.*>: This book is DUE on the last date stamped below 2 1025 MAR 81920 ocr s APR i 4 1932 AR 8 1943 ELEMENTARY TREATISE ON NATURAL PHILOSOPHY. BY A. PRIVAT DESCHANEL, FORMERLY PROFESSOR OF PHTSICS IN THE LTCEE LOUIS-LE -GRAND, INSPECTOR OF THE ACADEMY OF PARIS. TRANSLATED AND EDITED, WITH EXTENSIVE MODIFICATIONS, BY J. D. EVERETT, M. A., D. C. L., F. R. S., F. R. S. E., PROFESSOR OF NATURAL PHILOSOPHY IX THE QUEEN'S COLLEGE, BELFAST. IN FOUR PARTS. PART I. MECHANICS, HYDROSTATICS, AND PNEUMATICS, ILLUSTRATED BY 180 ENGRAVINGS ON WOOD, AND ONE COLORED PLATE. REVISED EDITION. NEW YORK : D. APPLETON AND COMPANY, 1, 3, AND 5 BOND STREET. 1884. 56207 AUTHOR'S PEEFACE. C^THE importance of the study of Physics is now generally acknowledged. Besides (f) the interest of curiosity which attaches to the observation of nature, the experi- rf> mental method furnishes one of the most salutary exercises for the mind ~~ constituting in this respect a fitting supplement to the study of the mathematical sciences. The method of deduction employed in these latter, while eminently adapted to form the habit of strict reasoning, scarcely affords any exercise for the critical faculty which plays so important a part in the physical sciences. In Physics we are called upon, not to deduce rigorous consequences from an ri absolute principle, but to ascend from the particular consequences which alone r^ are known to the general principle from which they flow. In this operation ^ there is no absolutely certain method of procedure, and even relative certainty can only be attained by a discussion which calls into profitable exercise all the faculties of the mind. Be this as it may, physical science has now taken an important place in educa- $ tion, and plays a prominent part in the examinations for the different university degrees. The present treatise is intended for the assistance of young men ^ preparing for these degrees; but I trust that it may also be read with profit ^5> by those persons who, merely for purposes of self-instruction, wish to acquire $ accurate knowledge of natxiral phenomena. Having for nearly twenty years been charged with the duty of teaching from the chair of Physics in one of the lyceums of Paris, I have been under the necessity of making continual efforts to overcome the inherent difficulties of this branch of study. I have endea- } voured to turn to account the experience thus acquired in the preparation of this j volume, and I shall be happy if I can thus contribute to advance the taste for a science which is at once useful aud interesting. L* I have made very limited use of algebra. Though calculation is a precious and often indispensable auxiliary of physical science, the extent to which it can be advantageously employed varies greatly according to circumstances. There are in fact some phenomena which cannot be really understood without having recourse to measurement: but in a multitude of cases the explanation of phenomena can be rendered evident without resorting to numerical expression. The physical sciences have of late years received very extensive developments. Facts have been multiplied indefinitely, and even theories have undergone great modifications. Hence arises considerable difficulty in selecting the most essential points and those which best represent the present state of science. I have done my best to cope with this difficulty, and I trust that the reader who attentively peruses my work, will be able to form a pretty accurate idea of the present position of physical science. TKANSLATOK'S PREFACE TO THE SIXTH EDITION. I DID not consent to undertake the labour of translating and editing the "TRAITS ELEMENTAIRE 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 steadily kept this use in view, believing that I was thus adopting the surest means of meeting the wants of teachers generally. In the original English edition, 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 IV. 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, and the order of tho chapters has been rearranged. The marks of distinction which were made in the earlier editions between new and old sections have now been dropped; but Professor Deschanel's foot-notes are still distinguished 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, but rough statements of quantity have generally been expressed in British units as being more familiar. A complete table for the conversion of French and English measures will be found at the end of the Table of Contents. In Part II, 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. TRANSLATOR S PREFACE. V 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 " Diff usivity " or " Thermometric Conductivity;" Some recent results on the conductivity of rocks, and on the conductivity of water ; A note on the mathematical discussion of periodical variations of under- ground 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 ; A description of the modern Gas Engine. Every chapter has been carefully revised, with a view to clearness, accuracy, and consolidation; and the result has been that, with the excep- tion of Melloni's experiments, and the Steam Engine, the treatment of nearly every subject has been materially changed. Part III. also contains extensive changes. In the electro-statics, the chapter on potential has been recast and made more demonstrative. There are also additions relating to Dr. Kerr's dis- coveries, charge by cascade, and some minor points. Under the head of Magnetism, investigations have been introduced relating to bih'lar suspension, and to the directive tendency of soft-iron needles. In the department of Current Electricity, there has been a complete rearrangement of subjects. The chemical relations of the current are discussed as early as possible, while thermo-electricity is reserved for a chapter on relations between electricity and heat. The chapter on induced currents, which was formerly the last of all, has been put next to that on electro-dynamics, and is followed by two chapters on telegraphs and other applications of electricity. Additional matter has been introduced under the following heads : General law for magnetic force due to current in given circuit; Helmholtz's galvanometer ; Swing produced by instantaneous current ; The galvanometer a true measurer of current ; Rowland's experiment on the motion of a charged body ; Flante's secondary battery ; Chemical relations of electro-motive force ; Resistance coils and boxes ; Wheatstone's bridge, and conjugate branches ; v i TRANSLATOR'S PREFACE. Clark's method for electro-motive force ; Thomson's method for resistance of galvanometer; Mance's method for resistance of battery ; Thermo-electric diagrams ; Convection of heat by electricity ; Pyro-electricity ; Effect of light on resistance of selenium ; Deduction of law of induced currents from electro-dynamic law; Superposition of tubes of force ; Stratified discharge from galvanic battery ; Siemens' and Gramme's magneto-electric machines ; Cowper's writing telegraph ; Duplex telegraphy ; Edison's electric pen ; The telephone, the microphone, and the induction balance. A collection of examples on electricity has been added. Part IV. contains no radical changes. The numbering of the chapters and sections has been altered to make it consecutive with the other three Parts, but there has been no rearrangement. Additions have been made under the following heads (those marked with an asterisk Avere introduced in a previous edition) : Mathematical note on stationary undulation ; Edison's phonograph ; Michelson's measurement of the velocity of light; Astronomical refraction ; *Eefraction at a spherical surface; Refraction through a sphere; Brightness of image on screen ; Field of view in telescope ; *Curved rays of sound ; *Retardatiou-gratings and reflection-gratings ; Kerr's magneto-optic discoveries; besides briefer additions and emendations which it would be tedious to enumerate. The whole volume has been minutely revised; and a copious collection of examples arranged in order, with answers, has been introduced at the end of each Part, in place of the " Problems " (translated from the French) which appeared in some of the earlier editions. The dates of revision of the four Parts were, October, 1879, November 1880, Docember, 1880, and May, 1881. J. D E BELFAST, September, 1SS1. CONTENTS-PAKT 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 viii 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. Resistance 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. OZrsted'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 CONTENTS. 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 Dome, 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 XVHI. 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. CHAPTKR 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. CHAPTER 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 CONTEXTS. 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, 2-13. Calculation of its effect, 244. Various contrivances for compressing air, 245. Practical applications of air-pump and compressing pump, 246. CHAPTER 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. Parallelogram of Velocities, and Parallelogram of Forces. Ex. 1-11, . . . P 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 ........ 042 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, ....... - . <, 4 g Density, and Principle of Archimedes. Ex. 124-159 ' 947 Capillarity. Ex. 160-164 .......... ' ~ ' Barometer, and Boyle's law. Ex. 165-181 Pumps, &c. Ex. 182-189, ...... 1 ."!."!.'.' ' 251 AXSWCBS TO EXAMPLES, FRENCH AND ENGLISH MEASURES. A DECIMETRE DIVIDED INTO CENTIMETRES AND MILLIMETRES. INCHES AND TENTHS. 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 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 '3156 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 = 15432 grains=2'205 Ibs. avoir. More accurately, the kilogramme is 2-20402125 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=2'5399772 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=2'59xl0 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 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 =447.04 cm. per seo, 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 # =981). (See p. 43.) Weight of 1 grain = 63 '57 dynes, nearly. 1 oz. avoir. =278 x lOMynes.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. WORK (assuming^ = 981). (See p. 48.) 1 foot-pound = 1 '356 x 1 7 ergs, nearly. 1 kilogrammetrs = 9'81 x 10 7 ergs, nearly. Work in a second ") by one theoretical |>:=7-46xl0 9 ergs, nearly, "'horse." STRESS (assuming <7=9S1). 1 Ib. per sq. ft. 479 dynes per sq. cin., nearly. 1 Ib. per sq. inch =6'9xl0 4 dynes per sq. cm., nearly. 1 kilog. per sq. cm. =9'81 x 10 3 dynes per sq. cm., nearly. 7GO mm. of mercury at 0C. = 1 '014 x 10 6 dynes per sq. cm. , nearly. 30 inches of mercury at C. = 1 "0163 x lU 6 dynes per sq. cm., nearly. 1 inch of mercury at C. r=3'38S x 10 4 dynes per sq. cm., nearly. TABLE OF DENSITIES, IN GRAMMES PER CUBIC CENTIMETRE. LIQUIDS. Pure water at 4 C., - - Sea water, ordinary, - - Alcohol, pure, - - - - proof spirit, - - - - - 1-000 - - - 1-026 - - - -791 --- -91G Ether, -716 Mercury at C., 13 "596 Naphtha, -848 SOLIDS. 7'8 to 8-4 wire, 8-54 Bronze, 8'4 Copper, cast, 8'6 sheet, 8-8 hammered. 8'9 Gold, 19 to 19-6 Iron, cast, 6-95 to 7 '3 wrought, 7'6 to 7'8 Lead, n . 4 Platinum, 21 to 22 er . 10-5 el, 7-8 to 7-9 . 7-3 to 7-5 Zinc, 6-8 to 7'2 Ice, -92 Basalt, 3-00 Brick, 2 to 2-17 Brickwork, 18 Chalk, 1-8 to 2-8 Clay, 1-92 Glass, crown, - -. 2 "5 flint, 3-0 Quartz (rock-crystal), 2'65 Sand, 1-42 Fir, spruce, -43 to '7 Oak, European, -69 to '99 Lignum-vitse, 'Co to 1 '33 Sulphur, octahedral, 2'05 prismatic, 1-93 GASES, at C. and a pressure of a million dynes per sq. cm. Air . dry, -0012759 Oxygen, -0014107 Nitrogen, -0012393 Hydrogen, '00008837 Carbonic acid, '00] 9509 ELEMENTARY TREATISE ON NATUBAL PHILOSOPHY. CHAPTER L 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 solids, liquids, or gases. Thus we have the three divisions, Mechanics, Hydrostatics, and Pneumatics. IT. THERMICS; the 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 FORGES. 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 whole 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. FIRST PRINCIPLES OF DYNAMICS. H 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 ot 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 m 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 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 force which would equilibrate the first Fig. 1. -Triangle of Forces. 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 the same rigid body in lines which meet in a point, be represented by tivo lines drawn from the point, and a parallelo- gram be constructed on these lines, the diagonal drawn from this point to the opposite corner Fi 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. W T e 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 P" 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 forces 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 g FIRST PRINCIPLES OF DYNAMICS. e nd of this crooked line. For by what precedes, if ABODE be a rooked line such that the straight lines AB EC CD, ^ repent four forces acting at a point, we may substitute for AB and ] Fig. 3. Gravesaiide'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 ADGv', &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. 9 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: Q 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 Fig - 5 - the distance between the lines of action of the other two; that is, we must have P _ Q _ R 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 lf P 2 (Fig. 6). Let O be any point on this line, and x lt x 2 - * the distances measured from to the f 4- i points of section, distances measured 2 in opposite directions being distin- Fig - 6 - guished by opposite signs, and forces in opposite directions being also distinguished by opposite signs. Also let R denote the resultant of Pj and P 2 , and x the distance from to its intersection with the line; then we shall have P! xi + P 2 x. t - R X. For, taking the standard case, as represented in Fig. G, in which all the quantities are positive, we have OA X = x l} OA 2 = x 2 , OB x, and by 19 or 20 we have Pi.A^Pa.BAj, that is, whence that is, Ri = P l a: 1 + P 3 a; > (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 1? P 2 , P 8 , &c., be the forces, R : the resultant of the first two forces PL P 2 , and R 2 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 O on this line be expressed according to the following scheme: Force PI P 2 P 3 ' R x R 2 Distance x, x , x 3 x, x MOMENT OF A FORCE. 11 Then, by equation (2) we have E! x 1 = P l x^ + P 2 x. i . Also since R 2 is the resultant of R : and P 3 , we have Ra^Raii + PaXs, and substituting for the term R x Hc v we have This reasoning can evidently be extended to any number of forces, so that we shall have finally KB = sum of such terms as P#, 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 = 2 (P) Kx=S (Px) whence =?-*) (3) S(P) 24. Moment of a Force about a Point. When the fixed line is at right angles to the parallel forces, the product 7x 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 force by the length of tJie 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 0, 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 Px 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- 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 ^=^ 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 two equal and parallel forces in opposite directions applied to the same body. The distance between 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 1 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 Rigid 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 GENERAL 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 wrench, 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 BO on this line; AC will represent the required component. CB represents the other component, which, along with 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- Fi g . o.-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 c language, and the component along AC will be F ^~ or in trigonometrical 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 betiveen 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 vertical 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 w l be the weight of the particle A, and w 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 W-L and w< or the centre of gravity of the two particles A and B. The magnitude of the resultant will be w 1 -\-w 2 , 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 combine this resultant with the weight of a third particle of the body, and shall thus obtain a resultant fr passing through a definite point in the line which joins J3 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 '=l%- 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. 