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" The subject is treated, so far as convenient, without mathematics this alone will be a boon to many readers while the descriptions of experiments, and accounts of practical appli- cations of the principles, impart to the work an interest that is sadly deficient in most purely mathematical introductions to this study." AtJienceum. "The treatise is remarkable for the vigour of its style, which specially commends it as a book for private reading ; but its leading excellence, as compared with the best works at present in use, is the thoroughly rational character of the information which it presents. . . . 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An Elementary Text-Book of Physics. By J. D. EVEBETT, M.A., D.C.L., Professor of Natural Philosophy in the Queen's College, Belfast; Editor of the English Edition of Deschanel's Natural Philosophy. Extra Foolscap 8vo, price 3s. 6d. This book is primarily intended as a Text-book for elementary classes of Physics. It aims at presenting, in brief space, those portions of Theoretical Physics which are most essential as a foundation for subsequent advances, while at the same time most fitted for exercising the learner in logical and consecutive thought. It does not give minute direc- tions for manipulation; but, avoiding details as much as possible, presents a connected out- line of the main points of theory. In order to place science upon an equal footing with the more established studies of ancient languages and mathematics, as a means of practically training the bulk of our youth to vigorous thought, it seems necessary that science text-books should be constructed upon such lines as these. It is not practicable to make the bulk of the boys in our public schools expert scientific manipulators; but it is practicable to ground them well in the main lines of scientific theory. The aim must be not so much to teach them many facts, as to teach them rightly to connect a few great facts together. Science must be taught them from a liberal, not from a technical stand-point. The selection of subjects has been governed by their educational value, and by their importance as a foundation for further advances. The order in which they are arranged is that which the Author has found, in his own experience, to be the best; having regard not only to lucidity of explanation, but also to the apparatus required for illustrating each lecture. The Student's English Dictionary, ETYMOLOGICAL, PRONOUNCING, AND EXPLANATORY. By JOHN OGILVIE, LL.D., Editor of the "Imperial" and the "Comprehensive" Dictionaries. Imperial 16mo, 830 pp., cloth, red edges, 7s. 6d. ; or, half-calf, 10s. 6d. Illustrated by about Three Hundred Engravings on Wood. I. Aim. The leading object of this Dictionary is to place the English Language, as far as possible, upon a sound Etymological basis, with the view of fixing the primary idea or root-meaning of each principal Word, after which the secondary meanings are arranged so as to follow in their proper order. The Etymologies of this Dictionary are original compilations, prepared expressly for this Work. II. Fulness. It comprises all thoroughly English words used by the best writers, excepting a number of easily, because analogically, formed derivatives. III. Etymology. It contains a full and satisfactory Etymology, in which the words are traced to their ultimate source, and the foreign vocables employed are generally trans- lated into English. In this department every principal word has been made the subject of distinct inquiry. IV. It gives accurate Root Meanings of the words, followed by the secondary Meanings, arranged in a natural sequence. The STUDENT'S ENGLISH DICTIONARY is the only English Dictionary possessing this very useful and important feature. V. The Definitions are peculiarly full and very concisely stated, and are fitted to meet at the same time the requirements of the junior and of the advanced student. VI. The Pronunciation has been adopted from the Comprehensive Dictionary, by the same editor, for which it was prepared by Richard Cull, F.S.A. VII. The method employed of exhibiting the pronunciation by re-writing the words and using accented letters, is simple and easily understood each page contains the key. VIII. Numerous Pictorial Illustrations have been introduced, which add to the utility of the Work, and convey accurate impressions where verbal definitions fail of their object. BLACKIE & SON: LONDON, GLASGOW, AND EDINBURGH. ELEMENTARY TREATISE ON NATURAL PHILOSOPHY BY A. PEIVAT DESCHANEL. it FORMERLY PROFESSOR OF PHYSICS IN THE LYC^E LOUIS-LE-GRAXD, INSPECTOR OF THE ACADEMY OF PARIS. TRANSLATED AND EDITED, WITH EXTENSIVE ADDITIONS, BY J. D. EVERETT, M.A., D.C.L., F.R.S.E., PROFESSOR OF NATURAL PHILOSOPHY IN THE QUEEN'S COLLEGE, BELFAST. ILLUSTRATED BY 760 ENGRAVINGS ON WOOD, AND THREE COLOURED PLATES. FIFTH LOXDON: BLACKIE & SON, OLD BAILEY, EC.; GLASGOW AND EDINBURGH. 1880. AH Rights Reserved. MAIN LIBRARY JOHN FRYER CHINESE LIBRARY QC GLASGOW: IV. G. BI.ACKIK AND CO., PRINTERS, VILLAFIELD. PEEFACE TO THE FIRST EDITION. 1 DID not consent to undertake the labour of translating and editing the " THAI-HE; ^LEMENTAIRE DE PHYSIQUE" of Professor Deschanel until a careful examination had convinced me that it was better adapted to the requirements of my own class of Experimental Physics than any other work with which I was acquainted; and in executing the translation I have steadily kept this use in view, believing that I was thus adopting the surest means of meeting the wants of teachers generally. The treatise of Professor Deschanel is remarkable for the vigour of its style, which specially commends it as a book for private read- ing. But its leading excellence, as compared with the best works at present in use, is the thoroughly rational character of the information which it presents. There is great danger in the present day lest science-teaching should degenerate into the accumulation of discon- nected facts and unexplained formulae, which burden the memory without cultivating the understanding. Professor Deschanel has been eminently successful in exhibiting facts in their mutual connec- tion; and his applications of algebra are always judicious. The peculiarly vigorous and idiomatic style of the original would be altogether unpresentable in English ; and I have not hesitated in numerous instances to sacrifice exactness of translation to effective rendering, my object being to make the book as useful as possible to English readers: For the same reason I have not scrupled to suppress or modify any statement, whether historical or philosophical, which I deemed erroneous or defective. In some instances I have endeavoured to simplify the reasonings by which propositions are established or formulae deduced. As regards weights and measures, rough statements of quantity have generally been expressed in British units; but in many cases the numerical values given in the original, and belonging to the metrical system, have been retained, with or without their English 751,587 IV PREFACE TO THE FIRST EDITION. equivalents; as it is desirable that all students of science should familiarize themselves with a system of weights and measures which affords peculiar facilities for scientific calculation, and is extensively employed by scientific men of all countries. For convenience of reference, a complete table of metrical and British equivalents has been annexed. The additions, which have been very extensive, relate either to subjects generally considered essential in this country to a treatise on Natural Philosophy, or to topics which have in recent years occupied an important place in physical discussions, though as yet but little known to the general public. The sections distinguished by a letter appended to a number are all new; as also are all foot-notes, except those which are signed with the Author's initial " D." In many instances the new matter is so interwoven with the old that it could not conveniently be indicated; and I have aimed at giving unity to the book rather than at preserving careful distinc- tions of authorship. Comparison with the original will however be easy, as the num- bering of the original sections has been almost invariably followed. The chief additions in Part I. (Chap, i.-xviii.) have been under the heads of Dynamics, Capillarity, and the Barometer. The chapter on Hydrometers has also been recast. PEEFACE TO THE PRESENT EDITION. IN the original English edition of this treatise, the earlier portions consisted of a pretty close translation from the French; but as the work progressed I found the advantage of introducing more considerable modifications; and Parts III. and IY. were to a great extent rewritten rather than translated. I have now, in like manner, rewritten Part I., and trust that in its amended form it will be found better adapted than before to the wants of English teachers. Several additional subjects have been introduced, the order of the chapters has been rearranged, and the collection of "Problems" (translated from the French), which appeared in the third, fourth, and fifth editions, has been replaced by a much larger number of " Examples " with answers appended. By pruning redundancies, the Part has been kept within its original limits of size. The marks of distinction between new and old sections have now been dropped; but Professor Deschanel's foot-notes are still dis- tinguished by the initial "D." The numbering of the sections is entirely new. All accurate statements of quantities have been given in the C.G.S. (Centimetre-Gramme-Second) system, which, by reason of its simplicity and of the sanction which it has received from the British Association, and the Physical Society of London, is coming every- day into more general use. BELFAST, October, 1879. CONTENTS-PAST I. (THE NUMBERS REFER TO THE SECTIONS.) CHAPTER I. INTRODUCTORY. Natural History and Natural Philosophy, 1, 2. Divisions of Natural Philosophy, 3. CHAPTER II. FIRST PRINCIPLES OF DYNAMICS, STATICS. Force, 4. Translation and rotation, 5, 6. Instruments for measuring force, 7. Gravita- tion units of force, 8. Equilibrium ; Statics and kinetics, 9. Action and reaction, 10. Specification of a force, point of application, line of action, 11. Rigid body, 12. Equilibrium of two forces, 13. Three forces in equilibrium at a point, 14. Resul- tant and components, 15. Parallelogram of forces, 16. Gravesande's apparatus, 17. Resultant of any number of forces at a point, 18. Equilibrium of three parallel forces, 19. Resultant of two parallel forces, 20. Centre of two parallel forces, 21. Moments of resultant and components equal, 22. Resultant of any number of parallel forces in one plane, 23. Moment of a force about a point, 24. Arithmetical lever, 25. Couple, 26. Composition of couples ; Axis of couple, 27. Resultant of force and couple in same plane, 28. General resultant of any number of forces ; Wrench, 29. Application to action and reaction, 30. Resolution, 31. Rectangular resolution ; Component of a force along a given line, 32. CHAPTER III. GRAVITY. Direction of gravity ; Neighbouring verticals nearly parallel, 33. Centre of gravity, 34. Centres of gravity of volumes, areas, and lines, 35. Methods of finding centres of gravity, 36. Centre of gravity of triangle, 37. Of pyramids and cones, 38, 39. Condition of standing or falling, 40. Body supported at one point, 41. Stability and instability, 42. Experimental determination of centre of gravity, 43, 44. Work done against gravity, 45. Centre of gravity tends to descend, 46. Work done by gravity, 47. Work done by any force, 48. Principle of work; Perpetual motions, 49. Criterion of stability, 50. Illustration, 51. Stability where forces vary abruptly, 52. Illustrations from toys, 53. Limits of stability, 54. CHAPTER IV. THE MECHANICAL POWERS. Enumeration, 55. Lever, 56-58. Mechanical advantage, 59. Wheel and axle, 60. Pulleys, 61-63. Inclined plane, 64-66. Wedge and screw, 67-69. CHAPTER V. THE BALANCE. General description, 70. Qualities requisite, 71. Double weighing, 72. Investigation of sensibility, 73. Advantage of weighing with constant load, 74. Details of con- struction, 75. Steelyard, 76. CHAPTER VI. FIRST PRINCIPLES OF KINETICS. Principle of Inertia, 77. Second law of motion, 78. Mass and momentum, 79. Proper selection of unit of force, 80. Relation between mass and weight, 81. Third law of Vlll TABLE OF CONTENTS. motion; Action and reaction, 82. Motion of centre of gravity unaffected, 83. Velocity of centre of gravity, 84. Centre of mass, 85. Units of measurement, 86. The C.G.S. system ; the dyne, the erg, 87. CHAPTER VII. LAWS OF FALLING BODIES. Fall in air and in vacuo, 88. Mass and gravitation proportional, 89. Uniform accelera- tion, 89. Weight of a gramme in dynes ; Value of g, 91. Distance fallen in a given time, 92. Work spent in producing motion, 93. Body thrown upwards, 94. Eesistance of the air, 95. Projectiles, 96. Time of flight, and range, 97. Morin's apparatus, 98. Atwood's machine, 99. Theory of Atwood's machine, 100. Uniform motion in a circle, 101. Deflecting force, 102. Illustrations, stone in sling, 103. Centrifugal force at the equator, 104. Direction of apparent gravity, 105. CHAPTER VIII. THE PENDULUM. Pendulum, 106. Simple pendulum, 107. Law of acceleration for small vibrations, 108. General law for period, 109. Application to pendulum, 110. Simple harmonic motion, 111. Experimental investigation of motion of pendulum, 112. Cycloidal pendulum, 113. Moment of inertia about an axis, 114. About parallel axes, 115. Application to compound pendulum, 116. Convertibility of centres, 117. Centre of suspension for minimum period, 118. Kater's pendulum, 119. Determination of g, 120. CHAPTER IX. ENERGY. Kinetic energy, 121. Static or potential energy, 122. Conservation of mechanical energy, 123. Illustration from pile-driving, 124. Hindrances to availability of energy ; Principle of the conservation of energy, 125. CHAPTER X. ELASTICITY. Elasticity and its limits, 126. Isochronism of small vibrations, 127. Stress, strain, coefficients of elasticity; Young's modulus, 128. Volume-elasticity, 129. (Ersted's piezometer, 130. CHAPTER XI. FRICTION. Friction, kinetical and statical, 131. Statical friction, limiting angle, 132. Coefficient^ tan 0', Inclined plane, 133. * CHAPTER XII. HYDROSTATICS. Hydrodynamics, 134. No statical friction in fluids, 135. Intensity of pressure, 136. Pressure the same in all directions, 137. The same at the same level, 138. Differ- ence of pressure at different levels, 139. Free surface, 140. Transmissibility of pressure; Pascal, 141. Hydraulic press, 142. "Principle of work" applicable, 143. Experiment on upward pressure, 144. Liquids in superposition, 145. Two liquids in bent tube, 146. Pascal's vases, 147. Resultant pressure on vessel, 148. Back pressure on discharging vessel, 149. Total and resultant pressures; Centre of pressure, 150. Construction for centre of pressure, 151. Whirling vessel; D'Alem- bert's principle, 152. CHAPTER XIII. PRINCIPLE OF ARCHIMEDES. Resultant pressure on immersed bodies, 153. Experimental demonstration, 154. Three cases distinguished, 155. Centre of buoyancy, 153, 155. Cartesian diver, 156. Stability of floating body, 157, 158. Floating of needles on water, 159. TABLE OF COJN TENTS. ix CHAPTER XIV. DENSITY AND ITS DETERMINATION. Absolute and relative density, 160. Ambiguity of the word "weight," 161. Determination of density from observation of weight and volume, 162. Specific gravity flask for solids, 163. Method by weighing in water, 164. With sinker, 165. Densities of liquids measured by loss of weight in them, 166. Measurement of volumes of solids by loss of weight, 167. Hydrometers, 168. Nicholson's, 169. Fahrenheit's, 170. Hydrometers of variable immersion, 171. General theory, 172. Beaume"'s hydro- meters, 173. Twaddell's, 174. Gay-Lussac's alcoholimeter, 175. Computation of densities of mixtures, 176. Graphical method of interpolation, 177. CHAPTER XV. VESSELS IN COMMUNICATION. LEVELS. Liquids tend to find their own level; Water-supply of towns, 178. Water-level; Levelling between distant stations, 179. Spirit-level and its uses, 180, 181. CHAPTER XVI. CAPILLARITY. General phenomena of capillary elevation and depression, 182. Influencing circum- stances, 183. Law of diameters, 184. Fundamental laws of capillary phenomena; Angle of contact ; Surface tension, 185. Application to elevation and depression in tubes, 186. Formula for normal pressure of film, 187. Film with air on both sides, 188. Drops, 189. Pressure in a liquid whose surface is convex or concave, 190. Interior pressure due to surface action when surface is plane, 191. Phenomena illus- trative of differential surface tensions; Table of tensions, 192. Endosmose and diffusion, 193. CHAPTER XVII. THE BAROMETER Expansibility of gases, 194. Direct weighing of air, 195. Atmospheric pressure, 196. Torricellian experiment, 197. Pressure of one atmosphere, 198. Pascal's experi- ment on Puy de D6me, 199. Barometer, 200. Cathetometer, 201. Fortin's Barometer ; Vacuum tested by metallic clink, 202. Float adjustment, 203. Baro- metric corrections; Temperature; Capillarity; Capacity; Index errors; Reduction to sea-level ; Intensity of gravity ; and reduction to absolute measure, 204. Siphon, wheel, and marine barometers, 205. Aneroid, 206. Counterpoised barometer; King's barograph ; Fahrenheit's multiple-tube barometer, 207. Photographic regis- tration, 208. CHAPTER XVIII. VARIATIONS OF THE BAROMETER, Measurement of heights by the barometer, 209. Imaginary homogeneous atmosphere, 210. Geometric law of decrease, 211. Computation of pressure-height, 212. For- mula for determining heights by the barometer, 213. Diurnal oscillation, 214. Irregular variations, 215. Weather charts, 216. CHAPTER XIX. BOYLE'S (OR MARIOTTE'S) LAW. Boyle's law, 217. Boyle's tube, 218. Unequal compressibility of different gases, 219, 220. Regnault's experiments, 221. Results, 222. Manometers or pressure gauges, 223. Multiple-branch manometer, 224. Compressed air manometer, 225. Metallic manometers, 226. Pressure of gaseous mixtures, 227. Absorption of gases by liquids and solids, 228. CHAPTEB XX. AIR-PUMP. Air-pump, 229. Theoretical rate of exhaustion, 230. Mercurial gauges, 231. Admission cock, 232. Double-barrelled pump, 233. Single barrel with double action, 234. English X TABLE OF CONTENTS. forms, 235. Experiments ; Burst bladder ; Magdeburg hemispheres ; Fountain, 236. Limit to action of pump and its causes, 237. Kravogl's pump, 238. Geissler's, 239, Sprengel's, 240. Double exhaustion, 241. Free piston, 242. Compressing pump, 213. Calculation of its effect, 244. Various contrivances for compressing air, 245. Practical applications of air-pump and compressing pump, 246. CHAPTEE XXI. UPWARD PRESSURE OF THE AIR. Baroscope, 247. Principle of balloons, 248. Details, 249. Height attainable by a given balloon, 250. Effect of air on apparent weights, 251. CHAPTER XXII. PUMPS FOR LIQUIDS. Invention of pump, 252. Reason of the water rising, 253. Suction pump, 254. Effect of untraversed space, 255. Force necessary to raise the piston, 256. Efficiency, 257. Forcing pump, 258. Plunger, 259. Fire-engine, 260. Double-acting pumps, 261. Centrifugal pumps, 262. Jet-pump, 263. Hydraulic press, 264. CHAPTER XXIII. EFFLUX OF LIQUIDS. Torricelli's theorem, 265. Froude's calculation of area of contracted vein, 266. Con- tracted vein for orifice in thin plate, 267. Apparatus for illustrating Torricelli's theorem, 268. Efflux from air-tight space, 269. Intermittent fountain, 270. Siphon, 271. Starting the Siphon, 272. Siphon for sulphuric acid, 273. Tantalus' cup, 274. Mariotte's bottle, 275. EXAMPLES. PAGE Parallelogram of Velocities, and Parallelogram of Forces. Ex. 1-11, . . . 239 Parallel Forces and Centre of Gravity. Ex. 10*-33, 239 Work and Stability. Ex. 34-43, 241 Inclined Plane, &c. Ex. 44-48, 242 Force, Mass, and Velocity. Ex. 49-59, 242 Falling Bodies and Projectiles. Ex. 60-83, 243 Atwood's Machine. Ex. 84-89 244 Energy and Work. Ex. 90-98, 245 Centrifugal Force. Ex. 99-101, 245 Pendulum, and Moment of Inertia. Ex. 101*-107, 246 Pressure of Liquids. Ex. 108-123, 246 Density, and Principle of Archimedes. Ex. 124-159, 247 Capillarity. Ex. 160-164, 249 Barometer, and Boyle's law. Ex. 165-181, 250 Pumps, &c. Ex. 182-189, 251 ANSWERS TO EXAMPLES, 252 FRENCH AND ENGLISH MEASURES. A DECI5IETRE DIVIDED INTO CENTIMETRES AND MILLIMETRES. INCHES AND TENTHS. REDUCTION OF FRENCH TO ENGLISH MEASURES. LENGTH. 1 millimetre == '03937 inch, or about -, inch. 1 centimetres '3937 inch. 1 decimetre =3 -937 inch. 1 metre=39'37 inch=3'281 ft. = l'0936 yd. 1 kilometre =1093 '6 yds., or about f mile. More accurately, 1 metre =39 '370432 in. =3-2808693 ft. = l '09362311 yd. AREA. 1 sq. millim. = '00155 sq. in. 1 sq. centim. = '155 sq. in. 1 sq. decim. =15 '5 sq. in. 1 sq. metre = 1550 sq. in. = 10764 sq. ft. = 1-196 sq. yd. VOLUME. 1 cub. millim. = -000061 cub. in. 1 cub. centim. = '061025 cub. in. 1 cub. decim. =61 '0254 cub. in. cub. metre =61025 cub. in. =35 '31 56 cub. ft. = 1-308 cub. yd. The Litre (used for liquids) is the same as the cubic decimetre, and is equal to 1'7617 pint, or '22021 gallon. MASS AND WEIGHT. 1 milligramme= '01543 grain. 1 gramme =15 '432 grain. 1 kilogramme=15432grains=2-205 Ibs. avoir. More accurately, the kilogramme is 2-20462125 Ibs. MISCELLANEOUS. 1 gramme per sq. centim. =2 '0481 Ibs. per sq. ft. 1 kilogramme per sq. centim. = 14 -223 Ibs. per sq. in. 1 kilogrammetre=7'2331 foot-pounds. 1 force de cheval=75 kilogrammetres per second, or 542^ foot-pounds per second nearly, whereas 1 horse-power (English) =550 foot- pounds per second. REDUCTION TO C.G.S. MEASURES. (See page 48.) [cm. denotes centimetre (s); ym. denotes gramme(s).] LENGTH. 1 inch =2'54 centimetres, nearly. 1 foot =30-48 centimetres, nearly. 1 yard =91 '44 centimetres, nearly. 1 statute mile =160933 centimetres, nearly. More accurately, 1 inch=2'5399772 centi- metres. AREA. 1 sq. inch =6-45 sq. cm., nearly. 1 sq. foot =929 sq. cm., nearly. 1 sq. yard =8361 sq. cm., nearly. 1 sq. mile =2'59 x 10 10 sq. cm., nearly. VOLUME. 1 cub. inch =16-39 cub. cm., nearly. 1 cub. foot =28316 cub. cm., nearly. 1 cub. yard =764535 cub. cm., nearly. 1 gallon =4541 cub. cm., nearly. MASS. 1 grain = '0648 gramme, nearly. 1 oz. avoir. = 28 '35 gramme, nearly. 1 Ib. avoir. =453'6 gramme, nearly. 1 ton =1 '016 x 10 6 gramme, nearly More accurately, 1 Ib. avoir. =453 '59265 gm. VELOCITY. 1 mile per hour =44 '04 cm. per sec. 1 kilometre per hour =27 "7 cm. per sec. DENSITY. 1 Ib. per cub. foot = '016019 gm. per cub, cm. 62 '4 Ibs. per cub. ft. =1 gm. per cub. cm. Xll FRENCH AND ENGLISH MEASURES. FORCE (assuming g = 981 ). (See p. 48. ) Weight of 1 grain = 63 -57 dynes, nearly. loz. avoir. =2'78 x 10 4 dynes, nearly. 1 Ib. avoir. = 4 '45 x 10 5 dynes,nearly. 1 ton = 9 '97 x 10 8 dynes,nearly. 1 gramme =981 dynes, nearly. 1 kilogramme = 9'81 x 10 5 dynes, nearly. WOBK (assuming rj- 981). (See p. 48.) 1 foot-pound =1-356 x 10 7 ergs, nearly. 1 kilogrammetre = 9'S1 x 10 7 ergs, nearly. Work in a second *) by one theoretical J.=7'46 xlO 9 ergs, nearly, "horse." STRESS (assuming ^=981). 1 1L>. per sq. ft. =479 dynes per sq. cm., nearly. 1 Ib. per sq. inch =6'9xl0 4 dynes per sq. cm., nearly. 1 kilog. per sq. cm. = 9'81 x 10 5 dynes per sq. cm., nearly. 760 mm. of mercury at 0C. = 1 '014 x 10 dynes per sq. cm. , nearly. 30 inches of mercury at C. = l'0163xl0 6 dynes per sq. cm., nearly. 1 inch of mercury at C. =3 '388 x 10 4 dynes per sq. cm., nearly. TABLE OF DENSITIES, IN GRAMMES PER CUBIC CENTIMETRE. LIQUIDS. Pure water at 4 C., l-QOO Sea water, ordinary, _ i -Q26 Alcohol, pure, -----... -791 proof spirit, -Q16 Ether, .715 Mercury at C., 13 '596 Naphtha, -848 SOLIDS. Brass, cast, 7'8to8'4 _ . wire, 8-54 Bronze, -----..... 8*4 Copper, cast, 8'6 sheet, g-8 ,, hammered, 8'9 Gold, 19 to 19-6 Iron, cast, 6-95 to 7'3 wrought, 7'6 to 7'8 Lead, YL'4 Platinum, 21 to 22 |jf v f' 10-5 Steel, 7-8 to 7'9 Tm > 7-3 to 7-5 Zmc, 6-8 to 7'2 Ice, -92 Basalt, 3-00 Brick, 2 to 217 Brickwork, ]_ Chalk, 1-8 to 2-8 Clay. - - - 1-92 Glass, crown, -> - - 2*5 flint, 3-0 Quartz (rock-crystal), 2 '65 Sand, j-42 Fir, spruce, '48 to '7 Oak, European, -69 to '99 Lignum-vitae, --..__ -65 to 1 '33 Sulphur, octahedral, - 2'05 ,, prismatic, -1'98 GASES, at C. and a pressure of a million dynes per sq. cm. Air ^y, -0012759 Oxygen, -0014107 Nitrogen, -0012393 Hydrogen, '00008837 Carbonic acid, '0019509 ELEMENTARY TREATISE ON NATUEAL PHILOSOPHY. CHAPTER I. INTRODUCTORY. 1. Natural Science, in the widest sense of the term, comprises all the phenomena of the material world. In so far as it merely describes and classifies these phenomena, it may be called Natural History; in so far as it furnishes accurate quantitative knowledge of the relations between causes and effects it is called Natural Philosophy. Many subjects of study pass through the natural history stage before they attain the natural philosophy stage; the phenomena being observed and compared for many years before the quantitative laws which govern them are disclosed. 2. There are two extensive groups of phenomena which are con- ventionally excluded from the domain of Natural Philosophy, and regarded as constituting separate branches of science in themselves; namely: First. Those phenomena which depend on vital forces; such phenomena, for example, as the growth of animals and plants. These constitute the domain of Biology. Secondly. Those which depend on elective attractions between the atoms of particular substances, attractions which are known by the name of chemical affinities. These phenomena are relegated to the special science of Chemistry. Again, Astronomy, which treats of the nature and movements of the heavenly bodies, is, like Chemistry, so vast a subject, that it forms a special science of itself; though certain general laws, which its phenomena exemplify, are still included in the study of Natural Philosophy. 2 INTRODUCTORY. 3. Those phenomena which specially belong to the domain of Natural Philosophy are called physical; and Natural Philosophy itself is called Physics. It may be divided into the following branches. I. DYNAMICS, or the general laws of force and of the relations which exist between- force, mass, and velocity. These laws may be -applied .to soJicls; liquids, or gases. Thus we have the three divisions, Mechanics? Jfydrostatics, and Pneumatics. '*' "J 7 L. TIIEP.MTCS; Ih-s -science of Heat. III. The science of ELECTRICITY, with the closely related subject of MAGNETISM. IV. ACOUSTICS; the science of Sound. V. OPTICS; the science of Light. The branches here numbered I. II. III. are treated in Parts I. II. III. respectively, of the present Work. The two branches numbered IV. V. are treated in Part IV. CHAPTER II. FIRST PRINCIPLES OF DYNAMICS. STATICS. 4. Force. Force may be defined as that which tends to produce motion in a body at rest, or to produce change of motion in a body which is moving. A particle is said to have uniform or unchanged motion when it moves in a straight line with constant velocity; and every deviation of material particles from uniform motion is due to forces acting upon them. 5. Translation and Rotation. When a body moves so that all lines in it remain constantly parallel to their original positions (or, to use the ordinary technical phrase, move parallel to themselves), its movement is called a pure translation. Since the lines joining the extremities of equal and parallel straight lines are themselves equal and parallel, it can easily be shown that, in such motion, all points of the body have equal and parallel velocities, so that the movement of the whole body is completely represented by the move- ment of any one of its points. On the other hand, if one point of a rigid body be fixed, the only movement possible for the body is pure rotation, the axis of the rotation at any moment being some straight line passing through this point. Every movement of a rigid body can be specified by specifying the movement of one of its points (any point will do) together with the rotation of the body about this point. 6. Force which acts uniformly on all the particles of a body, as gravity does sensibly in the case of bodies of moderate size on the earth's surface (equal particles being urged with equal forces and in parallel directions), tends to give the body a movement of pure translation. In elementary statements of the laws of force, it is necessary, for 4 FIRST PRINCIPLES OF DYNAMICS. the sake of simplicity, to confine attention to forces tending to produce pure translation. 7. Instruments for Measuring Force. We obtain the idea of force through our own conscious exercise of muscular force, and we can approximately estimate the amount of a force (if not too great or too small) by the effort which we have to make to resist it; as when we try the weight of a body by lifting it. Dynamometers are instruments in which force is measured by means of its effect in bending or otherwise distorting elastic springs, and the spring-balance is a dynamometer applied to the measure- ment of weights, the spring in this case being either a flat spiral (like the mainspring of a watch), or a helix (resembling a cork- screw). A force may also be measured by causing it to act vertically downwards upon one of the scale-pans of a balance and counter- poising it by weights in the other pan. 8. Gravitation Units of Force. In whatever way the measurement of a force is effected, the result, that is, the magnitude of the force, is usually stated in terms of weight; for example, in pounds or in kilogrammes. Such units of force (called gravitation units) are to a certain extent indefinite, inasmuch as gravity is not exactly the same over the whole surface of the earth; but they are sufficiently definite for ordinary commercial purposes. 9. Equilibrium, Statics, Kinetics. When a body free to move is acted on by forces which do not move it, these forces are said to be in equilibrium, or to equilibrate each other. They may equally well be described as balancing each other. Dynamics is usually divided into two branches. The first branch, called Statics, treats of the conditions of equilibrium. The second branch, called Kinetics, treats of the movements produced by forces not in equili- brium. 10. Action and Reaction. Experiment shows that force is always a mutual action between two portions of matter. When a body is urged by a force, this force is exerted by some other body, which is itself urged in the opposite direction with an equal force. When I press the table downwards with my hand, the table presses my hand upwards; when a weight hangs by a cord attached to a beam, the cord serves to transmit force between the beam and the weight, so that, by the instrumentality of the cord, the beam pulls the weight upwards and the weight pulls the beam downwards. Electricity EQUILIBRIUM OF TWO FORCES. 5 and magnetism furnish no exception to this universal law. When a magnet attracts a piece of iron, the piece of iron attracts the magnet with a precisely equal force. 11. Specification of a Force acting at a Point. Force may be applied over a finite area, as when I press the table with my hand; or may be applied through the w r hole substance of a body, as in the case of gravity; but it is usual to begin by discussing the action of forces applied to a single particle, in which case each force is supposed to act along a mathematical straight line, and the particle or material point to which it is applied is called its point of applica- tion. A force is completely specified when its magnitude, its point of application, and its line of action are all given. 12. Rigid Body. Fundamental Problem of Statics. A force of finite magnitude applied to a mathematical point of any actual solid body would inevitably fracture the body. To avoid this complication and other complications which would arise from the bending and yielding of bodies under the action of forces, the fiction of a perfectly rigid body is introduced, a body which cannot bend or break under the action of any force however intense, but always retains its size and shape unchanged. The fundamental problem of Statics consists in determining the conditions which forces must fulfil in order that they may be in equilibrium when applied to a rigid body. 13. Conditions of Equilibrium for Two Forces. In order that two forces applied to a rigid body should be in equilibrium, it is necessary and sufficient that they fulfil the following conditions: 1st. Their lines of action must be one and the same. 2nd. The forces must act in opposite directions along this common line. 3rd. They must be equal in magnitude. It will be observed that nothing is said here about the points of application of the forces. They may in fact be anywhere upon the common line of action. The effect of a force upon a rigid body is not altered by changing its point of application to any other point in its line of action. This is called the principle of the transmissi- bility of force. It follows from this principle that the condition of equilibrium for any number of forces with the same line of action is simply that the sum of those which act in one direction shall be equal to the sum of those which act in the opposite direction. 6 FIRST PRINCIPLES OF DYNAMICS. 14. Three Forces Meeting in a Point. Triangle of Forces. If three forces, not having one and the same line of action, are in equilibrium, their lines of action must lie in one plane, and must either meet in a point or be parallel. We shall first discuss the case in which they meet in a point. From any point A (Fig. 1) draw a line AB parallel to one of the two given forces, and so that in travelling from A to B we should be travelling in the same direction in which the force acts (not in the opposite direction). Also let it be understood that the length of AB repre- sents the magnitude of the force. From the point B draw a line BC c representing the second force in direc- tion, and on the same scale of magnitude on which AB represents the first. Then the line CA will represent both in direction and magnitude the third Fig. i._Triangie of Forces. force which would equilibrate the first two. The principle embodied in this construction is called the triangle of forces. It may be briefly stated as follows: The condition of equilibrium for three forces acting at a point is, that they be repre- sented in magnitude and direction by the three sides of a triangle, taken one way round. The meaning of the words " taken one way round " will be understood from an inspection of the arrows with which the sides of the triangle in Fig. 1 are marked. If the directions of all three arrows are reversed the forces represented will still be in equilibrium. The arrows must be so directed that it would be possible to travel completely round the triangle by moving along the sides in the directions indicated. When a line is used to represent a force, it is always necessary to employ an arrow or some other mark of direction, in order to avoid ambiguity between the direction intended and its opposite. In naming such a line by means of two letters, one at each end of it, the order of the letters should indicate the direction intended. The direction of AB is from A to B; the direction of BA is from B to A. 15. Resultant and Components. Since two forces acting at a point can be balanced by a single force, it is obvious that they are equiv- alent to a single force, namely, to a force equal and opposite to that which would balance them. This force to which they are equivalent EQUILIBRIUM OF THREE FORCES. 7 is called their resultant. Whenever one force acting on a rigid body is equivalent to two or more forces, it is called their resultant, and they are called its components. When any number of forces are in equilibrium, a force equal and opposite to any one of them is the resultant of all the rest. The " triangle of forces " gives us the resultant of any two forces acting at a point. For example, in Fig. 1, AC (with the arrow in the figure reversed) represents the resultant of the forces represented by AB and BC. 16. Parallelogram of Forces. The proposition called the "parallel- ogram of forces" is not essentially distinct from the "triangle of forces," but merely expresses the same fact from a slightly different point of view. It is as follows: If two forces acting upon ilie same rigid body in lines ^vh^ch meet in a point, be represented by tivo Ihws drawn from the point, and a parallelo- gram be constructed on these lines, the diagonal drawn from this point to the opposite corner ?>s- 2. -Parallelogram of of the parallelogram represents the resultant. For example, if AB, AC, Fig. 2, represent the two forces, AD will represent their resultant. To show the identity of this proposition with the triangle of forces, we have only to substitute BD for AC (which is equal and parallel to it). We have then two forces represented by AB, BD (two sides of a triangle) giving as their resultant a force represented by the third side AD. We might equally well have employed the triangle ACD, by substituting CD for AB. 17. Gravesande's Apparatus. An apparatus for verifying the par- allelogram of forces is represented in Fig. 3. ACDB is a light frame in the form of a parallelogram. A weight F' can be hung at A, and weights P, F can be attached, by means of cords passing over pulleys, to the points B, C. When the weights P, P', F' are proportional to AB, AC and AD respectively, the strings attached at B and C will be observed to form prolongations of the sides, and the diagonal AD will be vertical. 18. Resultant of any Number of Forces at a Point. To find the resultant of any number of f&rces whose lines of action meet in a point, it is only necessary to draw a crooked line composed of straight lines which represent the several forces. The resultant will be represented by a straight line drawn from the beginning to the 8 FIRST PRINCIPLES OF DYNAMICS. end of this crooked line. For by what precedes, if ABODE be a crooked line such that the straight lines AB, BC, CD, DE represent four forces acting at a point, we may substitute for AB and BC Fig. 3. Gravesande's Apparatus. the straight line AC, since this represents their resultant. We may then substitute AD for AC and CD, and finally AE for AD and DE. One of the most important applications of this construction is to three forces not lying on one plane. In this case the crooked line will consist of three edges of a parallelepiped, and the line which repre- sents the resultant will be the diagonal. This is evident from Fig. 4, in which AB, AC, AD represent three forces acting at A. The resultant of AB and AC is Ar, and the resultant of Ar and AD is Ar'. The crooked line whose parts represent the forces, may be either ABrr', or ABGr', or ADGr', &c., the total number of alternatives being six, since three things can be taken in six different orders. We have here an excellent illustration of the fact that the same final resultant is obtained, in whatever order the forces are combined. Fig. 4. Parallelepiped of Forces. PARALLEL FORCES. 19. Equilibrium of Three Parallel Forces. If three parallel forces, P, Q, R, applied to a rigid body, balance each other, the following conditions must be fulfilled: 1. The three lines of action AP, BQ, CR, Fig. 5, must be in one plane. 2. The two outside forces P, R, must act in the opposite direction to the middle force Q, and their sum must be equal to Q. p 3. Each force must be proportional to the distance between the lines of action of the other two; that is, we must have JL = JL - A (i) BC AC AB' The figure shows that AC is the sum of AB and BC; hence it fol- lows from these equations, that Q is equal to the sum of P and R, as above stated. 20. Resultant of Two Parallel Forces. Any two parallel forces being given, a third parallel force which will balance them can be found from the above equations; and a force equal and opposite to this will be their resultant. We may distinguish two cases. 1. Let the two given forces be in the same direction. Then their resultant is equal to their sum, and acts in the same direction, along a line which cuts the line joining their points of application into two parts which are inversely as the forces. 2. Let the two given forces be in opposite directions. Then their resultant will be equal to their difference, and will act in the direc- tion of the greater of the two forces, along a line which cuts the production of the line joining their points of application on the side of the greater force; and the distances of this point of section from the two given points of application are inversely as the forces. 21. Centre of Two Parallel Forces. In both cases, if the points of application are not given, but only the magnitudes of the forces and their lines of action, the magnitude and line of action of the resul- tant are still completely determined; for all straight lines which are drawn across three parallel straight lines are cut by them in the same ratio; and we shall obtain the same result whatever points of application we assume. If the points of application are given, the resultant cuts the line 10 FIRST PRINCIPLES OF DYNAMICS. joining them, or this line produced, in a definite point, whose posi- tion depends only on the magnitudes of the given forces, and not at all on the angle which their direction makes with the joining line. This result is important in connection with centres of gravity. The point so determined is called the centre of the two parallel forces. If these two forces are the weights of two particles, the "centre" thus found is their centre of gravity, and the resultant force is the same as if the two particles were collected at this point. 22. Moments of Resultant and of Components Equal. The follow- ing proposition is often useful. Let any straight line be drawn across the lines of action of two parallel forces P 1} P 2 (Fig. 6). Let be any point on this line, and x v x. 2 j^ ^ j^z the distances measured from to the / -/ / points of section, distances measured in opposite directions being distin- Fjg ' 6 - guished by opposite signs, and forces in opposite directions being also distinguished by opposite signs. Also let R denote the resultant of P x and P 2 , and "x the distance from to its intersection with the line; then we shall have P! j + P 2 a = B, x. For, taking the standard case, as represented in Fig. 6, in which all the quantities are positive, we have OA 1 = x it OA 2 = x 2 , OB x, and by 19 or 20 we have PI. A^^Pj.BAz, that is, P 1 (x-x l ) = P. 2 (x 2 -x), whence (Pj + P^x = ?!*! + P., as* that is, Ro; = P 1 a: 1 + P 2 x i . (2) 23. Any Number of Parallel Forces in One Plane. Equation (2) affords the readiest means of determining the line of action of the resultant of several parallel forces lying in one plane. For let P I} P 2 , P 3 , &c., be the forces, R x the resultant of the first two forces P 1( P 2 , and Ro the resultant of the first three forces P l5 P 2 , P 3 . Let a line be drawn across the lines of action, and let the distances of the points of section from an arbitrary point on this line be expressed according to the following scheme: Force P x P 2 P 3 R x R 2 Distance x x. x 3 ~x x MOMENT OF A FORCE. 11 Then, by equation (2) \ve have R! ! = ?!, + P 2 Xi. Also since R 2 is the resultant of R : and P :>> we have Ra X-^R! XI + PS x 3) and substituting for the term R L ~x v we have RS a^j P l Xi + Po x- 2 + P 3 x 3 . This reasoning can evidently be extended to any number of forces, so that we shall have finally Rx = sum of such terms as Pec, where R denotes the resultant of all the forces, and is equal to their algebraic sum ; while ~x denotes the value of x for the point where the line of action of R cuts the fixed line. It is usual to employ the Greek letter S to denote "the sum of such terms as." We may therefore write R=s (P) Ra;=S (Pz) whence {FX) (3) 24. Moment of a Force about a Point. When the fixed line is at right angles to the parallel forces, the product Px is called the moment of the force P about the point O. More generally, the moment of a force about a point is the product of the /orce by the length of the perpendicular dropped upon it from the point. The above equations show that for parallel forces in one plane, the moment of the resultant about any point in the plane is the sum of the moments of the forces about the same point. If the resultant passes through O, the distance x is zero; whence it follows from the equations that the algebraical sum of the moments vanishes. The moment of a force about a point measures the tendency of the force to produce rotation about the point. If one point of a body be fixed, the body will turn in one direction or the other according as the resultant passes on one side or the other of this point (the direction of the resultant being supposed given). If the resultant passes through the fixed point, the body will be in equi- librium. The moment T*x of any force about a point, changes sign with P and also with x; thereby expressing (what is obvious in itself) that 12 FIRST PRINCIPLES OF DYNAMICS. the direction in which the force tends to turn the body about the point will be reversed if the direction of P is reversed while its line of action remains unchanged, and will also be reversed if the line of action be shifted to the other side of the point while the direction of the force remains unchanged. 25. Experimental Illustration. Fig. 7 represents a simple appar- atus (called the arithmetical lever) for illustrating the laws of par- Fig. 7. Composition of Parallel Forces. allel forces. The lever AB is suspended at its middle point by a cord, so that when no weights are attached it is horizontal. Equal weights P, P are hung at points A and B equidistant from the centre, and the suspending cord after being passed over a very freely mov- ing pulley M, has a weight f hung at its other end sufficient to pro- duce equilibrium. It will be found that P' is equal to the sum of the two weights P together with the weight required to counter- poise the lever itself. In the second figure, the two weights hung from the lever are not equal, but one of them is double of the other, P being hung at B, and 2 P at C; and it is necessary for equilibrium that the dis- tance OB be double of the distance OC. The weight P' required COUPLES. 13 to balance the system will now be 3 P together with the weight of the lever. 26. Couple. There is one case of two parallel forces in opposite directions which requires special attention; that in which the two forces are equal. To obtain some idea of the effect of two such forces, let us first suppose them not exactly equal, but let their difference be very small compared with either of the forces. In this case, the resultant will be equal to this small difference, and its line of action will be at a great distance from those of the given forces. For in 19 if Q is very little greater than P, so that Q-P, or R is only a small fraction of P, the equation g7s=^g shows that AB is only a small fraction of BC, or in other words that BC is very large compared with AB. If Q gradually diminishes until it becomes equal to P, R will gradually diminish to zero; but while it diminishes, the product R . BC will remain constant, being always equal to P . AB. A very small force R at a very great distance would have sensibly the same moment round all points between A and B or anywhere in their neighbourhood, and the moment of R is always equal to the algebraic sum of the moments of P and Q. When Q is equal to P, they compose what is called a couple, and the algebraic sum of their moments about any point in their plane is constant, being always equal to P . AB, which is therefore called the moment of the couple. A couple consists of tu-o equal and parallel forces in opposite directions applied to the same body. The distance betiveen their lines of action is called the arm of the couple, and the 'product of one of the two equal forces by this arm is called the moment of the couple. 27. Composition of Couples. Axis of Couple. A couple cannot be balanced by a single force; but it can be balanced by any couple of equal moment, opposite in sign, if the plane of the second couple be either the same as that of the first or parallel to it. Any number of couples in the same or parallel planes are equiva- lent to a single couple whose moment is the algebraic sum of theirs. The laws of the composition of couples (like those of forces) can be illustrated by geometry. Let a couple be represented by a line perpendicular to its plane, marked with an arrow according to the convention that if an 14 FIRST PRINCIPLES OF DYNAMICS. ordinary screw were made to turn in the direction in which the couple tends to turn, it would advance in the direction in which the arrow points. Also let the length of the line represent the moment of the couple. Then the same laws of composition and resolution which hold for forces acting at a point will also hold for couples. A line thus drawn to represent a couple is called the axis of the couple. Just as any number of forces acting at a point are either in equilibrium or equivalent to a single force, so any number of couples applied to the same rigid body (no matter to what parts of it) are either in equilibrium or equivalent to a single couple. 28. Resultant of Force and Couple in Same Plane. The resultant of a force and a couple in the same plane is a single force. For the couple may be replaced by another of equal moment having its equal forces P, Q, each equal to the given force F, and the latter couple may ** then be turned about in its own plane and carried into such a position that one of its two forces destroys the force F, as represented in Fig. 8. There will then only remain the force P, which is equal and parallel to F. By reversing this procedure, we can show that a force P which does not pass through a given point A is equivalent to an equal and parallel force F which does pass through it, together with a couple; the moment of the couple being the same as the moment of the force P about A. 29. General Resultant of any Number of Forces applied to a Rig-id Body. Forces applied to a rigid body in lines which do not meet in one point are not in general equivalent to a single force. By the process indicated in the concluding sentence of the preceding section, we can replace the forces by forces equal and parallel to them, acting at any assumed point, together with a number of couples. These couples can then be reduced (by the principles of 27) to a single couple, and the forces at the point can be replaced by a single force; so that we shall obtain, as the complete resultant, a single force applied at any point we choose to select, and a couple. We can in general make the couple smaller by resolving it into two components whose planes are respectively perpendicular and parallel to the force, and then compounding one of these components (the latter) with the force as explained in 28, thus moving the UKXERAL RESULTANT. 15 force parallel to itself without altering its magnitude. This is the greatest simplification that is possible. The result is that we have a single force and a couple whose plane is perpendicular to the force. Any combination of forces that can be applied to a rigid body is reducible to a force acting along one definite line and a couple in a plane perpendicular to this line. Such a combination of a force and couple is called a wi^ench, and the " one definite line " is called the axis of the wrench. The point of application of the force is not definite, but is any point of the axis. 30. Application to Action and Reaction. Every action of force that one body can exert upon another is reducible to a wrench, and the law of reaction is that the second body will, in every case, exert upon the first an equal and opposite wrench. The two wrenches will have the same axis, equal and opposite forces along this axis, and equal and opposite couples in planes perpendicular to it. 31. Resolution the Inverse of Composition. The process of finding the resultant of two or more forces is called composition. The inverse process of finding two or more forces which shall together be equivalent to a given force, is called resolution, and the two or more forces thus found are called components. The problem to resolve a force into two components along two given lines which meet it in one point and are in the same plane with it, has a definite solution, which is obtained by drawing a triangle whose sides are parallel respectively to the given force and the required components. The given force and the required com- ponents will be proportional to the sides of this triangle, each being represented by the side parallel to it. The problem to resolve a force into three components along three given lines which meet it in one point and are not in one plane, also admits of a definite solution. 32. Rectangular Resolution. In the majority of cases which occur in practice the required components are at right angles to each other, and the resolution is then said to be rectangular. When " the component of a force along a given line" is mentioned, without anything in the context to indicate the direction of the other component or components, it is always to be understood that the resolution is rectangular. The process of finding the required component in this case is illustrated by Fig. 9. Let AB represent the given force F, and let AC be the line along which the com- ponent of F is required. It is only necessary to drop from B a 16 FIRST PRINCIPLES OF DYNAMICS. perpendicular BC on this line; AC will represent the required component. CB represents the other component, which, along with B AC, is equivalent to the given force. If the total number of rectangular components, of which AC represents one, is to be three, c then the other two will lie in a plane per- Fig. 9. Component along a given pendicular to AC, and they can be found by again resolving CB. The magnitude of AC will be the same whether the number of components be two or three, and the component along AC will be F -^ or in trigonometrical language, F cos . BAG. We have thus the following rule : The component of a given force along a given line is found by multiplying the force by the cosine of the angle bettveen its own direction and that of the required component. CHAPTER III. CENTRE OF GRAVITY. 33. Gravity is the force to which we owe the names "up" and " down." The direction in which gravity acts at any place is called the downward direction, and a line drawn accurately in this direc- tion is called vertical; it is the direction assumed by a plumb-line. A plane perpendicular to this direction is called horizontal, and is parallel to the surface of a liquid at rest. The verticals at different places are not parallel, but are inclined at an angle which is approximately proportional to the distance between the places. It amounts to 180 when the places are antipodal, and to about 1' when their distance is one geographical mile, or to about 1" for every hundred feet. Hence, when we are dealing with the action of gravity on a body a few feet or a few hundred feet in length, we may practically regard the action as consisting of parallel forces. 34. Centre of Gravity. Let A and B be any two particles of a rigid body, let u\ be the weight of the particle A, and iv. 2 the weight of B. These weights are parallel forces, and their resultant divides the line AB in the inverse ratio of the forces. As the body is turned about into different positions, the forces w l and w. 2 remain unchanged in magnitude, and hence the resultant cuts AB always in the same point. This point is called the centre of the parallel forces i^ and ^v. 2 , or the centre of gravity of the two particles A and B. The magnitude of the resultant will be w^ + w^, and we may substitute it for the two forces themselves; in other words, we may suppose the two particles A and B to be collected at their centre of gravity. We can now T combine this resultant with the weight of a third particle of the body, and shall thus obtain a resultant iv ) -}-w 2 -}-w 3 , passing through a definite point in the line which joins 18 CENTRE OF GRAVITY. the third particle to the centre of gravity of the first two. The first three particles may now be supposed to be collected at this point, and the same reasoning may be extended until all the particles have been collected at one point. This point will be the centre of gravity of the whole body. From the manner in which it has been ob- tained, it possesses the property that the resultant of all the forces of gravity on the body passes through it, in every position in which the body can be placed. The resultant force of gravity upon a rigid body is therefore a single force passing through its centre of gravity. 35. Centres of Gravity of Volumes, Areas, and Lines. If the body is homogeneous (that is composed of uniform substance throughout), the position of the centre of gravity depends only on the figure, and in this sense it is usual to speak of the centre of gravity of a figure. In like manner it is customary to speak of the centres of gravity of areas and lines, an area being identified in thought with a thin uniform plate, and a line with a thin uniform wire. It is not necessary that a body should be rigid in order that it may have a centre of gravity. We may speak of the centre of gravity of a mass of fluid, or of the centre of gravity of a system of bodies not connected in any way. The same point which would be the centre of gravity if all the parts were rigidly connected, is still called by this name whether they are connected or not. 36. Methods of Finding Centres of Gravity. Whenever a homo- geneous body contains a point which bisects all lines in the body that can be drawn through it, this point must be the centre of gravity. The centres of a sphere, a circle, a cube, a square, an ellipse, an ellipsoid, a parallelogram, and a parallelepiped, are ex- amples. Again, when a body consists of a finite number of parts whose weights and centres of gravity are known, we may regard each part as collected at its own centre of gravity. When the parts are at all numerous, the final result will most readily be obtained by the use of the formula - 2 ( p *) 2(P)' where P denotes the weight of any part, x the distance of its centre of gravity from any plane, and ~x the distance of the centre of gravity of the whole from that plane. We have already in 23 CENTRE OF GRAVITY OF A TRIANGLE. ID proved this formula for the case in which the centres of gravity lie in one straight line and x denotes distance from a point in this line; and it is not difficult, by the help of the properties of similar triangles, to make the proof general. 37. Centre of Gravity of a Triangle. To find the centre of gravity of a triangle ABC (Fig. 10), we may begin by supposing it divided into narrow strips by lines (such as be) parallel to BC. It can be shown, by similar triangles, that each of these strips is bisected 1-y the line AD drawn from A to D the middle point of BC. But each strip may be collected at its own centre of gravity, that is at its own middle point; hence the whole triangle may be collected on the line AD; its centre of gravity must therefore be situated upon this line. Similar reason- ing shows that it must lie upon the line Fig - 10 - BE drawn from B to the middle point of AC. It is therefore the intersection of these two lines. If we join DE we can show that the triangles AGB, DGE, are similar, and that AG _ AB GD ~ DE ~ DG is therefore one third of DA. The centre of gravity of a triangle therefore lies upon the line joining any corner to the middle point of the opposite side, and is at one-third of the length of this line from the end where it meets that side. It is worthy of remark that if three equal particles are placed at the corners of any triangle, they have the same centre of gravity as the triangle. For the two particles at B and C may be collected at the middle point D, and this double particle at D, together with the single particle at A, will have their centre of gravity at G, since G divides DA in the ratio of 1 to 2. 38. Centre of Gravity of a Pyramid. If a pyramid or a cone be divided into thin slices by planes parallel to its base, and a straight line be drawn from the vertex to the centre of gravity of the base, this line will pass through the centres of gravity of all the slices, since all the slices are similar to the base, and are similarly cut by this line. In a tetrahedron or triangular pyramid, if D (Fig. 11) be the centre of gravity of one face, and A be the corner opposite to this CENTRE OF GRAVITY. face, the centre of gravity of the pyramid must lie upon the line AD. In like manner, if E be the centre of gravity of one face, the centre of gravity of the pyramid must lie upon the line joining E with the oppo- site corner B. It must therefore be the intersection G of these two lines. That they do intersect is otherwise obvious, for the lines AE, BD meet in C, the middle point of one edge of the pyramid, E being found by taking CE one third of CA, and D by taking CD Fig. 11. Centre of Gravity of Tetral.elron. Qv>g third of CB If D, E be joined, we can show that the joining line is parallel to BA, and that the triangles AGB, DGE are similar. Hence AG GD AB DE BC DC That is, the line AD joining any corner to the centre of gravity of the opposite face, is cut in the ratio of 3 to 1 by the centre of gravity G of the triangle. DG is therefore one-fourth of DA, and the dis- tance of the centre of gravity from any face is one-fourth of the distance of the opposite corner. A pyramid standing on a polygonal base can be cut up into tri- angular pyramids standing on the triangular bases into which the polygon can be divided, and having the same vertex as the whole pyramid. The centres of gravity of these trian- gular pyramids are all at the same perpendicular distance from the base, namely at one-fourth of the distance of the vertex, which is therefore the distance of the centre of gravity of the whole from the base. The centre of gravity of any pyramid is there- fore found by joining the vertex to Fig. 12.-Centre of Gravity of Pyramid, fa e cen t re of gravity of the base, and cutting off one-fourth of the joining line from the end where it meets the base. The same rule applies to a cone, since a cone may be regarded as a polygonal pyramid with a very large number of sides. CENTRE OF GRAVITY OF PYRAMID. 21 Fig. 13. Equilibrium of a Body supported on a Horizontal Plane at three or more Points. 39. If four equal particles are placed at the corners of a triangular pyramid, they will have the same centre of gravity as the pyramid. For three of them may, as we have seen ( 37) be collected at the centre of gravity of one face; and the centre of gravity of the four particles will divide the line which joins this point to the fourth, in the ratio of 1 to 3. 40. Condition of Standing or Falling. When a heavy body stands on a base of finite area, and remains in equili- brium under the action of its own weight and the reaction of this base, the vertical through its centre of gravity must fall with- in the base. If the body is supported on three or more points, as in Fig. 13, we are to understand by the base the convex 1 poly- gon whose corners are the points of support; for if a body so supported turns over, it must turn about the line joining two of these points. 41. Body supported at one Point. When a heavy body supported at one point remains at rest, the reaction of the point of support equilibrates the force of gravity. But two forces cannot be in equilibrium unless they have the same line of action; hence the ver- tical through the centre of gravity of the body must pass through the point of support. If instead of being supported at a point, the heavy body is supported by an axis about which it is free to turn, the vertical through the centre of gravity must pass through this axis. 42. Stability and Instability. When the point of support, or axis of support, is vertically be loiv the centre of gravity, it is easily seen that, if the body were displaced a little to either side, the forces act- ing upon it would turn it still further away from the position of equilibrium. On the other hand, when the point or axis of sup- port is vertically above the centre of gravity, the forces which would 1 The word convex is inserted to indicate that there must be no re-entrant angles. Any points of support which lie within the polygon formed by joining the rest, must be left out of account. 22 CENTRE OF GRAVITY. act upon it if it were slightly displaced would tend to restore it. In the latter case the equilibrium is said to be stable, in the former unstable. When the centre of gravity coincides with the point of support, or lies upon the axis of support, the body will still be in equilibrium when turned about this point or axis into any other position. In this case the equilibrium is neither stable nor unstable but is called neutral. 43. Experimental determination of Cen- tre of Gravity. In general, if we suspend a body by any point, in such a manner that it is fjree to turn about this point, it will come to rest in a position of stable equilibrium. The centre of gravity will then be vertically beneath the point of Fig. 14. Experimental Determination Support. If W6 nOW SUSpend the body of Centre of Gravity. p ,-i ,-, n ., irom another point, the centre ot gravity will come vertically beneath this. The intersection of these two verticals will therefore be the centre of gravity (Fig. 14). 44. To find the centre of gravity of a flat plate or board (Fig. 15), we may suspend it from a point near its circumfer- ence, in such a manner that it sets itself in a ver- tical plane. Let a plumb-line be at the same time suspended from the same point, and made to leave its trace upon the board by chalking and "snap- ping" it. Let the board now be suspended from another point, and the operation be repeated. The two chalk lines will intersect each other at that point of the face which is opposite to the centre of gravity; the centre of gravity itself being of course in the substance of the board. 45. Work done against Gravity. When a heavy body is raised, work is said to be done against gravity, and the amount of this work is reckoned by multiplying together the weight of the body and the height through which it is raised. Horizontal movement does not count, and when a body is raised obliquely, only the vertical component of the motion is to be reckoned. Suppose, now, that we have a number of particles whose weights Fig. 15. Centre of Gravity of Board. WORK DONE AGAINST GRAVITY. 23 are io v u\ 2 , W- A &c., and let their heights above a given horizontal plane be respectively h 1} k. 2 , h s c. We know by equation (3), 23, that if h denote the height of their centre of gravity we have (Wi + ic 2 + &c.) h = ii'i ^ + w'2 h. + c. (4) Let the particles now be raised into new positions in which their heights above the same plane of reference are respectively H 1; H.,, H 3 &c. The height H of their centre of gravity will now r be such that (Wi + w.i + &c. ) H = w l H! + w.jt H 2 + &c. (5) From these two equations, we find, by subtraction (if- A) = M! (H^-AO + w, (H 2 -A,) + &c. (6) Now Hj li v is the height through which the particle of weight w-^ has been raised ; hence the work done against gravity in raising it is iv l (Hj Aj) and the second member of equation (6) therefore expresses the whole amount of w r ork done against gravity. But the first member expresses the work which would be done in raising all the particles through a uniform height H h, which is the height of the new position of the centre of gravity above the old. The work done against gravity in raising any system of bodies will therefore be correctly computed by supposing all the system to be collected at its centre of gravity. For example, the work done in raising bricks and mortar from the ground to build a chimney, is equal to the total w r eight of the chimney multiplied by the height of its centre of gravity above the ground. 46. The Centre of Gravity tends to Descend. When the forces which tend to move a system are simply the weights of its parts, w r e can determine whether it is in equilibrium by observing the path in which its centre of gravity would travel if movement took place. If we suppose this path to represent a hard frictionless surface, and the centre of gravity to represent a heavy particle placed upon it, the conditions of equilibrium will be the same as in the actual case. The centre of gravity tends to run down hill, just as a heavy particle does. There will be stable equilibrium if the centre of gravity is at the bottom of a valley in its path, and unstable equilibrium if it is at the top of a hill. When a rigid body turns about a horizontal axis, the path of its centre of gravity is a circle in a vertical plane. The highest and lowest points of this circle are the positions of the centre of gravity in unstable and stable equilibrium respectively; 24 CENTRE OF GRAVITY. except when the axis traverses the centre of gravity itself, in which case the centre of gravity can neither rise nor fall, and the equili- brium is neutral. A uniform sphere or cylinder lying on a horizontal plane is in neutral equilibrium, because its centre of gravity will neither be raised nor lowered by rolling. An egg balanced on its end as in Fig. 16, is in unstable equilibrium, because its centre of gravity is at- the top of a hill which it will descend when the egg rolls to one side. The position of equilibrium shown in Fig. 17 is stable as regards rolling to left or right, because the path of its centre of gravity in Fig. 16. Unstable Equilibrium. Fig. 17. Stable Equilibrium. such rolling would be a curve whose lowest point is that now occu- pied by the centre of gravity. As regards rolling in the direction at right angles to this, if the egg is a true solid of resolution, the equili- brium is neutral. 47. Work done by Gravity. When a heavy body is lifted, the lifting force does work against gravity. When it descends gravity- does work upon it; and if it descends to the same position from which it was lifted, the work done by gravity in the descent is equal to the work done against gravity in the lifting; each being- equal to the weight of the body multiplied by the vertical displace- ment of its centre of gravity. The tendency of the centre of gravity to descend is a manifestation of the tendency of gravity to do work ; and this tendency is not peculiar to gravity. 48. Work done by any Force. A force is said to do work when its point of application moves in the direction of the force, or in any direction making an acute angle with this, so as to give a component displacement in the direction of the force; and the amount of work done is the product of the force by this component. If F denote PRINCIPLE OF WORK. 25 the force, a the displacement, and the angle between the two, the work done by F is F a cos 0, which is what we obtain either by the above rule or by multiplying the whole displacement by the effective component of F, that is the component of F in the direction of the displacement. If the angle .0 is obtuse, cos is negative and the force F does negative work. If is a right angle F does no work. In this case F neither assists nor resists the displacement. When 8 is acute, F assists the dis- placement, and would produce it if the body were constrained by guides which left it free to take this displacement and the directly opposite one, while preventing all others. If is obtuse, F resists the displacement, and would produce the opposite displacement if the body were constrained in the manner just supposed. 49. Principle of Work. If any number of forces act upon a body which is only free to move in a particular direction and its opposite, we can tell in which of these two directions it will move by calcu- lating the work which each force would do. Each force would do positive work when the displacement is in one direction, and nega- tive work when it is in the opposite direction, the absolute amounts of work being the same in both cases if the displacements are equal. The body will upon the whole be urged in that direction which gives an excess of positive work over negative. If no such excess exists, but the amounts of positive and negative work are exactly equal, the body is in equilibrium. This principle (which has been called the principle of virtual velocities, but is better called the pi^inciple of work) is often of great use in enabling us to calculate the ratio which two forces applied in given ways to the same body must have in order to equilibrate each other. It applies not only to the "mechanical powers" and all combinations of solid machinery, but also to hydrostatic arrangements; for example to the hydraulic press. The condition of equilibrium between two forces applied to any frictionless machine and tending to drive it opposite ways, is that in a small movement of the machine they would do equal and opposite amounts of work. Thus in the screw-press (Fig. 30) the force applied to one of the handles, multiplied by the distance through which this handle moves, will be equal to the pressure which this force produces at the foot of the screw, multiplied by the distance that the screw travels. 26 CENTRE OF GRAVITY. This is on the supposition of no friction. A frictionless machine gives out the same amount of work which is spent in driving it. The effect of friction is to make the work given out less than the work put in. Much fruitless ingenuity has been expended upon contrivances for circumventing this law of nature and producing a machine which shall give out more work than is put into it. Such contrivances are called " perpetual motions." 50. General Criterion of Stability. If the forces which act upon a body and produce equilibrium remain unchanged in magnitude and direction when the body moves away from its position, and if the velocities of their points of application also remain unchanged in direction and in their ratio to each other, it is obvious that the equality of positive and negative work which subsists at the beginning of the motion will continue to subsist throughout the entire motion. The body will therefore remain in equilibrium when displaced. Its equilibrium is in this case said to be neutral. If the forces which are in equilibrium in a given position of the body, gradually change in direction or magnitude as the body moves away from this position, the equality of positive and negative work will not in general continue to subsist, and the inequality will increase with the displacement. If the body be displaced with a constant velocity and in a uniform manner, the rate of doing work, which is zero at first, will not continue to be zero, but will have a value, whether positive or negative, increasing in simple proportion to the displacement. Hence it can be shown that the whole work done is proportional to the square of the displacement, for when we double the displacement we, at the same time, double the mean working force. If this work is positive, the forces assist the displacement and tend to increase it; the equilibrium must therefore have been unstable. On the other hand, if the work is negative in all possible displace- ments from the position of equilibrium, the forces oppose the displacements and the equilibrium is stable. 51. Illustration of Stability. A good example of stable equili- brium of this kind is furnished by Gravesande's apparatus (Fig. 3) simplified by removing the parallelogram and employing a string- to support the three weights, one of them P" being fastened to it at a point A near its middle, and the others P, P' to its ends. The point A will take the same position as in the figure, and will return to it again when displaced. If \ve take hold of the point A and STABILITY. 27 move it in any direction whether in the plane of the string or out of it, we feel that at first there is hardly any resistance and the smallest force we can apply produces a sensible disturbance; but that as the displacement increases the resistance becomes greater. If we release the point A when displaced, it will execute oscillations, which will become gradually smaller, owing to friction, and it will .finally come to rest in its original position of equilibrium. The centre of gravity of the three weights is in its lowest position when the system is in equilibrium, and when a small dis- placement is produced the centre of gravity rises by an amount proportional to its square, so that a double displacement produces a quadruple rise of the centre of gravity. In this illustration the three forces remain unchanged, and the directions of two of them change gradually as the point A is moved. Whenever the circumstances of stable equilibrium are such that the forces make no abrupt changes either in direction or magnitude for small displacements, the resistance will, as in this case, be propor- tional to the displacement (when small), and the work to the square of the displacement, and the system will oscillate if displaced and then left to itself. 52. Stability where Forces vary abruptly with Position. There are other cases of stable equilibrium which may be illustrated by the example of a book lying on a table. If we displace it by lifting one edge, the force which we must exert does not increase with the displacement, but is sensibly constant when the displacement is small, and as a consequence the work will be simply proportional to the displacement. The reason is, that one of the forces concerned in producing equilibrium, namely, the upward pressure of the table, changes per saltum at the moment when the displacement begins. In applying the principle of work to such a case as this, we must employ, instead of the actual work done by the force which changes abruptly, the work which it would do if its magnitude and direction remained unchanged, or only changed gradually. 53. Illustrations from Toys. The stability of the "balancer" (Fig. 18) depends on the fact that, owing to the weight of the two leaden balls, which are rigidly attached to the figure by stiff wires, the centre of gravity of the whole is below the point of support. If the figure be disturbed it oscillates, and finally comes to rest in a position in which the centre of gravity is vertically under the toe on which the figure stands. CENTRE OF GRAVITY. The "tumbler" (Fig. 19) consists of a light figure attached to a hemisphere of lead, the centre of gravity of the whole being between the centre of gravity of the hemisphere and the centre of the sphere to which it belongs. When placed upon a level table, the lowest position of the centre of gravity is that in which the figure is upright, and it accord- ingly returns to this position when displaced. 54. Limits of Stability. In the foregoing discussion we have em- ployed the term "stability" in its strict mathematical sense. But there are cases in which, though small displacements would merely produce small oscillations, larger displacements would cause the body, when left to itself, to fall entirely away from the given position of equilibrium. This may Fig. is.-Baiancer. j^g ex p ressec l by saying that the equilibrium is stable for displacements lying within certain limits, but unstable for displacements beyond these limits. The equilibrium Fig 19. Tumblers. of a system is practically unstable when the displacements which it is likely to receive from accidental disturbances lie beyond its limits of stability. CHAPTER IV. THE MECHANICAL POWERS. 55. We now proceed to a few practical applications of the fore- gcing principles; and we shall begin with the so-called "mechanical powers," namely, the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. 56. Lever. Problems relating to the lever are usually most con- veniently solved by taking moments round the fulcrum. The general condition of equilibrium is, that the moments of the power and the weight about the fulcrum must be in opposite directions, and must be equal. When the power and weight act in parallel directions, the conditions of equilibrium are precisely those of three parallel forces ( 19), the third force being the reaction of the fulcrum. It is usual to distinguish three " orders " of lever. In levers of the first order (Fig. 20) the fulcrum is between the power and the 1 Fig. 20. Fig. 21. Fig. 22. Three Orders of Lever. weight. In those of the second order (Fig. 21) the weight is between the power and the fulcrum. In those of the third order (Fig. 22) the power is between the weight and the fulcrum. In levers of the second order (supposing the forces parallel), the weight is equal to the sum of the power and the pressure on the fulcrum; and in levers of the third order, the power is equal to the sum of the weight and the pressure on the fulcrum; since the middle one of three parallel forces in equilibrium must always be equal to the sum of the other two. 30 THE MECHANICAL POWERS. 57. Arms. The arms of a lever are the two portions of it inter- mediate, respectively, between the fulcrum and the power, and between the fulcrum and the weight. If the lever is bent, or if, though straight, it is not at right angles to the lines of action of the power and weight, it is necessary to distinguish between the arms of the lever as above denned (which are parts of the lever), and the arms of the poiver and weight regarded as forces which have moments round the fulcrum. In this latter sense (which is always to be understood unless the contrary is evidently intended), the arms are the perpendiculars dropped from the fulcrum upon the lines of action of the power and weight. 58. Weight of Lever. In the above statements of the conditions of equilibrium, we have neglected the weight of the lever itself. To take this into account, we have only to suppose the whole weight of the lever collected at its centre of gravity, and then take its moment round the fulcrum. We shall thus have three moments to take account of, and the sum of the two that tend to turn the lever one way, must be equal to the one that tends to turn it the opposite way. 59. Mechanical Advantage. Every machine when in action serves to transmit work without altering its amount; but the force which the machine gives out (equal and opposite to what is commonly called the weight) may be much greater or much less than that by which it is driven (commonly called the power). When it is greater, the machine is said to confer mechanical advantage, and the mechanical advantage is measured by the ratio of the weight to the power for equilibrium. In the lever, when the power has a longer arm than the weight, the mechanical advantage is equal to the quotient of the longer arm by the shorter. 60. Wheel and Axle. The wheel and axle (Fig. 23) may be regarded as an endless lever. The condition of equili- brium is at once given by taking moments round the common axis of the wheel and axle ( 24). If we neglect the thickness of the ropes, the condition is that the power multiplied by the radius of the wheel must equal the weight multiplied by the radius of the axle; but it is more exact to regard the lines of action of the F; e- - 3 - two forces as coinciding with the axes of the two ropes, so that each of the two radii should be increased by half the thick- ness of its own rope. If we neglect the thickness of the ropes, the PULLEYS. 31 mechanical advantage is the quotient of the radius of the wheel by the radius of the axle. 61. Pulley. A pulley, when fixed in such a way that it can only turn about a fixed axis (Fig. 24), confers no mechanical advantage. It may be regarded as an endless lever of the first order with its two arms equal. The arrangement represented in Fig. 25 gives a mechanical advantage of 2; for the lower or movable pulley may be regarded as an endless lever of the second order, in w r hich the arm of the power is the diameter of the pulley, and the arm of the weight is Fig. 24. Fig. 26. Fig. 27. half the diameter. The same result is obtained by employing the principle of work; for if the weight rises 1 inch, 2 inches of slack are given over, and therefore the power descends 2 inches. 62. In Fig. 26 there are six pulleys, three at the upper and three at the lower block, and one cord passes round them all. All por- tions of this cord (neglecting friction) are stretched with the same force, which is equal to the power; and six of these portions, parallel to one another, support the weight. The power is therefore one- sixth of the weight, or the mechanical advantage is 6. 63. In the arrangement represented in Fig. 27, there are three movable pulleys, each hanging by a separate cord. The cord which supports the lowest pulley is stretched with a force equal to half the weight, since its two parallel portions jointly support the weight. The cord which supports the next pulley is stretched with a force half of this, or a quarter of the weight; and the next cord with a force half of this, or an eighth of the weight; but this cord is directly attached to the power. Thus the power is an eighth of the M 32 THE MECHANICAL POWERS. weight, or the mechanical advantage is 8. If the weight and the block 1 to which it is attached rise 1 inch, the next block rises 2 inches, the next 4, and the power moves through 8 inches. Thus, the work done by the power is equal to the work done upon the weight. In all this reasoning we neglect the weights of the blocks them- selves; but it is not difficult to take them into account when necessary. 64. Inclined Plane. We now come to the inclined plane. Let AB (Fig. 28) be any portion of such a plane, and let AC and BC be drawn vertically and horizontally. Then AB is called the length, AC the height, and CB the base of the inclined plane. The force of gravity upon a heavy body M resting on the plane, may be represented by a vertical line MP, and may be resolved by the parallelogram of forces ( 16) into two components, MT, MN, the former parallel and the latter perpendicular to the plane. A force equal and oppo- site to the component MT will suffice to prevent the body from slip- ping down the plane. Hence, if the power act parallel to the plane, and the weight be that of a heavy body resting on the plane, the power is to the weight as MT to MP; but the two triangles MTP and ACB are similar, since the angles at M and A are equal, and the angles at T and C are right angles; hence MT is to MP as AC to AB, that is, as the height to the length of the plane. 65. The investigation is rather easier by the principle of work ( 49). The work done by the power in drawing the heavy body up the plane, is equal to the power multiplied by the length of the plane. But the work done upon the weight is equal to the weight multiplied by the height through which it is raised, that is, by the height of the plane. Hence we have Power x length of plane = weight X height of plane ; or power : weight : : height of plane : length of plane. 66. If, instead of acting parallel to the plane, the power acted parallel to the base, the work done by the power would be the product of the power by the base; and this must be equal to the product of the weight by the height; so that in this case the con- dition of equilibrium would be 1 The "pulley" is the revolving wheel. The pulley, together with the frame in which it is inclosed, constitute the "block." SCREW. 33 Power : weight : : height of plane : base of plane. 67. Wedge. In these investigations we have neglected friction. The wedge may be regarded as a case of the inclined plane; but its practical action depends to such a large extent upon friction and impact 1 that we cannot profitably discuss it here. 68. Screw. The screw (Fig. 29) is also a case of the inclined plane. The length of one convolution of the thread is the length of the corresponding inclined plane, the step of the screw, or distance between two successive convolutions (measured parallel to the axis of the screw), is the height of the plane, and the circumference of Fig 29. Fig. 30. the screw is the base of the plane. This is easily shown by cutting out a right-angled triangle in paper, and bending it in cylindrical fashion so that its base forms a circle. 69. Screw Press. In the screw press (Fig. 30) the screw is turned by means of a lever, which gives a great increase of mechanical advantage. In one complete revolution, the pressures applied to the two handles of the lever to turn it, do work equal to their sum multiplied by the circumference of the circle described (approxi- mately) by either handle (we suppose the two handles to be equi- distant from the axis of revolution); and the work given out by the machine, supposing the resistance at its lower end to be constant, is equal to this resistance multiplied by the distance between the threads. These two products must be equal, friction being neglected. 1 An impact (for example a blow of a hammer) may be regarded as a very great (and variable) force acting for a very short time. The magnitude of an impact is measured by the momentum which it generates in the body struck. CHAPTER Y. THE BALANCE. 70. General Description of the Balance. In the common balance (Fig. 31) there is a stiff piece of metal, A B, called the beam, which turns about the sharp edge O of a steel wedge form- ing part of the beam and resting upon two hard and smooth supports. There are two other steel wedges at A and B, with their edges upwards, and upon these edges rest the hooks for supporting the scale pans. The three edges (called knife-edges) are parallel to one another and perpen- dicular to the length of the beam, and are very nearly in one plane. 71. Qualities Requisite. The qualities requisite in a balance are: 1. That it be consistent with itself; that is, that it shall give the same result in successive weighings of the same body. This depends chiefly on the trueness of the knife-edges. 2. That it be just. This requires that the distances A 0, OB, be equal, and also that the beam remain horizontal when the pans are empty. Any inequality in the distances A O, OB, can be detected by putting equal (and tolerably heavy) weights into the two pans. This adds equal moments if the distances are equal, but unequal Fig. 31. Balance. SENSIBILITY OF BALANCK. 35 moments if they are unequal, and the greater moment will prepon- derate. 3. Delicacy or sensibility (that is, the power of indicating in- equality between two weights even when their difference is very small). This requires a minimum of friction, and a very near approach to neutral equilibrium ( 40). In absolutely neutral equilibrium, the smallest conceivable force is sufficient to produce a displacement to the full limit of neutrality; and in barely stable equilibrium a small force produces a large displacement. The condition of stability is that if the weights supported at A and B be supposed collected at these edges, the centre of gravity of the system composed of the beam and these two weights shall be below the middle edge 0. The equilibrium would be neutral if this centre of gravity exactly coin- cided with 0; and it is necessary as a condition of delicacy that its distance below O be very small. 4. Facility for weighing quickly is desirable, but must sometimes be sacrificed when extreme accuracy is required. The delicate balances used in chemical analysis are provided with a long pointer attached to the beam. The end of this pointer moves along a graduated arc as the beam vibrates; and if the weights in the two pans are equal, the excursions of the pointer on opposite sides of the zero point of this arc will also be equal. Much time is con- sumed in watching these vibrations, as they are very slow; and the more nearly the equilibrium approaches to neutrality, the slower they are. Hence quick weighing and exact weighing are to a certain ex- tent incompatible. 72. Double Weighing. Even if a balance be not just, yet if it be consistent with itself, a correct weighing can be made with it in the following manner: Put the body to be weighed in one pan, and counterbalance it with sand or other suitable material in the other. Then remove the body and put in its place such weights as are just sufficient to counterpoise the sand. These weights are evidently equal to the weight of the body. This process is called double weighing, and is often employed (even with the best balances) when the greatest possible accuracy is desired. 73. Investigation of Sensibility. Let A and B (Fig. 32) be the points from which the scale-pans are suspended, O the axis about which the beam turns, and G the centre of gravity of the beam. If when the scale-pans are loaded with equal weights, we put into one 36 THE BALANCE. of them an excess of weight p, the beam will become inclined, and will take a position such as A'B', turning through an angle which we will call a, and which is easily calculated. In fact let the two forces P and P + p act at A' and B' respec- tively, where P denotes the less of the two weights, including the weight of the pan. Then the two forces P destroy each other in conse- quence of the resistance of the axis O; there is left only the force p i applied at B', and the weight v of IB- the beam applied at G', the new position of the centre of gravity. |r+ P These two forces are parallel, and are in equilibrium about the axis O, that is, their resultant passes through the Fi s- 32 - point O. The distances of the points of application of the forces from a vertical through O are therefore inversely proportional to the forces themselves, which gives the relation v. G'R=j>. B'L. But if we call half the length of the beam I, and the distance OG r we have G'R = r sin a, B'L = i cos a. whence itr sin a = pi cos a, and consequently tan o = ?. (a) irr The formula (a) contains the entire theory of the sensibility of the balance when properly constructed. We see, in the first place, that tan a increases with the excess of weight p, which was evident be- forehand. We see also that the sensibility increases as I increases and as TT diminishes, or, in other words, as the beam becomes longer and lighter. At the same time it is obviously desirable that, under the action of the weights employed, the beam should be stiff enough to undergo no sensible change of shape. The problem of the balance then consists in constructing a beam of the greatest possible length and lightness, which shall be capable of supporting the action of given forces without bending. Fortin, whose balances are justly esteemed, employed for his beams bars of steel placed edgewise; he thus obtained great rigidity, but SENSIBILITY. 37 certainly not all the lightness possible. At present the makers of balances employ in preference beams of copper or steel made in the form of a frame, as shown in Fig 33. They generally give them the shape of a very elongated lozenge, the sides of which are connected by bars variously arranged. The determination of the best shape is, in fact, a special problem, and is an application on a small scale of that principle of applied mechanics which teaches us that hollow pieces have greater resisting power in proportion to their weight than solid pieces, and consequently, for equal resisting power, the former are lighter than the latter. Aluminium, which with a rigidity nearly equal to that of copper, has less than one-fourth of its density, seems naturally marked out as adapted to the construction of beams. It has as yet, however, been little used. The formula (a) shows us, in the second place, that the sensibility increases as r diminishes ; that is, as the centre of gravity approaches the centre of suspension. These two points, however, must not coin- cide, for in that case for any excess of weight, however small, the beam would deviate from the horizontal as far as the mechanism would permit, and would afford no indication of approach to equality in the weights. With equal weights it would remain in equilibrium in any position. In virtue of possessing this last property, such a balance is called indifferent Practically the distance between the centre of gravity and the point of suspension must not be less than a certain amount depending on the use for which the balance is designed. The proper distance is determined by observing what difference of weights corresponds to a division of the graduated arc along which the needle moves. If, for example, there are 20 divi- sions on each side of zero, and if 2 milligrammes are necessary for the total displacement of the needle, each division will correspond to an excess of weight of -$ or ^ of a milligramme. That this may be the case we must evidently have a suitable value of r, and the maker is enabled to regulate this value with precision by means of the screw which is shown in the figure above the beam, and which enables him slightly to vary the position of the centre of gravity. 74. Weighing with Constant Load. In the above analysis we have supposed that the three points of suspension of the beam and of the two scale-pans are in one straight line; in which case the value of tan a does not include P, that is, the sensibility is independent of the weight in the pans. This follows from the fact that the resultant of the two forces P passes through O, and is thus destroyed, because 38 THE BALANCE. the axis is fixed. This would not be the case if, for example, the points of suspension of the pans were above that of the beam; in this case the point of application of the common load is above the point O, and, when the beam is inclined, acts in the same direction as the excess of weight; whence the sensibility increases with the load up to a certain limit, beyond which the equilibrium becomes unstable. 1 On the other hand, when the points of suspension of the pans are below that of the beam, the sensibility increases as the load diminishes, and, as the centre of gravity of the beam may in this case be above the axis, equilibrium may become unstable when the load is less than a certain amount. This variation of the sensibility with the load is a serious disadvantage; for, as we have just shown, the displacement of the needle is used as the means of estimating weights, and for this purpose we must have the same displacement corresponding to the same excess of weight. If we wish to employ C.UAPLANTL. Fig. 33. Beam of Balance. either of the two above arrangements, we should weigh with a con- stant load. The method of doing so, which constitutes a kind of double weighing, consists in retaining in one of the pans a weight equal to this constant load. In the other pan is placed the same load subdivided into a number of marked weights. When the body 1 This is an illustration of the general principle, applicable to a great variety of philo- sophical apparatus, that a maximum of sensibility involves a minimum of stability ; that is, a very near approach to instability. This near approach is usually indicated by exces- sive slowness in the oscillations which take place about the position of equilibrium. BALANCES OF PRECISION. 39 to be weighed is placed in this latter pan, we must, in order to main- tain equilibrium, remove a certain number of weights, which evi- dently represent the weight of the body. We may also remark that, strictly speaking, the sensibility always depends upon the load, which necessarily produces a variation in the friction of the axis of suspension. Besides, it follows from the nature Fig. 34. Balance for Purposes of Accuracy. of bodies that there is no system that does not yield somewhat even to the most feeble action. For these reasons, there is a decided advantage in operating with constant load. 75. Details of Construction. A fundamental condition of the cor- rectness of the balance is, that the weight of each pan and of the load which it contains should always act exactly at the same point, and therefore at the same distance from the axis of suspension. This important result is attained by different methods. The arrange- ment represented in Fig. 33 is one of the most effectual. At the 40 THE BALANCE. extremities of the beam are two knife-edges, parallel to the axis of rotation, and facing upwards. On these knife-edges rests, by a hard plane surface of agate or steel, a stirrup, the front of which has been taken away in the figure. On the lower part of the stirrup rests another knife-edge, at right angles to the former, the two being together equivalent to a universal joint supporting the scale-pan and its contents. By this arrangement, whatever may be the position of the weights, their action is always reduced to a vertical force act- ing on the upper knife-edge. Fig. 34 represents a balance of great delicacy, with the glass case that contains it. At the bottom is seen the extremity of a lever, which enables us to raise the beam, and thus avoid wearing the knife-edge when not in use. At the top may be remarked an arrangement employed by some makers, consisting of a horizontal graduated circle, on which a small metallic index can be made to travel; its different displacements, whose value can be determined once for all, are used for the final adjustment to produce exact equilibrium. 76. Steelyard. The steelyard (Fig. 35) is an instrument for weighing bodies by means of a single weight, P, which can be hung at any point of a graduated arm O B. As P is moved further from the fulcrum O, its moment round O increases, and there- fore the weight which must be hung from the fixed point A to counterbalance it in- creases. Moreover, equal movements of p . 3g P along the arm pro- duce equal additions to its moment, and equal additions to the weight at A produce equal additions to the opposing moment. Hence the divisions on the arm (which indicate the weight in the pan at A) must be equidistant. CHAPTER VI. FIRST PRINCIPLES OF KINETICS. 77. Principle of Inertia. A body not acted on by any forces, or only acted on by forces which are in equilibrium, will not commence to move; and if it be already in motion with a movement of pure translation, it will continue its velocity of translation unchanged, so that each of its points will move in a straight line with uniform velocity. This is Newton's first law of motion, and is stated by him in the following terms: " Every body continues in its state of rest or of uniform motion in a straight line, except in so far as it is compelled by impressed forces to change that state." The tendency to continue in a state of rest is manifest to the most superficial observation. The tendency to continue in a state of uniform motion can be clearly understood from an attentive study of facts. If, for example, we make a pendulum oscillate, the amplitude of the oscillations slowly decreases and at last vanishes altogether. This is because the pendulum experiences resistance from the air which it continually displaces; and because the axis of suspension rubs on its supports. These two circumstances combine to produce a diminution in the velocity of the apparatus until it is completely annihilated. If the friction at the point of suspension is diminished by suitable means, and the apparatus is made to oscillate in vacua, the duration of the motion will be immensely increased. Analogy evidently indicates that if it were possible to suppress entirely these two causes of the destruction of the pendulum's velo- city, its motion would continue for an indefinite time unchanged. This tendency to continue in motion is the cause of the effects which are produced when a carriage or railway train is suddenly stopped. The passengers are thrown in the direction of the motion, 42 FIRST PRINCIPLES OF KINETICS. in virtue of the velocity which they possessed at the moment when the stoppage occurred. If it were possible to find a brake sufficiently powerful to stop a train suddenly at full speed, the effects of such a stoppage would be similar to the effects of a collision. Inertia is also the cause of the severe falls which are often received in alighting incautiously from a carriage in motion; all the particles of the body have a forward motion, and the feet alone being reduced to rest, the upper portion of the body continues to move, and is thus thrown forward. When we fix the head of a hammer on the handle by striking the end of the handle on the ground, we utilize the inertia of matter. The handle is suddenly stopped by the collision, and the head con- tinues to move for a short distance in spite of the powerful resist- ances which oppose it. 78. Second Law of Motion. Newton's second law of motion is that " Change of motion is proportional to the impressed force and is in the direction of that force." Change of motion is here spoken of as a quantity, and as a directed quantity. In order to understand how to estimate change of motion, we must in the first place understand how to compound motions. When a boat is sailing on a river, the motion of the boat relative to the shore is compounded of its motion relative to the water and the motion of the water relative to the shore. If a person is walk- ing along the deck of the boat in any direction, his motion relative to the shore is compounded of three motions, namely the two above mentioned and his motion relative to the boat. Let A, B and C be any three bodies or systems. The motion of A relative to B, compounded with the motion of B relative to C, is the motion of A relative to C. This is to be taken as the definition of what is meant by compounding two motions; and it leads very directly Fig. 36. -Composition of Motions ^O ^ e resu }t that two rectilinear motions are compounded by the parallelogram law. For if a body moves along the deck of a ship from O to A (Fig. 36), and the ship in the meantime advances through the distance OB, it is obvious that, if we complete the parallelogram OBCA, the point A of the ship will be brought to C, and the movement of the body in space will be from to C. If the motion along OA is uniform SECOND LAW OF MOTION. 4*.! and the motion of the ship is also uniform, the motion of the body through space will be a uniform motion along the diagonal OC. Hence, if two component velocities be represented by two lines drawn from a point, and a parallelogram be constructed on these lines, its diagonal will represent the resultant velocity. It is obvious that if OA in the figure represented the velocity of the ship and OB the velocity of the body relative to the ship, we should obtain the same resultant velocity OC. This is a general law; the interchanging of velocities which are to be compounded does not affect their resultant. Now suppose the velocity OB to be changed into the velocity OC, what are we to regard as the change of velocity? The change of velocity is that velocity which compounded with OB would give OC It is therefore OA. The same force which, in a given time, acting always parallel to itself, changes the velocity of a body from OB to OC, would give the body the velocity OA if applied to it for the same time commencing from rest. Change of motion, estimated in this way, depends only on the acting force and the body acted on by the force; it is entirely independent of any previous motion which the body may possess. No experiments on forces and motions inside a carriage or steamboat which is travelling with perfect smoothness in a straight course, will enable us to detect that it is travelling at all. We cannot even assert that there is any such thing as absolute rest, or that there is any difference between absolute rest and uniform straight movement of translation. As change of motion is independent of the initial condition of rest or motion, so also is the change of motion produced by one force act- ing on a body independent of the change produced by any other force acting on the body, provided that each force remains constant in magnitude and direction. The actual motion will be that which is compounded of the initial motion and the motions due to the two forces considered separately. If AB represent one of these motions, BC another, and CD the third, the actual or resultant motion will be AD. The change produced in the motion of a body by two forces act- ing jointly can therefore be found by compounding the changes which would be produced by each force separately. This leads at once to the " parallelogram of forces," since the changes of motion produced in one and the same body are proportional to the forces which produce them, and are in the directions of these forces. 44 FIRST PRINCIPLES OF KINETICS. In case any student should be troubled by doubt as to whether the "changes of motion" which are proportional to the forces, are to be understood as distances, or as velocities, we may remark that the law is equally true for both, and its truth for one implies its truth for the other, as will appear hereafter from comparing the formula for the distance s = |/Y 2 , with the formula for the velocity v = ft, since both of these expressions are proportional to /. 79. Explanation of Second Law continued. It is convenient to distinguish between the intensity of a force and the magnitude or amount of a force. The intensity of a force is measured by the change of velocity which the force produces during the unit of time; and can be computed from knowing the motion of the body acted on, without knowing anything as to its mass. Two bodies are said to be of equal mass when the same change of motion (whether as regards velocity or distance) which is produced by applying a given force to one of them for a given time, would also be produced by applying this force to the other for an equal time. If we join two such bodies, we obtain a body of double the mass of either; and if we apply the same force as before for the same time to this double mass, we shall obtain only half the change of velocity or distance that we obtained before. Masses can therefore be compared by taking the in- verse ratio of the changes produced in their velocities by equal forces. The velocity of a body multiplied by its mass is called the momen- tum of the body, and is to be regarded as a directed magnitude hav- ing the same direction as the velocity. The change of velocity, when multiplied by the mass of the body, gives the change of momentum; and the second law of motion may be thus stated: The change of momentum pwduced in a given time is propor- tional to the force which produces it, and is in the direction of this force. It is independent of the mass; the change of velocity in a given time being inversely as the mass. 80. Proper Selection of Unit of Force. If we make a proper selec- tion of units, the change of momentum produced in unit time will be not only proportional but numerically equal to the force which produces it; and the change of momentum produced in any time will be the product of the force by the time. Suppose any units of length, time, and mass respectively to have been selected. Then the unit velocity will naturally be denned as the velocity with which unit length would be passed over in unit time; the unit momentum will be the momentum of the unit mass moving with this velocity; UNIT OF FORCE. 45 and the unit force will be that force which produces this momentum in unit time. We define the unit force, then, as that force which acting for unit time upon unit mass produces unit velocity. 81. Relation between Mass and Weight. The weight of a body, strictly speaking, is the force with which the body tends towards the earth. This force depends partly on the body and partly on the earth. It is not exactly the same for one and the same body at all parts of the earth's surface, but is decidedly greater in the polar than in the equatorial regions. Bodies which, when weighed in a balance in vacuo, counterbalance each other, or counterbalance one and the same third body, have equal weights at that place, and will also be found to have equal weights at any other place. Experiments which we shall hereafter describe ( 89) show that such bodies have equal masses; and this fact having been established, the most convenient mode of comparing masses is by weighing them. A pound of iron has the same mass as a pound of brass or of any other substance. A pound of any kind of matter tends to the earth with different forces at different places. The weight of a pound of matter is therefore not a definite standard of force. But the pound of matter itself is a perfectly definite standard of mass. If we weigh one and the same portion of matter in different states; for instance water in the states of ice, snow, liquid water, or steam; or compare the weight of a chemical compound with the weights of its components; we find an exact equality; hence it has been stated that the mass of a body is a measure of the quantity of matter which it contains; but though this statement expresses a simple fact when applied to the compari- son of different quantities of one and the same substance, it expresses no known fact of nature when applied to the comparison of different substances. A pound of iron and a pound of lead tend to the earth with equal forces; and if equal forces are applied to them both their velocities are equally affected. We may if we please agree to mea- sure "quantity of matter" by these tests; but we must beware of assuming that two things which are essentially different in kind can be equal in themselves. 82. Third Law of Motion. Action and Reaction. Forces always occur in pairs, every exertion of force being a mutual action between two bodies. Whenever a body is acted on by a force, the body from which this force proceeds is acted on by an equal and opposite force. The earth attracts the moon, and the moon attracts the earth. A magnet attracts iron and is attracted by iron. When two 46 FIRST PRINCIPLES OF KINETICS. boats are floating freely, a rope attached to one and hauled in by a person in the other, makes each boat move towards the other. Every exertion of force generates equal and opposite momenta in the two bodies affected by it, since these two bodies are acted on by equal forces for equal times. If the forces exerted by one body upon the other are equivalent to a single force, the forces of reaction will also be equivalent to a single force, and these two equal and opposite resultants will have the same line of action. We have seen in 29 that the general resultant of any set of forces applied to a body is a wrench; that is to say it consists of a force with a definite line of action (called the axis), accompanied by a couple in a perpendicular plane. The reac- tion upon the body which exerts these forces will always be an equal and opposite wrench; the two wrenches having the same axis, equal and opposite forces along this axis, and equal and opposite couples in the perpendicular plane. 83. Motion of Centre of Gravity Unaffected. A consequence of the equality of the mutual forces between two bodies is, that these forces produce no movement of the common centre of gravity of the two bodies. For if A be the centre of gravity of a mass m^ and B the centre of gravity of a mass m 2 , their common centre of gravity C will divide AB inversely as the masses. Let the masses be originally at rest, and let them be acted on only by their mutual attraction or replusion. The distances through which they are moved by these equal forces will be inversely as the masses, that is, will be directly as AC and BC ; hence if A' B' are their new positions after any time, we have AC _ AA' _ ACAA' _ A/C BC ~~ EB' ~ BC BB' ~ B'C' The line A'B' is therefore divided at C in the same ratio in which the line AB was divided; hence C is still the centre of gravity. 84. Velocity of Centre of Gravity. If any number of masses are moving with any velocities, and in any directions, but so that each of them moves uniformly in a straight line, their common centre of gravity will move uniformly in a straight line. To prove this, we shall consider their component velocities in any one direction, let these component velocities be u x u 2 U 3 &c., the masses being m l m 2 m 3 &c., and the distances of the bodies (strictly speaking the distances of C.G.S. SYSTEM OF UNITS. 47 their respective centres of gravity) from a fixed plane to which the given direction is normal, be x l X 2 x% &c. The formula for the distance of their common centre of gravity from this plane is m l x l + m., j- 2 + &c. . - m,i + in^ + &c. In the time t, x l is increased by the amount uj, x 2 by u 2 t, and so on ; hence the numerator of the above expression is increased by mi MI t + m-i Wo t + &c., and the value of x is increased in each unit of time by mi u\ + m-2 u-2 + Sec. ,, mi + m-i + &c. which is therefore the component velocity of the centre of gravity in the given direction. As this expression contains only given constant quantities, its value is constant; and as this reasoning applies to all directions, the velocity of the centre of gravity must itself be constant both in magnitude and direction. We may remark that the above formula (2) correctly expresses the component velocity of the centre of gravity at the instant con- sidered, even when u lt u. 2 , &c., are not constant. 85. Centre of Mass. The point which we have thus far been speaking of under the name of " centre of gravity," is more appro- priately called the "centre of mass," a name which is at once suggested by formula (1) 84. When gravity act*s in parallel lines upon all the particles of a body, the resultant force of gravity upon the body is a single force passing through this point; but this is no longer the case when the forces of gravity upon the different parts of the body (or system of bodies) are not parallel. 86. Units of Measurement. It is a matter of importance, in scientific calculations, to express the various magnitudes with which we have to deal in terms of units which have a simple relation to each other. The British weights and measures are completely at fault in this respect, for the following reasons: 1. They are not a decimal system; and the reduction of a measurement (say) from inches and decimals of an inch to feet and decimals of a foot, cannot be effected by inspection. 2. It is still more troublesome to reduce gallons tocubic feet orinches. 3. The weight (properly the mass) of a cubic foot of a substance in Ibs., cannot be written down by inspection, when the specific gravity of the substance (as compared with water) is given. 48 FIRST PRINCIPLES OF KINETICS. 87. The C.G.S. System. A committee of the British Association, specially appointed to recommend a system of units for general adoption in scientific calculation, have recommended that the centimetre be adopted as the unit of length, the gramme as the unit of mass, and the second as the unit of time. We shall first give the rough and afterwards the more exact definitions of these quantities. The centimetre is approximately ^ of the distance of either pole of the earth from the equator; that is to say 1 followed by 9 zeros expresses this distance in centimetres. The gramme is approximately the mass of a cubic centimetre of cold water. Hence the same number which expresses the speci- fic gravity of a substance referred to water, expresses also the mass of a cubic centimetre of the substance, in grammes. The second is 2 4 x 60 x 60 ^ a mean so ^ ar day- More accurately, the centimetre is defined as one hundredth part of the length, at the temperature Centigrade, of a certain stand- ard bar, preserved in Paris, carefully executed copies of which are preserved in several other places; and the gramme is defined as one thousandth part of the mass of a certain standard which is preserved at Paris, and of which also there are numerous copies preserved elsewhere. For brevity of reference, the committee have recommended that the system of units based on the Centimetre, Gramme, and Second, be called the C.G.S. system. The unit of area in this system is the square centimetre. The unit of volume is the cubic centimetre. The unit of velocity is a velocity of a centimetre per second. The unit of momentum is the momentum of a gramme moving with a velocity of a centimetre per second. The unit force is that force which generates this momentum in one second. It is therefore that force which, acting on a gramme for one second, generates a velocity of a centimetre per second. This force is called the dyne, an abbreviated derivative from the Greek Su^a/ac (force). The unit of work is the work done by a force of a dyne working through a distance of a centimetre. It might be called the dyne- centimetre, but a shorter name has been provided and it is called the erg, from the Greek 'ipyov (work). CHAPTER VII. LAWS OF FALLING BODIES. 88. Effect of the Resistance of the Air. In air, bodies fall with unequal velocities; a sovereign or a ball of lead falls rapidly, a piece of down or thin paper slowly. It was formerly thought that this difference was inherent in the nature of the materials; but it is easy to show that this is not the case, for if we compress a mass of down or a piece of paper by rolling it into a ball, and compare it with a piece of gold-leaf, we shall find that the latter body falls more slowly than the former. The inequality of the velocities which we observe is due to the resistance of the air, which increases with the extent of surface exposed by the body. It was Galileo who first discovered the cause of the unequal rapidity of fall of different bodies. To put the matter to the test, he prepared small balls of different substances, and let them fall at the same time from the top of the tower of Pisa; they struck the ground almost at the same instant. On changing their forms, so as to give them very different extents of surface, he observed that they fell with very unequal velocities. He was thus led to the conclusion that gravity acts on all substances with the same intensity, and that in a vacuum all bodies would fall with the same velocity. This last proposition could not be put to the test of experiment in the time of Galileo, the air-pump not having yet been invented. The experiment was performed by Newton, and is now well known as the " guinea and feather " experiment. For this purpose a tube from a yard and a half to two yards long is used, which can be exhausted of air, and which contains bodies of various densities, such as a coin, pieces of paper, and feathers. When the tube is full of air and is inverted, these different bodies are seen to fall with very unequal velocities; but if the experiment is repeated after the tube 50 LAWS OF FALLING BODIES. has been exhausted of air, no difference can be perceived between the times of their descent. 89. Mass and Gravitation Proportional.- This experiment proves that bodies which have equal weights are equal in mass. For equal masses are denned to be those which, when acted on by equal forces, receive equal accelerations; and the forces, in this experiment, are the weights of the falling bodies. Newton tested this point still more severely by experiments with pendulums (Principia, book III. prop. vi.). He procured two round wooden boxes of the same size and weight, and suspended them by threads eleven feet long. One of them he filled with wood, and he placed very accurately in the centre of oscillation of the other the same weight of gold. The boxes hung side by side, and, when set swinging in equal oscillations, went and returned together for a very long time. Here the forces concerned in producing and checking the motion, namely, the force of gravity and the resistance of the air, were the same for the two pendulums, and as the move- ments produced were the same, it follows that the masses were equal. Newton remarks that a difference of mass amounting to a thousandth part of the whole could not have escaped detection. He experimented in the same way with silver, lead, glass, sand, salt, water, and wheat, and with the same result. He therefore infers that universally bodies of equal mass gravitate equally towards the earth at the same place. He further extends the same law to gravi- tation generally, and establishes the conclusion that the mutual gravitating force between any two bodies depends only on their masses and distances, and is independent of their materials. The time of revolution of the moon round the earth, considered in conjunction with her distance from the earth, shows that the relation between mass and gravitation is the same for the material of which the moon is composed as for terrestrial matter; and the same con- clusion is proved for the planets by the relation which exists between their distances from the sun and their times of revolution in their orbits. 90. Uniform Acceleration. The fall of a heavy body furnishes an illustration of the second law of motion, which asserts that the change of momentum in a body in a given time is a measure of the force which acts on the body. It follows from this law that if the same force continues to act upon a body the changes of momentum in successive equal intervals of time will be equal. When a heavy UNIFOKM ACCKLKHATION. 51 body originally at rest is allowed to fall, it is acted on during the time of its descent by its own weight and by no other force, if we neglect the resistance of the air. As its own weight is a constant force, the body receives equal changes of momentum, and therefore of velocity, in equal intervals of time. Let g denote its velocity in centimetres per second, at the end of the first second. Then at the end of the next second its velocity will be g + g, that is 2g; at the end of the next it will be 2g+g, that is 3g, and so on, the gain of velocity in each second being equal to the velocity generated in the first second. At the end of t seconds the velocity will therefore be tg. Such motion as this is said to be uniformly accelerated, and the constant quantity g is the measure of the acceleration. Accelera- tion is defined as the gain of velocity per unit of time. 91. Weight of a Gramme in Dynes. Value of g. Let m denote the mass of the falling body in grammes. Then the change of momentum in each second is mg, which is therefore the measure of the force acting on the body. The weight of a body of m grammes is therefore mg dynes, and the weight of 1 gramme is g dynes. The value of g varies from 97S'l at the equator to 983'1 at the poles; and 981 may be adopted as its average value in temperate latitudes. Its value at any part of the earth's surface is approximately given by the formula g = 980-6056 - 2-5028 cos 2\ - -000,003A, iii which \ denotes the latitude, and h the height (in centimetres) above sea-level. 1 In 79 we distinguished between the intensity and the amount of a force. The amount of the force of gravity upon a mass of m grammes is mg dynes. The intensity of this force is g dynes per gramme. The intensity of a force, in dynes per gramme of the body acted on, is always equal to the change of velocity which the force produces per second, this change being expressed in centimetres per second. In other words the intensity of a force is equal to the acceleration which it produces. The best designation for g is the intensity of gravity. 92. Distance fallen in a Given Time. The distance described in a given time by a body moving with uniform velocity is calculated by multiplying the velocity by the time; just as the area of a rect- angle is calculated by multiplying its length by its breadth. Hence if we draw a line such that its ordinates AA', BB', &c., represent the 1 For the method of determination see 120. 52 LAWS OF FALLING BODIES. velocities with which a body is moving at the times represented by OA, OB (time being reckoned from the beginning of the motion), it / can be shown that the whole distance B described is represented by the area OB'B bounded by the curve, the last ordinate, and the base line. In fact this area can be divided into narrow strips (one of which is shown at A A', Fig. 37) A each of which may practically be re- garded as a rectangle, whose height represents the velocity with which the body is moving during the very small interval of time represented by its base, and whose area therefore represents the distance described in this time. This would be true for the distance described by a body moving from rest with any law of velocity. In the case of falling bodies the law is that the velocity is simply proportional to the time; hence the ordinates AA', BB', &c., must be directly as the abscissae OA, OB ; this proves that the line A' B' must be straight ; and the figure OB' B is therefore a triangle. Its area will be half the product of OB and BB'. But OB represents the time t occupied by the motion, and BB' the velocity gt at the end of this time. The area of the triangle therefore represents half the product of t and gt, that is, represents ^gt z , which is accordingly the distance described in the time t. Denoting this distance by s, and the velocity at the end of time t by v, we have thus the two formulae v = gt, (1) 8 = to*', (2) from which we easily deduce gs = {*. (3) 93. Work spent in Producing Motion. We may remark, in pass- ing, that the third of these formulae enables us to calculate the work required to produce a given motion in a given mass. When a body whose mass is 1 gramme falls through a distance s, the force which acts upon it is its own weight, which is g dynes, and the work done upon it is gs ergs. Formula (3) shows that this is the same as ^v 2 ergs. For a mass of m grammes falling through a distance s, the work is |mi; 2 ergs. The work required to produce a velocity v (cen- timetres per second) in a body of mass m (grammes) originally at rest is ^mv z (ergs). 94. Body thrown Upwards. When a heavy body is projected ver- WORK IN PRODUCING MOTION. 53 tically upwards, the formulae (1) (2) (3) of 92 will still apply to its motion, with the following interpretations: v denotes the velocity of projection. t denotes the whole time occupied in the ascent. s denotes the height to which the body will ascend. When the body has reached the highest point, it will fall back, and its velocity at any point through which it passes twice will be the same in going up as in coming down. 95. Resistance of the Air. The foregoing results are rigorously applicable to motion in vacuo, and are sensibly correct for motion in air as long as the resistance of the air is insignificant in compari- son with the force of gravity. The force of gravity upon a body is the same at all velocities ; but the resistance of the air increases with the velocity, and increases more and more rapidly as the velocity becomes greater; so that while at very slow velocities an increase of 1 per cent, in velocity would give an increase of 1 per cent, in the resistance, at a higher velocity it would give an increase of 2 per cent., and at the velocity of a cannon-ball an increase of 3 per cent. 1 The formulae are therefore sensibly in error for high velocities. They are also in error for bodies which, like feathers or gold-leaf, have a large surface in proportion to their weight. 96. Projectiles. If, instead of being simply let fall, a body is pro- jected in any direction, its motion will be compounded of the motion of a falling body and a uniform motion in the direction of projection. Thus if OP (Fig. 38) is the direction of projection, and OQ the vertical through the point of pro- j jection, the body would move along OP keeping its original velocity unchanged, if it were not disturbed by gravity. To find Q Fig. 38. where the body w r ill be at any time t, we must lay off a length OP equal to V, V denoting the velocity of projec- tion, -and must then draw from P the vertical line PR downwards equal to ^gt z , which is the distance that the body would have fallen in the time if simply dropped. The point R thus determined, will be the actual position of the body. The velocity of the body at any time will in like manner be found by compounding the initial 1 This is only another way of saying that the resistance varies approximately as the velocity when very small, and approximately as the cube of the velocity for velocities like that of a cannon-ball. 54 LAWS OF FALLING BODIES. velocity with the velocity which a falling body would have acquired in the time. The path of the body will be a curve, as represented in the figure, OP being a tangent to it at O, and its concavity being down- wards. The equations above given, namely show that PR varies as the square of OP, and hence that the path (or trajectory as it is technically called) is a parabola, whose axis is vertical. 97. Time of Flight, and Range. If the body is projected from a point at the surface of the ground (supposed level) we can calculate the time of flight and the range in the following way. Let a be the angle which the direction of projection makes with the horizontal. Then the velocity of projection can be resolved into two components, V cos a and V sin a, the former being horizontal, and the latter vertically up\vard. The horizontal component of the velocity of the body is unaffected by gravity and remains constant. The vertical velocity after time t will be compounded of V sin a up- wards and gt downwards. It will therefore be an upw r ard velocity V sin a gt, or a downward velocity gt N sin a. At the highest point of its path, the body will be moving horizontally and the ver- tical component of its velocity will be zero; that is, we shall have ,,. . V sin a V sin a gt ; whence t - . y This is the time of attaining the highest point; and the time of flight will be double of this, that is, will be - y As the horizontal component of the velocity has the constant value V cos a, the horizontal displacement in any time t is V cos a multiplied by t. The range is therefore 2V 2 sin a cos a V 2 sin 2a or . 9 9 The range (for a given velocity of projection) will therefore be greatest when sin 2a is greatest, that is when 2a = 90 and n 45. We shall now describe two forms of apparatus for illustrating the laws of falling bodies. 98. Morin's Apparatus. Morin's apparatus consists of a wooden cylinder covered with paper, which can be set in uniform rotation about its axis by the fall of a heavy weight. The cord which sup- PROJECTILES 00 ports the weight is wound upon a drum, furnished with a toothed wheel which works on one side with an endless screw on the axis of the cylinder, and on the other drives an axis carrying fans which serve to regulate the motion. In front of the turning cylinder is a cylindro-conical weight of cast-iron carrying a pen- cil whose point presses against the paper, and having ears which slide on vertical threads, serv- ing to guide it in its fall. By pressing a lever, the weight can be made to fall at a chosen moment. The proper time for this is when the motion of the cylinder has become sensibly uniform. It fol- lows from this arrange- ment that during its vertical motion the pencil will meet in succession the different generating lines 1 of the revolving cylinder, and will conse- quently describe on its surface a certain curve, from the study of which we shall be able to gather the law of the fall of the body which has traced it. With this view, we describe (by turning the cylinder while the pencil is stationary) a circle passing through the commencement of the curve, and also draw a vertical line through this point. We cut the paper along this latter line and develop it (that is, flatten 1 A cylindric surface could be swept out or "generated" by a straight line moving round the axis and remaining always parallel to it. The successive positions of this generating line are called the "generating lines of the cylinder." Fig. i!9. Morin's Apparatus. 56 LAWS OF FALLING BODIES. it out into a plane). It then presents the appearance shown in Fig. 40. If we take on the horizontal line equal distances at 1, 2, 3, 4, 5 . . . , and draw perpendiculars at their extremities to meet the curve, it is evident that the points thus found are those which were traced by the pencil when the cylinder had turned through the dis- tances 1, 2, 3, 4, 5. ... The corresponding verticals represent the spaces traversed in the times 1, 2, 3, 4, 5. . . . Now we find, as the figure shows, that these spaces are represented by the numbers 1, 4, 9, 16, 25 . . . , thus verifying the principle that the spaces described are proportional to the squares of the times employed in their description. We may remark that the proportionality of the vertical lines to the squares of the horizontal lines shows that the curve is a parabola. The parabolic trace is thus the consequence of the law of fall, and from the fact of the trace being parabolic we can infer the proportionality of the spaces to the squares of the times. The law of velocities might also be verified separately by Morin's apparatus; we shall not describe the method which it would be necessary to employ, but shall content ourselves with remarking that the law of velocities is a logical consequence of the law of spaces. 1 99. Atwood's Machine. Atwood's machine, which affords great facilities for illustrating the effects of force in producing motion, consists essentially of a very freely moving pulley over which a fine cord passes, from the ends of which two equal weights can be sus- pended. A small additional weight of flat and elongated form is laid upon one of them, which is thus caused to descend with uni- form acceleration, and means are provided for suddenly removing 1 Consider, in fact, the space traversed in any time t ; this space is given by the formula s = K 2 ; during the time t + 6 the space traversed will be K( + 0) 2 = K 2 + 2Ki:0 + K0 2 , whence it follows that the space traversed during the time after the time t is 2K< + K0 2 . The average velocity during this time 6 is obtained by dividing the space by 0, and is 2K + K0, which, by making very small, can be made to agree as accurately as we please with the value 2K. This limiting value 2K must therefore be the velocity at the end of time t. D. ATWOODS MACHINK. 57 this additional weight at any point of the descent, so as to allow the motion to continue from this point onward with uniform velocity. The machine is re- presented in Fig. 41. The pulley over which the string passes is the largest of the wheels shown at the top of the apparatus. In order to give it greater freedom of movement, the ends of its axis are made to rest, not on fixed supports, but on the circumferences of four wheels (two at each end of the axis) called friction- wheels, because their office is to dim- inish friction. Two small equal weights are shown, suspended from this pulley by a string passing over it. One of them P' is represented as near the bottom of the supporting pillar, and the other P as near the top. The latter is resting upon a small platform, which can be suddenly dropped when it is desired that the motion shall commence. A little lower down and vertically beneath the platform, is seen a ring, large enough to let the weignt pass through it without danger of Fig. 41. Atwood's Machine. 58 LAWS OF FALLING BODIES. contact. This ring can be shifted up or down, and clamped at any height by a screw; it is represented on a larger scale in the margin. At a considerable distance beneath the ring, is seen the stop, which is also represented in the margin, and can like the ring be clamped at any height. The office of the ring is to intercept the additional weight, and the office of the stop is to arrest the descent. The up- right to which they are both clamped is marked with a scale of equal parts, to show the distances moved over. A clock with a pendulum beating seconds, is provided for measuring the time; and there is an arrangement by which the movable platform can be dropped by the action of the clock precisely at one of the ticks. To measure the distance fallen in one or more seconds, the ring is removed, and the stop is placed by trial at such heights that the descending weight strikes it precisely at another tick. To measure the velocity acquired in one or more seconds, the ring must be fixed at such a height as to intercept the additional weight at one of the ticks, and the stop must be placed so as to be struck by the descending weight at another tick. 100. Theory of Atwood's Machine. If M denote each of the two equal masses, in grammes, and m the additional mass, the whole moving mass (neglecting the mass of the pulley and string) is 2M + m, but the moving force is only the weight of m. The accel- eration produced, instead of being g, is accordingly only r^ g. In order to allow for the inertia of the pulley and string, a con- stant quantity must be added to the denominator in the above for- mula, and the value of this constant can be determined by observ- ing the movements obtained with different values of M and m. Denoting it by C, we have (A) as the expression for the acceleration. As m is usually small in comparison with M, the acceleration is very small in comparison with that of a freely falling body, and is brought within the limits of convenient observation. Denoting the acceleration by a, and using v and s, as in 92, to denote the velocity acquired and space described in time t, we shall have v = at, (I) s-^af, (2) ec* = jk (3) FORCE IN CIRCULAR MOTION. and each of these formulae can be directly verified by experiments with the machine. 101. Uniform Motion in a Circle. A body cannot move in a curved path unless there be a force urging it Fig. 42. towards the concave side of the curve. We shall proceed to in- vestigate the intensity of this force when the path is circular and the velocity uniform. We shall denote the velocity by v, the radius of the circle by r, and the intensity of the force by /. Let AB (Figs. 42, 43) be a small portion of the path, and BD a perpendicular upon AD the tangent at A. Then, since the arc AB is small in comparison with the whole circumference, it is sensibly equal to AD, and the body would have been found at D instead of at B if no force had acted Fig 43 - upon it since leaving A. DB is accordingly the distance due to the force; and if t denote the time from A to B, we have AD = vt (i) DB = 4/t'. (2) The second of these equations gives 2DB and substituting for t from the first equation, this becomes 2DB * " AD 2 (3) But if An (Fig. 43) be the diameter at A, and Bm the perpendicular upon it from B, we have, by Euclid, AD 2 = mB 2 =Am.mw=2r. Am sensibly, = 2r . DB. Therefore ~^z--' an d hence by (3) Hence the force necessary for keeping a body in a circular path without change of velocity, is a force of intensity - directed towards the centre of the circle. If m denote the mass of the body, the amount of the force will be . This will be in dynes, if m be in grammes, r in centimetres, and v in centimetres per second. If the time of revolution be denoted by T, and ir as usual denote the ratio of circumference to diameter, the distance moved in time 60 LAWS OF FALLING BODILS. T is 2wr; hence v = -|r, and another expression for the intensity of the force will be \ 2 ,,> * 102. Deflecting .Force in General. In general, when a body is moving in any path, and with velocity either constant or varying, the force acting upon it at any instant can be resolved into two components, one along the tangent and the other along the normal. The intensity of the tangential component is measured by the rate at which the velocity increases or diminishes, and the intensity of the normal component is given by formula (4) of last article, if we make r denote the radius of curvature. 103. Illustrations of Deflecting Force. When a stone is swung round by a string in a vertical circle, the tension of the string in the lowest position consists of two parts: (1) The weight of the stone, which is mg if m be the mass of the stone. (2) The force m - which is necessary for deflecting the stone from a horizontal tangent into its actual path in the neighbourhood of the lowest point. When the stone is at the highest point of its path, the tension of the string is the difference of these two forces, that is to say it is (?-.*) and the motion is not possible unless the velocity at the highest point is sufficient to make - greater than g. The tendency of the stone to persevere in rectilinear motion and to resist deflection into a curve, causes it to exert a force upon the string, of amount m -., and this is called centrifugal force. It is not a force acting upon the stone, but a force exerted by the stone upon the string. Its direction is from the centre of curvature, whereas the deflecting force which acts upon the stone is tcnuards the centre of curvature. 104. Centrifugal Force at the Equator. Bodies on the earth's surface are carried round in circles by the diurnal rotation of the earth upon its axis. The velocity of this motion at the equator is about 46,500 centimetres per second, and the earth's equatorial radius is about 6'38 x 10 8 centimetres. Hence the value of - is found to be about 3*39. The case is analogous to that of the stone APPARENT GRAVITY. 61 at the highest point of its patli in the preceding article, if instead of a string which can only exert a pull we suppose a stiff rod which can exert a push upon the stone. The rod will be called upon to exert a pull or a push at the highest point according as - is greater or less than g. The force of the push in the latter case will be and this is accordingly the force with which the surface of the earth at the equator pushes a body lying upon it. The push, of course, is mutual, and this formula therefore gives the apparent weight or apparent gravitating force of a body at the equator, mg denoting its true gravitating force (due to attraction alone). A body falling in vacuo at the equator has an acceleration 978'10 relative to the surface of the earth in its neighbourhood; but this portion of the surface has itself an acceleration of 3'39, directed towards the earth's centre, and therefore in the same direction as the acceleration of the body. The absolute acceleration of the body is therefore the sum of these two, that is 981 '49, which is accordingly the intensity of true gravity at the equator. 2 The apparent weight of bodies at the equator would be nil if - were equal to g. Dividing 3'39 into 981 '49, the quotient is approxi- mately 289, which is (17) 2 . Hence this state of things would exist if the velocity of rotation were about 17 times as fast as at present. Since the movements and forces which we actually observe depend upon relative acceleration, it is usual to understand, by the value of g or the intensity of gravity at a place, the apparent values, unless the contrary be expressed. Thus the value of g at the equator is usually stated to be 97810. 105. Direction of Apparent Gravity. The total amount of centri- fugal force at different places on the earth's surface, varies directly as their distance from the earth's axis ; for this is the value of r in the formula (5) of 101, and the value of T in that formula is the same for the whole earth. The direction of this force, being per- pendicular to the earth's axis, is not vertical except at the equator; and hence, when we- compound it with the force of true gravity, we obtain a resultant force of apparent gravity differing in direction as well as in magnitude from true gravity. What is always understood by a vertical, is the direction of apparent gravity; and a plane per- pendicular to it is what is meant by a horizontal plane. CHAPTER VIII. THE PENDULUM. 106. The Pendulum. When a body is suspended so that it can turn about a horizontal axis which does not pass through its centre of gravity, its only position of stable equi- librium is that in which its centre of gravity is in the same vertical plane with the axis and below it ( 42). If the body be turned into any other position, and left to itself, it will oscillate from one side to the other of the position of equilibrium, until the resistance of the air and the friction of the axis gradually bring it to rest. A body thus suspended, whatever be its form, is called a pendulum. It frequently consists of a rod which can turn about an axis (Fig. 44) at its upper end, and which carries at its lower end a heavy lens-shaped piece of metal M called the bob; this latter can be raised or lowered by means of the screw V. The applications of the pendulum are very impor- tant: it regulates our clocks, and it has enabled us to measure the intensity of gravity in different parts of the world; it is important then to know at least the fundamental points in its theory. For explaining these, we shall begin with the consideration of an ideal body called the simple pendulum. 107. Simple Pendulum. This is the name given to a pendulum consisting of a heavy particle M (Fig. 45) attached to one end of an inextensible thread without weight, the other end of the thread being fixed at A. When the thread is vertical, the weight of the particle Fig. 44,-penduium. acts in the direction of its length, and there is equilib- SIMPLE PENDULUM. Fig. 45. Motion of Simple Pendulum. riuni. But suppose it is drawn aside into another position, as AM. In this case, the weight MG of the particle can be resolved into two forces MC and MH. The former, acting along the prolongation of the thread, is destroyed by the resistance of the thread; the other, acting along the tangent MH, produces the motion of the particle. This effective com- ponent is evidently so much the greater as the angle of displacement from the vertical position is greater. The particle will there- fore move along an arc of a circle described from A as centre, and the force which urges it forward will continually diminish till it arrives at the lowest point M'. At M' this force is zero, but, in virtue of the velocity acquired, the particle will ascend on the opposite side, the effective component of gravity being now opposed to the direction of its motion; and, inas- much as the magnitude of this component goes through the same series of values in this part of the motion as in the former part, but in reversed order, the velocity will, in like manner, retrace its former values, and will become zero when the particle has risen to a point M" at the same height as M. It then descends again and performs an oscillation from M" to M precisely similar to the first, but in the reverse direction. It will thus continue to vibrate between the two points M, M" (friction being supposed excluded), for an indefinite number of times, all the vibra- tions being of equal extent and performed in equal periods. The distance through which a simple pendulum travels in moving from its lowest position to its furthest position on either side, is called its amplitude. It is evidently equal to half the complete arc of vibration, and is commonly expressed, not in linear measure, but in degrees of arc. Its numerical value is of course equal to that of the angle MAM', which it subtends at the centre of the circle. The complete period of the pendulum's motion is the time which it occupies in moving from M to M" and back to M, or more generally, is the time from its passing through any given position to its next passing through the same position in the same direction. What is commonly called the time of vibration, or the time of a single vibration, is the half of a complete period, being the time of THE PENDULUM. passing from one of the two extreme positions to the other. Hence what we have above defined as a complete period is often called a double vibration. When the amplitude changes, the time of vibration changes also, being greater as the amplitude is greater; but the connection between the two elements is very far from being one of simple proportion. The change of time (as measured by a ratio) is much less than the change of amplitude, especially when the amplitude is small; and when the amplitude is less than about 5, any further diminution of it has little or no sensible effect in diminishing the time. For small vibrations, then, the time of vibration is independent of the amplitude. This is called the law of isochronism. 108. Law of Acceleration for Small Vibrations. Denoting the length of a simple pendulum by I, and its inclination to the vertical at any moment by 6, we see from Fig. 45 that the ratio of the effective TVTTT component of gravity to the whole force of gravity is j^r, that is sin 0; and when is small this is sensibly equal to 6 itself as measured by ^ . Let s denote the length of the arc MM' inter- vening between the lower end of the pendulum and the lowest point of its swing, at any time; then is equal to -*, and the intensity of the effective force of gravity when is small is sensibly equal to gO, that is to ^. Since g and I are the same in all positions of the pendulum, this effective force varies as s. Its direction is always towards the position of equilib- rium, so that it accelerates the motion during the approach to this position, and retards it during the recess; the acceleration or retardation being always in direct pro- portion to the distance from the position of equilibrium. This species of motion is of extremely common occurrence. It is illus- trated by the vibration of either prong of a tuning-fork, and in general by the motion of any body vibrating in one plane in such a manner as to yield a simple musical tone. 109. General Law for Period. Suppose a point P to travel with uniform velocity round a circle (Fig. 46), and from its successive PS }'6 ps f, Fig. 46. Projection of Circular Motion. SIMPLE HARMONIC MOTION. 65 positions P I} P 2 , &c., let perpendiculars P^, P 2 p 2 , &c., be drawn to a fixed straight line in the plane of the circle. Then while P travels once round the circle, its projection p executes a complete vibration. The acceleration of P is always directed towards the centre of the circle, and is equal to (TJT) r ( 101). The component of this acceler- ation parallel to the line of motion of p, is the fraction - of the whole acceleration (x denoting the distance of p from the middle point of its path), and is therefore (-HT)* This is accordingly the accelera- tion of p, and as it is simply proportional to x we shall denote it for brevity by px. To compute the periodic time T of a complete vibration, we have the equation M= (TH-) > which gives T=^_. (l) V M 110. Application to the Pendulum. For the motion of a pendulum in a small arc, we have acceleration = | , where s denotes the displacement in linear measure. We must therefore put M = j, and we then have (2) which is the expression for the time of a complete (or double) vibra- tion. It is more usual to understand by the " time of vibration " of a pendulum the half of this, that is the time from one extreme position to the other, and to denote this time by T. In this sense we have T = ,r A /-. (3) To find the length of the seconds' pendulum we must put T = This gives If g were 987 we should have 1=100 centimetres or 1 metre. The actual value of g is everywhere a little less than this. The length of the seconds' pendulum is therefore everywhere rather less than a metre. 111. Simple Harmonic Motion. Rectilinear motion consisting of vibration about a point with acceleration fix, where x denotes 66 THE PENDULUM. distance from this point, is called Simple Harmonic Motion, or Simple Harmonic Vibration. The above investigation shows that Q such vibration is isochronous, its period being 7^- whatever the amplitude may be. To understand the reason of this isochronism we have only to remark that, if the amplitude be changed, the velocity at correspond- ing points (that is, points whose distances from the middle point are the same fractions of the amplitudes) will be changed in the same ratio. For example, compare two simple vibrations in which the values of /* are the same, but let the amplitude of one be double that of the other. Then if we divide the paths of both into the same number of small equal parts, these parts will be twice as great for the one as for the other; but if we suppose the two points to start simultaneously from their extreme positions, the one will constantly be moving twice as fast as the other. The number of parts described in any given time will therefore be the same for both. In the case of vibrations which are not simple, it is easy to see (from comparison with simple vibration) that if the acceleration in- creases in a greater ratio than the distance from the mean position, the period of vibration will be shortened by increasing the amplitude; but if the acceleration increases in a less ratio than the distance, as in the case of the common pendulum vibrating in an arc of moderate extent, the period is increased by increasing the amplitude. 112. Experimental Investigation of the Motion of Pendulums. The preceding investigation applies to the simple pendulum; that is to say to a purely imaginary existence; but it can be theoretically demonstrated that every rigid body vibrating about a horizontal axis under the action of gravity (friction and the resistance of the air being neglected), moves in the same manner as a simple pendu- lum of determinate length called the equivalent simple pendulum. Hence the above results can be verified by experiments on actual pendulums. The discovery of the experimental laws of the motion of pendu- lums was in fact long anterior to the theoretical investigation. It was the earliest and one of the most important discoveries of Galileo, and dates from the year 1582, when he was about twenty years of age. It is related that on one occasion, when in the cathedral of Pisa, he was struck with the regularity of the oscilla- tions of a lamp suspended from the roof, and it appeared to him CYCLOIDAL PENDULUM. 67 that these oscillations, though diminishing in extent, preserved the same duration. He tested the fact by repeated trials, which con- firmed him in the belief of its perfect exactness. This law of isochronism can be easily verified. It is only necessary to count the vibrations which take place in a given time with different amplitudes. The numbers will be found to be exactly the same. This will be found to hold good even when some of the vibrations compared are so small that they can only be observed with a telescope. By employing balls suspended by threads of different lengths, Galileo discovered the influence of length on the time of vibration. He ascertained that when the length of the thread increases, the time of vibration increases also; not, however, in proportion to the length simply, but to its square root. 113. Cycloidal Pendulum. It is obvious from 64 that the effective component of gravity upon a particle resting on a smooth inclined plane is proportional to the sine of the inclination. The accelera- tion of a particle so situated is in fact g sin , if a denote the inclina- tion of the plane. When a particle is guided along a smooth curve its acceleration is expressed by the same formula, a now denoting the inclination of the curve at any point to the horizon. This inclina- tion varies from point to point of the curve, so that the acceleration g sin a is no longer a constant quantity. The motion of a common pendulum corresponds to the motion of a particle which is guided to move in a circular arc; and if x denote distance from the lowest point, measured along the arc, and r the radius of the circle (or the length of the pendulum), the acceleration at any point is g sin - This is sensibly proportional to x so long as x is a small fraction of r; but in general it is not proportional to x, and hence the vibra- tions are not in general isochronous. To obtain strictly isochronous vibrations we must substitute for the circular arc a curve which possesses the property of having an inclination whose sine is simply proportional to distance measured along the curve from the lowest point. The curve which possesses this property is the cycloid. It is the curve which is traced by a point in the circumference of a circle which rolls along a straight- line. The cycloidal pendulum is constructed by suspending an ivory ball or some other small heavy body by a thread between two cheeks (Fig. 47), on which the thread winds as the ball swings to 68 THE PENDULUM. Fig 47. Cycloidal Pendulum. either side. The cheeks must themselves be the two halves of a cycloid whose length is double that of the thread, so that each cheek has the same length as the thread. It can be demonstrated 1 that under these circumstances the path of the ball will be a cycloid identical with that to which the cheeks belong. Ne- glecting friction and the rigidity of the thread, the acceleration in this case is proportional to dis- tance measured along the cycloid from its lowest point, and hence the time of vibration will be strictly the same for large as for small amplitudes. It will, in fact, be the same as that of a simple pendulum having the same length as the cycloidal pendulum and vibrating in a small arc. Attempts have been made to adapt the cycloidal pendulum to clocks, but it has been found that, owing to the greater amount of friction, its rate was less regular than that of the common pendu- lum. It may be remarked, that the spring by which pendulums are often suspended has the effect of guiding the pendulum bob in a curve which is approximately cycloidal, and thus of diminishing the irregularity of rate resulting from differences of amplitude. 114. Moment of Inertia. Just as the mass of a body is the measure of the force requisite for producing unit acceleration when the movement is one of pure translation; so the 'moment of inert in of a rigid body turning about a fixed axis is the measure of the couple requisite for producing unit acceleration of angular velocity. We suppose angle to be measured by so that the anle turned by the body is equal to the arc described by any point of it divided by the distance of this point from the axis; and the angular velocity of the body will be the velocity of any point divided by its distance from the axis. The moment of inertia of the body round the axis is numerically equal to the couple which would produce unit change of angular velocity in the body in unit time. We shall now show how to express the moment of inertia in terms of the masses of the particles of the body and their distances from the axis. Since the evolute of the cycloid is an equal cycloid. MOMENT OF INERTIA. 69 Let m denote the mass of any particle, r its distance from the axis, and the angular acceleration. Then 1 ^ is the acceleration of the particle m, and the force which would produce this acceleration by acting directly on the particle along the line of its motion is ni'i'ty. The moment of this force round the axis would be mr 9 <}> since its arm is r. The aggregate of all such moments as this for all the particles of the body is evidently equal to the couple which actually produces the acceleration of the body. Using the sign S to denote " the sum of such terms as," and observing that is the same for the whole body, we have Applied couple = 2 (mr*) = 02 (mr 2 ). (1) When $ is unity, the applied couple will be equal to 2 (mr~), which is therefore, by the foregoing definition, the moment of inertia of the body round the axis. 115. Moments of Inertia Round Parallel Axes. The moment .of inertia round an axis through the centre of mass is always less than that round any parallel axis. For if r denote the distance of the particle m from an axis not passing through the centre of mass, and x and y its distances from two mutually rectangular planes through this axis, we have r z =x z +y z . Now let two planes parallel to these be drawn through the centre of mass; let and ? be the distances of m from them, and p its distance from their line of intersection, which will clearly be parallel to the given axis. Also let a and b be the distances respectively between the two pairs of parallel planes, so that a 2 + 6 2 will be the square of the distance between the two parallel axes, which distance we will denote by h. Then we have x = g a y = 17 b x* - a 2 + f 2 2a f - 6 3 + if 26 17. 2 (mi*) - 2 {m (a 2 + 6 2 )J- + 2 |m (? + 77')} 2a 2 (m) 2b 2 (mij) - W 2m + 2 (mp 2 ) 2a 1 2m 26~^ 2m. where and i? are the values of 4 and i? for the centre of mass. But these values are both zero, since the centre of mass lies on both the planes from which and TJ are measured. We have therefore 2 (mr 3 ) - A 2 2? ft + 2 (mp 2 ), (2) that is to say, the moment of inertia round the given axis exceeds the moment of inertia round the parallel axis through the centre of 70 THE PENDULUM. mass by the product of the whole mass into the square of the dis- tance between the axes. 116. Application to Compound Pendulum. The application of this principle to the compound pendulum leads to some results of great interest and importance. Let M be the mass of a compound pendulum, that is, a rigid body free to oscillate about a fixed horizontal axis. Let h, as in the preceding section, denote the distance of the centre of mass from this axis; let denote the inclination of h to the vertical, and the angular acceleration. Then, since the forces of gravity on the body are equivalent to a single force M(/, acting vertically downwards at the centre of mass, and therefore having an arm h sin with respect to the axis, the moment of the applied forces round the axis is ~M.gh sin 0; and this must, by 114, be equal to ^2 (mr 2 ). We have therefore S (mr 3 ) _ g sin , g , MA $ ' If the whole mass were collected at one point at distance I from the axis, this equation would become MZ 2 _ , _ g sin ,,, Ml ~ and the angular motion would be the same as in the actual case if I had the value l - ^*. (5) - MA I is evidently the length of the equivalent simple pendulum. 117. Convertibility of Centres. Again, if we introduce a length k such that M/o 2 is equal to S (wp 2 ), that is, to the moment of inertia round a parallel axis through the centre of mass, we have Fig. 48. S (m?- 2 ) = 2 (m/> 2 ) + A 2 2m - M 2 + MA 3 , and equation (5) becomes P + A 2 t , 1 = *- = T + h ' or P = (I - A) A. . (7) In the annexed figure (Fig. 48) which represents a vertical section through the centre of niass, let G be the centre of mass, A the "centre COMPOUND PENDULUM. 71 of suspension," that is, the point in which the axis cuts the plane of the figure, and O the " centre of oscillation," that is, the point at which the mass might be collected without altering the movement. Then, by definition, we have I - AO, h = AG, therefore I - h, - GO, so that equation (7) signifies P = AG . GO. (8) Since k 2 is the same for all parallel axes, this equation shows that when the body is made to vibrate about a parallel axis through 0, the centre of oscillation will be the point A. That is to say; the centres of suspension and oscillation are interchangeable, and the product of their distances from the centre of mass is k 2 . 118. If we take a new centre of suspension A' in the plane of the figure, the new centre of oscillation 0' will lie in the production of A'G, and we must have A'G . GO' = F = AG . GO. If A'G be equal to AG, GO' will be equal to GO, and A'O' to AO, so that the length of the equivalent simple pendulum will be un- changed. A compound pendulum will therefore vibrate in the same time about all parallel axes which are equidistant from the centre of mass. When the product of two quantities is given, their sum is least when they are equal, and becomes continually greater as they depart further from equality. Hence the length of the equivalent simple pendulum AO or AG + GO is least when AG = GO = k, and increases continually as the distance of the centre of suspen- sion from G is either increased from k to infinity or diminished from k to zero. Hence, when a body vibrates about an axis which passes very nearly through its centre of gravity, its oscillations are exceed- ingly slow. 119. Eater's Pendulum. The principle of the convertibility of centres, established in 117, was discovered by Huygens, and affords the most convenient practical method of constructing a pendulum of known length. In Rater's pendulum there are two parallel knife-edges about either of which the pendulum can be made to vibrate, and one of them can be adjusted to any distance 72 THE PENDULUM. from the other. The pendulum is swung first upon one of these edges and then upon the other, and, if any difference is detected in the times of vibration, it is corrected by moving the adjustable edge. When the difference has been completely destroyed, the distance between the two edges is the length of the equivalent simple pendu- lum. It is necessary, in any arrangement of this kind, that the two knife-edges should be in a plane passing through the centre of gravity; also that they should be on opposite sides of the centre of gravity, and at unequal distances from it. 120. Determination of the Value of g. Returning to the formula for the simple pendulum T = v A/-> we easily deduce from it g = ^, whence it follows that the value of g can be determined by making a pendulum vibrate and measuring T and I. T is determined by counting the number of vibrations that take place in a given time; I can be calculated, when the pendulum is of regular form, by the aid of formulae which are given in treatises on rigid dynamics, but its value is more easily obtained by Kater's method, described above, founded on the principle of the convertibility of the centres of suspension and oscillation. It is from pendulum observations, taken in great numbers at different parts of the earth, that the approximate formula for the intensity of gravity which we have given at 91 has been deduced. Local peculiarities prevent the possibility of laying down any general formula with precision; and the exact value of g for any place can only be ascertained by observations on the spot. CHAPTER IX. CONSERVATION OF ENERGY. 121. Definition of Kinetic Energy. We have seen in 93 that the work which must be done upon a mass of m grammes to give it a velocity of v centimetres per second is |mv 2 ergs. Though we have proved this only for the case of falling bodies, with gravity as the working force, the result is true universally, as is shown in advanced treatises on mathematical physics. It is true whether the motion be rectilinear or curvilinear, and whether the working force act in the line of motion or at an angle with it. If the velocity of a mass increases from v 1 to v. 2 , the work done upon it in the interval is |m (v^v^); in other words, is the increase of |-mv 2 . On the other hand, if a force acts in such a manner as to oppose the motion of a moving mass, the force will do negative work, the amount of which will be equal to the decrease in the value of |mt> 2 . For example, during any portion of the ascent of a projectile, the diminution in the value of ^mv z is equal to gm multiplied by the increase of height ; and during any portion of its descent the increase in |mv 2 is equal to gm multiplied by the decrease of height. The work which must have been done upon a body to give it its actual motion, supposing it to have been initially at rest, is called the energy of motion or the kinetic energy of the body. It can be computed by multiplying half the mass by the square of the velocity. 122. Definition of Static or Potential Energy. When a body of mass m is at a height s above the ground, which we will suppose level, gravity is ready to do the amount of work gms upon it by making it fall to the ground. A body in an elevated position may therefore be regarded as a reservoir of work. In like manner a wound-up clock, whether driven by weights or by a spring, has 74 CONSERVATION OF ENERGY. work stored up in it. In all these cases there is force between parts of a system tending to produce relative motion, and there is room for such relative motion to take place. There is force ready to act, and space for it to act through. Also the force is always the same in the same relative position of the parts. Such a system possesses energy, which is usually called potential. We prefer to call it statical, inasmuch as its amount is computed on statical principles alone. 1 Statical energy depends jointly on mutual force and relative position. Its amount in any given position is the amount of work which would be done by the forces of the system in passing from this position to the standard position. When we are speaking of the energy of a heavy body in an elevated position above level ground, we naturally adopt as the standard position that in which the body is lying on the ground. When we speak of the energy of a wound-up clock, we adopt as the standard position that in which the clock has completely run down. Even when the standard position is not indicated, we can still speak definitely of the differ- ence between the energies of two given positions of a system; just as we can speak definitely of the difference of level of two given points without any agreement as to the datum from which levels are to be reckoned. 123. Conservation of Mechanical Energy. When a frictionless system is so constituted that its forces are always the same in the same positions of the system, the amount of work done by these forces during the passage from one position A to another position B will be independent of the path pursued, and will be equal to minus the work done by them in the passage from B to A. The earth and any heavy body at its surface constitute such a system; the force of the system is the mutual gravitation of these two bodies; and the work done by this mutual gravitation, when the body is moved by any path from a point A to a point B, is equal to the weight of the body multiplied by the height of A above B. When the system passes through any series of movements beginning with a given position and ending with the same position again, the algebraic total of work done by the forces of the system in this series of movements is zero. For instance, if a heavy body be carried by a roundabout path back to the point from whence it started, no work is done upon it by gravity upon the whole. Every position of such a system has therefore a definite amount 1 That is to say, the computation involves no reference to the laws of motion. TRANSFORMATIONS OF ENERGY. 75 of statical energy, reckoned with respect to an arbitrary standard position. The work done by the forces of the system in passing from one position to another is (by definition) equal to the loss of static energy; but this loss is made up by an equal gain of kinetic energy. Conversely if kinetic energy is lost in passing from one position to another, the forces do negative work equal to this loss, and an equal amount of static energy is gained. The total energy of the system (including both static and kinetic) therefore remains unaltered. An approximation to such a state of things is exhibited by a pendulum. In the two extreme positions it is at rest, and has there- fore no kinetic energy; but its statical energy is then a maximum. In the lowest position its motion is most rapid; its kinetic energy is therefore a maximum, but its statical energy is zero. The difference of the statical energies of any two positions, will be the weight of the pendulum multiplied by the difference of levels of its centre of gravity, and this will also be the difference (in inverse order) between the kinetic energies of the pendulum in these two positions. As the pendulum is continually setting the air in motion and thus doing external work, it gradually loses energy and at last comes to rest, unless it be supplied with energy from a clock or some other source. If a pendulum could be swung in a perfect vacuum, with an entire absence of friction, it would lose no energy, and would vibrate for an indefinite time without decrease of amplitude. 124. Illustration from Pile-driving. An excellent illustration of transformations of energy is furnished by pile-driving. A large mass of iron called a ram is slowly hauled up to a height of several yards above the pile, and is then allowed- to fall upon it. During the ascent, work must be supplied to overcome the force of gravity; and this work is represented by the statical energy of the ram in its highest position. While falling, it continually loses statical and gains kinetic energy; the amount of the latter which it possesses immediately before the blow being equal to the work which has been done in raising it. The effect of the blow is to drive the pile through a small distance against a resistance very much greater than the weight of the ram; the work thus done being nearly equal to the total energy which the ram possessed at any point of its descent. We say nearly equal, because a portion of the energy of the blow is spent in producing vibrations. 125. Hindrances to Availability of Energy. There is almost 76 CONSERVATION OF ENERGY. always some waste in utilizing energy. When water turns a mill- wheel, it runs away from the wheel with a velocity, the square of which multiplied by half the mass of the water represents energy which has run to waste. Friction again often consumes a large amount of energy; and in this case we cannot (as in the preceding one) point to any palpable motion of a mass as representing the loss. Heat, however, is pro- duced, and the energy which has disappeared as regarded from a gross mechanical point of view, has taken a molecular form. Heat is a form of molecular energy; and we know, from modern re- searches, what quantity of heat is equivalent to a given amount of mechanical work. In the steam-engine we have the converse process; mechanical work is done by means of heat, and heat is destroyed in the doing of it, so that the amount of heat given out by the engine is less than the amount supplied to it. The sciences of electricity and magnetism reveal the existence of other forms of molecular energy; and it is possible in many ways to produce one form of energy at the expense of another; but in every case there is an exact equivalence between the quantity of one kind which comes into existence and the quantity of another kind which simultaneously disappears. Hence the problem of constructing a self-driven engine, which we have seen to be impossible in mechanics, is equally impossible when molecular forms of energy are called to the inventor's aid. Energy may be transformed, and may be communicated from one system to another; but it cannot be increased or diminished in total amount. This great natural law is called the principle of the con- servation of energy. CHAPTEE X. ELASTICITY. 126. Elasticity and its Limits. There is no such thing in nature as an absolutely rigid body. All bodies yield more or less to the action of force; and the property in virtue of which they tend to recover their original form and dimensions when these are forcibly changed, is called elasticity. Most solid bodies possess almost per- fect elasticity for small deformations ; that is to say, when distorted, extended, or compressed, within certain small limits, they will, on the removal of the constraint to which they have been subjected, instantly regain almost completely their original form and dimen- sions. These limits (which are called the limits of elasticity) are different for different substances; and when a body is distorted beyond these limits, it takes a set, the form to which it returns being intermediate between its original form and that into which it was distorted. When a body is distorted within the limits of its elasticity, the force with which it reacts is directly proportional to the amount of distortion. For example, the force required to make the prongs of a tuning-fork approach each other by a tenth of an inch, is double of that required to produce an approach of a twentieth of an inch; and if a chain is lengthened a twentieth of an inch by a weight of 1 cwt., it will be lengthened a tenth of an inch by a weight of 2 cwt., the chain being supposed to be strong enough to experience no permanent set from this greater weight. Also, within the limits of elasticity, equal and opposite distortions, if small, are resisted by equal reactions. For example, the same force which suffices to make the prongs of a tuning-fork approach by a twentieth of an inch, will, if applied in the opposite direction, make them separate by the same amount. 78 ELASTICITY. 127. Isochronism of Small Vibrations. An important consequence of these laws is, that when a body receives a slight distortion within the limits of its elasticity, the vibrations which ensue when the constraint is removed are isochronous. This follows from 111, inasmuch as the accelerations are proportional to the forces, and arc therefore proportional at each instant to the deformation at that instant. 128. Stress, Strain, and Coefficients of Elasticity. A body which, like indian-rubber, can be subjected to large deformations without receiving a permanent set, is said to have wide limits of elasticity. A body which, like steel, opposes great resistance to deformation, is said to have large coefficients of elasticity. Any change in the shape or size of a body produced by the appli- cation of force to the body is called a strain; and an action of force tending to produce a strain is called a stress. When a wire of cross-section A is stretched with a force F, the pi longitudinal stress is r; this being the intensity of force per unit area with which the two portions of the wire separated by any cross-section are pulling each other. If the length of the wire when unstressed is L and when stressed L + , the lonitudinal strain is . A stress is always expressed in units of force per unit of area. A strain is always expressed as the ratio of two magnitudes of the same kind (in the above example, two lengths), and is therefore independent of the units employed. The quotient of a stress by the strain (of a given kind) which it produces, is called a coefficient or 'modulus of elasticity. In the above example, the quotient -^ is called Young's modulus of elasticity. As the wire, while it extends lengthwise, contracts laterally, there will be another coefficient of elasticity obtained by dividing the longitudinal stress by the lateral strain. It is shown, in special treatises, that a solid substance may have 21 independent coefficients of elasticity; but that when the substance is isotropic, that is, has the same properties in all directions, the number reduces to 2. 129. Volume-elasticity. The only coefficient of elasticity possessed by liquids and gases is elasticity of volume. When a body of volume V is reduced by the application of uniform normal pressure over its whole surface to volume V v, the volume-strain is =, and if this COEFFICIENTS OF ELASTICITY. effect is produced by a pressure of p units of force per unit of area, the elasticity of volume is the quotient of the stress p by the strain y, or is ~. This is also called the resistance to compression'-, and its reciprocal ^ is called the compressibility of the substance. In dealing with gases, p must be understood as a pressure super- added to the original pressure of the gas. Since a strain is a mere numerical quantity, independent of units, a coefficient of elasticity must be expressed, like a stress, in units of force per unit of area. In the C.G.S. system, stresses and coefficients of elasticity are expressed in dynes per square centimetre. The following are approximate values (thus expressed) of the two co- efficients of elasticity above defined: Young's Elasticity of Modulus. Volume. Glass (flintl, 60 x 10 10 40 x 10 10 Steel, 210 x 10 10 180 x 10 Iron (wrought), 190 x 10 10 140 x 10 10 Iron (cast), 130 x 10 10 96 x 10 i0 Copper, 120 x 10 l 160 x 10 l Mercury, 54 x 10 10 Water, 2 x 10 10 Alcohol, 1-2 xlO 10 130. (Ersted's Piezometer. The compression of liquids has been observed by means of (Ersted's piezometer, which is represented in Fig. 49. The liquid whose compression is to be observed is contained in a glass vessel b, resembling a thermometer with a very large bulb and short tube. The tube is open above, and a globule of mercury at the top of the liquid column serves as an index. This apparatus is placed in a very strong glass Vessel a full of water. When pressure is exerted by means of the piston Jdh, the index of mercury is seen to descend, showing a diminution of volume of the liquid, and showing moreover that this diminution of volume exceeds that of the containing vessel 6. It might at first Fig; 49. (Ersted's Piezometer. 80 ELASTICITY. sight appear that since this vessel is subjected to equal pressure within and without, its volume is unchanged; but in fact, its volume is altered to the same extent as that of a solid vessel of the same material; for the interior shells would react with a force precisely equivalent to that which is exerted by the contained liquid. CHAPTER XL FRICTION. 131. Friction, Kinetical and Statical. When two bodies are pressed together in such a manner that the direction of their mutual pressure is not normal to the surface of contact, the pressure can be resolved into two parts, one normal and the other tangential. The tangential component is called the force of friction between the two bodies. The friction is called kinetical or statical according as the bodies are or are not sliding one upon the other. As regards kinetical friction, experiment shows that if the normal pressure between two given surfaces be changed, the tangential force changes almost exactly in the same proportion; in other words, the ratio of the force of friction to the normal pressure is nearly constant for two given surfaces. This ratio is called the coefficient of kinetical friction between the two surfaces, and is nearly independent of the velocity. 132. Statical Friction. Limiting Angle. It is obvious that the statical friction between two given surfaces is zero when their mutual pressure is normal, and increases with the obliquity of the pressure if the normal component be preserved constant. The obliquity, however, cannot increase beyond a certain limit, depending on the nature of the bodies, and seldom amounting to so much as 45. Be- yond this limit sliding takes place. The limiting obliquity, that is, the greatest angle that the mutual force can make with the normal, is called the limiting angle of friction for the two surfaces; and the ratio of the tangential to the normal component when the mutual force acts at the limiting angle, is called the coefficient of statical friction for the two surfaces. The coefficient and limiting angle remain nearly constant when the normal force is varied. The coefficient of statical friction is in almost every case greater 6 82 FRICTION. than the coefficient of kinetical friction; in other words, friction offers more resistance to the commencement of sliding than to the continuance of it. A body which has small coefficients of friction with other bodies is called slippery. 133. Coefficient^ tan 0. Inclined Plane. If 6 be the inclination of the mutual force P to the common normal, the tangential com- ponent will be P sin 6, the normal component P cos 6, and the ratio of the former to the latter will be tan 6. Hence the coefficient of statical friction is equal to the tangent of the limiting angle, of friction. When a heavy body rests on an inclined plane, the mutual pressure is vertical, and the angle 6 is the same as the inclination of the plane. Hence if an inclined plane is gradually tilted till a body lying on it slides under the action of gravity, the inclination of the plane at which sliding begins is the limiting angle of friction between the body and the plane, and the tangent of this angle is the coefficient of statical friction. Again, if the inclination of a plane be such that the motion of a body sliding down it under the action of gravity is neither accelerated nor retarded, the tangent of this inclination will be the coefficient of kinetical friction. CHAPTER XII HYDROSTATICS. 134. Hydrodynamics. We shall now treat of the laws of force as applied to fluids. This branch of the general science of dynamics is called hydrodynamics (vdwp, water), and is divided into hydrostatics and hydrokinetics. Our discussions will be almost entirely confined to hydrostatics. FLUIDS. TRANSMISSION OF PRESSURE. The name fluid comprehends both liquids and gases. 135. No Statical Friction in Fluids. A fluid at rest cannot exert any tangential force against a surface in contact with it; its pressure at every point of such a surface is entirely normal. A slight tangen- tial force is exerted by fluids in motion; and this fact is expressed by saying that all fluids are more or less viscous. An imaginary perfect fluid would be perfectly free from viscosity; its pressure against any surface would be entirely normal, whether the fluid were in motion or at rest. 136. Intensity of Pressure. When pressure is uniform over an area, the total amount of the pressure, divided by the area, is called the intensity of the pressure. The C.G.S. unit of intensity of pressure is a pressure of a dyne on each square centimetre of sur- face. A rough unit of intensity frequently used is the pressure of a pound per square inch. This unit varies with the intensity of gravity, and has an average value of about 69,000 C.G.S. units. Another rough unit of intensity of pressure frequently employed is " an atmosphere " that is to say, the average intensity of pressure of the atmosphere at the surface of the earth. This is about 1,000,000 C.G.S. units. 84 HYDROSTATICS. The single word " pressure " is used sometimes to denote " amount of pressure" (which can be expressed in dynes) and sometimes " intensity of pressure" (which can be expressed in dynes per square centimetre). The context usually serves to show which of these two meanings is intended. 137. Pressure the Same in all Directions. The intensity of pressure at any point of a fluid is the same in all directions; it is the same whether the surface which receives the pressure faces upwards, downwards, horizontally, or obliquely. This equality is a direct consequence of the absence of tangential force between two contiguous portions of a fluid. For in order that a small triangular prism of the fluid (its ends being right sections) may be in equilibrium, the pressures on its three faces must balance each other. But when three forces balance each other, they are proportional to the sides of a triangle to which they are perpendicular; 1 hence the amounts of pressure on the three faces are proportional to the faces, in other words the inten- sities of these three pressures are equal. As we can take two of the faces perpendicular to any two given directions, this proves that the pressures in all directions at a point are of equal intensity. 138. Pressure the Same at the Same Level- In a fluid at rest, the pressure is the same at all points in the same horizontal plane. This appears from considering the equilibrium of a horizontal cylinder AB (Fig. 50), of small sectional area, its ends being right sections. The pressures on the sides are normal, and therefore give no component in the direction Flg>5 - of the length; hence the pressures on the ends must be equal in amount; but they act on equal areas; there- fore their intensities are equal. A horizontal surface in a liquid at rest may therefore be called a " surface of equal pressure." 139. Difference of Pressure at Different Levels. The increase of pressure with depth, in a fluid of uniform density, can be investi- gated as follows: Consider the equilibrium of a vertical cylinder mm' (Fig. 51), its ends being right sections. The pressures on its 1 This is an obvious consequence of the triangle of forces (art. 14) ; for if the sides of a triangle are parallel to three forces, we have only to turn the triangle through a right angle, and its sides will then be perpendicular to the forces. INCREASE OF PRESSURE WITH DEPTH. 85 Fig. 51. sides are normal, and therefore horizontal. The only vertical forces acting upon it are its own weight and the pressures on its ends, of which it is to be observed that the pressure on the upper end acts downwards and that on the lower end upwards. The pressure on the lower end therefore exceeds that on the upper end by an amount equal to the weight of the cylinder. If a be the sectional area, w the weight of unit volume of the liquid, and h the length of the cylinder, the volume of the cylinder is ha, and its weight wha, which must be equal to (pp) a if p,p' are the intensities of pressure on the lower and upper ends respectively. We have therefore p-p' wh, that is, the increase of pressure in descending through a depth h is wh. The principles of this and the preceding section remain appli- cable whatever be the shape of the containing vessel, even if it be such as to render a circuitous route necessary in passing from one of two points compared to the other; for this route can always be made to consist of a succession of vertical and horizontal lines, and the preceding principles when applied to each of these lines separ- ately, will give as the final result a difference of pressure wh for a difference of heights h. If d denote the density of the liquid, in grammes per sq. cm., the weight of a cubic cm. will be gd dynes. The increase of pressure for an increase of depth h cm. is therefore ghd dynes per sq. cm. If there be no pressure at the surface of the liquid, this will be the actual pressure at the depth h. 140. Free Surface. It follows from these principles that the free surface of a liquid at rest that is, the surface in contact with the atmosphere must be horizontal; since all points in this surface are at the same pressure. If the surface were not horizontal, but were higher at n than at n (Fig. 52), the pres- sures at the two points m, m' vertically beneath them in any horizontal plane AB would be unequal, for they would be due to the weights Fig. 52. 80 HYDROSTATICS. of unequal columns nm, n'm, and motion would ensue from m towards m'. The same conclusion can be deduced from considering the equili- brium of a particle at the surface, as M (Fig. 53). If the tangent plane at M were not horizontal there would be a component of gravity tending to make the particle slide down; and this tendency would produce motion, since there is no fric- tion to oppose it. 141. Transmissibility of Pressure in Fluids. Since the difference of the pressures at two points in a fluid can be determined by the foregoing prin- ciples, independently of any knowledge of the absolute intensity of either, it follows that when increase or diminution of pres- sure occurs at one point, an equal increase or diminution must occur throughout the whole fluid. A fluid in a closed vessel perfectly transmits through its whole substance whatever pressure we apply to any part. The changes in amount of pressure will be equal for all equal areas. For unequal areas they will be propor- tional to the areas. Thus if the two vertical tubes in Fig. 54 have sectional areas which are as 1 to 16, a weight of 1 kilo- Fig 53. JET. 16 gram acting on the surface of the liquid in the smaller tube will be balanced by 16 kilograms acting on the surface of the liquid in the larger. This principle of the perfect transmis- sion of pressure by fluids appears to have been first discovered and published by Stevinus; but it was rediscovered by Pascal a few years later, and having been made generally known by his writings is often called " Pascal's principle." In his celebrated treatise on the Equilibrium of Liquids, he says, " If a vessel full of water, closed on all sides, has two openings, the one a hundred times as large as the other, and if each be supplied with a piston which fits exactly, a man pushing the small piston will exert a force which will equilibrate that of a hundred men pushing the piston which is a hundred times as large, i'ig. 54. Principle of the Hydraulic Press. TRANSMISSIBILITY OF PRESSURE. 87 and will overcome that, of ninety-nine. And whatever may be the proportion of these openings, if the forces applied to the pistons are to each other as the openings, they will be in equilibrium." 142. Hydraulic Press. This mode of multiplying force remained for a long time practically unavailable on account of the difficulty of making the pistons water-tight. The hydraulic press was first successfully made by Bramah, who invented the cupped leather collar illustrated in Fig. 166, 264. Fig. 165 shows the arrangements of the press as a whole. Instead of pistons, plungers are employed; that is to say, solid cylinders of metal which can be pushed down into the liquid, or can be pushed up by the pressure of the liquid against their bases. The volume of liquid displaced by the advance of a plunger is evidently equal to that displaced by a piston of the same sectional area, and the above calculations for pistons apply to plungers as well. The plungers work through openings which are kept practically water-tight by means of the cup-leather arrange- ment. The cup-leather, w r hich is shown both in plan and section in Fig. 166, consists of a leather ring bent so as to have a semi- circular section. It is fitted into a hollow in the interior of the sides of the opening, so that water leaking up along the circumfer- ence of the plunger will fill the concavity of the leather, and, by pressing on it, will produce a packing which fits more tightly as the pressure on the plunger increases. 143. Principle of Work Applicable. In Fig. 54, when the smaller piston advances and forces the other back, the volume of liquid driven out of the smaller tube is equal to the sectional area multi- plied by the distance through which the piston advances. In like manner, the volume of liquid driven into the larger tube is equal to its sectional area multiplied by the distance that its piston is forced back. But these two volumes are equal, since the same volume of liquid that leaves one tube enters the other. The distances through which the two pistons move are therefore inversely as their sectional areas, and hence are inversely as the amounts of pressure applied to them. The work done in pushing forward the smaller piston is therefore equal to the work done by the liquid in pushing back the larger. This was remarked by Pascal, who says " It is, besides, worthy of admiration that in this new machine we find that constant rule which is met with in all the old ones, such as the lever, wheel and axle, screw, &c., which is that the distance is increased in proportion to the force; for it is evident that 88 HYDROSTATICS. as one of these openings is a hundred times as 'arge as the other, if the man who pushes the small piston drives it forward one inch, he will drive the large piston backward only one-hundredth part of that length." 