THIS BOOK NO LONGER PROPERTY Of SUTRO BRANCH CALIFORNIA STATE LIBRARY ENCYCLOPEDIA METROPOLITAN!; OE, JSgsiem 0f ttniteal Jhtofole&ge : y A METHODICAL PLAN PROJECTED BY SAMUEL TAYLOE COLEEIDGE. SECOND EDITION, REVISED. APPLIED MECHANICS. SERIES OF PRACTICAL MANUALS CIVIL ENGINEERING AND MECHANICS. Crown 8vo. In course of Publication. APPLIED MECHANICS; COMPRISING PRINCIPLES OF STATICS, CINEMATICS, AND DYNAMICS, AND THEORY OF STRUCTURES, MECHANISM, AND MACHINES. With numerous Illustrations, price 12s. 6d. THE STEAM ENGINE, AND OTHER PRIME MOVERS. With numerous Tables and Illustrations. CIVIL ENGINEERING; COMPRISING ENGINEERING SURVEYS, EARTHWORK, FOUNDATIONS, MASONRY, CARPENTRY, METAL- WORK, ROADS, RAILWAYS, CANALS, RIVERS, WATEE- WORKS, HARBOURS, &c. With numerous Tables and Illustrations. MACHINERY AND MILL WORK; COMPRISING GEOMETRY OP MACHINES, MOTIONS OP MACHINES, WORK OF MACHINES, STRENGTH OF MACHINES, CONSTRUCTION OP MACHINES, OBJECTS OF MACHINES, &c. With numerous Tables and Illustrations. USEFUL RULES A N ; D TABLES FOR ARCHITECTS, BUILDERS, CARPENTERS, ENGINEERS, FOUNDERS, MECHANICS, SHIPBUILDERS, AND SURVEYORS. A MANUAL OF a APPLIED MECHANICS! BY WILLIAM JOHN MACQDORN^AMINE, LL.D., C.E., F.R.SS. L. & E., &c. ; PRESIDENT OF THE INSTITUTION OF ENGINEEKS IN SCOTLAND, AND REGIUS PBOFESSOR OF CIVIL ENGINEERING AND MECHANICS IN THE UNIVERSITY OF GLASGOW. PB V4pD CAT FOR ENVIRON DESIGN LONDON AND GLASGOW: RICHARD GRIFFIN AND COMPANY, ^nblis^jrs to fyt tttnifrersiig of (^Issgofcr. 1858. \The Author reserves the right of Translation] 1NVIBOMEHTAL UBRABl CLASS 1 2 (3 PB 1 /4BD BUCK CAT FOR ENVIRON DESIGN GLASGOW: FEINTED BY BELL AND BAIN. PREFACE. THE object of this book is to set forth in a compact form those parts of the Science of Mechanics which are practically applicable to Structures and Machines. Its plan is sufficiently explained by the Table of Contents, by the Introduction, and by the initial articles of the six parts into which the body of the treatise is divided. This work, like others of the same class, contains facts and principles that have been long and widely known, mingled with others, of which some are the results of the labours of recent discoverers, some have been published only in scientific Transac- tions and periodicals, not generally circulated, or in oral lectures, and some are now published for the first time. I have endea- voured, to the best of my knowledge, to mention in their proper places the authors of recent discoveries and improvements, and to refer to scientific papers which have furnished sources of infor- mation. A branch of Mechanics not usually found in elementary treatises is explained in this work, viz., that which relates to the equili- brium of stress, or internal pressure, at a point in a solid mass, and to the general theory of the elasticity of solids. It is the basis of a sound knowledge of the principles of the stability of earth, and of the strength and stiffness of materials ; but, so far as I know, the only elementary treatise on it that has hitherto been published is that of M. Lame*, entitled Leqom sur la Theorie mathematique de V Elasticite des Corps solides. 00129 VI PREFACE. In treating of the stability of arches, the lateral pressure of the load is taken into account. So far as I know, the only author who has hitherto done so in an exact manner, is M. Yvon-Villarceaux, in the Memoires des Savans etrangers. The principle of the transformation of structures and its appli- cations have hitherto appeared in the Proceedings of the Royal Society alone. The correct laws of the flow of elastic fluids (first investigated by Dr. Joule and Dr. Thomson), and the true equations of the action of steam arid other vapours against pistons, as deduced from the principles of thermodynamics, by Professor Clausius and myself, contemporaneously, are now for the first time stated and applied in an elementary manual. Other portions of the work, which are wholly or partly new, are indicated in their places. In the arrangement of this treatise an effort has been made to adhere as rigidly as possible to a methodical classification of its subjects; and, in particular, care has been taken to keep in view the distinction between the comparison of motions with each other, and the relations between motions and forces, which was first pointed out by Monge and Ampere, and which Mr. Willis has so successfully applied to the subject of mechanism. The observing of that distinction is highly conducive to the correct understanding and ready application of the principles of Mechanics. W. J. M. R. GLASGOW UNIVERSITY, Afay, 1858. CONTENTS. PRELIMINARY DISSERTATION ON THE HARMONY OF THEORY AND PRACTICE IN MECHANICS, INTRODUCTION, Definition of General Terms, and Division of the bubject. Article Page 1. Mechanics, . . . . . 1 2. Applied Mechanics, . . .13 3. Matter, 13 4. Bodies, Solid, Liquid, Gaseous, 5. Material or Physical Volume, . 13 6. Material or Physical Surface, . 13 7. Line, Point, Physical Point, Mea- sures of Length, . . .13 Page 1 13 8. Rest, 14 9. Motion, 14 10. Fixed Point, ... 14 11. Cinematics, ... 15 12. Force, 13. Equilibrium or Balance, 14. Statics and Dynamics, 15. Structures and Machines, 16. General Arrangement of the Subject, 16 PART I. PRINCIPLES OF STATICS. CHAPTER I. BALANCE AND MEASUREMENT OF FORCES ACTING IN ONE STRAIGHT LINE. 17. Forces, how Determined, 17 18. Place of Application Point of Application, . . . .17 19. Supposition of Perfect Rigidity, . ] 20. Direction Line of Action, . . 18 21. Magnitude Units of Force, British and French, . . . .18 22. Resultant of Forces acting in one Line,18 23. Representation of Forces by Lines, 19 24. Pressure, 20 CHAPTER II. THEORY OF COUPLES AND OF THE BALANCE OF PARALLEL FORCES. SECTION 3. On Parallel Forces. 38. Balanced Parallel Forces in general, 25 SECTION 1. On Couples with the Same Axis. 21 25. Couples, ..... 26. Force of a Couple Arm or Lever- age Moment, . . . 27. Tendency of a Couple Plane and Axis of a CoupleRight-handed and Left-handed Couples, . 28. Equivalent Couples of equal Force and Leverage, . . . 29. Moment of a Couple, . . . 30. Addition of Couples of equal Force, 31. Equivalent Couples of equal Mo- ment, ..... 32. Resultant of Couples with the same Axis ...... 33. Equilibrium of Couples having the same Axis, . . . . 34. Representation of Couples by Lines, 23 SECTION 2. On Couples with Different Axes. 35. Resultant of two Couples with different Axes, . .24 36. Equilibrium of three Couples with different Axes in the same Plane, 25 37. Equilibrium of any number of Couples, . . . .25 21 39. Equilibrium of three Parallel Forces in one Plane Principle of Lever, 26 40. Resultant of two Parallel Forces, . 26 41. Resultant of a Couple, and a single Force in Parallel Planes, . . 27 42. Moment of a Force with respect to an Axis, . . . .27 43. Equilibrium of any system of Par- allel Forces in one Plane, . 28 44. Resultant of any number of Paral- lel Forces in one Plane, . . 28 45. Moments of a Force with respect to a pair of Rectangular Axes, 29 46. Equilibrium of any system of Par- allel Forces, . . . - 30 47. Resultant of any number of Paral- lel Forces, . . . .30 SECTION 4. On Centres of Parallel Forces. 48. Centre of a pair of Parallel Forces, 31 49. Centre of any system of Parallel Forces, 32 50. Co-ordinates of Centre of Parallel Forces, 32 51. Parallelogram of Forces, CHAPTER III BALANCE OF INCLINED FORCES. 52. Equilibrium of three Forces acting through one Point in one Plane, 53. Equilibrium of any system of 35 Forces acting through one Point, SECTION 1. Inclined Forces applied at One Point. Vlll CONTENTS. 54. Parallelepiped of Forces, 55. Resolution of a Force into two Components, . 56. Resolution of a Force into three Components, . 57. Rectangular Components, . CHAPTER IV ON Page 37 SECTION 2. Inclined Forces applied to a System of Points. 58. Forces acting in one Plane, Page Graphic Solution, ... 39 59. Forces acting in one Plane, Solu- tion by Rectangular Co-ordinates, 40 60. Any system of Forces, . . 41 PARALLEL PROJECTIONS IN STATICS. 61. Parallel Projection of a Figure de- fined, 45 62. Geometrical Properties of Parallel Projections, . . . .45 63. Application to Parallel Forces, . 46 64. Application to Centres of Parallel Forces, 46 65. Application to Inclined Forces acting through one Point, . 46 66. Application to any system of Forces, 47 CHAPTER V. ON DISTRIBUTED FORCES. G7. Restriction of the subject to Par- allel Distributed Forces, . . 48 68. Intensity of a Distributed Force, 48 SECTION I. Of Weight, and Centres of Gravity. 69. Specific Gravity, ... 49 70. Centre of Gravity, ... 49 71. Centre of Gravity of a Homoge- neous Body having a Centre of Figure, 49 72. Bodies having Planes or Axes of Symmetry, 49 50 51 53 53 54 54 56 57 58 61 63 68 68 SECTION 2. Of Stress, and its Resultants and Centres. 86. Stress its Nature and Intensity, 68 87. Classes of Stress, . . . 69 88. Resultant of Stress, its Magni- tude, .... 70 89. Centre of Stress, or of Pressure, 71 90. Centre of Uniform Stress, . 73 91. Moment of Uniformly -Varying Stress, Neutral Axis, . 73 92. Moment of Bending Stress, 75 93. Moment of Twisting Stress, 76 94. Centre of Uniformly - Varying Stress, Conjugate Axis, . 76 95. Moments of Inertia of a Surface, 77 73. System of Symmetrical Bodies, . 74. Homogeneous 'Body of any Figure, 75. Centre of Gravity found by Addi- tion, 76. Centre of Gravity found by Sub- traction, 77. Centre of Gravity altered by Trans- position, . . . . 78. Centres of Gravity of Prisms and Flat Plates, . . . . 79. Body with similar Cross-Sections, 80. Curved Rod, . . . . 81. Approximate Computation of In- tegrals, 82. Centre of Gravity found by Pro- jection, 83. Examples of Centres of Gravity, 84. Heterogeneous Body, ... 85. Centre of Gravity found experi- mentally, . . . . SECTION 3. Of Internal Stress: its Composition and Resolution. 96. Internal Stress in general, . . 82 97. Simple Stress and its Normal In- tensity, 83 98. Reduction of Simple Stress to an Oblique Plane, ... 83 99. Resolution of Oblique Stress into Nor- mal and Tangential Components, 84 100. Compound Stress, . . .84 101. Pair of Conjugate Stresses, . 85 102. Three Conjugate Stresses, . 85 103. Planes of Equal Shear, or Tan- gential Stress, . . .87 104. Stress on three Rectangular Planes, 88 105 Tetraedron of Stress, 106. Transformation of Stress, . 107. Principal Axes of Stress, . 108. Stress Parallel to one Plane, 109. Principal Axes of Stress Parallel to one Plane, .... 110. Equal Principal Stresses Fluid Pressure, .... 111. Opposite Principal Stresses com- posing Shear, .... 112. Ellipse of Stress, Problems, . 113. Combined Stresses in one Plane, 90 92 93 95 98 101 101 110 SECTION 4. Of the Internal Equilibrium of Stress and Weight, and the principles of Hydrostatics. 114. Varying Internal Stress, . 112 115. Causes of Varying Stress, . 112 116. General Problem of Internal Equi- librium, . . . .113 117. Equilibrium of Fluids, . 116 118. Equilibrium of a Liquid, . 118 119. Equilibrium of different Fluids in contact with each other, . . 118 120. Equilibrium of a Floating Body, 120 121. Pressure on an Immersed Body, 122 122. Apparent Weights, . . . 123 123. Relative Specific Gravities, . 124 124. Pressure on an Immersed Plane, 125 125. Pressure in an Indefinite Uni- formly Sloping S )n the Parallel Solid, . 126 126. On the Parallel Projection of Stress and Weight, . . .127 CONTENTS. CHAPTER VI. ON STABLE AND UNSTABLE EQUILIBRIUM. Page 127. Stable and Unstable Equilibrium of a Free Body, . 128 128. Stability of a Fixed Body, Page . 128 PART II. THEORY OF STRUCTURES. CHAPTER I DEFINITIONS AND GENERAL PRINCIPLES. 129. Structures Pieces Joints, 130. Supports Foundations, . 129 129 131. Conditions of Equilibrium of a Structure, .... 129 132. Stability, Strength, and Stiffness, 130 133. Resultant Gross Load, . .131 134. Centre of Resistance of a Joint, 131 135. Line of Resistance, . . . 131 136. Joints Classed, . . . .131 SECTION 1. Equilibrium and Stability of Frames. 137. Frame, 132 138. Tie, 132 139. Strut, 133 140. Treatment of the Weight of a Bar, 133 141. Beam under Parallel Forces, . 133 142. Beam under Inclined Forces, . 