19 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 EC. It can be shown, by similar triangles, that each of these strips is bisected by 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- - D 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 20 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 Tetrahedron. one 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 _ AB _ BC GD ~ DE ~ 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, tne cen t, re Q f g rav ity 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 below 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 ot 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 free 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. fr()m another pointj the centre Q f grav ity 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 iv l} w 2 , W 3 &c., and let their heights above a given horizontal plane be respectively h 1} h 2 , h 3 &c. We know by equation (3), 23, that if h denote the height of their centre of gravity we have (wi + w., + &c.) K=W! hi + w-i A.J + &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 2 , H 3 &c. The height H of their centre of gravity will now be such that (Wi + w.,, + &c.) H = Wi H! + Wi H 2 + &c. (5) From these two equations, we find, by subtraction to + wj + fcc.) (H-A)= 1 (H 1 -A l ) + w J (H,- A 2 ) + &c. (6) Now H 1 A! is the height through which the particle of weight u\ has been raised; hence the work done against gravity in raising it is 10! (Hj Aj) and the second member of equation (G) therefore expresses the whole amount of work done against gravity. But the first member expresses the work which would be done in raising all the particles through a uniform height H Ti, 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 weight 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, we 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 CENTKE 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 : J- M , 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 giavity 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 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 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 w r ere 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 w r ork are exactly equal, the body is in equilibrium. This principle (which has been called the principle of virtual velocities, but is better called the principle of work) is often of great use in enabling us to calculate the ratio which two forces applied in given w r ays 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 w r ork. 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 we 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. 28 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. i8.-Baiancer. be expressed by saying that the equilibrium is stable for displacements lying within certain limits, but unstable for displacements beyond these limits. The equilibrium Fig 19. Tur 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- going principles; and we shall begin with the so-called "mechanical powers," namely, the lever, the ivheel 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 h A * -o o 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. SO 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 defined (which are parts of the lever), and the arms of the power 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 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 which the arm of the power is the diameter of the pulley, and the arm of the weight is Fig. 24. Fig. 25. Fig. -26. Fig. 2T. 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 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 (Fio-. 28) be any portion of such a plane, and let AC and BC be drawn vertically and horizontally. Then AB t^-JLr is ca^d tne length, AC the height, and CB / /"-^-^^^ the base of the inclined plane. The force of c ii B gravity upon a heavy body M resting on the N ^ p plane, may be represented by a vertical line Fi s- 2S - MP, and may be resolved by the parallelogram of forces ( 1G) 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. CHAPTEK V, 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- I 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 O, B, be equal, and also that the beam remain horizontal when the pans are empty. Any inequality in the distances A 0, B, 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 BALANCE. 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 bo 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 O. The equilibrium would be neutral if this centre of gravity exactly coin- cided with O; and it is necessary as a condition of delicacy thnt 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, 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 30 THE BALANCE. of them an excess of weighty 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 applied at B', and the weight IT of the beam applied at G', the new position of the centre of gravity. p These two forces are parallel, and are in equilibrium about the axis 0, that is, their resultant passes through the Fig. 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 TT. G'K=p. B'L. But if we call half the length of the beam I, and the distance OG r we have G'R=rsino, B'L = 1 cos a. whence vr sin . Fig. 40. Parabolic Trace ATWOODS MACHINE. 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, Fl 's- -Atwor. Machine. large enough to let the weight pass through it without danger of 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 -^ 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 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 = at, (1) =4< (2) =W, (3) FORCE IN CIRCULAR MOTION. 59 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 Fi *- 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 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 (1) DB = i/c 2 . (2) The second of these equations gives /= and substituting for t from the first equation, this becomes f = S * (3) But if An (Fig. 43) be the diameter at A, and Bra the perpendicular upon it from B, we have, by Euclid, AD 2 = mB 2 Am.mn=2r. Am sensibly, -2r.DB. Therefore 33^= p 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 TT as usual denote the ratio of circumference to diameter, the distance mov^d in time 60 LAWS OF FALLING BODIES. T is 2wr; hence v = ^r, and another expression for the intensity of the force will be f=("ffr. M 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 T 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 v -, 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 towards 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 path 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 m\ g-- 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. The apparent weight of bodies at the equator would be nil if v - 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 O (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 K* 14. -Pendulum, acts in the direction of its length, and there is equilib- SIMPLE PENDULUM. G3 Fig. 45. Motion of Simple Pendulum. rium. 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 (54, 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 &, we see from Fig. 45 that the ratio of the effective component of gravity to the whole force of gravity is j^-, that is sin 6; 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 gd, 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 \ Fig. 46. Projection of Circular 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 SIMPLE HARMONIC MOTION. 65 positions P p 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 (Y) r ( 1 ^ 1 )- 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 \-f) x - 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 /*= (Y)*' which gives T=^. (1) V M / 110. Application to the Pendulum. For the motion of a pendulum in a small arc, we have acceleration =|a, where s denotes the displacement in linear measure. We must therefore put M - j, and we then have 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 To find the length of the seconds' pendulum we must put T = 1. This ives If g were 987 we should have Z = 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 px, where x denotes 5 QQ THE PENDULUM. distance from this point, is called Simple Harmonic Motion, or Simple Harmonic Vibration. The above investigation shows that such vibration is isochronous, its period being ^j 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 /z 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 a, 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, o 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 o 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 a? 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 08 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 inertia 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 ~j?^ so that the angle 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. 1 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 r 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 mr. The moment of this force round the axis would be mr ? 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 2 to denote " the sum of such terms as," and observing that $ is the same for the whole body, we have Applied couple = 2 (mi 3 <(>) = tf> "2, (mr 2 ). (!) When is unity, the applied couple will be equal to S (mr z ), 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 2 +y 2 . Now let two planes parallel to these be drawn through the centre of mass; let I and i) 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*+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 2 = a" -(- g 2 2a f , y 8 = 6" + if 26 17. 2 (wr 2 ) = 2 {m (a 8 + )} + 2 {m (f + i')} 2a 2 (m) 26 2 (mi)) = A 2 2?n. + I, (m,p*) 2a f 2m 26 77 I.m. where ^ and 77 are the values of and n for the centre of mass. But these values are both zero, since the centre of mass lies on both the planes from which and rj are measured. We have therefore 2 (mr*) = A 2 2m + 2 (mf), (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 Mg, 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#/i sin 0; and this must, by 114, be equal to 0S (rar 2 ). We have therefore 2 (mr 2 ) g sin 6 -SOT = -*- If the whole mass were collected at one point at distance I from the axis, this equation would become MZ S g sin 6, Mi = l = ~^T' and the angular motion would be the same as in the actual case if I had the value >**- I is evidently the length of the equivalent simple pendulum. 117. Convertibility of Centres. Again, if we introduce a length k such that M& 2 is equal to 2 (wp 2 ), that is, to the moment of inertia round a parallel axis through the centre of mass, we have Flg - 48 ' Zro = M z + MA, and equation (5) becomes In the annexed figure (Fig. 48) which represents a vertical section through the centre of mass, 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 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 i* = 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 O, 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 z . 118. If we take a new centre of suspension A' in the plane of the figure, the new centre of oscillation O' will lie in the production of A'G, and we must have A'G.GO' = P = 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 = Jc, 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. s 119. Kater'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 Kater'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^j, we easily deduce from it gr=|a 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 formulse which are given in treatises on rigid dynamics, but its value is more easily obtained by Rater'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 |rai> 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-^ to v 2 , the work done upon it in the interval is |m (v 2 z v a 2 ); 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 \mv z . For example, during any portion of the ascent of a projectile, the diminution in the value of %mv 2 is equal to gm multiplied by the increase of height; and during any portion of its descent the increase in ^mv z 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 70 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. CHAPTER 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 distoited 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, in'asmuch as the accelerations are proportional to the forces, and are 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 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+Z, the longitudinal strain is L. 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 Vv, the volume-strain is -, and if this COEFFICIENTS OF ELASTICITY. 79 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 ^, or is . This is also called the resistance to compression; and its reciprocal -y 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 denned: Voung's Elasticity of Modulus. Volume. Glass (flint), 60 x 10 10 40 x 10 10 Steel, 210 x 10 ISO x 10 l Iron (wrought), 190 x 10 10 140 x 10 M Iron (cast), 130 x 10 l 96 x 10 l Copper, 120 x 10 10 160 x 10 W Mercury, 54 x 10 10 Water, 2 x 10 10 Alcohol, 1-2 x 10 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 Wi, 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 b. It might at first Fig. 49.- its loW6r side B being at the bottom of the water and its upper side K at the top, the pressure is zero at R and goes on increasing uni- formly to B. The normals B6, Dd, Hfc, 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 65 _R 0tat i ng vessel tal plane in the liquid, the pressure at each point of of L 'i lud - 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. Fig. 66.-Principlo 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. G7); 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. 67. Principle of its weight, and acting in a line which passes Archimedes. . ,, ., TI? through its centre of gravity. It 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 weight 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 PKOOF. 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 Fig. CS. 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; if, on the contrary, it sinks, the pressure increases and tends to make it reascend. 156. Cartesian Diver. The experiment of the Cartesian diver, 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 O; 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. TO.- 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 last section, 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. 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 O 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. 103 is displaced, and takes the position shown in the figure, it will be seen that the effect of the two forces acting at O 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 Fig- 73 _ steel NeedlM Floath)g on 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 denned 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. 104 PRINCIPLE OF ARCHIMEDES. 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 Fig. 74. -insect walking on water. 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 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 V denote the volume of a substance, M its mass, and D its absolute density; then by definition, we have Mr=VD. If s denote the specific gravity of a substance, and d the absolute density of water in the standard condition, then ~D=sd and M= 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 C2-4 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. * According to the best determination yet published, the mass of a cubic centimetre of pure water at 4 is 1 '00001 3, 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 m 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 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 Fig 76. Specific Gravity Flask for Soli 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. 7S.-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 WEIGHING IN WATER. 109 Thus, from the observations Weight in air, 125 gm. Weight in water, 100 Loss of weight, 25 we deduce 1 or ^ ; = 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 w' 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-\- sw, and the specific gravity of the body is 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 gm Weight in liquid, 120 Weight in water, 110 we deduce Loss in liquid, 80. Loss in water, 90. Density of liquid, * = * 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 ww' 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 vr 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 >lson'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 tormer 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 om the French name of these instruments, artomtore. 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 hollo*v 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 in the dish be called w, and the weight of the instrument itself W, the weight of liquid displaced is now W + iv, whereas the weight of the same volume of water was W + 100; hence the specific gravity of the liquid is ^loo' 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^ 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. SO. Fahrenheit's Hydrometer. Fig. SI. 