certain (1) that the lens and plate are in contact at the centre ; 
 and (2) if they are in contact, that there is no deformation of the surfaces 
 
334 INTERFERENCE [ 133 
 
 at the point of contact due to the pressure employed to keep them in con- 
 tact. This difficulty is got over by measuring the diameters of two rings, 
 say, the #th and the (x + n)fh. In the case of the dark rings by trans- 
 mitted light we then have, if d Q and d n are the diameters, 
 
 Hence 
 
 in which expression it is no longer necessary to know the number of the 
 rings being used, but only the number of rings between the two of which 
 the diameters have been measured. 
 
 Similarly for reflected light we have 
 
 d n 2 - d$ n\R 
 
 . r^ ^ 
 
 4 cos p 
 
 and for the case of two lenses in contact 
 Transmitted 
 
 nX. = - n 7 I D~TV 
 Reflected 
 
 133. Measurement of the Wave-length of Light by Newton's 
 Rings. Either the transmitted or reflected system of rings can be used 
 to deduce the wave-length of the light. Since, however, it is difficult to 
 measure the angle of incidence /? if the reflected system is used, an 
 arrangement allowing normal incidence to be employed is advisable. The 
 advantage in using the reflected system is that the dark rings are much 
 darker than is the case in the transmitted system, for in this latter, owing 
 to the light which has suffered two reflections being very much more 
 feeble than that which has passed straight through the film, the inter- 
 ference is only partial, and the rings are by no means black. 
 
 An arrangement which will allow of the reflected system being ex- 
 amined with normal incidence is shown in Fig. 144. It consists of a small 
 wooden box A open on two sides and the top. A piece of plate glass B, 
 selected by the method described in 142, is cemented into a groove cut 
 in one of the sides so as to be at an angle of 45 with the horizontal. 
 
134] LIMIT OF RESOLUTION OF A TELESCOPE 335 
 
 The inside of the box is blacked, and the plane plate of glass c and 
 the lens D are placed on the bottom. A Bunsen flame s, coloured 
 with common salt, and a lens L serve to supply the light, which 
 is reflected downwards 
 by the glass plate B. 
 A travelling microscope 
 M serves to measure the 
 diameters of the rings. 
 The lens D may be 
 
 M 
 
 
 L 
 
 FIG. 144. 
 
 a piano - convex lens 
 having a focal length 
 of about 100 centi- 
 metres. 
 
 When performing 
 an experiment, first 
 measure the radius of 
 curvature of the lens 
 D, either with a sphero- 
 meter ( 15), an optical 
 lever ( 23), or by one 
 of the optical methods given in 142. Then very carefully clean the 
 surface of the lens and of the piece of plate glass c, and having placed 
 the lens in contact with the glass, fix them together with some soft wax 
 so that the centre of the rings seen is near the centre of the lens. 
 
 Since in this case the thickness of the air film used is small, and the 
 angle between the two surfaces enclosing the film is not zero, or even very 
 small, the rings appear as if located within the film ( 130). Hence if 
 the microscope M is focused on a speck of dust or mark on the upper 
 surface of the plate c the rings will be seen, and then the focus can be 
 adjusted till they appear most distinct. It will generally be found 
 advisable to place a screen to cut off stray light, and the lens L may with 
 advantage be placed over a hole in this screen. 
 
 Measure the diameters of one of the broad rings near the centre which 
 can easily be identified. Then count out either ten or twenty rings and 
 measure the diameter of this ring, and obtain the value of the wave-length 
 of the light by the formula given in the preceding section. 
 
 134. Diffraction through a Slit and Limit of Eesolution of a 
 Telescope. Suppose that a slit of breadth AB (Fig. 145) equal to a is 
 placed before the object glass L of a telescope, and that a parallel beam of 
 monochromatic light falls on the slit, the axis of the telescope OP being 
 parallel to the direction of the incident light. Then if a screen is placed 
 at the focal plane QPQ' of the lens, since all parts of the wave-front AB 
 will reach p in the same phase, there will be a bright band formed passing 
 through P. On either side of this bright band, which represents the 
 image of the slit of the collimator used to produce the parallel beam of 
 light, there will be a series of alternate dark and bright bands. Suppose 
 that Q is the centre of the first dark band. Then if QOM is drawn 
 
336 
 
 INTERFERENCE 
 
 134 
 
 through the centre of the lens, and AB' is drawn through A perpendicular 
 to MQ, if AB' were a wave-front all parts of this wave would reach Q in the 
 same phase. Hence if instead of being a wave-front so that all parts of 
 AB' are in the same phase the different parts of AB' are in different phases, 
 then the disturbances which reach Q from AB 7 will be in different phases, 
 but the relation between the phases at Q will be the same as that which 
 exists between the phases of the different parts of AB'. If, now, BB' is 
 
 FIG. 145. 
 
 equal to A, where A, is the wave-length of the light, the disturbances 
 which start from A and B' will be in the same phase, but the disturbances 
 which leave the intermediate points will differ from the phases of these 
 two points by amounts varying from to 2?r. Hence by dividing AB' 
 into two parts there will always be an element in one half which sends a 
 disturbance to Q, which is exactly opposite in phase to the disturbance 
 sent to Q by the corresponding element of the other half, so that these 
 disturbances will interfere at Q, and so a dark band will pass through this 
 point. Now the triangles ABB' and OPQ are similar. Hence 
 
 PQ 
 AB = 'OP' 
 
 Thus calling the distance QP between the first dark band and the central 
 bright band #, and the focal length OP of the lens F, we have 
 
 If we call the angle QOP 0, this relation may be written 
 
 *=<*e (i) 
 
 If in place of having a single beam of parallel light we have two beams 
 
134] LIMIT OF RESOLUTION OF A TELESCOPE 337 
 
 inclined at a small angle a, then each will produce its own central bright 
 image bordered on either side by diffraction bands. Now if is the angle 
 subtended at the centre of the lens by the interval between the centre of 
 the central bright band and that of the first dark band, it is evident that 
 if a is less than #, the central images due to the two sets of incident waves 
 will overlap, and it will be impossible to distinguish the two images. If 
 a is equal to 6, then the central images will just touch ; while if a is greater 
 than 0, the central bright images will be separated by one or more dark 
 bands, and thus it will be possible to distinguish the two images, and 
 hence detect the fact that the incident light consists of two distinct 
 systems of waves. In other words, when a is greater than 6 we are able 
 to detect the fact that we are observing a double source. If a is less than 
 6, however, we shall be unable to detect the fact that the source is double. 
 The condition 
 
 *-^ (2) 
 
 is called the limit of resolution, and it will be observed that it depends 
 alone on the width of the slit placed before the telescope objective and the 
 wave-length of the light. 
 
 The resolving power of a telescope with apertures of different sizes can 
 be obtained by setting up a piece of fine wire gauze containing about 
 twenty meshes to the centimetre before a source of light, which to avoid com- 
 plications, owing to imperfect achromatism of the lenses, may consist of a 
 sodium name, and having fitted a cap over the objective with an aperture 
 of the desired size, determining the distance at which the individual wires 
 of the gauze just cease to be distinguishable. If a rectangular aperture 
 is used, the gauze must be set up with one set of wires parallel to the 
 longer edge of the aperture, and it is the distance at which these wires 
 cease to be distinguishable which has to be obtained. As the telescope is 
 moved away from the gauze, the focus must be continually adjusted so 
 that the object is seen as distinctly as possible. 
 
 If (I is the distance between adjacent wires of the gauze, and D is 
 the distance between the objective of the telescope and the gauze when 
 resolution ceases, the angle a subtended by two adjacent sources is given 
 by a = djD. When the aperture used in front of the objective is a slit of 
 width a, it has been shown above that resolution will cease for a value of 
 a given by a = A/a. 
 
 Determine experimentally the value of a for slits of different widths, 
 and compare the results with the values calculated, that is, with the 
 angle subtended by the wave-length of the light at a distance equal 
 to the width of the slit. Slits 2 mm. and 4 mm. wide will be found 
 suitable. 
 
 Perform the same experiment, using circular apertures having dia- 
 meters of 5 mm. and 10 mm. 
 
 With a point source, such as a star, the radius of the first dark ring in 
 
 Y 
 
338 
 
 INTERFERENCE 
 
 [135 
 
 the case of a lens of diameter D and focal length F has been calculated 
 by Airy to be 
 
 I-22FX 
 D 
 
 Hence two point sources, which subtend an angle at the object glass 
 will be distinguishable as two sources when 
 
 e> 
 
 1-22A 
 D ' 
 
 It is of importance when making measurements on the resolving power 
 of telescopes to make certain that all the light which passes through the 
 object glass enters the pupil of the eye. Since the width of the pupil is 
 about 3 mm., if D cm. is the diameter of the object glass, the magnifying 
 power M of the telescope must be greater than D/*3 (see 145). If 
 the magnifying power is less than this, then the iris acts as a diaphragm, 
 and the peripheral portions of the object glass are not used, and thus the 
 effective aperture is reduced. (See Watson's Physics, 357.) 
 
 135. Resolving Power of a Spectroscope. (a) Prism Spectroscope. 
 Let ABC (Fig. 146) represent a section of a prism by a principal plane, 
 
 B 
 
 FIG. 146. 
 
 and FAG and OPQR the limiting rays of a rectangular pencil of parallel rays 
 traversing the prism. If the refracting angle is B, and the angles of 
 incidence and refraction at P are a and /3, the corresponding angles at Q 
 being a and ft', we have the following three equations : 
 
 sin a 
 sin P 
 
 sin a' 
 
135] KESOLYING POWER OF A SPECTROSCOPE 339 
 
 Suppose, now, keeping a, constant, we change the wave-length of the 
 incident light from A to A + <^A, as a result ft ft, a, and p will all vary. 
 Hence let us calculate the change da' in the angle between the emergent 
 beam and the normal. To do this we differentiate the equations (1) above, 
 regarding a and 6 as constants. Thus we obtain 
 
 = p cos ft . dft + sin /3 . dp 
 cos of . do! = p cos ft . dft' + sin ft' . dp 
 
 , 7 / /sin 8 cos ft' + cos 8 sin ft' 
 cos a', da' = ^ ^ 
 
 By eliminating d/3 and d/3' between these three equations, we get 
 
 /sin 8 cos ft' + cos 8 sin ft' \ 
 ( - ^ ^ y 
 
 \ cos/3 / ' 
 
 _ sin (/? + ft)d/x _ sin 6 . dp 
 ' cos ft cos /? 
 
 7 / sin $ 7 /0 \ 
 
 Hence da' = - , - -dp ......... (2) 
 
 cos a cos ft 
 
 When the prism is at minimum deviation a = a', ft = fi', and = 2ft. 
 Hence (2) reduces to 
 
 7 , sin 2ft 2 sin ft 7 /QX 
 
 da'= r - r dp= -- "rf/* ..... (3) 
 
 cos a cos ft cos a 
 
 Now if PD is drawn perpendicular to the incident light, and the width 
 of the incident or emergent beam is called a, so that PD = , we have 
 c'os a = a/PA. Further, if AE is drawn perpendicular to PQ, and PQ is 
 called ^, we have sin ft = t{2pA. Hence (3) reduces to 
 
 da'^-dfJL ......... (4) 
 
 a 
 
 which gives an expression for the separation, after passing through the 
 prism, between the two beams corresponding to wave-lengths A, and A + dX. 
 Now we have seen in 134 that if two beams of parallel rays of wave- 
 length A, and inclined at an angle d<$>, the section of the beams being 
 rectangular and of breadth a, are received in the object glass of a tele- 
 scope, the images formed will be separated if 
 
 d+= K ..... ..... (5) 
 
 Thus the slit of the collimator being of course adjusted to a very narrow 
 width so that (5) applies, we shall be able to distinguish the images 
 
340 INTERFERENCE [ 135 
 
 corresponding to the wave-lengths A and A-fc/A, i.e. the line will appear 
 double, if da' is greater than d(f>, that is, the limit is reached when 
 
 da' = d<}>,or*dp= ........ (6) 
 
 Here dp is the change in refractive index of the prism for a change 
 of wave-length dX. It is often convenient to replace dp by dX. in the 
 expression for the limit of resolution. To do this we require to know 
 how the refractive index changes with wave-length. In general it will 
 be sufficiently accurate to take Cauchy's formula 
 
 Differentiating 
 
 27? 
 
 d P = ~^ 
 
 Substituting in (6) 
 
 A 2tB 
 
 The quantity A/dfA is called the resolving power of the spectroscope, and 
 it depends on the wave-length of the light, the difference t in the length 
 of the path of the extreme rays in the prism, and a constant B depend- 
 ing on the kind of glass of which the prism is made. 
 
 The dispersion D of a prism is the ratio of the change in deviation 
 to the change in wave-length, or since the change in deviation is equal 
 to da 
 
 or substituting the value of da from (4) 
 
 Hence the dispersion varies inversely as the width of the beam. If the 
 prism is not at minimum deviation, then the above equation (8) still 
 holds if a is the width of the beam after its passage through the prism. 
 Hence if a prism is placed so that the deviation is not a minimum, then 
 in one position the dispersion will be greater, and in the other less, than 
 at minimum deviation, since although t will remain practically the same, 
 a will vary, as can easily be seen by drawing a figure. There will, 
 however, be very little change in resolving power, for this is equal to 
 
 and t does not change much. Using a prism at greater than 
 
 CtA 
 
 minimum deviation may therefore give greater angular separation between 
 
136] RESOLVING POWER OF A GRATING 341 
 
 the centres of two lines, but it does not give greater definition, the lines 
 themselves being broadened. 
 
 (b) Resolving Power of a Grating. If there are N lines per centi- 
 metre in the grating, and we observe in the rath order spectrum, the 
 light being incident normally, we have 
 
 sin = Nm\. 
 
 If, now, the wave-length of the light changes from X to X + dX, and the 
 angle 6 changes to 6 + dd, we obtain the connection between dB and dX 
 by differentiating, N and m being constant. Thus 
 
 cos 
 
 If I) is the width of the grating, the width a of the diffracted beam 
 of light is given by a b cos 0. Hence if the object glass of the tele- 
 scope is wider than a, the smallest angle d(f> between two beams of 
 light which shall produce images which are visibly separated is given by 
 
 7A X X 
 
 OO a* - = n * 
 
 a o cos u 
 Hence the limit of resolution is reached when dO = dcfr, or 
 
 that is, the resolving power of the grating is given by 
 
 ........ (11) 
 
 But IN is the total number of lines in the grating, and ra is the order of 
 the spectrum. Hence the resolving power is directly proportional to the 
 total number of lines in the grating, and to the order of the spectrum. 
 The dispersion D is defined, as in the case of the prism, by 
 
 
 Hence from (9) 
 
 - 
 
 D--"" (12) 
 
 cos 6 
 
 that is, the dispersion is directly proportional to the order . of the 
 spectrum, and directly proportional to the number of lines per centimetre. 
 
 Hence while the resolving power depends on the total number of 
 lines, and not on their closeness, the dispersion depends on the closeness 
 of the lines, and not on the total number. 
 
 136. Measurement of the Resolving Power of a Prism Spectro- 
 scope. Measure the refractive index of the prism for light of two or 
 
342 INTERFERENCE [ 137 
 
 more known wave-lengths, and then calculate the value of the constant 
 B in Cauchy's formula 
 
 from the observations taken in pairs. 
 
 Having adjusted the prism to minimum deviation for sodium light, 
 adjust two cards so as to cut down the aperture of the object glass of the 
 telescope till the two sodium lines just cease to be resolved. Make 
 certain that the beam of light which passes through the prism entirely 
 fills the portion of the object glass left uncovered, and then measure the 
 width of the rectangular aperture left between the cards. This width 
 gives the quantity a (see previous section), and t can be calculated by the. 
 formula 
 
 
 2a sin ^ 
 
 t = j 
 
 COS tt 
 
 where 6 is the angle of the. prism. Then calculate the resolving power 
 
 from equation (7), and compare your results with the value of - for 
 
 A 
 
 the two sodium lines. 
 
 Repeat the experiment, using the two yellow mercury lines. 
 
 Also calculate the full resolving power of the instrument, using the 
 full aperture of the prism. Two cases may occur. (1) The width of the 
 beam may be limited by the prism. In this case, if e is the width of the 
 face of the prism, then 
 
 (2) If the beam of light which passes through the prism entirely fills 
 the object glass of the telescope, then it can be shown l that 
 
 t= 
 
 cos a 
 
 where d is the diameter of the object glass. 
 
 137. Measurement of the Resolving Power of a Grating. Having 
 mounted and adjusted the grating as described in 126, bring the 
 sodium lines in the first spectrum into the field, and by means of two 
 cards reduce the width of the part of the grating being used till the 
 sodium lines just cease to be resolved. Then calculate the resolving 
 power by equation (10) of 135. 
 
 Repeat the experiment, using the second order spectrum, and compare 
 your results with the value of XjdX. for the sodium lines. 
 
 Calculate the resolving power of the whole grating, and find for what 
 value of t the prism used in the last section would have the same 
 resolving power. 
 
 1 Verdet, Lemons d'optique physique, vol. i. p. 301. 
 
138] 
 
 RAYLEIGH'S REFRACTOMETER 
 
 343 
 
 138. Rayleigh's Refractometer. Suppose that before the object 
 L (Fig. 147) of a telescope we place a screen pierced by two 
 parallel slits, each of width a, separated by a distance b, and that a 
 beam of parallel monochromatic light falls on the slits, the direction of 
 the incident light being parallel to the axis of the telescope. If QPQ' is 
 a screen placed at the focal plane of the lens L, the two systems of waves 
 which pass through the slits BC and DE will both reach P in the same 
 phase, and hence a bright band will be formed at this point. There 
 
 FIG. 147. 
 
 will also be produced diffraction bands by each of the slits, just as in the 
 case of the single slit considered in 134. In addition, however, there 
 will be a series of interference bands due to the mutual action of the 
 disturbances which have traversed the two slits. We now proceed to 
 determine the conditions under which these interference bands are 
 produced. 
 
 Let us consider the illumination at a point Q in the focal plane at a 
 distance x from the central bright band at P. Draw the straight line 
 QOM, and from a point A, such that AB is equal to the distance b between 
 the slits, draw AE' at right angles to QM. Then since AE is equal to 
 2(a + 6), we have that EE' = 2cc' and DD' = 2BB', and if we divide each of 
 the two slits into an equal number of elements, this relation holds for 
 
 each pair of elements. Hence if EE' = A, cc' = ~ , and the disturbance at 
 the points E' and c' will be in opposite phases, and each element of E'D' 
 
344 INTEREFERENCE [ 138 
 
 he corresponding e 
 
 . . . or(2?i+l)A, 
 
 will be in the opposite phase to the corresponding element of C'B'. The 
 same will occur when 
 
 while if 
 
 EE' = 2X, 4X . . . or 2nX, 
 
 the phases of corresponding elements of E'D' and C'B' will agree. 
 
 Hence, since any phase difference which exists along B'C', D'E' will 
 still exist when the disturbances reach Q, we see that there will be a 
 minimum of illumination at Q if 
 
 and a maximum of illumination at Q if 
 
 =2 
 
 pa x 
 
 Now -m = w = ~F =e > 
 
 where F is the focal length of the lens, and x is so small that 
 = tan 6. 
 
 Hence for a minimum of illumination at Q 
 _ 
 
 and for a maximum of illumination at Q 
 
 F2nX 
 
 x = 
 
 a 2nX 
 or 0= : 
 
 Thus the distance between consecutive bands, whether bright or dark, is 
 given by 
 
 XF 
 
 x = 
 
 or 
 
 a + V 
 X 
 
 The distance between the bands being inversely proportional to the 
 distance between the centres of the slits, and the wave-length of light 
 being very small, it is evident that unless the slits are very near together 
 the bands will be excessively fine. 
 
 Lord Rayleigh l has employed the bands produced by two slits to 
 
 1 Collected Works, vol. iv. p. 364. This arrangement had previously been 
 used by Arago, (Euvres Completes, x. p. 321. 
 
138] EAYLEIGH'S KEFRACTOMETER 345 
 
 compare the refractive index of two gases. The form of apparatus he 
 used is shown in Fig. 148. A collimator c, with a very fine slit, s, is 
 placed before a bright source of white light L. The slit of the collimator 
 may be made by making a fine scratch with a sharp knife on a piece of 
 silvered glass, so as to remove the silver along a fine line. The gases to 
 be compared are contained in two tubes A and B, which are of the same 
 length, and closed by two pieces of selected plate glass. A single piece 
 of glass is used at either end to close the two tubes, and this glass pro- 
 jects slightly above the tubes, as shown at F. The two slits are attached 
 to a cap which fits on the telescope T, and are at such a distance apart 
 that the light which enters through the lower part of one slit traverses 
 the tube A, while that which enters through the lower part of the other 
 traverses the tube B. 1 The light which passes over the tops of the tubes 
 
 enters the upper parts of the slits and gives a series of bands, the 
 position of which does not depend on the refractive index of the gases 
 within the tubes, and thus serves as a reference mark. 
 
 Since the bands produced are very fine, the magnification produced 
 by an ordinary eye-piece is not sufficient, and hence Lord Rayleigh used 
 as eye-piece a small cylindrical lens. This lens consisted of a cylinder 
 of glass having a diameter of 4 mm. fixed to a cork which replaced the 
 ordinary eye-piece of the telescope. The axis of the cylinder is adjusted 
 to be parallel to the slit s and the slits D, the final adjustment being per- 
 formed by turning the eye-piece till the bands are most distinct. One of 
 the tubes A and B is filled with the gas to be tested and the other with 
 air. The air tube is connected to a manometer, and an arrangement by 
 
 1 The tubes A and B can have a diameter of about 1 centimetre, the diameter 
 of the object glass of the telescope and collimator being about 3 centi- 
 metres. The slits D will then be 1 centimetre apart, and may have a width of 
 from 2 to 3 millimetres. The width of the field on which the bands are seen is 
 inversely proportional to the width of the individual slits. Thus if very narrow 
 slits are used a considerable field covered with bands may be seen, but the light 
 will be so feeble as to render measurement difficult. These slits can be made 
 by removing the silver from a piece of glass. Although Lord Rayleigh says that 
 he used cheap telescopes, the author has found that it is only with fairly good 
 lenses that satisfactory bands can be obtained. 
 
346 
 
 INTERFERENCE 
 
 [138 
 
 means of which the pressure of the air can be adjusted. Such an arrange- 
 ment is shown in Fig. 149, the tube A being connected to the tube (a), 
 Fig. 148. The pressure is adjusted by raising or lowering the mercury 
 
 reservoir B. This is done 
 by means of a string which 
 passes round a drum c, 
 the drum being rotated by 
 a tangent screw D. The 
 pressure is read off on the 
 manometer M. 
 
 Two series of bands 
 will be seen, one set formed 
 by light which has passed 
 over the top of the tubes, 
 and the other by light 
 which has passed through 
 the tubes. The upper 
 series is used as a refer- 
 ence mark, and by adjust- 
 ing the pressure in the air 
 tube the lower series of 
 bands are brought to form 
 continuations of the upper 
 bands. This can be per- 
 formed since, owing to a 
 white source of light being 
 used, the central band of 
 either system is the only 
 one which is free from 
 colour, and hence can be 
 identified. 
 
 Suppose the gas which 
 is being experimented upon 
 is at atmospheric pressure, 
 
 and when the bands are in their central position the pressure in the air 
 tube exceeds the external pressure by an amount p, the height of the 
 barometer being B, and the temperature t. Then since the refractive 
 index of the gas is equal to that of air at a pressure B +p, if ja ft is the 
 refractive index of air at a pressure B and temperature t , we have that the 
 refractive index p of the gas at this pressure and temperature is given by 
 
 FIG. 149. 
 
 For in the case of a gas we have 
 
 constant, 
 
 where p is the density. Hence if // is the density of air at a pressure 
 
138] RAYLEIGH'S REFRACTOMETER 347 
 
 B +p, and p the density at a pressure J5, and //, the refractive index at a 
 pressure B+p, we have 
 
 /*- 1 _p _B+p 
 
 fr-i ? m 'B 
 
 In the above we have supposed that the two tubes are of exactly 
 the same length, and when both are filled with air at the same pressure 
 the central band in the lower part of the field coincides with the central 
 band in the upper part of the field. This assumption may be avoided 
 by having a manometer and pressure-changing device connected with the 
 tube containing the gas which is being tested, and making two com- 
 parisons, one at a pressure nearly atmospheric, and the other at a con- 
 siderably different pressure. 
 
 Calling p - I the refractivity, let the refractivity of air at a pressure 
 p a be r m the density being p M and the corresponding quantities at a 
 pressure p' a be r'- a and p' a . Also let the corresponding quantities for the 
 gas be indicated by p g1 r g , p ffl and p' r' g , p' g . Then since the central 
 band is brought back to the same position when the refractivities are 
 r n r n and r' r' n we have 
 
 
 r.-r f -V.-f f (1) 
 
 r' a ~P' a a>[ r'g~p'g 
 
 Hence substituting in (1) 
 
 n t - f q = r a - ^ r , 
 ^ />(, 
 
 r a ~ Pa - P'g ' Pa ~Pg ~Pg Pa 
 
 Since r g and r a are measured at different pressures, to get the ratio 
 )f the refractivities we must reduce one to the pressure of the other. If 
 '* is the refractivity of the gas at a pressure p M we have 
 
 and hence J_o = Pa_Pa 
 
 r a P 3 -Pg 
 
 that is, the ratio of the refractivity of the gas to that of air is as the 
 change in pressure in the air tube is to the change in pressure in the 
 gas tube. 
 
CHAPTER XXI 
 
 LENSES AND MIREORS 
 
 139. Measurement of the Focal Length of a Thin Lens. It is 
 
 shown in books on optics (Watson's Physics, 352) that the focal 
 length/ of a lens is connected with the distances u and v of the conjugate 
 foci by the relation 1 
 
 111. 
 
 f v u 
 
 Hence by measuring the distances from the lens to any pair of conjugate 
 foci we can calculate the focal length of the lens. Thus if we place an 
 object on the axis of a lens at a measured distance from the centre of the 
 lens, and then determine the position of the image of the object, we have 
 
 
 i 
 
 Lr 
 
 FIG. 150. 
 
 the requisite data for obtaining the focal length. The chief practical 
 difficulty in carrying out this experiment is to determine the exact posi- 
 tion of the image. The usual method employed is to use as the object 
 a brightly illuminated and sharply denned body, and to move a screen 
 about till the image formed on the screen appears most sharply defined. 
 In practice, however, it is often extremely difficult to tell the exact posi- 
 tion of the screen for which the definition is the best, the image appearing 
 equally sharp over a considerable range of the position of the screen. A 
 very much better method is to make use of the parallax method to deter- 
 
 1 We suppose all distances measured from the lens, and that the positive 
 direction is opposite to that in which the light is travelling. 
 
 348 
 
139] FOCAL LENGTH OF A THIN LENS 349 
 
 mine the position of the image. The parallax method has the further 
 great advantage that it does not necessitate the use of a dark room. 
 
 As object, a scale consisting of fine lines on clear glass may be used. 
 Such a scale ought to have division lines about half a millimetre apart, 
 and can easily be prepared by photographing a suitable scale divided, 
 say, in tenths of an inch, by means of a camera. The image screen 
 consists of a plate of plate glass on which a fine vertical line has been 
 ruled. The object scale, the lens, and the glass to locate the image, are 
 mounted on the uprights of an optical bench, such as that shown in 
 Fig. 150. 1 These uprights can be moved along a wooden base, to which 
 is attached a scale AB 2 metres long. The base-plates which carry the 
 uprights have a short cylinder D attached at one side, which slides in a 
 groove in the board. The other end of the base rests on the scale, and 
 is fitted with a vernier v, by means of which the position of the 
 carriages can be read. A metal rod c with blunt points, and exactly 
 20 cm. long, is supported in a horizontal position by a similar carriage, 
 and serves for measuring the distance between the different pieces of 
 apparatus. The scale used as an object is illuminated by placing behind 
 it a flame, or even a brightly illuminated sheet of white paper. If a 
 very accurate measurement is being made, since in non-achromatised 
 lenses the focal length depends on the wave-length of the light, a source 
 of monochromatic light, such as a sodium flame, should be employed. 
 
 (a) Goncex Lens. First Method. In the case of a convergent lens 
 the eye is placed behind the glass plate having the single vertical line, 
 when an image of the scale will be seen on suitably moving the glass and 
 the eye. The surface of the glass which carries the vertical line, which 
 must be that surface which is turned towards the lens, is then adjusted 
 by the method of parallax so that it lies in the same plane as the image ; 
 that is, the position of the glass must be adjusted till on moving the 
 eye from side to side the line on the glass does not appear to move with 
 relation to the division lines of the image. 
 
 If the image is so small that it is difficult, with the unaided eye, to 
 see the division lines, then a short focus lens or positive eye-piece 2 may 
 be employed to examine the image and fiducial line. 
 
 When measuring a lens which is uncorrected for spherical aberration, 
 it will be found an advantage to stop down the lens so that the light 
 can only pass through the central portion, as in this way the image will 
 be much improved in definition. 
 
 Having fixed the position of the image, the distance between the scale 
 and the lens and between the lens and the image must be measured. 
 For this purpose the readings corresponding to the three carriages are 
 taken. Suppose that the reading for the carriage which carries the 
 scale is , that for the one \\hich carries the lens b, and for the glass 
 
 1 The lens, c., are held against the uprights by two small spring clips. 
 Each upright is pierced with a hole about 3 cm. in diameter, the centres of 
 all the holes being at the same height. 
 
 2 The thermometer reading glass shown in Fig. 67 is suitable for this purpose. 
 
350 LENSES AND MIRRORS [ 139 
 
 plate c. Next place one end of the rod c against the scale, and bring the 
 upright which carries the lens up till the point of the rod rests against 
 the surface of the lens, and let the reading for this carriage be b r . Next 
 bring the surface of the lens and the glass plate on which the image was 
 formed against the ends of the rod, and take the readings for the posi- 
 tions of the carriages b" and c', say. Finally, measure the thickness of 
 the lens, and let it be t. Now, when the difference between the readings 
 on the scale for the object and the lens was V - a, the true distance was 
 20 cm., i.e. the length of the rod c. 
 Hence 20 - (&' - a) + 1/2 
 
 will be the correction to be added to the difference between the readings 
 for the two carriages to give in any case the true difference between the 
 object and the middle of the lens. In the same way the quantity 
 20 - (c' - b"} + 1/2 has to be added to the difference of the readings for the 
 lens carriage and the image carriage to give the true distance between 
 the middle of the lens and the image. Applying these corrections to the 
 readings, obtain the values of v and u, and hence calculate the value of the 
 focal length of the lens. Repeat the measurements for different positions 
 of the lens, and take the mean for the true value of the focal length. 
 
 When employing this method it is advisable to choose the distance 
 between the object and image such that there is no very great difference 
 between the distances of the image and object from the lens. If one of 
 these distances is very much less than the other, then errors in measuring 
 the smaller distance will cause a larger proportional error in the focal 
 length deduced than would be the case if the two lengths measured were 
 nearly equal. 
 
 Second Method. If the screen on which the image is formed is 
 placed at a distance greater than four times the focal length of the lens 
 from the object, then there will be two positions of the lens in which 
 a sharp image will be formed. Let u and u be the two distances of 
 the lens from the object when a sharp image is formed on the screen, and 
 v and v the corresponding distances from the lens to the screen, then 
 
 1_1^1 
 
 / v u 
 
 - = !_ l . 
 
 f~v u'' 
 
 But from symmetry it is evident that, neglecting for the moment the signs, 
 
 u = v' and v - u' . 
 
 Also, if 1) is the distance between the object and image screen, and d is 
 the distance between the two positions of the lens 
 
 D+d 
 
 and -17 = 
 
139] FOCAL LENGTH OF A THIN LENS 351 
 
 1 1 1 
 
 Hence 
 
 / D-d + D+d 
 
 2 2 
 
 Thus by measuring the distance through which the lens has been moved, 
 which can be directly obtained by noting the difference in the readings 
 for the vernier attached to the support which carries the lens, and the 
 distance from the object to the screen, we can calculate the focal length 
 of the lens. As it is easier to measure the distance through which the 
 lens has been moved than to measure distances up to the lens, there are 
 distinct advantages in using this method. 
 
 Third Method. In the case of a very long focus lens, both of the 
 preceding methods would involve the use of an optical bench having a 
 length at least equal to four times the focal length, and hence might not 
 be available. In such a case we may make use of the fact that if a 
 luminous object is placed at the principal focus of a convex lens, and the 
 rays of light after passing through the lens are reflected back through 
 the lens by means of a plane mirror, then they will form an image of the 
 object at the principal focus, that is, alongside the object. 
 
 To apply this method the lens is mounted on the optical bench, and a 
 plane mirror is mounted on a second stand and placed immediately 
 behind the lens. A convenient form of object to use will be a needle 
 point mounted in one of the carriages, so that the point is on a level 
 with the centre of the lens. If the eye is placed on the side of the 
 needle remote from the lens an inverted image of the needle will be seen, 
 and by moving the carriage which supports the needle backwards and 
 forwards till there is no parallax between the needle point and its image, 
 the point can be adjusted to occupy the principal focus of the lens. 
 When making this adjustment it is advisable to so adjust the plane 
 mirror that the point of the image seems just opposite the point of the 
 needle, when any parallax will easily be detected. The distance between 
 the needle and the lens must then be measured by means of the measuring 
 rod, and this distance is the focal length of the lens. 
 
 (b) Concave Lens. The most generally convenient method of measuring 
 the focal length of- a concave lens is to combine it with a convex lens 
 of shorter focal length, so that the combination acts as a convex lens 
 capable of giving a real image, and determining by one of the methods 
 considered above the focal length F of the combination, and also the 
 focal length /' of the convex lens alone. The focal length / of the 
 concave lens is then giveji by 
 
 _ _ 
 T ~ F f 
 
 Another way of performing this experiment, which does not necessitate 
 
352 
 
 LENSES AND MIRRORS 
 
 [140 
 
 the concave and convex lenses being placed in contact is to arrange the 
 convex lens on the optical bench so as to give a real image at the point 
 o (Fig. 151), then to insert the concave lens at B, and determine the 
 position of the image I. Having obtained the~reading for the measuring 
 rod when in contact with the screen at i and the lens B, this lens is 
 removed, and without disturbing the position of the convex lens A, the 
 position of the image formed at o is determined. Then by obtaining the 
 
 FIG. 151. 
 
 reading for the measuring rod when in contact with the screen at o, 
 the distances BO and BI are at once deduced. Since o and i are con- 
 jugate foci for the concave lens, the focal length can be calculated by 
 means of the formula 
 
 140. Measurement of the Radius of Curvature of a Concave 
 Surface. The concave surface of which the radius of curvature has to be 
 measured is mounted on one of the carriages of the optical bench, while 
 a needle is mounted on another carriage, the point of the needle being 
 at the same level as the centre of the reflecting surface. On looking 
 towards the surface from beyond the needle an inverted image of the 
 latter will be seen. The needle is then moved till the needle and its 
 image are in the same plane, as tested by the parallax method. The 
 ease with which this adjustment can be made will be very much increased 
 if the position of the needle is so adjusted that the image appears to just 
 touch the point of the needle. A short focus lens or eye-piece may be 
 used to assist in seeing when the image is in the same plane as the 
 object, that is, when the needle point is at tfte centre of curvature of 
 the reflecting surface. The distance between the needle point and the 
 centre of the reflecting surface must then be measured by the measuring 
 rod, and this distance is the radius of curvature of the surface. 
 
 When measuring, by this method, the radius of curvature of the 
 
140] CURVATURE OF A CONCAVE SURFACE 353 
 
 surface of a lens, there will often be seen two images, one formed by 
 reflection at the front surface, and one at the back. It is generally easy 
 to distinguish which is which from the form of the two surfaces. If, 
 however, there is any ambiguity, by covering the back surface with a 
 piece of blotting paper, moistened with glycerine, or even water, the 
 intensity of the image formed by the back surface will be so much 
 reduced as to render it easily recognisable. 
 
 In place of the parallax method of determining when the object and 
 image are in the same plane, the following is sometimes more convenient, 
 especially when dealing with large mirrors, or when it is desired to 
 examine the accuracy with which the surface has been polished to a true 
 sphere : A small metal plate AB (Fig. 152) has a small circular hole 
 
 (a) 
 
 FIG. 152. 
 
 pierced at B, and a right-angled prism p or piece of mirror is placed so 
 as to reflect the light from a flame L through this hole. The edge next 
 B of the metal plate is ground to a smooth and sharp edge. The metal 
 plate with the prism is mounted on a stand as shown, and is placed on 
 the axis of the mirror M, so that the hole is a little to the right-hand 
 side of the axis. In this way an image of the hole will be formed alongside 
 the edge of the metal plate c. An eye placed at E will see the u-lwle 
 surface of the mirror illuminated. If, now, the metal plate is gradually 
 moved to the left by turning the screw D, the image at c will approach 
 the edge, and, if the image is in the plane of the edge, as the light is 
 intercepted by the edge, the mirror surface will appear to darken 
 uniformly all over. If, however, the edge is nearer to, or farther away 
 from, the mirror than the image, then the darkening of the surface of the 
 
354 LENSES AND MIRRORS [ 141 
 
 mirror will appear to sweep gradually across the mirror. If the edge 
 is nearer to the mirror than the image, then the darkening will appear 
 to sweep across the surface of the mirror in the same direction as that 
 in which the edge is being moved, while if the edge is farther away from 
 the mirror than the image the contrary will be the case, namely, the 
 darkening will appear to sweep across in the opposite direction to that in 
 which the edge is moved. The distance of the plate from the mirror is 
 therefore adjusted, by means of the micrometer screw M, till the mirror 
 seems to darken uniformly as the light is intercepted by the edge of the 
 plate. When this adjustment is complete, the edge of the plate and the 
 hole are both at the centre of curvature of the mirror. 
 
 If the edge of the metal plate, having been adjusted to lie at the 
 centre of curvature of the mirror, is very slowly moved so as to intercept 
 the light, then if the surface of the mirror is a true sphere, the surface 
 will appear of a uniform tint throughout. If, however, the surface is 
 not truly spherical, then the want of truth will be apparent just as the 
 greater part of the surface appears to become dark, for the irregularities 
 will appear bright. This method of testing the curvature of a spherical 
 surface, particularly in the case of a mirror of large radius of curvature, 
 is an excessively delicate one, and merely warming a portion of the mirror 
 by touching it with the tip of the finger will, owing to the expansion 
 caused by the heat communicated to the mirror, cause the figure to alter 
 by an amount which is at once apparent, the spot which has been touched 
 appearing bright and in marked relief. 
 
 In the case of a parabolic mirror, the radii of curvature of the various 
 zones can be measured by this method, cardboard screens being used so 
 that only the zone being tested is exposed. 
 
 141. Measurement of the Radius of Curvature of a Convex 
 Surface. In the case of a convex surface the image of a real object 
 is virtual, and hence the methods described in the last section are not 
 
 FIG. 153. 
 
 applicable. If, however, a virtual object is used, then a real image may 
 be formed, and can be received upon a screen. 
 
 Place the stand carrying the metal plate with the prism shown in 
 Fig. 152 on the optical bench at A (Fig. 153), and on another upright place 
 a convex lens L. The convex mirror M is mounted on a third upright, and 
 
141] CURVATURE OF A CONVEX SURFACE 355 
 
 the positions of L and M are adjusted so that the image is formed at A' in 
 the plane of the metal edge. Then, being careful not to disturb A or L, take 
 the reading on the optical bench scale corresponding to the mirror M. 
 Next remove M, and determine the position of the image o formed by the 
 lens L ; the distance between o and M will be the radius of curvature of 
 the mirror ; for when the mirror was in place, the image at A' was along- 
 side the object, that is, the rays of light which struck the mirror were 
 reflected straight back, that is, they were incident normally on the 
 mirror. But such normally incident rays would, if produced, pass 
 through the centre of curvature of the mirror, and when so produced, 
 or when the mirror is removed, they intersect at o, and hence o is the 
 centre of curvature of the mirror. 
 
 The radius of curvature of the surface of a thin convex lens can be 
 measured by adjusting the position of the lens so that the light reflected 
 from the back surface of the lens forms an image alongside the object. 
 If /is the focal length of the lens, and D the distance between the lens 
 and the object and image, the radius of curvature is given by 
 
 r - 
 
 We may obtain this expression as follows : Since the object and 
 image are alongside, it follows that the rays of light are incident normally 
 on the back surface of the lens at which they are reflected. 
 
 Hence the rays which are refracted at the first surface of the lens 
 proceed in the glass as if they would intersect (either actually or when 
 produced backwards) at the centre of curvature of the back surface. 
 Hence rays proceeding from an object at a distance D from the front 
 surface proceed in the glass after refraction at this surface as if they 
 came from a point at a distance r ; i.e. D and r are conjugate as far as 
 refraction in a single spherical surface are concerned. Hence if r is the 
 radius of the surface, and /A the refractive index, we have, by the known 
 formula- for refraction at a single spherical surface, 
 
 r D 
 
 which may be written 
 
 11 
 
 But if / is the focal length of the lens 
 
356 LENSES AND MIRKORS [ 142 
 
 1 1 1 
 
 ~= 
 
 
 142. To Test the Flatness of the Faces of a Piece of Glass. 
 It is often necessary to test whether the surface of a plate of glass or a 
 mirror is flat, and the following method will be found a convenient and 
 expeditious one : Prepare an artificial star by piercing a hole about half 
 a millimetre in diameter in a screen of metal or cardboard, and placing a 
 gas flame behind the screen. This star should be fixed up at as great a 
 distance as possible from the table on which observations are to be made. 
 Lay the reflecting surface flat on the table, and fix a telescope having a 
 fairly large objective in a retort stand with its object glass quite near to 
 the reflecting surface. Turn the telescope to view the artificial star 
 directly, and focus carefully. Then turn the telescope so that an image 
 is seen after reflection in the surface. If the image now seen in the 
 telescope is as sharply defined as when seen by direct light, then the 
 reflecting surface is plane. If the definition is impaired, this may be due 
 to two causes. In the first place, the surface, although it is not plane, 
 may be a regular surface of revolution, such as a sphere, in which case 
 the original definition can be regained by altering the focus of the 
 telescope. If, however, no alteration in the focus of the telescope 
 suffices to restore the definition, this indicates that the surface is 
 irregular. In this case we may test whether parts of the surface are 
 plane by covering the surface by a sheet of paper with a hole about a 
 centimetre in diameter pierced in the middle. The paper screen is 
 moved about so as to expose each part of the surface in turn, and by 
 watching the image in the telescope it will at once be seen whether any 
 part of the surface is plane. 
 
 In the case of a plate of glass the reflection at the second surface 
 may cause some confusion as to which is the image corresponding to the 
 upper surface. By placing a piece of wet blotting paper against the 
 back surface, the intensity of the light reflected at this surface will be so 
 much reduced that there will be no difficulty in distinguishing the two 
 images. 
 
 If a surface which is known to be optically plane is available, the 
 planeness of the surface of a glass plate can most quickly be tested by 
 means of interference fringes formed in the air film enclosed between the 
 two surfaces when they are placed in contact. Having cleaned the 
 standard surface and that to be tested, place them in contact, and examine 
 the interference fringes seen when a sodium flame is observed reflected 
 from the adjacent surfaces. If the surfaces are both plane, the inter- 
 ference fringes will be straight and parallel throughout. If, however, 
 either of the surfaces is at all irregular, then the fringes will be curved. 
 By making a drawing of the fringes we obtain a contour map of the 
 
142] 
 
 TEST OF PLANE GLASS PLATE 
 
 357 
 
 surface of the plate, the distance between adjacent contour planes being 
 equal to half a wave-length of sodium light. 
 
 By examining in this way a number of pieces of thick plate glass it 
 is generally possible in the course of a few minutes to pick out a few 
 pieces which have at least one surface as nearly plane as expensively 
 worked surfaces, and these selected pieces of plate glass will be found 
 suitable for use in many optical experiments, such as in Michelson's 
 interferometer. 
 
 It is sometimes necessary to measure the angle which the two plane 
 faces of a piece of glass make with one another. This can conveniently 
 be done by mounting the glass on the table of a spectrometer, the 
 telescope and collimator having been adjusted for parallel light. If the 
 two sides are exactly parallel, then only a single image will be seen when 
 the light from the collimator is reflected by the surface of the glass into 
 the telescope. If, however, the faces are not parallel, then two images 
 
 FIG. 154. 
 
 will be seen. To measure the angle, turn the glass round in its own 
 plane till the two images of the slit are as widely separated as possible, 
 and then by adjusting the telescope so that the intersection of the cross- 
 wires coincides first with one image and then with the other, and reading 
 the spectrometer circle, measure the angle 8 between the two images. If 
 < is the angle of incidence, and //, is the index of refraction of the glass, 
 the angle a included between the two faces is given by 
 
 8 cos <f> 
 2p cos t/'' 
 
 where 8 and a are supposed so small that we may take the angle as 
 equal to the sine and \[/ is the angle, such that 
 
 sn 
 
 sn 
 
 This formula can at once be obtained, for if AB, CD (Fig. 154) are 
 the traces of the faces enclosing the glass plate, and PQU and PQRST are 
 
358 
 
 LENSES AND MIRRORS 
 
 [143 
 
 two of the rays forming the two images, we have, using the notation 
 given in the figure for the angles 
 
 sin <' = /A sin (^ + 2a) 
 
 From (1) 
 
 sin </>' = jtx(sin ^ cos 2a + cos i// sin 2u), 
 
 2 CT - sin ^ ~ P sin ^ 
 
 ft COS \// 
 
 if a is so small that we may take sin 2a = 2a and cos 2a = 1. 
 From (2) in the same way 
 
 (1) 
 (2) 
 
 or 
 
 or since 8 is small 
 
 also 
 
 sin </>' = sin (8 + </>) 
 
 = sin 8 cos <^>+cos 
 
 sin <' = 8 cos < -f sin </>, 
 /A sin ^ = sin </>, 
 
 _ 8 cos <^ 
 2t cos ' 
 
 8 sin </>, 
 
 143. Thick Lenses and Systems of Lenses. Let FF' (Fig. 155) be 
 the axis of a thick lens or system of coaxial lenses, then if the media on 
 
 FIG. 155. 
 
 the two sides of the lens have the same refractive index, the following 
 are the cardinal points of the lens : 
 
 1. The principal focal points F and F' are such that a plane wave 
 incident on the lens at o is brought to a focus at F', while a plane wave 
 incident at o' is brought to a focus at F. Planes drawn through F and F' 
 perpendicular to the axis are called the principal focal planes. 
 
144] SYSTEMS OF LENSES 359 
 
 2. The points NN', called the principal points, which in the case 
 where there is the same medium on the two sides of the lens, coincide 
 with the nodal points. These points are such that if the direction of an 
 incident ray passes through one nodal point, then the direction of the ray 
 after passing through the lens will pass through the other nodal point, 
 and, further, will be parallel to the incident ray. These points are 
 conjugate foci ; that is, if an incident pencil is converging on N, then the 
 emergent pencil will diverge from N'. 
 
 The distances FN and F'N' are equal, and are called the focal length of 
 the lens or system. Planes drawn through N and N' perpendicular to 
 the axis of the lens are called the principal planes of the lens, and have 
 the property that, if an incident ray meets the first principal plane at a 
 point A, then the corresponding emergent ray i meets the second principal 
 plane at a point B, such that the line joining A and B is parallel to the 
 axis of the lens. 
 
 The distance u of an object from one of the principal points N, and 
 the distance v of the image from the other principal point N', are 
 connected with the focal length / by the relation 
 
 1 = 1_1 
 
 / v u 
 
 Further, the angle subtended by the object at the point N is equal to the 
 angle subtended by the image at the point N'. 
 
 If U is the distance of the object from the first principal focus, and V 
 is the distance of the image from the second principal focus, then the 
 following relation between these quantities and the focal length / will 
 sometimes be found of use 
 
 UV=f*. 
 
 144. To find the Focal Length and the Positions of the Prin- 
 cipal Planes of a Thick Lens or System of Lenses. Method 1. 
 Tn the case of a convex lens, or converging system, the following method 
 of determining the focal length, since it does not involve the use t)f a 
 micrometer, is often convenient : 
 
 If the image and object are of the same size, and u and v are the 
 distances of the object and image from the first and second nodal points 
 respectively, we have 
 
 image _ _ v 
 object u 
 
 Hence substituting in the general equation 
 
 v u~ 
 we get ^ = 2, 
 
 or v=2f, 
 
 also u = - 2/, 
 
360 LENSES AND MIRRORS [ 144 
 
 that is, the image and object are in either case at a distance equal to 
 twice the focal length from the corresponding nodal point. 
 
 If, now, we readjust the lens system so that the image is twice the size 
 of the object, and u f and v' are the new distances of object and image from 
 the nodal points, we get 
 
 = 2 = , 
 
 object u 
 
 and hence - = 3, 
 
 or /* = 3/, 
 
 and u=--f. 
 
 Thus in the second case (Fig. 156, b) the lens system has been moved 
 through a distance / away from the image P', and the object p has been 
 
 p' 
 
 (6) 
 
 FIG. 156. 
 
 moved through a distance //2. Hence if the distance through which the 
 lens system has been moved is measured this gives the focal length, the 
 image being formed in either case at the same point, p'. 
 
 To carry out the experiment it is necessary to have two transparent 
 scales having equal divisions. Such scales can easily be obtained by 
 photographing a suitably divided scale and then taking two positives on a 
 lantern plate from the negative. These two scales are mounted on two 
 of the uprights of the optical bench, with the film sides turned towards 
 each other, and the lens or lens system is mounted on a third carriage 
 between them. It is evident that the optical bench must have a length 
 at least four and a half times as great as the focal length of the lens 
 system. Having placed a light behind one of the scales, so that it is 
 well illuminated, the position of this scale and the lens must be adjusted 
 till the image is in the same plane as the other scale, and the divisions of 
 the image are of the same size as those of the scale. 
 

 144] SYSTEMS OF LENSES 361 
 
 It will be found an assistance when making the adjustments to fix a 
 low power microscope or reading lens immediately behind the scale on 
 which the image is to be received, and to focus this microscope on the 
 divisions of the scale. 
 
 The positions of the lens and the object having been so adjusted, the 
 position of the fiducial mark on the carriage which supports the lens is 
 recorded. 
 
 The object and the lens are then moved farther from the image scale, 
 care being taken not to disturb the position of this latter till the divisions 
 of the image are twice as far apart as those of the scale. The fact that 
 the image and the scale are in the same plane being tested in this case, as 
 in the previous measurement, by the parallax method. The reading for 
 the carriage which supports the lens is again taken, and the difference 
 between the readings for the two positions gives the focal length of the 
 system. 
 
 Determine by this method the focal length of a photographic objective, 
 a microscope or telescope eye-piece, and a microscope objective. 
 
 Having measured the focal length of the system, the positions of the 
 principal planes can now be found. To do this replace the lens and 
 object scale in the first position, namely, that in which the image and 
 object have the same size. In this position the focal planes are at a 
 distance of 2/ from the object and image respectively. Hence if a dis- 
 tance 2/ is measured from the object, the position of the principal plane 
 with reference to the mount which supports the lens is at once determined. 
 It will generally be found convenient to measure the distances between 
 the object and image and some well-defined point on the lens system, say, 
 one edge of the mount, and then the differences between these distances 
 and 2/ will give the distances of the principal planes from this point on 
 the mount. 
 
 Method 2. If when the distances of the image and object from the 
 nodal points are u and v respectively the magnification is m lt while when 
 the distances are u' and v' the magnification is ? 9 , we have 
 
 v i v f 
 
 = m, and . = w 9 ; 
 u u 
 
 hence substituting in the general equation we get 
 
 1=1-* 
 
 and 
 
 m 
 
 Thus -- - _ 
 
 /- 
 
362 LENSES AND MIRRORS [ 144 
 
 Now u - u is the difference in the distance between the object and the 
 first nodal point in the two positions of the lens, and its measurement does 
 not involve a knowledge of the position of the nodal point. Hence if, in 
 addition, we measure the two magnifications, we can calculate the focal 
 length. 
 
 In order to measure the magnification a micrometer similar to that 
 used in 127 will be required, and also a transparent scale to act as the 
 object. The lens system being mounted on the optical bench, it is placed 
 fairly close to the object, and the size of the image corresponding to a 
 convenient number of graduations of the scale is measured with the micro- 
 meter. Before measuring the size of the image, care must be taken that 
 the image is formed in the same plane as the cross-wire of the micrometer. 
 The position of the carriage which supports the lens is read, and the 
 measurement repeated with the lens at a considerably greater distance 
 from the object. The distance through which the carriage which supj>orts 
 the lens has been moved gives the quantity u f - u, while the magnifica- 
 tions m and m' are obtained by dividing the sizes of the images by the 
 corresponding size of the object. If the scale which is used for an object 
 is not one in which the distance between consecutive divisions is known, 
 it will be necessary to determine this quantity by means of a micrometer 
 microscope. 
 
 A modification of the above method consists in keeping the distance 
 between the object and the image fixed, the distance being chosen con- 
 siderably greater than four times the focal length, and measuring the 
 size of the image in the two positions of the lens which give an image in 
 the plane of the micrometer cross-wire. 
 
 If is the size of the object, and / and /' are the sizes of the image 
 in the two positions while the lens is moved through a distance I 
 between its two positions, we have 
 
 
 
 m j = -^. ana ni^ = -j,- 
 
 But from symmetry n^ = l//n.>. Hence = 
 and 
 
 Substituting these values in the expression used in the first method, 
 we get 
 
 /= 
 
 _ijn r 
 
 which does not involve a knowledge of the size of the object, and from 
 which the focal length can be calculated. 
 
145] TEST OF A PHOTOGRAPHIC LENS 363 
 
 Mefh&.l 3. In the case of a lens or system of short focal length, if 
 
 we take an object at a considerable distance, the image will be formed 
 practically at the principal focus. Thus in the case of a lens having a 
 focal length of a centimetre, if the object is at a distance of a metre the 
 image will be at only a tenth of a millimetre from the principal focal 
 plane. Hence if is the size of the object, and 11 is its distance from 
 the first principal plane, and / is the size of the image, we have 
 
 / 
 
 ru 
 
 Now if u is large, it will not appreciably alter the result if in place of 
 measuring u to the principal plane we measure it to the centre of the 
 lens. Thus the focal length is given by 
 
 f- 1 - 
 J ~ 0* 
 
 To apply this method, a brightly illuminated scale, or object of 
 known magnitude, is placed at a considerable distance from the lens, and 
 the size of the image is measured with a micrometer microscope. 
 
 If the lens is of such a long focus that it is not possible to place the 
 object at a sufficiently great distance, a collimator may be employed, 
 that is, the object is placed at the principal focus of an auxiliary lens. 
 A convenient arrangement is to use the collimator of a spectrometer 
 which, after being adjusted for parallel rays by one of the methods given 
 in 117, has the slit opened wide. The angle subtended by the two 
 edges of the slit must then be measured with the spectrometer telescope 
 in the ordinary manner. Let this angle be 0. The collimator must 
 then be removed and used as an object, while the size of the image, i.e. 
 the width of the image of the slit, must be measured with a micrometer 
 microscope. If the size of this image is /, then 
 
 /Wcotfl. 
 
 Another method of measuring the focal length of a system of lenses is 
 given in the next section. 
 
 145. Test of a Photographic Lens. The following method of 
 testing photographic lenses, which was developed by Major L. Darwin l 
 for the Kew Observatory, is both convenient and instructive. The 
 instrument used in making the tests is shown in Fig. 157, and can 
 easily be constructed in an ordinary laboratory workshop. The lens L 
 to be tested is mounted on a wooden block, and this block can be moved 
 backwards or forwards by a rack and pinion A. The lens and its 
 mounting can also be rotated about a horizontal axis, the amount of 
 the rotation being read by means of a pointer and divided circle. The 
 
 1 Proceedings of the Royal Society (1892), voL lii. p. 403. 
 
364 
 
 LENSES AND MIRRORS 
 
 [145 
 
 upright B, which carries the above lens fittings, is screwed to a wooden 
 board c, this board being pivoted at a point vertically under the 
 horizontal axis, about which the lens can turn. The board c moves 
 over a T-shaped base DE, and the amount of the rotation can be read 
 off on a scale of degrees engraved on the curved edge of the base. A 
 clamp F serves to fix c to the base, while two adjustable stops, D and E, 
 serve to limit the rotation. 
 
 On the board c slides a carriage G, which can be clamped in place 
 by the screw H. On the top of the carriage a second plate I can be 
 traversed through a distance of about 3 cm. by means of a micrometer 
 screw M. This plate I carries a divided scale J, the divisions (mm.) 
 being engraved on ground glass. The image formed by the lens is 
 received on this ground glass, and the position of the glass is roughly 
 
 FIG. 157. 
 
 adjusted by moving G, the final adjustment being performed by means 
 of the micrometer screw. A collimator N can be placed on two V's, 
 which are carried on an extension of the bed DE. The scale J can be 
 removed when desired, and replaced by a metal plate (a) in which a small 
 hole is pierced. 
 
 The first adjustment to be made is to move the lens backwards 
 or forwards by the pinion A till the horizontal axis about which the 
 lens can be turned, and hence also the vertical axis about which the 
 board c can turn, passes through the second nodal point of the lens. 
 To make this adjustment, focus the collimator N for parallel rays by the 
 method given in 117. Then set the collimator in place with the slit 
 horizontal, and having set up a bat's-wing burner behind the slit, focus 
 the image on the ground glass scale J by moving the carriage G. Next 
 rotate the lens about the horizontal axis, and move it backwards or forwards 
 by the pinion A till the image remains fixed in position. If the image 
 moves in the opposite direction to that in which the lens is rotated, then 
 
u- r >] 
 
 TEST OF A PHOTOGRAPHIC LENS 
 
 365 
 
 move the lens nearer the scale, and vice versa. When this adjustment is 
 complete, the axis about which the lens rotates passes through the second 
 nodal point, as will be at once evident from what has been said in 143. 
 
 1. Determination of the Effective Aperture of the Stops. Having 
 focused the lens on the ground glass, using the collimator as object, 
 replace the scale by the metal plate with a hole about a millimetre in 
 diameter, and place a flame behind the plate. The light from this hole, 
 which passes through the lens, will now issue as a parallel beam, and the 
 diameter of the beam will be the effective aperture of the stop which is 
 in place in the lens. This is at once evident, for if we imagine the 
 direction of the light reversed, it is only light within a cylinder of this 
 diameter which will be brought to a focus at a given point of the image. 
 For each of the stops provided with the lens measure the diameter of the 
 beam of parallel rays, using for the purpose the scale engraved on the 
 ground glass, which must be placed in the position previously occupied 
 by the collimator. 
 
 2. Determination of the Angle of the Cone of Illumination with the 
 Stops. Having replaced the collimator, look through the hole in the 
 metal plate, and turn the lens about a horizontal axis till all light is cut 
 off by the mount. Take the difference in 
 
 the readings of the circle attached to the 
 horizontal axis when the lens is rotated first 
 in one direction and then in the other. Half 
 the difference will give the angle of the 
 cone which includes all the rays which 
 traverse the lens. Since the illumination at 
 the edges of this outer cone falls off very 
 much, it is advisable to measure the angle of 
 the inner cone, which is limited by the con- 
 dition that we stop rotating the lens when 
 the edge of the mount is first seen to cut off 
 the light. By measuring this inner cone for 
 the different stops, determine the largest 
 plate which can be used with each stop, so 
 that the illumination should not fall off at 
 the edge, due to the plate extending beyond 
 the inner cone. 
 
 3. Determination of the Focal Length. ' 
 Using the collimator as object, put the ground 
 glass scale in place, and turn the board c 
 through an angle < (say, 20), and having 
 
 carefully focused, take the reading on the scale. Then rotate the 
 board c through an angle < on the opposite side, and again read the 
 position of the image on the scale. If x is the mean distance of the 
 image from the mid point of the scale, then 
 
 For if NjN., (Fig. 158) are the nodal points of the lens system, and when 
 
366 LENSES AND MIRRORS [ 145 
 
 the axis of the lens is rotated into the position DB, the axis of the 
 collimator lies along AN I? then the axis of the emergent pencil forming 
 the image will be along N 2 c, which is parallel to AN r But the lens has 
 been rotated about the point N O , and N O B is the. focal length F, while 
 the scale reading x is BC. Hence from the right-angled triangle CBN 2 
 (the scale CBC' remains perpendicular to N 2 B throughout) we have 
 
 F=~UCcot<f>. 
 
 4. Determination of the Curvature of the Principal Focal Plane. 
 The image of the slit of the collimator is focused as carefully as possible 
 when the axis of the lens is parallel to the axis of the collimator, and the 
 reading on the micrometer screw M is noted. The board c is then rotated 
 through 10, 20, 30, &c., and in each position the focus is adjusted, and 
 the micrometer screw is read. By plotting the difference between the 
 central reading and the peripheral readings against F tan <, where < is 
 the angle through which the board c has been rotated, the shape of a 
 meridian section of the principal focal " plane " will be obtained. 
 
 5. Determination of the Distortion of the Image of a Straight Line 
 near the Margin of 1he Plate. From the dimensions of the plate the lens 
 is intended to cover, calculate the angle /3 which half the breadth subtends 
 at a distance equal to the focal length of the lens. Place the slit of the 
 collimator horizontal, and turn the lens about the horizontal axis through 
 an angle /3, and measure the distance a between the image and the 
 horizontal line engraved on the scale. Next turn the board c through 
 such an angle that the image travels along the scale a distance equal to 
 half the length of the plate, and again measure the distance b of the 
 image from the horizontal line on the scale. Finally determine the dis- 
 tance c of the image from the line when the board c is rotated to an equal 
 amount on the other side of the axis. In this experiment we have, by the 
 rotation of the lens, obtained the same effect as if we had used a horizontal 
 linear source, and the image of the two ends and the middle of this line 
 are formed at distances b, c, and a from the straight line ruled on the scale. 
 Hence if the image is linear we must have 
 
 and the amount by which a differs from (&-}-c)/2 measures the dis- 
 tortion. 
 
 6. To Test the Achromatism of the Lens. To test the achromatism 
 it will be necessary to use a distant object, as it would not do to employ 
 the collimator, for any want of achromatism in its lens would vitiate the 
 result. As object a slit may be used, placed as far off as convenient. 
 The position of the focus is then determined when a red, yellow, and blue 
 glass is placed in succession behind the slit, a bright gas flame being used 
 as source. In this way any difference in focus for red, yellow, and blue 
 rays will be detected. 
 
146] MAGNIFYING POWER OF A TELESCOPE 367 
 
 7. To Examine the Lens for Astigmatism. A point source of light is 
 obtained by using the light reflected from a small glass bulb filled with 
 mercury or a silvered bead, when the bulb or bead is illuminated by a 
 small flame placed at a distance of about a metre. When the board c is 
 in the central position, the image will be a small circular disc. When, 
 however, the lens is tilted by rotating the board c, it will generally be 
 found that by altering the position of the ground glass the image in one 
 position is a fine radial, i.e. horizontal, line, and in another a fine line 
 at right angles to the first. Half-way between the position in which 
 these focal lines are formed the image is approximately circular, but 
 the diameter is greater than is the case when the image is on the 
 axis of the lens. 
 
 146. Measurement of the Magnifying Power of a Telescope. 
 The magnifying power of a telescope is the ratio of the angle which the 
 distant object subtends at the eye to the angle subtended by the image as 
 seen through the telescope. The most usual method adopted to measure 
 the magnifying power of a telescope is to place a scale at some distance 
 from the telescope, and to focus the telescope on the scale. The dis- 
 tance of the scale from the telescope must be so great that the length of 
 the telescope can be neglected compared to this distance. In addition to 
 the divisions, which must be of such a size that they can be seen clearly by 
 the telescope, two distinct marks must be attached to the scale of such a 
 nature that they can be distinctly seen with the naked eye from the posi- 
 tion where the telescope is set up. Looking through the telescope with 
 one eye, and at these two marks with the other unaided eye, the two images 
 will appear superposed. The number (n) of divisions of the scale, which 
 appear to correspond to the distance between the marks, must be noted. 
 If the distance between the marks is N, then the magnifying power (??/) of 
 the telescope is given by 
 
 N 
 
 m = 
 n 
 
 The angle subtended by the n divisions at the telescope is n/D, where 
 D is the distance of the scale from the telescope, while the angle sub- 
 tended by the N divisions at the eye is N/D, and these appear equal. 
 
 n N 
 
 Lence xm== 
 
 N 
 
 m= . 
 
 n 
 
 When making the comparison, the telescope must be so focused that the 
 two images do not appear to move relatively to one another when the eye 
 which receives light through the telescope is moved slightly from side to 
 
368 LENSES AND MIRRORS [147 
 
 side, which of course means that the image formed by the telescope is at 
 the same distance from the eye as the scale. In the case of a short- 
 sighted person, the spectacles habitually used when observing distant 
 objects must be worn when making the measurement. 
 
 When it is not convenient to place a scale at a sufficient distance, the 
 following modification of the method may be employed : Having focused 
 the telescope for a very distant object, a thin convex lens having a focal 
 length of about 2 metres is fixed immediately in front of the object 
 glass of the telescope. The focal length of this lens need not be known, 
 and a spectacle lens does quite well. Care being taken not to disturb 
 the adjustment of the focus of the telescope for distant objects, the tele- 
 scope is then placed at such a distance from a scale that the divisions 
 appear quite distinct. Then looking through the telescope with one eye, and 
 at the scale with the other eye, the number, n, of divisions of the scale, as 
 seen through the telescope, which appear to coincide with N divisions as 
 seen with the naked eye, must be noted. Then if d is the distance from 
 the scale to the object glass of the telescope, and D is the distance from 
 the scale to the eye, the magnifying power is given by 
 
 . 
 
 Dn 
 
 Since the telescope is focused for parallel rays, it follows that light 
 issuing from any given point on the scale after passing through the 
 auxiliary lens must form a parallel pencil. The angle included between 
 the two parallel pencils, which leave the two points of the scale at a dis- 
 tance n apart, is the same as the angle subtended by these two points at 
 the centre of the auxiliary lens. Hence this angle is n/d. The angle 
 subtended by the two divisions at a distance N apart at the eye is N/I), 
 and the two angles are the same, and hence 
 
 n N 
 d m = D> 
 
 Nd 
 
 or = m.--. 
 
 Dn 
 
 147. Measurement of the Magnifying Power of a Microscope. 
 
 The magnifying power of a microscope is the ratio of the angle sub- 
 tended by the image of a given object as seen through the microscope to 
 the angle which the same object would subtend if placed at the distance 
 of most distinct vision. This distance may be taken as 25 cm. A 
 finely divided scale is placed on the stage of the microscope, and a scale 
 divided in millimetres is supported at a distance of 25 cm. below the 
 eye-piece of the microscope. Then by looking through the microscope 
 with one eye, and at the other scale with the other, if it is found that the 
 two divisions of the scale as seen through the microscope, which are at 
 
147] MAGNIFYING POWER OF A MICROSCOPE 369 
 
 distance n apart, appear to coincide with N divisions of the scale seen 
 with the naked eye, then the magnifying power is given by 
 
 N 
 
 m - . 
 n 
 
 The comparison is made easier if a camera lucida is available, for then 
 the scale a.nd the image, as seen through the microscope, can be seen at 
 the same time with a single eye. 
 
 2 A 
 
CHAPTER XXII 
 
 POLAKISED LIGHT 
 
 148. Rotation of the Plane of Polarisation by Optically Active 
 Substances. If a beam of plane polarised light is passed through a plate 
 of quartz cut perpendicular to the optic axis, or through a solution of 
 cane sugar, the emergent light will be plane polarised. The plane of 
 polarisation of the emergent light will not, however, be the same as that 
 of the incident light. During the passage of the light through these 
 substances the plane of polarisation undergoes a rotation, which, for any 
 given substance, is proportional to the length of the substance traversed. 
 Substances which are capable of rotating the plane of polarisation are 
 said to be optically active. The rotation is said to be positive or right- 
 handed if, looking towards the source of light, the rotation takes place 
 in the direction of the hands of a clock. If the rotation takes place in 
 the opposite direction, the substance is said to possess negative or left- 
 handed rotatory power. 
 
 The rotation produced by a decimetre of a pure substance divided by 
 the density is called the specific rotation of the substance, and varies with 
 the wave-length of the light, being greater for light of short wave-length 
 than for light of longer wave-length. 
 
 If a solution of an active substance in an inactive solvent is used, the 
 rotation produced by a decimetre of the solution divided by the weight 
 of the active substance in unit volume of the solution is called the specific 
 rotation of the dissolved substance. Thus, suppose that x grams of a 
 substance are dissolved in an inactive solvent so that the volume of the 
 solution is 100 c.c., and that a length of 20 cm. of this solution rotates 
 the plane of polarisation through 6 degrees, then the specific rotation of 
 the dissolved body is given by 
 
 100(9 
 
 The specific rotation is not a constant, for not only does it depend upon 
 the wave-length of the light used and on the temperature, but it also 
 varies with the concentration of the solution used and the nature of the 
 inactive solvent. The variations are, however, small, and may often be 
 neglected. 
 
 The specific rotation multiplied by the molecular weight of the active 
 substance is called the molecular rotation of the substance. 
 
 370 
 
149] ROTATION OF THE PLANE OF POLARISATION 371 
 
 The following table gives the specific rotation of some substances for 
 sodium light and a temperature of 20, the concentration being c grams 
 of the salt in 100 c.c. of the solution : 
 
 Active Substance. 
 
 Solvent. 
 
 French turpentine {C 10 H 6 } . . 
 Ethyl tartrate {(C 2 H B )2C 4 H 4 6 } 
 Potassium tartrate {K 2 C 4 H 4 0(j} 
 Tartaric acid {C 4 H 6 O 6 } . . . 
 Potassium sodium tartrate 
 
 {KNaC 4 H 4 6 } 
 
 Camphor {C 10 H 16 O} .... 
 
 None 
 None 
 Water 
 Water 
 
 Water 
 
 Ethyl alcohol 
 
 Specific Rotation. 
 
 - 37-01 
 
 + 8-31 
 
 + 27-14 + 6-0992c-0-000938c 2 
 + 15-06 - 0-131c 
 
 + 29-73 - OO078c 
 
 + 41-982+ 0'11824c 
 
 In the case of cane sugar (C 12 H 22 O 11 ) dissolved in water, the specific 
 rotation for sodium light at a temperature t is given by l 
 
 [ a ]^ = 66-5- 0-0184(^-20). 
 
 For quartz, the rotation produced by 1 millimetre for different tem- 
 peratures is given by 
 
 [O]' D = 21-72[1 + 0-000147(- 20)]. 
 
 In the case of invert sugar (C^H.^Oio), the specific rotation changes 
 with the concentration much more than in the case of cane sugar. The 
 effect of temperature is also greater. Landolt gives the following expres- 
 sions for obtaining the specific rotation of invert sugar, where c is the 
 weight of invert sugar contained in 100 c.c. of solution, which holds for 
 values of c between 2 and 30 and from C. to 30 
 
 = 19-657 + 0-0361e' - 
 
 - 20). 
 
 If a solution is made up which contains c grams of cane sugar in 
 100 c.c., and then the cane sugar is converted into invert sugar, since the 
 molecular weights of the two sugars are not the same there will not be 
 c grams of invert sugar in 100 c.c. The molecular weight of cane sugar 
 is 342, and that of invert sugar is 360. Hence the concentration c of 
 the invert sugar solution is given by 
 
 , 360 
 
 149. Methods of Measuring the Rotation of the Plane of Polarisa- 
 tion. Theoretically, the simplest method of determining the plane of 
 polarisation of a beam of light is to examine it with a Nicol prism, for 
 when the prism is so turned that its principal plane is perpendicular to 
 
 Schonrock, Zeit. Instrumentenkunde (1900), xx. 97. 
 
372 
 
 POLAKISED LIGHT 
 
 [149 
 
 the plane of polarisation of the light, no light will pass through the prism. 
 In practice, however, such a simple prism forms a very insensitive arrange- 
 ment for determining the plane of polarisation, for it is impossible to tell 
 exactly when the light is exactly cut off, the field appearing dark while 
 the Nicol is being rotated through an appreciable angle. On this account 
 various arrangements have been devised to assist the setting of the Nicol 
 so that its principal plane is inclined at a definite angle to the plane of 
 polarisation of the incident light. Most of these arrangements consist in 
 the use of an auxiliary piece of apparatus, by means of which, instead of 
 using a beam of light which is polarised throughout in the same direc- 
 tion, the incident beam is divided into two parts, and the light in these 
 two parts is polarised in planes inclined at a small angle ft to one another. 
 When the analysing Nicol is so placed that its principal plane is perpen- 
 dicular to the plane of polarisation of half the field, that half of the field 
 will appear black. If, then, the analysing Nicol is turned through an 
 angle ft so that its plane is now perpendicular to the plane of polarisa- 
 tion of the other half of the field, this half will now appear black. When 
 the principal plane of the Nicol is equally inclined to the planes of polar- 
 isation of the two halves of the field, the whole field will appear of a 
 uniform tint. Now it is found that the eye can tell when the two halves 
 of the field are equally bright very much better than it can tell when the 
 field is quite dark. Since turning the analyser through an angle /3 
 changes from one half of the field being quite dark to the other half being 
 quite dark, the smaller we make ft, the smaller the angle through which 
 the analyser can be turned between these two limits, and it might at first 
 sight appear that by making ft very small the accuracy of setting could 
 be indefinitely increased. This is not the case, for if ft is very small, 
 then the intensity of the light in the two halves of the field is so small 
 that the eye is not able to judge whether the illumination is equal. The 
 
 smallest value of ft which it is advisable to 
 use depends on the intensity of the light 
 which enters the analyser, that is, on the 
 intensity of the source of light and on the 
 transparency of the substance being investi- 
 gated. Hence, since the intensity of the light 
 varies in different cases, it is an advantage 
 to be able to adjust the value of ft so that 
 maximum sensitiveness may be obtained in 
 any given condition. 
 
 The principal methods employed for pro- 
 ducing the difference in the plane of polari- 
 sation of the two halves of the field are 
 as follows : 
 
 1. A small plate of quartz cut parallel 
 to the axis is placed in half the beam of 
 
 light, the thickness being such that during the passage of the light 
 through the plate the ordinary ray gains in phase over the cxtra- 
 
 FIG. 
 
149] ROTATION OF THE PLANE OF POLARISATION 373 
 
 ordinary ray by 180, or A/2, where A is the wave-length of the light 
 used. Let ABC (Fig. 159) represent the quartz plate covering half the 
 field, the optical axis of the quartz being parallel to the edge AB. Let 
 the plane of polarisation of the light from the polarising Nicol be at right 
 angles to OP, so that the vibrations in the incident light take place along 
 OP. Then in the quartz the incident vibration is resolved into two com- 
 ponents, one taking place along OA, and the other at right angles to 
 the optic axis, i.e. along oc. On emergence from the quartz one of these 
 vibrations, namely, that in the direction OA, will be retarded by 180 
 more than that along oc, and hence when on leaving the quartz they 
 recombine to produce a single plane polarised beam, the vibration will 
 take place along OQ, which makes the same angle with the axis OA as does 
 the original direction of vibration OP, that is, the plane of polarisation 
 of the light which has traversed the quartz will be inclined to the original 
 plane of polarisation at an angle equal to twice the angle AOP (see 152). 
 By altering the angle AOP, that is, by altering the angle included between 
 the principal plane of the polarising Nicol and the optical axis of the 
 quartz plate, the angle between the planes of polarisation of the two halves 
 of the field can thus be altered. It will be noticed that such a half wave- 
 length plate of quartz can only be used with one kind of monochromatic 
 light, since it is only for light of a single wave-length that the difference 
 in phase between the two waves which have traversed the plate will be 
 exactly equal to 180, and hence it is only for such light that on recom- 
 bination we shall obtain a beam of plane polarised light. If white light 
 were used with a half wave-length plate of such a thickness that a differ- 
 ence of phase of 180 is produced with yellow light, then the yellow part 
 of the spectrum would on emergence be polarised in a plane making an 
 equal angle with the axis of the quartz to that made by the original plane 
 of polarisation. For red light, on the other hand, the difference of phase 
 would be less than 180, and hence the emergent light would be elliptically 
 polarised, and would not be completely extinguished for any position of 
 the analysing Nicol. This same remark applies to the green and blue, so 
 that the field would appear coloured, and no satisfactory match of the two 
 halves would be obtained. 
 
 2. Poynting has devised a simple arrangement for producing the field 
 having two halves polarised in slightly different direc- 
 tions, which consists of a small tube, the two ends of 
 
 which are closed by two plane glass plates A and B (Fig. 
 160). A semicircular plate of glass c is enclosed in the 
 cell, and the remainder of the cell is filled with a solution 
 of sugar. The sugar solution rotates the plane of polar- 
 isation, and the amount of the rotation is greater in the 
 case of the light which traverses the upper part of the 
 cell than in the case of the light which passes through 
 the lower part of the cell where part of the path con- IGt 160< 
 
 sists in the glass plate c. 
 
 3. One or more auxiliary Nicols can be placed after the analyser 
 
374 POLARISED LIGHT [ 149 
 
 SD as to partly cover the field, their principal planes being inclined at 
 a small angle to the principal plane of the polariser. Lippish uses 
 two auxiliary Nicols B and c (Fig. 161). Thus the field, as shown at 
 (a), consists of three parts. The central part A corresponds to light 
 
 which has passed through the analyser only 
 while the two outer parts B and c correspond 
 to light which has passed in addition through 
 the two auxiliary Nicols. The principal 
 planes of the auxiliary Nicols are parallel, but 
 slightly inclined to the principal plane of 
 the analyser. It is found that using a field 
 divided into three parts is an advantage, as 
 it facilitates judging when the whole field is 
 of a uniform brightness. 
 
 4. An arrangement called a biquartz 
 may be used. This consists of two semi- 
 circular plates of quartz of equal thickness 
 cut with the optic axis perpendicular to 
 the semicircular faces. One half of the 
 quartz is right handed, and the other half left handed. When plane 
 polarised white light is passed through the biquartz, the plane of 
 polarisation of the light in the two halves is rotated in the opposite 
 directions, and the different colours are rotated to a different extent. 
 If the thickness of the quartz plates is about 3 mm., then the yellow 
 light in each half is rotated through 90. Hence if the principal 
 plane of the analysing Nicol is turned so as to be perpendicular to 
 the plane of polarisation of the incident light, and therefore parallel 
 to the plane of polarisation to the yellow rays, the yellow rays will 
 be extinguished on either side, and both halves of the field will appear 
 a greyish- violet colour, called the tint of passage. On slightly rotating 
 the analysing Nicol the colour cut out will on one side be nearer the blue, 
 and on the other side nearer the red end of the spectrum, and hence one 
 half of the field will appear of a red tinge, and the other half of a blue 
 tinge. This change is fairly sudden, and hence it is possible to adjust 
 the analysing Nicol to the tint of passage with considerable accuracy. 
 The rotation measured when using the tint of passage corresponds 
 to yellow light near the Fraunhofer line E of the solar spectrum. 
 
 The general arrangement of the optical parts of most polarimeters, 
 as instruments for measuring the rotation of the plane of polarisation 
 are called, is shown in Fig. 162. The source of light s is placed at a 
 distance from the lens L equal to its focal length, so that an approxi- 
 mately parallel pencil of rays enters the polarising Nicol N r The half 
 wave-length plate Q is attached to a ring, so that the angle the optical 
 axis of the quartz makes with the principal plane of the Nicol can be 
 varied. The analysing Nicol N 2 and a low power telescope, consisting 
 of an object glass L. 2 and eye-piece H, are attached to a divided circle, 
 by means of which the angle through which they are rotated can be 
 
149] ROTATION OF THE PLANE OF POLARISATION 375 
 
 measured. The substance to be investigated is placed between the two 
 parts of the instrument. In the case of a liquid it is enclosed in a tube 
 A, which is closed with plates of optically worked glass. These plates 
 are held against the ends of the tube with metal caps, which screw on to 
 
 H 
 
 FIG. 162. 
 
 the ends of the tube. If it is desired to make accurate measurements of 
 rotation, it is essential that it should be possible to rotate the analysing 
 Nicol completely round so as to be able to make two settings, the Nicol 
 being rotated through 180 between the two. 
 
 In some forms of saccharimeter instead of directly measuring the 
 angle through which the plane of polarisation is rotated by means of a 
 divided circle attached to the analysing Nicol, the rotation is measured 
 by determining what thickness of some substance which rotates the plane 
 in the opposite direction must be interposed to neutralise the rotation 
 produced by the body under investigation. 
 