134 143. Load supported by three Parallel Forces, 135 144. Load supported by three Inclined Forces, 135 145. Frame of Two Bars Equilibrium, 136 146. Frame of Two Bars Stability, 136 147. Treatment of Distributed Loads, 137 148. Triangular Frame, . . .137 149. Triangular Frame under Parallel Forces, 150. Polygonal Frame Equilibrium, 151. Open Polygonal Frame, 152. Polygonal Frame Stability, 153. Polygonal Frame under Parallel Forces, 154. Open Polygonal Frame under Parallel Forces, . . .142 155. Bracing of Frames, . . .142 156. Rigidity of a Truss, . . .144 157. Variations of Load on Truss, . 144 158. Bar common to several Frames, 145 159. Secondary Trussing Examples from Roofs, .... 145 160. Compound Trusses, . . .148 161. Resistance of Frame at a Section, 150 162. Half-lattice Girder, any Load, 153 163. Half-lattice Girder, Uniform Load 156 164. Lattice Girder, any Load, . 160 165. Lattice Girder, Uniform Load, 161 166. Transformation of Frames, 162 SECTION 2. Equilibrium of Chains, Cords, JRibs, and Linear Arches 167. Equilibrium of a Cord, . 162 168. Cord under Parallel Loads, 164 169. Cord under uniform Vertical Load,164 170. Suspension Bridge with Vertical Rods, . . . .168 171. Flexible Tie, . 169 CHAPTER II. STABILITY. 172. Suspension Bridge with Sloping Rods, 171, 173. Extrados and Intrados, . .173. 174. Cord with Horizontal Extrados, 175 175. Catenary, ..... 177 176. Centre of Gravity of a Flexible Structure, . . . .180 177. Transformation of Cords and Chains, 180 178. Linear Arches or Ribs, . . 182 179. Circular Arch for Uniform Fluid Pressure, . . : .183 180. Elliptical Arches for Uniform Pressures, .... 184 181. Distorted Elliptic Arch, . 186 182. Arches for Normal Pressure in general, 189 183. Hydrostatic Arch (see also 319A), 190 184. Geostatic Arches, . . .196 185. Stereostatic Arch, . . .198 186. Pointed Arches, . . .203 187. Total Conjugate Thrust of Linear Arches Point and Angle of Rupture, . . . ,203 188. Approximate Hydrostatic and Geostatic Arches, . . .207 SECTION 3. On Fractional Stability. 189. Friction distinguished from Adhesion, . . . .209 190. Law of Solid Friction, . . 209 191. Angle of Repose, . . .210 192. Table of Co-efficients of Friction and Angles of Repose, . . 211 193. Frictional Stabilityof Plane Joints,211 194. Frictional Stability of Earth, . 212 195. Mass of Earth with Plane Surface,214 196. Principle of Least Resistance, . 215 197. Earth Loaded with its ownWeight,216 198. Pressure of Earth against a Ver- tical Plane, . . . .218 199. Supporting Power of Earth- Foundations, . . . .219 200. Abutting Power of Earth, . 220 201. Table of Examples, . . . 221 202. Frictional Tenacity or Bond of Masonry and Brickwork, . 222 203. Friction of Screws, Keys, and Wedges, . . . .226 204. Friction of Rest and Friction of Motion, 226 138 139 140 140 141 CONTENTS. SECTION 4. On the Stability of Abut- \ ments and Vaults. Page 205. Stability at a Plane Joint, . 226 206. Stability of a series of Blocks Line of Resistance Line of Pressures, . . . .230 207. Analogy of Blockwork and Framework, . . . .231 208. Transformation of Blockwork Structures, . . . .232 209. Frictional Stability of a Trans- formed Structure, . . 233 210. Structure not laterally pressed, 211. Moment of Stability, 212. Abutments classed, . 213. Buttresses in general, 214. Rectangular Buttress, 215. Towers and Chimneys, 216. Dams or Reservoir- Walls, '217. Retaining Walls in general, 218. Rectangular Retaining Walls. 219. Trapezoidal Walls, . 233 235 235 238 240 243 249 252 254 Page 220. Battering Walls of Uniform Thickness, . . . .254 221. Foundation Courses of Retaining Walls, 255 222. Counterforts, . . . .255 223. Arches of Masonry, . . .256 224. Line of Pressure in an Arch Condition of Stability, . . 257 225. Angle, Joint,and Point of Rupture, 259 226. Thrust of an Arch of Masonry, . 260 227. Abutments of Arches, . . 261 228. Skew Arches, . . . .261 229. Groined Vaults, . . .262 230. Clustered Arches, . . .263 231. Piers of Arches, . . .263 232. Open and Hollow Piers and Abutments, . . . .263 233. Tunnels, 264 234. Domes, 265 235. Strength of Abutments and Vaults,268 235A. Transformation of Structures in Masonry, . . . .268 CHAPTER III. STRENGTH AND STIFFNESS. SECTION 1. Summary of General 236. Theory of Elasticity, 237. Elasticity defined, Fo . 270 . 270 . 270 . 270 . 271 . 271 . 272 238. Elastic Force or Stress, 239. Fluid Elasticity, 240. Liquid Elasticity, 241. Rigidity or Stiffness, . 242. Strain and Fracture, 243. Perfect and Imperfect Elasticity Plasticity, - . . .272 244. Strength, Ultimate and Proof- Toughness Stiffness Spring, 273 245. Determination of Proof Strength, 274 246. Working Stress, . . .274 247. Factors of Safety, . . .274 248. Divisions of Mathematical Theory of Elasticity, . . . .275 249. Resolution and Composition of Strains, 275 250. Displacements, .... 276 251. Analogy of Stresses and Strains, 276 252. Potential Energy of Elasticity, . 277 253. Co-efficients of Elasticity, . . 277 254. Co-efficients of Pliability, . . 277 255. Axes of Elasticity, . . . 278 256. Isotropic Solid, . . .278 257. Modulus of Elasticity, . . 279 258. Examples of Co-efficients, . . 279 259. General Problem of Internal Equilibrium of an Elastic Solid, 280 SECTION 2. On Relations between Strain and Stress. 260. Ellipse of Strain, . . .280 261. Ellipsoid of Strain, . . . 283 262. Transverse Elasticity of an Iso- tropic Substance, . . . 284 263. Cubic Elasticity, . . .285 264. Fluid Elasticity, . . .285 SECTION 3. On Resistance to Stretching and Tearing. 265. Stiffness and Strength of a Tie-bar, 286 266. Resilience, or Spring of a Tie-Bar Modulus of Resilience , . 287 267. Sudden Pull, . . . .287 268. Explanation of Table of the Re- sistance of Materials to Stretch- ing and Tearing. (see Appen- dix), 288 269. Additional data Welded Joint Iron Wire Ropes Hempen Cables Leather Belts Chain Cables 288 270. Strength of Rivetted Joints, . 288 271. Thin Hollow Cylinders Boilers Pipes, 272. Thin Hollow Spheres, 273. Thick Hollow Cylinder, . 274. Cylinder of Strained Rings, 275. Thick Hollow Sphere, 276. Boiler Stays, . 277. Suspension - Rod of Uniform Strength, . . . .297 SECTION 4. On Resistance to Shearing. 278. Condition of Uniform Intensity, 298 279. Explanation of Table of Resist- ance of Materials to Shearing and Distortion (see Appendix) 299 280. Economy of Material in Bolts and Rivets, 299 281. Fastenings of Timber Ties, . 301 SECTION 5. On Resistance to Direct Compression and Crushing. 282. Resistance to Compression, . 302 283. Modes of Crushing- Splitting- Shearing Bulging Buckling Cross-breaking, . . .302 289 290 290 294 295 296 284. Explanation of Table of the Re- sistance of Materials to Crush- ing by a Direct Thrust (see Appendix), . . . .301 285. Unequal Distribution of the Pressure, . . . .304 286. Limitations, .... 306 287. Crushing and Collapsing of Tubes, 306 SECTION 6. On Resistance to Bending and Cross- Breaking. 288. Shearing Force and Bending Moment 307 289. Beams Fixed at one end only, . 310 290. Beams Supported at both ends, 310 291. Moments of Flexure in terms of Load and Length, . . . 311 292. Uniform Moment of Flexure- Railway Carriage Axles, . . 312 293. Resistance of Flexure, . . 312 294. Transverse Strength of Beams in General, .... 315 295. Transverse Strength in Terms of Breadth and Depth, . . 316 296. Explanation of Table of Moduli or Rupture (see Appendix), 317 297. Modulus of Rupture of Cast Iron Beams, 318 298. Section of Equal Strength for Cast Iron Beams, . . . 319 299. Beams of Uniform Strength, . 320 300. Proof Deflection of Beams, . 322 301. Deflection found by Graphic Con- struction, . . . .326 302. Proportion of the greatest Depth of a Beam to the Span, . . 327 303. Slope and Deflection of a Beam under any load, . . . 328 304. Deflection with Uniform Moment, 330 305. Resilience or Spring of a Beam, 330 306. Suddenlyapplied Transverse Load, 332 307. Beam Fixed at both ends, . . 332 308. Beam Fixed at one end, and Sup- ported at both, . . .338 309. Shearing Stress in Beams, . 338 310. Lines of Principal Stress in Beams, 341 311. Direct Vertical Stress, . . 342 312. Small effect of Shearing Stress upon Deflection, . . . 342 313. Partially-loaded Beam, . . 344 314. Allowance for Weight of Beam, 346 CONTENTS. Page XI Page . 347 315. Limiting Length of Beam, 316. Sloping Beam, .... 343 317. Originally Curved Beam, . . 348 318. Expansion and Contraction of Long Beams, . . . .348 319. Elastic Curve, . . . .349 319A. Hydrostatic Arch, . . .353 SECTION 7 On Resistance to Twisting and Wrenching. 320. Twisting Moment, . . .353 321. Strength of a Cylindrical Axle, 353 322. Angle of Torsion of a Cylindrical Axle, 356 323. Resilience of a Cylindrical Axle, 357 324. Axles not Circular in Section, . 358 325. Bending and Twisting combined ; Crank and Axle, . . . 358 326. Teeth of Wheels, - . . .359 SECTION 8. On Crushing by Bending. 327. Introductory Remarks, . . 360 328. Strength of Iron Pillars and Struts, 361 329. Connecting Rods, Piston Rods, . 363 330. Comparison of Cast and Wrought Iron Pillars, . . , . .363 331. Mr. Hodgkinson's Formulas for the Ultimate Strength of Cast Iron Pillars, . . . .363 332. Wrought Iron Framework, . 364 333. Wrought Iron Cells, . . . 364 334. Sides of Plate Iron Girders, . 365 335. Timber Posts and Struts, . . 365 SECTION 9. On Compound Girders, Frames, and Bridges. 336. Compound Girders in General, . 366 337. Plate Iron Girders, . . .366 338. Half-Lattice and Lattice Beams, 369 339. Bowstring Girder, . . .369 340. Stiffened Suspension Bridges, . 370 341. Ribbed Arches, . . .376 SECTION 10. Miscellaneous Remarks on Strength and Stiffness. 342. Effects of Temperature, . . 376 343. Effects of Repeated Meltings on Cast Iron, . . . .376 344. Effects of Ductility, . . .376 345. Internal Friction, . . . 377 346. Concluding remarks on Strength and Stiffness, . . . .377 PART III. PRINCIPLES OF CINEMATICS, OR THE COMPARISON OF MOTIONS. 347. Division of the Subject, 379 CHAPTER I. MOTIONS OF POINTS. SECTION 1 Uniform Motion of a Pair of Points. 348. Fixed andNearly Fixed Directions, 379 349. Motion of a Pair of Points, . 380 350. Fixed Point and Moving Point, 381 351. Component andResultant Motions, 381 352. Measurement of Time, . . 381 353. Uniform Velocity, . . .382 354. Uniform Motion, . . .382 SECTION 2. Uniform Motion of several Points. 355. Motion of Three Points, . . 383 356. Motions of a Series of Points, . 383 Xll CONTENTS. 357. Parallelepiped of Motions, 358. Comparative Motion, Page . 384 . 384 SECTION 3. Varied Motion of Points. 359. Velocity and Direction of Varied Motion, 385 360. Components of Varied Motion, . 386 361. Uniformly- Varied Velocity, . 386 Page 362. Varied Rate of Variation of Velo- city, 387 363. Uniform Deviation, . . .387 364. Varying Deviation, . . .388 365. Resultant Rate of Variation, . 388 366. Rates of Variation of Component Velocities, .... 388 367. Comparison of Varied Motions, 389 CHAPTER II MOTIONS OF RIGID BODIES. SECTION 1 Rigid Bodies, and their Translation. 368. Rigid Bodies how understood, 390 369. Translation or Shifting, . . 390 SECTION 2. Simple Rotation. 370. Rotation defined Centre of Rotation, . . . .390 371. Axis of Rotation, . . .390 372. Plane of Rotation Angle of Ro- tation, 391 373. Angular Velocity, . . .391 374. Uniform Rotation, . . .391 375. Rotation common to all parts of body, 391 376. Right and Left-handed Rotation, 391 377. Relative Motion of a pair of Points in a Rotating Body, . . 392 378. Cylindrical Surface of equal Ve- locities, 392 379. Comparative Motions of two Points relatively to an Axis, . 393 389. Components of Velocity of a Point in a Rotating Body, . . 393 SECTION 3. Combined Rotations and Translations. 381. Property of all Motions of Rigid Bodies, . . . . .394 382. Helical Motion, . . .394 383. To find the Motion of a Rigid Body from the Motions of three of its Points, . . . .395 384. Special cases, . . . .396 385. Rotation combined with Transla- tion in the same Plane, . . 397 386. Rolling Cylinder Trochoids, . 398 387. Plane rolling on Cylinder Spiral Paths, . . " . . .398 388. Combined Parallel Rotations, . 399 389. Cylinder rolling on Cylinder Epitrochoids, . . . .400 390. Curvature of Epitrochoids, . 401 391. Equal and Opposite Parallel Ro- tations combined, . . . 404 392. Rotations about IntersectingAxes combined, .... 404 393. Rolling Cones, . . . .405 394. Analogy of Rotations and Single Forces, 405 395. Comparative Motions in Com- pound Rotations, . . . 