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 Nd 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 Vd^V'd', OTd:d'::V:V, that is, the density varies inversely as the volume immersed. Let d lt d. 2 , c 3 ...be a series of densities, and V 1} V 2 , V 3 ...the corresponding volumes immersed, then we have 111 d lt d 2 , o 3 ... proportional to , , ... and V b V a V s ... proportional to --, - , ... 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 lt n. 2 , reckoning from the top downwards, and let the corresponding densities be d 1} d. 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 Tith 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-Wi) di = (c-ni) d-2, whence c = "' f | 1 ~ ^- . 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 - n,) rf t which gives D = "X- 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 1116, 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 whence 15x1-116 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 TOSS, 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.... Baume's Alcoholi- meters. and as we have now 7^ =10, the formulae give 1 10 128 =1, n. 2 =0, (7 2 = 1*08 174. 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 e is about 136, which would make the density corresponding to the zero of the scale about T074. 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 T005, 2 indicates 1O10, &c. 175. Gay-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 ss. 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 lf v 2 , v 3 . . . of the components be given, together with their densities d 1 , d 2 , d 3 . . . Then their masses (or weights) are v^d^ v 2 d. 2 , V 3 d 3 . . . The mass of the mixture is the sum of these masses, and its volume is the sum of the volumes v lt v 2 , % . . . ; hence its density is Secondly, let the weights or masses m v m z , m z . . . of the com- ponents be given, together with their densities d li d 2 , d z . . . 116 DENSITY AND ITS DETERMINATION. Then their volumes are ~\ ^, J The volume of the mixture is the sum of these volumes, and its mass is w, + m a + m s + . . . ; hence its density is 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 160, we must draw through the ^Oth division, on the line of equal parts, a horizontal fine to meet the curve, and, through the point thus found on the curve, I K L 1.4 R 1.6 B 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 |7 V\ ,^,J the principal vessel \1 if 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 to termin- ate ina narrowmouth, the liquid will be seen to spout up nearly to the level of that in the principal vessel. Fig. 87. Vessels in Communication. 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 Ib, bent at right angles at its extremities. These carry two glass tubes Fig. 88. Water-leveL aa, 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. Level! 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 x * r *"^^^^^^^^^^^^^^q it which are read off through the telescope. 180. Spirit-level. The Fig. 9o.-s P irit-ievei. 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 Fig. si. horizontal plane (Fig. 91). ... The air-bubble will take up a pos-faon MN at the highest part of the tube, such that the arcs MA and NB are equal. Hence it follows that if the level 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. 92. 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 Horizontality 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 Fig. 94. Spirit level with Telescope. 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 XVL 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. 9G), the surface is concave; on the contrary Fig. 96. Fig. 97. Fig. 98. 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 wiU 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 Fig. 100. . _ ft i in the case of the mercury. J 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 2979 mm in a tube 1 millimetre in diameter, alcohol rises 12'18 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 dimmish as the temperature increases; for example, the elevation of water, which in a tube of a certain diameter is equal to 132 mm 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 Diameters. 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 irxHw, 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. 127 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 irr~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 ir> s hw - 2irrT cos a ; whence h= r cf ^ a which, when the other elements are given, varies inversely as r, 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 Again, for two parallel vertical plates at distance u, the vertical force of capillarity for a unit of length is 2Tcoa, 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 ft _ 2Tcosa uw ' which agrees with the expression for the depression or elevation in a circu ar 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 herefore 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 two 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 ^ + ~ n 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 130 CAPILLARITY. where there can be no normal pressure towards either side of the surface, which implies that either the surface is plane, in which case each of the two terms is separately equal to zero, or else E = - K'; (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 jrp -, 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 which follows immediately from the formula for h in ISO. 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 water 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 however 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 (ftyviffKos, a crescent), which denotes a form approximately resembling that of a watch-glass. 132 CAPILLAEITY. 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 Sf&et 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, ] -88 Absolute alcohol, , 2'5 Olive-oil, 3-5 Mercury, 49-1 Bisulphide of carbon 3 -5 7 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 la, 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 SEPTA. 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 same 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 replace/! 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('i\\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. Endosmometer. 13C 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. CHAPTER 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 diminish 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 Fi - lOS.-ExpansibiUty 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 1G50 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 w r eighs 1*293 gramme. 2 Under these circum- stances, the ratio of the weight of a volume of air to that of an equal volume of water is l|S=-JL Air 1UUU / /o 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 1-43 gramme, a litre of carbonic hydrogen 0'089 gramme, &c. Fig. 104. Weight of Air. acid T97 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. Eegnault's determination of the weight of a litre of dry air at 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 Exj 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 intensity 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 mercury whose 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 a height of 76 centimetres of pure mercury at the temperature zero Centigrade, gravity being supposed to have 1 76 centimetres are 29'922 inches. Fig. 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 were 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 which sustains the mercurial column, was performed for the first time at Clermont, and on the top of the mountain Puy-de-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 UniverseJlc, article " Torricelli "). 2>. CISTERN BAROMETER. 143 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 : 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 Fig. 103. -cathetometer. 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- CATHETOMETER. 145 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 Fortin's Barometer. L4G THE BAROMETER. 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. Tbe 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 Utter 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 9 7 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 S T beyond the second on the vernier, and so on until the ninth on the scale, which ', ', \ \ 1 1 * ' f, PTTT3 n , . . . . ,T .u. 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 T 7 ^ of an inch. , If the vernier consisted of 19 inches divided into 20 equal parts, it would read to the ^ 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 -fa of a division on the circle, that is, -^ of a half degree, or one minute. D. PORTABLE BAROMETER. 147 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 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. Fig. 112. Barometer "with Tiipod 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 water which would exert the same pressure. Let h be the observed height at temperature t Centigrade, and h the height reduced to freezing- point. Then, if m be the coefficient of expansion of mercury per degree Cent., we have A (1 +m t) =h, whence h o = h-hmt nearly. The value of m is -^='00018018. For temperatures Fahrenheit, we have Fioatl&ent. where m denotes = -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 f 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 whence 1 + mt that is to say, if n be the height read off on the scale, it must be diminished by the correction n t (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 (< 32) (ml), where m I 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 CORKECTIONS. 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 a Fahr., is {! + *(* -62)}; and by equating this expression to h {l + m (t - 31)} we find A = {l -m (*-32)+Z(-62)} = n| 1 - (m-l)t+ (32m - 62Z) ] = K 1 1 - -00008967 t + -00255654 j- ; which, omitting superfluous decimals, may conveniently be put in the fi nn The correction vanishes when 09 t - 2-56 = 0; that is, when t=~ = 28-5. 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 ill 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 Oil 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 l-|- an( i ^ 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 + ^\x, 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 ratio - , 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 13'59G 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 Pig. 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 (a, rqpoc). This barometer depends upon the changes in the form of a thin metallic vessel partially exhausted Fig. 116. Aueroid 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 both lighter and less liable to injury. It is sometimes made small enough for the waistcoat 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 Xing's barograph, B is made greater than A by fixing a hollow iron drum round the lower end of the tube. 156 THE BAROMETER. 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 Baromeier. 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. 157 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 BAROMETER. lower extremity is partially supported by a counterpoise acting on a lio-ht 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 Reports of the Meteorological Office. CHAPTER XVIII. VARIATIONS 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 = g HD, which gives H = ^ (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 1GO 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. 7 211. 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 00,000 feet deep takes off aoioo 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 g = 783800000 nearly. Taking g as 981, we have H = ia{o -A** = 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 + -00366 t). ' (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. Denote height above a fixed level by x, and pressure by p. Then we have dx dp K = - P '> and if p l} p 2 are the pressures at the heights x l} x 2 , we deduce by in- tegration Xi - Xt = H (loge Pi - loge p. 2 ). 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* - *! = 1840000 (1 + -00366 t) (log Pl - log j*) 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 (v^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 (TTMTTW, 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 rariation, 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 11 162 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 Fig. 119. Curves of i>iurual Variation. 2 IB is 21 24 heights for the different hours, some indications of the law will be found. 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. 1C3 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 air 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 dfawn, 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 IQ4, 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 0-2 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 hillv 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 almost every instance, and is dependent on the earth's rotation. 2 The isobars frequently, as in the example here selected, form closed curves encircling a region of barometric depression. Two such centres are here exhibited one in the south of England and the other in the west of Russia. Great atmospheric disturbances are always accompanied by such centres of depression. The air, in fact, rushes in from all sides, usually with a spiral motion, towards these centres, the direction of rotation in the spiral being, for the northern hemisphere, opposite to the motion of the hands of a watch lying with its face upwards. The centrifugal force due to this rotation tends to increase the original central depression, and thus protracts the duration of the phenomenon. 1 The curves drawn upon this chart are isobaric lines, each corresponding to a particular barometric pressure, which is indicated by the numerals marked against it. These denote the pressure in millimetres diminished by 700. For example, the line which passes through the south of Spain corresponds to the pressure 770 millimetres; that through the north of Spain to 765 millimetres. The curves are drawn for every fifth millimetre. The smaller numerals, which are given to one place of decimals, indicate the pressures actually observed at the different stations, from which the isobaric lines are drawn by estimation. The other symbols refer to cloud, wind, and sea, and are explained at the left of the chart. 2 The influence of the earth's rotation in modifying the direction of winds is discussed in a paper " On the General Circulation and Distribution of the Atmosphere," by the editor of this work, in the Philosophical Magazine for September, 1871. ISOBARS AND WEATHER CHARTS. 165 These revolving storms are called cyclones. They attain their greatest violence in tropical regions, the West Indies being especially noted for their destructive effect. They frequently proceed from the Gulf of Mexico in a north-easterly direction, increasing in diameter as they proceed, but diminishing in violence. Their velocity of translation is usually from ten to twenty miles an hour. Storm-warnings are based partly upon information received by telegraph of storms that have actually commenced, and partly upon barometric gradients. 