 Solid's compensator consists of two wedges of quartz, ABC and DBF 
 (Fig. 163). These wedges have equal angles, and they are both cut so 
 that the optical axis of the quartz is perpendicular to the faces AB and FE. 
 These wedges are fixed in 
 frames, which by means of 
 a rack and pinion can be 
 caused to move one in one 
 direction and the other in 
 the opposite, the amount of 
 movement being indicated 
 by a scale. In this way 
 the thickness of quartz which 
 the light traverses before it 
 reaches the Nicol N., can be 
 varied. The azimuth of the 
 principal plane of the Nicol 
 N. 2 is so adjusted with refer- 
 ence to the plane of polarisa- 
 tion of the polarising Nicol, 
 that when the wedges are 
 
 placed immediately one FIG. 163. 
 
 behind the other the two 
 
 ives of the field appear equally dark. The reading on the scale is 
 len zero. When an optically active body is placed in the instru- 
 
376 POLARISED LIGHT [ 150 
 
 ment the wedges are moved so as either to increase or decrease 
 the thickness of quartz till the two halves of the field again appear 
 equally dark. Then from a previous calibration of the scale the rotation 
 produced by the substance is deduced. 
 
 150. Measurement of the Rotatory Power of Solutions of 
 Turpentine. As an illustration of the measurement of the rotatory 
 power of a solution we may take the case of solutions of turpentine in 
 paraffin. 
 
 The first operation is to obtain the zero reading of the instrument. 
 Focus * the eye-piece so that the line dividing the field into two parts is 
 seen sharply, and in the case of instruments using a biquartz adjust for 
 the sensitive tint. Then determine the reading when the two halves of 
 the field are equally illuminated or of the same tint, as the case may be. 
 Having carefully cleaned the tube and the glass plates which are used to 
 close the ends, place one of these plates on one end of the tube, then an 
 india-rubber ring, and finally the cap, which must not be screwed down 
 too tight. The reason for the introduction of the rubber ring between 
 the outside of the glass plate and the cap is that if glass is strained it 
 becomes birefringent, and would thus affect the plane of polarisation of 
 the light. The effect of strain is generally to produce elliptic polarisation, 
 so that the field appears patchy. Next fill the tube with pure turpentine 
 quite to the top, and slide on the other glass plate, and having wiped 
 off any superfluous liquid, fix the plate in place with a rubber washer and 
 the cap. 
 
 Place the tube in the instrument, and allow it to stand there for some 
 time so that the temperature may become uniform, and thus the striaB 
 due to differences of density may disappear. Since the rotatory power 
 changes with temperature, it is necessary to measure the temperature of 
 the substance. For this reason polarimeters which are intended for 
 accurate measurements are provided with water-jackets, which resemble 
 Liebig condensers, to go round the tube, the temperature of the liquid 
 being obtained by measuring the temperature of the water which is 
 passed through the jacket. In the absence of such a jacket a thermo- 
 meter must be placed alongside the tube, and time be allowed for the 
 thermometer and liquid within the tube to come to the same temperature. 
 Several settings of the instrument must be made for the analyser in 
 the two positions, separated by about 180, for which the field is equally 
 bright on either side. The mean of these, less the zero reading, will give 
 the rotation. 
 
 In the case of substances which have a large rotatory power, there 
 may be an uncertainty as to whether the rotation is to the right or to the 
 left. In such a case observations must be made with a shorter tube or 
 with a weaker solution, and these will at once show the sign of the 
 rotation. 
 
 1 When making this adjustment, it is advisable to place the tube filled with 
 water in the instrument. In this way the focus will probably be correct when 
 the liquid under observation is used. 
 
' 
 
 151] STRENGTH OF A SOLUTION OF SUGAR 377 
 
 Having determined the rotation produced by pure turpentine, make 
 up solutions containing different percentages of turpentine mixed with 
 paraffin oil. The easiest way of obtaining these solutions is to weigh 
 a 50 or 100 c.c. measuring flask empty, and then introduce about the 
 right quantity of turpentine, and again weigh. Finally, fill the flask up 
 to the mark with the paraffin. 
 
 Carefully measure the length of the tube, and then calculate the 
 rotatory power of the turpentine when pure and when in solution, and 
 see to what extent the rotatory power is constant. 
 
 151. Measurement of the Strength of a Solution of Sugar. To 
 determine the quantity of cane sugar in a solution, which may also 
 contain other substances which rotate the plane of polarisation, we 
 proceed as follows : 
 
 Take about 15 grams of the sugar to be tested and introduce it into 
 a weighed 100 c.c. flask, and again weigh. Then fill the flask up to its 
 mark with distilled water. Measure off 50 c.c. of this solution, and add 
 5 c.c. of strong hydrochloric acid, and warm this mixture to a temperature 
 of 70 C., keeping it at this temperature for about ten minutes, and then 
 allow to cool to the room temperature. By the action of the acid the 
 cane sugar will be converted into invert sugar. 
 
 The rotations produced by the original solution, and that in which 
 the sugar has been converted into levulose, must then be measured. It 
 must be remembered that 10 per cent, by volume of hydrochloric acid 
 has been added to one portion of the solution. Hence if w is the weight 
 of sugar originally taken, the unacted upon solution contains w grams of 
 sugar in 100 c.c. of the solution. The other portion, however, contains 
 only 10^/11 grams of the original sugar in 100 c.c. of the solution. 
 Thus if the same tube is used in the two measurements, the rotations 
 with the converted sugar must be multiplied by I'l to convert them to 
 what they would have been had not the solution been diluted with the 
 hydrochloric acid. To save having to make this correction two tubes are 
 sometimes supplied with saccharimeters, the length of one tube being 
 20 cm., and that of the other 22 cm. The weaker solution being placed 
 in the longer tube, the extra length compensates for the weakening of 
 the solution. The temperature of the solutions must be noted when 
 measuring the rotation, this being especially necessary in the case of the 
 invert sugar. 
 
 If a and a are the rotations produced by the two solutions, and t the 
 temperature at which the observations were made, and I the length of 
 the tube, we have ( 148) 
 
 a = { 66-5- 0-0184 (*- 20} 
 
 where c is the weight of cane sugar in w grains of the sugar being 
 tested, and is the quantity it is required to find, while /3 is the rotation 
 produced by substances other than cane sugar which may be present. 
 
378 
 
 Also a 
 where 
 
 POLARISED LIGHT 
 
 - f 19-657 + 0-0361c' - 0-304 (t - 20) j c l + ft 
 , 360 
 
 [152 
 
 Hence 
 
 a - a' = L J66-5 - 0-0184(* - 20)) c - {19-657 
 1 00 I ) 
 
 + 0-0380c - 0-304( - 20)}1'0526, 
 
 from which c can be calculated. 
 
 It will generally be most convenient to obtain the approximate value 
 for c, neglecting the term 0'0380c, and then introduce this approximate 
 value in the above term to get the final value for c. In the case of 
 fairly pure samples it is even sufficiently accurate to substitute w in place 
 of c in this small term. 
 
 152. Elliptically and Circularly Polarised Light. Suppose that 
 we have a plate of a doubly refracting crystal cut parallel to the optic 
 
 axis, and that the principal 
 planes of this plate are parallel 
 to the directions ox and OY 
 (Fig. 164), while a beam of 
 plane polarised light traverses 
 the plate normally, the plane of 
 polarisation of the incident light 
 being parallel to OQ. Then the 
 vibrations in the incident light 
 will take place in a direction 
 perpendicular to OQ, that is, 
 parallel to OP. The incident 
 vibration having an amplitude 
 equal to OP will give rise to a 
 vibration of amplitude ON or 
 FIG. 1G4. ^ cos a = a (say) parallel to 
 
 ox, and a vibration of amplitude 
 
 OM or A sin a = b (say) parallel to OY. At the face of incidence these 
 two vibrations will be in the same phase, but as they traverse the 
 crystal plate with different velocities, they will differ in phase when 
 they emerge. If the difference in the optical paths is 8, the period 
 of the incident light being T, and its wave-length in air A, while the 
 
 incident light vibration be represented by A cos 27r , then the vibrations 
 
 of the two components when they reach the second face of the plate will 
 be represented by . 
 
 X == a COS 27T 
 
 b cos 2?r 
 
 T A 
 
 respectively. 
 
152] ELLIPTIC ALLY POLARISED LIGHT 379 
 
 From these two equations the time t can be eliminated, when we 
 
 obtain 
 
 v 
 
 2xy cos 27T- 
 
 ab 
 
 which represents the resultant vibration on emergence. This expression 
 is the equation to an ellipse described in a rectangle having its sides 
 parallel to ox and OY, and of length 2a and 2b respectively. 
 
 & 
 
 If 8 = 0, then sin 2?r --=0, and cos 2?r- = 1. Thus the vibration is 
 A A 
 
 represented by 
 
 --| = ......... (2) 
 
 a b 
 
 which is the equation to a straight line parallel to OQ. Thus the 
 emergent light is plane polarised parallel to the incident light. 
 
 
 A $\ & 
 
 If 8 = -, then sin 2?r- = 0, and cos 27r- = - 1, and therefore 
 
 (3) 
 
 This represents plane polarised light, but the plane of vibration is 
 now parallel to OP', where OP' is inclined at the same angle to ox as is OP, 
 but on the opposite side of ox. 
 
 \ s ? 
 
 When 8 = -, then sin 2?r- = 1, and cos 27r- = 0, and therefore 
 4 A A 
 
 (4) 
 
 which represents an ellipse with its axes parallel to the principal axes of 
 the crystal. 
 
 If a = 45, that is, if the plane of polarisation of the incident light is 
 equally inclined to the principal axes, the resultant vibration reduces to 
 
 which represents a circle, and thus the emergent light is circularly 
 polarised. 
 
 It follows from the above that if we resolve an elliptic vibration into 
 components along the axes of the ellipse, then the difference in phase of the 
 
 components is - or -- . Hence if by any means we reduce the difference 
 
 of phase between the components by - or A/4, they will be in the same 
 
 2 
 
 phase, or differ in phase by 180, and will, if then recombined, produce 
 
380 POLARISED LIGHT [ 153 
 
 plane polarised light. If the angle which the vibrations in the plane 
 polarised light make with the major axis of the ellipse be a, then the 
 ratio of the axes of the ellipse is given by 
 
 . | -cot a (6) 
 
 153. Analysis of an Elliptical Vibration. The apparatus required 
 for this experiment is a spectrometer to which can be attached three 
 vertical graduated circles. Two of these fit over the object glasses of the 
 collimator and telescope, and the third fits in the eye-piece end of the tele- 
 scope. The circles on the collimator and at the eye-piece end of the telescope 
 are fitted with Nicol prisms ; arms are also attached, by means of which the 
 angles through which the Nicols are rotated can be read off on the circles. 
 A quarter wave-length plate is mounted on the circle attached to the 
 object glass of the telescope. 
 
 The quarter wave-length plate is a plate of crystal such that incident 
 plane polarised light is separated into two components polarised at right 
 angles, and the phase of one component is retarded by A/4 over that of 
 the other. Generally the plate is adjusted for sodium light. As has 
 been shown in the last section, when a difference of phase of A/4 is intro- 
 duced in the components into which elliptically polarised light can be 
 split, then these components will combine to form plane polarised light. 
 Hence by turning the quarter wave-length plate so that the directions in 
 which the vibrations within the plate take place are parallel to the axes 
 of the ellipse, the emergent light will be plane polarised, and can be 
 extinguished by the eye-piece Nicol. The position of the principal plane 
 of the quarter wave-length plate will then give the direction of the axes 
 of the elliptic vibration, and the cotangent of the angle the principal 
 plane makes with the principal plane of the eye-piece Nicol will give the 
 ratio of the diameters of the ellipse. 
 
 The telescope and collimator having been focused for parallel light, 
 the readings on the circle when the principal plane of the eye-piece Nicol 
 is horizontal must be obtained. To do this remove the polarising Nicol 
 and mount a piece of plate glass vertically on the prism table, or use the 
 face of a prism, and turn it so that the angle of incidence <f> of the light 
 from the collimator is the polarising angle, i.e. tan < = //.. Place the 
 telescope so as to receive the reflected light, and turn the eye-piece 
 Nicol till the light is completely extinguished. The principal plane 
 of the Nicol is then horizontal, and the reading on the circle must be 
 noted. 
 
 Next replace the Nicol on the collimator, and turn it till it is crossed 
 with the eye-piece Nicol, and hence its principal plane is vertical. The 
 quarter wave-length plate is then put in place and turned till the field is 
 dark, in which position its principal plane is either horizontal or vertical, 
 and the reading on the circle must be recorded. 
 
 The instrument is now in adjustment, and as an exercise a thin sheet 
 of mica may be introduced between the polariser and the quarter wave- 
 
153] ANALYSIS OF ELLIPTICAL VIBRATION 381 
 
 length plate. Unless the principal plane of the mica is horizontal or 
 vertical, elliptically polarised light will be produced. 
 
 To analyse this elliptically polarised light rotate the quarter wave- 
 length plate and the analysing Nicol till no light is transmitted. Then 
 read the circles for the plate and the analyser. The difference between 
 the two readings for the plate will give the angle made by the axes of 
 the elliptic vibration with the horizontal and vertical directions. The 
 difference in the two readings for the analyser will be the angle w T hose 
 tangent is equal to the ratio of the semi-diameters of the ellipse. 
 
CHAPTER XXIII 
 
 PHOTOMETRY AND COLOUR VISION 
 
 154. Standards of Light. The legal standard of light in Great 
 Britain is the sperm candle, which burns 120 grains (7*776 grams) of 
 spermaceti per hour, such a standard candle being said to give a 
 luminous intensity of one candle-power. Numerous investigations have, 
 however, shown that as a standard of light the candle is unsuitable and 
 unreliable. As alternative standards there have been a number suggested, 
 of which the following are the chief : 
 
 1. The Carcel standard, which consists of a colza lamp of fixed form 
 originally designed by Regnault and Dumas. This standard is used in 
 France. 
 
 2. The Amyl Acetat lamp, which is a lamp burning amyl acetate 
 under standard conditions, invented by von Hefner-Alteneck, and is 
 used as a standard in Germany. 
 
 3. A lamp burning normal pentane, invented by Vernon Harcourt, 
 and now used as the official standard for testing gas in London. 
 
 4. The light given out normally by a square centimetre of molten 
 platinum at its solidifying temperature. This unit of light was pro- 
 posed by Violle. 
 
 5. The light emitted by a square centimetre of solid platinum heated 
 by the passage of an electric current to such a temperature that 10 per 
 cent, of the radiation which it emits, as measured by a bolometer, can 
 pass through a layer of water 2 centimetres thick contained in a cell with 
 quartz sides. This standard was suggested by Lummer and Kurlbaum, 
 and is in use at the Reichsanstalt at Berlin. 
 
 As a secondary standard, that is, one of which the value is determined 
 once for all by comparison with one of the standards mentioned above, 
 Fleming has recommended the use of a special form of carbon incandescent 
 electric lamp working under fixed electrical conditions. 
 
 Of these various standards, the molten platinum and the Lummer- 
 Kurlbaum incandescent platinum strip standards require such delicate 
 manipulation that their description is beyond the scope of this book. 
 The Carcel is not a standard which in convenience in handling or accuracy 
 much exceeds the standard candle. The Hefner-Alteneck lamp suffers 
 from the defect that the colour of the light is much redder than the light 
 given, say, by an incandescent electric lamp. A description of this lamp 
 will be found in Elektrofeclmlsche Zeitschrift, vol. iii. p. 445, and vol. v. 
 p. 20. It must be noted that the light given by this standard varies 
 
155] PHOTOMETEKS 383 
 
 very considerably with the quantity of water vapour in the air and the 
 proportion of carbon dioxide present. A correction has also to be applied 
 for the effects of changes in the barometric pressure. 
 
 A description of the Vernon Harcourt pentane lamp will be found in 
 the Notification of the Gas Referees for the Year 1904, published by 
 Eyre & Spottiswoode. The light given by this standard also varies with 
 the moisture and carbon dioxide present in the air, and with the baro- 
 metric pressure. 
 
 The Fleming-Ediswan standard glow-lamp consists of a U-shaped 
 carbon filament which has been run in an exhausted bulb for fifty hours 
 at about 5 per cent, above its normal voltage. The filament was then 
 removed and sealed into a new bulb, which is very much larger than the 
 bulb of an ordinary incandescent lamp. Fleming T finds that such lamps, 
 if not run above their marked voltage, and not very much used, remain 
 constant for a long time. 
 
 155. Photometers. (1) The Bunsen Photometer. In its original 
 form the Bunsen photometer consisted of a disc of unglazed paper, with 
 a circular grease spot at the centre. The grease renders the paper more 
 translucent than the untreated portion of the disc. Hence when the 
 disc is placed between two lights practically no light is transmitted 
 through the ungreased portion of the paper, and the illumination of this 
 part of the disc depends on the light received from the source on that 
 side. The illumination of the spot, however, depends on the light which 
 is reflected at its surface, together with the light which traverses the spot 
 from the other side. If the illumination of the spot seems the same as 
 that of the surrounding paper, then just as much light traverses the spot 
 from right to left as is transmitted from left to right, so that the light 
 transmitted just makes up for the light which, in place of being reflected 
 at the surface, has passed through the paper. But if the amounts trans- 
 mitted by the spot in the two directions are equal, it means that the 
 illumination received on the two sides of the disc are equal. Hence if 
 we measure the distances of the sources of light from the disc, since the 
 illuminations at the disc are equal, we can immediately calculate the ratio 
 of the quantities of light given by the two sources in the directions 
 considered. 
 
 When using the Bunsen photometer it is necessary to be able to see 
 the two sides of the disc simultaneously ; hence it is usual to arrange two 
 mirrors inclined at 45 to the plane of the disc, and to examine the 
 images of the two sides of the disc as seen reflected in these two mirrors. 
 
 In practice it is found impossible to find a position of the disc at 
 which both sides of the disc seem quite uniformly illuminated. This is 
 particularly the case if the sources which are being compared do not give 
 light of exactly the same colour. This difficulty has been in a measure 
 reduced by Dibdin, who replaced the simple grease spot by a sheet of 
 tissue paper, placed between two sheets of white card in which star- 
 shaped openings have been cut. The two stars are placed exactly opposite 
 * Proceedings of the Institution of Electrical Ewjinetrs (1903), xxxii. p. 1. 
 
384 PHOTOMETRY [ 155 
 
 one another, and the tissue paper which stretches across the openings 
 plays the part of the grease spot. 
 
 (2) The Lummer-Brodhun Photometer. This photometer consists in 
 a system of reflecting prisms, arranged so as that the field of view of a 
 telescope is divided into two parts, one of which is illuminated by the 
 light diffused from one side of a slab of magnesium carbonate which is 
 exposed to the light from one of the sources, while the rest of the field is 
 illuminated by light received from the other side of the slab, which is 
 exposed to the second source of light. The arrangement of prisms is 
 shown in Fig. 165. The essence of the instrument consists in the double 
 prism CD, shown on a larger scale at (a). This prism consists of two 
 right-angled prisms placed with their hypotenuse faces in contact. The 
 hypotenuse face of the prism c is slightly ground away, except for a 
 
 circular portion in the 
 centre. The two prisms 
 are then cemented 
 together so that at the 
 centre light can pass 
 through the central 
 part from one prism 
 to the other, whatever 
 the angle at which it 
 is incident on the com- 
 mon face. AB is the 
 magnesia slab, and 
 light from the two 
 sides is reflected by 
 means of the prisms E 
 and F into the double 
 prism in the manner 
 shown by the arrows. 
 Of the light which 
 reaches the prism c from the left-hand surface of the slab, only that portion 
 which falls on the part of the prism in contact with the prism D will be able 
 to pass through and reach the telescope T. The rest of the light will be 
 reflected at the ground away portions of the hypotenuse face of the 
 prism c. Of the light which comes from the right-hand surface of the 
 slab, that which falls on the central part of the hypotenuse face of the 
 prism D will pass right through the prism c, and hence will not reach the 
 telescope. The light which falls on the outside portions of the hypotenuse 
 face of the prism D will, however, be totally reflected into the telescope. 
 Hence the central part of the field of the telescope will be illuminated 
 by light from the left-hand surface of the slab only, while the surround- 
 ing portions of the field will be illuminated by light coming from the 
 right-hand surface of the slab. It will be seen that both portions of 
 light have .traversed equal thicknesses of glass, so that absorption within 
 the glass will affect them equally. Thus if the whole field of the tele- 
 
 FIG. 165. 
 
PHOTOMETERS 
 
 385 
 
 155] 
 
 scope appears equally bright, it follows that the two sides of the slab AB 
 are equally illuminated. Arrangements are made so that the whole 
 instrument may be rotated through 180 with reference to the sources 
 of light, thus interchanging the surfaces of the slab exposed to the two 
 lights, so that any inequality in the surfaces may be eliminated. 
 
 The Lummer-Brodhun photometer is probably the most accurate form 
 of photometer at present in use, and with it comparisons of light, which 
 are of the same colour, can be made correct to about one-fifth per cent. 
 
 (3) The Rumford Photometer. In this photometer, which ought with 
 justice to be called the Lambert photometer, the two sources of light throw 
 shadows of a rod on a white screen, and the distances of the sources from the 
 screen are adjusted till the shadows appear equally dark. The distance of 
 the rod which casts the shadow from the screen is immaterial ; the edges 
 of the shadow which it casts must, however, be 
 sharp, so that in the case of sources of light of 
 considerable extent the rod must be placed fairly 
 close to the screen, and the lights at some distance. 
 The lights and rod must also be so adjusted that 
 the two shadows just touch, as in this case it is 
 much easier to tell when the shadows are equally 
 bright. This photometer has the merit of extreme 
 simplicity, and since the necessary apparatus can 
 be prepared in a few minutes, it is very con- 
 venient" for making rough measurements of the 
 relative intensity of two sources of light. 
 (4) The Jolly Paraffin Block Photometer. 
 
 D 
 
 YIG. 166. 
 
 photometer consists of two blocks of clear paraffin wax A and B (Fig. 166), 
 which are separated by a sheet of tin-foil. The faces c and D are turned 
 towards the sources of light, while the compound block is looked at 
 
 from the side AB. 
 When the illumi- 
 nation of the 
 faces c and D are 
 equal, the two 
 halves of the 
 block will appear 
 equally bright. 
 
 (5) The Pho- 
 t op ed. This 
 photometer i s 
 used by the Gas 
 Referees when 
 testing the illu- 
 minating power 
 of London gas. 
 
 It consists of a metal screen A (Fig. 167), pierced at the centre with a 
 square hole B, over which is stretched a piece of thin white paper. A 
 
 2B 
 
 B 
 
 FIG. 167. 
 
386 PHOTOMETRY [ 155 
 
 tube c is attached to the screen, and within this tube slides a second tube D. 
 A diaphragm E having a rectangular aperture is attached to D. The 
 distance of the diaphragm from the paper B is so adjusted that the two 
 patches of light due to the lights s x and s., just touch. When the two 
 patches appear equally bright, the illumination at B, due to the two lights, 
 is equal. 
 
 (6) The Ritchie Photometer. In this photometer two white surfaces 
 are inclined at equal angles to the light coming from the two sources. The 
 two surfaces meet along an edge, and the observer looks towards the edge, 
 thus seeing the two surfaces alongside. The sources of light are then 
 adjusted so that the two surfaces appear equally bright. In order to 
 
 FIG. 1(58. 
 
 make an accurate setting, it is of great importance that the edge, where 
 the two surfaces meet, should be sharp. 
 
 (7) The Flicker Photometer. This photometer consists of a white 
 screen A (Fig. 1 68) placed at an angle of 30 to the direction of the light 
 coming from one of the lights s r A second screen B is mounted on the 
 axle of a small electric motor. This screen is in the form of a Maltese cross, 
 as shown at E, the arms of the cross and the intervals being of equal width. 
 The plane of this cross makes an angle of 30 with the light from the second 
 source 2 . The observer's eye is placed at F, and looks down a blackened 
 tube D, and as the cross rotates he alternately sees the face of the cress 
 illuminated by the source s. 2 , and the screen A illuminated by the source 
 s r If the two illuminations are not equal, then when the cross rotates 
 a sense of flicker will be produced, which will vanish when the illumina- 
 tions are equal. This form of photometer is of special use when comparing 
 two sources of different colours, since the fact that the lights are of different 
 
156] DETERMINATION OF CANDLE-POWER 387 
 
 colours does not interfere with the vanishing of the sensation of flicker 
 when the intensities of the lights are suitably adjusted. 
 
 It is essential when making photometrical measurements that all 
 reflected light should be prevented from reaching the screen or photo- 
 meter. Thus the walls of the room in which the photometer is placed 
 must be painted a dead black, and the photometer itself placed at some 
 distance from the walls, for even such substances as black velvet reflect 
 an appreciable amount of light at large angles of incidence. Fleming 
 recommends that the minimum size of room should be 2 '5 metres wide, 
 2*5 metres high, and 6 metres long, with the photometer bar running 
 down the middle of the room. The photometer bar consists of a long 
 optical bench, with two carriages to support the lights, and a third carriage 
 to support the photometer. If a flame standard is used, it will be neces- 
 sary to provide means, by a fan or otherwise, to renew the air within the 
 room, otherwise, owing to the accumulation of carbon dioxide and moisture 
 in the air, the light given by the standard will decrease. 
 
 156. Determination of the Connection between the Candle-power 
 of an Incandescent Lamp and the Watts consumed for Different 
 Voltages. The lamp to be investigated is mounted on the carriage of 
 the photometer bar, and is connected through an ammeter and a variable 
 resistance with a source of E.M.F. slightly greater than the marked 
 voltage of the lamp. The terminals of the lamp are also connected to a 
 voltmeter. On the other carriage is mounted another lamp, which will be 
 taken as a standard. This lamp must either be permanently connected to 
 a voltmeter, or a switch must be arranged by means of which a voltmeter 
 can be connected to one or other of the lamps at will. An adjustable 
 resistance must also be placed in series with this standard lamp. Adjust 
 the resistance in series with the comparison lamp till the E.M.F. between 
 its terminals has the value for which the lamp is marked, or some whole 
 number of volts a little below this value. Also adjust the resistance in 
 series with the lamp to be tested so that it is at its marked voltage, and 
 then by moving the photometer or one of the lamps, secure uniformity of 
 illumination on the two sides of the photometer. Immediately this 
 balance is secured, read off the volts and amperes corresponding to the 
 lamp under test. The setting should be repeated several times, and the 
 mean taken. When adjusting the position of the photometer for equality 
 of illumination, it will be found advantageous to oscillate it backwards and 
 forwards fairly rapidly over a few millimetres, and to gradually reduce the 
 amplitude of the movement till finally a position of adjustment is arrived at. 
 
 Repeat the observations, using gradually decreasing voltages on the 
 lamp to be tested, but keeping the voltage on the comparison lamp con- 
 stant. Then plot a curve showing the connection between the voltage 
 and the light given by the lamp, taking the light given by the comparison 
 lamp as unit. Also calculate the watts consumed by the lamp, that is, 
 the product of the volts into the amperes, and divide the watts by the 
 intensity of the light, and plot the quotient, that is, watts per candle, 
 against the voltage. 
 
388 
 
 PHOTOMETRY 
 
 [ 
 
 157. The Spectro-photometer. It is sometimes necessary to compare 
 the intensities of the light given by two sources for light of the various 
 wave-lengths taken throughout the spectrum. For this purpose it is 
 necessary to form a spectrum with the light from each source, and then to 
 compare the brightness of corresponding parts of the spectra. Such a 
 comparison is made by means of a spectro-photometer. An ordinary 
 spectrometer may be converted into a spectro-photometer in the following 
 manner : In front of the lens of the collimator AB (Fig. 169) is placed a 
 double image prism c. This prism consists of a prism of calcite or quartz, 
 cut with its refracting edge parallel to the optic axis of the crystal, 
 cemented to a second prism of equal angle made of the same material, but 
 cut with the refracting edge perpendicular to the optic axis. A ray of 
 
 s, 
 
 B 
 
 A1 
 
 i 
 
 'H 
 , >O 
 
 1 
 
 B 
 
 
 -3te 
 
 
 C 
 
 (6) 
 
 FIG. 109. 
 
 light i (Fig. 169, 1) will pass through the prism AEG, in which the 
 refracting edge is perpendicular to the optic axis, parallel to the optic 
 axis, and therefore the ordinary and the extraordinary rays will coincide 
 in direction. On reaching the prism ACD, in which the refracting edge 
 is parallel to the optic axis, the ordinary ray o will be undeviated, while 
 the extraordinary ray E will be deviated, and thus after passing through 
 the double prism the ordinary and extraordinary rays will proceed in 
 directions inclined at a small angle. Thus the incident unpolarised light 
 by its. passage through the prism is divided into two rays of equal inten- 
 sity which are plane polarised, the planes of polarisation in the two rays 
 being at right angles. 
 
 The double image prism being placed in front of the collimator 
 object glass, on turning the telescope to view the slit, the prism D having 
 been removed, two images of the slit will be seen. The double image 
 
157] 
 
 THE SPECTRO-PHOTOMETER 
 
 389 
 
 prism must then be rotated till these two images form a single line. The 
 two images will overlap at the centre, and a piece of paper o(Fig. 169, a) 
 must be pasted over the central part of the slit, and its width so ad- 
 justed that of the four images now formed the two middle ones just 
 touch but do not overlap. Thus in the centre of the field there will be 
 two adjacent images of the slit, one formed by light which has passed 
 through the upper portion of the slit, and the other by light which has 
 passed through the lower portion, and since one of these will correspond 
 to an ordinary ray, and the other to an extraordinary ray, they will be 
 polarised in perpendicular planes. The uppermost and lowest of the 
 four images must then be cut out by a screen placed in the telescope 
 in the plane of the cross-wires. 
 
 The Nicol N (Fig. 169), with its divided circle, is then attached to 
 the telescope, and the Nicol is turned till one of the images vanishes, 
 when the reading of the circle is noted. This will give the zero reading, 
 and a rotation of the Nicol through 90 from this position will cause 
 the other image to vanish. 
 
 On placing the prism D on the table two spectra will be seen, one 
 formed with light from each half of the slit, arid these spectra will be 
 polarised in planes at right angles. 
 
 Suppose, then, that the lower part of the slit is illuminated by a 
 source of light s v while, by means of a small totally reflecting prism p, 
 the light from a second source S 2 is caused to illuminate the upper part 
 of the slit, then the two spectra will be due to the two lights. To 
 compare the brightness for different colours a diaphragm F, with a fairly 
 narrow vertical slit, is placed at the focal plane of the eye-piece of the 
 telescope so as to cut out all the spectrum except the portions which 
 are to be compared. The Nicol is then 
 turned till the two portions of the spectra 
 appear equally bright. Let the angle through 
 which the Nicol has to be turned from its 
 zero position be 8. Then 
 
 where I I and /., are the intensities of the two 
 sources for light of the wave-length being 
 compared. 
 
 Let AOB (Fig. 170) represent the zero 
 position of the Nicol, that is, the position of 
 its plane of polarisation when the upper 
 images is cut off, i.e. when the light from the 
 source s x is extinguished. Then oc will be the position of the plane 
 of polarisation of the Nicol when the light from the source S 2 is 
 extinguished. When the plane of polarisation of the Nicol is parallel 
 to OD, the intensity of the light from source s x transmitted will be 
 /! sin' J 0, while the intensity of the light due to the source S 2 will be 
 
390 PHOTOMETRY [ 158 
 
 7 2 cos 2 0. Hence since the intensities of the light transmitted by the 
 Nicol from the two sources are equal, we have 
 
 As an exercise in the use of the spectro-photometer, compare the 
 light given by an ordinary bat's-wing burner with that given by a 
 Welsbach incandescent mantle. It will, in the first place, be necessary 
 to draw a curve giving the deviations produced by the prism D (Fig. 169) 
 in terms of the wave-length by one of the methods described in 124. 
 
 Next place the two flames in such positions that they illuminate the 
 slit, one directly, and the other after reflection in the prism p. The exact 
 equality of the distances of the lights from the slit is not important, so 
 long as each of them entirely fills the angle obtained by producing the 
 lines drawn from the edge of the collimator lens through the slit. 
 
 Then determine the position of the Nicol when equality of illumina- 
 tion is secured for a number of points taken through the spectrum, and 
 plot a curve showing the connection between tan 2 B and the wave-length. 
 It will appear from this curve that the Welsbach is relatively richer in 
 the light of smaller wave-length. 
 
 158. Measurement of the Absorption Curve of a Solution. A 
 small glass cell containing a dilute solution of cyanine in alcohol is 
 placed over half of the slit of the spectro-photometer, and then the light 
 from a bright source of light, such as the filament of a Nernst lamp, 
 is allowed to pass directly through half of the slit, and after passing 
 through the solution, through the other half of the slit. Then deter- 
 mine the readings of the Nicol when the different portions of the two 
 spectra are equally bright. If 6 is the reading for the Nicol, then the 
 ratio of the light transmitted by the solution to the original light is 
 given by tan 2 6. Plot a curve showing the connection between the 
 proportion of light transmitted and the wave-length. 
 
 Similar curves may be obtained for solutions of permanganate of 
 potash and for various coloured glasses. When measuring the absorp- 
 tion with a cell containing the solution before part of 'the slit there 
 will be a certain amount of loss of light, owing to reflection at the glass 
 surfaces, as well as owing to absorption within the glass itself. Hence 
 it is often advisable to use a second cell with a smaller thickness of 
 solution before the other half of the slit, so that both beams of light 
 pass through a cell, but the thickness of solution in one case is greater 
 than in the other. In place of using two cells a single cell may be used, 
 but a plate of glass of measured thickness, say, 1 cm., may be placed in 
 the lower part of the cell, and the light which enters through the lower 
 half of the slit allowed to traverse this plate of glass, while the light 
 which enters through the upper part of the slit traverses the whole width 
 of the cell in the solution. The differences which are obtained in the 
 intensities of the transmitted light then correspond to the absorption 
 of a layer of the solution having a thickness equal to the thickness of the 
 glass plate. 
 
159] COLOUR MIXTURE WITH THE COLOUR TIP 391 
 
 If / is the intensity of the incident light, and I t is the intensity of 
 the light after passing through a thickness t of the substance, it is found 
 that the following relation holds, 
 
 where a is a constant depending on the colour of the light and the 
 nature of the substance, and e is the base of the Napierian system of 
 logarithms. Taking the Napierian logarithms of both sides, we have 
 
 In the case of logarithms to the base ten, then we have a similar rela- 
 tion, namely, 
 
 where A = a/2 '30. The quantity A is called the absorption coefficient 
 of the substance. 
 
 By making a series of observations with different thicknesses of 
 solution, and plotting on semi-logarithmic paper the relation between 
 7J//0 and the thickness, a straight line will be obtained, proving the 
 correctness of the expression assumed above for the relation between the 
 intensities of the incident and transmitted light. 
 
 159. Colour Mixture with the Colour Tip. The apparatus used 
 for this experiment consists of a small electric mctor, to the spirdle 
 of which is attached a circular plate and a milled clamping nut. The 
 nut allows of coloured cardboard discs being clamped against the face 
 of the plate. There are also cardboard discs of two sizes, coloured black 
 (lamp black), white (zinc white), red (vermilion), yellow (chrome yellow), 
 green (emerald green), and blue (French ultramarine). The larger disc 
 of each colour has a diameter of about 14 cm., while the smaller discs 
 have a diameter of about 8 cm. The discs have a hole punched at the 
 centre, 1 so that they will pass over the screwed end of the motor spindle. 
 Each disc is also slit radially, so that by arranging these discs in the 
 manner illustrated in Fig. 171, and then slipping round one disc with 
 reference to the other, the relative amounts of each colour exposed can 
 be altered. 
 
 Taking any five of the six colours, it will be found possible to match 
 the colour obtained by blending two of them taken in suitable propor- 
 tions with a blend of the other three. Thus, take a black and a 
 white disc of the larger size, and take smaller discs of the three colours, 
 
 1 It is essential that these coloured discs should be uniformly coated with 
 the paint, and further, that they should be cut truly circular and be accurately 
 centred. If the discs are not centred correctly, then when rotated a coloured 
 line will appear at the edge of the smaller discs, which will be found to make 
 the adjustment to an exact match very difficult. 
 
392 COLOUR VISION [ 159 
 
 red, green, and blue, and mount them all on the spindle of the 
 motor, and place the motor facing a window which has a northerly 
 aspect. Connect the motor to a switch, an adjustable carbon resistance, 
 and a sufficient number of accumulators, so that when the motor is 
 rotating the coloured discs appear to blend into one, and no sense of 
 flicker is apparent. 
 
 Starting with about a quarter of the white disc exposed, set the 
 motor in rotation, and observe whether the inner discs blend into a 
 grey tinge. If the tinge observed seems purple, increase the green 
 sector ; if yellow, increase the blue ; while, of course, if the appearance 
 is reddish or greenish, the red or green, as the case may be, is to be 
 reduced. Having adjusted the inner discs to produce a grey, adjust 
 the outer discs so that the luminosity of the grey they form is the same 
 as that of the inner discs. 
 
 When making the final match, it is advisable to observe the discs 
 from a distance of 2 or 3 metres, and not to stare at the discs too 
 
 FIG. 171. 
 
 long. If you cannot make up your mind as to whether the match is 
 complete, look away for a few seconds, and then, when the eyes are a bit 
 rested, look back at the discs. The quantities of the various colours 
 required to make a match depend on the character of the light by which 
 the discs are illuminated. Thus it will be found impossible to make a 
 consistent set of measurements if the light varies much, say, by clouds 
 drifting over a blue sky. For this reason it is often advisable to use the 
 light from an electric arc, the quality of which is much more constant 
 than is that of day-light. 
 
 To obtain the amounts of the different colours exposed, and which 
 form a match, a card annulus, of which the inner diameter is a trifle less 
 than the diameter of the smaller discs, and the external diameter is less 
 than the diameter of the larger discs, is used. The outside and inside 
 edges of this annulus are graduated in 100 parts, and the annulus being 
 placed over the discs, the angular aperture of each colour can at once 
 be read off. 
 
 Having made several settings with the above five discs, next take 
 
159] COLOUR MIXTURE WITH THE COLOUR TIP 393 
 
 large discs of white, black, and yellow, and small discs of green and red, 
 and adjust the green and red sectors till they blend into a yellow. Then 
 adjust the yellow, white, and black discs till they blend into a yellow 
 which matches the inner discs both as to colour and luminosity. 
 
 Then proceed to match the black and white against the green, yellow, 
 and blue ; the black, white, and red against the yellow and blue ; the 
 black, yellow, and blue against the white and green ; and finally, the 
 white, red, and green against the yellow and blue. 
 
 By the above experiments, six sets of matches will have been 
 obtained between five of the six colours. Now, according to Maxwell's 
 experiments, a match can be obtained between any four colours. The 
 reason we have had to take five colours in each of the above experiments 
 is that the rotating disc method of blending colours involves a special 
 condition, namely, that the whole circumference of both the circles must 
 be filled up by the colours. This is the reason for the inclusion of the 
 black disc, the effect of the black being to alter the luminosity of the 
 tint formed by the blending of the other colours with which it is 
 combined. 
 
 Using the symbols R, Y, G, 13, TF, and X to represent the colours 
 red, yellow, green, blue, white, and black, we may write the result of the 
 first match in the form of an equation as follows : 
 
 
 (1) 
 
 where r v c/ v b^ <fec., are the sizes of the sectors of the different colours 
 when a match is secured. 
 
 In the same way, for the second match we should obtain an equation 
 of the form 
 
 ..... (2) 
 
 In the same way, the third match will give 
 
 ..... (3) 
 
 Now if we multiply equation (1) through by r 2 , and equation (2) by 
 r x and subtract, we obtain 
 
 ( T 2fh ~ r lff*) G + r A /; + r i2/2 Y = ( T -2 W l ~ r i w z) W + ( r <2 X \ ~ r \ X 2) X ( 4 ) 
 
 that is, we get a relation between green, yellow and blue, and white 
 and black. But this is exactly what we have obtained by experiment 
 as expressed by equation (3). Proceeding in this manner, we can from 
 any two of the colour equations calculate the other four equations. 
 
 Choosing two of the matches which you consider the most accurate, 
 deduce from these the colour equations of the other combinations, and 
 compare the numbers obtained with those given by experiment. 
 
 In place of arbitrarily choosing two of the matches, and using these 
 to calculate the others, we may take the six colour equations obtained. 
 
394 COLOUR VISION [ 161 
 
 experimentally and calculate two normal equations between the six 
 quantities R, Y, G, 13, W, and X by the method described in 5. 
 Then from these two equations calculate the colour equations correspond- 
 ing to the six matches, when a comparison of the numbers thus obtained 
 with the experimental numbers will give a measure of the accuracy with 
 which the experiments have been carried out. 
 
 In the case of colour-blind observers, the matches obtained may 
 differ very considerably from those made by a normal-eyed observer. 
 
 160. Measurement of the Luminosity of Pigment Colours. To 
 measure the luminosity of the light reflected from different pigments 
 the piece of apparatus shown in Fig. 136 is arranged with an arc lamp 
 as the source of light, so that by rotating the prism a beam of mono- 
 chromatic light of different wave-lengths is obtained beyond the lens. 
 The pigment to be tested is painted on a card disc about 8 cm. in 
 diameter, and together with larger discs of white and black is mounted 
 on the axle of the motor used in the last section. The rotating discs are 
 then illuminated by the monochromatic light, and the white and black 
 discs are adjusted till the central and peripheral parts of the rotating 
 disc appear equally bright. When this adjustment is complete, the 
 number of divisions of the white sector gives the luminosity of the 
 pigment for the given coloured light in terms of the luminosity of the 
 white disc taken as 100. 
 
 Using discs of the colours employed in the last section, determine the 
 luminosity for different coloured lights taken throughout the spectrum. 
 By placing salts, such as lithium, sodium, and magnesium chlorides, in 
 the carbons of the arc, the lines corresponding to these metals will be 
 seen in the spectrum formed in the plane of the slit. Hence if the 
 prism is so adjusted that the red lithium line, the yellow sodium lines, 
 the green magnesium lines, and the blue lithium line are in turn on the 
 slit, observations 'may be made with light of these known wave-lengths, 
 and a curve may be plotted for each pigment showing the luminosity in 
 terms of the wave-length of the incident light. 
 
 A study of these curves, in connection with the character of the light 
 given by different sources, such as a gas flame, a sodium flame, &c., will 
 explain the different appearances of the different pigments when examined 
 by the light of these various sources. 
 
 161. Test for Colour-Blindness with Holmgren's Wools. Holm- 
 gren's wool test for colour-blindness consists in asking the candidate 
 to match certain test skeins by selecting from a large number a 
 different coloured skein. The skeins from which a selection has to be 
 made consist of several shades of red, orange, yellow, yellowish-green, 
 green, blue-green, blue, violet, purple, pink, brown, and grey. Varieties 
 of pinks, blues, violets, and light greys should be particularly well repre- 
 sented. The test skeins are three in number. The first is a very light 
 green, and is chosen of such a colour as to match the spectrum colour 
 which a red or green colour-blind person cannot distinguish from white 
 or g t *ey (see Watson's Physics, 401). A colour-blind person will 
 
161] TEST FOR COLOUR-BLINDNESS 395 
 
 therefore select, in addition to correct matches, some light greys or 
 drabs. 
 
 The second test skein is a light pink, which is very nearly a com- 
 plementary colour to the green of the first test. This colour in the 
 green-blind excites both the blue and the red sensations, as do also 
 greys and greens. Hence a green-blind person will select as matches 
 greys and greens in addition to the correct colours. To a red-blind 
 person the pink only excites the blue sensation, and hence he will include 
 in his selection blues and violets. 
 
 The third test skein is a vivid red containing a little blue. The 
 green-blind person will select shades of red slightly lighter than the test 
 skein. The red-blind, however, since he does not see the red, will match 
 this test against dark green, dark blue, and dark brown. 
 
 To carry out the test, the skeins are arranged in a heap on a sheet of 
 white paper in a good light, and the candidate is handed the first test skein 
 and asked to select from the heap all the skeins which are of the same 
 colour as the test skein. He is told to include not only skeins which 
 appear to him exactly the same colour, but also skeins which appear to 
 be slightly lighter and darker shades of the same colour. If no greys, 
 drabs, or light pinks are included in the selection, the candidate has 
 normal colour vision. 
 
 The second test skein is then treated in the same way, the skeins 
 selected during the first test having been returned to the heap. If, after 
 the correct matches have been removed from the selected skeins, the 
 prevalent colour of the incorrect skeins is blue, the candidate is red 
 colour-blind. If in addition to purple he selects greys and greens, he 
 is gi'een colour-blind. 
 
 The third test is made in the same manner as the first two, and 
 the character of the mistakes will confirm the results given by the 
 second test. 
 
 A person with normal colour vision may obtain an idea of the selection 
 which will be made by a colour-blind person by making a selection, 
 using either a blue-green or a purple glass before the eyes. Using a 
 blue-green glass he will make the mistakes made by a red colour-blind 
 person, while with the purple glass he will make the same mistakes 
 as a green colour-blind person. 
 
 For further particulars as to testing for colour-blindness, reference may 
 be made to the Proceedings of the Royal Society (1892), vol. li. p. 376. 
 
CHAPTER XXIV 
 
 MEASUREMENT OF THE EARTH'S MAGNETIC FIELD 
 
 162. Determination of the Direction of the Magnetic Meridian. 
 When determining the direction of the magnetic meridian it is not 
 sufficient simply to determine the direction in which the geometrical axis 
 of a freely suspended magnet points, since the magnetic axis of the 
 magnet seldom exactly coincides with the geometrical axis. If, how- 
 ever, after noting the direction of the geometrical axis when the magnet 
 is suspended in one position, the magnet is turned through an angle of 
 180 about its geometrical axis, suspended in this new position, and the 
 direction of the geometrical axis again noted, then the bisection of the 
 angle between the two positions occupied by the magnet is the magnetic 
 meridian. 
 
 In the Kew pattern magnetometer the magnet consists of a cylinder 
 of steel A (Fig. 172). This cylinder is fixed in a brass collar, to which 
 
 are attached two pegs 
 B and c. Either of 
 these pegs fits into a 
 brass collar which is 
 attached to the end of 
 the silk suspension 
 fibre. A short hollow 
 cylinder D, forming 
 part of the brass at- 
 tachment, serves for 
 the support of a 
 
 cylindrical bar when determining the moment of inertia of the magnet. 
 At one end of the hollow magnet is placed a convex lens L, while at 
 the other end is placed a disc of plane glass, on which is engraved a 
 fine scale. The focal length of the lens is equal to the distance between 
 the lens and the scale, so that the rays of light starting from any point 
 on the scale leave the lens as a parallel pencil. The line joining the 
 centre of the scale to the optical centre of the lens forms the geometrical 
 axis of the magnet, and this does not usually differ by more than about 
 five minutes of arc from the magnetic axis. 
 
 Since the magnetic axis of the suspended magnet will only point in 
 the direction of the magnetic meridian if the suspension thread is entirely 
 free from torsion, care has to be taken to remove as far as possible all 
 torsion from the silk suspension, and to allow for the effect of any residual 
 
 FIG. 172. 
 
163] MAGNETIC MERIDIAN 397 
 
 torsion. The suspension fibre usually consists of a bundle of unspun silk 
 fibres, and since the torsion of such a bundle of fibres is found to vary 
 with their dampness, it is advisable to ensure that they are always damp 
 by moistening the fibre with a little glycerine. To remove the torsion 
 the magnet is replaced by a plummet having the same weight as the 
 magnet, and if the fibre is new, the plummet is allowed to hang for 
 several hours. The plummet having come to rest, the torsion head, 
 from which the fibre hangs, is turned so that the clip is in such a position 
 that when the magnet is put in place the fibre will be free from torsion. 
 
 To make a measurement of declination with the Kew-pattern of mag- 
 netometer, the instrument is set up on a firm tripod stand or pillar and 
 levelled. The wooden box with the glass tube carrying the torsion head 
 is then attached, and the plummet suspended. The torsion having been 
 removed, the magnet is suspended in such a way that the scale, as seen 
 in the telescope, appears right side up. The motion of the magnet having 
 been checked with the arrestor, the whole instrument is turned till the 
 magnet swings approximately equally on the two sides of the zero of the 
 scale. The motion of the magnet must then be reduced to a fraction of a 
 scale division by alternately approaching and removing a feebly magnetised 
 piece of steel, such as the screw-driver supplied with the instrument. 
 Having finally adjusted the instrument so that the vertical cross-wire 
 coincides with the middle division of the scale, the time is noted, and the 
 two verniers on the horizontal circle of the instrument are read. The 
 magnet is then reversed, so that the scale appears inverted, and two 
 settings are made, the time and the readings of the verniers being noted 
 in each case. Finally, a second setting is made with the magnet in its 
 original position. The mean of all the vernier readings gives the reading 
 on the azimuth circle of the instrument corresponding to the magnetic 
 meridian at the mean time of the observations. 
 
 To obtain the declination, it is necessary to know the reading of the 
 azimuth circle corresponding to the geographical north. In the case of a 
 fixed observing station it is usual to have some fixed mark, the azimuth 
 of which is determined once for all. In the absence of such a fixed mark, 
 it will be necessary to make observations on the sun or stars to determine 
 the geographical meridian (see next section). In the case of the fixed 
 mark, the instrument is turned till the vertical cross-wire bisects the mark, 
 and the readings of the azimuth circle are taken. Then from the known 
 azimuth of the mark the reading corresponding to the geographical north 
 is obtained, and the difference between this reading and that obtained for 
 the magnetic meridian gives the declination. 
 
 163. Determination of the Geographical Meridian. If the geo- 
 graphical meridian is to be obtained by observations of the sun, then it 
 will be necessary to know (1) the latitude and longitude of the place of 
 observation, which may be obtained with sufficient accuracy from the 
 Ordnance map, and (2) the error of a chronometer. 
 
 Since in the Kew-pattern magnetometer the telescope is fixed in a 
 horizontal position, in order to obtain an image of the sun in the telescope 
 
398 MEASUREMENT OF EARTH'S MAGNETIC FIELD [ 163 
 
 a small mirror is used which can turn about a horizontal axis in two Vs. 
 The axis about which this mirror turns must fulfil three conditions : (1) 
 it must be exactly horizontal, (2) it must be parallel to the plane of the 
 mirror, and (3) it must be perpendicular to the optical axis of the telescope. 
 To perform the first adjustment a small striding level is provided^ and 
 one of the V's is adjustable in a vertical direction. This adjustable V 
 must be moved till, on reversing the level, the bubble remains in the same 
 position ; when this is the case, the axle of the mirror is horizontal. To 
 allow of the second and third adjustments being made, the eye-piece of 
 the telescope is provided with a lateral opening and an annular mirror 
 set at 45 to the axis. By means of this annular mirror the cross-wires 
 are illuminated, and if a blackened tube or other screen is placed between 
 the mirror and the telescope, so as to cut off stray light, on turning the 
 mirror an image of the cross-wires will be seen. The table which carries 
 the mirror V's must be turned about a vertical axis till the vertical cross- 
 wire coincides with its image. The mirror must then be reversed in its 
 V's, and if the image still coincides with the vertical cross- wire, the mirror 
 is in adjustment. If there is not coincidence, then the image must be 
 brought half-way back by rotating the table which carries the V's, and 
 the remainder by adjusting the mirror with reference to the axle by means 
 of a small screw placed at the back. In this way the adjustment of the 
 mirror can be made fairly accurately. Since, however, the observations are 
 so arranged as to allow of a correction being applied, so long as the 
 adjustment is not very bad, it is not necessary to attempt to make the 
 adjustment perfect. 
 
 The times of first and second contacts of the image of the sun with 
 the vertical cross-wire for four transits are then taken, a coloured glass 
 being interposed sa as to reduce the brightness of the image. For the 
 two first direct transits the observer faces the sun, the mirror being 
 reversed in its V's between the two transits, while for the second pair, or 
 back transits, the instrument is turned in azimuth through 180, so that 
 the observer has his back to the sun, the mirror being again reversed 
 between the two transits. After each transit the readings on the hori- 
 zontal azimuth circle are recorded as in the following example. 
 
 To calculate from the times of transit and the circle readings what 
 would be the reading if the telescope were pointed due geographical north, 
 it will be necessary to have a copy of the Nautical Almanac. The 
 method of performing the reduction will be seen by studying the example 
 in conjunction with the following notes : 
 
163] 
 
 GEOGRAPHICAL MERIDIAN 
 
 399 
 
 OBSERVATION OF SUN'S AZIMUTH. Date 23rd April 1891. 
 STATION, Kew. Lat. 51 28' 6". Long. 18' 46"'5 W.= m. 
 
 CHRONOMETER, Dent. 
 
 Error at Station, 45'0 seconds fast. 
 
 FRONT VIEW OF SUN. 
 
 Glass. 
 
 j Correction for 
 Glass. 
 
 ^Transit Bearing of Sun. i Mean of Verniers. 
 
 10 B 
 01 B 
 10 B 
 01 B 
 
 -30" 
 -30" 
 -30" 
 
 -30" 
 
 h. m. s. ' 
 4 50 37-0 309 26 
 53 25-6 129 26 40 
 54 36-0 310 13 20 
 57 26-0 130 14 20 
 
 Sums . . 
 Means . 
 Error of Chronometer . 
 GM.T 
 
 16 4-6 200 20 
 4 54 I'l 129 50 5 
 -45-0 -30 , 
 4 53 16-1 I 129 49 35 
 
 
 
 BACK VIEW OF SUN. 
 
 
 PM Correction for 
 Glass " Glass. 
 
 B - 30" 
 B - 30" 
 B - 3 >" 
 B - 30" 
 
 Time of Sun 
 Transit. 
 
 Bearing of Sun. 
 
 131 7 10 
 311 6 
 131 53 50 
 311 52 40 
 
 Mean of Verniers. 
 
 01 
 10 
 01 
 10 
 
 h. w. s. 
 4 59 7'5 
 5 1 56-5 
 5 3 6-0 
 5 5 55'3 
 
 1 II 
 
 Sums . . 
 Means . . 
 Error of Chronometer . 
 G.M.T. . .... 
 
 10 5-3 
 5 2 31-3 
 -45-0 
 5 1 46-3 
 
 119 40 
 131 29 55 
 -30 
 131 29 25 
 
 
 
 a ~ Polar distance of Sun. c = J (a -b). d \ (a + b). 
 b = Co-Latitude of place of observation. 
 C and C" = Hour angles of the Sun at the time of direct and back observations. 
 P and Q = Angles, such that P+Q = A. 
 A and A' = Azimuths of Sun from north at time of direct and back observations. 
 
 O = Altitude of Sun. 
 
 Observer, W. W. 
 
400 
 
 MEASUREMENT OF EARTH'S MAGNETIC FIELD 
 
 [16 
 
 j 
 
 HO Trai? " EQUATION 
 
 OF TIME. 
 
 DECLINATION. 
 
 At 
 
 Noon. 
 
 Variation. 
 
 At Noon. Variation. 
 
 h. m. s. I m. 
 
 18' 1 12 i +1 
 46" -5 31 ! 
 
 s. 
 
 41-74 
 
 + 2-35 
 
 48 
 4-9 
 
 49-9 
 4-9 
 
 
 192 
 43 
 
 12 31 55 1996 
 + 4 4-5 449 
 
 
 -0 1 15-1 ! +1 
 
 441 
 
 2-35 
 
 12 36 2445 
 
 HOUR ANGLE. 
 
 TIME INTO ARC. 
 
 Direct View. 
 
 Back View. 
 
 4h. 60 5h. 75 
 53m. 13 15 2m. 
 451s. 11 17 15 -3s. 
 
 ( 
 30 ( 
 3 5( 
 
 m. s. 
 Eq. of Time . . +1 441 
 Longitude 1 1K-1 
 
 
 
 
 33 5( 
 
 Correction ... + 29 '0 
 G.M.T 4 53 161 
 
 + 29-0 
 5 1 46-3 
 
 
 ALTITUDE. 
 
 log sin C= 1-98159 
 + log sin a - 1 '98941 
 
 Diffce. fiom S. . 4 
 C= 73 
 %C=D= 36 
 
 53 45-1 
 
 26' 17" 
 43 8 
 
 5 2 15-3 
 
 75 33' 50" 
 37 46 55 
 
 MERIDIAN 
 
 90 
 Dec.= 12 36 
 
 READINGS. 
 
 90 
 Lat.= 51 28 6 
 
 f-97100 
 -log sin S= 1-99834 
 
 log cos 0= T-97266 
 = 20 7' J 0=10 3' 
 log sin 10= 1-24181 
 2 
 
 a= 77 24 
 b= 38 31 54 
 
 a-6= 38 52 6 
 c= 19 26 3 
 
 b= 
 
 a 
 
 a + b 
 
 38 31 54 
 
 77 24 
 
 115 55 54 
 57 57 57 
 
 log sin 2 \0= 2^48362 
 + log(M l ~M z ) = '30103 
 
 Lcosc= 1-97453 
 L cos d= 1-72462 
 
 L sin c= 1-52209 
 L sin d= 1-92826 
 
 L Iin d= T ' 59383 
 + LcotZ>= 12733 
 
 Correction = -06 
 
 cos d~ si' 1 d~ 
 + L cot D' '11060 + L cot D'= 
 
 1-593S 
 HOf 
 
 ""cos d~ 
 + LcotD= 12733 
 
 L tan P= -37724 
 
 L tan Q= 1-72116 
 
 P= 67 14-5 
 
 Q= 27 45-2 
 
 L tan P 1 - -36051 L tan Q' = 
 
 P'= 66 26-6 
 0/= 26 51-3 
 
 P' + Q' 93 17'9= 
 Angle from S~ 86 42 1- 
 
 1:7044 
 
 P+Q= 94 59-7=4 
 Angle from 5= 85 0'3=S 
 
 Sun's Bearing^ 129 49'6 
 M! = Meridian Reading = 44 49 '3 
 Correction 1 
 Corrected Reading = 44 49 '2 
 
 Sun's Bearing= 131 29'4 
 A/2 = Meridian Reading = 44 47'3 
 M!-M= 2-0 
 
 
163] 
 
 GEOGRAPHICAL MERIDIAN 
 
 401 
 
 Take the mean of the four times of contact for the " front " transits, 
 and apply the chronometer correction so as to obtain the Greenwich mean 
 time corresponding to the mean of the times of contact. Also take the 
 mean of the vernier readings, applying where necessary a correction to 
 allow for any error due to the dark glass employed being prismatic. The 
 value of this correction is generally supplied with the instrument. It 
 can easily be obtained by looking with the telescope at a bright spot of 
 light, and seeing how much the telescope has to be rotated to bring the 
 image back on the cross-wire when the glass is interposed. 
 
 Obtain the same means for the back observations. 
 
 Convert the longitude into time, allowing four minutes for each 
 degree * i.e. calculate how long after noon at Greenwich, noon will occur 
 at the place of observation. Of course if the station is east of Greenwich, 
 noon will occur earlier than at Greenwich. 
 
 We have next to allow for the fact that owing to the eccentricity of 
 the earth's orbit the true sun is not in general on the meridian at neon by 
 mean time. The time before or after mean noon when the sun crosses 
 the meridian is called the equation of time, and is given for noon of 
 each day in the Nautical Almanac. What is required is the equation 
 of time at the mean time of the 
 " front " observations of transit. 
 This is obtained by multiplying 
 the variation of the equation of 
 time per hour by the number of 
 hours between the mean time of 
 transit for the front observations 
 and noon. The value of the sun's 
 declination at the mean time of 
 transit for the front observation 
 is also calculated from the quanti- 
 ties given in the Almanac. 
 
 Let o (Fig. 173) be the position 
 of the observer,^ the north pole of 
 the earth, and PP' the axis about 
 which the earth turns, cutting the 
 celestial sphere at the points P and FIG. 173. 
 
 P', z the zenith, and 2 the sun. 
 
 Then the great circle HGB is the celestial equator, and the great circle 
 DFE is the observer's horizon, and the angle to be calculated is DQF, for 
 the mean of the circle readings corresponds to the point F, where the 
 vertical circle through the sun at the time of the transit observations cuts 
 the horizon. To obtain this angle we must first calculate the hour angle 
 zp2, which is the angle the great circle p2p' through the sun makes 
 with the meridian PZP'. 
 
 To obtain this angle we have to calculate what is the local solar time 
 
 1 1 = 4 minutes; l' = 4 seconds; 15" = 1 second. Tables for this conversion 
 are given in Mathematical Tables, c/. Chambers, p. 434. 
 
 2 C 
 
 
402 MEASUREMENT OF EARTH'S MAGNETIC FIELD [ 163 
 
 corresponding to the mean time of the front transition observations. 
 Now this time will differ from Greenwich mean time (G.M.T.) on account 
 (1) of the equation of time, and (2) of the longitude of the place being 
 different from the longitude of Greenwich. The correction for equation 
 of time has already been calculated, and is positive if the true sun is 
 ahead of the imaginary mean-time sun. In the example this is the case, 
 for on April 23 the true sun crosses the meridian at Greenwich before 
 noon, as indicated by a mean-time clock. Hence solar time is ahead of 
 mean time, and we have to add the equation of time. 
 
 If the station is east of Greenwich, the sun will cross the meridian 
 before noon, as indicated by a clock keeping Greenwich solar time, and 
 hence the correction for longitude is positive. If the station is west of 
 Greenwich, the correction for longitude is negative, as in the example. 
 
 Applying the sum of these two corrections, we get the local solar time 
 corresponding to the time of transit for the mean of the front observa- 
 tions, and if the back observations were only a few minutes later, the 
 same correction may be used in their case. The difference between this 
 corrected time and noon gives the angle ZpS in time. This has to be 
 converted into degrees, &c., by means of the table given in the Nautical 
 Almanac, two small spaces on the form being set apart for the purpose. 
 
 We now know two sides, and the included angle of the spherical 
 triangle zp2, and have to calculate the value of the angle pz2. The side 
 2p, or the polar distance of the sun, is 90 - o2, that is, 90 - sun's 
 declination, the declination being taken plus when the sun is above the 
 celestial equator. The side ZP is the co-latitude of the place of observa- 
 tion, i.e. 90 - latitude. Calling the polar distance of the sun a, the 
 co-latitude &, the hour angle C, the angle pz2 A, and the angle p2z J5, 
 we have, by the usual formulae for the solution of spherical triangles, 
 
 a-b 
 A + B C S "IT C 
 
 C0t 
 
 -2- - 2' 
 cos - 
 
 a-b 
 sin n 
 
 and tan ~~ B = - L_ cot ~. 
 
 2 . a+b 2 
 
 sin _ 
 
 Calling - P and Q, we get from the above equations the 
 
 values c f P and (2, and then 
 
 A=P+Q 
 
 Now, when making the declination observations the telescope looks 
 at the A 7 end of the needle. Hence what we require is the reading o; 
 
 ng on 
 
164] HORIZONTAL COMPONENT OF EARTH'S FIELD 403 
 
 the horizontal circle when the telescope points due south, that is, we 
 require the angle Dz2, which is given by 
 
 This gives the actual angle through which the telescope must be turned 
 from its position when the transits were taken to that in which it points 
 due south. If the transits take place before noon, then this angle must 
 be added to the "sun's bearing," i.e. the mean of the vernier readings 
 corresponding to the transits, and in the afternoon subtracted from this 
 bearing. The result is the meridian reading of the circle. Two values 
 of this meridian reading are obtained, one corresponding to the front 
 observations, and the other to the back. If these agree to within a 
 minute, then the mirror was in such good adjustment that no correction 
 is necessary, the value given by the front observations being alone used 
 when calculating the declination. It can be shown that the correction to 
 be applied to the front reading is the difference between the front and 
 back readings multiplied by the square of the sine of half the sun's 
 altitude. This correction has to be applied so as to bring the corrected 
 result nearer the result given by the back observations. The method of 
 calculating the sun's altitude and the correction is sufficiently clearly 
 indicated in the example. 
 
 164. Measurement of the Horizontal Component of the Earth's 
 Field. In order to measure the horizontal component of the earth's field, 
 we have to determine the period of vibration of a suspended magnet, and 
 then determine the deflection which this magnet gives when placed at a 
 known distance from an auxiliary suspended magnet. 
 
 Suitable magnets may be made by taking lengths of Stubb's drawn 
 steel, having a diameter of about 3 mm. and a length of about 10 cm., 
 and carefully squaring off the ends. A number of such pieces are bound 
 together with iron wire and then heated in a furnace to a bright red 
 heat, and, holding them in a vertical position, they are plunged into cold 
 water. Those of the steel rods which remain straight after the harden- 
 ing are then heated in boiling water for eight or ten hours, and are then 
 magnetised to saturation by placing them inside a coil through which 
 an electric current is sent. After .magnetisation the magnets are again 
 maintained at a temperature of 100 for about five hours. In this way 
 magnets are obtained which only lose their magnetism very slowly. 
 Care, however, must be taken not to jar the magnets or allow them to 
 touch one another or any piece of iron. Also they must be handled with 
 care, as glass-hard steel is very brittle. 
 
 One of the magnets is suspended by a length of about 30 cm. of 
 unspun silk of such a thickness that it will just safely carry the weight 
 of the magnet. The upper end of the fibre is made into a single loop and 
 the lower into a double loop, the method of tieing such a double loop 
 being shown in Fig. 174. The upper loop is put on a hook, and the 
 magnet rests in the double loop, so that its axis is horizontal. The 
 
40 1 MEASUREMENT OF EARTH'S MAGNETIC FIELD [164 
 
 magnet will have to be protected from the disturbing effects of draughts by 
 
 enclosing it within a box with glass sides, such as that shown in Fig. 46. 
 
 Having removed all other magnets and iron from the neighbourhood 
 
 of the suspended magnet, this magnet is brought to rest, and then two 
 
 small strips of paper are gummed to the 
 front and back of the box, so that they lie 
 on the prolongation of the axis of the magnet, 
 or a telescope may be set up at a little dis- 
 tance, and in such a position that the end of 
 the magnet, when at rest, appears to coincide 
 with the vertical cross- wire. 
 
 The magnet having been set swinging 
 through an arc not greater than 10 by 
 bringing near another magnet, and this 
 deflecting magnet having been removed, the 
 period of the suspended magnet must be 
 determined either by timing 100 vibrations 
 with a stop-watch, or by the eye and ear 
 method described in 43. 
 
 A convenient form of magnetometer to 
 use in the deflection experiment is shown in 
 Fig. 175. It consists of a boxwood scale A, 
 1 metre long, which is attached to a circular 
 base of wood H. An upright of wood B has 
 a groove in one face, in which the suspended 
 system hangs. This system consists of two 
 short magnetised pieces of watch spring 
 cemented on opposite sides of a 'piece of 
 mica c. This piece of mica also carries a 
 light galvanometer mirror E. The front of 
 
 the groove in which the suspended system hangs is closed by a piece of 
 plate glass G. A notch is cut in the boxwood scale so that the sus- 
 pended magnets can be placed at a level with the upper surface of the 
 scale, while a V groove cut along the upper edge of the scale permits of 
 the deflecting magnet being supported on the scale. The mica to which 
 the magnets and mirror are attached serves as a damper to check the 
 vibrations of the system. 
 
 A line having been drawn in the east and west magnetic direction 
 through the position occupied by the magnet in the vibration experiment, 
 either by suspending a long magnetised needle or with a prismatic com- 
 pass, the magnetometer is placed with the scale immediately over this 
 east and west line. A telescope and scale (Fig. 23) are then set up, and 
 the image of the scale seen by reflection in the magnetometer is focused, 
 care being taken that the telescope scale is at right angles to the scale of 
 the magnetometer. 
 
 The magnet used in the vibration experiment is then placed on the 
 west side of the magnetometer needle, with its centre at 30 cm. from the 
 
 FIG. 174. 
 
164] HORIZONTAL COMPONENT OF EARTH'S FIELD 405 
 
 needle, and the reading on the telescope scale is noted. The magnet is 
 then reversed, and the reading again taken. Two similar readings are then 
 taken with the magnet placed with its centre at 30 cm. on the east side 
 of the magnetometer needle. The differences between the pair of readings 
 taken on either side gives twice the angle through which the needle has 
 
 FIG. 175. 
 
 been deflected, and the mean of the two values obtained on the two sides 
 is taken when calculating H from the results. If this mean at any given 
 distance is d, while the distance between the mirror and the telescope 
 scale is , the angle 6 through which the needle has been deflected is 
 given by 
 
 tan 20 = i 
 
 Similar sets of deflections must be mads with the centre of the 
 deflecting magnet at 35 and 40 cm. from the magnetometer needle. 
 
 To reduce the vibration observations it is necessary to know the 
 moment of inertia of the magnet. This quantity can be calculated if the 
 
406 MEASUREMENT OF EARTH'S MAGNETIC FIELD [ 165 
 
 length /, diameter 2r, and mass m of the magnet are measured, when the 
 moment of inertia is given by 
 
 *& 
 
 Special care must be taken when measuring the length, as the propor- 
 tional error produced by an error in the length is much greater than 
 that produced by an error in the diameter. 
 
 If T is the period of the magnet, and M its magnetic moment, then 
 (see Watson's Physics, 430, 431) 
 
 where D is the distance between the centre of the deflecting magnet and 
 the deflected needle, and 2L is the distance between the poles of the 
 deflecting magnet. This distance may be taken as 2/3 of the length of 
 the magnet. 
 
 Now if the length of the magnet is 10 cm., so that L = 5, and the 
 smallest value of the distance D employed in the experiments is 30 cm., 
 
 (T 2\ 2 
 1 - - j by the binomial theorem and 
 
 neglect all the terms but the first two, so that the formula reduces to 
 M 1 
 
 a form which is more convenient v for calculation, since the correction 
 term 2L-/D 2 cm easily be evaluated on a slide rule. 
 
 165. Comparison of the Values of " H " at different parts of a 
 Building. It is sometimes necessary to measure the ratio of the values 
 of the horizontal component of the earth's field at two spots in the same 
 room or building, that is, in two neighbouring points. This may be 
 done most readily by datermining the times of vibration of a suspended 
 magnet in the two positions. The method of making the observation is 
 similar to that adopted when making the vibration experiment described 
 in the previous section. Having made a determination of period at A, 
 and noted the temperature of the inside of the box in which the magnet 
 was suspended, a similar determination is made at B. A second de- 
 termination is then made at B, and then a second at A. In this way the 
 mean time of the observations at the two stations is the same, and thus 
 the effects of diurnal change and disturbances are minimised. Since the 
 horizontal force is, owing to diurnal change, near a minimum at 11 A.M., 
 it is advisable not to time the observations so that the observations at 
 station A are one before and the other after 11 A.M. If two observers 
 
166] MEASUREMENT OF "H" 407 
 
 are available, then the effects of all changes in H during the course of 
 the comparison can be eliminated by the two observers making deter- 
 minations of period simultaneously with two magnets, one at either 
 station, and then interchanging stations. 
 
 If the temperature differs at the two stations, then a correction will 
 have to be applied, since the moment of a magnet changes with tempera- 
 ture. The temperature coefficient of the moment of the magnet can be 
 determined in the manner described in 167. If 7\ and T 9 are the 
 periods of the magnet at the two stations, the temperature at one station 
 being ^ and that at the other t. 2 . Then if a is the temperature co- 
 efficient of the magnet, and M the moment at the lower temperature, 
 say, t v the moment at t 2 is M{\ - a(t 2 - ^)}. Hence 
 
 A 
 
 Therefore 
 
 a 
 
 H l {l- a (*2-*l 
 
 -1 * 
 
 2 
 " 
 
 If a really accurate comparison is required, then further corrections 
 to allow for the rate of the chronometer, effect of temperature on the 
 moment of inertia, &c., must be taken into account, as described in 166. 
 
 166. Measurement of "H" with the Kew-pattern Unifilar Magneto- 
 meter. The determination of the horizontal component of the earth's 
 magnetic field with the Kew-pattern unifilar magnetometer consists in 
 two distinct operations. One of these is the determination of the period 
 of the collimator magnet which was used in determining the magnetic 
 meridian when it is vibrating in a horizontal plane, and the other is the 
 determination of the angle through which an auxiliary suspended magnet 
 is deflected by the collimator magnet. The first experiment gives the 
 product MH y where M is the magnetic moment of the collimator magnet, 
 and the other the quotient MjH, so that from the two observations both 
 H and M can be calculated. 
 
 1. Vibration Experiment. The collimator magnet is suspended as 
 in the declination experiment, and the vertical cross-wire brought into 
 coincidence with the centre of the scale. By means of the magnetised 
 screw-driver the magnet is then deflected, so that it vibrates over the 
 whole extent of the scale. The temperature as indicated by the thermo- 
 meter attached to the box containing the magnet, and the number of 
 divisions of the scale over which the magnet is swinging, are then noted, 
 and the observations of times of transit commenced. The method em- 
 ployed is the eye and ear method described in 43, a chronometer 
 beating half seconds being used. The time of transit of the centre 
 division of the scale when moving from left to right is noted down. 
 The magnet being of such size that where // has a value of about 0'18 a 
 
408 MEASUREMENT OF EARTH'S MAGNETIC FIELD [ 166 
 
 half vibration takes place in about four seconds, about twenty seconds 
 after the first transit the magnet will have completed five half vibrations, 
 and will then transit, moving from left to right. Thus about fifteen 
 seconds after the first transit the time is again taken up from the chrono- 
 meter, and the time of the transit from left to right, which occurs about 
 twenty seconds after* the first transit, is noted. This is repeated till the 
 times of six transits in either direction have been noted, the results 
 being entered as in the following example. The difference between the 
 times of transit of vibrations and 50 and 5 and 55 respectively give 
 the time occupied by fifty half vibrations. These times added to the 
 times of the 50th and 55th vibrations give the times to expect the 100th 
 and 105th transits. Just before the time at which tlie 100th transit is 
 expected the time is taken up from the chronometer, and the observation 
 of every fifth transit continued up to the 150th and 155th transit. In 
 this way twelve independent observations of the time of 100 half vibra- 
 tions are obtained, and the mean is taken in deducing the time of a 
 single vibration. Immediately after the observation of the time of the 
 last transit the amplitude of the vibrations must be noted as well as the 
 temperature. 
 
 Since the suspension fibre is never, so thin as to possess negligible 
 rigidity, the observed period has to be corrected to allow for the fact 
 that it is smaller than it would be supposing the fibre were entirely free 
 from rigidity. In order to be able to apply such a correction, it is neces- 
 sary to determine the magnitude of the couple brought into play when 
 the magnet is deflected due to the torsion of the fibre. To obtain the 
 necessary data, the number of scale divisions through which the magnet 
 is deflected when the torsion head, from which the suspension fibre is 
 supported, is turned through 180 is observed. The deflection is noted 
 when the torsion head is turned first in one direction and then in the 
 other, and the mean is taken. 
 
 The observed period has to be corrected to allow for (1) the rate of 
 the chronometer, (2) the fact that the arc of swing is not infinitely small, 
 (3) the torsion of the fibre, (4) the variation of the magnetic moment of 
 the magnet with temperature, and (5) the increase of the moment of the 
 magnet owing to the inductive action of the earth's field. To assist in 
 the application of these different corrections, a series of tables is supplied 
 with the instrument. 
 
 If the chronometer gains s seconds a day, one second as given by the 
 chronometer is really 1 - s/86400 true seconds, and hence the observed 
 period has to be multiplied by (1 - s/86400). If the chronometer loses 
 s seconds a day, the factor will be (l+s/86400). The values of the 
 factor for different rates are given in Table 8. 
 
 To reduce to infinitely small arc, the initial and final arcs must be 
 reduced from scale divisions to minutes of arc. The value of a scale 
 division in minutes is given in the table of corrections supplied with the 
 instrument. If it is not known, then it can be obtained by noting tht 
 angle through which the instrument has to be turned to cause the vertical 
 
166] 
 
 MEASUREMENT OF "H" 
 
 409 
 
 cross- wire of the telescope to pass from one end of the scale to the other. 
 If a and a are the initial and final amplitudes in radians the correction is 
 
 m , so that to correct for both rate and finite arc the observed time has 
 
 /, s aa'\ aa 
 
 to be multiplied by ( 1 - Sfiinn - -./. ). The values of , for different 
 
 values of a and a are given in Table 8. 
 
 If F is the couple called into play owing to the rigidity of the 
 suspension fibre when one end is twisted through an angle of one radian, 
 then if a deflection of the magnet of d (radians) is produced when the 
 torsion head is turned through 90, the couple due to the torsion is 
 F(ir/'2 - d). Since d is small, the couple due to the earth's field tending 
 to bring the magnet back into the meridian is MH<1. Hence 
 
 or 
 
 -d 
 
 
 Thus the force of restitution for a small rotation 6 is (MH+F)0, or 
 
 (7f \ 
 1 + ), so that the correction factor to reduce the square of 
 MHJ 
 
 MH 
 
 the periods is 1 + 
 
 The values of this factor for different values of 
 
 d are given in the following table : 
 
 VALUES OF 1 + FOR DIFFERENT VALUES OF THE DEFLECTION 
 JP 
 
 PRODUCED BY A TWIST OF 90 OF THE SUSPENSION FlBRE. 
 
 3tof90 l + Mff 
 
 Effect of 90 
 
 MH 
 
 wist. F ' 
 
 Twist. 
 
 { '~F" 
 
 r 
 
 1-00019 
 
 8' 
 
 1-00148 
 
 2' 
 
 1-00037 
 
 9' 
 
 1-00167 
 
 3' 
 
 1-00056 
 
 10' 
 
 1-00185 
 
 4' 
 
 1-00074 
 
 11' 
 
 1-00204 
 
 5' 
 
 1-00093 
 
 12' 
 
 1-00223 
 
 6' 
 
 1-00111 
 
 13' 1-00241 
 
 7' 
 
 1-00130 
 
 14' 
 
 1-00260 
 
 If the temperature of the magnet were the same during the vibration 
 and the deflection experiments, then the value obtained for H would be 
 independent of the temperature of the magnet. In general, however, 
 
410 MEASUREMENT OF EARTH'S MAGNETIC FIELD [ 166 
 
 there is a small difference in temperature between the two experiments, 
 and hence a correction has to be applied on this account. It is usual 
 to reduce the period and the deflection to what they would be if the 
 temperature of the magnet were C. A table of the values of the 
 magnetic moment at different temperatures, the moment at C. being 
 taken as unity, is supplied with each magnetometer, and serves for this 
 reduction. Since the magnetic moment decreases with rise of tempera- 
 ture, this correction is negative for temperatures above C. 
 
 Finally, the magnetic moment has to be corrected for the increase in 
 the moment produced by the inductive action of the earth's field. On 
 this account we have to add a quantity which is the product of a con- 
 stant factor /A, which is determined once for all, into HIM. The value 
 of HfM is obtained from the deflection experiment, while that of /A is 
 given in the tables of corrections. 
 
 The manner of applying the various corrections is shown in the 
 example which follows. The corrected value of the square of the 
 period has to be divided into ir 2 K to give the value of MH. Since the 
 moment of inertia of the magnet varies with the temperature, the value 
 of K varies with the temperature, and a table for different temperatures 
 is given amongst the constants of the instrument. 
 
 OBSERVATIONS OF VIBRATION. April 23, 1891. 
 
 STATION, Kew. Lat. ' ". 
 
 Chronometer, Dent. Error at Station = 
 Magnet (70 A) suspended, q 
 Effects of 90 of Torsion = 1 '2. Div. = 2' '2. 
 
 Long. ' ". 
 
 . Daily Rate (*)=+!!. 
 
 Log. M = 1-74630. 
 One Div. of Scale = 1' '8. 
 
 At Commencement ) Mean Time ) 11 h. 28 m. ( Semiarc ) 68' ( Temp, of n2 0> 7 ) 1<y , 
 At End f at Station f 11 h. 38 m. \ of Vib. f 20' ( Magnet f 13'l J Correction - -5 
 
 12-4 
 
 Scale moving apparently to the Right. 
 
 Scale moving apparently to the Left. 
 
 No. 
 of 
 Vib. 
 
 Time of 
 Centre 
 passing Wire. 
 
 No. 
 
 of 
 Vib. 
 
 Time of Time 
 Centre of . 
 passing Wire. 100 Vib. 
 
 No. 
 of 
 Vib. 
 
 5 
 . 15 
 25 
 35 
 45 
 55 
 
 Time of 
 Centre 
 passing Wire. 
 
 No. 
 of 
 Vib. 
 
 Time of 
 Centre 
 passing Wire. 
 
 77?. S. 
 
 35 18'5 
 35 58'5 
 36 38-6 
 37 18-5 
 37 58'5 
 38 38-5 
 
 Mean (2) = 
 
 Time 
 of 
 
 100 Vib. 
 
 
 10 
 20 
 30 
 40 
 50 
 
 h. m. s. 
 11 28 19-0 
 28 58-9 
 29 38-9 
 30 18-8 
 30 58-7 
 31 38-7 
 
 100 
 110 
 120 
 130 
 140 
 150 
 
 m. s. m. s. 
 34 58-5 6 39-5 
 35 38'5 -6 
 36 18-6 -7 
 36 58-5 -7 
 37 38-5 -8 
 38 18-5 -8 
 
 h. m. s. 
 11 28 39-0 
 29 18-9 
 29 58-8 
 30 38-7 
 31 18-7 
 31 587 
 
 105 
 115 
 125 
 135 
 145 
 155 
 
 m. s. 
 6 39-5 
 6 
 8 
 8 
 8 
 8 
 
 >iff. 
 
