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 The new physics 
 
 Cornell University Library 
 
 3 1924 031 231 388 
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 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924031231388 
 
THE 
 
 NEW PHYSICS 
 
 4 MANUAL OF EXPERIMENTAL STUDY FOR HIGH 
 
 SCHOOLS AND PREPARATORY SCHOOLS 
 
 FOR COLLEGE. 
 
 BY 
 
 JOHN TROWBRIDGE, 
 
 PROFESSOR OF PHYSICS, HARVARD UNIVERSITY. 
 
 KEW YORK: 
 D. APPLETON AND COMPANY, 
 
 I, 3, AND 5 BOND STREET. 
 1884. 
 
COPYKIGHT, 1884, 
 
 By D. APPLETON AND COMPANY. 
 
PREFACE. 
 
 The experience of those who have made a care- 
 ful study of the methods of instruction in physical 
 science in the secondary schools, and who have 
 judged of the results of these methods by the intel- 
 lectual training manifested by students who pre- 
 sent themselves for college, is unfavorable to the 
 lecture or recitation system unsupported by lab- 
 oratory work. While the student's knowledge of 
 language in general has been acquired through 
 many years of patient and long-continued work 
 in what may be termed a literary laboratory, in 
 which the implements or apparatus consist of 
 grammars and dictionaries, his acquaintance with 
 the subject of physics has often been gained by 
 superficial study in a few weeks. 
 
 One of the most important factors in intellect- 
 ual growth is the long and persistent exercise of 
 the mind upon a subject sufficiently broad to afford 
 this continuous effort. It is believed that the study 
 of physical science, in a laboratory, affords an 
 
iv PREFACE. 
 
 opportunity for this strenuous intellectual effort. 
 The literary habit of mind is acquired by long 
 study of language ; and the scientific habit or in- 
 stinct does not require less cultivation than the 
 literary instinct. 
 
 The teacher may grant the comparatively small 
 result which is obtained from the study of physics 
 by the method of lectures and recitations ; but he 
 will immediately ask, " How can we obtain the time 
 for laboratory work in physics — crowded as we are 
 with so many studies ? " Moreover, the subject of 
 physics is unlike that of chemistry or botany. In 
 these subjects the cost of the apparatus and mate- 
 rial is comparatively small. Each student can have 
 his own tubes, his own reagents, and his own mi- 
 croscope. In physics it is very different. The ap- 
 paratus is expensive. A spectroscope, for instance, 
 would cost nearly as much as the entire apparatus 
 in a chemical laboratory. These are strong objec- 
 tions, and need careful consideration. The time, 
 however, which is now given to recitation from 
 text-books could at least be devoted to laboratory 
 work ; and the lectures could be reduced in num- 
 ber and only given to direct the students in their 
 laboratory work. More can be done with cheap 
 apparatus than is generally supposed. It is one of 
 the objects of my treatise to show this. 
 
 I have endeavored to put into the hands of the 
 teacher a manual which will stand in the same re- 
 
PREFACE. y 
 
 lation to physics that many of the excellent man- 
 uals in chemistry stand to the instruction of chem- 
 istry in the secondary schools. I have endeavored 
 also to arrange the experiments so that there may 
 be a logical connection between them. I have 
 early in the treatise directed the mind of the stu- 
 dent toward the doctrine of the conservation of 
 energy, and to the observation of the truth that 
 our knowledge is relative, and that there are not 
 different kinds of forces. Thus, I have endeavored 
 to give a clear conception of what may be termed 
 the new physics, so that even the general reader, 
 in glancing over the character and the mutual re- 
 lation of the experiments, can see the tendency of 
 physical science to-day. I have not hesitated to 
 avail myself early of electrical methods to illus- 
 trate problems of motion, and I have treated the 
 magnetic pendulum together with the gravitation 
 pendulum. 
 
 It is thought by some that a student can not 
 begin the study of mechanics, and physics in gen- 
 eral, without a good knowledge of trigonometry. 
 The writer believes that the necessary amount of 
 geometry and trigonometry can be taught almost 
 at one sitting. On page 58 will be found an ex- 
 position of the amount of trigonometry which is 
 necessary in order to understand this treatise. It 
 is related in the life of De Morgan, the mathema- 
 tician, that his teachers found -him dull in mathe- 
 
vi PREFACE. 
 
 matics. One day his parents saw him occupied 
 with a pair of compasses, and when he was asked 
 what he was doing, he repHed that he was " draw- 
 ing mathematics." It is not derogatory to the 
 dignity of mathematics to approach the subject 
 by means of a pair of compasses or a foot-rule or 
 a metre-scale, instead of by the stately process of 
 proposition-lemmas and corollaries ending by Q. E. 
 D. If we examine our own progress in geometry 
 we shall find that we accept many things as the 
 result of the experience of measurement, and have 
 forgotten the philosophical proofs. Thus we can 
 by actual measurement ascertain the truth that 
 the square on the hypotenuse of a right-angled tri- 
 angle is equal to the sum of the squares on the 
 base and altitude of the triangle, and, knowing this 
 fact, we can make a great and rapid advance in our 
 knowledge of the configurations about us. Yet if 
 our advance must be checked until we can prove 
 rigidly by geometry this proposition, the concep- 
 tions of the world of motion, which our minds 
 long to seize, must wait upon a cast-iron meth- 
 od which is not always the best calculated for 
 all minds. How many of us to-day — having for 
 gotten the proof in Euclid of the Pythagorean 
 proposition — can sit down and prove it without 
 considerable effort? Yet we use it every day in 
 extending our knowledge of physics. The same 
 remark can be applied to our use of ir, the ratio 
 
PREFACE. 
 
 Vil 
 
 of the circumference of a circle to its diameter. 
 How many of us can go through the steps of the 
 demonstration without effort ? It may be said 
 that we work to-day by means of the old disci- 
 pline in rigid geometry. This is true to a certain 
 extent. We also work in philology by means of 
 our exact grammatical training ; but the teachers 
 in languages are everywhere insisting upon the 
 advantages of reading at sight, accepting certain 
 grammatical forms as true without carefully prov- 
 ing them at first ; judging by experience — or what 
 corresponds to our measurement b)' the compasses 
 — of the truth of these forms. 
 
 I have treated trigonometry as a mere branch 
 of our conception of motion on an inclined plane. 
 
 It may be urged that the strict geometrical 
 proofs of the law of pulleys, of the law of the 
 conical pendulum, of the law of the simple pendu- 
 lum, are far more conclusive and exact than any 
 experimental proofs, which must necessarily be 
 imperfect, on account of friction of the apparatus 
 and the difficulty of properly applying the forces. 
 This is true ; but the mind may easily grasp the 
 geometrical proofs, and proceed with facility to 
 the Q.-E. D. of the proposition, yet may not have 
 a full conception of the magnitude of the forces 
 which are brought into play and their relative im- 
 portance. Thus, one who understands the theory 
 of centripetal force might be incompetent to esti- 
 
viii PREFACE. 
 
 mate the proper size for the fly-wheel of an engine. 
 For he has not handled apparatus, and has not 
 gained a certain balance of judgment which comes 
 from actual knowledge of the relative magnitude 
 of the forces in question. It is only by actually 
 trying experiments in mechanics that a student 
 realizes the importance of the units which must 
 be adopted. One can readily follow a geometrical 
 proof of the law of the screw, but at the same time 
 fail to seize the point that all distances must be 
 expressed in the same unit, and that the force and 
 the weight must also be expressed in terms of a 
 consistent unit. The cultivated scientific instinct, 
 moreover, will often save one the expenditure of 
 much time and labor on unprofitable experiments 
 in many departments of human effort. 
 
 This practical knowledge, it may be said, should 
 follow the theoretical knowledge. Thus the young 
 lawyer gains first his theoretical knowledge of law, 
 and obtains his balance of judgment in actual prac- 
 tice. The young medical student and the engineer 
 should have laboratory practice, however, before 
 they experiment upon human life. It is necessary 
 for the student of science, whether he is to be a 
 physicist, a chemist, a botanist, or a geologist, or 
 whether he is to be a physician or an engineer, to 
 obtain a certain balance of judgment, and to culti- 
 vate a certain scientific instinct; which comes from 
 putting theories into practice. The mind, more- 
 
PREFACE. ix 
 
 over, must rest upon physical laws for a compara- 
 tively long period in order to understand their true 
 significance. Five minutes might enable one to 
 grasp the geometrical proofs of the laws of centrip- 
 etal force, yet the reflection of days may be neces- 
 sary to see their physical importance. A physical 
 experiment is valuable as an intellectual exercise, 
 compelling one to exercise judgment in the disposi- 
 tion of material, in the avoidance of errors, and 
 leading one to study the influence of laws other 
 than those which we are striving to prove. In 
 the elementary study of physics, however, the 
 proposition proved geometrically makes but a 
 slight impression upon the mind unless its sig- 
 nificance is seen from an industrious and patient 
 application. 
 
 Many teachers also object to apparatus which 
 will not give the most accurate results that are 
 obtainable in the present state of science. The 
 writer believes, however, from long experience, that 
 a careful study of the sources of error of an imper- 
 fect piece of apparatus often brings to the attention 
 of the student problems in mechanics which a 
 nicely adjusted piece of apparatus would not sug- 
 gest to him. Mathematical problems in physics 
 have their place in teaching ; but physics should 
 not be made a means of teaching mathematics. I 
 have, therefore, substituted experimental problems 
 for the mathematical problems which are usually 
 
X PREFACE. 
 
 given in treatises on natural philosophy, in the hope 
 of cultivating the scientific instinct. 
 
 The natural progress of our study of any subject 
 is from the qualitative, or the comparatively rough 
 evidence of our senses, to the quantitative, or exact 
 determination of how much there is of anything. I 
 have given both qualitative and quantitative ex- 
 periments, endeavoring to lead the way from the 
 first rough trials to the more exact methods. I 
 have also often given several methods of deter- 
 mining the same quantity, endeavoring to point 
 out that any intelligent student can cultivate self- 
 reliance by trying his experiment in different ways 
 and, by his control methods, ascertain how near to 
 the truth his results are without appealing to the 
 authority of a teacher. This cultivation of a men- 
 tal fiber is a valuable result of laboratory work in 
 physics. This treatise can, therefore, be used by 
 the teacher to illustrate what the subject of mod- 
 ern physics is by a course of lectures, in which the 
 experiments are performed before the class. A 
 certain number of the same experiments should 
 afterward be performed by each student. With- 
 out much expense, each student can have his own 
 specific-gravity bottle, his own spiral spring-bal- 
 ance, his own vernier gauge, his own battery, and 
 his own galvanometer. The more expensive instru- 
 ments can be assigned in rotation to different stu- 
 dents. 
 
PREFACE. 
 
 XI 
 
 The author recommends that from one to two 
 lectures be given during the week. In these lect- 
 ures the experiments should be performed which 
 the students afterward perform themselves in the 
 laboratory ; the complete calculations should be 
 given, and attention be called to the precautions 
 to be taken. In this way the class is carried over 
 the same ground at the same time, and less indi- 
 vidual instruction has to be given by the instruct- 
 or in the laboratory. This method requires a 
 certain amount of duplication of apparatus. It is 
 best that the class should be divided into sets of 
 six or ten students, and a sufficient number of 
 vernier gauges, of specific-gravity bottles, of lenses, 
 of galvanometers, should be provided to enable the 
 men to keep pace with each other. This, how- 
 ever, is not an essential point. 
 
 An elementary laboratory of physics for a 
 school, with experiments properly selected — for it 
 is not necessary to cover the whole ground of ex- 
 perimental physics in order to gain a large amount 
 of intellectual discipline — need not cost more than 
 a chemical laboratory such as is now provided in 
 many high-schools. I have endeavored to describe 
 only simple and inexpensive apparatus. The teach- 
 er can readily invent simple contrivances which in 
 many cases will be better than those I have recom- 
 mended. My endeavor has been to point out the 
 way to a more rational method of studying physics. 
 
xii PREFACE. 
 
 In the appendix will be found additional direc- 
 tions for constructing simple apparatus^ 
 
 It is not supposed that the student will perform 
 all the experiments in this treatise during the time 
 that can be devoted to physics in the secondary 
 schools. The choice of experiments by the teacher 
 must necessarily depend upon the apparatus on 
 hand which can be modified for the purposes of this 
 treatise, and upon the amount of appropriation 
 which can be devoted to cheap apparatus. 
 
 It is best that the student should see a greater 
 amount of work than he can accomplish by an easy 
 performance. In the study of language it is not ex- 
 pected that the student in the preparatory schools 
 shall be able to master all the difficulties in the 
 grammar set before him. It is hoped that this 
 treatise will show that the subject of physics 
 also affords great opportunity for intellectual train- 
 ing, and can not be compassed in a cram of a few 
 weeks. In the preparation of this manual I have 
 found valuable. suggestions in the treatises of Pick- 
 ering, of Worthington, and also of Thresh ; and I 
 am also indebted, to Mr. Harold Whiting, assistant 
 in the Physical Laboratory of Harvard University, 
 for interesting modifications of certain experiments. 
 
CONTENTS. 
 
 CHAPTER I. 
 
 GENERAL IDEAS OF LENGTH AND VOLUME. 
 EXP. PAGE 
 
 I. — Measurement of length by pacing .... 
 
 2. — Measurement of length in French and English measure 
 
 3. — Measurement of circumference of circles 
 
 4. — Measurement of volume of cylinders .... 
 
 5. — Measurement of volume of cylinders . 
 
 6. — Measurement of volume of irregular body . 
 
 CHAPTER II. 
 
 SPECIFIC GRAVITY. 
 
 7. — Spiral-spring balance 7 
 
 8. — Calculation of dimensions from weight .... 7 
 
 9. — Specific gravity of irregular body 8 
 
 10. — Loss of weight in water ....... 8 
 
 II. — Density of alcohol 9 
 
 12. — Hydrometers 10 
 
 13. — Nicholson's hydrometer 11 
 
 14. — Weighing 16 
 
 15. — Hydrostatic balance 17 
 
 16. — Specific-gravity bottle 18 
 
 17. — Specific gravity of alcohol 18 
 
 18. — Linear and cubical expansion ...... 20 
 
 CHAPTER IIL 
 
 PRESSURE OF THE AIR. 
 
 19. — Pressure of the air 21 
 
 20. — Weight of the air 22 
 
xiv CONTENTS. 
 
 EXP. PAGE 
 
 21. — Pressure of the air . 22 
 
 22. — Barometer 23 
 
 23. — Pressure of the air . . . ..... 24 
 
 24. — Pressure of fluids 25 
 
 25. — Hydrostatic press ..... ... 26 
 
 26. — Constant pressure ......... 27 
 
 27. — Sprengel air-pump ....... .28 
 
 28. — Density of mercury and water 29 
 
 29. — Measurement of pressure of air ...... 30 
 
 30. — Boyle's or Mariotte's law . . .... 30 
 
 31. — Volumenometer 32 
 
 32. — Reduction to standard pressure 33 
 
 33. — Boyle's law for pressures greater than one atmosphere . . 34 
 
 34. — Law of Charles 35 
 
 35. — Air-thermometer . . 36 
 
 36. — Standard temperature and pressure 38 
 
 CHAPTER IV. 
 
 MEASUREMENT OF DIMENSIONS. 
 
 37. — ^Vernier 40 
 
 38. — Measurement with vernier 41 
 
 39. — Micrometer .......... 42 
 
 40. — Graduation .......... 44 
 
 41. — Spherometer 44 
 
 42. — Measurement with vernier gauge and cathetometer . . 45 
 
 43. — Elasticity 48 
 
 44. — Limit of elasticity . 49 
 
 45. — Co-efficient of elasticity 49 
 
 46. — Measurement of internal diameter 49 
 
 CHAPTER V. 
 
 MOTION. 
 
 47. — Trigonometrical relations 60 
 
 48. — Trigonometrical relations 60 
 
 49. — Laws of falling bodies 66 
 
 50. — Final velocity of falling body 70 
 
 CHAPTER VL 
 
 PENDULUM MOVEMENTS AND THE MEASURE OF FORCE. 
 51. — Relation between pendulum and tuning-fork . . .72 
 
 52. — Torsion pendulum yj 
 
CONTENTS. 
 
 XV 
 
 EXP. 
 
 53- — Center of gravity .... 
 
 54- — Center of gravity of a triangle 
 
 55- — Parallel forces ... 
 56. — Parallel forces . . 
 
 57. — Parallel forces 
 
 58. — Parallel forces 
 
 59. — Parallel forces 
 
 60. — Parallel forces 
 
 61. — Parallelogram of forces .... 
 62. — Parallelogram of forces .... 
 
 63. — Couples 
 
 64. — Laws of inclined planes 
 
 65. — Laws of inclined planes 
 
 66. — Law of pendulum 
 
 67. — Employment of pendulum in obtaining laws of falling bodies 
 
 68. — Co-efficient of friction 
 
 69. — Co-efficient of friction 
 
 70- — Acceleration of gravitation 
 
 CHAPTER VIL 
 
 PAGE 
 
 74 
 
 75 
 75 
 75 
 76 
 
 76 
 77 
 77 
 77 
 78 
 
 79 
 81 
 
 83 
 84 
 85 
 89 
 89 
 90 
 
 MOMENTS OF INERTIA. 
 
 71. — Uniform field of magnetic force and chronograph 
 
 72. — Laws of falling bodies . 
 
 73. — Laws of falling bodies .... 
 
 74. — Laws of falling bodies . 
 
 75. — Laws of falling bodies .... 
 
 76. — Time of pendulum .... 
 
 77. — Centripetal force 
 
 78. — Conical pendulum .... 
 79. — Compound pendulum .... 
 80. — Ballistic pendulum and moment of inertia 
 81. — Reduction of moments of inertia 
 82. — Measurement of^magnetism 
 
 97 
 103 
 104 
 105 
 106 
 III 
 112 
 114 
 
 "5 
 119 
 124 
 127 
 
 CHAPTER VIII. 
 
 MEASUREMENT OF MAGNETIC FORCE. 
 
 83. — Dipping-needle 129 
 
 84. — Measurement of magnetic force and angular measurement 
 
 with telescope and scale 131 
 
 85. — Bifilar suspension 139 
 
 86. — Horizontal component of the earth's magnetism . . . 142 
 
xvi CONTENTS. 
 
 EXP. '■*™ 
 
 87. — Electrical attraction ^47 
 
 88.— Leyden-jar ^48 
 
 89. — Electro-magnet ^52 
 
 go. — Magnetic curves ^53 
 
 gi. — Electro-magnetic force '55 
 
 CHAPTER IX. 
 
 POTENTIAL AND WORK. 
 
 92. — Contour-line ^^^ 
 
 93. — Equipotential lines 1^4 
 
 94. — Electrometer . , 1^9 
 
 95. — Laws of pulley and screw I77 
 
 CHAPTER X. 
 
 WORK AND HEAT. 
 
 96. — Friction-brake 179 
 
 97. — Friction-brake 182 
 
 98. — Mechanical equivalent of heat 184 
 
 99. — Fixed points of thermometer 186 
 
 100. — Unit of heat 188 
 
 loi. — Unit of heat 188 
 
 102. — Loss of heat i8g 
 
 103. — Specific capacity for heat igo 
 
 104. — Specific capacity for heat of thermometer .... igo 
 
 105. — Mechanical equivalent of heat . -- 192 
 
 106. — Specific heat ... ig4 
 
 107. — Latent heat of ice ig7 
 
 108. — Latent heat of steam igg 
 
 log. — Co-efficient of expansion of glass 201 
 
 no. — Co-efficient of expansion of alcohol 202 
 
 CHAPTER XI. ^ 
 
 ELECTRICAL WORK AND ELECTRICAL MEASUREMENTS. 
 
 iti. — Magnetism 207 
 
 112. — Magnetic induction 208 
 
 113. — Voltaic action 211 
 
 114. — ^Voltaic action 211 
 
 115. — Gravity cell 212 
 
 ii6.-^Water-battery . 213 
 
 117. — Heat and electricity 215 
 
CONTENTS. xvii 
 
 ='='•• PAGE 
 
 n8. — Heat and electricity 216 
 
 119. — Ohm's law 216 
 
 120. — Specific electrical resistance 218 
 
 121. — Wheatstone's bridge 219 
 
 122. — Wheatstone's bridge . 220 
 
 123. — Resistance coils 221 
 
 124. — Resistance of galvanometer 222 
 
 125. — Electrical resistance and temperature . . . 223 
 
 126. — Fall of electrical potential . 224 
 
 i27.^-Work of electrical current 227 
 
 128. — Cutting lines of force 229 
 
 129. — Strength of magnetic field and determination of the ohm . 230 
 
 130. — Electro-magnetic induction 236 
 
 131. — Measurement of electrical current in amperes . . . 241 
 
 132. — Measurement of electro-motive force 242 
 
 133. — Measurement of electro-motive force ..... 243 
 
 134. — Measurement of electro-motive force . . . . 244 
 
 135. — Kirchhoff's laws 246 
 
 136. — Electro-motive force in absolute units .... 247 
 
 137. — Measurement of reduction factor of a galvanometer . . 247 
 
 138.— rMeasurement of reduction factor of a galvanometer . . 247 
 
 i39.^Measurement of electro-motive force of a large battery . 249 
 
 i40.^Large resistance galvanometer 249 
 
 141. — Resistance of a battery 257 
 
 142. — Electrical polarization 261 
 
 143. — ^Voltameter 261 
 
 144. — Measurement of horizontal intensity of earth's magnetism 262 
 
 CHAPTER XII. 
 
 THERMO-ELECTRICITY. 
 
 145. — Thermo-electric junction 265 
 
 146. — Measurement with a thermo-electric junction . . . 266 
 
 147.— Efficiency of a thermo-pile 268 
 
 CHAPTER XIII. 
 
 SOUND. 
 
 148. — Sound in rarefied air 
 
 149, 
 150. 
 151 
 152 
 
 — Comparison of tuning-forks 
 
 , — Comparison of tuning-fork and wire . 
 
 Comparison of tuning-forks 
 — Measurement of transverse vibration of rods 
 
 270 
 270 
 271 
 271 
 272 
 
xviii CONTENTS. 
 
 EXP. ""AGE 
 
 153. — Vibration of strings 272 
 
 154. — Time of vibration of tuning-fork 273 
 
 155. — Time of vibration of tuning-fork 274 
 
 156. — Transverse vibrations of wires 275 
 
 157. — Propagation of sound-waves 278 
 
 158. — Propagation of sound-waves 279 
 
 159. — Simple harmonic vibration 280 
 
 160. — Interference of sound ....... 281 
 
 161.— Beats 282 
 
 162. — Velocity of sound in a gas 2S3 
 
 163. — Change of temperature of air due to rarefaction and com- 
 pression 285 
 
 164. — Longitudinal vibrations, Kundt's method .... 287 
 
 165. — Longitudinal vibrations, Kundt's method .... 289 
 
 166. — Modulus of elasticity from sound 289 
 
 167. — Velocity of sound in air 290 
 
 168. — Velocity of sound in air 291 
 
 169. — Velocity of sound in wood 292 
 
 170. — Harmonics 292 
 
 i7i.-;-Hannonics 293 
 
 172. — Harmonics 293 
 
 173. — Resonators 295 
 
 174. — Composition of vibrations 297 
 
 175. — Lissajons' curves ........ 298 
 
 176. — Lissajons' curves 299 
 
 177. — Comparison of intensities of two sounds .... 301 
 
 178. — Sympathetic vibrations 302 
 
 179. — Sympathetic vibrations 303 
 
 CHAPTER XIV. 
 
 LIGHT. 
 
 180. — Angles of incidence and reflection ..... 307 
 181. — Angles of incidence and reflection ..... 307 
 182. — Comparison of intensities of illumination at different dis- 
 tances 308 
 
 183. — Rumford's photometer ....... 309 
 
 184. — Bunsen's photometer ........ 309 
 
 185. — Pickering's photometer 311 
 
 186. — Measurement of index of refraction 315 
 
 187. — Minimum deviation and measurement of angle of prism . 317 
 
 1.88. — Principal focus 321 
 
CONTENTS. 
 
 
 
 xix 
 
 EXP. PAGE 
 
 189. — Conjugate foci 321 
 
 190. — Compound lenses 
 
 
 
 
 323 
 
 191. — Focal length of concave lens 
 
 
 
 
 324 
 
 192. — Index of refraction of a liquid 
 
 
 
 
 324 
 
 193. — Size of images . 
 
 
 
 
 325 
 
 194. — Concave lens 
 
 
 
 
 326 
 
 195. — Concave lens 
 
 
 
 
 326 
 
 ig6. — Plano-concave lens 
 
 
 
 
 327 
 
 197. — Projection of solar spectrum 
 
 
 
 
 328 
 
 198. — Spectroscope 
 
 
 
 
 331 
 
 199. — Spectroscope 
 
 
 
 
 332 
 
 200. — Spectroscope 
 
 
 
 
 332 
 
 201. — Absorption spectra 
 
 
 
 
 332 
 
 202. — Absorption spectra 
 
 
 
 
 333 
 
 203. — Absorption spectra 
 
 
 
 
 333 
 
 204. — Interference of light . 
 
 
 
 
 334 
 
 205. — Interference of light . 
 
 
 
 
 334 
 
 2o5. — Diffraction of light . 
 
 
 
 
 335 
 
 207. — Diffraction of light . 
 
 
 
 
 337 
 
 208. — Newton's rings . 
 
 
 
 
 ■ 337 
 
 209. — Distribution of energy in solar spectrum . 
 
 
 
 344 
 
 210. — Distribution of energy in solar s 
 
 Dectrum . 
 
 
 
 346 
 
NEW PHYSICS. 
 
 CHAPTER I. 
 
 GENERAL IDEAS OF LENGTH AND VOLUME. 
 
 Our knowledge of the world about us depends 
 upon measurements. These measurements can be 
 of two kinds — qualitative and quantitative. The 
 child generally makes qualitative measurements 
 in order to ascertain the nature and extent of its 
 surroundings. Thus, one learns that one body is 
 heavier than another, that the distance between 
 two bodies is greater than that between two other 
 bodies, that one body is hotter than another, by 
 what are called qualitative tests. As we grow old- 
 er and more mature, we find that it is necessary to 
 know exactly how much heavier one body is than 
 another, and exactly how far one body is from 
 another. The parts of a steamship must be fitted 
 together with the utmost nicety, otherwise the ocean 
 never could be crossed. Whenever we make care- 
 ful measurements of anything, our measurements 
 cease to be qualitative and become quantitative. 
 In qualitative tests of the nature of things we do 
 not need accurate standards of length or weight. 
 We can ascertain without a balance that lead is 
 heavier than cork. We can pace off a certain dis- 
 
2 NEIV PHYSICS. 
 
 tance and discover that it is greater than another 
 distance without actually knowing how long one 
 of our paces is. If we wish, however, to ascertain 
 whether one coin is gold and another half gold and 
 copper, we must make quantitative measurements, 
 and need a standard of weight. If we wish to or- 
 der a carpet for a room, we must do more than pace 
 it. We must have a foot-rule or a metre-scale, and 
 we must measure it exactly. 
 
 Our increase in knowledge can therefore be 
 tested by our care in making measurements. The 
 difference between the scientific man and the un- 
 scientific man is that one makes accurate measure- 
 ment while the other is satisfied with general 
 knowledge and does not ask the question, " How 
 much ? " In order to answer the question " How 
 much ? " in regard to distance, we must settle upon 
 some unit which will not change with time, which 
 will be the same a hundred years from now as it is 
 to-day. The paces of a man alter from year to 
 year, and different men have different paces. It is 
 very important to the advance of science that the 
 measure of length should be invariable in all coun- 
 tries. The French measure of length is called a 
 metre, and is about three feet, or exactly 3.280 feet. 
 The advantage of using the metre is that it is di- 
 vided into decimal parts — a decimetre is a tenth of 
 a metre — a centimetre, one-hundredth of a metre — 
 a millimetre, one-thousandth of a metre — so that to 
 reduce, for instance, 3,600 millimetres to metres you 
 have merely to move the decimal-point three places 
 to the left, or, in other words, divide by one thou- 
 sand. To reduce inches to feet, in English measure, 
 it is necessary to divide by twelve, and this takes 
 
IDEAS OF LENGTH AND VOLUME. 3 
 
 time and mental effort. There is, therefore, econ- 
 omy in the use of the French or decimal system of 
 measure of length. 
 
 Experiment i. — Ascertain the average length 
 of one of your paces by measuring a certain dis- 
 tance and then pacing it and dividing the distance 
 by the number of your paces. Then pace a room 
 to ascertain how many yards of carpet will be neces- 
 sary to cover it. Suppose the carpet is three dollars 
 a yard. Estimate its cost. 
 
 Experiment 2. — Measure the length and breadth 
 of your room in both French and English measure. 
 
 Instructions. — Record each measure as follows, 
 in fractions of a centimetre and also of an inch : 
 
 Measures of 
 Length. Length. Breadth. Breadth. 
 
 Cm. Inches. Cm. Inches. 
 
 Mean _ Mean _ Mean _ Mean _ 
 
 length ~ ■ ■ ■ ■ length ~ ' ' ' ' breadth ~ ' ' ' ' breadth — • • • • 
 The mean or average length is obtained by dividing the sum 
 of the observations by the number. 
 
 Calculate the expense of carpeting, and compare 
 the results of Experiment i and Experiment 2. 
 
 Experiment 3. — Measure the circumference of 
 two circles, and find the numbeir you must multiply 
 the radius of the circles by in order to obtain the 
 circumferences. 
 
 Instructions. — Draw two circles of different radii 
 upon stiff paper. Cut these circles out carefully 
 with scissors, and, having drawn a straight line 
 upon the page of your note-book, cause the cir- 
 
4 N£IV PHYSICS. 
 
 cles to roll along the straight line and measure the 
 spaces passed over. These spaces are evidently the 
 circumferences of the circle. Measure the radii of 
 the circles and divide the length of the circumfer- 
 ences of the circles by the values of the radii. In 
 this way you will obtain the number by which you 
 must multiply the radius of any circle by to obtain 
 the length of the circumference. This number is 
 generally wrritten 2 tt, in which ir =3.1416. 
 
 Hence, one half of the number you obtain ex- 
 perimentally should be equal to 3.1416. 
 
 Having settled upon our unit of length, we can 
 determine how many cubic metres or feet could 
 be placed in a given room. A cubic foot measures 
 one foot in breadth, one foot in length, and one foot 
 in height. A cubic metre measures one metre in 
 breadth, one metre in length, and one metre in 
 height. Moreover, one cubic foot contains 1,728 
 cubic inches, and one cubic metre, 1,000,000 cubic 
 centimetres. Knowing the volume of water which 
 certain standard vessels contain, we can ascertain 
 the volume of any irregular body. It is found most 
 convenient to adopt as unit of volume a cubic cen- 
 timetre — that is, the volume occupied by a cube, 
 each side of which is a centimetre, or one-hun- 
 dredth of a metre. 
 
 Experiment 4. — Measure the volume of water 
 which two hollow cylindrical vessels will contain in 
 quarts and in litres, also in cubic inches and cubic 
 centimetres. 
 
 Instructions. — One vessel graduated to cubic 
 centimetres is necessary. From a table of French 
 and English measures calculate the number of cm. 
 in a litre and in a quart. 
 
IDEAS OF LENGTH AND VOLUME. 
 
 5 
 
 Experiment 5.— Compare the volume of two 
 cylinders of the same height. 
 
 Instructions. — Proceed as in the previous experi- 
 ment, and obtain the volumes in cubic cm. Since 
 the cylinders have the same height, the variation in 
 the volumes must be due to the radii of the bases of 
 the cylinders. Measure the radii of the bases, and 
 see whether the volumes of the cylinders vary as 
 these radii, or as the square of these radii. Then 
 find by trial the value of the number C by which 
 you must multiply the radius ; or the radius squared 
 in order to obtain the area of the bases of the cylin- 
 ders. The volume of a cylinder is equal to the area 
 of its base multiplied by its altitude. Prove this by 
 your measurement, and compare the value of the 
 number C with the value of it obtained in Experi- 
 ment 2. 
 
 Experiment 6. — Measure the volume of an ir- 
 regular piece of stone. 
 
 Instructions. — Suspend the stone by a fine string 
 or wire in a glass vessel filled with water to a cer- 
 tain height, so that the displacement of water by 
 the stone will cause the water to flow out of the 
 vessel through an orifice. Measure the amount of 
 water that overflows, and this will give you the 
 cubic contents of the stone. 
 
CHAPTER II. 
 
 SPECIFIC GRAVITY. 
 
 Our experiments thus far would enable us to 
 measure distances, and to ascertain the volume or 
 space occupied by bodies. Besides the ideas of 
 space, which we obtain first by rough estimates and 
 afterward by careful measurements with a standard 
 of length, we obtain ideas of the relative weight of 
 bodies. We perceive by our muscular sensations 
 that it is harder to lift one body than another, and 
 we could obtain equal volumes of different sub- 
 stances and compare the muscular sensations we 
 should have in lifting these equal volumes in com- 
 parison with lifting an equal volume of water. To 
 perceive how erroneojus such a qualitative test 
 would be, it would only be necessary to blindfold a 
 person and give him the unit of volume made of 
 different substances and ask him how much more 
 or less they weighed than an equal volume of water. 
 Long practice might give a certain skill, but no one 
 would choose to have his gold or silver estimated 
 in this way. We could, however, assume a weight 
 for a cubic centimetre of water at o° C. or 32° Fahr., 
 since this is a fixed temperature, and compare the 
 weight of other bodies with this standard. 
 
 Experiment 7. — Coil a fine steel wire around 
 
SPECIFIC GRAVITY. 7 
 
 a smooth cane or other cylinder, so that, when the 
 wire is slipped ofi the cylinder, it shall form a spiral. 
 Suspend this spiral verticall}'- in front of a long 
 piece of looking-glass. Arrange a bit of wire upon 
 the spiral in any suitable position, so that it may 
 indicate by its movement up or down the change of 
 weight of a body that is put in a little scale-pan, 
 which is attached to the lower end of the spiral. 
 The object of the mirror is to prevent the error 
 which would arise from the different position of the 
 observer's eyes. When the pointer covers its re- 
 flection in the mirror, the eyes, the pointer, and the 
 reflection are all on the same line. With this in- 
 strument compare the weight of a cubic centimetre 
 of water, which weighs one gramme, with that of a 
 cubic centimetre of alcohol. 
 
 Note. — In the first place, test the accuracy of the spring with dif- 
 ferent weights, and be careful not to permanently stretch the spring. 
 
 Experiment 8.* — You are given six straight 
 wires and one twisted one, all cut from the same piece, 
 and are required to find, by measuring and weighing 
 the straight pieces, the average weight of a centi- 
 metre of the wire, and thence to calculate the length 
 of the twisted piece. 
 
 Instructions. — The observations are to be record- 
 ed in the following form. 
 
 Finish all the weighing and measuring before 
 doing any calculation. 
 
 The calculated weight W' of each wire is found 
 by taking the mean value of the weight of one cen- 
 timetre and multiplying this by the length. 
 
 * From Professor Hinrich's " Elements of Physics." 
 
NEW PHYSICS. 
 
 NO. OF 
 WIRE. 
 
 L, length in 
 
 cm. 
 
 W, weight in 
 grammes. 
 
 p, weight of 
 I cm. 
 
 W, calc. wt. 
 of wire. 
 
 E, error of 
 W-W.' 
 
 No. I. 
 
 
 
 
 
 
 No. 2. 
 
 
 
 
 
 No. 3. 
 
 
 
 
 
 
 No. 4. 
 
 
 
 
 
 
 No. 5. 
 
 
 
 
 
 
 No. 6, 
 
 
 
 
 
 
 Bent wire, 
 
 Mean, .... 
 Weight, .... Calc. length, 
 
 'Experiment 9. — Obtain the specific gravity of 
 the piece of stone of which you have obtained the 
 volume in Experiment 6. 
 
 Instructions. — The specific gravity of a body, or 
 its density, is its weight compared with the weight 
 of an equal volume of water. Since, in French 
 measure, a cubic cm. of water weighs i gramme, if 
 one should find that the stone weighs 100 grammes, 
 and has a volume of 10 cubic cm., each cubic cm. 
 will weigh 10 grammes, or 10 times as much as a 
 cubic cm. of water, and hence the specific gravity of 
 the stone will be 10. You must therefore obtain 
 the weight of the stone. To do this, suspend it 
 from an ordinary spring-balance. 
 
 Experiment 10. — The upward pressure on a 
 body immersed in water is equal to the weight of 
 the volume of water displaced. 
 
 Instructions. — Suspend the stone used in Experi- 
 ment 6 by a fine string from the hook of a spring- 
 
SPECIFIC GRAVITY. 9 
 
 balance, and immerse the stone in water. Compare 
 the weight indicated by the balance when the stone 
 is weighed in air and afterward in water, and com- 
 pare the loss of weight with the weight of the water 
 which was displaced by the stone (Experiment 6). 
 
 Experiment i i . — Find the density of acohol by 
 weighing in the liquid. ^ 
 
 Instructions. — Suspend the stone used in Experi- 
 ment 6 from a spring-balance and immerse it in a 
 vessel containing alcohol. Observe the loss of 
 weight, and compare it with the loss of weight of 
 the stone when it is immersed in water. Since the 
 loss of weight is the weight of an equal volume of 
 liquid, we can compare the weights of the equal 
 volumes of water and alcohol, and this comparison 
 will give us the specific gravity of the alcohol. 
 
 The advantage of the term " specific gravity " 
 can be perceived from the preceding experiments. 
 It tells us what we must multiply a volume of water 
 equal to that of the given substance by in order to 
 obtain its weight. The adoption of water as a stand- 
 ard of volume has great convenience, especially in 
 the French measure, for the weight of one cubic cen- 
 timetre of water at d° C. is one gramme, and, when 
 we know the volume of any vessel, we also know 
 what weight of water it contains, and if it contains 
 any other liquid we can also obtain its weight by 
 ascertaining its specific gravity. The specific grav- 
 ity of a man is not far from that of water ; therefore, 
 a man should be able to float in water with very 
 slight exertion. 
 
 There is a large class of instruments called hy- 
 drometers which determine the specific gravity of a 
 liquid by the degree to which they sink in this liquid 
 
10 NEW PHYSICS. 
 
 compared with their depth in water. It is very 
 easy, for instance, to attach a weight to a hollow 
 tube closed at one end so that it will float vertically 
 in water at a certain depth, and at another depth in 
 pure alcohol, and to graduate the distance between 
 the lines which mark these two depths, so that the 
 graduation shall give the percentage of alcohol in 
 a mixture of water and alcohol. These are rela- 
 tive instruments — that is, they give relative indica-- 
 tions. 
 
 Experiment 12. — Subdivide the tube of a hy- 
 drometer so that the scale will indicate percentages 
 of alcohol in a mixture of alcohol and water. The 
 word subdivision is used to denote the operation of 
 dividing a scale into portions which shall represent 
 — for instance, in this case — the immersion of the 
 hydrometer-tube with different percentages of alco- 
 hol. Thin glass tubes suitable for hydrometers can 
 be readily obtained. An ordinary piece of glass 
 tubing, can be drawn out at one end into a bulb ; in 
 this bulb can be placed a suitable number of fine 
 shot, in order to make the tube float upright. A 
 paper scale can be pasted inside the tube, and the 
 value of its scale divisions, in percentages of alcohol, 
 can be entered in a note-book. These percentages 
 can be made by mixing known volumes of alcohol 
 with known volumes of water. 
 
 We can employ water in general as a medium 
 by which the weights of different bodies can be 
 compared. It is only necessary to place the body 
 to be weighed upon a float and see how far it de- 
 presses the float, and then replacing the weight by 
 known weights until the same point of immersion 
 is reached. The weight of the body will then be 
 
SPECIFIC GRA VITY. 
 
 TI 
 
 equal to the weights which produced the same im- 
 mersion. Attaching the body afterward beneath 
 the float in the water and placing weights upon the 
 float until the same fixed point of immersion is 
 reached, we know that these weights balance the 
 upward pressure of the water which the body dis- 
 places, and are equal to the number of cubic centi- 
 metres in the volume of the body. 
 
 Experiment 13. — A hydrometer, called Nichol- 
 son's, can be readily made by any 
 tinman. It consists of a cyhndri- 
 cal float, to the top of this is at- 
 tached a stout wire that supports 
 a plate upon which the weights 
 and the body to be weighed can 
 be placed. At the other extremi- 
 ty of the float is attached a little 
 scale-pan in which the body can 
 be immersed in the liquid. The 
 float is weighted so that it re- 
 mains upright, and sinks to a cer- 
 tain mark c upon the wire. 
 
 The body is first placed upon 
 the plate with additional weights 
 in order to sink the hydrometer 
 to a fixed mark upon the stem. The body is then 
 removed, and the increase of weight necessary to 
 again sink the hydrometer to the same mark will 
 give the weight of the body in air. The latter is 
 then placed in the lower scalcrpan. The weights in 
 the upper plate, or pan, necessary to sink the hy- 
 drometer to the same level, will be greater than 
 when the body was placed in the upper pan. The 
 difference of the weighing of the body in the two 
 
 Fig. I. 
 
12 NEW PHYSICS. 
 
 pans gives the loss of weight of the body in water, 
 or, in other words, the upper pressure of the water. 
 This upward pressure is equal to the weight of the 
 water displaced by the body. Knowing this weight 
 in grammes, we know the volume of the body in 
 cubic centimetres, and dividing the weight in air 
 by the loss of weight in water will, therefore, give 
 us its specific gravity. If we call W the weight of 
 the body in air, and Wj its loss of weight in water, 
 
 we have sp. gr. = rrr^- 
 
 Our measurements of weight and volume thus 
 far have been conducted without reference to the 
 air about us. This air, however, has weight. We 
 are immersed in it, together with all our apparatus. 
 It is as if we were at the bottom of a sea, and were 
 conducting our measures of weight with reference 
 to a lake of mercury, which answers to the water 
 when we are in the air. We might compare the 
 weights of bodies with the weight of a cubic centi- 
 metre of mercury, and obtain our specific gravities 
 by weighing the body in the water which surrounds 
 us instead of the air, and afterward observing the 
 loss of weight in the mercury. Suppose, howevfer, 
 that the glass vessel with which we make our meas- 
 urements of volume should contain air instead of 
 water, it is evident that it would not weigh as much 
 as an equal vessel filled with water. In making our 
 calculations we should have to make allowances for 
 the weight of water which fills the vessel, for we 
 are balancing this vessel, with its contained water, 
 against certain weights which do not displace the 
 same volume of water as the vessel. 
 
 Above the water and in our atmosphere we are 
 
SPECIFIC GRAVITY. 13 
 
 also surrounded by a sea of air which exerts press- 
 ure just as the water does when we are beneath the 
 ocean. The common experiment of pressing a disk 
 of wet leather against the floor, and endeavoring to 
 pull it away by a string which is attached to its 
 middle, illustrates the weight of the atmosphere. 
 In endeavoring to pull up the disk, we are endeav- 
 oring to lift a column of air which extends from 
 the floor to the extreme limits of our atmosphere. 
 This column has a weight equivalent to fifteen 
 pounds on every square inch. If our disk were 
 just one square inch, and, if we should attach a 
 spring-balance to the string which is connected 
 with the disk, the balance would register a pull of 
 fifteen pounds just as the disk is pulled from the 
 floor. The higher we go up in our atmosphere, the 
 less pull we should have to exercise. It is evident, 
 therefore, that if we make delicate weighings we 
 must take into account the weight of the air in the 
 vessel whose volume we wish to measure. 
 
 Our experiments hitherto upon the relative 
 masses of two bodies have been performed by the 
 aid of given weights. Now, the weight of a body 
 depends upon the pull of gravitation upon every par- 
 ticle in the body. Bodies differ from each other in 
 weight because there is more or less matter to be 
 attracted to the earth, and to also exercise attrac-_ 
 tion upon the earth, in the one than in the other. 
 The force of attraction, or the pull of gravitation, 
 varies at different places on the globe. A pound 
 of tea at Canton is attracted with a certain force. 
 When we carry this pound of tea and measure the 
 pull of gravitation upon it in Newfoundland we find 
 that it weighs more than a pound. If, however. 
 
14 
 
 NEW PHYSICS. 
 
 in Newfoundland we balance it against the same 
 weights that we used at Canton, it will still balance 
 with a pound-weight, for the pull on the weight is 
 the same as the pull on the pound of tea. The force 
 of attraction between the particles of the tea and the 
 earth and the particles of the standard weights and 
 the earth, is less at Canton than in Newfoundland, 
 yet the amount of matter in the volumes of the two 
 substances remains the same. In our operation of 
 weighing we really compare, therefore, the amount 
 of matter in two volumes. This, is what we call 
 mass of the bodies.* We determine by weighing 
 the masses of two bodies and not their absolute 
 weight, or the force by which they are attracted to 
 the center of the earth. In order to determine the 
 weight, we must know the exact value of the force of 
 attraction of the earth at the place whe;-e we make 
 our experiment. This force has been carefully de- 
 termined at the latitude of Paris, and our gramme- 
 weights will represent at Paris the absolute weight 
 of one gramme ; in any other latitude one-gramme 
 weight has not an absolute value, and we must cor- 
 rect it by ascertaining the ratio between, the force 
 of gravitation at that latitude and at the latitude of 
 Paris. In practice, however, it is not necessary to 
 know the absolute weight, and in weighing we de- 
 termine the masses of bodies. The methods that 
 have been given in this treatise for determining the 
 masses of bodies are sufficiently accurate for many 
 purposes ; but when it is necessary to determine the 
 mass of a body (what is commonly termed weight) 
 to the hundredth and thousandth place of decimals, 
 or, in other words, to centigraimmes and to milli- 
 
 * Mass will be defined by reference to inertia later. 
 
SPECIFIC GRAVITY. 15 
 
 grammes, we use the chemical balance. This is ex- 
 plained in all text-books of physics, and it is not 
 necessary to enter into a full description here. In 
 its elements it is a horizontal bar balanced upon a 
 knife-edge. It has also a knife-edge at each extrem- 
 ity, upon which are hung the scale-pans. A pointer 
 shows when the balance-arm is horizontal, or, in 
 other words, when the masses in the two scale-pans 
 balance each other. We test the equality of the 
 length of the arms of the balance by placing a body 
 first in one pan and then in the other, and see if the 
 same weights are necessary to bring the balance 
 into a horizontal position in the two cases. The 
 center of gravit)^ of the balance-arm should be 
 slightly below the knife-edges. If it is too low, the 
 balance loses in sensitiveness ; if too high, the bal- 
 ance becomes unstable ; and if it is on the line with 
 the knife-edges, the balance-arm will not return to 
 its horizontal position after it has been disturbed. 
 The three knife-edges must be in line. These pre- 
 cautions, it will be observed, arise from the action 
 of gravitation upon the particles in the balance-arm. 
 When the particles are equally distributed along 
 the balance-arm, the pull of gravitation on one side 
 of the middle of the balance-arm is equal to the pull 
 on the other side. A pointer is attached to the bal- 
 ance-arm, and this should point to zero on the little 
 scale attached to the balance. When only moder- 
 ate accuracy is needed, we can take the mean of 
 two successive points at which the pointer or index- 
 hand turns when the balance-arm is swinging, and 
 take the mean of these for the zero-point. If we 
 can not make the pointer return to a zero-point 
 when we are weighing, on account of want of small 
 
i6 NEW PHYSICS. 
 
 weights, we can observe hojv much the pointer 
 moves for a centigramme or a milligramme, and 
 determine how much of this space the pointer lacks 
 from the zero when the balance is unloaded. In 
 the appendix will be found directions for weighing 
 when great accuracy is desired. 
 
 Experiment 14. — The general precautions are 
 as follow : The object, if perfectly dry and non- 
 corrosive, can be placed directly in the scale-pans. 
 If it will attack the scale-pan, it is usual to place it 
 in a watch-glass or other protecting vessel, which is 
 then rested upon the scale-pan. The balance-arm 
 being off its knife-edges, place one of the weights 
 in the other scale-pan, and carefully turn the milled 
 head so as to lower the balance-arm upon its knife- 
 edge. This must be done very carefully and slowly. 
 If the' beam falls suddenly upon the knife-edges they 
 are apt to be ruined. When large weights are used, 
 a very small amount of lowering will show which 
 pan of the balance contains the larger weights. The 
 smallest weights must not be touched with the fin- 
 gers ; a pair of forceps is always provided for them. 
 Weights should be deposited either in the scale-pan 
 or in their proper places in the box. If they are 
 placed on the table, they become dirty, and are gen- 
 erally swept off by the arm of the weigher and' 
 lost. To save time in weighing, proceed as fol- 
 lows: Place the body in one scale-pan, and, for in- 
 stance, a 6-gramme weight in the other. The point- 
 er moves toward the 6 grammes. This shows that 6 
 grammes is too little. Replace by twice 6 grammes 
 or 12 grames. This is too much, and the pointer 
 moves in the opposite direction, halve the additional 
 6 grammes— that is, add 3 grammes, making in all 
 
SPECIFIC GRAVITY. 
 
 17 
 
 9 grammes. This is too light. Add one half of 3 
 grammes, or one gramme and five decigrammes, 
 making in all 10.5. This is too heavy. Subtract seven 
 decigrammes ; this is still too light. The weight 
 is between 9.8 and 10.5. Proceed in the same way, 
 and do not add small weights by guess, for much 
 time will be lost. Make a set of small weights out 
 of tin-foil by weighing a rectangle of the foil of a 
 suitable size, and, having computed its area by mul- 
 tiplying its length by its breadth, cut it up into 
 parts which will represent proportional parts of the 
 ascertained weight of the rectangle. The weights 
 of these parts thus found by calculation can be com- 
 pared with their actual weight in the balance-pan. 
 It is well to turn up one corner of these weights 
 in order to take them up readily with the pin- 
 cers. The amount of weight they represent can be 
 punched upon them. 
 
 By means of a delicate balance we can repeat 
 many of the preceding experiments and obtain 
 closer results. By attaching the body, whose spe- 
 cific gravity we wish to obtain, by means of a fine 
 string to a hook beneath one scale-pan of a balance, 
 and weighing the body in water, and afterward in 
 air, we can obtain its specific gravity by dividing 
 its weight in air by its loss of weight in water. A 
 balance adapted for this is called a hydrostatic bal- 
 ance. 
 
 Experiment 15. — Determine the specific grav- 
 ity of lead by the hydrostatic balance. Allowance 
 should be made for the fine wire by which the body 
 is attached to the bottom of one of the scale-pans. 
 If it is moistened with a little spirits of wine, the pull 
 of the water upon it can be very much lessened. 
 
1 8 NEW PHYSICS. 
 
 Experiment i6. — Determine the specific grav- 
 ity of lead-shot by a specific-gravity bottle. Obtain 
 of the druggist, a glass vial with a ground-glass 
 stopper. The capacity of the vial should be about 
 30 cubic centimetres. Determine the capacity by 
 weighing the bottle first empty, and afterward full 
 of water. The difference between these two weigh- 
 ings is the weight of the volume of water which the 
 vessel contains; it will be expressed in grammes 
 and decimal parts of a gramme ; and the number of 
 grammes will also give the number of cubic centi- 
 metres, since a cubic centimetre of water at 0° 
 weighs one gramme. Then, having weighed a cer- 
 tain amount of the lead-shot, pour out some of the 
 water of the flask and pour in the shot. Shake the 
 bottle gently to drive out the air-bubbles, fill up 
 with water, and weigh again. 
 
 Let W r= weight of shot in air. 
 
 " w = " " bottle and water. 
 
 " Wj = " " bottle, water, and shot. 
 
 " d = density of water at the temperature of 
 the experiment. 
 
 The weight of the water displaced is W -|- w — 
 
 dW 
 Wj ; hence the specific gravity = 8 = :r== —— - 
 
 W -j- w — Wj. 
 
 Experiment 17. — Determine the specific grav- 
 ity of alcohol, and its percentage strength. Having 
 ascertained the capacity of the specific-gravity bot- 
 tle, pour out the water and carefully rinse the vessel 
 with small amounts of alcohol several times. Then 
 fill with alcohol and weigh again, taking care that 
 the alcohol is at the same temperature as the water 
 which we used to obtain the capacity of the bottle, 
 to obviate corrections for variations in the capacity 
 
SPECIFIC GRAVITY. ig 
 
 of the flask, since a change of temperature would 
 cause the bottle to expand or contract. The weight 
 of the water divided by that of the alcohol will give 
 the specific gravity of the alcohol. 
 
 In our determination of the specific gravity of 
 bodies we speedily discover that the temperature 
 of a body greatly influences our results. All bod- 
 ies, with few exceptions, expand under the influ- 
 ence of heat. The point of rest of our spring- 
 balances which were used in Experiment 7 will 
 change with the temperature of the room. Air 
 heated in a tightly-corked flask will burst the flask. 
 A tightly-fitting glass stopper can often be removed 
 by placing a hot wet cloth around the neck of the 
 bottle. When a body expands, its volume is larger, 
 although its mass is the same ; it will, therefore, dis- 
 place a larger amount of water. If, then, we divide 
 the weight in air by the loss of weight or the vol- 
 ume of displaced water, at a temperature above 
 zero, our specific gravity would be too small for 
 thte specific gravity at 0°. The greater the expan- 
 sion of our substance, the less would be its specific 
 gravity. Hence, to refer the specific gravity to 0° 
 C, we must correct for the cubical expansion for 
 each degree of temperature. 
 
 Let us see what the cubical expansion is. The 
 coefficient of expansion of a long rod, for instance, 
 under the effect of t° would be found' by (knowing 
 the length of the rod at 0°) dividing the elongation 
 expressed in fractions of a metre by the length of 
 the rod in metres, and by the number of degrees of 
 temperature above 0°. Or the volume at the tem- 
 perature / would be /, the length at 0° plus the in- 
 crease in length at t°. 
 
20 NEW PHYSICS. 
 
 We can express this increase as a certain part 
 of / multiplied by the rise in temperature t or kit 
 in which k is called the coefficient of expansion. 
 Hence the volume at t° = I -\- kit = I {i -\- kt). 
 This is the formula for linear expansion. Let us 
 take now the expansion of a cube. Each side of the 
 cube expands to / (i + kt). 
 
 The volume of the cube is the product of its 
 base by its altitude or /* (i -\-ktf. If we expand 
 (i -|- >^^)^ knowing that ^ is a very small quantity, 
 we can neglect the squares and cubes of k, and can 
 take I + 3y^/ as the value of i -\-ikt-{- ^Bt^ + '^^^ 
 Hence 3^, the coefficient of cubical expansion, is 
 three times the coefficient of linear expansion. Let 
 us place 3^ = /8. Now we can reduce our values 
 of specific gravity obtained at t° to 0°. 
 
 Experiment 18. — If sp. gr. and (sp. gr.)' are 
 the specific gravities of lead at t and t'^ degrees, and 
 W the weight of the lead, V the volume at 0°, and 
 yS the coefficient of cubical expansion, we have the 
 volumes at t^ and t°. 
 
 V(i + ^t) and V(i + /S/*) and 
 
 sp-gr- = „, , - - (sp. gr.y 
 
 V(i J^^t)^^ ^ ' V(i + /i/S) 
 
 also^ = fP-^'-t7^^P-g"^ 
 (sp. gr.)'/i - (sp. gr.)/. 
 
 This method, therefore, enables us to obtain the lin- 
 ear expansion, for this is one third of /8. 
 
CHAPTER III. 
 
 PRESSURE OF THE AIR. 
 
 What is called the force of buoyancy arises 
 from a difference of pressure. The upward press- 
 ure, for instance, on a surface at c c' (Fig-. 2) exceeds 
 the downward pressure on b b' by the weight of the 
 column of liquid b c c' b' . What is often called the 
 amount of buoyancy is the weight of the volume 
 of fluid which is displaced by the 
 body. What is erroneously called 
 the force of suction is also a mani- 
 festation of the pressure of the at- 
 mosphere. We remove the pressure 
 of the atmosphere from one end of a 
 tube by drawing out the inclosed 
 air, and, the pressure being thus re- 
 moved at this end, the atmosphere 
 forces itself, or any fluid, into the other end. 
 
 Experiment 19. — Bend a glass tube which is 
 open at both ends into the form of a U, one arm 
 being somewhat longer than the other ; fill the U- 
 tube with water or any liquid up to the level of the 
 shorter arm. Close the end of the shorter arm 
 tightly with the finger, and endeavor to suck out 
 the liquid. Why do you not succeed ? 
 
 Experiment 20. — In order to obtain the weight 
 
 Fig. 
 
22 NEW PHYSICS. 
 
 of air, we can proceed as follows : Take a Florence 
 flask of about 300 cubic centimetres, fill it one quar- 
 ter full of water, place it over a sand-bath, and boil 
 the water. The steam will expel the air in the flask. 
 After the boiling has been conducted for some time, 
 remove the flask from the sand-bath, and, when the 
 boiling has ceased, loosely cork the flask and cool it 
 by putting a wet cloth around the upper portion. 
 This will cause the water to boil again. Then tightly 
 cork the flask with a rubber cork, and, the tempera- 
 ture having become that of the room, weigh the 
 flask and the water that remains in it. The pecul- 
 iar water-hammer sound that the flask gives when 
 shaken will show that the air which the flask held 
 has been expelled. Then take out the cork and 
 weigh the flask again. It is now filled with air and 
 water, the amount of water being the same in both 
 w^eighings. The difference in the weighings will 
 give the weight of the number of cubic centimetres 
 of air above the water in the flask. Knowing the 
 capacity of the flask, we have the weight of one 
 cybic centimetre of air (in contact with water). The 
 weight of one cubic centimetre of dry air at 0° and 
 at 760 millimetres pressure is .001293 of a gramme. 
 A very good approximation to the true value can 
 be obtained with ease in this experiment. In tightly 
 corking the flask, it is well to protect the hands by 
 means of a thick towel from the possible breakage 
 of the flask. 
 
 Experiment 21. — Close a glass tube at one end, 
 grind the edges of the other end upon emery-paper 
 or vipon a grindstone, so that the tube may be 
 closed tightly by means of a piece of glass or a 
 sheet of cardboard; fill the tube with water and 
 
PRESSURE OF THE AIR. 23 
 
 place the glass or cardboard over the open end, and 
 invert the tube. The water will not run out. The 
 pressure of the atmosphere, or, in other words, its 
 weight upon the end of the tube, holds the weight 
 of water in the tube. In this way, a tube thirty feet 
 long, filled with water, could be supported with 
 merely a thin plate upon the open end to distribute 
 the pressure equally. 
 
 Experiment 22. — Close the end of a stout glass 
 tube (see Appendix), which is about one centimetre 
 in internal diameter and 80 centimetres long, with 
 mercury. Place the thumb closely over the open 
 end, and invert it in a vessel filled with mercury. 
 The mercury-column will stand at a height of about 
 760 millimetres. The weight of this column of mer- 
 cury will be found to be fifteen pounds, or about 
 seven kilogrammes. This inverted tube filled with 
 mercury constitutes a barometer. 
 
 Let us not forget that we are at the bottom of 
 an ocean of air, and that all the experiments on 
 weight and pressure of water are equally applicable 
 in the ocean of air. 
 
 It has been shown by Andrews and others that 
 by sufficiently great pressure and at sufficiently low 
 temperature we can compress all gases into the 
 liquid form, and that there is a continuous grada- 
 tion from the state of a liquid to the state of a gas. 
 A beautiful experiment consists in inclosing a cer- 
 tain amount of carbonic-acid gas in a little glass 
 cylinder, and then, by great pressure, showing that 
 the line of separation between the liquid produced 
 and the carbonic acid left above the liquid is a 
 shadowy one, and that the Uquid fades insensibly 
 into the gas. 
 
24 
 
 NEW PHYSICS. 
 
 The simple experiment of blowing soap-bubbles 
 shows that the pressure of a fluid is exerted equally 
 in all directions. If this were not so, the bubble 
 would not be spherical. 
 
 Experiment 23. — Pour a small amount of liquid 
 glycerine soap (see Appendix) into a shallow ves- 
 sel. Connect a tobacco-pipe, by means of a piece 
 of rubber tubing, to a gas-jet, and allow the gas 
 (street or hydrogen gas) to flow into the pipe. In 
 this way bubbles filled with hydrogen gas can be 
 formed. The weight of the hydrogen gas is less 
 than that of the atmosphere, and these ascend, just 
 as a thin bulb of glass filled with air would ascend 
 from the bottom of the sea. 
 
 The atmosphere in which we are immersed has 
 its waves and its commotions similar to those of the 
 ocean. These commotions are shown by the varia- 
 tions in the height of a barometer. When the crest 
 of a wave is over a barometer, there is greater 
 weight of air, and consequently it takes a greater 
 height or weight of mercury to balance it ; the de- 
 pression of the wave is followed by a low barome- 
 ter. The weight of the air depends upon the 
 amount of moisture it holds. The more moisture it 
 holds the heavier it is. Dry, warm air can take up 
 more moisture than cold. Besides the actual 
 weight of any fluid, we find that the pressure due 
 to this weight upon a body immersed in a fluid is 
 equally exerted in all directions. 
 
 A man does not feel the weight of the atmos- 
 phere, for the pressure within him at all points is 
 equal to that on the outside of his body. 
 
 Experiment 24. — Bend two glass tubes as in 
 Fig. 3 ; pour a certain amount of mercury into 
 
PRESSURE OF THE AIR. 
 
 25 
 
 f3 
 
 Fig. 3. 
 
 them and immerse them in water. The difference 
 in height between the surfaces of the mercury- 
 columns gives the pressure of the water upon the 
 ends of the tube. It will be noticed 
 that the vertical pressure of the 
 water in one tube is equal to its 
 lateral pressure in the other, and 
 this will be true as we turn the 
 tubes about a vertical axis, keeping 
 their orifices always at the same 
 level. By immersing either tube 
 farther, we find that the column 
 of mercury supported is greater, 
 and therefore that the pressure in- 
 creases with the depth. This is 
 necessarily so, since the pressure is simply the 
 weight of a column of water resting upon the lower 
 surface of the mercury. 
 
 This fact, when we reason upon it, shows that 
 the pressure on the bottom of any vessel, which is 
 
 immersed in a fluid, 
 depends merely upon 
 the area of the base 
 of the vessel and the 
 depth of each element 
 of the base below the 
 surface of the fluid. 
 
 Experiment 25. — 
 Obtain from a tinman 
 a cylindrical tube. A, 
 and a tin vessel made 
 m the form of the 
 frustum of a cone, both vessels being open at each 
 end (Fig. 4), the end A being equal to the end B, with 
 
 Fig. 4. 
 
26 
 
 NEW PHYSICS. 
 
 a small flange of tin at one end. Cover this end with 
 a well-fitting piece of leather, to the middle of which 
 is attached a string. Immerse the vessels in succes- 
 sion to the same depth, and pour in water until the 
 leather is released. This will happen when the 
 level of the liquid in the vessel is equal to the level 
 of the water outside. But the pressure on the end 
 A of the cylinder is equal to the weight of the col- 
 umn of water resting upon it. A and B being at 
 the same depth, and having the same area, the 
 pressure is the same upon them, and is independent 
 of the form of the vessels. By inverting the vessel 
 B we can prove that the pressure on the base can 
 be greater than the weight of water which it can 
 hold. 
 
 This behavior of fluids shows that we can lift 
 very heavy bodies by the exertion of a very small 
 pressure. 
 
 For, suppose we exert a pressure of a kilogramme 
 upon a piston through one 
 centimetre, and commu- 
 nicate by means of a fluid 
 this pressure to every unit 
 of area of the larger pis- 
 ton. The pressure mul- 
 tiplied by the distance 
 passed through is the work 
 we exert. The piston P 
 (Fig. 5) moves up a certain 
 height under the influence 
 of the pressure we exert 
 on the smaller piston. We shall therefore have the 
 pressure p on the small piston multiplied by the 
 distance through which it is exerted, or s equal to 
 
 Fig. s. 
 
PRESSURE OF THE AIR. 
 
 27 
 
 the same pressure/ on each unit of area A of the 
 base of the larger piston P multiplied b)'^ the dis- 
 tance through which the larger piston moves ; or — 
 
 p=pAs' 
 
 in order to make these two values of work equal s' 
 must be very small compared with s. 
 
 In order to keep the level of water constant, the 
 following simple contrivance is used, which also 
 illustrates the pressure of the atmosphere : 
 
 Experiment 26. — Fill a small bottle (Fig. 6) and 
 invert its nozzle under the water whose level we wish 
 to keep constant. As 
 the water in the lower 
 vessel evaporates, the 
 nozzle of the suspend- 
 ed bottle is uncovered, 
 and a bubble of air 
 enters and drives out 
 water enough to cover 
 the nozzle again to the 
 same amount. 
 
 In order to drive 
 out the air from a ves- 
 sel, we can exhaust it 
 by means of an air- 
 pump. Instead of us- 
 ing pistons, the mod- 
 ern laboratory-pump employs the mechanical action 
 of falling water or mercury. 
 
 Experiment 27.— Cut a thick bottle in two (see 
 
 Appendix). Fit it tightly with a cork in which a 
 
 short tube of glass is inserted. To this tube join 
 
 hermetically tight a short piece of rubber tubing, 
 
 3 
 
 Fig. 6. 
 
28 
 
 NEIV PHYSICS. 
 
 upon which is sHpped a pinch-cock, P (Fig. 7). Con- 
 nect this rubber tubing with a three-way joint, /, of 
 
 glass or of iron gas-tubing. 
 Insert in one of the ways 
 of this joint a piec'e of glass 
 tubing about a metre long ; 
 one end of this dips in a 
 mercury - vessel, M. Con- 
 nect the vessel to be ex- 
 hausted by a short piece 
 of glass tubing with the 
 other way of the joint. Fill 
 A with mercury and allow 
 the mercury to fall slowly 
 through the long tube, the 
 internal diameter of which 
 should not be more than 
 three millimetres. The mer- 
 cury will fall in detached 
 portions, separated by air- 
 columns, which come from 
 the vessel that we are ex- 
 hausting. This air rushes 
 out to fill the more or less rarefied portions of the 
 vertical tube, the mercury in falling having dis- 
 placed the air. The amount of exhaustion is shown 
 by the height of the mercury-column above the 
 level of mercury in the lower vessel M. The Flor- 
 ence flask F can be removed by applying a jet of 
 gas at a. 
 
 Where a head of water can be obtained, this can 
 ■>e substituted for a head of mercury. An appa- 
 ratus called Richards's aspirator can be screwed 
 upon a water-faucet, and is very useful as a substi- 
 
 FlG. 7. 
 
PRESSURE OF THE AIR. 
 
 29 
 
 tute for an air-pump. In this instrument, the ve- 
 locity of water flowing through a narrow orifice 
 serves to drive down small columns of water sepa- 
 rated by columns of air ; the weight of mercury 
 accomplishes the same object in Experiment 27. 
 (See Appendix.) 
 
 No matter what medium is used in the pump, it 
 is usual to employ the 
 apparatus shown in 
 Fig. 8. The fluid en- 
 ters by the tube a, and 
 falls through the nar- 
 rowed portion of the 
 tube b. The tube T, 
 coming from the ves- 
 sel which we desire to 
 exhaust, is provided 
 with a tube, U, which 
 contains mercury, and 
 serves to show how 
 far the exhaustion has 
 proceeded. Instead of 
 allowing the water to 
 fall through a, it can 
 be blown through, or caused to issue with consid- 
 erable velocity, and thus inclose bubbles of air and 
 draw them down with it. The form of the tube at 
 b is found to be the best for entangling the air-bub- 
 bles. The pinch-cock 5 regulates the supply of 
 mercury or water. 
 
 Experiment 28.— Measure the relative density 
 of mercury and water by means of the atmospheric 
 
 Fig. 8. 
 
 pressure. 
 
 Bore two holes in a cork. 
 
 Bend two glass tubes 
 
30 
 
 NEW PHYSICS. 
 
 twice at right angles, and fit them into these holes 
 in the cork, arranging these glass tubes so that 
 when the cork is fitted to a bottle or flask the por- 
 tions of the glass tubes that are outside the bottle 
 can dip into vessels containing the mercury and the 
 water. Heat the air contained in the bottle. A 
 portion of the air will be expelled in bubbles 
 through the liquid. Then allow the bottle or flask 
 to cool, and measure the relative heights of the two 
 hquid columns. The column of mercury will be 
 found to be 13.59 times that of the water. This 
 number will be the specific gravity of the mercury. 
 Repeat the experiment with alcohol instead of mer- 
 cury. The cork must fit the flask tightly. 
 
 Experiment 29. — Ascertain the pressure of air 
 in a flask from which the air has been partly ex- 
 pelled. 
 
 Allow both of the tubes inserted in the apparatus 
 described in Experiment 28 to dip into vessels con- 
 taining mercury. Heat the air in the flask until a 
 certain portion of it is expelled in bubbles through 
 the mercury. Then allow the flask to cool. Meas- 
 ure the heights of the mercury-columns, and com- 
 pare with the height of the mercury-column in Ex- 
 periment 22, which gives the atmospheric pressure. 
 
 Experiment 30. — Verify Boyle's or Mariotte's 
 law. 
 
 This law states that the volume of air contained 
 in a closed tube decreases in proportion as the 
 pressure upon the air is increased. Calling the 
 pressure P and the volume V, we have P x V= a. 
 constant quantity. Close a glass tube, of about one 
 metre in length, at one end. Paste a thin strip of 
 paper along the tube, beginning at the closed end. 
 
PRESSURE OF THE AIR. 31 
 
 Graduate the tube in cubic centimetres by pour- 
 ing in mercury from a vessel graduated to cubic 
 centimetres. This operation is called calibration. 
 Having finished the calibration, pour in mercury 
 until the tube is about half full. Putting your 
 thumb over the open end of the tube, invert it in 
 the receptacle (see description of receptacle be- 
 low). Then lift the tube through varying heights 
 and measure the column of mercury in the tube, 
 and also the volume of air above the mercury. The 
 column of mercury and the pressure of the in- 
 closed air evidently balance the pressure of the at- 
 mosphere. Hence, subtracting the heights of the 
 mercury-columns from the height of the mercury- 
 column in Experiment 22, which measures the 
 pressure of the atmosphere, we obtain the pressure 
 of the inclosed air. Then multiply these values by 
 the corresponding volumes. Take four different 
 observations, and arrange the results thus : 
 
 Mean 
 
 {Receptacle. — This can be made out oi a piece of 
 iron gas-pipe about i metre long and 2 centimetres 
 internal diameter. This pipe should be closed by a 
 nut at one end. Then cut a bottle in two (see Ap- 
 pendix), and fit the mouth of the bottle on the open 
 end of the tube by means of a cement (see Appendix). 
 This receptacle should then be mounted upright at 
 the side of your working table.) 
 