1 1 For fuller information respecting the laws of storms, which is a purely modern subject, and is continually receiving fresh developments, we would refer to Mr. Buchan's llandy Book of Meteorology. See also the last chapter of Part II. of the present Work. CHAPTER XIX. BOYLE'S (OR MARIOTTE'S) LAW. 1 217. Boyle's Law. All gases exhibit a continual tendency to ex- pand, and thus exert pressure against the vessels in which they are confined. The intensity of this pressure depends upon the volume which they occupy, increasing as this volume diminishes. By a number of careful experiments upon this point, Boyle and Mariotte independently established the law that this pressure varies inversely as the volume, provided that the temperature remain constant. As the density also varies inversely as the volume, we may express the law in other words by saying that at the same temperature the density varies directly as the pressure. If V and V be the volumes of the same quantity of gas, P and F, D and D', the corresponding pressures and densities, Boyle's law will be expressed by either of the equations P V P D F~V' F = D'' 218. Boyle's Tube. The correctness of this law may be verified by means of the following apparatus, which was employed by both the experimenters above named. It consists (Fig. 120) of a bent tube with branches of unequal length; the long branch is open, and the short branch closed. The tube is fastened to a board provided with two scales, one by the side of each branch. The 1 Boyle, in his Defence of the Doctrine touching the Spring and Weight of the Air against the Objections of Francisc.us Linus, appended to New Experiments, Physico-mechanical, &c. (second edition, 4to, Oxford, 1662), describes the two kinds of apparatus represented in Figs. 120, 121 as having been employed by him, and gives in tabular form the lengths of tube occupied by a body of air at various pressures. These observed lengths he compares with the theoretical lengths computed on the assumption that volume varies reciprocally as pressure, and points out that they agree within the limits of experimental error. Manotte's treatise, De la Nature de I' Air, is stated in th Bioyraphie UnuerseUe to have been published in 1679. (See Preface to Tait's Thermodynamics, p. iv.) BOYLE'S TUBE. 167 graduation of both scales begins from the same horizontal line through 0, 0. Mercury is first poured in at the extremity of the long branch, and by inclining the apparatus to either side, and cautiously adding more of the liquid if required, the mercury can be made to stand at the same level in both branches, and at the zero of both scales. Thus we have, in the short branch, a quantity of air separated from the external air, and at the same pressure. Mercury is then poured into the long branch, so as to reduce the volume of this inclosed air by one-half; it will then be found that the difference of level of the mercury in the two branches is equal to the height of the barometer at the time of the experiment; the compressed air therefore exerts a pressure equal to that of two atmospheres. If more mercury be poured in so as to reduce the volume of the air to one-third or one- fourth of the original volume, it will be found that the difference of level is respectively two or three times the height of the barometer; that is, that the compressed air exerts a pressure equal respectively to that of three or four atmospheres. This ex- periment therefore shows that if the volume of the gas becomes two, three, or four times as small, the pressure becomes two, three, or four times as great. This is the principle expressed in Boyle's law. The law may also be verified in the case where the gas expands, and where its pressure conse- quently diminishes. For this purpose a barometric tube (Fig. 121), partially filled with mercury, is* inverted in a tall vessel, containing mercury also, and is held in such a position that the level of the liquid is the same in the tube and in the vessel. The volume occupied by the gas is marked, and the tube is raised; the gas expands, its pressure diminishes, and, in virtue of the excess of the atmospheric pressure, a column of mercury ab rises in the tube, such that its height, added to the pressure of the expanded air, is equal to the atmospheric pressure. It will then be seen that if the volume of air becomes double w T hat it was before, the height of the column raised is one-half that of the barometer; that is, the 1C8 BOYLE'S (OR MARIOTTE'S) LAW. expanded air exerts a pressure equal to half that of the atmosphere. If the volume is trebled, the height of the column is two-thirds that of the barometer; that is, the pressure of the expanded air is one-third that of the atmosphere, a result in accordance with Boyle's law. 219. Despretz's Experiments. The simplicity of Boyle's law, taken in conjunction with its apparent agreement with facts, led to its general acceptance as a rigorous truth of nature, until in 1825 Despretz published an account of experiments, showing that different gases are unequally com- pressible. He inverted in a cistern of mercury several cylindrical tubes of equal height, and filled them with different gases. The whole apparatus was then inclosed in a strong glass vessel filled with water, and having a screw piston as in CErsted's piesometer ( 130). On pressure being applied, the mercury rose to unequal heights in the different tubes, carbonic acid for example being more reduced in volume than air. These experiments proved that even supposing Boyle's law to be true for one of the gases employed, U could not be rigorously true for more than one. In 1829 Dulong and Arago undertook a laborious series of experi- ments with the view of testing the accuracy of the law as applied to air; and the results which they obtained, even when the pressure was increased to twenty-seven atmospheres, agreed so nearly with it as to confirm them in the conviction that, for air at least, it was rigor- ously true. When re-examined, in the light of later researches, the results obtained by Dulong and Arago seem to point to a different conclusion. 220. Unequal Compressibility of Different Gases. The unequal compressibility of different gases, which was first established by Despretz's experiments above described, is now usually exhibited by the aid of an apparatus designed by Pouillet (Fig. 122). A is a cast- iron reservoir, containing mercury surn^unted by oil. In this latter liquid dips a bronze plunger P, the upper part of which has a thread cut upon it, and works in a nut, so that the plunger can be screwed up or down by means of the lever L. The reservoir A communicates NOT RIGOROUSLY EXACT. 169 by an iron tube with another cast-iron vessel, into which are firmly fastened two tubes TT about six feet in length and -j^th of an inch in internal diameter, very carefully calibrated. Equal volumes of two gases, perfectly dry, are introduced into these tubes through their upper ends, which are then hermetically sealed. The plunger is then made to descend, and a gradually increasing pressure is exerted, the volumes occupied by the gases are measured, and it is ascertained that no two gases follow precisely the same law of compression. The difference, however, is almost insensible when the gases employed are those which are very difficult to liquefy, as air, oxygen, hydrogen, nitrogen, nitric oxide, and marsh -gas. But when we compare any one of these with one of the more liquefiable gases, such as carbonic acid, cyanogen, or ammonia, the difference is rapidly and distinctly manifested. Thus, under a pressure of twenty-five atmospheres, carbonic acid occupies a volume which is only ths of that occupied by air. 221. Regnault's Experiments. Boyle's law, therefore, is not to be considered as rigorously exact; but it is so nearly exact that to demon- strate its inaccuracy for one of the more permanent gases, and still more to determine the law of deviation for each gas, very precise methods of measurement are necessary. Jn ordinary experiments on compression, and even in the elaborate investigations of Dulong and Arago, a definite portion of gas is taken and successively diminished in volume by the application of continually increasing pres- sure. In experiments of this kind, as the pressure increases, the volume under measurement becomes smaller, and the precision with which it can be measured consequently diminishes. Fig. 122.-Po .ibUity of Ditfe ing Unequal 170 BOYLE'S (OB MARIOTTE'S) LAW. Regnault adopted the plan of operating in all cases upon the same volume of gas, which being initially at different pressures, was always reduced to one-half. The pressure was observed before and after this operation, and, if Boyle's law were true, its value should be found to be doubled. In this way the same precision of measurement is obtained at high as at low pressures. A general view of Regnault's apparatus is given in Fig. 123. There is an iron reservoir containing mercury, furnished at the top with a force-pump for water. The lower part of this reservoir communicates with a cylinder which is also of iron, and in which are two openings to admit tubes. Communication between the reservoir and the cylinder can be established or interrupted by means of a stop-cock R, of very exact workmanship. Into one of the openings is fitted the lowest of a series of glass tubes A, which are placed end to end, and firmly joined to each other by metal fittings, so as to form a vertical column of about twenty-five metres in height. The height of the mercurial column in this long mano- metric tube could be exactly Fig. 1C3. Regnault's Apparatus for Testing Boyle's Law. REGNAULT'S INVESTIGATIONS. 171 determined by means of reference marks placed at distances of about '95 of a metre, and by the graduation on the tubes forming the upper part of the column. The mean temperature of the mercurial column was given by thermometers placed at different heights. Into the second opening in the cylinder fits the lower extremity of the tube B, which is divided into millimetres, and also gauged with great accuracy. This tube has at its upper end a stop- cock r which can open communication with the reservoir V, into which the gas to be operated on is forced and compressed by means of the pump P. An outer tube, which is not shown in the figure, envelops the tube B, and, being kept full of water, which is continually renewed, enables the operator to maintain the tube at a temperature sensibly constant, which is indicated by a very delicate thermometer. Before fixing the tube in its place, the point corresponding to the middle of its volume is carefully ascertained, and after the tube has been per- manently fixed, the distance of this point from the nearest of the reference marks is observed. 1 After these explanatory remarks we may describe the mode of conducting the experiments. The gas to be operated on, after being first thoroughly dried, was introduced at the upper part of the tube B, the stop-cock of the pump being kept open, so as to enable the gas to expel the mercury and occupy the entire length of the tube. The force-pump was then brought into play, and the gas was reduced to about half of its former volume; the pressure in both cases being ascertained by observing the height of the mercury in the long tube above the nearest mark. It is important to remark that it is not at all necessary to operate always upon exactly the same initial volume, and reduce it exactly to one-half, which would be a very tedious operation; these two conditions are approximately fulfilled, and the graduation of the tube enables the observer always to ascertain the actual volumes. > 222. Results. The general result of the investigations of Regnault 1 Regnault's apparatus was fixed in a small square tower of about fifteen metres in height, forming part of the buildings of the College de France, and which had formerly been built by Savart for experiments in hydraulics. The tower could therefore contain only the lower part of the manometric column ; the upper part rose above the platform at the top of the tower, resting against a sort of mast which could be ascended by the ob- server. The readings inside the tower could be made by means of a cathetometer, but this was impossible in the upper portion of the column, and for this reason the tubes forming this portion were graduated. D. 172 BOYLE'S (OR MARIOTTE'S) LAW. is, that Boyle's law does not exactly represent the compressibility even of air, hydrogen, or nitrogen, which, with carbonic acid, were the gases operated on by him. He found that for all the gases on which he operated, except hydrogen, the product VP of the volume and pressure, instead of remaining constant, as it would if Boyle's law were exact, diminished as the compression was increased. This diminution is particularly rapid in the cases of the more liquefiable gases, such as carbonic acid, at least when the experiments are con- ducted at ordinary atmospheric temperatures. The lower the tem- perature, the greater is the departure from Boyle's law in the case of these gases. For hydrogen, he found the departure from Boyle's law to be in the opposite direction; the product VP increased as the gas was more compressed. 223. Manometers or Pressure-gauges. Manometers or pressure- gauges are instruments for measuring the elastic force of a gas or vapour contained in the interior of a closed space. This elastic force is generally expressed in units called atmospheres ( 198), and is often measured by means of a column of mercury. When one end of the column of mercury is open to the air, as in Regnault's experiments above described, the gauge is called an open mercurial gauge. The open mercurial pressure-gauge is often used in the arts to measure pressures which are not very considerable. Fig. 124 repre- sents one of its simplest forms. The apparatus consists of a box, generally of iron, at the top of which is an opening closed by a screw stopper, which is traversed by the tube 6, open at both ends, and dipping into the mercury in the box. The air or vapour whose elastic force is to be measured enters by the tube a, and presses upon the mercury. It is evident that if the level of the liquid in the box is the same as in the tube, the pressure in the box must be exactly equal to that of the atmosphere. If the mercury in the tube rises above that in the box, the pressure of the air in the box must exceed that of the atmosphere by a pressure corresponding to the height of the column raised. The pressures are generally marked in atmo- spheres upon a scale beside the tube. -, 224. Multiple Branch Manometer. When the pressures to be mea- sured are considerable, as in the boiler of a high-pressure steam- engine, the above instrument, if employed at all, must be of a length corresponding to the pressure. If, for instance, the pressure in ques- tion is eight atmospheres, the length of the tube must be at least MANOMETERS. 173 8 X 30 inches= 20 feet. Such an arrangement is inconvenient even for stationary machines, and is entirely inapplicable to movable machines. Without departing from the principle of the open mercurial pres- sure-gauge, namely, the balancing of the pressure to be observed against the weight of a liquid increased by one atmosphere, we may reduce the length of the instrument by an artifice already employed by Fahrenheit in his barometer ( 207). The apparatus for this purpose consists of an iron tube ABCD Fig. 124. Open Mercurial Manometer. Fig 125. Multiple Branch Manometer. (Fig. 125) bent back upon itself several times. The extremity A communicates with the boiler by a stop-cock, and the last branch CD is of glass, with a scale by its side. The first step is to fill the tube with mercury as far as the level MN. At this height are holes by which the mercury escapes when it reaches them, and which are afterwards hermetically sealed. The upper portions are filled with water through openings which are also stopped after the tube has been filled. If the mercury in the first tube, which is in communication with the reservoir of gas, falls through a distance h, it will alternately rise and fall through the same distance in the other tubes. The difference of pressure between the two ends of the gauge is represented by the weight of a column of mercury of height 10k diminished by the weight of a column of water of height 8h. Reduced to mercury, the difference of pressure is therefore 10A - = Q'4h. 174 BOYLE'S (OR MARIOTTE'S) LAW. 225. Compressed-air Manometer. This instrument, which may as- sume different forms, sometimes consists, as in Fig. 126, of a bent tube AB closed at one end a, and containing within the space Act a quantity of air, which is cut off from external communication by a column of mercury. The apparatus has been so constructed, that when the pressure on B is equal to that of the atmosphere, the mer- cury stands at the same height in both branches; so that, under these circumstances, the inclosed air is exactly at atmospheric pres- sure. But if the pressure increases, the mercury is forced into the left branch, so that th'e air in that branch is compressed, until equi- librium is established. The pressure exerted by the gas at B is then equal to the pressure of the compressed air, together with that of a column of mercury equal to the difference of level of the liquid in the two branches. This pressure is usually expressed in atmospheres on the scale ab. The graduation of this scale is effected empiri- cally in practice, by placing the manometer in communication with a reservoir of compressed air wnose pressure is given by an open mercurial gauge, or by a standard manometer of any kind. If the tube AB be supposed cylindrical, the graduation can be calculated by an application of Boyle's law. Let I be the length of the tube occupied by the inclosed air when its pressure is equal to that of one atmosphere; at the point to which the level of the mercury rises is marked the number 1. It is required to find what point the end of the liquid column should reach when a pressure of n atmospheres is exerted at B. Let x be the height of this point above 1; then the volume of the air, which was originally I, has become I - x, and its pressure is therefore equal to H -^ H being the mean height of the barometer. This pressure, together with that due to the difference of level 2x, is equivalent to n atmospheres. We have thus the equation H r^ + 2 * = n11 ' whence 2**-(nH + 20*+ (n-l)Hi = 0. r _ nH + -a Let us take, for instance, a piece of sulphur whose weight has been found to be 100 grammes, the weights being of copper, the density of which is 8*8. The density of sulphur is 2. We have, by applying the formula, 7o (i - Fa) ! = 100 ' 05 grammes ' We see then that the difference is not altogether insensible. It varies in sign, as the formula shows, according as d or 3 is the greater. When the density of the body to be weighed is less than 14 210 UPWARD PRESSURE OF THE AIR. that of the weights used, the real weight is greater than the apparent weight; if the contrary, the case is reversed. If the body to be weighed were of the same density as the weights used, the real and apparent weights would be equal. We may remark, that in determining the ratio of the weights of two bodies of the same density, by means of standard weights which are all of one material, we need not concern ourselves with the effect of the upward pressure of the air; as the correcting factor, which has the same value for both cases, will disappear in the quotient. CHAPTER XXII. PUMPS FOR LIQUIDS. 