 :>r50 
 
 3 197 
 
 41 
 
 for 50 
 
 3 19-7 
 
 43 
 
 00 at 
 
 34 58-4 
 
 Mean(l)^ 6 39 '68 
 
 105 at 
 
 35 18-4 
 
 6 39-75 
 
166] 
 
 MEASUREMENT OF "H" 
 
 411 
 
 s aa 
 86400 ~ 16 
 
 = 1-00000 
 
 86400 
 
 -== -ooooi 
 
 ID 
 
 S aa 1 -00001 
 "86400 16 ~ 
 
 l + M ^= 1-00039 
 -q(t Q ~t)= '00503 
 + /*-= -00110 
 
 //o 
 
 ~ 
 
 Mean (1) for 1 Vib.=3'9968 
 
 (2) ,, = 72 
 
 T =3 '9970 Log= -60173 
 
 Log = 1-99999 
 Log= -60172 
 
 2 
 
 Log = 1-20344 
 
 i-S-g? '99646 
 
 M Log =1-74630 
 |? Log =2 -70491 
 
 Log =1-99846 
 
 T2 Log = 1-20190 
 *K Log =2 -42247 
 
 M-^ ^ -00110 Log=3'04139 
 
 OBSERVATION OF TORSION. 
 Circle turned. Scale. Mean. I Diff. 
 
 = 40-0 
 + 180 = 42-5 40-0 2-5 
 
 
 = 40-0 
 -180 = 37-6 
 = 40-0 
 
 40-0 
 
 2-5 
 
 90= 1-2 divisions. 
 
 Observer, W. W. 
 
 2. Deflection Experiment. When making the deflection experiment 
 the wooden box which surrounded the vibration magnet is removed, the 
 glass suspension tube being screwed into the metal base of the instru- 
 ment, and a small auxiliary magnet, fitted with a mirror, is suspended. 
 A metal bar, having divisions on one surface, fits immediately below the 
 compartment in which the mirror magnet hangs. This bar serves to 
 suppDrt a small carriage, on the V's of which the deflecting magnet is 
 placed. This carriage is placed on the right-hand side of the mirror 
 magnet when looking towards the face of the mirror, and with its centre 
 at a distance of 30 cm. from the centre of the instrument. A small 
 sighting tube is then placed on the V's of the carriage, and the mirror 
 magnet adjusted so that its centre lies along the prolongation of this 
 sighting tube, so that the centre of the magnet will lie in the same 
 horizontal plane as the deflecting magnet when placed in the carriage. 
 
 To determine the position of the mirror magnet a small telescope 
 with an ivory scale attached is screwed to the base below the telescope 
 
412 MEASUREMENT OF EARTH'S MAGNETIC FIELD [ 166 
 
 used in the vibration experiment, and when making a deflection experi- 
 ment the cross- wire of this telescope is brought to the middle division of 
 the scale, as seen after reflection in the mirror attached to the magnet. 
 A thermometer is attached to the bar which supports the magnet, the 
 bulb being placed as near the magnet as possible. 
 
 The collimator magnet having been placed in the carriage with its 
 north end pointing towards the east, the whole instrument is turned till 
 the cross-wire corresponds with the centre division of the scale. The 
 temperature and time are then noted, and the verniers attached to the 
 azimuth circle are read. The carriage is then moved to 40 cm., and the 
 deflected reading obtained. The magnet is then reversed in the carriage, 
 and another deflection is obtained. Another deflection is then obtained 
 with the carriage at 30 cm., but with the magnet in the reverse position 
 to that when the first deflection was obtained. Finally, an exactly 
 similar series of deflections are obtained with the magnet placed on the 
 other side of the mirror magnet. The order of making the observations, 
 as well as the method of recording them, will be clear from a study of 
 the example given. 
 
 OBSERVATIONS OF DEFLECTION. April 23, 1891. 
 
 STATION, Kew. Lat. ' ". Long. ' ". 
 
 Mean Time at Station ; commencing 11 b. 57 m. , ending 12 h. 23 m. 
 
 Magnet (70 A) deflecting ; (70 C) suspended. One Div. of Scale = 60". 
 
 Deflecting Magnet. 
 
 Readings 
 of 
 Verniers. 
 
 Scale 
 Reading. 
 
 Correction 
 to Middle 
 of Scale. 
 
 Mean of 
 Verniers. 
 
 Corrected Means 
 Circle and 
 Reading. Differences. 
 
 Distance. 
 
 N. End. | Temp. 
 
 EAST. 
 
 
 o 
 
 1 II 
 
 Div. 
 
 / 
 
 o / n 
 
 1 It 1 II 
 
 Centimetre. 
 
 E. 
 
 13-3J 
 
 159 59 10 
 339 57 50 
 
 
 
 
 159 58 30 160 8 48 
 
 M 
 
 W. 
 
 13-4 1 
 
 129 1 20 
 309 20 
 
 
 
 116 4 25 
 129 50 '44TT23' 
 
 M 
 
 E. 13 -6 1 
 
 147 8 
 
 327 7 20 
 
 
 
 22 2 Hi 
 
 
 -, , 
 
 W. 
 
 13 -8 1 
 
 116 10 
 295 59 50 
 
 116 
 
 
 
 
 
 
 
 WEST. 
 
 
 
 
 I 
 
 Centimetre. 
 
 W. 
 
 14 -2 1 
 
 116 9 
 296 8 40 
 
 116 8 50 147 10 40 
 
 ,, 
 
 E. 
 
 14 -5 1 
 
 147 14 
 327 13 20 
 
 
 
 129 '2 17 
 147 13 40 ~18"" 23 
 
 ., 
 
 W. 
 
 14 "7 1 
 
 129 4 20 
 309 3 10 
 
 
 
 129 345 
 
 
 
 E. 
 
 15 'OJ 
 
 160 19 50 
 340 18 20 
 
 
 160 19 5 
 
 Mean 
 
 <= 141 
 
 
 Correction 
 
 -1 
 
 
 
 14-0 
 
166] 
 
 MEASUREMENT OF "H" 
 
 413 
 
 r=30 
 
 r'=40 
 
 9u 
 
 1+=1-00041 1-00017 
 
 + (t 9 -t)q= -00571 
 
 >v 
 
 00571 
 1-00588 
 
 r=30 
 
 ij-3 Log =3-13066 
 Sin M Log =1-57425 
 
 Log= -00265 
 
 rft 
 
 Log =9 -99547 
 
 ~ Log =2 -70303 
 MH Log =1-22057 
 
 H Log =1-25877 
 H= -18145 
 
 Log M= 1-961 80 
 Jf =91 -580 
 
 3-50548 
 1-19765 
 
 2-70313 
 00254 
 
 2-70567 
 9-99746 
 
 2-70313 
 1-22057 
 
 2-51744 
 
 1 -25872 
 18143 
 1-96185 
 91-590 
 
 W. W 
 
 f> 
 To reduce the observations, the means of the vernier readings in each 
 position are first entered. Then the mean of the two deflections at 
 30 cm. in the anti-clockwise direction is deducted from the mean of 
 the two deflections in the clockwise direction, the difference being twice 
 the deflection u (} . A similar operation with the deflections taken with 
 the magnet at 40 cm. gives the deflection u' at this distance. 
 
 The values of log ?' 3 /2 for different temperatures is given in the 
 table of constants, and this is looked out and written down. Then 
 log sin UQ is added. Corrections have now to be applied for induction 
 and temperature. These are looked out in the tables, added together, 
 and their logarithm added to log ?' 3 /2 . sin u . 
 
 The next correction has to be applied to allow for the fact that the 
 
 TT 
 
 3 sin UQ only holds if the magnets concerned are 
 To allow for the finite length of the magnets, the value 
 be multiplied by (1-P/r 2 ), where P 
 
 formula 
 
 
 infinitely short. 
 obtained for 
 
 has to 
 
 s 
 
 obtained from the values of log M'jH* obtained at the two distances 
 by means of the following formula : ! 
 
 A- A 
 
 A A" 
 
 ,.2 ~'2 
 
 P = 
 
 where A and A' are the values of M ' JH' at the distances r and r. 
 
 1 For a graphical method of calculating P and log (1 - P/r 2 ), see Terrestrial 
 Magnetism (1901), vol. vi. p. 70. 
 
414 MEASUREMENT OF EARTH'S MAGNETIC FIELD [ 167 
 
 To obtain a satisfactory value for P, it is necessary to take the mean 
 of a number of observations and use this mean value when reducing all 
 the observations. Since the value of P only changes very slowly with 
 the age of the magnets, it is generally sufficient to use a mean value 
 for P obtained from all the observations taken in any year. 
 
 Having added log (1 - P/r 2 ) to the value of log M'/H', the value 
 of log MH is copied down from the vibration experiment, and the values 
 of H and of M are deduced. The value of H corresponds to the mean 
 of the values of this quantity during the vibration and the deflection 
 experiments, and would require to be corrected for diurnal variation and 
 any accidental disturbance by means of the record of a self-recording 
 magnetograph, such as those set up at a magnetic observatory. 
 
 167. Determination of the Temperature Coefficient of a Magnet. 
 The magnetometer described on p. 404 may be used in measuring the 
 change in the moment of a magnet with change in temperature. The 
 magnet is supported on V's within a glass or porcelain vessel, and is 
 placed due north or south of the magnetometer, with its axis east and 
 west, and at such a distance that the deflection produced is nearly the 
 full extent of the telescope scale. The magnet must be firmly fixed in 
 the V's, and the vessel fixed in position with the magnet in the same 
 horizontal plane as the magnetometer needle. 
 
 The vessel containing the magnet having been filled with a mixture 
 of ice and water, the magnet is allowed to take up the temperature, and 
 then the deflection is read off on the scale. The water is then syphoned 
 off and replaced by water at about 20 C. and the deflection again 
 read, the temperature being read on a thermometer placed in the water, 
 which must be kept well stirred. By gradually syphoning off the water 
 and adding hot water a series of readings are thus obtained every 
 20 up to nearly 100 C. Then by adding first cold water and then ice 
 a second series at as nearly the same temperatures as the first set, but in 
 the reverse order, is taken, and finally the magnet is removed and a 
 second zero is obtained. The means of the temperatures and deflections 
 at each temperature are then taken, and the values of their deflections are 
 plotted on a curve against the temperature. If the points obtained lie on 
 a straight line, then the expression for the moment M at a temperature t 
 is given by an equation of the form 
 
 in which the value of the quantity a can at once be deduced from the 
 straight line drawn on the curve (see 5). In general, the moment 
 decreases a little faster at high temperatures than at low, and an expres- 
 sion of the form 
 
 has to be employed. In such a case the values of the constants a and {$ 
 
168] 
 
 MEASUREMENT OF THE DIP 
 
 415 
 
 must be obtained from the curve by one of the methods given in 
 
 5, 
 
 168. Measurement of the Dip with the Kew-pattern Circle. 
 
 The Kew-pattern dip circle is shown in Fig. 176. It consists of a metal 
 base on levelling screws, the upper part of the instrument being capable 
 of rotation about a vertical axis with reference to this base, the 
 azimuth being indicated by a divided circle and vernier H. The needle 
 AB rests on two agate knife-edges K, K', and the position of the ends of the 
 
 needle are observed by two small microscopes M, M' and the attached 
 verniers which read on the vertical circle. These microscopes and verniers 
 can be slowly moved by means of the tangent screw T. Two V's L, L' 
 (Fig. 176, ), actuated by a screw head E, serve to lift the needle from 
 the knife-edges and replace it so that the axis coincides with the axis of 
 the vertical circle. A level L serves to show when the vertical axis 
 of the instrument is vertical, in which position the agate knife-edges are 
 horizontal. A small wooden block, with a clip to protect the axle of the 
 needle, is used to support the needle while it is being magnetised. 
 
 The circle having been set up and levelled, the needle is placed in the 
 
. 416 MEASUREMENT OF EARTH'S MAGNETIC FIELD [ 168 
 
 magnetising clip, and taking one of the magnets supplied in each hand, 
 that in the right hand having the north pole downwards, and that in the 
 left the south pole downwards, the needle is stroked ten times, starting at 
 the middle and drawing the two magnets apart till they are past the ends 
 of the needle. The needle is then reversed in the clip, and the other side 
 stroked ten times in the same manner. In this way the needle is 
 magnetised. Since it is of importance that the needle should be 
 magnetised to the same extent each time, it is essential that the stroking 
 should be performed in exactly the same manner throughout. 
 
 The needle having been removed from the clip, in which operation, 
 and on all other occasions when the needle is handled, care being taken 
 not to touch the axles, the axles must be cleaned by pressing them into a 
 piece of cork or dry pith. They must then be dusted with a clean camel's- 
 hair brush. The agate knife-edges having been rubbed lightly with the 
 pith are also dusted, and then the needle is placed in the instrument, 
 with the side marked AB turned away from the vertical circle. The two 
 verniers are then adjusted to read 90, that is, the line joining the two 
 wires of the microscopes is vertical, and the whole instrument is turned in 
 azimuth till the upper end of the needle coincides with the wire of the 
 upper microscope. The needle, while this adjustment is being made, 
 should be raised and lowered several times by means of the screw E, and 
 the adjustment made so that the end appears to swing equally on the 
 two sides of the wire. It is always best to have the needle swinging 
 over a small arc, as in this way the effects of dust on the axle or knife- 
 edges is more easily observed, when, if necessary, the axle can again be 
 cleaned in the manner already employed. The adjustment being com- 
 plete, the vernier on the azimuth circle is read and recorded. A similar 
 reading is then obtained when the lower end of the needle is made to 
 coincide with the wire of the lower microscope. 
 
 The whole upper part of the instrument is then turned through 
 1 80, and the settings for the upper and lower ends of the needle repeated. 
 The needle is then reversed, so that the face lettered AB is turned towards 
 the vertical circle, and a setting for each end of the needle made, first 
 with the instrument in its present position, and then when turned through 
 180. The mean of all the readings obtained gives the azimuth reading 
 when the axle of the needle points east and west, that is, when the plane 
 in which the needle swings is perpendicular to the magnetic meridian. 
 The azimuth circle is so divided that if the vernier is set to this reading 
 in the next quadrant, then the plane in which the needle swings will 
 be that of the magnetic meridian. 
 
 The instrument having been turned with its face towards the east, 
 and having been set to the correct azimuth reading, a stop attached to 
 the base plate is adjusted, so that after rotation the instrument can be 
 replaced in the same azimuth. The needle having been set swinging, the 
 upper microscope is adjusted on the upper end of the needle and the 
 corresponding vernier is read. Two independent settings are then made 
 with the lower microscope, the needle being raised from the agates 
 
169] COMPARISON OF MAGNETIC MOMENTS 417 
 
 between the two, and then another setting with the upper. If these 
 settings differ by more than two minutes of arc, then both the axle of 
 the needle and the knife-edges must be cleaned, and this must be repeated 
 till the readings agree to within this limit. The instrument is then 
 turned through 180, the vernier H being adjusted to the correct azimuth 
 reading, and a second stop is fixed in position, and then readings are 
 made for the two ends of the needle as before. The needle is then 
 reversed in the V's and another set of readings taken, first with the 
 instrument facing west, and then when it faces east. 
 
 Next the needle is removed and remagnetised with its polarity 
 reversed, and a similar set of readings obtained. The mean of all the 
 readings gives the dip at the time of observation. 
 
 The series of readings is so arranged as to eliminate errors due to the 
 centre of gravity of the needle not coinciding with the centre of the axle, 
 and the agate planes not being exactly horizontal, or the line joining the 
 90 marks on the vertical circle truly vertical (see Watson's Physics, 
 434). The example on p. 418 illustrates the method of recording the 
 observations. 
 
 169. Comparison of the Magnetic Moments of Two Magnets. 
 1. Deflection Method.- The magnetic moments of two magnets may 
 be compared by means of the magnetometer shown in Fig. 175. The 
 magnetometer having been set up as described on p. 404, the two magnets 
 are placed on the cross-bar, one on either side of the needle, and with 
 like poles towards one another. Their distances are then so adjusted 
 that the needle is undetected. The positions of the centres of the 
 magnets having been noted, the magnets are turned through 180, and 
 leaving one of the magnets, say, A, at the same distance as before, the 
 other is moved till the deflection is again reduced to zero. The two 
 magnets are then interchanged, and keeping the magnet A at a fixed 
 distance from the needle, the position of the other magnet is adjusted to 
 give no deflection, first, when the two north poles are turned towards the 
 needle, and secondly, when the two south poles are so turned. The 
 mean of all the readings for the centre of magnet B is then taken, when 
 the ratio of the moments is obtained by means of the expression 
 
 M t Df 
 
 where D l and D 2 are the distances of the centres of the magnets from the 
 needle. 
 
 If the two magnets differ much in length, or if the distances from the 
 needle at which they have to be placed differ considerably, then account 
 will have to be taken of the distance between the poles of the magnets in 
 deducing the ratio of their moments. It will generally be sufficient if 
 the distance between the poles is taken as 2/3 of the total length of the 
 magnets, and the following expression, in which L^ and L 9 are the distances 
 
 2 D 
 
418 MEASUREMENT OF EARTH'S MAGNETIC FIELD [ 169 
 
 MAGNETIC DIP. 
 
 STATION, Kew. 
 
 Circle No. 83. 
 
 Setting of Azimuth Circle, 32 37'. 
 
 Remarks. 
 
 Time, 2 '20 to 2 '35. 
 
 Date, 23rd of April 1891. 
 Needle No. 2 (94). 
 
 2 -37 to 2 -47. 
 
 Face of Needle to face of Instrument. 
 
 Poles direct, A dipping. 
 
 Poles reversed, B dipping. 
 
 Face 
 of 
 Instr. 
 
 Readings of Needle.. 
 
 Readings of Needle. 
 
 Lower end. 
 
 Upper end. 
 
 Mean. 
 
 Lower end. 
 
 Upper end. 
 
 Mean. 
 
 I 
 
 67 39 
 37 
 37 
 
 67 28 
 25 
 24 
 
 33-5 
 31-0 
 30-5 
 
 67 26 
 
 26 
 
 67 9 
 
 7 
 
 17-5 
 16-5 
 
 Sum 
 Mean = a = 
 
 5-0 
 
 Sum 
 
 
 31-7 
 
 17-0 
 
 WEST. 
 
 67 30 
 31 
 30 
 
 67 11 
 14 
 12 
 
 20-5 
 22-5 
 21'0 
 
 67 57 
 56 
 
 67 43 
 41 
 
 50-0 
 48-5 
 
 Sum 
 Mean = a' 
 
 1-0 
 
 Mean 
 
 Sum 
 
 
 21-3 
 
 49-3 
 
 Face of Needle reversed. 
 
 WEST. 
 
 67 52 
 51 
 
 67 35 
 
 37 
 
 43'5 
 44-0 
 
 67 36 
 36 
 
 67 20 
 21 
 
 28-0 
 28-5 
 
 Sum 
 Mean = a" 
 
 
 Mean 
 
 Sum 
 = &" = 
 
 
 43-8 
 
 28-2 
 
 I 
 
 67 20 
 
 20 
 
 67 3 
 67 1 
 
 11-5 
 10-5 
 
 67 44 
 43 
 
 67 27 
 29 
 
 35'5 
 36-0 
 
 I 
 
 Sum 
 
 lean = a'" 
 a" 
 a' 
 a 
 
 67 ll'O 
 43-8 
 21-3 
 317 
 
 B 
 
 Mean of 
 Do. 
 
 Sum 
 
 b" 
 b' 
 b 
 
 Sum 
 
 Means =/3 
 do. = a 
 
 67 35-8 
 28-2 
 49-3 
 
 
 Sum 
 
 107-8 
 
 130-3 
 
 Means of 
 
 Means = a 
 
 67 26-9 
 
 67 82-6 
 
 Observer, W. W. 
 
 .+=D 
 
 ip = 67 29-8 
 
169] COMPARISON OF MAGNETIC MOMENTS 419 
 
 between the poles, is immediately deduced from that given in 429 of 
 Watson's Physics : 
 
 -vy 
 
 DJ) 
 
 l 
 
 or since l and _* are generally fairly small, 
 
 1 2 
 
 2. Vibration Method. If the moments of inertia of the magnets 
 are known, or can be calculated, then the ratio of the moments can be 
 obtained by determining the periods of vibration when they are in 
 succession suspended by a fine silk thread at the same spot. A deter- 
 mination of period with magnet A should first be made, then two with 
 magnet B, and finally a second with magnet A, so that the mean time of 
 the experiment may be the same for each magnet, and thus to a certain 
 extent the effects of alterations in the earth's field eliminated. If 7\ and 7 T 9 
 are the mean periods, and K l and K 2 the moments of inertia, we have 
 
 M K T^ 
 
 III! =- til j 
 
 " ' ' 
 
CHAPTER XXV 
 
 ADJUSTMENT AND USE OF GALVANOMETERS 
 
 170. Measurement of Angles of Rotation by means of a Mirror and 
 
 Scale. A method much used in Physics of measuring the angle through 
 
 which a body has rotated is to attach a mirror to the body and observe 
 the movement of a beam of light reflected from the mirror. There are 
 two arrangements commonly employed. In one an image of an illumi- 
 nated slit is formed by the mirror in a divided scale, and the rotation of 
 the mirror is deduced from the displacement of the spot of light. In the 
 
 B 
 
 FIG. 177. 
 
 other arrangement a telescope is used, and an image of a scale seen by 
 reflection in the mirror is observed, the rotation being deduced from the 
 change in the reading of the scale corresponding to the cross- wire of the 
 telescope. 
 
 The only advantage of the lamp and slit is that the image on the 
 scale can be examined by more than one observer at a time. The 
 disadvantages are that the room has to be partly darkened in order to 
 allow of the image being seen, further it is difficult to read the position 
 of the image, and to focus the image it is in general necessary to move 
 
 420 
 
170] MIRROR AND SCALE 421 
 
 the scale, a procedure which will upset the adjustment of this latter. 
 With the telescope, of course, only one person can observe ; this limita- 
 tion, however, is generally of no importance. Since the focusing is done 
 by the telescope independent of the scale, the focus can always be 
 adjusted without disturbing the scale. With a telescope of about an 
 inch aperture, and^ a reasonably good mirror, it is possible to - read the 
 position of the cross-wire on a scale placed at a metre from the mirror 
 to within '05 of a millimetre, while with the lamp and scale to read 
 to '2 mm. is difficult. Hence where it is of importance to read the 
 deflection accurately, the telescope method should always be employed. 
 
 We will first suppose that the plane of the mirror M (Fig. 177) is 
 parallel to the axis about which the rotation takes place. Let AB be the 
 scale, and o the foot of the perpendicular from the axis about which 
 the mirror turns to the scale, the perpendicular distance between the 
 mirror and the scale being D. We will suppose that when the plane of 
 the mirror is parallel to the scale, the image of the point o on the scale 
 is on the vertical cross- wire of the telescope. Then if the mirror turns 
 through an angle 6 so that the image of the point E of the scale now 
 coincides with the vertical cross- wire, and we call the distance OE d, we 
 have, since the angle through which the reflected ray is turned is twice 
 
 angle through which the mirror turns, 
 
 tan 26 = -, 
 
 from which 6 can be calculated. 
 
 When 6 is fairly small, it is possible to obtain an expression for 6 and 
 the other functions of 6 commonly required, such as tan 6, sin 6, and 
 
 sin 6/2, in terms of powers of --' Thus using the symbol p for d/D, we 
 have, by Gregory's series, 1 
 
 i i i ^ 
 
 (i) 
 
 If p is not greater than '2, i.e. if we are dealing with deflections d less 
 than 200 scale divisions when the distance between the mirror and scale 
 is equal to 1000 scale divisions, the term in p 6 in the bracket is negligible. 
 For values of p less than '1, in the same way the term in p 4 is negligible, 
 while for values of p less than '01 the term in /o 2 does not affect the 
 results to more than one part in 30,000. If we are able to read the 
 deflection on the scale only to one part in 500, then p may be as great as 
 '05 without it becoming necessary to consider terms in p 2 or higher 
 powers ; this corresponds to a deflection of 50 scale divisions when the 
 distance between the mirror and scale amounts to 1000 divisions. 
 
 1 The value obtained for 6 is in circular measure. To reduce to degrees we 
 must multiply the value obtained for by 57 s '296, or 57 17' 45". 
 
422 ADJUSTMENT AND USE OF GALVANOMETERS [ 170 
 In Table 24 are given for different values of p the values of 
 
 the terms p 2 - -p 4 , so that the correction to be applied in any given 
 a 5 
 
 case can be obtained by interpolation. Thus if /a='15, the table gives 
 '00740 as the value of the correction terms, so that 
 
 = (l- 0-00740 
 
 By inserting the value of 6 given by equation (1) in the expressions 
 for the sine, tangent, &c., we obtain 
 
 and the corresponding corrections are given in Table 24. 
 
 If the reading on the scale in the undeflected position does not 
 coincide with the foot of the perpendicular drawn from the axis of 
 rotation on to the scale, but to some point such as F (Fig. 177), then 
 calling OF d$, we have 
 
 tan EMO = d/D, and tan FMO = d Q /D. 
 Hence angle EMO^{ I- \(ff + J(g* -*, } 
 
 angle 
 And 0, the angle through which the mirror has turned, is given by 
 
 Now, in practice the adjustment of the apparatus is always so nearh 
 ect that d is small, and hence the terms involving d within th' 
 
 correct 
 
 bracket may be neglected, so that 
 
 lfd\* 
 
 d-d,f \(d\ 
 ~2D~\ 3\D' 
 
 and in the correction terms an approximate value for d will give suff 
 ciently accurate results, the value of the correction being obtained froi 
 Table 24. 
 
MIRROR AND SCALE 
 
 423 
 
 170] 
 
 If the plane of the mirror is not parallel to the axis about which it 
 turns, then a correction will have to be applied. If F is the height 
 above the horizontal plane through the mirror of the point of intersection 
 of the axis of the telescope with the vertical plane through the scale, and 
 AS' is the height above the same horizontal plane of the scale, then the 
 distance between the mirror and scale to be used in computing the 
 deflection is 
 
 where D is the horizontal distance between the mirror and the vertical plane 
 through the scale. For the proof of the formula, the reader must refer 
 to a paper by F. Kohlrausch in Wiedemann's Annalen (1887), vol. xxxi. 
 p. 96. 
 
 The following corrections, in addition to those considered above, have 
 to be made when great accuracy is aimed at : 
 
 (a) For the thickness of the glass window. If the mirror is enclosed 
 in'a box which is provided with a plate glass window of thickness t, the 
 refractive index of the glass being /z, then the distance between the 
 mirror and scale must be reduced by the amount 
 
 where in general p may be taken as 3/2, so that the correction amounts 
 to t/3. 
 
 The above expression is obtained as follows : Let MB (Fig. 178) be a 
 reflected ray incident 
 on the glass plate at B. 
 This ray is refracted 
 along BC, and after 
 leaving the glass 
 travels along CF, meet- 
 ing the scale OA at F. 
 If there had been no 
 glass plate, the 
 
 ray 
 
 would have met the 
 scale at A. If FF' is 
 drawn perpendicular 
 to the scale, then the 
 reading on a scale O'F', M 
 if there were no glass 
 plate, would be the 
 same as the actual FIG. 178. 
 
 reading on the scale 
 
 OA when the plate is present, so that we virtually have to do with a 
 scale at a distance MO' from the mirror, and o'o is the correction to 
 allow for the effect of the glass. 
 
424 ADJUSTMENT AND USE OF GALVANOMETERS [ 171 
 
 If CN is drawn normal to the surface of the glass, CP is equal to F'F 
 or o'o. But if a is the angle BMO, the angle of incidence at u is a. 
 Hence calling the angle of refraction at B ft, we have 
 
 NB Q NB BG 
 
 But if, as is always the case, a is small, BP is very nearly equal to NP, 
 and BC very nearly equal to BC. Thus 
 
 NO 
 
 But PC=NC-NP. 
 
 Hence PC=NC- NC . 
 
 Thus 
 
 If the mirror consists of a piece of glass silvered on the lack, then 
 the distance between the front surface of the mirror and the scale must 
 be increased by 8//z, where 8 is the thickness of the glass of the mirror. 
 This formula follows immediately from the discussion given above. 
 
 171. Adjustment of a Telescope and Scale to measure a Rotation 
 by means of a Mirror. The telescope and scale shown in Fig. 23 can 
 be used for measuring rotations, and if, as is generally the case, the axis 
 about which the rotation takes place is vertical, the scale is placed hori- 
 zontally. The numbers on the scale are printed as follows : 
 
 'S'i 5 
 
 so that when seen by the telescope, which is not fitted with an erecting 
 eye-piece, after reflection in the mirror they appear the right way up. 
 
 For definiteness suppose the body of which the rotation is to be 
 measured is the needle of a galvanometer, a thin plane mirror being 
 attached to the needle system. 
 
 Having focused the eye-piece of the telescope on the cross-wires, and 
 adjusted one of these vertical, focus the telescope on an object at twice 
 the distance at which the scale is to be placed from the galvanometer. 
 Then hang a small plummet in front of the object glass of the telescope 
 so that it bisects the object glass, and having placed the telescope and 
 scale in front of the galvanometer at the desired distance, which may 
 conveniently be a metre, adjust the position till an image of the scale is 
 seen in the field. Then move the telescope and scale till the image of 
 the thread of the plummet where it crosses the scale coincides with the 
 vertical cross- wire. 
 
172] NEEDLE REFLECTING GALVANOMETER 425 
 
 It is next required to adjust the scale so that it is at right angles to 
 the axis of the telescope, that is, that the perpendicular from the mirror 
 on the scale shall intersect the thread of the plummet. In the case of 
 the telescope and scale shown in Fig. 23, the axis of the telescope is per- 
 manently fixed perpendicular to the scale, so that when the adjustment 
 described above is complete the scale must necessarily be perpendicular to 
 the line joining the mirror to the centre of the object glass of the telescope. 
 
 In those cases where the telescope and scale are capable of relative 
 motion, the easiest way of making the adjustment is to attach a small 
 piece of plane mirror to the scale behind the plummet and push in the 
 draw-tube of the telescope till an image of the galvanometer is seen. 
 This image is formed by reflection in the mirror attached to the scale 
 and the galvanometer mirror. Then rotate the scale about a vertical 
 axis till the image of the galvanometer mirror is bisected by the vertical 
 cross-wire. If when the telescope is again focused on the scale the 
 image of the plummet thread no longer coincides with the vertical wire, 
 the first adjustment must be repeated, and then the second, till both 
 adjustments are complete. 
 
 Having in this way completed the adjustment, it remains to measure 
 the distance between the mirror and the scale, To do this support a 
 metre scale in a retort stand at a higher level than the mirror and scale, 
 and by means of a level adjust it horizontal. Then hang two plummets, 
 and adjust one so that it just grazes the surface of the scale, and the 
 other so that it hangs in front of the galvanometer. A small pointed 
 rod of brass about 10 cm. long must then be held in a clamp, and care- 
 fully moved till the point just touches the surface of the mirror. Having 
 then adjusted the plummet nearer the galvanometer so that its thread just 
 touches the other extremity of this piece of brass, read the metre scale 
 where the two threads of the plummets cross. Then the difference of 
 these readings plus the length of the brass rod will give the distance 
 between the mirror and the scale. 
 
 If necessary, this distance must be corrected to allow for the effect of 
 the thickness of the mirror and the cover glass enclosing the galvanometer 
 mirror, as described in the last section. 
 
 172. Adjustment of a Suspended Needle Reflecting Galvano- 
 meter. First remove the controlling magnet, and then, if it is not 
 already known, determine whether the magnetism of the astatic system 
 is such that the instrument requires to be set up facing east or west. 
 Suppose that it requires to face the east. By means of a prismatic 
 compass or a long suspended magnetised needle draw a line north and 
 south, and then draw a line through the position to be occupied by the 
 galvanometer at right angles to this north and south line. Adjust the 
 galvanometer so that the coils are parallel to the magnetic meridian, and 
 level the instrument till the suspension hangs quite free. Whether the 
 suspension is free can usually be discovered by lightly tapping the case, 
 for if any part of the suspended system is touching the case, the system 
 will be seen to give a sudden jump when the case is tapped. 
 
426 ADJUSTMENT AND USE OF GALVANOMETERS [ 172 
 
 In order to observe the rotation of the suspended system, use may 
 be made either of a lamp and scale, or of a telescope and scale. The 
 telescope and scale is in general to be preferred, both on account of its 
 being the more accurate, and because it does not necessitate a partly 
 darkened room. Place the telescope and scale on the east and west line, 
 so that the scale is at some definite distance, such as 100 centimetres, 
 from the mirror of the galvanometer, and adjust its position in the 
 manner described in the last section. 
 
 The next adjustment which has to be made is that of setting the 
 magnets of the suspended system parallel to the coils, and of so placing 
 the directing magnet that the resultant couple, due to the earth's field 
 and the field of this magnet, when the needle is deflected should be 
 sufficiently small so that the required sensitiveness shall be attained. 
 The suspended system is generally so magnetised that the upper magnet, 
 in the case where the directing magnet is above, and the lower when the 
 directing magnet is below the galvanometer, shall be the stronger of 
 the astatic pair or sets of magnets. Thus to weaken the resultant 
 couple the directing magnet must be turned so that its north pole points 
 towards the north. The sensitiveness of the galvanometer being pro- 
 portional to the square of the period of the suspended system, we may 
 make use of observations of period to judge how the adjustment for 
 sensitiveness is proceeding. It will in general be found advisable to use 
 a period of about ten seconds, since a much longer period makes working 
 with the instrument tedious, and often involves changes of zero. The 
 desired sensitiveness having been attained, the director magnet is turned 
 till the central division of the scale coincides with the vertical cross-wire. 
 For most purposes we may now assume that the magnets of the sus- 
 pended system are parallel to the plane of the coils, since the suspended 
 system is made with the magnets and mirror in the same plane. In 
 some cases, however, it is necessary to arrange that the axis of the 
 magnet is accurately parallel to the galvanometer coil. This is best 
 
 done by setting up a circuit con- 
 sisting of a constant cell, such as 
 a Daniell or a small accumulator, 
 a very high resistance, a com- 
 mutator, and the galvanometer. A 
 suitable form of commutator is the 
 Pohl commutator, shown in Fig. 
 179. It consists of a base made 
 of some insulating material, such 
 as ebonite, on which are fixed 
 four mercury cups A, B, c, D. Two 
 metal uprights L, M pivot about 
 metal blocks attached to the base, 
 and to these uprights are soldered two semicircular pieces of strong 
 copper wire I and J, while a piece of glass or ebonite K connects the 
 upper ends of the uprights. The mercury cups A and D are 
 
 FIG. 179. 
 
173] SUSPENDED COIL GALVANOMETER 427 
 
 connected together by a copper wire, as are also the cups B and c. 
 Four binding screws E, F, G, H are connected as shown, and serve for 
 joining the commutator to the circuit. If the battery terminals 
 are connected to G and H, and the ends of the circuit containing the 
 resistance and the galvanometer are connected to E and F, then when 
 the movable part is rocked into the position shown, H is connected to 
 E, and G to F. If, however, the movable part is rocked towards the left, 
 then H is connected to F, and G to E, that is, the direction of the current 
 in the galvanometer is reversed. 
 
 A simpler form of commutator, which can be made in a few minutes, 
 but which is not quite so convenient, is shown in Fig. 180. It consists 
 of a "block of ebonite or paraffin wax, with four 
 small hollows filled with mercury, which are 
 either connected with binding screws as shown, 
 or into which the ends of the wires forming 
 the different parts of the circuit simply dip. 
 The mercury cups can be connected by two 
 small U-shaped pieces of copper wire, one of 
 which is so bent as to clear the other when 
 the cups are connected crosswise, as shown in 
 the figure. A and B being connected to the 
 battery, and c and D to the circuit, then by FlG - 18 - 
 
 putting the connectors straight across from A 
 
 to c and B to D, or crosswise as shown, the current can be sent through 
 the galvanometer in either direction. 
 
 To adjust the needle parallel to the coils, the resistance in the circuit 
 must be adjusted so that the deflection is about 10 centimetres, and the 
 directing magnet rotated till on reversing the direction of the current 
 the deflections on the two sides of the zero are equal. In the case of a 
 galvanometer without controlling magnet, it would be necessary to rotate 
 the whole galvanometer when making this adjustment, and hence it 
 should be performed before the final adjustments of the scale are made. 
 
 173. Adjustment of a Suspended Coil Galvanometer. In places 
 where the earth's field rs disturbed by the neighbourhood of moving 
 masses of magnetised matter, such as lifts or dynamos, or by the presence 
 of stray electric currents, such as exist in the vicinity of electric railways 
 or tramways, the ordinary form of suspended needle galvanometer cannot 
 be used, owing to the continual disturbance of zero which is produced. 
 In such a case a suspended coil galvanometer will be found of service. 
 In this type of galvanometer the current to be measured is passed 
 through a small coil which is suspended between the poles of a strong 
 permanent magnet, so that when undeflected the plane of the coil lies 
 parallel to the lines of force due to the magnet. The current is led into 
 and out from the coil by means of two thin ribbons of phosphor bronze. 
 The upper ribbon is generally straight, and serves as a suspension fibre 
 for the coil, while the lower ribbon is wound into the form of a helix, so 
 as to reduce its torsional rigidity to a minimum. 
 
428 ADJUSTMENT AND USE OF GALVANOMETERS [ 174 
 
 The azimuth in which such a galvanometer is set up is immaterial, 
 a point which is sometimes of importance. 
 
 The instrument having been levelled, so that the coil hangs freely, 
 the adjustment of the telescope and scale, or of the lamp and scale, is 
 made as in the case of the needle galvanometer. The sensitiveness, 
 however, cannot be varied by altering the field strength, unless the 
 permanent magnet is replaced by an electromagnet, in which case the 
 sensitiveness can be altered within certain limits by altering the strength 
 of the current in the electromagnet. 
 
 When using a suspended coil galvanometer it is advisable to have 
 a key by means of which the terminals of the galvanometer can be 
 connected together, and thus the suspended coil short-circuited. This 
 short-circuiting key serves to bring the coil to rest when it has been set 
 swinging, for, owing to the movement of the coil in the magnetic field, 
 currents are induced which rapidly damp out the oscillations of the coil. 
 On open circuit, however, the E.M.F. induced in the coil, owing to its 
 motion in the magnetic field, cannot produce a current, and hence very 
 little damping is produced, and the coil continues swinging for some time. 
 If the coil is of fairly high resistance, it is advisable to insert some resist- 
 ance in series with the short-circuiting key, as otherwise the motion of 
 the coil is so much damped that a very considerable time elapses before 
 the zero position is reached. The best results are attained when the 
 coil just passes beyond its position of rest and then comes to rest at the 
 second passage. 
 
 174. Determination of the Figure of Merit of a Galvanometer. 
 By the figure of merit of a galvanometer is meant the current in amperes 
 required to produce a deflection of one division of the scale. Since the 
 sensitiveness depends on the resultant field in which the suspended system 
 swings, in the case of needle galvanometers the figure of merit will depend 
 on the position of the directing magnet. It will also depend on the dis- 
 tance at which the scale is placed. For this reason it is usual to calculate 
 the figure of merit corresponding to a period of the needle of ten seconds 
 and for a scale distance equal to 1000 scale divisions, that is, in the case 
 of a scale divided into millimetres for a distance of 1 metre. Further, 
 since the strength of the deflecting field produced by the coils, when the 
 galvanometer is traversed by a current, depends not only on the strength 
 of the current, but also on the number of turns in the coils, and on the 
 position of these turns with respect to the needles, it is evident that a 
 galvanometer having a very great number of turns will, for a given current, 
 be likely to give a larger deflection than another containing a few turns, 
 and this although the latter may be the better galvanometer. 
 
 The only really satisfactory way of comparing the performance of 
 different galvanometers, of which the resistance varies, is to compare the 
 amount of electrical energy which has to be supplied to the galvanometer 
 when the period is ten seconds and the scale is at 1000 scale divisions 
 to maintain a deflection of one scale division. The electrical power is 
 obtained by multiplying the resistance of the galvanometer by the square 
 
174] FIGURE OF MERIT OF A GALVANOMETER 429 
 
 of the current required to give a deflection of one division under standard 
 conditions. 
 
 In order to find the figure of merit of a galvanometer it is connected 
 up with a battery and a series of resistances, as shown 
 in Fig. 181. Here B is the battery, which should 
 preferably consist of a single accumulator, which 
 after being fully charged has been allowed to 
 partly discharge through a resistance. K is a 
 make-and-break key, and may consist of a piece 
 of copper wire, the ends of which dip into two 
 mercury cups scooped in a piece of paraffin wax. 
 p and Q are two resistance boxes, and, if one is 
 available, it is of advantage that p should be 
 capable of variation by tenths of an ohm. R is 
 a high resistance, generally 10,000 or 100,000 
 ohms, while G is the galvanometer. The sum of 
 the two resistances P and Q is kept constant, and 
 may conveniently be made a thousand times the 
 E.M.F. of the battery. Thus if an accumulator is used, P+ Q may 
 be 2100 ohms; while if a Daniell cell is used, their sum may be 1080 
 ohms. 
 
 If E is the E.M.F. of the battery, and B its resistance, and P, Q, R, and 
 G represent the resistances of the various parts of the circuit, the E.M.F. 
 between the terminals of the resistance P when the galvanometer' circuit 
 is broken is given by 
 
 EP 
 
 P+Q + B' 
 
 If P is very small compared to K + G, as is generally the case, then this 
 E.M.F. when the galvanometer circuit is closed will be sensibly the same. 
 Hence the current in the galvanometer is 
 
 Generally B is so small compared to P + Q, that the expression for the 
 current may be written 
 
 
 _ 
 (P+Q)(R+G) 
 
 If we take into account the effect of the galvanometer circuit in reducing 
 the resistance between the terminals of the resistance P, since the resist- 
 ance of P and R + G in parallel is 
 
 P(R+G) 
 
430 ADJUSTMENT AND USE OF GALVANOMETERS [ 175 
 the E.M.F. between the terminals of P is 
 
 P(R + G)\ 
 
 L (2) 
 
 and the current in the galvanometer is this quantity divided by (R + G), 
 which is the same as the expression (1), where P(R + G)/(P + R + G) has 
 
 been written in place of P. Now P(R + Gr)j(P+R+G) may be written 
 
 (p \ 
 1 - ), so that unless P/(P+G + fl) is greater than, say, 
 P + Q> + R/ 
 
 001, we need not employ the more cumbersome expression for finding the 
 current in the galvanometer. 
 
 Determine for different positions of the directing magnet the period 
 and the current required to produce a deflection of, say, 5 or 6 cm. 
 In performing this experiment, adjust the resistance P so that a deflec- 
 tion of about the desired amount is produced, and then read the deflection. 
 The accuracy will be slightly increased if a commutator is included in the 
 galvanometer circuit, and deflections on either side of the zero are taken. 
 Plot on squared paper the current required to produce a deflection of 
 one division against the square of the period, and see how nearly the 
 points lie on a straight line. 
 
 Also determine the currents necessary for producing different deflec- 
 tions right up to the end of the scale, and plotting current against deflection 
 test whether the two are proportional. 
 
 175. Comparison of Electromotive Forces 
 with the Galvanometer. The two cells of 
 which the E.M.F.'s are to be compared, B X and 
 B 2 (Fig. 182), are connected in the manner 
 shown to a set of mercury cups in a block of 
 paraffin, the two middle cups being connected 
 to the galvanometer G- and a high resistance R. 
 A resistance s is placed as a shunt on the 
 galvanometer, so that the sensitiveness may 
 be reduced if required. The resistance R, 
 together with that of the galvanometer, ought 
 to be at least 10,000 ohms. 
 
 The shunt resistance is adjusted till with 
 the cell of higher electromotive force the 
 FIG. 182. deflection is almost to the end of the scale, and 
 
 then the deflections when first one and then the 
 
 other cell is connected with the galvanometer circuit are read. The 
 deflection experiment should be repeated several times, and the mean 
 taken. Also by slightly altering the shunt resistance, readings can be 
 taken with different deflections. If D l and D. 2 are the mean deflections, 
 then the ratio of the electromotive forces, so long as the resistance in 
 
175] COMPARISON OF ELECTROMOTIVE FORCES 431 
 
 the galvanometer circuit is very high compared to the resistance of either 
 of the cells, is given by 
 
 If G 1 and C 2 are the currents in the circuit in the two cases, we have 
 G= E *- 
 
 where 7? x and J9 2 are the resistances of the cells. 
 
 Hence if the deflections are proportional to the currents l 
 
 If B l and B^ are both small compared to R + G, the factor which 
 multiplies E-^j K^ is sensibly unity, and hence 
 
 a =5 
 
 A *V 
 
 To test whether we are justified in making the above assumption, we 
 have 
 
 G + v 2 B*-BI 
 
 G+'B l + R+G + B^ 
 
 T) jy 
 
 Hence if 2 1 ~- is negligible compared to unity, the approximation 
 R + (JT + JD-^ 
 
 is sufficiently accurate. 
 
 1 The experiments made in the preceding section will show whether this 
 assumption is justified. If the currents are not proportional to the deflections, 
 then the value of the ratio CyCa deduced from the curve obtained in 174 must 
 be used in place of Z>j/Z> 2 . 
 
CHAPTER XXVI 
 
 MEASUREMENT OF RESISTANCE 
 
 176. The Wheats-bone's Network of Conductors. That particular net- 
 work consisting of six conductors, 
 joined up as shown in Fig. 183, which 
 is called a Wheatstone's network or 
 bridge, will be found of such wide 
 application in the following pages 
 that a few words as to its general 
 properties may be of service. Let 
 p, Q, R, s, G, and b be the resistances of 
 the arms AC, CB, AD, DB, DC, and AB 
 respectively. If we suppose that 
 there is a battery of E.M.F. E placed 
 in the arm AB, while the resistances 
 p, Q, R, and s are so adjusted that a 
 galvanometer placed in the arm CD is 
 undeflected, then it at once follows 
 
 that PS = RQ. For the fact that there is no current in the arm DC shows 
 that the points D and c are at the same potential. Hence 
 
 drop of potential along AD 
 drop of potential along DB 
 
 drop of potential along A C 
 drop of potential along CB 
 
 But as there is no current in the arm DC, the current through AD is the 
 same as the current through DB. Hence the drop of potential along AD 
 is to the drop of potential along DB as the resistance of AD is to the 
 resistance of DB. Similarly the drop of potential along AC and CB are to 
 one another as the resistances of AC to CB. Hence 
 
 ^ 
 
 Q~ S 
 
 (1) 
 
 In exactly the same way we can prove that the condition that when an 
 E.M.F. is placed in the arm CD there shall be no current in AB is 
 
 P JB 
 
 a s' 
 
 that is, the same condition as before. The two arms AB and CD, such 
 
 432 
 
176] WHEATSTONE'S BRIDGE 433 
 
 that an E.M.F. in one produces no current in the other, are said to be 
 con jugute. 
 
 An important property of conjugate arms is that if we insert an E.M.F. 
 in any other arm of the network, then the current produced by this E.M.F. 
 in one of the conjugate arms is independent of the resistance of the other. 
 For suppose that an E.M.F. inserted in one of the arms, say, AC, produces a 
 current in AB. Then we may, if we please, introduce an additional E.M.F. 
 in the arm AB, and adjust its magnitude till the current it sends is just 
 equal and opposite to the current in AB, due to the E.M.F. in the arm 
 AC. Now the E.M.F. introduced in AB will produce no current in the 
 arm CD, since AB and CD are conjugate, and hence the introduction of 
 the E.M.F. in AB will not have affected the current in CD ; its effect, how- 
 ever, has been to reduce the current in AB to zero, and hence we may 
 make the resistance of the arm AB whatever value we like without affecting 
 the current in CD. 
 
 Hence to calculate the current in CD, produced by an E.M.F. e r 
 introduced in the arm AC,- we may take the resistance of the arm AB as 
 being infinite. This being the case, the current in AC is 
 
 e r 
 
 Q + S+G 
 Hence the current in CD is 
 
 e 
 
 _ 
 P , R+ Q + S+G* 
 
 Q + S+G 
 
 e r (Q + S) 
 or _ - x '- 
 
 (P + It) (Q + X + G) + (Q + S)G' 
 
 But the resistances P, Q, R, S are subject to the condition PS = RQ. 
 Hence the current in CD is 
 
 S 
 
 Exactly the same expression would be obtained for the current in CD, due 
 to the introduction of an E.M.F. e p in the arm AD. 
 
 Next suppose that we introduce an E.M.F. e s in the arm DB. The 
 current produced in CD is 
 
 P + R+G 
 
 2 E 
 
434 MEA8CFBEMEKT OF RESISTANCE 
 
 ~3) 
 
 - 
 
 n tJMJ OfWI a 
 
 e* It will he noticed tkrt the 
 re the same, 
 
 If the ewreirt sent throngi the arm ci>, when an E,M,F, 
 in A/J or AD, w the same a that sent when an E,M.F. #, is placed in CB or 
 KD, then 
 
 
 ftince whether the galvanometer is connected between c and i>, and 
 
 i ttery between A and u, or the galvanometer connects A and B, and 
 
 .es r; and i>, there will he no current in the galvanometer if 
 
 ft follow* that in any given owe we have a choice as to the 
 
 way in which the battery and galvanometer are connected up, a 
 
 -cessary to decide which is the better arrangement. If the 
 reitaricf;M are all of atxmt the name magnitude, then it is quite im- 
 material which way the connectif/n i made. If, however, the resist- 
 of very different rnagnitnde, then if 5 and S are the two 
 larger, arid, aft lit tutially the ease, the resistance of the galvanometer it 
 greater than that of the battery (including the battery leads and anjf 
 :riee which rray }& inserted in neries with the battery to reduce the 
 magnitude of the currents sent through the resistances), the galvano- 
 meter ought, to \K-. eonrifjct/jd to the point where the resistance* h and S 
 meet., arid t/> the f^int where the two lower resistances P and Q. meet. 
 If the resistance of the battery is greater than that of the galvanometer, 
 then the battery should connect, the jioirit where the two higher 
 
 meet to the point where the two lower ones meet. 
 Sine*; the passage of the currents through the resistances heats the 
 resistances, and h^nce alters their resistance, if any of the resistan*-- 
 of small section, r>r constructed of a material of which the res.' 
 ehangfrs rapidly with temjHrrature, then considerations of heat develoj>- 
 rnent, rathr-.r than the. aliovf:, ought to settle the manner in which the 
 Iwitte.ry in connected up. 
 
 Let us suppose that the. values of the. resistances are P== 100 ohm*, 
 Q 50 ohms, It 10 ohms, and ,S 5 ohms. Then when the }*atter-y is 
 connected, as in Fig. IN4 (a), we have 
 
 A.fo !! 10 
 
 0, y^+.v 15 
 
 If, however, tin; battery is connected, as shown in Fig. 1*4 (h), and 
 
METBE FORM OF WHRATSIUMKS BRIDGE 
 
 :,: . -1,- :: -.-. ,~__ : v- 
 
 LC-- 
 
 t\ P+M 110 _ 
 
 -.'..-: 55 
 
 are ffcrceoifer 
 
 Tbe gape at M and s cant, 
 
 and m whfle 
 
 nmns B, an m 
 d tke <*der to a 
 
 contact ** Made mitk UK 
 
 ffefe, Udk ha it> length placed at right 
 
 
 ft fa, 
 
436 MEASUREMENT OF RESISTANCE [ 177 
 
 slider by means of which the position of the contact is read off on the 
 scale. 
 
 In the simplest method of using this form of bridge, the two resist- 
 ances to be compared are connected to the terminals PJP., and Q X Q 9 . 
 Balance is obtained by moving the slider D, and then if "the wire is 
 
 TO BATTERY 
 
 TO GALVANOMETER 
 
 P, 
 
 -jmfflLVSW - GALVANOMETER 
 TO BATTERY 
 
 FIG. 185. 
 
 uniform, and the resistance of the connecting pieces F and H can be 
 neglected, and the slider divides the wire into portions of length p and q 
 respectively, we have 
 
 where A is the value of the resistance between Y I and P.,, and B that 
 between Q x and Q 2 . 
 
 If the resistances of the end connections of the wire, including the 
 soldered junction, are not negligible, suppose that the added resistance at 
 one end is equivalent to the resistance of a length \ p of the wire, the 
 corresponding quantity for the other end being A a , then the expression 
 giving the ratio of the two resistances which are being compared is 
 
 A p + ^ t 
 B q + X q ' 
 
 In order to determine the values of the end corrections A 7> and A (/ , 
 two resistances, the ratio of which are known, are inserted in the gaps 
 p and Q. Let the ratio of these two resistances be n, so that BiA = n. 
 Then 
 
178] CALIBRATION OF A BRIDGE WIRE 437 
 
 Next reverse the two resistances, placing A in the Q gap and R in 
 the P gap, and let the point of balance divide the wire into lengths of 
 p and q. Then 
 
 ' ' 
 
 Subtracting one of the equations from the other 
 q-q = n(p + \,) - ^(p + \ p ) 
 or 
 
 But since the whole length of the wire is 1000, <? = 1000-/?, and 
 q = 1000 -p'j therefore 
 
 Similarly 
 
 - 
 
 When performing the experiment, n should be large, say, about 100. 
 A convenient arrangement is to make one resistance 1 ohm, and the 
 other 101 ohms. Then w=101, and w-1, which occurs in the denomi- 
 nator of each of the expressions, is 100. 
 
 178. Calibration of a Bridge Wire by Carey Foster's Method. 
 Since the resistance of a unit of length of a wire is never quite uniform 
 throughout the whole length of a wire, it becomes necessary to calibrate 
 the wire of a Wheatstone's bridge, that is, to determine the manner in 
 which the resistance per unit length varies at different parts of the 
 length. The resistance of the whole of the bridge wire is either measured 
 by means of another bridge, or it may be obtained sufficiently accurately 
 in the following way : Two approximately equal resistances, say, 1 ohm 
 each, are placed in the gaps P and Q, while a 10-ohm coil is placed in 
 each of the gaps M and N, a box of coils being arranged as a shunt on 
 the coil in the M gap. The value of the shunting resistance is then 
 altered till a point of balance is obtained near the end c of the wire. 
 This point of balance having been determined, the coils in M and N, 
 together with the shunt, are interchanged, and the point of balance near 
 the end A of the wire is determined. 
 
 If i\ is the value of the resistance in one gap, and r. 7 the value of 
 that in the other, while the length of the wire between the two points 
 of balance is d, the resistance of unit length of the wire is given by 
 
 
 ~ d 
 
 he proof of this expression is given in 181. 
 
438 MEASUREMENT OF RESISTANCE [ 178 
 
 Suppose that it is desired to calibrate the wire by twenty steps, that 
 is, to divide the wire into twenty segments of equal resistance. A gauge 
 piece is prepared, consisting of a short length of thick German silver or 
 manganine wire, having a resistance equal to, or very slightly less than, 
 that of 50 mm. of the bridge wire, soldered to two copper tags suitable 
 for clamping in the gaps of the bridge. If the resistance of the gauge is 
 made slightly less than the desired value, it can be adjusted by filing so 
 as to reduce the diameter of the wire. 
 
 A stretched wire, which may conveniently be the wire of a second 
 bridge, is connected to the bridge in the manner shown in Fig. 186. 
 
 A battery is connected 
 to c and D, while the 
 galvanometer is con- 
 nected to the movable 
 
 ^ A | [h ^. contact E on the wire 
 
 p Q H Q Cr) ^ ^ e bridge, and to a 
 
 [L, Q Q t-jj y^ second movable contact 
 
 F on the auxiliary Avire. 
 
 The copper connector 
 belonging to the bridge 
 FIG. 186. being placed in the gap 
 
 N, and the % gauge in the 
 
 gap M, the slider E is placed as near the end of the wire as 'possible, and 
 slider F moved till, on completing the galvanometer circuit, no deflection 
 is produced. The reading for the slider E having been recorded, the 
 gauge and connector are interchanged, and the slider F remaining 
 fixed in position, balance is obtained by moving the slider E. The gauge 
 is then replaced in N, and balance obtained by moving the slider F, and 
 so on till the end of the wire AB is reached. When the gauge is in the gap M, 
 the slider E is moved to obtain balance, while, when the gauge is in N, 
 the slider F on the auxiliary wire is moved. 
 
 The portions of the wire AB (as also of the auxiliary Avire), between 
 the successive positions of the slider, have resistances equal to the differ- 
 ence between the resistance of the gauge and the connector used to 
 bridge the other gap. That this is the case is at once evident, for, 
 suppose that when the gauge is in N, the positions of balance are as shown 
 in Fig. 186. Then the fact that there is no current in the galvanometer 
 shows that the two points E and F are at the same potential. When 
 the gauge is in M, let the position of balance be at E', that is, the points 
 E' and F are now at the same potential. We then have that the resist- 
 ances of the gauge, the strip of copper at the end of the bridge, and the 
 portion BE of the wire is to the resistance of the connector, the strip of 
 copper at the end A and the portion AE of the wire as the resistance of the 
 connector, the copper strip at B, and the portion BE' of the wire is to the 
 resistance of the gauge, the strip of copper at A, and the portion AE' of 
 the wire ; that is, if G and c are used to indicate the resistances of the 
 gauge and connector respectively, while \ B is the resistance of the copper 
 
178] CALIBRATION OF A BRIDGE WIRE 439 
 
 connecting strips at the end B, that is, the resistance of the strips between 
 the point H, where the battery circuit divides, and the end B of the wire, 
 exclusive of the resistance in the gap N, X^ being the corresponding 
 quantity at the other end, and the resistances of the portions of the wire 
 are indicated by BE, &c., we have 
 
 or, subtracting the denominators from the numerators for new numerators, 
 and adding denominators and numerators for a new denominator, 
 
 G - C+ A,, - \ A + BE -AE C -G + \ B - 
 
 G + C+X A + \ B + BE + AE~ G+G+X A + X B + BE' + AE' 
 
 But BE + AE = BE' + AE' . 
 
 Hence 
 
 2(G -C) = BE- BE' + AE-AE' = 2EE' 
 or EE' = G-C. 
 
 In the same way, it can be shown that the steps on the auxiliary wire 
 have each a resistance equal to G - C. 
 
 When performing the experiment, care must be taken to protect the 
 wires and movable contacts from becoming heated by the presence of the 
 observer, otherwise thermo-currents will affect the results. The effects 
 of thermo-currents at the two movable contacts E and F may be eliminated 
 by having a key in the battery circuit, and then, having closed the two 
 contacts at E and F, and waited till the galvanometer deflection becomes 
 constant, closing the battery key, and noting whether any permanent 
 change of the deflection takes place. There may be a slight kick on 
 closing the battery circuit owing to induction in the circuits, but this will 
 in general be so small as to be unobservable. The effects of thermo- 
 currents at the ends of the wires can be eliminated by having a com- 
 mutator in the battery circuit, and obtaining the point of balance with 
 the current in either direction. 
 
 If l v / 2 , / 3 , &c., are the lengths of the bridge wire intercepted between 
 consecutive points of balance, then each of these represents the length of 
 the wire having a resistance equal to the difference in the resistances 
 of the gauge and connector. Then if the difference between the resist- 
 ance of the gauge and the connector is r, the mean .resistance of unit 
 length of the wire over the first step is r/l v that over the second 
 step r// 2 , and so on. The value of r can be at once obtained from 
 the observations, if the resistance R of the whole wire has been 
 determined. 
 
440 MEASUREMENT OF RESISTANCE [ 179 
 
 For if S/ represents the sum of the lengths of the steps, and L is the 
 total length of the wire, there being n steps, we have 
 
 from which r can be obtained, and hence the mean resistance per unit 
 length at the different parts of the bridge wire calculated. For the 
 purposes of interpolation, it is advisable to draw a curve in which the 
 ordinates are the readings on the bridge wire, and the abscissae are the 
 resistances of unit length. This curve is obtained by plotting the values 
 for the mean resistance of unit length at the reading of the wire corre- 
 sponding to the middle point of each step, and drawing an even curve 
 through the points thus obtained. Instead of plotting the actual resist- 
 ances per unit length, in the case where great accuracy is required, paper 
 may be saved by plotting the differences between the actual resistance of 
 unit length at the different parts of the wire, and the mean resistance 
 of unit length taken over all the wire, that is, the quantity RjL. 
 
 The objection to the above method of calibration lies in the fact that 
 the positions of the gauge and connector are continually being changed, 
 and hence if there are any differences in the resistances of the contacts 
 between the terminals of the gauge, or the connector, and the binding 
 screws of the bridge, errors will be produced. For this reason it is 
 necessary to make the terminals of the gauge of a flat piece of copper, 
 
 with a slot cut in it, so as to fit the screw 
 of the terminals, and to carefully clean 
 these copper plates as well as the binding 
 screws. 
 
 179. Calibration of a Wire by 
 means of a Double Circuit. The 
 following method of calibrating a wire 
 due to Griffiths 1 has the advantage over 
 Carey Foster's method that the only 
 contacts which are movable are potential 
 contacts, that is, points between which 
 the potential is compared, no current 
 being allowed to flow through the con- 
 tact. In such a case the resistance of 
 the contact has no effect on the position 
 of the point of balance. 
 
 The wire to be treated AB (Fig. 
 187) is connected in series with a re- 
 sistance RJ and a storage cell B r A 
 second similar cell B 2 is connected 
 
 to an auxiliary circuit, consisting of a portion of stretched wire, IJ, of 
 thickness at least as great as that of the wire AB, and a resistance R 2 . 
 The two cells must have a considerable capacity, and after being fully 
 charged they must be partly discharged before the measurements are 
 1 Proceedings of the Cambridge Philosophical Society, vol. vii. p. 2G9. 
 
180] CALIBRATION OF A WIRE 441 
 
 commenced. The resistances R X and R. 2 must be adjusted so that the 
 resistances of the two circuits are approximately equal, and such that the 
 current sent by each cell is not greater than a tenth of an ampere. If 
 these precautions are taken, it will be found that the E.M.F. of the cells 
 varies very little, and what variation there is takes place at the same rate 
 in the two cells, and thus produces no effect. 
 
 Two contact makers are placed on the wire AB, at a distance approxi- 
 mately equal to the step which is to be employed in the calibration. 
 One of these contact makers E is connected to a high resistance galvano- 
 meter G, and the other terminal of the galvanometer is taken to a point H 
 on the wire u, where it is soldered in place. The other contact maker D 
 must then be connected to a second point F on the wire u, and the 
 point F moved till the galvanometer remains undeflected. The contact at 
 F should then be soldered, so that accidental movement is impossible. 
 
 Starting with one contact near the end A of the wire, the other is 
 moved till there is no deflection of the galvanometer. The left-hand 
 contact D is then moved to the right of E, and the commutator c reversed. 
 Then keeping E fixed, the position of F is altered till there is no galvano- 
 meter deflection. When this is the case, the resistance of the portion of 
 the wire included between the two contacts is equal to the resistance of 
 that portion of the wire which was included between the two contacts in 
 their previous position. The contact D is then left in place, and E moved 
 over to the right of D, the commutator is reversed and the position of 
 no deflection obtained as before, and this process is continued till the end 
 B of the wire is reached. The whole proceeding is then repeated, starting 
 from B. The wire having been divided into steps having equal resistance, 
 the calibration curve may be prepared in the manner described in the 
 preceding section. 
 
 This method lends itself to the performance of a rapid test of the 
 -uniformity of a wire, which is sometimes necessary when choosing a wire 
 for use in a wire bridge or a potentiometer. For this purpose the two 
 contacts D and E consist of two sharpened pieces of metal cemented to 
 a piece of ebonite at a convenient distance apart. They are placed in 
 contact with the wire to be tested, and the position of the contacts F and 
 H on the auxiliary wire adjusted in the manner already described. Then 
 the ebonite carrying the two contacts is moved along the wire, and if the 
 wire is perfectly uniform, the galvanometer will remain undeflected, 
 and any deflections obtained will indicate the extent of the departure of 
 the wire from uniformity. 
 
 180. Calibration of a Wire by Strouhal and Barus's Method. A 
 convenient arrangement to use when a number of similar wires have to 
 be calibrated, or where, owing to possible wear, it will be necessary to 
 frequently recalibrate the same wire, is that used by Strouhal and Barus. 1 
 Ten approximately equal resistances, each about a tenth of the re- 
 sistance of the w r ire, are prepared. These resistances may with advan- 
 tage consist of the same kind and gauge of wire as that to be tested, 
 when they may be adjusted by simply measuring off the required length 
 
 i Wied. Annalen (1880), x. p. 326. 
 
442 
 
 MEASUREMENT OF RESISTANCE 
 
 [181 
 
 of wire. If the wire to be calibrated is 100 cm. long, then cut off ten 
 pieces of the same wire each 12 cm. long, and solder the end centimetre 
 of each piece to a strip of copper A (Fig. 188), bent at right angles as 
 shown. These strips of copper are screwed to a small block of wood c, 
 which has been soaked in melted paraffin wax, and then the ends of the 
 
 copper which project below 
 the wood are amalgamated. 
 A row of eleven holes, at a 
 distance apart equal to the 
 distance between A and B, are 
 then bored in a piece of wood, 
 each hole being about a centi- 
 metre deep. These holes are 
 filled half full with mercury, 
 and into these holes the ter- 
 minals of the resistances dip, 
 as shown in Fig. 188. Mer- 
 cury cups 1 and 11 are con- 
 nected to the ends of the wire 
 FH by thick copper straps, 
 while a battery is connected 
 to F and H. One terminal of a 
 galvanometer G is then placed 
 in mercury cup 1, and the 
 position of the slider s on the 
 wire is adjusted till the gal- 
 vanometer is undeflected. The 
 position of s having been re- 
 
 corded, the galvanometer terminal is moved to mercury cup 2, and the 
 position of s for no galvanometer deflection again determined. The- 
 resistances a and b are then interchanged, and the positions of s on the 
 wire determined for which the galvanometer is undeflected when con- 
 nected in turn to mercury cups 2 and 3. The length of wire now 
 intercepted has the same resistance as that previously intercepted between 
 the readings for s when the resistance a was between the cups 1 and 2. 
 The resistances a and c are next interchanged, and so on till the resist- 
 ance a is finally placed between the cups 10 and 11. In this way the 
 wire FH is divided into ten segments which have 'equal resistances, and 
 the calibration curve can be calculated, as in 178. 
 
 181. Comparison of Two nearly equal Resistances by Carey 
 Foster's Method. Let two nearly equal resistances p and Q be inserted 
 in the P and Q gaps of the bridge (Fig. 185), while the two resistances X 
 and I 7 , which have to be compared, are inserted in the gaps at M and N. 
 Then when X is in the gap M, the condition for balance is 
 
 FIG. 188. 
 
 ^ 
 
 Q Y+\ q + q 
 the symbols having the same meaning as in 177. 
 
181] CAREY FOSTER'S METHOD 443 
 
 If, now, the resistances X and Y are interchanged, the new condition 
 for balance is 
 
 Q X+K~+? 
 From these equations we have 
 
 P-Q 
 P+Q 
 
 or since p 4- q =p' + q, and p-p' = q - q, 
 
 X- Y=p-p' = q'-q. 
 
 Thus the difference of the resistance of X and Y is equal to the resistance 
 of the portion of the wire intercepted between the points of contact in 
 the two positions, and since the resistance of this portion of the wire 
 is known from the calibration, we thus obtain the difference in the 
 resistance of the two coils. It will be observed that, as the resistance 
 of the coils placed in the gaps P and Q are not involved in the expression 
 for the difference in the resistance of the two coils being measured, we 
 do not require to know their resistance. What is, however, necessary 
 is that their ratio should remain constant throughout the measurement. 
 Hence their temperature must either remain quite constant, or, if they 
 are made of the same material, their temperature, although it may vary, 
 must at every instant be the same. For this reason these two coils are 
 often wound on the same bobbin, and hence, as the wires lie side by 
 side, their temperature will always remain the same. 
 
 A form of bridge suitable for comparing resistance coils with a 
 standard by the above method is illustrated in Fig. 189, in which the 
 lettering is similar to that adopted in Fig. 185. The bridge wire AC 
 is only 10 centimetres long, and is connected to the rest of the bridge 
 by thick copper wires dipping into two mercury cups L. Thus bridge 
 wires of different thicknesses can be employed, according to the difference 
 in the resistances of the two coils being compared. The two coils to be 
 compared have thick copper terminals which dip in the mercury cups M 
 and N. The ratio coils (a) are wound on the same bobbin, and their 
 terminals, which are also copper rods, dip in the mercury cups p and Q. 
 
 Each of the copper blocks, which carry the mercury cups M, N, P, Q, 
 and L has a second mercury cup, and over these cups there is an ebonite 
 disc, fitted with four copper connectors, F, H, J, and K, as shown. The 
 ends of these connectors dip in the mercury cups, and the arrangement 
 forms a commutator, by means of which the coils in the gaps M and N 
 can be interchanged with reference to the rest of the bridge when they 
 are being compared by Carey Foster's method. When the commutator 
 is in the position shown, the coil in M is connected between c and B r 
 When, however, the commutator is rotated through 180, so that the 
 copper connectors occupy the positions shown dotted, the coil N is now 
 
444 
 
 MEASUREMENT OF RESISTANCE 
 
 [181 
 
 between c and BI . Thus the interchange of the coils is obtained without 
 having to move them, a great convenience, since, to prevent errors due 
 to changes of temperature, the coils have to be immersed in well stirred 
 
 FIG. 189. 
 
 oil baths, and either lifting the coils out of the baths or interchanging 
 the baths is an awkward operation. 
 
 The galvanometer contact D moves on a divided rod s, on which a 
 scale is engraved, and the length of wire, of which the resistance is equal 
 to the difference in the resistances of the coils in M and N, is read off on 
 this scale. 
 
 Generally three pairs of proportional coils to go in the gaps P and Q 
 are supplied, having resistances of 1, 10, and 100 ohms respectively. 
 
181] CAREY FOSTER'S METHOD 445 
 
 The pair should be chosen of which the resistance most nearly approaches 
 the resistance of the coils to be compared. 
 
 Since the difference in the resistance of the coils is obtained in terms 
 of the resistance of a portion of the bridge wire, it is necessary to measure 
 the resistance of unit length of the wire, and, where great accuracy is 
 desired, calibrate the wire. 
 
 Having carefully amalgamated the terminals of the proportional coils 
 to be used and those of two standard resistances, set up the bridge with 
 the standard coils in oil baths. An accumulator can with advantage be 
 used as battery, but if the coils to be compared are 1 or 10 ohm coils, 
 some resistance R must be included in the battery circuit. It is also 
 advantageous to place a commutator X in the battery circuit. Deter- 
 mine which of the coils X and Fin the gaps M and N has the higher 
 resistance by seeing in which direction the galvanometer contact has to 
 be moved when the coils are interchanged by rotating the bridge com- 
 mutator. Then connect a resistance box in parallel with the coil of 
 higher resistance, say, X, using thick copper leads which dip in the same 
 mercury cups. Now adjust this resistance so that the galvanometer 
 contact, when balance is obtained, is near the centre of the wire, and 
 let the shunting resistance be S v Then reverse the resistances by rotat- 
 ing the commutator, and let the distance through which the galvano- 
 meter contact has to be moved to preserve the balance be b r Next 
 adjust the shunt so that the point of balance is near the end of the wire, 
 and let the new value of the shunting resistance be S 2 , and the change 
 in reading for the galvanometer contact on reversal be now b 2 , />., cor- 
 responding to nearly the whole length of the wire. If, then, r is the 
 resistance of unit length of the wire, we have 
 
 XS, 
 
 'ti-x^-r* 
 
 XS 
 
 ^ = 
 
 Hence, subtracting 
 
 from which r can be calculated. In general, however, this expression may 
 be much simplified, for if X and Y are very nearly equal, and S l is very 
 large, also if, as usual, the wire has a small resistance per unit length, 
 then S. 2 will also be large compared to X. Thus 
 
 When making the balances, thermo-currents at the galvanometer 
 contact D are liable to give trouble. In such a case it will be found 
 
446 MEASUREMENT OF RESISTANCE [ 182 
 
 advisable to make the contact at D, and after the galvanometer has 
 come to rest, to reverse the battery current and note whether the galvano- 
 meter deflection remains unaffected. 
 
 A note must be made of the temperature of the wire, which may be 
 obtained by placing a small thermometer alongside. A piece of cotton 
 wool placed over the wire and thermometer will be found of service in 
 protecting both from the heat radiated from the hand when adjusting the 
 contact. 
 
 The procedure when comparing two coils will be evident from what 
 has been said above. It is of great importance that the temperatures of 
 the coils should be known accurately, particularly in the case of coils 
 wound with platinum-silver, which has a very high temperature co- 
 efficient. If the temperature of the room is very constant, and care is 
 taken that the testing current is not too large, and is kept on for only a 
 short time, then the temperature of the oil baths in which the coils stand 
 may be taken as giving the temperatures of the coils themselves. If, 
 however, the temperature is changing, or the testing current has to be 
 kept on for some time, then the baths must be kept well stirred. This 
 can best be accomplished by using a small propeller in each bath, driven 
 by a small electric motor by means of a light cord (see Fig. 72). 
 
 182. To Determine the Ratio of Two nearly equal Resistances. 
 It is frequently necessary to determine the ratio of two nearly equal 
 resistances. The two coils P and Q to be compared are placed in the 
 PQ gaps of the bridge (Fig. 185), while in the two outer gaps M and N are 
 placed two known resistances A and B, which are approximately equal. 
 If the resistances of the two parts of the bridge wire on either side of the 
 slider, when there is no current through the galvanometer, are p and q, 
 while the resistance of the whole wire is p, we have 
 
 If, now, P and Q are interchanged, and the new point of balance divides 
 the wire into portions having resistances of p f and q', we have 
 
 = = 
 
 Q ~ X p + A + p' ~ X p + A +_#' 
 
 Adding the denominators for a new denominator and the numerators for 
 a new numerator on either side of these two equations, we get 
 
 P = X 1 , + X (J + A 
 Q X + X + A 
 
 where R is written for (\ p + X u + A + B + p), and 8 for the resistance of 
 the portion of the bridge wire between the two points of balance. When 
 
182] RATIO OF TWO NEARLY EQUAL RESISTANCES 447 
 
 8 is small compared to It, that is, when p and Q are nearly equal and A 
 and B are fairly large, we have as a very near approximation 
 
 a R 
 
 Since the maximum value we can have for S is p, that is, the two 
 points of balance are at the extreme ends of the bridge wire, the maxi- 
 mum ratio of P to Q which it is possible to measure by this method is 
 
 Q R-p 
 
 or since A.,, and \ are always small compared to A and B, the maximum 
 ratio is given by 
 
 A + B 
 
 _ 
 
 A + B 
 
 Suppose, then, that the total resistance of the bridge wire is 1 ohm, 
 and that we calculate the maximum ratio of P to Q for A and B being 
 each equal to 1, 5, 10, 100 ohms, we obtain the following numbers : 
 
 
 p 
 
 Alteration in the Ratio . x 
 
 Value of A or B. 
 
 Maximum Rutio 
 of P to Q. 
 
 Q 
 
 Corresponding to an Error 
 of 1 mm. in the Wire 
 
 
 
 in Measuring d. 
 
 1 
 
 2 
 
 0007 
 
 5 
 
 1-2 
 
 0002 
 
 10 
 
 1-1 
 
 0001 
 
 100 
 
 1-01 
 
 0000 1 
 
 Thus by increasing the value of the resistances placed in the end 
 gaps of the bridge, we rapidly limit the range of departure from equality 
 in P and Q which we are able to measure. Although the range gets more 
 and more limited as the resistances placed in the end gaps get greater 
 and greater, the accuracy with which we are able to read the value 
 of the ratio on the wire increases. Thus in the last column of the 
 above table is given the alteration in the ratio of P to Q corresponding 
 to an error of one scale division (that is, 1 mm. in a wire 1000 mm. 
 long) in the determination of 8. When there is no resistance in the 
 end gaps, a millimetre corresponds to an error in the ratio of P to Q, 
 if P and Q are nearly equal, of about "004, that is, about 4 parts in 
 1000. When A and B are each 100 ohms, a millimetre, however, corre- 
 sponds to -00001, or 4 in 400000. 
 
 Of course we are unable to make use of the increased sensitiveness 
 
448 
 
 MEASUREMENT OF RESISTANCE 
 
 [183 
 
 obtained with high resistances in the end gaps, unless the galvano- 
 meter is sufficiently sensitive to indicate a readable departure from the 
 correct position of balance. Increased sensitiveness may be obtained 
 by increasing the battery power, but care must be taken not to use 
 currents of such a magnitude as to appreciably heat the resistances. 
 For this reason the current should not be kept on longer than neces- 
 sary. Further, if A and B are either much larger or smaller than p and 
 Q, then the positions of the battery and galvanometer should be so 
 chosen that the current divides approximately equally between the two 
 arms of the bridge (see p. 434). 
 
 183. The Post Office form of Wheatstone Bridge. A convenient 
 form of bridge for rapidly making measurements of resistance where no 
 very great degree of accuracy is aimed at is shown in Fig. 190, and 
 is known as the Post Office form of resistance box. It consists of a 
 
 FIG. 190. 
 
 number of resistance coils wound on bobbins placed under an ebonite top, 
 and connected to brass blocks screwed to the upper surface of the ebonite. 
 By means of brass plugs which fit into conical holes between the blocks, 
 the resistance coil which connects the two blocks can be short-circuited. 
 The resistances of the different coils are indicated by the numbers placed 
 alongside the holes for the plugs. The resistance to be measured is con- 
 nected to the two binding screws A and E, which are generally marked 
 "line" and "earth" respectively, as this form of box was originally 
 designed for testing the resistance of telegraph lines. The battery is 
 connected to the two binding screws E and j, while the galvanometer 
 is connected to A and I. By means of a wire beneath the ebonite top, 
 when the key H is depressed connection is made with the point B, while 
 on depressing the key F connection is made with the point c. The 
 resistances between A and B and B and c consist of coils of 10, 100, and 
 1000 ohms respectively, while by removing the necessary plugs the 
 
183] POST OFFICE FORM OF WHEATSTONE BRIDGE 449 
 
 resistance between c and E can be made of any integral value between 
 and 10,000 ohms. 
 
 The way in which this arrangement of coils forms a Wheatstone's 
 bridge is indicated diagrammatically in Fig 129 (a), where the different 
 branches are lettered similarly to what they are in the figure of the 
 bridge. If the resistances in the branches AB and BC are equal, then to 
 obtain balance the resistance of the branch DE must be equal to the 
 resistance being measured. If, however, the resistance of the branch AB 
 is made ten times the resistance of the branch BC, the resistance in DE 
 required to produce balance is only a tenth of the resistance which is 
 being measured. Finally, if the resistance of the branch BC is ten times 
 the resistance of the branch AB, the resistance in CE required to produce 
 balance is ten times the resistance which is being measured. Similarly, 
 by making the ratio arms AB and BC 1 to 100 or 100 to 1, the resistance 
 in the branch CE will be 100 times or one hundredth respectively of the 
 resistance being measured. 
 
 The following is the procedure to be adopted when measuring a 
 resistance with the Post Office bridge : The resistance having been con- 
 nected to the terminals A and E, the plugs corresponding to 10 ohms are 
 taken out of each of the proportional arms AB and BC. The battery 
 key H having been depressed, the galvanometer key H is momentarily 
 depressed, and the direction of the resulting galvanometer throw noted. 
 If no deflection is produced, it probably indicates that there is a break in 
 the battery circuit. The infinity plug, marked " inf.," is then removed, 
 and the direction of the galvanometer deflection again noted, which 
 should be opposite to that which occurred previously. These tests will 
 have indicated the direction of the deflection corresponding to the 
 resistance in the branch CE being too small. Resistance is then gradually 
 unplugged in the branch CE till two values are discovered, which differ 
 from one another by 1 ohm, and for which the galvanometer is deflected 
 in opposite directions, indicating that the resistance being measured lies 
 between these two values. Suppose that in this way it is found that the 
 resistance lies between 14 and 15 ohms. If, now, 100 ohms is unplugged 
 in the arm BC, we shall have to increase the resistance in the arm CE to at| 
 least 140 ohms, since it will be ten times the resistance being measured. 
 Proceeding as before, it is found that 146 ohms is too small and 147 ohms 
 is too large, showing that the resistance lies between 14 '6 and 14 '7 ohms. 
 Next making the resistance in the arm BC 1 000 ohms, it was found that 
 the resistance which had to be unplugged in CE was between 1462 and 
 1463 ohms, indicating that the resistance being measured lies between 
 14*62 and 14'63 ohms. To obtain a nearer value to the resistance, 
 recourse must be had to galvanometer deflections. Thus when the 
 resistance in CE was 1462 ohms the galvanometer was deflected through 
 thirteen divisions to the right, while when the resistance was 1463 the 
 deflection was twenty-five divisions to the left. Thus a change in the 
 resistance in CE of 1 ohm produces a deflection of thirty-eight divisions, 
 while to produce equilibrium we should have to increase the resistance in 
 
 2 F 
 
450 
 
 MEASUREMENT OF RESISTANCE 
 
 [ 
 
 CE so as to produce a deflection of thirteen divisions ; that is, we 
 should have to increase the resistance by 13/38 of an ohm, or O34 ohm. 
 Hence the resistance in the arm CE required to give balance is 1462 '34 
 ohms, and the resistance being measured is therefore 14'6234 ohms. 
 