32 
 
 NEW PHYSICS. 
 
 A practical application of Boyle's law is called 
 a volumometer. 
 
 Experiment 31. — Instead of the barometer-tube 
 closed at one end, used in Experiment 22, terminate 
 an open tube by means of half a bottle, the edges of 
 which have been carefully ground (see Appendix), 
 so that it can be closed hermetically by a ground- 
 glass plate, /"(Fig. 9). First immerse the glass tube, 
 when the plate P is removed, 
 until the level of mercury inside 
 at d is the same as that outside. 
 Afterward put on the plate P, 
 so that the apparatus is hermet- 
 ically closed, and lift the appa- 
 ratus. The mercury will then 
 rise to some point, d^- Call the 
 height d"^ = h. The original 
 volume V of the air, which was 
 under the pressure of the at- 
 mosphere, is now increased to 
 V-\-v. Calling the pressure of 
 the atmosphere at the time of 
 our experiment P, we shall have 
 
 V P-h ^ 
 
 •7; = — r= — Let us now 
 
 K -f- V F 
 
 place a body, B, whose volume 
 we wish to find, in the appara- 
 tus. It is evident that it will displace its own vol- 
 ume of air. We repeat our observations, calling 
 the volume of the body B ^ x. 
 
 V-x 
 
 Fig. 9. 
 
 We shall then have ■ 
 
 P- hi 
 
 From 
 
 {V—x) + v P 
 
 this and the preceding equation, we get x=:{V+ v) 
 
(-i)- 
 
 PRESSURE OF THE AIR. 33 
 
 We can determine the volume F+ z- by 
 /^/ 
 
 making the experiment with a cylinder of known 
 volume. It is, of course, necessary to close the 
 nozzle of the bottle, where the glass tube enters, 
 perfectly air-tight ; and to be sure that the plate P 
 also closes the apparatus hermetically. 
 
 The law of Boyle is found to be only approxi- 
 mately true for very high pressures ; for ordinary 
 experiments in laboratories, however, it can be 
 taken to be sufficiently exact. It is evident that 
 all observations on the volumes of gases must be 
 made at the same pressure of the atmosphere, or, 
 in other words, at the same height of the barometer, 
 in order to be compared. We can reduce our ob- 
 servations made at different barometric heights to 
 the same standard by Boyle's law, for, since the 
 volumes are inversely as the pressures, we have 
 
 V P 
 
 — = — , and, if we know the volume Fat the press- 
 
 ure P, we can obtain V-^ at the pressure Pj. 
 
 If we wish to obtain also the density of a gas in 
 terms of its density at a standard pressure, we pro- 
 ceed as follows : 
 
 Experiment 32. — Let M be the mass of the gas ; 
 
 V is volume, and D its density at one pressure, and 
 Fi and D^ the corresponding values at another press- 
 
 ure; then VD = M=V^D,; or^ = ^- = ^ = ^; 
 
 and, if we take H^ = 760 millimetres as the standard 
 
 D T-f 
 pressure, we have D — — i— where D^ is density at 
 
 standard pressure. 
 
34 iV£-fF PHYSICS. 
 
 Experiment 33. — Boyle's and Mariotte's law for 
 pressures greater than one atmosphere. 
 
 Close a long glass tube at one end. Then bend 
 this end into a U-shape, making one branch of the U 
 much longer than the other. Graduate the shorter 
 branch in cubic centimetres, by pouring in mercury 
 and moving the tube so as to allow the imprisoned 
 air to escape. Having graduated the tube, allow 
 the mercury to run out, and then refill until the 
 mercury stands at the same level in both branches 
 of the U-tube. The pressure of the inclosed air is 
 then the same as that of the atmosphere. Now 
 pour in mercury into the longer branch and meas- 
 ure the height of the mercury in the longer tube, 
 above the level of the mercury in the shorter 
 tube. This mercury-height measures, the pressure 
 of the imprisoned air. Knowing the volumes of 
 the imprisoned air and the corresponding mercury- 
 heights, apply the same calculation as in Experi- 
 ment 30. 
 
 The law of Boyle, which our experiments illus- 
 trate, is extremely important, for by means of it we 
 can calculate the volume of a gas when it is sub- 
 jected to different pressures, if we know its volume 
 at some standard pressure. For instance, if we call 
 the volume of a gas at 0° C. and 760 mm. of mer- 
 cury pressure V, and we wish to know its volume 
 
 Fi at a pressure of P^ = 765 mm'., we have — = -^ 
 
 Fi 760 
 
 Vx = — — ; that is, we multiply V by the ratio of 
 
 the barometric heights ^. We perceive that, how- 
 ever we may change the pressure P\ the product 
 
PRESSURE OF THE AIR. 
 
 35 
 
 of ViPi = Vy6o; that is, ViP^ is always equal to a 
 constant quantity F760. 
 
 Now, there is another law, called the law of 
 Charles, which states that " the volume of a gas 
 under constant pressure expands when raised from 
 the freezing to the boiling temperature by the same 
 fraction of itself, whatever be the nature of the gas." 
 
 Experiment 34. — Take a glass tube about 16 cc. 
 long, open at both ends, and draw into it, by prop- 
 erly inclining it, a little index of mercury about 10 
 mm. long. Move the index until it occupies a posi- 
 tion near the middle of the tube. Then close one 
 end of. the tube by holding the end horizontally in 
 a Bunsen burner and turning it quickly between 
 the fingers. Mount this tube vertically upon a 
 piece of graduated wood or ivory, such as is often 
 provided with thermometer-scales, place the closed 
 end in melting ice, and note the position of the 
 index. Then suspend the apparatus in a Florence 
 flask, or other receptacle, which is filled with 
 steam, arising from water boiling in the lower por- 
 tion of the flask. Read again the new position of 
 this index. We can find in this way the ratio of 
 the volume of air at 0° to that at 100°. Afterward 
 remove the tube from the steam and observe the 
 position of the index after the inclosed air has 
 taken the temperature of the room. Find also the 
 ratio between the volume of air at 0° and that at 
 the temperature of the room. Find the amount of 
 dilatation of the air for each degree between 0° and 
 100°, and each degree between 0° and the tempera- 
 ture of the room. The amount of dilatation should 
 be -^ for each degree centigrade, or the air in ex- 
 panding from 0° to 100° increases from i to 1.36. 
 
36 
 
 NEW PHYSICS. 
 
 In this experiment we have supposed that the bore 
 of the tube is uniform. It is necessary to ascertain 
 the actual volumes of air in the tube for different 
 positions of the index by means of calibration (see 
 Experiment 30). 
 
 The accurate value of this amount of expansion 
 is 1.3665. Now, if we combine this law of dilata- 
 tion of gases with Boyle's law, we see that the in- " 
 creased volume must be equal to the volume at any 
 standard temperature multiplied by 1.3665. In or- 
 der to understand clearly the use of this 
 property of gas, let us examine the phe- 
 nomenon more closely. 
 
 Experiment 35. — Blow a bulb at 
 the end of a thermometer-tube of com- 
 paratively large bore (Fig. 10). Inclose 
 a certain amount of air in the bulb and 
 tube by means of a little index of mer- 
 cury, a, as in Experiment 34. Place the 
 bulb in melting ice, and notice the posi- 
 tion of one end of the mercury-index. 
 Afterward place the bulb in steam, and 
 read the same end of the index, the 
 pressure acting on the confined air re- 
 FiG. 10. maining constant. It will be quickly 
 seen that we can use this instrument to 
 observe temperatures, instead of a mercury-ther- 
 mometer. It constitutes what is called an air- 
 thermometer. Supposing that the bore of the tube 
 is uniform, we can graduate the space between the 
 freezing and the boiling points into one hundred 
 parts, and carry this graduation below the freezing, 
 or 0° (Fig. 10) point, and above the 100°, or boiling- 
 point. How far below the 0° point shall we go ? 
 
100 
 
 h. 
 
 PRESSURE OF THE AIR. 37 
 
 We know from our preceding experiment that 
 the distance of the freezing-point from the bot- 
 tom of the tube will be to the distance of 
 the boiling-point from the bottom as i is to 
 1.3665. Or, if we call the volume of air in 
 the tube when the index is at 0° V, and the 
 volume when it is at 100° Fi, we shall have 
 V I 
 
 y.^T^- " ^^ ^^^^ ^ (^'^- ") *^^ o- 
 
 length of the tube of uniform bore from the 
 bottom to 0°, the volume of air in this tube 
 up to this point would be the section of the 
 tube s multiphed by k or /ts. The volume 
 contained in the tube when the index is at 
 100° will in the same way be equal to (A + 
 106) s^, since the distance from 0° to 100° is 
 100 scale divisions more than k scale divi- 
 sions ; we shall therefore have — 
 
 V Aj_ h i_ 
 
 V-^ {h + 100) s ~ k + 100 ~~ 1.3665 ' 
 we shall find that the value of k, which makes this 
 equality true, is 272.85°, or, as it is generally taken, 
 273". This point is called the absolute zero on 
 the air-thermdmeter scale. The point 273° is merely 
 a useful limit, and it is found convenient to reckon 
 temperatures from this point, for at this point the 
 air is supposed to be devoid of heat. 
 
 Experiment 36. — Take the apparatus of Experi- 
 ment 33, and, having measured the volume of the 
 gas in the U-tube at 0°, and the pressure upon it, 
 which is the height of the barometer plus the differ- 
 ence in level of the two surfaces of mercury in the 
 tube, submit it to the temperature of steam, and 
 also change the pressure by a known amount by 
 
 Fig. II. 
 
38 NEW PHYSICS. 
 
 pouring mercury into the longer arm of the tube ; 
 
 estimate this new pressure as before. We shall 
 
 then find that, calling V the volume at o°, and P the 
 
 pressure at o°, and V^ the volume at ioo° and P^ the 
 
 „ VP 273" 
 
 pressure at 100 , -rrr; = 7, 'r. ! or, 11 we call 
 
 VxPi 273° + 100" 
 
 the distance of the freezing-point and the thermom- 
 eter reading from the absolute zero T and T^, we 
 
 VP T ViPi T 
 
 have — -— = — , or F= — , which will enable 
 
 us to reduce a gas to a standard pressure and a 
 standard temperature, if we know its volume at any- 
 other temperature and pressure. A bent tube very 
 much smaller than that used in Experiment 33 can 
 be employed, so that the tube can be easily insert- 
 ed in or placed over a vessel in which steam is 
 generated. 
 
CHAPTER IV. 
 
 MEASUREMENT OF DIMENSIONS. 
 
 In our experiments thus far we have used prin- 
 cipally a unit of mass, the gramme and its subdi- 
 visions. It is now necessary to investigate more 
 closely the dimensions of bodies. It is not conven- 
 ient to divide a measure of length much finer than 
 into millimetres. There are about twenty-five mil- 
 limetres to an inch, and it will be readily perceived 
 that one twenty-fifth of an inch is a small quantity. 
 Still, it is necessary to measure spaces much smaller 
 than a millimetre. At first sight fine measurements 
 seem a useless refinement, but as we study nature 
 we find that we have to deal with extremely mi- 
 nute distances, which, although small in themselves, 
 nevertheless influence our final results to a great 
 extent. Take, for instance, the measurements con- 
 ducted by the Coast Survey from which the area of 
 a State is computed, and the positions of its great 
 features are fixed by one line, which is called the 
 base-line. This line is sometimes only a few hun- 
 dred feet long. It must be measured with the ut- 
 most accuracy, for an error of a millimetre would 
 make a large error in the line E F oi a. triangle 
 (Fig. 12), which is derived from the triangle B C B^, 
 of which B Bi is the base. E F may be several miles 
 
40 
 
 NEW PHYSICS. 
 
 Fig. 12. 
 
 long. An error of a millimetre in such a base-line 
 might cause the center lines of two borings for a 
 tunnel to diverge many feet. 
 
 In order to subdivide our smallest unit of length, 
 
 the millimetre, we 
 use an instrument 
 called a vernier. 
 
 Suppose we _ 
 should slip along- 
 side a millimetre- 
 scale a piece of pa- 
 per upon which a 
 space equal to nine 
 millimetres has been divided into ten equal parts. 
 It will be seen that each one of the divisions on the 
 paper falls short of a millimetre by one tenth. We 
 therefore look along the vernier until we find some 
 line of its graduation which is exactly opposite a 
 line on the millimetre-scale. The num- 
 ber of divisions on the vernier-scale 
 up to this line will express the num- 
 ber of tenths of a millimetre, in addi- 
 tion to the whole number of millime- 
 tres. Thus, in Fig. 13, the reading is 
 five centimetres, two millimetres, and 
 five tenths of a millimetre, or .0525 of 
 a metre. 
 
 ExPERUVfENT 37. — Make a vernier. 
 Take nine millimetres on a fixed scale, 
 and divide the space of nine millime- 
 tres into ten parts, by means of a 
 scale * of proportional parts. It is evident that one 
 of these ten divisions will fall short of a millimetre 
 
 * Such as is used by engineers and surveyors. 
 
 z: 6 
 
 Fig. 13. 
 
MEASUREMENT OF DIMENSIONS. 
 
 41 
 
 MERNIER 
 
 11 
 
 WOOD 
 
 Fig; 14. 
 
 by one tenth of a millimetre. Attach this vernier 
 
 (Fig. 14) to a piece of wood which can slip beside 
 
 the metre-scale. Measure the diameter of a lead 
 
 ball, by placing a 
 
 piece of wood per- scale: 
 
 pendicular to the 
 
 metre - scale — al- . 
 
 lowing the ball to 
 
 be inclosed by this 
 
 piece of wood, the 
 
 metre - scale, and 
 
 the vernier, which 
 
 lies beside the scale, and opposite to the piece of 
 
 wood. (See figure.) 
 
 Experiment 38. — Find the average thickness of 
 a piece of wire, by measuring it with a vernier- 
 gauge in six different places, and taking the aver- 
 age of these measurements. Compare this result 
 by that obtained as follows: Obtain the loss of 
 weight of the wire in water, this gives the volume 
 of the wire. The volume 
 
 = area of cross-section X length 
 
 , vol 
 
 It is evident that, if we knew the pitch of a 
 screw — or, in other words, the distance between 
 the threads — and if this pitch or distance were uni- 
 form, we could mount a screw in a long nut, and 
 interpose the object whose dimensions we wish to 
 measure between the end of th€ screw and a fixed 
 
42 
 
 NEW PHYSICS. 
 
 support ; and then ascertain how many whole turns 
 and fractions of a turn of the screw are necessary 
 to make the end of the screw pass over the space 
 occupied by the body. Unless the screw is made 
 with great care, the pitch is not uniform. With 
 care in selection, however, one can be obtained, from 
 a hardware-store or a mechanician, the pitch of which 
 is sufficiently uniform through short distaitces. 
 
 Experiment 39. — Obtain a screw with a com- 
 paratively fine thread. Solder upon the head of the 
 screw {P, Fig. 15) a cii-cular tin plate, and glue to 
 
 Fig. 15. 
 
 this a paper or horn protractor. Pass the screw 
 through a metallic nut, and imbed this nut in one 
 end of a suitable box or frame. Affix to the frame 
 two scales, 5 S, in order to determine how many 
 whole turns the screw makes. The fractions of a 
 turn are given by the protractor or divided head, 
 MN. 
 
 It is necessary to place a piece of brass against 
 tlie end, A. The object to be measured is inclosed 
 between A and B, the end of the screw. This ap- 
 paratus can be converted into a dividing engine, 
 by fixing the screw between A and C (Fig. 16), so 
 that it can not move forward or backward, and can 
 
MEASUREMENT OF DIMENSIONS. 
 
 43 
 
 only turn. In this case the nut N N' is allowed to 
 move on the screw along a suitable groove, and a 
 collar at C confines the screw. 
 
 Fig. i6. 
 
 A little sharpened pointer attached to the mov- 
 able nut will enable one to graduate scales or glass 
 tubes. If divisions are to be made on glass, the 
 glass is first covered with a thin coating of wax or 
 parafifine. After the divisions are made, the glass 
 tube can be immersed for a moment in hydrofluoric 
 acid. In order to do this, take a piece of lead pipe 
 of a suitable length and diameter. Close one end 
 by hammering. Pour into the vessel thus made a 
 sufiBcient amount of hydrofluoric acid (commonly 
 called white acid by etchers on glass). A few pre- 
 liminary trials with a roughly graduated piece of 
 glass will enable one to determine how long the 
 carefully divided glass tube should be immersed, in 
 order that the divisions, and their appropriate fig- 
 ures or designations, drawn through the paraffine, 
 may be etched. Two or three minutes generally 
 will be sufficient. 
 
 Experiment 40. — Graduate a piece of paper 
 into millimetres, and also into -^ of an inch. 
 
44 
 
 NEH^ PHYSICS. 
 
 Suppose that the pitch of the screw of your 
 micrometer, or dividing engine, is -^ of an inch. 
 It is evident that the head of the screw should make 
 a complete revolution in order to move your pointer 
 through ^Jg- of an inch. Since the head of the screw 
 is generally divided into one hundred parts, you 
 must reduce ^ of an inch to a fraction of a milli- 
 metre, and calculate what fraction of a turn the 
 screw-head will give. Move the pointer or pencil 
 of the graduating apparatus over a millimetre. If 
 the screw-head is divided into 360°, a turn through 
 one degree will evidently move the screw through 
 t^XtJt=^sW of an inch. 
 
 Experiment 41. — A cheap spherometer can be 
 made in the following manner: File one end of a 
 screw which has a fine thread to a point. Solder 
 a round, metallic disk to the other end. Paste upon 
 this disk a paper protractor. Sink the nut of the 
 screw into a wooden base. Upon this base erect 
 two standards 180° apart. Graduate these standards 
 in whole turns of the screw. Provide the base with 
 three metallic legs which are pointed. Three long 
 brass screws, pointed at the end, will answer for 
 these legs. In turning the head of the screw, and 
 therefore the protractor, the half circle of the pro- 
 tractor should always just glide by one of the stand- 
 ards, and the reading of the whole turns, and the 
 fractions of a turn, can therefore be taken at one of 
 the standards. In sinking the screw-nut into the 
 wooden base, a hole should, of course, be made 
 through the wood below the nut, so as to allow the 
 screw to pass through the base. The spherometer 
 is then placed upon a piece of plane glass, and the 
 screw turned until it just touches the plane glass. 
 
MEASUREMENT OF DIMENSIONS. 
 
 4S 
 
 This will be when the reflection in the glass appears 
 to touch the screw. This is the zero of the scale 
 on the head. The instrument is then placed upon a 
 curved surface, such as a lens, and the screw ele- 
 vated until it just touches the curved surface. The 
 number of divisions on the head which pass by the 
 fixed standard, or index, gives the difference in 
 height between that of the plane surface and that of 
 the curved one. 
 
 Obtain in this way the thickness of plates of 
 mica, and compare the values with those obtained 
 by a vernier-gauge. (See Appendix.) 
 
 Experiment 42. — Obtain the specific gravity of 
 an iron cylinder by measuring its diameter and its 
 length and weighing it. The measures of length 
 give its volume, and therefore the weight of the 
 water which it would displace. Care should be 
 taken that the cylinder should always be inclosed 
 by the arms of the gauge at the same place. If the 
 ends of the cylinder are concave, a small piece of 
 sheet-brass or platinum can be placed in the ends 
 of the cylinder, and from the length obtained the 
 width of the brass can be subtracted. A spherom- 
 eter can be used to make these measurements. 
 
 With an accurate vernier-gauge we can measure 
 very small subdivisions of a millimetre. Another 
 device for accomplishing the same purpose is called 
 a micrometer. It consists of a screw, the head of 
 which is enlarged into a circle, which is divided 
 generally into one hundred parts. If the pitch of 
 the screw — that is, the distance between any two 
 threads — is one millimetre, it is evident that a por- 
 tion of a turn can be read in hundredths by observ- 
 ing how many of the divisions of the head of the 
 
46 
 
 NEW PHYSICS. 
 
 screw pass a fixed point outside the screw. A mi- 
 crometer can be supported in various ways. If it 
 is used to measure the curvature of a surface, it is 
 called a spherometer. 
 
 Micrometers are generally attached to the eye- 
 pieces of microscopes and telescopes. In this case 
 they are used to push a fine cross-hair across the 
 field of view, and thus measure the diameter of a 
 
 small object, or small dis- 
 tances between two objects. 
 Thus, let O (Fig. 17) be a 
 fixed circle or aperture at 
 the focus of the eye-piece of 
 a microscope or telescope, 
 and let the screw P move a 
 frame, A, B, C, D, E, F, pro- 
 vided with a cross-hair, B E, 
 across this aperture. The 
 springs n n serve to push the 
 frame against the screw. The 
 distance that the movable 
 .hair B E passes over will be 
 given in fractions of the pitch 
 of the screw by the head of 
 the screw M N. In this way 
 astronomers can measure the 
 diameter of a small planet.. 
 Such a micrometer is also necessary in measuring 
 the expansion of a bar of metal under the effect of 
 heat. This is too small to observe with the naked 
 eye at ordinary temperatures, and hence micro- 
 scopes provided with micrometers in the eye-piece 
 have to be used. The small expansion of metals 
 when heated becomes a very important considera- 
 
 FlG. 17. 
 
MEASUREMENT OF DIMENSIONS. 
 
 47 
 
 tion when a base-line is to be measured or a piston- 
 rod is to be fitted into the cylinder of a steam-en- 
 gine. A bar of brass one foot in length increases 
 in length only 0.0000187 of itself for one degree 
 rise in temperature, yet the force exerted is enor- 
 mous, and the expansion must be carefully esti- 
 mated. 
 
 The microscopes and verniers we have described 
 can be attached 
 to any instru- 
 ment with which 
 we wish to meas- 
 ure the dimen- 
 sions of small ob- 
 jects. 
 
 In order to 
 attach such a mi- 
 crometer to a 
 telescope, obtain 
 a brass or tin 
 tube (A, Fig. 18), place a small lens of about twelve 
 inches focus (see Appendix) in a piece of cork, and 
 slip it into the end L of this tube. Screw the other 
 end of the tube upon one side of the small box con- 
 taining the micrometer M. Screw another piece of 
 tin tubing, b, upon the other side of the microme- 
 ter-box. Into this last tube slip an eye-piece con- 
 taining two small lenses. The interior of the tin 
 tubes and the micrometer-box should be blackened. 
 The telescope should be provided with a little 
 spirit-level at C. The telescope can be made to slip 
 up and down a vertical rod, similar to that which 
 is used by photographers for a head-rest. Such an 
 instrument is useful in measuring, for instance, the 
 
 Fig. 18. 
 
48 
 
 NEW PHYSICS. 
 
 height of a mercury-column. In order to do this a 
 metre-scale is suspended vertically by means of a 
 plumb-line beside the column to be measured, and 
 the whole divisions are read from this scale, and 
 the small interval between the last division on the 
 scale and the top of the mercury-column is read by 
 means of the micrometer. 
 
 The cross-hair should be a fiber of silk, obtained 
 by untwisting a piece of silk thread, and taking the 
 ultimate fiber. This cross-hair should 
 be at the common focus of the lens at 
 L, and the eye-piece. The metre-rod 
 (Fig. 19), suspended in a vertical posi- 
 tion by means of a plumb-line, rests 
 against a vertical wire, E F, of which 
 one end, F, just touches the surface 
 of liquid in the reservoir R. Knowing 
 the length of E F, we can add it to the 
 indications of the metre-rod.* 
 
 Experiment 43. — Ascertain wheth- 
 er an India-rubber strip is perfectly 
 elastic. A substance is perfectly elas- 
 tic when it recovers its original form 
 after any distortion. Measure a cer- 
 tain length, stretch it a certain distance, 
 and then measure again, and so on, in- 
 creasing the amount of stretching. 
 From your observations deduce the 
 limit of elasticity. An ordinary scale 
 will answer for this experiment. 
 
 Experiment 44. — When a substance is stretched 
 
 * It is well for the student to construct a micrometer such as I 
 have described, for the practical knowledge that is thus obtained in 
 optics. 
 
 Fig. 19. 
 
MEASUREMENT OF DIMENSIONS. 49 
 
 beyond a certain limit, the amount of stretching does 
 not increase with the weight appHed. Up to this 
 point, however, two grammes produce twice as 
 much elongation as one gramme, three grammes 
 three times as much as one gramme, and so on. In 
 other words, up to a certain limit the ratio between 
 the increase in length and the weight is constant. 
 Find this limit for India-rubber: 
 
 Experiment 45. — We compare the elasticity of 
 different substances by observing the elongations 
 which a weight of one gramme will produce in a 
 cylindrical or prismatic bar of the substance one 
 centimetre long and one square centimetre in sec- 
 tion,—^ : — expresses the difficulty of stretching. 
 
 elongation 
 
 By measuring the elongation within the limits of 
 elasticity of a rod or cord of suitable length and 
 section, we can readily find by proportion how 
 much a rod or cord one centimetre long and one 
 square centimetre in section will be elongated. 
 The co-efficient of elasticity is the ratio of increase 
 of a bar of unit section when it is stretched with 
 the unit force to the entire length of the bar. When 
 the elongation of a metal wire is observed, it is best 
 to pass it through the center of a thin brass tube, 
 which is clamped to the wire by a screw (5, Fig. 20). 
 The length of this brass tube remains unaffected by 
 the stretching weight W, and the yielding of the 
 support M will be obviated if we measure from the 
 end b of the brass tube to a mark, a, on the wire. 
 Measurements with metal wires require a micro- 
 scope or telescope to read the small elongation. 
 
 Experiment 46. — Find the internal diameter of 
 a given glass tube. 
 
50 
 
 NEW PHYSICS. 
 
 The volume of a cylinder is the area of its base 
 multiplied by its altitude. We have seen that the 
 area of a circle is expressed by irr', in 
 M which IT is 3. 141 6, and r is the radius of 
 
 the circle. Close the tube at one end by 
 a cork, and fill the tube with water. 
 Weigh this water, and the number of 
 grammes will give the volume of the 
 cylinder in cubic centimetres ; from this 
 volume obtain the radius of the cylinder. 
 Repeat this experiment with mercury. 
 
 If the internal diameter is small, draw 
 into it a thread of mercury, two or three 
 inches long ; measure its length at differ- 
 ent points, moving it along by inclining 
 the tube. Knowing the weight of the 
 mercury, by placing the glass tube in a 
 balance, and subtracting the weight of 
 the glass tube from that of the glass tube 
 with the thread of mercury, we can ob- 
 tain the radius of the bore of the tube 
 from the expression W=/7r/j, in which W is the 
 weight of the mercury, / the length of the mercury- 
 column, 77- is 3.1416, r the radius of the tube, and s the 
 specific gravity of mercury. This operation of deter- 
 mining the internal radius of a tube at different parts 
 is called calibration, and is necessary whenever we 
 wish to graduate the scale of a thermometer ; for, if 
 the bore is not uniform, it is evident that the divis- 
 ions on the thermometer-scales must be unequal. 
 With the instruments we have described we can now 
 ascertain the variations of bodies under the influence 
 of heat or of any force of distortion. The measure- 
 ment of these variations is extremely important, for 
 
 Fig. 20. 
 
MEASUREMENT OF DIMENSIONS. 
 
 51 
 
 upon these measurements is based the great doc- 
 trine of work, which expresses the law that the force 
 multiplied by the space passed through gives the 
 work which is done. This will be explained later. 
 All our experiments are directed toward obtaining 
 a clear idea of the doctrine of work, and it is neces- 
 sary, in the first place, by accurate measurement, 
 to obtain an idea of the change of dimensions of 
 bodies under the effect of weight or of pressure. 
 
 We have said that our operation of weighing is 
 really a determination of mass. The mass is the 
 volume of a body in cubic centimetres multiplied by 
 the density of the substance, and hence we have 
 
 M=Vd. 
 
 Now, the weight of a body is the mass multiplied 
 by the pull of gravitation upon every particle, hence 
 the weight is the mass multiplied by the force of 
 gravitation, or 
 
 W=Mg. Eq. (i), 
 
 where g is the force or pull of gravitation on every 
 unit of mass. If, then, the weight of a body is 
 
 W 
 
 given, we can always determine the mass for M=: — 
 
 from Equation i. 
 
CHAPTER V. 
 
 MOTION. 
 
 We have devoted considerable space to meas- 
 urement of space of volume and of mass ; for the 
 ideas we first gain in childhood are ideas of dis- 
 tances, of space more or less occupied with matter. 
 To the child the earth appears to be standing still 
 in the heavens, and the conception that all bodies 
 are in motion, although they may seem to be at rest, 
 is one of maturity. We perceive after a time that 
 there is a science which treats of the relative posi- 
 tion and motion of bodies, and of the regular suc- 
 cession of events in Nature. Day follows night. 
 The temperature at noonday changes with the 
 season of the year. The height of the tide alters 
 with the relative position of the sun and moon. 
 The frost-figures on the window-pane vary with 
 the amount of moisture in the air and the condi- 
 tion of the window-pane. Physical science has 
 been defined as that department of knowledge 
 which relates to the order of Nature, or, in other 
 words, to the regular succession of events. 
 
 In order to fix our ideas, we speak of a material 
 particle. This is a body so small that we can 
 neglect the distances between its various parts. 
 When we investigate the law of attraction between 
 
MOTION. 55 
 
 a small stone and the earth, we consider the stone 
 as a material particle, for we are not able to take 
 into consideration the irregularity of the stone, and 
 the different effect of its various parts. We are 
 obliged to assume that there are molecules or ulti- 
 mate particles of every substance, in order to fix 
 our ideas and to simplify our calculations; but, 
 when we consider the rotation of a particle about 
 some axis, we can not neglect the different effect 
 of its various parts: for the effect of the rotation 
 of a part far from the axis differs greatly from that 
 of one close to the axis, the farther particles mov- 
 ing with a greater velocity. 
 
 The universe is so extended and so complicated 
 that it is necessary for us to restrict our measure- 
 ments to a certain area, or what is called a con- 
 figuration. In order to get a clear idea of a State 
 like Ohio, we must have a map of it, in which cer- 
 tain parts are connected with others by lines which 
 have been extended from some base-line. In order 
 to obtain a conception of a complicated piece of 
 machinery, we must have a diagram of the motion 
 of its various parts. In order to study motion we 
 must carefully examine our thoughts in regard to 
 space and time. For the purpose of physical 
 science, the object of which is to investigate suc- 
 cessive events, these ideas must be definite. Our 
 ideas of space and time are relative, and our ideas 
 of motion are relative, since they are based upon 
 ideas of space and time. 
 
 If we should at any instant desire to fix the posi- 
 tion of three points on three circles of card-board, 
 which are revolving near each other with the same 
 velocity, with reference to any outside point, a dia- 
 
54 NEW PHYSICS. 
 
 gram would be necessary. Knowing the rate of 
 revolution of the circles, we could construct on a 
 diagram the position of the points with reference 
 to an outside point at any subsequent time. We 
 could do the same if the circles revolve with differ- 
 ent velocities ; we could apply the same method of 
 observation to the motion of the parts of a machine. 
 From the results of our observation of different con- 
 figurations, or diagrams, we could write a treatise 
 on mechanics; such treatises are called works on 
 kinematics. Some knowledge of geometry and 
 trigonometry would be necessary to simplify our 
 observations, but we should not have to study any 
 change of position of the circles or the machines, 
 or to investigate what causes the movement of the 
 various parts. When we take into account the 
 mutual action between bodies, the science of mo- 
 tion is termed kinetics; and when we investigate 
 force, we embody our results under the head of 
 dynamics. 
 
 We shall find that our observations in dynamics 
 resemble those of a business-man who (see Maxwell's 
 " Matter and Motion," page 54) opens an account 
 with another business-man. The same transaction is 
 called by one man buying and by the other selling ; 
 and it is called trade when we take both parties 
 into consideration. These transactions are entered 
 by the two merchants on opposite sides of their re- 
 spective ledgers ; and the accountant, in comparing 
 the ledgers, " must remember in whose interest 
 each book is made up." In dynamical inquiries we 
 must remember also which of two bodies we are 
 dealing with, and set down to the respective bodies 
 the forces which belong to them. We shall see also 
 
MOTION. 
 
 55 
 
 that our idea of force is gained by the rate of 
 change of motion. 
 
 Before beginning our study of motion, let us 
 ask ourselves what is the object of our study. We 
 can not investigate the origin of force. We can 
 not know what is the ultimate cause of anything — 
 why the planets were set in revolution, and why 
 there is a force of attraction which acts instantane- 
 ously throughout the universe. We must content 
 ourselves in studying the changes and variations of 
 motion. We say in general that a force or forces 
 produce motion and a change in motion. We can 
 only study these changes by noticing their rate of 
 change, just as the political economist studies the 
 movement of gold by the rate of exchange. The 
 gold represents a certain amount of energy. We 
 think of force through the manifestation of ex- 
 changes. The rate of the exchange may be high 
 or low ; and we therefore speak of the intensity of 
 a force, and we get our conceptions from noticing 
 the rate of change of motion. It is necessary in 
 unscientific language to speak of forces ; thus, we 
 have the expressions, the forces of Nature, the 
 force of magnetism, the force of electricity, the vital 
 forces, etc. Modern physics no longer treats of 
 different forces ; it does not strive to ascertain what 
 electricity is, but it endeavors to ascertain how the 
 manifestations of the exchanges of energy in elec- 
 trical phenomena are connected with the light and 
 heat phenomena that result, and how these heat 
 phenomena are related to the work done by a stone 
 falling under the influence of gravitation. If we 
 knew what electricity is, we should know what the 
 ultimate cause of motion is. All that we can expect 
 
56 NEW PHYSICS. 
 
 to discover is the relation between the exchanges 
 of energy which are manifested by electrical phe- 
 nomena, and the exchanges in heat and in motion 
 in general. 
 
 A treatise on mechanics could be written in 
 which the law of attraction of gravitation might 
 not be mentioned. The law of attraction of mag- 
 netism might take its place ; and we could prove 
 the law of the parallelogram of forces, the law of 
 couples, and, indeed, study every change of con- 
 figuration of a system submitted to magnetic force, 
 just as we commonly study dynamics by the aid of 
 the law of gravitation. The effects of the attraction 
 of gravitation, however, can be more universally 
 observed, and the measurements can be more con- 
 veniently made. 
 
 Up to this point we have not considered any 
 problems which arise when the motion of bodies is 
 considered. When we reflect that no particle on 
 the earth can be said to be at rest ; that we are 
 whirling through space ; that what is above us by 
 day is below us by night — we see the need of fixing 
 our ideas in regard to motion in general, and the 
 necessity of having some standard which will en- 
 able us to estimate the motion of all bodies, just as 
 we investigate the relative masses of different bod- 
 ies by reference to a fixed standard. 
 
 The laws of motion are three, and were first 
 clearly enunciated by Sir Isaac Newton. They are 
 as follow : 
 
 Law I. — Every body perseveres in its state of 
 rest or of moving uniformly in a straight line, except 
 in so far as it is made to change that state by ex- 
 ternal forces. 
 
MOTION. 
 
 57 
 
 Law II. — Change of motion is proportional to 
 the impressed forces, and takes place in the direc- 
 tion in which the force is impressed ; or, as it is 
 sometimes expressed — 
 
 The change of momentum of a body is numeri- 
 cally equal to the impulse which produces it, and 
 is in the same direction. 
 
 Law III. — Reaction is always equal and opposite 
 to action — that is to say, the actions of two bodies 
 upon each other are always equal and in opposite 
 directions. 
 
 Let us begin our study of motion by investigat- 
 ing the fall of a stone, or any body, to the earth. 
 The stone falls toward the earth because there is 
 an attraction between it and the earth. It starts 
 from a state of rest and acquires a certain velocity 
 at the end of a certain time. If it rolls down an 
 inclined plane, it moves with a certain velocity in 
 a horizontal direction, and with another velocity 
 in a vertical direction. If a body is shot up an 
 inclined plane against the force of gravitation, it 
 will move a certain space in a horizontal direction 
 and a certain space in a vertical direction. To reach 
 the top of the plane a person can either walk up the 
 plane, or walk a distance in a horizontal direction 
 equal to the base of the inclined plane, and then climb 
 up a ladder equal to the vertical height of the incHned 
 plane. Suppose two persons should start from the 
 same point, one desirous of reaching the vertex of 
 an inclined plane at the same time that the other 
 reaches the foot of the vertical line of the incHned 
 plane, it is evident that the man ;«^ (Fig. 21), running 
 along the base, should always be directly under the 
 man, m, running up the plane. If we represent the 
 
58 
 
 NEW PHYSICS. 
 
 Fig. 21. 
 
 velocity with which m is running up the plane by the 
 length of the line m a, the velocity which the man tn^ 
 
 must have will be m b, 
 which is found by drop- 
 ping a perpendicular 
 from the extremity of the 
 line representing m's ve- 
 locity to a horizontal line, 
 m b. In other words, 
 we project the actual ve- 
 locity by perpendiculars upon the line along which 
 we wish to obtain the relative velocity. In the same 
 manner, if we wish to find how fast a man must climb 
 the vertical height B C oi the plane in order to reach 
 the height B at the same time that another man, m, 
 ascends the hypothenuse of the plane, we project 
 the velocity m a by perpendiculars upon the height 
 B C, and the line a b would represent the corre- 
 sponding velocity that the man ascending the plane 
 must have, at any time. 
 
 In order to denote the rates of movement along 
 the height and base of an inclined plane in terms of 
 the rate along the hypothe- 
 nuse, we must adopt some 
 convention which will ab- 
 breviate such an account as 
 we have just given. 
 
 In a circle with a radius 
 of I (Fig. 22), let us draw any 
 angle, a. In the small in- 
 clined plane ^ ^ ^, we have 
 taken the hypothenuse yi ^ as 
 I. Let us call for convenience the base A c, the 
 cosine of a, generally written (cos a), and the alti- 
 
 FlG. 22. 
 
MOTION. 5g 
 
 tude be, the sine of a (sin a). These are merely con- 
 venient designations for the sides of an inclined 
 plane. Let us also draw a tangent to the circle at 
 e, and call the line d e tangent of a (tang. a). By 
 
 actual measurement we shall find that = . 
 
 Ae Ac 
 
 or (since we have taken the radius of the circle = 
 
 . . d e sin a 
 
 I, or A e = \), — = = tan. a. Also we find 
 
 I cos a 
 
 Ac AC 
 by actual measurement that — — = — — , or (since A b 
 
 A b A B 
 
 A C 
 = i),Ac = cosa= — -, or ^ C = ^ ^ cos a. Also 
 
 be B C , . B C 
 
 A~b = -AB' °'" ^^ = ^'" «==Z:g' ^' BC = AB 
 sin a. Hence we can always find the projection of 
 a line which represents a force or a velocity, upon 
 another line, by multiplying the line representing 
 the force or velocity by the cosine or the sine of the 
 angle between the two lines. The values of the 
 conventional sine and cosine, or, in other words, the 
 base and height of inclined planes having acute an- 
 gles at the center of a circle whose radius is one, 
 have been calculated and tables formed (see Appen- 
 dix). 
 
 The mathematical conceptions which we employ 
 to render our thoughts more concise when we in- 
 vestigate motion serve, therefore, to express the ve- 
 locity or the force along any line which is inclined 
 to the direction of the motion. 
 
 For the purposes of this treatise we need only to 
 comprehend the signification of sine and cosine and 
 tangent of an angle, and this comprehension can be 
 
6o NEW PHYSICS. 
 
 obtained without any previous knowledge of mathe- 
 matics save that of elementary algebra and the ele- 
 mentary ideas of proportion. 
 
 Experiment 47. — Draw an inclined plane, meas- 
 ure the angle at the base by means of a protractor, 
 and taking the values of the cosine and the sine 
 from a table of natural cosines and sines, multiply 
 the length of the hypothenuse by these values re- 
 spectively, and show that we thus obtain the base 
 and the altitude of the inclined plane. 
 
 Experiment 48. — In order to obtain an idea of 
 the relations of our conventions in regard to angles 
 and the sides of a triangle, draw a circle with a 
 radius of one foot ; lay off various angles at the cen- 
 ter of the circle. One side of each of these angles 
 being a horizontal diameter of the circle, draw 
 the sine and cosine for each of these angles. The 
 
 cosine is by definition equal to —-7- = , and the 
 
 Ji. U I 
 
 sine equal to -— r = — Measure A c and b c for 
 ^ Ab \ 
 
 the different angles, and find their ratio to the radius 
 of your circle, or to i, and compare the results you 
 obtain with those corresponding to the same angles 
 in a table of natural sines and cosines. Do the same 
 for the tangents. 
 
 In our measurement of masses in obtaining spe- 
 cific gravity, we have supposed that the attraction 
 of gravitation is the same upon the weights as upon 
 the body to be weighed, and thus instead of weights 
 we have compared masses, for our weights are only 
 correct for the latitude of Paris. We can also meas- 
 ure equal masses in another way without any refer- 
 ence to the force of gravitation. Suppose we should 
 
MOTION. 6 1 
 
 attach a certain mass to an India-rubber cord, swing 
 it around in a horizontal circle, and observe the 
 elongations of the rubber cord. Our experiments 
 on elasticity show us that the same force elongates 
 the rubber to the same amount as long as we keep 
 within certain limits of elasticity. H we swing the 
 two bodies around Avith the same length of cord 
 and with the same velocity, and find that the elonga- 
 tion of the rubber cord which serves as a radius of 
 the horizontal circle is the same for both circles, 
 we know that the bodies have the same mass. We 
 have substituted in this experiment the tension along 
 the cord for the pull of gravitation. If it were not 
 for this tension, or, as it is sometimes called, centrip- 
 etal force, and if it were not for the force of gravi- 
 tation and the resistance of the air, the mass would 
 continue to move in a horizontal plane, for the sec- 
 ond law of motion states that " change of motion is 
 proportional to the impressed force, and takes place 
 in the direction in which the force is impressed." 
 Newton understood by the word motion what we 
 call momentum, that is, the quantity of matter 
 moved, multiplied by its velocity, and the term " im- 
 pressed force " is what is understood by the word 
 impulse, " in which " (Maxwell's " Matter and Mo- 
 tion," p. 62) " the time during which a force acts 
 is taken into account as well as the intensity of the 
 force." 
 
 In dynamics, we say that two bodies have equal 
 mass when equal forces applied to them will pro- 
 duce equal changes of velocity in equal times. 
 
 The unit of mass in England and America is 
 I pound ; one seven-thousandth part of a pound 
 avoirdupois is called a grain. The French stand- 
 
62 J^EIV PHYSICS. 
 
 ard of mass is called a kilogramme ; it is equal to 
 15,432.34874 grains, or 2.2 pounds. 
 
 When a mass is attracted toward the earth, we 
 notice that it moves faster and faster ; it strikes with 
 a certain blow, or, as we say in common language, 
 it strikes with a certain force. The greater the dis- 
 tance through which it falls, the greater the force of 
 its blow. When we lift a weight, we feel that we 
 are overcoming a certain force. We could com- 
 pare the muscular force with which a person strives 
 to push us, with the pull that we must give to lift a 
 given mass against the force of gravitation. In a 
 certain sense a weight can be taken as a force ; it is 
 the evidence of a pull or force. We speak some- 
 times of a force of forty pounds, thus comparing 
 weight to force. This method of estimating force 
 is termed gravitation measure. Unless we measure 
 the force of gravitation at the place where we esti- 
 mate force, our results can not be compared with 
 those obtained at any other place on the surface of 
 the globe, for the pull of gravitation varies with our 
 position on the globe. 
 
 We therefore take as a unit of force that force 
 which acting on the unit of mass will give it a unit 
 velocity in a unit of time. The force of gravitation 
 acting on a gramme will give it a velocity of 980 
 centimetres (in the latitude of Paris) at the end of 
 one second ; hence the weight of one gramme can 
 be represented by 980 when the unit of weight is 
 the gramme, the unit of distance the centimetre, 
 and the unit of time is one second. 
 
 In the French system, the force which will give 
 to one gramme a velocity of one centimetre per 
 second is called a dyne. 
 
MOTION. 63 
 
 Let us now investigate the motion which results 
 from various impulses. Motion is of two kinds: 
 uniform motion and accelerated motion. 
 
 The motion of the second-hand of a watch is 
 uniform. It is evident that, if it moved faster one 
 minute than it did during another minute, the min- 
 utes would not be of the same length, and there- 
 fore we could not estimate time. A second-hand 
 makes sixty ticks in a minute. The second-hand 
 passes round the circumference of a circle once in a 
 minute. If we divide the circumference of a circle 
 into sixty parts, the second-hand will pass over sixty 
 of these parts in a minute. It takes one second to 
 pass over one of these divisions. If the length of 
 one of these divisions were one inch, it would pass 
 over one inch in one second. We can estimate its 
 speed, therefore, and say that it is one inch per 
 second. It will pass over just one inch in a second 
 at any point on the circumference of the circle. 
 Suppose that it passes over two inches in a second, 
 its velocity will then be two inches per second. 
 We estimate the speed or velocity by the space the 
 moving point passes over in one second. 
 
 What would be the space passed over by the 
 end of a second-hand of a clock in one minute if the 
 space, passed over in one second is three inches? 
 This would be three inches, or the space passed over 
 in one second — which is the velocity — multiplied by 
 the number of seconds in one minute, which is 
 sixty. That is, the whole space passed over would 
 be 3 X 60 = 180 inches. We express this by saying 
 that in uniform motion space = velocity multi- 
 plied by the time in seconds, or s = vt. 
 
 What would be the space passed over by the 
 
64 NEW PHYSICS. 
 
 end of a second-hand of a clock in one hour, if it 
 passes over one tenth of an inch in one second ? 
 
 If the second-hand of a clock went faster and 
 faster, we could have no fixed standard of time, 
 unless we compared our clock with some clock 
 which never varied from a uniform rate. Suppose 
 that the second-hand of my clock passed over one 
 inch the first second, two inches the second second, 
 three inches the third second, and so on. This 
 would be a case of what is called accelerated mo- 
 tion. The second-hand is going faster and faster, 
 and it would evidently be very difficult to tell the 
 time by such a clock. The second-hand would gain 
 fast upon the second-hand of the standard clock 
 which is marking the time uniformly. We could, 
 however, calculate the time from our clock, whose 
 second-hand is going faster and faster, by compari- 
 son with that of our standard uniform clock. 
 
 Most of the examples of motion which attract 
 our attention are not uniform. The motion of the 
 earth around its axis, giving us night and day, is 
 uniform ; and this motion affords us our common 
 standard of uniform time. It is very difficult, how- 
 ever, to obtain uniform motion. This is attained by 
 great care and skill in the manufacture of watches 
 and clocks. It is found very difficult to make an 
 engine of any kind run uniformly. The human 
 heart, which is a species of engine, beats from sixty 
 to eighty times a minute ; at times it beats very 
 uniformly, so that we could estimate the time of a 
 boat-race by it, but our eagerness and excitement 
 would speedily make its beats increase, so that it 
 would cease to be a standard of uniform time. The 
 kinds of motion which we see in every-day life are 
 
MOTION. 65 
 
 not uniform ; indeed, if there were many cases of 
 uniform motion we should not have to pay so much 
 for our watches and clocks. 
 
 The commonest case of accelerated motion is 
 that of a falling stone. When we let a body fall, its 
 velocity steadily increases as the time increases. 
 We say that it is attracted to the center of the 
 earth by the force of gravitation. Place a small 
 magnet, made from a piece of watch-spring, on a 
 bit of cork, and float it on the surface of water; 
 bring a strong magnet within a short distance. The 
 small magnet will be attracted toward the large 
 and stronger magnet, and will move with accelerat- 
 ed motion. 
 
 . Suspend a light pith-ball, or a light feather, by 
 a string, and, having rubbed a piece of sealing-wax 
 with silk, bring the sealing-wax into the neighbor- 
 hood of the pith-ball ; it will move toward the seal- 
 ing-wax with accelerated motion. 
 
 It is necessary to obtain clear ideas in regard to 
 the accelerated motion produced by these various 
 manifestations of attracting force. We shall see 
 that work is measured by the force acting through 
 a certain space. Since our spaces are passed 
 through, in general, not at a uniform rate, the im- 
 portance of being able to estimate the space passed 
 through in a given time is very great. Before 
 showing in a rigorous manner that work done in 
 any case is proportional to the space through which 
 the force acts, it is only necessary to ask whether 
 you would prefer to Uft a heavy bucket from a well 
 sixty feet in depth, turning the windlass faster and 
 faster each second, or from a depth of forty feet in 
 the same manner ; or if you would prefer to lift the 
 
66 NEW PHYSICS. 
 
 same bucket from those depths, turning the windlass 
 at a uniform rate of speed. It is evident that the 
 effort would be different in each case, and that we 
 must have some standard in order to compare the 
 different amounts of work we do in the different 
 methods of working. We shall find that the work 
 done in lifting a weight is the same whether we 
 take a day or a year to lift it a certain height. The 
 total amount of work, however, we do in a given 
 time depends upon the number of times we lift the 
 weight. 
 
 Let us first try simple methods, and afterward 
 proceed to more accurate and scientific methods of 
 experiment. 
 
 Experiment 49. — To ascertain the Laws of Fall- 
 ing Bodies. — A and B are two grooved wheels placed 
 upon a board and connected by a string. When A 
 is turned by a crank C, the smaller wheel B revolves 
 more rapidly than the wheel A. If the circumfer- 
 ence of the wheel A is twenty-four inches and that 
 of B three inches, and if A revolves once in one sec- 
 ond, it is evident that B will in one second measure 
 off its circumference as many times as three inches 
 is contained in twenty-four inches, or eight times. 
 
 Fig, 23. 
 
 The small wheel B will therefore revolve eight 
 times in one second. Let us now place a large, flat 
 piece of wood upon the axle of B. This flat circle 
 
MOTION. 67 
 
 of wood will also go round eight times in one sec- 
 ond, or will make one revolution in f^ of a second. 
 Let us cover this flat circle with a thin layer of 
 moist clay or dough rolled out to a suitable consist- 
 ency. If we could make the wheel B revolve uni- 
 formly eight times a second, or once in -J of a sec- 
 ond, we could let two small bullets or lead shots 
 fall from heights separated by a measured interval, 
 and measure the distance of the dints of their blows 
 in the clay or dough from each other. If the lead 
 balls were suspended vertically, one of them just 
 above the other, at a distance of one foot, and if they 
 were released at the same instant, the wheel would 
 revolve through a considerable angle after the first 
 ball had struck the clay and before the second ball 
 could reach it. Suppose this angle is measured, and 
 is found to be \ of the four right angles contained 
 about the center of B, or, in other 
 words, one right angle or J of a 
 whole turn, we should know that 
 the interval between the time of 
 striking of the first ball and that of 
 the second is J of |- of a second 
 (since the wheel turns entirely 
 around in -J of a second), or ^ of a 
 second. Suppose instead of two 
 lead balls we have but one, we 
 could not tell the time it has taken 
 to fall through a measured height 
 unless we could record on the 
 wheel the exact instant we released IT 
 
 the ball : for we must measure from ^"^- '^• 
 
 some starting-point on the wheel. 
 Let us, however, make an approximation in this ex- 
 
 M 
 
68 
 
 NEW PHYSICS. 
 
 periment and afterward see how much this approx- 
 imation is — in other words, how much of an error 
 we shall make. Suspend a lead ball over the flat 
 circle B (Fig. 24), a convenient distance out from its 
 center. Provide the ball with a light string to 
 which a weight, M, is attached. The weight M is 
 pointed and remains at a very small distance from 
 the surface of the clay or dough upon the wheel B. 
 Set the wheel B in rotation, and, when this rotation 
 has become uniform, burn at N the string which sus- 
 tains the ball O. The pointed weight M will strike 
 the clay approximately at the instant of burning the 
 thread, and the time that it takes the ball O to fall 
 through the distance O M will be measured by the 
 part that the angle subtended between the mark 
 made by M and the mark made by O in the clay on 
 the circle B is of the whole angle about B, or four 
 right angles. 
 
 In order to maintain the revolution of B uniform, 
 suspend a heavy weight at the end of a strong 
 string of a suitable length, per- 
 haps one metre or three feet, and 
 make it revolve around a circle 
 drawn on the floor, the center of 
 this circle being directly beneath 
 the heavy weight when it is at 
 rest. This apparatus is called a 
 conical pendulum. You will find 
 by timing its revolutions with 
 your watch that it executes a 
 complete revolution always in 
 the same time, although of course 
 the weight does not keep always on the same circle. 
 Having ascertained the time of the pendulum, fol- 
 
 FlG. 25 
 
MOTION. 
 
 69 
 
 low its motions with the crank arm attached to the 
 circle A. A little practice will enable you to main- 
 tain the circle A in uniform rotation wi±h the coni- 
 cal pendulum. The circle B can be made to run at 
 any suitable rate by increasing or diminishing its 
 size, and also that of A. 
 
 Now let an assistant drop the ball O from differ- 
 ent heights, and tabulate the results thus : 
 
 Spaces or heights. 
 
 Time of falling. 
 
 Square of time of falling. 
 
 other as the squares of the times, or 7 — ^. 
 
 It will be found that the heights will be to each 
 
 k~ t; 
 
 In any one case, therefore, the height must be 
 proportional to the squares of the times it has taken 
 to fall through this height. Hence h = Ci^, in which 
 C is a number we must multiply by the square of the 
 time in order to obtain the height. It is evident 
 that the time squared must always be multiplied by 
 the same number, otherwise the heights would not 
 be proportional to the squares of the times alone, 
 but also to this number if it varied. Since it is a 
 constant multiplier, it cancels out when we obtain 
 the ratio between any two heights through which 
 the weight falls. By trial, taking the height fallen 
 through in feet and the time in seconds, we find that 
 this number by which we must multiply the square 
 of the time in order to obtain the height, is sixteen 
 feet (about 16.08 for New York). 
 
 We can therefore write — 
 
70 
 
 NEW PHYSICS. 
 
 Space fallen through = i6 multiplied by time 
 squared, or j = i6f. 
 
 Hence, if we make t =■ i, or one second, we find 
 that a body will fall i6 feet in the first second. If 
 we make ^ = 2 seconds, the space fallen through 
 will be 32 feet. When / = 3, the space will be 64 
 feet, and so on. 
 
 The principal source of error in this experi- 
 ment arises from the difficulty of maintaining the 
 rotation of A uniform. A little practice, however, 
 will give fairly uniform rotation. 
 
 Another source of error is in neglecting the 
 slight distance of the pointed weight M from the 
 surface of the clay. This distance, however, can 
 be made very small in comparison with the distance 
 through which the ball O falls. 
 
 Experiment 50.— To obtain the final velocity of 
 a falling body. 
 
 Suspend two balls, at different distances apart, 
 
 by passing the connecting 
 thread over a wire sup- 
 port, as in Fig. 26. Place 
 the revolving circle B 
 with its level surface of 
 clay or dough beneath, so 
 that the balls F and E 
 snail lie on the same di- 
 ameter of B (as in figure). 
 Set B in uniform rotation, 
 and then let an assistant 
 burn the string at n with 
 Fig. 26. • a match. The velocity at- 
 
 tained by E during the 
 last few inches of its fall will be approximately that 
 
MOTION. 
 
 n 
 
 of F. If F is sufficiently near to E, this velocity will 
 be measured by the relative positions of the blows 
 of F and E on the clay of the circle B (Fig. 26). 
 
 The part that the angle between the diameters 
 upon which E and F have fallen is of the whole 
 angle, or a^ RA, will give the veloc- 
 ity of E and F during the last inch 
 or inches of its fall. By arranging 
 the weight E so that it may fall 
 through sixteen feet, and placing the 
 weight F slightly above it, we shall 
 find that the falling bodies are mov- 
 ing at the rate of thirty-two feet per 
 second. In one second, therefore, a falling body 
 under the influence of gravitation will acquire a 
 velocity of thirty-two feet per second. 
 
 Fig. 27. 
 
CHAPTER VI. 
 
 PENDULUM MOVEMENTS AND THE MEASURE OF 
 FORCE. 
 
 Pendulum movements may be said to constitute 
 our chief study when we investigate the laws of 
 motion in the subjects of sound and light. The pen- 
 dulum also plays an important part in our investi- 
 gation of the velocity of moving bodies. The sim- 
 ple gravitation-pendulum is a weight suspended at 
 the end of a string or rod. A weight suspended by 
 a slightly twisted wire or rod and the magnetic 
 compass constitute horizontal pendulums. An elas- 
 tic rod set into transverse vibration can be used as 
 a rapidly moving pendulum. A tuning-fork is used 
 when we wish to employ a very rapidly moving 
 pendulum. The shortest gravitation - pendulum 
 which we could use conveniently would not swing 
 faster than ten times a second ; and, therefore, by its 
 use we could only measure -^ of a second. A tun- 
 ing-fork can be made which will execute many hun- 
 dred swings in one second. To show that the mo- 
 tion of a tuning-fork and that of an ordinary grav- 
 ity-pendulum are similar, proceed as follows : 
 
 Experiment 51. — Suspend a little tin can by a 
 string or strings about one metre long. Make a 
 small hole in the bottom of the can; fill the can 
 with fine sand. Set the pendulum to swinging, and 
 
PENDULUM MOVEMENTS. 73 
 
 draw a board or stiif piece of cardboard beneath 
 the pendulum-bob, at right angles to its direction 
 of swing. In this way we shall get a sinuous line 
 on the board, formed by the sand. Then mount a 
 large tuning-fork horizontally in a vise. Provide 
 one of its prongs with a small bit of wire, which 
 shall just touch a plate of smoked glass. Set the 
 fork into vibration by striking its prongs with a 
 wooden mallet, and quickly draw the smoked 
 glass beneath the prong. You will obtain a sinuous 
 line which is similar to that obtained by means of 
 the gravity-pendulum. 
 
 Experiment 52. — Torsioit-Pendulum. — Suspend a 
 wire vertically from a vise screwed to a firm sup- 
 port — a horizontal rafter or beam. Attach, the low- 
 er end firmly to a cylindrical vessel ; a tin pail will 
 answer. Fill this pail with lead bullets. Paste a 
 slip of paper, which has been graduated into milli- 
 metres or tenths of an inch, around the rim of the 
 pail, and provide two vertical wires which will 
 serve to direct your eye always in the same vertical 
 plane. Twist the suspending wire, and you will per- 
 ceive the graduations on the rim of the pail pass 
 these vertical wires to and fro. Observe how many 
 of these graduations pass the wires while a clock is 
 beating seconds, or while your seconds-pendulum is 
 beating seconds. You will in this way obtain the 
 value of one graduation in fractions of a second. 
 Then, when the seconds-clock beats a second, ob- 
 serve how far in graduations the zero-mark (which is 
 opposite the vertical wires when the pendulum is at 
 rest) is from the vertical wires. The time of cross- 
 ing will then be one second, plus the fraction of a 
 second indicated by these graduations. 
 
74 
 
 NEW PHYSICS. 
 
 The experiments we have performed thus far in 
 motion give us only comparatively rough values of 
 the acceleration of bodies under the influence of a 
 stress, such as is exercised by gravitation. It is 
 evident that a more exact method must be devised. 
 The pendulum gives us an accurate method of esti- 
 mating time, and, as will also be seen, of measuring 
 force. 
 
 Before we can understand pendulum move- 
 ments it is necessary to comprehend what is called 
 the resolution of forces. Hitherto we have had 
 to do only with the force of gravitation, and the 
 motion we have studied has been in the direction 
 of the pull of this force. Upon every particle of 
 a body a vertical force acts which attracts the body 
 to the earth ; for the sum of the attraction upon 
 each unit of mass, one force can be substituted 
 which will pass through what is called the center 
 of gravity of the body. 
 
 Experiment 53. — Find the center of gravity of 
 an irregular sheet of card-board. Attach a small 
 piece of lead to a thread, stick a pin through one 
 corner of the card-board, making the hole so large 
 that the card-board will readily move around the 
 pin. Stick the pin thoroughly into a vertical sup- 
 port, so that the sheet of card-board will hang ver- 
 tically. Suspend the little plumb-line you have 
 made with the thread, and the bit of lead, from the 
 pin, and draw a line on the card-board along the 
 plumb-line. Suspend the card-board in the same 
 way from another corner, and draw the new course 
 of the plumb-line. Where these lines intersect will 
 be the center of gravity of the card-board. Verify 
 this by balancing the card-board on a vertical point. 
 
PENDULUM MOVEMENTS. 
 
 75 
 
 \\N 
 
 Experiment 54. — Find in the same way the cen- 
 ter of gravity of a triangular piece of card-board, 
 and discover how far the center 
 of gravity of any triangle is from 
 its base and from its vertex. 
 
 Experiment 55. — Suspend a 
 rod from its center of gravity by 
 a string which passes over a little 
 smooth pulley and is weighted at 
 its end (Fig. 28). It will be found 
 that the weight W, necessary to 
 keep the system in equilibrium, 
 is equal to the weight of the rod. ^w 
 Then place equal weights at the fig. 28. 
 
 ends of the rod. The additional 
 weights at the end of the string necessary to keep 
 the system in equilibrium will be equal to the sum 
 of the weights W-^ and W^. The weights W^^ and J^ 
 represent equal forces acting vertically upon the ends 
 of the rod. The pull on the suspending thread, there- 
 fore, is equivalent to the pull of these two vertical 
 forces, together with the weight of the rod. Since 
 the pulley is smooth, the tension on the suspending 
 thread is everywhere the same, and is equal to the 
 pull of the force due to the weight W. We see 
 that the resultant of two equal parallel forces bi- 
 sects the distance between these forces. 
 
 Experiment 56. — Repeat the preceding experi- 
 ment, placing unequal weights at the ends of the 
 rod. It will then be found necessary to move the 
 point of attachment of the thread nearer the heavier 
 weight. Ascertain the relation between the dis- 
 tances of this point of attachment from the points 
 of attachment of the two weights. In this case 
 S 
 
76 NEW PHYSICS. 
 
 also it will be seen that the resultant of the two 
 vertical forces is also a vertical force, and is equal 
 to the sum of the two vertical forces. 
 
 Experiment 57. — Balance a light stiff rod hori- 
 zontally upon a knife-edge, then attach equal weights 
 to its two ends : the rod remains horizontal. The 
 resultant passes through the knife-edge, for we can 
 replace the pressure on the knife-edge by the pull 
 on the string in Experiment 55. Place unequal 
 weights at the end of the rod and it will be neces- 
 sary to move the knife-edge or fulcrum nearer the 
 greater weight. When the system is balanced on 
 the knife-edge, it is evident that we can again re- 
 place the pressure on the fulcrum by the tension on 
 the string in Experiment 56. The relation between 
 the length of the lever-arms and the weights must 
 therefore be the same as that 
 3 obtained in Experiment 56. 
 
 - A Experiment 58. — Re- 
 
 ^^.^ place the straight rod in Ex- 
 
 f. ^,y"^ \ ^*V^^ periment 55 by two rods 
 
 -'^--— ■■ -^^^-.. jointed firmly together at A 
 
 T (Fig- 29), If the rods are of 
 
 ^, I equal length and of equal 
 
 Fig. 29. weight, the direction of the 
 
 resultant will bisect the line 
 
 joining Cand D, and a straight rod CD can evidently 
 
 replace the bent rod CAD. Make A D longer than 
 
 A C, and place unequal weights at the ends C and D 
 
 (Fig. 30). Find what would be the length of the 
 
 straight horizontal rod which can replace the bent 
 
 rod CAD. Then measure by means of a protractor 
 
 the angles a and /3 which the two portions CA and 
 
 A D make with a horizontal rod, and prove numeri- 
 
PENDULUM MOVEMENTS. yy 
 
 cally that A D X cos a = A E, and that A C X 
 cos^ =A E'. 
 
 Experiment 59.— Obtain a piece of watch-spring 
 from a jeweler. Bend it 
 straight. Cut it into two I 
 
 equal portions about six ^ ' 1 £■ 
 
 centimetres long. Magnet- j Jj^''^^^^-^-:.^..^^ 
 ize these strips equally. cQ 
 This can be done by strok- 
 ing each strip from the ,, 
 center to each end by a bar- pj^ 
 
 magnet, using one pole for 
 
 one direction and the other for the opposite direc- 
 tion. Paste the two magnets on a circle of card- 
 board so that the middle of each steel strip will lie 
 on the center of the card-board. Suspend the card- 
 board by means of a silk strand free from torsion. 
 Such a strand can be obtained from a silk thread by 
 untwisting it and moistening one of the constituent 
 strands. If the card-board does not lie in a horizon- 
 tal plane, balance it by small bits of wax. It will be 
 noticed that the line bisecting the line joining the 
 two poles of the same name points in the same di- 
 rection as a compass-needle. Here we have substi- 
 tuted the magnetic pull of the earth on each of the 
 poles for the pull of gravitation in Experiment 58. 
 
 Experiment 60. — Magnetize the needles un- 
 equally by stroking one a greater number of times 
 than the other, and observe the relation between 
 this experiment and Experiment 58. 
 
 Experiment 61. — Screw two smooth pulleys, 
 which can be obtained at any hardware-store, into 
 a board placed vertically. Pass a string over these 
 pulleys. Weight the string at each end, and also 
 
78 NEW PHYSICS. 
 
 in the middle (Fig. 31). Lay off on the line A B s. 
 line, Am, to represent the weight W^ (if W^ is ten 
 
 grammes, the line A m can be 
 taken equal to ten centime- 
 tres) ; and in the same man- 
 ner a line ^w to represent the 
 weight W. On completing 
 the parallelogram, the diago- 
 nal will be found to be equal 
 to the weight W^. Attach two 
 spring balances to an ordi- 
 nary wooden hoop in such 
 a manner that they can b& 
 clamped upon the hoop in 
 any position (Fig. 32). Con- 
 nect a weight, W, with both 
 springs by means of cords, and vary the position of 
 the spring balances SS^, so that the junction Cof the 
 cords shall always pass through the center of the 
 circle O. Find the relation between the weight W 
 and the pull upon each bal- 
 ance. The hoop should be 
 placed against a vertical wall, 
 and a paper protractor, with 
 its center at O, the center of 
 the circle, will give the angles 
 made by the inclination of the 
 cords at O. 
 
 Experiment 62. — Attach 
 a weight, w, to one end of a 
 horizontal lever supported at 
 
 its middle, .^4, on a knife-edge (Fig. 33). Attach an- 
 other weight, Wj, to the other end by means of a 
 string, which passes over a smooth pulley. Main- 
 
 FiG. 32. 
 
nL"* 
 
 A A' 
 
 "7^ 
 
 PENDULUM MOVEMENTS. yg 
 
 tain the lever in a horizontal position by attaching 
 another weight, w^, at A'. Ascertain the relations 
 
 between the weights 
 Wi, w, and w^ and 
 the distances A B, 
 "^^t AC, and A A'. 
 
 Experiment 63. 
 j/^^ ^ur. -Support a mag- 
 
 FiG. 33. net, about ten centi- 
 
 metres long, upon a 
 cork, and place the cork upon the surface of water 
 in the middle of a large dish. Turn the magnet so 
 that its poles point east and west, and then release 
 it. It will be noticed that the magnet will rotate 
 until its poles point north and south. It will not 
 move forward. Hence, two equal and opposite 
 forces acting upon the magnet will produce rotation 
 but not translation. In other words, the cork will 
 simply revolve on its center, and this center will not 
 move along the surface of the water. Two equal 
 and opposite forces, acting at the end of a lever, 
 constitute what is called a couple. 
 
 Pendulums are of various kinds. The simplest 
 is the vertical pendulum, which is actuated by the 
 force of gravitation. 
 
 Hang up a weight by a string and set it to swing- 
 ing. Count the number of times it crosses the low- 
 est point of its swing in a minute, and it will be 
 found that, as long as the length of the string re- 
 mains the same, the pendulum makes the same num- 
 ber of swings in a minute, and therefore can serve 
 as a measure of time. 
 
 Suspend a magnet by a string attached to its 
 middle, and set it into vibration by turning the end 
 
8o ^E^ PHYSICS. 
 
 that points north a little out of the north and south 
 line, or what is called the magnetic meridian, and 
 count the number of times it crosses the middle 
 point of its swing, or the magnetic meridian. It 
 will be found that this number is invariable for the 
 same magnet, if it is not allowed to lose its magnet- 
 ism. A magnet can, therefore, serve as a measure 
 of time.* 
 
 In the case of the gravity-pendulum, we have to 
 consider the force that draws the weight to the 
 earth. This force is a vertical one. Hang up a 
 number of pendulums, and they will be seen to be 
 all parallel to each other. A spring-balance will 
 show the same pull, in the room where you experi- 
 ment, when the same weight is attached to it, at the 
 same height, wherever it is placed. You can im- 
 agine, therefore, the room to be filled with a vast 
 number of vertical lines of force, which pass through" 
 every inch of the floor of the room. Such a room 
 constitutes what is called a field of force — a uni- 
 form field of force, in which all the lines of force are 
 parallel, and the force acting along these lines is 
 everywhere the same, and equal to a definite pull 
 on a spring-balance. In the case of the gravity- 
 pendulum, we have this vertical force acting upon 
 every particle of the weight, tending to make it 
 move along the lines of force toward the earth ; it 
 would fall vertically if it were not for the string. 
 Hang the string to a spring-balance. When the 
 string is vertical, the pull on the balance represents 
 what we call the weight of the pendulum — that 
 
 * The magnetism of the earth changes slightly from year tp year, 
 and varies with the locality. It can be taken as invariable during sev- 
 eral months. Methods of measuring it are given later. 
 
PENDULUM MOVEMENTS. 8 1 
 
 is, the united pull of the force of gravitation on 
 every particle of the mass of the pendulum-weight. 
 When we draw the pendulum-bob out from the ver- 
 tical by a string attached to the bob, and which is 
 pulled in a horizontal direction, we perceive that 
 it is necessary to exert a pull in order to hold the 
 pendulum and prevent it falling to the vertical posi- 
 tion. At the same time we perceive that the pull 
 on the spring-balance is less than it was when the 
 pendulum was in the vertical position. Since the 
 force of gravitation is unaltered — for the same num- 
 ber of lines of equal force pass through the pendu- 
 lum-bob as before — the force which balances the 
 earth's pull must be the resultant of the force along 
 the string, or its tension and the force that is exerted 
 in a horizontal direction. These forces can be laid 
 off, as in Experiment 6i, and the resultant will be 
 found equal to the pull of the earth on the pendu- 
 lum-bob. When the horizontal pull is released, the 
 pendulum-bob is urged along the arc of a circle by 
 the resultant of the earth's force and the pull on the 
 string which sustains the pendulum-bob. It is then 
 falling down an inclined plane. 
 