406 SECTION 4. Varied Rotation. 396. Variation of Angular Velocity, . 406 397. Change of the Axis of Rotation, 407 398. Components of Varied Rotation, 407 CHAPTER III. MOTIONS OP PLIABLE BODIES, AND OF FLUIDS. 399. Division of the Subject, 408 SECTION I. Motions of Flexible Cords. 400. General Principles, . . .408 401. Motions classed, . . .409 402. Cord guided by Surfaces of Revo- lution, 409 SECTION 2. Motions of Fluids of Constant Density. 403. Velocity and Flow, . . .410 404. Principle of Continuity, . . 411 405. Flow in a Stream, . . .411 406. Pipes, Channels, Currents, and Jets, 411 407. Radiating Current, . . .412 408. Vortex, Eddy, or Whirl, . . 412 409. Steady Motion, .... 412 410. Unsteady Motion, . . .413 411. Motion of Piston, . . .413 412. General Differential Equations of Continuity, .... 413 413. General Differential Equations of Steady Motion, . . .414 414. General Differential Equations of Unsteady Motion, . . .415 415. Equations of Displacement, . 415 416. Wave, 416 417. Oscillation, . . . .416 SECTION 3. Motions of Fluids of Varying Density. 418. Flow of Volume and Flow of Mass, .... 417 419. Principle of Continuity, . 417 420. Stream, .... 418 421. Steady Motion, ... 419 422. Pistons and Cylinders, . 419 423. General Differential Equa- tions, 419 424. Motions of Connected Bodies, . 420 CONTENTS. PART IV. THEORY OF MECHANISM. Xlll CHAPTER I. DEFINITIONS Page 425. Theoryof Pure Mechanism defined,421 426. General Problem of Mechanism stated, 422 427. Frame Moving Pieces Con- nectors, ..... 422 428. Bearings, 422 429. Motions of'PrimaryMoving Pieces, 423 430. Motions of secondaryMovingPieces,423 AND GENERAL PRINCIPLES. Page 431. Elementary Combinations in Me- chanism, .... 423 432. Line of Connection, . . . 424 | 433. Principle of Connection, . . 424 434. Adjustments of Speed, . . 424 435. Train of Mechanism, . . 425 436. Aggregate Combinations in Me- chanism, .... 425 CHAPTER II. ON ELEMENTARY COMBINATIONS AND TRAINS OP MECHANISM. SECTION 1. Rolling Contact. 437. Pitch-Surfaces, . . .426 438. Smooth Wheels, Rollers, Racks, 426 439. General Conditions of Rolling Contact, . . . .426 440. Circular Cylindrical Wheels, . 427 441. Straight Rack and CircularWheel, 427 442. Bevel Wheels, . . . .428 443. Non-Circular Wheels, . . 428 SECTION 2. Sliding Contact. 444. Skew-Bevel Wheels, . . 430 445. Grooved Wheels, or Frictional Gearing, .... 431 446. Teeth of Wheels Definitions and General Principles, . . 432 447. Pitch and Number of Teeth, . 433 448. Hunting Cog, . . . .434 449. Trains of Wheelwork, . . 434 450. Principle of Sliding Contact, . 436 451. Teeth of Spur-Wheels and Racks General Principle, . . 438 452. Teeth described by rolling curves, 438 453. Sliding of a pair of Teeth (see also 455, 458, 462A), . . 439 454. Arc of Contact on Pitch Lines, . 440 455. Length of a Tooth; Slidingof Teeth,440 456. Inside Gearing, . . . .441 457. Involute Teeth for CircularWheel s, 441 458. Sliding of Involute Teeth, . . 443 459. Addendum for Involute Teeth, . 443 460. Smallest Pinion with Involute Teeth, 443 461. Epicycloidal Teeth least Pinion, 444 462. Addendum for Epicycloidal Teeth, 445 462A. Sliding of Epicycloidal Teeth, 445 463. Approximate Epicycloidal Teeth, 445 464. Teeth of Wheel and Trundle, . 447 465. Dimensions of Teeth, . . 447 466. Mr. Sang's process for Describing Teeth, 448 467. Teeth of Bevel Wheels, . . 448 468. Teeth of Skew-Bevel Wheels, . 449 469. Teeth of Non-Circular Wheels, . 449 470. Cam, or Wiper, . . .449 471. Screws Pitch, . . . .449 472. Normal and Circular Pitch, . 450 473. Screw Gearing, . . .451 474. Hooke's Gearing, . . .451 475. Wheel and Screw, . . .452 476. Relative Sliding of Pair of Screws, 453 477. Oldham's Coupling, . . .453 SECTION 3. Connection by Bands. 478. Bands classed: Belts,Cords,Chains,454 479. Principle of Connection by Bands, 454 480. Pitch Surface of a Pulley or Drum, 455 481. Circular Pulleys and Drums, . 455 482. Length of an Endless Belt, . 456 483. Speed Cones, . . . .457 SECTION 4. Linkwork. 484. Definitions, . . . .458 485. Principles of Connection, . . 458 486. Dead Points, . . . .458 487. Coupling of Parallel Axes, . 459 488. Comparative Motion of Connected Points, 459 489. Eccentric, . . '. .460 490. Length of Stroke, . . .460 491. Hooke's UniversalJoint, . . 461 492. Double Hooke's Joint, . . 462 493. Click, 462 SECTION 5. Reduplication of Cords. 494. Definitions, . . . .462 495. Velocity-Ratio, . . .463 496. Velocity of any Ply, . . .463 497. White's Tackle, . . .463 SECTION 6. Hydraulic Connection. 498. General Principle, . . . 464 499. Valves,Pumps, Working Cylinder, 464 500. Hydraulic Press, . . .464 501. Hydraulic Hoist, . . .465 SECTION 7. Trains of Mechanism. 502. Trains of Elementary Combina- tions, 465 CHAPTER III. ON AGGREGATE COMBINATIONS. 503. General Principles, . . .466 504. Differential Windlass, . . 466 505. Compound Screws, . . . 467 506. Link Motion, . 5C7. Parallel Motions, 508. Epicyclic Trains, 469 473 XIV CONTENTS. PART V. PRINCIPLES OF DYNAMICS. 509. Division of the Subject, Page . 475 CHAPTER I. UNIFORM MOTION UNDER BALANCED FORCES. Page 510. First Law of Motion, . . 476 511. Effort, Resistance, Lateral Force, 476 512. Conditions of Uniform Motion, . 477 513. Work, 477 514. Energy, 477 515. Energy and Work of Varying Forces, . . . . . 477 516. Dynamometer, or Indicator, . 478 517. Energy and Work of Fluid Pres- sure, 478 518. Conservation of Energy, . . 478 519. Principle of Virtual Velocities, . 479 520. Energy of Component Forces and Motions, . . . .480 CHAPTER II. ON THE VARIED TRANSLATION OF POINTS AND RIGID BODIES. SECTION 3. Transformation of Energy. 547. Actual Energy defined, Vis-Viva, 499 548. Components of Actual Energy, 499 549. Energy of Varied Motion, . 499 550. Energy Stored and Restored, 501 551. Transformation of Energy, 501 552. Conservation of Energy in Varied Motion, .... 501 553. Periodical Motion, . . 501 554. Measures of Unbalanced Force, 501 555. Energy due to Oblique Force, 502 556. Reciprocating Force, . 503 557. Total Energy Initial Energy, 503 SECTION 1 521. Mass, or Inertia, . . .482 522. Centre of Mass, . . .482 523. Momentum, . . . .482 524. Resultant Momentum, . . 482 525. Variations and Deviations of Mo- mentum, . . . .483 526. Impulse, . ^ . . 483 527. Impulse, Accelerating, Retarding, Deflecting, . . . .483 528. Relations between Impulse, Energy, and Work, . . 484 SECTION 2. Law of Varied Translation. 529. Second Law of Motion, . . 484 530. General Equations of Dynamics, 484 531. Mass in terms of Weight, . . 485 532. Absolute Unit of Force, . . 486 533. Motion of a Falling Body, . . 486 534. Projectile, Unresisted, . . 487 535. Motion along an Inclined Path, . 489 536. Uniform Effort, or Resistance, . 490 537. Deviating Force, . . .491 538. Centrifugal Force, . . .491 539. Revolving Simple Pendulum, . 492 540. Deviating Force in terms of An- gular Velocity, . . .492 541. Rectangular Components of De- viating Force, . . .493 542. Straight Oscillation, . . .494 543. Elliptical Oscillations, or Revo- lutions, 495 544. Simple Oscillating Pendulum, . 496 545. Cycloidal Pendulum, . . 497 546. Residual Forces, . . .498 SECTION 4. Varied Translation of a System of Bodies. 558. Conservation of Momentum, . 505 559. Motion of Centre of Gravity, . 505 560. Angular Momentum Defined, . 505 561. Angular Impulse Defined, . 506 562. Relations of Angular Impulse and Angular Momentum, . . 506 563. Conservation of Angular Momen- tum, 506 564. Actual Energy of a System of Bodies, 507 565. Conservation of Internal Energy, 508 566. Collision, . . . .508 567. Action of Unbalanced External Forces on a. System. General Equations, .... 510 568. Determination of Internal Forces D'Alembert's Principle, . 511 569. Residual External Forces in a System of Bodies, . . .511 CHAPTER III. ROTATIONS OP RIGID BODIES. 570. Motion of a Rigid Body in General, 513 SECTION 1 On Moments of Inertia, Radii of Gyration, Moments of Devia- tion, and Centres of Percussion. 571. Moment of Inertia Defined, . 514 572. Moment of Inertia of a System of Physical Points, . . . 514 573. Moment of Inertia of a Rigid Body,514 574. Radius of Gyration, . . .515 575. Components of Moments of In- ertia, 515 576. Moments of Inertia round Pa- rallel Axes compared, . .516 577. Combined Moments of Inertia, . 517 578. Examples of Moments of Iner- tia and Radii of Gyration, . 51~ 579. Moments of Inertia found Division and Subtraction, 580. Moments of Inertia found Transformation, by by 519 519 581. Centre of Percussion Centre of Gyration, . . . .520 582. No Centre of Percussion, . . 522 583. Moments of Inertia about Inclined Axes, 522 584. Principal Axes of Inertia, . . 524 585. Ellipsoid of Inertia, . . .526 586. Resultant Moment of Deviation, 528 SECTION 2. On Uniform Rotation. 587. Momentum, . . . .529 588. Angular Momentum, . . 529 589. Actual Energy of Rotation, . 532 590. Free Rotation, . . 533 591. Uniform Rotation about a Fixed Axis, 535 592. Deviating Couple Centrifugal Couple, 535 593. Energy and Work of Couples, . 537 SECTION 3. On Varied Rotation. 594. Law of Varied Rotation, . . 538 CONTENTS. XV Page 595. Varied Rotation about a fixedAxis,540 596. Analogy of Varied Rotation and Varied Translation, . . 541 597. Uniform Variation. . . .541 598. Gyration, or Angular Oscillation, 542 599. Single Force applied to a Body with a Fixed Axis, . . 543 SECTION 4. Varied Rotation and Trans- lation Combined. 600. General Principles, . . .543 601. Properties of the Centre of Per- cussion, ..... 544 602. Fixed Axis, . . . .545 603. Deviating Force, . . .545 604. Compound Oscillating Pendulum Centre of Oscillation, . . 546 605. Compound Revolving Pendulum, 547 606. Rotating Pendulum, . . 547 607. Ballistic Pendulum, . . .548 CHAPTER IV. MOTIONS OP PLIABLE BODIES. 608. Nature of the Subject Vibration, 552 609. Isochronous Vibration. Condi- tion of Isochronism, . . 553 610. Vibrations of a Mass held by a light Spring, .... 554 611. Superposition of Small Motions, 555 612. Vibrations not Isochronous, . 557 613. Vibrations of an Elastic Body in General, . . . .557 614. Waves of Vibration, . . .562 615. Velocity of Sound, . . .563 616. Impact and Pressure; Pile-driving,564 CHAPTEK V. MOTIONS OF FLUIDS HYDRODYNAMICS. 617. Division of the Subject, SECTION 1. Motion of Liquids without Friction. 618. General Equations, . . .567 619. Dynamic Head, . . . 568 620. General Dynamic Equations in terms of Dynamic Head, 621. Law of Dynamic Head for Steady Motion, 622. Total Energy, 623. Free Surface, 624. Surface of Equal Pressure, 625. Motion in Plane Layers, 626. Contracted Vein, 627. Vertical Orifices, 628. Surfaces of Equal Head, 629. Radiating Current, . 630. Free Circular Vortex, 631. Free Spiral Vortex, . 632. Forced Vortex, . 633. Combined Vortex, . 634. Vertical Revolution, . 568 569 . 570 . 570 . 570 . 572 . 572 . 573 . 574 . 574 . 576 . 576 . 576 . 578 SECTION 2. Motions of Gases without Friction. 635. Dynamic Head in Gases, . .579 636. Equation of Continuity for a Steady Stream of Gas, . . 581 637. Flow of Gas from an Orifice, . 581 637A. Maximum Flow of Gas, . . 582 SECTION 3. Motions of Liquids with Friction. 638. General Laws of Fluid Friction, 584 566 639. Internal Fluid Friction, . . 585 640. Friction in an Uniform Stream Hydraulic Mean Depth, . 586 641. Varying Stream, . . .587 642. Friction in a Pipe running full, . 588 643. Resistance of Mouthpieces, . 589 644. Resistance of Curves and Knees, 589 645. Sudden Enlargement of Chan- nel, 589 646. General Problem, . . .590 SECTION 4. Flow of Gases with Friction. 647. General Law, . . . .590 SECTION 5. Mutual Impulse of Fluids and Solids. 648. Pressure of a Jet against a fixed Surface, 591 649. Pressure of a ,7 et against a moving Surface, 593 650. Pressure of a Forced Vortex against a Wheel, . . .595 651. Centrifugal Pumps and Fans, . 597 652. Pressure of a Current, . . 698 653. Resistance of Fluids to Float- ing and Immersed Bodies, . 598 654. Stability of Floating Bodies Metacentre of a Ship, . . 600 655. Oscillations of Floating Bodies, . 603 656. Action between a Fluid and a Piston Work of Air Work of Steam, 604 CONTENTS. PART VI. THEORY OF MACHINES. 657. Nature and Division of the Subject, CHAPTER I. WORK OF MACHINES WITH Page SECTION 1 General Principles. 658. Useful and Lost Work, . . 