 If the resistance to be measured is greater than 11,111 ohms it will 
 be impossible to obtain a balance with equal ratio arms, and the resistance 
 of AB must be made 10 or 100 times the resistance of BC. 
 
 It will sometimes be found more sensitive to use a 1 to 1 or a 10 to 1 
 ratio for the ratio arms rather than a 10 to 1 or a 100 to 1 ratio as the 
 case may be, and to deduce the fractions of an ohm by galvanometer 
 deflections. In such a case we have a choice as to the way in which we 
 make up the desired ratio. Thus in the case of equal ratio arms we may 
 make the resistance in either of the branches AB or BC 10, 100, or 
 1000 ohms. The best value to take in any particular case is that which 
 will make the resistance of each of the ratio arms most nearly equal 
 to the resistance being measured. 
 
 Since the resistance of the wire of which the coils are constructed 
 varies with the temperature, it is only at some fixed temperature that the 
 coils of the bridge have the value assigned to them. The temperature at 
 which the coils have their nominal values is generally marked on the 
 box by the makers. When the temperature of the box is different from 
 this value a correction will be required, the magnitude of which depends 
 on the material of which the coils are made. The following table gives 
 the temperature coefficients of the alloys which are commonly employed 
 for the construction of resistance coils. It must, however, be remembered 
 that the temperature coefficient of different samples of what is nominally 
 the same alloy differ considerably, so that this table must be used with 
 circumspection, and in every case where accurate measurements are being 
 made either care must be taken to observe with the coils at the tempera- 
 ture at which they were compared with a standard, or in each case the 
 actual temperature coefficient of the wire used in the coils must be 
 measured. 
 
 Alloy. 
 
 Approximate Composition 
 (parts by weight). 
 
 Specific 
 Resistance. 
 
 Temperature 
 Coefficient. 
 
 Platinum silver . 
 
 2 Ag ; 1 Pt 
 
 31xlO- 
 
 00026 
 
 Platinum indium 
 
 Pt with trace of Ir 
 
 27xlO- 6 
 
 0017 
 
 German silver 
 
 10Cu;25Zn; 14Ni;0'3Fe 
 
 
 00027 
 
 Platinoid . . . < 
 
 German silver with trace 
 of tungsten 
 
 J34xlO- 8 
 
 0002 
 
 Manganine . . . 
 
 84 Cu ; 12 Mn ; 4 Ni 
 
 42xlO- 6 -00004 
 
 Constantin . . . 
 
 60 Cu ; 40 Ni 
 
 49xlO- 6 
 
 
 184. The Dial-pattern Wheatstone's Bridge. Although the Post 
 Office pattern of bridge is very compact and fairly inexpensive to make, 
 
184] THE DIAL-PATTERN WHEATSTONE'S BRIDGE 451 
 
 considering the great range of resistances which it is capable of measur- 
 ing, for accurate work it suffers from one very serious defect ; this is, 
 that in the arm DE (Fig. 190, p. 448) there are always a large number of 
 plug contacts, and the resistances of these contacts are neither negligible 
 nor are they constant. This inconstancy is not only due to the fact that 
 the cleanliness of the plugs and of the holes into which they fit varies, 
 but also to the fact that the insertion of any one plug affects the fit of 
 the plugs in the neighbouring holes. This is due to the displacement of 
 the brass blocks, either owing to a slight give in the screws by which 
 they are attached, or owing to bending of the ebonite top of the box. It 
 is thus practically impossible to allow for the resistance of the plugs, and 
 all we can do is to reduce the resistance of these contacts to a minimum 
 by using well-fitting plugs 1 and keeping them as clean as possible. 
 
 To reduce the plug difficulty to a minimum, the form of box shown in 
 Fig. 191 is used. This pattern of box is known as the dial pattern, and 
 
 FIG. 191. 
 
 the example shown in the figure contains ten coils of 1 ohm each, ten of 
 10 ohms each, ten of 100 ohms each, and ten of 1000 ohms each, the 
 manner in which the coils are connected up being indicated by the dotted 
 lines. There is a single plug in each of the dials, and it will be seen that 
 if all the plugs are opposite the blocks marked o, then none of the coils 
 are included between the terminals A and E, the only resistance between 
 these two points consisting of the connecting pieces between the dials 
 and the resistance corresponding to four plug contacts. If, now, the plug 
 in the units dial is moved to the hole marked 1, the resistan e included 
 between the terminals E and A will consist of 1 ohm, together with the 
 resistance of the connecting wires and the four plug contacts. If the 
 units plug is moved to the hole marked 2, there will be 2 ohms included 
 
 1 As the plugs belonging to various boxes, although these boxes are of the 
 same pattern, are generally not interchangeable, all plugs belonging to any one 
 box ought to bear a distinguishing mark, such as a letter stamped on the head. 
 This mark should be stamped on the top of the box, and care be taken that the 
 plugs are always replaced in their own box. 
 
452 
 
 MEASUREMENT OF RESISTANCE 
 
 [185 
 
 in addition to the connecting and plug resistances. Proceeding in this 
 way, it will be seen that it is possible to obtain a resistance consisting of 
 any whole number of ohms up to 11,110 ohms, and whatever the resist- 
 ance there will be only four plug contacts. Further, the insertion of a 
 plug has no influence on the fit of any of the other plugs, and so it is 
 possible to keep the resistance of the plug contact very much more con- 
 stant than in the case where the plugs are arranged as shown in Fig. 190. 
 In addition to the dials, a set of ratio coils are provided in the arms BC, CD. 
 
 The resistance to be measured is connected between A and B, the 
 plug at F being removed and that at G inserted, or between E and D, 
 the plug at G being removed and that at F inserted. 
 
 It will be observed that the tenth coil in each of the dials, that 
 is, the coil between the block marked 9 and the central block, is not 
 absolutely necessary in order to obtain any combination up to 10,000 
 ohms. These additional coils are, however, of much use when testing 
 the box, since with them the sum of the resistances in any dial is equal 
 to the resistance of a single coil in the dial next above. Hence when 
 testing the box the unit coils are first compared together, then the sum 
 of these coils is compared with one of the tens. The tens are then com- 
 pared amongst themselves, and their sum is compared to the first hundred 
 coil, and so on. 
 
 When using the bridge, if the ratio arms are taken equal, any error 
 due to their ratio not being exactly unity can be eliminated by placing 
 the resistance to be measured first between A and B and then between 
 D and E and taking the mean of the two values obtained. 
 
 185. Calibration of a Dial-pattern Wheat stone's Bridge. The 
 dial bridge is connected by thick copper straps to a mercury cup a (Fig. 
 192) and one terminal of a resistance box r. The other terminal of this 
 
 FIG. 192. 
 
 box is connected to a second mercury cup b. Two other resistance boxes, 
 s and s, are connected as shunts on the mercury cups and the resistance 
 box r respectively. Finally, a resistance R of 1 ohm is placed connecting 
 the mercury cups, and a thick U-shaped piece of copper is provided by 
 means of which the mercury cups can be connected, and thus the resist- 
 ance R short-circuited. 
 
185] CALIBRATION OF A WHEATSTONE'S BRIDGE 453 
 
 The resistance R must be very slightly larger than 1 ohm, and may 
 consist of a thick piece of manganine wire soldered to two stout copper 
 strips, the mercury cups a and b consisting of holes made in these strips. 
 To adjust the value of R, the mercury cup b should be connected direct 
 to E by means of a copper strap, and using 10 ohms in each of the arms 
 BC and CD, and 1 ohm in the dial arm, the resistance of R is adjusted till 
 the bridge is very nearly balanced. It will be found most convenient to 
 adjust the resistance of u by means of the soldered junctions till its 
 resistance is a little too small. Then by filing the wire the resistance 
 can be made a little greater than 1 ohm. 
 
 Next, placing the plug at 1 in the units dial, and still using 10 ohms 
 in each of the arms BC, CD, adjust the shunt s till the bridge is balanced. 
 Then if R and S a are the resistances of R and the shunt s, and a is the 
 resistance of the first unit coil, including the resistance of the wires 
 joining the different dials, we have 
 
 RS a 
 
 but since R is very nearly unity and S a is large, we may, without appre- 
 ciable error, replace R in the denominator by 1. Then 
 
 Next place the U-shaped connector between the cups a and b and connect 
 up the resistance boxes r and s in parallel between b and E, as shown in 
 the figure. Keeping the plug in the units dial at 1, adjust r and s till 
 the bridge is balanced. To obtain this balance make r 1 ohm and s 
 infinity. If the resistance between D and E is now too small, replace the 
 copper strap joining E to r by a short piece of thick copper wire, and 
 thus make the resistance in this arm too great. Then a balance can be 
 obtained by adjusting the resistance of the shunt , and since r is only a 
 little too great, the resistance unplugged in s will be large, and thus the 
 adjustment can be made with accuracy. 
 
 Next remove the connection between the mercury cups and move the 
 dial plug to 2, and then adjust the bridge by altering the shunt s. If b 
 is the resistance of the second dial coil, and S b the value of s when 
 balance is secured, we now have 
 
 *_ RS* 
 
 in which, as before, we may put R = 1 in the denominator, and thus 
 
 7 &h rt 
 
 Proceeding in this way we obtain the resistance of each of the coils of 
 the unit dial in terms of the resistance R. Or if we like we may express 
 them all in terms of the resistance a of the first coil. 
 
454 MEASUREMENT OF RESISTANCE [ 185 
 
 To compare the coils of the tens dial, we put the plug at 10 in the 
 units, i.e. remove the plug altogether (see Fig. 191), and adjust the 
 resistances r and s (the connector being inserted between a and I) till 
 the bridge is balanced. Then place the tens plug at 1, i.e. at 10 ohms, 
 and the unit plug at 0, and adjust s till the bridge is again balanced. 
 Then let s } be the value of s when the 10 V ohm coil is in, ,s., the value 
 when the ten 1 ohm coils are in, r the value of ?*, a 10 the resistance 
 of the 10 ohm coil, and 2 the sum of the resistances of the units coils. 
 Thus 
 
 + r n 2 + r 
 s n 
 
 Next remove the units plug and balance by altering r and s, and let 
 the value of these quantities be r' and s\. Then move the tens plug to 
 2 and the unit plug to 0, and balance by altering s, and let the new 
 value be s' 2 . If, then, & 10 is the resistance of the second 10 ohm coil, 
 we have 
 
 Proceeding in this way, we obtain the difference between the resistance 
 of each of the 10 ohm coils and the sum of the 1 ohm coils, and since 
 we have found the resistances of these in terms of the resistance a of 
 the first coil, we can express the resistance of the 10 ohm coils in terms 
 of the resistance of the coil a. 
 
 Proceeding in the same way, each of the hundred coils is compared 
 with the resistance of the sum of the 10 ohm coils, and each of the 
 thousand coils with the sum of the 100 ohm coils. 
 
 Having in this way expressed each of the coils in terms of the 
 resistance of a, we now require to determine the value of this quantity. 
 To do this the mercury cup b is connected direct to E, and a standard 
 resistance coil is placed between the mercury cups. If possible, this coil 
 ought to be a 1000 ohm coil. The bridge is then balanced, and the 
 resistance of the coils used on the dials expressed in terms of a is 
 equated to the resistance of the standard, and thus the value of a is 
 deduced. If only a 1 ohm coil is available, then the plug in the units 
 dial being at 1, balance must be obtained by either altering the shunt s, 
 or if the resistance in the arm is already too low, a resistance box must 
 be connected as a shunt on the dials, and balance be obtained by adjust- 
 ing this shunt. In this way the resistance of a is directly obtained 
 in terms of the value of the unit and the shunt resistance. 
 
 The same method as that described above can be employed to cali- 
 brate an ordinary Post Office pattern resistance box, or, by using two 
 auxiliary coils as ratio arms, a simple box of resistance coils. 
 
186] CALLENDAR AND GRIFFTHS PATTERN BRIDGE 455 
 
 It is of importance when making such a calibration as that described 
 above that the temperature should be fairly constant, particularly if 
 the material of which the coils are wound has a large temperature 
 coefficient. 
 
 186. The Callendar and Griffiths Pattern Bridge. A form of 
 Wheatstone's bridge which is particularly well suited for measuring the 
 change of resistance of platinum resistance thermometers has been devised 
 by Callendar and Griffiths. A simple form of this bridge, which is suit- 
 able for measurements in which the greatest accuracy is not aimed at, is 
 shown somewhat diagrammatically in the upper part of Fig 193, the 
 scheme of connections being shown in the lower part of the figure. The 
 
 FIG. 193. 
 
 bridge wire AB is stretched alongside a second wire G 2 H of the same 
 material, which is connected to the galvanometer, and a movable slider 
 carries a cross-piece F of the same wire, which serves to make contact 
 between the bridge wire and the galvanometer. The object of making 
 the two wires and the cross-piece all of the same wire is to reduce the 
 thermo-electromotive forces at the contacts, for if the contacts are between 
 exactly the same materials, then differences of temperature will not pro- 
 duce thermo-electromotive forces. One end A of the wire is connected to 
 a series of resistance coils DC. 2 . The first of these has a resistance equal 
 to that of 20 cm. of the bridge wire, the next to 40, and so on, the 
 resistance of each coil being double that of the preceding one. 
 
 Connected to the binding screw c x is one terminal of the ratio coil p, 
 the other ratio coil Q being connected to the binding screw P r The two 
 
456 MEASUREMENT OF RESISTANCE [ 186 
 
 battery terminals BJ and B 2 are connected to the points GJ and p. 2 , while 
 the galvanometer terminal G I is connected to the point between the ratio 
 coils. The ratio coils are wound together so as to always have the same 
 temperature, and they are adjusted to equality. 
 
 The thermometer T with its leads is connected, to the screws p^, 
 and the compensating leads are connected to C 1 and C 2 . This pair of 
 leads are joined together at the end, and are of exactly the same resistance 
 as the thermometer leads, and are placed alongside these latter. 
 
 Let r be the resistance of the thermometer leads or the compensator 
 leads, T the resistance of the thermometer coil, and R the resistance un- 
 plugged in the coils, and let the galvanometer contact be at G 2 in the lower 
 figure when balance is secured. If p is the resistance per centimetre of 
 the bridge wire, we have, since P= Q, 
 
 or 
 
 where o is the centre of the bridge wire. Hence 
 
 T=R + 2p.OU 2 (1) 
 
 Thus the resistance of the thermometer is equal to the resistance of the 
 coils unplugged plus twice the resistance of the part of the bridge wire 
 intercepted between the galvanometer contact and the centre of the wire. 
 If the galvanometer contact is to the left of o, then twice the resistance 
 of the portion of the wire intercepted between the contact and the centre 
 must be subtracted from the resistance unplugged. 
 
 It will be observed that any change in the resistance of the thermo- 
 meter leads, so long as the same change takes place in the compensating 
 leads, will not affect the point of balance in the wire. 
 
 It is very convenient to arrange the resistance of the bridge wire so 
 that 1 centimetre has a resistance of -^^ of an ohm. In this case each 
 centimetre that the galvanometer contact is away from the centre of the 
 bridge wire represents "01 of an ohm. The first coil, since it has a 
 resistance equal to that of 20 cm. of the wire, has therefore a resistance of 
 O'l of an ohm. The remaining coils have resistances of '2, *4, '8, 1*6, 
 3'2, 6'4, arid 12*8 ohms. Thus the bridge will measure up to 25 6 ohms. 
 If a platinum thermometer is used of which the resistance in ice is about 
 12*8 ohms, then its change in resistance when heated from to 100 (its 
 fundamental interval, see 208) will be 5 ohms, and hence 1 cm. on 
 the bridge wire will represent two hundredths of a degree. Such a ther- 
 mometer is suitable for work at comparatively low temperatures. For 
 high temperatures a thermometer having a fundamental interval of 1 ohm 
 may be used, in which case each centimetre of the bridge wire represents 
 a tenth of a degree. From what has been said above, it is apparent that 
 the adjustment of the resistance of the bridge wire to a particular 
 resistance per centimetre is of considerable practical convenience. As, 
 
187] CALLENDAR-GRIFFITHS BRIDGE 457 
 
 however, it is difficult to draw a wire such that its resistance per centi- 
 metre has any predetermined value, Professor Callendar has adopted the 
 device of shunting the wire. A wire is chosen of which the resistance 
 per centimetre is slightly greater than the value required, and then a 
 resistance s (Fig. 193) is placed as a shunt on the wire, the magnitude of 
 the resistance being so chosen that the combined resistance of the wire 
 and s in parallel divided by the total length of the wire has the desired 
 value, viz. 1/200 of an ohm. The presence of the shunt has no effect on 
 the reading of the bridge, except that the resistance of the wire must now 
 be assumed to be that of the wire and shunt in parallel. For if c s is the 
 current passing in the shunt, and c to that in the wire, I the total length of 
 the wire, and p its resistance per centimetre, then 
 
 Cw _ ^ C w _ S /gv 
 
 c,~ pi c w + c a ~ S + pl 
 The difference of potential between B x and G 2 is 
 
 where r is the resistance of the compensating leads. 
 The difference of potential between G 2 and B 2 is 
 
 Hence since the ratio coils P and Q are equal, these two differences of 
 potential are equal. Thus 
 
 w + c t ) (r + R) c w p(AO +~M 2 ) = (c, v + c s ) (r+T) + c w p(AO - 
 
 Substituting from (2) we get 
 
 But - is the resistance of the wire and shunt in parallel, and this 
 
 o + pi 
 
 quantity divided by I is the equivalent resistance per centimetre of the 
 shunted wire. Hence calling this equivalent resistance per centimetre 
 p, we have 
 
 T=JR + '2 P '.OG ........ (4) 
 
 the same expression as before obtained, except that p replaces />. 
 
 187. Calibration of a Callendar-Griffiths Bridge. The calibration of 
 the bridge described in the last section consists of (1) determination of 
 the electrical mid-point of the bridge wire ; (2) determination of the 
 ratio of the resistances of the two ratio coils ; (3) calibration of the 
 
458 MEASUREMENT OF RESISTANCE [ 187 
 
 bridge wire ; (4) determination of the resistance of each of the coils 
 in terms of the resistance of 1 centimetre of the bridge wire (the unit 
 used in the bridge) ; and (5) if the bridge is to be used for other 
 purposes besides platinum thermometry, a determination of the value 
 of the bridge unit in ohms. 
 
 (1) To determine the centre of the bridge wire the gaps c^ and 
 p^ (Fig. 193) are closed by copper strips, and the position of the 
 slider is adjusted till there is no galvanometer deflection. The reading 
 for this point gives the mid-point o of the wire. 
 
 (2) Next two nearly equal resistances, X and Y, are placed in the 
 gaps ?iPy c i c o- These resistances ought to be about 10 ohms each. 
 Let the reading on the wire when the resistance X is in the gap c be 
 ]_, and that when the resistance X is in the gap P be x 2 . If the 
 ratio coils p and Q are equal, then (x l + x 2 )/2 will be the same as the 
 reading for the point 0. The difference between (x l + x z )/2 and is 
 a measure of the difference of the resistances p and Q, and the ques- 
 tion arises whether such a difference will appreciably affect the 
 measurements made with the bridge. If the bridge is to be used 
 exclusively for platinum thermometer work, then such differences as 
 are likely to be found between p and Q will probably not appreciably 
 affect the measurements. The reason for this is that if for the moment 
 we suppose that the thermometer leads and compensating leads have 
 zero resistance, then all the resistance measurements will require to be 
 multiplied by the same factor (P/Q), and hence in the expression used 
 in platinum thermometry, namely (see 208), 
 
 we may divide out the numerator and denominator by this correction 
 factor. 
 
 A slight error will, however, be produced if the leads have appre- 
 ciable resistance. This error will in general be negligible, for the 
 resistance of the leads is small compared to the resistance of the 
 thermometer, and the correction on account of the error in the ratio 
 coils is a small correction on this small resistance, i.e. is of the second 
 order. 
 
 If the bridge is to be used for other measurements besides thermo- 
 metric measurements, then it will be advisable to adjust p and Q to exact 
 equality. This may be done by moving the point M (Fig. 193) where the 
 galvanometer lead is attached, the adjustment being tested by reversing 
 the coils x and Y and seeing whether the mean of the points of balance 
 coincides with the point of balance when the gaps are closed with copper 
 connectors. 
 
 (3) The calibration of the bridge wire is performed by placing two 
 pieces of copper strip A and B (Fig. 194) in the terminals GJ and c. 2 . 
 These strips have two mercury cups D, formed by cementing small 
 
187] CALLENDAR-GRIFFITHS BRIDGE 459 
 
 pieces of ebonite having a central hole on the top of the copper. A 
 piece of wire E, preferably of the same material as the bridge .wire, 
 is soldered to the copper strips, arid its resistance is adjusted so as to 
 be equal to the resistance of 2 centimetres of the bridge wire. In 
 the gap Pjp^j are placed two resistance 
 boxes connected in parallel, one of which 
 ought if possible to be variable by tenths 
 of an ohm. 
 
 A copper connector being placed 
 between the mercury cups D, the coils 
 10, 320, and 640 are unplugged (hence 
 # = 9-7 ohms), and 10 ohms is taken 
 out of one of the boxes in the gap P. 
 The galvanometer contact is then moved FIG. 194. 
 
 to near the extreme right-hand end of the 
 
 wire, and balance l is obtained by altering the resistance unplugged in 
 the other box in p. The position of the slider having been noted, the 
 copper connector between the mercury cups D is removed, and the slider 
 moved till the balance is restored. The difference between the slider 
 readings gives the length of bridge wire which corresponds to the resist- 
 ance oi the wire E, i.e. has a resistance half that of E. The connector is 
 then placed between the mercury cups, and balance obtained by altering 
 the resistance box in the gap P. The connector is then removed and 
 balance restored by moving the slider, and so on. In this way the bridge 
 wire is divided up into segments of equal resistance. When balancing by 
 altering the resistance in the gap p, it will be found best to get approxi- 
 mate balance by altering the resistances and to obtain the final balance by 
 moving the slider. In this way the end of one segment into which the 
 wire is divided will not exactly coincide with the commencement of the 
 next, but so long as the part left out, or the overlap, as the case may be, 
 is not very great, then no appreciable error will arise on this account. 
 The reduction of the numbers is performed just as in the case of the cali- 
 bration of a mercury thermometer described in 59. 
 
 (4) Calibration of the Coils. The gap c being closed by a copper con- 
 nector, and the coil 10 unplugged, a resistance of O'l ohm is unplugged 
 in one of the resistance boxes in p and the point of balance obtained. 
 Coil 10 is then cut out by inserting the corresponding plug, and the 
 slider moved till balance is restored. The difference in the readings of 
 the slider gives the resistance in terms of the bridge wire, and this, by 
 means of the calibration, is expressed in terms of the mean bridge wire, 
 the value being 'Z v say. Coil 20 is then unplugged, and coil 10 cut out 
 by inserting the plug, while by altering the resistances in P the point of 
 balance is brought near the centre of the wire. Coil 20 is then cut out 
 and 10 inserted, and the slider moved till balance is again secured. Lat 
 the resistance of the portion of the wire between the two points when 
 
 1 The final balance can if necessary be made by slightly moving the position 
 of the galvanometer contact. 
 
460 
 
 MEASUREMENT OF RESISTANCE 
 
 [188 
 
 allowance has been made for the calibration be Z^. Next insert coil 40, 
 cutting out 10, and bring the point of balance near the centre by altering 
 the resistance in P., then remove coil 40 and insert coils 10 and 20, and 
 move slider, and so on. 
 
 Proceeding in this way we obtain the following series of equations, 
 where the symbol (10) stands for the resistance of coil 10 in terms of 
 1 cm. of the mean bridge wire, and so on : 
 
 (10) = Z! 
 
 (40)-(20)-(10) = Z 8 
 (80) -(40) -(20)- (10) = Z 4I 
 &c. &c. 
 
 Then 
 
 and so on. 
 
 By arranging that Z lt Z 2 , Z 3 , &c., all correspond to nearly the same 
 portion of the bridge wire, errors in the calibration have very little 
 influence on the results. 
 
 (5) To determine the value of the resistance of the unit used in the 
 above calibration, viz. the mean value of 1 cm. in the bridge wire, a 
 
 resistance of known value must be inserted 
 in the gap P, and its value in terms of 
 the bridge resistances measured. Then 
 since the coils are all expressed in terms 
 of the bridge unit, the value of this unit 
 can at once be determined. 
 
 188. Measurement of High Resist- 
 ances. In the case of resistances of a 
 megohm (10 6 ohms) and over the Wheat- 
 stone bridge method is generally not 
 applicable, and a direct deflection method 
 is usually -adopted, that is, the current a 
 battery of given E.M.F. can send through 
 a known high resistance is measured by 
 means of a galvanometer and compared 
 with the current the same battery can 
 send through the unknown resistance. 
 
 A convenient arrangement is shown in Fig. 195, where B is a 
 battery, G a high resistance galvanometer, and c a commutator, con- 
 sisting of six small mercury cups scooped out in a block of paraffin. The 
 known resistance R, and the resistance to be measured X, are connected 
 to these mercury cups as shown, so that by means of two U-shaped 
 
 FIG. 195. 
 
189] MEASUREMENT OF INSULATION RESISTANCE 461 
 
 pieces of copper one or other can be included in the circuit. A resistance 
 S is connected so as to act as a shunt on the galvanometer. 
 
 Suppose that the E.M.F. of the battery is E, and that the resistance 
 S being infinite, that is, the galvanometer unshunted, the deflection pro- 
 duced with the resistance X in series is d r Then we have 
 
 where K is a constant depending on the galvanometer, and expresses the 
 factor by which the deflection has to be multiplied to give the value of 
 the current passing through the galvanometer. 
 
 The resistance R is then introduced into the circuit, and the resist- 
 ance of the shunt altered till the deflection is of about the same magnitude 
 as in the previous case. If the deflection is c? 2 , we have 
 
 (2) 
 
 From the expressions (1) and (2) we can eliminate E and K, when 
 we get 
 
 -G-B. . . . (3) 
 
 S 
 
 In general, the battery resistance B is small compared to R and G, 
 and may be neglected, so that the expression for X becomes 
 
 The reason for reducing the galvanometer deflection in the second 
 experiment to nearly the same value as in the first, is that in general the 
 currents are not proportional to the galvanometer deflections, i.e. K is not 
 a constant for all deflections. If, however, the deflections are nearly the 
 same, we are justified in assuming that the same value of K is applicable 
 in the two cases. 
 
 189. Measurement of the Insulation Resistance of a Piece of 
 Rubber Covered Cable. The coil of cable E (Fig. 196) is placed either 
 in a metal tub, or, if a wooden or earthenware vessel is used, a sheet of 
 zinc D is placed at the bottom. The tub is filled with water to well over 
 the cable, except the two ends which project above the water, and which 
 may be conveniently clamped between two pieces of wood which rest on 
 the rim of the tub. The insulation must then be pared away at the two 
 ends of the cable with a clean and sharp knife so as to leave a conical, 
 fresh-cut surface of the rubber, as shown at (a). The copper core of 
 the cable is connected by a wire to one of the terminals F of the 
 galvanometer, while the metal plate in the tub is connected to one 
 
462 
 
 MEASUREMENT OF RESISTANCE 
 
 [190 
 
 pole of a battery of cells, the other terminal of the battery being con- 
 nected to the other terminal of the galvanometer. The wire joining 
 c and F should be an air-wire, that is, it should not touch any body 
 between c and P. A piece of fine uncovered copper wire is then wound 
 round the middle of the conical, fresh-cut portions of the insulation and 
 connected to the terminal E of the galvanometer. The object of this wire 
 is to allow of the electricity which leaks over the surface of the insulation to 
 pass back to the battery without going through the galvanometer. In the 
 absence of this guard- wire, the use of which is due to Price, 1 the deflection 
 of the galvanometer would be due to the sum of the current which passes 
 through the rubber coating of the cable, and the current which passes 
 from the water along the surface of the rubber to the copper core at 
 A or c. Owing to the fact that a film of moisture almost immediately 
 forms, even on a freshly cut surface, this surface leakage has quite an 
 
 (Ct) 
 
 FIG. 196. 
 
 appreciable magnitude compared to the current which passes through 
 the insulating sheath when this sheath is composed of a good insulator. 
 Since the wire H and the core of the cable are very nearly at the same 
 potential, there is no appreciable leakage over the portion of the cleaned 
 rubber between the wire H and the core. There will be some leakage 
 between the water and the wire H along the surface of the sheath, but 
 this leakage current, since it does not pass through the galvanometer, 
 will have no influence on the galvanometer deflection. 
 
 The procedure is the same as in the last section, the value of the 
 galvanometer constant K being derived from an experiment made with a 
 known high resistance, the galvanometer Veing shunted. 
 
 190. Measurement of Low Resistances Mathissen and Hocking's 
 Method of the Projection of Equal Potentials.- When the conductor 
 of which the resistance has to be determined has a very low resistance, 
 the Wheatstone bridge method is not applicable, owing to the fact that 
 
 1 Electrical Review (1895), vol. xxxvii. p. 702. 
 
190] MEASUREMENT OF LOW RESISTANCES 463 
 
 the resistances at the places where the conductor is connected to the 
 bridge would bear an appreciable ratio to the resistance to be measured. 
 For this reason, methods of measuring the resistance have to be employed 
 in which the contacts which are employed are potential contacts, and 
 hence variations in the resistance of these contacts will not affect the 
 result. 
 
 Suppose that it is required to determine the resistance of a certain 
 length, say, 50 cm., of a piece of thick copper wire. The wire CD (Fig. 
 197) is stretched straight, and two contact pieces, E and F, are slipped 
 on the wire and 
 placed at a distance 
 of 50 cm. apart. 
 The construction of 
 one of the contact 
 pieces is shown 
 at (a). It consists 
 of a wooden block, 
 with a V groove cut 
 in its lower face. In 
 a slot in the middle 
 of the block is 
 placed a brass disc 
 c, pierced with a cir- 
 cular hole, through 
 which the rod B, 
 which is being 
 measured, passes. 
 To the upper edge 
 of the disc is 
 soldered a screwed 
 upright which car- 
 ries three nuts. The 
 lower of these, E, 
 serves to clamp the 
 contact maker down 
 
 on the rod by drawing the disc c upwards. The other two, F and o, 
 serve to clamp the connecting wire which goes to the galvanometer. In 
 addition a calibrated wire, such as the wire of a metre bridge, must 
 be placed alongside the copper rod CD. A standard resistance coil of 
 O'Ol ohm will be required. In the usual form of such a coil there are 
 two thick copper terminals with binding screws J, H, so placed that 
 when a current is passed through the coil, entering at the end of one of 
 the terminals and leaving by the end of the other, then the resistance 
 between the points where these binding screws are attached is exactly 
 O'Ol ohm. 
 
 The manner in which the copper rod, the bridge wire, and the resist- 
 ance coil R are connected to a battery B (preferably a storage cell) is 
 
 FIG. 197. 
 
464 MEASUREMENT OF RESISTANCE [ 190 
 
 shown in the figure. Connecting wires are run from the points E, F, j, H 
 to four mercury cups in a block of varnished wood or paraffin, while one 
 terminal of a low resistance galvanometer G is connected to a fifth cup, 
 and by means of a U-shaped copper connector can be connected to either 
 of the other cups. The second terminal of the galvanometer is connected 
 to the slider i on the bridge wire. A resistance, R', of an ohm or two 
 will generally be found necessary in the battery circuit to prevent the 
 cell sending too great a current. 
 
 The battery circuit having been closed by means of the key o, the 
 galvanometer is connected to the point E, and the slider I moved till the 
 galvanometer is undeflected. The galvanometer is then connected in 
 turn with the points F, j, and H, and the positions of the slider for no 
 deflection determined. When making these adjustments it is as well to 
 roughly determine the positions for balance of all the points, and then going 
 through the different points again to determine the exact positions, keep- 
 ing the battery current on only just long enough to see whether there is a 
 permanent deflection after each adjustment of the slider. The reason for 
 not keeping the current on longer than is absolutely necessary is that 
 in order to obtain sensitiveness it is necessary to use a fairly large current, 
 and hence if this current is allowed to flow for any length of time, the 
 conductors will become appreciably heated. 
 
 If the length of the wire included between the points which are at the 
 same potential as E and F is Z 15 while the length between the points at 
 the same potential as the points J and H is 1 2 , while the resistance of the 
 coil E, is called R, the resistance of the rod CD between the points E and 
 F is given by 
 
 In this expression the lengths ^ and Z 2 are the reduced lengths of the 
 corresponding portions of the wire ; that is, they are the lengths, deduced 
 from the calibration curve, of the mean uniform wire, which would have 
 the same resistance as the actual portion of the wire used. 
 
 The above expression for the resistance of the portion of the rod can 
 at once be obtained as follows : If C is the current in the rod, and c 
 that in the wire AB, the difference of potential between E and F is XC, 
 while the difference of potential between J and H is RG. Also, the 
 difference of potential between the two points of balance on the wire 
 corresponding to E and F is l^-c, and that corresponding to the points G 
 and H is / 2 ?v, where r is the resistance of unit length of the bridge wire. 
 Hence we have the two equations 
 
 RC=L 2 rc, 
 
 X L 
 
 = 
 
191] MEASUREMENT OF LOW RESISTANCE 465 
 
 If the mean diameter of the rod between the points E and F is d, 
 while the distance between E and F is L, the specific resistance p of the 
 material of which the rod is composed is given by 
 
 This method may also be used for comparing the resistance of two 
 rods, and hence comparing the specific resistances of the materials of 
 which the rods are composed. For this purpose it is only necessary 
 to replace the resistance coil R by the second rod fitted with contact 
 pieces similar to those at E and F, proceeding exactly as before. It is 
 important to make the junction between the two rods in such a way that 
 the resistance of the junction does not vary during the course of the 
 experiment, otherwise the positions of the points of balance on the bridge 
 wire will also vary, and it will be impossible to obtain satisfactory 
 measurements. A satisfactory way of joining the rods is either to solder 
 them together or to bend about an inch at the end of either rod at right 
 angles, amalgamate these bent portions, and then bind them together 
 with uncovered copper wire, and dip the bent portions into a mercury 
 cup. 
 
 191. Measurement of Low Resistance The Method of Auxiliary 
 Conductors, or Kelvin's double Bridge. The two conductors which are to 
 be compared are stretched 
 out straight and connected 
 together at c (Fig. 198), 
 either by soldering or 
 with a mercury cup, as 
 described in the last 
 section. Two contacts 
 will be required on each 
 rod ; those on one rod, 
 as they can be kerjt at a A 
 fixed distance apart, may 
 be both attached to a 
 wooden rod, and the dis- 
 tance between the points 
 
 of contact determined K ^ N 
 
 once for all. The con- 
 tact makers are connected, 
 as shown in the figure, 
 
 to two resistances u and KL. These resistances consist of resistance coils 
 having a resistance of 2 or 3 ohms, and means is provided for connecting 
 the middle point of each resistance to the galvanometer G'. The two 
 resistance coils are placed close alongside in one box, so that they 
 shall always have the same resistance, and the ends and the mid- point 
 may conveniently be soldered to three binding screws, as shown in 
 Fig. 199. 
 
466 
 
 MEASUREMENT OF RESISTANCE 
 
 [191 
 
 The contacts on one of the rods, say, G and H, being kept fixed in 
 position, the distance between the contacts on the other rod is varied till 
 
 there is no deflection of the galvanometer. 
 
 When making the adjustment the battery 
 circuit must only be closed for the time that 
 is absolutely necessary, so as to reduce the 
 heating effect of the current to a minimum. 
 In order to minimise the effects of thermo- 
 currents, balance should be obtained with 
 the current in either direction. 
 
 Let jo be the resistance of that portion 
 of the rod AC intercepted between the points 
 E and F, while y is the resistance of the 
 rod CD between the points G and H. Further, 
 let the resistance of the segments of the split 
 resistances KL and u be called r v r 2 , r 3 , and 
 r 4 respectively, these quantities to include 
 
 the resistances of the leads to the points E, F, G, and H respectively. 
 Then the condition for no galvanometer deflection is 
 
 FIG. 199. 
 
 The theory of the method is most easily obtained graphically. Suppose 
 that we plot a diagram on which the ordinates represent the potentials 
 of the different points along the two rods ACD. Such a diagram is shown 
 in Fig. 200 (a), where the points are lettered to correspond with the 
 points on Fig. 198. Next on a base of convenient length OQ (Fig. 200, b) 
 let us plot as ordinates at o the potentials of the points E and F, and as 
 ordinates at Q the potentials of the points G and H. Thus OE represents 
 the potential of the point E, and QH that of the point H, while the 
 straight line EH will represent the fall of the potential along the EKNLH 
 (Fig. 198) if we suppose that abscissae represent the resistance of this- 
 branch measured from the point E. In the same way the straight line FG 
 will represent the fall of potential along the branch FIMJG. Now the onlj 
 point which is common to these two straight lines is the point p where 
 they intersect, so that if the galvanometer is to have no deflection il 
 must be connected between the points on the two branches which an 
 represented by the point z. Now the two triangles EPF and HPG are 
 similar, hence 
 
 OZ 
 
 GH PH 
 
 Now EF represents the difference of potential between the points E and 
 while HG represents the difference of potential between the points 
 and H. But the current which flows through the portion of the one ro< 
 between E and F must be the same as the current which flows througl 
 
191] MEASUREMENT OF LOW RESISTANCE 467 
 
 that portion of the other rod included between G and H, for no current 
 passes across from M to N through the galvanometer. Hence 
 
 ^l = x^OZ 
 GH y ZQ 
 
 Now OZ/ZQ represents the ratio in which the point p divides either of 
 
 FIG. 200. 
 
 the branch resistances, and since this ratio is the same for either branch, 
 we have 
 
 ZQ r 2 r 4 
 
 that is, the condition that there should be no current through the gal- 
 vanometer is 
 
 If i\ = )\, and ?' 3 = r 4 , then x = y. 
 
 In general, the condition ^ 1 /^ 2 = r 3/ r 4 w ^ no * exactly be fulfilled, so 
 that the question arises if rj/r 2 is not exactly equal to ?* 3 /?* 4 , which of 
 these quantities, or what function of them, must be equated to xjy. This 
 can at once be answered by a study of Fig. 200 (6). It is obvious that 
 the slope of the line EH must always be greater than that of the line FG, 
 
468 
 
 MEASUREMENT OF RESISTANCE 
 
 [191 
 
 for the difference of potential between E and H must be greater than 
 that between F and G. Suppose, then, that r^ : ? % 2 as oz' : Z'Q, and ?' 3 : r 3 
 as oz : ZQ, so that the points between which the galvanometer is con- 
 nected are represented by P and P', evidently there would be quite a 
 considerable difference of potential, and to obtain balance we should have 
 to adjust, say, the contact H to a point where the potential was repre- 
 sented by QH', so that the straight line joining E and H' should pass 
 through the point P" where z'p" = ZP. Evidently, then, if s is the point 
 where the lines EH' and FG intersect, we have 
 
 y GH sii 1 
 
 But from the figure, since the line FG is nearly horizontal, because the 
 resistance between the points F and G (Fig. 198) is small, it follows that 
 
 ES . EP" 
 
 the ratio ==- r is nearly the same as or 
 
 ,^, 
 ^> that is, as r^fr^ V\ e 
 
 r 3 /V 4 is not exactly fulfilled 
 
 r ft 
 
 thus see that even when the condition rjr 2 
 
 that, so long as the line FG is nearly horizontal, that is, that the resistance 
 between the points where the contacts F and G touch the rods is small, 
 we have 
 
 y r 2 
 
 The above investigation shows the importance of making the con- 
 tacts F and G of the branch circuit as near the point of junction of the 
 rods as possible. 
 
 By replacing one of the rods by a standard O'Ol ohm coil the 
 resistance of a certain length of the rod under experiment can be 
 
 determined in absolute 
 measure, and hence the 
 specific resistance of 
 the material can be 
 calculated if the cross- 
 section of the rod is 
 measured. The dis- 
 advantage of using a 
 coil is that such coils 
 are generally made of 
 some alloy, such as 
 manganine or con- 
 stantin, which has a 
 FIG. 201. small temperature co- 
 
 efficient, while generally 
 
 the material being measured is copper, which has a fairly large tem- 
 perature coefficient. Thus if the temperature varies, the ratio of the 
 
192] 
 
 THE DIFFERENTIAL GALVANOMETER 
 
 469 
 
 resistance of a given length of the specimen to that of the coil varies also. 
 If, however, a rod of copper is employed as the standard, then so long 
 as the temperature of this standard and that of the specimen being 
 tested vary together the ratio of their resistances remains the same. 
 
 The above investigation shows that particular care should be taken in 
 measuring the ratio r-^/r^ which must include not only the portions of the 
 divided resistance, but also the wires used to connect the resistance to 
 the contacts E and F. This ratio can be determined by the method given 
 in 182, the connections being made as shown in Fig. 201. 
 
 192. The Differential Galvanometer. A differential galvanometer 
 is one in which the coils are wound double, that is, two wires are wound 
 on side by side, the ends of the two wires being brought to separate 
 terminals. If the two circuits were exactly the same, then if a current is 
 passed one way through one of the circuits and in the opposite direction 
 through the other their effects will be exactly equal and opposite, and the 
 needle will remain undeflected. In general, such equality is not exactly 
 secured, and a small coil, which is in series with one of the circuits, 
 is arranged so that it can be moved towards or away from one of the sets 
 of magnets, and the position of this coil is adjusted till there is no 
 deflection when the same current is passed through the two circuits in 
 opposite directions. In many cases what is required is that there should 
 be no deflection when the E.M.F. between the terminals of the two 
 circuits is the same. To make the balance in such a case the two coils 
 are connected in parallel, but in such a manner that the current in the 
 two coils is in opposite directions, and the adjusting coil is moved till 
 there is no deflection. It must be remembered that the adjustment, 
 whether for equal currents or for equal potential differences, depends on 
 the exact position of the magnet 
 system with reference to the 
 coils, and hence on the level of 
 the galvanometer. ' For this 
 reason each time the galvano- 
 meter is moved the adjustment 
 must be repeated. 
 
 When the galvanometer has 
 to be adjusted for equal 
 potentials, Crawley l has pro- 
 posed to use an adjustable 
 resistance placed in series with 
 the coil which has the greater 
 magnetic effect on the needle, 
 this resistance being adjusted till when the coils are in parallel there is 
 no deflection. The arrangement employed when balancing is shown in 
 Fig. 202. M and N are the two coils of the galvanometer which have 
 very nearly the same resistance, a convenient value being 100 ohms 
 each. The coil M is connected to the terminal 1 on the base of the 
 1 Jour, Inst. Elec. Eng., April 1901. 
 
470 
 
 MEASUREMENT OF RESISTANCE 
 
 [ 193 
 
 galvanometer through a resistance r of Ol ohm, a stretched wire CD, and 
 a sliding contact F. A resistance K can, if necessary, be inserted in the 
 circuit of the coil N so as to make it possible to balance on the slide wire. 
 A key K when depressed short-circuits the resistance r, and hence alters 
 the resistance of one of the circuits by a known amount, and the corre- 
 sponding deflection serves to determine the value of a scale division of the 
 galvanometer. It is advisable to have a short flexible lead permanently 
 attached to each terminal of the coil so that these leads are included 
 in the circuits when balancing. 
 
 193. Comparison of two Low Resistances with the Differential 
 Galvanometer. The differential galvanometer forms an easy, and at the 
 same time fairly accurate, means of comparing two nearly equal resist- 
 
 ances, whether they are in the 
 form of resistance coils with 
 potential terminals, or rods 
 with sliders. The manner in 
 which the connections are made 
 is shown in Fig. 203. The 
 potential terminals of the two 
 resistances P and Q which are 
 to be compared are connected 
 by means of the flexible leads 
 with the four terminals of the 
 differential galvanometer, and 
 a current from a battery B is 
 passed through the resistances, 
 which are placed in series. If 
 the resistances are very nearly 
 the same, or if they can be 
 made the same, as in the case 
 of rods in which the distance 
 between the contacts on one of 
 
 the rods can be altered, the galvanometer deflections are generally 
 sufficient to determine the ratio of the resistances, since the resistance 
 r and the key K enable the resistance in one of the galvanometer 
 circuits to be decreased by a known fraction of the resistance of the 
 galvanometer coil. 
 
 If the difference between the resistance of P and Q is such that a 
 deflection of more than a few scale divisions is produced, then a resistance 
 box s is placed as a shunt on whichever of the two resistances is the 
 greater, and the value of s adjusted till there is no deflection. When 
 this is the case we have 
 
 P + S' 
 
 Although the effect of the two coils are equal on the needle, it does 
 not follow that they have equal resistances. Jf this difference were very 
 
194] THE SHUNT POTENTIOMETER METHOD 471 
 
 great, then, since the resistances p and Q are shunted by the galvanometer 
 coils and a current is passed through these coils, the current through P 
 would not be exactly the same as that through Q, and hence the above 
 expression would not hold, for it has been obtained on the supposition 
 that the same current flows through p and Q. The correction to be 
 applied in such a case can at once be obtained. In the absence of 
 the shunt s the resistance between the points p and p f is 
 
 MP 
 
 where M is the resistance of the coil M including the leads and 
 adjusting resistance. Similarly, the resistance between the points 
 
 NQ, 
 q and q' is -^ ^. Now the differential galvanometer shows that these 
 
 resistances are equal. Hence 
 
 MP NQ 
 
 M + p ~ N+Q' 
 
 P P(N-M) 
 
 U=* + - NM 
 
 Now N-M is the difference between the resistance of the two 
 galvanometer coils, and is always small compared to either N or M. 
 Thus if N and M are each 100 ohms, and N- M is even as much as 
 1 ohm, while P and Q are each about a tenth of an ohm, the correction 
 
 P(N M} 
 factor flV-r> has the value '00001, or the correction only amounts 
 
 to 1 part in 100,000. Thus as long as the difference N- M is not very 
 large, and pare is taken to always have M and N large compared to 
 P and Q, no correction on account of this effect will be required. In 
 any case in which the correction becomes appreciable it can immediately 
 be calculated if the difference between the resistance of the two galvano- 
 meter circuits is measured. 
 
 194. The Shunt Potentiometer Method of Comparing two Low 
 Resistances. The method described in 179 for calibrating a wire may 
 be applied to the comparison of two low resistances. 1 Suppose that the 
 two resistances to be compared, A and B (Fig. 204), are O'l and O'Ol 
 ohm, and that as usual they are provided with potential leads #, a' and 
 b, //. The resistances A and B are connected as shown in series with 
 a battery B and an adjustable resistance E r ' A second circuit consists 
 of two standard coils p and Q, which in the case considered would have 
 resistances of 10 and 1 ohm respectively, connected in series through an . 
 adjustable resistance n. 2 and a switch c. 2 with a battery B 2 . A resistance 
 box s is arranged as a shunt on one or other of the resistances P and Q 
 
 1 Campbell, Proc. Physical Soc. of London (1903), xviii. 480. 
 
472 MEASUREMENT OF RESISTANCE [ 194 
 
 according as the ratio A/B is greater or less than the ratio P/Q. A switch 
 c is prepared, consisting of twelve mercury cups, which are connected in 
 the manner shown to the potential leads on the resistances A, B, P, and 
 Q, and the galvanometer G. The cups 1 and 2 are permanently con- 
 nected together, and by means of a block of ebonite with four U-shaped 
 pieces of copper, shown at Y, the two circuits can be connected in two 
 separate manners. When the connector Y is between the left-hand set of 
 cups and the middle, the points a and p are connected together, and the 
 points a and p are connected to the galvanometer. When, however, the 
 
 B, 
 
 FIG. 204. 
 
 \ 
 
 connector is moved into the right-hand position the points b and q are 
 connected together, and the points 6' and q' are connected to the 
 galvanometer. 
 
 'jTo make the measurement, the current in the circuit containing 
 A and B is kept constant, and the resistance R 2 is adjusted till the galva- 
 nometer is undetected when the connector is in the left-hand position. 
 The connector is then placed in the right-hand position, and balance 
 obtained by varying the shunt s. Since a change in s will slightly alter 
 the resistance in the circuit of the battery B 2 , the first adjustment will 
 be slightly upset. By, however, making the two adjustments alternately, 
 a condition will soon be obtained in which there is balance in both 
 positions. The effect of a change in s on the adjustment made when the 
 connector is to the left will be decreased by making the E.M.F. of the 
 battery B 2 larger, and correspondingly increasing the resistance R. 2 . This 
 resistance may conveniently consist of a box of coils from a tenth of 
 an ohm upwards and an adjustable carbon resistance, such as that shown 
 
194] 
 
 THE SHUNT POTENTIOMETER METHOD 
 
 473 
 
 in Fig. 205, placed in series. The carbon resistance consists of a number 
 of plates of gas carbon about 7 centimetres square and 5 millimetres thick 
 held in a frame. These plates can be compressed by turning the screw A, 
 and in this way the resistance between the copper plates BC is reduced. 
 The carbon plates are insulated from the frame by vulcanised fibre 
 tubes, which cover the rods by means of which the ends of the frame 
 are connected. With fifty plates the resistance can be varied from about 
 0*3 ohm to 3 ohms. This combination of a box of coils reading down to 
 a tenth of an ohm and up to about 500 ohms and a carbon resistance 
 will be found of much use in many cases. The coils of the resistance 
 box need not be carefully adjusted, but they ought to be wound with 
 
 FIG. 205. 
 
 fairly thick wire, and to consist of some material, such as manganine, 
 which has a very small temperature coefficient, so that the resistance 
 of the coils shall not vary much owing to the heat developed by the 
 current, and, further, a fairly strong current may be passed without fear 
 of damaging the coils. 
 
 The balance is obtained with the batteries connected up in one 
 direction, and then they are both reversed by means of the switches c x 
 and C 9 , and a balance is again obtained, the mean of the two values 
 obtained for s being used in the calculation. 
 
 The resistance between the points q and q is 
 
 QS 
 
474 MEASUREMENT OF RESISTANCE [ 194 
 
 Hence since the drop of potential between a and a is the same as that 
 between p and p', while that between b and V is the same as that 
 between q and q ', if c t is the current through A and B, while c 2 is the 
 current through P and Q, we get 
 
 or 
 
 Ac^Pc, 
 QS 
 
 QS 
 
 In the case where the resistances A and B (Fig. 206), which are to 
 be compared, are nearly equal, a modification of the above method may be 
 
 employed. In this 
 case the auxiliary 
 resistance only con- 
 tains a single resist- 
 ance P, in addition 
 to the variable resist- 
 ance R. 2 , which is used 
 to adjust the current 
 in the auxiliary cir- 
 cuit. The resistance 
 P need not be known. 
 A shunt s is placed 
 between the terminals 
 of whichever of the 
 two resistances A or 
 B is the larger. A 
 switch similar to that 
 
 shown in Fig. 205 is employed to join the galvanometer and a 
 connecting wire between ad and pp, or between bb' and pp. Balance 
 is obtained by altering s and R 2 as before, when 
 
 BS 
 
 
 FIG. 206. 
 
 or 
 
 Of the various methods of comparing small resistance described, 
 Mathissen and Hocking's method is distinctly the least accurate, while 
 the accuracy which can be attained with the others, when suitable pre- 
 cautions are taken, is about the same, and they are capable of showing a 
 difference of about 1 in 100,000. 
 
CHAPTER XXVII 
 
 THE RESISTANCE OF ELECTROLYTES 
 
 195. Mance's Method of Measuring the Resistance of a Battery. 
 
 In this method the battery E to be tested is placed in one of the gaps 
 of a Wheatstone's bridge, the position generally occupied by the battery 
 being occupied by a make-and-break key K, as shown in Fig. 207. The 
 resistances P, Q, R are then adjusted, till completing the circuit at K 
 does not alter the galvanometer de- 
 flection. When this is the case, the 
 resistance of the battery is given by 
 
 B = RQ 
 
 ' p ' > 
 
 This at once follows from the 
 property of congregate conductors ; 
 see p. 433. 
 
 The current which is passing 
 through the galvanometer all the 
 time will either produce a deflection 
 right off the scale, or at any rate the 
 deflection will be so great that the 
 galvanometer is not at all sensitive to 
 
 small changes in the current, and hence considerable changes in the 
 resistances in the branches can be made without appreciably altering the 
 deflection. This difficulty can be got over by reducing the galvanometer 
 deflection to zero by some external means. One method is to adjust 
 the controlling magnet so that the needle occupies its central position 
 when the current is passing. In many galvanometers, however, the 
 controlling magnet is not sufficiently strong for this purpose, and an 
 auxiliary magnet must be used. A better plan, however, is to reduce 
 the deflection to zero by acting on the galvanometer needle by means of 
 the field produced by an auxiliary current. If a differential galvano- 
 meter ( 192) is available, one circuit can be used in connection with the 
 bridge, while a current from an auxiliary battery is passed through the 
 other circuit and an adjustable resistance, and the current in this 
 circuit is so adjusted that the deflection is small. In the absence 
 of a differential galvanometer, a Thomson galvanometer, in which the 
 two sets of coils are brought to separate terminals, may be employed, 
 
 476 
 
476 THE RESISTANCE OF ELECTROLYTES [ 196 
 
 one pair of coils being used for the bridge, and the other for the auxiliary 
 current. 
 
 Another method of avoiding the difficulty, and one which avoids all 
 chance of damaging the galvanometer by the passage of an excessively 
 large current, has been recommended by Lodge. This consists in placing 
 a condenser in series with the galvanometer. The presence of the 
 condenser prevents any steady current passing through the galvanometer, 
 which may therefore be made very sensitive, and should preferably have 
 a high resistance. If, however, the difference of potential between the 
 points c and D changes when the key K is closed, the charge in the 
 condenser will alter, and this change in the charge of the condenser takes 
 place through the galvanometer, which will therefore give a momentary 
 kick. The experiment consists, therefore, in adjusting the resistances 
 till there is no galvanometer kick when the key K is opened or closed. 
 The condenser ought to have a capacity of about J microfarad. 
 
 In the case of cells, such as Daniell cells, which have a fairly high 
 resistance, and in which polarisation does not readily occur, the resistances 
 used in the arms of the bridge need not be very much larger than the 
 resistance of the battery being tested. If, however, a cell, such as a 
 Leclanche, which polarises easily, is being tested, the resistances in the 
 arms of the bridge should be kept large, otherwise polarisation of the 
 cell will occur and interfere with the measurements. 
 
 196. Beetz's Method of Measuring the Resistance of a Battery. 
 In Mance's method of measuring the resistance of a cell it is necessary 
 to allow the cell to send a current continuously while the measurement 
 
 is being made, and hence it 
 
 1 l Ipj} is only applicable to so-called 
 constant cells. In the case 
 of inconstant cells, a method 
 devised by Beetz, in which 
 it is only necessary to close 
 the battery circuit moment- 
 arily, can be employed. In 
 addition to the cell B (Fig. 
 208), of which the resistance 
 is required, this method re- 
 FIG. 208. quires a second cell b, which 
 
 must of necessity have a 
 
 lower E.M.F. than that of the cell B. These two cells are connected up 
 with two resistance boxes R X and R 2 , a stretched wire AD, which may be 
 the wire of a metre bridge, a galvanometer G, and a special form of key 
 K. This key is so constructed that on pressing it down it first completes 
 the circuit joining RJ to B, and then immediately after completes the 
 connection with the cell b. The experiment consists of adjusting the 
 resistances R X and R 2 and the position of the slider c on the wire, till on 
 pressing the key the galvanometer remains undeflected. 
 
 If the E.M.F. of the cell B is E, since there is no current through the 
 
197] RESISTANCE OF ELECTROLYTES 477 
 
 galvanometer, the current through the circuit BKADB is E/(R l + R 2 + B + p), 
 where B is the resistance of the cell B, and p is the resistance of the whole 
 bridge wire. Hence if a is the resistance of the portion AC of the bridge 
 wire, the difference of potential between the points K and c of this 
 circuit is 
 
 Z + B + P 
 
 But since there is no current through the galvanometer, the difference of 
 potential between the ends of the branch KGC, due to the current in the 
 other branch, must be equal and opposite to the E.M.F. e of the cell b. 
 Hence 
 
 The resistances R x and R 9 are now altered, and a new position of balance 
 is found. Using accented letters to indicate the values of the various 
 resistances which occur when this second balance is being obtained, we 
 have 
 
 
 Hence from (1) and (2) 
 
 P _ (R\ + a')(R l + Ro + p) - (R l + a)(R\ 
 
 from which the resistance of the cell B can be obtained. 
 
 When making the adjustment, it will often be found that on pressing 
 the key the galvanometer gives a sharp movement, and then on keeping 
 the key down a slowly increasing deflection is obtained, which may be in 
 the same or the opposite direction as that of the original kick. The first 
 kick indicates that the balance is incomplete, while the slow drift is due 
 to the fact that the E.M.F. of the cell B is changing, due to the passage 
 of the current. Hence in making the adjustment it is the initial kick 
 which must be reduced to nothing, so that the galvanometer starts out 
 on its slow drift without any preliminary waver. 
 
 197. Measurement of the Resistance of Electrolytes. Owing to 
 the phenomenon of polarisation which takes place at the surface of the 
 electrodes used to connect an electrolytic resistance to the rest of the cir- 
 cuit, the measurement of the resistance of electrolytes is a matter of some 
 little difficulty. Hence the methods used in measuring the resistance of 
 metallic conductors have to be modified in the case of electrolytes in such 
 a manner as to eliminate the effects of this polarisation. There are three 
 methods commonly employed for this purpose, which depend on the 
 following principles : 
 
 (1) Balancing the polarisation in the cell containing the electrolyte 
 
478 
 
 THE KESISTANCE OF ELECTROLYTES 
 
 [198 
 
 being measured by the polarisation in an auxiliary cell with similar 
 electrodes, and included in the opposite arm of the Wheatstone's bridge. 
 At the same time by using high resistances in the arms of the bridge, and 
 a large E.M.F. in the battery, swamping the residual effects of polarisation 
 due to the two cells not being exactly alike. 
 
 (2) Using a potentiometer method, in which no current is allowed 
 to pass through the electrodes used to connect the electrolyte to the 
 measuring apparatus. 
 
 (3) Using a rapidly alternating current, in which case it is found 
 that since it requires the passage of a certain quantity of electricity to 
 develop the polarisation, the effect of polarisation may be reduced to an 
 inappreciable amount if suitable precautions are taken to use electrodes 
 of large area and not to have a large alternating current. 
 
 We shall proceed to describe a typical arrangement for carrying out 
 an experiment by each of the above methods. 
 
 198. Measurement of Electrolytic Resistance by the Passage of a 
 Direct Current. The connections for this method of measuring the 
 resistance of an electrolyte are shown in Fig. 209. The two arms A and B 
 
 consist of two equal resistances of 
 about 1000 ohms, while in the 
 other arms are placed two similar 
 electrolytic cells, except that while 
 in P the tube containing the electro- 
 lyte is long, in the cell Q the tube 
 is short. An adjustable resistance 
 E, is placed in series with Q, and 
 the bridge is balanced by varying 
 this resistance. The galvanometer 
 G ought to be a high resistance 
 one, and the battery E should have 
 an electromotive force of from 10 
 to 30 volts. 
 
 The electrolytic cells consist of 
 two stout-walled tubes A, A' (Fig. 
 210), with a side tubulure near the 
 bottom of each, and a narrow-bore 
 
 tube B, which passes through corks in these tubumres. In the case of 
 electrolytes which would be contaminated by contact with a cork, either 
 wood or india-rubber, the end of the narrow tube can either be ground into 
 the tubulure, or it can be fused into the side of the tube A. Each electrode 
 consists of a plate of platinum foil D, which is welded to a short length of 
 platinum wire, this wire being fused through the bottom of a short length 
 of glass tube c. By pouring mercury into this tube, connection between 
 the cell and the rest of the bridge can be made by means of stout cop pel- 
 wires, the ends of which are amalgamated and dip into the mercury. 
 These electrodes are held in place by means of corks, through which the 
 glass tubes c pass. 
 
 FIG. 209. 
 
198] MEASUREMENT OF ELECTROLYTIC RESISTANCE 479 
 
 The electrodes must be platinised by placing them in a solution of 
 3 parts of platinic chloride in 100 parts of water, to which a drop or 
 two of lead acetate solution has been added (0*02 parts), and passing a 
 current. The strength of the current must be so adjusted that a free but 
 regular evolution of bubbles takes place, the direction of the current being 
 reversed from time to time. The electrodes must then be well washed 
 and allowed to soak in water for several hours. 
 
 The horizontal tubes B should be made from stout-bore thermometer 
 tubing, a piece of tubing about 35 cm. long being chosen, of which the 
 bore is fairly uniform. To test the uniformity of the bore a thread of 
 mercury 2 or 3 cm. long is sucked into the tube and the length of the 
 
 FIG. 210. 
 
 thread measured at different parts of the tube, just as was done in the 
 case of the calibration of a mercury thermometer ; see 59. A suitable 
 piece of tubing having been found, 5 cm. is cut off from one end to form 
 the tube of the smaller electrolytic cell. 
 
 If L is the length of the tube P, while I is that of Q, then if they had 
 exactly the same bore the resistance R when balance is secured would be 
 equal to the resistance of a column of the electrolyte of length L - I. In 
 general, however, the bore is not quite uniform, and as a next approxima- 
 tion we may suppose that while the bore of either P or Q is uniform, yet 
 they are not exactly the same. If the radius of P is A, and that of Q is a, 
 the volume of the bore of P is irA 2 L, and that of Q is Tra, 2 /. Hence if W 
 is the weight of mercury of density 8, which will just fill the bore of P, 
 and w is the corresponding quantity for Q, we have 
 
 and 
 
 TF 
 
 10 
 
480 THE RESISTANCE OF ELECTROLYTES [ 198 
 
 Suppose, now, that the two tubes are filled with an electrolyte of specific 
 resistance p, then the resistance of the cell P is given by 
 
 In the same 
 
 way 
 
 Hence P - Q-p L- 
 
 2 ' 
 
 L L*S I PS 
 
 -- 
 
 Hence B- P- Q- 
 
 
 The factor by which ^? is multiplied is a constant for any pair of tubes, 
 and having been determined once for all, the specific resistance of the 
 electrolyte is obtained by multiplying R by this factor. 
 
 To determine the value of the reduction factor for a pair of tubes, the 
 ends of the tubes must be ground off perpendicular to the axis of the 
 tubes. The tubes having been well cleaned and dried, a piece of rubber 
 tubing is attached to one end and mercury drawn up to the top of the 
 tube. The lower end is then closed by pressing with the finger a small 
 disc of glass against the ground end of the tube. The rubber tube is 
 then removed and, if necessary, mercury added till the tube is quite full 
 and a small globule of mercury stands above the end of the tube. A 
 second glass plate is then slipped on so as to remove the excess mercury, and 
 the mercury in the tube is allowed to flow into a weighed watch glass, 
 and its weight is determined. The lengths of the tubes have to be 
 measured with some care, since they are involved to the second power. 
 The best method to employ is the comparator method, described in 19. 
 
 In place of obtaining the values of the constant of the tubes by direct 
 measurement, it may be obtained by filling the cells with an electrolyte of 
 known conductivity and determining the corresponding value of R for 
 balance In Table 25 will be found the specific resistance of solutions of 
 certain salts for different concentrations as determined by Kohlrausch, 
 and these may be employed to standardise the tubes. It is advisable not 
 to use too weak a solution, otherwise the errors in making up the solution 
 may have too much influence on the result. 
 
 To perform an experiment the cells are filled with the electrolyte, the 
 electrodes placed in the end tubes, and the cells placed side by side in an 
 
199] SPECIFIC RESISTANCES OF ELECTROLYTES 481 
 
 oil bath, the level of the oil being above the horizontal tubes. A ther- 
 mometer is placed in the bath with its bulb alongside the horizontal 
 tubes, and the bath must be kept well stirred during the measurements, 
 as the conductivity of electrolytes varies considerably with the tempera- 
 ture. It is advisable to make settings with the current passing in either 
 direction, for which purpose a commutator is placed in the battery circuit. 
 199. Comparison of the Specific Resistances of Electrolytes by the 
 Potentiometer Method. The theory of the method is the same as that 
 used in the case of low resistances, described in 190. The solutions to 
 be compared are contained in two tubes x and Y (Fig. 211). The ends 
 of these tubes are bent at right angles to the length of the tube and dip 
 
 C, D, 
 
 (a) 
 
 into two tubes B and c, forming mercury electrodes. The form of one 
 of these tubes is shown in Fig. 211 (a). They can be easily made by 
 drawing down a piece of glass tube having a bore of about a centimetre, 
 and then bending the drawn-down portion in the manner shown. At 
 either end of each tube there is a second electrode vessel, which is con- 
 nected with the first by means of a U tube, which is filled with the same 
 solution as the main tube. The battery E I} which is used to send a 
 current through the tubes, is connected to A X and A 2 , the connecting 
 wires dipping into the drawn-down limbs of the vessels. The vessels 
 B I} Cp B 2 , c 2 are connected, by wires dipping into the mercury in the 
 small side tubes, with a set of mercury cups F. A second cell E 2 is 
 connected to two resistance boxes s and R placed in series, and wires 
 are taken from the terminals of R through a high resistance galvano- 
 meter G to the middle pair of mercury cups of F. The sum of the 
 
482 THE RESISTANCE OF ELECTROLYTES [ 199 
 
 resistances E, and s is kept constant, say, at 10,000 ohms, and the 
 resistance in B, required to give no deflection of the galvanometer 
 determined when the secondary circuit is first connected to B X and c l5 
 and, secondly, when connected to B. 2 and C 2 . 
 
 The object of the additional electrode vessels is to reduce the effects 
 of polarisation to a minimum, since, except when the point of balance 
 is being sought, no current passes at the electrodes B I} B 2 and C 1} C 2 . 
 Polarisation takes place at the electrodes A I} A 2 , D p D 2 , but the effect is 
 only to slightly reduce the current in the circuit, an effect which, as it 
 affects equally the current in the two tubes, does not influence the result. 
 
 If R x and R y are the resistances required for balance when the 
 secondary circuit is connected to the ends of X and Y respectively, then 
 
 X R r 
 
 Y R 
 
 If p x is the specific resistance of the solution in X, the resistance of 
 the solution in the tube will be given by X = p a x J where x is a constant 
 depending on the length and bore of the tube. Similarly for the tube Y 
 we have Y=p y y. 
 
 Hence Px_^ ^_^ y_ 
 
 p y ~ x ' Y~ R y ' x 
 
 In order to determine the value of the factor y/x, an experiment must 
 be made with the same solution in the two tubes. Let r x and r y be the 
 values of R required for balance. Then 
 
 y r.. 
 
 or = -?. 
 
 x r x 
 
 Thus the value of the factor required to obtain the ratio of the specific 
 resistances of the two solutions is obtained, this factor being a constant 
 as long as the same pair of tubes is employed. If the specific resistance 
 of one of the solutions is known, then that of the other can immediately 
 be calculated. 
 
 In the case of salts of the alkali metals, mercury electrodes such 
 as described are quite satisfactory. The polarisation may, however, be 
 reduced by covering the mercury surface in the electrode vessels with 
 calomel (mercurous chloride) if the solutions being compared are chlorides, 
 or with mercurous sulphate if they are sulphates. A little of the mer- 
 cury salt is placed on the surface of the mercury before the solutions 
 are poured in. 
 
 When performing the experiment the tubes and electrode vessels 
 must be carefully cleaned and dried, filled with the solution, and the 
 approximate value of R required for balance determined. The tubes 
 
200] MEASUREMENT OF ELECTROLYTIC RESISTANCE 483 
 
 should then be left for some time, so that the temperature may become 
 uniform throughout, this temperature being read by means of a thermo- 
 meter placed alongside. To assist in the temperature of the two tubes 
 being the same, they ought to be placed close together. The current in 
 the main circuit should be made as small as is consistent with sensitive- 
 ness, and should not be kept passing longer than is necessary for making 
 a reading. It is often advisable to use a battery of larger E.M.F. than 
 is absolutely required, and to reduce the current to the required value 
 by means of a resistance Q. Since it is assumed that the current in both 
 circuits remains constant throughout, the test balance should be made 
 first with X then with Y, and finally with X again, the mean of the two 
 values with X being taken. In this way a steady change in the currents 
 will not affect the results. 
 
 200. Measurement of Electrolytic Resistance by Kohlrausch's 
 Method with Alternating Currents. Kohlrausch has shown that the 
 E.M.F. of polarisation in an electrolytic cell is proportional to the 
 quantity of electricity which has passed through the cell, as long as 
 the quantity of electricity which has passed is not too great. Thus the 
 difference of potential between the electrodes of an electrolytic cell, 
 through which a current i is passing, when the resistance of the electro- 
 lyte is 7?, is given by 
 
 E=Ri+Pfidt, 
 
 where P is a constant which depends on the area of the electrodes and 
 on the nature of the electrodes and the electrolyte. 
 
 Suppose, now, that an alternating E.M.F. of frequency p/2ir is 
 applied to the terminals of the cell. Then if the applied E.M.F. follows 
 the simple harmonic law, it may be represented by E sin pt, and as 
 this must always be equal to the E.M.F. existing between the electrodes 
 of the cell, we have 
 
 Ri + Pj'idt = E Q sin pt. 
 Differentiating this expression with respect to the time 
 
 Rd i 
 
 The integral of this equation is 
 
 where the angle 6 is such that 
 
 tan 6 = - . 
 Rp 
 
484 THE RESISTANCE OF ELECTROLYTES [ 200 
 
 If the electrodes were such that there was no polarisation, so that P = 0, 
 the current would be given by 
 
 Jf 
 
 i =. - sin pt. 
 
 Thus the effect of the polarisation is to change the phase of the current 
 with reference to that of the applied E.M.F., and to decrease the ampli- 
 tude. Both these effects vanish when the fraction PjRp is vanishing 
 small. Hence to decrease the effect of polarisation, P should be small 
 and p great, that is, the area of the electrodes should be as great as 
 is convenient, and they should be platinised, also the frequency of the 
 alternations of the current should be great. 
 
 In order to supply the alternating currents, a small induction coil is 
 used. Quite a small coil, which will only give a microscopic spark, is 
 much the best, and it ought to be provided with a very rapid trembler. 
 The terminals of the secondary of the coil are connected to the battery 
 terminals of the Wheatstone's bridge, and the galvanometer is replaced 
 by a telephone. It will be found a great convenience to use two tele- 
 phones placed in series and fixed to a spring. This spring rests on the 
 top of the observer's head and keeps one of the telephones pressed 
 against either ear, thus allowing the hands to be free for manipulating 
 the contact on the bridge. It is also a good thing to place the induction 
 coil at some distance from the bridge, so that the noise of the interrupter 
 is not heard. 
 
 If the resistance coils employed to balance the bridge have either 
 self-induction or capacity, then there will be no arrangement of the 
 bridge which will give complete silence in the telephone, and the point 
 of minimum sound will be difficult to determine. Now, although the 
 ordinary bifilar method of winding resistance coils almost completely 
 avoids self-induction, yet coils wound in this way, especially those which 
 contain very many turns of wire, have very appreciable capacity. To 
 get over this difficulty, Chaperon has devised a form of winding in 
 which the capacity is much reduced. Since, however, coils wound in 
 this special way are seldom available, it is important to reduce the 
 number of coils to a minimum, and not to use coils having a great 
 number of turns, that is, coils of very high resistance. Hence the 
 stretched wire form of bridge is better for use with this method than 
 the Post Office form or dial form, and the resistances in the arms ought 
 never to exceed 500 ohms. 
 
 If it is desired to measure the absolute specific resistance of an 
 electrolyte, then the form of cell shown in Fig. 210 may be employed. 
 It will, however, be advisable to use a fairly short and wide-bore tube, 
 so that the resistance is not greater than 500 ohms. The electrodes 
 ought to be platinised in the manner described on p. 479. 
 
 When the relative specific resistances of two solutions has to be 
 determined, or where a solution of known specific resistance is available 
 
201] 
 
 PREPARATION OF SOLUTIONS 
 
 485 
 
 for standardising the electrolytic cell, the forms of electrode described 
 below may be employed. 
 
 In addition to the forms of cell described in the preceding sections, 
 there are other forms which in special circumstances present advantages. 
 A selection are shown in Fig. 212. The cell shown at a (Fig. 212) will 
 be found convenient when measuring the change of resistance with 
 temperature of badly conducting solutions. The solution having been 
 heated up to the highest temperature required, a series of measurements 
 
 (d) 
 
 of resistance are made as the solution cools, the temperature being in- 
 dicated by a thermometer with its bulb near the electrodes. 
 
 The cells shown at (b) and (c) are such that they can be introduced 
 within the liquid contained in a bottle or flask, and so long as the liquid 
 covers the upper hole A, the resistance is independent of the quantity of 
 liquid outside the cell. In the case of bad conductors the form (b) is 
 used, while for fairly good conductors the form (c) is better. 
 
 In the form of cell shown at (d) the liquid is sucked up into the bulb, 
 and then the tap is turned. 
 
 In the case of all these cells they may be standardised by using solu- 
 tions of known conductivity (Table 25), the method of preparing such 
 solutions being described in the following section. 
 
 201. The Preparation of Solutions for the Determination of Resist- 
 ance. In the preparation of solutions for conductivity determinations, 
 particularly if dilute solutions are to be used, it is important that the 
 water employed to make up the solutions should be as free from dissolved 
 electrolytes as possible. Ordinary distilled water, particularly if it has 
 stood about in glass bottles, has quite an appreciable conductivity. Its 
 conductivity may be much reduced by placing some distilled water in 
 
486 THE RESISTANCE OF ELECTROLYTES [ 201 
 
 a carefully cleaned beaker and boiling the water. The water is then 
 thrown away and a fresh supply poured into the beaker, and the beaker 
 is placed in a freezing mixture. The water in the beaker is not stirred, 
 so that a coating of ice forms round the walls of the beaker. When 
 about a quarter of the water has been frozen, the remaining water is 
 poured away and the ice allowed to melt. The water produced from the 
 melting of the ice will be found to have a much smaller conductivity 
 than that of the original water. Such water must, however, be used at 
 once, as if it is kept it will dissolve salts from the glass of the vessel in 
 which it is kept. The rate at which water dissolves glass depends very 
 considerably on the nature of the glass. The firm of Schott of Jena 
 make a glass which is particularly resistant to the action of water, and 
 can therefore be recommended for use when preparing solutions for resist- 
 ance measurements and for the construction of the vessels in which the 
 resistance of the solutions is measured. 
 
 The concentration of the salt used in the solutions is generally 
 expressed in terms of the number of gram-molecules of the salt con- 
 tained in a litre of the solution. By a gram-molecule is meant a number 
 of grams of the salt equal to the molecular weight of the salt. Thus in 
 the case of sodium chloride (NaCl), of which the molecular weight is 
 58*4, the normal solution, that is, one containing 1 gram-molecule per 
 litre, will contain 58*4 grams of the salt per litre. In the case of a salt, 
 such as copper sulphate, the molecule of which contains five molecules 
 of water of hydration, the molecular weight being 63*2 + 32 + 16x4 
 + 5(2 + 16), or 249'2, then the normal solution contains 249 '2 grams of 
 the hydrated salt per litre. 
 
CHAPTER XXVIII 
 
 MEASUREMENT OF ELECTROMOTIVE FORCE 
 
 202. The Construction of Standard Cadmium (Weston) Cells. 
 The cell, consisting of an amalgam of cadmium as the positive pole, 
 mercury as the negative pole, cadmium sulphate as electrolyte together 
 with mercurous sulphate as a depolariser, has many advantages over the 
 Clark cell, . in which zinc replaces the cadmium. These ad- 
 vantages are a very much smaller temperature coefficient, 
 greater permanence, and more rapid recovery from the effects 
 of the passage of an excessive current. 
 
 The following description of the method of preparing 
 cadmium cells is compiled from a specification prepared by 
 Mr. F. E. Smith of the National Physical Laboratory. 1 If 
 the work for which the cells are required does not necessitate 
 an accuracy greater than about 1 or 2 parts in 10,000, and 
 the cells are not required to last for several years, then a 
 simplified method of preparation may be employed, as in- 
 dicated below. 
 
 (A) Preparation of the Materials for a Standard Cadmium 
 Cell. (1) Mercury. The mercury is allowed to fall in a 
 very thin stream, obtained by attaching a very fine capillary 
 to a separation funnel, through a long (1 metre) glass tube 
 containing dilute nitric acid (1 of acid to 6 of water). A 
 convenient form of tube is shown in Fig. 213. The lower 
 part of the tube is drawn down and bent into an S form, as 
 shown. The mercury collects at the bottom of the tube, and 
 then flows over into a bottle c. Having been passed twice 
 through the acid, the mercury is passed through a column 
 of water contained in a similar tube. 
 
 The mercury is then twice distilled in vacuo. There are 
 a number of patterns of mercury still on the market, and 
 that shown in Fig. 214, called the Clark still, may be taken 
 as a type. At the same time, it is a pattern which the 
 author can recommend as requiring very little attention. A fairly, 
 .wide tube AB has a bulb at the upper end, and a side tube B near the 
 bottom. Inside this tube is a second tube AC, which is open at the 
 top, and has an S bend at the bottom. The side tube B is con- 
 nected by a length of thick-walled rubber tube with a Woolf's bottle F 
 
 1 British Association Report, 1905, p. 98. 
 
 487 
 
 L 
 
 FIG. 213. 
 
488 MEASUREMENT OF ELECTROMOTIVE FORCE [ 202 
 
 by means of a glass T-piece DB. A funnel G, which serves for the 
 introduction of the mercury, and a tube H, fitted with a tap, pass through 
 two well-fitting rubber corks in the upper openings of the bottle. 
 
 A tap K serves to stop the flow of 
 mercury when adjusting the pump. A 
 ring burner L below the bulb A serves 
 to heat the mercury. The object of the 
 tube DE is to keep the head of mercury 
 constant. Thus after mercury has been 
 introduced into the bottle, during which 
 operation the cock K must be closed, air 
 is sucked out through the tube H till 
 bubbles enter through D. The tap H is 
 then closed, and on opening K the head of 
 mercury is measured from D. As the 
 mercury flows out of the bottle into the 
 still, air enters through ED, but the level 
 of the mercury at D remains unaltered. 
 
 To start such a still, supposing it 
 empty, the bottle F having been filled 
 with mercury, it is raised up till the 
 mercury fills the tube AB and flows down 
 the tube AC. The rate of flow of the 
 mercury is adjusted so that the mercury 
 passes down AC in separate short columns, 
 which carry with them the air remaining 
 in the bulb. The still during this opera- 
 tion acts as an ordinary Sprengel pump. 
 When the exhaustion is complete, which 
 is indicated by the sharp click heard as 
 each mercury drop falls down the tube 
 AC, the bottle F is lowered on to a shelf 
 provided, the height of which is adjusted 
 so that the mercury about half fills the 
 bulb A. The burner L is then lighted, 
 the gas at first being turned very low. 
 When the mercury gets hot it boils, the 
 temperature at w^hich ebullition takes 
 place being fairly low since the bulb is 
 exhausted, and the vapour passes down 
 the tube AC, where it becomes condensed 
 by the cold mercury which is on its way 
 up to the bulb. The first pound or so 
 of mercury which is collected must be 
 returned to the bottle F, since it contains the mercury which filled the 
 bottom of the tube AC, and has therefore not been distilled. 
 
 If an efficient shield is provided to protect the flame from draughts, 
 
 FIG. 214. 
 
STANDARD CADMIUM CELLS 
 
 489 
 
 202] 
 
 and, further, if a regulator is placed to keep the gas pressure constant, 
 such a still may be kept working day and night without more attention 
 than is required to fill up the bottle F when necessary. The mercury 
 which is left in the tube AB contains all the metallic impurities, and 
 hence must be run out from time to time. 
 