 Experiment 64. — Suspend two sfnooth and 
 small lead balls, one inch from each other (see Fig. 
 34), at the top of a smooth inclined plane. Set the 
 circle B (see Experiment 49), in uniform rotation, 
 and burn the thread at a. Measure the angular in- 
 terval between the impressions of the ball on the 
 surface of the revolving circle B. Afterward sus- 
 pend the two balls at M and N. Allow them to 
 drop in the same manner, preserving the same uni- 
 form rotation of the wheel B, and measure the an- 
 gular interval between their impressions on B. It 
 
82 
 
 NEW PHYSICS. 
 
 Fig. 34. 
 
 will be found that these intervals are sensibly the 
 same, which shows that the balls are moving with 
 
 the same velocity, 
 whether they fall 
 down the inclined 
 plane or down the 
 height of the in- 
 clined plane. Al- 
 lowance must, of 
 course, be made for 
 the retarding effect 
 of the friction of the 
 balls on the inclined 
 plane. It will take, 
 however, a longer 
 time for the balls to fall down the inclined plane than 
 to fall down the height of the plane. This will be 
 made evident by suspending two equal balls, one 
 so that it rests on an inclined plane, and the other 
 hangs freely, so that it may fall down the height of 
 the same plane, and burning the string between 
 them, letting them in this manner fall upon a tin 
 pan or other sounding material. It will be noticed 
 that there is an interval of time after the ball which 
 falls down the height of the plane has struck, and 
 before the sound of the blow of the ball which has 
 fallen down the inclined plane is heard. This also 
 can be proved by employing an inclined plane of 
 a sharp vertical angle, and allowing the two balls 
 of the previous experiment to fall upon the revolv- 
 ing-wheel B. It will be found that there is an angu- 
 lar interval which represents the difference of time 
 between faUing down the hypotenuse, or length of 
 the inclined plane, and the height of the plane. 
 
PENDULUM MOVEMENTS. 
 
 83 
 
 Experiment 65.— Prove experimentally that a 
 body will fall down the vertical diameter of a ver- 
 tical circle in the same time that it will fall down 
 any chord of the circle which can be drawn from 
 the upper end of such a diameter. 
 
 Or, Fig. 35, to prove that a ball will fall down 
 the diameter to Z>, starting from A, in the same 
 time that it will fall down A C or A C : 
 
 Arrange the revolving-table B so that A D, the 
 hypothenuse of an inclined plane, may be a vertical 
 diameter of a circle, of which ^ C is a chord. Burn 
 the thread which holds the balls, after having set 
 the wheel B in revolution. It 
 will be seen that the impressions 
 of the blows of the two balls are 
 along the same diameter of the 
 circle B, there is no angular in- 
 terval between them, and there- 
 fore no interval of time. A body 
 will also fall down the chords 
 D C and D C in the same time 
 it will descend the diameter .^4 D. 
 In the gravity-pendulum, the 
 weight of the pendulum in fall- fig. 35. 
 
 ing down the 'small arc of a 
 circle, which is always approximately equal to the 
 length of an inclined plane, acquires the same ve- 
 locity as if it had fallen down the height of this plane. 
 When it arrives at the bottom of the plane, or at the 
 point where the pendulum rod or string is in the 
 same line with the direction of the earth's attraction, 
 it has the velocity it has acquired in falling down 
 the height it has been lifted above this point. This 
 velocity would carry it again up another inclined 
 
84 
 
 NEW PHYSICS. 
 
 plane equal in height to the first one, if there were 
 no resistance to be overcome, and then it would fall 
 again, and so its motion would continue forever. 
 The resistance of the air, however, and the friction 
 of the thread upon its point of support brings it 
 after a time to rest. It will be noticed that the 
 work done, when there are no resistances in lifting 
 the ball up the inclined plane against the pull of 
 gravitation, is equivalent to the work done by let- 
 ting the ball fall down the same inclined plane. 
 This fact will be enlarged upon later. 
 
 Experiment 66. — To show that a pendulum 
 moves through different lengths of arc in the same 
 time : Suspend two equal pendulums side by side 
 (Fig. 36). Provide the bobs of each with a small, 
 flexible pointer, which shall just touch a surface cov- 
 ered with lamp-black. This surface should be that 
 
 17 
 
 1 
 
 Fig. 36. 
 
 of a cylinder. Set the cylinder in rotation, and let 
 the pendulums descend at the same instant from 
 unequal heights. It will be found that the lines 
 
PENDULUM MOVEMENTS. 8 5 
 
 drawn on the blackened cylinder will indicate that 
 the pendulums passed together, and at the same 
 instant, through the lowest point of their swing. 
 Lines of unequal length will be drawn upon the re- 
 volving cylinder. These lines are proportional to 
 the velocity acquired by the pendulums in falling 
 through their arcs. Measure the length of these 
 lines, and compare them with the velocities a body 
 would acquire in falling vertically through the 
 heights the pendulums have descended. 
 
 Another method of studying the law of falling 
 bodies requires us to know the law of pendulums. 
 This law is discussed later. We can anticipate it 
 so far as to use the pendulum as a simple indicator 
 of time. 
 
 Experiment (yj. — Suspend a lead ball at the end 
 of a string, set it to swinging, and count, by means 
 of a watch, its number of swings in a certain number 
 of ticks of the watch. A vibration of a pendulum 
 is from the extreme point of the swing on one side 
 of a vertical line passing through the point of sus- 
 pension of the pendulum to the extreme point 
 reached by the pendulum on the other side of this 
 line. Having obtained the time of the pendulum, 
 arrange the experiment (Fig. 37) so that the fall- 
 ing body will reach the horizontal plane passing 
 through the lowest point of the swing of the pen- 
 dulum in the same time that the pendulum-bob 
 reaches it. This can be ascertained by quickly 
 drawing a board, covered on its upper surface with 
 a sheet of tin smoked with lamp-black, beneath 
 this lowest point. In order to insure that the fall- 
 ing body and the pendulum-bob shall be released 
 at the same instant, it is well to make them both 
 
86 
 
 NEW PHYSICS. 
 
 of iron, and to place them at the poles of two elec- 
 tro-magnets, AB. A thin piece of paper should 
 
 Fig. 37. 
 
 prevent their contact with the iron cores of the 
 electro-magnets. On breaking the circuit at K, the 
 balls will be released practically at the same instant. 
 
 (For description of electro-magnets and battery, 
 see Appendix.) 
 
 With this apparatus one can find the relation 
 between the spaces fallen through and the times of 
 
PENDULUM MOVEMENTS. 
 
 87 
 
 falling. The times of swing of the pendulums of 
 different lengths can be ascertained by counting 
 the number of swings during a sufficiently long in- 
 terval of time, and dividing the number of seconds 
 by the number of swings. 
 
 With this apparatus prove that s=i—, in which 
 
 s = space in feet, / = time, and g — 32.2 feet. 
 
 Find, also, the relation between the distance 
 the body ^ falls and the length, CB, of the pendu- 
 lum. 
 
 In order to obtain the final velocity of the falling 
 body, suspend two small lead balls with a small 
 piece of iron or steel driven into them — one ver- 
 tically above the other, by means of two electro- 
 magnets, and distant from each other by a small 
 measured interval. Arrange the revolving cylinder 
 of Experiment 66 beneath them, so that, when the 
 electrical circuit which contains both the electro- 
 magnets is broken, the balls will strike the revolv- 
 ing cylinder one after the other. Arrange a tuning- 
 fork of known number of vibrations, which shall 
 record its rate upon the revolving cylinder. The 
 number of the vibrations of this fork between the 
 marks made by the two falling bodies will give the 
 rate of the uppermost ball while traversing the 
 known distance between the two falling bodies. 
 
 The prong of the tuning-fork should be provided 
 with a little flexible wire, and the fork should be 
 mounted so that, when the fork is excited by a bow, 
 this little wire shall draw a sinuous line on the re- 
 volving c)dinder. If we know the number of vibra- 
 tions that the fork makes per second, by counting 
 the sinuosities or waves between the marks made 
 
88 NEW PHYSICS. 
 
 by the falling bodies, we shall obtain the time of 
 arrival of one ball and that of the other, and in this 
 way obtain the rate of acceleration during the last 
 second of the descent. In this way prove that 
 1^ = 2gs, in which v is velocity, g is 32.2 feet, and s 
 is space fallen through in feet. 
 
 We have thus far verified the following expres- 
 sions : 
 
 In uniform motion, space = vel. x time, or j = 
 vt ; velocity = acceleration x time, or v = gt. 
 
 In accelerated motion s = gf 
 
 2 
 
 "" = /2gS 
 
 W 
 
 Force = mass x acceleration, or F = — x /, m 
 
 which IV ^ weight, and g = acceleration of gravi- 
 tation = 32.2 feet per second = 981 centimetres per 
 second. Thus far we have measured only the ac- 
 celeration of gravitation. Let us now examine the 
 question of acceleration in general. 
 
 In order to do this, we generally are obliged to 
 modify the motion of bodies by certain mechanical 
 arrangements which involve friction and the idea of 
 inertia. 
 
 The ratio of the friction to the pressure, perpen- 
 dicular to a surface, is called the co-efficient of fric- 
 tion. For example, if our weight is 100 pounds and 
 the friction amounts to 10 pounds, the co-efficient of 
 
 friction is C = = — . To draw a sled weighing 
 
 100 10 & & 
 
 100 pounds along a surface of snow, the co-efficient 
 
 of friction being C = 0.04, we require a force oi F= 
 
 0.04 X 100 = 4 pounds. 
 
MEASURE OF FORCE. 
 
 89 
 
 -Arrange the following experi- 
 
 EXPERIMENT 68.- 
 mentr 
 
 A weight, W is drawn along a horizontal oiled 
 or greased surface, by means of the tension T, on 
 a thread which passes over 
 a small pulley. Ascertain, 
 in the first place, the coeffi- 
 cient of friction between the 
 weight W and the greased 
 surface. This can be done by 
 attaching a spring-balance to 
 the weight W^and ascertain- 
 ing what pull is needed to 
 just move it. This pull, or 
 
 force, is equal to the friction, and the ratio between 
 this force and the weight W is called the co-efficient 
 of friction. 
 
 Experiment 69. — Attach two boards together 
 at one end by a hinge (Fig. 38). Place one board on 
 a table and elevate the other until a weight, W, shall 
 just begin to slide down the greased inclined plane. 
 Measure the angle a of this inclined plane by means 
 of a protractor, or by finding the ratio of the height 
 MN to the base OM; this will be the natural tan- 
 gent of a, and can, therefore, be obtained from a 
 table of natural tangents. 
 
 The pressure on the perpendicular plane will be 
 a b cos a (since the angle abcis equal to a, for ab \s 
 perpendicular to OM and be is perpendicular to O N. 
 Two angles are equal whose sides are mutually per 
 pendicular). The friction is proportional to this 
 pressure, and opposes the component of W, or a c, 
 which would urge the body down the inclined 
 plane. 
 
go NEW PHYSICS. 
 
 Hence we have C W cos a = F, or C = 
 
 Wcos 
 
 a. 
 
 . F ac MN . „ ^ 
 
 But the ratio -jy^ = — 7 = ;^-t7 = sm a. Hence 6 = 
 W ab ON 
 
 sin a 
 
 = tan a. 
 
 cos a 
 
 This value of C should agree with that obtained by 
 measuring the force necessary to just move the 
 weight along a horizontal surface which is covered 
 with the same substance as the inclined plane. It 
 is well to test the truth of the trigonometrical rela- 
 tion, cosine d and sin a, by actual measurements. 
 
 Experiment 70. — Let F be the force necessary 
 to draw the weight W along the greased surface. 
 The motive force will then heF= F — C W,\n which 
 C is the co-eflScient of friction. 
 
 FA-W . 
 
 The mass moved will be , in which g is the 
 
 g 
 acceleration of gravitation. The force, however, is 
 always the mass moved multiplied by the accelera- 
 tion, or F= Mf. The mass, however, 
 
 _ weight P-\-W 
 
 acceleration of gravitation g 
 
 Hence /^=(^/,or/>-C^=(^/,or 
 
 ^ F-CW „^ , , f^ 2s ^ 
 
 2s F-CW 
 
 Ascertain from direct experiment what value of 
 g will satisfy this equality. To do this, suspend a 
 simple pendulum, whose time of vibration you have 
 ascertained, so that it shall be in the same vertical 
 
MEASURE OF FORCE. 
 
 91 
 
 plane with the string to which the weight P is at- 
 tached. Attach a small bit of wire to the bob of the 
 pendulum, arrange the position of P so that it will 
 fall through a certain height in the same time that 
 the pendulum executes one half its swing.* In this 
 
 Fig. 39. 
 
 case the weight Pand the pendulum-bob will touch 
 a plate of smoked glass or paper, which is drawn 
 horizontally by the hand beneath the lowest point 
 of the swing of the pendulum, at the same time ; the 
 time of the pendulum will then give us the value of 
 t in the formula, which is one half the time of the 
 pendulum, and the space fallen through by the 
 weight P in this time is what had been arranged. 
 In this experiment we neglect the friction and in- 
 ertia of the pulley a, and also the rigidity of the 
 
 * This experiment requires two persons — one to release the pen- 
 dulum and the weight P at the same time, and another to draw the 
 smoked surface. 
 
92 
 
 NEW PHYSICS. 
 
 string. The friction of the pulley should be re- 
 duced by oiling, and a strand should be used to 
 connect W and P. Two electro-magnets, however, 
 can be used, as in Experiment 67. 
 
 Our experiments on the motion of bodies falling 
 under the influence of the force of gravitation illus- 
 trate the second law of motion, which states that 
 change of motion is proportional to the impressed 
 force, and takes place in the direction in which the 
 force is impressed. 
 
 By motion is understood momentum, or the 
 quantity of matter moved multiplied by the rate at 
 which it moves. 
 
 In order to clear our ideas, certain definitions 
 are necessary. We can not define force without 
 having glanced over the whole subject of motion. 
 It is necessary for us to perceive what is termed the 
 relativity of all phenomena. In physics we are con- 
 stantly comparing the state of one body with the 
 state of another, both in regard to volume and in re- 
 gard to motion. We get our ideas of force by the 
 work which is necessary to do in order to resist a 
 change of position of the material particles of a sys- 
 tem. We have defined the " unit of force to be that 
 force which, acting on the unit of mass during the 
 unit of time, generates the unit of velocity." We 
 also know that the total effect of a force in commu- 
 nicating velocity to a body is proportional to the 
 force, and to the time during which the force acts. 
 We have spoken of an impressed force as- an im- 
 pulse. By impulse is meant the product of the 
 force by the time during which it acts — under- 
 standing by the word force its mean intensity. 
 
 By the definition of a unit force, we perceive that 
 
MEASURE OF FORCE. 
 
 93 
 
 the force is expressed as a velocity. Since our ex- 
 act knowledge of nature depends upon measure- 
 ments, it is very important that we should have a 
 common standard for the measurement of length, 
 and also of mass, and of force. The system which 
 we shall use is called the centimetre-gramme-sec- 
 ond system, or the C. G. S. system. The advantage 
 of the metric system in determining distance and 
 volume we have already dwelt upon. Since this 
 system often involves the use of large numbers, it 
 is usual, to express the results by certain numbers 
 multiplied by ten raised to a power ; thus, 45,000,000 
 = 45 X lO^ 
 
 The C. G. S. unit of force is called a dyne. A 
 dyne is the force which, acting upon one gramme 
 during one second, generates a velocity of one cen- 
 timetre per second. 
 
 The C. G. S. unit of work is called an er£: It 
 represents the work done by one dyne through a 
 distance of one centimetre. 
 
 The unit of energy is also the erg. 
 
 Work is expressed in gravitation measure as fol- 
 lows : I gramme-centimetre is equal to g ergs (^ be- 
 ing the acceleration of gravitation, or the standard 
 by which we measure force, £■ = 980 centimetres for 
 latitude of Paris). One kilogrammetre is equal to 
 100,000 ergs. One foot-pound is 13.825 £■ ergs, or 
 taking g-= 980, is 1.355 X 10' ergs. The unit rate 
 of working is i erg per second ; a watt is 5 50 foot- 
 pounds per second. 
 
 The word force is often used in a wrong sense 
 for energy,. and also for the power of doing work. 
 An engine is estimated by the rate at which it will 
 lift a given weight,. We can speak of the power of 
 
94 
 
 NEW PHYSICS. 
 
 a. steam-engine or the power of a horse, but we can 
 not use the expression " force of a steam-engine," for 
 in our idea of force we do not also include the exer- 
 tion of the force through a space. When we do in- 
 clude this latter idea we are really speaking of work. 
 
 We define work as follows : 
 
 " Work is the act of producing a change of con- 
 figuration in a system in opposition to force which 
 resists that change." 
 
 Energy is the capacity of doing work. 
 
 " When the nature of a material system is such 
 that, if, after the system has undergone any series of 
 changes, it is brought back to its original state, the 
 whole work done by external agents on the system 
 is equal to the whole work done by the system in 
 overcoming external forces, the system is called a 
 conservative system." 
 
 General Statement of the Principle of the Conserva- 
 tion of Energy. — "The total energy of any material 
 system is a quantity which can neither be increased 
 nor diminished by any action between the parts of 
 the system, though it may be transformed into 
 any of the forms of which energy is susceptible " 
 (Maxwell's " Matter and Motion," p. 103). 
 
 The word stress is much used in modern phys- 
 ics to denote the mutual action between two por- 
 tions of matter. The tension on a rope is a stress, 
 or the pressure between two bodies. Stress im- 
 plies action and reaction, for there must always 
 be an opposite effect opposing tension. Suppose 
 that two persons pull against each other by means 
 of a rope, there is a stress upon the rope. Take 
 away one of the persons, and the reaction disap- 
 pears. The pull on the rope is the force exercised 
 
MEASURE OF FORCE. gj 
 
 by only one of the persons, for we could replace 
 one of the persons by the resistance opposed by a 
 post to which the rope may be attached. The stress 
 or tension is the same throughout the rope. 
 
 In the case of gravitation, electricity, and mag- 
 netism, the bodies act upon each other without any 
 visible means of connection. The stress appears to 
 be exercised by means of some medium in which 
 the bodies are placed. Even in the case of the at- 
 traction or repulsion of magnets we find that action 
 and reaction are equal to each other. 
 
 If instead of a unit mass we suppose that equal 
 forces are applied to every unit of mass of the sys- 
 tem, the total force acting upon the whole mass 
 would be the mass multiplied by the velocity which 
 any one of the equal forces would give one of the 
 units of mass. We express this by the equation 
 F =: MV. This is often called the equation of mo- 
 mentum. J/ represents the number of units of mass, 
 and Fthe number of units of velocity. The direc- 
 tion of the momentum is that of the velocity, and, 
 since we estimate velocity by a reference to two 
 epochs, we must also estimate momentum by a 
 reference to some point. Force is sometimes said 
 to be momentum. Momentum can only be esti- 
 mated by its changes. We conceive of something 
 which produces these rates of change, and this 
 something is what we call force. 
 
 Thus we say that " the change of momentum of 
 a body is numerically equal to the impulse which 
 produces it, and is in the same direction " (Max- 
 well's " Matter and Motion," p. 75). 
 
CHAPTER VII. 
 
 MOMENTS OF INERTIA. 
 
 In the case of the magnetic pendulum we have 
 the same species of motion as in the case of the ver- 
 tical or gravity pendulum. The magnetic pendu- 
 lum, however, is not a vertical one. It moves in a 
 horizontal plane. The force acting upon it is the 
 pull that the north pole exerts upon the south pole 
 of the suspended magnet. An equal pull is exerted 
 on the north pole of the magnet by the south pole 
 of the earth. These pulls upon magnets suspended 
 in the same room beyond each other's influences 
 are all in the same direction; for all ships' com- 
 passes, free from the influences of iron, and at the 
 same place, point to the north pole. This is so far 
 from our habitable regions that the lines of mag- 
 netic force converging to it are approximately 
 parallel. To prove that the magnetic force is equal 
 everywhere along these lines of magnetic force, as in 
 the same room, where there are no masses of iron, 
 make a magnet, suspend it and set it in vibration. 
 It will be found that it will make the same number 
 of vibrations wherever it is placed in the same 
 room (where there are no other masses of iron or 
 steel). We can therefore picture to ourselves lines 
 of magnetic force which are everywhere parallel, 
 
MOMENTS OF INERTIA. 97 
 
 and along which the magnetic pull is the same. 
 Great care is taken in magnetic measurements to 
 have the magnetic pendulum vibrate in such a field 
 of uniform magnetic force. There is little difficulty 
 in obtaining a uniform field of gravitative force in 
 which to swing our gravity-pendulum, but it is gen- 
 erally very difficult to obtain a field of uniform 
 magnetic force. If the field of magnetic force is 
 not uniform, the time of the magnetic pendulum 
 would vary, for at one time it would be falling 
 down an inclined plane under the action of one 
 pull, and at another time down another inclined 
 plane under another pull, and all uniformity would 
 be lost. 
 
 If the earth's pull on a pendulum should in- 
 crease, our gravitation-pendulums would move fast- 
 er, for the force urging them down their respective 
 heights would be greater. 
 
 If the earth's magnetic force should increase, 
 our magnetic pendulum would swing faster. We 
 have no method of increasing the earth's gravita- 
 tive force, and therefore there is no way of increas- 
 ing the time of swing of a gravity-pendulum of 
 which the length is maintained constant. 
 
 We can not increase the earth's magnetic force, 
 but we can make a field of magnetic force of any 
 strength we choose, in which we can place our mag- 
 netic pendulum and increase or decrease its time of 
 swing at pleasure. This artificial field, however, 
 must be a uniform one, such as we have previously 
 described. 
 
 Experiment 71. — To make a uniform field of 
 magnetic force in which a magnetic pendulum swings 
 faster than in the earth's magnetic field. 
 
98 ^EW PHYSICS. 
 
 Having sprinkled spme fine iron-filings upon a 
 piece of card-board, place two magnets in such a 
 manner that the iron-filings will arrange themselves 
 in straight lines between the poles, as in the-figure. 
 Suspend a short magnet, A (Fig. 40), made of a bit 
 of watch-spring, between the poles of these mag- 
 nets : it will be in an approximately uniform field of 
 
 Fig. 40. 
 
 magnetic force. On setting it into vibration by 
 suddenly bringing another large magnet near it, 
 it will be seen that it swings much more rapidly 
 than in the earth's field. Instead of the permanent 
 magnets, two electro-magnets can be used. 
 
 Second Metliod of ascertaining the Law of Falling 
 Bodies. — If we could make a gravity-pendulum or 
 a magnetic pendulum vibrate very fast, and could 
 also count these vibrations, we could use such a 
 pendulum to estimate small intervals of time. The 
 shortest gravity-pendulum we can make, however, 
 will not enable us to estimate tenths' of a second ; 
 and we can not follow the movements of a quicker 
 moving magnetic pendulum by the eye. A tuning- 
 fork which sounds a comparatively low note, as the 
 middle C on the piano, affords us a convenient 
 means of measuring small fractions of a second. 
 
 If, now, we could arrest a swiftly-moving mag- 
 
MOMENTS OF INERTIA. 
 
 99 
 
 netic needle, which has been put in a strong mag- 
 netic field, at any point of its swing, and if we knew 
 what interval of time this portion of the swing cor- 
 responded to, we could measure the time of falling 
 of a body by setting a magnetic needle free, which 
 is held at a certain deflection at the instant the body 
 begins to fall, and stopping the needle in its swing 
 the instant the body has fallen through a certain 
 height. This can be accomplished as follows : 
 
 The tuning-fork F (Fig. 41) is firmly fixed in a 
 vise, V, and adjusted 
 until the style, or light 
 pointer attached to 
 one of its prongs, just 
 touches a plate of 
 smoked glass, which 
 can slide beneath it. 
 This plate of glass is 
 placed upon a piece 
 of board, which slides 
 between guides, G G. 
 This piece of board also 
 carries a piece of stiff 
 brass, P, which slides 
 with it, and touches m 
 or m' merely with its 
 point. The extent of 
 the slide of P, and the 
 board to which it is attached, is limited by the stop 
 5 5. The point of the piece of brass rests upon a 
 piece of brass, M, when the slide is drawn back. 
 The brass pointer leaves the brass M, crosses an in- 
 terval of wood or paper, or other non-metallic sub- 
 stance, and rests upon another brass plate, M^. If, 
 6 
 
 Fig. 41. 
 
lOO N£W' PHYSICS. 
 
 now, the tuning-fork should be set mto vibration by 
 a bow, and the slide should be drawn back to the 
 stop s s, the tuning-fork will draw a sinuous curve 
 upon the smoked glass, whose length will evidently 
 be equal to the distance the point of the brass piece 
 P has traveled. By laying out on this sinuous line 
 the distance of the starting-point of the point of P 
 from the edges of the brass pieces M, M^, we can 
 evidently obtain, by counting the number of vibra- 
 tions registered on the smoked glass between these 
 brass pieces, the time it has taken to cross the in- 
 terval from one brass piece, M, to 
 the other, M^. In the accompa- 
 nying Fig. 42, we see that there 
 are two complete vibrations, and 
 a fraction of another one, which 
 must be carefully estimated. 
 
 Let us now connect the two 
 brass pieces M, M^ with one pole 
 Fig 42 of a battery, B, and the brass 
 
 pointer with the other pole of the 
 same battery, passing the current through a gal- 
 vanoscope, G. It is evident that, when the brass 
 pointer touches the brass M, the current is closed, 
 and a current passes through the circuit. This cir- 
 cuit is broken when the pointer leaves the brass M, 
 and is made again when it has crossed the interval 
 between M and M^ and touches M^. By regulating 
 the interval between M and M^, one can make the 
 interval what one chooses, and the tuning-fork will 
 give the time of crossing the interval. 
 
 Let us now look at the action of the galvano- 
 scope G. This is made as follows : A piece of board, 
 about five inches wide, one inch and a quarter thick, 
 
MOMENTS OF INERTIA. 
 
 lOI 
 
 and eight inches long, has two grooves cut in it. 
 Four rounded cleats, c c, are glued above these 
 grooves, as in Fig. 43. Over these cleats are wound 
 several turns of wire continuously, as if one were 
 winding thread on a spool, leaving a space, however. 
 
 Fig. 43. 
 
 between the two equal windings, W, W^ (Fig. 47). 
 The ends of the windings are brought to binding- 
 screws at B and B^. Suspend a small magnet, M, 
 made from a bit of watch-spring, and provided with 
 a light, long pointer, P, P* (which is at right angles 
 to the magnet), by a silk fiber in the middle of the 
 space between the two windings, as is shown in the 
 accompanying figures. 
 
 Place two equal magnets north and south of 
 the little suspended magnet, in order to increase 
 the magnetic field in which it swings. Place a 
 plate of looking-glass under the long pointer, at one 
 
I02 
 
 NEIV PHYSICS. 
 
 side, and cut out the graduation of a paper protract- 
 or, or any graduated circle, and paste it upon the 
 looking-glass, so that the center of the circle from 
 which it was drawn shall be directly at the center 
 of the two windings, and therefore under the center 
 of the suspended magnet. The looking-glass is to 
 avoid parallax in reading the excursions of the 
 pointer attached to the magnet. This galvanoscope 
 should be placed under a box provided with a glass 
 lid, L (Fig. 44), in order to protect the needle from 
 currents of air. 
 
 When the circuit of the battery is closed through 
 
 ^ 
 
 this galvanoscope, the needle is deflected, and the 
 pointer attached to the magnet indicates a certain 
 angle on the graduated circle. This is the case 
 when the pointer rests upon the brass piece M. 
 When the pointer leaves the brass M, at that instant 
 the circuit is broken, and the needle rapidly swings 
 toward its position of rest. When, however, the 
 brass pointer touches the brass M^, its fall is ar- 
 
MOMENTS OF INERTIA. 
 
 103 
 
 rested, it receives a sudden check, and swings back 
 to its first deflection. The time of its fall, from this 
 first deflection to the point where it receives a 
 check, is registered by the tuning-fork in the man- 
 ner we have indicated above. With a suitably con- 
 stant cell, like a Daniell cell (see Appendix), the 
 times of small breaks in an electrical circuit can be 
 read off from the angles through which the needle 
 of the galvanoscope falls, having first ascertained 
 the value of these -angles in time by means of the 
 tuning-fork. 
 
 Let us now apply this apparatus to determining 
 the laws of falling bodies, not only in the hopes of 
 reaching more accurate results than by our former 
 apparatus, but also to obtain a knowledge of the use 
 of electrical apparatus. 
 
 Experiment 72. — To find the time it takes a 
 body to fall under the influence of gravitation : 
 
 Suspend a ball, C, from a bit of spring brass at 
 A (shown at a, in Fig. 45), 
 which completes the electri- ^ 
 
 cal circuit through G and the 
 battery B. When the spring 
 which holds the falling body C 
 is burned, the latter in falling ji^ 
 
 breaks the circuit at A (Fig. 
 46), and, in striking at D, closes it again. 
 
 The break at D should be arranged so that it 
 will be kept closed after the falling body strikes 
 it, otherwise the needle of the galvanometer will 
 not receive a decided check in its fall to a position 
 of rest. The height through which the body falls 
 can then be compared with the times of falling by 
 knowing the values of the angular deflections of 
 
I04 
 
 NEW PHYSICS. 
 
 Fig. 46. 
 
 the magnet in times. This can be determined pre- 
 viously, in the manner we have shown, by the tun- 
 ing-fork, provided that 
 the electrical current 
 is maintained uniform. 
 This will not be diffi- 
 cult, if a Daniell cell is 
 employed. 
 
 Experiment 73. — 
 Prove that the squares 
 of the velocities at- 
 tained by a body fall- 
 ing through different 
 heights are proportional to the heights. 
 
 This can be done by arranging the electrical 
 connections so that the falling body shall break the 
 circuit after it has been falling a certain time, and 
 shall make it again at any suitable time. Thus, 
 when the falling body strikes / (Fig. 47), it breaks 
 the electrical circuit, and, 
 after falling through the 
 distance /, /*, it makes the 
 circuit again at /', and 
 keeps it permanently 
 closed. The breaking at / 
 should not sensibly impede 
 the fall of the heavy falling 
 body. This can be accom- 
 plished by making a deli- 
 cate connection at /. 
 
 In one case the electrical circuit is I C G B. 
 When the falling body rests on /^, the circuit is 
 closed through I^ C G B. The distances 11^ should 
 be small, for the velocity is not uniform ; but we 
 
 Fig. 47. 
 
MOMENTS OF INERTIA. 
 
 I05 
 
 suppose that the velocity during the last two or 
 
 three inches of a fall is uniform. This is approxi- 
 
 v^ h 
 mately true. In this way we can prove that — 2=t-, 
 
 ,in which v and v^ are velocities, and h and h-^ spaces 
 or heights. 
 
 There is ^till another simple apparatus by which 
 the same law can be tested ; the error arising from 
 friction, however, has to be carefully estimated. It 
 is as follows : 
 
 Experiment 74. — Adjust a tuning-fork of known 
 rate of vibration in a vise (Fig. 48). Allow the tuning- 
 fork and the vise to slide 
 in vertical guides, which 
 are carefully greased so 
 as to avoid friction as 
 much as possible. Ar- 
 range a smoked glass at 
 suitable distances, so 
 that the stylus or point 
 upon the tuning - fork 
 shall just touch the 
 smoked glass in the last 
 few instants of its fall, 
 and register its vibra- 
 tions for a measured 
 space. If the slide con- 
 taining the vise and tun- 
 ing-fork is sufficiently 
 weighted, and the fric- 
 tion in the guides is made 
 
 small, the final velocity of a falling body, which, in 
 this case, is the apparatus for measuring the time, 
 can be readily measured. The final record will be 
 
 I 
 
 .,1 
 
 ■ , 
 
 ,", 
 
 1^ 
 
 ; 1 
 
 /I 
 
 Fig. 48. 
 
I06 NE^ PHYSICS. 
 
 of the nature indicated in the accompanying figure. 
 By laying off a measure-line upon this sinuous curve, 
 the number of vibrations of the tuning-fork, made 
 while it passed through this distance, can be ascer- 
 tained, and from this the velocity. Instead of a tun- 
 ing-fork an elastic rod can be employed, and its rate 
 of vibration be determined afterward by a tuning- 
 fork of known rate.* 
 
 Experiment 75. — Laws of Inclined Planes. — By 
 inclining the guides, the laws of the fall of bodies 
 down inclined planes can also be tested. 
 
 We have given various methods of determining 
 the laws of falling bodies, both to obtain practice in 
 physical experimentation and to obtain clear ideas 
 of energy and of work. We have seen how impor- 
 tant it is to have a uniform standard of time, in or- 
 der to know how long an interval elapses between 
 one occurrence and another. But why do we wish 
 to know the time between two events ? Is it not 
 because some effort must be made — some work must 
 be done, either by ourselves or by others, during 
 these intervals of time ? A steamer sometimes runs 
 for several days in a dense fog, the number of revo- 
 lutions of the engine is recorded, and the time dur- 
 ing which these revolutions have been maintained 
 is ascertained. If sixty revolutions of the screw per 
 minute will drive the steamer ten miles in an hour, 
 the distance the steamer has gone in several days 
 can be approximately ascertained. This is called 
 dead-reckoning. The velocity of the screw and the 
 rate of time of the clock or watch are both sup- 
 posed to be uniform. If either varied, the steamer 
 might be driven upon dangerous rocks in a fog. 
 * See MuUer's " Physik," p. 254. 
 
MOMENTS OF INERTIA. 107 
 
 Hence the necessity for a uniform standard of 
 time. Behind this estimate of time and velocity, 
 however, is an (unconscious, perhaps) estimate of 
 the work that the engines can do during the same 
 interval of time. It is this work that conveys the 
 steamer. When we carry a weight up a steep hill, 
 the time we take, and therefore the velocity, becomes 
 of great importance, when we consider the weight 
 which we carry. We should prefer to engage to 
 roll small cannon-balls up to a certain point on an 
 inclined plane, rather than cannon-balls three times 
 as great, at the same velocity. In order to shoot 
 an arrow vertically, so that it may attain a certain 
 height, we must exert a certain pull upon the bow- 
 string. In order to increase this height, we must 
 increase our pull on the string. Now, the arrow, 
 having attained a certain height, will return in a 
 certain time and with a certain velocity. If Ave 
 neglect the resistance of the air, the arrow or the 
 rifle-ball will return with the same velocity with 
 which it was shot forth. We have seen that the 
 square of this final velocity is proportional to the 
 height through which the body falls, or z^ = 2gh, 
 
 or h = — . 
 
 In order therefore to shoot an arrow to a certain 
 height, we must do a certain amount of work upon 
 the b5w-string, or burn a certain amount of powder 
 in a rifle in order to communicate the velocity which 
 will carry the missile to a certain height in a certain 
 time. Behind the exhibition of velocity, therefore, 
 is an exertion of work. We could measure the 
 muscular work done by different people in pulling 
 a bow by the height to which the missile rises, or, 
 
I08 NEW PHYSICS. 
 
 by the velocity the missile acquires in falling from 
 this height. In carrying a heavy weight up a 
 flight of stairs, we are conscious of working against 
 the force of gravitation at every instant. If we 
 carry a ten-pound weight up a flight of stairs 
 twenty feet high, we feel that the work we do is 
 proportional to the number of feet we rise, for the 
 force of gravitation is a vertical one, and we are al- 
 ways advancing along lines of vertical force against 
 the pull along these lines. In rising above the earth, 
 the work we do is proportional to the mass we lift, 
 and to the pull along these Unes of force, or what 
 we term the force of gravitation, and to the distance 
 we pull against this force, or the height we lift the 
 weight. 
 
 The work is proportional to nothing else. It 
 does not matter whether we are all day or only one 
 instant in lifting the weight. 
 
 Hence we say that the work done is equal to the 
 mass lifted, multiplied by the pull of the force of 
 gravitation upon every particle of it, in a uniform 
 field of gravitative force, and finally multiplied by 
 the distance we pull against gravitation, or the 
 height through which the mass is lifted. To ex- 
 press this in letters, let w = work, let m = mass, let 
 g — gravitation, pull, or force, let h = height or dis- 
 tance passed through against the vertical pull of 
 gravitation: then W=mgk. 
 
 Now, we have said that it does not make any dif- 
 ference whether we take a day to lift the weight or 
 one minute, as far as regards the work we do in lift- 
 ing this weight against the force of gravitation. If 
 we let this weight fall from the height of twenty 
 feet it will acquire the same velocity when it reaches 
 
MOMENTS OF INERTIA. 109 
 
 the earth, whether we take an hour to lift it or only 
 one minute. It would take a certain muscular pull 
 if this weight were in the shape of an arrow, to pull 
 a bow-string in order to give the weight the velocity 
 which would enable it to reach the height h. The 
 work we put into this muscular pull to overcome 
 the pull of gravitation, through this space of h, is 
 evidently the equivalent of the work that must be 
 done against this pull in mounting the stairs, and 
 conveying the weight to the height of h. In esti- 
 mating the work of carrying this weight once to the 
 height h against gravitation, we find that by run- 
 ning up with it or proceeding slowly, we can not do 
 more than is expressed by the muscular pull on the 
 bow-string which will urge the same weight to the 
 same height. If, however, it is necessary to carry 
 the same weight up twenty feet sixty times a min- 
 ute, or to shoot an equivalent weight from a bow 
 sixty times a minute, the total amount of work will 
 be sixty times the work that is done in conveying 
 the same weight once up the same height. 
 
 Let us see now what work will be done by shoot- 
 ing a ball up a certain vertical height, h, with a cer- 
 tain velocity, v. 
 
 Since the body in falling through this same 
 
 height will require the same velocity, v, with which 
 
 1? 
 it started, we have 1? = igh, or ^ = — . 
 
 But the work w — mgh. If, therefore, we mul- 
 tiply hhj mg, we have work, 
 
 1? mi? 
 
 w = mgh = mg -^— = 
 
 2^ 2 
 
 Hence we see that the work is proportional to the 
 
 mass moved and to the square of the velocity with 
 
no NEW PHYSICS. 
 
 which it is shot forth. On the other hand, we see 
 that the body, in falling through the height h from a 
 position of rest, will acquire a velocity of v. If it could 
 be arranged that this falling weight— supposed to 
 be inelastic — should meet a similar inelastic weight 
 just being shot forth from a bow at the foot of the 
 height, neither would move — since both are mov- 
 ing with the same velocity in opposite directions, 
 and the effort of one weight in rising would be ex- 
 actly counterbalanced by the energy which the 
 other has acquired in falling. Hence the work 
 done by the falling weight would be equal to the 
 muscular pull given to the bow, acting through a 
 certain distance, to urge the same weight up the 
 height h. 
 
 We see, therefore, that a body falling down an 
 inclined plane would acquire the same velocity as if 
 it fell from the height of the plane, for the work of 
 carrying a body up the length of the inclined plane 
 is the same as lifting it up the height of the plane. 
 
 The action of a pendulum consists in alternately 
 lifting a weight up an inclined plane and letting it 
 fail down this plane. The work that it does consists 
 in lifting the pendulum up the height of the inclined 
 planes whose lengths are the small chords of the arc 
 in which it moves, against the pull of gravitation. 
 If h represents the height to which the pendulum- 
 bob is lifted at the end of one excursion, the work 
 
 m v^ 
 done, or w = mgh = , this would be the work 
 
 2 
 
 developed by the pull of the pendulum-bob down 
 the height of the inclined plane, or down to the foot 
 of the length of the inclined plane. 
 
 Since the gravity-pendulum is employed to ob- 
 
MOMENTS OF INERTIA. m 
 
 tain our standard of uniform time, and can be used 
 to measure the work done by falling bodies, and the 
 magnetic pendulum, as we shall see later, is used 
 to measure electrical energy, we shall examine the 
 motion of pendulums more closely. 
 
 Experiment 76. — Hang up a ball of suitable 
 weight by a string — a leaden ball, for instance, by a 
 fine black silk thread — and count the time of its vi- 
 brations. 
 
 In order to obtain the time of a pendulum, count 
 the beats of a clock (or notice the number of seconds 
 by a watch) between the time when the pendulum 
 passes through the lowest point of its arc at the 
 same instant that the clock beats and the next time 
 when the swing and the beat are again together. 
 Thus, suppose that this interval is 50 seconds. It has 
 taken the slower pendulum this length of time to 
 
 lose one vibration, hence its time will be — = 1.020 
 
 49 
 seconds. Take the mean of a number of such ob- 
 servations. This method of coincidences, it will be 
 noticed, is a very accurate one, for, if the mean had 
 
 52 
 been 52 instead of 50, we should have had — = 1.019, 
 
 which differs from the preceding only by ^ of one 
 per cent. 
 
 Then alter the length of the pendulum, and again 
 obtain its time of swing. Prove in this way the 
 
 ./} 
 
 formula ^ — tt y — in which t is time of swing. 
 g 
 IT = 3.1416, /= length in centimetres, ^= force of 
 
 gravitation (980.8 in the latitude of Cambridge, 
 
 Massachusetts, expressed in C. G. S. units). 
 
112 
 
 NEW PHYSICS. 
 
 The formula ^ = tt y — can also be proved in 
 g 
 another way. The method demands a knowledge 
 of the action of forces directed toward a fixed cen- 
 ter. The motion of a w^eight whirled around at the 
 end of a string is the commonest illustration of a 
 body moving in a curve under the action of a force 
 constantly directed toward a fixed central point. 
 
 When we swing a stone, attached to a string, 
 around our head, we are conscious of a pull on the 
 string, the stone by Newton's first and third law of 
 motion tends to preserve the direction of its motion 
 at any instant, and we are compelled, in order that 
 the stone shall move in a circle, to exert a tension 
 on the string. The reaction to this pull or tension 
 is what is often called centrifugal force. The ten- 
 sion on the string is the 
 centripetal force. It 
 will be seen that the 
 centrifugal force is 
 merely the tendency of 
 the stone to continue in 
 its line of motion, away 
 from the hand, and is 
 the reaction to the cen- 
 tripetal force. 
 
 Experiment t]. — 
 Suspend a weight at the 
 end of a wire, about two 
 metres long (Fig. 49). 
 Slip a small glass rod 
 over a certain portion 
 of the wire, and confine it to the wire, so that it may 
 move as a part of it. Slip upon this glass rod a small 
 
 Fig. 49. 
 
MOMENTS OF INERTIA. 
 
 "3 
 
 ring, which is supported by two small spiral springs. 
 Attach to this ring suitable weights, and fix the spi- 
 ral springs to a suitable support. Blacken the glass 
 rod by carefully moving it over a lamp which is 
 smoking. Draw the pendulum from its point of 
 rest to a measured height, hold the springs by means 
 of two strings connected at A, and then burn the 
 thread at A with a match. The small weights will 
 acquire the velocity of the pendulum at a certain 
 point, which is the lowest point marked on the 
 blackened glass by the ring. This is when the pen- 
 dulum is passing through the lowest point of its 
 swing. The centripetal force elongates the spiral, 
 and the amount of this force can be measured by 
 ascertaining what weight will extend the spring to 
 the mark on the blackened glass. The weight of 
 the pendulum-bob B should be much greater than 
 the weights attached to the spirals. 
 
 Prove that this force is proportional to the mass, 
 the velocity being the same. 
 
 Also to the square of the velocity, the mass be- 
 ing the same. 
 
 Also varies inversely as the radius of the arc 
 
 through which the weight swings — 
 
 ■mi? 
 
 or i^= . 
 
 r 
 
 The velocity of any point of the pendulum B, at 
 the lowest point of its swing, can be obtained by 
 previous methods. The length of the pendulum B 
 can be changed by lengthening the wire to which 
 the bob is attached. 
 
 We can now prove the formula t = ir\ — as 
 
 follows : 
 
114 
 
 NEW PHYSICS. 
 
 Experiment 78. — Suspend a heavy ball from a 
 wire about one metre long, and set it to swinging 
 around a circle drawn with chalk upon the floor be- 
 neath it. It thus constitutes a conical pendulum which 
 will be found to execute all its swings approximately 
 in the same time. There is, therefore, some simple 
 pendulum which will execute its two swings in the 
 same time that this conical pendulum moves once 
 around its circle. The conical pendulum can be 
 made to move around this circle uniformly. The 
 space described is 2irr, or the circumference of the 
 circle around which it moves. In uniform motion 
 we always have — 
 
 Space = velocity x time, 
 
 , . space 2 IT r 
 or velocity, = -f- = 
 
 time 
 
 2T 
 
 {T being the time of a simple pendulum, which 
 makes two swings while the coni- 
 cal pendulum makes one). 
 
 For small circles of swing the 
 tension along the wire of the coni- 
 cal pendulum does not differ sen- 
 sibly from its tension when the 
 wire A Bis vertical. This can be 
 proved by attaching a spring-bal- 
 ance to C (Fig. 50), and connecting 
 this balance with a cord, which 
 passes over a smooth pulley, and 
 has a weight, W, attached to its 
 other end. This tension is equal 
 to the pull of gravitation, or the 
 mass multiplied by the force of 
 gravitation, or m g. 
 
 Fig. so. 
 
MOMENTS OF INERTIA. 115 
 
 We have proved that the centripetal force tend- 
 ing to draw a body revolving in a circle of radius r 
 
 toward the center of the circle is This is bal- 
 
 r 
 
 anced. by the reaction, or, as it is often called, the 
 
 centrifugal force, which can be represented by the 
 
 line A W-^, \i A C represents the tension or weight 
 
 m g. We can prove directly that 
 
 ^ C X sin ^ = mg x , (since angle A C W-^= C 
 
 BD) 
 
 . ^^, m 1? 
 = A W\ = , 
 
 , , 2 7rr irr 
 
 but V — — ^=^77-, 
 
 and v' = ~Y2~ ; 
 
 , r mi^r^ ^ /~T 
 
 hence mg~r^= ^r, or r= Try __. 
 
 L r 1 r g 
 
 In our experiments thus far, we have supposed 
 the whole weight of the pendulum to be in the bob, 
 and we have neglected the weight of the suspend- 
 ing wire or rod. In many cases, however — like the 
 swinging magnet, for instance — the weight of the 
 pendulum is distributed throughout its entire length. 
 The movement of such a pendulum is very different 
 from a pendulum in which all the weight is collected 
 in the bob, as in the pendulums with which we have 
 experimented. 
 
 Experiment 79. — Suspend an iron or brass rod 
 (Fig. 51), two feet in length, from a very short string 
 attached to One end, and having made a lead weight, 
 or having filled a tin vessel with sand, so that the 
 weight of the lead, or of the filled vessel, shall equal 
 
ii6 
 
 NEW PHYSICS. 
 
 Fig. si. 
 
 the weight of the metalHc rod, suspend this equiva- 
 lent weight by a light string or wire, and lengthen 
 or shorten this suspension until 
 "~ the two pendulums swing to- 
 gether in the same time. It will 
 be found that the equivalent 
 pendulum, or simple pendulum, 
 in which all the weight is col- 
 lected in the bob, has a shorter 
 length than the metallic pendu- 
 lum. 
 
 This is indirectly an illustra- 
 tion of the conservation of ener- 
 gy ; for the work the metallic- 
 rod pendulum does in each 
 swing is the weight of each little 
 particle multiplied by its fall toward the earth. 
 Each small particle falls a different distance. The 
 work done by all the particles must lie between the 
 work done by the fall of the particles near the lower 
 end of the rod and those at the upper end. There 
 is, therefore, some point where, if the weight of the 
 entire rod were collected, its work in descending an 
 arc, or the height of an inclined plane, would be 
 equal to the work of all the particles in descend- 
 ing various different arcs. The length of a simple 
 pendulum, whose time is equivalent to that of a 
 compound pendulum — like a uniform metallic cyl- 
 inder, or any irregularly shaped body — gives the 
 radius of the arc, down which the total collected 
 weight of the compound pendulum can descend, 
 and perform the same work that all the little par- 
 ticles of the compound pendulum perform in de- 
 scending their various arcs. 
 
MOMENTS OF INERTIA. 
 
 117 
 
 The length of the equivalent simple pendulum 
 is called the radius of gyration. Instead of finding 
 this radius of gyration by experiment, as we have 
 done, it can be calculated. This calculation re- 
 quires the use of a term called the moment of 
 inertia, which is a very important term, and re- 
 quires careful explanation. Remembering, from 
 the illustrations just given, that we obtain the 
 length of a simple pendulum, whose time of vi- 
 bration is equal to that of a compound pendulum, 
 for the purpose of estimating the work done by 
 all the particles of a compound pendulum in rising 
 and falling against the force of gravity, we shall 
 see that the term " moment of inertia " is neces- 
 sary to estimate the work done by all moving 
 bodies. 
 
 We know, from the law of the lever, that Fl = 
 F' /i, where F and F' are parallel forces, and / and 
 /j are the distances of their points of application from 
 the fulcrum A (Fig. 52). The products Fl and F' l^ 
 axe called the moments of the forces about the ful- 
 crum A. The term "mo- 
 ment " is a useful one ; for, 
 suppose we wish to esti- 
 mate the effect of a weight If ^r' 
 attached to a horizontal pj^ ^^ 
 lever-arm at a distance of 
 
 /j, = two feet, from the fulcrum A, and again at a 
 distance of three feet. We know by experiment 
 that the effect of these weights, placed at these'dis- 
 tances, is proportional to the weight, and also to 
 the distances that the weight is placed from the ful- 
 crum. Our standard for estimating the effects of 
 the weight applied at different distances must evi- 
 
 I 
 
 A 
 
 ?l 
 
 
 A 
 
 
 'F 
 
 
 ' 
 
ii8 
 
 NEW PHYSICS. 
 
 dently be some pull or force applied at the opposite 
 side of the fulcrum at our unit distance, whatever 
 we decide that this unit distance shall be. 
 
 Suppose that we call this unit distance one foot. 
 We shall, therefore, have the unit force F at the 
 distance of one foot, balancing some other pull or 
 weight at some other distance on the opposite side 
 of the fulcrum. Let this second pull or weight be 
 F' , and let it be placed at the distance /i, we shall 
 have ^ X i = i^' x /; or, our standard force at the 
 distance of / is equal to the product of another bal- 
 ancing force at a distance of l^- In order, there- 
 fore, to find the effect of the application of a pull or 
 a weight at different distances from a fulcrum, in 
 terms of a known force or pull at a distance of one 
 from the same fulcrum, but on the opposite side, we 
 must make use of the product of the pull or weight 
 by its distance from the fulcrum. 
 This product is called, for con- 
 venience, the moment of such a 
 pull or force. 
 
 Suppose that the compound 
 pendulum A ^(Fig. 53) is struck 
 by a force F, at the point B, at a 
 distance of / from the fulcrum A, 
 it is evident that the moment F I 
 would be equal to some standard 
 force applied at a distance of uni- 
 ty from the fulcrum A. If the 
 pendulum should be struck at C, 
 at a distance oi A C from yi, in order to communicate 
 the same swing to the pendulum that the blow at B 
 does, this force, F^, multiplied by its arm /^ must be 
 equal to the force F multiplied by its arm /, sincq 
 
 Fig. 53. 
 
MOMENTS OF INERTIA. 
 
 119 
 
 they are both compared by the same force multi- 
 plied by the arm unity. 
 
 F I 
 Hence we have F^ /* — F I, or F^ = -t^. 
 
 Experiment 80.— Suspend a small lead ball by 
 a silk thread of a suitable length (Fig. 54). Place a 
 comparatively stiff piece of steel spring, one end of 
 which is confined in a vise, directly beneath the ver- 
 tical pendulum. Draw the 
 unconfiined end of the flexi- 
 ble steel out of the vertical 
 through a certain angle, at- 
 taching a thread to it and to 
 a point, A. On burning the 
 thread at A, the steel will 
 strike the weight of the pen- 
 dulum, and urge the latter up 
 a height, h. The work the 
 steel does in striking the pen- 
 dulum-weight ^is measured 
 by the weight lifted through the height h, or M gh 
 = W h. The force with which the piece of steel 
 strikes the weight W\% the same as that with which 
 the weight W, on its fall down the arc or down the 
 height h, would strike the steel spring when it is 
 unbent, and in the same vertical line as the weight 
 W when it is at the lowest point of its swing. The 
 velocity with which the bit of steel strikes W is also 
 approximately the same as would be acquired by the 
 weight W in falling down the height h. We say 
 approximately, for the resistance of the air and the 
 friction of the point of support of the pendulum 
 use up a portion of the energy of the steel spring. 
 In order to find the force with which a bit of 
 
 Fig. 54. 
 
I20 NEW PHYSICS. 
 
 steel strikes a blow, and compare the force with 
 
 the lengths of the piece of steel employed, using 
 
 the apparatus just described, we must measure the 
 
 height h. This can be done by placing a piece of 
 
 paper beside the pendulum, so that the height to 
 
 which the pendulum -bob ascends can be noted. 
 
 The work done in lifting the pendulum to the 
 
 height h is then — 
 
 ■}? Wv 
 Fh^ Wh = mg—= — , 
 
 ^ Wi? 
 or F = - — J. 
 2gh 
 
 Compare the force with which pieces of steel of 
 different lengths will strike the pendulum. This 
 apparatus in a modified form is called a ballistic 
 pendulum, and is used to find the velocity and also 
 the force with which cannon- 
 balls strike any obstacle. 
 Suppose that a pendulum 
 .' , is made so that, when the ball 
 / strikes it, the ball should be- 
 "p' ^,--'' come imbedded in it, as in 
 
 " Fig. 55. The energy of the 
 
 Fig. ss. ^^^^ ^^ consumed in lifting the 
 
 weight of the ball and that of 
 the pendulum-bob through a certain distance against 
 the pull of gravitation. 
 
 In these examples it is evident that we should 
 know what the length of the simple pendulum is 
 which is equivalent to the compound pendulum, 
 which, from the necessity of strength and solidity, 
 is generally employed instead of a simple pendu- 
 lum. Thus it would be difficult to make a sim- 
 ple pendulum which would receive the great blow 
 
 \ 
 
MOMENTS OF INERTIA. 121 
 
 of a cannon-ball. The pendulum for this purpose 
 must be a solid structure in which the weight is 
 not all collected at the end of the suspending 
 arm. 
 
 Let us now return for a moment to the case of a 
 lever. We know that a certain weight, F, at the end 
 of an arm, /, will balance another weight, F' at the 
 end of an arm, /j, or Fl— F'l^ (Fig. 52). Now, sup- 
 pose that the lever 
 
 moves up and down , ^^,i»' 
 
 in a vertical plane, ^^ 
 
 the space through „ .A,-:::::^.. 
 
 which the force F \ ^^ 
 
 moves is the arc a a' "' j ^ 
 
 (Fig. 56), and the >^ 
 
 space through which Fig. 56. 
 
 F' moves is b b' . If 
 
 the angle corresponding to these arcs is a, we shall 
 
 have I a^ aa', and /j a = 3 b', for we can prove by 
 
 actual measurement, with a pair of dividers and a 
 
 1 • la a a' 1 , , , 
 
 string, that in any case — - = — — ; and the value of 
 
 l^a b b 
 
 the force F = — — . 
 
 Let us suppose that the lever a b' moves in a 
 horizontal plane, to obtain the work done by the 
 force F in moving through the arc «rtS we must 
 multiply the force by the space passed through, or 
 the arc ad^, and, to obtain the equivalent work done 
 by the force F' in moving through the arc b b', 
 we must multiply the force F' by this arc. Hence, 
 F'l^l^a F'l^a 
 
 Fla 
 
 I I 
 
 By comparing the work done by different 
 
122 
 
 NEW PHYSICS. 
 
 weights or forces attached to the lever and moving 
 through different arcs, we find — 
 
 Fla = work 
 
 -, etc., 
 
 / / / 
 
 ox F' l^ = F"lr,\Qtc., 
 or the different expressions for the work done are 
 proportional to the products of the different forces 
 by the squares of the length of the arms. These 
 products are called " moments of inertia." They 
 enable us to estimate the effort or work that is re- 
 quired to move a body through a certain arc or a 
 certain distance. 
 
 Let us illustrate the use of the term " moment 
 of inertia " by the tilt-hammer. 
 
 This apparatus is often employed in iron-mills to 
 give great blows to masses of heated metal. It con- 
 
 FlG. 57. 
 
 sists of a heavy weight (Fig. 57), W,2X the extremity 
 of a lever, A B ; regular blows are given at one ex- 
 tremity of this lever by means of a revolving wheel, 
 M. The work that the weight W does is propor- 
 tional to F', the force exerted at C, multiplied by the 
 square of the arm /', or the distance of the weight 
 from the fulcrum B. If the mass of the weight at C 
 is m, we have F'^ = Mg — weight at C. Hence work 
 
MOMENTS OF INERTIA. 123 
 
 done by the fall of this weight through the angle a 
 is equal to the energy which was used to force the 
 end A through the same angle but over a different 
 
 arc, or Fla = work = — — = — . 
 
 / / 
 
 Here we see that the product M l^ occurs ; this, 
 as we have said, is called the moment of inertia, and 
 we shall use the letter K to denote it in future. If 
 the weight of the trip-hammer is not all collected at 
 its extremity — and it never is in practice — it is evi- 
 dent that every particle of the arm of the trip-ham- 
 mer obeys the same law as the weight of the simple 
 trip-hammer which we have been using as an illus- 
 tration. If we call such a particle m, and its distance 
 from the fulcrum x, we shall find that the total work 
 done by the fall of the trip-hammer will be the sum 
 of the work done by each particle falling through 
 its own arc. We should find that, in summing up 
 the work of each particle, the whole work, or 
 
 mgi^aAr m g xfa -^nhg x^a, etc., 
 p la ^ 
 
 Mg f^a Kg a 
 ^ 7 ^^T" 
 (If we put M to represent the whole mass, and r 
 the radius of gyration, or the length of the simple 
 pendulum which is equal to that of the compound 
 one.) In general the moments of inertia are calcu- 
 lated. A table of these for simple bodies will be 
 found in the Appendix. 
 
 Masses will therefore have the saihe effect upon 
 the conditions of motion of a revolving body, if 
 they posses equal moments of inertia, and we can 
 therefore replace one mass by another at a suitable 
 distance, without altering the conditions of motion. 
 7 
 
124 
 
 NEW PHYSICS. 
 
 Fig. s8. 
 
 Experiment 8i. — Obtain two brass rods of equal 
 length (Fig. 58), one of which screws into the oth- 
 er; suspend this com- 
 pound rod by a wire, one 
 end of which is screwed 
 between the rods A and 
 B. On twisting the rod 
 through a small angle 
 in the horizontal plane, it 
 will swing to and fro un- 
 der the influence of the 
 torsion of the suspension 
 wire, and the apparatus 
 constitutes what is called 
 a torsion pendulum. On 
 the extremities of the horizontal bar of this pendu- 
 lum, slip equal hollow cylinders of wood or metal. 
 If the mass of one of these cylinders is M, and its 
 distance from the axis of suspension is b, its mo- 
 ment of inertia will heMb^. Remove these weights 
 and slip on two others, each of mass M^. If these 
 are placed so that 
 
 Mb^ = M^b,\ or b,^ = -j^, 
 
 the time of swing will not be altered. 
 
 Verify the law, and also ascertain the laws of the 
 torsion-pendulum, by altering the length of the sus- 
 pension, and changing the weights on the horizontal 
 rod. 
 
 Experiment 81 enables us to reduce a weight 
 from one point to another, so as not to alter the 
 conditions of motion of any revolving body. Let 
 us take the case of the wheel and axle. Suppose 
 that the weight P acts on the wheel with the arm 
 
MOMENTS OF INERTIA. 
 
 125 
 
 A B, and the weight Q on the axle with the arm 
 CD (Fig. 59). Suppose, also, that the strings, by 
 means of which these weights act, 
 are perfectly flexible. If the mo- 
 ment of the weight P is greater 
 than that of Q, the weight P will 
 descend. The weight Q] acting 
 with its arm D C ox b, is equiv- 
 
 alent to a force 
 
 Qb Qb 
 
 act- 
 
 FlG. 59. 
 
 AB~ a 
 ing at B, which is opposite to the 
 force due to the weight P. Hence 
 
 /O h 
 
 we have a resultant force P acting at B. The 
 
 mass — is reduced from — to — 2 by removing it 
 from the distance b to that of a. Hence the mass 
 
 is Y»2 = I y -t- 
 
 a 
 
 moved by Z' 1- is M ■- 
 
 (-+^-^- 
 
 If the moment of inertia of the wheel and axle 
 
 = — —, its inert mass reduced to B = — — j. 
 
 and we 
 
 then have 
 
 M: 
 
 Q^_^Gf 
 
 )- 
 
 a' ' a' 
 Pa'-{-QP-{rGf 
 
 Or the acceleration of the weight P 
 __ moving force _ / P— — 
 
 "^^^ \Pd'-\-Ql?^Gf 
 
 _ / Pa 
 
 \gc^ 
 
 :?]sa- 
 
 Ql^+Gy 
 
 » See Weisbach's " Mechanics "—wheel and axle. 
 
126 NEW PHYSICS. 
 
 Prove this last equation by the method employed 
 in Experiment 67. 
 
 Turning now to the compound pendulum, let 
 us find an expression for its time of vibration. 
 We know that the time of a simple pendulum is 
 
 ^ = TT V —. This is the .time of each particle of a 
 
 g 
 compound pendulum, such as m. Suppose we mul- 
 tiply both numerator and denominator under the 
 square root by m I, we shall then have — 
 
 t = 'iry J = 7ry — > = Ty 77-, 
 
 ' gml ^ mgl Dg 
 
 if we put ml = D. Z> is evidently the moment of 
 the force which sets the pendulum to swinging, and 
 // stands for radius of gyration. 
 
 This discussion is necessary for the compre- 
 hension of the magnetic pendulum, and its use in 
 measuring the work done by a magnet in its 
 swing. The pendulum, which acts under the in- 
 fluence of gravitation, serves to give us our esti- 
 mate of time, and also of the earth's gravitation- 
 force. In the same way the swinging magnet can 
 give us a standard of time in terms of the earth's 
 magnetic force ; and also, knowing this time, we can 
 obtain the magnetic force of the earth, and the 
 work a magnetic pendulum does in swinging under 
 the influence of this force. 
 
 We have seen that a magnet, which is suspended 
 horizontally by its middle, is a horizontal pendulum 
 which moves under the influence of the magnetic 
 force of the earth. It is also a compound pendu- 
 lum ; for we can not put all the magnetic attracting 
 matter at one place, at a definite distance from the 
 point of suspension. Each particle of magnetic 
 
MOMENTS OF INERTIA. 127 
 
 matter y» acts at a certain arm, /. Instead of the ex- 
 
 /~~K~ ' 
 
 pression T = tt y — j- which gives the time of a 
 
 compound pendulum vibrating under the influence 
 of the directing force g of the earth, we shall have 
 
 t ■=ir\ , in which /t is the attracting mass of 
 
 the magnet, /* / its moment, T the directive force of 
 the earth's magnetism, which now takes the place 
 of the earth's gravitation-force. Taking the square 
 
 of t, we have r = —pf^ or y = — j-^. If, there- 
 fore, we knew the product /it /, we could obtain the 
 earth's magnetic force. Methods will be given 
 later to determine the value of fi.1. The impor- 
 tance of the measurement of T will be understood, 
 when it is seen that all the measurements in elec- 
 tro-magnetism depend upon a knowledge of the 
 value of this force — ^just as all measurements of 
 work and force in mechanical units depend upon 
 our knowledge of g, or the earth's gravitation- 
 force. 
 
 For the present, therefore, we will content our- 
 selves with obtaining the value of iilT= M T {hy 
 putting /il = M, in which M represents the sum of 
 fi I for every particle of the magnet) for different 
 magnets. 
 
 Experiment 82. — Obtain the value oi M T for 
 several cylindrical magnets of the same section, but 
 of different lengths. 
 
 The value of isTcan be calculated from the length 
 of the magnet and its diameter expressed in centi- 
 metres, and its weight in grammes. (See table of 
 moments of inertia in Appendix.) 
 
128 NEW PHYSICS. 
 
 In obtaining the time of swing, the magnets 
 should not be allowed to swing through large arcs, 
 and they should be suspended by long strands of 
 silk, which are freed from torsion. This can be ac- 
 complished by unwinding a silk thread, and taking 
 one of the strands and moistening it slightly be- 
 tween the fingers, in order to straighten it. 
 
CHAPTER VIII. 
 
 MEASUREMENT OF MAGNETIC FORCE. 
 
 Let us now obtain an idea of the magnitude of 
 the attracting forces with which we deal in physics. 
 It is not difficult to obtain a conception of the 
 amount of the earth's gravitation-pull. A pound- 
 weight exerts what we call a certain force. It 
 will break a strand of silk by which it is suspend- 
 ed. It will cause a certain deflection on a spring- 
 balance. Its constant pull on a falling body will 
 give it a velocity of about thirty-two feet at the 
 end of the first second of its fall, and will make 
 it pass over sixteen feet during the first second. 
 The earth's pull is exercised upon every particle 
 in a body, and the whole pull, therefore, increases 
 with the mass of the body. The effect, therefore, 
 of the fall of a ton-weight through thirty-two feet 
 would be very great. Turning our attention, how- 
 ever, to the attracting force between a magnet 
 and the earth, we find that the force is very feeble 
 compared with that of the gravitation-pull of the 
 earth. 
 
 Experiment 83. — Magnetize an ordinary knit- 
 ting-needle, by stroking the north pole of a strong 
 magnet upon one half, and the south pole of the 
 same magnet along the other half. Repeat this half 
 
I30 
 
 NEW PHYSICS. 
 
 a dozen times. The knitting - needle will then be 
 found to be a magnet. Endeavor to balance it upon 
 a knife-edge at or near its middle. It will be found 
 to be impossible. The earth, therefore, attracts one 
 end (making it dip), and repels the other end. Slip 
 a very small piece of cork along the magnet to its 
 middle ; run a very fine needle through this piece of 
 cork, as near as possible to the knitting-needle, and 
 at right angles to it (Fig. 60). It will be found that 
 the needle dips from the horizon through an angle of 
 
 about 74° (in latitude of 
 Cambridge, Massachu- 
 setts). It tends, there- 
 fore, to put itself along 
 a line of magnetic force ; 
 and these lines, there- 
 fore, make this angle 
 with the horizon. The 
 accompanying diagram 
 shows that the earth's 
 magnetic force can be resolved into a horizontal 
 force, H, and a vertical force, V. The horizontal 
 magnetic pendulum, which we have considered, 
 moves under the influence of the horizontal direct- 
 ing pull H. It will be seen from the figure that 
 the vertical pull of the earth's magnetism is much 
 greater than the horizontal force, H. To obtain an 
 idea of the amount of the pull of this vertical 
 force, attach a very fine thread to the upper end 
 of the knitting-needle N, as shown in Fig. 60, and 
 ascertain how much weight must be attached to the 
 thread to bring the needle into a horizontal position. 
 When it is in this horizontal position, the repelling 
 force of the vertical component of the earth's mag- 
 
 FlG. 60. 
 
MEASUREMENT OF MAGNETIC FORCE. 
 
 131 
 
 Fig. 6i. 
 
 netism on N is balanced by the downward pull of 
 the attached weight, or the earth's gravitation. It 
 will be found that a very small weight will bring 
 the dipping needle into a horizonal position. Know- 
 ing the angle between the direction of the dipping 
 needle and the horizontal (74°), and having found 
 the pull of the vertical component V, in terms of the 
 earth's gravitation, we can construct the parallelo- 
 gram of forces, and find the value of 
 H in terms also of the earth's gravita- 
 tion (Fig. 61). 
 
 Experiment 84. — It will be found 
 that the readiest way to bring a dip- 
 ping needle into a horizontal position, 
 or to deflect a compass-needle through 
 a certain angle, is to bring another 
 magnet near it. We can not conveniently compare 
 
 the gravitation pull 
 of the earth with 
 either the vertical 
 magnetic pull or 
 the horizontal mag- 
 netic pull. It is 
 necessary to em- 
 ploy magnets to as- 
 certain the value of 
 the magnetic at- 
 tracting force. 
 
 We find by our 
 previous experi- 
 ment that we can 
 not measure the 
 magnetic pull 
 
 Fig. 62. which a magnet 
 
132 
 
 NEIV PHYSICS. 
 
 N^S^ exerts upon another, NS, as in Fig. 62, by 
 attaching a thread to 5, and passing it over a pul- 
 ley and attaching a weight, W, to it. If the mag- 
 net S^ N^ is a feeble one, we can counteract the 
 effect of S'^ N^ upon 5 N, by putting another mag- 
 net so that its pull upon 5iVmay be opposed to 
 that of S^ N^. We can then estimate all mag- 
 netic pulls and pushes in terms of the effect of 
 this last standard magnet. It is found, however, 
 that no magnet remains the same for any length of 
 time, and our magnetic standard would vary from 
 time to time. It would be difficult, moreover, as 
 we have seen by the previous experiment, to com- 
 pare our magnetic standard with the earth's gravi- 
 tation-pull, which is our invariable standard of 
 force and of work. We must, therefore, find some 
 method of expressing the earth's magnetic pull 
 in the units which are based upon our standards 
 of time and of force, and which we employ in me- 
 chanics to estimate force and work. 
 
 The effect of one magnet on 
 another can be seen in two 
 ways. Suspend a magnet by a 
 \^ single fiber, without torsion, in 
 
 -g ^ front of another magnet, and 
 L set it to swinging (Fig. 63). It 
 will be found that the number 
 of vibrations is determined by 
 the strength and distance of the 
 larger magnet, NS. When TV 5 
 is placed directly north of the 
 
 V 
 
 
 Co 
 
 Fig. 63. small suspended magnet n s, and 
 
 near it, its effect upon the small 
 
 magnetic pendulum is much greater than that of the 
 
MEASUREMENT OF MAGNETIC FORCE. 133 
 
 earth, as can be seen by comparing the number of 
 vibrations oi ns before and after NS is placed be- 
 fore it. One effect, therefore, of one magnet upon 
 another can be detected and measured by the mag- 
 netic pendulum. 
 
 The time of this little pendulum is / = Try — — =, 
 
 in which 7i is the strength of the magnetic pull 
 upon the small suspended magnet, and, in the case 
 just considered, is the sum of the pull of the earth 
 and of the large magnet N S. Without the large 
 magnet the little pendulum swings under the influ- 
 ence of the earth's magnetic pull alone. 
 