610 659. Useful and Prejudicial Resistance, 610 660. Efficiency, . . . .610 661. Power and Effect : Horse Power, 610 662. Driving Point : Train : Work- ing Point, . . . .610 663. Points of Resistance, . .610 664. Efficiencies of Pieces of a Train, 610 665. Mean Efforts and Resistances, . 666. General Equations, . . - . 667. Equations in terras of Compara- tive Motion, 611 611 612 668. Reduction of Forces and Couples, 612 SECTION 2. On the Friction of Machines. 669. Co-efficients of Friction, . 612 670. 671. 672. 673. 674. 675. 676. 677. 678. 679. 680. 681. 682. 683. 684. 685. Page 609 UNIFORM OR PERIODIC MOTION. Unguents, .... 613 Limit of Pressure between Rub- bing Surfaces, . . .613 Friction of a Sliding Piece, . 614 Moment of Friction, . . . 614 Friction of an Axle, . . .614 Friction of a Pivot, . . .616 Friction of a Collar, . . .616 Friction of Teeth, . . .617 Friction of a Band, . . .617 Frictional Gearing, . . .618 Friction Couplings, . . .618 Stiffness of Ropes, . . .619 Rolling Resistance of Smooth Surfaces, .... 619 Resistance of Carriages on Roads, 619 Resistance of Railway Trains, . 620 Heat of Friction, . . .620 CHAPTER II. VARIED MOTIONS OF MACHINES. 686. Centrifugal Forces and Couples, 621 687. Actual Energy of a Machine, . 621 688. Reduced Inertia, 689. Fluctuations of Speed, 690. Fly-Wheel, CHAPTER III. ON PRIME MOVERS. 692. Prime Mover defined, 693. Regulators Governors, 694. Prime Movers Classed, 695. Muscular Strength, . 696. Water- Pressure Engines, 697. Water Wheels in General, 698. Classes of Water Wheels, 699. Overshot and Breast Wheels, 622 621 691. Starting and Stopping Brakes, 624 . 625 . 625 . 625 . 625 4 . 626 . 627 . 628 Is, . 628 700. Undershot Wheels, . . .628 701. Turbines, 629 702. Windmills, . . ^ . .629 703. Efficiency of Heat Engines in General, . . . .629 704. Steam Engines, . . .630 705. Electrodynamic Engines: Science of Energetics, . . . .630 APPENDIX. I. Table of the Resistance of Materials to Stretching and Tearing, II. Table of the Resistance of Materials to Shearing and Distortion, III. Table of the Resistance of Materials to Crushing, IV. Table of the Resistance of Materials to Breaking Across, V. Comparative Tables of British and French Measures, . VI. Table of Specific Gravities of Materials, .... VII. Dimensions and Stability of the Great Chimney of St. Rollox, List of Errata, ; 631 633 633 634 636 637 640 640 PRELIMINARY DISSERTATION HARMONY OF THEORY AND PRACTICE IN MECHANICS.* THE words, tlieory and practice, are of Greek origin : they carry our thoughts back to the time of those ancient philosophers by whom they were contrived ; and by whom also they were con- trasted and placed in opposition, as denoting two conflicting and mutually inconsistent ideas. In geometry, in philosophy, in poetry, in rhetoric, and in the fine arts, the Greeks are our masters ; and great are our obligations to the ideas and the models which they have transmitted to our times. But in physics and in mechanics their notions were very generally pervaded by a great fallacy, which attained its complete and most mischievous development amongst the mediaeval school- men, and the remains of whose influence can be traced even at the present day the fallacy of a double system of natural laws; one theoretical, geometrical, rational, discoverable by contemplation, applicable to celestial, setherial, indestructible bodies, and being an object of the noble and liberal arts ; the other practical, mechanical, empirical, discoverable by experience, applicable to terrestrial, gross, destructible bodies, and being an object of what were once called the vulgar and sordid arts. The so-called physical theories of most of those whose under- standings were under the influence of that fallacy, being empty dreams, with but a trace of truth here and there, and at variance with the results of every-day observation on the surface of the planet we inhabit, were calculated to perpetuate the fallacy. The stars were celestial, incorruptible bodies ; their orbits were circular and their motions perpetual ; such orbits and motions being charac- teristic of perfection. Objects on the earth's surface were terrestrial * This Dissertation contains the substance of a discourse, " De Concordia inter Scientiai-um Machinalium Contemplationem et Usum," read before the Senate of the University of Glasgow on the 10th of December, 1855, and of an. inaugural lec- ture, delivered to the Class of Civil Engineering and Mechanics in that University on the 3d of January, 1 856. B 2 PRELIMINARY DISSERTATION. and corruptible ; their motions being characteristic of imperfection, were in mixed straight and curved lines, and of limited duration. Rational and practical mechanics (as Newton observes in his preface to the Principia) were considered as in a measure opposed to each other, the latter being an inferior branch of study, to be cultivated only for the sake of gain or some other material advantage. Archytas of Tarentum might illustrate the truths of geometry by mechanical contrivances ; his methods were regarded by his pupil Plato as a lowering of the dignity of science. Archi- medes, to the character of the first geometer and arithmetician of his day, might add that of the first mechanician and physicist, he might, by his unaided strength acting through suitable machinery, move a loaded ship on diy land, he might contrive and execute deadly engines of war, of which even the Roman soldiers stood in dread, he might, with an art afterwards regarded as fabulous till it was revived by Button, burn fleets with the concentrated sunbeams ; but that mechanical knowledge, and that practical skill, which, in our eyes, render that great man so illustrious, were, by men of learning, his contemporaries and successors, regarded as accomplishments of an inferior order, to which the philosopher, from the height of geometrical abstraction, condescended, with a view to the service of the State. In those days the notion arose that scientific men were unfit for the business of life, and various facetious anecdotes were contrived illustrative of this notion, which have been handed down from age to age, and in each age applied, with little variation, to the eminent philosophers of the time. That the Romans were eminently skilful in many departments of practical mechanics, especially in masonry, road-making, and hydraulics, is clearly established by the existing remains of their magnificent works of engineering and architecture, from many of which we should do well to take a lesson. But the fallacy of a supposed discordance between rational and practical, celestial and terrestrial mechanics, still continued in force, and seems to have gathered strength, and to have attained its full vigour during the middle ages. In those ages, indeed, were erected those incom- parable ecclesiastical buildings, whose beauty, depending, as it does, mainly on the nice adjustment of the form, strength, and position of each part, to the forces which it has to sustain, evinces a pro- found study of the principles of equilibrium on the part of the architects. But the very names of those architects, with few and doubtful exceptions, were suffered to be forgotten ; and the prin- ciples which guided their work remain unrecorded, and were left to be re-discovered in our own day ; for the scholars of those times, despising practice and observation, were occupied in developing and magnifying the numerous errors, and in perverting and obscur- PRELIMINARY DISSERTATION. 3 ing the much more numerous truths, which are to be found in the writings of Aristotle ; and those few men who, like Roger Bacon, combined scientific with practical knowledge, were objects of fear and persecution, as supposed allies of the powers of darkness. At length, during the great revival of learning and reformation of science in the fifteenth, sixteenth, and seventeenth centuries, the system falsely styled Aristotelian was overthrown : so also was the fallacy of a double system of natural laws ; and the truth began to be duly appreciated, that sound theory in physical science con- sists simply of facts, and the deductions of common sense from them, reduced to a systematic form. The science of motion was founded by Galileo, and perfected by Newton. Then it was estab- lished that celestial and terrestrial mechanics are branches of one science ; that they depend on one and the same system of clear and simple first principles ; that those very laws which regulate the motion and the stability of bodies on earth, govern also the revolutions of the stars, and extend their dominion throughout the immensity of space. Then it came to be acknowledged, that no material object, however small, no force, however feeble, no phenomenon, however familiar, is insignificant, or beneath the attention of the philosopher ; that the processes of the workshop, the labours of the artizan, are full of instruction to the man of science ; that the scientific study of practical mechanics is well worthy of the atten- tion of the most accomplished mathematician. Then the notion, that scientific men are unfit for business, began to disappear. It was not court favour, not high connection, not Parliamentary in- fluence, which caused Newton to be appointed Warden, and after- wards Master, of the Mint ; it was none of these' ; but it was the knowledge possessed by a wise minister of the fact, that Newton's skill, both theoretical and practical, in those branches of knowledge which that office required, rendered him the fittest man in all Britain to direct the execution of a great reform of the coinage. Of the manner in which Newton performed the business entrusted to him, we have the following account in the words of Lord Macaulay, an author who cannot be accused of undue partiality to speculative science or its cultivators : " The ability, the industry, and the strict uprightness of the great philo- sopher, speedily produced a complete revolution throughout the depart- ment which was under his direction. He devoted himself to the task with an activity which left him no time to spare for those pursuits in which he had surpassed Archimedes and Galileo. Till the great work was com- pletely done, he resisted firmly, and almost angrily, every attempt that was made by men of science, here or on the Continent, to draw him away from his official duties."* * Vol. iv., p. 703. 4 PRELIMINARY DISSERTATION. Then the historian proceeds to detail the results of Newton's exertions, and shows, that within a short time after his appoint- ment, the weekly amount of the coinage of silver was increased to- eightfold of that which had been looked upon as the utmost practi- cable amount by his predecessors. The extension of experimental methods of investigation, has caused even manual skill in practical mechanics, when scientifically exercised, to be duly honoured, and not (as in ancient times) to be regarded as beneath the dignity of science. As a systematically avowed doctrine, there can be no doubt that the fallacy of a discrepancy between rational and practical me- chanics came long ago to an end ; and that every well-informed and sane man, expressing a deliberate opinion upon the mutual relations of those two branches of science, would at once admit that they agree in their principles, and assist each other's progress, and that such distinction as exists between them arises from the differ- ence of the purposes to which the same body of principles is applied. If this doctrine had as strong an influence over the actions of" men as it now has over their reasonings, it would have been unne- cessary for me to describe, so fully as I have done, the great scienti- fic fallacy of the ancients. I might, in fact, have passed it over in silence, as dead and forgotten ; but, unfortunately, that discrepancy between theory and practice, which in sound physical and mechani- cal science is a delusion, has a real existence in the minds of men ; and that fallacy, though rejected by their judgments, continues to exert an influence over their acts. Therefore it is that I have endeavoured to trace the prejudice as to the discrepancy of theory and practice, especially in Mechanics, to its origin ; and to show that it is the ghost of a defunct fallacy of the ancient Greeks and of the mediaeval schoolmen. This prejudice, as I have stated, is not to be found, at the present day, in the form of a definite and avowed principle : it is to be traced only in its pernicious effects on the progress both of specula- tive science and of practice, and sometimes in a sort of tacit influ- ence which it exerts on the forms of expression of writers, who have assuredly no intention of perpetuating a delusion. To exem- plify the kind of influence last referred to, I shall cite a passage from the same historical work which I recently quoted for a differ- ent purpose. Lord Macaulay, in treating of the Act of Toleration of William III., compares, metaphorically, the science of politics to that of mechanics, and then proceeds as follows : "The mathematician can easily demonstrate that a certain power, ap- plied by means of a certain lever, or of a certain system of pulleys, will suffice to raise a certain weight. But his demonstration proceeds on the supposition that the machinery is such as no load will bend or break. If PRELIMINARY DISSERTATION. ."