252. Machines for raising water have been known from very early ages, and the invention of the common pump is pretty generally ascribed to Ctesibius, teacher of the celebrated Hero of Alexandria; but the true theory of its action was not understood till the time of Galileo and Torricelli. 253. Reason of the Rising of Water in Pumps. Suppose we take a tube with a piston at the bottom (Fig. 154),and immerse the lower end of it in water. The raising of the piston tends to produce a vacuum below it, and the atmospheric pressure, acting upon the external surface of the liquid, compels it to rise in the tube and follow the upward motion of the piston. This upward move- ment of the water would take place even if some air were interposed between the piston and the water; for on raising the piston, this air would be rarefied, and its pressure no longer balancing that of the atmosphere, this latter pressure would cause the liquid to ascend in a column whose weight, added to the pressure of the air below the piston, would be equal to the atmospheric pressure. This is the principle on which water rises in pumps. These in- p struments have a considerable variety of T ^=- forms, of which we shall describe the most Fig. i54.-PnncipieofSuction-pump. important types. y 254. Suction-pump. The suction-pump (Fig. 155) consists of a 212 PUMPS FOR LIQUIDS. cylindrical pump-barrel traversed by a piston, and communicating by means of a smaller tube, called the suction-tube, with the water in the pump-well. At the junction of the pump-barrel and the tube is a valve opening upward, called the suction-valve, and in the piston is an opening closed by another valve, also opening upward. Suppose now the suction-tube to be filled with air at the atmo- spheric pressure, and the water consequently to be at the same level inside the tube and in the well. Suppose the piston to be at the end of its downward stroke, and to be now raised. This motion tends to produce a vacuum below the pis- ton, hence the air contained in the suc- tion-tube will open the suction-valve, and rush into the pump-barrel. The elastic force of this air being thus diminished, the atmospheric pressure will cause the water to rise in the tube to a height such that the pressure due to this height, increased by the pressure of the air inside, will ex- actly counterbalance the pressure of the atmosphere. If the piston now descends, the suction- valve closes, the water remains at the level to which it has been raised, and the air, being compressed in the barrel, opens the piston-valve and escapes. At the next stroke of the piston, the water will rise still further, and a fresh portion of air will escape. If, then, the length of the suction-tube is less than about 30 feet, the water will, after a certain number of strokes of the piston, be able to reach the suction-valve and rise into the pump-barrel. When this point has been reached the action changes. The piston in its downward stroke compresses the air, which escapes through it, but the water also passes through, so that the piston when at the bottom of the pump-barrel will have above it all the water which has previously risen into the barrel. If the piston be now raised, supposing the total height to which it is raised to be not more than 34 feet above the level of the water in the well, as should always be the case, the water will follow it in its upward movement, and will fill the Fig. 155. Suction-pump. SUCTION-PUMP. 213 pump-barrel. In the downward stroke this water will pass up through the piston-valve, and in the following upward stroke it will be discharged at the spout. A fresh quantity of water will by this time have risen into the pump-barrel, and the same operations will be repeated. We thus see that from the time when the water has entered the pump-barrel, at each upward stroke of the piston a volume of water is ejected equal to the contents of the pump-barrel. In order that the water may be able to rise into the pump-barrel, the suction- valve must not be more than 34 feet above the level of the water in the well, otherwise the water would stop at a certain point of the tube, and could not be raised higher by any farther motion of the piston. Moreover, in order that the working of the pump may be such as we have described, that is, that at each upward stroke of the piston a quantity of water may be removed equal to the volume of the pump-barrel, it is necessary that the piston when at the top of its stroke should not be more than 34 feet above the water in the well. 255. Effect of untraversed space. If the piston does not descend to the bottom of the barrel, it is possible that the water may fall short of rising to the suction-valve, even though the total height reached by the piston be less than 34 feet. When the piston is at the end of its downward stroke, the air below it in the barrel is at atmospheric pressure; and when the limit of working has been reached, this air will expand during the upward stroke until it fills the barrel. Its pressure will now be the same as that of the air in the top of the suction-tube; and if this pressure be equivalent to h feet of water, the height to which water can be drawn up will be only 34 A feet. Example. The suction- valve of a pump is at a height of 27 feet above the surface of the water, and the piston, the entire length of whose stroke is 7*8 inches, when at the lowest point is 31 inches from the fixed valve; find whether the water will be able to rise into the pump-barrel. When the piston is at the end of its downward stroke, the air below it in the barrel is at the atmospheric pressure; when the piston is raised this air becomes rarefied, and its pressure, by Boyle's law, becomes ^ that of the atmosphere; this pressure can therefore 214 PUMPS FOR LIQUIDS. balance a column of water whose height is 34 x ^ 9 feet, or 9'67 feet. Hence, the maximum height to which the water can attain is 34 - 9-67 feet = 24'33 feet; and consequently, as the suction-tube is 27 feet long, the water will not rise into the pump-barrel, even supposing the pump to be perfectly free from leakage. Practically, the pump-barrel should not be more than about 25 feet above the surface of the water in the well; but the spout may be more than 34 feet above the barrel, as the water after rising above the piston is simply pushed up by the latter, an operation which is independent of atmospheric pressure. Pumps in which the spout is at a great height above the barrel are commonly called lift-pumps, but they are not essentially different from the suction- pump. 256. Force necessary to raise the Piston. The force which must be expended in order to raise the piston, is equal to the weight of a column of water, whose base is the section of the piston, and whose height is that to which the water is raised. Let S be the section of the piston, P the atmospheric pressure upon this area, h the height of the column of water which is above the piston in its present position, and h' the height of the column of water below it; then the upper surface of the piston is subjected to a pressure equal to P + S A; the lower face is subjected to a pressure in the opposite direction equal to P Sh', and the entire downward pressure is represented by the difference between these two, that is, by S (h + h'). The same conclusion would be arrived at even if the water had not yet reached the piston. In this case, let I be the height of the column of water raised; then the pressure below the piston is P S 1; the pressure above is simply the atmospheric pressure P, and, consequently, the difference of these pressures acts downward, and its value is S I. ' 257. Efficiency of Pumps. From the results of last section it follows that the force required to raise the piston, multiplied by the height through which it is raised, is equal to the weight of water discharged multiplied by the height of the spout above the water in the well. This is an illustration of the principle of work ( 49). As this result has been obtained from merely statical con- siderations, and on the hypothesis of no friction, it presents too favourable a view of the actual efficiency of the pump. EFFICIENCY OF PUMPS. 215 Besides the friction of the solid parts of the mechanism, there is work wasted in generating the velocity with which the fluid, as a whole, is discharged at the spout, and also in producing eddies and other internal motions of the fluid. These eddies are especially pro- duced at the sudden enlargements and contractions of the passages through which the fluid flows. To these drawbacks must be added loss from leakage of water, and at the commencement of the opera- tion from leakage of air, through the valves and at the circum- ference of the piston. In com- mon household pumps, which are generally roughly made, the effi- ciency may be as small as '25 or 3; that is to say, the product of the weight of water raised, and Fig. It Suction-pump. the height through which it is raised, may be only '25 or '3 of the work done in driving the pump. In Figs. 156 and 157 are shown the means usually employed for working the piston. In the first figure the upward and downward 216 PUMPS FOR LIQUIDS. movement of the piston is effected by means of a lever. The second ficrure represents an arrangement often employed, in which the alternate motion of the piston is effected by means of a rotatory motion. For this purpose the piston-rod T is joined by means of the connecting-rod B to the crank C of an axle turned by a handle attached to the fly-wheel V. . 258. Forcing-pump. The forcing-pump consists of a pump-barrel dipping into water, and having at the bottom a valve opening up- ward. In communication with the pump-barrel is a side- tube, with a valve at the point of junction, opening from the barrel into the tube. A solid piston moves up and down the pump-barrel, and it is evident that when this piston is raised, water enters the barrel by the lower valve, and that when the piston descends, this water is forced into the side-tube. The greater the height of this tube, the greater will be the force required to push the piston down, for the resistance to be overcome is the pressure due to the column of water raised. The forcing-pump most frequently has a short suction-pipe leading from the reservoir, as represented in Fig. 159. In this case the water is raised from the reservoir into the barrel by atmospheric pressure during the up-stroke, and is forced from the barrel into the ascending pipe in the down-stroke. 259. Plunger. When the height to which the water is to be forced is very considerable, the different parts of the pump must be very strongly made and fitted together, in order to resist the enormous pressure produced by the column of water, and to prevent leakage. In this case the ordinary piston stuffed with tow or leather washers cannot be used, but is replaced by a solid cylinder of metal called a plunger. Fig. 1GO represents a section of a pump thus constructed. The plunger is of smaller section than the barrel, and passes through a stuffing-box in which it fits air-tight. The volume of water which enters the barrel at each up-stroke, and is expelled in the down- stroke, is the same as the volume of a length of the plunger equal Fig. 158. Forcing-pump. FORCE-PUMPS. 217 to the length of stroke; and the hydrostatic pressure to be overcome is proportional to the section of the plunger, not to that of the barrel. As the operation proceeds, air is set free from the water, and would eventually impede the working of the pump were it not permitted to escape. For this purpose the plunger is pierced with a narrow passage, which is opened from time to time to blow out the air. The drainage of deep mines is usually effected by a series of pumps. The water is first raised by one pump to a reservoir, into which dips the suction- tube of a second pump, which sends the water up Fig. 159. p; g . ico. - . , Suction and Force Pump. to a second reservoir, and so on. The piston-rods of the different pumps are all joined to a Fig. 101. Fire engine. single rod called the spear, which receives its motion from a steam- engine. 218 PUMPS FOR LIQUIDS. 260. Fire-engine. The ordinary fire-engine is formed by the union of two forcing-pumps which play into a common reservoir, contain- ing in its upper portion (called the air-chamber) air compressed by the working of the engine. A tube dips into the water in this reservoir, and to the upper end of this tube is screwed the leather hose through which the water is discharged. The piston-rods are jointed to a lever, the ends of which are raised and depressed alter- nately, so that one piston is ascending while the other is descending. Water is thus continually being forced into the common reservoir except at the instant of reversing stroke, and as the compressed air in the air-chamber performs the part of a reser- voir of work (nearly analogous to the fly-wheel), the discharge of water from the nozzle of the hose is very steady. The engine is sometimes supplied with water by means of an attached cistern (as in Fig. 162) into which water is poured; but it is more usually furnished with a suction-pipe which renders it self -feeding. 261. Double-acting Pumps. These pumps, the invention of which is due to Delahire, are often employed for household purposes. They consist of a pump-barrel VV (Fig. 162), with four open- ings in it, A, A', B, B'. The openings A and B' are in communication with the suction-tube C; A' and B are in communication with the ejec- tion-tube C'. The four openings are fitted with four valves opening all in the same direction, that is, from right to left, whence it follows that A and B' act as suction- valves, and A' and B as ejection- valves, and, consequently, in whichever direction the piston may be moving, the suction and ejection of water are taking place at the same time. 262. Centrifugal Pumps. Centrifugal pumps, which have long been used as blowers for air, and have recently come into extensive use for purposes of drainage and irrigation, consist mainly of a flat casing or box of approximately circular outline, in which the fluid is made to revolve by a rotating propeller furnished with fans or blades. These extend from near the centre outwards to the circum- ference of the propeller, and are usually curved backwards. The Fig. Ifi2. Double-action Pump. CENTRIFUGAL AND JET PUMPS. 219 fluid between them, in virtue of the centrifugal force generated by its rotation, tends to move outwards, and is allowed to pass off through a large conduit which leaves the case tangentially. Fig. 163.-Centrifugal Pump. The first part of Fig. 1G3 is a section of the propeller and casing, C being "a central opening at which the fluid enters, and D the conduit through which it escapes. The second part of the figure represents a small pump as mounted for use. The largest class of Fig. 164. Jet Pump. centrifugal pumps are usually immersed in the water to be pumped, and revolve horizontally. 263. Jet-pump. The jet-pump is a contrivance by Professor 220 PUMPS FOR LIQUIDS. James Thomson for raising water by means of the descent of other water from above, the common outfall being at an intermediate level. Its action somewhat resembles that of the blast-pipe of the locomo- tive. The pipe corresponding to the locomotive chimney must have a narrow throat at the place where the jet enters, and must thence widen very gradually towards its outlet, which is immersed in the outfall water so as to prevent any admission of air during the pumping. The water is drawn up from the low level through a suction-pipe, terminating in a chamber surrounding the jet-nozzle. Fig. 1G4 represents the pump in position, the jet-nozzle with its surroundings being also shown separately on a larger scale. The action of the jet-pump is explained by the following consider- ations. Suppose we have a horizontal pipe varying gradually in sectional area from one point to another, and completely filled by a liquid flowing steadily through it. Since the same quantity of liquid passes all cross-sections of the pipe, the velocity will vary inversely as the sectional area. Those portions of the liquid which are passing at any moment from the larger to the smaller parts of the pipe are being accelerated, and are therefore more strongly pushed behind than in front; while the opposite is the case with those which are passing from smaller to larger. Places of large sectional area are therefore places of small velocity and high pressure, and on the other hand, places of small area have high velocity and low pressure. Pressure, in such discussions as this, is most conveniently expressed by pressure-height, that is, by the height of an equivalent column of the liquid. Neglecting friction, it can be shown that if v l} v 2 be the velocities at two points in the pipe, and h lt h 2 the pressure- heights at these points, g denoting the intensity of gravity. The change in pressure-height is therefore equal and opposite to the change in ^-' Tnis is for a horizontal pipe. In an ascending or descending pipe, there is a further change of pressure-height, equal and opposite to the change of actual height. Let H be the pressure-height at the free surfaces, that is, the height of a column of water which would balance atmospheric pres- sure; HYDRAULIC PEESS. 