 The disadvantage of the Clark mercury still, and all stills of this kind, 
 is that since zinc, cadmium, &c., are all volatile, some of these impurities 
 will be found in the distillate. It has, however, been shown by Hulett 
 and Minchin 1 that if air is passed through the mercury in the still, then 
 the zinc, cadmium, and other more oxidisable impurities become oxi- 
 dised, and since the oxides are not volatile, the distillate is free from 
 these impurities. The form of still they employ is shown in Fig. 215. 
 It consists of a distilling flask A 
 with a round bottom, having a 
 long neck. To the side tube B 
 is fused a thin-walled tube BC, 
 about 2 centimetres in diameter 
 and 50 centimetres long. The 
 end of this tube is narrowed down, 
 and passes through a wood cork 
 in the receiving flask D. To a 
 side tube (or if the flask has 
 no side tube, to a tube passing 
 through the cork) is attached a 
 T-piece E, having a cock F on one 
 branch. The third branch of the 
 T is connected by thick-walled 
 rubber tube to an aspirator (water 
 pump). The cork in the flask D 
 may be made air-tight with 
 sealing-wax. 
 
 Through the neck of the flask 
 A passes a narrow glass tube i, 
 
 drawn down to a fine capillary at the bottom. The joint between this 
 tube and the flask is made air-tight by a short piece of rubber tubing, 
 which slips tightly over both tubes. A short length of rubber tube is 
 attached to the top of I, and a p : ich-cock on this tube serves to adjust 
 the flow of air. 
 
 The flask A stands in a sand bath, and is surrounded by a shield J, 
 made of asbestos board. 
 
 The mercury having been introduced, the aspirator is started and the 
 flask heated with a Bunsen burner, the pinch-cock G being so adjusted 
 that the air bubbles through the mercury at the rate of one or two 
 bubbles per second. If the distillation becomes too rapid, as it has a 
 tendency to do just at first, it may be checked by admitting a little 
 air through the tap F. 
 
 1 Physical Review (1905), xxi. p. 388. 
 
 FIG. 215. 
 
FIG. 216. 
 
 490 MEASUREMENT OF ELECTROMOTIVE FORCE [ 202 
 
 (2) Preparation of the Amalyam. The amalgam used contains 12*5 
 per cent, of cadmium, and is prepared by electrolysis. To prepare this 
 amalgam some of the pure distilled mercury is placed in a glass vessel, 
 and above the mercury is poured a solution of cadmium sulphate, which 
 has been acidified with a few drops of sulphuric acid. A glass tube A 
 (Fig. 216), with a platinum wire fused through the end, serves for 
 making electrical connection with the mercury. The anode consists of 
 a rod of pure commercial cadmium B, to which the positive pole of a 
 battery is connected. The lower end of the cadmium rod rests in a 
 
 small crystallising dish c. The object 
 of this dish is to prevent the slime 
 which forms on the cadmium reaching 
 the mercury. The resistance in the 
 circuit is adjusted so that the fall of 
 potential between the anode and 
 cathode does not exceed 0'3 volt. 
 
 The cadmium anode is weighed 
 before the electrolysis and also after, 
 the difference of weight giving the 
 weight of cadmium in the amalgam. 
 
 The mercury also having been weighed, the quantity of mercury necessary 
 to reduce the strength of the amalgam to 12*5 per cent, is added. 
 
 The amalgam is then heated over a water bath, the surface being still 
 covered with a little of the cadmium sulphate solution, and well stirred. 
 The amalgam having been cooled, the acid cadmium sulphate solution is 
 removed, and neutral cadmium sulphate added. This neutral solution 
 consists of the saturated solution prepared as described in (3), diluted 
 with its own bulk of distilled water. The amalgam is now ready for use. 
 It is entirely liquid at about 60 C. 
 
 (3) Preparation of Cadmium Sulphate Crystals and Solution. Com- 
 mercially pure cadmium sulphate is dissolved in about 1'25 times its 
 weight of distilled water, which will require continual agitation for 
 some six hours, or periodical agitation for two or three days. This 
 solution is filtered, and is then placed in crystallising dishes, which 
 must be protected from dust, and allowed to slowly evaporate, the 
 temperature not exceeding 35 C. The crystals formed should be pre- 
 vented from adhering together by removing the liquid to a fresh dish 
 as soon as the crystals have grown to such an extent that they are 
 mostly in contact. In this way about five-sixths of the liquid may 
 be evaporated, and the mother liquid can be used for washing the 
 mercurous sulphate, as described in (4). 
 
 The cadmium sulphate crystals must be washed with successive 
 small quantities of distilled water, till the liquid that drains away 
 shows no trace of acidity after testing for five minutes with congo red. 
 The crystals while still moist must then be transferred to a stock 
 bottle. 
 
 The saturated solution is prepared from the crystals by prolonged 
 
202] 
 
 STANDARD CADMIUM CELLS 
 
 491 
 
 agitation with distilled water at a temperature of about 20 to 
 25 C. 
 
 (4) Preparation of the Mercurous Sulphate. This is prepared by 
 electrolysis, pure distilled mercury forming the anode, and a piece of 
 platinum foil the cathode, the electrolyte being dilute sulphuric acid 
 (1 volume of concentrated acid to 5 of water). The mercury is placed at 
 the bottom of a large flat-bottomed glass dish and covered with about 
 twenty times its volume of the dilute acid. The current is conveyed to 
 .the mercury as in the case of the preparation of the amalgam, and the 
 platinum cathode is suspended in the upper portion of .the acid. A 
 current density of about 0*01 amperes per square centimetre of the 
 mercury surface may be employed. During the electrolysis it is most 
 important that the electrolyte near the surface of the mercury shall be kept 
 in continual agitation. This may be done by using an L-shaped glass 
 stirrer, with the horizontal bar near the mercury surface. 
 
 The salt obtained in this way ought to have a grey colour ; if it is 
 tinged yellowish, this shows that the stirring has not been sufficiently 
 vigorous. Such yellowish salt must not be employed to construct 
 cells. 
 
 The mercurous sulphate is agitated with dilute sulphuric acid (1 
 volume of acid to 6 of water) and distilled mercury, and is then filtered, 
 and the greater part of the mercury is removed. The salt is then washed 
 with small quantities of cadmium sulphate solution till free from acid, 
 the mother liquid being 
 first used, but the final 
 washings being made 
 with the neutral solu- 
 tion prepared from the 
 crystals. 
 
 (5) Preparation of 
 the Mercurous Sulphate 
 Paste. Some cadmium 
 sulphate crystals are 
 ground in an agate 
 mortar with a little 
 cadmium sulphate solu- 
 tion, and about one 
 quarter the bulk of 
 distilled mercury and 
 twice the bulk of mer- 
 curous sulphate. The 
 
 whole must be well FIG. 217. 
 
 ground together, the 
 
 quantity of cadmium sulphate solution being so adjusted that a thin 
 
 paste is formed. 
 
 (B) Filling the Cell. In Fig. 217 are shown two types of cell. That 
 on the left (a) is hermetically sealed, while that on the right (b) is 
 
492 MEASUREMENT OF ELECTROMOTIVE FORCE [ 202 
 
 closed by means of corks and marine glue. The glass tube for the 
 (b) form is rather simpler to make, since the platinum wires have not 
 to be fused through the sides. This type is, however, slightly more 
 difficult to fill, since the glass tubes which protect the wires leading down 
 to the amalgam and mercury respectively are apt to become displaced 
 during the process of filling. In either figure M is mercury, A the 
 amalgam, p the mercurous sulphate paste, c cadmium sulphate crystals, 
 and s saturated solution of cadmium sulphate. 
 
 When setting up the cells, the first operation is to amalgamate the 
 platinum wires used to make contact with the mercury and amalgam. 
 In the case of the (1} form this can be done by heating the platinum red 
 hot, and then dipping it in mercury. In the (a) form the amalgamation 
 can be performed by electrolysing a solution of mercuric nitrate. 
 
 The corks K, used to close the (b) pattern tubes, must be well boiled 
 in water, and then soaked in the cadmium sulphate solution. In addition 
 to the hole bored to allow the passage of the electrodes, a second hole 
 must be bored through which the paste, crystals, &c., can be introduced. 
 
 The amalgam having been melted over a water bath, the surface 
 being covered with some dilute cadmium sulphate solution, some of the 
 amalgam is sucked into a heated pipette. The end of the pipette being 
 introduced into the limb of the H tube, the amalgam is allowed to flow 
 down into the tube, sufficient being introduced to well cover the platinum 
 wire. After the amalgam has solidified, the limb containing it should be 
 washed out with a little cadmium sulphate solution. Into the other 
 limb is then introduced, first, the mercury, and then the paste, a small 
 pipette made by drawing down a piece of glass tube being used, care 
 being taken that the paste does not spread up the side of the tube. 
 Next a layer of finely powdered cadmium sulphate crystals is placed in 
 each limb, and, finally, some saturated cadmium sulphate solution is 
 added. 
 
 The cells are then to be stored in a warm room for a week or more, 
 to allow some of the liquid to evaporate, the new crystals deposited 
 serving to bind the original crystals together. In the case of the (a) 
 type the glass tubes are sealed off before the blow-pipe, care being taken 
 not to heat the contents of the cell. In the cells of type (b), hot marine 
 glue is poured on the top of the corks, the small central holes having 
 been closed by means of little pieces of cork. 
 
 When in use, cells of type (a) should be immersed in an oil bath, 
 while cells of type (b) may be partly immersed either in an oil or 
 water bath. 
 
 In many cases a cell which is correct to about one part in two 
 thousand is amply accurate enough, in which circumstances a very 
 much simplified method of preparation may be adopted. The mercury 
 used should be distilled in vacuo as described, but the amalgam may be 
 prepared by dissolving commercially pure cadmium in the mercury. The 
 cadmium sulphate solution may be prepared by simply dissolving the 
 bought crystals in distilled water, while the crystals themselves may be 
 
204] COMPARISON OF E.M.F.'S WITH POTENTIOMETER 493 
 
 used in the cell. Bought mercurous sulphate is more apt to produce 
 differences ; if, however, Karlbaum's is obtainable, no very serious differ- 
 ence will be produced by using the salt as supplied. If this salt is 
 unobtainable, then it will be advisable to prepare the salt by the method 
 described above. 
 
 The E.M.F. at a temperature t of a cadmium cell, prepared as described 
 in the specification given above, may be taken l as given by 
 
 E t = 1-0186 - 3-8 x W~\t - 20) - 0'065 x 10- 5 (* - 20) 2 
 
 between the limits of 10 and 30. It is not advisable to use the cell at 
 temperatures outside these limits. In Table 26 will be found the value 
 of the E.M.F. of the cadmium cell at different temperatures calculated 
 from the above formula. 
 
 203. The Construction of the Standard Clark Cell. This cell 
 differs from the cadmium cell in that the cadmium is replaced by zinc. 
 Thus the positive pole consists of an amalgam containing 10 per cent, 
 of zinc, while the cadmium sulphate crystals and solution are replaced 
 by zinc sulphate crystals and solution. 
 
 The procedure described with reference to the cadmium cell may 
 mutatis mutandis be applied to the construction of the Clark cell, and 
 hence need hardly be repeated. This is particularly the case, since it is 
 likely that shortly the Clark cell will be entirely replaced as a legal 
 standard by the cadmium cell. 
 
 The E.M.F. of a Clark cell, at a temperature t } is given in'-inter- 
 national volts by 2 
 
 #,= 1-4328 - 119 x lQ- 5 (t - 15) - 0'7 x 10- 5 (- 15) 2 . 
 
 204. Comparison of E.M.F. 's with the Potentiometer. By far 
 the most accurate and convenient method of comparing the E.M.F. of 
 cells is the potentiometer. There are various arrangements for carrying 
 out the method, which will be described in the next section, but the 
 principle of all is the same. Suppose that a cell E (Fig. 218), of which 
 the E.M.F. is larger than that of the cells to be compared, sends a 
 current through a stretched wire AB, and to the end A of this wire, 
 which is connected to the positive pole of E, is connected the positive 
 pole of one of the cells e v which are to be compared. Then a wire 
 connected to the other pole of the cell e l will be at a lower potential 
 than the potential at A, and as by supposition the cell E has a larger 
 E.M.F. than that of e lt there will be some point along the wire AB at 
 which the potential, owing to the flow of the current in the wire, is as 
 much below that at A as is the lead attached to the negative pole of e r 
 Hence if we bring the lead in contact with this point on the stretched 
 
 1 Jaeger and Kahle, Wied. Annalen (1898), Ixv. 942. The E.M.F. is expressed 
 in international volts. The E.M.F. of a cadmium cell in true volts (10 8 c.g.s. 
 units) is probably 1-0184 Jit 20. 
 
 2 Jaeger and Kahle, Wied. Annalen (1898), Ixv. p. 941. 
 
494 MEASUREMENT OF ELECTROMOTIVE FORCE [ 205 
 
 wire, since they are at the same potential, no current will pass from the 
 lead to the stretched wire ; in other words, a galvanometer G will show 
 no deflection. Hence if C is the current in the wire AB, and the resist- 
 ance of the portion of this wire between A and c is r 1? we have that the 
 E.M.F. of the cell e l is given by 
 
 e.-r.C. 
 
 If, now, a second cell, of which the E.M.F. is e. 2 , is used in place of e l9 
 and the resistance of the portion of the wire between A and the new 
 position c', for which the galvanometer indicates no current, is r , then 
 
 Hence 
 
 If the wire is uniform, and the distances of the points c and c' from A 
 are ^ and 1 2 respectively, this result may be written 
 
 Such a potentiometer may be made direct reading by using a standard 
 cell, say, a cadmium cell, the E.M.F. of which at the temperature of the 
 observation is known. Thus suppose that the temperature is 20 C., 
 so that the E.M.F. is 1'0184 volts. If the wire is divided into 2000 
 divisions, the movable contact is placed at the reading 1018'4, counting 
 from the end A, and an adjustable resistance R in the circuit of the cell 
 E is adjusted till the galvanometer is undeflected. When this is the 
 case, we know that the difference of potential between A and the 101 8 '4 
 division is 1'0184 volts, and hence, if the wire is uniform, that the 
 difference of potential between the point A and the 1000th division is 
 1 volt. If, then, keeping the resistance K' constant we replace the 
 cadmium cell by a cell of unknown E.M.F., and the point of balance 
 is at the with division, the E.M.F. of the cell will be m/1000 volts. 
 Of course this relation will only hold so long as the current flowing 
 through the wire remains constant. Hence it is necessary to test this 
 by replacing the standard cell from time to time, arid if necessary 
 readjust the resistance ft'. If the battery E consists of one or two 
 accumulators of a capacity of about twenty ampere hours, and, after fully 
 charging, they are partly discharged and allowed to send a current 
 through the wire for a little time before the final adjustment is made, 
 it will be found that the changes in the current are extremely slow, so 
 that the adjustment remains satisfactory for a very considerable time. 
 
 205. Potentiometers. (1) Stretched Wire Potentiometer. A 
 stretched wire potentiometer, such as that shown diagrammatically in 
 
205] 
 
 POTENTIOMETERS 
 
 495 
 
 FIG. 218. 
 
 Fig. 218, may be made up by means of a metre bridge. The cell E 
 ought to be one or more accumulators, and the two cells to be com- 
 pared may conveniently be connected to a commutator, such as that 
 
 shown at c (Fig. 219), by means of which 
 the change from one to the other may be 
 rapidly effected. It is advisable to use a 
 high resistance galvanometer, so that when 
 the position of balance is being sought only 
 a small current should pass through the 
 cell, a matter of considerable importance 
 when using standard cells, such as Clark 
 cells or cadmium cells. For this same reason it is advisable to insert a 
 resistance of about 10,000 ohms in series with the galvanometer, a key 
 being arranged to short-circuit this resistance when balance has nearly 
 been secured. 
 
 (2) Ray lei gh' a Form of Potentiometer. Lord Kayleigh, in his work 
 on Clark cells, used a form of potentiometer which is very convenient, 
 and, where the utmost accuracy is not desired, does not involve the use 
 of any elaborate apparatus. The arrangement is shown in Fig. 219. 
 Here the stretched wire is replaced 
 by two resistance boxes p and Q, 
 each capable of giving a resistance 
 of 10,000 ohms, and when in use 
 the sum of the two resistances p and Q 
 is kept constant and equal to 10,000 
 ohms. The positive poles of the two 
 cells e\ and e 2 , which are being com- 
 pared, are connected to one end of the 
 resistance P and the galvanometer G to 
 the other end, while by means of a 
 commutator c either cell can be con- 
 nected to the galvanometer at will. 
 
 If PI and P 2 are the resistances 
 which have to be unplugged in P to 
 
 produce balance when first one cell and then the other are introduced, 
 we have 
 
 FIG. 219. 
 
 This arrangement may be used to give the E.M.F. of a cell being 
 tested direct by making the resistance of P 1000 times the E.M.F. of 
 a standard cell, and then adjusting Q till, when the standard is in 
 circuit, the galvanometer is undeflected. The resistance of P, when any 
 other cell is used, will then be numerically equal to the E.M.F. of the 
 cell multiplied by 1000. Of course the sum of the resistances P and Q 
 must be kept constant after the adjustment with the standard cell has 
 been made, and this adjustment must be tested from time to time 
 
496 MEASUREMENT OF ELECTROMOTIVE FORCE [ 205 
 
 to allow for changes in the current through the resistance boxes. It 
 is advisable to have a resistance R of about 10,000 ohms in series with 
 the galvanometer when making the preliminary adjustments, and to cut 
 this resistance out of the circuit for the final adjustments. 
 
 (3) The Kelvin and Varley Slide. The objection to the slide wire 
 form of potentiometer is that, in the first place, to obtain accuracy of 
 reading the wire must be long, and it must be calibrated. Further, owing 
 to the wire being exposed throughout its whole length and the action of 
 the sliding contact it is liable to damage, which will necessitate recali- 
 bration. In Rayleigh's form of potentiometer these objections are 
 
 101 COILS OF 1000 OHMS EACH 
 
 100 COILS OF20 OHMS EACH 
 
 98 99 100 
 
 N 
 
 FIG. 220. 
 
 removed, but where great accuracy is required the resistance boxes 
 require to be carefully calibrated, an operation in the usual form of box 
 which involves very considerable labour. Further, it is important that 
 the coils in the two boxes should have the same actual values, so that 
 removing one coil from one box and introducing a coil of the same 
 nominal value in the other should not alter the total resistance in the 
 circuit. An arrangement of resistances called the Kelvin and Varley slide 
 overcomes these objections. The plan on which the slide works is shown 
 diagrammatically in Fig. 220. A hundred and one coils AB, each having a 
 resistance of 1000 ohms, are connected in series with a contact stud between 
 each of the coils. A hundred equal coils MN, each having a resistance 
 of 20 ohms, are also connected in series with contact studs between the 
 
205] POTENTIOMETERS 497 
 
 coils. The terminals of this second set of coils are connected to two 
 sliding contacts KL, which are fixed at such a distance apart that they 
 always bridge over two coils of the first set in the manner shown in the 
 figure. Now when the sliders are in the position shown in the figure the 
 resistance from A to K is 4000 ohms, and the resistance from L to B is 
 95,000 ohms. The resistance from K to L is 1000 ohms, since the coils 
 UN, which have a combined resistance of 2000 ohms, are in parallel with 
 two of the coils of the AB series. Hence the combined resistance between 
 A and B, whatever the position of the sliders KL, is 100,000 ohms. 
 
 A single slider moves over the contacts of the 100 coils of 20 
 ohms each, and to this contact is connected the galvanometer and one 
 pole of the cell e, which is being tested. Suppose, now, that the other 
 pole of this cell is connected to A, while a battery E is connected to A and 
 B, and that by first moving the double slider KL approximate balance is 
 
 FIG. 221. 
 
 obtained, and that final balance is obtained by moving the slider c, the 
 position of balance being as shown in the figure. If, now, we call the 
 whole drop of potential between A and B 100, the drop of potential be- 
 tween A and K will be 4. Also the drop of potential between K and L 
 will be 1, that is, the drop of potential between M and N will be 1. The 
 drop of potential between M and c will, therefore, be 3/100 of 1, or 0'03. 
 Hence the whole drop of potential between A and c will be 4 + '03, or 
 4 '03, which, it will be noticed, is at once given by the numbers opposite 
 the sliders L and c. If, now, another cell is used, and the sliders moved 
 till the galvanometer is again undeflected, the readings corresponding to 
 L and c will give the drop of potential equal to the E.M.F. of this cell in 
 terms of the drop of potential between A and B. Hence if this drop of 
 potential has remained constant, that is, if the E.M.F. of the cell E has 
 remained constant, the ratio of the E.M.F.'s of the two cells will be equal 
 to the ratio of the readings of the two slides in the two cases. 
 
 ' 2i 
 
498 MEASUREMENT OF ELECTROMOTIVE FORCE [ 206 
 
 It will be noticed that the arrangement virtually amounts to a long 
 stretched wire, in which in place of being able to make contact at every 
 part we are restricted to making contact at 100 x 100, or ten thousand 
 points, which are distributed evenly along the wire. 
 
 The usual form which the Kelvin and Varley slide takes is shown in 
 Fig. 221, and the arrangement will be at once clear by comparison with 
 the diagram in Fig. 220. When using the slide to compare E.M.F.'s the 
 connections are similar to those shown in Fig. 220, and the method of 
 conducting the experiment is in every way the same. 
 
 Owing to the fact that all the coils in each dial have the same 
 resistance, the adjustment and testing of such a set of resistances is 
 particularly easy. 
 
 206. The Capillary Electrometer. A convenient form of capillary 
 
 electrometer is shown in Fig. 222. 
 It consists of a vertical upright tube 
 AB, alongside which is fixed a scale, 
 while to a side tube B is connected a 
 mercury reservoir D by an india-rubber 
 tube. This reservoir is supported 
 by a string, which passes over a 
 pulley E on to a drum F. The drum 
 can be rotated rapidly for the rough 
 adjustment by a small handle, and 
 slowly, for the final adjustment, by 
 the tangent screw G. To the lower 
 end of AB is attached by a short 
 sleeve of rubber tube a piece of glass 
 tube drawn down at the end to a fine 
 capillary. The end of the capillary 
 dips beneath the surface of an electro- 
 lyte contained in a glass vessel, the 
 lower part of the vessel being filled 
 with mercury. A platinum wire I, 
 fused through the side of the tube 
 AB, and a second wire J, enclosed in a 
 piece of glass tube, serve to join the 
 electrometer to the circuit. The 
 position of the mercury meniscus in 
 the capillary is observed by means 
 of a microscope M, which is fitted 
 with an eye-piece scale. 
 
 The position of the vessel contain- 
 ing the electrolyte should be so ad- 
 FIG. 222. justed that the capillary is near the 
 
 wall next the microscope, otherwise 
 
 a clear image of the meniscus will not be obtained. The end of the 
 eye-piece scale should always be brought to coincide with the end of the 
 
 
 
 ' 
 
 E 
 
 
 -A 
 
 
 I 
 
 
 Q 
 
 ! 
 
 
 
 
 \ 
 
 J 
 
 
 
 \ 
 
 L 
 
 hi 
 
 -x. _ 
 
 . kV 
 
CAPILLARY ELECTROMETER 
 
 499 
 
 207] 
 
 capillary, so that if the microscope is accidentally displaced it may be 
 put back into its original position. The mercury meniscus, when the 
 wires i and J are joined together, so that no difference of potential 
 is applied to the meniscus, should be near the middle of the scale, and 
 by altering the pressure, by raising or lowering D, the meniscus should be 
 brought to coincidence with one of the long divisions of the scale. If the 
 meniscus is always brought to this division, then the changes in the read- 
 ing for the upper end of the mercury column in AB will give the variation 
 in the surface tension. 
 
 The electrolyte generally used is a dilute solution of sulphuric acid 
 containing 1 volume of acid to 7 of water. The solution is saturated 
 with mercurous sulphate by placing a little of this salt on the top of the 
 mercury. 
 
 The capillary employed should be of such a bore that the surface 
 tension supports a head of mercury of about 20 cm. With such a capil- 
 lary the application of 10~ 4 volt will cause the meniscus to move 
 appreciably when a microscope magnifying fifty diameters is used. 
 
 For a description of the theory of the capillary electrometer, the reader 
 may consult Watson's Physics, 552. 
 
 207. To Examine the Relation between the Surface Tension of 
 Mercury and an Electrolyte and the E.M.F. acting across the Surface. 
 Connect a single secondary cell E (Fig. 223) to two resistance boxes 
 R and s, the sum of the two 
 resistances in circuit being 
 always kept equal to 10,000 
 ohms. The two ends of the 
 resistance s are connected to 
 two mercury cups, 1 and 2, in 
 a block of paraffin. The 
 terminal i of the capillary 
 electrometer is connected to 2, 
 and the terminal J to a third 
 cup 3. 
 
 By altering the resistance s, 
 any difference of potential may 
 be applied to the terminals of 
 the electrometer. Thus if E 
 is the E.M.F. of the battery, 
 and S is the resistance un- 
 plugged in s, R + S being 
 10,000, the applied E.M.F. is 
 S#/10,000 volts. 
 
 First join the cups 2 and 3, and adjust the position of the meniscus 
 in the capillary, so that it coincides with some fixed division of the eye- 
 piece scale, and read the height of the mercury column. Then starting 
 with the resistance S low, and having connected the cups 1 and 3, take 
 the readings of the heights of the mercury column required to bring the 
 
 FIG. 223. 
 
500 MEASUREMENT OF ELECTROMOTIVE FORCE [ 207 
 
 meniscus back to the fixed mark for different values of S. If the resist- 
 ance S be increased beyond a certain value, bubbles of hydrogen will 
 appear in the capillary, and it will be found impossible to proceed beyond 
 this point. 
 
 By reversing the battery, so as to make the mercury in the capillary 
 positive, it will be found possible to obtain a few more points. As, 
 however, S gets above 100 ohms, the mercury surface becomes sluggish, 
 owing to oxidation of the mercury, and the experiment must be stopped. 
 The oxidised surface may be got rid of by raising the mercury reservoir, 
 so as to cause some of the mercury to flow out of the capillary. 
 
 Plot a curve showing the connection between the surface tension and 
 the applied E.M.F. The scale readings for the top of the mercury 
 column may be taken as ordinates, and the values of S as abscissae. 
 The curve will be found to have a maximum for an applied E.M.F. of 
 about 1 volt. To determine the E.M.F. corresponding to the maximum 
 surface tension, which probably represents the contact electromotive force 
 between mercury and the electrolyte (see Watson's Physics, 552), it is 
 necessary to measure the E.M.F. of the cell E. This may best be per- 
 formed by comparing it with a standard Clark or cadmium cell by the 
 method given in 204, in which comparison the capillary electrometer 
 may be used in place of the galvanometer. To obtain the exact point 
 of the curve corresponding to the maximum, choose points in the two 
 sides of the maximum at which the surface tension is the same, and join 
 each pair of points. If each of these lines be bisected, the middle points 
 will be found to lie very nearly on a vertical straight line, and the point 
 where this line cuts the curve will give the maximum surface tension, 
 and the abscissa of this point is the E.M.F. required. 
 
CHAPTER XXIX 
 
 RESISTANCE THERMOMETERS AND THERMO-JUNCTIONS 
 
 208. Determination of the Constants of a Platinum Thermometer. 
 
 It has been found that the resistance of a wire of pure platinum 
 at a temperature t is given in terms of the resistance R Q at by an 
 expression of the form 
 
 where a and I are constants. This formula is found to hold over a very 
 large range of temperatures ; it is, however, not a convenient one to use, 
 since to deduce t from the values of R t and R involves the solution of a 
 quadratic equation. Professor Callendar has shown that if the resistance 
 of the wire at 100 is R L , and we define the platinum temperature pt, 
 corresponding to the resistance R t , by 
 
 
 then the difference between the temperature on the air scale and that on 
 the platinum scale is given by 
 
 
 where 8 is a constant for any particular sample of wire. Thus the right- 
 hand side of equation (2) gives the correction which has to be applied 
 to the platinum temperature, deduced from the resistance measurements 
 by means of equation (1), to obtain the corresponding temperature on 
 the air thermometer scale. 
 
 It will be seen that the correction involves t, the temperature on the 
 air thermometer, i.e. the quantity which we wish to obtain. This diffi- 
 culty may be removed, in the case of fairly low temperatures, by calcu- 
 lating an approximate value of t by replacing t in the right-hand side by 
 pt, and then using this approximate value of t.to calculate a more accurate 
 value of the correction. Griffiths has described another method of avoid- 
 ing this difficulty, which depends on the observation that if very pure 
 platinum is used, then the value of 8 does not differ much from 1'5. 
 A table, such as that given below, is first constructed, giving the value 
 
 501 
 
502 THERMOMETERS AND THERMO-JUNCTIONS [ 208 
 
 of the correction, assuming that 8 has the value of 1 '5 for different values 
 of pt, and these corrections are plotted on a large scale on a curve. 
 
 RELATION BETWEEN PLATINUM TEMPERATURES AND AIR 
 THERMOMETER TEMPERATURES WHEN 8 = 1 '5. 
 
 Platinum 
 Scale. 
 
 Correction. 
 
 Air Thermo- 
 meter Scale. 
 
 Platinum 
 Scale. 
 
 Correction. 
 
 Air Thermo- 
 meter Scale. 
 
 - 100 
 
 + 2'9 
 
 o 
 
 -97-2 
 
 400 
 
 + 20-2 
 
 420'2 
 
 -50 
 
 4-1*1 
 
 -48-9 
 
 450 
 
 + 27-0 
 
 477-0 
 
 
 
 o-o 
 
 o-o 
 
 500 
 
 + 34-9 
 
 534-9 
 
 + 50 
 
 -0-4 
 
 49-6 
 
 550 
 
 + 44-0 
 
 594-0 
 
 100 
 
 o-o 
 
 100-0 
 
 600 
 
 + 54-4 
 
 654-4 
 
 150 
 
 + 1-2 
 
 151-2 
 
 650 
 
 + 66-2 
 
 716-2 
 
 200 
 
 + 3-1 
 
 203-1 
 
 700 
 
 + 79-4 
 
 779-4 
 
 250 
 
 + 6-0 
 
 256-0 
 
 750 
 
 + 94-2 
 
 844-2 
 
 300 
 
 + 9-8 
 
 309-8 
 
 800 
 
 + 110-7 
 
 910-7 
 
 350 
 
 + 14-5 
 
 364-5 
 
 900 
 
 + 149-4 
 
 1049-4 
 
 
 
 
 1000 
 
 + 197-0 
 
 1197-0 
 
 If, then, the value of pt has been determined with a thermometer for 
 which the coefficient 8 has the value 8', ascertain from the curve the 
 value of the correction d corresponding to the platinum temperature pt, 
 then the correction in the case of the actual thermometer is 
 
 so that 
 
 In order to determine the value of 8 for any particular thermometer 
 the resistance must be measured at three known temperatures, the three 
 chosen being usually the melting point of ice, the boiling point of water, 
 and the boiling point of sulphur. 
 
 If the height of the barometer when the boiling point of sulphur is 
 determined is 760 + h millimetres, the temperature of the vapour over 
 boiling sulphur is given by 
 
 t, = 444-53 + 0-082/2,, 
 provided h is small. 
 
 Substituting this value of / s , in the expression 
 
 15-32 + 0-0065^. 
 
 , we get 
 
208] CONSTANTS OF A PLATINUM THERMOMETER 503 
 
 Hence if pt s is the temperature of the sulphur vapour in the platinum 
 scale, we have 
 
 t, -pt s = SJ15-32 
 
 or 
 
 444-53 + 0-082A 
 "15 
 
 (3) 
 
 The most convenient form of bridge to use with a platinum thermo- 
 meter is that described in 186. The leads connected to the thermometer 
 are attached to the binding screws P I and P 2 , while the compensating 
 leads are attached to c x and C 2 . Since the passage of the 
 current will of necessity heat the thermometer wire, it is 
 necessary to use as small a current as possible, and hence 
 it is advisable to place a resistance of from 50 to 100 ohms 
 in the battery circuit, a single accumulator being used to 
 send the current. 
 
 A convenient form of platinum thermometer is shown 
 in -Fig. 224. The wire is wound in a double spiral on a 
 mica frame, and the leads, as well as the compensating 
 leads, which are connected together at the lower end, are 
 kept apart by being threaded through a number of small 
 mica discs. For temperatures up to about 300 C., the 
 thermometer can be enclosed in a glass tube; for higher 
 temperatures, a glazed porcelain tube must be employed. 
 
 When determining the resistance in ice and steam the 
 apparatus used in obtaining the fixed points of a mercury 
 thermometer is employed, the precautions given in 59, 
 60 being observed. 
 
 To determine the resistance in sulphur vapour, the piece 
 of apparatus shown in Fig. 225 may be employed. It 
 consists of a glass 1 tube AB, with a bulb at the bottom, 
 such as is used with Victor Meyer's vapour density 
 apparatus. The bulb fits tightly in a hole in a piece of 
 asbestos board CD, while a tube of asbestos board F sur- 
 rounds the upper part of the glass tube, the intervening 
 space being packed with asbestos wool. A cone of asbestos board F 
 protects the lower part of the apparatus from draughts, and the top of 
 the tube is closed by a sheet of asbestos, through which the thermometer 
 passes. A cylindrical screen G, made of tinned iron, surrounds the 
 " bulb " of the thermometer, and serves to protect it from the loss of 
 heat by radiation. This screen is shown developed at H. A piece of thin 
 tinned iron having been cut into this shape, it is bent into a cylinder, 
 and the three upper tags are bound to the thermometer stem with iron 
 wire. The lower tags are bent to meet under the thermometer. 
 
 1 This glass tube may be replaced by a piece of thin si eel tube, closed at the 
 bottom by an iron cap. Thick iron gas-pipe is not suitable, as too much heat is 
 conducted along the walls. 
 
 (a) 
 
504 THERMOMETERS AND THERMO-JUNCTIONS [ 209 
 
 Enough sulphur is introduced to fill the bulb to about 3 centi- 
 metres above the board CD, and the apparatus is heated over a Bunsen 
 
 burner, the flame being adjusted so 
 that the vapour reaches about half- 
 way up the part of the tube left ex- 
 posed at the top. If the sulphur has 
 previously been melted in the tube, 
 then it will be necessary to push the 
 tube down so that the sulphur may 
 be melted, starting at the top. When 
 the whole of the sulphur has been 
 melted, then the bulb can be raised 
 up into the position shown in the 
 figure. 
 
 When the resistance of the thermo- 
 meter has become constant it must 
 be noted, as also the height of the 
 barometer. If R Q is the resistance in 
 ice, and R l the resistance in steam, 
 the temperature of which deduced 
 from the barometer pressure is t v 
 then the change of resistance for a rise 
 of temperature from to 100, which 
 is called the fundamental interval 
 (fl), is given by 
 
 FIG. 225. 
 
 00 
 
 Then if J^ y is the resistance in sulphur, 
 
 Then, finally, the value of 8 is obtained by equation (3), given above. 
 Knowing the fundamental interval and the value of 8, we have the data 
 for deducing the temperature from measurements of the resistance of the 
 thermometer. 
 
 209. Measurement of the E.M.F. of a Thermocouple. The most 
 satisfactory way of measuring the E.M.F. of a thermocouple is 
 the potentiometer method, described in 204, 205. A suitable 
 arrangement is shown in Fig. 226. The main circuit consists of the 
 accumulator E, a wire AB stretched alongside a scale and calibrated, a 
 resistance box K capable of giving a resistance up to 1100 ohms, and a 
 resistance box r, which is adjustable by tenths of an ohm. In the 
 absence of such a box, one adjustable by ohms, together with a carbon 
 variable resistance placed in series, may be used. 
 
 One lead from the hot junction H is taken to A, the lead from the 
 cold junction c being brought to the mercury cup 3. The galvanometer 
 
210] STANDARDISATION OF A THERMOCOUPLE 505 
 
 G, which ought to have a low resistance, is connected to a movable 
 contact on the wire and to the mercury cup 1. A standard cadmium 
 cell ( 202) is connected to the cup 2 and to the point F. 
 
 The wire AB ought to be at least a metre long, and it is a convenience 
 if its resistance is exactly 1 ohm. If the resistance of the wire is a little 
 over an ohm, then a 
 
 resistance s may be E 
 
 connected in parallel 
 with the wire, and 
 adjusted so that the 
 resistance of the two 
 in parallel is 1 ohm 
 (see p. 457). Sup- 
 pose that the tem- 
 perature of the 
 cadmium cell is 15, 
 so that its E.M.F. is 
 1-0188, then the re- 
 sistance R is made 
 
 1018 ohms, so that 
 the resistance be- 
 tween A and F is 
 
 1019 ohms. The mercury cups 1 and 3 are then connected together, and 
 the galvanometer contact moved to the end A of the wire. The resist- 
 ance r is then adjusted till the galvanometer is undeflected. When 
 this is the case, the difference of potential between A and F is 1'019 
 volt, and hence the difference of potential between the ends of the wire 
 is '001 volt, and each millimetre of the wire corresponds to a microvolt. 
 Hence if, when the mercury cups 1 and 3 are connected, the galvano- 
 meter contact has to be made at a point n millimetres from A, so that 
 there shall be no galvanometer deflection, then the E.M.F. of the thermo- 
 couple is n microvolts. 
 
 210. Standardisation of a Thermocouple. For high temperatures 
 (300 to 1000) the most satisfactory thermocouple is one consisting of 
 pure platinum and platinum containing 10 per cent, of rhodium, the 
 wire being about '6 mm. in diameter. If one junction of such a thermo- 
 couple is kept at C., and the other is heated to a temperature t, the 
 E.M.F. developed in the circuit can be expressed by a formula of the 
 form 
 
 FIG. 226. 
 
 It is found, however, that such an expression only holds at high tempera- 
 tures, 300 C. and over. 
 
 In the case of the Reichsanstalt standard Pt - PtR junction, the 
 E.M.F. is given in microvolts (10 ~ 6 ) by 
 
 E t = - 310 + 8-048* + 0-00172* 2 , 
 
506 THERMOMETERS AND THERMO-JUNCTIONS [ 210 
 
 and it is at once evident that such a formula cannot hold at low tem- 
 peratures ; for instance, at 0, i.e. when the two junctions are at the same 
 temperature, it gives an E.M.F. of - 310 microvolts. Over the range 
 300 C. to 1100 C. the formula, however, gives good results. 
 
 In standardising such a thermocouple, observations of the melting 
 points of various metals are employed, the following being frequently 
 used : 
 
 Metal. 
 
 Melting Point. 1 
 
 Zinc . ... 
 Antimony .... 
 Silver (in graphite crucible with 
 graphite lid) .... 
 Copper (in air) . . ,. . 
 
 419-0 C. 
 630-5 
 
 961-5 
 1065-0 
 
 The form of oven used by Holborn and Day for standardising thermo- 
 couples is shown in Fig. 227. It consists of a graphite crucible B, in 
 
 which the metal is heated, and in the case 
 of silver, where it is necessary to protect 
 the molten metal from the air, this is 
 covered by a second crucible D through 
 a hole in which passes a thin-walled 
 porcelain tube. The heat is supplied by 
 a coil of uncovered nickel wire wound 
 on a fireclay 2 tube. The wire has a 
 diameter of about 1 mm., and when the 
 oven has to be heated to a temperature of 
 1000, about 600 watts must be supplied 
 to the circuit. The heating coil is packed 
 round with asbestos, and the whole 
 enclosed by thick-walled fireclay slabs. 
 The thermo-element wires are threaded 
 down thin porcelain tubes, which are 
 contained within an outside porcelain 
 tube 5 mm. in diameter, having walls 1;5 
 mm. thick. The end of this tube dips 
 about 4 cm. below the surface of the 
 
 molten metal . 
 
 When performing an experiment the 
 metal is melted and then the heating 
 
 current is slightly reduced, so that the temperature slowly falls, and 
 the E.M.F. of the thermo-element is noted at equal short intervals 
 of time. When solidification takes place the rate of cooling is reduced, 
 
 1 Holborn and Day, Drude Annalen (1905), ii. 505. 
 
 2 Such tubes are sold under the trade name of " Morganite." 
 
 FIG. 227. 
 
210] STANDARDISATION OF A THERMOCOUPLE 507 
 
 and the E.M.F. at the melting point is obtained by plotting the E.M.F.'s 
 observed against the time, as was done in 104. 
 
 For comparatively low temperatures, say, below 300, platinum- 
 platinum-rhodium thermocouple is unsuited, the thermo-electromotive 
 force being so small as to make it impossible to measure temperatures 
 with the accuracy that is generally desired at these temperatures. In 
 such a case a junction consisting of pure iron and pure nickel will be 
 found suitable. Such a couple gives about 30 microvolts per degree, so 
 that even this couple is comparatively insensitive, as it is difficult to 
 measure the E.M.F. to much less than 1 microvolt, and in most cases 
 a platinum thermometer would be more suitable. If, however, such 
 a thermocouple is used, then it may be standardised by heating to 
 known temperatures, such as those given in 98, and a curve may be 
 plotted showing the connection between E.M.F. and temperature by 
 means of which the temperature corresponding to any particular E.M.F. 
 may be read off. 
 
CHAPTER XXX 
 
 MEASUREMENT OF CURRENT 
 
 211. The Tangent Galvanometer. The magnetic field F at the centre 
 of a circular coil of mean-radius R, and containing n turns, when 
 traversed by a current of A amperes, the cross-section of the coil being 
 very small compared to R, is given by 
 
 T-, irnA 
 
 1* = --- -~- gausses. 1 
 
 Hence if such a coil is placed with its plane in the magnetic meridian 
 at a place where the horizontal component of the earth's field is H, a 
 freely suspended magnetic needle of very small length will set itself 
 inclined at an angle 6 to the meridian such that 
 
 irn 
 
 (1) 
 
 Hence if the dimensions R and n of the coil are known, and H and the 
 angle are measured, we can obtain the value of the current which 
 traverses the coil. 
 
 The above expression only holds within the limitations mentioned; 
 namely, that the cross-section of the coil is small compared to the radius, 
 and that the needle is very short. If these conditions are not fulfilled, 
 then the expression connecting F and R requires modification. If the 
 section of the coil is a rectangle, the axial breadth, that is, the dimension 
 parallel to the axis of the coil being b, and the radial depth being <1, 
 and the distance 2 between the poles of the needle I, then the value 
 of the current is given by (see Maxwell's Electricity and Magnetism, 
 vol. ii. chapter xv.) 
 
 1 V* I <L 3 72 
 
 1 The unit of magnetic force on the c.g.s. system is called a gauss. 
 
 2 As an approximation the distance between the poles may be taken as equal 
 to 2/3 of the length of the needle. 
 
 508 
 
211] THE TANGENT GALVANOMETER 509 
 
 By suitably choosing the dimensions of the cross-section the term 
 
 1> 2 d? 
 - - - - . 2 can be made to vanish. This occurs when 
 
 that is, when 
 
 = 
 8 12' 
 
 v /2 
 
 The value of the term 2 (1 - 5 sin 2 #), since it depends on the 
 
 angle through which the needle is deflected, varies with the deflection. 
 The following table shows the value of the factor (1-5 sin 2 0) for some 
 values of 6 : 
 
 e 
 
 1-5 sin2 e 
 
 10 
 
 + 0-85 
 
 20 
 
 + 0-42 
 
 26 34' 
 
 + 0-00 
 
 30 
 
 -0-25 
 
 40 
 
 -1-07 
 
 50 
 
 -1-93 
 
 The value of the correction factor, and also the magnitude of the error 
 produced by any want of adjustment of the needle- to the centre of the 
 coil, can be materially reduced by employing a double coil. This arrange- 
 ment, which is due to Helmholtz, consists of two equal coils placed 
 coaxially and with their planes at a distance apart equal to the radius 
 of either, the needle being placed on the common axis half-way between 
 the coils. 
 
 If r is the mean radius of the coils, 2x the distance between their 
 mean planes, and the total number of turns on the two coils is N, then 
 the axial component of the field at a point in the plane perpendicular to 
 the common axis half-way between the mean planes of the coils, at a 
 distance y from the axis, is given by 
 
 F= Z 
 
 6a 7 
 
 Sa 11 
 
 + <fec. 
 
 . (3) 
 
510 ., MEASUREMENT OF CURRENT [ 212 
 
 where I is the axial breadth of the cross-section of each coil, d is the 
 radial depth, and a 2 = r 2 + x 2 , and C is the current in c.g.s. units. 
 
 If 2// is the distance between the poles of the needle suspended at 
 the centre of the double coil, then the above expression can be used 
 to calculate the couple acting on the needle, and hence if the deflection 
 is measured, and H is known, to calculate the value of the current 
 traversing the coils. In general, it will be found that the terms within 
 the brackets, except the first, are excessively small, and may be neglected. 
 Thus if r = 30 cm., x= 15 cm., b= *6 cm., d= '35, we have 
 
 F= 27r^C{0-02365 - 0'0 r 5 + 0'0 6 1 1?/ 2 - } . 
 
 212. Adjustment of a Tangent Galvanometer. Before a tangent 
 galvanometer can be used to measure a current, it is necessary that the 
 following adjustments be made : 
 
 (1) The plane of the coil must be vertical. 
 
 (2) The needle must be at the centre of the coil. 
 
 (3) The axis of the needle, when no current is passing, must be 
 
 parallel to the plane of the coils. 
 
 (4) The scale must be parallel to the plane of the mirror when the 
 
 needle is undeflected. 
 
 In order to make the first adjustment, a plumb-line may be set up 
 near the instrument, and nearly in the same plane as the coil. The 
 levelling screws are then adjusted till the face of the coil is parallel to the 
 plumb line. The coil is generally provided with marks, by means of 
 which we may see when the needle is at the centre of the coil, and it is 
 important to verify this adjustment. 
 
 To adjust the axis of the needle parallel to the plane of the coil, a 
 cell, preferably a secondary cell, must be connected to the galvanometer 
 through a resistance and a reversing key. The 
 E.M.F. of the cell must be so large that a measur- 
 able deflection is produced when the added resist- 
 ance is at least 10 ohms, otherwise changes in the 
 resistance of the commutator contacts may produce 
 considerable errors. The coils must then be turned 
 about a vertical axis till the deflection obtained is 
 the same whichever way the current is passing. 
 The theory of this adjustment is as follows : 
 Suppose that when no current is passing the needle 
 makes an angle a with the plane of the coil, and let 
 F be the strength of the magnetic field at the centre 
 produced when a current G traverses the coil, the 
 deflection of the needle produced being 0. In Fig. 228 
 let NS be the magnetic meridian, that is, the direction of the undeflected 
 needle, AB the plane of the coil, and m the needle when deflected. Then 
 the magnetic field F, due to the coil, is along OF. Hence if m is the strength 
 . of the pole n of the needle, we have a force mH parallel to ON, and a 
 
213] THE SINE GALVANOMETER 511 
 
 force mF parallel to OF. The moment of the first of these is mH' Dn 
 and that of the second is mF OG, and since there is equilibrium we have 
 
 00. 
 But if I is half the length of the needle, 
 
 Wn = I sin 0, and ~OC = I cos (6 + a). 
 Hence H sin 6 = F cos (8 + a). 
 
 Next suppose that the current is reversed, the deflection being now 0', we 
 should have by exactly the same reasoning 
 
 Hence if the deflections are the same, i.e. if = 8', we get 
 
 cos (0 + a) = cos (6 - a), 
 
 which only holds if a = 0, that is, if the plane of the coil coincides with 
 the axis of the needle in its undeflected position. 
 
 The manner in which the fourth adjustment is made has already been 
 described in 172. 
 
 213. The Sine Galvanometer. If a galvanometer similar to a 
 tangent galvanometer, that is, one with a coil the radius of which is 
 large compared to the size of the needle, is so mounted that it can be 
 turned about a vertical axis, and the angle through which it is turned can 
 be accurately measured, then a very convenient method of working is as 
 follows : The current being passed in one direction, the whole galvano- 
 meter is turned till the needle lies in the plane of the coils, a mirror 
 attached to the needle, and a telescope and scale attached -to, and turning 
 with, the coils being used to tell when this adjustment is complete. The 
 azimuth reading having been noted, the current is reversed, and another 
 azimuth reading is obtained when the needle is again in the plane of the 
 coils. If 20 is the difference between the azimuth readings, then using 
 the same notation as in 211, the current A (in amperes) is given by 
 
 _ 1 ^ _ J_ <P 3 P } 
 
 = irn \ * 8 r 2 12 r 2 + 16 r 2 / ' 
 
 It will be observed that the correction depending on the length of the 
 needle is the same for all values of 8 ; this is owing to the fact that 
 when taking a reading the axis of the needle is always at right angles to 
 the field due to the coil. Further, since the needle and coil have always 
 the same relative positions, the torsion of the suspension fibre does not 
 come in. The correction for the size of the needle can be reduced by 
 adopting the Helmholtz arrangement for the coils. 
 
512 
 
 MEASUREMENT OF CURRENT 
 
 [214 
 
 
 214. Measurement of a Current by the Deposition of Copper. 
 
 Where a current has to be measured, or a current measuring instrument 
 calibrated, and an accuracy of about 1 part in 500 is sufficient, this can 
 most conveniently be done by the electrolysis of a copper sulphate solu- 
 tion between copper electrodes. The conditions necessary for satisfactory 
 working have been determined by T. Gray, 1 and the following method 
 is based on his results : 
 
 A convenient form of electrolytic cell for use with currents up to a 
 few amperes is shown in Fig. 229. The anode consists of two copper 
 
 plates A and A', which are 
 held in clips fixed to a circular 
 disc of wood, the wood having 
 been soaked in hot paraffin 
 wax. The two clips are 
 connected together on the 
 upper side of the disc, and 
 | a binding screw H serves to 
 attach the connecting wire. 
 : The cathode c consists of a 
 similar plate of copper, which 
 fits in a clip, to which is at- 
 tached a binding screw G. 
 A slit in the wooden disc 
 A C A' allows of the cathode being 
 removed for weighing. The 
 disc rests on the top of a 
 beaker, in which copper sul- 
 phate is placed. The elec- 
 trode plates are made from 
 
 high-conductivity sheet copper, the cathode being about 0'3 millimetre 
 thick (No. 30 standard wire-gauge), and the anode plates about a 
 millimetre thick (No. 19 standard wire-gauge). The area of the cathode 
 should be such that there are at least 50 square centimetres of surface 
 per ampere of current passing through the cell. The edges and corners 
 of the electrodes should be carefully rounded, otherwise there is a 
 tendency for the deposit to be very thick and rough at the edges. 
 
 The solution consists of copper sulphate dissolved in water till the 
 density of the solution is between 1-15 and 1*18, that is, contains 
 between 20 and 25 grams of crystallised copper sulphate to 80 grams of 
 water. To this solution is added 1 per cent, of strong sulphuric acid. 
 
 When performing an electrolysis the cathode plate is removed and 
 well polished with sand or sand-paper (not emery paper), washed in water, 
 and dipped for a few minutes in a bath of water containing a little sulphuric 
 acid, the object of the acid being to remove any oxide not removed by 
 the sand-paper. Having been rinsed in pure water, the plate is placed 
 in the electrolytic cell, and a current passed for a short time. While this 
 
 i Phil. Mag., October 1886. 
 
 FIG. 229. 
 
215] 
 
 CALIBRATION OF AN AMMETER 
 
 513 
 
 current is passing, the resistance of the circuit can be adjusted so that 
 the current which passes is of the desired magnitude. The plate is then 
 removed and well rinsed in pure water, allowed to drain on to a piece of 
 white blotting-paper, and then dried by heating the tag with a spirit- 
 lamp. This heating must be very carefully performed, otherwise the 
 copper will oxidise. The electrode having cooled, it is weighed, and only 
 placed in the electrolytic cell immediately before the current is passed, since 
 copper is slightly soluble in copper sulphate solution. After the electrolysis 
 is completed, the plate is removed, rinsed in water, to which a few drops of 
 sulphuric acid has been added ; then well rinsed in water free from acid, 
 allowed to drain on to a sheet of clean white blotting-paper, and dried as 
 before, and then weighed. The increase in weight in grams divided by 
 the time the current passed and by 0-0003294, gives the current in 
 amperes. 
 
 215. Calibration of an Ammeter by Copper Deposition. A circuit 
 is arranged as shown in Fig. 230. The copper voltameter v is connected 
 to a switch c, by means of which either the 
 voltameter or a resistance r can be placed 
 in circuit. The circuit also contains the 
 battery E, the ammeter A, a resistance box 
 K, and an adjustable carbon resistance P. 
 In order that changes in the resistance of the 
 voltameter may not produce much variation 
 in the strength of the current, it is advisable 
 to use a battery having an E.M.F. consider- 
 ably greater than is absolutely necessary to 
 send the desired current and to include re- 
 sistance in the circuit. 
 
 The cathode having been cleaned and 
 placed in the cell, the current is allowed to 
 pass and the resistance R adjusted till the 
 desired deflection of the ammeter is ob- 
 tained. The switch is then turned over so as 
 to place r in the circuit, and this resistance is adjusted till, whether the 
 voltameter or the resistance are in circuit, the current is the same. The 
 resistance r may conveniently consist of a length of uncovered manganine 
 or platinoid wire, the resistance being adjusted .by drawing the wire 
 through the binding screws of the commutator. 
 
 The adjustment of r being complete, the cathode is removed, washed, 
 dried, and weighed. During this operation the current can be allowed 
 to pass through the resistance r. The cathode is then replaced, and 
 at a noted time, either on a chronometer or a clock, the current is 
 switched over into the voltameter. The reading of the ammeter is 
 noted, and the current kept constant, by adjusting the resistances R and 
 P, for from half-an-hour to an hour, and then at a noted time the 
 current is broken, the cathode removed, washed, and dried as before, 
 and again weighed, 
 
 2k 
 
514 
 
 MEASUREMENT OF CURRENT 
 
 [216 
 
 If w is the increase in weight of the cathode in grams, and the 
 current flowed for t seconds, the current in amperes is given by 
 
 w 
 f ?4MXX)S294*' 
 
 Readings ought to be made at three or four points on the scale of the 
 ammeter approximately equally distributed throughout the range of the 
 instrument. In the case of the lower readings, if the instrument is a 
 long range one, the current must be passed for a longer time than for the 
 upper points. Since there is a small loss of weight when a copper plate 
 is allowed to stand in copper sulphate solution, and this loss will be much 
 more important at the lower current readings, a correction ought to be 
 applied. It has been found that with a solution of the strength recom- 
 mended above the loss amounts to about O'OOS milligrams per square 
 centimetre of the cathode surface per hour. Hence if S is the area of the 
 cathode, and the electrolysis has lasted for x hours, the observed increase 
 of weight must be increased by '000005 Sx grams. 
 
 216. The Silver Voltameter. If suitable precautions are taken, a 
 current can be measured by means of the deposition of silver to within 
 1 or 2 parts in 10,000. The most convenient form of voltameter is 
 
 that used by Ray- 
 leigh. In it the 
 anode consists of 
 a plate or rod of 
 pure silver wrapped 
 in Swedish filter 
 paper, and the 
 cathode consists of 
 a platinum bowl 
 or crucible. The 
 platinum dish c 
 (Fig. 231) rests in 
 a hole in a copper 
 plate A, which is 
 itself supported on 
 a wooden base, the 
 wood having been 
 treated with hot 
 paraffin. A bind- 
 ing screw G serves 
 for the attachment 
 of the negative 
 
 terminal of the circuit. An upright ebonite rod B carries a horizontal 
 rod D, the height of which can be adjusted with a set screw. The 
 anode F is suspended from a binding screw attached to the end of the 
 arm by four platinum wires, and the positive wire of the circuit in also 
 attached to E, 
 
216] THE SILVER VOLTAMETER 515 
 
 The anode F consists of a square plate of pure silver, of which the 
 corners have been turned up and holes have been drilled for the sus- 
 pension wires. This anode is thoroughly cleaned by rubbing with sea- 
 sand and washing, first in ordinary water, and finally in distilled water, 
 and is then wrapped round with a piece of filter paper, so that the black 
 powder of disintegrated silver which forms at the anode may not fall on 
 the deposited silver in the bowl. 
 
 The platinum bowl which is to form the cathode is well cleaned, 
 rinsed in distilled water, dried by heating over a spirit flame, and placed 
 in a desiccator to cool. It is then weighed. 
 
 The silver nitrate solution is obtained by dissolving from 15 to 20 
 grams of pure recrystallised silver nitrate in 100 c.c. of distilled water. 
 The solution ought to be neutral to litmus paper. If acid, digestion for 
 some time with silver oxide will remove the acidity. 
 
 The method of conducting the electrolysis is similar to that described 
 in the preceding section. After the deposition is complete, the solution is 
 carefully poured off into a clean beaker and examined for specks of silver 
 which may have become detached. If any are discovered, they must be 
 replaced in the bowl. The bowl is then rinsed out by filling up with 
 distilled water twice, the water in each case being poured into a clean 
 beaker so that any detached particles of silver may be discovered. 
 Finally the bowl is filled with water and allowed to stand for an hour. 
 This water having been poured off, the bowl is heated over a small spirit 
 flame till quite hot, then placed in a desiccator to cool, and finally 
 weighed. 
 
 The current used ought not to exceed 0'03 ampere per square centi- 
 metre of cathode surface. 
 
 If the solution used in the electrolysis has been freshly prepared, or 
 at any rate there has not been deposited from it more than 3 grams 
 of silver per 100 c.c. of the solution, the electro-chemical equivalent may 
 be taken as O'OOlllS. Hence if w is the weight of silver deposited 
 in t seconds, the current is given in international amperes 1 by 
 
 A = 
 0-001118*' 
 
 If the same solution is used for some time, it is found that the weight 
 of silver deposited by a given quantity of electricity gradually increases, 
 the increase amounting finally to as much as 1 part in 1000. 
 
 Where great accuracy is aimed at, it must be remembered that it will 
 probably be necessary to correct w for the errors in the weights, and, 
 further, that it is the absolute errors that are required, and not, as is often 
 
 1 The international ampere is denned as the current which in one second will 
 deposit O001118 grams of silver. A true ampere, i.e. a tenth of a c.g.s. unit 
 current, is probably slightly larger than the international ampere, the weight of 
 silver which it will deposit in one seiond being O0011183 grams (see Phil. Trans, 
 Royal Society (1902), vol. cxcviii. p. 456). 
 
516 MEASUREMENT OF CURRENT [217 
 
 the case, simply the relative errors. For this reason it is advisable, where 
 possible, to so adjust the time of deposition or the current, or both, that 
 the weight of silver deposited amounts to very nearly an integral number 
 of grams, so that as few separate weights as possible may be used when 
 determining the weight of silver. It is also advisable to counterpoise 
 the empty bowl by means of a separate set of weights, these weights 
 remaining unchanged throughout the weighings. Allowance must also 
 be made for. want of equality in the balance arms (see 26), and for 
 buoyancy of the air (see 28). 
 
 217. The Potentiometer Method of Measuring a Current. When 
 a cell of which the E.M.F. is accurately known is available, then the 
 most convenient and accurate method of measuring a current is to 
 balance the difference of potential produced between the terminals of a 
 known resistance when traversed by the current against the E.M.F. of 
 
 the cell. The most convenient cell to use for 
 this purpose is a cadmium cell, the construc- 
 tion of which is described in 202. 
 
 The arrangement of connections is shown 
 in Fig. 232. Suppose that the graduations 
 of an ammeter A are to be tested. A battery 
 E, preferably of accumulators, is connected 
 so as to send a current through the ammeter 
 and two adjustable resistances R and p. Of 
 these the absolute values of the resistances 
 in the box P must be known, while since R 
 is only used to adjust the current to the 
 desired value, the absolute values are in this 
 case immaterial. The standard cell e is con- 
 nected directly to one terminal of the re- 
 FIG. 232. sistance P and to the other terminal through 
 
 a key K I} a resistance of about 100,000 
 
 ohms R', and a high resistance galvanometer G. A key K 9 allows of the 
 resistance R' being cut out when the adjustment is nearly complete. The 
 object of this resistance is to prevent any but very minute currents being 
 passed through the standard cell. Since the E.M.F. of the cell e, 
 particularly if it is a Clark cell, depends on the temperature, this cell 
 should be placed in a water or oil bath, the temperature of which can 
 be obtained by means of a thermometer. 
 
 The key K t being open, the current in the main circuit is adjusted till 
 the ammeter shows the deflection required. The key K X is then closed, 
 K being open, and the resistance P is adjusted till the galvanometer is 
 deflected in opposite directions by the change of P by the smallest step 
 which the box allows. The higher of these two resistances being un- 
 plugged in P, Q is adjusted till, even when the key K 2 is shut, the 
 galvanometer is undeflected or, if it is impossible to obtain perfect 
 balance, the resistance which would have to be removed from Q to pro- 
 duce balance is obtained by galvanometer deflections. 
 
217] POTENTIOMETER METHOD 517 
 
 If A is the current in the main circuit, the difference of potential 
 in volts between the terminals of the resistance P is given by 
 
 PQ A 
 P + Q ' 
 
 and this is equal to the E.M.F. of the cell e. Hence if e t is the E.M.F. 
 of this cell l at the temperature of the observation, 
 
 ' 
 
 Where the utmost accuracy is required, and the exact value of the 
 current is immaterial, the resistance P can consist of a single standard 
 resistance coil, such as that shown in Fig. 197, the terminals of the 
 standard cell circuit being connected to the binding screws J and H. In 
 this case, in addition to the resistance R used to roughly adjust the 
 current in the main circuit, there must be included one or more adjust- 
 able carbon resistances by means of which the current can be adjusted so 
 that the galvanometer shows no deflection. When this is the case, the 
 current A is given by 
 
 A = e t /P. 
 
 By using one or more standard cells in the auxiliary circuit and 
 different resistances P, it is possible to apply this method for measuring 
 both small and large currents. Thus if a standard O'Ol ohm resistance 
 is used, currents from a little over 100 amperes up to 1000 amperes 
 can be measured by using one to ten cadmium cells in the auxiliary 
 circuit. 
 
 1 The E.M.F. 's of Clark and cadmium cells at different temperatures are 
 given in Table 26. 
 
CHAPTER XXXI 
 
 THE BALLISTIC GALVANOMETER AND MEASUREMENT OF CAPACITY 
 
 218. The Ballistic Galvanometer. A ballistic galvanometer is one 
 which is used for measuring the quantity of electricity which passes, the 
 whole duration of the current being small compared to the time of swing 
 of the galvanometer needle. The conditions which a ballistic galvano- 
 meter has to fulfil are that the time of vibration of the needle shall 
 be fairly long, and that the damping shall not be very great. Subject 
 to these conditions, any form of galvanometer can be used ballistically. 
 
 When a transient current is sent through a galvanometer, the 
 duration of the current being such that the needle has not had time to 
 move appreciably before the current has stopped, then it can be shown 
 that the quantity of electricity Q, which has passed through the galvano- 
 meter is given by 
 
 where a is the amplitude of the first swing of the needle, T is the time 
 of a complete vibration of the needle, G is the field produced at the 
 centre of the coils when they are traversed by unit current, and hence 
 is a constant for any particular galvanometer, A is the logarithmic decre- 
 ment of the oscillations of the needle, and H is the strength of the 
 magnetic field in which the needle swings. If G is the field produced 
 by a current of 1 ampere (10" 1 c.g.s. units), then Q is given in the 
 above formula in coulombs (10" 1 c.g.s. units of quantity). 
 
 First suppose that the damping is zero. Then if M is the magnetic 
 moment of the needle and 7 its moment of inertia, the couple acting 
 on the needle when the galvanometer is traversed by a current C, the 
 needle being in its undeflected position is MGC. The impulse due to 
 the couple acting during the time dt is MGCdt. Hence if the whole of 
 the transient current passes before the needle has appreciably moved 
 from its position of rest, the total momentum communicated to the 
 needle is MGJ'Cdt, or MGQ,. The angular velocity with which the 
 
 needle starts moving is , and the momentum it possesses owing to 
 
 CLT* 
 
 this angular velocity is f . Hence equating the two expressions 
 
 Cvv 
 
 518 
 
218] THE BALLISTIC GALVANOMETER 519 
 
 we have obtained for the momentum, we get 
 
 at 
 or da_MGQ 
 
 dt~~r 
 
 The kinetic energy possessed by the needle at the start of its swing is 
 
 IJM 7 J , and as the needle swings round this kinetic energy is con- 
 
 2i \cit / 
 
 verted into potential energy, due to the work done against the magnetic 
 field in which the needle moves. When the deflection is a, the couple 
 due to the field If is MH sin a. The work done while a increases to 
 a + da. is therefore MH sin a . <ia, and the total work done up to the elonga- 
 
 f a 
 tion a is MH I sin a . da, or MH (1 - cos a). 
 
 Equating the kinetic energy at the start to the work done against the 
 field up to the instant when the needle comes to rest at the elongation, 
 we get 
 
 Substituting for - from (2), and making use of the relation 1 -cos a 
 
 CLT 
 
 = 2 sin* 
 
 
 Now if T is the period of the galvanometer needle, we have (see Watson's 
 Physics, 430) 
 
 Substituting the value of obtained from this expression in (3), we get 
 
 HT sin" 
 
 When the motion of the needle of the galvanometer is damped, the 
 logarithmic decrement being A, then it can be shown that the light- 
 hand side of (4) must be multiplied by (1 + A/2) (see Maxwell's Electricity 
 and Magnetism^ vol. ii. p. 356). 
 
 If the throw is measured by means of a telescope, or lamp, and scale, 
 
520 THE BALLISTIC GALVANOMETER [219 
 
 the deflection being d divisions of the scale, and the distance of the 
 mirror from the scale D divisions of the scale, then if a is small we 
 may to a near approximation take 
 
 sin -- = a/2, and - = tan 2a = 2a, 
 
 a cl 
 S1U = 
 
 Hence 
 
 where K is a constant so long as the conditions under which the 
 galvanometer is set up are not changed. 
 
 If the throw d is not small, then, as is shown in 170, we have 
 
 where 8 is a quantity which depends on the value of d/D, and can be 
 obtained from Table 24. In this case 
 
 For ordinary measurements with the ballistic galvanometer it will 
 in general be sufficient if the period T is between ten and twenty seconds. 
 When, however, the transient currents are produced by the change in 
 induction in electromagnets and such like, where the time constant is 
 very large, it will be always necessary to assure oneself that the whole 
 current is over in a time small compared to the period of the needle, and 
 it may be necessary to load the needle and so increase its period. 
 
 One of the assumptions made in deducing the formula for the 
 ballistic galvanometer must be carefully borne in mind, namely, that 
 the axis of the needle before the passage of the electricity must be 
 parallel to the plane of the galvanometer coils. It is essential when 
 using a ballistic galvanometer that this adjustment be carefully attended 
 to at the start and verified periodically during the course of the measure- 
 ments. The method by which the needle is adjusted parallel to the 
 plane of the coils is given on pp. 426, 510. 
 
 219. Determination of the Constant of a Ballistic Galvano- 
 meter. Arrange a cell E (Fig. 233), preferably an accumulator, to send 
 a current through the galvanometer G, and a resistance R of at least 
 
219] CONSTANT OF A BALLISTIC GALVANOMETER 521 
 
 100,000 ohms, the galvanometer being shunted by a resistance s. The 
 
 value of s is adjusted till a deflection nearly to the end of the scale 
 
 is obtained. This deflection having 
 
 been noted, the current is reversed by 
 
 the commutator c, and the deflection 
 
 again observed. Let d' be the mean of 
 
 the two deflections, the scale distance 
 
 being D. Then, if the deflections are 
 
 small, we have, if 6 is the angle 
 
 through which the needle has been 
 
 deflected, = cT/2Z); or more nearly 
 
 6 = 9 ' (1-8), where the value of 8 is 
 
 obtained from Table 24. If T is the 
 resistance of the galvanometer, and S 
 and R the resistances of the shunt and 
 the resistance in the box E, respectively, 
 E is the E.M.F. of the cell and B its 
 resistance. We have that the current 
 sent by the battery is 
 
 FIG. 233. 
 
 E+ p + R + 
 
 TS 
 
 and the current through the galvanometer is 
 
 E S ES 
 
 JJ + R + 
 
 TS 
 
 H 
 
 But the current through the galvanometer is also given by l ^ tan 6. 
 Hence 2 
 
 Hd' ES 
 
 2GD~(B + R)(T + S) + TS' 
 
 The period T is obtained by observing the time of a number of 
 oscillations with a stop-watch or chronometer in the manner described 
 
 jr 
 
 1 The expression -.^ tan d for the current through the galvanometer only 
 
 holds so long as the field due to the current in the galvanometer coils is uniform 
 throughout the space through which the needle moves. Whether it is justifiable 
 to use this expression can be determined by making a calibration of the scale as 
 described in 174, and seeing whether over the range employed the current is 
 proportional to the tangent of the angle of deflection. 
 
 ' 2 If E is measured in volts and the resistances in ohms, then G corresponds to 
 the field produced at the centre of the coil when traversed by 1 ampere. 
 
522 THE BALLISTIC GALVANOMETER [ 220 
 
 in 43. The logarithmic decrement A is obtained by observing the 
 amplitude of successive elongations in the manner described in 56. 
 
 The resistance of the battery, if an accumulator is used, will be 
 negligibly small compared to R, and hence may be neglected. If, 
 however, a cell having considerable resistance is used, its resistance 
 must be measured. The E.M.F. must be measured by comparison with 
 a standard cell. 
 
 Another way of proceeding is to use a standard cadmium cell in 
 place of E, in which case the E.M.F. will be known ; but since the resist- 
 ance of such cells is considerable, the value of B will have to be deter- 
 mined, which is best done by the alternating current method. Since, 
 however, if a single cell were placed in one arm of the bridge it would be 
 able to send an appreciable current, the resistance of the arms not being 
 very great, and as such a current would affect the E.M.F. of the cell, it is 
 best to place two cells in series with like poles connected, so that their 
 E.M.F.'s. are opposed, and to measure the sum of their resistances. 
 When making the deflection experiment with the ballistic galvano- 
 meter the two cells are used in series, but now with unlike poles 
 connected. 
 
 220. Standardisation of a Ballistic Galvanometer by means of a 
 Solenoid Inductor. Suppose that a long, straight cylinder uniformly 
 wound with insulated wire, the number of turns per centimetre being n, 
 is traversed by a current of A amperes, then, neglecting a small correction 
 for the effect of the ends, the strength of the magnetic field near the 
 centre of the coil is given by (Watson's Physics, 516) 
 
 F=Q'4c7rnA gausses.i 
 
 If F is the field strength at any point on the axis of the coil, then it can 
 be shown that, making allowance for the effects of the ends, 
 
 F--= '2TrnA (cos ^ + cos ^ 2 ), 
 
 where ^i and ^ are the angles subtended at the given point by a radius 
 of the coil at either end. Hence if r is the radius of the coil, and '21 the 
 length, we have for a point at the centre of the coil 
 
 or approximately, 
 
 1 r 
 
 If the length of the coil is 100 cm., so that Z = 50 cm., then the following 
 1 The name gauss has been given to the c.g.s. unit of magnetic force. 
 
220] CONSTANT OF A BALLISTIC GALVANOMETER 523 
 table shows the value of the correcting factor (1 s fr) for different 
 
 values of r : 
 
 Radius of Coil. 
 
 Value of (l - }^\ torl~- 
 
 = 50 cm. 
 
 5 cm. 
 
 0-9950 
 
 
 4 
 
 0-9968 
 
 
 3 
 
 0-9982 
 
 
 2 
 
 (V9992 
 
 
 1 
 
 0-9998 
 
 
 It will be seen that for a coil 100 centimetres long and 4 centimetres in 
 diameter the correction amounts to 0'08 per cent., that is, less than 1 
 part in 10,000, an amount which can in general be neglected. In 
 what follows we shall suppose that the coil is so long compared with its 
 diameter that, at or near the centre, the correction for the ends can be 
 neglected. In any case, where a correction may be necessary it is quite 
 easy to apply it. 
 
 The total induction through a cross-section of the coil taken near the 
 centre is F . TD 2 or 
 
 0'4:Tr 2 )^nA maxwells. 1 
 
 Hence, since there is no field outside such a coil, at any rate if the 
 coil is very long, the induction through a loop of wire which surrounds 
 the coil near the centre will be 
 
 Thus if a secondary coil having JV turns be wound round the centre of 
 the long coil, the total induction through this secondary will be 
 
 If, now, the current A in the primary be either started or stopped, the 
 change in induction in the secondary will have the value given above, or 
 if the current is simply reversed, the change in induction in the secondary 
 will be 
 
 Hence if R is the resistance of the secondary and any conductors, such as 
 leads and a ballistic galvanometer, included in the secondary circuit, the 
 
 1 The name maxwell is given to the unit of magnetic induction or flux in the 
 c.g.s. system of electromagnetic units. 
 
524 THE BALLISTIC GALVANOMETER [ 220 
 
 quantity of electricity which will traverse the secondary when a current 
 of A amperes is reversed in the primary will be given by 
 
 D c.g.s. units 
 
 - 
 coulombs, 
 
 and if d is the throw of the galvanometer, the scale distance being D, and 
 the logarithmic decrement A,, we get 
 
 where the throw is supposed small,, and K is the constant (218) of the 
 ballistic galvanometer. This equation allows of the ballistic constant K 
 of the galvanometer being calculated. 
 
 A convenient form of coil is one 100 centimetres long with a 
 diameter of 4 centimetres, formed by winding two layers of number 
 20 S.W.G. double-silk covered copper wire on a length of glass tube. 
 The object of having two layers is that the two ends of the coil should be 
 at the same end. If for any reason a single layer is used, then the wire 
 should be brought along the axis of the coil, the two terminals being 
 placed close together at one end. Each layer after being wound must 
 be given a coating of shellac varnish. The number of turns must be 
 counted, and the mean diameter over each layer measured. The mean 
 radius of the wire can be obtained by dividing the length of the coil by 
 the number of turns, and the mean radius of each layer of the coil is the 
 external radius less the radius of the wire. The secondary consists of 
 
 two layers contain- 
 ing 100 turns each 
 of number 28 S.W.G. 
 double-silk covered 
 wire, wound uni- 
 formly over the 
 middle part of the 
 FIG. 234. primary. It is con- 
 
 venient to wind the 
 
 secondary in separate sections, the lower layer of 100 turns forming one 
 section, and the upper layer being divided into sections of 10, 20, 30, 
 and 40 turns respectively, the points between the ends of the sections 
 being brought to binding screws attached to the base on which the coil 
 is fixed, as shown diagrammatically in Fig. 234. In this way it is 
 possible to use a secondary containing various numbers of turns accord- 
 ing to the quantity of electricity it is required to send through the 
 galvanometer. 
 
 PRIMARY TERMINALS 
 
221] COMPARISON OF CAPACITIES 525 
 
 221. To Compare the Capacities of two Condensers with the 
 Ballistic Galvanometer. When making experiments on condensers a 
 discharge key, such as that shown in Fig. 235, will be found of much use. 
 The metal arm AB is pivoted to A, and there is a spring which tends to 
 lift the arm up. At the end B there are two steps, such that when the 
 catch on the top of the arm o is in the lower step, then the arm rests 
 against a stop E, while 
 when the catch is in the 
 middle step the arm does 
 not touch either D or E. 
 If the catch is held back 
 so as to miss the lower 
 step, then the spring 
 forces the arm AB against 
 the stop D. Suppose that 
 one pole of a battery x 
 (Fig. 235, a) is connected 
 to one plate of a con- 
 denser c, and the other 
 pole to the stop E of the 
 key, while the same plate 
 of the condenser is joined 
 through the galvanometer 
 
 G to the stop D, the arm AB being connected to the other plate of the 
 condenser. Then on pressing the arm AB down, the plates of the con- 
 denser are connected to the opposite poles of the battery, and hence the 
 condenser is changed. On moving the catch so that AB is in its central 
 position, the charged condenser is disconnected from the battery, while 
 if the catch is released the arm AB strikes against the stop D, and the 
 condenser discharges through the galvanometer. 
 
 Each of the condensers which have to be compared is connected to the 
 key in turn, and the throw of the ballistic galvanometer is obtained when 
 the condenser is discharged after being charged by the same battery. If 
 E is the E.M.F. of the battery, a quantity which it is not necessary to 
 know so long as it remains constant, and d l and d 2 are the throws pro- 
 duced by the discharge of two condensers of capacity C l and C 2 , then the 
 charges of the condensers are EC l and EC 2 respectively, and since the 
 throws of the galvanometer are proportional to the charges, we have, if 
 the throws are small, 
 
 S. .4 
 
 cy d,' 
 
 A nearer approximation is obtained by using the correction factors for 
 sin 0/2, given in Table 24, when 
 
 FIG. 235. 
 
526 THE BALLISTIC GALVANOMETER [ 222 
 
 If the capacity of one condenser is very much greater than that of the 
 other, so that if the sensitiveness of the galvanometer or the E.M.F. of 
 the charging battery is made large enough to give a measurable kick 
 with the smaller condenser, the kick obtained with the larger is off the 
 scale, a modification will be necessary. 
 
 In such a case the simplest, and at the same time the most accurate, 
 method of procedure is to use a number of standard cells to charge the 
 smaller condenser, and a single cell to charge the larger. 
 
 In the absence of standard cells, other cells or accumulators may be 
 used, and the E.M.F.'s used with the two condensers can be compared by 
 one of the methods given in 204, 205. 
 
 Where a ballistic galvanometer of very long period is available, instead 
 of using two batteries of different E.M.F.'s we may shunt the galvano- 
 meter when using the condenser of larger capacity. If the shunt is 
 adjusted so that the kicks with the two galvanometers are exactly the 
 same, then the ratio of the capacities is equal to (G + S)fS, where G is 
 the resistance of the galvanometer, and S the resistance of the shunt. 
 
 When a discharge, such as that of a condenser, is passed through 
 a shunted ballistic galvanometer, owing to the fact that the shunt is 
 generally wound non-inductively, while the coils of the galvanometer must 
 necessarily be wound inductively, the currents in the galvanometer and 
 shunt do not bear a constant ratio to one another throughout the passage 
 of the discharge. While the current is increasing, owing to the induction 
 of the galvanometer, the fraction of the current passing through the shunt 
 is greater than would be the case if the current were steady. While the 
 current is decreasing, on the other hand, a larger proportion of the current 
 passes through the galvanometer than when the current is steady. It 
 can be shown, however, that the total quantity of the discharge which 
 passes through the galvanometer bears the same ratio to the total dis- 
 charge as does the fraction of the current which passes in the case of 
 steady currents. If, however, the needle has time to appreciably move 
 during the discharge, since any movement will decrease the impulse due 
 to the passage of a given quantity of electricity, the effect of the delay in 
 the current in the galvanometer, due to the induction of the coils, is to 
 reduce the deflection produced by the fraction of the discharge which 
 passes through the galvanometer. Wlien the period of the needle is so 
 great compared to the time occupied by the discharge that the whole dis- 
 charge has passed through before the needle has moved, there is no error 
 produced on this account. This effect is particularly likely to cause errors 
 when the condenser employed is one in which there is much residual 
 charge, so that the discharge is prolonged by this soakage effect, that is, 
 with condensers with paraffined paper as the dielectric, or when measur- 
 ing the capacity of a telegraph cable, in which case other methods must 
 be employed ; see 226 
 
 222. Comparison of Electromotive Forces with the Ballistic Gal- 
 vanometer. The E.M.F.'s of two cells or batteries can be compared by 
 using each in turn to charge the same condenser, and then obtaining the 
 
223] MEASUREMENT OF A MAGNETIC FIELD 527 
 
 throws when the discharge is passed through the ballistic galvanometer. 
 The E.M.F.'s are to one another as the throws produced, if the deflections 
 are small. When the throws are not small, then they must be corrected 
 by means of the corrections for sin 0/2, given in Table 24. 
 
 It is advisable to take the throw with the cell of smaller E.M.F. first, 
 as otherwise, if the condenser employed gives any residual charge, the 
 residual charge left after the charge to the higher potential may appreci- 
 ably affect the result obtained with the lower E.M.F. When using a 
 condenser, it is always advisable to keep the armatures in metallic con- 
 nection except when actually making a measurement, so as to allow any 
 residual charge to escape. 
 
 223. Measurement of a Magnetic Field with the Earth Inductor. 
 If a coil is placed with its plane perpendicular to the lines of force of 
 a magnetic field, and is then rotated through an angle of 180 about an 
 axis in its own plane, the quantity of electricity in c.g.s. units which will 
 flow through a circuit connected to the coil is given by 
 
 where a is the inductive area of the coil, that is, the sum of the areas 
 enclosed by all the turns of wire on the coil, R is the total resistance in 
 c.g.s. units of the circuit in which the coil is included, and H is the 
 strength of the magnetic field. If R is measured in ohms (10 9 c.g.s. units), 
 then 
 
 If a ballistic galvanometer is included in the coil circuit, so that R 
 includes the resistance of the coil, the leads and the galvanometer, then 
 if d is the throw when the coil is reversed, we have, if d is small, 1 
 
 TF TT KdR i /, x 
 
 H = 1 + ..... ' 0) 
 
 from which, if we know the constant K ( 218) of the ballistic galvano- 
 meter, the value of the strength of the field can be obtained. 
 