> the engineer who has to lift a great mass of real granite by the instru- mentality of real timber and real hemp, should absolutely rely on the pro- positions which he finds in treatises on Dynamics, and should make no allowance for the imperfection of his materials, his whole apparatus of beams, wheels, and ropes, would soon come down in ruin, and with all his geometrical skill, he would be found a far inferior builder to those painted barbarians who, though they never heard of the parallelogram of forces, managed to pile up Stonehenge."* It is impossible to read this passage without feeling admiration for the force and clearness (and I may add, for the brilliancy and wit) of the language in which it is expressed ; and those very qualities of force and clearness, as well as the author's eminence, render it one of the best examples that can be found to illustrate the lurking influence of the fallacy of a double set of mechanical laws, rational and practical. In fact, the mathematical theory of a machine, that is, the body of principles which enables the engineer to compute the arrange- ment and dimensions of the parts of a machine intended to perform given operations, is divided by mathematicians, for the sake of convenience of investigation, into two parts. The part first treated of, as being the more simple, relates to the motions and mutual actions of the solid pieces of a machine, and the forces exerted by and upon them, each continuous solid piece being treated as a whole, and of sensibly invariable figure. The second and more intricate part relates to the actions of the forces tending to break or to alter the figure of each such solid piece, and the dimensions and form to be given to it in order to enable it to resist those forces : this part of the theory depends, as much as the first part, on the general laws of mechanics; and it is, as truly as the first part, a subject for the reasonings of the mathematician, and equally requisite for the completeness of the mathematical treatise which the engineer is supposed to consult. It is true, that should the engineer implicitly trust to a pretended mathematician, or an incomplete treatise, his apparatus would come down in ruin, as the historian has stated : it is true also that the same result would follow, if the engineer was one who had not qualified himself, by experience and observation, to distinguish between good and bad materials and workmanship ; but the passage I have quoted conveys an idea different from these ; for it proceeds on the erroneous sup- position, that the first part of the theory of a machine is the whole theory, and is at variance with something else which is independent of mathematics, and which constitutes, or is the foundation of, practical mechanics. The evil influence of the supposed inconsistency of theory and Vol. iii., p. 84. PRELIMINARY DISSERTATION. practice upon speculative science, although, much less conspicuous than it was in the ancient and middle ages, is still occasionally to be traced. This it is which opposes the mutual communication of ideas between men of science and men of practice, and which leads scientific men sometimes to employ, on problems that can only be regarded as ingenious mathematical exercises, much time and mental exertion that would be better bestowed on questions having some connection with the arts, and sometimes to state the results of really important investigations on practical subjects in a form too abstruse for ordinary use ; so that the benefit which might be derived from their application is for years lost to the public; and valuable practical principles, which might have been anticipated by reasoning, are left to be discovered by slow and costly experience. But it is on the practice of mechanics and engineering that the influence of the great fallacy is most conspicuous and most fatal. There is assuredly, in Britain, no deficiency of men distinguished by skill in judging of the quality of materials and work, and in directing the operations of workmen, by that sort of skill, in fact, which is purely practical, and acquired by observation and experience in business. But of that scientifically practical skill which produces the greatest effect with the least possible expendi- ture of material and work, the instances are comparatively rare. In too many cases we see the strength and the stability, which ought to be given by the skilful arrangement of the parts of a structure, supplied by means of clumsy massiveness, and of lavish expenditure of material, labour, and money ; and the evil is increased by a perversion of the public taste, which causes works to be admired, not in proportion to their fitness for their purposes, or to the skill evinced in attaining that fitness, but in proportion to their size and cost. With respect to those works which, from unscientific design, give way during or immediately after their erection, I shall say little ; for, with all their evils, they add to our experimental know- ledge, and convey a lesson, though a costly one. But a class of structures fraught with much greater evils exists in great abundance throughout the country : namely, those in which the faults of an unscientific design have been so far counteracted by massive strength, good materials, and careful workmanship, that a temporary stability has been produced, but which contain within themselves sources of weakness, obvious to a scientific examination only, that must inevi- tably cause their destruction within a limited number of years. Another evil, and one of the worst which arises from the separa- tion of theoretical and practical knowledge, is the fact that a large number of persons, possessed of an inventive turn of mind and of considerable skill in the manual operations of practical mechanics, PRELIMINARY DISSERTATION. 7 are destitute of that knowledge of scientific principles which is requisite to prevent their being misled by their own ingenuity. Such men too often spend their money, waste their lives, and it may be lose their reason, in the vain pursuit of visionary inventions, of which a moderate amount of theoretical knowledge would be sufficient to demonstrate the fallacy;' and for want of such know- ledge, many a man who might have been a useful and happy member of society, becomes a being than whom it would be hard to find anything more miserable. The number of those unhappy persons to judge from the patent- lists, and from some of the mechanical journals must be much greater than is generally believed. The most absurd of all their delusions, that commonly called the perpetual motion, or to speak more accurately, the inexhaustible source of power, is, in various forms, the subject of several patents in each year. The ill success of the projects of misdirected ingenuity has very naturally the effect of driving those men of practical skill who, though without scientific knowledge, possess prudence and common sense, to the opposite extreme of caution, and of inducing them to avoid all experiments, and to confine themselves to the careful copying of successful existing structures and machines : a course which, although it avoids risk, would, if generally followed, stop the progress of all improvement. A similar course has sometimes, indeed, been adopted by men possessed of scientific as well as practical skill : such men having, in certain cases, from deference to popular prejudice, or from a dread of being reputed as theorists, considered it advisable to adopt the worse and customary design for a work in preference to a better but unusual design. Some of the evils which are caused by the fallacy of an incom- patibility between theory and practice having been described, it must now be admitted, that at the present time those evils show a decided tendency to decline. The extent of intercourse, and of mutual assistance, between men of science and men of practice, the practical knowledge of scientific men, and the scientific knowledge of practical men, have been for some time steadily increasing ; and that combination and harmony of theoretical and practical knowledge that skill in the application of scientific principles to practical purposes, which in former times was confined to a few remarkable individuals, now tends to become more generally diffused. With a view to promote the diffusion of that kind of skill, Chairs were instituted at periods of from fifteen to ten years ago, in the two Colleges of the University of London, in the University of Dublin, in the three Queen's Colleges of Belfast, Cork, and Galway, and in this University of Glasgow. For the sake of a parallel, it may here be worth while to refer 8 PRELIMINARY DISSERTATION. to another branch of practical science that of Medicine. From the time of the first establishment of Medical Schools in Universities, there have existed, not only Chairs for the teaching of the purely scientific departments of Medical Science, such as Anatomy and Physiology, but also Chairs for instruction in the art of applying scientific principles to practice, such as those of Surgery, the Practice of Physic, and others. The institution of a Chair of Mechanics and Engineering in a University where there have long existed Chairs of Mathematics and Natural Philosophy, is an endeavour to place Mechanical Science on the same footing with that of Medicine. Another parallel may be found in an Institution, which, though not a University, and though established as much for the advance- ment as for the diffusion of knowledge, has had a most beneficial effect in promoting the appreciation of science by the public, I mean the British Association. When that body was first instituted, both the theoretical advancement and the practical application of Mechanics, and the several branches of Physics, were allotted to a single section, called Section A. The business before that Section soon became so excessive in amount, and so multifarious in its character, that it was found necessary to institute Section G, for the purpose of considering the practical application of those branches of science to whose theoretical advancement Section A was now devoted; and notwithstanding this separation, those two Sections work harmoniously together for the promotion of kindred objects ; and the same men are, in many instances, leading members of both. What Section G is to Section A in the British Association, this class of Engineering and Mechanics is to those of Physics and Mathematics in the University. It being admitted, that Theoretical and Practical Mechanics are in harmony with each other, and depend on the same first prin- ciples, and that they differ only in the purposes to which those principles are applied, it now remains to be considered, in what manner that difference affects the mode of instruction to be followed in communicating those branches of science. Mechanical knowledge may obviously be distinguished into three kinds : purely scientific knowledge, purely practical knowledge and that intermediate knowledge which relates to the application of scientific principles to practical purposes, and which arises from understanding the harmony of theory and practice. The objects of instruction in purely scientific mechanics and physics are, first, to produce in the student that improvement of the understanding which