221 k the difference of level between the jet-nozzle and the free surface above it. I the difference of level between the jet-nozzle and the free surface of the water which is to be raised. v the velocity with which the liquid rushes through the jet- nozzle, then the pressure-height at the jet-nozzle may be taken as H + k -; and if this be less than H I the water will be sucked up. The condition of working is therefore that II - I be greater than H + Te - |-, or v 1 - greater than k + I, where it will be observed that k + 1 is the difference of levels of the highest and lowest free surfaces. 264. Hydraulic Press. The hydraulic press (Fig. 1G5) consists of a suction and force pump aa worked by means of a lever turning about an axis O. The water drawn from the reservoir BB is forced alonor Fig. 1G5. Bramah Pr the tube CO into the cistern V. In the top of the cistern is an open- ing through which moves a heavy metal plunger AA. This carries on its upper end a large plate B'B', upon which are placed the objects to be pressed. Suppose the plunger A to be in its lowest position when the pump begins to work. The cistern first begins to fill with water; then the pressure exerted by the plunger of the pump is transmitted, according to the principles laid down in 141, to the bottom of the plunger A; which accordingly rises, and the objects to 222 PUMPS FOR LIQUIDS. be pressed, being intercepted between the plate and the top of a fixed frame, are subjected to the transmitted pressure. The amount of this pressure depends both on the ratio of the sections of the pistons, and on the length of the lever used to work the force-pump. Sup- pose, for instance, that the distance of the point m, where the hand is applied, from the point O, is equal to twelve times the distance 10, and suppose the force exerted to be equal to fifty pounds. By the principle of the lever this is equivalent to a force of 50 x 12 at the point I; and if the section of the piston A be at the same time 100 times that of the piston of the pump, the pressure trans- mitted to A will be 50 x 12 x 100 = 60,000 pounds. These are the ordinary conditions of the press usually employed in workshops. By drawing out the pin which serves as an axis at O, and introducing it at O', we can increase the mechanical advantage of the lever. Two parts essential to the working of the hydraulic press are not represented in the figure. These are a safety-valve, which opens when the -pressure attains the limit which is not to be exceeded; and, secondly, a tap in the tube C, which is opened when we wish to put an end to the action of the press. The water then runs off, and the piston A descends again to the bottom of the cistern. The hydraulic press was clearly described by Pascal, and at a still earlier date by Stevinus, but for a long time remained practically useless; because as soon as the pressure began to be at all strong, the water escaped at the surface of the piston A. Bramah invented the cupped leather cottar, which prevents the liquid from escaping, and thus enables us to utilize all the power of the machine. It con- sists of a leather ring A A (Fig. 1GC), bent so as to have a semicir- cular section. This is fitted into a hollow in the interior of the sides of the cistern, so that water passing between the piston and cylinder will fill the concavity of the cupped leather collar, and by pressing on it will produce a packing which fits more tightly as the pressure on the piston increases. The hydraulic press is very extensively employed in the arts. Fig. ICO.-Cup-leathir. HYDRAULIC PRESS. 223 It is of great power, and may "be constructed to give pressures of two or three hundred tons. It is the instrument generally employed in cases where very great force is required, as in testing anchors or raising very heavy weights. It was used for raising the sections of the Britannia tubular bridge, and for launching the Great Eastern. CHAPTER XXIII. EFFLUX OF LIQUIDS. TORRICELLl'S THEOREM. 265. If an opening is made in the side of a vessel containing water, the liquid escapes with a velocity which is greater as the surface of the liquid in the vessel is higher above the orifice, or to employ the usual phrase, as the head of liquid is greater. This point in the dynamics of liquids was made the subject of experiments by Torricelli, and the result arrived at by him was that the velocity of efflux is equal to that which would be acquired by a body falling freely from the upper surface of the liquid to the centre of the orifice. If h be this height, the velocity of efflux is given by the formula This is called Torricelli's theorem. It supposes the orifice to be small compared with the horizontal section of the vessel, and to be exposed to the same atmospheric pressure as the upper surface of the liquid in the vessel. It may be deduced from the principle of conservation of energy; for the escape of a mass m of liquid involves a loss mgh of energy of position, and must involve an equal gain of energy of motion. But the gain of energy of motion is %mv z ; hence we have ^mv 1 = mgh, i? = 2gh. The form of the issuing jet will depend, to some extent, on the form of the orifice. If the orifice be a round hole with sharp edges, in a thin plate, the flow through it will not be in parallel lines, but the outer portions will converge towards the axis, producing a rapid narrowing of the jet. The section of the jet at which this conver- gence ceases and the flow becomes sensibly parallel, is called the contracted vein or vena contracta. The pressure within the jet at this part is atmospheric, whereas in the converging part it is greater CONTRACTED VEIN. 225 than atmospheric; and it is to the contracted vein that Torricelli's formula properly applies, v denoting the velocity at the contracted vein, and h the depth of its central point below the free surface of the liquid in the vessel. 266. Area of Contracted Vein. Froude's Case. A force is equal to the momentum which it generates in the unit of time. Let A denote the area of an orifice through which a liquid issues horizon- tally, and a the area of the contracted vein. From the equality of action and reaction it follows 'that the resultant force which ejects the issuing stream is equal and opposite to the resultant horizontal force exerted on the vessel. The latter may be taken as a first approximation to be equal to the pressure which would be exerted on a plug closing the orifice, that is ^ f to <7/iA if the density of the liquid be +^ ^* taken as unity. k \ / \ The horizontal momentum geneY- ated in the water in one second is the product of the velocity v and the mass ejected in one second. The volume ejected in one second is va. This is equal to the mass, since the density is unity, and hence the momentum is v z a, that is, 2gha. Equating this last expression for the momentum to the foregoing expres- sion for the force, we have - yhA. Fig. 167. that is, the area of the contracted vein is half the area of the orifice. Mr. Froude has pointed out that this reasoning is strictly correct when the liquid is discharged through a cylindrical pipe projecting inwards into the vessel and terminating with a sharp edge (Fig. 167); and he has verified the result by accurate experiments in which the jet was discharged vertically downwards. The direction of flow in different parts of the jet is approximately indicated by the arrows and dotted lines in the figure; and, on a larger, scale by those in Fig. 1G8, in which the sections of the orifice and of the contracted vein are also indicated by the lines marked D and d. We may remark that since liquids press equally in all directions, there can 15 Vr L than on Fig. 103. their convex 226 EFFLUX OF LIQUIDS. TOREICELLl'S THEOREM. be no material difference between the velocities of a vertical and of a horizontal jet at the same depth below the free surface. 267. Contracted Vein for Orifice in Thin Plate. When the liquid is simply discharged through a hole cut in the side of the vessel and bounded by a sharp edge, the direc- tion of flow in different parts of the stream is shown by the arrows and dotted lines in Fig. 169. The pres- sure on the sides, in the neighbour- hood of the orifice, is less than that due to the depth, because the curved form of the lines of flow implies (on the principles of centrifugal force) a smaller pressure on their concave side. The pressure around the orifice is therefore less than it would be if the hole were plugged. The unbalanced horizontal pressure on the vessel (if we suppose the side containing the jet to be vertical) will therefore exceed the statical pressure on the plug ghA, since the removal of the plug not only removes the pressure on the plug but also a portion of the M pressure on neighbouring parts. This unbalanced force, If which is greater than ghA, is necessarily equal to the ]y momentum generated per second in the liquid, which is still represented by the expression v*a or 2gha; hence 2gha is greater than ghA, or a is greater than A. s'"# f ...-~ Reasoning similar to this applies to all ordinary forms of j( orifice. The peculiarity of the case investigated by Mr. 1 Froude consists in the circumstance that the pressure on H the parts of the vessel in the neighbourhood of the orifice Fig. 169. i s norma i to the direction of the jet, and any changes in its amount which may be produced by unplugging the orifice have therefore no influence upon the pressures on the vessel in or opposite to the direction of the jet. 1 263. Apparatus for Illustration. In the preceding investigations, 1 This section and the preceding one are based on two communications read before the Philosophical Society of Glasgow, February 23d and March 31st, 1876; one being an extract from a letter from Mr. Froude to Sir William Thomson, and the other a com- munication from Professor James Thomson, to whom we are indebted for the accompany- ing illustrations. APPARATUS FOR ILLUSTRATION. 227 no account is taken of friction. When experiments are conducted on too small a scale, friction may materially diminish the velocity; and further, if the velocity be tested by the height or distance to which the jet will spout, the resistance of the air will diminish this height or distance, and thus make the velocity appear less than it really is. Fig. 170 represents an apparatus frequently employed for illustrat- Fig. 170. Apparatus for verifying Torricelli's Theorem. ing some of the consequences of Torricelli's theorem. An upright cylindrical vessel is pierced on one side with a number of orifices in the same vertical line, which can be opened or closed at pleasure. A tap placed above the vessel supplies it with water, and, with the help of an overflow pipe, maintains the surface at a constant level, which is as much above the highest orifice as each orifice is above that next below it. The liquid which escapes is received in a trough, the edge of which is graduated. A travelling piece with an index 228 EFFLUX OF LIQUIDS. TORRICELLl'S THEOREM. line engraved on it slides along the trough; it carries, as shown in one of the separate figures, a disc pierced with a circular hole, and capable of being turned in any direction about a horizontal axis pass- ing through its centre. In this way the disc can always be placed in such a position that its plane shall be at right angles to the liquid jet, and that the jet shall pass freely and exactly through its centre. The index line then indicates the range of the jet with considerable precision. This range is reckoned from the vertical plane containing the orifices, and is measured on the horizontal plane passing through the centre of the disc. The distance of this latter plane below the lowest orifice is equal to that between any two consecutive orifices. The jet, consisting as it does of a series of projectiles travelling in the same path, has the form of a parabola. Let a be the range of the jet, b the height of the orifice above the centre of the ring, and v the velocity of discharge, which we assume to be horizontal. Then if t ba the time occupied by a particle of the liquid in passing from the orifice to the ring, we have to express that a is the distance due to the horizontal velocity v in the time t, and that b is the vertical distance due to gravity acting for the same time. We have therefore But according to Torricellf s theorem, if h be the height of the sur- face of the water above the orifice, we have v 2 = 2gh; and comparing this with the above value of v- we deduce One consequence of this last formula is, that if the values of b and h be interchanged, the value of a will remain unaltered. This amounts to saying that the highest orifice will give the same range as the lowest, the highest but one the same as the lowest but one, and so on; a result which can be very accurately verified. If we describe a semicircle on the line b+h, the length of an ordi- 3 erected at the point of junction of b and h is ^, and since <*<= 274. Cup of Tantalus. The siphon may be employed to produce the intermittent flow of a liquid. Suppose, for instance, that we have a cup (Fig. 178) in which is a bent tube rising to a height n, and with the short branch termi- nating near the bottom of the cup, while the long branch passes through the bottom. If liquid be poured into the cup, the level will gradually rise in the short branch of the bent tube, till it reaches the summit of the bend, when the siphon will begin to discharge the liquid. If the liquid then escapes by the siphon faster than it is poured into the vessel, the level of the liquid in the cup will gradu- ally fall below the termination of the shorter branch. The siphon will then empty itself, and will not recommence its action till the liquid has again risen to the level of the bend. The siphon may be concealed in the interior of the figure of a man whose mouth is just above the top of the siphon. If water be poured in very slowly, it will continually rise nearly to his lips and then descend again. Hence the name. Instead of a bent tube we may employ, as in the first figure, a straight tube covered by a bell- glass left open below; in this case the space between the tube and the bell takes the place of the shorter leg of the siphon. It is to an action of this kind that natural intermittent springs are generally attributed. Suppose a reservoir (Fig. 179) to communicate with an outlet by a bent tube forming a siphon, and suppose it to be fed by a stream of water at a slower rate than the siphon is able to discharge it. When the water has reached the bend, the siphon will become charged, and the reservoir will be emptied; flow will then cease until it becomes charged again. - 275. Mariotte's Bottle. This is an apparatus often employed to ob- tain a uniform flow of water. Through the cork at the top of the bottle (Fig. 180) passes a straight vertical tube open at both ends, and 236 EFFLUX OF LIQUIDS. TORRICELLI'S THEOREM. in one side of the bottle near the bottom is a second opening furnished with a horizontal efflux tube b at a lower level than the lower end of the vertical tube. Suppose that both the bottle and the vertical tube are in the first instance full of water, and that the efflux tube is then opened. The liquid flows out, and the vertical tube is rapidly emptied. Air then enters the bottle through the vertical tube, and bubbles up from its lower end a through the liquid to the upper part of the bottle. As soon as this process begins, the velocity of efflux, which up to this point has been rapidly diminishing (as is shown by the diminished range of the Fig. 179. -Intermittent Spring. jet), becomes constant, and continues so till the level of the liquid has fallen to a, after which it again diminishes. During the time of constant flow, the velocity of efflux is that due to the height of a above 6, and the air in the upper part of the bottle is at less than atmospheric pressure, the difference being measured by the height of the surface of the liquid above a. Strictly speaking, since the air enters not in a continuous stream but in bubbles, there must be slight oscillations of velocity, keeping time with the bubbles, but they are scarcely perceptible. Instead of the vertical tube, we may have a second opening in the MARIOTTE S BOTTLE. 