 Now, place the large magnet directly to the east 
 of the small suspended magnet ns. Its effect is to 
 deflect the small magnet n s through an angle, a. 
 There are two effects, therefore, that a stationary 
 magnet, N S, can exert upon a movable or sus- 
 pended magnet — one is to increase the number of 
 its vibrations, and the other to deflect it out of the 
 magnetic meridian. We have already obtained the 
 value of the time of swing of a suspended magnet. 
 Let us now see if we can estimate the quantitative 
 effect of one magnet upon another. 
 
 Suspend a small magnet about one centimetre 
 long, made from a bit of watch-spring, and pro- 
 vided with a long, light pointer, by means of a silk 
 fiber without torsion. Place a paper graduated 
 circle, or two paper protractors suitably joined, be- 
 neath the magnet, so that the center of the gradu- 
 ated circle shall be immediately beneath the middle 
 of the suspended magnet. 
 
 Instead of providing a magnet with an index or 
 pointer, which moves over a graduated circle, we 
 
W4 
 
 NEW PHYSICS. 
 
 can place a little mirror upon the magnet, and re- 
 flect a spot of light upon a strip of paper which is 
 graduated into millimetres and centimetres. When 
 the magnet moves, the beam of light is reflected 
 along the scale. In this case we have a long index- 
 arm without weight. In Fig. 64, B represents a 
 box in which a little magnet, n s, provided with a 
 small, plane mirror (a piece of thin looking-glass 
 
 will answer) is suspended, the magnet is placed at 
 right angles to the mirror. 5 is a small clear spot 
 in a blackened lamp-chimney. The light after 
 reflection from the mirror m is received upon a 
 strip of ground glass or oiled paper, upon which 
 a suitable scale is placed. A paper scale can be 
 pasted upon the ground glass, or a wooden scale 
 can be suitably placed against the glass. The 
 glass allows the spot of light to be seen through 
 
MEASUREMENT OF MAGNETIC FORCE. 
 
 135 
 
 the glass by the eye, which is placed in front of the 
 scale. 
 
 It is necessary to place a lens of long focus in 
 front of the plane mirror M (unless one uses a con- 
 cave mirror, see Appendix), in order to obtain a 
 defined image of the slit 
 5 upon the ground gla 
 In order to understa I 
 this, arrange in a dz I 
 room a lens, /, and a lai | 
 with a slit, s (Fig. 65), 
 that an image of the sli 
 may be formed by ll 
 lens at F. Then place 
 piece of looking-glass 1 
 hind the lens, and asc 
 tain the position F' of t 
 reflected image. It is i 
 necessary that the s 
 pended magnet should 
 placed in a dark room : fwx , 
 
 in a room only moderate- 
 ly darkened, the eye when placed in line with the 
 reflected beam can perceive an image of the slit 
 upon the ground glass or oiled paper. It will be 
 found necessary to obtain the deflection of the little 
 suspended magnet in terms of the tangent of the 
 angle through which it is deflected by any magnet 
 N S, placed to the east or west of it. In the first 
 place, arrange the ground glass with its scale so 
 that it shall be parallel to the mirror. This can be 
 done by setting the magnet in motion, and by mov- 
 ing the scale until the deflections first on one side of 
 the zero-line and then on the other are equal. When 
 
136 
 
 NEW PHYSICS. 
 
 Fig. 66. 
 
 the mirror turns through an angle, a, the angle 
 (the tangent of which is n, or the number of scale 
 
 deflections divided 
 by d, the distance 
 of the mirror from 
 the scale) is twice 
 the angle a. This 
 can readily be 
 proved by setting 
 up a piece of look- 
 ing-glass on a pro- 
 tractor, and ob- 
 serving, by means 
 of a telescope or 
 eye placed at /, the reflections of the scale from 
 this mirror when it is turned through a small angle. 
 Since, however, TM and MA are perpendiculars, 
 or normals to the mirror in its two positions, the 
 angles between these normals is the same as the 
 angle between the two positions of the mirrors. 
 Moreover, a beam of light passing along B M, it is 
 known, will make equal angles with a normal, MA, 
 to the reflecting surface.* Hence the angle 6 = 2a. 
 The tangent oi = tangent of 2 a, 
 
 _ scale reading _ n 
 
 distance of scale to mirror d' 
 
 I n 
 
 2 2d' 
 
 If a higher degree of accuracy is desired, the 
 
 value of tan a in terms of tan 2 a can be obtained 
 
 from trigonometrical tables. 
 
 With this apparatus let us now ascertain how 
 
 the force of attraction between magnets varies. 
 
 * Prove this experimentally. 
 
 For very small angles, tan o = - tan 2 a 
 
MEASUREMENT OF MAGNETIC FORCE. 
 
 137 
 
 With the object of finally obtaining a measure of 
 this force in gravitation units, place a comparatively 
 large magnet, which is five centimetres long and 
 one centimetre in diameter, directly to the east of 
 the suspended magnet, and at a distance of at least 
 ten centimetres. 
 
 If this latter magnet is very strong, or if it is 
 placed very near the suspended magnet, the lat- 
 ter will point directly toward it, in the east and 
 west line in which the large magnet is placed. 
 The further the large magnet is removed from the 
 suspended magnet, the less does the suspended 
 magnet point toward the stationary or larger 
 magnet. This is because the magnetism of the 
 earth exerts its effect ; and the suspended magnet 
 comes to rest at a cer- 
 tain angle, a, from the 
 north and south line, un- 
 der the influence of the 
 two pulls. Why does it 
 come to rest at a certain 
 angle? In order to an- 
 swer this question, try 
 the following experi- 
 ment: Weight a rod 
 heavily at one end (Fig. 
 67). Suspend the rod 
 
 vertically by slipping it through a hole at A, upon 
 a smooth peg, so that the weights and the connect- 
 ing rod can readily revolve about A. Attach a 
 spring-balance to the lower weight and pull it out 
 horizontally through a vertical angle of a. The 
 product of the pull in pounds expressed by the 
 spring-balance and the projection of the half length 
 
 Fig. 67. 
 
138 
 
 NEW PHYSICS. 
 
 AB = I oi the rod on the vertical, or A C, will be 
 found to be equal to the product of the weight at 
 B expressed in pounds multiplied by the projection 
 of ^ ^ on the horizontal, or A D. Now we have 
 shown that the projection A C =^ AB cos o, and the 
 projection A D ^ A B sina. 
 
 Hence, if we call the horizontal pull on B = F, 
 and the vertical pull of the earth on ^ = F', we 
 shall have FAB cos a — F' A B sin a, or F = 
 
 F' sin a 
 
 cos a 
 
 F' tan a. 
 
 Returning now to our magnetic forces, we find 
 that the pull F in our last example on the spring- 
 balance is replaced by the magnetic pull F, and the 
 earth's gravitation pull F' by the earth's magnetic 
 pull F' = MT. 
 
 Hence, we shall have here also F = F' tan a 
 = MTtan a. We place F' = M T, for the earth's 
 pull is proportional to its own magnetic effect, T, 
 and to the strength of magnetism, M, of the sus- 
 
 FiG. 68. 
 
 pended magnet, and to nothing else. We find, 
 also, by placing the magnet NS (Fig. 68) at differ- 
 
MEASUREMENT OF MAGNETIC FORCE. 139 
 
 ent distances to the east of the movable magnet, 
 N' S', that the tangent of the angle varies in a certain 
 way with the distance. It also varies as the strength 
 of the suspended magnet and as the strength of the 
 large magnet, N S. 
 
 Prove that the force /^varies inversely as the 
 cube of the distances from the center of the magnet 
 NS to the center, C, of the suspended magnet, or 
 
 I 
 , F MTtana tana d} d^ ^ 
 ^^^^ Fr MTt^na, = teiT^ =^ T = Z^" ^"'<=^ 
 
 the force also varies as the strength MM-^^ of the 
 two magnets, we shall have 3 ^ = M T tan a for 
 
 any distance, d, of the magnet M^ or NS from the 
 
 Ml 
 magnet M or ns. From this we find that -= = 
 
 d^ tan a. 
 
 We shall now give another method of ascer- 
 taining the relation between F and F', with the 
 purpose not only of determining this relation, but 
 also of illustrating the effect of couples in me- 
 chanics. 
 
 Experiment 85. — A small bolt (Fig. 69, A) has 
 a nut and a washer. Between the nut and the 
 washer a long pointer, P, is confined. This pointer 
 can move over a graduated circle or the arc of a 
 protractor when the bolt is turned in its support. 
 To the lower end of the bolt is fixed a small cross- 
 arm, B, to which a small magnet, N S, can be at- 
 tached by two strands of silk, 5 5, free from torsion. 
 Fix the suspension-head, BA, in the top of a suit- 
 able box, with a plate of glass so that the position of 
 
I40 
 
 NEIV PHYSICS. 
 
 '(f 
 
 rest of the magnet can be observed. For this 
 purpose the suspended magnet should be pro- 
 vided with a little bristle or index which should 
 point to a suitable mark on 
 a bit of mirror beneath it 
 when the suspension-threads 
 are parallel and in the same 
 vertical plane. Place a large 
 magnet outside the box and 
 east or west of the sus- 
 pended magnet. It will 
 then be found necessary to 
 turn the suspension - head 
 Fig. 69. through a certain number 
 
 of degrees in order to bring 
 the index of the suspended magnet back to its 
 zero-mark, or to its resting-place when the outer 
 magnet is removed. 
 
 A body hung up by a bifilar suspension consti- 
 tutes a pendulum. Its time of swing can be ob- 
 tained by the methods we have given ; or this time 
 
 can be calculated from the formula t = "Try -j^, in 
 
 which K is the moment of inertia of the sus- 
 pended body, and Z> is the directive force of the 
 suspension. The tension on the suspending fibers 
 caused by the weight of the suspended body pro- 
 duces a couple, which tends to bring the sus- 
 pended body back to its position of rest. The 
 bifilar suspension, therefore, constitutes a hori- 
 zontal pendulum. Suppose that A B (Fig. 70) 
 represents a magnet which is suspended by two 
 fibers, A C and B D. (For the sake of clearness 
 the suspension apparatus and magnet are much 
 
MEASUREMENT OF MAGNETIC FORCE. 141 
 
 enlarged.) When another magnet, N S, is brought 
 near the suspended magnet, it will turn about a 
 vertical axis, E E'. In order to keep it in the 
 
 Fig. 70. 
 
 original position, A B, we must turn the suspen- 
 sion head, C D, through a certain angle, 6. If, now, 
 the stationary magnet N S '\s removed, the sus- 
 pended magnet A B will move under the action 
 of the couple D sin (in which D is the directive 
 force, or F' I), into the plane A' CD' B', and will 
 vibrate from one side to the other of this plane 
 until it finally comes to rest. In the position A B 
 we should therefore have the couple due to the 
 mutual action of NS and A B, or Fl = D sin B, and 
 for another position of the magnet N S along the 
 same line perpendicular to A B, F' I =^ D sin 6', or 
 
142 NEIV PHYSICS. 
 
 F sin 5 d? , 
 
 F' ^ ih^ ^ d^ ^^^^ Experiment 84). 
 
 Instead of using a graduated circle with an in- 
 dex, or pointer, a little plane mirror can be placed 
 at E, with a small lens in front of it, as in Experi- 
 ment 84. In this case we shall obtain the tangent 
 of 2 0, and from a table of natural sines and tangents 
 (see Appendix) we can easily obtain sin 6. In this 
 experiment the stationary magnet must be much 
 larger than the suspended magnet, and must be 
 placed at a considerable distance from the latter. 
 The suspended magnet should not be more than 
 two centimetres in length, and the stationary mag- 
 net NS should be at least fifteen centimetres distant. 
 
 Experiment 86. — To find the value of the earth's 
 magnetic force, T, or the horizontal component of 
 the earth's magnetism, suspend the magnet which 
 we have used in deflecting the suspended magnet 
 (Experiment 85) in a position where there are no 
 other magnets or masses of iron, and determine the 
 time of its swing under the influence of the earth's 
 magnetism. Then calculate its moment of inertia, 
 measuring its length and diameter by non-metaUic 
 or brass measuring instruments. In this way we 
 
 ■/^ 
 
 shall obtain the time / = tt y -jrpp or M T — 
 
 — ^ (i). Now place the same magnet about thirty 
 
 centimetres to the east or west of a small suspended 
 
 magnet which we used in the previous experiment, 
 
 M 
 and obtain the value — = a^' tan a (2). Then divide 
 
 eq. (i) by eq. (2), and we shall obtain T^ — ^ .g , 
 
MEASUREMENT OF MAGNETIC FORCE. 143 
 
 •K ./ K M 
 
 or Z = — y — -T— . The measurement of '^ 
 
 t d^ tan a 1 
 
 should be made at equal distances to the east and 
 west of the suspended magnet, and the mean of the 
 distance of the points where the center of N S is 
 placed should be taken, on account of the difficulty 
 of measuring from either to the suspended magnet. 
 The mean of the angles of deflection should also be 
 taken. The measurements of length should be in 
 centimetres, the weight in grammes, and the time 
 in seconds. The value of T for Cambridge, Massa- 
 chusetts, in 1884, was .174 C. G. S. It varies, of 
 course, with the latitude of the place, and varies 
 slightly also for the same latitude from year to year. 
 We have proved by experiment that the force 
 of attraction between two magnets varies as the 
 product of the strength M and M^ of the two mag- 
 nets, and inversely as the cube of the distance be- 
 tween their centers. It is difficult to ascertain by 
 direct experiment how the force of attraction varies 
 between two poles — a north and south pole, for in- 
 stance — for we can never separate the two poles of 
 a magnet. The effect of a magnet, N S, in the last 
 experiment upon a small suspended magnet, ns, is 
 due to the combined effect of both poles oi N S 
 upon both poles of ns. We have found that this 
 combined effect is proportional to the strength of 
 the magnets N S and ns, and to the inverse cube 
 of the distance between their centers. This effect 
 must be due to both the north pole N and the south 
 pole S of the magnet N S. What, therefore, must 
 be the effect of the north pole iV alone upon the 
 south pole of the suspended magnet, and the effect 
 of the south pole 5 alone upon the same pole s ? 
 
144 ^^^ PHYSICS. 
 
 Suppose we assume that the mutual effect of these 
 poles is inversely as the square of the distance be- 
 tween them (in Fig. 68). Calling 2 / the length of 
 the magnet N S, and d the distance of the center 
 of NS from the pole N'^ of N'^ S^, we shall have the 
 
 force of attraction between 5 and iV* = F= y^ r-j 
 
 {d — l) 
 
 (calling /i and ^^ the magnetic strengths of N^ and S). 
 
 In the same way we shall have the force between 
 
 the pole iVand the pole N'^, F^ = ,/_/*/ \y This force 
 
 is in the opposite direction to F. Hence, the re- 
 sultant pull between the two magnets must be the 
 difference between F and F-^ ; the first, or F, being 
 a stronger pull than the push of F^, since 5 is nearer 
 N^ than N. 
 
 ThedUere„ce^-/-. = j£L.__Jta^^ 
 
 _ / I I \ _ 2j//ii.* 
 
 - ^/^ \(d- If ~ {d+ If) - d^ 
 
 Hence we see that the force of attraction between 
 a north pole and a south pole really varies inversely 
 as the square of the distance between them, since 
 by assuming this law we are led to the same result 
 as is given by direct experiment. 
 
 This is the same law as that of gravitation. The 
 sun and the earth attract each other directly as the 
 
 # /__l l^\ _ ^djjtjui 
 
 ^dlixfil 4//l;Ui 2 M fli 
 
 4-3 
 
 /^2~ d' 
 
 reducing to a common denominator, neglecting the square of /, and 
 placing 2 l/i = Af. 
 
MEASUREMENT OF MAGNETIC FORCE. 
 
 I4S 
 
 product of their masses, and inversely as the square 
 of the distance between them. There is this marked 
 difference, however, between the attraction of gravi- 
 tation and that of magnetism, that the force of at- 
 traction of gravitation increases as the masses of the 
 two bodies, and can be made indefinitely great. 
 The attraction, for instance, between the earth and 
 a steamship is enormously greater than that be- 
 tween a pound-weight and the earth. The force of 
 attraction, however, between a magnet weighing 
 half a pound and another magnet weighing half a 
 pound is not necessarily increased when the weight 
 of these magnets is increased. A piece of iron or 
 steel can only retain a certain amount of what is 
 called magnetism, or what we have termed M. It 
 is a fallacy to suppose that, if we increase the size 
 of a magnet indefinitely, we shall increase the force 
 of attraction which it exerts upon another magnet. 
 We now see, if we could place one pole of a 
 magnet free from the other, in a uniform mag- 
 netic field, that it would move along the lines of 
 uniform magnetic force, just as a falling body moves 
 along the lines of uniform gravitative force. Such 
 a magnet would move with an accelerated velocity, 
 and would strike with a certain force. Its veloci- 
 ty would be (if it fell vertically along lines of mag- 
 netic force, of strength T) v = y 2 {M T -\- g)h, 
 
 instead oiv =^ a/ 2gh. 
 
 A small being, like a Lilliputian, placed in a mag- 
 netic field and surrounded by particles of matter, 
 all of which possessed one magnetic pole, would 
 obtain the same laws for falling bodies that we ob- 
 
146 NEW PHYSICS. 
 
 tain upon the earth. A magnetic body, however, 
 placed in a uniform magnetic field, always possesses 
 two poles. The law of the fall of such bodies along 
 lines of magnetic force must be modified, since one 
 pole is attracted while the other is repelled. 
 
 We have supposed that the movements of our 
 magnetic pendulums have taken place in a uniform 
 magnetic field — that is, a space in which all the 
 lines of force are parallel and of the same strength. 
 It is hard to obtain a uniform magnetic field. We 
 exist, however, at the same place on the earth's sur- 
 face in a uniform gravitation field in which the 
 lines of force are all sensibly vertical and equal. 
 It must be remjembered that our investigation of 
 the law of bodies falling under the influence of 
 gravitation takes place very near to the surface of 
 the earth. If we should represent the earth by a 
 circle one foot in diameter, the highest tower now 
 built would not be appreciable if we should at- 
 tempt to draw it on the circumference of such a 
 circle. Therefore, when a body falls to the earth, 
 we consider that the same force acts upon it 
 through the whole of its fall. This force is propor- 
 tional to the mass of the stone and to the mass of 
 the earth, and inversely proportional to the square 
 of the distance between the center of the body and 
 the center of the earth. This distance really varies 
 during the fall of the body, but its variation, even 
 if the body could fall from the top of a tower 100 
 feet vertically to the base, would be inappreciable 
 compared with the distance between the top or bot- 
 tom of the tower and the center of the earth. The 
 radius of the earth is about 21,120,000 feet. It will 
 be seen that 100 feet, or even 1,000 feet, is very 
 
MEASUREMENT OF MAGNETIC FORCE. 
 
 H7 
 
 small compared with a distance of 21,120,000 feet. 
 The other manifestations of forces of attraction 
 which we commonly see are due to electricity. 
 
 Experiment 87. — Hang up by a fine silk fiber a 
 pith-ball, and, having rubbed a piece of sealing-wax 
 upon the coat-sleeve, present it to the pith-ball : it 
 is immediately attracted to the wax, and after re- 
 ceiving a charge from the wax it is quickly re- 
 pelled. We find by experience that bodies charged 
 by what is called the same kind of electricity — that 
 is, two pith-balls electrified in the same manner by 
 a piece of sealing-wax — will repel each other. 
 
 We also know that we can induce an opposite 
 charge on a non-electrified body by bringing it 
 near an electrified body, for the non-electrified 
 body will be attracted to the charged body. The 
 body A (Fig. 71) charged with positive electricity 
 will attract a negative charge 
 to the end d, and repel a posi- 
 tive charge to the end c of 
 a neighboring conductor c d. 
 This phenomenon is called elec- (-4. _) O "^ 
 trostatic induction. ■* 
 
 A thunder - cloud fig- 71- 
 
 charged positively 
 
 will attract a negative charge beneath 
 it on the earth, and repel a positive 
 charge to the other side of the earth. 
 If we should coat the inside of a tum- 
 bler with tin-foil (Fig. 72), and also the 
 pjg 2 outside, keeping the coatings separate, 
 we should find that the positive charge 
 on the inner coating will induce a negative charge 
 on the inside surface of the outer coating of tin-foil, 
 
148 NEW PHYSICS. 
 
 and repel a positive charge to the outside surface 
 of the outer coating. By connecting the outer 
 coating with the ground, the positive charge passes 
 into the ground, and we have left a negative charge 
 held on the outer tin-foil by a positive charge on 
 the inner tin-foil. The two electrifications are then 
 said to be bound, and the tumbler constitutes what 
 is called a Leyden-jar. By such an arrangement we 
 can therefore obtain a definite charge of electricity. 
 
 It is important to obtain a clear idea of electri- 
 cal attraction and of its magnitude compared with 
 the attraction of gravitation and of magnetism. 
 We shall, therefore, refer the student to general 
 treatises on the various phenomena of electro-statics, 
 and confine ourselves to the fundamental experi- 
 ments which are necessary in order to obtain a 
 clear conception of the relation between the vari- 
 ous forms of energy. 
 
 Experiment 88. — Coat the exterior of a smooth, 
 large tumbler, or similar glass vessel (Fig. 73), with 
 tin-foil to within two inches of its rim. Fill it to 
 within this dist-ance of the top with strong sulphuric 
 acid. The acid absorbs moisture and also answers 
 for the inner coating of a Leyden-jar. It will be 
 found that this vessel thus filled can be charged with 
 electricity by connecting the tin-foil with the ground 
 and passing sparks from an electrical machine into 
 the sulphuric acid. A special contrivance, by means 
 of which this can be done conveniently, will be 
 described later. Suspend from a torsion-head simi- 
 lar to that in Fig. 6g, but one much smaller, and 
 made with more delicacy, a light, flat disk of pa- 
 per, covered with tin-foil, and balanced at the end 
 of a bright metaUic pointer, which is connected 
 
MEASUREMENT OF MAGNETIC FORCE. 
 
 149 
 
 with the tin-foil on the flat disk. The suspension 
 should consist of two fibers of silk — the finest or ul- 
 timate fibers of a white silk thread (cobwebs can be 
 used). These two fibers should be a small distance 
 apart. Suspend from immediately beneath the 
 point of attachment of these silk threads a very fine 
 platinum wire, with a small platinum or lead weight, 
 
 {ororunAl 
 
 Fig. 73. 
 
 /, at the bottom. Let this disk, with its index, hang 
 vertically over the tumbler, and the platinum weight 
 and wire be immersed one or two inches in the sul- 
 phuric acid. When the sulphuric acid is electrified 
 as we. have described, the disk will also be electri- 
 fied by the same kind of electricity, but will not 
 move unless some body is brought near it. The 
 torsion-head should be placed in the top of a box, 
 the inside of which should be coated with tin-foil, 
 with the exception of one of the vertical sides, 
 which should be of glass. A metallic ball, stuck by 
 
ISO 
 
 NEW PHYSICS. 
 
 a bit of wax to the end of a liollow glass rod, Pg, 
 and connected through the center of this glass rod 
 by a bit of wire to another metallic ball at g, can be 
 electrified at g. The attraction or repulsion be- 
 tween the electrified ball P and the disk d will re- 
 quire that the suspension-head should be turned 
 through a certain angle in order to maintain the 
 disk at zero. By placing the electrified body Pd± dif- 
 ferent distances from the position of rest of the disk 
 d, it will be found that the suspension-head must be 
 turned through different degrees which are propor- 
 tional to the force between the disk d and the elec- 
 trified body P, and also inversely proportional to 
 the square of the distance between d and P. It is 
 convenient to electrify the sulphuric acid of the 
 tumbler by passing a fine platinum wire or rod 
 through a hollow glass tube, T, which is passed 
 through the side of the box. The glass rods serve 
 to insulate the charging-rods from the box. The 
 tumbler should be placed upon a metallic plate, and 
 this plate should be connected with the ground. 
 
 By connecting the inner coating of a Leyden- 
 jar, L, with the rod Pg, and also with the charg- 
 ing-rod T, we shall have approximately the same 
 charge upon d as upon P, and we can therefore 
 measure the repulsion between two similarly elec- 
 trified bodies, by means of the method of Experi- 
 ment 85, the electrical attraction taking the -place 
 of the force F of magnetic attraction. In this case, 
 ascertain the relation between the sines of the an- 
 gles through which we must turn the suspension- 
 head in order to balance the couples due to the 
 attracting body P at the different distances of P 
 from d. The interior of the box should be per- 
 
MEASUREMENT OF MAGNETIC FORCE. 
 
 ISI 
 
 fectly dry, and the charging-rod T, and the charged 
 body P, should be carefully insulated from the box 
 by means of glass tubes which are frequently 
 cleaned with alcohol. It is best to use a plane mir- 
 ror upon the suspension-head, with a stationary lens, 
 and a ground-glass scale, according to the method 
 described in Experiment 84. 
 
 By a method similar to the one we have just de- 
 scribed, it has been proved that the force of elec- 
 trical attraction varies inversely as the square of 
 the distance between the electrified bodies and di- 
 rectly as the product of the charge of electricity 
 upon the bodies. 
 
 Here we have another example of attracting 
 force, which varies inversely as the square of the 
 distance between the bodies. The bodies which 
 are attracted move along lines of electrical attract- 
 ing force. These lines are not straight, in general, 
 between two attracting pith-balls. By electrifying 
 two plane metallic disks, we can obtain a uniform 
 field of attracting force at the center of these disks. 
 At the edges, however, the lines of force are 
 curved. A small being, placed between the two 
 disks, if he could measure the laws of falling bodies, 
 would find laws similar to those we have discov- 
 ered in the case of the attracting force of gravita- 
 tion (if all the matter about him were electrified by 
 the same kind of electrification). Electricity differs 
 from magnetism in this respect. We can charge a 
 pith-ball or a piece of steel with one kind of elec- 
 tricity, and thus observe the effect of, we will say, 
 a positive charge upon another positive or a nega- 
 tive charge, whereas we can not separate one mag- 
 netic pole from another and opposite one. On the 
 
152 
 
 NEW PHYSICS. 
 
 Other hand, electricity resembles magnetism in this, 
 that a positive charge, A (Fig. 71), induces two 
 states — a negative and a positive — in all neighbor- 
 ing bodies. A positive magnetic pole, or a north 
 pole, induces a south and a north pole in all sur- 
 rounding magnetic matter. In the latter respect, 
 electricity and magnetism are closely analogous, 
 while they both differ from the effect of gravitation 
 upon different masses. 
 
 We do not know whether electricity or magnet- 
 ism is some peculiarity of matter — some grouping 
 of the particles — some quick or slow action of mole- 
 cules with respect to other particles or molecules. 
 It is probable that we shall never know what elec- 
 tricity is, or what magnetism is. Indeed, there is 
 as much mystery in regard to the force of gravita- 
 tion as in regard to electrical and magnetic force. 
 What we can do, however, is to observe and meas- 
 ure the relations of electrical and magnetic force 
 to gravitation force, and to study, as we shall see, 
 the great law called the conservation of energy, 
 which connects all the manifestations of force. We 
 see from the preceding" examples how to make 
 measurements of the attracting force of gravitation, 
 of electricity, and of magnetism. There is still 
 another manifestation of electricity, closely allied 
 to that of magnetism, by which attractive force is 
 shown. 
 
 Experiment 89. — Having formed a spool by 
 gluing to the ends of a hollow cylinder of wood — 
 five centimetres long, with an external diameter of 
 two centimetres, and an internal diameter one and 
 a half centimetres — two circles of wood, in order to 
 form the ends ; wind upon this spool from four to 
 
MEASUREMENT OF MAGNETIC FORCE. 
 
 153 
 
 six layers of No. 1 8 insulated copper wire. Connect 
 the ends of this spool or coil of wire with the poles 
 of a Daniell cell, and place the coil to the east of a 
 small suspended magnet. It will be found that one 
 end of the coil acts like the north pole of a magnet 
 upon the suspended magnet, while the other end 
 acts like the south pole. We could, therefore, use 
 a coil of wire through which a current of electricity 
 passes as a magnet. Place a cylinder of iron in the 
 center of the spool, making the length of the iron 
 nearly equal to that of the spool. It will be found 
 that the effect of the coil is now greatly increased. 
 Repeat Experiment 84, replacing the stationary 
 magnet by the coil provided with an iron core. 
 This coil is called an electro-magnet. It is there- 
 fore evident that we can make a strong and uniform 
 magnetic field, or field of force, by means of a cur- 
 rent of electricity which is made to pass through 
 a coil of wire. This field exists without the use of 
 an iron core. The latter, however, strengthens it 
 greatly. 
 
 Experiment 90. — Support the electro-magnetic 
 coil vertically, and, having placed a sheet of stiff 
 paper upon one end, sprinkle or sift some line iron- 
 filings upon this paper. On tapping the paper the 
 filings will be seen to arrange themselves in lines, 
 exactly similar to those seen when a permanent 
 steel magnet replaces the coil. These magnetic 
 figures can be obtained when a coil is used without 
 a magnetic core, especially if a strong current is 
 used. Place two electro-magnetic coils vertically 
 face to face, so that the north end of one is opposite 
 the south end of the other. Slip a piece of card- 
 board between these poles, not placing the poles 
 
154 ^EW PHYSJCS. 
 
 more than two inches apart. On sifting iron-filings 
 upon this card-board they will be seen to arrange 
 themselves in approximately straight lines between 
 the poles. The space between the poles is a field 
 of force, and all our previous remarks and experi- 
 ments in regard to fields of attractive force apply 
 also to this field of force. 
 
 In order to make measurements of force, it is 
 essential that the field of force should be uniform — 
 that is, the lines of force should be straight, and 
 the pull or push along them should be the same at 
 any point in the field. If the lines of force are 
 not straight, a pendulum in swinging across these 
 curved lines of force would not be uniformly acted 
 upon by equal parallel forces. It would move un- 
 der the influence of one force in one direction, and 
 at another time under the influence of another in a 
 different direction. There could be no uniformity 
 in its motion in different parts of such a curved 
 field of force. In all our experiments, therefore, 
 we endeavor to make our measurements in fields 
 of uniform force, whether these fields be gravita- 
 tion fields, magnetic, or electrical fields. We gen- 
 erally think of the force of gravitation as a force 
 which is always the same in the same place, and as 
 always acting along a line directed to the center of 
 the earth. Since the center of the earth is so re- 
 mote, the lines of gravitative force are considered 
 parallel, and their divergence from parallelism is so 
 slight that it can not be detected in a laboratory. 
 Suppose, however, that a meteor should be falling 
 in outer space somewhere between the earth and 
 the sun. It would evidently move in a curved line 
 of force under the effect of the gravitating force of 
 
MEASUREMENT OF MAGNETIC FORCE. 155 
 
 the sun and of the earth. All the meteors near 
 each other in space would move approximately 
 along the same curved lines of force. The field of 
 force, however, would not be uniform, but would 
 vary at different points along these curved lines. 
 It is only when the meteors are close to the surface 
 of the earth that they move along straight lines of 
 force, and in a uniform field of force, for then the 
 influence of the sun becomes inappreciable. 
 
 We have no difficulty in obtaining uniform 
 fields of gravitative force, for we are like the mi- 
 nutest speck of magnetic matter infinitely near a 
 magnetic pole : the moment that we become of a 
 size comparable with the attracting mass, or find 
 ourselves between different attracting masses, at a 
 considerable distance from each, we cease to move 
 in uniform fields of force. In electricity and mag- 
 netism the little horizontal pendulums which we use 
 to measure the direction and the amount of the 
 attracting force are comparatively large bodies, and 
 are not insensible, compared with the size of the at- 
 tracting magnets or the electrified bodies whose 
 attracting force we wish to measure. Hence, we 
 must arrange our electro-magnets or our magnets 
 and electrified surfaces so that there shall be a uni- 
 form field of attracting force where we swing our 
 little measuring pendulums. 
 
 We have seen that a current of electricity in 
 -passing through a wire exerts an attracting effect 
 upon a neighboring magnet. Let us examine this 
 new attraction still further. 
 
 Experiment 91. — Place a wire immediately over 
 or under a compass, so that this wire will run north 
 and south. Pass a current of electricity through 
 
156 
 
 NEW PHYSICS. 
 
 f MT 
 
 this wire (Fig. 74). It will be seen that the needle 
 of the compass will make an angle with this wire. 
 There is, therefore, a force of repulsion at A acting 
 upon the poles of the magnet, repelling it from the 
 wire. By changing the poles of the battery, so that 
 
 the current may 
 flow through the 
 wire W in oppo- 
 site directions, it 
 will be found that 
 the magnetic nee- 
 dle changes the 
 direction of its de- 
 flections. It will 
 be observed that 
 the needle is only 
 deflected through 
 a certain angle, 
 and that this an- 
 gle varies with the 
 strength of the current sent through W. We know 
 that the earth's magnetism acts upon each pole of 
 the needle in the north and south line along whfch 
 the wire Wruns. The deflection of the needle is 
 due to the earth's magnetic pull, M T (proportional 
 to M, the strength of the little magnet which is 
 used, and to the earth's magnetism, T), and to 
 the repulsive force, F, due to the current in the 
 wire W. 
 
 Thus we see that the force F of the current on 
 the magnetic pole of the needle multiplied by the 
 projection of half the length of the needle on the 
 wire, or /cos 8, /being the half-length of the needle, 
 is equal to the earth's pull, M T, multiplied by the 
 
 Fig. 74. 
 
MEASUREMENT OF MAGNETIC FORCE. 
 
 157 
 
 projection of the half-length of the needle on the 
 horizontal line perpendicular to the wire. 
 
 By lifting the wire, through which the current 
 of electricity passes, vertically above the needle 
 (Fig. 75), still keeping it parallel to its first north 
 and south direction, we find that the needle is still 
 deflected, but through smaller and smaller angles. 
 The force from the current through the wire is still 
 perpendicular to a vertical line passing through the 
 center of the magnet ; that is, the force F acting 
 upon the magnet A A, tending to turn it out of the 
 plane mnn' m' , is perpendicular to the line m n, or 
 to its parallel m' n'. 
 
 If, therefore, we could have a long strip of steel 
 which had but one north pole, this strip of steel 
 would wind itself like a 
 serpent around a wire 
 which conveyed an elec- 
 trical current ; for, since 
 the force of the current 
 acts everywhere per- 
 pendicular to the line 
 joining any part of the 
 magnetic pole and the 
 nearest portion of the 
 wire conveying the cur- 
 rent, the mechanical ef- 
 fect would be to whirl the magnetic pole around 
 the current. 
 
 By means of a circle of wire through which a 
 current passes, and so arranged that the needle 
 placed at the center shall always move in a uniform 
 field of force, we can evidently measure the strength 
 of an electrical current. For the force, or push, or 
 
 Fig. 75. 
 
158 NEW PHYSICS. 
 
 pull which the current exerts upon a needle is evi- 
 dently proportional to the strength of the current 
 which passes through the circle of wire and to the 
 radius of this wire ; for the stronger the current the 
 larger the deflection of the magnetic needle, and 
 the smaller the radius of the circle the larger the 
 deflection. Every portion of the circle carrying 
 the current exerts an electro-magnetic effect. The 
 length of the circumference of a circle is 2 tt « ; the 
 whole electro-magnetic effect is, therefore, propor- 
 tional X.o2'jraS, where 5 is strength of current pass- 
 ing through the wire, and to the strength of the 
 magnetic pole, m, and inversely proportional to the 
 square of the radius of the circle of wire. Hence 
 
 , _ 2ir a Sm 
 
 we have F = -. . 
 
 cr 
 
 This force acts with the arm A in cos Q (Fig. 74), 
 
 and balances the earth's magnetic effect M T acting 
 
 with the arm A m sin Q. Hence we have 
 
 7. A in cos =. M TA m sin ^, or 5 = — tan 6. 
 
 If there are n turns of wire instead of one turn, we 
 
 shall have 5' = tan Q. 
 
 Ztrn 
 
 Since we can measure T (see Experiment 86), we 
 shall have the strength of the current expressed in 
 terms of distances or lengths, of moments of inertia 
 and of weight. Hence it is but a step to express 
 the strength of an electrical current in terms of me- 
 chanical work. Before doing this, let us gain clearer 
 ideas of work. 
 
CHAPTER IX. 
 
 POTENTIAL AND WORK. 
 
 We have seen that a body faUing under the in- 
 fluence of gravitation will do an amount of work 
 which depends upon the height fallen through. 
 The greater the height, the greater the amount of 
 work that can be obtained from a falling body. In 
 the case of gravitation, we know that the falling 
 body descends along vertical lines of force under 
 the influence of the force which acts toward the 
 center of the earth. In order to lift a body through 
 a height, we do work constantly against this force 
 of gravitation. In lifting a body, and in thus doing 
 work against the force of gravitation, we know that 
 at any moment we can let the body fall, and thus 
 regain by its blow the energy we exert in lifting. 
 The moment that we have let the body fall and it 
 has exerted its effort in driving a pile or in mold- 
 ing a mass of hot iron, we can get no. more work 
 out of it unless we expend work upon it by lifting 
 it again to a height above the point where it is to 
 strike. 
 
 Our ideas of work done and of work that can be 
 done by falling bodies depend, therefore, upon the 
 height we lift the body. How can we compare the 
 energy which can be obtained from two weights by 
 
l6o NEW PHYSICS. 
 
 letting them fall upon a certain place ? Suppose 
 one of the weights weighs two pounds, and is at a 
 height of ten feet above the ppint we wish to strike ; 
 suppose another weight of four pounds is ten feet 
 below the same place : it is evident that we must 
 raise the latter weight above the level of the point 
 we desire to strike, in order to get the same work 
 from it that we can get from the two-pound weight 
 by simply letting it fall. 
 
 The work of the two-pound weight can be called 
 plus work, and the work that has to be expended 
 on the four-pound weight to simply lift it to the 
 height of the point where the two-pound weight 
 strikes can be called negative work. If we keep on 
 lifting the four-pound weight we know that we can 
 get back the work we do by letting it fall to the 
 point where the two-pound weight strikes, but we 
 can not get back at the point A which we wish to 
 strike the negative work we have done on the four- 
 pound weight in lifting it simply to .^. The height, 
 therefore, of a body above or below the point at 
 which we wish to do work is an important consid- 
 eration in our estimate of the work necessary to be 
 done in lifting bodies to a certain height and letting 
 them fall upon some intermediate point, A. From 
 the whole positive work we shall have to subtract 
 the negative work in order to estimate the expense 
 or the loss of energy. If bodies are being lifted to 
 various heights from various depths, and allowed to 
 fall to some intermediate point, ^, and this operation 
 is going on at different points on the earth's surface, 
 there would be great confusion in the note-books of 
 the different observers unless some uniform level 
 should be assumed, to which all the heights could 
 
POTENTIAL AND WORK. 
 
 I6l 
 
 be referred. The level of the sea is commonly 
 taken as the level to which we refer all our heights. 
 Since most of our operations are confined to points 
 above this level, the convenience of such a plane of 
 reference as the sea-level in 
 all our estimates of heights, 
 and therefore of the work 
 done against and in the di- 
 rection of gravitation, can be 
 illustrated by the operation 
 of leveling. This consists in 
 obtaining the difference in 
 height between two places 
 in order to see, for instance, 
 if water can be made to flow 
 from one point to another. 
 The leveling instrument (Fig. 
 76) consists merely of a tel- 
 escope which is provided 
 with a long spirit-level, L, 
 which is placed parallel to the 
 axis of the telescope. This 
 telescope can turn around 
 horizontally on a vertical 
 axis. It is placed on a tripod 
 directly over some mark. A, 
 and adjusted so that the line 
 of sight CD through the tele- 
 scope shall be horizontal. A 
 vertical graduated rod, B M, 
 is placed at some point, B, and 
 
 the height C above B, where the line of horizontal 
 sight strikes the rod, is read. The rod BM is then 
 moved forward a measured distance to B', and the 
 
1 62 NEW PHYSICS. 
 
 intersection of the same horizontal line of sight with 
 the vertical rod is found at D. Then the leveling-in- 
 strument is carried forward to another position, A', 
 and what is called a back-sight is taken at C upon 
 the vertical rod placed at B', and a fore-sight upon 
 the vertical rod when it is carried to B", and so on. 
 I The height or depth of the point ^" above or below 
 B can be found by comparing the different readings 
 upon the vertical rod. It is evident that we should 
 have to register in our note-books plus heights or 
 heights above B, and negative heights or heights 
 below B, in order to decipher our notes. We should 
 have great difficulty unless we called certain heights 
 which were above B -\- heights, and those that 
 were below B — heights. In order to prevent con- 
 fusion, it is customary to take an arbitrary plane 
 twenty or more feet below the lowest depression 
 that can possibly occur in going from B to B". 
 Suppose the distance of this arbitrary reference 
 plane is sixty feet below C, if we simply subtract 
 from sixty feet the difference of the back-sight and 
 the fore-sight for the point B', we shall have the 
 height of C, and, knowing B' C, the height of the 
 contour-line B A B' A' B" at B' above this reference 
 plane, and our result will always be positive. It is 
 evident that this plane can be taken at any depth 
 so long as it is below the lowest valley that is likely 
 to occur. The height of B" can be obtained, know- 
 ing that of C, and so on. 
 
 Experiment 92. — Lay off on paper the follow- 
 ing line of country between B and B" from this 
 page of an engineer's note-book : 
 
 Distance of reference plane below point B, forty 
 feet ; distance between the stations, one hundred feet. 
 
POTENTIAL AND WORK. 
 
 163 
 
 Station. 
 
 Back-sights. 
 
 Station. 
 
 Fore-sights. 
 
 B 
 
 S feet. 
 
 B 
 
 2 feet. 
 
 B, 
 
 7 " 
 
 B, 
 
 10 " 
 
 B„ 
 
 I " 
 
 Bii 
 
 5 " 
 
 Bill 
 
 I " 
 
 Bin 
 
 15 " 
 
 Bi, 
 
 2 " 
 
 B,v 
 
 2 " 
 
 In the same way the work that is necessary to 
 be expended in rolling a weight over the line of 
 country represented by this contour-line, can best 
 be estimated by referring all the heights and valleys 
 to some reference plane beneath the lowest valley. 
 
 Let us now consider the work we do in moving 
 a body about so as to keep it constantly at the same 
 height above a reference plane. It is evident that, 
 since we do not lift it against the force of gravi- 
 tation, we do no work against this pull. Since 
 the weight is not allowed to fall along a line of 
 gravitating force, it does no work in the direction 
 of the line of gravitation. The work of the moving 
 weight, therefore, with respect to gravitation, is noth- 
 ing. There is work done upon the body in moving 
 it against friction from place to place ; but this work 
 is estimated by the horizontal pull upon the body, 
 and the distance through which this pull must be 
 exercised. It has nothing to do with the height of 
 the plane along which the body is moved above the 
 plane of reference or the sea-level, except that the 
 friction of the body depends upon the pressure it 
 exerts upon the plane along which it is moved, and 
 this pressure depends upon the force of gravitation. 
 As long as the force of gravitation remains sensibly 
 the same, it does not matter how high the plane is 
 along which a train of cars moves ; whereas, the 
 
i64 NEW PHYSICS. 
 
 force with which a weight falls to the level of the 
 sea, and therefore the work that is done, varies 
 greatly with every foot in increase of height. 
 
 If we should cut a mountain by successive planes 
 (Fig. "jf), all of which are parallel to the sea, we see 
 from the preceding that a train of cars passing 
 around the mountain along any line cut by any one 
 plane, A B, would do no work with respect to gravi- 
 tation ; for it would always move at the same level 
 above the sea-level. If it should fall from the level 
 AB to the level .^'^', it would do a certain amount 
 of work by its fall, which would depend upon the 
 
 SEA LEVEL 
 Fig. 77. 
 
 difference of level oi A B and A^ ff-. These planes 
 and their bounding lines around the mountain are 
 called with respect to gravitation equipotential 
 planes and equipotential lines. 
 
 Experiment 93. — Draw the equipotential lines 
 or contour-lines represented by the different heights 
 of the corners of the squares (Fig. 78) formed by in- 
 tersecting Hues of level which have been taken by 
 an engineer. It is evident that one of these contour- 
 lines of height 2 will run from 2 on line C D m the 
 manner shown in Fig. 78. This curved line will 
 represent the boundary of a horizontal section of 
 the country, A B C D, taken at the level 2. Draw in 
 this way the other contour-lines, then represent the 
 
POTENTIAL AND WORK. 165 
 
 section along a line of railway, E F. The sides 
 of the little squares may be taken as one hundred 
 feet. 
 
 The term 
 equipotential is 
 of great signifi- 
 cance in phys- 
 ics. If we should 
 measure the 
 work done by 
 a body in falling 1 
 to the level of 
 
 the sea from dif- fig. 78. 
 
 ferent heights, 
 
 we could designate these heights and the planes 
 passing through them as points where the poten- 
 tial has a certain value, or the planes as equipo- 
 tential planes. 
 
 When a body strikes a blow after having fallen 
 from a certain height, we call the energy with which 
 it strikes kinetic energy : it is energy that results 
 from the motion or fall of the body. Before the 
 body fell, it had, by virtue of its position, or height 
 above the point struck, a certain advantage. It 
 could strike a certain blow if it fell. It has what 
 we call potential energy, or energy of position. This 
 energy it sacrifices in falling. It gains kinetic en- 
 ergy but loses potential energy, so that we have the 
 sum of the potential energy and the kinetic energy 
 always equal to a constant. Let us lift a certain 
 weight vertically above the sea-level. We are do- 
 ing work against the force of gravitation ; on the 
 other hand, we are gaining potential energy. Let 
 us now let the bodj slip from our grasp and fall a 
 
i66 
 
 NEW PHYSICS. 
 
 certain distance. During its fall it is losing poten- 
 tial energy, but is constantly gaining kinetic energy. 
 Therefore, with reference to the sea-level, the falling 
 body loses potential energy but gains kinetic en- 
 ergy. When it strikes at some point above the 
 sea-level, it has manifested kinetic energy which 
 can be used in driving a pile or striking any blow, 
 but it has lost potential energy with respect to the 
 sea-level. We have gained the energy of the blow, 
 but we have lost our advantage of position. Sup- 
 pose we have an elastic string (Fig. 79) attached 
 horizontally at A and B ; if we pluck this by the 
 
 middle and al- 
 low it to swing 
 back, it is evi- 
 dent that a par- 
 ticle at m will 
 be losing poten- 
 tial energy with 
 respect to the 
 line of level A B, but it will be gaining kinetic 
 energy which will be proportional to the distance 
 m m^. The potential energy at Mi will be only pro- 
 portional to the height m^ mi, whereas before it was 
 proportional to the height m m^. The bob of a 
 pendulum loses potential energy on one side of its 
 swing, and gains kinetic energy. On the opposite 
 side it loses kinetic energy and gains potential 
 energy. 
 
 The remarks we have made upon the expres- 
 sions equipotential lines, and planes, or surfaces, 
 with reference to gravitation, apply equally well to 
 the lines and surfaces in magnetic and electrical 
 regions. It is evident that a pendulum will swing 
 
 Fig. 79. 
 
POTENTIAL AND WORK. 167 
 
 in the same time at all points on the same equipo- 
 tential plane, for the force of gravitation is the same 
 at every point on this equipotential plane. We 
 could therefore determine different contour-lines 
 around a high mountain, cut out by imaginary equi- 
 potential planes, by swinging a pendulum at differ- 
 ent points and passing the planes through these 
 points; these planes would then cut out contour- 
 lines or equipotential lines along which the force of 
 gravitation would be the same. This method of 
 determining equipotential planes is difficult to carry 
 out on the sur- 
 
 faceoftheearth, \ \ f 
 
 for the force of \ ,-X ,-'' 
 
 gravitation is 'v','-' '''.--'^ 
 
 sensibly con- ,.-'-'r--''',V--'' ,\ ' 
 
 stant for con- " 
 
 
 
 ::-;- — 1« 
 
 
 V- ie 
 
 
 
 ----"■^-11 
 
 ----tiT'-f- 
 
 "---I ---..."" 
 
 -v- ---i-- 
 
 ^*^^^ 
 
 ;__>~.,~'-~i,,_ 
 
 
 
 
 / "v X - 
 
 / 
 
 \/ \ 
 
 
 
 Fig. 80. 
 
 
 siderable varia- 
 tions in height. "^ 
 No difference in 
 the length of a 
 seconds-pendu- 
 lum would be 
 detected (ex- 
 cept by very 
 delicate experi- 
 ments) between its length at the sea-level and its 
 length at the height of a mountain one thousand feet 
 above the sea. We can, however, use this method in 
 electricity and magnetism. Let 5 (Fig. 80) be a strong 
 electro-magnet, with a large pole or extended sur- 
 face, a b. The lines of force will be approximately 
 straight across the section ab ; move a little magnet, 
 N S, or compass, along a line, mn, so that its vibra- 
 
i68 NEW PHYSICS. 
 
 tions will be the same at every part of m n. This line 
 m n will be perpendicular to the lines of force emanat- 
 ing from a b, and will be an equipotential line. There 
 will be no tendency of the magnet to move along the 
 equipotential line m n. The only way that one mag- 
 netic pole could do work under the influence of the 
 attracting pole S, is to move under the influence of 
 the attracting pole from one equipotential line mn to 
 another, m' n' . It will do work in moving from one 
 equipotential line to another, just as a body under the 
 influence of the force of gravitation will do work in 
 .falling from one equipotential line or plane to an- 
 other. We have, therefore, magnetic potential ener- 
 gy as well as gravitative potential energy, and it is 
 evident that we can replace the magnetic pole S'\n the 
 above example by an electrified body, and the small 
 magnet i\^5by another electrified movable body, and 
 find that we could arrive at the conception of elec- 
 trical equipotential lines and surfaces and electrical 
 potential energy, just as we measure potential en- 
 ergy due to gravitation by reference to a plane or 
 a zero equipotential plane above which we must lift 
 a body in order to get any work out of it. In mag- 
 netism we can assume any reference plane for this 
 purpose. Since the earth's magnetic field is feeble 
 compared to that which we can make by means of 
 electro-magnets, we can often neglect it and perform 
 our experiments in any suitable magnetic field. 
 
 In experiments, however, upon the attractive or 
 repulsive force of electricity, our results are gener- 
 ally much influenced by the proximity of neighbor- 
 ing objects ; for it is well known that an electrical 
 charge upon any body will induce a similar charge 
 upon all objects in the neighborhood. The earth's 
 
POTENTIAL AND WORK. 
 
 169 
 
 atmosphere, moreover, contains a certain varying 
 electrical charge. There are electrical fields of va- 
 rying strength all around us, and we are forced to 
 adopt some reference plane by which we can esti- 
 mate the work necessary to be done in lifting bodies 
 against an electrical attraction up to this plane, or 
 the work that can be gained by letting bodies fall 
 to this reference plane. We take the electrification 
 of the ground as zero ; but the earth may have a 
 certain charge and the moon another. We know 
 that the electrical charge of the surface of the earth 
 is constant, as far as our senses can perceive. We 
 can, therefore, take the equipotential surface of the 
 earth as our plane of reference in electricity, just as 
 we take the level of the sea as our plane of reference 
 in regard to heights on the surface of the earth, and 
 to the potential energy at different heights. 
 
 We speak, therefore, of a certain electrically 
 charged body. A, as having a certain potential with 
 respect to another body. We mean that we must do 
 a certain amount of work to remove it beyond the in- 
 fluence of the second body, and this amount of work 
 is measured by reference to the potential of the earth. 
 
 The best illustration of the use of the potential 
 of the earth, as a reference for all electrical poten- 
 tials, is in the action of an ordinary cell of an elec- 
 trical battery with reference to the earth. 
 
 Experiment 94. — To prove that in any galvanic 
 cell one metal or pole, has one charge of electricity, 
 or plus charge, and the other an opposite or negative 
 charge. The instrument we have used in Experi- 
 ment 88 can be modified so as to show this. The 
 movable index is replaced by an aluminum needle, 
 shaped as is shown in Fig. 81. 
 
I/O 
 
 NEW PHYSICS. 
 
 Above the needle, and rigidly connected with it 
 by a piece of aluminum, is a light concave mirror, 
 m. A platinum wire is attached to the needle which 
 dips into the sulphuric acid 
 of the vessel, V, as before. 
 This needle can be electri- 
 fied by electrifying the sul- 
 phuric acid. When it is 
 placed in a dark room, a 
 beam of light from a point 
 in the soot - blackened chim- 
 ney of a kerosene-lamp can 
 be thrown upon the mirror, 
 and its reflection can be re- 
 ceived upon a piece of ground 
 glass placed at the focus of 
 the mirror, as is shown in 
 Fig. .72. 
 
 This arrangement will be 
 found to be very sensitive, 
 and will show the nature of 
 an electrical charge. The 
 poles of a battery might be 
 presented one after the oth- 
 er, at a certain distance from 
 this needle, and their different electrical effects ob- 
 served. But the electricity on the hand or clothes 
 of the observer would be likely to mask any effect 
 that the poles of the battery^ might have. The in- 
 terior of the box, therefore, which protects the nee- 
 dle from currents of air, is coated with tin-foil, and 
 is connected with the outside of the little glass tum- 
 bler or vessel F'(Fig. 82), which is also coated on 
 the outside with tin-foil. This coating of tin-foil is 
 
 Fig. 81. 
 
POTENTIAL AND WORK. 
 
 171 
 
 then connected with the ground, in order that it 
 shall all be at the potential or charge of the earth, 
 and shall remain constant, since it then becomes a 
 part of the earth. Four hollow quadrants of brass, 
 one of which can be pushed in and out, are then 
 placed upon glass pillars, and the aluminum needle 
 is hung at the center of the circle which these quad- 
 rants form. 
 
 around 
 
 The quadrant A is then connected by a fine 
 platinum wire with B, and the quadrant C'va the same 
 way with D. The charge which A receives will then 
 be communicated to ^, and also any which Z> re- 
 ceives will be shared by C. Connect at first all the 
 quadrants together and with the ground by means 
 9 
 
1^2 NEW PHYSICS. 
 
 of the metallic rods, which run from two adjacent 
 quadrants beneath them, and are carefully kept from 
 touching the tin-foil or any part of the inclosing 
 box. By thus connecting the quadrants with the 
 ground they can all be kept at the constant potential 
 of the earth, and the needle will not move, since it is 
 equally attracted on all sides. Connect now one set 
 of quadrants with the ground and with one pole of 
 a battery, and the other set with the other pole of 
 the battery, as is shown in the accompanying figure. 
 It will be found that the spot of light will move in 
 one direction ; by reversing the poles of the battery, 
 or, in other words, connecting different poles with 
 the same set of quadrants, it will be found that the 
 needle will move in the opposite direction. Thus it 
 will be seen that the poles of a battery have opposite 
 charges, and are, therefore, at different potentials 
 with respect to that of the earth. 
 
 Compare the difference of the charge on the 
 poles of one battery and that on two which are con- 
 nected, the positive pole of one with the negative 
 pole of the other ; also, when the positive of one is 
 connected with the positive of the other. 
 
 There is thus a difference of potential between 
 the poles of a battery ; or, in other words, the 
 charge upon one of the metals or poles of a battery- 
 cell is below that of the earth. It is in the condi- 
 tion of a weight upon which work must be done in 
 order to lift it to the earth's electrical level. It is in 
 a position analogous to a weight which is below the 
 level of the sea. The charge upon the other pole 
 has a potential above that of the earth. It is in a 
 condition analogous to a weight above the level of 
 the sea, and it can do work by falling to the electri- 
 
POTENTIAL AND WORK. 
 
 173 
 
 fication of the earth. If we, therefore, connect both 
 poles with the ground we can conceive of the electri- 
 fication of the earth flowing to the negative pole and 
 the electrification of the positive pole flowing to the 
 lesser charge of the earth. This happens in the case 
 of a cell when the circuit is completed through the 
 ground. We expect, from our analogy in the case 
 of mechanical work, that work must be done by this 
 rise and fall of electrical potential ; for, wherever 
 there is a difference of potential, work can be done. 
 When there is no difference of potential, then we 
 have the case of two heights standing at the sea- 
 level, and no work can be obtained from them with 
 reference to this level. In the case of the cell we 
 do not know why there should be this difference of 
 potential between two different metals. It is not 
 necessary at present to enter into theories in regard 
 to the cause of the difference of potential which pro- 
 duces what is called electro-motive force. All we 
 wish to know at present is, how we can connect the 
 phenomena of energy, and measure them all by the 
 same units of time, space, and mass. We can, how- 
 ever, think of a battery as an engine which is pump- 
 ing up water through one pipe and delivering it 
 through another. Suppose that the engine is in- 
 
 
 r''" 
 
 A 
 
 1 1 : '''■?.. 
 
 ■ ->-Z3 
 
 
 
 . *p' 
 
 Fig. 83. 
 
 closed at A (Fig. 83), and that the water is lifted 
 through one pipe p and discharged through another, 
 
174 
 
 NEW PHYSICS. 
 
 p'. Suppose that we can not study the mechanism of 
 the engine at A. All we can see and measure is the 
 difference and direction of the pressure in the two 
 pipes / and /'. This difference of pressure is anal- 
 ogous to the difference of potential between the poles 
 of an electrical battery or cell, which maintains in 
 some unknown way an electro-motive force along 
 the wire. We do not know whether there is any 
 current in a wire like that produced by the flow of 
 water in a pipe. We only know that there is a dif- 
 ference of level, and that there are conditions for the 
 performance of work in an electrical circuit. If we 
 could measure this work by the same units with 
 which we measure mfechanical work, or the work 
 that is done, for instance, by a weight falling a cer- 
 tain distance under the effect of gravitation, we could 
 compare electrical work or electrical engines with 
 mechanical work, or the work done by machines. 
 
 Before doing this, let us look more carefully at 
 the way we measure work. We have seen that we 
 can measure the work done by a body falling verti- 
 cally under the influence of gravitation, or by a body 
 falling down an inclined plane. The work of a man 
 could be estimated by the weight he can lift above 
 the level of the sea. The unit which is generally 
 adopted in England and America for measuring 
 work is a horse-power, which is taken to be equiva- 
 lent to raising 33i,DOO pounds one foot above the level 
 of the sea. It does not matter whether it takes one 
 minute to raise this weight or one year, the work 
 done is the same. Five or six men are roughly 
 equal to one horse. What are called the mechani- 
 cal powers enable us to distribute the work done in 
 various ways. For instance, with the simple levef 
 
POTENTIAL AND WORK. 
 
 175 
 
 l\<- 
 
 
 m^ 
 
 -a 
 
 \ 
 
 a 
 
 ^ . 
 
 
 
 A' 
 
 
 
 *w, 
 
 (Fig. 84), by means of a small weight, W, moving 
 through a considerable distance, BB',yfe can move 
 a much larger weight through a small distance, 
 A A'. We see here an illustration of the conserva^ 
 tion of energy, for the small weight at^', in falling 
 through B B', can do a certain amount of work by- 
 its blow, at B. The 
 energy of this blow, 
 if properly redirect- 
 ed, could raise an 
 equal weight to the 
 same height from 
 which it fell. The 
 law of the lever is 
 merely the statement 
 that the energy de- 
 veloped by two fall- 
 ing bodies is propor- 
 tional to the weights 
 
 of those bodies, and to the distances through which 
 they fall. These distances in the case of the lever 
 are the sines of the angles, described by the points on 
 the lever to which the weights are attached : 
 
 Or, W^A' m sin a = WmB' sin a (Fig. 84). 
 
 Since sine a cancels out, we have the law of the 
 
 ly, in B' 
 
 lever: W^A' m= Wm B', or— = The law of 
 
 JV A'm. 
 
 the compound lever can also be shewn to be a state- 
 ment of the conservation of energy. A compound 
 lever is merely a collection of simple levers, by 
 means of which the pressure at one point is dis- 
 tributed always in accordance with the law that the 
 distance through which the force or weight which 
 is applied travels, multiplied by this pressure or 
 
 Fig. 84. 
 
176 N£:iV PHYSICS. 
 
 weight, must be equal to the distance through which 
 the weight to be Ufted traverses, multiplied by the 
 weight ; or, in other words, the work done by the 
 falling body must be equal to the work done on the 
 body which is lifted. 
 
 AH forms of levers can be resolved into simple 
 levers by an application of the law of work. 
 
 The law of the simple pulley can also be imme- 
 diately deduced from the law of work. If the weight 
 W is pulled up a^ certain height by the descent of 
 _ Wx (Fig. 85), we know that the work 
 
 done by the fall of W-^ through its dis- 
 tance d, must be equal to that done 
 on f^ through its rise d' . 
 
 If the pull on the rope of the pul- 
 ley is not vertical but inclined at an 
 angle on opposite sides of the pulley, 
 we know that the work done on W 
 can be compared to that done by W^ 
 by comparing the vertical heights 
 through which W rises and H^i falls. 
 
 No matter how many pulleys we 
 may have, the law of the combination 
 can be deduced by ascertaining how 
 the vertical distance through which the pressure or 
 pull is exerted compares with that through which 
 the weight is lifted. In Fig. 86 the pull on the rope 
 passing around the lowest pulley, ^, is ^ that of the 
 pull exerted by W, since it is supported one half at 
 m, and one half by the next pulley, B. The pull on 
 the rope passing around j5 is ^ of ^ of Wiox the same 
 reason, and the pull on the rope passing around the 
 pulley C'\s\W. If, therefore, we wish to balance 
 the work done by the weight P in falling a vertical 
 
 
 k 
 
 Fig. 85. 
 
POTENTIAL AND WORK. 
 
 177 
 
 
 ax 
 
 distance, h, we must allow a weight, W, to be lifted 
 only i of k. 
 
 The popular adage, that what we gain in power 
 we lose in time, is a statement of the conservation of 
 energy, or of the equivalence of 
 work. We see in the case of the -^ 
 compound pulley that in order to 
 lift a weight W by one J as great, 
 we must let the smaller weight 
 pass over a space four times great- 
 er than that of the larger weight, 
 and accordingly occupy a time in 
 doing this four times longer than 
 if we lifted a weight simply equal ^.^^ 
 to the falling weight, or J of W. (Tj 
 
 The law of the screw can be ^^ 
 deduced from the principle of 
 work. The force P is exerted 
 through a space equal to the cir- 
 cumference of the circle, whose radius is equal to 
 the distance of the point of application of the force 
 from the axis of the screw. At the same time the 
 
 screw is forced through 
 the distance between two 
 threads, and lifts a weight 
 through this distance. 
 
 If r = radius of circle 
 around which P, the force, 
 moves, W^ weight lifted, 
 and h = distance between 
 two threads, we have 
 
 Fig. 87. 2 TT rP= W h. 
 
 Experiment 95. — Ar- 
 range the experiment as follows : Around the wood- 
 
 w 
 
 Fig. 86. 
 
178 
 
 NEIV PHYSICS. 
 
 en circle (Fig. 87) which forms the head of a screw 
 pass a silk thread. Pass this thread horizontally 
 around a smooth pulley, and attach a weight, P, to 
 its free end. The tension on the thread will be equal 
 to the weight and to the force exerted at the cir- 
 cumference of the screw. Place a weight upon the 
 head of the screw, and prove the truth of the 
 formula given for the screw. 
 
 In this way, by the application 
 of the parallelogram of forces and 
 by the law of work, the laws of 
 the mechanical powers can read- 
 ily be obtained. By the princi- 
 ple of work we can also deduce 
 the laws which govern the mo- 
 tion of fluids. For instance : Why 
 should a jet of water (Fig. 88) 
 strive to rise to the level of the 
 reservoir which supplies it with water ? 
 
 We know that any particle or drop of the water 
 in falling from the top of the reservoir to the bottom, 
 where the jet issues, would, by the law of falling 
 bodies, acquire a velocity v = i^2zh. 
 
 This would be sufficient, if we neglected the fric- 
 tion at the orifice and the resistance of the air, to 
 carry this particle again to the height from which 
 it fell. Hence this action of fluids is another illus- 
 tration of the conservation of energy. 
 
 FlO. 88. 
 
CHAPTER X. 
 
 WORK AND HEAT. 
 
 It is evident that we could measure the work 
 done by a man, a horse, or a steam-engine on any 
 piece of mechanism, if we could compare the weight 
 lifted and the space through which it is lifted with 
 the force applied and the space through which it is 
 exerted. In many cases this can be done. In most 
 cases, however, the motion of the parts is too rapid 
 and the spaces passed through too great for us to 
 measure conveniently. For instance, suppose we 
 wish to measure the work an engine does in turning 
 an axle which revolves sixty times a minute. By 
 grasping a slowly-revolving axle with the hand, 
 which is protected by a thick glove, we find that we 
 can alter the velocity of the moving axle (provided 
 that small power is applied to it) by tightening our 
 grasp, still allowing the axle to turn. The friction 
 ihat arises between our glove and the axle is pro-_ 
 portioned to the pressure we exert in grasping the 
 axle. 
 
 Experiment 96. — Obtain two iron cog-wheels, 
 one twelve inches, and the other two or three inches 
 in diameter. Provide them with axles, and mount 
 them £ts in Fig. 89. The larger cog-wheel will thus 
 be in the position of the stone of an ordinary grind- 
 
i8o 
 
 NEW PHYSICS. 
 
 Stone, and will gear into the smaller wheel placed 
 at its circumference. Arrange upon the axle of the 
 latter the apparatus represented in Fig. 89. This 
 consists of a lever of wood, one end of which carries 
 a balance-pan ; the other is in the form of a ring cut 
 in two, the upper half of which is hinged so that it 
 can be lifted and clasped around an axle and then 
 be bolted tight to this axle, as is shown in the fig- 
 
 FlG. 8g. 
 
 ure. When the axle is turned, it is evident that this 
 apparatus, which is called a friction-brake, will tend 
 to revolve with the axle. If, however, we place a 
 suitable weight in the balance-pan, at the end of the 
 lever, we can maintain the end of the lever in a hori- 
 zontal position between the stops S S, when we turn 
 the axle with a certain velocity. If we turn with a 
 lesser or a greater velocity, the end of the lever does 
 not remain between the two stops. When it does 
 
WORK AND HEAT. 
 
 181 
 
 remain there, we know that the weight in the balance- 
 pan, multiplied by its length of arm, which is meas- 
 ured to the center of the revolving axle, is balanced 
 by the friction which arises on the surface of the 
 axle, between the wood and the iron of the axlei 
 multiplied by the radius of the axle or the arm at 
 which the friction acts. When the lever arm is hori- 
 zontal, and the brake is screwed to the axis, and a cer- 
 tain number of revolutions of the axis is produced, 
 with the weight W in the balance-pan, we know 
 that the whole mechanical effect produced is ex- 
 pended in overcoming the friction between the shaft 
 and the brake, and this mechanical effect is equal 
 to the work of the revolving shaft. The friction 
 acting in the direction of the revolution counter- 
 balances the effect of the weight W at the end of 
 its arm. 
 
 If we call the lever A C oi the weight a, the 
 moment of this, weight = Wa, and this is equal to 
 the moment of the friction, or the friction acting 
 with the arm of unity. If <o represents the an- 
 gular velocity of the shaft, the mechanical effect 
 E-=F V ^^ Wa ft) per second. If we call n the 
 number of the revolutions of the shaft per min- 
 ute, we shall have w = -^ — = — , and the work 
 ' 60 30 
 
 requiredZ = ^^-^^. (This is in foot-pounds.) The 
 
 weight fF includes the weight of the apparatus re- 
 duced to the point of suspension of the balance-arm. 
 To ascertain this, the apparatus is placed upon 
 a knife-edge atiT, and a string, attached to C, and to 
 a spring-balance, will give the required weight. 
 In order to measure the velocity with which you 
 
l82 
 
 NEW. physics: 
 
 turn the large cog-wheel, hang up a pendulum, and 
 move the handle of the axle in unison with the swings 
 of the pendulum. This can be done by practice. 
 Knowing the circumference of two cog-wheels, and 
 the rate of one, we can readily determine that of the 
 other. 
 
 A little practice will enable one to turn the axle 
 quite uniformly once a second and twice a second, 
 and also at some intermediate rate. In this way 
 measure the work at different velocities. 
 
 Experiment 97. — The work done can also be 
 measured in the following way : Attach one end of a 
 leather band to a fixed support at A (Fig. 90), pass 
 
 -Fig. 90. 
 
 Fig. 91. 
 
 the band over the axle, and place a suitable weight 
 in a balance-pan, which is attached to the other 
 end of the band. Ascertain the weight that is 
 necessary to keep the band, tight when a certain 
 velocity of rotation is given to the axle against the 
 pull of the weight. Notice also what weight must 
 be put in the pan to tighten the band when the 
 axle revolves in the same direction (Fig. 91) as the 
 pull of the weight. One experiment will then give 
 
IVOJi/ir AND HEAT. 183 
 
 US (2 = P-\- F, and the other Q = P, in which F is 
 the friction. 
 
 By these methods, therefore, we can obtain the 
 work that is done by transmitting power from one 
 axle to another. If we call the work that can be 
 done by the original machine W, and the work that 
 can be done by the second machine W^, the per- 
 centage or proportion of work that we can trans- 
 
 mit will be -|tf = «• The loss of power that is per- 
 ceived is due to the slipping of the connecting belts 
 by which the power is transmitted, and to the heat- 
 ing of the axles. That a part of the power is em- 
 ployed in heating the axles is a common fact of ex- 
 perience. It is a most important fact, and the whole 
 science of modern physics is based upon it. That 
 friction produces heat is evidenced by the heating 
 of car-axles, by the heating of tools driven by lathes, 
 and by the friction of an ordinary saw while it is 
 being driven through a plank. This heating was 
 known from the earliest times. Indeed, it is said 
 that Indians have obtained fire by rubbing dry sticks 
 together. No advance, however, was made in our 
 knowledge of the relation between work and heat 
 until an exact measurement was made of the amount 
 of heat produced by a measured amount of work. 
 When this measurement was made, it was discovered 
 that mechanical work could be converted into an 
 exact equivalent of heat, and it was further discov- 
 ered that heat itself was a form of motion. We are 
 in a condition now to make a comparatively exact 
 measurement of work; It is only necessary for 
 us to know how we can measure an amount of 
 heat. Let us, first, make a comparatively rough 
 
1 84 ^EW PHYSICS. 
 
 experiment, and afterward proceed to a more ex- 
 act one. 
 
 Experiment 98. — Slip upon the axle of the 
 smaller cog-wheel, Fig. 89, the open end of a thin 
 brass tube (Fig. 92), having first closed the tube by 
 a plug of plaster at a suitable point, P. Screw the 
 brass tube upon the axle A of the smaller cog-wheel, 
 placing leather washers between the brass tube and 
 the screws, and also between the brass tube and the 
 
 axle, to prevent conduc- 
 
 A _^ P h tion of heat to the axle. 
 