results from the cultivation of natural knowledge, and that elevation of mind which flows from the con- templation of the order of the universe ; and secondly, if possible, PRELIMINARY DISSERTATION. to qualify him to become a scientific discoverer. In this branch of study exactness is an essential feature ; and mathematical difficulties inust not be shrunk from when the nature of the subject leads to them. The ascertainment and illustration of truth are the objects ; and structures and machines are looked upon merely as natural bodies are : namely, as furnishing experimental data for the ascer- taining of principles, and examples for their illustration. Instruction in purely practical knowledge is that which the student acquires by his own experience and observation of the transaction of business. It enables him to judge of the quality of materials and workmanship, and of questions of convenience and commercial profit, to direct the operations of workmen, to imitate existing structures and machines, to follow established practical rules, and to transact the commercial business which is connected with mechanical pursuits. The third and intermediate kind of instruction, which connects the first two, and for the promotion of which this Chair was estab- lished, relates to the application of scientific principles to practical purposes. It qualifies the student to plan a structure or a machine for a given purpose, without the necessity of copying some existing example, and to adapt his designs to situations to which no existing example affords a parallel. It enables him to compute the theo- retical limit of the strength or stability of a structure, or the efficiency of a machine of a particular kind, to ascertain how far an actual structure or machine fails to attain that limit, to dis- cover the causes of such shortcomings, and to devise improvements for obviating such causes; and it enables him to judge how far an established practical rule is founded on reason, how far on mere custom, and how far on error. There are certain characteristics in the mode of treating the subjects, by which this practical-scientific instruction ought to be distinguished from instruction for purely scientific purposes. In the first place it will be universally admitted, that as far as is possible, mathematical intricacy ought to be avoided. In the original discovery of a proposition of practical utility, by deduction from general principles and from experimental data, a complex algebraical investigation is often not merely useful, but indispensable; but in expounding such a proposition as a part of practical science, and applying it to practical purposes, simplicity is of the first importance : and, in fact, the more thoroughly a scien- tific man has studied the higher mathematics, the more fully does he become aware of this truth, and, I may add, the better qualified does he become to free the exposition and application of scientific principles from mathematical intricacy. I cannot better support this view than by referring to Sir John Herschel's Outlines of 10 PRELIMINARY DISSERTATION. Astronomy a work in which one of the most profound mathema- ticians in the world has succeeded admirably in divesting of all mathematical intricacy the explanation of the principles of that natural science which employs the higher mathematics most. In fact, the symbols of algebra, when employed in abstruse and complex theoretical investigations, constitute a sort of thought- saving machine, by whose aid a person skilled in its use can solve problems respecting quantities, and dispense with the mental labour of thinking of the quantities denoted by the symbols, except at the beginning and end of the operation. In treating of the practical application of scientific principles, an algebraical formula should only be employed when its shortness and simplicity are such as to render it a clearer expression of a proposition or rule than common language would be, and when there is no difficulty in keeping the thing represented by each symbol constantly before the mind. Another characteristic by which instruction in practical science should be distinguished from purely scientific instruction, is one which appears to me to possess the advantage of calling into opera- tion a mental faculty distinct from those which are exercised by theoretical science. It is of the following kind : In theoretical science, the question is Wliat are we to think ? and when a doubtful point arises, for the solution of which either experimental data are wanting, or mathematical methods are not sufficiently advanced, it is the duty of philosophic minds not to dis- pute about the probability of conflicting suppositions, but to labour for the advancement of experimental inquiry and of mathematics, and await patiently the time when these shall be adequate to solve the question. But in practical science the question is What are we to do ? a a question which involves the necessity for the immediate adoption of some rule of working. In doubtful cases, we cannot allow our machines and our works of improvement to wait for the advance- ment of science ; and if existing data are insufficient to give an exact solution of the question, that approximate solution must be acted upon which the best data attainable show to be the most probable. A prompt and sound judgment in cases of this kind is one of the characteristics of a PRACTICAL MAN, in the right sense of that term. In conclusion, I will now observe, that the cultivation of the Harmony between Theory and Practice in Mechanics of the application of Science to the Mechanical Arts besides all the benefits which it confers on us, by promoting the comfort and prosperity of individuals, and augmenting the wealth and power of the nation confers on us also the more important benefit of raising the character of the mechanical arts, and of those who practise them. A great mechanical philosopher, the late Dr. Robison of PRELIMINARY DISSERTATION. 11 Edinburgh, after stating that the principles of Carpentry depend on two branches of the science of Statics, remarks "It is this which makes Carpentry a liberal art." So also is Masonry a liberal art, so is the art of working in Iron, so is every art, when guided by scientific principles. Every structure or machine, whose design evinces the guidance of science, is to be regarded not merely as an instrument for promoting con- venience and profit, but as a monument and testimony that those who planned and made it had studied the laws of nature ; and this renders it an object of interest and value, how small soever its bulk, how common soever its material. For a century there has stood, in a room in this College, a small, rude, and plain model, of appearance so uncouth, that when an artist lately introduced its likeness into a historical painting, those who saw the likeness, and knew nothing of the original, wondered what the artist meant by painting an object so unattractive. But the artist was right ; for ninety-one years ago a man took that model, applied to it his knowledge of natural laws, and made it into the first of those steam engines that now cover the land and the sea; and ever since, in Reason's eye, that small and uncouth mass of wood and metal shines with imperishable beauty, as the earliest embodiment of the genius of James Watt. Thus it is that the commonest objects are by science rendered precious; and in like manner the engineer or the mechanic, who plans and works with understanding of the natural laws that regulate the results of his operations, rises to the dignity of a Sage. INTRODUCTION. DEFINITION OF GENERAL TERMS AND DIVISION OF THE SUBJECT. ART. 1. Mechanics is the science of rest, motion, and force. The laws, or first principles of mechanics, are the same for all bodies, celestial and terrestrial, natural and artificial. The methods of applying the principles of mechanics to particular cases are more or less different, according to the circumstances of the case. Hence arise branches in the science of mechanics. 2. Applied Mechanics. The branch to which the term " APPLIED MECHANICS" has been restricted by custom, consists of those consequences of the laws of mechanics which relate to works of human art. A treatise on applied mechanics must commence by setting forth those first principles which are common to all branches of mechanics ; but it must contain only such consequences of those principles as are applicable to purposes of art. 3. Matter (considered mechanically) is that which fills space. 4. Bodies are limited portions of matter. Bodies exist in three conditions the solid, the liquid, and the gaseous. Solid bodies tend to preserve a definite size and shape. Liquid bodies tend to preserve a definite size only. Gaseous bodies tend to expand inde- finitely. Bodies also exist in conditions intermediate between the solid and liquid. 5. A Material or Physical Volume is the space occupied by a body or by a part of a body. 6. A Material or Physical Surface is the boundary of a body, or between two parts of a body. 7. Line, Point, Physical Point, Measure of Length. In mechanics, as in geometry, a LINE is the boundary of a surface, or between two 14 INTRODUCTION. parts of a surface ; and a POINT is the boundary of a line, or be- tween two parts of a line ; but the term " Physical Point" is some- times used by mechanical writers to denote an immeasurably small body a sense inconsistent with the strict meaning of the word " point ;" but still not leading to error, so long as it is rightly under- stood. In measuring the dimensions of bodies, the standard British unit of length is the yard, being the length at the temperature of 62 Fahrenheit, and at the mean atmospheric pressure, between two marks on a certain bar which is kept in the office of the Exchequer, at Westminster. In computations respecting motion and force, and in expressing the dimensions of large structures, the unit of length commonly employed in Britain is the foot, being one-third of the yard. In expressing the dimensions of machinery, the unit of length commonly employed in Britain is the inch, being one-thirty-sixth part of the yard. Fractions of an inch have hitherto been stated by mechanics and other artificers in halves, quarters, eighths, six- teenths, and thirty-second parts ; but according to a resolution of the Institution of Mechanical Engineers, passed at the meeting held at Manchester in June, 1857, the practice is to be introduced of expressing fractions of an inch in decimals. The French unit of length is the metre, being TOOUOOOTJ of the earth's circumference, measured round the poles. (See table at the end of the volume.) 8. Rest is the relation between two points, when the straight line joining them does not change in length nor in direction. A body is at rest relatively to a point, when every point in the body is at rest relatively to the first mentioned point. 9. motion is the relation between two points when the straight line joining them changes in length, or in direction, or in both. A body moves relatively to a point when any point in the body moves relatively to the first mentioned point. 10. Fixed Point. When a single point is spoken of as having motion or rest, some other point, either actual or ideal, is always either expressed or understood, relatively to which the motion or rest of the first point takes place. Such a point is called a fixed point. So far as the phenomena of motion alone indicate, the choice of a fixed point with which to compare the positions of other points appears to be arbitrary, and a matter of convenience alone ; but when the laws of force, as affecting motion, come to be considered, STRUCTURES AND MACHINES. 15 it will be seen that there are reasons for calling certain points fixed, in preference to others. In the mechanics of the solar system, the fixed point is what is called the common centre of gravity of the bodies composing that system. In applied mechanics, the fixed point is either a point which is at rest relatively to the earth, or (if the structure or machine under consideration be moveable from place to place on the earth), a point which is at rest relatively to the structure, or to the frame of the machine, as the case may be. Points, lines, surfaces, and volumes, which are at rest relatively to a fixed point, are fixed. 