 A form of coil suitable for applying this method to the measurement 
 of the earth's field is shown in Fig. 236. It consists of a square coil AB 
 containing a number of turns of insulated wire, the area included by the 
 turns on each layer having been measured while the coil was being wound. 
 This coil can be rotated about an axis ss', two stops limiting the rotation 
 
 1 If d is not small, then we must take Q = Kd(l - d), the value of 5 being ob' 
 tained from Table 24 corresponding to sin 0/2. 
 
528 
 
 THE BALLISTIC GALVANOMETER 
 
 >< 223 
 
 to exactly 180. The ends of the coil are connected by short lengths of 
 
 flexible wire with the binding screws c and D, by means of which the coil 
 
 is connected to the rest of the circuit. It 
 will be seen that the coil stand which carries 
 the coil can be placed either with the axis 
 about which the rotation takes place 
 vertical or horizontal. 
 
 By means of a prismatic compass or 
 long suspended magnetic needle draw a line 
 on the table magnetic east and west, and 
 arrange the coil so that when against one 
 of the stops the plane of the coil is at right 
 angles to the magnetic meridian. Then 
 determine the throw ^ of the ballistic 
 galvanometer when the coil is rotated 
 through 180. Next place the stand on 
 its side so that the axis about which the 
 coil can turn is horizontal, and starting 
 
 with the coil in the horizontal plane determine the throw d 2 when the coil 
 
 is rotated through 180. 
 
 Now if H and V are the horizontal and the vertical components 
 
 respectively of the earth's field at the spot where the coil is placed, we 
 
 have 
 
 
 H= 
 
 FIG. 236. 
 
 ^ 
 
 - and V= - h - . 
 2a 2a 
 
 Hence if is the magnetic dip, 
 
 If the value of the horizontal component of the earth's field is known, 
 then an earth inductor forms a convenient means of determining the con- 
 stant K of a ballistic galvanometer, for if we know H, then the equation 
 (1) above will allow of A' being calculated. 
 
 When measuring the strength of a magnetic field, the value of the 
 constant K may with advantage be determined by including in the 
 galvanometer circuit the secondary of the solenoid inductor ( -20), 
 and taking throws with the earth inductor and the solenoid inductor 
 alternately. 
 
 Place the secondary of the solenoid inductor in series with the earth 
 inductor and a ballistic galvanometer, care being taken to place the 
 solenoid at such a distance from the inductor and the galvanometer that 
 the solenoid when traversed by a current has no appreciable direct 
 magnetic effect on either. This can easily be tested by breaking the 
 galvanometer circuit and cutting out the secondary by joining together 
 the ends of the leads which come from the galvanometer and earth 
 
STRENGTH OF A MAGNETIC FIELD 
 
 , and seeing whether reversing the current in the primary of the 
 produces any kick of the galvanometer. If only a very slight 
 roduced, this can often be reduced to aero by turning the solenoid 
 to some definite azimuth, which must be found by triaL When 
 ustment is being node, the earth inductor most occupy the 
 it will occupy either at the commencement or end of its 
 :.:. 
 throw of the galvanometer must then be determined, (1) when 
 
 inductor is rotated through 180, and (2) when a current of 
 es is reversed in die primary of the solenoid inductor.. If H is 
 in which the earth inductor is turned, <i is the inductive area of 
 
 inductor, and f, and <L* are the throws, 
 
 then 
 
 .-.-... 
 
 where E is measured in c.g.s. units (10 9 ohms). From these equations K 
 can be eliminated, and hence H may be calculated 
 
 The current J, or the number of turns in the secondary 3T, should be 
 so adjusted that d+ is nearly the same as d^. When this condition is 
 fulfilled, it is not necessary to apply a correction to reduce to the sine of 
 half the angle of throw, since the correction would affect 7j and </.-, in the 
 same ratio. 
 
 The current A must be measured either with a sensitive ammeter, 
 or by the potentiometer method described in 217. For reversing the 
 current a Pohl commutator, such as that shown in Fig. 179, will be found 
 most convenient. Since it is important that the current should not vary 
 n the time when it is measured and the instant when the throw is 
 taken, it is advisable to use accumulators, and to use a higher E.M.F. than 
 is absolutely necessary, reducing the current to the desired amount by 
 means of an added resistance. 
 
 224. Measurement of the Strength of a Magnetic Field with an 
 Exploring Coil. The principle of the earth inductor may be employed 
 to measure other magnetic fields besides that due to the earth. As most 
 such fields which have to be measured are not uniform, the coil, by the 
 rotation of which the induced current is obtained, is made much smaller 
 than that used in the earth inductor. The induction coil used for ex- 
 ploring the magnetic fields near electromagnets is attached to a handle, 
 and is made narrow, for the coil has often to be introduced into narrow 
 :he necessary inductive area being obtained by using a large number 
 of turns of very fine wire. The throw can be caused in two ways. Where 
 there is sufficient room, the coil can be turned through 180. For this 
 
530 
 
 MEASUREMENT OF CAPACITY 
 
 [ 225 
 
 purpose there is fitted a spring and catch, and on releasing the catch and 
 holding the handle firm the coil is rapidly rotated through 180 by the 
 spring. If the space in which the field is to be measured is confined, 
 then the kick is measured when the coil is suddenly removed from the 
 given position to some other position where the field is negligible com- 
 pared to that being measured. 
 
 As an exercise in this method, the strength of the field between the 
 pole-pieces of an electromagnet for different currents in the filed coils and 
 for different distances between and shapes of the pole-pieces may be 
 measured. 
 
 Such an exploring coil may be standardised by using two equal 
 solenoids placed with their ends close together, only just allowing space 
 for the insertion of the exploring coil, and determining the throw when a 
 known current is reversed in the solenoids. If, at the same time, a throw 
 is taken with a secondary wound on one of the solenoids, so that the 
 current in the two cases is the same, it is possible to calculate the induc- 
 tive area of the inductor coil, making use of the expressions given in the 
 preceding sections. 
 
 225. Comparison of Capacities by de Sauty's Method. The two 
 condensers to be compared, GJ and C 2 , are placed in the adjacent arms of 
 a Wheatstone bridge (see Fig. 237). The re- 
 maining arms consist of the two non-inductive 
 resistances p and Q. A high resistance 
 galvanometer G is connected between B and D, 
 while a battery, which ought to have a rather 
 high E.M.F., but since very little current is 
 taken need not have the power of sending a 
 large current, is connected to A and to a key 
 K. This key is so arranged that when the 
 knob is pressed it makes contact at F, and 
 the condensers are charged to the E.M.F. E 
 of the battery. When the key is released, 
 contact is made at D, and the condensers 
 discharge. 
 
 The resistances p and Q are adjusted till 
 there is no galvanometer kick when the key 
 is depressed or released. When this condition 
 is fulfilled we have 
 
 C Q 
 
 a P 
 
 The resistances P and Q should be large, and 
 
 the E.M.F. of the battery be increased till sufficient sensitiveness is 
 obtained. This method is not applicable when the absorption of one of 
 the condensers is very different from that of the other. 
 
 The charges of the condensers, when the key makes contact at F, are 
 and C 2 ti, E being the E.M.F. of the battery. The portion of the 
 
226] 
 
 COMPARISON OF CAPACITIES 
 
 531 
 
 charge of C\ which passes through the galvanometer when the key makes 
 contact at n is 
 
 P 
 
 In the same way, the portion of the charge of C 2 which passes through the 
 galvanometer is 
 
 CJS. 
 
 P+Q+G : 
 
 Hence, since these discharges pass through the galvanometer in opposite 
 directions, and the galvanometer is undenected, we have 
 
 or 
 
 226. Comparison of Capacities by the Method of Mixtures. One 
 
 coating of either condenser is connected to the middle pair of mercury 
 cups of a Pohl commutator K (Fig. 
 238), from which the cross con- E 
 
 nectors between the end cups have 
 been removed. The other coatings 
 of the condensers are connected to 
 one terminal of a high resistance 
 galvanometer G, and also to earth. 
 The testing battery E is connected 
 to the terminals of two adjustable 
 high resistance boxes P and Q. The 
 points A and B are connected to the 
 one set of outside mercury cups of 
 the Pohl commutator, while the 
 other pair are connected through 
 a key k to the galvanometer. A 
 wire is also run between the point B 
 and that terminal of the galvano- 
 meter to which the condensers are 
 directly connected. 
 
 When the key K is placed so as FIG. 238. 
 
 to connect the mercury cups 1 to 3 
 
 and 2 to 4, the condensers are charged. If the current passing through 
 P and Q is i t the charge of C l will be iPC l for the difference of potential 
 between A and B, and therefore also between the terminals of the con- 
 denser c is iP. In the same way the charge of (7 2 is iQC. 2 . When the 
 key K is placed so as to connect 3 to 5 and 4 to 6, the charges in 
 the condensers can mix, and the resulting charge in the two is equal to 
 
 EARTH 
 
532 MEASUREMENT OF CAPACITY [ 226 
 
 i(PC l - QC 2 ). When the key k is closed, this charge passes through the 
 galvanometer. Hence if on closing k there is no galvanometer kick, we 
 conclude that there was no charge left after the condensers have been 
 connected. 
 
 Hence PC-QC.O. 
 
 
 Thus the experiment consists in altering the resistances p and Q till on 
 mixing and depressing the key k there is no galvanometer deflection. 
 
 When applying this method, particular care must be taken that the 
 battery is well insulated. 
 
 A modification of this method, described by Young, 1 is particularly 
 applicable in the case of long telegraphic cables in which there is appreci- 
 able dielectric conductivity, and where also the great length of the core 
 
 causes a considerable time to elapse before 
 the cable becomes charged. The con- 
 nections are represented in Fig. 239, where 
 GJ is the condenser to be tested, between 
 the coatings of which there is leakage, 
 which in the figure is represented by a 
 resistance R b placed in parallel with the 
 condenser. The standard condenser C 2 is 
 supposed to have a very high dielectric 
 resistance, and an adjustable high resistance 
 R 2 is placed across its terminals. The 
 FIG. 239. approximate values of P and q for a 
 
 balance having been found by the method 
 
 of mixtures as in the previous method, the key K 2 is kept depressed and 
 the resistance E 2 is adjusted so that on making the battery circuit by 
 depressing the key K 15 and keeping the current on for, say, one minute, 
 there is no steady galvanometer deflection. The absence of this steady 
 deflection indicates that the bridge is balanced for steady currents, so 
 that RJR^PIQ. 
 
 The resistances p and Q are then finally adjusted, the procedure 
 being as follows : The key K 2 being closed, the battery key iq is closed 
 for one minute. The key K 2 is then opened, and then the battery key 
 KJ. The charges of the condensers are allowed to mix for from fifteen 
 to thirty seconds, and then the key K 9 is closed and the galvanometer 
 observed. When p and Q are so adjusted that there is no galvanometer 
 kick, the following relation holds, as in the previous method 
 
 C 1= Q 
 1 Journal of the Institution of Electrical Engineers (1889), vol. xxviii. p. 485. 
 
CAPACITY IN ABSOLUTE UNITS 
 
 533 
 
 227] 
 
 227. Measurement of a Capacity in Electromagnetic Units in 
 terms of a Resistance and a Time. A condenser c (Fig. 240) is 
 arranged so that one armature is permanently connected with one corner 
 of a Wheatstone bridge, while by means of a commutator the other 
 armature is alternately connected to the points a and Z>, so that the 
 condenser is alternately charged and discharged. The commutator is 
 shown at (a), 1 and consists of an ebonite cylinder having two stepped 
 pieces of brass on the cylindrical surface. The brushes a, 6, and c press 
 against the brass, and as the commutator rotates the brush (c) is 
 
 FIG. 240. 
 
 alternately connected to a and to b. The commutator is driven by a 
 motor, and the speed is measured by means of a tuning-fork and a 
 stroboscopic disc c (see 109). 
 
 If C is the capacity of the condenser, and the commutator charges 
 and discharges the condenser n times per second, while the resistances 
 of the arms P, Q, and R are so adjusted that the galvanometer G is 
 undeflected, we have (see J. J. Thomson's Electricity and Magnetism, 
 260). 
 
 Q \ 
 
 where G is the resistance of the galvanometer arm, and B that of the 
 battery arm. Generally the resistance of the battery is so small com- 
 pared to the other resistances that it can be neglected, so that the 
 expression reduces to 
 
 nC 
 
 1 A description of a good form of commutator will be found in a paper by 
 Fleming and Clinton, Philosophical Magazine (1903), v. p. 493. 
 
534 MEASUREMENT OF CAPACITY [ 227 
 
 If the capacity of the condenser is small, then R will have to be taken 
 very large, and hence Q will also be very large compared to P. In this 
 case the expression reduces to 
 
 nC~ P'' 
 
 and it will be noticed that the condenser and commutator produce the 
 same result as if they were replaced by a resistance 1/nO. 
 
 If the resistances are expressed in ohms, then the capacity C will be 
 expressed in farads (10~ 9 c.g.s. units). To obtain the capacity in micro- 
 farads, the capacity expressed in farads must be divided by 10*>. 
 
 When measuring a fairly large capacity, say, of the order of a 
 microfarad, a Post Office bridge may be used, the ratio arms P and Q 
 being 10 to 1000. When very small capacities are being measured, it 
 is necessary to employ large resistances, and a battery having a large 
 E.M.F., say, 50 or 100 volts. For an air condenser having plates of 
 about 30 cm. in diameter at a distance of about a millimetre, R may be 
 a megohm (1,000,000 ohms) and Q about 300,000 ohms, balance being 
 obtained by using a Post Office or equivalent resistance box for P. When 
 using high resistances and high E.M.F.'s, it is very important to care- 
 fully insulate the various wires, while one point, preferably one terminal 
 of the galvanometer, may be connected to earth by being connected to 
 the water or gas pipes. 
 
 By determining by this method the capacity in electromagnetic units 
 of a condenser of such a shape that the capacity in electrostatic units 
 can be calculated from the dimensions, the ratio of the units, F, may be 
 determined. A suitable condenser can be built up of two silvered sheets 
 of plate glass, the silver being removed for about 2 cm. round the edge. 
 The plates are placed with their silvered sides towards one another, and 
 separated by three small pieces of ebonite or glass, the distance between 
 the plates being obtained by measuring the thickness of these distance 
 pieces with a screw gauge. In order to make contact, a thin strip of 
 the silvering is left on each plate leading up to one of the corners, a 
 piece of the glass at one of the other corners being cut off, and the 
 plates are so arranged that the silver strip on one plate is opposite the 
 trimmed corner of the other. Contact may be made by clamping a wire 
 down on the silver, a few thicknesses of thin tin-foil being interposed 
 to ensure good contact. 
 
 If A is the area of either of the plates, and d is the distance between 
 them, then, neglecting the effect of the edges, the capacity in electro- 
 static units is A/4:7rd. 
 
 Another method of conducting the experiment, which eliminates the 
 effect of the edge of the condenser plates, is to have two cylindrical 
 condensers of the same diameter but different lengths, say, one twice as 
 long as the other. The capacities of these condensers are measured, and 
 let us suppose the numbers obtained are G l and C y the lengths being 
 
227] MEASUREMENT OF A CAPACITY. 535 
 
 /! and 7 2 respectively, and the radius of the outer surface of the inner 
 tube being r lt and that of the inner surface of the outer tube being r 2 . 
 Then since, neglecting the effect of the ends, the capacity per unit 
 length of such a condenser is 
 
 1 
 
 2 log 
 
 we have 
 
 Cylindrical condensers suitable for this experiment may be made up 
 out of "triblet" drawn brass tube, i\ being about 2*5 cm., and r 2 = 2*6, 
 the lengths being 15 cm. and 30 cm. The brass tubes have their ends 
 turned off square, and are fitted at the ends into shallow grooves cut in 
 discs of ebonite. On one of the discs are screwed two binding screws 
 connected by wires with the two tubes. 
 
CHAPTER XXXII 
 
 FIG. 241. 
 
 MEASUREMENT OF SELF AND MUTUAL INDUCTION 
 
 228. Measurement of the Coefficient of Self-induction of a Coil 
 by Rayleigh's Method. The coefficient of self-induction of a coil which 
 does not contain any magnetisable material, and which has a fairly large 
 
 coefficient (10 7 cm.), 
 
 A can be measured by 
 
 a method described by 
 Maxwell and modified 
 by Rayleigh. The 
 coil L (Fig. 241) is 
 * placed in one arm of 
 a Wheatstone's bridge, 
 and the bridge having 
 been balanced for 
 steady currents, the 
 kick a of a ballistic 
 galvanometer G ob- 
 tained when the cur- 
 rent is reversed is noted. Next the resistance of the arm CB is altered 
 by an amount dR, and the steady deflection 6 of the galvanometer 
 produced, owing to the want of balance, is noted. If, then, T is the 
 period of the galvanometer needle, A its logarithmic decrement, and 
 C/! and C 2 the currents traversing the arms CB and BD respectively when 
 the balance is upset by the change of resistance, it can be shown l that 
 the coefficient of self-induction L of the coil is given by 
 
 T8RC,f, A\ sin a/2 . 
 L = -^-\ 1 + 2 ha^T henr ^" 
 
 In the practice of this method it will be found advantageous to make 
 the two ratio coils P and Q equal, and each as nearly equal in resistance 
 to the coil as possible. Then in order to be able to obtain the exact 
 balance required when making the observations of throw, an adjustable 
 resistance r is placed in series with the coil, and the branch CB consists 
 of two resistance boxes placed in parallel. The adjustable resistance r 
 
 1 J. J. Thomson, Electricity and Magnetism, 248. 
 
 A henry is 10 9 cm. , and the value obtained for L is given in henrys if 8R is 
 measured in ohms. 
 
228] RAYLEIGH'S METHOD 537 
 
 may either consist of a short length of uncovered German silver or 
 manganine wire, the length of which is adjusted by slipping it through 
 the binding screw by which the coil is connected to the bridge, or by an 
 adjustable carbon resistance. So long as the carbon resistance is kept 
 fairly tightly screwed up, it will be found quite satisfactory and very 
 much more easily adjusted than the wire. In addition to the com- 
 mutator F, it is advisable to have a key in both the battery and the 
 galvanometer circuits. If a Post Office bridge is used, these keys are 
 provided. The object of the keys is to allow of the battery being 
 connected only for as short a time as possible. If the current is allowed 
 to flow continuously, the balance will be found to be continually altering. 
 This is due to the change in temperature of the resistances, particularly 
 that of the coil, for the coil generally consists of copper wire, for which 
 the temperature coefficient is very much greater than is the case for the 
 alloys used to construct the resistance coils. 
 
 In order that it may be possible to adjust the resistance of the arm 
 CB to the exact value required for balance, it is necessary that one of the 
 resistances E X and R 2 shall be large, that is, the other resistance must 
 only slightly exceed the resistance in the arm BD. Hence the resistance, 
 R 2 say, being infinite, R x is adjusted till it is the nearest unit higher 
 than the resistance of the coil L. The adjustable resistance r is then 
 altered till balance is very nearly secured, care being taken, however, to 
 have Rj a little too great. The final balance is then obtained by altering 
 R O . Immediately balance for steady currents has been obtained, the 
 throw obtained by reversing the current with the commutator F is 
 observed, the battery and galvanometer keys being of course kept closed. 
 The resistance R 2 is then altered by such an amount that the steady 
 deflection is approximately the same as the throw and the deflection is 
 noted. Balance for steady currents is then again secured, and the whole 
 series of operations repeated several times, the interval between the 
 several observations being made as small as possible. 
 
 The time of vibration of the galvanometer must be determined, as 
 also the logarithmic decrement. When determining the logarithmic 
 decrement the galvanometer key must be kept closed, for the decrement 
 on open circuit is not as great as that on closed circuit, owing to the 
 currents induced in the coils by the motion of the needle when the circuit 
 is closed. 
 
 The value of the factor C^C Z can generally be calculated on the 
 assumption that none of the current goes through the galvanometer, 
 since with a sensitive galvanometer the current required to produce the 
 steady deflection is very small. Hence since fi + SR is the resistance of 
 the branch CB when the steady deflection is being obtained, and the 
 resistance of the branch BD is the same as the resistance R of the branch 
 CB when balanced for steady currents, since P = Q,, we have 
 
 C l E+Q 
 
538 
 
 SELF AND MUTUAL INDUCTION 
 
 [229 
 
 In this case, as in all others where a ballistic galvanometer is used, 
 an auxiliary coil should be placed near the galvanometer with a cell and 
 tapping key connected, so that by momentarily passing a current through 
 this coil the observer may be able to check the swing of the galvanometer 
 needle. 
 
 229. Use of a Rotating Commutator when making Measurements 
 of Induction (Ayrton and Perry's Secohmmeter). In the case of coils 
 having small coefficients of self-induction, the method described in the 
 last section is not sufficiently sensitive to give accurate results. In order 
 to increase the sensitiveness by sending a rapid succession of induction 
 currents through the galvanometer, Ayrton and Perry l have designed a 
 commutator which they call a secohmmeter. This commutator performs 
 two functions. In the first place, it makes the battery circuit, the 
 
 FIG. 242. 
 
 galvanometer being short-circuited. Next it removes the short-circuit 
 from the galvanometer, and then breaks the battery circuit, so that the 
 induction current can pass through the galvanometer. Next short- 
 circuits the galvanometer and afterwards makes the battery circuit, the 
 short-circuit on the galvanometer preventing any of the make induced 
 current passing through the galvanometer. This cycle of operations is 
 repeated at a speed which is only limited by the necessity that the 
 current should have attained its final state before the galvanometer is 
 made or broken. The arrangement is shown diagrammatically in Fig. 242. 
 
 1 Journal of the Society of Telegraph Engineers (1887), vol. xvi. p. 292. 
 
SECOHMMETER 
 
 539 
 
 o, 
 
 229] 
 
 The commutator consists of two parts, one o p being connected to the 
 galvanometer G, and the other, o 2 , to the battery E. In reality these 
 two parts would be fixed to the same axle, so that they rotate at the 
 same speed. The shaded parts of the commutators are supposed to be 
 metal, and the unshaded parts non-conducting, so that when the shaded 
 portion is opposite the brushes N X or N 2 the galvanometer is short-cir- 
 cuited, or the battery circuit completed, as the case may be, while when 
 the unshaded portion of the cylinder is opposite the brushes, the galvano- 
 meter is in circuit or the battery circuit broken. One form of commu- 
 tator is shown in Fig. 243, the 
 arrangement of which will be 
 obvious from the drawing. In 
 another form the brushes are 
 lifted up to break the circuits 
 by means of cams which are 
 attached to a rotating shaft. 
 Since it is necessary to know 
 the speed at which the com- 
 mutator is rotating, some form 
 of speed indicator must be 
 attached. In the absence of 
 a direct reading speed indicator 
 a stroboscopic disc may be 
 attached to the commutator 
 and the speed of rotation ad- 
 justed, so that when viewed 
 
 through a pair of slits attached to the prongs of a tuning-fork one 
 of the rows of dots appears stationary, in the manner described 
 in 109. 
 
 The method of using the secohmmeter is as follows : Using the 
 bridge in the ordinary manner, balance for steady currents. Then rotate 
 the commutator at a speed of n-^ revolution per second, and adjust the 
 resistance R till the galvanometer is undeflected, and let the change in 
 R necessary be SR^ Next rotate the commutator at some other speed, 
 n 2 , and again determine the change in R, say, SR> 2 , from the value for 
 balance with steady currents necessary to restore balance. Then if, 
 within the limits of accuracy of the settings, we have 
 
 Ni 
 
 GALV/WOMETEf? 
 
 FIG. 243. 
 
 it shows that the frequency of the commutator has not been so great that 
 the steady state has not had time to become established before the 
 galvanometer is either put into circuit or cut out. If, however, the 
 ratios are different, then a slower speed of the commutator must be tried, 
 till a constant value for the ratio is obtained. 
 
540 
 
 SELF AND MUTUAL INDUCTION 
 
 [230 
 
 <A value of SR/n which does not vary with the speed having been 
 obtained, the coefficient of self-induction of the coil is given by l 
 
 K8R, 
 L = - henrys, 
 
 n 
 
 where K is a constant depending on the setting of the two portions of 
 the commutator with reference to one another. This constant is the 
 ratio of the angle 6 through which the commutator turns between the 
 instant when the battery circuit is closed and the instant when the 
 galvanometer is short-circuited to 360. The angle 9 (Fig. 243) may be 
 determined experimentally by throwing the bridge out of adjustment, 
 and having attached a divided circle to the commutator, rotating this 
 latter by hand and noting the angle through which it must be rotated 
 between the production of a deflection, owing to the closing of the 
 battery circuit, and the return of the needle to zero owing to the short- 
 circuiting of the galvanometer. The most satisfactory method of deter- 
 mining K is, however, to use the commutator to measure a known 
 self-induction, and to deduce the value of K from the equation 
 
 K- UL 
 
 ~ BR 
 
 230. Measurement of Self-induction by Comparison with a Capa- 
 
 city (Rimington's 
 Method 2 ). A Wheat- 
 stone's bridge is arranged 
 with the coil in one of 
 the branches, and balance 
 for steady currents is ob- 
 tained. A condenser K is 
 then arranged as a shunt 
 on part of the resistance 
 of the arm AC (Fig. 244), 
 which is conjugate to the 
 arm containing the coil. 
 The ratio of the resistances 
 p and p', into which the 
 point 7i', where one arma- 
 ture of the condenser is 
 connected, divides the 
 resistance forming the 
 arm AC, is varied, till on making or breaking the battery circuit there is 
 
 1 For a proof of this formula the reader must refer to the paper by Professors 
 Ayrton and Perry, Zoc. cit. 
 
 2 Philosophical Magazine (1887), xxiv. p. 54. 
 
230] RIMINGTON'S METHOD 541 
 
 no galvanometer kick. When this is the case, the coefficient of self-induc- 
 tion of the coil is given by 
 
 L = Kp 2 S/(p +2'') henrys, 
 
 where K is the capacity of the condenser in farads, and S is the resistance 
 of the arm BD in which the coil is placed. 
 
 If x is the current flowing through the arm BD, i.e. through the coil, 
 and y the current flowing through the arm AC when the currents are 
 steady, then the charge in the condenser is Kpy, and on breaking the 
 battery circuit the quantity of electricity which traverses the coil owing 
 to self-induction is LxlX, where X is the resistance of the arm BD to- 
 gether with the resistance of the remainder of the network between the 
 points B and D when the battery circuit is broken. 1 Thus 
 
 where P is written for the resistance of the arm AC, and G is the resist- 
 ance of the galvanometer. Hence the quantity of electricity which 
 traverses the galvanometer due to the self-induction of the coil is 
 
 Lx P+Q LxP 
 
 77T x ~-F^~r , Or ^7 
 
 + 
 
 since PS= RQ as the bridge is balanced for steady currents. 
 
 In the same way the quantity of electricity which traverses the galvano- 
 meter (in the reverse direction) due to the discharge of the condenser is 
 
 T> 
 
 
 Since the galvanometer is unaffected, these two quantities must be equal. 
 Hence 
 
 or T _ Kp 2 R y 
 
 P ' x 
 
 But y _ S 
 
 x 
 
 so that 
 
 1 Since the bridge is balanced for steady currents, the battery and galvano- 
 meter arms are conjugate. Hence the current in the galvanometer produced by 
 the E.M.F. of induction is independent of the resistance of the battery. Thus 
 the expression obtained when the battery arm is broken will also hold when 
 the battery is connected to the network (see 176). 
 
542 
 
 SELF AND MUTUAL INDUCTION 
 
 230 
 
 K 
 
 The arrangement of resistance boxes used in practice is shown in 
 Fig. 245. The sum of the two resistances p and p being made equal to 
 1000 ohms, the values of the resistances Q and R are adjusted so that 
 there is balance for steady currents, an adjustable resistance r in series 
 with the coil being used if necessary for the final adjustment. The 
 resistance p is then adjusted, care being taken to keep the sum p + ;/ 
 
 constant, till there is no induc- 
 tive kick of the galvanometer 
 on making or breaking the 
 battery circuit. 
 
 The sensitiveness of the 
 arrangement can be much in- 
 creased by the use of a com- 
 mutator, such that the battery 
 circuit is rapidly reversed, while 
 between the battery reversals 
 the galvanometer connections 
 are also reversed, so that any 
 residual current due to imperfect 
 balance passes through the 
 galvanometer in the same direc- 
 tion. In this case, as in all cases 
 FIG. 245. where a commutator is used to 
 
 increase the sensitiveness of a 
 
 zero method, it is not necessary to know either the number of rotations 
 per second or the lead of the brushes. It is, however, always advisable 
 to observe with the commutator running at different speeds, so as to 
 make certain that the speed used is not too great. If the commutator 
 is running too fast, it generally happens that the discharge due to the 
 condenser is over before the reversal of the galvanometer, while the 
 inductive discharge due to the coil may not have finished. This is 
 particularly the case when using coils with iron cores. In the case of 
 circuits having large time constants it is impossible to obtain balance, 
 since the inductive discharge lasts so much longer than the condenser 
 discharge, so that although the correct balance may be attained such 
 that the quantities of electricity which pass through the galvanometer 
 due to these two effects may be equal, yet a small kick is first obtained in 
 one direction due to the condenser, and then there is a kick in the other 
 direction due to the self-induction of the coil. In such a case the con- 
 denser is removed, and the bridge having been balanced for steady currents, 
 the kick d^ obtained when the current is reversed is observed. The coil 
 is then replaced by two resistance boxes in series, with a condenser of 
 capacity K placed as a shunt on one of them. The resistances unplugged 
 from these boxes are adjusted till the bridge is again balanced for steady 
 currents, showing that the sum of these resistances, p + ;/, is equal to 
 the resistance of the same arm when the coil was in place. The 
 resistance p in parallel with the condenser is adjusted till the throw of 
 
231] 
 
 ANDERSON'S METHOD 
 
 543 
 
 the galvanometer when the battery is reversed is about the same as with 
 the coil, and the value of this throw rf., is observed. Then 
 
 L ^ 
 Kp- d^ 
 
 Since it is assumed that the E.M.F. and resistance of the battery 
 remain constant, it is advisable to use accumulators, and to repeat 
 the observations several times, alternating the kicks with the coil and the 
 condenser. 
 
 231. Measurement of the Self-Induction of a Coil by Comparison 
 with a Capacity (Anderson's Method). 1 A Wheatstone's bridge is built 
 up, as shown in Fig. 246, p, Q, 
 and R being non-inductive resist- 
 ances, and L the coil of which the 
 self-induction is to be measured. 
 The condenser K X is connected to 
 the point B and to one terminal 
 of a galvanometer G and tele- 
 phone T, and the three wires are 
 joined to the point A through an 
 adjustable non-inductive resist- 
 ance r. The other terminals of 
 the galvanometer and telephone 
 are connected to a two-way 
 switch k, by means of which 
 either can be put in communica- 
 tion with c. The battery E is 
 connected to a key k. 2 and a 
 buzzer z. This latter consists 
 of a small electromagnet, the 
 armature forming a break, such 
 as is usually fitted to an induc- 
 tion coil. 
 
 The bridge is first balanced for steady currents, using the galvano- 
 meter G, the key k. 2 being closed. 
 
 The key k 9 is then opened so that the buzzer works, producing a 
 rapidly interrupted current in the network of conductors, and the 
 telephone is used in place of the galvanometer. The resistance r is then 
 adjusted till the sound in the telephone is a minimum. When this 
 adjustment is complete we have 
 
 FIG. 246. 
 
 where if K is measured in farads and the resistances in ohms, the self- 
 induction is obtained in henrys. 
 
 1 Philosophical Magazine (1891), xxxi. p. 334. See also Fleming, Philosophical 
 Magazine (1904), vii. p. 586. 
 
544 SELF AND MUTUAL INDUCTION [ 231 
 
 We may suppose that the potential of the point N is always equal to 
 that of the point c, and let x, y, and z be the total quantities of electricity 
 which have traversed the arms BA, BC, and BK respectively, from the 
 starting of the currents up to the time t. Since the current in the arm 
 
 BA at the time t is and so on, we have the difference of potential 
 
 (it 
 
 between B and A measured along the arm p is P , and the difference 
 
 dt 
 
 of potential between these same points measured along BNA is + r ] 
 
 K df 
 
 for the difference of potential between the poles of the condenser K is 
 equal to the charge z divided by the capacity K. Hence equating the 
 two expressions for the difference of potential between B and A we get 
 
 pdx^z dz 
 dt K dt 
 
 Similarly, since N and c are at the same potential 
 
 The difference of potential between c and D is equal to the sum of the 
 
 differences of potential due to the steady current, viz. $-^, and that 
 
 dt 
 
 due to the self-induction, viz. L ( ~ 
 
 dt \dt 
 
 The difference of potential between c and A reckoned along CNAD is 
 dz 
 
 Hence 
 
 dz Q(^ X , dz\ _ ^ly jd?y 
 
 dt \tft dt)~ dt dp (j 
 
 Substituting in (3) the values for -|, -^, and , deduced from (1) 
 
 (.IT Civ (it 
 
 and (2), we get 
 
 Now, if there is no steady deflection the coefficient of z in (4) must be 
 zero, which gives the usual condition 
 
 ^ 
 
 R P' 
 
232] COMPARISON WITH VARIABLE STANDARD 545 
 
 If, further, there is to be no variable current, then the coefficient of 
 
 dt 
 must be zero, that is, 
 
 R S 
 
 which gives, since -= -, 
 r Qi 
 
 When performing the experiment, the resistance p, Q, and R may con- 
 sist of the arms of an ordinary Post Office Wheatstone's bridge, in which 
 a resistance box is placed in series with the telephone or galvanometer 
 in the galvanometer gap, the condenser being connected as shown in 
 Fig. 246. 
 
 232. Measurement of Self-Induction by means of a Variable 
 Standard of Self-induction. Where a variable standard of self-indue^ 
 tion is available, of which the maximum 
 value is at least as great as that of the 
 coil to be measured, the following 
 method, which is a modification of one 
 devised by Maxwell, is convenient. 
 The coil to be measured, L (Fig. 247), 
 and the adjustable standard N, are 
 placed in adjacents arms of a Wheat- 
 stone's bridge, the other arms being 
 formed by non-inductive resistances. 
 It will generally be found convenient 
 to place an adjustable resistance, such 
 as a bare wire or a carbon resistance 
 r, in series with either L or N to allow 
 of the final adjustment for steady 
 currents being made. 
 
 The bridge having been adjusted 
 for steady currents, the standard of 
 self-induction is adjusted till there is 
 no galvanometer kick on reversal of 
 
 the battery current. When this is the case we have, if N is the value of 
 the self-induction of the standard, 
 
 L = R = P 
 
 N~ S~ Q' 
 
 In the absence of a variable standard of self-induction, or when 
 comparing two fixed self-inductions, the above method is inapplicable. 
 Maxwell has shown that when two coils having self-induction are arranged 
 as are L and N in Fig. 247, then the bridge will be balanced both for 
 
 2 M 
 
546 
 
 SELF AND MUTUAL INDUCTION 
 
 [233 
 
 steady currents, and also during the time when the current is varying, if 
 the following relation between the resistances of the arms holds, 
 
 L_R_P 
 N~S~"Q' 
 
 The application of this method is, however, so very tedious, since it in- 
 volves a double adjustment, as to render it unpractical. A modification 
 of the method has been devised by Niven, 1 in which the necessity for the 
 double adjustment is avoided. The method is not, however, a very con- 
 venient one. A fairly rapid way of making the comparison, but one 
 involving the use of a ballistic galvanometer, is as follows : Arrange 
 the two coils having self-induction as in Fig. 247, and having balanced 
 for steady currents note the swing d l when the current is reversed. 
 Next remove one of the coils, say L, and substitute a non-inductive re- 
 sistance, and having adjusted till the bridge is again balanced for steady 
 currents, measure the throw d 2 when the current is again reversed. 
 
 When the two coils are used we have, if k is a constant depending on 
 the galvanometer, Q>L - PN=kdi. 
 
 When the coil N only is included, we have, in the same way, 
 
 Hence 
 
 = 
 
 N Q 
 
 d 
 
 Since it is assumed that the current remains constant throughout, it 
 
 is important to use accumulators, and 
 to alternate the two measurements 
 several times. 
 
 233. To Compare the Mutual- 
 Induction between two Coils with 
 the Self-Induction of one of them. 
 Suppose the coil L is that of which 
 the coefficient of self-induction L has to 
 be compared with the coefficient of 
 mutual-induction M between this coil 
 and another coil N. The coil L is in- 
 cluded in one arm of the Wheatstone's 
 bridge, and the coil N is placed in 
 the battery circuit, as shown in Fig. 
 248. The bridge having been adjusted 
 for steady currents, during which ad- 
 justment the key K may be open, this 
 key is closed, and the resistance r is 
 adjusted till there is no kick on reversing 
 the battery current. This adjustment 
 is only possible if the coefficient of self- 
 induction of L is greater than the coefficient of mutual-induction be- 
 
 1 Philosophical Magazine (1887), vol. xxiv. p. 227. 
 
233] MUTUAL-INDUCTION AND SELF-INDUCTION 547 
 
 tween the two coils. If this condition is not fulfilled, then the modi- 
 fication given below must be employed. When the adjustment for both 
 steady and variable currents is complete, we have 
 
 M Pr 
 
 L r(PVJB) 
 
 the minus sign indicating that the coil N must be so connected up as to 
 give an opposite induced E.M.F. in the branch BC to that produced by L. 
 The fact that the galvanometer gives no kick when the battery is 
 reversed shows that the integral E.M.F. induced in the arm BC, due to 
 the reversal of the current in the coil N, is equal to the integral E.M.F. 
 induced in this arm, due to the self-induction of the coil L. Hence if x 
 is the steady current in M, and y that in N, we have 
 
 
 But x is the total current in the battery branch, and the ratio of the 
 current in the arm BC to the total current is given by 
 
 (P+Q)r 
 ?/_ P+Q + 'r (P+Q)r 
 
 * (P+Q)r 
 
 P+Q+T+ 
 
 
 
 But Q = - P, hence making this substitution 
 
 y _ 
 
 S " , 3T , SWT" SP , \ r(P + ^)4-P(^4-^) 
 
 Substituting this value for ^ in (1), we get 
 
 M JL?_ 
 
 L~ r(P + R) 
 
 When M is greater than L, the coil N can be included in the branch r 
 in place of being in the battery arm. If r is the value of the total 
 resistance in the r branch when there is balance for variable currents, 
 we have 
 
 L 
 
 Both the above methods allow of the use of a rotating commutator, 
 
548 
 
 SELF AND MUTUAL INDUCTION 
 
 [ 234 
 
 N 
 
 such as the secohmmeter, by means of which the sensitiveness may be 
 
 very materially increased. 
 
 234. Measurement of the Coefficient of Mutual Induction between 
 two Coils by means of a Variable Standard 
 of Self-induction. The two coils L L and L.,, 
 of which the coefficient of mutual-induction M 
 is to be determined, are placed in one arm of 
 a Wheatstone bridge, as shown in Fig. 249, 
 and the variable standard of self-induction 
 N is placed in the adjacent arm. The bridge 
 having been balanced for steady currents, the 
 standard N is adjusted till there is no kick on 
 reversing the battery current. Suppose that 
 the coils L! and L 2 are so connected that the 
 mutual-induction assists the self-induction, 
 and that the value of the standard when the 
 adjustment for variable currents is complete 
 is NI. Then 
 
 FIG. 249. 
 
 If, now, the connections of one of the coils 
 
 or L 2 is reversed, so that the effect of the mutual-induction is opposed to 
 the effects of the self-inductions of the separate coils, and N, 2 is the new 
 value of the variable self-induction when the bridge is balanced for 
 variable currents, we have 
 
 Hence 
 
 235. The Comparison of two Mutual-Inductances. The two pairs 
 of coils are arranged, as shown in Fig. 
 
 250, one coil of each pair being in the 
 battery circuit. The other two coils are 
 connected to a galvanometer through the 
 resistances P and Q. The values of p and 
 Q are then adjusted till there is no galvano- 
 meter kick on reversing the current in the 
 battery circuit. When this adjustment is 
 complete, we have, if Sj and s 2 are the 
 resistances of the coils connected to the 
 galvanometer, 
 
 JJ^ = S l +~Q' FlG ' 230 ' 
 
 236. The Measurement of Self Induction with Alternating 
 Currents (Wien-Dolezalek Method). A very accurate method of 
 
236] 
 
 WIEN-DOLEZALEK'S METHOD 
 
 549 
 
 _i_ -Tannr^ 
 
 measuring self-induction, particularly small inductances, has been 
 described by Wien, 1 in which an alternating current is employed. Two 
 coils, LJ and L 2 (Fig. 251), having self-induction are placed in adjacent 
 
 arms of a Wheatstone's 
 bridge. One of these coils 
 may conveniently have its 
 self - induction adjustable, 
 though the value of the 
 self-induction need not be 
 known. A non-inductive 
 resistance / is placed as a 
 shunt on the coil L 15 and 
 another non-inductive re- 
 sistance s is placed in series 
 with L. 2 . 
 
 The two remaining re- 
 sistances, P and Q, may be 
 formed by a bridge wire. 
 The source of current is 
 FIG. 251. a sma ll induction coil M, 
 
 the make - and - break of 
 
 which consists of a microphone attached to a telephone diaphragm. 
 This method of obtaining a senusoidal current is due to Dolezalek. 
 The arrangement used, called a hummer, is shown in Fig. 252. It 
 consists of an iron disc AB, to the centre of which is attached a 
 small carbon disc c. To this disc is attached a small silk bag D filled 
 with carbon grains, forming the microphone contact. One terminal of 
 a battery E is attached to AB, and the 
 other to the primary P of a small induc- 
 tion coil, the other terminal of the 
 primary being connected to D by a fine 
 flexible wire. A hollow cylindrical steel 
 magnet G is placed near the disc AB, and 
 on the outside of this magnet is wound 
 a coil F, which is in series with the 
 secondary s of the induction coil. The 
 terminals a and b are connected to the 
 bridge. When the diaphragm AB has 
 
 been set in vibration, the microphone contacts at D vary in resist- 
 ance, and hence the primary current varies in intensity. Hence 
 there is an induced secondary current produced, and this current 
 passing through the coil F causes the magnetism of the magnet G to vary, 
 and in this way keeps up the vibrations of the disc. In place of the 
 galvanometer a special telephone is employed, which is so adjusted that 
 its diaphragm lias the same natural period as that of the diaphragm of 
 the hummer. The result is that the telephone practically only responds 
 1 Wied. AnnaZcn (1891), xliv. p. 701. 
 
 FIG. 252. 
 
550 SELF AND MUTUAL INDUCTION [ 236 
 
 to alternating currents of the one period, and the results are the same as 
 if a simple sine-form alternating current was used in the bridge. 
 
 The resistance r (Fig. 251) being given some value about equal to 
 that of the coil L P the resistances s, P and Q are adjusted till there is no 
 sound in the telephone T. 
 
 Since we are dealing with simple sine-form alternating currents, it 
 can be shown (see Wien, lot', cit.) that we may apply the ordinary 
 condition for steady current balance of the bridge if we replace the 
 resistances 1 of the arms by the operator r + ipL, where r is the resistance, 
 and L the self-induction of the arm, p is 2vr times the frequency of the 
 alternating current, and i is ,J 1. 
 
 In the case of two inductors in series, or parallel, the value of the 
 operator r + ipL for the combination is obtained from the values for the 
 individual conductors by the same rules as is the resistance of two 
 conductors in series or in parallel, as the case may be. 
 
 Hence, since the resistance r is non-inductive (i.e. L = Q), the 
 operator in the case of the arm AC (Fig. 251) becomes 
 
 r + ipL l + r' 
 The condition for balance of the bridge is 
 
 where S is the resistance of the arm CB. By multiplying across, the 
 above equation reduces to 
 
 (r + ipL l )Q = P{S(r + r') -p^L^ + i[SpL^ + (r + r')pL 2 ]}. 
 Equating the real parts of this equation, we get 
 
 
 1 If the arm consists of an inductive resistance and a condenser of capacity 
 
 C in series, then the operator has the form r + ipL-i , and this operator can 
 
 pL> 
 
 be used to replace the resistance in obtaining the conditions of balance of 
 a bridge in which condensers are included when alternating currents are 
 employed. 
 
236] WIEN-DOLEZALEK'S METHOD 551 
 
 In the same way, equating the imaginary parts, we get 
 
 ^ + Pp(r + r')L 2 , 
 
 
 Equations (1) and (2) enable us to calculate both L l and Z/ 2 . 
 
 To obtain the quantity p (2?r times the frequency), the frequency of 
 the alternating current must be measured. This quantity may be 
 obtained by comparing the pitch of the note given, by the telephone 
 with that of a tuning-fork of known pitch. The comparison can be 
 made either by means of a monochord ( 110), or by using a resonance 
 column ( 112). 
 
 Several measurements ought to be made, using different values for 
 the shunt resistance /. 
 
CHAPTER XXXIII 
 
 PERMEABILITY 
 
 237. Measurement of the Permeability of Iron by the Magnetometer 
 Method. If a rod of magnetic material, such as iron, is placed with its 
 length parallel to the lines of force of a magnetic field of strength f\ the 
 rod becomes magnetised by induction. If the strength of the poles 
 induced is m, and the distance between them is /, the magnetic moment 
 of the rod is ml. If the volume of the rod is v, the intensity of 
 magnetisation / of the iron is given by 
 
 T _ m ^ 
 
 V 
 
 In general, the intensity of magnetisation is not quite uniform throughout 
 the whole length of the rod, there being a falling off near the ends. If, 
 however, the rod is very long compared to its cross-section s, the effects 
 of the ends is very small, and then the distance between the poles may 
 be taken as equal to the length of the rod without appreciable error, 
 and we have 
 
 The number, of lines of force which proceed from a pole of strength m 
 being 4?rm, the number of lines which leave the north pole of the rod and 
 enter the south pole are 47rw, and hence the number which cross unit 
 area of the cross-section of the iron is 47rm/s. Since in addition to the 
 lines due to the magnetism induced in the rod we have the lines due to 
 the magnetising field, the total number of lines which cross unit area is 
 4:7riu/s + F. The total number of lines which cross unit area is called the 
 induction J3, hence since I=m/8, 
 
 The ratio of the induction B to the inducing field F is called the per- 
 meability /x, so that /J, = B/F. The ratio of the intensity of magneti- 
 sation / to the inducing field is called the magnetic susceptibility 7r, 
 so that k = I/F, and hence 
 
 In the above relations the magnetising field F is the strength of the 
 
 552 
 
MAGNETOMETER METHOD 
 
 553 
 
 237] 
 
 field within the iron itself, and this is not in general equal to the strength 
 of the field before the introduction of the iron, owing to the fact that the 
 poles induced at the ends of the rod produce a field which is in the 
 opposite direction to the original field. If, however, the rod is very long 
 and thin the poles are weak, and are at such a great distance from the 
 greater part of the iron that the demagnetising field due to the poles may 
 be neglected, so that in such a case we may take the field which exists 
 before the introduction of the iron as the magnetising field F. The 
 method by which, in the case of a short cylinder, the effect of the 
 demagnetising field due to the ends may be allowed for is described 
 later ( 238). 
 
 In the magnetometer method of measuring the permeability of iron 
 the material is taken in the form of a long thin wire DG (Fig. 253), and 
 is magnetised by being placed within a long solenoid B?, through which 
 
 G 
 
 no 
 
 
 FIG. 253. 
 
 a current is passed. The induced magnetism is then measured by noting 
 the deflection which the iron causes to a magnetometer needle M placed in 
 the neighbourhood, the magnetic field produced at the needle of the 
 magnetometer by the direct action of the solenoid being compensated by 
 means of an auxiliary coil F, through which the current which passes 
 through the solenoid is also passed. 
 
 If the solenoid is placed with its axis E. and W. (magnetic), the earth's 
 field will have no component parallel to the length of the wire, and when 
 long thin wires are used, the effect of the transverse magnetisation due to 
 the earth's field may be neglected. 
 
 i Let the magnetising solenoid BC be placed in the magnetic east and 
 west position, with its axis passing through the needle of the magneto- 
 meter, the mid-point of the wire being at a distance D from the needle, 
 while the length of the wire is 2Z>, and its cross-section is s. If the 
 intensity of magnetisation of the iron is /, the magnetic moment of the 
 iron is 2sLL If M' is the moment of the magnetometer needle, the 
 
554 PERMEABILITY [ 237 
 
 couple, exerted by the iron on the needle, if we suppose the poles to be 
 at the ends of the wire, when the needle has been deflected through an 
 angle 6 from the magnetic meridian is * 
 
 4 M'sLID 
 
 If H is the value of the horizontal component of the field in which the 
 magnetometer needle hangs, the couple acting on the needle tending to 
 bring it back into the meridian is M'H sin 0. When the needle is in 
 equilibrium, these two couples must be equal, and hence 
 
 
 If is small compared to unity, then this expression may be written 
 
 (2) 
 
 where the second term within the brackets is often so small as to be 
 negligible. Hence if H and D are known, / can be calculated from the 
 magnetometer deflection. 
 
 If the magnetising solenoid contains n turns per centimetre, and is 
 traversed by a current of C c.g.s. units, the field at the centre of a long 
 solenoid is equal to irnC. Hence if the iron wire is so long that we 
 may neglect the demagnetising effects of the field due to the poles, the 
 magnetising field F acting on the iron is given by 
 
 F= 4:7rnC gausses. 
 
 If the current is A amperes, then, since an ampere is one-tenth of a 
 c.g.s. unit, 
 
 ,., SimA 
 
 F- -- gausses. 
 o 
 
 The current is obtained from the battery E, which may with advan- 
 tage consist of accumulators, and passes through a variable resistance R, 
 an ammeter A, and a commutator x. 
 
 A suitable magnetising coil can be made by winding No. 22 
 (standard wire-gauge) double-silk covered copper wire on a tube 50 cm. 
 long and 1 cm. outside diameter. If the coil is only to be used for 
 
 1 Watson's Text-Book of Physics, 429. 
 
237] MAGNETOMETER METHOD 555 
 
 magnetometer experiments, then it may be wound on a brass tube. When, 
 however, it is to be used in addition for ballistic experiments, it is better 
 wound on a non-conducting tube, such as glass or vulcanised fibre. If 
 eight layers of wire are wound on the tube, the end of each alternate 
 layer being taken to binding screws attached to the base on which the 
 coil is mounted, then by using different numbers of layers, varying 
 magnetising forces can be obtained without using a very great range of 
 currents. The wire must be wound on very regularly, and the number 
 of turns in each layer noted. It is also a good thing to note the mean 
 diameter of each layer, and to give each layer a coating of shellac 
 varnish. Each layer of such a coil will have a resistance of about 2 
 ohms, and with a current of 2 amperes the whole coil will give a 
 magnetising field of. about 360 c.g.s. units. 
 
 In order to vary the magnetising field when carrying the iron round 
 a cycle, some means of varying the current by varying the resistance in 
 the circuit will have to be adopted. This adjustable resistance must be 
 such that it can be either increased or decreased by suitable steps in a 
 perfectly regular manner. Thus when the current is being increased, the 
 resistance must never, even temporarily, be decreased, and vice versa, or 
 owing to hysteresis the curve obtained will be irregular. The steps by 
 which the resistance is altered should not be equal ones, but should get 
 greater and greater as the resistance in the circuit increases, so that the 
 changes in the current, and hence also in the magnetising field, are 
 approximately equal. If r is the resistance of the whole circuit when the 
 largest current is passing, and it is required to obtain n steps, then the 
 resistances which have to be added to r are 
 
 nr 
 
 nr 
 
 r- &C. 
 
 Thus if the value of r is 9 ohms and we require ten steps, the resist- 
 ances to be added to r are 
 
 1; 2-25; 3'86; 6; 8; 13'5; 21; 36; 81 ohms. 
 
 In general, it will be found convenient to prepare a special set of resist- 
 ances to suit any given coil, for 
 they need not be adjusted at all 
 carefully, and in fact the required 
 lengths of wire can be measured 
 off with sufficient accuracy, the 
 resistance of a metre of the wire 
 having been measured. 
 
 In place of the usual plug 
 contact, which is expensive to 
 make and difficult to keep in 
 proper condition, the arrange- F 1G . 254. 
 
 ment 'shown in Fig. 254, which 
 has been designed by Professor Callendar, may be used. A strip of 
 
556 PERMEABILITY [ 237 
 
 copper or brass is taken, and a number of equally spaced holes about 
 0'6 cm. in diameter are bored. This strip is fastened to a piece of wood 
 by two screws between each pair of holes, this piece of wood forming the 
 top of the box in which the coils will be contained. The strip of copper 
 is then removed and sawn across through each of the large holes, and the 
 separate pieces again attached to the wood. An ordinary binding screw 
 terminal A, the screw -down portion of which is of greater diameter than the 
 holes, is then fastened, as shown, at the centre of each of the holes. The ends 
 of the wires are brought up through the wood and soldered to the pieces 
 of copper, the wire itself being coiled in the form of a double spiral, either 
 loose or round a wooden peg. When the binding screws are screwed 
 down they make contact with the copper strips on either side, and the 
 coil which is attached to these strips is short-circuited. When the 
 binding screw is raised the current must, however, pass through the coil. 
 This form of contact is easy to make, and can be cleaned without 
 difficulty. 
 
 The apparatus is set up in the following manner : The magneto- 
 meter having been set up and levelled, the magnetising coil and com- 
 pensating coils are placed with their axes in the magnetic east and west 
 positions, and at such a height that their axes pass through the needle 
 of the magnetometer ; and the ammeter, adjustable resistance, and a 
 battery are connected up, as shown in Fig. 253. A current is then passed 
 through the circuit, no iron being in the coils, and the position of the 
 compensating coil is adjusted till the magnetometer is undeflected when 
 the current is reversed. The iron, in the form of a wire about 30 cm. 
 long and 1 to 2 mm. in diameter, is straightened and demagnetised by 
 one of the methods described in 240. In the case of fine wire it 
 will be found convenient to thread it through a fine bore glass tube, 
 fixing it in position with a little sealing-wax. This glass tube is then 
 supported axially within the magnetising coil by means of two corks, 
 and is adjusted so that the wire lies symmetrically with respect to the 
 coil. If the glass tube is made so long that it projects beyond the 
 magnetising coil, and the distance of the end of the wire is measured 
 from the end of the tube, the longitudinal adjustment may be made by 
 measuring from the end of the tube. 
 
 Having inserted the largest resistance in the circuit, the current is 
 made, and the deflected reading of the magnetometer and the value of 
 the current are read and recorded. The resistance is then decreased and a 
 new reading obtained, and so on till the maximum current is reached. 
 In this way the magnetisation curve for an unmagnetised piece of iron 
 is obtained. 
 
 Before proceeding to take a complete cycle of magnetisation, it will 
 be found advisable to take the iron ten or twenty times round the 
 proposed cycle. Then starting with the current at its maximum value 
 in one direction, the current is reduced by inserting the resistances in 
 the circuit, and the values of the current and the magnetometer readings 
 are recorded. When all the resistance is in the circuit the current is 
 
237] MAGNETOMETER METHOD 557 
 
 broken, and the magnetometer reading taken with no magnetising force. 
 The current is then reversed, and the readings corresponding to gradually 
 increasing currents in this reverse direction are obtained. Next the 
 current is gradually reduced to zero, reversed and increased to the 
 maximum value in the original direction, readings of the ammeter and 
 magnetometer being taken at each step. 
 
 To reduce the results the dimensions of the iron will be required, as 
 well as the distance of the iron from the magnetometer needle and the 
 value of the horizontal component of the earth's magnetic field at the 
 place where the magnetometer needle hangs. This latter can most con- 
 veniently be obtained by deflecting the magnetometer by passing a 
 measured current through a coil of known dimensions. If the dimensions 
 of the magnetising coil and of the compensating coil are the same, the 
 best way to proceed is to reverse the connections on one of them, so that 
 they now tend to deflect the needle in the same direction, and to read the 
 current which must be passed to obtain a deflection nearly up to the end 
 of the scale. If is the angle through which the magnetometer needle 
 is deflected when the coils are traversed by a current C c.g.s. units, each 
 of the coils containing n turns of wire per centimetre, the mean diameter 
 of the windings being r. and the length of each coil 2Z, while the 
 distances of the mid-points of the coils from the magnetometer needle 
 is D, then 
 
 G 
 
 ' 
 
 since the magnetic moment of each coil is 2wnr 2 LC. 
 
 If the dimensions of the magnetising coil are alone known, then the 
 earth compensating coil must be cut out of circuit, and the magnetising 
 coil alone used. The expression for calculating H is half that given above. 
 The advantage of using the two coils is, that if we measure the distance 
 between the centres of the coils, and take half this distance for D, then, 
 so long as the magnetometer needle is placed fairly accurately half-way 
 between the coils, errors due to the difficulty of accurately measuring to 
 the magnetometer needle will be practically eliminated. 
 
 The results of the experiments with the iron must be plotted, taking the 
 magnetising force F as abscissa, and either the magnetic induction or the 
 intensity of magnetisation as ordinate. In the case" of the long thin 
 cylinder, such as is used in this experiment, the magnetising force may 
 be taken as equal to the magnetic field within the coil before the intro- 
 duction of the iron, the demagnetising field due to the poles at the ends 
 of the wire being neglected. 
 
 As will be seen by a study of Fig. 255, both the induction and the 
 intensity of magnetisation increase very rapidly during the portion AB 
 of the curve, and hence if the deflection of the magnetometer is to be 
 kept on the scale, it is necessary to have the magnetometer at such a 
 distance that the deflections produced, when small values of the mag- 
 netising field are used, are small, and hence the accuracy with which the 
 
558 PERMEABILITY [ 237 
 
 corresponding B or I can be determined is small. This difficulty may be 
 met by placing the compensating coil nearer the magnetometer than the 
 position where it exactly neutralises the field produced by the magnetising 
 coil. The magnetometer deflections then give the difference between the 
 effect of the iron and the excess of magnetic action of the compensating 
 coil, and this excess can be determined by making a series of readings of 
 the magnetometer deflection produced when no ,iron is contained in the 
 magnetising coil. 
 
 Another method of arranging the specimen is to place the magnetising 
 coil in the magnetic east and west direction, with its centre on the 
 
 1600 
 
 12000 
 
 8000 
 
 4000 
 
 24 
 
 4 8 12 16 20 
 
 MAGNETIZING FORCE IN C.C.S. UNITS 
 
 FIG. 255. 
 
 magnetic meridian passing through the magnetometer, as shown in Fig. 
 256, the compensating coil F being placed in a symmetrical position on 
 the other side of the magnetometer. 
 
 If D is the perpendicular distance of the magnetometer needle M from 
 the iron wire GD, the couple acting on the magnetometer needle when 
 the deflection is is 
 
 M'sLI 
 
 Hence 
 
 or 
 
 sL 
 
 The advantage of this position over the end-on position lies in the fact 
 that the uncertainty as to the location of the poles of the magnetised 
 
237] 
 
 MAGNETOMETER METHOD 
 
 559 
 
 iui 
 
 iron produces less error thaii in the case where one end of the specimen 
 is comparatively close to the magnetometer. Further, in the end-on 
 
 position a displacement of the pole 
 along the specimen, since it directly 
 
 _ L _ alters the distance between the pole 
 
 and the magnetometer, and the deflec- 
 tion varies inversely as the cube of this 
 distance, may produce a very consider- 
 able error. In the side-on position, 
 however, if the poles are displaced, as 
 one pole approaches the magnetometer 
 the other pole recedes from the magneto- 
 meter, and hence for small displacements 
 these two effects will compensate for 
 one another. 
 
 Both the above methods of ob- 
 servation, however, suffer from the 
 serious objection that we have to 
 assume a knowledge of the position of 
 
 the poles in the iron, and in deducing the formula? we have supposed 
 that the poles are at the ends of the wire. There is no doubt that this 
 assumption is wrong, and a somewhat better result will be obtained if 
 we assume that the distance between the poles is 2/3 of the length of the 
 wire. 
 
 In this case the formula in the first case, when the axis of the iron 
 passes through the magnetometer, becomes 
 
 FIG. 256. 
 
 tan 0, 
 
 M 
 
 and in the second case 
 
 /4 \3 
 
 /= ^L_ I tan ft 
 
 N 
 
 s 
 
 This uncertainty as to the position of the N g 
 
 poles can in a measure be removed by adopt- 
 ing a third arrangement, which is shown B 
 diagrammatically in Fig. 257. The magnetis- FIG. 257. 
 ing coil is here placed vertical, and due 
 
 east or west of the magnetometer needle M, and the iron is adjusted so 
 that the upper pole is in the same horizontal plane as the needle. This 
 adjustment can be made by passing a current through the magnetising 
 coil, and then moving the iron wire up or down till the deflection of the 
 magnetometer is a maximum. In this way the position of the pole N, 
 with reference to the wire, is found with sufficient accuracy, and since the 
 
560 PERMEABILITY [ 237 
 
 wire is symmetrical, we therefore know the position of the pole s and the 
 distance between the poles. Thus the distances D and T> 1 of the poles 
 from the magnetometer needle can be calculated. If the strength of either 
 of the poles is m, then the horizontal component of the field at M due to 
 the two poles is 
 
 m in *T*r 
 
 & - 772 cos NM *> 
 
 1 
 
 or ni{- n - 
 
 But m 
 
 Hence the horizontal component is 
 
 , ( 1 
 
 and if 6 is the deflection of the magnetometer, 
 
 or H tan 8 HD i 
 
 /= ,TT-^\ =77-^1 
 
 '{& fitf I /VJ 
 
 The disadvantage of this method is that in addition to the magnetising 
 field of the coil in which the iron wire is placed, there is the vertical 
 component of the earth's field acting on the iron. It is possible to 
 neutralise the earth's field within the coil if an additional layer of wire 
 is wound and a current of suitable amount is passed through this wind- 
 ing. The adjustment is, however, troublesome, and, further, since it is 
 impossible to introduce the wire into the coil without getting it mag- 
 netised, it becomes necessary to demagnetise the iron in place. In 
 general, it will be found most convenient not to attempt to neutralise 
 the earth's field, but to apply a. correction afterwards. This correction 
 will only be appreciable in the case of soft iron, in which material the 
 permeability for small fields is fairly large. 
 
 Suppose that Fig. 258 represents the result of plotting B against the 
 magnetising field, on the supposition that the wire is at the start un- 
 magnetised, and the magnetising field F is that deduced from the current 
 in the magnetising coil, i.e. we neglect the effect of the vertical component 
 of the earth's field. It will be seen that the curve cuts the axis of F at 
 two points a and c, which are not symmetrical with reference to the 
 origin ; the points of intersection corresponding to the values of F+ 1 P 3 
 and - 2'2. Hence if the origin is moved to the point - 0'45, the curve 
 
238] CORRECTION OF FINITE LENGTH OF SPECIMEN 561 
 
 will be symmetrical about the new axis of B, the quantity + '45 repre- 
 senting the value of the vertical component of the earth's "field. In the 
 same way the points I and d, where the curve cuts the axis of B, or the 
 points ef, where B is a maximum for the fields used, are not symmetrical, 
 and to obtain symmetry we must move the origin down to the point - 200, 
 
 -8 -6 -4-2 O2 
 
 the value B = 200 representing the induction produced in the iron by the 
 earth's field. Hence by using these new axes we obtain a curve in 
 which both B and C represent the actual values of these quantities, and 
 the effect of the earth's field has been taken into account. 
 
 238. Correction on Account of the Finite Length of the Speci- 
 men. In the magnetometer method we assume that the magnetisation 
 throughout the rod is uniform, and that the magnetising field is also 
 uniform. Now this is certainly not true, for the magnetising field is not 
 exactly the same as the field which would exist if the iron were removed, 
 owing to the field produced by the magnetisation developed in the 
 specimen. Now, although this demagnetising effect of the induced 
 magnetisation can be calculated in the case of an ellipsoid, such a shape 
 is not practicable, and although a very long and thin cylinder may, as 
 a first approximation, be taken as being equivalent to an ellipsoid, yet 
 
 2N 
 
562 
 
 PERMEABILITY 
 
 [238 
 
 experiment has shown that even in such a case the magnetisation is 
 very far from uniform. 1 Hence where accurate measurements of the 
 magnetic properties of a sample of iron are required, it is necessary to 
 employ some other method in which the iron is magnetised uniformly. 
 Such a method is described in the next section. 
 
 Experiments have, however, been made to obtain the correction which 
 has to be applied on account of the demagnetising field due to the poles 
 developed at the ends of the specimen, and the results obtained may be 
 applied to obtain an approximate correction in such cases where only 
 comparatively short cylindrical rods of the iron are available. 
 
 If F' is the field at any point of a cylinder of iron due to the mag- 
 netism induced in the iron, then this quantity varies from one part of 
 the bar to another, being a minimum at the centre of the bar. The 
 mean value of F', however, is proportional to the mean intensity of 
 magnetisation /, and the ratio F /I is called the demagnetising factor N 
 for the bar. The value of this factor N depends on the dimension 
 ratio m of the bar, that is, on the ratio of the length to the diameter. 
 The following table gives the values of N for different values of m : 
 
 DEMAGNETISING FACTORS FOR CYLINDRICAL RODS. 
 
 m 
 
 N 
 
 m 
 
 if 
 
 10 
 
 0-2160 
 
 80 
 
 0-0069 
 
 20 
 
 0775 
 
 100 
 
 0045 
 
 30 
 
 0393 
 
 150 
 
 0020 
 
 40 
 
 0328 
 
 200 
 
 0011 
 
 50 
 
 0162 
 
 300 
 
 0005 
 
 60 
 
 0118 
 
 500 
 
 00018 
 
 
 
 1000 
 
 00005 
 
 Assuming that we know the demagnetising factor for a particular 
 specimen, the easiest way of applying the correction is as follows : 
 Plot the results of the measurements of / and F, on the assumption that 
 the demagnetising field is zero, thus obtaining the curve OA (Fig. 259). 
 We have now to reduce the values of F by an amount F' or NI 9 where N 
 is the demagnetising factor for the specimen. Tq do this draw the 
 straight line oc, making an angle < with the Y axis, where 
 
 tan <j> = N. 
 
 Then if we take the line oc as a new [axis of Y, the values of the 
 magnetising field measured from this new axis will be corrected for the 
 demagnetising field. For consider a point p on the curve. The intensity 
 of magnetisation is OM, and the demagnetising field is therefore N . OM. 
 But NM = OM tan < = N . OM. Hence the demagnetising field is repre- 
 1 See Lamb, Philosophical Magazine, September 1899. 
 
239] 
 
 BALLISTIC METHOD 
 
 sented by NM, and the actual magnetising field acting on the iron" is 
 OQ - NM, that is, represented by the line NP. Thus to obtain the true 
 
 o Q * 
 
 MAGNETISING FIELD F 
 
 FIG. 259. 
 
 curve OB, we decrease the abscissa of every point on the observed curve OA 
 by an amount equal to the intersept between the line oc and the axis of Y. 
 239. The Ballistic Method of Measuring Permeability. If a ring 
 of iron of which the dimensions are in- 
 dicated by the symbols shown in Fig. 260 
 is wrapped round with a secondary coil 
 containing N turns, and over this is wound 
 a magnetising coil of M turns, while the 
 magnetising coil is traversed by a current 
 C. Then the mean magnetising field over 
 the' cross-section of the iron is l F= 2MC/r. 
 The field at the inner edge of the ring is 
 greater than this mean and that near the 
 outer is less, but if the radial width 
 of the iron t\ r 9 is kept small compared to the mean radius r, the 
 
 1 If we suppose the iron removed without disturbing the primary coil, and 
 then carried a unit magnetic pole once round the anchor ring thus left, the work 
 done would be lirMC. For we should have carried the pole once round M circuits, 
 through each of which a current C is flowing. (See Watson's Physics, 478.) If 
 F is the field along the middle of the ring, i.e. along a circle of radius r, the 
 work done when the unit pole is carried once round this circle is ItrrF. Hence 
 equating the two expressions for the work, we get 
 
 FIG. 260. 
 
564 
 
 PERMEABILITY 
 
 [239 
 
 magnetising field may be assumed to be uniform and equal to the 
 mean value. 1 
 
 If B l is the,induction within the iron when the magnetising current is (7, 
 then the total induction, or the number of tubes of force which cross any 
 cross-section of the iron, is sB^ where s is the cross-section of the iron. 
 Hence since there are N turns in the secondary, the total induction 
 through the secondary is sB^N. If the secondary is connected to a ballistic 
 galvanometer, the resistance of the secondary circuit being R, and the 
 magnetising current is changed to (?,, the magnetic induction in the 
 iron becoming B^ the quantity of electricity which flows through the 
 secondary circuit will be given by 
 
 Q 
 
 R 
 
 Hence if the throw of the ballistic galvanometer is a, and the constant 
 of the galvanometer is K, we have 
 
 R 
 
 or 
 
 RK 
 
 jr. D . 0, / 
 
 VA-isr-.* 1 !- 1 ( 
 
 Thus a change of the magnetising field from 2MCJr to 2MC<,!r has 
 caused a change in the induction of B l - # 2 , 2 which can immediately be 
 calculated from the throw of the ballistic galvanometer if the constant of 
 the instrument is known. 
 
 The general arrangement of the apparatus is shown in Fig. 261. 
 The secondary coil of the iron anchor ring is connected to a ballistic 
 galvanometer G and to a secondary coil D wound on a long solenoid 
 FH. The object of this coil D is to determine the constant of the gal- 
 vanometer by noting the throw when a known current in the primary is 
 
 1 The errors produced in the case of iron by assuming that the field is uniform 
 throughout the cross-section for different values of the external radius r- 2 , the 
 width of the ring, namely, r- 2 - r } , being unity, are as follows : 
 
 
 Values of ?- 2 for the Error to be 
 
 
 1%. 
 
 2%. 
 10 
 
 4%. 
 
 1-0%. 
 t>-7 
 
 i-o 
 
 20 
 
 8 
 
 2 Where JS z - l is given in maxwells if R is measured in c.g.s. units 
 - 9 ohms). 
 
BALLISTIC METHOD 
 
 565 
 
 239] 
 
 reversed (see 220). The primary coil of the anchor ring is connected 
 through a two-way key K 2 to a commutator c. This commutator is one 
 designed by Ewing for use in such measurements, and may consist of a 
 Pohl commutator, in which the connection between the cups 3 and 4 
 has been removed. To the cups 1 and 4 are connected an ammeter J, a 
 battery E, and an adjustable resistance R r To the cups 3 and 4 are 
 connected a key K l and a resistance box R 2 , joined in parallel. The 
 primary of the solenoid FH is connected to one cup of the two-way key 
 K 2 and to the cup 5 of the commutator c, as shown in the figure. 
 
 When the key KJ is closed, then throwing over the commutator c 
 simply reverses the current in the primary of the ring, without altering 
 the magnitude of the current. When, however, the key KJ is open, then 
 putting the commutator 
 over to the left not only 
 reverses the current, but 
 also reduces it, since 
 the additional resistance 
 R. 2 is now inserted in 
 the battery circuit. 
 
 When the constant 
 of the ballistic galvano- 
 meter is to be deter- 
 mined, key K 2 is turned 
 to the right and the 
 key K X is closed, and 
 the throw of the galvano- 
 meter noted when the 
 commutator c is put 
 across. The method of 
 deducing the constant 
 is that given in 220. 
 
 The method of ob- 
 taining the BF curve 
 is as follows : The ring- 
 having been demagnetised by reversals ( 240), the resistance R l is 
 adjusted to give the maximum magnetising field to be used, and the 
 key KJ being closed, the commutator c is worked backwards and forwards 
 some twenty times, so as to establish a cyclical state in the iron. The 
 commutator then being to the left, the current passing is noted and the 
 key KJ is opened, thus producing a drop in the magnetising force, and 
 the kick of the galvanometer is noted, as well as the new value of the 
 current. We thus obtain the change in B produced by a certain drop 
 in F. 
 
 The key K! is then closed and the commutator c worked for about 
 twenty times to re-establish the initial conditions, and R 2 is given a larger 
 value. The current having been read, the key KJ is opened and the 
 throw noted, as well as the new value of the current. In this way the 
 
 FIG. 261. 
 
566 PERMEABILITY [ 239 
 
 change in^B corresponding to a greater change in F is obtained. In this 
 way a number of points are obtained, the final one corresponding to the 
 resistance E 2 being infinite, i.e. the kick corresponding to the change in 
 induction when the magnetising field F changes from its maximum value 
 to zero. 
 
 To obtain the portion of the curve corresponding to negative values 
 of F, the key K t is closed and the commutator reversed twenty times, 
 being finally left to the right. R 2 having been given a large value, the 
 key iq is opened and the commutator c moved to the left. In this way 
 the current is reversed, and at the same time is reduced to a small value, 
 the throw corresponding to a change of F from its maximum positive 
 value to a small negative value. Proceeding in this way R 2 is gradually 
 decreased till it becomes zero, when the throw corresponds to the 
 change in B, produced by reversing the maximum magnetising field. 
 Half this change in B gives the value of B corresponding to the ^maxi- 
 mum value of F, and hence having plotted this point the other points in 
 the curve can at once be set out, for we know the difference between the 
 value of B for each of the steps in F and the value of B for the maximum 
 value of F. 
 
 It is unnecessary to observe the other half of the cycle, since the curve 
 obtained will be similar to the first half. 
 
 By changing the value of the resistance RJ cycles with different maxi- 
 mum values of B can be obtained. 
 
 Having plotted the B-F curve, the hysteresis may be calculated by 
 measuring the area of the loop enclosed by the curve. If A is the area 
 of the loop, then the energy dissipated per cycle in each cubic centimetre 
 of the iron is A/^TT (see Watson's Physics, 509). If the I-F loop is 
 plotted, then the hysteresis loss per cubic centimetre per cycle is numeri- 
 cally equal to the area of the loop. 
 
 When calculating the hysteresis, it must be remembered that the area 
 in square centimetres will only represent the hysteresis numerically if 
 1 centimetre represents unity in both B and F. Since B is in general 
 enormously greater than F, the scale adopted for B must be taken much 
 smaller than that taken for F. If 1 cm. on the curve represents / units 
 of F and b units of B, then 1 square centimetre of the loop represents 
 fb ergs. 
 
 The advantage of the anchor ring is that since there are no free poles 
 developed there is no demagnetising field, and hence the magnetising 
 force is the same as the magnetic field which would exist inside the 
 magnetising coil supposing the iron were absent. There is, however, a 
 practical objection to this form in that every sample of iron which is to 
 be tested requires winding with a primary and a secondary coil, and 
 since the winding has perforce to be done by hand, the process is one 
 which takes some time. The ballistic method may be applied to the case 
 of samples in the form of long rods by using as the magnetising coil a 
 helix similar to that used in 220, and surrounding the centre of 
 the specimen with a secondary coil, which is connected up to a ballistic 
 
240] DEMAGNETISATION OF A SPECIMEN OF IRON 567 
 
 galvanometer, as in the case* of thev\ anchor ring. This secondary coil may 
 either be wound direct on the iron, or it can be wound on a glass or 
 vulcanised fibre tube, which is slipped over the iron. If the secondary is 
 wound on a tube, the diameter of the tube should be as small as possible, 
 so that the space between the secondary coil and the iron may be as 
 small as possible. A correction will have to be applied |for the effect of 
 the lines of force which thread through the secondary coil between the 
 coil and the iron. If the cross-section of the iron is s, and that of the 
 secondary coil is cr, while there are N turns in the -secondary, then when 
 the magnetising field is F the induction through the secondary, due to 
 the lines of force of the magnetising field which are included between 
 the secondary and the iron, is (a- - s)NF. Hence when the magnetising 
 field is changed by an amount 8F, the change in induction through the 
 secondary, due to the field between the secondary circuit and the iron, 
 will be (a- - s)N8F, and hence the measured induction, as deduced from 
 the galvanometer throw, will have to be reduced by this amount to give 
 the change in induction through the iron. The advantage this method 
 of testing a long thin rod has over the magnetometer method is that we 
 are only here concerned with the intensity of magnetisation of the iron 
 near the middle of the rod, which, if the rod is long and thin, is very 
 nearly independent of the effect of the ends. 
 
 --2 40. To Demagnetise a Specimen of Iron. Since the induction in 
 a sample of iron at any given instant depends not only on the magnetising 
 force which exists at the time, but also to some extent on the previous 
 magnetic history of the specimen, it is of importance to be able to reduce 
 the iron to some definite magnetic condition which shall be independent 
 of all previous magnetisation. A specimen of iron to which this process 
 has been applied is said to be demagnetised. 
 
 In the case of thin wires, perhaps the most convenient method of 
 demagnetisation is to heat the wires to redness. This can be done by 
 slowly passing the wire through a Bunsen flame, care being taken to hold 
 the wire horizontal and in an east and west direction, so that it is 
 unacted upon by the earth's field. 
 
 In many cases, however, as, for example, with an anchor ring 
 wound with its coils, heating to redness is impossible. The procedure 
 here adopted is to pass an alternating current through the primary 
 winding, and to gradually reduce the strength of this alternating current 
 to zero. It is of importance to start with a sufficiently strong current, 
 otherwise the effects of previous magnetisations may not entirely be 
 removed. The current employed should be such as to give an intensity 
 of magnetisation considerably greater than that corresponding to the 
 point where the IF curve bends sharply over, i.e. the field should be 
 sufficient to magnetise the iron to saturation. In the case of soft iron, if 
 the magnetising field at the start amounts to about 10 c.g.s. units 
 (gausses), the demagnetisation will be complete. 
 
 If an alternating current is available, then this may be used to 
 produce the demagnetisation, otherwise a small commutator driven by 
 
568 PERMEABILITY [ 240 
 
 a small motor must be used to convert a direct current into an alternating 
 one. A resistance, consisting of a trough about 50 cms. long and 5 
 cm. wide and 8 cm. deep, filled with zinc sulphate solution, should be 
 included in the circuit. The electrodes consist of two amalgamated 
 plates of zinc. One of these plates is fixed, and the other is movable. 
 The trough is tilted, so that the depth of the solution at the end farthest 
 from the fixed electrode is only about a centimetre. Starting with the 
 movable electrode close to the fixed one, the movable electrode is slowly 
 moved to the shallow end of the trough, by which means the current is 
 gradually reduced to zero. 
 
CHAPTER XXXIV 
 
 THE QUADRANT ELECTROMETER 
 
 241. The Quadrant Electrometer. The original Kelvin pattern 
 quadrant electrometer was a very elaborate instrument, troublesome to 
 set up, and difficult 
 to keep in adjust- 
 ment. Of recent 
 years, however, a 
 more simple form 
 of quadrant electro- 
 meter has been 
 found just as accu- 
 rate, and very much 
 easier to use. The 
 best known pattern 
 of quadrant electro- 
 meter of this more 
 simple form is that 
 designed by ' Dole- 
 zalek, 1 and is shown 
 in Fig. 262. The 
 needle is made of 
 two pieces of silvered 
 paper of the form 
 shown at (a) at- 
 tached to a thin 
 stem of aluminium, 
 which also carries a 
 mirror. The needle 
 system is suspended 
 by a thin strip of 
 phosphor bronze or 
 a quartz fibre, 
 rendered conducting 
 by being smeared 
 with a solution of 
 calcium chloride, or coated with a thin layer of platinum. 
 
 The needle should be charged to between 100 and 200 volts, and for 
 
 1 Zeit. fur Instrumentcnkunde (1901), xxi. 345. 
 
 FIG. 262. 
 
570 
 
 THE QUADRANT ELECTROMETER 
 
 [241 
 
 this purpose it is connected to one pole of a battery of about eighty small 
 accumulators, the other pole of the battery, as well as the outside case of 
 the instrument, being earthed by being joined to the gas or water pipes. 
 
 A convenient form of battery of small accumulators for use with an 
 electrometer is shown in Fig. 263. The cells consist of glass tubes having 
 a diameter of about 1 inch, and a length of 6 inches. Forty of these glass 
 tubes are imbedded in paraffin wax contained in a shallow wooden tray, 
 the upper inch of the tubes also being coated with paraffin wax to pre- 
 vent the acid creeping. This coating is easily performed after the tubes 
 are fixed in the tray by dipping them into some melted wax contained in 
 a shallow dish. The electrodes consist of strips of lead bent into a U 
 
 FIG. 263. 
 
 shape, each strip dipping into two adjacent glass tubes. A small strip of 
 glass is placed between the lead plates in each tube, so as to prevent short- 
 circuiting by the lead plates coming in contact. The glass tubes are 
 filled to within about'an inch of the top with a solution of sulphuric acid, 
 made by adding 1 part of pure sulphuric acid to 8 parts of distilled water. 
 If such a box containing forty cells is connected to a 100 volt lighting 
 circuit through an ordinary 25 candle-power 100 volt incandescent lamp, 
 the cells will be charged at the right rate. If the voltage of the electric 
 supply is 200, then two such boxes may be charged in series. When the 
 cells are first set up, it will be necessary to charge them first in one 
 direction and then in the other some twenty or thirty times, so as to 
 " form " the plates. After the plates have been formed in this way, the 
 
1242] CAPACITY OF A QUADRANT ELECTROMETER 571 
 
 cells should receive a good charging in one direction, and then care must 
 be taken to always charge them in this direction. 
 