 Drill a hole in the brass 
 tube at h, sufficiently 
 Fig. 92. large to insert a ther- 
 
 mometer. Fill the brass 
 tube nearly full with water, and insert a thermom- 
 eter in the water. After the water has arrived 
 at a stationary temperature, remove the thermom- 
 eter, close the little hole tightly, and put a friction- 
 brake upon the metallic tube, having placed a suit- 
 able weight in the balance-pan, in order to keep 
 the bar of the friction-brake in a horizontal posi- 
 tion, while maintaining a comparatively rapid rate 
 of rotation of the small cog-wheel and its axis. 
 After a measured time it will be found that the water 
 has risen in temperature. It is evident that a cer- 
 tain amount of heat has been absorbed by the sides 
 of the metallic vessel, and a certain amount will be 
 absorbed by the bulb of the thermometer when it is 
 placed in the hot water, after the cog-wheel has 
 ceased to turn. Knowing the work that has been 
 expended in maintaining a certain velocity of rota- 
 tion against the friction-brake, the heat that has 
 been produced by this can be estimated in term of 
 
WORK AND HEAT. 
 
 185 
 
 the weight of the water in pounds and the number 
 of degrees of temperature that it has been raised. 
 It has been found that 772 pounds lifted one foot 
 high is the equivalent of one pound of water raised 
 one degree Fahrenheit in temperature. You can 
 not expect to obtain this exact number by your 
 experiment, for it is difficult to maintain the rota- 
 tion of the cog-wheel uniform. Moreover, you have 
 neglected the amount of heat that has been ab- 
 sorbed by the sides of the metallic vessel and by the 
 thermometer-bulb, and also the amount that has 
 radiated into space while you have been making 
 your observation. The result you will obtain, how- 
 ever, will be a fair approximation for the first result. 
 In order to obtain a closer result, it is necessary to 
 examine the methods by which we can estimate the 
 quantity of heat. We shall then apply these more 
 refined methods. 
 
 In the first place, let us examine our thermome- 
 ter. The effect of heat upon a mercury thermome- 
 ter is a matter of common experience. We know 
 that on warming the bulb, the little thread of mer- 
 cury rises in the fine capillary tube of the ther- 
 mometer, and that on cooling it sinks. On sur- 
 rounding the bulb of the thermometer with melting 
 ice we find that the column of mercury remains at 
 a fixed point in the tube of the thermometer, which 
 is marked freezing-point (0° on the centigrade ther- 
 mometer and 32° on Fahrenheit). On the other 
 hand, if we place the bulb and tube of the ther- 
 mometer in steam, carefully protecting the bulb 
 from the bubbling water, we find that there is 
 another stationary point, which is the heat of steam. 
 This is marked 100° on the centigrade thermome- 
 
1 86 
 
 NEW PHYSICS. 
 
 ter and 212° on the Fahrenheit. These points are 
 approximately invariable at the same place on the 
 earth's surface. If we ascend a high mountain, we 
 find that the boiling-point is not so high as at the 
 level of the sea. We know, therefore, that the boil- 
 ing-point is affected by the diminished or increased 
 pressure of the column of air resting upon the boil- 
 ing water. The freezing-point is also affected by dif- 
 ferences of pressure. Moreover, the glass changes 
 from time to time its molecular structure when 
 submitted to changes of temperature, and these 
 two fixed points, therefore, the freezing-point and 
 the boiling-point, are not really invariable, though 
 enough so for ordinary measurements. These two 
 points, however, should be carefully examined be- 
 fore any thermometer is used. ^ 
 
 Experiment . 99. — To test the 
 freezing-point, the thermometer-bulb 
 is placed in melting ice. The ice 
 should be packed around the stem 
 as far as the top of the mercury- 
 column. 
 
 To test the boiling-point, it is 
 necessary to hang the thermometer 
 so that it may be entirely in the 
 steam, and the steam should not be 
 allowed to condense on the bulb. 
 A simple arrangement for accom- 
 plishing this is represented in Fig. 
 93, Pass the thermometer through 
 a hole in the cork, until the bulb 
 jnearly touches the surface, of the 
 water. A glass tube, also passing through' the 
 cork, and bent at an obtuse angle, ailoXys the steani 
 
 Fig. 93. 
 
WORK AND HEAT. 187 
 
 to escape ; otherwise it is evident that there would 
 be an explosion. 
 
 After these fixed points are obtained, the space 
 between them is divided into 100° for the centi- 
 grade thermometer and 1 80" for the Fahrenheit ther- 
 mometer. One hundred centigrade degrees are, 
 therefore, equal to one hundred and eighty Fahren- 
 heit degrees, or one Fahrenheit degree is \^, or ^ 
 of a centigrade degree. It will be noticed that a 
 centigrade degree is nearly twice as large as a 
 Fahrenheit degree. In order to reduce frdm the 
 Fahrenheit scale to the centigrade, we must first 
 subtract 32° from the reading of the Fahrenheit 
 scale, since the freezing-point on this scale is 32° 
 above its zero ; and then take -| of the remainder. 
 In dividing the space between the freezing-point 
 and the boiling-point into equal degrees in a ther- 
 mometer, it is assumed that the capillary bore of 
 the thermometer is of the same size throughout. 
 If it were not, it is evident that a degree at one 
 place would be more or less than at another place. 
 If the tube were large at one place, the mercury 
 would move through a less space for the same in- 
 crease of heat than it would if the bore were nar- 
 rower. It is necessary, therefore, to carefully ex- 
 amine this question if great accuracy is required 
 (see Appendix). In our present experiment the 
 other errors of experimentation will probably be 
 greater than the errors in the lengths of the degrees 
 of an ordinary Fahrenheit thermometer. If the 
 fixed points have changed, we must allow for the 
 change. For instance, if the freezing-point is low- 
 ered, we must subtract the amount of fall from each 
 reading. 
 
1 88 NEW PHYSICS. 
 
 The unit of measurement in heat is called the 
 unit of heat. 
 
 Definition. — The unit of heat is the amount re- 
 quired to raise one gramme of water i ° C. If a gramme 
 of water loses a unit of heat, it will fall i° in tempera- 
 ture. 
 
 It is necessary to prove that the unit of heat is 
 constant for all temperatures of water — that is, it 
 will take as much heat to raise one gramme of water 
 from 5° to io° as from 15" to 20°, or from 35° to 40°, 
 etc. Otherwise our standard would change, and 
 we should no longer have a standard unit of meas- 
 urement. 
 
 Experiment 100. — Pour some water into a beak- 
 er, and, having carefully dried the outside of the 
 beaker, place it upon a piece of wire gauze over a 
 gas-burner (or alcohol-lamp), and heat the water to 
 some convenient temperature — say, 30° C. Take 
 an equal quantity of water in another beaker, which 
 is at the temperature of the air — about 10° C. — and 
 mix the two together, taking care to pour the hot 
 water into the cold water, to avoid breakage. It 
 will be found that the resulting temperature will be 
 midway between 30° and 10°, or 20°, which shows 
 that the heat which the hot water lost in falling 
 from 30° to 20° is just equivalent to raise the colder 
 water through the same number of degrees (10°). 
 That is, the unit has the same value between 10° 
 and 20° as between 20° and 30°. 
 
 It is well to vary this experiment in the follow- 
 ing way : 
 
 Experiment ioi. — Take a weight of water, W, 
 at a temperature, T, and a different weight, W^, at 
 a lower temperature, t ; then, if \ is the resulting 
 
WOJiX- AND HEAT. 
 
 189 
 
 temperature of the mixture, we shall find that the 
 heat lost by the hot water = heat gained by the cold 
 water, ^F(r— \) = W^i(\ — /). 
 
 In both these experiments it should be observed 
 that if the water is quite hot there will be consider- 
 able loss of heat while the hot and cold water are 
 mixed and are being stirred, so that the tempera- 
 ture of the mixture shall be everywhere the same. 
 This loss must be carefully estimated, by noticing 
 the rate of cooling of the water. 
 
 It is well to repeat the preceding experiments, 
 bearing in mind this correction for loss in cooling. 
 
 Experiment 102. — With a watch observe the 
 time of mixture, the time of taking the tempera- 
 ture of the mixture, the time it takes the mixture 
 to cool 1°, the time it takes to cool 2°, and so on. 
 It will be found that the intervals of time between 
 its falling through successive degrees — for instance, 
 through three or four degrees — will be nearly uni- 
 form. If this interval is 10" for the fall of one de- 
 gree, and if it has taken 20" to observe the temper- 
 ature of the mixture, we know that the loss would 
 
 20 
 be — = 2°, and the true temperature of the mixture 
 
 would be \ -f 2°, instead of X which is observed. 
 
 Another source of error, which is common to all 
 our experiments thus far in heat, is the loss of heat 
 which comes from the absorption of heat by the 
 containing vessels. For instance, when we pour 
 the hot water into the cold water, a certain amount 
 of the heat of the hot water goes into the vessel 
 which contains the mixture. When this vessel is 
 glass or thin metal, this loss is not great ; if, how- 
 ever, the amount of water is great, and it is poured 
 
igO NEW PHYSICS. 
 
 into a thick copper vessel, this loss may be very 
 large. In all cases, however, we must determine 
 how much this loss of heat is. 
 
 Experiment 103. — Take some water at a tem- 
 perature of T° (about 50°) and pour it into the empty 
 vessel used in the previous experiments, which is 
 at the temperature of the air of the room in which 
 we experiment. Find the resulting temperature 
 and the rate of cooling, and correct as in the pre- 
 vious experiment for the rate of cooling. Call the 
 weight of the water which is added, W. We then 
 have: the heat lost to the vessel is W{T — \) units, 
 and the heat gained by the vessel is \ — 1°. Hence, 
 
 to raise the vessel through 1° we need — r 
 
 units of heat. This number is called the specific 
 capacity of the vessel. It is usual to divide this 
 number by the weight of the vessel in grammes if 
 it conducts heat uniformly throughout, as in the 
 case of thin metal vessels. In this way we obtain 
 what is called the specific heat of a substance, 
 which is the heat required to raise one gramme of 
 the substance through i" C. 
 
 There is another source of error which arises 
 from the use of the thermometer. This also ab- 
 sorbs a certain amount of heat while it is used in 
 gently stirring the mixture and in observing its 
 temperature. It also has a specific capacity which 
 must be measured. 
 
 Experiment 104. — Pour a sufficient amount of 
 water into the vessel, whose specific capacity we 
 have found, to just cover the bulb of the thermom- 
 eter. Call the weight of this water J^ grammes, 
 and observe its temperature t°. Heat the bulb of 
 
WORK AND HEAT. igr 
 
 the thermometer to ioo° C. in steam ; dry it and 
 hold it just above the water, and notice its tem- 
 perature T. Then plunge it into the water of the 
 vessel and notice the temperature. The thermom- 
 eter in falling {T — t)° raises the water and the ves- 
 sel through (\ — t)° — that is, it gives up (\ — t) 
 tmits of heat to each gramme of water, and (\ — t) 
 X specific capacity of vessel to the vessel. Calling 
 this specific capacity S, we perceive that the loss 
 W (;k — t) + <^ — t)S, or i;K — t){W+ S) units of 
 heat, lowers the temperature of the thermometer 
 {T — \)° ; hence the specific capacity of the ther- 
 mometer, or the quantity of heat necessary to raise 
 
 It 1°, IS '^ 4^^^ — - units. 
 
 I — A. 
 
 We are now in a condition to estimate the losses 
 of heat in our first rough measurement of the me- 
 chanical equivalent of heat. Having carefully made 
 all the corrections, upon a repetition of the experi- 
 ment, we shall find that we approximate nearer 
 to the true value, which, in English measure, we 
 have said is as follows : 772 pounds raised one foot 
 is the equivalent of the heat necessary to raise one 
 pound of water 1° Fahr. ; or, in French measure, 
 424 grammes raised one kilometre is the equivalent 
 of the heat necessary to raise one gramme of water 
 I'-C. 
 
 Our first experiment, however, is affected by 
 the error in measuring the power consumed. Let 
 us proceed to a more careful measurement, bearing 
 in mind the fact that it is very difficult to measure 
 heat on account of its rapid escape by radiation 
 into the air, and by conduction into neighboring 
 masses of matter. 
 
192 
 
 NEW PHYSICS. 
 
 The following experiment was performed by 
 Hirn : * 
 
 Experiment 105. — Suspend an iron bar, A (Fig. 
 94), weighing 350 kilogrammes, b}' wires three me- 
 
 FiG. 94. 
 
 tres long ; suspend also a block of stone, B, weigh- 
 ing 941 kilogrammes. (This is called the anvil.) 
 This is provided with an iron plate, C. Between 
 the hammer and the anvil a cylindrical piece of 
 lead, P, weighing 2.948 kilogrammes, is 
 r-| held by a wooden fork. A cavity is made 
 
 I in the lead, as is shown in Fig. 95. The 
 
 ' temperature of this cavity = 7.87° is ob- 
 
 FiG. 95. tained by means of the thermometer, and 
 also the specific capacity of the lead. 
 The hammer A is let fall from a height of i.i66 
 metres. 
 
 The height of the recoil after the blow is 0.087 
 metres. 
 
 The height of the recoil of anvil is 0.103 metres. 
 The work of the blow of the hammer is Z = 
 
 * Muller's " Lehrbuch der Physik," ii, p. 896. 
 
WORK AND HEAT. 1^3 
 
 weight X height = 350 x 1.166 = 408.100 kilo- 
 gramme-metres. 
 
 The work done in the recoil of the hammer and 
 the anvil is / = 0.103 (941 -f- 2.95) + 0.087 X 3So = 
 127.677 metre-kilogrammes. 
 
 The work used up in the compression of the 
 lead is Z — / = 408.100 — 127.677 = 280.423 metre- 
 kilogrammes. 
 
 In the cavity of the compressed lead were placed 
 18.5 grammes of water at 0°. 
 
 Four minutes after the blow, the temperature 
 was 12.10°. 
 
 Eight minutes after the blow, the temperature 
 was 11.75". 
 
 Therefore, during this time there was a cooling 
 of 0.35"; and we have approximately, 11.75° • o-35° 
 = 12.10° : jr, or ;ir = 0.36°, on the supposition that 
 we have made in Experiment 102. Hence, the tem- 
 perature of the lead at the moment of compression 
 is 12.10° + 0.36° = 12.46°. 
 
 The heating of the water by the blow = 12.46° 
 — 7.87° = 4-59°. Hence, since the specific heat of 
 lead = 0.03145 (see next experiment), 4.59 x 2.948 
 X 0.03145 + 12.46 X 0.0185 = 0.659 units of heat. 
 
 (The factor 0.0185 is the amount of water, 18.5 
 grammes, reduced to a kilogramme.) • 
 
 ,T^ , , . 280.423 , ., 
 
 We thus obtam — -^-7- = 427 kilogramme-me- 
 tres. 
 
 This experiment can be performed with good 
 results by taking much smaller masses for both the 
 hammer and the anvil. Suspend a small piece of 
 lead before the anvil, by means of a thread, and let 
 it drop quickly into a little calorimeter made of 
 
194 
 
 NEW PHYSICS. 
 
 sheet-brass, Holding a small known quantity of water 
 (see following experiment). 
 
 When a closer approximation is made in regard 
 to the rate of cooling, 425.2 can be obtained. 
 
 Instead of taking the specific capacity of the lead 
 vessel, we have used the specific heat of lead, which 
 can be obtained from a table. It is always best to 
 test the specific heat of the specimen of metal which 
 is used. The following experiment is called the 
 method of obtaining specific heat by mixtures : 
 
 Experiment 106. — Take a very thin copper or 
 brass vessel, about one quarter the size of an ordi- 
 nary tumbler, and find its specific capacity, which 
 we will call S. Pour a known weight of water, W, 
 into the calorimeter, and place a known weight, W, 
 of the metal divided into small pieces in a thin glass 
 test-tube; raise this metal to 100° C, by holding 
 the test-tube in boiling water, then pour the metal 
 quickly into the water which is in the calorimeter 
 and at the temperature t of the room ; then test the 
 temperature with a thermometer whose specific ca- 
 pacity has been measured, and make ai correction 
 for cooling. 
 
 If we denote by x the specific heat of the metal, 
 or the number of units of heat necessary to change 
 the temperature of one gramme of the metal i" C, 
 we have heat lost by the metal = heat gained by 
 the water, -f- the heat gained by the calorime- 
 ter -(-the heat gained by the thermometer, or, 
 W^ (100 -\)x= W{X- i)^S{\-t)-{-S,{\- t) 
 
 Hence, x = (^ — t) ^-rff — i— ^, in which 5i is spe- 
 cific capacity of the thermometer, and \ is the tem- 
 
IVOJiX- AND HEAT. 
 
 195 
 
 perature of the mixture. Obtain in this way the 
 specific heat of lead and of copper. 
 
 The results of Experiments 98 and 105 show us 
 that there is an exact equivalence between me- 
 chanical motion and what we call heat. Wherever 
 the energy resulting from the motion of bodies dis- 
 appears, heat takes its place. Motion is therefore 
 being continually changed into heat in all the 
 operations of machines, and in the case of all falling 
 bodies, and the heat is the equivalent of the work. 
 
 We have tested this great law, which is called 
 the conservation of energy, as far as motion and 
 heat are concerned. It is necessary to see how far 
 it applies to other forms of energy, such as light, 
 sound, electricity, and magnetism. Our experi- 
 ments have also shown that perpetual motion is im- 
 possible, for we can not get any more work out of 
 any combination of mechanical powers than we put 
 into it. The term perpetual motion is often misun- 
 derstood. A water-wheel placed under Niagara 
 Falls, if the materials would not wear out, might run 
 forever. This, however, would not be perpetual 
 motion, for the work done 
 by the wheel is the equiva- 
 lent of the work done by the 
 water upon it. Neglecting 
 the resistance of the air, a 
 cannon-ball might be fired 
 from a mountain at the 
 equator with a sufficient ve- 
 locity to make it revolve Fig. g6. 
 forever around the earth as 
 
 a satellite. A velocity of five miles a second would 
 accomplish this ; for at the end of the first second 
 
196 NEW PHYSICS. 
 
 the ball would have fallen toward the center of 
 the earth (Fig-. 96) just as much as the earth curves 
 away from a tangent line through the mountain, 
 and there being no resistance, and consequently no 
 heat as the result of motion, this motion would 
 give up no part of itself, and the velocity of the 
 ball would remain the same always. This is not an 
 example of perpetual motion, for the motion hav- 
 ing been once started, there is nothing to oppose it, 
 a condition which is never met with in the case of 
 any human mechanism. 
 
 We should obtain perpetual motion if we could 
 arrange a series of inclined planes, so that a body 
 after falling down one should rise along another to 
 a greater height than that from which it fell, and so 
 on. Taking the impossibility of perpetual motion 
 as an axiom, we can prove that the velocity which 
 a body acquires in falling down an inclined plane is 
 equal to that which it would acquire if it fell down the 
 height of the plane, for it would be just sufficient to 
 carry the body up the height, neglecting friction. 
 
 If we could cut off the force of attraction be- 
 tween a pile-driver and the earth, we could lift it as 
 we could a feather, without any effort, and letting 
 it fall we could get any amount of work from it. A 
 machine that could do this in an automatic way 
 would be a perpetual-motion machine. The appli- 
 cation of the principle of the conservation of energy 
 and the impossibility of perpetual motion enable us 
 to ascertain many laws of nature, and these laws are 
 receiving every day fresh confirmation of their truth. 
 We can prove, for instance, from the conservation 
 of energy, that ice must melt under pressure, a fact 
 which it would be difficult to test in a laboratory. 
 
WORK AND HEAT. 157 
 
 Water, it is well known, expands when it freezes. 
 Suppose, therefore, we put some water in a cylinder, 
 place a piston upon it, and heavily load this piston. 
 Put a certain amount of freezing mixture around the 
 cylinder. The water in freezing will lift this weight 
 through a certain distance. Double the weight 
 upon the piston : will the ice hft double the weight 
 through the same distance as it did half of this 
 weight ? If it should do this, we should be getting 
 out of the same freezing mixture an unlimited 
 amount of work. Since we can not conclude that 
 the amount of energy in a given amount of a freez- 
 ing mixture is inexhaustible, we are forced to the 
 conclusion that the freezing-point of water is low- 
 ered by pressure, or, in other words, that ice melts 
 under pressure. 
 
 We have assumed in this example that the freez- 
 ing mixture, which could consist of salt and pounded 
 ice, can do a certain amount of work. Indeed, we 
 know this from its effect in lifting a weight against 
 the force of gravitation, by, the expansion of ice 
 which it forms. We commonly say that the freez- 
 ing mixture abstracts heat from the water. In order 
 to do work, there must be a difference of potential : 
 we can say, therefore, that the heat potential of the 
 water is higher than that of the salt and ice, and in 
 the equalization of the difference of potential a cer- 
 tain amount of work is done. We are led to think 
 of heat as a form of motion, since it is equivalent to 
 motion and to mechanical work. Let us examine 
 more carefully the amount of motion or heat, or the 
 fall of heat potential when water congeals into ice. 
 
 Experiment 107. — Place in a calorimeter of 
 known specific capacity, S, a known weight of water 
 
198 NEW PHYSICS. 
 
 fF, at a known temperature T (about 50° C); put into 
 the water small pieces of ice which have been care- 
 fully dried with filter-paper ; stir quickly until the 
 ice is entirely melted, and observe the temperature 
 of the water ; then weigh the mixture ; this will give 
 the amount of ice, W^, that has been added. 
 
 A certain amount of heat has evidently been 
 used to change the ice at 0° into water at \°. 
 
 Hence, heat lost by the water and calorimeter 
 in falling from T° to X° = heat required to convert 
 W^ grammes of ice into water at 0°, and afterward 
 to raise W^ grammes of water from 0° to \°, or 
 (5-f- W){T-\)= WiL-\- W^ \. Hence, 
 
 L = ^^ — ^ \ ~ — ^- -^ is called the latent 
 
 heat of water, and is the number of units of heat re- 
 quired to convert a unit weight of ice at zero into 
 water at zero. 
 
 One must carefully make corrections for the 
 cooling of the water if its temperature is higher 
 than the air of the room before the ice is put in, and 
 also for the heating of the mixture by the air if its 
 resulting temperature is less than that of the room. 
 
 This latent heat is looked upon as the expression 
 of a certain internal movement among the particles 
 of the water. 
 
 The efficiency of a mixture of salt and ice in 
 freezing neighboring bodies of water comes from its 
 withdrawal of latent heat from the water in order 
 that the liquefaction of the salt and ice can take 
 place. Thus, when salt is placed upon sidewalks to 
 melt the ice, it melts the ice in contact with it, but 
 also freezes a certain amount of water in neighbor- 
 ing cracks and crevices. - 
 
WOX/C AND HEAT. igg 
 
 We have spoken of the work that can be ob- 
 tained from a freezing mixture ; by freezing water 
 and thus lifting weights, we cause in this way a dif- 
 ference of potential. By boring holes in granite and 
 filling them with water, the fall of heat potential 
 when the water freezes is sufficient to split the 
 rock. 
 
 The great agent, however, which we use to work 
 for us is steam. Let us examine the amount of work 
 that is necessary to convert a unit weight of water 
 at ioo° C. into steam at ioo° C. 
 
 Experiment io8. — Lead steam from a flask in 
 which water is kept boiling, by a short tube, to pre- 
 vent condensation, into another flask which 
 contains a known weight, W, of water. To 
 prevent the water from the condensing 
 steam in the connecting tube entering the 
 latter flask, provide the tube when it en- 
 ters this flask with an arrangement similar 
 to that in the figure, which consists mere- 
 ly of a larger piece of glass tubing which 
 serves to catch the water coming from 
 condensation at the end of the entering 
 tube. 
 
 Let t be the temperature of the water 
 and of the air, also \ the temperature of fig. 97. 
 the mixture. After it has been corrected 
 for the rate of cooling, weigh the water in the 
 flask ; the amount of steam that has entered can 
 thus be found ; call this Wy The latent heat of 
 steam will then be found by making the heat lost 
 by the steam in condensing and in then falling from 
 100° to \° = the heat gained by the water in rising 
 from t° to \°, 
 
20O NEW PHYSICS. 
 
 or, J^i I Z + (lOO - X) I = PF(\ - t), 
 
 °^' ^ = -w, ■ 
 
 Precaution. — The flask into which the steam is 
 led should not be tightly corked. A way of es- 
 cape should be allowed the excess of steam that is 
 not condensed. 
 
 Since so much heat, or, in other terms, work is 
 required to convert water int6 steam, we cati con- 
 clude that more work can be got out of steam than 
 out of boiling water. We are also led to suppose 
 that there is greater movement of the particles of 
 water in the shape of steam than when they are in 
 the condition of particles in boiling water. More- 
 over, we must admit, when steam expands, and 
 thus lifts a weight against the force of gravitation, 
 for instance, that in doing this work it must part 
 with a portion of its heat, otherwise we might 
 obtain unlimited work from the same amount of 
 heat. The amount of work we can get from steam 
 at a certain pressure and temperature depends upon 
 the fall of potential that we can obtain, just as in the 
 case of a pile-driver, the amount of work we can 
 get depends upon its height above the point where 
 we wish to do the work ; or, in the case of a water- 
 fall, upon the vertical height through which the 
 weight of water can descend. We could get all the 
 work out of steam if we could deprive it of all its 
 heat and utilize this heat completely. 
 
 The work that is done on a metallic bar, in com- 
 municating a certain amount of heat to it, can be 
 made to have its equivalent in the work done in lift- 
 ing a weight, W, against gravitation, and in doing 
 
IVOHIC AND HEAT. 20 1 
 
 internal work among the molecules of the iron. It 
 is evident, from the conservation of energy, that if 
 we keep increasing the pressure against the end 
 of a vertical rod, by increasing the weight upon it, 
 the same amount of heat applied to the rod will not 
 lift an indefinite weight. Otherwise, by merely ex- 
 pending a few foot-pounds of work in the shape of 
 heat, we might get in return far more than we ex- 
 pended. We conclude, therefore, that a mass of iron 
 inclosed under great pressure in a furnace will not 
 expand so much by the application of a given quan- 
 tity of heat as it would if the pressure were removed, 
 and, consequently, would not melt at the same tem- 
 perature as it would if the pressure were removed. 
 
 A mass of metal changes its dimensions with 
 every change of temperature. By delicate means, 
 even the influence of the approach of a person's 
 body to within three feet of a metre-bar, a square 
 centimetre in section, can be detected, and all very 
 exact measurements of length must be conducted in 
 a room of constant temperature.. This molecular 
 change, or evidence of work done among the parti- 
 cles of a body, evidently must greatly influence the 
 glass and the mercury by which we make our heat- 
 measurements. 
 
 Experiment 109. — To determine the co-efficient 
 of expansion of glass of which a thermometer has 
 been made, we proceed as follows : Blow a strong 
 bulb about an inch in diameter on a strong piece of 
 thermometer-tubing; cut the tube off four inches 
 from the bulb ; let the weight of the mercury in the 
 bulb and tube be W, and observe the tempera- 
 ture of the air, t, which is the same as that of the 
 mercury. Suspend the bulb in a large beaker, so 
 
202 NEW PHYSICS. 
 
 that the end or mouth of its tube projects over the 
 side of the beaker. Then heat the water in the 
 beaker to 75° C. or 80° C. (call this temperature 
 7"°) and catch the mercury that is expelled by ex- 
 pansion from the bulb. Weigh this amount of mer- 
 cury, and call it zv. If we now neglect the expan- 
 sion of the glass vessel, we perceive that the weight 
 W—w of mercury that is left in the bulb at t°, be- 
 comes at T° of the same volume as W. Therefore, 
 W— w, on heating from t to T°, will increase by 
 
 w 
 
 w. One volume will therefore increase^T? , and 
 
 W— w 
 
 w 
 w 
 
 through one degree will expand by ™ — —5 of -ht— 
 
 or TF5 To rjy This is the co-efficient of ap- 
 
 I — t IV — w. ^ 
 
 parent expansion of mercury in a vessel made of this 
 kind of glass. Let us call this co-efficient A. We 
 shall find, from tables of expansion, that the co-effi- 
 cient of real expansion of mercury is 0.0018 { = B). 
 Hence we have B — A ^ co-efficient of expansion 
 of the glass. This instrument is called a weight- 
 thermometer. 
 
 Experiment no. — Determine the co-efficients of 
 the absolute expansion of alcohol, between the tem- 
 perature of the room and that of ice. 
 
 Weigh the weight-thermometer filled with the liq- 
 uid at the temperatures 0° and T°. Call the weight 
 of the liquid at T° = W, and the loss of weight at 
 T"^ =:w. We then have the apparent co-efficient of 
 
 w 
 expansion C = 777™. We must add to this the co- 
 efficient of the glass. A further study of the phe- 
 nomena of heat leads us to the modern views of 
 
WORK AND HEAT. 
 
 203 
 
 molecular motion. In order to understand new 
 theories, it is necessary to study first what has been 
 well established. We shall, therefore, not dwell 
 upon molecular theories, but proceed to show the 
 relation between electrical work and heat and 
 mechanical work, leaving to higher and more ex- 
 tended treatises the exposition of molecular theo- 
 ries. 
 
CHAPTER XI. 
 
 ELECTRICAL WORK AND ELECTRICAL MEASURE- 
 MENTS. 
 
 We began our study of motion with the law of 
 attraction of gravitation, and we have found that 
 the attraction between magnetic poles and between 
 electrified bodies, or electrical currents, is expressed 
 by a law similar to that of the attraction of gravita- 
 tion. 
 
 In order to study the relation between mechani- 
 cal work and electrical energy, we employ an in- 
 strument called a galvanometer. An instrument 
 has been used in Experiment 7i^not to measure 
 an electrical current, but merely to indicate one. 
 The instrument described in Experiment 71 is 
 not suitable for the measurement of electrical cur- 
 rents ; for it does not follow accurately the law of 
 tangents. We have shown, in Chapter X, that the 
 strength of a current passing through n turns of 
 wire coiled upon a circular hoop is measured by 
 the tangents of the deflection of a small needle 
 placed at the center of the coil thus formed ; or, 
 
 •S' = tan a, in which a is the mean radius of the 
 
 21771 
 
 coil. T is the horizontal component of the earth's 
 magnetism, n the number of turns, and tt = 3.1416. 
 
ELECTRICAL WORK. 
 
 205 
 
 Such a tangent galvanometer can be very easily made 
 by winding five to ten turns of No. 20 insulated 
 wire upon a small hoop or wooden circle, such as 
 the cover of a circular box, twelve to fifteen inches 
 in diameter ; fastening this hoop vertically to the 
 end of a box ; suspending a little magnet, provided 
 with a plane mirror, by means of a silk fiber, at the 
 center of the circle, and covering the circle with a 
 plate of glass. A little lens, placed in front of the 
 magnet, can be used to bring a spot of light to focus 
 upon a piece of oiled paper or ground glass, accord- 
 ing to the method described in Experiment 84. Rub- 
 ber bands will serve to keep the plate of glass in 
 position. The ends of the wire can be brought 
 to the end of the box at B B', Fig. 98. The box 
 can be made steady by loading it inside. 
 
 -Myjy]^ 
 
 Fig. 98. 
 
 Fig. 99. 
 
 It is often desirable to be able to read the deflec- 
 tion of the magnet directly in degrees. For this 
 purpose a cheap galvanometer can be constructed 
 by placing the coil vertical in the middle of a box 
 (Fig. 99) ; running a strip of board along its horizon- 
 tal diameter, boring a hole in the middle of this 
 
2o6 NEW PHYSICS. 
 
 board, inserting a glass tube in this hole, suspending 
 a magnet, with a pointer at right angles to it, by 
 means of a silk fiber which runs through the glass 
 tube, and is attached to a cork in the top of this 
 tube. The pointer of the magnet moves over a 
 graduated circle (of paper or of metal) which is 
 mounted directly beneath the magnet. The box is 
 then covered with two plates of glass, one on each 
 side of the wooden strip which carries the suspen- 
 sion. The ends of the coil are brought to the sides 
 of the box at ^5*.* 
 
 Tangent galvanometers are usually made with 
 only comparatively few turns of wire in a coil of 
 large radius. The magnetic needle is made very 
 short in order that its movements may be confined 
 to the small space at the center of the galvanometer- 
 coil, where the field of force due to the coil is uni- 
 form. When a sensitive instrument is desired, which 
 shall merely detect a feeble electric current, a gal- 
 vanometer similar to that employed in Experiment 
 71 can be employed without the stationary magnets. 
 Another little magnet should be hung upon the 
 suspension, directly above the central magnet and 
 outside the coils. The north pole of this outside 
 magnet should be over the south pole of the lower 
 magnet, as represented in Fig. 100, in which C rep- 
 resents an end-view of the coils. This arrangement 
 is called an astatic system. The influence of the 
 earth's magnetism is very feeble, since it acts in oppo- 
 site directions upon the two magnets, and the lower 
 magnet is ready to turn under the influence of feeble 
 currents. The magnets should be rigidly connected 
 
 * The figure shows only the half of the coil which is outside the box. 
 
ELECTRICAL WORK. 
 
 207 
 
 together. Two sewing-needles, carefully magnet- 
 ized and stuck through a piece of stiff straw, a b, will 
 answer. A fiber of silk, 
 without torsion, can 
 then be attached at a. 
 By suitably increasing 
 the turns of wire upon 
 the coils C, the instru- 
 ment can be made very 
 sensitive. We shall call 
 it an astatic galvanome- 
 ter, to distinguish it from 
 a tangent galvanometer. 
 
 The metals which exhibit the peculiar property 
 of magnetism are iron, steel, cobalt, and nickel. 
 The earth itself can be likened to a great magnet. 
 The position of the magnetic poles of the earth 
 changes during a period of years. The angle which 
 the compass-needle makes with the geographical 
 meridian is called the angle of declination. This 
 varies with the place and the time. Every piece 
 of iron and steel is more or less magnetic. Any ob- 
 ject made of the magnetic metals is magnetized to a 
 certain degree by the earth's influence. 
 
 Experiment i i i. — Examine the magnetic condir 
 tion of a soft iron bar by presenting its ends to a gal- 
 vanometer or compass-needle. Then hold it in the 
 direction of the dipping-needle (see Experiment 83), 
 strike it a sharp blow with a hammer, and examine 
 its condition again. Repeat the operation, placing 
 the bar in a horizontal position, in a line at right 
 angles to the direction of the compass-needle. State 
 the results of your investigation. 
 
 Direct experiment will show that there is no sub- 
 
2o8 NEW PHYSICS. 
 
 stance which will intercept or cut off the line of mag- 
 netic force. It is true that a piece of iron placed 
 between a compass-needle and a magnet modifies 
 the deflection of the needle, but this is due to the 
 effect of the additional magnets formed by induction 
 in the iron plate. 
 
 Experiment 112. — Study the effect of magnetic 
 induction by placing a magnet at right angles to the 
 magnetic meridian, and observing its effect upon a 
 compass-needle when a sheet of soft iron is moved 
 at regular distances between the pole of the magnet 
 and the compass. Then place the compass inside a 
 hollow cylinder, formed by bending the sheet-iron, 
 and move the outside magnet. State the result of 
 your investigation. 
 
 All bodies, and even gases, are more or less sub- 
 ject to magnetic influences, and are divided into two 
 classes — paramagnetic, like iron and steel ; and dia- 
 magnetic, like bismuth. Substances of the latter 
 class tend to set themselves at right angles to the 
 line joining two strong north and south poles of an 
 electro-magnet. Diamagnetism is supposed to be 
 due to a difference of effect upon the medium in 
 which the substance is suspended, and upon the 
 substance itself. 
 
 AmpSre showed that the magnetism of a steel 
 magnet could be explained by the action of electrical 
 currents, which can be supposed to circulate around 
 the axis of a magnet ; for the action of a magnet 
 can be imitated by a current passing through a 
 wire coiled in the form of a spiral, thus constituting 
 what is called a solenoid. This we have seen from 
 Experiment 89. The direction of the current de- 
 termines the nature of the magnetic poles — i. e., 
 
ELECTRICAL WORK. 
 
 209 
 
 whether north or south. The poles of a solenoid, 
 or of an electro-magnet, can always be determined 
 by means of a small compass. 
 
 If a man were to place himself, in imagination, 
 in the position of a very long, straight wire carry- 
 ing an electric current, the current entering by his 
 head and departing by his feet, a magnet suspend- 
 ed freely before him would turn, under the action 
 of the current, the pole which points north toward 
 his right hand. 
 
 We have shown that an electro-magnet resem- 
 bles a steel magnet. It has two poles, which de- 
 pend upon the direction of the current through 
 the coil. Two parallel coils, therefore, in which 
 the current passes in the same direction should at- 
 tract each other, for the north pole of one will 
 attract the south pole of the other. Two parallel 
 coils in which the current passes in opposite direc- 
 tions will repel each other, for poles of the same 
 name will be brought opposite each other. We 
 conclude, in general, that parallel currents in the 
 same direction attract each other, and parallel cur- 
 rents in opposite directions repel each other. 
 
 We have, therefore, a general relation between 
 electricity and magnetism. We can produce mag- 
 netism by an electrical current. We shall show 
 subsequently that we can produce electricity by 
 the movement of a magnet, and we shall find an 
 equivalence between the electricity produced and 
 the movement which produces it, thus finally bring- 
 ing our study of electricity and magnetism back 
 again to the study of motion in general. 
 
 In the first place, it may be stated as a general 
 law that any change in the relative position of the 
 
210 NEW PHYSICS. 
 
 particles of two different substances is accompanied 
 by a manifestation of electricity. We can think, 
 to fix our minds, upon a certain modification of the 
 forces of attraction between the particles of two 
 different substances which is always accompanied 
 not only by heat, but also by a difference of elec- 
 trical potential. Thus, friction produces this differ- 
 ence of potential ; and the difference of action of a 
 liquid upon two metals dipped in the same liquid 
 modifies the attraction between the particles of the 
 metal, and we have also a modification of the forces 
 of attraction between the molecules of the liquid 
 and the molecules of the metals. We have treated 
 electrical force and magnetic force, together with 
 gravitation force, simply as what are called forces 
 of attraction, the general law of attraction being of 
 the same form ; the force in each case varying in- 
 versely as the square of the distance between the 
 attracting configurations. In proceeding to study 
 magnetic and electrical phenomena, we shall see 
 that we are still studying potential and kinetic ener- 
 gy, due to the force of attraction. Although we do 
 not know what magnetism is and what electricity is, 
 we must also reflect that we do not know what 
 gravitation is. We can, however, study the changes 
 or transformations of energy which take place, and 
 the relations between these various manifestations of 
 attraction. 
 
 We have already seen that a current of elec- 
 tricity can be considered as the evidence of the fall 
 from one electric level to another, and is the evi- 
 dence of work being performed. A difference of 
 electrical level can always be produced by means 
 of two different metals, or, in general, by means of 
 
ELECTRICAL WORK. 211 
 
 any two substances brought in contact with each 
 other, either directly, or by means of a liquid or 
 liquids. Whenever the forces of attraction be- 
 tween the particles of two dissimilar bodies are 
 modified in any way, a difference of electrical level 
 is produced. 
 
 Experiment 113. — Place a strip of sheet-zinc, 
 about eight by four centimetres, in acidulated water 
 formed by adding one part of sulphuric acid to 
 eight of water. Bubbles of hydrogen gas will col- 
 lect on the zinc, and the zinc v/ill be gradually con- 
 sumed. 
 
 Place a copper strip, of the same size as the zinc 
 strip, also in the liquid, but not touching the zinc, 
 and note what happens. 
 
 Now connect the zinc and copper strips outside 
 the liquid by bending them until they touch. Ob- 
 serve what happens. Weigh the zinc and copper 
 electrodes or plates after an interval of an hour. 
 
 Amalgamate a zinc plate by carefully cleaning 
 it first in acidulated water, and rubbing mercury 
 upon it with an old tooth-brush. Replace it in the 
 water, opposite the copper, and observe what hap- 
 pens. 
 
 Weigh the amalgamated zinc before it is put in 
 the acidulated water in company with the copper, 
 and after an hour has elapsed. (In weighing, be 
 careful that the amalgamated zinc does not come in 
 contact with the metal of the balance-pans ; place a 
 piece of paper beneath it.) 
 
 Experiment 114. — Connect the zinc with the 
 copper by means of the terminals of a galvanometer. 
 The electrical current will be shown by a deflection 
 of the magnetic needle. 
 
212 NEW PHYSICS. 
 
 The number of different forms of galvanic cells 
 is very great. The following are those which are 
 generally used in laboratories : 
 
 Experiment 115. — Make a gravity-cell in the fol- 
 lowing manner : Coil one end of a bare copper wire 
 into a flat spiral (Fig. loi). Make the plane of this 
 spiral perpendicular to the straight portion of the 
 wire. Cover this straight portion with some liquid- 
 proof insulator, like gutta-percha and tar, except at 
 
 its point of connection with 
 
 t the spiral. The spiral, which 
 IHJ^I is merely to obtain consid- 
 ^9 erable surface of copper, is 
 ^H then laid at the bottom of 
 |jH| a glass vessel, which can 
 VMM be made by cutting off the 
 ^g^ top of an ordinary bottle. 
 ^^ The spiral is then covered 
 
 Fig. ioi. with Crystals of sulphate of 
 
 copper to the height of two 
 or three inches, and the vessel is filled with water. 
 In the water, about four inches from the layer of 
 copper, is hung a piece of zinc with a suitable con- 
 nection for attaching a wire to it. After a while 
 the sulphate of copper dissolves, and sulphate of 
 zinc is also formed. The difference in specific grav- 
 ity of the two liquids keeps them separated. In- 
 stead of the zinc, one can use a piece of iron. In 
 this case it is best to partially fill the jar with sul- 
 phuric acid and water — one part of acid to twelve 
 of water — and afterward drop into this crystals of 
 sulphate of copper. 
 
 The Daniell cell, which is most commonly used 
 for delicate electrical measurements, differs from 
 
ELECTRICAL WORK. 
 
 213 
 
 the gravity-battery in having the piece of zinc and 
 the surrounding sulphate of zinc separated from 
 the sulphate of copper by a porous jar. The zinc 
 is carefully amalgamated by cleaning it with dilute 
 sulphuric acid and immersing it in mercury, to 
 prevent unequal action of the sulphuric acid upon 
 the zinc. , 
 
 The Bunsen cell consists of an amalgamated 
 cylinder of zinc immersed in sulphuric acid and 
 water (one part of acid to twelve or fourteen of 
 water), and a stick of carbon immersed in concen- 
 trated nitric acid contained in a porous jar, the 
 porous jar being placed inside the zinc cylinder. 
 This cell is very objectionable, on account of the 
 fumes of nitrous acid which arise when the cell is 
 working. These fumes quickly injure all apparatus 
 made of brass, and also tools. It is usual, there- 
 fore, to replace the nitric acid by a saturated solu- 
 tion of bichromate of potash and a certain amount 
 of sulphuric acid. The ordinary proportion is : 
 two ounces of bichromate of potash dissolved in a 
 quart of water, to which is added two fluid ounces 
 of strong sulphuric acid. 
 
 The Leclanche cell is in common use for ringing 
 bells and for the telephone service. It consists of 
 a cylinder of zinc immersed in a solution of sal- 
 ammoniac, and a bar of carbon in a porous jar, sur- 
 rounded by black oxide of manganese and pow- 
 dered carbon. It is a very inconstant cell, and its 
 strength decreases rapidly when the poles are con- 
 nected by short, thick wire. When it is used, the 
 circuit should be closed only for a moment. 
 
 Experiment 116. — Make a water-battery. This 
 consists of zinc and copper elements immersed in 
 
214 
 
 NEW PHYSICS. 
 
 pure or distilled water. Obtain a gross of small 
 bottles of a druggist. Two and a half mches in 
 height, with a mouth one inch in diameter, is a con- 
 venient size. Dip the mouths of the bottles in 
 melted paraffine to a distance of one inch, and place 
 them in a box provided with a cover, to keep put 
 the dust. Solder a strip of zinc four inches broad 
 to a strip of copper of the same breadth. Cut the 
 united strips into small couples like C Z, Fig. 102. 
 Bend these couples and place the zinc end in one 
 
 cz 
 
 c z 
 
 1 
 
 [] 
 
 (] 
 
 Fig. 102. 
 
 bottle and the copper in the adjoining one, and so 
 on. A large number of these cells can be quickly 
 and cheaply made, and a great difference of electrical 
 potential can thus be obtained. From eighty to one 
 hundred of these small cells will serve to charge 
 Coulomb's balance (Experiment 88), and also the 
 electrometer (Experiment 94). To employ the bat- 
 tery for this purpose; connect the last copper strip 
 with the inside of the Leyden-jar, and the last zinc 
 strip with the outside of this jar. This use of a 
 water-battery shows that the poles of a battery 
 have opposite electrical charges — one positive and 
 the other negative. 
 
 We have seen that we can measure the strength 
 of an electrical current by the expression 5 = 
 
ELECTRICAL WORK. 
 
 215 
 
 Ta 
 2ir n 
 
 tan a. The question arises, How can we meas- 
 
 ure the work which a current of strength 5 does ? 
 It is not difficult to show that a wire becomes heat- 
 ed through which a current of electricity passes. 
 
 Experiment 117. — Connect four Bunsen cells in 
 series — that is, the carbon of one cell with the zinc of 
 the next, and so on (Fig. 103). Make the conductor 
 which connects the last zinc and the last carbon of 
 thick copper wire, except at one place, where the 
 ends of the thick copper wire are connected with 
 a very fine platinum wire about ten centimetres in 
 length. On making the 
 circuit, the platinum 
 wire will glow. Place 
 it in a small copper ca- 
 lorimeter similar to 
 those which were used 
 in the determination of 
 specific heat, taking care 
 that the platinum wire 
 does not touch the sides of the calorimeter. Fill the 
 vessel with alcohol and place a thermometer in it. 
 Place another thermometer in the liquid of the bat- 
 tery, next the carbon element. In a short time both 
 of these thermometers will indicate heat. It will 
 be found that whenever, by changing the length of 
 the platinum wire, the heat is decreased in the cell, 
 it is increased between the poles of the battery — 
 or, in other words, in the calorimeter. The short, 
 thick copper conductors are not sensibly heated. 
 
 This heat is an evidence of work, and has its 
 equivalent in work, which in the case of a galvanic 
 battery is due to the difference of potential created 
 
 Fig. 103. 
 
2i6 NEW PHYSICS. 
 
 in some unknown way between two different metals. 
 The amount of heat developed in the circuit out- 
 side the battery depends upon what is called the 
 resistance of the circuit. This consists of the ex- 
 tent of liquid through which the electrical action 
 takes place in the cells, and also of the length and 
 section of the interpolar wire. In order to obtain 
 the work of an electrical current we must first ob- 
 tain an idea of electrical resistance. 
 
 Experiment ii8. — Arrange a galvanic cell so 
 that the resistance of the cell can be neglected. In 
 a vessel containing bichromate of potash immerse a 
 plate of zinc and a plate of carbon at a certain dis- 
 tance from each other, and connect them by a cer- 
 tain interpolar wire of which the wire of a galva- 
 nometer forms a part. Place the galvanometer so 
 that the plane of the coil is vertical and the mag- 
 netic needle is in the plane of the coil. The strength 
 
 of the current will be expressed by 5 = tan a. 
 
 When the interpolar wire is long, it will be found 
 that the distance of the plates in the liquid can be 
 doubled without affecting the strength of the cur- 
 rent, as is indicated by the galvanometer. We there- 
 fore conclude that this liquid resistance can be 
 made inappreciable in our experiment by experi- 
 menting with considerable interpolar resistance, in 
 comparison with which the battery or liquid re- 
 sistance can be neglected. 
 
 Experiment 119. — Therefore, taking suitable 
 lengths of copper wire, interpose them in a circuit 
 consisting of a battery and a galvanometer, and 
 observe the relation between the strength of the 
 currents and the length of the wires. It will be 
 
ELECTRICAL WORK. 
 
 217 
 
 found that the tangents of the angles of deflection 
 are inversely as the length of the wires of the same 
 section. This shows that — 
 
 R' 
 
 s 
 
 Ta E 
 
 = tan a = — 
 
 27r« R 
 
 
 Ta ^ . E 
 
 tan a^ 77 
 
 2 7r« i?i 
 
 T n 
 
 for the factor is constant for the same sral- 
 
 2irn ^ 
 
 vanometer, and for the same magnetic field of the 
 
 earth. It will, therefore, cancel out when we take 
 
 the ratio of the strength of the currents. We know 
 
 also that the expression for the strength of the 
 
 current must be a constant quantity E divided by 
 
 the resistance, for the ratio between the currents is 
 
 that of the inverse ratio of the length of the wire. 
 
 The constant factor is called the reduction fac- 
 
 2'irn 
 
 tor of the galvanometer, because it enables us to re- 
 duce our expression for the strength of an electrical 
 current to absolute units — that is, to distances in cen- 
 timetres, and to weights in grammes. An observer 
 at any point on the earth's surface, knowing the re- 
 duction factor of his galvanometer, can compare his 
 results with those of an observer situated at some 
 other point of the earth's surface. The constant 
 factor E evidently depends upon the difference of 
 potential of the particular cell that is employed. 
 The strength of a current, therefore, we can con- 
 
 E 
 elude is expressed as S = -5, in which the resist- 
 ance of the battery is neglected in comparison with 
 the interpolar resistance, and the resistance of the 
 
2i8 ^EW PHYSICS. 
 
 interpolar wire depends merely upon its length, for 
 we have supposed that its section is uniform. 
 
 Experiment 120. — Repeat the previous experi- 
 ment, keeping the length of a wire constant, but 
 using wires of different cross-section. We shall find 
 that the greater the section the greater the current, 
 
 or -FT = = — , in which r and r, are the radii 
 
 Si tan Oi Ti 
 
 of the cross-sections. 
 
 Hence, we conclude that the strength of the cur- 
 rents varies inversely as the resistances interposed 
 in the circuit, and that these resistances vary di- 
 rectly as the length of the wire, and inversely as 
 
 their section ; or, the resistance R = C—, C being a 
 
 constant factor depending upon the kind of wire 
 that is used. In obtaining the ratio between the re- 
 sistances of any lengths of the same kind of wire, it 
 is evident that this factor will cancel out. If wires 
 of different metals are employed, this factor will be 
 different for the different metals. It is called the 
 specific resistance of a metal. In order to obtain a 
 standard by means of which we can estimate spe- 
 cific resistance, we must determine what our unit 
 of resistance shall be. It will be remembered that, 
 in order to determine specific gravity, we had to 
 know the weight of a cubic centimetre of water at 
 0° C, and in order to determine the specific heat of a 
 metal we were obliged to know the amount of heat 
 necessary to raise a gramme of water 1° C. In the 
 same way we must base specific resistance upon 
 some unit which will be universally recognized. 
 The simplest unit of resistance is that proposed by 
 Siemens. It consists of a metre-length of mercury 
 
ELECTRICAL WORK. 
 
 219 
 
 contained in a tube one square millimetre in sec- 
 tion. This is called Siemens's mercury-unit. Its 
 value is -^ of the B. A. unit, or British Associa- 
 tion unit, which is the one usually adopted.* The 
 explanation of how the B. A. unit is measured is 
 somewhat difficult, and we shall defer it until we 
 have seen how resistances in general can be meas- 
 ured ; and how resistance finally can be expressed 
 in absolute measure — that is, in units of space, time, 
 and mass. Our final object is to connect the phe- 
 nomena of electricity and magnetism with those of 
 motion and of heat, 
 and to illustrate 
 the conservation of " 
 energy. 
 
 Experiment 
 121. — Experiment 
 1 1 7 shows that the 
 amount of heat de- 
 veloped in an elec- 
 trical circuit de- 
 pends upon the 
 length of wire or 
 resistance. In or- 
 der^ therefore, to 
 determine the 
 amount of heat, 
 and in this way 
 
 measure the mechanical equivalent of it, we must 
 be able to determine the resistance of the cir- 
 cuit. The simplest method is by what is called 
 
 * At the Electrical Conference held in Paris, April, 1884, a column 
 of mercury 106 centimetres in length and one square millimetre in sec- 
 tion was adopted as the ohm. 
 II 
 
 Fig. 104. 
 
220 
 
 NEW PHYSICS. 
 
 the Wheatstone bridge. Arrangie a parallelogram, 
 
 A B CD (Fig. 104), of uniform copper wire, in the 
 
 manner shown in the figure, leading the poles of 
 
 the battery to the points A and C, and attaching 
 
 the wires of the galvanometer at B and D. It will 
 
 be found by experiment that the length A B will be 
 
 A B B C 
 to the length A D 2i^ B C'xsto D C, or -j-^ = fij; or 
 
 BC 
 
 AB 
 
 DC 
 
 A D. If, therefore, we knew the elec- 
 
 trical resistance of A D, we could obtain that of 
 A B, for the ratio of the resistance of B C io D C\s 
 evidently the same as that of their length, since the 
 section of the copper wire is everywhere the same ; 
 and these can be readily measured. The resistance 
 ol A D can evidently be a Siemens mercury-unit, 
 and we can ohtdiin A B in terms of it. 
 
 Fig. 105. 
 
 Experiment 122.— A simple Wheatstone bridge 
 can be constructed as follows: Screw three flat 
 pieces of brass or copper plate, C, D, and E, to a 
 board, as in Fig. 105. Solder a German-silver wire, 
 S 5S to C and E. Provide binding-screws at m, n, 
 
ELECTRICAL WORK. 22 1 
 
 X, o, and /. One pole of the battery is attached 
 
 permanently at the point x, while the other, by 
 
 means of a block of wood sliding on a guide, G G, 
 
 can be made to touch 5 5* at any point. When no 
 
 current flows through the galvanometer G, we have 
 
 SI IS^ SI R ^^ 
 
 -^ = ^— , or j^ = ^. Knowmg the ratio of 5 / 
 
 to /5^ and knowing also the value of R^, we can ob- 
 tain R. 
 
 A table of the electrical resistance of copper 
 wire of various diameters is given in the Appendix. 
 From this table a set of resistance-coils can be con- 
 structed which will be accurate enough for ordi- 
 nary experiments. When great accuracy is re- 
 quired, these resistances must be compared with 
 some standard B. A. coils. Other resistances, based 
 upon these calculated ones, can then be made by 
 the Wheatstone bridge. 
 
 Experiment 123. — Make a set of resistainces. 
 Solder a piece of copper wire of large diameter to 
 one end of a fine copper wire. Measure a metre of 
 the wire and interpose it as a resistance to be meas- 
 ered in the Wheatstone bridge. Having measured 
 the required resistance, solder a large copper wire 
 to the end of the wire which you cut, so that both 
 ends of the resistance may now be provided with 
 these large wires, which are useful for connecting 
 with the binding-cups. Wind these resistances upon 
 spools, and slip pieces of rubber tubing over the 
 spools, in order to confine the wire, ft is useful to 
 obtain an idea of the average resistance of the wire 
 you are employing to make these resistances, so that 
 you may not cut the wires unnecessarily short or 
 long. 
 
222 
 
 NEW PHYSICS. 
 
 Experiment 124. — Instead of placing the galva- 
 nometer between B and D, as is usual in the Wheat- 
 stone bridge, place it between A and D (Fig. 106) 
 
 as a resistance to be de- 
 termined, and put a key 
 in its place between B 
 and D. Obtain the ratio 
 
 AD DC ^^, ... 
 
 = When this 
 
 AB BC. 
 
 ratio is true, the deflec- 
 tion of the galvanometer 
 should not change when 
 the circuit BD is made or 
 broken. Therefore the re- 
 sistance of the galvanom- 
 
 DC 
 eter, or AD=——,AB. 
 Fig. 106. B 6 
 
 It is generally necessary 
 
 to place a magnet outside the galvanometer to 
 bring the deflection of the needle of the latter 
 within range (if the galvanometer is a reflecting 
 instrument). 
 
 The electrical resistance, R, of a cylindrical con- 
 ductor is directly proportional to its length, /, and 
 
 inversely to its sectional area, q j ov, R ^ k — The 
 
 factor k is called the specific resistance of the con- 
 ductor. The specific resistances of various bodies 
 can be conveniently referred to that of Siemens's 
 mercury-unit, which consists of a metre-length of 
 mercury, one square millimetre in section, at 0° 
 C. The specific resistance of this unit at this 
 temperature is taken as i. Table VI, Appendix, 
 gives the galvanic resistances of the common met- 
 
ELECTRICAL WORK. 223 
 
 als compared with that of a column of mercury 
 at 0° C. 
 
 Experiment 125. — Determine the specific resist- 
 ance of a length of copper wire at the temperature 
 of boiling water. 
 
 An increment of temperature of 1° C, at a mean 
 temperature, produces an increased resistance in 
 pure, solid metals of about 0.4 per cent — German- 
 silver about 0.04 per cent, and mercury about 0.08 
 per cent. 
 
 Example. — The resistance of five metres of pure 
 copper wire 0.4 of a square millimetre in section, at 
 
 0° = i— — '- = 0.2 Siemens, and at 10° C, 0.2+ 
 
 0.4 
 
 0.2 X 0.004 X 10 = 0.208 Siemens. 
 
 The length of copper wire should be attached 
 to large connecting pieces of copper, the resistance 
 of which can be neglected, and immersed in a beaker 
 of water which had first been raised to the boiling- 
 point. Ascertain how nearly the actual resistance 
 of a piece of copper wire agrees at the temperature 
 of 100° C. with the calculated value. 
 
 The arrangement called Wheatstone's bridge, 
 after its inventor. Sir Charles Wheatstone, can be 
 
 Fig. 107. 
 
 explained by the theory of the fall of potential. 
 Let us recur to the analogy of the engine pumping 
 water, and suppose that a pipe should connect some 
 
224 
 
 NEW PHYSICS. 
 
 point, A (Fig. 107), on the main through which the 
 water is rushing from the engine E, with some point 
 B, another branch of the main through which the 
 water also issues. If the pressure at A is the same 
 as the pressure at B, or, in other words, if there is 
 no difference of level between A and B, or no differ- 
 ence of potential, there will be no flow from A to B, 
 or from B to A. A small float placed in the water 
 which fills the pipe A B, would be stationary wher- 
 
 Groxmcl 
 
 . A'i 
 
 1 
 
 QrounA 
 
 ever it is placed between 
 A and B. The moment, 
 however, that B is moved 
 from this position of equi- 
 librium — A remaining the 
 same — there will be a flow 
 through A B, since there 
 Fig. 108. is now a difference of level. 
 
 We must not infer from this 
 analogy that there is a flow of electricity like that 
 of water. This analogy, however, leads us to try 
 the following experiment : 
 
 Experiment 126. — We have seen that there is a 
 difference of potential between the two poles of a 
 battery. We should therefore expect to find a dif- 
 ference of electrical level between the zinc pole Z 
 (Fig. 108), which is connected with the ground, and 
 
ELECTRICAL WORK. 
 
 225 
 
 any point A along the conductor which connects the 
 positive pole C with the ground. With the elec- 
 trometer which is used in Experiment 94, test the 
 difference of potential between the zinc pole Z and 
 any point, such 2& A. To do this, connect one set 
 of the quadrants with Z and the other with the 
 point A, and notice the swijig of the needle. Lay 
 off perpendicularly to A with any convenient unit, 
 the distance A D equal to the deflection of the 
 electrometer, laying off also the distances A C, A' C, 
 equal to the lengths of wire or resistances between 
 the pole C and the points A, A', etc. On connect- 
 ing the points D, D' , etc., it will be found that they 
 will lie on a gradient which rises from the point 
 where the line is grounded at E to the positive pole 
 of the battery. It is evident, therefore, that this 
 method would enable us to discover a complete 
 break in a cable ; for, having obtained several points 
 on the gradient, such as D and Z>', on each side of 
 certain resistances interposed between the end of the 
 cable and the positive pole of the battery, we can 
 lay off these resistances and deflections on a suitable 
 scale on paper, and extend the line of gradient until 
 it strikes the point on the extended line of resistance, 
 or cable, where the contact with the water, or, in 
 other words, the earth, occurs. This experiment 
 requires five or six cells. 
 
 The fall of electrical potential can also be shown 
 by means of an electrical machine. Descriptions 
 of the many forms of electrical machines can be 
 found in any treatise on natural philosophy. For 
 our present purpose we merely wish to show the 
 close analogy between the kind of electricity pro- 
 duced by the motion of the hand in turning such a 
 
226 
 
 NEW PHYSICS. 
 
 machine, and the electricity produced by the mere 
 contact of dissimilar metals in a battery. 
 
 Connect the terminals of the machine (Fig. 109) 
 with the ground. Interpose certain resistances at 
 A, and charge a Leyden-jar, similar to that which 
 
 GROUND 
 
 forms a part of the elec- 
 trometer, Experiment 94 
 (for this purpose it is better to 
 substitute an internal coating of 
 tin-foil for the sulphuric acid). To 
 charge the jar, connect the internal coating with one 
 pole — the negative — of the machine, and the other 
 with a point such as ^. If we should now connect 
 the charged coatings of this Leyden-jar with our 
 electrometer terminals, the needle of the instrument 
 would receive too great an impulse. We shall, there- 
 fore, have to connect one set of the quadrants with 
 a metallic plate, which is mounted on a glass insulat- 
 ing stand, readily made from a piece of glass tubing 
 and a wooden base. Another exactly similar insu- 
 lated metallic plate is placed directly opposite the 
 first, at a suitable distance, this distance being regu- 
 lated by the effect upon the electrometer. In this 
 way the charge of the metallic plate connected with 
 the Leyden-jar induces a charge upon the opposed 
 metallic plate, which is proportional to the charge 
 
ELECTRICAL WORK. 
 
 227 
 
 of the Leyden-jar, and also to the difference of poten- 
 tial between one pole of the electrical machine and 
 the point touched upon the line running from the 
 other pole to the earth. Laying off the distances 
 and deflections, as in the experiment with the bat- 
 tery, we find the same law of fall. On connecting 
 the terminals of a Leyden-jar with each other, we 
 perceive that a spark passes. We know in this case 
 that a difference of potential exists in the charged jar, 
 and we perceive, by the light and heat (for this spark 
 can ignite gunpowder), that work is done by the 
 sudden fall of this potential. If we had the means 
 of measuring the heat of the spark, we could see 
 whether the mechanical work done in producing 
 the eleclrical difference of potential is equivalent 
 to the heat produced. Let us now measure the 
 heating effect of an electrical current produced by 
 an ordinary battery. We have shown that this 
 heating effect depends upon the resistance of the 
 circuit. It is proved by direct experiment that 
 the heating effect, H, is equal to the square of the 
 current, S, multiplied by the resistance, i?, of the 
 circuit, and multiplied by the time, t, or H = 
 
 Experiment 127. — To prove this, connect four 
 Bunsen cells, or their equivalent, by thick copper 
 strips, interposing a galvanometer, and certain^ 
 lengths — from three to ten centimetres — of fine 
 platinum wire, of which the resistances have been 
 measured. These lengths of platinum wire are 
 immersed in a calorimeter filled with distilled 
 water or alcohol (if alcohol is used its specific heat 
 must be known, or the amount of heat absorbed 
 by the alcohol compared with that of an equal 
 
228 NEW. PHYSICS. 
 
 weight of water). If we call the weight of the 
 water heated W, and ft its temperature, we shall 
 
 find — = = — s^; in which we 
 
 /IVA mHan'a, Ri( S^'Rit, 
 
 have called the mechanical equivalent of heat // m, 
 the reduction-factor of the galvanometer employed 
 (whose coil should be one or two turns of copper wire 
 about twelve centimetres radius), and a and Oj, the dif- 
 ferent deflections with different lengths, R and Ri, of 
 wire heated, i and ii being the times during which 
 the current runs. In proving this, it is evident that 
 we do not need to know the mechanical equivalent of 
 heat,y,or the units in which we estimate the strength, 
 S, of the current, or the unit of resistance, since the 
 numerical values of these units would cancel out in 
 the above ratio. If we knew the value of the unit of 
 resistance expressed in terms of the units by which 
 we express motion, we could obtain the value of the 
 
 r w ii 
 current in the same absolute currents, for 5^= — - — 
 
 Rt ' 
 
 in which everything is expressed in absolute units. 
 
 Moreover, since 5' = --5 (Ohm's law), we have 
 
 R 
 
 B JWh E-2 JWhR xj ^,. 
 
 -s-=^;— ,ox E?^^± trom this we per- 
 
 R* Rt t ^ 
 
 ceive that we can also obtain the electro-motive force 
 E in absolute units if we know the unit of resist- 
 ance. Conversely, if we could measure the unit of 
 resistance exactly, we could obtain the mechanical 
 equivalent of heat,y. To express, therefore, our 
 electrical measurements in the same units that we 
 have used in the expression of mechanical work and 
 its equivalent heat, we must know the value of the 
 
ELECTRICAL WORK. 229 
 
 electrical unit of resistance. To obtain an idea of 
 the method employed, let us first see what effect will 
 be produced by moving a conductor, of which the 
 ends are connected, in a magnetic field. We shall 
 take, in the first place, the magnetic field of the 
 earth. 
 
 Experiment 128. — Wind around an ordinary 
 wooden hoop a number of turns of moderately fine 
 insulated copper wire, tying the wire at certain inter- 
 vals to the hoop by strings. Leave the ends of this 
 wire long, and connect them with an astatic galva- 
 nometer of low resistance. Lay the hoop horizon- 
 tally upon a table and turn it quickly completely 
 over, so that it may rest upon what was before its 
 upper side. The needle of the galvanometer will be 
 deflected, thus showing that an electrical current is 
 produced by the motion of the conductor in the 
 earth's magnetic field. Notice how much this de- 
 flection is, then place the hoop vertical, and perpen- 
 dicular to the direction of the magnetic needle. 
 Turn the coil quickly around a vertical axis, through 
 180°, and note the deflection. By the first operation 
 we have cut the vertical lines of the earth's magnet- 
 ism twice (each half-circumference cutting them all), 
 and by the second we have cut the horizontal lines 
 twice. We shall find that if we divide the deflec- 
 tion produced by cutting the vertical lines of force 
 by a given coil by that obtained from cutting the 
 horizontal lines of force by the same coil, we shall 
 obtain the tangent of the angle of dip, or the 
 inclination of the total magnetic force to the hori- 
 zon. We can conclude from this that the strength 
 of a magnetic field can be measured by cutting 
 its lines of force by a coil of a known number 
 
230 
 
 NEW'PHYSICS. 
 
 of coils, and a known fadius. That different coils 
 produce different effects, wlien revolved in the above 
 manner in the earth-fields, can be readily learned 
 from experiment. There must, therefore, be a 
 constant, vsrhich belongs to each coil. To obtain 
 the strength of any magnetic field, proceed as fol- 
 lows : 
 
 Experiment 129. — Make a small bobbin of fine 
 insulated copper wire (twenty turns on a circle of 
 wood about one centimetre in diameter). Connect 
 the terminals of this coil with an astatic galvanom- 
 eter .of small resistance, and place the coil parallel 
 to the plane of the pole of an electro-magnet, the 
 area of which is considerably larger than that of 
 the circle of the small bobbin. Quickly reverse 
 (see Appendix) the current in the electro-magnet, 
 and notice the swing of the galvanometer-needle. 
 Afterward, suddenly revolve the bobbin through 
 180° around an axis perpendicular to the axis of the 
 electro-magnet, while the current is constantly flow- 
 ing through the electro-magnet. It will be noticed 
 that the induction-currents are the same in both 
 cases. Hence, the effect produced by reversing the 
 current is the same as cutting the lines of mag- 
 netic force twice by the bobbin. This shows us 
 how we can refer our measures of the strength 
 of a magnetic field to the strength of the earth's 
 field, and therefore to absolute measure. Inclose 
 in the circuit of the small bobbin a large hoop cov- 
 ered with twenty turns of insulated wire. Mount 
 this hoop vertically, with its plane east and west. 
 Turn it quickly through 180°. The induction-cur- 
 rent thus produced will be due to cutting the 
 horizontal lines of force of the earth's magnetism 
 
ELECTRICAL WORK. 23 1 
 
 twice ; for each semi - circumference cuts them 
 through 180°. 
 
 The operation of turning a coil of a certain ra- 
 dius and a certain number of turns of wire in the 
 earth's magnetic field, we perceive, will give us 
 an electrical current through a certain resistance. 
 Since we know from Ohm's law that the current is 
 equal to the electro-motive force divided by the re- 
 sistance of the circuit, the resistance of the circuit not 
 being altered when we turn a coil in a magnetic field, 
 the electro-motive force must be proportional to the 
 strength of the field, and to the number of lines of 
 force cut by the revolving coil. If the area in- 
 closed by one turn of wire of a coil is ir I<?, then 
 2irl^ X H, where H is the strength of the field, 
 represents the number of lines of force cut by turn- 
 ing this coil through 180°. If there are n turns of 
 wire in the coil, the number of lines of force cut 
 will he 2 TT I^ x n X H. 
 
 Hence, if we should inclose two coils in the 
 same circuit with a galvanometer, and, placing the 
 plane of the larger of the two horizontal, should 
 quickly turn this through 180°, and, having placed 
 the smaller with its plane perpendicular to the 
 lines of force in another uniform magnetic field 
 like that between two magnetic poles, the lines of 
 force being straight, should also turn this through 
 180°, we should have — 
 
 2irF? y. n x H _a _^ 
 
 2'itR^ X rf- xW^ ~^~ 
 
 number of lines of force of earth inclosed 
 
 number of lines of force of magnetic field inclosed. 
 
 * In this equation a and j8 are the sines of one half the angle of 
 deflection of the galvanometer, which for small deflections can be 
 
232 
 
 NEW PHYSICS. 
 
 Knowing the strength of the vertical component 
 H of the earth's field, we can readily obtain the 
 strength H^ of any other field. 
 
 The method of determining the unit of electri- 
 cal resistance which is called the ohm depends 
 upon a process of cutting the lines of force of the 
 earth. A circular coil of radius K, and including 
 a length L of wire, is rotated around a vertical axis 
 which is perpendicular to the horizontal lines of 
 force of the earth's magnetism. 
 