144. Experiment on Upward Pressure. The upward pressure exerted by a liquid against a horizontal surface facing down- wards can be exhibited by the following experiment. Take a tube open at both ends (Fig. 55), and keeping the lower end covered with a piece of card, plunge it into water. The liquid will press the card against the bottom of the tube with a force which increases as it is plunged deeper. If water be now poured into the tube, the card will remain in its place as long as the level of the liquid is lower within the tube than with- out; but at the moment when equality of levels is attained it will become detached, 145. Liquids in Superposition. When one liquid rests on the top of another of different density, the foregoing principles lead to the result that the surface of demarcation must be horizontal. For the free surface of the upper liquid must, as we have seen, be horizontal. If now we take two small equal areas u and ri (Fig. 56) in a horizontal layer of the lower liquid, they must be subjected to equal pressures. But these pressures are measured by the weights of the liquid cylinders nrs, n'tl; and these latter cannot be equal unless the points r and t at the junction of the two liquids are at the same level. All points in the surface of demarca- tion are therefore in the same horizontal plane. The same reasoning can be extended downwards to any number of liquids of unequal densities, which rest one upon another, and shows that all the surfaces of demarcation between them must be horizontal. Fig. 55. Upward Pressure. Fig. 56. LIQUIDS IN SUPERPOSITION. 89 Fig. 57. Phial of the Four Elements. An experiment in illustration of this result is represented in Fig. 57. Mercury, water, and oil are poured into a glass jar. The mercury, being the heaviest, goes to the bottom; the oil, being the lightest, floats at the top; and the surfaces of contact of the liquids are seen to be horizontal. Even when liquids are employed which gradually mix with one another, as water and alcohol, or fresh water and salt water, so that there is no definite surface of demarcation, but a gradual increase of density with depth, it still remains true that the density at all points in a horizontal plane is the same. 146. Two Liquids in Bent Tube. If we pour mercury into a bent tube open at both ends (Fig. 58), and then pour water into one of the arms, the heights of the two liquids above the surface of junction will be very unequal, as shown in the figure. The general rule for the equilibrium of any two liquids in these circum- stances is that their heights above the surface of junction must be in- versely as their densities, since they correspond to equal pressures. 147. Experiment of Pascal's Vases. Since the amount of pressure on a horizontal area A at the depth h in a liquid is whA., where w denotes the weight of unit volume of the liquid, it follows that the pressure on the bottom of a vessel containing liquid is not affected by the breadth or narrowness of the upper part of the Fig. 58. Equilibrium of Two Fluids in Communicating Vessels. 90 HYDROSTATICS. vessel, provided the height of the free surface of the liquid be given. Pascal verified this fact by an experiment which is frequently ex- hibited in courses of physics. The apparatus employed (Fig. 59) is a tripod supporting a ring, into which can be screwed three vessels of different shapes, one widened upwards, another cylindrical, and the third tapering upwards. Beneath the ring is a movable disc Fig. 59. Experiment of Pascal's Vases. supported by a string attached to one of the scales of a balance. Weights are placed in the other scale in order to keep the disc pressed against the ring. Let the cylindrical vase be mounted on the tripod, and filled up with water to such a level that the pressure is just sufficient to detach the disc from the ring. An indicator, shown in the figure, is used to mark the level at which this takes place. Let the experiment be now repeated with the two other vases, and the disc will be detached when the water has reached the same level as before. In the case of the cylindrical vessel, the pressure on the bottom is evidently equal to the weight of the liquid. Hence in all three PRESSURE ON VESSEL. 91 Fig. 60. Total Pressure. cases the pressure on the bottom of the vessel is equal to the weight of a cylindrical column of the liquid, having the bottom as. its base, and having the same height as the liquid in the vessel. 148. Resultant Pressure on Vessel. The pressure exerted by the bottom of the vessel upon the stand on which it rests, consists of the weight of the vessel itself, together with the resultant pressure of the contained liquid against it. The actual pressure of the liquid against any portion of the vessel is normal to this portion, and if we resolve it into two components, one vertical and the other hori- zontal, only the vertical component need be attended to, in computing the resultant; for the horizontal components will always destroy one another. At such points as n, n' (Fig. 60) the vertical component is downwards; at s and s' it is upwards; at r and r there is no vertical component; and at AB the whole pressure is vertical. It can be demonstrated mathematically that the resultant pressure is always equal to the total weight of the contained liquid; a conclusion which can also be deduced from the consideration that the pressure exerted by the vessel upon the stand on which it rests must be equal to its own weight together with that of its contents. Some cases in which the proof above indicated becomes especially obvious, are represented in Fig. 61. In the cylindrical vessel ABDC, it is evident that the only pressure trans- mitted to the stand is that exerted upon the bottom, which is equal to the weight of the liquid. In the case of the vessel which is wider at the top, the stand is subjected to the weight of the liquid column ABSK, which presses on the bottom AB, together with the columns GHKC, RLDS, pressing on GH and RL; the sum of which weights composes the total weight of liquid contained in the vessel. Finally, in the third case, the pressure on the bottom AB, which is equal to the weight of a liquid column ABSK, must be diminished by the K S 1 F~ =-- i i It ^ T--E- BL i - T KC ABA 61. Hydrostatic Paradox. 92 HYDROSTATICS. upward pressures on HG and RL. These last being represented by liquid columns HGCK, RLSD, there is only left to be transmitted to the stand a pressure equal to the weight of the water in the vessel. 149. Back Pressure in Discharging Vessel. The same analysis which shows that the resultant vertical pressure of a liquid against the containing vessel is equal to the weight of the liquid, shows also that the horizontal components of the pressures destroy one another. This conclusion is in accordance with everyday experience. How- ever susceptible a vessel may be of horizontal displacement, it is not found to acquire any tendency to horizontal motion by being filled with a liquid. When a system of forces are in equilibrium, the removal of one of them destroys the equilibrium, and causes the resultant of the system to be a force equal and opposite to the force removed. Accordingly if we remove an element of one side of the containing vessel, leaving a hole through which the liquid can flow out, the remaining pressure against this side will be insufficient to preserve equilibrium, and there will be an excess of pressure in the opposite direction. This conclusion can be directly verified by the experiment repre- sented in Fig. 62. A tall floating- vessel of water is fitted with a hori- zontal discharge-pipe on one side near its base. The vessel is to be filled with water, and the discharge-pipe opened while the vessel is at rest. As the water flows out, the vessel will be observed to acquire a velocity, at first very slow, but continually increasing, in the opposite direction to that of the issuing stream. This experiment may also be re- garded as an illustration of the law of action and reaction, which asserts that momentum cannot be imparted to any body without equal and opposite momentum being imparted to some other body. The water in escaping from the vessel acquires horizontal momentum in one direction, and the vessel with its remaining contents acquires horizontal momentum in the opposite direction. Fig. 62. Backward Movement of Discharging Vessel. BACKWARD MOVEMENT. 93 The movements of the vessel in this experiment are slow. More marked effects of the same kind can be obtained by means of the hydraulic tourn- iquet (Fig. 63), which when made on a larger scale is called Barker's mill. It consists of a vessel of water free to rotate about a vertical axis, and having at its lower end bent arms through which the water is discharged hori- zontally, the direction of dis- charge being nearly at right angles to a line joining the dis- charging orifice to the axis. The unbalanced pressures at the bends of the tube, opposite to the openings, cause the apparatus to revolve in the opposite direction to the issuing liquid. 150. Total and Resultant Pressures. Centre of Pressure. The intensity of pressure on an area which is not horizontal is greatest on those parts which are deepest, and the average intensity can be shown to be equal to the actual intensity at the centre of gravity of the area. Hence if A denote the area, h the depth of its centre of gravity, and w the weight of unit volume of the liquid, the total pressure will be w Ah. Strictly speaking, this is the pressure due to the weight of the liquid, the transmitted atmospheric pressure being left out of account. In attaching numerical values to w, A, and h, the same unit of length must be used throughout. For example, if h be expressed in feet, A must be expressed in square feet, and w must stand for the weight of a cubic foot of the liquid. When we employ the centimetre as the unit of length, the value XfiMl'L. net. 94 HYDEOSTATICS. of w will be sensibly 1 gramme if the liquid be water, so that the amount of pressure in grammes will be simply the product of the depth of the centre of gravity in centimetres by the area in square centimetres. For any other liquid, the pressure will be found by multiplying this product by the specific gravity of the liquid. These rules for computing total pressure hold for areas of all forms, whether plane or curved; but the investigation of the total pressure on an area which is not plane is a mere mathematical exercise of no practical importance; for as the elementary pressures in this case are not parallel, their sum (which is the total pressure) is not the same thing as their resultant. For a plane area, in whatever position, the elementary pressures, being everywhere normal to its plane, are parallel and give a resul- tant equal to their sum; and it is often a matter of interest to determine that point in the area through which the resultant passes. This point is called the Centre of Pressure. It is not coincident with the centre of gravity of the area unless the pressure be of equal intensity over the whole area. When the area is not hori- zontal, the pressure is most intense at those parts of it which are deepest, and the centre of pressure is accordingly lower down than the centre of gravity. For a horizontal area the two centres are coincident, and they are also sensibly coincident for any plane area whose dimensions are very small in comparison with its depth in the liquid, for the pressure over such an area is sensibly uniform. 151. Construction for Centre of Pressure. If at every point of a plane area immersed in a liquid, a normal be drawn, equal to the depth of the point, the normals will represent the intensity of pressure at the respective points, and the volume of the solid con- stituted by all the normals will represent the total pressure. That normal which passes through the centre of gravity of this solid will be the line of action of the resultant, and will there- fore pass through the centre of pressure. Thus, if RB (Fig. 64) be a rectangular surface (which we may suppose to be the surface of a flood-gate or of the side of a dam), its lower side B beino- at the Fig. 64. Centre of Pressure. ' bottom of the water and its upper side R at the top, the pressure is zero at R and goes on increasing uni- formly to B. The normals B6, Dd, Hh, LI, equal to the depths of a CENTRE OF PRESSURE. 95 series of points in the line BR will have their extremities b, d, h, I, in one straight line. To find the centre of pressure, we must find the centre of gravity of the triangle RB6 and draw a normal through it. As the centre of gravity of a triangle is at one-third of its height, the centre of pressure will be at one-third of the height of BR. It will lie on the line joining the middle points of the upper and lower sides of the rectangle, and will be at one-third of the length of this line from its lower end. The total pressure will be equal to the weight of a quantity of the liquid whose volume is equal to that of the triangular prism constituted by the aggregate of the normals, of which prism the triangle RB6 is a right section. It is not difficult to show that the volume of this prism is equal to the product of the area of the rectangle by the depth of the centre of gravity of the rectangle, in accordance with the rule above given. 152. Whirling Vessel. D'Alembert's Principle. If an open vessel of liquid is rapidly rotated round a vertical axis, the surface of the liquid assumes a concave form, as represented in Fig. 65, where the dotted line is the axis of rota- tion. When the rotation has been going on at a uniform rate for a sufficient time, the liquid mass rotates bodily as if its particles were rigidly connected together, and when this state of things has been attained the form of the surface is that of a paraboloid of revolution, so that the section represented in the figure is a parabola. We have seen in 101 that a particle moving uniformly in a circle is acted on by a force directed towards the centre. In the present case, therefore, there must be a force acting upon each particle of the liquid urging it towards the axis. This force is supplied by the pressure of the liquid, which follows the usual law of increase with depth for all points in the same vertical. If we draw a horizon- Fig es.-Rotating Vessel tal plane in the liquid, the pressure at each point of of Liqmd - it is that due to the height of the point of the surface vertically over it. The pressure is therefore least at the point where the plane is cut by the axis, and increases as we recede from this centre. Consequently each particle of liquid receives unequal pressures on two opposite sides, being more strongly pressed towards the axis than from it. 96 HYDROSTATICS. Another mode of discussing the case, is to treat it as one of statical equilibrium under the joint action of gravity and a fictitious force called centrifugal force, the latter force being, for each par- ticle, equal and opposite to that which would produce the actual acceleration of the particle. This so-called centrifugal force is therefore to be regarded as a force directed radially outwards from the axis; and by compounding the centrifugal force of each particle with its weight we shall obtain what we are to treat as the resul- tant force on that particle. The form of the surface will then be determined by the condition that at every point of the surface the normal must coincide with this resultant force; just as in a liquid at rest, the normals must coincide with the direction of gravity. The plan here adopted of introducing fictitious forces equal and opposite to those which if directly applied to each particle of a system would produce the actual accelerations, and then applying the conditions of statical equilibrium, is one of very frequent appli- cation, and will always lead to correct results. This principle was first introduced, or at least systematically expounded, by D'Alem- bert, and is known as D'Alembert's Principle. CHAPTER XIII. PRINCIPLE OF ARCHIMEDES. 153. Pressure of Liquids on Bodies Immersed. When a body is immersed in a liquid, the different points of its surface are sub- jected to pressures which obey the rules laid down in the preceding chapter. As these pressures increase with the depth, those which tend to raise the body exceed those which tend to sink it, so that the resultant effect is a force in the direction opposite to that of gravity. By resolving the pressure on each element into horizontal and vertical components, it can be shown that this resultant upward force is exactly equal to the weight of the liquid displaced by the body. The reasoning is particularly simple in the case of a right cylinder (Fig. 66) plunged vertically in a liquid. It is evident, in the first place, that if we consider any point on the sides of the cylinder, the normal pressure on that point is horizontal and is destroyed by the equal and contrary pressure at the point dia- metrically opposite; hence, the horizontal pres- sures destroy each other. As regards the vertical pressures on the ends, one of them, that on the upper end AB, is in a downward direction, and equal to the weight of the liquid column ABNN; the other, that on the lower end CD, is in an upward direction, and equal to the weight of the liquid column CNND ; this latter pressure exceeds the former by the weight of the liquid cylinder ABDC, so that the resultant effect of the pressure is to raise the body with a force equal to the weight of the liquid displaced. v \ T Fig. 66. Principle of Archimedes. 98 PRINCIPLE OF ARCHIMEDES. By a synthetic process of reasoning, we may, without having recourse to the analysis of the different pressures, show that this conclusion is perfectly general. Suppose we have a liquid mass in equilibrium, and that we consider specially the portion M (Fig. 67) ; this portion is likewise in equilibrium. If we suppose it to become solid, without any change in its weight or volume, equilibrium will still subsist. Now this is a heavy mass, and as it does not fall, we must conclude that the effect of the pressures on its surface is to produce a resultant upward pressure exactly equal to Fig. er. Principle of its weight, and acting in a line which passes Archimedes. ' . through its centre of gravity. If we now suppose M replaced by a body exactly occupying its place, the exterior pressures will remain the same, and their resultant effect will therefore be the same. The name centre of buoyancy is given to the centre of gravity of the liquid displaced, that is, if the liquid be uniform, to the centre of gravity of the space occupied by the immersed body; and the above reasoning shows that the resultant pressure acts vertically upwards in a line which passes through this point. The results of the above explanations may thus be included in the following pro- position: Every body immersed in a liquid is subjected to a resul- tant pressure equal to the weight of the liquid displaced, and acting vertically upwards through the centre of buoyancy. This proposition constitutes the celebrated principle of Archimedes. The first part of it is often enunciated in the following form : Every body immersed in a liquid loses a portion of its ^ueight equal to the weight of the liquid displaced; for when a body is immersed in a liquid, the force required to sustain it will evidently be diminished by a quantity equal to the upward pressure. 154. Experimental Demonstration of the Principle of Archimedes. The following experimental demonstration of the principle of Archi- medes is commonly exhibited in courses of physics : From one of the scales of a hydrostatic balance (Fig. 68) is sus- pended a hollow cylinder of brass, and below this a solid cylinder, whose volume is equal to the interior volume of the hollow cylinder; these are balanced by weights in the other scale. A vessel of water is then placed below the cylinders, in such a position that the lower cylinder shall be immersed in it. The equilibrium is immediately EXPERIMENTAL PROOF. 99 destroyed, and the upward pressure of the water causes the scale with the weights to descend. If we now pour water into the hollow cylinder, equilibrium will gradually be re-established; and the beam Z.LAPt-AHTt. Fig. 68. Experimental Verification of Principle of Archimedes. will be observed to resume its horizontal position when the hollow cylinder is full of water, the other cylinder being at the same time completely immersed. The upward pressure upon this latter is thus equal to the weight of the water added, that is, to the weight of the liquid displaced. 155. Body Immersed in a Liquid. It follows from the principle of Archimedes that when a body is immersed in a liquid, it is subjected to two forces: one equal to its weight and applied at its centre of gravity, tending to make the body descend; the other equal to the weight of the displaced liquid, applied at the centre of buoyancy, and tending to make it rise. There are thus three different cases to be considered : (1.) The weight of the body may exceed the weight of the liquid displaced, or, in other words, the mean density of the body may be 100 PRINCIPLE OF ARCHIMEDES. greater than that of the liquid; in this case, the body sinks in the liquid, as, for instance, a piece of lead dropped into water. (2.) The weight of the body may be less than that of the liquid displaced; in this case the body will not remain submerged unless forcibly held down, but will rise partly out of the liquid, until the weight of the liquid displaced is equal to its own weight. This is what happens, for instance, if we immerse a piece of cork in water and leave it to itself. (3.) The weight of the body may be equal to the weight of the liquid displaced; in this case, the two opposite forces being equal, the body takes a suitable position and remains in equilibrium. These three cases are exemplified in the three following experi- ments (Fig. 69): (1.) An egg is placed in a vessel of water; it sinks to the bottom Fig. 69. Egg Plunged in Fresh and Salt Water. of the vessel, its mean density being a little greater than that of the liquid. (2.) Instead of fresh water, salt water is employed; the egg floats at the surface of the liquid, which is a little denser than it. (3.) Fresh water is carefully poured on the salt water; a mixture of the two liquids takes place where they are in contact; and if the egg is put in the upper part, it will be seen to descend, and, after a few oscillations, remain at rest at such a depth that it displaces its own weight of the liquid. In speaking of the liquid displaced in this case, we must imagine each horizontal layer of liquid surrounding the egg to be produced through the space which the egg occupies; and by the centre of buoyancy we must understand the centre of "LIQUID DISPLACED" DEFINED. 101 gravity of the portion of liquid which would thus take the place of the egg. We may remark that, in this position the egg is in stable equilibrium; for, if it rises, the upward pressure diminish- ing, its weight tends to make it descend again jiiiV pa. "the contrary, it sinks, the pressure increases and tends to .analf e it reascen(L , * ; ; 156. Cartesian Diver. The experiment ^^\!k4te#i^*fato&, which is described in old treatises on physics, shows each of the different cases that can present themselves when a body is immersed. The diver (Fig. 70) consists of a hollow ball, at the bottom of which is a small opening 0; a little porcelain figure is attached to the ball, and the whole floats upon water contained in a glass vessel, the mouth of which is closed by a strip of caoutchouc or a blad- der. If we press with the hand on the bladder, the air is compressed, and the pressure, trans- mitted through the different horizontal layers, condenses the air in the ball, and causes the entrance of a portion of the liquid by the open- ing O; the floating body becomes heavier, and in consequence of this increase of weight the diver descends. When we cease to press upon the bladder, the pressure becomes what it was before, some water flows out and the diver ascends. It must be observed, however, that as the diver continues to descend, more and more water enters the ball, in conse- quence of the increase of pressure, so that if the depth of the water exceeded a certain limit, the diver would not be able to rise again from the bottom. Fig. 70. Cartesian Diver. 102 PRINCIPLE OF ARCHIMEDES. If we suppose that at a certain moment the weight of the diver becomes exactly equal to the weight of an equal volume of the liquid, there, will be equilibrium ; but, unlike the equilibrium in the experi- ment {3) of \>at;;,sectiyn, this will evidently be unstable, for a slight \ movement either, upwards or downwards will alter the resultant force' so as to- produce further movement in the same direction. As a consequence of this instability, if the diver is sent down below a certain depth he will not be able to rise again. 157. Relative Positions of the Centre of Gravity and Centre of Buoyancy. In order that a floating body either wholly or partially immersed in a liquid, may be in equilibrium, it is necessary that its weight be equal to the weight of the liquid displaced. This condition is however not sufficient; we require, in addition, that the action of the upward pressure should be exactly opposite to that of the weight; that is, that the centre of gravity and the centre of buoyancy be in the same vertical line; for if this were not the case, the two contrary forces would compose a couple, the effect of which would evidently be to cause the body to turn. In the case of a body completely immersed, it is further necessary for stable equilibrium that the centre of gravity should be below the centre of buoyancy; in fact we see, by Fig. 71, that in any other Fig. 71. F'g- 72. Relative Positions of Centre of Gravity and Centre of Pressure. position than that of equilibrium, the effect of the two forces applied at the two points G and would be to turn the body, so as to bring the centre of gravity lower, relatively to the centre of buoyancy. But this is not the case when the body is only partially immersed, as most frequently happens. In this case it may indeed happen that, with stable equilibrium, the centre of gravity is below the centre of pressure; but this is not necessary, and in the majority of instances is not the case. Let Fig. 72 represent the lower part of a floating body a boat, for instance. The centre of pressure is at 0, the centre of gravity at G, considerably above; if the body STABILITY OF FLOATATION. 108 is displaced, and takes the position .shown in the figure, it will be seen that the effect of the two forces acting at and at G is to restore the body to its former position. This difference from what takes place when the body is completely immersed, depends upon the fact that, in the case of the floating body, the figure of the liquid displaced changes with the position of the body, and the centre of buoyancy moves towards the side on which the body is more deeply immersed. It will depend upon the form of the body whether this lateral movement of the centre of buoyancy is sufficient to carry it beyond the vertical through the centre of gravity. The two equal forces which act on the body will evidently turn it to or from the original position of equilibrium, according as the new centre of buoyancy lies beyond or falls short of this vertical. 1 158. Advantage of Lowering the Centre of Gravity. Although stable equilibrium may subsist with the centre of gravity above the centre of buoyancy, yet for a body of given external form the stability is always increased by lowering the centre of gravity; as we thus lengthen the arm of the couple which tends to right the body when displaced. It is on this principle that the use of ballast depends. 159. Phenomena in Apparent Contradiction to the Principle of Archimedes. The principle of Archimedes seems at first sight to be contradicted by some well- known facts. Thus, for instance, if small needles are placed carefully on the surface of water, they will remain there in equilibrium (Fig. 73). It is on a similar principle 1 If a vertical through the new centre of buoyancy be drawn upwards to meet that line in the body which in the position of equilibrium was a vertical through the centre of gravity, the point of intersection is called the metacentre. Evidently when the forces tend to restore the body to the position of equilibrium, the metacentre is above the centre of gravity ; when they tend to increase the displacement, it is below. In ships the dis- tance between these two points is usually nearly the same for all amounts of heeling, and this distance is a measure of the stability of the ship. We have defined the metacentre as the intersection of two lines. When these lines lie in different planes, and do not intersect each other, there is no metacentre. This indeed is the case for most of the displacements to which a floating body of irregular shape can be subjected. There are in general only two directions of heeling to which metacentres correspond, and these two directions are at right angles to each other. Fig. 73. Steel Needles Floating on Water. 104 PRINCIPLE OF ARCHIMEDES. Fig. 74. Insect Walking on Water. that several insects walk on water (Fig. 74), and that a great number of bodies of various natures, provided they be very minute, can, if we may so say, be placed on the surface of a liquid with- out penetrating into its interior. These curious facts depend on the circumstance that the small bodies in question are not wetted by the liquid, and hence, in virtue of principles which will be explained in connection with capillarity (Chap, xvi.), depressions are formed around them on the liquid surface, as represented in Fig. 75. The curvature of the liquid surface in the neighbourhood of the body is very distinctly shown by observing the shadow cast by the floating body, when it is illumined by the sun; it is seen to be bordered by luminous bands, which are owing to the refraction of the rays of light in the portion of the liquid bounded by a curved surface. The existence of the depression about the floating body enables us to bring the condition of equilibrium in this special case under the general enunciation of the principle of Archimedes. Let M (Fig. 75) be ~&... _~~^PJ the body, CD the region of the depression, and AB the corresponding portion of any horizontal Fig. 75. layer; since the pressure at each point of AB must be the same as in other parts of the same horizontal layer, the total weight above AB is the same as if M did not exist and the cavity were filled with the liquid itself. We may thus say in this case also that the weight of the floating body is equal to the weight of the liquid displaced, understanding by these words the liquid which would occupy the whole of the depression due to the presence of the body. CHAPTER XIV. DENSITY AND ITS DETERMINATION. 160. Definitions. By the absolute density of a substance is meant the mass of unit volume of it. By the relative density is meant the ratio of its absolute density to that of some standard substance, or, what amounts to the same thing, the ratio of the mass of any volume of the substance in question to the mass of an equal volume of the standard substance. Since equal masses gravitate equally, the com- parison of masses can be effected by weighing, and the relative den- sity of a substance is the ratio of its weight to that of an equal volume of the standard substance. Water at a specified tempera- ture and under atmospheric pressure is usually taken as the standard substance, and the density of a substance relative to water is usually called the specific gravity of the substance. Let Y denote the volume of a substance, M its mass, and D its absolute density; then by definition, we have M=VD. If 8 denote the specific gravity of a substance, and d the absolute density of water in the standard condition, then D sd and M= Vsd When masses are expressed in Ibs. and volumes in cubic feet, the value of d is about 62'4, since a cubic foot of cold water weighs about 62-4 Ibs. 1 In the C.G.S. system, the value of d is sensibly unity, since a cubic centimetre of water, at a temperature which is nearly that of the maximum density of water, weighs exactly a gramme. 2 The gramme is defined, not by reference to water, but by a standard kilogramme of platinum, which is preserved in Paris, and 1 In round numbers, a cubic foot of water weighs 1000 oz., which is 62'5 Ibs. 2 According to the best determination yet published, the mass of a cubic centimetre of pure water at 4 is 1 -000013, at 3 is 1-000004, and at 2 is -999982. 106 DENSITY AND ITS DETERMINATION. of which several very carefully made copies are preserved in other places. In the above statements (as in all very accurate statements of weights), the weighings are supposed to be made in vacuo; for the masses of two bodies are not accurately proportional to their apparent gravitations in air, unless the two bodies happen to have the same density. 161. Ambiguity of the word " Weight." Properly speaking, " the weight of a body " means the force with which the body gravitates towards the earth. This force, as we have seen, differs slightly according to the place of observation. If 7n denote the mass of the body, and g the intensity of gravity at the place, the weight of the body is mg. When the body is carried from one place to another without gain or loss of material, m will remain constant and g will vary; hence the weight mg will vary, and in the same ratio as g. But the employment of gravitation units of force instead of absolute units, obscures this fact. The unit of measurement varies in the same ratio as the thing to be measured, and hence the numerical value remains unaltered. A body weighs the same number of pounds or grammes at one place as at another, because the weights of the pound and gramme are themselves proportional to g. Expressed in absolute units, the weight of unit mass is g, and the weight of a mass m is mg. The latter is m times the former; hence when the weight of unit mass is employed as the unit of weight, the same number m which denotes the mass of a body also denotes its weight. What are usually called standard weights that is, standard pieces of metal used for weighing are really standards of mass; and when the result of a weighing is stated in terms of these standards, (as it usually is,) the " weight," as thus stated, is really the mass of the body weighed. The standard " weights " which we use in our balances are really standard masses. In discussions relating to density, weights are most conveniently expressed in gravitation measure, and hence the words mass and weight can be used almost indiscriminately. 162. Determination of Density from Weight and Volume. The absolute density of a substance can be directly determined by weighing a measured volume of it. Thus if v cubic centimetres of it weigh m grammes, its density (in grammes per cubic centimetre) is . This method can be easily applied to solids of regular geometrical forms; since their volumes can be computed from their SPECIFIC GRAVITY BOTTLE. 107 linear measurements. It can also be applied to liquids, by employ- ing a vessel of known content. The bottle usually employed for this purpose is a bottle of thin glass fitted with a perforated stopper, so that it can be filled and stoppered without leaving a space for air. The difference between its weights when full and empty is the weight of the liquid which fills it; and the quotient of this by the volume occupied (which can be determined once for all by weighing the bottle when filled with water) is the density of the liquid. The advantage of employing a perforated stopper is that it enables us to ensure constancy of volume. If a wide-mouthed flask were employed, without a stopper, it would be difficult to pronounce when the flask was exactly full. This source of inaccuracy would be diminished by making the mouth narrower: but when it is very narrow, the filling and emptying of the flask are difficult, and there is danger of forcing in bubbles of air with the liquid. When a per- forated stopper is employed, the flask is first filled, then the stopper is inserted and some of the liquid is thus forced up through the perforation, overflowing at the top. When the stopper has been pushed home, all the liquid outside is carefully wiped off, and the liquid which remains is as much as just fills the stoppered flask including the perforation in the stopper. In accurate work, the temperature must be observed, and due allowance made for its effect upon volume. 163. Specific Gravity Flask for Solids. The volume and density of a solid body of irregular shape, or consisting of a quantity of small pieces, can be de- termined by put- ting it into such a bottle (Fig. 76), and weighing the water which it displaces. The most convenient way of doing this is to observe ! (1) the weight of the Solid; (2) the Fig. 76.-Spetific Gravity Flask for Solids. weight of the bottle full of water; (3) the weight of the bottle when it contains the solid, together with as much water as will fill it up. If the 108 DENSITY AND ITS DETERMINATION. third of these results be subtracted from the sum of the first two, the remainder will be the weight of the water displaced; which, when expressed in grammes, is the same as the volume of the body expressed in cubic centimetres. The weight of the body, divided by this remainder, is the density of the body. 164. Method by Weighing in Water. The methods of determining density which we are now about to describe depend upon the prin- ciple of Archimedes. One of the commonest ways of determining the density of a solid body is to weigh it first in air and then in water (Fig. 77) the Fig. 77. Specific Gravity of Solids. Fig. 78. Specific Gravity of Liquids. counterpoising weights being in air. Since the loss of weight due to its immersion in water is equal to the weight of the same volume of water, we have only to divide the weight in air by this loss of weight We shall thus obtain the relative density of the body as compared with water in other words, the specific gravity of the body. WEIGHING IN WATER. 109 Thus, from the observations Weight in air, 125 ym, Weight in water, 100 Loss of weight, 25 \ve deduce 125 _ = 5 = density. A very fine and strong thread or fibre should be employed for sus- pending the body, so that the volume of liquid displaced by this thread may be as small as possible. 165. Weighing in Water, with a Sinker. If the body is lighter than water, we may employ a sinker that is, a piece of some heavy material attached to it, and heavy enough to make it sink. It is not necessary to know the weight of the sinker in air, but we must observe its weight in water. Call this s. Let w be the weight of the body in air, and w' the weight of the body and sinker together in water. Then \v' will be less than s. The body has an apparent upward gravitation in water equal to sw', showing that the resultant pressure upon it exceeds its weight by this amount. Hence the weight of the liquid displaced is w + s w', and the specific gravity of the body is ,. f J w + s - w If any other liquid than water be employed in the methods described in this and the preceding section, the result obtained will be the relative density as compared with that liquid. The result must therefore be multiplied by the density of the liquid, in order to obtain the absolute density. 166. Density of Liquid Inferred from Loss of Weight. The densities of liquids are often determined by observing the loss of weight of a solid immersed in them, and dividing by the known volume of the solid or by its loss of weight in water. Thus, from the observations Weight in air, 200 ym Weight in liquid, 120 Weight in water, 110 we deduce Loss in liquid, 80. Loss in water, 90. Density of liquid, = -. yu y A glass ball (sometimes weighted with mercury, as in Fig. 78) is the solid most frequently employed for such observations. 110 DENSITY AND ITS DETERMINATION. 167. Measurement of Volumes of Solids by Loss of Weight. The volume of a solid body, especially if of irregular shape, can usually be determined with more accuracy by weighing it in a liquid than by any other method. If it weigh w grammes in air, and w' grammes in water, its volume is ww cubic centimetres, since it displaces w iu' grammes of water. The mean diameter of a wire can be very accurately determined by an observation of this kind for volume, combined with a direct measurement of length. The volume divided by the length will be the mean sectional area, which is equal to irr 2 , where r is the radius. 168. Hydrometers. The name hydrometer is given to a class of instruments used for determining the densities of liquids by observ- ing either the depths to which they sink in the liquids or the Kig. 79. Nioholsoii's Hydrometer. weights required to be attached to them to make them sink to a given depth. According as they are to be used in the latter or the former of these two ways, they are called hydrometers of constant or of variable immersion. The name areometer (from apcuoc, rare) is used as synonymous with hydrometer, being probably borrowed from the French name of these instruments, artfometre. The hydro- NICHOLSON'S HYDROMETER. Ill meters of constant immersion most generally known are those of Nicholson and Fahrenheit. 169. Nicholson's Hydrometer. This instrument, which is repre- sented in Fig. 79, consists of a hollow cylinder of metal with conical ends, terminated above by a very thin rod bearing a small dish, and carrying at its lower end a kind of basket. This latter is of such weight that when the instrument is immersed in water a weight of 100 grammes must be placed in the dish above in order to sink the apparatus as far as a certain mark on the rod. By the principle of Archimedes, the weight of the instrument, together with the 100 grammes which it carries, is equal to the weight of the water dis- placed. Now, let the instrument be placed in another liquid, and the weights in the dish above be altered until they are just sufficient to make the instrument sink to the mark on the rod. If the weights o in the dish be called w, and the weight of the instrument itself W, the weight of liquid displaced is now W + w, whereas the weight of the same volume of water was W + 100; hence the specific gravity of the liquid is ^^^ This instrument can also be used either for weighing small solid bodies or for finding their specific gravities. To find the weight of a body (which we shall suppose to weigh less than 100 grammes), it must be placed in the dish at the top, together with weights just sufficient to make the instrument sink in water as far as the mark. Obviously these weights are the difference between the weight of the body and 100 grammes. To find the specific gravity of a solid, we first ascertain its weight by the method just described; we then transfer it from the dish above to the basket below, so that it shall be under water during the observation, and observe what additional weights must now be placed in the dish. These additional weights represent the weight of the water displaced by the solid; and the weight of the solid itself divided by this weight is the specific gravity required. 170. Fahrenheit's Hydrometer. This instrument, which is repre- sented in Fig. 80, is generally constructed of glass, and differs from Nicholson's in having at its lower extremity a ball weighted with mercury instead of the basket. It resembles it in having a dish at the top, in which weights are to be placed sufficient to sink the instrument to a definite mark on the stem. 112 DENSITY AND ITS DETERMINATION. Hydrometers of constant immersion, though still described in text-books, have quite gone out of use for practical work. 171. Hydrometers of Variable Immersion. These instruments are usually of the forms represented at A, B, C, Fig. 81. The lower end is weighted with mercury in order to make the instrument sink to a convenient depth and preserve an upright position. The stem is cylindrical, and is graduated, the divisions being frequently marked Fig. 80. Fahrenheit's Hydrometer. Fig 81. Forms of Hydrometers. upon a piece of paper inclosed within the stem, which must in this case be of glass. It is evident that the instrument will sink the deeper the less is the specific gravity of the liquid, since the weight of the liquid displaced must be equal to that of the instrument. Hence if any uniform system of graduation be adopted, so that all the instruments give the same readings in liquids of the same densi- ties, the density of a liquid can be obtained by a mere immersion of the hydrometer an operation not indeed very precise, but very easy of execution. These instruments have thus come into general use for commercial purposes and in the excise. 172. General Theory of Hydrometers of Variable Immersion. Let V be the volume of a hydrometer which is immersed when the in- strument floats freely in a liquid whose density is d, then Vd repre- HYDROMETERS. 113 sents the weight of liquid displaced, which by the principle of Archi- medes is the same as the weight of the hydrometer itself. If V, d' be the corresponding values for another liquid, we have therefore that is, the density varies inversely as the volume immersed. Let di, d. 2 , dg Hence, if we wish the divisions to indicate equal differences of den- sity, we must place them so that the corresponding volumes im- mersed form a harmonical progression. This implies that the dis- tances between the divisions must diminish as the densities increase. The following investigation shows how the density of a liquid may be computed from observations made with a hydrometer gradu- ated with equal divisions. It is necessary first to know the divisions to which the instrument sinks in two liquids of known densities. Let these divisions be numbered n v n. 7 , reckoning from the top downwards, and let the corresponding densities be d l , cZ 2 . Now if we take for our unit of volume one of the equal parts on the stem, and if we take c to denote the volume which is immersed when the instrument sinks to the division marked zero, it is obvious that when the instrument sinks to the nth division (reckoned downwards on the stem from zero) the volume immersed is cn, and if the corre- sponding density be called d, then (c n) d is the weight of the hydrometer. We have therefore (C-HI) d, = (c-n<>) d, whence c = . ~ '- -. This value of c can be computed once for all. Then the density D corresponding to any other division N can be found from the equation (c-N) D = (c-nO d,. which gives D = ^^^. 173. Beaume's Hydrometers. In these instruments the divisions are equidistant. There are two distinct modes of graduation, accord- ing as the instrument is to be used for determining densities greater or less than that of water. In the former case the instrument is 8 114 DENSITY AND ITS DETERMINATION. called a salimeter, and is so constructed that when immersed in pure water of the temperature 12 Cent, it sinks nearly to the top of the stem, and the point thus determined is the zero of the scale. It is then immersed in a solution of 15 parts of salt to 85 of water, the density of which is about 1*116, and the point to which it sinks is marked 15. The interval is divided into 15 equal parts, and the graduation is continued to the bottom of the stem, the length of which varies accord ing to circumstances; it generally terminates at the degree 66, which corresponds to sulphuric acid, whose density is commonly the greatest that it is required to determine. Referring to the formulae of last section, we have here Fig. 82. Baumd's Sali- meter. whence 15x1-116 ... ._ 144 c- -^ =144, D = . 116 '144-N When the instrument is intended for liquids lighter than water, it is called an alcoholimeter. In this case the point to which it sinks in water is near the bottom of the stem, and is marked 10; the zero of the scale is the point to which it sinks in a solution of 10 parts of salt to 90 of water, the density of which is about T085, the divisions in this case being numbered upward from zero. In order to adapt the formulae of last section to the case of graduations numbered upwards, it is merely necessary to reverse the signs of %, n 2 , and N; that is we must put Fig. 83. Fig. 84. Baume's Alcoholi- meters. n. 2 d. 2 nidi and as we have now 7^ =10, the formulae give 1 ^rl, n. 2 =0, c? 3 =l*085 128 174. Twaddell's Hydrometer. In this instrument the divisions are 1 On comparing the two formulae for D in this section with the tables in the Appendix to Miller's Chemical Physics, I find that as regards the salimeter they agree to two places of decimals and very nearly to three. As regards the alcoholimeter, the table in Miller implies that c is about 136, which would make the density corresponding to the zero of the scale about 1'074. DENSITY OF MIXTURES. 115 placed not as in Beaume's, at equal distances, but at distances corresponding to equal differences of density. In fact the specific gravity of a liquid is found by multiplying the reading by 5, cutting off three decimal places, and prefixing unity. Thus the degree 1 indicates specific gravity TOOo, 2 indicates I'OIO, &c. 175. G-ay-Lussac's Centesimal Alcoholimeter. When a hydrometer is to be used for a special purpose, it may be convenient to adopt a mode of graduation different in principle from any that we have described above, and adapted to give a direct indication of the proportion in which two ingredients are mixed in the fluid to be examined. It may indicate, for example, the quantity of salt in sea-water, or the quantity of alcohol in a spirit consisting of alcohol and water. Where there are three or more ingredients of different specific gravities the method fails. Gay-Lussac's alcoholimeter is graduated to indicate, at the temperature of 15 Cent., the percentage of pure alcohol in a specimen of spirit. At the top of the stem is 100, the point to which the instrument sinks in pure alcohol, and at the bottom is 0, to which it sinks in water. The position of the intermediate degrees must be determined empirically, by placing the instrument in mix- tures of alcohol and water in known proportions, at the Fig. so. temperature of 15. The law of density, as depending on Aicohoii- the proportion of alcohol present, is complicated by the fact that, when alcohol is mixed with water, a diminution of volume (accompanied by rise of temperature) takes place. 176. Specific Gravity of Mixtures. When two or more substances are mixed without either shrinkage or expansion (that is, when the volume of the mixture is equal to the sum of the volumes of the components), the density of the mixture can easily be expressed in terms of the quantities and densities of the components. First, let the volumes v ly v 2 , v 3 . . . of the components be given, together with their densities d l , d 2> d 3 . . . Then their masses (or weights) are v^, v. 2 d 2 , v s d 3 . . . The mass of the mixture is the sum of these masses, and its volume is the sum of the volumes v 1} i> 2 , v s . . . ; hence its density is Secondly, let the weights or masses f m l , m 2 , -m 3 . . . of the com- ponents be given, together with their densities d l , d 2 , d a . . . 116 DENSITY AND ITS DETERMINATION. C.80 rrn il 1 m l W1 2 W^3 Then their volumes are -r, -r, -r &l Ui-2 (43 The volume of the mixture is the sum of these volumes, and its mass is m^ + m 2 + w< 3 + . . . ; hence its density is m\ + wi-2 + 5* . fjtt 0^1 rfj 177. Graphical Method of Graduation. When the points on the stem which correspond to some five or six known densities, nearly equidifferent, have been determined, the intermediate graduations can be inserted with tolerable accuracy by the graphical method of interpolation, a method which has many applications in physics besides that which we are now considering. Suppose A and B (Fig. 86) to represent the extreme points, and I, K, L, R intermediate points, all of which correspond to known densities. Erect ordinates (that is to say, per- pendiculars) at these points, proportional to the respective densities, or (which will serve our purpose equally well) erect ordinates II', KK', LL', RR', BC proportional to the excesses of the densities at I, K, L, R, B above the den- sity at A. Any scale of equal parts can be employed for laying off the ordinates, but it is convenient to adopt a scale which will make the greatest ordinate BC not much greater nor much less than the base-line AB. In the figure, the density at B is supposed to be 1*80, the density at A being 1. The difference of density is therefore '80, as indicated by the figures 80 on the scale of equal parts. Having erected the ordinates, we must draw through their extremities the curve AI'K'L'R C, making it as free from sudden bends as possible, as it is upon the regu- larity of this curve that the accuracy of the interpolation depends. Then to find the point on the stem AB at which any other density is to be marked say T60, we must draw through the 60th division, on the line of equal parts, a horizontal line to meet the curve, and, through the point thus found on the curve, -10 L 1.4 R 1.6 Fig. 86. Graphical Method of Graduation. GRAPHICAL INTERPOLATION. 117 draw an ordinate. This ordinate will meet the base-line AB in the required point, which is accordingly marked 1'6 in the figure. The curve also affords the means of solving the converse problem, that is, of finding the density corresponding to any given point on the stem. At the given point in AB, which represents the stem, we must draw an ordinate, and through the point where this meets the curve we must draw a horizontal line to meet the scale of equal parts. The point thus determined on the scale of equal parts indi- cates the density required, or rather the excess of this density above the density of A. CHAPTER XV. VESSELS IN COMMUNICATION LEVELS. 178. Liquids tend to Find their own Level. When a liquid is contained in vessels communicating with each other, and is in equilibrium, it stands at the same height in the different parts of the system, so that the free surfaces all lie in the same horizontal plane. This is obvious from the considerations pointed out in 138, 139, being merely a particular case of the more general law that points of a liquid at rest which are at the same pressure are at the same level. In the apparatus represented in Fig. 87, the liquid is seen to stand at the same height in the principal vessel and in the variously shaped tubes com- municating with it. If one of these tubes is cut off at a height less than that of the liquid in the principal vessel, and is made totermin- | ateinaiiarrowmouth, B the liquid will be seen to spout up nearly to the level of that in Fig. 87. -Vessels in Communication. fa Q p r i nc jp a l vessel. This tendency of liquids to find their own level is utilized for the water-supply of towns. Water will find its way from a reservoir through pipes of any length, provided that all parts of them are below the level of the water in the reservoir. It is necessary how- WATER SUPPLY. 119 ever to distinguish between the conditions of statical equilibrium and the conditions of flow. If no water were allowed to escape from the pipes in a town, their extremities might be carried to the height of the reservoir and they would still be kept full. But in practice there is a continual abstraction of energy, partly in the shape of the kinetic energy of the water which issues from taps, often with considerable velocity, and partly in the shape of work done against friction in the pipes. When there is a continual draw- ing off from various points of a main, the height to which the water will rise in the houses which it supplies is least in those which are most distant from the reservoir. 179. Water-level. The instrument called the water-level is another illustration of the same principle. It consists of a metal tube bb, bent at right angles at its extremities. These carry two glass tubes r> Fig. 88. Water-level. eta, very narrow at the top, and of the same diameter. The tube rests on a tripod stand, at the top of which is a joint that enables the observer to turn the apparatus and set it in any direction. The tube is placed in a position nearly horizontal, and water, generally coloured a little, is poured in until it stands at about three-fourths of the height of each of the glass tubes. By the principle of equilibrium in vessels communicating with each other, the surfaces of the liquid in the two branches are in the same horizontal plane, so that if the line of the observer's sight just grazes the two surfaces it will be horizontal. This is the principle of the operation called levelling, the object of which is to determine the difference of vertical height, or difference of level, between two given points. Suppose A and B to be the two points (Fig. 89). At each of these points is fixed a levelling-staff, 120 VESSELS IN COMMUNICATION LEVELS. Fig. 89. Levelling. that is, an upright rod divided into parts of equal length, on which slides a small square board whose centre serves as a mark for the observer. The level being placed at an intermediate station, the observer directs the line of sight towards each levelling-staff, and the mark is raised or lowered till the line of sight passes through its centre. The marks on the two staves are in this way brought to the same level. The staff in the rear is then carried in advance of the other > the level is again placed between the two, and an- other observation taken. In this way, by noting the division of the staff at which the sliding mark stands in each case, the difference of levels of two distant stations can be deduced from observations at a number of intermediate points. For more accurate work, a telescope with attached spirit-level ( 181) is used, and the level- ling staff has divisions upon it which are read off through the telescope. 180. Spirit-level. The spirit-level is composed of a glass tube slightly curved, containing a liquid, which is generally alcohol, and which fills the whole extent of the tube, except a small space occupied by an air- bubble. This tube is inclosed in a mounting which is firmly sup- ported on a stand. Suppose the tube to have been so constructed that a vertical section of its upper surface is an arc of a circle, and suppose the instrument placed upon a horizontal plane (Fig. 91). The air-bubble will take up a position MN at the highest part of the tube, such that the arcs MA and NB are equal. Hence it follows that if the level Fig. 90. Spirit-level. Fig. 91. SPIRIT-LEVEL. 121 be reversed end for end, the bubble will occupy the same position in the tube, the point N coming to M, and vice versa. This will not be the case if AB is inclined to the horizon (Fig. 92), for then the bubble will always stand nearest to that end of the tube which is highest, and will therefore change its place in the tube when the Fig. 02 latter is reversed. The test, then, of the horizontality of the line on which the spirit-level rests is, that after this operation of reversal the bubble should remain between the same marks on the tube. The maker marks upon the tube two points equidistant from the centre, the distance between them being equal to the usual length of the bubble; and the instrument ought to be so adjusted that when the line on which it stands is horizontal, the ends of the bubble are at these marks. In order that a plane surface may be horizontal, we must have two lines in it horizontal. This result may be attained in the Fig. 93. Testing the Horizon tality of a Surface. following manner: The body whose surface is to be levelled is made to rest on three levelling-screws which form the three vertices of an isosceles triangle; the level is first placed parallel to the base of the triangle, and, by means of one of the screws, the bubble is brought between the reference-marks. The instrument is then placed perpendicularly to its first position, and the bubble is brought between the marks by means of the third screw; this second opera- tion cannot disturb the result of the first, since the plane has only been turned about a horizontal line as hinge. 181. Telescope with Attached Level. In order to apply the spirit- level to land-surveying, an apparatus such as that represented in 122 VESSELS IN COMMUNICATION LEVELS. Fig. 94 is employed. Upon a frame AA, movable about a vertical axis B, are placed a spirit-level nn, and a telescope LL, in positions parallel to each other. The telescope is furnished at its focus with two fine wires crossing one another, whose point of intersection deter- mines the line of sight with great precision. The appar- atus, which is provided with levelling-screws H, rests on a tripod stand, and the observer is able, by turning it about its axis, to command the dif- ferent points of the horizon. By a process of adjustment which need not here be described, it is known that when the bubble is between the marks the line of sight is horizontal. By furnishing the instrument with a graduated horizontal circle P, we may obtain the azimuths of the points observed, and thus map out contour lines. Divisions are sometimes placed on each side of the reference- marks of the bubble, for measuring small deviations from horizon- tality. It is, in fact, easy to see, by reference to Fig. 91, that by tilting the level through any small angle, the bubble is displaced by a quantity proportional to this angle, at least when the curvature of the instrument is that of a circle. For determining the angular value corresponding to each division Pig. 94. Spirit level with Telescope. Fig. 95. Graduation of Spirit-level. of the tube, it is usual to employ an apparatus opening like a pair of compasses by a hinge C (Fig. 95), on one of the legs of which rests, by two V-shaped supports, the tube T of the level. The com- SPIRIT-LEVEL. 123 pass is opened by means of a micrometer screw V, of very regular action; and as the distance of the screw from the hinge is known, as well as the distance between the threads of the screw, it is easy to calculate beforehand the value of the divisions on the micrometer head. The levelling-screws of the instrument serve to bring the bubble between its reference-marks, so that the micrometer screw is only used to determine the value of the divisions on the tube. CHAPTER XVI. CAPILLARITY. 182. Capillarity General Phenomena. The laws which we have thus far stated respecting the levels of liquid surfaces are subject to remarkable exceptions when the vessels in which the liquids are contained are very narrow, or, as they are called, capillary (capillus, a hair) ; and also in the case of vessels of any size, when we consider the portion of the liquid which is in close proximity to the sides. 1. Free Surface. The surface of a liquid is not horizontal in the neighbourhood of the sides of the vessel, but presents a very decided curvature. When the liquid wets the vessel, as in the case of water in a glass vessel (Fig. 96), the surface is concave; on the contrary Fig. 93. when the liquid does not wet the vessel, as in the case of mercury in a glass vessel (Fig. 97), the surface is, generally speaking, convex. 2. Capillary Elevation and Depression. If a very narrow tube of glass be plunged in water, or any other liquid that will wet it (Fig. 98), it will be observed that the level of the liquid, instead of remaining at the same height inside and outside of the tube, stands perceptibly higher in the tube; a capillary ascension takes place, which varies in amount according to the nature of the liquid and GENERAL PHENOMENA. 125 the diameter of the tube. It will also be seen that the liquid column thus raised terminates in a concave surface. If a glass tube be dipped in mercury, which does not wet it, it will be seen, by bringing the tube to the side of the vessel, that the mercury is depressed in its interior, and that it terminates in a convex surface (Fig. 99). 3. Capillary Vessels in Communication with Others. If we take two bent tubes (Fig. 100), each having one branch of a considerable diameter and the other extremely narrow, and pour into one of them a liquid which wets it, and into the other mercury, the liquid will be observed in the former case to stand higher in the capillary than in the prin- cipal branch, and in the latter case to stand lower; the free surfaces being at the same time concave in the case of the liquid which wets the tubes, and convex in the case of the mercury. 183. Circumstances which influence Capillary Elevation and Depres- sion. In wetted tubes the elevation depends upon the nature of the liquid; thus, at the temperature of 18 Cent., water rises 29'79 mm in a tube 1 millimetre in diameter, alcohol rises 1218 mm , nitric acid 22-57 mm , essence of lavender 4'28 mm , &c. The nature of the tube is almost entirely immaterial, provided the precaution be first taken of wetting it with the liquid to be employed in the experiment, so as to leave a film of the liquid adhering to the sides of the tube. Capillary depression, on the other hand, depends both on the nature of the liquid and on that of the tube. Both ascension and depression diminish as the temperature increases; for example, the elevation of water, which in a tube of a certain diameter is equal to 132 nim at Cent, is only 106 mm at 100. 184. Law of Diameters. Capillary elevations and depressions, when all other circumstances are the same, are inversely propor- tional to the diameters of the tubes. As this law is a consequence of the mathematical theories which are generally accepted as ex- plaining capillary phenomena, its verification has been regarded as of great importance. The experiments of Gay-Lussac, which confirmed this law, have been repeated, with slight modifications, by several observers. The 126 CAPILLARITY. method employed consists essentially in measuring the capillary elevation of a liquid by means of a cathetometer (Fig. 101). The telescope II is directed first to the top n of the column in the tube, and then to the end of a pointer b, which touches the surface of the Fig. 101. Verification of Law of Di; liquid at a point where it is horizontal. In observing the depression of mercury, since the opacity of the metal prevents us from seeing the tube, we must bring the tube close to the side of the vessel e. The diameter of the tube can be measured directly by observing its section through a microscope, or we may proceed by the method employed by Gay-Lussac. He weighed the quantity of mercury which filled a known length I of the tube; this weight w is that of a cylinder of mercury whose radius x is determined by the equation 13'59 irxH=w, where x and I are in centimetres, and w in grammes. The result of these different experiments is, that in the case of wetted tubes the law is exactly fulfilled, provided that they be pre- viously washed with the greatest care, so as to remove all foreign matters, and that the liquid on which the experiment is to be per- formed be first passed through them. When the liquid does not wet the tube, various causes combine to affect the form of the surface in which the liquid column terminates; and we cannot infer the depres- sion from knowing the diameter, unless we also take into considera- tion some element connected with the form of the terminal surface, such as the length of the sagitta, or the angle made with the sides FUNDAMENTAL PRINCIPLES. ] '27 of the tube by the extremities of the curved surface, which is called the angle of contact 185. Fundamental Laws of Capillary Phenomena. Capillary phe- nomena, as they take place alike in air and in vacuo, cannot be attri- buted to the action of the atmosphere. They depend upon molecular actions which take place between the particles of the liquid itself, and between the liquid and the solid containing it, the actions in question being purely superficial that is to say, being confined to an extremely thin layer forming the external boundary of the liquid, and to an extremely thin superficial layer of the solid in contact with the liquid. For example, it is found in the case of glass tubes, that the amount of capillary elevation or depression is not at all affected by the thickness of the sides of the tube. The following are some of the principles which govern capillary phenomena. 1. For a given liquid in contact with a given solid, with a definite intimateness of contact (this last element being dependent upon the cleanness of the surface, upon whether the surface of the solid has been recently washed by the liquid, and perhaps upon some other particulars), there is (at any specified temperature) a definite angle of contact, which is independent of the directions of the surfaces with regard to the vertical. 2. Every liquid behaves as if a thin film, forming its external layer, were in a state of tension, and exerting a constant effort to contract. This tension, or contractile force, is exhibited over the whole of the free surface (that is, the surface which is exposed to air); but wherever the liquid is in contact with a solid, its existence is masked by other molecular actions. It is uniform in all directions in the free surface, and at all points in this surface, being dependent only on the nature and temperature of the liquid. Its intensity for several specified liquids is given in tabular form further on ( 192) upon the authority of Van der Mensbrugghe. Tension of this kind must of course be stated in units of force per linear unit, because by doubling the width of a band we double the force required to keep it stretched. Mensbrugghe considers that such tension really exists in the superficial layer; but the majority of authors (and we think with more justice) regard it rather as a convenient fiction, which accurately represents the effects of the real cause. Two of the most eminent writers on the cause of capillary phenomena are Laplace and Dr. Thomas Young. The subject presents difficulties which have not yet been fully surmounted. 128 CAPILLARITY. 186. Application to Elevation in Tubes. The law of diameters is a direct consequence of the two preceding principles; for if a denote the external angle of contact (which is acute in the case of mercury against glass), T the tension per unit length, and r the radius of the tube, then 2irrT will be the whole amount of force exerted at the margin of the surface; and as this force is exerted in a direction making an angle a with the vertical, its vertical component will be 2irrT cos a, which is exerted in pulling the tube upwards and the liquid downwards. If w be the weight of unit volume of the liquid, then -n-r^w is the weight of as much as would occupy unit length of the tube; and if h denote the height of a column whose weight is equal to the force tending to depress the liquid, we have irr^hw - 27D > Tcos a ; whence h= ^A which, when the other elements are given, varies r.w inversely as T, the radius of the tube. Having regard to the fact that the surface is not of the same height in the centre as at the edges, it is obvious that h denotes the mean height. If a be obtuse, h will be negative that is to say, there will be elevation instead of depression. In the case of water against a tube which has been well wetted with that liquid, a is 180 that is to say, the tube is tangential to the surface. For this case the formula for h gives 2T elevation = rw Again, for two parallel vertical plates at distance u, the vertical force of capillarity for a unit of length is 2Tcoe a, which must be equal to whu, being the weight of a sheet of liquid of height h, thickness u, and length unity. We have therefore , 2Tcosa h = , uw which agrees with the expression for the depression or elevation in a circular tube whose radius is equal to the distance between these parallel plates. The surface tension always tends to reduce the surface to the smallest area which can be inclosed by its actual boundary; and therefore always produces a normal force directed from the convex to the concave side of the superficial film. Hence, wherever there is PRESSURE EXERTED BY FILM. 129 capillary elevation the free surface must be concave; wherever there is depression it must be convex. 187. It follows from a well-known proposition in statics (Tod- hunter's Statics, 194), that if a cylindrical film be stretched with a uniform tension T (so that the force tending to pull the film asunder across any short line drawn on the film, is T times the length of the line), the resultant normal pressure (which the film exerts, for ex- ample, against the surface of a solid internal cylinder over which it is stretched) is T divided by the radius of the cylinder. It can be proved that a film of any form, stretched with uniform tension T, exerts at each point a normal pressure equal to the sum of the pressures which would be exerted by tw r o overlapping cylin- drical films, whose axes are at right angles to one another, and whose cross sections are circles of curvature of normal sections at the point. That is to say, if P be the normal force per unit area, and r, r the radii of curvature in two mutually perpendicular normal .sections at the point, then At any point on a curved surface, the normal sections of greatest and least curvature are mutually perpendicular, and are called the prin- cipal normal sections at the point. If the corresponding radii of curvature be R, R', we have or the normal force per unit area is equal to the tension per unit length multiplied by the sum of the principal curvatures. In the case of capillary depressions and elevations, the superficial film at the free surface is to be regarded as pressing the liquid in- wards, or pulling it outwards, according as this surface is convex or concave, with a force P given by the above formula. The value of P at any point of the free surface is equal to the pressure due to the height of a column of liquid extending from that point to the level of the general horizontal surface. It is therefore greatest at the edges of the elevated or depressed column in a tube, and least in the centre; and the curvature, as measured by ^ + ^ must vary in the same proportion. If the tube is so large that there is no sensible elevation or depression in the centre of the column, the centre of the free surface must be sensibly plane. 188. Another consequence of the formula is, that in circumstances 9 130 CAPILLARITY. where there can be no normal pressure towards either side of the .surface, K + K. = > < 21 which implies that either the surface is plane, in which case each of the two terms is separately equal to zero, or else R = - R'; (3) that is, the principal radii of curvature are equal, and lie on opposite sides of the surface. The formulae (2), (3) apply to a film of soapy water attached to a loop of wire. If the loop be in one plane, the film will be in the same plane. If the loop be not in one plane, the film cannot be in one plane, and will in fact assume that form which gives the least area consistent with having the loop for its boundary. At every point it will be observed to be, if we may so say, concave towards both sides, and convex towards both sides, the concavity being precisely equal to the convexity that is to say, equation (3) is satisfied at every point of the film. In this case both sides of the film are exposed to atmospheric pressure. In the case of a common soap-bubble the outside is ex- posed to atmospheric pressure, and the inside to a pressure somewhat greater, the difference of the pressures being balanced by the ten- dency of the film to contract. Formula (1) becomes for either the outer or inner surface of a spherical bubble but this result must be doubled, because there are two free surfaces; hence the excess of pressure of the inclosed above the external air is ^5-, R denoting the radius of the bubble. The value of T for soapy water is about 1 grain per linear inch ; hence, if we divide 4 by the radius of the bubble expressed in inches, we shall obtain the excess of internal over external pressure in grains per square inch. The value of T for any liquid may be obtained by observing the amount of elevation or depression in a tube of given diameter, and employing the formula (4) which follows immediately from the formula for h in 186. 189. It is this uniform surface tension, of which we have been DROPS. 131 speaking, which causes a drop of a liquid falling through the air either to assume the spherical form, or to oscillate about the spheri- cal form. The phenomena of drops can be imitated on an enlarged scale, under circumstances which permit us to observe the actual motions, by a method devised by Professor Plateau of Ghent. Olive- oil is intermediate in density between water and alcohol. Let a mixture of alcohol and w r ater be prepared, having precisely the density of olive-oil, and let about a cubic inch of the latter be gently introduced into it with the aid of a funnel or pipette. It will as- sume a spherical form, and if forced out of this form and then left free, will slowly oscillate about it; for example, if it has been com- pelled to assume the form of a prolate spheroid, it will pass to the oblate form, will then become prolate again, and so on alternately, becoming how r ever more nearly spherical every time, because its movements are hindered by friction, until at last it comes to rest as a sphere. 190. Capillarity furnishes no exception to the principle that the pressure in a liquid is the same at all points at the same depth. When the free surface within a tube is convex, and is consequently depressed below the plane surface of the external liquid, the pres- sure becomes suddenly greater on passing downwards through the superficial layer, by the amount due to the curvature. Below this it increases regularly by the amount due to the depth of liquid passed through. The pressure at any point vertically under the con- vex meniscus 1 may be computed, either by taking the depth of the point below the general free surface, and adding atmospheric pres- sure to the pressure due to this depth, according to the ordinary principles of hydrostatics, or by taking the depth of the point below that point of the meniscus which is vertically over it, adding the pressure due to the curvature at this point, and also adding atmo- spheric pressure. When the free surface of the liquid within a tube is concave, the pressure suddenly diminishes on passing downwards through the superficial layer, by the amount due to the curvature as given by formula (1); that is to say, the pressure at a very small depth is less than atmospheric pressure by this amount. Below this depth it goes on increasing according to the usual law, and becomes equal to 1 The convex or concave surface of the liquid in a tube is usually denoted by the name meniscus (nyvlffKos, a crescent), which denotes a form approximately resembling that of a watch-glass. 132 CAPILLARITY. atmospheric pressure at that depth which corresponds with the level of the plane external surface. The pressure at any point in the liquid within the tube can therefore be obtained either by subtract- ing from atmospheric pressure the pressure due to the elevation of the point above the general surface, or by adding to atmospheric pressure the pressure due to the depth below that point of the meniscus which is on the same vertical, and subtracting the pressure due to the curvature at this point. These rules imply, as has been already remarked, that the curva- ture is different at different points of the meniscus, being greatest where the elevation or depression is greatest, namely at the edges of the meniscus; and least at the point of least elevation or depres- sion, which in a cylindrical tube is the middle point. The principles just stated apply to all cases of capillary elevation and depression. They enable us to calculate the force with which two parallel ver- tical plates, partially immersed in a liquid which wets them, are urged towards each other by capillary action. The pressure in the portion of liquid elevated between them is less than atmospheric, and therefore is insufficient to balance the atmospheric pressure which is exerted on the outer faces of the plates. The average pres- sure in the elevated portion of liquid is equal to the actual pressure at the centre of gravity of the elevated area, and is less than atmo- spheric pressure by the pressure of a column of liquid whose height is the elevation of this centre of gravity. Even if the liquid be one which does not wet the plates, they will still be urged towards each other by capillary action; for the inner faces of the plates are exposed to merely atmospheric pressure over the area of depression, while the corresponding portions of the ex- ternal faces are exposed to atmospheric pressure increased by the weight of a portion of the liquid. These principles explain the apparent attraction exhibited by bodies floating on a liquid which either wets them both or wets neither of them. When the two bodies are near each other they behave somewhat like parallel plates, the elevation or depression of the liquid between them being greater than on their remote sides. If two floating bodies, one of which is wetted and the other un- wetted by the liquid, come near together, the elevation and depres- sion of the liquid will be less on the near than on the remote sides, and apparent repulsion will be exhibited. APPARENT ATTRACTIONS. 133 In all cases of capillary elevation or depression, the solid is pulled downwards or upwards with a force equal to that by which the liquid is raised or depressed. In applying the principle of Archi- medes to a solid partially immersed in a liquid, it is therefore neces- sary (as we have seen in 159), when the solid produces capillary depression, to reckon the void space thus created as part of the dis- placement; and when the solid produces capillary elevation, the fluid raised above the general level must be reckoned as negative displace- ment, tending to increase the apparent weight of the solid. 191. Thus far all the effects of capillary action which we have mentioned are connected with the curvature of the superficial film, and depend upon the principle that a convex surface increases and a concave surface diminishes the pressure in the interior of the liquid. But there is good reason for maintaining that whatever be the form of the free surface there is always pressure in the interior due to the molecular action at this surface, and that the pressure due to the curvature of the surface is to be added to or subtracted from a definite amount of pressure which is independent of the curvature and depends only on the nature and condition of the liquid. This indeed follows at once from the fact that capillary elevation can take place in vacuo. As far as the principles of the preceding paragraphs are concerned, we should have, at points within the elevated column, a pressure less than that existing in the vacuum. This, however, cannot be; we cannot conceive of negative pressure existing in the interior of a liquid, and we are driven to conclude that the elevation is owing to the excess of the pressure caused by the plane surface in the containing vessel above the pressure caused by the concave surface in the capillary tube. There are some other facts which seem only explicable on the same general principle of interior pressure due to surface action, facts which attracted the notice of some of the earliest writers on pneumatics, namely, that siphons will work in vacuo, and that a column of mercury at least 75 inches in length can be sustained as if by atmospheric pressure in a barometer tube, the mercury being boiled and completely filling the tube. 192. We have now to notice certain phenomena which depend on the difference in the surface tensions of different liquids, or of the same liquid in different states. Let a thin layer of oil be spread over the upper surface of a thin sheet of brass, and let a lamp be placed underneath. The oil will be 134 CAPILLARITY. observed to run away from the spot directly over the flame, even though this spot be somewhat lower than the rest of the sheet. This effect is attributable to the excess of surface tension in the cold oil above the hot. In like manner, if a drop of alcohol be introduced into a thin layer of water spread over a nearly horizontal surface, it will be drawn away in all directions by the surrounding water, leaving a nearly dry spot in the space which it occupied. In this experiment the water should be coloured in order to distinguish it from the alcohol. Again, let a very small fragment of camphor be placed on the sur- face of hot water. It will be observed to rush to and fro, with frequent rotations on its own axis, sometimes in one direction and sometimes in the opposite. These effects, which have been a frequent subject of discussion, are now known to be due to the diminution of the surface tension of the water by the camphor which it takes up. Superficial currents are thus created, radiating from the fragment of camphor in all directions; and as the camphor dissolves more quickly in some parts than in others, the currents which are formed are not equal in all directions, and those which are most powerful prevail over the others and give motion to the fragment. The values of T, the apparent surface tension, for several liquids, are given in the following table, on the authority of Van der Mens- brugghe, in milligrammes (or thousandth parts of a gramme) per millimetre of length. They can be reduced to grains per inch of length by multiplying them by '392; for example, the surface ten- sion of distilled water is 7'3 X '392 = 2'86 grains per inch. Distilled water at 20 Cent 7'3 Sulphuric ether, 1 -88 Absolute alcohol, 2 '5 Olive-oil, 3'5 Mercury, 49'1 Bisulphide of carbon, 3 '57 Solution of Marseilles soap, 1 part of soap to 40 of water, 2'83 Solution of saponine, 4'67 Saturated solution of carbonate of soda 4-28 Water impregnated with camphor, . 4 '5 193. Endosmose. Capillary phenomena have undoubtedly some connection with a very important property discovered by Dutrochet, and called by him endosmose. The endosmometer invented by him to illustrate this phenomenon consists of a reservoir v (Fig. 102) closed below by a membrane ba, and terminating above in a tube of considerable length. This reser- voir is filled, suppose, with a solution of gum in water, and is kept DIFFUSION THROUGH SEFfA. 135 immersed in water. At the end of some time the level of the liquid in the tube will be observed to have risen to n, suppose, and at the - ame time traces of gum will be found in the water in which the reservoir is immersed. Hence we conclude that the two liquids have penetrated through the membrane, but in different proportions ; and this is what is called endosmose. If instead of a solution of gum we employed water containing albumen, sugar, or gelatine in solution, a similar result would ensue. The membrane may be replaced by a slab of wood or of porous clay. Physiologists have justly attached very great importance to this discovery of Dutrochet. It explains, in fact, the interchange of liquids which is continually taking place in the tissues and vessels of the animal system, as well as the absorption of water by the spongioles of roots, and several similar phenomena. As regards the power of passing through porous diaphragms, Graham has divided substances into two classes crystalloids and colloids (K('j\\r) glue). The former are sus- ceptible of crystallization, form solutions free from viscosity, are sapid, and possess great powers of diffusion through porous septa. The latter, including gum, starch, albumen, &c., are characterized by a remarkable slug- gishness and indisposition both to diffusion and to crystallization, and when pure are nearly tasteless. Diffusion also takes place through col- loidal diaphragms which are not porous, the diaphragm in this case acting as a solvent, and giving out the dissolved mate- rial on the other side. In the important process of modern chemistry called dialysis, saline ingredients are separated from or- ganic substances with which they are blended, by interposing a colloidal dia- phragm (De La Rue's parchment paper) between the mixture and pure water. The organic matters, being colloidal, remain behind, while the salts pass through, and can be obtained in a nearly pure state by evaporating the water. Gases are also capable of diffusion through diaphragms, whether Fig. 102. Endoemometer. 136 CAPILLARITY. porous or colloidal, the rate of diffusion being in the former case inversely as the square root of the density of the gas. Hydrogen diffuses so rapidly through unglazed earthenware as to afford oppor- tunity for very striking experiments; and it shows its power of traversing colloids by rapidly escaping through the sides of india- rubber tubes, or through films of soapy water. CHAPTEK XVII. THE BAROMETER. 194. Expansibility of Gases. Gaseous bodies possess a number of properties in common" with liquids; like them, they transmit pres- sures entire and in all directions, according to the principle of Pascal; but they differ essentially from liquids in the permanent repulsive force exerted between their molecules, in virtue of which a mass of gas always tends to expand. This property, called the expansibility of gases, is commonly illus- trated by the following experiment: A bladder, nearly empty of air, and tied at the neck, is placed under the receiver of an air-pump. At first the air which it contains and the external air oppose each other by their mutual pressure, and are in equilibrium. But if we proceed to exhaust the receiver, and thus dimmish the external pressure, the bladder gradually be- comes inflated, and thus manifests the tendency of the gas which it con- tains to OCCUpy a greater K * lOS.-Expansibility of Gases. volume. However large a vessel may be, it can always be filled by any quantity whatever of a gas, which will always exert pressure against 138 THE BAROMETER. the sides. In consequence of this property, the name of elastic fluids is often given to gases. 195. Air has Weight. The opinion was long held that the air was without weight; or, to speak more precisely, it never occurred to any of the philosophers who preceded Galileo to attribute any influence in natural phenomena to the weight of the air. And as this influence is really of the first importance, and comes into play in many of the commonest phenomena, it very naturally happened that the discovery of the weight of air formed the commencement of the modern revival of physical science. It appears, however, that Aristotle conceived the idea of the possibility of air having weight, and, in order to convince himself on this point, he weighed a skin inflated and collapsed. As he obtained the same weight in both cases, he relinquished the idea which he had for the moment entertained. In fact, the experiment, as he performed it, could only give a negative result; for if the weight of the skin was increased, on the one hand, by the intro- duction of a fresh quantity of air, it was diminished, on the other, by the corresponding increase in the upward pressure of the air displaced. In order to draw a certain conclusion, the experiment should be performed with a vessel which could receive within it air of different degrees of density, without changing its own volume. Galileo is said to have devised the experiment of weighing a globe filled alternately with ordinary air and with compressed air. As the weight is greater in the latter case, Galileo should have drawn the inference that air is heavy. It does not appear, however, that the importance of this conclusion made much impression on him, for he did not give it any of those developments which might have been expected to present themselves to a mind like his. Otto Guericke, the illustrious inventor of the air-pump, in 1650 performed the following experiment, which is decisive: A globe of glass (Fig. 104), furnished with a stop-cock, and of a sufficient capacity (about twelve litres), is exhausted of air. It is then suspended from one of the scales of a balance, and a weight sufficient to produce equilibrium is placed in the other scale. The stop-cock is then opened, the air rushes into the globe, and the beam is observed gradually to incline, so that an additional weight is required in the other scale, in order to re-establish equilibrium. If the capacity of the globe is 12 litres, about 15*5 grammes will be WEIGHT OF AIR. 139 needed, which gives 1*3 gramme as the approximate weight of a litre (or 1000 cubic centimetres) of air. 1 If, in performing this experiment, we take particular precautions to insure its precision, as we shall explain in the book on Heat, it will be found that, at the temperature of freezing water, and under the pressure of one atmosphere, a litre of perfectly dry air weighs 1*293 gramme. 2 Under these circum- stances, the ratio of the weight of a volume of air to that of an equal l-293_ 1000 ~ 773* volume of water is =**. Air is thus 773 times lighter than water. By repeating this experiment with other gases, we may determine their weight as compared with that of air, and the absolute weight of a litre of each of them. Thus it is found that a litre of oxygen weighs T43 gramme, a litre of carbonic hydrogen - 089 gramme, &c. Fig. 104. Weight of Air. acid 1'97 gramme, a litre of 1 A cubic foot of air in ordinary circumstances weighs about an ounce and a quarter. 2 In strictness, the weight in grammes of a litre of air under the pressure of 760 millimetres of mercury is different in different localities, being proportional to the inten- sity of gravity not because the force of gravity on the litre of air is different, for though this is true, it does not affect the numerical value of the weight when stated in grammes, but because the pressure of 760 millimetres of mercury varies as the intensity of gravity, so that more air is compressed into the space of a litre as gravity increases. ( 201, 6.) The weight in grammes is another name for the mass. The force of gravity on a litre of air under the pressure of 760 millimetres is proportional to the square of the intensity of gravity. This is an excellent example of the ambiguity of the word weight, which sometimes denotes a mass, sometimes a force ; and though the distinction is of no practical importance so long as we confine our attention to one locality, it cannot be neglected when different localities are compared. Regnault's determination of the weight of a litre of dry air at 0" Cent, under the pressure of 760 millimetres at Paris is 1-293187 gramme. Gravity at Paris is to gravity at Greenwich as 3456 to 3457. The corresponding number for Greenwich is therefore 1-293561 140 THE BAROMETER. 196. Atmospheric Pressure. The atmosphere encircles the earth with a layer some 50 or 100 miles in thickness; this heavy fluid mass exerts on the surface of all bodies a pressure entirely analogous both in nature and origin to that sustained by a body wholly immersed in a liquid. It is subject to the fundamental laws men- tioned in 137-139. The pressure should therefore diminish as we ascend from the surface of the earth, but should have the same value for all points in the same horizontal layer, provided that the air is in a state of equilibrium. On account of the great compressi- bility of gas, the lower layers are much more dense than the upper ones; but the density, like the pressure, is constant in value for the Fig. 105. Torricellian Experiment. same horizontal layer, throughout any portion of air in a state of equilibrium. Whenever there is an inequality either of density or pressure at a given level, wind must ensue. PRESSURE OF ONE ATMOSPHERE. 141 We owe to Torricelli an experiment which plainly shows the pressure of the atmosphere, and enables us to estimate its intensit}* with great precision. This experiment, which was performed in 1643, one year after the death of Galileo, at a time when the weight and pressure of the air were scarcely even suspected, has immor- talized the name of its author, and has exercised a most important influence upon the progress of natural philosophy. 197. Torricellian Experiment. A glass tube (Fig. 105) about a quarter or a third of an inch in diameter, and about a yard in length, is completely filled with mercury; the extremity is then stopped with the finger, and the tube is inverted in a vessel containing mercury. If the finger is now removed, the mercury will descend in the tube, and after a few oscillations will remain stationary at a height which varies according to circumstances, but which is gen- erally about 76 centimetres, or nearly 30 inches. 1 The column of mercury is maintained at this height by the pres- sure of the atmosphere upon the surface of the mercury in the vessel. In fact, the pressure at the level ABCD (Fig. 106) must be the same within as without the tube; so that the column of mercury BE exerts a pressure equal to that of the atmosphere. Accordingly, we conclude from this experiment of Torricelli that every surface exposed to the atmosphere sustains a normal pressure equal, on an average, to the weight of a column of mercui^y ivhose base is this surface, and whose height is 30 inches. It is evident that if we performed a similar experi- ment with water, whose density is to that of mercury as 1 : 13*59, the height of the column sustained would be 13*59 times as much; that is, 30xl3'59 inches, or about 34 feet. This is the maximum height to which water can be raised in a pump; as was observed by Galileo. In general the heights of columns of different liquids equal in weight to a column of air on the same base, are inversely proportional to their densities. 198. Pressure of one Atmosphere. What is usually adopted in accurate physical discussions as the standard " atmosphere " of pres- sure is the pressure due to aJheight of 76 centimetres of pure mercury at the temperature zero Centigrade, gravity being supposed to have 1 76 centimetres are 29'922 inches. Fi S 106. 142 THE BAROMETER. the same intensity which it has at Paris. The density of mercury at this temperature is 13'596; hence, when expressed in gravitation measure, this pressure is 76 X 13'596 = 1033*3 grammes per square centimetre. 1 To reduce this to absolute measure, we must multiply by the value of g (the intensity of gravity) at Paris, which is 980'94; and the result is 1013600, which is the intensity of pressure in dynes per square centimetre. In some recent works, the round number a million dynes per square centimetre has been adopted as the standard atmosphere. 199. Pascal's Experiments. It is supposed, though without any decisive proof, that Torricelli derived from Galileo the definite conception of atmospheric pressure. 2 However this may be, when the experiment of the Italian philosopher became known in France in 1644, no one was capable of giving the correct explanation of it, and the famous doctrine that " nature abhors a vacuum," by which the rising of water in a pump was accounted for, was generally accepted. Pascal was the first to prove incontestably the falsity of this old doctrine, and to introduce a more rational belief. For this purpose, he proposed or executed a series of ingenious experiments, and discussed minutely all the phenomena which were attributed to nature's abhorrence of a vacuum, showing that they \vere necessary consequences of the pressure of the atmosphere. We may cite in particular the observation, made at his suggestion, that the height of the mercurial column decreases in proportion as we ascend. This beautiful and decisive experiment, which is repeated as often as heights are measured by the barometer, and which leaves no doubt as to the nature of the force \vhich sustains the mercurial column, was performed for the first time at Clermont, and on the top of the mountain Puy-cle-D6me, on the 19th September, 1648. 200. The Barometer. By fixing the Torricellian tube in a perman- 1 This is about 147 pounds per square inch. 2 In the fountains of the Grand-duke of Tuscany some pumps were required to raise water from a depth of from 40 to 50 feet. When these were worked, it was found that they would not draw. Galileo determined the height to which the water rose in their tubes, and found it to be about 32 feet; and as he had observed and proved that air has weight, he readily conceived that it was the weight of a column of the atmosphere which maintained the water at this height in the pumps. No very useful results, however, were expected from this discovery, until, at a later date, Torricelli adopted and greatly extended it. Desiring to repeat the experiment in a more convenient form, he conceived the idea of substituting for water a liquid that is 14 times as heavy, namely, mercury, rightly imagining that a column of one-fourteenth of the length would balance the force which sustained 32 feet of water (Biot, Biographic Universette, article "Torricelli"). D. CISTKRX BAROMETER. ent position, we obtain a means of measuring the amount of the atmospheric pressure at any moment; and this pressure may be ex- pressed by the height of the column of mercury which it supports. Such an instrument is called a barometer. In order that its indica- tions may be accurate, several precautions must be observed. In the first place, the liquid used in different barometers must be identical; for the height of the column supported naturally depends upon the density of the liquid employed, and if this varies, the obser- vations made with different instruments will not be comparable. The mercury employed is chemically pure, being generally made so by washing with a dilute acid and by subsequent distillation. The baro- metric tube is filled nearly full, and is then placed upon a sloping furnace, and heated till the mer- cury boils. The object of this process is to expel the air and moisture which may be contained in the mercurial column, and which, without this pre- caution, would gradually ascend into the vacuum above, and cause a downward pressure of un- certain amount, which would prevent the mercury from rising to the proper height. The next step is to fill up the tube with pure mercury, taking care not to introduce any bubble of air. The tube is then inverted in a cistern likewise containing pure mercury recently boiled, and is firmly fixed in a vertical position, as shown in Fig. 107. We have thus a fixed barometer; and in order to ascertain the atmospheric pressure at any moment, it is only necessary to measure the height of the top of the column of mercury above the surface of the mercury in the cistern. One method of doing this is to employ an iron rod, working in a screw, and fixed vertically above the surface of the mercury in the dish. The extremities of this rod are pointed, and the lower extremity being brought down to touch the surface of the liquid below, the distance of the upper extremity from the top of the column of mercury is measured. Adding to this the Fig. 107.-- Barometer in its simplest form. 144 THE BAROMETER. length of the rod, which has previously been determined once for all, we have the barometric height. This measurement may be effected with great precision by means of the cathetometer. 201. Cathetometer. This instrument, which is so frequently em- ployed in physics to measure the vertical distance between two points, was invented by Dulong and Petit. It consists essentially (Fig. 108) of a vertical scale divided usually into half millimetres. This scale forms part of a brass cylinder capable of turning very easily about a strong steel axis. This axis is fixed on a pedestal provided with three levelling screws, and with two spirit-levels at right angles to each other. Along the scale moves a sliding frame carry- ing a telescope furnished with cross- wires, that is, with two very fine threads, usually spider lines, in the focus of the eye-piece, whose point of intersection serves to determine the line of vision. By means of a clamp and slow-motion screw, the telescope can be fixed with great precision at any required height. The telescope is also provided with a spirit-level and adjusting screw. When the apparatus is in correct adjustment, the line of vision of the telescope is horizontal, and the graduated scale is vertical. If then we wish to measure the difference of level between two points, we have only to sight them successively, and measure the distance passed over on the scale, which is done by means of a vernier attached to the sliding frame. 202. Fortin's Barometer. The barometer just described is intended to be fixed; when portability is required, the construction devised by Fortin (Fig. 109) is usually employed. It is also frequently em- Fig. 108. Cathetometer. CATHETOMETER. 14.-) ployed for fixed barometers. The cistern, which is formed of a tube of boxwood, surmounted by a tube of glass, is closed below by a piece of leather, which can be raised or lowered by means of a screw. This screw works in the bottom of a brass case, which incloses the cistern except at the middle, where it is cut away in front and at the back, so as to leave the surface of the mercury open to view. The barometric tube is encased in a tube of brass with two slits at opposite sides (Fig. 110); and it is on this tube that the divisions are engraved, the zero point from which they are reckoned being the lower extremity of an ivory point fixed in the covering of the cistern. The tem- perature of the mercury, which is required for one of the corrections mentioned in next section, is given by a thermometer with its bulb resting against the tube. A cylindrical sliding piece (shown in Fig. 110) furnished with a vernier, 1 moves along the tube and enables us to determine the height with great precision. Its lower edge is the zero of the vernier. The way in which the barometric tube is fixed upon the cistern is worth notice. In the centre of the upper surface of the copper casing there is an opening, from which rises a short tube of the same metal, lined with a tube of box- wood. The barometric tube is pushed inside, and fitted in with a piece of chamois leather, which prevents the mercury from issuing, but does not exclude the air, which, passing through the pores of the leather, penetrates into the cistern, and so transmits its pressure. Before taking an observation, the surface of the mercury is ad- 1 The vernier is an instrument very largely employed for measuring the fractions of a unit of length on any scale. Suppose we have a scale divided into inches, and another scale containing nine inches divided into ten equal parts. If now we make the end of this 10 Fig. 110. Upper portion of Barometer. Fig. 109. Cistern of Fortiu's Barometer. 146 THE BAEOMETER. justed, by means of the lower screw, to touch the ivory point. The observer knows when this condition is fulfilled by seeing the extremity of the point touch its image in the mercury. The sliding piece which carries the vernier is then raised or lowered, until its base is seen to be tangential to the upper surface of the mercurial column, as shown in Fig. 110. In making this adjustment, the back of the instrument should be turned towards a good light, in order that the observer may be certain of the position in which the light is just cut off at the summit of the convexity. When the instrument is to be carried from place to place, precau- tions must be taken to prevent the mercury from bumping against the top of the tube and breaking it. The screw at the bottom is to be turned until the mercury reaches the top of the tube, and the instrument is then to be inverted and carried upside down. We may here remark that the goodness of the vacuum in a bar- ometer, can be tested by the sound of the mercury when it strikes the top of the tube, which it can be made to do either by screwing latter scale, which is called the vernier, coincide with one of the divisions in the scale of inches, as each division of the vernier is T ^ of an inch, it is evident that the first division on the scale will be ^ of an inch beyond the first division on the vernier, the second on the scale T \ beyond the second on the vernier, and so on until the ninth on the scale, which Fig. 111. Vernier. will exactly coincide with the tenth on the vernier. Suppose next that in measuring any length we find that its extremity lies between the degrees 5 and 6 on the scale; we bring the zero of the vernier opposite the extremity of the length to be measured, and observe what division on the vernier coincides with one of the divisions on the scale. We see in the figure that it is the seventh, and thus we conclude that the fraction required is ^ of an inch. If the vernier consisted of 19 inches divided into 20 equal parts, it would read to the -jV of an inch ; but there is a limit to the precision that can thus be obtained. An exact coin- cidence of a division on the vernier with one on the scale seldom or never takes place, and we merely take the division which approaches nearest to this coincidence; so that when the difference between the degrees on the vernier and those on the scale is very small, there may be so much uncertainty in this selection as to nullify the theoretical precision of the instrument. Verniers are also employed to measure angles ; when a circle is divided into half degrees, a vernier is used which gives ^ of a division on the circle, that is, -$$ of a half degree, or one minute. D. PORTABLE BAROMETER. up or by inclining the instrument to one side. If the vacuum is good, a metallic clink will be heard, and unless the contact be made very gently, the tube will be broken by the sharpness of the col- lision. If any air be present, it acts as a cushion. In making observa- tions in the field, a barometer is usually suspended from a tri- pod stand (Fig. 112) by gimbals 1 , so that it always takes a vertical position. 203. Float Adjust- ment. In some barom- eters the ivory point for indicating the proper level of the mercury in the cistern is replaced by a float. F (Fig. 113) is a small ivory piston, having the float at- tached to its foot, and moving freely up and down between the two ivory guides I. A hori- zontal line (interrupted by the piston) is en- graved on the two guides, and another is Fig. 112. Barometer with Tripod Stand. engraved on the piston, at such a height that the three lines form one straight line when the surface of the mercury in the cistern stands at the zero point of the scale. 204. Barometric Corrections. In order that barometric heights 1 A kind of universal joint, in common use on board ship for the suspension of com- passes, lamps, &c. It is seen in Fig. 112, at the top of the tripod stand. 148 THE BAROMETER. may be comparable as measures of atmospheric pressure, certain cor- rections must be applied. 1. Correction for Temperature. As mercury expands with heat, it follows that a column of warm mercury exerts less pressure than a column of the same height at a lower temperature; and it is usual to reduce the actual height of the column to the height of a column at the temperature of freezing w T ater which would exert the observed height at temperature f same pressure. Let h be the Centigrade, and h the height reduced to freezing- point. Then, if m be the coefficient of expansion of mercury per degree Cent., we have h a (1 + m t) =h, whence k o = h hmt nearly. The value of m is g^ = 4 00018018. For temperatures Fahrenheit, we have h {l + m, (<-32) } -h,h o -U-lim (-32), where m denotes g^='0001001. But temperature also affects the length of the divisions on the scale by which the height of the mercurial column is measured. If these divisions be true inches at Cent., then at t the length of n divisions will be n (1 + It) inches, I denoting the coefficient of linear expansion of the scale, the value of which for brass, the usual material, is '00001878. If then the observed height h amounts to n divisions of the scale, we have h o (1 +mt)=h = n (1 + It); whence Fig. 113. Float Adjustment. ft.- n (1 + 1 1) 1 + mt = n - nt (m - I), nearly ; that is to say, if n be the height read off on the scale, it must be diminished by the correction nt (m l), t denoting the temperature of the mercury in degrees Centigrade. The value of m l is 0001614. For temperatures Fahrenheit, assuming the scale to be of the correct length at 32 Fahr., the formula for the correction (which is still subtractive), is n (t 32) (m l), where m l has the value 00008967. 1 1 The correction for temperature is usually made by the help of tables, which give its amount for all ordinary temperatures and heights. These tables, when intended for CORRECTIONS. 149 '2. Correction for Capillarity. In the preceding chapter we have seen that mercury in a glass tube undergoes a capillary depression: whence it follows that the observed barometric height is too small, and that we must add to it the amount of this depression. In all tubes of internal diameter less than about f of an inch this correction is sensible ; and its amount, for which no simple formula can be given, has been computed, from theoretical considerations, for various sizes of tube, by several eminent mathematicians, and recorded in tables, from which that given below is abridged. These values are appli- cable on the assumption that the meniscus which forms the summit of the mercurial column is decidedly convex, as it always is when the mercury is rising. When the meniscus is too flat, the mercury must be lowered by the foot-screw, and then screwed up again. It is found by experiment, that the amount of capillary depression is only half as great when the mercury has been boiled in the tube as when this precaution has been neglected. For purposes of special accuracy, tables have been computed, giving the amount of capillary depression for different degrees of convexity, as determined by the sagitta (or height) of the meniscus, taken in conjunction with the diameter of the tube. Such tables, however, are seldom used in this country. 1 English barometers, are generally constructed on the assumption that the scale is of the correct length not at 32 Fahr., but at 62 J Fahr., which is (by act of Parliament) the temperature at which the British standard yard (preserved in the office of the Exchequer) is correct. On this supposition, the length of n divisions of the scale at temperature t Fahr., is ujl + J(*-62)j-; and by equating this expression to A |l +m (-32)j- we find = n|l - (m-l) t+ (32m- 62Z)} = n |l - -00008967 t + -00255654 ] ; which, omitting superfluous decimals, may conveniently be put in the form The correction vanishes when 09 t - 2-56 = 0; that is, when < = ^ = 28'5. 9 For all temperatures higher than this the correction is subtractive. 1 The most complete collection of meteorological and physical tables, is that edited by Professor Guyot, and published under the auspices of the Smithsonian Institution, Wash- ington. 150 THE BAROMETER. TABLE OF CAPILLARY DEPRESSIONS IN UNBOILED TUBES. (To be halved for Soiled, Tubes.) Diameter of tube iii inches. Depression in inches. Diameter. Depression. Diameter. Depression. 10 140 20 058 40 015 11 126 22 050 42 013 12 114 24 044 44 on 13 104 26 038 46 009 14 094 28 033 48 008 15 086 30 029 50 007 16 079 32 026 55 005 17 073 34 023 60 004 18 068 36 020 65 003 19 063 38 017 70 002 3. Correction for Capacity. When there is no provision for ad- justing the level of the mercury in the cistern to the zero point of the scale, another correction must be applied. It is called the cor- rection for capacity. In barometers of this construction, which were formerly much more common than they are at present, there is a certain point in the scale at which the mercurial column stands when the mercury in the cistern is at the correct level. This is called the neutral point. If A be the interior area of the tube, and C the area of the cistern (exclusive of the space occupied by the tube and its contents), when the mercury in the tube rises by the amount x, the mercury in the cistern falls by an amount y = -^x; for the volume of the mercury which has passed from the cistern into the tube is C y = A x. The change of atmospheric pressure is correctly measured by x + y=(l-\-Q\x; and if we now take x to denote the distance of the summit of the mercurial column from the neutral point, the cor- rected distance will be ( l + ?j) #> and the correction to be applied to the observed reading will be ^ x, which is additive if the observed reading be above the neutral point, subtractive if below. It is worthy of remark that the neutral point depends upon the volume of mercury. It will be altered if any mercury be lost or added; and as temperature affects the volume, a special temperature- correction must be applied to barometers of this class. The investi- gation will be found in a paper by Professor Swan in the Philo- sophical Magazine for 1861. In some modern instruments the correction for capacity is avoided, by making the divisions on the scale less than true inches, in the CORRECTIONS. 151 Q ratio . c , and the effect of capillarity is at the same time compen- sated by lowering the zero point of the scale. Such instruments, if correctly made, simply require to be corrected for temperature. 4. Index Errors. Under this name are included errors of gradua- tion, and errors in the position of the zero of the graduations. An error of zero makes all readings too high or too low by the same amount. Errors of graduation (which are generally exceedingly small) are different for different parts of the scale. Barometers intended for accurate observation are now usually examined at Kew Observatory before being sent out; and a table is furnished with each, showing its index error at every half inch of the scale, errors of capillarity and capacity (if any) being included as part of the index error. We may make a remark here once for all respecting the signs attached to errors and corrections. The sign of an error is always opposite to that of its correction. When a reading is too high the index error is one of excess, and is there- fore positive; whereas the correction needed to make the reading true is subtractive, and is therefore negative. 5. Reduction to Sea-level. In comparing barometric observations taken over an extensive district for meteorological purposes, it is usual to apply a correction for difference of level. Atmospheric pressure, as we have seen, diminishes as we ascend; and it is usual to add to the observed height the difference of pressure due to the elevation of the place above sea-level. The amount of this correc- tion is proportional to the observed pressure. The law according to which it increases with the height will be discussed in the next chapter. 6. Correction for Unequal Intensity of Gravity. When two barometers indicate the same height, at places where the intensity of gravity is different (for example, at the pole and the equator), the same mass of air is superincumbent over both; but the pressures are unequal, being proportional to the intensity of gravity as measured by the values of g ( 91) at the two places. If h be the height, in centimetres, of the mercurial column at the temperature Cent., the absolute pressure, in dynes per square centimetre, will be gh X 13'596; since 13'596 is the density of mercury at this temperature. 205. Other kinds of Mercurial Barometer. The Siphon Barometer, which is represented in Fig. 114, consists of a bent tube, generally 152 THE BAROMETER. of uniform bore, having two unequal legs. The longer leg, which must be more than 30 inches long, is closed, while the shorter leg is open. A sufficient quantity of mercury having been introduced to fill the longer leg, the instrument is set upright (after boiling to expel air), and the mercury takes such a position that the difference of levels in the two legs represents the pressure of the atmosphere. Supposing the tube to be of uniform section, the mer- cury will always fall as much in one leg as it rises in the other. Each end of the mercurial column therefore rises or falls through only half the height corresponding to the change of atmospheric pressure. In the best siphon barometers there are two scales, one for each leg, as indicated in the figure, the divisions on one being reckoned upwards, and on the other down- wards, from an intermediate zero point, so that the sum of the two readings is the difference of levels of the mercury in the two branches. Inasmuch as capillarity tends to depress both extrem- ities of the mercurial column, its effect is generally neglected in siphon barometers; but practically it causes great difficulty in obtaining accurate observations, for according as the mercury is rising or falling its ex- tremity is more or less convex, and a great deal of tapping is usually required to make both ends of the column assume the same form, which is the condition necessary for annihilating the effect of capillary action. Wheel Barometer. The wheel barometer, which is in more gen- eral use than its merits deserve, consists of a siphon barometer, the two branches of which have usually the same diameter. On the surface of the mercury of the open branch floats a small piece of iron or glass suspended by a thread, the other extremity of which is fixed to a pulley, on which the thread is partly rolled. Another thread, rolled parallel to the first, supports a weight which balances the float. To the axis of the pulley is fixed a needle which moves on a dial. When the level of the mercury varies in either direction, the float follows its movement through the same distance; by the action of the counterpoise the pulley turns, and with it the needle, the extremity of which points to the figures on the dial, marking the barometric heights. The mounting of the dial is usually placed Fig. 114. Siphon Barometer. SIPHON AND WHEEL BAROMETERS. 153 in front of the tube, so as to conceal its presence. The wheel barometer is a very old invention, and was introduced by the celebrated Hooke in 1683. The pulley and strings are sometimes replaced by a rack and pinion, as represented in the figure (Fig. 115). Besides the faults incidental to the siphon barometer, the wheel Fig. 115. Wheel Barometer. barometer is encumbered in its movements by the friction of the additional apparatus. It is quite unsuitable for measuring the exact amount of atmospheric pressure, and is slow in indicating changes. Marine Barometer. The ordinary mercurial barometer cannot be used at sea on account of the violent oscillations which the mercury would experience from the motion of the vessel. In order to meet this difficulty, the tube is contracted in its middle portion nearly to 154 THE BAROMETER, capillary dimensions, so that the motion of the mercury in either direction is hindered. An instrument thus constructed is called a marine barometer. When such an instrument is used on land it is always too slow in its indications. 206. Aneroid Barometer (, ^poc). This barometer depends upon the changes in the form of a thin metallic vessel partially exhausted Fig. 116. Aneroid Barometer. of air, as the atmospheric pressure varies. M. Vidie was the first to overcome the numerous difficulties which were presented in the con- struction of these instruments. We subjoin a figure of the model which he finally adopted. The essential part is a cylindrical box partially exhausted of air, the upper surface of which is corrugated in order to make it yield more easily to external pressure. At the centre of the top of the box is a small metallic pillar M, connected with a powerful steel spring R. As the pressure varies, the top. of the box rises or falls, transmitting its movement by two levers I and m, to a metallic axis r. This latter carries a third lever t, the extremity of which is attached to a chain s which turns a drum, the axis of which bears the index needle. A spiral spring keeps the chain constantly stretched, and thus makes the needle always take a position corre- ANEROID. 155 spending to the shape of the box at the time. The graduation is performed empirically by comparison with a mercurial barometer. The aneroid barometer is very quick in indicating changes, and is much more portable than any form of mercurial barometer, being- bo th lighter and less liable to injury. It is sometimes made small enough for the M^aistcoat pocket. It has the drawback of being affected by temperature to an extent which must be determined for each instrument separately, and of being liable to gradual changes which can only be checked by occasional comparison with a good mercurial barometer. In the metallic barometer, which is a modification of the aneroid, the exhausted box is crescent-shaped, and the horns of the crescent separate or approach according as the external pressure diminishes or increases. 207. Old Forms Revived. There are two ingenious modifications of the form of the barometer, which, after long neglect, have recently been revived for special purposes. Counterpoised Barometer. The invention of this instrument is attributed to Samuel Morland, who constructed it about the year 1680. It depends upon the following principle: If the barometric tube is suspended from one of the scales of a balance, there will be required to balance it in the other scale a weight equal to the weight of the tube and the mercury contained in it, minus the upward pressure due to the liquid displaced in the cistern. 1 If the atmo- spheric pressure increases, the mercury will rise in the tube, and consequently the weight of the floating body will increase, while the sinking of the mercury in the cistern will diminish the upward pressure due to the displacement. The beam will thus incline to 1 A complete investigation based on the assumption of a constant upward pull at the top of the suspended tube shows that the sensitiveness of the instrument depends only on the internal section of the upper part of the tube and the external section of its lower part. Calling the former A and the latter B, it is necessary for stability that B be greater than A (which is not the case in the figure in the text) and the movement of the tube will be to that of the mercury in a standard barometer as A is to B - A. The directions of these movements will be opposite. If B - A is very small compared with A, the instrument will be exceedingly sensitive; and as B-A changes sign, by passing through zero, the equilibrium becomes unstable. A curious result of the investigation is that the level of the mercury in the cistern re- mains constant. In the instrument represented in the figure, stability is probably obtained by the weight of the arm which carries the pencil. In King's barograph, B is made greater than A by fixing a hollow iron drum round the lower end of the tube. 156 THE BAEOMETER. the side of the barometric tube, and the reverse movement would occur if the pressure diminished. For the balance may be substi- tuted, as in Fig. 117, a lever carrying a counterpoise; the variations of pressure will be indicated by the movements of this lever. Such an instrument may very well be used as a barograph or re- cording barometer; for this purpose we have only to attach to the lever an arm with a pencil, which is con- stantly in contact with a sheet of paper moved uniformly by clock-work. The result will be a continuous trace, whose form corresponds to the variations of pressure. It is very easy to deter- mine, either by calcula- tion or by comparison with a standard baro- meter, the pressure cor- responding to a given position of the pencil on the paper; and thus, if the paper is ruled with twenty-four equidistant lines, corresponding to the twenty-four hours of the day, we can see at a glance what was the pressure at any given time. An arrangement of this kind has been adopted by the Abbe Secchi for the meteorograph of the observatory at Rome. The first successful employment of this kind of barograph appears to be due to Mr. Alfred King, a gas engineer of Liverpool, who invented and constructed such an instrument in 1853, for the use of the Liverpool Observatory, and subsequently designed a larger one, which is still in use, furnishing a very perfect record, magnified five-and-a-half times. Fahrenheit's Barometer. Fahrenheit's barometer consists of a tube bent several times, the lower portions of which contain mercury; the upper portions are filled with water, or any other liquid, usually Fig. 117. Counterpoised Barometer. PHOTOGRAPHIC REGISTRATION. Fig. 118. Fahrenheit's Barometer. coloured. It is evident that the atmospheric pressure is balanced by the sum of the differences of level of the columns of mercury, dimin- ished by the sum of the corresponding differences for the columns of water; whence it follows that, by employing a considerable number of tubes, we may greatly reduce the height of the barometric column. This circum- stance renders the instrument interesting as a scientific curiosity, but at the same time diminishes its sensitiveness, and renders it unfit for purposes of precision. It is therefore never used for the measurement of atmospheric pressure; but an instrument upon the same prin- ciple has recently been employed for the measurement of very high pressures, as will be explained in Chap. xix. 208. Photographic Registration. Since the year 1847 various meteorological instruments at the Royal Observatory, Greenwich, have been made to yield continuous traces of their indications by the aid of photography, and the method is now generally employed at meteorological observatories in this country. The Greenwich system is fully described in the Greenwich Magnetical and Meteorological Observations for 1847, pp. Ixiii.-xc. (published in 1849). The general principle adopted for all the instruments is the same. The photographic paper is wrapped round a glass cylinder, and the axis of the cylinder is made parallel to the direction of the move- ment which is to be registered. The cylinder is turned by clock- work, with uniform velocity. The spot of light (for the magnets and barometer), or the boundary of the line of light (for the ther- mometers), moves, with the movements which are to be registered, backwards and forwards in the direction of the axis of the cylinder, while the cylinder itself is turned round. Consequently (as in Morin's machine, Chap, vii.), when the paper is unwrapped from its cylindrical form, there is traced upon it a curve of which the abscissa is proportional to the time, while the ordinate is proportional to the movement which is the subject of measure. The barometer employed in connection with this system is a large siphon barometer, the bore of the upper and lower extremities of its arms being about I'l inch. A glass float in the quicksilver of the 158 THE BAROMETEK. lower extremity is partially supported by a counterpoise acting on a light lever (which turns on delicate pivots), so that the wire support- ing the float is constantly stretched, leaving a definite part of the weight of the float to be supported by the quicksilver. This lever is lengthened to carry a vertical plate of opaque mica with a small aper- ture, whose distance from the fulcrum is eight times the distance of the point of attachment of the float-wire, and whose movement, therefore ( 205), is four times the movement of the column of a cis- tern barometer. Through this hole the light of a lamp, collected by a cylindrical lens, shines upon the photographic paper. Every part of the cylinder, except that on which the spot of light falls, is covered with a case of blackened zinc, having a slit parallel to the axis of the cylinder; and by means of a second lamp shining through a small fixed aperture, and a second cylindrical lens, a base line is traced upon the paper, which serves for reference in subsequent measurements. The whole apparatus, or any other apparatus which serves to give a continuous trace of barometric indications, is called a barograph; and the names thermograph, magnetograph, anemograph, &c., are similarly applied to other instruments for automatic registration. Such registration is now employed at a great number of observa- tories; and curves thus obtained are regularly published in the Quarterly Keports of the Meteorological Office. CHAPTER XVIII. VAKIATIONS OF THE BAROMETER. 209. Measurement of Heights by the Barometer. As the height of the barometric column diminishes when we ascend in the atmo- sphere, it is natural to seek in this phenomenon a means of measuring heights. The problem would be extremely simple, if the air had everywhere the same density as at the surface of the earth. In fact, the density of the air at sea-level being about 10,500 times less than that of mercury, it follows that, on the hypothesis of uniform density, the mercurial column would fall an inch for every 10,500 inches, or 875 feet that we ascend. This result, however, is far from being in exact accordance with fact, inasmuch as the density of the air diminishes very rapidly as we ascend, on account of its great compressibility. 210. Imaginary Homogeneous Atmosphere. If the atmosphere were of uniform and constant density, its height would be approxi- mately obtained by multiplying 30 inches by 10,500, which gives 26,250 feet, or about 5 miles. More accurately, if we denote by H the height (in centimetres) of the atmosphere at a given time and place, on the assumption that the density throughout is the same as the observed density D (in grammes per cubic centimetre) at the base, and if we denote by P the observed pressure at the base (in dynes per square centimetre), we must employ the general formula for liquid pressure ( 139) P P -g HD, which gives H = -g- (1) The height H, computed on this imaginary assumption, is usually called the height of the homogeneous atmosphere, corresponding to the pressure P, density D, and intensity of gravity g. It is some- times called the pressure-height. The pressure-height at any point 160 VARIATIONS OF THE BAROMETER. in a liquid or gas is the height of a column of fluid, having the same density as at the point, which would produce, by its weight, the actual pressure at the point. This element frequently makes its appearance in physical and engineering problems. The expression for H contains P in the numerator and D in the denominator; and by Boyle's law, which we shall discuss in the ensu- ing chapter, these two elements vary in the same proportion, when the temperature is constant. Hence H is not affected by changes of pressure, but has the same value at all points in the air at which the temperature and the value of g are the same. 211. Geometric Law of Decrease. The change of pressure as we ascend or descend for a short distance in the actual atmosphere, is sensibly the same as it would be in this imaginary " homogeneous atmosphere;" hence an ascent of 1 centimetre takes off g of the total pressure, just as an ascent of one foot from the bottom of an ocean 60,000 feet deep takes off 60 ^ 00 of the pressure. Since H is the same at all heights in any portion of the air which is at uniform temperature, it follows that in ascending by successive steps of 1 centimetre in air at uniform temperature, each step takes off the same fraction g of the current pressure. The pressures there- fore form a geometrical progression whose ratio is 1 g. In an at- mosphere of uniform temperature, neglecting the variation of g with height, the densities and pressures diminish in geometrical progres- sion as the heights increase in arithmetical progression. 212. Computation of Pressure-height. For perfectly dry air at Cent., we have the data ( 195, 198), D = -0012932 when P = 1013600; which give P - = 783800000 nearly. Taking g as 981, we have H = lligonjuia = 799000 centimetres nearly. This is very nearly 8 kilometres, or about 5 miles. At the temper- ature t Cent., we shall have H = 799000 (1 -f -00366 *). (2) Hence in air at the the temperature Cent., the pressure diminishes by 1 per cent, for an ascent of about 7990 centimetres or, say, 80 metres. At 20 Cent., the number will be 86 instead of 80. HYPSOMETRIC AL FORMULA. 161 213. Formula for determining Heights by the Barometer. To obtain an accurate rule for computing the difference of levels of two stations from observations of the barometer, we must employ the integral calculus. L>enote height above a fixed level by x, and pressure by p. Then we have dx _ dp H - - p ' and if p l} p 2 are the pressures at the heights cc : , X 2 , we deduce by in- tegration 2 - Xi = H (loge Pi - loge Pi). Adopting the value of H from (2), and remembering that Napierian logarithms are equal to common logarithms multiplied by 2'3026, we finally obtain x^ - Xj, - 1840000 (1 + -00366 1) (log p l - log pj as the expression for the difference of levels, in centimetres. It is usual to put for t the arithmetical mean of the temperatures at the two stations. The determination of heights by means of atmospheric pressure, whether the pressure be observed directly by the barometer, or in- directly by the boiling-point thermometer (which will be described in Part II.), is called hypsometry (ut/'oc, height). As a rough rule, it may be stated that, in ordinary circumstances, the barometer falls an inch in ascending 900 feet. 214. Diurnal Oscillation of the Barometer. In these latitudes, the mercurial column is in a continual state of irregular oscillation; but in the tropics it rises and falls with great regularity according to the hour of the day, attaining two maxima in the twenty-four hours. It generally rises from 4 A.M. to 10 A.M., when it attains its first maximum; it then falls till 4 P.M., when it attains its first minimum; a second maximum is observed at 10 P.M., and a second minimum at 4 A.M. The hours of maxima and minima are called the tropical hours (r/)7rw, to turn), and vary a little with the season of the year. The difference between the highest maximum and lowest minimum is called the diurnal 1 range, and the half of this is called the ampli- 1 The epithets annual and diurnal, when prefixed to the words variation, range, ampli- tude, denote the period of the variation in question ; that is, the time of a complete oscilla- tion. Diurnal variation does not denote variation from one day to another, but the varia- tion which goes through its cycle of values in one day of twenty-four hours. Annual 11 102 VARIATIONS OF THE BAROMETER. tude of the diurnal oscillation. The amount of the former does not exceed about a tenth of an inch. The character of this diurnal oscillation is represented in Fig. 119. The vertical lines correspond to the hours of the day; lengths have been measured upwards upon them proportional to the barometric heights at the respective hours, diminished by a constant quantity; and the points thus determined have been connected by a continuous curve. It will be observed that the two lower curves, one of which relates to Cumana, a town of Venezuela, situated in about 10 north latitude, show strongly marked oscillations corresponding to the maxima and minima. In our own country, the regular diurnal oscilla- tion is masked by irregular fluctuations, so that a single day's observations give no clue to its existence. Nevertheless, on taking observations at regular hours for a number of consecutive days, and comparing the mean j 3 e 9 12 15 is 21 24 heights f or the different hours, some indications Curves of Dfuma^ Variation. f the laW wil1 be f UIld - A month's observa- tions will be sufficient for an approximate indication of the law; but observations extending over some years will be required, to establish with anything like precision the hours of maxima and the amplitude of the oscillation. The two upper curves represent the diurnal variation of the baro- meter at Padua (lat. 45 24') and Abo (lat. 60 56'), the data having been extracted from Kaemtz's Meteorology. We see, by inspection of the figure, that the oscillation in question becomes less strongly marked as the latitude increases. The range at Abo is less than half a millimetre. At about the 70th degree of north latitude it becomes insensible; and in approaching still nearer to the pole, it appears from observations, which however need further confirmation, that the oscillation is reversed; that is to say, that the maxima here are contemporaneous with the minima in lower latitudes. There can be little doubt that the diurnal oscillation of the barometer is in some way attributable to the heat received from the sun, which produces expansion of the air, both directly, as a mere range denotes the range that occurs within a year. This rule is universally observed by writers of high scientific authority. A table, exhibiting the values of an element for each month in the year, is a table of annual (not monthly) variation ; or it may be more particularly described as a table of variations from month to month. PREDICTION OF WEATHER. 16o consequence of heating, and indirectly, by promoting evaporation: but the precise nature of the connection between this cause and the diurnal barometric oscillation has not as yet been satisfactorily established. 215. Irregular Variations of the Barometer. The height of the barometer, at least in the temperate zones, depends on the state of the atmosphere; and its variations often serve to predict the changes of weather with more or less certainty. In this country the baro- meter generally falls for rain or S.W. wind, and rises for fine weather or N.E. wind. Barometers for popular use have generally the words Set fair. Fair. Change. Rain. Much rain. Stormy. marked at the respective heights 30-5 30 29'5 29 28'5 28 inches. These words must not, however, be understood as absolute predic- tions. A low barometer rising is generally a sign of fine, and a high barometer falling of wet weather. Moreover, it is to be borne in mind that the barometer stands about a tenth of an inch lower for every hundred feet that we ascend above sea-level. The connection between a low or falling barometer and wet weather is to be found in the fact that moist air is specifically lighter than dry, even at the same temperature, and still more when, as usually happens, moist air is warmer than dry. Change of wind usually begins in the upper regions of the aii and gradually extends downwards to the ground; hence the baro- meter, being affected by the weight of the whole superincumbent atmosphere, gives early warning. 216. Weather Charts. Isobaric Lines. The extension of tele- graphic communication over Europe has led to the establishment of a system of correspondence, by which the barometric pressures, at a given moment, at a number of stations which have been selected for meteorological observation, are known at one or more stations appointed for receiving the reports. From the information thus furnished, curves (called isobaric lines, or isobars) are drawn, upon a chart, through those places at which the pressure is the same. The barometric condition of an extensive region is thus rendered intelligible at a glance. Plate I. is a specimen of one of these 1G4 VARIATIONS OF THE BAROMETER. charts, 1 prepared at the observatory of Paris; it refers to the 22d of January, 1868. Besides the isobaric lines, the charts indicate, by the system of notation explained at the left of the figure, the general state of the weather, the strength of wind, and state of the sea. The isobaric curves correspond to differences of five millimetres (about ; 2 inch) of pressure, and according as they are near together or far apart the variation of pressure in passing from one to another is more or less sudden (or to use a very expressive modern phrase, the barometric gradient is more or less steep), just as the contour lines on a map of hilly ground approach each other most nearly where the ground is steepest. Charts on the same general plan are issued daily from the Meteorological Office in London. A steep barometric gradient tends to produce a strong wind. It will be observed, however, from the arrows on the chart, that the direction of the wind, instead of being coincident with the line of steepest descent from each isobar to the next below it, generally makes a large angle, considerably exceeding 45, to the right of it. In the southern hemisphere the deviation is to the left instead of to the right. This law, known as Buys Ballot's, is found to hold in , y of the whole quantity or mass M. The quantity remaining after the second stroke is y> . y of that after the / V \ 2 / V \ n first, or is f y , y j M; and after n strokes ( y , y j M. Hence the density and (by Boyle's law) the pressure are each reduced by n strokes to (^-,-y ) of their original values. This calculation gives the theoretical rate of exhaustion for a perfect pump. Ordinary pumps come nearly up to this standard during the earlier part of the process of exhaustion; but as further progress is made, the imperfections of the apparatus become more sensible, and set a limit to the exhaustion attainable. 231. Mercurial Gauges. To enable the operator to observe the progress of the exhaustion, the instrument is usually provided with a mercurial gauge. Sometimes, as in Fig. 130, this consists of a short siphon-barometer, the difference of levels between its two columns being the measure of the pressure in the receiver. Another plan is to have a straight tube open at both ends, and more than 30 KATE OF EXHAUSTION. 181 inches long; its upper end being connected with the receiver, while its lower end dips into a cistern of mercury. As exhaustion pro- ceeds, the mercury rises in this tube, and its height above the mercury in the cistern measures the difference between the pressure in the receiver and that in the external air. 232. Admission Stop-cock. After the receiver has been exhausted of air, if it were required to raise it from the plate, a very consider- able force would be necessary, amounting to as many times fifteen pounds as the base of the receiver contained square inches. This difficulty is obviated by having an admission stop-cock R, which is shown in section above. It is perforated by a straight channel, which, when the machine is being worked, forms part of the com- municating passage. At 90 from the extremities of this channel is another opening O, forming the mouth of a bent passage, leading to the external air. When we wish to admit the air into the receiver, we have only to turn the stop-cock so as to bring the opening to the side next the receiver; if, on the contrary, we turn it towards the pump-barrel, all communication between the pump and the receiver is stopped, the risk of air entering is diminished, and the vacuum remains good for a greater length of time. This precaution is taken when we wish to leave bodies in a vacuum for a consider- able time. Another method is to employ a separate plate, which can be detached so as to leave the machine available for other pur- poses. 233. Double-barrelled Air-pump. The machine just described has only a single pump-barrel; air-pumps of this kind are sometimes employed, and are usually worked by a lever like a pump-handle. With this arrangement, it is evident that no air is expelled in the down-stroke; and that the piston, after having expelled the air from the barrel in the up-stroke, must descend idle in order to prepare for the next stroke. Double-barrelled pumps are more frequently used. An idea of their general arrangement may be formed from Figs. 131, 182, and 133. Fig. 133 gives the machine in perspective, Fig. 131 is a section through the axes of the pump-barrels, and Fig. 132 shows the manner in which communication is established between the receiver and the two barrels. It will be observed that the two passages from the barrels unite in a single passage to the centre of the plate p. Two racks carrying the pistons CO work with the pinion P. This pinion is turned by a double-handed lever, which is moved alter- 182 AIR-PUMP. nately in opposite directions. In this arrangement, when one piston ascends the other descends, and consequently in each single stroke the air of the receiver passes into one or other pump-barrel. A vacuum is thus produced by half the number of strokes which would be required with a single-barrelled pump. It has besides another advantage, as compared with the single-barrelled pump above described. In that pump the force required to raise the piston increases as the exhaus- tion proceeds, and when it is nearly completed there is the resistance of almost an atmosphere to be overcome. In the Double-barrelled Air-pump. Fig. 132. double-barrelled pump, with the same construction of barrel, the force opposing the ascent of one piston is precisely equal, at the "beginning of each stroke, to that which assists the descent of the other. This equality, however, exists only at the beginning of the stroke; for the air below the descending piston is compressed, and its tension increases till it becomes equal to that of the atmo- sphere and raises the piston valve. During the remainder of the stroke, the resistance to the ascent of the other piston is entirely uncompensated, and up to this point the compensation has been gradually diminishing. But the more nearly we approach to a perfect vacuum, the later in the stroke does this compensation occur. DOUBLE-BARRELLED PUMP. 183 The pump, accordingly, becomes easier to work as the exhaustion proceeds. 234. Single-barrelled Pumps with Double Action. We do not, however, require two pump-barrels in order to obtain double action, Fig. 133. Air-pump. as the same effect may be obtained with a single barrel. An arrange- ment for this purpose was long ago suggested by Delahire for water- pumps; but the principle has only lately been applied to the con- struction of air-pumps. Fig. 134 represents the single barrel of the double-acting pump of Bianchi. It will be seen that the piston- valve opens into the hollow piston-rod; a second valve, also opening upwards, is placed at the top of the pump-barrel. Two other openings, one above, the other below, serve to establish communication, by means of a bent vertical tube, between the pump-barrel and the passage to the plate. These openings are closed alternately by two conical stoppers at the two extremities of a metal rod passing through the piston, and carried with it in its vertical movement by means of friction. When the AIR-PUMP. piston ascends, as in the figure, the upper opening is closed and the lower one is open; when the piston begins to descend, the opposite effect is immediately produced. Accordingly we see that, whichever be the direction in which the piston is moving, the receiver is being exhausted of air. In fact, when the pis- ton ascends, air from the receiver will enter by the lower opening, and the air above the piston will be gradually com- pressed, and will finally escape by the valve above. In the descending move- ment, air will enter by the upper opening, and the compressed air beneath the piston will escape by the piston-valve. The movement of the piston is produced by a peculiar arrangement shown in Fig. 135, which gives a general view of the ap- paratus. The pump-barrel, which is composed entirely of cast-iron, oscillates about an axis passing through its base. On the top are guides in which the end of a crank travels. The pump is worked by turning a heavy fly- wheel of cast-iron, on the axis of which is a pinion which drives a toothed wheel on the axis of the crank. The end of the crank is attached to the extremity of the piston-rod. It is evident that on turning the fly-wheel the pump-barrel will oscillate from side to side, following the motions of the crank, and the piston will alternately ascend and descend in the barrel, the length of which should be equal to the diameter of the circle described by the end of the crank. 235. English forms of Air-pump. Some of the drawbacks to the single- barrelled pump are obviated by inserting a valve, opening upwards, in the top of the barrel as at U, Fig. 136. The top of the piston is thus relieved from atmospheric pressure, and the operation of pumping does not become more laborious as Fig. 134. Barrel of Biaiichi's Air pump. Fig. 136 186 AIR-PUMP. the exhaustion proceeds, but less laborious, the difference being most marked when tho receiver is small. In the up-stroke, the piston- valve V keeps shut, and the air above the piston is pushed out of the barrel through the valve U. In the down-stroke, TJ is kept closed by the preponderance of atmospheric pressure outside, and V opens, allowing the air to pass up through it as the piston descends to the bottom of the barrel. When the exhaustion is far advanced, U does not open till the piston has nearly reached the top. This is a simple and good form of pump. Another form very much in use in this country is the double-act- ing pump of Professor T. Tate, the working parts of which are shown in Fig. 137. CD is the barrel; A and B are two solid pistons rigidly connected by a rod, and moved by the piston-rod AH, which passes through a stuffing- box S. W are valves in the two ends of the barrel, both opening outwards, and II is a passage leading from the middle of the cylinder to the receiver. The distance between the extreme faces of the pistons is about f ths of an inch less than half the length of the cylinder. The volume of air expelled at each single stroke is thus about half the volume of the cylinder. This figure and description are in accordance with the original account of the pump given by the inventor in the Philosophical Magazine. It is now usual to replace the two pistons by a single piston of great thickness, its two faces being as far apart as the extreme faces of the two pistons in the figure. It is also usual to make the barrel horizontal. The valves of these pumps, and of most English pumps are " silk valves." They consist of a short and narrow slit in a thin plate of brass, with a flap of oiled silk secured at both ends to the plate, in such a position that its central portion covers the slit. When the pressure of the air is greater on the further side of the plate than on the side where the silk is, the flap is slightly lifted and the air gets through ; but excess of pressure on the near side presses the flap down over the slit and makes it air-tight. 236. Various Experiments with the Air-pump. At the time when the air-pump was invented, several experiments were devised to .show the effects of a vacuum, some of which have become classieal and are usually repeated in courses of experimental physics. Burst Bladder. On the plate of an air-pump (Fig. 138) is EXPERIMENTS. 187 placed a glass cylinder open at the bottom, and having a piece of Madder or thin indian-rubber tightly stretched over the top. As the exhaustion proceeds, this bends inwards in consequence of the atmospheric pressure above it, and finally bursts with a loud report. Magdeburg Hemispheres. We take two hemispheres (Fig. 139), which can be exactly fitted on each other; their exact adjustment is further assisted by a projecting internal rim, which is smeared with lard. The apparatus is exhausted of air through the medium of the stop- cock attached to one of the hemispheres ; and when a vacuum has been pro- duced, it will be found that a considerable force is required to separate the two parts, this force increasing with the size of the hemispheres. This resistance to sep- aration is due to the normal exterior pressure of the air on every point of the surface, a pressure which is counterbalanced by only a very feeble pressure from the interior. In order to estimate the resultant effect of these different pressures, let us suppose that one hemisphere is vertically over the other, and that the external surface is cut into a series of steps, that is to say, of alternate vertical and horizontal elements. It is evident that the pressure urging either hemisphere towards the other will be simply the sum of the pressures upon its horizontal elements; and this sum is identical with the pressure which would be exerted upon a cir- cular area equal to the common base of the hemispheres. For example, if this area is 10 square inches, and the external pressure exceeds the internal by 14 Ibs. to the inch, the hemispheres will be pressed together with a force of 140 Ibs. Fountain in Vacuo. The apparatus for this experiment consists of a bell-shaped vessel of glass (Fig. 140), the base of which is pierced by a tube fitted with a stop-cock which enables us to exhaust the vessel of air. If, after a vacuum has been produced, we place the Fig. 138. Burst Bladder. Fig. 139. Magdeburg Hemisphere. 188 AIR-PUMP. lower end of the tube in a vessel of water, and open the stop-cock, the liquid, being pressed externally by the atmosphere, mounts up the tube and ascends in a jet into the interior of the vessel. This experiment is often made in the opposite manner. Under the receiver of the air-pump is placed a vial partly filled with water, and having its cork pierced by a tube open at both ends, the lower end being beneath the surface of the water. As the exhaustion proceeds, the air in the vial, by its excess of pres- sure, acts upon the liquid and makes it issue in a jet. 237. Limit to the Action of the Air- pump. We have said above ( 230) that the air-pump does not continue the process of rarefaction indefi- nitely, but that at a certain stage its effect ceases, and the pressure of the air in the receiver undergoes no further diminution. If the pump is very badly made, this pres- sure is considerable; but even with the most perfect machines it is always sensible. A pump such as we have described may be con- sidered good if it reduces the pressure of the air in the receiver to a tenth of an inch of mercury. A fiftieth of an inch is perhaps the lowest limit. LEAKAGE. This limit to the action of the machine is due to vari- ous causes. In the first place, there is frequently leakage at different parts of the apparatus; and although at the beginning of the opera- tion the quantity of air which thus enters is small in comparison with that which is pumped out, still, as the exhaustion proceeds, the air enters faster, on account of the diminished internal pressure, and at the same time the quantity expelled at each stroke becomes less, Fig. 140. - Fountain in Vacuo. LIMITS TO ACTION. 189 so that at length a point is reached at which the inflow and outflow are equal. In order to prevent leakage as far as possible, the plate of the pump and the base of the receiver must be truly plane so as to fit accurately; the base of the receiver must be ground (that is rough- ened) and must be well greased before pressing it down on the plate. The piston must also be well lubricated with oil. SPACE UNTRA VERSED BY PISTON. Another reason of imperfect exhaustion is that, after all possible precautions, a space is still left between the bottom of the pump-barrel and the lower surface of the piston when the latter is at the end of its downward stroke. It is evident that at this moment the air contained in this untraversed xpcice is of the same tension as the atmosphere. On raising the piston, this air is indeed rarefied; but it still preserves a certain tension, and it is evident that when the air in the receiver has been brought to this stage of rarefaction, the machine will cease to pro- duce any effect. If v is the volume of this space, and V the volume of the pump- barrel, the air, which at volume v has a pressure H equal to that of the atmosphere, will have, at volume V, a pressure H ^- This gives the limit to the action of the machine as deduced from the consider- ation of the untraversed space. AIR GIVEN OUT BY OIL. Finally, perhaps the most important cause, and the most difficult to remedy, is the absorption of air by the oil used for lubricating the pistons. This oil is poured on the top of the piston, but the pressure of the external air forces it be- tween the piston and the barrel, whence it falls in greater or less quantity to the bottom of the barrel, where it absorbs air, and par- tially yields it up at the moment when the piston begins to rise, thus evidently tending to derange the working of the machine. It has been attempted to get rid of untraversed space by employing a kind of piston of mercury. This has also the advantage of fitting the barrel more accurately, and thus preventing the entrance of air. The use of oil is at the same time avoided, and we thus escape the injurious effects mentioned above. We proceed to describe two machines founded upon this principle. 238. Kravogl's Air-pump. This contains a hollow glass cylinder AB (Fig. 141) tapering at the upper end, and surmounted by a kind of funnel. The piston is of the same shape as the cylinder, and is 190 AIR-PUMP covered with a layer of mercury, whose depth over the point of the piston is about -Vth of an inch when the piston is at the bottom of its stroke, but is nearly an inch when the piston rises and fills the Fig. 141. Kravogl's Air-pump. funnel-shaped cavity in which the pump-barrel terminates. A small interval, filled by the liquid, is left between the barrel and the piston; but at the bottom of the barrel the piston passes through a leather box carefully made, so as to be perfectly air-tight. The air from the receiver enters through the lateral opening e, and KHAVOGL'S PUMP. 191 is driven before the mercury into the funnel above. With the air passes a certain quantity of mercury, which is detained by a steel valve c at the narrowest part of the funnel. This valve rises auto- matically when the surface of the mercury is at a distance of about half an inch from the funnel, and falls back into its former position when the piston is at the end of its upward stroke. In the down- ward stroke, when the mercury is again half an inch from the funnel, the valve opens again and allows a portion of the mercury to pass. The effect of this arrangement is easily understood; there is no " untraversed space," the presence of the mercury above and around the piston causes a very complete fit, and excludes the external air; and hence the machine, when well made, is very effective. When this is the case, and when the mercury used in the apparatus is perfectly dry, a vacuum of about -g^th of an inch can be obtained. The dryness of the mercury is a very important condition, for at ordinary temperatures the elastic force of the vapour of water has a very sensible value. If we wish to employ the full powers of the machine, we must have, between the vessel to be exhausted of air and the pump-barrel, a desiccating apparatus. The arrangement of the valve e is peculiar. It is of a conical form, so as, in its lowest position, to permit the passage of air coming from the receiver. Its ascent is produced by the pressure of the mercury, which forces it against the conical extremity of the passage, and the liquid is thus prevented from escaping. The figure represents a double-barrelled machine analogous to the ordinary air-pump. Besides the pinion working with the racks of the pistons, there is a second smaller pinion, not shown in the figure, which governs the movements of the valves c. All the parts of this machine, as the stop-cocks, valves, pipes, &c., must be of steel, to avoid the action which the mercury would have upon any other metal. 239. Geissler's Machine. Geissler, of Bonn, invented a mercurial air-pump, in which the vacuum is produced by communication of the receiver with a Torricellian vacuum. Fig. 142 represents this machine as constructed by Alvergniat. It consists of a vertical tube, serving as a barometric tube, and communicating at the bottom, by means of a caoutchouc tube, with a globe which serves as the cistern. At the top of the tube is a three-way stop-cock, by which com- munication can be established either with the receiver to the left, or AIll-PUMP. with a funnel to the right, which latter has an ordinary stop-cock at the bottom. By means of another stop-cock on the left, com- munication with the receiver can be opened or closed. These stop- cocks are made entirely of glass. The machine works in the following- manner; communication being established with the funnel, the globe which serves as cistern is raised, and placed, as shown in the figure, at a higher level than the stop-cock of the funnel. By the law of equili- brium in communicat- ing vessels, the mercury fills the barometric tube, the neck of the funnel, and part of the funnel itself. If the communi- cation between the fun- nel and tube be now stopped, and the globe lowered, a Torricellian vacuum is produced in the upper part of the vertical tube. Communication is now opened with the re- ceiver; the air rushes into the vacuum, and the column of mercury falls Fig. i42.-Gei.ssier-s Machine. a little. Communication is now stopped between the tube and receiver, and opened between the tube and the funnel, the simple stop-cock of the funnel being, however, left shut. If at this moment the globe is replaced in the position shown in the figure, the air tends to escape by the funnel, and it is easy to allow it to do so. Thus, a part of the air of the receiver has been removed, GElSSLEll's AND SPllENGEL's. 193 and the apparatus is in the same position as at the beginning. The operation described is equivalent to a stroke of the piston in the ordinary machine, and this process must be repeated till the receiver is exhausted. As the only mechanical parts of this machine are glass stop-cocks, which are now executed with great perfection, it is capable of giving very good results. With dry mercury a vacuum of -s-yrrth of an inch may very easily be obtained. The working of the machine, how- ever, is inconvenient, and becomes exceedingly laborious when the receiver is large. It is therefore employed directly only for pro- ducing a vacuum in very small vessels; when the spaces to be exhausted of air are at all large, the operation is begun with the ordinary machine, and the mercurial air-pump is only employed to render the vacuum thus obtained more perfect. 240. Sprengel's Air-pump. This instrument, which may be re- garded as an improvement upon Geissler's, is represented in its simplest form in Fig. 143. cd is a glass tube longer than a baro- meter tube, down which mercury is allowed to fall from the funnel A. Its lower end dips into the glass vessel B, into which it is fixed by means of a cork. This vessel has a spout at its side, a few milli- metres higher than the lower end of the tube. The first portions of mercury which run down will consequently close the tube, and prevent the possibility of air entering it from below. The upper part of cd branches off at x into a lateral tube communicating with the receiver R, which it is required to exhaust. A convenient height for the whole instrument is 6 feet. The funnel A is supported by a ring as shown in the figure, or by a board with a hole cut in it. The tube cd consists of two parts, connected by a piece of india-rubber tubing, which can be compressed by a clamp so as to keep the tube closed when desired. As soon as the mercury is allowed to run down, the exhaustion begins, and the whole length of the tube, from x to d, is seen to be filled with cylinders of mercury separated by cylinders of air, all moving downwards. Air and mercury escape through the spout of the bulb B, which is above the basin H, where the mercury is collected. This has to be poured back from time to time into the funnel A, to pass through the tube again and again until the exhaustion is completed. As the exhaustion is progressing, it will be noticed that the inclosed air between the mercury cylinders becomes less and less, until the lower part of cd presents the aspect of a continuous column of mer- 13 194. AIR-PUMP. cury about 30 inches high. Towards this stage of the operation a considerable noise begins to be heard, similar to that of a shaken water-hammer, and common to all liquids shaken in a vacuum. The operation may be considered completed when the column of mercury does not inclose any air, and when a drop of mercury falls upon the top of this column without inclosing the slightest air-bubble. The height of this column now corresponds exactly with the height of the column of mer- cury in a barometer; or, what is the same, it represents a barometer whose vacuum is the receiver R and connecting O tube. Dr. Sprengel recommends the employment of an auxiliary air-pump of the ordinary kind, to commence the exhaustion, when time is an object, as with- out this from 20 to 30 minutes are required to exhaust a receiver of the capacity of half a litre. As, however, the em- ployment of the auxiliarypump involves additional connections and increased leakage, it should be avoided when the best pos- sible exhaustion is desired. The fall tube must not exceed about a tenth of an inch in diameter, and special precautions must be employed to make the india- (See Chemical Journal for 1865, p. 9.) th of atmo- Fig. 143. - Sprengel's Air-pump. i a o o o o o' rubber connections air-tight. By this instrument air has been reduced to spheric density, and the average exhaustion attainable by its use is about one-millionth, which is equivalent to "00003 of an inch of mercury. 241. Double Exhaustion. In the mercurial machines just described there is no " untra versed space," as the liquid completely expels all the air from the pump-barrel. These machines are of very recent DOUBLE EXHAUSTION. 195 invention. Babinet long before introduced an arrangement for the purpose, not of getting rid of this space, but of exhausting it of air. For this purpose, when the machine ceases to work with the ordi- nary arrangement, the communication of the receiver with one of the pump-barrels is shut off, and this barrel is employed to exhaust the air from the other. This change is effected by means of a stop- cock at the point of junction of the passages leading from the two barrels (Fig. 144). The stop-cock has a T-shaped aperture, the point of intersection of the two branches being in constant communication with the receiver. In a dif- ferent plane from that of the T-shaped aperture is another aperture mn, which, by means of the tube I, establishes communication between the pump-barrel B and the com- municating passage of the pump-barrel A. From this explanation it will be seen that if the stop-cock be turned as shown in the first figure, the two pump-barrels both com- municate with the receiver, and the operation proceeds in the Ordinary manner. But if Fig. 144. Babinet's Doubly-exhausting Stop-cock. the stop-cock be turned through a quarter of a revolution, as shown in the second figure, the pump- barrel B alone communicates with the receiver, while it is itself exhausted of air by the barrel A. It is easy to express by a formula the effect of this double exhaus- tion. Suppose the pump to have ceased, under the ordinary method of working, to produce any farther exhaustion, the air in the receiver has therefore reached a tension nearly equal to H^. ( 237). At this moment the stop-cock is turned into its second position. When the piston B descends, the piston A rises, and the air of the "untraversed space" in B is drawn into A and rarefied. During the inverse operation, the air in A is prevented from returning to B, and thus the rarefied air from B, becoming still further rarefied, will draw a fresh quantity of air from the receiver. This air will then be driven 196 AIR-PUMP. into A, where it will be compressed by the descending movement of the piston, and will find its way into the air outside. 1 This double exhaustion will itself cease to work when air ceases to pass from the pump-barrel B into the pump-barrel A. Now when the piston in this latter is raised, the elastic force of the air which was contained in its " untraversed space " is equal to H^, for, on the last opening of the valve, the air in this space escaped into the atmo- sphere. On the other hand, when the piston in B is at the end of its upward stroke, the tension of the air is the same as in the receiver. Let this be denoted by x. When the piston in B descends, the air is compressed into the " untraversed space " and the passage leading to A. Let the volume of this passage be I Then the tension will increase, and become x j-^j? When the machine ceases to produce any farther effect, this tension cannot be greater than that in the pump-barrel A, which is Hyj we have thus, to determine the limit to the action of the pump, the equation V + 1 TT v , x -j- = H y, whence x H -v . and 32 Ibs. weight respectively. Show that the resultant is equal to 13 Ibs. weight. 9. Five equal forces act at a point, in one place. The angles between the first and second, between the second and third, between the third and fourth, and between the fourth and fifth, are each 60. Find their resultant. 10. If 6 be the angle between the directions of two forces P and Q acting at a point, and E be their resultant, show that" R = P + Q2 + 2PQ cos ft 11. Show that the resultant of two equal forces P, acting at an angle 6, is IP cos \6. PARALLEL FORCES, AND CENTRE OF GRAVITY. 10*. A straight rod 10 ft. long is supported at a point 3 ft. from one end. What weight hung from this end will be supported by 12 Ibs. hung from the other, the weight of the rod being neglected 1 11*. Weights of 15 and 20 Ibs. are hung from the two ends of a straight rod 70 in. long. Find the point about which the rod will balance, its own weight being neglected. 2-tO EXAMPLES. 12. A weight of 100 Ibs. is slung from a pole which rests 011 the shoulders of two men, A and B. The distance between the points where the pole presses their shoulders is 10 ft., and the point where the weight is slung is 4 ft. from the point where the pole presses on A's shoulder. Find the weight borne by each, the weight of the pole being neglected. 13. A uniform straight lever 10 ft. long balances at a point 3 ft. from one end, when 12 Ibs. are hung from this end and an unknown weight from the other. The lever itself weighs 8 Ibs. Find the unknown weight. 14. A straight lever 6 ft. long weighs 10 Ibs., and its centre of gravity is 4 ft. from one end. What weight at this end will support 20 Ibs. at the other, when the lever is supported at 1 ft. distance from the latter? 15. Two equal weights of 10 Ibs. each are hung one at each end of a straight lever 6 ft. long, which weighs 5 Ibs.; and the lever, thus weighted, balances about a point 3 in. distant from the centre of its length. Find its centre of gravity. 16. A uniform lever 10 ft. long balances about a point 1 ft. from one end, when loaded at that end with 50 Ibs. Find the weight of the lever. / 17. A straight lever 10 ft. long, when unweighted, balances about a point 4 ft. from one end ; but when loaded with 20 Ibs. at this end and 4 Ibs. at the other, it balances about a point 3 ft. from the end. Find the weight of the lever. 18. A lever is to be cut from a bar weighing 3 Ibs. per ft. What must be its length that it may balance about a point 2 ft. from one end, when weighted at this end with 50 Ibs.? (The solution of this question involves a quadratic equa- tion.) 19. A lever is supported at its centre of gravity, which is nearer to one end than to the other. A weight P at the shorter arm is balanced by 2 Ibs. at the longer ; and the same weight P at the longer arm is balanced by 18 Ibs. at the shorter. Find P. 20. Weights of 2, 3, 4 and 5 Ibs. are hung at points distant respectively 1, 2, 3 and 4 ft. from one end of a lever whose weight may be neglected. Find the point about which the lever thus weighted will balance. (This and the following questions are best solved by taking moments round the end of the lever. The sum of the moments of the four weights is equal to the moment of their resul- tant.) 21. Solve the preceding question, supposing the lever to be 5 ft. long, uniform, and weighing 2 Ibs. 22. Find, in position and magnitude, the resultant of two parallel and oppo- sitely directed forces of 10 and 12 units, their lines of action being 1 yard apart. 23. A straight lever without weight is acted on by four parallel forces at the following distances from one end : At 1 ft., a force of 2 units, acting upwards. At 2 ft., 3 downwards. At 3 ft., ,, 4 upwards. At 4 ft., 5 downwards. Where must the fulcrum be placed that the lever may be in equilibrium, and what will be the pressure against the fulcrum ? 24. A straight lever, turning freely about an axis at one end, is acted on by four parallel forces, namely KXAMPLKS. 241 A downward force of 3 Ibs. at 1 ft. from axis. A downward force of o 3 ft. An upward force of 4 2 ft. An upward force of ti" 4 ft. What must be the weight of the lever that it may be in equilibrium, its centre of gravity being 3 ft. from the axis? 25. In a pair of nut-crackers, the nut is placed one inch from the hinge, and the hand is applied at a distance of six inches from the hinge. How much pressure must be applied by the hand, if the nut requires a pressure of 13 Ibs. to break it, and what will be the amount of the pressure on the hinges? 26. In the steelyard, if the horizontal distance between the fulcrum and the knife-edge which supports the body weighed be 3 in., and the movable weight be 7 Ibs., how far must the latter be shifted for a difference of 1 Ib. in the body weighed / 27. The head of a hammer weighs 20 Ibs. and the handle 2 Ibs. The distance between their respective centres of gravity is 24 inches. Find the distance of the Centre of gravity of the hammer from that of the head. 28. One of the four triangles into which a square is divided by its diagonals is removed. Find the distance of the centre of gravity of the remainder from the intersection of the diagonals. 29. A square is divided into four equal squares and one of these is removed. Find the distance of the centre of gravity of the remaining portion from the centre of the original square. 30. Find the centre of gravity of a sphere 1 decimetre in radius, having in its interior a spherical excavation whose centre is at a distance of 5 centimetres from the centre of the large sphere and whose radius is 4 centimetres. 31. Weights P, Q, E, S are hung from the corners A, B, C, D of a uniform square plate whose weight is W. Find the distances from the sides AB, AD of the point about which the plate will balance. 32. An isosceles triangle stands upon one side of a square as base, the altitude of the triangle being equal to a side of the square. Show that the distance of the centre of the whole figure from the opposite side of the square is of a side of the square. 33. A right cone stands upon one end of a right cylinder as base, the altitude of the cone being equal to the height of the cylinder. Show that the distance of the centre of the whole volume from the opposite end of the cylinder is \% of the height of the cylinder. WORK AND STABILITY. 34. A body consists of three pieces, whose masses are as the numbers 1, 3, 9; and the centres of these masses are at heights of 2, 3, and 5 cm. above a certain level. Find the height of the centre of the whole mass above this level. 35. The body above-mentioned is moved into a new position, in which the heights of the centres of the three masses are 1, 3, and 7 cm. Find the new height of the centre of the whole mass. 36. Find the work done against gravity in moving the body from the first position into the second ; employing as the unit of work the work done in raising the smallest of the three pieces through 1 cm. 16 24*2 EXAMPLES. 37. Find the portions of this work done in moving each of the three pieces. 38. The dimensions of a rectangular block of stone of weight W are AB = a, AC = b, AD = c, and the edges AB, AC are initially horizontal. How much work is done against gravity in tilting the stone round the edge AB until it balances. 39. A chain of weight "W and length I hangs freely by its upper end which is attached to a drum upon which the chain can be wound, the diameter of the drum being small compared with I. Compute the work done against gravity in winding up two-thirds of the chain. 40. Two equal and similar cylindrical vessels with their bases at the same level contain water to the respective heights h and H centimetres, the area of either base being a, sq. cm. Find, in gramme-centimetres, the work done by gravity in equalizing the levels when the two vessels are connected. 41. Two forces acting at the ends of a rigid rod without weight equilibrate each other. Show that the equilibrium is stable if the forces are pulling outwards and unstable if they are pushing inwards. 42. Two equal weights hanging from the two ends of a string, which passes over a fixed pulley without friction, balance one another. Show that the equili- brium is neutral if the string is without weight, and is unstable if the string is heavy. 43. Show that a uniform hemisphere resting on a horizontal plane has two positions of stable equilibrium. Has it any positions of unstable equilibrium '? INCLINED PLANE, &c. 44. On an inclined plane whose height is \ of its length, what power acting parallel to the plane will sustain a weight of 112 Ibs. resting on the plane without friction ? 45. The height, base, and length of an inclined plane are as the numbers 3, 4, 5. What weight will be sustained on the plane without friction by a power of 100 Ibs. acting (a) parallel to the base, (b) parallel to the plane? 46. Find the ratio of the power applied to the pressure produced in a screw- press without friction, the power being applied at the distance of 1 ft. from the axis of the screw, and the distance between the threads being \ in. 47. In the system of pulleys in which one cord passes round all the pulleys, its different portions being parallel, what power will sustain a weight of 2240 Ibs. without friction, if the number of cords at the lower block be 6? 48. A balance has unequal arms, but the beam assumes the horizontal position when both scale-pans are empty. Show that if the two apparent weights of a body are observed when it is placed first in one pan and then in the other, the true weight will be found by multiplying these together and taking the square root. FORCE, MASS, AND VELOCITY. The motion is supposed to be rectili-iiear. 49. A force of 1000 dynes acting on a certain mass for one second gives it a velocity of 20 cm. per sec. Find the mass in grammes. 50. A constant force acting on a mass of 12 gm. for one sec. gives it a velocity of 6 cm. per sec. Find the force in dynes. EXAMI'I.KS. 24)3 51. A force of 490 dynes acts on a mass of 70 gm. for one sec. Find the velocity generated. 52. In the preceding example, if the time of action be increased to 5 sec., what will be the velocity generated! In the following examples the unit of momentum referred to is the momentum of a gramme moving with a velocity of a centimetre per second. 53. What is the momentum of a mass of 15 gm. moving with a velocity of translation of 4 cm. per sec.? 54. What force, acting upon the mass for 1 sec., would produce this velocity ? 55. What force, acting upon the mass for 10 sec., would produce the same velocity? 56. Find the force which, acting on an unknown mass for 12 sec., would pro- duce a momentum of 84. 57. Two bodies initially at rest move towards each other in obedience to mutual attraction. Their masses are respectively 1 gm. and 100 gm. If the force of attraction be T ^j of a dyne, find the velocity acquired by each mass in 1 sec. 58. A gun is suspended by strings so that it can swing freely. Compare the velocity of discharge of the bullet with the velocity of recoil of the gun ; the masses of the gun and bullet being given, and the mass of the powder being neglected. 59. A bullet fired vertically upwards, enters and becomes imbedded in a block of wood falling vertically overhead ; and the block is brought to rest by the im- pact. If the velocities of the bullet and block immediately before collision were respectively 1500 arid 100 ft. per sec., compare their masses. FALLING BODIES AND PROJECTILES. Assuming that a falling body acquires a velocity of 980 cm. per sec. by falling for 1 sec., find : 60. The velocity acquired in ^ of a second. 61. The distance passed over in ^ sec. 62. The distance that a body must fall to acquire a velocity of 980 cm. per sec. 63. The time of rising to the highest point, when a body is thrown vertically upwards with a velocity of 6860 cm. per sec. 64. The height to which a body will rise, if thrown vertically upwards with a velocity of 490 cm. per sec. 65. The velocity with which a body must be thrown vertically upwards that it may rise to a height of 200 cm. 66. The velocity that a body will have after ^j sec., if thrown vertically up- wards with a velocity of 300 cm. per sec. 67. The point that the body in last question will have attained. 68. The velocity that a body will have after 2| sees., if thrown vertically up- wards with a velocity of 800 cm. per sec. 69. The point that the body in last question will have reached. Assuming that a falling body acquires a velocity of 32 ft. per sec. by falling for 1 sec., find : 70. The velocity acquired in 12 sec. 71. The distance fallen in 12 sec. EXAMPLES. 72. The distance that a body must fall to acquire a velocity of 10 ft. per sec. 73. The time of rising to the highest point, when a body is thrown vertically upwards with a velocity of 160 ft. per sec. 74. The height to which a body will rise, if thrown vertically upwards with a velocity of 32 ft. per sec. 75. The velocity with which a body must be thrown vertically upwards that it may rise to a height of 25 ft. 76. The velocity that a body will have after 3 sec., if thrown vertically up- wards with a velocity of 100 ft. per sec. 77. The height that the body in last question will have ascended. 78. The velocity that a body will have after 1^ sec., if thrown vertically down- wards with a velocity of 30 ft. per sec. 79. The distance that the body in last question will have described. 80. A body is thrown horiz< ntally from the top of a tower 100 m. high with a velocity of 30 metres per sec. When and where will it strike the ground? 81. Two bodies are successively dropped from the same point, with an interval of i of a second. When will the distance between them be one metre ? 82. Show that if x and y are the horizontal and vertical co-ordinates of a pro- jectile referred to the point of projection as origin, their values after time t are x = ~Vt cos a, y = ~Vt sin a \ gt 2 . 83. Show that the equation to the trajectory is ^ = * tana -2V^oIV and that if V and can be varied at pleasure, the projectile can in general be made to traverse any two given points in the same vertical plane with the point of projection. ATWOOJD'S MACHINE. Two weights are connected by a cord passing over a pulley as in Atwood's machine, friction being neglected, and also the masses of the pulley and cord ; find: 84. The acceleration when one weight is double of the other. 85. The acceleration when one weight is to the other as 20 to 21. Taking g as 980, in terms of the cm. and sec., find : 86. The velocity acquired in 10 sec., when one weight is to the other as 39 to 41. 87. The velocity acquired in moving through 50 cm., when the weights are as 19 to 21. 88. The distance through which the same weights must move that the velocity acquired may be double that in last question. 89. The distance through which two weights which are as 49 to 51 must move that they may acquire a velocity of 98 cm. per sec. EXAMPLES. 245 ENERGY AND WORK. 90. Express in ergs the kinetic energy of a mass of 50 gm. moving with a velocity of 60 cm. per sec. 91. Express in ergs the work done in raising a kilogram through a height of 1 metre, at a place where g is 981. 92. A mass of 123 gm. is at a height of 2000 cm. above a level floor. Find its energy of position estimated with respect to the floor as the standard level (g being 981). 93. A body is thrown vertically upwards at a place where g is 980. If the velocity of projection is 9800 cm. per sec. and the mass of the body is 22 gm., find the energy of the body's motion when it has ascended half way to its maximum height. Also find the work done against gravity in this part of the ascent. 94. The height of an inclined plane is 12 cm., and the length 24 cm. Find the work done by gravity upon a mass of 1 gm. in sliding down this plane (g being 980), and the velocity with which the body will reach the bottom if there be no friction. 95. If the plane in last question be not frictionless, and the velocity on reaching the bottom be 20 cm. per sec., find how much energy is consumed in friction. 96. Find the work expended in discharging a bullet whose mass is 30 gm. with a velocity of 40,000 cm. per sec.; and the number of such bullets that will be discharged with this velocity in a minute if the rate of working is 7460 million ergs per sec. (one horse-power). 97. One horse-power being defined as 550 foot-pounds per sec. ; show that it is nearly equivalent to 8'8 cubic ft. of water lifted 1 ft. high per sec. (A cubic foot of water weighs 62^ Ibs. nearly. A foot-pound is the work done against gravity in lifting a pound through a height of 1 ft.) 98. How man}' cubic feet of water will be raised in one hour from a mine 200 ft. deep, if the rate of pumping be 15 horse-power? CENTRIFUGAL FORCE. 99. What must be the radius of curvature, that the centrifugal force of a body travelling at 30 miles an hour may be one-tenth of the weight of the body ; dlstance f rom A I> WTF^TE Ex. 34. 4^ cm. Ex. 35. 5 T % cm. Ex. 36. 17. Ex. 37. - 1, 0, + 18. Ex. 38. fW (V(6 2 + c2)-c). Ex. 39. fWl. Ex. 40. \ EXAMPLES. 253 Ex. 44. 14 Ibs. Ex. 45. (a) 133 Ibs.; (b) 166f Ibs. Ex. 46. 1 to 603 nearly. Ex. 47. 373. Ex. 49. 50. Ex. 50. 72. Ex. 51. 7 cm. per sec. Ex. 52. 35. Ex. 53. 60. Ex. 54. 60 dynes. Ex. 55. 6 dynes. Ex. 56. 7 dynes. Ex. 57. Smaller mass T J TT , larger TGU&Q cm - P er sec - Ex. 58. Inversely as masses of bullet and gun. Ex. 59. Mass of bullet is ^ of mass of block. Ex. 60. 98 cm. per sec. Ex. 61. 4'9 cm. Ex. 62. 490 cm. Ex. 63. 7 sec. Ex. 64. 122^ cm. Ex. 65. 626 cm. per sec. Ex. 66. 6 cm. per sec. upwards. Ex. 67. 45'9 cm. above point of projection. Ex. 68. 1650 cm. per sec. downwards. Ex. 69. 1062| cm. below starting point. Ex. 70. 384 ft. per sec. Ex. 71. 2304 ft. Ex. 72. l$f ft. Ex. 73. 5 sec. Ex. 74. 16 ft. Ex. 75. 40 ft. per sec. Ex. 76. 4 ft. per sec. upwards. Ex. 77. 156 ft. Ex. 78. 78 ft. per sec. Ex. 79. 81 ft Ex. 80. After 4'52 sec. At 135'6 m. from tower. Ex. 81. After '41 sec. from dropping of second body. Ex. 84. g. Ex. 85. -fa g. Ex. 86. 245 cm. per sec. Ex. 87. 70 cm. per sec. Ex. 88. 200 cm. Ex. 89. 245 cm. Ex. 90. 90,000 ergs. Ex. 91. 98,100,000 ergs. Ex. 92. 241,326,000 ergs. Ex. 93. 528,220,000 ergs each. Ex. 94. 11,760 ergs ; V 23520= 153'4 cm. per sec. Ex. 95. 11,560 ergs. Ex. 96. 24 x 10 9 ergs in each discharge. Not quite 19 discharges per min. Ex. 98. 2376 nearly. Ex. 99. 18330 cm. or about 600 ft. Ex. 100. 2 V^a, V ~bgct. Ex. 101*. 223-679 cm. Ex. 102. ME 2 , Ex. 103. |a; mass of rod multi- plied by Y\ a2 - Ex. 104. At either of the two points distant j= from centre ; at either of the two points distant ^ from centre. Ex. 106. (2 a-) 2 = 39'48. Ex. 107. (102-:*-) 2 = 10350000. Ex. 108. 102500. Ex. 109. 883'35. Ex. 110. 31'53. Ex. 111. 149'5. Ex. 112. 12217. Ex. 113. 13338. Ex. 114. 29901. Ex. 115. 126'5. Ex. 116. 30. Ex. 117. 12ft. 7 in. Ex. 118. 980-68. Ex. 119. 4579. Ex. 120. 7342. Ex. 121. 994. Ex. 122. 125. Ex. 123. Ex. 124. 2-357. Ex. 125. T8. Ex. 126. '932. Ex. 127. '0046 sq. cm. Ex. 128. 6-25. Ex:. 129. 4J. Ex. 130. 1'9125. Ex. 131. l^ r . Ex. 132. f J. Ex. 133. 10 cub. cm., 78 gm., 7'8. Ex. 134. 104. Ex. 135. 40'83. Ex. 136. 117-5. Ex. 137. 19-3, 18-3, 57. Ex. 138. '393 sq. cm., '354 cm. Ex. 139. 18, 8777, 00835 sq. cm., '0516 cm. Ex. 140. 60'48, 52'62. Ex. 141. 900 c.c. Ex. 142. 5-56 cm. Ex. 143. 50 c.c. Ex. 145. 3, 70 c.c. Ex. 146. 400 gm. Ex. 147. 4^y = 4-185. Ex. 148. 3^ = 3-6115. Ex. 149. 36'6 c.c. Ex. 150. 257'7 gm. Ex. 151. 1-0033. Ex. 152. Jf of the iron. Ex. 153. 1 lit. of first, 2 lit. of second. Ex. 154. | of a litre. Ex. 155. l-/^ decim. = '158 cm. Ex. 156. ty 1 *O -lo'O 935 cm. Ex. 157. Gold: silver :: *,-p : ^-^ Ex. 158. (a) 577, (6) 10'6. Ex . 159. A - 254 EXAMPLES. Ex. 161. 6-6 cm. nearly. Ex. 162. 177 cm. Ex. 165. if. Ex. 166. f in. Ex. 168. 1010564, 1015730. Ex. 169. 77'3832. Ex. 170. 75-93. Ex. 171. '076. Ex. 172. 30 in. Ex. 173. 2J. Ex. 174. -0693. Ex. 175. (a) -001308, (b) '001635. Ex. 176. (a) -0012961, (b) '0012895, (c) '0012933. Ex. 177. d varies as gr, and therefore gd varies as g z . Ex. 178. Its top must be 76-5 = 71 cm. deep. Ex. 179. 760 m. Ex. 180. 33 to 5. Ex. 181. 454 - cm. Ex. 182. 30 tons. Ex. 183. 4608. Ex. 184. 459500 nearly. Ex. 185. (a) 970. (b) 990 gm. wt. per sq. cm. Ex. 186. -$ of an atmosphere, nearly. Ex. 187. 3 atmospheres. INDEX TO PAET I. Absorption of gases, 177. Acceleration defined, 51. Air, weight of, 138, 139. pump, 179. chamber, 218. film, adherent, 177. Alcoholimeters, 114, 115. Amplitude of vibration, 63. Aneroid, 154. Annual and diurnal variations, 161. Archimedes' principle, 97. Aristotle's experiment, 138 Arithmetical lever, 12. Ascent in capillary tubes, 124, 125, 128. Atmosphere, 140. standard of pressure, 141. Attractions, apparent, 133. Atwood's machine, 57. Axis of couple, 14. of wrench, 15. Babinet's air-pump, 196. Back-pressure on discharging ves- sel, 92, 225, 226. Balance, 34-40. Balloons, 204-208. Barker's mill, 93. Barographs, 156, 158. Barometer, 142. , corrections of, 148-151. Barometric measurement of heights, 159-161. prediction, 163. Baroscope, 204. Beaume's hydrometers, 113. Bianchi's air-pump, 183. Bladder, burst, 187. Bourdon's gauge, 175. Boyle's law, 166. tube, 1 66. Bramah press, 222. Bubbles filled with hydrogen, 205. , tension and pressure in, 130. Buoyancy, centre of, 98, 100. Buys Ballot's law, 164. Caissons, 202. Camphor, movements of, 134. Capillarity, 124-134. Cartesian diver, 101. Cathetometer, 144. Centre of buoyancy, 98, 100. of gravity, 17-21. by experiment, 22. , velocity of, 46. of mass, 47. of oscillation, 71. of parallel forces, 10, 17. of pressure, 93. Centrifugal force, 60, 95. pump, 219. C.G.S. system, 48. Change of momentum, 42. of motion, 42. Charts of weather, 163. Circular motion, 59. Clearance, see untraversed space, 189, 213. Coefficients of elasticity, 79. of friction, 81. Colloids, 135. Communicating vessels, 118, 125. Component along a line, 16. Components, 7. Compressed-air machines, 202. Compressibility, 79. Compressing pump, 199. Conservation of energy, 74-76. Constant load, weighing with, 37. Contracted vein, 225. Contractile film, 127-130, 133, 134. Convertibility of centres, 70. Corrections of barometer, 148-151. Counterpoised barometer, 155. Couple, 13. Crystalloids, 135. Cupped-leather collar, 222. Cycloidal pendulum, 67. Cyclones, 165. D'Alembert's principle, 96. Deflecting force, 60. Deleuil's air-pump, 196. Density, absolute and relative, 105. , determination of, 106-112. , table of, xii. Depression, capillary, 124, 125, 128. Despretz*s experiments on BoyleV law, 168. Dialysis, 135. Diameters, law of, 125. Diffusion, 135. Displaced liquid defined, 100. Diurnal barometric curve, 161. Diver, Cartesian, 101. Double-acting pumps, 183, 218. Double-barrelled air-pump, 181. Double exhaustion, 194. weighing, 35. Drops, 131. Dynamics, 2. Dynamometer, 4. Dyne, 48. Efficiency of pumps, 214. Efflux of liquids, 224. from air-tight spaces, 229. Egg in water, 100. Elasticity, 77-80. Elevation, capillary, 124-128. Endosmose, 134. Energy, conservation of, 7476. Energy, kinetic, 73. , static or potential, 73. English air-pumps, 184. Equilibrium, 4. Equivalent simple pendulum, 66. Erg, 48. Errors and corrections, signs of, 151. Exhaustion, limit of, 188. , rate of, 180. Expansibility of gases, 137. Fahrenheit's barometer, 156. hydrometer, in. Fall in vacuo, 49. Falling bodies, 52. Film of air on solids, 177. Films, tension in, 127-130, 133, 134. Fire-engine, 218. Float-adjustment of barometer, '47; Floatation, 102. Floating needles, 103. Fluid, perfect, 83. Force, 3. , amount of, 44. , intensity of, 44. , unit of, 44, 48. Forcing-pump, 216. Fortin's barometer, 144. Fountain in vacuo, 187. , intermittent, 230. Free-piston air-pump, 196. Friction, 81, 82. in connection with conservation of energy, 76. Froude on contracted vein, 225. Galileo on falling bodies, 49. on suction by pumps, 142. Gases, expansibility of, 137. Geissler"s air-pump, 191. Geometric decrease of pressure upwards, 160. Gimbals, 147. Gradient, barometric, 164. Gramme, 105. Graphical interpolation, 116. Gravesande's apparatus, 7. Gravitation units of force, 4, 106. Gravity, apparent and true, 61. , centre of, 17-21. , its velocity, 46. , formula for its intensity, 51. measured by pendulums, 7?. proportional to mass, 50. Guinea-and-feather experiment, 49- Head of liquid, 224. Heights measured by barometer, 159-161. Hemispheres, Magdeburg, 187. Homogeneous atmosphere, 159, 1 60. " Horizontal " defined, 17. Horse-power, xi. Hydraulic press, 87, 221. tourniquet, 93. Hydrodynamics, 83. Hydrogen, bubbles filled with, 205. 256 INDEX TO PART I. Hydrokinetics, 83. Hydrometers, 110-117. Hydrostatics, 83. Hypsometric formula, 161. Immersed bodies, 98. Inclined plane, 32. Index errors and corrections, 151. Inertia, 41. , moment of, 68. Inexhaustible bottle, 230. Insects walking on water, 104. Intermittent fountain, 230. Isobars, 163. Isochronous vibrations, 66, 78. Jet-pump, 219. Jets, liquid, 224. Kater's pendulum, 71. Kinetic energy, 73. Kinetics, 4. King's barograph, 155, 156. Kravogl's air-pump, 190. Laws of motion, 41-45. Levels, 119-123. Lever, 29. Limit to action of air-pump, 188. Liquids find their own level, 118. in superposition, 88. Magdeburg hemispheres, 187. Magic funnel, 229. Manometers, 172-175. Marine barometer, 153. Mariotte's bottle, 235. law, 1 66. tube, 166. Mass, 44, 45. and gravitation proportional, SO- , centre of, 47. Mechanical advantage, 30. powers, 29-33. Mechanics, 2. Meniscus, 131. Metacentre, 103. Metallic barometer, 155. Mixtures, density of, 115. of gases, 176. Moduli of elasticity, 78. Moment of couple, 13. offeree about point, n. of inertia, 68. Momentum, 44. Morin's apparatus, 55. Motion, laws of, 41-45. Motions, composition of, 42. Mountain-barometer, theory of, 159-161. Multiple-tube barometer, 157. manometer, 172. Natural history and natural phi- losophy, i. Needles floating, 103. Newton's experiments with pen- dulums, 50. ' laws of motion, 41-45. Nicholson's hydrometer, in. CErsted's piezometer, 79. Oscillation, centre of, 71. Parachute, 207. Paradox, hydrostatic, 91. Parallel forces, 9-14. Parallelogram of forces, 7, 43. of velocities, 43. Parallelepiped of forces, 8. Pascal's mountain experiment, 142. principle, 86. vases, 89. Pendulum, 62. , compound, 70. , cycloidal, 67. , isochronism of, 64. , simple, 62. , time of vibration of, 65. Period of vibration, 63. " Perpetual motion," 26. Phial of four elements, 89. Photographic registration, 157. Piezometer, 79. Pile-driving, 75 Pipette, 229. Plateau's experiments, 131. Platinum causing ignition of hydrogen, 177. Plunger, 216. Pneumatic despatch, 202. Potential energy, 73. Pressure, centre of, 93. , hydrostatic, 84. , intensity of, 83. on immersed surfaces, 93. , reduction of, to absolute mea- sure, 151. Pressure-gauges, 172-175. Pressure-height denned, 159, 220. Pressure in air computed, 160. least where velocity is greatest, 220. Principle of Archimedes, 97. Projectiles, 53. Pulleys, 31. Pump, forcing, 216. , suction, 211. Pumps, efficiency of, 214. Quantity of matter, 45. Range and amplitude, 161. Rarefaction, limit of, 188. , rate of, 180. Reaction, 4, 15, 45. of issuing jet, 92, 225, 226. Rectangular components, 15. Regnault's experiments on Boyle's law, 169-172. Resistance of the air, 49, 53. Resolution, 15. Resultant, 7. Rigid body, 5. Rotating vessel of liquid, 95. Screw, and screw-press, 33. Second law of motion, 42. Sensibility and instability, 38. Sensibility of balance, 35. Simple-harmonic motion, 65. Simple pendulum, 62. Siphon, 23r. for sulphuric acid, 234. Siphon-barometer, 151. Specific gravity, 105. by weighing in water, 108. flask, 107. , table of, xii. Spirit-levels, 120-123. Sprengel's air-pump, 193. Spring-balance, 4. Stability, 21-28, 38. Standard kilogramme, 105. Statics, 4. Steelyard, 40. Suction, 211. pump, 211. Sugar-boiling, 202. Superposed liquids, 88. Surface of liquids level, 85. Surface-tension, 127-130, 133, 134. , table of, 134. Tantalus' cup, 274. Tale's air-pump, 186. Torricellian experiment, 141. Torricelli's theorem on efflux, 224. Tourniquet, hydraulic, 93. Trajectory, 54. Translation and rotation, 3. Transmission of pressure in fluids, 86. Triangle of forces, 6. Twaddell's hydrometer, 114. Uniform acceleration, 50. Unit of force, 44, 48. of work, 48. Units of measurement, 47. , C.G.S., 48. Unstable equilibrium, 21-28, 38. Untraversed space, 189, 213. Upward pressure in liquids, 88. Vena contracta, 225. Vernier, 145. " Vertical" defined, 17. Vessels in communication, 118, 125. Vibrations, 66. , when small, isochronous, 78. Volumes measured by weighing in water, no. Water, compressibility of, 79. level, irg. supply of towns, 118. Wedge, 33. Weighing, double, 35. in water, 108. with constant load, 37. Weight affected by air, 209. "Weight" ambiguous, 106. Wheel and axle, 30. Wheel-barometer, 152. Whirling vessel of liquid, 95. Work, 22-25. in producing motion, 52. , principle of, 25. Wrench, 15. Young's modulus, 78. Zero, errors of, 151. ELEMENTARY TREATISE ON NATURAL PHILOSOPHY BY A. PRIVAT DESCHANEL, FORMERLY PROFESSOR OF PHYSICS IN THE LYCEE LOUIS-LE-GRAND, INSPECTOR OF THE ACADEMY OF PARIS. TRANSLATED AND EDITED, WITH EXTENSIVE MODIFICATIONS, BY J. D. EVEEETT, M.A., D.C.L., F.R.S., F.R.S.E., PROFESSOR OF NATURAL PHILOSOPHY IN THE QUEEN'S COLLEGE, BELFAST. Part IL HEAT. ILLUSTRATED BY 151 ENGRAVINGS ON WOOD. SIXTH EDITION. LONDON: BLACKIE & SOX OLD BAILEY, EC.; GLASGOW, EDINBURGH, AND DUBLIN. 