 It will be found advantageous to enclose the trays containing the 
 cells in wooden boxes fitted with lids, as in this way the cells are protected 
 from dust. The leads from the cells may be taken through holes in the 
 sides of these boxes, the holes being bushed with paraffin wax. 
 
 Two boxes containing forty cells in each can be used in series to charge 
 the needle of the electrometer, the boxes being charged by being con- 
 nected in parallel if the lighting circuit is at 100 volts. 
 
 One pair of quadrants of the electrometer is permanently connected 
 to earth by being joined to the case of the instrument. The other pair 
 is connected to the apparatus being employed, and to a key by which 
 these quadrants also may be earthed when required. A simple form of 
 key for this purpose is shown in 
 Fig. 264. It consists of a block of 
 paraffin wax A, in which is scooped 
 a mercury cup B, the mercury being 
 connected to the electrometer by the 
 wire c. A piece of clock spring D 
 is screwed to the base, and to the 
 side of this spring is soldered a 
 piece of wire E. The spring is con- 
 nected to earth through a wire G, 
 and when the spring' is bent by 
 pulling on a string F, the point of 
 E dips into the mercury, and the FIG. 264. 
 
 electrometer is earthed. 
 
 Having set up the electrometer, and adjusted a telescope and scale to 
 measure the movement of the needle, connect the needle and both sets of 
 quadrants to earth. Having made a note of the scale reading, charge 
 the needle, keeping the quadrants earthed. If there is a change in the 
 reading, it shows that the needle is not symmetrical with respect to the 
 quadrants. In some forms of electrometer one of the quadrants is made 
 adjustable, and in this case this quadrant must be moved till the needle 
 comes back to the position it occupied before it was charged. In most 
 cases, however, it is very much easier to adjust the needle by tilting 
 the whole instrument by means of the levelling screws. The scale read- 
 ing having been brought back to its original value by tilting the instru- 
 ment, it will be advisable to discharge the needle and make certain that 
 the "uncharged" reading has not altered. 
 
 When the needle has been adjusted to a symmetrical position in this 
 way, the deflections on the two sides of the zero should, for equal charges, 
 be the same, also the position of the zero will not alter if the potential 
 of the needle changes. 
 
 242. Determination of the Capacity of a Quadrant Electrometer. 
 For many experiments it is necessary to know the capacity of a 
 quadrant electrometer, for often this capacity is comparable in magnitude 
 
572 THE QUADRANT ELECTROMETER [ 242 
 
 with the capacity of the condensers, &c., with which the instrument is 
 connected. 
 
 The electrometer A (Fig. 265) is connected to the key p, which is of 
 the pattern shown in Fig. 264, and to a small mercury cup G in a block 
 of paraffin. A second mercury cup in this block is connected to one 
 coating of a small condenser. This condenser consists of two concentric 
 brass tubes. The outer tube D is connected with earth, while the inner 
 
 EARTH 
 
 tube is insulated by being supported on two small ebonite blocks. If 
 I is the length of the tubes, a the external diameter of the inner tube, 
 and b the internal diameter of the outer tube, the capacity is given in 
 electrostatic units by 
 
 I 
 
 C = 
 
 2h *.a 
 
 This expression neglects the effect of the ends, but if the space between 
 the tubes is small, the error produced on this account will not be sufficient 
 to invalidate the results. 
 
 The condenser being disconnected from the electrometer, the battery E 
 is joined to the electrometer by closing the key F, and the deflection d l is 
 observed. The key F having been opened, the electrometer is caused to 
 share its charge with the condenser by inserting the small copper bridge 
 B in the mercury cups G. If the deflection is now d 2 , we have, if c is the 
 capacity of the electrometer, 
 
 For if we assume that the deflection is proportional to the potential, the 
 original charge on the electrometer is o.cd v where a is a constant factor, 
 while the charge on the electrometer and condenser is a(c + C)d f) . Hence, 
 since these charges are equal, 
 
 from which the formula given above at once follows. 
 
243] 
 
 SATURATION CURRENT 
 
 573 
 
 243. Determination of the Saturation Current. If a layer of 
 some radio-active substance, such as uranium oxide, is placed on an 
 insulated metal plate c (Fig. 266), and this plate is connected to one pole 
 of a battery E, the other pole being earthed, then an electrometer A 
 connected to a second insulated plate B will gradually be deflected, 
 showing that a current is passing through the gas between the plates 
 c and B. If the difference of potential between the plates B and c 
 is gradually increased, then the current which passes also gradually 
 increases. This increase is at first rapid, but becomes slower and slower, 
 till finally a fairly moderate further increase of E.M.F. produces no further 
 increase in the current. This maximum current is called the saturation 
 
 EAKTH 
 
 FIG. 266. 
 
 current, and is such that the ions being produced between the plates due 
 to the radio-active material are being removed by this current as fast as 
 they are being produced (see Watson's Physics, 566). 
 
 The metal plates B and c (Fig. 266) are supported on ebonite pillars 
 inside a metal box D, and the radio-active substance is spread on the 
 plate c. The distance between the plates ought to be about a centimetre, 
 when, with uranium oxide, the saturation current will be obtained with 
 between 100 and 200 volts. 
 
 If c is the capacity in electrostatic units of the electrometer and 
 its connections, including the plate B (this quantity can be determined in 
 the manner described in the last section), d the number of scale divisions 
 passed over in a second when the needle has settled down to its uniform 
 speed of rotation, and D the deflection produced by 1 volt, that is, 
 1/300 electrostatic units, then the rise in potential in one second is 
 
 _d_ 
 
 300D 
 
 units. 
 
 The charge on the electrometer and plate B has therefore increased in 
 one second by 
 
 cd 
 
 electrostatic units. 
 
574 THE QUADRANT ELECTROMETER [ 243 
 
 But the increase in the charge in one second is equal to the current. 
 Hence the current is 
 
 cd 
 
 electrostatic units, 
 
 or Q-~ x 10~ n amperes ...... (1) 
 
 To perform the experiment, the small accumulators described in 241 
 are used, and starting with a few cells, the rate at which the electrometer 
 deflection increases when the earthing key F (Fig. 266) is opened is 
 determined. To obtain this rate, the time taken for the deflection to 
 increase by 100 scale divisions is noted on a stop-watch, care being taken 
 not to start observing till the needle has moved some little distance from 
 its position of rest, so that its motion has had time to become uniform. 
 When using an electrometer in this way to measure a current, it is 
 important that the needle should be almost dead-beat. If the damping 
 is too small, then the needle will move with an oscillatory movement 
 superimposed on the steady movement. If, however, the damping is too 
 great, the needle will take some time before its motion becomes uniform. 
 
 If the rate of movement of the needle is too great to be accurately 
 observed, then a condenser may be put in parallel with the electrometer, 
 in which case the quantity c in (1) is the sum of the capacities of the 
 condenser and the electrometer. 
 
 Determine the current passing between the plates c and B for 
 different E.M.F.'s by the above method, and plot your results in the 
 form of a curve, taking the current as ordinate, and the applied E.M.F. as 
 abscissa. If you have been able to apply a sufficiently high E.M.F. 
 to obtain the saturation current, the curve will be almost horizontal for 
 the higher E.M.F.'s, and the current corresponding to this horizontal 
 portion of the curve will be the saturation current. 
 
 Alter the distance between the plates B and c, and repeat the observa- 
 tions. In this way prove that as the distance between the plates 
 increases the current produced by a given difference of potential in- 
 creases, in this respect differing from what occurs when an E.M.F. is 
 applied to an electrolyte. 
 
APPENDIX 
 
 1. Grlass Blowing. The art of performing simple glass blowing opera- 
 tions is so essential, that although it is almost impossible to impart it by 
 description, yet it may be worth while to devote a few pages l to giving a 
 few hints. In glass blowing success can only be obtained after much 
 practice, and wherever possible the student is advised to obtain some 
 lessons from a competent worker. 
 
 The blow-pipe to be used is an important adjunct, but the exact 
 pattern is not of very much importance. There are many forms of blow- 
 pipe on the market, some of which are complicated and expensive. The 
 student is, however, recommended to use a simple pattern, having a single 
 tap on the gas-supply, and a joint by which the inclination of the jet can 
 be varied. The chief fault of most blow-pipes as supplied by the makers 
 is that the nipple at the end of the gas-supply pipe is much too fine. The 
 bore of the hole in the nipple ought to be between 0'5 and 2 mm., a very 
 useful size being 1*4 mm. A foot bellows of ample size will be required 
 to supply the air, and this bellows must be fastened to the blow-pipe 
 table, so that when in use it cannot move away from the operator's foot, 
 as invariably occurs at a critical moment if it simply rests on the floor 
 alongside the table. Fletcher's universal blow-pipe, number 3, on a stand, 
 with a tap to the gas-supply, and his foot blower, number 3 or 5, will be 
 found suitable for all but the heaviest glass blowing required in an 
 ordinary physical laboratory. 
 
 Since soda-glass is almost invariably used for making physical appa- 
 ratus, and, further, its manipulation is simpler than that of lead-glass, the 
 following notes only apply to the former : The glass used should be of a 
 good quality, and should bear heating in the blow-pipe flame for some 
 considerable time without showing any signs of becoming frosted. 
 
 Other accessories which will be required in addition to the blow-pipe 
 and bellows are : (1) some small triangular files foivcutting off glass 
 tube ; (2) some pieces of wood charcoal, sharpened at the end into 
 conical points of different sizes ; (3) a wire nail, about 2 inches long, 
 fitted in a handle : the top of the nail is cut off, and this end is forced into 
 the handle so that the point projects for about an inch ; (4) a small 
 mouth-piece made by first expanding the end of a piece of glass tube of 
 
 1 A small book in which there are many hints on glass blowing is Shenstone's 
 The Methods of Glass Bloivitig (Longmans). Many hints on glass blowing and 
 many other matters of interest to the worker in a physical laboratory will be 
 found in Laboratory Art, by Threlfall (Macmillan). 
 
 575 
 
576 APPENDIX 
 
 about 7 mm. diameter, and then, when the glass is still hot, pinching the 
 tube together near the end. This mouth-piece is attached to about a 
 metre of small-bore rubber tubing, and is used for blowing into pieces of 
 apparatus which cannot conveniently be brought directly to the mouth, 
 or where pressure has to be exerted on the work while being heated in 
 the flame. 
 
 (a) Bending Glass Tube, Glass tube up to about a centimetre in 
 diameter can most conveniently be bent by heating the place where the 
 bend is to occur over an ordinary bat's-wing burner. While the tube is 
 being heated it must be continually rotated, being supported by the hands 
 placed as nearly as possible at the centre of gravity of the two portions 
 of the tube on opposite sides of the heated portion. When it is felt that 
 the tube has become quite bendable, remove it from the flame, and slowly 
 and steadily bend the two sides upwards. Throughout the bending 
 operation the tube should be kept in a vertical plane, otherwise the weight 
 of the heated portion of the tube will cause the bend to be on one side. 
 
 To bend tube having a diameter greater than a centimetre without 
 reducing the bore is a difficult operation. One end of the tube must be 
 stopped with a cork, and in the other is placed a cork, through which 
 passes a short length of quill tubing. This piece of tube is connected to 
 the rubber-tube attached to the mouth-piece described above. The tube 
 to be bent is heated in a very large blow-pipe flame, and as the glass 
 softens the hands are very slightly pressed together, so as to slightly thicken 
 the glass. If the glass during this operation shows a tendency to con- 
 strict the bore, it must be removed from the flame, and must be gently 
 blown out. When about 4 or 5 cm. of the tube has been thickened in 
 this way, having thoroughly and uniformly heated the thickened portion, 
 remove the tube from the flame and gently draw it out, and immediately 
 after, or rather almost during the drawing out, bend the tube into the 
 required shape. During the drawing out and bending blow gently into 
 the tube, so as to prevent the sides from collapsing. 
 
 If constricting the tube at the bend is no disadvantage, then the 
 following method is quite easy : Heat the tube in a moderate sized 
 flame, and gradually thicken the glass by very gently moving the hands 
 towards one another. When about 4 cm. of the tube has been thickened, 
 so that the bore is about two-thirds of its original size without the out- 
 side being much reduced in size, remove the glass from the flame, and 
 very slowly draw the ends apart. When the drawn-down portion is 
 about 8 cm. long, bend the two ends upwards. The glass at the drawn- 
 down portion being thick will retain enough heat to remain flexible for 
 some considerable time. Hence it will be easy to adjust the bend to the 
 desired shape. The thickness of the walls also "prevents the sides of the 
 tube collapsing at the bend. 
 
 (b) To Close the End of a Tube.Rold the end of the tube in the 
 blow-pipe flame, and when the edge of the glass is soft, take a small 
 piece of waste glass and push the walls of the tube together. Having 
 bunched the end of the tube together, allow the waste piece of glass to 
 
APPENDIX 577 
 
 adhere, and then heat the tube a little beyond the bunched in portion 
 with a small name. The tube must be kept in continuous rotation 
 during this operation, and a gentle outward strain kept on the piece of 
 glass which was fixed on the end. In this way a narrow neck is pro- 
 duced, and by continuing the heating this neck is gradually closed, and 
 the end piece of glass is drawn off. The tube will thus be closed, but 
 the closed end will be conical, ending in a sharp point where the glass 
 was drawn off. The conical end is then heated pretty strongly, and 
 after removal from the flame, the end is rounded by blowing gently into 
 the tube. To successfully round off the tube, the heating and blowing 
 require the exercise of some judgment. If, however, the following is 
 kept in mind, success will be obtained after a little practice : When the 
 tube is in the flame the thinner portions get hottest, and hence if you 
 blow into the tube immediately after its removal from the flame, these 
 hot portions will be most blown out, and hence the thin parts will 
 become thinner. If, however, you wait a little after removing the glass 
 from the flame, the thin parts will cool most rapidly, and hence if you 
 then blow into the tube you will blow out the thicker portions, since 
 these will have retained the heat. In this way the thickness of the 
 glass is gradually made more uniform. It will in general be necessary 
 to heat up the end two or three times, blowing it out between each heating, 
 before a nicely rounded end, having a uniform thickness, is obtained. 
 
 (c) To Blow a Bulb on the End of a Tube. Having closed the end 
 of the tube in the manner described above, hold the tube in the right 
 hand, with the closed end slightly higher than the open end, and so that 
 the flame plays on the tube a little inside the end. Keep the tube 
 rotating backwards and forwards by rolling it with the thumb over the 
 extended fingers. When the glass has thickened and the bore been 
 reduced to about half, start heating a little farther along the tube. 
 The rate of turning must be so adjusted that it just counteracts the 
 tendency of the heated portion to droop under the action of gravity. 
 If the heated portion becomes very irregular, remove the tube from the 
 flame, and gently blow into the tube. By alternate heating and blowing 
 obtain a pear-shaped mass of glass, the bore of the tube, however, 
 running almost to the end. It is most important that the bore should 
 extend almost to the end of the lump of glass, and this was the reason 
 why the first heating was made just inside the end. When the pear- 
 shaped mass of glass has been obtained it must be heated to a bright 
 red, the heating being as uniform as possible. Then remove the glass 
 from the flame, and holding the tube as nearly vertical as possible, the 
 heated end being downwards, very gently blow into the tube. It is 
 advisable to keep the tube slowly rotating while you are blowing out 
 the bulb. If the bulb shows a marked inclination to bulge out in one 
 direction, hold the tube horizontal with the bulge turned downwards. 
 The uprush of cold air produced by convection will in this position 
 slightly cool the lower portion of the bulb, and hence this portion will 
 not be blown out so easily. 
 
 2 6 
 
578 APPENDIX 
 
 If the first attempt results in a very irregular bulb, by turning on 
 the gas full and using a large flame with little air the bulb may slowly 
 be reduced in size. Whenever the surface becomes crinkled, gently blow 
 into the tube. This reducing process must be continued till the pear- 
 shaped mass of glass, with a small hole down the centre, is again 
 obtained. It is useless to attempt to again blow up the bulb after 
 only slightly reducing its size. 
 
 (d) To Blow a Bulb in the Middle of a Tube. Having closed one 
 end of the tube, either by drawing it out in the flame or by means of 
 a cork, hold the tube on the two sides of the place where it is desired 
 to make the bulb in the two hands. The two hands must be held in 
 the same way, either with the tube held on the outstretched fingers by 
 means of the thumbs, the backs of the hands being downwards, or 
 rolling on the thumb and first finger, the back of the hands being turned 
 upwards. Unless both hands are held in the same way, it will be found 
 very difficult to turn the two halves of the tube at exactly the same rate. 
 Heat the tube in a 'moderately large flame, keeping up a to-and-fro 
 
 rotatory motion, and slowly press 
 
 Apr* the two halves together. When 
 
 ^^^^^ r ~, -zr--*r"~r-r *ke glass has thickened, blow out 
 
 - ~~* " the mass slightly as at A (Fig. 
 "' "" ''sax*^asm*K&8s=ss*~ i ~** a * 267). Then "heat the tube along- 
 side the thickened portion, and 
 having thickened the glass, slightly 
 blow it out. Eepeat the process 
 again if a large bulb is required, 
 FIG. 267. so that the tube has the form 
 
 shown in the figure. Next 
 
 using a large flame, heat the thickened portion, and by alternately 
 heating and gently blowing, work the glass into the form shown at D. 
 Finally, heat the collected glass to a uniform bright red, and then 
 removing the tube from the flame, blow the bulb of the required size. 
 The bulb must be blown slowly, and any tendency to a bulge on one 
 side checked by turning this side down. If an oval bulb is required, 
 the two portions of the tube must be pulled apart as the bulb is reaching 
 its full size. 
 
 (e) To Join together Two Pieces of Tube. If the two pieces of tube 
 have the same diameter, then having seen that the ends to be joined are 
 both smooth and at right angles to the axis of the tube, close the other 
 end of one piece. Then using a fairly large flame, heat the extreme ends 
 of the tubes, and when the glass is quite hot and plastic remove from the 
 flame, and immediately press the two ends together, taking care that they 
 are in contact all round. If the tubes are of small diameter, turn the 
 flame down and thoroughly heat the joint all round till the glass has 
 fallen together enough to appreciably reduce the bore. Remove the tube 
 from the flame and gently blow into the open tube, and at the same time 
 draw the two halves of the tube apart. The pressure exerted and the 
 
APPENDIX 579 
 
 amount of traction must be so adjusted that the diameter of the tube at 
 the joint becomes the same as that of the rest of the tube, while the 
 thickness of the glass at the joint is also the same as that of the rest of the 
 tube. The joint must then be well annealed by rotating it over a small 
 flame, obtained without using any air in the blow-pipe. In this snioky 
 flame the glass will become coated with soot. After heating the glass 
 for a minute or two over the smoky flame, wrap the joint round with 
 cotton wool, and set it aside to cool. 
 
 If the tubes have a diameter greater than about 7 mm., then after 
 making them adhere as described above, turn the flame down so that a 
 very small pointed jet is obtained. With the point of this jet heat a 
 small portion of the joint till the glass is quite fluid. Then remove the 
 glass from the flame, and by gently blowing into the tube blow out the 
 heated portion of glass which will have caved in under the action of the 
 flame. Next heat in the same manner the adjacent portion of the joint, 
 and proceeding in this way gradually work completely round the joint. 
 By this procedure the glass at the joint will be thoroughly fused 
 together, and during the process the portions of the joint, which are not 
 being strongly heated, will be rigid, and so will assist in preventing the 
 joint becoming irregular. Having welded the glass well together by the 
 above method, the whole joint must be heated by rotating the tube in a 
 slightly larger flame, and then by blowing and gentle traction the joint 
 must be brought to the same diameter as the rest of the tube. Finally, 
 the joint must be annealed as described above. 
 
 If the tubes to be joined are of small diameter, it will be found of 
 advantage to enlarge them at the ends to be joined. This can be done 
 by heating the end of the tube and then inserting the iron nail fitted to a 
 handle and rotating the tube, the nail being pressed against the inside of 
 the tube. 
 
 If the tubes to be joined are of different diameters, then the larger 
 tube must be drawn down till the drawn-down portion is of the same 
 diameter as the smaller tube. The junction between this drawn-down 
 portion and the other tube is then effected as in the case of two tubes of 
 equal diameter. 
 
 (/) To Make a T -piece. To make a T-piece, close one end of the 
 glass A, which is to form the top of the T, and also one end of the glass 
 B, which is to form the stem. Then using a very small pointed flame, 
 heat a single spot on the tube A at the point where the side tube is to be 
 attached. During this heating the glass must not be rotated. When the 
 spot of glass is red-hot, blow gently into the tube, and thus raise a small 
 pimple on the glass. The diameter of this pimple ought to be the same 
 as the diameter of the tube B. Next strongly heat the extreme top of 
 the pimple, and blow strongly into the tube. In this way a large kidney- 
 shaped bulb of very thin glass will be produced. This bulb is broken 
 away, leaving a hole in the side of the tube A with a raised edge, the 
 diameter of the hole being approximately the same as the diameter of 
 the tube B. 
 
580 APPENDIX 
 
 Taking the tube A in the left hand and the tube B in the right, heat 
 the hole in A and the end of B in a fairly large flame, and when the edges 
 are fairly fluid bring them in contact, being care- 
 ful that no gaps are left. 
 
 Next attach the rubber tube and mouth-piece 
 to the open end of the tube A, and using a very 
 
 small jet heat a single point of the joint till the 
 
 glass flows together, after which gently blow out 
 
 this heated portion. Proceeding bit by bit, heat 
 
 FIG. 268. a n round the joint in this way, being particularly 
 
 careful to round off the corners A, B (Fig. 268). 
 
 Finally, give the whole joint, including the portion c of the tube A, a 
 
 good general heating in a fairly large flame, and having carefully annealed 
 
 in a smoky flame, pack the joint in cotton wool and allow it to cool. 
 
 (g) To Seal a Platinum Wire through the Side of a Glass Tube. To 
 make the hole in the glass tube through which the platinum wire is to 
 pass, heat a small spot on the tube, and when the glass is red-hot, touch 
 the place with a piece of platinum wire. The glass will adhere to the 
 end of the wire, and by quickly drawing the wire away the glass will be 
 drawn out into a fine capillary tube. Break off this capillary near the 
 surface of the tube and, if necessary, trim the end flush with the side of 
 the tube with a file. Draw down a piece of the same length of glass tube 
 as that into which the wire is to pass to a capillary with walls about a 
 millimetre thick, and of such a bore that the platinum wire fits fairly 
 closely. Cut off about a centimetre of this capillary and thread it on 
 the platinum wire, and then heat one end of the capillary so that it flows 
 round the wire. By means of 
 another portion of the drawn- 
 down tube wind a little glass 
 round the fused end of the 
 capillary so as to form a little . 
 
 bead of glass, as shown at (a), ' ' 
 
 Fig. 269. Pass the wire and the 
 capillary through the hole in 
 the tube, and with a small FIG. 269. 
 
 pointed flame strongly heat the 
 
 bead of glass and the portion of the tube in the immediate vicinity. 
 During the heating move the wire about a little by means of a pair of 
 small forceps, and occasionally blow down the tube. In this way the 
 seal is to be brought to the form shown at (b), Fig. 269. Such a seal 
 must be very carefully annealed by being heated for several minutes in 
 a smoky flame, and then being packed round with cotton wool. 
 
 Wherever possible, a platinum wire should be sealed through the 
 end of a glass tube, rather than through the side. In fact it is quite 
 worth while, in the case where an electrode is required to pass through 
 the side of a glass tube, to seal on a small side tube, the wire passing 
 through the end of this tube. 
 
APPENDIX 
 
 581 
 
 FIG. 270. 
 
 The disadvantage of the form of electrode shown in Fig. 269 is that 
 the loop of platinum on the outside is apt to get broken off. A form of 
 electrode which will often be found better is 
 shown in Fig. 270. A short length of narrow 
 tubing has its bore contracted, and is then 
 attached to the side of the tube, forming a 
 T-piece. The platinum wire is then inserted, 
 and the thickened part of the side tube is heated 
 till the glass flows round the middle of the 
 wire. Finally, if necessary, the side tube is bent 
 round at right angles. When in use a little 
 mercury is placed in the side tube, and the 
 connecting wire dips into this mercury. 
 (h) To Cut Large Sore Glass Tube,G\nss tube up to about 2 cm. 
 in diameter can be cut satisfactorily by means of a file ; in the case of 
 larger tubes, however, the only really satisfactory method of making a 
 neat square cut is by making a circular scratch on the inside of the tube 
 with a diamond. The instrument used to make such a scratch is shown 
 in Fig. 271. It consists of a steel rod AB, having a small cutting 
 diamond let in at A. A metal disc c attached to a collar can be clamped 
 by means of a screw at any desired point along the rod AB. The disc c 
 having been set so that the distance between it and the diamond point 
 is equal to the length of tube to be cut off, the diamond is lightly 
 pressed against the 
 inside of the tube, 
 while the tube is 
 rotated. In this 
 way a very fine 
 scratch will be made 
 round the tube. If 
 in place of a fine 
 
 scratch a wide rough scratch is obtained, it shows that the diamond 
 is not acting properly, and generally means that too much pressure is 
 being exerted. Having made a satisfactory scratch, try whether the tube 
 will break when you apply a fairly strong longitudinal pull to the tube. 
 If such a pull will not cause the tube to break, bring the scratched part 
 quickly into a small flame, rotating the tube the while. The sudden 
 heating will cause the tube to crack at the place where the diamond 
 made the scratch. 
 
 2. Working Fused Silica Manufacture of Quartz Fibres. 
 Amorphous silica, obtained by the fusion of pure quartz, has many pro- 
 perties which render it very valuable in the physical laboratory. The 
 chief of these are, first, that its coefficient of thermal expansion is 
 excessively small (about '0000001), so that a piece of the material may 
 be suddenly introduced into the oxy-hydrogen flame, or when red-hot 
 plunged into cold water, without cracking. Secondly, fibres of this 
 material are not only very strong for their cross-section, but what is very 
 
 FIG. 271. 
 
582 APPENDIX 
 
 much more valuable, they are almost entirely free from torsional elastic 
 fatigue. It is this property which makes quartz fibres so very valuable for 
 supporting galvanometer and electrometer needles, for it is only with this 
 material that the position of the zero does not move after the needle has 
 been deflected. With all other materials there is always a movement of 
 the zero, owing to the imperfect recovery of the fibre from the effects of 
 the torsion to which it has been subjected. 
 
 Since the manipulation necessary to make and mount such quartz 
 fibres as are ordinarily required for suspension purposes in the laboratory 
 is easily acquired, it is hoped that the following hints will encourage all 
 students to familiarise themselves with the processes : 
 
 Since quartz only fuses at a temperature higher than that attainable 
 with the ordinary air-gas blow-pipe, it is necessary to use either an 
 oxygen and coal-gas or oxygen and hydrogen jet. Personally, the author 
 prefers hydrogen to coal-gas, since a smaller and hotter name can be 
 obtained. The coal-gas, however, has cheapness to recommend it. The 
 
 blow-pipe to be used is 
 such as is employed for 
 the limelight, and what 
 is called a mixed jet is 
 to be preferred. In this 
 type of jet the gases are 
 mixed in a small chamber 
 before they reach the 
 jet. A suitable form of 
 jet is shown in Fig. 272. 
 The bottle containing 
 FIG. 272. . the oxygen, also the one 
 
 containing the hydrogen, 
 
 if hydrogen is used, should be fitted with a pressure regulator, so that 
 the amount of gas in the flame can be regulated by the taps ^on the 
 jet. If no pressure regulators are used, then the adjustment must be 
 performed by means of the cocks on the bottles, for if the taps on the jet 
 are used, the rubber tubing will burst. 
 
 The supply of the gases must be so adjusted that a narrow flame 
 about 20 cm. long is obtained which emits only a very slight noise so 
 long as the jet is at rest, but gives a roar when the jet is moved fairly 
 rapidly from side to side. 
 
 A pair of very dark smoked spectacles will be required, since the hot 
 quartz is much too bright to be looked at with the naked eye. It is 
 advisable to remember that although the flame is quite invisible through 
 such glasses, yet it is still there, and a very nasty burn will result if the 
 fingers are inadvertently placed in the flame. 
 
 The most troublesome process when working fused silica is to fuse 
 the quartz into sticks, since when a crystal of quartz is heated it 
 immediately flies into small fragments. By far the cheapest method of ob- 
 taining sticks of fused quartz is to buy them from Messrs. Baird and 
 
APPENDIX 583 
 
 Tatlock of 14 Cross Street, Hatton Garden, London, since the cost in gas 
 of preparing such sticks will probably amount to more than the price. 1 
 If, however, it is desired to prepare the sticks of fused quartz, the 
 following method, described by Shenstone, will be found fairly easy : 
 Small pieces of pure clear quartz are placed in a platinum crucible, and 
 heated over an ordinary blow-pipe to a bright red. The quartz is then 
 tipped into a large beaker containing cold water, and, as a result of the 
 innumerable small cracks produced by the sudden cooling, acquires a 
 milk white colour. Two fairly large pieces of this milky quartz are held 
 in two small forceps, and the edges are heated in oxy-hydrogen flame and 
 brought into contact. They adhere together, and then holding one piece 
 in the forceps the other is melted, or at any rate partly melted. A 
 third lump is then added, and this is melted, and so on. When about an 
 inch of very rough rod has been built up in this way, heat the rod in 
 the middle strongly, and then slowly draw it out, forming a fairly 
 uniform rod having a diameter of about 2 mm. Cut this rod in half, 
 and then using these smooth portions as handles, bring the rough ends 
 together in the flame, and work the quartz, by alternate heating and 
 drawing down, to a uniform rod. Next cut this rod in half, and fuse a 
 fresh lump of milky quartz to one end of one half, and then fuse on the 
 other half of the rod and work up the fresh quartz. Proceeding in this 
 way, with a little practice very fairly uniform rods of the fused silica 
 can be built up. Having once got started, always add fresh quartz by 
 cutting the stick and inserting the new material in the middle. In this 
 way the portions of the stick already prepared serve as handles. 
 
 Before starting to make quartz fibres it will be necessary to prepare 
 some frames on which to 
 
 store the fibres. A con- | 
 
 venient form of frame is f^ y~l O* Q 
 
 shown in Fig. 273, and for J ^ 
 
 ordinary purposes may con- 
 veniently be 'about 30 centi- FIG. 273. 
 metres long. Four or five 
 
 of these frames may be stored together in a wooden box, into which 
 they fit, each frame resting on the one below, the raised sides of the 
 frames keeping the ends, to which the fibres are attached by shellac 
 varnish, from touching. 
 
 The bow and arrow used to draw out the fibre are shown in Fig. 274. 
 The support A for the arrow consists of a piece of wood 25 centimetres 
 long, with a narrow groove running along its upper surface. The bow 
 itself, BC, consists of a piece of straight-grained yellow pine 108 centi- 
 metres long. The section at the middle is a square of 1*2 centimetre 
 side, and the section gradually tapers down to a circle of about 0*8 centi- 
 metre in diameter at the ends. The string consists of a piece of ordinary 
 thin twine, and is of such a length that when the bow is unbent, 
 
 1 The present price is about 20s. per ounce. A quarter of an ounce will make 
 all the fibres required in a laboratory for very many years. 
 
584 
 
 APPENDIX 
 
 the string is almost tight. The trigger D consists of a U-shaped 
 piece of brass wire, the limbs of the U passing through two small 
 holes in the end of A, as shown in the section at (a). This trigger is 
 attached by a piece of string to a treadle board E, so that by pressing 
 the foot on E the trigger is pulled down, and the string of the bow, 
 
 which was held by the 
 portions of the U which 
 project above A, is re- 
 leased. 
 
 The arrow (/>) consists 
 of a piece of straw F about 
 13 centimetres long, having 
 an ordinary sewing-needle 
 G cemented into one end 
 by means of sealing-wax. 
 The other end H is plugged 
 (a) with a little sealing-wax, 
 
 and a notch is burnt to 
 receive the string of the 
 bow. A piece of the fused 
 quartz i, about 8 centi- 
 metres long and 1 to 2 
 millimetres in diameter, 
 is cemented to the arrow 
 FIG. 274. with sealing - wax, as 
 
 shown. 
 
 When shooting an arrow, the base A of the bow is held in a small 
 vice fixed to a table, and the trigger D being in place, the bow is bent. 
 The arrow is then placed in the groove with the notch against the 
 string between the two uprights of the trigger. Then holding the end i 
 of the quartz in the left hand, and the oxy-hydrogen jet in the right, 
 direct the flame on the point J of the quartz. The jet ought to be nearly 
 horizontal and at right angles to the length of the quartz. When the 
 quartz is seen to have become quite soft, press down the treadle E, 
 at the same instant raising the flame and tightening your grasp on the 
 end of the quartz. If the operations have been correctly timed, the 
 arrow will be shot out, and the heated portion of the quartz will be 
 drawn out into a fine fibre. 
 
 The thickness of the fibre depends in a measure on the thickness of 
 the original quartz rod, but chiefly on the temperature to which the 
 quartz was raised when the arrow was fired. Practice will soon indicate 
 when to fire in order to obtain fibres of any desired thickness. 
 
 Having produced a fibre, its collection and mounting presents some 
 little difficulty. To assist in the collection it will be found advisable to 
 set up a board at about 6 to 8 metres from the bow, into which board 
 the arrow will embed itself. Then between the bow and the board place 
 a number of chairs in pairs, each pair to be back to back, and separated 
 
APPENDIX 585 
 
 by about a metre. Between the backs of each pair of chairs place a 
 clean length of glass tubing, the tube being at right angles to the direc- 
 tion of motion of the arrow, and about a foot below the path along 
 which the arrow will travel. These glass tubes serve to keep the fibre 
 off the floor. 
 
 If a fairly thick fibre has been produced, it will probably stretch from 
 the arrow to the piece of quartz left in the hand. In such a case break 
 oft' the fibre from the quartz in your hand, and fix a small piece of 
 gummed paper to the end of the fibre. This can be performed by fold- 
 ing a piece of gummed paper, and having placed the fibre in the fold, 
 pressing the sides together. Having in this way marked one end of 
 the fibre, take one of the frames (Fig. 273) and coat the ends A and B with 
 shellac varnish. Then going to the arrow, break off the fibre, and place 
 the end against the varnish near one side of the frame. The frame 
 must now be slowly turned so as to wind on the fibre, during which 
 process the paper attached at the free end will serve as a weight to keep 
 the fibre taut. 
 
 If the fibre is fine, it will probably have broken near the bow, and a 
 somewhat different procedure is necessary. Take a piece of glass tube 
 or rod about 30 centimetres long, and starting at the arrow, allow the 
 fibre to run over the rod as you move slowly towards the bow. Watch 
 the fibre carefully so as to see when you get near the end. When the 
 glass rod is about a foot from the end, take a folded piece of gummed 
 paper between the finger and thumb of the left hand, and catch the fibre 
 near the end in the fold. Having attached the paper, drop it, and start- 
 ing at the arrow, proceed as described above to wind the fibre on a 
 frame. 
 
 To attach a quartz fibre to, say, a galvanometer needle system, first 
 measure the length the fibre has to be, and then draw a line on a clean 
 sheet of paper, and on this line make two transverse marks, showing the 
 length the fibre is to have. Having selected a frame containing a fibre 
 of suitable diameter, attach a piece of gummed paper to one end, and 
 then break the fibre between this paper and the end. Next grasp the 
 fibre near the fixed end between the finger and thumb of the left hand, 
 and pull it away from the frame. Take the pin to which the top of the 
 fibre is to be attached and put a spot of shellac varnish on the end, 
 placing the pin at the edge of the table, with the point projecting over 
 the edge of the table. The above must be done without letting go the 
 fibre, or may be done before the fibre is detached from the frame. Next 
 taking a sewing-needle in the right hand, guide the end of the fibre, 
 which will be seen projecting above the thumb and finger of the left 
 hand, with the needle so that it touches the shellac. The shellac varnish 
 must now be heated by holding a piece of heated copper wire below, the 
 heating being continued till the varnish froths up. When the heating is 
 complete, release the fibre by opening the fingers, so that the piece of 
 paper at the end hangs down. liaise the pin to which the fibre is now 
 attached, and gently draw the paper at the free end over the sheet of 
 
586 APPENDIX 
 
 paper on which the length of the suspension fibre has been marked. 
 Stop when the end of the pin is opposite the further mark, the fibre 
 lying along the longitudinal line. Now draw a sharp knife firmly across 
 the fibre at about half a centimetre beyond the mark which shows the 
 length of the fibre. Having placed a drop of shellac varnish on the end 
 of the wire which carries the magnets and mirror, support this system 
 with the end projecting over the edge of the table, and taking the pin to 
 which the fibre is attached in the left hand, cause the end of the fibre to 
 touch the varnish. A needle held in the right hand may be used to 
 direct the fibre. Then boil off" the alcohol from the varnish, as was done 
 before. 
 
 Having attached the fibre, prop the needle system up almost vertical 
 before attempting to lift it by means of the fibre, for quartz fibres will 
 not support a bending stress. 
 
 If difficulty is found in seeing the fibres, it will be advisable to either 
 work over a piece of clean mirror, or to work at a table before a window, 
 the table being fitted with a drawer. If the drawer is opened a few 
 inches and a piece of black velvet is placed in it, a dark background will 
 be provided against which it will be found easy to see the fibres. 
 
 For ordinary galvanometer suspensions, fibres having diameters from 
 002 to '006 millimetre will be found suitable. 
 
 For further particulars of the methods of manipulating quartz fibres 
 the reader must be referred to the following : On Laboratory Arts, 
 Threlfall ; Boys, Philosophical Magazine, p. 498, June 1887, and p. 463, 
 1894. 
 
 3. Making Divided Scales. Divided scales on glass can best be 
 made by etching with hydrofluoric acid. The surface of the glass must 
 be protected with some kind of varnish, the division lines being drawn 
 through the coating of varnish. One of the most satisfactory varnishes 
 is obtained by diluting Brunswick black with its own volume of benzene. 
 The glass must be thoroughly cleaned and the varnish poured on the 
 surface, being made to flow evenly all over by suitably tilting the glass. 
 When every part of the surface has been coated, allow as much of the 
 varnish as will drain off in half a minute to flow back into the bottle, 
 and then lay the glass on a horizontal surface. After from ten to twenty 
 minutes the varnish will have become tacky, and the drawing of the 
 division lines may be proceeded with. The varnish will remain 
 sufficiently soft for several hours. If, however, for any reason the 
 division lines are not drawn within an hour or two of coating the glass, 
 it will be advisable to dissolve off the varnish with benzene and give the 
 glass a fresh coating. 
 
 In the case of a tube, the varnish must be applied by means of a 
 fairly large soft camel's hair brush. 
 
 Having made the division lines by one of the methods described 
 below, cut a piece of clean white blotting-paper a little larger than the 
 divided portion of the surface, and lay this paper over the divisions. 
 Then by means of a straw moisten the paper with hydrofluoric acid, 
 
APPENDIX 587 
 
 taking care that no air bubbles are enclosed between the paper and the 
 glass. The exclusion of air bubbles will be facilitated if the hydrofluoric 
 acid is applied regularly, starting from one end of the paper. The acid 
 must be left on from one to two minutes, the exact time depending 
 on the strength of the acid and the character of the division lines 
 required. It is always advisable to make a trial on a waste piece of 
 glass, for which purpose several groups of three or four division lines 
 may be ruled, the acid being left on one group for half a minute, on the 
 next for one minute, and so on. 
 
 To make the division lines and figures visible, rub over with some 
 black paint, to which a little varnish has been added. Allow the paint to 
 remain on for a few minutes, and then rub off the excess with ordinary 
 writing paper. When removing the excess of paint, only rub in a 
 direction perpendicular to the division lines. 
 
 A similar process can be employed to etch scales on brass, the 
 hydrofluoric acid being replaced by dilute nitric acid. 
 
 In place of using the Brunswick black, beeswax may be used ; it 
 will, however, be found very much more difficult to obtain a uniform 
 coating with the wax. The same remark applies even more to paraffin 
 wax, so that on the whole it will be found to save time in the long run 
 to use Brunswick black. 
 
 If very fine division lines are required on glass, then it will be 
 advisable to rule them. A splinter of diamond is generally employed 
 for the purpose, but a small crystal of carborundum (carborundum 
 crystals only cost a few shillings the pound) will be found to give much 
 better lines. With a diamond the edges of the lines are very apt to 
 be splintered. If a very light pressure is used with carborundum, the 
 edges of the lines will be quite sharp and free from all splintering. 
 
 If a dividing engine is available, then making the graduations 
 presents no difficulty. The tool used to make the scratches on the 
 varnish is a sharp point of steel, fairly light pressure being used to 
 press the point against the work. 
 
 In the absence of the dividing engine, a screw-cutting lathe may be 
 employed to make scales which are correct to about a twentieth of a 
 millimetre, so long as the leading screw of the lathe is a good one. It 
 will be advisable to fit the traversing carriage of the lathe with an 
 arrangement to carry the tracing point by which the length of the 
 divisions can be adjusted, and the fifth and tenth divisions are auto- 
 matically made longer than the other graduations. With a leading screw 
 having eight threads to the inch, the following change wheels give a very 
 close approximation to a millimetre per turn : Headstock, 28 ; stud, 
 32, 36; screw, 100. 
 
 A scale may be copied by hand by means of the arrangement shown 
 in Fig. 275. The scale to be copied, AB, must be one with fairly deep 
 division marks, preferably of steel. It is fixed to a block of wood about 
 2 cm. high, this block being fixed to a base board c about 130 cm. long. 
 The glass to be divided, DE, is cemented to the base board, or held down 
 
588 
 
 APPENDIX 
 
 by clips, so that the line along which the graduations are to be made is a 
 prolongation of the scale AB. To transfer the graduations from the scale 
 to the glass, a beam compass FG is employed. This beam compass consists 
 of a piece of light brass tube about 70 cm. long and 1 *5 cm. in diameter. 
 At either end is soldered a small tube at right angles to the axis of FG, 
 and through these cross tubes pass two steel rods, the rods being held in 
 place by set screws. Both the steel rods are pointed at the lower end, 
 the rod H serving as a scriber to make the graduations on the glass. The 
 end of the rod i is placed in the division lines of the scale AB. Attached 
 to the tube FG, and alongside the tube which carries I, is a brass rod KL, 
 bent as shown. The side of this rod rests against the edge of the scale, 
 and thus ensures that the point of i always occupies the same relative 
 position along the division lines. The end L of the rod is always allowed 
 to rest against the top of the base board c, and i is so adjusted that when 
 
 
 FIG. 275. 
 
 its point is on the scale, and L rests on the base, the tracing point H is 
 vertical. It will be found of assistance to have a knob on the top of I, 
 and to hold this knob in the left hand. A knob G is fixed to the end of 
 the tube FG and serves as a handle, by means of which the scribing point 
 is guided with the right hand. 
 
 Starting with the point at a whole centimetre division near the left- 
 hand extremity of the scale, a division line is made by lightly drawing 
 G towards one. The point i is then moved to the next division to the 
 right, H being lifted off the glass, and a second division line is made, and 
 so on. Some little practice will be required before the lines will be 
 obtained of uniform width, and also equally spaced. A very light pressure 
 must be exerted with either hand, and no attempt must be made to guide 
 the point. It will be advisable to fix the base board c at about 20 cm. 
 from the edge of a table, and to rest the elbows of both arms on the 
 table top. 
 
APPENDIX 
 
 589 
 
 An arrangement suitable for dividing short scales, and which can 
 easily be fitted up in the laboratory, is shown in Fig. 276. A round 
 steel rod A is fixed about 1 cm. above a base board, and there are 
 two carriages, B and c, which can slide on this rod, the rod passing 
 through a split tube D (Fig. 276, a). This split tube can be pinched 
 together by the milled-headed screw E, so that the carriage is clamped to 
 the rod A. A point F on each carriage rests on a piece of plate glass G, 
 so that the upper surfaces of the carriages are horizontal. On the top of 
 carriage c is clamped a micrometer screw-gauge H, while on B is screwed 
 an arm K, which has a steel sphere J at the end, this sphere coming 
 between the jaws of the screw-gauge. The plate on which the scale is to 
 
 be divided is fixed to the carriage B, while the scriber L, used to make the 
 division lines, consists of a pointed steel rod attached to the end of a long 
 rod which is pivoted at M. 
 
 To use this instrument the carriages are adjusted so that the sphere J 
 is in contact with the left-hand jaw of H, and the screw is brought up to 
 touch the sphere. The jaws are then opened by an amount equal to the 
 distance between the division lines on the scale which is to be divided. 
 Starting with both carriages in such a position that the scriber is at the 
 left-hand end of the plate on which the scale is to be divided, clamp 
 carriage B, and bring carriage c as far to the left as the right-hand jaw of 
 the micrometer will allow, and clamp. Then rule the first graduation line 
 with the scriber. Next unclamp B and move it to the left as far as the 
 left-hand jaw of the micrometer will allow, and clamp. Rule the second 
 line. Unclamp c and move to the left, then clamp. Unclamp B, move 
 to the left, clamp and rule a division line, and so on. In this way 
 
590 APPENDIX 
 
 the scale is produced by a step by step process, the length of the step 
 being regulated by the distance through which the sphere J can travel 
 between the jaws of the micrometer. 
 
 Such a dividing engine is unsuited for any but quite short scales, and 
 it is more difficult to produce uniform results that would perhaps at first 
 sight be imagined. It will, however, be found of considerable service 
 when making verniers, the scale on which the vernier is to be used being 
 preferably made by the beam-compass method. 
 
 4. Silvering Glass. There are a number of methods of silvering 
 glass, each of which in practised hands is capable of giving good 
 results. The following method, due to Brasheur, will, however, prob- 
 ably be found to give the most consistent results, and has the further 
 advantage that some of the solutions employed will keep, and hence a stock 
 can be made at one time. 
 
 The following solutions will be required : 
 
 . A. The reducing solution 
 
 Loaf sugar or sugar candy . . . .90 grams. 
 Strong, pure nitric acid (sp. gr. 1*22) . . 4 c.c. 
 Alcohol (absolute) . . . . .175 c.c. 
 Distilled water 1000 c.c. 
 
 The sugar is dissolved in the water, and then the alcohol and nitric 
 acid are added. The solution will require to be kept for at least a week 
 before it is used, and must be kept in a stoppered bottle. 
 
 B. Silver solution 
 
 Silver nitrate ...... 1 gram. 
 
 Distilled water 100 c.c. 
 
 C. Ammonia solution 
 
 Ammonia (sp. gr. '880) . . . . 1 c.c. 
 
 Distilled water . . . . . . 10 c.c. 
 
 D. Potash solution 
 
 Caustic potash (prepared by alcohol) . . 0'5 gram. 
 Distilled water . . . . . . 50 c.c. 
 
 Solutions B and C will keep for some months if contained in well stoppered 
 bottles and kept in the dnrk. Solution D will only keep for a few 
 weeks. 
 
 The above amounts will suffice for silvering a mirror having an area 
 of about 50 square cm. if used in a dish such that most of the silver is 
 deposited on the mirror, and not on the sides of the dish. In smaller or 
 larger mirrors, proportional quantities of the ingredients can be taken. 
 
 To prepare the solution for silvering, take about four-fifths of the 
 silver solution tt and place it in a beaker which has been washed out 
 with strong nitric acid and then with distilled water, and add the 
 ammonia solution C drop by drop, stirring all the while, till the pre- 
 
APPENDIX 591 
 
 cipitate is partly dissolved, the solution being a light brown colour. 
 Then very carefully add drop after drop of the ammonia, stirring for 
 nearly a minute between each drop, till the solution attains a light straw 
 colour. If by accident too much ammonia is added, so that after stirring 
 for some time the solution becomes colourless, add a drop or two of the 
 silver nitrate solution. It is essential to success that no excess of 
 ammonia is present. 
 
 Next add the potash solution D, and then having diluted the 
 remainder of the ammonia to about twice its bulk, add this solution 
 drop by drop, till the precipitate formed when the potash was added is 
 nearly all dissolved. A few particles of the precipitate may refuse to 
 dissolve ; these should be neglected, and the addition of ammonia stopped 
 when the bulk of the liquid becomes colourless. Now add some of the 
 reserve silver nitrate solution drop by drop till the solution, even after 
 stirring for two or three minutes, does not become clear, but remains of 
 a light brown or yellow colour. The solution may then be filtered, and 
 is ready for use. 
 
 The glass to be silvered must be cleaned in the following manner. 
 Since to obtain a satisfactory deposit it is essential that the glass surface 
 should be perfectly clean, the student is advised to follow out the 
 following directions implicitly : 
 
 1. Remove any wax or oil on the glass by means of turpentine. 
 
 2. Dissolve off any pre-existing silver with nitric acid, and then 
 
 rinse in water. 
 
 3. Immerse in a strong solution of caustic potash, and rub the 
 
 surface to be coated with a swab of cotton wool tied to the end 
 of a glass rod. Care must be taken not to scratch the surface 
 with the glass rod. The cotton wool should be carefully tied 
 over the end of the rod. Rinse in water. 
 
 4. Immerse in strong nitric acid, and again rub the surface with a 
 
 swab, as in 3. 
 
 5. Wash thoroughly with distilled water, giving the surface a rub 
 
 with the swab during the first few washings. 
 
 6. Leave the objects covered with distilled water till they are wanted 
 
 for silvering. 
 
 The objects to be silvered can either be placed at the bottom of a 
 glass dish, the surfaces to be silvered being turned upwards, or they can 
 be attached by a speck of wax at the back to a plate of glass, the glass 
 resting on the top of the vessel containing the solution, so that the 
 surfaces to be coated dip below the surface of the solution. In this 
 case the silver is deposited upwards. The advantage of this latter 
 arrangement is that any solid specks in the solution do not settle on 
 surfaces being silvered, and hence do not produce holes in the film, as 
 occurs when the surfaces are at the bottom. 
 
 The glass having been cleaned, take the solution prepared as described 
 above and add 6 c.c. of the reducing solution A for each gram of silver 
 nitrate taken, and having well mixed the solutions, pour over the 
 
592 APPENDIX 
 
 surfaces to be silvered. The solution will first turn almost black, and 
 then the silver will start depositing, and the solution will gradually turn 
 first to a light brown, and then after fifteen to thirty minutes it will 
 become almost clear, when the deposition is complete. The rate at which 
 the deposition goes on depends in a great measure on the temperature, and 
 it is an advantage' to work in a room at a temperature of about 20 C. 
 If the temperature is much below 20, it is advisable to heat the solution 
 to 20 before adding the reduced solution. 
 
 After the deposition is complete remove the glass, being careful not 
 to touch the silvered surface, and thoroughly wash with distilled water. 
 Then stand the mirror on its edge on a piece of clean filter paper, and 
 allow the film to become thoroughly dry. 
 
 If the mirror is to be used with the silvering on the back of the 
 glass, the film of silver will require protecting by a coat of varnish. 
 The best varnish to use is gold lacquer, as thin a coating as possible 
 being employed. 
 
 If the silver film has to be polished, so as to obtain reflection direct 
 from the silver, as will be the case with the mirrors used in Michelson's 
 interferometer, prepare two or three pads by stuffing some pieces of clean 
 soft chamois leather with cotton wool. First lightly rub the silver film 
 with one of the pads, using circular strokes. Then place a piece of 
 rouge about the size of a pea on a clean sheet of paper, and rub one of 
 the pads on the paper, so as to work the rouge well into the leather. 
 The mirror is then polished with this pad, using light circular strokes. 
 
 When silvering, always remember (1) an excess of ammonia must be 
 avoided, and (2) the surfaces to be silvered must be absolutely clean. 
 
 5. Mounting Cross-wires in Telescopes and Microscopes. An 
 operation which occurs with some frequency in a physical laboratory is 
 the replacement of broken cross-wires in the eye-pieces of microscopes 
 and telescopes. In the case of low power eye-pieces, and where it is 
 not necessary to read very accurately, such as for use with the telescope 
 and scale method of measuring rotations, cross-wires made of a single 
 thread of silk are quite satisfactory. 
 
 The ring on which the cross-wires are mounted having been removed 
 
 from the telescope, the remains of the cement used to attach the old 
 
 cross-wires is dissolved off with alcohol or amyl-acetate. In general, two 
 
 lines at right angles, used to define the positions of the 
 
 A wires, will be found scribed on the ring. If no such 
 
 C / f- lines exist, it will be advisable to put them on, using 
 
 CZ^> < \' a sharp needle as a scriber. 
 
 The satisfactory attachment of the fibre will be 
 very much facilitated if a piece of brass wire is bent 
 
 Fro. 277. into the shape shown in Fig. 277, the distance between 
 
 the arms AB being about twice the diameter of the 
 
 ring on which the fibre is to be mounted. Having selected a fibre of 
 
 unspun silk, place it on a piece of black velvet, or on a piece of mirror 
 
 glass, and by means of a needle and a pair of forceps, proceed to split 
 
APPENDIX 593 
 
 it into the two fibres of which each single thread generally consists. 
 Having separated a sufficient length of fibre, coat the end of one of the 
 prongs of the bent wire with a little solution of celluloid in amyl-acetate. 
 Such solution is sold as a transparent varnish for coating silver goods, 
 and is known as silico enamel. In a few seconds the varnish will 
 become tacky, when the end of the wire may be pressed against the end 
 of the silk fibre. The fibre having become attached to the cement, give 
 the wire a turn or two, so as to wind the fibre round the prong. Next 
 coat the other prong with the varnish, and having slightly pressed 
 the two prongs together with the left hand, wind the silk fibre round 
 the second prong. After a few seconds, when the varnish has dried, the 
 pressure on the prongs may be released, when the spring of the wires 
 will keep the fibre taut. 
 
 Next gently lay the fibre in place on the ring, allowing the end c of 
 the wire frame to rest on the table, the other end being supported by the 
 fibre as it rests on the top of the ring. Then with a pin or very small 
 brush place a small drop of the varnish at either end of the fibre, and allow 
 the arrangement to remain till the varnish is quite hard. When the varnish 
 has set, the superfluous fibre can be cut away with a sharp pair of scissors. 
 
 The second cross- wire is cemented in place in exactly the same way, 
 the lines ruled on the ring being used as a guide to ensure that the two 
 fibres are at right angles. 
 
 In place of silk, very fine nickel wire may be used for cross-wires. 
 Such wire, having a diameter of 0'02 millimetre, can be obtained from 
 Messrs. Moul & Co., 7678 York Street, Westminster, London, and costs 
 one shilling per metre. 
 
 For use in the eye-piece of a micrometer, silk cross-wires are in 
 general too coarse, and no better material seems to have been dis- 
 covered than spider web. The most suitable web is that produced by 
 the Diadema spider, a spider with a large body and comparatively short 
 legs. In the absence of the Diadema, any spider can be made to give 
 fairly satisfactory threads. 
 
 Having obtained a spider, he can generally be persuaded to make 
 a thread by gently tilting him off a piece of card, the thread being 
 left attached to the card as the spider falls. This thread is then 
 wound on a light wire frame of the shape shown in Fig. 277. In the 
 case of a fresh spun web, it will be unnecessary to use any varnish to 
 attach the thread to the wire. Having obtained the thread on the wire, 
 the attachment to the ring is performed in exactly the same manner as in 
 the case of the silk; 
 
 Quartz fibres are quite unsuited for cross-wires, as it is almost im- 
 possible to keep them sufficiently tight and yet not break them when 
 replacing the ring in the microscope. 
 
 6. On the Use of Manganine Wire for the Construction of Resist- 
 ance Coils. The alloy of copper, nickel, and manganese, called man- 
 ganine, is very suitable for the construction of resistance coils, since it has 
 an extremely small temperature coefficient, and is fairly cheap. 
 
 2 P 
 
594 
 
 APPENDIX 
 
 In order that a resistance made of manganine may retain its resistance 
 unaltered, the following must be attended to : 
 
 1. The wire must be attached to the terminals by means of silver 
 
 solder. The tin in ordinary soft solder appears to exert an 
 influence on the wire, causing its resistance to alter. 
 
 2. .The wire, after being wound in the coil, must be annealed at a 
 
 temperature of 140 C. for at least twenty-four hours. 
 
 3. The wire must be protected from the air by being coated with 
 
 shellac. 
 
 The following table gives suitable sizes of wire for coils of various 
 resistance : 
 
 Resistance. 
 
 Size of Wire. 
 
 Maximum Current. 
 
 Ohms. 
 
 Standard Wire-Gauge. 
 
 Amperes. 
 
 o-i 
 
 Two of 16 in parallel. 
 
 32 
 
 1 
 
 19 
 
 10 
 
 10 
 
 25 
 
 3-2 
 
 50 
 
 27 
 
 1-5 
 
 100 
 
 33 
 
 1-0 
 
 500 
 
 37 
 
 0-4 
 
 1000 
 
 38 
 
 o-i 
 
 In the last column are given the maximum currents which ought to 
 be passed through the coils if of the form described below, when they are 
 
 immersed in an oil bath. 
 Where measurements of pre- 
 cision are made, the currents 
 ought, however, not to exceed 
 a tenth of the values given. 
 
 The form of coil recom- 
 mended by the Reichsanstalt 
 is shown in Fig. 278. The 
 wire, double-silk covered, is 
 wound on a brass cylinder 
 A, which is covered with a 
 layer of silk, the silk having 
 been thoroughly coated with 
 shellac varnish before the 
 wire is wound on. The ends 
 of the wire are silver soldered 
 to small copper discs, and 
 FIG. 278. these discs are attached by 
 
 screws and soft solder to the 
 
 ends of the thick copper rods B and c. The rods are fastened to a 
 thick disc of ebonite, in which a central hole D is pierced for the passage 
 
APPENDIX 595 
 
 of a thermometer. A length of brass tube B, open at the bottom, serves 
 to protect the coil from blows. 
 
 When making a coil, sufficient wire is cut off to give a resistance about 
 4 per cent, more than the resistance it is wished to construct, 1 and the 
 silk having been removed for about 4 centimetres from one end, the wire 
 is thoroughly cleaned with sand-paper. A small copper disc, 1 centimetre 
 diameter and about 4 millimetres thick, has a central hole bored, and a V 
 notch filed across one face. The notched face is covered with a paste of 
 borax and water, as is also the end of the wire. The disc is placed on a 
 block of charcoal, a small piece of silver solder (silver 2 parts, brass 1 part) 
 is put in the notch, and the disc is heated with a blow-pipe till the 
 solder is quite fluid. The end of the wire is then placed in the notch, 
 and almost simultaneously the flame is removed. The disc must be made 
 sufficiently hot before, the wire is put in place, so that it is not neces- 
 sary to continue heating with the flame. If the wire is heated in the 
 flame the manganese will be burnt out, and the properties of the wire 
 will be altered. Immediately the solder has solidified, plunge the disc in 
 water. Then clean off the borax, and polish up the side opposite to that 
 on which the wire is attached, and give it a coating of soft solder with an 
 ordinary soldering iron. The wire must then be doubled on itself and 
 wound uniformly on the brass cylinder A, being held in place by a few 
 turns of silk. Next give the wire a thorough coating of shellac varnish, 
 made by dissolving selected shellac in absolute alcohol, and place the coil 
 in an air bath heated to 140 C., leaving it there for twenty-four hours. 
 During the annealing it is advisable to give the coil two or three addi- 
 tional coats of varnish. 
 
 The coil having been annealed, its resistance must be adjusted to the 
 desired value. Screw the copper disc which is attached to one end of 
 the coil to one of the copper connectors (B or c), and then heat the 
 junction with a soldering iron, and run a little soft solder between the 
 disc and the rod, care being taken to get no solder on the face of the disc 
 to which the manganine wire is attached. 
 
 The other end of the wire, having been well cleaned with sand-paper, 
 is clamped between two copper discs, which are screwed to the end of the 
 other connector, and the resistance of the coil is compared with a standard 
 coil. The wire is adjusted between the copper discs till the resistance 
 is a little greater than the desired value. The position is marked, which 
 may most conveniently be done by sharply bending the wire at the edge 
 of the discs. This end of the wire is then soldered to a copper disc 
 exactly as was the other end, and the resistance of the coil is carefully 
 measured, and ought to be a very little too great. Next calculate by 
 what lenytfi the coil would have to be reduced to give the correct resist- 
 ance, and then measure off three times this length from one end, and 
 having bared the wire at the point, solder another piece of the same wire 
 at this point. This wire must be attached with silver solder, and the 
 
 1 The resistance of a manganine wire decreases from 2 to 3 per cent, when the 
 wire is annealed. 
 
596 APPENDIX 
 
 best method of procedure is as follows : Having carefully cleaned the 
 portions of wire to be joined, give the additional piece two turns round 
 the other wire, and place a little powdered borax on the joint. Then 
 heat a small bead of solder, together with a little borax, on a piece of 
 charcoal, and when the solder is heated well above the melting point, 
 place the junction on the bead, removing the flame immediately the 
 solder flows. A very small flame ought to be used when making this 
 joint. Next adjust the length of the added wire so that the resistance of 
 the whole coil, with this wire shunting the end portion of the main wire, 
 is a very little too small, and then attach a copper disc to the added wire 
 with silver solder. The two copper discs must now be screwed to the 
 connector, and the junctions between the discs made good with soft 
 solder. If the operations have been successfully performed, the resist- 
 ance of the coil will now be slightly too small, and the added wire is then 
 reduced in diameter by careful filing till the resistance is correct. 
 
 In the case of coils wound with fine wire, it will be found easiest to 
 attach a portion of thick wire in series with the fine wire, and then 
 having adjusted the length till the coil has a slightly too low resistance, 
 to adjust finally by filing the thick wire. 
 
 The above descriptions apply to the construction of a manganine 
 resistance coil where the utmost accuracy in adjustment and permanence 
 is desired. For many purposes in the laboratory such refinement is 
 unnecessary. In such a case, having cut off a length slightly in excess 
 of that required to give the desired resistance, coil the wire into a double 
 coil, coat it with shellac, and anneal at 140. The annealing even may 
 be omitted, in which case the coils will slowly change in resistance, and 
 a fairly frequent recalibration will be necessary. The coils are then 
 attached to the copper connectors or the contact blocks, as the case may 
 be, with soft solder, and adjusted by altering the length ; for which 
 purpose the wire is drawn through the solder, which is made fluid by 
 placing a hot soldering iron in the neighbourhood. 
 
APPENDIX 597 
 
 TABLE II. 
 
 Approximate Formulae. 
 
 If a, b, c, and d are small compared to unity, then to a first 
 approximation we have 
 
 1. (la)(l6)(lc). .. = 1 
 
 2. (la) 2 =l2a. 
 
 3. (1 a) n = 1 ?za. 
 
 4. ^/(l a) = 1 - 
 
 5. 
 
 1 
 
 6 = 1 + a. 
 
 la 
 
 7 ' (1 a)* = [ + 2' 
 1 =l+7za. 
 
 ^ ) = l + 
 
 10. e= 
 
 11. =1 +a log e ic. 
 
 12. lo 
 
 13. sin (6 a) = sin a cos ^. 
 
 14. cos (9 a) = cos 6a sin ^. 
 
 In 13 and 14 a is measured in radians, and if it does not exceed 2, 
 the above expressions are correct to three decimal places, and if a does 
 not exceed 30', to four places. 
 
598 
 
 APPENDIX 
 
 TABLE III. 
 Reduction of Weighings to Vacuo. 
 
 If D is the density of the body, and 8 is that of the weights, and 
 a- the density of the air, the correction to the weight of the body is 
 
 wo- ( - i\ where w is the apparent weight of the body. The following 
 \D *./ 
 
 table gives the values of the factors ^(jrj^j) calculated b J using tne 
 
 mean value *0012 for o-. The tabulated number multiplied by w in 
 grams gives the correction in milligrams, and is to be added to the 
 apparent weight if the sign is positive : 
 
 
 Correction in Milligrams. 
 
 Density of Body. 
 D. 
 
 Brass Weights. 
 = 8-4. 
 
 Platinum Weights. 
 8-21-5. 
 
 Quartz or Aluminium 
 Weights. 
 5 = 2-65. 
 
 50 
 
 -i-2'26 
 
 + 2-34 
 
 + 1-95 
 
 55 
 
 204 
 
 2-13 
 
 1-73 
 
 60 
 
 1-86 
 
 1-94 
 
 1-55 
 
 65 
 
 1-70 
 
 1-79 
 
 1-39 
 
 70 
 
 1-57 
 
 1-66 
 
 1-26 
 
 75 
 
 1-46 
 
 1-55 
 
 1-15 
 
 80 
 
 1-36 
 
 1-44 
 
 1-05 
 
 85 
 
 1-27 
 
 1-36 
 
 96 
 
 90 
 
 1-19 
 
 1-28 
 
 88 
 
 95 
 
 1-12 
 
 1-21 
 
 81 
 
 10 
 
 1-06 
 
 1-14 
 
 75 
 
 11 
 
 95 
 
 1-04 
 
 64 
 
 1-2 
 
 86 
 
 94 
 
 55 
 
 1-3 
 
 78 
 
 87 
 
 47 
 
 1-4 
 
 71 
 
 80 
 
 40 
 
 1-5 
 
 66 
 
 75 
 
 35 
 
 1-6 
 
 61 
 
 69 
 
 30 
 
 1-7 
 
 56 
 
 65 
 
 25 
 
 1-8 
 
 52 
 
 62 
 
 21 
 
 1-9 
 
 49 
 
 58 
 
 18 
 
 2-0 
 
 46 
 
 54 
 
 15 
 
 2-5 
 
 *34 
 
 43 
 
 + 03 
 
 3-0 
 
 26 
 
 34 
 
 -05 
 
 3-5 
 
 20 
 
 29 
 
 11 
 
 4-0 
 
 16 
 
 24 
 
 15 
 
 5-0 
 
 10 
 
 19 
 
 21 
 
 6-0 
 
 06 
 
 14 
 
 25 
 
 8-0 
 
 + 01 
 
 09 
 
 30 
 
 10-0 
 
 -02 
 
 06 
 
 33 
 
 15-0 
 
 06 
 
 03 
 
 37 
 
 20-0 
 
 -08 
 
 004 
 
 39 
 
APPENDIX 
 
 TABLE IV. 
 Density of Dry Air at different Temperatures and Pressures. 
 
 Tempera- 
 ture. 
 
 Pressure in Millimetres. 
 
 Proportional Parts. 
 
 720 
 
 730 
 
 740 
 
 750 
 
 760 
 
 770 
 
 
 
 o-oo 
 
 1225 
 
 o-oo 
 
 1242 
 
 o-oo 
 
 1259 
 
 o-oo 
 
 1276 
 
 o-oo 
 
 1293 
 
 o-oo 
 
 1310 
 
 17 
 
 1 mm. 2 
 2 =3 
 3 =5 
 
 4 =7 
 
 50 
 
 4 
 
 o-i=o 
 
 2=1 
 3=1 
 4 = 2 
 
 1 
 
 2 
 
 3 
 4 
 
 1221 
 1216 
 1212 
 1207 
 
 1237 
 1233 
 
 1228 
 1224 
 
 1254 
 1250 
 1245 
 1241 
 
 1271 
 1267 
 1262 
 1258 
 
 1288 
 1284 
 1279 
 1274 
 
 1305 
 1301 
 1296 
 1291 
 
 5 
 
 1203 
 
 1220 
 
 1236 
 
 1253 
 
 1270 
 
 1286 
 
 6 =10 
 
 5 Z 
 
 6 = 2 
 
 6 
 
 7 
 8 
 9 
 
 1199 
 1194 
 1190 
 1186 
 
 1215 
 1211 
 1207 
 1202 
 
 1232 
 1227 
 1223 
 1219 
 
 1249 
 1244 
 1240 
 1235 
 
 1265 
 1261 
 1256 
 1252 
 
 1282 
 1277 
 1273 
 1268 
 
 \.& 
 8 =14 
 9 =15 
 
 16 
 
 1 mm. = 2 
 2 =3 
 3 =5 
 4 =6 
 5 =8 
 6 =10 
 7 =11 
 8 =13 
 9 =14 
 
 15 
 
 1 mm.= 1 
 2 =3 
 3 =4 
 
 16 
 
 8 = 3 
 9 = 4 
 
 5 
 
 0-1 = 1 
 2=1 
 3 = 2 
 4-2 
 
 10 
 
 11 N 81 
 
 1198 
 
 1214 
 
 1231 
 
 1247 
 
 1264 
 
 11 
 12 
 13 
 14 
 
 1177 
 1173 
 1169 
 1165 
 
 1194 
 1190 
 1185 
 1181 
 
 1210 
 1206 
 1202 
 1198 
 
 1227 
 1222 
 1218 
 1214 
 
 1243 
 1238 
 1234 
 1230 
 
 1259 
 1255 
 1250 
 1246 
 
 15 
 
 1161 
 
 1177 
 
 1193 
 
 1209 
 
 1226 
 
 1242 
 
 1237 
 1233 
 1229 
 1225 
 
 1220 
 
 5 = 3 
 6 = 3 
 7 = 4 
 
 8 = 4 
 9 = 5 
 
 16 
 17 
 18 
 19 
 
 1157 
 1153 
 1149 
 1145 
 
 1173 
 11(59 
 11(55 
 1161 
 
 1189 
 1185 
 1181 
 1177 
 
 1205 
 1201 
 1197 
 1193 
 
 1221 
 1217 
 1213 
 1209 
 
 20 
 
 1141 
 
 1157 
 
 1173 
 
 1189 
 
 1205 
 
 21 
 22 
 23 
 24 
 
 1137 
 1133 
 1130 
 1126 
 
 1153 
 1149 
 1145 
 1141 
 
 1169 
 1165 
 1161 
 1157 
 
 1185 
 1181 
 1177 
 1173 
 
 1200 
 1196 
 1192 
 1188 
 
 1216 
 1212 
 1208 
 1204 
 
 25 
 
 1122 
 
 1138 
 
 1153 
 
 1169 
 
 1184 
 
 1200 
 
 o 
 5 =8 
 
 26 
 27 
 28 
 
 29 
 
 1118 
 1115 
 1111 
 1107 
 
 1134 
 1130 
 1126 
 1123 
 
 1149 
 1146 
 1142 
 1138 
 
 1165 
 1161 
 1157 
 1153 
 
 1180 
 1176 
 1173 
 1169 
 
 1196 
 1192 
 1188 
 1184 
 
 y 
 7 =11 
 
 8 =12 
 9 =13 
 
 30 
 
 1103 
 
 1119 
 
 1134 
 
 1149 
 
 1165 
 
 1180 
 
600 
 
 APPENDIX 
 
 TABLE V. Density of Water at various Temperatures measured 
 on the Hydrogen Scale. 
 
 Degrees 
 
 Tenths. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 0-99987 
 
 87 
 
 88 
 
 89 
 
 89 
 
 90 
 
 90 
 
 91 
 
 92 
 
 92 
 
 1 
 
 2 
 3 
 4 
 
 93 
 
 97 
 99 
 1-00000 
 
 93 
 97 
 99 
 00 
 
 94 
 
 97 
 99 
 00 
 
 94 
 
 98 
 00 
 00 
 
 95 
 98 
 00 
 00 
 
 95 
 98 
 00 
 00 
 
 95 
 99 
 00 
 00 
 
 96 
 99 
 00 
 00 
 
 96 
 99 
 00 
 00 
 
 97 
 99 
 00 
 00 
 
 5 
 
 0-99999 
 
 99 
 
 97 
 93 
 
 87 
 80 
 
 99 
 
 99 
 
 98 
 
 98 
 
 98 
 
 98 
 
 97 
 
 97 
 
 6 
 
 7 
 8 
 9 
 
 97 
 
 93 
 
 88 
 81 
 
 96 
 92 
 86 
 79 
 
 96 
 92 
 
 86 
 79 
 
 95 
 91 
 
 85 
 
 78 
 
 95 
 90 
 
 84 
 
 77 
 
 95 
 90 
 
 84 
 76 
 
 94 
 
 89 
 83 
 75 
 
 94 
 89 
 
 82 
 
 74 
 
 93 
 
 88 
 82 
 74 
 
 10 
 
 73 
 
 72 
 
 71 
 
 70 
 
 69 
 
 68 
 
 67 
 
 66 
 
 65 
 
 64 
 
 11 
 12 
 13 
 14 
 
 63 
 53 
 40 
 27 
 
 62 
 51 
 39 
 
 26 
 
 61 
 50 
 38 
 24 
 
 60 
 49 
 37 
 23 
 
 59 
 
 48 
 35 
 22 
 
 58 
 47 
 34 
 20 
 
 57 
 45 
 33 
 19 
 
 56 
 44 
 31 
 17 
 
 55 
 43 
 30 
 16 
 
 54 
 42 
 29 
 14 
 
 15 
 
 13 
 
 11 
 
 10 
 
 08 
 
 06 
 
 05 
 
 03 
 
 02 
 
 00 
 
 99 
 
 16 
 17 
 18 
 19 
 
 20 
 
 0-99897 
 80 
 62 
 43 
 
 95 
 
 78 
 60 
 41 
 
 94 
 77 
 59 
 39 
 
 92 
 75 
 57 
 37 
 
 90 
 73 
 55 
 35 
 
 15 
 
 89 
 71 
 53 
 33 
 
 87 
 70 
 51 
 31 
 
 85 
 68 
 49 
 2<) 
 
 84 
 66 
 47 
 27 
 
 82 
 64 
 45 
 25 
 
 23 
 
 21 
 
 19 
 
 17 
 
 13 
 
 91 
 
 68 
 45 
 20 
 
 11 
 
 08 
 
 06 
 
 04 
 
 21 
 22 
 23 
 24 
 
 02 
 0-99780 
 56 
 32 
 
 00 
 77 
 54 
 30 
 
 98 
 75 
 
 52 
 
 27 
 
 95 
 73 
 49 
 25 
 
 93 
 71 
 47 
 22 
 
 89" 
 66 
 42 
 17 
 
 86 
 64 
 40 
 15 
 
 84 
 61 
 37 
 12 
 
 8~2 
 59 
 35 
 10 
 
 25 
 
 07 
 
 05 
 
 02 
 
 99 
 
 97 
 
 94 
 
 92 
 
 89 
 
 86 
 
 84 
 
 26 
 27 
 28 
 29 
 
 0-99681 
 54 
 26 
 99597 
 
 78 
 51 
 23 
 94 
 
 76 
 48 
 20 
 91 
 
 73 
 
 46 
 17 
 
 88 
 
 70 
 43 
 15 
 
 85 
 
 67 
 40 
 12 
 
 82 
 
 65 
 37 
 09 
 79 
 
 62 
 34 
 06 
 76 
 
 59 
 32 
 03 
 73 
 
 57 
 
 29 
 00 
 70 
 
 30 
 
 67 
 
 64 
 
 61 
 
 31 
 99 
 66 
 33 
 
 58 
 
 55 
 
 52 
 
 49 
 
 46 
 
 43 
 
 40 
 
 31 
 
 32 
 33 
 34 
 
 37 
 
 05 
 0-99473 
 40 
 
 34 
 02 
 70 
 36 
 
 02 
 
 27 
 96 
 63 
 30 
 
 24 
 92 
 
 60 
 26 
 
 21 
 89 
 56 
 23 
 
 18 
 86 
 53 
 20 
 
 15 
 83 
 50 
 16 
 
 12 
 79 
 46 
 13 
 
 08 
 76 
 43 
 09 
 
 35 
 
 06 
 
 9l) 
 
 95 
 
 92 
 
 89 
 
 85 
 
 82 
 
 78 
 
 75 
 
APPENDIX 601 
 
 TABLE VI. Correction for Buoyancy in Density Measurements. 
 
 If D' is the density uncorrected for buoyancy, then the corrected density is 
 D' + <r(l - D'), and in the following table are given the values of the correction 
 term <r(l - D') for various values of D', the mean value '0012 being assumed for a: 
 
 D' 
 
 Correction. 
 
 D' 
 
 Correction. 
 
 Proportional 
 Parts. 
 
 5 
 
 + "0006 
 
 12-0 
 
 - -0132 
 
 0012 
 
 1-0 
 
 
 14-0 
 
 - -0156 
 
 2-0 
 
 - -0012 
 
 16-0 
 
 - -0180 -1 = -0001 
 
 3-0 
 
 - -0024 
 
 18-0 
 
 -0204 "2 = '0002 
 
 4-0 
 
 - -0036 
 
 20-0 
 
 -0228 '3 ='0004 
 
 5-0 
 
 - -0048 
 
 
 
 4 = -0005 
 
 6-0 
 
 - '0060 
 
 
 
 5 = -0006 
 
 7-0 
 
 - -0072 
 
 
 
 6 = -0007 
 
 8-0 
 
 - -0084 
 
 
 
 7 = '0008 
 
 9-0 
 
 - -0096 
 
 
 
 8 = -0010 
 
 10-0 
 
 - -0108 
 
 
 
 9 = -0011 
 
 1 
 
 
 
 
 
 TABLE VII. Moments of Inertia. 
 
 The following table contains the squares of the radii of gyration, and these 
 numbers multiplied by the mass of the body will give the moment of inertia : 
 
 Shape of Body. 
 
 Position of Axis about 
 which Rotation takes 
 place. 
 
 Square of 
 Radius 
 of Gyration. 
 
 
 Solid cylinder 
 
 Perpendicular to axis 
 of cylinder through 
 centre of gravity 
 
 12 + 4 
 
 r = radius. 
 = length. 
 
 Solid cylinder 
 
 Coincident with axis 
 of cylinder 
 
 2 
 
 r radius. 
 
 Hollow cylinder . 
 Hollow cylinder . 
 
 Coincident with axis 
 of cylinder 
 
 Through centre of 
 gravity and per- 
 pendicular to axis 
 of cylinder 
 
 Jp + r * 
 
 1R external radius. 
 ,r = internal radius. 
 l/i = length. 
 
 2 
 
 *-*-+& 
 
 Sphere 
 
 Through centre 
 
 2r 2 
 
 T 
 
 r = radius. 
 
 Hollow sphere 
 
 Through centre 
 
 3 7i? r 3 
 
 R = external radius. 
 r = internal radius. 
 
 Rectangular bar . 
 
 Through centre of 
 two opposite faces 
 
 ~12~ 
 
 a and b are adjacent 
 edges of the faces 
 through which 
 the axis passes. 
 
 Anchor ring 
 Anchor ring 
 
 About a diameter 
 
 Through centre, and 
 perpendicular to 
 plane of ring 
 
 A )2 + 5r 2 
 
 + 4 
 
 R = radius of anchor 
 ring. 
 
 tr = radius of circu- 
 lar cross section. 
 
 If the square of the radius of gyration about a given axis is tr 2 , then the square 
 
TABLE VIII. ReducMon of Periodic, Time to infinitely small Arc, 
 
 and Correction on Account of Chronometer Rate. 
 Reduction to infinitely small Arc. If 6 and 6' are the initial and final 
 
 f\/\ 
 
 amplitudes of the vibration, the correction is . 
 
 16 
 
 Amplitude at 
 
 Amplitude at End in Minutes of Arc. 
 
 Commencement 
 
 
 in Minutes 
 
 
 
 
 
 
 
 of Arc. 
 
 80' 
 
 70' 
 
 60' 
 
 50' 
 
 40' 
 
 30' 
 
 100' 
 
 0-00004 
 
 0-00004 
 
 0-00003 
 
 0-00003 
 
 0-00002 
 
 0-00002 
 
 90 
 
 00004 
 
 00003 
 
 00003 
 
 00002 
 
 00002 
 
 00001 
 
 80 
 
 00003 
 
 00003 
 
 00003 
 
 00002 
 
 00002 
 
 00001 
 
 70 
 
 
 00003 
 
 00002 
 
 00002 
 
 00001 
 
 00001 
 
 60 
 
 
 
 00002 
 
 00002 
 
 00001 
 
 00001 
 
 50 
 
 
 
 
 00001 
 
 00001 
 
 00001 
 
 
 
 
 
 
 00001 
 
 00001 
 
 Correction for Chronometer Rate. If s is the daily rate of the chronometer, 
 
 the factor required is 1 - - . 
 86400 
 
 Daily Rate, 
 Seconds. 
 
 Chronometer 
 Gaming. 
 
 Chronometer 
 Losing. 
 
 5 
 
 0-99994 
 
 1-00006 
 
 10 
 
 88 
 
 12 
 
 15 
 
 83 
 
 17 
 
 20 
 
 77 
 
 23 
 
 25 
 
 71 
 
 29 
 
 30 
 
 65 
 
 35 
 
 35 
 
 59 
 
 41 
 
 40 
 
 54 
 
 46 
 
 45 
 
 48 
 
 52 
 
 50 
 
 42 
 
 58 
 
 
 
 
 TABLE IX. Correction of the Sun's or a Star's Apparent Altitude 
 to allow for Parallax and Refraction. 
 
 The correction is to be subtracted, and is the difference between the refraction 
 and the parallax. 
 