 In one complete revolution each half of the coil 
 
 will cut the horizontal lines of force twice. We 
 
 can satisfy ourselves by experiment that a certain 
 
 electro-motive force is generated by cutting the 
 
 lines of magnetic force, for we perceive that we 
 
 obtain an electrical current by this means through 
 
 E , 
 
 a. certain resistance. Ohm's law, or 5 = -5, applies 
 
 to this current. The factor R is fixed by the length 
 of wire we use in our experiment. The factor E, 
 or the electro-motive force, can only depend upon 
 the strength of the magnetic force, or, in other 
 words, the number of lines of magnetic force which 
 are cut by the revolution of the coil, and the num- 
 ber of revolutions. Calling the radius of our coil 
 K, it is evident that the number of lines of force 
 which pass through one coil of this radius would 
 be IT K^ H, where H is the strength of the earth's 
 field or the number of lines of force to the unit 
 area. If this coil revolved once, each half of the 
 
 taken as proportional to the swings themselves. If the small coil is 
 
 quickly removed from the magnetic field parallel to itself, the ex- 
 
 , 2, n IP v. n y. H a 
 pression becomes — ^--= ; — — . 
 
ELECTRICAL WORK. 
 
 233 
 
 coil would cut this number of lines of force twice, 
 and therefore the whole coil would cut them four 
 times. If n were the number of turns of the coil 
 in a second, we should, therefore, cut 4X1"^'' 
 y, H X n lines of force. If now we have N coils 
 instead of one coil, and if K is the mean radius 
 of all the coils, each coil will evidently cut the 
 lines of force four times, and, therefore, the whole 
 number will be ^nr K'^ HnN. This is the elec- 
 tro-motive force which produces the current 5 
 through the resistance R, which includes the re- 
 sistance of the galvanometer and the revolving coil 
 and connecting wires. The current will then be 
 
 5 = 5 This current, on the other hand, 
 
 will produce a magnetic field at the center of the 
 revolving coil. It will be noticed, in turning the 
 coil of the preceding experiment from east to west, 
 that the deflection of the galvanometer is in one di- 
 rection, and in the opposite when the revolution is 
 completed from west to east. The current, there- 
 fore, reverses its direction with each half-revolu- 
 tion of the coil. If a compass were placed at the 
 center of a stationary vertical coil, and an electrical 
 current were quickly sent through it, first in one 
 direction and then in another, we find that the nee- 
 dle would remain at rest, for the north pole of the 
 magnet is acted upon alternately by a north and 
 south pole of the electro-magnetic coil constituting 
 the galvanometer, and, being thus solicited by op- 
 posite forces, would remain at rest. Suppose, how- 
 ever, we should hang at the center of the revolving 
 coil, by a fine silk fiber which a large number of 
 revolutions would not twist sensibly, a small mag- 
 
234 NEW PHYSICS. 
 
 net provided with a mirror. If, now, each time the 
 current should reverse, we should accompany this 
 reversal by a corresponding turn of the coil, it is 
 evident that the little magnet would be deflected 
 from the magnetic meridian, and the action of the 
 forces upon this little magnet would be the same as 
 
 in the ordinary galvanometer, or 5 — -p^ — fi I cos <j> 
 
 = H 11 1 sin <^ ; or, substituting the value of .? = 
 
 ^tr K^nHN , 4'irK^nHN 2irKNu, 
 
 , we have X I 
 
 R R K^ 
 
 cos ^ = H iL I sin ^ (calling H the intensity of the 
 
 earth's magnetism, /* the magnetism of the magnet, 
 
 / the half-length of the magnet). In this equation R 
 
 is the only quantity that is unknown. On solving, 
 
 we find n = tan d>,OTR= — ^ — . The 
 
 R ^ tan </) 
 
 whole length of wire, Z = 2 tt KN (that is, N times 
 
 the average circumference of each coil). Hence, 
 
 we can express R in terms of this length, or R = 
 
 2 I^ tl 
 
 ■^77 — -;. It will be seen from this expression that 
 
 the electrical unit of resistance is expressed as a 
 velocity, for n is the velocity of revolution, and 
 
 j^. , IS a co-efficient which can be made unity by 
 
 properly taking the length of wire and the radius 
 of the coil. 
 
 We see, therefore, that from the motion of a 
 wire in a magnetic field of given strength we have 
 the means of obtaining a unit of resistance which is 
 absolute — that is, it depends upon the same meas- 
 ures by which we estimate velocity in mechanics, 
 
ELECTRICAL WORK. 
 
 235 
 
 or is ultimately referred to the motion of a pendu- 
 lum, and therefore to the attraction of gravitation. 
 We know from the preceding that we can also 
 express the unit of electro-motive force in absolute 
 measure ; for suppose we have a rectilinear conduct- 
 or of length /(Fig. no), perpendicular to the lines 
 of magnetic force, and moving in a uniform magnetic 
 field of the intensity H, perpendicularly to the lines 
 of force. Let this wire move along two parallel 
 rails, A B and CD, with a velocity v. By this mo- 
 tion we obtain an electro-motive force E ^ ITv, 
 and, therefore, a current 
 through B G D. This 
 shows that our unit of 
 electro-motive force can 
 be taken as that force 
 which would be induced 
 in a rectilinear conductor 
 of unit length, moving 
 with a unit velocity across 
 a field of unit magnetic 
 strength, in a direction 
 perpendicular to itself 
 and to the lines of mag- 
 netic force. Thus we see that we can also base 
 our electro-motive force upon what we have called 
 absolute measure. If the intensity of the mag- 
 netic field is taken as one unit in the centimetre- 
 gramme-second system of units, the distance be- 
 tween the rails A B and CZ> as i centimetre, and 
 the velocity of the conductor / as i centimetre 
 per second, the electro-motive force or difference 
 of potential produced will be i centimetre-gramme- 
 second unit of electro-motive force. This unit is 
 
 Fig. iio. 
 
236 NEW PHYSICS. 
 
 too small for practical purposes, and the difference 
 of potential produced by such a slider or conduct- 
 or moving with a velocity of 100,000,000 centime- 
 tres per second is taken as a practical unit, and is 
 called a volt. It is very nearly the electro-motive 
 force of a Daniell cell. 
 
 The work to produce an electro-motive force in 
 a closed conductor when it is moved across the 
 lines of magnetic force is very evident to the senses 
 when the field of force is a strong one. It requires 
 a steam-engine of several horse-power to revolve a 
 coil in a strong magnetic field in order to produce 
 the electric light. A consideration of the work 
 done in producing a given current will enable us to 
 understand how the units of electro-motive force 
 and of current are based upon the same units as 
 those we have adopted in mechanics and in heat. 
 In the first place, let us see how far the doctrine of 
 the conservation of energy can already be applied. 
 
 Experiment 130. — Connect the poles of a bat- 
 tery with an ordinary electro-magnet such as is used 
 on telegraphic sounders. Make a small coil of very 
 fine copper wire, similar to those used upon the 
 magnet of the Bell telephone, and connect this small 
 coil with an astatic galvanometer. On quickly 
 bringing one face of this coil to one end of the elec- 
 tro-magnet, a current will be induced in it in one 
 direction ; on removing the small coil along the ex- 
 tended axis of the electro-magnet, a current will be 
 induced in the opposite direction. We know that if 
 the conservation of energy is true in regard to elec- 
 tro-magnetism, the direction of these induced cur- 
 rents should be such as to oppose our motion in both 
 cases, for these currents in the small coil make it for 
 
ELECTRICAL WORK. 
 
 237 
 
 an instant a little electro-magnet having attracting 
 and repelling poles. Suppose that, in moving the 
 small coil toward the north pole of the large station- 
 ary electro-magnet, we should induce a current in the 
 small coil in such direction as to make the end of the 
 small coil toward the north pole of the stationary 
 magnet a south pole ; the attraction between these 
 poles would then draw the small coil to the larger, 
 and would aid our effort to move them together ; 
 at the same time we should gain a certain amount 
 of heat from the current which has been induced 
 in the small coil. Suppose we should fire a cannon- 
 ball from a high elevation vertically downward. 
 The energy with which the ball strikes the earth 
 would depend upon the velocity with which it 
 starts and the velocity gained under the attraction 
 of the earth. The energy expended upon it in the 
 shape of the combustion of powder has its equiva- 
 lent in the work it can do with the velocity given 
 to it. In the case of the small moving coil we exert 
 a certain amount of energy upon it to give it a mag- 
 netic potential with respect to the attracting sta- 
 tionary magnet. The work that it does in falling, 
 or in approaching the attracting magnet, not only 
 is equivalent to the energy gained, but we have in 
 addition the heat produced by the electrical in- 
 duced current which has been produced from 
 nothing. This is evidently impossible, and we must 
 therefore conclude that the currents induced in the 
 coil when it is moved toward the electro-magnet, 
 and when it is moved away along the same path, 
 are such as to form in the first case a repelling pole, 
 and in the second case an attracting one. The di- 
 rection of induced current, therefore, can always 
 
238 
 
 NEW PHYSICS. 
 
 be learned from the application of the doctrine of 
 the conservation of energy. 
 
 Since we find that currents of electricity are 
 produced in coils which are moved in a magnetic 
 field, it is evident that we could make an apparatus 
 by which, with the expenditure of mechanical work, 
 we could produce strong currents of electricity. 
 Suppose we should wind a ring of iron with wire 
 continuously (Fig. 1 1 1), and then revolve this ring 
 between the poles of a permanent horseshoe mag- 
 net, the plane 
 of the ring be- 
 ing perpendic- 
 ular to the axis 
 of revolution, 
 which in turn 
 is perpendicu- 
 lar to the plane 
 in which the 
 poles of the permanent magnet are placed. While 
 one of the spires of the wire on the ring is ap- 
 proaching the north pole, a similar spire is ap- 
 proaching the south pole of the permanent mag- 
 net. We know, from experiment and from the 
 preceding reasoning, that the currents induced in 
 the wire of the ring will be opposite in direction, 
 and will neutralize each other. Suppose, however, 
 we should remove the covering of the wire along 
 the periphery of the ring, and allow the brass spring 
 to rub against this periphery at the points A and B. 
 These springs, which are called commutators, will 
 collect the opposite currents in the two halves of 
 the ring, and the united currents will then flow from 
 A through the outside circuit to B. On the way 
 
 Fig. III. 
 
ELECTRICAL WORK. 
 
 239 
 
 to B, however, it is customary to wind this wire in 
 equal coils upon the poles of the magnet between 
 which the ring revolves. It will then be seen that 
 the current produced by the motion of the ring 
 can be made to strengthen the magnetism of the 
 poles of the permanent magnet. The stronger the 
 current becomes in the wire of the ring, the stronger 
 becomes the magnetic field in which it revolves, 
 and the more work it requires to turn the ring. 
 This is the principle of the ordinary dynamo-elec- 
 tric machine which is used to produce the electric 
 light. Instead of a permanent magnet, coils of wire 
 with ordinary soft-iron cores are used ; for the 
 slightest amount of magnetism in the iron, together 
 with the magnetic effect of the induced current 
 produced by the revolution of the ring, is sufficient 
 to start the machine. Its law of increase is, then, 
 like that of compound interest. There is a certain 
 limit, however, to its efficiency, for iron can not be 
 magnetized to an indefinite amount. On estimat- 
 ing the work done in turning the ring, or, as it 
 is called, the armature of a dynamo-machine, and 
 comparing it with the heat produced in the circuit, 
 we find, on estimating the losses that take place, 
 that there is an exact equivalence between the two. 
 If we call H the amount of heat evolved by the cur- 
 rent 5, in a conductor of resistance R, in the time 
 t, we find H —S"^ Rt. Eq. (i). The practical unit of 
 
 electrical resistance when the length R = j^ — ^ 
 
 (p, 234) is expressed in centimetres, is 10', or 
 1,000,000,000; having determined upon this, we can 
 find the value that 5 must have in the equation 
 H=S^Rt, in order that the unit of work shall 
 
240 I^EW PHYSICS. 
 
 be one gramme raised one metre, or loo centime- 
 tres high. 
 
 We know that the unit of force adopted in me- 
 chanics is that force which in unit time will com- 
 municate a unit velocity to a unit mass. 
 
 The general expression for force is F ^ Mv. 
 
 „. space -f . r- ^^ 
 
 bmce V = —7 — = --, we have /* = -—- . 
 time i t 
 
 To obtain the force in the unit of time, we must 
 
 divide by t. 
 
 • rr Ms 
 
 Hence, the unit of force = --tj-. 
 
 Ms 
 The expression — jp is called the dimensions of 
 
 force. The force exerted on M grammes by the 
 earth's attraction = 980.6 M. If M is one gramme, 
 our unit of force becomes in the centimetre-gramme- 
 second system, 980.6 (latitude of Paris). 
 
 In the metre-gramme-second system it is 9.806. 
 This, it will be remembered, is the same as g, the 
 acceleration produced by gravitation. The value 
 of the earth's force of gravitation and the accelera- 
 tion imparted by it to one gramme are the same. 
 
 Work is performed when the point of applica- 
 tion of the force is moved by it. Work is equal to 
 the force multiplied by the distance through which 
 
 M ^ 
 
 it acts, or W= Fs. Since the value oi F = — g-, we 
 
 have W = 
 
 To raise, for instance, one gramme one metre, 
 we must do the work 9.806, since in this case 
 W= Fs — 9.806 X I = 9.806 ; or, if the centimetre- 
 gramme-second system is used, W = 980.6 X i X 
 
ELECTRICAL WORK. 
 
 241 
 
 100 = 980.6 X 10'. The value of current, therefore, 
 which, working through the unit of resistance R = 
 10' in one second, will correspond in units of work 
 to nearly 102 grammes raised one metre high in one 
 second, or 102 gramme-metres, will be 102 x 980.6 
 
 X id^ = S^ R t = S^ 10^ X 1, or S = — . This is 
 
 10 
 
 called an ampere. 
 
 We can, therefore, form the following expression : 
 
 S^Rtc — hW X 772, or 5 = y /?/ , in which 
 
 W = weight of water in pounds, h = increase of 
 temperature in Fahrenheit degrees, 772 = mechani- 
 cal equivalent of heat, t *= time in seconds, c = 
 .7373 = equivalent in foot-pounds of one ampere 
 per ohm per second. 
 
 Experiment 131. — Measure the strength of an 
 electrical current in amperes by running the current 
 through a fine piece of iron or platinum wire, which 
 is immersed in a given calorimeter such as is used 
 in Experiment 88, which contains a given amount 
 of water. Be careful that the wire does not touch 
 the sides of the calorimeter. 
 
 Having decided upon the practical value of the 
 unit of current S, we have from Ohm's law 5 = 
 
 -75, and — = — -9, or E = icf. 
 R 10 10' 
 
 The unit of current — in the centimetre-gramme- 
 second system,- C 6^ 5 system, is called an ampfere, 
 from Ampere, the great French physicist. The unit 
 of electro-motive force, E = IO^ is called a volt, 
 from Volta, the Italian physicist. 
 
 These units appear arbitrary at first, but the 
 
242 
 
 NEW PHYSICS. 
 
 unit of resistance being expressed and determined 
 as a velocity, we can express our unit of current in 
 the same decimal system, so that it may have as 
 near as possible the value of loo metre-grammes. 
 It, moreover, happens that, taking the unit of resist- 
 ance = lo' and the unit of current — , or io~S which 
 
 in turn give the unit of electro-motive force lo*, 
 a Daniell cell is found to have an electro-motive 
 force of about one volt. The Daniell cell being the 
 most constant cell which is in common use, this 
 system adapts itself readily to practice. Let us 
 now see how electro-motive force can be measured 
 in these absolute units. 
 
 The electro-motive force of a cell does not de- 
 pend upon the size of the metal plates of the bat- 
 tery. 
 
 Experiment i 32. — Obtain the electro-motive 
 force of two gravity-cells, in one of which the plate 
 of zinc or iron is twice as large as in the other. 
 
 The internal resistance of the cell, however, is 
 much modified by the size of the plates, for the resist- 
 ance of a layer of liquid follows the same law as 
 that of a metallic conductor, varying directly as the 
 thickness of the layer and inversely as its section. 
 The greater the size of the metallic plates of the 
 battery, therefore, the less the resistance of the 
 liquid between them. The operation of coupling 
 batteries for quantity consists merely in reducing 
 or increasing the battery resistance in comparison 
 with that of the exterior circuit, by coupling a cer- 
 tain number of the zinc plates together and the 
 same number of the carbon or copper plates to- 
 gether, so as to form larger plates, and therefore in- 
 
ELECTRICAL WORK. 243 
 
 creasing the section of the liquid and reducing the 
 resistance of the hquid. 
 
 Fig. 103 shows four cells arranged for intensity ; 
 that is, so that both the electro-motive force and 
 the battery resistance are increased, the strength of 
 
 current being- 5 = — %—. — . 
 
 Fig. 112 shows the four cells arranged for quan- 
 
 2 F 
 tity. In this case we have 5* = — ?;— ; — . 
 ^ 2B -\-r 
 
 Experiment 133. — To 
 
 determine the electromo- 
 tive force of a Bunsen cell 
 in terms of a Daniell, con- 
 nect the Daniell cell through 
 a galvanometer and also 
 through a certain resist- 
 ance, R. Call the resist- 
 ance of the galvanometer ^^^ 
 g ; this can be measured 
 
 by the Wheatstone bridge. If K is the reduc- 
 tion factor of the galvanometer employed, we shall 
 have for two different values of the resistance R 
 
 E 
 
 {B being the battery resistance) 6" = -^-. j — ~ = 
 
 B-\-g-[-R 
 
 Kt^na, S, = -BJ^fj^, = ^tana,, or ^+^+^j 
 
 tan a , . „ , . , 
 
 = 7 . rrom this we can obtain B, the internal 
 
 tanoi 
 
 resistance of the battery. 
 
 Obtain also the internal resistance of the Bunsen 
 cell. Then compare the deflections produced by 
 the two cells through known resistances (which in- 
 clude the battery resistance, the galvanometer re- 
 
244 
 
 NEW PHYSICS. 
 
 sistance, and the resistance R, interposed together 
 with the connecting wires). 
 
 We shall then have 5 = ^i\j^ = -^ tan a 
 
 Ex 
 for the Daniell cell, S^ = ^ i ^.^ = K tan a, for 
 
 .u TJ n ^ B^g+R tana 
 
 the Bunsen cell, or -gr = B^^g^R^ tliT^' 
 
 The deflections should not exceed 45°, for a 
 greater deflection moves the needle out of a uni- 
 form magnetic field. 
 
 Experiment 134. — Another method is to con- 
 nect the cells together and obtain their combined 
 effect upon the galvanometer, or 
 
 E-,-\-E 
 S -\- Sx — -5—; — 5 — :— 5 = isTtan o, and afterward 
 
 run them against each other and obtain the differ- 
 
 E,—E 
 
 ence of their effects, or 5, — 5 = -ttt — s — i i — n 
 
 B-\-Bx-\-g-\-R 
 
 Ex + E tan a 
 
 = A tan a,, or -7= = = r . 
 
 " Ex — E tan oi 
 
 The denominator B -\- Bx -{- g -\- R is the total 
 resistance of the circuit, which is the same for both 
 cases, and therefore cancels out. E being the elec- 
 tro-motive force of the Daniell cell. Ex can be ob- 
 tained in terms of it. 
 
 This method forms a control method to that 
 employed in the preceding experiment, but this 
 method can only be regarded as approximative, 
 since the stronger cell affects the weaker by what 
 is called polarization, which will be explained later. 
 
 Perhaps the most instructive method of deter- 
 mining electro-motive force is that of Poggendorff. 
 Suppose we should have two water-engines, E' be- 
 
ELECTRICAL WORK. 
 
 245 
 
 ing the stronger. E' takes in water from a reservoir 
 and forces it through the pipe ABC back to the 
 same reservoir. If a smaller engine, £ (Fig. 113), 
 endeavors to also force wa- 
 ter into the pipe at B, its 
 ability to do so will depend 
 upon the pressure it can 
 exert at B. The difference 
 of level between B and C 
 will be just equal to the 
 pressure the engine E can 
 maintain if no water flows 
 through BEC. 
 
 Substituting galvanic 
 qells for our water-engines, 
 and connecting the similar 
 poles in the manner shown in the figure, by placing 
 a galvanometer, G, in the circuit of the lesser cell, 
 we shall find, by properly adjusting the resistances 
 R and R^, that no current will pass through the gal- 
 vanometer G which is placed in the circuit of the 
 weaker cell. The current through ABC will be C = 
 
 E' 
 
 — — , B^ being the resistance of the battery E' . 
 
 By + R + K-i 
 
 The current from ^ to C is the same as would 
 
 be maintained by the battery E through the re- 
 
 sistance R^ ; hence we shall also have C = -5-, or 
 
 Fig. 113. 
 
 E' 
 
 E_ 
 
 E> B^ + R + R^ 
 or 7^ = - 
 E 
 
 Ri 
 
 Bi + R + Ri~ R^'^^ E ~ R^ 
 
 This is an application of the general law (called 
 Kirchhofl's law) that the current between any two 
 points depends only upon the difference of potential 
 or level maintained between its two ends, and also 
 
246 NEW PHYSICS. 
 
 upon the resistance of the conductor between these 
 two points ; or, calling the strength of the current 
 
 C, we have C = -^, or C R = E, E being the elec- 
 tro-motive force which is directly proportional to 
 the difference of potential between A and B. 
 
 Experiment 135. — Ascertain by a galvanometer 
 the truth of Kirchhoff's law for a circuit. Form 
 a divided circuit at the point ^ and C (Fig. 114), 
 The current will divide at B, and after flowing 
 around both branches, B QC and BmC, will unite 
 again at C. The current C 
 through B G C, plus that 
 through B in C or C^, will be 
 equal to the whole current 5 
 between A and B or C ■\- Ci 
 = S. Placing a galvanome- 
 ter, therefore, in one of the 
 branches, B G C, another G^ 
 being placed in the circuit 
 between A and B, or at G^, 
 we shall find that C multi- 
 plied by the resistance be- 
 tween B and C or C {G + R^ will be equal to the 
 electro-motive force of the battery or C {G + R^ = 
 E. We conclude from symmetry that the same law 
 must apply to the circuit B m C or C^ (R^) = E, R^ 
 being the resistance of BmC. Knowing, however, 
 the strength of the current 5 = C -|- Cj, and having 
 obtained C, we can show that Kirchhoff's law also 
 holds for the circuit BmC. We therefore see that 
 
 C R 
 
 -pr = ^ . „ , or the current divides inversely as the 
 
 resistances of the divided circuit. 
 
ELECTRICAL WORK. 
 
 247 
 
 These methods of determining electro-motive 
 force are relative, and not direct. Let us now ex- 
 amine more direct methods and obtain the exact 
 value, as far as our apparatus permits, of the elec- 
 tro-motive force of a Daniell cell. 
 
 Experiment 136. — Pass the current from a 
 Daniell cell through a known resistance, R, and a 
 galvanometer, having obtained the resistance, B, of 
 the cell, and G, that of the galvanometer. Know- 
 ing the reduction factor of the galvanometer = 
 
 a T E 
 
 = A, we then have 5 = -tj— — 7^— — rr = J^ 
 
 2«7r B + G -^^ R 
 
 tan a, and £■ x 10' = (5 -f- C-f- i?) X 10' x A' tan o, or 
 
 (B^G-^R)\o ^K\.z.na ,, . ^, . . 
 
 E = ■ ■ ^^8 , 10" being the unit of 
 
 electromotive force and 10' that of resistance. 
 
 Many of our experiments thus far do not require 
 a knowledge of the reduction factor of the galvanom- 
 eter. We have shown how this factor is obtained 
 from the dimensions of the coil, and from the value 
 of the earth's magnetic field. In certain cases it is dif- 
 ficult to obtain the dimensions of the coils. If we know 
 the reduction factor of one galvanometer, we can 
 readily obtain thatof another in thefollowing manner: 
 
 Experiment 137. — Pass the same current 
 through two galvanometers, the reduction factor 
 of one being C, and the other C C being known, 
 we shall then have from one instrument 5 = 
 strength of current = C tan a, and from the other 
 5 = C* tan *a. Hence we have C tan a — O tan oS 
 
 or -;=rr = . Knowing C, we can obtain Ci. 
 
 C^ tan a 
 
 Experiment i 38. — To obtain the reduction factor 
 of a galvanometer of which we can not obtain the 
 
248 NEW PHYSICS. 
 
 number of turns. Ascertain accurately the electro- 
 motive force of a Daniell cell, connect it with the gal- 
 vanometer, and interpose a very large resistance in 
 
 E 
 
 the circuit. We shall then have 5 = — = Ctan a, 
 
 1%. 
 
 E Ey. 10? . ^ „ ^ . 
 
 or C = = 1 m C. G. S. system. 
 
 R tan a R x lo^ tan a 
 
 We have thus far used the expression reduction 
 
 factor for the number by which we must multiply 
 
 the tangent of the deflection of a galvanometer to 
 
 obtain the strength of the current. It is usual to 
 
 separate this factor into two quantities, one of which 
 
 is T, the horizontal intensity of the earth's magnet- 
 
 2 IT fZ 
 
 ism, and the other is , which depends merely 
 
 upon the number of turns of wire and the radius of 
 
 the coil ; this latter factor is always constant for the 
 
 same instrument, and hence is called the constant 
 
 of the galvanometer. The strength of a current 
 
 T' . . 2 IT n 
 
 will then be 5 = — tan a, in which in = . 
 
 m a 
 
 Let us now return to the expression for the work 
 done by any current of electricity. This is II = S'^Rt. 
 
 Instead of determining the value of 5 by means 
 of a galvanometer, we can use the value of the elec- 
 
 E 
 tro-motive force, for S = —, and substituting this, 
 
 R 
 
 we have H = — --. By the use of an electrom- 
 R 
 
 eter, we can determine E in terms of a standard 
 Daniell cell, for the ratio of the deflections produced 
 by the Daniell cell, and the cell whose electro-mo- 
 tive force is to be determined, is the same as the 
 
ELECTRICAL WORK. 249 
 
 ratio of their difference of potential, and this in turn 
 is the ratio of the electro-motive forces. 
 
 Experiment 139. — To determine the electro-mo- 
 tive force of a battery or a dynamo-electric machine 
 by comparison with one or a number of Daniell 
 cells, we have merely to connect the terminals of 
 the electrometer to the poles of the battery or to 
 the terminals of the dynamo-machine, having inter- 
 posed a sufficiently large resistance in the circuit 
 between these poles. Since we must observe the 
 deflections of the mirror of the electrometer by re- 
 flection of a spot of light upon a scale, having ob- 
 tained the values of tan a for the cells which we 
 
 wish to compare, we then have -— = , E being: 
 
 ^1 tan oi 
 
 the electro-motive force of the standard cell. 
 
 In general, however, quantitative measurements 
 with an electrometer are more difficult than with a 
 galvanometer, on account of the care that must be 
 taken to keep a constant charge on the needle of the 
 electrometer. This can be done, however, with a 
 water-battery. Instead of an electrometer, we can 
 use a galvanometer of a very high resistance to de- 
 termine electro-motive force. Such a galvanometer 
 can be made very simply if the secondary coil of an 
 induction-coil can be separated from its primary. 
 
 Experiment 140. — Cut a bottle in two, place a 
 small bit of magnetized watch-spring upon the back 
 of a small mirror, and hang it up by a single cocoon- 
 fiber in the center of the lower half of the bottle ; 
 push this half-bottle, with the mirror suspended 
 in it, into the axis of the secondary coil. Having 
 mounted the coil horizontally on a firm support, a 
 spot of light can then be used to observe the de- 
 
250 NE^ PHYSICS. 
 
 flections of the magnet when a current is passed 
 through the coil. If a plain mirror is used, the de- 
 flections can be observed with a telescope and scale. 
 Discharge a Leyden-jar which has been only mod- 
 erately charged through this galvanometer (there 
 being no danger of destroying the insulation be- 
 tween the coils of this galvanometer). It will be 
 found that this sudden discharge or fall of electrical 
 potential will give evidence of an electrical current ; 
 the effect being the same as if we should send for 
 an instant a current from a battery through the in- 
 strument. It is evident, therefore, that we can 
 measure the quantity of electricity in a Leyden-jar 
 by such an instrument. 
 
 In order to understand the effect of a sudden 
 current in a coil upon a magnetic needle at its cen- 
 ter, it is necessary to return to the subject of me- 
 chanics. We have seen how the work of a projec- 
 tile can be estimated by the height a ballistic pen- 
 dulum is lifted against the force of gravitation. We 
 can liken the sudden creation of a field of force at 
 the center of a coil, and the sudden impulse given 
 to the magnet, to the circumstances of the blow and 
 the effect of the projectile on the ballistic pendulum. 
 Let us first examine our ideas of angular motion. 
 
 In the case of the ballistic pendulum, we have 
 the bob of the pendulum moving through a certain 
 arc with a certain velocity. The angle traced by 
 the bob in one second measures this velocity and is 
 called the angular velocity. 
 
 In general we define angular velocity as the 
 angle traced in one second by a line fixed in the 
 body and moving at right angles to the axis of rota- 
 tion of the body. 
 
ELECTRICAL WORK. 
 
 251 
 
 Angular velocity can be uniform, as we have 
 supposed in our first experiments upon the fall of 
 bodies ; or it can be variable. In the case of the 
 ballistic pendulum it is variable. 
 
 If 6) is the angular velocity of a point on the arc 
 of a circle whose radius is one, then the arc traced 
 by the point in one second measures the angular ve- 
 locity. Any other point on the continuation of this 
 radius at a distance, r, from the center of the circle 
 will describe an arc, ra, in the same time that the 
 arc ft) is described. 
 
 The energy of a body rotating with the angular 
 velocity w '\%\mi^(i?,r(o being substituted for v in 
 
 VI IT . " 
 
 the expression . We perceive that mr^ \& the 
 
 expression we have called moment of inertia in our 
 discussion of the compound pendulum. We can 
 therefore write the above expression \ K<o^, in which 
 K is the moment of inertia. 
 
 In the case of the pendu- 
 lum acted upon by gravita- 
 tion, the pendulum has a cer- 
 tain angular velocity w at Z> 
 (Fig. 115) and another angu- 
 lar velocity a>i at C and no 
 angular velocity at B, the 
 highest point of its swing. 
 Hence, if we call F the force 
 acting upon the pendulum- 
 bob at a distance of r from 
 the point of suspension A, we 
 
 have the work done in lifting the bob from D to 
 C through the height DE = FDE-Fr{i- 
 cos G). 
 
252 I^EW PHYSICS. 
 
 The energy at Z> = i Ko^ and that at C = ^Km^ ; 
 hence the work between D and C will be Fr (i — 
 cosd) = iir(«^-i»i='). 
 
 If we call the angle B A D = a, we shall have, 
 when the bob reaches its highest point B, wi = o, or 
 Fr (i — cos a) = i Kw^. 
 
 From a table of trigonometrical relations we find 
 
 a ./Fr 
 
 that I — cos a = 2 sin — . Hence, <b = 2 sin , 
 
 2 2 K 
 
 In the case of magnetism, the force acting upon 
 the magnetic pendulum is yi,Hr instead of Fr ; the 
 strength of the pole being /t, that of the field being H. 
 
 If we call fL r the magnetic moment, and denote 
 it by G, we shall have in the case of the magnetic 
 
 pendulum ■ = H G {i — cos a). 
 
 or, Q) = 2 sm y — -—. 
 2 K 
 
 The work done in turning a body through a 
 small angle, Q, is the force multiplied by the dis- 
 tance passed over, or the arc. Hence, 
 work = i^X Id. 
 If ft) and ftji are the angular velocities at the be- 
 ginning and end of the movement, we have the 
 energy expressed by 
 
 i ^ (o)" - a»i«) =Fy^ie = Fl<oi T, 
 in which t is the time during which the force acts, 
 J (ft)* — a^ = ^ (ft) — ft)j) (ft) 4- ft)j) 
 f (ft) + ft)i) = ft)i, approximately. 
 Hence, K{(o — (o^ = FIt. 
 
 In the case of a galvanometer we have 
 
 2 TT « ? /* . 
 
 -r' = (« being strength of current). 
 
ELECTRICAL WORK. 
 
 253 
 
 2 TT ft 
 
 If we denote by C, and fil hy G, we have 
 
 K{w — ©i) = CGiT, but iT = ^ = quantity of elec- 
 tricity transmitted. Hence, ir(s) — <»i) = C G q. 
 
 If the magnet moves through a very small an- 
 gle (and this can be accomplished if the discharge 
 through the coil takes place very quickly), we can 
 suppose that the force is always perpendicular to 
 the magnet, and can take K<o = C G Q,\x\ which Q 
 is the whole quantity of electricity transmitted ; but 
 
 a./GH 
 
 we have m = 2 sin-y —-—■ - Hence, 
 2 A 
 
 Xa> = 2Ksm-y^ = C G Q, 
 2 JC 
 
 ande = 2-y^^sm-. 
 If T is the time of vibration of the magnet, we 
 
 • = ^/J 
 
 have found previously, T -^ .. , 
 
 „ ^ ^HT . a ^^ 
 
 Hence, Q — -—, sm -. 
 
 L ir 2 
 
 We always have Q^ St, where 5 is the current 
 
 J • .,. .• ^ ^ 2HT . a 
 
 and / IS the time ; or, Q = --t = —p. sm -. 
 
 K L TT 2 
 
 From this expression we can obtain the electro- 
 motive force if we know the time during which the 
 discharge lasts, or we can obtain the ratio between 
 two electro-motive forces if we make the times of 
 discharge equal, for t will then cancel out in the 
 ratio of the electro-motive forces. 
 
 With the aid of a galvanometer of high resist- 
 ance, we can obtain the electro-motive force of a 
 series of batteries or of a dynamo-machine. It is 
 
254 ^ElV PHYSICS. 
 
 often important to be able to do this, for the accu- 
 rate measurement of current depends upon the com- 
 paratively small deflection of a magnetic needle in 
 a uniform magnetic field, and these conditions are 
 not always easily and accurately obtained. For in- 
 stance, a strong current from an electrical machine, 
 if passed through a galvanometer of one coil, will 
 ordinarily give a deflection of more than 45°. In 
 this case the field usually ceases to be uniform, and 
 the deflections are not accurately proportional to 
 the tangents of the angles. If we proceed to shunt 
 the galvanometer — that is, to direct a part of the cur- 
 rent through a wire of resistance m (Fig. 1 14), con- 
 necting the binding-screws of the galvanometer — we 
 shall find that even a small error in the determina- 
 tion of the amount of wire we bind into the binding- 
 screws of the instrument, will make a large error in 
 our determination of the current. 
 
 „ , m C KXSina _ GiTtana, 
 For we have — = — = — - — , or, 61 = 
 
 where G is the resistance of the galvanometer and 
 
 C and Cx are the portions of the current passing 
 
 through the two branches of the divided circuit. 
 
 Since we have also 
 
 „_ ^ „ _ Kmt2xva-\- GKtB.na 
 o — C -j- Li — , 
 
 m 
 
 or, 5= ^tana^ — ■ — , 
 in 
 
 we see that a small error in the determination of m 
 affects the whole value of S, G being usually very 
 small, and R being less than G. It is, however, diffi- 
 cult to obtain an accurate value of a small resistance. 
 With a high resistance galvanometer we can de- 
 termine the electro-motive force of a large battery 
 
ELECTRICAL WORK. 255 
 
 or dynamo-machine by connecting one terminal of 
 the galvanometer with one pole of the machine, and 
 touching the other for an instant only to the other 
 pole. In this case the sudden swings of the galva- 
 nometer (or, more accurately, the sines of one half 
 the angles) are proportional to the difference of po- 
 tential of these poles. Do the same with a series 
 of Daniell cells upon a circuit of known resistance. 
 
 We shall then have -— = -. — znr, in which E is the 
 E<^ sm \ 0^ 
 
 desired electro-motive force, and £1 that of the stand- 
 ard or Daniell cells. 
 
 Let us now make a measurement of the work 
 done in producing a current of electricity by means 
 of a dynamo-machine. Although these machines 
 are usually run by a steam-engine, we will suppose 
 that we have one which is worked by two men. 
 The small dynamo-machine is run by a band which 
 passes over a large fly-wheel which is turned by 
 the men. A friction-brake is put upon the axle 
 of this large wheel, and the work done in turning 
 this with a certain velocity is measured. The elec- 
 trical current, in the first place, is passed through a 
 galvanometer of a known resistance in order to de- 
 termine by the steadiness of its deflection whether 
 the current is constant or not. The difference of 
 potential between the binding-screws or poles of the 
 machine is then measured by comparison with a 
 certain number of standard cells whose electro-mo- 
 tive force has been determined. The work done by 
 the electrical current will then be in metre-grammes, 
 
 ^ R X 980.6 ie X 10* X 980.6 
 
 980.6 
 
 W. 
 
256 NEW PHYSICS. 
 
 If we call the work done in turning the fly-wheel, 
 
 expressed in metre-granimes, W^, the efficiency of 
 
 W 
 the dynamo-electric machine will be — — A. 
 
 It is supposed that the belt connecting the re- 
 volving axles does not slip. The same method can 
 of course be applied to the case where a steam- 
 engine turns the dynamo-electric machine. 
 
 Instead of measuring the current by the indirect 
 method, suppose we adopt the direct method and 
 use the deflection of a galvanometer. In this case 
 it will be necessary to shunt the galvanometer, 
 
 and the whole current will he S = K tan a — --^. 
 
 K 
 
 The whole resistance of the circuit will include 
 that of the battery or dynamo-machine, and the re- 
 sistance outside or interpolar resistance. This in- 
 terpolar resistance is made up of the connecting 
 wires, of whatever resistance is interposed, and that 
 of the shunted galvanometer. The use of a shunt 
 alters the resistance of an instrument, as can readily 
 be seen by measuring the divided circuit, as a whole, 
 by means of a Wheatstone bridge. By considering 
 
 the difference of po- 
 tential between the 
 two points A and B 
 (Fig. 116), we shall 
 see that some wire 
 of resistance -y^ = com- 
 bined resistance of the 
 two wires, R and ^1, 
 can be substituted between A and B, and the dif- 
 ference of potential or level A D — B D^ can be 
 maintained. Hence this difference of potential or 
 
 Fig. 116. 
 
ELECTRICAL WORK. 
 
 257 
 
 electro-motive force must be equal to the resistance 
 
 X multiplied by the current S,ovC-\- Ci. 
 
 R C 
 Now, we know that -^= -^, which is the same 
 
 ratio as -^±_^ =..^ = ^ = ^=^ 
 ^1 <^ <^ X x' 
 
 E 
 
 R 
 Hence the combined resistance x = ^^ . V> • We 
 
 shall then have W = work in metre-grammes, 
 ^_S^_Rot^ (Sj\fRoio't ^ ^^. ^j^g gj^^jj.^ j.g_ 
 
 980.6 980.6 " ^ 
 
 sistance of the circuit, and the efficiency can be 
 obtained as before. This method is open to the ob- 
 jection of the error produced by shunting. 
 
 If a battery is employed instead of a dynamo- 
 machine, the process of obtaining the work it does 
 in heat-units has already been described. The de- 
 termination of the internal resistance of the battery 
 should in general be made unnecessary by the ar- 
 rangement of the experiment, for it is a very uncer- 
 tain quantity. A simple method has already been 
 given by which it can be measured. The following 
 method is often useful, and is given to illustrate the 
 simple laws of electricity which we have used in the 
 preceding experiments: 
 
 Experiment 141. — To obtain the internal resist- 
 ance of a battery, or a single cell, pass the current 
 through a galvanometer and a known resistance, and 
 note the deflection of the galvanometer. Call this 
 
 deflection d. We then have = Ktan 0, 
 
 Jo -\-R -\- (jr 
 
 in which B is the resistance of the battery, G that of 
 
258 NEW PHYSICS. 
 
 the galvanometer, and R is the interposed resistance. 
 Now shunt the battery by a wire of known resistance, 
 M; in order to produce the same deflection as before, 
 it will be necessary to change the resistance R to 
 some other resistance, R^ ■ 
 
 We shall then have . — —. C and Q being 
 
 G-i-R^ Ci' IS 
 
 the strengths of currents through G + Ri and M, 
 
 respectively. The whole current S= C-\- C^, and 
 
 E 
 by the conditions of the experiment C — „, „, ^ . 
 
 Hence, 
 
 E E G+Ri E 
 
 B-\- M{Ri-{-G) B-{-R-\-G'^ M B + R-\-G 
 
 M+{R,+ G) 
 
 Solving for B, we obtain B = M I ^^j. 
 
 \Ri 4- G/ 
 
 This method is open to the objection that any 
 change of resistance in the outer circuit necessarily 
 modifies the electro-motive force and internal resist- 
 ance of the battery. It is difficult to avoid this by 
 any method. 
 
 Instead of a battery, or of a dynamo-machine, 
 we can employ the ordinary Holtz electrical machine 
 to obtain the mechanical equivalent of heat. This 
 machine can be likened to a water-battery of a very 
 great number of cells. The internal resistance is 
 very great, and the difference of potential maintained 
 by mechanical work is also very great. Accurate 
 measurements of the electrical work done by the 
 electrical machine are difficult on account of the 
 tendency of electricity of high potential to escape 
 from conductors. The method employed, however, 
 is extremely instructive, since it leads us to connect 
 
ELECTRICAL WORK. 
 
 259 
 
 the manifestations of electricity of high tension, such 
 as is produced by the common electrical machines, 
 with the manifestation of electrical potential exhib- 
 ited by batteries. The Holtz electrical machine is 
 run by some small motor, such as a water-engine. 
 The mechanical work consumed can be estimated 
 by some of the methods already given. The poles 
 of the electrical machine are connected by wires to 
 the terminals of a galvanometer of high resistance 
 and of great insulation. 
 
 In order to obtain the reduction factor of our 
 galvanometer, since we can not in general measure 
 with accuracy the mean radius of the coil, or count 
 the number of turns, we make use of a Daniell cell, 
 knowing the electro-motive force of this cell. We 
 
 E 
 shall have — = K tan a. Making R very large, and 
 R 
 
 including in it the large resistance of the gal- 
 vanometer, we can neglect the resistance of the 
 
 E lo^ 
 
 Daniell cell. We shall then have K^ — . . in 
 
 io'i?tana 
 
 which K is expressed in C. G. S. units. If we then 
 call S the current produced by a definite amount of 
 work performed upon the Holtz machine, we shall 
 have S^Rt^ heat equivalent, =y" /T = work equiv- 
 alent. From this expression we can obtain /, the 
 mechanical equivalent of heat. 
 
 We have thus far measured the magnetic and 
 heating effects of the current. There are other 
 phenomena, which have important bearings upon 
 the doctrine of work, especially of molecular work. 
 Among these phenomena are the effects of the 
 electrical current in decomposing certain liquids. 
 If we should attempt to measure the resistance of 
 
26o JVEW PHYSICS. 
 
 a column of water by inclosing it in a tube, dipping 
 platinum wires into the water at the extremities of 
 this tube and connecting them with Wheatstone's 
 bridge, we should obtain very inconstant results. 
 This is due to the fact that the electrical current, in 
 passing through the liquid, decomposes it, produc- 
 ing hydrogen at one of the electrodes or wires dip- 
 ping into it, and oxygen at the other. These layers 
 of gas, in turn, produce a difference of potential, as 
 if another battery were interposed in the circuit, and 
 the law of the Wheatstone bridge is no longer main- 
 tained. It would evidently be impossible to meas- 
 ure the internal resistance of a cell by putting it into 
 one of the branches of a Wheatstone bridge, for, by 
 constructing a diagram of the fall of potential, it 
 would be seen that the proportionality between the 
 adjacent sides of the bridge exists no longer, and 
 the simple law of the bridge is destroyed. All liquids 
 which can be decomposed by the electrical current 
 are called electrolytes, and all metals immersed in 
 them, conveying an electrical current from one 
 point to another, become polarized — that is, are 
 made -f- or — to each other. This polarizing effect 
 is almost nothing when the electrodes conveying 
 the current to and from the liquid form the base of 
 the liquid ; that is, when the liquid is a salt of these 
 metals. For instance, zinc electrodes in sulphate 
 of zinc, or copper electrodes in sulphate of copper, 
 exert very little of this polarizing effect. It would 
 be impossible, however, to measure the electrical 
 resistance even of these non-polarizable liquids by 
 introducing them into the Wheatstone bridge, for 
 the current decomposes them— depositing the metal 
 upon one electrode, the negative electrode. 
 
ELECTRICAL WORK. 261 
 
 The effect of the polarization of an electrical cur- 
 rent can be shown in a very striking way by means 
 of what is called a storage-battery. 
 
 Experiment 142. — Place two strips of sheet-lead, 
 about six by eight centimetres, at a distance of six 
 centimetres from each other, in a vessel containing 
 water acidulated with sulphuric acid — one part of 
 acid to ten of water. Pass the current from one 
 Bunsen cell through the liquid from one plate to the 
 other, placing also a low-resistance galvanometer in 
 the circuit. It will be found that the deflection of 
 the galvanometer grows less and less, and finally 
 becomes steady. When this latter point has been 
 reached, disconnect the battery and connect the lead 
 plates with the galvanometer. A current will be 
 obtained which is opposite to that of the battery. 
 This is a polarization current, and is due to genera- 
 tion of oxygen on one of the lead plates and of. 
 hydrogen on the other, thus producing a difference 
 of potential. It is evident, from the conservation of 
 energy, that the resultant current must be opposed to 
 that which produced it. Find the relation between 
 the strength of the current of polarization and that 
 of the current which produces it. 
 
 The lead cell we have described is called a Plant6 
 cell, from the name of its inventor. The Faure cell 
 consists merely of lead plates covered with a layer 
 of red oxide of lead, which increases the polariza- 
 tion. It is evident that electricity is not stored in 
 these cells. A difference of chemical constitution 
 is produced, which in turn gives a difference of 
 electrical level. 
 
 Experiment 143. — Pass a current through a gal- 
 vanometer and through two copper strips immersed 
 
262 ^EW PHYSICS. 
 
 at a fixed distance from each other in a bath of sat- 
 urated solution of sulphate of copper. AUdw the 
 current to run for a given length of time, and then 
 weigh the negative electrode. Do the same with a 
 different strength of current (obtained by altering 
 the resistance of the circuit). It will be found that 
 the amounts of the copper deposited in the same 
 time will be proportional to the current strengths. 
 If Wand. W^ are the weights of copper deposited, 
 
 W S 
 and 5 and S, are the currents, we have — = — 
 
 Hence, the amounts being proportional to the time 
 
 W 
 the current runs, — = iT tan a will express the 
 
 strength of the current in the amount of copper 
 deposited in one second. It has been found that a 
 current of one ampere {S = -^j^) will deposit .01972 of 
 a gramme of copper in one minute, or .0033 of a 
 gramme in one second. 
 
 Experiment 144. — Determine the horizontal in- 
 tensity of the earth's magnetism by means of the 
 deposition of copper. This method is not considered 
 so accurate as the method previously given, but it 
 is often employed as a control method. Knowing 
 the constant of a galvanometer, pass the current 
 through it and a suitable vessel which contains the 
 solution of copper. The current will vary during 
 the experiment on account of the increase of resist- 
 ance in the liquid produced by the deposition of 
 copper. It is best to observe the deflections of the 
 galvanometer at regular intervals, and to take the 
 mean of the deflections. The angle of deflection 
 should not exceed 45°. A simple method of form- 
 ing a voltameter (the terra applied to the vessel 
 
ELECTRICAL WORK. 263 
 
 filled with the electrolyte) is to replace the copper 
 pole of an ordinary Daniell cell by a spiral of cop- 
 per wire, which is hung in the saturated solu- 
 tion of sulphate of copper. In this case the battery 
 itself constitutes the voltameter. We shall have 
 
 W r T 
 
 = tan a. This will eive us the current in 
 
 .0033^ 2irn 
 
 amperes, and by solving we can obtain the factor T. 
 The electrode upon which the copper is deposited 
 should be carefully washed and dried before weigh- 
 ing. Table VII, Appendix, enables us to reduce the 
 values we obtain from the use of an electrolyte to 
 absolute measure, or magnetic measure. If we 
 should collect the gases formed by the decomposi- 
 tion of water, we should find that the volume of 
 hydrogen would be twice that of oxygen. Finding 
 the amount of water decomposed by one ampere, we 
 could also use this method for determining the value 
 of T ; or, knowing T, to determine the value of the 
 reduction factor of the galvanometer. Our measure- 
 ment, however, would be inaccurate, owing to the 
 absorption of the gases in the water over which they 
 are collected. This method is not often employed 
 in a laboratory, because it requires a comparatively 
 strong current — from two to four Bunsen cells. 
 
CHAPTER XII. 
 
 THERMO-ELECTRICITY, 
 
 The experiments which we have given thus far 
 show how the value of electrical work can be ob- 
 tained in terms of mechanical work and illustrate 
 the conservation of energy. Let us not lose sight 
 of our effort to connect all the manifestations of 
 motion, heat, electricity, magnetism, and light to- 
 gether, as illustrating the doctrine of the conserva- 
 tion of energy. We have treated the galvanic cell 
 merely as a motor which maintains a difference of 
 level or difference of potential. We have perceived 
 by direct experiment that any two metals immersed 
 in an electrolyte will give a difference of electrical 
 level, or potential. This difference of electrical po- 
 tential is capable of doing work, as all difference of 
 potential is capable of doing. We have measured 
 this work in heat and then in mechanical units ; 
 and we have shown that the ordinary electrical 
 machine can be likened to a galvanic battery made 
 of a very great number of cells with large internal 
 resistances. It is not necessary to know whether 
 the electrical state is one of molecular motion or is 
 due to the flow of fluids, in order to study the sub- 
 ject from our present point of view. We have oc- 
 cupied ourselves merely with the question of the 
 
THERMO-ELECTRICITY. 265 
 
 work done by a fall in potential, and we have studied 
 electricity in the same manner and with the same 
 units that we have employed in the study of me- 
 chanics. 
 
 Let us pursue the subject from the same point 
 of view somewhat further. We have seen that any 
 two different metals will, on being placed in con- 
 tact or in an electrolyte, exhibit a difference of po- 
 tential. We shall find also that the junction of two 
 different metals, on being heated, will also show a 
 difference of electrical level or potential. 
 
 Experiment 145. — Solder a piece of German- 
 silver to each end of a piece of iron wire. Connect 
 the ends of the German-silver wire to the terminals 
 of an astatic galvanometer. Place one of the junc- 
 tions of wire and German-silver in melting ice or 
 in water of a known temperature, and heat the other 
 junction by placing it in a beaker of water and 
 heating the beaker gradually. Place a thermome- 
 ter also in the latter beaker. It will be found that 
 the deflections of the galvanometer indicate an 
 electrical current, which arises from the difference 
 of heat potential which is maintained between 
 the junctions of iron and German-silver. If we lay 
 off the degrees with a suitable unit of temperature 
 indicated by the thermometer on a horizontal line, 
 O X, and the deflections of the galvanometer parallel 
 to the line O V, we shall find that the rise in tempera- 
 ture is uniform for small ranges of temperature — in 
 other words, a straight line represents the rise. This 
 indication of a uniform rise in temperature affords 
 us a method of testing the accuracy of graduation 
 of an ordinary mercury thermometer. To do this, 
 obtain the deflection when one junction is at 0°, 
 
266 iV^JF PHYSICS. 
 
 or the temperature of melting ice, and the other 
 at that of steam ; and draw the line O B — the ab- 
 scissa O D representing ioo° 
 on the centigrade scale. Any 
 intermediate temperature, O C, 
 can be obtained by comparing 
 the indication of the mercurial 
 thermometer when the strength 
 of current indicated by B' C is 
 obtained. 
 
 It is found that two differ- 
 ent metals are not needed to produce a thermo- 
 electric current. Any difference of temperature, 
 even in a circuit of one metal, will manifest itself 
 by an electrical current. 
 
 Experiment 146. — Connect a piece of iron wire 
 with the terminals of a galvanometer, and heat it 
 at its middle point : a thermo-electric current will 
 be produced. Here the two junctions of iron with 
 the terminals of the galvanometer are at the same 
 temperature, that of the room, and the current is 
 produced by a difference of temperature between 
 different portions of the iron wire. Any change in 
 structure, such as is produced by twisting or bend- 
 ing, will also produce an electrical current. This 
 thermal electrical current is generally regarded as 
 the evidence of a difference of potential created by 
 different rates of internal changes among the mole- 
 cules of the different metals. Instead of the mercury 
 thermometer which we employed in our heat exper- 
 iments, a junction of iron and German-silver wire 
 can be employed, one junction being immersed in 
 melting ice and the other in the hquid whose tem- 
 perature we wish to measure. It is best to com- 
 
THERMO-ELECTRICITY. 267 
 
 pare the indications of the galvanometer with those 
 of a standard mercury thermometer for certain 
 temperatures ; for the indications of the thermo- 
 electric junctions are subject to sudden fluctua- 
 tions, which have to be carefully guarded against. 
 By the use of a thermo-electrical junction, the ex- 
 periment to test the law // = S^Rt can be con- 
 ducted with the use of a comparatively small bat- 
 tery. 
 
 All the experiments on heat which have been 
 given should now be repeated, a thermo-electric 
 junction replacing the mercury thermometer. For 
 this purpose a sufficiently sensitive astatic galva- 
 nometer should be made. The coil for a galvanom- 
 eter of very low resistance can be made by cutting 
 a long strip, about five millimetres wide, from a 
 thin sheet of copper or brass, and coiling it into a 
 small flat spiral, interposing oiled paper between 
 the spires, and soldering thick connecting wires to 
 the ends of the spiral. Suspend in front of this 
 spiral, which is mounted vertically on the end of a 
 box, an astatic combination of needles provided 
 with a little plane or concave mirror. If a plane 
 mirror is used, a lens of long focus should be placed 
 directly in front of it, in the manner we have al- 
 ready described. The lower magnet of the astatic 
 combination should be made from a piece of watch- 
 spring, and should be one centimetre long and two 
 millimetres wide. This magnet is placed very close 
 to the center of the spiral, leaving it, however, suf- 
 ficient space to turn slightly. The magnet with 
 opposite poles should be placed near the periphery 
 of the spiral, directly above the lower magnet, and 
 rigidly connected with it by a hght, stiff filament of 
 13 
 
268 NEW PHYSICS. 
 
 glass (obtained by melting a glass tube and quickly 
 drawing it out into a narrow thread). Thick liquid 
 shellac is useful for connecting the magnets to the 
 glass filament. The suspension fiber should be ten 
 centimetres long. The entire galvanometer can be 
 inclosed in another box provided with a glass end. 
 This instrument can be easily made to measure one 
 fiftieth of a centigrade degree. 
 
 The use of the thermo-electric element in meas- 
 uring heat leads us to the subject of the distribu- 
 tion of energy in light of different colors, and thus 
 leads us indirectly to the application of the doctrine 
 of the conservation of energy to the subject of light. 
 
 By increasing the number of junctions of two 
 different metals, and maintaining one set of junc- 
 tions at one temperature — that of melting ice — and 
 submitting the other corresponding set to the tem- 
 perature we wish to measure, we can greatly in- 
 crease the sensitiveness of the arrangement, which 
 we then call a thermopile. It is found that junc- 
 tions of bismuth and antimony give a comparatively 
 large electro-motive force. 
 
 Experiment 147. — To obtain the efficiency of 
 a thermopile, we must measure the amount of fuel 
 in heat units we expend in producing a measured 
 strength of current, and then, by the application of 
 the law J H = S^ Rt, obtain the value of the cur- 
 rent in units of work. If the thermopile is heated 
 by a gas-burner, we must measure the amount of 
 gas burned, and compute its value in heat units. 
 
CHAPTER XIII. 
 
 SOUND. 
 
 The close relationship of light and heat to elec- 
 tricity having been seen, it is simpler to begin our 
 study of wave-motion by means of the phenomena 
 of sound ; and afterward, through the mechanical 
 conceptions that we have thus gained, to turn to 
 light as another manifestation of wave-motion. 
 
 Let us in the first place examine the motion of a 
 tuning-fork. When we press together the prongs 
 of the fork by means of the fingers, and suddenly 
 release them, the prongs spring back not only to 
 their original position, but to a point beyond, and 
 continue to vibrate to and fro, as we have seen in 
 the experiments in mechanics, in which we have 
 used the tuning-fork as a pendulum. The time that 
 is occupied in swinging from one side to the other, 
 and then back again, is called the periodic time, or 
 the period of vibration ; and the distance that any 
 point on the prong of the fork travels in executing 
 a vibration is called the amplitude of vibration for 
 this point. 
 
 Sound is a sensation in the ear, produced by the 
 vibratory motion of bodies upon some elastic me- 
 dium which transmits the vibration to the tym- 
 panum or membrane of the ear. The medium of 
 
2-jo NEW PHYSICS. 
 
 communication is usually the air. A person un- 
 der water, however, can hear sounds transmitted 
 through the water. Sounds can be heard farther 
 through water than through air. The more elastic 
 the medium, the swifter is the propagation of sound. 
 A vacuum does not transmit sound. 
 
 Experiment 148. — Attach a little toy bell to 
 the rubber cork of Experiment 20, and, having ex- 
 pelled the air by steam, incline the flask in order to 
 ring the bell. It will be found that the sound is 
 very faint. On letting in air, the ringing will be 
 plainly heard. 
 
 Experiment 149. — Compare two tuning-forks. 
 Clamp the two tuning-forks in two vises side by 
 side. Provide them both with little pointers. 
 Smoke a piece of glass and place it upon a horizon- 
 tal support beneath the prongs of the tuning-forks. 
 Adjust the forks in the vises until the pointers 
 just touch the smoked glass. Then, having set the 
 tuning-forks in- vibration with a violoncello-bow, 
 draw the smoked glass rapidly away from the 
 pointers and along the horizontal support. To do 
 this uniformly, the glass should have a narrow strip 
 of wood glued or cemented to one side, and this 
 wooden strip should be pressed against the edge of 
 the board which forms the horizontal support. 
 
 This experiment will show that the intensity or 
 pitch of the tuning-fork depends upon the number 
 of vibrations it niakes per second. It will be found 
 that the number of vibrations of the lower note are 
 less than those of the higher note. 
 
 Experiment 150. — Compare the vibrations of a 
 wire with those of a tuning-fork. Stretch a steel 
 wire between two points of a horizontal support, 
 
SOUND. 
 
 271 
 
 A B (Fig. 118). Attach the wire rigidly to one of its 
 supports, A, and weight the wire at its other end, 
 B, by allowing the wire to pass over a little cylinder 
 and allowing the stretching weights to hang ver- 
 tically. Place a little pointer at the middle of the 
 
 Fig. 118. 
 
 wire, m, and allow it to just touch the surface of 
 mercury in the little cup below it. Place an elec- 
 tro-magnet, E, above the middle of the half-length 
 of the steel wire. The wire will then vibrate. 
 Place another electro-magnet, E}, in the circuit, 
 and ascertain what length of wire A B will cause 
 a tuning-fork, F, placed very near the end of the 
 electro-magnet E^, to respond to the wire. It is 
 well to cover the surface of the mercury-cup with 
 a little alcohol and glycerine, to keep the surface 
 clean. 
 
 Experiment 151. — Make one tuning-fork re- 
 spond to another. Provide the prong of one tun- 
 ing-fork with a little pointer. Allow this prong to 
 just touch the surface of mercury in a little dish. 
 Place an electro-magnet above the other prong, and 
 arrange the electrical circuit so that it shall be 
 
272 
 
 NEIV PHYSICS. 
 
 closed when the pointer attached to the prong 
 touches the surface of the mercury. The tuning- 
 fork will then be attracted by the electro-magnet, 
 and the circuit will be broken, and the elalsticity of 
 the steel will quickly cause it to be made again. In 
 this way the vibration of the tuning-fork can be 
 maintained. Now arrange the second tuning-fork 
 with another electro-magnet, placed in the same 
 electrical circuit as the first electro-magnet. If 
 the two tuning-forks give the same note, they will 
 respond to each other. 
 
 Experiment 152. — Comparison of the vibration 
 of a tuning-fork with the vibration of a straight 
 rod. Clamp a tuning-fork beside a straight steel or 
 wooden rod. Provide both the fork and the rod 
 with metallic pointers, and adjust these pointers so 
 that they shall just rest upon a plate of smoked 
 glass. Set both the tuning-fork and the rod into 
 vibration, and draw the smoked glass away. Com- 
 pare the vibrations produced when the length 
 of the rod is varied, and ascertain the law of the 
 vibration of straight rods. Repeat the experiment 
 with a rod of different section, and in this way 
 study the influence of the section upon the rate of 
 vibration. It is evident that the rate of the tuning- 
 fork need not be known, since it is used relatively. 
 
 Experiment 153. — Investigate the laws of the 
 vibration of strings by means of a tuning-fork. 
 Clamp a tuning-fork by means of a vise in a hori- 
 zontal position, and arrange the prongs so that the 
 vibrations of the fork shall take place in a vertical 
 plane. Attach a silk thread to one prong of the 
 tuning-fork and a little tin cyUnder to the other end 
 of the string. Fill this cylinder with water, and 
 
SOUND. 
 
 273 
 
 place it in a larger jar filled with water. By draw- 
 ing off the water from the larger jar, either by a 
 siphon or a stop-cock, it is evident that the tension 
 on the string can be varied at will and by known 
 amounts. On exciting the fork with a bow, the 
 string will divide up into nodes and segments. 
 Vary the length of the string and also the tension 
 upon it, and ascertain its laws of vibration. 
 
 Experiment i 54. — Obtain the time of vibration 
 of a tuning-fork. Adjust the tuning-fork so that 
 the pointer with which one of its prongs is pro- 
 vided shall touch the blackened surface of a re- 
 volving cylinder. Arrange the pointer attached to 
 a pendulum-bob so that when the pendulum swings 
 the pointer shall also draw a line upon the black- 
 ened cylinder. Excite the tuning-fork and set the 
 pendulum in motion, and count the number of 
 vibrations of the tuning-fork between the swings 
 of the pendulum as they are indicated by the breaks 
 in the straight line drawn on the cylinder by the 
 pointer of the pendulum. Knowing the time of 
 swing of the pendulum, or the interval between the 
 breaks indicated by the pointer, we can obtain the 
 number of vibrations of the tuning-fork in a second. 
 The following apparatus will be necessary : 
 
 Obtain a wooden cylinder which is turned true 
 about an iron rod that serves as its axis. The 
 diameter of the cylinder should be about one foot, 
 its length six inches. Two holes, about four inches 
 deep and two inches wide, should be bored in the 
 cylinder, parallel to its axis, and filled with lead. 
 The iron axis of the cylinder should turn upon well- 
 oiled supports. A little handle screwed upon one 
 end of the cylinder, perpendicular to its axis, will 
 
274 
 
 NEW PHYSICS. 
 
 enable one to give a rapid motion to the cylinder. 
 The cylinder should be covered with a sheet of 
 well-glazed writing-paper, and then revolved gently 
 over a smoking keroseiie-lamp. On a stand placed 
 at the side of the cylinder, and across the top of the 
 box on which the axle of the cylinder rests, should 
 be screwed a little iron vise, in which the tuning- 
 fork can be adjusted. 
 
 Experiment 155.— Find the rate of a given 
 
 n 
 
 n 
 
 r 
 
 
 V 
 
 
 
 
 
 
 S 
 
 
 fi 
 
 
 * 
 
 Y 
 
 Fig. 
 
 119. 
 
 fork.* A silk 
 fiber stretches 
 from the prong 
 of the fork to a 
 slender vertical 
 spring, s (Fig. 
 119), fixed at its 
 lower end, while 
 its upper end 
 rests against a set 
 screw, ^. The two 
 surfaces in contact are platinized. Every vibra- 
 tion of the fork is transmitted to the fiber, which 
 causes the spring to open and close an electrical 
 
 * "Journal of the Franklin Institute," vol. Ixxii. 
 
 Fig. 120. 
 
SOUND. 27s 
 
 circuit in unison with the fork. Place this circuit- 
 breaker in the same circuit with a heavy pendu- 
 Uim which swings about twice a second. Ar- 
 range this pendulum so that it closes the circuit 
 during the time of a complete swing. This can be 
 done as' is shown in Fig. 120. La B is a bent lever 
 of metal, one arm of which is connected by a light 
 fiber of silk with the pendulum-rod, and the other 
 end, B, just rests upon a surface of mercury when 
 the pendulum is at rest. 
 
 Arrange in another part of the circuit a flexible 
 wire, c, which closes the circuit at d. Draw a strip 
 of paper, moistened with a mixture of potassium 
 iodide and starch, beneath this wire. While the cir- 
 cuit is closed by the bent lever connected with the 
 pendulum, a blue line is drawn on the paper. If, 
 however, the tuning-fork is vibrating, this blue line 
 will be broken up into dots, and the number of dots 
 gives the number of vibrations of the fork during the 
 time of swing of the pendulum. It is best to have 
 a little piece of platinum-foil beneath the strip of 
 paper on the surface d. It is evident that the rate 
 of movement of the paper must be such as to sepa- 
 rate the dots so that they can be counted. 
 
 This method can also be used to investigate the 
 transverse vibrations of wires. 
 
 Experiment 156. — Instead of a tuning-lork, use 
 the apparatus of Experiment 150, and prove the 
 following laws : 
 
 1. The number of vibrations in a given time 
 varies inversely as the lengths of the wire. 
 
 2. Their number varies also as the square root 
 of the tension of the wire. 
 
 In our study of the movement of pendulums we 
 
276 
 
 NEW PHYSICS. 
 
 have considered a vibration as the movement of the 
 pendulum-bob from its highest position on one side 
 of a vertical line to its highest position on the other 
 side of this line. In discussions of wave-motion, a 
 complete vibration is double this. Certain prelimi- 
 nary conceptions of harmonic motion are necessary 
 to our study of sound. 
 
 It will be noticed that the point A (Fig. 121) on a 
 rolling wheel draws a sinuous curve ; and, moreover, 
 
 the distance between one summit, A, and the next, 
 ^S is divided into parts AB,B C, CD, DA^, which 
 are equal and are reversed in position. The dis- 
 tance A O \s 
 
 €k 
 
 I 
 
 W 
 
 l| 
 
 111 fl, 
 
 the amplitude 
 of the vibra- 
 tion, as can be 
 seen by refer- 
 ence to the mo- 
 tion in a circle, 
 and the dis- 
 tance traveled 
 along the line 
 X X^, while 
 the point on 
 the wheel has moved over the entire circumference, 
 is called a wave-length ; thus, the distance A A^ 
 represents a wave-length. 
 
 Fig. 
 
SOUND. 
 
 277 
 
 The motion of particles on a circle illustrates the 
 expression " difference of phase." The position of 
 the point B at the end of the radius A ^(Fig. 122) can 
 be fixed by the angle it makes with a fixed line, C A. 
 The points B and B' make different angles, d and 6' , 
 with C A. The difference of phase is therefore pro- 
 portional to ^ — Q'. When the particles B and B' 
 pass through K at the same time, the difference of 
 phase becomes nothing. When one particle passes 
 through E while the other is passing through E' , 
 the difference of phase is 180°. The period of a 
 vibration is the time of moving from C completely 
 around the circle to C again. If t represents the 
 time of describing the angle 6, and T the period, 
 
 we have — = "t^- To an eye, /, placed in the plane 
 
 of a vertical circle upon which B moves, the parti- 
 cle B appears to move up and down a vertical di- 
 ameter E E'. 
 
 Suppose that the lines aa^, (5 ^1, etc., represent 
 equal parallel lines, and that we have simply vi- 
 bration of particles 
 along the lines a a^, 
 bb^, etc. (Fig. 123). 
 The position of any 
 particle at any time 
 with respect to any 
 quadrant of the cir- 
 cle whose radius 
 
 represents the amplitude of the vibration is called 
 the phase. The particle r and particle s have a dif- 
 ference of phase, since, on the supposition that both 
 are moving from left to right, the particle r will pass 
 through E from the quadrant C E to E D (Fig. 122) 
 
 a b 
 
 ^irm^ KT 
 
 K^J 
 
 a, K 
 
 Fig. 123. 
 
2/8 ^ElV PHYSICS. 
 
 later than the particle s. If the periods of vibra- 
 tion of the particles are all equal, and the difference 
 of phases also all equal, we have a simple harmonic 
 undulation. The wave-length may be stated as the 
 horizontal distance between one particle and the 
 next when they are in the same phase. A particle 
 at the highest part of its vibration is at the crest of 
 a wave, and a particle at the lowest part of its vi- 
 bration is in the trough of the wave. These particles 
 will again assume these positions when another wave 
 replaces the first wave, for the particles themselves 
 are not carried along by the waves ; they rise and 
 fall like boats as the waves of the sea pass under 
 them. 
 
 The distance that the waves travel in one period 
 is called a wave-length. If v denotes the velocity 
 of the propagation of a sound-wave, \ the wave- 
 length, and T the period, we have for uniform mo- 
 tion X = v T. 
 
 If we express 7" as — of the unit of time, we have 
 
 n 
 
 v = n\ or the distance traveled by the sound-waves 
 in a unit of time is n wave-lengths. The general 
 remarks that we have made in respect to wave- 
 motion apply whether the motion of the particle is 
 transverse to the direction of the wave or in the 
 direction of the wave. Sound-waves consist of lon- 
 gitudinal vibrations, while light is produced by 
 transverse vibrations. The propagation of a sound- 
 wave can be illustrated in the following manner : 
 
 Experiment 157. — Wind a brass wire around a 
 wooden cylinder of about one centimetre diameter, 
 making the spires very near to each other. In this 
 way make a spiral spring of about three metres long. 
 
SOUND. 
 
 279 
 
 Attach one end of this spiral spring to a thin, hollow- 
 box which is fixed to a table ; hold the other end of 
 the spiral in the hand and force the spirals togeth- 
 er through a short space near the hand. A wave 
 made of spirals compressed together, and of spirals 
 stretched apart, will travel along the bar and will be 
 reflected from the box back to the hand. The com- 
 pression of the spirals can represent the condensa- 
 tion, and the extension the rarefaction, of the air, 
 when a sound-wave travels through air. In the 
 brass wire the elasticity of the metal aids the propa- 
 gation of the wave. In air the elasticity of the air 
 performs the same function. 
 
 Experiment 158. — Hang up ten or twelve mag- 
 netized bars of steel, each by a bifilar suspension, 
 in line, with their opposite poles very near each 
 other but not touching each other. Set the end- 
 magnet to swinging in the direction of its length, 
 and its vibration will be transmitted from magnetic 
 particle to magnetic particle. In this experiment 
 the magnetism of each pole represents the elasticity 
 of the air. 
 