1880. AH Rights Iff served. GLASGOW : W. O. BLACKIR /,-ND CO., PEISTEKS, V1LLAKIELD. PREFACE TO THE PRESENT EDITION OF PART H. The present edition of this Part has been completely recast, and the treatment of several subjects will be found more complete and consecutive than in previous editions. The subject of Heat as a measurable Quantity is introduced at a much earlier stage than before, the chapter on Calorimetry being placed immediately after those on Thermometry and Expansion. Latent Heat and Heat of Combination are not now included in this chapter, but are treated later in connection with the subjects of Fusion, Vaporization, and Thermo-dynamics. Among the new matter, may be mentioned: An investigation of the temperature of minimum apparent volume of water in a glass envelope; An account of Guthrie's results on the freezing of brine: A proof that the pressure of vapour in the air at any time is equal to the maximum pressure for the dew-point; Descriptions of Dines' hygrometer, and of Symons' Snowdon rain-gauge; A full explanation of " Ditfusivity " or " Thermometric Conduc- tivity;" Some recent results on the conductivity of rocks, and on the con- ductivity of water; A note on the mathematical discussion of periodical variations of underground temperature ; A proof of the formula for the efficiency of a perfect thermo- dynamic engine; Several investigations relating to the two specific heats of a gas, and to adiabatic changes in gases, liquids, and solids; iv PREFACE. A description of the modern Gas Engine. Every chapter has been carefully revised, with a view to clear- ness, accuracy, and consolidation; and the result has been that, with the exception of Melloni's experiments, and the Steam Engine, the treatment of nearly every subject has been materially changed. The collection of Examples at the end of the volume has been enlarged and re-arranged; and the answers are appended. J. D. E. BELFAST, November, 1880. NOTE PREFIXED TO FIRST EDITION. In the present volume, the chapter on Thermo-dynamics is almost entirely the work of the editor. Large portions of the chapters on Conduction and on Terrestrial Temperatures have also been re- written; and considerable additions have been made in connection with Hygrometry, the Theory of Exchanges, the Specific Heats of Gases, and the Motion of Glaciers. Minor additions and modifica- tions have been numerous, and will easily be detected by comparison with the similarly numbered sections in the original. The nomenclature of units of heat which has been adopted, is borrowed from Prof. G. C. Foster's article "Heat" in Waits Dictionary of Chemistry. CONTENTS-PART II. HEAT, THE NUMBERS REFER TO THE SECTIONS. CHAPTER XXIV. THERMOMETRY. Heat and Cold, 276. Temperature, 277. Expansion, 278. General idea of thermo- meter, 279. Choice of thermometric substance, 280. Construction of mercurial thermometer, 281. Adjustment of the quantity of mercury, 282. Thermometric scales, 283. Displacement of zero, 284. Sensibility of thermometer, 285. Alcohol thermometer, 286. Self-registering thermometers, 287. Thermograph, 288. Me- tallic thermometers, 289. Pyrometers, 290. Differential thermometer, 291, pp. 257-276. CHAPTER XXV. MATHEMATICS OF EXPANSION. Expansion. Factor of expansion, 292. Linear and cubical expansion, 293. Change of density, 294. Real and apparent expansion, 295. Degree of mercurial thermo- meter, 296. Comparability of mercurial thermometers, 297. Steadiness of zero in spirit thermometers, 298. Length of a degree on the stem, 299. Weight thermo- meter, 300. Expansion of gases, 301. General definition of coefficient of expan- sion, 302, pp. 277-282. CHAPTER XXVI. EXPANSION OF SOLIDS. Observations of linear expansion, 303. Compensating pendulum, 304. Force of expan- sion, 305, . . pp. 283-286. CHAPTER XXVII. EXPANSION OF LIQUIDS. Method of equilibrating columns, 306. Experiments of Dulong and Petit, and of Reg- nault, 307. Expansion of glass, 308. Expansion of any liquid, 309, 310. Formulae for expansion of liquids, 311. Maximum density of water, 312. Saline solutions, 313. Apparent expansion of water, 314. Density of water at various temperatures, 315. Expansion of iron and platinum, 316. Convection currents, 317. Heating of build- ings by hot water, 318 pp. 287-296. CHAPTER XXVIII. EXPANSION OF GASES. Experiments of Guy-Lussac, 319. Regnault's apparatus, 320. Results, 321. Reduction to Fahrenheit scale, 322. Air-thermometer, 323. Perfect gas, 324. Absolute tem- perature by air-thermometer, 325. Pyrometers, 326. Density of gases, 327. Mea- surement of relative density of a gas, 328. Absolute densities, 329. Draught of chimneys, 330. Stoves, 331, pp. 297-309. CHAPTER XXIX. CALORIMETRY. Quantity of heat, 332. Principles assumed, 333. Cautions, 334. Unit of heat, 335. Thermal capacity, 336. Specific thermal capacities, 337. Method of mixtures, 338. VI TABLE OF CONTENTS. Practical application, 339. Corrections, q 40. Regnault's apparatus, 341. Great specific heat of water, 342. Specific heat of gases, 343. Dulong and Petit's la\\ , 344. Ice calorimeters, 345. Method of cooling, 346 pp. 310-319. CHAPTER "XJTX. FUSION AND SOLIDIFICATION. Fusion, 347. Definite temperature, 348. Latent heat of fusion, 349. Measurement of heat of fusion, 350. Conservative power of water, 351. Solution, 352. Freezing- mixtures, 353, 354. Solidification or congelation, 355. Heat set free in congelation, 356. Crystallization, 357. Ice-flowers, 358. Supersaturation, 359. Change of volume in congelation ; Expansive force of ice, 360. Melting-point altered by hydro- static pressure, 361. By stress of any kind, 362. Bottomley's experiment, 363. Regelation of ice, 364. Apparent plasticity of ice ; motion of glaciers, 365, pp. 320-336. CHAPTEB XXXI. EVAPORATION AND CONDENSATION. Transformation into vapour, 366. Vapour and gas, 367. Pressure of vapours; Maximum pressure and density, 368. Influence of temperature on maximum density and pressure, 369. Mixture of gas and vapour; Dalton's law, 370. Liquefaction of gases, 371. Faraday's method, 372. Latent heat of evaporation; Cold produced, 373. Leslie's experiment, 374. Cryophorus, 375. Freezing of water by evaporation of ether, 376. Freezing of mercury by means of sulphurous acid, 377. Carry's appa- ratus for freezing by ammonia, 378. Solidification of carbonic acid, 379. Con- tinuity of the liquid and gaseous states ; Critical temperature, 380. Liquefaction and solidification of oxygen and hydrogen, 381, pp. 337-354. CHAPTER XXXII. EBULLITION. Ebullition, 382. Its laws, 383. Its definition, 384. Effect of pressure upon the boiling- point, 385. Franklin's experiment, 386. Determination of heights by boiling-point, 387. Papin's digester, 388. Boiling-point of saline solutions, 389. Of liquid mix- tures, 390. Difficulty of boiling without air. Experiments of Donny and Dufour, 391. Spheroidal state, 392. Freezing of water and mercury in a red-hot crucible, 393. The metal not in contact with the liquid, 394. Distillation, 395. Circumstances which influence rapidity of evaporation, 396 pp. 355-369. CHAPTER XXXIII. QUANTITATIVE MEASUREMENTS RELATING TO VAPOURS. Experiments on pressure of steam, 397. Dalton's, 398. Regnault's, 399, 400. Results expressed by a curve, 401. Empirical formula, 402. Pressure of other vapours, 403. Gravitation measure and absolute measure of pressure, 404. Laws of combination by volume, 405. Relation of vapour-densities, to chemical equivalents, 406. Deter- mination of vapour-densities; Dumas' method, 407, 408, 409. Limiting values of relative densities, 410. Gay-Lussac's method, 411. Meyer's method, 412. Volume of vapour formed by a given mass of water, 413. Latent heat of evaporation; Despretz's apparatus, 414. Regnault's experiments, 415, .... pp. 370-388. CHAPTER XXXIV. HYGROMETRY. Relative humidity, 416. Simultaneous changes of the dry and vaporous constitutents, 417. Dew-point, 418, 419. Hygroscopes, 420. Hygrometers, 421. De Saussure's, 122. Dew-point hygrometers, 423. Dines', 424. DanielPs, 425. Regnault's, 426. TABLE OF CONTENTS. vii Wet and dry bulb hygrometer, 427, 428. Chemical hygrometer, 429. Weight of a given volume of moist air, 430. Saturated air at different temperatures and pres- sures, 431. Aqueous meteors, 432. Cloud and mist, 433. Varieties of cloud, 434. Causes of formation of cloud, 435. Ram, 436. Snow and hail, 437. Dew, 438, pp. 389-411. CHAPTER XXXV. CONDUCTION OF HEAT. Conduction, 439. Variable and permanent states, 440. Conductivity and diffusivity, 441. Their definitions, 442. Effect of change of units, 443. Illustrations of con- duction, 444. Metals the best conductors, 445. Davy lamp, 446. Walls of houses, 447. Norwegian cooking box, 448. Experimental determination of conductivity, 449. Of diffusivity, 450. Conductivity of rocks, 451. Of liquids, 452. Of water, 453, 454. Of gases, 455. Of hydrogen, 456, pp. 412-425. Note A. Differential equation of linear flow of heat, p. 425. Note B. Flow of heat in a bar, p. 426. Note C. Deduction of diffusivity from observations of underground temperature, p. 426. CHAPTEB XXXVI. RADIATION. Radiation distinguished from conduction, 457. A ponderable medium not essential, 458. Radiant heat travels in straight lines, 459. Surface conduction often co-operates with radiation, 460. Newton's law of cooling, 461. Dulong and Petit's law of cooling, 462. Consequences of this law, 463. Theory of exchanges, 464. Law of inverse squares, 465. Law of reflection of heat, 466. Burning mirrors, 467. Con- jugate mirrors, 468. Reflection, diffusion, absorption, and transmission, 469. Co- efficient of absorption and of emission, 470. Limit to radiating power, 471, pp. 428-439. CHAPTER XXXVII. RADIATION (Continued). Thermoscopic apparatus for radiant heat, 472. Comparison of emissive powers, 473. Of absorbing powers, 474. Variation of absorption with the source, 475. Reflecting power, 476. Diffusive power, 477. Peculiar property of lamp-black, 478. Diather- mancy, 479. Diathermancy of different substances, 480. Influence of thickness, 481. Relation between radiant heat and light, 482. Selective emission and absorption, 483, pp. 440-456. CHAPTER XXXVIII. THERMO-DYNAMICS. Connection between heat and work ; Fire-syringe, 484. Heat produced by friction, 485. Foucault's rotating disc, 486. Joule's determination of mechanical equivalent of heat, 487. First law of thermo-dynamics, 488. Heat lost in expansion of gases, 489. Difference of the two specific heats, 490. Thermic engines, 491. Carnot's investigations, 492. Examples of reversibility, 493. Second law of thermo-dynamics, 494. In- vestigation of formula for efficiency of reversible engine, 495. Thomson's absolute scale of temperature, 496. Heat required for change of volume and temperature, 497. Adiabatic changes of gases ; Ratio of the two specific heats, 498. Relations between adiabatic changes of volume, temperature, and pressure, 499. Numerical value of the ratio K, 500. Rankine's prediction of specific heat of air, 501. Cooling of air by ascent; Convective equilibrium, 502. Adiabatic compression of liquids and solids, 503. Adiabatic extension of a wire, 504. Adiabatic coefficients of elasticity, 505. Freezing of water which has been cooled below 0, 506. Lowering of freezing-point by pressure, 507. Heat of chemical combination, 508. Observations on it, 509. Ani- viii TABLE OF CONTENTS. mal heat and work, 510. Vegetable growth, 511. Solar heat, 512. Its sources, 513. Sources of energy available to man, 514. Dissipation of energy, 515, pp. 457-490. CHAPTER XXXIX. STEAM AND OTHEE HEAT ENGINES. Heat-engines, 516. Stirling's air-engine, 517. The steam-engine; its history, 518. Prin- ciple of the double-acting engine, 519. Arrangement for admitting the steam, 520. Movement of the slide-valve, 521. Air-pump of the condenser, 522. Governor-balls, 523. Use of the fly-wheel, 524. General description of Watt's engine, 525. Work- ing expansively, 526. Modification of slide-valve for expansive working, 527. Compound engines, 528. Surface condensers, 529. Classification of steam-engines, 530. Form and arrangement of the several parts, 531. Rotatory engines, 532. Boilers, 533. Boilers with the fire inside, 34. Bursting of boilers; Safety-valves, 535. Causes of explosion, 536. Feeding of the boiler; Giffard's injector, 537. Locomotive, history, 538. Description of a locomotive, 539. Apparatus for reversing: link-motion, 540. Gas-engines, 542, pp. 491-517. CHAPTER XL. TERRESTRIAL TEMPERATURES AND WINDS. Temperature of the air, 543. Mean temperature of day, month, and year, 544. Isothermals, 545. Insular and continental climates, 546. Temperature of the soil at different depths, 547. Increase of temperature downwards in the earth, 548. Decrease upwards in the air, 549. Causes of winds : general principle, 550. Land and sea breezes; Monsoons, 551. Trade-winds ; General atmospheric circulation, 552. Origin of cyclones, 553. Anemometers, 554. Ocean currents, 555, . pp. 518-530. EXAMPLES. PAGE Temperature the Three Scales. Ex. 1-12, 531 Expansion. Ex. 13-31, 531 Densities of Gases. Ex. 32-43, 532 Thermal Capacity. Ex. 44-50 534 Latent Heat. Ex. 51-59, 534 Various. Ex. 60, 61 535 Conduction. Ex. 62-65 535 Hygrometry. Ex. 66, 67, 536 Thermodynamics. Ex. 68-74, 53(5 Adiabatic Compression and Extension. Ex. 75-78, 53(5 ANSWERS TO EXAMPLES 537 FRENCH AND ENGLISH MEASURES. DECIMETRE DIVIDED INTO CENTIMETRES AND MILLIMETRES. INCHES AND TENTH! REDUCTION" OF FRENCH TO ENGLISH MEASURES. LENGTH. 1 millimetre = '03937 inch, or about ^ inch. 1 centimetre = '3937 inch. 1 decimetre =3 -937 inch. 1 metre =39 -37 inch =3 '281 ft. = l'0936 yd. 1 kilometre =1093 '6 yds., or about ! mile. More accurately, 1 metres 39 '370432 in. =3-2808693 ft. = l '09362311 yd. AREA. 1 sq. millim. ='00155 sq. in. 1 sq. centim. = '155 sq. in. 1 sq. decim. =15'5 sq. in. 1 sq. metre = 1550 sq. in. = 10764 sq. ft. = 1-196 sq. yd. VOLUME. 1 cub. millim. = -000061 cub. in. 1 cub. centim. = '061025 cub. in. 1 cub. decim. =61 '0254 cub. in. cub. metre =61025 cub. in. =35 '3156 cub. ft. = 1-308 cub. yd. The Litre (used for liquids) is the snme as the cubic decimetre, and is equal to 1'7617 pint, or -22021 gallon. MASS AND WEIGHT. 1 milligramme = '01543 grain. 1 gramme =15 '432 grain. 1 kilogramme=15432grains=2-205 Ibs. avoir. More accurately, the kilogramme is 2-20462125 Ibs. MISCELLANEOUS. 1 gramme per sq. centim. =2 '0481 Ibs. per sq. ft. 1 kilogramme per sq. centim. =14-223 Ibs. per sq. in. 1 kilogrammetre=7'2331 foot-pounds. 1 force de cheval=75 kilogrammetres per second, or 542^ foot-pounds per second nearly, whereas 1 horse-power (English) =550 foot- pounds per second. REDUCTION TO C.G.S. MEASURES. (See page 48.) [cm. denotes centimetre (s); gm. denotes gramme(s).] LENGTH. 1 inch =2 - 54 centimetres, nearly. 1 foot =30 "48 centimetres, nearly. 1 yard =91 '44 centimetres, nearly. 1 statute mile =160933 centimetres, nearly. More accurately, 1 inch =2 '53997 72 centi- metres. AREA. 1 sq. inch =6'45 sq. cm., nearly. 1 sq. foot =929 sq. cm., nearly. 1 sq. yard =8361 sq. cm., nearly. 1 sq. mile =2'59 x 10 10 sq. cm., nearly. VOLUME. 1 cub. inch =16-39 cub. cm., nearly. 1 cub. foot =23316 cub. cm., nearly. 1 cub. yard=764535 cub. cm., nearly. 1 gallon =4541 cub. cm., nearly. MASS. 1 grain = '0648 gramme, nearly. 1 D7,. avoir. = 28 - 35 gramme, nearly. 1 Ib. avoir. = 453'6 gramme, nearly. 1 ton =1 '016 x 10 6 gramme, nearly More accurately, 1 Ib. avoir. =453 '59265 gm. VELOCITY. 1 mile per hour =44 7.04 em. per sec. 1 kilometre per hour =27 7' cm. per sec. DENSITY. 1 Ib. per cub. foot = '016019 gm. per cub. cm. fl'2'4 Ibs. per cub. ft. =1 gm. per cub. cm. FRENCH AND ENGLISH MEASURES. FORCE (assuming <7=981). (Seep. 48.) Weigh t of 1 grain =63 '57 dynes, nearly. loz. avoir. = 2'7S x 10 4 dynes, nearly. 1 Ib. avoir. = 4'45 x 10 3 dynes,nearly. 1 ton =9'97 x!0 8 dynes, nearly. 1 gramme = 981 dynes, nearly. 1 kilogramme = 9-81 x 10 5 dynes, nearly. WORK (assuming (7=981). (See p. 48.) 1 foot-pound =1-356 x 10 7 ergs, nearly. 1 kilogrammetre =9 '81 x 10 7 ergs, nearly. Work in a second "j by one theoretical V 1 '46 x 10 9 ergs, nearly, "horse." STRESS (assuming #=981). 1 Ib. per sq. ft. =479 dynes per sq. cm., nearly. 1 Ib. per .sq. inch =6'9xlO" 4 dynes per sq. cm., nearly. 1 kilog. per sq^ cm. =9'81 x 10 5 dynes per sq. cm., nearly. 760 mm. of mercury at 0C. = 1 '014 x 10 6 dynes per sq. cm., nearly. 30 inches of mercury at C. = 1'0163 x 10 U dynes per sq. cm., nearly. 1 inch of mercury at C. =3'388 x 10 4 dynes per sq. cm., nearly. TABLE OF CONSTANTS. The velocity acquired in falling for one second in vacuo, in any part of Great Britain, is about 32'2 feet per second, or 9'81 metres per second. Tho pressure of one atmosphere, or 760 millimetres (29'922 inches) of mercury, is 1'033 kilogramme per sq. centimetre, or 14'73 Ibs. per square inch. The weight of a litre of dry air, at this pressure (at Paris) and 0" C., is 1-293 gramme. The weight of a cubic centimetre of water is about 1 gramme. The weight of a cubic foot of water is about 62' 4 Ibs. The equivalent of a unit of heat, in gravitation units of energy, is 772 for 1 the foot and Fahrenheit degree. 1390 for the foot and Centigrade degree. 424 for the metre and Centigrade degree. 42400 for the centimetre and Centigrade degree. In absolute units of energy, the equivalent is 41 - 6 millions for the centimetre and Centigrade degree ; or 1 gramme-degree is equivalent to 41 '6 million ergs. HEAT. CHAPTEE XXIV. THERMOMETRY. 276. Heat Cold. The words heat and cold express sensations so well known as to need no explanation; but these sensations are modified by subjective causes, and do not furnish an invariable cri- terion of objective reality. In fact, we may often see one person suffer from heat while another complains of cold. Even for the same person the sensations of heat and cold are comparative. A tempera- ture of 50 Fahr. suddenly occurring amid the heat of summer pro- duces a very decided sensation of cold, whereas the same temperature in winter has exactly the opposite effect. We may mention an old experiment upon this subject, which is at once simple and instructive. If we plunge one hand into water at 32 Fahr., and the other into water at about 100; and if after having left them some time in this position we immerse them simultaneously in water at 70, they will experience very different sensations. The hand which was formerly in the cold water now experiences a sensation of heat; that which was in the hot water experiences a sensation of cold, though both are in the same medium. This plainly shows that the sensations of heat and cold are modified by the condition of the observer, and conse- quently cannot serve as a sure guide in the study of calorific phe- nomena. Recourse must therefore be had to some more constant standard of reference, and such a standard is furnished by the ther- mometer. 277. Temperature. If several bodies heated to different degrees are placed in presence of each other, an interchange of heat takes place between them, by which they undergo modifications of opposite kinds; those that are hottest grow cooler, and those that are coldest grow warmer; and after a longer or shorter time these inverse phe- nomena cease to take place, and the bodies come to a state of mutual 17 258 THERMOMETRY. equilibrium. They are then said to be at the same temperature. If a source of heat is now brought to act upon them, their temperature is said to rise; if they are left to themselves in a colder medium, they all grow cold, and their temperature is said to fall. Two bodies are said to have the same temperature if when they are placed in contact no heat passes from the one to the other. If when two bodies are placed in contact heat passes from one to the other, that which gives heat to the other is said to have the higher temperature. Heat always tends to pass from bodies of higher to those of lower tem- perature. 278. Expansion. At the same time that bodies undergo these changes in temperature, which may be verified by the different impressions which they make upon our organs, they are subjected to other modifications which admit of direct measurement, and which serve as a means of estimating the changes of temperature themselves. These modifications are of different kinds, and we shall have occasion to speak of them all in the course of this work; but that which is especially used as the basis of thermometric measurement is change of volume. In general, when a body is heated, it increases in volume; and, on the other hand, when it is cooled its volume diminishes. The expansion of bodies under the action of heat may be illustrated by the following experiments. 1. Solid Bodies. We take a ring through which a metal sphere Fig. 181. Gravesande's King. just passes. This latter is heated by holding it over a spirit-lamp, and it is found that after this operation it will no longer pass through the ring. Its volume has increased. If it is now cooled by immer- sion in water, it resumes its former volume, and will again pass EXPANSION. 259 through the ring. If, while the sphere was hot, we had heated the ring to about the same degree, the ball would still have been able to pass, their relative dimensions being unaltered. This little apparatus is called Gravesandes Ring. 2. Liquids. A liquid, as water for instance, is introduced into the Ib2. Expansion of Liquids. Fig 183. Expansion of Gases. apparatus shown in Fig. 182, so as to fill at once the globe and a portion of the tube as far as a. The instrument is then immersed in a vessel containing hot water, and at first the extremity of the liquid column descends for an instant to 6; but when the experiment has continued for some time, the liquid rises to a point a at a con- siderable height above. This twofold phenomenon is easily explained. The globe, which receives the first impression of heat, increases in volume before any sensible change can take place in the temperature of the liquid. The liquid consequently is unable to fill the entire 260 THERMOMETRY. capacity of the globe and tube up to the original mark, and thus the extremity of the liquid column is seen to fall. But the liquid, receiving in its turn the impression of heat, expands also, and as it passes the original mark, we may conclude that it not only expands, but expands more than the vessel which contains it. 3. Gases. The globe in Fig. 183 contains air, which is separated from the external air by a small liquid index. We have only to warm the globe with the hands and the index will be seen to be pushed quickly upwards, thus showing that gases are exceedingly expansible. 279. General Idea of the Thermometer. Since the volume of a body is changed by heat, we may specify its temperature by stating its volume. And the body will not only indicate its own tempera- ture by this means, it will also exhibit the temperature of the bodies by which it is surrounded, and which are in equili- brium of temperature with it. Any body which gives quantitative indications of temperature may be called a thermometer. 280. Choice of the Thermometric Substance. As the expansions of different substances are not exactly proportional to one another, it is necessary to select some one substance or combination of sub- stances to furnish a standard; and the standard usually adopted is the apparent expansion of mer- cury in a graduated glass vessel. The instrument which exhibits this expansion is called the mercu- rial thermometer. It consists essentially, as shown in Fig. 184, of a tube of very small diameter, called the stem, terminating in a reservoir which, whatever ||; its shape, is usually called the bulb. The reservoir and a portion of the tube are filled with mercury. Q^ II If the temperature varies, the level of the liquid will rise or fall in the tube, and the points at which it stands can be identified by means of a scale attached to or engraved on the tube. 281. Construction of the Mercurial Thermometer. The construction of an accurate mercurial thermometer is an operation of great delicacy, and comprises the following processes. 1. Choice of the Tube. The first object is to procure a tube of as uniform bore as possible. In order to test the uniformity of the thermometers. CONSTRUCTION O* THERMOMETER. 261 bore, a small column of mercury is introduced into the tube, and the length which it occupies in different parts of the tube is measured. If these lengths are not equal, the tube is not of uniform bore. When Fig. 185. Introduction of the Mercury. a thermometer of great precision is required, the tube is calibrated; that is, divided into parts of equal volume, by marking upon it the lengths occupied by the column in its different positions. When a suitable tube has been obtained, a reservoir is either blown Fig. 186. Furnace for heating Thermometers. at one end or attached by melting, the former plan being usually preferable. 2. Introduction of the Mercury. At the upper end of the tube a temporary bulb is blown, and drawn out to a point, at which there is 2G2 THERMOMETRY. a small opening. This bulb, and also the permanent bulb, are gently heated, and the point is then immersed in a vessel containing mer- cury (Fig. 185). The air within the instrument, growing cold, diminishes in expansive force, so that a quantity of mercury is forced into the temporary bulb by the pressure of the atmosphere. The instrument is then set upright, and by alternate heating and cooling of the permanent bulb, a large portion of the mercury is caused to descend into it from the bulb at the top. The instrument is theii laid in a sloping position on a special furnace (Fig. 186) till the mer- cury boils. The vapour of the boiling mercury drives out the air, and when the mercury cools it forms a continuous column, filling the permanent bulb and tube. If any bubbles of air are seen, the operation of boiling and cooling is repeated until they are expelled. 3. Determination of the Fixed Points. The instrument, under these conditions, and with any scale of equal parts marked on the tube, would of course indicate variations of temperature, but these indications would be arbitrary, and two thermometers so constructed would in general give different indications. IP order to insure that the indications of different thermometers may be identical, it has been agreed to adopt two standard temperatures, which can easily be reproduced and main- tained for a considerable time, and to denote them by fixed numbers. These two temperatures are the freezing- point and boiling-point of water; or to speak more strictly, the temperature of melting ice, and the temperature of the steam given off by water boil- ing under average atmospheric pres- sure. It has been observed that if the thermometer be surrounded with melt- ing ice (or melting snow), the mercury, under whatever circumstances the ex- periment is performed, invariably stops at the same point, and remains stationary there as long as the melting continues. This then is a fixed temperature. On the Centigrade scale it is called zero, on Fahrenheit's scale 32. In order to mark this point on a thermometer, it is surrounded by melting ice, which is contained in a perforated vessel, so as to allow Fig. 187. Determination of Freezing- point. FIXED POINTS. 2G3 the water produced by melting to escape. When the level of the mercury ceases to vary, a mark is made on the tube with a fine diamond at the extremity of the mercurial column. This is frequently called for brevity the freezing-point. Fig. 188. Determination of Boiling-point. It has also been observed that if water be made to boil in an open metallic vessel, under average atmospheric pressure (76 centimetres, or 29'922 inches), and if the thermometer be plunged into the steam, the mercury stands at the same point during the entire time of ebullition, provided that the external pressure does not change. This second fixed temperature is called 100 in the Centigrade scale (whence 264 THERMOMETRY. the name), and 212 on Fahrenheit's scale. In order to mark this second point on the thermometer, an apparatus is employed which was devised by Gay-Lussac, and perfected by Regnault. It consists of a copper boiler (Fig. 188) containing water which is raised to ebullition by means of a furnace. The steam circulates through a double casing, and escapes by a tube near the bottom. The ther- mometer is fixed in the interior casing, and when the mercury has become stationary, a mark is made at the point at which it stops, which denotes what is commonly called for brevity the boiling-point. A small manometric tube, open at both ends, serves to show, by the equality of level of the mercury in its two branches, that the ebullition is taking place at a pressure equal to that which prevails externally, and consequently that the steam is escaping with suffi- cient freedom. It frequently happens that the external pressure is not exactly 760 millimetres, in which case the boiling-point should be placed a little above or a little below the point at which the mer- cury remains stationary, according as the pressure is less or greater than this standard pressure. When the difference on either side is inconsiderable, the position of the boiling-point may be roughly cal- culated by the rule, that a difference of 27 millimetres in the pres- sure causes a difference of 1 in the temperature of the steam pro- duced. We shall return to this point in Chap, xxxii. It now only remains to divide the portion of the instrument between the freezing and boiling points into equal parts corresponding to .single degrees, and to continue the division beyond the fixed points. Below the zero point are marked the numbers 1, 2, 3, &c. These temperatures are expressed with the sign . Thus the tempera- ture of 17 below zero is written 17. 282. Adjustment of the Quantity of Mercury. In order to avoid complicating the above explanation, we have omitted to consider an operation of great importance, which should precede those which we have just described. This is the determination of the volume which Fig. 189. must be given to the reservoir, in order that the instrument may have the required range. When the reservoir is cylindrical, this is easily effected in the following manner. Suppose we wish the thermometer to indicate temperatures comprised between 20 and 130 Cent., so that the range is to be 150; the reservoir is left open at O (Fig. 189), SCALES OF TEMPERATURE. 265 j| P P and is filled through this opening, which is then hermetically sealed. The instrument is then immersed in two baths whose temperatures differ, say, by 50, and the mercury rises through a distance m m'. This length, if the quantity of mercury in the reservoir be exactly sufficient, should be the third part of the length of the stem. The quantity of mercury in the reservoir is always taken too large at first, so that it has only to be reduced, and thus the space traversed by the liquid is at first too great. Suppose it to be equal to f ths of the length of the stem. The degrees will then be too long, in the ratio f :=f; that is, the reservoir is |- of what it should be. We therefore measure off ths of the length of the reservoir, beginning at the end next the stem; this distance is marked by a line, and the end is then broken and the mercury suiFered to escape. The glass is then melted down to the marked line, and the reservoir is thus brought to the proper dimensions. It only remains to regulate the quantity of mercury admitted, by making it fill the tube at the highest temperature which the instrument is intended to indicate. If the reservoir were spherical, which is a shape generally ill adapted for delicate thermometers, the foregoing process would be inapplicable, and it would be necessary to determine the proper size by trial. 283. Thermometric Scales. In the Centigrade scale the freezing-point is marked 0, and the boil- ing-point 100. In Reaumur's scale, which is still popularly used on the Continent, the freezing-point is also marked 0, but the boiling-point is marked 80. Hence, 5 degrees on the former scale are equal to 4 on the latter, and the reduction of temperatures from one of these scales to the other can be effected by multiplying by % or . For example, the temperature 75 Centigrade is the same as 60 Reaumur, since 75 xf =60; and the temperature 36 Re'aumur is the same as 45 Centigrade, since 36 x=45. The relation between either of these scales and that of Fahrenheit is rather more complicated, inasmuch as Fahrenheit's zero is not at freezing-point, but at 32 of his degrees below it. Fig. 190 Thermometric Scales. 266 THERMOMETRY. As regards intervals of temperature, 180 degrees Fahrenheit are equal to 100 Centigrade, or to 80 Reaumur, and hence, in lower terms, 9 degrees Fahrenheit are equal to 5 Centigrade, or to 4 Reaumur. The conversion of temperatures themselves (as distinguished from intervals of temperature) will be best explained by a few examples. Example 1. To find what temperatures on the other two scales are equivalent to the temperature 50 Fahrenheit. Subtracting 32, we see that this temperature is 18 Fahrenheit degrees above freezing-point, and as this interval is equivalent to 18 x , that is 10 Centigrade degrees, or to IS x f , that is 8 Re'aumur degrees, the equivalent temperatures are respectively 10 Centigrade and 8 Reaumur. Example 2. To find the degree on Fahrenheit's scale, which is equivalent to the temperature 25 Centigrade. An interval of 25 Centigrade degrees is equal to 25 x f, that is 45 Fahrenheit degrees, and the temperature in question is above freezing-point by this amount. The number denoting it on Fahren- heit's scale is therefore 32 + 45, that is 77. The rules for the conversion of the three thermometric scales may be summed up in the following formulae, in which F, C, and R denote equivalent temperatures expressed in degrees of the three scales: C = |R=f (F-32). R=|C=(F-32). It is usual, in stating temperatures, to indicate the scale referred to by the abbreviations Fahr., Cent., Reau., or more briefly by the initial letters F., C., R. 284. Displacement of the Zero Point. A thermometer left to itself after being made, gradually undergoes a contraction of the bulb, leading to a uniform error of excess in its indications. This pheno- menon is attributable to molecular change in the glass, which has, so to speak, been tempered in the construction of the instrument, and to atmospheric pressure on the exterior of the bulb, which is unre- sisted by the internal vacuum. * The change is most rapid at first, and usually becomes insensible after a year or so, unless the thermo- meter is subjected to extreme temperatures. Its total amount is usually about half a degree. On account of this change it is advis- able not to graduate a thermometer till some time after it has been sealed. ALCOHOL THERMOMETER. 267 285. Sensibility of the Thermometer. The power of the instrument to detect very small differences of temperature may be regarded as measured by the length of the degrees, which is proportional to the capacity of the bulb directly and to the section of the tube inversely ( 299). Quickness of action, on the other hand, requires that the bulb be small in at least one of its dimensions, so that no part of the mercury shall be far removed from the exterior, and also that the glass of the bulb be thin. Quickness of action is important in measuring temperatures which vary rapidly. It should also be observed that, as the thermometer, in coming to the temperature of any body, necessarily causes an inverse change in the temperature of that body, it follows that when the mass of the body to be investigated is very small, the thermo- meter itself should be of extremely small dimensions, in order that it may not cause a sensible variation in the temperature which is to be observed. 286. Alcohol Thermometer. In the construction of thermometers, other liquids may be introduced instead of mercury; and alcohol is very frequently employed for this purpose. Alcohol has the disadvantage of being slower in its action than mercury, on account of its inferior conductivity; but it can be em- ployed for lower temperatures than mercury, as the latter congeals at 39 Cent. ( 38 Fahr.), whereas the former has never congealed at any temperature yet attained. If an alcohol thermometer is so graduated as to make it agree with a mercurial thermometer (which is the usual practice), its degrees will not be of equal length, but will become longer as we ascend on the scale. If mercury is regarded as expanding equally at all temperatures, alcohol must be described as expanding more at high than at low temperatures. 287. Self-registering Thermometers. It is often important for meteorological purposes to have the means of knowing the highest or the lowest temperature that occurs during a given interval. In- struments intended for this purpose are called maximum and mini- mum thermometers. The oldest instrument of this class is Six's (Fig. 191), which is at once a maximum and a minimum thermometer. It has a large cylindrical bulb C filled with alcohol, which also occupies a portion of the tube. The remainder of the tube is partly filled with mercury, 268 THEKMOMETRY. which occupies a portion of the tube shaped like the letter U, one extremity of the mercurial column being in contact with the alcohol already mentioned, while the other extremity is in contact with a second column of alcohol; and beyond this there is a small space occupied only with air, so as to leave room for the expansion of the liquids. When the alcohol in the bulb ex- pands, it -pushes the mercurial column before it, and when it contracts the mercurial column follows it. The extreme points reached by the two ends of the mercurial column are registered by a pair of light steel indices c, d (shown on an enlarged scale at K), which are pushed before the ends of the column, and then are held in their places by springs, which are just strong enough to prevent slipping, so that the indices do not follow the mercury in its retreat. One of the indices d registers the maximum and the other c the minimum temperature which has occurred since the instrument was last set. The setting consists in bringing the indices into contact with the ends of the mercurial column, and is usually effected by means of a magnet. This instrument is now, on account of its complexity, little used. It possesses, however, the advantages of being equally quick (or slow) in its action for maximum and minimum temperatures, which is an important property when these tem- peratures are made the foundation for the computation of the mean temperature of the interval, and of being better able than most of the self -registering thermometers to bear slight jolts without disturbance of the indices. Rutherford's self -registering thermometers are frequently mounted together on one frame, as in Fig. 192, but are nevertheless distinct instruments. His 'minimum thermometer, which is the only mini- mum thermometer in general use, has alcohol for its fluid, and is always placed with its tube horizontal, or nearly so. In the fluid column there is a small index n of glass or enamel, shaped like a dumb-bell. Fig. 191. Six's Self -registering Thermometer. SELF-REGISTERING THERMOMETERS. 269 When contraction occurs, the index, being wetted by the liquid, is forced backwards by the contractile force of the superficial film which forms the extremity of the liquid column ( 185); but when expansion takes place the index remains stationary in the interior of the liquid. Hence the minimum temperature is indicated by the position of the Fig. 192. Rutherford's Maximum and Minimum Thermometers. forward end of the index. The instrument is set by inclining it so as to let the index slide down to the end of the liquid column. The only way in which this instrument is liable to derangement, is by a portion of the spirit evaporating from the column and becom- ing condensed in the end of the tube, which usually terminates in a small bulb. When the portion thus detached is large, or when the column of spirit becomes broken into detached portions by rough usage in travelling, "let the thermometer be taken in the hand by the end farthest from the bulb, raised above the head, and then forcibly swung down towards the feet; the object being, on the prin- ciple of centrifugal force, to send down the detached portion of spirit till it unites with the column. A few throws or swinging strokes will generally be sufficient; after which the thermometer should be placed in a slanting position, to allow the rest of the spirit still adherino- to the sides of the tube to drain down to the column. But O another method must be adopted if the portion of spirit in the top of the tube be small. Heat should then be applied slowly and cautiously to the end of the tube where the detached portion of spirit is lodged ; this being turned into vapour by the heat will condense on the sur- face of the unbroken column of spirit. Care should be taken that the heat is not too quickly applied. . . . The best and safest way to apply the requisite amount of heat, is to bring the end of the tube slowly down towards a minute flame from a gas-burner; or if gas is not to be had, a piece of heated metal will serve instead." 1 Rutherford's maximum thermometer is a mercurial thermometer with the stem placed horizontally, and with a steel index c in the tube, outside the mercurial column. When expansion occurs, the 1 Buchan's Handy Book of Meteorology, p. 62. 270 THERMOMETRY. index, not being wetted by the liquid, is forced forwards by the con- tractile force of the superficial film which forms the extremity of the liquid column ( 185); but when contraction takes place, the index remains stationary outside the liquid. Hence the maximum tem- perature is indicated by the position of the backward end of the index. The instrument is set by bringing the index into contact with the end of the liquid column, an operation which is usually effected by means of a magnet. This thermometer is liable to get out of order after a few years' use, by chemical action upon the surface of the index, which causes it to become wetted by the mercury, and thus renders the instrument useless. Phillips' maximum thermometer (invented by Professor Phillips, the eminent geologist, and made by Casella) is recommended for use in the official Instructions for Taking Meteorological Observations, drawn up by Sir Henry James for the use of the Royal Engineers. It is a mercurial thermometer not deprived of air. It has an exceed- Fig. 193. Phillips' Maximum Thermometer. ingly fine bore, and the mercurial column is broken by the insertion of a small portion of air. The instrument is set by reducing this portion of ' air to the smallest dimensions which it can be made to assume, and is placed in a horizontal position. When the mercury expands, it pushes forwards this intervening air and the detached column of mercury beyond it; but when contraction takes place the intervening air expands, and the detached column remains unmoved. The detached column is not easily shaken out of its place, and when the bore of the tube is made sufficiently narrow the instrument may even be used in a vertical position, a property which is often of great service. In Negretti and Zambra's maximum thermometer (Fig. 194), which is employed at the Royal Observatory, Greenwich, there is an obstruction in the bent part of the tube, near the bulb, which barely leaves room for the mercury to pass when forced up by expansion, and is sufficient to prevent it from returning when the bulb cools. DEEP-WELL THERMOMETER 271 The objection chiefly urged against this thermometer is the extreme mobility of the detached column, which renders it very liable to Fig. 194. Negretti's Maximum Thermometer. accidental displacement; but in the hands of a skilful observer this is of no moment. Dr. Balfour Stewart (Elementally Treatise on Heat, p. 20, 21), says: "When used, the stem of this instrument ought to be inclined downwards. ... It does not matter if the column past the obstruction go down to the bottom of the tube; for when the instrument is read, it is gently tilted up until this detached column flows back to the obstruction, where it is arrested, and the end of the column will then denote the maximum temperature. In resetting the instrument, it is necessary to shake the detached column past the obstruction in order to fill up the vacancy left by the contraction of the fluid after the maximum had been reached." DEEP-SEA AND WELL THERMOMETERS. Self-registering thermometers intended for observing at great depths in water should be inclosed in an outer case of glass herme- tically sealed, the intervening space being occupied wholly or partly by air, so that the pressure outside may not be transmitted to the thermometer. A thermometer not thus protected gives too high a reading, because the compres- sion of the bulb forces the liquid up the tube. The instru- ment represented in Fig. 195 was designed by Sir Wm. Thomson for the Committee on Underground Temperature appointed by the British Association. A is the protecting case, B the Phillips' thermometer inclosed in it, and sup- ported by three pieces of cork ccc. A small quantity of spirit s occupies the lower part of the case; d is the air- bubble characteristic of Phillips' thermometer, and serving to separate one portion of the mercurial column from the ^omson's rest. In the figure this air-bubble is represented as ex- Protected . Thermometer. panded by the descent of the lower portion ot mercury, while the upper portion remains suspended by adhesion. This in- strument has been found to register correctly even under a pressure of 2 1 tons to the square inch. 272 THERMOMETRY. The use of the spirit s is to bring the bulb more quickly to the temperature of the surrounding medium. Another instrument, designed, like the foregoing, for observations in wells and borings, is Walferdin's maximum thermometer (Fig. 196). Its tube terminates above in a fine point opening into a cavity of considerable size, which contains a sufficient quantity of mercury to cover the point when the instrument is inverted. The instrument is set by placing it in this inverted position and warming the bulb until the mercury in the stem reaches the point and becomes connected with the mercury in the cavity. The bulb is then cooled to a temperature lower than that which is to be observed; and during the operation of cooling, mercury enters the tube so as always to keep it full. The instrument is then lowered in the erect position into the bore where observations are to be made, and when the temperature of the mercury rises a portion of it overflows from the tube. To ascertain the maximum temperature which has been experienced, the instrument may be im- mersed in a bath of known temperature, less than that of the boring, and the amount of void space in the upper part of the tube will indicate the excess of the maximum tem- perature experienced above that of the bath. If the tube is not graduated, the maximum temperature can be ascertained by gradually raising the temperature of the bath till the tube is just full. If the tube is graduated, the graduations can in strictness only indicate true degrees for some one standard temperature of setting, since the length of a true degree is proportional to the quantity of mercury in the bulb and tube; but a difference of a few degrees in the temperature of setting is immaterial, since 10 Cent, would only alter the length of a degree by about one six-hundredth part. 288. Thermograph. A continuous automatic record of the indica- tions of a thermometer can be obtained by means of photography, and this plan is now adopted at numerous observatories. The fol- lowing description relates to the Royal Observatory, Greenwich. A sheet of sensitized paper is mounted on a vertical cylinder just behind the mercurial column, which is also vertical, and is protected from the action of light by a cover of blackened zinc, with the exception of a narrow vertical strip just behind the mercurial column. A strong beam of light from a lamp or gas flame is concentrated by a cylindric Fig. 196. THERMOGRAPH. 273 lens, so that if the thermometer were empty of mercury a bright vertical line of light would be thrown on the paper. As this beam of light is intercepted by the mercury in the tube, which for this purpose is made broad and flat, only the portion of the paper above the top of the mercurial column receives the light, and is photographi- cally affected. The cylinder is made to revolve slowly by clock-work, and if the mercury stood always at the same height, the boundary between the discoloured and the unaffected parts of the paper would be straight and horizontal, in consequence of the horizontal motion of the paper itself. In reality, the rising and falling of the mercury, combined with the horizontal motion of the paper, causes the line of separation to be curved or wavy, and the height of the curve above a certain datum-line is a measure of the temperature at each instant of the day. 1 The whole apparatus is called a thermograph, and ap- paratus of a similar character is employed for obtaining a continuous photographic record of the indications of the barometer 2 and mag- netic instruments. 289. Metallic Thermometers. Thermometers have sometimes been constructed of solid metals. Breguet's thermometer, for example (Fig. 197), consists of a helix carrying at its lower end a horizontal needle which traverses a dial. The helix is composed of three metallic strips, of silver, gold, and platinum, soldered together so as to form a single ribbon. The silver, which is the most expansible, is placed in the interior of the helix; the platinum, which is the least expansible, on the exterior; and the gold serves to connect them. When the temperature rises, the helix unwinds and produces a deflection of the needle; when the temperature falls, the helix winds up and deflects the needle in the opposite direction. Fig. 198 represents another dial-thermometer, in which the ther- mometric portion is a double strip composed of steel and brass, bent into the form of a nearly complete circle, as shown by the dotted lines in the figure. One extremity is fixed, the other is jointed to the 1 Strictly speaking, the temperatures corresponding to the various points of the curve are not read off by reference to a single datum-line, but to a number of datum -lines which represent the shadows of a set of horizontal wires stretched across the tube of the ther- mometer at each degree, a broader wire being placed at the decades, and also at 32, 52, and 72. In order to give long degrees, the bulb of the thermometer is made very large eight inches long, and *4 of an inch in internal diameter. (Greenwich Observations, 1847.) 5 See 208. 18 274 THERMOMETRY. shorter arm of a lever, whose longer arm carries a toothed sector. This latter works into a pinion, to which the needle is attached. &*$ Fig. 197. Breguet's Thermometer. Fig. 198. Metallic Thermometer. It may be remarked that dial-thermometers are very well adapted for indicating maximum and minimum temperatures, it being only necessary to place on opposite sides of the needle a pair of movable indices, which could be pushed in either direction according to the variations of temperature. Generally speaking, metallic thermometers offer great facilities for automatic registration. In Secchi's meteorograph, for example, the temperature is indi- cated and registered by the expansion of a long strip of brass (about 17 metres long) kept constantly stretched by a suitable weight; this expansion is rendered sensible by a system of levers connected with the tracing point. The thermograph of Hasler and Escher consists of a steel and a brass band connected together and rolled into the form of a spiral. The movable extremity of the spiral, by acting upon a projecting arm, produces rotation of a steel axis which carries the tracer. 290. Pyrometers. Metallic thermometers can generally be em- ployed for measuring higher temperatures than a mercurial ther- mometer could bear; but there is great difficulty in constructing any instrument to measure temperatures as high as those of furnaces. Instruments intended for this purpose are called pyrometers. Wedgwood, the famous potter, invented an apparatus of this kind, consisting of a gauge for measuring the contraction experienced by a piece of baked clay when placed in a furnace; and Brongniart DIFFERENTIAL THERMOMETERS. 275 introduced into the porcelain manufactory at Sevres the instrument represented in Fig. 199,. consisting of an iron bar lying in a groove in a porcelain slab, with one end abutting against the bottom of the groove, and the other projecting through the side of the fur- nace, where it gave motion to an indicator. Neither of these instruments has, how- ever, been found to furnish consistent in- dications, and the only instrument that is now relied on for the measurement of very high temperatures is the air-thermometer. 291. Differential Thermometer. Leslie of Edinburgh invented, in the beginning of the present century, the instrument shown in Fig. 200, for detecting small differences of temperature. A column of USs Fig. 199. Brongniart's Pyrometer. Fig. 200. Leslie's Differential Thermometer. Fig. 201. Rumlord's Thennoscope. sulphuric acid, coloured red, stands in the two branches of a bent tube, the extremities of which terminate in two equal bulbs contain- ing air. When both globes are at the same temperature, whatever that temperature may be, the liquid, if the instrument is in order, stands at the same height in both branches. This height is marked zero on both scales. When there is a difference of temperature be- tween them, the expansion of the air in the warmer bulb produces a 276 THERMOMETRY. depression of the liquid on that side and an equal elevation on the other side. The differential thermometer is an instrument of great sensibility, and enabled Leslie to conduct some important investigations on the subject of the radiation of heat. It is now, however, superseded by the thermo-pile invented by Melloni. This latter instrument will be described in another portion of this work. Rumford's thermoscope (Fig. 201) is analogous to Leslie's differential thermometer. It differs from it in having % the horizontal part much longer, and the vertical branches shorter. In the horizontal tube is an alcohol index, which, when the two globes are at the same temperature, occupies exactly the middle. CHAPTER XXY. MATHEMATICS OF EXPANSION. 292. Expansion. Factor of Expansion. When a body expands from volume V to volume V + v, the ratio ^ is called the expansion of volume or the cubical expansion of the body. In like manner if the length, breadth, or thickness of a body in- creases from L to L + 1, the ratio j- is called the linear expansion. The ratio -^ will be called, in this treatise, the factor of cubical expansion, and the ratio -jj- the factor of linear expansion. In each case the factor of expansion is unity plus the expansion. Similar definitions apply to expansion of area or superficial expan- sion; but it is seldom necessary to consider this element in thermal discussions. 293. Relation between Linear and Cubical Expansion. If a cube, whose edge is the unit length, expands equally in all directions, the length of each edge will become 1 + 1, where I is the linear expan- sion; and the volume of the cube will become (l + ) 3 or In the case of the thermal expansion of solid bodies I is always very small, so that l z and I 3 can be neglected, and the expansion of volume is therefore SI; that is to say, the cubical expan- sion is three times the linear expansion. This is illustrated geometrically by Fig. 202, which Fig 202 represents a unit cube with a plate of thickness I and therefore of volume I applied to each of three faces; the total volume added being therefore SI. Similar reasoning shows that the superficial expansion is double ttie linear expansion. 278 MATHEMATICS OF EXPANSION. These results have been deduced from the supposition of equal expansion in all directions. If the expansions of the cube in the directions of three conterminous edges be denoted by a, b, c, the angles being supposed to remain right angles, the volume will become (l + a)(l + 6)(l + c) or 1 + a+b + c + ab + ac+bc+abc, which, when a, b and c are so small that their products can be neglected, becomes l_j_ a _l_& + C ; S o that the expansion of volume is the sum of the expansions of length, breadth, and thickness. 294. Variation of Density. Since the density of a body varies inversely as its volume, the density after expansion will be obtained by dividing the original density by the factor of expansion. In fact, if V, D denote the volume and density before, and V, D' after expansion, the mass of the body, which remains unchanged, is equal to V D, and also to V D'. We have therefore ^ = ^-^ where e denotes the expansion of volume, and therefore 1-f e the factor of expansion. Since j is 1 e + e 2 e 3 +&c., it is sensibly equal to le when e is small. We have therefore D'=D(1 e). 295. Real and Apparent Expansion. When the volume of a liquid is specified by the number of divisions which it occupies in a gradu- ated vessel, it is necessary to take into account the expansion of the vessel, if we wish to determine the true expansion of the liquid. Let a denote the apparent expansion computed by disregarding the expansion of the vessel and attending only to the number of divisions occupied. Then if n be the number of divisions occupied before, and n after expansion, we have n' = n (1 +a). Let g denote the real expansion of the containing vessel; then if d be the volume of each division before, and d' after expansion, we have Let m denote the real expansion of the liquid. Then if v denote the real volume of the liquid before, and v' after expansion, we have i/=v (l + m). But since the volume v consists of n parts each having the volume d, we have v nd, and in like manner REAL AND APPARENT EXPANSION. 279 Substituting for n and d' in this last equation, we have v' = n(l+a)d(l+g)=v(I + a) But t/ = v(l + m). Hence we have that is, the factor" of real expansion of the liquid is the product of the factor of real expansion of the vessel and the factor of apparent expansion. Multiplying out, we have and as the term ag, being the product of two small quantities, is usually negligible, we have sensibly that is, the expansion of the liquid is the sum of the expansion of the glass and the apparent expansion. This investigation is applicable to the mercurial thermometer when the capacity of the bulb has been expressed in degrees of the stem. Similar reasoning applies to the apparent expansion of a bar of one metal as measured by means of a graduated bar of a less expan- sible metal. The real expansion of the bar to be measured will be sensibly equal to the sum of the expansion of the measuring bar and the apparent expansion. In adopting the mercurial thermometer as the standard of tem- perature (the tube being graduated into equal parts), we virtually adopt the apparent expansion of mercury in glass as our standard of uniform expansion. 296. Physical Meaning of the Degrees of the Mercurial Thermometer. Since the stem of a mercurial thermometer is divided into degrees of equal capacity, we can express the capacity of the bulb in degrees. Let the capacity of the bulb together with as much of the stem as is below the freezing-point be N degrees, and let the interval from freezing to boiling point be n degrees; then ^ is the apparent ex- pansion of the mercury from freezing to boiling point. When the Centigrade scale is employed, this apparent expansion is -^-, and the apparent expansion from zero to t is ^. Hence the apparent expan- sion from zero to t is ^ of the apparent expansion from zero to 100. This last statement constitutes the definition of the temperature t when the mercurial thermometer is regarded as the standard. 280 MATHEMATICS OF EXPANSION. 297. Comparability of Mercurial Thermometers. If two mercurial thermometers, each of them constructed so as to have its degrees rigorously equal in capacity, agree in their indications at all tem- peratures, the above investigation shows that the apparent expan- sions of the mercury in the two instruments must be exactly propor- tional. But we have shown in 295 that the apparent expansion a is equal to f mg, f m denoting the real expansion of the mercury, and g that of the glass. Mercury, being a liquid and an elementary sub- stance, can always be obtained in the same condition, so that m will have the same value in the two thermometers; but it is difficult to ensure that two specimens of glass shall be exactly alike; hence g has different values in different thermometers. The agreement of the two thermometers does not, however, require identity in the valves of mg, but only proportionality; in other words it requires that the fraction (where g^ and g z are the values of g for the two instruments) shall have the same value at all temperatures. The average value of g is about $ of that of m. In other words mercury expands about 7 times as much as glass. 298. Steadiness of Zero in Spirit Thermometers. It is obvious from 296 that the volume of a degree can be computed by multi- plying the capacity of the bulb by the number which denotes the apparent expansion for one degree. Alcohol expands about 6 times as much as mercury, and its apparent expansion in glass is about 7 times that of mercury. Hence with the same size of bulb, the degrees of an alcohol thermometer will be about 7 times as large as those of a mercurial thermometer, and a contraction of the bulb which produces a change of one degree in the reading of a mercurial thermometer, would only produce a change of one-seventh of a degree in the reading of an alcohol thermometer. This is the reason, or at all events one reason, why displacement of the zero point ( 284) is insignificant in spirit thermometers. 299. Length of a Degree on the Stem. Since the length of a degree upon the stem of a thermometer is equal to the volume of a degree divided by the sectional area of the tube, the formula for this length is ^-, where a denotes the apparent expansion for one degree, C the capacity of the bulb with as much of the stem as is below zero, and WEIGHT THERMOMETER. 281 s the sectional area of the stem. The value of a for the mercurial Centigrade thermometer is about ^-^ D-toO 300. Weight Thermometer. In the weight thermometer (Fig. 203) the apparent expansion of mercury is observed by comparing the weight of the mercury which passes the zero point with that of the mercury which remains below it. The tube is open, and its mouth is the zero point. The instrument is first filled with mercury at zero, and is then exposed to the tempera- ture which it is required to measure. The mercury which overflows is caught and weighed, and the weight of the mercury which remains in the instrument is also deter- mined usually by subtracting the weight of the overflow from that of the original contents. The weight of the overflow, divided by the weight of what remains, is equal to the apparent expansion; for it is the same as the ratio of the volume of mercury above the zero point to the volume below it in an ordinary thermometer. In order to measure temperatures in degrees, with this thermo- meter, the apparent expansion from to 100 C. must be determined once for all and put on record. One hundredth part of this must be divided into the apparent expansion observed at the unknown temperature t, and the quotient will be t. 301. Expansion of Gases. In the case of solids and liquids the expansions produced by heat are usually very small, so that it is not important to distinguish between the value of ^ and the value of y3_- ( 292). But in the case of gases much larger expansions occur, and it is essential to attend to the above distinction. By general agreement, the volume of a gas at zero (Centigrade) is taken as the standard with which the volume at any other temperature is to be compared. We shall denote the volume at zero by V , and the volume at temperature t by V t . Then, if the pressure be the same at both temperatures, we shall write Vi=V,(l+at) where a is called the mean coefficient of expansion between the temperatures and t. Experiment has shown that when tempera- tures are measured by the mercurial thermometer, graduated in the manner which we have already described, a is practically the same at all temperatures which lie within the range of the mercurial 282 MATHEMATICS OF EXPANSION. thermometer. In other words, the expansions of gases are sensibly proportional to the apparent expansion of mercury in glass. More- over, the coefficient a is not only the same for different temperatures, but it is also the same for different gases; its value being always very approximately 00366 or 2^- By Boyle's law, the product of the volume and pressure of a gas remains constant when the temperature is constant. We have been supposing the pressure to remain constant, so that the product in question is proportional to the volume only. If the volume is kept constant the pressure will vary in proportion to l+a, so that we shall have P< = P (l+aO, P and P t denoting the pressures at and t respectively. If we remove all restriction, we have where (V P) , (V P) t denote the products of volume and pressure at and t respectively. Hence the value of the expression VP l + at will be the same for all values of V, P and t. Since the mass is unchanged, the density D varies inversely as the volume, and there- fore P D (l + at) is also constant. 302. General Definition of Coefficient of Expansion. If V denote the volume of any substance at temperature (Centigrade), V t its volume under the same pressure at temperature t, and V^ its volume at a higher temperature t', the mean coefficient of expansion a. be- tween the temperatures t and t' is denned by the equation V f -V|=V,a(f-l) and the coefficient oj expansion at the temperature t is the limit to which a approaches as t' approaches t; that is, in the language of the differential calculus, it is ! dy V df If we make V unity, the coefficient of expansion at temperature t will be simply dy df B \B' CHAPTER XXVI. EXPANSION OF SOLIDS. 303. Observations of Linear Expansion. Laplace and Lavoisier determined the linear expansion of a great number of solids by the following method. The bar AB (Fig. 204) whose expansion is to be determined, has one end fixed at A, while the other can move freely, pushing before it the lever OB, which is movable about the point O, and carries a tele- scope Whose line of Sight prtocipie O f the Meth^of Laplace and Lavoirier. is directed to a scale at some distance. A displacement BB' corresponds to a considerably greater length CC' on the scale, the ratio of the former to the latter being the same as that of OB to 00. The apparatus employed by Laplace and Lavoisier is shown in Fig. 205. The trough C, in which is laid the bar whose expansion is to be determined, is placed between four massive uprights of hewn stone N. One of the extremities of the bar rests against a fixed bar B', firmly joined to two of the uprights; the other extremity, which rests upon a roller to give it greater freedom of movement, pushes the bar B, which produces the rotation of the axis aa'. This axis carries with it in its rotation the telescope LL', which is directed to the scale. The first step is to surround the bar with melting ice, and take a reading through the telescope when the bar is at the tempera- ture zero. The temperature of the trough is then raised, and read- 284 EXPANSION OF SOLIDS. ings are taken, which, by comparison with the first, give the increase of length. Fig. 20 The following table contains the most important results thus ob- tained: COEFFICIENTS OF LINEAR EXPANSION. Gold, Paris standard, annealed, 0'000015153 unannealed, 0'000015515 Steel not tempered, . . . 0-000010792 Tempered steel reheated to 65, 0-000012395 Silver obtained by cupellation, 0'000019075 Soft wrought iron, .... Round iron, wire drawn, . English flint-glass, .... Gold, procured by parting, . Platina, 0-000012204 0-000012350 0-000008116 0-000014660 0-000009318 Silver, Paris standard, . . 0-000019086 Lead. 0-000088483 Copper, 0-000017173 French glass with lead . 0-000008715 Brass, 0-000018782 0-000029416 Malacca tin, '00001 9376 Forged zinc, ...... 0-000031083 Falniouth tin, . 0'000021729 The coefficient of expansion of a metal is not precisely the same at all temperatures, but it is sensibly constant from to 100 C. A simpler and probably more accurate method of observing expan- sions was employed by Ramsden and Boy. It consists in the direct observation of the distances moved by the ends of the bar, by means of two microscopes furnished with micrometers, the microscopes themselves being attached to an apparatus which is kept at a constant temperature by means of ice. 304. Compensated Pendulum. The rate of a clock is regulated by the motion of its pendulum. Suppose the clock to keep correct time at a certain temperature. Then at higher temperatures the pendu- lum will be too long and will therefore vibrate too slowly, so that COMPENSATED PENDULUMS. 285 the clock will lose. At lower temperatures, on the other hand, the clock will gain. To obviate or, at least, diminish this source of irregularity, the following methods of compensation are employed. 1. Harrisons Gridiron Pendulum. This consists of four oblong frames, the uprights of which are alternately of steel F and of brass Fig. 206. Han of Gridiron Pendulum. Fig. 207. Gridiron Pendulum. Fig. 208. Graham's Mercurial Pendulum. C (Fig. 206), so arranged that the bob will rise or fall through a distance equal to the difference between the total expansion of 3 steel rods and that of 2 brass rods. As the coefficients of expansion of these metals are nearly as 2 to 3, it is possible to make the com- pensation nearly exact. 2. Graham's Mercurial Pendulum. This consists of an iron rod 286 EXPANSION OF SOLIDS. carrying at its lower end a frame, in which are fixed one or two glass cylinders containing mercury. When the temperature rises, the lengthening of the rod lowers the centre of gravity and centre of oscillation of the whole; but the expansion of the mercury pro- duces the contrary effect; and if there is exactly the right quantity of mercury the compensation will be nearly perfect. 305. Force of Expansion of Solids. The force, of expansion is often very considerable, being equal to the force necessary to compress the body to its original dimensions. Thus, for instance, iron when heated from to 100 increases by '0012 of its original length. In order to produce a corresponding change of length in a rod an inch square by mechanical means, a force of about 15 tons would be required. This is accordingly the force necessary to prevent such a rod from ex- panding or contracting when heated or cooled through 100. This force has frequently been utilized for bringing in the walls of a building when they have settled outwards. For this purpose the walls are first tied together by iron rods, which pass through the walls, and are furnished at the ends with screws and nuts. All the nuts having been tightened against the wall, alternate bars are heated; and while they are hot, the nuts upon them, which have been thrust away from the wall by the expansion, are screwed home. As these bars cool, they draw the walls in and allow the nuts on the other bars to be tightened. The same operation is then repeated as often as may be necessary. Iron cannot with safety be used in structures, unless opportunity is given it to expand and contract without doing damage. In laying a railway, small spaces must be left between the ends of the rails to leave room for expansion; and when sheets of lead or zinc are em- ployed for roofing, room must be left for them to overlap. CHAPTER XXVII. EXPANSION OF LIQUIDS. 306. Method of Equilibrating Columns. Most of the methods em- ployed for measuring the expansion of liquids depend upon a previous knowledge of the expansion of glass, the observation itself consisting in a determination of the apparent expansion of the liquid relative to glass. There is, however, one method which is not liable to this objection, and it has been employed by Dulong and Petit, and after- wards by Regnault, for measuring the expansion of mercury an element of great importance for many physical applications. It depends upon the hydrostatic principle that A ^ the heights of two liquid columns which produce equal pressures are inversely as their densities ( 146). Let A and B (Fig. 209) be two tubes con- taining mercury, and communicating with each other by a very narrow horizontal tube CD at the bottom. If the tempera- ture of the liquid be uniform, the mercury will stand at the same height in both branches; but if one column be kept at c " and the other be heated, their densities will Fig - 2oa , Principle of Dulong s Method. be unequal. Let d d' be their densities, and h h' their heights. Then since their pressures at the bottom are equal, we must have hd=h'd'. But if v and v denote the volumes of one and the same mass of liquid at the two temperatures, we have v d = t/ d'. From these two equations, we have 288 EXPANSION OF LIQUIDS. v : v' : : h ; h', so that the expansion of volume is directly given by a comparison of the heights. Denoting this expansion by m, we shall have h'-h Strictly speaking, the mercury in this experiment is not in equi- librium. There will be two very slow currents through the hori- zontal tube, the current from hot to cold being above, and the current from cold to hot below. Equilibrium of pressure will exist only at the intermediate level that of the axis of the tube, and it is from this level that h and h' should be measured. 307. The apparatus employed by Dulong and Petit for carrying out this method is represented in Fig. 210. The two upright tubes Fig. 210. Apparatus of Dulong and Petit. A, B, and the connecting tube at their base, rest upon a massive support furnished with levelling screws, and with two spirit-levels at right angles to each other, for insuring horizontality. The tube B is surrounded by a cylinder containing melted ice. The other tube A is surrounded by a copper cylinder filled with oil, which is heated by a furnace connected with the apparatus. In making an observation, the first step is to arrange the apparatus so that, when EXPANSION OF MERCURY. 289 the oil is heated to the temperature required, the mercury in the tube A may just be seen above the top of the cylinder, so as to be sighted with the telescope of a cathetometer; this may be effected by adding or taking away a small quantity of mercury. The ex- tremity of the column B is next sighted, which gives the difference of the heights Ji and h. The absolute height h is determined by means of a fixed reference mark i near the top of the column of mercury in the tube B. This reference mark is carried by an iron rod surrounded by the ice, and its distance from the axis of the hori- zontal connecting tube has been very accurately measured once for all. The temperature of the oil is given by the weight thermometer t, and by the air thermometer r, which latter we shall explain here- after. By means of this method Dulong and Petit ascertained that the expansion of mercury is nearly uniform between and 100 C., as compared with the indications of an air-thermometer, and that though its expansion at higher temperatures is more rapid, the differ- ence is less marked than in the case of other liquids. They found the mean coefficient of expansion from to 100 to be ^ ; from to 200, -L ; and from to 300, JL Regnault, without altering the principle of the apparatus of Dulong and Petit, introduced several improvements in detail, and added greatly to the length of the tubes A and B, thereby rendering the apparatus more sensitive. His results are not very different from those of Dulong and Petit. For example, he makes the mean coeffi- cient from to 100 to be -^ ; from to 200, -^ ; and from to 300, jrgTry His experiments show that the mean coefficient from to 50 is j^^, a value almost identical with ^^. 554/ 5550 308. Expansion of Glass. The expansion of mercury being known, we can find the expansion of any kind of glass by observing the apparent expansion of mercury in a weight thermometer ( 300) constructed of this glass, and subtracting this apparent expansion from the real expansion of the liquid; or more rigorously, by divid- ing the factor of real expansion of the liquid by the factor of ap- parent expansion ( 295), we shall obtain the factor of expansion of the glass. Dulong and Petit found fl ^ as the mean value of the coefficient blot) 19 290 EXPANSION OF LIQUIDS of apparent expansion of mercury in glass, and ^^ as the coefficient of real expansion of mercury. The difference of these two fractions is approximately 8 _ QO , which may therefore be taken as the coefficient of expansion of glass. It is about one-seventh of the coefficient of expansion of mercury. 309. Expansion of any Liquid. The expansion of the glass of which a thermometer is made being known, we may use the instru- ment to measure the expansion of any liquid. For this purpose we must measure the capacity of the bulb and find how many divisions of the stem it is equal to. We can thus determine how many divi- sions the liquid occupies at two different temperatures, that is, we can determine the apparent expan- sion of the liquid; and by adding to this the expansion of the glass, we shall obtain the real expansion of the liquid. Or more rigorously, we shall obtain the factor of real expansion of the liquid by mul- tiplying together the factor of ap- parent expansion and the factor of expansion of the glass. M. Pierre has performed an ex- tensive series of experiments by this method upon a great num- ber of liquids. The apparatus em- ployed by him is shown in Fig. 211. The thermometer containing the given liquid is fixed beside a mercurial thermometer, which marks the temperature. The re- servoir and a small part of the tube are immersed in the bath contained in the c vlinder below. The upper parts of the stems are inclosed in a second anl smaller cylinder, the water in which is maintained at a sensibly constant temperature indicated by a very delicate thermometer. From these experiments it appears that the expansions of liquids are in general much greater than those of solids; also that their ex- Fig. 211. Pierre's Apparatus. EXPANSIONS OF VARIOUS LIQUIDS. 291 pansion does not proceed uniformly, as compared with the indications of a mercurial thermometer, but increases very perceptibly as the temperature rises. This is shown by the following table: Volume at 0*. Volume at 10. Volume at 40* Water 1 1-000146 1-007492 Alcohol 1 1-010661 1-044882 Ether 1 1-015408 1-066863 Bisulphide of carbon... 1 1-011554 1*049006 Wood-spirit 1 1-012020 1 '050509 310. Other Methods. Another method of determining the apparent expansion of a liquid, with a view to deducing its real expansion, consists in weighing a glass bottle full of the liquid at different temperatures. This is virtually employing a weight thermometer. A third method consists in observing the loss of weight of a piece of glass when weighed in the liquid at different temperatures. Time must be given in each case for the glass to take the temperature of the liquid; and when this condition is fulfilled, the factor of expan- sion will be equal to the loss of weight at the lower temperature, divided by the loss of weight at the higher. For if the volume of the glass at the lower temperature be called unity, and its volume at the higher temperature 1 -}- whence we can find x. 317. Convection of Heat in Liquids. When different parts of a liquid or gas are heated to different temperatures, corresponding differences of density arise, leading usually to the formation of cur- rents. This phenomenon is called convection. Thus, for instance, if we apply heat to the bottom of a vessel con- taining water, the parts immediately subjected to the action of the heat expand and rise to the surface; they are replaced by colder portions, which in their turn are heated and ascend; and thus a con- tinual circulation is maintained. The ascending and descending currents can be rendered visible by putting oak sawdust into the water. 318. Heating of Buildings by Hot Water. This is a simple applica- tion of the principle just stated. One of the most common arrange- ments for this purpose is shown in Fig. 215. The boiler C is heated by a fire below it, and the products of combustion escape through the chimney A B. At the top of the house is a reservoir D, com- municating with the boiler by a tube. From this reservoir the liquid flows into another reservoir E in the story immediately below, thence into another reservoir F, and so on. Finally, the last of these 296 EXPANSION OF LIQUIDS. reservoirs communicates with the bottom of the boiler. The boiler, tubes, and reservoirs are all completely filled with water, with the Fig. 215. Heating by Hot Water. exception of a small space left above in order to give room for the expansion of the liquid. An ascending current flows through the left-hand tube, and the circulation continues with great regu- larity, so long as the temperature of the water in the boiler remains constant. CHAPTER XXVIII. EXPANSION OF GASES. 319. Experiments of Gay-Lussac. Gay-Lussac conducted a series of researches on the expansion of gases, the results of which were long regarded as classical. He employed a thermometer with a large reservoir A, containing the gas to be operated on; an index of mer- cury mn separated the gas from the external air, while leaving it full liberty to expand. The gas had previously been dried by pass- Fig. 216. Gay-Lussac's Apparatus. ing it through a tube containing chloride of calcium, or some other desiccating substance. The thermometer was first placed in a vessel filled with melting ice, and when the gas had thus been brought to C., the tube was so adjusted that the index coincided with the opening through which the thermometer passed. The tube and reservoir having been previously gauged, and the former divided into parts of equal capacity, the apparent volume of the gas (expressed in terms of these divisions) is indicated by the position of the index; let the apparent volume observed at C. be called n, and let H denote the external pressure as indicated by a 298 EXPANSION OF GASES. barometer. The apparatus is then raised to a known temperature t by means of the furnace below the vessel, and the stem of the ther- mometer is moved until the index reaches the edge of the opening. Let n be the apparent volume of the gas at this new temperature, and as the external pressure may have varied, let it be denoted by H'. The real volumes of the gas will be as n to n (1 +gt), where g denotes the mean coefficient of expansion of the glass; and the pro- ducts of volume and pressure will be as n H to ri (1 +gt) H'. Hence if a denote the mean coefficient of expansion of the air, we have n'R(l + at)=n'(l+gt) H'; from which equation a can be determined. By means of this method Gay-Lussac verified the law previously announced by Sir Humphrey Davy for air, that the coefficient of expansion is independent of the pressure. He also arrived at the result that this coefficient is sensibly the same for all erases. He / O found its value for dry air to be '00375. This result, which was for a long time the accepted value, is now known to be in excess of the truth. Rudberg, a Swedish philosopher, was the first to point out the necessity for using greater precautions to insure the absence of moisture, which adheres to the glass with great tenacity at the lower temperature, and, by going off into vapour when heated, adds to the volume of the air at the higher temperature. He found that the last traces of vapour could only be removed by repeatedly ex- hausting the vessel with an air-pump when heated, and refilling it with dried air. Another weak point in the method employed by Gay-Lussac was the shortness of the mercurial index, which, in con- junction with the fact that mercury does not come into close contact with glass (as proved by the fact of its not wetting it), allowed a little leakage in both directions. These imperfections have been remedied in later investigations, of which the most elaborate are those of Regnault. He employed four distinct methods, of which we shall only describe one. 320. Renault's Apparatus. The glass vessel BC (Fig. 217) con- taining the air to be experimented on, is connected with the T-shaped piece El, the branch I of which communicates, through desiccating tubes, with an air-pump, and is hermetically closed with a blow-pipe after the vessel has received its charge of dry air; while the branch ED communicates with the top of a mercurial manometer. A mark is made at a point 6 in the capillary portion of the tube, and in every REGNAULT'S OBSERVATIONS. 299 observation the mercury in the manometer is made to reach exactly to this point, either by pouring in more mercury at the top M' of the other tube of the manometer, or by allowing some of the liquid to escape through the cock R at the bottom. The air under experi- ment is thus always observed at the same apparent volume, and the observation gives its pressure. The vessel B is inclosed within a Fig. 217. Regnault's Apparatus. boiler, which consists of an inner and an outer shell, with a space be- tween them, through which the steam circulates when the water boils. In reducing the observations, the portion of the glass vessel within the boiler is regarded as having the temperature of the water in the boiler, while the portion of the tube external to the boiler is regarded as having the temperature of the surrounding air. In this mode of operating, the volume, or at least the apparent volume, is constant, so that the coefficient a which is determined is substantially denned by the equation 300 EXPANSION OF GASES. P and P t denoting the pressures at constant volume. The coefficient thus defined should be called the coefficient of increase of pressure. It is often called the "coefficient of expansion at constant volume," which is a contradiction in terms. In another mode of operating Regnault observed the expansion at constant pressure, and thus determined the coefficient of expansion properly so called. A small but steady difference was found between the two. If Boyle's law were exact they would be identical. As a matter of fact, the coefficient of increase of pressure was found, in the case of air and all gases except hydrogen, to be rather less than the coefficient of expansion. In other words, the product of volume and pressure at one and the same temperature t was found to be least when the volume was least; a result which accords with Regnault's direct observations on Boyle's law. 321. Results. The following table contains the final results for the various gases which were submitted to experiment: Coefficient of increase Coefficient of increase of pressure at con- of volume at con- stant volume. stant pressure. Air 0-003665 0'003670 Nitrogen 0'003668 Hydrogen 0'003667 0'003661 Carbonic oxide 0-003667 0-003669 Carbonic acid 0'003688 0-003710 Nitrous oxide 0'003676 0'003720 Cyanogen 0'003829 0'003877 Sulphurous acid 0'003845 0'003903 It will be observed that the largest values of the coefficients belong to those gases which are most easily liquefied. We may add that the coefficients increase very sensibly with the pressure ; thus between the pressures of one and of three atmospheres the coefficient of expansion of air increases from '00367 to '00369. This increase is still more marked in the case of the more liquefiable gases. 322. Reduction to the Fahrenheit Scale. The coefficient of expan- sion of any substance per degree Fahrenheit is f of the coefficient per degree Centigrade; the volume at 32 F. being made the standard from which expansions are reckoned, so that if V denote the volume at this temperature and V the volume at t F., the coefficient of expansion a is defined by the equation V=V AIR-THERMOMETER. 301 323. Air-thermometer. The close agreement between the expan- sions of different gases, and between the expansions of the same gas at different pressures, is a strong reason for adopting one of these bodies as the standard substance for the measurement of temperature by expansion, rather than any particular liquid. Moreover, the expansion of gases being nearly twenty times as great as that of mercury, the expansion of the containing vessel will be less important; the apparent expansion will be nearly the same as the real expansion, and differences of quality in the glass will not sensibly affect the comparability of different thermometers. Air-thermometers have accordingly been often used in delicate investigations. They consist, like other thermometers, of a reservoir and tube; but the latter, instead of being sealed, is left open. This open end, in one form of the instrument, is pointed downwards, and immersed in a liquid, usually mercury, which rises to a greater or less distance up the tube as the air in the thermometer contracts or expands. As variations of pressure in the surrounding air will also affect the height of this column of liquid, it is necessary to take readings of the barometer, and to make use of them in reducing the indications of the air-thermometer. Even if the barometer continues steady, it is still necessary to apply a correction for changes of pres- sure; since the difference between the pressure in the air-thermo- meter and that of the external air is not constant, but is proportional to the height of the column of liquid. In the form of air-thermometer finally adopted by Regnault, the air in the instrument was kept at constant (apparent) volume, and its variations of pressure were measured, the apparatus employed being precisely that which we have described in 320. 324. Perfect Gas. In discussions relating to the molecular consti- tution of gases, the name perfect gas is used to denote a gas which would exactly fulfil Boyle's law; and molecular theories lead to the conclusion that for all such gases the coefficients of expansion would be equal. Actual gases depart further from these conditions as they are more compressed below the volumes which they occupy at atmo- spheric pressure; and it is probable that when very highly rarefied they approach the state of "perfect gases" very closely indeed. 325. Absolute Temperature by Air-thermometer. Absolute temper- ature by the air-thermometer is usually defined by the condition that the temperature of a given mass of air at constant pressure is to be regarded as proportional to its volume. If the difference of 302 EXPANSION OF GASES. temperature between the two ordinary fixed points be divided into a hundred degrees, as in the ordinary Centigrade thermometer, the two fixed points themselves will be called respectively 273 and 373; since air expands by ^ of its volume at the lower fixed point for each degree, and therefore by ^ of this volume for a hundred degrees. There is some advantage in altering the definition so as to make the temperature of a given mass of air at constant volume propor- tional to its pressure. The two fixed points will then be 273 and 373 as above, and the zero of the scale will be that temperature at which the pressure vanishes. The advantage of the second form of definition is that it enables us to continue our scale down to this point called absolute zero without encountering any physical impossibility, such as the concep- tion of reducing a finite quantity of air to a mathematical point, which would be required according to the first form of definition. Practically, "absolute temperatures by air- thermometer" are com- puted by adding 273 to ordinary "temperatures by air- thermometer," these latter being expressed on the Centigrade scale. We shall employ the capital letter T to denote absolute temperature, and the small letter t to denote ordinary temperature. We have and the general law connecting the volume, pressure, and temperature of a gas is VP 1 m = constant; or, introducing the density D instead of the pressure P, p DT = constant. As above explained, these laws, though closely approximate in ordi- nary cases, are not absolutely exact. 326. Pyrometers. The measurement of high temperatures such as those of furnaces is very difficult. Instruments for this purpose are called pyrometers. One of the best is the air-thermometer em- ployed by Deville and Troost, having a bulb of hard porcelain. 327. Density of Gases. The absolute density of a gas that is, its mass per unit volume which is denoted by D in the above formula, p is proportional, as the formula shows, to ^ and may therefore DENSITY OF GASES. 303 undergo enormous valuation. In stating the relative density of a gas as compared with air, the air and the gas are supposed to be at the same pressure and temperature. For purposes of great accuracy this pressure and temperature must be specified, since, as we have seen, there are slight differences in the changes produced in different gases by the same changes of pressure and temperature. The com- parison is generally supposed to be made at the temperature C., and at the pressure of one standard atmosphere. 328. Measurement of the Relative Density of a Gas. The densities of gases have been the subject of numerous investigations; we shall describe only the method employed by Regnault. The gas is inclosed in a globe, of about 12 litres capacity (Fig. 218), furnished with a stop-cock leading to a three-way tube, one of whose branches is in communication with a ma- nometer, and the other with an air-pump. The globe is exhausted several times, and each time the gas is dried on its way to the globe by passing through a number of tubes containing pieces of pumice-stone moistened with sulphuric acid. When all moisture has been re- moved, the globe is sur- rounded with melting ice, and is kept full of gas at the pressure of the atmosphere till sufficient time has been given for its contents to assume the temperature of the melting ice. The stop-cock is then closed, the globe is taken out, carefully dried, and allowed to take the temperature of the atmo- sphere. It is then weighed with a delicate balance. The experiment is repeated, with no change except that by means of the air-pump the gas in the globe is reduced to as small a pressure as possible. Let this pressure be denoted by h, and the atmospheric pressure in the previous experiment by H. Then the difference of the two weights is the weight of as much gas at temperature and pressure H h as would fill the globe. Let w denote this difference, and let w' be the difference between two weighings made in the same Fig. 218. Measurement of Density of Gases. 304 EXPANSION OF GASES. manner with dry air in the globe at pressures H' and ti. relative density of the gas will be Then the _ w' H -h' We must now describe a special precaution which was employed by Regnault (and still earlier by Dr. Prout) to avoid errors in weighing arising from the varying weight of the external air displaced by the globe. A second globe (Fig. 219) of precisely the same external volume as the first, made of the same glass, and closed air-tight, was used as a counterpoise. The equality of external volumes was ensured in the following way. The globes were filled with water, hung from the two scales of a balance, and equilibrium was brought about by putting a sufficient quantity of some material into one scale. Both globes, thus hanging from the scales in equilibrium, were then immersed in water, and if this operation disturbed the equilibrium it was known that the external volumes were not equal. Let p be the weight which must be put into one scale to restore equili- brium; then this weight of water represents the difference of the two external volumes; and the next operation was to prepare a small piece of glass tube closed at the ends which should lose p when weighed in water. The larger of the two globes was used for containing the gases to be weighed, and the smaller globe along with this piece of tube constituted the counterpoise. Since the volume of the gas globe was exactly the same as that of the counterpoise, the pressure of the external air had no tendency to make either preponderate, and variations in the condition of this air, whether as regards pressure, temperature, or humidity, had no dis- turbing effect. 329. Absolute Densities. In order to convert the preceding relative determinations into absolute determinations, it is only necessary to know the precise internal volume of the globe at the temperature C. In order to determine this with the utmost possible exactness the following operations were performed. The globe was first weighed in air, with its stop-cock open, the temperature of the air and the height of the barometer being noted. Fig. 219. Compensating Globe. DENSITY OF AIR. 305 It was then filled with water, special precautions being taken to expel every particle of air; and was placed for several hours in the midst of melting ice, to insure its being filled with water at C. The stop-cock was then closed, and the globe was left for two hours in a room which had a very steady temperature of 6. It was then weighed in this room, the height of the barometer being at the same time observed. The difference between this weight and that of the globe before the introduction of the water, was the weight of the water minus the weight of the same volume of air, subject to a small correction for change of density in the external air between the two weighings, which, with the actual heights of the barometer and thermometer, was insensible. The weight of water at which the globe would hold at was therefore known; and hence the weight of water at 4 (the tempera- ture of maximum density) which the globe would hold at was calculated, from the known expansion of water. This weight, in grammes, is equal to the capacity in cubic centimetres. The result thus obtained was that the capacity of the globe at was 9881 cubic centimetres; and the weight of the dry air which filled it at and a pressure of 7GO mm was 12778 grammes. Hence the weight (or mass) of 1 cubic centimetre of such air is '0012932 gramme. This experiment was performed at Paris, where the value of g (the intensity of gravity) is 980'94; and since the density of mercury at is 13'596, the pressure of 76 centimetres of mercury was equiva- lent to 76 x 13-596 x 980-94 = 1-0136 x 10 6 dynes per square centimetre. If we divide the density just found by 1'0136, we obtain the den- sity of air at and a pressure of a million dynes per square centi- metre, which is a convenient standard for general reference; we have thus 0012932-7-1-0136 = -0012759. A litre or cubic decimetre contains 1000 cubic centims. Hence the weight of a litre of air in the standard condition adopted by Regnault is T2932 gramme. o o The following table gives the densities of several gases at C. at a pressure of 7 GO millimetres of mercury at Paris. 20 306 EXPANSION OF GASES. Name of Gas. Relative Density. Air 1 1-2932 Oxygen 1-10563 1-4298 Hydrogen -06926 '08957 Nitrogen -97137 1-25615 Chlorine 2'4216 3'1328 Carbonic oxide -9569 1-2344 Carbonic acid 1-52901 1'9774 Protoxide of nitrogen 1-5269 1*9697 Binoxide of nitrogen 1-0388 1-3434 Sulphurous acid 21930 2'7289 Cyanogen 1-8064 2-3302 Marsh-gas -559 727 Olefiantgas -985 1-274 Ammonia -5967 7697 330. Draught of Chimneys. The expansion of air by heat pro- duces the upward current in chimneys, and an approximate expres- sion for the velocity of this current may be obtained by the applica- tion of Torricelli's theorem on the efflux of fluids from orifices (Chap, xxiii.). Suppose the chimney to be cylindrical and of height h. Let the air within it be at the uniform temperature t' Centigrade, and the external air at the uniform temperature t. According to Torricelli's theorem, the square of the linear velocity of efflux is equal to the product of 2g into the head of fluid, the term head of fluid being employed to denote the pressure producing efflux, expressed in terms of depth of the fluid. In the present case this head is the difference between h, which is the height of air within the chimney, and the height which a column of the external air of original height h would have if expanded upwards, by raising its temperature from t to t'. This latter height is h j- ^; a denoting the coefficient of expansion '00306; and the head is , 1 + at? ha (t'- t) 1 + at ~ 1 + at Hence, denoting by v the velocity of the current up the chimney, we have a _1gha (tf-t) l + at ' This investigation, though it gives a result in excess of the truth, from neglecting to take account of friction and eddies, is sufficient to explain the principal circumstances on which the strength of draught FIREPLACES. 307 depends. It shows that the draught increases with the height h of the chimney, and also with the difference t' t between the internal and external temperatures. The draught is not so good when a fire is first lighted as after it has been burning for some time, because a cold chimney chills the Fig. 220. Rumford's Fireplace. air within it. On the other hand, if the fire is so regulated as to keep the room at the same temperature in all weathers, the draught will be strongest when the weather is coldest. The opening at the lower end of the chimney should not be too wide nor too high above the fire, as the air from the room would then enter it in large quantity, without being first warmed by passing through the fire. These defects prevailed to a great extent in old chimneys. Rumford was the first to attempt rational improve- ments. He reduced the opening of the chimney and the depth of the fireplace, and added polished plates inclined at an angle, which serve both to guide the air to the fire and to reflect heat into the room (Fig. 220). The blower (Fig. 221) produces its well-known effects by compel- ling all air to pass through the fire before entering the chimney. This at once improves the draught of the chimney by raising the Fig. 221. Fireplace with Blower. 308 EXPANSION OF GASES. temperature of the air within it, and quickens combustion by in- creasing the supply of oxygen to the fuel. 331. Stoves. The heating of rooms by open fireplaces is effected almost entirely by radiation, and much even of the radiant heat is. wasted. This mode of heating then, though agreeable and healthful, is far from economical. Stoves have a great advantage in point of economy; for the heat absorbed by their sides is in great measure given out to the room, whereas in an ordinary fireplace the greater part of this heat is lost. Open fireplaces have, however, the advan- tage as regards ventilation; the large opening at the foot of the chimney, to which the air of the room has free access, causes a large body of air from the room to ascend the chimney, its place being supplied by fresh air entering through the chinks of the doors and windows, or any other openings which may exist. Stoves are also liable to the objection of making the air of the room too dry, not, of course, by re- moving water, but by raising the temperature of the air too much above the dew-point (Chap, xxxiv.). The same thing occurs with open fireplaces in frosty weather, at which time the dew-point is un- usually low. This evil can be re- medied by placing a vessel of water on the stove. The reason why it is more liable to occur with stoves than with open fireplaces, is mainly that the former raise the air in the room to a higher temperature than the latter, the defect of air- temperature being in the latter case compensated by the intensity of the direct radiation from the glowing fuel. Fire-clay, from its low conduct- ing power, is very serviceable both for the backs of fireplaces and for the lining of stoves. In the former situation it prevents the wasteful escape of heat backwards into the chimney, and keeps the back of the fire nearly as hot as the centre. 836? Fig. 222. Ventilating Stove. STOVES. 309 As a lining to stoves, it impedes the lateral escape of heat, thus answering the double purpose of preventing the sides of the stove from overheating, and at the same time of keeping up the tempera- ture of the fire, and thereby promoting complete combustion. Its use must, however, be confined to that portion of the stove which serves as the fire-box, as it would otherwise prevent the heat from being given out to the apartment. The stove represented in Fig. 222 l belongs to the class of what are called in France caloriferes, and in England ventilating stoves, being constructed with a view to promoting the circulation and renewal of the air of the apartment. G is the fire-box, over which is the feeder U, containing unburned fuel, and tightly closed at top by a lid, which is removed only when fresh fuel is to be introduced. The ash-pan F has a door pierced with holes for admitting air to support com- bustion. The flame and smoke issue at the edge of the fire-box, and after circulating round the chamber which surrounds the feeder, enter the pipe T which leads to the chimney. The chamber O is surrounded by another inclosure L, through which fresh air passes, entering below at A, and escaping into the room through perforations in the upper part of the stove as indicated by the arrows. The amount of fresh air thus admitted can be regulated by the throttle- valve P. 1 With the exception of the ventilating arrangement, this stove is identical with what is known in this country as Walker's self -feeding stove. CHAPTER XXIX. CALORIMETRY. 332. Quantity of Heat. We have discussed in previous chapters the measurement of temperature, and have seen that it is to a great extent arbitrary, since intervals of temperature which are equal as measured by the expansion of one substance are not equal as mea- sured by the expansion of another. The measurement of quantities of heat stands upon an entirely different footing. There is nothing arbitrary or conventional in asserting the equality or inequality of two quantities of heat. 333. Principles Assumed. The two following principles may be regarded as axiomatic. (1) The heat which must be given to a body to raise it through a given range of temperature at constant pressure, is equal to that which the body gives out in falling through the same range of tem- perature under the same pressure. For instance, the heat which must be given to a gramme of water, to raise its temperature from 5 to 10, is equal to that which is given out from the same water when it falls from 10 to 5. (2) In a homogeneous substance equal portions require equal quan- tities of heat to raise them from the same initial to the same final temperature; so that, for example, the heat required to raise two grammes of water from 5 to 10 is double of that which is required to raise one gramme of water from 5 to 10. 334. Cautions. We are not entitled to assume that the quantities of heat required to raise a given body through equal intervals of temperature for example, from 5 to 10, and from 95 to 100 are equal. Indeed we have already seen that the equality of two intervals of temperature is to a considerable extent a matter of mere UNIT OF HEAT. 311 convention; temperature being conventionally measured by the expansion (real or apparent) of some selected substance. It would, however, be quite possible to adopt a scale of tempera- ture based on the elevation of temperature of some particular sub- stance when supplied with heat. We might, for instance, define a degree (at least between the limits and 100) as being the elevation of temperature produced in water of any temperature by giving it one hundredth part of the heat which would be required to raise it from to 100. Experiments which will be described later show that if air or any of the more permanent gases were selected as the standard substance for thus defining equal intervals of temperature, the scale obtained would be sensibly the same as that of the air-thermometer; and the agreement is especially close when the gases are in a highly rarefied condition. 335. Unit of Heat. We shall adopt as our unit, in stating quan- tities of heat, the heat required to raise a gramme of cold water through one degree Centigrade. This unit is called, for distinction, the gramme degree. The kilogramme-degree and the pound-degree are sometimes employed, and are in like manner defined with refer- ence to cold water as the standard substance. There is not at present any very precise convention as to the tem- perature at which the cold water is to be taken. If we say that it is to be within a few degrees of the freezing-point, the specification is sufficiently accurate for any thermal measurements yet made. 336. Thermal Capacity. If a quantity Q of heat given to a body raises its temperature from ti to 2 , the quotient Q <*-7~.go2 ~ -804246 = =1698 cubic centimetres. Hence we see that water at 4 gives about 1700 times its volume of vapour at 100 C. The latent heat of evaporation is doubtless connected with this increase of volume; and it may be remarked that both these elements appear to be greater for water than for any other substance. 414. Heat of Evaporation. The latent heat oi evaporation of \vater, and of some other liquids, can be determined by means of Despretz's apparatus, which is shown in Fig. 265. The liquid is boiled in a retort C, which is connected with a worm S surrounded by cold water, and terminating in the reservoir R. The vapour is condensed in the worm, and collects in the reservoir, whence it can be drawn by means of the stop-cock r. The tube T, which is fitted with a ctop-cock r', serves to establish communication between the reservoir and the atmosphere, or between the reservoir and a space where a fixed pressure is maintained, so as to produce ebulli- tion at any temperature required, as indicated by the thermometer t. A is an agitator for keeping the water at a uniform temperature, indicated by the thermometer t'. In using the apparatus, the first step is to boil the liquid in the 25 386 QUANTITATIVE MEASUREMENTS RELATING TO VAPOURS. retort, and when it is in active ebullition, it is put in communication with the worm. The temperature of the calorimeter has previously been lowered a certain number of degrees below that of the surround- ing air, and the experiment ceases when it has risen to the same number of degrees above. The compensation may thus be considered as complete, since the rate of heating is nearly uniform. If W be the equivalent of the calorimeter in water, t its initial temperature, its final temperature; then the quantity of heat gained by it is W (0 t). This heat comes partly from the latent heat dis- engaged at the moment of condensation of the vapour, partly from the loss of temperature of the condensed water, which sinks from T, the boiling-point of the liquid, to the temperature of the calorimeter. If, then, x denote the latent heat of evaporation, w the weight of the Fig. 265. Despretz's Apparatus liquid collected in the box R, and c its specific heat, we have the equation W (6 - t)=wx + wc(T- 6). This experiment is exposed to some serious causes of error. The calorimeter may be heated by radiation from the screen F which protects it from the direct radiation of the furnace. Heat may also be propagated by means of the neck of the retort. Again, the vapour is not dry when it passes into the worm, but carries with it small I.ATKXT HEAT OF STKAM. 387 drops of liquid. Finally, some of the vapour may be condensed at the top of the retort, and so pass into the worm in a liquid state. This last objection is partly removed by sloping the neck of the retort upwards from the fire, but it sometimes happens that this precaution is not sufficient. 415. Regnault's Experiments. The labours of Begnault in con- nection with the subject of latent heat are of the greatest importance, and have resulted in the elaboration of a method in which all these sources of error are entirely removed. The results obtained by him are the following: The quantity of heat required to convert a gramme of water at 100 into vapour, without change of temperature, is 537 gramme- degrees. If the water were originally at zero, the total amount of heat required to raise it to 100 and then convert it into vapour would evidently be 637 gramme-degrees; and it is this total amount which is most important to know in the application of heat in the arts. In general, if Q denote the total quantity of heat 1 required to raise water from zero to the temperature T, and then convert it into vapour at this temperature, the value of Q may be deduced with great exactness from the formula Q = 606-5 + -305T. (a) From what we have said above, it will be seen that if X denote the latent heat of evaporation at temperature T, we must have whence, by substituting for Q in (a), we have X = 606-5 --695T. (b) Hence it appears that latent heat varies in the opposite direction to temperature. This fact had been previously discovered by Watt; but he went too far, and maintained that the increase of the one was equal to the diminution of the other, or, in his own words, that "the sum of the sensible and latent heats" (that is T + A) "is con- stant." From (a) we can find the total heat for any given tempera- ture, and from (6) the latent heat of evaporation at any given tem- 1 Called by Regnault the total heat of saturated vapour at T, or the total heat of vaporization at T". 888 QUANTITATIVE MEASUREMENTS RELATING TO VAPOURS. perature. The results for every tenth degree between and 230 are given in the following table : Temperatures Centigrade. Latent Heat. Total Heat. Temperatures Centigrade. Latent Heat. Total Heat. 606 606 120 522 642 10 600 610 130 515 645 20 593 613 140 508 648 30 586 616 150 501 651 40 579 619 160 494 654 50 572 622 170 486 656 60 565 625 180 479 659 70 558 628 190 472 662 80 551 631 200 464 664 90 544 634 210 457 667 100 537 637 220 449 669 110 529 639 230 442 672 To reduce latent heat and total heat from the Centigrade to the Fahrenheit scale, we must multiply by f . Thus the latent and total heat of steam at 212 F. are 966'6 and 1146'6. Total heat is here reckoned from 32 F. If we reckon it from F., 32 must be added. The following table taken from the researches of Favre and Sil- bermann, gives the latent heat of evaporation of a number of liquids at the temperature of their boiling-point, referred to the Centigrade scale : Boiling- point. Latent Heat. Boiling- point. Latent Heat. Wood-spirit, . Absolute alcohol,. . Valeric alcohol, . . Ether, 66-5 78 78 38 264 208 121 91 Acetic acid, . . . Butyric acid, . . . Valeric acid, . . . Acetic ether, . 120 164 175 74 102 115 104 100 Ethal 38 58 156 69 Valeric ether, . . . Formic acid, . . . 113-5 100 113-5 169 Essence of citron, 165 70 CHAPTER XXXIV. HYGROMETRY. 416. Humidity. The condition of the air as regards moisture involves two elements: (1) the amount of vapour present in the air, and (2) the ratio of this to the amount which would saturate the air at the actual temperature. It is upon the second of these elements that our sensations of dryness and moisture chiefly depend, and it is this element which meteorologists have agreed to denote by the term humidity; or, as it is sometimes called, relative humidity. It is visually expressed as a percentage. The words humid and moist, as applied to air in ordinary lan- guage, nearly correspond to this technical use of the word humidity; and air is usually said to be dry when its humidity is considerably below the average. In treatises on physics, "dry air" usually denotes air whose humidity is zero. The air in a room heated by a hot stove contains as much vapour weight for weight as the open air outside; but it is drier, because its capacity for vapour is greater. In like manner the air is drier at noon than at midnight, though the amount of vapour present is about the same ; and it is for the most part drier in summer than in winter, though the amount of vapour present is much greater. It is to be borne in mind that a cubic foot of air is able to take up the same amount of vapour as a cubic foot of empty space; and "relative humidity" may be defined as the ratio of the mass of vapour actually present in a given space, to the mass which would saturate tlie space at the actual temperature. Since aqueous vapour fulfils Boyle's law, these masses are propor- tional to the vapour-pressures which they produce, and relative humidity may accordingly be defined as the ratio of the actual 390 HY GEOMETRY. vapour-pressure to the maximum vapour-pressure for the actual temperature. 417. Simultaneous Changes in the Dry and Vaporous Constituents. When a mixture of air and vapour is subjected to changes of tem- perature, pressure, or volume which do not condense any of its vapour, the two constituents are similarly affected, since they have both the same coefficient of expansion, and they both obey Boyle's law. If the volume of the whole be reduced from v\ to v 2 at constant pressure, both the densities will be multiplied by , and hence, by Boyle's law, the pressures will also be multiplied by -. If, on the other hand, the temperature be altered from t t to t 2 without change of volume, both the pressures will be multiplied by ^ ~~. The ratio of the vapour-pressure to the dry-air pressure remains unchanged in both cases. If the changes of volume and temperature are effected simultane- ously, each of the pressures will be multiplied by l ~ ^ ~f, and the total pressure will be multiplied by the same factor. If the total pressure remains unchanged, as is the case when there is free com- munication between the altered air and the general atmosphere, both the dry-air pressure and the vapour-pressure will therefore remain unchanged. 418. Dew-point. When a mixture of dry air and vapour is cooled down at constant pressure until the vapour is at saturation, the temperature at which saturation occurs is called the dew-point of the original mass; and if the mixture be cooled below the dew-point, some of the vapour will be condensed into liquid water or solid ice. The reasoning of the preceding section shows that the process of cooling down to the dew-point does not alter the vapour-pressure. The actual vapour-pressure in any portion of air is therefore equal to the maximum vapour-pressure at the dew-point. When air is confined in a close vessel, and cooled at constant volume, its pressure and density at any given temperature, and the pressures and densities of its dry and vaporous constituents, will be less than if it were in free communication with the atmosphere. Hence its vapour will not be at saturation when cooled down to what is above defined as the dew-point of the original mass, but a lower temperature will be requisite. 419. These conclusions can also be established as follows: HYGROSCOPES. 391 Let P denote the pressure of the mixture, p the pressure of the vaporous constituent, V the volume, T the temperature reckoned from absolute zero on the air thermometer. Then for all changes which do not condense any of the vapour VP Vp -m is constant, and -j^- is constant. When P is also constant, we have -^ constant, and therefore p con- stant. On the other hand, when V is constant, p will vary as T, and will diminish as T diminishes. 420. Hygroscopes. Anything which serves to give rough indica- tions of the state of the air as regards moisture may be called a hygroacope (vypoc, moist). Many substances, especially those which are composed of organic tissue, have the property of absorbing the moisture of the surrounding air, until they attain a condition of equilibrium such that their affinity for the moisture absorbed i exactly equal to the force with which the latter tends to evaporate. Hence it follows that, according to the dampness or dryness of the air, such a substance will absorb or give up vapour, either of which processes is always attended with a variation in the dimensions of the body. The nature of this variation depends upon the peculiar struc- ture of the substance; thus, for instance, bodies formed of filaments exhibit a greater increase in the direction of their breadth than of their length. Membranous bodies, on the other hand, such as paper or parchment, formed by an interlacing of fibres in all directions, expand or contract almost as if they were homogeneous. Bodies composed of twisted fibres, as ropes and strings, swell under the action of moisture, grow shorter, and are more tightly twisted. The opposite is the case with catgut, which is often employed in popular hygroscopes. 421. Hygrometers. Instruments intended for furnishing precise measurements of the state of the air as regards moisture are called hygrometers. They may be divided into four classes: 1. Hygrometers of absorption, which should rather be called hygroscopes. 2. Hygrometers of condensation, or dew-point instruments. 3. Hygrometers of evaporation, or wet and dry bulb thermo- meters. 392 HYGROMETRY. 4. Chemical hygrometers, for directly measuring the weight of vapour in a given volume of air. 422. De Saussure's Hygrometer. The best hygrometer of absorp- tion is that of De Saussure, consisting of a hair deprived of grease, which by its contractions moves a needle (Fig. 266). When the hair relaxes, the needle is caused to move in the opposite direction Fig. 266. Be Saussure's Hygi-oscope. Fig. 267. Monnier's Hygroacope. by a weight, which serves to keep the hair always equally tight. The hair contracts as the humidity increases, but not in simple pro- portion, and Regnault's investigations have shown that, unless the most minute precautions are adopted in the construction and gradua- tion of each individual instrument, this hygrometer will not furnish definite numerical measures. Fig. 267 represents Monnier's modification of De Saussure's hygrometer, in which the hair, after passing over four pulleys, is attached to a light spring, which serves instead of a weight, and gives the advantage of portability. These instruments are never employed for scientific purposes in this country. 423. Dew-point Hygrometers. These are instruments for the direct observation of the dew-point, by causing moisture to be con- densed from the air upon the surface of a body artificially cooled to a known temperature. DEW-POINT INSTRUMENTS. 393 Fig. 268. Dines' Hygrometer. SEC The dew-point, which is itself an important element, gives directly, as we have seen in 418, the pressure of vapour; and if the temper- ature of the air is at the same time observed, the pressure requisite for saturation is known. The ratio of the former to the latter is the humidity. 424. Dines' Hygrometer. One of the best dew-point hygrometers is that recently invented by Mr. Dines, shown both in perspective and in section in Figs. 268, 269. Cold water, with ice, if neces- sary, is put into the reser- voir A, and by turning on the tap B this water is allowed to flow through the pipe C into the small double cham- ber D, the top of which, E, is formed of thin black glass, on which the smallest film of dew is easily perceived. After flowing under the black glass and around the bulb of a thermometer which lies immediately below it, the water escapes through a discharge pipe, and can be received in a vessel, from which it may again be poured into the reservoir A. As soon as any dew is seen on the black glass, the thermometer should be read, and the tap turned off", or partly off, until the dew disappears, when a second reading of the thermometer should be taken. The mean of the two will be approximately the dew-point; and in order to obtain a good determination, matters should be so managed as to make the temperatures of appearance and disappear- ance nearly identical. 425. DanielFs Hygrometer. Daniell's hygrometer has been very extensively used. It consists of a bent tube with a globe at each end, and is partly filled with ether. The rest of the space is occu- pied with vapour of ether, the air having been expelled. One of the globes A is made of black glass, and contains a thermometer t The method of using the instrument is as follows: The whole of the liquid is first passed into the globe A, and then the other globe B, which is covered with muslin, is moistened externally with ether. Fig. 269. Dines' Hygrometer. 394 HYGROMETRY. The evaporation of this ether from the muslin causes a condensa- tion of vapour of ether in the interior of the globe, which produces a fresh evaporation from the surface of the liquid in A, thus lowering the temperature of that part of the instrument. By carefully watching the surface of the globe, the exact moment of the deposition of dew may be ascertained. The temperature is then read on the inclosed thermometer. If the instrument be now left to itself, the exact moment of the disappearance of the dew may be observed; and the usual plan is to take the mean between this temperature and that first observed. The temperature of the given by a thermometer t' attached to the Fig. 270. Daniell's Hygrometer. surrounding air is stand. 426. Regnault's Hygrometer. Regnault's hygrometer consists (Fig. 271) of a glass tube closed at the bottom by a very thin silver cap D. The opening at the upper end is closed by a cork, through which Fig. 271. Regnault's Hygrometer. passes the stem of a thermometer T, and a glass tube t open at both ends. The lower end of the tube and the bulb of the thermometer dip into ether contained in the silver cap. A side tube establishes communication between this part of the apparatus and a vertical PSYCHROMETER. 395 tube U V, which is itself connected with an aspirator 1 A, placed at a convenient distance. By allowing the water in the aspirator to escape, a current of air is produced through the ether, which has the effect of keeping the liquid in agitation, and thus producing uniform- ity of temperature throughout the whole. It also tends to hasten evaporation; and the cold thus produced speedily causes a deposition of dew, which is observed from a distance with a telescope, thus obviating the risk of vitiating the observation by the too close proximity of the observer. The observation is facilitated by the contrast offered by the appearance of the second cap, which has no communication with the first, and contains a thermometer for giving the temperature of the external air. By regulating the flow of liquid from the aspirator, the temperature of the ether can be very nicely controlled, and the dew can be made to ap- pear and disappear at temperatures nearly identical. The mean of the two will then very accurately represent the dew-point. The liquid employed in Regnault's hygro- meter need not be ether. Alcohol, a much less volatile liquid, will suffice. This is an important advantage; for, since the boiling- point of ether is 36 C. (97 F.), it is not easy to preserve it in hot climates. 427. Wet and Dry Bulb Hygrometer. This instrument, which is also called Mason's hygrometer, and is known on the Continent as August's psychrometer, consists (Fig. 272) of two precisely similar thermometers, mounted at a short distance from each other, the bulb of one of them being covered with muslin, which is kept moist by means of a cotton wick leading from a vessel of water. The evaporation which takes place from the moistened bulb produces a depres- sion of temperature, so that this thermometer reads lower than the other by an amount which increases with the dryness of the air. The instrument must be mounted in such a way that the air can circulate 1 An aspirator is a vessel into which air is sucked at the top to supply the place of water which is allowed to escape at the bottom ; or, more generally, it is any appaiatus for sucking in air or gas. Fig. 272. Wet and Dry Thermometers. 396 HYGROMETRY. very freely around the wet bulb; and the vessel containing the water should be small, and should be placed some inches to the side. The level of this vessel must be high enough to furnish a supply of water which keeps the muslin thoroughly moist, but not high enough to cause drops to fall from the bottom of the bulb. Unless these precautions are observed, the depression of temperature will not be sufficiently great, especially in calm weather. In frosty weather the wick ceases to act, and the bulb must be dipped in water some time before taking an observation, so that all the water on the bulb may be frozen, and a little time allowed for evaporation from the ice, before the reading is taken. The great facility of observation afforded by this instrument has brought it into general use, to the practical exclusion of other forms of hygrometer. As the theoretical relation between the indications of its two thermometers and the humidity as well as the dew-point of the air is rather complex, and can scarcely be said to be known with certainty, it is usual, at least in this country, to effect the re- duction by means of tables which have been empirically constructed by comparison with the indications of a dew-point instrument. The tables universally employed by British observers were constructed by Mr. Glaisher, and are based upon a comparison of the simultaneous readings of the wet and dry bulb thermometers and of Daniell's hygrometer taken for a long series of years at Greenwich observatory, combined with SOHIQ similar observations taken in India and at Toronto. 1 According to these tables, the difference between the dew-point and the wet-bulb reading bears a constant ratio to the difference between the two thermometers, when the temperature of the dry- bulb thermometer is given. When this temperature is 53 F., the dew-point is as much below the wet-bulb as the wet-bulb is below the temperature of the air. At higher temperatures the wet-bulb reading is nearer to the dew-point than to the air-temperature, and the reverse is the case at temperatures below 53. 428. In order to obtain a clue to the construction of a rational formula for deducing the dew-point from the indications of this in- strument, we shall assume that the wet-bulb is so placed that its temperature is not sensibly affected by radiation from surrounding objects, and hence that the heat which becomes latent by the 1 The first edition of these Tables differs considerably from the rest, and is never used; but there has been no material alteration since the second edition (1856). WET AND DRY BULBS. 397 evaporation from its surface is all supplied by the surrounding air. When the temperature of the wet-bulb is falling, heat is being con- sumed by evaporation faster than it is supplied by the air; and the reverse is the case when it is rising. It will suffice to consider the case when it is stationary, and when, consequently, the heat con- sumed by evaporation in a given time is exactly equal to that supplied by the air. Let t denote the temperature of the air, which is indicated by the dry -bulb thermometer; t' the temperature of the wet-bulb; T the temperature of the dew-point, and let /) /', F be the vapour-pressures corresponding to saturation at these three temperatures. Then, as shown in 418, the tension of the vapour present in the air at its actual temperature t is also equal to F. We shall suppose that wind is blowing, so that continually fresh portions of air come within the sphere of action of the wet-bulb. Then each particle of this air experiences a depression of temperature and an increase of vapour-pressure as it comes near the wet-bulb, from both of which it afterwards recovers as it moves away and mixes with the general atmosphere. If now it is legitimate to assume 1 that this depression of tempera- ture and exaltation of vapour-pressure are always proportional to one another, not only in comparing one particle with itself at different times, but also in comparing one particle with another, we have the means of solving our problem; at all events, if we may make the additional assumptions that a portion of the air close to the wet- bulb is at the temperature of the wet-bulb, and is saturated. On these assumptions the greatest reduction of temperature of the air is t t', and the greatest increase of vapour-pressure is f F, and the corresponding changes in the whole mass are proportional to these. The three temperatures t, t', T must therefore be so related, that the heat lost by a mass of air in cooling through the range t t', is just equal to the heat which becomes latent in the formation of as much vapour as would raise the vapour-pressure of the mass by the amount/' F. 1 The assumption which Dr. Apjohn actually makes is as follows : " When in the moist- bulb hygrometer the stationary temperature is attained, the caloric which vaporizes the water is necessarily exactly equal to that which the air imparts in descending from the temperature of the atmosphere to that of the moistened bulb ; and the air which has undergone this reduction becomes saturated with moisture" (Trans. R.I.A. Nov. 1834). This implies that all the air which is affected at all is affected to the maximum extent a very harsh supposition ; but August independently makes the same assumption. 398 HYGROMETRY. Let h denote the height of the barometer, s the specific heat of air, D the relative density of vapour ( 406), L the latent heat of steam, and let the vapour-pressures be expressed by columns of mercury. Then the mass of the air is to that of the vapour required to pro- duce the additional tension, as h to D (f F), and we are to have or which is the required formula, enabling us, with the aid of a table of vapour-pressures, to determine F., and therefore the dew-point T, when the temperatures t, t' of the dry and wet bulb, and the height k of the barometer, have been observed. The expression for the rela- F tive humidity will be j Properly speaking, s denotes the specific heat not of dry air but of air containing the actual amount of vapour, and therefore depends to some extent upon the very element which is to be determined; but its variation is inconsiderable. L also varies with the known quantity t', but its variations are also small within the limits which occur in practice. The factor ^ may therefore be regarded as con- stant, and its value, as adopted by Dr. Apjohn 1 for the Fahrenheit scale, is ^^-f. or 57; x -. . We thus obtain what is known as ^OIU OU o/ formula, When the wet-bulb is frozen, L denotes the sum of the latent heats of liquefaction and vaporization, and the formula becomes F =/'-* . (3) J 96 30 1 This value was founded on the best determinations which had been made at the time, the specific heat of air being taken as '267, the value obtained by Delaroche and Berard. The same value was employed by Regnault in his hygrometrical investigations. At a still later date Regnault himself investigated the specific heat of air and found it to be "237. When this correct value is introduced into Regnault's theoretical formula (which is substantially the same as Apjohn's), the discrepancies which he found to exist between calculation and observation are increased, and amount, on an average, to about 25 per cent of the difference between wet-bulb temperature and dew-point. The inference is that the assumptions on which the theoretical formulae are based are not accurate ; and the discrepancy is in such a direction as to indicate that diffusion of heat is more rapid than diffusion of vapour. CTIEMK 'A I. HYGHOMKTER. 390 In calm weather, and also in very dry weather, the humidity, as deduced from observations of wet and dry thermometers, is generally too great, probably owing mainly to the radiation from surrounding objects on the wet-bulb, which makes its temperature too high. 429. Chemical Hygrometer. The determination of the quantity of aqueous vapour in the atmosphere may be effected by ordinary chemical analysis in the following manner: An aspirator A, of the capacity of about 50 litres, communicates at its upper end with a system of U-tubes 1, 2, 3, 4, 5, 6, filled with Fig 273. Chemical Hygrometer. pieces of pumice soaked in sulphuric acid. The aspirator being full of water, the stop-cock at the bottom is opened, and the air which enters the aspirator to take the place of the water is obliged to pass through the tubes, where it leaves all its moisture behind. This moisture is deposited in the first tubes only. The last tube is intended to absorb any moisture that may come from the aspirator. Suppose w to be the increase of weight of the first tubes 4, 5, 6 ; this is evidently the weight of the aqueous vapour contained in the air which has passed through the apparatus. The volume V of this air, which we will suppose to be expressed in litres, may easily be found by measuring the amount of water which has escaped. This air has been again saturated by contact with the water of the aspirator, and the aqueous vapour contained in it is consequently at the maxi- mum pressure corresponding to the temperature indicated by a ther- mometer attached to the apparatus. Let this pressure be denoted 400 HYGROMETRY. by /. The volume occupied by this air when in the atmosphere, where the temperature is T, is known by the regular formulae to have been V * 1 + ttT H - x 1 + at x denoting the pressure of the aqueous vapour in the atmosphere, and H the total atmospheric pressure as indicated by the barometer; and, since the relative density of steam is '622, and the weight of a litre of air at temperature C. and pressure 760 mm. is 1*293 gramme, the weight of vapour which this air contained must have been V . ? / 1 + aT x 1-293 x -622 . H-35 l + at 760 1 + aT which must be equal to the known weight w, and thus we have an equation from which we find x _ w (1 + at) 760 H ~ V (H -/) x -622 x 1-293 + w (1 + at) 760 ' A good approximation will be obtained by writing w= Vx 1-293 x '622^' whence w a;=945 y- This method has all the exactness of a regular chemical analysis, but it involves great labour, and is, besides, incapable of showing the sudden variations which often occur in the humidity of the atmosphere. It can only give the mean quantity of moisture in a given volume of air during the time occupied by the experiment. Its accuracy, however, renders it peculiarly suitable for checking the results obtained by other methods, and it was so employed by Regnault in the investigations to which we have referred in the foot- note to the preceding section. 430. Weight of a given Volume of Moist Air. The laws of vapours and the known formulas of expansion enable us to solve a problem of very frequent occurrence, namely, the determination of the weight of a given volume of moist air. Let V denote the volume of this air, H its pressure, / the pressure of the vapour of water in it, and t its temperature. The entire gaseous mass may be divided into two parts, a volume V of dry air at the temperature t and the pressure H f, whose weight is, by known formulae, WEIGHT OF MOIST AIR. 401 and a volume V of aqueous vapour at the temperature t and the pressure /; the weight of this latter is 5 1 / ' The sum of these two weights is the weight required, viz. l H ~8^ Vx 1-293 x 1 + oT 760 431. Ratio of the Volumes occupied by the same Air when saturated at Different Temperatures and Pressures. Suppose a mass of air to be in presence of a quantity of water which keeps it always saturated ; let H be the total pressure of the saturated air, t its temperature, and V its volume. At a different temperature and pressure t' and H', the volume occupied V will in general be different. The two quantities V and V may be considered as the volumes occupied by a mass of dry air at temperatures t and t' and pressures H /and H' /'; we have then the relation V'~W^f ' rw' In passing from one condition of temperature and pressure to another, it may be necessary, for the maintenance of saturation, that a new quantity of vapour should be formed, or that a portion of the vapour should be condensed, or again, neither the one nor the other change may take place. To investigate the conditions on which these alter- natives depend, let D and D' be the maximum densities of vapour at the temperatures t and t' respectively. Suppose we have t >t, and that, without altering the pressure /, the temperature of the vapour is raised to t', all contact with the generating liquid being prevented. The vapour will no longer remain saturated; but, on increasing the pressure to /', keeping the temperature unchanged, saturation will again be produced. This latter change does not alter the actual quantity of vapour, and if we suppose its coefficient of expansion to be the same as that of air, we shall have (2) and, by multiplying together equations (1) and (2), we have "VD Wf-fl* (3) 26 402 HYGROMETRY. From this result the following particular conclusions may be de- duced: 1. If H'/=H/', VD = V'D', that is, the mass of vapour is the same in both cases; consequently, neither condensation nor. evaporation takes place. 2. If H'/>H/', VD>V'D', that is, partial condensation occurs. 3. If H'/ x and v 2 are very nearly equal to each other and to v. The fact that . conductivity varies with temperature was discovered by Forbes. He found that a specimen of iron which had a conductivity '207 at C. had only a conductivity '124 at 275 C. 443. Effect of Change of Units. In the C.G.S. (Centimetre-Gramme- Secorid) system, which we have explained in Part I. 87, A is ex- pressed in square centimetres, x in centimetres, and Q in gramme- degrees. It is immaterial whether the degree be Centigrade or Fahrenheit; for a change in the length of the degree will affect the numerical values of Q and of v 2 Vi alike, and will leave the numer- ical value of , and hence of - . . . . or k unaltered. ft Vi A ( t>2 Vi ) t ' To find the effect of changes in the units of length and time, we must note that if the unit of length be x centimetres, the unit of area will be a? square centimetres, and the unit of mass, being the mass of unit volume of cold water, will be a? grammes. The new unit of heat will therefore be a? gramme-degrees. IN ITS OF CONDUCTIVITY. 415 The new unit of conductivity will be the conductivity of a substance such that a? gramme-degrees of heat flow in the new unit of time which we will call t seconds through a? sq. cm. of a plate x cm. thick, with a difference of 1 between its faces. The conductivity of such a plate, when expressed in C.G.S. units, would be found by putting Q=x*, Aa^, v t -Vi=-l in the formula Q* A fa - vi) t' and would be ^- or . Hence to reduce conductivities from the new scale to the C.G.S. scale we must multiply them by yj and the same rule will apply to diffusivities, since the quantity c in equation (3) being the ratio of the thermal capacity of the substance to that of water, bulk for bulk, is independent of units. 444. Illustrations of Conduction. The following experiments are often adduced in illustration of the different conducting and diffusing powers of different metals. Two bars of the same size, but of different metals (Fig. 282), are placed end to end, and small wooden balls are attached by wax to Fig. 282. Balls Melted off. their under surfaces at equal distances. The bars are then heated at their contiguous ends, and, as the heat extends along them, the balls successively drop off. If the conditions are in other respects equal, the balls will begin to drop off first from that which has the greater diffusivity, and the greatest number of balls will ultimately drop off from that which has the greater conductivity. The well-known experiment of Ingenhousz (Fig. 283) is of the same kind. The apparatus consists of a box, with a row of holes in one of its sides, in which rods of different metals can be fixed. The rods having previously been coated with wax, the box is filled with 41G CONDUCTION OF HEAT. Fig. 283. Ingenhousz's Apparatus. boiling water or boiling oil, which comes into contact with the inner ends of the rods. The wax gradually melts as the heat travels along the rods. The order in which the melting begins is the order of the dif- fusivities of the metals employed, and when it has reached its limit (if the temperature of the liquid be maintained con- stant) the order of the lengths melted is the order of their conductivities. 445. Metals the Best Conductors. Metals, though differing con- siderably one from another, are as a class greatly superior both in conductivity and diffusivity to other substances, such as wood, marble, brick. This explains several familiar phenomena. If the hand be placed upon a metal plate at the temperature of 10 C., or plunged into mercury at this temperature, a very marked sensation of cold is experienced. This sensation is less intense with a plate of marble at the same temperature, and still less with a piece of wood. The reason is that the hand, which is at a higher temperature than the substance to which it is applied, gives up a portion of its heat, which is conducted away by the substance, and consequently a larger por- tion of heat is parted with, and a more marked sensation of cold experienced, in the case of the body of greater conducting power. 446. Davy Lamp. The conducting power of metals explains the curious property possessed by wire-gauze of cutting off a flame. If, for example, a piece of wire - gauze be placed above a jet of gas, the flame is pre- vented from rising above the gauze. If the gas be first al- lowed to pass through the gauze, and then lighted above, the flame is cut off from the burner, and is unable to extend itself to the under surface of the gauze. These facts depend upon the conducting power of Fig. 284. Action of Wire-gauze on Flame. WIRE-GAUZE. 417 Fig. 285 Davy Lamp. The explosive gases pass metallic gauze, in virtue of which the heat of the flame is rapidly dissipated at the points of contact, the result being a diminution of temperature sufficient to prevent ignition. This property of metallic gauze has been turned to account for various pur- poses, but its most useful application is in the safety-lamp of Sir Humphry Davy. It is well known that a gas called fire- damp is often given off in coal-inines. It is a compound of carbon and hydrogen, and is a large ingredient in ordinary coal-gas. This fire-damp, when mixed with eight or ten times its volume of air, explodes with great violence on coming in contact with a lighted body. To obviate this danger, Davy invented the safety - lamp, which is an ordinary lamp with the flame inclosed by wire-gauze, through the gauze, and burn in- side the lamp, in such a manner as to warn the miner of their presence; but the flame is unable to pass through the gauze. 447. Walls of Houses. The knowledge of the relative con- ducting powers of different bodies has several important practical applications. In cold countries, where the heat produced in the interior of a house should be as far as pos- sible prevented from escaping, the walls should be of brick or wood, which have feeble con- ducting powers. If they are of stone, which is a better conduc- rig. sse. -ice-house, tor, a greater thickness is re- quired. Thick walls are also useful in hot countries in resisting the power of the solar rays during the heat of the day. 27 418 CONDUCTION OF HEAT. We have already alluded ( 331) to the advantage of employing fire-brick, which is a bad conductor, as a lining for stoves. The feeble conducting power of brick has led to its employment in the construction of ice-houses. These are round pits (Fig. 286), generally from 6 to 8 yards in diameter at top, and somewhat nar- rower at the bottom, where there is a grating to allow the escape of water. The inside is lined with brick, and the top is covered with straw, which, as we shall shortly see, is a bad conductor. In order to diminish as much as possible the extent of surface exposed to the Fig. 287. Norwegian Cooking-box. action of the air, the separate pieces are dipped in water before depositing them in the ice-house, and, by their subsequent freezing together, a solid mass is produced, capable of remaining unmelted for a very long time. 448. Norwegian Cooking-box. A curious application of the bad conducting power of felt is occasionally to be seen in the north of Europe in a kind of self-acting cooking-box. This is a box lined RELATIVE CONDUCTIVITIES. 419 inside with a thick layer of felt, into which tits a metallic dish with a cover. The dish is then covered with a cushion of felt, so as to be completely surrounded by a substance of very feeble conducting power. The method of employing the apparatus is as follows : The meat which it is desired to cook is placed along with some water in the dish, the whole is boiled for a short time, and then transferred from the fire to the box, where the cooking is completed without any further application of heat. The resistance of the stuffing of the box to the escape of heat is exceedingly great; in fact, it may be shown that at the end of three hours the temperature of the water has fallen by only about 10 or 15 C. It has accordingly remained during all that time sufficiently high to conduct the operation of cooking. 449. Experimental Determination of Conductivity. Several ex- perimenters have investigated the conductivity of metals, by keeping one end of a metallic bar at a high temperature, and, after a sufficient lapse of time, observing the permanent temperatures assumed by different points in its length. If the bar is so long that its further end is not sensibly warmer than the surrounding air, and if, moreover, Newton's law of cooling ( 461) be assumed true for all parts of the surface, and all parts of a cross section be assumed to have the same temperature, the con- ductivity being also assumed to be independent of the temperature, it is easily shown that the temperatures of the bar at equidistant points in its length, beginning from the heated end, must exceed the atmospheric temperature by amounts forming a decreasing geometric series. Wiedemann and Franz, by the aid of the formula to which these assumptions lead, 1 computed the relative conducting powers of several of the metals, from experiments on thin bars, which were steadily heated at one end, the temperatures at various points in the length being determined by means of a thermo-electric junction clamped to the bar. The following were the results thus obtained: EELATIVE CONDUCTING POWERS. SUver 100 Copper, 77'6 Gold, 53-2 Brass, 33 Zinc, 19-9 Tin, 14-5 Steel, 12 Iron, 11-9 Lead, 8'5 Platinum 8'2 Palladium 6 '3 Bismuth, ...... 1'9 1 See note B at the end of this chapter. 420 CONDUCTION OF HEAT. The absolute conductivity of wrought iron was investigated with great care by Principal Forbes, by a method which avoided some of the questionable assumptions above enumerated. The end of the bar was heated by a bath of melted lead kept at a uniform tempera- ture, screens being interposed to protect the rest of the bar from the heat radiated by the bath. The temperatures at other points were observed by means of thermometers inserted in small holes drilled in the bar, and kept in metallic contact by fluid metal. In order to determine the loss of heat by radiation at different temperatures, a precisely similar bar, with a thermometer inserted in it, was raised to about the temperature of the bath, and the times of cooling down through different ranges were noted. The conductivity of one of the two bars experimented on, varied from -01337 at C. to '00801 at 275 C., and the corresponding- numbers for the other bar were '00992 and '00724, the units being the foot, the minute, the degree (of any scale), ai.d the foot-degree 1 (of the same scale). In both instances, the conductivity decreased regularly with increase of temperature. To reduce these results to the C.G.S. scale, we must (as directed 2 in 443) multiply them by -, where x denotes the number of centi- metres in a foot, or 30'48, and t the number of seconds in a minute; will therefore be t , or 15-48. 60 The reduced values will therefore be as follows: At 0'. At 275 1st bar, -207 '1240 2dbar, -1536 -1121 450. Experimental Determination of Diffusivity. Absolute deter- minations of the diffusivity K or - for the soil or rock at three local- ities in or near Edinburgh were made by Principal Forbes and Sir Win. Thomson. They were derived from observations on the tem- perature of the soil as indicated by thermometers having their bulbs buried at depths of 3, 6, 12 and 24 French feet. The annual range of temperature diminished rapidly as the depth increased, and this diminution of range was accompanied by a retardation of the times of maximum and minimum. The greater the diffusivity the more slowly will the range diminish and the less will be the retardation 1 See 335. ABSOLUTE CONDUCTIVITIES. 421 of phase. By a process described in note C at the end of this chapter the value of K was deduced; and by combining this with the value of c (the product of specific heat and density), which was determined by Regnault, from laboratory experiments, the value of k or CK was found. The following are the results, expressed in the C.G.S. scale: Diffusivity. Conductivity. Trap rock of Calton Hill, .... .................... '00786 ......... '00415 Sand of Experimental Garden, .................. '00872 ......... '00262 Sandstone of Craigleith Quarry, ................ '02311 ......... '01068 Similar observations made at Greenwich Observatory, and reduced by the editor of the present work, gave "01249 as the diffusivity of the gravel of Greenwich Observatory Hill. A method based upon similar principles has since been employed by Angstrom and also by Neumann for laboratory experiments; a bar of the substance under examination being subjected to regular periodical variations of temperature at one end, and" the resulting periodic variations at other points in its length being observed. These gave the means of calculating the diffusivity, and then obser- vations of the specific heat and density gave the conductivity. The following conductivities were thus obtained by Neumann: Conductivity in C.G.S. units. Copper, ......................................... 1-108 Brass ............................................ -302 Zinc, ................................ . ........... -307 Iron, ............................................. -164 German silver ................................. '109 451. Conductivity of Rocks. The following values of thermal and thermometric conductivity in C.G.S. units are averages based on the experiments of Professor Alexander Herschel. k k. c - Granite, ................................ '0053 ..... . .. -015 Limestone, .............................. '005 ........ . '009 Sandstone, dry, ........................ '0056 ......... -012 Sandstone, thoroughly wet, ......... '0060 ......... '010 Slate, along cleavage ................. '0060 ......... -010 Slate, across cleavage, ................ '0034 ......... -006 Clay, sun-dried, ........................ -0022 ......... -0048 Eed brick, ... .......................... -0015 ......... -0044 Plate-glass, .............................. -0023 ......... '0040 452. Conducting Powers of Liquids. With the exception of mer- cury and other melted metals, liquids are exceedingly bad conduc- 422 CONDUCTION OF HEAT. tors of heat. This can be shown by heating the upper part of a column of liquid, and observing the variations of temperature below. These will be found to be scarcely perceptible, and to be very slowly produced. If the heat were applied below (Fig. 288), we should have the process called convection of heat; the lower layers of liquid would rise to the surface, and be replaced by others which would rise in their turn, thus producing a circulation and a general heating of the liquid. On the other hand, when heat is applied above, the expanded Fig. 238. Liquid heated from below. Fig. 289. Boiling of Water over Ice. layers remain in their place, and the rest of the liquid can be heated by conduction and radiation only. The following experiment is one instance of the very feeble con- ducting power of water. A piece of ice is placed at the bottom of a glass tube (Fig. 289), which is then partly filled with water; heat is applied to the middle of the tube, and the upper portion of the water is readily raised to ebullition, without melting the ice below. 453. Conducting Power of Water. The power of conducting heat possessed by water, though very small, is yet quite appreciable. This was established by Despretz by the following experiment. He took a cylinder of wood (Fig. 290) about a yard in height and eight inches in diameter, which was filled with water. In the side of this CONDUCTIVITY OF WATER. 423 cylinder were arranged twelve thermometers one above another, their bulbs being all in the same vertical through the middle of the liquid column. On the top of the liquid rested a metal box, which was filled with water at 100, frequently renewed during the course of the experiment. Under these circumstances Despretz observed that the temperature of the thermometers rose gradually, and that a long time about 30 hours was required before the permanent state was Fig. 290. Despretz's Experiment assumed. Their permanent differences, which formed a decreasing geometric series, were very small, and were inappreciable after the sixth thermometer. The increase of temperature indicated by the thermometers might be attributed to the heat received from the sides of the cylinder, though the feeble conducting power of wood renders this idea some- what improbable. But Despretz observed that the temperature w T as higher in the axis of the cylinder than near the sides, which proves that the elevation of temperature was due to the passage of heat downwards through the liquid. From experiments by Professor Guthrie, 1 it appears that water conducts better than any other liquid except mercury. 454. Absolute Measurement of Conductivity of Water. The abso- 1 B, A. Report, 1868, and Trans. R. S. 1869. 424 CONDUCTION OF HEAT. lute value of k for water has been determined by Mr. J. T. Bottom- ley. Hot water was gently placed on the top of a mass of water nearly filling a cylindrical wooden vessel. Readings were taken from time to time of two horizontal thermometers, one of them a little lower than the other, which gave the difference of temperature between the two sides of the intervening stratum. The quantity of heat conducted in a given time through this stratum was known from the rise of temperature of the whole mass of water below, as in- dicated by an upright thermometer with an exceedingly long cylindrical bulb extending downwards from the centre of the stratum in question nearly to the bottom of the vessel. A fourth thermometer, at the level of the bottom of the long bulb, showed when the increase of temperature had extended to this depth, and as soon as this occurred (which was not till an hour had elapsed) the experiment was stopped. The result of these experiments is that the value of k for water is from '0020 to '0023, which is nearly identical with its value for ice, this latter element, as determined by Professor George Forbes, being 00223. The conductivity of water seems to be much greater than that of wood. 455. Conducting Power of Gases. Of the conducting powers of gases it is almost impossible to obtain any direct proofs, since it is exceedingly difficult to prevent the interference of convection and direct radiation. However, we know at least that they are exceed- ingly bad conductors. In fact, in all cases where gases are. inclosed in small cavities where their movement is difficult, the system thus formed is a very bad conductor of heat. This is the cause of the feeble conducting powers of many kinds of cloth, of fur, eider-down, felt, straw, saw-dust, &c. Materials of this kind, when used as articles of clothing, are commonly said to be warm, because they hinder the heat of the body from escaping. If a garment of eider-down or fur were compressed so as to expel the greater part of the air, and to reduce the substance to a thin sheet, it would be found to be a much less warm covering than before, having become a better con- ductor. We thus see that it is the presence of air which gives these substances their feeble conducting power, and we are accordingly justified in assuming that air is a bad conductor of heat. 456. Conductivity of Hydrogen. The conducting power of hydrogen is much superior to that of the other gases a fact which agrees CONDUCTIVITY OF HYDROGEN. 425 with the view entertained by chemists, that this gas is the vapour of a metal. The good conductivity of hydrogen is shown by the following experiments : 1. Within a glass tube (Fig. 291) is stretched a thin platinum wire, which is raised to incandescence by the passage of an electric current. When air, or any gas other than hydrogen, is passed through the tube, the incandescence continues, though with less vividness than in vacuo; but it disappears as soon as hydrogen is employed. 2. A thermometer is placed at the bottom of a vertical tube, and heated by a vessel containing boiling water which is placed at the top of the tube. The tube is exhausted of air, and different gases are successively admitted. In each case the indication of the thermometer is found to be lower than for vacuum, except when the gas is hydrogen. With this gas, the difference is in the opposite direction, showing that the diminution of radiation has been more than compensated by the conducting power of the hydrogen. NOTE A. DIFFERENTIAL EQUATION FOR LINEAR FLOW OF HEAT. The mode of obtaining differential equations for the variation of temperature at each point of a body during the variable stage, may be illustrated by considering the simplest case, that in which the isothermal surfaces (surfaces of equal temperature) are parallel planes, and therefore the lines of flow (which must always be normal to the isothermal surfaces) parallel straight lines. Let x denote distance measured in the direction in which heat is flowing, v the teiu perature at the time t at a point specified by x, k the conductivity, and c the thermal capacity per unit volume (both at the temperature v). Then the flow of heat per unit time past a cross section of area A is - k A , and the flow past an equal and parallel section ax further on by the small distance 5x is greater by the amount Fig. 291. Cooling by Contact of Hydrogen. dx \ dx) 426 CONDUCTION OF HEAT. This latter expression therefore represents the loss of heat from the intervening prism A8x, and the resulting fall of temperature is the quotient of the loss by the thermal capacity cA6x, which quotient is 1 ( -k dv \ c dx\ dx) This, then, is the fall of temperature per unit time, or is - -=- If the variation of k is die "* insensible, so that -5- can be neglected, the equation becomes dv _Tc cPv di~~c dx*' which applies approximately to the variations of temperature in the soil near the surface of the earth, x being in this case measured vertically. For the integral of this equation, see Note C. NOTE B. FLOW OF HEAT IN A BAR ( 449). If p and s denote the perimeter and section of the bar, k the conductivity, and h the coefficient of emission of the surface at the temperature v, the heat emitted in unit time from the length 5z is hvpSx, if we assume as our zero of temperature the temperature of the surrounding air. But the heat which passes a section is - sk =- , and that which passes a section further on by the amount S.c d?v is less by the amount sk -f. Sx; and this difference must equal the amount emitted from W . 7 the intervening portion of the surface. Hence we have the equation -= 2 = ^- v, the in- (/.'' KS tegral of which for the case supposed is - /** v = Ve V kg , V denoting the temperature at the heated end. NOTE C. DEDUCTION OF DIFFUSIVITY FROM OBSERVATIONS OF UNDERGROUND TEM- PERATURE ( 450). Denoting the diffusivity by K, the equation of Note A is dv cPv -77 ~ K 7-5- W dt dxr This equation is satisfied by v=e~ a sm(pt-ax), (5) where a and /3 are any two constants connected by the relation for we find, by actual differentiation, -T- = e~ MX {-a sin (pt-ax)-acos (fit -ax)} ; -rl = e ~* x { a 2 sin (fit - ax) + a 2 cos (j8* - ax) + a? cos (fit - ax) - a 2 sin (/3 - ax) } = e~* X la? cos (fit-ax); dv -a.x o io t \ P &V More generally, equation (4) will be satisfied by making v equal to the sum of any VARIATIONS OF SOIL-TEMPERATURE. 427 number of terras similar to the right-hand member of (5), each multiplied by any constant and a constant term may be added In fact we may have sin (ft<- a 1 x + 'E 1 ) + A a e ~*~ & sin (ft*-o l +]^)+Aa' sin s ) + &c., (7) where AO, A 1} EI, &c., are any constants. Let x be measured vertically downwards frjm the surface of the ground (supposed horizontal); then at the surface the above expression becomes &c. (8) Now, if T denote a year, it is known that the average temperature of the surface at any time of year can be expressed, in terms of t the time reckoned from 1st of January or any stated day, by the following series : &c., (9) where AO is the mean temperature of the whole year, and A 1? A 2 , AS, &c., which are called the amplitudes of the successive terms, diminish rapidly. The term which contains A t and EI (called the annual term), completes its cycle of values in a year, the next term in half a year, the next in a third of a year, and so on. The annual term is much larger, and more regular in its values from year to year than any of those which follow it. Each term affords two separate determinations of the diffusivity. Thus, for the annual term, we have, by comparing (8) and (9) 0!= * whence, by (6), a '~ \/YK~ Vfic" At the depth x, the amplitude of this term will be A a.\X l , the logarithm of which is log Aj - dtX. Hence a t can be deduced from a comparison of the annual term at two different depths, by dividing the difference of the Napierian logarithms of the amplitudes by the difference of depth. But o x can also be determined by comparing the values of ft t - a t x + E t at two depths for the same value of t, and taking their difference (which is called the retardation of phase, since it expresses how much later the maximum, minimum, and other phases, occur at the lower depth than at the upper). This difference, divided by the difference of depth, will be equal to a t . These two determinations of a t ought to agree closely, and K will then be found by the equation 01 = RADIATION. 457. Radiation distinct from Conduction. When two bodies at different temperatures are placed opposite to each other, with no- thing between them but air or some other transparent medium, the hotter body gives heat to the colder by radiation. It is by radia- tion that the earth receives heat from the sun and gives out heat to the sky; and it is by radiation that a fire gives heat to a person sit- ting in front of it. Radiation is broadly distinguished from conduction. In conduc- tion, the transmission of heat is effected by the warming of the in- tervening medium, each portion of which tends to raise the succeed- ing portion to its own temperature. On the other hand heat transmitted from one body to another by radiation does not affect the temperature of the intervening medium. The heat which we receive from the sun has traversed the cold upper regions of the air; and paper can be ignited in the focus of a lens of ice, though the temperature of ice cannot exceed the freezing- point. Conduction is a gradual, radiation an instantaneous process. A screen interposed between two bodies instantly cuts off radiation between them; and on the removal of such a screen radiation in- stantly attains its full effect. Radiant heat, in fact, travels with the velocity of light, and it is subject to laws similar to the laws of light ; for example, it is usually propagated only in straight lines. Strictly speaking, radiant heat, like latent heat, is not heat at all, but is a form of energy which is readily converted into heat. Its nature is precisely the same as that of light, the difference between them being only a difference of degree, as will be more fully ex- plained in treating of the analysis of light by the prism and spectro- RADIATION DISTINGUISHED FROM CONDUCTION. 429 scope. The present chapter will contain numerous instances of the analogy between the properties of non-luminous radiant heat and well-known characteristics of light. 458. A Ponderable Medium not Essential. The transmission of the sun's heat to the earth shows that radiation is independent of any ponderable medium. But since the solar heat is accompanied by light, it might still be questioned whether dark heat could be propagated through a vacuum. This was tested by Rumford in the following way: He con- structed a barometer (Fig. 292), the upper part of which was ex- panded into a globe, and contained a thermometer hermetically sealed into a hole at the top of the globe, so that the bulb of the thermometer was at the centre of the globe. The globe was thus a Torricellian vacuum -chamber. By melting the tube with a blow-pipe, the globe was separated, and was then immersed in a vessel containing hot water, when the thermometer was immediately observed to rise to a temperature evidently higher than could be due to the conduction of heat through the stem. The heat had therefore been communi- cated by direct radiation through the vacuum be- tween the sides of the globe and the bulb a of the thermometer. 459. Radiant Heat travels in Straight Lines. In a uniform medium the radiation of heat takes place in straight lines. If, for instance, between a ther- mometer and a source of heat, there be placed a number of screens, each pierced with a hole, and if the screens be so arranged that a straight line can be drawn without interruption from the source to the thermo- meter, the temperature of the latter immediately rises ; if a different arrangement be adopted, the heat is stopped by the screens, and the thermometer indicates no effect. Hence we can speak of rays of heat just as we speak .of rays of light. Thus we say that rays of heat issue from all points of the surface of a heated body, or that such a body emits rays of heat. The word ray when thus used scarcely admits of precise definition. It is a popular rather than a scientific term; for no finite quantity of heat or light can travel along a mathematical line. In a mere Fig. 292. Kumford's Experiment. 430 RADIATION. geometrical sense the rays are the lines which indicate the direction of propagation. It is now generally admitted that both heat and light are due to a vibratory motion which is transmitted through space by means of a fluid called ether. According to this theory the rays of light and heat are lines drawn in all directions from the origin of motion, and along which the vibratory movement advances. 460. Surface Conduction. The cooling of a hot body exposed to the air is effected partly by radiation, and partly by the conduction of heat from the surface of the body to the air in contact with it. The activity of the surface-conduction is greatly quickened by wind, which brings continually fresh portions of cold air into contact with the surface, in the place of those which have been heated. The cooling of a body in vacuo is effected purely by radiation, except in so far as there may be conduction through its supports. 461. Newton's Law of Cooling. In both cases, if the body be ex- posed in a chamber of uniform temperature, the rate at which it loses heat is approximately proportional to the excess of the temper- ature of its surface above that of the chamber, and the proportion- ality is sensibly exact when the excess does not exceed a few degrees. If the body be of sensibly uniform temperature throughout its whole mass, as in the case of a thin copper vessel full of water which is kept stirred, its fall of temperature is proportional to its loss of heat, and hence the rate at which its temperature falls is proportional to the excess of its temperature above that of the chamber. Prac- tically if the body be a good conductor and of small dimensions say a copper ball an inch in diameter, or an ordinary mercurial ther- mometer the fall of its temperature is nearly in accordance with this law, which is called Newton's law of cooling. The observed fact is that when the readings of the thermometer are taken at equal intervals of time, their excesses above the temperature of the inclosure (which is kept constant) form a diminishing geometrical progression. To show that this is equivalent to Newton's law, let denote the excess of temperature at time t', then, in the notation of the differ- ential calculus, ~ is the rate of cooling; and Newton's law asserts that this is proportional to 0, or that -=A0, (1) LAW OF COOLING. 431 where A is a constant multiplier. This is equivalent to -~=Adt, (2) which asserts that for equal small intervals of time the differences between the temperatures are proportional to the temperatures. But if the differences between the successive terms of a series are propor- tional to the terms themselves, the series is geometrical; for if we have 01 0> _0-2 0$ _0s 0^ 0\ v% 63 we obtain, by subtracting unity from each member, __ a a * that is, b d^, 63, 64 are in geometrical progression. j n The expression - - & in equation (2) is, by the rules of the differ- ential calculus, equal to cHog0; hence equation (2) shows that log diminishes by equal amounts in equal times. Log 6 here denotes the Napierian logarithm of d; and since common logarithms are equal to Napierian logarithms multiplied by a constant factor, the common logarithm of will also diminish by equal amounts in equal times. The constant A in equation (1) or (2) will be deter- mined from the experimental results by dividing the decrement of log by the interval of time. We have been assuming that the body is hotter than the chamber or inclosure; but a precisely similar law holds for the warming of a body which is colder than the inclosure in which it is placed. 462. Dulong and Petit's Law of Cooling. Newton's law is sensibly accurate for small differences of temperature between the body and the inclosure. Dulong and Petit conducted experiments on the cool- ing of a thermometer by radiation in vacuo with excesses of temper- ature varying from 20 to 240 C., from which they deduced the formula de it i\ dt= ca (a - 1); or, as it may be otherwise written, -^--cla v+B -a v ) dt ( '' where v denotes the temperature of the walls of the inclosure, which was preserved constant during each experiment, v-{-6 the tempera- ture of the thermometer, and 1 -,~ the rate of cooling. The other letters, c and a, denote constants. When the temperatures are Centi- 432 RADIATION. grade, the constant a is 1*0077; when they are Fahrenheit it is 1*0043, the form of the expression for the rate of cooling being un- affected by a change of the zero from which temperatures are reck- oned. The value of c depends upon the size of the bulb and some other circumstances, and is changed by a change of zero. 463. Consequences of this Law. The formula in its first form shows that, for the same excess 0, the cooling is more rapid at high than at low temperatures. Employing the Centigrade scale, we have a= 1*0077, whence log a ='0077 nearly, and since a 6 = I + e log a + }(0 log a? + 1 (0 log a) 3 + &c., Dulong and Petit's formula, in its first form, gives -^=e(l-0077) t '{-0077tf+i(-00770) t + &c.}; which shows that, for a given temperature of the inclosure, the rate of cooling is not strictly proportional to 0, but is equal to multi- plied by a factor which increases with 0, this factor being proportional to l + K-0077 0) + i-(-o077 0) 2 + &c. When is small enough for *0077 to be neglected in comparison with unity, the factor will be sensibly constant, in accordance with Newton's law. 464. Theory of Exchanges. The second form of Dulong and Petit's formula, namely -=.+->. suggests that an unequal exchange of heat takes place between the thermometer and the walls, the thermometer giving to the walls a quantity of heat ca v+e (where v + d denotes the temperature of the thermometer), and the walls giving to the thermometer the smaller quantity cd. This is the view now commonly adopted with respect to radiation in general. It has been fully developed by Professor Balf our Stewart under the name of the theory of exchanges. Its original promulgator, Prevost of Geneva, called it the theory of mobile equilibrium of temperature. The theory asserts that all bodies are constantly giving out radiant heat, at a rate depending upon their substance and temperature, but independent of the substance or temperature of the bodies which surround them; and that when a body is kept at a uniform temper- ature, it receives back just as much heat as it gives out. LAW OF INVERSE SQUARES. 433 According to this view, two bodies at the same temperature, ex- posed to mutual radiation, exchange equal amounts of heat; but if two bodies have unequal temperatures, that which is at the higher temperature gives to the other more than it receives in exchange. 1 465. Law of Inverse Squares. If we take a delicate thermometer and place it at successively increasing distances from a source of heat, the temperature indicated by the instrument will exceed that of the atmosphere by decreasing amounts, showing that the intensity of radiant heat diminishes as the distance increases. The law of varia- tion may be discovered by experiment. In fact, when the excess of temperature of the thermometer becomes fixed, we know that the heat received is equal to that lost by radiation ; but this latter is, by Newton's law, proportional to the excess of temperature above that of the surrounding air; we may accordingly consider this excess as the measure of the heat received. It has been found, by experiments at different distances, 2 that the excess is inversely proportional to the square of the distance; we may therefore conclude that the intensity of the heat received from any source of heat varies inversely as the square of the distance. The following experiment, devised by Tyndall, supplies another simple proof of this fundamental law: The thermometer employed is a Melloni's pile, the nature of which we shall explain in 472. This is placed at the small end of a hollow cone, blackened inside, so as to prevent any reflection of heat from its inner surface. The pile is placed at S and S' in front of a vessel filled with boiling water, and coated with lamp-black on the side next the pile. It will now be observed that the temperature in- dicated by the pile remains constant for all distances. This result proves the law of inverse squares. For the arrangement adopted prevents the pile from receiving more heat than that due to the area of A B in the first case, and to the area A' B' in the second. These are the areas of two circles, whose radii are respectively proportional to S and S' O ; and the areas are consequently proportional to the squares of S O and S'C. Since, therefore, these two areas communi- 1 For a full account of this subject see "Report on the Theory of Exchanges," by Bal- four Stewart, in British Association Report, 1861, p. 97; and Stewart on Heat, book ii. chap. iii. 2 The dimensions of the source of heat must be small in comparison with the distance of the thermometer, as otherwise the distances of different parts of the source of heat from the thermometer are sensibly different. In this case, the amount of heat received varies directly as the solid angle subtended by the source of heat. 28 434 RADIATION. cate the same quantity of heat to the pile, the intensity of radiation must vary inversely as the squares of the distances S O and S'O. The law of inverse squares may also be established a priori in the following manner: Suppose a sphere of given radius to be described about a radiating particle as centre. The total heat emitted by the particle will be received by the sphere, and all points on the sphere will experience Fig. 203. Law of Inverse Squares. the same calorific effect. If now the radius of the sphere be doubled, the surface will be quadrupled, but the total amount of heat remains the same as before, namely, that emitted by the radiating particle. Hence we conclude that the quantity of heat absorbed by a given area on the surface of the large sphere is one-fourth of that absorbed by an equal area on the small sphere; which agrees with the law stated above. This demonstration is valid, whether we suppose the radiation of heat to consist in the emission of matter or in the emission of energy; for energy as well as matter is indestructible, and remains unaltered in amount during its propagation through space. 466. Law of the Reflection of Heat. When a ray of heat strikes a polished surface, it is reflected in a direction determined by fixed laws. If, at the point of incidence, that is, the point where the ray meets the surface, a line be drawn normal or perpendicular to the surface, the plane passing through this line and the incident ray is called the plane of incidence. With this explanation we proceed to give the laws of the reflection of heat: 1. When a ray of heat is reflected by a surface, the line of reflec- tion lies in the plane of incidence. BURNING MIRRORS. 435 2. The angle of reflection is equal to the angle of incidence; that is, the reflected and incident rays make equal angles with the normal to the surface at the point of incidence. 467. Burning-mirrors. These laws, which hold good for light also, , Fig. 294. Focus of Concave Mirror. can be verified by experiments with concave mirrors. These are usually either spherical or parabolic. All rays, either of heat or light, falling on a parabolic mirror in directions parallel to its axis (AC, Fig. 294) are reflected ac- curately to its focus F, and all rays from F falling on the mirror are reflected parallel to the axis. A spherical concave mirror is a small portion of a sphere, and rays parallel to its axis are re- flected so as approximately to pass through its "principal focus" F (same figure), which is midway between A, the central point of the mirror, and C, the centre of the sphere. When the axis of a concave mirror, of either form, is directed towards the sun, intense heat is produced at the focus, especially if the mirror be large. Fig. 295 represents such a mirror suitably Fig. 295. Burning Mirror, mounted for producing ignition of combustible substances. Tschirnhausen's mirror, which was con- structed in 1687, and was about 6i- feet in diameter, was able to 436 RADIATION. melt copper or silver, and to vitrify brick. Instead of curved mir- rors, Buffon employed a number of movable plane mirrors, which were arranged so that the different pencils of heat-rays reflected by them converged to nearly the same point. In this way he obtained an extremely powerful effect, and was able, for instance, to set wood on fire at a distance of between 80 and 90 yards. This is the method which Archimedes is said to have employed for the destruction of the Roman fleet in the siege of Syracuse; and though the truth of the story is considered doubtful, it is not altogether absurd. 468. Conjugate Mirrors. Fig. 296 represents an experiment which is said to have been first performed by Pictet of Geneva. Two large parabolic mirrors are placed facing each other, at any convenient distance, with their axes in the same straight line. In Fig. 296. Conjugate Mirrors. the focus of one of them is placed a small furnace, or a red-hot cannon-ball, and in the focus of the other some highly inflammable material, such as phosphorus or gun-cotton. On exciting the furnace with bellows, the substance in the other focus immediately takes fire. With two mirrors of 14 inches diameter, gun-cotton may thus be set on fire at a distance of more than 30 feet. The explanation is very easy. The rays of heat coming from the focus of the first mirror are reflected in parallel lines, and, on impinging upon the COEFFICIENT OF ABSORPTION. 437 surface of the second mirror, converge again to its focus, and are thus concentrated upon the inflammable material placed there. Careful adjustment is necessary to the success of the experiment, and the adjustment is most easily made by first placing a source of light (such as the flame of a candle) in one focus, and forming a luminous image of it in the other. We have thus a convincing proof that heat and light obey the same law as regards direction of reflection. 469. Reflection, Diffusion, Absorption, and Transmission. Sup- pose a quantity of heat denoted by unity to be incident upon the surface of a body. This quantity will be divided into several distinct parts. 1. A portion will be regularly reflected according to the law given above. If the fraction of heat thus reflected be denoted by -, then is the measure of the reflecting power of the surface. 2. A portion -^ will be irregularly reflected, and will be scattered or diffused through space in all directions. Thus ^ is the measure of the diffusive power of the surface. 3. A portion -- will penetrate into the body so as to be absorbed by it, and to contribute to raise its temperature; is therefore the measure of the absorption. 4. Finally, we shall have, in many cases, a fourth portion -g , which passes through the body without contributing to raise its tempera- ture. This fraction, which exists only in the case of diathermanous bodies, is the measure of the transmission. The sum of these fractional parts must evidently make up the original unit; that is The amount of the transmission, where it exists, will generally vary with the thickness of the substance ; and what is lost in transmission by increasing the thickness is gained in absorption. When there is no transmission, the absorption may be called the absorbing power of the surface. 470. Coefficient of Absorption and Coefficient of Emission. Apply- ing Newton's law ( 461), let be the small difference of temperature between the surface of the body and the inclosure, and S the area 438 RADIATION. of this surface, which we suppose to have no concavities, then the quantity of heat gained or lost by the body per unit of time is ex- pressed by the formula AS 6, where A is a constant depending on the nature of the body and more especially on the nature of its surface. This constant A may be called indifferently the coefficient of emission or the coefficient of absorption, inasmuch as it has the same value (the temperature of the body being given) whether the inclosure be colder or warmer than the body. Experiments conducted by Mr. M'Farlane under the direction of Sir W. Thomson, have shown that when the surface of the body (a copper ball) and the walls of the inclosure are both covered with lamp-black, the inclosure being full of air at atmo- spheric pressure, the value of the coefficient A in C.G.S. units is about xtrtnrj that is to say ^-jrrnr of a gramme degree of heat is gained or lost per second for each square centimetre of surface of the body, when there is 1 of difference between its temperature and that of the walls of the inclosure. When the surface of the body (the copper ball) was polished, the walls of the inclosure being blackened as before, the coefficient had only T V of its former value. It is estimated that of the value -JTRTO f r blackened surfaces, one-half is due to atmospheric contact and the other half to radiation. As the excess of temperature of the body above that of the walls increased from 5 to 60, the quantity of heat emitted, instead of being in- creased only twelve-fold, was increased about sixteen-fold for the blackened and fifteen-fold for the polished ball. When air is excluded, and the gain or loss of heat is due to pure radiation between the body and the walls, the coefficient A repre- sents, according to the theory of exchanges, the difference between the absolute emission at the temperature of the body and at a tem- perature 1 higher or lower. 471. Limit to Radiating Power. It is obviously impossible for a body to absorb more radiant heat than falls upon it. There must ? therefore, be a limiting value of A applicable to a body whose absorbing power - is unity, and such a body must also be regarded as possessing perfect emissive power for radiant heat. Hence it appears that good radiation depends rather upon defect of resistance than upon any positive power. A perfect radiator would be a sub- stance whose surface offered no resistance to the passage of radiant PERFECT RADIATOR. 439 heat in either direction; while an imperfect radiator is one whose surface allows a portion to be communicated through it, and reflects another portion regularly or irregularly. The reflecting and diffusive powers of lamp-black are so insigni- ficant, at temperatures below 100, that this substance is commonly adopted as the type of a perfect radiator, and the emissive and ab- sorptive powers of other substances are usually expressed by com- parison with it. CHAPTER XXXVII. EADIATION (CONTINUED). 473. Thermoscopic Apparatus employed in researches connected with Radiant Heat. An indispensable requisite for the successful study of radiant heat is an exceedingly delicate thermometer. For this purpose Leslie, about the beginning of the present century, invented the differential thermometer, with which he conducted some very important investigations, the main results of which are still acknowledged to be correct. Modern investigators, as Melloni, Laprovostaye, &c., have exclusively employed Nobili's thermo-multi- plier, which is an instrument of much greater delicacy than the differential thermometer. The thermo-pile, invented by Nobili, and improved by Melloni, consists essentially of a chain (Fig. 297) formed of alternate elements of bismuth and antimony. If the ends of the chain be connected by a wire, and the alternate joints slightly heated, a thermo-electric current will be produced, as will be Fig. 2or. Nobiii's Thermo-electric Series. explained hereafter. The amount of current increases with the num- ber of elements, and with the difference of temperatures of the oppo- site junctions. In the pile as improved by Melloni, the elements are arranged side by side so as to form a square bundle (Fig. 298), whose opposite ends consist of the alternate junctions. The whole is contained in a copper case, with covers at the two ends, which can be removed when it is desired to expose the faces of the pile to the action of heat. Two metallic rods connect the terminals of the thermo-electric series THERMO-MULTIPLIER. 441 with wires leading to a galvanometer, 1 so that the existence of any current will immediately be indicated by the deflection of the needle. The amounts of current which correspond to different deflections are known from a table compiled by a method which we shall explain hereafter. Consequently, when a beam of radiant heat strikes the pile, an electric current is produced, and the amount of this current Fig. 203. Melloni's Thermo-multiplier. is given by the galvanometer. We shall see hereafter, when we come to treat of thermo-electric currents, that within certain limits, which are never exceeded in investigations upon radiant heat, the current is proportional to the difference of temperature between the two ends of the pile. As soon as all parts of the pile have acquired their permanent temperatures, the quantity of heat received during any interval of time from the source of heat will be equal to that lost to the air and surrounding objects. But this latter is, by New- ton's law, proportional to the excess of temperature above the sur- rounding air, and therefore to the difference of temperature between the two ends of the pile. The current is therefore proportional to the quantity of heat received by the instrument. We have thus in Nobili's pile a thermometer of great delicacy, and admirably adapted 1 The pile and galvanometer together constitute the thermo-multiplier. 442 HADIATION. to the study of radiant heat; in fact, the immense progress which has been made in this department of physics is mainly owing to this invention of Nobili. 473. Measurement of Emissive Powers. The following arrangement was adopted by Melloni for the comparison of emissive powers. A graduated horizontal bar (Fig. 299) carries a cube, the different sides of which are covered with different substances. This is filled with water, which is maintained at the boiling-point by means of a spirit- lamp placed beneath. The pile is placed at a convenient distance, Fig. 299. Measurement of Emissive Powers. and the radiation can be intercepted at pleasure by screens arranged for the purpose. The whole forms what is called Melloni's apparatus. If we now subject the pile to the heat radiated from each of the faces in turn, we shall obtain currents proportional to the emissive powers of the substances with which the different faces are coated. From a number of experiments of this kind it has been found that lamp-black has the greatest radiating power of all known substances, while the metals are the worst radiators. Some of the most impor- tant results are given in the following table, in which the emissive powers of the several substances are compared with that of lamp- black, which is denoted by 100: RELATIVE EMISSIVE POWERS AT 100 C. Lamp-black, 100 White-lead, 100 Paper 98 Glass 90 Indian ink, 85 Shellac 72 Steel 17 Platinum, 17 Polished brass, 7 Copper, 7 Polished gold, ..... 3 Polished silver, 3 ABSORBING POWERS. 443 474. Absorbing Power. The method which most naturally suggests itself for comparing absorbing powers, is to apply coatings of different substances to that face of the pile which is exposed to the action of the source of heat. But this would involve great risk of injury to the pile. The method employed by Melloni was as follows: He placed in front of the pile a very thin copper disc (Fig. 300), coated with lamp- black on the side next the pile, and on the other side with the sub- stance whose absorbing power was required. The disc absorbed heat Fig. 300. Measurement of Absorbing Powers. by radiation from the source, of amount proportional to the absorb- ing power of this coating, and at the same time emitted heat from both sides in all directions. When its temperature became stationary, the amounts of heat absorbed and emitted were necessarily equal, and its two faces had sensibly equal temperatures. Let E and E' denote the coefficients of emission of lamp-black and of the substance with which the front was coated, and the excess of temperature of the disc above that of the air; then (E + E')0 is the heat emitted in unit time, if the area of each face is unity, and this must be equal to the heat absorbed in unit time. But the indications of the thermo-pile are proportional to the heat radiated from the back alone, that is, to E0. The heat absorbed is therefore represented by the indication of the pile multiplied by E-KE' E ' In this way the absorbing powers given in the following list have been calculated from experiments of Melloni, the source of heat being a cube filled with water at 100 C. 444 RADIATION. RKLATIVE ABSORBING POWERS AT 100 C. Lamp-black 100 White-lead, 100 Isinglass, 91 Indian ink, 85 Shellac 72 Metal, 13 It will be observed that these numbers are identical with those which represent the emissive powers of the same substances. 475. Variation of Absorption with the Source. The absorbing power varies according to the source of heat employed. In estab- lishing this important fact, Melloni employed the following sources of heat: 1. Locatelli's lamp, a small kind of oil-lamp, in which the level of the oil remains invariable, and which has a square-cut solid wick. As a source of heat it is of tolerably constant action, and it has been employed in most of the experiments upon diathermancy. It is shown in Fig. 300. 2. Incandescent platinum. This is a spiral of platinum wire (Fig. 301) suspended over a spirit-lamp so as to envelop the flame. The Fig. 301. Incandescent Platinum. Fig. 302. Copper heated to 400' . metal is heated to a bright white heat; and since the radiating powers of the flame are very feeble, the metal may be regarded as the sole source of radiation. The flame, in fact, is scarcely distin- guishable. 3. Copper heated to about 400 C. This is effected by placing a spirit-lamp behind a curved copper plate (Fig. 302). 4. Copper covered with lamp-black at 100 C. This is a cube con- REFLECTING POWERS. 445 taining boiling water (Fig. 303) similar to that already described in connection with the measurement of emissive powers. The face whose radiation is employed is covered with lamp-black. If these different sources of heat be severally used in measuring absorbing powers, it will be found that these powers vary consider- ably according to the particular source of heat employed, and that if we denote the absorption of lamp-black in each case by 100, the relative absorbing powers of other substances are in general greater as the temperature of the source is lower. In establishing this important principle by experiment, the sources of heat are first placed at such distances that the direct radiation upon the pile shall be the same for each, and the pile is then replaced by the disc. The following table contains some of the results obtained by Melloni: SUBSTANCES. Locatelli's Lamp. Incandescent Platinum. Heated Copper. Hot-water Cube. Lamp-black, 100 100 100 100 Indian ink, 96 95 87 85 White-lead, 53 56 89 100 Isinglass, . 