11. Cinematics. The comparison of motions with each other, without reference to their ca,uses, is the subject of a branch of geometry called " Cinematics" 12. Force is an action between two bodies, either causing or tending to cause change in their relative rest or motion. The notion of force is first obtained directly by sensation; for the forces exerted by the voluntary muscles can be felt. The ex- istence of forces other than muscular tension is inferred from their effects. 13. Equilibrium or Balance is the condition of two or more forces which are so opposed that their combined action on a body produces no change in its rest or motion. The notion of balance is first obtained by sensation; for the forces exerted by voluntary muscles can be felt to balance some- times each other, and sometimes external pressures. 14. Statics and Dynamics. Forces may take effect, either by balancing other forces, or by producing change of motion. The former of those effects is the subject of Statics ; the latter that of Dynamics; these, together with Cinematics, already defined, form the three great divisions of pure, abstract, or genera! mechanics. 15. Structures and machines. The works of human art to which the science of applied mechanics relates, are divided into two classes, according as the parts of which they consist are intended to rest or to move relatively to each other. In the former case they are called Structures ; in the latter, Machines. Structures are sub- jects of Statics alone ; Machines, when the motions of their parts are considered alone, are subjects of Cinematics; when the forces acting on and between their parts are also considered, machines are subjects of Statics and Dynamics. 1 6 INTRODUCTION. 16. General Arrangement of the Subject. The subject of the pre- sent treatise will be arranged as follows : I. FIRST PRINCIPLES OF STATICS. II. THEORY OF STRUCTURES. III. FIRST PRINCIPLES OF CINEMATICS. IV. THEORY OF MECHANISM. V. FIRST PRINCIPLES OF DYNAMICS. VI. THEORY OF MACHINES. PART I. PRINCIPLES OF STATICS. CHAPTER I. BALANCE AND MEASUREMENT OF FORCES ACTING IN ONE STRAIGHT LINE. 17. Forces how Determined. Although every force (as has been stated in Art. 12) is an action between two bodies, still it is con- ducive to simplicity to consider in the first place the condition of one of those two bodies alone. The nature of a force, as respects one of the two bodies between which it acts, is determined, or made known, when the following three things are known respecting it : first, the place, or part of the body to which it is applied; secondly, the direction of its action ; thirdly, its magnitude. 1 8. Place of Application Point of Application. The place of the application of a force to a body may be the whole or part of its in- ternal mass ; in which case the force is an attraction or a repulsion, according as it tends to move the bodies between which it acts towards or from each other; or the place of application may be the surface at which two bodies touch each other, or the bounding surface between two parts of the same body, in which case the force is a tension or pull, a thrust or push, or a lateral stress, according to circumstances. Thus every force has its action distributed over a certain space, either a volume or a surface; and a force concentrated at a single point has no real existence. Nevertheless it is necessary, in treating of the principles of statics, to begin by demonstrating the properties of such ideal forces, conceived to be concentrated at single points. It will afterwards be shown how the conclusions so arrived at re- specting single forces (as they may be called), are made applicable to the distributed forces which really act in nature. In illustrating the principles of statics experimentally, a force concentrated at a single point may be represented with any required degree of accuracy by a force distributed over a very small space, if that space be made small enough. c 18 PRINCIPLES OF STATICS. 19. Supposition of Perfect Rigidity. In reasoning respecting forces concentrated at single points, they are assumed to be applied to solid bodies which are perfectly rigid, or incapable of alteration of figure under any forces which can be applied to them. This also is a supposition not realized in nature. It will afterwards be shown how its consequences are applied to actual bodies. 20. Direction Line of Action. The DIRECTION of a force is that of the motion which it tends to produce. A straight line drawn through the point of application of a single force, and along its direction, is the LINE OF ACTION of that force. 21. Magnitude Unit of Force. The magnitudes of two forces are equal, when being applied to the same body in opposite direc- tions along the same line of action, they balance each other. The magnitude of a force is expressed arithmetically by stating in numbers its ratio to a certain unit or standard of force, which is usually the weight (or attraction towards the earth), at a certain latitude, and at a certain level, of a known mass of a certain material. Thus the British unit of force is the standard pound avoirdupois; which is the weight in the latitude of London of a certain piece of platinum kept in the Exchequer office (See the Act 18 and 19 Viet., cap. 72; also a paper by Professor W. H. Miller, in the Philosophical Transactions for 1856). For the sake of convenience or of compliance with custom, other units of force are occasionally employed in Britain, bearing certain ratios to the standard pound ; such The grain = Tinnr of a pound avoirdupois. The troy pound = 5,760 grains = 0-822857 14 pound avoirdupois. The hundredweight = 112 pounds avoirdupois. The ton = 2,240 pounds avoirdupois. The French standard unit of force is the gramme, which is the weight, in the latitude of Paris, of a cubic centimetre of pure water, measured at the temperature at which the density of water is greatest, viz., 4'l centigrade, or 39'4 Fahrenheit, and under the pressure which supports a barometric column of 760 millimetres of mercury. A comparison of French and British measures of force and of size is given in a table at the end of this volume. 22. Resultant of Forces Acting in One Straight Line. The RE- SULTANT of any number of given forces applied to one body, is a .single force capable of balancing that single force which balances the given forces ; that is to say, the resultant of the given forces is equal and directly opposed to the force which balances the given forces ; and is equivalent to the given forces so far as the balance of REPRESENTATION OF FORCES. 10 the body is concerned. The given forces are called components of their resultant. The resultant of any number of forces acting on one body in the same straight line of action, acts along that line, and is equal in magnitude to the sum of the component forces ; it being under- stood, that when some of the component forces are opposed to the others, the word " simi " is to be taken in the algebraical sense \ that is to say, that forces acting in the same direction are to be added to, and forces acting in opposite directions subtracted from each other. 23. Representation of Forces by rim * A single force may be represented in a drawing by a straight line ; an extremity of the line indicating the point of application of the force, the direction of the line, the direc- tion of the force, and the length of the line, the magnitude of the force, according to an arbitrary scale. Fig lp For example, in fig. 1, the fact that the body B B B B is acted upon at the point O x by a given force, may be expressed by drawing from C^ a straight line Oj F! in the direction of the force, and of a length representing the magnitude of the force. If the force represented by O l F l is balanced by a force applied either at the same point, or at another point 2 (which must be in the line of action L L of the force to be balanced), then the second force will be represented by a straight line O 2 F 2 , opposite in direc- tion, and equal in length to C^ F 1? and lying in the same line of action L L. If the body B B B B (fig. 2), be balanced by several forces acting in the same straight line LL, applied at points O x O 2 , &c., and re- E resented by lines O l Fj, O 2 F 2 , &c. ; then either direction in the ne L L (such as the direc- tion towards + L) is to be ^^ - _ considered as positive, and the opposite direction (such as the direction towards L) as negative ; and if the sum of all the lines repre- senting forces which point positively be equal to the pj 2 sum of all those which point negatively, the algebraical sum of all the forces is nothing, and the body is balanced. 20 PRINCIPLES OF STATICS. 24. Pressure. Most writers on mechanics, in treating of the first principles of statics, use the word "pressure" to denote any balanced force. In the popular sense, which is also the sense generally employed in applied mechanics, the word pressure is used to denote a force, of the nature of a thrust, distributed over a surface ; in other words, the kind of force with which a body tends to expand, or resists an effort to compress it. In this treatise care will be taken so to employ the word " pressure " that the context shall show in what sense it is used. 21 CHAPTER II. THEORY OF COUPLES AND OF THE BALANCE OF PARALLEL FORCES. SECTION 1. On Couples with the Same Axis. 25. Couples. Two forces of equal magnitude applied to the same body in parallel and opposite directions, but not in the same line of action, constitute what is called a " couple" 26. Force of a Couple Arm or Leverage. Theforce of a couple is the common magnitude of the two equal forces ; the arm or leverage of a couple is the perpendicular distance between the lines of action of the two equal forces. 27. Tendency of a Couple Plane of a Couple Right-handed and Left-handed Couples. The tendency of a couple is to turn the body to which it is applied in the plane of the couple that is, the plane which contains the lines of action of the two forces. (The plane in which a body turns, is any plane parallel to those planes in the body whose position is not altered by the turning). The axis of a couple is any line perpendicular to its plane. The turning of a body is said to be right-handed when it appears to a spectator to take place in the same direction with that of the hands of a watch, and left-handed when in the opposite direction; and couples are desig- nated as right-handed or left-handed according to the direction of the turning which they tend to pro- duce. Thus in fig. 3, the equal and opposite forces O : F b O 2 F 2 , whose leverage is L, L 2 , form a right- handed couple ; and the equal and opposite forces O 3 F 3 , O 4 F 4 , form a left-handed couple. 28. Equivalent Couples of l.qnal Force and Leverage. In Order that two couples similar in direction, and of equal force and lever- age, may be exactly alike or equivalent in their tendency to turn the body, it is necessary and sufficient that their planes should be either identical or parallel. Fig. 3. PRINCIPLES OF STATICS. Two couples applied to the same body in the same plane, or in parallel planes, of equal force and leverage, but opposite in direction, balance each other; and if for either of the two an equivalent couple be substituted, the equilibrium will not be disturbed. 29. Moment of a Couple. The moment of a couple means the product of the magnitude of its force by the length of its arm. If the force be a certain number of pounds, and the arm a certain number of feet, the product of those two numbers is called the moment in foot-pounds, and similarly for other measures. 