 Thus, in investigating wave-motion, as it is illus- 
 trated by the phenomena of sound, we need to 
 fix our ideas by the ideas of time and space and 
 velocity. To a person standing above a tuning-fork, 
 who could also see the to-and-fro excursions of the 
 fork, the particles would move along a line the 
 length of which gradually decreases as the sound 
 dies away. The same person lying on his back, and 
 looking up at a pendulum which is also executing 
 its swing to and fro, would see the pendulum-bob 
 move to and fro along a line which is also continu- 
 ally decreasing in length. Let one also look edge- 
 
28o NEW PHYSICS. 
 
 wise at a point on a wheel which is revolving uni- 
 formly : this point will move up and down a line 
 which is the diameter of the wheel. In this case 
 the amplitude of motion of the point is the radius. 
 We can fix the position of this point on the circum- 
 ference of the wheel at any time by knowing its 
 angular distance from the center of the circle, esti- 
 mating the angle from any fixed line or diameter of 
 the circle we may choose. The time that it takes 
 the point to travel around the circle from one point 
 back to this point again, passing through 360°, we 
 have seen, is called the period. 
 
 Experiment 159. — Cut a rectangular slit in a 
 piece of card-board ; move a pencil to and fro in this 
 slit in time with any pendulum, and at the same 
 time draw a sheet of paper beneath the pencil : we 
 shall obtain a sinuous curve. 
 
 Now we have seen in the case of pendulums that 
 the time of swing is independent of the amplitude ; 
 that is, a pendulum executes its swings, if they are 
 not very large, in the same time. The elasticity of 
 the prong of a tuning-fork serves to exhibit pendu- 
 lum-movements. We know that, under certain lim- 
 its, an elastic body when distorted returns to its 
 original position ; moreover, that the force of resti- 
 tution, or the force necessary to bring the body back 
 to its original position after a force has distorted it, 
 is directly proportional to the amount of distortion. 
 When a tuning-fork or any rod is vibrating, the 
 forces of elasticity urge the prongs toward the po- 
 sition of equilibrium. They vanish when the prong 
 passes through this position of equilibrium, and 
 they increase gradually as the prong advances on 
 the opposite side of the position of equilibrium. 
 
SOUND. 
 
 281 
 
 If we consider the manner of propagation of 
 waves, we perceive that one wave can interfere 
 with another. Let two waves of the same length 
 proceed in the same direction ; if they coincide in 
 their phases, they strengthen each other, for their 
 crests and troughs would then coincide. If, how- 
 ever, one wave has a phase one half a wave-length 
 behind the other, the waves neutralize each other 
 and no sound is produced. 
 
 Experiment 160. — Arrange glass or smooth tin 
 tubes, as shown in Fig. 124, connecting the T-joints 
 
 Fig. 124. 
 
 by rubber tubing. On sounding a tuning-fork at F, 
 no sound will be heard at E, if the length A B Cis 
 half the wave-length produced by the fork. 
 
 If the length ABC could be easily varied, and 
 if there were no errors due to irregularities and 
 friction of air in the tubes, this method would en- 
 able us to measure the wave-length of sound. Com- 
 pare the wave-length obtained in this way with that 
 obtained in Experiment 156. 
 
 If two waves proceed in the same direction, the 
 waves differing in wave-length, it is evident that 
 
282 TVSfT PHYSICS. 
 
 at some part of their path the crests of the two 
 waves may partially coincide and at another part 
 the phases may differ by a fraction of a wave-length. 
 Thus the two waves might coincide at A (Fig. 125) 
 and interfere at B C, just as two men walking be- 
 side each other with steps of unequal length, -and 
 
 Fig. 125. 
 
 not endeavoring to keep step, will fall into step and 
 out of step at regular intervals. The ear, listening 
 to two sources of sound giving forth waves of dif- 
 ferent lengths, will detect what are called beats, 
 which are produced by these coincidences and in- 
 terferences. 
 
 Experiment 161. — Support two equal tuning- 
 forks, which give about 250 vibrations per second, 
 over two vertical glass vessels. Sound the tuning- 
 forks and gradually fill the glass vessels with water. 
 The forks being equal, they will both be re-enforced 
 when the water stands at the same level in the two 
 equal vessels. Now load the prongs of one of the 
 forks with a bit of wax. The column of water 
 which will respond to the loaded fork must now be 
 changed. When it re-enforces the loaded fork, beats 
 will be heard, and a condensation from one fork 
 will reach the ear at the same time that a rarefac- 
 tion comes from thejather. If one fork should vi- 
 brate 100 times a second, and the other loi, there 
 would be one beat a second. The number of beats 
 
SOUND. 283 
 
 is equal to the difference in the number of vibrar 
 tions of the forks per second. 
 
 Experiment 162. — Velocity of sound in air or 
 in a gas. A tall glass cylindrical vessel is filled 
 with gas by displacement. Carbonic-acid gas can 
 be easily used, or even common street gas. While 
 the gas is flowing into the cylinder gently through 
 a rubber tube attached to a receptacle of the 
 gas, sound a tuning-fork over the mouth of the 
 vessel, and pour in water until the note of the fork 
 is greatly strengthened. The length of the column 
 of gas in the tube, multiplied by four, gives the 
 theoretical length of the sound-waves of the note of 
 the fork, in the gas which is employed. It is found 
 necessary to add to this one fourth of the diameter 
 
 of the tube, or v = 4n (l-\ ),in which (3? is diame- 
 ter of the tube, / length of column of gas, n rate of 
 vibration of fork. A tuning-fork giving from 250 to 
 300 vibrations per second can be employed. Take 
 the revolving wheels we have used in Experiment 49 
 and replace the more rapidly revolving one by a thin 
 disk of sheet-iron or card-board, and punch holes 
 about five millimetres in diameter in this disk around 
 concentric circles. Arrange a glass tube, one end 
 of which is connected with a rubber tube, opposite 
 any row of holes, and blow through the rubber tube 
 while the wheel is rapidly rotated. The pulses of air 
 thus produced will give a note which rises in pitch as 
 we increase the velocity of revolution of the perfo- 
 rated wheel. Place opposite to the row of holes 
 through which we blow, a glass tube open at one end 
 and closed at the other by a cork. Place this cork 
 so that the column of air in the tube shall re-enforce 
 
284 iVSW' PHYSICS. 
 
 the sound given by the pulses of air. We can main- 
 tain the revolution of the perforated wheel fairly 
 uniform by maintaining the loudest effect of the 
 re-enforcement. In this way we can compare the 
 sound of a tuning-fork with that of this apparatus, 
 which is called a siren. Knowing the number of 
 holes through which we blow, and ascertaining the 
 rate of the wheel which we turn, we can compute 
 the number of pulses of air which are sent into the 
 resonating tube in a second. This will give the 
 number of vibrations of the tuning-fork which pro- 
 duces the same note. It will be noticed that the 
 effect on the air is the same as if a vibrating 
 body was setting it in motion. The pulse of air 
 driven through an opening causes a condensation, 
 and the elasticity of the air in turn produces a 
 rarefaction. 
 
 A description of a more complicated instrument 
 called a siren can be found in any work on natural 
 philosophy. The instrument we have described 
 embodies all the principles of a siren. 
 
 It has been proved that v ^= y jy,v being the 
 
 velocity of sound, E the elasticity of the medium, 
 and D its density. This formula is sensibly true 
 for substances which radiate heat readily, but air 
 and most gases are very poor radiators ; and La- 
 place showed that the value for v must be multi- 
 plied by a factor which is the ratio of the specific 
 heat at constant pressure to that at constant volume. 
 The compression of the atmosphere in front of a 
 sound-wave increases its temperature, while the 
 rarefaction or extension behind it lowers the tem- 
 perature. This increase of temperature increases 
 
SOUND. 
 
 285 
 
 the elasticity in front, and the diminution of tem- 
 perature diminishes the elasticity of the air behind, 
 the wave-surface. The velocity depends upon this 
 difference of stress, and is consequently greater 
 than if the heat were quickly radiated. The ratio 
 of the two specific heats is 1.41. Hence the ve- 
 
 fE 
 locity of sound in a gas h,v =■ \ -^1.41. 
 
 Experiment 163. — The change of temperature 
 due to compression of air and its rarefaction can be 
 shown by the following apparatus: A tube, of 
 smooth internal bore, is 
 bolted air-tight upon a 
 support (Fig. 1 26), a leath- 
 er washer being placed 
 beneath the flange of the 
 tube and the support. 
 Pass through the sup- 
 port two wires leading 
 from a thermo-electric 
 junction ; the other junc- 
 tion is immersed in wa- 
 ter of a constant tem- 
 perature outside. Con- 
 nect these junctions with 
 a low-resistance astatic 
 
 galvanometer, G. On quickly compressing the air 
 in the tube, by means of the air-tight piston, it will 
 be seen that the air within the tube is heated, and 
 on lifting the piston it is cooled. 
 
 In this experiment, suppose that we have a tun- 
 ing-fork instead of the piston, and a resonating tube ; 
 it is evident that we must have heat attending the 
 compression produced in front of the sound-wave, 
 
 Fig. 126. 
 
286 NEW PHYSICS. 
 
 and loss of heat accompanying the rarefaction be- 
 hind the wave. 
 
 The apparatus of Experiment 163 suggests how 
 the mechanical equivalent of heat might be obtained 
 by its use. Suppose the apparatus, before it is bolted 
 to a support, should be placed in water, and the pis- 
 ton should be moved rapidly backward and forward 
 one hundred strokes, and the increase of tempera- 
 ture of the water produced by the friction of the 
 piston against the sides of the tube be measured. 
 Then replace the apparatus upon its support, and 
 measure, by means of the thermo-electric junctions, 
 the increase in temperature after one hundred 
 strokes. After subtracting the amount of heat due 
 to friction, we have the heat due to the work done 
 in compressing the air. The mechanical equivalent 
 of heat has been obtained in this way. If we could 
 make a tuning-fork vibrate in a closed tube and 
 could measure the increased temperature of the air, 
 and also the work done on the tuning-fork and its 
 increased temperature, we could also measure the 
 mechanical equivalent of heat. 
 
 Sound is transmitted through metals much more 
 rapidly than through air or gases. The wire or 
 string telephone is a familiar illustration of this. 
 Connect the centers of the bottom of two tin cups 
 by means of an iron wire thirty to a hundred feet 
 long, supporting the wire if necessary at one or two 
 points by fine thread. On speaking into one cup, a 
 listener at the other can hear even a whisper. The 
 interval between sounds spoken at one end and 
 heard at the other, both through the iron wire and 
 the air, is easily distinguishable when the distance 
 is considerable. 
 
SOUND. 287 
 
 We are familiar with the difference between elas- 
 tic and non-elastic substances. The pendulum-move- 
 ments of the particles of an iron wire must be very 
 rapid ; and, moreover, if the wire is very elastic, the 
 particles in executing this movement must come to 
 rest in their original position. The disturbance 
 caused by a wave of sound in metals is very small, 
 usually far within the limit of any permanent dis- 
 tortion of the metal. By a suitable weight attached 
 to a wire, we can easily stretch an iron wire so that 
 it will not contract to its original length. It will 
 then be found to transmit sound less quickly than 
 before : its elasticity has been altered. The elas- 
 ticity is therefore connected with the relative po- 
 sition of the particles of the metal. The time of 
 swing of these little particles considered as pendu- 
 lums has been lengthened. In order to estimate 
 elasticity, therefore, our experiments must be made 
 on the metal within the limits of its deformation. 
 In other words, the particles must be able, after 
 swinging or being pulled out of their position of 
 rest, to return to this position when the motion im- 
 parted to them has ceased. It is customary, there- 
 fore, to measure elasticity by the weight in kilo- 
 grammes which is necessary to hang on a wire of 
 one square millimetre section in order to double its 
 length. This weight is called the modulus of elas- 
 ticity. 
 
 Experiment 164. — Another method of showing 
 the propagation of longitudinal vibrations is that 
 devised by Kundt (Fig. 127). This consists of a 
 glass tube five or six feet long and one and a half 
 inch in diameter, and another tube of glass of the 
 same length, but one half an inch in diameter. One 
 
288 NEW PHYSICS. 
 
 end of the smaller tube is closed with a piece of 
 cork just large enough to slip along the wider tube; 
 the smaller tube also passes through a rubber cork 
 which closes one end of the larger tube ; the other 
 
 end of the large tube is closed by another rubber 
 cork, the position of which can be regulated. The 
 smaller glass tube is then pushed exactly half its 
 length into the larger tube, and clamped at its 
 middle point. A small quantity of lycopodium 
 powder is sprinkled in the free portion, A B, of the 
 tube. On exciting the free end, m, of the smaller 
 glass tube by means of a dampened or resined cloth, 
 it gives a note corresponding to its length ; the lyco- 
 podium collects in little heaps which are separated 
 by spaces of glass from which the powder is thrown ; 
 the heaps represent a node where there is no vibra- 
 tion. The rubber cork A is moved until no dust 
 collects at its position. The column, A B, of air is 
 then measured, and the number of spaces between 
 the nodes counted. We then compare the note 
 given by the small glass tube with that of a pitch- 
 pipe or a siren. We shall then have, if Z = length 
 of the column of air, n the number of spaces between 
 the nodes, Wj the vibration note of the glass, the 
 
 2 L 
 wave-length in air = — -, and the velocity = wave- 
 
 " 2 Ln^ 
 
 length X number of vibrations = 
 
 n 
 
 Care must be taken in this experiment to stroke 
 
SOUND. 289 
 
 the smaller glass tube exactly in the direction of the 
 axis of the larger tube, otherwise a pressure will be 
 brought upon the end of the larger tube which will 
 break it. The same methods can be employed to 
 obtain the velocity of sound in solids. 
 
 Experiment 165.— Instead of the glass rod of 
 the previous experiment, a rod of the substance in 
 which we wish to determine the velocity is substi- 
 tuted for the small glass rod. It is better, if possi- 
 ble, to make the new rod of the same length as the 
 small glass rod. The new rod is excited in the same 
 manner. 
 
 If we call / the length of the rod, l^ the length of 
 
 the vibrating column of air, n the number of spaces 
 
 between the nodes, v the velocity of sound in air at 
 
 the time of the experiment, v^ the velocity in the 
 
 2 L V I n 
 
 rod, we shall have v w-^^ — : 2 /, or, z^i = — — -. 
 
 n /i 
 
 Experiment 166.— Another method of obtain- 
 ing the velocity of sound by longitudinal vibrations 
 consists in altering the length of a wire which is 
 stretched between two points until the note excited 
 by rubbing with a cloth covered with resin (or rub- 
 bing with the thumb and forefinger covered with 
 powdered resin) is equal to that of a tuning-fork or 
 of a given note of a pitch-pipe. The note excited 
 should not be a high one, for it is difficult to distin- 
 guish the fundamental of a high note from its oc- 
 tave. 
 
 The method of longitudinal vibrations gives us 
 also the means of obtaining the modulus of elasticity 
 
 
 of a substance, for we have z^ = y-- = 2 n L. 
 Hence, E = i^r? 1} D. Since weight W — volume 
 
290 
 
 NEW PHYSICS. 
 
 multiplied b}' density multiplied by g the attraction 
 
 W 
 of gravitation, or, W = VD g, we see that D= — ; 
 
 if, then, we call W the weight of a unit volume of 
 
 W 
 the substance, we have D = — {v bemg equal to 
 
 . , „ ^ A.n'DW ^. 
 unity). Hence E = ■- bince, in expressions 
 
 for the modulus, the millimetre has been chosen as 
 the unit of length, and the kilogramme as the unit 
 of weight, it will be necessary to express Was the 
 weight of a cubic millimetre in kilogrammes. 
 
 We have hitherto measured the velocity acquired 
 by bodies falling under the influence of gravitation. 
 The same methods as those already adopted can be 
 used to determine the time between the passage of 
 a body between any two points, moving with any 
 measurable velocity. Thus, the velocity of a rifle- 
 ball can be found by a suitable chronograph which 
 registers the time of its leaving the rifle and the time 
 it arrives at a given point. It can be made to break 
 an electrical circuit at the moment of its departure, 
 and make the circuit again on its arrival at a given 
 place. The length of time it takes a wave of water 
 to reach a given point, distant a measured interval 
 from the point of disturbance, can be readily meas- 
 ured in the same manner. The interval which 
 elapses between the actual discharge of the rifle and 
 the sensation of its report can also be measured. 
 This interval is evidently the time that it has taken 
 the sound-wave to travel through the air. 
 
 Experiment 167. — Measure the velocity of the 
 sound-wave in air. For this experiment the galva- 
 nometer used in Experiment 71 can be employed. 
 
SOUND. 
 
 291 
 
 At a measured distance a sounding-board can be 
 struck, the blow making a sound and breaking at 
 the same time an electrical circuit connected with 
 the galvanometer of the experiment. The moment 
 the sound is heard at the instrument, the circuit can 
 be made again, and the time between breaking and 
 making the circuit can be estimated from the swing 
 of the needle. The personal error of the observer 
 must be estimated. This is the error that each indi- 
 vidual makes in not closing the current on the instant 
 of the arrival of the sound to his ears. It is called the 
 personal error, and can be eliminated by receiving 
 the sound from different distances. Another method 
 of measuring the velocity of sound is as follows : 
 
 Experiment 168. — Hang up a pendulum of suit- 
 able length. Place a graduated arc behind it so 
 that the pointer connected with the pendulum may 
 pass over this arc. Station an observer near this 
 pendulum. At a distance of several hundred feet 
 place another observer, provided with a telescope, 
 a hammer, and a board. At the instant this latter 
 observer perceives tlirough the telescope the pend- 
 ulum swinging over the lowest point of the arc, let 
 him strike the board with the hammer. When the 
 observer near the pendulum hears the blow, the 
 pendulum will be at some other point of its swing. 
 Now let the first observer move toward or away 
 from the pendulum until the sound of his blow 
 comes to the observer, when the pendulum is again 
 at the lowest point of its swing. It has then taken 
 the sound just one swing of the pendulum to trav- 
 el over the distance between the two observers. 
 Knowing the time of the pendulum and this distance, 
 the velocity of sound can be obtained from the ex- 
 '4 
 
292 
 
 NEW PHYSICS. 
 
 pression S=vt, in which 5 is distance, v the veloci- 
 ty of sound, and t the time. 
 
 Experiment leg.^Obtain the velocity of -sound 
 in bars of wood, for instance, from longitudinal 
 vibrations. Clamp a long bar of wood, of somewhat 
 small section, in a horizontal position, by means 
 of a vise, at its middle point. Attach a small, flex- 
 ible stylus at right angles to its extremity. Adjust 
 this stylus so that it just touches a plate of smoked 
 glass. By its side also adjust the stylus connected 
 with a fork of known number of vibrations. On strok- 
 ing the other half of the wooden rod by a cloth 
 or piece of leather covered with resin, longitudinal 
 vibrations can be excited at the same time that the 
 tuning-fork is sounded. On slipping the plate of 
 smoked glass beneath the two vibrating bodies, and 
 comparing their vibrations through the same spaces, 
 we can obtain the number of vibrations of the wood- 
 en bar, and hence the velocity of sound in it. 
 
 In order to analyze the sounds of a tuning- 
 fork, we have made it draw its vibrations on a 
 rapidly -moving plate of smoked glass. By properly 
 exciting the fork we can superpose upon the funda- 
 mental vibrations of the fork other vibrations, which 
 are called harmonics. This is often accomplished as 
 follows : 
 
 Experiment 170. — Procure a fork of considera- 
 ble amplitude of vibration, so that its record is read- 
 ily obtained. Ex- 
 cite it by drawing 
 the bow across the 
 prongs near their 
 liiii^^ijjj ' I L ' /j' i I ends, and, while it 
 
 Fig. 128. is still sounding. 
 
 
SOUND. 
 
 293 
 
 draw the bow also across the base of the V-shaped 
 portion of the fork. The fundamental vibration 
 will have harmonics superposed upon it, similar to 
 those represented in Fig. 128. 
 
 Another way of showing the character of a sound- 
 wave is by means of a rotating mirror ; the method 
 of using this can be illustrated by an apparatus 
 which illustrates the character of the beat of the hu- 
 man pulse. 
 
 Experiment 171. — Stretch a string or wire 
 tightly between two vertical supports. Twist a stiff 
 light pointer around this string, so that one end of 
 a little cross-bar at the shorter end of the pointer 
 shall press somewhat tightly upon the pulse of the 
 wrist ; provide the other end of the pointer with a 
 mirror ; reflect a beam of light from the mirror to 
 a little vertical mirror placed at the center of the 
 whirling table used in the experiments on accelera- 
 tion of gravity, and study the appearance of the 
 sinuous curve in the revolving mirror. 
 
 Instead of studying the impulses or beats of the 
 heart, suppose we should communicate the impulses 
 of the air, produced by a source of sound, to a mem- 
 brane whose movements to and fro, synchronous 
 with the impulse of the air, should be made manifest 
 by reflecting a beam of light from a little mirror 
 placed upon the membrane to the same revolving 
 mirror which we have used in the preceding experi- 
 ment. This method might be used, but a better one 
 is the following — a simple form of which is given in 
 Mayer's " Sound," page 157: 
 
 Experiment 172. — " Take a piece of pine board, 
 A, one inch (25 millimetres) thick, one and a half 
 inch (38 millimetres) wide, and nine inches (22.8 
 
294 
 
 NEW PHYSICS. 
 
 centimetres) long. One inch from its top bore a 
 shallow hole, one inch in diameter and one eighth 
 of an inch deep. Bore a like shallow hole in the 
 
 Fig. 129. 
 
 block B, which is three fourths of an inch thick, 
 one and a half inch wide, and two inches (51 
 millimetres) long. Place a half-inch center-bit in 
 the center of the shallow hole in A, and bore with 
 
SOUND. 
 
 295 
 
 it a hole through the wood. Into this fit a glass 
 or metal tube, as shown at E.^ Bore a three-six- 
 teenth-inch (five miUimetres) hole obliquely into the 
 shallow hole in B, and into this fit the glass tube C. 
 Then bore another three-sixteenth-inch hole directly 
 into the shallow hole in B. Into this hole fit a glass 
 tube, bent at right angles, and one end drawn out 
 into a fine tube. Get a small piece of the thinnest 
 sheet-rubber you can find, or a piece of thin linen 
 paper, and having rubbed glue on the board A, 
 around the shallow hole, stretch the thin rubber or 
 paper over the hole, and glue it there. Then rub 
 glue on the block B, and place the shallow hole in 
 this block directly over the shallow hole in A, and 
 fasten ^ to ^ by wrapping twine around these 
 blocks. Thus you will have made a little box di- 
 vided into two compartments by a partition of thin 
 rubber. Fasten the rod A to the side of a small 
 board, so that it may stand upright. Attach a piece 
 of large-sized rubber tube to the glass, tube E, and 
 into the other end of the tube stick a cone, made by 
 rolling up a piece of card-board, G, so as to form a 
 cone eight inches long, and with a mouth two inch- 
 es (5 1 millimetres) in diameter." On singing into 
 this cone and revolving the mirror, the little flame 
 will be seen to respond to the impulses upon the 
 membrane of rubber, or the partition of paper, and 
 the form of different impulses can be studied by ob- 
 serving the serrated band of light in the revolving 
 mirror. With this apparatus study the sound of 
 do, re, mi, fa, sol, la, si, do. 
 
 Experiment 173. — A suitable resonator can be 
 made cheaply by making a tin tube, T (Fig. 130), 
 slide into another, T' , and providing T' with a rub- 
 
296 
 
 NEW PHYSICS. 
 
 Fig. 130. 
 
 ber tube and ear-piece. The length of the column 
 of air which re-enforces any note sounded over T 
 can be read ofE on the side of the tube T. A series 
 
 of these resonators, of dif- 
 ferent diameters, can then 
 be used to analyze the har- 
 monics or overtones of any 
 source of sound. 
 
 In our study of sound 
 ■ m t u ttr-m-i ^^ have thus far treated it 
 
 !Ar^^ as simply a question of ve- 
 il locities of propagation of 
 wave-motion, and we have 
 employed the simple formu- 
 las of uniform motion, the 
 principal one of which is 
 space = velocity x time, or S =^ vt. In mechanics, 
 after treating the subject of velocity in reference to 
 the space passed through, we dealt with the prob- 
 lems of the combination of several velocities, either 
 acting in the same line or in lines inclined at differ- 
 ent angles. In the study of wave-motion of sound 
 we have also to consider the composition of veloci- 
 ties. 
 
 The motion of a pendulum can be likened to the 
 motion of a nail in the rim of a wheel which is spun 
 with uniform velocity upon a fixed axle. The swing 
 of the pendulum would then correspond to the di- 
 ameter of the wheel. In the case of the wheel we 
 can estimate the position of the nail at any time by 
 its angular position with reference to the center of 
 the wheel : this we call its phase. If we had two 
 wheels with two nails, one in each rim, we could 
 estimate their position relative to each other by 
 
SOUND. 
 
 297 
 
 measuring their angular distance from the horizontal 
 diameter of the wheels. If these angular distances 
 are equal and of the same sign, the nails would be 
 in the same phase. 
 
 If we could see the motions of the prongs of a 
 tuning-fork, looking vertically down upon its ends, 
 we should see a bright bead upon the fork, moving 
 in the same way as the nail upon our wheel, and by 
 comparing this motion with that of a point on a cir- 
 cle, of which the diameter represents the swing of 
 the tuning-fork, we could compare the periods of 
 two tuning-forks. 
 
 Instead of lying on our back and watching the 
 movement of a pendulum, we can make a pendulum 
 draw -its own movement ; and, 
 by combining two pendulums, 
 we can obtain their combined 
 movement. 
 
 Experiment 174. — To the 
 middle of the loop formed by 
 a string (Fig. 131), the ends of 
 which, a a', are attached at some 
 distance apart, attach another 
 string, s. At the extremity of 
 this latter string tie a tin cup 
 provided with a small pin-hole 
 in its bottom. This apparatus 
 can be made to move so as to 
 describe two lines at right an- 
 gles with each other, which rep- 
 resent the amplitude of the pen- 
 dulum whose length is de, and 
 the pendulum whose length is n d. If the vessel at 
 d is drawn a little out of the direction of the swing 
 
298 
 
 NEW PHYSICS. 
 
 of the pendulum de, the sand will describe figures 
 which represent the composition of the two pendu- 
 lum movements. 
 
 It is evident that, if we could combine the move- 
 ments of two tuning-forks in this manner, we could 
 obtain similar figures. Before showing how this 
 can be done, let us turn to the methods of mechanics 
 and devise a machine by which we can draw the 
 curves which result from the combination of two 
 uniform motions or pendulum vibrations at right 
 angles with each other. 
 
 Experiment 175. — Upon a board (Fig. 132), pro- 
 vided with supports, so that it can be made to stand 
 
 vertically, fix 
 two wheels, A, 
 B, upon axles. 
 Connect these 
 wheels by a 
 belt, so that 
 the movement 
 of one is ac- 
 companied by 
 a movement of 
 the other. To 
 these wheels 
 attach two 
 crank-arms or 
 piston-rods. 
 The rod at- 
 tached to the wheel B moves a stage, s, upon 
 which a plate of smoked glass, g, is mounted ver- 
 tically, to and fro through a suitable amplitude. 
 The rod attached to ^, in the same way, moves a 
 slide, 0, in guides, n n^, to and fro in a direction at 
 
 Fig. 132. 
 
SOUND. 
 
 299 
 
 right angles with the movement of the rod con- 
 nected with B. A stylus attached to the slide 
 will then draw on the smoked glass the figures re- 
 sulting from the different rates of movements of the 
 wheels A and B. The amplitudes can be changed 
 by altering the point of attachment of the piston- 
 rods, and different wheels can be slipped upon the 
 two axles. A little wheel, connected with a slide at 
 m, can be clamped in different positions, and serves 
 to tighten the belt. 
 
 Experiment 176. — A small lens, about one centi- 
 metre in diameter, is fixed by bees- wax to the prong 
 of one tuning-fork F (Fig. 133), which is made to 
 vibrate in a ver- 
 tical plane. The 
 prong of anoth- 
 er tuning-fork, 
 F', which is 
 made to vibrate 
 in a horizontal 
 
 plane, is black- iteT^. 
 
 ened,andasmall 
 bright dot is made on the blackened surface by a 
 scratch. On looking through the lens, a curve will 
 be seen, which is due to the composition of the 
 vibrations of the two forks. Load the prongs of 
 one fork by sticking small weights to them until the 
 curve is a figure of 8. In this case, one fork gives 
 the octave of the other. Prove this by comparing 
 the vibrations of the fork by Experiment 154. 
 
 Our study of sound-waves, therefore, resembles 
 our study of velocity and of motion. We have been 
 concerned principally with the question. How many 
 vibrations a second does a sounding body make? 
 
300 
 
 NEW PHYSICS.. 
 
 We have found it necessary to use certain terms, 
 such as pitch, amplitude, phase, and wave-length, in 
 order to fix our ideas, and to represent what takes 
 place when waves interfere with each other. The 
 graphical methods which we have used illustrate 
 the combination of periodic motion, and the sound- 
 ing-flames ''illustrate the fact that evei'y wave of 
 sound consists of a fundamental wave upon which 
 are superposed smaller waves, just as a long wave 
 on the surface of water can carry smaller waves or 
 crispations with it. The discussion of the charac- 
 ter and influence of these superposed waves upon 
 the nature of the sound given forth by a vibrating 
 body would carry us into the subject of music, and 
 we have, therefore, merely indicated the method of 
 study which would serve to give us the relative 
 value of sounds. 
 
 Since we are studying motion and energy, the 
 question now arises, How can we obtain the energy 
 of a vibrating body ? This is the question we asked 
 ourselves as soon as we were able to measure the 
 velocity of a body acted upon by a force. 
 
 If we strike together two pebbles in air; waves 
 of sound spread out from the source of sound in all 
 directions, forming spheres, the surfaces of .which 
 pass through the alternate condensation and rare- 
 faction of the air. These spheres are continually 
 enlarging. Each wave carries with it its original 
 amount of energy, and since this vibratory energy 
 is shot forth, so to speak, in radiating lines, a cone 
 of radiating lines would be cut by the continually 
 enlarging spheres, and the same amount of energy 
 would be carried through the successive intersec- 
 tions of the surfaces of the spheres and the cone. 
 
SOUND. 
 
 301 
 
 These surfaces are inversely as the squares of the 
 radii of the spheres. The intensit)'^ of sound propa- 
 gated through equal areas on any two spheres 
 would therefore be inversely as the squares of the 
 distances of those equal areas from the source of 
 sound. 
 
 Experiment 177. — Arrange a microphone by 
 gluing a piece of smooth carbon to the center of a 
 thin circular disk of paper or sheet-metal (Fig. 1 34), 
 and, suspending against this another piece of car- 
 
 FlG. 134. 
 
 bon, connect these two pieces of carbon with a tele- 
 phone and a battery. Move the two sources of 
 sound until no sound is heard in the telephone : the 
 intensities of the source of sound will then be equal 
 at this distance of the microphone. We can thus 
 compare one source of sound with another whose 
 intensity is known. 
 
 A similar method to the above is used to com- 
 pare the intensity of two lights. 
 
 Although it is comparatively easy to compare 
 the relative intensity of two sounds, it is difficult to 
 
302 ^EW- PHYSICS. 
 
 obtain an absolute value of the amount of energy- 
 due to vibration. The work done by pendulums 
 moving under the influence of gravitation can be 
 readily estimated. Most vibrating bodies, however, 
 are complex in their structure. The work done 
 may be called internal work, and we can not meas- 
 ure it so simphr as we have done in the case of the 
 simple pendulum. 
 
 If we take any particle of a string which is 
 vibrating transversely, we see that, as it swings to 
 and fro across a horizontal line, on one side this 
 line it is gaining kinetic energy and losing poten- 
 tial energy, and on the other side of the horizontal 
 line this operation is reversed. The analogy be- 
 tween the vibrations of a particle of the string and 
 that of a pendulum is complete. There are, how- 
 ever, various considerations of elasticity of the 
 vibrating body which make the calculation of the 
 transformation of energy in the case of vibrating 
 bodies, like this string or the prongs of a tuning-fork, 
 somewhat difficult. These 
 calculations would lead us 
 into the subject of molecular 
 physics. We soon perceive a 
 difference in the transmission 
 of energy by means of vibra- 
 tions. We can apply the force 
 directly to keep up the energy 
 of vibration, or we can set a 
 body in vibration by transmit- 
 irie_ i35_ ting the energy through the 
 
 intervening media. 
 Experiment 178.— Hang up two equal balls (Fig. 
 135), by strings of the same length, from the same 
 
SOUND. 
 
 303 
 
 string which is stretched between two points of sup- 
 port ; hold one ball, and set the other to swinging ; 
 then remove your hand from the first ball : it will be 
 noticed that in a short time the second ball will give 
 up its energy to the first ball and come to rest. In 
 turn the first ball will communicate its energy to 
 the second and come to rest, and this operation will 
 be repeated until the friction and air resistance will 
 consume all the energy that has been communi- 
 cated to the system. 
 
 Experiment 179. — Hang up two long magnets 
 by equal bifilar suspensions, so that their opposite 
 poles may be very near but not touching each other. 
 Set one magnet to swinging, and hold the other for 
 a moment : we shall perceive here, again, a transfor- 
 mation of energy from one magnet to the other by 
 sympathetic vibration, principally through the me- 
 dium which transmits magnetic pulls and pushes. 
 
CHAPTER XIV. 
 
 LIGHT. 
 
 The phenomena of heat and light are so closely 
 allied .that it is impossible to regard them as sepa- 
 rate manifestations of energy. A heated substance 
 gradually increases in luminosity as more work is 
 done upon it, and finally emits a bright light. We 
 can measure the heat produced in terms of the work 
 that is done; there is an exact equivalence between 
 the two. We seem, therefore, to have the light 
 without the expenditure of any external work. The 
 light of a candle can be seen several miles, but its 
 heat can be measured only within a few feet of the 
 flame. By passing an electrical current through a 
 fine platinum wire or filament of carbon, we can 
 produce a light equivalent to twenty candles. We 
 can measure the heat equivalent of the current, and 
 we find that it is equivalent to the work done in 
 producing the current. In this case also we seem 
 to have the light without the production of work. 
 This is merely because our means of measuring the 
 energy in the shape of light are not delicate enough 
 to measure it in comparison with the enormous 
 amount of energy sensible in a body raised to the 
 temperature of red heat. 
 
 The common experiment of bringing the sun's 
 rays to a focus by means of a lens, and burning 
 
LIGHT. 
 
 305 
 
 paper at this focus, shows us that light and heat 
 invariably accompany each other in the case of the 
 energy transmitted to us from the sun. The sim- 
 plest mechanical analogy that occurs to us is that of 
 the blow of a six-hundred-pound shot against a bal- 
 listic pendulum, accompanied by the blow of a small 
 rifle-ball, both coming with the same velocity. It is 
 evident that the ballistic pendulum which would 
 measure the energy developed by the blow of the 
 six - hundred - pound shot would not be sensitive 
 enough to measure that of the accompanying rifle- 
 ball, although the effects produced by the rifle-ball 
 might be perfectly sensible in other ways than in 
 producing a swing of the large mass of the pendu- 
 lum. In the same way a large wave in the ocean 
 may bear upon its surface many smaller waves 
 which move with it and with the same velocity. 
 The large wave can lift a steamship of many tons 
 weight ; while the smaller waves, producing no ap- 
 preciable effect upon the steamship, are yet doing 
 work in lifting the particles of the water. 
 
 The conception that the sun's energy comes to 
 us in the form of waves seems to be the only admis- 
 sible one. The conception of wave-motion requires 
 also that of some medium in which this wave-motion 
 can take place. Thus we iare led to the hypothesis 
 of an ether which fills all space, and is found even 
 between the particles of solid bodies. The energy 
 of the sun is transmitted to us by wave-motion, the 
 large waves giving more of the sensation of heat 
 than of light, while the shorter waves give more the 
 sensation of light than of heat, both the sensations 
 of light and heat depending upon the length of 
 waves which are propagated in the ether, accept- 
 
3o6 NEW PHYSICS. 
 
 ing the hypothesis of an ether and the establishment 
 of waves in it by the sun. 
 
 In our study of light we are necessarily occu- 
 pied at first in tracing the path of the rays of light 
 from various objects to our eyes, and also in observ- 
 ing the effect of the bending of these rays of light 
 by different media. Thus, our study at first is a 
 study of diagrams or configurations, and is neces- 
 sarily geometrical.- It is analogous to our study of 
 diagrams of velocities, and at first does not intro- 
 duce the idea of energy. Since we use lenses and 
 prisms to concentrate or to break up beams of light 
 in order to study the distribution of energy in a 
 beam of light, it is necessary to have clear ideas in 
 regard to the action of lenses and prisms. 
 
 In the first place, we notice that rays of light 
 travel in straight lines ; that a smooth surface, like a 
 mirror or unruffled water, reflects the ray that im- 
 pinges upon the surface, just as a hard wall reflects 
 an elastic ball that strikes it at an angle. If the sur- 
 face is roughened, each little facet of the surface 
 diffuses the light by reflection, and only certain par- 
 ticles reflect rays to the eye. If all surfaces were 
 brilliantly polished, we should not be able to see the 
 form of objects distinctly, for we should notice 
 merely lines of great brilliancy such as are seen on a 
 polished metallic cylinder in the sunlight. Since pol- 
 ished surfaces reflect light in different directions, a 
 person on one mountain can converse with another 
 person on a distant peak by flashing the image of 
 the sun, by means of a mirror, to the eye of the dis- 
 tant observer, having previously arranged an alpha- 
 bet of short and long duration of flashes, similar to 
 the Morse telegraphic alphabet. A piece of plate- 
 
LIGHT. 
 
 30; 
 
 glass will give one reflection from its first surface 
 and several from its second surface. One ray, be- 
 coming imprisoned, so to speak, between the two 
 surfaces of the glass, bounds backward and forward, 
 giving what is called multiple reflections. A simi- 
 lar case of internal reflections in a tube is afforded 
 by the chimney of a student-lamp. Concentric 
 rings of light and shade will be seen above the 
 chimney on the wall. Holding a candle in front of 
 a mirror and looking at its reflection at a considera- 
 ble angle of reflection, several images of the candle 
 will be seen ; these images are due to the reflections 
 from the inner surface of the glass. 
 
 Experiment 180. — To prove that the angle of 
 incidence is equal to the angle of reflection, place a 
 piece of looking-glass in a vertical position at the 
 center of a large graduated circle (a paper or horn 
 protractor, such as surveyors use, can be employed), 
 or a circle divided into equal parts by a pair of 
 compasses. Attach a long pointer to the little mir- 
 ror, and observe that, when the mirror is turned 
 through a certain angle, a beam of reflected light 
 moves through twice this angle. 
 
 Experiment 181. — Place a screen, B (Fig. 136), 
 in front of a lamp, with a little pin-hole in the black- 
 ened lamp-chimney. Move a lens (a spectacle-lens 
 will answer) between the lamp and the screen, B, 
 until an image of this illuminated pin-hole is seen 
 sharply defined on the screen. Then place a little 
 mirror, a in, between the lamp and B, and move 
 the lamp along a line perpendicular to the direc- 
 tion of the mirror, making the distance C£ = 
 D E. A sharp image will still be seen on the screen 
 B. This shows that light reflected from a mirror 
 
308 
 
 NEW PHYSICS. 
 
 appears to come from a source as far behind the 
 mirror as the real source is in front. 
 
 \^/ 
 
 c/ 
 
 FlG. 136. 
 
 Experiment 182.^ — Compare the areas of the 
 three screens A , B, and C, which in turn intercept all 
 the light that comes from a small opening in a 
 screen placed in front of a lamp, and also see how 
 these areas are related to their distances from the 
 hole D in the screen. Let A be one foot from the 
 screen, B two feet, and C three feet. The screen 
 at the distance of one foot intercepts all the light 
 that falls upon it. 
 
 If we call the whole amount of light that is dis- 
 tributed over the area of ajiy one circle M, we shall 
 
 M 
 have in each square centimetre of A, — , = /, and on 
 
 n/r T^ 
 
 B, 
 
 M 
 irr:- 
 
 = /i, since the same amount of light is re- 
 
 ceived by both circles, but less per unit of area in 
 the two cases. 
 
 Hence 
 
 / 
 
 -^ ^= J- We can also prove by meas- 
 
 d,' 
 
 urement that -r = -h, or ^ 
 
 
LIGHT. 309 
 
 If, now, we could measure the quantity M in 
 absolute measure, we could obtain an absolute 
 standard for the estimation of the intensities of 
 different lights. Unfortunately, there is great diffi- 
 culty in obtaining such an absolute standard. It is, 
 however, very easy to measure the relative intensi- 
 ties of different lights. 
 
 Experiment 183. — Place in front and quite near 
 a vertical screen made of tissue-paper or of porce- 
 lain, a vertical rod. Place the two lights to be 
 compared on each side of the perpendicular to the 
 screen passing through the rod, so that the horizon- 
 tal lines drawn through them and the rod will form 
 equal angles with this perpendicular, and throw shad- 
 ows of equal breadth upon the screen. Move one 
 light away from or toward the screen, until the shad- 
 ows are of the same intensity. The intensity of one 
 
 hght IS -jg, and that 01 the other -r^. iiiese intensi- 
 ties are made equal by the experiment ; hence ^ = 
 
 -t4, or ttt = -Tn. This method is due to Rumford. 
 d^' Ml d{ 
 
 Experiment 184. — Cover a circular disk of pa- 
 per, except a small circle about two centimetres in 
 diameter in the center, with a solution of wax in 
 benzine. This will render the paper transparent. 
 Mount this disk so that it can slide horizontally 
 along a graduated bar ; place a standard candle — a 
 sperm-candle weighing one sixth of a pound and 
 burning one quarter of an ounce per hour — at one 
 end of this bar, and the light to be compared with 
 it at the other end. Slide the disk along the bar 
 until the disk is equally illuminated on both sides. 
 
3IO NEW PHYSICS. 
 
 When this is done, the white-paper spot at the 
 
 center of this disk will no longer be seen. In this 
 
 case we have made the intensities of the two lights 
 
 M d^ 
 equal, and here we also have -j^ = -7-^. This method 
 
 is due to Bunsen. 
 
 It will be observed that the methods thus far 
 given for obtaining the intensities of two lights are 
 simply relative. There is no absolute standard of 
 light. The lights produced by different materials 
 are all different in tint, and it is found very diffi- 
 cult in practice to compare the intensities of light of 
 different colors. The best method of connecting 
 our measurements of the intensity of light with our 
 measurements of energy in general would seem to 
 be by comparing all lights with an electric incan- 
 descent light produced by passing an electrical cur- 
 rent of known strength through a known resistance. 
 In this case we should know how much energy is 
 necessary to produce the standard light, and other 
 lights could then be compared with such a stand- 
 ard. Even in this case, however, we do not know 
 the mechanical equivalent of the light-waves. 
 Moreover, we should still be confronted with the 
 difficulty of comparing lights of different colors. It 
 is also difficult to maintain a constant light by an 
 electrical current. The molecular work done by 
 light of different wave-lengths in photography might 
 give us a standard if we were able to estimate this 
 work in absolute units. If we had also a medium, 
 in passing through which a definite amount of light 
 would be absorbed, and if we could estimate the 
 amount of internal work done by the waves of 
 energy in passing through such a medium, we 
 
LIGHT. 
 
 3" 
 
 could use this medium as a means of estimating the 
 strengths of lights. Our means of measuring the 
 intensities of light by absorption are only relative. 
 In order to measure the amount of light that is ab- 
 sorbed by a medium, this experimental determina- 
 tion having a certain analogy to the determination 
 of the amount of energy consumed in the transmis- 
 sion of mechanical power, we can proceed as in the 
 following experiment: 
 
 Experiment 185. — Two mirrors, m in! (Fig. 137), 
 are placed upon a graduated bar. A greased disk 
 can slide on this bar between the mirrors. A light, 
 L, suitably placed behind the angle of a screen, can 
 be reflected by 
 
 the two rair- L 
 
 rors upon this ,-^\ 
 
 disk. This /y^ 
 
 happens when «/ 
 
 the latter is at 
 
 the middle of \,''' "\ ^ 
 
 the graduated ~m/^ '^ 7w~ 
 
 bar, the mir- fig. 137. 
 
 rors at an an- 
 gle of 45° with the bar, and the distance from 
 the disk to the light equal to its distance from 
 each mirror. No spot can then be seen upon the 
 disk. Interpose, however, the absorbing material 
 at either side of the disk, as at yi, and it will be 
 found that the disk g must be moved nearer the 
 mirror m. It is as if a light of feebler intensity, 
 /i, placed at a distance gm L along the graduated 
 bar, is to be compared with a light of intensity 
 I at a distance gm! L in the opposite direction 
 (since to the eye placed at g the light at L appears 
 
312 NEW PHYSICS. 
 
 to come from behind the mirror w at a distance 
 along the scale equal to m L. If we call the distance 
 m' L = d, and the distance between the mirrors = 
 a, and supposing this latter distance to be divided 
 into centimetres, the readings, n, running each way 
 
 from the center of a, we shall have 7-7-1 r — Ja 
 
 ' (d -\-a-\-nf 
 
 /I 
 
 which will give the ratio of the in- 
 
 tensities of the two lights ; or, in other words, the 
 proportion of the light L that is absorbed by pass- 
 ing through the given medium. 
 
 There are many analogies between sound and 
 light, since the phenomena of both are due to wave- 
 motion. We can also measure the relative intensity 
 of two sounds, but we have no mechanical standard 
 or absolute standard of sound intensity. , The nicely- 
 trained eye can estimate the equality of two lights 
 of the same tint to a very close approximation. The 
 well-trained ear can also decide whether two notes 
 sounded on two instruments are the same. But 
 there is no absolute standard for the estimation of 
 the intensity of sound. Therefore, there has been 
 during successive years a change in the pitch to which 
 concert instruments are tuned, for, having no abso- 
 lute standard for the estimation of the intensity of 
 sound, our relative standards necessarily fluctuate. 
 In the same way our eyes get accustomed to strong 
 light or weak light, and, having no unchangeable 
 standard, the scale by which we estimate the in- 
 tensity of light is subject to a similar variation to 
 that which our ears experience. 
 
 Assuming that the sensations of heat and light 
 are produced by wave-motion, we can apply the 
 
LIGHi. wx 
 
 laws of mechanics to the question of the movement 
 of those waves, and the law of the conservation of 
 energy to the transformation of the energy of wave- 
 motion into that of molecular vibration. The first 
 question that meets us in the discussion of the prop- 
 agation of energy from the sun, is the same as that 
 in regard to motion in general. What is the veloci- 
 ty of its propagation ? It is found by experiments 
 on wave-motion in different media that the law 
 
 ^=y — -, in which V ^ velocity, E — elasticity, 
 
 I) = density, is a true one. If we could find the 
 elasticity and density of the ether, we could then cal- 
 culate the velocity of light. This, however, seems 
 to be impossible at present. We therefore have to 
 resort to direct experiments to determine the ve- 
 locity of light. The principal methods are those of 
 Fizeau and Foucault, descriptions of which can be 
 found in most works on natural philosophy. The 
 best values of the velocity of light seem to be those 
 of Cornu and Michelson, which are, respectively, 
 300,400,000 and 299,940,000 metres per second. The 
 methods of the later observers depend upon the 
 measurement of the small interval of time which 
 elapses between the reflection of light from one 
 station to another and its return to its starting-point. 
 This gives the time that it has taken the light to pass 
 over twice the distance between the two mirrors. 
 
 When the velocity of light is known in one me- 
 dium, it can be easily obtained in another by what is 
 called Huygens's construction for the wave-front. 
 
 When a wave passes from one medium to another, 
 the points where each point of the front of the wave 
 strikes the line of demarkation between the two me- 
 
314 
 
 NEW PHYSICS. 
 
 dia become the center of propagation of new waves. 
 Let A BAiB^ be the path of a beam of light (Fig. 
 138), and suppose that A £> is the wave-front, which 
 is supposed to be so small as to be approximately 
 straight. Let^ A^ be the surface of separation be- 
 tween two media, 
 and let the velocity 
 of light in the up- 
 per medium be V, 
 and in the lower Fj. 
 If the lower medi- 
 um is the denser, 
 the light will trav- 
 el slower in it, and 
 will be refracted 
 toward the per- 
 pendicular to line 
 or surface of sepa- 
 ration AA\ The 
 following illustra- 
 tion of Herschel 
 connects this prop- 
 erty of light with our ideas of work : Suppose that 
 A B or DB^ represent the line of advance of a col- 
 umn of soldiers, and let ^ Z> be the front rank. 
 When the column reaches a plowed field, those first 
 striking the field at A will move more slowly, with 
 the same expenditure of energy, than those at D, for 
 instance. In order to preserve the order of march- 
 ing, the front rank must therefore take a line A^ m, 
 and the new line of march must be inclined at a dif- 
 ferent angle to the line between the smooth field 
 and the plowed one. If v is the velocity of the 
 soldiers along A^ D, we shall have A^ I) = v i; 
 
LIGHT. 
 
 315 
 
 and if v-^ is the velocity of the soldiers in the 
 
 plowed field, or along A m, we shall have A m=. 
 
 ^ u A^D V t ^ . ViAiD 
 
 Vi t. Hence -^ — = ■, or A m = -i. — ?_; or, pass- 
 
 A in v^t V 
 
 ing to the case of light, a wave starting from A will 
 
 have advanced to m through a space — — ^ — , while 
 
 the original wave advances from D to A^. If we 
 now draw a circle from A with a radius A m, and 
 then draw a line from A > tangent to this circle, we 
 shall have the new wave-front, and the lines A * 
 C^ A C, perpendicular to this new front, will be the 
 new direction of the ray. The angle which A B 
 or A^ £^, the original ray, makes with the normals 
 ^ «, ^ 1 «i is a, or the angle of incidence. The 
 angle of refraction, or CA O — 0,is also equal to MA^ 
 yj, since their sides are perpendicular to each other. 
 Hence we shall have, A A ' cos (90° — a) = AA^ sin 
 a = A^ I) = v i; A A^ sin d== AM— V it. Hence, 
 
 -. — ^— = a. The ratio between the velocities in 
 
 sm Vi 
 
 the two media is called the index of refraction fi, 
 and we see that this is the ratio between the sines 
 of the angles of incidence and of refraction. Our 
 determination, therefore, of indices of refraction is 
 really a measurement of velocities. Our subsequent 
 work upon the relations between heat and light re- 
 quires the determination of this ratio of velocities. 
 
 Experiment 186. — Obtain a small cylindrical 
 bottle of clear glass, a homoeopathic vial for in- 
 stance, holding about twenty centimetres, selecting 
 one that is circular in section. Grind one side away 
 and cement on a piece of plate-glass. Mount this 
 vial at the center of a graduated paper circle, or 
 15 
 
3i6 
 
 NEW PHYSICS. 
 
 horn protractor (Fig. 139), and fill it with water. 
 An illuminated pin-hole in a screen placed at B, 
 will be seen by the eye at A. 
 The angle which the inci- 
 dent ray makes with the per- 
 pendicular Be io the surface 
 of the liquid, is called the 
 angle of incidence, and the 
 angle which the refracted ray 
 makes with this perpendicu- 
 lar the angle of refraction, 
 and the ratio of the sines of 
 these angles, or, in other words, the ratio of the 
 perpendiculars B D and A m, is called the index 
 
 of refraction, or ^ = — : — . Having obtained the 
 
 Fig. 139. 
 
 sm^- 
 
 value of n for water, obtain it also for alcohol. 
 
 In order to experiment upon light, we must be 
 able to make the rays of light parallel, after they 
 have passed through a small slit, the image of 
 which we use in studying the effects of refraction 
 and dispersion. By reflecting sunlight into a dark 
 room, through a pin-hole or narrow slit, we quick- 
 ly perceive that the rays are no longer parallel 
 after passing through the slit, but diverge, form- 
 ing a cone or wedge of rays. It is evident that 
 we can not experiment upon refraction with such 
 a cone of rays. It is necessary to have a beam 
 of which the rays are approximately parallel. We 
 can accomphsh this by using a lens of a long fo- 
 cus. For some distance the rays from such a lens, 
 for instance an ordinary spectacle-lens, will be ap- 
 proximately parallel, and if a prism be placed in 
 the path of this beam, we can easily trace its pas- 
 
LIGHT. 
 
 317 
 
 sage through the prism or refracting apparatus we 
 employ. 
 
 In order to understand more refined apparatus, 
 it is often useful to dissect, so to speak, such appa- 
 ratus and form rough models of it. The following 
 apparatus will give approximate results, and is use- 
 ful to enable us to comprehend an instrument called 
 a spectrometer, by means of which we finally meas- 
 use the length of the waves of light. 
 
 Experiment 187. — Having made the rays of a 
 beam of light parallel by means of a narrow slit and 
 a lens, as above described, place a glass prism or a 
 
 Fig. 140. 
 
 liquid prism in its path (Fig. 140), and receive the 
 refracted beam upon a piece of oiled paper, or a 
 ground-glass screen. The beam of light will be re- 
 fracted to some point, E, which can be found by 
 placing a lens, /, in the path of the refracted rays, 
 thus bringing the rays proceeding from the slit, S, 
 to a focus at E, and forming there an image of this 
 slit. The tangent of the angle of deviation can then 
 
3i8 
 
 NEW PHYSICS. 
 
 be read in terms of the division of the proper scale, 
 knowing the distance of the scale from the center of 
 the prism. On turning the prism around a vertical 
 axis passing through its center A , it will be found 
 that the image E travels at first in the direction in 
 which we turn the prism ; but presentl)^ while we 
 continue to turn, the image ceases to turn in the 
 same direction, and moves in the opposite direction. 
 
 Fig. 141. 
 
 The position of the prism, when the refracted ray 
 ceases to follow the motion of the prism around a 
 vertical axis, is called the position of minimum de- 
 viation. When the prism is in this position the inci- 
 dent ray and the emergent ray make equal angles 
 with the faces of the prism. For measurements the 
 prism is usually placed in this position of minimum 
 deviation. 
 
 It is necessary also to obtain the refracting angle 
 
LIGHi* ^ I Q 
 
 of a prism ; this is the acute angle, away from which 
 the Hght is refracted. In order to do this, place the 
 prism so that its refracting edge shall be on the line 
 passing through the slit (Fig. 141)7 which is also per- 
 pendicular to the screen. In this position, the beam 
 of parallel light striking on the two edges of the 
 prism which intersect at A, are reflected so as to 
 make equal angles with the normals A n, and strike 
 the screen at m m', the positions of which can be 
 found by moving a lens, /, until an image of the slit 
 is seen at m and m' ; m A m' is then the tangent of 
 twice the refracting angle A of the prism. PA D' = 
 90° — angle of incidence, m' A D' =: 90° — angle ,of 
 reflection ; hence, PA Z?' =m' A D'. In the same 
 way mAD = PAD, or m' A D' -\-mA D = D AD' , 
 orm' Am = 2DAD'. 
 
 In case that the points m and in}- are too far apart 
 to be received on one screen, place a smaller screen, 
 o 0^, at m or m^, and drop a perpendicular from m to 
 
 MP 
 line A P; then— —will be the tangent of the angle 
 A ± 
 
 of the prism, and the angle can be readily obtained 
 
 from a table of natural tangents. 
 
 The index of refraction of a substance is gener- 
 ally determined by means of a prism. If the sub- 
 stance is a specimen of glass, it is made into a prism. 
 If the substance is a liquid, it is poured into a hollow 
 prism of glass. 
 
 When a beam of light is passed through a prism 
 it is bent or refracted from its course. Let us ex- 
 amine the directions of the incident and emergent 
 rays. 
 
 At the point B (Fig. 142), the angle of incidence 
 of the ray A B is t{Bn being the normal at B). The 
 
320 
 
 NEW PHYSICS. 
 
 angle of refraction is D B c or r. The divergence 
 of the ray B D from A B is D B m or i — r, since 
 the angle mBc = A B n. If we arrange the prism 
 so that the emergent ray D E makes the same angle 
 
 1} 
 
 Fig. 142. 
 
 with the normal that A B does, we shall have angle 
 c B D or r^BDc. Hence, Wj c w = r -f- ^ = angle 
 a, since the normals are respectively perpendicular 
 to the sides of the angle a, or a = 2 r. In the same 
 way from the triangle O B D we have t — r-}-i—r, 
 or 2 i — 2r = E Om, or S; S being the angle of devia- 
 tion of the ray. Hence, 2 t — a = S, or i = . 
 
 sin i a 
 
 But - — ■ = u, and a = 2 r, or r = ~. Hence, 
 smr 2 
 
 sin (a + S) 
 
 — - — =/*• 
 
 sm a 
 
 2 
 
 If we use a prism of very small angle, 
 
 since the sine equals the arc in this case, or 
 S — (jji— i)a. 
 
 S 
 
 2 
 
 ■ 11, 
 
LIGHT. 
 
 321 
 
 A lens can be considered as made up of a num- 
 ber of prisms (Fig. 143) ; the rays of light coming 
 from any source are bent by the prisms, and are 
 
 I 
 
 r I I'l I 
 III 
 
 II 
 
 h 
 
 I I i| 
 
 I ' I 
 
 I 'I, ' 
 
 iiihh 
 
 1 - . I. I.. ! I. .II. .1 
 
 'i,,i '1 
 
 1' III 
 
 ill 
 
 Fig. 143. 
 
 collected at a point which is called the focus. The 
 principal axis of the lens is line A B. 
 
 The rays of light coming from the sun are par- 
 allel. The point on the principal axis of the lens 
 where these parallel rays meet is called the princi- 
 pal focus. 
 
 Experiment 188. — Place a convex lens in the 
 sunlight, and measure the distance of the principal 
 focus from the center of the lens. 
 
 Experiment 189. — Substitute for the light of the 
 sun a kerosene-light or a gas-flame, and receive the 
 image on a ground-glass screen. It will be found 
 that the distances of the images of the light vary 
 when the lens is moved to or from the light. It will 
 I I I 
 
 — — + — expresses the law of the 
 Where F is the principal focus obtained by 
 
 be found that 
 lens, 
 
322 
 
 NEW PHYSICS. 
 
 preceding experiment, d and d^ are the distances 
 from the light to the lens and from the lens to the 
 screen respectively. 
 
 This law of a lens can be obtained very simply 
 by geometry. The propositions necessary to prove 
 it are the following : 
 
 1. The exterior angle of a triangle is equal to 
 the sum of the opposite interior angles. 
 
 2. Two angles are equal when their sides are 
 mutually perpendicular. 
 
 3. The sine of an acute angle at the base of a 
 right-angled triangle is equal to the altitude of the 
 triangle divided by the hypothenuse. 
 
 4. The sine of a very small angle is equal to the 
 arc which measures the angle. 
 
 Let ^ ^ be a section of the lens (Fig. 144), CD 
 its principal axis, C EFD a ray which passes from 
 
 
 I 
 
 "•Fig. 144. 
 
 one conjugate focus, C, to the other atZ), /i^and 
 KE the normals at F and E, Jm ^r, Km = r^, C E 
 =/, D F= p^. Since yi^ and KEdire perpendicular 
 to the sides of the lens, we have EmJ ^ EAF^d. 
 S = (/t - I) ^, = (/* - I) {m/K-{- m KJ). 
 Let fall a perpendicular from E upon CD; since 
 
LIGHT. 323 
 
 the angle of the lens is small, this perpendicular 
 may be considered equal to and coincident with one 
 
 let fall from i^ upon CD, or D = (u,— i)( 1 ), 
 
 EO EO 
 
 and being the sines of the angles mj K and 
 
 m KJ, and the sines being taken equal to the angles 
 for small angles. 
 
 We also have 8 = E C 0-\-FD O = ^ + ~ 
 
 P A 
 (the sines being taken equal to the angles) 
 
 ^-■)C-+^) = ^j;-<'- 
 
 When one of the foci D is at an infinite di 
 
 is- 
 
 tance, — = o, and we then have —=- = (/*— i) 
 A . p F ^ ' 
 
 f- -J J, eq. 2. From eq. i and 2 we see that 
 
 - = - + -■ 
 F p^ p. 
 
 Experiment 190. — Place two convex lenses in 
 front of a source of light, and let / be the focal 
 length of one lens. A, and f^ that of the other, B. 
 Let d be the distance apart of the lenses A and B. 
 Then, calling the distance from the lens B to the 
 
 image D, we have ^ = -- -|- ? • If the lenses 
 
 are close together, we have d ^ o, or j^ ^ y-{- -j. 
 
 If the lenses have the same curvature, then / = 
 /Sor-i=|. 
 
 If the second lens is a concave lens of focus f, 
 
324 
 
 NEW PHYSICS. 
 
 we then have 
 
 D 
 
 I 
 
 ; . , and if the lenses are 
 
 -d /' 
 
 I 
 
 I 
 
 £i 
 
 in close proximity, j^ = — — 7. 
 ■^ fi f 
 
 Experiment 191. — Determine the focal distance 
 of a concave lens by placing it in front of a convex 
 lens of greater focus, and measuring the focal lengths 
 
 of the combination. In this 
 case the conjugate foci of 
 the combination must be 
 real. 
 
 Experiment 192. — The 
 index of refraction of a 
 liquid can also be obtained 
 in the following way : Place 
 a concave lens of known ra- 
 dius of curvature upon a 
 drop of the liquid on a plate 
 of glass (Fig. 145). Reflect 
 the light from an illuminated 
 pin-hole, by means of an in- 
 clined mirror, through this 
 combination of a plano-con- 
 cave and convex lenses, to a 
 screen, S, made of ground glass. The focal distance 
 of the combination can thus be obtained ; and we can 
 obtain the focal length of the liquid lens from the 
 
 formula k = i- — :?, since we have -= (u — i)-from 
 D A f fx r 
 
 eq. 2 by making r^ the radius of the plane surface 
 
 of one side of a convex lens equal to infinity, and 
 
 hence — = <?). 
 
 In order to ascertain the position and the size of 
 
 Fig. 145. 
 
LIGHT. 
 
 325 
 
 an image formed by a convex lens, it is only neces- 
 sary to draw two rays of light from each extremity 
 of the object, and find where these intersect. One 
 of these rays is generally drawn through the center 
 O of the lenses, and the other can be drawn to any 
 part C of the lens ; thus the image of A (Fig. 146) 
 will be found at A' . In the same way B' can be 
 found, and the size oi A' B' and its distance from O 
 can be readily measured. 
 
 Fig. 146. 
 
 Experiment 193. — Ascertain the size of the im- 
 ages formed by a convex lens, with varying dis- 
 tances. 
 
 A bit of wire moved along the principal axis A B 
 (Fig. 147) of a mirror will be brilliantly illuminated 
 at the principal focus. 
 
 Prove that the lengths of image and object are 
 directly as their distances from the mirror, and that 
 their areas vary directly as the square of the dis- 
 tances. 
 
 The images produced by convex mirrors are not 
 real images — that is, they can not be projected on 
 the screen, and are only received by the eye. 
 
326 
 
 NEW PHYSICS. 
 
 Experiment 194.— Take a small Florence flask, 
 or a small spherical bulb, and fill it with clean mer- 
 
 M 
 
 -vi=^ 
 
 --F 
 
 Fig. 148. 
 
 HS 
 
 Fig. 147. 
 
 cury, and show that the distances of the virtual 
 images and of the object from the center of curva- 
 ture vary directly 
 as their lengths 
 and as the square 
 root of their areas. 
 Let O be the 
 center of curva- 
 ture of a concave 
 mirror of small 
 angular aperture 
 (Fig. 147). A B its 
 principal axis, and 
 F its principal fo- 
 cus. This focus 
 can be found thus : 
 Experiment 195. — Select a small evaporating- 
 dish which is approximately a portion of a spher- 
 ical surface, and press it down upon the surface 
 
LIGHT. 327 
 
 of clean mercury, confining it by suitable press- 
 ure. 
 
 Incline a mirror at an angle of 45° (Fig. 148), so as 
 to throw the image of a slit illuminated by the sun 
 upon the concave mirror which is formed by the 
 surface of mercury. A little alcohol should be first 
 poured upon the mercury. 
 
 Experiment 196. — Make a concave lens, one 
 side of which is plane — or, in other words, has a 
 radius of infinite length — by pressing a glass evap- 
 orating-dish, whose surface is spherical, upon the 
 surface of water which is contained in a vessel with 
 a piece of thin plate-glass for a bottom. Keep the 
 evaporating-dish in position by a suitable arrange- 
 ment, and view some object beneath the lens thus 
 formed. Prove that the formula in this case is 
 I _ I I 
 
 By means of the simple apparatus described in 
 Experiment 187, we can determine the index of re- 
 fraction of a prism. This simple apparatus is really 
 a rude spectrometer, the slit S and the lens near 
 it forming what is called a collimator — that is, an 
 apparatus to make the rays of light parallel. It is 
 usual to form this collimator by making a slit at 
 one end of a long brass tube which is blackened 
 inside. At the other end is placed the lens of 
 long focus. The collimator is really a darkened 
 room or gallery. The observing telescope is also 
 a brass tube, one end of which is closed by a lens 
 of moderate focus ; and the other end is provided 
 with an eye-piece suitable for observing the image 
 of the slit formed by the lens /. The screen is 
 replaced in the spectrometer by an observing tele- 
 
328 
 
 NEW PHYSICS. 
 
 scope, which moves around a graduated circle of 
 brass. 
 
 Let us now return for a moment to the simple 
 apparatus described in Experiment 187. Allow the 
 refracted beam to fall upon a white screen. In this 
 way we shall form a spectrum of the sun. This 
 consists of a series of images of the slit in different 
 colors. To form a conception of the effect of the 
 prism in separating the beam of white light into 
 light of different colors, let us suppose that a flock 
 of birds, all advancing with the same velocity — red 
 birds moving with long, flapping wings, and yellow 
 birds with somewhat less powerful wings, and there- 
 fore flapping their wings of tener to keep up with the 
 red birds, blue birds, green birds, violet birds, and 
 so on — should fly toward us. All together, their col- 
 ors, properly mingled as in the case of a beam of white 
 light, would be undistinguishable. On meeting a 
 dense, wedge-shaped medium, the more powerful 
 birds would be less diverted from their path than 
 the smaller birds, whose wings move much faster 
 than those of the larger or red birds. In the effort 
 to maintain their advance with the red birds, the 
 birds whose wings move more rapidly are crowded 
 away from the slower and more powerful birds and 
 are dispersed, so that a gunner on the other side of 
 this wedge-shaped medium would see the birds ad- 
 vancing in the order of the colors of the spectrum. 
 
 Experiment 197. — In order to project the solar 
 spectrum upon a screen, deflect the sunlight into a 
 darkened room through a narrow slit, about four 
 centimetres long and two millimetres wide. Form 
 an image of this slit on a screen by means of a lens 
 of long focus. 
 
LIGHT. 329 
 
 Place a prism in the path of the convergent 
 beams not far from the lens. On moving the 
 screen, still keeping it at the same focal distance 
 from the lens, the spectrum will be formed upon it, 
 and, if the slit is of suitable width, the dark solar 
 lines will be seen on the screen. 'The dark line 
 (really two dark lines) in the yellow is called the 
 sodium light, and is due to the absorption of the sun- 
 light in passing through a layer of sodium vapor on 
 the sun's surface. To illustrate this, and to make the 
 line visible to a large audience, obtain some metallic 
 sodium, and, having placed it in a small cavity in a 
 piece of chalk, burn it in the flame of a Bunsen 
 burner or spirit-lamp, which is placed directly in 
 front of the slit, inside the darkened room. (The 
 prism should be put in the position of minimum 
 deviation.) 
 
 By means of a spectrometer the spectrum is seen 
 to be crossed by a great number of fine, dark lines, 
 which belong to the elements. These lines can be 
 made bright lines by burning in front of the slit 
 the different metals and cutting off the sunlight by a 
 screen. The bright lines occupy the exact position 
 of certain dark lines in the solar spectrum, and are 
 reversed into dark lines in passing through their 
 own vapor in the atmosphere of the sun. It is not 
 possible to project on the screen, by means of the 
 last experiment, the bright line of sodium, for the 
 light is not strong enough. It is necessary for this 
 purpose to burn the sodium in the electric light. 
 In order that the bright line shall be reversed into a 
 dark one, it is necessary that the sodium rays shall 
 pass through the vapor of sodium, which is at a lower 
 temperature than the source of the rays. A certain 
 
330 
 
 NEW PHYSICS. 
 
 portion of the energy of the waves of sodium light is, 
 therefore, absorbed in striving to heat the sodium 
 vapor which is at a lower temperature. The mech- 
 anism of this endeavor can be comprehended by 
 a reference to pendulum movements. In order to 
 increase the height to which a pendulum swings, 
 it is evident that our blows must be timed so as 
 to coincide in time with the swings of the pendu- 
 lum. A heavy ball can be set in movement and 
 maintained in vibration by attaching a fine thread to 
 it and timing our impulses so that they coincide with 
 the rate of swing of the pendulum. If, therefore, 
 the molecules of sodium vapor, whose to-and-fro 
 movements are like those of a pendulum, are to have 
 their energy increased, it must be by impulses timed 
 to their own rate of vibration. 
 
 This is the principle of a method of sending many 
 messages over one telegraph-wire at the same instant: 
 A number of tuning-forks or reeds are employed. 
 Electro-magnets are then placed opposite the prongs 
 of the tuning-forks, and the latter pick out of the 
 impulses sent over the telegraph-wire those that cor- 
 respond to their own rate of vibration. Returning 
 now to the dark lines in the spectrum, we see that 
 an analogy to this phenomenon would be the action 
 of a harp, the strings of which are between us and a 
 certain tuning-fork which is vibrating. If one of the 
 strings is tuned to vibrate to the note of the fork, the 
 note of the fork will be weakened to a very sensitive 
 ear placed near the string, for the energy of the tun- 
 ing-fork is absorbed in setting this string into vibra- 
 tion. If our means were sensitive enough, we could 
 thus determine by means of a harp the existence of 
 a number of tuning-forks on the side of the harp op- 
 
LIGHT. 
 
 331 
 
 posite to ourselves, which are too distant to affect 
 our sense of hearing by their own vibrations. 
 
 In order to see the bright 
 lines of the elements, it is slit 
 necessary to employ an ob- 
 serving telescope, for the va- 
 por of sodium, for instance, 
 would not give a sufficiently 
 powerful light to project an 
 image on the screen. The ar- 
 rangement of the lenses in a 
 spectroscope can be seen in 
 Fig. 149. The beam, passing 
 through the slit S, is rendered 
 nearly parallel by the lens. A, 
 of long focus. It strikes upon 
 the prism and is refracted and 
 dispersed, and the refracted ^ye piece "^P 
 
 rays, entering the lens B, pro- fig. 149. 
 
 duce an image of the slit, 
 
 which in turn is magnified by the combination of 
 lenses in the eye-piece. 
 