52 54 64 91 Shellac, . . 48 47 70 72 Metallic surface, 14 13-5 13 13 476. Reflecting Power. The reflecting power of a surface is mea- sured by the proportion of incident heat which is regularly reflected from it. This subject has been investigated by Melloni, and by Laprovostaye and Desains. The arrangement used for the purpose is shown in Fig. 304. The substance under investigation is placed upon the circular plate D, which is graduated round the circumference. The pile E is carried by the horizontal bar HH', which turns about the pillar supporting the plate D. This bar is to be so adjusted as to make the reflected rays impinge upon the pile, the adjustment being made by the help of the divisions marked on the circular plate. In making an observation, the bar HH' is first placed so as to coincide with the prolongation of the principal bar, and the intensity of direct radiation is thus observed. The pile is then placed so as to receive the reflected rays, and the ratio of the intensity thus obtained to the intensity of direct radiation is the measure of the reflecting power. 446 RADIATION. The following are some of the results obtained by Laprovostaye and Desains. the source of heat employed being a Locatelli lamp : Reflecting Power. Silver plate, '97 Gold, -95 Brass, '93 Speculum metal, .... '86 Tin, . -85 Reflecting Power. Polished platinum, . . . '80 Steel, -83 'Zinc, -81 Iron, -77 Laprovostaye and Desains have also shown that, in the case of diathermanous substances, the reflecting power varies considerably. . 304. Measurement of Reflecting Power. increasing with the angle of incidence, which is also the case for luminous rays. In the case of metals, the change in the reflecting power produced by a change in the angle of incidence is not nearly so great; the reflecting power remains almost constant till about 70 or 80, and when the angle of incidence exceeds this limit, the reflecting power decreases, whereas the opposite is the case with diathermanous bodies. Finally, Laprovostaye and Desains have shown that, contrary to what was previously supposed, the reflecting power varies according to the source of heat. Thus the reflecting power of polished silver, which is '97 for rays from a Locatelli lamp, is only '92 for solar rays. In either case it will be seen that the reflecting powers of polished silver are very great; and since experiment has shown that luminous and calorific rays from the same source are reflected in nearly equal DIFFUSIVE REFLECTION. 447 proportions, the advantages attending the use of silvered specula in telescopes can easily be understood. 477. Diffusive Power. 1 Diffusion is the irregular reflection of heat, doubtless owing to the minute inequalities of surface which are met with on even the most finely-polished bodies. The existence of this power may very easily be verified. We have only to let a beam of radiant heat fall upon any dead surface, for example on carbonate of lead. On placing the pile before the surface in any position, a deviation of the galvanometer is observed, which cannot be attri- buted to radiation from the surface, since in that case the effect, instead of instantly attaining its maximum, as it actually does, would increase gradually as the substance became warmed by the heat falling upon it. Moreover the heat thus diffused, when the source of heat is a body at high temperature, such as a lamp-flam e,, is found to agree in its properties with the heat radiated from a body at high temperature, and to be altogether different from that which the diffusing surface is capable of radiating at its actual temperature. The diffused heat, for example, passes through a plate of alum without undergoing much absorption. The diffusive power of powders, especially if white, is very con- siderable, as is shown by the following table taken from the results published bv Laprovostaye and Desains: DIFFUSIVE POWER. White-lead, '82 Powdered silver, '76 Chromate of lead, "66 The knowledge of this property enables us to explain the intense heat which is felt in the neighbourhood of a white wall lighted up by the sun. Diffusion takes place in different proportions according to the direction, and is a maximum for points near the direction of the regularly-reflected ray. The intensity of the diffused rays varies very considerably accord- ing to the source of heat employed. This was shown by Melloni in the following manner: He directed a ray of heat upon the surface of a disc of very thin copper covered with a substance capable of diffusing the rays. The 1 There is no connection whatever between this "diffusive power" and the "diffusivity " which we have discussed in the chapter on Conduction. 448 RADIATION. back of the disc was coated with lamp-black. When the different parts had acquired their permanent temperatures, the pile was placed in symmetrical positions first in front of, and then behind the plane of the disc, so as to receive the heat due to radiation and diffusion from the front in the first case, and that due to radiation from the back in the second. It was found that the ratio of the two indica- tions of the pile in these two positions varied very much according to the source of heat, the general rule being that the ratio of the diffused to the radiated heat was greatest when the source of heat was luminous, and at a high temperature. 478. Peculiar Property of Lamp-black. If a similiar experiment be performed with a disc covered on both sides with lamp-black, it will be found that the difference between the indications of the pile in the two positions is very small. This difference, such as it is, may be accounted for by a slight difference of temperature between the two faces of the disc. We may therefore conclude that the whole of the heat has been absorbed by the lamp-black. This important result has been confirmed by direct experiments, which have failed to discover any trace of reflecting or diffusive power in this substance. Further, in the above experiment, the ratio of the indications in the two positions of the pile remains constant for all sources of heat; whence we see that the absorption of rays of heat by lamp-black is independent of the nature of the source. We thus see the advantage of applying a coating of lamp-black to all thermoscopic apparatus intended for the absorption of radiant heat. 479. Diathermancy. It has long been known that some of the heat from an intensely luminous body, like the sun, could pass through certain transparent substances, such as glass; but it was formerly supposed that this could not happen in the case of dark, or even feebly luminous rays. Pictet, of Geneva, was the first to establish the fact of diathermancy for radiant heat in general. He showed that a thermometer rose in temperature when exposed to radiation from a source of heat, not- withstanding the interposition of a transparent lamina; and the idea that this could be owing to the absorption and subsequent radiation of heat from the lamina was completely exploded by PreVost, who showed that the effect occurred even when the interposed substance was a sheet of ice. It is to Melloni, however, that we are indebted for the principal results which have been obtained in connection with this subject. DIATHERMANCY. 449 480. Influence of the Nature of the Substance. The arrangement adopted by Melloni for testing the diathermancy of a solid body is that shown in Fig. 305. The Locatelli lamp A radiates its heat upon the pile E when the screen B is lowered ; the hole in the screen C is for the purpose of limiting the pencil of rays. Direct radiation is first allowed to take place, and the resulting current as indicated by the galvanometer G is noted. The diathermanous plate D is then interposed between the lamp and the pile, and the current is again Fig. 305. Measurement of Diathermancy. measured; the ratio of the latter current to the former is the expres- sion of the diathermancy of the plate. In the case of liquids, Melloni employed narrow troughs with sides of very thin glass ; the rays were first transmitted through the empty vessel, and then through the same vessel filled with liquid; the dif- ference of the two results thus obtained being the measure of the heat stopped by the liquid. Specimens of the results are given in the following table: HEAT TRANSMITTED BY DIFFERENT SUBSTANCES FROM AN ARGAND LAMP. (The direct heat is represented by 100.) CRYSTALLIZED BODIES. (Thickness 3 '62 mm. A plate of glass of the same thickness gives C2.) Colourless. Rock-salt 92 Iceland-spar, 12 Rock-crystal, 57 Brazilian topaz, 54 Carbonate of lead, 52 Borate of soda 28 Sulphate of lime, .,,,,,, 20 Citric acid, 15 IlDck alum, 12 Coloured. Smoky quartz (brown), 57 Aqua-marina (light blue), .... 29 Yellow agate 29 Green tourmaline, 27 Sulphate of copper (blue), .... C 29 450 RADIATION. SOLIDS. Colourless Glass. (Thickness 1 '88 mm.) Flint-glass from 67 to 64 Plate-glass, 62 to 59 Crown-glass (French), 58 Crown-glass (English), . ... 49 Window-glass, 54 to 50 Coloured Glass. (Thickness 1 85 mm.) Deep violet, . . 53 Pale violet, 45 Very deep blue, 19 Deep blue, 33 Light blue, 42 Mineral green, 23 Apple green, 26 Deep yellow, 40 Orange, . . 44 Yellowish red, 53 Crimson, . 51 LIQUIDS. (Thickness 9'21 mm. A plate of glass of the same thickness gives 53.) Colourless Liquids. Distilled water, 11 Absolute alcohol, 15 Sulphuric ether, 21 Sulphide of carbon, 63 Spirits of turpentine, 31 Pure sulphuric acid, 17 Pure nitric acid, 15 Solution of sea-salt, 12 Solution of alum, 12 Solution of sugar, .12 Solution of potash, 13 Solution of ammonia, 15 Coloured Liquids. Nut-oil (yellow) 31 Colza-oil (yellow) 30 Olive-oil (greenish yellow), .... 30 Oil of carnations (yellowish), ... 26 Chloride of sulphur (re Idish brown), . 63 Pyroligneous acid (brown), . . . . 12 White of egg (slightly yellow), ... 11 It will be seen from this table that though diathermancy and transparency for light usually go together, the one is far from being a measure of the other. We see, for instance, that colourless nitric acid is much less diathermanous than strongly-coloured chloride of sulphur; and perfectly colourless alum allows much less heat to pass than deeply-coloured glass of the same thickness. Tyndall has shown that a solution of iodine in sulphide of carbon, though excessively opaque to light, allows heat to pass in large quantity. The substance possessing the greatest diathermanous power is rock- salt, which allows the passage of '92 of the incident heat. The diathermancy of gases has been investigated by Tyndall. The gases were contained in a long metallic tube with rock-salt ends; and, in order to obtain greater sensitiveness, a compensating cube filled with hot water was employed. This cube was placed at such a distance from one end of the thermo-pile as exactly to counter- balance the effect of the radiation from the principal source of heat when the tube was vacuous, so that the needle of the galvanometer in these circumstances stood at zero. The tube was then filled with different gases in turn, the compensating cube remaining unmoved ; and the indications of the galvanometer were found to vary accord- ing to the gas employed. Compound gases stopped more than simple RADIANT HEAT AND LIGHT. 4<51 ones; the vapours of aromatic substances increased the absorptive power of dry air from 30 to 300 fold, and a similar effect was pro- duced by the vapour of water, air more or less charged with aqueous vapour being found to exercise from 30 to 70 times the absorption of pure dry air. It is probable that the aqueous vapour which is always present in the atmosphere greatly mitigates the heat of the solar rays, and also greatly retards the cooling of the earth by radiation at night. On the other hand, vapour being a better absorber is also a better radiator than dry air, a circumstance which conduces to the cooling and condensation of the upper portions of masses of vapour in the atmosphere, and the consequent formation of cloud. 481. Influence of Thickness. From the experiments of Jamin and Masson, it appears that, when heat of definite ref rangibility passes through a plate, the amount transmitted decreases in geometrical progression as the thickness increases in arithmetical progression; a result which may also be expressed by saying, that if a plate be divided in imagination into laminae of equal thickness, the ratio of the heat absorbed to the heat transmitted is the same for them all. In the case of mixed radiation, such as is emitted by nearly all available sources of heat, we must suppose this law to hold for each separate constituent; but some of these are more easily absorbed than others, and as these accordingly diminish in amount more rapidly than the others, the beam as it proceeds on its way through the plate acquires a character which fits it for transmission rather than absorp- tion. Hence the foremost layers absorb much more than the later ones, if the plate be of considerable thickness. In the case of bodies which are opaque to heat, absorption and radiation are mere surface-actions. But in diathermanous substances, as we have seen, absorption goes on in the interior, so that a thick plate absorbs more heat than a thin one. The same thing is true as regards radiation: a diathermanous substance radiates from its interior as well as from its surface, as proved by the fact that a thick plate radiates more heat than a thin one at the same temperature. 482. Relation between Radiant Heat and Light. The property in virtue of which particular substances select particular kinds of heat for absorption and other kinds for transmission, was called by Melloni thermochrose (literally heat-colour), from its obvious analogy to what we call colour in the case of light. A piece of coloured glass, for example, selects rays of certain wave-lengths for absorption, and 452 RADIATION. transmits the rest; what we call the colour of the glass being deter- mined by those which it transmits. It is now believed that ther- mochrose and colour are not merely analogous but essentially identical. Prismatic analysis shows that rays exist of refrangibilities much greater and much less than those which compose the luminous spec- trum. The spectrum of the electric light, for example, extends on both sides of the visible spectrum to distances considerably exceed- ing the length of the visible spectrum itself. The invisible ultra- violet rays can be detected by their chemical action, or by causing them to fall upon certain substances (called fluorescent) which become luminous when exposed to their action, but have exceedingly small heating effect. The heat becomes considerable in the yellow portion of the spectrum, stronger in the red, and goes on increasing in the invisible portion beyond the red, up to a certain point, beyond which it gradually diminishes till it becomes inappreciable. It would, however, be an error to suppose that there is a heat spectrum consisting of distinct rays from those which form the lumin- ous spectrum, and that the two spectra are superimposed one upon the other. There is every reason for believing that the contrary is the fact, and that the radiations which constitute heat and light are essentially identical. In operating upon rays of definite refrangi- bility, it is never found possible to diminish their heating and illu- minating powers in unequal proportions; an interposed plate of any partially transparent material, if it stops half the light, also stops half the heat. It is true that the most intense heat is not found in the most luminous portion of the spectrum; but it is probable that the eye, like the ear, is more powerfully affected by quick than by slow vibrations when the amount of energy is the same; and as a treble note contains far less energy than a bass note which strikes the ear as equally loud, so a blue ray contains much less energy than a red ray if they strike the eye as equally bright. The invisibility at least to human eyes of the ultra-red and ultra-violet rays may be due either to the absorption of these rays by the humours of the eye before they can reach the retina, or to the inability of our visual organs to take up vibrations quicker than the violet or slower than the red. A body at a low temperature (say 100 C.) emits only dark heat. As the temperature rises, the emission of dark heat becomes more RADIATION AT DIFFERENT TEMPERATURES. 453 energetic, and at the same time rays of a more refrangible character are added. This strengthening of the rays formerly emitted, with the continual addition of new rays of higher refrangibility, goes on as long as the temperature of the body continues to rise. The lumi- nosity of the body begins with the emission of the least refrangible of the visible rays, namely the red, and goes on to include rays of other colours as it passes from a red to a white heat. Tyndall, by thus gradually raising the temperature of a platinum spiral, obtained the following measures of the heat received in a definite position in the dark portion of the spectrum: Appearance of Spiral. Heat Received. Dark, 1 Dark, 6 Faint red, 10 Dull red, 13 Red, 18 Appearance off " Heat Received Spiral. Full red, ' . 27 Orange, 60 Yellow, 93 Full white, 122 Generally speaking, the rays which fall within the limits of the visible spectrum are the most transmissible, and the extreme rays at both ends of the complete spectrum are the soonest absorbed. This is probably the reason why the invisible portion of the solar spec- trum, though extending to a considerable distance in both directions, is less extensive than that of the electric light. The extreme rays have probably been absorbed by the earth's atmosphere. Ordinary glass is comparatively opaque to both classes of dark rays. Rock -salt surpasses all other substances in its transparency to the dark rays beyond the red; and quartz (rock-crystal) is very- transparent to the dark rays beyond the violet. Alum is remarkable as a substance which is exceedingly opaque to the ultra-red rays, though exceedingly transparent to visible rays; and Tyndall has found that a solution of iodine in sulphide of carbon is, on the con- trary, highly transparent to the ultra-red and opaque to the luminous rays. Great interest was excited some years ago by Stokes' discovery that the ultra-violet rays, when they fall upon fluorescent substances, undergo a lowering of refrangibility which brings them within the limits of human vision. Akin subsequently proposed the inquiry whether it was possible, by a converse change, to transform the ultra-red into visible rays, and Tyndall, by taking advantage of this peculiar property of the solution of iodine, succeeded in effecting the transformation. He brought the rays of the electric lamp to a focus 454 EADIATION. by means of a reflector, and, after stopping all the luminous rays by interposing a vessel with rock-salt sides, containing the solution of iodine, he found that a piece of platinum foil, when brought into the focus, was heated to incandescence, and thus emitted light as well as heat. To this transformation of dark radiant heat into light he gave the name of calorescence. 483. Selective Emission and Absorption. In order to connect together the various phenomena which may be classed under the general title of selective radiation and absorption, it is necessary to ' form some such hypothesis as the following. The atoms or mole- cules of which any particular substance is composed, must be sup- posed to be capable of vibrating freely in certain periods, which, in the case of gases, are sharply defined, so that a gas is like a musical string, which will vibrate in unison with certain definite notes and with no intermediate ones. The particles of a solid or liquid, on the other hand, are capable of executing vibrations of any period lying between certain limits; so that they may perhaps be compared to the body of a violin, or to the sounding-board of a piano; and these limits (or at all events the upper limit) alter with the temperature, so as to include shorter periods of vibration as the temperature rises. These vibrations of the particles of a body are capable of being excited by vibrations of like period in the external ether, in which case the body absorbs radiant heat. But they may also be excited by the internal heat of the body; for whenever the molecules expe- rience violent shocks, which excite tremors in them, these are the vibrations which they tend to assume. In this case the particles of the body excite vibrations of like period in the surrounding ether, and the body is said to emit radiant heat. One consequence of these principles is that a diathermanous body is particularly opaque to its own radiation. Rock-salt transmits 92 per cent, of the radiation from most sources of heat; but if the source of heat be another piece of rock-salt, especially if it be a thin plate, the amount transmitted is much less, a considerable proportion being absorbed. The heat emitted and absorbed by rock-salt is of exceedingly low refrangibility. Glass largely absorbs heat of long period, such as is emitted by bodies whose temperatures are not sufficiently high to render them luminous, but allows rays of shorter period, such as compose the luminous portion of the radiation from a lamp-flame, to pass almost EQUALITY OF EMISSION AND ABSORPTION. 455 entire. Accordingly glass when heated emits a copious radiation of non-luminous heat, but comparatively little light. Experiment shows that if various bodies, whether opaque or trans- parent, colourless or coloured, are heated to incandescence in the interior of a furnace, or of an ordinary coal-fire, they will all, while in the furnace, exhibit the same tint, namely the tint of the glowing coals. In the case of coloured transparent bodies, this implies that the rays which their colour prevents them from transmitting from the coals behind them are radiated by the bodies themselves most copiously; for example, a glass coloured red by oxide of copper per- mits only red rays to pass through it, absorbing all the rest, but it does not show its colour in the furnace, because its own heat causes it to radiate just those rays which it has the power of absorbing, so that the total radiation which it sends to the eye of a spectator, con- sisting partly of the radiation due to its own heat, and partly of rays which it transmits from the glowing fuel behind it, is exactly the same in kind and amount as that which comes direct from the other parts of the fire. This explanation is verified by the fact that such glass, if heated to a high temperature in a dark room, glows with a green light. A plate of tourmaline cut parallel to the axis has the property of breaking up the rays of heat and light which fall upon it into two equal parts, which exhibit opposite properties as regards polarization. One of these portions is very largely absorbed, while the other is transmitted almost entire. When such a plate is heated to incan- descence, it is found to radiate just that description of heat and light which it previously absorbed; and if it is heated in a furnace, no traces of polarization can be detected in the light which comes from it, because the transmitted and emitted light exactly complement each other, and thus compose ordinary or unpolarized light. Spectrum analysis as applied to gases furnishes perhaps still more striking illustrations of the equality of selective radiation and absorp- tion. The radiation from a flame coloured by vapour of sodium for example, the flame of a spirit-lamp with common salt sprinkled on the wick consists mainly of vibrations of a definite period, corre- sponding to a particular shade of yellow. When vapour of sodium is interposed between the eye and a bright light yielding a continuous spectrum, it stops that portion of the light which corresponds to this particular period, and thus produces a dark line in the yellow portion of the spectrum. 456 RADIATION. An immense number of dark lines exist in the spectrum of tne sun's light, and no doubt is now entertained that they indicate the presence, in the outer and less luminous portion of the sun's atmo- sphere, of gaseous substances which vibrate in periods corresponding to the position of these lines in the spectrum. CHAPTER XXXVIII. THERMODYNAMICS. 484. Connection between Heat and Work. That heat can be made to produce work is evident when we consider that the work done by steam-engines and other heat-engines is due to this source. Conversely, by means of work we can produce heat. Fig. 306 represents an apparatus called the fire-syringe or pneumatic tinder-box, consisting of a piston working tightly in a glass barrel. If a piece of cotton wool moist- ened with bisulphide of carbon be fixed in the cavity of the piston, and the air be then suddenly compressed, so much heat will be developed as to produce a visible flash of light. A singular explanation of this effect was at one time put forward. It was maintained that heat or caloric was a kind of imponderable fluid, which, when introduced into a body, produced at once an increase of volume and an eleva- tion of temperature. If, then, the body was compressed, the caloric which had served to dilate it was, so to speak, squeezed out, 1 and hence the development of heat. An immediate consequence of this theory is that heat cannot be increased or diminished in quantity, but that any addi- tion to the quantity of heat in one part of a system must be compensated by a corresponding loss in another part. But we know that there are cases in which heat is pro- duced by two bodies in contact, without our being able to observe any traces of this compensating process. An Fj 3C instance of this is the production of heat by friction. Fire-syringe. 1 In other words, the thermal capacity of the body was supposed to be diminished, so that the amount of heat contained in it, without undergoing any increase, was able to raise it to a higher temperature. 458 THERMODYNAMICS. 485. Heat produced by Friction. Friction is a well-known source of heat. Savages are said to obtain fire by rubbing two pieces of dry wood together. The friction between the wheel and axle in railway-carriages frequently produces the same effect, when they have been insufficiently greased; and the stoppage of a train by applying a brake to the wheels usually produces a shower of sparks. The production of heat by friction may be readily exemplified by the following experiment, due to Tyndall. A glass tube containing water (Fig. 307), and closed by a cork, can be rotated rapidly about Fig. 307. Heat produced by Friction. its axis. While thus rotating, it is pressed by two pieces of wood, covered with leather. The water is gradually warmed, and finally enters into ebullition, when the cork is driven out, followed by a jet of steam. Friction, then, may produce an intense heating of the bodies rubbed together, without any corresponding loss of heat else- where. At the close of last century, Count Kumford (an American in the service of the Bavarian government) called attention to the enormous amount of heat generated in the boring of cannon, and found, in a special experiment, that a cylinder of gun-metal was raised from the temperature of 60 F. to that of 130 F. by the friction of a blunt steel borer, during the abrasion of a weight of metal equal to about ^ of the whole mass of the cylinder. In another experiment, he sur- rounded the gun by water (which was prevented from entering the CALORIC THEOEY. 459 bore), and, by continuing the operation of boring for 2| hours, he made this water boil. In reasoning from these experiments, he strenuously maintained that heat cannot be a material substance, but must consist in motion. The advocates of the caloric theory endeavoured to account for these effects by asserting that caloric, which was latent in the metal when united in one solid mass, had been forced out and rendered sensible by the process of disintegration under heavy pressure. This supposition was entirely gratuitous, no difference having ever been detected between the thermal properties of entire and of comminuted metal; and, to account for the observed effect, the latent heat thus supposed to be rendered sensible in the abrasion of a given weight of metal, must be sufficient to raise 950 x 70, that is 66,500 times its own weight of metal through 1. Yet, strange to say, the caloric theory survived this exposure of its weakness, and the, if possible, still more conclusive experiment of Sir Humphry Davy, who showed that two pieces of ice, when rubbed together, were converted into water, a change which involves not the evolution but the absorption of latent heat, and which cannot be explained by diminution of thermal capacity, since the specific heat of water is much greater than that of ice. Davy, like Rumford, maintained that heat consisted in motion, and the same view was maintained by Dr. Thos. Young; but the doctrine of caloric nevertheless continued to be generally adopted until about the year 1840, since which time, the experiments of Joule, the eloquent advocacy of Mayer, and the mathematical deductions of Thomson, Rankine, and Clausius, have completely established the mechanical theory of heat, and built up an accurate science of thermo- dynamics. 486. Foucault's Experiment. The relations existing between elec- trical and thermal phenomena had considerable influence in leading to correct views regarding the nature of heat. An experiment de% ised by Foucault illustrates these relations, and at the same time furnishes a fresh example of the production of heat by the performance of mechanical work. The apparatus consists (Fig. 308) of a copper disc which can be made to rotate with great rapidity by means of a system of toothed wheels. The motion is so free that a very slight force is sufficient to maintain it. The disc rotates between two pieces of iron, constituting the armatures of one of those temporary magnets which are obtained 460 THERMODYNAMICS. by the passage of an electric current (called electro-magnets). If, while the disc is turning, the current is made to pass, the armatures become strongly magnetized, and a peculiar action takes place between them and the disc, consisting in the formation of induced currents in the latter, accompanied by a resistance to motion. As Fig. 308. Foucault's Apparatus. long as the magnetization is continued, a considerable effort is necessary to maintain the rotation of the disc; and if the rotation be continued for two or three minutes, the disc will be found to have risen some 50 or 60 C. in temperature, the heat thus acquired by the disc being the equivalent of the work done in maintaining the motion. It is to be understood that, in this experiment, the rotating disc does not touch the armatures; the resistance which it experiences is due entirely to invisible agencies. JOULES EXPERIMENT. 461 The experiment may be varied by setting the disc in very rapid rotation, while no current is passing, then leaving it to itself, and immediately afterwards causing the current to pass. The result will be, that the disc will be brought to rest almost instantaneously, and will undergo a very slight elevation of temperature, the heat gained being the equivalent of the motion which is destroyed. 487. Mechanical Equivalent of Heat. The first precise determina- tion of the numerical relation subsisting between heat and mechani- cal work was obtained by the following experiment of Joule. He constructed an agitator which is somewhat imperfectly represented in Fig. 309, consisting of a vertical shaft carrying several sets of paddles revolving between stationary vanes, these latter serving to Fig. 309. Determination of the Mechanical Equivalent of Heat. prevent the liquid in the vessel from being bodily whirled in the direction of rotation. The vessel was filled with water, and the agitator was made to revolve by means of a cord, wound round the upper part of the shaft, carried over a pulley, and attached to a weight, which by its descent drove the agitator, and furnished a measure of the work done. The pulley was mounted on friction- wheels, and the weight could be wound up without moving the paddles. When all corrections had been applied, it was found that the heat communicated to the water by the agitation amounted to one pound-degree Fahrenheit for every 772 foot-pounds of work spent in producing it. This result was verified by various other forms of experiment, and may be assumed to be correct within about 462 THERMODYNAMICS. one foot-^ound. The experiments were made at Manchester, where g is 32'194, and it is to be borne in mind that a foot-pound does not denote precisely the same amount of work at all places on the earth's surface, but varies in direct proportion to the intensity of gravity. The difference in its value in passing from one place to another on the earth is, however, not greater than the probable error of the number 772. We may therefore, with about as much accuracy as is warranted by the present state of our knowledge, assert that the energy comprised in one-pound degree Fahrenheit is about 772 ter- restrial foot-pounds. 1 The mechanical equivalent of the pound-degree Centigrade is -? of this, or about 1390 foot-pounds. The number 772 or 1390, according to the scale of temperature adopted, is commonly called Joules equivalent, and is denoted in formulae by the letter J. If we take the kilogramme-degree Centi- grade for unit of heat, and the kilogrammetre for unit of work, the value of J will be 424, and the same value will be given by the gramme-degree and gramme-metre. The gramme-degree and gramme-centimetre will give 42,400, and the gramme-degree and erg will give the product of this number by 981, which is 41 '6 millions. This is accordingly the value of J in the C.G.S. system. 488. First Law of Thermo-dynamics. Whenever work is per- formed by the agency of heat, an amount of heat disappears equi- valent to the work performed; and whenever mechanical work is spent in generating heat, the heat generated is equivalent to the work thus spent; that is to say, we have in both cases WrrJH; W denoting the work, H the heat, and J Joule's equivalent. This is called the first law of thermo-dynamics, and it is a particular case of the great natural law (Chap, ix.) which asserts that energy may be transmuted, but is never created or destroyed. It may be well to remark here that work is not energy, but is rather the process by which energy is transmuted, amount of. work being measured by the amount of energy transmuted. Whenever work is done, it leaves an effect behind it in the shape of energy of 1 In British absolute units of work (called foot-pounflals], of which a foot-pound con- tains g, the equivalent of a pound-degree Fahrenheit is 772 x 32'194 = 24854, which is within less than 1 per cent, of 25,000. Hence the heat-equivalent of the kinetic energy of a mass of m pounds moving with a velocity of v feet per second is approximately | me* -^ 25000, or 2mi> 2 -M 00000. FIRST LAW OF THERMODYNAMICS. 4G3 some kind or other, equal in amount to the energy consumed in per- forming the work, or, in other words, equal to the work itself. As regards the nature of heat, there can be little doubt that heat properly so called, that is sensible as distinguished from latent heat, consists in some kind of motion, and that quantity of heat is quan- tity of energy of motion, or kinetic energy ( 121), whereas latent heat consists in energy of position or potential energy ( 122). We have already had in the experiments of Rumford, Davy, Fou- cault, and Joule, some examples of transmutation of energy; but it will be instructive to consider some additional instances. When a steam-engine is employed in hauling up coals from a pit, an amount of heat is destroyed in the engine equivalent to the energy of position which is gained by the coal. In the propulsion of a steam-boat with uniform velocity, or in the drawing of a railway train with uniform velocity on a level, there is no gain of potential energy, neither is there, as far as the vessel or train is concerned, any gain of kinetic energy. In the case of the steamer, the immediate effect consists chiefly in the agitation of the water, which involves the generation of kinetic energy; and the ultimate effect of this is a warming of the water, as in Joule's experi- ment. In the case of the train, the work done in maintaining the motion is spent in friction and concussions, both of which operations give heat as the ultimate effect. Here, then, we have two instances in which heat, after going through various transformations, reappears as heat at a lower temperature. In starting a train on a level, the heat destroyed in the engine finds its equivalent mainly in the energy of motion gained by the train; and this energy can again be transformed into heat by turning off the steam and applying brakes to the wheels. When a cannon-ball is fired against an armour plate, it is heated red-hot if it fails to penetrate the plate, the energy of the moving ball being in this case obviously converted into heat. If the plate is penetrated, and the ball lodges in the wooden backing, or in a bank of earth, the ball will not be so much heated, although the total amount of heat generated must still be equivalent to the energy of motion destroyed. The ruptured materials, in fact, receive a large portion of the heat. The heat produced in the rupture of iron is well illustrated by punching and planing machines, the pieces of iron punched out of a plate, or the shavings planed off it, being so hot that they can scarcely be touched, although the movements of the 464 THERMODYNAMICS. punch and plane are exceedingly slow. The heat gained by the iron is, in fact, the equivalent of the work performed, and this work is considerable on account of the great force required. 489. Heat Lost in Expansion. The difference between the specific heat of a gas at constant pressure and at constant volume, is almost exactly the equivalent of the work which the gas at constant pres- sure performs in pushing back the surrounding atmosphere. Joule immersed two equal vessels in water, one of them containing highly- compressed air, and the other being exhausted; and when they were both at the temperature of the water he opened a stop-cock which placed the vessels in communication. The compressed air thus expanded to double its volume, but the temperature of the surround- ing water was unaltered, the heat converted into energy of motion by the expansion being, in fact, compensated by the heat generated in the destruction of this motion in the previously vacuous vessel. This experiment shows that, when air expands without having to overcome external resistances, its temperature is not sensibly changed by the expansion. The work done by a gas in expanding against uniform hydrostatic or pneumatic pressure may be computed by multiplying the increase of volume by the pressure per unit area. For, if we suppose the expanding body to be immersed in an incompressible fluid without weight, confined in a cylinder by means of a movable piston under constant pressure, the work done by the expanding body will be spent in driving back the piston. Let A be the area of the piston, x the distance it is pushed back, and p the pressure per unit area. Then the increment of volume is A x, and the work done is the product of the force pA by the distance x, which is the same as the product of p by A x. 490. Difference of the two Specific Heats. Let a gramme of air, occupying a volume V cub. cm. at the absolute temperature T, be raised at the constant pressure of P grammes per sq. cm. to the temperature T + 1. It will expand by the amount ^-, and will do VIP work to the amount -^- in pushing back the surrounding resist- VT* ances. Now the value of -^,- is ( 325) the same for all pressures and temperatures. But at C. and 760 mm. we have T=273, P=1033, and since the volume of T293 grammes is 1 litre or 1000 cub, cm., we have 1000 TWO SPECIFIC HEATS. 465 and VP 1000 1033 000 ,. ~T~ = T : 293 X ^73~ = gramme-centimetres. This is the work done in the expansion of 1 gramme of air at any constant pressure when raised 1 C. in temperature, and its ther- mal equivalent is the excess of the specific heat at constant pressure above the specific heat at constant volume. In the above calculation, the only factor which is peculiar to air is 1-293 in the denominator. Hence, if we multiply the result by T293, that is, by the mass of a litre of air, we shall obtain a product which would be the same for all gases at least for all which have the coefficient of expansion ^. But the product of the specific heat of a substance by the mass of a given volume of it, is the ther- mal capacity of that volume. Hence, the difference of the two ther- mal capacities of a given volume is the same for all gases at the same pressure and temperature. Assuming Regnault's value of the specific heat of air at constant pressure, '2375, the specific heat at constant volume will be 2375 --0690= -1685. The heat required to produce a given change of temperature in a gas, when its volume changes in any specified way, may be com- puted to a very close approximation by calculating the work done by the gas against external resistances during its change of volume, and adding the heat- equivalent of this work to the heat which would have produced the same change of temperature at constant volume. The above calculation of the difference of the two specific heats rests upon the previously known value of Joule's equivalent. Con- versely, from the work done in the expansion of air at constant pressure, combined with the observed value of the specific heat of air at constant pressure, the value of Joule's equivalent can be com- puted. A calculation of this kind, but with an erroneous value of the specific heat of air, was made by Mayer, before Joule's equiva- lent had been determined. 491. Thermic Engines. In every form of thermic engine, work is obtained by means of expansion produced by heat, the force of 30 466 THERMODYNAMICS, expansion being usually applied by admitting a hot elastic fluid to press alternately on opposite sides of a piston travelling in a cylinder. Of the heat received by the elastic fluid from the furnace, a part leaks out by conduction through the sides of the containing vessels, another part is carried out by the fluid when it escapes into the air or into the condenser, the fluid thus escaping being always at a temperature lower than that at which it entered the cylinder, but higher than that of the air or condenser into which it escapes; but a third part has disappeared altogether, and ceased to exist as heat, having been spent in the performance of work. This third part is the exact equivalent of the work performed by the elastic fluid in driving the piston, 1 and may therefore be called the heat utilized, or the heat converted. The efficiency of an engine may be measured by the ratio of the heat thus converted to the whole amount of heat which enters the engine; and we shall use the word efficiency in this sense. 492. Carnot's Investigations. The first approach to an exact science of thermo-dynamics was made by Carnot in 1824. By rea- soning based on the theory which regards heat as a substance, but which can be modified so as to remain conclusive when heat is regarded as a form of energy, he established the following prin- ciples: I. The thermal agency by which mechanical effect may be obtained is the transference of heat from one body to another at a lower tem- perature. These two bodies he calls the source and the refrigerator. Adopting the view generally received at that time regarding the nature of heat, he supposed that all the heat received by an engine was given out by it again as heat; so that, if all lateral escape was prevented, all the heat drawn by the engine from the source was given by the engine to the refrigerator, just as the water which by its descent turns a mill-wheel, runs off" in undiminished quantity at a lower level. We now know that, when heat is let down through an engine from a higher to a lower temperature, it is diminished in amount by the equivalent of the work done by the engine against external resistances. He further shows that the amount of work which can be obtained by letting down a given quantity of heat or, as we should say with our present knowledge, by partly letting it down and partly con- 1 If negative work is done by the fluid in any part of the stroke (that is, if the piston presses back the fluid), the algebraic sum of work is to be taken. CARNOT'S PRINCIPLE. 4-G7 suming it in WOI*K, is increa,sed by raising the temperature of the source, or by lowering the temperature of the refrigerator; and estab- lishes the following important principle: II. A perfect thermo-dynamic engine is such that, whatever amount of mechanical effect it can derive from a certain thermal agency; if an equal amount be spent in working it backwards, an equal reverse thermal effect will be produced. This is often expressed by saying that a completely reversible engine is a perfect engine. By a perfect engine is here meant an engine which possesses the maximum of efficiency compatible with the given temperatures of its source and refrigerator; and Carnot here asserts that all completely reversible engines attain this maximum of efficiency. The proof of this important principle, when adapted to the present state of our knowledge, is as follows: Let there be two thermo-dynamic engines, A and B, working between the same source and refrigerator; and let A be completely reversible. Let the efficiency of A be m, so that, of the quantity Q of heat which it draws from the source, it converts m Q into mechan- ical effect, and gives Q m Q to the refrigerator, when worked f or- wards. Accordingly, wlaen worked backwards, with the help of work mQ applied to it from without, it takes Q mQ, from the refrigerator, and gives Q to the source. In like manner, let the efficiency of B be m', so that, of heat Q' which it draws from the source, it converts m'Q' into mechanical effect, and gives Q' m'Q' to the refrigerator. Let this engine be worked forwards, and A backwards. Then, upon the whole, heat to the amount Q' Q is drawn from the source, heat m'Q' mQ is converted into mechanical effect, and heat Q' Q (m'Q' mQ) is given to the refrigerator. If we make m'Q'=mQ, that is, if we suppose the external effect to be nothing, heat to the amount Q' - Q or f -, - 1 j Q is carried from the source to the refrigerator, if m be greater than m', that is, if the reversed engine be the more efficient of the two. If the other engine be the more efficient, heat to the amount (i-- ,) Q is trans- \ Vfl / ferred from the refrigerator to the source, or heat pumps itself up from a colder to a warmer body, and that by means of a machine which is self-acting, for B does work which is just sufficient to drive A. Such a result we are entitled to assume impossible, therefore B cannot be more efficient than A 468 THERMO-DYNAMICS. Another proof is obtained by making Q':=Q. The source then neither gains nor loses heat, and the refrigerator gains (m - m') Q, which is derived from work performed upon the combined engine from without, if A be more efficient than B. If B were the more efficient of the two, the refrigerator would lose heat to the amount (mf - m) Q, which would yield its full equivalent of external work, and thus a machine would be kept going and doing external work by means of heat drawn from the coldest body in its neighbourhood, a result which cannot be admitted to be possible. 493. Examples of Reversibility. The following may be mentioned as examples of reversible operations. When a gas expands at constant temperature, it must be supplied from without with a definite amount of heat; and when it returns, at the same temperature, to its original volume, it gives out the same amount of heat. When a gas expands adiabatically (that is to say, without inter- change of heat with other bodies), it falls in temperature; and when it is compressed adiabatically from the condition thus attained to its original volume, it regains its original temperature. When water at freezes, forming ice at 0, under atmospheric pressure, it expands and does external work in pushing back the atmosphere. It also gives out a definite quantity of heat called the latent heat of liquefaction. This ice can be melted at the same pressure and temperature, and in this reverse process it must be supplied with heat equal to that which it formerly gave out. Also, since the shrinkage will be equal to the former expansion, the pres- sure of the surrounding atmosphere will do work equal to that for- merly done against it. On the other hand, conduction and radiation of heat are essentially irreversible, since in these operations heat always passes from the warmer to the colder body, and refuses to pass in the opposite direc- tion. 494. Second Law of Thermo-dynamics. It follows, from the prin- ciple thus established, that all reversible engines with the same tem- peratures of source and refrigerator have the same efficiency, whether the working substance employed in them be steam, air, or any other material, gaseous, liquid, or solid. Hence we can lay down the fol- lowing law, which is called the second law of thermo-dynamics: the 'efficiency of a completely reversible engine is independent of the nature of the working substance, and depends only on the temperatures at EFFICIENCY OF REVERSIBLE ENGINE. 469 which the engine takes in and gives out heat; and the efficiency of such an engine is the limit of possible efficiency for any engine. As appendices to this law it has been further established: 1. That when one of the two temperatures is fixed, the efficiency is simply proportional to the difference between the two, provided this difference is very small. This holds good for all scales of tem- perature. 2. That the efficiency of a reversible engine is approximately T 1" -m , T denoting the upper and T' the lower temperature between which the engine works, reckoned from absolute zero ( 325), on the air-thermometer. This is more easily remembered when stated in the following more symmetrical form. Let Q denote the quantity of heat taken in at the absolute temperature T, Q' the quantity given out at the absolute temperature T', and consequently Q - Q' the heat converted into mechanical effect, then we shall have approximately 495. Proof of Formula for Efficiency. This important proposition may be established as follows : Let the volume and pressure of a given portion of gas be re- presented by the rectangular co-ordinates of a movable point, which we will call " the indicating point," horizontal distance representing volume, and vertical distance pressure. When the temperature is constant, the curve which is the locus of the indicating point is called an isothermal, and the relation between the co-ordinates is vp C, where C is a constant, depending upon the given temperature, and in fact proportional to the absolute temperature by air-thermometer. When the changes of volume and pressure are adiabatic ( 497), a given change of volume will produce a greater change of pressure than when they are isothermal, and the curve traced by the indicat- ing point is called an adiabatic line. Whenever the given gas gains or loses heat by interchange with surrounding bodies, the indicating point will be carried from one adiabatic line to another; and by successive additions or subtractions of small quantities of heat we can get any number of adiabatic lines as near together as we please. By drawing a number of adiabatic lines near together, and a number 470 THERMODYNAMICS. of isothermals near together, we shall cut up our diagram into a number of small quadrilaterals which will be ultimately parallelo- grams. Let A B C D (Fig. 310) be one of these parallelograms, and let the gas be put through the series of changes represented by A B, B C, C D, D A, all of which, it will be observed, are reversible. In A B the gas expands at constant temperature. Let this temperature, expressed on the absolute scale of the air or gas thermometer, be T, let the small increase of volume be v, and the mean pressure P, so that the work done against external resistances is Pv. In B C the gas expands adiabatically and falls in temperature. Let the fall of temperature be T. In C D it is compressed at the constant temperature T T. In D A it is compressed adiabatically, and ends by being in the same state in which it was at the commencement of this cycle of four operations. Since the external work done by a gas is equal to the algebraic sum of such terms as pressure x increase of volume, it is easily shown that the algebraic sum of external work done by the gas in the above cycle is represented by the area of the par- allelogram A B C D. Through A and B draw verticals A F, B E, which, by construction, represent diminution of pressure at constant volume; and produce D to meet A F in F. Then the area A B C D is equal to the area A B E F (since the parallelograms are on the same base and between the same parallels), that is to A F multiplied by the perpendicular distance between AF and BE. But this perpendicular distance represents v, the increase of volume from A to B ; and A F represents the difference (at constant volume) between the pressure at T and the pressure at T + r. This difference is P T T' hence the work done in the cycle is * rf- PROOF OF FORMULA FOR EFFICIENCY. 471 But the work done in the operation A B was Pv, and this work, being performed at constant temperature, is known ( 489) to be sensibly equivalent to the whole heat supplied to the gas in the performance of it. This is the only operation in which heat is received from the source, and C D is the only operation in which heat is given out to the refrigerator. Hence we have heat converted T r heat from source Pv T* or, if Qj represent the heat received from the source, Q 2 the heat given to the refrigerator, T l the temperature of the source, and T 2 the temperature of the refrigerator, This proves the law for any reversible engine with an indefinitely small difference of temperature between source and refrigerator. Now, let there be a series of reversible engines, such that the first acts as source to the second, the second as source to the third, and so on; and let the notation be as follows: The first receives heat Q t at temperature T v and gives to the second heat Q 2 at temperature T 2 . The second gives to the third heat Q 3 at temperature T 3 , and so on. Then supposing the excess of each of these temperatures above the succeeding one to be very small, we have, from above. Q. 2 T 2 1-1 Q* TV Q, T,' ' ' Q B "T, Whence, by multiplying equals, 9l = Ti therefore ^^ = -''--" The law is therefore proved for the engine formed by thus combin- ing all the separate engines. But this engine is reversible, and there- fore ( 494) the law is true for all reversible engines. 496. Thomson's Absolute Scale of Temperature. In ordinary thermometers, temperatures are measured by the apparent expansion of a liquid in a glass envelope. If two thermometers are constructed, one with mercury and the other with alcohol for its liquid, it is 472 THERMODYNAMICS. obviously possible to make their indications agree at two fixed temperatures. If, however, the volume of the tube intervening be- tween the two fixed points thus determined be divided into the same number of equal parts in the two instruments, and the divisions be numbered as degrees of temperature, the two instruments will give different indications if plunged in the same bath at an intermediate temperature, and they will also differ at temperatures lying beyond the two fixed points. It is a simple matter to test equality of temperature, but it is far from simple to decide upon a test of equal differences of temperatures. Different liquids expand not only by different amounts but by amounts which are not proportional, no two liquids being in this respect in agreement. In the case of permanent gases expanding under constant pressure, the discordances are much less, and may, in ordinary circumstances, be neglected. Hence gases would seem to be indicated by nature as the proper substances by which to measure temperature, if differences of temperature are to be measured by differences of volume. It is also possible to establish a scale of temperature by assuming that some one substance rises by equal increments of temperature on receiving successive equal additions of heat; in other words, by making some one substance the standard of reference for specific heat, and making its specific heat constant by definition at all temperatures. Here, again, the scale would be different according to the liquid chosen. A mixture of equal weights of water at C. and 100 C. will not have precisely the same temperature as a mixture of equal weights of mercury at these temperatures. If, however, we resort to permanent gases, we find again a very close agreement, so that, if one gas be assumed to have the same specific heat at all tempera- tures (whether at constant volume or at constant pressure), the specific heat of any other permanent gas will also be sensibly in- dependent of temperature. More than this; the measurement of temperature by assuming the specific heats of permanent gases to be constant, agrees almost exactly with the measurement of tem- perature by the expansion of permanent gases. For, as we have seen ( 343), a permanent gas under constant pressure has its volume increased by equal amounts on receiving successive equal additions of heat. The air-thermometer, or gas-thermometer, then, has a greatly superior claim to the mercury thermometer to be considered as fur- nishing a natural standard of temperature. HEAT REQUIRED FOR A GIVEN CHANGE. 473 But a scale which is not only sensibly but absolutely independent of the peculiarities of particular substances, is obtained by defining temperature in such a sense as to make appendix (2) to the second law of thermo-dynamics rigorously exact. According to this system (which was first proposed by Sir Wm. Thomson), the ratio of any two temperatures is the ratio of the two quantities of heat which would be drawn from the source and supplied to the refrigerator by a completely reversible thermo-dynamic engine working between these temperatures. This ratio will be rigorously the same, what- ever the working substance in the engine may be, and whether it be solid, liquid, or gaseous. 497. Heat required for Change of Volume and Temperature. The amount of heat which must be imparted to a body to enable it to pass from one condition, as regards volume and temperature, to another, is not a definite quantity, but depends upon the course by which the transition is effected. It is, in fact, the sum of two quan- tities, one of them being the heat which would be required if the transition were made without external work as in Joule's experi- ment of the expansion of compressed air into a vacuous vessel and the other being the heat equivalent to the external work which the body performs in 'making the transition. As regards the first of these quantities, its amount, in the case of permanent gases, depends almost entirely upon the difference between the initial and final tem- peratures, being sensibly independent of the change of volume, as Joule's experiment shows. In the case of liquids and solids, its amount depends, to a very large extent, upon the change of volume, so that, if the expansion which heat tends to produce is forcibly prevented, the quantity of heat required to produce a given rise of temperature is greatly diminished. This contrast is sometimes ex- pressed by saying that expansion by heat involves a large amount of internal work in the case of liquids and solids, and an exceedingly small amount in the case of gases ; but the phrase internal work has not as yet acquired any very precise meaning. As an illustration of the different courses by which a transition may be effected, suppose a quantity of gas initially at C. and a pressure of one atmosphere, and finally at 100 C. and the same pressure, the final volume being therefore T366 times the initial volume. Of the innumerable courses by which the transition may be made, we will specify two: 1st. The gas may be raised, at its initial \ olume, to such a tern- 474 THERMODYNAMICS. perature that, when afterwards allowed to expand against pressure gradually diminishing to one atmosphere, it falls to the temperature 100 C. Or, 2d. It may be first allowed to expand, under pressure diminishing from one atmosphere downwards, until its final volume is attained, and may then, at this constant volume, be heated up to 100. In both cases it is to be understood that no heat is allowed to enter or escape during expansion. Obviously, the first course implies the performance of a greater amount of external work than the second, and it will require the communication to the gas of a greater quantity of heat, greater by the heat-equivalent of the difference of works. When a body passes through changes which end by leaving it in precisely the same condition in which it was at first, we are not entitled to' assume that the amounts of heat which have entered and quitted it are equal. They are not equal unless the algebraic sum of external work done by the body during the changes amounts to zero. If the body has upon the whole done positive work, it must have taken in more heat than it has given out, otherwise there would be a creation of energy; and if it has upon the whole done negative work, it must have given out more heat than it has taken in, other- wise there would be a destruction of energy. In either case, the difference between the heat taken in and given out must be the equivalent of the algebraic sum of external work. These principles are illustrated in the following sections. 498. Adiabatic Changes. Heating by Compression, and Cooling by Expansion. When a gas is compressed in an absolutely non-conduct- ing vessel, or, more generally, when a gas alters its volume without giving heat to other bodies or taking heat from them, its changes are called adiabatic [literally, without passage across]. Let unit volume of gas, at pressure P and absolute temperature T, receive heat which raises its temperature to T + r at constant pres- sure. The increase of volume will be ^, and the work done by the gas against external resistance will be -j. Next let the gas be compressed to its original volume without entrance or escape of heat, and let the temperature at the end of this second operation be denoted by T + KT, so that the elevation of temperature produced by the compression is (K - 1) T. The pressure will now be P m* 7 ', as appears by comparing the final condition of the gas CHANGES OF VOLUME, TEMPERATURE, AND PRESSURE. 475 with its original condition at the same volume. This may be written P l+, and the mean pressure during the second operation may be taken as half the sum of the initial and final pressures, that is, as P (i + irjr). The work done upon the gas by the external compress- ing forces in the second operation is therefore or, to the first order of small quantities, P^, which is the same as the work done by the gas in the first operation. Hence, to the first order of small quantities, the heat which has been given to the gas is the same as if the gas had been brought without change of volume from its initial to its final condition. That is to say, the heat which produces an elevation r of temperature, at constant pressure, would produce an elevation KT at constant volume. Hence ' Specific heat at constant pressure Specific heat at constant volume _ ~ where K may be defined as the ratio of the elevation of temperature produced by a small adiabatic compression to the elevation of tem- perature which would be required to produce an equal expansion at constant pressure. 499. Relations between Adiabatic Changes of Volume, Temperature, and Pressure. For the sake of greater clearness, we will tabulate the values of volume, temperature, and pressure, at the beginning and end of the adiabatic compression above discussed. At beginning. At end. Change. Volume, ........................ l+ 1 -^, Temperature, ................. T + T T + KT (K-!)T Pressure, P P (l + ,KT T Denoting volume, temperature, and pressure by V, T, P, and their changes by cZV, cT, dP, we have, to the first order of small quan- tities, dV__r dT_( K -l)T ^P_' C I IT" ~T' T ~ T ' P ~ T ' dV dT dP ,, p ,. , , V "T" P are therefore proportional to - 1, K- 1, *; that is _ K - 1 476 THERMODYNAMICS. whence, if Vj, T 1} PI are one set of corresponding values, and V 2 , T 2 , P 2 another set, we have W Pi' 500. Numerical Value of*. Since ^divided by -^ is K,dP divided by =- (which, as shown in 129, is the coefficient of elasticity of the gas), is equal to PK. Now the square of the velocity of sound in a gas can be proved to be equal to the coefficient of elasticity divided by the density, and hence from observations on the velocity of sound the value of K can be determined. It is thus found that r=l-408 for perfectly dry air; and its value is very nearly the same for all other gases which are difficult to liquefy. 501. Rankine's Prediction of the Specific Heat of Air. Let Sj denote the specific heat of air at constant pressure, and S 2 its specific heat at constant volume. Then ( 408, 500) we have 5=1-408. -i But we have proved ( 489) by thermo-dynamic considerations, in- dependent of any direct observation of specific heat, that !-, = -0690. From these two equations we have 2 (1-408-1) = -0690 In this way the correct values of the two specific heats of air were calculated by Rankine, before any accurate determinations of them had been made by direct experiment. 502. Cooling of Air by Ascent. Convective Equilibrium. When a body of air ascends in the atmosphere it expands, in consequence of being relieved of a portion of its pressure, and the foregoing prin- ciples enable us to calculate the corresponding fall produced in its temperature. For we have COOLING OF AIR BY ASCENT. 477 But ( 213) if x denote height above a fixed level, and H "pressure height," or " height of homogeneous atmosphere," we have dP_dx P ~H ; also H is proportional to T, so that if H , T , denote the values of rii H, T at the freezing-point, we have Hr=H T . Thus we have _dT_K-- dxT^ dT_K-l T T ~ K H T" OI dx~ K Hj Expressing height in metres, the value of H will be 7990, and d T ~ S the larger piston, Fig. 321. Compound-cylinder Arrangement. working in the larger cylinder A'B'C'D'. In the up-stroke, the passage DA' is closed, and CB' is open. The small piston is forced up by the high-pressure steam beneath, while the steam above it, instead of escaping to a condenser, expands into the large cylinder, and there raises the piston P, the upper part of the large cylinder being connected with the condenser. In the down-stroke, the passage C B' is closed, and D A' is open. 502 STEAM AND OTHER HEAT ENGINES. The two piston-rods are connected with the same end of the beam, and rise and fall together. The distribution of the steam is effected by means of two slide-valves, each governed by an eccentric. Compound engines have been adopted for several lines of ocean- steamers, where it is of primary importance to obtain as much work as possible from a limited quantity of fuel. 529. Surface Condensation. In several modern engines, the con- denser consists of a number of vertical tubes of about half an inch diameter, connected at their ends, and kept cool by the external contact of cold water. The steam, on escaping from the cylinder, enters these tubes at their upper ends, and becomes condensed in its passage through them, thus yielding distilled water, which is pumped back to feed the boiler. The same water can thus be put through the engine many times in succession, and the waste which occurs is usually repaired by adding from time to time a little distilled water prepared by a separate apparatus. 530. Classification of Steam-engines. The distinctions which exist between different kinds of stationary engines relate either to the pressure of the steam, or to its mode of action, or to the arrangement of the mechanism, especially as regards the mode in which the move- ment of the piston is transmitted to the rest of the machinery. On the first of these heads, it must be remarked that the terms low-pressure and high-pressure are no longer equivalent to con- densing and non-condensing as they once were, and merely express differences of degree. When the pressures employed are very low there is little risk of explosion and little wear and tear; but the engine must be very large in proportion to its power, and expansive working cannot be em- ployed. Low-pressure engines are always condensing, though the converse is not true. With regard to the mode of action of the steam, engines may be classed as condensing or non-condensing, as expansive or non-expan- sive. Condensation increases the quantity of work obtainable from a given consumption of fuel, and is almost always employed for stationary engines where the supply of water is abundant, and also for marine engines, but it is dispensed with in locomotives and agricultural engines. Expansive working is also conducive to efficiency, as is obvious from 526. Assuming the temperature of the steam to remain con- stant during the expansion, the following table, calculated by an HIGH-PRESSURE ENGINE. 503 application of Boyle's law, exhibits the relative amounts of work Fig. 322. High-pressure Engine with Vertical Cylinder, working Expansively and without Condeusf tion. A, steam-pipe through which the steam arrives from the boiler. Z, valve-chest. B, sliae- vaive. C, cylinder. GG, guides fixed to the cylinder and to the frame of the engine at K and H. EF, connecting-rod. J, crank. W, fly-wheel. L, eccentric governing the slide-valve. N, eccentric of the exhaust-pump P. D, outlet pipe for the steam. M, lever of throttle-valve, regulated by centrifugal governor. obtained from the same weight of steam with different ranges of expansion: Fraction of the stroke completed before the Work shutting-nit of steam. done. 1-0 1-000 9 1-105 3 1-223 7 ....... 1-357 6 . 1-509 Fraction of the stroke completed before the Work shutting-off of steam. done. 5 1-693 4 1-916 3 2-201 2 2-609 1 . . 3-302 Expansive working is often combined with the superheating of steam, that is to say, heating the steam after it has been formed, so 504 STEAM AND OTHER HEAT ENGINES. as to raise its temperature above the point of saturation. This increases the difference of temperatures, to which, according to the second law of thermo-dynamics, the maximum efficiency is propor- tional; and experience has shown that an actual increase of efficiency is thus obtained. 531. Form and Arrangement of the Several Parts. As regards their mechanism, the arrangement of steam-engines is considerably varied. In stationary condensing engines, the beam and parallelogram are usually retained; but the arrangement of high-pressure non-con- densing engines is generally simpler. The piston-rod frequently travels between guides, and drives the crank by means of a connect- ing-rod. The cylinder may be either vertical or horizontal, or even inclined at an angle. An engine of this kind is represented in Fig. 322. Oscillating Engines. The space occupied by the engine may be lessened by jointing the piston-rod directly to the crank without any connecting-rod. In this case the cylinder oscillates around two gudgeons, one of which serves to admit the steam, the other to let it escape. The distribution of the steam is effected by means of a slide-valve whose movements are governed by those of the cylinder. Oscillating engines are very common in steam-boats, and usually produce an exceedingly smooth motion. 532. Rotatory Engines. Numerous attempts have been made to dispense with the reciprocating movement of a piston, and obtain rotation by the direct action of steam. Watt himself devised an engine on this plan in 1782. Hitherto, however, the results obtained by this method have not been encouraging. Behren's engine, which we now proceed to describe, is one of the best examples. Fig. 323 is a perspective view of the engine, and Fig. 324 a cross- section of the cylinders, showing the mode of action of the steam. C and C' are two parallel axes, connected outside by two toothed wheels, so that they always turn in opposite directions. One of these axes is the driving-shaft of the engine. These two axes are surrounded by fixed collars c and c, which fit closely to the cylin- drical sectors E and E'; these latter, which are rigidly connected with the axes C, C', are capable of moving in the incomplete cylinders A and A', and act as revolving pistons. In the position represented in the figure, the steam enters at B, and will escape at D ; it is acting only upon the sector E, and pushes it in the direction indicated by the arrow; the shaft C is thus turned, and causes the shaft C' to turn ROTATORY ENGINE. 505 in the opposite direction, carrying with it E', to which it is attached. After half a revolution the sector E' will be in a position correspond- ing, left for right, to that which E now occupies; it will then be Fig. 323. Behren's Rotatory Engine. urged by the steam, so as to continue the motion in the same direc- tion for another half-revolution, when the two sectors will have resumed the position represented in the figure. 533. Boilers. There are many forms of boiler in use. That which 506 STEAM AND OTHER HEAT ENGINES. is represented in Fig. 325 is the favourite form in France, and is also extensively used in this country, where it is called the French boiler, or the cylindrical boiler with heaters. The main boiler-shell A is cylindri- cal with hemispherical ends. B B are two cylindrical tubes called heaters, of the same length as the main shell, and connected with it by vertical tubes d, d, of which there are usually three to each heater. A horizontal brick partition, a little higher than the centres of the heaters, extends along their whole length; and a vertical partition runs along the top of each heater, except where interrupted by the vertical tubes. The flame from the furnace is thus compelled to travel in the first instance back- wards, beneath the heaters; then forwards, through the intermediate space between the heaters, the vertical tubes, and the main shell; and lastly, backwards, through the side passages C C, which lead to the Fig 324 Section of Behren's Engine. Fig. 325. Boiler with Heaters. chimney. By thus compelling the flame to travel for a long dis- tance in contact with the boiler, the quantity of heat communicated to the water is increased. The level of the water is shown at A in the left-hand figure. The relative spaces allotted to the steam and the water are not always the same; but must always be so regulated that the steam shall arrive in the cylinder as dry as possible, that is to say, that it shall BURSTING OF BOILERS. 507 not carry with it drops of water. Before being used, boilers should always be tested by subjecting them to much greater pressures than they will have to bear in actual use. Hydraulic pressure is com- monly employed for this purpose, as it obviates the risk of explosion in case of the boiler giving way under the test. 534. Boilers with the Tire Inside. When it is required to lessen the weight of the boiler, without much diminishing the surface exposed to heat, as in the case of marine engines, the method adopted is to place the furnace inside the boiler, so that it shall be completely surrounded with water except in front. The flame passes from the furnace, which is in the front of the boiler, into one or two large tubes, leading to a cavity near the back, whence it returns through a number of smaller tubes traversing the boiler, and finally escapes by the chimney. 535. Bursting of Boilers: Safety-valves. Notwithstanding the tests to which boilers are subjected before being used, it too often happens that, owing either to excessive pressure or to weakening of the boiler, very disastrous explosions occur. Excess of pressure is guarded against by gauges, which show what the pressure is at any moment, and by safety-valves, which allow steam to escape whenever the pressure exceeds a certain limit. Various kinds of manometer or pressure-gauge have been described in Chap. xix. That which is most commonly employed in connection with steam-boilers is Bourdon's ( 226). A thermometer, specially protected against the pressure and con- tact of the steam, is also sometimes employed, under the name of thcrmo-manometer, on the principle that the pressure of saturated steam depends only on its temperature. The safety-valve, represented in ths upper part of Fig. 325, consists of a piece of metal, having the form either of a truncated cone or of a flat plate, fitting very truly into or over an opening in the boiler. The valve is pressed down by a weighted lever; the weight and the length of the lever being calculated, so that the force with which the valve is held down shall be exactly equal to the force with which the steam would tend to raise it when at the limiting pressure. In movable engines, the weighted lever is replaced by a spring, the tension of which can be regulated by means of a screw. Safety-valves afford ample protection against the danger arising from gradual increase of pressure; but they are liable to fail in cases 508 STEAM AND OTHER HEAT ENGINES. where there is a sudden generation of a large quantity of steam. This explosive generation of steam may occur from various causes. If, for instance, the water in the boiler is allowed to fall too low. the sides of the boiler may be heated to so high a temperature that, when fresh water is admitted, it will be immediately converted into steam on coming in contact with the metal. Hence it is of great importance to provide that the water in the boiler shall not fall below a certain level, depending on the shape of the boiler and furnace. The following are the means employed for securing this end: 1. Two cocks are placed, one a little below the level at which the water should stand, and the other a little above it; these are opened from time to time, when water should issue from the first, and steam from the second. 2. The water-gauge is a strong vertical glass tube, having its ends fitted into two short tubes of metal, proceeding one from the steam- space and the other from the water-space. The level of the water is therefore the same in the gauge as in the boiler, and is constantly visible to the attendant. The metal tubes are furnished with cocks, which can be closed if the glass tube is accidentally broken. 536. Causes of Explosion. Another cause of the explosive genera- tion of steam is the incrustation of the boiler with a hard deposit, due to the impurities of the water employed. This crust is a bad conductor, and allows the portion of the boiler covered with it to become overheated; when, if water should find its way past the crust, and come in contact with the hot metal, there is great danger of explosion. The best preventive of incrustation is the employment of distilled water in connection with surface condensation ( 529). In default of this, portions of the water in the boiler must be blown off from time to time, so as to prevent it from becoming too highly concen- trated. This is especially necessary when the boiler is fed with salt water. Among the causes of the bursting of boilers, we may also notice undue smallness of the vertical tubes in boilers with heaters ( 533). When this fault exists, the steam which is generated is not imme- diately replaced by water, and overheating is liable to occur. Another cause of explosions is probably to be found in a property of water which has only recently been recognized. It has been shown that, when water is deprived of air, it does not begin boiling GIFFARD S IXJECTOR. 509 till it has acquired an abnormally high temperature, and then bursts into steam with explosive violence ( 391, Donny's experiment). This danger is to be apprehended when a boiler, which has been allowed to cool after being for some time in use, is again brought into action without the addition of a fresh supply of water. But it appears that the most frequent cause of boiler explosions is the gradual eating away of some portion of the boiler by rust, so as to render it at last too weak to withstand the pressure of the steam within it. The only general remedy for this danger is periodical and enforced inspection. 537. Feeding of the Boiler: Giffard's Injector. The feeding of the boiler is usually effected by means of a pump driven by the engine Fig. 826. Giffard's Injector. itself. Of late years this plan has been largely superseded by Giffard's invention of an apparatus by means of which the boiler is supplied with water by the direct action of its own steam. This very curious apparatus contains a conical tube 1 1 (Fig. 326), by which the steam issues when .the injector is working; the steam from the boiler comes through the tube V V, and enters the tube 1 1 through small holes in its circumference. On issuing from the cone 1 1, the steam enters another cone c c, where it meets the water which is to feed the boiler, and which comes through the tube E E, The 510 STEAM AND OTHER HEAT ENGINES. contact of the water and the steam produces two results: (1) the steam, which possesses a great velocity due to the pressure of the boiler, com- municates part of its velocity to the water; (2) at the same time the steam is condensed by the low temperature of the water, so that at the extremity of the cone as far as ee the entire space is occupied by water only, with the exception of a few bubbles of steam which remain in the centre of the liquid vein. The liquid, on issuing from the cone c c, traverses an open space for a little distance before entering the divergent opposite cone d d, through which it is conducted to the boiler by the pipe M. The water will not enter the boiler unless it possess a sufficient velocity to produce in the divergent cone a greater pressure than that which exists in the boiler; when this is the case, the excess of pressure opens a valve, and water enters the boiler from the injector. We may complete this brief description by pointing out one or two arrangements by which the action of the apparatus is regulated. It is useful to be able to vary the volume of steam issuing through the cone it, as required by the pressure in the boiler; this is easily effected by means of the pointed rod a a, which is called the needle, and is screwed forwards or backwards by turning a handle. It is also necessary to be able to regulate the volume of water which enters the cone c c from the supply-pipe E; this is done by means of a lever, which is not shown in the figure, and which moves the tube and cone 1 1 forwards or backwards. The tube E dips into a bath containing the, feed- water; and A T is the overflow pipe. It appears at first sight paradoxical that steam should be able, as in Giffard's injector, to overcome its own pressure, and force water into the boiler against itself; but it must be remembered that the water which is forced in is less bulky than the steam which issues, so that the exchange, though it produces an increase of mass in the contents of the boiler, involves a diminution of pressure, as well as a fall of temperature. 538. Locomotive: History. The following sketch of the history of the locomotive is given by Professor Rankine. 1 " The application of the steam-engine to locomotion on land was, according to Watt, sug- gested by Robison in 1759. In 1784 Watt patented a locomotive- engine, which, however, he never executed. About the same time Murdoch, assistant to Watt, made a very efficient working model of 1 Manual of the Steam-engine, pp. xxv-xxvii, edition 1866. LOCOMOTIVE. 511 a locomotive-engine. In 1802 Trevithick and Vivian patented a locomotive-engine, which was constructed and set to work in 1804 or 1805. It travelled at about 5 miles an hour, with a net load of ten tons. The use of fixed steam-engines to drag trains on railways l>y ropes, was introduced by Cook in 1808. " After various inventors had long exerted their ingenuity in vain to give the locomotive-engine a firm hold of the track by means of rack work -rails and toothed driving-wheels, legs and feet, and other contrivances, Blackett and Hedley, in 1813, made the important dis- covery that no such aids are required, the adhesion between smooth wheels and smooth rails being sufficient. To adapt the locomotive- engine to the great and widely- varied speeds at which it now has to travel, and the varied loads which it now has to draw, two things are essential that the rate of combustion of the fuel, the original source of the power of the engine, shall adjust itself to the work which the engine has to perform, and shall, when required, be capable of being increased to many times the rate at which fuel is burned in the furnace of a stationary engine of the same size; and that the surface through which heat is communicated from the burning fuel to the water shall be very large compared with the bulk of the boiler. The first of these objects is attained by the blast-pipe, in- vented and used by George Stephenson before 1825; the second by the tubular boiler, invented about 1829, simultaneously by Seguin in France and Booth in England, and by the latter suggested to Stephenson. On the 6th October, 1829, occurred that famous trial of locomotive-engines, when the prize offered by the directors of the Liverpool and Manchester Railway was gained by Stephenson's engine the ' Rocket,' the parent of the swift and powerful locomo- tives of the present day, in which the blast-pipe and tubular boiler are combined. Since that time the locomotive-engine has been varied and improved in various details, and by various engineers. Its weight now ranges from five tons to fifty tons; its load from fifty to five hundred tons; its speed from ten miles to sixty miles an hour." 539. Description of a Locomotive. A section of a locomotive is represented in Fig. 327. The boiler is cylindrical. Its forward end abuts on a space beneath the chimney, called the smoke-box. At its other end is a larger opening, surrounded above and on the two sides by the boiler, and called the fire-box. The fuel is heaped up on the bars which form the bottom of the fire-box, and the cinders fall on the line. The fire-box and smoke-box are connected by brass tubes, 512 STEAM AND OTHER HEAT ENGINES. firmly ri vetted to the ends of the boiler; and the products of combus- tion escape by traversing these from end to end. The tubes are very numerous, usually from 150 to 180, thus affording a very large heat- ing surface. The water in the boiler stands high enough to cover all Fig. 327. Section of Locomotive. the tubes, as well as the top of the fire-box. Its level is indicated in the same way as in stationary engines ; and water is pumped in from the tender as required; its amount being regulated by means of a stop-cock in the pipe e. The steam escapes from the boiler by ascending into a dome, which forms its highest part, and thence descending the tube p, this ar- rangement being adopted in order to free the steam from drops of water. It then passes through a regulator q, which can be opened to a greater or less extent, into the pipe s, which leads to the valve- chests and traverses the whole length of the boiler. There are two cylinders, one on each side of the engine, each having a valve-chest and slide-valve, by means of which steam is admitted alternately before and behind the pistons, The steam escapes from the cylinde^ REVERSING. 513 through the blast-pipe v, up the chimney, and thus increases the draught of the fire, a is one of the pistons, 6 the piston-rod, cc' the connecting-rod, which is jointed to the crank d on the axle of the driving-wheel m. The cranks of the two driving-wheels, one on each side of the engine, are set at right angles to each other, so that, when one is at a dead point, the other is in the most advantageous position. w is a spring safety-valve, and J the steam whistle. 540. Apparatus for Reversing: Link-motion. The method usually employed for reversing engines is known as Stephenson's link- Fig. 328. Link-motion. motion, having been first employed in locomotives constructed by Robert Stephenson, son of the maker of the " Rocket." The merit of the invention belongs to one or both of two workmen in his employ Williams, a draughtsman, who first designed it, and Howe, a pattern-maker, who, being employed by Williams to construct a model of his invention, introduced some important improvements. The link-motion, which is represented in Fig. 328, serves two pur- poses; first, to make the engine travel forwards or backwards at pleasure; and, secondly, to regulate the amount of expansion which shall take place in the cylinder. Two oppositely placed eccentrics, A and A', have their connecting-rods jointed to the two extremities of the link B B', which is a curved bar, having a slit, of uniform 33 514 STEAM AND OTHER HEAT ENGINES. width, extending along nearly its whole length. In this slit travels a stud or button C, forming part of a lever, which turns about a fixed point E. The end D of the lever D E is jointed to the connecting- rod D N, which moves the rod P of the slide-valve. The link itself is connected with an arrangement of rods L I K H, 1 which enables the engine-driver to raise or lower it at pleasure by means of the handle G H F. When the link is lowered to the fullest extent, the end B of the connecting-rod, driven by the eccentric A, is very near the runner C which governs the movement of the slide-valve; this valve, accordingly, which can only move in a straight line, obeys the eccen- tric A almost exclusively. When the link is raised as much as pos- sible, the slide-valve obeys the other eccentric A', and this change reverses the engine. When the link is exactly midway between the two extreme positions, the slide-valve is influenced by both eccen- trics equally, and consequently remains nearly stationary in its middle position, so that no steam is admitted to the cylinder, and the engine stops. By keeping the link near the middle position, steam is admitted during only a small part of the stroke, and con- sequently undergoes large expansion. By moving it nearer to one of its extreme positions, the travel of the slide-valve is increased, the ports are opened wider and kept open longer, and the engine will accordingly be driven faster, but with less expansion of the steam. As a means of regulating expansion, the link-motion is far from per- fect, but its general advantages are such that it has come into very extensive use, not only for locomotives but for all engines which need reversal. 541. Gas-engines. This name includes engines in which work is obtained by the expansion of a mixture of coal-gas and air, on igni- tion or explosion. In the engine of Otto and Langen (Fig. 329), a true explosive mixture is introduced beneath the piston, and is exploded by means of a lighted jet, which is brought into contact with the mixture by means of a hole in a movable plate of metal, driven by an eccentric. The upward movement of the piston thus produced is too violent to admit of being directly communicated to machinery. The piston-rod is a rack, working with a pinion, which is so mounted that it can slip round on the shaft when the piston ascends, but carries the shaft 1 1 is a fixed centre of motion, and the rods K I, M L are rigidly connected at right angles to each other. M is a heavy piece, serving to counterpoise the link and eccentric rods. Fig. 329. Gas-engine of Otto and Langen. 516 STEAM AND OTHER HEAT ENGINES. with it when it turns in the opposite direction during the descent of the piston, this descent being produced by the pressure of the atmo- sphere, as the steam resulting from the explosion condenses, and the unexploded gases cool. The vessel shown on the right contains cold water, which is employed to cool the cylinder by circulating round the lower part of it. This engine, which works with much jarring and noise, has been almost completely superseded by the " Otto Silent Gas Engine," which runs as smoothly as a steam-engine. 542. Otto's Silent Gas-engine. A dilute mixture of gas and air (about one part in twelve being gas) is admitted into the cylinder, and, after being compressed to about three atmospheres, is ignited Fig. 330. Otto's Silent Gas-engine. by instantaneous communication with a small jet of gas kept con- stantly burning. The effect is something intermediate between ignition and explosion ; the maximum pressure in the early part of the stroke being 10 or 12 atmospheres, and the mean pressure in the whole stroke 4 or 5. In the return stroke, the products of combus- tion escape at atmospheric pressure, this return stroke being effected by the momentum of the ny-wheel, which also carries the piston through another forward stroke during which the charge of gas and air is admitted, and through another backward stroke in which it is compressed previous to ignition as above described. This is the ordinary cycle of operations when the engine is working up to the full power for which it is intended; but a centrifugal GAS-ENGINE. 517 governor is provided which prevents the gas from being admitted oftener than is necessary for keeping up the standard number of revolutions per minute; so that in working far below its full power the gas is only admitted at every third, fourth, or fifth stroke, the intervening strokes being maintained by the fly-wheel. The governor can be regulated to give any speed required, the most usual being 170 revolutions per minute; and the difference of speed between full work and running idle is only one or two revolutions. The general appearance of the engine is shown in Fig. 330. A is the cylinder, with a jacket round it through which a convective circulation of water is maintained by means of two pipes, not shown in the figure, connecting it with a tank at a higher level. This is necessary to prevent overheating. C is the centrifugal governor. B, D are two vessels containing oil with automatic lubricators, B lubricates the piston, and D the slide which controls the ignition of the charge. E is a chimney, in the lower part of which the gas jet is kept burning. F is a spring fastening, which keeps the slide strongly pressed home so as to prevent leakage. The connecting-rod, crank, and heavy fly-wheel speak for themselves. Gas-engines have a great advantage in being constantly ready for use without the tedious process of getting up steam. They are started by lighting the gas jet and giving one turn to the fly-wheel by hand; and are stopped by turning out the jet. The usual sizes are from \ to 20 horse-power. They are easily kept in order, the principal trouble consisting in the removal of a hard deposit of carbon which forms in certain places. CHAPTER XL. TERRESTRIAL TEMPERATURES AND WINDS. 543. Temperature of the Air. By the temperature of a place meteorologists commonly understand the temperature of the air at a moderate distance (5 or 10 feet) from the ground. This element is easily determined when there is much wind; but in calm weather, and especially when the sun is shining powerfully, it is often diffi- cult to avoid the disturbing effect of radiation. Thermometers for observing the temperature of the air must be sheltered from rain and sunshine, but exposed to a free circulation of air. 544. Mean Temperature of a Place. The 'mean temperature of a day is obtained by making numerous observations at equal intervals of time throughout the day (24 hours), and dividing the sum of the observed temperatures by their number. The accuracy of the deter- mination is increased by increasing the number of observations; as the mean temperature, properly speaking, is the mean of an infinite number of temperatures observed at infinitely short intervals. If the curve of temperature for the day is given, temperature being represented by height of the curve above a horizontal datum line, the mean temperature is the height of a horizontal line which gives and takes equal areas; or is the height of the middle point of any straight line (terminated by the extreme ordinates of the curve) which gives and takes equal areas. Attempts have been made to lay down rules for computing the mean temperature of a day from two, three, or four observations at stated hours; but such rules are of very limited application, owing to the different character of the diurnal variation at different places; and at best they cannot pretend to give the temperature of an individual day, but merely results which are correct in the long run. Observations at 9 A.M. and 9 P.M. are very usual in this country; and MEAN TEMPERATURE. 519 the half-sum of the temperatures at these hours is in general a good approximation to the mean temperature of the day. The half -sum of the highest and the lowest temperature in the day, as indicated by maximum and minimum thermometers, is often adopted as the mean temperature. The result thus obtained is usually rather above the true mean temperature, owing to the circumstance that the extreme heat of the day is a more transient phenomenon than the extreme cold of the night. The employment of self -registering thermometers has, however, the great advantage of avoiding errors arising from want of punctuality in the observer. The correction which is to be added or subtracted in order to obtain the true mean from the mean of two observations is called a correction for diurnal range. Its amount differs for different places, being usually greatest where the diurnal range itself ( 214) is greatest. The mean temperature of a calendar month is computed by adding the mean temperatures of the days which compose it, and dividing by their number. The mean temperature of a year is usually computed by adding the mean temperatures of the calendar months, and dividing by 12; but this process is not quite accurate, inasmuch as the calendar months are of unequal length. A more accurate result is obtained by adding the mean temperatures of all the days in the year, and dividing by 305 (or in leap-year by 366). 545. Isothermals. The distribution of temperature over a large region is very clearly represented by drawing upon the map of this region a series of isothermal lines; that is, lines characterized by the property that all places on the same line have the same temperature. These lines are always understood to refer to mean annual temper- ture unless the contraiy is stated; but isothermals for particular months, especially January and July, are frequently traced, one serving to show the distribution of temperature in winter, and the other in summer. The first extensive series of isothermals was drawn by Humboldt in 1817, on the basis of a large number of observations collected from all parts of the world; and the additional informa- tion which has since been collected has not materially altered the forms of the lines traced by him upon the terrestrial globe. They are in many places inclined at a very considerable angle to the parallels of latitude; and nowhere is this deviation from parallelism more observable than in the neighbourhood of Great Britain, Nor- way, and Iceland places in this region having the same mean 520 TERRESTRIAL TEMPERATURES. annual temperature as places in Asia or America lying from 10 to 20 further south. 546. Insular and Continental Climates. We have seen that the specific heat of water, the latent heat of liquid water, and the latent heat of aqueous vapour are all very large. The presence of water accordingly exerts a powerful effect in moderating the extremes both of heat and cold, and a moist climate will in general have a smaller range of temperature than a dry climate. Moreover, since earth and rock are opaque to radiant heat, while water is to a con- siderable extent diathermanous, the surface of the ground is much more quickly heated and cooled by radiation than the surface of water. This difference is increased by the continual agitation of the surface of the ocean. Large bodies of water thus act as equal- izers of temperature, and the most equable climates are found on oceanic islands or on the ocean itself; while the greatest difference between summer and winter is found in the interior of large con- tinents. It is common to distinguish in this Tsense between conti- nental climates on the one hand, and insular or marine climates on the other. Some examples of both kinds are given in the following table. The temperatures are Centigrade: Summer. Difference. 11-60 7'70 11 -92 7 '87 15 -08 9 -49 15 -83 8 79 16 -00 9 -81 CONTINENTAL CLIMATES. St. Petersburg, - 870 15'96 24-66 Moscow, -10-22 17-55 27 '77 Kasan, -13 '66 17 '35 31 '01 Slatoust, . . . . -16-49 16-08 32 -57 Irkutsk, -17-88 16-00 33 -88 Jakoutsk -38-90 17 '20 56-10 547. Temperature of the Soil at Different Depths. By employing thermometers with their bulbs buried in the earth, and their stems projecting above, numerous observations have been made of the tem- perature from day to day at different depths from 1 inch to 2 or 3 feet; and at a few places observations of the same kind have been made by means of gigantic spirit-thermometers with exceedingly strong Faroe Islands, .... MARINE CLIMATES. Winter. 3-90 Isle of Unst (Shetland), Isle of Man, .... 4-05 5 -59 Penzance, 7 -04 Helston, . 6 -19 , UNDERGROUND TEMPERATURE. 521 bulbs, at depths extending to about 25 feet. It is found that varia- tions depending on the hour of the day are scarcely sensible at the depth of 2 or 3 feet, and that those which depend on the time of year decrease gradually as the depth increases, but still remain sen- sible at the depth of 25 feet, the range of temperature during a year at this depth being usually about 2 or 3 Fahrenheit. It is also found that, as we descend from the surface, the seasons lag more and more behind those at the surface, the retardation amounting usually to something less than a week for each foot of descent; so that, at the depth of 25 feet in these latitudes, the lowest temperature occurs about June, and the highest about December. Theory indicates that 1 foot of descent should have about the same effect on diurnal variations as V365 that is 19 feet on annual varia- tions; understanding by sameness of effect equal absolute amounts of lagging and equal ratios of diminution. As the annual range at the surface in Great Britain is usually about 3 times greater than the diurnal range, it follows that the diurnal range at the depth of a foot should be about one-third of the annual range at the depth of 19 feet. The variations of temperature at the surface are, as every one knows, of a very irregular kind; so that the curve of surface tem- perature for any particular year is full of sinuosities depending on the accidents of that year. The deeper we go, the more regular does the curve become, and the more nearly does it approach to the char- acter of a simple curve of sines, whose equation can be written y = a sin. x. Neglecting the departures of the curve from this simple character, theory indicates that, if the soil be uniform, and the surface plane, the annual range (which is equal to 2 a) goes on diminishing in geo- metrical progression as the depth increases in arithmetical; and observation shows that, if 10 feet be the common difference of depth, the ratio of decrease for range is usually about \ or \. To find a range of a tenth of a degree Fahrenheit, we must go to a depth of from 50 to 80 feet in this climate. At a station where the surface range is double what it is in Great Britain, we should find a range of about two-tenths of a degree at a depth and in a soil which would here give one-tenth. These remarks show that the phrase " stratum of invariable tem- perature," which is frequently employed to denote the supposed 522 TERRESTRIAL TEMPERATURES. lower boundary of the region in which annual range is sensible, has no precise significance, inasmuch as the boundary in question will vary its depth according to the sensitiveness of the thermometer employed. 548. Increase of Temperature Downwards. Observations in all parts of the world show that the temperature at considerable depths, such as are attained in mining and boring, is much above the surface temperature. In sinking a shaft at Rose Bridge Colliery, near Wigan, which is the deepest mine in Great Britain, the temperature of the rock was found to be 94 F. at the depth of 2440 feet. In cutting the Mont Ce'nis tunnel, the temperature of the deepest part, with 5280 feet of rock overhead, was found to be about 85 F. The rate of increase downwards is by no means the same every- where; but it is seldom so rapid as 1 F. in 40 feet, or so slow as 1 F. in 100 feet. The observations at Rose Bridge show a mean rate of increase of about 1 in 55 feet; and this is about the average of the results obtained at other places. This state of things implies a continual escape of heat from the interior of the earth by conduction, and the amount of this loss per annum can be approximately calculated from the absolute values of conductivity of rock which we have given in Chap. xxxv. There can be no reasonable doubt that the decrease of temperature upwards extends to the very surface, when we confine our attention to mean annual temperatures, for all the heat that is conducted up through a stratum at any given depth must also traverse all the strata above it, and heat can only be conducted from a warmer to a colder stratum. Professor Forbes found, at his three stations near Edinburgh, increases of 1'38, 0> 96, and 0< 19 F. in mean temperature, in descending through about 22 feet, that is, from the depth of 3 to the depth of 24 French feet. The mean annual temperature of the surface of the ground is in Great Britain a little superior to that of the air above it, so far as present observations show. The excess appears to average about 1 F. 549. Decrease of Temperature Upwards in the Air. In comparing the mean temperatures of places in the same neighbourhood at dif- ferent altitudes, it is found that temperature diminishes as height increases, the rate of decrease for Great Britain, as regards mean annual temperature, being about 1 F. for every 300 feet. A decrease of temperature upwards is also usually experienced in balloon ascents, and numerous observations have been taken for the purpose of deter- TEMPERATURE ALOFT. 523 mining its rate. Mr. Glaisher's observations, which are the most numerous as well as the most recent, show that, upon the whole, the decrease becomes less rapid as we ascend higher; also, that it is less rapid with a cloudy than with a clear sky. The following table exhibits a few of Mr. Glaisher's averages: Decrease of Temperature TJpwarda. Height. With clear sky. With cloudy sky. From to 1000 feet, . . . 1 F. in 139 feet. 1 F. in 222 feet. From to 10,000 ft. ... 1 F. in 288 feet. 1 F. in 33i feet From to 20,000 ft. ... 1 F. in 365 feet. 1 F. in 468 feet. These rates may be taken as representing the general law of decrease which prevails in the air over Great Britain in the daytime during the summer half of the year; but the results obtained on different days differ widely, and alternations of increase and decrease are by no means uncommon in passing upwards through successive strata of air. Still more recent observations by Mr. Glaisher, relating chiefly to the first 1000 feet of air, show that the law varies with the hour of the day. The decrease upwards is most rapid soon after midday, and is at this time, and during daytime generally, more rapid as the height is less. About sunset there is a uniform decrease at all heights if the sky is clouded, and a uniform temperature if the sky is clear. From a few observations which have been taken after sunset, it appears that, with a clear sky, there is an increase upwards at night. That an extremely low temperature exists in the interplanetary spaces, may be inferred from the experimental fact recorded by Sir John Herschel, that a thermometer with its bulb in the focus of a reflector of sufficient size and curvature to screen it from lateral radiation, falls lower when the axis of the reflector is directed upwards to a clear sky than when it is directed either to a cloud or to the snow- clad summits of the Alps. The atmosphere serves as a protection against radiation to these cold spaces, and it is not surprising that, as we in- crease our elevation, and thus diminish the thickness of the coating of air above us, the protection should be found less complete. But pro- bably the principal cause of the diminution of temperature upwards is the cooling of air by expansion, which we have discussed in 502. 550. Causes of Winds. The influences which modify the direction and intensity of winds are so various and complicated that anything like a complete account of them can only find a place in treatises specially devoted to that subject. There is, however, one fundamental 524 TERRESTRIAL TEMPERATURES. principle which suffices to explain the origin of many well-known winds. This principle is plainly illustrated by the following experi- ment, due to Franklin. A door between two rooms, one heated, and the other cold (in winter), is opened, and two candles are placed, one at the top, and the other at the bottom of the doorway. It is found that the flame of the lower candle is blown towards the heated room, and that of the upper candle away from it. The principle which this experiment illustrates may be stated as follows : When two neighbouring regions are at different tempera- tures, a current of air flows from the warmer to the colder in the upper strata of the atmosphere; and in the lower strata a current flows from the colder to the warmer. The reason is that variation of pressure with height is greater in the cold than in the hot region; so that if there be one level at which the pressure is the same in both, the pressure in the cold region will preponderate at lower and that in the hot region at higher levels. We proceed to apply this prin- ciple to the land and sea breezes, the monsoons, and the trade-winds. 551. Land and Sea Breezes. At the sea-side during calm weather a wind is generally observed to spring up at about eight or nine in the morning, blowing from the sea, and increasing in force until about two or three in the afternoon. It then begins gradually to die away, and shortly before sunset disappears altogether. A few hours after- wards, a wind springs up in the opposite direction, and lasts till nearly sunrise. These winds, which are called the sea-breeze and land-breeze, are exceedingly regular in their occurrence, though they may sometimes be masked by other winds blowing at the same time. Their origin is very easily explained. During the day the land grows warmer than the water; hence there results a wind blowing towards the warmer region, that is, towards the land. During the night the land and sea both grow colder, but the former more rapidly than the latter ; and, accordingly, the relative temperatures of the two elements being now reversed, a breeze blowing from the land towards the sea is the consequence. Monsoons. The same cause which, on a small scale, produces the diurnal alternation of land and sea breezes, produces, on a larger scale, the annual alternation of monsoons in the Indian Ocean, and the seasonal winds which prevail in some other parts of the world. The general direction of these winds is towards continents in summer, and away from them in winter. 552. Trade-winds: General Atmospheric Circulation. The trade- TRADE WINDS. 525 winds are winds which blow constantly from a north-easterly quarter over a zone of the northern hemisphere extending from a litcle north of the tropic of Cancer to within 9 or 10 degrees of the equator; and from a south-easterly quarter over a zone of the southern hemisphere extending from about the tropic of Capricorn to the equator. Their limits vary slightly according to the time of year, changing in the same direction as the sun's declination. Between them is a zone some 5 or 6 wide, over which calms and variable winds prevail. The cause of the trade-winds was first correctly indicated by Hadley. The greater power of the sun over the equatorial regions causes a continual ascent of heated air from them. This flows over to both sides in the upper regions of the atmosphere, and its place is supplied by colder air flowing in from both sides below. If the earth were at rest, we should thus have a north wind sweeping over the earth's surface on the northern side of the equatorial regions, and a south wind on the southern side. But, in virtue of the earth's rotation, all points on the earth's surface are moving from west to east, with velocities proportional to their distances from the earth's axis. This velocity is nothing at the poles, and increases in approach- ing the equator. Hence, if a body on the earth's surface, and origi- nally at rest relatively to the earth, be urged by a force acting along a meridian, it will not move along a meridian, but will outrun the earth, or fall behind it, according as its original rotational velocity was greater or less than those of the places to which it comes. That is to say, it will have a relative motion from the west if it be approach- ing the pole, and from the east if it be approaching the equator. This would be true, even if the body merely tended to keep its original rotational velocity unchanged, and the reasoning becomes still more forcible when we apply the principle of conservation of angular momentum, in virtue of which the body tends to increase 1 its absolute rotational velocity in approaching the pole, and to dim- inish it in approaching the equator. Thus the currents of air which flow in from both sides to the equatorial regions, do not blow from due north and due south, but from north-east and south-east. There can be little doubt that, not- withstanding the variable character of the winds in the temperate and frigid zones, there is, upon the whole, a continual interchange of air between them and the intertropical regions, brought about by the permanent excess of temperature of the latter. Such an interchange, 1 The tendency is for velocity to vary inversely as distance from the axis of rotation. 526 TERRESTRIAL TEMPERATURES. when considered in conjunction with the difference in the rotational velocities of these regions, implies that the mass of air over an equatorial zone some 50 or 60 wide, must, upon the whole, have a motion from the east as compared with the earth beneath it; and that the mass of air over all the rest of the earth must, upon the whole, have a relative motion from the west. This theoretical conclusion is corroborated by the distribution of barometric pressure. The barom- eter stands highest at the two parallels which, according to this theory, form the boundaries between easterly and westerly winds, while at the equator and poles it stands low. This difference may be accounted for by the excess of centrifugal force possessed by west winds, and the defect of centrifugal force in east winds. If the air simply turned with the earth, centrifugal force combined with gravity would not tend to produce accumulation of air over any particular zone, the ellipticity of the earth being precisely adapted to an equable distribution. But if a body of air or other fluid is moving with sensibly different rotational velocity from the earth, the difference in centrifugal force will give a tendency to move towards the equator, or from it, according as the differential motion is from the west or from the east. The easterly winds over the equatorial zone should therefore tend to remove air from the equator and heap it up at the limiting parallels; and the westerly winds over the remainder of the earth should tend to draw air away from the poles and heap it up at the same limiting parallels. This theore- tical consequence exactly agrees with the following table of mean barometric heights in different zones given by Maury: 1 North Latitude. I to 5 Jarometer. 29-915 29-922 29-964 30-018 30-081 30-149 30-210 30-124 30-077 30-060 29-99 29-88 29-759 Meteorology o South Latitude. to 5 . . Barometer. . . . 29'940 5 to 10 5 to 10 . . . . 29-981 10 to 15 10 to 15 . . . . 30'028 15 to 20 15 to 20 . . . . 30-060 20 to 25 20 to 25 30-102 25 to 30 25 to 30 30-095 30 to 35 30 to 36 . 30-052 35 to 40" 42 53' . . 29-90 40 to 45 45 0' . . 29-66 45 to 50 49 8' . . . . . 29-47 51 29' 51 33' . . . . . . 29-50 59 51' 54 26' . . . . 29-35 78 37' 55 52' . . . . . . 99-36 1 Physical Geography and 60 0' . . . . . . 29-11 66 0' . . . . . . 29-08 74 0' . . . . 28'93 fthe Sea, p. 180, art. 362, edition 1860. INDRAUGHT TO POLES. 527 This table shows that the barometric height falls off regularly on both sides frpm the two limiting zones 30 to 35 N. and 20 to 25 S., the fall continuing towards both poles as far as the observations extend, and continuing inwards to a central minimum between and 5 N. 1 If the bottom of a cylindrical vessel of water be covered with saw- dust, and the water made to rotate by stirring, the saw-dust will be drawn away from the edges, and heaped up in the middle, thus showing an indraught of water along the bottom towards the region of low barometer in the centre. It is probable that, from a similar cause (a central depression due to centrifugal force), there is an indraught of air along the earth's surface towards the poles, under- neath the primary circulation which our theory supposes; the diminu- tion of velocity by friction against the earth, rendering the lowest portion of the air obedient to this indraught, which the upper strata are enabled to resist by the centrifugal force of their more rapid motion. This, according to Professor James Thomson, 2 is the ex- planation of the prevalence of south-west winds in the north tem- perate zone; their southerly component being due to the barometric indraught and their westerly component to differential velocity of rotation. The indraught which also exists from the limiting parallels to the region of low barometer at the equator, coincides with the current due to difference of temperature; and this coincidence may be a main reason of the constancy of the trade-winds. 553. Origin of Cyclones. In the northern hemisphere a wind which would blow towards the north if the earth were at rest, does actually blow towards the north-east; and a wind which would blow towards the south blows towards the south-west. In both cases, the earth's rotation introduces a component towards the right with reference to a person travelling with the wind. In the southern hemisphere it introduces a component towards the left. Again, a west wind has an excess of centrifugal force which tends to carry it towards the equator, and an east wind has a tendency to move towards the pole; so that here again, in the northern hemi- 1 The explanation here given of the accumulation of air towards the limiting parallels, as due to excess and defect of centrifugal force, appears to have been first published by Mr. W. Ferrel, a gentleman connected with the American Nautical Almanack. His later treatise (1860), reprinted from vols. i. ii. of the Mathematical Monthly, is the most com- plete exposition we have seen of the theory of general atmospheric circulation. * Brit. Assoc. Report, 1857. 528 TERRESTRIAL TEMPERATURES. sphere the deviation is in both cases to the right, and in the southern hemisphere to the left. We have thus an explanation of cyclonic movements. In the northern hemisphere, if a sudden diminution of pressure occurs over any large area, the air all around for a considerable distance receives an impetus directed towards this area. But, before the converging streams can meet, they undergo deviation, each to its own right, so that, instead of arriving at their common centre, they blow tangeii- tially to a closed curve surrounding it, and thus produce an eddy from right to left with respect to a person standing in the centre. This is the universal direction of cyclonic rotation in the northern hemisphere; and the opposite rule holds for the southern hemisphere. The former is opposite to, the latter the same as the direction of motion of the hands of a watch lying with its face up. In each case the motion is opposite to the apparent diurnal motion of the sun for the hemisphere in which it occurs. 554. Anemometers. Instruments for measuring either the force or the velocity of the wind are called anemometers. Its force is usually measured by Osier's anemometer, in which the pressure of the wind is received upon a square plate attached to one end of a spiral spring (with its axis horizontal), which yields more or less according to the force of the wind, and transmits its motion to a pencil which leaves a trace upon paper moved by clock-work. It seems that the force received by the plate is not rigorously propor- tional to its size, and that a plate a yard square receives rather more than 9 times the pressure of a plate a foot square. The anemometer which has yielded the most satisfactory results is that invented by the Rev. Dr. Robinson of Armagh, which is represented in Fig. 331, and which indicates the velocity of the wind. It con- sists of four hemispherical cups attached to the ends of equal hori- zontal arms, forming a horizontal cross, which turns freely about a vertical axis. By means of an endless screw carried by the axis, a train of wheel- work is set in motion; and the indication is given by a hand which moves round a dial; or, in some instruments, by several hands moving round different dials like those of a gas-meter. The anemometer can also be made to leave a continuous record on paper, for which purpose various contrivances have been successfully employed. It was calculated by the inventor, and confirmed by his own experiments both in air and water, as well as by experi- ments conducted by Prof. C. Piazzi Smyth at Edinburgh, and more ANEMOMETER 529 recently by the astronomer-royal at Greenwich, that the centre of each cup moves with a velocity which is almost exactly one-third of that of the wind. This is the only velocity - anemometer whose indications are exactly proportional to the velocity itself. Dr. Whewell's anemo- meter, which resembles a small windmill, is very far from ful- filling this condition, its varia- tions of velocity being much less than those of the wind. The direction of the wind, as indicated by a vane, can also be made to leave a continuous record by various contrivances; one of the most common bein Fig. 331. - Robinson's Anemometer. a pinion carried by the shaft of the vane, and driving a rack which carries a pencil. But perhaps the neatest ar- rangement for this purpose is a large screw with only one thread composed of a metal which will write on paper. A sheet of paper is moved by clock-work in a direction perpendicular to the axis of the screw, and is pressed against the thread, touching it of course only in one point, which travels parallel to the axis as the screw turns, and comes back to its original place after one revolution. When one end of the thread leaves the paper, the other end at the same instant comes on. The screw turns with the vane, so that a com- plete revolution of the screw corresponds to a complete revolution of the wind. This is one of the many ingenious contrivances devised and executed by Mr. Beckley, mechanical assistant in Kew Observatory. 555. Oceanic Currents. The general principle of 550 applies to liquids as well as to gases; though the effects are usually smaller, owing to their smaller expansibility. The warm water in the equatorial regions overflows towards the poles, and an under-current of cold water which has descended in the polar regions flows towards the equator. Recent observations 34 530 TERRESTRIAL TEMPERATURES. have shown that a temperature not much above C. prevails at the bottom of the ocean even between the tropics. A very gradual circulation is thus produced on a very large scale. The rapid currents which are observed on some parts of the sur- face of the ocean are probably due to wind. Among these may be mentioned the Gulf Stream. This current of warm water forms a kind of immense river in the midst of the sea, differing in the tem- perature, saltness, and colour of its waters from the medium in which it flows. Its origin is in the Gulf of Mexico, whence it issues through the straits between the Bahamas and Florida, turns to the north- west, and splits into two branches, one of which goes to warm the coasts of Ireland and Norway, the other gradually turns southwards, traverses the Atlantic from north to south, and finally loses itself in the regions of the equator. " The Gulf Stream is a river in the ocean ; in the severest droughts it never fails, and in the mightiest floods it never overflows; its banks and its bottom are of cold water, while its current is of warm ; it takes its rise in the Gulf of Mexico, and empties into Arctic seas. There is on earth no other such majestic flow of waters. Its current is more rapid than the Mississippi or the Amazon, and its volume more than a thousand times greater. Its waters, as far out from the Gulf as the Carolina coasts, are of indigo blue. They are so distinctly marked that their line of junction with the common sea- water may be traced by the eye. Often one-half of the vessel may be perceived floating: in Gulf Stream water, while the other half is in common O water of the sea, so sharp is the line." (Maury, Physical Geography of the Sea.) It would appear that an accumulation of water is produced in the Gulf of Mexico by the trade-wind which blows steadily towards it over the South Atlantic, and that the elevation of level thus occa- sioned is the principal cause of the Gulf Stream. EXAMPLES. [The Centigrade Scale is employed, except where otJierwise stated.] SCALES OF TEMPERATURE. 1. The difference between the temperatures of two bodies is 30 F. Express this difference in degrees Cent, and in degrees Reau. 2. The difference between the temperatures of two bodies is 12 C. Express this difference in degrees Rau. and in degrees Fahr. 3. The difference between the temperatures of two bodies is 25 R. Express this difference in the Cent, and Fahr. scales. 4. Express the temperature 70 F. in the Cent, and Reau. scalos. 5. Express the temperature 60 C. in the Reau. and Fahr. scales. 6. Express the temperature 30 R. in the Cent, and Fahr. scales. 7. Air expands by '00366 of its volume at the freezing-point of water for each degree Cent. By how much does it expand for each degree Fahr. ? 8. The temperature of the earth increases by about one degree Fahr. for every 50 feet of descent. How many feet of descent will give an increase of 1 Cent., and how many centimetres of descent will give an increase of 1 Cent., the foot being 30-48 cm.? 9. The mean annual range of temperature at a certain place is 100 F. What is this in degrees Cent. ? 10. Lead melts at 326 C., and in melting absorbs as much heat as would raise 5'37 times its mass of water 1 C. What numbers will take the place of 326 and 5'37 when the Fahrenheit scale is employed? 11. Show that the temperature -40 C. and the temperature -40 F. are identical. 12. What temperature is expressed by the same number in the Fahr. and Ri-au. scales? EXPANSION. The following coefficients of expansion can be used :- Linear. Cubical. Steel, -0000116 Copper, -0000172 Brass, '0000188 Glass -0000080 Glass, -000024 Mercury, -000179 Alcohol, -001050 Ether, -00152C 13. The correct length of a steel chain for land measuring is 66 ft. Express, as a decimal of an inch, the difference between the actual lengths of such a chain at 0" and 20. 532 EXAMPLES. 14. One brass yard-measure is correct at and another at 20. Find, as a decimal of an inch, the difference of their lengths at the same temperature. 15. A lump of copper has a volume 258 cc. at 0. Find its volume at 100. 16. A glass vessel has a capacity of 1000 cc. at 0. What is its capacity at 10? 17. A weight-thermometer contains 462 gm. of a certain liquid at and only 454 gm. at 20. Find the mean relative expansion per degree between these limits. 18. A weight-thermometer contains 325 gm. of a liquid at zero, and 5 gm. run out when the temperature is raised to 12. Find the mean coefficient of apparent expansion. 19. If the coefficient of relative expansion of mercury in glass be ^o^, what mass of mercury will overflow from a weight-thermometer which contains 650 gm. of mercury at when the temperature is raised to 100? 20. The capacity of the bulb of a thermometer together with as much of the stem as is below zero is - 235 cc. at 0, and the section of the tube is ^innr S( l- cm - Compute the length of a degree (1), if the fluid be mercury; (2), if it be ether. 21. The bulb, together with as much of the stem as is below the zero-point, contains 3'28 gm. of mercury at zero, and the length of a degree is '1 cm. Com- pute the section of the tube, the density of mercury being about 13'6. 22. What will be the volume at 300 of a quantity of gas which occupies 1000 cc. at 0, the pressure being the same? 23. What will be the volume at 400 of a quantity of gas which occupies 1000 cc. at 100, the pressure being the same? 24. What will be the pressure at 30 of a quantity of gas which at has a pressure of a million dynes per sq. cm., the gas being confined in a close vessel whose expansion may be neglected ? 25. A thousand cc. of gas at T0136 million dynes per sq. cm. are allowed to expand till the pressure becomes a million dynes per sq. cm., and the temperature is at the same time raised from its initial value to 100. Find the final volume. 26. A gas initially at volume 4500 cc., temperature 100, and a pressure repre- sented by 75 cm. of mercury, has its pressure increased by 1 cm. of mercury and its temperature raised to 200. Find its final volume. 27. At what temperature will the volume of a gas at a pressure of a million dynes per sq. cm. be 1000 cc., if its volume at temperature and pressure 1'02 million dynes per sq. cm. be 1200 cc. 1 28. What temperature on the Fahrenheit scale is the absolute zero of the air- thermometer ? 29. Find the coefficient of expansion of air per degree Fahrenheit, when F. is the starting-point. 30. Express the freezing-point and boiling-point of water as absolute tempera- tures Fahrenheit. 31. What is the interior volume at C. of a glass bulb which at 25 C. is exactly filled by 53 grammes of mercury ? FOR DENSITIES or GASES SEE p. 297. 32. At what temperature does a litre of dry air at 760 mm. weigh 1 gramme? 33. At what temperature will the density of oxygen at the pressure 0'20 u>. be the same as that of hydrogen at C., at the pressure T60 m. I EXAMPLES. 533 DT [The tabulated densities are proportional to the values of -p- for the different gases.] 34. What must be the pressure of air at 15, that its density may be the same as that of hydrogen at and 760 mm.? 35. A mercurial barometer with brass scale reads at one time 770 mm. with a temperature 85, and at another time 760 mm. with a temperature 5. Find the ratio of the former pressure to the latter. 36. The normal density of air being '000154 of that of brass, what change is produced in the force required to sustain a kilogramme of brass in air, when the pressure and temperature change from 713 mm. and 19 to 781 mm. and +36? 37. A cylindrical tube of glass is divided into 300 equal parts. It is loaded with mercury, aud sinks to the 50th division from the top in water at 10. To what division will it sink in water at 50, the volumes of a given mass of water at these temperatures being as T000268 to 1*01205 1 38. A closed globe, whose external volume at is 10 litres, is immersed in air at 15 and at a pressure of 0'77 m. Eequired (1) the loss of weight which it experiences from the action of the air; (2) the change which this loss would undergo if the pressure became 0'768 m. and the temperature 17. 39. A brass tube contains mercury, with a piece of platinum immersed in it; and the level of the liquid is marked by a scratch on the inside of the tube. On applying heat, it is found that the liquid still stands at this mark. Deduce the ratio of the weight of the platinum to that of the mercury, assuming the density of mercury to be 21'5, and its linear expansion '00001 per degree. 40. A glass tube, closed at one end and drawn out at the other, is filled with dry air, and raised to a temperature x at atmospheric pressure. It is then her- metically sealed. When it has been cooled to the temperature 100 C., it is inverted over mercury, and its pointed end is broken off beneath the surface of the liquid. The mercury rises to the height of 19 centimetres in the tube, the external pressure remaining at 76 cm. as at the commencement of the experiment. The tube is re-inverted, and weighed with the mercury which it contains. The weight of this mercury is found to be 200 grammes ; when completely full it con- tains 300 grammes of mercury. Deduce the temperature x. 41. A glass tube, whose interior is a right circular cylinder, 2 millimetres in diameter at C., contains a column of mercury, whose length at this temperature is 2 decim. What will be the length of this column of mercury when the tem- perature is 80 C.? 42. Some dry air is inclosed in a horizontal thermometric tube, by means of an index of mercury. At C. and 0760 m. the air occupies 720 divisions of the tube, the tube being divided into parts of equal capacity. At an unknown temperature and pressure the same air occupies 960 divisions. The tube being immersed in melting ice, and the latter pressure being still maintained, the air occupies 750 divisions. Required the temperature and pressure. 43. A Graham's compensating pendulum is formed of an iron rod, whose length at C. is I, carrying a cylindrical vessel of glass, which at the same temperature has an internal radius r, and height h. Find the depth x of mercury at C. which is necessary for compensation, supposing that the compensation consists in keeping the centre of gravity of the mercury at a constant distance from the axis of suspension. 534 EXAMPLES. THERMAL CAPACITY. The following values of specific heat can be used : Iron, -1098 Copper, '0949 Platinum, '0335 Sand, -215 Ice '504 Mercury, ...... '033 Alcohol, -548 Ether, '529 Air, at constant pressure, "2375 44. 17 parts by mass of water at 5 are mixed with 23 parts at 12. Find the resulting temperature. 45. 200 gm. of iron at 300 are immersed in 1000 gm. of water at 0. Find the resulting temperature. 46. Find the specific heat of a substance 80 gm. of which at 100, when im- mersed in 200 gm. of water at 10 give a resulting temperature of 20. 47. 16 parts by mass of sand at 75, and 20 of iron at 45 are thrown into 50 of water at 4. Find the temperature of the mixture. 48. 300 gm. of copper at 100 are immersed in 700 gm. of alcohol at 0. Find the resulting temperature. 49. If the length, breadth, and height of a room are respectively 6, 5, and 3 metres, how many gramme-degrees of heat will be required to raise the tempera- ture of the air which fills the room by 20, the pressure of the air being constant, and its average density '00128 gm. per cubic centimetre? 50. Find the thermal capacities of mercury, alcohol, and ether per unit vol- ume, their densities being respectively 13'6, "791, and '716. LATENT HEAT. The following values of latent heat can be used : In Melting. In Evaporation at Atmospheric Pressure. Water, 80 Lead, 5'4 Steam, 536 51. Find the result of mixing 5 gm. of snow at with 23 gm. of water at 20. 52. Find the result of mixing 6 parts (by mass) of snow at with 7 of water at 50. 53. Find the result of mixing 3 parts by mass of snow at - 10 with 8 of water at 40. 54. Find the result of mixing equal masses of snow at 10 and water at G0. 55. Find the temperature obtained by introducing 10 gm. of steam at 100 into 1000 gm. of water at 0. 56. Lead melts at 326. Its specific heat is '0314 in the solid, and '0402 in the liquid state. Find what mass of water at will be raised one-tenth of a degree by dropping into it 100 gm. of melted lead at 350. 57. What mass of mercury at will be raised 1 by dropping into it 150 gm. of lead at 400? 58. A litre of alcohol, measured at C., is contained in a brass vessel weigh- ing 100 grammes, and after being raised to 58 C., is immersed in a kilogramme EXAMPLES. 535 of water at 10 C., contained in a brass vessel weighing 200 grammea. The tem- perature of the water is thereby raised to 27. What is the specific heat of alcohol? The specific gravity of alcohol is 0'8 ; the specific heat of brass is O'l. 59. A copper vessel, weighing 1 kilogramme, contains 2 kilogr. of water. A thermometer composed of 100 grammes of glass and 200 gr. of mercury, is com- pletely immersed in this water. All these bodies are at the same temperature, C. If 100 grammes of steam at 100 C. are passed into the vessel, and con- densed in it, what will be the temperature of the whole apparatus when equili- brium has been attained, supposing that there is no loss of heat externally. The specific heat of mercury is 0'033 ; of copper, 0'095; of glass, 0'177. VARIOUS. 60. A truly conical vessel contains a certain quantity of mercury at C. To what temperature must the vessel and its contents be raised that the depth of the liquid may be increased by T J 5 of itself? 61. There is a bent tube, terminating at one end in a large bulb, and simply closed at the other. A column of mercury stands at the same height in -the two branches, and thus separates two quantities of air at the same pressure. The air in the bulb is saturated with moisture ; that in the opposite branch is perfectly dry. The length of the column of dry air is known, and also its initial pressure, the temperature of the whole being C. Calculate the displacement of the mercurial column when the temperature of the apparatus is raised to 100 C. The bulb is supposed to have enough water in it to keep the air constantly saturated ; and is also supposed to be so large that the volume of the moist air is not sensibly affected by the displacement of the mercurial column. CONDUCTION. (Units the centimetre, gramme, and second.) 62. How many gramme-degrees of heat will be conducted in an hour through each sq. cm. of an iron plate '02 cm. thick, its two sides being kept at the respec- tive temperatures 225 and 275, and the mean conductivity of the iron between these temperatures being '12? 63. Through what thickness of copper would the same amount of heat flow as through the "02 cm. of iron in the preceding question, with the same tempera- tures of its two faces, the mean conductivity of the copper between these tem- peratures being unity? 64. How much heat will be conducted in an hour through each sq. cm. of a plate of ice 2 cm. thick, one side of the ice being at and the other at - 3, and its conductivity being '00223 ; and what volume of water at would be converted into ice at by the loss of this quantity of heat? 65. How much heat will escape in an hour from the walls of a building, if their area be 80 sq. metres, their thickness 20 cm., their material sandstone of conductivity '01, and the difference of temperature between outside and inside 15? What quantity of carbon burned per hour would generate heat equal to this loss? 536 EXAMPLES. HYGROMETRY. 66. A cubic metre of air at 20 is found to contain 11*56 grammes of aqueous vapour. What is the relative humidity of this air, the maximum pressure of vapour at 20 being 17'39 mm.? 67. Calculate the weight of 15 litres of air saturated with aqueous vapour at 20 and 750 mm. THERMODYNAMICS. For the value of Joule's equivalent see 487. For heats of combustion see 509. 68. The labour of a horse is employed for 3 hours in raising the temperature of a million grammes of water by friction. What elevation of temperature will be produced, supposing the horse to work at the rate of 6 x 10 9 ergs per second? 69. From what height (in cm.) must mercury fall at a place where g is 980, in order to raise its own temperature 1 by the destruction of the velocity acquired, supposing no other body to receive any of the heat thus generated? 70. With what velocity (in cm. per sec.) must a leaden bullet strike a target that its temperature may be raised 100 by the collision, supposing all the energy of the motion which is destroyed to be spent in heating the bullet? 71. What is the greatest proportion of the heat received by an engine at 200 that can be converted into mechanical effect, if the heat which is given out from the engine is given out at the temperature 10? 72. If a perfect engine gives out heat at 0, at what temperature must it take in heat that half the heat received may be converted? 73. What mass of carbon burned per hour would produce the same quantity of heat as the work of one horse for the same time, a horse-power being taken as 75 x 10 8 ergs per second. 74. A specimen of good coal contains 88 per cent, of carbon and 4 per cent, of hydrogen not already combined with oxygen. How many gramme-degrees of heat are generated by the combustion of 1 gm. of this coal; and with what velocity must a gramme of matter move that the energy of its motion may be equal to the energy developed by the combustion of the said gramme of coal? ADIABATIC COMPRESSION AND EXTENSION. 75. Find the rise of temperature produced in water at 10 C. by an atmosphere of additional pressure, an atmosphere being taken as a million dynes per sq. cm., and the coefficient of expansion at this temperature being '000092. 76. Find the ratio of the adiabatic to the isothermal resistance of water at 10 to compression, the value of the latter being 2'1 x 10 10 dynes per sq. cm. 77. Find the fall of temperature produced in a wrought iron bar by applying a pull of a million dynes per sq. cm. of section, the coefficient of expansion being 0000122. 78. Find the ratio of the adiabatic to the isothermal resistance of the bar to extension, the value of the latter being T96 x 10 12 dynes per sq. cm. EXAMPLES. 537 ANSWERS TO EXAMPLES. Ex. 1. 16 C., 13 E. Ex. 2. 9f C., 21f F. Ex. 3. 31 C., 56J F. Ex. 4. 21 C., 16f E. Ex. 5. 48 E., 140 F. Ex. 6. 37 C., 99| F. Ex. 7. '00203. Ex. 8. 90 ft., 2743 cm. Ex. 9. 55| C. Ex. 10. 619, 9'666. Ex. 12. -25'6. Ex. 13. '184 in. Ex. 14. '0135 in. Ex. 15. 259'33 cc. Ex. 16. 1000'24 cc. Ex. 17. -000881. Ex. 18. '001302. Ex. 19. <^ = 9'85 gm. Ex. 20. (1) '084 cm., (2) -714 cm. Ex. 21. -000432 sq. cm. Ex. 22. 2098 cc. Ex. 23. 1804 cc. Ex. 24. M098 million. Ex. 25. 1385 cc. Ex. 26. 5631 cc. Ex. 27. -50. Ex. 28. -459. Ex. 29. ^. Ex. 30. 491, 671. Ex. 31. 3-913 cc. Ex. 32. 80 C. Ex. 33. 272. Ex. 34. 55'5 mm. Ex. 35. 759'53 : 759'39. Ex. 36. '155 - -140 = '015 grammes of increase in the apparent weight. Ex. 37. 47'36. Ex. 38. Loss of 12'42 grn., diminished by '12 gm. Ex. 39. The ratio of the platinum to the mercury is4'7 to 1 by volume, and 7'5 to 1 by weight. Ex. 40. 1219. Ex. 41. 2-0248 decim. Ex. 42. 76'5, '7296 m. Ex. 43. '150Z + '103A. Ex. 44. 9'02. Ex. 45. 6'44. Ex. 46. ^ = '3125. Ex. 47. 10. Ex. 48. 6'91. Ex. 49. 547200. Ex. 50. '449, '433, '379. Ex. 51. Water at 2|. Ex. 52. If part snow, llf water, all at zero. Ex. 53. Water at 5'9. Ex. 54. '313 snow, 1-687 water, all at zero. Ex. 55. Water at 6'3. Ex. 56. 16600 gm. nearly. Ex. 57. 84400 gm. nearly. Ex. 58. '687. Ex. 59. 28'7. Ex. 60. 88. Ex. 61. The displacement x is given by the equation 2.r = 753*7 -f- ,p and I being the given pressure and 273 I x length. Ex. 62. 1080000. Ex. 63. cm. = '1666 cm. Ex. 64. 11'88 gm.-deg., -149 cc. Ex. 65. 21600000 gm.-deg., 2700 gm. Ex. 66. 67 per cent. Ex. 67. 17'68 gm. Ex. 68. l-54. Ex. 69. 1414 cm. Ex. 70. 16240 cm. per sec. Ex. 71. }& = 4 nearly. Ex. 72. 273. Ex. 73. 80'36 gm. Ex. 74. 8570 gm.-deg., 848400 cm. per sec. nearly. Ex. 75. -000626. Ex. 76. 1'0012. Ex. 77. 0'00009. Ex. 78. 1-002. INDEX TO PAET II. Absolute temperature and abso- lute zero by air-thermometer, 301. by thermo - dynamic scale, 473- Absorbing powers, 443. Absorption and emission, 437. equality of, 454. Actinometer, 486. Adiabatic changes in gases, 474. liquids and solids, 477- 480. Air, cooling of, by ascent, 476. density of dry, 305. of moist, 400. temperature of, 518, 523. Air-engine, 491. Air-thermometer, 301. Alcohol at low temperatures, 350. thermometers, 267, 280. Alum, its small diathermancy, 450, 453- Andre ws'calorimetric experiments, 483. on continuity of liquid and gaseous states, 351. Anemometers, 528. Animal heat and work, 485. Apjohn's Formula, 397. Ascent, cooling of air by, 476. Aspirator, 395. Atmosphere, distribution of, over the earth, 526. Atmospheric circulation, general, 525- Atomic weight inversely as specific heat, 318. August's psychrometer, 395. Bar, flow of heat in, 426. Barometric variation with lati- tude, 526. Boiler of steam-engine, 506-508. Boiling, 355, 357. by bumping, 365. explosive, 363. promoted by presence of air, 362. Boiling-points, affected by pres- sure, 358. heights determined by, 359. of solutions, 362. table of, 356. Bottomley's ice experiment, 333. Boutigny's experiments, 366. Breezes, land and sea, 524. Breguet's thermometer, 274. Burning mirrors, 435. Bursting by freezing, 330. Bursting of boilers, 507. Cagniard de Latour's experiments on vaporization, 351. Calibration. 261. Calorescence, 454. Caloric theory, 457. Calorimeter, 313. Calorimetry, 310. Capacity, thermal, 311. specific, 312. Carbonic acid, solidification of, 35- Carnot's principles, 467. Carre's two freezing-machines, 347, 349- Centigrade scale, 265. Centrifugal governor, 498. Chemical combination, 482. hygrometer, 399. Cherra Ponjee, rainfall at, 406. Chimneys, draught of, 306. Climates, insular and continental. 520. Clothing, warmth of, 424. Clouds, 402-406. Coal, origin of, 486. Coefficient of expansion, 282. table of, 284. Cold of evaporation, 345. Combination by volume, 378. Combination, heat of, 482. table of, 484. Combustion, heat of, 484. Comparability of thermometers, 280. Compensated pendulums, 284. Compound engines, 501. Condensation, 342. Condenser of steam-engine, 498, 502. Conduction of heat, 412. in gases, 424. in liquids, 422. Conductivity defined, 413. determinations of, 419-424. Congelation, 325. at temperatures below freezing, 326. Conjugate mirrors, 436. Continental climates, 520. Continuity of gaseous and liquid states, 351. Convection of heat, 295, 422, 524. Convective equilibrium of air, 476. Cooling, law of, 430-432. method of, 319. of air by ascent, 476. Critical temperature, Andrews', 352- Cryophorus, 348. Crystallization, 327. Currents, marine, 529. Cyclones, 527. Dalton's experiments on vapours, 370. laws of vapours, 342. Daniell's hygrometer, 393. Davy lamp, 416. on friction of ice, 459. Dead points, 496 Deep-water thermometers, 271. Degree of thermometer, 279, 280. Delaroche's value of specific heat of air, 398. Delicacy of thermometer, 267. Density, see Air, Vapour. correction of, for temperature, 278. of air, 305, 400. of gases, 302. table of, 306. of vapours, 379. De Saussure's hygrometer, 392. Despretz's experiments on alcohol at low temperatures, 350. Dew, 409. point, 390. computation of, 396. Diathermancy, 448. table of, 449. Differential equations for flow of heat, 425, 426. thermometer, 275. Difficulty of commencing change of state, 364. Diffusive reflection, 437, 447. Diffusivity, 413, 414. deduced from underground temperatures, 426. Digester, Papin's 360. Dines' hygrometer, 393. Dissipation of energy, 489. Distillation, 368. Donny's experiment, 363. Draught of chimneys, 306. Drion's experiments, 351. Dufour's experiment, 363. Dulong & Petit's law, 318. law of cooling, 431. Dumas' method for vapour den- sities, 379. Ebullition, 355, 357. Eccentric of slide-valve, 497. Efficiency of thermic engine, 466- 471 ; reversible, 469. Emission, coefficient of, 438. Emissive power, 442. Energy, available sources of, 488. dissipation of, 489. Engines, thermic, see Steam- engine, 465, 491. Equilibrating columns of liquid, 287. Equivalents of heat and work, 462. Evaporation, 337. cold of, 345. latent heat of, 385-388. Exchanges, theory of, 432. Expansion, apparent and real, of liquids, 278. INDEX TO PART II. 530 Expansion by heat, 258. coefficient of, 282. cubic and linear, 277. force of. 286. heat lost in, 464. in freezing, 330. linear, modes of observing, 283. table of, 284. mathematics of, 277. of gases, 281. table of, 300. of liquids, table of, 291, 292. of mercury, 287-289. of solids, 283. Expansion factor, 277. Expansive working in steam-en- gine, 500, 503. Explosion of boilers, 507. Factor of expansion, 277. Fahrenheit's scale of temperature, 265. Faraday's experiments on lique- faction of gases, 343, 345. on solidification of gases, 350. Favre and Silbermann's calori- meter, 482. Fire-places, 307. Fire-syringe, 457. Flowers of ice, 327-329. Fluorescence, 453. Fly-wheel, 498. Forbes' experiments on conduc- tivity, 420. observations on glacier motion, 334- Foucault's magneto - thermic ex- periment, 459. Franklin's experiment on ebulli- tion, 358. Freezing at abnormally low tem- peratures, 326, 480. by evaporation, 346-350. by the spheroidal state, 366. expansion in, 330. mercury in red-hot crucible, 366. mixtures, 324. Freezing-point lowered by pres- sure, 331, 481. ; computation, 481. by stresses, 332. Friction, heat of, 458. Frost, hoar, 411. Fusion, 320. latent heats of, 322. temperatures of, 320. Gas-engines, 514-517. Gases, table of densities of, 306. conducting power of, 424. their expansion by heat, 281, 297-300. two specific heats of, 317. Gay-Lussac's experiments on ex- pansion of gases, 297. method for vapour densities, 382- _ Giffard's injector, 509. Glaciers, motion of, 334. Glaisher's balloon-ascents, 523. tables, 396. Glass, expansion of, 289. Governor balls, 498. Gramme-degree, 311. Gridiron-pendulum, 285. Gulf-stream, 530. Hail, 409. Harrison'sgridiron-pendulum, 285. Head, 306. Heat, mechanical equivalent of, 461. of combustion, table of, 484. quantity of, 310. required for a cyclic change, 474. for change of volume and temperature, 473. units, 311. Heating by hot water, 295. Heights measured by boiling-point, 359- High-pressure engines, 502. Him on animal heat, 485. Hoar-frost, 411. Hope's experiment, 292. Howard'scloud nomenclature, 404. Humidity of air, 389. Hydrogen, conductivity of, 424. heat of combustion of, 484. Hygroscopes and hygrometers 39!-399- Hypsometer, 359. Ice-calorimeter, 319. flowers, 327-329. houses, 418. plasticity of, 334. regelation of, 334. Ingenhousz's experiment, 415. Injector, Giffard's, 509. Insular climates, 520. Inverse squares, 433. Iodine, solution of, in bisulphide of carbon, 453. Isothermal lines, 469, 519. Joule's equivalent, 462. experiment in stirring water, 461. Lamp-black as absorber, 448. radiator, 439. Land and sea breezes, 524. Laplace and Lavoisier's experi- ments, 283. Latent heat effusion, 321. of vaporization, 385-388. of water, 322, 324. Leidenfrost's phenomenon, 366. Leslie's differential thermometer, 275- experiment (freezing by evapo- ration), 346. Linear flow of heat, 425. Link-motion, 513. Liquefaction of gases, 342, 353. of oxygen and hydrogen, 353. of solids, see Fusion. Liquids, expansion of, 287-296. Liquid and gaseous states continu- ous, 351. Locomotive, 510. Mason's hygrometer, 395. Maximum thermometers, 267. Mean temperature, 518. Mechanical equivalent of heat, 461. Melloni's experiments, 441-450. Melting-points, table of, 320. Mercury, expansion of, 287-289. Metallic thermometers, 273. Meteoric theory of sun's heat, 487. Meteorology, 402. Meyer's method for vapour den- sities, 384. Mirrors, conjugate, 436. Mist, 402, 407. Mixture of gases and vapours, 342. 1'ixtures, boiling-point of, 362. method of, 312. Moist air, density of, 400. Monsoons, 524. Mousson's experiment, 331. Negretti's maximum thermometer, 270. Newton's law of cooling, 430. Nobili's thermo-pile, 440. Norwegian cooking-box, 418. Obscure radiation, 453. Oceanic currents, 529. Oscillating engines, 504. Oxyhydrogen blow-pipe, 484. Papin's digester, 360. Parabolic mirrors, 435. Pendulum, compensated, 284. Perfect gas, 301. Phillips' maximum thermometer, 270. Plasticity of ice, 334. Platinum, expansion of, 294. Pluviometer, see Rain-gauge. Pneumatic tinder-box, 457. Pressure, correction of, for gravity, 377- Prevost's theory of radiation, 432. Psychrometer, 395. Pyrheliometer, 486. Pyrometer, 274, 302. Quantity of heat, 310. Quartz transparent to ultra-violet rays, 453. Radiant heat and light, 451. Radiation, 428. at different temperatures, 453. selective, 454. Rain, 407. Rainfall, British, 409. Rain-gauge, 407-408. Ramsden and Roy's experiments, 284. Rankine's prediction of specific heat, 476. Real and apparent expansion, 278. Reaumur's scale, 265. Red-hot ball in water, 368. Reflecting power,437 ; table of, 445. Reflection of heat, 434. Refrangibility, change of, 453. Regelation, 334. Regnault's hygrometer, 394. hypsometer, 359. 540 INDEX TO PART II. Regnault's experiments on expan- sion of gases, 298. on specific heat, 315. on vapour-pressures, 371. Reversal of bright lines, 455. Reversible engine, perfect, 467. efficiency of, 469. Reversing of locomotive, sis- Rock-salt, its diathermancy, 453, 454- Rotation of earth as affecting wind, 525- Rotatory engines, 504. Roy and Ramsden's measures of expansion, 284. Rumford on heat of friction, 458. on radiation in vacuo, 429. Rumford's thermoscope, 275. Rutherford's self-registering ther- mometers, 268. Safety-valve, 361, 507. Saline solutions, boiling-point of, 361. Saturated air, weight of, 400. vapour, 339. Scales, thermometric, 265. Scattered rays, 437, 447. , Sea-breeze and land-breeze, 524. Selective emission and absorption, 454- Self-registering thermometers, 267. Sensibility of thermometer, 267. Six's thermometer, 267. Slide-valve, 497. Snow, 408-410. Soil, temperature of, 520. Solar heat, 486. sources of, 487. Solidification, change of volume in, 33- of gases, 350, 353. . of liquids, 325. Solution, 324. Solutions, boiling-points of, 362. Sources of energy, 488. Specific gravity, correction of, for temperature, 278. Specific heat, 312. of. gases, 317. See Two Specific Heats. tables of, 316. Spheroidal state, 365. Spirit thermometer, 267, 280. Squares, inverse, 433. Steam, pressure of, 375. volume of, 385. Steam-engine, 493. locomotive, 510. Still, 368. Stirling's air-engine, 491 Stoves, 308. Norwegian, 418. Sulphate of soda, 329. Sulphuric acid, boiling of, 365. Sun, see Solar. Superheated vapour, 341. Superheating of steam, 503. Supersaturated solutions, 329 Surface-condensers, 502. Surface conduction, 430. Syringe, pneumatic, 457. Temperature, 257. absolute, 301, 471. mean, 518. of a place, 518. of the air, 518. decrease upwards, 522. of the soil, 521. increase downwards, 522. scales of, 265. Tension of vapour, 338. Terrestrial temperatures, 518. Theory of exchanges, 432. Thermic engines, 465. Thermochrose, 451. Thermo-dynamics, 457. first law of, 462. second law of, 468. Thermographs, 272. Thermo'meter, 257-275. alcohol, 267, 280. differential, 275. metallic, 273. self-registering, 267. Thermo-pile, 440. Thilorier's apparatus, 344. Thomson, J., on glacier motion, 332. 335- on lowering of freezing-point, 33 1 - Two specific heats, difference of, 464. Two specific heats, ratio of, 475, 476. Tyndall on moulding of ice, 335. Underground temperature, 520- 522. diffusivity deduced from, 426. Units of heat, 311. Vapour, 337. apparatus to illustrate, 339. at maximum tension, 338. Vapour-density, 379-385. related to chemical combina- tion, 378. Vapour-pressure, measurement of, 370-37S- pressures of various liquids, 376. Vegetable growth, 485. Vesicular state, 403. Volume, change of, in congelation, 33- in vaporization, 385. Walferdin's maximum thermo- meter, 272. Water, conductivity of, 423. conservatism of, 323. equivalent of a body, 312. expansion of, 294. maximum density of, 292. specific heat of, 317. Watt's steam-engine, 495-499. Weight-thermometer, 281. Well-thermometers, 271. Wet and dry bulb, 395. Wiedemann and Franz's experi- ments, 419. Williams', Major, experiment with ice, 330. Wind, causes of, 524. measurement of, 528. trade, 525. Wire-gauze and flame, 416. Work spent in generating heat, 458-464. Zero, absolute, of temperature, 3!. 473- displacement of, in thermome- ters, 266, 280. ON NATURAL PHILOSOPHY. BY A. PKIVAT DESCHANEL, FORMERLY PROFESSOR OF PHYSICS IN THE LTC^E LOUIS-LE GRAND, INSPECTOR OF THE ACADEMY OF PARIS. TRANSLATED AND EDITED, WITH EXTENSIVE ADDITIONS, BY J. D. EVERETT, M.A., D.C.L., F.R.S.E., PROFESSOR OF NATURAL PHILOSOPHY IN THE QUEEN'S COLLEGE, BF.LFAST. Part III. -ELECTRICITY AND MAGNETISM. ILLUSTRATED BY 241 ENGRAVINGS ON WOOD, AND ONE COLOURED PLATfc. FIFTH EDITION. LONDON: BLACKIE & SON, OLD BAILEY, E.G., GLASGOW AND EDINBURGH. 1880. All Rights Reserved. NOTE PREFIXED TO FIRST EDITION. THE accurate method of treating electrical subjects which has been established in this country by Sir \Vm. Thomson and his coadjutors, has not yet been adopted in France ; and some of Faraday's electro- magnetic work appears to be still very imperfectly appreciated by French writers. The Editor has accordingly found it necessary to recast a considerable portion of the present volume, besides intro- ducing two new chapters (XXXIX A . and XLI A .) and an Appendix. Potential and lines of force are not so much as mentioned in the original. The elements of the theory of magnetism have been based on Sir Wm. Thomson's papers in the Philosophical Transactions; and the description of the apparatus used in magnetic observatories has been drawn from the recently published work of the Astronomer Royal. The account of electrical units given in the Appendix is mainly founded on the Report of the Electrical Committee of the British Association for the year 1863. M. Deschanel's descriptions of apparatus, of which some very elaborate examples occur in the present volume, left little to be desired in point of clearness. In no instance has it been found necessary to resort to the mere verbal rendering of unintelligible details. This Part has been thoroughly revised since the publication of the first edition. Important changes and additions have been made at pages 572, 582, 642-644, 675-677, 773*, 778-784, and an Alphabetical Index has been added. J. D. E. BELFAST, December, 1873. CONTENTS-PABT III. ELECTRICITY. CHAPTER XXXV. INTRODUCTORY PHENOMENA. Fundamental phenomena. Conductors and non-conductors. Duality of electricity. Electric pendulum. Electricities of opposite kind. Both excited at once. Two-fluid and one-fluid theories, pp. 505-512. CHAPTEB XXXVI. ELECTRICAL INDUCTION. Induction. Charging by induction. Faraday's theory of induction by contiguous particles. Attraction of unelectrified bodies. Induction favours attraction. Repulsion the safer test of kind. Electroscopes. Pith-ball. Gold-leaf electroscope, . . pp. 513-518. CHAPTER XXXVII. MEASUREMENT OF ELECTRICAL FORCES. Coulomb's torsion-balance. Repulsion. Law of inverse squares. Fallacious objections. Attraction. Force proportional to amount of charge. Electricity resides on external surface. Experimental proofs. Limitations of the rule. Currents. Electricity induced on internal surface. Ice-pail experiment. No force within a conductor. Faraday's cubical box. Inference regarding law of inverse squares. Electrical density and distribution. Coulomb's experiments. Density on points and edges. Dissipation of charge, pp. 519-532. CHAPTER XXXVIII. ELECTRICAL MACHINES. Early history. Ramsden's machine. Limit of charge. Quadrant electroscope. Amal- gam for rubbers. Nairne's machine. "Winter's machine. Armstrong's hydro-electric machine. Holtz's machine. Electrophorus. Bertsch's machine, . pp. 533-545. CHAPTER XXXIX. VARIOUS EXPERIMENTS WITH THE ELECTRICAL MACHINE. Electric spark. Brush. Why crooked. Preceded by polar tension. Duration of spark. Wheatstone's experiment with revolving mirror. Spark in rarefied air. Electric egg. Discharge in Torricellian vacuum. Nodischarge in perfect vacuum. Colour of spark. Spangled tube and pane. Electric shock. Tickling sensation. Mechanical effects. Kinnersley's thermometer. Heating effects. Inflammation of coal-gas. Explosion of gaseous mixture. Volta's pistol. Decomposition of ammonia. Wind from points. Electric whirl. Electric watering-pot, pp. 546-558. i CHAPTER XXXIX*. ELECTRICAL POTENTIAL, AND LINES OF ELECTRIC FORCE. Introductory remarks on potential. Relation of potential to force. Line of force. Intensity of force equal to rate of variation of potential. Relation between potential and work. Equipotential surfaces. Tubes of force. Force varies inversely as section of tube. Analogy to filaments of a flowing liquid. Cases of conical tubes and cylindric tubes. Force proportional to number of tubes per unit area. Force just outside a iv TABLE OF CONTENTS. charged conductor is iwp. Relation of induction to lines and tubes of force. Potential equal to sum of quotients of quantity by distance. Potential of sphere is charge divided by radius. Capacity of a conductor. Capacity of sphere is equal to radius. Capacity varies as linear dimensions. Connection between potential and induced distri- bution. A hollow conductor screens its interior from external influence. Electrical images, pp. 559-5G6. CHAPTER XL. ELECTRICAL CONDENSERS. Condensation. Collecting and condensing plate. Capacity of condenser. Discharge of condenser. Jointed discharger. Invention of Leyden jar. Energy which runs down in discharge. Residual charge. Jar with movable coatings. Discharge by alternate contacts. Condensing power. Riess' experiments. Free and bound electricity. Influence of dielectric. Specific inductive capacity. Faraday's determinations. Polarization of dielectric. Thickness of dielectric. Volta's condensing electroscope. Leyden battery. Lichtenberg's figures. Charge by cascade, . . . pp. 567-582. CHAPTER XLI. EFFECTS PRODUCED BY THE DISCHARGE OF CONDENSERS. Shock to a number of persons. Coated pane. Universal discharger. Heating of metallic threads. Electric portrait. Velocity of electricity. Watson's experiment. Wheat- stone's determination. Trials with Atlantic cable. Unit-jars of Lane and Harris. Perforation of card and glass. Explosion of mines, pp. 583-590. CHAPTER XLI A . ELECTROMETERS. Electrometers measure potential. Attracted -disc electrometers. Absolute electrome- ter. Pu i table electrometer. Quadrant electrometer. Replenisher. Cage electro- meter, pp. 591-598. CHAPTER XLII. ATMOSPHERIC ELECTRICITY. Franklin's discovery. Duration of lightning. Thunder. Shock by influence. Lightning- conductors. Use of point. Ordinary electricity of the atmosphere. Methods of obtaining indications. Arrow, burning-match, conducting- ball, water-jet. Action of match and jet explained. Interpretation of indications. They measure density of electricity on earth's surface. This is induced by electricity overhead. Results of observation. At Kew Observatory. At Windsor, Nova Scotia. At Brussels anil Kreuznach. Conjectures regarding the sources of atmospheric electricity. Volta's theory of hail. Theories regarding waterspouts, pp. 599-611. MAGNETISM. CHAPTER XLIII. GENERAL STATEMENT OF FACTS AND LAWS. fjodestone and magnetic iron ore. Artificial magnets. Force greatest at ends. Poles and neutral part. Lines formed by filings. Curve of force-intensity. Magnetized needle. Azimuth. Meridian. Magnetic declination. Dip, or inclination. Mutual action of poles. Names of poles. North-seeking and south-seeking, or austral and boreal. Am- biguity of terms north and south. Magnetic induction. Magnetic chain. Polarity of broken pieces of magnet. Imaginary magnetic matter of two opposite kinds. Magnetic potential and lines of magnetic force. Uniform magnetization. Direction of mag- netization. Ideal simple magnet. Strength of pole. Magnetic field. Moment of magnet. Terrestrial couple on magnet. Moment of uniformly magnetized bar is sum of moments of its parts. Intensity of magnetization. Actual magnets. Their mag- netization is weakest at the ends. Their moment defined, .... pp. 612-022. TABLE OF CONTENTS. V CHAPTER XLIV. EXPERIMENTAL DETAILS. The earth's force simply directive. Horizontal, vertical, and total intensities. Torsion- balance. Observation of declination. Declination theodolite. Declination magnet. Observation of dip. Dip-circle. Kew dip-circle. Observation of intensity. By vibra- tions, and statically. Absolute determinations. Bifilar magnetometer. Balance magnetometer. Magnetic meridians and lines of equal dip. The earth as a magnet. Biot's hypothesis of a short central magnet. Changes of declination and dip. Magnetic storms. Ship's compass. Methods of magnetization. Consequent points. Lifting power. Compound magnet. Molecular changes accompanying magnetization. All bodies either paramagnetic or diamagnetic. Magneto- cry stallic action, pp. 623-641. CUKKENT ELEGTEICITY. CHAPTER XLV. GALVANIC BATTEEY. Voltaic electricity. Voltaic element. Fundamental principles of contact. Electricity. Battery. Galvani's discovery. Volta's pile. Couronne de tasses. Cruickshank's trough. Wollaston's battery. Hare's deflagrator. Polarization of plates. Daniell's battery. Bunsen's and Grove's. Amalgamated zinc. Sawdust battery. Dry pile. No current without consumption. Bohnenberger's electroscope. Thermo-electric cur- rents. Thermo-electric order. Comparison of electro-motive forces. Reversal at high temperatures. Thermo-pile. Thermo- electric observation of temperature, pp. 642-655. CHAPTER XLVI. GALVANOMETER. (Ersted's discovery of deflection of needle by current. Ampere's rule. Lines of magnetic force due to current. Force on current in magnetic field. Numerical estimate of cur- rents. Galvanometers. Sine galvanometer. Tangent galvanometer. Schweiger'e multiplier. Differential galvanometer. Astatic needle. Thomson's mirror galvano- meter. Reduction of galvanometer indications to proportional measure, pp. 6o6-G6J. CHAPTER XL VII. OHM'S LAW. Statement of Ohm's law. Meaning of "electro-motive force." Meaning of "resistance." Resistances of wires. Specific resistance. Pouillet's experimental proofs. Reduced length. Rheostat. Electrical and thermal conductivities proportional. Resistance of liquids. Resistance in battery cells. Advantage of large plates. Arrangement of cells in battery. Divided circuits. Wheatstone's bridge. Potential in different points of battery and connecting wire. Mea=yrfiment of resistance of battery. Measurement of electro- motive force, pp. 665-679. CHAPTER XL VIII. ELECTRO-DYNAMICS. Meaning of "electro-dynamics." Ampere's stand. Three elementary laws. Continuous rotation produced by a circular current. Action of an indefinite rectilinear current. Action upon a rectangular current. Sinuous currents. Mutual action of two elements of currents. Magneto-electric explanation. Maxwell's rule. Action of the earth on currents. Solenoids. Their declination and dip. Their mutual action. Action be- tween solenoid and magnet. Astatic circuits. Ampere's theory of magnetism. Rota- tion of a magnet on its axis. Magnetization of iron and steel by currents. Electro- magnets. Residual magnetism, . . pp. 680-698. CHAPTER XLIX. HEATING EFFECTS OF CURRENTS. Heating of wires. Joule's law. Relation of heat in circuit to chemical action in batterv Distribution of heat in circuit. Mechanical work done by current diminishes heat. VI TABLE OF CONTENTS. Electric light. Changes in the carbons. Properties of the voltaic arc. Intensity of the light. Applications. Duboscq's regulator of the electric light. Foucault's regulator. Thermal effect at junctions, pp. 699-709. CHAPTER L. ELECTRO-MOTORS TELEGRAPHS. Electro- magnetic engines. Bourbouze's. Froment's. Electric telegraph. History of its invention. Batteries. Wires. Return wire dispensed with. Single-needle tele- graph. Dial telegraphs. Breguet's. Alarum. Wheatstone's universal telegraph.- Morse's telegraph. Receiving instrument. Digney's ink- writer. Key. Telegraphic alphabet. Relay. Hughes' printing telegraph. Bain's electro-chemical telegraph. Caselli's autographic telegraph. Submarine telegraphs. Retardation by induction. Thomson's receiving instruments. Wheatstone's automatic system. Specimen of message. Limits of speed. Application of electricity to clocks. Jones' system of control, i . . pp. 710-737. CHAPTER LI. ELECTRO-CHEMISTRY. Decomposition by passage of a current Voltameter. Transport of elements. Anion and cation. Grotthus' hypothesis. Electrolysis of binary compounds. Electrolysis of salts. Secondary actions. Electrolysis of water. Definite laws of electrolysis. Polariza- tion of electrodes. Gas-battery. Secondary pile. Electrolytes never conduct without decomposition. Electro-metallurgy. Electro -gilding and electro-plating. Electro- type. Applications of electrotype, pp. 738-749. CHAPTER LII. INDUCTION OF CURRENTS. Currents induced by commencement or cessation of neighbouring currents. By variations of strength. By variations of distance. By movement of a magnet. By change of strength in a magnet. Direction of induced current specified by Lenz's law. By refer ence to lines of magnetic force. Quantitative statement by reference to number of force-tubes cut through. Relation of induced current to work done. Movement of lines of force with change of magnetization. Motion in uniform field. Unit of resist- ance defined. Movement of lines of force with change of strength. Induction of cur- rents by means of terrestrial magnetism. Delezenne's circle. British Association ex- periment. Induction of a current on itself. Extra currents. Ruhmkorff's induction- coil. Spark from induction-coil. Discharge in rarefied gases. Geissler's tubes. Action of magnets on luminous discharge. Magneto- electric machines. Pixii's. Clarke's. Machines for lighthouses. Siemens' armature. Wilde's machine. Sie- mens' and Wheatstone's. Accumulation by successive action, and accumulation by mutual action. Ladd's machine. Gramme's machine. Currents in Wheatstone's dial telegraph. Arago's rotations and Faraday's explanation. Copper dampers. Faraday's experiment of the copper cube. Electro-medical machines. No hypothesis assumed in using lines of force, pp. 750-778. ADDITIONS IN 1878. Loop test. Measurement of electro-motive force. Jablochkoff s system of electric light- ing. Telephone. Microphone, pp. 778-784* APPENDIX. ON ELECTRICAL AND MAGNETIC UNITS. Mutual relations of units of different kinds. Derived units and their dimensions. Electro-static system of derived units. Electro-magnetic system. Dimensions of the same quantity different in the two systems. Ratio of the two units of quantity of electricity is equal to velocity of light, pp. 779-783. FRENCH AND ENGLISH MEASURES. A DECIMETRE DIV1DKD INTO CESTIMLTRES AND MILLIMETRES. INCHES AND TENTHS. TABLE FOR THE CONVERSION OF FRENCH INTO ENGLISH MEASURES. 1 Millimetre = 1 Centimetre 1 Decimetre = 1 Metre 1 Kilometre = MEASURES OF AREA. MEASURES OF LENGTH. 039370432 inch, or about j-s inch. 39370432 inch. 3.9370432 inches. 39.370432 inches, or 3.2809 feet nearly. 39370.432 inches, or 1093'6 yards nearly. 1 sq. millimetre = '00155003 square inch. 1 sq. centimetre = '155003 square inch. 1 sq. decimetre = 15'5003 square iuches. 1 sq. metre = 1550 '03 square inches, or 107641 square feet. MEASURES OF VOLUME. 1 cubic centimetre = '0610254 cubic inch. 1 cubic decimetre = 61 '0254 cubic inches. 1 cubic metre = 61025 '4 cubic inches, or 35 '3156 cubic feet. The Litre (used for liquids) is the same as the cubic decimetre, and is equal to 1 '76172 imperial pint, or '220215 gallon. MEASURES OF WEIGHT (or MASS). 1 milligramme 1 centigramme 1 decigramme 1 gramme 1 kilogramme 015432349 grain. 15432349 grain. 1-5432349 grain. 15-432349 grains. 15432'349 grains, or 2'20462125 Ibs. avoir. MEASURES INVOLVING REFERENCE TO TWO UNITS. Lbs ^ square font. 1 gramme per sq. centimetre ~- 2 "048 124 1 kilogramme per sq. metre - '2048124 1 kilogramme per sq. millimetre = 204812'4 1 kilogrammetre = 7 '23307 foot-pounds. 1 force de cheval 75 kilogram metres per second, or 542^ foot-pounds per second nearly, whereas 1 horse-power (English) 550 foot- pounds per second. TABLE FOR THE CONVERSION OF ENGLISH INTO FRENCH MEASURES. MEASURES OF LENGTH. 1 inch =25'39977 millimetres. 1 foot = -30479726 metre. 1 yard= '9143918 metre. 1 mile =1-60933 kilometre. MEASURES op AREA. 1 sq. inch =645-148 sq. millimetres. 1 sq. foot = -0929014 sq. metre. 1 sq. yard ='8361124 sq. metre. 1 sq. mile=2'589942 sq. kilometres. SOLID MEASURES. 1 cubic inch=16386'6 cubic millimetres. 1 cubic foot = '028:3161 cubic metre. 1 cubic yard= '7645343 cubic metre. MEASURES OF CAPACITY. Ipint ='5676275 litre. 1 gallon =4-54102 litres. 1 bushel =36 -32816 litres. Vlll FRENCH AND ENGLISH MEASURES. MEASURES OF WEIGHT. 1 grain = '064799 gramme. 1 oz. avoir. =28 '3496 grammes. 1 Ib. avoir. = "453593 kilogramme. 1 ton = 1 '01605 tonne = 1016 '05 kilos. MEASURES INVOLVING REFERENCE TO TWO UNITS. 1 Ib. per sq. foot =4 '88252 kilos, persq. metre. 1 Ib.per sq. inch= '0703083 kilos, persq. centi- metre. 1 foot-pound = '138254 kilogrammetre. TABLE OF CONSTANTS. The velocity acquired in falling for one second in vacuo, in any part of Great Britain, is about 32 '2 feet per second, or 9 '81 metres per second. The pressure of one atmosphere, or 760 millimetres (29'922 inches) of mercury, is 1'033 kilogramme per sq. centimetre, or 14'73 Ibs. per square inch. The weight of a litre of dry air, at this pressure (at Paris) and C., is 1'293 gramme. The weight of a cubic centimetre of water is about 1 gramme. The weight of a cubic foot of water is about fi2'4 Ilia. ELECTKICITY. CHAPTER XXXV. INTRODUCTORY PHENOMENA. 408. Fundamental Phenomena. If a glass tube be rubbed with a silk handkerchief, both tube and rubber being very dry, the tube will be found to have acquired the property of attracting light bodies. If the part rubbed be held near to small scraps of paper, pieces of Fig. 332. Attraction of Light Bodies by au Electrified Body. cut straw, sawdust, &c., these objects will move to the tube ; some- times they remain in contact with it, sometimes they are alternately attracted and repelled, the intensity as well as the duration of these effects varying according to the amount of friction to which the tube has been subjected. If the tube be brought near the face, the result is a sensation similar 33 506 INTRODUCTORY PHENOMENA. to that produced by the contact of a cobweb. If the knuckle be held near the tube, a peculiar crackling noise is heard, and a bright spark passes between the tube and knuckle. The tube then has acquired peculiar properties by the application of friction. It is said to be electrified, and the name of electricity is given to the agent to which the various phenomena just described are attributed. Glass is not the only substance which can be electrified by friction ; the same property is possessed also by resin, sulphur, precious stones, amber, &c. The Greek name of this last substance (yXeKrpov) is the root from which the word electricity is derived. At first sight it appears that this property of becoming electrified by friction is not common to all bodies ; for if a bar of metal be held in the hand and rubbed with wool, it does not acquire the properties Fig. 333. Electrification of a Metal by Friction. of an electrified body. But we should be wrong in concluding that metals cannot be electrified by friction; for if the bar be fitted on to a glass rod, and, while held by this handle, be struck with flannel or catskin, it may be very sensibly electrified. There is therefore no basis for the distinction formerly made between electrics and non- electrics, that is, between substances capable and incapable of being electrified by friction ; for all bodies, as far as at present known, are capable of being thus excited. There is, however, an important dif- ference of another kind between them, which was first pointed out by Stephen Grey in 1729. 409. Conductors and Non-conductors. In certain bodies, such as glass and resin, electricity does not spread itself beyond the parts of the surface where it has been developed ; while in other bodies, such as metals, the electricity developed at any point immediately spreads itself over the whole body. Thus, in the last-mentioned experiment, the signs of electricity are immediately manifested at the end of the metal bar which is farthest from the glass rod, if the end next the rod be submitted to friction. Bodies of the former kind, such as glass, resin, &c., are said to be non-conductors. Metals are said to be good conductors. A non-conductor is often called an insulator, and a conductor supported by a non-conductor is said to be in- sulated. The appropriateness of these expressions is evident. No substance is perfectly non-conducting, but the difference in conduct- CONDUCTORS AND NON-CONDUCTORS. 507 ing power between what are called non-conductors and good con- ductors, is enormous. The following are lists of conductors and non-conductors, arranged, at least approximately, in order of their conducting powers. In the list of conductors, the best conductors are put first ; in the list of non-conductors, the worst conductors (or best insulators) are put first. CONDUCTORS. All metals. Metallic ores. Living vegetables. Well- burned charcoal. Animal fluids. Flax. Plumbago. Sea water. Hemp. Concentrated acids. Spring water. Living animals. Dilute acids. Rain water. Flame. Saline solutions. Snow. Moist earth and stones. NON-CONDUCTORS. Shellac. Gems. Leather. Amber. Ebonite. Baked wood. Resins. Caoutchouc. Porcelain. Sulphur. Gutta-percha. Marble. Wax. Silk. Camphor. Jet. Wool Chalk. Glass. Feathers. Lime. Mica. Dry paper. Oils. Diamond. Parchment. Metallic oxides. The human body is a good conductor of electricity. If a person standing on a stool with glass legs be struck with a catskin, he becomes electrified in a very perceptible degree, and sparks may be drawn from any part of his body. When an insulated and electrified conductor is allowed to touch another conductor insulated but not electrified, it is observed that, after the contact, both bodies possess electrical properties, electricity having been communicated to the second body at the expense of the first. If the second body be much the larger of the two, the electri- city of the first is greatly diminished, and may become quite insen- sible. This explains the disappearance of electricity when a body is put in connection with the earth, which, together with most of the objects on its surface, may be regarded as constituting one enormous conductor. On account of its practicaUy inexhaustible capacity for furnishing or absorbing electricity, the earth is often called the com- mon reservoir. It will now be easily understood why it is not possible to electrify a metal rod by rubbing it while it is held in the hand; since the 508 INTRODUCTORY PHENOMENA. electricity, as fast as it is generated, passes off through the body into the earth. Air, when thoroughly dry, is an excellent insulator ; and electrified conductors exposed to it, and otherwise insulated, retain their charge with very little diminution for a considerable time. Dampness in the air is, however, a great obstacle to insulation, mainly, or (as it would appear from Sir W. Thomson's experiments) entirely, by reason of the moisture which condenses on the insulating supports. Electrical experiments are accordingly very difficult to perform in damp wea- ther. The difficulty is sometimes met by employing a stove to heat the air in the neighbourhood of the supports, and thus diminish its relative humidity. Sir W. Snow Harris employed heating-irons, which were heated in a fire, and then fixed near the insulating sup- ports; and thus succeeded in exhibiting electrical experiments to an audience in the most unfavourable weather. Sir W. Thomson, by keeping the air in the interior of his electrometers dry by means of sulphuric acid, causes them to retain their charge with only a small percentage of loss in twenty-four hours. Dry frosty days are the best for electrical experiments, and next perhaps to these, is the season of dry cutting winds in spring. 410. Duality of Electricity. The elementary phenomena which we have mentioned in the beginning of this chapter may be more accu- rately studied by means of the electric pendulum, which consists of a pith-ball suspended by a silk fibre from an insulated support. When an electrified glass rod is brought near the insulated ball, the latter is attracted ; but as soon as it touches the glass tube, the attraction is changed to repulsion, which lasts as long as the ball retains the electricity which it has acquired by the contact. A similar experi- ment can be shown by employing, instead of the glass tube, any other body which has been electrified by friction, for example, a piece of resin which has been rubbed with flannel If, while the pendulum exhibits repulsion for the glass, the electri- fied resin is brought near, it is attracted by the latter ; and conversely, when it is repelled by the resin, it is attracted by the glass. These phenomena clearly show that the electricity developed on the resin is not of the same kind as that developed on the glass. They exhibit opposite forces towards any third electrified body, each attracting what the other repels. They have accordingly received names which indi- cate opposition. The electricity which glass acquires when rubbed with silk, is called positive; and that which resin acquires by friction DUALITY OP ELECTRICITY. 509 with flannel, negative. The former is also called vitreous, and the latter resinous. On repeating the experiment with other substances, Fig. 334. Electric Pendulum. it is found that all electrified bodies behave either like the glass or like the resin. 410 A. Without making any assumption as to what electricity is, we may speak of an electrified body as being charged with electricity, and we may compare quantities of electricity by means of the attrac- tions and repulsions exerted. Bodies oppositely electrified must then be spoken of as charged with electricities of opposite kind, or of opposite sign ; and experiment shows that, whenever electricity of the one kind is developed, whether by friction or by any other means, electricity of the opposite sign is always developed in exactly equal quantity. If a conductor receives two charges of electricity of equal quantity but opposite sign, it is found to exhibit no traces of electri- city whatever. Electricities of like sign repel one another and those of unlike sign attract one another. The magnitude of the force exerted upon each other by two electrified bodies, is not altered in amount by reversing the sign of the electricity of one or both of them, provided that the quantities of electricity, and their distribution over the two 510 INTRODUCTORY PHENOMENA. bodies, remain unchanged. If the sign of one only be changed, the mutual force is simply reversed, and if the signs of both be changed, the force is not changed at all. 411. The simultaneous development of both kinds of electricity is illustrated by the following experiment: Two persons stand on stools with glass legs, and one of them strikes the other with a cat- skin. Both of them are now found to be electrified, the striker posi- tively, and the person struck negatively, and from both of them sparks may be drawn by presenting the knuckle. The kind of electricity which a body obtains by friction with another body, evidently depends on the nature of their surfaces. If, for example, we take two discs, one of glass, and the other of metal, and, holding them by insulating handles, rub them briskly together, we shall find that the metal becomes negatively, and the glass posi- tively electrified ; but if the metal be covered with a catskin, and the experiment repeated, it will be the glass which will this time be negatively electrified. In the subjoined list, the substances are arranged in such order that, generally speaking, each of them be- comes positively electrified by friction with those which follow it, and negatively with those which precede it. Fur of cat. Feathers. Silk. Polished glass. Wood. Shellac. Woollen stuffs. Paper. Rough glass. 411 A. Hypotheses regarding the Nature of Electricity. Two theories regarding the nature of electricity must be described on account of the historical interest attaching to them. The two-fluid theory, originally propounded by Dufaye, reduced to a more exact form by Symmer, and still very extensively adopted, maintains that the opposite kinds of electricity are two fluids. Posi- tive electricity is called the vitreous fluid, and negative electricity the resinous fluid. Fluids of like name repel, and those of unlike name attract each other. The union of equal quantities of the two fluids constitutes the neutral fluid which is supposed to exist in very large quantity in all unelectrified bodies. When a body is electri- fied, it gains an additional quantity of the one fluid, and loses an equal quantity of the other, so that the total amount of electric fluid in a body is never changed ; and (as a consequence of this last con- dition) when a current of either fluid traverses a body in any direc- tion, an equal current of the other fluid traverses it in the opposite direction. THE NATURE OF ELECTRICITY. 511 This theory is in complete agreement with all electrical phenomena so far as at present known ; but as it is conceivable that the two electricities, instead of being two kinds of matter, may be two kinds of motion, or, in some other way, may be opposite states of one and the same substance, it is more philosophical to avoid the assumption involved in speaking of two electric fluids, and to speak rather of two opposite electricities. They may be distinguished indifferently by the names vitreous and resinous, or positive and negative. The one-fluid theory, as originally propounded by Franklin, main- tained the existence of only one electric fluid, which unelectrified bodies possess in a certain normal amount. A positively electrified body has more, and a negatively electrified body less than its normal amount. The particles of this fluid repel one another, and attract the particles of other kinds of matter, at all distances. ^Epinus, in developing this theory more accurately, found it necessary to intro- duce the additional hypothesis that the particles of matter repel one another. Thus, according to ^Epinus, the absence of sensible force between two bodies in the neutral condition, is due to the equilibrium of four forces, two of which are attractive, and the other two repul- sive. Calling the two bodies A and B, the electricity which A pos- sesses in normal amount, is repelled by the electricity of B, and attracted by the matter of B. The matter of A is attracted by the electricity of B, and repelled by the matter of B. These four forces are all equal, and destroy one another; but, without the supplemen- tary hypothesis of ^Epinus, one of the four forces is wanting, and the equilibrium is not easily explained. To reconcile ^Epinus's addition with the Newtonian theory of gravitation, it is necessary to suppose that the equality between the four forces is not exact, the attractions being greater by an infinitesimal amount than the repulsions. The one-fluid theory in this form is, like the two-fluid theory, con- sistent with the explanation of all known phenomena. But it is to be remarked that there is no sufficient reason, except established usage, for deciding which of the two opposite electricities should be regarded as corresponding to an excess of the electric fluid. Franklin was the author of the terms positive and negative to denote the two opposite kinds of electrification ; but the names can legitimately be retained without accepting the one-fluid theory, understanding that opposite signs imply forces in opposite directions, and that the connection between the positive sign and the forces exhibited by vitreous electricity is merely conventional. 512 INTRODUCTORY PHENOMENA. 411 B. In speaking of electric currents, the language of the one- fluid theory is almost invariably employed. Thus, if A is a con- ductor charged positively, and B a conductor charged negatively; when the two are put in connection by a wire, we say that the direction of the current is from A to B ; whereas the language of the two-fluid theory would be, that a current of vitreous or positive electricity travels from A to B, and a current of resinous or negative from B to A. CHAPTER XXXVI. ELECTRICAL INDUCTION. 412. Induction. In the preceding chapter we have spoken of move- ments of material bodies caused by electrical attractions and repul- sions. We have now to treat of the movement of electricity itself in obedience to the attractions or repulsions exerted upon it by other electricity. This kind of action is called induction. It may be illustrated by means of the arrangement shown in Fig. 336. The apparatus consists of a sphere C which is electrified posi- tively, suppose, and of a conducting insulated cylinder A B placed near it. From this latter are suspended at equal distances a few pairs of pith - balls. When the cylinder is brought near the sphere, the balls are observed to diverge. The divergence of the different pairs is not the same, but goes on decreasing from the pair nearest the cylinder until a point M is reached, where there is no divergence. Beyond this the divergence goes on increasing. The neutral point M does not exactly bisect the length of the cylinder, but is nearer the end A than the end B, and the former end is found to be more strongly electrified than the latter. It is easy to show that the two ends of the cylinder are charged with opposite kinds of electricity ; the end A being negatively, and 34 Fig. 336. Electrification by Influence. 514 ELECTEICAL INDUCTION. the end B positively electrified. We have only to bring an electrified stick of resin near the pith-balls at A, when these will be found to be repelled ; if, on the contrary, it be held near those at B, they will be attracted. The explanation is, that the positive electricity with which C is charged attracts the negative electricity of AB to the end A, and repels the positive to the end B. This action is more powerful at A than at B, on account of the greater proximity of the influencing body, and for the same reason the effect falls off more rapidly in the portion AM than in MB. If the cylinder be brought closer to the sphere, the divergence of the balls increases ; if it be removed farther from it, the divergence diminishes. Finally, all signs of electricity disappear if the sphere be taken away, or connected with the earth. If, while the cylinder is under the influence of the electricity of C, the end B is connected with the earth, the pith-balls at this end Fig. 337. Successive Induction. immediately collapse, while the divergence of those at A increases. The explanation is that the electricity which was repelled to the end B escapes to the earth, and thus affords an opportunity for a fresh exercise of induction on the part of the sphere, which increases the accumulation of negative electricity at A. We may also remark that the whole of the cylinder is now negatively electrified, the neutral line being pushed back to the earth. If the earth-connection be now broken, and the sphere C be then removed, the cylinder will remain negatively electrified, and will be in the same condition as if it had been touched by a negatively-electrified body. This mode ATTRACTION AND REPULSION. 515 of giving a charge to a conductor is called charging by induction, and the charge thus given is always opposite to that of the inducing body C. If a series of such conductors as AB be placed in line, without contact, and the positively-electrified body C be placed opposite to one end of the series, all the conductors will be affected in the same manner as the single conductor in the last experiment. They will all be charged with negative electricity at the end next C, and with positive electricity at the remote end, the effect, however, becoming feebler as we advance in the series. In this experiment each of the conductors acts inductively upon those next it ; for example, if there be two conductors AB, A'B', as in Fig. 337, the development of electricity at A' and B' is mainly due to the action of the positive electricity in MB. If the conductor AB be removed, the pith-balls at A' and B' will diminish their divergence. The molecules of a body may be regarded as such a series of con- ductors, or rather as a number of such series. When an electrified body is brought near it, each molecule may thus become positive on one side and negative on the other. In the case of good conductors, this polarization is only instantaneous, being destroyed by the dis- charge of electricity from particle to particle. Good insulators are substances which are able to resist this tendency to discharge, and to maintain a high degree of polarization for a great length of time. This is Faraday's theory of " induction by contiguous particles." 413. Electrical Attraction and Repulsion. The attraction which is observed when an electrified is brought near to an un electrified body, is dependent upon induction. Suppose, for instance, that a body C, which is positively electrified, is brought near to an insulated and uncharged pith-ball. Negative electricity is induced on the near side of the pith-ball, and an equal quantity of positive on the further side. The former, being nearer to the bod}'- C, Fig. sss.-Eiectncai Attraction. is more strongly attracted than the other is repelled. The ball is therefore upon the whole attracted. If the pith-ball, instead of being insulated, is suspended by a con- ducting thread from a support connected with the earth, it will be more strongly attracted than before, as it is now entirely charged with negative electricity. In the case of any insulated conductor, the algebraic sum of the 516 ELECTRICAL INDUCTION. electricities induced upon it by the presence of a neighbouring elec- trified body must be zero. If the pith-ball be insulated, and have an independent charge of either kind of electricity, the total force exerted on the pith -ball is the algebraic sum 1 of the two following quantities : (1) The force which the ball would experience, if it had no independent charge. This force, as we have just seen, is always attractive. (2) The force due to the independent charge when distributed over the ball as it would be if C were removed. This second force is attractive or repulsive, according as the independent charge is ot unlike or like sign to that of C. In the latter case, repulsion will generally be observed at distances exceeding a certain limit and attraction at nearer distances, the reason being that the force (1) due to the induced distribution increases more rapidly than the other as the distance is diminished. It is important to remember this in testing, by the electric pen- dulum, or by any other electroscope, the kind of electricity with which a body is charged. In bringing the body towards the elec- troscope, the first movement produced is that which is to be observed, and repulsion is in general a more reliable test of kind of electricity than attraction. 415. Electroscopes. An electroscope is an apparatus for detecting the presence of electricity, and determining its sign. The insulated electric pendulum is an electroscope. If the pith- ball, when itself uncharged, is attracted by a body brought near it, we know that the body is electrified. To determine the kind of electricity, the body is allowed to touch the pith-ball, which is then repelled. At this moment an excited glass tube is brought near. If it repels the ball, this latter, as well as the body which touched it, must be electrified positively. If the glass tube attracts it, or, still more decisively, if excited resin or sealing-wax repels it, the ball and the body which touched it are electrified negatively. The loss of electricity from the pith-ball is often so rapid as to render this test of sign somewhat uncertain. The gold-leaf electroscope (Fig. 339) is constructed as follows : 1 We here suppose C to be a non-conductor, so that the distribution of its electricity ia flot affected by the presence of the pith-ball. If C be a conductor, the effect of induction upon it will be to favour attraction, so that an attractive force must be added to the two forces specified in the text. ELECTROSCOPES. 517 Pig. 339. Gold-leaf Electroscope. Two small gold-leaves are attached to the lower end of a metallic rod, which passes through an. opening in the top of a bell-glass, and terminates in a ball. The metallic rod is sometimes, for the sake of better insulation, inclosed in a glass tube secured by sealing-wax or some other non-conducting cement, and, for the same purpose, the upper part of the bell-glass is often varnished with shellac, which is less apt than glass to acquire a deposit of moisture from the air. The bell-glass is attached below to a metallic base, which ex- cludes the external air. For the gold- leaves are sometimes substituted two straws, or two pith-balls suspended by linen threads; we have thus the straw-electroscope and the pith-ball electroscope. To test whether a body is electri- fied, it is brought near the ball at the top of the electroscope. The like electricity is repelled into the leave?, and makes them diverge, while the unlike is attracted into the ball. The sign of the body's charge may be determined in the following manner: While the leaves are divergent under the in- fluence of the body, the operator touches the ball with his finger. This causes the leaves to collapse, and gives to the insulated con- ductor composed of leaves, rod, and ball, a charge opposite to that of the influencing body. The finger must be removed while the influencing body remains in position, as the amount of the induced charge depends upon the position of the influencing body at the instant of breaking connection. On now withdrawing the influencing body, the charge of unlike electricity is no longer attracted to the ball, but spreads over the whole of the conductor, and causes the leaves to diverge. If, while this divergence continues, an excited glass tube, when gradually brought towards the ball, diminishes the divergence, we know that the body in question was electrified posi- tively. If it increases the divergence, the body was electrified nega- tively. Great caution must be used in bringing electrified bodies near the gold-leaf electroscope, as the leaves are very apt to be ruptured by 518 ELECTRICAL INDUCTION. quick movements. If they diverge so widely as to touch the sides of the bell-glass, it is often difficult to detach them from the glass without tearing. To prevent this contact, two metallic columns are interposed, communicating with the ground. If the leaves diverge too widely, they touch these columns and lose their electricity. CHAPTER XXXVII. MEASUREMENT OF ELECTRICAL FORCES. 