237 side of the bottle, at a higher level than the first; as shown in Fig. 180. Air will enter through the pipe a, which is fitted in this upper opening, and the liquid will issue at the lower pipe 6, with a constant velocity due to the height of a above b. Mariotte's bottle is sometimes used in the laboratory to produce Fig. ISO. Mariotte's Bottle. the uniform flow of a gas by employing the water which escapes to expel the gas. We may also draw in gas through the tube of Mariotte's bottle; in this case, the flow of the water is uniform, but the flow of the gas is continually accelerated, since the space occupied by it in the bottle increases uniformly, but the density of the gas in this space continually increases. EXAMPLES. PARALLELOGRAM OF VELOCITIES, AND PARALLELOGRAM OF FORCES. I. A ship sails through the water at the rate of 10 miles per hour, and a ball rolls across the deck in a direction perpendicular to the course, at the same rate. Find the velocity of the ball relative to the water. , 2. The wind blows from a point intermediate between N. and E. The nor- thei-ly component of its velocity is 5 miles per hour, and the easterly component is 12 miles per hour. Find the total velocity. 3. The wind is blowing due N.E. with a velocity of 10 miles an hour. Find the northerly and easterly components. 4. Two forces of 6 and 8 units act upon a body in lines which meet in a point and are at right angles. Find the magnitude of their resultant. 5. Two equal forces of 100 units act upon a body in lines which meet in a point and are at right angles. Find the magnitude of their resultant. 6. A force of 100 units acts at an inclination of 45 to the horizon. Resolve it into a horizontal and a vertical component. 7. Two equal forces act in lines which meet in a point, and the angle between their directions is 120. Show that the resultant is equal to either of the forces. 8. A body is pulled north, south, east, and west by four strings whose direc- tions meet in a point, and the forces of tension in the strings are equal to 10, 15, 20, 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 betwteii 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 R be their resultant, show that E 2 = P s + Q 2 + 2PQ cos 0. II. Show that the resultant of two equal forces P, acting at an angle 6, is 2P cos 4J. 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 ? 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. 240 EXAMPLES. 12. A weight of 100 Ibs. is slung from a pole which rests on 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 EXAMPLES. 241 A downward force of 3 Ibs. at 1 ft. from axis. A downward force of 5 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, R, 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 llie 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 |J 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; aud 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. 16 242 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 ^ 39*1 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 1 INCLINED PLANE, &c. 44. On an inclined plane whose height is J 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, (6) 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 1 ? 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 le rectilinear. 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. EXAMPLES. 243 51. A force of 490 dynes acts on a mass of 70 gin. 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 and 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 ^ 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 crn. 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. 244 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 horizcntally 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 \ 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*. 83. Show that the equation to the trajectory is y = * tana -^k' and that if V and a, 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. ATWOOD'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 (y 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 many 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 ; g being 981, and a mile an hour being 44'7 cm. per sec.? 100. A heavy particle moves freely along a frictionless tube which forms a vertical circle of radius a. Find the velocity which the particle will have at the lowest point, if it all but comes to rest at the highest. Also find its velocity at the lowest point if in passing the highest point it exerts no pressure against the tube. [Use the principle that what is lost in energy of position is gained in energy of motion.] 101. Show that the total intensity of centrifugal force due to the earth's rotation, at a place in latitude A, is w 2 E cos X, u denoting ^, and E the earth's radius ; that the vertical component (tending to diminish gravity) is w 2 E cos 2 A, and that the horizontal component (directed from the pole towards the equator) is w 2 E cos A sin A. 24G EXAMPLES. PENDULUM, AND MOMENT OF INERTIA. 101*. The length of the seconds pendulum at Greenwich is 99'413 cm.; find the length of a pendulum which makes a single vibration in lj sec. 102. The weight of a fly-wheel is M grammes, and the distance of the inside of the rim from the axis of revolution is E centims. Supposing this distance to be identical with k ( 117), find the moment of inertia. If a force of F dynes acts steadily upon the wheel at an arm of a centims., what will be the value of the angular velocity - after the lapse of t seconds from the commencement of motion ? 103. For a uniform thin rod of length a, swinging about a point of suspension at one end, the moment of inertia is the mass of the rod multiplied by |a 2 . Find the length of the equivalent simple pendulum ; also the moment of inertia round a parallel axis through the centre of the rod. 104. At what point in its length must the rod in last question be suspended to give a minimum time of vibration : and at what point must it be suspended to give the same time of vibration as if suspended at one end? 105. Show that if P be the mass of the pulley in Atwood's machine, r its radius, and P 2 its moment of inertia, the value of C in 100 will be P JJ plus the mass of the string. [The mass of the friction-wheels is neglected.] 106. A body moves with constant velocity in a vertical circle, going once round per second ; and its shadow is cast upon level ground by a vertical sun. Find the value of ft, ( 111) for the shadow, using the centimetre and second as units. 107. "What is the value of ft for one of the prongs of a C tuning-fork which makes 512 complete vibrations per second? PRESSURE OF LIQUIDS. Find, in gravitation measure (grammes per sq. cm.), atmospheric pressure being neglected : 108. The pressure at the depth of a kilometre in sea-water of density 1'025. 109. The pressure at the depth of 65 cm. in mercury of density 13-59. 110. The pressure at the depth of 2 cm. in mercury of density 13'59 sur- mounted by 3 cm. of water of unit density, and this again by li cm. of oil of density '9. Find, in centimetres of mercury of density 13'6, atmospheric pressure being included, and the barometer being supposed to stand at 70 cm.: 111. The pressure at the depth of 10 metres in water of unit density. 112. The pressure at the depth of a mile in sea-water of density 1-026 a mile being 160933 cm. Find, in dynes per square centimetre, taking g as 981 : 113. The pressure due to 1 cm. of mercury of density 13-596. EXAMPLES. 247 114. The pressm-e due to a foot of water of unit density, a foot being 30-48 cm. 115. The pressure due to the weight of a layer a metre thick, of air of density 00129. 116. At what depth, in brine of density I'l, is the pressure the same as at a depth of 33 feet in water of unit density ? 117. At what depth, in oil of density '9, is the pressure the same as at the depth of 10 inches in mercury of density 13'596? 118. With what value of g will the pressure of 3 cm. of mercury of density 13-596 be 4 x 10 4 1 Find, in grammes weight, the amount of pressure (atmospheric pressure being neglected) : 119. On a triangular area of 9 sq. cm. immersed in naphtha of density '848; the centre of gravity of the triangle being at the depth of 6 cm. 120. On a rectangular area 12 cm. long, and 9 cm. broad, immersed in mercury of density 13-596 ; its highest and lowest corners being at depths of 3 cm. and 7 cm. respectively. 121. On a circular area of 10 cm. radius, immersed in alcohol of density "791, the centre of the circle being at the depth of 4 cm. 122. On a triangle whose base is 5 cm. and altitude 6 cm., the base being at the uniform depth of 9 cm., and the vertex at the depth of 7 cm., in water of unit density. 123. On a sphere of radius r centimetres, completely immersed in a liquid of density d; the centre of the sphere being at the depth of h centimetres. [The amount of pressure in this case is not the resultant pressure.] DENSITY, AND PRINCIPLE OF ARCHIMEDES. Densities are to le expressed in grammes per cullc centimetre. 124. A rectangular block of stone measures 86 x 37 x 16 cm., and weighs 120 kilogrammes. Find its density. 125. A specific-gravity bottle holds 100 gm. of water, and 180 gm. of sulphuric acid. Find the density of the acid. 126. A certain volume of mercury of density 13'6 weighs 216 gm., and the same volume of another liquid weighs 14'8 gm. Find the density of this liquid. 127. Find the mean section of a tube 16 cm. long, which holds 1 gm. of mercury of density 13'6. 128. A bottle filled with water, weighs 212 gm. Fifty grammes of filings are thrown in, and the water which flows over is removed, still leaving the bottle just filled. The bottle then weighs 254 gm. Find the density of the filings. 129. Find the density of a body which weighs 58 gm. in air, and 46 gm. in water of unit density. 130. Find the density of a body which weighs 63 gm. in air, and 35 gm. in a liquid of density "85. 248 EXAMPLES. 131. A glass ball loses 33 gm. when weighed in water, and loses G gm. more when weighed in a saline solution. Find the density of the solution. 132. A body, lighter than water, weighs 102 gm. in air; and when it is im- mersed in water by the aid of a sinker, the joint weight is 23 gm. The sinker alone weighs 50 gm. in water. Find the density of the body. 133. A piece of iron, when plunged in a vessel full of water, makes 10 grammes ran over. When placed in a vessel full of mercury it floats, displacing 78 grammes of mercury. Required the weight, volume, and specific gravity of the iron. 134. Find the volume of a solid which weighs 357 gm. in air, and 253 gm. in water of unit density. 135. Find the volume of a solid which weighs 458 gm. in air, and 409 gm. in brine of density 1'2. 136. How much weight will a body whose volume is 47 cubic cm. lose, by weighing in a liquid whose density is 2'5? 137. Find the weights in air, in water, and in mercury, of a cubic cm. of gold of density 19'3. 138. A wire 1293 cm. long loses 508 gm. by weighing in water. Find its mean section, and mean radius. 139. A copper wire 2156 cm. long weighs 158 gm. in air, and 140 gm. in water. Find its volume, density, mean section, and mean radius. 140. What will be the weights, in air and in water, of an iron wire 1000 cm. long and a millimetre in diameter, its density being 7'7? 141. How "much water will be displaced by 1000 c.c. of oak of density '9, floating in equilibrium] 142. A ball, of density 20 and volume 3 c.c., is surmounted by a cylindrical stem, of density 2'5, of length 12 cm., and of cross section | sq. cm. What length of the stem will be in air when the body floats in equilibrium in mercury of density 13'6l 143. A hollow closed cylinder, of mean density '4 (including the hollow space), is weighted with a ball of volume 5, and mean density 2. What must be the volume of the cylinder, that exactly half of it may be immersed, when the body is left to itself in water? 144. A long cylindrical tube, constructed of flint glass of density 3, is closed at both ends, and is found to have the property of remaining at whatever depth it is placed in water. If the mass of the ends can be neglected, show that the ratio of the internal to the external radius is */ -r- 145. A glass bottle provided with a stopper of the same material weighs 120 gm. when empty. When it is immersed in water, its apparent weight is 10 gm., but when the stopper is loosened and the water let in, its apparent veight is 80 gm. Find the density of the glass and the'capacity of the bottle. 146. A hydrometer sinks to a certain depth in a fluid of density -8; and if 100 gm. be placed upon it, it sinks to the same depth in water. Find the weight of the hydrometer. 147. Find the mean density of a combination of 8 parts by volume of a sub- stance of density 7, with 19 of a substance of density 3. EXAMPLES. 249 148. Find the mean density of a combination of 8 parts by weight of a sub- stance of density 7, with 19 of a substance of density 3. 149. "What volume of fir, of density '5, must be joined to 3 c.c. of iron, of density 7'1, that the mean density of the whole may be unity? 150. What mass of fir, of density '5, must be joined to 300 gm. of iron, of density 7'1, that the mean density of the whole may be unity? 151. Two parts by volume of a liquid of density '8, are mixed with 7 of water, and the mixture shrinks in the ratio of 21 to 20. Find its density. 152. A piece of iron of density 7'5 floats in mercury of density 13'5, and is completely covered with water which rests on the top of the mercury. How much of the iron is immersed in the mercury? 153. Two liquids are mixed. The total volume is 3 litres, with a sp. gr. of 0-9. The sp. gr. of the first liquid is 1'3, of the second 07. Find their volumes. 154. What volume of platinum of density 21 '5 must be attached to a litre of iron of density 7'5 that the system may float freely at all depths in mercury of density 13'5? 155. What must be the thickness of a hollow sphere of platinum with an ex- ternal radius of 1 decim., that it may barely float in water? 156. A sphere of cork of density "24, 3 cm. in radius, is weighted with a sphere of gold of density 19'3. What must be the radius of the latter that the system may barely float in alcohol of density '8? 157. An alloy of gold and silver has density D. The density of gold is d, that of silver d'. Find the proportions by weight of the two metals in the alloy, sup- posing that neither expansion nor contraction occurs in its formation. 158. A mixture of gold, of density 19'3, with silver, of density 10'5, has the density 18. Assuming that the volume of the alloy is the sum of the volumes of its components, find how many parts of gold it contains for one of silver (a) by volume; (b) by weight. 159. A body weighs t ^M dynes in air of density A, gm in water, and gx in vacuo. Find x in terms of M, m, and A. CAPILLARITY. 1GO. A horizontal disc of glass is held up by means of a film of water between it and a similar disc of the same or a larger size above it. If E, denote the radius of the lower disc, d the distance between the discs, which is very small compared with E, T the surface tension of water, show that the weight of the lower disc together with that of the water between the discs is approximately equal to -j [The disc of water will be concave at the edge, and the radius of curvature of the concavity may be taken as %d.] 161. The surface-tension of water at 20 C. is 81 dynes per linear centim. How high will water be elevated by capillary action in a wetted tube whose dia- meter is half a millimetre? 250 EXAMPLES. 162. Uow much will mercury be depressed by capillary action in a glass tube of half a millimetre diameter, the surface-tension of mercury at 20 C. being 41 dynes per cm., its density 13'54, and the cosine of the angle of contact '703 163 Show by the method of 186 that the capillary elevation or depression will be the same in a square tube as in a circular tube whose diameter is equal a side of the square. 164. Two equal discs in a vertical position have a film of water between them sustained by capillary action. Show that if the water at the lowest point is at atmospheric pressure, the water at the centre of the discs is at a pressure less than atmospheric by rg dynes per sq. cm., r being the common radius of the discs in cm.; and that the discs are pressed together with a force of * 1*g dynes. BAROMETER, ASD BOYLE'S LAW. 165. A bent tube, having one end open and the other closed, contains mercury which stands 20 cm. higher in the open than in the closed branch. Compare the pressure of the air in the closed branch with that of the external air; the baro- meter at the time standing at 75 cm. 166. The cross sections of the open and closed branches of a siphon barometer are as 6 to 1. What distance will the mercury move in the closed branch, when a normal barometer alters its reading by 1 inch? 167. If the section of the closed limb of a siphon barometer is to that of the open limb as a to b, show that a rise of 1 cm. in the mercury in the closed limb corresponds to a rise of ^^ cm. of the theoretical barometer. 168. Compute, in dynes per sq. cm., the pressure due to the weight of a column of mercury 76 cm. high at the equator, where g is 978, and at the pole, where g is 983. 169. The volimes of a given quantity of mercury at C. and 100 C. are as 1 to 1-0182. Compute the height of a column of mercury at 100, which will produce the same pressure as 76 cm. of mercury at 0. 