 Experiment 198. — Make a small spectroscope 
 by obtaining tin tubes from the tin-man and insert- 
 ing lenses, which can be obtained from any optician. 
 The interior of the tubes should be blackened to 
 prevent internal reflections. The combination of 
 lenses for an eye-piece can be seen by examining 
 any small telescope. A dark box should be fitted 
 over the prism, with holes for the ends of the two 
 telescopes. The distance of the slits from the object- 
 glass A (Fig. 149) must be regulated so that the rays 
 proceeding from a slit, after emerging from the 
 glass A, shall be approximately parallel. In order 
 
332 
 
 NEW PHYSICS. 
 
 to accomplish this, focus the telescope, which has the 
 object-glass B, upon some distant object. Then re- 
 move the prism and place the collimator and tele- 
 scope in the same line, and move the tube contain- 
 ing the slit 5 until a distinct image is seen in the ob- 
 serving telescope B ; the rays coming from 5 will 
 then emerge from the object-glass, yi, parallel. 
 
 Experiment 199. — Having reflected the sun- 
 light upon the slit so that the ray is horizontal and 
 will pass through the axes of the two telescopes, 
 place the prism in the position of minimum devia- 
 tion, and focus the telescope upon the dark solar lines 
 in the yellow. Then cut off the sun, and place a 
 wire, dipped in salt, in a flame before the slit. A 
 yellow line, called the sodium line, will appear ex- 
 actly in the place of the darkest solar line in the yel- 
 low (the observing telescope should be provided 
 with cross-hairs ; see Appendix). It is best to use a 
 platinum wire, one end of which is bent into a loop, 
 which is dipped into fine salt. The other end of the 
 platinum wire can be fused into a glass tube which 
 will- serve as a holder. 
 
 Experiment 200. — Repeat the above experiment 
 with chloride of lithium and potassium hydrate, and 
 locate the lines with reference to the colors of the 
 solar spectrum. 
 
 Experiment 201. — Take a thin Florence flask ; 
 pour into it half a teaspoonful of nitric acid ; drop a 
 bit of copper wire into the acid, and close the flask. 
 In a few mohients the flask will be filled with the 
 fumes of nitrous acid. Place the flask before the 
 slit of a spectroscope and examine the sunlight 
 which has passed through the vapor. The solar 
 spectrum will be found to be crossed by a great. 
 
LIGHT. 
 
 333 
 
 number of vertical bands which constitute the ab- 
 sorption spectrum of nitrous acid. 
 
 Experiment 202. — Make a vessel with parallel 
 glass sides, the sides being eight centimetres apart. 
 Fill the cell with a solution of sulphate of copper 
 and place it in front of the slit. The solar spectrum 
 will be absorbed except in the green and blue. 
 Place a second cell in front of the one at the slit, so 
 that the sun's rays will pass through both cells. 
 It will be found that the light which has already 
 passed through the solution of copper, and has 
 lost its red and yellow rays, passes without further 
 absorption through another layer of sulphate of 
 copper. 
 
 Experiment 203. — Draw diagrams of the por- 
 tions of the solar spectrum absorbed by a solution 
 of indigo. 
 
 \.&t AB C , 
 D (Fig. 150) 
 be a section 
 of a thin film | 
 like that of a ' 
 soap - bubble. . 
 m B, a ray of 
 light striking 
 the surface of 
 the bubble at 
 ^, will be part- 
 ly refracted in [ 
 
 the direction Fig. 150. 
 
 B D, and will 
 
 emerge in the direction D R. It will also suffer 
 
 internal reflection at D, O, and E, giving rise to 
 
 rays O I, Tp, and to E r. The luminiferous surface 
 
334 
 
 NEW PHYSICS. 
 
 S, perpendicular to the incident ray m B, will cor- 
 respond to a luminiferous surface at S' after reflec- 
 tion. The corresponding luminiferous surface per- 
 pendicular to the ray O I will be behind that at S' 
 by the distance B D plus D O, and therefore in- 
 terference can take place. The following experi- 
 ment will show the brilliant colors due to inter- 
 ference : 
 
 Experiment 204. — Into a pint bottle, half filled 
 with distilled water, put one ounce of shavings of 
 white Castile soap. Shake the bottle well and add 
 water until the soap completely dissolves and the 
 solution is clear. Then add a gill of glycerine 
 and strain the mixture. If liquid glycerine soap 
 can be obtained, a suitable solution of this in 
 distilled water will answer. Dip the mouth of an 
 ordinary tumbler into this solution, and having 
 placed the tumbler so that the film is vertical 
 and reflects the light from a window, you will 
 observe beautiful bands of brilliant colors crossing 
 the film. 
 
 Experiment 205. — If a beam of light can be ob- 
 tained in a dark room, receive it upon the film and 
 reflect it upon a white screen. The phenomenon 
 can thus be shown to a large class. 
 
 From Fig. 1 50 we see that the path of the rays 
 reflected from the front surface of the film and the 
 path of those reflected from the back surface differ 
 by twice the thickness of the film of water, and when- 
 ever the difference of path brings the crest of one 
 wave directly over the trough of the other, we have 
 complete interference or darkness. This happens 
 whenever the path of the beam in the film is an 
 even number of half wave-lengths. 
 
LIGHT. 
 
 335 
 
 The dark lines in the solar spectrum are called 
 Fraunhofer's lines, since he first observed them. 
 Their wave-length can be obtained by a diffraction 
 grating. The most prominent lines are given in the 
 following table. Calling the wave-length \, we have 
 _ velocity 
 
 number of vibrations* 
 is 300,000,000 metres per second 
 
 The velocity of light 
 
 Portions of the- Solar Spectrum. 
 
 Dark red A 
 
 Orange C 
 
 Yellow D 
 
 Green E 
 
 El- \l 
 
 Violet H 
 
 Length of waves in 
 
 Number of waves 
 
 millimetres. 
 
 per second. 
 
 .000750 
 
 395 X 10" 
 
 .000656 
 
 458 X 10" 
 
 .000589 
 
 510 X 10''' 
 
 .000527 
 
 570 X 10" 
 
 .000486 
 
 618 y. io'« 
 
 .000431 
 
 697 X 10" 
 
 .000397 
 
 760 X 10" 
 
 Thus it will be seen that the longest wave- 
 lengths of light which belong to the red and ultra- 
 red rays are approximately -j o^oooo ^^ ^ millimetre, 
 and 395,000,000,000,000 of these waves strike the eye 
 in one second, while the tuning-fork which gives 
 256 vibrations per second, and which corresponds 
 to the middle C of the piano-forte, sends forth 
 wave-lengths 1.30 metres in length. The velocity 
 of sound is about 333 metres per second in air. 
 
 Our experiments with soap-films show us some 
 of the most striking phenomena of interference of 
 light. The phenomena of diffraction also exhibit 
 the interference of waves. 
 
 Experiment 206. — Take a common spectacle- 
 lens, A (Fig. 1 5 1), of two or three feet focus, and an 
 eye-lens, B, of one or two inches focus. Mount them 
 
336 NEW PHYSICS. 
 
 by surrounding them by a ring of cork and pushing 
 them into pasteboard or tin tubes — the tube contain- 
 ing the eye-piece slipping within the tube containing 
 
 ^[I 
 
 D 
 
 Fig." 151 
 
 the object-glass A. The interior of the tubes should 
 be blackened, and the lenses set perpendicular to 
 the axis of the telescope. Mount the telescope thus 
 made upon a suitable stand. An eye-hole should 
 be placed in front of B, a little within its focus. 
 Draw the eye-tube out until an image of a distant 
 luminous point is seen distinctly in the telescope. 
 Then place the support of the apparatus at C, be- 
 tween the object-glass A and the luminous point. 
 
 The light should be at 
 a considerable distance. 
 Fora distance from twelve 
 f ^ f ~\ to eighteen feet the ob- 
 ject-glass should have a 
 focal length of one to one 
 and a half feet. As a 
 ^^°- ^52. luminous point the sun's 
 
 rays striking a convex 
 mirror can be used. Cover one half the object- 
 glass by a sheet of tin-foil (Fig. 152), running the 
 straight thin edge of the foil across the middle, a b, 
 
LIGHT. 
 
 337 
 
 of the glass. The image of the luminous point / is 
 now spread out by diffraction into the hne cd. The 
 luminiferous surface is unaltered in the direction 
 parallel to the edge a b, and the lens brings these 
 rays to a focus at /, but the surface will be formed 
 of arcs nearly circular in a direction perpendicular 
 to the edge a b, and the diffracted rays originating 
 at this edge will not be brought to a focus by the 
 object-glass, and hence are opened out into a line 
 of light. 
 
 Make a narrow slit about one fiftieth of an inch 
 broad, and place it between the object-glass A and 
 the distant source of light. The central image will 
 
 o 
 o 
 o 
 o 
 o 
 
 f 
 
 b 
 
 r"""\ 
 
 • ; 
 
 ■v_y 
 
 
 Fig. 153. 
 
 Fig. 154. 
 
 be elongated, and colored spectra will be seen on 
 both sides of /(Fig. 153). 
 
 Experiment 207. — Study the effects produced 
 by giving the aperture different forms, such as a tri- 
 angle, a square aperture, a number of small circular 
 holes punched in a thin metallic plate ; also use a 
 fine wire gauze. 
 
 Experiment 208.— Place a spectacle-lens of long 
 focus on a flat piece of glass, having carefully cleaned 
 them both (Fig. 154). Place this combination be- 
 tween two pieces of wood, and by means of screws 
 at three points press the lens and glass together, 
 
338 NEW PHYSICS. 
 
 and rings of color will be observed around the cen- 
 ter of the lens: these are called Newton's rings. 
 The explanation is the same as that of the interfer- 
 ence produced by soap-films. View these rings by 
 homogeneous light. Homogeneous light can be 
 produced by burning salt in the flame of a Bunsen 
 burner, or of a spirit-lamp. With this light the col- 
 ors of the rings disappear, and we have merely dark 
 rings. 
 
 The experiments which we have now tried upon 
 diffraction and the interference of light prepare us 
 to understand how the energy of the solar spectrum 
 can be measured. 
 
 In the case of radiant energy the movements of 
 the ether are perpendicular to the direction of prop- 
 agation of the wave ; that is, a particle of the ether 
 moves like a boat at anchor — ^up and down — while 
 the wave passes beneath it. This motion is also, it 
 will be observed, a pendulum motion. It is the 
 same as that made by the prong of a tuning-fork 
 observed by an eye placed directly in line with the 
 vibrating prong, and can be represented also by 
 the ordinary pendulum by connecting a string to 
 the pendulum-bob, passing it through an eyelet E, 
 in the line of rest of the pendulum, and attaching a 
 weight w to the other end ; as the pendulum-bob 
 swings to and fro, the weight w executes periodic 
 movements similar to a particle of ether when a 
 wave of light passes through it. 
 
 In the case of sound-waves, which are propagated 
 through ordinary air and gases and solids, and not 
 through or by the ether, the vibration of the parti- 
 cles is in the direction of the propagation of the 
 wave. Perhaps the simplest illustration of this is 
 
LIGHT. 
 
 339 
 
 the phonograph, which consists of a circular metal- 
 lic disk supported around its edges, and provided 
 with a point or stylus at its center. This point rests 
 upon a grooved cylinder covered with tin-foil.* 
 Speak into the mouth-piece surrounding the disk and 
 turn the cylinder : the sound-waves will be seen to 
 make little dots upon the ' tin-foil. These dots are 
 due to impulses perpendicular to the disk, and there- 
 fore in the direction of the sound-waves. On re- 
 volving the cylinder in the opposite direction, the 
 little dots or depressions will communicate impulses 
 to the disk, which in turn will give sound-waves. 
 It is evident that sound-waves can be refracted as 
 well as light-waves, and Herschel's illustration of 
 the march of soldiers across a plowed field can 
 also be used in reference to the front of a sound- 
 wave, which impinges upon the line of separation 
 between two media. The striking difference be- 
 tween the waves of sound and those of light is due 
 to the difference in velocity of their propagation, the 
 velocity of sound-waves in air being about i ,000 feet 
 per second, while that of light is 180,000 miles per 
 second, and also the difference in their size ; the long- 
 est visible wave of light is' not more than .000760 of 
 a millimetre, while the longest wave of sound sensi- 
 ble to us as a musical sound is about sixteen feet. 
 Our apparatus for measuring heat- and light-waves 
 must therefore be capable of measuring very much 
 smaller quantities than our sound apparatus. 
 
 This difference in dimensions is shown in the 
 phenomenon of diffraction. On looking through a 
 very narrow slit at a flame, we perceive bright and 
 
 * The qrlinder moves along its axis of revolution by means of a 
 screw, and the stylus therefore draws a helix upon the tin-foil. 
 16 
 
340 
 
 NEW PHYSICS. 
 
 dark spaces on each side of the light of the slit. 
 The light rays appear to be bent out of their straight 
 course by the edges of the slit. On sending a sound- 
 wave through a slit, and using suitable apparatus to 
 detect spaces where sound can be heard and where 
 it can not be heard on each side of the slit, we find 
 that with a suitable sized slit the phenomena of dif- 
 fraction of sound can also be observed. The size of 
 the slit, however, is enormous compared with that 
 which is used to obtain the phenomena of the dif- 
 fraction of light. In both cases the slit must be of 
 the order of dimension of the waves which are to 
 be affected in their passage. There is very little 
 difficulty in getting the length of the waves of 
 sound, but we naturally infer, from the size of the 
 waves of light, that it is much more difficult to ob- 
 tain their dimensions. The method of diffraction is 
 employed for this purpose. 
 
 Let us return to the analogy we have used in 
 the case of the refraction of light, and suppose that 
 our flock of birds of various colors and with differ- 
 ent rate of wing-movements, all advancing with the 
 same velocity, should endeavor to fly through a nar- 
 row slit between high rocks. The sportsman on the 
 side toward which the birds are flying would then 
 see a number of birds of all colors flying directly 
 through the middle of the opening and not diverted 
 by the sides of the slit. On each side, however, of 
 this line passing perpendicularly through the mid- 
 dle of the slit, and in the horizontal plane passing 
 through the middle of the slit, there will be spaces 
 where there will be no birds, and where there are 
 birds advancing in an extended front ; the red birds 
 being most diverted from the central line, and the 
 
LIGHT. 
 
 341 
 
 violet birds, having the shortest length of wing, the 
 least diverted. The disappearance of birds from 
 this horizontal plane can be explained by the clash- 
 ing of wings of two birds of the same length of 
 wing, one bird moving its wings upward while the 
 bird above it moves its wings downward. In this 
 way the work the birds do in striving to maintain 
 their- common velocity in passing through- the slit, 
 gives a spectrum, so to speak, of birds separated by 
 spaces where there are no birds, on each side of the 
 central line passing perpendicularly through the 
 middle of the slit. Passing now to the case of light- 
 waves, we see that two waves of the same length, 
 counting the length of a wave as the distance from 
 the height of an elevation to the depth of a depres- 
 sion, advancing together, will superimpose them- 
 selves on each other, and the resulting height and 
 depression will be due to their combined movement. 
 If, however, the following wave should be one half 
 a wave-length behind, the upward movement of one 
 will be opposed by ia downward movement of the 
 other, and there will be no motion, and consequent- 
 ly no light. 
 
 If, therefore, we can make our slit so small that 
 the difference in the paths of the deflected rays of 
 light can in certain cases be whole wave-lengths and 
 half-waves, we shall have spectra of light separated 
 by dark spaces, the spectra being produced by the 
 superposition of waves of the same length and of 
 the same path, while the dark spaces are due to the 
 interference of waves of the same length but whose 
 paths differ by one half a wave-length. 
 
 The best way to form slits is found to be to rule 
 very fine lines on glass or speculum-metal, the light 
 
342 
 
 NEW PHYSICS. 
 
 in the case of glass passing through it, and in the 
 case of lines ruled on speculum-metal being reflect- 
 ed from the polished surface of the metal ; in both 
 cases the inequality in the surfaces affording a 
 slight difference of path for the different waves of 
 light. Let us first take the case of the glass diffrac- 
 tion grating. Let S (Fig. 155) be 
 the distance between any two fine 
 lines. Let the angle a represent 
 the angular interval between the 
 direction of the line passing direct- 
 ly through the slit, and the direc- 
 tion of the ray of any color whose 
 wave-length we wish to measure. 
 From our preceding illustrations it 
 is evident that light of this color is 
 found in one of the diffraction spec- 
 tra when the difference in position 
 between the wave-front of the light 
 striking at B, and that wave-front of the same col- 
 ored light arriving at C, after striking at ^, is a 
 whole wave-length. If this difference were half a 
 wave-length, the two waves would interfere, and 
 no light would be seen at this angular interval. 
 Hence, 5 sin a = \ = wave-length. Since A B C = 
 a, the sides being mutually perpendicular. 
 
 It is customary to measure the angular interval 
 between the same points in the two spectra which 
 are formed on both sides of the line perpendicular 
 to the grating which also passes through the slit 
 employed to throw a beam of parallel light upon 
 the grating. 
 
 The apparatus for measuring wave-lengths there- 
 fore is a spectrometer, in which the prism of Ex- 
 
 FlG. 155. 
 
LIGHT. 
 
 343 
 
 periment 187 is replaced by a diffraction grating 
 mounted at the middle of a graduated circle ; the 
 angular intervals between the line of light passing 
 through the collimator perpendicular to the grating 
 and any line of light or reversed solar line in any 
 spectra being measured on this graduated circle. 
 The spectrum formed by rays the least diverted is 
 called a spectrum of the first order ; then follows a 
 dark space, and then a spectrum of the second order, 
 and so on. 
 
 These spectra are called normal spectra, since 
 they are not distorted by a medium such as that of 
 a prism. 
 
 Modern measurements of wave-lengths are there- 
 fore made with a refined spectrometer which is 
 capable of measuring angles with great precision, 
 and a diffraction grating, which is generally ruled 
 on highly polished speculum-metal. The difference 
 in path of the waves in this case is produced by re- 
 flection from the grating on the metal. The sub- 
 stance of the metal absorbs a portion of the energy 
 of the light falling upon it, but this portion is ex- 
 tremely small, and is not far from the order of errors 
 which are due to the most refined apparatus we 
 employ in measuring the wave-lengths. The dis- 
 tance 5 between the lines of the grating can be 
 measured under a microscope. It is generally re- 
 corded by the dividing engine which rules the grat- 
 ing. On measuring by means of the thermopile the 
 distribution of heat in the spectrum, it is found that 
 the maximum energy occurs in the orange. There 
 is, then, a great field beyond the red invisible to the 
 eye, in which nevertheless the waves of energy can 
 affect the molecules of matter. 
 
344 
 
 NEW PHYSICS. 
 
 In the direction of the ultra-violet there is less 
 energy manifested as heat, and less as light. The 
 very short waves of the ultra-violet, moving with 
 great velocity, are capable, however, of disturbing 
 the equilibrium of the grouping of molecules in 
 many substances, especially in the salts of silver. 
 Thus we have at this end of the spectrum great pho- 
 tographic sensitiveness. 
 
 Experiment 209. — Having formed a solar spec- 
 trum, receive it upon a screen S (Fig. 1 56). Place this 
 screen so as to cut off half the visible spectrum, and 
 
 I I li 
 
 I i 
 
 hi. I 
 
 I ri'' I 
 
 if , f| u I'f I Hiflt 
 
 Fig. 156. 
 
 allow the rays of the remainder, brought to a focus 
 by a lens, to fall upon a thermopile, T, made of a great 
 number of junctions of bismuth and antimony, which 
 is connected with a very sensitive galvanometer, the 
 deflections of which are observed by means of a 
 telescope and scale. It will be found by moving 
 the screen that the two halves of the spectrum thus 
 collected by the lens do not give the same effect ; 
 and that a less portion near the red end is equiva- 
 lent to the larger portion of the remainder of the 
 spectrum. If the instruments are sensitive enough; 
 it will be found that there are dark rays of heat be* 
 
LIGHT. 345 
 
 yond the red, which can affect the thermopile, but 
 are not able to affect the eye of the observer. If we 
 should represent by a curve the distribution of heat 
 in the solar spectrum, produced by means of a dif- 
 fraction grating, it will be found that the maximum 
 heat energy is in the red and orange. / 
 
 Spectrum analysis is a study of the conservation 
 of energy. The application of a spectrometer to the 
 detection of metals by the existence of their charac- 
 teristic bright lines, or to the mapping of the dark 
 lines in the solar spectrum, is merely an application 
 of the principle of the conservation of energy. 
 
 To show that the light-waves carry also heat, 
 and that the two are merely manifestations of ener- 
 gy, and that we can not separate our consideration 
 of light from that of heat, a study of the distribu- 
 tion of heat in the solar spectrum is necessary. This 
 study requires very sensitive apparatus, for the 
 amount of heat in any one portion of the spectrum 
 is very small. Formerly, this study was conducted 
 entirely by means of the spectra produced by 
 glass or liquid prisms. It was soon discovered, 
 however, that these prisms distorted the spectrum, 
 giving a great distribution of heat near the red end 
 of the spectrum and very little in other parts. Be- 
 fore explaining the more refined methods of subse- 
 quent observers, let us explain the old methods. 
 Instead of a thermopile, electrical phenomena can 
 be used in another way to illustrate the distribution 
 of energy. The electrical current heats a wire, and 
 in thus heating a conductor internal work is done 
 upon the wire. In turn, if we apply external heat 
 to the wire, the electrical resistance of the wire is 
 increased. 
 
346 
 
 NEW PHYSICS. 
 
 If we should, therefore, form a Wheatstone's 
 bridge, and make one of the branches very sensitive 
 to the rays of heat, that is, so that a. slight change in 
 temperature will produce a great change in electri- 
 cal resistance, we could use this bridge to measure 
 temperature as well as electrical resistance. 
 
 Experiment 2io.~7Measure the variations in 
 temperature in the neighborhood of a gas-flame. 
 
 Make one of the arms, 
 7? (Fig. 157), of a bridge 
 of fine platinum wire, 
 and cover it with lamp- 
 black, lamp-black being 
 the best absorbent of 
 heat ; place a similar 
 R""^\ ^n' ^ coil, R' , in the opposite 
 
 branch, in water, in 
 which is also a ther- 
 mometer. The connec- 
 
 FiG. 157. 
 
 tions of R and R' with 
 
 the corners of the bridge should be made of thick 
 copper wire, of which the change of resistance can 
 be neglected. As we move the wire R toward the 
 flame, its resistance will change, and we must heat 
 the water in which R! is immersed in order to re- 
 store the balance of the bridge. The increase of 
 temperature is indicated by the thermometer in 
 the water surrounding R'. This method can evi- 
 dently be used to ascertain the temperature of 
 places where a mercury thermometer can not be 
 conveniently used, such as the depth of the sea, or 
 the interior of furnaces. It has certain advantages 
 over a thermo-electric junction, for a thin wire can 
 be moved across the solar spectrum and will indicate 
 
LIGHT. 
 
 347 
 
 by a change in electrical resistance the temperature 
 of different portions. 
 
 The extreme sensitiveness of metals to changes 
 of temperature is illustrated by this use of the Wheat- 
 stone bridge, in which electricity makes these changes 
 evident. This sensitiveness can also be manifested 
 by what is called the radiophone. A beam of light 
 is interrupted a great number of times a second, and 
 its pulsations are received upon a thin metallic plate 
 which closes one end of an ordinary speaking-tube. 
 On placing the other end of the speaking-tube to the 
 ear, the note produced by the number of interrup- 
 tions of the beam of light will be heard. In this case 
 the metallic disk changes its form in unison with the 
 increase and diminution of temperature. On replac- 
 ing this thin metallic disk with various non-metallic 
 "Substances, the same phenomenon is observed. Dif- 
 ferent gases, through which such an intermitted beam 
 is made to pass, also give forth the note caused by the 
 wheel. The apparatus necessary for this experiment 
 is very simple. The chief difficulty is in making the 
 interrupting wheel revolve noiselessly at high speed. 
 
 These experiments illustrate the conversion of ra- 
 diant energy, which reaches us in a beam of light, into 
 molecular agitation and sensible heat, and then into 
 sound ; the wave-motion of light in these experiments 
 being finally changed into the wave-motion of sound. 
 
 In pursuing the study of physics we find that the 
 principle of the conservation of energy is constantly 
 exemplified, and that even in the subjects of light 
 and electricity we are studying motion, and need a 
 sound knowledge of mechanics in order to test mo- 
 lecular theories. The subject of molecular physics 
 is beyond the scope of this treatise. 
 
APPENDIX. 
 
 GENERAL DIRECTIONS. 
 
 The laboratory should be provided with certain reference- 
 books. The following will be found extremely useful : 
 
 Daniell's " Principles of Physics." 
 
 Everett's " PhysicEil Constants." 
 
 Pickering's " Physical Manipulations." 
 
 Ganot's "Physics." 
 
 Kohlrausch's " Physical Measurements." 
 An elementary physical laboratory requires merely some suit- 
 able tables, with plenty of light. At one or two of the tables gas 
 should be introduced. If gas can not be obtained, the various 
 forms of spirit-lamps, such as are used in little portable cooking- 
 arrangements, can take the place of Bunsen gas-burners. A 
 water-tap is also necessary, and a soapstone sink, sufficiently large 
 to contain the batteries employed. It is convenient to have a large 
 closet, or a portion of the laboratory inclosed, which may be dark- 
 ened, for mirror galvanometers, and for experiments with lenses. 
 
 When the teacher has the requisite leisure and ingenuity, 
 much physical apparatus can be constructed with comparatively 
 little money. The student can readily construct his own galva- 
 nometer and make resistance-coils with the aid of one standard 
 box of coils. Instead of making electro-magnets, one can gen- 
 erally purchase at a low price suitable ones which have been dis- 
 carded by manufacturing electricians. Gravity batteries can be 
 readily constructed by the student, and also bichromate of potash 
 batteries. See pages 212-213. 
 
 A list of essential apparatus for a class of six boys is appended, 
 with an approximate estimate of its cost : 
 
35° 
 
 APPENDIX. 
 
 A balance weighing to five milligrammes, to- 
 gether with the weights $60 00 
 
 Three Bunsen burners, or spirit-lamps 4 00 
 
 Three hydrometers i 5° 
 
 Six specific-gravity bottles (bottles with ground- 
 glass stoppers, ten cents apiece) 60 
 
 Set of cork-borers 2 00 
 
 Mercury 10 00 
 
 Air-pump (Richards' aspirator can be obtained 
 
 of Prang & Co., Boston) 2 50 
 
 Thermometers and vernier gauges (Alvergniat 
 Frferes, Paris, France ; Brown & Sharp, Provi- 
 dence, Rhode Island) 20 00 
 
 One box of resistance-coils (Charles Williams & 
 
 Co., electricians, Boston) 50 00 
 
 Four galvanometers 10 00 
 
 Two electrometers 50 00 
 
 Four tuning-forks 8 00 
 
 Four lenses 8 00 
 
 Four prisms 8 00 
 
 Tools, corks, wire, calorimeters, alcohol, rubber 
 
 tubing, glass, general supplies 50 00 
 
 $284 60 
 
 With the above apparatus a class of twelve boys can be well 
 employed, for the time generally devoted to physics in the high 
 schools, working in sets of three. It is best to gradually duplicate 
 the balances and sets of resistance-coils. The student should 
 make his own sets of small weights and set of small resista,nce- 
 coils. 
 
 USEFUL APPLIANCES. 
 
 Cement. — Take of gutta-percha (sheets cut into shreds) one 
 part, and of pine (not coal-tar) pitch one part. 
 
 Melt at 100° C. for some time, stirring occasionally till homo- 
 geneous. This cement is useful as a substitute for sealing-wax 
 in fitting up apparatus, and fuses at so low a temperature that 
 tight joints may be made even in the presence of water, and it im- 
 proves every time it is heated. 
 
APPENDIX. 
 
 351 
 
 When a large hole is to be made with a " cork-borer" in a 
 small cork, the danger of splitting can be avoided by wrapping 
 the cork tightly with twine. 
 
 Paraffine is an excellent lubricator when thermometers, etc., 
 are to be thrust through tightly-fitting holes in corks. 
 
 The interior of a wet bottle can be dried by passing a blast of 
 air from a bellows through a piece of glass tubing which projects 
 into the bottle. It can be more quickly dried by heating the air 
 as it passes through the glass tube by placing a Bunsen burner 
 under the glass tube and heating it to a low red heat. 
 
 A simple method of calibrating a thermometer-tube is to sep- 
 arate a thread of mercury (by shaking the thermometer^ about 
 50° long. Run it to the upper end of the thermometer-stem, 
 placing one end at the 100° mark. Then read the other end. 
 Then run it down the stem until one end rests at the 50° mark, 
 and read the other end, and estimate how much too long or too 
 short the divisions are between 50° and 100° compared with those 
 between 0° and 50°. Then separate a thread 25° long, and re- 
 peat the operation, using the 25° and the 50° marks in the same 
 way that the 50° and 100° marks were used, and distribute the 
 errors of graduation by comparing all the graduations with the 
 graduations from 0° to 25° as standards, and so on. 
 
 Glass-working requires much practice in order to do well. 
 The following hints may be useful : 
 
 To smooth the Ends of Tubes. — Warm the end in a flame, and 
 then hold it obliquely in the flame, turning it till the edges are 
 rounded. 
 
 To bend Tubes. — Gradually warm the tube in the flame ; then 
 hold it a little obliquely, turning it between the thumbs and the 
 fore-fingers until it is yellow. Then remove it from the flame and 
 bend it with one motion. 
 
 To dose the End of a Tube. — Soften it at the end and pull 
 the end out with another piece of glass. Keep removing the 
 small tail that is left till the latter is very slender ; then heat the 
 latter, turning the tube in the fingers. 
 
 To fuse Platinum into a Glass Tube. — Heat the glass tube 
 thoroughly at the portion where the platinum wire is to be in- 
 
352 
 
 APPENDIX. 
 
 serted. Then thrust the wire into the glass and pull out the lat- 
 ter into a fine tube. Cut this tube off near the main tube and 
 round the edges. Then insert the red-hot platinum wire and 
 turn it around until perfect contact is obtained. 
 
 To join Tubes at Right Angles. — Close one end of each tube, 
 and then direct a pointed flame from a blow-pipe upon the side 
 of one of the tubes. Blow into the tube and gradually produce a 
 swelling on the side of the tube. Finally, by a quick puff, force 
 this swelling open. Round the edges of this opening in the flame. 
 Heat the edges of the other tube and then fuse it to the opening. 
 
 To blow a Bulb. — First close the end of a tube by turning it 
 for some time in a flame. After the end has become very hot, 
 blow gently into the other end. 
 
 Holes can be bored in glass readily by employing a very 
 small rat-tail file, the end of which has been broken off with a 
 pair of pincers. This broken end is pressed against "the glass and 
 kept moistened with a paste made of spirits of turpentine and gum- 
 camphor. The camphor acts in a mechanical way, holding the 
 grinding particles in the cavity made by the end of the file. The 
 end of the file should be broken off slightly from time to time 
 with the pincers. 
 
 Glass bottles can be cut in two by running a little pointed 
 flame from a crack started by a file-scratch in the surface of the 
 glass. In order to produce the pointed flame, pull out a small 
 tube of glass to a fine point and break off the end of this point 
 and insert the wide end of the tube in a piece of rubber tub- 
 ing connected with a gas-cock. The size of the flame can be 
 -regulated by the fineness of the point of the glass. The flame 
 should be very small and very pointed. 
 
 A soldering-iron, with a comparatively fine point, can be used 
 instead of the small gas-flame. 
 
 The edges of a glass vessel can be ground smooth by rubbing 
 the edges on emery-paper moistened with water, or by rubbing 
 the edges on a flat surface of iron covered with emery-powder 
 suitably moistened. 
 
 Experiments involving the use of mercury should be per- 
 formed upon a table, the top of which is provided with ledges. 
 The table should be slightly inclined, in order that the stray mer- 
 cury may collect at one side. 
 
APPENDIX. 
 
 353 
 
 If millimetre scales can not be obtained, the box-wood scales 
 employed by engineers, which are divided into tenths and twen- 
 tieths of an inch, can be used. 
 
 Accurate vernier gauges or micrometer gauges can be ob- 
 tained of Alvergniat Fr^res, of Paris, or of Brown & Sharp, Provi- 
 dence, Rhode Island. 
 
 A simple dividing engine is described in Pickering's " Physical 
 Manipulation," vol. i, page 59, which can be constructed at slight 
 expense, and will enable one to graduate paper scales, which are 
 of great service in elementary physical laboratories. 
 
 Insertion of Cross-Hairs. — Cut from card-board a little ring 
 with projecting flanges. Stretch two fine fibers of silk — the finest 
 fibers of a silk thread — at right angles to each other. Bend the 
 flanges so that the ring can be slipped along the tube of a tele- 
 scope and be placed at the focus of the eye-piece. 
 
 The telescope and micrometer should be made by the student. 
 The apparatus needed consists merely of tin or brass tubes, and 
 corks to fit the lenses to the tubes or the tubes to each other. 
 
 A large tuning-fork, suitable for many of the experiments on 
 motion, can be made by any blacksmith. The length of its prongs 
 should be about fifteen inches, the distance of these apart about 
 four inches, and the section half an inch by a quarter of an 
 inch. It is well that the handle should be long and heavy, in 
 order to afford a counterpoise to the prongs of the fork. A smaller 
 fork made in the same way, of about one half the size of the larger 
 fork, will also be useful. The price of such tuning-forks need not 
 exceed one dollar and a half apiece. Small vises suitable for 
 clamping such forks can be obtained at hardware-stores. These 
 vises can be screwed to loaded boxes or to any suitable supports. 
 
 Plane or convex mirrors suitable for galvanometers can be ob- 
 tained by mail from Elliott Brothers, 449 Strand, London, W. C. 
 The average price of these mirrors is about fifty cents. 
 
 If a telescope is used, a plane mirror should be affixed to the 
 magnet ; if a spot of light, a convex mirror. A lens of long focus, 
 such as an ordinary spectacle-lens, placed directly in front of a 
 plane mirror, will enable one to use a spot of light. It is there- 
 
354 
 
 APPENDIX. 
 
 fore better to order plane mirrors, for they can be used as con- 
 cave mirrors by employing a lens placed in front of them, or they 
 can be used with a telescope and scale. The lens, moreover, en- 
 ables one to vary the distance of the spot of light on a screen, and 
 thus increase or diminish the index beam of light. 
 
 Sets of resistance-coils suitable for the experiments in this 
 book can be obtained of Charles Williams & Co., electricians, 
 Boston, Mass. The price is about fifty dollars. With one box 
 of coils other sets of resistances can be made by the students 
 themselves. For many experiments the estimation of resistances 
 by means of Table VIII is sufficient. The correct value of such 
 estimated resistances can be obtained subsequently. 
 
 Simple break and make keys can be made by screwing a 
 piece of spring-brass to a board, connecting a wire to the screw, 
 and another wire to a screw upon which the brass can be pressed. 
 There are many forms of reversing-keys. A simple one can be 
 made as follows : Screw four square pieces of brass at the four 
 angles of a square piece of board, having first boiled the wood in 
 oil or covered it with paraffine. The pieces of brass should be 
 about one inch square and half an inch apart, Solder wires to 
 each of these brass plates, or screw binding-screws 
 A B upon them. Connect A and D with the galvanometer 
 
 E or electrometer, and C and B with the battery. If, then, 
 C D ■we. connect C and A, and B and D, the current may 
 flow from C to A, then to the galvanometer, and may 
 return to B and D, and then to the battery. If, on the other 
 hand, we connect B and A, and D and C, the current passes 
 through the galvanometer in the reverse direction. A suitable 
 bar, provided at its ends with pieces of spring-brass insulated 
 from each other, and capable of revolving about E, can serve to 
 perform the operation indicated. 
 
 It is a useless expense for the purposes of elementary labora- 
 tory instruction to buy an expensive galvanometer. The student 
 should construct his own galvanometers. With very little care 
 he can make instruments which are as exact and as sensitive as 
 those made by instrument-makers. In making astatic gcdvanora- 
 eters, the principal difficulty consists in making the two magnets, 
 which are placed one above the other with poles reversed, exactly 
 equal. A little care will accomplish this. The astatic combina- 
 
APPENDIX. 
 
 355 
 
 tion should then have a very slow time of vibration. It is neces- 
 sary to employ a magnet outside the galvanometer to bring the 
 astatic combination back to a zero. The astatic galvanometer 
 described in the chapter on thermo-electricity need not cost 
 more than three dollars, and is extremely sensitive to thermo-elec- 
 tric currents. 
 
 TABLES. 
 
 The following tables embrace only the data necessary for the 
 performance of the experiments in this treatise, and are taken 
 mainly from Kohlrausch's " Physical Measurements.'' The stu- 
 dent is referred to Everett's " Physical Constants " for more ex- 
 tended tables. 
 
 TABLE I. 
 
 To convert millimetres into inches, multiply by .03937. 
 
 To convert metres into inches, multiply by 39.37. 
 
 To convert grammes into grains, multiply by 15.44, 
 
 To convert kilogrammes into pounds, multiply by 2.205. 
 
 I United States Gquid quart = 0.946 litre, a little^less than i 
 litre. 
 
 I United States dry quart = l.ioi litre, a little more than i 
 litre. 
 
 I United States gallon = 3.785 litres, or about Jrh litres. 
 
 I avoirdupois ounce = 0.02835 kilo, or rather less than 30 
 grammes. 
 
 I Troy and apothecaries' ounce = 0.031 10 kilo, or rather more 
 than 30 grammes. 
 
 I avoirdupois pound = 0.45359 kilo, or about ^t kilo. 
 
 TABLE II. 
 
 Table of Specific Gravities. 
 
 Solids. 
 
 Standard for solids and liquids is distilled water at 0° C. 
 
 Antimony 6.712 Glass, flint 3.400 
 
 Beech 0.852 Gold 19.360 
 
 Bismuth 9.822 Human body. 0.890 
 
 Brass 8.380 Ice 0.920 
 
 Cork 0.240 Iridium 23.000 
 
 Diamond 3-530 Iron, cast 7.210 
 
 Ebony 1.187 Iron, bar 7-78o 
 
356 APPENDIX. 
 
 Lead, cast i i-3So Saltpeter 1.900 
 
 Oak 0.845 Silver, cast 10.470 
 
 Pine .• . . . 0.650 Sulphur, natural 2.033 
 
 Platinum 22.069 Tin, cast 7.290 
 
 Quartz 2.650 Zinc, cast 6.860 
 
 Rock-salt 2.257 
 
 Liquids. 
 
 Alcohol, absolute 0.800 Nitric acid 1.420 
 
 Bisulphide of carbon .. . 1.293 Oil of turpentine 0.870 
 
 Ether 0.723 Olive-oil o-9iS 
 
 Hydrochloric acid i .240 Sea-v\rater 1.026 
 
 Mercury 13-598 Sulphuric acid 1.841 
 
 Milk 1.032 Water, 4° C, distilled .. . i.ooo 
 
 Naphtha 0.847 Water, 0° C, distilled . . . 0.999 
 
 Gases. 
 Standard : air at 0° C. ; barometer, 76™. 
 
 Air i.oooo Hydrogen 0.0693 
 
 Ammonia . .~ 0.5367 Nitrogen 0.9714 
 
 Carbonic acid 1.5290 Oxygen 1.1057 
 
 Chlorine 34400 Sulphuretted hydrogen. 1.1912 
 
 Hydrochloric acid 1.2540 Sulphurous acid 2.2474 
 
 TABLE III. 
 Coefficients of Expansion for \° C. 
 The length Z of a body is increased by /3Z for each degree of 
 increased temperature, and its volume V by 3/3 V. 
 
 Lead 0.0000285 Brass 0.000019 
 
 Iron o.ooooi 2 Platinum 0.000009 
 
 Glass 0.0000085 Silver 0.000019 
 
 Gold O.OOOOI 5 Zinc 0.000029 
 
 Copper 0.0000175 Tin 0.000022 
 
 The volume V of quicksilver increases 0.000181 5 V for 1°. 
 
 TABLE IV. 
 
 Moments of Inertia. 
 Thin bar of length / and weight m. The thickness uniform 
 and very small compared with /. Referred to an axis at right 
 angles to the length /, and passing through its center : 
 
APPENDIX. 357 
 
 „_ nt P 
 
 12 
 
 Right-angled parallelopipedon — a and b two adjacent edges. 
 Axis passing through the center of gravity and parallel to the 
 
 third edge-: K-m /'^!-±i!V 
 
 Cylinder of radius r and length /, referred to the axis of the 
 
 cylinder : K = — . 
 
 Cylinder of radius r and length /, referred to the axis perpen- 
 
 (P r^\ 
 1 — )• 
 
 Hollow cylinder, r = inner radius, and r, = outer radius, re- 
 
 — ' )• 
 
 Sphere of radius r, referred to a diameter : IC — — mr'. 
 
 S 
 Experimental Determination of Moment of Inertia. 
 The time of oscillation of the body is observed about the axis, 
 and then the moment of inertia increased by a known amount 
 without changing the field of force, or the directive force, and the 
 time of oscillation again observed. 
 
 li t =: time of oscillation of the body alone, 
 t, = time of oscillation of the body with added weight, 
 i = moment of inertia : 
 We have t; : f = (K + i) : K, 
 
 The oscillations must be through very small arcs. 
 TABLE V. 
 
 Specific Heats. 
 
 Lead 0.0314 Silver 0.0570 
 
 Iron 0.114 Zinc 0.0955 
 
 Glass 0.19 Tin 0.0562 
 
 Gold 0.0324 Ether at 17° 0.516 
 
 Copper 0.0951 Alcohol at 17° 0.615 
 
 Brass 0.094 Quicksilver 0.0333 
 
 Platinum 0.0324 Oil of turpentine at 17°. 0.426. 
 
358 
 
 APPENDIX. 
 
 Water at o° i.oooo Water at 30° 1.0020 
 
 Water at 10° 1.0005 Water mean between 0° 
 
 Water at 20° 1.0012 and 100° 1.0050 
 
 TABLE VI. 
 
 Specific Resistances of Metals compared with that 
 OF a Column of Mercury at 0°. 
 
 Lead (pressed) 0-i99 
 
 Mercury i.oooo 
 
 Platinum (soft) 0.0918 
 
 Silver (soft) 0.0153 
 
 Silver (hard) 0.0166 
 
 Tin 0.134 
 
 Zinc 0.0571 
 
 Antimony (pressed) 0.360 
 
 Bismuth (pressed) 0.133 
 
 Brass (hard) 0.051 
 
 Copper (hard) 0.0166 
 
 Copper (soft) 0.0162 
 
 German-silver (hard) ... 0.2x2 
 
 Gold 0.0209 
 
 Iron (soft) 0.0986 
 
 If we call the specific resistance of a metal s, the resistance 
 .ff of a length / of section q expressed in square millimetres will be 
 
 q 
 
 The numbers given in the table are only approximately correct 
 for the commercial metals. 
 
 TABLE vn. 
 
 Comparison of Measures of Electric Current- 
 Strength. 
 
 
 Must be multiplied by the following numbers to reduce it to — 
 
 a current-strength 
 
 which is measured 
 
 in — 
 
 Cubic cm. 
 
 water-gases 
 per min. 
 
 Mgr. water 
 per minute. 
 
 Mgr. copper 
 per minute. 
 
 Mgr. silver 
 per minute. 
 
 Magnetic 
 
 measure 
 
 Mm.* mgr.i 
 
 
 Sec." 
 
 Cubic cm. water- 
 gases peir min. 
 Mgr. water per 
 
 1.865 
 
 0.5294 
 
 0.1555 
 
 1.044 
 
 0.5363 
 
 0.2839 
 0.0834 
 
 0.5599 
 
 1.889 
 3.522 
 0.2937 
 
 1.972 
 
 6.432 
 
 11.99 
 3.405 
 
 6.714 
 
 0.9579 
 
 1.786 
 
 0.5071 
 
 0.1489 
 
 Mgr. copper. . . . 
 
 Mgr. silver 
 
 Magnetic meas. 
 
 Mm.i mgr.i 
 
 Sec." 
 
APPENDIX. 
 
 359 
 
 table viii. 
 Birmingham Wire Gauge. 
 
 Temperature 15° C. 
 
 B W G 
 
 Diameter 
 
 Resistance per 
 
 B W G 
 
 Diameter 
 
 Resistance per 
 
 
 in inches. 
 
 yard in ohms. 
 
 
 in inches. 
 
 yard in ohms. 
 
 
 
 •340 
 
 .ooo2,^^ 
 
 22 
 
 .028 
 
 .0400 
 
 2 
 
 .284 
 
 .cxx)389 
 
 24 
 
 .022 
 
 .0649 
 
 4 
 
 .238 
 
 .000554 
 
 26 
 
 .018 
 
 .0969 
 
 6 
 
 .203 
 
 .000762 
 
 28 
 
 .014 
 
 .160 
 
 8 
 
 .165 
 
 .00115 
 
 30 
 
 .012 
 
 .218 
 
 10 
 
 •134 
 
 .00175 
 
 32 
 
 .009 
 
 .388 
 
 12 
 
 .109 
 
 .00264 
 
 33 
 
 .008 
 
 .491 
 
 14 
 
 .083 
 
 .00456 
 
 34 
 
 .007 
 
 .641 
 
 16 
 
 .065 
 
 .00743 
 
 35 
 
 .005 
 
 1.26 
 
 18 
 
 .049 
 
 .0131 
 
 36 
 
 .004 
 
 1.96 
 
 20 
 
 •03 s 
 
 .0256 
 
 
 
 
 TABLE IX. 
 
 Electro-motive Force of Constant Batteries (J. C. 
 Maxwell). 
 
 
 
 
 Concentrated 
 
 
 
 
 ~>~-~. 
 
 solution of 
 
 Volt. 
 
 Daniell I. 
 
 Amalgamated zinc. 
 
 HjSOi + 4 Aq. 
 
 CuSOi Copper. 
 
 1.079 
 
 " IL 
 
 
 H5SO4 + 12 Aq. 
 
 CuSOi " 
 
 0.978 
 
 " in. 
 
 
 HjSO, + 12 Aq. 
 
 CuNOs " 
 
 i.oo 
 
 Bunsen I. 
 
 
 HjSO, + 12 Aq. 
 
 HNOs Carbon. 
 
 1.964 
 
 " n. 
 
 
 HjSOi + 12 Aq. 
 
 sp. grr. 1.38 " 
 
 1.888 
 
 Grove. 
 
 
 H5SO4 + 4 Aq. 
 
 HNOs Platinum. 
 
 1-956 
 
 TABLE X. 
 
 Mean Indices of Refraction and Dispersions of 
 several Bodies. 
 
 Crown glass (mean) 
 Flint glass (mean). . 
 
 Water 
 
 Alcohol 
 
 Carbon disulphide. . 
 Canada balsam .... 
 Air 
 
 Index of 
 refraction. 
 
 Dispersion. 
 
 1-53 
 
 0.022 
 
 1.60 
 
 0.042 
 
 1-336 
 
 0.0132 
 
 1.372 
 
 0.0133 
 
 1.68 
 
 0.0837 . 
 
 1-54 
 
 
 1.000294 
 
 
360 APPENDIX. 
 
 TABLE XI. 
 
 Numbers frequently used. 
 Coefficient of expansion of gases, 0.003665 (^f|^). 
 Latent heat of water = 79.4. 
 Latent heat of steam at 100° C. = 540. 
 Mechanical equivalent of heat : 
 
 I pound water heated 1° F. = 772 foot-pounds. 
 
 I pound water heated 1° C. =: 1390 foot-pounds. 
 
 I gramme water heated 1° C. = 424 gramme-metres. 
 77 = 3.1415926. 
 Square root of 2 — 1.4142. 
 Square root of 3 — 1.7320. 
 
 Force of gravity at equator in centimetres, 978.0. 
 Force of gravity at pole in centimetres, 983.2. 
 Cubic inch in centimetres, 16.387. 
 
 Weight of I cubic inch of water in grains at 62° F. = 252.458. 
 Foot-pound in kilogramme-metres, .13825. 
 Dyne in grammes, approximate, .00102. 
 Erg in gramme-centimetres, approximate, .00102. 
 Velocity of sound in air, in centimetres at 0° C. = 33030. 
 Latent heat of fusion of ice, 79.4. 
 
APPENDIX. 
 
 table xii. 
 Triqonometrical Functions. 
 
 361 
 
 Angle. 
 
 Sine. 
 
 Tangent. 
 
 Cotangent. 
 
 Cosine. 
 
 
 0° 
 
 I 
 
 o.ooo 
 0.017 
 
 17 
 18 
 
 0.000 
 0.017 
 
 17 
 18 
 
 00 
 57-29 
 
 
 I.OOO 
 I.OCO 
 
 
 
 90° 
 89 
 
 2 
 
 3 
 
 0-035 
 0.052 
 
 17 
 18 
 
 0.035 
 0.052 
 
 17 
 18 
 
 28.64 
 i9.oci 
 
 
 0.999 
 0.999 
 
 
 
 I 
 
 88 
 
 4 
 
 0.070 
 
 
 0.073 
 
 17 
 18 
 18 
 18 
 
 14-30 
 
 
 0.998 
 
 86 
 
 7 
 
 0.0S7 
 0.105 
 0.122 
 
 17 
 18 
 
 17 
 17 
 
 0.0S7 
 0.105 
 0.123 
 
 11-43 
 9.514 
 8.144 
 
 
 0-996 
 0.99s 
 O.S93 
 
 2 
 
 I 
 
 2 
 
 3 
 
 85 
 84 
 83 
 
 8 
 
 0.139 
 
 0.141 
 
 17 
 i3 
 
 7-115 
 
 811 
 643 
 
 
 82 
 
 9 
 
 0.156 
 
 17 
 18 
 
 0.158 
 
 6.314 
 
 0.988 
 
 2 
 
 3 
 
 81 
 
 lo- 
 
 0.174 
 
 
 0.175 
 
 
 S-671 
 
 
 0.985 
 
 
 80° 
 
 II 
 12 
 13 
 
 0.191 
 
 O.208 
 
 0.225 
 
 17 
 17 
 17 
 17 
 17 
 
 il 
 17 
 17 
 16 
 
 0.194 
 0.213 
 0.231 
 
 18 
 
 19 
 18 
 18 
 
 5-145 
 4- 70s 
 4-331 
 
 526 
 440 
 
 374 
 320 
 
 0.982 
 0.978 
 0.974 
 
 3 
 4 
 4 
 4 
 
 79 
 78 
 77 
 
 14 
 
 0.242 
 
 0.249 
 
 16 
 
 4.01 1 
 
 0.970 
 
 76 
 
 15 
 
 0.259 
 
 0.26S 
 
 3-732 
 
 279 
 
 0.966 
 
 4 
 
 75 
 
 16 
 17 
 
 0.276 
 0.292 
 
 0.287 
 0.336 
 
 19 
 19 
 19 
 19 
 23 
 
 3-487 
 3-271 
 
 245 
 216 
 
 0.961 
 0.956 
 
 S 
 S 
 S 
 
 74 
 73 
 
 18 
 19 
 
 0.339 
 0.325 
 
 0.325 
 0-344 
 
 3-078 
 2.904 
 
 193 
 174 
 157 
 
 0.951 
 0.946 
 
 72 
 71 
 
 20° 
 
 0.342 
 
 
 0.364 
 
 
 2.747 
 
 
 0.940 
 
 
 70° 
 
 21 
 22 
 23 
 24 
 
 0.358 
 0.375 
 0.391 
 0.407 
 
 16 
 
 17 
 16 
 16 
 16 
 
 0.384 
 0.434 
 0.424 
 0.445 
 
 23 
 20 
 20 
 21 
 
 2.605 
 2-475 
 2.356 
 2.246 
 
 142 
 130 
 119 
 no 
 
 0.934 
 0.927 
 
 0.y2I 
 0.914 
 
 6 
 
 7 
 6 
 
 7 
 8 
 
 1 
 
 25 
 
 0.423 
 
 15 
 16 
 
 0.455 
 
 
 2.145 
 
 
 0.906 
 
 
 65 
 
 26 
 
 0.438 
 
 0.48S 
 
 22 
 
 2.050 
 
 95 
 
 82 
 
 0.899 
 
 7 
 g 
 
 64 
 
 27 
 
 0.4S4 
 
 15 
 16 
 
 15 
 
 0.510 
 
 22 
 
 1-963 
 
 0.891 
 
 8 
 8 
 9 
 
 63 
 
 28 
 
 0.469 
 
 0.532 
 
 1.881 
 
 0.883 
 
 62 
 
 29 
 
 0.48s 
 
 0.554 
 
 23 
 
 1.804 
 
 77 
 73 
 
 0.87s 
 
 61 
 
 30° 
 
 0.500 
 
 
 0.577 
 
 
 1.732 
 
 
 0.866 
 
 
 6o° 
 
 31 
 
 o-SiS 
 
 IS 
 IS 
 IS 
 
 14 
 
 IS 
 
 0.631 
 
 24 
 
 1.664 
 
 60 
 
 57 
 55 
 
 0-857 
 
 9 
 
 S9 
 
 32 
 
 0.530 
 
 0.625 
 
 24 
 
 1.6^ 
 
 0.848 
 
 9 
 9 
 10 
 10 
 
 S8 
 
 33 
 
 O.S45 
 
 0.6+9 
 
 24 
 26 
 
 25 
 
 1.540 
 
 0.839 
 
 57 
 
 3t 
 3S 
 
 O.S59 
 0.574 
 
 0.67s 
 
 0.700 
 
 1.483 
 1.428 
 
 0.829 
 0.819 
 
 56 
 
 55 
 
 36 
 37 
 
 0.588 
 o.6d2 
 
 14 
 
 14 
 14 
 13 
 
 i4 
 
 0.727 
 0.754 
 
 27 
 27 
 
 1-376 
 1-327 
 
 52 
 49 
 
 0.809 
 0.799 
 
 10 
 10 
 
 54 
 S3 
 
 3^ 
 39 
 
 0.616 
 0.629 
 
 0.731 
 0.810 
 
 27 
 29 
 29 
 
 1.280 
 1-235 
 
 47 
 45 
 43 
 
 0.788 
 0.777 
 
 II 
 II 
 II 
 
 52 
 
 SI 
 
 40° 
 
 0.643 
 
 
 0.839 
 
 
 1. 192 
 
 
 0.766 
 
 
 50° 
 
 41 
 42 
 
 43 
 
 0.656 
 0.669 
 0.682 
 
 13 
 13 
 13 
 
 0.869 
 0.900 
 0.933 
 
 30 
 
 31 
 33 
 32 
 35 
 
 1.150 
 i.iii 
 1.072 
 
 42 
 39 
 39 
 
 36 
 
 0.7S5 
 0.743 
 0.731 
 
 II 
 12 
 12 
 12 
 12 
 
 49 
 48 
 47 
 
 44 
 
 0.655 
 
 13 
 12 
 
 o.g65 
 
 1.035 
 
 0.719 
 
 46 
 
 45° 
 
 0.707 
 
 
 0.000 
 
 I.OOO 
 
 0.707 
 
 
 45° 
 
 
 Cosine. 
 
 Cotangent. 
 
 Tangent. j 
 
 Sine. 
 
 Angle. 
 
INDEX, 
 
 Absolute zero, 37. 
 Absorption spectra, 333. 
 Accelerated motion, 65. 
 
 equations of, 87, 88, 
 Action, voltaic, 211. 
 Air-pump, 28, 29. 
 Air-thermometer, 36. 
 Alcohol, density of, 9, 356. 
 Ampere, 241. 
 Ampfere's hypothesis, 208. 
 Amplitude of vibration, 276. 
 Angle of prism, determination of, 
 
 318. 
 Angular velocity, 250. 
 Aspirator, 28. 
 
 Astatic galvanometers, 207. 
 Atmosphere, pressure of, 12, 21. 
 Attraction, electrical, 147. 
 
 force of electrical, 151. 
 
 Balance, 15. 
 
 hydrostatic, g, 17. 
 Ballistic pendulum, 120. 
 Barometer, 23. 
 Base-lines, 39. 
 Batteries, electro-motdve force of, 
 
 359- 
 
 resistance of, 257. 
 
 storage, 261. 
 
 water, 213. 
 Beats, 282. 
 
 Bifilar suspension, 140. 
 Boiling-point on thermometer, 186. 
 Bolometer, 346. 
 Boyle's law, 30. 
 Brake, friction, 179, 182. 
 
 17 
 
 Bridge, Wheatstone's, 220. 
 Bunsen cell, 213. 
 Bunsen's photometer, 310. 
 
 Calibration, 351. 
 Cathetometer, 47. 
 Cell, Bunsen, 213. 
 
 Daniell, 212. 
 
 Faure, 261. 
 
 gravity, 212. 
 
 l.eclanche, 213. 
 
 Plants, 261. 
 Cement, 350. 
 Center of gravity, 74. 
 Centigrade thermometer, 187. 
 Centrifugal force, 112. 
 Centripetal force, 112. 
 C. G. S. system, 93, 241. 
 Charles's law, 35. 
 Circuits, divided, 246. 
 Co-efficient of expansion of glass, 
 201. 
 
 of friction, 88, 89. 
 Collimator, 327. 
 Commutator, 354. 
 Comparison of tuning-forks, 270. 
 Composition of pendulum move- 
 ments, 297, 299. 
 Compound lenses, 323. 
 
 pendulum, 116, 118, 126. 
 Concave lenses, 324. 
 
 mirrors, 326. 
 Conical pendulum, 1 14. 
 Conjugate foci, 323. 
 Conservation of energy, 94. 
 Contour-lines, 162, 164. 
 
3^4 
 
 INDEX. 
 
 Convex lens, 321. 
 
 Couples, 79, I3g. 
 
 Cross-hairs, 353. 
 
 Cubical expansion, 19, 356. 
 
 Current strengths, comparison of, 
 
 358. 
 Curves, Lissajous', 297, 299. 
 
 Daniell cell, Z12. 
 Density of alcohol, g, 356. 
 Determination of horizontal inten- 
 sity of earth's magnetism, 262. 
 
 of Ohm, 232. 
 
 of reduction factor, 247, 250. 
 Deviation, minimum, 318. 
 Difference of phase, 277. 
 
 of potential, 173, 223. 
 Diffraction gratings, 343. 
 Diffraction of light, 335. 
 Dipping-needle, 130. 
 Dispersion of light, 328, 359. 
 Divided circuits, 246. 
 Dynamo-electric machine, 238. 
 Dyne, 62, 93. 
 
 Earth's magnetism, horizontal 
 component of, 131, 142. 
 
 vertical component of, 130. 
 Efficiency of electrical machines, 
 
 256. 
 Elasticity, 49. 
 
 modulus of, 287, 289. 
 Electrical attraction, 147. 
 
 attraction, force of, 151. 
 
 current, strength of, 217. 
 
 induction, 147. 
 
 keys, 354. 
 
 machines, efficiency of, 256. 
 
 polarization, 261. 
 
 resistance, specific, 218. 
 
 resistance and temperature, 223. 
 
 units, 241. 
 
 work and electrical measure- 
 ments, 204, 228. 
 Electricity, quantity of, 253. 
 
 and heat, 215. 
 Electro-magnet, 152. 
 Electro-magnetic force, 156, 158. 
 
 induction, Z36. 
 Electrometer, 170. 
 Electro-motive force, 228, 229, 
 235. 359- 
 
 Electro-motive force, measure- 
 ments of, 242, 249. 
 
 Poggendorff's method, 244. 
 Energy, 94. 
 
 conservation of, 94. 
 
 in solar spectum, 338. 
 
 kinetic, 165, 168. 
 
 potential, 165. 
 Equation of momentum, 95. 
 Equations of accelerated motion, 
 
 87, 88. 
 Equipotential lines, 164. 
 Erg, 93. 
 Expansion, cubical, ig, 356. 
 
 linear, 20, 356. 
 
 Factor, reduction, 217, 247, 259. 
 Fahrenheit thermometer, 187. 
 Faure cell, 261. 
 Field of magnetic force, 97, 230, 
 
 231. 
 Foot-pound, 93. 
 Force, centrifugal, 112. 
 
 centripetal, 112. 
 
 electro-magnetic, 156, 158. 
 
 electro-motive, 228, 229, 235, 
 
 359- 
 
 of electrical attraction, 151. 
 
 unit of, 62. 
 Forces, 55. 
 
 parallel, 75. 
 
 parallelogram of, 78. 
 Freezing-point on thermometer, 
 
 186. 
 Friction-brake, I7g, 182. 
 
 co-efficient of, 88, 8g. 
 Functions, trigonometric, 361. 
 
 Galvanometer mirrors, 353. 
 Galvanometers, astatic, 207. 
 
 tangent, 204. 
 Gases, table of specific gravity, 
 
 356. 
 Glass, co-efficient of expansion of, 
 
 201. 
 Glass-vcorking, 351. 
 Gratings, diffraction, 343. 
 Gravitation, 13. 
 
 pendulum, iii. 
 Gravity-cell, 212. 
 Gravity, center of, 74. 
 
 specific, 6, 356. 
 
INDEX. 
 
 365 
 
 Hammer, tilt, 122. 
 Harmonic vibrations, 293. 
 
 simple, 280. 
 Heat and electricity, 215. 
 
 mechanical equivalent of, 184, 
 191. 
 
 specific, igo. 
 
 specific capacity for, 190. 
 
 unit of, 188. 
 Homogeneous light, 338. 
 Horizontal component of earth's 
 magnetism, 131, 142, 262. 
 
 intensity of earth's magnetism, 
 determination of, 142, 262. 
 Horse-power, 174. 
 Huygens's construction for wave- 
 front, 313. 
 Hydrometer, Nicholson's, 11. 
 Hydrometers, 10. 
 Hydrostatic balance, 9, 17. 
 
 press, 26. 
 
 Ice, latent heat of, 198. 
 
 Images, size of, formed by lenses, 
 
 325- 
 Impulse, 92. 
 
 Inclined plane, laws of, 106. 
 Index of refraction, 315, 320, 324, 
 
 359- 
 Induction, electrical, 147. 
 
 electro-magnetic, 236. 
 
 magnetic, 208. 
 Inertia, moments of, 123, 356. 
 Intensity of light, measurement of, 
 
 309- 
 of sound, 301. 
 Interference of light, 334, 335. 
 of sound-waves, 281. 
 
 Jar, Leyden, 148. 
 
 Keys, electrical, 354. 
 Kilogramme-metre, 93. 
 Kinematics, 54. 
 Kinetic energy, 165, 168. 
 KirchhofFs law, 245. 
 Kundt's method of determining 
 velocity of sound, 287. 
 
 Latent heat of ice, 198. 
 
 of steam, 199. 
 Lavir, Kirchhoff's, 245. 
 
 Law, Mariotte's, 30. 
 
 Ohm's, 217, 232. 
 Laws of inclined planes, 106. 
 
 of motion, 56. 
 Leclanche cell, 213. 
 Lens, convex, 321. 
 
 plano-concave, 327. 
 
 principal axis of, 321, 
 Lenses, compound, 323. 
 
 concave, 324. 
 Lever, 117, 175. 
 Leveling, 161. 
 Leyden-jar, 148. 
 Light, 304. 
 
 diffraction of, 335. 
 
 homogeneous, 338. 
 
 interference of, 334, 335. 
 
 reflection of, 307. 
 
 velocity of, 313. 
 
 wave-lengths of, 335, 342. 
 Linear expansion, 20, 356. 
 Lines, base, 39. 
 
 contour, 162, 164. 
 
 equipotential, 164. 
 Liquids. Table of specific gravity, 
 
 356. 
 Lissajous' curves, 297, 299. 
 Longitudinal vibrations, 292. 
 
 Machine, dynamo-electric, 238. 
 Magnetic field, strength of, 97, 
 230, 231. 
 
 earth's, 130, 131, 142. 
 
 figures, 153. 
 
 force, measurement of, 129. 
 
 induction, 208. 
 
 metals, 207. 
 
 pendulum, 126. 
 Mariotte's law, 30. 
 Mass, 51, 61. 
 
 unit of, 61. 
 Measurement of electro-motive 
 force, 242, 249. 
 
 of intensity of light, 309. 
 
 of intensity of sound, 301. 
 
 of magnetic force, 129. 
 
 of wave-lengths of light, 342. 
 Mechanical equivalent of heat, 
 
 184, 191. 
 Mercury table, 352. 
 
 thermometer, 185. 
 
 unit, Siemens's, 219. 
 
366 
 
 INDEX. 
 
 MetalSj magnetic, 207. 
 
 specific resistance of, 358. 
 Metre, 2. 
 Micrometers, 42. 
 Minimum deviation, 318. 
 Mirror and scale, 136. 
 Mirrors, concave, 326. 
 
 galvanometer, 353. 
 Mixtures, specific heat by, 194. 
 Modulus of elasticity, 287, 289. 
 Moments of inertia, 123, 356. 
 
 reduction of, 123. 
 Momentum, 61. 
 
 equation of, 95. 
 Motion, 52. 
 
 accelerated, 65. 
 
 laws of, 56. 
 
 perpetual, 193. 
 
 uniform, 63. 
 Movements, pendulum, 72. 
 
 Needle, dipping, 130. 
 Nicholson's hydrometer, 11. 
 Normal spectra, 343. 
 
 Ohm, 2ig. 
 
 Ohm's law, 217, 232. 
 
 Parallel forces, 75. 
 Parallelogram of forces, 78. 
 Pendulums, 7g. 
 Pendulum, ballistic, 120. 
 
 compound, 116, 118, 126. 
 
 conical, 114. 
 
 gravitation, III. 
 
 magnetic, 126. 
 
 movements, 72. 
 
 movements, composition of, 297, 
 299. 
 
 torsion, 73, 124. 
 Perpetual motion, 195. 
 Phase, difference of, 277. 
 Phonograph, 339. 
 Photometer, Bunsen's, 310. 
 
 Pickering's, 311. 
 
 Rumforas, 309. 
 Plane, inclined, 106. 
 Plano-concave lens, 327. 
 Plants cell, 261. 
 Poggendorff s method of measuring 
 
 electro-motive force, 244. 
 Polarization, electrical, 261. 
 
 Potential and work, 159. 
 
 difference of, 173, 223. 
 
 energy, 165. 
 Power, horse, 174. 
 Press, hydrostatic, 26. 
 Pressure of atmosphere, 12, 21. 
 Principal axis of lens, 321. . 
 Prism, determination of angle of, 
 
 318. 
 Projection of solar spectrum, 328. 
 Pulleys, 176, 177. 
 Pump, air, 28, 29. 
 
 Sprengel, 28. 
 
 Quantities of electricity, 253. 
 
 Reduction factor, 217, 247, 259. 
 
 of moments of inertia, 125. 
 Reflection of light, 307. 
 Refraction, index of, 315, 320, 324, 
 
 359- 
 Resistance-coils, 354. 
 Resistance of a battery, 257. 
 
 specific, 218. 
 Rumford's photometer, 309. 
 
 Scale and mirror, 136. 
 
 Screw, 177. 
 
 Shunts, 254. 
 
 Siemens's mercury-unit, 219. 
 
 Simple harmonic vibration, 280. 
 
 Size of images formed by lenses, 
 
 325. 
 Solar spectrum, energy in, 338. 
 
 projection of, 328. 
 Solids, Table of specific gravity, 
 
 355- 
 Sound, 269. 
 
 intensity of, 3or. 
 
 velocity of, 283, 284, 287, 291. 
 
 wave-lengths of, 276. 
 
 waves, 278. 
 
 waves, interference of, 281. 
 Specific capacity for heat, 190. 
 
 capacity of heat of a thermome- 
 ter, 190. 
 
 electrical resistance, 218. 
 
 gravity, 6, 355. 
 
 gravity bottle, 18. 
 
 heat by mixtures, 194. 
 Spectra, normal, 343. 
 
 of absorption, 333. 
 
INDEX. 
 
 367 
 
 Spectrometer, 327. 
 Spectroscope, 331. 
 Spherometer, 44, 46. 
 Sprengel, 28. 
 
 Steam, latent heat of, 199. 
 Storage-battery, 261. 
 Strength of electrical current, 217. 
 of magnetic field, 97, 230, 231. 
 Stress, 94. 
 
 Suspension, bifilar, 140. 
 Sympathetic vibrations, 302. 
 System, C. G. S.,'93, 241. 
 
 Table of Birmingham wire gauge, 
 
 359-. 
 
 co-efficients of expansion, 356. 
 
 comparison of cui'rent strength, 
 358. 
 
 electro-motive force of batteries, 
 
 ' 359- 
 
 mean indices of refraction and 
 dispersion, 359. 
 
 moments of inertia, 356. 
 
 specific gravity of gases, 356. 
 
 specific gravity of liquids, 356. 
 
 specific gravity of solids, 356. 
 
 specific resistances of metals, 
 .358. 
 
 trigonometrical functions, 361. 
 Tangent galvanometers, 204. 
 Temperature and electrical resist- 
 ance, 223. 
 Thermo-electricity, 264. 
 Thermometer, air, 36. 
 
 boiling-point, 186. 
 
 centigrade, 187. 
 
 Fahrenheit, 187. 
 
 freezing-point, 186. 
 
 mercury, 185. 
 
 weight, 202. 
 Thermopile, 268. 
 Tilt-hammer, 122. 
 Torsion pendulum, 73, 124. 
 
 Trigonometry, 58, 361. 
 Tuning-forks, comparison of, 270. 
 
 Uniform motion, 63. 
 Units, electrical, 241. 
 
 of force, 62. 
 
 of heat,- 188. 
 
 of mass, 6l. 
 
 Velocity, angular, 250. 
 
 of light, 313. 
 
 of sound, 283, 284, 287, 291. 
 
 of sound, Kundt's method, 287. 
 Verniers, 40, 353. 
 Vertical component of earth's mag- 
 netism, 130. 
 Vibrations, amplitude of, 276. 
 
 haimonic, 293. 
 
 longitudinal, 292. 
 
 sympathetic, 302. 
 Volt, 236. 
 Voltaic action, 2H. 
 Voltameter, 262. 
 Volumometer, 32. 
 
 Water-battery, 213. 
 Watt, 93. 
 
 Wave-front, Huygens's construc- 
 tion for, 313. 
 Wave-length, 278. 
 
 of light, 335. 342. 
 
 of sound, 276. 
 Wave-lengths, measurements of, 
 
 342- 
 Weighing, 16. 
 Weight, 51. 
 
 thermometer, 202. 
 Wheatstone's bridge, 220. 
 Wire-gauge, Birmingham, 359. 
 Work, 93, 94, 108, 228. 
 
 and potential, 159. 
 
 Zero, absolute," 37. 
 
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