416. Coulomb's Torsion-balance. Coulomb, who was the first to make electricity an accurate science, employed in his researches an instrument which is often called after his name, and which is still extensively employed. It depends on the principle that the torsion of a wire is simply proportional to the twisting couple. We shall first describe it, and then point out some of its applications. It consists of a cylindrical glass case A A (Fig. 340), from the upper end B of which rises another glass cylinder DD of much smaller dia- meter. This small cylinder is fitted at the top with a brass cap a, carry- ing an index C. Outside of this, and capable of turning round it, is another cap 6, the top of which is divided into 360 equal parts. In the centre of the cap 6 is an opening through which passes a small metal cylinder d, capable of turning in the opening with moderate fric- tion, and having at its lower end a notch or slit. When the cap b is turned, the cylinder d turns with it; but the latter can also be turned separately, so as not to change the reading. These parts com- pose the torsion-head. A very fine metallic wire is held by the notch, and supports a small piece of metal, through which passes a light needle of shellac /, carrying at one end a small gilt ball g. A circular Fig. 340. Coulomb's Torsion-balance. 520 MEASUREMENT OF ELECTRICAL FORCES. scale runs round the outside of the large cylinder in the plane of the- needle. Finally, opposite the zero of this scale, there is a fixed ball g' of some conducting material, supported by a rod/ of shellac, which passes through a hole in the cover of the cylindrical case. 417. Laws of Electric Repulsion. To illustrate the mode of em- ploying this apparatus for electrical measurements, we shall explain the course followed by Coulomb in investigating the law according to which electrical repulsions and attractions vary with the distance. The index is set to the zero of the scale. The inner cylinder d is then turned, until the movable ball just touches the fixed ball without any torsion of the wire. The fixed ball is then taken out, placed in communication with an electrified body, and replaced in the apparatus. The electricity with which it is charged is commu- nicated to the movable ball, and causes the repulsion of this latter through a number of degrees indicated by the scale which surrounds- the case. In this position the force of repulsion is in equilibrium with the force of torsion tending to bring back the ball to its original position. The graduated cap b is then turned so as to oppose the repulsion. The movable ball is thus brought nearer to the fixed ball, and at the same time the amount of torsion in the wire is- increased. By repeating this process, we obtain a number of dif- ferent positions in which repulsion is balanced by torsion. But we know, from the laws of elasticity, that the force (strictly the couple 1 ) of torsion is proportional to the angle of torsion. Hence we have only to compare the total amounts of torsion with the distances of the two balls. By such comparisons Coulomb found that the force of electrical repulsion varies inversely as the square of the distance. The following are the actual numbers obtained in one of the experiments. The original deviation of the movable ball being 36, it was found that, in order to reduce this distance to 18, it was necessary to turn the head through 1 26, and, for a farther reduction of the deviation to 8 0> 5, an additional rotation through 441 was required. It will thus be perceived that at the distances of 36, 18, and 8 0< 5, which may be practically considered as in the ratio of 1, \, and , the forces of repulsion were equilibrated by torsions of 30, 1 The repulsive force on the movable ball is equivalent to an equal and parallel force acting at the centre of the needle (the point of attachment of the wire), and a couple whose arm is the perpendicular from this centre on the line joining the balls. This couple must be equal to the couple of torsion. The other component produces a small deviation of the suspending wire from the vertical. EQUATION OF EQUILIBRIUM. 521 126 + 18= 144, and 441 + 126 + 8-5 = 575-5 respectively. Now 144 is 36x4, and 575'5 may be considered as 576, or 36x16. Hence we perceive that, as the distance is divided by 2, or by 4, the force of repulsion is multiplied by 4 or by 16, which precisely agrees with the law enunciated above. 418. Equation of Equilibrium. We must, however, observe that in this mode of reducing the obser- vations two inaccurate assumptions are made. First, the distance be- tween the balls is regarded as being equal to the arc which lies between them, whereas it is really the chord of that arc. Secondly, the force of repulsion is regarded as acting always at the same arm, whereas its arm, being the perpendicular from the centre on the chord, dimi- nishes as the distance increases. The following investigation is more rigorous. Let AOB (Fig. 341), the angular distance of the balls, be denoted by a, and let I be the length of the radius OA. Then the chord A B is 21 sin a, and the arm O K is I cos a. Let / denote the force of repulsion at unit distance, and n the couple of torsion for 1. Then the force of repulsion in the given position is -j -. a -T- if the law of inverse squares be true, and the moment of this about the centre is -rji which must be equal to nA., if A be the number Fig. 341. of derees of torsion. Hence we have = A sin a tan and as the first member of this equation is constant, the second mem- I -er must be constant also for different values of A and o, if the law of inverse squares be true. The degree of constancy is shown by the following table : 1st experiment, 36 2d experiment, 18 3d experiment, 8 '5 Supposed case, 9 A 36 144 575-5 576 A sin o tan \ a 3-614 3-568 3-169 3-557 The difference between the first and second numbers of the last 522 MEASUREMENT OF ELECTRICAL FORCES. column is insignificant. That between the second and third is more considerable, 1 but in reality only corresponds to an error of half a degree in the measurement of the arc. 419. Case of Attraction. The law of attractions may be investigated by a similar method. The index is set to zero, and the central piece is turned so as to place the movable ball at a known distance from the fixed ball. The two balls are then charged with electricity of different kinds. The movable ball is accordingly attracted towards the other, and settles in a position in which attraction is balanced by torsion. By altering the amount of torsion, different positions of the ball can be obtained. On comparing the distances with the corre- sponding torsions, it is found that the same law holds as in the case of repulsion. The experiment, however, is difficult, and is only pos- sible when the balls are very feebly electrified. To prevent the contact of the two balls, Coulomb fixed a silk thread in the instru- ment, so as to stop the course of the movable ball. 420. Law of Attraction and Repulsion as depending on Amount of Charge. We may assume as evident, that when an electrified ball is placed in contact with a precisely equal and similar ball, the charge will be divided equally between them, so that the first will retain only half the charge which it had before contact. Suppose that an observation on repulsion has just been made with the torsion-balance, and that we touch the fixed ball with another exactly equal insulated ball, which we then remove. It will be found that the amount of torsion requisite for keeping the movable ball in its observed position is just half what it was before. The 1 We have already seen that the mutual induction of two conductors tends to diminish their mutual repulsion, and that this inductive action becomes more important as the distance is diminished. Hence the repulsion at distance 9 should be less than a quarter of that at distance 18. The apparent error thus confirms the law. Many persons have adduced, as tending to overthrow Coulomb's law of inverse squares, experimental results which really confirm it. Except when the dimensions of the charged bodies are very small in comparison with the distance, the observed attraction or repulsion is the resultant of an infinite number of forces acting along lines drawn from the different points of the one body to the different points of the other. The law of inverse squares applies directly to these several components, and not to the resultant which they yield. The latter can only be computed by elaborate mathematical processes. It is incorrectly assumed in the text that the law ought to apply directly to two spheres, when by their distance we understand the distance between their nearest points. It is not obvious that the distance of the nearest points should give a better result than the distance between the centres. The strongest evidence for the rigorous exactness of the law of inverse squares is indirect; see 4'21 C. LAW OF ATTRACTION AND REPULSION. same result will be obtained by touching the movable ball with a ball of its own size. We conclude that, if the charge of either body be altered, the attractive or repulsive force between the bodies ;it given distance will be altered in the same ratio. The law is not rigorously true for bodies of finite size, unless the distribution of the electricity on the two bodies remains unchanged. When the two bodies are very small in all their dimensions in comparison with the distance between them, their mutual force is represented by the expression ff q and q' denoting their charges, and D the distance. If this expression has the positive sign, the force is repulsive, if negative, attractive. 421. Electricity resides on the Surface. Electricity (subject to the Fig. 342. Blot's Experiment. exceptions mentioned below) resides exclusively on the external surface of a conductor. This is perhaps implied in the experimental fact frequently observed by Coulomb, that when a solid and a hollow sphere of equal external diameter are allowed to touch each other, any charge possessed by either is divided equally between them. A 524 MEASUREMENT OF ELECTRICAL FORCES. direct demonstration is afforded by the following experiment of Biot: We take an insulated sphere of metal, charge it with electricity, and cover it with two hemispheres furnished with insulating handles, which fit the sphere exactly (Fig. 34)2). If the two hemispheres be quickly removed, and presented to an electric pendulum, they will be found to be electrified, while the sphere itself will show hardly any traces of electricity. We must, however, remark that this experiment is rarely successful, and that generally the sphere remains very sensibly electrified. The reason of this is, that it is very difficult to remove the hemispheres so steadily, as not to permit their edges- to touch the sphere after the first separation. The following is a much surer form of the experiment: A. hollow insulated sphere, with an orifice in the top, is charged with electricity (Fig. 343). A proof-plane, consisting of a small disc of gilt paper insu- lated by a thin handle of shellac, is then ap- plied to the interior surface of the sphere, and, when tested by an electric pendulum or an electroscope, is- found to exhibit no trace of electricity. But if, on the contrary, tht- disc be applied to the external surface of the sphere, it will be found to be electrified, and capable of attracting light bodies. Faraday varied this experiment, by substituting a cylinder of wire-gauze for the sphere. This cylinder rested on an insulated disc of metal. The disc was charged with electricity, and it was found that no trace of the electricity could be detected by applying the proof-plane to the interior surface of the cylinder. Fig. 343. Proof-plane and Hollow Sphere. ELECTRICITY CONFINED TO EXTERNAL SURFACE. Olio The following experiment is also due to Faraday. A metal ring is fixed upon an insulating stand (Fig. 344). To this ring is attached a cone-shaped bag of fine linen, which is a conductor of electricity. A silk thread, attached to the apex of the cone, and extending both Fig. 344. Faraday's Experiment. ways, enables the operator to turn the bag inside out as often as required, without discharging it. When the bag is electrified, the application of the proof-plane always shows that there is electricity on the outer, but not on the inner surface. When the bag is turned inside out, the electricity therefore passes from one surface of the linen to the other. 421 A. Limitations of the Rule. There are two exceptions to the rule that electricity is confined to the external surface of a conductor. 1. It does not hold for electric currents. We shall see hereafter in connection with galvanic electricity, that the resistance which a wire of given length opposes to the passage of electricity through it, depends not upon its circumference but upon its sectional area. A hollow wire will not conduct electricity so well as a solid wire of the same external diameter. 2. Electricity may be induced on the inner surface of a hollow conductor by the presence of an electrified body insulated from the conductor itself. If an insulated body charged with electricity be introduced into the interior of a hollow conductor, so as to be com- pletely surrounded by it, but still insulated from it, it induces upon the inner surface a quantity equal to its own charge, but of opposite sign. If the conductor is insulated, an equal quantity, but of the same sign as the charge of the inclosed body, is repelled to the outside, and 526 MEASUREMENT OF ELECTRICAL FORCES. this is true whether the conductor has an independent charge of its own or not. In this case, then, we have electricity residing on both the external and the internal surfaces of a hollow conductor, but it still resides only on the surfaces. If a conducting body connected with the earth be introduced into the interior of a hollow charged conductor, so as to be partially sur- rounded by it, the body thus introduced will acquire an opposite charge by induction, and, by the reciprocal action of this charge, electricity will be induced on the inner at the expense of the outer surface of the hollow conductor, just as in the preceding case. 421 B. Ice-pail Experiment. The effect of introducing a charged body within a hollow conductor is well illustrated by the following experiments of Faraday. Let A (Fig. 344 A) represent an insulated pewter ice-pail, ten and a half inches high and seven inches in diameter, connected by a wire with a delicate gold leaf electro- scope E, and let C be a round brass ball insulated by a dry thread of white silk, three or four feet in length, so as to remove the influence of the hand holding it from the ice-pail below. Let A be perfectly discharged, and let C, after being charged at a distance, be introduced into A as in the figure. If C be positive, E also will diverge positively; if C be taken away, E will collapse perfectly, the apparatus being in good order. As C enters the vessel A, the divergence of E will increase until C is about three inches below the edge of the vessel, and will remain quite steady and unchanged for any greater depression. If C be made to touch the bottom of A, all its charge is communicated to A, and C, upon being withdrawn and examined, is found perfectly discharged. Now Faraday found that at the moment of contact of C with the bottom of A, not the slightest change took place in the divergence of the gold-leaves. Hence the charge previously developed by induc- tion on the outside of A must have been precisely equal to that acquired by the contact, that is, must have been equal to the charge of C. E Fig. 344A. Ice-pail Experiment. EXPERIMENT WITH ICE-PAILS. 527 He then employed four ice-pails (Fig. 344 B), arranged one within the other, the smallest innermost, insulated from each other by plates of shellac at the bottom, the outermost pail being connected with the electroscope. When the charged carrier- ball C was introduced within the innermost pail, and lowered until it touched the bot- tom, the electrometer gave precisely the same indications as when the outermost pail was employed alone. When the inner- most was lifted out by a silk thread after being touched by C, the gold-leaves col- lapsed perfectly. When it was introduced again, they opened out to the same extent as before. When 4 and 3 were connected by a wire let down between them by a silk thread, the leaves remained unchanged, and so they still remained when 3 and 2 were connected, and finally when all four pails were connected. 421 c. No Force within a Conductor. When a hollow conductor is electrified, however strongly, no effect is produced upon pith-balls, gold-leaves, or any other electroscopic apparatus in the interior, whether connected with the hollow conductor, or insulated from it, provided, in the latter case, that they have no communication with bodies external to the hollow conductor. Faraday constructed a cubical box, measuring 12 feet each way, covered externally with copper wire and tin-foil, and insu- lated from the earth. He charged this box very strongly by outside communication with a powerful electrical machine ; but a gold-leaf electrometer within showed no effect. He says, "I went into the cube and lived in it, using lighted candles, electrometers, and all other tests of electrical states. I could not find the least influence upon them, or indication of anything particular given by them, though all the time the outside of the cube was powerfully charged, and large sparks and brushes were darting off from every part of its outer surface." The fact that electricity resides only on the external surface of a conductor, combined with the fact that there is no electrical force in the space inclosed by this surface, affords a rigorous proof of the law Fig. 344 B. Experiment with Four Ice-pails. 528 MEASUREMENT OF ELECTRICAL FORCES. of inverse squares. For if the conductor be a sphere removed from the influence of external bodies, its charge must be distributed uniformly over its surface. Now it admits of proof, and is well known to mathematicians, that a uniform spherical shell exerts no attraction at any point of the interior space, if the law of attraction be that of inverse squares, and that the internal attraction does not vanish for any other law. 421 D. Electrical Density and Distribution. When the proof-plane is applied to different parts of the surface of a conductor, the quan- tities of electricity which it carries off are not usually equal. But the electricity carried off by the proof-plane is simply the electricity which resided on the part of the surface covered by it, for the proof- plane during the time of its contact is virtually part of the surface of the conductor. We must therefore conclude that equal areas on different parts of the surface of a conductor have not equal amounts of electricity upon them. It is also found that if the charge of the conductor be varied, the electricity resident upon any specified portion of the surface is changed in the same ratio. The ratio of the quantities of electricity on two specified portions of the surface is in fact independent of the charge, and depends only on the form of the conductor. This is expressed by saying that distribution is inde- pendent of charge, and that the distribution of electricity on the surface of a conductor depends on its form. By the average electrical density on the whole or any specified portion of the surface of a conductor, is meant the quantity of elec- tricity upon it, divided by its area. By the electrical density at a specified point on the surface of a conductor, is meant the average electrical density on an exceedingly small area surrounding it, in other words, the quantity of electricity per unit area at the point. The name is appropriate, from the analogy of ordinary material density, which is mass per unit volume, and is not intended to imply any hypothesis as to the nature of electricity. The name was intro- duced by Coulomb, who first investigated the subject in question, and is generally employed by the best electricians in this country. The term thickness of electrical stratum, which was introduced by Poisson, is much used in France, but is more open to objection from the coarse assumptions which it seems to involve. The following are some of Coulomb's results. The dotted line in each of the figures is intended to represent, by its distance from the outline of the conductor, the electric-density at each point of the ELECTRIC DENSITY. 529 latter. In all cases it is to be understood that the conductor is so far removed from external bodies as not to be influenced by them : 1. Sphere (Fig. 345). The electric density is the same for all points on the surface of a spherical conductor. 2. Ellipsoid (Fig. 346). The density is greatest at the ends of the Fig. 345. Distribution on Sphere. Fig. 346. Distribution on Ellipsoid. longest, and least at the ends of the shortest axis; and the densities at these points are simply proportional to the axes themselves. 1 3. Flat Disc (Fig. 347). The density is almost inappreciable over the whole of both faces, except close to the edges, where it increases almost per saltum. 4. Cylinder with Hemispherical Ends (Fig. 348). The density is Fig. 347. Distribution on Disc. Fig. 348. Distribution on Cylinder with rounded end*. a minimum, and nearly uniform, at parts remote from the ends, and attains a maximum at the ends. The ratio of the density at the ends to that at the sides increases as the radius of the cylinder diminishes, the length of the cylinder remaining the same. 5. Spheres in Contact In the case of equal spheres, the charge, which is nothing at the point of contact, and very feeble up to 30 from that point, increases very rapidly from 30 to 60, less rapidly from 60 to 90, and almost insensibly from 90 to 180. When the spheres are of unequal size, the charge at any point on the smaller 1 More generally, the density at any point on the surface of an ellipsoid is proportional to the length of a perpendicular from the centre of the ellipsoid on a tangent plane at the point. If an ellipsoid, similar and nearly equal to the given one, be placed so that the corre- sponding axes of the two are coincident, we shall have a thin ellipsoidal shell, whose thick- ness at any point exactly represents the electric density at that point. Such a shell, if composed of homogeneous matter attracting inversely as the square of the riistance, would exercise no force at points in its interior. 35 530 MEASUREMENT OF ELECTRICAL FORCES. sphere is greater than at the corresponding point on the larger onej anil as the smaller sphere is continually diminished, the other remaining the same, the ratio of the densities at the extremities of the line of centres tends to become 2:1. 422. Method of Experiment. The preceding results were obtained by Coulomb in the following manner. He touched the electrified body at a known point with the proof-plane, and then put the plane in the place of the fixed ball of the torsion-balance, the movable ball having previously been charged with electricity of the same- sign. Repulsion was thus produced, and the amount of torsion necessary to keep the balls at a certain distance asunder was observed. He then repeated the experiment with electricity taken from a dif- ferent point of the body under examination, and the ratio of the densities at the two points was given by the ratio of the torsions necessary to keep the balls at the same distance. By way of checking the accuracy of this mode of experimentation, Coulomb electrified an insulated sphere, and measured the electric density on its surface by the method described above. He then touched the sphere with another precisely equal sphere, and on again applying the proof-plane he found that the charge carried off by the plane was just half what it had been before. 423. Alternate Contact. The above experiments naturally require some time, during which the body tinder investigation is gradually losing its charge. The consequence is, that the densities indicated by the balance, if taken singly, do not correctly represent the electric distribution. This source of error was avoided by Coulomb- in the following manner. He touched two points on the body suc- cessively, and determined the electric density at each; and then, after an interval equal to that between the two experiments, he touched the first point again, and obtained a second measure of its density, -vhich was less than the first, on account of the dissipation of elec- tricity. If the densities thus observed be denoted by A and A', and the density observed at the second point by B, it is evident that g is- greater, and -g less than the ratio required. Coulomb adopted, a A + A' the correct value, their arithmetic mean \ - R . 424. Power of Points. The distribution of electricity on a con- ductor of any form may be roughly described, by saying that the density is greatest on those parts of the surface which project most. DISSIPATION OF CHARGE. 531 or which have the sharpest convexity, and that in depressions or concavities it is small or altogether insensible. Theory shows that at a perfectly sharp edge, such, for example, as is formed by two planes meeting at any angle however obtuse, but not rounded off, the density must be infinite, and a fortiori it must be infinite at a perfectly sharp point, for example at the apex of a cone, however obtuse, if not rounded off. Practically, the points and edges of bodies are always rounded off; the microscope shows them merely as places of very sharp convexity (that is, of very small radius of curva- ture), and hence the electric density at those places is really finite; but it is exceedingly great in comparison with the density at other parts, and this is especially true of very acute points, such as the point of a fine needle. The consequence is, that if a pointed con- ductor is insulated and charged, the concentration of a large amount of repulsive force within an exceedingly small area pro- duces very rapid escape of electricity at the points. Conductors intended to retain a charge of electricity must have no points or edges, and must be very smooth. If of considerable length in proportion to their breadth, they are usually made to terminate in large knobs. 425. Dissipation of Charge. When an insulated conductor is charged and left to itself, its charge is gradually dissipated, and at length completely disappears. This loss takes place partly through the supports, and partly through the air. As regards the supports, the loss occurs chiefly at their surface, especially if (as is usually the case) they are not perfectly dry. It is diminished by diminishing their perimeter, and by increasing their length; for example, a long fibre of glass or raw silk is an excellent insulator. As rp^nrds the air, we must distinguish between conduction and convection. Very hot air and highly rarefied air probably act as conductors; but air in the ordinary condition acts chiefly by con- tact and convection. Successive layers of air become electrified by contact with the conductor, and are then repelled, carrying off the electricity which they have acquired. It is by an action of this kind that electricity escapes into the air from points, as is proved by the wind which passes off from them (444). Particles of dust present in the air, in like manner, act as carriers, being attracted to the conductor, charged by contact with it, and then repelled. They also frequently adhere by one end to the conductor, 632 MEASUREMENT OF ELECTRICAL FORCES. and thus constitute pointed projections through which electricity is discharged into the air. Coulomb deduced from his observations on dissipation of charge, a law precisely analogous to Newton's law of cooling, namely, that when all other circumstances remain the same, the rate of loss is simply proportional to the charge, so that the charges at equal intervals of time form a decreasing geometric series. Subsequent experience has confirmed this law, as approximately true for moderate charges of the same sign. Negative charges are, however, dissipated more rapidly than positive. CHAPTER XXXVIII. ELECTRICAL MACHINES. 426. Electrical Machines. The first electrical machine was invented "by Otto Guericke, to whom, as we have already seen ( 129), science is indebted for the invention of the air-pump. It consisted of a ball of sulphur which was turned upon its axis by one person, while another held his hands upon the ball, thus causing the friction necessary for the production of electricity. The result was that the globe was negatively electrified, and the positive electricity escaped into the earth through the hands of the operator. This machine, however, was capable of producing only very feeble effects, and the sparks obtained from it were visible only in the dark. An English philosopher, Hawksbee, substituted a globe of glass for the globe of sulphur; the electricity thus obtained was positive, and the sparks obtained by the new machine were of considerable bright- ness. The machine, however, was for the time superseded by the use of glass tubes, which continued to be the favourite instruments for generating electricity until the middle of the eighteenth century, when a German philosopher, Boze, professor of physics at Wittem- berg, revived and perfected Hawksbee's machine, which became universally adopted. Fig. 349, which is taken from the Lemons de Physique of the Abb Nollet, published in 1767, shows the arrangement of the machine adopted by this celebrated philosopher. It consists of a large wheel, round which is passed an endless cord, which, passing also round a pulley, serves to turn a glass globe when the wheel is set in motion. The electricity thus produced is collected on a conductor suspended from the ceiling by silk cords. It will be observed that, in the figure, the friction is produced by the hand. This mode of applying friction, which is evidently rude RAMSDEN S MACHINE. 535 and defective, was nevertheless long used for want of a better, though many attempts were made to replace it by the use of rubbers of leather, stuffed with hair, and pressed against the globe by means of regulating screws. The shape of the globe rendered the use of these very difficult, and it was not until a cylinder was substituted for the globe that they were generally adopted. 427. Ramsden's Machine. The kind of machine most commonly employed at present is the plate-machine, invented by Ramsden about 1768, and only slightly changed and improved since. The most usual form of this machine is shown in Fig. 350. It Fig. 350. Ramsden's Electrical Machine. has a circular plate of glass, which turns on an axis supported by two wooden uprights. On each side of the plate, at the upper and lower parts of the uprights, are two cushions, which act as rubbers when the plate is turned. In front of the plate are two metallic conductors supported on glass legs, and terminating in branches, which are bent round the plate at the middle of its height, 536 ELECTRICAL MACHINES. and are studded with points projecting towards it. The plate becomes charged with positive electricity by friction against the cushions, and gives off its electricity through the points to the two conductors, or, what amounts to the same thing, the conductors give off negative electricity through the points to the positively- electrified plate. In order to avoid loss of electricity from that portion of the plate which is passing from the cushions to the points, sector-shaped pieces of oiled silk are placed so as to cover it on both sides. The cushions become negatively electrified by the friction ; and the machine will not continue working unless this negative electricity is allowed to escape. The cushions are accordingly connected with the earth by means of metal plates let into their supports. 428. Limit of Charge. As the conductors become more highly charged, they lose electricity to the air more rapidly, and a time soon arrives when they lose electricity as fast as they receive it from the plate. After this, if the machine continues to be worked uniformly, their charge remains nearly constant. This limiting amount of charge depends very much upon the condition of the air ; and in damp weather the machine often refuses to work unless special means are employed to keep it dry. The rubbers are covered with a metallic preparation, of which several different kinds are employed. Sometimes it is the compound called aurum musivum (bisulphide of tin), but more frequently an amalgam. Kienmeier's amalgam consists of one pan of zinc, one of tin, and two of mercury. The amalgam is mixed with grease to make it adhere to the leather or silk which forms the face of the cushion. Before using the machine, the glass legs which support the conductors should be wiped with a warm dry cloth. The plate must also be cleaned Fig. 35i.-Quadrant Electroscope. from any dust or portions of amalgam which may adhere to it, and lastly, dried with a hot cloth "or paper. When these precautions are taken, the machine, if standing near a fire, will always work; but the charging of Leyden jars, and especially of batteries, may be rendered impossible by bad weather. The variations of charge are indicated by the quadrant electroscope (Fig. 351), which is attached to one of the conductors It consists NAIRNE S MACHINE. 537 of an upright conducting stem, supporting a quadrant, or more com- monly a semicircle, of ivory, at whose centre a light needle of ivory is jointed, carrying a pith-ball at its end. When there is no charge in the conductor, this pendulum hangs vertically, and as the charge increases it is repelled further and further from the stem. In damp weather it will be observed to return to the vertical position almost immediately on ceasing to turn the machine, while in very favour- able circumstances it gives a sensible indication of charge after two or three minutes. 429. Nairne's Machine. Ramsden's machine furnishes only positive electricity. In order to obtain negative electricity, it is necessary to Fig. 352. Nairne's Electrical Machine. insulate the cushions from the ground, and to place them in com- munication with an insulated conductor. An arrangement of this kind is adopted in Nairne's machine. In this machine a large cylinder of glass revolves between two separately insulated conductors. One of these has a row of points projecting towards the glass, and collects positive electricity. The other is connected with the rubber, and collects negative. If one kind of electricity only is required, the conductor which furnishes the other must be connected with the ground. 430. Winter's Machine. Winter, of Vienna, has introduced some modifications in Ramsden's machine. 533 ELECTRICAL MACHINES. Instead of four cushions, there are, as will be seen by the figure (Fig. 353), only two, which are in communication with a spherical conductor, supported on a glass pillar. This may be used to collect negative electricity, in the same way as the negative conductor in Nairne's machine. The chief or positive conductor consists of an insulated sphere, on the top of which is often another sphere of smaller size. The positive electricity is collected from the plate by Fig. 353. Winter's Electrical Machine. means of two rings opposite to each other, one on each side of the plate. On the side next the plate, they have a groove, which is lined with metal, and studded with points. They are supported by an arm which is inserted in the positive conductor. The size of the positive conductor is often increased by the addition of a very large ring (3 or 4 feet in diameter) which is supported on the top of the large sphere. The ring consists of very stout brass wire inclosed in well- polished mahogany. Winter's machine appears to give longer sparks than the ordinary machine under the same circumstances. This circumstance is owing, partly at least, to the considerable distance between the rubber and the positive conductor, which prevents the occurrence of discharges between them. HYDRO-ELECTRIC MACHINE. 539 431. Hydro-electric Machine. About the year 1840, Mr. (now Sir) W. Armstrong invented an electric machine, in which electricity was generated by the friction of steam against the sides of orifices, through which it is allowed to escape under high pressure. It consists of a Fig. 354. Armstrong's Hydro-electric Machine. boiler with the fire inside, supported on four glass legs. The steam, before escaping, passes through a number of tubes which traverse a cooling-box containing water, into which dip meshes of cotton, which are led over the tubes, and passed round them. The cooling thus produced in the tubes, causes partial condensation of the steam. This has been found to be an indispensable condition, the friction of per- 540 ELECTRICAL MACHINES. fectly dry steam being quite inoperative. Speaking strictly, it is the friction of the drops of water against the sides of the orifice, which generates the electricity, and the steam merely furnishes the means of applying the friction. The jet of steam is positively, and the boiler negatively electrified. The positive electricity is collected by directing the jet of steam upon a metal comb communicating with an insulated conductor. The form of the outlet by which the steam escapes is shown in Fig. 355. The steam is checked in its course by a tongue of metal, round which it has to pass, before it can enter the wooden tube through which it escapes into the air. This machine, in order to work well, requires a pressure of several atmospheres. The water in the boiler should be distilled water. If a saline solution be intro- duced into the tube through which the steam escapes, all traces of electricity immediately _^ disappear. The generation of electricity varies both in sign and degree, according to the substance of which the escape-tube is Fig. sss.-outiet of steam. composed, and according to the liquid whose particles are carried out by the steam. Thus, when a small quantity of oil of turpentine is introduced into the jet of steam, the boiler becomes positively, and the steam negatively electrified. The hydro-electric machine is exceedingly powerful. At the Polytechnic Institution in London, there was one with a boiler 78 inches long and 42 in diameter, and with 46 jets. Sparks were obtained from the conductor at the distance of 22 inches. The machine is, however, very inconvenient to manage. A long time is required to get up the requisite pressure of steam. The boiler must be carefully washed with a solution of potash, after each occasion of its use; and, finally, the working of the machine is necessarily accom- panied by the disengagement of an enormous quantity of steam, which, besides causing a deafening noise, has the mischievous effect of covering with moisture everything within reach. Accordingly, though very interesting in itself, it is by no means adapted to the general purposes of an electrical machine. 432. Holtz's Machine. In the machines just described, electricity is produced by the friction of one substance against another. Quite recently, several machines have been invented of quite a different kind, in which a body is electrified once for all, and made to act by HOLTZS MACHINE. induction upon a movable system, in such a way as to produce a continual generation of electricity. The most successful of these is that invented by Holtz of Berlin in 1865. It contains two thin circular plates of glass, one of which, A, is fixed, while the other, B, which is rather smaller, can be made to revolve very near it. In the fixed plate there are two large openings called windows near the extremities of its horizontal diameter. Fig. 356. Holtz's Electrical Machine. Adjacent to these are glued two paper bands or armatures //', each having a sharp tongue of card projecting through the window, and pointing the opposite way to that in which the revolving plate turns. Two rows of brass points P, P' are placed opposite the armatures, on the other side of the revolving plate, and are connected with two insulated conductors terminating in the knobs n, m, which may be called the poles or electrodes of the machine. These knobs can be set at any distance asunder. In starting the machine, they are placed in contact, and one of the armatures, suppose /, is electrified by holding against it an excited sheet of vulcanite, or by leading to 542 ELECTRICAL MACHINES. it a wire from the conductor of a frictional machine. A peculiar sizzling sound is almost immediately heard, and the knobs may then be separated to a gradually increasing distance, brilliant discharge all the time taking place between them. In the circumstances sup- posed, the knob n is the negative, arid the knob m the posi- tive electrode. The best machines are made double, havino- two 7 O revolving plates, with two fixed plates between them. The Holtz machine, when well made, far surpasses the frictional machine in power. Its action is as follows : The negative electricity of the arma- ture /, acting inductively on the opposed conductor, from which it is separated by the revolving plate, causes this conductor to discharge positive electricity, through its points, upon the face of the plate, and thus to acquire a negative charge ; when the part of the plate which has been thus affected comes opposite the tongue of the other arma- ture, the latter is affected inductively, and discharges negative elec- tricity upon the back of the plate, thus becoming itself positively electrified. Positive electricity from the front of the plate is imme- diately afterwards given off to the points P', an equal quantity of negative being of course discharged, from the conductor to which they belong, upon the face of the plate. In the subsequent stages of the process, the negative electricity thus discharged upon the face of the plate exceeds the positive which was previously there, so that the face of the plate passes on with a negative charge. When the por- tion of the plate which we are considering again comes opposite/, it increases the negative electrification both of the armature and the conductor, inasmuch as it has more of negative or less of positive elec- tricity upon both its surfaces than it had when it last moved away from that position. Both armatures thus become more and more strongly electrified, until a limit is attained which depends on the goodness of the insulation ; and as the electrification of the armatures increases, the conductors also become more powerfully affected, and are able to discharge to each other by the knobs m n at a continually increasing distance. The inventor has recently introduced a modified form of his machine. The plates are placed horizontally (Fig. 357), they have neither windows nor armatures, and they both revolve, but in oppo- site directions. Two conductors furnished with rows of points are placed above the upper plate at the extremities of one diameter, and two others below the lower plate at the extremities of another HOLTZS MACHINE. 543 diameter perpendicular to the former. Each of the upper conductors is connected with one of the lower, so that there are virtually only two conductors. In starting the machine, a sector of electrified vul- canite *s held over the upper plate, opposite one of the lower comba Fig. 357. HoHz's Machine with Horizontal Plates. When the machine has been turned for a few seconds, the sector may l>e removed, and a continual discharge of sparks takes place between the two knobs which are connected with the two conducting systems. Frequently, as in the figure, a comb is placed above, opposite to the lower comb, and this arrangement appears to increase the efficiency of the machine. The action in this form of the machine also depends upon induc- tion, the conductors performing the duty of armatures as well. We shall not enter into details, but merely remark that, in both forms of the machine, work is spent in turning the plates in opposition to elec- trical attractions and repulsions; and that the mechanical energy thus consumed produces an equivalent in the form of electrical energy. 433. Electrophorus. When electricity is required in comparatively sinall quantities, it is readily supplied by the simple apparatus called 544 ELECTRICAL MACHINES. the electrophorus. This consists (Fig. 358) of a disc of resin, or some other material easily excited by friction, and of a polished metal disc B with an insulating handle CD. The resin disc is electrified by striking or rubbing it with catskin or flannel, and the metal plate is then laid upon it. In these circumstances, the upper plate does not receive a direct charge from the lower, but, if touched with the finger (to connect it with the earth), receives an opposite charge by induction. On lifting it away by its insulating handle, Fig 358.-Eiectn>phonis. ifc is found to be charged, and will give a spark. It may then be replaced on the lower plate (touching it at the same time with the finger), and the process repeated an indefinite number of times, without any fresh excitation, if the weather is favourable. The resinous plate has usually a base or sole of metal, which is in connection with the earth while the electrophorus is being worked. This sole, by the mutual induction which takes place between it and the upper plate or cover, increases the capacity of the latter (see Chap, xl.), and thus increases the charge acquired. When the cover receives its positive charge on being connected with the earth, the sole at the same time receives from the earth a negative charge, and as the cover is gradually lifted this negative charge gradually returns to the earth. The most convenient form of the electrophorus is that of Professor Phillips, in which the cover, when placed upon the resinous plate, comes into metallic connection with the metal plate below. That this arrangement is allowable is evident, when we reflect that, when the upper plate is touched with the finger, it is in fact connected with the lower plate, since both are connected with the earth ; and it effects a great saving of time when many sparks are required in quick succession, for the cover may be raised and lowered as fast as we please, coining alternately into contact with the resinous plate and the body which we wish to charge. 434. Bertsch's Electrical Machine. A machine which has been called a rotatory electrophorus has recently been invented by Bertsch, and is represented in Fig. 3GO. A circular plate of ebonite D can be made to revolve rapidly. A sector of the same material, previously BERTSCHS ELECTRICAL MACHINE. 545 excited by friction, is fixed opposite the lower portion of the plate ; and on the other side, immediately opposite to this, is a metallic comb N forming the extremity of a conductor connected with the earth. / At the upper part is another comb /*x M connected with the conductor A. \ Under the influence of the electrified im V J Ilk %x tive electricity on the plate through the comb N. In passing the comb Al, a portion of this electricity is col- - lected by the points, and charges Fig. 359. Electrified Sector. the conductor A. The effect is in- creased by connecting A with another conductor E of very large dimensions. This machine differs from that of Holtz in furnishing no means for CO. Bertsch's Electrical Machine increasing, or even sustaining, the charge of the armature. In this respect it resembles the ordinary electrophorus. CHAPTER XXXIX. VARIOUS EXPERIMENTS WITH THK ELECTRICAL MACHINE. 435. Electric Spark. The spai-k furnished by an electrical machine >f small dimensions is short, and usually straight. Powerful machines sometimes give sparks of the length of a foot. Such sparks have usually a zig-zag form, like flashes of lightning. One of the readiest means of obtaining long sparks consists in placing, opposite to one of the small knobs of the conductor of the machine, a large conductor, having good earth connection, and present- ing a polished and slightly con- vex surface towards the knob. A more powerful effect will be obtained by connecting this conductor with the rubber or the negative conductor of the machine, instead of with the earth. Very frequently, when the spark is a foot or more in length, finer ramifications proceed from its main track, as shown in Fig. 362. 436. Brush. When a powerful machine is working in a very dry atmosphere, the rubbers being in good order, and the machine being turned rapidly, a characteristic sound is heard, which is an indication of continuous discharge into the air. In the dark, luminous appear- ances, called brushes are seen on the projecting parts of the con- ductors. They may be rendered very conspicuous by presenting a large conducting surface at a distance a little too great for a spark to pass. It will then be observed that the brush consists (Fig. 363) Fig. 361. Electric Syark. 1'HE BRUSIf. 547 ot a short foot-stalk, with A multitude of rays di- verging from it like a fan, And with other smaller ramifications proceeding from these. Positive elec- tricity gives larger and finer brushes than nega- tive. We may add, that, when the machine is working well, brilliant sparks continually leap across the plate, consist- ing of discharges between the cushions and the near- est part of the conductor. The conductor itself is also surrounded with lu- minosity. In the dark, the brilliant spectacle pre- sented by these combined appearances, with the con- tinual crackling which ac- companies them, is very impressive, and furnished an inexhaustible subject of curiosity to the elec- tricians of last century. It is probable that the passage of a spark is always preceded by a very high degree of polar ten- sion in all the particles ol air in and about its track, and that the spark occurs when this tension any- where exceeds what the particles are able to bear. The frequent crookedness of the spark is probably SO'. 1 . Spark with Ramifications. 5*8 EXPERIMENTS WITH THE ELECTRICAL MACHINE. due to the presence of conducting particles of dust, which serve as stepping-stones, and render a crooked course the easiest. 437. Duration of the Spark. We can form no judgment of the duration of the electric spark from what we see with the unaided Fig. 363. Electric Brush, after Vau Alarum. eye ; for impressions made upon the retina remain uneffaced for some- thing like ^ of a second, and the duration of the spark is incompar- ably less than this. Wheatstone, in a classical experiment, succeeded DUKAT1ON OF THE S TAJIK. 549 in measuring its duration by means of a revolving mirror; an exped- ient which has since been employed with great advantage in many other researches, especially in determining the velocity of light. Let run (Fig. 364*) be a mirror revolving with great velocity about an axis passing through c, and suppose that, during the rotation, an electric spark is pro- duced at a. An eye stationed at o will see an image in the sym- metrical position a. If the spark is strictly instantaneous, its image will be seen as a luminous point at a, notwithstanding the rota- tion of the mirror; but if it has a finite duration, the image will move from a' to a", while the mirror moves from ee' to tt' } the latter being its position when the spark ceases. What is actually seen in the mirror will therefore not be a point, but a luminous track a a". The length of this image will be double of the arc et ; for the angle ect at the centre is equal to the angle a'aa" at the circum- ference, the sides of the one being perpendicular to those of the other. In Wheatstone's experiment, the mirror made 800 turns in a second, and the image a a" was an arc of 24; the mirror therefore turned through 12, or ^ of a revolution, while the spark lasted. The dura- tion of the spark was therefore ^- Q of g^, that is, 24000 f a second. By examining the brush in the same way, Wheatstone found it to consist of a succession of sparks. 438. Spark in Rarefied Gases. The appearance of the spark is greatly modified by rarefying the air in which it is taken. To show this, an apparatus is employed which is called the electric egg. It is an oval glass vessel, which can be exhausted by means of a stop-cock at its lower end. Its upper end is closed by a cap, in which slides a brass rod terminated by a knob, which can be adjusted to any dis- tance from another knob connected with a cap at the lower end. When the egg contains air at atmospheric pressure, a spark passes Fig. 364. Duration of S^ark. 550 EXPERIMENTS WITH THE ELECTRICAL MACHINE. in the ordinary way between the two knobs ; but, as the pressure is- diminished, the aspect of the spark changes. At a pressure of six centimetres of mercury ( jg of an atmosphere), a sort of ramified sheaf proceeds from the positive knob, some of the rays terminating at a Fig. 365. Electric Egg. Fig. 366. Spark in Rarefied Air. small distance from their origin, while others extend to the negative knob. The latter is surrounded with a violet glow; the rays are also violet, but with a reddish tinge. The light at the positive knob is of a reddish purple. As the pressure is gradually reduced to a few millimetres, the rays become less distinct, and finally coalesce into an oval cloud of pale violet light, extending from one knob to the other, with a reddish tint at the positive and a deep violet at the negative end. In performing this experiment with the ordinary electrical machine, the upper knob is connected with the conductor, and the lower one with the ground. Holtz's machine can be very advantageously SPARK IN RAREFIED GASES. 551 employed in experiments of this kind, its two poles being connected with the two knobs. When, instead of the electric egg, we employ a long tube, such as- is employed for showing the fall of bodies in vacuo, the whole length of the tube is filled with violet light, which exhibits continual flicker- ing, and suggests the idea of undulations travelling in the same direction as the positive electricity. In all these experiments, as we dimmish the density of the air, we diminish the resistance to dis- charge, and at the same time diminish the intrinsic brightness of the spark. In the Torricellian vacuum, electric discharge is accompanied by a perceptible though very feeble luminosity, as may be shown by an arrangement due to Cavendish, and represented in Fig. 3G7. Two barometric tubes, united at the top, are plunged in two cups of mercury. The mercury in one cup is connected with the conductor of the machine, while that in the other is connected with the earth. In these circum- stances, the vacuum-space is filled with luminosity, which is brighter as the temperature is higher, pro- bably on account of the greater den- sity of the mercurial vapour which serves as the medium of discharge. The experiments of Gassiot and others have shown that electricity traverses a space occupied by a gas with continually increasing facility as the density of the gas is dimi- nished, until a certain limit is attained; but that when special means are employed to render the vacuum as nearly perfect as possible, this limit can be exceeded, and the resistance may increase so much as to prevent discharge. This latter point is illustrated by the apparatus represented in Fig, 368, which is constructed by Alvergniat. T is a tube which has been exhausted as completely as possible by a Geissler's pump. It has then been heated, and maintained for some time near the tempera- Fig. 3ti". Discharge ill Torricellian Vacuum. 552 EXPERIMENTS WITH THE ELECTRICAL MACHINE. P P' ture of fusion of glass, in order to produce absorption of the remain- ing air. Two platinum wires have been previously sealed in the ends of the tube, and approach within JQ of a millimetre of each other. The two poles of a Holtz's machine are connected with the binding- screws B and B', which are in communication with these two wires, and also with two rods whose extremities pp' are at a moderate strik- ing distance from each other in air. As long as the machine works, sparks pass between these latter, while, in spite of the very much closer proximity of the platinum wires, no lu- minosity is perceptible between them. Instead of being placed a small distance apart in air, p and p' may be fitted into the ends of a tube Fig. 368. Non-conductivity of Perfect Vacuum. of Considerable length containing rarefied air. It will be found that discharge can take place at greater distance as the air is more rarefied, till we attain a limit far beyond the reach of ordinary air-pumps. 439. Colour of the Spark. The colour of the spark or other lumin- ous discharge depends partly on the material of the conductors between which it passes, and partly on the gaseous medium which it traverses. The former influence predominates when the spark is strong, the latter when it is weak. The effect of the metal seems to depend upon the vaporization of a portion of it, for, on examining the spark by the spectroscope, bright lines are seen which are known to indicate the presence of metallic vapour. For studying the effect of the gaseous medium, the discharge is taken between two platinum wires sealed into the ends of glass tubes, containing the gases in a MULTIPLICATION OF THE ELECTRIC SPARK. 553 rarefied condition. The wires are connected either with the poles of a Holtz's machine, or of a RuhmkorfFs coil, which we shall describe in Chap. lii. It is found that the colour in air or oxygen is white with a tinge of blue, in nitrogen blue, in hydrogen red, and in carbonic acid green. 440. Multiplication of the Electric Spark. The old electricians contrived several pieces of apparatus for multiplying the electric spark. The principle of all is the same. Small squares of tin-foil are arranged in series at a small distance from each other on an insulating surface. The first of the series is connected with a metallic knob which can be brought near the electrical machine; and the last of them is connected with another knob which is in communication with the earth. By allowing a discharge to pass through the series, sparks can be simultaneously obtained at all the intervals between the successive squares. In the spangled tube (Fig. 370) the squares of tin-foil are arranged spirally along a cylin- drical glass tube which has a brass cap at each end. One cap is put in communication with the machine, and the other with the earth. Sometimes a glass globe is substituted for the cylinder. We have thus the spangled globe (Fig. 371). In the sparkling pane a long strip of tin-foil is disposed in one continuous crooked line (consisting of parallel strips connected at alternate ends) from a knob at the top to another knob at the bottom of the pane. A pattern is then traced by scratching away the tin-foil in numerous places with a point, and when the spark passes, it is seen at all these places, so as to render the pattern luminous (Fig. 372). 441. Physiological Effects of the Spark: Electric Shock. When a strong spark is drawn by presenting the hand to the conductor of a very large and powerful machine, a peculiar sensation is experi- enced. With ordinary machines the same effect can be obtained by Fig. 369. Tulxs for Rarefied Gases. Fig. 370. Spangled Tube. 554; EXPERIMENTS WITH THE ELECTRICAL MACHINE. employing a Leyden jar. The sensation is difficult to describe, and only capable of being produced by electrical agency. It is a painful shock, felt especially in the arm, and causing an involuntary bending of the elbow. At the distance of a few feet from a machine in powerful action > a tickling sensation is felt on the exposed parts of the body, due to the movement of the hairs in obedience to electrical force. These phenomena are exhibited in a still more marked manner when a Fig. 371. Spangled Globe. Fig. 372. Spangled Pane. person stands on a stool with glass legs, and keeps his hand upon the conductor. He thus becomes highly charged with electricity. His hair stands on end, and is luminous if seen in the dark. If a conductor connected with the earth is presented to him, a spark passes, and his hair falls again. Electricity has frequently been resorted to for medical purposes. The electrical machine was first employed, and afterwards the Leyden jar, but both have now been abandoned in favour of magneto-electric machines and induction coils, which we shall describe in a later chapter (Chap, lii.) PROPERTIES OF THE SPARK. 555 442. Mechanical and Physical Properties of the Spark. The electric spark produces a violent commotion in the medium in which it occurs. This is easily shown by means of Kinnersley's thermometer (Fig. 373), which consists of two glass tubes of unequal diameters, the smaller being open at the top, while the larger is completely closed, with the exception of a side passage, by which it communi- cates with the smaller. The caps which close the ends of the large tube are traversed by rods terminating in knobs, and the upper one can be raised and lowered to vary the distance between the knobs. Both tubes are filled, to a height a little below the lower knob, with a very mobile liquid such as alcohol. When the spark passes between the knobs, the liquid is projected with great violence, and may rise Fig. 3~:i. Kinnersley's Thermometer. Fig. 374. Electric Mortar. to a height of several yards if the spark is very strong. The same property of the spark is exhibited in the experiment of the electric mortar, which is sufficiently explained by the figure (Fig. 374). The spark may be obtained in the interior of a non-conducting liquid, which it agitates in a similar manner. If the liquid is con- tained in a closed vessel, this is often broken. The spark can also traverse thin non-conducting plates, producing in this case perfora- tion of the plates ; but the experiment usually requires very powerful discharges, such as can only be obtained by means of apparatus described in the next chapter. The luminosity of the electric spark is probably due to the very high temperature which is produced in the particles traversed by the 556 EXPERIMENTS WITH THE ELECTRICAL MACHINE. discharge. Coal-gas is easily inflamed, by a person standing on a stool with glass legs holding one hand on the conductor of the machine, and giving sparks from a finger of the other hand to the burner from which the gas is issuing. Kinnersley regarded elevation of temperature as the cause of the movement of the liquid in his apparatus ; hence the name which it bears. Heating may also occur in the case of conductors. This is shown by the influence of the metal upon the colour of the spark, and it may be more directly proved by arranging a conductor in communication Fig. 375. Volta's Pistol. with the earth, and connected by an exceedingly fine metallic wire with another conductor. When the latter is presented to a very powerful electrical machine, so that a strong spark passes, the fine wire is sometimes heated to redness. 443. Chemical Properties of the Spark. The electric spark is able to produce very important chemical effects. When it occurs in an explosive mixture of two parts of hydrogen with one of oxygen, it causes these gases instantly to combine. This experiment is usually shown by means of Volta's pistol (Fig. 375), which is a metallic vessel, containing the mixture, and closed by a cork. Through one side PROPERTIES OF THE SPARK. 557 passes an insulated metallic rod with a knob at each end, that at the inner end being at a short distance from the opposite side of the vessel, so that, if a spark is given to the exterior knob, a spark also passes in the interior, and inflames the mixture. This effect is accom- panied by a violent detonation, and the cork is projected to a dis- tance. The electric spark often produces a reverse effect that is to say, the decomposition of a compound body; but the action in this case is gradual, and a great number of sparks must be passed before the full effect is obtained. Thus, if a succession of sparks be passed in Fig. 376. Wind from Points. the interior of a mass of ammonia, contained in a vessel inverted over mercury, the volume of the gas is observed to undergo a gradual increase, until at length, if kept at constant pressure, the volume is exactly doubled. It then consists of a mechanical mixture of nitrogen and hydrogen, the constituents of ammonia. Composition and decomposition are often both produced at once. Thus, if a spark is passed in a mixture of carburetted hydrogen and a certain proportion of oxygen, the former gas is decomposed, its hydrogen combining with a portion of the oxygen to form water, and its carbon combining with another portion to form carbonic acid. 558 EXPERIMENTS WITH THE ELECTRICAL MACHINE. Vessels intended for taking the electric spark in gases are extensively used in chemistry, and are called eudiometers. 444. Wind from Points. If a metallic rod terminating in a point be attached to the conductor of the electrical machine, electricity escapes in large quantity from the point, which, accordingly, when viewed in the dark, is seen to be crowned with a tuft of light. A layer of air in front of the point is electrified by contact, and then repelled, to make way for other portions of air, which are in their turn repelled. A continuous current of air is thus kept up, which is quite perceptible to the hand, and produces a very visible effect on the flame of a taper (Fig. 376). The electric whirl (Fig. 377) consists of a set of metallic arms, radiating horizontally from a common centre about which they can turn freely, and bent, all in the same direction, at the ends, which are pointed. When the central support is mounted on the conductor of the machine, the arms revolve in a direction opposite to that in which their ends point. This effect is due to the mutual repulsion between the pointed ends and the electrified air which flows off from them. It is instructive to remark that if, by a special arrangement, the rotating part be inclosed in a well-insulating glass case, the rotation soon ceases, because, in these circumstances, the inclosed air quickly attains a state of permanent electrification. 445. Electric Watering-pot. Let a vessel containing a liquid, and furnished with very fine discharge tubes, be suspended from the conductor of the machine. When the vessel is not electrified, the liquid comes out drop by drop; but when the machine is turned, it issues in continuous fine streams. It has, however, been observed that the quantity discharged in a given time is sensibly the same in both cases. This must be owing to the equality of action and reaction between different parts of the issuing stream. Fig. 377. Electric Whirl. Fig. 378. Electric Bucket. CHAPTER XXXIX A ELECTRICAL POTENTIAL, AND LINES OF ELECTRIC FORCE. 445 A. Introductory Remarks on Potential. Electrical reasonings are, in many cases, greatly facilitated by employing the conception denoted by the name electrical potential. Potential essentially depends upon forces (whether attractive or repulsive) mutually exerted upon each other by particles at a dis- tance, and has been advantageously employed in the theories of gravitation and magnetism, as well as of electricity. We shall at present confine our attention to electrical potential. All the space in the neighbourhood of electrified bodies is in a certain sense pervaded by their influence. This influence is com- pletely specified by stating the numerical values, with proper sign, of electrical potential, at the different points of the space. Electrical potential never changes its value per saltum in passing from one point to the next. Moreover, if it is constant in value throughout any finite portion of a space not containing electricity, it is constant throughout the whole of this space. 445 B. Relation of Potential to Force. When electrical potential is constant throughout a given space, there is no electrical force 1 in that space ; and, conversely, if there be an absence of electrical force in a given space, the potential throughout that space must be uniform. These propositions apply to the space within a hollow conductor. They also apply to the whole substance of a solid conductor, and to the whole space inclosed within the outer surface of a hollow con- 1 DEFINITION. There is electrical force at a point in the air, if an electrified particle placed there would experience force tending to move it in virtue of electrical attraction or repulsion. There is electrical force at a point in a conductor, when electricity flows through the point. A conductor is said to be in electrical equilibrium, when there is no electrical force at any part of it; in other words, when it is completely free from currents of electri- city. 560 ELECTRICAL POTENTIAL LINES OF ELECTRIC FORCE. duetor. Whenever a conductor is in electrical equilibrium, it has the same potential throughout the whole of its substance, and also through any completely inclosed hollows which it may contain. When a conductor is not in electrical equilibrium, currents set in, tending to restore equilibrium ; and the direction of such currents is always from places of higher to places of lower potential. In like manner, when a small positively electrified body experiences elec- trical force tending to move it, the direction of this force is from higher to lower potential When flow of electricity is compared with flow of heat, potential is the analogue of temperature. Heat flows from places of higher to places of lower temperature. The precise direction of the force exerted upon a positively electri- fied particle (or upon an element of positive electricity), when brought to a place where potential has not a constant value from point to point, is the direction in which potential diminishes most rapidly. A negatively electrified particle (or an element of negative electricity) will be urged in the opposite direction, which is the direction in which potential increases most rapidly. We here use the words increase and decrease in the algebraical sense. 445 c. Line of Force. The direction thus defined (especially by re- ference to the force on the positive particle) is called the direction oj resultant electrical force at the point where the particle is placed. If a line be traced such that every small portion of it (small enough to be regarded as straight) is the direction of resultant electrical force at the points which lie upon it, it is called a line of force; in other words, a line of force is a line whose tangent at any point is the direction of resultant electrical force at that point. We may express this briefly by saying that lines of electrical force are the lines along which resultant electrical force acts. It is evident that lines of force cannot cut one another, since we cannot have two different directions of resultant force at a point. 445 D. Intensity of Force is Equal to Rate of Variation of Potential The intensity of resultant force at a point is equal to the rate at which potential diminishes in the direction in which the diminution is most rapid, namely, along a line of force at the point. Let V denote the potential at the point, and V 2V the potential at a neighbouring point on the same line of force, at a distance 3s from the first point; then ^- is the intensity of force at either point, or, more strictly, is RELATION BETWEEN POTENTIAL AND WORK. 561 the average intensity along the short line $s, and the direction of the force (for a positive particle) is from the first point towards the second. A similar proposition applies to two neighbouring points not situated on the same line of force ; the component force, in the direc- tion of a line joining them, is equal to j^, where Ix denotes the length of the joining line, and 3V the difference of the potentials of the two points. This proposition is usually expressed by saying that the rate of variation of potential in any direction is equal to the component force in that direction. 445 E. Relation between Potential and Work. The work done by or against electrical force in carrying a unit of electricity through this distance $x is the product of force by distance, and is therefore simply 3V. More generally, the work done by or against electrical force in carrying a unit of electricity from one point to another, is equal to the difference of potentials of the two points ; and the work done in carrying any quantity of electricity is the product of this quantity by the difference of potentials. An analogy is thus suggested between different potentials and different levels. Positive electricity tends to run down from higher to lower potential, and, when it does so run down, there is a loss of potential energy equal to the product of the quantity which runs down, and the difference of potential through which it runs down (see note 1, p. 784). When the quantity which runs down is unity, the loss of potential energy is equal to the loss of potential. It is usual to assume, as the zero of potential, the potential of the earth at the place of observation; but this assumption is not rigorously consistent with itself, since the existence of earth- currents demon- strates that different potentials may exist at different parts of the earth. Electrical potential is rigorously zero at places infinitely distant from all electricity. 445 F. Equipotential Surfaces. An equipotential surface is a surface over the whole of which there is the same value of potential. It is obvious, from the latter part of 445 D, that there is no tangential force at any point of such a surface. The direction of resultant force is everywhere normal to the surface, or equipotential surfaces every- where cut lines of force at right angles. An equipotential surface is the analogue of a level surface. If two equipotential surfaces are given, their potentials being Vj and V 2 , the work done in carrying 37 562 ELECTRICAL POTENTIAL LINES OF ELECTRIC FORCE. a unit of electricity from any point of the one to any point of the other, is constant, and equal to the difference of V x and V 2 . If we consider two equipotential surfaces very near to one another, so that the portions which they intercept on the lines of force may be regarded as straight, the intensity of force at different points of the intermediate space will vary inversely as the distance between the two equipotential surfaces ; for, when equal amounts of work are done in travelling unequal distances, the forces must be inversely a& the distances. 445 G. Tubes of Force. If we conceive a narrow tube bounded on all sides by lines of force, and call it a tube of force, we can lay down the following remarkable rules 1 for the comparison of the forces which exist at different points in its length. (1) In any portion of a tube of force not containing electricity, the intensity of force varies inversely as the cross-section of the tube, or the product of intensity of force by section of tube, is constant. 2 (2) When a tube of force cuts through electricity, this product changes, from one side of this electricity to the other, by the amount 4 \ 445 p. Condensers. The process called condensation of electricity consists in increasing the capacity of a conductor by bringing near it another conductor connected with the earth. The two conductors are usually thin plates or sheets of metal, placed parallel to one another, with a larger plate of non-conducting material between them. Let A and B (Fig. 379) be the two conducting plates, of which A, called the collecting plate, is connected with the conductor of the machine, and B, called the condensing plate, with the earth ; and let C be the non-conducting plate (or dielectric) which separates them. Then, if the machine has been turned until the limit of charge is attained, the surface of B which faces towards A is covered with negative electricity, drawn from the earth, and held by the attrac- tion of the positive electricity of A ; and, conversely, the surface of A which faces towards B, is covered with positive elec- tricity, held there by the attraction of the negative of B, in addition to the charge which would reside upon it if the conductor were at the existing potential, and B and C were absent. In fact, the electrical density on the face of A, as well as the whole charge of A, would, in this latter case, be almost inappreciable, in comparison with those which exist in the actual circumstances. By condensation of electricity, then, we are to understand increase usually enormous increase Fig. 379. Theoretical Condenser. 568 ELECTRICAL CONDENSERS. of electrical density on a given surface, attained without increase of potential. If two conducting plates, in other respects alike, but one with, and the other without a condensing plate, be connected by a wire, and the whole system be electrified, the two plates will have the same potential, but nearly the whole of the charge will reside upon the face of that which is accompanied by a condensing plate. 445ft. Calculation of Capacity of Condenser. The lines of force between the two plates A and B are everywhere sensibly straight and perpendicular to the plates, with the exception of a very small space round the edge, which may be neglected. The tubes of force ( 445 G) are therefore cylinders, and the intensity of force is con- stant at all parts of their length. Also, since the potential of the plate B is zero, if we take V to denote the potential of the plate A, which is the same as the potential of the conductor, and t to denote the thickness of the intervening plate C, the rate at which potential Y varies along a line of force is -j, which is therefore ( 445 D) the expression for the force at any point between the plates A, B. The whole space between the plates may be regarded as one cylindrical tube of force of cross-section S equal to the area of either plate, the two ends of the tube being the inner faces of the plates. The quan- tities of electricity +. Q residing on these faces are therefore equal, but of opposite sign ( 445 j) ; and as the force changes from nothing y" to in passing from one side to the other of the electricity which resides on either of these surfaces, we have ( 445 G) Hence the capacity of the plate A, being, by definition, equal to ^, is equal to We should, however, explain that, if the intervening plate C is a solid or liquid, we are to understand by t not the simple thickness, but the thickness reduced to an equivalent of air, in a sense which will be explained further on ( 453). This reduced thickness is, in the case of glass, about half the actual thickness. If s denote an element of area of A, and q the charge residing upon it, it is evident, from considering the tube of force which has 8 for one of its ends, that V - .s = 4 TT q ; DISCHARGE OF CONDENSER. 569 and the electric density on the element is equal to -* t , which is constant over the whole face of the plate. To give a rough idea of the increase of capacity obtained by the employment of a condensing plate, let us compare the capacity of a circular disc of 10 inches diameter, accompanied by a condensing plate at a reduced distance of ^V of an inch, with the capacity of a globe of the same diameter as the disc. The capacity of the globe is equal to its radius, and may therefore be denoted by 5. The 25 T capacity of the disc is 4ff x ^ = 125, or 25 times the capacity of the globe. It is, in fact, the same as the capacity of a globe 250 inches (or 20 ft. 10 in.) in diameter. 446. Discharge of Condenser. If, by means of a jointed brass dis- charger (Fig. 380) with knobs M N at the ends, and with glass Pig. 380. -Discharge of Condenser. Fig. 381. Discharger without Handles. handles, we put the two plates A and B in communication, a brilliant spark is obtained, resulting from the combination of the positive charge of A with the negative of B, and the condenser is discharged. When the quantity of electricity is small, the glass handles are un- necessary, and the simpler apparatus represented in Fig. 381 may be employed, consisting simply of two brass rods jointed together, and with knobs at their ends, care being taken to touch the plate B, which is in communication with the earth, before the other. The operator will then experience no shock, as the electricity passes in preference through the brass rods, which are much better conductors than the human body. If, however, the operator discharges the 570 ELECTRICAL CONDENSERS. condenser with his hands by touching first the plate B, and then also the plate A, the whole discharge takes place through his arms and chest, and he experiences a severe shock. If he simply touches the plate A, while B remains connected with the earth by a chain, as in Fig. 379, he receives a shock, but less violent than before, because the discharge has now to pass through external bodies which con- sume a portion of its energy. If, instead of a chain, B is connected with the earth by the hand of an assistant touching it, he too will receive a shock when the operator touches A. 447. Discovery of Cuneus. The invention of the Leyden jar was brought about by a shock accidentally obtained. Some time in the Fig. 382. Experiment of Cuneus. year 1746, Cuneus, a pupil of Muschenbroeck, an eminent philosopher of Leyden, wishing to electrify water, employed for this purpose a wide-mouthed flask, which he held in his hand, while a chain from the conductor of the machine dipped in the water (Fig. 382). When the experiment had been going on for some time, he wished to dis- connect the water from the machine, and for this purpose was about to lift out the chain; but, on touching the chain, he experienced a shock, which gave him the utmost consternation, and made him let THE LEYDEN JAK. 571 fall the flask. He took two days to recover himself, and wrote to Reaumur that he would not expose himself to a second shock for the crown of France. The news of this extraordinary experiment spread over Europe with the rapidity of lightning, and it was eagerly repeated everywhere. Improvements were soon introduced in the arrangement of the flask and its contents, until it took the present form of the Ley den Phial or Ley den Jar. It is easy to see that the effect obtained by Cuneus depended on condensation of electricity, the water in the vessel serving as the collecting plate, the hand as condensing plate, and the vessel itself as the dielectric. When he touched the chain, the two oppositely charged conductors were put in communication through the operator's body, and he received a shock. 448. Leyden Jar. The Leyden jar, as now usually constructed, Fig. 383. Leyden Jar. Fig. 384. Discharge of Leyden Jar. consists of a glass jar coated, both inside and out, with tin-foil, for about four- fifths of its height. The mouth is closed by a cork, through which passes a metallic rod, terminating above in a knob, 572 ELECTRICAL CONDENSERS. and connected below with the inner coating, either by a chain de- pending from it, or by pieces of metallic foil with which the jar is filled. The interior of the jar must be thoroughly dry before it is closed, and the cork and neck are usually covered with sealing-wax, and shellac varnish, which is less hygroscopic than glass. The Leyden jar is obviously a condenser, its two coatings of tin-foil performing the parts of a collecting plate and a condensing plate. If the inner coating is connected with the electrical machine, and the outer coat- ing with the earth, the former acquires a positive, and the latter a negative charge. On connecting them by a discharger, as in Fig. 384, a spark is obtained, whose power depends on the potential of the inner coating, and on its electrical capacity. If these be denoted respectively by V and C, and if Q denote the quantity of electricity residing on either coating, the amount of electrical energy which runs down and undergoes transformation when the jar is discharged, is J QV= CV 2 =i (See note 1, p. 784.) The quantities Q, V, C, which are, properly speaking, the charge, potential, and capacity of the inner coating, are usually called the charge, potential, and capacity of the jar. 449. Residual Charge. When a Leyden jar has been discharged by connecting its two coatings, if we wait a short time we can obtain another but much smaller spark by again connecting them, and other sparks may sometimes be obtained after further intervals. These are called secondary discharges, and the electricity which thus remains after the first discharge is called the residual charge. Faraday's experiments leave little room for doubt that they depend mainly upon a gradual penetration of electricity from both sides into the -substance of the glass, to a very small depth, but sufficient to prevent the electricity which has so entered from at once escaping to the earth when connection is made. Faraday appeals to this phenomenon as strongly confirming his view that the difference between con- ductors and non-conductors is only one of degree, this penetration being only an extremely slow process of conduction. A small part of the residual charge also consists of electricity spread over the sur- face of the glass beyond the edges of the coatings. The whole charge of the outer coating, and all except an insigni- ficant portion of the charge of the inner coating, resides on the side of the foil which is in contact with the glass, or, more probably, on the surfaces of the glass itself, the mutual attraction of the two DISCHARGE BY ALTERNATE CONTACTS. 573 opposite electricities causing them to approach as near to each other as the glass will permit. This is illustrated by Franklin's experi- ment of the jar with movable coatings (Fig. 385). The jar is charged in the ordinary way, and placed on an insulating stand. The inner coating is then lifted out by a glass hook, and touched with the hand to dis- charge it of any electricity which it may retain. The glass is then lifted out, and the outer coating also discharged. The jar is then put together again, and is found to give nearly as strong a spark as it would have given originally. 450. Discharge by Alternate Contacts. In- stead of discharging a Leyden jar at once by connecting its two coatings, we may gradually discharge it by alternate contacts. To do this, we must set it on an insulating stand (or other- wise insulate both coatings from the earth), and then touch the two coatings alternately. At every contact a small spark will be drawn. The coating last touched has always rather less elec- tricity upon it than the other, but the difference is an exceedingly small fraction of the whole charge, and, after a great number of sparks have been drawn by these alternate con- tacts, we may still obtain a powerful discharge by connecting the two coatings. The quantities of electricity thus alternately discharged from the two coatings form two decreasing geometric series, one for each coating. In fact, if ra and mf be two proper fractions such that, when the outer coating is connected with the earth, the ratio of its charge to that of the inner is m; and, when the inner coating is connected with the earth, the ratio of its charge to that of the outer is m', we have the following series of values: Fig. 385. Jar with Movable Coatings. On inner coating. Original charges, + Q After 1st contact, . . . . + m'm Q 2d + m'm Q 3d + ro' a m a Q &c. On outer coating. - mQ. - mQ, - m'm a Q - m'm 2 Q &c. The quantities discharged from the inner coating are, successively (1 m'm) Q , ra'ra (1 m'm) Q , m' 9 m* (1 ra'm) Q, &c. ; and the 574 ELECTRICAL CONDENSERS. quantities successively discharged from the outer, neglecting sign, are, m (lm'm) Q , m'm 2 (1 m'm) Q, m' 2 m 8 (1 m'm) Q, &c. The quantity (1 m'm) Q discharged at the first contact, repre- sents that portion of the charge 1 which is not due to condensation ; so that the actual capacity of the Leyden jar is to the capacity of the inner coating if left to itself as 1 : lm'm. The discharge by alternate contacts can be effected by means of a carrier suspended between two bells, as in Fig. 386. The rod from the inner coating ter- minates in a bell, and the outer coating is connected, by means of an arm of tin, with another bell supported on a metallic column. An insulated metallic ball is suspended be- tween the two. This is first attracted by the positive bell. Then, being repelled by this and attracted by the other, it carries its charge of positive elec- tricity to the negative bell, and receives a charge of negative, which it carries to the positive bell, and so on alternately. The whole apparatus stands upon an insulating support. It is not, however, necessary that the carrier should be insulated from the earth, but it must be insulated from both coatings. 451. Condensing Power. By the condensing power of a given arrangement is meant the ratio in which the capacity of the collecting plate is increased by the presence of the condensing plate, which ratio, as we have seen in last section, is equal to the frac- tion l _ m > m - Riess has investigated its amount experimentally under varying conditions, by means of the apparatus represented in Fig. 387, which is a modification of the condenser of ^Epinus. It consists Fig. 386. Alternate Discharge. 1 This portion of the original charge is said to be free, and the remaining portion to be bound, dissimulated, or latent. These terms are applicable to all cases of condensation. CONDENSING POWER. 575 of two metallic plates A and B, supported on glass pillars, and travelling on a rail, so that they can be adjusted at different distances. Between them is a large glass plate C. A is charged from the machine, B being at the same time touched to connect it with the ' O ground. The electrical density on the anterior face of A was ob- Fig. 387. Condenser of JSpioui. served by means of Coulomb's proof-plane and torsion-balance. Riess' experiments are completely in agreement with the theory laid down in the preceding sections of this chapter; for example, he found, among other results, that the condensing power was independent of the absolute charge, and that it varied nearly in the inverse ratio of the distance. 453. Influence of the Dielectric. Faraday discovered that the amount of condensation obtained in given positions of the two con- ducting plates, depended upon the material of the intervening non- conductor or dielectric. The annexed figure (Fig. 388) represents a modification of one of Faraday's experiments. A is an insulated metallic disc, with a charge, which we will suppose to be positive. B and C are two other insulated metallic discs at equal distances from A, each having a small electric pendulum suspended at its back. Let B and C be touched with the hand; they will become negatively 576 ELECTRICAL CONDENSERS. electrified by induction, but their negative electricity will reside only on their sides which face towards A, and the pendulums will hang vertically. If, while matters are in this condition, we move B nearer to A, we shall see both the pendulums diverge, and, on testing, we shall find that the pendulum B diverges with positive, and C with negative electricity. The reason is obvious. The approach of B to A causes increased induction be- tween them, so that more negative is drawn to the face of B, and positive is driven to its back; at the same time the symmetrical distribution of electricity on A is disturbed, a portion being accu- mulated on the side next B at the expense of the side next C. The inductive action of A upon C is thus diminished, and a portion of the negative charge of C is left free to spread itself over the back, and affect the pith-ball. If, while the discs are in their initial position, B and C being equi- distant from A, and the pendulums vertical, we interpose between B and A a plate of sulphur, shellac, or any other good insulator, the same effect will be produced as if B had been brought nearer to A We see, then, that the insulating plate of a condensing arrangement serves not only to prevent discharge, but also to increase the induc- tive action and consequent condensation, as compared with a layer of air of the same thickness ; inductive action through a plate of sulphur or shellac of given thickness, is the same as through a thinner plate of air. The numbers in the subjoined table denote the thick- ness of each material which is equivalent to unit thickness of air. For example, the mutual induction through 2-24 inches of sulphur is the same as through 1 inch of air. These numbers are called SPECIFIC INDUCTIVE CAPACITIES. Fig. 388. Change of Distance. Air or any gas, .... 1 '00 Spermaceti, 1'45 Glass, 176 Resin. 177 Pitch, 1-80 Wax, 1-86 Shellac, 2'00 Sulphur, 2-24 The quotient of the actual thickness of the plate by the specific FARADAY'S DETERMINATIONS. 577 inductive capacity of its material may appropriately be called the thickness reduced to its equivalent of air, or simply the reduced thickness. 454. Faraday's Determinations. Faraday, to whom the name and discovery of specific inductive capacity are due, operated by com- paring the capacities of condensers, alike in all other respects, but differing in the materials employed as dielectrics. One of his con- densers is represented in Fig. 389. It is a kind of Leyden jar, containing a metallic sphere A, attached to the rod M, and forming with it the inner conductor. The outer conductor consists of the hollow sphere B, divided into two hemi- spheres which can be detached from each other. The interval between the outer and inner conductor can be filled, either with a cake of solid non-conducting material, or with gas, which can be introduced by means of the cock R. The method of observation and reduction will be best understood from an example. The interval being occupied by air, the apparatus was charged, and a carrier-ball, having been made to touch the summit of the knob M, was introduced into a Coulomb's torsion-balance, and found to be charged with a quantity of electricity represented by 250 of torsion. When the second apparatus was precisely similar to the first, it was found that, on contact of the two knobs, the charge divided itself equally, and the carrier-ball, if applied to either knob, took a charge repre- sented very nearly by 125. The conditions were then changed in the following way. The first jar still containing air, the interval between the two conductors in the second was filled with shellac. It was then found that the air- jar, being charged to 290, was reduced, by contact of its knob with that of the shellac-jar, to 114, thus losing 176. If no allowance be made for dissipation, the capacities of the air-jar and shellac-air would therefore be as 114 : 176, or as 1 : 1'54, and the specific inductive capacity of shellac would be 1'54. 455. Polarization of the Dielectric. As the interposed non-con- ductor, or dielectric, modifies the mutual action of the two electri- 38 Fig. 389. Apparatus for Specific Inductive Capacity. 57S ELECTKICAL CONDENSERS. cities which it separates, and does not play the mere passive part which was attributed to it before Faraday's experiments, it is natural to conclude that the dielectric must itself experience a peculiar modification. According to Faraday, this modification consists in a polarization of its particles, which act inductively upon each other along the lines of force, and have each a positive and a negative side, the positive side of each facing the negative side of the next. This polarization is capable of being sustained for a great length of time in good non-conductors ; but in good conductors it instantly leads to discharge between successive particles, and the opposite electricities appear only at the two surfaces. The polarization of dielectrics is clearly shown in the following experiment. In a glass vessel (Fig. 390) is placed oil of turpentine, containing filaments of silk 2 or 3 millimetres long. Two metallic rods, A, B, each terminating within in a point, are connected, one Fig. 390. Polarization of Dielectric. with the ground, and the other with an electric machine. On work- ing the machine, the little filaments are seen to arrange themselves in a line between the points, and, on endeavouring to break the line with a glass rod, it will be found that they return to this position with considerable pertinacity. On stopping the machine, they imme- diately fall to the bottom. An experiment of Matteucci's demonstrates this polarization still more directly. A number of thin plates of mica are pressed strongly together between two metallic plates. One of the metallic plates is charged, while the other is connected with the ground; and, on removing the metallic plates by insulating handles, it is found that all the mica plates are polarized, the face turned towards the positive metal plate being covered with positive electricity, and the other face with negative. 459. Limit to Thinness of Interposed Plate. We have seen ( 445 Q) that the capacity of a condenser varies inversely as the distance VOL1AS CONDEJSSi.NG ELECTROSCOPE. 571) between the collecting and the condensing plate. But if this dis- tance is very small, the resistance of the interposed dielectric (which varies directly as its thickness) may be insufficient to prevent dis- charge, and it will not be practicable to establish a great difference of potential between the two plates. We may practically distinguish two sorts of condensers, one sort having a very thin dielectric and very great condensing power, but only capable of being charged to feeble 1 potential; the other having a dielectric thick enough to resist the highest tensions attainable by the electrical machine. The Leyden jar comes under the second category. The first includes the electrophorus (except in so far as its action is aided by the metallic sole), and the condenser of Volta's electroscope. 460. Volta's Condensing Electroscope. This instrument, which has rendered very important services to the science of electricity, differs from the simple gold-leaf electro- scope previously described ( 415), in having at its top two metal plates, of which the lower one is connected with the gold-leaves, and is covered on its upper face with insulating var- nish, while the upper is varnished on its lower face, and furnished with a glass handle. These two plates con- stitute the condenser. In using the instrument, one of the two plates (it matters not which) is charged by means of the body to be tested, while the other is connected with the earth. They thus receive oppo- site and sensibly equal charges. The upper plate is then lifted off, and the higher it is raised the wider do the gold-leaves diverge. The separation of the plates dimin- ishes the capacity, and strengthens the potential of both, one becoming more strongly positive, and the other more strongly negative. This involves increase of - potential energy, which is represented by the amount of work done against electrical attrac- 1 Strong potential is potential differing very much from zero either positively or nega- tively. Feeble potential is potential not differing much from zero. Tension is measured by difference of potential; and when the earth is one of the terms of the comparison, tension becomes identical with potential. Fig. 391. Condensing Electroscope. 580 ELECTRICAL CONDENSERS. tion in separating the plates. No increase in quantity of electricity is produced by the separation ; hence the instrument is chiefly ser- viceable in detecting the presence of electricity which is available in large quantity but at weak potential. The glass handle of the upper plate is by no means essential, as it is only necessary that the lower plate should be insulated. The charge may be given by induction; in which case one plate must be connected with the earth while the inducing body is held near it, and the other plate must be kept con- nected with the earth while the influencing body is withdrawn. The plates will then be left charged with opposite electricities, that which was more remote from the influencing body having acquired a charge similar to that of the body. For inductive charges, how- ever, the condensing arrangement serves no useful purpose, beyond enabling the electroscope to retain its charge for a longer time, the Fig. 392. Battery of Ley den Jars. effect finally obtained on separating the plates being no greater than would have been obtained by employing only the lower plate. 461. Ley den Battery. The Leyden battery consists of a number of THE LEYDEN BATTERY. 581 Leyden jars, placed in compartments of a box lined with tin-foil, which serves to establish good connection between their outer coat- ings, while their inner coatings are connected by brass rods. It is advisable that the outer coatings should have very free communica- tion with the earth. For this purpose a metallic handle, which is in metallic communication with the lining of the box, should be con- nected, by means of a chain, with the gas or water pipes of the building. The capacity of a Leyden battery is the sum of the capacities of Fig. 393. Lichtenberg's Figures the jars which compose it. The charge is given in the ordinary way, by connecting the inner coatings with the conductor of the machine. In bad weather this is usually a very difficult operation, on account 582 ELECTiUCAL CONDENSERS. of the large quantity of electricity required for a full charge, and the large surface from which dissipation goes on. Holtz's machine can be very advantageously employed for charging a battery, one of its poles being connected with the inner, and the other with the outer coatings. In dry weather it gives the charge with surprising quickness. 462. Lichtenberg's Figures. An interesting experiment devised by Lichtenberg serves to illustrate the difference between the physical properties of positive and negative electricity. A Leyden jar is charged, and the operator, holding it by the outer coating, traces a design with the knob on a plate of shellac or vul- canite. He then places the jar on an insulating stand, to enable him to transfer his hold to the knob, and traces another pattern on the cake with the outer coating. A mixture of flowers of sulphur and red-lead, which has previously been well dried and shaken together, is then sprinkled over the cake. The sulphur, having become nega- tively electrified by friction with the red-lead, adheres to the pattern which was traced with positive electricity, while the red-lead adheres to the other. The yellow and red colours render the patterns very conspicuous. The particles of sulphur (represented by the .inner pattern in Fig. 893) arrange themselves in branching lines, while the red-lead (shown in the outer pattern) forms circular spots ; whence it would appear that positive electricity travels along the surface more easily than negative. A similar difference has already been pointed out between positive and negative brushes. 462 A. Charge by Cascade. Let there ben jars, all precisely alike, and let the inner coating of the first be charged directly from the machine, while the outer coating of each is connected with the inner coating of its follower, the outer coating of the last being connected with the ground. Then the charge given to the first produces by induction an equal charge in each of the others, and the jars are said to be charged by cascade. The charge of each of the jars when thus arranged is only - of its ordinary charge, and the difference of potentials between its coatings is only - of that obtainable in the ordinary way. Its spark has therefore only ^ of the energy of its ordinary spark. CHAPTER XLI. EFFECTS PRODUCED BY THE DISCHARGE OF CONDENSERS. 463. Discharge of Batteries. The effects produced by the discharge of a Leyden jar or battery differ only in degree from those of an ordinary electric spark. The shock, which is smart even with a smn 1 1 jar, becomes formidable with a large jar, and still more with a battery of jars. If a shock is given to a number of persons at once, they must form a chain by holding hands. The person at one end of the chain Fig. 394. Coated Pane. must place his hand on the outer coating of a charged jar, and the person at the other end must touch the knob. The shock will be felt by all at once, but somewhat less severely by those in the centre. The coated pane, represented in Fig. 394, is simply a condenser, consisting of a pane of glass, coated on both sides, in its central por- 584 EFFECTS PRODUCED BY DISCHARGE OF CONDENSERS. tion, with tin-foil. Its lower coating is connected with the earth by a chain, and a charge is given to its upper coating by the machine. When it is charged, if a person endeavours to take up a coin laid upon its upper face, he will experience a shock as soon as his hand comes near it, which will produce involuntary contraction of his arm, and prevent him from taking hold of the coin. 464. Heating of Metallic Threads. The discharge of electricitj^ through a conducting system produces elevation of temperature, the amount of heat generated being the equivalent of the potential energy which runs down in the discharge, and which is jointly pro- portional to quantity of electricity and difference of potential. The incandescence of a fine metallic thread can be easily produced by the discharge of a battery. The thread should be made to connect the knobs a b of an apparatus called a, universal discharger; these knobs Fig. 395. Universal Discharger. being the extremities of two metallic arms supported on glass stems. One of the arms is connected with the external surface of the battery,, arid the other arm is then brought into connection with the internal surface by means of a discharger with glass handles. At the instant of the spark passing, the thread becomes red-hot, melts, burns, or volatilizes, leaving, in the latter cases, a coloured streak on a sheet of paper c placed behind it. When the thread is of gold, this streak is purple, and exactly resembles the marks left on walls when bell- pulls containing gilt thread are struck by lightning. 465. Electric Portrait. The volatilization of gold is employed in producing what are called electric portraits. The outline of a por- VELOCITY OF ELECTRICITY. 585 trait of Franklin is executed in a thin card by cutting away narrow strips. Two sheets of tin-foil are gummed to opposite edges of the card, which is then laid between a gold-leaf and another card. The whole is then placed in a press (Fig. 396), the tin-foil being allowed Fig. 396. Press for Portrait. to protrude, and strong pressure is applied. The press is placed on the table of the universal discharger, and the two knobs of the latter are connected with the two sheets of tin-foil. The discharge is then passed, the gold is volatilized, and the vapour, passing through the slits to the white card at the back, leaves purple traces which reproduce the design. 466. Velocity of Electri- city. Soon after the in- vention of the Leyden jar, various attempts were made to determine the velocity with which the discharge travels through a Conductor Fig. 39r. Arrangement for Portrait. connecting the two coat- ings. Watson, about 1748, took two iron wires, each more than a mile long, which he arranged on insulating supports in such a way that all four ends were near together. He held one end of each wire in his hands, while the other ends were connected with the two coat- ings of a charged jar. Although the electricity had more than a mile to travel along each wire before it could reach his hands, he could never detect any interval of time between the passage of the spark 586 EFFECTS PRODUCED BY DISCHARGE OF CONDENSERS. from the knob of the jar and the shock which he felt. The velocity was in fact far too great to be thus measured. Wheatstone, about 1836, investigated the subject with the aid of the revolving mirror of which we have spoken above ( 437). He connected the two coatings of a Ley den jar by means of a conductor which had breaks in three places, thus giving rise to three sparks. When the sparks were taken in front of the revolving mirror, the positions of the images indicated a retardation of the middle spark, as compared with the other two, which were taken near the two coatings of the jar, and were strictly simultaneous. The middle break was separated from each of the other two by a quarter of a mile of copper wire. He calculated that the retardation of the middle 1 ' "* spark was 1 Ig9 000 of a second, which was therefore the time occupied in travelling through a quarter of mile of copper wire. This is at the rate of 288,000 miles per second, a greater velocity than that of light, which is only about 184,000 miles per second. Since the introduction of electric telegraphs, several observations have been taken on the time required for the transmission of a signal. For instance, trials in Queenstown harbour, in July, 1856, when the two portions of the first Atlantic cable, on board the Agamemnon and Niagara, were for the first time joined into one conductor, 2500 miles long, gave about If seconds as the time of transmission of a signal from induction coils, corresponding to a velocity of only 1400 miles per second. In 1858, before again proceeding to sea, a quicker and more sensitive receiving instrument Thomson's mirror galvanometer gave a sensible indication of rising current at one end of 3000 miles of cable about a second after the application of a Daniell's battery at the other. It seems to be fully established by experiment that electricity has no definite velocity, and that its apparent velocity depends upon various circumstances, being greater through a short than through a long line, greater (in a long line) with the greater intensity and suddenness of the source, greater with a copper than with an iron wire, and much greater in a wire suspended in air on poles than in one surrounded by gutta-percha and iron sheathing, and buried under ground or under water. In a long submarine line, a short sharp signal sent in at one end, comes out at the other as a signal gradually increasing from nothing to a maximum, and then gradually dying away. THE UNIT-JAR. 5b7 467. Unit-jar. For quantitative experiments on the effects of discharge, Lane's unit-jar has frequently been employed. One of its forms is represented in Fig. 398. It consists of a very small Leyden phial, having two knobs a, b, one connected with each coating, the distance between them being adjustable by means of a sliding rod. To measure the charge given to a jar or battery, the latter is placed upon an insulating sup- port, its inner coating is connected with the con- ductor of the machine, and its outer coating is connected with the inner coating of the unit-jar. The outer coating of the unit-jar must be in con- nection with the ground. When the machine is Fig. 393. unit jar. worked, sparks pass be- tween a and b, each spark being produced by the escape of a definite quantity of electricity from the outer coating of the bat- tery, and indicating the addition of a definite amount to the charge of the inner coating. The charge is measured by counting the sparks. Snow Harris modified the arrangement by insulating the unit-jar instead of the battery. One coating of his unit-jar is connected with the battery, and the other with the conductor of the machine. The battery thus receives its charge through the unit-jar 1 by a succession of discharges between the knobs a, b, each representing a definite quantity of electricity. Both arrangements, as far as their measuring power is concerned, depend upon the assumption that discharge between two given conductors, in a given relative position, involves the transfer of a definite quantity of electricity. This assumption implies a constant condition of the atmosphere. It may be nearly fulfilled during a short interval of time in one day, but is not true from one day to another. Moreover, it is to be remembered that, as dissipation is continually going on, the actual charge in the battery at any time is less than the measured charge which it has received. 1 Lane's arrangement might have been described by saying that the outer coating of the battery receives its negative charge from the earth through the unit- jar. 588 EFFECTS PRODUCED BY DISCHARGE OF CONDENSt.RS. 468. Mechanical Effects. The effects of discharge through bad conductors are illustrated by several well-known experiments. 1. Puncture of card. A card is placed (Fig. 399) between two points connected with two conductors which are insulated from one another by means of a glass stem. The lower conductor having been con- nected with the outer coating of a Leyden jar which is held in the hand, the knob of the jar is brought near the upper conductor. A spark passes, and another spark at the same instant passes between the two points, and punctures the card. In performing this experi- Fig. 399. Puncture of Card. menfc it is observed that, if the points are not opposite each other, tlie perforation is close to the negative point. This want of symmetry appears to be due to the properties of the air. When arrangements are made for exhausting the air, it is found that, as the density of the air is diminished, the perforation takes place nearer to the centre. The piercing of a card can very easily be effected by Holtz's machine. Its two conductors are connected with the two coatings of a small Leyden jar. The discharges between the poles will then consist of powerful detonating sparks in rapid succession ; and if a sheet of paper or card be interposed, every spark will puncture a minute hole in it. 2. Perforation of glass. To effect the perforation of glass, a pane MECHANICAL EFFECTS OF DISCHARGE. 589 of glass is suppoi'ted on one end of a glass cylinder in whose axis there is a metallic rod terminating in a point which just touches the pane. Another pointed rod exactly over this, and insulated from it, is lowered until it touches the upper face of the pane. A powerful spark from a Leyden jar or battery is passed between the two points, and, if the experiment succeeds, a hole is produced by pulverization of the jrlass. Fig. 400. Puncture of Glass. The experiment sometimes fails, by discharge taking place round t!ie edge of the glass instead of through its substance. To prevent this, a drop of oil is placed on the upper face of the pane at the point where the hole is to be made; but this precaution does not alwa}'s insure success, and, when the experiment has once failed, it is useless to try it again with the same piece of glass, for the electricity is sure to follow in the course which the first discharge has marked out for it. 469. Explosion of Mines. If a strongly charged Leyden jar be discharged by means of a jointed discharger which has one of its knobs covered with gun-cotton, when the spark passes between the jar and this knob, the gun-cotton will be inflamed. Ordinary 590 EFFECTS PRODUCED BY DISCHARGE OF CONDENSERS. cotton mixed with powdered resin can be kindled in the same way. A similar arrangement is often used for exploding mines. A fuse is employed containing two wires embedded in gutta-percha, but \\ ith their ends unprotected and near together. One of these wires is Fig. 401. Gun-cotton Fired. connected with the outer coating of a condenser, and the other is brought into communication with the inner coating. The discharge is thus made to pass between the ends within the fuse, and to ignite a very inflammable compound by which they are surrounded. Some- times one of the wires, instead of being connected with the outer coating, is connected with the earth by means of a buried wire. CHAPTER XLI A . ELECTROMETERS. 469 A. Object of Electrometers. Electrometers are instruments for the measurement of differences of electrical potential. The gold-leaf electroscope, the straw- electroscope, and other instruments of the same type, afford rough indications of the difference of potential between the diverging bodies and the air of the apartment, and more measurable indications are given by the electrometers of Peltier and Dellmann ; but none of these instruments are at all comparable in precision to the various electrometers which have been invented from time to time by Sir Wm. Thomson. 469 B. Attracted-disc Electrometers, or Trap-door Electrometers. We shall first describe what Sir Wm. Thomson calls "Attracted-disc Electrometers." These instruments, one of which is represented in Figs. 401 A, 401 B, contain two parallel discs of brass g, h, with an aperture in the centre of one of them, nearly filled up by a light trap-door of aluminium /, which is supported in such a way as to admit of its electrical attraction towards the other disc being resisted by a mechanical force which can be varied at pleasure. The trap- door and the perforated plate surrounding it must have their faces as nearly as possible in one plane when the observation is taken, and, as they are electrically connected, they may then be regarded as forming one disc of which a small central area is movable. There is always attraction between the two parallel discs, except when they are at the same potential. Let their potentials He denoted by V and V, the electrical densi- ties on their faces by p and p, and their mutual distance by D. We have seen ( 445 Q) that, in such circumstances, p and p are constant (except near the edges of the discs), opposite in sign, and equal, and that the intensity of force in the space between them is everywhere 592 ELECTROMETERS. the same, and equal at once to ^ and to 4nrp. This force is jointly due to attraction by one plate and repulsion by the other, each of these having the intensity 2 v p, or half the total intensity. Let A denote the area of the trap-door. The quantity of electricity upon it will be p A, and the force of attraction which this experiences will bepAx2-p = 2xp'A, which we shall denote by F. Then from the equations (1) we find, by eliminating p, ' OT v - v ' - 469 c. Absolute Electrometer. In the absolute electrometer, which somewhat resembles Fig. 401 B turned upside down, the force of electrical attraction on the trap-door is measured by direct com- parison with the gravitating force of known weights. This is done by first observing what weights must be placed on the trap-door to bring it into position when no electrical force acts (the plates being electrically connected), and by then removing the weights, allowing electrical force to act, and adjusting the plates at such a distance from one another, by the aid of a micrometer screw, that the trap-door shall again be brought into position. Then, in equation (2), F, A, and D are known, and the difference of potentials V V can be determined. In the absolute electrometer, the perforated disc h is uppermost, so that the direction of electrical attraction on the trap- door is similar to the direction of the gravitating force of the weights. The reverse arrangement is usually adopted in the portable electro- meter, which we shall next describe. In both instruments, the trap- door constitutes one end of a very light lever fil of aluminium, balanced on a horizontal axis. 469 D. Portable Electrometer. In the portable electrometer (Figs. 401 A, 401 B) this axis passes very accurately through the centre of gravity of the lever, the suspension being effected by means of a fine platinum wire ww tightly stretched, which is secured at its centre to the lever in such a manner that, when the trap-door comes into position, the wire is under torsion tending to draw back the disc from the attracting plate g. This torsion (except in so far as it is affected by causes of error such as temperature and gradual loss of elasticity) is always the same when the disc is in position, and as it PORTABLE ELECTROMETER. 593 is to be balanced in every observation by electrical attraction, the latter must also be always the same ; that is to say, the quantity F in equations (2) is constant for all observations with the same instru- ment ; whence it is obvious that V V is directly proportional to D, the distance between the plates. The observation for difference of potential therefore consists in altering this distance until the trap- door comes into position. This is done by turning the micrometer Fig. 401 A. Portable Electrometer. Fig. 401 B. Parallel Discs. screw, by means of the milled head m. The divided circle of the micrometer indicates the amount of turning for small distances, and whole revolutions are read off on the vertical scale traversed by the index carried by the arm d. The correct position is very accurately identified by means of two sights, one of them being attached to a fixed portion of the instrument, and the other to one end I of the lever. One of these sights moves up and down close in front of the other, and they are viewed through a lens o in front of both. This arrangement is also adopted in the absolute electrometer. 39 51H ELECTROMETERS. One of the two parallel plates h is connected with the inner coat- ing of a Leyden jar, 1 which, being kept dry within by means of pumice p wetted with sulphuric acid, retains a sufficient charge for some weeks. The other plate g is in communication, by means of the spiral wire r, with the insulated umbrella c, which can be con- nected with any external conductor; and, in order to determine the Fig. 401 C. Quadrant Electrometer. potential of any conductor which we wish to examine, two observa- tions are taken, one of them giving the difference of potential between this conductor and the Leyden jar, and the other the dif- ference between the earth and the jar. We thus obtain, by subtrac- 1 The use of the Leyden jar is to give constancy of potential. Its capacity is so much greater than that of the disc with which it is connected that the electricity which enters or leaves the latter in consequence of the inductive action of the other disc is no sensible fraction of its whole charge, and produces no sensible change in its potential. Its great capacity in comparison with the extent of surface exposed likewise tends to prevent rapid loss of potential by dissipation of charge. QUADRANT ELECTROMETER. 593 tion, the difference of potential between the conductor in question and the earth. 469 E. Quadrant Electrometer. The most sensitive instrument yet invented for the measurement of electrical potential is the quadrant electrometer, which is represented in front view in Fig. 401 c, some of its principal parts being shown on a larger scale in Figs. 40 ID, 401 E. In this instrument, the part whose movements give the indica- tions is a thin flat piece of aluminium u, narrow in the middle and broader towards the ends, but with all corners rounded off. This piece, which is called the needle, and is represented by the dotted line in Fig. 401 D, is inclosed almost completely in what may be described as a shallow cylindrical box of brass, cut into four quadrants, c, d, c'd'. These parts are shown in plan in Fig. 401 D, and in front view in Fig. 401 c. The needle u is attached to a stiff platinum wire, which is supported by a silk fibre hanging vertically. The same wire carries a small concave mirror t (Fig. 401 c) for reflecting the light from an illuminated vertical slit. An image of O the slit is thus formed at the distance of about a yard, and is received upon a paper scale of equal parts, by reference to which the move- ments of the image can be measured. The movements of the image depend upon the movements of the mirror, which are precisely the same as those of the needle. We have now to explain how the movements of the needle are produced. One pair of opposite quadrants c c' are connected with each other, and with a stiff wire I projecting above the case of the instrument. The other quadrants d d' are in like manner connected with the other projecting wire m. The projecting parts I m are called the chief electrodes, and are to be connected respectively with the two con- ductors whose difference of potential is required, one of which is usually the earth. Suppose the needle to have a positive charge of its own, then if the potential of c and c' be higher (algebraically) than that of d and d', one end of the needle will experience a force urging it from c to d, and the other end will experience a force urging it from c' to d'. These two forces constitute a couple tending to turn Fig. 401 D. Needle and Quadrants. 596 ELECTROMETERS. the needle about a vertical axis. If the potential of c and c' be lower than that of d and d', the couple will be in the opposite direction. To prevent the needle from deviating too far under the action of this couple, and to give it a definite position when there is no elec- trical couple acting upon it, a small light magnet is attached to the back of the mirror, and by means of controlling magnets outside the case the earth's magnetism is overpowered, so that, whatever position be chosen for the instrument, the needle can be made to assume the proper zero position. In some instruments recently constructed, the magnets are dispensed with, and a bifilar suspension is substituted for the single silk fibre. The permanent electrification of the needle is attained by connecting it, by means of a descending platinum wire, with a quantity of strong sulphuric acid, which occupies the lower part of the containing glass jar. The acid, being an excellent conductor, serves as the inner coating of a Leyden jar, the outside of the glass opposite to it being coated with tin-foil, and connected with the earth. The acid at the same time serves the purpose of keeping the interior of the apparatus very dry. The charge is given to the jar through the charging electrode p, which can be thrown into or out of connection at pleasure. As the sensibility of the instrument increases with the potential of the jar, a gauge and replenishes are provided for keeping this potential constant. The gauge is simply an "attracted-disc electrometer," in which the distance between the parallel discs is never altered, so that the aluminium square only comes into position when the potential of one of the discs, which is connected with the acid in the jar, differs by a certain definite amount from the potential of the other, which is connected with the earth. A glance at the gauge shows, at any moment, whether the potential of the jar has the normal strength. If it has fallen below this point, the replenisher is employed to increase the charge. This apparatus, which is separately represented, dissected, in Fig. 401 E, and is for simplicity omitted in Fig. 401 c, consists of a vertical stem of ebonite s, which can be rapidly twirled with the finger by means of a milled head y, and which carries two metal wings or carriers, b, b, insulated from each other. In one part of their revolu- tion, these come in contact with two light steel springs //, which simply serve to connect them for the instant with each other. In another part of their revolution, they come in contact with two other springs e e, connected respectively with the acid of the jar and with the earth. The first of these contacts takes place just before THE REPLENISHED 597 Fig. 401 E. Replenish. the wings emerge from the shelter of the larger metallic sectors or inductors a a, of which one is connected with the acid, and the other with the earth. Suppose the acid to have a positive charge Then, at the instant of contact, an inductive movement of electricity takes place, producing an accumula- tion of negative electricity in the carrier which is next the positive inductor, and an accumulation of positive in the other. The next contacts are effected when the car- rier which has thus acquired a posi- tive charge is well under cover of the positive inductor, to which ac- cordingly it gives up its electricity, for, being in great part surrounded by this inductor, and being con- nected with it by the spring, the carrier may be regarded as forming a portion of the interior of a concave conductor, and the electricity accordingly passes from it to the external surface, that is to the inductor, and to the acid con- nected with it, which form the lining of the jar. The negative elec- tricity on the other carrier is, in like manner, given off to the other inductor, and so to the earth. The jar thus receives an addition to its charge once in every half- revolution of the replenisher; and, as these increments are very small, it is easy to regulate the charge so that the gauge shall indicate exactly the normal potential. If the charge is too strong, it can be diminished by turning the replenisher in the reverse direction. 469 F. Cage-electrometer. In another form of electrometer, which has some advantages of its own, though now but little used, the ob- servation for difference of potential consists in applying torsion to a glass fibre until the needle (a straight piece of aluminium wire) which it carries, is forced, against electrical repulsion, to assume a definite position marked by sights. The repulsion, which acts upon the two nds of the needle so as to produce a couple, is exerted by two ver- tical brass plates, which are connected with the needle by means of fine platinum wires dipping in sulphuric acid at the bottom of a Leyden jar. The needle and the plates which repel it are thus at the potential of the jar. The repulsion between them is modified by 598 ELECTROMETERS. the influence of a cage of brass wire, which surrounds them, and which is connected with the conductor whose potential is to be ex amined. If this conductor has the same potential as the jar, there is no repulsion. If its potential differs either way from that of the jar, the couple of repulsion is proportional to the square of this difference of potentials. 1 The difference of potential is therefore obtained by taking the square root of the number of degrees of torsion of the fibre. 1 In a given position of the needle, the quantities of electricity upon it and upon the plates which repel it are both proportional to this difference of potentials, and the distribu- tion is invariable. Hence ( 420) the force of repulsion is proportional to the product of the two Quantities, that is to the sqnaro of either of them. CHAPTER XLII. ATMOSPHERIC ELECTRICITY. 470. Resemblance of Lightning to the Electric Spark. The resem- blance of the effects of lightning to those of the electric spark struck the minds of many of the early electricians. Lightning, in fact, ruptures and scatters non-conducting substances, inflaming those which are combustible ; heats, reddens, melts, and volatilizes metals ; and gives shocks, more or less severe, and frequently fatal, to men and animals; all of these being precisely the effects of the electric spark with merely a difference of intensity. We may add that lightning leaves behind it a characteristic odour precisely similar to that which is observed near an electrical machine when it is working, and which we now know to be due to the presence of ozone. More- over, the form of the spark, its brilliancy, and the detonation which attends it, all remind one forcibly of lightning. To Franklin, however, belongs the credit of putting the identity of the two phenomena beyond all question, and proving experi- mentally that the clouds in a thunder-storm are charged with elec- tricity. This he did by sending up a kite, armed with an iron point with which the hempen string of the kite was connected. To the lower end of the string a key was fastened, and to this again was attached a silk ribbon intended to insulate the kite and string from the hand of the person holding it. Having sent up the kite on the approach of a storm, he waited in vain for some time even after a heavy cloud had passed directly over the kite. At length the fibres of the string began to bristle, and he was able to draw a strong spark by presenting his knuckle to the kej 7 ". A shower now fell, and, by wetting the string, improved its conducting power, the silk ribbon being still kept dry by standing under a shed. Sparks in rapid succession were drawn from the key, a Leyden jar was charged by it, and a shock given. 600 ATMOSPHERIC ELECTRICITY. Fig. 402. Electric Chimes. Shortly before this occurrence, Dalibard, acting upon a published suggestion of Franklin, had erected a pointed iron rod on the top of a house near Paris. The rod was insulated from the earth, and could be connected with various electrical apparatus. A thunder-storm having occurred, a great number of sparks, some of them of great power, were drawn from the lower end of the rod. These experiments were repeated in various places, and Richmann of St. Petersburg, while conducting an in- vestigation with an apparatus somewhat resembling that of Dalibard, received a spark which killed him on the spot. 472. Electric Chimes. Franklin devised an apparatus for giving warning when the insulated rod is charged with electricity. It consists (Fig. 402) of a metal bar, carry- ing three bells with two clappers between them. The two extreme bells are hung from the bar by metallic chains. The middle one is hung by a silk thread, and connected with the ground. The clappers are also hung by silk threads. When the bar is electrified, the clap- pers are first attracted by the two extreme bells, and then repelled to the middle bell, through which they discharge themselves, to be again attracted and repelled, thus keeping up a continual ringing as long as the bar remains electrified. 473. Duration of Lightning. It appears that thunder-clouds must be regarded as charged masses of considerable conducting power. The discharges which produce lightning and thunder occur sometimes between two clouds, and sometimes between a cloud and the earth. The duration of the illumination produced by lightning is certainly less than the ten-thousandth of a second. This has been established by observing a rapidly rotating disc (Fig. 403) divided into sectors alternately black and white. If viewed by daylight, the disc appears of a uniform gray; and if lightning, occurring in the dark, renders the separate sectors visible, the duration of its light must be less than the time of revolving through the breadth of one sector. The experi- ment has been tried with a disc divided into 60 sectors, and making 180 revolutions per second, so that the time of turning through the Fig. 403. Duration of Flash. DURATION OF LIGHTNING. 601 o Fig. 404. Simultaneous Explosions. space occupied by one sector is -^ of -j-^ of a second, that is, T0 \ 00 . When the disc, turning with this velocity, is rendered visible by lightning, the observer sees black and white sectors with gray ones between them. For the black and white sectors to be seen sharply defined, without intermediate gray, it would be necessary that the illumination should be absolutely instan- taneous. 476. Thunder. Thunder frequently consists of a number of reports heard in succession. This can be explained by sup- posing that (as in the experiment of the spangled tube, 440) discharge occurs at several places at once. The reports of these explosions will be heard in the order of their distance from the observer. If, for example, the lines of discharge form the zig-zag MN (Fig. 404), an observer at will hear first the explosion at a, then, a little later, the five explosions at m, n, r t s, t\ he will conse- quently observe an increase in the intensity of the sound. 477. Shock by Influence. Persons near whom a flash of lightning passes, frequently experience a severe shock by induction. This is analogous to the phenomenon, first observed by Galvani, that a skinned frog in the neighbourhood of an electrical machine, although dead, exhibits convulsive movements every time a spark is drawn from the conductor. In like manner, if Volta's pistol ( 443) be placed on the wooden supports of an electrical machine, and its knob be connected with the ground by a chain, on drawing a spark from the machine, another spark will pass in the interior of the pistol, and fire it off. 478. Lightning-conductors. Experience having shown that elec- tricity travels in preference through the best conductors, it is easy to understand that, if a building be fitted with metallic rods termi- nating in the earth, lightning will travel through these instead of striking the building. But further, if these rods terminate above in a point, they may exercise a preventive influence by enabling the earth and clouds to exchange their opposite electricities in a gradual way, just as the conductor of a machine is prevented from giving powerful sparks b} r presenting to it a sharp point connected with the earth. 602 ATMOSPHERIC ELECTRICITY. While the electrical machine is working powerfully, and the quad- rant electroscope shows a strong charge, let a pointed metallic rod be presented, as in Fig. 405 ; the pith-ball will immediately fall back to the vertical position, and it will be found impossible to draw a spark Fig. 405. Conductor Discharged by presenting a Point. from any part of the conductor. If the experiment is performed in the dark, the point will be seen to be tipped with light; and a similai appearance is sometimes observed on the tops of lightning-rods and of ships' masts. In the latter position it is known to sailors as St. Elmo's fire. 479. Construction of Lightning-conductors. A badly constructed lightning-conductor may be a source of danger, instead of a protec- tion. The following conditions should always be complied with : 1. The connection with the ground should be continuous. 2. The conductor must be everywhere of so large a section that it will not be melted by lightning passing through it. The French Academy of Sciences recommend that the section for iron rods should be nowhere less than 2'25 centimetres, or T ^j- of an inch. 3. The earth contact must be good. The conductor may be con nected at its base with the iron pipes which supply the neighbour hood with water or gas; or it may terminate in the water of a well or pond. Failing these, it should be provided with branches travers ing the soil in different directions, and surrounded by coke, which is a good conductor. 4. At no part of its course above ground should it come near to the metal pipes which supply the house with water or gas, nor to any large masses of metal in the house. All large masses of metal on the outside of the house, such as lead roofing, should be well connected with the conductor. LIGHTNING-CONDUCTORS. 603 5. The extreme point should be sharp. A former commission of the Academy recommended a platinum point, which should be con- nected with the iron by welding. But as this construction is both difficult and expensive, later directions have been issued recommending a gilded copper cone, screwed on to the iron, as shown in Fig. 407, which is half the actual size. This form of termination is better than a needle point, because less liable to fusion. The general arrangement is represented in Fig. 406. The rod has a diameter of 2 or it inches at its base, and gradually tapers upwards to the place where the point is screwed on. The descending portion b is connected with the base of this rod by the broad band II'. 480. Ordinary Electricity of the Atmosphere. The presence of electricity in the upper re- gions of the air is not confined to thunder- clouds, but can be detected at all times. In fine weather this electricity is almost invariably I positive, but in showery or stormy weather nega- tive electricity is as fre- quently met with as posi- tive; and it is in such weather that the indica- tions of electricity, whe- ther positive or negative, are usually the strongest. 480 A. Methods of ob- taining Indications. One of the early methods of ob- serving atmospheric elec- tricity consisted in shoot- ing up an arrow, attached to a conducting thread, having at its lower end a ring, which was laid upon the top of a gold-leaf electroscope. As the arrow ascends higher, the leaves diverge more and more with elec- tricity of the same sign as that overhead ; and they remain diverg- Fig. 406. Lightning- conductor. Fig. 407. Gilded Copi>ci Point. 601 ATMOSPHERIC ELECTRICITY. ent after the ring has been lifted off by the movement of the arrow. Sometimes, instead of the arrow, a point on the top of the electro- scope is employed to collect electricity from the air, as in Fig. 408. Both these methods are very uncertain in their action. A better method of collecting electricity from the air was long ago devised by Volta, who employed for this purpose a burning match attached to the top of a rod connected with the gold-leaves or straws of his electro- scope. If there is positive electricity over- head, its influence causes negative electricity to collect at the upper end of the rod, whence it passes off by convection in the products of combustion of the match, leaving the whole conducting system positively electrified. In like manner, if the electricity overhead be ne- gative, the system will be left negatively elec- trified. Another method which, in the hands of Pel- tier, Quetelet, and Dellmann, has yielded good results, consists in first exposing, in an elevated position such as the top of a house, a conducting ball supported on an insulating stand, and, while exposed, connecting it with the earth; then insulating it, and examining the charge which it has acquired. This charge, being ac- quired from the earth by the inductive action of the electricity overhead, is opposite in sign to the inducing electricity. Another method, which in principle re- sembles that of Volta, but is speedier in its action, has been intro- duced by Sir W. Thorn- Fig. 408 A. Water dropping Collector. SOU. It. Consists m al- lowing a fine stream of water to flow, from an insulated metallic vessel, through a pipe, which projects through an open window or other aperture in the Fig. 408. Early Form of Electrometer. f, THE WATER-DROPPING COLLECTOR. 605 wall of a house, so that the nozzle from which the water Hows is in the open air. The apparatus for this purpose, called the water- dropping collector, is represented in Fig. 408 A. a is a copper can, containing water, which can be discharged through the brass pipe b by turning a tap. The mode of insulation is worthy of notice. The can is supported on a glass stem c, which is surrounded, without con- tact, by a ring or rings of pumice dd, moistened with sulphuric acid. These are protected by an outer case of brass ee, having a hole in its top rather larger than the glass stem, the brass being separated from the moist pumice by an inner case of gutta-percha. The acid needs renewal about once in two months. In severe frost, burning matches can be used instead of water, and are found to give identical indications. Whether water or match be used, the principle of action 1 is that, as long as any differ- ence of potential exists between the insulated conductor and the point of the air where the issuing stream (whether of water or smoke) ceases to be one continuous conductor, and begins to be a non-con- ductor or a succession of detached drops, so long will each drop or portion that detaches itself carry off either positive or negative elec- tricity, and thus diminish the difference of potential. This is an application of the principle of 445 B, that electricity tends to travel from places of higher to places of lower potential. The time required to reduce the system to the potential which exists at the point above specified, is practically about half a minute with the water jet, and from half a minute to a minute or more, according to the strength of the wind, with a match. The water-dropper is the most convenient collecting apparatus when the observations are taken always in the same place. For 1 The following quotation from an article by Sir W. Thomson puts the matter very clearly: "If, now, we conceive an elevated conductor, first belonging to the earth, to be- come insulated, and to be made to throw off, and to continue throwing off, portions from an exposed part of its surface, this part of its surface will quickly be reduced to a state of no electrification, and the whole conductor will be brought to such a potential as will allow it to remain in electrical equilibrium in the air, with that portion of its surface neutral. In other words, the potential throughout the insulated conductor is brought to be the same as that of the particular equi- potential surface in the air, which passes through the point of it from which matter breaks away. A flame, or the heated gas passing from a burning match, does precisely this : the flame itself, or the highly heated gas close to the match, being a conductor which is constantly extending out, and gradually becoming a non-con- ductor. The drops [into which the jet from the water- dropper breaks] produce the same effects, with more pointed decision, and with more of dynamical energy to remove the re- jected matter, with the electricity which it carries, from the neighbourhood of the fixed conductor." NickoPt Cyclopedia, second edition, art. "Electricity, Atmospheric." G06 ATMOSPHERIC ELECTRICITY. portable service, Sir Wm. Thomson employs blotting-paper, steeped in solution of nitrate of lead, dried, and rolled into matches. The portable electrometer carries a light brass rod or wire projecting up- wards, to the top of which the matches can be fixed. 480 B. Interpretation of Indications. We have seen that the col- lecting apparatus, whether armed with water-jet or burning match, is merely an arrangement for reducing an insulated conductor to the potential which exists at a particular point in the air. An electro- meter will then show us the difference between this potential and that of any other given conductor, for example the earth. The earth offers so little resistance to the passage of electricity, that any tem- porary difference of potential which may exist between different parts of its surface, must be very slight in comparison with the dif- ferences of potential which exist between different points in the non- conducting atmosphere above it. As there is no possible method of determining absolute potential, since all electric phenomena would remain unchanged by an equal addition to the potentials of all points, it is convenient to assume, as the zero of potential, that of the most constant body to which we have access, namely the earth ; and under the name earth we include trees, buildings, animals, and all other con- ductors in electrical communication with the soil. Now we find that, as we proceed further from the earth's surface, whether upwards from a level part of it, or horizontally from a verti- cal part of it, such as an outer wall of a house, the potential of points in the air becomes more and more different from that of the earth, the difference being, in a broad sense, simply proportional to the dis- tance. Hence we can infer 1 that there is electricity residing on the surface of the earth, the density of this electricity, at any moment, in the locality of observation, being measured by the difference of potential which we find to exist between the earth and a given point in the air near it. Observations of so-called atmospheric elec- tricity 2 made in the manner we have described, are in fact simply 1 By 445 1, if p denote the quantity of electricity per unit area on an even part of the earth's surface, the force in the neighbouring air is 4wp. This must be equal to the change of potential in going unit distance ( 445 D). If potential increases positively, p is negative. 1 No good electrical observations have yet been made in balloons, and very little is known regarding the distribution of electricity at different heights in the air. A method of gauging this distribution by balloon observations is suggested by the principles of 445 G, which show that, when the lines of force are vertical, and the tubes of force consequently cylindrical, the difference of electrical force at different heights is proportional to the quantity of electricity which lies between them. RESULTS OF OBSERVATION. 607 determinations of the quantity of electricity residing on the earth's surface at the place of observation. The results of observations so made are however amply sufficient to show that electricity residing in the atmosphere is really the main cause of the variations observed. A charged cloud or body of air induces electricity of the opposite kind to its own on the parts of the earth's surface over which it passes; and the variations which we find to occur in the electrical density at the parts of the surface where we observe, are so rapid and considerable, that no other cause but this seems at all adequate to account for them. We may therefore safely assume that the differ- ence of potential which we find, in increasing our distance from the earth, is mainly due to electricity induced on the surface of the earth by opposite electricity in the air overhead. As electrical density is greater on projecting parts of a surface than on those which are plane or concave, we shall obtain stronger indica- tions on hills than in valleys, if our collecting apparatus be at the same distance from the ground in both cases. Under a tree, or in any position excluded from view of the sky, we shall obtain little or no effect. 480c. Results of Observation. The only regular series of observa- tions which have as yet been taken 1 with Sir Win. Thomson's instru- ments, consist of two years' continuous observations with self-recording apparatus at Kew Observatory; and two years' observations, at three stated times daily, and at other irregular times, at Windsor in Nova Scotia (lat. 45 N.) The electrometer used at Kew was an earlier form of the quadrant electrometer already described ; and the auto- graphic registration was effected by throwing the image of a bright point (a small hole with a lamp behind it) upon a sheet of photo- graphic paper drawn upwards by clock-work, whereas the movements of the image, formed by means of the mirror attached to the needle, were horizontal. The curves thus obtained give very accurate infor- mation respecting the potential of the air at the point of observation, when of moderate strength ; but fail to record it when of excessive strength, as the image on these occasions passed out of range. The Windsor observations were taken with the cage-electrometer, of which two forms were employed, one being much more sensitive than 1 The observations at Windsor, N.S., and at Kew, are described in three papers by the editor of this work, Proc. R. S., June 1863, January 1865, and Trans. R. S., December 1867. Dellraann's observations at Kreuznach, which were taken with apparatus devised by himself, are described in Phil. Mag. June 1858. Quetelet's observations (taken with Peltier's apparatus) are described in his volume Sur le Climat de la Belgique (Brussels, 1849). 008 ATMOSPHERIC ELECTRICITY. the other. The more sensitive form was usually employed. When the potential became inconveniently strong, the first step was to shorten the discharging pipe by screwing off' some of its joints. This reduced the strength of potential in about the ratio of 3 : 1 ; but even this reduction was often not enough for the more sensitive instru- ment, and on such occasions the other (which was intended as a port- able electrometer) was employed instead. As the ratio of the indica- tions of the two instruments was known, a complete comparison of potentials in all weathers was thus obtained. The results are as follows. Employing a unit in terms of which the average fine-weather potential for the year was +4, the potential was seldom so weak as 1, though on rare occasions it was for a few minutes as low as O'l. In wet weather, especially with sudden heavy showers, the potential was often as strong as 20 to 30, and it was fully as strong during hail. With snow, the average strength was about the same as with heavy rain, but it was less variable, and the sign was almost always positive. Occasionally, with high wind accompanying snow, during very severe frost, it was from +80 to +100, or even higher. With fog, it was always positive, averaging about +10. In thunder- storms it frequently exceeded +100, and on a few occasions ex- ceeded 200. There was usually a great predominance of negative potential in thunder-storms. Change of sign was a frequent accom- paniment of a flash of lightning or a sudden downpour of rain. At all times, there was a remarkable absence of steadiness as compared with most meteorological phenomena, wind-pressure being the only element whose fluctuations are at ah 1 comparable, in magnitude and suddenness, with those of electrical potential. Even in fine weather, its variations during two or three minutes usually amount to as much as 20 per cent. In changeable and stormy weather they are much greater; and on some rare occasions it changes so much from second to second that, notwithstanding the mitigating effect of the collecting process, which eases off' all sudden changes, the needle of the electro- meter is kept in a continual state of agitation. 480 D. Annual and Diurnal Variations. Observations everywhere 1 concur in showing that the average strength of potential is greater in winter than in summer; but the months of maxima and minima appear to differ considerably at different places. The chief maximum occurs in one of the winter months, varying at different places from 1 The remarks in this section express the results of observation at places all of which are in the north temperate zone. ANNUAL AND DIURNAL VARIATIONS. 609 the beginning to the end of winter : and the chief minimum occurs O O ' everywhere in May or June. Both Kew and Windsor show dis- tinctly two maxima in the year, but Brussels, and apparently Kreuznach, show only one. The ratio of the highest monthly aver- age to the lowest is at Kew about 2*5, at Windsor ]*9, and at Kreuznach 2'0. The Kew observations, being continuous, are specially adapted to throw light on the subject of diurnal variation. They distinctly in- dicate for each month two maxima, which in July occur at about 8 A.M. and 10 P.M., in January about 10 A.M. and 7 P.M., and in spring and autumn about 9 and 9. The result of the Brussels obser- vations is about the same. 481. Causes of Atmospheric Electricity. Various conjectures have been hazarded regarding the sources of atmospheric electricity; but little or no certain knowledge has yet been obtained on this subject. Evaporation has been put forward as a cause, but, as far as laboratory experiments show, whenever electricity has been generated in connec- tion with evaporation, the real source has been friction, as in Arm- strong's hydro-electric machine. The chemical processes involved in vegetation have also been adduced as causes, but without any suffi- cient evidence. It is perhaps not too much to say that the only natural agent which we know to be capable of electrifying the air is the friction of solid and liquid particles against the earth and against each other by wind. The excessively strong indications of electricity observed during snow accompanied by high wind, favour the idea that this may be an important source. Without knowing the origin of atmospheric electricity, we may, however, give some explanation of the electrical phenomena which occur both in showers and in thunder-storms. Very dry air is an excellent non-conductor ; very moist air has, on the other hand, con- siderable conducting power. When condensation takes place at several centres, a number of masses of non-conducting matter are transformed into conductors, and the electricity which was diffused through their substance passes to their surfaces. These separate con- ductors influence one another. If one of them is torn asunder while under influence, its two portions may be oppositely charged ; and if rain falls from the under surface of a cloud which is under the in- fluence of electricity above it, the rain which falls may have an opposite charge to the portion which is left suspended. The coalescence of small drops to form large ones, though it in- 40 610 ATMOSPHERIC ELECTRICITY. creases the electrical density on the surfaces of the drops, does not increase the total quantity of electricity, arid therefore ( 445 K) cannot directly influence the observed potential. Thunder-storms and other powerful manifestations of atmospheric electricity seem to be accompaniments of very sudden and complete condensation which gives unusually free scope to the causes of irregular distribution just indicated. 483. Hail. Hail has sometimes been ascribed to an electrical Fig. 409. Electric HaiL origin, and a singular theory was devised by Volta to account for the supposed fact that hailstones are sustained in the air. He imagined that two layers of cloud, one above the other, charged with opposite electricities, kept the hailstones continually moving up and down by alternate attraction and repulsion. An experiment called electric hail is sometimes employed to illustrate this idea. Two metallic plates are employed (Fig. 409), the lower one connected with the earth, and the upper one with the conductor of the electrical machine ; and pith-balls are placed between them. As the machine is turned, the balls fly rapidly backwards and forwards from one plate to the other. WATERSPOUTS. 611 484. Waterspouts. Waterspouts, being often accompanied by strong manifestations of electricity, have been ascribed by Peltier and Fig. 410. Waterspouts. others to an electrical origin ; but the account of them given in the subjoined note appears more probable. 1 1 "On account of the centrifugal force arising from the rapid gyrations near the centre of a tornado, it must frequently be nearly a vacuum. Hence when a tornado passes over a building, the external pressure, in a great measure, is suddenly removed, when the atmosphere within, not being able to escape at once, exerts a pressure upon the interio^ of perhaps nearly fifteen pounds to the square inch, which causes the parts to be thrown in every direction to a great distance. For the same reason, also, the corks fly from empty bottles, and everything with air confined within explodes. When a tornado happens at sea, it generally produces a waterspout. This is generally first formed above, in the form of a cloud shaped like a funnel or inverted cone. As there is less resistance to the motions in the upper strata than near the earth's surface, the rapid gyratory motion commences there first. . . . This draws down the strata of cold air above, which, coming in contact with the warm and moist atmosphere ascending in the middle of the tornado, condenses the vapour and forms the funnel-shaped cloud. As the gyratory motion becomes more violent, it gradually overcomes the resistances nearer the surface of the sea, and the vertex of the funnel- shaped cloud gradually descends lower, and the imperfect vacuum of the centre of the tornado reaches the sea, up which the water has a tendency to ascend to a certain height, and thence the rapidly ascending spiral motion of the atmosphere carries the spray upward, until it joins the cloud above, when the waterspout is complete. The upper part of a waterspout is frequently formed in tornadoes on land. When tornadoes happen on sandy plains, instead of waterspouts they produce the moving pillars of sand which are often seen on sandy deserts." W. Ferrel, in Mathematical Monthly. See note 406. MAGNETISM. CHAPTER XLIIL GENERAL STATEMENT OF FACTS AND LAWS. 485. Magnets, Natural and Artificial. Natural magnets, or lode- stones, are exceedingly rare, although a closely allied ore of iron, capable of being strongly acted on by magnetic forces, and hence called magnetic iron-ore, is found in large quantity in Sweden and elsewhere. Artificial magnets are usually pieces of steel, which have been permanently endowed with magnetism by methods which we shall hereafter describe. Magnets are chiefly characterized by the property of attracting iron, and by the tendency to assume a par- ticular orientation when freely suspended. 486. Force Greatest at the Ends. The property of attracting iron is very unequally manifested at different points of the surface of a magnet. If, for example, an ordinary bar-magnet be plunged in iron -filings, these cling in large quantity to the terminal portions, and leave the middle bare, as in the lower diagram of Fig. 411. If the magnet is very thick in pro- portion to its length, we may have filings adhering to all parts of it, but the quantity diminishes rapid- ly towards the middle. The name poles is used, in a somewhat loose sense, to denote the two terminal portions of a magnet, or to denote two points, not very accurately defined, situated in these portions. The middle portion, to which the filings refuse to adhere, is called neutral. 487. Lines Formed by Filings. If a sheet of card is laid horizon- tally upon a magnet, and wrought- iron filings are sifted over it, these will, with the assistance of a few taps given to the card, arrange Pig. 411. Magnets dipped in Filings. CURVE OF INTENSITIES. 613 themselves in a system of curved lines, as shown in Fig. 412. These lines give very important indications both of the direction and inten- sity of the force produced by the magnet at different points of the Fijr. 412. Magnetic Curves. space around it. 1 They cluster very closely about the two poles p p, and thus indicate the places where the force is most intense. 488. Curve of Intensities. Some idea may be obtained of the rela- tive intensities of magnetic force at different points in the length of a magnet, by measuring the weights of iron which can be supported at them. Much better determinations can be obtained either by the use of the torsion-balance, or by counting the number of vibrations made by a small magnetized needle when suspended opposite different parts of the bar, the bar being in a vertical position, and the vibra- tions of the needle being horizontal. The intensity of the force is nearly as the square of the number of vibrations ; on the same prin- ciple that the force of gravity at different places is proportional to the square of the number of vibrations of a pendulum ( 47). Both these methods of determination were employed by Coulomb, who was the first to make magnetism an accurate science ; and the results which he obtained are represented by the curve of intensities AMB (Fig. 413). M is the middle of the bar, O one end of it, and the ordinates 1 The lines formed by the filings may be called the lines of effective force for particles only free to move in the plane of the card. The lines of total force cut the card at various angles, and are at some places perpendicular to it, as shown by the filings standing on eixl For the definition of lines of magnetic force, see 494 A. 614 GENERAL STATEMENT OF FACTS AND LAWS. of the curve (that is, the distances of its points from the line OX) represent the intensities of force at the different points in its length. The curve was constructed from observations of the force at several points in the length ; but in dealing with the observa- tion made opposite the very end, the force actually ob- served was multiplied by 2. Perfect symmetry was found between the intensities over Fig. 413. Curve of Intensities. the two halves of the length. In the figure we have in- verted the curve for one- half, in order to indicate an opposition of properties, which we shall shortly have to describe. The curves of intensities for two magnets of different sizes but of the same form are usually similar. 489. Magnetic Needle. Any magnet freely suspended near its centre is usually called a magnetic needle, or more properly a magne- tized needle. One of its most usual forms is that of a very elongated rhombus of thin steel, having, very near its centre, a concavity or cup by means of which it can be bal- anced on a point. When it is thus balanced horizontally, it does not, like a piece of ordinary matter, re- main in equilibrium in all azi- muths, 1 but assumes one particular direction, to which it always conies back after displacement. In this position of stable equilibrium, one of its ends points to magnetic north, and the other to magnetic south, which differ in general by several degrees from geographical (or true) north and south. This is the principle on which compasses are constructed. 1 All lines in the same vertical plane are said to have the same azimuth. Azimuthal angles are angles between vertical planes, or between horizontal lines. The azimuth of a line when stated numerically, is the angle which the vertical plane containing it makes with a vertical plane of reference, and this latter is usually the plane of the meridian. Fig. 414. Magnetized Needle, DECLINATION AND INCLINATION. 615 Fig. 415. Declination. 490. Declination. The difference between magnetic and true north, or the angle between the magnetic meridian and the geographical meridian, is called magnetic declination. 1 It is very different at different places, and at a given place undergoes a gradual change from year to year, besides smaller changes, backwards and forwards, which are continu- ally taking place. At Greenwich, at the pre- sent time, its value is about 20 W., that is, magnetic north is west of true north by this amount. For the British Isles generally its value is from 20 to 30 W. 491. Inclination or Dip. If, before mag- netizing a needle, we mount it on an axis passing through its centre of gravity, and support the ends of the axis, as in Fig. 416, by a thread without torsion, the needle will remain in equilibrium in any position in which it may be placed. If it be then magnetized, it will no longer be indifferent, but will place itself in a particular vertical plane called the magnetic meri- dian, and will take a particular direction in this plane. This di- rection is not horizontal, but in- clined, generally at a considerable angle, to the horizon; and this angle is called dip or inclina- tion. Its value at Greenwich is about 67, the end which points to the north pointing at the same time downwards. In the north- ern hemisphere generally, it is the north end of the needle which dips, and in the southern hemi- sphere it is the end which points south. Fig. 416. Dip. Some readers may be glad to be reminded that by the plane of the meridian is meant a vertical plane passing through the place of observation, and through or parallel to the earth's axis. A horizontal line in this plane is a meridian line. The magnetic meridian is the vertical plane in which a magnetized needle, when freely suspended, tends to place itself. 1 The nautical name for magnetic declination is variation ; but it is most inconvenient and confusing to denote the element itself by the same name as the variations of the element. 016 GENERAL STATEMENT OF FACTS AND LAWS. It follows that, if a magnetized needle is to be balanced in a hori- zontal position, the point or axis of support must not be in the same vertical with the centre of gravity, but must be between the centre of gravity and the end which tends to dip. Needles thus balanced, as in the ordinary mariner's compass, are called declination needles. 492. Mutual Action of Poles. On presenting one end of a magnet to one end of a needle thus balanced, we obtain either repulsion or attraction, according as the pole which is presented is similar or dis- similar to that to which it is presented. Poles of contrary name attract each other ; poles of the same name repel each other. This property furnishes the means of distinguishing a body which is merely magnetic (that is, capable of temporary magnetization) from a permanent magnet. The former, a piece of soft iron for example, is always attracted by either pole of a magnet; while a body which has received permanent magnetization has, in ordinary cases, two poles, of which one is attracted where the other is repelled. Mag- netic attractions and repulsions are exerted without modification through any body which may be interposed, provided it be not magnetic. 492 A. Names of Poles. The phenomena of declination and inclina- tion above described, evidently require us to regard the earth, in a broad sense, as a magnet, having one pole in the northern and the other in the southern hemisphere. Now since poles which attract one another are dissimilar, it follows that the magnetic pole of the earth which is situated in the northern hemisphere is dissimilar to that end of a magnetized needle which points to the north. Hence great confusion of nomenclature has arisen, the usage of the best writers being opposite to that which generally prevails. We shall call that end or pole of a needle which seeks the north, the north- seeking end or pole, and the other the south-seeking end or pole. Sir Wm. Thomson calls the north-seeking pole the south pole, and the other the north pole, because the former is similar to the south, and the latter to the north pole of the earth. In like manner most French writers call the north-seeking pole of a needle the austral, and the other the boreal pole. Popular usage in this country calls the north-seeking end the north, and the other the south pole, a nomenclature which introduces great confusion whenever we have to reason respecting the earth regarded as a magnet. Faraday, to avoid the ambiguity which has attached itself to the names north and south pole, calls the north-seeking end the marked, and the other MAGNETIC INDUCTION. 617 the unmarked pole. Airy, for a similar reason, employs, in his recent Treatise on Magjietism, the distinctive names red and blue to denote respectively the north-seeking and south-seeking ends, these names, as well as those employed by Faraday, being purely conventional, and founded on the custom of marking the north-seeking end of a magnet with a transverse notch or a spot of red paint. Maxwell and Jenkin, in a report to the British Association, 1 call the south-seeking pole of a needle positive, and the north-seeking pole negative. 493. Magnetic Induction. When a piece of iron is in contact with Fig. 417. Induced Magnetism. a magnet, or even when a magnet is simply brought near it, it becomes itself, for the time, a magnet, with two poles and a neutral portion between them. If we scatter filings over the iron, they will adhere to its ends, as shown in Fig. 417. If we take away the influencing maornet, the filings will fall off, and the iron will retain either no O ' o traces at all or only very faint ones of its magnetization. If we apply similar treatment to a piece of steel, we obtain a result similar in some respects, but with very important differences in degree. The steel, while under the influence of the magnet, exhibits much weaker effects than the iron ; it is much more difficult to magnetize than iron, and does not admit of being so powerfully magnetized ; but, on the other hand, it retains its magnetization after the influencing magnet has been withdrawn. This property of retaining magnetism when once imparted has been (somewhat awkwardly) named coercive force. Steel, especially when very hard, possesses great coercive force; iron, especially when very pure and soft, scarcely any. In magnetization by influence, which is also called magnetic induction, it will be found, on examination, that the pole which is next the inducing pole is of contrary name to it ; and it is on account of the mutual attraction of dissimilar poles that the iron is attracted 1 Report of Electrical Standards Committee, Appendix C. 1863. G18 GENERAL STATEMENT OF FACTS AND LAWS. by the magnet. The iron can, in its turn, support a second piece of iron ; this again can support a third, and so on through many steps. A magnetic chain can thus be formed, each of the com- ponent pieces having two poles. An action of this kind takes place in the clus- ters of filings which attach themselves to one end of a magnetized bar, these clus- ters being composed of nu- merous chains of filings. In comparing the pheno- mena of magnetic induction with those of electrical in- duction, we find both points of resemblance and points of difference. In the case of electricity, if the influenc- ing and influenced body are allowed to come in contact, the former loses some of its own charge to the latter. In the case of magnetism there is no such loss, a magnet after touching soft iron is found to be as strongly magnet- ized as it was before. 494. Effect of Rupture on a Magnet. If a magnet is broken into any number of pieces, every piece will be a complete magnet with Fig. 418. Magnetic Chain. Fig. 419. Broken Masmet. poles of its own. In the case of an ordinary bar-magnet or needle, the similar poles of the pieces will all be turned the same way, as in Fig. 419, which represents a magnet AB broken into four pieces. The ends a, a, a, a are of one name, and the ends b, b, b, b of the opposite name. 494 A. Imaginary Magnetic Fluids: Magnetic Potential. All mutual forces between magnets can be reduced to attractions and repulsions between different portions of two imaginary fluids, 1 one of which 1 Poisson, following Coulomb, spoke of two magnetic fluids, and laid down a theory of MAGNETIC POTENTIAL. C19 may be called positive and the other negative. Neither fluid can exist apart from the other ; every magnet possesses equal quantities of both ; quantity being measured by force of attraction or repulsion at given distance, just as in the case of electricity, like portions repelling, and unlike portions attracting each other inversely as the square of the distance. Equal quantities of the two fluids, when coexisting at the same place, produce no resultant effect, and may be regarded as destroying each other. With reference to these imaginary fluids, magnetic potential can be defined in the same way as electrical potential, and magnetic lines of force possess the same properties as electrical lines of force ( 445 A 445 K). The direction of magnetic force at a point can either be defined as the direction in which a pole of a magnet would be urged if brought to the point, or as the direction in which a small magnet- ized needle, if brought to the point and balanced at its centre of gravity, would place its line of poles; and lines of magnetic force are lines to which this direction is everywhere tangential It is impor- tant to remark that a linear piece of soft iron, though it sets its length along a line of force, does not travel along a line of force, but deviates towards the concave side. This is easily shown by tapping the card represented in Fig. 412. It will be found that filings placed on the line m m move along that line, and therefore at right angles to the lines of force. The force which is specified by magnetic "lines of force" is the force which one pole of a permanent magnet would experience; and it is the same in intensity, but opposite in direction, for dissimilar poles. 494s. Specification of Magnetization. A piece of steel is said to be uniformly magnetized, if equal and similar portions, cut in parallel directions from all parts of it, are precisely alike in their magnetic properties. If a piece of magnetized steel be suspended at its centre of gravity, so as to be free to turn all ways about it, the effect of the earth's magnetism upon it consists in a tendency for a particular line, through this centre of gravity, to take a determinate direction, which is the direction of terrestrial magnetic force. When the line is placed in any other position, the couple tending to bring it back is propor- their action. Sir W. Thomson, avoiding the hypothetical parts of Poisson's theory, speaks of imaginary marjnetic matter of two dissimilar kinds. We have retained the more familiar name fluid, simply because it is more convenient to speak of two fluids than of two kinds of matter. It is to be noted that we cannot speak of two magnetisms, the name magnetism having been already appropriated in a different sense. 620 GENERAL STATEMENT OF FACTS AND LAWS. tional to the sine of the angle between the two positions, and is the same for all directions of deviation. The line which possesses this property is the magnetic axis of the body, and the name is sometimes given to all lines parallel to it. If the piece of steel be uniformly magnetized, this axis is the direction of magnetization; or the direc- tion of magnetization is the common direction of all those lines which tend to place themselves along lines of force in a field 1 where the lines of force are parallel. 494 c. Ideal Simple Magnet: Thin Bar, uniformly and longitudinally Magnetized. The mutual actions of magnets admit of very accurate expression when the magnets are very thin in comparison with their length, uniform in section, and uniformly magnetized in the direc- tion of their length. Such bars, which may be called simple magnets, behave as if their forces resided solely in their ends, which may there- fore in the strictest sense be called their poles. The two poles of any one such bar are equal in strength ; that is to say, one of them attracts a pole of another simple magnet with the same force with which the other repels it at the same distance. In the language of the two- fluid theory, the two fluids destroy one another except at the two ends, and the quantities which reside at the ends are equal but of opposite sign. The same number which denotes the quantity of fluid at either pole, denotes the strength of the pole, or, as it is often called, the strength of the magnet. Its definition is best expressed by saying that the force between a pole of one simple magnet and a pole of another, is the product of their strengths divided by the square of the distance between them. 2 The force which a pole of a simple magnet experiences in a mag- netic field, is the product of the strength of the pole and the intensity of the jield. This rule applies to the force which a pole experiences from the earth's magnetism, the intensity of the field being in this case the intensity of terrestrial magnetic force ; and, from the uni- formity of the field, the forces on the two poles are in this case equal, constituting a couple, whose arm is the line joining the poles multi- 1 A field of force is any region of space traversed by lines of force ; or, in other words, any region pervaded by force of attraction or repulsion. A magnetic field is any region pervaded by magnetic force. All space in the neighbourhood of the earth is a magnetic field, and within moderate distances the lines of force in it may be regarded as parallel, unless artificial magnets or pieces of iron are present to produce disturbance. * We here, and throughout the remainder of this chapter, ignore the existence of induc- tion, which, however, is not altogether absent even in the hardest steel. The effect of induction is always to favour attraction. The attractions will therefore be somewhat stronger, and the repulsions somewhat weaker, than our theory supposes. MOMENT OF MAGNET. 621 plied by the sine of the angle which this line makes with the li of force. The product of the line joining the two poles by the strength of either pole is called the moment of the magnet, and it is evident, from what has just been said, that the continued product of the moment of the magnet, the intensity of terrestrial magnetic force, xnd the sine of the angle between the length of the magnet and the lines of force, is equal to the moment of the couple which the earth's magnetism exerts upon the magnet. 494 D. Compound Magnet of Uniform Magnetization. Any magnet which is not a simple magnet in the sense defined in 494 c may be called a compound magnet. It is convenient to define the moment of a compound magnet by the condition stated in the concluding words of that section, so that the moments of different magnets, whether simple or compound, may be compared by comparing the couples exerted on them by terrestrial magnetism when their axes are equally inclined to the lines of force. If a number of simple magnets of equal strength be joined end to end, with their similar poles pointing the same way, there will be mutual destruction of the two imaginary fluids at every junction, and the system will constitute one simple magnet of the same strength as any one of its components ; but its moment will evidently be the sum of their moments. If any number of simple magnets be united, either end to end or side to side, provided only that they are parallel, and have their similar poles turned the same way, the resultant couple exerted upon the whole system by terrestrial magnetism will ( 14) be the sum of the separate couples exerted on each simple magnet, and the moment of the system will be the sum of the moments of its parts. But any piece of uniformly magnetized material may be regarded as being thus built up, and hence, if different portions be cut from the same uniformly magnetized mass, their moments will be simply propor- tional to their volumes. The quotient of moment by volume, for any uniformly magnetized mass, is called intensity of magnetization. 494 E. Actual Magnets. The definitions and laws of simple magnets are approximately applicable to actual magnets, when magnetized in the usual manner. If an actual bar-magnet in the form of a rectangular parallelepiped were magnetized with perfect uniformity, and in the direction of its length, it might be regarded as made up of a number of simple 622 GENERAL STATEMENT OF FACTS AND LAWS. magnets laid side by side, and its behaviour would be represented by supposing a complete absence of magnetic fluid from all parts of it except its ends (in the strict mathematical sense). One of these ter- minal faces would be covered with positive, and the other with negative fluid, and if the magnet were broken across at any part of its length, the quantities of positive and negative fluid on the broken ends would be the same as on the ends of the complete magnet. The observed fact that magnets behave as if the fluids were distributed through a portion of their substance in the neighbourhood of the ends, and not confined to the ends strictly so called, indicates a falling off in magnetization towards the extremities, and is approximately repre- sented by conceiving of a number of short magnets laid end to end, and falling off in strength towards the two extremities of the series. 1 The resultant force due to the imaginary magnetic fluids which are distributed through the terminal portions of an actual bar-magnet is, in the case of actions at a great distance, sensibly the same as if the two portions of fluid were collected at their respective centres of gravity. These two centres of gravity are the poles of the magnet for all actions between the magnet and other magnets at a great distance, and more especially between the magnet and the earth. The' moment of any magnet, however irregular in its magnetiza- tion, may be defined by reference to the expression given in 494 c for the couple exerted on the body by terrestrial magnetism. This couple is M I sin a, where I denotes the intensity of terrestrial mag- netic force, a the inclination of the magnetic axis of the body to the lines of the earth's magnetic force, and M the moment which we are defining. 1 Thus the last magnet at the positive end being weaker than its neighbour, its negative pole will be weaker than its neighbour's positive pole, so that there will be an excess of positive fluid at this junction. Similar reasoning applies to all the junctions near the ends. There will be an excess of positive fluid at all junctions near the positive end, and an excess of negative at all junctions near the negative end. CHAPTER XLIV. EXPERIMENTAL DETAILS. 495. The Earth's Force simply Directive. The forces which pro- duce the orientation of a magnet depend upon causes of which very little is known. They are evidently connected in some way with the earth, and are accordingly referred to TERRESTRIAL MAGNETISM. We have already stated ( 494 B) that the combined effect of the forces exerted by terrestrial magnetism upon a magnetized needle is equi- valent to a couple tending to turn the needle into a particular direc- tion, and ( 494 E) that in the case of needles magnetized in the ordinary way, there are two definite points or poles (near the two ends of the needle) which may be regarded as the points of applica- tion of the two equal forces which constitute the couple. The fact that terrestrial magnetic force simply tends to turn the needle, and not to give it a movement of translation, in other words, that the resultant force, (as distinguished from couple} is zero, is com- pletely proved by the two following experiments: (1) If a bar of steel is weighed before and after magnetization, no change is found in its weight. This proves that the vertical com- ponent is zero. (2) If a bar of steel, not magnetized, is suspended by a long and fine thread, the direction of the thread is of course vertical. If the bar is then magnetized, the direction of the thread still remains ver- tical. The most rigorous tests fail to show any change of its position. This proves that the horizontal component is zero, a conclusion which may be verified by floating a magnet on water by means of a cork. It will be found that there is no tendency to move across the water in any particular direction. 496. Horizontal, Vertical, and Total Intensities. If S denote the strength of a magnet, and I the intensity of terrestrial magnetic force, 624 EXPERIMENTAL DETAILS. each pole of the rnagnet experiences a force SI, and if L denote the distance between the poles (often called the length of the magnet), the distance between the lines of action of these two parallel and opposite forces may have any value intermediate between L and zero, according to the position in which the needle is held. It will be zero when the line of poles is that of the dipping-needle; it will be L when the line of poles is perpendicular to the dipping-needle; and will be L sin o when the line of poles is inclined at any angle a to the dipping-needle. The force S I upon either pole of the magnet acts in the direction of the dipping-needle ; in other words, in the direction of the lines of force due to terrestrial magnetism. Let 3 denote the dip, that is the inclination of the lines of force to the horizon, then the force S I can be resolved into SI cos 5 horizontal, and SI sin ef vertical. Hence the horizontal and vertical intensities H and V are connected with the total intensity and dip I and S by the two equations H = I cos 5 , V = I sin 5 (1) which are equivalent to the following two g = tan5 , V+H 2 =P. (2) 497. Torsion-balance. Coulomb, in investigating the laws of the mutual action of magnets, employed a torsion-balance scarcely dif- fering from that which he used in his electrical researches. The suspending thread carried, at its lower end, a stirrup on which a magnetized bar was laid horizontally. The torsion- head was so adjusted that one end of the magnet was opposite the zero of the divisions on the glass case when the supporting thread was without torsion. In order to effect this adjustment, the magnet was first suspended by a thread whose torsional power was inconsiderable, so that the magnet placed itself in the magnetic meridian. The case was then turned till its zero came to this position. The torsionless thread was then replaced by a fine metallic wire, and the magnet was replaced by a copper bar of the same weight. The head was then turned till this bar came into the magnetic meridian, and lastly the magnet was put in the place of the bar. Fig. 420 shows the arrangement adopted for observing the repul- sion or attraction between one pole of the suspended magnet and one pole of another magnet placed vertically. Before the insertion of the latter, the suspended magnet was acted on by no horizontal TORSION- BALANCE. G25 forces except the horizontal component of terrestrial magnetism and the torsion of the wire. It was then found that the torsion requisite for keeping the magnet in any position was proportional to the sine of the displacement from the meridian. This result is evidently in accordance with the principles stated Fig. 420. Torsion-balance. above, for the two equal horizontal forces on the two poles being constant for all positions, the couple which they compose is propor- tional to the distance between their lines of action, and this distance is evidently L sin 0, L denoting the constant distance between the poles, and the deviation of the needle from the meridian. 499. Measurement of Declination. Magnetic declination has been observed with several different forms of apparatus. At sea, the most common method of determining it has consisted in observing the magnetic bearing of the rising or setting sun, and comparing this with its true bearing as calculated by a well-known astronomical method. For more accurate determination on land, the declination compass or declination theodolite 1 (Fig. 422) has been frequently employed. 1 A theodolite consists of a telescope mounted so as to have independent motions in 41 626 EXPERIMENTAL DETAILS. When the instrument is set by the help of astronomical observations, so that the vertical plane in which the telescope LL' (or more accurately its line of col- limation) moves, coincides with the geographical meridian, the ends of the needle indicate the declination on the graduated circle over which they move. This circle in fact turns with the telescope, the line of and 180 ns being always in the same vertical plane with the line of collimation of the tele- scope. The external divided circle P Q is used for setting the instrument in the meridian. Fig. 422. Declination Theodolite. At fixed observatories more accurate methods of observation are employed. Fig. 422 A shows the arrangement adopted at Green- wich. A bar-magnet B carries at one end a cross of fine threads C, and Fig. 422 A. Declination Magnet. azimuth and altitude, the amounts of these motions being indicated by divided circles or arcs of circles. It does not differ essentially from the larger instrument called th altazimuth. MEASUREMENT OF DECLINATION. 627 at the other a lens D, the distance between them being equal to the focal length of the lens, thus forming a kind of inverted telescope, whose line of collimation is the line joining the cross to the optical centre of the lens. The bar is suspended by means of a stirrup from a torsionless thread, and sets its magnetic axis in the magnetic meridian. The telescope E, with theodolite mounting, is stationed opposite the end which carries the lens, and is so adjusted at each observation that its line of collimation is parallel to that of the inverted telescope carried by the magnet, an adjustment which is identified by seeing the cross C coincident with a similar cross fixed in the interior of the telescope E. When the observation has been made with the magnet in one position, it must be repeated with the magnet turned upside down as shown in the figure. Error of parallelism between the magnetic axis of the bar and the line of collimation of the inverted O telescope which it carries, will affect these two observations to the same extent in opposite directions, and will therefore disappear from their mean. The readings are taken on a horizontal circle corre- sponding to the outer circle in Fig. 422, and astronomical observa- tions must be made once for all to determine what reading corre- sponds to the geographical meridian. Another very accurate method consists in rigidly attaching to the bar, instead of the lens an4 cross, a small vertical mirror. This can either be viewed through a telescope, so as to show the reflection of a horizontal scale of equal parts, which will appear to travel across the field of view of the telescope as the magnet turns, or it can be employed to throw the image of a spot of light either upon a screen viewed by the observer, or still better upon photographic paper drawn by clock- work, which leaves a permanent record of continuous changes. Both these methods of employing mirrors for the observation of small movements of rotation are now extensively employed in many appli- cations. They appear to have been first introduced by Gauss, who employed them for the purpose which we are now considering. 500. Measurement of Dip. The dip-circle or inclination compass is represented in Fig. 423. It consists essentially of a magnetized needle, very accurately and delicately mounted on a horizontal axis through its centre of gravity, in the centre of a vertical circle on which the positions of the two ends of the needle can be read off. This circle can be turned with the needle into any azimuth, the amount of rotation being indicated by a horizontal circle. It is obvious that, if the vertical circle is placed in the plane of the mag- 628 EXPERIMENTAL DETAILS. netic meridian, the needle, being free to move in this plane, will directly indicate the dip. On the other hand, if the vertical circle is placed in a plane perpendicular to the magnetic meridian, the horizontal component of terrestrial magnetism is prevented from moving the needle, which, accordingly, obeys the vertical component only, and takes a vertical position. In intermediate positions of the vertical circle, the needle will assume positions intermediate between the vertical and the true angle of dip. In fact, if 6 be the angle which the plane of the vertical circle makes with the magnetic meridian, the component H sin of terrestrial magnetism, being per- pendicular to this plane, merely tends to produce pressure against the supports, and the horizontal component influencing the position of the needle is only H cos 0, which lies in the plane of the circle. As none of the vertical force is de- stroyed, the tangent of the appar- The Fig. 42 ent dip will be ^ a=- a- H cos cos most accurate method of setting the vertical circle in the magnetic meridian consists in first adjust- ing it so that the needle takes a vertical position, and then turning it through 90. The instrument having thus been set, and a reading taken at each < ~J O end of the needle, it should be turned in azimuth through 180, and another pair of readings taken. By employing the mean of these two pairs of readings, several sources of error are eliminated, includ- ing non-coincidence of the axis of magnetization with the line joining the ends of the needle. One important source of error deviation of the centre of gravity from the axis of suspension in a direction parallel to the length of the needle, is, however, not thus corrected. It can only be eliminated by remagnetizing the needle in the reverse direction so as to interchange its poles. The mean of the results obtained before and after the reversal of its magnetization will be the true dip. A better form of instrument, known as the Kew dip-circle, is now OBSERVATIONS OF MAGNETIC FORCE. 629 employed. Its essential parts are represented in Fig. 423 A. There is no metal near the needle, and the readings are taken on a circle round which two telescopes travel. In each observation the telescopes are directed to the two ends of the needle. 500 A. Measurement of In- tensity of Terrestrial Mag- netic Force. The complete specification of the earth's magnetic force at any place involves three independent elements. For example, if declination, dip, and hori- zontal force are determined by observation, vertical force and total force can be cal- culated by the formulae of 4-96. Observations of magne- tic force are made either by $ Fig. 423 A. Kew Dip-circle. counting the number of vi- brations executed in a given time, or by statical mea- surements. If a magnet executes small horizontal vibrations under the influence of the earth's magnetism, the square of the number of vibrations in a given time is proportional to , H denoting the horizontal intensity, M the moment of the magnet, and /u its moment of inertia about the centre of suspension. Hence it is easy to observe the variations of horizontal intensity which occur from time to time, if we can insure that our magnet itself shall undergo no change, or if we have the means of correcting for such changes as it undergoes. To obtain absolute determinations of horizontal intensity, the fol- lowing method is employed. First, observe the time of vibration of a freely-suspended horizon- tal magnet under the influence of the earth alone, this will give the product of the earth's horizontal intensity and the moment of the magnet. Secondly, employ this same magnet to act upon another also freely 630 EXPERIMENTAL DETAILS. suspended, and thus compare its influence with that of the earth, this will give the ratio of the same two quantities whose product was found before. Hence the two quantities themselves can easily be computed. 500 B. Bifilar and Balance Magnetometers. The changes of horizontal intensity are measured statically by means of the bifilar magnetometer. This consists of a bar-magnet (Fig. 423 B) suspended by two threads, which would be parallel if the bar were unmagne- tized, but matters are so arranged that, un- der the combined action of the pull of the threads, the weight of the bar, and the earth's magnetism, the bar is kept in a posi- tion nearly perpendicular to the magnetic meridian. The only changes which occur in its position from time to time are those due to changes in the intensity of the earth's Tf Fig. 423 u.-Bifijar Magnetometer, horizontal force, changes in the direction of this force, to the extent of a few minutes of angle, having no sensible effect, on account of the near approach to perpendicularity. The changes of vertical intensity are measured by the balance- magnetometer, which consists of a bar-magnet placed in the magnetic meridian and suspended on knife-edges like the beam of an ordinary balance. Its deviations from horizontality are measures of the changes of vertical intensity. Both these instruments have mirrors attached to the magnet, which produce a photographic record of the movements of the magnet, on principles above explained. The moment of a magnet varies with temperature, being dimin- ished by something like one ten-thousandth part of itself for each degree Fahr. of increase, and increasing again at the same rate when the temperature falls. Hence magnetic observatories must be kept at a nearly uniform temperature. They must also be completely free from iron. No iron nails are allowed to be used in their construc- tion, copper being employed instead. 500 c. Results of Observation. The annexed figures 1 contain an 1 For Figs. 422 A, 423 A, B, c, D, we are indebted to the publishers of Airy's Treatise on Magnetism. Fig. 423 c. Northern Hemisphere. Fig. 423D. Southern Hemisphere. MAGNETIC MERIDIANS AND LINES OF EQUAL Dip. 632 EXPERIMENTAL DETAILS. approximate representation of the magnetic meridians and lines of equal dip over both hemispheres of the earth. These two systems of lines combined, furnish a complete specification of the direction of magnetic force at all parts of the earth's surface ; but they indicate nothing as to intensity. The curves of equal total intensity have a general resemblance to the lines of equal dip, the intensity being greatest near the poles, and least near the equator ; but their arrange- ment is somewhat more complicated, there being two north poles of greatest intensity, one in Canada, and the other in the northern part of Siberia. Speaking roughly, the intensity near the poles is about double of the intensity near the equator. Curves of equal total intensity are often called isodynamic lines ; curves of equal dip are often called isoclinic lines ; curves of equal declination are often called iogonic lines; curves cutting the magnetic meridians at right angles are often called magnetic parallels. They are the lines which would be traced by continually travelling in the direction of magnetic east or west. 500 D. The Earth as a Magnet. 