 Apparent 
 Altitude. 
 
 Correction. 
 
 Apparent 
 Altitude. 
 
 Correction. 
 
 20 
 
 2' 30" 
 
 32 
 
 1' 25" 
 
 21 
 
 2 22 
 
 34 
 
 1 19 
 
 22 
 
 2 15 
 
 36 
 
 1 13 
 
 23 
 
 2 8 
 
 38 
 
 1 7 
 
 24 
 
 2 2 
 
 40 
 
 1 3 
 
 25 
 
 1 56 
 
 45 
 
 52 
 
 26 
 
 1 51 
 
 50 
 
 43 
 
 27 
 
 1 46 
 
 55 
 
 36 
 
 28 
 
 1 41 
 
 60 
 
 29 
 
 29 
 
 1 37 
 
 70 
 
 18 
 
 30 
 
 1 33 
 
 80 
 
 9 
 
APPENDIX 603 
 
 TABLE ^.Reduction of Barometer Readings to C. 
 
 In the following table are given the corrections in millimetres to be 
 
 applied to reduce the measured height of the barometer to what it would 
 
 be if the mercury and scale were both at C., that is, the values of the 
 
 (-) 
 
 quantity i- 
 
 tne correction in every case being subtractive : 
 
 Temperature 
 
 C. 
 
 Glass Scale. 
 
 Brass Scale. 
 
 !i 
 
 Uncorrected Height. 
 
 rncorrected Height. 
 
 740 
 
 750 
 
 mm. 
 13 
 26 
 39 
 52 
 
 760 
 
 mm. 
 13 
 26 
 39 
 53 
 
 770 
 
 mm. 
 13 
 27 
 40 
 53 
 
 780 
 
 mm. 
 13 
 27 
 41 
 54 
 
 740 
 
 750 
 
 760 
 
 770 
 
 780 
 
 1 
 2 
 3 
 4 
 
 mm. 
 13 
 26 
 38 
 51 
 
 mm. 
 12 
 24 
 36 
 
 48 
 
 mm. 
 12 
 25 
 37 
 49 
 
 61 
 
 73 
 
 86 
 98 
 110 
 
 mm. 
 12 
 25 
 37 
 50 
 
 mm. 
 13 
 25 
 38 
 50 
 
 mm. 
 13 
 25 
 38 
 51 
 
 12 
 01=-01 
 2= '02 
 3 ='03 
 
 5 
 
 64 
 
 65 
 
 66 
 
 67 
 
 68 
 
 60 
 
 62 
 
 63 
 
 64 
 
 4 (JO 
 
 . '5= -06 
 6 =-07 
 
 7 ='08 
 8= 10 
 9=11 
 
 13 
 
 oi=-oi 
 
 2= -03 
 3 ='04 
 
 6 
 7 
 8 
 9 
 
 77 
 90 
 1-02 
 1-15 
 
 78 
 91 
 1-04 
 1-17 
 
 79 
 92 
 1-05 
 1-18 
 
 80 
 93 
 1-07 
 1-20 
 
 81 
 95 
 1-08 
 1-21 
 
 72 
 
 85 
 97 
 1-09 
 
 '74 
 87 
 99 
 112 
 
 75 
 88 
 1-01 
 113 
 
 76 
 89 
 1-02 
 115 
 
 10 
 
 1-28 
 
 1-30 
 
 1-31 
 
 1-33 
 
 1-35 
 
 1-21 
 
 1-22 
 
 124 
 
 1-26 
 
 1-27 
 
 11 
 12 
 13 
 14 
 
 1-41 
 1-53 
 1T>6 
 1-79 
 
 1-43 
 1-56 
 1-69 
 1-81 
 
 1-45 
 1-58 
 1-71 
 1-84 
 
 1-46 
 1-60 
 1-73 
 1-86 
 
 1-48 
 1-62 
 1-75 
 1-89 
 
 1-33 
 1-45 
 1-57 
 1-69 
 
 1-35 
 1-47 
 1-59 
 1-71 
 
 1-36 
 1'49 
 1-61 
 1-73 
 
 1-38 
 1-51 
 1-63 
 1-76 
 
 1-40 
 1-53 
 1-65 
 
 178 
 
 15 
 
 1-92 
 
 1-94 
 
 1-97 
 
 2-00 
 
 2-02 
 
 1-81 
 
 1-83 
 
 1-86 
 
 1-88 
 
 1-91 
 
 5 =-07 
 fi -n& 
 
 16 
 17 
 18 
 19 
 
 2-05 
 2-17 
 2-30 
 2-43 
 
 2-07 
 2-20 
 2-33 
 246 
 
 2-10 
 2-23 
 2-36 
 2-49 
 
 2-13 
 2-26 
 2-39 
 2-53 
 
 2-16 
 2'29 
 2-43 
 2-56 
 
 1-93 
 2-05 
 2-17 
 2-29 
 
 1-96 
 2-08 
 2-20 
 2'32 
 
 1-98 
 210 
 2-23 
 2-35 
 
 2-01 
 213 
 
 2-26 
 2-38 
 
 2-03 
 216 
 219 
 2-41 
 
 7 ='09 
 8=10 
 9=12 
 
 14 
 
 01='01 
 2= -03 
 3= -04 
 
 4 'Oft 
 
 20 
 
 21 
 
 22 
 23 
 24 
 
 2-56 
 
 2-59 
 
 2-62 
 
 2-66 
 
 2'69 
 
 2-41 
 
 2-44 
 
 2'47 
 
 2'51 
 
 2-54 
 
 2-68 
 2-81 
 2-94 
 3-06 
 
 272 
 2-85 
 2-98 
 311 
 
 2-76 
 
 2-89 
 3-02 
 315 
 
 3-28 
 
 2'79 
 2-92 
 3-06 
 319 
 
 2-83 
 2'96 
 3-10 
 3'23 
 
 2-53 
 2-65 
 
 2-77 
 2-89 
 
 2'56 
 2-69 
 2-81 
 2-93 
 
 2-60 
 2-72 
 
 2-84 
 2-97 
 
 2-63 
 2-76 
 2-88 
 3-01 
 
 2'67 
 2-79 
 2'92 
 3-05 
 
 25 
 
 319 
 
 3-23 
 
 3-32 
 
 3-36 
 
 3-50 
 3-63 
 3-77 
 3-90 
 
 3-01 
 
 3-13 
 3-25 
 3-37 
 3-49 
 
 3-05 
 
 ,V09 
 
 313 
 
 317 
 
 5= -07 
 fi -OS 
 
 26 
 27 
 28 
 29 
 
 30 
 
 3-32 
 3-45 
 3-57 
 370 
 
 3'36 
 3-49 
 3-62 
 3-75 
 
 3-41 
 3-54 
 3-67 
 
 3-80 
 
 3-45 
 3'59 
 3-72 
 3-85 
 
 317 
 3-29 
 3-41 
 3-54 
 
 3-21 
 3-34 
 3-46 
 3-58 
 
 3-26 
 3'38 
 3'51 
 3'63 
 
 3-30 
 3-42 
 3'55 
 3-H8 
 
 7=10 
 8=11 
 9=13 
 
 3-83 
 
 3-88 
 
 3-93 
 
 3-98 
 
 4-03 
 
 3-61 
 
 3-66 
 
 371 
 
 3'75 
 
 3-80 
 
 31 
 32 
 33 
 
 34 
 
 3-95 
 
 4-08 
 4-21 
 4-33 
 
 4-01 
 
 4-14 
 4-26 
 4-39 
 
 4-06 
 419 
 4-32 
 4-45 
 
 411 
 4-25 
 4-38 
 4-51 
 
 4-17 
 4-30 
 4'43 
 4-57 
 
 3-73 
 3-85 
 3-97 
 4-09 
 
 3'78 
 3-90 
 4-02 
 414 
 
 3'83 
 3'95 
 4-07 
 4'20 
 
 3'88 
 4-00 
 413 
 4-25 
 
 3-93 
 4-05 
 418 
 4-31 
 
 35 
 
 4-46 
 
 4-52 
 
 4'58 
 
 4-65 
 
 4'71 
 
 4-21 
 
 4-20 
 
 4-32 
 
 4-38 
 
 4-43 
 
DU4: 
 
 TABLE XI. 
 
 Correction to the Height of the Barometer to allow for the 
 effect of Capillarity. 
 
 The correction, which is given in millimetres, has always to be 
 added to the observed height. 
 
 Diameter 
 of Tube, 
 mm. 
 
 Height of the Meniscus in Millimetres. 
 
 4 
 
 6 
 
 8 
 
 1-0 
 
 1-2 
 
 1-4 
 
 1-6 
 
 1-8 
 
 4 
 
 83 
 
 1-22 
 
 1-54 
 
 1-98 
 
 2-37 
 
 
 
 
 5 
 
 47 
 
 65 
 
 86 
 
 1-19 
 
 1-45 
 
 1-80 
 
 
 
 6 
 
 27 
 
 41 
 
 56 
 
 78 
 
 98 
 
 1-21 
 
 1-43 
 
 
 7 
 
 18 
 
 28 
 
 40 
 
 53 
 
 67 
 
 82 
 
 97 
 
 1-13 
 
 8 
 
 
 20 
 
 29 
 
 38 
 
 46 
 
 56 
 
 65 
 
 77 
 
 9 
 
 
 15 
 
 21 
 
 28 
 
 33 
 
 40 
 
 46 
 
 52 
 
 10 
 
 
 
 15 
 
 20 
 
 25 
 
 29 
 
 33 
 
 37 
 
 11 
 
 
 
 10 
 
 14 
 
 18 
 
 21 
 
 24 
 
 27 
 
 12 
 
 
 
 07 
 
 10 
 
 13 
 
 15 
 
 18 
 
 19 
 
 13 
 
 
 
 
 07 
 
 10 
 
 12 
 
 13 
 
 14 
 
APPENDIX 
 
 TABLE XII. 
 
 Reduction of the Volume of a Gas to Standard Pressure. 
 
 73 
 
 The table gives the values of the logarithms of the factors - for 
 
 760 
 
 different values of B, the height of the barometer at the time the 
 volume of the gas was measured. 
 
 B 
 
 mm. 
 
 Los m 
 
 Proportional Parts. 
 
 B 
 
 mm. 
 
 ^m 
 
 720 
 
 1-97652 
 
 60 57 
 
 750 
 
 1-99425 
 
 1 
 
 2 
 3 
 4 
 
 97712 
 97772 
 
 97832 
 
 97892 
 
 1 mm. = 6 '1 mm. = 5'7 
 2 =12 -2 =11-4 
 3 =18 '3 =17-1 
 4 =24 -4 =22-8 
 
 \ 
 
 3 
 4 
 
 99483 
 99540 
 99598 
 99656 
 
 5 
 
 97952 
 
 5 =30 -5 =28'5 
 
 5 
 
 99713 
 
 6 
 
 7 
 8 
 9 
 
 98012 
 98072 
 98132 
 98191 
 
 6 =36 -6 =34-2 
 7 =42 7 =39-9 
 8 =48 -8 =45-6 
 9 =54 -9 =51-3 
 
 6 
 
 7 
 8 
 9 
 
 99771 
 99828 
 99886 
 99943 
 
 730 
 
 98251 
 
 f)Q Kfi 
 
 760 
 
 o-ooooo 
 
 1 
 2 ' 
 3 
 4 
 
 98310 
 98370 
 98429 
 
 98488 
 
 1 mrn.= 5-9 *1 mm. = 5*6 
 2 =11-8 '2 =11-2 
 3 =17-7 '3 =16-8 
 4 =23-6 -4 =22-4 
 
 1 
 
 2 
 3 
 4 
 
 00057 
 00114 
 00171 
 
 00228 
 
 5 
 
 98547 
 
 5 =29-5 -5 =28-0 
 
 5 
 
 00285 
 
 6 
 
 7 
 8 
 9 
 
 98606 
 98665 
 
 98724 
 98783 
 
 6 =35-4 -6 =33-6 
 7 =41-3 -7 =392 
 8 =47'2 -8 =44-8 
 9 =53-1 -9 =50-4 
 
 6 
 
 7 
 8 
 9 
 
 00342 
 00398 
 00455 
 00511 
 
 740 
 
 98842 
 
 58 
 
 770 
 
 00568 
 
 1 
 
 2 
 3 
 
 4 
 
 98900 
 98959 
 99018 
 99076 
 
 1 mm.= 5-8 
 2 =11-6 
 3 =17-4 
 4 =23-2 
 
 1 
 2 
 3 
 4 
 
 00624 
 00680 
 00737 
 00793 
 
 5 
 
 99134 
 
 5 = 29-0 
 
 5 
 
 00849 
 
 6 
 
 7 
 8 
 9 
 
 99193 
 99251 
 99309 
 99367 
 
 6 ^34-8 
 7 =40-6 
 8 =464 
 9 =52-2 
 
 6 
 
 7 
 8 
 9 
 
 00905 
 00961 
 01017 
 01072 
 
 750 
 
 99425 
 
 
 780 
 
 01128 
 
606 
 
 APPENDIX 
 
 TABLE XIII. 
 
 Values of (I + at) for reducing the Volume of Gases to Standard Tem- 
 perature for Values of t from to 40 C. (a =--0-003670). 
 
 Temperature 
 
 l + at. 
 
 Temperature 
 
 l + at. 
 
 Proportional 
 
 Parts. 
 
 1 
 
 1-00367 
 
 21 
 
 1-07707 
 
 367 
 
 2 
 
 1-00734 
 
 22 
 
 1-08074 
 
 0-l = 36-7 
 
 3 
 
 1-01101 
 
 23 
 
 1-08441 
 
 2 = 73-4 
 
 4 
 
 1-014G8 
 
 24 
 
 1-08808 
 
 3 = 110-1 
 
 5 
 
 1-01835 
 
 25 
 
 1-09175 
 
 4 = 146-8 
 
 6 
 
 1-02202 
 
 26 
 
 1-09542 
 
 5=1835 
 
 7 
 
 1-025G9 
 
 27 
 
 1-09909 
 
 6 = 220-2 
 
 8 
 
 1-02936 
 
 28 
 
 1-10276 
 
 7^256-9 
 
 9 
 
 1-03303 
 
 29 
 
 1-10643 
 
 8 = 293-6 
 
 10 
 
 1-03670 
 
 30 
 
 1-11010 
 
 9=330*3 
 
 11 
 
 1-04037 
 
 31 
 
 1-11377 
 
 
 12 
 
 1-04404 
 
 32 
 
 1-11744 
 
 
 13 
 
 1-04771 
 
 33 ' 
 
 1-12111 
 
 
 14 
 
 1-05138 
 
 34 
 
 1-12478 
 
 
 15 
 
 1-05505 
 
 35 
 
 1-12845 
 
 
 16 
 
 1-05872 
 
 36 
 
 1-13212 
 
 
 17 
 
 1-06239 
 
 37 
 
 1-13579 
 
 
 18 
 
 1-06606 
 
 38 
 
 1-13946 
 
 
 19 
 
 1-06973 
 
 39 
 
 1-14313 
 
 
 20 
 
 1-07340 
 
 40 
 
 1-14680 
 
 
APPENDIX 
 
 607 
 
 TABLE XIV. Tension of Water Vapour and Mass of Water 
 
 in a Cubic Metre of Saturated Air. 
 
 (1) Vapour Tension of Water for every Tenth of a Degree between 
 and 35 G, 
 
 
 Decree. 
 
 
 
 Vapour Tension in Millimetres. 
 
 Tenths. 
 
 o 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 4-57 
 
 4-60 
 
 464 
 
 4-67 
 
 4-70 
 
 474 
 
 4-77 
 
 4-80 
 
 4-84 
 
 4-87 
 
 1 
 
 2 
 3 
 
 4 
 
 4-91 
 5-27 
 5-66 
 0-07 
 
 4-94 
 5-31 
 570 
 6-11 
 
 4-98 
 5-35 
 5-74 
 6-15 
 
 5-02 
 5-39 
 5'78 
 6-20 
 
 5-05 
 5-42 
 
 5-82 
 6-24 
 
 5-09 
 5-46 
 5-86 
 6-28 
 
 5-12 
 5-50 
 5-90 
 6-33 
 
 516 
 5-54 
 5-94 
 6-37 
 
 5-20 
 
 5-58 
 5-99 
 6-42 
 
 5-23 
 5-62 
 6-03 1 
 6-46 
 
 5 
 
 6-51 
 
 6-55 
 
 6-60 
 
 6-64 
 
 6-69 
 
 7-17 
 7-67 
 
 8-21 
 8-78 
 
 6-74 
 
 6-78 
 
 6-83 
 
 6-88 
 
 6-92 
 
 6 
 7 
 8 
 9 
 
 6-97 
 7-47 
 
 7'99 
 8-55 
 
 7-02 
 7-52 
 8-05 
 8-61 
 
 7-07 
 7-57 
 810 
 8-66 
 
 712 
 7'62 
 815 
 
 8-72 
 
 7-22 
 7-72 
 8-27 
 8-84 
 
 7-26 
 
 778 
 8-32 
 8-90 
 
 7-31 
 
 7-83 
 8-38 
 8-96 
 
 7-36 
 7-88 
 8-43 
 9-02 
 
 7-42 
 794 
 8-49 
 9-08 
 
 10 
 
 914 
 
 9-20 
 
 9-26 
 
 9-32 
 
 9-39 
 
 9-45 
 
 9-51 
 
 9-58 
 
 9-64 970 
 
 11 
 12 
 13 
 14 
 
 9-77 
 10-43 
 11-14 
 11-88 
 
 9-83 
 10-50 
 11-21 
 11-96 
 
 9-90 
 10-57 
 11-28 
 12-04 
 
 9-96 
 10-64 
 11-36 
 1212 
 
 10-03 
 10-71 
 11-43 
 12-19 
 
 10-09 
 10-78 
 11-50 
 12-27 
 
 10-16 10-23 10-30 
 10-85 10-92 10-99 
 11-58 11-66 11-73 
 12-35 12-43 ; 12-51 
 
 10-36 
 11-07 
 11-81 
 12-59 
 
 15 
 
 1267 
 
 12-76 
 
 12-84 12-92 13-00 
 
 13-09 
 
 13-17 13-25 13-34 13'42 
 
 16 
 17 
 18 
 19 
 
 13-51 
 14-40 
 15-33 
 16-32 
 
 17'36 
 
 13-60 
 14-49 
 15-43 
 16-42 
 
 17-47 
 
 13-68 13-77 13-86 
 14-58 14-67 14-76 
 15-52 15-62 15-72 
 16-52 16-63 1673 
 
 13-95 
 14-86 
 15-82 
 16-83 
 
 17-91 
 
 14-04 1412 
 14-95 15-04 
 15-92 16-02 
 16-94 17-04 
 
 14-21 
 15-14 
 16-12 
 1715 
 
 14-30 
 15-23 
 16-22 
 17-26 
 
 18-35 
 
 20 
 
 JL7-58 17-69 17 '80 
 
 18-02 1813 
 
 18-24 
 
 21 
 
 22 
 23 
 24 
 
 18-47 
 19-63 
 20-86 
 22-15 
 
 18-58 
 1975 
 20-98 
 22-29 
 
 18-69 18-81 18-92 
 19-87 19-99 20-11 
 21-11 21-24 21-37 
 22-42 22-55 22 '69 
 
 19-04 
 20-24 
 21-50 
 22-83 
 
 1916 
 20-36 
 21-63 
 22-96 
 
 19-27 
 20-48 
 21-76 
 2310 
 
 19-39 
 20-61 
 21-89 
 23-24 
 
 19-51 
 20-73 
 22-02 
 23-38 
 
 25 
 
 23-52 
 
 23-66 
 
 23-80 23-94 24'08 
 
 24-23 
 
 24-37 
 
 24-52 24-66 24 '81 
 
 26 
 27 
 28 
 29 
 
 24-96 
 26-47 
 28-07 
 29-74 
 
 2510 
 26-63 
 
 28-23 
 29-92 
 
 25-25 
 
 26-78 
 28-39 
 30-09 
 
 25-40 25-55 
 26-94 | 27-10 
 28-56 28-73 
 30-26 30-44 
 
 25-70 
 27-26 
 28-89 
 30-62 
 
 25-86 
 27-42 
 29-06 
 30-79 
 
 26-01 26-16 i 26-32 
 27-58 27*74 27-90 
 29-23 29-40 29 '57 
 30-9713115 31-33 
 
 30 
 
 31-51 
 
 31-69 
 
 31-87 32-06 32-24 
 
 32-43 
 
 32-61 32-80 ! 32-99 3318 
 
 31 
 
 32 
 33 
 34 
 
 33-37 
 35-32 
 37-37 
 39-52 
 
 33-56 
 35-52 
 37-58 
 39-74 
 
 33-75 33-94 34 '14 
 35-72 35-92 3613 
 37-79 I 38-00 38 '22 
 39-97 40-19 40-41 
 
 34-33 
 36-33 
 38-43 
 40-64 
 
 34-53 
 36-54 
 38-65 
 40-86 
 
 4319 
 
 34-72 
 36-74 
 38-87 
 41-09 
 
 43-43 
 
 34-92 
 36-95 
 39-08 
 41-32 
 
 3512 
 3716 
 39-30 
 41-55 
 
 35 
 
 4178 
 
 42-02 
 
 42-25 
 
 42-48 42-72 
 
 42-96 
 
 43-67 ; 43-92 
 
 
 o 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 -7 -8 -9 
 
 i 
 
608 
 
 APPENDIX 
 
 TABLE XIV. (continued} (2) Mass of Water contained in a Culric Metre 
 of Air at a Pressure of 76 cm. of Mercury, and Saturated at variou* 
 Temperatures between and 35 C. 
 
 Degrees. 
 
 
 1 
 
 Mass of Water in Grams. 
 
 Tenths. 
 
 9 
 
 o 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 5-07 
 
 8 
 
 
 
 4-84 
 
 4-87 
 
 4-90 
 
 4-93 
 
 4-97 
 
 5-00 
 
 5-04 
 
 511 
 
 514 
 
 1 
 
 2 
 3 
 4 
 
 5 
 
 5-18 
 5-54 
 5-'J2 
 G-33 
 
 5-2L 
 5-57 
 5-96 
 6-37 
 
 5-25 
 5-61 
 6-00 
 6-41 
 
 5-28 
 5-65 
 6-04 
 6-46 
 
 5-32 
 5-69 
 6-08 
 6-50 
 
 5-35 
 5-73 
 6-12 
 6-54 
 
 5-39 
 5-77 
 6-16 
 6-59 
 
 5-43 
 
 5-80 
 6-20 
 6-63 
 
 5'46 
 . 5-84 
 6-25 
 6-67 
 
 5-50 
 5-88 
 6-29 
 6-72 
 
 6-76 
 
 6-81 
 
 6-85 
 
 6-90 
 
 6-94 
 
 6-99 
 
 7-03 
 
 7-08 
 
 712 
 
 717 
 
 6 
 
 7 
 8 
 9 
 
 7-22 
 7-70 
 8-21 
 8-76 
 
 9-33 
 
 7-27 
 7-75 
 8-27 
 8-81 
 
 7-31 
 
 7-80 
 8-32 
 8-87 
 
 7'36 
 
 7-85 
 8-37 
 8-92 
 
 7-41 
 
 7-90 
 8-43 
 898 
 
 7-46 
 
 7-95 
 8-48 
 9-04 
 
 7-51 
 8-01 
 8-54 
 910 
 
 7'55 
 8-06 
 8-59 
 9-15 
 
 7-60 
 8-11 
 8-65 
 9-21 
 
 7-65 
 8-16 
 870 
 9-27 
 
 10 
 
 9-39 
 
 9-45 
 
 9-51 
 
 9-57 
 
 9-63 
 
 9-69 
 
 9-75 
 
 9-81 
 
 9-87 
 
 11 
 
 12 
 13 
 14 
 
 9-93 
 10-57 
 11-25 
 11-96 
 
 10-00 
 
 10-64 
 11-32 
 1203 
 
 10-06 
 10-71 
 11-39 
 12-11 
 
 10-12 
 1077 
 11-46 
 12-18 
 
 10-19 
 10-84 
 11-53 
 12-26 
 
 10-25 
 10-91 
 11-60 
 12-33 
 
 10-31 
 10-97 
 11-67 
 12-41 
 
 10-38 
 11-04 
 11-74 
 
 12-48 
 
 10-44 
 1111 
 11-82 
 12-56 
 
 13-34 
 
 10-51 
 11-18 
 11-89 
 12-64 
 
 15 
 
 12-71 
 
 12-79 
 
 12-87 
 
 12-95 
 
 1302 
 
 13-10 
 
 1318 
 
 13-26 
 
 13-42 
 
 16 
 
 17 
 18 
 19 
 
 20 
 
 13-50 
 14-34 
 15-22 
 16-14 
 
 13-58 
 14-42 
 15-31 
 16-24 
 
 13-67 
 14-51 
 15-40 
 16-33 
 
 13-75 
 14-60 
 15-49 
 16-43 
 
 13-83 
 14-68 
 15-58 
 16-53 
 
 13-92 
 1477 
 15-67 
 16-62 
 
 14-00 
 1 4-8(5 
 15-77 
 1672 
 
 14-08 
 14-95 
 15-86 
 16-82 
 
 14-17 
 15-04 
 15-95 
 16-92 
 
 1425 
 15-13 
 16-05 
 17-02 
 
 17-12 
 
 17-22 
 
 17-32 
 
 17-42 
 
 17-52 
 
 17-62 
 
 17-73 
 
 17-83 
 
 17 93 
 
 18-04 
 
 19-11 
 
 20-24 
 21-42 
 22-67 
 
 23-97 
 
 21 
 22 
 23 
 24 
 
 18-14 
 19-22 
 20-35 
 21-54 
 
 18-25 
 19-33 
 
 20-47 
 21-67 
 
 18-35 
 19-44 
 20-59 
 21-79 
 
 18-46 
 19-55 
 20-70 
 21-91 
 
 18-57 
 19-67 
 20-82 
 22-04 
 
 18-67 
 19-78 
 20-94 
 22-16 
 
 18-78 
 19-89 
 21-06 
 22-29 
 
 18-89 
 20-01 
 21-18 
 22-41 
 
 23-71 
 
 19-00 
 20-12 
 21-30 
 22-54 
 
 25 
 
 22-80 
 
 2411 
 25-49 
 26-93 
 28-45 
 
 22-92 
 
 23-05 
 
 2318 
 
 23-31 
 
 24-65 
 26-06 
 27-53 
 29-07 
 
 23-44 
 
 23-58 
 
 23-84 
 
 26 
 27 
 28 
 29 
 
 24-24 
 25-63 
 
 27-08 
 28-60 
 
 24-38 
 25-77 
 27-23 
 
 28-76 
 
 24-51 
 25-91 
 
 27-38 
 28-92 
 
 24-79 
 26-20 
 27-68 
 29-23 
 
 24-93 
 26-35 
 
 27-83 
 29-39 
 
 25-07 
 26-49 
 27-99 
 29-55 
 
 25-21 
 26-64 
 28-14 
 29-71 
 
 25-35 
 
 26-78 
 28-29 
 29-87 
 
 30 
 
 30-04 
 
 30-20 
 
 30-36 
 
 30-53 
 
 30-69 
 
 30-86 
 
 31-03 
 
 3119 
 
 31-36 
 
 31-53 
 
 31 
 32 
 33 
 34 
 
 31-70 
 33-45 
 35-27 
 3718 
 
 31-87 
 33-63 
 35-46 
 37-38 
 
 32-04 
 33-81 
 35-65 
 37-58 
 
 32-22 
 33-99 
 35-84 
 37-77 
 
 32-39 
 3417 
 36-03 
 37-97 
 
 32-56 
 34-35 
 
 36-22 
 3817 
 
 32-74 
 34-53 
 36-41 
 38-37 
 
 32-91 
 34-72 
 36-60 
 38-57 
 
 33-09 
 34-90 
 36-79 
 
 38-78 
 
 33-27 
 35-09 
 36-99 
 38-98 
 
 35 
 
 3918 
 
 39-39 
 
 39-59 
 
 39-80 
 
 40-01 
 4 
 
 40-22 
 
 40-43 
 
 40-64 
 
 40-85 
 
 41-06 
 
 
 o 
 
 1 
 
 2 
 
 3 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
APPENDIX 
 
 609 
 
 TABLE XV. Boiling Point of Water. 
 
 The following table contains the boiling point of water under different 
 pressures between 700 mm. and 799 mm. The numbers given are due 
 to Wieb (Zeitschrift fur Instrumentenkunde (1893), xiii. 329 ; see also 
 Landolt & Bernstein's Tabelleri). 
 
 I: 
 
 Millimetres. 
 
 1 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 70 
 
 97-714 
 
 753 
 
 792 
 
 832 
 
 871 
 
 910 
 
 949 
 
 989 
 
 r 028 
 
 ^067 
 
 71 
 
 98-106 
 
 145 
 
 184 
 
 223 
 
 261 
 
 300 
 
 339 
 
 378 
 
 416 
 
 455 
 
 72 
 
 98-493 
 
 532 
 
 570 
 
 609 
 
 647 
 
 686 
 
 724 
 
 762 
 
 800 
 
 838 
 
 73 
 
 98-877 
 
 915 
 
 953 
 
 991 
 
 ^029 
 
 W7 
 
 T04 
 
 7 142 
 
 180 
 
 7 2l8" 
 
 74 
 
 99-255 
 
 293 
 
 331 
 
 368 
 
 406 
 
 443 
 
 481 
 
 518 
 
 555 
 
 592 
 
 75 
 
 99-630 
 
 667 
 
 704 
 
 741 
 
 778 
 
 815 
 
 852 
 
 889 
 
 926 
 
 963 
 
 76 
 
 100-000 
 
 037 
 
 074 
 
 110 
 
 147 
 
 184 
 
 220 
 
 257 
 
 293 
 
 330 
 
 77 
 
 100-366 
 
 403 
 
 439 
 
 475 
 
 511 
 
 548 
 
 584 
 
 620 
 
 656 
 
 692 
 
 78 
 
 100-728 
 
 764 
 
 800 
 
 836 
 
 872 
 
 908 
 
 944 
 
 979 
 
 : 0l5 
 
 ^051 
 
 79 
 
 101-087 
 
 122 
 
 158 
 
 193 
 
 229 
 
 264 
 
 300 
 
 335 
 
 370 
 
 406 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 PROPORTIONAL PARTS. 
 
 mm. 
 
 039 
 
 038 
 
 037 
 
 036 
 
 035 
 
 o-i 
 
 004 
 
 004 
 
 004 
 
 004 
 
 004 
 
 0'2 
 
 008 
 
 008 
 
 007 
 
 .007 
 
 007 
 
 0-3 
 
 012 
 
 on 
 
 on 
 
 on 
 
 on 
 
 0-4 
 
 016 
 
 015 
 
 015 
 
 014 
 
 014 
 
 0-5 
 
 020 
 
 019 
 
 019 
 
 018 
 
 018 
 
 6 
 
 023 
 
 023 
 
 022 
 
 022 
 
 021 
 
 0-7 
 
 027 
 
 027 
 
 026 
 
 025 
 
 025 
 
 0-8 
 
 031 
 
 030 
 
 030 
 
 029 
 
 028 
 
 0-9 
 
 035 
 
 034 
 
 033 
 
 032 
 
 032 
 
 
 1 
 
 
 
 
610 APPENDIX 
 
 TABLE XVI. Vapour Pressure of Liquids suitable for Use in 
 Vapour Jackets. 
 
 Carbon Bisulphide. 
 
 Ethyl Alcohol. 
 
 Chlorobenzene. 
 
 Temperature. 
 
 Vapour Pressure 
 mm. 
 
 Temperature. 
 
 Vapour Pressure 
 mm. 
 
 L'emperature. 
 
 Vapour Pressure 
 mm. 
 
 10 
 
 198-5 
 
 45 
 
 172-2 
 
 75 
 
 119-5 
 
 11 
 
 207-0 
 
 46 
 
 181-0 
 
 76 
 
 124-2 
 
 12 
 
 215-8 
 
 47 
 
 190-1 
 
 77 
 
 129-1 
 
 13 
 
 225-0 
 
 48 
 
 199-2 
 
 78 
 
 134-2 
 
 14 
 
 234-4 
 
 49 
 
 209-6 
 
 79 139-4 
 
 15 
 
 244-2 
 
 50 
 
 220-0 
 
 80 
 
 144-8 
 
 16 
 
 254-3 
 
 51 
 
 230-8 
 
 81 
 
 150-3 
 
 17 
 
 265-7 
 
 52 
 
 242-1 
 
 82 
 
 156-1 
 
 18 
 
 275-4 
 
 53 
 
 253-8 
 
 83 
 
 162-0 
 
 19 
 
 286-6 
 
 54 
 
 265-9 
 
 84 
 
 168-0 
 
 20 
 
 298-1 
 
 55 
 
 278-6 
 
 85 174-3 
 
 21 
 
 309-9 
 
 56 
 
 291-9 
 
 86 181-7 
 
 22 
 
 322-1 
 
 57 
 
 305-7 
 
 87 
 
 187-3 
 
 23 
 
 334-7 
 
 58 
 
 320-0 
 
 88 
 
 1941 
 
 24 
 
 347-7 
 
 59 
 
 334-9 
 
 89 
 
 201-2 
 
 25 
 
 361-1 
 
 60 
 
 350-.5 
 
 90 
 
 208-4 
 
 26 
 
 375-0 
 
 61 
 
 356-4 
 
 91 
 
 215-8 
 
 27 
 
 389-2 
 
 62 
 
 3831 
 
 92 
 
 223-5 
 
 28 
 
 403-9 
 
 63 400-4 
 
 93 
 
 221-3 
 
 29 
 
 409-0 
 
 64 418-4 
 
 94 
 
 239-4 
 
 30 
 
 434-6 
 
 65 
 
 437-0 
 
 95 
 
 247-7 
 
 31 
 
 450-7 
 
 66 
 
 456-4 
 
 96 
 
 256-2 
 
 32 
 
 467-2 
 
 67 
 
 476-5 
 
 97 
 
 265-0 
 
 33 
 
 484-2 
 
 68 
 
 497*3 
 
 98 
 
 274-0 
 
 34 
 
 501-7 
 
 69 
 
 518-9 
 
 99 
 
 283-3 
 
 35 
 
 519-7 
 
 70 
 
 541-2 
 
 100 
 
 292-8 
 
 36 
 
 538-2 
 
 71 
 
 564-4 
 
 101 
 
 302-5 
 
 37 
 
 5o7'2 
 
 72 
 
 588-4 
 
 102 
 
 312-5 
 
 38 
 
 576-8 
 
 73 
 
 613-2 
 
 103 
 
 322-8 
 
 39 
 
 596-9 
 
 74 
 
 639-0 
 
 104 
 
 H33-4 
 
 40 
 
 617-5 
 
 75 
 
 665-6 
 
 105 
 
 344-2 
 
 41 
 
 638-7 
 
 76 693-1 
 
 106 
 
 355-3 
 
 42 
 
 660-5 
 
 77 721-6 
 
 107 
 
 366-7 
 
 43 
 
 682-9 
 
 78 
 
 751-0 
 
 108 
 
 378-3 
 
 44 
 
 705-9 
 
 79 
 
 781-5 
 
 109 
 
 390-3 
 
 45 
 
 7295 
 
 
 
 110 
 
 402-6 
 
 46 
 
 753-8 
 
 
 
 111 
 
 415-1 
 
 47 
 
 778-6 
 
 
 
 112 
 
 428-0 
 
 48 
 
 804-1 
 
 
 
 113 
 
 441-2 
 
 49 
 
 830-3 
 
 
 
 114 
 
 456-7 
 
 50 
 
 857-1 
 
 
 
 115 
 
 468-5 
 
 
 
 
 
 116 
 
 482-7 
 
 
 
 
 
 117 497-2 
 
 
 
 
 
 118 512-1 
 
 
 
 
 
 119 527-3 
 
 
 
 
 
 120 542-8 
 
 
 
 
 
 121 558-7 
 
 
 
 
 
 122 5751 
 
 
 
 
 
 123 591-7 
 
 
 
 
 
 124 60S -8 
 
 
 
 
 
 125 626-2 
 
 
 
 
 
 126 
 
 644-0 
 
 
 
 
 
 127 
 
 662-2 
 
 
 
 
 
 128 
 
 680-8 
 
 
 
 
 
 129 699-7 
 
 
 
 
 
 130 719-0 
 
 
 
 
 
 131 
 
 738-7 
 
 
 
 
 
 132 
 
 758-8 
 
APPENDIX 
 
 611 
 
 TABLE XVII. 
 
 Vapour Pressure of Mercury. 
 
 The numbers in the following table were obtained by Ramsay and 
 Young (Journal of the Chemical Society (1886), xlix. p. 37) : 
 
 Temperature. 
 
 Vapour Pressure. 
 mm. 
 
 Temperature. 
 
 Vapour Pressure. 
 mm. 
 
 40 
 
 0-0008 
 
 180 
 
 8-535 
 
 50 
 
 015 
 
 190 
 
 12-137 
 
 60 
 
 029 
 
 200 
 
 17-015 
 
 70 
 
 052 
 
 210 
 
 23-482 
 
 80 
 
 092 
 
 220 
 
 31-957 
 
 90 
 
 160 
 
 230 
 
 42-919 
 
 100 
 
 270 
 
 240 
 
 56-919 
 
 110 
 
 445 
 
 250 
 
 74-592 
 
 120 
 
 719 
 
 260 
 
 96-661 
 
 130 
 
 1-137 
 
 270 
 
 123-905 
 
 140 
 
 1-763 
 
 280 
 
 157-378 
 
 150 
 
 2-684 
 
 290 
 
 198-982 
 
 160 
 
 4-013 
 
 300 
 
 246-704 
 
 170 
 
 5-904 
 
 
 
 TABLE XVIII. 
 Depression of Zero. 
 
 The numbers for verre dur are the means of those obtained by 
 Guillaume, Thiesen, and Schloesser ; those for 16'" the means of those 
 obtained by Thiesen, Schloesser, and Bottcher; those for 56'" were 
 obtained by Thiesen. 
 
 Temperature. 
 
 Verre dur. 
 
 Jena 16'". 
 
 Jena 59"'. 
 
 10 
 
 0-008 
 
 0'005 
 
 0--005 
 
 20 
 
 017 
 
 on 
 
 009 
 
 30 
 
 027 
 
 017 
 
 014 
 
 40 
 
 037 
 
 024 
 
 017 
 
 50 
 
 048 
 
 031 
 
 021 
 
 60 
 
 060 
 
 039 
 
 024 
 
 70 
 
 071 
 
 048 
 
 027 
 
 80 
 
 084 
 
 057 
 
 030 
 
 90 
 
 097 
 
 066 
 
 033 
 
 100 
 
 111 
 
 077 
 
 035 
 
612 
 
 APPENDIX 
 
 TABLE XIX. 
 
 Corrections to reduce Readings on Mercury -in-Glass Thermometers 
 to the Hydrogen or Air Thermometer. 
 
 The numbers for verre dur are from the results published by the 
 Bureau International ; those for the Jena glass from tables published 
 by Griitzmacher in Wied. Annalen (1899), Ixviii. p. 769. The original 
 numbers are, in most cases, given to an additional place. Since, however, 
 different observers have obtained numbers which differ by as much as 
 0'01, it seems unnecessary to include the third place. 
 
 Temperature. 
 
 Verre dur. 
 
 Jena 16"'. 
 
 Jena 59'". 
 
 TH-T v .a. 
 
 T H - 7*16 
 
 T H -T 59 
 
 10 
 
 -05 
 
 -06 
 
 -02 
 
 20 
 
 -08 
 
 -09 
 
 -04 
 
 30 
 
 -10 
 
 -11 
 
 -04 
 
 40 
 
 -11 
 
 -12 
 
 -04 
 
 50 
 
 -10 
 
 -11 
 
 -03 
 
 60 
 
 -09 
 
 -10 
 
 -02 
 
 70 
 
 -07 
 
 -08 
 
 - 01 
 
 80 
 
 -05 
 
 -06 
 
 + 00 
 
 90 
 
 -03 
 
 -03 
 
 00 
 
 
 
 Tair-T u 
 
 Tair - TW 
 
 110 
 
 + 04 
 
 + 03 
 
 + 00 
 
 120 
 
 + 06 
 
 + -05 
 
 02 
 
 130 
 
 + 07 
 
 + 07 
 
 -04 
 
 140 
 
 + 07 
 
 + 09 
 
 - 08 
 
 150 
 
 + 06 
 
 + 10 
 
 -13 
 
 160 
 
 + 03 
 
 + 10 
 
 -19 
 
 170 
 
 + .00 
 
 + 08 
 
 -28 
 
 180 
 
 -04 
 
 + 06 
 
 -39 
 
 190 
 
 -09 
 
 + 02 
 
 -'52 
 
 200 
 
 -13 
 
 - 04 
 
 -67 
 
 220 
 
 
 -21 
 
 
 240 
 
 
 -47 
 
 
 260 
 
 
 -83 
 
 
 280 
 
 
 -1-30 
 
 
 300 
 
 
 -1-91 
 
 
APPENDIX 
 
 613 
 
 TABLE XX. 
 
 Coefficients of Expansion of Water and Mercury. 
 
 The tables contain the volumes of a gram of water or mercury at 
 different temperatures, and allow of the mean coefficient of expansion 
 between any two temperatures being calculated. Thus if v 1 and v 2 are 
 the volumes of a gram at the temperature t 1 and t 2 , the mean coefficient of 
 
 expansion is -2 J~ The numbers given in the table are due to 
 
 *W2 ~~ *l) 
 Thiesen, Scheel, and Diesselhorst (Wiss. Abh. d. Phys. Tech. Rdchsan- 
 
 stalt, vol. ii., 1895 ; vol. iii., 1900; and vol iv., 1904). 
 
 Volume of one Gram of Water. 
 
 Tempera- 
 ture. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 1-00013 
 
 007 
 
 003 
 
 001 
 
 000 
 
 001 
 
 003 
 
 007 
 
 012 
 
 019 
 
 10 
 
 027 
 
 037 
 
 048 
 
 060 
 
 073 
 
 087 
 
 103 
 
 120 
 
 138 
 
 157 
 
 20 
 
 177 
 
 199 
 
 221 
 
 244 
 
 268 
 
 294 
 
 320 
 
 347 
 
 376 
 
 405 
 
 30 
 
 435 
 
 466 
 
 497 
 
 530 
 
 563 
 
 598 
 
 633 
 
 669 
 
 706 
 
 743 
 
 40 
 
 782 
 
 821 
 
 861 
 
 901 
 
 943 
 
 985 
 
 028 
 
 072 
 
 116 
 
 162 
 
 50 
 
 1-01207 
 
 254 
 
 301 
 
 349 
 
 398 
 
 448 
 
 498 
 
 548 
 
 600 
 
 652 
 
 60 
 
 705 
 
 758 
 
 812 
 
 867 
 
 923 
 
 979 
 
 036 
 
 093 
 
 151 
 
 210 
 
 70 
 
 1 -02270 
 
 330 
 
 390 
 
 452 
 
 514 
 
 576 
 
 639 
 
 703 
 
 768 
 
 833 
 
 80 
 
 899 
 
 965 
 
 032 
 
 099 
 
 168 
 
 237 
 
 306 
 
 376 
 
 447 
 
 518 
 
 90 
 
 1 -03590 
 
 663 
 
 736 
 
 810 
 
 884 
 
 959 
 
 035 
 
 111 
 
 188 
 
 265 
 
 100 
 
 1-04343 
 
 422 
 
 501 
 
 
 
 
 
 
 
 
 Volume of one Gram of Mercury. 
 
 Tempera- 
 ture. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 073554 
 
 567 
 
 581 
 
 594 
 
 608 
 
 621 
 
 634 
 
 648 
 
 661 
 
 674 
 
 10 
 
 688 
 
 701 
 
 714 
 
 728 
 
 741 
 
 755 
 
 768 
 
 781 
 
 795 
 
 808 
 
 20 
 
 821 
 
 835 
 
 848 
 
 862 
 
 875 
 
 888 
 
 902 
 
 915 
 
 928 
 
 942 
 
 30 
 
 955 
 
 969 
 
 982 
 
 995 
 
 009 
 
 022 
 
 035 
 
 049 
 
 062 
 
 076 
 
 40 
 
 074089 
 
 102 
 
 116 
 
 129 
 
 143 
 
 156 
 
 169 
 
 183 
 
 196 
 
 210 
 
 50 
 
 223 
 
 236 
 
 250 
 
 263 
 
 277 
 
 290 
 
 304 
 
 317 
 
 330 
 
 344 
 
 60 
 
 357 
 
 371 
 
 384 
 
 398 
 
 411 
 
 424 
 
 438 
 
 452 
 
 465 
 
 478 
 
 70 
 
 492 
 
 505 
 
 519 
 
 532 
 
 545 
 
 559 
 
 572 
 
 586 
 
 599 
 
 613 
 
 80 
 
 626 
 
 640 
 
 653 
 
 667 
 
 680 
 
 694 
 
 707 
 
 721 
 
 734 
 
 748 
 
 90 
 
 761 
 
 775 
 
 788 
 
 802 
 
 815 
 
 829 
 
 842 
 
 856 
 
 869 
 
 883 
 
 100 
 
 896 
 
 
 
 
 
 
 
 
 
 
614 
 
 APPENDIX 
 
 TABLE XXI. 
 
 Specific Heat of Water. 
 
 The numbers in this table are due to Callendar (Phil. Trans. Royal 
 Society (1902), vol. cxcix. p. 144), and are expressed in terms of- the 
 calorie at 20 C. 
 
 Temperature. 
 
 Specific Heat. 
 
 Temperature. 
 
 Specific Heat. 
 
 
 
 1-0094 
 
 50 
 
 9987 
 
 5 
 
 1-0054 
 
 55 
 
 9992 
 
 10 
 
 1-0027 
 
 60 
 
 1-0000 
 
 15 
 
 1-0011 
 
 65 
 
 1-0008 
 
 20 
 
 1-0000 
 
 70 
 
 1-0016 
 
 25 
 
 9992 
 
 75 
 
 1-0024 
 
 30 
 
 9987 
 
 80 
 
 1-0033 
 
 35 
 
 .9983 
 
 85 
 
 1-0043 
 
 40 
 
 9982 
 
 90 
 
 1-0053 
 
 45 
 
 9983 
 
 95 
 
 1-0063 
 
 
 
 100 
 
 1-0074 
 
 TABLE XXII. 
 
 Refractive Indices for Sodium Light (A = 589/x,/u). 
 
 Substance. 
 
 Temperature. 
 
 Refractive Index. 
 
 Fluorspar 
 Iceland spar .... 
 
 Quartz 
 
 18 
 18 
 
 18 
 
 1 '4339 
 1-6584 ordinary. 
 1*4864 extraordinary. 
 1*5442 ordinary. 
 
 Kock-salt 
 
 Water 
 
 18 
 20 
 
 1'5533 extraordinary. 
 1-5443 
 1-3329 
 
 Carbon bisulphide . . 
 Benzene 
 Ethyl alcohol ... 
 
 20 
 21-6 
 20 
 
 1-6277 
 1-5004 
 1-3617 
 
APPENDIX 
 
 615 
 
 TABLE XXIII. 
 
 Wave-lengths. 
 
 The following wave-lengths are in air at a temperature of 15 C. 
 and a pressure of 760 mm. : 
 
 Substance. 
 
 How caused to 
 Emit Light. 
 
 Wave-length in 
 10- 8 cm. 
 
 Colour. 
 
 Sodium .... 
 
 Bunsen flame 
 
 58902 
 
 Orange. 
 
 
 
 5896-2 
 
 M 
 
 Potassium .... 
 
 > 
 
 7668-5 
 
 Red. 
 
 
 
 7701-9 
 
 ?> 
 
 
 
 4044-3 
 
 Violet. 
 
 
 
 4047-4 
 
 
 Lithium .... 
 
 > >? 
 
 6708-2 
 
 Red. 
 
 
 
 6103-8 
 
 Orange. 
 
 Thallium . . 
 
 5> 
 
 5350-7 
 
 Green. 
 
 Strontium .... 
 
 
 4607-5 
 
 Blue. 
 
 Calcium . . . . 
 
 
 4226-9 
 
 
 Rubidium .... 
 
 55 V 
 
 4202-0 
 
 Violet. 
 
 
 
 4215-7 
 
 j? 
 
 Hydrogen .... 
 
 Vacuum tube 
 
 .65630 
 
 Red. 
 
 
 
 4861-5>- 
 
 Blue-green. 
 
 
 
 4340-7 
 
 Violet. 
 
 Helium .... 
 
 .5 
 
 7065-2 
 
 Red. 
 
 
 
 6678-1 
 
 . 
 
 
 
 5875-6 
 
 Yellow. 
 
 
 
 5015-7 
 
 Green. 
 
 
 
 4921-9 
 
 Blue. 
 
 
 
 4713-2 
 
 M 
 
 
 
 4471-5 
 
 Violet. 
 
 Mercury . . 
 
 Mercury lamp 
 
 6232-0 
 
 Red. 
 
 
 
 5790-7 
 
 Yellow. 
 
 
 
 5769-6 
 
 5> 
 
 
 
 5460-7 
 
 Green. 
 
 
 
 4959-7 
 
 Green-blue. 
 
 
 
 4916-4 
 
 Blue. 
 
 
 
 4358-3 
 
 
 
 
 
 4078-1 
 
 Violet. 
 
 
 
 4046-8 
 
 
 Cadmium .... 
 
 Vacuum tube 
 
 6438-5 
 
 Red. 
 
 
 
 5085-8 
 
 Green. 
 
 
 
 4799-9 
 
 Blue. 
 
 Zinc 
 
 
 6362-3 
 
 Red. 
 
 
 
 4810-5 
 
 Blue. 
 
 
 
 4722-2 
 
 M 
 
 
 
 46801 
 
 
 
616 
 
 APPENDIX 
 
 TABLE XXIV. 
 
 Correction to Scale Readings when a Mirror and Scale is used to 
 Measure a Rotation. 
 
 If d is the deflection read off on a scale at a distance D, and is the 
 angle through which the mirror has turned, then we have to a first 
 approximation 
 
 In the following table are given for values of d/D between "01 and *2 
 a quantity 8, such that to within terms in d 5 /IP, we have 
 
 unction. 
 
 d/D='Ql 
 
 02 
 
 -03 
 
 04 
 
 05 
 
 06 
 
 07 -08 
 
 09 
 
 10 
 
 e 
 
 00003 
 
 1 
 00013 '00030 
 
 00053 
 
 00083 
 
 00120 
 
 00163 
 
 00212 
 
 00269 
 
 00331 
 
 Jin . 
 
 00004 
 
 00015 
 
 00034 
 
 00060 
 
 00094 
 
 00135 
 
 00184 
 
 00239 
 
 00302 
 
 00373 
 
 Jin 0/2 
 
 00003 
 
 00014 
 
 00031 
 
 00055 
 
 00086 
 
 00124 
 
 00168 
 
 00219 
 
 00277 
 
 00342 
 
 L'an.0 . 
 
 00002 
 
 00010 
 
 00022 
 
 00040 
 
 00062 
 
 00090 
 
 00122 
 
 00160 
 
 00201 
 
 00249 
 
 
 d/D='ll 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 
 19 
 
 20 
 
 e 
 
 00400 
 
 0047G -00557 
 
 00645 
 
 00740 
 
 00840 
 
 00946 
 
 01059 
 
 01177 
 
 01301 
 
 line . 
 
 00450 
 
 00535 
 
 00627 
 
 00726 
 
 00832 
 
 00944 
 
 01064 
 
 01190 
 
 01323 
 
 01461 
 
 Sin 8/2 
 
 00413 
 
 00491 
 
 00575 
 
 00666 
 
 00763 
 
 00867 
 
 00976 
 
 01092 
 
 01214 
 
 01341 
 
 Fan0 . 
 
 00300 
 
 00357 
 
 00418 
 
 00485 
 
 00556 
 
 00632 
 
 00712 
 
 00797 
 
 00886 
 
 00980 
 
APPENDIX 
 
 61 
 
 TABLE XXV. 
 
 Conductivity of Electrolytes. 
 
 In the table p is the weight . of the anhydrous electrolyte in 
 100 grams of the solution, m is the number of gram molecules of 
 the electrolyte in a litre of the solution, d is the density of the 
 solution at 18 C., and a- is the conductivity of the solution in ohm -1 
 centimetre ~ l at 18 C. 
 
 Substance. 
 
 P 
 
 m 
 
 d 
 
 <rXlO* 
 
 Substance. 
 
 P 
 
 m 
 
 d 
 
 trXlO* 
 
 KC1 
 
 5 
 
 0-691 
 
 1-0308 
 
 690 
 
 ZnSO 4 
 
 15 
 
 2 169 
 
 1-1675 
 
 415 
 
 
 10 
 
 1-427 
 
 1-0638 
 
 1359 
 
 
 20 
 
 3-053 
 
 1-2323 
 
 468 
 
 
 15 
 
 2-208 
 
 1-0978 
 
 2020 
 
 
 25 
 
 4-040 
 
 1-3045 
 
 480 
 
 
 20 
 
 3-039 
 
 1-1335 
 
 2677 
 
 
 30 
 
 5-124 
 
 1-3788 
 
 444 
 
 NaCl 
 
 5 
 
 0-884 
 
 1-0345 
 
 672 
 
 CuS0 4 
 
 5 
 
 0-658 
 
 1-0531 
 
 189 
 
 
 10 
 
 1 -830 
 
 1-0707 
 
 1211 
 
 
 10 
 
 1-387 
 
 1-1073 
 
 320 
 
 
 15 
 
 2-843 
 
 1-1087 
 
 1642 
 
 
 15 
 
 2-194 
 
 1-1675 
 
 421 
 
 
 20 
 
 3-924 
 
 1-1477 
 
 1957 
 
 
 
 
 
 
 
 25 
 
 5 '085 
 
 1-1898 
 
 2135 
 
 f!rlSO. 
 
 
 0*504 
 
 1 -0486 
 
 14fi 
 
 CuCl 2 
 
 5 
 10 
 15 
 20 
 25 
 
 0-938 
 1-957 
 3-059 
 4-253 
 5-545 
 
 1-0409 
 1-0852 
 1-1311 
 1-1794 
 1-2305 
 
 643 
 1141 
 
 1505 
 1728 
 1781 
 
 
 10 
 15 
 20 
 25 
 30 
 
 1-060 
 1 674 
 2-354 
 3-112 
 3-958 
 
 1-1026 
 1-1607 
 1-2245 
 1-2950 
 1-3725 
 
 247 
 325 
 388 
 430 
 436 
 
 
 30 
 
 6-945 
 
 1-2841 
 
 1658 
 
 
 
 
 
 
 
 35 
 
 8-468 
 
 1-3420 
 
 1366 
 
 
 
 
 
 
 
 
 
 
 
 H 2 SO 4 
 
 5 
 
 1-053 
 
 1-0331 
 
 2085 
 
 
 
 
 
 
 
 10 
 
 2-176 
 
 1-0673 
 
 3915 
 
 KN0 3 
 
 5 
 
 0-509 
 
 1-031 
 
 454 
 
 
 15 
 
 3-376 
 
 1-1036 
 
 5432 
 
 
 10 
 
 1-051 
 
 1-063 
 
 839 
 
 
 20 
 
 4-655 
 
 1-1414 
 
 6527 
 
 
 15 
 
 1-626 
 
 1-097 
 
 1186 
 
 
 25 
 
 6-019 
 
 1-1807 
 
 7171 
 
 
 20 
 
 2-240 
 
 1-113 
 
 1505 
 
 
 30 
 
 7-468 
 
 1-2207 
 
 7388 
 
 
 
 
 
 
 
 35 
 
 9-011 
 
 1-2625 
 
 7243 
 
 
 
 
 
 
 
 40 
 
 10-649 
 
 1-3056 
 
 6800 
 
 ZnS0 4 
 
 5 
 
 0-651 
 
 1-0509 
 
 191 
 
 
 45 
 
 12-396 
 
 1-3508 
 
 6164 
 
 
 10 
 
 1-371 
 
 1-1069 
 
 321 
 
 
 50 
 
 14-258 
 
 1-3984 
 
 5405 
 
618 
 
 APPENDIX 
 TABLE XXVI. 
 
 E.M.F. of Clark and Cadmium Cells at Different Temperatures. 
 
 The values in the following table were calculated by Marck (Drude Ann. 
 (1900), i. p. 617) from the formulae 
 
 Clarkcell. . . . ^ = 1-4328 - ll-9(- 15)10- *_0'07(- 15) 2 10- 4 , 
 Cadmium cell . . #,= 1-0186 - 38'0( - 20)10- 6 - 0'65(< - 20) 2 10-, 
 
 the E.M.F. being expressed in International volts. 
 
 E.M.F. OF CLARK CELL. 
 
 Degrees C. 
 
 Tenths. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 i 
 
 8 
 
 9 
 
 
 
 1-4491 
 
 90 
 
 89 
 
 88 
 
 87 
 
 86 
 
 85 
 
 84 
 
 83 
 
 82 
 
 1 
 
 81 
 
 80 
 
 79 
 
 78 
 
 77 
 
 76 
 
 75 
 
 73 
 
 72 
 
 71 
 
 2 
 
 70 
 
 69 
 
 68 
 
 67 
 
 66 
 
 65 
 
 64 
 
 63 
 
 62 
 
 61 
 
 3 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 54 
 
 53 
 
 52 
 
 51 
 
 4 
 
 50 
 
 49 
 
 48 
 
 47 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 5 
 
 40 
 
 39 
 
 38 
 
 37 
 
 36 
 
 35 
 
 33 
 
 32 
 
 31 
 
 30 
 
 6 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 24 
 
 23 
 
 22 
 
 20 
 
 19 
 
 7 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 12 
 
 11 
 
 10 
 
 09 
 
 8 
 
 07 
 
 06 
 
 05 
 
 04 
 
 03 
 
 02 
 
 01 
 
 00 
 
 99 
 
 98 
 
 9 
 
 1-4397 
 
 96 
 
 94 
 
 93 
 
 92 
 
 91 
 
 90 
 
 89 
 
 88 
 
 87 
 
 10 
 
 86 
 
 85 
 
 83 
 
 82 
 
 81 
 
 80 
 
 79 
 
 78 
 
 77 
 
 75 
 
 11 
 
 74 
 
 73 
 
 72 
 
 71 
 
 70 
 
 68 
 
 67 
 
 66 
 
 65 
 
 64 
 
 12 
 
 63 
 
 61 
 
 60 
 
 59 
 
 58 
 
 57 
 
 56 
 
 55 
 
 53 
 
 52 
 
 13 
 
 51 
 
 50 
 
 49 
 
 48 
 
 46 
 
 45 
 
 44 
 
 43 
 
 42 
 
 41 
 
 14 
 
 40 
 
 38 
 
 37 
 
 36 
 
 35 
 
 34 
 
 33 
 
 31 
 
 30 
 
 29 
 
 15 
 
 28 
 
 27 
 
 26 
 
 24 
 
 23 
 
 22 
 
 21 
 
 19 
 
 18 
 
 17 
 
 16 
 
 16 
 
 15 
 
 13 
 
 12 
 
 11 
 
 10 
 
 08 
 
 07 06 
 
 05 
 
 17 
 
 04 
 
 02 
 
 01 
 
 00 
 
 99 
 
 97 
 
 96 
 
 95 
 
 94 
 
 92 
 
 18 
 
 1 -4291 
 
 90 
 
 89 
 
 88 
 
 86 
 
 85 
 
 84 
 
 83 
 
 81 
 
 80 
 
 19 
 
 79 
 
 78 
 
 77 
 
 75 
 
 74 
 
 73 
 
 72 
 
 70 69 
 
 68 
 
 20 
 
 67 
 
 65 
 
 64 
 
 63 
 
 62 
 
 60 
 
 59 
 
 58 56 
 
 55 
 
 21 
 
 54 
 
 53 
 
 51 
 
 50 
 
 49 
 
 47 
 
 46 
 
 45 ' 43 
 
 42 
 
 22 
 
 41 
 
 40 
 
 38 
 
 37 
 
 36 
 
 34 
 
 33 
 
 32 
 
 30 
 
 29 
 
 23 
 
 28 
 
 27 
 
 25 
 
 24 
 
 23 
 
 21 
 
 20 
 
 19 
 
 18 
 
 16 
 
 24 
 
 15 
 
 14 
 
 12 
 
 11 
 
 10 
 
 08 
 
 07 
 
 06 
 
 05 
 
 03 
 
 25 
 
 02 
 
 01 
 
 99 
 
 98 
 
 97 
 
 95 
 
 94 
 
 92 
 
 91 
 
 90 
 
 26 
 
 1-4188 
 
 87 
 
 86 
 
 84 
 
 83 
 
 82 
 
 80 
 
 79 
 
 77 
 
 76 
 
 27 
 
 75 
 
 73 
 
 72 
 
 71 
 
 69 
 
 68 
 
 67 
 
 65 
 
 64 
 
 62 
 
 28 
 
 61 
 
 60 
 
 58 
 
 57 
 
 56 
 
 54 
 
 53 
 
 51 
 
 50 
 
 49 
 
 29 
 
 47 
 
 46 
 
 45 
 
 43 
 
 42 
 
 41 
 
 39 
 
 38 
 
 36 
 
 35 
 
 30 
 
 1-4134 
 
 32 
 
 31 
 
 29 
 
 28 
 
 27 
 
 25 
 
 24 
 
 22 
 
 21 
 
 E.M.F. OF CADMIUM CELL. 
 
 Temperature. 
 
 E.M.F. 
 
 Temperature. 
 
 E.M.F. 
 
 Temperature. 
 
 E.M.F. 
 
 10 
 
 1-0189 
 
 17 
 
 1-0187 
 
 24 
 
 1-0184 
 
 11* 
 
 9 
 
 18 
 
 7 
 
 25 
 
 4 
 
 12 
 
 9 
 
 19 
 
 6 
 
 26 
 
 3 
 
 13 
 
 8 
 
 20 
 
 6 
 
 27 
 
 3 
 
 14 
 
 8 
 
 21 
 
 6 
 
 2S 
 
 3 
 
 15 
 
 8 
 
 22 
 
 5 
 
 29 
 
 2 
 
 16 
 
 7 
 
 23 
 
 5 
 
 
 
APPENDIX 
 
 619 
 
 TABLE XXVII. Connection between the Units in the C.G.S. System and 
 the Practical System. (See Watson's "Physics" 539, 540, 541.) 
 
 Quantity. 
 
 Name of Unit in 
 C.G.S. System. 
 
 Dimensions. 
 
 Name of Unit in 
 Practical System. 
 
 Value of Practical 
 Unit in C.G.S. 
 System. 
 
 Length . . . 
 
 Centimetre 
 
 L 
 
 Quadrant 
 
 10 9 cm. 
 
 Mass .... 
 
 Gram 
 
 M 
 
 
 IO- 11 gram. 
 
 Time .... 
 
 Second 
 
 T 
 
 Second 
 
 1 second. 
 
 Force . . . * 
 
 Dyne 
 
 LMT- 2 
 
 
 
 Work . . . . 
 
 Erg 
 
 VMT-* 
 
 Joule 
 
 IO 7 ergs. 
 
 Power .... 
 
 Erg per sec. 
 
 VMT-* 
 
 Watt 
 
 IO 7 ergs per sec. 
 
 Resistance . . 
 
 
 LT-^ 
 
 Ohm 
 
 IO 9 cm./sec. 
 
 Current . . . 
 
 
 ^Miz-V-i 
 
 Ampere 
 
 IO- 1 . 
 
 Quantity . . . 
 
 
 zfcrtH 
 
 Coulomb 
 
 10-1. 
 
 E.M.F 
 
 
 L*M*T^ 
 
 Volt 
 
 IO 8 . 
 
 Magnetic field . 
 
 Gauss 
 
 Lr^M^T^I^ 
 
 None 
 
 io- 10 . 
 
 Magnetic flux 
 density or in- 
 duction 
 
 Maxwell 
 
 L~%M^T~ I I$ 
 
 None 
 
 IO-". 
 
 Inductance . . 
 
 Centimetre 
 
 L 
 
 Henry 
 
 IO 9 . 
 
 Capacity . . . 
 
 
 zr^V 
 
 Farad 
 Microfarad 
 
 IO- 9 . 
 
 io- 15 . 
 
 Ratios of Electrostatic Units to Electromagnetic Units. 
 
 Unit in Electrostatic System. 
 
 Value in Electromag- 
 netic Units (c.g.s.). 1 
 
 Value in Practical 
 Units. 
 
 Quantity. 
 
 Dimensions. 
 
 Resistance . . . 
 
 IrW 
 
 72 
 
 IO- 9 F 2 ohms. 
 
 Current .... 
 
 itMiT*** 
 
 F- 1 
 
 10 F- 1 amperes. 
 
 Quantity . . . 
 E.M.F 
 
 LK 
 
 V 
 
 F- 2 
 
 10 F- 1 coulombs. 
 IO- 8 F volts. 
 
 10 9 F-2 farads, 
 or 10 15 F 2 microfarads. 
 
 Capacity . . . 
 
 The value of V is 3'OOXlO 10 . 
 
620 
 
 APPENDIX 
 
 TABLE XXVIII. 
 
 Substance. 
 
 Density. 
 
 Temperature. 
 
 Hydrogen .... 
 
 00009004 
 
 C. 
 
 Oxygen .... 
 
 0014292 
 
 
 
 Nitrogen .... 
 
 0012542 
 
 
 
 Carbon dioxide 
 
 0019652 
 
 
 
 Anilin 
 
 1-0342 
 
 4 
 
 Carbon bisulphide 
 
 1-2922 
 
 
 
 Carbon tetrachloride . 
 
 1-5947 
 
 
 
 Ethyl alcohol 
 
 0-7937 
 
 15 
 
 Methyl alcohol 
 
 0-810 
 
 
 
 Benzene .... 
 
 0-8799 
 
 20 
 
 Ether . 
 
 0-736 
 
 
 
 Xylene . 
 
 0-8903 
 
 4 
 
 
 
 
 Substance. 
 
 Density. 
 
 Aluminium 
 
 2-6 
 
 Lead . . .--.. 
 
 11-4 
 
 Gold 
 
 19-3 
 
 Iridium 
 
 22-4 
 
 Copper 
 
 8-9 
 
 Nickel 
 
 8-9 
 
 Platinum 
 
 21-5 
 
 Sulphur (Rhombic) .... 
 
 2-07 
 
 Silver 
 
 10-5 
 
 Bismuth 
 
 9-8 
 
 Zinc 
 
 7'1 
 
 Tin 
 
 7'3 
 
 Iceland spar 
 
 2-7 
 
 Quartz ...... 
 
 2'65 
 
 Iron 
 
 7-8 
 
 Steel 
 
 7-6-7-8 
 
 Fluorspar ...... 
 
 3-2 
 
 Ice(0) 
 
 9167 
 
INDEX 
 
 Absorption hygrometer, 249 
 Absorption of light, 390 
 Accumulators, small, 570 
 Adjustable carbon resistance, 473 
 Air, density of, 77, 599 
 Air thermometer, 203 
 
 constant pressure, 209 
 
 constant volume, 204 
 
 reduction of readings of mercury 
 
 thermometer to, 182, and Table XIX. 
 Algebraical expression, to find an, to 
 
 represent a series of results, 10 
 Alternating current met hod of measur- 
 ing self-induction, 548 
 Alternating current method of measur- 
 ing the resistance of electrolytes, 
 483 
 
 Ammeter, calibration of, 513, 516 
 Analysis, harmonic, 35 
 Anderson's method of measuring self- 
 induction, 543 
 
 Angle, measurement of, of prism, 290 
 Apparent expansion of a liquid, 191, 
 
 201 
 
 Approximate formulae, 597 
 Area, measurement of, 32 
 Arithmometer, 29 
 Aspirator, 249 
 Atmospheric pressure, measurement of, 
 
 155 
 Ayrton and Perry's secohmmeter, 538 
 
 B 
 
 Balance, 63 
 ratio of arms of, 68 
 
 Ballistic galvanometer, 518, 520, 522 
 
 Ballistic method of measuring- permea- 
 bility, 563 
 
 Barometer, 155 
 
 corrections to readings, 157, 603 
 604, and Table I. 
 
 Battery resistance, Beetz's method, 476 
 - Mance's method, 475 
 
 Baume, hydrometer scale, 95 
 
 Beats of two notes, 277, 281 
 
 Beckmann's apparatus for measuring 
 the boiling point, 264 
 
 apparatus for measuring the 
 
 freezing point, 258 
 
 thermometer, 262 
 
 Beetz's method of measuring the re- 
 sistance of a cell, 476 
 
 Berthelot's method of measuring latent 
 heat, 237 
 
 Biprism, Fresnel's, 316, 320 
 
 Bird call for producing high note, 286 
 
 Brasheur's method of silvering o-lass 
 590 
 
 Bunsen photometer, 383 
 
 Bunsen's ice calorimeter, 230 
 
 Buoyancy, correction for, 76, 79, 601 
 
 Burette, calibration of, 97 
 
 C 
 
 Cable, measurement of insulation re- 
 sistance of, 461 
 
 Cadmium cell, 487 
 - E.M.F. of, 618 
 
 Calculating machines, 29 
 
 Calibration of ammeter, 513, 516 
 
 of a Callendar and Griffiths bridge, 
 
 of a dial-pattern Wheatstone's 
 bridge, 452 
 
 of a spectroscope, 311, 312 
 
 - of a wire, 437, 440, 441 
 of burette, 97 
 
 of tube of thermometer, 162 
 Callendar and Griffiths pattern Wheat- 
 stone's bridge, 455 
 
 Calorimeter, 214 
 Calorimetry, 213 
 Candle-power of incandescent lamp 
 
 387 
 
 Capacity, absolute measurement of, 533 
 comparison of, by de Sauty's 
 
 method, 530 
 
 - comparison of, by method of 
 mixture, 531 
 
 comparison of, with ballistic gal- 
 vanometer, 525 
 
 621 
 
622 
 
 INDEX 
 
 Capacity of quadrant electrometer, 571 
 Capillarity, 139 
 Capillary electrometer, 498 
 Carbon resistance, adjustable, 473 
 Carey Foster's method of calibrating a 
 wire, 437 
 
 method of comparing resistances, 
 442 
 
 Cathetometer, 53 
 
 Chemical (absorption) hygrometer, 249 
 Clark cell, E.M.F. of, 493, 618 
 Clement and Desorme's method of 
 
 measuring the ratio of the specific 
 
 heats, 2(57 
 
 Clock, determination of rate of, 131, 135 
 Coincidence, method of, when deter- 
 mining a period, 124 
 Collimation, adjustment for, 54 
 Colour-blindness, 394 
 Colour mixtures, 391 
 
 top, 392 
 
 Commutator, Pohl's, 426 
 
 Cool column, correction of thermometer 
 
 readings for, 180 
 Cooling, specific heat by the method 
 
 of, 224 
 
 Comparator, 51 
 
 Comparison of thermometers, 174 
 Comptometer, 32 
 Condensers, comparison of capacity of, 
 
 525, 530, 531, 571 
 Conductivity for heat, 270, 272 
 
 of electrolytes, 475, 617 
 
 Copper, deposition of, for current 
 
 measurement, 512 
 
 voltameter, 512 
 
 Corrections to observed quantities, 3 
 Cross- wires, mounting, 592 
 
 Curves, application of, to reduction of 
 
 observations, 5 
 drawing, with spline, 9 
 
 D 
 
 Declination, 396 
 
 Demagnetisation of iron, 567 
 
 Demagnetising factor, 562 
 
 Density, 79 
 
 correction for buoyancy of air, 
 
 79, 601 
 of air, 77, Table IV. 
 
 of ice, 228 
 
 - of liquids, 87, 89, 92, 94, 96 
 
 of solids, 82, 84, 85 
 
 of various substances, 620 
 
 Depression of zero in mercury thermo- 
 meters, 182, 185, 611 
 
 Depression of freezing point, 258, 261 
 
 de Sauty's method of comparing capa- 
 city, 530 
 
 Dew point, 247 
 
 Dial-pattern Wheatstone's bridge, 450, 
 452 
 
 Differential galvanometer, 469 
 
 Diffraction fringes, 322 
 
 grating, 314 
 
 through a slit, 335 
 Dilatometer, volume, 197 
 weight, 193 
 
 Dip, 415 
 
 Discharge key, 525 
 Dispersion, 305 
 Dividing scales on glass, 586 
 Dolezalek's method of measuring self- 
 induction, 548 
 
 quadrant electrometer, 569 
 Double bridge for measuring low resist- 
 ances, 465 
 
 E 
 
 Earth inductor, 527 
 Electrolytes, conductivity of, 617 
 
 resistance of, 475 
 Electrometer, capillary, 498 
 
 quadrant, 569 
 
 Elliptically polarised light, 378, 380 
 Emergent column, correction of ther- 
 mometer readings for, 180 
 E.M.F. of Clark and cadmium cells, 618 
 
 standards of, 487 
 E.M.F.'s, comparison of, 493, 526 
 
 comparison of, by galvanometer 
 deflections, 430 
 
 Expansion, apparent, of a liquid, 191, 201 
 
 cubical, 191 
 
 cubical, with hydrostatic balance, 
 
 200 
 
 cubical, with weight thermometer, 
 193 
 
 cubical, with volume dilatometer, 
 197 
 
 linear, 187, 188 
 
 Exploring coil, 529 
 
 Eye and ear method of measuring a 
 
 period, 107 
 
 Eye-piece, adjustment of, 49, 289 
 , Gauss, 290 
 
 Figure of merit of galvanometer, 428 
 Filler for specific gravity bottles and 
 
 dilatometers, 90 
 Fixed points, auxiliary, 178 
 of thermometer, 169, 172 
 
INDEX 
 
 623 
 
 Flatness, test of, 356 
 Flicker photometer, 386 
 Focal length of lenses, 348, 359 
 
 length of mirrors, 352 
 
 Fourier analysis, 35 
 
 Freezing point of solution, 258 
 
 Frequency of tuning fork, 275, 277,279, 
 
 281 
 
 Fresnel's biprism, 316, 320 
 Fundamental interval, correction for, 
 
 173 
 of platinum thermometer, 504 
 
 G 
 
 "gr," determination of, by Borda's 
 pendulum, 122 
 
 by Rater's pendulum, 128 
 Galvanometer, ballistic, 518 
 
 differential, 469 
 
 figure of merit of, 428 
 
 sine, 511 
 
 suspended coil pattern, 427 
 
 suspended needle pattern, 425 
 
 tangent, 508 
 
 Gas, reduction of volume of, 605, 606 
 Gauss eye-piece, 290 
 Geographical meridian, 397 
 Glass blowing, 575 
 Grating, resolving power of, 341 
 
 measurement of wave - length 
 with, 314 
 
 H 
 
 Hare's apparatus for density of liquids > 
 
 92 
 
 Harmonic analysis, 35 
 Heat, conductivity for, 270, 272 
 
 of solution, 241 
 
 Heater for calorimeter, 215, 233 
 High note, wave-length of, 286 
 High resistance, measurement of, 460 
 Holmgren's test wools, 394 
 Horizontal component of the earth's 
 
 field, 403, 406, 407 
 Hydrogen scale, reduction of readings 
 
 of mercury thermometer to, 182, and 
 
 Table XIX. 
 Hydrometer, 94 
 
 Nicholson's, 96 
 
 Hydrostatic balance, measurement of 
 
 cubical expansion with, 200 
 Hygrometer, 247, 248, 249 
 
 Ice, density of, 228 
 latent heat of, 233 
 
 Ice calorimeter (Bunsen's), 230 
 Illumination of divisions of standard 
 
 scale, 50 
 Incandescent lamp, candle-power of, 
 
 387 
 
 Inertia, moments of, 107, 601 
 Insulation resistance, 461 
 Interference fringes, localisation of, 324 
 Interferometer, Michelson's, 327 
 Kayleigh's, 343 
 
 Iron, permeability of, 552 
 
 Jolly photometer, 385 
 Joly's steam calorimeter, 239 
 
 K 
 
 Kelvin and Varley slide, 496 
 
 Kelvin's double bridge, 465 
 
 Key, discharge, 525 
 
 for electrometer, 571 
 - Pohl's, 426 
 
 Kohlrausch's method of measuring the 
 resistance of electrolytes, 483 
 
 Kundt's method of measuring the velo- 
 city of sound, 284 
 
 Lag of thermometer, correction for, 220 
 Latent heat, by Berthelot's method, 237 
 
 of ice, 233 
 
 of steam, 235 
 
 Length, correction of measurements of, 
 to allow for effects of temperature, 52 
 measurement of, 41 
 
 standards of, 41 
 Lenses and mirrors, 348 
 
 Light, sources of, for dispersion 
 
 measurements, 305 
 standards of, 382 
 Light filter, 310 
 
 Liquid, specific heat of, 223, 224 
 Localisation of interference fringes, 
 
 324 
 Logarithmic decrement, 149 
 
 paper, 11 
 
 Low resistances, measurement of, 462, 
 
 465, 470, 471 
 
 Luminosity of pigments, 394 
 Luramer-Brodhun photometer, 384 
 
 M 
 
 Magnet, temperature coefficient of, 414 
 Magnetic field, measurement of, with 
 earth inductor, 527 
 
624 
 
 INDEX 
 
 Magnetic field, measurement of, with 
 exploring coil, 529 
 
 Magnetic meridian, 396 
 
 moments, comparison of, 417 
 
 Magnetometer, 404, 407 
 
 - method of measuring permea- 
 bility, 553 
 
 Magnifying power of microscope, 368 
 
 telescope, 367 
 
 Mance's method of measuring the re- 
 sistance of a battery, 475 
 
 Manganine wire resistances, 593 
 
 Manometer for very small pressures, 144 
 
 Mathematical tables, 25 
 
 Mathissen and Hocking's method of 
 measuring low resistances, 462 
 
 Melting point by the method of cooling, 
 267 
 
 Melting point of metals, 506 
 
 Mercury arc, 308 
 
 still, 487, 489 
 
 vapour pressure of, 611 
 
 Metals, melting point of, 506 
 
 Metastatic thermometer, 262 
 
 Michelson's interferometer, 327 
 
 Micrometer microscope, 48 
 
 screw, 44 
 
 Microscope, magnifying power of, 368 
 
 Mirror and scale for measuring rota- 
 tions, 420, 616 
 
 Mirrors, radius of curvature of, 352 
 
 Mixture, method of, for comparing 
 capacity, 531 
 
 Mixture, method of, in calorimetry, 213 
 
 Molecular weight from the depression 
 of the freezing point of solutions, 261 
 
 Moments of inertia, comparison of, 107 
 table of, 601 
 
 Movable zero method of using a mer- 
 cury thermometer, 183 
 
 Mutual induction, 546, 548 
 
 N 
 
 Newton's rings, 332 
 Normal equations, 21 
 
 Optical bench, 317, 348 
 lever, 59 
 
 Parallax of sun or star, 602 
 Pendulum, Borda's, 122 
 compound, 126 
 
 Pendulum, effect of arc on period, 118, 
 
 and Table VIII. 
 Kater's reversible, 128 
 
 simple, 118 
 Period, determination of, by method of 
 
 coincidence, 124 
 measurement of, by eye and ear 
 
 method, 107 
 Permeability, 552 
 
 by the ballistic method, 563 
 
 by the magnetometer method, 553 
 
 Photographic lens, test of, 363 
 
 Photometers, 383 
 
 Photoped, 385 
 
 Pigments, luminosity of, 394 
 
 Pitch comparison of, of two forks, 281 
 
 of tuning fork, 275, 277, 279 
 
 Planimeter, 33 
 
 Platinum thermometer, 501 
 
 Pohl commutator, 426 
 
 Polarisation, rotation of the plane of, 
 
 370 
 Potentiometer method of measuring 
 
 current, 516 
 Potentiometers, 494 
 Poynting's polariscope, 373 
 Prism, measurement of angle of, 290 
 Pyknometer, 85 
 
 Q 
 
 Quadrant electrometer, 569 
 
 capacity of, 571 
 
 Quartz fibres, 581 
 
 Radius of curvature of mirrors, 352 
 of curvature with optical lever, 61 
 
 of curvature with spherometer, 46 
 Rate of clock, 131,135,602 
 
 Ratio of specific heats (Clement and 
 Desorme's method), 267 
 
 of specific heats from velocity of 
 
 sound, 286 
 
 Rayleigh's method of measuring self- 
 induction, 536 
 
 refractometer, 343 
 
 Reduction of barometer readings, 157, 
 
 and Tables I., X., and XI. 
 Refraction of sun or star, 602 
 Refractive index by total reflection, 
 
 297, 299, 303 
 
 index of gases, 345 
 index of prism, 295, 302 
 
 index of various substances, 614 
 Refractometer, Rayleigh's, 343 
 Regnault's hygrometer, 247 
 Resistance coils, construction of, 593 
 
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