170. The volumes of a given mass of mercury, at and 20, are as 1 to 1'0036. Find the height reduced to 0, when the actual height (in true centimetres), at a temperature of 20, is 76'2. 171. In performing the Torricellian experiment a little air is left above the mercury. If this air expands a thousandfold, what difference will it make in the height of the column of mercury sustained when a normal barometer reads 76 cm.? 172. In performing the Torricellian experiment, an inch in length of the tube is occupied with air at atmospheric pressure, before the tube is inverted. After the inversion, this air expands till it occupies 15 inches, while a column of mercury 28 inches high is sustained below it. Find the true barometric height. 173. The mercury stands at the same level in the open and in the closed branch of a bent tube of uniform section, when the air confined at the closed end is at the pressure of 30 inches of mercury, which is the same as the pressure of the external air. Express, in atmospheres, the pressure which, acting on the surface of the mercury in the open branch, compresses the confined air to half its original EXAMPLES. 251 volume, and at the same time maintains a difference of 5 inches in the levels of the two mercurial columns. 174. At what pressure (expressed in atmospheres) will common air have the same density which hydrogen has at one atmosphere ; their densities when com- pared at the same pressure being as 1276 to 88'4? 175. Two volumes of oxygen, of density -00141, are mixed with three of nitrogen, of density '00124. Find the density of the mixture (a) if it occupies five volumes; (b) if it is reduced to four volumes. 176. The mass of a cub. cm. of air, at the temperature C., and at the pressure of a million dynes to the square cm., is '0012759 gramme. Find the mass of a cubic cm. of air at C., under the pressure of 76 cm. of mercury (a) at the pole, where g is 983 - l; (b) at the equator, where g is 978'1 ; (c) at a place where g is 981. 177. Show that the density of air at a given temperature, and under the pressure of a given column of mercury, is greater at the pole than at the equator by about 1 part in 196 ; and that the gravitating force of a given volume of it is greater at the pole than at the equator by about 1 part in 98. 178. A cylindrical test-tube, 1 decim. long, is plunged, mouth downwards, into mercury. How deep must it be plunged that the volume of the inclosed air may be diminished by one-half? 179. The pressure indicated by a siphon barometer whose vacuum is defective is 750 mm., and when mercury is poured into the open branch till the barometric chamber is reduced to half its former volume, the pressure indicated is 740 mm. Deduce the true pressure. 180. An open manometer, formed of a bent tube of iron whose two branches are parallel and vertical, and of a glass tube of larger size, contains mercury at the same level in both branches, this level being higher than the junction of the iron with the glass tube. What must be the ratio of the sections of the two tubes, that the mercury may ascend half a metre in the glass tube when a pres- sure of 6 atmospheres is exerted in the opposite branch? 181. A curved tube has two vertical legs, one having a section of 1 sq. cm., the other of 10 sq. cm. "Water is poured in, and stands at the same height in both legs. A piston, weighing 5 kilogrammes, is then allowed to descend, and press with its own weight upon the surface of the liquid in the larger leg. Find the elevation thus produced in the surface of the liquid in the smaller leg. PUMPS, &c. 182. The sectional area of the small plunger in a Bramah press is 1 sq. cm., and that of the larger 100 sq. cm. The lever handle gives a mechanical advan- tage of 6. What weight will the large plunger sustain when 1 cwt. is hung from the handle? 183. The diameter of the small plunger is half an inch; that of the larger 1 foot. The arms of the lever handle are 3 in. and 2 ft. Find the total mechan- ical advantage. 184. Find, in grammes weight, the force required to sustain the piston of a suction-pump without friction, if the radius of the piston be 15 cm., the depth 252 EXAMPLES. from it to the surface of the water in the well 600 cm., and the height of the column of water above it 50 cm. Show that the answer does not depend on the size of the pipe which leads down to the well. 185. Two vessels of water are connected by a siphon. A certain point P in its interior is 10 cm. and 30 cm. respectively above the levels of the liquid in the two vessels. The pressure of the atmosphere is 1000 grammes weight per sq. cm. Find the pressure which will exist at P (a) if the end which dips in the upper vessel be plugged ; (6) if the end which dips in the lower vessel be plugged. 186. If the receiver has double the volume of the barrel, find the density of the air remaining after 10 strokes, neglecting leakage, &c. 187. Air is forced into a vessel by a compression pump whose barrel has j^th of the volume of the vessel. Compute the density of the air in the vessel after 20 strokes. 188. In the pump of Fig. 136 show that the excess of the pressure on the upper above that on the lower side of the piston, at the end of the first up-stroke, is y y y-, of an atmosphere [in the notation of 230]; and hence that the first stroke is more laborious with a small than with a large receiver. 189. In Tate's pump show that the pressure to be overcome in the first stroke is nearly equal to an atmosphere during the greater part of the stroke ; and that, when half the air has been expelled from the receiver, the pressure to be over- come varies, in different parts of the stroke, from half an atmosphere to an atmo- sphere. ANSWEES TO EXAMPLES. Ex. 1. 14-14. Ex. 2. 13. Ex. 3. 7'07 each. Ex. 4. 10. Ex. 5. 141-4. Ex. 6. 70'7 each. Ex. 7. Introduce a force equal and opposite to the resultant. Then we have three forces making angles of 120 with each other. Ex. 9. Equal to one of the forces. Ex. 10*. 28. Ex. 11*. 40 in. from smaller weight. Ex. 12. 60 Ibs. by A, 40 Ibs. by B. Ex. 13. 2? Ibs. Ex. 14. 2 Ibs. Ex. 15. 15 in. from centre. Ex. 16. 121 Ibs. Ex. 17. 32 Ibs. Ex. 18. 10'4 ft. nearly. Ex. 19. 6 Ibs. Ex. 20. 2? ft. from end. Ex. 21. 2}f. Ex. 22. 2 units acting at distance of 5 yards trom the greater force. Ex. 23. 6 ft. from the end ; pressure 2 units. Ex. 24. 4| Ibs Ex. 25. 2fr Ibs, lOf Ibs. Ex. 26. f in. Ex. 27. 2ft- in. Ex. 28. I of side of square. Ex. 29. & of diagonal of large square. Ex. 30. wVV cm. from +T+ p s here ' Ex ' 3L Denoting side of s( i uare b y a > distance from AZ distance froi n Ex. 34. 4& cm. Ex. 35. 5& cm. Ex. 3G. 17. Ex. 37. -1, 0, + 18. Ex. 38. i\V (V(& 2 + c*)-c). Ex. 39. fWL Ex. 40. a (II-/,)*. EXAMPLES. 253 Ex. 44. 14 Ibs. Ex. 45. (a) 133 J Ibs.; (6) 166f Ibs. Ex. 46. 1 to 603 nearly. Ex. 47. 373J. Ex. 49. 50. Ex. 50. 72. Ex. 61. 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 r fa, larger y^s cm. per sec. Ex. 58. Inversely as masses of bullet and gun. Ex. 59. Mass of bullet is j^ 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. 1062i cm. below starting point. Ex. 70. 384 ft, per sec. Ex. 71. 2304 ft. Ex. 72. 1& 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. ^ 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", V5^*7 Ex. 101*. 223-679 cm. Ex. 102. ME 2 , ~ Ex. 103. fa; mass of rod multi- plied by ^a 2 . Ex. 104. At either of the two points distant -p from centre ; at either of the two points distant -j from centre. Ex. 106. (^) 2 = 39-48. Ex. 107. (1024 w) 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. 12 ft. 7 in. Ex. 118. 980-68. Ex. 119. 4579. Ex. 120. 7342. Ex. 121. 994. Ex. 122. 125. Ex. 123. 4*r 2 M Ex. 124. 2-357. Ex. 125. 1'8. Ex. 126. '932. Ex. 127. '0046 sq. cm. Ex. 128. 6-25. Ex. 129. 4|. Ex. 130. 1'9125. Ex. 131. 1ft. Ex. 132. f|. Ex. 133. 10 cub. cm., 78 gra., 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^V = 4-185. Ex. 148. 3ft 6 T = 3'6115. Ex. 149. 36'6 c.c. Ex. 150. 2577 gm. Ex. 151. 1-0033. Ex. 152. if of the iron. Ex. 153. 1 lit. of first, 2 lit. of second. Ex. 154. | of a litre. Ex. 155. !-%/*- decim. = -158 cm. Ex. 156. ^4^1 = v 4d 18 D 935 cm. Ex. 157. Gold: silver : : --^ ; : g-J- Ex. 158. (a) 577, (6) 10'6. Ex. 159. M ,""! A - 254 EXAMPLES. Ex. 161. 6'6 cm. nearly. Ex. 162. 1*77 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 g, and therefore gd varies as # 2 . Ex. 178. Its top must be 76-5 = 71 cm. deep. Ex. 179. 760 m. Ex. 180. 33 to 5. Ex. 181. 454j 6 r cm. Ex. 182. 30 tons. Ex. 183. 4608. Ex. 184. 459500 nearly. Ex. 185. (a) 970. (b) 990 gin. wt. per sq. cm. Ex. 186. ^ of an atmosphere, nearly. Ex. 187. 3 atmospheres. INDEX TO PAKT I. Absorption of gases, 177. Charts of weather, 163. Errors and corrections, signs of, Acceleration defined, 51. Circular motion, 59. 151- Air, weight of, 138, 139. Clearance, see untraversed space, Exhaustion, limit of, 188. pump, 179. 189, 213. , rate of, 180. chamber, 218. Coefficients of elasticity, 79. Expansibility of gases, 137. film, adherent, 177. of friction, 8 1. Alcoholimeters, 114, 115. Colloids, 135. Fahrenheit's barometer, 156. Am-plitude of vibration, 63. Communicating vessels, 118, 125. hydrometer, in. Aneroid, 154. Component along a line, 16. Fall in vacuo, 49. Annual and diurnal variations, 161. Components, 7. Falling bodies, 52. Archimedes' principle, 97. Compressed-air machines, 202. Film of air on solids, 177. Aristotle's experiment, 138. Compressibility, 79. Films, tension in, 127-130, 133, 134. Arithmetical lever, 12. Compressing pump, 199. Fire-engine, 218. Ascent in capillary tubes, 124, Conservation of energy, 74-76. Float-adjustment of barometer, 125, 128. Constant load, weighing with, 37. 147. Atmosphere, 140. Contracted vein, 225. Floatation, 102. standard of pressure, 141. Contractile film, 127-130, 133, 134. Floating needles, 103. Attractions, apparent, 133. Convertibility of centres, 70. Fluid, perfect, 83. Atwood's machine, 57. Corrections of barometer, 148-151. Force, 3. Axis of couple, 14. Counterpoised barometer, 155. , amount of, 44. of wrench, 15. Couple, 13. , intensity of, 44. Crystalloids, 135. , unit of, 44, 48. Babinet's air-pump, 196. Back-pressure on discharging ves- Cupped-leather collar, 222. Cycloidal pendulum, 67. Forcing-pump, 216. Fortin's barometer, 144. sel, 92, 225, 226. Cyclones, 165. Fountain in vacuo, 187. Balance, 34-40. , intermittent, 230. Balloons, 204-208. D'Alembert's principle, 96. Free-piston air-pump, 196. Barker's mill, 93. Deflecting force, 60. Friction, 81, 82. Barographs, 156, 158. Deleuil's air-pump, 196. in connection with conservation Barometer, 142. Density, absolute and relative, 105. of energy, 76. , corrections of, 148-151. , determination of, 106112. Froude on contracted vein, 225. Barometric measurement of , table of, xii. heights, 159-161. Depression, capillary, 124, 125, Galileo on falling bodies, 49. prediction, 163. 128. on suction by pumps, 142. Baroscope, 204. Despretz's experiments on Boyle'? Gases, expansibility of, 137. Beaume's hydrometers, 113. law, 168. Geissler's air-pump, 191. Bianchi's air-pump, 183. Dialysis, 135. Geometric decrease of pressure Bladder, burst, 187. Diameters, law of, 125. upwards, 160. Bourdon's gauge, 175. Diffusion, 135. Gimbals, 147. Boyle's law, 166. Displaced liquid defined, 100. Gradient, barometric, 164. tube, 166. Diurnal barometric curve, 161. Gramme, 105. Bramah press, 222. Diver, Cartesian, 101. Graphical interpolation, 116. Bubbles filled with hydrogen, 205. , tension and pressure in, 130. Double-acting pumps, 183, 218. Double-barrelled air-pump, i8t. Gravesande's apparatus, 7. Gravitation units of force, 4, 106. Buoyancy, centre of, 98, 100. Double exhaustion, 194. Gravity, apparent and true, 61. Buys Ballot's law, 164. weighing, 35. , centre of, 17-21. Drops, 131. , its velocity, 46. Caissons, 202. , formula for its intensity, 51. Camphor, movements of, 134. Dynamometer, 4. measured by pendulums, 72. Capillarity, 124-134. Dyne, 48. proportional to mass, 50. Cartesian diver, 101. Guinea-and-feather experiment. Cathetometer, 144. Efficiency of pumps, 214. 49- Centre of buoyancy, 98, 100. Efflux of liquids, 224. of gravity, I7 -2i. by experiment, 22. from air-tight spaces, 229. Egg in water, 100. Head of liquid, 224. Heights measured by barometer, , velocity of, 46. Elasticity, 77-80. 159-161. of mass, 47. Elevation, capillary, 124-128. Hemispheres, Magdeburg, 187. of oscillation, 71. Endosmose, 134. Homogeneous atmosphere, 159, of parallel forces, 10, 17. Energy, conservation of, 74-76. 1 60. of pressure, 93. Energy, kinetic, 73. " Horizontal " defined, 17. Centrifugal force, 60, 95. , static or potential, 73. Horse-power, xi. pump, 219. English air-pumps, 184. Hydraulic press, 87, 221. C.G.S. system, 48. Equilibrium, 4. tourniquet, 93. Change of momentum, 42. of motion, 42. Equivalent simple pendulum, 66. Erg, 48. Hydrodynamics, 83. Hydrogen, bubbles filled with, 205. 256 INDEX TO PART I. Hydrokinetics, 83. Hydrometers, 110-117. CErsted's piezometer, 79. Oscillation, centre of, 71. Siphon, 231. for sulphuric acid, 234. Siphon-barometer, 151. Hydrostatics, 83. Hypsometric formula, 161. Parachute, 207. Paradox, hydrostatic, 91. Specific gravity, 105. by weighing in water, 108. Immersed bodies, 98. Inclined plane, 32. Index errors and corrections, 151. Parallel forces, 9-14. Parallelogram of forces, 7, 43. of velocities, 43. Parallelepiped of forces, 8. flask, 107. , table of, xii. Spirit-levels, 120-123. Sprengel's air-pump, 193. u , 4 . Pascal's mountain experiment, 142. Spring-balance, 4. , moment ot, DO. Inexhaustible bottle, 230. Insects walking on water, 104. Intermittent fountain, 230. principle, 86. vases, 89. Pendulum, 62. , compound, 70. Stability, 21-28, 38. Standard kilogramme, 105. Statics, 4- Steelyard, 40. Isochronous vibrations, 66, 78. , cycloidal, 67. , isochronism of, 64. Suction, 211. pump, 211. Jet-pump, 219. Jets, liquid, 224. , simple, 62. , time of vibration of, 65. Period of vibration, 63. Sugar-boiling, 202. Superposed liquids, 83. Surface of liquids level, 85. Kater's pendulum, 71. " Perpetual motion," 26. Surface-tension, 127-130, 133, 134. Kinetic energy, 73. Phial of four elements, 89. , table of, 134. Kinetics, 4. Photographic registration, 157. King's barograph, 155, 156. Piezometer, 79. Tantalus' cup, 285. Kravogl's air-pump, 190. Pile-driving, 75 Torricellian experiment, 141. Laws of motion, 41-45. Levels, 119-123. Plateau's experiments, 131. Platinum causing igniiion of Torricelli's theorem on efflux, 224. Tourniquet, hydraulic, 93. Lever, 29. Limit to action of air-pump, 188. Liquids find their own level, 118. hydrogen, 177. Plunger, 216. Pneumatic despatch, 202. Trajectory, 54. Translation and rotation, 3. Transmission of pressure in fluids, in superposition, 88. Magdeburg hemispheres, 187. Potential energy, 73. Pressure, centre of, 93. , hydrostatic, 84. 86. Triangle of forces, 6. Twaddell's hydrometer, 114. Magic funnel, 229. Manometers, 172-175. Marine barometer, 153. , intensity of, 83. on immersed surfaces, 93. , reduction of, to absolute mea- Uniform acceleration, 50. Unit of force, 44, 48. Mariotte's bottle, 235. law, 166. tube, 166. Mass, 44, 45. and gravitation proportional, 50- Pressure-gauges, 172-175. Pressure-height defined, 159, 220. Pressure in air computed, 160. least where velocity is greatest, Units of measurement, 47. , C.G.S., 48. Unstable equilibrium, 21-28, 38. Untraversed space, 189, 213. Upward pressure in liquids, 88. , centre of, 47. Mechanical advantage, 30. Principle of Archimedes, 97. Projectiles, 53. Vena contracta, 225. Vernier, 145. powers, 29-33. Mechanics, 2. Meniscus, 131. Pulleys, 31. Pump, forcing, 216. , suction, 211. "Vertical "denned, 17. Vessels in communication, 118, 125. Metacentre, 103. Pumps, efficiency of, 214. Vibrations, 66. Metallic barometer, 155. , when small, isochronous, 78. Mixtures, density of, 115. of gases, 176. Quantity of matter, 45. Volumes measured by weighing in Moduli of elasticity, 78. Range and amplitude, 161. ' Moment of couple, 13. Rarefaction, limit of, 188. Water, compressibility of, 79. of force about point, n. , rate of, 180. level, 119. of inertia, 68. Reaction, 4, 15, 45. supply of towns, 118. Momentum, 44. of issuing jet, 92, 225, 226. Wedge, 33- Morin's apparatus, 55. Rectangular components, 15. Weighing, double, 35. Motion, laws of, 41-45. Regnault's experiments on Boyle's in water, 108. Motions, composition of, 42. law, 169-172. with constant load, 37. Mountain-barometer, theory of, Resistance of the air, 49, 53. Weight affected by air, 209. 159-161. Resolution, 15. "Weight" ambiguous, 106. Multiple-tube barometer, 157. Resultant, 7. Wheel and axle, 30. manometer, 172. Rigid body, 5. Wheel-barometer, 152. Natural history and natural phi- Rotating vessel of liquid, 93. Whirling vessel of liquid, 95. Work, 22-25. losophy, i. Needles floating, 103. Screw, and screw-press, 33. Second law of motion, 42. in producing motion, 52. , principle of, 25. Newton's experiments with pen- Sensibility and instability, 38. Wrench, 15. dulums, 50. laws of motion, 41-45. Nicholson's hydrometer, in. Sensibility of balance, 35. Simple-harmonic motion, 65. Simple pendulum, 62. Young's modulus, 78. Zero, errors of, 151. REGIONAL LIBRARY FA 000 940 772 7 -'"CALIFORNIA, LIBRARY