1 The intensity and direction of terres- trial magnetic force at different places may be roughly represented by sup- posing that there is a magnet TT IT' (Fig. 421) at the earth's centre, having a length very small in comparison with the earth's radius, and making an angle of about 20 with the earth's axis of rotation. The points A and B obtained by producing this magnet longitudin- ally to meet the surface, would be the magnetic poles, and at any other place the magnetic meridian would be the vertical plane containing the magnetic axis AB. At places situated on the great circle whose plane contains both the axis of rotation and the magnetic axis, the magnetic meridian would coincide with the geographical meridian, and the declination would be zero. At any other place M, the two meridians would cut each other at an angle which would be the angle of declination. At all places on the great circle e E' whose plane is perpendicular to the magnetic axis, a needle 1 This section corresponds to 498 in the original. The hypothesis which it describes is known as Blot's hypothesis. See note 2, p. 784. Fig. 421. Biot's Hypothesis. TERRESTRIAL MAGNETIC FORCE. 633 suspended at its centre of gravity would place itself parallel to this axis, and consequently the dip would be zero. This circle would be the magnetic equator. 1 It would cut the geographical equator at an angle of 20. Proceeding from the magnetic equator towards the north magnetic pole B, the needle would dip more and more, until at B it became vertical A declination needle at B would remain indifferently in all positions. Similar phenomena would be observed at the other magnetic pole A. The end of the needle which would dip at B, and which at other parts of the earth would point to magnetic north, is that which is similar to the southern pole *' of the terrestrial magnet * v, and the pole which is similar to * would dip at A The actual phenomena of terrestrial magnetism are not in very close agreement with the results which would follow from the pre- sence of such a magnet as we have described in the earth's interior, nor do they agree well with the hypothesis of two interior magnets inclined at an angle to each other, which has also been proposed. It would rather appear that the earth's magnetism is distributed in a manner not reducible to any simple expression. 501. Changes of Declination and Dip. Declination and dip vary greatly not only from place to place, but also from time to time. Thus at the date of the earliest recorded observations at Paris, 1580, the declination was about 11 30' E. In 1663 the needle pointed due north and south, so that Paris was on the line of no declination. Since that time the declination has been west, increasing to a maximum of 22 34', which it attained in 1814. Since then it has gone on diminishing to the present time, its present value being about 19 W. As to dip, its amount at Paris has continued to diminish ever since it was first observed in 1671. From 75 it has fallen to 66, its pre- sent value. As its variations since 1863 have been scarcely sensible, it would seem to have now attained a minimum, to be followed by a gradual increase. 501 A. Magnetic Storms. Besides the gradual changes which occur in terrestrial magnetism, both as regards direction and intensity of force, in the course of long periods of time, there are minute fluctua- 1 If latitude reckoned from the magnetic equator be called magnetic latitude, and de- noted by X, it can be shown that we should have, on this theory, tan 8 = 2 tan X ; I = E *J cos" * + 4sin a X, E denoting the intensity at the magnetic equator. 634 EXPERIMENTAL DETAILS. tions continually traceable. To a certain extent these are dependent on the varying position of the sun, and, to a much smaller extent, of the moon, with respect to the place of observation; but over and above all regular and periodic changes, there is a large amount of irregular fluctuation, which occasionally becomes so great as to con- stitute what is called a magnetic storm. Magnetic storms " are not connected with thunder-storms, or any other known disturbance of the atmosphere ; but they are invariably connected with exhibitions of aurora borealis, and with spontaneous galvanic currents in the ordinary telegraph wires ; and this connection is found to be so cer- tain, that, upon remarking the display of one of the three classes of phenomena, we can at once assert that the other two are observable (the aurora borealis sometimes not visible here, but certainly visible in a more northern latitude)." 1 They are sensibly the same at stations many miles apart, for example at Greenwich and Kew, and they affect the direction and amount of horizontal much more than of vertical force. 502. Ship's Compass. In a ship's compass, the box cc which contains the needle is weighted below, and hung on gimbals, which consist of two rings so arranged as to admit of motion about two independent horizontal axes tt, uu at right angles to each other. This arrange- ment prevents it from being tilted by the pitching and rolling of the ship. The needle a 6 is firmly at- tached to the compass-card, which is a circular card marked with the 32 points of the compass, as in Fig. 425, and also usually divided at its circumference into 360 degrees. The card with its attached needle is accurately balanced on a point at its centre. The needle, which, in actual use, is concealed from view, lies along the line NS. The box contains a vertical mark in its interior on the side next the ship's bow; and this mark serves as 1 Airy on Magnetism, p. 204. Fig 424. Ship's Compass. METHODS OF MAGNETIZATION. 635 an index for reading off on the card the direction to which the ship's head is turned. Sometimes a reflector is employed, as m in the first figure, in such a position that an observer looking in from behind can read oft' the indi- cated direction by reflection, and can at the same time sight a distant object whose mag- netic bearing is required. The origin of the compass is very obscure. The ancients were aware that the loadstone at- tracted iron, but were ignor- ant of its directing property. The instrument came into use in Europe some time in the course of the thirteenth cen- tury. 503. Methods of Magnetization. The usual process of magnetizing a bar consists in rubbing it with or against a bar already magnetized. Different methods of doing this, called single touch, double touch, &c., have been devised, in which magnetized bars of steel were the mag- netizing agents. Much greater power can, however, be obtained by means of electro-magnetism ; and the two following methods are now almost exclusively employed by the makers of magnets. 1. A fixed electro-magnet (Fig. 427) is employed, and the bar to Pig. 425. Compass card. Fig. 427. Methods of Magnetization. Fig. 428. be magnetized is drawn in opposite directions over its two poles. Each stroke tends to develop at the end of the bar at which the motion ceases, the opposite magnetism to that of the pole which is 636 EXPERIMENTAL DETAILS. in contact with it. Hence strokes in opposite directions over the two contrary poles tend to magnetize the bar the same way. 2. When very intense magnetization is to be produced, the electro- magnet must be very powerful, and the bar then adheres to it so strongly that the operation above described becomes difficult of exe- cution, besides scratching the bar. Hence it is more convenient to move along the bar, as in Fig. 428, a coil of wire through which a current is passing. This was the method employed by Arago and Ampere. A bar of steel is said to be magnetized to saturation, when its magnetization is as intense as it is able to retain without sensible loss. It is possible, by means of a powerful magnet, to magnetize a bar considerably above saturation; but in this case it rapidly loses intensity. Pieces of iron and steel frequently become magnetized temporarily or permanently by the influence of the earth's magnetism, and this action is the more powerful as the direction of their length more nearly coincides with that of the dipping-needle. If fire-irons which have usually stood in a nearly vertical position be examined by their influence on a needle, they will generally be found to have acquired some permanent magnetism, the lower end being that which seeks the north. It sometimes happens that, either from some peculiarity in the structure of a bar, or from some irregularity in the magnetizing pro- .1 Puin.s. cess, a reversal of the direction of magnetization occurs in some part or parts of the length as compared with the rest. In this case the magnet will have not only a pole at each end, but also a pole at each point where the reversal occurs. These intermediate poles are called consequent points. Fig. 429 represents the arrangement of iron- filings about a bar-magnet which has two consequent points a', b'. METHODS OF MAGNETIZATION. 637 The whole bar may be regarded as consisting of three magnets laid end to end, the ends which are in contact being similar poles. Thus the two poles at a' and the one pole at a are of one kind, while the two poles at b' and the one pole at b are of the opposite kind. The lifting power (or portative force) of a magnet generally increases with its size, but not in simple proportion, small magnets being usually able to sustain a greater mul- tiple of their own weight than large ones. ^ --:' ^ __- Hence it has been found advantageous to dg construct compound magnets, consisting of a number of thin bars laid side by side, with their similar poles all pointing the same way. Fig. 430 represents such a compound magnet composed of twelve ele- mentary bars, arranged 4 x 3. Their ends are in- serted in masses of soft iron, the extremities of which constitute the poles of the system. Fig. 431 represents a compound horse-shoe mag- net, whose poles N and S support a keeper of soft iron, from which is hung a bucket for holding weights. By continually adding fresh weights day after day, the magnet may be made to carry a much greater load than it could have supported originally; but if the keeper is torn away from the magnet, the addi- tional power is instantly lost, and the magnet is only able to sustain its original load. Much attention was at one time given to methods of obtaining steel magnets of great power. These researches have now been superseded by electro-magnetism, which affords the means of obtain- ing temporary magnets of almost any power we please. 503 A. Molecular Changes accompanying Magnetization. Joule has shown that, when a bar of iron is magnetized longitudinally, it Fig. 430. Compound Magnet. Fig. 431. Compound Horse-shoe Magnet. 638 EXPERIMENTAL DETAILS. acquires a slight increase of length, compensated, however, by trans- verse contraction, so that its volume undergoes no change. If the magnetization is effected suddenly, by completing an electric circuit, an ear close to the bar hears a clink, and another clink is heard when the current is stopped. These phenomena have been accounted for by the hypothesis that, when iron is magnetized, its molecules place their longest dimensions in the direction of magnetization. The effect of heat in diminishing the strength of a magnet is another instance of the connection between magnetism and other molecular conditions. In ordinary cases, this diminution is merely transient; but if a steel magnet is raised to a white-heat, it is per- manently demagnetized. 504. Action of Magnetism on all Bodies. It has long been known that iron and steel were not the only substances which could be acted on by magnetism. Nickel and cobalt especially were known to be attracted by a magnet, though very much more feebly than iron, while bismuth and antimony were repelled. Faraday, by means of a powerful electro-magnet, showed that all or nearly all substances in nature, whether solid, liquid, or gaseous, were susceptible of mag- netic influence, and that they could all be arranged in one or the other of two classes, characterized by opposite qualities. This opposi- tion of quality is manifested in two ways. 1. As regards attraction and repulsion, iron and other paramag- netic bodies are attracted by either pole of a magnet, or more gene- rally, they tend to move from places of weaker to places of stronger force. On the other hand, bismuth and other diamagnetic bodies are repelled by either pole of a magnet, and in general tend to move from places of stronger to places of weaker force. 2. As regards orientation, a paramagnetic body when suspended between the poles of a magnet tends to set axially ; that is to say, tends to place its length along the line joining the poles, or more generally, when placed in any magnetic field, tends to place its length along the lines of force. Hence the name paramagnetic. 1 A dia- magnetic body, on the other hand, when suspended between the poles, sets equatorially; that is to say, places its length at right angles to 1 The nomenclature here adopted was proposed by Faraday in 1850 (Researches, 2790), and is eminently worthy of acceptance. Many writers, however, continue to employ magnetic in the exclusive sense of paramagnetic. To be consistent, they should call the other class aw^magnetic, not cfo'amagnetic. "The word magnetic ought to be general, and include all the phenomena and effects produced by the power." ACTION OF MAGNETISM. 639 the line joining the poles, or, more generally, tends to place its length at right angles to magnetic lines of force. Fig. 432 represents the apparatus commonly employed for experi- ments on this subject. B, B are two large coils of stout copper wire, wound on massive hollow cylinders of soft iron. These latter form portions of the heavy frames F, F, which can be slid to or from each other, and fixed firmly at any distance by means of the screws E, E. Fig. 432. Apparatus for Diamagnetism. The armatures P, which can be screwed on or off, have the form of rounded cones, and produce a great concentration of force at their extremities. The action of magnetism upon a solid can be examined by suspend- ing a small bar of it a b, by means of a special support RS, between the poles P. When a current is passed through the coils, the bar immediately exhibits a preference either for the axial or the equa- torial position. Attraction and repulsion are most easily tested by suspending a small ball of the substance at the level of the central line of poles, but a little beside it, the poles having first been brought very near together. On passing the current through the coil, the ball will move inwards towards the line of poles if paramagnetic, and outwards if diamagnetic. It is important, however, to remark, that experiments of this kind, unless performed in vacuo, are merely differential they merely indicate that the suspended body is, in the one case, more para- magnetic or less diamagnetic ; in the other case more diamagnetic 640 EXPERIMENTAL DETAILS. or less paramagnetic, than the medium in which it moves, the com- parison being made between equal volumes. Oxygen is paramag- netic, and nitrogen is nearly or quite indifferent. Air is accordingly paramagnetic, and a body suspended in air appears less paramagnetic or more diamagnetic than it really is. If more feebly paramagnetic than air, it will appear to be diamagnetic. Thus heated air, in consequence probably of its rarefaction, appears diamagnetic when surrounded by cold air, and the flame of a taper is repelled down- wards and outwards from the axial line. If, on the other hand, the body under examination is suspended in water, it will appear more paramagnetic than it really is, by reason of the diamagnetism of water. The following metals are paramagnetic: iron, nickel, cobalt, man- ganese, chromium, titanium, cerium, palladium, platinum, osmium. The following are diamagnetic : bismuth, antimony, lead, tin, mercury, gold, silver, zinc, copper. The following substances are also diamagnetic : water, alcohol, flint glass, phosphorus, sulphur, resin, wax, sugar, starch, wood, ivory, beef (whether fresh or dried), blood (whether fresh or dried), leather, apple, bread. 504 A. Magneto-crystallic Action. The orientation of crystals in a magnetic field presents some remarkable peculiarities, which were extremely perplexing to investigators until Tyndall and Knoblauch discovered the principle on which they depend. This principle is, that crystals are susceptible of magnetic induction to different degrees in different directions. Every crystal (except those belonging to the cubic system) has either one line or one plane along which induction takes place more powerfully than in any other direction ; and it is this line or plane which tends to place itself axially or equatorially according as the crystal is paramagnetic or diamagnetic. The direc- tions of most powerful and least powerful induction are found to be closely related to the optic axes of crystals, and also to their planes of cleavage. When a sphere cut from a crystal is brought near to one pole of a magnet, it is attracted or repelled (according as it is para- or dia-magnetic) with the greatest force when the direction of most powerful induction coincides with the direction of the force. Directions of unequal induction can be produced artificially in non-crystalline substances by applying pressure. " Bismuth is a brittle metal, and can readily be reduced to a fine powder in a mortar. Let a tea-spoonful of the powdered metal be wetted with MAGNETO-CRYSTALLIC ACTION. 64-1 gura-water, kneaded into a paste, and made into a little roll, say an inch long and a quarter of an inch across. Hung between the excited poles, it will set itself like a little bar of bismuth equatorial. . Place the roll, protected by bits of pasteboard, within the jaws of a ' vice, squeeze it flat, and suspend the plate thus formed between the poles. On exciting the magnet, the plate will turn, with the energy of a magnetic substance, into the axial position, though its length may be ten times its breadth. " Pound a piece of carbonate of iron into fine powder, and form it into a roll in the manner described. Hung between the excited poles, it will stand as an ordinary [parajmagnetic substance axial. Squeeze it in the vice, and suspend it edgeways, its position will be immediately reversed. On the development of the magnetic force, the plate thus formed will recoil from the poles, as if violently repelled, and take up the equatorial position." 1 In these experiments the direction of most powerful induction is a line transverse to the thickness, and this is also the direction in which pressure has been applied. Tyndall accordingly concludes that " if the arrangement of the component particles of any body be such as to present different degrees of proximity in different direc- tions, then the line of closest proximity, other circumstances being equal, will be that chosen by the respective forces for the exhibition of their greatest energy. If the mass be [parajmagnetic, this line will stand axial; if diamagnetic, equatorial." 2 1 Tyndall on Diamcupictifm, p. 18. * Ibid. p. 23. 42 CURRENT ELECTRICITY. CHAPTER XLV. GALVANIC BATTERY. 505. Voltaic Electricity. Towards the close of last century, when the discovery of the various phenomena of frictional electricity had been followed by Coulomb's investigations, which first reduced them to an accurate theory, a new instrument was brought to light destined to effect a complete revolution in 'electrical science. In place of an element difficult to manage, capricious and uncertain in its behaviour, and constantly baffling investigation by the rapiditx 1 of its dissipation, the galvanic battery furnished a steady source of electricity, constantly available in all weathers, and requiring no special precautions to prevent its escape. Moreover, the electricity thus developed exhibited an entirely new set of phenomena, and opened up the way to such various and important applications, that frictional electricity at once fell into the second place, and the new agent became the main object of interest with all electrical inves- tigators. 506. Galvanic Element. If two plates, one of zinc and the other of copper (Fig. 433), are immersed in water acidulated by the addi- tion of sulphuric acid, and are not allowed to touch each other within the acid, but are connected outside it, either by direct contact, or by a metallic wire M and binding screws, as in the figure, a continuous current of electricity flows round the circuit thus formed, the direc- tion of the positive current being from copper to zinc in the portion external to the liquid, and from zinc to copper through the liquid. Chemical action at the same time takes place, the zinc being gradually dissolved by the acid, and hydrogen being given out at the copper plate. If, instead of employing two metals and a liquid, we form a circuit with any number of metals alone, no current will be gene- GALVANIC ELEMENT. 643 rated, provided that the whole circuit be kept at one temperature. If, however, some of the junctions be kept hot and others cold, a current will in general be produced. The principles which underlie these phenomena appear to be as follows: (1). When two dissimilar substances touch each other, they have not exactly the same potential at the point of contact. For instance, when zinc is in contact with copper, it is at higher potential than the copper. (2). The difference is in general greater for two metals than for a metal and a non-metal or two non-metals. (3). The difference depends not only on the nature of the two substances, but also on their temperatures. (4). The difference of potentials between two metals is the same when they are in direct contact as when they are connected by one or more intervening metals; all the metals being still supposed to be at the same temperature. (5). When two metals are connected by a conducting liquid which is susceptible of decomposition, their difference of potential is much smaller than when they are in direct contact. Thus, if the connecting wire M (Fig. 433) be of copper, and we break its connection with the copper plate, the difference of po- tential between the two plates will be less than the difference between the zinc plate and the copper wire. The zinc plate is positive with re- spect to the copper wire; hence the copper plate is positive with re- spect to the copper wire. On com- pleting the circuit, positive elec- tricity accordingly flows from the copper plate into the copper wire. As the difference of potentials at the junction of the dissimilar metals is permanent, the current is permanently maintained. Chemical combination at the same time goes oh; and the potential energy of chemical affinity which thus runs down, is the source of the energy of the current. Every electric current may be regarded as a flow of positive elec- tricity in one direction, and of negative electricity in the opposite Fig. 433. Voltaic Element. 644 GALVANIC BATTERY. direction. The direction in which the positive electricity flows is always spoken of as the direction of the current. 508. Galvanic Battery. By connecting the plates of successive elements in the manner represented in Fig. 434, we obtain a battery. The copper of the first cell on the left hand is connected with the Fig. 434. Battery of Four Elements. zinc of the second; the copper of the second with the zinc of the third ; and so on to the end of the series. If two wires of the same metal be connected, one with the first zinc and the other with the last copper, the difference of potential between these wires is independent of the particular metal of which they are composed, and is called the electro-motive force of the battery. Its amount can be measured by means of Thomson's quadrant electrometer ; and in applying this test, it is not necessary that the wires which connect the battery with the electrometer should be of the same metal ; for, whatever metals these wires may be composed of, the quadrants of the electrometer will (by law (4) above) assume the same potentials as if in direct contact with the plates of the battery. The zinc of the first and the copper of the last cell (or wires pro- ceeding from them) are called the electrodes or poles of the battery, the zinc being the negative and the copper the positive electrode. The current flows through the connecting wire from the positive to the negative electrode, and is forced through the battery from the negative to the positive. 509. Galvani's Discoveries. About the year 1 780, Galvani, professor of anatomy at Bologna, had his attention called to the circumstance that some recently skinned frogs, lying on a table near an electrical machine, moved as if alive, on sparks being drawn from the machine. GALVANl'S DISCOVERIES. Fig. 435. Experiment with Frog. Struck with the apparent connection thus manifested between elec- tricity and vital action, he commenced a series of experiments on the effects of electricity upon the animal system. In the course of these experiments, it so happened that, on one occasion, several dead frogs were hung on an iron balcony by means of cop- per hooks which were in con- tact with the lumbar nerves, and the legs of some of them were observed to move con- vulsively. He succeeded in obtaining a repetition of these movements by placing one of the frogs on a plate of iron, and touching the lumbar nerves with one end of a cop- per wire, the other end of which was in contact with the iron plate. Another mode of obtaining the result is represented in Fig. 435, two wires of differ- ent metals being employed which touch each other at one end, while their other ends touch respectively the lumbar nerves and the crural muscles. Every time the contact is completed, the limb is convulsed. Galvani's explanation was, that at the junction of the nerves and muscles there is a separation of the two electricities, the nerve being positively, and the muscle negatively electrified, and that the con- vulsive movements are due to the establishment of communication between these two electricities by means of the connecting metals. Volta, professor of physics at Pavia, disproved this explanation by showing that the movements could be produced by merely connect- ing two parts of a muscle by means of an arc of two metals; and he referred the source of electricity not to the junction of nerve and muscle, but to the junction of the two metals. Acting on this belief, he constructed in the year 1 800 a voltaic pile. 511. Voltaic Pile. This consisted of a series of discs of copper, zinc, and wet cloth, c, z, d, Fig. 436, arranged in uniform order, thus copper, zinc, cloth, copper, zinc, cloth . . . the lowest plate of all being copper and the highest zinc. The wet cloth was intended 646 GALVANIC BATTERY. merely to serve as a conductor, and prevent contact between each zinc and the copper above it. All the contacts between zinc and copper were between a copper below and a zinc above, so that they all tended, according to Volta's theory, to pro- duce a current of electricity in the Fig. 436. Structure of Pile. Fig. 437. Complete Pile. same direction. The effects obtained from the pile were so power- ful as to excite extraordinary interest in the scientific world. Fig. 438. Couronne de Tosses. 513. Couronne de Tasses. He shortly afterwards invented the TROUGH BATTERY. 64-7 couronne de tasses (crown of cups), consisting of a series of cups arranged in a circle, each containing salt water with a plate of silver or copper and a plate of zinc immersed in it, the silver or copper of each cup being connected with the zinc of the next, with the excep- tion of the extreme plates. The last plate in liquid at each end of the series was connected with a plate of the other metal in air. These two plates in air are now known to be useless, and are omitted in the figure 514. Trough Battery. More convenient arrangements, equivalent Fig. 439. Cruickshank'g Trough. to the couronne de tasses, were soon introduced. One of these, devised by Cruickshank, is represented in Fig. 439, consisting of a rectangular box, called a trough, of baked wood, which is a non- conductor of electricity, divided into compartments by partitions each consisting of a plate of zinc and a plate of copper soldered together. Dilute acid is poured into these compartments. 515. Wollaston's Battery. In Wollaston's battery, the plates were suspended from a single horizontal bar, by means of which they could all be let down into the acid, or lifted out of it together. The liquid was con- tained either in compartments rig. 440. Wollaston's ceil of a trough of glazed earthen- ware, with partitions of the same material, or in separate vessels as shown in Fig. 441. The plates were double-coppered; that is to say, 648 GALVANIC BATTERY. they consisted of a zinc plate with a copper plate bent round it on Fig. 441. Wollaston's Battery. both sides (Fig. 440), contact between them being prevented by pieces of wood or cork. 517. Hare's Deflagrator. For some purposes it is more important to diminish the resistance of a cell, or, in other words, to facilitate the conduction of electricity between the zinc and the copper plate, than to increase the elec- tro-motive force by multi- plying cells. The helical arrangement devised by Hare of Philadelphia (Fig. 443) is specially adapted to such purposes. It con- sists of two very large plates of zinc and copper rolled upon a central cylin- der of wood, and prevent- ed from touching each other by pieces of cloth or twine inserted between Fig^HH^e'TDeflagrator. tllem - Ifc is plunged in a tub of acidulated water, as represented in the figure. From the remarkably powerful heat- ing effects which can be obtained by the use of this cell, it is called Hare's deflagrator. DANIELL'S BATTERY. 649 518. Polarization of Plates. All the forms of battery which we have thus far described, are liable to a rapid decrease of power, owing to causes which are partly chemical and partly electrical. The chemical action which takes place in each cell consists primarily in the formation of sulphate of zinc, at the expense of the zinc plate, the sulphuric acid, and the oxygen of the water with which the acid is diluted, the hydrogen of the water being thus liberated. As this action proceeds, the liquid becomes continually less capable of acting powerfully on the zinc. Again, a portion of the zinc which has been dissolved becomes deposited on the copper plate, thus tending to make the two plates alike, and so to destroy the current, which essentially depends on the difference between them. But the most important cause of all is to be found in what is called the polarization of the copper plate; that is to say, in the deposition of a film of hydrogen on the surface of the plate. This film not only interposes resistance by its defect of conductivity, but also brings to bear an electro- motive force in the direction opposed to that of the current. These obstacles to the maintenance of a constant current were first overcome by Daniell. 519. Daniell's Bat- tery. In the cell de- vised by Daniell, there is a porous partition of unglazed earthenware, separating the two li- quids, which are in con- tact one with the zinc, and the other with the copper plate. These two liquids are not precisely alike, that which is in contact with the copper being not simply dilute sulphuric acid like the other, but containing also as much sulphate of copper as it will take up. For the purpose of keeping it saturated, crystals of sul- Fig. 444. Daniell's Cell. 650 GALVANIC BATTERY. phate of copper are suspended in it near its surface by means of a wire basket of copper. The effect of this arrangement is, that the hydrogen is intercepted before it can arrive at the copper plate, and the deposit which takes place on the copper plate is a deposit of copper, the hydrogen taking the place of this copper in the saturated solution. The current given by a battery of these cells remains nearly con- stant for some hours. In the figure, the copper plate C is represented as a cleft cylinder occupying the interior, with the crystals of sulphate of copper piled up round it. The entire cylinder surrounding these is the porous partition, outside of which is the cleft cylinder of zinc Z, the whole being contained in a vessel of glass. It is more usual in this country to dispense with the glass vessel, and interchange the places of the zinc and copper in the figure, the copper plate being a cylindrical vessel of copper containing the saturated solution. In this is immersed the porous vessel containing the other fluid with the zinc plate immersed in it. The cells thus constructed are usu- ally arranged in square compartments in a wooden box. 520. Bunsen's Bat- tery. The battery which is now perhaps most extensively used for class experiments is that which was in- vented by Bunsen in 1813, being substanti- ally identical with one previously invented by Grove, except that carbon is substituted for platinum. The usual construc- tion of its cells is very The cleft cylinder is the zinc plate, Within this is the Fig. 4-15. Bunseu's CelL clearly represented in Fig. 445. which is immersed in dilute sulphuric acid. porous cylinder, similar to Daniell's, containing strong nitric acid, BUNSEN'S BATTERY. 651 in which is immersed a rectangular prism, of a very dense kind of charcoal, obtained from the interior of the retorts at gas-works, being deposited there in the manufacture of gas. Fig. 446. Bunsen's Battery. In this cell the hydrogen is intercepted on its way to the carbon plate by the nitric acid, with which it forms nitrous acid. Grove's battery possesses some advantages over Bunsen's ; but its first cost is much greater. 521. Amalgamated Zinc. When the poles of a battery are insulated from one another, there ought to be no chemical action in the cells. Any action which then goes on is wasteful, and is an indication that unproductive consumption of zinc goes on when the current is pass- ing, in addition to the consumption which is necessary for producing the current. This wasteful action, which is called local action, goes on largely when the zinc plates are of ordinary commercial zinc, but not when they are of perfectly pure zinc. In this respect amal- gamated zinc behaves like pure zinc, and it is accordingly almost universally employed. The amalgamation, which must be often renewed in the case of a battery in constant use, is performed by first cleaning the zinc plates with dilute acid, and then rubbing them with mercury. 522. Dry Pile: Bohnenberger's Electroscope. For telegraphic pur- poses in this country, a battery is very commonly employed in which sand or sawdust, moistened with acidulated water, separates the zinc and copper plates of each cell. The other forms of battery which have been devised are exceed- ingly numerous, and new forms are continually being introduced. A dry pile, built up on the general plan of Volta's moist pile, was 652 GALVANIC BATTERY. devised by De Luc, and improved by Zamboni. In Zamboni's con- struction, sheets of paper are prepared by pasting finely laminated zinc or tin on one side, and rubbing black oxide of manganese on the other. Discs are punched out of this paper, and piled up into a column, with their similar sides all facing the same way, to the number of a thousand or upwards, and are well pressed together. The difference of potential between the two ends is sufficient to produce sensible divergence of the gold-leaves of an electroscope, but the quantity of electricity which can be developed in a given time is exceedingly small. No pile or battery can generate a sensible cur- rent, except by a sensible consumption of its materials in the shape of chemical action. A very delicate gold-leaf electroscope was devised by Bohnen- berger, consisting of a single leaf suspended between the two polos of a dry pile, which for this purpose is arranged in two columns connected below, so that the poles are at the summits. If their lower ends, which form the middle of the series, be connected witli the earth, one pole will always have positive, and the other negative potential. A very slight charge, positive or negative, given to the gold-leaf by means of the knob at the top of the case, suffices to make it move to the negative or the positive pole. 523. Thermo-electric Currents. Electric currents can be produced by applying heat or cold to one of the junctions in a circuit composed of two different metals. This was first shown by Seebeck of Berlin in 1821. It may be illus- trated by employing a rectan- gular frame (Fig. 448), having three sides formed of a copper plate, and the fourth of a cylin- der of bismuth. It must be .^^^^=^^^==^=^^^=^^^^^ placed in the magnetic meridian, with a magnetized needle in its Fig. 448. -Thermo-electric Current. interior. On heating one of the junctions with a spirit-lamp, the needle will be deflected in such a direction as to indicate the existence of a current, which, in the copper portion of the circuit flows from the hot to the cold junction, and in the bismuth portion from the cold to the hot. If cold instead of heat be applied to one junction, the direction of the current will still be from the warmer junction THERMO-ELECTRIC ORDER. 653 Fig. 449. Current with one Metal. through the copper to the colder junction, and from this through the bismuth to the warmer junction. Antimony, if employed instead of copper, gives a still more powerful effect. 524. Though a circuit composed of bismuth and antimony is specially susceptible of thermo-electric excitation, the property is possessed, in a more or less marked degree, by every circuit composed of two metals, and even by cir- cuits composed of the same metal in different states. If, for ex- ample, a knot or a helix (as in Fig. 449), be formed in a piece of platinum wire, and heat applied at one side of it, a current will be indicated by a delicate galvanometer. In metals which are usually heterogeneous in their structure, such as bismuth, it is not uncom- mon to find currents produced by heating parts which appear quite uniform. If the ends of two copper wires be bent into hooks, and one of them be heated, on placing them in contact, a current will be produced due to the presence of a thin film of oxide on the heated wire. With two platinum wires, no such effect is obtained. 525. Thermo-electric Order. According to Becquerel's experiments, the metals may be ranged in the following order, as regards the direction of the current produced by heating a junction of any two of them: Bismuth, platinum, lead, tin, copper, silver, zinc, iron, antimony ; that is to say, if a junction of any two of these metals be heated, the direction of the current at the junction in question will be from that which stands first in the list to the other. His ' experiments have also established the important fact that the cur- rent obtained by heating all the junctions B, C, D, E, F, of a chain of dissimilar metals to one com- mon temperature, is the same as that obtained by uniting the two extreme bars AB, FG, directly to each other, and heating their junction to the same temperature. 1 526. Comparison of Electro-motive Forces. By employing a chain 1 The more accurate statement is, that the electro- motive force is the same in the two cases. The current will be sensibly the same if the resistance in BCDEF is insignificant in comparison with the rest of the circuit. In order that there may be a current, the 654 GALVANIC BATTERY. composed of wires of different metals soldered together, with its two extremities connected with a galvanometer, and heating one junction to 20 C., while the rest were kept at C., Becquerel obtained cur- rents proportional to the following numbers: Junction heated. Current. Iron -silver, 26'20 Iron copper, .... 27'96 Iron -tin, 31 '24 Iron platinum, . . . . 36'07 Junction heated. Current. Copper - platinum, . . . 8'55 Copper -tin, 3'50 Silver copper, .... 2 '00 Zinc -copper, . . . . TOO On comparing these numbers, it will be found that they are in approximate agreement with the law above stated. Thus the electro- motive force of a silver-platinum circuit comes out 10'55 by adding 2-00 to 8-55, and 9'87 by subtracting 26'20 from 36 07. The electro- motive force of copper-platinum is 8'55 as observed directly, and 8'11 as computed by taking the difference of iron-copper and iron- platinum. The deviation from precise agreement is not more than may fairly be ascribed to errors of observation. 527. Neutral Point. For every two metals there is a particular temperature called their neutral point, such that a circuit composed of these metals will give no current when one junction is just as much above the neutral point as the other is below it. This defini- tion holds in every case when the difference of temperature between the junctions is small, and it generally holds as far as differences of some hundreds of degrees. If one junction is kept at a constant temperature lower than the neutral point, while the other, initially at the same temperature, is steadily raised, the current first increases to a maximum, which it attains when the neutral point is reached, then decreases to zero (according to the above definition), and then becomes reversed, with continually increasing strength. These changes can be shown with copper and iron wire. Let a piece of iron wire be joined at both ends to copper wires, and let the copper wires be led to a delicate galvanometer. By gently heating one of the two junctions a current will be produced which will Deflect the needle in one direction; but if the heating is continued to redness, the needle comes back, and is still more strongly deflected in the opposite direction. circuit must of course be completed, and not left open as in Fig. 451. In the case of an open circuit, the result of the heating will simply be to produce difference of potential between the extremities, A, G. This difference of potential is the measure of the electro- motive force, and will accordingly be the same in the two cases. THERMO-ELECTRIC PILE. 655 528. Thermo-electric Pile. If a thermo-electric chain be composed of two metals occurring alternately (as in Fig. 452), no effect will be Fig. 452. Pouillet's Thermo-pile. obtained by equally heating tivo consecutive junctions; for the current which would be generated by heating the one is in the opposite direction to that due to the heating of the other. If we number the junctions in order, we shall obtain a current in one direction by heating any junction which bears an odd number, and in the opposite direction by heating any one that bears an even number. The thermo- electric pile, or thermo-pile, whose use has been already described in connection with experiments on radiant heat ( 313), is an arrange- ment of this kind, in which all the odd junctions are presented together at one end, and all the even junctions at the other, the two metals composing the pile being antimony and bismuth. The electro- motive force obtained with a given difference of temperature between the ends of the pile is proportional to the number of junctions, except in so far as accidental differences may exist between different junc- tions. 529. Application to Measurement of Temperature. Thermo-electric currents may be employed either in testing equality of temperatures, or in comparing small differences of temperature. As an example of the former application, suppose a circuit to be formed of two long wires, one of iron and the other of copper, connected at both ends, and covered with gutta-percha or some other insulator except at the two junctions. Let one junction be lowered to the bottom of a boring, or any other inaccessible place whose temperature we wish to ascertain, and let the other junction be immersed in a vessel of water containing a thermometer. If one of the wires be carried round a galvanometer, the direction in which the needle is deflected will indicate whether the upper or lower junction is the warmer, and if we alter the temperature of the water in the vessel till the deflec- tion is reduced to zero, we know that the two junctions are at the same temperature, which we can read off by the thermometer immersed- in the water. CHAPTER XLVI. GALVANOMETER. 530. (Ersted's Experiment. The discovery by the Danish philo- sopher (Ersted, in 1819, that a magnetized needle could be deflected by an electric current, was justly regarded with intense interest by the scientific world, as affording the first indication of a definite rela- tion existing between magnetism and electricity. CErsted's experiment can be repeated by means of the apparatus represented in Fig. 456. Two insulated metallic wires are placed in the magnetic meridian, one of them above, and the other below a magnetized needle. If a current be sent through one of these wires, the needle will be deflected; and if the current be strong, the deflection will nearly amount to a right angle. The direction of the deflection will be re- versed if the current be passed through the lower instead of the upper wire. It will also be reversed by reversing the direction of the current. In the figure, the current is supposed to be passing above the needle from south to north. In this case the north end of the needle moves to the west, and the south end to the east. On making the current pass in various directions, either horizontally, vertically, or obliquely, near one pole of the needle, it will be found that deviation is always produced except when the plane containing the pole and current is perpendicular to the length of the needle. Fig. 456. CErsted's Experiment. AMPERES RULE. 657 531. Ampere's Rule. The direction in which either pole of a needle is deflected by a current, whatever their relative positions may be, is given by the following rule, which was first laid down by Ampere. Fig 457. Ampere's Law. Fig. 458. Imagine an observer to be so placed that the current passes through him, entering at his feet and leaving at his head, then the deflection of a north-seeking pole will be to the left as seen by him. The deflec- tion of a south-seeking pole will be in the opposite direction. The two figures 457, 458 illustrate the application of this rule to the two cases just considered. The current is supposed, in both cases, to be flowing from south to north. A is the austral or north-seeking pole of the needle, and B the boreal or south-seeking pole. 531 A. Lines of Magnetic Force due to Current. The relation between currents and magnetic forces may be more precisely expressed by saying that a cur- rent flowing through a straight wire pro- duces circular lines of force, having the wire for their common axis. A pole of a magnet placed anywhere in the neighbour- hood of the wire, experiences a force tend- ing to urge it in a circular path round the wire, and the direction of motion round the wire is opposite for opposite poles. Fig. 458 A represents three of the lines of force for a north-seeking pole, due to a current flowing through a straight wire from the end marked 4- to the end marked . The lines of force are circles (shown in perspec- tive as ellipses), having their centre at a point C in the wire, and having their plane perpendicular to the length of the wire. The arrows indicate the direction in which a north- 43 Fig. 458 A. Lines of Force due to Current. 658 GALVANOMETER. seeking pole will be urged. This direction is from right to left round the wire as seen from the wire itself by a person with his feet to- wards + and his head towards , according to Ampere's rule. The figure may be turned upside down, or into any other position, and will still remain true. 531 B. Reaction of Magnet on Current. While the wire, in virtue of the current flowing up through it, urges an austral pole from A towards A' (Fig. 458 B), it is itself urged in the opposite direction CO'. If an observer be in imagination identified with the wire, the current being supposed, as in Ampere's rule, to enter at his feet, and come out at his head, the force which he will experi- ence from a north-seeking pole directly in front of him will be a force to his right. It will be noted that the magnetic influ- ence which thus urges him to the right, would urge a north-seeking pole from his front to his back. A conductor conveying a current is not urged along lines of mag- netic force, but in a direction whic/t is at right angles to them, and at the same time at right angles to its own length. 532. Numerical Estimate of Currents. The numerical measure of a current denotes the quantity of electricity which flows across a section of it in unit time. It is sometimes called strength of current, sometimes, especially by French writers, intensity of current, some- times simply current or amount of current. If a thin and a thick wire are joined end to end, it has the same value for them both; just as the same quantity of water flows through the broad as through the contracted parts of the bed of a stream. Hence the name inten- sity is obviously inappropriate, for, with the same total quantity of electricity flowing through both, the current is, properly speaking, more intense in the thin than in the thick wire. Currents may be measured experimentally by various tests, which are found to agree precisely. The most convenient of these for general purposes is the deflection of a magnetized needle. The force which a given pole experiences in a given position with respect to a wire conveying a current, is simply proportional to the current. Hence the name strength of current admits of being interpreted in a Fig. 458 B. Reaction oil Current. SINE GALVANOMETER. 659 sense corresponding to that in which we speak of the strength of a pole. Instruments for measuring currents by means of the deflec- tions which they produce in a magnetized needle are called yalvano- nieters. 533. Sine Galvanometer. The sine galvanometer, which was invented by Pouillet, is represented in Fig. 459. The current which is to be measured traverses a copper wire, wrapped round with silk for insula- tion, which is carried either once or several times round a vertical circle; and this circle can be turned into any position in azimuth, the amount of turning be- ing indicated on a horizon- tal circle. In the centre of the vertical circle, a decli- nation needle is mounted, surrounded by a horizontal circle for indicating its position, this circle being rigidly attached to the ver- tical circle. Suppose that, before the current is allowed to pass, both the needle and the vertical circle are in the magnetic meridian, and that the needle consequently points at zero on its horizontal circle. On the current passing, the needle will move away. The vertical circle must then be turned until it overtakes the needle; that is, until the needle again points at zero. This implies turning the circles through an angle a equal to that by which the needle finally deviates from the magnetic meridian. In this position the terrestrial couple tending to bring back the needle to the meridian is proportional to sin a ( 498), The forces exerted upon the two poles by the current are perpendicular to the plane of the vertical circle, and are simply proportional to the current. Hence, in com- paring different observations made with the same instrument, the amounts of current are proportional to the sines of the deviations. Fig. 46'J. Sinu Galvanometer. 660 GALVANOMETER Fig. 460. Principle of Tangent Galvanometer. 534. Tangent Galvanometer. The tangent galvanometer, which is simpler in its construction and use, arid is much more frequently employed, consists of a declination needle mounted in the centre of a vertical circle whose plane always coincides with the magnetic meridian, the length of the needle being small in comparison with the radius of the circle. Let o (Fig. 460) be the centre of suspen- sion, a b the initial position of the needle, and a' 6' its deflected position. The force F exerted on either pole by the current is sen- sibly the same at a as at a on account of the smallness of the needle, and it acts in the direction Ik, while the horizontal force of the earth upon the pole acts along a'm; and these two forces give a resultant along oa'. Hence, taking the triangle ola as the triangle of forces, 1 the force exerted by the current is to the hori- zontal force exerted by the earth as la' to ol, or as tan a to unity; that is, the current is proportional to the tangent of the deflection. In order to permit the deviations of the short needle to be accur- ately read, a long pointer is attached to it, usually at right angles, the two ends of which move along a fixed horizontal circle. 535. Multiplier. The idea of carrying a wire several times round a needle in a vertical plane is due to Schweiger. The form of apparatus de- signed by him, called Schweiger's mul- tiplier, is represented in Fig. 461. The difference between the rectangular and the circular form is merely a matter of detail. The name multiplier is derived from the fact that, if the current is not sensibly diminished by increasing the number of convolutions of wire through which it has to pass, the force exerted on the needle is n times as great with n convolutions as with only 1, since each convolution exerts its own force on the needle indepen- dent of the rest. Cases, however, frequently occur in which the increased resistance introduced by increasing the number of corivolu- 1 The parallelogram of forces is divided by its diagonal into two triangles, either of which may be called the triangle of forces. Fig. 461. Schweiger's Multiplier. DIFFERENTIAL GALVANOMETER. 061 tions outweighs the advantage'of multiplication, so that a short thick wire with few convolutions gives a more powerful effect than a long thin wire with many. This is especially the case with thermo-electric currents. The names multiplier and galvanometer are commonly used as equivalent. The difference between the rectangular and the circular form is merely a matter of detail. Whichever form be adopted, all parts of the coil contribute to make the needle deviate in the same direction. For instance, in Fig. 402, if the current proceeds in the direction indicated by the arrows, the application of Ampere's rule to any one of the four sides of the rectangle shows that the austral pole a will be urged towards the front of the figure. When the coil is circular, and the needle so small that each pole is nearly in the centre, equal lengths of the current, in whatever parts of the circle they may be situated, exert equal forces upon the needle, and ah 1 alike urge the poles in directions perpendicular to the plane of the coil. 535 A. Differential Galvanometer. The coil of a galvanometer some- times consists of two distinct wires, having the same number of con- volutions, and connected with separate binding-screws. This arrange- ment allows of currents from two distinct sources being sent at the same time round the coil either in the same or in opposite directions. Tn the latter case, the resultant effect upon the needle will be that due to the difference of the two currents; and if they are not exactly equal, the direction of the deflection will indicate which of them is the greater. An instrument thus arranged is called a differential (jalvanometer. 536. Astatic Needle. The sensibility of the galvanometer is greatly increased by employing what is called an astatic needle. It consists of a combination of two magnetized needles ivith their poles turned opposite ways. The two needles are rigidly attached at different heights to a vertical stem, and the system is usually suspended by a silk fibre, which gives greater freedom of motion than support upon a point. On account of the opposition of the poles, the directive action of the earth on the system is very feeble. If the magnetic moments of the two needles were exactly equal, the resultant moment would be zero, and the system would remain indifferently in all azimuths. 662 GALVANOMETER. Pi-. -163. One of the needles a b (Fig. 463) is nearly in the centre of the coil CDEF through which the current passes. The other a b' is just above the coiL When a current traverses the coil in the direction of the arrows, the action of all parts of the current upon the lower needle tends to urge the austral pole a towards the back of the figure, and the boreal pole b to the front. The upper needle a b' is affected principally by the current in the upper part CD of the coil, which urges the austral pole a' to the front of the figure and the boreal pole b' to the back. Both needles are thus urged to rotate in the same direc- tion by the current, and as the opposing action of the earth is greatly enfeebled by the combination, a much larger deflection is obtained than would be given by one of the needles if employed alone. If the two needles had rigorously equal mo- ments, the system would be said to be perfectly astatic. The smallest current in the coil would then suffice to set the needles at right angles to the meridian, and no measure would be ob- tained of the amount of current. Fig. 464 represents an i astatic galvanometer, as I usually constructed. The ? coil is wound upon an ivory frame, which sup- ports the divided circle in whose centre the up- per needle is suspended. The ends of the coil are connected with two binding-screws for O making connection with the wires which convey the current to be measured. The needles are usually two sewing-needles, and the upper one often carries a light pointer. The suspending fibre is attached at its upper end to a hook, which can be raised or lowered, Fig. 464. Astatic Galvanometer. MIRROR GALVANOMETER. 663 and when the instrument is not in use this is lowered till the upper needle rests upon the plate beneath it, so as to relieve the fibre from strain. In using the instrument, care must be taken to adjust the three levelling-screws so that the needle swings free. 536 A. Thomson's Mirror Galvanometer. The most sensitive galvano- meter as yet invented is the mirror galvanometer of Sir W. Thomson. Its needle, which is very short, is rigidly attached to a small light concave mirror, and suspended in the centre of a vertical coil of very small diameter by a silk fibre. A movable magnet is provided for bringing the needle into the plane of the coil when the latter does not coincide. with the magnetic meridian. A divided scale is placed a Fig. 464 A. Mirror and Scale. in a horizontal position in front of the mirror, at the distance of about a yard, and the image of an illuminated slit, which is thrown by the mirror upon this scale, serves as the index. The arrangement of the mirror and scale, which is the same as in the case of the quadrant electrometer described in a previous chapter, is exhibited in Fig. 464 A. M is the mirror of silvered glass, slightly concave, with a small piece of magnetized watch-spring attached to its back, the two together weighing only a grain and a half, and suspended by a few fibres of unspun silk. A A is a divided scale forming an arc of a horizontal circle about the mirror as centre. Immediately below the centre of this scale is a circular opening S with a fine wire stretched vertically at the back of it. A paraffine lamp L is placed directly behind this opening, so as to shine through it upon the mirror, which is at such a distance as to throw upon the screen a bright image of the opening with a sharply-defined dark image of the wire in its centre. The image of the wire is employed as the index in taking the readings. 664 GALVANOMETER. For use at sea, the galvanometer is modified by fastening the supporting fibre of silk at both ends, so as to keep it tight, with the needle and mirror attached at its centre, care being taken to make the direction of the fibre pass through the common centre of gravity of the needle and mirror, in order that the rolling of the ship may not tend to produce rotation. In this form it is called the marine galvanometer. 537. Calibration of Galvanometer. The deviations of the needle of a galvanometer are not in general proportional to the currents which produce them. In order to be able to translate the indications of the instrument into proportional measure, a preliminary investigation must be made, and its results embodied in a table. This has been done in several ways. We shall merely indicate the method em- ployed by Melloni for deducing from the deflections of his galvano- meter the amounts of heat received by his thermo-pile. He placed two sources of heat opposite the two ends of the pile, and allowed them to radiate to it, first one at a time, and then both together. One of them produced a deviation, say of 5, and the other of 10, and when the two were acting jointly the deviation was 5. Since the latter number is the difference of the other two, the infer- ence is that up to 10 the deflections are proportional to the amounts of heat received. Melloni thus established that the proportionality subsisted up to 20. When the two sources separately produced deflections of 20 and 25, and a deflection of 6'5 jointly, he inferred that a deflection of 25 indicated an amount of heat represented by 26 0< 5 ; for the heat which produced the deflection of 25 was the sum of the two amounts represented separately by 20 and 6 0- 5. By a succession of steps of this kind, the calibration 1 (as this process is called) can be extended nearly to 90. Tli is mode of investigation covers any want of proportionality which may exist in the production of thermo-electric currents, as well as in the proportionality of these currents to the deflections. Another method of calibrating a galvanometer will be described in the next chapter. 1 The application of the name calibration to this process is, we believe, due to Professor Tyndall. Its analogy to the calibration of a thermometer is obvious ; the object in both cases being to reduce observed differences to pro'portional measure. It is often called the graduation of a galvanometer ; but, in point of fact, the galvanometer is graduated, by dividing its circle into 360 degrees, before the process begins. CHAPTER XL VII. OHMS LAW. 538. Statement of Ohm's Law. The strength of the current which traverses a circuit depends partly on the electro- motive force of the source of electricity, and partly on the resistance of the circuit. For equal resistances, it is proportional to the whole electro-motive force tending to maintain the current, and for equal electro-motive forces it is inversely as the whole resistance in the circuit. Hence, when proper units are chosen for expressing the current C, the resist- ance R, and the electro-motive force E, we have E 0g, or the current is equal to the electro-motive force divided by the resistance. This is Ohm's law, so called from its discoverer. 539. Explanation of the term Electro -motive Force. When a steady current is flowing through a galvanic circuit, there must be a gradual fall of potential in every uniform conductor which forms part of the circuit; since, in such a conductor, the direction of a current must necessarily be from higher to lower potential. These gradual falls are exactly compensated by the abrupt rises (diminished by the abrupt falls, if any) which occur at the various places of contact of dissimilar substances. Recent experiments by Sir W. Thomson seem to prove that by far the most important of these abrupt transitions occur at the junctions of dissimilar metals, a view which was originally propounded by Volta, who appears, however, to have overlooked the essential part played by chemical combination in supplying the necessary energy. If we imagine a large and deep trough of water of annular form, divided into compartments by transverse partitions ; and suppose a constant difference of level to be maintained on opposite sides of each 666 OHM'S LAW. partition, by steady pumping of water from each compartment to the next ; we have a rough representation of the distribution of potential in the cells of a battery ; the rise of level in passing across a partition being analogous to the rise of potential in traversing a metallic junction. The electro-motive force of a galvanic battery may be defined as the algebraic sum of the abrupt differences of potential which occur at the junctions of dissimilar substances. In a battery consisting of a number of similar cells arranged in series, it is of course pro- portional to the number of cells. In like manner, in a thermo-electric circuit, there is difference of potential probably at each junction, whatever its temperature may be; and the algebraic sum of these sudden differences (a rise of potential being called positive and a fall negative, in travelling with the current) is the whole electro-motive force of the thermo- pile. When the faces of the pile are at equal temperatures, the opposite electro-motive forces are equal, and destroy one another; when the temperatures are unequal, the positive electro-motive forces exceed the negative, and the total or resultant electro-motive force O ' is the measure of this excess. 540. Explanation of the term Resistance. When the current of a circuit is taken through the coil of a galvanometer, it is found that, by introducing different lengths of connecting wire, very different amounts of deflection can be obtained. The longer the wire which connects either pole of the battery with the galvanometer, the smaller is the deflection ; and a small deflection indicates a feeble current. The current is in like manner weakened by introducing a fine instead of a stout wire, if their length and material be the same, or by introducing an iron wire instead of a copper wire of the same dimensions. These diffei-- ences in the properties of the different wires are expressed by saying that they have different resistances. The apparatus represent- ed in Fig. 465 can be em Fig. 465.-Com P arisou of Resistances. ployed for comparing resist- ances in this way. The cur- rent given by a battery P passes through a wire to the galvanometer B, and after traversing its coil is led on by another wire to the cup EXPERIMENTAL PROOFS. 667 of mercury a, thence through the connecting wire m to the other cup of mercury b, and back to the battery through another wire. The circuit can also be completed as shown in the figure, without passing through m, by means of a broad conducting plate whose resistance may be neglected. In changing the wire m, it is found that, to produce no change in the deflection, the length of the wire must vary directly as its cross- section; that is to say, if I, I', I" .... be the lengths of different wires employed, and s s s" . . . . their sectional areas, their resist- ances will be equal, if This is on the supposition that the wires are all of precisely the same material. Every substance has its own specific resistance, the recip- rocal of which is its electrical conductivity and is precisely analogous to thermal conductivity. Denoting specific resistances by r, r, r", .... the condition of equal resistances, when the materials are different, is rl rFr"l" and the resistance of any wire is expressed by the formula , I denoting its length, s its sectional area, and r the specific resistance of its material. 541. Experimental Proofs of Ohm's Law. Pouillet, who conducted numerous experiments bearing on Ohm's law, investigated the con- nection between cur- rents and resistances in the following ways: 1 . For thermo electric curren ff>, he employed two thermo-electric ele- ments, each consisting of a StOUt Cylinder of Fig. 466. Pouillet's Comparison. bismuth with its ends lient down and soldered to copper wires. The two elements were arranged side by side as in Fig. 466, and the junctions at one end were immersed in hot water, those at the other end being kept in ice. The hot and cold junctions of the one were connected 668 OHM'S LAW. by a wire which was carried round a galvanometer needle. Those of the other were connected by a wire ten times as long, which made ten times as many turns round the same needle in the opposite direc- tion, so that the two currents opposed each other in their action on the needle. It was found that the needle remained at zero, showing 7 O that the current in the short wire was ten times as strong as the other, for one of its convolutions was able to balance ten convolutions of the other. As the resistance in the stout bars of bismuth was inappreciable, it followed that the currents in the two circuits were inversely as the resistances. 2. For voltaic currents, he first sent the current of a battery through a galvanometer without any interposed resistance, and observed the strength of current C. He then introduced, successively, known lengths of uniform wire Z 1( 1 2 , 1 3 , and observed the currents obtained. Denoting these by Cj, C 2 , C 3 , and taking x to denote the length of wire which would be equivalent to the unknown resistance of the original circuit consisting only of the battery and the galvanometer, we should have e _ x + Z, c _ x + l 9 c _ x + l s From any one of these three equations x can be determined, and Ohm's law is verified if they all give the same value of x. This Pouillet found to be the case. By repeating the experiment with a different kind of wire, a new value of x will be obtained, and thus the resistances of equal lengths of the two wires can be compared. 542. Reduced Length: Total Resistance of Circuit. To express, in terms of the equivalent length of one wire, the resistance of a circuit composed of several, we can employ the relation ( 540) rl r'l' , 7 s r' T , - =-r- ; whence I = I, s s ' s r * I denoting the length of one kind of wire equivalent to the length I' of the other. The length I is called the reduced length of the wire whose actual length is I'. 543. Rheostat. Wheatstone's rheostat is a very convenient instru- ment for the comparison of resistances. It consists (Fig. 467) of two cylinders, one of brass, and the other of non-conducting material, so arranged that a copper wire can be wound off the one on to the other by turning a handle. The surface of the non-conducting cylinder B RHEOSTAT. 669 Fig. 467. Rheostat. has a screw-thread cut iu it, for its whole length, in which the wire lies, so that its successive convolutions are well insulated from each other. Two binding-screws are provided for introducing the rheostat into a circuit ; and the resistance which is thus introduced depends on the length of wire which is wrapped upon the non- conducting cylinder, for the brass cylinder A has so large a section that its resistance may be neglected. The amount of resistance can thus be varied as gradually as we please by winding on and off. The handle can be shifted from one cylinder to the other. The figure shows it in the position for winding wire off A on to B. The number of convolutions of wire on B can be read off on a graduated bar provided for the purpose, and parts of a revolu- tion are indicated on a circle at one end. Fig. 468 represents a very direct mode of measuring resistances by the rheostat. The current traverses a galvanometer B, a rheostat R, and the conductor ra, whose resistance is to be measured, the whole of the wire of the rheostat being wound on the brass cylinder. The deflection of the galvanometer hav- ing been observed, the conductor ra is taken out of circuit, the two wires at a and b are directly connected, and as much of the rheostat wire is brought into circuit as suffices to reduce the deflection to its former O amount. 543 A. Specific Resistances and Conductivities. Numerous experi- menters have compared the specific resistances of the different metals. Though the results thus obtained exhibit some diversity, they all agree in making silver, gold, and copper the three best conductors. Slight impurities, especially in the case of copper, have a very great effect in diminishing conductivity, or, in other words, in increasing Fig. 468. Measurement of Resistance. 670 OHM S LAW. resistance. Resistance is also increased, in the case of metals, by increase of temperature. Forbes has pointed out that the order of the metals as regards their conductivity for heat is the same as for electricity. The effects of impurity and of change of temperature are also alike in the two cases, as has been recently shown by Professor Tait. The following are E. Becquerel's determinations of specific elec- trical resistance at the temperature 15 G, the resistance of silver at G being denoted by 100:- SPECIFIC RESISTANCES AT 15 C. Silver, 107 Copper, 112 Gold, 155 Cadmium, 407 Zinc, 414 Tin 734 Palladium, 715 Iron, 825 Lead, 1213 Platinum, 1243 Mercury, 5550 On comparing this list with the list of thermal conductivities, 333, it will be observed that the order is precisely the same as far as the comparison extends, and that the numerical values are nearly in inverse proportion, showing that electrical and thermal conductivi- ties are nearly in direct proportion. 544. Resistance of Liquids. The resistance of liquids can be determined on similar prin- ciples, the current being transmitted between two parallel plates of metal immersed in the liquid. One form of apparatus for this pur- pose is represented in Fig. 469. Care must be taken to employ metals which will not give rise to electro- motive force by chemical action. The resistance even of the best conducting liquids, except mercury, is enormously greater than that of metals. For instance, in round numbers, the resistance of dilute sulphuric acid is a million times, and that of solution of sulphate of copper ten million times greater than that of pure silver. The resistance of pure water is very much greater than either of these. In the cells of a galvanic battery, the current has to traverse liquid conductors, and the resistance of these is sometimes a large part of Fig. 469. Resistance of Liquids. CALIBRATION OF GALVANOMETER. 671 the whole resistance in circuit. It is diminished by bringing the plates nearer together, and by increasing their size, since the former change involves diminution of length, and the latter increase of O o sectional area in the liquid conductor to be traversed. This is the only advantage of large plates over small ones, the electro-motive force being the same for both. The advantage of the double coppers in Wollastons battery ( 515) is similarly explained, the resistance with this arrangement being about half what it would be with copper on only one side of the zinc, at the same distance. 545. Calibration of Galvanometer by the Rheostat. The rheostat can be employed for determining the relative values of the deflec- tions of a galvanometer. For this purpose the two instruments are to be introduced into the circuit of a batter}'-, and in the first instance all the wire of the rheostat is to be on the non-conducting cylinder. The deflection, which will then be comparatively small, is to be noted. Successive lengths of the rheostat wire are then to be wound off, so as to diminish the resistance, and the deflections are to be noted in each case. If the current, when the whole length I of rheostat wire was in circuit, be denoted by C, and the currents with lengths 1^1% . . in circuit by G l C a . . . , and if r denote the resistance of the battery and galvanometer, which can be determined by methods already explained, we -shall have , = C Hence the ratios of the currents C, C,, C, . . . corresponding to the observed deflections are known. 546. Arrangement of Cells in Battery. Suppose that we have a number n of precisely similar cells, each having electro-motive force e and resistance r, and that we connect them in a series, as in Figs. 434-, 446, with a conductor of resistance R joining their poles. The whole electro-motive force in the circuit will then be ne, and the whole resistance will be n r + R ; hence the strength of current will be C= _^-. nr + R This formula shows that, if the external resistance R is much greater than the resistance in the battery nr, any change in the number of cells will produce a nearly proportional change in the current; but that when the external resistance is much less than that of one cell, 672 OHM'S LAW. as is the case when the poles are connected by a short thick wire, a change in the number of cells affects numerator and denominator almost alike, and produces no sensible change in the current. It is impossible, by connecting any number of similar cells in a series, to obtain a current exceeding , which is precisely the current which one of the cells would give alone if its plates were well connected by a short thick wire. It is possible, however, by a different arrangement of the cells, to obtain a current about n times stronger than this, namely, by con- Fig. 470. Cells with similar Plates connected. necting all the zinc plates to one end of a conductor, and all the carbons or coppers to the other end, as in Fig. 470. In the arrange- ment of three cells here figured, the current which passes through the spiral connecting wire is the sum of the currents which the three cells would give separately. The arrangement is equivalent to a single cell with plates three times as large superficially, and at the same distance apart. The electro-motive force with n cells so arranged is simply e, but the resistance is only ^ + R, so that the current is r + nR This system of arrangement may be called arranging the cells as element. It has sometimes been called the arrangement for quantity, the arrangement in a series being called the arrangement for intensity. If in Fig. 470 we substitute for each of the three cells a series consisting of four cells, the electro-motive force in circuit will be 4 e, A * and the resistance in circuit will be -- o for each series has DIVIDED CIRCUITS. 673 resistance of 4 r, and three parallel series connected at the ends are equivalent to a single series, of the same electro-motive force as one of the component series, and of one-third the resistance. The curren< will therefore be 4e 12 e e C = + R 3 The question often arises, What is the best manner of grouping a given number of cells in order to give the strongest possible current through a given external conductor? The answer is, they should be so grouped that the internal and external resistance should be as nearly as possible equal; for example, if we have 12 cells as above, and the resistance R in the given conductor is g- of the resistance of one of these cells, the arrangement just described is the best. 1 547. Divided Circuits. When two or more wires are connected in line, that is so as to form one continuous wire, the resistance of the whole is the sum of the resistances of the wires composing it. On the other hand, when two or more wires are arranged side by side, and connected at each end, so as to constitute so many indepen- dent channels of communication between the ends, the joint resist- ance is evidently less than the resistance of any one of the wires. When such an arrangement occurs in any part of a circuit, the circuit is said to be divided. If the several wires are of the same length and material, they act as one wire having a section equal to the sum of their sections, and the joint resistance is the quotient of the resist- ance of one of the wires by the number of wires. More generally, if the reciprocal of the resistance of a conductor be called its con- ducting power, the conducting power of a system of wires thus con- nected at both ends is the sum of the conducting powers of the several wires which compose it. Thus, in Fig. 471, if r 1? r a denote the resistances of the wires acb, adb, their joint resistance R will be given by the equation 1 Instead of 3 and 4, put x for the number of series, and y for the number of cells in a T> series. Then the current will be r R, and will vary inversely as _ + _ . Now the pro- + x y r R X y duct of and is given, being the quotient of r R by the whole number of cells ; and when the product of two variables is given, their sum is least when they are equal, and increases as they are made more and more unequal. As x and y must be integers, exact equality cannot generally be obtained. 44 074 OHM S LAW. E whence E = 547 A. Wheatstone's Bridge. In any wire through which a current is flowing steadily, without leakage or lateral offshoots, the amount Fig. 471. Divided Circuit. of the current is equal to the difference of potential between the ends of the wire, divided by the resistance of the wire, the units employed E being the same as those which make C = ^ for the whole circuit. The same thing is true for any portion of the length of such a wire, and, still more generally, for any portion of a circuit, whether single or divided, terminated by equipotential cross-sections, provided that no source of electro-motive force occurs in it. It follows that, in travelling along such a wire with the current, the fall of potential is proportional to the resistance travelled over, or equal falls of poten- tial occur in traversing equal resistances. This rule does not apply to the comparison of the two independent channels of a divided cir- cuit, unless equal currents are passing through them. It applies tc the comparison of any two wires which are conveying equal currents, and it is not applicable to the comparison even of different portions of the same wire if, owing to leakage, the current is unequal at dif- ferent parts of its length. Equality of potential in two points of a divided circuit can be tested by observing whether, when they are connected by a cross- channel, any current passes between them. This principle has been applied by Wheatstone, Thomson, and others, to the measurement of resistances, and the apparatus employed for the purpose is generally known as Wheatstone's bridge. It is typically represented in Fig. 471 A. The poles P, N of a battery are connected by two independent WHEATSTONE'S BRIDGE. 075 channels of communication AC B, A D J E B. The former is a uniform wire; the latter consists of the wire D, whose resistance is to be deter- mined, and of a standard resistance-coil E. The observation has for its immediate object to find what point in the uniform wire AB has the same potential as the junction J of the other two. When this point C is found, and connected with J through a galvanometer G, no current will pass across, and the needle of the galvanometer Fig 471 A ._ wll8at8tolie .. Bridge . will not move. If a point C x on the positive side of C were connected with J, a current would run from GX to J, and if a point C 2 on the negative side were connected, the current would be from J to C 2 . The deflection diminishes as the right point C is approached, and becomes reversed in passing it. When it is found, we know that the resistances in AC and CB have the same ratio as those of D and E, each of those ratios being in fact equal to the fall of potential between A and JC divided by the fall between JC and B. As the resistance of E is known, and the resistances of AC, CB are as their lengths, which are indicated on a divided scale, the resistance of D can be computed by simple proportion. In Wheatstone's original arrangement, the resistances of the two portions AC, CB were equal, and the resistances of the other two portions A D J, J E B were made equal by the help of a rheostat. 547f. Distribution of Potential in a closed Voltaic Circuit. When the electrodes of a battery are not connected, their difference of potential, supposing them to be of the same metal, is a measure of the electro-motive force of the battery. On joining them by a con- necting wire, their difference of potential will be diminished, and will be the same fraction of the whole electro-motive force that the resistance in the connecting wire is of the whole resistance. This follows at once from the principle that the gradual falls of potential in different portions of the same single circuit are directly as their resistances. In a battery of four cells, like that represented in Fig. 431, when the extreme plates are connected by a wire whose resistance is double that of the battery, the fall of potential in the connecting wire will 676 OHM S LAW. be two-thirds, and the fall of potential in the battery will be one- third, of the whole electro-motive force. To avoid fractions, let the electro-motive force of each cell be denoted by 3. Then the total Fig. 471 B. Curve of Potential for Closed Circuit. electro-motive force will be 12, the fall of potential in the connect- ing wire will be 8, in the battery 4, and in each cell 1. The distribution of potential, both before and after making con- nection, is exhibited in the two columns subjoined, the connecting wire being supposed to be of copper, and to be connected with the earth close to its junction with the first zinc plate, so that this end of the wire will always be at zero potential. We may suppose con- nection to be broken by disconnecting the other end of the copper wire from the last copper plate. CONNECTION BROKEN. Potentials. Copper Wire, ( Zinc plate, .... 3 1st cell ] Acid, 3 ( Copper plate, ... 3 ( Zinc plate, .... 6 2d cell j Acid, 6 ' Copper plate, ... 6 ( Zinc plate, .... 9 3d cell ] Acid, 9 ( Copper plate, .... 9 Zinc plate, . . . .12 4th cell Acid, . Copper plate, . 12 12 CONNECTION MADE. Potentials. Copper Wire, ..... 8 to ( Zinc plate, ... 3 1st cell ] Acid, ..... 3 to 2 ( Copper plate, . . 2 i Zinc plate, ... 5 2d cell ] Acid, ..... 5 to 4 ( Copper plate, . . 4 I Zinc plate, ... 7 3d cell ] Acid, ..... 7 to 6 ( Copper plate, . . 6 Zinc plate, ... 9 Acid, ..... 9 to 8 4th cell Copper plate, 8 The distribution of potential when connection is made is graphi- cally represented by the crooked line A3254769C (Fig. 471 B) ; resistances being represented by horizontal, and potentials by vertical distances. A C represents the total resistance in circuit ; A B being CHOICE OF GALVANOMETER. 677 the resistance of the battery, and B C that of the connecting wire. A D represents the total electro- motive force. The points C and A are to be regarded as identical; in other words, the diagram ought to be bent round a cylinder so as to make one of these points fall upon the other. 547 c. Measurement of Resistance of Battery. The resistance of a battery may be measured in various ways, of which we shall only describe one. Let the poles of the battery be directly connected with a galvano- meter whose resistance is either very small or accurately known, and let the deflection be noted. Then let a wire of known resistance be introduced into the circuit, and the deflection again noted. The two currents thus measured will be inversely as the resistances, since the electro-motive force is the same in both cases. Let the resistance of the galvanometer coil be denoted by G, that of the wire introduced in the second case by W, and that of the battery by x. Then if the amounts of current be denoted by C 1} C 2 , we have c ~ = x + X + G ; whence x can be determined. 548. Choice of Galvanometer. The circumstances which should influence the choice of a galvanometer coil for a particular purpose, will now be intelligible. If stout wire is employed, the resistance is small, but it is not practicable to multiply convolutions to any great extent. Short coils of thick wire are accordingly employed in con- nection with therm o-piles, the resistance in the pile itself being so small that the total resistance in circuit is nearly proportional to the number of convolutions. When, on the other hand, the resistance in the other parts of the circuit is very considerable, the resistance of the galvanometer coil becomes comparatively immaterial, so that, within moderate limits, the deflection of the needle is nearly proportional to the number of convolutions, and a coil composed of a great length of wire will give the maximum effect. In both cases, for a given length and diameter of wire, the sen- sibility increases with the conductivity of the metal composing the wire. Copper is the metal universally employed, and its purity is of immense importance for purposes of delicacy, as impurities often increase its resistance by 50 or even 100 per cent. 549. Measurement of Electro-motive Force. The most direct mode of comparing the electro-motive forces of cells of different kinds, would C78 OHM S LAW. be to observe how many cells of the one kind arranged in series must be opposed to a given number of the other kind, in order that the resultant electro-motive force may be nil as indicated by the absence of deflection in a galvanometer forming part of the circuit. For example, if two Daniell's cells and one Grove's cell be connected with each other and with a galvanometer, in such a manner that the cur- rent due to the Daniell is in one direction, and that due to the Grove is in the opposite direction, the current actually produced will be in the direction of the greater electro-motive force. It will thus be shown whether the electro-motive force of a Grove's cell is more or less than double that of a Daniell's. This method has not been much used. Another method of comparison consists in first connecting the two cells to be compared, so that their electro-motive forces tend the same way, and then again connecting them, so that they tend opposite ways, the resulting current being observed in both cases with the same galvanometer. The resistance in circuit is the same in both cases, being the resistance of the galvanometer plus the sum of the resistances of the cells ; hence the currents will be simply as the electro-motive forces, that is to say, as Ej-t-Ej to Ej E 2 , if E : and E 2 denote the electro-motive forces of the cells. Hence the ratio of E! to E 2 is easily computed. Another method, which has been employed by Jules Regnault, is Fig. 472. Jules Regnault's Apparatus. illustrated by Fig. 472. It consists in balancing the electro-motive force of the cell P which is to be tested, by that of a series of thermo- MEASUREMENT OF ELECTRO-MOTIVE FORCES. 679 electric elements, the number of which can be varied at pleasure. A is a thermo-electric pile, consisting of sixty elements of bismuth and copper, with their opposite junctions maintained at and 100 C. Any number of these can be included in the circuit by moving the slider a, and the direction of the current which they tend to produce is opposite to that due to the cell P. As sixty thermo-electric ele- ments would not be enough to balance one ordinary cell, some auxiliary cells o o of feeble electro-motive force, which has been pre- viously determined, are employed to assist in opposing the cell P. It has thus been found that one Daniell's cell has the electro-motive force of about 174 of these thermo-electric elements. Electro-motive force may also be measured statically by means of Thomson's quadrant electrometer, the poles of the battery being connected with the two chief electrodes of the instrument, in which arrangement no current will pass, and the electro-motive force will be directly indicated by the difference of potential observed. According to Latimer Clark, the electro-motive forces of a cell of Grove, Bunsen, Daniell, and Wollaston are approximately as 100, 98, 56, and 46; but the last of these, being a one-fluid battery, is liable to fall off 50 per cent, or more, owing to the deposition of hydrogen on the copper plata CHAPTER XLVTII. ELECTRO-DYNAMICS. 551. Meaning of Electro-dynamics. A wire through which a cur- rent is passing, is found to be capable of producing movements in other wires also conveying currents. The theory of these move- ments, or more generally, of the mechanical actions of currents upon one another, constitutes a distinct branch of electrical science, and is called electro-dynamics. It stands in very close relation to electro- magnetism ; and if the laws of either of the two sciences are given, those of the other may be deduced as consequences. The science of electro-dynamics was founded by Ampere. Figs. 473, 474 represent an arrangement which he devised for rendering a conductor movable with- out interruption of the cur- rent conveyed by it. A wire is bent into the form of a nearly complete rectangle, and its two ends terminate in points, one above the other, so arranged that a vertical through the centre of gravity passes through them both. Ac- cordingly, if either or both of these points be supported, the wire can turn freely about this vertical as axis. The points dip into two small metallic cups x y containing mercury, and the weight is usually borne by the upper point alone, which touches the bottom of its cup. The cups are attached to two horizontal arms of metal, supported on metallic pillars, which can be con- Fig. 473. Ampere's Stand. MUTUAL FORCES BETWEEN CURRENTS. 681 Fig. 474. Action of Magnet on Movable Circuit. nected with the two terminals of a battery. The wire thus forms part of the circuit, the current being down one side of the rectangle and up the other. Instead of the rectangular the circular form may be employed, as in Fig. 475. If a magnet be placed be- neath, as in Fig. 474, the wire : rame will set its plane perpen- licular to the length of the magnet, the relative position nssumed being the same as if the wire frame were fixed, and the magnet freely suspended, if we neglect the disturbing effect of the earth's mag- netism. 552. Mutual Forces between Conductors conveying Currents. The following elementary laws, regarding the mutual forces exerted be- tween conductors through which currents are pass- ing, were established by Ampere. For brevity of expression, it is usual to speak, in this sense, of the mutual forces between currents, or of the mutual mechanical action of currents. I. Successive portions of the same rectilinear current repel one another. 1 This is proved by the aid of two troughs of mer- cury separated by a partition (Fig. 476). A var- nished wire is bent into such a form that two portions of it can float on the surface of the mercury in the two troughs, while connected with each other by an arc passing over the partition. The only por- tions without varnish are the ends. \Vhen the terminals of a battery are inserted in the mercury, opposite the ends, as shown in the figure, the circuit is completed through the wire, and repulsion is exhibited, the wire moving away to the further end of the vessel. 1 This first law is not universally accepted, and can scarcely be regarded as resting on the same sure foundation as the rest. Fig. 475. Fig. 476. Repulsion of Successive Portions. 682 ELFCTRO-DYNAMICS. II. Parallel currents, if in the same direction, attract, and if in the opposite direction, repel each other. The apparatus employed for demonstrating this twofold proposi- tion, consists of two metallic pillars t, v (Fig. 477), which are respec- tively connected at their upper ends with the two cups of mercury x, y. The rectangular conductor abode is suspended with its terminal points in these cups so as to complete the circuit between the pillars. When the current is passed, this movable conductor always places ,3r t d __. 1 r Fig. 477. Attraction of Parallel Currents. Fig. 478. Apparatus for Repulsion. itself so that its plane coincides with that of the two pillars, and so that currents in the same direction in the pillars and in the wire are next each other, as shown in the figure. For establishing repulsion, a slightly different form of wire is employed, which is represented in Fig. 478. When this is hung from the cups, in the position which the figure indicates, the currents in the pillars are in opposite directions to those in the neighbouring portions of the movable conductor, and the latter accordingly turns away until it is stopped by the collision of the wires above. III. Currents whose directions are inclined to each other at any angle, attract each other if they both flow towards the vertex of the angle, 1 or if they both flow from it, and repel each other if one of them flows toivards the angle, and the other from it. A consequence of this law is that two currents, as A B, DC (Fig. 479), crossing one another near O in different planes, tend to set themselves parallel, and so that their directions shall be the same. 1 If the currents are not in the same plane, we must substitute the feet of their common perpendicular for the vertex of the angle, in the enunciation of this law. CONTINUOUS ROTATION OF RADIAL CURRENT Fig. 479. Tendency to set Parallel. For there is attraction between the portions AO and DO, and also I letween the portions B and O C ; whereas there is repulsion between A O and O C, and between O B and O D. Accordingly, if the movable conductor of Fig. 477 or 478 be traversed by a current, and another wire carrying a current be placed horizontally at any angle un- derneath its lower side, the movable conductor will turn on its point of suspension till it becomes parallel to the wire below it ; and in the position of stable equilibrium the current in its lower side will have the same direction as that in the influencing wire. 553. Continuous Rotation produced by a Circular Current. Suppose we have a current flowing round a circle (Fig. 480), and also a current flowing along OA, which is approximately a radius of this circle. First let the current in OA be from the centre towards the circumference, as indi- cated in the figure. Then, by law III., OA is attracted on one side and repelled on the other, both forces combining to make OA sweep round the circle in the opposite direc- tion to that in which the circular current is flowing. If the current in OA were from circumference to centre, the tendency would be for OA to sweep round the circle in the same direction as the circular current The reasoning still holds if A is in a plane parallel to that of the circular current, O being a point on the axis of the circle and the length of OA being not greater than the radius. A circular current may also produce continuous rotation in a con- ductor parallel to the axis of the circle, and movable round that axis. Fig. 481 represents an arrangement for obtaining this effect. A coil of wire through which a current can be sent, is wound round o the copper basin EF, its extremities being connected with the bind- ing-screws m, o. From the centre of the basin rises the little O * metallic pillar A, terminating above in a cup containing mercury. This pillar is connected with the binding-screw n. The basin, which is connected with the binding-screw p, contains water mixed with a Fig. 480. Continuous Rotation of Radial Current. ELECTRO- DYNAMICS. little acid to improve its conducting power, and a movable conductor BC rests, by a point, on the bottom of the cup of mercury, while its lowest portion, which consists of a light hoop, dips in the acidulated water. By connecting m and n a single circuit is obtained, of which Fig. 481. -Apparatus for Continuous Rotation. o and p are the terminals, so that if o is connected with the positive and p with the negative pole of a battery, the current entering at o first traverses the wire coil, then ascends the pillar A, returns down the sides B, C to the floating ring and liquid, and so escapes to p. As soon as these connections have been completed, the movable con- ductor commences continuous rotation in the direction opposite to that of the current in the coil. If, instead of connecting m and n, we connect n and o, and lead the positive wire from the battery to p and the negative wire to o, the course of the current will be from p to the acid, thence up the sides B, C, and inwards along the top of the movable conductor to the mercury cup, then down the pillar to n, thence to o, and through the coil from o to m in the same direction as in the former experiment; but the rotation of the movable conductor will now occur in the opposite direction to that before observed, and therefore in the same direction as the current. 554. Action of an Indefinite 1 Rectilinear Current upon a Finite Cur- rent movable around one Extremity. A finite current movable about one extremity may also be caused to rotate continuously about this extremity by the action of an indefinite rectilinear current. This is clearly indicated by Fig. 482. In the right-hand diagram, the cur- 1 The word indefinite, in this application, simply means of great length in comparison with the distance and length of the movable current. ROTATION OF RADIAL CURRENT. G85 Fig. 482. Eotation of Radial Current. rent OA flowing outwards from the centre of motion O, and acted on by the indefinite current MN, is first attracted into the position OA'. In this new position it is repelled by uN, and attracted by MTI. It is thus brought successively into the positions OA", OA '", OA IV . In this last- mentioned position, the two currents being parallel and opposite, there is repulsion; and after passing it, there is again repulsion on one side and attraction on the other, till it is carried round to its first position OA. It is thus kept in con- tinual rotation. If the movable current flows inwards to the centre of motion O, as in the left-hand diagram, while the direction of the indefinite current is the same as before, the direction of rotation will be reversed. 555. Action of an Indefinite Rectilinear Current on a Finite Current Perpendicular to it. Let M N, in the upper half of Fig. 483, be an indefinite rectilinear current, and AD a portion of another current either in the same or in any other plane. In the latter case let DC be the common perpendicular. Then, if the currents have the directions represented by the ar- rows, an element at p will attract an element at m with a force which we may represent by a line mf; and an element at^>' equal to that at p and situated at the same distance from C on the other side, will repel the element at m with an equal force, represented by m f. Constructing the parallelogram of forces, the resultant force of these two elements upon m is repre- sented by the diagonal raF, which is parallel to MN and in the opposite direction to the indefinite current. As this reasoning applies Fig. 483. Translation Parallel to Indefinite Current. oau ELECTRO DYNAMICS. to all the elements of both currents, it follows that the current AB will experience a force tending to give it a motion of translation parallel to MN. This motion will be opposite to the direction of the indefinite current when the direction of the finite current is towards the common perpendicular DC, as in the upper diagram, and will be in the same direction as the indefinite current when the direction of the finite current is from the common perpendicular, as in the lower diagram. 556. Action upon a Rectangular Current movable about an Axis Perpendicular to an Indefinite Current. It follows from the preceding section that if a finite current AB (Fig. 484), perpendicular to an Fig 481. Position assumed by Perpendicular Current. indefinite current, is movable round an axis 0' parallel to itself, the plane A B O O' will place itself parallel to the indefinite current, and AB will place itself in advance or in rear of the axis according as the current in A B is from or towards the indefinite current. If a pair of parallel and opposite currents BA, A'B', rigidly con- nected together, and movable round the axis 0' lying between them, are submitted to the action of the indefinite current, the forces upon them will conspire to place the system in the position indicated in the figure. If the two currents A B, A'B' are both in the same direction, their tendencies to revolve round the axis 0' will counteract each other. 557. Action upon a Rectangular Current movable round an Axis Perpendicular to an Indefinite Current. If a rectangular current (Fig. 485) is movable round an axis oo' per- pendicular to the direction of the indefinite rectilineal current, we have just seen that the action upon the two sides of the rectangle which are perpendicular to the latter, tends to place the system so that its plane shall be Fig. 4S. 1 ! Position nssnmed by Rectangular (Junent. POSITION ASSUMED BY VERTICAL CURRENT. 087 parallel to the indefinite current, and that the side which carries the receding current shall be in advance of the other. The action upon the near side of the rectangle contributes to produce the same effect, since this side tends to set itself parallel to the influencing current, and so that the directions of the two shall be the same. The action upon the further side of the rectangle tends to produce an opposite effect; but, in consequence of the greater distance, this action is feebler than that upon the near side. The system accord- ingly tends to take the position of stable equilibrium represented in the right-hand half of the figure. The diagram on the left hand represents a position of unstable equilibrium. What is here proved for a rectangular current, is true for any closed plane circuit movable round an axis of symmetry perpendicular to an indefinite rectilinear current; that is to say, any such circuit tends to place itself so that the current in the near side of it is in the same direction as the indefinite current. The results of 556 can be verified experimentally by the aid of the apparatus represented in Fig. 486. C C, D D are two cups (shown in section) surrounding the me- tallic pillar AB at its upper and lower ends, and containing u con- ducting liquid. The lower cup is insulated from the pillar, and con- nected with the binding-screw g. The liquid in the upper cup CO is connected with the upper end of the pillar by the bent arm d m. oK is a light horizontal rod sup- ported on a point at B, and carry- ing a counterpoise K at one end, while the other carries a wire mnop, whose two ends nm and op descend vertically into the two cups, the middle portion of the wire being wrapped tightly round the rod. The binding-screw / is con- nected with the lower end of the pillar. If a current enters at / and leaves at g, its direction in the long vertical wire op will be descending; and it will be ascending, if the connections are reversed. By sending a current at the same time through a long horizontal wire in the neighbourhood of the system, movements will be obtained in accord- ance with the foregoing conclusions. 558. Sinuous Currents. A sinuous current exhibits the same action Fig. 486. Position assumed by Vertical Current. 688 ELECTRO-DYNAMICS. as a rectilinear current, provided that they nowhere deviate far from each other. This principle can be exemplified by bringing near to a movable conductor (Fig. 487) another conductor consisting of a wire doubled back upon itself, having one of its portions straight, and the other sinuous, but very near the first. A current sent through this double wire traverses the straight and the sinuous portions in oppo- site directions, and it will be found that their joint effect upon the movable conductor is inappreciable. This principle holds not only for rectilinear currents but for cur- rents of any form, and is very extensively employed in the analytical investigations of electro-dynamics. In computing the action exer- cised by or upon a conductor of any form, it is generally convenient Fig. 487. Sinuous Currents. Fig. 483. to substitute for the conductor itself an imaginary conductor, nearly coincident with it, and consisting of a succession of short straight portions at right angles to one another (Fig. 488). 559. Mutual Action of Two Elements of Currents. Ampere based his analytical investigations on the assumption that the action exer- cised by an element (i.e. a very short portion) of one current upon an element of another, consists of a single force directed along the join- ing line. This assumption conducted him to a formula for the amount of this force, which has been found to give true results in every case capable of being tested by experiment. Nevertheless, it is by no means certain that either Ampere's formula or his fundamental assumption is true. Other assumptions have been made, leading to other formulae in contradiction to that of Ampere, which also give true results in every case capable of being experimentally tested. MUTUAL ACTIONS BETWEEN ELEMENTS. 689 The fact is that experiments can only be performed with complete circuits, and the contradictions which subsist between the different assumptions, in the case of the several parts of a circuit, vanish when the circuit is considered as a whole. All the formulae, however, agree in making the mutual force or forces between two elements vary inversely as the square of their distance, and directly as the products of the currents which pass through them. Professor Clerk Maxwell 1 discards all assumptions as to mutual actions between elements at a distance, and employs the principle that a circuit conveying a current always tends to move in such a manner as to increase the number of magnetic force-tubes (in the sense of 445 H) which pass through it. The work done in any displacement is measured by the number of tubes thus added ; but tubes which cross the circuit in the opposite direction to those due to the current in the circuit are to be regarded as negative. We have seen ( 531 A, B) that the lines of magnetic force due to a current are circles surrounding it ; and also that, when a line of magnetic force cuts a current, the latter experiences a force tending to move it at right angles to the plane of itself and the line of force. In the case of two parallel currents, each is cut at right angles by the lines of magnetic force due to the other; the direction of the force experienced by either current is therefore directly to or from the other current ; and the criterion of 531 B will be found to indicate attraction when the directions of the currents are the same, and repulsion when they are opposite. In Fig. 480 the lines of magnetic force cut OA in a direction perpendicular to the plane of the diagram, OA accordingly experi- ences a force perpendicular to its own length in the plane of the diagram ; and the same remarks apply to A B in Fig. 484. All the experimental facts above detailed are in fact thus explicable. In the experiment of Fig. 476, where the application is scarcely so obvious as in the other cases, the observed motion may be deduced from the direction in which the bridge or arc connecting the two side-wires is cut by the lines of force. 2 560. Action of the Earth on Currents. In virtue of terrestrial magnetism, movable circuits, when left to themselves, take up de- finite positions having well-marked relations to the lines of terrestrial 1 Maxwell "On Faraday's Lines of Force." Camb. Trans. 1858, p. 50. 8 Some further remarks on the forces experienced by currents in magnetic fields will be found in Chap. lii. 45 690 ELECTRO-DYNAMICS. magnetic force. For example, in the apparatus of Fig. 486, the ver- tical wire op will place itself to the west or east (magnetic) of the pillar A B, according as the current in op is ascending or descending. This effect is due to the horizontal component of terrestrial mag- netism. In the apparatus of Fig. 481, if the current be sent only through the movable portion, continuous rotation will be produced, which will be with or against the hands of a watch according as the current in the top wires is inwards or outwards. This effect is due to the ver- tical component of the earth's magnetism, acting on the currents in the horizontal wires. Vertical lines of magnetic force falling on a horizontal current give the latter a tendency to move perpendicular to its own length in a horizontal plane. 561. Solenoids. If we suspend from Ampere's stand (Fig 473) a plane circuit, whether rectangular or circular, it will place itself perpendicular to the magnetic meridian, in such a manner that the current in its lower side is from east to west; or, in other words, so that the ascending current is in its western and the descending cur- rent in its eastern side ; this effect being due to the action of the horizontal component of terrestrial magnetism upon the ascending and descending parts of the current. If, then, we have a number of such circuits, rigidly connected together at right angles to a com- mon axis, and with their currents all circulating the same way, their common axis will tend to place itself in the magnetic meridian, like II the axis of a magnet. Such a j lf ^ . AJ[A system was called by Ampere a solenoid (aw\i]v, a tube), and was realized by him in the following manner. Imagine a wire bent into such a shape as to consist of a number of rings united to each other by straight portions. It will differ from a theoretical solenoid only Fig. 49i.-soienoi d8 . b J having currents in these straight portions; but if the two ends of the wire be carried back till they nearly meet in the middle of the length, as shown at A and B (Fig. 491), the currents in these returning portions, being opposite to those in the other straight portions, will destroy their effect, and the re- ORIENTATION OF SOLENOID. 691 North. KtiBt sultant electro-dynamic action of the system will be simply due to the currents in the rings. The same effect is more conveniently obtained by substituting for the rings and intermediate straight portions, a helix, which, by the principles of sinuous currents, is equivalent to them. Each spire of the helix represents a circle perpendicular to the axis, together with a straight portion parallel to the axis and equal to the distance between two spires. The effect of all the straight portions is exactly destroyed by the wires which return from the ends of the helix and meet in the middle. This arrangement, which is re- presented at C, is that which is universally adopted, the returning wires being sometimes in the axis, and sometimes on the outside of the helix. If a solenoid, thus con- structed, be suspended on an Ampere's stand, as in Fig. 492, and a current sent through it, it will immediately place its axis parallel to a declination needle. It may accord- ingly be said to have poles. In Fig. 493, A represents the austral or north- seeking, B the boreal or south-seeking pole of the solenoid; that is to say, the direction of the current is against or with the hands of a watch according as the austral or boreal pole is presented to the observer. The same difference is illustrated by Fig. 492. 562. Dip of Solenoid. If a solenoid could be balanced so as to be perfectly free to move about its centre of gravity, it would place its axis parallel to the dipping-needle. The experiment would be scarcely practicable with a solenoid properly so called, on account of its weight; but it can be performed with a single plane circuit, such as that shown in Fig. 494. If such a circuit is nicely balanced about South. Fig. 492. Orientation of Solenoid. Fig. 493. Poles of Solenoid. 692 ELECTRO-DYNAMICS. an axis through its centre of gravity, and placed so that it can turn freely in the plane of the magnetic meridian, the passing of a current through it will cause it to set its plane perpendicular to the directioi t of a dipping-needle. This effect is due to the action of terrestrial magnetism on the upper and lower sides of the rectangle. The plane of the rectangle is re- presented in the figure as coinciding with the direction of dip. In this position the action of terrestrial magnet- ism urges the upper side backwards, and the lower side forwards, and stable equilibrium will be attained when Fig. 494. Dip of Element of Solenoid. the rectangle has turned through 90. 563. Mutual Actions of Solenoids. Solenoids behave like magnets not only as regards the forces which they experience from terrestrial magnetism, but also as regards the actions which they exert upon one another. The similar poles of two solen- oids repel, and the unlike poles attract each other, as we may easily prove by suspending one solenoid from an Ampere's stand and bringing another near it. The reason of these attractions and re- pulsions is illustrated by Fig. 495. If two austral poles are placed opposite each other, as in the upper part of the figure, the cur- rents are circulating round them in opposite directions, and, by the laws of parallel cur- rents, should therefore repel each other; whereas if two dissimilar poles be placed face to face, the currents which circulate round them are in the same direction, and attraction should therefore ensue. Fig. 495. Mutual Action of Solenoids. ACTION OF MAGNET ON SOLENOID. 693 Lastly, if one pole of an ordinary magnet be brought near one pole of a suspended solenoid, as in Fig. 496, repulsion or attraction will be exhibited according as the poles in question are similar or dis- similar. In the position represented in the figure, this action is mainly due to the action of the boreal pole of the magnet upon the descending currents in the near side of the solenoid. This action consists in a force to the left hand, nearly parallel to the axis of the solenoid, which tends to make the solenoid rotate about its supports, and thus to bring the end A of the solen- oid into contact with the end B of the magnet. It may be shown, by the aid of Ampere's formula for the mutual force between two elements, that the mu- tual action of two solenoids is equi- valent to four forces, directed along lines joining the poles of the solenoids, and varying inversely as the squares of the distances between the poles; the forces between similar poles being repulsive, and the other two attractive. The analogy between solenoids and magnets is thus complete. 564. Astatic Circuits. When it is desired to eliminate the influence Fig. 496. Action of Magnet on Solenoid. Fig. 497. Astatic Circuit*. netization is greatest when the soft-iron magnet is horizontal, vanishes when it is vertical, and in passing through the vertical position undergoes reversal. If we call one direction of magnetization positive and the opposite direction negative, the strongest positive magnetization corresponds to one of the two horizontal positions, and the strongest negative to the other, the two positions differing by 180. While the magnet, then, is revolv- ing from one horizontal position to the other, its magnetization is changing from the strongest positive to the strongest negative, and this change produces a current in one definite direction in the sur- rounding coil. During: the next half-revolution the magnetization o o <-? is again gradually reversed, and an opposite current is generated in the coil. If we examine the direction of the currents due to the cutting across of the lines of force of the permanent magnet by the convolutions of the coil, we shall find that they concur with those due to the action of the cores. The current in the coils circulates in one direction as long as the electro-magnet is moving from one horizontal position to the other, and changes its direction at the instant when the cores come opposite the poles of the steel magnet. By the aid of the commutator represented in Fig. 554, the currents may be made to pass always in the same direction through an external circuit, r and r' are two contact-springs bearing against the two metal pieces E, E', which are the terminals of the coil. At the instant when the current in the coil is reversed, these springs CLARKE'S MACHINE. 769 are in contact with intermediate insulating pieces which separate the metallic pieces E, E'. When the current in the coil is in one direc- tion (say from E to E 7 ), r is in contact with E, and r' with E'. When the current in the coil is in the opposite direction (E' to E), r is in contact with E', and r' with E ; thus in each case r is the positive and r the negative spring, and the current will be from r to r' in an external connecting wire. OO, O'O', are metallic pieces insulated from each other, and connected with the springs rr' respectively. Binding-screws are provided for attaching wires through which the current is to be passed. With this machine water can be decomposed, wire heated to red- ness, or soft iron magnetized; but these effects are usually on a small scale on account of the small dimensions of the machine. For giving shocks, two wires furnished with metallic handles are attached to the binding-screws, and a third spring is employed which puts the terminals E E' in direct connection with each other twice in each revolution, by making contact with two plates q. When these contacts cease, the current is greatly diminished by having to pass through the body of the person holding the handles, and the extra- current thus induced gives the shock. To obtain the strongest effect, the hands should be moistened with acidulated water before grasping V the handles. 622. Magneto-electric Machines for Lighthouses. Very powerful effects can be obtained from magneto-electric machines of large size driven rapidly. Such machines were first suggested by Professor Nollet of Brussels; and they have been constructed by Holmes of London and the Compagnie 1'AUiance of Paris. It is by means of these machines that the electric light is maintained in lighthouses; they have also been employed to some extent in electro-metallurgy. Fig. 555 represents the pattern adopted by the French company. It has eight rows of compound horse-shoe magnets fixed symmetri- cally round a cast-iron frame. They are so arranged that opposite poles always succeed each other, both in each row and in each cir- cular set. There are seven of these circular sets, with of course six intervening spaces. Six bronze wheels, mounted on one central axis, revolve in these intervals, the axis being driven by steam-power transmitted by a pulley and belt. The speed of rotation is usually about 350 revolutions of the axis per minute. Each of the six bronze wheels carries at its circumference sixteen coils, corresponding to the number of poles in each circular set. The core of each coil is 50 770 INDUCTION OF CURRENTS. a cleft tube of soft iron, this form having been found peculiarly favourable to rapid demagnetization. Each core has its magnetism reversed sixteen times in each revolu- Fig. 5.15. Lighthouse Machine. tion, by the influence of the sixteen successive pairs of poles between which it passes, and the same number of currents in alternately opposite directions are generated in the coils. The coils can be con- nected in different ways, according as great electro-motive force or small resistance is required. The positive ends are connected with the axis of the machine, which thus serves as the positive electrode, and a concentric cylinder, well insulated from it, is employed as the negative electrode. When the machine is employed for the production of the electric light, the currents may be transmitted to the carbon points in alter- nate directions, as they are produced. For electro-metallurgical purposes they are brought into one constant direction by a com- mutator, as in Clarke's machine above described. The driving-power required for lighthouse purposes is about three horse-power. 623. Siemens' Armature. An important improvement in Clarke's machine was introduced by Siemens of Berlin in 1854. It consists in the adoption of a peculiar form of electro-magnet, which is repre- SIEMENS ARMATURE. 771 sented in Fig. 556. The iron portion is a cylinder with a very deep and wide groove cut along a pair of opposite sides, and continued round the ends. The coil is wound in this groove like thread upoii a shuttle. Regarded as an electro-magnet, the poles are not the ends of the cylinder, but are the two cylindrical faces which have not been cut away. In Fig. 557, a 6 is a section of the armature with the coil wound upon it. ABMN is a socket within which the armature revolves, the portions AB being of iron, and M N of brass. The advantage of Siemens' armature is that, on account of the small space required for its rotation, it can be kept in a region of very intense magnetic force by the use of comparatively small magnets. Its form is also eminently favourable to rapid rotation. It is placed between the opposite poles of a row of horse-shoe magnets which be- stride it along the whole of its length, as shown at the top of Fig. 559, and is rotated by means of a driving-band passing over the pulley shown at the lower end of Fig. 556. The polarity of the electro-magnet is reversed at each half-revolution as in Clarke's arrangement, and the alter- nately opposite currents generated are reduced to a com- mon direction by a commutator nearly identical with Clarke's, and represented in Figs. 556, 558. Siemens' Fig. sse. machines are much more powerful than Clarke's when of 8l ^u 8 re Ar the same size. 624. Accumulation by Successive Action: Wilde's Machine. By Fig. 557. Section of Siemens' Armature. Fig. 558. Commutator. employing the current from a Siemens' machine to magnetize soft 772 INDUCTION OF CURRENTS. iron, we can obtain an electro-magnet of much greater power than the steel magnets from whose induction the current was derived. By causing a second coil to rotate between the poles of this electro- magnet, we can obtain a current of much greater power than the Fig. 559. Wilde's Machine. primary current. This is the principle of Wilde's machine, which is represented in Fig. 559. It consists of two Siemens' machines, one above the other. The upper machine derives its inductive action from a row of steel magnets M, whose poles rest on the soft- iron masses m, n, forming the sides of the socket within which a Siemens' armature r rotates. The currents generated in the coil, after being SIEMENS' AND WHEATSTONE'S MACHINE. 773 reduced to a uniform direction by a commutator, flow to the l>inding-screws p, q. These are the terminals of the coil of the large electro-magnet AB, through which accordingly the current circulates. The core of this electro magnet consists of two large plates of iron, connected above by another iron plate, which supports the primary machine. Its lower extremities rest, like those of the primary magnets, on two iron masses T, T, separated by a mass of brass i; ;md a second Siemens' armature F, of large size, revolving within this system, furnishes the currents which are utilized externally. Wilde's machine produces calorific and luminous effects of remark- able intensity ; but the speed of rotation required is very great, being sometimes 1500 revolutions a minute for the large, and 2000 for the small armature. This great speed involves serious inconveniences; and the machine does not appear to have been used for lighthouses, or other practical purposes. Wilde's principle can be carried further. The current of the second armature can be employed to animate a second electro-magnet of greater power than the first, with a third Siemens' armature revolving between its poles. This has actually been done by Wilde. By means of the current from this triple machine, driven by 15 horse- power, the electric light was maintained between two carbons as thick as a man's finger, and a bar of platinum 2 feet long and a quarter of an inch in diameter was quickly melted. This system of accumulation could probably be carried several steps further, but always with the expenditure of a proportionately large amount of energy in driving it. In no magneto-electric machine can the electrical energy obtained exceed the mechanical energy expended in producing it. 625. Accumulation by Mutual Action: Siemens' and Wheatstone's Machine. Siemens and Wheatstone nearly simultaneously proposed the construction of a magneto-electric machine in which the induced currents are made to circulate round the soft-iron magnet which pro- duced them. Iron has usually some traces of permanent magnetism, especially if it has once been magnetized. This magnetism serves to induce very feeble currents in a revolving armature. These cur- rents are sent round the iron magnet, thus increasing its magnetiza- tion. This again produces a proportionate increase in the induced currents; and thus, by a successive alternation of mutual actions, very intense magnetization and very powerful currents are speedily obtained. In the machine as exhibited by Siemens in 1867, the INDUCTION OF CURRENTS. current was diverted into an external circuit, at regular intervals, by an automatic arrangement. 626. Ladd's Machine. Ladd in 1867 constructed a machine based on the principle of mutual action just described; but, instead of utilizing the current by occasional interruptions, he employed a second revolving armature whose coil was in permanent connection with the external circuit. B, B' (Fig. 560) are two plates of iron surrounded by coils which are connected at the right-hand end so as to form but one circuit. The other ends are attached to two binding-screws connected with Fig. 560. Ladd's Machine. the ends of the coil of a Siemens' armature a'. The direction of winding of the two large coils BB' is the same as for a horse-shoe magnet, so that the two poles at either end are of opposite sign. The ends of the cores are let into masses of soft iron MM, N N, between which two armatures a a' rotate. The coil of the armature a is connected with the external circuit containing, for example, two carbon points for exhibiting the electric light. On the principle of mutual action, the electro-magnets B,B', which we may suppose to have at first only a trace of magnetism, are soon GRAMMES MACHINE. 773 raised to very intense magnetization by the rapid rotation of the armature a, and as long as the rotation continues, the magnetization is maintained. The rapid rotation of the other armature a between the poles thus strongly excited, produces a very powerful current which can be utilized externally. Ruhmkorff has modified the arrangement by using a single rotating armature with two coils wound upon it, one of them being connected with the electro- magnet, and the other with the external circuit. The efficiency of machines of this description, regarded as means for the transformation of mechanical into electrical energy, is un- doubtedly very considerable ; nevertheless it is not perfect^ a large amount of energy being wasted in generating heat. On account of the high velocity necessary for efficient, working, and the small size of the apparatus in comparison with the currents obtained, the elevation of temperature is often so great as to prove a source of much annoyance. 626bis. Gramme's Machine. The most efficient magneto -electric machine yet invented is that of Mons. Gramme, which has come exten- sively into use in recent years. Let c D E F (Fig. 560A) be a ring of soft iron, wrapped round with insulated copper wire, and revolving in its own plane between the poles P, p' of a fixed magnet. The ring will, at any given instant, con- sist virtually of two semicircular magnets, F C D, FED, having a pair of similar poles at F, and the other pair at D, these being the points directly opposite the poles of the fixed magnet. Since the poles of the ring remain fixed in space, the electric effect in the copper wire is the same as if the wire coil alone rotated, its core re- maining stationary. The effect of this rotation would be, that in the portion CFE of the coil there would be electro-motive force tending to produce a current in one direction, say the direction CFE; while in the other half, c D E, there would be electro- motive force tending to produce a current in the opposite direction that is the direction c D E. The effects in the two halves are opposite as regards the current which they tend to produce in the coil as a whole; but they are the same as regards the electro-motive force between the oppo- site points c and E; and if the two ends of an external conductor be maintained in rubbing contact with the coil at these two points, Fig. 560 A. Magnet and Ring. 774* INDUCTION OF CURRENTS. a permanent current will flow through it in virtue of this electro- motive force. The above reasoning may be put in the following form. Nearly all the tubes of force which run from one pole to the other of the permanent magnet are concentrated in the substance of the iron ring, one half traversing the upper and the other the lower half-ring- Each convolution of the coil, in ascending from its lowest position E by way of F to its highest position c, cuts each of these tubes once, and all in the same direction, namely, from below to above. In descending on the other side by way of D to E, the same tubes are cut, each once, in the opposite direction, namely, from above to below. Hence the movement in E F C generates electro-motive force in one direction through the wire composing the coil, and the movement in c D E generates electro-motive force in the opposite direction ; both parts of the motion conspiring to produce difference of potential between the convolution at c and that at E. The details of the armature of Gramme's machine are shown in- Fig. 560 B, in which different parts are represented in different stages of construction. The ring or core consists of a bundle of iron wires, shown in sec- tion at A. The copper wire, covered as usual with an insu- lating material, is divided into a number of separate coils, as B B. The two ends of each coil are respectively connected to two thick pieces of copper (one of which is marked R R in the figure), against which the rub- bing contact above described takes place, the number of these Fig. sees. coppers being equal to the num- ber of separate coils. In pass- ing the two points most remote from the poles, these coppers rub against two contact-springs, (each consisting of a flexible bundle of copper wires,) connected respectively with two binding-screws, one forming the positive and the other the negative electrode of the machine. As each bundle makes contact with two or more coppers at the same time, the current is never interrupted, and undergoes but small fluctuations of strength. WHEATS-TONE'S TELEGRAPHIC CURRENTS. 775 In consequence of the great steadiness of the current thus obtained, the machine can be used instead of a galvanic battery for nearly all the purposes to which batteries are commonly applied. Instead of producing a current by turning the machine, the machine may be turned by means of a current for example, by connecting the two binding-screws to the poles of a battery. This is a conse- quence of the tendency of the convolutions of the coil, when a current is flowing through them, to move across the tubes of mag- netic force. The direction of the rotation produced by a current is opposite to the direction of the rotation which would produce the current. The possibility of employing a ring-shaped armature, as in Gramme's machine, was first pointed out by Dr. A. Pacinotti of Florence, in an article published in 1865; but he appears to have made an important oversight, in consequence of which the machine, as constructed by him, was of little value. 1 626 A. Wheatstone's Telegraphic Currents. In Wheats! one's Univer- sal Telegraph, which has been partially described in a previous chapter, the magneto-electric currents which give the signals are produced by causing a small flat bar of soft iron to rotate rapidly before the poles of a steel horse-shoe magnet, which has two con- nected coils of wire wound upon it in the same manner as upon electro-magnets. It is in these coils that the currents are generated, the iron bar being a temporary magnet, and thus influencing the coils nearly in the same manner as if it were a permanent magnet. A current is induced in one direction as it approaches the poles, and in the opposite direction as it recedes from them, so that altogether four currents are generated in each complete revolution. On account of the lightness of the bar, it can be rotated with great rapi- dity. 627. Arago's Rotations. Fara- day successfully applied his dis- covery of magneto-electric induc- tion to account for a phenomenon first observed by Arago in 1824, and subsequently investigated by Babbage and Sir John HerscheL A horizontal disc of copper b b, 1 See an article by Dr. Andrews in Nature, voL xii. pages 90-92. Fig. 561. Arago's Rotations. 776 INDUCTION OF CURRENTS. placed in the interior of a box, is set in rapid rotation by turning a handle. Just over the copper disc, but above the thin plate which forms the top of the box, a magnetized needle a a is balanced horizon- tally. When the disc is made to rotate, the needle is observed to 'deviate from the meridian in the direction of the rotation. When the speed of rotation exceeds a certain limit, the needle is not only deflected, but carried round in continuous rotation in the same direction as the disc. The explanation is to be found in the currents which are induced in the disc by its motion in the vicinity of the magnetized needle. The forces between these currents and the needle are (by Lenz's law) such as to urge the disc backwards; and, from the universal relation which subsists between action and reaction, they must be such as to urge the needle forwards; hence the motion. The direction of the induced current at any instant is in fact along that diameter of the disc which is directly under the needle, the circuit being completed through the lateral portions of the disc ; and it is evident that a current thus flowing parallel to the needle underneath it tends to produce deflection. If the continuity of the disc is interrupted by radial slits, the observed effect is considerably weakened inasmuch as the return circuit is broken. Faraday succeeded in directly demonstrating the existence of currents in a disc rotating near a fixed magnet, by exploring its surface with the amalgamated ends of two wires connected with a galvanometer. The experiment performed by Arago may be reversed by setting the magnet in rotation, and observing the effect produced on the disc. The latter, if delicately suspended, will be found to rotate in the same direction as the magnet. This experiment was first performed by Babbage and Herschel. Its explanation is identical with that just given. In both cases the induced rotation must be slower than that of the body turned by hand, as the existence of the induced currents depends upon the motion of the one body relative to the other. When an iron disc is used instead of a copper one, magnetism is induced in the portions which pass under the poles of the magnet; and as this requires a sensible time for its disappearance, there is always attraction between the poles of the needle and the portions of the disc which have just moved past. The needle is thus drawn forwards by magnetic attraction, and the observed effect is similar to COPPER DAMPERS. 777 that obtained with the copper disc, though the cause 1 is altogether different. 627A. Copper Dampers. Precisely similar to the above is the explanation of the utility of a copper disc in checking the vibrations of a magnetized needle under which it is fixed. As the needle swings to either side, its motion induces currents in the copper which ur^e the needle in the opposite direction to that in which it is moving. When it rests for an instant at the extremity of its swing, the cur- rents cease; and as soon as it begins to return, the currents again resist its motion. A copper plate thus used is called a damper, and the vibrations thus resisted and destroyed are said to be damped. The name is applied to any other means for gradually destroying vibrations, and is probably based on the analogy between this action and the steadying action of a liquid upon a suspended body immersed in it. The resistance which induced currents oppose to the motion pro- ducing them is well illustrated by Faraday's experiment of the copper cube. A cube of copper is suspended by a thread, and set spinning by twisting the thread and then allowing it to untwist. If, while spinning, it is held between the poles of a powerful magnet, like than represented in Fig. 432, it is instantly brought to rest. If the poles are brought very near together, so as to heighten the intensity of the field, and a thin sheet of copper is inserted between them and moved rapidly in its own plane, the operator feels its motion resisted by some invisible influence. The sensation has been compared to that of cutting cheese. Foucault's apparatus for the heating of a copper disc by rotating it between the poles of a magnet ( 356), is another illustration of the same principle. In all cases where induced cur- rents are generated, and are not called upon to perform external work, they yield their full equivalent of heat. The advantage of employing copper in experiments of this kind arises from its superior conductivity, to which the induced currents are proportional. 628. Electro-medical Machines. The application of electricity is often resorted to for certain nervous affections and local paralyses. Many different forms of apparatus are employed for this purpose. 1 That is to say, the main cause; for there must be induced currents in the iron as well as in the copper, though inferior in strength, on account of the inferior conductivity of the former metal 778 INDUCTION OF CURRENTS. One of the most convenient is represented in Fig. 562. Two small coils connected with each other, and furnished with a vibrating contact-breaker, are traversed by the current from a miniature bat- tery. The coils are surrounded by hollow cylinders of copper or brass, in which induced currents are gen- erated as often as the current in the coils is established or interrupted. This action diminishes the energy of the extra- currents on which the shock depends, and the operator can accordingly regulate its strength at pleasure by sliding the cylinders on or off. 628A. Caution regarding Lines of Force. After the very extensive use which has been made in this volume of lines and tubes of force, we think it right to caution the reader against supposing that these conceptions depend upon any doubtful hypothesis. They merely serve, like meridians and parallels of latitude, to map out space in a mode convenient for the statement of phvsical laws. Fig. 5(52. Electro-medical Machine. ADDITIONS IN 1878. 628a. Loop Test. (See p. 675.) The following method of finding the position of a fault in a telegraph wire is an application of the principle of Wheatstone's bridge. We suppose the fault to consist in loss of insulation at some point of the wire, so that the resistance between this point and the ground is much less than it ought to be, though it may still be as great as that of some miles of wire. The fault is known to be between two given stations. At one of these stations let the end of the faulty wire be joined to the end of another wire; and at the other station let the ends A, B, of the same two wires be put in connection with the two poles of a battery LOOP TEST. 779* (Fig. 562A). Also let A and B be connected by a circuit con- taining two variable resistances D, E, and let an intermediate point C be connected, through a galvanometer G, with the ground. Let the resistances D, E, be made such that no current goes through the galvanometer. Then we know that the point C has the same in Earth Earth Fig 562 A. potential as the earth, and these circumstances the faulty point J of the wire will also be at the potential of the earth; for if there were a current flowing through the fault to the earth the battery would be steadily giving off elec- tricity of one sign while having no outlet for electricity of the oppo- site sign, and this cannot be. The points J and C are therefore at the same potential, like the points J and C in Fig. 47lA; and a com- parison of the two figures shows that the same reasoning applies to both. The loop A J B formed by the two telegraph wires is there- fore divided by the point J in the ratio of the two known resistances D and E. That is, we have <* A J resistance of B J = -fy This determines Earth Earth Fig. 562 a the position of the point J. The positions of the battery and galvanometer may be inter- changed, as in Fig. 562s, and the equation above obtained will still apply; for when no current flows through the gal- vanometer in this new arrangement, the two paths C D A J and C E B J, which lead from C to J, must be divided proportionally at A and B. The interchangeableness of the galvanometer and battery is a general property of Wheatstone's bridge. 628 c. Measurement of Electro-motive Force. (See p. 679.) The following mode of determin- ing the electro -motive force of a battery is very convenient in practice. A circuit is formed containing the battery B and two variable resistances R and S (Fig. 562c). Fig. 562 c. At points P and Q in this circuit, one on each side of S, wires are led off to complete 780* MEASUREMENT OF ELECTRO-MOTIVE FORCE. a branch circuit through a galvanometer G and a standard cell C. The resistance R having first been fixed, it is possible, by increasing or diminishing the resistance S, to make the current flow in either direction through the galvanometer, and a certain value of S will bring the needle to zero. Let this value of S be noted, together with the * O simultaneous value of R. Then let R be increased by any convenient amount r, and let S be increased by an amount s just sufficient to bring the needle again to zero. Then, if B denote the resistance of the battery, E its electro-motive force, 6 the resistance of the standard cell, and e its electro-motive force, we know in each case, from the absence of current through the branch, that there is uniform potential along the wire leading from Q, through the galvanometer, to one pole of the cell, and also uniform potential along the wire connecting P with the other pole. If these two wires were respective!} 7 connected with the two electrodes of an electrometer, the difference of potential indicated would be the electro- motive force of the cell. Hence the difference of potential between P and Q is equal to e. Again, since there is no current through the branch, the presence of the branch does not affect the condition of the main circuit B P S Q R, through which a current is passing due to the electro-motive force E overcoming the resistance B + S + R, the resistances of the other parts of the circuit being C supposed negligible. The strength of the current is therefore B + S + R But again, since the difference of potential between P and Q is e, and the intermediate resistance S, the strength of the current is ^' This expression may therefore be equated to the former, and we deduce E _ B + S + R ... e ~~ S Similar reasoning holds for the second experiment, in which R is replaced by R+r, and S by S + s. Hence we have E _ . e ' S + s By taking the differences of numerators and of denominators, we have an equation which determines the value of E, since e, s, and r are known. JABLOCHKOFF'S SYSTEM OF LIGHTING. 781* These experiments also suffice for determining the resistance of the battery, for B can now be found from equation (1). 628D. Jablochkoff's System of Electric Lighting. (See p. 704.) During the recent summer (1878) some of the streets of Paris have been lighted by electric lamps constructed on a plan devised by M. Jablochkoff. Instead of placing the two carbons end to end, and providing mechanism for keeping them at the proper distance, he dispenses with mechanism, and places them side by side, with an insulating substance between them, which is gradually consumed. A A (Fig. 562 D) are the two carbons, separated by a stick of plaster of Paris B. The heat produced by the electric current fuses the plaster of Paris between the points of the carbons, and the fused portion acts as a conductor of high resistance, becoming brightly incandescent. To light the lamp, a piece of carbon, held by an insulator, is laid across the two carbon points until the light appears, and is then removed. The lower ends of the carbons are inserted in copper or brass tubes C C, separated from each other by asbestos; and these tubes are connected by binding-screws with the two wires which convey the current. When the current employed flows always in the same direction, the positive carbon is made twice as large in section as the negative, because it is consumed about twice as fast. When the current is alternating, which is the preferable arrangement, they are made equal The light, when used for street lamps, is surrounded by a globe of opal glass, which serves to diffuse its intensity and prevent dazzling. The current is furnished by a magneto-electric machine, either an ordinary Gramme machine, which gives a current always in one direction, or a Gramme machine specially modified for giving cur- rents in alternate directions. The machine is driven by a small steam or gas engine of as many horse-power as there are lamps to be supplied; sixteen lamps being sometimes supplied in one circuit by a single machine. 628E. Telephone. The articulating telephone invented by Professor Graham Bell is represented in Figs. 562E, 562F. D D is a steel magnet, C a coil of very fine silk-covered copper wire, surrounding the magnet close to one end, and having its terminals in permanent connection with the two binding-screws E E. B B is a thin disc 782* TELEPHONE. B of soft iron, (usually one of the ferrotype plates prepared for photo- graphers,) tightly clamped, in its circumferential portion, between the two parts of the wooden case H H, which are held together by screws, while its central portion is left free and nearly touches the end of the magnet. A A is the mouth-piece, through which the speaker directs his voice upon the iron disc. Two telephones must be employed, one for transmitting, and the other for receiving, one binding-screw of each being connected with the line wire, and the other with the earth or with a return wire, so that their coils form parts of one arid the same circuit, and every current generated in the one traverses the other. The mouth-piece A of the receiving telephone is held to the ear of the listener, and he is able to hear the words which are spoken into the transmitting telephone. There is a great falling Fig. 562 E. -Telephone. ff i n loudness, and a decided nasal twang is imparted, but so much of the original character is preserved that familiar voices can be recognized. Conversations have thus been carried on through 60 or 70 miles of submarine telegraph cable, and through as much as 200 miles of wire suspended in the air on poles. These results, which have come upon the scientific world as a most startling surprise, must be explained as follows. The voice of the speaker produces changes of pressure in the air in front of the iron disc, and thus causes the disc alternately to advance and recede, its movements keeping time with the sonorous vibrations, and the am- plitudes of its movements being approximately proportional to those of the particles of air which convey the sound. Now a piece of soft iron, when brought near a magnet, exercises a quasi attraction upon the lines of force, causing them to be more closely aggregated in its own neighbourhood, and more widely separated in the other parts of the field. Hence when the disc approaches the magnet, it causes Fig. 562 F. Section of Telephone. TELEPHONE. 783* the lines of force to move in towards the axis, and when it recedes it causes them to open out again. The lines of force thus cut the convolutions of the coil in opposite directions, according as the disc is approaching or receding, and give rise to alternate currents. These currents passing through the coil of the receiving telephone, render this coil a magnet, and cause it, according to the direction of the current, to assist or to oppose the attraction of the steel magnet for the iron disc. The disc is accord- ingly set in vibration, and imitates on a diminished scale the move- ments of the disc of the transmitter. Thus the original sonorous vibrations, having first been converted into undulating currents of electricity, are reproduced as sonorous vibrations. The currents are excessively feeble, probably millions of times feebler than ordi- nary telegraphic currents ; but on the other hand the ear is extremely sensitive to movements however small which recur periodically. Lord Rayleigh has made experiments from which it appears, that the note of a whistle is audible at a distance at which the amplitude of the vibrating particles of air is less than a millionth of a millimetre. When the telephone is employed for conversing through one of a number of telegraphic wires suspended on the same poles, it is found that messages sent by ordinary telegraphic instruments along the other wires are audible in the telephone as a succession of loud taps, so loud in fact as seriously to interfere with the telephonic con- versation. This is an illustration of the principle, that the starting or stopping of a current in one wire gives rise to an induced current in a neighbouring wire ; but the induced currents in this case, though so loudly audible in the telephone, have never been detected by any other receiving instrument. The telephone appears likely to sup- plant the galvanometer as a means of detecting feeble currents. 628F. Microphone. Fig. 562c represents one of the best forms of the microphone of Professor Hughes, the inventor of the print- ing telegraph which we have de- scribed in 591. A is a stick of carbon about an inch long, sharpened at both ends, which rest in cavities in the two horizontal supports B B, also of carbon. The upper end of A 50* Fig. 562 o. Microphone. 78-i* MICROPHONE. is free to rattle about in the cavity which contains it, but not to fall away. The two wires E E are in connection respectively with the two supports B B, and are used for putting the instrument into circuit with a receiving telephone at another station. A battery, usually consisting of two or three very small cells, is also introduced into the circuit. The back C in which the supports B B are fixed, and the base D, are of wood, and, besides insulating the carbons, serve to convey to them the sonorous vibrations of the air or of surround- ing bodies. These vibrations produce alternate increase and diminu- tion of pressure at the points of contact of the carbons with one another, and as increase of pressure gives closer contact and conse- quently diminished resistance, the current in the circuit undergoes corresponding changes of strength. These changes act upon the receiving telephone, and cause it to emit sounds which are often much louder than the originals. The microphone in fact acts as a relay, turning on and off the current of the battery, like the Morse relay described on p. 725. The action is improved by employing carbon which has been "metallized" by heating it white hot, and then plunging it in mercury. The back C should be attached to the base D by a pivot which permits it to be inclined to one side. The best results for speech are usually obtained with an inclination of some 20 or 30 degrees. When the slope is too small there is an increase of noise in the receiving telephone, but a loss of distinctness. A microphone of the above kind transmits spoken sounds with as much distinctness as a telephone, and with much greater loudness. It has also a sur- prising power of transmitting very faint sounds produced by rubbing or striking the base or back with light bodies. Sounds of this kind which are quite inaudible at the place where they are produced, are easily heard by a person with his ear to the receiving telephone. APPENDIX. ON ELECTRICAL AND MAGNETIC UNITS. (1). The numerical value of a concrete quantity is its ratio to a particular unit of the same kind; the selection of this unit being always more or less arbitrary. (2). One kind of quantity may, however, be so related to two or more others, as to admit of being specified in terms of units of these other kinds. For example, of the three kinds of quantity, called dis- tance (or length), time, and velocity, any one is capable of being expressed in terms of the other two. Velocity can be specified (as regards amount) by stating the distance passed over in a specified time. Distance can be specified by stating the velocity required for describing it in a specified time, and time can be specified by stating the distance described with a specified velocity. Force, distance, and work are in like manner three kinds of quan- tity, of which any two are just sufficient to specify the third. (3). Calculation is greatly facilitated by employing as few original or underived units as possible. These should be of kinds admitting of easy and accurate comparison; and all other units should be derived from them by the simplest modes of derivation which are available. (4). Velocity is proportional directly to distance described, and inversely to the time of its description; and is independent of all other elements. This is expressed, by saying that the dimensions , 7 .. distance length of velocity are -^- or Again, if we define the unit of velocity to be that with which unit distance would be described in unit time, the real magnitude of the unit of velocity will depend upon the units of length and time selected, being proportional directly to the real magnitude of the former, and inversely to the real magnitude of the latter. This is 780 APPENDIX. expressed by saying that the dimensions of the unit of velocity are - . Both forms of expression are convenient; and the ideas which they are intended to express are logically equivalent. (5). All electrical and magnetic units can be derived from units of length, mass, and time. We shall denote length by I, mass by m, and time by t. (6). The unit of velocity is the velocity with which unit length is described in unit time. Its dimensions are j. (7). The unit of acceleration is the acceleration which gives unit increase of velocity in unit time. Its dimensions are t - me I or-p (8). The unit force is that which acting on unit mass produces unit acceleration. Its dimensions are mass X acceleration, or -p-. (9). The unit of work is the work done by unit force working m P through unit length. Its dimensions are force X length, or -,- (10). The unit of kinetic energy is the kinetic energy of two units of mass moving with unit velocity (according to the formula m l z Its dimensions are mass X (velocity) 2 , or -^-, and are the same as the dimensions of work. It might appear simpler to make it the energy of one unit of mass moving with unit velocity ; but if this change were made, it would be necessary either to halve the unit of work, or else to make kinetic energy double of the work which produced it. Either of these alternatives would involve greater inconvenience and complexity than the selection made above. UNITS OF STATICAL ELECTEICITY. (H). Let q denote quantity of electricity measured statically, so that the mutual repulsion of two equal quantities q at distance I, is ^ a . This being equal to a force, the dimensions of an( ^ as e l ec fcro-motive force is difference of potential, these are also the dimensions of potential. (23). The capacity of a conductor is the quotient of quantity of elec- t z tricity by potential; its dimensions are therefore -y. E (24). The resistance R of a circuit is, by Ohm's law, equal to ^. Its dimensions are therefore or -, and are the same as the dimensions of velocity. (25). On comparing the dimensions of the same element as mea- sured according to the two systems, it will be observed that they are not identical. The dimensions of quantity of electricity, for example, in the first system, are to its dimensions in the second, as I to t ; and the dimensions of capacity are as I 2 to t 2 . Notwithstanding this difference of dimensions, two quantities of electricity which are equal when compared statically, are also equal when compared magnetically, or if one be double of the other when compared statically, it will also be double of the other when compared magnetically. (26). An illustration from a somewhat more familiar department may assist the reader in convincing himself that it is possible for one and the same kind of quantity to have different dimensions according to the line of derivation employed. It is well known that uniform spheres attract each other with a force which is directly as the pro- duct of their masses, and inversely as the square of the distance ELECTRO-MAGNETIC UNITS. 783 between their centres. If this law were made to furnish the unit of force, the dimensions of force would be ^, instead of ^-, as pre- viously found. The ambiguity depends partly on the fact that I in the one formula denotes distance between attracting centres, and in the other distance moved over. It is only when the mode of deriva- tion is distinctly specified, or is too obvious to need specification, that the dimensions of a quantity admit of being determinately stated. As the definition of a derived unit necessarily involves a specification of the mode of its derivation, there is some advantage in speaking of the dimensions of a unit, rather than of the dimensions of the quantity which the unit serves to measure. (27). Derived units are often called absolute units; but it seems an abuse of language to define a unit by its relation to other arbitrary units, and then call it absolute. (28). A committee of the British Association have recommended that the centimetre, gramme, and second be adopted as the general basis of all derived units ; and that the units thence derived be dis- tinguished by the initial letters C. G. S. prefixed. (29). Let the units of length, mass, and time in any other system be respectively equal to I centimetres, m grammes, t seconds. Then the new electro-magnetic unit of quantity will be m* l times the C. G. S. electro-magnetic unit ; and the new electro-static unit of quantity will be m* I* t~ l times the C. G. S. electro-static unit. If the two new units of quantity are equal, we shall have the following relation between the two C. G. S. units, namely m* # electro-magnetic units = m* II t' 1 electro-static units; that is, C. G. S. electro-magnetic unit _ I. C. G-. S. electro-static unit ~ t But - is clearly the value, in centimetres per second, of that velocity which would be called unity in the new system. This is a definite concrete velocity; and its numerical value will always be equal to the ratio of the electro-magnetic to the electro-static unit of quantity, whatever units of mass, length, and time are employed. From numerous experiments in which the same quantity of elec- tricity was measured both statically and magnetically, it appears that this velocity is (within the limits of experimental error) identi- 784 APPENDIX. cal with the velocity of light. Professor Clerk Maxwell maintains that light, electricity, and magnetism are all affections of one and the same medium; that light is an electro-magnetic phenomenon, and that its laws can be deduced from those of electricity and mag- netism. NOTE 1, p. 561, 572. The total work done in charging a conductor (or the total energy which runs down in discharging it) is half the product of potential and charge. For if we suppose the charge to be communicated in a numerous succession of equal parts, it is only the later parts that will be raised through the full difference of potential ; and the mean dif- ference of potential through which the successive parts are raised will be the half of this. NOTE 2, p. 632 If the earth were a uniformly^ magnetized sphere, its effect would be the same as that of a small magnet, of the same moment, at the centre. For if we have a sphere built up of a number of equal and similar small magnets, with their poles pointing the same way, we may suppose all the imaginary fluid at their northern ends to be collected at one central point, and all the imaginary fluid at their southern ends at another central point, the distance between the two central points being equal to the common length of the small magnets. INDEX TO PAET III. Absolute unit of force, 780. of work, 780. units, 783. Accumulation by mutual action, 773- Alarum, telegraphic, 721. vibrating, 721. Alphabet, telegraphic, 724. Alternate contact, 530. discharge by, 573. Amalgam for rubbers, 536. Amalgamated zinc, 651. Ampere's electro - dynamic for- mula, 688. rule for deflection, 657. stand, 680. theory of magnetism, 694. Anode, 739. Arago's rotations, 775. Arc, voltaic, 703. Armstrong's hydro - electric ma- chine, 539. Arrangement of cells in battery, 671. Astatic circuits, 693. galvanometer, 662. needle, 661. Atlantic cable, 733. velocity through, 586. Atmospheric electricity, 599-611. modes of observing, 603-606. results of observation, 607. Attraction, electrical, laws of, 520- 523- magnetic, laws of, 619. Aurora borealis, 634. Aurum musivum, 536. Austral pole, 616. Autographic telegraph, 730. Automatic system, Wheatstone's. 735- Axis, magnetic, 620. Azimuth, 614. Babbage & Herschel's rotations, 776. Bain's electro-chemical telegraph, 730. Balance, torsion, 519, 624. Battery, galvanic, 644. Bunsen's, 650. Cruickshank's, 650. Daniell's, 649. Grove's, 651. Hare's, 648. telegraphic, 715. Wollaston's, 647. Battery of Leyden jars, 580. discharge of, 583. Bertsch's electrical machine, 545. Bifilar magnetometer, 630. Biot's hypothesis of terrestrial magnetism, 632. Boreal pole, 616. Bourbouze's electro-magnetic en- gine, 711. Breguet's telegraph, 718. Bridge, Wheatstone's, 674. British Association unit of resist- ance, 760. Broken magnet, 618. Brush, electric, 548. Bucket, electric, 558. Bunsen's cell, 650. c Cage electrometer, 597. Calibration of thermo - multiplier, 664. Capacity, electric, 565. of condenser, 568. specific inductive, 576. Carbon melted, 703. points, image of, 704. Cascade, charge by, 582. Caselli's telegraph, 730. Cathode, 739. Cells,arrangementof,for maximum current, 671. Charge by cascade, 582. residual, 572. Charts of magnetic lines, 631. Chemical action necessary to cur- rent, 652. Chimes, electric, 600. Clarke's machine, 767. Clink accompanying magnetiza- tion, 638. Clocks, electrically controlled, 736. Coatings, jar with movable, 573. Coercive force, 617. Coil, Ruhmkorff's induction, 761. Commutator, 763. Compass, ship's, 634. Compound magnet, 621, 637. Condensers, electric, 567. capacity of, 568. discharge of, 569. Condensing electroscope, 579. power, 574. Conductivity, comparison of ther- mal and electrical, 670. electrical, see Resistance. Conductors, electrical, list of, 507. lightning, 601-603. Consequent points, 636. Contact-electricity, 643. Contiguous particles, induction \\, 5i5, 578. Convection of electricity, 531, 604 Copper-cube experiment, 777. Coulomb's torsion-balance.s 19,624. Couronne de lasses, 646. Cruickshank's trough, 647. Current, deflected by magnetic force, 658. direction of, in battery, 643. induced by motion across lines of force, 752760. numerical estimate of, 658. Cushions of electrical machine. 535, 536. Cyclones, 611. D Dampers, copper, 777. Daniell's battery, 649. Declination magnet, 626. magnetic, 615. changes of, 633. theodolite, 626. Deflagrator, Hare's, 648. Delezenne's circle, 759. Density, electric, 528. Despretz"s experiments on heat of voltaic arc, 703. Dial telegraphs, 718, 722. Diamagnetic bodies, 638; their coefficient of induction nega- tive, 781. Dielectric, influence of, 575. polarization of, 578. Differential galvanometer, 661. Dimensions of units, 779. Dip, 615. Dip-circle, 628. Discharge in rarefied gases, 549- 552, 765- Discharger, jointed, 569. universal, 584. Dissipation of charge, 531. Distribution of electricity on con- ductors, 528. Divided circuits, 673. Dry pile, 651. Duality of electricity, 508. Duboscq's regulator, 706. Dynamo-electric machines, see Accumulation by Mutual Ac- tion. E Earth, action of, on currents, 689. as a massnet, 632. 50' 786* INDEX TO PART III. Earth-currents, 634. Efficiency of engines, 710. Electrical force at a point defined, 559- machines, 533, see Machine. Electric chimes, 600. egg, 55- light, 702, 769, 781*. pendulum, 509. spark, 546, see Spark. telegraph, 713-736. whirl, 558. Electricity, 505. atmospheric, 599. voltaic, 642. Electrodes of battery, 644, 739. Electro-dynamics, 680. gilding, 746. -magnetic engines, 710. magnets, 697. medical machines, 778. motors, 710. Electrolysis, 738-744. Electrolytes, conduction in, 746. Electrometer, absolute, 592. attracted disc, 591. cage, 507. portable, 593. quadrant, 595. Electrometers, 591-598. Electro-motive force,665, 677, 779*. its value for different bat- teries, 679. Electrophorus, 544. Electro-plating, 746. Electroscope, 517. Bohnenberger's, 652. condensing, 579. Electrotype, 747. Elements of currents, mutual ac- tion of, 688. Ellipsoid, distribution of electri- city on, 529. Elmo's fire, St., 602. Equipotential surfaces, 561. Extra current, 761. F Faraday's experiments within elec- trified box, 527. views regarding electro-static induction, 578, 515. Field, magnetic, 620. intensity of, 620. uniform, 757. Filings, lines formed by, 612. Fluids, electric, theories of, 510. imaginary magnetic, 618. Force, lines and tubes of, 560-563. their movement, 757. their relation to induced currents, 754-760. unit of, 780. Foucault's regulator, 707. Franklin's experiment on light- ning, 599- Frog, experiment with, 645. Froment's engine, 712. Fuse, Statham's, 764. G Galvani, 644. Galvanic battery, 644. electricity, 642. jalvanometers, 659-664. choice of, 677. ias-battery, 745. eissler's tubes, 765. imbals, 634. rold-leaf electroscope, 517. Tamme's machine, 773*. irotthus' hypothesis, 739. Grove's battery, 651. H Hail, Volta's theory of, 610. Hare's deflagrator, 648. Heat, effects of, on magnets, 638. produced by discharge of Ley- den jars, 584, 590. by electric currents, 699. Holtz's electrical machine, 541. Houdin's regulator, 708. Hughes' printing telegraph, 726. Hydro-electric machine, 539. Ice-pail experiment, 526, 564. Images, electric, 566. Imaginary magnetic matter, 619. Inclination, magnetic, 615. Induced currents, 750-760. Induction coil, 761. electric-static, 513-527- relation to force-tubes, 563. magnetic, 617, coefficientof,78i. Inductive capacity, specific, 576. Insulators, list of, 507. Intensity, horizontal, vertical, and total, 623. of field, 620, 781. of magnetization, 621, 781. Isoclinic and other magnetic lines, 632. J Jones' controlled clocks, 736. Joule's law for energy of current, 699-702. K Key, Morse's telegraphic, 724. Kienmayer's amalgam, 536. Kinnersley's thermometer, 555. Ladd's machine, 774. Lenz's law, 753. Leyden battery, 580. jar, 571. capacity of, 568. with movable coatings, 573. Lichtenberg's figures, 581. Light, electric, 702. for lighthouses, 769. Lightning, 599. conductors, 601. duration of, 600. Lines, isoclinic, isodynamic, iso gonic, 632. Lines of force, 560. caution regarding, 778. due to current, 657, 689. magnetic, 619. shown by filings, 613. Local action, 651. Lodestone, 612. M Machine, electrical, 531. Bertsch's, 545. Guericke's, 533. Holtz's, 541. Nairne's, 537. Ramsden's, 535 Winter's, 538. hydro - electric, Armstrong's, 539- Machines, magneto-electric, 766- 774- Magnet, ideal simple, 620. moment of, 621, 622. natural, 612. Magnetic attraction and repulsion. 619. charts, 631. curves formed by filings, 613. fluids, imaginary, 618. meridian, 615, 631. potential, 619. storms, 634. variations, 633. Magnetism, remanent or residual, 698. Magnetization.methods 0^635,696. specification of, 619. Magneto-crystallic action, 640. electric machines, 766-775. Magnetometers, 630. Matches for collecting electricity, 605. Maxwell's rule for action between circuits, 689. Melloni's method of evaluating deflections, 664. Meridian, 615. Meridians, chart of magnetic, 631. Microphone, 783*. Mines, firing by electricity, 590. Mirror electrometer, 594. galvanometer, 663. Moment of magnet, 621-622. Morse's telegraph, 722. telegraphic alphabet, 724. Mortar, electric, 555. N Nairne's electrical machine, 537. Needle, magnetized, 614. o CErsted's experiment, 656. Ohm as unit of resistance, 758-760. Ohm's law, 665. Parallel currents, 682. Paramagnetic bodies, 638, 781. Peltier effect, 708. Pendulum, electric, 509. electrically controlled, 737. Perforation by electric discharge, 588. Phillips' electrophorus, 544. Pile, dry, 651. Volta's, 645. Pistol, Volta's, 556. Pixii's machine, 767. Points discharge electricity, 530. wind from, 557. INDEX TO PART III. 787' Polarization in batteries, 649, 675. of dielectric, 578. Poles of battery, 644. of magnet, 612. -- their names, 616. Portable electrometer, 592. Portative force, 637. Portrait, electric, 585. Potential, 559. analogous to level, 561. curve of, in battery, 676. equal to sum of quotients, 564. its relation to force and work, strong and feeble, 579. Proof-plane, 524. Puncture by electric discharge, 588. Q Quadrant electrometer, 594. - electroscope, 536. R Ramsden's electrical machine, 535. Rarefaction by Alvergniat's meth- od, 551. Rarefied gases, discharge in, 765. Regulators for electric light, 705- 708. Relay, 725. Remanent magnetism, 698. Replenisher, 597. Repulsion, see Attraction. a more reliable test than attrac- tion, 516. Residual charge, 609. magnetism, 698. Resistance, electrical, 666. -- and thermal, compared, 670. in battery, 677. of wires, 667. specific, 667. table of, 670. unit of, 758, 782. Rheostat, 668. Rotations, electro-dynamic, 683. electro-magnetic, 695. Rubbers of electrical machine, 53 = -536. Ruhmkorff's coil, 761. Rupture of magnet, 618. S Saturation, magnetic, 636. Sawdust battery, 651. Schweiger's multiplier, 660. Secondary coil, 762. pile, 746. Series, arrangement of cells in, 672. Siemens' armature, 771. and Wheatstone's machine; 773- Simple magnet, ideal, 620. Sine-galvanometer, 659. Sinuous currents, 688. Solenoids, 690. Spangled tube, 553. Spark, electric, 546. colour of, 552. duration of, 549. heating effects of, 556. in rarefied air, 550. Specific inductive capacity, 576. Sphere, electric capacity ot, 565. Squares, inverse, in electricity, 520-528. Statham's fuse, 764. Steel, its magnetic properties, 617. Step-by-step telegraphs, 718-722. Storms, magnetic, 634. Stratification in electric discharge, 765- Strength of pole, 620. of current, 658. Submarine telegraphs, 733. the Atlantic cables, 733. inductive action, 734. Surface, electricity resides on, 523. Tangent galvanometer, 660. Telegraph, autographic, 730. automatic, 735. dial, 718, 722. electric, 713-736. electro-chemical, 730. Morse's, 722. printing, 726. single-needle, 716. submarine, 733. Telegraphic alarum, 721. alphabet, 724. Telephone, 781*. Tension, electric, 579, Testing for faults, 778. Thermo-electricity, 652-655. Thermo-pile, 654. Thunder, 601. Tickling by electricity, 554. Tornadoes, 6n, 503. Torsion balance, 519, 624. Transport of elements, 739. Tubes of force, 562. -- movement of, 757. -- relation of, to induce cur- rents, 754-760. Tyndall on magneto -crystallic action, 640. u Unit-jar, 587. Unit of resistance, B.A., 760. Units and their dimensions, 779- Variation of magnetic elements, 633. Velocity of electricity, 585. Vitreous and resinous electricity, 510. Volta, 645. Voltaic arc, 703. electricity, 642. element, 643. Voltameter, 738, 742. w Water-dropping collector, 604. -- spouts, 611. Watering-pot, electric, 558. Wheatstone's automatic system, 735- bridge, 674, 778. rotating mirror, 549, 586. universal telegraph, 721, 775. and Cooke's telegraphs, 716. Whirl, electric, 558. Wilde's machine, 772. Wind, from points, 557. Winter's electrical machine, 538. Wires, telegraphic, 716. Wollaston's battery, 647. Work done by current, 699-702. Zamboni's pile, 652. ELEMENTARY TREATISE ON NATURAL PHILOSOPHY. BY A. PEIVAT DESCHANEL, FORMERLY PROFESSOR OF PHYSICS IN THE LYCEE LOUIS-LE-GRAND, INSPF.CTOR OF THE ACADEMY OF PARIS. TRANSLATED AND EDITED, WITH EXTENSIVE ADDITIONS, BY J. D. EVEKETT, M.A., D.C.L., F.R.S.E., PROFESSOR OF NATURAL PHILOSOPHY IN THE QUEEN'S COLLEGE, BELFAST. Part IV.-SOUND AND LIGHT. ILLUSTRATED BY 187 ENGRAVINGS ON WOOD, AND ONE COLOURED PLATE. FIFTH EDITION. LONDON: BLACKIE & SON, 49 OLD BAILEY, E.C.; GLASGOW AND EDINBURGH. 1879. All Rights Reserved, NOTE PREFIXED TO FIRST EDITION. IN the present Part, the chapters relating to Consonance and Dissonance, Colour, the Undulatory Theory, and Polarization, are the work of the Editor ; besides numerous changes and additions in other places. The numbering of the original sections has been preserved .only to the end of Chapter LX. ; the two last chapters of the original having been transposed for greater convenience of treatment. With this exception, the announcements made in the "Translator's Preface," at the beginning of Part I., are applicable to the entire work. The present edition has been carefully revised and corrected. New matter has been introduced at pages 826, 827, 864, 875, 911, 916, 941, 998, 1023, 1024, 1029. J. D. E. QUEEN'S COLLEGE, BELFAST. May, 1879. CONTENTS-PART IV. ACOUSTICS. CHAPTER LIII. PRODUCTION AND PROPAGATION OF SOUND. Sound results from vibratory movement. Vibration of straight spring. Single and double vibration. Period. Amplitude. Isoclironisin. Analogy of pendulum. Vibration of bell. Of plate. Nodal lines. Opposite behaviour of sand and lycopodium. Vibration of string. Of air in a pipe. Chemical harmonica. Trevelyan experiment. Dis- tinctive character of musical sound. Periodicity. Vehicle of sound. Not transmitted through vacuum. Liquids and solids can convey sound. Mode of propagation. Transmission of condensations and rarefactions. Graphical representation. Relation between period, wave-length, and velocity. Nature of undulatory movement. Its geometrical possibility illustrated. Longitudinal and transverse vibrations. Veloci- ties greatest at centres of condensation and rarefaction. Propagation in an open space. Law of inverse squares. Regnault's experiments with sewer-pipes. Transformations of energy in undulation. Dissipation of sonorous energy. Conversion into heat. Velocity of sound in air. Mode of determining. Results. Theoretical computation of velocity. Newton's theory and Laplace's modification. Velocity in gases gene- rally. In liquids. Colladon's experiment at Lake of Geneva. Theoretical computa- tion. Allowance for heat of compression. Velocity in solids. Biot's experiment. Wertheim's experiments and results. Theoretical computation. Reflection of sound. Illustrations. Conjugate mirrors. Echo. Refraction of sound. Speaking-trum- pet. Ear-trumpets. Interference of sound-waves. Experimental illustrations. Interference of direct and reflected waves. Nodes and antinodes. Acoustic pendulum. Beats produced by interference, pp. 785-813. Note A. Rankine's investigation, pp. 813, 814. Note B. Usual investigation of velocity of sound, pp. 814, 815. CHAPTER LIV. NUMERICAL EVALUATION OF SOUND. Ijoudness, pitch, and character. Pitch depends on frequency. Period and frequency are reciprocals. Wave-length is distance travelled in one period. Velocity is wave-length multiplied by frequency. Character or timbre. Musical intervals. Gamut. Tem- perament. Absolute pitch. Limits of musical pitch. Minor scale. Pythagorean scale. Methods of counting vibrations. Syren. Vibroscope for writing vibrations. Phonautograph. Tonometer. Pitch modified by relative motion, . pp. 816-827. CHAPTER LV. MODES OF VIBRATION. Longitudinal and transverse vibrations. Transverse vibrations of strings. Their velocity of propagation and frequency. Sonometer. Harmonics. Overtones not always har- monics. Strings vibrating in segments. Segmental vibration of strings. Sympathetic vibration or resonance. Sounding-boards. Longitudinal vibrations of strings. Stringed instruments. Transversal vibrations of rigid bodies. Plates and bells. iv TABLE OF CONTENTS. Tuning-fork. Affected by temperature. Mounted fork. Law of linear dimensions. Organ-pipes. Mouth-piece. Pitch depends on column of air. Overtones of open and stopped pipes. Nodes and antinodes. Stationary undulations. Wave-length of fun- damental note. Wave-lengths of overtones. Analogous laws for certain vibrations of rods and strings. Application to measurement of velocity of sound in various sub- stances. Reed-pipes. Opposite effect of temperature. Wind-instruments. Mano- metric flames, pp. 828-846. CHAPTER LVI. ANALYSIS OF VIBRATIONS. CONSTITUTION OF SOUNDS. Optical examination of sonorous vibrations. Lissajous' experiment. Composition of two simple vibrations in perpendicular directions. Unison gives an ellipse which can be inscribed in a given rectangle. General equations to Lissajous' figures. Optical tuning. Other modes of exhibiting the composition of rectangular vibrations. Black- burn's pendulum. Elastic rod. Character. Form of vibration. Resolution of perio- dic motions by Fourier's theorem. Every periodic vibration consists of a fundamental simple vibration and its harmonics. Every musical note consists of a fundamental note and its harmonics. Constitution of a vibration defined. Corresponds to character of resulting sound. The harmonics which are present in a note may or may not have their origin in segmental vibrations of the instrument. Combinations of stops in organs. Helmholtz's resonators. Adaptation to manometric flames. Human voice. Vowel sounds. Experiments of Willis, Wheatstone, and Helmholtz, . . . pp. 847-858. CHAPTER LVH. CONSONANCE, DISSONANCE, AND RESULTANT TONES. Concord and Discord. Examples. Consonant intervals can be more accurately identified than dissonant. Dissonance depends on the jarring effect of beats not too slow nor too rapid. 33 beats per second give a maximum of discomfort. Proof that all beats are due to imperfect unison. Beating notes must be near in pitch. Helmholtz's calcula- tion of amounts of dissonance. Discordant elements in an imperfect concord. Result- ant tones. Difference-tones discovered by Sorge and Tartini. Erroneously attri- buted to coalescence of beats. Summation-tones. Resultant tones occur when small quantities of the second order are sensible. Beats of resultant tones. Edison's phon- ograph, PP- 859-864. OPTICS. CHAPTER LVII. PROPAGATION OF LIGHT. Light. Hypothesis of cether capable of propagating transverse vibrations. Excessive shortness and excessive frequency of luminous waves. Strength of shadows. Recti- linear propagation. Diffraction an exception. Images produced by small apertures. Images of sun. Shadows ; umbra and penumbra. Velocity of light. Seven and a half circumferences of the earth per second. Flzeau's experiment with toothed- wheel. Cornu's determination, 300.4 million metres per second, or 186,700 miles per second. Foucalt's experiment with rotating mirror. Ingenious method of keeping rate of rota- tion constant. Mode of reducing the observations. Velocity deduced from eclipses of Jupiter's satellites. From aberration of stars. Sun's distance deduced from Foucault's determination of velocity. Photometry, principles of. Photometers of Bouguer, Rumford, Foucault, and Bunsen, pp. 865-882. CHAPTER LVIII. REFLECTION OF LIGHT. Plane of incidence and reflection. Angles of incidence and reflection equal. Apparatus TABLE OF CONTENTS. V For verification. Artificial horizon. Regular and irregular reflection. Looking- glasses. Speculum metal. Silvered specula. Plane mirrors. Position and size of image. Images by successive reflections. Parallel mirrors. Mirrors at right angles. Kaleidoscope. Pepper's ghost. Deviation produced by rotating a mirror. Hadley's sextant. Spherical mirror. Centre of curvature. Principal and secondary axes. Principal focus of concave mirror. Parabolic mirrors. Spherical aberration. Con- jugate foci. Formula for conjugate focal distances. Formation of real images. Position of principal focus. March of conjugate foci. Construction for position and size of image. Calculation of size of image. Phantom bouquet. Images on screen. Image as seen directly. Caustic surface. Primary and secondary foci. Primary and secondary focal lines on a screen. Virtual image in concave mirror. Distinction between real and virtual images. Convex mirrors. Cylindric mirrors. Anamor- phosis. Ophthalmoscope and laryngoscope, pp. 883-907. CHAPTER LIX. REFRACTION. Sudden change of direction. Sunbeam entering water. Coin in basin. Stick appears broken. Refractive powers of different media. The denser usually the more refrac- tive. Law of sines. Apparatus for verification. Airy's apparatus. Index of refrac- tion. Table of indices. Critical angle and total reflection. Camera lucida. Image by refraction at a plane surface. Caustic by refraction at a plane surface, and apparent position of virtual image. Refraction through parallel plate. Multiple images in plate. Candle in looking-glass. Prism or wedge. Refraction through it. Displacement of objects seen through it. Investigation of formulae. Geometrical construction for deviation, and proof of minimum deviation. Conjugate foci with respect to prism in position of minimum deviation. Double refraction. Iceland-spar. Ordinary and extraordinary image, pp. 908-928. CHAPTER LX. LENSES. Forms of lenses. Converging and diverging, or convex and concave. Principal axis. Principal focus. Optical centre. Secondary axes. Conjugate foci. Comparative sizes of object and image. Whether image will be erect or inverted. Investigation of formulae for focal length and conjugate focal distances. Conjugate foci on secondary axis. March of conjugate foci. Minimum distance between object and real image is four tunes focal length. Construction for position and size of real image. Calculation of size. Example. Image on cross- wires. Cross-wires at conjugate foci. Aberration of lenses. Virtual images, and formulae relating to them. Concave lenses. Foco- meter. Refraction at a single spherical surface. Camera obscura. Photographic camera. Example of photographic processes. Projection of experiments on screens. Solar microscope. Magic-lantern. Photo-electric microscope, . . pp. 929-945. CHAPTER LXL VISION AND OPTICAL INSTRUMENTS. Description of the eye. Adaptation to different distances. Binocular vision. Data for judgment of distances. Perception of relief. Stereoscope. Visual angle (plane) or apparent length. Apparent area or solid visual angle. Magnifying power. Spec- tacles. Magnifying lens. Visual angle in different positions of lens. Simple micro- scope. Compound microscope. Magnifying power computed and observed. Astrono- mical telescope. Magnifying power computed and observed. Finder. Bright spot. Magnifying power deduced from comparison of object-glass with bright spot. Terres- trial eye-piece. Galilean telescope. Its peculiarities. Opera-glass. Reflecting tele- scopes. Herschelian and Newtonian. Magnifying power. Gregorian and Oasse- granian. Silvered specula. Measure of brightness. Intrinsic and effective. Iiitrin- VI TABLE OF CONTENTS. sic brightness is -S. Surfaces are equally bright at all distances. Image formed by s gramme. The weight of a cubic centimetre of water is about 1 gramme. The weight of a cubic foot of water is about 62 '4 Ibs. ACOUSTICS. CHAPTER LIU. PRODUCTION AND PROPAGATION OF SOUND. 629. Sound is a Vibration. Sound, as directly known to us by the sense of hearing, is an impression of a peculiar character, very broadly distinguished from the impressions received through the rest of our senses, and admitting of great variety in its modifications. The at- tempt to explain the physiological ac- tions which constitute hearing forms no part of our present design. The business of physics is rather to treat of those external actions which constitute sound, considered as an objective existence ex- ternal to the ear of the percipient. It can be shown, by a variety of ex- periments, that sound is the result of vibratory movement. Suppose, for ex- ample, we fix one end C of a straight spring C D (Fig. 563) in a vice A, then draw the other end D aside into the position D', and let it go. In virtue of its elasticity ( 23), the spring will return to its original position ; but the kinetic energy which it acquires in returning is sufficient to carry it to a nearly equal Fig. sea. vibration oi straight spring, distance on the other side ; and it thus swings alternately from one side to the other through distances very gradually diminishing, until at last it comes to rest. Such move- ment is called vibratory. The motion from D' to D", or from D" to D', is called a single vibration. The two together constitute a double 51 786 PRODUCTION AND PROPAGATION OF SOUND. or complete vibration; and the time of executing a complete vibra- tion is the period of vibration. The amplitude of vibration for any point in the spring is the distance of its middle position from one oi its extreme positions. These terms have been already employed ( 44) in connection with the movements of pendulums, to which indeed the movements of vibrating springs bear an extremely close resemblance. The property of isochronism, which approximately characterizes the vibrations of the pendulum, also belongs to the Fig. 564. Vibration of Bell. spring, the approximation being usually so close that the period may practically be regarded as altogether independent of the amplitude. When the spring is long, the extent of its movements may gene- rally be perceived by the eye. In consequence of the persistence of impressions, we see the spring in all its positions at once ; and the edges of the space moved over are more conspicuous than the central parts, because the motion of the spring is slowest at its extreme positions. As the spring is lowered in the vice, so as to shorten the vibrating portion of it. its movements become more rapid, and at the same time VIBRATION OF A BELL. more limited, until, when it is very short, the eye is unable to detect any sign of motion. But where sight fails us hearing comes to our aid As the vibrating part is shortened more and more, it emits a musical note, which continually rises in pitch ; and this effect con- tinues after the movements have become much too small to be visible. It thus appears that a vibratory movement, if sufficiently rapid, may produce a sound. The following experiments afford additional illustration of this principle, and are samples of the evidence from which it is inferred that vibratory movement is essential to the pro- duction of sound. Vibration of a Bell. A point is fixed on a stand, in such a posi- Fig. 565. Vibration of Plate. tion as to be nearly iu contact with a glass bell (Fig. 564-). If a rosined fiddle-bow is then drawn over the edge of the bell, until a musical note is emitted, a series of taps are heard due to the striking of the bell against the point. A pith-ball, hung by a thread, is driven out by the bell, and kept in oscillation as long as the sound continues. By lightly touching the bell, we may feel that it is vibrating ; and if we press strongly, the vibration and the sound will both be stopped. Vibration of a Plate. Sand is strewn over the surface of a hori- zontal plate (Fig. 565), which is then made to vibrate by drawing a 788 PRODUCTION AND PROPAGATION OF SOUND. bow over its edge. As soon as the plate begins to sound, the sand dances, leaves certain parts bare, and collects in definite lines, which are called nodal lines. These are, in fact, the lines which separate portions of the plate whose movements are in oppo- site directions. Their position changes whenever the plate changes its note. The vibratory condition of the plate is also mani- fested by another phenomenon, opposite so to speak to that just described. If very fine powder, such as lycopodium, be mixed with the sand, it will not move with the sand to the nodal lines, but will form little heaps in the centre of the vibrating segments; and these heaps will be in a state of violent agitation, with more or less of gyratory movement, as long as the plate is vibrating. This phe- nomenon, after long baffling explanation, was shown by Faraday to be due to indraughts of air, and ascending currents, Vibra J? n 5 f 66 striug . brought about by the move- ments of the plate. In a moderately good vacuum, the lycopodium goes with the sand to the nodal lines. Vibration of a String. When a note is produced from a musical string or wire, its vibrations are often of sufficient amplitude to be detected by the eye. The string thus assumes the appearance of an elongated spindle (Fig. 566). Vibration of the Air. The sonorous body may sometimes be air, as in the case of organ- pipes, which we shall describe in a later chap- ter. It is easy to show by experiment that when a pipe speaks, the air within it is vibra- ting. Let one side of the tube be of glass, O o ' and let a small membrane m, stretched over Fig 567-vibrationof A*. a frame > be strewed with sand, and lowered into the pipe. The sand will be thrown into violent agitation, and the rattling of the grains, as they fall back on the membrane, is loud enough to be distinctly heard. SINGING FLAMES. 789 Singing Flames. An experiment on the production of musical sound by flame, has long been known under the name of the chemi- cal harmonica. An appara- tus for the production of hydrogen gas (Fig. 568) is furnished with a tube, which tapers oft* nearly to a point at its upper end, where the gas issues and is lighted. When a tube, open at both ends, is held so as to surround the flame, a musical tone is heard, which varies with the dimensions of the tube, and often attains considerable power. The sound is due to the vibration of the air and products of combustion with- in the tube; and on observ- ing the reflection of the flame in a mirror rotating about a vertical axis, it will be seen that the flame is alternately rising and falling, its successive imao-es, as drawn out into a horizontal series by the O / rotation of the mirror, resembling a number of equidistant tongues of flame, with depressions between them. The experiment may also be performed with ordinary coal-gas. Trevelyan Experiment. A fire-shovel (Fig. 569) is heated, and balanced upon the edges of two sheets of lead fixed in a vice ; it is then seen to execute a series of small oscillations each end being alternately raised and depressed and a sound is at the same time emitted. The oscillations are so small as to be scarcely perceptible in themselves ; but they can be rendered very obvious by attaching to the shovel a small silvered mirror, on which a beam of light is directed. The reflected light can be. made to form an image upon a screen, and this image is seen to be in a state of oscillation as long as the sound is heard. The movements observed in this experiment are due to the sudden expansion of the cold lead. When the hot iron comes in contact Fig. 568. Chemical Harmonica. 790 PllODUCTION AND PROPAGATION OF SOUND. with it, a protuberance is instantly formed by dilatation, and the iron is thrown up. It then comes in contact with another portion Fig. 569. Trevelyan Experiment. of the lead, where the same phenomenon is repeated while the first point cools. By alternate contacts and repulsions at the two points, the shovel is kept in a continual state of oscillation, and the regular succession of taps produces the sound. The experiment is more usually performed with a special instru- ment invented by Trevelyan, and called a rocker, which, after being heated and laid upon a block of lead, rocks rapidly from side to side, and yields a loud note. 630. Distinctive Character of Musical Sound. It is not easy to draw a sharp line of demarcation between musical sound and mere noise. The name of noise is usually given to any sound which seems unsuited to the requirements of music. This unfitness may arise from one or the other of two causes. Either, 1. The sound may be unpleasant from containing discordant ele- ments which jar with one another, as when several consecutive keys on a piano are put down together. Or, 2. It may consist of a confused succession of sounds, the changes being so rapid that the ear is unable to identify any particular note. This kind of noise may be illustrated by sliding the finger along a violin-string, while the bow is applied. All sounds may be resolved into combinations of elementary musi- cal tones occurring simultaneously and in succession. Hence the study of musical sounds must necessarily form the basis of acoustics. Every sound which is recognized as musical is characterized by what may be called smoothness, evenness, or regularity; and the physical cause of this regularity is to be found in the accurate VEHICLE OF SOUND. 791 periodicity of the vibratory movements which produce the sound. By periodicity we mean the recurrence of precisely similar states at equal intervals of time, so that the movements exactly repeat them- selves; and the time which elapses between two successive recur- rences of the same state is called the period of the movements. Practically, musical and unmusical sounds often shade insensibly into one another. The tones of every musical instrument are accom- panied by more or less of unmusical noise. The sounds of bells and drums have a sort of intermediate character; and the confused as- semblage of sounds which is heard in the streets of a city blends at a distance into' an agreeable hum. 631. Vehicle of Sound. The origin of sound is alwaj- s to be found in the vibratory movements of a sonorous body; but these vibratory movements cannot bring about the sensation of hearing unless there be a medium to transmit them to the auditory apparatus. This medium may be either solid, liquid, or gaseous, but it is necessary that it be elastic. A body vibrating in an absolute vacuum, or in a medium utterly destitute of elasticity, would fail, to excite our sensations of hearing. This assertion is justified by the following experiments : 1. Under the receiver of an air-pump is placed a clock-work arrangement for producing a number of strokes on a bell. It is placed on a thick cushion of felt, or other inelastic material, and the air in the receiver is exhausted as com- pletely as possible. If the clock-work is then started by means of the handle g, the hammer will be seen to strike the bell, but the sound will be scarcely audi- ble. If hydrogen be introduced into the vacuum, and the receiver be again exhausted, the sound will be much more completely extinguished, being heard pig. 570. Souud in Exhausted Receiver, with difficulty even when the ear is placed in contact with the receiver. Hence it may fairly be con- cluded that if the receiver could be perfectly exhausted, and a per- fectly inelastic support could be found for the bell, no sound at all would be emitted. 792 PRODUCTION AND PROPAGATION OF SOUND. 2. The experiment may be varied by using a glass globe, furnished with a stop-cock, and having a little bell suspended within it by a thread. If the globe is exhausted of air, the sound of the bell will be scarcely audible. The globe may be filled with any kind of gas, or with vapour either saturated or non-saturated, and it will thus be found that all these bodies transmit sound. Sound is also transmitted through liquids, as may easily be proved by direct experiment. Experiment, however, is scarcely necessary for the establishment of the fact, seeing that fishes are provided with audi- ^I'fl 1 ' tory apparatus, and have often an acute sense of Globe with Stop-cock. ' _ hearing. As to solids, some well-known facts prove that they transmit .sound very perfectly. For example, light taps with the head of a pin on one end of a wooden beam, are distinctly heard by a person with his ear applied to the other end, though they cannot be heard at the same distance through air. , This property is sometimes em- ployed as a test of the soundness of a beam, for the experiment will not succeed if the intervening wood is rotten, rotten wood being very inelastic. The stethoscope is an example of the transmission of sound through solids. It is a cylinder of wood, with an enlargement at each end, and a perforation in its axis. One end is pressed against the chest of the patient, while the observer applies his ear to the other. He is thus enabled to hear the sounds produced by various internal actions, such as the beating of the heart and the passage of the air through the tubes of the lungs. Even simple auscultation, in which the ear is applied directly to the surface of the body, implies the transmission of sound through the walls of the chest. By applying the ear to the ground, remote sounds can often be much more distinctly heard ; and it is stated that savages can in this way obtain much information respecting approaching bodies of enemies. We are entitled then to assert that sound, as it affects OUT organs of hearing, is an effect which is propagated, from a vibrating body, through an elastic and ponderable medium. 632. 1 Mode of Propagation of Sound. We will now endeavour to explain the action by which sound is propagated. 1 The numbering of 632-638 does not correspond with the original, some transposi- tions having been made. MODE OF PROPAGATION OF SOUND. 793 Let there be a plate a vibrating opposite the end of a long tube, and let us consider what happens during the passage of the plate from its most backward position a", to its most advanced position a. This movement of the plate may be divided in imagination into a number of successive parts, each of which is communicated to the layer of air close in front of it, which is thus compressed, and, in its Fig. 573. Propagation of Sound. endeavour to recover from this compression, reacts upon the next layer, which is thus in its turn compressed. The compression is thus passed on from layer to layer through the whole tube, much in the same way as, when a number of ivory balls are laid in a row, if the first receives an impulse which drives it against the second, each ball will strike against its successor and be brought to rest. The compression is thus passed on from layer to layer through the tube, and is succeeded by a rarefaction corresponding to the back- ward movement of the plate from a' to a". As the plate goes on vibrating, these compressions and rarefactions continue to be propa- gated through the tube in alternate succession. The greatest com- pression in the layer immediately in front of the plate, occurs when the plate is at its middle position in its forward movement, and the greatest rarefaction occurs when it is in the same position in its backward movement. These are also the instants at which the plate is moving most rapidly. 1 When the plate is in its most advanced position, the layer of air next to it, A (Fig. 574) will be in its natu- ral state, and another layer at A : , half a wave-length further on, will also be in its natural state, the pulse having travelled from A to A 1( while the plate was moving from a" to a. At intervening points between A and A 1; the layers will have various amounts of compression corres- ponding to the different positions of the plate in its forward move- 1 See 632 A, also Note A at the end of this chapter. Fig. 574. Graphical Representation. 794 PRODUCTION AND PROPAGATION OF SOUND. ment. The greatest compression is at C, a quarter of a wave-length in advance of A, having travelled over this distance while the plate was advancing from a to a'. The compressions at D and Dj repre- sent those which existed immediately in front of the plate when it had advanced respectively one-fourth and three-fourths of the dis- tance from a" to a, and the curve AC' Aj is the graphical represen- tation both of condensation and velocity for all points in the air between A and A a . If the plate ceased vibrating, the condition of things now existing in the portion of air A A x would be transferred to successive portions of air in the tube, and the curve A C' A x would, as it were, slide onward through the tube with the velocity of sound, which is about 1100 feet per second. But the plate, instead of remaining perman- ently at a', executes a backward movement, and produces rarefactions and retrograde velocities, which are propagated onwards in the same manner as the condensations and forward velocities. A complete wave of the undulation is accordingly represented by the curve AE'A 1 C'A 2 (Fig. 575), the portions of the curve below the line of Fig. 575. Graphical Representation of Complete Waye. abscissas being intended to represent rarefactions and retrograde velo- cities. If we suppose the vibrating plate to be rigidly connected with a piston which works air-tight in the tube, the velocities of the particles of air in the different points of a wave-length will be iden- tical with the velocities of the piston at the different parts of its motion. The wave-length A A 2 is the distance that the pulse has travelled while the vibrating plate was moving from its most backward to its most advanced position, and back again. During this time, which is called the period of the vibrations, each particle of air goes through its complete cycle of changes, both as regards motion and density, The period of vibration of any particle is thus identical with that of the vibrating plate, and is the same as the time occupied by the waves in travelling a wave-length. Thus, if the plate be one leg of a common A tuning-fork, making 435 complete vibrations per second, the period will be T -^ T th of a second, and the undulation will travel in NATURE OF UNDULATIONS. 795 this time a distance of VW f ee t> or 2 feet 6 inches, which is there- fore the wave-length in air for this note. If the plate continues to vibrate in a uniform manner, there will be a continual series of equal and similar waves running along the tube with the velocity of sound. Such a succession of waves constitutes an undulation. Each wave consists of a condensed portion, and a rarefied portion, which are distinguished from each other in Fig. 573 by different tints, the dark shading being intended to represent condensation. 632 A. Nature of Undulations. The possibility of condensations and rarefactions being propagated continually in one direction, while each particle of air simply moves backwards and forwards about its original position, is illustrated by Fig. 575 A, which represents, in an A B C D E F ^ A B C D E F a/ AB C D E F a/ A BC D E F a/ A BC D E F a/ A B CD E F a/ A B CD E F < K A B C DE F ^ A B C D E F 0, A B C D EF ^ A B C D EF^ A B C D E F ci/ A B C D E F & Fig. 575 A. Longitudinal Vibration. exaggerated form, the successive phases of an undulation propagated through 7 particles A B C D E F a originally equidistant, the dis- tance from the first to the last being one wave-length of the undula- tion. The diagram is composed of thirteen horizontal rows, the first and last being precisely alike. The successive rows represent the positions of the particles at successive times, the interval of time from each row to the next being y^-th of the period of the undulation. In the first row A and a are centres of condensation, and D is a centre of rarefaction. In the third row B is a centre of condensa- tion, and E a centre of rarefaction. In the fifth row the con- 796 PRODUCTION AND PROPAGATION OF SOUND. densation and rarefaction have advanced by one more letter, and so on through the whole series, the initial state of things being reproduced when each of these centres has advanced through a wave- length, so that the thirteenth row is merely a repetition of the first. The velocities of the particles can be estimated by the comparison of successive rows. It is thus seen that the greatest forward velocity is at the centres of condensation, and the greatest backward velocity at the centres of rarefaction. Each particle has its greatest veloci- ties, and greatest condensation and rarefaction, in passing through its mean position, and comes for an instant to rest in its positions o1 greatest displacement, which are also positions of mean density. The distance between A and a remains invariable, being always a wave-length, and these two particles are always in the same phase. Any other two particles represented in the diagram are always in different phases, and the phases of A and D, or B and E, or C and F, are always opposite ; for example, when A is moving forwards with the maximum velocity, D is moving backwards with the same velocity. The vibrations of the particles, in an undulation of this kind, are called longitudinal; and it is by such vibrations that sound is pro- pagated through air. Fig. 575 B illustrates the manner in which an undulation may be propagated by means of transverse vibrations, that is to say, by vibrations executed in a direction perpendicular to that in which the undulation advances. Thirteen particles AB C D EFGHIJKLa are represented in the positions which they occupy at successive times, whose interval is one-sixth of a period. At the instant first considered, D and J are the particles which are furthest displaced. At the end of the first interval, the wave has advanced two letters, so that F and L are now the furthest displaced. At the end of the next interval, the wave has advanced two letters further, and so on, the state of things at the end of the six intervals, or of one complete period, being the same as at the beginning, so that the seventh line is merely a repetition of the first. Some examples of this kind of wave-motion will be mentioned in later chapters. 633. Propagation in an Open Space. When a sonorous disturb- ance occurs in the midst of an open body of air, the undulations to which it gives rise run out in all directions from the source. If the disturbance is symmetrical about a centre, the waves will be spheri- cal; but this case is exceptional. A disturbance usually produces condensation on one side, at the same instant that it produces rare- PROPAGATION IN AN OPEN SPACE. 797 faction on another. This is the case, for example, with a vibrating plate, since, when it is moving towards one side, it is moving away from the other. These inequalities which exist in the neighbour- hood of the sonorous body, have, however, a tendency to become less marked, and ultimately to disappear, as the distance is increased. Fig. 576 represents a diametral section of a series of spherical waves. Their mode of propagation has some analogy to that of the circular B , , D H . , A tf J K L a- p G H | j E d K A B C D L * H I J H , A G L a B C D E F A B > E r G H A B C D E L 6 H I H , j k L Fig. 575 B. Transverse Vibration. waves produced on water by dropping a stone into it; but the par- ticles which form the waves of water are elevated and depressed : whereas those which form sonorous waves merely advance and retreat, their lines of motion being always coincident with the di- rections along which the sound travels. Tn both cases it is im- portant to remark that the undulation does not involve a movement of transference. Thus, when the surface of a liquid is traversed by waves, bodies floating on it rise and fall, but are not carried onward. This property is characteristic of undulations generally. An undu- 798 PRODUCTION AND PROPAGATION OF SOUND. lation may be defined as a system of movements in which the several particles move to and fro, or round and round, about definite Fig. 576. Propagation in Open Space. points, in such a manner as to produce the continued onward transmission of a condition, or series of conditions. There is one important difference between the propagation of sound in a uniform tube and in an open space. In the former case, the layers of air corresponding to successive wave-lengths are of equal mass, and their movements are precisely alike, except in so far as they are interfered with by friction. Hence sound is transmitted through tubes to great distances with but little loss of intensity, especially if the tubes are large. 1 1 Eegnault, in his experiments on the velocity of sound, found that in a conduit '108 of a metre in diameter, the report of a pistol charged with a gramme of powder ceased to be heard at the distance of 1T50 metres. In a conduit of "3 m , the distance was 3810. In the great conduit of the St. Michel sewer, of l m '10, the sound was made by successive reflections to traverse a distance of 10,000 metres without becoming inaudible. D. DISSIPATION OF SONOROUS ENERGY. 791) In an open space, each successive layer has to impart its own condition to a larger layer; hence there is a continual diminution of amplitude in the vibrations as the distance from the sotirce increases. This involves a continual decrease of loud ness. An undulation involves the onward transference of energy ; and the amount of energy which traverses, in unit time, any closed surface described about the source, must be equal to the energy which the source emits in unit time. Hence, by the reasoning which we employed in the case of radiant heat ( 308), it follows that the intensity of sonorous energy diminishes according to the law of inverse squares. The energy of a particle executing simple vibrations in obedience to forces of elasticity, varies as the square of the amplitude of its excursions; for, if the amplitude be doubled, the distance worked through, and the mean working force, are both doubled, and thus the work which the elastic forces do during the movement from either extreme position to the centre is quadrupled. This work is equal to the energy of the particle in any part of its course. At the ex- treme positions it is all in the shape of potential energy ; in the middle position it is all in the shape of kinetic energy; and at intermediate points it is partly in one of these forms, and partly in the other. If we sum the potential energies of all the particles which consti- tute one wave, and also sum their kinetic energies, we shall find the two sums to be equal. 1 633 A. Dissipation of Sonorous Energy. The reasoning by which we have endeavoured to establish the law of inverse squares, assumes that onward propagation involves no loss of sonorous energy. This assumption is not rigorously true, inasmuch as vibration implies friction, and friction implies the generation of heat, at the expense of the energy which produces the vibrations. Sonorous energy must therefore diminish with distance somewhat more rapidly than ac- cording to the law of inverse squares. All sound, in becoming extinct, becomes converted into heat. This conversion is greatly promoted by defect of homogeneity in 1 In the case of one of the particles, the potential energy at distance y from the position of equilibrium is half the product of force by distance, and may be denoted by - y 3 ; the kinetic energy is ~- (a 2 y-), a being the amplitude. The former of these quantities may 2t be written a 2 cos 2 6, and the latter will be -- a 2 sin 2 0. In dealing with the series of 2, 2 particles which form one wave, is equicrescent from particle to particle, and its limiting values differ by an entire circumference. Under these conditions, it is obvious that the mean values of cos 2 and sin 2 are equal, and that each of them is equal to . 800 PRODUCTION AND PROPAGATION OF SOUND. the medium of propagation. In a fog, or a snow-storm, the liquid or solid particles present in the air produce innumerable reflections, in each of which a little sonorous energy is converted into heat. 634. Velocity of Sound in Air. The propagation of sound through an elastic medium is not instantaneous, but occupies a very sensible time in traversing a moderate distance. For example, the flash of a gun at the distance of a few hundred yards is seen some time before the report is heard. The interval between the two impressions may be regarded as representing the time required for the propagation of the sound across the intervening distance, for the time occupied fry the propagation of light across so small a distance is inappreciable. It is by experiments of this kind that the velocity of sound in air has been most accurately determined. Among the best determina tions may be mentioned that of Lacaille, and other members of a commission appointed by the French Academy in 1738; that ot Arago, Bouvard, and other members of the Bureau de Longitudes, in 1822; and that of Moll, Vanbeek, and Kuy tenbrouwer in Holland, in the same year. All these determinations were obtained by firing cannon at two stations, several miles distant from each other, and noting, at each station, the interval between seeing the flash and hearing the sound of the guns fired at the other. If guns were fired only at one station, the determination would be vitiated by the effect of wind blowing either with or against the sound. The error from this cause is nearly eliminated by firing the guns alternately at the two stations, and still more completely by firing them simultaneously. This last plan was adopted by the Dutch observers, the distance of the two stations in their case being about nine miles. Regnault has quite recently repeated the investigation, taking advantage of the important aid afforded by modern electrical methods for registering the times of observed phenomena. All the most careful determina- tions agree very closely among themselves, and show that the velo- city of sound through air at 0C. is about 332 metres, or 1090 feet per second. 1 The velocity increases with the temperature, being 1 A recent determination by Mr. Stone at the Cape of Good Hope is worthy of note as being based on the comparison of observations made through the sense of hearing alone. It had previously been suggested that the two senses of sight and hearing, which are con- cerned in observing the flash and report of a cannon, might not be equally prompt in receiving impressions (Airy on Sound, p. 131). Mr. Stone accordingly placed two ob- servers one near a cannon, and the other at about three miles distance ; each of whom, on hearing the report, gave a signal through an electric telegraph. The result obtained was in precise agreement with that stated in the text. VELOCITY OF SOUND IN AIR. 801 proportional to the square root of what we have called in 21 9 A the absolute temperature. If t denote the ordinary Centigrade tem- perature, and a the coefficient of expansion -00366, the velocity of sound through air at any temperature is given by the formula 332 >/ 1 + a i in metres per second, or 1090 ^/1 + atin feet per second. The actual velocity of sound from place to place on the earth's sur- face is found by compounding this velocity with the velocity of the wind. There is some reason, both from theory and experiment, for be- lieving that very loud sounds travel rather faster than sounds of moderate intensity. 635. Theoretical Computation of Velocity. By applying the prin- ciples of dynamics to the propagation of undulations, 1 it is computed that the velocity of sound through air must be given by the formula * = V* (1) D denoting the density of the air, and E its elasticity, as measured by the quotient of pressure applied by compression produced. Let P denote the pressure of the air in units of force per unit of area; then, if the temperature be kept constant during compression, a small additional pressure p will, by Boyle's law, produce a com- pression equal to^-, and the value of E, being the quotient of p by this quantity, will be simply P. On the other hand, if no heat is allowed either to enter or escape, the temperature of the air will be raised by compression, and addi- tional resistance will thus be encountered. In this case the compres- sion (^in 347 A) will be p /i + g)> 1 -f-/3 denoting the ratio of the two specific heats, which for air and simple gases is about 1*41 ; and the value of E will be P (1+/3). It thus appears that the velocity of sound in air cannot be less than ./^- nor greater than A/1 '41 ^. Its actual velocity, as determined by observation, is nearly identical with the latter of these limiting values. It is probable that the compressions and extensions which the particles of air undergo in transmitting sound are of too brief duration to allow of any sensible transference of heat from particle to particle. 1 See Note B at the end of this chapter. 62 802 PRODUCTION AND PROPAGATION OF SOUND. The following is the actual process of calculation for perfectly dry air at 0G, the centimetre, gramme, and second being taken as the units of length, mass, and time. The density of dry air at 0, under the pressure of 1033 grammes per square centimetre, at Paris, is "001293 of a gramme per cubic centimetre. But the gravitating force of a gramme at Paris is 981 (38). The density -001293 therefore corresponds to a pressure ot 1033x981 units of force per unit of area; and the expression for the velocity in centimetres per second is v = /l-4lZ= /1-41 1033x981 =33210 nearly: V D V -001293 that is, 332'4 metres per second, or 1093 feet per second. 635 A. Effects of Pressure, Temperature, and Moisture. The velo- city of sound is independent .of the height of the barometer, since changes of this element (at constant temperature) affect P and D in the same direction, and to the same extent. For a given density, if P denote the pressure at 0, and a the coefficient of expansion of air, the pressure at t Centigrade is P (1 +a f), the value of a being about 273- Hence, if the velocity at be 1090 feet per second, the velocity at f will be 1090A1 + ^- At the temperature 50 F. or 10 G, which is approximately the mean annual temperature of this country, the value of this expression is about 1110, and at 86 F. or 30 C. it is about 1148. The increase of velocity is thus about a foot per second for each degree Fahrenheit. The humidity of air has some influence on the velocity of sound, inasmuch as aqueous vapour is lighter than air, but the effect is comparatively trifling, at least in temperate climates. At the tem- perature 50 F., air saturated with moisture is less dense than dry air by about 1 part in 220, and the consequent increase of velocity cannot be greater than about 1 part in 440, which will be between 2 and 3 feet per second. The increase should, in fact, be somewhat less than this, inasmuch as the value of 1 +0 (the ratio of the two specific heats) appears to be only T31 for aqueous vapour. 1 635 B. Newton's Theory, and Laplace's Modification. The earliest theoretical investigation of the velocity of sound was that given by Newton in the Principia (book 2, section 8). It proceeds on the 1 Rankine on the Steam Enyine, p. 320. VELOCITY IN GASES GENERALLY. 803 tacit assumption that uo changes of temperature are produced by the compressions and extensions which enter into the constitution of a sonorous undulation; and the result obtained by Newton is equivalent to the formula or since ( 111 A) g=:^H, where H denotes the height of a homo- geneous atmosphere, and the velocity acquired in falling through any height s is */2gs, the velocity of sound in air is, according to Newton, the same as the velocity which would be acquired by falling in vacuo through half the height of a homogeneous atmosphere This in fact, is the form in which Newton states his result. 1 Newton himself was quite aware that the value thus computed theoretically was too small, and he throws out a conjecture as to the cause of the discrepancy; but the true cause was first pointed out by Laplace, as depending upon increase of temperature produced by compression, and decrease of temperature produced by expansion. 635c. Velocity in Gases generally. The same principles which appty to air apply to gases generally ; and since for all simple gases the ratio of the two specific heats is 1-41, the velocity of sound in any simple gas is,^/! 41 ^, D denoting its absolute density at the pressure P. Comparing two gases at the same pressure, we see that the velocities of sound in them will be inversely as the square roots of their absolute densities; and this will be true whether the tern peratures of the two gases are the same or different. 636. Velocity of Sound in Liquids. The velocity of sound in water was measured by Colladon, in 182G, at the Lake of Geneva. Two boats were moored at a distance of 13,500 metres (between 8 and 9 miles). One of them carried a bell, weighing about 140 Ibs., immersed in the lake. Its hammer was moved by an external lever, so arranged as to ignite a small quantity of gunpowder at the instant of striking the bell. An observer in the other boat was enabled to hear the sound by applying his ear to the extremity of a trumpet- shaped tube (Fig. 572), having its lower end covered with a mem- brane and facing towards the direction from which the sound pro- 1 Newton's investigation relates only to simple waves ; but if these have all the same velocity (as Newton shows), this must also be the velocity of the complex wave which they compose. Hence the restriction is only apparent. 804 PRODUCTION AND PROPAGATION OF SOUND. ceeded. By noting the interval between seeing the flash and hearing the sound, the velocity with which the sound travelled through the water was determined. The velocity thus com- puted was 1435 metres per second, and thp temperature of the water was 8'l 0. Formula (1) of 633 holds for liquids as well as for gases, and is easily applied to the case of water if we neglect the changes of temperature produced by compression and extension. We have stated in 22 (Part I.) that the compressi- bility of water, as determined by the most re- cent experiments, is 0000457 per atmosphere, at the temperature of 15 C. The value of E in terms of the units employed in 635 is therefore . O ooo~457 ' an( ^ -^> ^ ne mass f a cubit- centimetre of water expressed in grammes, is unity. We have therefore 1033x981 1l(onoA , ^00457- = which, reduced to metres per second, is 1489 '2. This computation applies to water at 1 5, which is 7 warmer than the water of the lake. As the elasticity of water is known to increase with its temperature, the difference between the two results is in the right direction. The agreement is sufficiently close to show that the increase of elasticity from the instantaneous changes of temperature produced by sonorous undulations is insignificant in the case of water. 1 1 Sir W. Thomson has investigated, on thermo- dynamic principles, the additional pres- sure required to produce a given diminution of volume, when the heat of compression is not allowed to escape. He computes that the elasticity of a fluid under these circum p 2m stances is to its elasticity at constant temperature as 1+ - to 1, E denoting th. J O elasticity at constant temperature, a the coefficient of expansion of the fluid per degree Centigrade, T the absolute temperature of the fluid, or the common Centigrade temperature increased by 273, C the thermal capacity of unit volume of the liquid, and J Joule's equivalent for a unit of heat. If E be expressed in absolute units of force per unit of area. J must be expressed in absolute units of work, and will be42400x981 if the centimetre, the gramme, and the second be the units of length, mass, and time. For water at 15 C. the coefficient of expansion o is about '00015, T is 288, and C is -p 2 rp unity. The value of will be found to be about '003, so that the heat of compres- J C sion and cold of expansion increase the effective elasticity by 3 parts in a thousand, and therefore increase the velocity of sound by 1 4 part in a thousand. The same formula applies to solids if we make E denote Young's modulus, and a the coefficient of linear expansion. For iron it gives, according to Sir "W. Thomson (Pioc. R. S. E. 1865-6), an increase of about J per cent, in Young's modulus, and therefore of J per cent, in the velocity of sound. VELOCITY OF SOUND IN SOLIDS. 805 Wertheim has measured the velocity of sound in some liquids by an indirect method, which will be explained in a later chapter. He finds it to be 1160 metres per second in ether and alcohol, and 1900 in a solution of chloride of calcium. 637. Velocity of Sound in Solids. The velocity of sound in cast- iron was determined by Biot and Martin by means of a connected series of water-pipes, forming a conduit of a total length of 951 metres. One end of the conduit was struck with a hammer, and an observer at the other end heard two sounds, the first transmitted by the metal, and the second by the air, the interval between them being 2-5 seconds. Now the time required for travelling this dis- tance through air, at the temperature of the experiment (11C.), is 2'8 seconds. The time of transmission through the metal was there- fore -3 of a second, which is at the rate of 3170 metres per second. It is, however, to be remarked, that the transmitting body was not a continuous mass of iron, but a series of 376 pipes, connected to- gether by collars of lead and tarred cloth, which must have consid- erably delayed the transmission of the sound. But in spite of this, the velocity is about nine times greater than in air. Wertheim, by the indirect methods above alluded to, measured the velocity of sound in a number of solids, with the following results, the velocity in air being taken as the unit of velocity: Lead, 3-974 to 4'120 Tin, 7-338 to 7 '480 Gold, 5-603 to 6-424 Silver, 7"903 to 8'057 Zinc, 9-863 to 1T009 Copper, 11-167 Steel, 14-361 to 15'108 Iron, 15-108 Brass, 10'224 Glass, 14-956 to 16759 Flint Glass, . . H'890 to 12-220 Oak, 9-902 to 12-02 Platinum, .... 7'823 to 8'467 I Fir ....... 12'49 to 17 '26 638. Theoretical Computation. The formula A/JJ serves for solids as well as for liquids and gases; but as solids can be subjected to many different kinds of strain, whereas liquids and gases can be subjected to only one, we may have different values of E, and dif- ferent velocities of transmission of pulses for the same solid. This is true even in the case of a solid whose properties are alike in all directions (called an isotropic solid) ; but the great majority of solids are very far from fulfilling this condition, and transmit sound more rapidly in some directions than in others. When the sound is propagated by alternate compressions and extensions running along a substance which is not prevented from 806 PRODUCTION AND PROPAGATION OF SOUND. extending and contracting laterally, the elasticity E becomes iden- tical 1 with Young's modulus ( 23). On the other hand, if uniform spherical waves of alternate compression and extension spread out- wards, symmetrically, from a point in the centre of an infinite solid, lateral extension and contraction will be prevented by the symmetry of the action. The effective elasticity is, in this case, greater than Young's modulus, and the velocity of sound will be increased accord- By the table on p. 29 the value of Young's modulus for copper is 12,558 kilogrammes per square millimetre, or 1,255,800,000 grammes per square centimetre, and by the table on p. 89 the density of copper in grammes per cubic centimetre is 8 '8. Hence, for the velo- city of sound through a copper rod, in centimetres per second, we have /E _ /1255800000 x 981 ~ VD V 88 = 374150 nearly, or 3741 - 5 metres per second. This is about 11 '2 times the velocity in air. 639. Reflection of Sound. When sonorous waves meet a fixed obstacle they are reflected, and the two sets of waves one direct, Fig. 577. Reflection of Sound. and the other reflected are propagated just as if they came from two separate sources. If the reflecting surface is plane, waves di- verging from any centre in front of it are reflected so as to diverge 1 Subject to the small correction mentioned in the foot-note to 636. REFLECTION OF SOUND. 807 from a centre O' symmetrically situated behind it, and an ear at any point M in front bears the reflected sound as if it came from O'. The direction from which a sound appears to the hearer to proceed is determined by the direction along -which the sonorous pulses are propagated, and is always normal to the waves. A normal to a set of sound-waves may therefore conveniently be called a ray of sound. O I is a direct ray, and I M the corresponding reflected ray ; and it is obvious, from the symmetrical posi- tion of the points 0', that these t two rays are equally inclined to the surface, or the angles of inci- dence and reflection are equal. 640. Illustrations of Reflection of Sound. The reflection of sonorous Fig 6T8 ._ Reflection fl . om Elliptic Roof . waves explains some well-known phenomena. If aba be an elliptic dome or arch, a sound emitted from either of the foci // will be reflected from the elliptic surface Fig. 579. Reflection of Sound from Conjugate Mirrors. in such a direction as to pass through the other focus. A sound emitted from either focus may thus be distinctly heard at the other, even when quite inaudible at nearer points. This is a consequence 808 PRODUCTION AND PROPAGATION OF SOUND. of the property, that lines drawn to any point on an ellipse from the two foci are equally inclined to the curve. The experiment of the conjugate mirrors ( 311) is also applicable to sound. Let a watch be hung in the focus of one of them (Fig. 579), and let a person hold his ear at the focus of the other; or still better, to avoid intercepting the sound before it falls on the second mirror, let him employ an ear-trumpet, holding its open end at the focus. He will distinctly hear the ticking, even when the mirrors are many yards apart. 1 641. Echo. Echo is the most familiar instance of the reflection of sound. In order to hear the echo of one's own voice, there must be a distant body capable of reflecting sound directly back, and the number of syllables that an echo will repeat is proportional to the distance of this obstacle. Reckoning ^ of a second as the time of pronouncing a syllable, the space traversed by sound in this time is Fig. 580. - Eclio. about 200 feet, and an obstacle must be at half this distance in order that it may be able to send back a single syllable. The sounds reflected to the speaker have travelled first over the distance A (Fig. 580) from him to the reflecting body, and then back from A to O. Supposing five syllables to be pronounced in a second, and taking the velocity of sound as 1100 feet per second, a distance of 1 Sondhaus has shown that sound, like light, is capable of being refracted. A spherical balloon of collodion, filled with carbonic acid gas, acts as a sound-lens. If a- watch IK. hung at some distance from it on one side, an ear held at the conjugate focus on the other side will hear the ticking. See also 819 A. HEARING AND SPEAKING TRUMPETS. 809 550 feet from the speaker to the reflecting body would enable the speaker to complete the fifth syllable before the return of the first; this is at the rate of 110 feet per syllable. At distances less than about 100 feet there is not time for the distinct reflection of a single syllable ; but the, reflected sound mingles with the voice of the speaker. This is particularly observable under vaulted roofs. Multiple echoes are not uncommon. They are due, in some cases, to independent reflections from obstacles at different distances; in others, to reflections of re- flections. A position exactly midway between two parallel walls, at a sufficient distance apart, is favour- able for the observance of this latter phenomenon. One of the most frequently cited instances of multiple echoes is that of the old palace of Simonetta, near Milan, which forms three sides of a quadrangle. According to Kircher, it repeats forty times. 642. Speaking and Hearing Trumpets. The complete explanation of the action of these instruments presents considerable difficulty. The speaking-trumpet (Fig. 581) consists of a long tube (sometimes 6 feet long), slightly tapering towards the speaker, furnished at this end with Speak fng.trampet. a hollow mouth-piece, which nearly fits the lips, and at the other with a funnel-shaped enlargement, called the bell, opening out to a width of about a foot. It is much used at sea, and is found very effectual in making the voice heard at a distance. The expla- nation usually given of its action is, that the slightly conical form of the long tube produces a series of reflec- tions in directions more and more nearly parallel to the axis; but this explanation fails to account for the utility of the bell, which experience has shown to be considerable. Ear-trumpets have various forms, as represented in Fig. 582; Fig. 582. Ear trumpets. 810 PRODUCTION AND PROPAGATION OF SOUND. having little in common, except that the external opening or bell is much larger than the end which is introduced into the ear. Mem- branes of gold-beaters' skin are sometimes stretched across their interior, in the positions indicated by the dotted lines in Nos. 4 and 5. No. G consists simply of a bell with such a membrane stretched across its outer end, while its inner end communicates with the ear by an mdian-rubber tube with an ivory end-piece. These light membranes are peculiarly susceptible of impression from aerial vibrations. In Regnault's experiments above cited, it was found that membranes were affected at distances greater than those at which sound was heard. 643. Interference of Sonorous Undulations. When two systems of waves are traversing the same matter, the actual motion of each particle of the matter is the resultant of the motions due to each system separately. When these component motions are in the same direction the resultant is their sum; when they are in opposite directions it is their difference ; and if they are equal, as well as opposite, it is zero. Very remarkable phenomena are thus produced when the two undulations have the same, or nearly the same wave- length; and the action which occurs in this case is called interference. When two sonorous undulations of exactly equal wave length and amplitude are traversing the same matter in the same direction, their phases must either be the same, or must everywhere differ by the same amount. If they are the same, the amplitude of vibration for each particle will be double of that due to either undulation separately. If they are opposite in other words, if one undulation be half a wave-length in advance of the other the motions which they would separately produce in any particle are equal and oppo- site, and the particle will accordingly remain at rest. Two sounds will thus, by their conjoint action, produce silence. In order that the extinction of sound may be complete, the rare- fied portions of each set of waves must be the escccct counterparts of the condensed portions of the other set, a condition which can only be approximately attained in practice. The following experiment, due to M. Desains, affords a very direct illustration of the principle of interference. The bottom of a wooden box is pierced with an opening, in which a powerful whistle fits. The top of the box has two larger openings symmetrically placed with respect to the lower one. The inside of the box is lined with felt, to prevent the vibrations from being communicated to the box, INTERFERENCE OF SONOROUS UNDULATIONS. 811 and to weaken internal reflection. When the whistle is sounded, if a membrane, with sand strewn on it, is held in various positions in the vertical plane which bisects, at right angles, the line joining the two openings, the sand will be agitated, and will arrange itself in nodal lines. But if it is carried out of this plane, positions will be found, at equal distances on both sides of it, at which the agitation is scarcely perceptible. If, when the membrane is in one of these positions, we close one of the two openings, the sand is again agitated, clearly showing that the previous absence of agitation was due to the interference of the undulations proceeding from the two orifices. In this experiment the proof is presented to the eye. In the fol- lowing experiment, which is due to M. Lissajous, it is presented to the ear. A circular plate, supported like the plate in Fig. 565, is made to vibrate in sectors separated by radial nodes. The num- ber of sectors will always be even, and adjacent sectors will vi- brate in opposite directions. Let a disk of card-board of the same size be divided into the same number of sectors, and let alternate sectors be cut away, leaving only enough near the centre to hold the remaining sectors together. If the card be now held just over the vibrating disk, in such a manner that the sectors of the one are exactly over sectors of the other, a great increase of loudness will be observed, consequent on the suppression of the sound from alternate sectors ; but if the card-board disk be turned through the width of half a sector, the effect no longer occurs. If the card is made to rotate rapidly in a continuous manner, the alternations of loudness will form a series of beats. It is for a similar reason that, when a large bell is vibrating, a person in its centre hears the sound as only moderately loud, while within a short distance of some portions of the edge the loudness is intolerable. 644. Interference of Direct and Reflected Waves. Nodes and Anti- nodes. Interference may also occur between undulations travelling in opposite directions ; for example, between a direct and a reflected sys- tem. When waves proceeding along a tube meet a rigid obstacle, form- ing a cross section of the tube, they are reflected directly back again, the motion of any particle close to the obstacle being compounded of that due to the direct wave, and an equal and opposite motion due to the reflected wave. The reflected waves are in fact the images (with reference to the obstacle regarded as a plane mirror) of the waves which would exist in the prolongation of the tube if the 812 PRODUCTION AND PROPAGATION OF SOUND. obstacle were withdrawn. At the distance of half a wave-length from the obstacle the motions due to the direct and reflected waves will accordingly be equal and opposite, so that the particles situated at this distance will be permanently at rest ; and the same is true at the distance of any number of half wave-lengths from the obstacle. The air in the tube will thus be divided into a number of vibrating segments separated by nodal planes or cross sections of no vibra- tion arranged at distances of half a wave-length apart. One of these nodes is at the obstacle itself. At the centres of the vibrating seg- ments that is to say, at the distance of a quarter wave-length plus any number of half wave-lengths from the obstacle or from any node the velocities due to the direct and reflected waves will be equal and in the same direction, and the amplitude of vibration will ac- cordingly be double of that due to the direct wave alone. These o * are the sections of greatest disturbance as regards change of place. We shall call them antinodes. On the other hand, it is to be remem- bered that motion with the direct wave is motion against the re- flected waves, and vice versa, so that ( 632) at points where the velocities due to both have the same absolute direction they corre- spond to condensation in the case of one of these undulations, and to rarefaction in the case of the other. Accordingly, these sections of maximum movement are the places of no change of density ; and on the other hand, the nodes are the places where the changes of density are greatest. If the reflected undulation is feebler than the direct one, as will be the case, for example, if the obstacle is only imper- fectly rigid, the destruction of motion at the nodes and of change of density at the antinodes will not be complete; the former will merely be places of minimum motion, and the latter of minimum change of density. Direct experiments in verification of these principles, a wall being the reflecting body, were conducted by Savart, and also by Seebeck. the latter of whom employed a testing apparatus called the acoustic- pendulum. It consists essentially of a small membrane stretched in a frame, from the top of which hangs a very light pendulum, with its bob resting against the centre of the membrane. In the middle por- tions of the vibrating segments the membrane, moving with the air on its two faces, throws back the pendulum, while it remains nearly free from vibration at the nodes. Regnault made extensive use of the acoustic pendulum in his ex- periments on the velocity of sound. The pendulum, when thrown BEATS PRODUCED BY INTERFERENCE. 813 back by the membrane, completed an electric circuit, and thus effected a record of the instant when the sound arrived. 644A. Beats Produced by Interference. When two notes, which are not quite in unison, are sounded together, a peculiar palpitating effect is produced ; we hear a series of bursts of sound, with inter- vals of comparative silence between them. The bursts of sound are called beats, and the notes are said to beat together. If we have the power of tuning one of the notes, we shall find that as they are brought more nearly into unison, the beats become slower, and that, as the departure from unison is increased, the beats become more rapid, till they degenerate first into a rattle, and then into a discord. The effect is most striking with deep notes. These beats are completely explained by the principle of interfer- ence. As the wave-lengths of the two notes are slightly different, while the velocity of propagation is the same, the two systems of waves will, in some portions of their course, agree in phase, and thus strengthen each other; while in other parts they will be opposite in phase, and will thus destroy each other. Let one of the notes, for example, have 100 vibrations per second, and the other 101. Then, if we start from an instant when the maxima of condensation from the two sources reach the ear together, the next such conjunction will occur exactly a second later. During the interval one of the systems of waves has been gradually falling behind the other, till, at the end of the second, the loss has amounted to one wave-length. At the middle of the second it will have amounted to half a wave- length, and the two sounds will destroy each other. We shall thus have one beat and one extinction in each second, as a consequence of the fact that the higher note has made one vibration more than the lower. In general, the frequency of beats is the difference of the frequencies of vibration of the beating notes. NOTE A. 632. That the particles which are moving forward are in a state of compression, may be shown in the following way: Consider an imaginary cross section travelling forward through the tube with the same velocity as the undulation. Call this velocity v, and the velocity of any particle of air u. Also let the density of any particle be denoted by p. Then M and p remain constant for the imaginary moving section, and the mass of air which it traverses in its motion per unit time is (v u) p. As there is no permanent transfer of air in either direction through the tube, the mass thus traversed must be the same as if the air were at rest at its natural density. Hence the value of (v-u)p is the same for 814 PRODUCTION AND PROPAGATION OF SOUND. all cross sections ; whence it follows, that where u is greatest p must be greatest, and where u is negative p is less than the natural density. If p a denote the natural density, we have (v u) p = v p a , whence = ^- P- ; that is to say, v p the ratio of the velocity of a particle to the velocity of the undulation is equal to the conden- sation existing at the particle. If u is negative that is to say, if the velocity be retrograde its ratio to v is a measure of the rarefaction. From this principle we may easily derive a formula for the velocity of sound, bearing in mind that u is always very small in comparison with v, and that consequently the ratio of p to p is very nearly unity. For, consider a thin lamina of air, whose natural thickness is ox, and let 8p and Sp be the differences between the densities and pressures respectively on its two faces. Then the equation above investigated leads to the condition = But the time which the mov- 8p p 5 x ing section occupies in traversing the lamina is approximately > and in this time the velocity of the lamina changes by the amount Su. The force producing this change of velocity is op, or I '41-? dp, and must be equal to the quotient of change of momentum by P time, that is to pox. o -=- or to pv ou. Hence _l-41- . Equating this to the v 5 p p v other expression for we have o p v p This investigation is due to Professor Eankine, Phil. Trans. 1869. NOTE B. 635. The following is the usual investigation of the velocity of transmission of sound through a uniform tube filled with air, friction being neglected : Let x denote the original distance of a particle of air from the section of the tube at which the sound originates, and x + y its distance at time t, so that y is the displacement of the particle from the position of equili- brium. Then a particle which was originally at distance x + dx will at time t be at the distance x + Sx + y + Sy; and the thickness of the intervening lamina, which was originally 8 x, is now 8 x + 8 y. Its compression is therefore - -^ or ultimately - ^, and if P denote ox dx the original pressure, the increase of pressure is 1'41 P-~- The excess of pressure d x behind a lamina 3x above the pressure in front is -- -fl'41 P-,-^ ox, or 1'41 P -r^ox; d x ^ dx' dx and if D denote the original density of the air, the acceleration of the lamina will be the quotient < equation d? y quotient of this expression by D.8x. But this acceleration is -J. Hence we have the a t = dt* D dx 2 ' the integral of which is y = (x-vt)+f(x + vt); where v denotes /I '41 --> and F, /denote any functions whatever. TRANSMISSION OF SOUND. 815 The term F (x-vt) represents a wave, of the form y = F (x), travelling forwards with velocity v ; for it has the same value for t l + S t and x l +v . 8 t as for t l and x t . The term f (x + vt) represents a wave, of the form y =f (x), travelling backwards with the same velocity. In order to adapt this investigation, as well as that given in Note A, to the propagation of longitudinal vibrations through any elastic material, whether solid, liquid, or gaseous, we have merely to introduce E in the place of 1/41 P, E denoting the coefficient of elasticity of the substance, as defined by the condition that a compression i is produced by a force (per unit area) of E ^1'- d x CHAPTER LIV. NUMERICAL EVALUATION OF SOUND. 645. Qualities of Musical Sound. Musical tones differ one from another in respect of three qualities ; loudness, pitch, and character. Loudness. The loudness of a sound considered subjectively is the intensity of the sensation with which it affects the organs of hearing. Regarded objectively, it depends, in the case of sounds of the same pitch and character,, upon the energy of the aerial vibrations in the neighbourhood of the ear, and is proportional to the square of the amplitude. Our auditory apparatus is, however, so constructed as to be more susceptible of impression by sounds of high than of low pitch. A bass note must have much greater energy of vibration than a treble note, in order to strike the ear as equally loud. The intensity of sonorous vibration at a point in the air is therefore not an absolute measure of the intensity of the sensation which will be received by an ear placed at the point. The word loud is also frequently applied to a source of sound, an when we say a loud voice, the reference being to the loudness as heard at a given distance from the source. The diminution of loud- ness with increase of distance according to the law of inverse squares is essentially connected with the proportionality of loudness to square of amplitude. Pitch. Pitch is the quality in respect of which an acute sound differs from a grave one ; for example, a treble note from a bass note. All persons are capable of appreciating differences of pitch to some extent, and the power of forming accurate judgments of pitch con- stitutes what is called a musical ear. Physically, pitch depends solely on frequency of vibration, that is to say, on the number of vibrations executed per unit time. In QUALITIES OF MUSICAL SOUND. 817 ordinary circumstances this frequency is the same for the source of sound, the medium of transmission, and the drum of the ear of the person hearing; and in general the transmission of vibrations from one body or medium to another produces no change in their fre- quency. The second is universally employed as the unit of time in treating of sonorous vibrations; so that frequency means number of vibrations per second. Increase of frequency corresponds to eleva- tion of pitch. Period and frequency are reciprocals. For example, if the period of each vibration is y^j- of a second, the number of vibrations per second is 100. Period therefore is an absolute measure of pitch, and the longer the period the lower is the pitch. The wave-length of a note in any medium is the distance which sound travels in that medium during the period corresponding to the note. Hence wave-length may be taken as a measure of pitch, pro- vided the medium be given ; but, in passing from one medium to another, wave-length varies directly as the velocity of sound. The wave-length of a given note in air depends upon the temperature of the air, and is shortened in transmission from the heated air of a concert-room to the colder air outside, while the pitch undergoes no change. If we compare a series of notes rising one above another by what musicians regard as equal differences of pitch, their frequencies will not be equidifferent, but will form an increasing geometrical pi-o- gression, and their periods (and wave-lengths in a given medium) will form a decreasing geometrical progression. Character. Musical sounds may, however, be alike as regards pitch and loudness, and may yet be easily distinguishable. We speak of the quality of a singers voice, and the tone of a musical instrument; and we characterize the one or the other as rich, sweet, or mellow, on the one hand ; or as poor, harsh, nasal, &c , on the other. These epithets are descriptive of what musicians call timbre a French word literally signifying stamp. German writers on acoustics denote the same quality by a term signifying sound-tint. It might equally well be called sound-flavour. We adopt character as the best English designation. Physically considered, as wave-length and wave-amplitude fall under the two previous heads, character must depend upon the only remaining point in which aerial waves can differ namely their form, meaning by this term the law according to which the velo- 53 818 NUMERICAL EVALUATION OF SOUND. cities and densities change from point to point of a wave. This subject will be more fully treated in Chapter Ivi. Every musical sound is more or less mingled with non-musical noises, such as puffing, scraping, twanging, hissing, rattling, &c. These are not compre- hended under timbre or character in the usage of the best writers on acoustics. The gradations of loudness which characterize the commencement, progress, and cessation of a note, and upon which musical effect often greatly depends, are likewise excluded from this designation. In distinguishing the sounds of different musical in- struments, we are often guided as much by these gradations and extraneous accompaniments as by the character of the musical tones themselves. 646. Musical Intervals. When two notes are heard, either simul- taneously OF in succession, the ear experiences an impression of a special kind, involving a perception of the relation existing between them as regards difference of pitch. This impression is often recog- nized as identical where absolute pitch is very different, and we express this identity of impression by saying that the musical inter- val is the same. Each musical interval, thus recognized by the ear as constituting a particular relation between two notes, is found to correspond to a particular ratio between their frequencies of vibration. Thus the octave, which of all intervals is that which is most easily recognized by the ear, is the relation between two notes whose frequencies are as 1 to 2, the upper note making twice as many vibrations as the lower in any given time. It is the musician's business so to combine sounds as to awaken emotions of the peculiar kind which are associated with works of art. In attaining this end he employs various resources, but musical Intervals occupy the foremost place. It is upon the judicious employ- ment of these that successful composition mainly depends. 647. Gamut. The gamut or diatonic scale is a series of eight notes 'having certain definite relations to one another as regards fre- quency of vibration. The first and last of the eight are at an inter- val of an octave from each other, and are called by the same name ; and by taking in like manner the octaves of the other notes of the series, we obtain a repetition of the gamut both upwards and downwards, which may be continued over as many octaves as we please. The notes of the gamut are usually called by the names THE GAMUT. 819 Do Re Mi Fa Sol La Si Do 2 and their vibration-frequencies are proportional to the numbers i i $ * f V s 2 or, clearing fractions, to 24 27 30 32 36 40 45 48 The intervals from Do to each of the others in order are called a second, a major third, a fourth, a fifth, a sixth, a seventh, and an octave respectively. The interval from La to Do 2 is called a minor third, and is evidently represented by the ratio -. The interval from Do to Re, from Fa to Sol. or from La to Si, is represented by the ratio -f-, and is called a major tone. The interval from Re to Mi, or from Sol to La, is represented by the ratio V> and is called a minor tone. The interval from Mi to Fa, or from Si to Do 2 , is represented by the ratio ^i, and is called a limma. As the square of ^-?- is a little greater than f , a limma is rather more than half a major tone. The intervals between the successive notes of the gamut are ac- cordingly represented by the following ratios 1 : Do Re Mi Fa Sol La Si Do, fio 10 9 10 9 10 "9" T5" U" 7 ~5 Do (with all its octaves) is called the key-note, or simply the key, of the piece of music, and may have any pitch whatever. In order to obtain perfect harmony, the above ratios should be accurately main- tained whatever the key-note may be. 648. Tempered Gamut. A great variety of keys are employed in music, and it is a practical impossibility, at all events in the case of instruments like the piano and organ, which have only a definite set of notes, to maintain these ratios strictly for the whole range of pos- sible key-notes. Compromise of some kind becomes necessary, and different systems of compromise are called different temperaments or different modes of temperament. The temperament which is most in favour in the present day is the simplest possible, and is called equal temperament, because it favours no key above another, but makes the tempered gamut exactly the same for all It ignores the . ' The logarithmic differences, which are accurately proportional to the intervals, are approximately as under, omitting superfluous zeros. Do Re Mi Fa Sol La Si Do 51 46 28 51 46 51 28 820 NUMERICAL EVALUATION OF SOUND. difference between major and minor tones, and makes the limma exactly half of either. The interval from Do to Do 2 is thus divided into 5 tones and 2 semitones, a tone being of an octave, and a semitone ^ of an octave. The ratio of frequencies corresponding to a tone will therefore be the sixth root of 2, and for a semitone it will be the 12th root of 2. The difference between the natural and the tempered gamut for the key of C is shown by the following table, which gives the number of complete vibrations per second for each note of the middle octave of an ordinary piano: Tempered Gamut. Natural, Gamut. C . . 2587 2587 D . . 290-3 291-0 B . . 325-9 323-4 F . 345-3 344-9 Tempered Gamut. Natural Gamut G . . 387-6 .388-0 A . . 435-0 431-1 B . . 488-2 485-0 C 517-3 517-3 The absolute pitch here adopted is that of the Paris Conservatoire, and is fixed by the rule that A (the middle A of a piano, or the A string of a violin) is to have 435 complete vibrations per second in the tempered gamut. This is rather lower than the concert-pitch which has prevailed in this country in recent years, but is probably not so low as that which prevailed in the time of Handel. It will be noted that the number of vibrations corresponding to C is ap- proximately equal to a power of 2 (256 or 512). Any power of 2 accordingly expresses (to the same degree of approximation) the number of vibrations corresponding to one of the octaves of C. The Stuttgard congress (1834) recommended 528 vibrations per second for C, and the C tuning-forks sold under the sanction of the Society of Arts are guaranteed to have this pitch. By multiplying the numbers 24, 27 ... 48, in 647, by 11, we shall obtain the frequencies of vibration for the natural gamut in C corresponding to this standard. What is generally called concert-pitch gives C about 538. The C of the Italian Opera is 546. Handel's C is said to have been 499f 649. Limits of Pitch employed in Music.^-The deepest note re- gularly employed in music is the C of 32 vibrations per second which is emitted by the longest pipe (the 16-foot pipe) of most organs. Its wave-length in air at a temperature at which the velo- city of sound is 1120 feet per second, is m^ = 35 f ee t. The highest note employed seldom exceeds A, the third octave of the A above defined. Its number of vibrations per second is 435 X 2 3 = 3480, and MINOR AND PYTHAGOREAN SCALES. 821 its wave-length in air is about 4 inches. Above this limit it is difficult to appreciate pitch, but notes of at least ten times this num- ber of vibrations are audible. The average compass of the human voice is about two octaves. The deep F of a bass-singer has 87, and the upper G of the treble 775 vibrations per second. Voices which exceed either of these limits are regarded as deep or high. 650. Minor Scale and Pythagorean Scale. The difference between a major and minor tone is expressed by the ratio f-^, and is called a comma. The difference between a minor tone and a limma is ex- pressed by the ratio ---, and is the smallest value that can be assigned to the somewhat indefinite interval denoted by the name semitone, the greatest value being the limma itself (). The signs $ and > (sharp and flat) appended to a note indicate that it is to be raised or lowered by a semitone. The major scale or gamut, as above given, is modified in the following way to obtain the minor scale: Do Re Mib Fa Sol Lab Sib Do a 1 6 109 16 9 10 15 15" "9 the numbers in the second line being the ratios which represent the intervals between the successive notes. It is worthy of note that Pythagoras, who was the first to attempt the numerical evaluation of musical intervals, laid down a scheme of values slightly different from that which is now generally adopted. According to him, the intervals between the successive notes of the major scale are as follows: Do Re Mi Fa Sol La Si Do 9 9 256 999 256 "8 TTT5 W ' 9 9 "2~T3 This scheme agrees exactly with the common system as regards the values of the fourth, fifth, and octave, and makes the values of the major third, the sixth, and the seventh each greater by a comma, while the small interval from mi to fa, or from si to do, is diminished by a comma. In the ordinary system, the prime numbers which enter the ratios are 2, 3, and 5 ; in the Pythagorean system they are only 2 and 3; hence the interval between any two notes of the Pythagorean scale can be expressed as the sum or difference of a certain number of octaves and fifths. In tuning a violin by making the intervals between the strings true fifths, the Pythagorean scheme is virtually employed. 822 NUMERICAL EVALUATION OF SOUND. 651. Methods of Counting Vibrations. Siren. The instrument which is chiefly employed for counting the number of vibrations corresponding to a given note, is called the siren, and was devised by Cagniard de Latour. It is represented in Figs. 583, 58t, the former being a front, and the latter a back view. There is a small wind-chest, nearly cylindrical, having its top pierced with fifteen holes, disposed at equal distances round the circumference of a circle. Just over this, and nearly touching it, is a movable circular plate, pierced with the same number of holes Fig. 583. Fig. 584. similarly arranged, and so mounted that it can rotate very freely about its centre, carrying with it the vertical axis to which it is attached. This rotation is effected by the action of the wind, which enters the wind-chest from below, and escapes through the holes. The form of the holes is shown by the section in Fig. 584. They do not pass perpendicularly through the plates, but slope contrary ways, so that the air when forced through the holes in the lower plate impinges upon one side of the holes in the upper plate, and thus blows it round in a definite direction. The instrument is driven by means of the bellows shown in Fig. 595 ( 66-t). As the rotation of one plate upon the other causes the holes to be alternately opened and closed, the wind escapes in successive puffs, whose frequency THE SIREN. 823 depends upon the rate of rotation. Hence a note is emitted which rises in pitch as the rotation becomes more rapid. The siren will sound under water, if water is forced through it instead of air; and it was from this circumstance that it derived its name. In each revolution, the fifteen holes in the upper plate come opposite to those in the lower plate 15 times, and allow the com- pressed air in the wind-chest to escape; while in the intervening positions its escape is almost entirely prevented. Each revolution thus gives rise to 1 5 vibrations ; and in order to know the number of vibrations corresponding to the note emitted, it is only necessary to have a means of counting the revolutions. This is furnished by a counter, which is represented in Fig. 581. The revolving axis carries an endless screw, driving a wheel of 100 teeth, whose axis carries a hand traversing a dial marked with 100 divisions. Each revolution of the perforated plate causes this hand to advance one division. A second toothed-wheel is driven inter- mittently by the first, advancing suddenly one tooth whenever the hand belonging to the first wheel passes the zero of its scale. This second wheel also carries a hand traversing a second dial ; and at each of the sudden movements just described this hand advances one division. Each division accordingly indicates 1 00 revolutions of the perforated plate, or 1 500 vibrations. By pushing in one of the two buttf ns which are shown, one on each side of the box containing the toothed-wheels, we can instantaneously connect or disconnect the endless screw and the first toothed-wheel. In order to determine the number of vibrations corresponding to any given sound which we have the power of maintaining steadily, we fix the siren on the bellows, the screw and wheel being dis- connected, and drive the siren until the note which it emits is judged to be in unison with the given note. We then, either by regulating the pressure of the wind, or by employing the finger to press with more or less friction against the revolving axis, contrive to keep the note of the siren constant for a measured interval of time, which we observe by a watch. At the commencement of the interval we sud- denly connect the screw and toothed-wheel, and at its termination we suddenly disconnect them, having taken care to keep the siren in unison with the given sound during the interval. As the hands do not advance on the dials when the screw is out of connection with the wheels, the readings before and after the measured interval of 824 NUMERICAL EVALUATION OF SOUND. time can be taken at leisure. Each reading consists of four figures, o o ' indicating the number of revolutions from the zero position, units and tens being read off on the first dial, and hundreds and thousands on the second. The difference of the two readings is the number of revolutions made in the measured interval, and when multiplied by ]5 gives the number of vibrations in the interval, whence the num- ber of vibrations per second is computed by division. 652. Graphic Method. In the hands of a skilful operator, with a good musical ear, the siren is capable of yielding very accurate deter- minations, especially if, by adding or subtracting the number of beats, Fig. 585. Vibroscope. correction be made for any slight difference of pitch between the siren and the note under investigation. The vibrations of a tuning-fork can be counted, without the aid of the siren, by a graphical method, which does not call for any exer- cise of musical judgment, but simply involves the performance of a mechanical operation. The tuning-fork is fixed in a horizontal position, as shown in Fig. 585, and has a light style, which may be of brass wire, quill, or bristle, attached to one of its prongs by wax. To receive the trace, a piece of smoked paper is gummed round a cylinder, which can be turned by a handle, a screw cut on the axis causing it at the same THE PHONAUTOGRAPH. 825 time to travel endwise. The cylinder is placed so that the style barely touches the blackened surface. The fork is then made to vibrate by bowing it, and the cylinder is turned. The result is a wavy line traced on the blackened surface, and the number of wave- forms (each including a pair of bends in opposite directions) corres- ponds to the number of vibrations. If the experiment lasts for a measured interval of time, we have only to count these wave-forms, and divide by the number of seconds, in order to obtain the number of vibrations per second for the note of the tuning-fork. By plunging the paper in ether, the trace will be fixed, so that the paper may be laid aside and the vibrations counted at leisure. The apparatus is called the vibroscope, and was invented by Duhamel. M. Leon Scott has invented an instrument called the phonauto- graph, which is adapted to the graphical representation of sounds in Fig. 586. Traces by Phonautograph. general. The style, which is very light, is attached to a membrane stretched across the smaller end of what may be called a large ear- trumpet. The membrane is agitated by the aerial waves proceeding from any source of sound, and the style leaves a record of these agitations on a blackened cylinder, as in Duhamel's apparatus. Fig. 586 represents the traces thus obtained from the sound of a tuning-fork in three different modes of vibration. 653. Tonometer. When we have determined the frequency of vibration for a particular tuning-fork, that of another fork, nearly in unison with it, can be deduced by making the two forks vibrate simultaneously, and counting the beats which they produce. Scheibler's tonometer, which is constructed by Koenig of Paris, consists of a set of 65 tuning-forks, such that any two consecutive forks make 4 beats per second, and consequently differ in pitch by S26 NUMERICAL EVALUATION OF SOUND. -i vibrations per second. The lowest of the series makes 256 vibra- tions, and the highest 512, thus completing an octave. Any note within this range can have its vibration-frequency at once deter- mined, with great accuracy, by making it sound simultaneously with the fork next above or below it, and counting beats. With the aid of this instrument, a piano can be tuned with cer- tainty to any desired system of temperament, by first tuning the notes which come within the compass of the tonometer, and then proceeding by octaves. In the ordinary methods of tuning pianos and organs, tempera- ment is to a great extent a matter of chance ; and a tuner cannot O ' attain the same temperament in two successive attempts. 653 A. Pitch modified by Relative Motion. We have stated in 645 that, in ordinary circumstances, the frequency of vibration in the source of sound, is the same as in the ear of the listener, and in the intervening medium. This identity, however, does not hold if the source of sound and the ear of the listener are approaching or receding from each other. Approach of either to the other produces increased frequency of the pulses on the ear, and consequent elevation of pitch in the sound as heard; while recession has an opposite effect. Let n be the number of vibrations performed in a second by the source of the sound, v the velocity of sound in the medium, and a the relative velocity of approach. Then the number of waves which reach the ear of the listener in a second, will be n plus the number of waves which cover a length a, that is (since n waves cover a length v), will be n + n or n. The following investigation is more rigorous. Let the source make n vibrations per second. Let the observer move towards the source with velocity a. Let the source move away from the observer with velocity a'. Let the medium move from the observer towards the source with velocity m, and let the velocity of sound in the medium be v. Then the velocity of the observer relative to the medium is a m towards the source, and the velocity of the source relative to the medium is a' m away from the observer. The velocity of the sound relative to the source will be different in different directions, its greatest amount being v + a m towards the observer, and its least being v a' + m away from the observer. The length of a wave will vary with direction, being - of the velocity of the sound MODIFICATION OF PITCH. 827 relative to the source. The length of those waves which meet the observer will be - ^ rn , and the velocity of these waves relative to the observer will be v + a m; hence the number of waves that meet him in a second will be - T ~^n. v + a, -m Careful observation of the sound of a railway whistle, as an express train dashes past a station, has confirmed the fact that the sound as heard by a person standing at the station is higher while the train is approaching than when it is receding. A speed of about 40 miles an hour will sharpen the note by a semitone in approaching, and flatten it by the same amount in receding, the natural pitch being heard at the instant of passing. 1 1 The best observations of this kind were those of Buys Ballot, in which trumpeters, with their instruments previously tuned to unison, were stationed, one on the locomotive, and others at three stations beside the line of railway. Each trumpeter was accompanied by musicians, charged with the duty of estimating the difference of pitch between the note of his trumpet and those of the others, as heard before and after passing. CHAPTER LV. MODES OF VIBRATION. 654. Longitudinal and Transverse Vibrations of Solids. Sonorous vibrations are manifestations of elasticity. When the particles of a solid body are displaced from their natural positions relative to one another by the application of external force, they tend to return, in virtue of the elasticity of the body. When the external force is removed, they spring back to their natural position, pass it in virtue of the velocity acquired in the return, and execute isochronous vibra- tions about it until they gradually come to rest. The isochronism of the vibrations is proved by the constancy of pitch of the sound emitted ; and from the isochronism we can infer, by the aid of mathe- matical reasoning, that the restoring force increases directly as the displacement of the parts of the body from their natural relative position ( 53 A, B, c). The same body is, in general, susceptible of many different modes of vibration, which may be excited by applying forces to it in dif- ferent ways. The most important of these are comprehended under the two heads of longitudinal and transverse vibrations. In the former the particles of the body move to and fro in the direction along which the pulses travel, which is always regarded as the longitudinal direction, and the deformations produced consist in alternate compressions and extensions. In the latter the particles move to and fro in directions transverse to that in which the pulses travel, and the deformation consists in bending. To produce longi- tudinal vibrations, we must apply force in the longitudinal direction. To produce transverse vibration, we must apply force transversely. 655. Transverse Vibrations of Strings. To the transverse vibra- tions of strings, instrumental music is indebted for some of its most O ' precious resources. In the violin, violoncello, but also < = _*_ hence V=--. V 27T71' 6 Now the distance between the two images (corresponding to a, a' respectively) at the back of the revolving mirror is (I -{I') 8, and is also 161 ( 705 A). Hence 0= .. 8 TTJlP and V = /> . /A * The observed distance o a! between the two images is equal to the di- tance between a, a', that is to r 5. Calling this distance d, we have finally, SirnPr PROPAGATION OF LIGHT. The constant rate of revolution is maintained by comparison with a clock. A wheel with 400 teeth, driven by the clock, makes exactly one revolution per second. A tooth and a space alternately cover the part of the field where the image of the wire-grating (which has been substituted for the single wire) is formed. The same instan- taneous flashes of light from the revolving mirror which form the image, also illuminate the rim of the wheel. If the wheel advances exactly one tooth and space between consecutive flashes, its illumi- nated positions are undistinguishable one from another, and the wheel accordingly appears stationary. When this is the case, it is known that the mirror is making exactly 400 turns per second. A slight departure from this rate either way, makes the wheel appear to be slowly revolving either forwards or backwards, and the bellows must be regulated until the stationary appearance is presented. By means of this admirable combination, Foucault has made what must be regarded as the best determination yet obtained of the velo- city of light. The value thus found, namely, 298 million metres per second, is smaller than that which, until a few years ago, was gene- rally received ; but recent astronomical discussions have shown that the sun's distance is somewhat less than was previously supposed ; and when this correction is made, the astronomical determinations of the velocity of light agree well with that of Foucault. 688. Velocity of Light deduced from Observations of the Eclipses of Jupiter's Satellites. The fact that light occupies a sensible time in travelling over celestial distances, was first established about 1675, by Roemer, a Danish astronomer, who also made the first computa- tion of its velocity. He was led to this discovery by comparing the observed times of the eclipses of Jupiter's first satellite, as contained in records extending over many successive years. The four satellites of Jupiter revolve nearly in the plane of the planet's orbit, and undergo very frequent eclipse by entering the cone of total shadow cast by Jupiter. The satellites and their eclipses are easily seen, even with telescopes of very moderate power ; and being visible at the same absolute time at all parts of the earth's surface at which they are visible at all, they serve as signals for comparing local time at different places, and thus for determining longitudes. The first satellite (that is, the one nearest to Jupiter), from its more rapid motion and shorter time of revolution, affords both the best and the most frequent signals. The interval of time between two successive eclipses of this satellite is about 42| hours, VELOCITY OF LIGHT. 879 Fig. 620. Earth and Jupiter. but was found by Roemer to vary by a regular law according to the position of the earth with respect to Jupiter. It is longest when the earth is increasing its distance from Jupiter most rapidly, and is shortest when the earth is diminishing its distance most rapidly. Starting from the time when the earth is nearest to Jupiter, as at T, J (Fig. 620), the intervals between successive eclipses are always longer than the mean value, until the greatest distance has been attained, as at T', J', and the sum of the excesses amounts to 10 min. 26 '6 sec. From this time until the nearest distance is again attained, as at T", J", the inter- vals are always shorter than the mean, and the sum of the defects amounts to 16 min. 26'6 sec. It is evident, then, that the eclipses are visible 16 m. 26'6 s. earlier at the nearest than at the remotest point of the earth's orbit; in other words, that this is the time required for the propagation of light across the diameter of the orbit. Taking this diameter as 183 millions of O miles, 1 we have a resulting velocity of about 185,500 miles per second. 688 A. Velocity of Light deduced from Aberration. About fifty years after Roemer's discovery, Bradley, the English astronomer, employed the velocity of light to explain the astronomical pheno- menon called aberration. This consists in a regular periodic displace- ment of the stars as seen from the earth, the period of the displace- ment being a year. If the direction in which the earth is moving in its orbit at any instant be regarded as the forward direction, every star constantly appears an the forward side of its true place, so that, as the earth moves once round its orbit in a year, each star describes in this time a small apparent orbit about its true placa The phenomenon is explained in the same way as the familiar fact, that a shower of rain falling vertically, seems, to a person run- ning forwards, to be coming in his face. The relative motion of the rain-drops with respect to his body, is found by compounding the actual velocity of the drops (whether vertical or oblique) with a 1 The sun's mean distance from the earth was, until recently, estimated at 95 millions of miles. It is now estimated at 92 or 91^ millions. 880 PROPAGATION OF LIGHT. velocity equal and opposite to that with which he runs. Thus if AB (Fig. 620 A) represents the velocity with which he runs, and C A the true velocity of the drops, the apparent velo- city of the drops will be represented by DA. If a tube pointed along AD moves forward parallel to itself with the velocity A B, a drop entering at its upper end will pass through its whole length without wetting its sides ; for while the drop is falling along D B (we suppose with uniform velocity) the tube moves along A B, so that the lower end of the tube reaches B at the same time as the rain-drop. In like manner, if A B is the velocity of the earth, and C A the velocity of light, a telescope must be Fig.620A. pointed along AD to see a star which really lies in Aberration. l ' J the direction of AC or B D produced. When the angle BAG is a right angle (in other words, when the star lies in a direction perpendicular to that in which the earth is moving), the angle CAD, which is called the aberration of the star, is 20"'5, and the tangent of this angle is the ratio of the velocity of the earth to the velocity of light. Hence it is found by computation that the velocity of light is about ten thousand times greater than that with which the earth moves in its orbit. The latter is easily computed, if the sun's distance is known, and is about 18 J miles per second. Hence the velocity of light is about 185,000 miles per second. It will be noted that both these astronomical methods of computing the velocity of light, depend upon the knowledge of the sun's distance from the earth, and that, if this distance is overestimated, the com- puted velocity of light will be too great in the same ratio. Conversely, the velocity of light, as determined by Foucault's method, can be employed, in connection either with aberration or the eclipses of the satellites, for computing the sun's distance; and the first correct determination of the sun's distance was, in fact, that deduced by Foucault from his own results. 689. Photometry. Photometry is the measurement of the relative amounts of light emitted by different sources. The methods em- ployed for this purpose all consist in determinations of the relative distances at which two sources produce equal intensities of illumina- tion. The eye would be quite incompetent to measure the ratio of two unequal illuminations; but a pretty accurate judgment can be formed as to equality or inequality of illumination, at least when the BOUGUER'S PHOTOMETER. 881 surfaces compared are similar, and the lights by which they are illu- minated are of the same colour. The law of inverse squares is always made the basis of the resulting calculations ; and this law may itself be verified by showing that the illumination produced by one candle at a given distance is equal to that produced by four candles at a distance twice as great 690. Bouguer's Photometer. Bouguer's photometer consists of a semi-transparent screen, of white tissue paper, ground glass, or thin white porcelain, divided into two parts by an opaque partition at right angles to it. The two lamps which are to be compared are Fig. 621. Bouguer's Photometer. placed one on each side of this partition, so that each of them illu- minates one-half of the transparent screen. The distances of the two lamps are adjusted until the two portions of the screen, as seen from the back, appear equally bright. The distances are then mea- sured, and their squares are assumed to be directly proportional to the illuminating powers of the lamps. 691. Rumford's Photometer. Rumford's photometer is based on the comparison of shadows. A cylindric rod is so placed that each of the two lamps casts a shadow of it on a screen; and the distances are adjusted until the two shadows are equally dark. As the shadow thrown by one lamp is illuminated by the other lamp, the compari- son of shadows is really a comparison of illuminations. 692. Foucault's Photometer. The two photometers just described 67 882 PllOPAGATION OF LIGHT. are alike in principle. In each of them the two surfaces compared are illuminated each by one only of the sources of light. In Rum- ford's the remainder of the screen is illuminated by both. In Bouguer's it consists merely of an intervening strip which is illumi- nated by neither. If the partition is movable, the effect of moving it further from the screen will be to make this dark strip narrower until it disappears altogether ; and if it be advanced still further, the two illuminated portions will overlap. In Foucault's photometer there is an adjusting screw, for the purpose of advancing the parti- tion so far that the dark strip shall just vanish. The two illuminated portions, being then exactly contiguous, can be more easily and certainly compared. Fig. 622. Rumford's Photometer. 693. Bunsen's Photometer. Bunsen's photometer consists of a screen of white paper with a grease-spot in its centre. The lights to be compared are placed on opposite sides of this screen, and their distances are so adjusted that the grease-spot appears neither brighter nor darker than the rest of the paper, from whichever side it is viewed. When the distances have not been correctly adjusted, the grease-spot will appear darker than the rest of the paper when viewed from the side on which the illumination is most intense, and lighter than the rest of the paper when viewed from the other side. CHARTER LVIIL REFLECTION OF LIGHT. 694. Reflection. If a beam of the sun's rays A B (Fig. 623) be admitted through a small hole in the shutter of a dark room, and allowed to fall on a polished plane surface, it will be seen to continue its course in a different direction B C. This is an example of reflec- tion. A B is called the incident beam, and B C the reflected beam. The angle A B D contained between an incident ray and the normal is called the angle of incidence ; and the angle C B D contained between the corresponding reflected ray and the normal is called the angle of reflection. The plane A B D containing the incident ray and the normal is called the plane of incidence. 695. Laws of Reflection. The reflection of light from polished surfaces takes place according to the following laws : ]. The reflected ray lies in the plane of incidence. 884 REFLECTION OF LIGHT. 2. The angle of reflection is equal to the angle of incidence. These laws may be verified by means of the apparatus represented in Fig. 624. A vertical divided circle has a small polished plate fixed at its centre, at right angles to its plane, and two tubes travelling on its circumference with their axes always directed towards the centre. The zero of the divisions is the highest point of the circle, the plate being horizontal. A source of light, such as the flame of a candle, is placed so that its rays shine through one of the tubes upon the plate at the centre. As the tubes are blackened internally, no light passes through except in a direction almost Fig. 621. -verification of Laws precisely parallel to the axis of the tube. The of Reflection. * r observer then looks through the other tube, and moves it along the circumference till he finds the position in which the reflected light is visible through it. On examining the graduations, it will be found that the two tubes are at the same distance from the zero point, on opposite sides. Hence the angles of incidence and reflection are equal. Moreover the plane of the circle is the plane of incidence, and this also contains the reflected rays. Both the laws are thus verified. 696. Artificial Horizon. These laws furnish the basis of a method of observation which is frequently employed for determining the altitude of a star, and which, by the consistency of its results, fur- nishes a very rigorous proof of the laws. A vertical divided circle (Fig. 625) is set in a vertical plane by proper adjustments. A telescope movable about the axis of the circle is pointed to a particular star, so that its line of collimation I'S' passes through the apparent place of the star. Another tele- scope, 1 similarly mounted on the other side of the circle, is directed downwards along the line I' R towards the image of the star as seen in a trough of mercury I. Assuming the truth of the laws of reflec- tion as above stated, the altitude of the star is half the angle between the directions of the two telescopes ; for the ray S I from the star to the mercury is parallel to the line ST, by reason of the excessively great distance of the star; and since the rays S I, I R are equally inclined to the normal I N, which is a vertical line, the lines I' S', I' R are also equally inclined to the vertical, or, what is the same thing, 1 In practice, a single telescope usually serves for both observations. ARTIFICIAL HORIZON. 885 are equally inclined to a horizontal plane. A reflecting surface of mercury thus used is called a mercury horizon, or an artificial Fig. 625. Artificial Horizon. horizon. Observations thus made give even more accurate results than those in which the natural horizon presented by the sea is made the standard of reference. 697. Irregular Reflection. The reflection which we have thus far been discussing is called regular reflection. It is more marked as the reflecting surface is more highly polished, and (except in the case of metals) as the incidence is more oblique. But there is an- other kind of reflection, in virtue of which bodies, when illuminated, scud out light in all directions, and thus become visible. This is called irregular reflection or diffusion. Regular reflection does not render the reflecting body visible, but exhibits images of surrounding objects. A perfectly reflecting mirror would be itself unseen, and 886 REFLECTION OF LIGHT. actual mirrors are only visible in virtue of the small quantity of diffused light which they usually emit. The transformation of in- cident into diffused light is usually selective; so that, though the incident beam may be white, the diffused light is usually coloured. The power which a body possesses of making such selection consti- tutes its colour. The word reflection is often used by itself to denote what we have here called regular reflection, and we shall generally so employ it. 698. Mirrors. The mirrors of the ancients were of metal, usually of the compound now known as speculum-metal. Looking-^asses date from the twelfth century. They are plates of glass, coated at the back with an amalgam of quicksilver and tin, which forms the reflecting surface. This arrangement has the great advantage ot excluding the air, and thus preventing oxidation. It is attended, however, with the disadvantage that the surface of the glass and the surface of the amalgam form two mirrors ; and the superposition of the two sets of images produces a confusion which would be in- tolerable in delicate optical arrangements. The mirrors, or specula as they are called, of reflecting telescopes are usually made of specu- lum-metal, which is a bronze composed of about 32 parts of copper to 15 of tin. Lead, antimony, and arsenic are sometimes added. Of late years specula of glass coated in front with real silver have been extensively used; they are known as silvered specula. A coating of platinum has also been tried, but not with much success. The mirrors employed in optics are usually either plane or spherical. 699. Plane Mirrors. By a plane mirror we mean any plane reflect- ing surface. Its effect, as is well- known, is to produce, behind the mir- ror, images exactly similar, both in form and size, to the real objects in front of it. This phenomenon is easily explained by the laws of reflection. Let M N (Fig. 626) be a plane mir- ror, and S a luminous point. Rays S I, S I', S 1" proceeding from this point give rise to reflected rays I O, I' 0', I" 0" ; and each of these, if produced backwards, will meet the normal S K in a point S', which is at the same distance behind the mirror that S is in front of Fig. 626. Plane Mirror. PLANE MIRRORS. 887 Fig. 627. Image of Candle. it. 1 The reflected rays have therefore the same directions as if they had come from S', and the eye receives the same impression as if S' were a luminous point. Fig. 627 represents a pencil of rays emitted by the highest point of a candle-flame, and re- flected from a plane mir- ror to the eye of an ob- server. The reflected rays are divergent (like the in- cident rays), and if pro- duced backwards would meet in a point, which is the position of the image of the top of the flame. As an object is made up of points, these principles show that the image of an object formed by a plane mirror must be equal to the object, and symmetrically situated with respect to the plane of the mirror. For example, if A B (Fig. 628) is au object in front of the mirror, an eye placed at will see the image of the point A at A', the image of B at B', and so on for all the other points of the ob- ject. The position of the image A'B' depends only on the posi- tions of the object and of the mirror, and remains stationary as the eye is moved about. It is possible, however, to find positions from which the eye will not see the im^ge at all, the conditions of visibility being the same as if the image were a real object, and the mirror were an opening through which it could be seen. The images formed by a plane mirror are erect. They are not however exact duplicates of the objects from which they are formed, 1 This is evident from the comparison of the two triangles S K I, S' K I, bearing in mind that the angle N I S is equal to the alternate angle I S K, and N 1 to K S' I. Fig. 62S. Incident and Reaected Pencil*. 888 REFLECTION OF LIGHT. but differ from them precisely in the same way as the left foot or hand differs from the right. The image of a printed page is like the appearance of the page as seen through the paper from the back, or like the type from which the page was printed. 700. Images of Images. When rays from a luminous point m have been reflected from a mirror AB (Fig. 629), their subsequent course is the same as if they had come from the image m' at the back of the mirror. Hence, if they fall upon a second mirror C D, an image m" of the first image will be formed at the back of the second mirror. If, after this, they undergo a third- reflection, an image of m" will be formed, and so on indefinitely. The figure shows the actual paths of two rays mirs, mi' r s. They diverge first Pig. 629. Reflection from two Mirrors. Fig. 630. Parallel Mirrors. from m, then from m', and lastly from m". This is the principle of the multiple images formed by two or more mirrors, as in the following experiments. 701. Parallel Mirrors. Let an object be placed between two PARALLEL MIRRORS. 889 parallel mirrors which face each other, as in Fig. 630. The first reflections will form images a x Oj. The second reflections will form images 0-2 o 2 of the first images; and the third reflections will form images a 3 o 3 of the second images. The figure represents an eye receiving the rays which foi*m the third images, and shows the paths which these rays have taken in their whole course from the object to the eye. The rays by which the same eye sees the other images are omitted, to avoid confusing the figure. A long row of images o O O can thus be seen at once, becom- ing more and more dim as they recede in (he distance, inasmuch as each reflection involves a loss of light. If the mirrors are truly parallel, all the images will be ranged in Fig. 631. Mirrors at Right Angles. one straight line, which will be normal to the mirrors. If the mirrors are inclined at any angle, the images will be ranged on the circumference of a circle, whose m" Fig. 632. Mirrors at Right Angles. centre is on the line in which the reflecting surfaces would intersect if produced. This principle is sometimes employed as a means of adjusting mirrors to exact parallelism. 702. Mirrors at Right Angles. Let two mirrors A, B (Fig. 631), 890 REFLECTION OF LIGHT. be set at right angles to each other, facing inwards, and let m be a luminous point placed between them. Images m' m" will be formed by first reflections, and two coincident images will be formed at m"' by second reflections. No third reflection will occur, for the point m'", being behind the planes of both the mirrors, cannot be reflected in either of them. Counting the two coincident images as one, and also counting the object as one, there will be in all four images, placed at the four corners of a rectangle. Fig. 632 will give an idea of the appearance actually presented when one of the mirrors is vertical and the other horizontal. When both the mirrors are verti- cal, an observer sees his own image constantly bisected by their com- mon section, in a way which appears at first sight very paradoxical. 703. Mirrors Inclined at 60 Degrees. A symmetrical distribution of images may be obtained by placing a pair of mirrors at any angle which is an aliquot part of 360. If, for example, they oe inclined at 60 to each other, the number of images, counting the object itself as one, will be six. Their position is illustrated by Fig. 633. The object is placed in the sector A C B. The images formed by first reflections are situated in the two neighbouringsectors B C A ', A CB'; Fig. 633.-ige. >>i Kaleidoscope. tbe images formed by second reflec- tions are in the sectors B'CA", A' C B", and these yield, by third reflections, two coincident images in the sector B" C A", which is vertically opposite to the sector A C B in which the object lies, and i, therefore behind the planes of both mirrors, so that no further reflection can occur. 704. Kaleidoscope. The symmetrical distribution of images, ob- tained by two mirrors inclined at an angle which is an aliquot part of four right angles, is the principle of the kaleidoscope, an optical toy invented by Sir David Brewster. It consists of a tube containing two glass plates, extending along its whole length, and inclined at an angle of 60. One end of the tube is closed by a metal plate, with the exception of a hole in the centre, through which the observer looks in ; at the other end there are two plates, one of ground and the other of clear glass (the latter being next the eye), with a number of little pieces of coloured glass lying loosely between them. These THE KALEIDOSCOPE. 891 coloured objects, together with their images in the mirrors, form sym- metrical patterns of great beauty, which can be varied by turning or shaking the tube, so as to cause the pieces of glass to change their positions. A third reflecting plate is sometimes employed, the cross-section of the three forming an equilateral triangle. As each pair of plates produces a kaleidoscopic pattern, the arrangement is nearly equiva- lent to a combination of three kaleidoscopes. The kaleidoscope is capable of rendering important aid to designers. Fig 634. Kaleidoscopic Pattern. Fig. 634 represents a pattern produced by the equilateral arrange- ment of three reflectors just described. 705. Pepper's Ghost. Many ingenious illusions have been con- trived, depending on the laws of reflection from plane surfaces. We shall mention two of the most modern. In the magic cabinet, there are two vertical mirrors hinged at the two back corners of the cabinet, and meeting each other at a right angle, so as to make angles of 45 with the sides, and also with the back. A spectator seeing the images of the two sides, mistakes them for the back, which they precisely resemble; and performers may be concealed behind the mirrors when the cabinet appears empty. If one of the persons thus concealed raises his head above the mirrors, it will appear to be suspended in mid-air without a body. 892 REFLECTION OF LIGHT. The striking spectral illusion known as Pepper's Ghost is produced by reflection from a large sheet of unsilvered glass, which is so ar- ranged that the actors on the stage are seen through it, while other actors, placed in strong illumination, and out of the direct view of the spectators, are seen by reflection in it, and appear as ghosts on the stage. 705 A. Deviation produced by Rotation of Mirror. Let AB (Fig. 634 A) represent a mirror perpendicular to the plane of the paper, and capable of being rotated about an axis through C, also perpendicular to the paper ; and let I C represent an incident ray. When the mirror is in the position A B, perpendicular to I C, the ray will be re- flected directly back upon its course ; but when the mirror is turned through the acute angle A C A', the reflected ray will take the direction C R, making with the Fig.634x.-EffectofrotatingaMirro, ^^ QN an angle ^ Q ^ ^^ ^ ^ angle of incidence NCI. The deviation I C R of the reflected ray, produced by rotating the mirror, is therefore double of the angle I C N or A C A', through which the mirror has been turned ; and if, starting from the position A B', we turn the mirror through a further angle 0, the reflected ray CR will be turned through a further angle 2 0. It thus appears, that, when a plane mirror is rotated in the plane of incidence, the direction of the reflected ray is changed by double the angle through ivhich the mirror is turned. Con- versely, if we assign a constant direc- tion C I to the reflected ray, the direction of the incident ray R C must vary by double the angle through which the mirror is turned. 705 B. Hadley's Sextant. The above principle is illustrated in the nautical instrument called the sextant or quadrant, which was in- vented by Newton, and reinvented by Hadley. It serves for mea- suring the angle between any two distant objects as seen from the station occupied by the observer. Its essential parts are represented in Fior. 634 B. Fig. 634 B. Sextant. HADLEY'S SEXTANT. 893 It has two plane mirrors A, B, one of which, A, is fixed to the frame of the instrument, and is only partially silvered, so that a dis- tant object in the direction A H can be seen through the unsilvered part. The other mirror B is mounted on a movable arm B I, which carries an index I, traversing a graduated arc P Q. When the two mirrors are parallel, the index is at P, the zero of the graduations, and a ray H' B incident on B parallel to H A, will be reflected first along B A, and then along A T, the continuation of H A. The ob- server looking through the telescope T thus sees, by two reflections, the same objects which he also sees directly through the unsilvered part of the mirror. Now let the index be advanced through an angle 0; then, by the principles of last section, the incident ray S B makes with H' B, or H A, an angle 2 B. The angle between S B and H A would therefore be given by reading off the angle through which the index has been advanced, and doubling ; but in practice the arc P Q is always graduated on the principle of marking half degrees as whole ones, so that the reading at I is the required angle 20. In using the instrument, the two objects which are to be observed are brought into apparent coincidence, one of them being seen directly, and the other by successive reflection from the two mirrors. This coincidence is not disturbed by the motion of the ship; but unpractised observers often find a difficulty in keeping both objects in the field of view. Dark glasses, not shown in the figure, are provided for protecting the eye in observations of the sun, and a vernier and reading microscope are provided instead of the pointer I. 706. Spherical Mirrors. By a spherical mirror is meant a mirror Fig. 635. Principal Focus. whose reflecting surface is a portion (usually a very small portion) of the surface of a sphere. It is concave or convex according as the inside or outside of the spherical surface yields the reflection. The centre of the sphere (C, Fig. 635) is called the centre of curvature of 894 REFLECTION OF LIGHT. the mirror. If the mirror has a circular boundary, as is usually the case, the central point A of the reflecting surface may conveniently be called the pole of the mirror. Centre of the mirror is an ambigu- ous phrase, being employed sometimes to denote the pole, and some- times the centre of curvature. The line A C is called the principal axis of the mirror, and any other straight line through C which meets the mirror is called a secondary axis. When the incident rays are parallel to the principal axis, the re- flected rays converge to a point F, which is called the principal focus. This law is rigorously true for parabolic mirrors (generated by the revolution of a parabola about its principal axis). For sphe- rical mirrors it is only approximately true, but the approximation is very close if the mirror is only a very small portion of an entire sphere. In grinding and polishing the specula of large reflecting telescopes, the attempt is made to give them, as nearly as possible, the parabolic form. Parabolic mirrors are also frequently employed Fig. 636. Theory of Conjugate Foci. to reflect, in a definite direction, the rays of a lamp placed at the focus. Rays reflected from the circumferential portion of a spherical mir- ror are always too convergent to concur exactly with those reflected from the central portion. This deviation from exact concurrence is called spherical aberration. 707. Conjugate Foci. Let P (Fig. 636) be a luminous point situ- ated on the principal axis of a spherical mirror, and let P I be one of the rays which it sends to the mirror. Draw the normal I, which is simply a radius of the sphere. Then I P is the angle of incid- CONJUGATE FOCI. 895 ence, and the angle of reflection I P' must be equal to it ; hence I bisects an angle of the triangle P I P', and therefore we have IP _ OP If OP" Let p, p denote AP, A P' respectively, and let r denote the radius of the sphere. Then, if the angular aperture of the mirror is small, I P is sensibly equal to p, and I F to p. Substituting these approxi- mate values, the preceding equation becomes P p- r p r-p' or, dividing by pp' r t whence p r + p' r = 2 p p'; l - + - f = - (a) p p' r This formula determines the position of the point P', in which the reflected ray cuts the principal axis, and shows that it is, to the accuracy of our approximation, independent of the position of the point I ; that is to say, all the rays which P sends to the mirror are reflected to the same point P'. We have assumed P to be on the principal axis. If we had taken it on a secondary axis, as at p (Fig. 636), we should have found, by the same process of reasoning, that the reflected rays would all meet in a point p' on that secondary axis. The distinction between primary and secondary axes, in the case of a spherical mirror, is in fact merely a matter of convenience, not representing any essential difference of property. Hence we can lay down the following general proposition as true within limits of error corresponding to the approximate equalities which we have above assumed as exact : Rays proceeding from any given point in front of a concave spherical mirror, are reflected so as to meet in another point; and the line joining the two points passes through the centre of the sphere. It is evident that rays proceeding from the second point to the mirror would be reflected to the first. The relation between them is therefore mutual, and they are hence called conjugate foci. By a focus in general is meant a point in which a number of rays meet (or would meet if produced) ; and the rays which thus meet, taken collectively, are called a pencil. Fig. 637 represents two pencils of rays whose foci Ss are conjugate, so that, if either of them be re- garded as an incident pencil, the other will be the corresponding reflected pencil 896 REFLECTION OF LIGHT. We can now explain the formation of images by concave mirrors. Each point of the object sends a pencil of rays to the mirror, which converge, after reflection, to the conjugate focus. If the eye of the observer be placed beyond this point of concourse, and in the path of the rays, they will present to him the same appearance as if they Fig. 637. Conjugate Foci. had come from this point as origin. The image is thus composed of points which are the conjugate foci of the several points of the object. 708. Principal Focus. If, in formula (a) of last section, we make p increase continually, the term will continually decrease, and will vanish as p becomes infinite. This is the case of rays parallel to the principal axis, for parallel rays may be regarded as coming from a point at infinite distance. The formula then becomes 1 A 1 f 7 . = whence = , p' r ' 2 that is to say, the principal focal distance is half the radius of cur- vature. This distance is often called the focal length of the mirror. If we denote it by /, the general formula becomes i + i = j> <*) P p f 709. Discussion of the Formula. By the aid of this formula we can easily trace the corresponding movements of conjugate foci. If p is positive and very large, p' is a very little greater than /; that is to say, the conjugate focus is a very little beyond the princi- pal focus. As p diminishes, p' increases, until they become equal, in which case each of them is equal to r or 2/; that is to say, the conjugate foci move towards each other till they coincide at the centre of cur- vature. This last result is obvious in itself; for rays from the centre DISCUSSION OF THE FORMULA. 897 of curvature are normal to the mirror, aud are therefore reflected directly back. As p continues to diminish, the two foci, as it were, change places ; the luminous point advancing from the centre of curvature to the principal focus, while the conjugate focus moves away from the centre of curvature to infinity. As the luminous point continues to approach the mirror, is greater than j* and hence ' and therefore also p', must be nega- tiva The physical interpretation of this result is that the conjugate focus is behind the mirror, as at s (Fig. 038), and that the reflected Fig. 638. Virtual Focus. rays diverge as if they had come from this point. Such a focus is called virtual, while a focus in which rays actually meet is called real. As the luminous point moves up from F to the mirror, the conjugate focus moves up from an infinite distance at the back, and meets it at the surface of the mirror. If S is a real luminous point sending rays to the mirror, it must of necessity lie in front of the mirror, and p therefore cannot be nega- tive ; but when we are considering images of images this restriction no longer holds. If an incident beam, for example, converges to- wards a point s at the back of the mirror, it will be reflected to a point S in front. In this case p is negative, and p' positive. The conjugate foci S s have in fact changed placea It appears from the above investigation that there are two prin- cipal cases, as regards the positions of conjugate foci of a con cave mirror. 1 . One focus between F and C ; and the other beyond C. 2. One focus between F and the mirror; and the other behind the mirror. In the former case, the foci move to meet each other at C ; in the latter, they move to meet each other at the surface of the mirror. 58 898 REFLECTION OF LIGHT. 710. Formation of Images. We are now in a position to discuss the formation of images by concave mirrors. Let A B (Fig. 639) be an object placed in front of a concave mirror, at a distance greater than its radius of curvature. All the rays which diverge from A will be reflected to the conjugate focus a. Hence this point can be found by the following construction. Draw through A the ray A A' parallel to the principal axis, and draw its path after reflection, which must of necessity pass through the principal focus. The intersection of this reflected ray with the secondary axis through A will be the point required. A similar construction will give the conjugate focus Fig. 639. Formation of Image. corresponding to any other point of the object; b, for example, 1 is the focus conjugate to B. Points of the object lying between A and B will have their conjugate foci between a and b. An eye placed behind the object A B will accordingly receive the same impression from the reflected rays as if the image a b were a real object. 711. Size of Image. As regards the comparative sizes of object and image, it is obvious, from similar triangles, that their linear di- mensions are directly as their distances from C the centre of curvature. Again, since C F and A A' are parallel, we have oF _ aC _ ab t FA' ~ CA ~ AB* length of image .]_ distance of image from principal focus . , . length of object focal length and by a similar construction we can prove that 1 It is only by accident that b happens to lie on A A' in the figure EXPERIMENT OF THE PHANTOM BOUQUET. 81)9 length of object _ distance of object from principal focus length of image focal length (d) This last formula affords the readiest means of calculating the size of the image when the size and position of the object are given. Both the formulae (c) and (d) are perfectly general, both for concave and convex mirrors. They show that the object and image will be equal when they coincide at the centre of curvature, and that as they move away from this point, in opposite directions, that which moves away from the mirror continually gains in size upon the other. Since the lines joining corresponding points of object and image cross at the point C, which lies between them when the image is real, a real image formed by a concave mirror is always inverted. 712. Experiment of the Phantom Bouquet. Let a box open on one Fig. 640. Experiment of Pbautom Bouquet. side be placed in front of a concave mirror, at a distance about equal to its radius of curvature, and let an inverted bouquet be suspended within it, the open side of the box being next the mirror. By giving a proper inclination to the mirror, an image of the bouquet will be obtained in mid-air, just above the top of the box. As the bouquet 900 KEFLECTION OF LIGHT. is iaverted, its image is erect, and a real vase may be placed in such a position that the phantom bouquet shall appear to be standing in it. The spectator must be full in front of the mirror, and at a suf- ficient distance for all parts of the image to lie between his eyes and the mirror. When the colours of the bouquet are bright, the image is generally bright enough to render the illusion very complete. 713. Images on a Screen. Such experiments as that just described can only be seen by a few persons at once, since they require the spectator to be in a line with the image and the mirror. When an image is projected on a screen, it can be seen by a whole audience Fig. 641. Image on Screen. at once, if the room be darkened and the image be large and bright. Let a lighted candle, for example, be placed in front of a concave mirror, at a distance exceeding the focal length, and let a screen be placed at the conjugate focus; an inverted image of the candle will be depicted on the screen. Fig. 641 represents the case in which the candle i's at a distance less than the radius of curvature, and the image is accordingly magnified. By this mode of operating, the formula for conjugate focal dis- tances can be experimentally verified with considerable rigour, care being taken, in each experiment, to place the screen in the position which gives the most sharply defined image. IMAGES ON SCREEN AND IN MID-AIR. 901 714. Difference between Image on Screen, and Image as seen in Mid-air. Caustics. For the sake of simplicity we have made some statements regarding visible images which are not quite accurate; and we must now indicate the necessary corrections. Images thrown on a screen have a determinate position, and are really the loci of the conjugate foci of the points of the object ; but this is not rigorously true of images seen directly. They change their position to some extent, according to the position of the observer. The actual state of things is explained by Fig. 611 A. The plane of the figure 1 is a principal plane (that is, a plane containing the prin- cipal axis) of a concave hemispherical mirror, and the incident rays Fig. 641 A. Position of Image in Oblique Reflection. are parallel to the principal axis. All the rays reflected in the plane of the figure touch a certain curve called a caustic curve, which has a cusp at F, the principal focus ; and the direction in which the image is seen by an eye situated in the plane of the figure is deter- mined by drawing from the eye a tangent to this caustic. If the eye be at E, on the principal axis, the point of contact will be F; but when the rays are received obliquely, as at E', it will be at a point a not lying in the direction of F. For an eye thus situated, a is called the primary focus, and the point where the tangent at a cuts the principal axis is called the secondary focus. When the eye is moved in the plane of the diagram, the apparent position of the 1 Figs. 641 A and 657 A are borrowed, by permission, from Mr. Osmund Airy's Geometri- cal Optics. 902 REFLECTION OF LIGHT. image (as determined by its remaining in coincidence with a cross of threads or other mark) is the primary focus; and when the eye is moved perpendicular to the plane of the diagram, the apparent posi- tion of the image is the secondary focus. 1 If we suppose the diagram to rotate about the principal axis, it will still remain true in all positions, and the surface generated by this revolution of the caustic curve is the caustic surface. Its form and position vary with the position of the point from which the incident rays proceed ; and it has a cusp at the focus conjugate to this point. There is always more or less blurring, in the case of images seen obliquely (except in plane mirrors), by reason of the fact that the point of contact with the caustic surface is not the same for rays entering different parts of the pupil of the eye. A caustic curve can be exhibited experimentally by allowing the rays of the sun or of a lamp to fall on the concave surface of a strip of polished metal bent into the form of a circular arc, as in Fig. 642, the reflected light being received on a sheet of white paper on which the strip rests. The same effect may often be observed on the surface of a cup of tea, the reflector in this case being the inside of the tea-cup. The image of a luminous point received upon a screen is formed by all the rays which touch the corresponding caustic surface. The brightest and most distinct image will be formed at the cusp, which is, in fact, the conjugate focus; but there will be a border of fainter light surrounding it. This source of indistinctness in images is an example of spherical aberration ( 707). 714 A. Image on a Screen by Oblique Reflection. If we attempt to throw upon a screen the image of a luminous point by means of a concave mirror very oblique to the incident rays, we shall find that no image can be obtained at all resembling a point ; but that there are two positions of the screen in which the image becomes a line. 1 Since every ray incident parallel to the principal axis, is reflected through the principal axis. If the incident rays diverged from a point on the principal axis, they would still be reflected through the principal axis. Fig. 642. Caustic by Reflection. IMAGE OX SCREEN BY OBLIQUE REFLECTION. 903 Fig. 641 B. Formation of Focal Lines. Tliis is called the primary In the annexed figure (Fig. 641 B), which represents on a larger scale a portion of Fig. 64-1 A, ac,bd are rays from the highest and lowest points of the portion R S of the hemispherical mirror, which portion we suppose to be small in both its dimensions in comparison with the radius of curvature ; and we may suppose the rest of the hemisphere to be re- moved, so that R S will repre- sent a small concave mirror re- ceiving a pencil very obliquely. Then, if a screen be held perpendicular to the plane of the diagram, at m, where the section of the pencil by the plane of the diagram is nar- rowest, a blurred line of light will be formed upon it, the length of the line being per- pendicular to the plane of the diagram. focal line. The secondary focal line is c d, which, if produced, passes through the centre of curvature of the mirror, and also through the point from which the incident light proceeds. This line is very sharply formed upon a screen held so as to coincide with c d and to be per- pendicular to the plane of the diagram. Its edges are much better defined than those of the primary line ; and its position in space is also more definite. If the mirror is used as a burning-glass to collect the sun's rays, ignition will be more easily obtained at one of these lines than in any intermediate position. 1 Focal lines can also be seen directly. In this case a small element of the mirror sends all its reflected rays to the eye, the rays from opposite sides of the element crossing each other at the focal lines, before they reach the eye. It is possible, in certain positions of the eye, to see either focal line at pleasure, by altering the focal adjust- ment of the eye ; or the two may be seen with imperfect definition 1 The "elongated figure of 8" which is often mentioned in connection with the secondary focal line, is obtained by turning the screen about n the middle point of cd, so as to blur both ends of the image by bad focussing. It will be observed, from an inspection of the diagram, that cd is very oblique to the reflected rays. If we neglect the blurring of the primary line, we may describe the part of the pencil lying between the two lines as a tetrahedron, of which the two lines are opposite edges. 904- REFLECTION OF LIGHT. Fig. 643. Formation of Virtual Image. crossing each other at right angles. The experiment is easily made by employing a gas flame, turned very low, as the source of light. One line is in the plane of incid- ence, and the other is nor- mal to this plane. 715. Virtual Image in Concave Mirror. Let an object be placed, as in Fig. 643, in front of a concave mirror, at a distance less than that of the principal focus. The rays incident on the mirror from any point of it, as A, will be reflected as a divergent pencil, the focus from which they diverge being a point b at the back of the mirror. To find this point, we may trace the course of a ray through A parallel to the principal axis. Such a ray will be reflected to the principal focus F, and by producing this reflected ray backwards till it meets the secondary axis C A, the point b, which is the conju- gate focus of A, is deter- mined. We can find in the same way the position of a, the conjugate focus of B, and it is obvious that the image of A B will be erect and magnified. 716. Remarks on Virtual Images. A virtual image cannot be projected on a screen; for the rays which produce it do not actually pass through its place, but only seem to do so. A screen placed at a b would obviously receive none of the reflected light whatever. Fig. 644. Virtual Image in Concave Mirror. CONVEX MIRRORS. 9<)j The images seen in a plane mirror are virtual ; and any spherical mirror, whether concave or convex, is nearly equivalent to a plane mirror, when the distance of the object from its surface is small in comparison with the radius of curvature. 717. Convex Mirrors. It is easily shown, by a simple construction, that rays incident from any luminous point upon a convex mirror, diverge after reflection. The principal focus, and the foci conjugate to all points external to the sphere, are therefore virtual. To adapt formulae (a) and (6) of the preceding sections to the case 2 1 of convex mirrors, we have only to alter the sign of the term r or -, > so that for a convex mirror we shall have 1 + 4 = _ 1 = _ 1; (c) p p f r r and /being here regarded as essentially positive. From this formula it is obvious that one at least of the two dis- tances p,p must be negative ; that is to say, one at least of any pair of conjugate foci must lie behind the mirror. The construction for an image (Fig. 645) is the same as in the case of concave mirrors. Through any selected point of the object Fig. 645. Formation of Image in Convex Mirror. draw a ray parallel to the principal axis; the reflected ray, if pro- duced backwards, must pass through the principal focus, and its intersection with the secondary axis through the selected point deter- mines the corresponding point of the image. The image of an ex- ternal object will evidently be erect, and smaller than the object. Repeating the same construction when the object is nearer to the mirror, we see that the image will be larger than before. The linear dimensions of an object and its image, whether in the 906 REFLECTION OF LIGHT. case of a convex or a concave mirror, are directly proportional to their distances from the centre of curvature. The image is inverted or erect according as this centre does or does not lie between the object and its image. In the case of a convex mirror the centre never lies between them (if the object be real), and therefore the image is always erect. Convex mirrors are very seldom employed in optical instruments. The silvered globes which are frequently used as ornaments, are examples of convex mirrors, and present to the observer at one view an image of nearly the whole surrounding landscape. As the part of the mirror in which he sees this image is nearly an entire hemi- sphere, the deformation of the image is very notable, straight lines being reflected as curves. 718. Anamorphosis. Much greater deformations are produced by Fig. 646. Anamorphosis. cylindric mirrors. A cylindric mirror, when the axis of the cylinder is vertical, behaves like a plane mirror as regards the angular magni- tude under which the height of the image is seen, and like a spherical mirror as regards the breadth of the image. If it be a convex cylin- der, it causes bodies to appear unduly contracted horizontally in pro- portion to their heights. Distorted pictures are sometimes drawn upon paper, according to such a system that when they are seen MEDICAL APPLICATIONS. DO? reflected in a cylindric mirror properly placed, as in Fig. 640, the distortion is corrected, and while the picture appears a mass of con- fusion, the image is instantly recognized. This restoration of true proportion in a picture is called anamorphosis. 719. Medical Applications. Concave mirrors are frequently used to concentrate light upon an object for the purpose of rendering it more distinctly visible. The ophthalmoscope is a small concave mirror, with a small hole in its centre, through which the observer looks from behind, while he directs a beam of reflected light from a lamp into the pupil of the patient's eye. In this way (with the help sometimes of a lens) the retina can be rendered visible, and can be minutely examined. The laryngoscope consists of two mirrors. One is a small plane mirror, with a handle attached, at an angle of about 45 to its plane. This small mirror is held at the back of the patient's mouth, so that the observer, looking into it, is able by reflection to see down the patient's throat, the necessary illumination being supplied by a con- cave mirror, strapped to the observer's forehead, by means of whHi the light from a lamp is reflected upon the plane mirror, which again reflects it down the throat. CHAPTER LIX. REFRACTION. 720. Refraction. When a ray of light passes from one transparent medium to another, it undergoes a change of direction at the surface of separation, so that its course in the second medium makes an angle with its course in the first. This changing of direction is called re- fraction. The phenomenon can be exhibited by admitting a beam of the sun's rays into a dark room, and receiv- ing it on the surface of water con- tained in a rectangular glass vessel. The path of the beam will be easily traced by its illumination of the small solid particles which lie in its course. The following experiment is a well-known illustration of refrac- tion : A coin ra n (Fig. 648) is laid at the bottom of a vessel with opaque sides, and a spectator places himself so that the coin is just hidden from him by the side of the vessel ; that is to say, so that the line m A in the figure passes just above his eye. Let water now be poured into the vessel, care being taken not to displace the coin. The bottom of the vessel will appear to rise, and the coin will come into sight. Hence a pencil of rays from m must have entered the spectator's eye. The pencil in fact undergoes a sudden bend at the surface of the water, and thus reaches the eye by a crooked course, Fig. 647. I REFRACTIVE POWERS OF DIFFERENT MEDIA. 909 Fig. 648. Experiment of Coin in Basiu. in which the obstacle A is evaded. If the part of the pencil in air be produced backwards, its rays will approximately meet in a point m', which is therefore the image of m. Its position is not correctly indicated in the figure, being placed too much to the left ( 727 A). The broken appearance presented by a stick (Fig. 649) when partly immersed in water in an oblique position, is similarly ex- plained, the part beneath the water being lifted up by refraction. 721. Refractive Powers of Different Media. In the experiments of the coin and stick, the rays, in leaving the water, are bent away from the normals Z I N, Z' I' N' at the points of emergence; in the experiment first described (Fig. 647), on the other hand, the rays, in passing from air into water, are bent nearer to the normal. In every case the path which the rays pursue in going is the same as they would pur- sue in returning; and of the two media concerned, that in which the ray makes the smaller angle with the normal is said to have greater refractive power than the other, or to be more highly refracting. Liquids have greater refractive power than gases, and as a general rule (subject to some exceptions in the comparison of dissimilar sub- stances) the denser of two substances has the greater refracting power. Hence it has become customary, in enunciating some of the laws of optics, to speak of the denser medium and the rarer medium, when the more correct designations would be more refractive and less refractive. 722. Laws of Refraction. The quantitative law of refraction was not discovered till quite modern times. It was first stated by Snell, a Dutch philosopher, and was made more generally known by Des- cartes, who has often been called its discoverer. Fig. 649. Appearance of Stick in Water. 910 . REFRACTION. Let R 1 (Fig. 650) be a ray incident at I on the surface of separa- tion of two media, and let I S be the course of the ray after refrac- tion. Then the angles which R I and I S make with the normal are called the angle of incidence and the angle of refraction respec- tively ; and the first law of refraction is that these angles lie in the same plane, or the plane of refraction is the same as the plane of incidence. The law which connects the mag- O nitudes of these two angles, and which was discovered by Snell, can only be stated either by reference to a geo- metrical construction, or by employing the language of trigonometry. De- scribe a circle about the point of in- cidence I as centre, and drop perpen- Fig. 650. Law of Refraction. diculars, from the points where it cuts the rays, on the normal. The law is that these perpendiculars R' P', S P, will have a constant ratio ; or the sines of the angles of incidence and refraction are in a constant ratio. It is often referred to as the law of sines. The angle by which a ray is turned out of its original course in undergoing refraction is called its deviation. It is zero if the inci- dent ray is normal, and always increases with the angle of incid- ence. 723. Verification of the Law of Sines. These laws can be verified by means of the apparatus represented in Fig. 651, which is very similar to that employed by Descartes. It has a vertical divided circle, to the front of which is attached a cylindrical vessel, half-filled with water or some other transparent liquid. The surface of the liquid must pass exactly through the centre of the circle. I is a movable mirror for directing a reflected beam of solar light on the centre O. The beam must be directed centrally through a short tube attached to the mirror, and to facilitate this adjustment the tube is furnished with a diaphragm with a hole in its centre. The arm a is movable about the centre of the circle, and carries a ver- nier for measuring the angle of incidence. The ray undergoes refrac- tion af; O ; and the angle of refraction is measured by means of a second arm R, which is to be moved into such a position that the diaphragm of its tube receives the beam centrally. No refraction AIRYS APPARATUS. 911 occurs at emergence, since the emergent beam is normal to the sur- faces of the liquid and glass; the position of the arm accordingly indicates the direction of the refracted ray. The angles of inci- dence and refraction can be read off at the verniers carried by the two arms; and the ratio of their sines will be found con- stant. The sines can also be directly mea- sured by employing sliding-scales as indi- cated in the figure, the readings being taken at the extre- mity of each arm. It would be easy to make a beam of light enter at the lower side of the apparatus, in a radial direction ; and it would be found that the ratio of the sines was precisely the same as when the light entered from above. This is merely an instance of the general law, that the course of a returning ray is the same as that of a direct ray. 723 A. Airy's Apparatus. The following apparatus for the same pur- pose was invented, some fifty years ago, by the present astronomer royal. B' is a slider travelling up and down a vertical stem. AC' and BC are two rods pivoted on a fixed point B of the vertical stem. C' B' and CB' are two other rods jointed to the former at C' and C, and pivoted at their lower ends on the centre of the slider. B C is equal to B' C', and 1. Apparatus for Verifying the Law. Fig 651 A. Airy's Apparatus. 912 REFRACTION. B C' to B' C. Hence the two triangles B C B', B' C' B are equal to one another in all positions of the slider, their common side BB' being variable, while the other two sides of each remain unchanged in length though altered in position, B C B' C' The ratio ^^, or ^^ is made equal to the index of refraction of the liquid in which the observation is to be made. For water this ratio will be 3. Then, if the apparatus is surrounded with water up to the level of B, ABC will the path of a ray, and a stud at C will appear in the same line with studs at A and B; for we have SinC'BB' sinC'BB' C'B' _4 ~3' Sin C B B' sin C' B' B C'B' C'B 724. Indices of Refraction. The ratio of the sine of the angle of incidence to the sine of the angle of refraction when a ray passes from one medium into another, is called the relative index of refraction from the former medium to the latter. When a ray passes from vacuum into any medium this ratio is always greater than unity, and is called the absolute index of refraction, or simply the index of refraction, for the medium in question. The relative index of refraction from any medium A into another B is always equal to the absolute index of B divided by the absolute index of A. The absolute index of air is so small that it may usually be neglected in comparison with those of solids and liquids: but strictly speaking, the relative index for a ray passing from air into a given substance must be multiplied by the absolute index for air, in order to obtain the absolute index of refraction for the substance. The following table gives the indices of refraction of several sub- stances: INDICES OF REFRACTION.* Diamond 2'44 to 2'755 Sapphire, 1794 Flint-glass, 1-576 to T642 Crown-glass, T531 to 1'563 Rock-salt, 1-545 Canada balsam, 1-540 Bisulphide of carbon, .... T678 Linseed oil (sp. gr. '932), . . . 1'482 Oil of turpentine (sp. gr. -885), . T47S Alcohol, 1-372 Aqueous humour of eye, .... l - 337 Vitreous humour, 1'339 Crystalline lens, outer coat, . . . T337 under coat, . . . l - 379 central portion, . 1'400 Sea water, 1'343 Pure water, T336 Air at 0. C. and 760mm . . . 1-000294 725. Critical Angle. We see, from the law of sines, that when the incident ray is in the less refractive of the two media, to every possible angle of incidence there is a corresponding angle of refraction. This, 1 The index of refraction IB always greater for violet than for red (see Chap. Ixii.) The numbers in this table are to be understood as mean values. CRITICAL ANGLE. 913 however, is not the case when the incident ray is in the more refractive of the two media. Let S 0, S'O, S"0 (Fig. G52) be incident rays in the less refractive medium, and R, E', R" the corresponding refracted rays. There will be a particular direction of refraction L corresponding to Fig. 652. Critical Angle. the angle of incidence of 90. Conversely, incident rays RO, R'O, R"0, in the more refractive medium, will emerge in the directions S, S', S", and the direction of emergence for the incident ray L will be B, which is coincident with the bounding surface. The angle L N is called the critical angle, and is easily computed when the relative index of refraction is given. For let /* denote this index (the incident ray being supposed to be in the less refractive medium), then we are to have sin 90 1 = n, whence sin x = ; sin x that is, the sine of the critical angle is the reciprocal of the index of refraction. When the media are air and water, this angle is about 48 30'. For air and different kinds of glass its value ranges from 38 to 41. If a ray, as 1 0, is incident in the more refractive medium, at an angle greater than the critical angle, the law of sines becomes nugatory, and experiment shows that such a ray undergoes internal reflection in the direction I', the angle of reflection being equal to the angle of incidence. Reflection occurring in these circumstances is nearly perfect, and has received the name of total reflection. Total reflection occurs when rays are incident in the more refractive medium at an angle greater than the critical angle. The phenomenon of total reflection may be observed in several familiar instances. For example, if a glass of water, with a spoon in it (Fig. 653), 59 91 4- REFRACTION. is held above the level of the eye, the under side of the surface of the water is seen to shine like a brilliant mirror, and the lower part of the Fig. 653. Total Reflection. spoon is seen reflected in it. Beautiful effects of the same kind may be observed in aquariums. 727. Camera Lucida. The camera lucida is an instrument sometimes employed to facilitate the sketching of objects from nature. It acts by total reflection, and may have various forms, of which that proposed by Wollaston, and represented in Fig. 656, 657, is one of the commonest. The essential part is a totally -reflecting prism with four angles, one of which is 90, the opposite one 135, and the other two each 67 30'. One CAMERA LUCIDA. 915 of the two faces which contain the right angle is turned towards the objects to be sketched. Rays incident normally on this face, as xr, make an angle greatly exceeding the critical angle with the face c d, and are totally reflected from it to the next face da, whence they are again totally reflected to the fourth face, from which they emerge normally. 1 An eye Fig. 656. Section of 1'risin. Fig. 657. Camera Luoida. placed so as to receive the emergent rays will see a virtual image in a direction at right angles to that in which the object lies. In practice, ,- the eye is held over the angle a of the prism, in such a position that one-half of the pupil receives these reflected rays, while the other half receives light in a parallel direction outside the prism. The observer thus sees the reflected image projected on a real back -ground, which consists of a sheet of paper for sketching. He is thus enabled to pass a pencil over the outlines of the image, pencil, image, and paper being simultaneously visible. It is very desirable that the image should lie in the plane of the paper, not only because the pencil point and the image will then be seen with the same focussing of the eye, but also because parallax is thus obviated, so that when the observer shifts his eye the pencil point is not displaced on the image. As the paper, for conveni- ence of drawing, must be at a distance of about a foot, a concave lens, with a focal length of something less than a foot, is placed close in front of the prism, in drawing distant objects. By raising or lowering the prism in its stand (Fig. 657), the image of the object to be sketched may be made to coincide with the plane of the paper. The prism is mounted in such a way that it can be rotated either about a horizontal or a vertical axis ; and its top is usually covered with a mov- able plate of blackened metal, having a semicircular notch at one edge, for the observer to look through. 727 A. Images by Refraction at a Plane Surface. Let (Fig. 657 B) 1 The use of having two reflections is to obtain an erect image. An image obtained by one reflection would be upside down. 916 REFRACTION. be a small object in the interior of a solid or liquid bounded by a plane surface A B. Let B C be the path of a nearly normal ray, and let B C (the portion in air) be produced backwards to meet the normal in I. Then, since A IB and AOB are the inclinations of the two portions of the ray to the normal, we have (if /z be the index of refraction from air into the sub- stance) sin AIB sin AOB OB IB* Fig. 657s. Image by Refraction. But B is ultimately equal to A, and I B to I A. Hence, if we make A I equal to AO , all the emergent rays of a small and nearly normal pencil emitted by will, if produced backwards, intersect OA at points indefinitely near to the point I thus determined. If the eye of an observer be situated on the production of the normal A, the rays by which he sees the object constitute such a pencil. He accordingly sees the image at I. As the 4 3 value of /i is | for water, and about ^ for glass, it follows that the 3 apparent depth of a pool of clear water when viewed vertically is 7 of the true depth, and that the apparent thickness of a piece of plate- 2 glass when viewed normally is only g of the true thickness. 727 B. When the incident pencil (Fig. 65 7 A) is not small, but includes Fig 6574. Caustic by Refraction. rays of all obliquities, those of them which make angles with the normal PARALLEL PLATE. 9J7 less than the critical angle N Q R will emerge into air; and the emergent rays, if produced backwards, will all touch a certain caustic surface, which has the normal Q N for its axis of revolution, and touches the surface at all points of a circle of which N R is the radius. Wherever the eye may be situated, a tangent drawn from it to the caustic will be the direction of the visible image. If the observer sees the image with both eyes, both being equidistant from the surface and also equidistant from the normal, the two lines of sight thus determined (one for each eye) will meet at a point on the normal, which will accordingly be the apparent position of the image. If, on the other hand, both eyes are in the same plane con- taining the normal, the two lines of sight will intersect at a point between the normal and the observer. The image, whether seen with one eye or two, approaches nearer to the surface as the direction of vision becomes more oblique, and ultimately 3 coincides with it. The apparent depth of water, which is only 7- of the Q real depth when seen vertically, is accordingly less than j when seen obliquely, and becomes a vanishing quantity as the direction of vision approaches to parallelism with the surface. The focus I determined in the preceding section is at the cusp of the caustic. 728. Parallel Plate. Rays falling normally on a uniform trans- parent plate with parallel faces, keep their course unchanged; but this is not the case with rays incident obliquely. A ray S I (Fig. 658), incident at the angle SIN, is refracted in the direction I R. The angle of incidence at R is equal to the angle of refraction at I, Fig. 658. -Parallel Plate. Fig. 659. Vision through Plate. and hence the angle of emergence S' R N' is equal to the original angle of incidence SIN. The emergent ray R S' is therefore parallel to the incident ray S I, but is not in the same straight line with it 918 REFRACTION. Objects seen obliquely through a plate are therefore displaced from their true positions. Let S (Fig. 659) be a luminous point which sends light to an eye not directly opposite to it, on the other side of a parallel plate. The emergent rays which enter the eye are parallel to the incident rays; but as they have undergone lateral displabement, their point of concourse 1 is changed from S to S', which is accordingly the image of S. The displacement thus produced increases with the thickness of the plate, its index of refraction, and the obliquity of incidence. It furnishes one of the simplest means of measuring the index of refrac- tion of a substance, and is thus employed in Pichot's refractometer. 729. Multiple Images produced by a Plate. Let S (Fig. G60) be a luminous point in front of a transparent plate with parallel faces. Of the rays which it sends to the plate, some will be reflected from the front, thus giving rise to an image S'. Another por- tion will enter the plate, un- dergo reflection at the back, and emerge with refraction at the front, giving rise to a second image S. Another portion will undergo internal reflection at the front, then again at the back, and by emerging in front will form a third image Sj. The same pro- cess may be repeated several times; and if the luminous object be a candle, or a piece of bright metal, a number of images, one behind another, will be visible to an eye properly placed in front. All the successive images, after the first two, continually diminish in brightness. If the glass be silvered at the back, the second image is much brighter than the first, when the incidence is nearly normal, but as the angle of incidence increases, the first image gains upon the second, and ultimately surpasses it. This is due to the fact that the re- flecting power of a surface of glass increases with the angle of incidence. 1 The rays which compose the pencil that enters the eye will not exactly meet (when produced backwards) in any one point. There will be two focal lines, just as in the case of spherical mirrors (714, 71 4 A). Fig. 660. Multiple Images in Plate. REFRACTION THROUGH PRISMS. 919 If the luminous body is at a distance which may be regarded a* infinite, if it is a star, for example, all the images should coincide, and form only a single image, occupying a position which does not vary with the position of the observer, provided that the plate is perfectly homogeneous, and its faces perfectly plane and par- allel. A severe test is thus furnished of the fulfilment of these conditions. Plates are sometimes tested, for parallelism and uniformity, by sup- porting them in a horizontal position on three points, viewing the image of a star in them with a telescope fur- nished with cross wires, and observ- Fi g . eei.-image.of Candle in Looking-gja*, ' m S whether the image is displaced on the wires when the plate is shifted into a different position, still resting on the same three points. 730. Refraction through a Prism. For optical purposes, any por- Fig. 662. Equilateral Prism. Fig. 663. Priam mounted on Stand. tion of a transparent body lying between two plane faces which are 920 II EFR ACTION. not parallel may be regarded as a prism. 1 The line in which these faces meet, or would meet if produced, is called the edge of the prism, and a section made by a plane perpendicular to them both is called a principal section. The prisms chiefly employed are really prisms in the geometrical sense of the word. Their principal sections are usually triangular, and are very frequently equilateral, as in Fig. 662. The stand usually employed for prisms when mounted separately is represented in Fig. 663. It contains several joints. The uppermost is for rotating the prism about its own axis. The second is for turn- ing the prism so that its edges shall make any required angle with the vertical. The third gives motion about a vertical axis, and also furnishes the means of raising and lowering the prism through a range of several inches. Let S I (Fig. 664) be an incident ray in the plane of a principal section of the prism. If the external medium be air, or any other substance of less refractive power than the prism, the ray in entering the prism will be bent nearer to the normal, taking such a course as I E, and in leaving the prism will be bent away from the normal, taking the course E B. The effect of these two refractions is, there- fore, to turn the ray away from the edge (or refracting angle) of the prism. In practice, the prism is usually so placed that I E, the path of the ray through the prism, makes equal angles with the two faces at which refraction occurs ( 731). If the prism is turned very far from this position, the course of the ray may be alto- gether different from that repre- sented in the figure; it may, for Fig 664 _ Refraction through Prim . example, enter at one face, be in- ternally reflected at another, and come out at the third; but we at present exclude such cases from consideration. The direction of deviation is easily shown experimentally, by admitting a narrow beam of sunlight into a dark room, and intro- ducing a prism in its course. It will be found that the refracted beam, in the circumstances represented in Fig. 664, is turned aside some 40 or 50 from its original course. 2 1 This amounts to saying that the word prism in optics means wedye. 8 The phenomena here described are complicated in practice by the unequal refrangibi- REFRACTION THROUGH PRISMS. 921 Since the rays which traverse a prism are beat away from the edge, the object from which they proceed will appear, to an observer looking through the prism, to be more nearly in the direction of the edge than it really is. If, for example, he looks at the flame of a candle through a prism placed so that the edge which corresponds Fig. 665. Vision through Prism. to the refracting angle is at the top (Fig. 665), the apparent place of the flame will be above its true place. 731. Formulae for Refraction through Prisms. Minimum Deviation. Let S I (Fig. 666) be an incident ray in the plane of a principal section A B C of a prism. Let i be the angle of incidence SIN, and r the angle of refraction M 1 1'. Then, denoting the index of refrac- tion by p, we have sin i = p. sin r. In like manner, putting r' for lity of rays of different colours (Chap. Ixii.) The complication may be avoided by em- ploying homogeneous light, of which a spirit-lamp, with common salt sprinkled on the wick, affords a nearly perfect example. 922 REFRACTION. the angle of internal incidence on the second face 1 1' M, and i' for the angle of external refraction N' I' R,, we have sin i' = p. sin r'. The deviation produced at I is i r, and that at I' is i' r', so that Fig. 666. Refraction through Prism. the total deviation, which is the acute angle D contained between the rays S I, R I/, when produced to meet at o, is D=i-r + i'-/. (1) But if we drop a perpendicular from the angular point A on the ray 1 1', it will divide tfce refracting angle BAG into two parts, of which that on the left will be equal to r, and that on the right to r', since the angle contained between two lines is equal to that contained between their perpendiculars. We have therefore A=r + r', and by substitution in the above equation D = t + ?-A. (2) When the path of the ray through the prism 1 1' makes equal angles with the two faces, the whole course of the ray is symmetrical with respect to a plane bisecting the refracting angle, so that we have Equation (2) thus becomes D = 2 t A, whence i = , A + D. (3) sin 2 A + D andM= 8 ! 1J =_ sin r j^ sin- CONSTRUCTION FOB DEVIATION. 923 This last result is of great practical importance, as it enables us to calculate the index of refraction / from measurements of the refract- ing angle A of the prism, and of the deviation D which occurs when the ray passes symmetrically. When a beam of sunlight in a dark room is transmitted through a prism, it will be found, on rotating the prism about its axis, that there is a certain mean position which gives smaller deviation of the transmitted light than positions on either side of it; and that, when the prism is in this position, a small rotation of it has no sensible effect on the amount of deviation. The position determined experi- mentally by these conditions, and known as the position of mini- mum deviation, is the position in which the ray passes symmetrically. 731 . Construction for Deviation. The following geometrical con- struction furnishes a very sim- ple method of representing the variation of deviation with the angle of incidence : 1. When the refraction is at a single surface, describe two circular arcs about a common centre O (Fig. 666 A), the ratio of their radii being the index of refraction. Then if the in- cidence is from rare to dense, draw a radius O A of the smaller circle to represent the direction of the incident ray, and let N A B be the direction of the normal to the surface at the point of incidence, so that O A N is the angle of incidence. Join O B. Then O B N is the angle 'of refraction, since ?| n -P>?=!= index of refraction; hence B is parallel to the sin B N O A refracted ray. If the incidence is from dense to rare, we must draw B to represent the incident ray, make O B N equal to the angle of incidence, and join O A. In either case the angle A O B is the deviation, and it evidently increases with the angle of incidence O A N, attaining its greatest value when this angle (O A N" in the figure) is a right angle, in which case the angle of refraction B""N" is the critical angle. 2. To find the deviation in refraction through a prism, describe two concentric circular arcs as before (Fig. 666 B), the ratio of their radii being the index of refraction. Draw the radius O A of the smaller circle to represent the incident ray, N B to represent the Fig. 666 A. General Construction for Deviation. 924- REFRACTION. normal at the first surface, B N' the normal at the second surface. Then O B represents the direction of the ray in the prism, O A' the direction of the emergent ray, and A O A' is accordingly the total deviation. In fact we have O A N = angle of incidence at first surface. B N = refraction O B N' = incidence at second sur- face. OA'N'= refraction A B = deviation at first surface. B A' = second A B A' = angle between normals = angle of prism. Fig. 666 A.-General Construction for Deviation. A g ain > the Deviation A A', being the angle at the centre of a circle, is measured by the arc A A', which subtends it. To obtain the minimum deviation, we must so arrange matters that the angle ABA' being given (=angle of prism), the arc A A' shall be a mini- mum. Let A B A', a B a' (Fig. 666 c), be two consecutive positions, B A' and B a being greater than B A and B a. Then, since the small angles A B a, A' B a' are equal, it is obvious, for a double reason, that the small arc A' a' is greater than A a, and hence the B' Fig. 666s. Application to Prism. Fig. 666 c. Proof of Minimum Deviation. whole arc a a is greater than A A'. The deviation is therefore in- creased by altering the position in such a way as to make B A and B A' depart further from equality, and is a minimum when they are equal. 731 B. Conjugate Foci for Minimum Deviation. When the angle of incidence is nearly that corresponding to minimum deviation, a small change in this angle has no sensible effect on the amount of deviation. Hence a small pencil of rays sent in this direction from a luminous point, and incident near the refracting edge, will emerge with their DOUBLE REFRACTION. 925 divergence sensibly unaltered, so that if produced backwards they would meet in a virtual focus at the same distance (but of course not in the same direction) as the point from which they came. In like manner, if a small pencil of rays converging towards a point, are turned aside by interposing the edge of a prism in the position of minimum deviation, they will on emergence converge to another point at the same distance. We may therefore assert that, neglecting the thickness of a prism, conjugate foci are at the same distance from it, and on the same side, when the deviation is a 'minimum. 732. Double Refraction. Thus far we have been treating of what is called single refraction. We have assumed that to each given incident ray there corresponds only one refracted ray. This is true when the refraction is into a liquid, or into well -annealed glass, or into a crystal belonging to the cubic system. On the other hand, when an incident rav is refracted m into a crystal of any other than the cubic system, or into glass which is unequally stretched or compressed in different directions ; for example, into unannealed glass, it gives rise in general to two refracted rays which take different paths; and this pheno- menon is called double refraction. Attention was first called to it in 1670 by Bartholin, who observed it in the case of Iceland-spar, and its laws for this substance were accurately determined by Huyghens. 733. Phenomena of Double Refraction in Iceland-spar. Iceland-spar or calc-spar is a form of crystallized carbonate of lime, and is found in large quantity in the country from which it derives its name. It is usually found in rhombohedral form, as represented in Figs. 667, 668. To observe the phenomenon of double refraction, a piece of the spar may be laid on a page of a printed book. All the letters seen through it will appear double, as in Fig. 668 ; and the depth of their Fig. 667. Iceland-spar. 926 REFRACTION. blackness is considerably less than that of the originals, except where the two images overlap. In order to state the laws of the phenomena with precision, it is necessary to attend to the crystalline form of Iceland-spar. At the corner which is represented as next us in Fig. 6G7 three equal obtuse angles meet ; and this is also the case at the opposite Fig. 668. Double Refraction of Iceland-spar. corner which is out of sight. If a line be drawn through one of these corners, making equal angles with the three edges which meet there, it or any line parallel to it is called the axis of the crystal ; the axis being properly speaking not a definite line but a definite direction. The angles of the crystal are the same in all specimens; but the lengths of the three edges (which may be called the oblique length, breadth, and thickness) may have any ratios whatever. If the crystal Fig. 669. Axis of the Crystal. is of such proportions that these three edges are equal, as in the first part of Fig. 669, the axis is the direction of one of its diagonals, which is represented in the figure. DOUBLE REFRACTION IN ICELAND-SPAR. 927 Any plane containing (or parallel to) the axis is called a principal plane of the crystal. If the crystal is laid over a dot on a sheet of paper, and is made to rotate while remaining always in contact with the paper, it will be observed that, of the two images of the dot, one remains un- moved, and the other revolves round it. The former is called the ordinary, and the latter the extraordinary image. It will also be observed that the former appears nearer than the latter, being more lifted up by refraction. The rays which form the ordinary image follow the ordinary law of sines ( 722). They are called the ordinary rays. Those which form the extraordinary image (called the extraordinary rays) do not follow the law of sines, except when the plane of incidence is perpen- dicular to the axis of the crystal, and in this case their index of refraction (called the extraordinary index) is different from that of bhe ordinary rays. The ordinary index is T66, and the extraordinary 1-52. When the plane of incidence is parallel to the axis, the extra- ordinary ray lies in this plane, but the ratio of the sines of the angles of incidence and refraction is variable. When the plane of incidence is oblique to the axis, the extra- ordinary ray generally lies in a different plane. We shall recur to the subject of double refraction in the concluding chapter of this volume. CHAPTER LX. LENSES. 735. Forms of Lenses. A lens is usually a piece of glass bounded by two surfaces which are portions of spheres. There are two prin- cipal classes of lenses. 1. Converging lenses or convex lenses, which have one or other of the three forms represented in Fig. 670. The first of these is called double convex, the second plano-convex, and the third concavo- convex. This last is also called a converging meniscus. All three Fig. 670. Converging Lenses. Fig. 671. Diverging Trfmw*. are thicker in the middle than at the edges. They are called con- verging, because rays are always more convergent or less divergent after passing through them than before. 2. Diverging lenses or concave lenses (Fig. 671) produce the opposite effect, and are characterized by being thinner in the middle than at the edges. Of the three forms represented, the first is double concave, the second plano-concave, and the third convexo-concave (also called a diverging meniscus). 60 930 LENSES. Fig. 672. Principal Focus of Convex Lens. From the immense importance of lenses, especially convex lenses, in practical optics, it will be necessary to explain their properties at some length. 736. Principal Focus. A lens is usually a solid of revolution, and the axis of revolution is called the axis of the lens, or sometimes the principal axis. When the surfaces are spherical, it is the line join- ing their centres of curvature. When rays which were originally parallel to the principal axis pass through a convex lens (Fig. 672), the ef- fect of the two refrac- tions which they un- dergo, one on entering and the other on leav- ing the lens, is to make them all converge ap- proximately to one point F, which is called the principal focus. The dis.tance AF of the principal focus from the lens is called the principal focal distance, or more briefly and usually, the focal length of the lens. There is another prin- cipal focus at the same distance on the other side of the lens, cor- responding to an incid- ent beam coming in the opposite direction. The focal length depends on the convexity of the sur- faces of the lens, and also on the refractive power of the material of which it is composed, being shortened either by an increase of refractive power or by a diminution of the radii of cur- vature of the faces. In the case of a concave lens, rays incident parallel to the prin- cipal axis diverge after passing through ; and their directions, if produced backwards, would approximately meet in a point F, which is still called the principal focus. It is only a virtual focus, inasmuch as the emergent rays do not actually pass through it, whereas the principal focus of a converging lens is real. Fig. 673. Principal Focus of Concave Lens. OPTICAL CENTRE OF A LENS. 931 737. Optical Centre of a Lens. Secondary Axes. Let and 0' be the centres of the two spherical surfaces of a lens. Draw any two parallel radii 01, O'E to meet these surfaces, and let the joining line IE represent a ray passing through the lens. This ray makes equal angles with the normals at I and E, since these latter are parallel by construction; hence the incident and emergent rays S I, E R also make equal angles with the normals, and are there- fore parallel. In fact, if tangent planes (indicated by the dotted lines in the figure) are drawn at I and E, the whole course of the ray SIER will be the same as if it had passed through a plate bounded by these planes. Let C be the point in which the line I E cuts the principal axis, and let R, R' denote the radii of the two spherical surfaces. Then, from the similarity of the triangles OCI, O'CE, we have Fig. 674. Centre of Lens. oc CO' (1) which shows that the point C divides the line of centres 0' in a definite ratio depending only on the radii. Every ray whose direc- tion on emergence is parallel to its direction before entering the lens, must pass through the point C in traversing the lens ; and conversely, every ray which, in its course through the lens, traverses the point C, has parallel directions at incidence and emergence. The point C which possesses this remarkable property is called the centre, or optical centre, of the lens. In the case of a double convex or double concave lens, the optical centre lies in the interior, its distances from the two surfaces being directly as their radii In plano-convex and plano-concave lenses it is situated on the convex or concave surface. In a meniscus of either kind it lies outside the lens altogether, its distances from the surfaces being still in the direct ratio of their radii of curvature. 1 1 These consequences follow at once from equation (1) when applied to the several cases. 932 LENSES. In elementary optics it is usual to neglect the thickness of the lens. The incident and emergent rays SI, ER may then be re- garded as lying in one straight line which passes through C, and we may lay down the proposition that rays which pass through the centre of a Lens undergo no deviation. Any straight line through the centre of a lens is called a secondary axis. The approximate convergence of the refracted rays to a point, when the incident rays are parallel, is true for all directions of incidence ; Fig. 675. Principal Focus on Secondary Axis. and the point to which the emergent rays approximately converge (/, Fig. 675) is always situated on the secondary axis (a cf) parallel to the incident rays. The focal distance is sensibly the same as for rays parallel to the principal axis, unless the obliquity is consider- able. 738. Conjugate Foci. When a luminous point S sends rays to a Fig. 676. Conjugate Foci, both Real. lens (Fig. 676), the emergent rays converge (approximately) to one The distances of C from the two faces are respectively the difference between R and C, and the difference between R' and O'C, and we have R _ OC = R^0 R' O'C R' - O'C' FORMULAE RELATING TO LENSES. 933 point S'; whence it follows that rays sent from S' to the lens would converge (approximately) to S. Two points thus related are called conjugate foci of the lens, and the line joining them always passes through the centre of the lens ; in other words, they must either be both on the principal axis, or both on the same secondary axis. The fact that rays which come from one point go to one point is the foundation of the theory of images, as we have already explained in connection with mirrors ( 707). The diameters of object and image are directly as their distances from the centre of the lens, and the image will be erect or inverted according as the object and image lie on the same side or on opposite sides of this centre ( 711). There is also, in the case of lenses, the same difference between an image seen in mid-air and an image thrown on a screen which we have pointed out in 714. It is to be remarked that the distinction between principal and secondary axes has much more significance in the case of lenses than of mirrors; and images produced by a lens are more distinct in the neighbourhood of the principal axis than at a distance from it. 739. Formulae relating to Lenses. The deviation produced in a ray by transmission through a lens will not be altered by substituting Fig 677 Diagram showing Path of Hay, and Normals. for the lens a prism bounded by planes which touch the lens at the points of incidence and emergence ; and in the actual use of lenses, the direction of the rays with respect to the supposed prism is such as to give a deviation not differing much from the minimum. The expression for the minimum deviation ( 731) is 2i 2r or 2i A; and when the angle of the prism is small, as it is in the case of ordinary lenses, we may assume -^ = ^^ = /i; so that 2i becomes 2pr or n A, and the expression for the deviation becomes (M-l)A, (1) 934 LENSES. A being the angle between the tangent planes (or between the nor- mals) at the points of entrance and emergence. Let x l and # 2 denote the distances of these points respectively from the principal axis, and TJ, r 2 the radii of curvature of the faces on which they lie. Then x ~> * 2 are the sines of the angles which the normals make with the axis, and the angle A is the sum or differ- ence of these two angles, according to the shape of the lens. In the case of a double convex lens it is their sum, and if we identify the sines of these small angles with the angles themselves, we have A=- J +^. (2) 'l r 2 But if PI, p 2 denote the distances from the faces of the lens to the points where the incident and emergent rays cut the principal axis, - are the sines of the angles which these rays make with the PI p* axis, and the deviation is the sum or difference of these two angles, according as the conjugate foci are on opposite sides or on the same side of the lens. In the former case, identifying the angles with their sines, the deviation is + x -> and this, by formula (1), is to be equal to fc-1) A, that is, to (>-l) - 1+ ' If the thickness of the lens is negligible in comparison with p lt p 2 , we may regard x and x 2 as equal, and the equation (3) Pi Pz i r 2 will reduce to 1 + 1 =(,-1) (1 + 1). (4) Pi Pz Vl r 2/ If p l is infinite, the incident rays are parallel, and p z is the principal focal length, which we shall denote by /. We have therefore and l*!-r <> PI PI f 740. Conjugate Foci on Secondary Axis. Let M be a luminous CONJUGATE FOCI. 935 point on the secondary axis M O M', being the centre of the lens, and let M.' be the point in which an emergent ray corresponding to the incident ray MI cuts this axis. Let x denote x 1 orx 2 , the distances of the points of incidence and emergence from the principal axis, and Fig. 678. Conjugate Foci on Secondary Axis. the obliquity of the secondary axis; then x cos is the length of the perpendicular from I upon M M', and -^ , X M*I' are ^ ie s ^ nes ^ tne angles O M I, O M' I respectively. But the deviation is the sum of these angles ; hence, proceeding as in last section, we have a; cos "MT" and when 6 is small, its cosine is sensibly equal to unity; 1 in which case the equation reduces to + = 1 (S) The fact that x does not appear in equations (6) and (8) shows that, for every position of a luminous point, there is a conjugate focus lying on the same axis as the luminous point itself, at least in so far as the approximate assumptions which we have made are allowable. 741. Discussion of the Formula for Convex Lenses. It thus appears that the formula - + -, = \ (9) p P f 1 The quantity -= + - - is in fact rather greater than ; for in the first place, we M I M I / have taken cos as equal to unity, and in the second place we have assumed that the actual deviation is equal to the minimum deviation . Correcting these inexact assumptions, we see that - + ^-p-^ is really equal to a quantity rather greater than , multiplied by sec 6, which is also rather greater than unity. The effective focal length is therefore rather less for oblique than for direct pencils. 936 LENSES. applies both to direct and oblique pencils, / denoting the principal focal length of the lens (supposed convex), and p, p' the distances of a pair of conjugate foci from the lens, on opposite sides of it. This formula being identical with equation (6) of 708, leads to results similar to those already deduced in the case of concave mirrors. As one focus advances from infinite distance to a principal focus, its conjugate moves away from the other principal focus to infinite distance on the other side. The more distant focus is always moving more rapidly than the nearer, and the least distance between them is accordingly attained when they are equidistant from the lens; in which case the distance of each of them from the lens is 2/, and their distance from each other 4/. If either of the distances, as p, is less than /, the formula shows that the other distance p' is negative. The meaning is that the two foci are on the same side of the lens, and in this case one of them (the more distant of the two) must be virtual. For example, in Fig. 679, if S, S' are a pair of conjugate foci, one of them S being be- Fig. 679.-Conjugate Foci, one Real, one Virtual. twe6D the principal focus F and the lens, rays sent to the lens by a luminous point at S, will, after emergence, diverge as if from S'; and rays coming from the other side of the lens, if they converge to S' before incidence, will in reality be made to meet in S. As S moves towards the lens, S' moves in the same direction more rapidly ; and they become coincident at the surface of the lens. The formula in fact shows that if - is very great in comparison with j> and positive, -j must be very great and negative ; J P that is to say, if p is a very small positive quantity, p is a very small negative quantity. 742. Formation of Eeal Images. Let A B (Fig. 680) be an object in front of a lens, at a distance exceeding the principal focal length. It will have a real image on the other side of the lens. To deter- mine the position of the image by construction, draw through any point A of the object a line parallel to the principal axis, meeting FORMATION OF REAL IMAGES. 937 the lens in A.'. The ray represented by this line will, after refrac- tion, pass through the principal focus F; and its intersection with the secondary axis A O determines the position of a, the focus conju- gate to A. We can in like manner determine the position of b, the focus conjugate to B, another point of the object; and the joining line a b will then be Fig 680 ._ Real and Dimini8hed Image . the image of the line AB. It is evident that if a b were the object, AB would be the image. Figs. 680, 681 repiesent the cases in which the distance of the object is respectively greater and less than twice the focal length of the lens. In each case it is evident that -^ = -Q-^ =- or the linear dimensions of object and image are directly as their distances from the cen- tre of the lens. Again, since F O is parallel to a side of the triangle a A' A, we have OA = FA' = / a F a P' ~f Fig. 6S1. Real and Magnified Image. And by making a similar construction with respect to the other principal focus, we can prove that ^ ,= Pit. tt / We have therefore / denoting the focal length of the lens, and p, p the distances of A'B, a b respectively from the lens. 938 LENSES. 743. Example. A straight line 25 mm long is placed perpendi- cularly on the axis, at a distance of 35 centimetres from a lens of 1 5 centimetres' focal length ; what are the position and magnitude of the image? To determine the distance p we have For the length of the image we have 74:3 A. Image on Cross-wires. The position of a real image seen in mid-air can be tested by means of a cross of threads, or other con- venient mark, so arranged that it can be fixed at any required point. The observer must fix this cross so that it appears approximately to coincide with a selected point of the image. He must then try whether any relative displacement of the two occurs on shifting his eye to one side. If so, the cross must be pushed nearer to the lens, or drawn back, according to the nature of the observed displacement, which follows the general rule of parallactic displacement, that the more distant object is displaced in the same direction as the ob- server's eye. The cross may thus be brought into exact coincidence with the selected point of the image, so as to remain in apparent coincidence with it from all possible points of view. When this coincidence has been attained, the cross is at the focus conjugate to that which is occupied by the selected point of the object. By employing two crosses of threads, one to serve as object, and the other to mark the position of the image, it is easy to verify the fact that when the second cross coincides with the image of the first the first cross also coincides with the image of the second. 744. Aberration of Lenses. In the investigations of 739, 740, we made several assumptions which were only approximately true. The rays which proceed from a luminous point to a lens are in fact not accurately refracted to one point, but touch a curved surface called a caustic. The cusp of this caustic is the conjugate focus, and is the point at which the greatest concentration of light occurs. It is accordingly the place where a screen must be set to obtain the brightest and most distinct image. Rays from the central parts 'of VIRTUAL IMAGES. 939 the lens pass very nearly through it ; but rays from the circumferen- tial portions fall short of it. This departure from exact concurrence is called spherical aberration. The distinctness of an image on a screen is improved by employing an annular diaphragm to cut off all except the central rays; but the brightness is of course diminished. By holding a convex lens in a position very oblique to the incident light, a primary and secondary focal line can be exhibited on a screen, just as in the case of concave mirrors ( 714A). The experiment, however, is ratbsr more difficult of performance. 745. Virtual Images. Let an object AB be placed between a con- vex lens and its prin- cipal focus. Then the foci conjugate to the points A, B are virtual, and their positions can be found by construc- tion from the consid- eration that rays through A, B, parallel to the principal axis, will be refracted to F, the principal focus on the other side. These refracted rays, if pro- duced backward, must meet the secondary axes OA, OB in the required points. An eye placed on the other side of the lens will accordingly see a virtual image erect, magnified, and at a greater distance from the lens than the object. This is the principle of the simple microscope. The formula for the distances D, d of object and imao-e from the lens, when both are on the same side, is Fig. 682. Virtual Image formed by Convex Lens. 1 /' (11) / denoting the principal focal length. 746. Concave Lens. For a concave lens, if the focal length be still regarded as positive, and denoted by /, and if the distances D, d be on the same side of the lens, the formula becomes (12) which shows that d is always less than D ; that is, the image is nearer to the lens than the object. 940 LENSES. In Fig. 683, AB is the object, and ab the image. Rays incident from A and B parallel to the principal axis will emerge as if they came from the principal focus F. Hence the points a b are determined by the in- tersections of the dot- ted lines in the figure Avith the secondary axes A, OB. An eye on the other side of the lens sees the image a b, which is always virtual, Fig. 683. Virtual Image formed by Concave Lens. erect and diminished. 747. Focometer. Silbermann's focometer (Fig. 684) is an instrument for measuring the focal lengths of convex lenses, and is based on the principle ( 741) that, Fig. 684. Silbermann's Focometer. when the object and its image are equidistant from the lens, their distance from each other is four times the focal length. It consists of a graduated rule carrying three runners M, L, M'. The middle one L is the support for the lens which is to be examined; the other two, M M', contain two thin plates of horn or other translucent mate- rial, ruled with lines, which are at the same distance apart in both. The sliders must be ad- justed until the image of one of these plates is thrown upon the other plate, without enlarge- ment or diminution, as tested by the coincidence of the ruled lines of the image with those of the plate on which it is cast. The distance between M and M' is then read off, and divided by 4. Fig. 684 A. SINGLE SPHERICAL SURFACE. 941 747 A. Refraction at a Single Spherical Surface. Suppose a small pencil of rays to be incident nearly normally upon a spherical surface which forms the boundary between two media in which the indices are H L and /*., respectively. Let C (Fig. 684 A) be the centre of curvature, and C A the axis. Let Pj be the focus of the incident, and P 2 of the refracted rays. Then for any ray PjB, CBP,^ is the angle of incidence and CBP 2 the angle of refraction. Hence by the law of sines we have Mi sin C B P, = /ji. 2 sin C B P 2 . Dividing by sin B C A, and observing that ~ ! C Pt C P! , suTBCA = BP; = Ap! ultimatel y; sin C B P 2 C P., C P 2 = BPJ = AF S ultimatel y; we obtain the equation - 1 - M *, (13) AP, AP/ which expresses the fundamental relation between the positions of the conjugate foci. Let AC=r, AP X = p lt AP 2 =p z > then equation (13) becomes or, dividing by r, "(s- Again, let C A = p, CP X = q lf C P 2 = and the intrinsic brightness I. We may therefore write q = I e - 2 = I s ^; and if we put w for the solid angle ~ which the pupil subtends at s, we have q=Is<*>. The intrinsic brightness of a small area s is therefore measured by -, where q denotes the quantity of light which s emits per unit time, in directions limited by the small solid angle of divergence <>. 768. Applications. One of the most obvious consequences is that surfaces appear equally bright at all distances in the same direction, provided that no light is stopped by the air or other intervening medium ; for q and w both vary inversely as the square of the dis- tance. The area of the image formed on the retina in fact varies directly as the amount of light by which it is formed. Images formed by Lenses. Let A B (Fig. 707) be an object, and a b its real image formed by the lens C D, whose centre is 0. Let BRIGHTNESS OF IMAGES. 967 S denote a small area at A, and Q the quantity of light which it sends to the lens; also let s denote the corresponding area of the image, and q the quantity of light which traverses it. Then q would be identical with Q if no light were stopped by the lens; the areas S, s, are directly as the squares of the conjugate focal distances A, a ; and the solid angles of divergence ii and w for Q and q, being the solid angles subtended by the lens at A and a (for the plane angle cad in the figure is equal to the Fig 707 _ Brightne8S of Image vertical angle C a D), are inversely as the squares of the conjugate focal distances. We have accordingly = * and S & = *<> The intrinsic brightness of the image is S therefore equal to the intrinsic brightness ^ of the object, except in so far as light is stopped by the lens. Precisely similar reasoning applies to virtual images formed by lenses. 1 In the case of images formed by mirrors, 1 and w are the solid angles subtended by the mirror at the conjugate foci, and are in- versely as the squares of the distances from the mirror ; while S and s are directly as the squares of the distances from the centre of cur- vature; but these four distances are proportional ( 407), so that the same reasoning is still applicable. If the mirror only reflects half the incident light, the image will have only half the intrinsic bright- ness of the object. If the pupil is filled with light from the image, the effective brightness will be the same as the intrinsic brightness thus computed. If this condition is not fulfilled, the former will be less than the latter. When the image is greatly magnified as compared with the object, the angle of divergence is greatly diminished in comparison with the angle which the lens or mirror subtends at the object, and often becomes so small that only a small part of the pupil is utilized This is the explanation of the great falling off of light which is ob- 1 For refraction out of a medium of index ^ into another of index /j, 2 , we have by 747 A, equation (13), ^ : ^ : : 7S ^- 1 : xp^. But since a,, *, are the areas of corres- O AI O Xa ponding parts of object and image, we have i : s 2 : : C P t J : CPj J , and since u lt w, are the solid angles subtended at P] , P 2 by one and the same portion of the bounding surface, we have u^. w 2 : : A P a a : A P^. Therefore -?- : -2_: : Ml : *. The intrinsic brieht- 1 Wl fi Wj nesses of a succession of images in different media are therefore directly as the squares of the absolute indices. 968 VISION AND OPTICAL INSTRUMENTS. served in the use of high magnifying powers, both in microscopes and telescopes. 769. Brightness of Image in a Telescope. It has been already pointed out ( 762) that in most forms of telescope (the Galilean being an exception), there is a certain position, a little behind the eye-piece, at which a well-defined bright spot is formed upon a screen held there while the telescope is directed to any distant source of light. It has also been pointed out that this spot is the image, formed by the eye-piece, of the opening which is filled by the object- glass, and that the magnifying power of the instrument is the ratio of the size of the object-glass to the size of this bright spot. Let s denote the diameter of the bright spot, o the diameter of the object-glass, e the diameter of the pupil of the eye; then is the linear magnifying power. We shall first consider the case in which the spot exactly covers the pupil of the observer's eye, so that s = e. Then the whole light which traverses the telescope from a distant object enters the eye; and if we neglect the light stopped in the telescope, this is the whole light sent by the object to the object-glass, and is (\ times that which would be received by the naked eye. The magni- fication of apparent area is (- J , which, from the equality of s and e, is the same as the increase of total light. The brightness is therefore the same as to the naked eye. Next, let s be greater than e, and let the pupil occupy the central part of the spot. Then, since the spot is the image of the object- glass, we may divide the object glass into two parts a central part whose image coincides with the pupil, and a circumferential part whose image surrounds the pupil All rays from the object which traverse the central part, traverse its image, and therefore enter the pupil; whereas rays traversing the circumferential part of the object- glass, traverse the circumferential part of the image, and so are wasted. The area of the central part (whether of the object-glass or of its image) is to the whole area as e 2 : s 2 ; and the light which the object sends to the central portion, instead of being (^V times that which would be received by the naked eye, is only - times. But (-) 2 is the magnification of apparent area. Hence the bright- ness is the same as to the naked eye. In these two cases, effective and intrinsic brightness are the same. BRIGHTNESS OF IMAGE IN TELESCOPE. 969 Lastly (and this is by far the most common case in practice), let s be less than e. Then no light is wasted, but the pupil is not filled. The light received is (\ times that which the naked eye would receive; and the magnification of apparent area is (-Y- The effec- tive brightness of the image, is to the brightness of the object to the naked eye, as (-) : (-?) 2 ; that is, as s 2 : e 2 ; that is, as the area of the bright spot to the whole area of the pupil. To correct for the light stopped by reflection and imperfect trans- parency, we have simply to multiply the result in each case by a proper fraction, expressing the ratio of the transmitted to the incident light. This ratio, for the central parts of the field of view, is about 0-85 in the best achromatic telescopes. In such telescopes, therefore, the brightness of the image cannot exceed 0'85 of the brightness of the object to the naked eye. It will have this precise value, when the magnifying power is equal to or less than -; and from this C point upwards will vary inversely as the square of the linear mag- nification. The same formulae apply to reflecting telescopes, o denoting now the diameter of the large speculum which serves as objective; but the constant factor is usually considerably less than 0'85. It may be accepted as a general principle in optics, that while it is possible, by bad focussing or instrumental imperfections, to obtain a confused image whose brightness shall be intermediate between the brightest and the darkest parts of the object, it is impossible, by any optical arrangement whatever, to obtain an image whose brightest part shall surpass the brightest part of the object. 770. Brightness of Stars. There is one important case in which the foregoing rules regarding the brightness of images become nuga- tory. The fixed stars are bodies which subtend at the earth angles smaller than the minimum visibile, but which, on account of their excessive brightness, appear to have a sensible angular diameter. This is an instance of irradiation, a phenomenon manifested by all bodies of excessive brightness, and consisting in an extension of their apparent beyond their actual boundary. What is called, in popular language, a bright star, is a star which sends a large total amount of light to the eye. Denoting by a the ratio of the transmitted to the whole incident light, a ratio which, as we have seen, is about 0'85 in the most 1)70 VISION AND OPTICAL INSTRUMENTS. favourable cases, and calling the light which a star sends to the naked eye unity, the light perceived in its image will be a (j) , or a X square of linear magnification, if the bright spot is as large as the pupil. When the eye-piece is changed, increase of power dimin- ishes the size of the spot, and increases the light received by the eye, until the spot is reduced to the size of the pupil. After this, any further magnification has no effect on the quantity of light received, its constant value being a (j) 8 . The value of this last expression, or rather the value of a o 2 , is the measure of what is called the space-penetrating power of a telescope ; that is to say, the power of rendering very faint stars visible ; and it is in this respect that telescopes of very large aperture, notably the great reflector of Lord Rosse, are able to display their great superiority over instruments of moderate dimensions. We have seen that the total light in the visible image of a star remains unaltered, by increase of power in the eye-piece above a certain limit. But the visibility of faint stars in a telescope is pro- moted by darkening the back-ground of sky on which they are seen. Now the brightness of this back-ground varies directly as s 2 , or in- versely as the square of the linear magnification (s being supposed less than e). Hence it is advantageous, in examining very faint stars, to employ eye-pieces of sufficient power to render the bright spot much smaller than the pupil of the eye. 771. Images on a Screen. Thus far we have been speaking of the brightness of images as viewed directly. Images cast upon a screen are, as a matter of fact, much less brilliant. Their brightness depends greatly on the nature of the screen, and can never exceed the bright- ness which the surface of the screen would exhibit if held very near the source of light. When a condensing lens is used to collect the rays of a lamp, an eye placed at the conjugate focus sees the whole lens full of light of uniform brightness, which, neglecting reflection and absorption, can be shown to be the same as that of the flame itself. 1 The illumination of a screen placed in the focus, is therefore jointly proportional to the solid angle which the lens subtends at the focus, and to the brightness of the flame; and is the same as if the screen were directly illuminated by the flame, 1 This is strictly true of intrinsic brightness, which is all that our reasoning requires. It is true for effective brightness, if the image is large enough to cover the pupil, and if the lens is at a proper distance for distinct vision. CROSS- WIRES OF TELESCOPES. 971 at a distance at which the flame itself would subtend the same solid angle. 772. Cross-wires of Telescopes. We have described in 743 A a mode of marking the place of a real image by means of a cross of threads. When telescopes are employed to assist in the measure- ment of angles, a contrivance of this kind is almost always intro- duced. A cross of silkworm threads, in instruments of low power, or of spider threads in instruments of higher power, is stretched across a metallic frame just in front of the eye-piece. The observer must first adjust the eye-piece for distinct vision of this cross, and must then (in the case of theodolites and other surveying instru- ments) adjust the distance of the object-glass until the object which is to be observed is also seen distinctly in the telescope. The image of the object will then be very nearly in the plane of the cross. If it is not exactly in the plane, parallactic displacement will be observed when the eye is shifted, and this must be cured by slightly altering the distance of the object-glass. When the adjustment has been completed, the cross always marks one definite point of the object, however the eye be shifted. This coincidence will not be disturbed by pushing in or pulling out the eye-piece ; for the frame which carries the cross is attached to the body of the telescope, and the coincidence of the cross with a point of the image is real, so that it could be observed by the naked eye, if the eye- piece were re- moved. The adjustment of the eye-piece merely serves to give dis- tinct vision, and this will be obtained simultaneously for both the cross and the object. 773. Line of Collimation. The employment of cross-wires (as these crossing threads are called) enormously increases our power of making accurate observations of direction, and constitutes one of the greatest advantages of modern over ancient instruments. The line which is regarded as the line of sight, or as the direction in which the telescope is pointed, is called the line of collimation. If we neglect the curvature of rays due to atmospheric refraction, we may define it as the line joining the cross to the object whose image falls on it. More rigorously, the line of collimation is the line joining the cross to the optical centre of the object-glass. When it is desired to adjust the line of collimation, for example, to make it truly perpendicular to the horizontal axis on which the telescope is mounted, the adjustment is performed by shifting the frame which carries the wires, slow-motion screws being provided for this purpose. VISION AND OPTICAL INSTRUMENTS. Telescopes for astronomical observation are often furnished with a number of parallel wires, crossed by one or two in the transverse direction ; and the line of collimation is then denned by reference to an imaginary cross, which is the centre of mean position of all the actual crosses. 774. Micrometers. Astronomical micrometers are of various kinds, some of them serving for measuring the angular distance between two points in the same field of view, and others for measuring their apparent direction from one another. They generally consist of .spider threads placed in the principal focus of the object-glass, so as to be in the same plane as the images of celestial objects, one or more of the threads being movable by means of slow-motion screws, furnished with graduated circles, on which parts of a turn can be read off. One of the commonest kinds consists of two parallel threads, which can thus be moved to any distance apart, and can also be turned round in their own plane. CHAPTER LXII. DISPERSION. STUDY OF SPECTRA. 775. Newtonian Experiment. In the chapter on refraction, we have postponed the discussion of one important phenomenon by which it is usually accompanied, and which we must now proceed to explain. The following experiment, which is due to Sir Isaac Newton, will furnish a fitting introduction to the subject. On an extensive background of black, let three bright strips be laid in line, as in the left-hand part of Fig. 708, and looked at through a prism with its refracting edge parallel to the strips. We Fig. 708. Spectra of White and Coloured Strips. shall suppose the edge to be upward, so that the image is raised above the object. The images, as represented in the right-hand part of Fig. 708, will have the same horizontal dimensions as the strips, but will be greatly extended in the vertical direction ; and each image, instead of having the uniform colour of the strips from which it is derived, will be tinted with a gradual succession of colours from top to bottom. Such images are called spectra. If one of the strips (the middle one in the figure) be white, its spectrum will contain the following series of colours, beginning at the top : violet, blue, green, yellow, orange, red. 974 DISPERSION. STUDY OF SPECTRA. If one of the strips be blue (the left-hand one in the figure), its image will present bright colours at the upper end; and these will be identical with the colours adjacent to them in the spectrum 01 white. The colours which form the lower part of the spectrum of white will either be very dim and dark in the spectrum of blue, or will be wanting altogether, being replaced by black. If the other strip be red, its image will contain bright colours at the lower or red end, and those which belong to the upper end of the spectrum of white will be dim or absent. Every colour that occurs in the spectrum of blue or of red will also be found, and in the same horizontal line, in the spectrum of white. If we employ other colours instead of blue or red, we shall obtain analogous results; every colour will be found to give a spectrum which is identical with part of the spectrum of white, both as regards colour and position, but not generally as regards brightness. We may occasionally meet with a body whose spectrum consists only of one colour. The petals of some kinds of convolvulus give a spectrum consisting only of blue, and the petals of nasturtium give only red. 776. Composite Nature of Ordinary Colours. This experiment shows that the colours presented by the great majority of natural bodies are composite. When a colour is looked at with the naked eye, the sensation experienced is the joint effect of the various elementary colours which compose it. The prism serves to resolve the colour into its components, and exhibit them separately. The experiment also shows that a mixture of all the elementary colours in proper pro- portions produces white. 777. Solar Spectrum. The coloured strips in the foregoing experi- ment may be illuminated either by daylight or by any of the ordinary sources of artificial light. The former is the best, as gas-light and candle-light are very deficient in blue and violet rays. Colour, regarded as a property of a coloured (opaque) body, is the power of selecting certain rays and reflecting them either exclusively or in larger proportion than others. The spectrum presented by a body viewed by reflected light, as ordinary bodies are, can thus only consist of the rays, or a selection of the rays, by which the body is illuminated. A beam of solar light can be directly resolved into its constituents by the following experiment, which is also due to Newton, and was the first demonstration of the composite character of solar light. SOLAR SPECTRUM. 975 Let a beam of sun-light be admitted through a small opening into a dark room. If allowed to fall normally on a white screen, it pro- duces ( 683) a round white spot, which is an image of the sun. Now let a prism be placed in its path edge-downwards, as in Fig. 709; the Fig. 709. Solar Spectrum by Newton's Method. beam will thus be deflected upwards, and at the same time resolved into its component colours. The image depicted on the screen will be a many-coloured band, resembling the spectrum of white described in 775. It will be of uniform width, and rounded off at the ends, being in fact built up of a number of overlapping discs, one for each kind of elementary ray. It is called the solar spectrum. The rays which have undergone the greatest deviation are the violet. They occupy the upper end of the spectrum in the figure. Those which have undergone the least deviation are the red. Of all visible rays, the violet are the most, and the red the least refrangible ; and the analysis of light into its components by means of the prism is due to difference of refrangibility. If a small opening is made in the screen, so as to allow rays of only one colour to pass, it will be found, 976 DISPERSION. STUDY OF SPECTRA. on transmitting these through a second prism behind the screen, as in Fig. 709, that no further analysis can be effected, and the whole of the image formed by receiving this transmitted light on a second screen will be of this one colour. 778. Mode of obtaining a Pure Spectrum. The spectra obtained by the methods above described are built up of a number of overlapping images of different colours. To prevent this overlapping, and obtain each elementary colour pure from all admixture with the rest, we must in the first place employ as the object for yielding the images a very narrow line; and in the second place we must take care that the images which we obtain of this line are not blurred, but have the greatest possible sharpness. A spectrum possessing these character- istics is called pure. The simplest mode of obtaining a pure spectrum consists in looking through a prism at a fine slit in the shutter of a dark room. The edges of the prism must be parallel to the slit, and its distance from the slit should be five feet or upwards. The observer, placing his eye close to the prism, will see a spectrum; and he should rotate the prism on its axis until he has brought this spectrum to its smallest angular distance from the real slit, of which it is the image Let E (Fig. 710) be the position of the eye, S that of the slit. Then the extreme red and violet images of the slit will be seen at R, V, at distances from the prism sensibly equal to the real distance of S ( 731 B); and the other images, which compose the remainder of the spectrum, will occupy posi- tions between R and V The spectrum, in this mode of operat- ing, is virtual. To obtain a real spectrum in a state of purity, a convex lens must be employed. Let the lens L (Fig. 711) be first placed in such a position as to throw a sharp image of the slit S upon & Screen at I. Next let a prism P be introduced between the lens and screen, and rotated on its axis till the position of minimum devia- tion is obtained, as shown by the movements of the impure spectrum which travels about the walls of the room. Then if the screen be moved into the position RV, its distance from the prism being tin- Fig.TlO.-ArrangementforBeeingapureS^ctrum. PURE SPECTRUM. 977 same as before,, a pure spectrum will be depicted upon it. A similar result can be obtained by placing the prism between the lens and the slit, but the adjustments are rather more troublesome. Direct Fig. 711. Arrangement for Pure Spectrum on Screen. sun-light, or sun-light reflected from a mirror placed outside the shutter, is necessary for this experiment, as sky-light is not suffi- ciently powerful. It is usual, in experiments of this kind, to em- ploy a movable mirror called a heliostat, by means of which the light can be reflected in any required direction. Sometimes the move- ments of the mirror are obtained by hand; sometimes by an ingeni- ous clock-work arrangement, which causes the reflected beam to keep its direction unchanged notwithstanding the progress of the sun through the heavens. The advantage of placing the prism in the position of minimum deviation is two-fold. First, the adjustments are facilitated by the equality of conjugate focal distances, which subsists in this case and in this only. Secondly and chiefly, this is the only position in which the images are not blurred. In any other position it can be shown 1 that a small cone of homogeneous incident rays is no longer a cone (that is, its rays do not accurately pass through one point) after transmission through the prism. The method of observation just described was employed by Wollaston, in the earliest observations of a pure spectrum ever obtained. Fraunhofer, a few years later, independently devised the same method, and carried it to much greater perfection. Instead of looking at the virtual image with the naked eye, he viewed it through a telescope, which greatly magnified it, and revealed several features never before detected. The prism and telescope were at a distance of 24 feet from the slit. 1 Parkinson's Optict, 96. Cor. 2. 63 DISPERSION. STUDY OF SPECTRA. 779. Dark Lines in the Solar Spectrum. When a pure spectrum of t^olar light is examined by any of these methods, it is seen to be traversed by numerous dark lines, constituting, if we may so say, dark images of the slit. Each of these is an indication that a par- ticular kind of elementary ray is wanting 1 in solar light. Every elementary ray that is present gives its own image of the slit in its own peculiar colour ; and these images are arranged in strict con- tiguity, so as to form a continuous band of light passing by perfectly gradual transitions through the whole range of simple colour, except at the narrow intervals occupied by the dark lines. Fig. 1, Plate III., is a rough representation of the appearance thus presented. If the slit is illuminated by a gas flame, or by any ordinary lamp, instead of by solar light, no such lines are seen, but a perfectly continuous spectrum is obtained. The dark lines are therefore not characteristic of light in general, but only of solar light. Wollaston saw and described some of the more conspicuous of them. Fraunhofer counted about 600, and marked the places of 354 upon a map of the spectrum, distinguishing some of the more con- spicuous by the names of letters of the alphabet, as indicated in fig. 1. These lines are constantly referred to as reference marks for the accurate specification of different portions of the spectrum. They always occur in precisely the same places as regards colour, but do not retain exactly the same relative distances one from another, when prisms of different materials are employed, different parts of the spectrum being unequally expanded by different refracting sub- tances. 2 The inequality, however, is not so great as to introduce any difficulty in the identification of the lines. The dark lines in the solar spectrum are often called Fraunhofer's lines. Fraunhofer himself called them the " fixed lines." 780. Invisible Bays of the Spectrum. The brightness of the solar spectrum, however obtained, is by no means equal throughout, but is greatest between the dark lines D and E ; that is to say, in the yellow and the neighbouring colours orange and light green ; and falls off gradually on both sides. The heating effect upon a small thermometer or thermopile in- creases in going from the violet to the red, and still continues to increase for a certain distance beyond the visible spectrum at the red end. Prisms and lenses of rock-salt should be employed for this 1 Probably not absolutely wanting, but so feeble as to appear black by contrast. * This property is called the irrationality o/ dispersion. SPKCTHA <>K VARIOUS SOURCES <>K LIGHT. n. 2 The- Sun's edge'. 3. Sodhun. ^.Paiassiujr^. 5. Lithium. 6. Caesium., l.fiubidium. B.TT 9. Calc-ium. 10. Strontium*. ^Barium.. 12. Indium,. 13. Phosfihorus. 14. Hydrogen*. BIACK1E t SON. LONDON. GLASGOW & BD1NBUROH PHOSPHORESCENCE. 979 investigation, as glass largely absorbs the invisible rays which lie beyond the red. When the spectrum is thrown upon the sensitized paper employed in photography, the action is very feeble in the red, strong in the blue and violet, and is sensible to a great distance beyond the violet end. When proper precautions are taken to insure a very pure spectrum, the photograph reveals the existence of dark lines, like those of Fraunhofer, in the invisible ultra-violet portion of the spec- trum. The strongest of these have been named L, M, N, O, P. 781. Phosphorescence and Fluorescence. There are some sub- stances which, after being exposed in the sun, are found for a long time to appear self-luminous when viewed in the dark, and this Fig. 712. Becquerel'i Phosphoroscope. without any signs of combustion or sensible elevation of temperature. Such substances are called phosphorescent. Sulphuret of calcium and sulphuret of barium have long been noted for this property, and have hence been called respectively Canton's phosphorus, and Bologn& 980 DISPERSION. STUDY OF SPECTRA. The phenomenon is chiefly due to the action of the violet and ultra-violet portion of the sun's rays. More recent investigations have shown that the same property exists in a much lower degree in an immense number of bodies, their phosphorescence continuing, in most cases, only for a fraction of a second after their withdrawal from the sun's rays. E. Becquerel has contrived an instrument, called the phosphoroscope, which is ex- tremely appropriate for the observation of this phenomenon. It is represented in Fig. 712. Its most characteristic feature is a pair of rigidly connected discs (Fig. 713), each pierced with four openings, those of the one being not opposite but midway between those of the other. This pair of discs can be set in very rapid rotation by means of a series of M r heels and pinions. The body to be examined is attached to a fixed stand between the two discs, so that it is alternately exposed on opposite sides as the discs rotate. One side is turned towards the sun, and the other towards the observer, who accordingly only sees the body when it is not exposed to the sun's rays. The cylindrical case within Fig. 713.-Di 8 csof Phosphoroecope. Wnlch the disCS TCVOlve, is fitted itlto a hole in the shutter of a dark room, and is pierced with an opening on each side exactly opposite the position in which the body is fixed. The body, if not phosphorescent, will never be seen by the observer, as it is always in darkness except when it is hidden by the intervening disc. If its phosphorescence lasts as long as an eighth part of the time of one rotation, it will become visible in the darkness. Nearly all bodies, when thus examined, show traces of phosphores- cence, lasting, however, in some cases, only for a ten-thousandth of a second. The phenomenon of fluorescence, which is illustrated in Plate II. accompanying 618, appears to be essentially identical with phos- phorescence. The former name is applied to the phenomenon, if it is observed while the body is actually exposed to the source of light, the latter to the effect of the same kind, but usually less intense, which is observed after the light from the source is cut off. Both forms of the phenomenon occur in a strongly-marked degree in the same bodies. Canarj'-glass, which is coloured with oxide of uranium, is FLUORESCENCE. 981 a very convenient material for the exhibition of fluorescence. A thick piece of it, held in the violet or ultra-violet portion of the solar spectrum, is filled to the depth of from i to ^ of an inch with a faint nebulous light. A solution of sulphate of quinine is also frequently employed for exhibiting the same effect, the luminosity in this case being bluish. If the solar spectrum be thrown upon a screen freshly washed with sulphate of quinine, the ultra-violet portion will become visible by fluorescence; and if the spectrum be very pure, the pre- sence of dark line.s in this portion will be detected. The light of the electric lamp is particularly rich in ultra-violet rays, this portion of its spectrum being much longer than in the case of solar light, and about twice as long as the spectrum of luminous rays. Prisms and lenses of quartz should be employed for this pur- pose, as this material is specially transparent to the highly-refrangible rays. Flint-glass prisms, however, if of good quality, answer well in operating on solar light. The luminosity produced by fluorescence has sensibly the same tint in all parts of the spectrum in which it occurs, and depends upon the fluorescent substance employed. Pris- matic analysis is not necessary to the exhibition of fluorescence. The phenomenon is very conspicuous when the electric discharge of a Holtz's machine or a Ruhmkorffs coil is passed near fluorescent substances, and it is faintly visible when these substances are examined in bright sunshine. The light emitted by a fluorescent substance is found by analysis not to be homogeneous, but to consist of rays having a wide range of refrangibility. The ultra-violet rays, though usually styled invisible, are not altogether deserving of this title. By keeping all the rest of the spectrum out of sight, and carefully excluding all extraneous light, the eye is enabled to perceive these highly refrangible rays. Their colour is described as lavender-gray or bluish white, and has been attributed, with much appearance of probability, to fluorescence of the retina. The ultra-red rays, on the other hand, are never seen ; but this may be owing to the fact, which has been established by experiment, that they are largely, if not entirely, absorbed before they can reach the retina, 782. Recomposition of White Light. The composite nature of white light can be established by actual synthesis. This can be done in several ways. 1. If a second prism, precisely similar to the first, but with its refracting edge turned the contrary way, is interposed in the path of 982 DISPERSION. STUDY OF SPECTRA. the coloured beam, very near its place of emergence from the first prism, the deviation produced by the second prism will be equal and opposite to that produced by the first, the two prisms will produce the effect of a parallel plate, and the image on the screen will be a white spot, nearly in the same position as if the prisms were re- moved. 2. Let a convex lens (Fig. 714) be interposed in the path of the coloured beam, in such a manner that it receives all the rays, and Fig. 714. Recomposition by Lens. that the screen and the prism are at conjugate focal distances. The image thus obtained on the screen will be white, at least in its cen- tral portions. 3. Let a number of plane mirrors be placed so as to receive the successive coloured rays, and to reflect them all to one point of a Fi?. 715. Recomposition by Mirrors. screen, as in Fig. 715. The bright spot thus formed will be white, or approximately white. More complete information respecting the mixture of colours will be given in the next chapter. SPECTROSCOPE. 983 783. Spectroscope. When we have obtained a pure spectrum by any of the methods above indicated, we have in fact effected an analysis of the light with which the slit is illuminated. In recent years, many forms of apparatus have been constructed for this pur- pose, under the name of spectroscopes. A spectroscope usually contains, besides a slit, a prism, and a telescope (as in Fraunhofer's method of observation), a convex lens called a collimator, which is fixed between the prism and the slit, at the distance of its principal focal length from the latter. The effect of this arrangement is, that rays from any point of the slit emerge parallel, as if they came from a much larger slit (the virtual image of the real slit) at a much greater distance. The prism (set at minimum deviation) forms a virtual image of this image at the same distance, but in a different direction, on the principle of Fig. 711. Fig. 716. Spectroscope. To this second virtual image the telescope is directed, being focussed as if for a very distant object. Fig. 716 represents a spectroscope thus constructed. The tube of 984 DISPERSION. STUDY OF SPECTRA. the collimator is the further tube in the figure, the lens being at the. end of the tube next the prism, while at the far end, close to the lamp flame, there is a slit (not visible in the figure) consisting of an opening between two parallel knife-edges, one of which can be moved to or from the other by turning a screw. The knife-edges must be very true, both as regards straightness and parallelism, as it is often necessary to make the slit exceedingly narrow. The tube on the left hand is the telescope, furnished with a broad guard to screen the eye from extraneous light. The near tube, with a candle opposite its end, is for purposes of measurement. It contains, at the end next the candle, a scale of equal parts, engraved or photographed on glass. At the other end of the tube is a collimating lens, at the distance of its own focal length from the scale ; and the collimator is set so that its axis and the axis of the telescope make equal angles with the near face of the prism. The observer thus sees in the telescope, by reflection from the surface of the prism, a magnified image of the scale, serving as a standard of reference for assigning the positions of the lines in any spectrum which may be under examination. This arrangement affords great facilities for rapid observation. Another plan is, for the arm which carries the telescope to be movable round a graduated circle, the telescope being furnished with cross- wires, which the observer must bring into coincidence with any line whose position he desires to measure. Arrangements are frequently made for seeing the spectra of two different sources of light in the same field of view, one half of the length of the slit being illuminated by the direct rays of one of the sources, while a reflector, placed opposite the other half of the slit, supplies it with reflected light derived from the other source. This method should always be employed when there is a question as to the exact coincidence of lines in the two spectra. The re- Fig. 7i7. fleeter is usually an equilateral prism. The light enters Reflecting Prism. * . n normally at one of its faces, is totally reflected at another, and emerges normally at the third, as in the annexed sketch (Fig. 717, where the dotted line represents the path of a ray. A one-prism spectroscope is amply sufficient for the ordinary pur- poses of chemistry. For some astronomical applications a much greater dispersion is required. This is attained by making the light pass through a number of prisms in succession, each being set in the proper position for giving minimum deviation to the rays which have USE OF COLLIMATOH. 985 Fig. 718. Train of Prisms. passed through its predecessor. Fig. 718 represents the ground plan of such a battery of prisms, and shows the gradually increasing width of a pencil as it passes round the series of nine prisms on its way from the collimator to the telescope. The prisms are usual- ly connected by a special ar- rangement, which enables the observer, by a single movement, to bring all the prisms at once into the proper position for giving minimum deviation to the parti- cular ray under examination, a po- sition which differs considerably for rays of different refrangibilities. 784. Use of Collimator. The introduction of a collimating lens, to be used in conjunction with a prism and observing telescope, is due to Professor Swan. 1 Fraun- hofer employed no collimator; but his prism was at a distance of 24 feet from the slit, whereas a distance of less than 1 foot suffices when a collimator is used. It is obvious that homogeneous light, coming from a point at the distance of a foot, and falling upon the whole of one face of a prism say an inch in width, cannot all have the incidence proper for minimum deviation. Those rays which very nearly fulfil this con- dition, will concur in forming a tolerably sharp image, in the posi- tion which we have already indicated. The emergent rays taken as a whole, do not diverge from any one point, but are tangents to a virtual caustic ( 714). An eye receiving any portion of these rays, will see an image in the direction of a tangent from the eye to the caustic; and this image will be the more blurred as the deviation is further from the minimum. When the naked eye is employed, and the prism is so adjusted that the centre of the pupil receives rays of minimum deviation, a distance of five or six feet between the prism and slit is sufficient to give a sharp image; but if we employ an observing telescope whose object-glass is five times larger in diameter than the pupil of the eye, we must increase the distance between the 1 Trans. Roy. Soc. Edinburgh, 1847 and 1856. 986 DISPERSION. STUDY OF SPECTRA. prism and slit five-fold to obtain equally good definition. A colli- mating lens, if achromatic and of good quality, gives the advantage of good definition without inconvenient length. When exact measures of deviation are required, it confers the further advantage of altogether dispensing with a very troublesome correction for parallax. 785. Different Kin is of Spectra. The examination of a great variety of sources of light has shown that spectra may be divided into the following classes: 1. The solar spectrum is characterized, as already observed, by a definite system of dark lines interrupting an otherwise continuous succession of colours. The same system of dark lines is found in the spectra of the moon and planets, this being merely a consequence of the fact that they shine by the reflected light of the sun. The spectra of the fixed stars also contain systems of dark lines, which are different for different stars. 2. The spectra of incandescent solids and liquids are completely continuous, containing light of all refrangibilities from the extreme red to a higher limit depending on the temperature. 3. Flames not containing solid particles in suspension, but merely emitting the light of incandescent gases, give a discontinuous spec- trum, consisting of a finite number of bright lines. The continuity of the spectrum of a gas or candle flame, arises from the fact that nearly all the light of the flame is emitted by incandescent particles of solid carbon, particles which we can easily collect in the form of soot. When a gas-flame is fed with an excessive quantity of air, as in Bunsen's burner, the separation of the solid particles of carbon from the hydrogen with which they were combined, no longer takes place; the combustion is purely gaseous, and the spectrum of the flame is found to consist of bright lines. When the electric light is produced between metallic terminals, its spectrum contains bright lines due to the incandescent vapour of these metals, together with other bright lines due to the incandescence of the oxygen and nitro- gen of the air. When it is taken between charcoal terminals, its spectrum is continuous; but if metallic particles be present, the bright lines due to their vapours can be seen as well. The spectrum of the electric discharge in a Geissler's tube consists of bright lines characteristic of the gas contained in the tube. 786. Spectrum Analysis. As the spectrum exhibited by a com- pound substance when subjected to the action of heat, is frequently SPECTRUM ANALYSIS. 987 found to be identical with the spectrum of one of its constituents, or to consist of the spectra of its constituents superimposed, 1 the spec- troscope affords an exceedingly ready method of performing qualita- tive analysis. If a salt of a metal which is easily volatilized is introduced into a Bunsen lamp-flame, by means of a loop of platinum wire, the bright lines which form the spectrum of the metal will at once be seen in a spectroscope directed to the flame; and the spectrum of the Bunsen flame itself is too faint to introduce any confusion. For those metals which require a higher temperature to volatilize them, electric discharge is usually employed. Geissler's tubes are com- monly used for gases. Plate III. contains representations of the spectra of several of the more easily volatilized metals, as well as of phosphorus and hydro- gen; and the solar spectrum is given at the top for comparison. The bright lines of some of these substances are precisely coincident with some of the dark lines in the solar spectrum. The fact that certain substances when incandescent give definite bright lines, has been known for many years, from the researches of Brewster, Herschel, Talbot, and others; but it was for a long time thought that the same line might be produced by different sub- stances, more especially as the bright yellow line of sodium was often seen in flames in which that metal was not supposed to be present. Professor Swan, having ascertained that the presence of the 2,500,000th part of a grain of sodium in a flame was sufficient to produce it, considered himself justified in asserting, in 1856, that this line was always to be taken as an indication of the presence of sodium in larger or smaller quantity. But the greatest advance in spectral analysis was made by Bunsen and Kirchhoff, who, by means of a four-prism spectroscope, obtained accurate observations of the positions of the bright lines in the spectra of a great number of substances, as well as of the dark lines in the solar spectrum, and called attention to the identity of several of the latter with several of the former. Since the publication of their researches, the spectroscope has come into general use among chemists, and has already led to the discovery of four new metals, cesium, rubidium, thallium, and indium. 787. Reversal of Bright Lines. Analysis of the Sun's Atmosphere. 1 These appear to be merely examples of the dissociation of the elements of a chemical compound at high temperatures. 988 DISPERSION. STUDY OF SPECTRA. It may seem surprising that, while incandescent solids and liquids are found to give continuous spectra, containing rays of all refran- gibilities, the solar spectrum is interrupted by dark lines indicating the absence or relative feebleness of certain elementary rays. It seems natural to suppose that the deficient rays have been removed by selective absorption, and this conjecture was thrown out long since. But where and how is this absorption produced? These questions have now received an answer which appears completely satisfactory. According to the theory of exchanges, which has been explained in connection with the radiation of heat ( 31 2 c, 326), every sub- stance which emits certain kinds of rays to the exclusion of others, absorbs the same kind which it emits; and when its temperature is the same in the two cases compared, its emissive and absorbing power are precisely equal for any one elementary ray. When an incandescent vapour, emitting only rays of certain definite refrangibilities, and therefore having a spectrum of bright lines, is interposed between the observer and a very bright source of light, giving a continuous spectrum, the vapour allows no rays of its own peculiar kinds to pass ; so that the light which actually comes to the observer consists of transmitted rays in which these particular kinds are wanting, together with the rays emitted by the vapour itself, these latter being of precisely the same kind as those which it has refused to transmit. It depends on the relative brightness of the two sources whether these particular rays shall be on the whole in excess or defect as compared with the rest. If the two sources are at all comparable in brightness, these rays will be greatly in excess, inasmuch as they constitute the whole light of the one, and only a minute fraction of the light of the other; but the light of the electric lamp, or of the lime-light, is usually found sufficiently powerful to produce the contrary effect ; so that if, for example, a spirit-lamp with salted wick is interposed between the slit of a spectroscope and the electric light, the bright yellow line due to the sodium appears black by contrast with the much brighter back-ground which belongs to the continuous spectrum of the charcoal points. By employing only some 10 or 15 cells, a light may be obtained, the yellow portion of which, as seen in a one-prism spectroscope, is sensibly equal in brightness to the yellow line of the sodium flame, so that this line can no longer be separately detected, and the appearance is the same whether the sodium flame be interposed or removed. The dark lines in the solar spectrum would therefore be accounted TELESPECTROSCOPE. 989 for by supposing that the principal portion of the sun's light comes from an inner stratum which gives a continuous spectrum, and that a layer external to this contains vapours which absorb particular rays, and thus produce the dark lines. The stratum which gives the continuous spectrum might be solid, liquid, or even gaseous, for the experiments of Frankland and Lockyer have shown that, as the pressure of a gas is increased, its bright lines broaden out into bands, and that the bands at length become so wide as to join each other and form a continuous spectrum l Hydrogen, potassium, sodium, calcium, barium, magnesium, zinc, iron, chromium, cobalt, nickel, copper, and manganese have all been proved to exist in the sun by the accurate identity of position of their bright lines with certain dark lines in the sun's spectrum. The strong line D, which in a good instrument is seen to consist of two lines near together, is due to sodium ; and the lines C and F are due to hydrogen. No less than 450 of the solar dark lines have been identified with bright lines of iron. 788. Telespectroscope. Solar Sierra. For astronomical investiga- tions, the spectroscope is usually fitted to a telescope, and takes the place of the eye-piece, the slit being placed in the principal focus of the object-glass, so that the image is thrown upon it, and the light which enters it is the light which forms one strip (so to speak) of the image, and which therefore comes from one strip of the object. A telescope thus equipped is called a telespectroscope. Extremely interesting results have been obtained by thus subjecting to exami- nation a strip of the sun's edge, the strip being sometimes tangential to the sun's disk, and sometimes radial. When the former arrange- O ment is adopted, the appearance presented is that depicted in fig. 2, Plate III., consisting of a few bright lines scattered through a back- ground of the ordinary solar spectrum. The bright lines are due to an outer layer called the sierra or chromosphere, which is thus proved to be vaporous. The ordinary solar spectrum which accompanies it, is due to that part of the sun from which most of our light is derived. This part is called the photosphere, and if not solid or liquid, it must consist of vapour so highly compressed that its pro- perties approximate to those of a liquid. When the slit is placed radially, in such a position that only a 1 The gradual transition from a spectrum of bright lines to a continuous spectrum may be held to be an illustration of the continuous transition which can be effected from the condition of ordinary gas to that of ordinary liauid ( 2 4 6 A). 990 DISPERSION. STUDY OF SPECTRA, small portion of its length receives light from the body of the sun, the spectra of the photosphere and chromosphere are seen in imme- diate contiguity, and the bright lines in the latter (notably those of hydrogen, No. 14, Plate III.) are observed to form continuations of some of the dark lines of the former. The chromosphere is so much less bright than the photosphere, that, until a few years since, its existence was never revealed except during total eclipses of the sun, when projecting portions of it (from which it derives its name of sierra) were seen extending beyond the dark body of the moon. The spectrum of these projecting portions, which have been variously called "prominences," "red flames," and "rose-coloured protuberances," was first observed during the "Indian eclipse" of 1868, and was found to consist of bright lines, including those of hydrogen. From their excessive brightness, M. Janssen, who was one of the observers, expressed confidence that he should be able to see them in full sunshine ; and the same idea had been already conceived and published by Mr. Lockyer. The expectation was shortly afterwards realized by both these observers, and the chromo- sphere has ever since been an object of daily observation. The visi- bility of the chromosphere lines in full sunshine, depends upon the principle that, while a continuous spectrum is extended, and there- fore made fainter, by increased dispersion, a bright line in a spectrum is not sensibly broadened, and therefore loses very little of its in- trinsic brightness ( 791). Very high dispersion, attainable only by the use of a long train of prisms, is necessary for this purpose. Still more recently, by opening the slit to about the average width of the prominence-region, as measured on the image of the sun which is thrown on the slit, it has been found possible to see the whole of an average-sized prominence at one view. This will be understood by remembering that a bright line as seen in a spectrum is a mono- chromatic image of the illuminated portion of the slit, or when a tele- spectroscope is used, as in the present case, it is a monochromatic image of one strip of the image formed by the object-glass, namely, that strip which coincides with the slit. If this strip then contains a prominence in which the elementary rays C and F (No. 2, Plate III.) are much stronger than in the rest of the strip, a red image of the prominence will be seen in the part of the spectrum corresponding to the line C, and a blue image in the place corresponding to the line F. This method of observation requires greater dispersion than is necessary for the mere detection of the chromosphere lines; the DISPLACEMENT OF LINES. 991 dispersion required for enabling a bright-line spectrum to predominate over a continuous spectrum being always nearly proportional to the width of the slit ( 791). Of the nebulae, it is well known that some have been resolved by powerful telescopes into clusters of stars, while others have as yet proved irresolvable. Huggins has found that the former class of nebulae give spectra of the same general character as the sun and the fixed stars, but that some of the latter class give spectra of bright lines, indicating that their constitution is gaseous. 789. Displacement of Lines consequent on Celestial Motions. Ac- cording to the undulatory theory of light, which is now universally accepted, the fundamental difference between the different rays which compose the complete spectrum, is a difference of wave- frequency, and, as connected with this, a difference of wave-length in any given medium, the rays of greatest wave- frequency or shortest wave-length being the most refrangible. Doppler first called attention to the change of refrangibility which must be expected to ensue from the mutual approach or recess of the observer and the source of light, the expectation being grounded on reasoning which we have explained in connection with acoustics (653 A). Doppler adduced this principle to explain the colours of the fixed stars, a purpose to which it is quite inadequate; but it has rendered very important service in connection with spectroscopic research. Displacement of a line towards the more refrangible end of the spec- trum, indicates approach, displacement in the opposite direction indi- cates recess, and the velocity of approach or recess admits of being calculated from the observed displacement. When the slit of the spectroscope crosses a spot on the sun's disc, the dark lines lose their straightness in this part, and are bent, some- times to one side, sometimes to the other. These appearances clearly indicate uprush and downrush of gases in the sun's atmosphere in the region occupied by the spot Huggins has observed a displacement of the F line towards the red end, in the spectrum of Sirius, as compared with the spectrum of the sun or of hydrogen. The displacement is so small as only to admit of measurement by very powerful instrumental appliances; but, small as it is, calculation shows that it indicates a motion of recess at the rate of about 30 miles per second. 1 1 The observed displacement corresponded to recess at the rate of 41 '4 miles per second ; 992 DISPERSION. STUDY OF SPECTRA. 790. Spectra of Artificial Lights. The spectra of the artificial lights in ordinary use (including gas, oil-lamps, and candles) differ from the solar spectrum in the relative brightness of the different colours, as well as in the entire absence of dark lines. They are comparatively strong in red and green, but weak in blue ; hence all colours which contain much blue in their composition appear to disadvantage by gas-light. It is possible to find artificial lights whose spectra are of a com- pletely different character. The salts of strontium, for example, give red light, composed of the ingredients represented in spectrum No. 10, Plate III., and those of sodium yellow light (No. 3, Plate III.) If a room is illuminated by a sodium flame (for example, by a spirit- lamp with salt sprinkled on the wick), all objects in the room will appear of a uniform colour (that of the flame itself), differing only in brightness, those which contain no yellow in their spectrum as seen by day-light being changed to black. The human countenance and hands assume a ghastly hue, and the lips are no longer red. A similar phenomenon is observed when a coloured body is held in different parts of the solar spectrum in a dark room, so as to be illuminated by different kinds of monochromatic light. The object either appears of the same colour as the light which falls upon it, or else it refuses to reflect this light and appears black. Hence a screen for exhibiting the spectrum should be white. 791. Brightness and Purity. The laws which determine the bright- ness of images generally, and which have been expounded at some length in the preceding chapter, may be applied to the spectroscope. We shall, in the first instance, neglect the loss of light by reflection and imperfect transmission. Let A denote the prismatic dispersion, as measured by the angular separation of two specified monochromatic images when the naked eye is applied to the last prism, the observing telescope being re- moved. Then, putting m for the linear magnifying power of the but 12'0 of this must be deducted for the motion of the earth in its orbit at the season of the year when the observation was made. The remainder, 29 "4, is therefore the rate at which the distance between the sun and Sirius is increasing. In a more recent paper, read while this volume was going through the press, Dr. Hug- gins gives the results of observations with more powerful instrumental appliances. The recess of Sirius is found to be only 20 miles per second. Arcturus is approaching at the rate of 50 miles per second. Community of motion has been established in certain sets of stars ; and the belief previously held by astronomers, as to the direction in which the solar system is moving with respect to the stars as a whole, is fully confirmed. BRIGHTNESS AND PURITY telescope, mA is the angular separation observed when the eye is applied to the telescope. We shall call m A the total dispersion. Let denote the angle which the breadth of the slit subtends at the centre of the colliraating lens, and which is measured by focal length of lens' Then Q * s also the a PP arent breadth of any absolutely monochromatic image of the slit, formed by rays of minimum devia- tion,as seen by an eye applied either to the first prism, the last prism, or any one of the train of prisms. The change produced in a pencil of monochromatic rays by transmission through a prism at minimum deviation, is in fact simply a change of direction, without any change of mutual inclination; and thus neither brightness, nor apparent size is at all affected. In ordinary cases, the bright lines of a spectrum may be regarded as monochromatic, and their apparent breadth, as seen without the telescope, is sensibly equal to 0. ' Strictly speaking, the effect of prismatic dispersion in actual cases, is to increase the apparent breadth by a small quantity, which, if all the prisms are alike, is proportional to the number of prisms ; but the increase is usually too small to be sensible. Let I denote the intrinsic brightness of the source as regards any one of its (approximately) monochromatic constituents; in other words, the brightness which the source would have if deprived of all its light except that which goes to form a particular bright line. Then, still neglecting the light stopped by the instrument, the bright- ness of this line as seen without the aid of the telescope will be I; and as seen in the telescope it will either be equal to or less than this, according to the magnifying power of the telescope and the effective aperture of the object-glass ( 769). If the breadth of the slit be halved, the breadth of the bright line will be halved, and its brightness will be unchanged. These conclusions remain true so long as the bright line can be regarded as practically monochromatic. The brightness of any part of a continuous spectrum follows a very different law. It varies directly as the width of the slit, and inversely as the prismatic dispersion. Its value without the ob- a serving telescope, or its maximum value with a telescope, is A i, where i is a coefficient depending only on the source. The purity of any part of a continuous spectrum is properly mea- sured by the ratio of the distance between two specified mono- chromatic images to the breadth of either, the distance in question being measured from the centre of one to the centre of the other. 64 994) DISPERSION. STUDY OF SPECTRA. This ratio is unaffected by the employment of an observing telescope, j A and is -5 . The ratio of the brightness of a bright line to that of the adjacent portion of a continuous spectrum forming its back-ground, is -^-' assuming the line to be so nearly monochromatic that the increase of its breadth produced by the dispersion of the prisms is an insigni- ficant fraction of its whole breadth. As we widen the slit, and so increase 0, we must increase A in the same ratio, if we wish to preserve the same ratio of brightness. As -j is increased indefinitely, the predominance of the bright lines does not increase indefinitely, but tends to a definite limit, namely, to the predominance which they would have in a perfectly pure spectrum of the given source. The loss of light by reflection and imperfect transmission, increases with the number of surfaces of glass which are to be traversed; so that, with a long train of prisms and an observing telescope, the actual brightness will always be much less than the theoretical bright- ness as above computed. The actual purity is always less than the theoretical purity, being greatly dependent on freedom from optical imperfections; and these can be much more completely avoided in lenses than in prisms. It is said that a single good prism, with a first-class collimator and telescope, (as originally employed by Swan,) gives a spectrum much more free from blurring than the modern multiprism spectroscopes, when the total dispersion m A is the same in both the cases com- pared. 792. Chromatic Aberration. The unequal refrangibility of the different elementary rays is a source of grave inconvenience in con- nection with lenses. The focal length of a lens depends upon its index of refraction, which of course increases with refrangibility, the focal length being shortest for the most refrangible rays. Thus a lens of uniform material will not form a single white image of a white object, but a series of images, of all the colours of the spectrum, arranged at different distances, the violet images being nearest, and the red most remote. If we place a screen anywhere in the series ot images, it can only be in the right position for one colour. Every other colour will give a blurred image, and the superposition of them all produces the image actually formed on the screen. If the object be a uniform white spot on a black ground, its image on the screen ACHROMATISM. 995 will consist of white in its central parts, gradually merging into a coloured fringe at its edge. Sharpness of outline is thus rendered impossible, and nothing better can be done than to place the screen at the focal distance corresponding to the brightest part of the spec- trum. Similar indistinctness will attach to images observed in mid- air, whether directly or by means of another lens. This source of confusion is called chromatic aberration. 793. Possibility of Achromatism. In order to ascertain whether it was possible to remedy this evil by combining lenses of two different materials, Newton made some trials with a compound prism com- posed of glass and water (the latter containing a little sugar of lead), and he found that it was not possible, by any arrangement of these two substances, to produce deviation of the transmitted light without separation into its component colours. Unfortunately he did not extend his trials to other substances, but concluded at once that an achromatic prism (and hence also an achromatic lens) was an impos- sibility ; and this conclusion was for a long time accepted as indis- putable. Mr. Hall, a gentleman of Worcestershire, was the first to show that it was erroneous, and is said to have constructed some achromatic telescopes; but the important fact thus discovered did not become generally known till it was rediscovered by Dollond, an eminent London optician, in whose hands the manufacture of achro- matic instruments attained great perfection. 794. Conditions of Achromatism. The conditions necessary for achromatism are easily explained. The angular separation between the brightest red and the brightest violet ray transmitted through a prism is called the dispersion of the prism, and is evidently the differ- ence of the deviations of these rays. These deviations, for the position of minimum deviation of a prism of small refracting angle A, are (// 1) A and (//' 1) A, // and //' denoting the indices of refraction for the two rays considered 739, equation (1) and their difference is (ft" fi) A. This difference is always small in comparison with either of the deviations whose difference it is, and its ratio to either of them, or more accurately its ratio to the value of (p. 1) A for the brightest part of the spectrum, is called the dispersive power of the substance. As the common factor A may be omitted, the formula for the dis- persive power is evidently ^-^y- If this ratio were the same for all substances, as Newton supposed, achromatism would be impossible ; but in fact its value varies greatly, and is greater for flint than for crown glass. If two prisms of these 996 DISPERSION. STUDY OF SPECTRA. substances, of small refracting angles, be combined into one, with their edges turned opposite ways, they will achromatize one another if (/z"-/z')A, or the product of deviation by dispersive power, is the same for both. As the deviations can be made to have any ratio we please by altering the angles of the prisms, the condition is evidently possible. The deviation which a simple ray undergoes in traversing a lens, at a distance x from the axis, is -^> / denoting the focal length of the lens ( 739), and the separation of the red and violet constituents of a compound ray is the product of this deviation by the dispersive power of the material. If a convex and concave lens are combined, fitting closely together, the deviations which they produce in a ray traversing both, are in opposite directions, and so also are the dis- persions. If we may regard x as having the same value for both (a supposition which amounts to neglecting the thicknesses of the lenses in comparison with their focal lengths) the condition of no resultant dispersion is that j dispersive power x -^ has the same value for both lenses. Their focal lengths must there- fore be directly as the dispersive powers of their materials. These latter are about '033 for crown and -052 for flint glass. A converg- ing achromatic lens usually consists of a double convex lens of crown fitted to a diverging meniscus of flint. In every achromatic com- bination of two pieces, the direction of resultant deviation is that due to the piece of smaller dispersive power. The definition above given of dispersive power is rather loose. To make it accurate, we must specify, by reference to the " fixed lines," the precise positions in the spectrum of the two rays whose separa- tion we consider. Since the distances between the fixed lines have different propor- tions for crown and flint glass, achromatism of the whole spectrum is impossible. With two pieces it is possible to unite any two selected rays, with three pieces any three selected rays, and so on. It is considered a sign of good achromatism when no colours can be brought into view by bad focussing except purple and green. 795. Achromatic Eye-pieces. The eye-pieces of microscopes and astronomical telescopes, usually consist of two lenses of the same kind of glass, so arranged as to counteract, to some extent, the spherical and chromatic aberrations of the object-glass. The positive eye-piece, THE RAINBOW. 997 invented by Ramsden, is suited for observation with cross-wires or micrometers; the negative eye-piece, invented by Huyghens, is not adapted for purposes of measurement, but is preferred when distinct vision is the sole requisite. These eye-pieces are commonly called achromatic, but their achromatism is in a manner spurious. It con- sists not in bringing the red and violet images into true coincidence, but merely in causing one to cover the other as seen from the posi- tion occupied by the observer's eye. In the best opeva-glasses ( 764), the eye-piece, as well as the ob- ject-glass, is composed of lenses of flint and crown so combined as to he achromatic in the more proper sense of the word. 796. Rainbow. The unequal refrangibility of the different ele- mentary rays furnishes a complete explanation of the ordinary phe- nomena of rainbows. The explanation was first given by Newton, who confirmed it by actual measurement. It is well known that rainbows are seen when the sun is shining on drops of water. Sometimes one bow is seen, sometimes two, each of them presenting colours resembling those of the solar spectrum. When there is only one bow, the red arch is above and the violet below. When there is a second bow, it is at some distance outside of this, has the colours in reverse order, and is usually less bright. Rainbows are often observed in the spray of cascades and fountains, when the sun is shining. In every case, a line join- ing the observer to the sun is the axis of the bow or bows; that is to say, all parts of the length of the bow are at the same angu- lar distance from the sun. The formation of the pri- mary bow is illustrated by Fig. 719. A ray of solar light, falling on a spherical drop of water, in the direc- tion S I, is refracted at I, then reflected internally from the back of the drop, and again refracted into the air in the direction I' M. If we take different points of incidence, we shall obtain differ- ent directions of emergence, so that the whole light which emerges Fig. 719. Production of Primary Bow. 998 DISPERSION. STUDY OF SPECTRA. from the drop after undergoing, as in the figure, two refractions and one reflection, forms a widely- divergent pencil. Some portions of this pencil, however, contain very little light. This is especially the case with those rays which, having been incident nearly normally, are returned almost directly back, and also with those which were almost tangential at incidence. The greatest condensation, as re- gards any particular species of elementary ray, occurs at that part of the emergent pencil which makes the smallest obtuse angle or the greatest acute angle with the direction of incidence. As the ob- tuse angle is the measure of the deviation, the direction of greatest condensation is the direction of minimum deviation. It is by means of rays which have undergone this minimum deviation, that the observer sees the corresponding colour in the bow ; and the devia- tion which they have undergone is evidently equal to the angular distance of this part of the bow from the sun. The minimum deviation will be greatest for those rays which are most refrangible. If the figure, for example, be supposed to represent the circumstances of minimum deviation for violet, we shall obtain smaller deviation in the case of red, even by giving the angle I A I' the same value which it has in the case of minimum deviation for violet, and still more when we give it the value which corresponds to the minimum deviation of red. The most refrangible colours are accordingly seen furthest from the sun. The effect of the rays which undergo other than minimum deviation, is to produce a border of white light on the side remote from the sun ; that is to say, on the inner edge of the bow. 1 The condensation which accompanies minimum deviation, is merely a particular case of the general mathematical law that magnitudes remain nearly constant in the neighbourhood of a maximum or minimum value. A small parallel pencil S I incident at and around the precise point which corresponds to minimum deviation, will thus 1 When the drops are very uniform in size, a series of faint supernumerary bow, alternately purple and green, is sometimes seen beneath the primary bow. These bows are produced by the mutual interference of rays which have undergone other than minimum deviation, and tlie interference arises in the following way. Any two parallel directions of emergence, for rays of a given refrangibility, correspond in general to two different points of incidence on any given drop, one of the two incident rays being more nearly normal, and the other more nearly tangential to the drop than the ray of minimum deviation. These two rays have pursued dissimilar paths in the drop, and are in different phases when they reach the observer's eye. The difference of phase may amount to one, two, three, or more exact wave-lengths, and thus one, two, three, or more supernumerary bows may be formed. The distances between the super- numerary bows will be greater as the drops of water are smaller. This explanation is due to Dr. Thomas Young. A more complete theory, in which diffraction is taken into account, is given by Airy in the Cambridge Transactions for 1838 ; and the volume for the following year contains an experimental verification by Miller. It appears from this theory that the maximum of intensity is less sharply marked than the ordinary theory would indicate, and does not correspond to the geometrical minimum of deviation, I nit to a deviation sensibly greater. Also that the region of sensible illumination extends beyond this geometrical minimum and shades off gradually. THE RAINBOW. 999 Fig. 720. Production of Secondary Bow. have deviations which may be regarded as equal, and will accord- ingly remain sensibly parallel at emergence. A parallel pencil in- cident on any other part of the drop, will be divergent at emer- gence. The indices of refraction for red and violet rays from air into water are respectively l -/-^ and ^j- 9 , and calculation shows that the distances from the centre of the sun to the parts of the bow in which these colours are strongest should be the supplements of 42 2' and 40 17' respectively. These results agree with observation. The angles 42 2' and 40 1 7' are the distance from the antisolar point, which is always the cen- tre of the bow. The rays which form the secondary bow have undergone two internal reflections, as repre- sented in Fig. 720, and here again a special con- centration occurs in the direction of minimum deviation. This devia- tion is greater than 1 80, and is greatest for the most refrangible rays. The distance of the arc thus formed from the sun's centre, is 360 min- us the deviation, and is accordingly least for the most refrangible rays. Thus the violet arc is nearest the sun, and the red furthest from it, in the secondary bow. Some idea of the relative situations of the eye, the sun, and the drops of water in which the two bows are formed, may be obtained from an inspection of Fig. 721. \ Fig. 721 .Relative Positions. CHAPTER LXIII. COLOUR. 797. Colour as a Property of Opaque Bodies. A body which reflects (by irregular reflection) all the rays of the spectrum in equal propor- tion, will appear of the same colour as the light which falls upon it ; that is to say, in ordinary cases, white or gray. But the majority of bodies reflect some rays in larger proportion than others, and are therefore coloured, their colour being that which arises from the mixture of the rays which they reflect. A body reflecting no light would be perfectly black. Practically, white, gray, and black differ- only in brightness. A piece of white paper in shadow appears gray, and in stronger shadow black. 798. Colour of Transparent Bodies. A transparent body, seen by transmitted light, is coloured, if it is more transparent to some rays than to others, its colour being that which results from mixing the transmitted rays. No new ingredient is added by transmission, but certain ingredients are more or less completely stopped out. Some transparent substances appear of very different colours according to their thickness. A solution of chloride of chromium, for example, appears green when a thin layer of it is examined, while a greater thickness of it presents the appearance of reddish brown. In such cases, different kinds of rays successively disappear by selec- tive absorption, and the transmitted light, being always the sum of the rays which remain unabsorbed, is accordingly of different com- position according to the thickness. When two pieces of coloured glass are placed one behind the other, the light which passes through both has undergone a double process of selective absorption, and therefore consists mainly of those rays which are abundantly transmitted by both glasses; or to speak broadly, the colour which we see in looking through the combination COLOURS OF MIXED POWDERS. 1001 is not the sum of the colours of the two glasses, but their common part. Accordingly, if we combine a piece of glass coloured red with oxide of copper, and transmitting light which consists almost entirely of red rays, with a blue or violet glass of about the same depth of tint, and transmitting hardly any red, the combination will be almost black. The light transmitted through two glasses of different colour, and of the same depth of tint, is always less than would be trans- mitted by a double thickness of either; and the colour of the trans- mitted light is in most cases a colour which occupies in the spectrum an intermediate place between the two given colours. Thus, if the two glasses are yellow and blue, the transmitted light will, in most cases, be green, since most natural yellows and blues when analyzed by a prism show a large quantity of green in their composition. Similar effects are obtained by mixing coloured liquids. 799. Colours of Mixed Powders. "In a coloured powder, each par- ticle is to be regarded as a small transpai*ent body which colours light by selective absorption. It is true that powdered pigments when taken in bulk are extremely opaque. Nevertheless, whenever we have the opportunity of seeing these substances in compact and homogeneous pieces before they have been reduced to powder, we find them transparent, at least when in thin slices. Cinnabar, chroraate of lead, verdigris, and cobalt glass are examples in point "When light falls on a powder thus composed of transparent par- ticles, a small part is reflected at the upper surface ; the rest penetrates, and undergoes partial reflection at some of the surfaces of separation between the particles. A single plate of uncoloured glass reflects -^ of normally incident light; two plates T *j, and a large number nearly the whole. In the powder of such glass, we must accordingly con- clude that only about -^ of normally incident light is reflected from the first surface, and that all the rest of the light which gives the powder its whiteness comes from deeper layers. It must be the same with the light reflected from blue glass; and in coloured powders generally only a very small part of the light which they reflect comes from the first surface; it nearly all comes from beneath. The light reflected from the first surface is white, except when the reflection is metallic. That which comes from below is coloured, and so much the more deeply the further it has penetrated. This is the reason why coarse powder of a given material is more deeply col- oured than fine, for the quantity of light returned at each successive reflection depends only on the number of reflections and not on the 1002 COLOUR. thickness of the particles. If these are large, the light must pene- trate so much the deeper in order to undergo a given number of reflections, and will therefore be the more deeply coloured. "The reflection at the surfaces of the particles is weakened if we interpose between them, in the place of air, a fluid whose index of refraction more nearly approaches their own. Thus powders and pigments are usually rendered darker by wetting them with water, and still more with the more highly refracting liquid, oil. "If the colours of powders depended only on light reflected from their first surfaces, the light reflected from a mixed powder would be the sum of the lights reflected from the surfaces of both. But most of the light, in fact, comes from deeper layers, and having had to traverse particles of both powders, must consist of those rays which are able to traverse both. The resultant colour therefore, as in the case of superposed glass plates, depends not on addition but rather on subtraction. Hence it is that a mixture of two pigments is usually much more sombre than the pigments themselves, if these are very unlike in the average refrangibility of the light which they reflect. Vermilion and ultramarine, for example, give a black-gray (showing scarcely a trace of purple, which would be the colour obtained by a true mixture of lights), each of these pigments being in fact nearly opaque to the light of the other." l 800. Mixtures of Colours. By the colour resulting from the mix- ture of two lights, we mean the colour which is seen when they both fall on the same part of the retina. Propositions regarding mixtures of colours are merely subjective. The only objective differences of colour are differences of refrangibility, or if traced to their source, differences of wave-frequency. All the colours in a pure spectrum are objectively simple, each having its own definite period of vibra- tion by which it is distinguished from all others. But whereas, in acoustics, the quality of a sound as it affects the ear varies with every change in its composition, in colour, on the other hand, very different compositions may produce precisely the same visual im- pression. Every colour that we see in nature can be exactly imitated by an infinite variety of different combinations of elementary rays. To take, for example, the case of white. Ordinary white light consists of all the colours of the spectrum combined; but any one of the elementary colours, from the extreme red to a certain point in yellowish green, can be combined with another elementary colour 1 Translated from Helmholtz's Physiological Optics, 20. MIXTURE OF COLOURS. 1003 on the other side of green in such proportion as to yield a perfect imitation of ordinary white. The prism would instantly reveal the differences, but to the naked eye all these whites are completely undistinguishable one from another. 801. Methods of Mixing Colours. The following are some of the best methods of mixing colours (that is coloured lights) : 1. By combining reflected and transmitted light; for example, Ly looking at one colour through a piece of glass, while another colour is seen by reflection from the near side of the glass. The lower sash of a window, when opened far enough to allow an arm to be put through, answers well for this purpose. The brighter of the two coloured objects employed should be held inside the window, and seen by reflection ; the second object should then be held outside in such a position as to be seen in coincidence with the image of the first. As the quantity of reflected light increases with the angle of incidence, the two colours may be mixed in various proportions by shifting the position of the eye. This method is not however adapted to quantitative comparison, and can scarcely be employed for combining more than two colours. 2. By employing a ro- tating disc (Fig. 722) composed of differently coloured sectors. If the disc be made to revolve rapidly, the sectors will not be separately visible, but their colours will appear blended into one on account of the per- sistence of visual impres- sions. The proportions can be varied by varying the sizes of the sectors. Coloured discs of paper, each having a radial slit, are very convenient for this purpose, as any moderate number of such discs can be combined, and the sizes of the sectors exhibited can be varied at pleasure. The mixed colour obtained by a rotating disc is to be regarded as Fig. 722. Rotating Disc. 1004 COLOUR. a mean of the colours of the several sectors a mean in which each of these colours is assigned a weight proportional to the size of its sector. Thus, if the 360 degrees which compose the entire disc consist of 100 of red paper, 100 of green, and 160 of blue, the intensity of the light received from the red when the disc is rotating will be only - of that which would be received from the red sector when seen at rest; and the total effect on the retina is represented by 7 ^-- of the intensity of the red, plus -$ of the intensity of the green, plus -^f- of the intensity of the blue ; or if we denote the colours of the sectors by their initial letters, the effect may be symbolized by the formula - + + - . Denoting the resultant colour by C, we have the symbolic equation and the resultant colour may be called the mean of 10 parts of red, 10 of green, and 16 of blue. Colour-equations, such as the above, are frequently employed, and may be combined by the same rules as ordinary equations. 3. By causing two or more spectra to overlap. We thus obtain mixtures which are the sums of the overlapping colours. If, in the experiment of 778, we employ, instead of a single straight slit, a pair of slits meeting at an angle, so as to form either an X or a V, we shall obtain mixtures of all the simple colours two and two, since the coloured images of one of the slits will cross those of the other. The display of colours thus obtained upon a screen is exquisitely beautiful, and if the eye is placed at any point of the image (for example, by looking through a hole in the screen), the prism will be seen filled with the colour which falls on this point. 802. Experiments of Helmholtz and Maxwell. Helmholtz, in an excellent series of observations of mixtures of simple colours, em plcyed a spectroscope with a V-shaped slit, the two strokes of the V being at right angles to one another; and by rotating the V he was able to diminish the breadth and increase the intensity of one of the two spectra, while producing an inverse change in the other. To isolate any part of the compound image formed by the two over- lapping spectra, he drew his eye back from the eye-piece, so as to limit his view to a small portion of the field. But the most effective apparatus for observing mixtures of simple colours is one devised by Professor Clerk Maxwell, by means of which any two or three colours of the spectrum can be combined in MIXTURE OF COLOURS. 1005 any required proportions. In principle, this method is nearly equi- valent to looking through the hole in the screen in the experiment above described. Let P (Fig. 723) be a prism, in the position of minimum deviation; L a lens ; E and R conju- gate foci for rays of a particular refrangibility, say red; E and V conju- gate foci for rays of an- other given refrangibi- lity, say violet. If a slit is opened at R, an eye at E will receive only red rays, and will see the lens filled with red light. If this slit be closed, and a slit opened at V, the eye, still placed at E, will see the lens filled with violet light. If both slits be opened, it will see the lens filled with a uniform mixture of the two lights; and if a third slit be opened, between R and V, the lens will be seen filled with a mixture of three lights. Again, from the properties of conjugate foci, if a slit is opened at E, its spectral image will be formed at R V, the red part of it being at R, and the violet part at V. The apparatus was inclosed in a box painted black within. There was a slit fixed in position at E, and a frame with three movable slits at R V. When it was desired to combine colours from three given parts of the spectrum, specified by reference to Fraunhofer's lines, the slit E was first turned towards the light, giving a real spectrum in the plane R V, in which Fraunhofer's lines were visible, and the three movable slits were set at the three specified parts of the spectrum. The box was then turned end for end. so that light was admitted (reflected from a large white screen placed in sunshine) at the movable slits, and the observer, looking in at the slit E, saw the resultant colour. 803. Results of Experiment. The following are some of the prin- cipal results of experiments on the mixture of coloured lights : 1. Lights which appear precisely alike to the naked eye yield identical results in mixtures; or employing the term similar to express apparent identity as judged by the naked eye, the sums of similar lights are themselves similar. It is by reason of this phy- sical fact, that colour-equations yield true results when combined according to the ordinary rules of elimination. 1006 COLOUR. In the strict application of this rule, the same observer must be the judge of similarity in the different cases considered. For 2. Colours may be similar as seen by one observer, and dissimilar as seen by another ; and in like manner, colours may be similar as seen through one coloured glass, and dissimilar as seen through another. The reason, in both cases, is that selective absorption depends upon real composition, which may be very different for two merely similar lights. Most eyes are found to exhibit selective absorption of a certain kind of elementary blue, which is accord- ingly weakened before reaching the retina. 3. Every colour, except purple, is similar to a colour of the spec- trum either pure or diluted, and all purples are similar to mixtures of red and blue with or without dilution. By diluting a colour we mean mixing it with white, gray, or black. Brown colours are obtained by diluting red, orange, or yellow of feeble intensity. 4. Between any four colours, given in intensity as well as in kind, one colour-equation subsists; expressing the fact that, when we have the power of varying their intensities at pleasure, there is one defi- nite way of making them yield a match, that is to say, a pair of similar colours. Any colour (intensity included) can therefore be completely specified by three numbers, expressing its relation to three arbitrarily selected colours. This is analogous to the theorem in statics that a force acting at a given point can be specified by three numbers denoting its components in three arbitrarily selected directions. 5. Between any five colours (intensity included) a match can be made in one definite way by taking means; 1 for example, by mount- ing the colours on two rotating discs. If we had the power of illu- minating one disc more strongly than the other in any required ratio, four colours would be theoretically sufficient ; and we can, in fact, do what is nearly equivalent to this, by employing black as one of our five colours. Taking means of colours is analogous to finding centres of gravity. In following out the analogy, a colour (given in kind merely) must be represented by a material point given in position merely, and the intensity of the colour must be represented by the mass of the material point. The means of two given colours will be represented by points in the line joining two given points. The means of three given colours will be represented by points lying 1 Propositions 4 and 5 are not really independent, but represent different aspects of one physical (or rather physiological) law. CONE OF COLOUR. 1007 within the triangle formed by joining three given points, and the means of four given colours will be represented by points within a tetrahedron whose four corners are given. When we have five colours given, we have five points given, and of these generally no four will lie in one plane. Call them A, B, C, D, E. Then if E lies within the tetrahedron A B C D, we can make the centre of gravity of A, B, C, and D coincide with E, and the colour E can be matched by a mean of the other four colours. If E lies outside the tetrahedron, let the planes which contain the tetrahedron be produced indefinitely. Then if E lies in the external solid angle which is vertical to the solid angle A of the tetrahedron, the point A lies within the tetra- hedron E B C D, and the colour A is the match. Lastly, let E lie in the external space which is separated from the tetrahedron by the plane BCD. Then the point where this plane is cut by the line joining A E represents the match, for it is a mean of A, E, and is also a mean of B, C, D. With six given colours, combined five at a time, six different matches can be made, and six colour-equations will thus be obtained, the consistency of which among themselves will be a test of the accuracy both of theory and observation, as only three of the six can be really independent. Experiments which have been conducted on this plan have given very consistent results. 804. Cone of Colour. All combinations of colour (intensity in- cluded) can be represented geometrically by means of a cone or pyramid within which all possible colours will have their definite places. The vertex will represent total blackness, or the complete absence of light; and colours situated on the same line passing through the vertex will differ only in intensity of light. Any cross- section of the cone will contain all colours, except so far as intensity is concerned, and the colours residing on its perimeter will be the colours of the spectrum ranged in order, with purple to fill up the interval between violet and red. It appears from Maxwell's experi- ments, that the true form of the cross-section is approximately trian- gular, 1 with red, green, and violet at the three corners. When all the colours (intensity included) have been assigned their proper places in the cone, a straight line joining any two of them passes through colours which are means of these two ; and if two lines are drawn from the vertex to any two colours, the parallelogram constructed 1 The shape of the triangle is a mere matter of convenience, not involving any question of fact. 1008 COLOUR. on these two lines will have at its further corner the colour which is the sum of these two colours. A certain axial line of the cone will contain white or gray at all points of its length, and may be called the line of white. It is convenient to distinguish three qualities of colour which may be called hue, depth, and brightness. Brightness or intensity of light is represented by distance from the vertex of the cone. Depth depends upon angular distance from the line of white, and is the same for all points on the same line through the vertex. Paleness or lightness is the opposite of depth, and is measured by angular nearness to the line of white. Hue or tint is that which is often par excellence termed colour. If we suppose a plane, containing the line of white, to revolve about this line as axis, it will pass succes- sively through different tints ; and in any one position it contains only two tints, which are separated from each other by the line of white, and are complementary. Red is complementary to Bluish green. Orange ,, ,, Sky blue. Yellow ,, Violet blue. Greenish yellow Violet. Green ., Pink. Any two colours, of complementary tint, give white when mixed in proper proportions; and any three colours can be mixed in such proportions as to yield white, unless they are all on the same side of a plane drawn through the line of white. According to Maxwell, the orange and yellow of the spectrum can be exactly reproduced by mixtures of red and green, and the extreme colours of the spectrum (crimson and violet) can be reproduced (approximately at least) by mixtures of red and blue. 805. Three Primary Colour-sensations. All authorities are now agreed in accepting the doctrine, first propounded by Dr. Thomas Young, that there are three elements of colour-sensation ; or, in other words, three distinct physiological actions, which, by their various combinations, produce our various sensations of colour. Each is excitable by light of various wave-lengths lying within a wide range, but has a maximum of excitability for a particular wave-length, and is affected only to a slight degree by light of wave-length very different from this. The cone of colour is theoretically a triangular pyramid, having for its three edges the colours which correspond to these three wave-lengths ; but it is probable that we cannot obtain THREE PRIMARY COLOUR-SENSATIONS. 1009 one of the three elementary colour-sensations quite free from admix- ture of the other two, and the edges of the pyramid are thus practi- cally rounded off. One of these sensations is excited in its greatest purity by the green near Fraunhofer's line b, another by the extreme red, and the third by a part of the spectrum lying somewhere in violet or deep blue, its precise position being difficult to determine by reason of the feebleness of the light at this end of the spectrum. Helmholtz ascribes these three actions to three distinct sets of nerves, having their terminations in different parts of the thickness of the retina a supposition which aids in accounting for the approxi- mate achromatism of the eye, for the three sets of nerve-terminations may thus be at the proper distances for receiving distinct images of red, green, and violet respectively, the focal length of a lens being shorter for violet than for red. Light of great intensity, whatever its composition, seems to pro- duce a considerable excitement of all three elements of colour-sensa- tion. If a spectroscope, for example, be directed first to the clouds and then to the sun, all parts of the spectrum appear much paler in the latter case than in the former. The popular idea that red, yellow, and blue are the three prima- ries, is quite wrong as regards mixtures of lights or combinations of colour- sensations. The idea has arisen from facts observed in con- nection with the mixture of pigments and the transmission of light through coloured glasses. We have already pointed out the true interpretation of observations of this nature, and have only now to add that in attempting to construct a theory of the colours obtained by mixtures of pigments, the law of substitution of similars cannot be employed. Two pigments of similar colour will not in general give the same result in mixtures. 806. Accidental Images. If we look steadily at a bright stained- glass window, and then turn our eyes to a white wall, we see an image of the window with the colours changed into their com- plementaries. The explanation is that the nerves which have been strongly exercised in the perception of the bright colours have had their sensibility diminished, so that the balance of action which is necessary to the sensation of white no longer exists, but those elements of sensation which have not been weakened preponderate. The sub- jective appearances arising from this cause are called negative acci- dental images. Many well-known effects of contrast are similarly explained. White paper, when seen upon a background of any one 65 1010 COLOUR. colour, often appears tinged with the complementary colour; and stray beams of sunlight entering a room shaded with yellow holland blinds, produce blue streaks when they fall upon a white table- cloth. In some cases, especially when the object looked at is painfully bright, there is a positive accidental image; that is, one of the same colour as the object; and this is frequently followed by a negative image. A positive accidental image may be regarded as an extreme instance of the persistence of impressions. 807. Colour-blindness. What is called colour-blindness has been found, in every case which has been carefully investigated, to consist in the absence of the elementary sensation corresponding to red. To persons thus affected the solar .spectrum appears to consist of two decidedly distinct colours with white or gray at their place of junc- tion, which is a little way on the less refrangible side of the line F. One of these two colours is doubtless nearly identical with the normal sensation of blue. It attains its maximum about midway between F and G, and extends beyond G as far as the normally visible spectrum. The other colour extends a considerable distance into what to normal eyes is the red portion of the spectrum, attaining its maximum about midway between D and E, and becoming deeper and more faint till it vanishes at about the place where to normal eyes crimson begins. The scarlet of the spectrum is thus visible to the colour-blind, not as scarlet but as a deep dark colour, probably a kind of dark green, orange and yellow as brighter shades of the same colour, while bluish-green appears nearly white. It is obvious from this account that what is called " colour-blind- ness" should rather be called dichroic vision, normal vision being distinctively designated as trichroic. To the dichroic eye any colour can be matched by a mixture of yellow and blue, and a match can be made between any three (instead of four) given colours. Objects which have the same colour to the trichroic eye have also the same colour to the dichroic eye. 808. Colour and Musical Pitch. As it is completely established that the difference between the colours of the spectrum is a difference of vibration-frequency, there is an obvious analogy between colour and musical pitch ; but in almost all details the relations between colours are strikingly different from the relations between sounds. The compass of visible colour, including the lavender rays which lie beyond the violet, and are perhaps visible not in themselves, but by COLOUR AND MUSICAL PITCH. 1011 the fluorescence which they produce on the retina, is, according to Helmholtz, about an octave and a fourth ; but if we exclude the lavender, it is almost exactly an octave. Attempts have been made to compare the successive colours of the spectrum with the notes of the gamut; but much forcing is necessary to bring out any trace of identity, and the gradual transitions which characterize the spectrum, and constitute a feature of its beauty, are in marked contrast to the transitions per saltum which are required in music. CHAPTER LXIV. WAVE THEORY OF LIGHT. 809. Principle of Huygens. 1 The propagation of waves, whether of sound or light, is a propagation of energy. Each small portion of the medium experiences successive changes of state, involving changes in the forces which it exerts upon neighbouring portions. These changes of force produce changes of state in these neighbouring por tions, or in such of them as lie on the forward side of the wave, and thus a disturbance existing at any one part is propagated onwards. Let us denote by the name wave-front a continuous surface drawn through particles which have the same phase ; then each wave-front advances with the velocity of light, and each of its points may be regarded as a secondary centre from which disturbances are continu- ally propagated. This mode of regarding the propagation of light is due to Huygens, who derived from it the following principle, which lies at the root of all practical applications of the undulatory theory: The disturbance at any point of a wave-front is the resultant (given by the parallelogram of motions) of the separate disturbances which the different portions of the same wave-front in any one of its earlier positions, would have occasioned if acting singly. This principle involves the physical fact that rays of light are not affected by crossing one another; and its truth, which has been experiment- ally tested by a variety of consequences, must be taken as an indica- tion that the amplitudes of luminiferous vibrations are infinitesimal in comparison with the wave-lengths. A similar law applies to the resultant of small disturbances generally, and is called by writers on dj^namics the law of "superposition of small motions." It is analo- gous to the arithmetical principle that, when a and b are very small fractions, the product of l+o- and l-j-6 may be identified with 1 For the spelling of this name see remarks by Lalande, Mimoires de V Academic, 1773. RECTILINEAR PROPAGATION. 1013 1 + a + ^> the term a 6, which represents the mutual influence of two small changes, being negligible in comparison with the sum a + b of the small changes themselves. 810. Explanation of Rectilinear Propagation. In a medium in which light travels with the same velocity in all parts and in all directions, the waves propagated from any point will be concentric spheres, having this point for centre, and the lines of propagation, in other words the rays of light, will be the radii of these spheres. It can in fact be shown that the only part of one of these waves which needs to be considered, in computing the resultant disturbance of an external point, is the part which lies directly between this external point and the centre of the sphere. The remainder of the wave-front can be divided into small parts, each of which, by the mutual interference of its own subdivisions, gives a resultant effect of zero at the given point. We express these properties by saying that in a homogeneous and isotropic medium the wave-surface is a sphere, and the rays are normal to the wave-fronts. This class of media includes gases, liquids, crystals of the cubic system, and well- annealed glass. If a medium be homogeneous but not isotropic, disturbances emanating from a point in it will be propagated in waves which will retain their form unchanged as they expand in receding from their source, but this form will not generally be spherical. The rays of light in such a medium will be straight, proceeding directly from the centre of disturbance, and any one ray will cut all the wave-fronts at the same angle; but this angle will generally be different for different rays. In this case, as in the last, the disturbance produced at any point may be computed by merely taking into account that small portion of a wave-front which lies directly between the given point and the source, in other words, which lies on or very near to the ray which traverses the given point. A disturbance in such a medium usually gives rise to two sets of waves, having two distinct forms, and these remarks apply to each set separately. The tendency of the different parts of a wave-front to propagate disturbances in othsr directions besides the single one to which such propagation is usually confined, is manifested in certain phenomena which are included under the general name of diffraction. The only wave-fronts with which it is necessary to concern our- selves are those which belong to waves emanating from a single 1014 WAVE THEORY OF LIGHT. point, that is to say, either from a surface really very small, or from a surface which, by reason of its distance, subtends a very small solid angle at the parts of space considered. 811. Application to Refraction. When waves are propagated from one medium into another, the principle of Huygens leads to the following construction: Let A E (Fig. 724) represent a portion of the surface of separation between two media, and A B a portion of a wave-front in the first medium ; both portions being small enough to be regarded as plane. Fig. 724. Huygens' Construction for Wave-front Then straight lines C A, D B E, normal to the wave-front, represent rays incident at A and E. From A as centre, describe a wave-surface, of such dimensions that light emanating from A would reach this surface in the same time in which light in air travels the distance BE, and draw a tangent plane (perpendicular to the plane of incid- ence) through E to this surface. Let F be the point of contacl (which is not necessarily in the plane of incidence). Then the tan gent plane E F is a wave-front in the second medium, and A F is a ray in the second medium ; for it can be shown that disturbances propagated from all points in the wave-front A B will just have reached E F when the disturbance propagated from B has reached E. For example, a ray proceeding from m, the middle point of the line A B, will exhaust half the time in travelling to the middle point a of AE, and the remaining half in travelling through af, equal and parallel to half of A F. When the wave-surfaces in both media are spherical, the planes ot incidence and refraction ABE, A F E coincide, the angle B A E (Fig. 725) between the first wave-front and the surface of separation is the same as the angle between the normals to these surfaces, that APPLICATION TO REFLECTION. 1015 is to say, is the angle of incidence ; and the angle A E F between the surface of separation and the second wave-front is the angle of refraction. The sine of the former is 5J?. and the sine of the latter Jli A But B E and A F are the Fig. 725. Wave-front in Ordinary Retraction. is ^|. The ratio ^ is therefore distances travelled in the same time in the two media. Hence the sines of the angles of in- cidence and refraction are di- rectly as the velocities of pro- pagation of the incident and refracted light. The relative index of refraction from one medium into another is there- fore the ratio of the velocity of light in the first medium to its velocity in the second; and the absolute index of refraction of any medium is inversely as the velocity of light in that medium. 812. Application to Reflection. The explanation of reflection is precisely similar. Let C A, D E (Fig. 72G) be parallel rays incident at A and E ; A B the wave- front. As the successive points of the wave-front arrive at the reflecting surface, hemispherical waves di- verge from the points of inci- dence; and by the time that B reaches E, the wave from A will have diverged in all direc- tions to a distance equal to B E. If then we describe in the plane of incidence a semi- circle, with centre A and radius equal to B E, the tangent E F to this semicircle will be the wave - front of the reflected light, and A F will be the reflected ray corresponding to the incident ray CA From the equality of the right-angled triangles ABE, E F A, it is evident that the angles of incidence and reflection are equal. 813. Newtonian Explanation of Refraction. In the Newtonian theory, the change of direction which a ray experiences at the bound E A Fig. 726. Wave-front in Reflection. 1016 WAVE THEORY OF LIGHT. ing surface of two media, is attributed to the preponderance of the attraction of the denser medium upon the particles of light. As the resultant force of this attraction is normal to the surface, the tan- gential component of velocity remains unchanged, and the normal component is increased or diminished according as the incidence is from rare to dense or from dense to rare. Let p. denote the relative index of refraction from rare to dense. Let v,v' be the velocities of light in the rarer and denser medium respectively, and i, i' the angles which the rays in the two media make with the normal. Then the tangential components of velocity in the two media are v sin i, v sin i' respectively, and these by the Newtonian theory are equal; whence -= 8 !-4=u; whereas according to the undulatory theory - = -. In v sin t r ' J v jj. the Newtonian theory, the velocity of light in any medium is di- rectly as the absolute index of refraction of the medium; whereas, in the undulatory theory, the reverse rule holds. The main design of Foucault's experiment with the rotating mir- ror ( 687), in its original form, was to put these opposite conclusions to the test of direct experiment. For this purpose it was not neces- sary to determine the velocity of the rotating mirror, since it affected both the observed displacements alike. The two images were seen in the same field of view, and were easily distinguished by the green- ness of the water-image. In every trial the water-image was more displaced than the air-image, indicating longer time and slower velo- city; and the measurements taken were in complete accordance with the undulatory theory, while the Newtonian theory was conclusively disproved. 814. Principle of Least Time. The path by which light travels from one point to another is in the generality of cases that which occupies least time. For example, in ordinary cases of reflection (except from very concave 1 surfaces), if we select any two points, one on the in- cident and the other on the reflected ray, the sum of their distances from the point of incidence is less than the sum of their distances from any neighbouring point on the reflecting surface. In this case, since only one medium is concerned, distance is proportional to time. When a ray in air is refracted into water, if we select any two points, 1 Suppose an ellipse described, having the two selected points for foci, and passing through the point of incidence. If the curvature of the reflecting surface in the plane of incidence is greater than the curvature of this ellipse, the length of the path is a maximum, if less, a minimum. This follows at once from the constancy of the sum of the focal distances in an ellipse. VRINCIPLE OF LEAST TIME. 1017 one on the incident and the other on the refracted ray, and call their distances from any point of the refracting surface s, s' respectively, and the velocities of propagation in the two media v, v', then the sum ./ of - and is generally less when s and s' are measured to the point of incidence than when they are measured to any neighbouring point on the surface. - is evidently the time of going from the first point to the refracting surface, and ^ the time from the refracting surface to the second point. The proposition as above enunciated admits of certain exceptions, the time being sometimes a maximum instead of a minimum. The really essential condition (which is fulfilled in both these opposite cases) is that all points on a small area surrounding the point of incidence give sensibly the same time. The component waves sent from all parts of this small area will be in the same phase, and will propagate a ray of light by their combined action. When the two points considered are conjugate foci, and there is no aberration, this condition must be fulfilled by all the rays which pass through both; and the time of travelling from one focus to the other is the same for all the rays. Spherical waves diverging from one focus will, after incidence, become spherical waves converging to or diverging from its conjugate focus. An effect of this kind can be beautifully exhibited to the eye by means of an elliptic dish contain- ing mercury. If agitation is produced at one focus of the ellipse by dipping a small rod into the liquid at this point, circular waves will be seen to converge towards the other focus. A circular dish exhi- bits a similar result somewhat imperfectly ; waves diverging from a point near the centre will be seen to converge to a point symmetri- cally situated on the other side of the centre. When the second point lies on a caustic surface formed by the reflection or refraction of rays emanating from the first point, all points on an area of sensible magnitude in the neighbourhood of the point of incidence would give sensibly the same time of travelling as the actual point of incidence, so that the light which traverses a point on a caustic may be regarded as coming from an area of sensible magnitude instead of (as in the case of points not on the caustic) an excessively small area. An eye placed at a point on a caustic will see this portion of the surface filled with light As the velocity of light is inversely proportional to the index of 1018 WAVE THEORY OF LIGHT. refraction p, the time of travelling a distance s with constant velocity may be represented by pS, and if a ray of light passes from one point to another by a crooked path, made up of straight lines s 1? s 2 , s s , . . . . lying in media whose absolute indices are ju x , p , /u 3 , . . . , the expres- sion P! ! + A* 2 s 2 +/u s s 3 + ... represents the time of passage. This expression, which may be called the sum of such terms as ps, must therefore fulfil the above condition ; that is to say, the points of incidence on the surfaces of separation must be so situated that this sum either remains absolutely constant when small changes are sup- posed to be made in the positions of these points, or else retains that approximate constancy which is characteristic of maxima and minima. Conversely, all lines from a luminous point which fulfil this condition, will be paths of actual rays. 815. Terrestrial Refraction. 1 The atmosphere may be regarded as homogeneous when we confine our attention to small portions of it, and hence it is sensibly true, in ordinary experiments where no great distances are concerned, that rays of light in air are straight, just as it is true in the same limited sense that the surface of a liquid at rest is a horizontal plane. The surface of an ocean is not plane, but approximately spherical, its curvature being quite sensible in ordinary nautical observations, where the distance concerned is merely that of the visible sea-horizon ; and a correction for curvature is in like manner required in observing levels on land. If the observer is standing on a perfectly level plain, and observing a distant object at precisely the same height as his eye above the plain, it will appear to be below his eye, for a horizontal plane through his eye will pass above it, since a perfectly level plain is not plane, but shares in the general curvature of the earth. It is easily proved that the apparent depression due to this cause is half the angle between the verticals at the positions of the observer and of the object observed. But experience has shown that this apparent depression is to a consider- able extent modified by an opposite disturbing cause, called terres- trial refraction. When the atmosphere is in its normal condition, a ray of light from the object to the observer is not straight, but is slightly concave downwards. This curvature of a nearly horizontal ray is not due to the curva- ture of the earth and of the layers of equal density in the earth's atmosphere, as is often erroneously supposed, but would still exist, 1 For the leading idea which is developed in 815-817, the Editor is indebted to suggestions from Professor James Thomson. CURVATURE OF RAY. 1019 and with no sensible change in its amount, if the earth's surface were plane, and the directions of gravity everywhere parallel. It is due to the fact that light travels faster in the rarer air above than in the denser air below, so that time is saved by deviating slightly to the upper side of a straight course. The actual amount of curvature (as determined by surveying) is from to -^ of the curvature of the earth ; that is to say, the radius of curvature of the ray is from 2 to 10 times the earth's radius. 816. Calculation of Curvature of Bay. In order to calculate the radius of curvature from physical data, it is better to approach the subject from a somewhat different point of view. The wave-fronts of a ray in air are perpendicular to the ray ; and if the ray is nearly horizontal, its wave-fronts will be nearly ver- tical. If two of these wave-fronts are produced downwards until they meet, the distance of their intersection from the ray will be the radius of curvature. Let us consider two points on the same wave- front, one of them a foot above the other ; then the upper one being in rarer air will be advancing faster than the lower one, and it is easily shown that the difference of their velocities is to the velocity of either as 1 foot is to the radius of curvature. Put (> for the radius of curvature in feet, v and v + $v for the two velocities, p and p. I p. for the indices of refraction of the air at the two points. Then we have L= ?? = ^ = J ft nearly. (1) p v fj. Now it has been ascertained, by direct experiment, that the value of /u 1 for air, within ordinary limits of density, is sensibly pro- portional to the density (even when the temperature varies), and is 0002943 or 34 1 00 at the density corresponding to the pressure 760 mm (at Paris) and temperature 0C. The difference of density at the two points considered, supposing them both to be at the same tempera- ture, will be to the density of either as 1 foot is to the "height of the homogeneous atmosphere" in feet, which call H ( 1 1 1 A). Then -^ will be g, and the value of - in (1) may be written 7 = /rri^- 1)= S ( ^ 1) = S mo- ( 2 ) Hence p is 3400 times the height of the homogeneous atmosphere. But this height is about 5 miles, or -$%-$ of the earth's radius. The 1020 WAVE THEORY OF LIGHT. value of p is therefore about 4|- radii of the earth. This is on the assumptions that the barometer is at 760 mm , the thermometer at 0C., and that there is no change of temperature in ascending. If we depart from these assumptions, we have the following consequences: I. If the barometer is at any other height^ the factor g remains unaltered, and the other factor /j, 1 varies directly as the pressure. II. If the temperature is t Centigrade, H is changed in the direct ratio of l+a, a denoting the coefficient of expansion. The first factor g is therefore changed in the inverse ratio of 1 + a t. The second factor is changed in the same ratio. The curvature of the ray therefore varies inversely as (1 +o) 2 . III. Suppose the temperature decreases upwards at the rate of - of a degree Centigrade per foot. The expansion due to - of a degree .-, . . 1 mi <- i r> Su difference of density -11 Centigrade is -^^. The first factor fj, or - density -- -, will therefore become g 273^* which, if we put n = 540 (corresponding to 1 Fahr. in 300 feet), and reckon H as 26,000, is approximately 26000 ~ 147000 or H 67' ^ ie secon( ^ factor of the expression for - is unaffected. It appears, then, that decrease of temperature upwards at the rate of 1C. in 540 feet, or 1 F. in 300 feet (which is the gene- rally-received average), makes the curvature of the ray five-sixths of what it would be if the temperature were uniform. 1 Combining this correction with correction II., it appears that, with a mean temperature of 10C. or 50 F., and barometer at 760 mm , the curvature of a nearly horizontal ray (taking the earth's curvature as unity) is This is in perfect agreement with observation, the received average (obtained as an empirical deduction from observation) being 1 or . 817. Curvature of Inclined Rays. Thus far we have been treating of nearly horizontal rays. To adapt our formula for - ( (2) 816) to the case of an oblique ray, we have merely to multiply it by cos 6, 1 If the temperature decreases upwards at the rate of 1C. in n feet, or 1F. in n' feet the first factor of the expression for- (which would be g at uniform temperature) becomes 1 /, 96\ 1 /, 53\ approximately g (1 --) or - ( 1 _ _) MIRAGE. 1021 denoting the inclination of the ray to the horizontal, or the inclina- tion of the wave-front to the vertical. For, if we still compare two points a foot apart, on the same wave-front, and in the same vertical plane with each other and with the ray, their difference of height will be the product of 1 foot by cos 0, and - will therefore be less than before in the ratio cos 0. Hence it can be shown that the earth's curvature, so far from being the cause of terrestrial refraction, rather tends in ordinary cir- cumstances to diminish it, by increasing the average obliquity of a ray joining two points at the same level. . The general formula for the curvature of a ray (lying in a vertical plane) at any point in its length, may be written (3) n denoting the number of feet of ascent which give a decrease of 1C., and n' the number of feet which give a decrease of 1 F. The unit of length for H and p may be anything we please. 818. Astronomical Refraction. Astronomical refraction, in virtue of which stars appear nearer the zenith than they really are, can be reduced to these principles ; but it is simpler, in the case of stars not more than 70 or 80 from the zenith, to regard the earth and the layers of equal density in the atmosphere as plane, and to assume that the final result is the same as if the rays from the star were refracted at once out of vacuum into the horizontal stratum of air in which the observer's eye is situated. If z be the apparent and z the true zenith distance of the star, we shall thus have sin z'=p. sin z, whence it may be shown that the value of z' z, in terms of- ^-> ^ approximately (/* !) tan z. 819. Mirage. An appearance, as of water, is frequently seen in sandy deserts, where the soil is highly heated by the sun. The observer sees in the distance the reflection of the sky and of terres- trial objects, as in the surface of a calm lake. This phenomenon, which is called 'mirage, is explained by the heating and conse- quent rarefaction of the air in contact with the hot soil. The den- sity, within a certain distance of the ground, increases upwards, 1022 WAVE THEORY OF LIGHT. and rays traversing this portion of the air are bent upwards (Fig. 655), in accordance with the general rule that the concavity must be turned towards the denser side. Rays which were descending at a very slight inclination before entering this stratum of air may have their direction so much changed as to be bent up to an observer's eye, and Fig. 655. Theory of Mirage. the change of direction will be greatest for those rays which have descended lowest; for these will not only have travelled for the greatest distance in the stratum, but will also have travelled through that part of it in which the change of density is most rapid. Hence, if we trace a pencil of rays from the observer's eye, we shall find that those of them which lie in the same vertical plane cross each other in traversing this stratum, and thus produce inverted images. If the stratum is thin in comparison with the height of the observer's eye, the appearance presented will be nearly equivalent to that pro- duced by a mirror, while the objects thus reflected are also seen erect by higher rays which have not descended into the stratum where this action occurs. A kind of inverted mirage is often seen across masses of calm water, and is called looming, images of distant objects, such as ships or hills, being seen in an inverted position immediately over the objects themselves. The explanation just given of the mirage of the desert will apply to this phenomenon also if we suppose at a certain height, greater than that of the observer's eye, a layer of rapid tran- sition from colder and denser air below to warmer and rarer air above. An appearance similar to mirage may be obtained by gently CURVED RAYS OF SOUND. 1023 depositing alcohol or methylated spirit upon water in a vessel with plate-glass sides. The spirit, though lighter, has a higher index of refraction than the water, and rays traversing the layer of transition are bent upwards. This layer accordingly behaves like a mirror when looked at very obliquely by an eye above it. 1 819A. Curved Rays of Sound. The reasoning of 814-816 can be applied, with a slight modification, to the propagation of sound. Sound travels faster in warm than in cold air. On calm sunny afternoons, when the ground has become highly heated by the sun's rays, the temperature of the air is much higher near the ground than at moderate heights; hence sound bends upwards, and may thus become inaudible to observers at a distance by passing over their heads. On the other hand, on clear calm nights the ground is cooled by radiation to the sky, and the layers of air near the ground are colder than those above them; hence sound bends downwards, and may thus, by arching over intervening obstacles, become audible at distant points, which it could not reach by rectilinear propagation. This influence of temperature, which was first pointed out by Pro- fessor Osborne Reynolds, is one reason why sound from distant sources is better heard by night than by day. A similar effect of wind had been previously pointed out by Pro- fessor Stokes. It is well known that sound is better heard with the wind than against it. This difference is due to the circumstance that wind is checked by friction against the earth, and therefore increases in velocity upwards. Sound travelling with the wind, therefore, travels fastest above, and sound travelling against the wind travels fastest below, its actual velocity being in the former case the sum, and in the latter the difference, of its velocity in still air and the velocity of the wind. The velocity of the wind is so much less than that of sound, that if uniform at all heights its influence on audibility would scarcely be appreciable. 819s. Calculation. To calculate the curvature of a ray of sound due to variation of temperature with height, we may employ, as in 816, the formula -= , where Sv denotes the difference of velocity for a difference of 1 foot in height. The value of v varies as V(l -f at), or approximately as 1+^ at, t denoting temperature, and a the co- 1 A more complete discussion of the optics of mirage will be found in two papers by the editor of this work in the Philosophical Magazine for March and April, 1873, and in Nature for Nov. 19 and 26, 1874. 1024 WAVE THEORY OF LIGHT. efficient of expansion, which is ^g- Hence if the velocity at be denoted by 1, the value at t will be denoted by 1 + -| a t; and if the temperature varies by of a degree per foot, the value of at tem- peratures near zero will be /-, that is, , , and the radius of curva- 2 n' ' 546 n ture will be 546 n feet. This calculation shows that the bending is much more considerable for rays of sound than for rays of light. 820. Diffraction Fringes. When a beam of direct sunlight is admitted into a dark room through a narrow slit, a screen placed at any distance to receive it will show a line of white light, bordered with coloured fringes which become wider as the slit is narrowed. They also increase in width as the screen is removed further off. If they are viewed through a piece of red glass which allows only red rays to pass, they will appear as a succession of bands alternately bright and dark. To explain their origin, we shall suppose the sun's rays (which may be reflected from an external mirror) to be perpendicular to the plane of the slit, 1 so that the wave-fronts are parallel to this plane, and we shall, in the first instance, confine our attention to light of a particular wave-length ; for example, that of the light transmitted by the red glass. Then, if the slit be uniform through its whole length, the positions of the bright and dark bands will be governed by the fol- lowing laws: 1. The darkest parts will be at points whose distances from the two edges of the slit differ by an exact number of wave-lengths. If the difference be one wave-length, the light which arrives at any instant from different parts of the width of the slit is in all possible phases, and the disturbance produced by the nearer half of the slit cancels that produced by the remoter half. If the difference be n wave-lengths, we can divide the slit into n parts, such that the effect due to each part is thus nil. 2 1 That is, to the plane of the two knife-edges by which the slit is bounded. This condition can only be strictly fulfilled for a single point on the sun's disk. Every point on the sun's surface sends out its own waves as an independent source; and waves from one point cannot interfere with waves from another. In the experiment as described in the text the fringes due to different parts of the sun's surface are all produced at once on the screen, and overlap each other. 2 The following explanation will serve to establish the legitimacy of the reasoning here employed : Each element of the length of the slit tends to produce a system of circular rings (the screen being supposed parallel to the plane of the slit). If the width of the slit is uniform, DIFFRACTION. 1025 2. The brightest parts will be at points whose distances from the two edges of the slit differ by an exact number of wave-lengths plus a half. Let the difference be n + \; then we can divide the slit into n inefficient parts and one efficient part, this latter having only half the width of one of the others. Each colour of light has its own alternate bands of brightness and darkness, the distance from band to band being greatest for red and least for violet. The superposition of all the bands constitutes the coloured fringes which are seen. This experiment furnishes the simplest answer to the objection formerly raised to the undulatory theory, that light is not able, like sound, to pass round an obstacle, but can only travel in straight lines. In this experiment light does pass round an obstacle, and turns more and more away from a straight line as the slit is narrowed. When the slit is not exceedingly narrow, the light sent in oblique directions is quite insensible in comparison with the direct light, and no fringes are visible. "We have reason to think that when sound passes through a very large aperture, or when it is reflected from a large surface (which amounts nearly to the same thing), it is hardly sensible except in front of the opening, or in the direction of reflection." 1 There are several other modes of producing diffraction fringes, which our limits do not permit us to notice. We proceed to describe the mode of obtaining a pure spectrum by diffraction. 821. Diffraction by a Grating. If a piece of glass is ruled with parallel equidistant scratches (by means of a dividing engine and diamond point) at the rate of some hundreds or thousands to the inch, we shall find, on looking through it at a slit or other bright line (the glass being held so that the scratches are parallel to the slit), that a number of spectra are presented to view, ranged at nearly equal distances, on both sides of the slit. If the experiment is made under favourable circumstances, the spectra will be so pure as to show a number of Fraunhofer's lines. Instead of viewing the spectra with the naked eye, we may with advantage employ a telescope, focussed on the plane of the slit; or we may project the spectra on a screen, by first placing a convex lens so as to form an image of the slit (which must be very strongly these systems will be precisely alike, and will have for their resultant a system of straight bands, parallel to the slit and touching the rings. These are the bands described in the text. Hence, to determine the illumination of any point of the screen, it is only necessary to attend, as in the text, to the nearest points of the two edges of the slit. 1 Airy, Undulatory Theory. Art. 28. 1026 WAVE THEORY OF LIGHT. illuminated) on the screen, and then interposing the ruled glass in the path of the beam. A piece of glass thus ruled is called a grating. 1 A grating for diffraction experiments consists essentially of a number of parallel strips alternately transparent and opaque. The distance between the "fixed lines" of the spectra, and the distance from one spectrum to the next, are found to depend on the distance of the strips measured from centre to centre, in other words, on the number of scratches to the inch, but not at all on the relative breadths of the transparent and opaque strips. This latter circumstance only affects the brightness of the spectra. Diffraction spectra are of great practi- cal importance 1. As furnishing a uniform standard of reference in the comparison of spectra. 2. As affording the most accurate method of determining the wave-lengths of the different elementary rays of light 822. Principle of Diffraction Spectrum. Let GG (Fig. 728) be a grating, re- ceiving light from an infinitely 2 distant point lying in a direction perpendicular to the plane of the grating, so that the wave-fronts of the incident light are parallel to this plane. Let a convex lens L be placed on the other side of the grating, and let its axis make an acute angle 6 with the rays incident on the grating. Then the light collected at its principal focus F consists of all the light incident upon the lens parallel to its axis. Let s denote the distance between the rulings, measured from centre to centre, so that if, for 1 Engraved glass gratings of sufficient size for spectroscopic purposes (say an inch square) are extremely expensive and difficult to procure. Lord Rayleigh has made numerous pho- tographic copies of such gratings, and the copies appear to be equally effective with the originals. a It is not necessary that the source should be infinitely distant (or the incident rays parallel); but this is the simplest case, and the most usual case in practice. Fig. 728. Principle of Diffraction Spectrum. DIFFRACTION SPECTRUM. 1027 example, there are 1 000 lines to the inch, s will be 10 1 00 of an inch ; and suppose first that s sin is exactly equal to the wave-length X of one of the elementary kinds of light. Then, of all the light which falls upon the lens parallel to its axis, the left-hand portion in the figure is most retarded (having travelled farthest), and the right-hand portion least, the retardation, in comparing each transparent interval with the next, being constant, and equal to s sin 0, as is evident from an inspection of the figure. Now, for the particular kind of light for which X=s sin0, this retardation is exactly a wave-length, and all the transparent intervals send light of the same phase to the focus F; so that, if there are 1000 such intervals, the resultant amplitude of vibration of F is 1000 times the amplitude due to one interval alone. For light of any other wave-length this coincidence of phase will not exist. For example, if the difference between X and s sin is 10 1 00 X, the difference of phase between the lights received from the 1st and 2d intervals will be 10 1 00 X, between the 1st and 3d 10 2 00 X, between the 1st and 501st i^nnr X, or just half a wave-length, and so on. The 1st and 501st are thus in complete discordance, as are also the 2d and 502d, &c. Light of every wave-length except one is thus almost completely destroyed by interference, and the light collected at F consists almost entirely of the particular kind defined by the condition X = shi 9. (I) The purity of the diffraction spectrum is thus explained. If a screen be held at F, with its plane perpendicular to the prin- cipal axis, any point on this screen a little to one side of F will receive light of another definite wave-length, corresponding to an- other direction of incidence on the lens, and a pure spectrum will thus be depicted on the screen. 823. Practical Application. In the arrangement actually em- ployed for accurate observation, the lens L L is the object-glass of a telescope with a cross of spider-lines at its principal focus F. The telescope is first pointed directly towards the source of light, and is then turned to one side through a measured angle 0. Any fixed line of the spectrum can thus be brought into apparent coincidence -with the cross of spider-lines, and its wave-length can be computed by the formula (1). The spectrum to which formula (1) relates is called the spectrum of the first order. 1028 WAVE THEORY OF LIGHT. There is also a spectrum of the second order, corresponding to values of nearly twice as great, and for which the equation is 2 X = sin 6. (2) For the spectrum of the third order, the equation is 3X = sin0; (3) and so on, the explanation of their formation being almost precisely the same as that above given. There are two spectra of each order, one to the right, and the other at the same distance to the left of the direction of the source. In Angstrom's observations, 1 which are the best yet taken, all the spectra, up to the sixth inclusive, were observed, and numerous independent determinations of wave-length were thus obtained for several hundred of the dark lines of the solar spectrum. The source of light was the infinitely distant image of an illumi- nated slit, the slit being placed at the principal focus of a collimator, and illuminated by a beam of the sun's rays reflected from a mirror. The purity of a diffraction spectrum increases with the number of lines on the grating which come into play, provided that they are exactly equidistant ; and may therefore be increased either by in- creasing the size of the grating, or by ruling its lines closer together. The gratings employed by Angstrom were about f of an inch square, the closest ruled having about 4500 lines, and the widest 1500. As regards brightness, diffraction spectra are far inferior to those obtained by prisms. To give a maximum of light, the opaque inter- vals should be perfectly opaque, and the transparent intervals per- fectly transparent ; but even under the most favourable conditions, the whole light of any one of the spectra cannot exceed about -^ of the light which would be received by directing the telescope to the slit. The greatest attainable intrinsic brightness in any part of a diffraction spectrum is thus not more than -^ of the intrinsic brightness in the same part of a prismatic spectrum, obtained with the same slit, collimator, and observing telescope, and with the same angular separation of fixed lines. The brightness of the spectra partly depends upon the ratio of the breadths of the transparent and opaque intervals. In the case of the spectra of the first order, the best ratio is that of equality, and equal departures from equality in opposite directions give identical results ; for example, if the breadth 1 Angstrom, Recherches sur la Spectre Solaire. Upsal, 1868. STANDARD SPECTRUM. 1029 of the transparent intervals is to the breadth of the opaque either as 1 : 5 or as 5 : 1, it can be shown that the quantity of light in the first spectrum is just a quarter of what it would be with the breadths equal. When a diffraction spectrum is seen with the naked eye, the cornea and crystalline of the eye take the place of the lens L L, and form a real image on the retina at F. 823 A. Retardation Gratings. If, instead of supposing the bars of the grating to be opaque, we suppose them to be transparent, but to produce a definite change of phase either by acceleration or retarda- tion, the spectra produced will be the same as in the case above discussed, except as regards brightness. We may regard the effect as consisting of the superposition of two exactly coincident sets of spectra, one due to the spaces and the other to the bars. Any one of the resultant spectra may be either brighter or less bright than either of its components, according to the difference of phase between them. If the bars and spaces are equally transparent, the two super- imposed spectra will be equally bright, and their resultant at any part may have any brightness intermediate between zero and four times that of either component. 823 B. Reflection Gratings. Diffraction spectra can also be obtained by reflection from a surface of speculum metal finely ruled with parallel and equidistant scratches. The appearance presented is the same as if the geometrical image of the slit (with respect to the grating regarded as a plane mirror) were viewed through the grating regarded as transparent. 824. Standard Spectrum. The simplicity of the law connecting wave-length with position, in the spectra obtained by diffraction, offers a remarkable contrast to the "irrationality" of the dispersion produced \)y prisms. Diffraction spectra may thus be fairly regarded as natural standards of comparison ; and, in particular, the limiting form (if we may so call it) to which the diffraction spectra tend, as sin 6 becomes small enough to be identified with 6, so that devia- tion becomes simply proportional to wave-length, is generally and deservedly accepted by spectroscopists as the absolute standard of reference. This limiting form is often briefly designated as "the diffraction spectrum;" it differs in fact to a scarcely appreciable extent from the first, or even the second and third spectra furnished in ordinary cases by a grating. The diffraction spectrum differs notably from prismatic spectra in 1030 WAVE THEORY OF LIGHT. the much greater relative extension of the red end. Owing to this circumstance, the brightest part of the diffraction spectrum of solar light is nearly in its centre. The first three columns of numbers in the subjoined table indicate the approximate distances between the fixed lines B, D, E, F, G in certain prismatic spectra, and in the standard diffraction spectrum, the distance from B to G being in each case taken as 1000: B to D, . . D to E, . . E to F, . . F to G, . . Flint-glass. Angle of 60. Bisulphide of Carbon. Angle of 60. Diffraction, or Difference of Wave-length. Difference of Wave-frequency. 220 214 192 374 194 206 190 410 381 243 160 216 278 232 184 306 1000 1000 1000 1000 In the standard diffraction spectrum, deviation is simply propor- tional to wave-length, and therefore the distance between two colours represents the difference of their wave-lengths. It has been sug- gested that a more convenient reference -spectrum would be con- structed by assigning to each colour a deviation proportional to its wave-frequency (or to the reciprocal of its wave-length), so that the distance between two colours will represent the difference between their wave- frequencies. The result of thus disposing the fixed lines is shown in the last column of the above table. It differs from pris- matic spectra in the same direction, but to a much less extent than the diffraction spectrum. It has been suggested by Mr. Stoney as extremely probable, that the bright lines of spectra are in many cases harmonics of some one fundamental vibration. Three of the four bright lines of hydrogen have wave- frequencies exactly proportional to the numbers 20, 27, and 32; and in the spectrum of chloro-chromic acid all the lines whose positions have been observed (31 in number) have wave-frequencies which are multiples of one common fundamental. 825. Wave-lengths. Wave-lengths of light are commonly stated in terms of a unit of which 10 10 make a metre, hence called the tenth-metre. The following are the wave-lengths of some of the principal "fixed lines" as determined by Angstrom: 1 1 The wave-lengths of the spectral lines of all elementary substances will be found in Dr. W. M. Watts' Index of Spectra. NEWTON'S RINGS. 1031 WAVE-LENGTHS IN TENTH-METKE8. A ... 7604 E B 6867 F C 6562 G D 2 5895 H t . D! 5889 H a . 5269 4861 4307 3933 The velocity of light as determined by Cornu is 300 million metres per second, or 300 x 10 16 tenth-metres per second. The number of waves per second for any colour is therefore 300 x 1 16 divided by its wave-length as above expressed. Hence we find approximately: For A 395 millions of millions per second. D 510 H 760 826. Colours of Thin Films. Newton's Rings. If two pieces of glass, with their surfaces clean, are brought into close contact, coloured fringes are seen surrounding the point where the contact is closest. They are best seen when light is obliquely reflected to the eye from the surfaces of the glass, and fringes of the complementary colours may be seen by transmitted light. A drop of oil placed on the surface of clean water spreads out into a thin film, which exhibits similar fringes of colour; and in general, a very thin film of any transparent substance, separating media whose indices of refraction are different from its own, exhibits colour, especially when viewed by obliquely reflected light. In the first experiment above-mentioned, the thin film is an air-film separating the pieces of glass. In soap-bubbles or films of soapy water stretched on rings, a similar effect is produced by a small thickness of water separating two portions of air. The colours, in all these cases, when seen by reflected light, are produced by the mutual interference of the light reflected from the two surfaces of the thin film. An incident ray undergoes, as ex- plained in 729, a series of reflections and refractions; and we may thus distinguish, for light of any given refrangibility, several systems of waves, all of which originally came from the same source. These systems give by their interference a series of alternately bright and dark fringes; and when ordinary white light is employed, the fringes are broadest for the colours of greatest wave-length. Their super- position thus produces the observed colours. The colours seen by transmitted light may be similarly explained. The first careful observations of these coloured fringes were made by Newton, and they are generally known as Newton's rings. CHAPTER LXV. POLARIZATION AND DOUBLE REFRACTION. Fig. 729. Tourmaline Plates. 827. Polarization. When a piece of the semi-transparent mineral called tourmaline is cut into slices by sections parallel to its axis, it is found that two of these slices, if laid one upon the other in a particuiai relative position, as A, B (Fig. 729), form an opaque combination. Let one of them, in fact, be turned round upon the other through various angles (Fig. 729). It will be found that the combination is most transparent in two posi- tions differing by 180, one of them a b being the natural position which they originally occupied in the crystal; and that it is most opaque in the two positions at right angles to these. It is not necessary that the slices should be cut from the same crystal. Any two plates of tourmaline with their faces parallel to the axes of the crystals from which they were cut, will exhibit the same phenomenon. The experiment shows that light which has passed through one such plate is in a peculiar and so to speak unsymmetrical condition. It is said to be plane-polarized. According to the undulatory theory, a ray of common light contains vibrations in all planes passing through the ray, and a ray of plane- polarized light contains vibrations in one plane only. Polarized light cannot be distinguished from common light by the naked eye ; and for all experiments in polarization two pieces of apparatus must be employed one to produce polarization, and the other to show it. The former is called the polarizer, the latter the analyzer; and every apparatus that serves for one of these purposes will also serve for the other. In the experiment above described, the plate next the eye is POLARIZATION BY REFLECTION. 1033 the analyzer. The usual process in examining light with a view to test whether it is polarized, consists in looking at it through an analyzer, and observing whether any change of brightness occurs as the analyzer is rotated. When the light of the blue sky is thus examined, a difference of brightness can always be detected accord- ing to the position of the analyzer, especially at the distance of about 90 from the sun. In all such cases there are two positions; differing by 180, which give a minimum of light, and the two posi- tions intermediate between these give a maximum of light. The extent of the changes thus observed is a measure of the com- pleteness of the polarization of the light. 828. Polarization by Reflection. Transmission through tourmaline is only one of several ways in which light can be polarized. When a beam of light is reflected from a polished surface of glass, wood, ivory, leather, or any other non-metallic substance, at an angle of from 50 to 60 with the normal, it is more or less polarized, and in like manner a reflector composed of any of these substances may be employed as an analyzer. In so using it, it should be rotated about an axis parallel to the incident rays which are to be tested, and the observation consists in noting whether this rotation produces changes in the amount of reflected light. Malus' Polariscope (Fig. 730) consists of two reflectors A, B, one serving as polarizer and the other as analyzer, each consisting of a pile of glass plates. Each of these reflectors can be turned about a horizontal axis ; and the upper one (which is the analyzer) can also be turned about a vertical axis, the amount of rotation being mea- sured on the horizontal circle C C. To obtain the most powerful ef- fects, each of the reflectors should be set at an angle of about 33 to the vertical, and a strong beam of common light should be allowed to fall upon the lower pile in such a direction as to be reflected vertically upwards. It will thus fall upon the centre of the upper pile, and the angles of incidence and reflection on both the piles will be about 57. The observer looking into the upper pile, in such a r> Fig. 730. Mains' Polariscope. 1034 POLARIZATION AND DOUBLE REFK ACTION. direction as to receive the reflected beam, will find that, as the upper pile is rotated about a vertical axis, there are two positions (differing by 180) in which he sees a black spot in the centre of the field of view, these being the positions in which the upper pile refuses to reflect the light reflected to it from the lower pile. They are 90 on either side of the position in which the two piles are parallel ; this latter, and the position differing from it by 180, being those which give a maximum of reflected light. For every reflecting substance there is a particular angle of in- cidence which gives a maximum of polarization in the reflected light. It is called the polarizing angle for the substance, and its tangent is always equal to the index of refraction of the substance ; or what amounts to the same thing, it is that particular angle of incidence which is the complement of the angle of refraction, so that the refracted and reflected rays are at right angles. 1 This important law was discovered experimentally by Sir David Brewster. The reflected ray under these circumstances is in a state of almost complete polarization; and the advantage of employing a pile of plates consists merely in the greater intensity of the reflected light thus furnished. The transmitted light is also polarized ; it diminishes in intensity, but becomes more completely polarized, as the number of plates is increased. The reflected and the transmitted light are in fact mutually complementary, being the two parts into which common light has been decomposed ; and their polarizations are accordingly opposite, so that, if both the transmitted and reflected beams are examined by a tourmaline, the maxima of obscuration will be obtained by placing the axis of the tourmaline in the one case parallel and in the other perpendicular to the plane of incidence. It is to be noted that what is lost in reflection is gained in trans- mission, and that polarization never favours reflection at the expense of transmission. 829. Plane of Polarization. That particular plane in which a ray of polarized light, incident at the polarizing angle, is most copiously reflected, is called the plane of polarization of the ray. When the polarization is produced by reflection, the plane of reflection is the 1 Adopting the indices of refraction given in the table 724, we find the following values for the polarizing angle for the undermentioned substances : Diamond, . . Flint-glass, . Crown-glass, . 67 43' to 70 3' 57 36' to 58 40' 56 51' to 57 23' Pure Water, 53 11' Air, 45 THEORY OF DOUBLE REFRACTION. 1035 i plane of polarization. According to Fresnel's theory, which is that generally received, the vibrations of light polarized in any plane are perpendicular to that plane ( 841). The vibrations of a ray reflected at the polarizing angle are accordingly to be regarded as perpendi- cular to the plane of incidence and reflection, and therefore as parallel to the reflecting surface. 830. Polarization by Double Refraction. We have described in 732 some of the principal phenomena of double refraction in uniaxal crystals. We have now to mention the important fact that the two rays furnished by double refraction are polarized, the polarization in this case being more complete than in any of the cases thus far dis- cussed. On looking at the two images through a plate of tourmaline, or any other analyzer, it will be found that they undergo great varia- tions of brightness as the analyzer is rotated, one of them becoming fainter whenever the other becomes brighter, and the maximum brightness of either being simultaneous with the absolute extinction of the other. If a second piece of Iceland-spar be used as the analyzer, four images will be seen, of which one pair become dimmer as the other pair become brighter, and either of these pairs can be extin- guished by giving the analyzer a proper position. 831. Theory of Double Refraction. The existence of double refrac- tion admits of a very natural explanation on the undulatory theory. In uniaxal crystals it is assumed that the elasticity of the luminifer- ous aether is the same for all vibrations executed in directions perpen- dicular to the axis ; and that, for vibrations in other directions, the elasticity varies solely according to the inclination of the direction of vibration to the axis. There are two classes of doubly- refracting uniaxal crystals, called respectively positive and negative." In the former the elasticity for vibrations perpendicular to the axis is a maximum ; in the latter it is a minimum. Iceland-spar belongs to the latter class ; and as small elasticity implies slow propagation, a ray propagated by vibrations perpendicular to the axis will, in this crystal, travel with minimum velocity ; while the most rapid pro- pagation will be attained by rays whose vibrations are parallel to the axis. Consider any plane oblique to the axis. Through any point in this plane we can draw one line perpendicular to the axis ; and the line at right angles to this will have smaller inclination to the axis than any other line in the plane. These two lines are the directions of least and greatest resistance to vibration ; the former is the direc- 1036 POLARIZATION AND DOUBLE REFRACTION. tion of vibration for an ordinary, and the latter for an extraordinary ray. The velocity of propagation is the same for the ordinary rays in all directions in the crystal, so that the wave-surface for these is spherical ; but the velocity of propagation for the extraordinary rays differs according to their inclination to the axis, and their wave- surface is a spheroid whose polar diameter is equal to the diameter of the aforesaid sphere. The sphere and spheroid touch one another at the extremities of this diameter (which is parallel to the axis of the crystal), and the ordinary and extraordinary rays in this par- ticular direction coincide, so that the double refraction becomes single. The course of the two rays produced in the crystal b a given ray incident on a plane face, may be determined by Huygens' construc- tion, which has been described in 811. The ordinary index is the ratio of the velocity in air to the velocity of the ordinary ray. The extraordinary index (so called) is the ratio of the velocity in air to the velocity of the slowest extraordinary rays in the case of positive crystals, or to the velocity of the swiftest extraordinary rays in the case of negative crystals. In both cases the extraordinary index is ,, if sine of incidence i i i-/v> c ,-\ T that value of s i ne O f refraction wnic ' 1 differs most from the ordinary index. The extraordinary index is applicable to refraction at a plane surface parallel to the axis, when the plane of incidence is perpendicular to the axis. Tourmaline, like Iceland-spar, is a nega- tive uniaxal crystal ; and its use as a pol- arizer depends on the property which it possesses of absorbing the ordinary much more rapidly than the extraordinary ray, so that a thickness which is tolerably transparent to the latter is almost com- pletely opaque to the former. 832. Nicol's Prism. One of the most convenient and effective contrivances for polarizing light, or analyzing it when pol- arized, is that known, from the name of its inventor, as Nicol's prism. It is made by slitting a rhomb of Iceland-spar along a diagonal plane acbd (Fig. 731), and cementing the two pieces together in their natural position by Canada balsam, a substance whose refractive index S. o Fig. 731. Nicol's Priam. ELLIPTIC POLARIZATION. 1037 is intermediate between the ordinary and extraordinary indices of the crystal. 1 A ray of common light S I undergoes double refraction on entering the prism. Of the two rays thus formed, the ordinary ray is totally reflected on meeting the first surface of the balsam, and passes out at one side of the crystal, as o ; while the extraordinary ray is transmitted through the balsam as through a parallel plate, and finally emerges at the end of the prism, in the direction e E, parallel to the original direction SI. This apparatus has nearly all the convenience of a tourmaline plate, with the advantages of much greater transparency and of complete polarization. In Foucaillt's prism, which is extensively used instead of Nicol's, the Canada balsam is omitted, and there is nothing but air between the two pieces. This change has the advantage of shortening the prism (because the critical angle of total reflection depends on the relative index of refraction of the two media), but gives a smaller field of view, and rather more loss of light by reflection. 833. Colours produced by Elliptic Polarization. Very beautiful colours may be produced by the peculiar action of polarized light. For example, if a piece of selenite (crystallized gypsum) about the thickness of paper, is introduced between the polarizer and analyzer of any polarizing arrangement, and turned about into different directions, it will in some positions appear brightly coloured, the colour being most decided when the analyzer is in either of the two critical positions which give respectively the greatest light and the greatest darkness. The colour is changed to its complementary by rotating the analyzer through a right angle ; but rotation of the piece of selenite, when the analyzer is in either of the critical positions, merely alters the depth of the colour without changing its tint, and in certain critical positions of the selenite there is a complete absence of colour. Thicker plates of selenite restore the light when ex- tinguished by the analyzer, but do not show colour. 834. Explanation. The following is the explanation of these appearances. Let the analyzer be turned into such a position as to produce complete extinction of the plane-polarized light which comes to it from the polarizer; and let the plane of polarization and the plane perpendicular thereto (and parallel to the polarized rays) be 1 a and b are the corners at which three equal obtuse angles meet ( 733). The ends of the rhomb which are shaded in the figure are rhombuses. Their diagonals drawn through a and 6 respectively will lie in one plane, which will contain the axis of the crystal, and will cut the plane of section a c b d at right angles. The length of the rhomb is about three and a half times its breadth. 1038 POLARIZATION AND DOUBLE REFRACTION. called the two planes of reference. Let the slice of selenite be laid so that the polarized rays pass through it normally. Then there are two directions, at right angles to each other, which are the directions of greatest and least elasticity in the plane of the slice. Unless the slice is laid so that these directions coincide with the two planes of reference, the plane -polarized light which is incident upon it will be broken up into two rays, one of which will traverse it more rapidly than the other. Referring to the diagram of Lissajous' figures (Fig. 604), let the sides of the rectangle be the directions of greatest and least elasticity, and let the diagonal line in the first figure be the direction of the vibrations of an incident ray, this diagonal accord- ingly lies in one of the two planes of reference. In traversing the slice, the component vibrations in the directions of greatest and least elasticity will be propagated with unequal velocities ; and if the incident ray be homogeneous, the emergent light will be elliptically polarized ; that is to say, its vibrations, instead of being rectilinear, will be elliptic, precisely on the principle 1 of Blackburn's pendulum ( 677 A). The shape of the ellipse depends, as in the case of Lis- sajous' figures, on the amount of retardation of one of the two com- ponent vibrations as compared with the other, and this is directly proportional to the thickness of the slice. The analyzer resolves these elliptic vibrations into two rectilinear components parallel and per- pendicular to the original direction of vibration, and suppresses one. of these components, so that only the other remains. Thus if the ellipse in the annexed figure (Fig. 732) represent the vibrations of the light as it emerges from the selenite, and CD, EF be tan- gents parallel to the original direction of vibration, the perpendicular distance between these tangents, AB, is the component vibration which is not sup- Fig. 732. colours of pressed when the analyzer is so turned that all the Selenite Plates. f. J , , light would be suppressed if the selenite were re- moved. By rotating the analyzer, we shall obtain vibrations of vari- ous amplitudes, corresponding to the distances between parallel tan- gents drawn in various directions. For a certain thickness of selenite the ellipse will become a circle, 1 The principle is that, whereas displacement of a particle parallel to either of the sides of the rectangle calls out a restoring force directly opposite to the displacement, displace- ment in any other direction calls out a restoring force inclined to the direction of displace- ment, being in fact the resultant of the two restoring forces which its two components parallel to the sides of the rectangle would call out. COLOURS OF SELENITE PLATES. 1039 arid we have thus what is called circularly-polarized light, which is characterized by the property that rotation of the analyzer produces no change of intensity. Circularly-polarized light is not however identical with ordinary light; for the interposition of an additional thickness of selenite converts it into elliptically (or in a particular case into plane) polarized light ( 840). The above explanation applies to homogeneous light. When the incident light is of various refrangibilities, the retardation of one component upon the other is greatest for the rays of shortest wave- length. The ellipses are accordingly different for the different elemen- tary colours, and the analyzer in any given position will produce unequal suppression of different colours. But since the component which is suppressed in any one position of the analyzer, is the com- ponent which is not suppressed when the analyzer is turned through a right angle, the light yielded in the former case plus the light yielded in the latter must be equal to the whole light which was incident on the selenite. 1 Hence the colours exhibited in these two positions must be complementary. It is necessary for the exhibition of colour in these experiments that the plate of selenite should be very thin, otherwise the retarda- tion of one component vibration as compared with the other will be greater by several complete periods for violet than for red, so that the ellipses will be identical for several different colours, and the total non-suppressed light will be sensibly white in all positions of the analyzer. Two thick plates may however be so combined as to produce the effect of one thin plate. For example, two selenite plates, of nearly equal thickness, may be laid one upon the other, so that the direc- tion of greatest elasticity in the one shall be parallel to that of least elasticity in the other. The resultant effect in this case will be that due to the difference of their thicknesses. Two plates so laid are said to be crossed. 835. Colours of Plates perpendicular to Axis. A different class of appearances are presented when a plate, cut from a uniaxal crystal by sections perpendicular to the axis, is inserted between the polar- izer and the analyzer. Instead of a broad sheet of uniform colour, 1 We here neglect the light absorbed and scattered; but the loss of this does not sensibly affect the colour of the whole. It is to be borne in mind that the intensity of light is mea- sured by the square of the amplitude, and is therefore the simple sum of the intensities of its two components when the resolution is rectangular. 1040 POLARIZATION AND DOUBLE REFRACTION. we have now a system of coloured rings, interrupted when the analyzer is in one of the two critical positions, by a black or white cross, as at A, B (Fig. 733). 836. Explanation. The following is the explanation of these ap- Fifc. 733. Rings and Cross. pearances. Suppose, for simplicity, that the analyzer is a plate of tourmaline held close to the eye. Then the light which comes to the eye from the nearest point of the plate under examination (the foot of a perpendicular dropped upon it from the eye), has traversed the plate normally, and therefore parallel to its optic axis. It has therefore not been resolved into an ordinary and an extraordinary ray, but has emerged from the plate in the same condition in which it entered, and is therefore black, gray, or white according to the position of the analyzer, just as it would be if the plate were re- moved. But the light which comes obliquely to the eye from any other part of the plate, has traversed the plate obliquely, and has undergone double refraction. Let E (Fig. 734) be the position of the eye, E O a perpendicular on the plate, P a point on the circumference of a circle described about as centre. Then, since E is parallel to the axis of the plate, the direc- tion of vibration for the ordinary ray at P is perpendicular to the plane E P, and is tangen- tial to the circle. The direction of vibration for the extraordinary ray lies in the plane E P, is nearly perpendicular to E (or to the axis), if the angle E P is small, and deviates more from perpendicularity to the axis as the angle E P increases. Both for this reason, and also on account of the greater thickness traversed, the retardation of one ray upon the other is greater as P is taken further from 0; and from the symmetry of the circumstances, it Fig. 734. Theory of Rings and Cross. THEORY OF RINGS AND CROSS. 1041 must be the same at the same distance from all round. In con- sequence of tliis retardation, the light which emerges at P in the di- rection P E is elliptically polarized ; and by the agency of the analyzer it is accordingly resolved into two components, one of which is sup- pressed. With homogeneous light, rings alternately dark and bright would thus be formed at distances from corresponding to retarda- tions of 0, |, 1, 1, 2, 2|, . . . complete periods; and it can be shown that the radii of these rings would be proportional to the numbers 0, VI, V2, V3, V4, Vo, V6: . . The rings are larger for light of long than of short wave-length ; and the coloured rings actually exhibited when white light is employed, are produced by the superposition of all the systems of monochromatic rings. The monochromatic rings for red light are easily seen by looking at the actual rings through a piece of red glass. Let 0, P, Fig. 735, be the same points which were denoted by these letters in Fig. 734, and let A B be the direction of vibration of the light incident on the crystal at P. Draw A C, D B parallel to O P, and complete the rectangle A C B D. Then the length and breadth of this rect- angle are approximately the direc- tions of vibration of the two com- Fig 735 ._ Theory of Ringa and Cro88 . ponents, one of which loses upon the other in traversing the crystal. The vibration of the emergent ray is represented by an ellipse inscribed in the rectangle A C B D ( 676, note 2) ; and when the loss is half a period, this ellipse shrinks into a straight line, namely, the diagonal CD. Through C and D draw lines parallel to AB; then the distance between these parallels represents the double amplitude of the vibration which is trans- mitted when there has been a retardation of half a period, and is greater than the distance between the tangents in the same direc- tion to any of the inscribed ellipses. A retardation of another half period will again reduce the inscribed ellipse to the straight line A B, as at first. The position D C corresponds to the brightest and A B to the darkest part of any one of the series of rings for a given wave-length of light, the analyzer being in the position for sup- pressing all the light if the crystal were removed. When the analyzer is turned into the position at right angles to this, A B corresponds to the brightest, and D C to the darkest parts of the rings. It is to 67 1042 POLARIZATION AND DOUBLE REFRACTION. be remembered that amount of retardation depends upon distance from the centre of the rings, and is the same all round. The two diagonals of our rectangle therefore correspond to different sizes of rings. If the analyzer is in such a position with respect to the point P considered, that the suppressed vibration is parallel to one of the sides of the rectangle (in other words, if O P, or a line perpendi- cular to P, is the direction of suppression) the retardation of one component upon the other has no influence, inasmuch as one of the two components is completely suppressed and the other is completely transmitted. There are, accordingly, in all positions of the analyzer, a pair of diameters, coinciding with the directions of suppression and non-suppression, which are alike along their whole length and free from colour. Again if P is situated at B or at 90 from B, the corner C of the rectangle coincides with B or with A, and the rectangle, with all its inscribed ellipses, shrinks into the straight line AB. The two diameters coincident with and perpendicular to A B are therefore alike along their whole length and uncoloured. The two colourless crosses which we have thus accounted for, one of them turning with the analyzer and the other remaining fixed with the polarizer, are easily observed when the analyzer is not near the critical positions. In the critical positions, the two crosses come into coincidence ; and these are also the positions of maximum black- ness or maximum whiteness for the two crosses considered separ- ately. Hence the conspicuous character of the cross in either of these positions, as represented at A, B, Fig. 733. As the analyzer is turned away from these positions, the cross at first turns after it with half its angular velocity, but soon breaks up into rings, some- what in the manner represented at C, which corresponds to a posi- tion not differing much from A. 837. Biaxal Crystals. Crystals may be divided optically into three classes: 1. Those in which there is no distinction of different directions, as regards optical properties. Such crystals are said to be optically isotropic. 2. Those in which the optical properties are the same for all direc- tions equally inclined to one particular direction called the optic axis, but vary according to this inclination. Such crystals are called uniaxal. BIAXAL CRYSTALS. 1043 3. All remaining crystals (excluding compound and irregular for- mations) belong to the class called biaxal. In any homogeneous elastic solid, there are three cardinal directions called axes of elasti- city, possessing the same distinctive properties which belong to the two principal planes of vibration in Blackburn's pendulum ( 677 A) ; that is to say, if any small portion of the solid be distorted by for- cibly displacing one of its particles in one of these cardinal directions, the forces of elasticity thus evoked tend to urge the particle directly back ; whereas displacement in any other direction calls out forces whose resultant is generally oblique to the direction of displacement, so that when the particle is released it does not fly back through the position of equilibrium, but passes on one side of it, just as the bob of Blackburn's pendulum generally passes beside and not through the lowest point which it can reach. In biaxal crystals, the resistances to displacement in the three cardinal directions are all unequal ; and this is true not only for the crystalline substance itself, but also for the luminiferous aether which pervades it, and is influenced by it. 1 The construction given by Fresnel for the wave-surface in any crystal is as follows: First take an ellipsoid, having its axes parallel to the three cardinal direc- tions, and of lengths depending on the particular crystalline sub- stance considered. Then let any plane sections (which will of course be ellipses) be made through the centre of this ellipsoid, let normals to them be drawn through the centre, and on each normal let points be taken at distances from the centre equal to the greatest and least radii of the corresponding section. The locus of these points is the complete wave-surface, which consists of two sheets cutting one another at four points. These four points of intersection are situated upon the normals to the two circular sections of the ellipsoid, and the two optic axes, from which biaxal crystals derive their name, are closely related to these two circular sections. The optic axes are the directions of single wave-velocity, and the normals to the two circular sections are the directions of single ray-velocity. The direction of advance of a wave is always regarded as normal to the front of the wave, whereas the direction of a ray (defined by the condition of traversing two apertures placed in its path) always passes through the centre of the wave- surface, and is riot in general normal to the front. Both these pairs of directions of single velo- 1 The cardinal directions are however believed not to be the same for the aether as for the material of the crystal. 1044 POLARIZATION AND DOUBLE REFRACTION. city are in the plane which contains the greatest and least axes of the ellipsoid. When two axes of the ellipsoid are equal, it becomes a spheroid, and the crystal is uniaxal. When all three axes are equal, it be- comes a sphere, and the crystal is isotropic. Experiment has shown that biaxal crystals expand with heat unequally in three cardinal directions, so that in fact a spherical piece of such a crystal is changed into an ellipsoid 1 when its tem- perature is raised or lowered. A spherical piece of a uniaxal crystal in the same circumstances changes into a spheroid ; and a spherical piece of an isotropic crystal remains a sphere. It is generally possible to determine to which of the three classes a crystal belongs, from a mere inspection of its shape as it occurs in nature. Isotropic crystals are sometimes said to be symmetrical about a point, uniaxal crystals about a line, biaxal crystals about neither. The following statement is rather more precise : If there is one and only one line about which if the crystal be rotated through 90 or else through 120 the crystalline form remains in its original position, the crystal is uniaxal, having that line for the axis. If there is more than one such line, the crystal is isotropic, while, if there is no such line, it is biaxal. Even in the last case, if there exist a plane of crystalline symmetry, such that one half of the crystal is the reflected image of the other half with respect to this plane, it is also a plane of optical symmetry, and one of the three cardinal directions for the aether is perpendicular to it. 2 Glass, when in a strained condition, ceases to be isotropic, and if inserted between a polarizer and an analyzer, exhibits coloured streaks or spots, which afford an indication of the distribution of strain through its substance. The experiment is shown sometimes with unannealed glass, which is in a condition of permanent strain, sometimes with a piece of ordinary glass which can be subjected to force at pleasure by turning a screw. Any very small portion of a piece of strained glass has the optical properties of a crystal, but different portions have different properties, and hence the glass as a whole does not behave like one crystal. The production of colour by interposition between a polarizer and 1 This fact furnishes the best possible definition of an ellipsoid for persons unacquainted with solid geometry. 2 The optic axes either lie in the plane of symmetry, or lie in a perpendicular plane and are equally inclined to the plane of symmetry. For the precise statement here given, the Editor is indebted to Professor Stokes. ROTATION OF PLANE OF POLARIZATION. 1()4 promoted by presence of air, 341- Boiling points, affected by pres- sure, 336. heights determined by, 338. of solutions, 340. table of, 335. Boreal pole, 616. Bottle, inexhaustible, 232. Mariotte's, 239. Bourbouze's apparatus for falling bodies, 46. electro-magnetic engine, 711. Bourdon's gauge, 181. Boutigny's experiments, 345. Boyle's law, 170-182. Bramah press, 224. Breezes, land and sea, 499. Breguet's telegraph, 718. thermometer, 261. Bridge, Wheatstone's, 674. Brightness, 965-970. intrinsic and effective, 966. of spectra, 992. Bright spot behind eyepiece, 960. British Association unit of resist- ance, 760. Brocot's pendulum, 273. Broken magnet, 618. Brush, electric, 548. Bubbles, filled with hydrogen, 209. pressure in, 132. Bucket, electric,558. Bunsen & Kirchhoff's researches, 441- Bunsen's cell, 650. Buoyancy, centre of, 105. Burning mirrors, 392. Bursting of boilers, 483. Buys Ballot's experiment on sound, 827. law, 169. {"""AGE electrometer, 597. Cagniard de Latour*s experi- ments on vaporization, 325. siren, 822. Caissons, 206. Calibration, 245. of thermo-multiplier, 664. Calorescence, 410. Caloric theory, 445. Calorimeter, 430. Calorimetry, 426-444. Camera lucida, 914. obscura, 942. photographic, 943. Camphor, movements of, 137. Canton's phosphorus, 979. Capacity, electric, 565. of condenser, 568. specific inductive, 576. Capacity, thermal, 427. Capillarity, 127-138. Carbonic acid, solidification of, 333- Carbon melted, 703. points, image of, 704. Carnot's principles, 454. Carre's two freezing apparatus, 329, 332. Cartesian diver, 108. Cascade, charge by, 582 (zd edi- tion). Caselli's telegraph, 730. Cassegranian telescope, 965. Cathetometer, 146. Cathode, 739. Caustics, 901, 917, 1017. Cavendish experiment, 67. Cells.arrangement of, for maximum current, 671. Centesimal alcoholimeter, 119. Centigrade scale, 250. Centre of buoyancy, 105. of gravity, 33-39. of inertia, 72. - of lens, 931. of mass, 72. of mirror, 894. of oscillation, 60. of parallel forces, 17. of percussion, 76. Centrifugal force, 62. pump, 222 (sd edition). theory of atmospheric circula- tion, 501. Character of a musical note, 817, 853- Charge by cascade, 582 (2d edition). residual, 572. Charts of magnetic lines, 631. of weather, 168. Chemical action necessary to cur- rent, 652. combination, 442, 462, 435. harmonica, 789. hygrometer, 373. Cherra Ponjee, rainfall at, 380. Chimes, electric, 600. Chimneys, draught of, 298. Chromatic aberration, 792. Chromosphere, 989. Circular polarization, 1039, 1047. Clarke's machine, 767. Clearance, see Untraversed Space, 193. Climates, insular and continental, 495- Clink accompanying magnetiza- tion, 638. Clocks, electrically controlled, 736. Clothing, warmth of, 423. Clouds, 377-380. Coal, origin of, 462. Coatings, jar with movable, 573. Coefficient of expansion, 264. Coercive force, 617. Coil, Ruhmkorff's induction, 761. Cold of evaporation, 328. Colladon's experiment at Lake of Geneva, 803, 867. Collimation, line of, 971. Collimator of spectroscope, 983. 985. Colloids, 139. Colour, looo-ioii. and music, 1010. blindness, 1010. by polarized light, 1037-1045, cone, 1007. equations, 1004. mixture of, 1002-1008. of thin films, 1030. Combination, heat of, 442 Combustion, heat of, 443. table, 444. Comma, 821. Commutator, 763. Compass, ship's, 634. Compensated pendulums, 271. Complementary colours, 1008. Compound engines, 477. magnet, 621, 637. Compressed-air engines, 207. Compressibility, 25. of water, 26. Concave mirrors, 893-904. Concord, 859. Condensation, 322. Condenser of steam-engine, 474. Condensers, electric, 567. capacity of, 568. discharge of, 569. Condensing electroscope, 579. power, 574. pump, 202. Conduction of heat, 414-425. in gases, 423-425. in liquids, 421-423. Conductivity, comparison of ther- mal and electrical, 670. defined, 415. determinations of absolute, 420- 421. electrical, see Resistance. table of, 420. Conductors, electrical, list of, 507. lightning, 601-603. Cone of colour, 1007. Congelation, 306. Conjugate foci, 894, 932. mirrors, 393, 807. Consequent points, 636. Conservation of energy, 79. motion of centre of mass, 74. Constitution of compound vibra- tions, 854. Contact-electricity, note on, 784 (2d edition). Contiguous particles, induction by, S'S. 578. Continental climates, 495. Continuity of gaseous and liquid states, 325. Contracted vein, 229. Contractile force in liquids, 131. Convection of heat, 284. of electricity, 531, 604. Convertibility of centres in pendu- lum, 60. Convex mirrors, 905. Cooling, law of, 386. of air by ascent, 498. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IV., p. 785-1050. INDEX. 1003 Copper-cube experiment, 777. Depressions, capillary, 128. Duboscq's regulator, 706. Cornu's experiments on velocity of De Saussure's hygrometer, 366. Dufour's experiment, 342. light, 875. Despretz's experiments on Boyle's Duhamel's vibroscope, 824. Corti's organ, 861. law, 172. Dulong & Petit's law, 435. Coulomb's torsion-balance.519,624. on alcohol at low tempera- law of cooling, 388. Counterpoised barometer, 159. tures, 333. Dumas' method for vapour den- Couples, 16. on heat of voltaic arc, 703. sities, 359. Couronne de tasses, 646. Deviation, constructions for, Dynamics of rigid bodies, 7277. Critical angle, 912. 9 2 3 9 2 4- Dynamometer, 30. temperature, Andrews', 327. by rotation of mirror, 892. Dynamo-electric machines, see Cross-wires of telescope, 971. minimum, 97 3-925. Accumulation by Mutual Ac- Cruickshank's trough, 647. Dew, 412. tion. Cryophorus, 330. point, 365. Crystallization, 307. computation of, 371. ft AR, how affected by discord, Crystalloids, 139. Dial telegraphs, 718, 722. ^ 861. Crystals, optical, classification of, Dialysis, 139 (36 edition). Earth, action of, on currents, 689. 1042. Diamagnetic bodies, 638 ; their as a magnet, 632. Cup-leathers, 225. coefficient of induction nega- mean density of, 67. Current, deflected by magnetic tive, 781. Earth-currents, 634. force, 658. Diathermancy, 405. Ebullition, 334. direction of, in battery, 643. table of, 406. Eccentric of slide-valve, 473. induced by motion across lines Dielectric, influence of, 575. Echo, 808. of force, 752760. polarization of, 578. Edison's phonograph, 864. numerical estimate of, 658. Difference-tones, 862. Efficiency of engines, 710 ; of Currents, marine, 284. Differential galvanometer, 661. pumps, 218; of thermic engine, Curvature in connection with ca- thermometer, 263. 453; reversible, 454. pillarity, 128, 133. Difficulty of commencing change Efflux of liquids, 226. of rays in air, 1019. of state, 306, 343. Elasticity, 27. Cushions of electrical machine, Diffraction, 1013. Young's modulus of, 29. 535, 536- by grating, 1025. Electrical force at a point defined, Cycloidal pendulum, 71. fringes, 1024. 559- Cyclones, 502, 611. spectrum, 1025. machines, 533, see Machine. Cylindric mirror, 906. Diffusion, 139. Electric chimes, 600. Digester, Papin's, 339. egg, 55- T~)ALTON'S experiments on Dimensions of units, 779. light, 702, 769. vapours, 349. Dip, 615. pendulum, 509. laws of vapours, 322. Dip-circle, 628. spark, 546, see Spark. Dampers, copper, 777. Direction of vibration in polarized telegraph, 713-736. Daniell's battery, 649. light, 1048. whirl, 558. hygrometer, 368. Discharge in rarefied gases, 549- Electricity, 505. Dark ends of spectrum, 978. 552, 765- atmospheric, 599. lines in spectrum, 978. Discharger, jointed, 569. voltaic, 642. Davy lamp, 418. universal, 584. Electrodes of battery, 644, 739. - on friction of ice, 447. Discord, 859. Electro-dynamics, 680. Dead points, 472. Dispersion, chromatic, 973. gilding, 746. Declination magnet, 626. in spectroscope, 992. magnetic engines, 710. - magnetic, 615. Displacement of spectral lines by magnets, 697. changes of, 633. motion, 991. -medical machines, 778. theodolite, 626. Dissipation of charge, 531. motors, 710. Deep-water thermometers, 258. of energy, 466. Electrolysis, 738-744. Deflagrator, Hare's, 648. of sonorous energy, 799. Electrolytes, conduction in, 746. Degree of thermometer, 250. Distance, adaptation ot eye to. Electrometer, absolute, 592. physical meaning of, 251. 948. attracted disc, 591. Delezenne's circle, 759. judgment of, 949. cage, 597. Delicacy of thermometer, 252. Distillation, 347. portable, 593. Density, 85. Distribution of electricity on con- quadrant, 595. by hydrometers, 114. ductors, 528. Electrometers, 591-598 by specific gravity bottle, 88. Diurnal variations defined, 165. Electro-motive force, 665, 677. by weighing in water, 113. Diver, Cartesian, 108. its value for different bat- correction of, for temperature Divided circuits, 673. teries, 679. 266. Divisibility, 23. Electrophorus, 544. electric, 528. Donny's experiment, 341. Electro-plating, 746. of air, 141, 296, 375. Doppler's principle, 991. Electroscope, 517. of gases, 294-297. Double-action air-pump, 189. Bohnenberger's, 652. table of, 297. water-pump, 222. condensing, 579. of mixtures, 120. Double refraction, 925, 1035. Electrotype, 747. of vapours, 357-363. Doubly-exhausting air-pump, 199. Elementary tones, 863. table of, 88. Draught of chimneys, 298. Elements of currents, mutual ac- see Air, Vapour, Earth. Orion's experiments, 326. tion of, 688. Depolarization, see Elliptic Polar- Dry pile, 651. Ellicott's pendulum, 272. ization. Duality of electricity, 508. Ellipsoid, 1044. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IV., p. 785-1050. 1064- INDEX. Ellipsoid, distribution of electri- Films, colours of, 1030. Friction in connection with con- city on, 529. tension in, 131-138. servation of energy, 79. Elliptic polarization, 1037. Fire-engine, 221. Fringes, diffraction, 1024. Elmo's fire, St., 602. Fizeau's measurement of velocity Frog, experiment with, 645. Emissive power, 394. of light, 873. Froment's engine, 712. Endosmose, 138. Flames, manometric, 846, 857. Frost, hoar, 413. Energy, available sources of, 465. singing, 789. Fuse, Statham's, 764. conservation of, 79. Flexure, resistance to, 29. Fusion, 302. dissipation of, 466. Floating, conditions of, 107. latent heats of, 439. of motion, 76. Floating bodies, attraction be- temperatures of, 302. of position, 78. tween, 136. of rotation, 75. of sonorous vibrations, 799. stability of, 109. Floating needles, 1 10. QALILEAN telescope, 962. Galileo's experiments on fall- transformation of, 79. Flowers of ice, 308. ing bodies, 40. Engines, thermic, 453, see Steam- Flue-pipe, 837. explanation of suction -pump. engine. Fluids, 21. 144. Equipotential surfaces, 561. electric, theories of, 510. Galvani, 644. Equivalent simple pedulum, 60. imaginary magnetic, 618. Galvanic battery, 644. Equivalents of heat and work, 449. Fluorescence, 980, 410. electricity, 642. Errors and corrections, signs of, Flute mouthpiece, 837. Galvanometers, 659-664. !53- Fly-wheel, 75, 474. choice of, 677. Evaporation, 317. Focal lines, 903. Gamut, 819. cold of, 328. Foci, conjugate, 894, 932. Gas-battery, 745. latent heat of, 441. explained by wave theory, 1017. engine, 490. Exchanges, theory of, 396. primary and secondary, 901. Gases distinguished from liquids, Exhaustion, calculation of, 185. principal, 893, 930. 21. limit of, 193. virtual, 897. table of densities of, 297. Expansion, apparent and real, of Focometer, 940. their expansion by heat, 287. liquids, 275. Forbes" experiments on conduc- their tendency to expand, 22. by heat, 242, 264. tivity, 420. two specific heats of, 435, 451. coefficient of, 264. observations on glacier motion, Gauss' unit of force, 54. cubic and linear, 265. SM- Gay-Lussac's experiments on ex- force of, 273. Force, n. pansion of gases, 287. formulae relating to, 264. lines and tubes of, 560-563. method for vapour densities, heat lost in, 431. their movement, 757. 362. in freezing, 311. their relation to induced Geissler's air-pump, 195 ; tubes, 765. linear, modes of observing, 269. currents, 754-760. Giffard's injector, 485. table of, 270. unit of, 54, 780. Gimbals, 634, 149. of gases, 287. Force-pump, 220. Glaciers, motion of, 314. table of, 292. Fortin's barometer, 147. Glaisher's balloon-ascents, 497. of liquids, table of, 277, 280. Foucault's experiment on velocity tables, 371. of mercury, 280. of light, 875, 1016. Glass, expansion of, 276. Expansion-factor, 264. magneto -thermic experiment, strained, exhibits colours, 1044. Expansive working in steam-en- 448. Gold-leaf electroscope, 517. gine, 476. prism, 1037. Governor balls, 474. Explosion of boilers, 484. regulator, 707. Gradient, barometric, 168. Extra current, 761. Fountains, 230. Gramme's machine, 773*. Extraordinary index, 927. intermittent, 233. Gramme-degree, 427. rays, 927, 1035. in vacuo, 192. Graphical interpolation, 120. Eye, 946. Fourier's theorem, 853. Gratings for diffraction, 1026. Eye-pieces, 996. Franklin's experiment on ebulli- photographic, 1026. tion, 337. reflection, 1029. pAHRENHEIT'S barometer, on lightning, 599. Fraunhofer's lines, 978. retardation, 1029. Gravesande's apparatus, 13. hydrometer, 116. Free-piston air-pump, 200. Gravitation, universal, 66. scale of temperature, 250. Free-reed, 845. Gravity, centre of, 33. Falling bodies, laws of, 49. Freezing at abnormally low tem- formula for variation, 61. Fall in vacuo, 41. peratures, 306, 460. proportional to mass, 55. Faraday's experiments within elec- by evaporation, 328-333. terrestrial, 31. trified box, 527. by the spheroidal state, 345. Gregorian telescope, 964. on liquefaction of gases, 323. expansion in, 311. Gridiron-pendulum, 271. on solidification of gases, 333. mercury in red-hot crucible,345. Grotthus' hypothesis, 739. views regarding electro-static mixtures, 305. Grove's battery, 651. induction, 578, 515. Freezing-point lowered by pres- Gulf stream, 285. Favre and Silbermann's calori- sure, 312. meter, 442. ; computation, 459. tTADLEY'S sextant, 892. Field, magnetic, 620. by stresses, 313. 1 L Hail, 383. intensity of, 620. Frequency, 817. Volta's theory of, 610. uniform, 757. Fresnel's rhomb, 1047. Hare's deflagrator, 648. Filings, lines formed by, 612. wave-surface, 1043. Harmonics, 832, 854, see Over- Film of air, adherent, 183. Friction, heat of, 446. tones. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IV., p. 785-1050. INDEX. 1065 Harrison's gridiron-pendulum, 271. Index of refraction, table of, 912. Leslie's differential thermometer, Head, 226, 298. of air, 1019. 263. Heat, effects of, on magnets, 638. Induced currents, 750-760. experiment (freezing by evapo- mechanical equivalent of, 449. Induction coil, 761. ration), 328. of combustion, table of, 444. electro-static, 513-527. Levelling, 124. polarization of, 1049. its relation to force-tubes, corrections in, 1018. produced by discharge of Ley- 563- Lever, 15. den jars, 584, 590. magnetic,6i7, coefficient of,78i. Leyden battery, 580. by electric currents, 699. Inductive capacity, specific, 576. jar, 571. quantity of, 426. Inertia, 9. capacity of, 568. required foracyclicchange,459. Inexhaustible bottle, 232. with movable coatings, 573. for change of volume and Ingenhousz's experiment, 416. Lichtenberg"s figures, 581. temperature, 457. Injector, Giffard's, 485. Light, 865-1050. units, 427. Insects walking on water, no. electric, 702. Heating by hot water, 28 1 . Insular climates, 495. for lighthouses, 769. Heights measured by barometer, Insulators, list of, 507. Lightning, 599. 162. Intensity, horizontal, vertical, and conductors, 601. by boiling point, 338. total, 623. duration of, 600. Heliostat, 977. of field, 620, 781. Limma, 819. Helmholtz's colour -observations. of magnetization, 621, 781. Linear dimensions, in sound, 836, 1004. Interference, 810, see Diffraction. 839. resonators, 856. Intervals, musical, 818. Line of collimation, 971. theory of dissonance, 860. Iodine, solution of, in bisulphide Lines, isoclinic, isodynamic, iso- Hemispheres, Magdeburg, 191. of carbon, 410. gonic, 632. Herschelian telescope, 963. Isobaric lines and charts, 168. Lines of force, 560. High-pressure engines, 478. Isochronism, condition of, 70. caution regarding, 778. Him on animal heat, 461. of pendulum, 58. due to current, 657, 689. Hoar-frost, 413. Isoclinic and other magnetic lines, magnetic, 619. Holtz's electrical machine, 541. 632. shown by filings, 613. Homogeneous atmosphere, height Isothermal lines, 494. Link-motion, 489. of, 162. Liquefaction of gases, 322-328. Hope's experiment, 279. T ET-PUMP, 223 ( 3 d edition). of solids, see Fusion. Horse-power, 19. J Jets, liquid, 227. Liquefiable and non - liquefiable Houdin's regulator, 708. Jones' controlled clocks, 736. gases, 173. Howard's cloud nomenclature's. Joule's equivalent, 450, 452. Liquid and gaseous states continu- Hughes' printing telegraph, 726. experiment in stirring water,44g. ous, 325-328. Humidity of air, 364. law for energy of current, 699- Liquids, 21. Huygens' construction for wave- 702. Lissajous" curves, 849. front, 1014. Jupiter's satellites, eclipsesof, 878. equations to, 850. principle, 1012. experiments, 847. Hydraulic press, 93, 224. tourniquet, 101. L^ALEIDOSCOPE, 890. Rater's pendulum, 60. Local action, 651. Locomotive, 486. Hydro-electric machine, 539. Key, Morse's telegraphic, 724. Lodestone, 612. Hydrogen, conductivity of, 425. Kienmayer's amalgam, 536. Longitudinal vibrations of rods heat of combustion of, 444. King's barograph, 160. and strings, 843. soap-bubbles filled with, 209. Kinnersley's thermometer, 555. Looking-glasses, 886. Hydrometers, 113-121. Konig's manometric flames, 846 I-oudness, 816. Hygroscopes and hygrometers, 857. Luminiferous aether, 865, 1043. 365-374- Kravogl's air-pump, 194. Lycopodium on vibrating plate. Hypsometer, 338. 788. Hypsometry, 163. T ADD'S machine, 774. Land and sea breezes, 495. 1V/T ACHINE, electrical, 531. TCE-CALORIMETER, 429. flowers, 308. Lantern, magic, 945. Laplace and Lavoisier's experi- Bertsch's, 545. Guericke's, 533. pail experiment, 526, 564. ments, 269. Holtz's, 541. regelation of, 314. Laplace's correction of sound-velo- Nairne's, 537. Iceland-spar, 92^. city, 803. Ramsden's, 535. Images, accidental, 1009. Laryngoscope, 907. Winter's, 538. brightness of, 967. Latent heat of fusion, 303. hydro - electric, Armstrong's, electric, 566. of steam, 441. 539- formation of, 898. of vaporization, 328. Machines, magneto-electric, 766- in mid air, 901. of water, 304. 774- on screen, 900. below freezing-point, 460. Magdeburg hemispheres, 191. produced by small apertures, Latitude, 33. Magic funnel, 232. 868. its influence on gravity, 61. lantern, 945. size of, 898, 937. Least time, principle of, 1016. Magnet, ideal simple, 620. Imaginary magnetic matter, 6iq. Lcidenfrost's phenomenon, 345. moment of, 621, 622. Inclination, magnetic, 615. Lenses, 929. natural, 612. Inclined plane, 41. centre of lens, 931. Magnetic attraction and repulsion, Index errors, 153. Lenz's law, 753. 619. Index of refraction, 912. Le Roy's hygrometer, 367 charts, 631. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IV., p. 785-1050. 1066 INDEX. Magnetic curves formed by filings, 613. fluids, imaginary, 618. meridian, 615, 631. potential, 619. storms, 634. variations, 633. Magnetism, remanent or residual, 698. Magnetization, methods 0^635,696. specification of, 619. Magneto-crystallic action, 640. -electric machines, 766-775. optic rotation, 1045. Magnetometers, 630. Magnification, 951. by lens, 954. by microscope, 959. by telescope, 959, 961, 964. Malus" polariscope, 1033. Manometers, 177. Manometric flames, 846, 857. Marine barometer, 156. Wariotte's bottle, 239. law, 170. tube, 171. Mason's hygrometer, 370. Mass, 54. centre of, 72. Matches for collecting electricity, 605. Maximum thermometers, 254. Maxwell's colour-box, 1005. rule for action between circuits, 689. Mean temperature, 493. Mechanical equivalent of heat, 449- Mechanics, n. Melloni's experiments, 399-406. method of evaluating deflec- tions, 664. Melting-points, table of, 302. Meniscus, 135, 929. Mercury, density of, 88. expansion of, 280, 253. Meridian, 615. Meridians, chart of magnetic, 631. Metacentre, no. Metallic barometers, 157. thermometers, 261. Meteoric theory of sun's heat, 464. Mica plates for circular polariza- tion, 1047. Micrometers, 972. Microscope, compound, 956. electric, 945. simple, 955. solar, 944. Mines, firing by electricity, 590. Minimum deviation by prism, 923-925. Mirage, 1021-1023. Mirror electrometer, 594, galvanometer, 663. Mirrors, 886. concave, 893. conjugate, 393, 807. convex, 905. cylindric, 906. parabolic, 894 Mirrors, plane, 886. Mist, 377. Mixture of colours, 1002-1008. of gases and vapours, 181, 321. Mixtures, density of, 120. method of, 429. Modulus of elasticity, 29. Moist air, density of, 375. Moment of couple, 16. of force about axis, 75. of inertia, 74. of magnet, 621-622. Momentum, 76. angular, 75. Monochord, see Sonometer. Monochromatic light, 992. Monsoons, 499. Morin's apparatus, 47. Morse's telegraph, 722. telegraphic alphabet, 724. Mortar, electric, 555. Motions, composition of, 52. Mountain - barometer, theory of, 162. Mousson's experiment, 312. Mouth-pieces of organ-pipes, 837, 844- Multiple images, 888, 918. Multiple-tube barometer, 160. manometer, 170. Musical sound, 790. XTAIRNE'S electrical machine, 537- Needle, magnetized, 614. Negretti's maximum thermometer, 257- Newtonian telescope, 964. Newton's law of cooling, 386. rings, 1030. spectrum experiment, 975. theory of refraction, 1015. Nicholson's hydrometer, 115. Nicol's prism, 1036. Nobili's thermo-pile, 397. Nodal lines on plate, 788. Nodes and antinodes in air, 811. in pipes, 840. Noise and musical sound, 790. Non-liquefiable gases, 173. Notes in music, 820. QBSCURE radiation, 409, 978. CErsted's experiment, 656. piezometer, 26. Ohm as unit of resistance, 758- 760. Ohm's law, 665. Opera-glass, 963. Ophthalmoscope, 907. Optical centre of lens, 931. examination of vibrations, 847- 851. Optic axes in biaxal crystals, 1043. axis in uniaxal crystals, 926, 1042. Order, thermo-electric, 654. Ordinary and extraordinary image, 927. index, 927. rays, 1035. Organ-pipes, 837-842. effect of temperature on, 84S. overtones of, 840-843, 855. Oscillating engines, 480. Overtones, 832-846. Oxyhydrogen blow-pipe, 444. DAPIN'S digester, 339. Parabolic mirrors, 894, 393, 807. Parachute, 211. Paradox, hydrostatic, 100. Parallel currents, 682. forces, 14. mirrors, 889. Parallelogram of forces, 12. Paramagnetic bodies, 638, 781. Pascal's experiment at Puy- de- Dome, 145. principle, 91. vases, 97. Peltier effect, 708. Pendulum, 56. compensated, 271. compound, 60. convertibility of centres in, 60. cycloidal, 71. electric, 309. electrically controlled, 737. isochronism of, 58, 70. time of vibration of, 58. Penumbra, 872. Pepper's ghost, 892. Percussion, centre of, 76. Perforation by electric discharge, 588. Period of vibration, 57, 70, 786. Permanent gases, 173. Perpetual motion schemes, 20. Person on specific heat of ice, 460. Phantom bouquet, 899. Phial of four elements, 112. Phillips' electrophorus, 544. maximum thermometer, 257. Phonautograph, 825. Phonograph, Edison's, 864. Phosphorescence, 979. Phosphoroscope, 979. Photographic registration, 160. Photography, 943. Photometers, 881. Piezometer, 26. Pile, dry, 651 ; Volta's, 645. Pipes, vibration of air in, 788; see Organ-pipes. Pipette, 231. Pistol, Volta's, 556. Pitch, 816. modified by motion, 826. standards of, 820. Pixii's machine, 767. Plane mirrors, 886. Plane of polarization, 1034, 1048. Plasticity of ice, 314. Plateau's experiments, 134. Plates, refraction through, 917. vibration of, 787, 835. Plumb-line, 31. Plunger, 220. Pluviometer, see Rain-gauge. Part I., p. 1-240. Part II., p. 241-594. Part III., p. 505-784. Part IV., p. 785-1050. INDEX. 1067 Pneumatic despatch, 206. tinder-box, 445. Points discharge electricity, 530. wind from, 557. Polarization by absorption, 1032. by double refraction, 1035. by reflection and transmission .1033- circular, 1047. elliptic, 1037. in batteries, 649, 675. of dark rays, 1050 of dielectric, 578. of light, 1032. plane of, 1034. Polarizer, 1032. Poles of battery, 644. of magnet, 612. their names, 616. Porosity, 25. Portable electrometer, 592. Portative force, 637. Portrait, electric, 585. Potential, 539. analogous to level, 561. curve of, in battery, 676. energy, 78. equal to sum of quotients, 564. its relation to force and work, 559-561. strong and feeble, 579. Pouillet's apparatus for compress- ing gases, 173. Pound, a standard of mass, 54. Pressure, centre of, 102. hydrostatic, 90-101. intensity of, 95. reduction of, to absolute mea- sure, 154. total amount of, 103. Pressure and volume when no heat - enters or escapes, 436. Pressure-gauges, 177-182. Prevost's theory of radiation, 396. Primary colour-sensations, 1008. Principal focus, 893, 930. Principle of Archimedes, 104. of Huyghens, 1012. of Pascal, 91. Prism in optics, 019-925. Nicol's and Foucault's, 1036. Problems in Acoustics and Optics, IOS7- Dynamics and Hydrostatics, 1051. Heat, 1055. Projectiles, motion of, 50. Projection by lenses, 944. Proof-plane, 524. Propagation of light, 1012. of sound, 792. Psychrometer, 370. Pumps, centrifugal, 222 (3d edit.) for air, 184; see Air-pump. forcing, 220. for liquids, 215. Galileo on, 144. jet, 223 (3d edition). suction, 216. Puncture by electric discharge, 588. Pure spectrum, 976. Purity numerically measured, 993. Pyrheliometer, 463. Pyrometer, 262, 293. Pythagorean scale, 821. QUADRANT electrometer, 594. ~ electroscope, 536. Quantity of heat, 426. Quarter-wave plates, 1047. Quartz rotates plane of polariza- tion, 1045. transparent to ultra-violet rays, 410, 981. D ADIANT heat and light, 408. Radiation, 385. coefficient of, 394. selective, 410. Rain, 381. Rainbow, 997. Rainfall, British, 382. Rain-gauge, 382. Ramsden and Roy's experiments, 270. Ramsden's electrical machine, 535. Rarefaction by Alvergniat's meth- od, 551. in air-pump, 185. in Sprengel's air-pump, 198. Rarefied gases, discharge in, 765. Rayleigh's (Lord) gratings, 1026. Reaction of issuing jet, 101. Real and apparent expansion, 275. Reaumur's scale, 250. Recomposition of white light, 982. Rectilinear propagation, 866, 1013. Reed-pipes, 844. Reflecting power, 393; table of, 403. Reflection of heat, 390. of light, 883. irregular, 885. total, 913. of sound, 806. Refraction, 908. at plane surface, 916. at spherical surface, 941. atmospheric, 1018. double, 925, 1035. Newtonian explanation of, 1015. of sound, 808. table of indices of, 912. undulatory explanation of, 1014. Refrangibility, change of, 410, 981. Regelation, 314. Regnault's hygrometer, 369. hypsometer, 338. experiments on Boyle's Iaw,i73. on expansion of gases, 288. on sound, 798. on specific heat, 432. on vapour-tensions, 350. Regulators for electric light, 705- 708. Relay, 725. Remanent magnetism, 698. Replenisher, 597. Repulsion, see Attraction. Repulsion a more reliable test than attraction, 516. Residual charge, 609. magnetism, 698. Resistance, electrical, 666. and thermal, compared, 670. in battery, 677. of wires, 667. specific, 667. table of, 670. unit of, 758, 782. Resonance, 833. Resonators, 856. Resultant, 12. tones, 862. Reversal of bright lines, 412, 988. Reversible engine, perfect, 454. Reversing of locomotive, 489. Rheostat, 668. Rhomb, Fresnel's, 1047. Rings by polarized light, 1040. Newton's, 1031. Rock-salt, its diathermancy, 407, 411. Rods, vibrations of, 843. Rotating vessel of liquid, 96. Rotation of earth as affecting wind, 500. plane of polarization, 1045. Rotations, electro-dynamic, 683. electro-magnetic, 695. Rotatory engines, 480. Roy and Ramsden's measures of expansion, 370. Rubbers of electrical machine, 535 -536. RuhmkorfFs coil, 761. Rumford on heat of friction, 446. on radiation in vacuo, 385. Rumford's thermoscope, 263. Rupture of magnet, 618. Rutherford's self-registering ther- mometers, 255. CACCHARINE solutions, by polarized light, 1045. Safety-valve, 483. Saturated vapour, 318. Saturation, magnetic, 636. Sawdust battery, 651. Scales measure mass, 30. Scales, musical, 818. thermometric, 250. Scattered light, 393. Schiehallien experiment, 67. Schweiger's multiplier, 660. Sea-breeze and land-breeze, 499. Secondary axis, 894, 932. coil, 762. pile, 74& Segmental vibration, 832. Selective emission and absorption, 410. Selenite by polarized light, 1037. Semitone, 820. Sensibility of balance, 86 of thermometer, 252. Series, arrangement of cells in, 672. Sextant, 892. Shadows, 870. Siemens' armature, 771 and Wheatstone's machine, 773- Simple harmonic motion, 68, 70, magnet, ideal, 620. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IV., p. 785-1050. 1068 INDEX. Simple tones arise from simple vibrations, 863. vibrations, 68, 70. Sine-galvanometer, 659. Sines, law of, 910. Singing flames, 786. Sinuous currents, 688. Siphon, 234. barometer, 154. temperature correction of, 'S3- Siren, 822. Sirius, motion of, 991. Six's thermometer, 254. Slide-valve, 473. Snow, 383. Soap-bubbles, pressure within, 134. with hydrogen, 209. films, 133. Sodium line, 987. Solar heat, 462. sources of, 464. microscope, 944. spectrum, 975, 978. Solenoids, 690. Solidification, change of vol. in, 310. of gases, 333; of liquids, 306. Solution, 304. Solutions, boiling points of, 340. Sondhaus' experiment, 808. Sonometer, 831. Sound, 785-864. in exhausted receiver, 791. propagation of, 792. reflection of, 807. refraction of, 808. shadows in water, 867. curved rays of, 1023. calculation of, 1023, Sources of energy, 465. Spangled tube, 553. Spark, electric, 546. colour of, 552. duration of, 549. heating effects of, 556. in rarefied air, 550. Speaking-trumpet, 808. Specific gravity, 86. correction of,for temperature, 266. for weight of air, 213. determination of, by hydro- meters, 114. by weighing in water, 113. flask, 88. of mixtures, 120. table of, 88. Specific heat, 427-436. at constant pressure and con- stant volume, 435, 451. tables of, 434, 439. Specific inductive capacity, 576. Spectacles, 952. Spectra, 986-994. brightness and purity of, 791. by diffraction, 1026-1030. Spectroscope, 983. Spectrum analysis, 986. Specula, silvered, 965. Speculum-metal, 886. Sphere, electric capacity of, 565. Spherical mirrors, 893-905. aberration, 894. Spheroidal state, 344. Spirit-level, 124. thermometer, 254. Sprengel's air-pump, 197. Springs and spring-balances, 30. vibration of, 785. Squares, inverse, 389. in electricity, 520-528. Stable equilibrium, f>. Stars, brightness of, 969 motion of, 991. spectra of, 986. Statham's fuse, 764. Stationary undulations, 841 Steam, volume of, 363. Steam-engine, 469-490. locomotive, 486. Steel, its magnetic properties, 617. Step-by-step telegraphs, 718-722. Stereoscope, 949. Still, 347- Stirling's air-engine, 468. Storms, magnetic, 634. Storm-warnings, 169, Stoves, 299. - Norwegian, 424. Strained glass, by polarized light, 1044. Stratification in electric discharge, 765. Strength of pole, 620. of current, 658. Striking reed, 845. Stringed instruments, 835. Strings, overtones of, in longi- tudinal vibration, 843. vibration of, 788, 829-835, 854. Submarine telegraphs, 733. the Atlantic cables, 733. inductive action, 734. Successive reflections, 888. Suction, 215. pump, 216. Sulphate of soda, 310. Summation-tones, 862. Sun, atmosphere of, 987. distance of, 879. see Solar. Superheating of steam, 480. Supersaturated solutions, 310. Surface, electricity resides on, 523. Surface-condensers, 478. tensions, table of, 138. Sympiesometer, 156. Synthesis of sounds, 858. Syringe, pneumatic, 445. Swan on the sodium line, 987. '"TANGENT galvanometer, 660. Tantalus' vase, 237. Tartini's tones, 863. Telegraph, autographic, 730. automatic, 735. dial, 718, 722. electric, 713-736. electro-chemical, 730. Morse's 722. printing, 726. single-needle, 716. Telegraph, submarine, 733. Telegraphic alarum, 721. alphabet, 724. Telescopes, 958-964. Telespectroscope, 989. Temperament, 819. Temperature, 241. absolute, 293, 456. mean, 493. of a place, 493. of the air, 493. decrease upwards, 497. of the soil, 421, 495. increase downwards, 496. scales of, 250. Tempering of metals, 29. Tension, electric, 579. Terrestrial refraction, 1018. temperatures, 493. Thermochrose, 408. Thermo-dynamics, 445. first law of, 450. second law of, 455. Thermo-electricity, 652-655. Thermographs, 260. Thermometer, 244-252. alcohol, 254. differential, 262. metallic, 260. self-registering, 254. Thermopile, 397, 654. Thilorier's apparatus, 324. Thin films, colours of, 1630. Thomson, J . ,on glacier motion, 3 1 5. on lowering of freezing-point, 312. Thomson's galvanometer, 663. Thunder, 601. Tickling by electricity, 554. Timbre, 817. Tones, major and minor, 819. resultant, 862. Tonometer, 825. Tornadoes, 611, 503. Torricellian experiment, 143. Torricelli's theorem on efflux, 226. Torsional rigidity, 29. Torsion-balance, 519, 624. Total reflection, 913. Tourmalines, 1032. Tourniquet, hydraulic, 101. Transmission of sound, 793. Transport of elements, 739. Transverse and longitudinal vibra- tions, 795, 828. Trevelyan experiment, 789. Trumpet, speaking and hearing, 809. Tubes of force, 562. movement of, 757. relation of, to induced cur- rents, 754-760. Tuning-fork, 836. Twaddell's hydrometer, 119. Tyndall on magneto - crystallic action, 640. on moulding of ice, 315. T T MBRA and penumbra, 872. Unannealed glass, by polar- ized light, 1044. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IV., p. 785-1050. INDEX. 1069 Undergiound temperature, 495, Vibrations, simple, 68. Weighing in water, 113. 421. single and double, 785. with constant load, 84. Undulation, definition of, 798. transverse and longitudinal, 795, Weight-thermometer, 253 nature of, 795, 1012. 828. Well-thermometers, 258. stationary, 841. Vibroscope, 824. Wertheim's experiments on velo- Uniaxal crystals, 1042, 1044, 926. Virtual images, 904, 939, 940. city of sound, 844. Uniform acceleration, 52. Vision, 948. Wet and dry bulb, 370. field, 757. Visual angle, 951. Wheatstone's automatic system, Unit-jar, 587. Vitreous and resinous electricity, 735- Unit of resistance, B. A., 760. 510. bridge, 674. Units and their dimensions, 779- Volta, 645. rotating mirror, 549, 586. 783- Voltaic arc, 703. universal telegraph, 721, 775. of heat, 427. electricity, 642. and Cooke's telegraphs, 716. Unstable equilibrium, 36. element, 643. Wheel-barometer, 155. Untraversed space, 193. Voltameter, 738, 742. Whirl, electric, 558. Volume, change of, in congelation, Wiedemann and Franz's experi- WAPOUR, 317. apparatus toillustrate,3i9. at maximum tension, 318. Vapour-density, 357363. related to chemical combina- 310. in vaporization, 363. and pressure, changes of, when no heat enters or escapes, 437. Vowel-sounds, 857. ments, 419. Wilde's machine, 772. Williams', Major, experiment with ice, 311. Wind, causes of, 499. from points, 557. tion, 357. Vapour-tension, measurement of, 349-356. "UfALFERDIN'S maximum thermometer, 259. measurement of, 503. trade, 500. Wind-chest, 838. Variation of magnetic elements. Water, compressibility of, 26. instruments, 845. 633. -dropping collector, 604. Winter's electrical machine, 538. Vegetable growth, 462. Velocity of electricity, 585. of light, 873-880. equivalent of calorimeter, 431. level, 123. maximum density of, 278. Wires, telegraphic, 716. Wollaston's battery, 647. Work, 1 8. of sound in air, 800. specific heat of, 434. done by current, 699-702. in gases, 803, 844. spouts, 6u. principle of, 19. in liquids, 803. in solids, 805, 844. Watering-pot, electric, 558. Watt's improvements in steam-en- spent in generating heat, 445- mathematically investi- gine, 470. 453- gated, 814. Vena contracta, 229. Wave-front, 1012. lengths of light, 1030. yOUNG'S modulus, 29. Vernier, 148. of sound, 794-817. Vertical, 31. Vesicular state, 377. relation of, to velocity and frequency, 794, 866. 7 AM BON I'S pile, 652. Zero, absolute, of tempera- Vessels in communication, 122. surface, 1013-1014, 1043. ture, 293, 456. with two liquids, 127. theory of light, 1012. displacement of, in thermome- Vibrations of ordinary light, 1049. Weighing, double, 81. ters, 252. of plane polarized light, 1048. in air, 213. error of, 153. Part I., p. 1-240. Part II., p. 241-504. Part III., p. 505-784. Part IV., p. 785-1050. THE END. GLASGOW": W. G. BLACKIE AND CO., PRINTERS, V1LLAFIELD. ru 02319 NON-CIRCULATING BOOK The conditions under which this book was acquired place certain restrictions upon its use. It may not be taken from the Library building, nor may it be reserved in the Reserved Book room, in a seminar room or elsewhere. 751 587 ' W UNIVERSITY OF CALIFORNIA LIBRARY