30. Addition of Couples of Equal Force. LEMMA. Two COUples of equal force acting in the same direction, with the same axis, are equiva- lent to a couple whose moment is the sum of their moments. Let the two couples be denoted by A and B ; let F A = F B be their equal forces; let L A and L B be their respective arms ; then F A L A and F B L B are their moments, which, as their forces are equal, are pro- portional to the arms. In fig. 4, let the forces F A constituting A be applied in lines passing through a and c, a c or L A being perpen- dicular to the lines of action of ^*' 4- the forces; and if the forces con- stituting B be not already applied as shown in the figure, sub- stitute for B an equivalent couple of equal force and arm, having its forces F B applied in lines parallel to the lines of action of the forces F A , and passing one through the point c and the other through b, so that the arm c b or L B shall be in the same straight line with a c or L A . Then the equal and opposite forces F A , F B , applied at c, balance each other, and there remain only the equal and opposite forces F A , F B , applied at a and b, which form a couple whose force is F A = F B , and its arm a b = L A + L B , being the sum of the arms of the couples A and B ; so that its moment is the sum of their moments ; and this couple is equivalent to the two couples A and B. 31. Equivalent Couples of Equal Moment. THEOREM. If the mo- ments of two couples acting in the same direction and with the same axis are equal, those couples are equivalent. Let one of the couples be called A, and let its force, arm, and moment be respectively F A , L A , and F A L A ; let the other couple be called B, and let its force, arm, and moment be respectively F B , L B , and F B L B . The equality of the moments of those couples is expressed by the equation FT T? T A L A = -b B -L B . If the forces and arms of the two couples be commensurable, so that REPRESENTATION OF COUPLES. 23 F A : F B : : L B : L A : : m : n (m and n being two whole numbers), let /-&'-&, w- m n and l=Z .. **, m H Then the couple A is equivalent to m n couples of the moment fl ; and so also is the couple B ; therefore the couples A and B are equivalent to each other. If the forces and arms are incommensurable, it is always possible to find forces and arms which shall be commensurable, and shall differ from the given forces and arms by differences less than any given quantity ; so that if the theorem were in error for incommen- surable forces and arms, it would also be in error for certain com- mensurable forces and arms \ but this is impossible ; therefore the theorem is true for incommensurable as well as for commensurable forces and arms. 32. Resultant of Couples with the Same Axis. COROLLARY. A combination of any number of couples having the s^me axis is equiva- lent to a couple whose moment is the algebraical sum of the moments of tlie combined couples. 33. Equilibrium of Couples having the Same Axis. Two Opposite couples of equal moment, having the same axis, balance each other. Any number of couples, having the same axis, balance each other when the moments of the right-handed couples are together equal to the moments of the left-handed couples ; in other words, when the resultant moment is nothing. 34. Representation of Couples by Lines. The nature and amount of the tendency of a couple to turn a body are completely known when the moment and direction of the couple, and the position of its axis, are known. These circum- stances are expressed by means of a line in the following manner. In fig. 5, from any point O draw a straight line OM, parallel to the axis (that is, perpendicular to the plane) of the couple to be represented, and in such F . a direction, that to an observer looking from M towards O the couple shall seem right-handed ; and make the length of the line O M represent the moment of the couple, according to any assigned scale. '24: PRINCIPLES OF STATICS. SECTION 2. On Couples with Different Axes. 35. Resultant of Two Couples with Different Axes. THEOREM. If the two sides of a parallelogram represent the positions of the axes, and the directions and moments, of two couples acting on tlie same body, the diagonal of the parallelogram will in like manner represent the position of the axis, the direction and the moment of the resultant couple, which is equivalent to those two. In fig. 6, let the plane of the paper represent a plane which con- tains the axes of the two couples, and is therefore perpendicular to both their planes. Let a c, c b be parts of the lines in which the planes of the couples A B respectively intersect the plane of the paper. If the couples are not already of equal force, reduce them to equiva- lent couples of equal force ; let F denote the common magnitude of their forces, and let L A , , L B denote the respective arms of the couples. From c, the intersection of the three planes already mentioned, take c a = L A , c b = L B , and join a b. Conceive the couple A (or an equivalent couple) to consist of the force + F acting forwards at a, and the equal and opposite force F acting backwards at c ; also conceive the couple B (or an equivalent couple) to con- sist of the force 4- F acting forwards at c, and the equal and opposite force F acting back- wards at b. The forces + F, F, at c balance each other ; and there are left the equal and opposite forces + F at a, and F at b, forming the resultant couple, which is equivalent to the two couples A and B, and has for its arm the third side a b L of the triangle a be. Now from any point O draw O M A perpendicular to a c, and O M B perpendicular to b c, and representing the axes, directions, and moments of the couples A and B : complete the parallelogram of which those lines are the sides, and draw its diagonal O M c . This diagonal will be perpendicular to a b, and will therefore re- present the axis and direction of the resultant couple ; and because of the similarity of the triangles a be, O M c M B , the following pro- portions will exist : Oll A : Oll B : O M c , : : L A : L B : L c ; and consequently O M c will also represent the moment of the re- sultant couples. Q. E. D. PARALLEL FORCES. 20 36. Equilibrium of Three Couples with Different Ax-cs in the Same Plane. COROLLARY. A couple equal and opposite to that represented by the diagonal O M c balances the couples represented by the sides O M A , O M B . In other words, three couples represented by the three sides of a triangle balance each other. 37. Equilibrium of any Number of Couples. COROLLARY. If a number of couples acting on the same body be represented by a series of lines joined end to end, so as to form sides of a polygon, and if the polygon is closed, these couples balance each other. To fix the ideas let there be five couples, whose moments are respectively M], M 2 , M 3 , M 4 , M 5 ; and let them be represented by the sides of the polygon in fig. 7 in such a manner that M! is represented by A, and seems right-handed looking from A towards 0. M 2 A~B, from B towards A. M 3 B~C, from C towards B. M 4 C D, from D towards C. M 5 TTO, from towards D. Then by the theorem of Article 35, the resultant of M : and M 2 is B ; the resultant of this and M 3 is O C ; the resultant of this and M 4 is O D, right-handed in looking from D towards O, and con- sequently equal and opposite to M 5 , which last couple balances it, and reduces the final resultant to nothing. Q. E. D. This proposition evidently holds for any number of couples, and whether the closed polygon be plane or gauche (that is to say, not plane). The resultant of the couples represented by all the sides of the polygon, except one, is equal and opposite to the couple represented by the excepted side. SECTION 3. On Parallel Forces. 38. Balanced Parallel Forces in General. A balanced system of parallel forces consists either of pairs of directly opposed equal forces, or of couples of equal forces, or of combinations of such pairs and couples. Hence the following propositions as to the relations amongst the magnitudes of systems of parallel forces are obvious : I. In a balanced system of parallel forces, the sums of the forces acting in opposite directions are equal ; in other words, the alge- PRINCIPLES OF STATICS. braical sum of the magnitudes of all the forces taken with their proper signs is nothing. II. The magnitude of the resultant of any combination of par- allel forces is the algebraical sum of the magnitudes of the forces. The relations amongst the positions of the lines of action of balanced parallel forces remain to be investigated ; and in this inquiry, all pairs of directly opposed equal forces may be left out of consideration ; for each such pair is independently balanced what- soever its position may be ; so that the question in each case is to be solved by means of the theory of couples. 39. Equilibrium of Three Parallel Forces in One Plane. Prin- ciple of the JLevev. THEOREM. If three parallel forces applied to one body balance each other, they must be in one plane; the two extreme forces must act in tJie same direction; the middle force must act in the opposite direc- tion; and the magnitude of each force must be proportional to the distance between the lines of action of the other two. Let a body (fig. 8) be maintained in equilibrio by two opposite couples having the same axis, and of equal moments, according to the notation already used ; and let those couples be so applied to the body that the lines of action of two of these forces, - F A , F B , which act in the same direction, shall coincide. Then those two forces are equivalent to the single middle force F c = (F A 4- F B ), equal and opposite to the sum of the extreme forces + F A , + F B , and in the same plane with them ; and if the straight line A C B be drawn perpendicular to the lines of action of the forces, then ~ = L B ; AB = L A Fig. 8. L B ; and consequently F A : F B :F C : : AC" : C~B : A^; so that each of the three forces is proportional to the distance between the lines of action of the other two ; and if any three parallel forces balance each other, they must be equivalent to two couples, as shown in the figure. 40. Resultant of Two Parallel Forces The resultant of any two of the three forces F A , F B , F c , is equal and opposite to the third. Hence the resultant of two parallel forces is parallel to th MOMENT OF A FORCE. 27 and in the same plane ; if they act in the same direction, then their resultant is their sum, acts in the same direction, and lies between them if they act in opposite directions, their resultant is their difference, acts in the direction of, and lies beyond, the prepon- derating force ; and the distance between the lines of action of any two of those three forces the resultant and its two components is proportional to the third force. In order that two opposite parallel forces may have a single resultant, it is necessary that they should be unequal, the resultant being their difference. Should they be equal, they constitute a couple, which has no single resultant. 41. Resultant of a Couple and a Single Force in Parallel Planes. Let M denote the moment of a couple applied to a body (fig. 9) ; and at a point O let a single force F be applied, in a plane parallel to that of the couple. For the given couple substitute an equivalent couple, consisting of a force F equal and directly opposed to F at O, and a force F applied at A, the arm A M , _ being = -, and of course par- # Fig. 9. allel to the plane of the couple M. Then the forces at balance each other, and F applied at A is the resultant of the single force F applied at O, and the couple M j that is to say, that if to a single force F there be added a couple M whose plane is parallel to the force, the effect of that addition is to shift the line of action of the force parallel to itself through a M distance O A = -=-; to the left if M is right- Jb handed to the right if M is left-handed. 42. Moment of a Force with respect to an Axis. Let the straight line F represent a force ap- plied to a body. Let O X be any straight line perpendicular in direction to the line of action of the force, and not intersecting it, and let A B be the common perpendicular of those two lines. At B conceive a pair of equal and directly op- posed forces to be applied in a line of action parallel to F, viz. : F = F, and - F = - F. The supposed application of such a pair of balanced forces does not alter the statical condition of the Fig. 10. body. Then the original single force F, applied in a line tra- PRINCIPLES OF STATICS. versing A, is equivalent to the force F' applied in a line traversing B, the point in O X which is nearest to A, combined with the couple composed of F and F', whose moment is F AB. This is called the moment of the force F relatively to the axis O X, and sometimes also, the moment of the force F relatively to the plane which contains O X, and is parallel to the line of action of the force. If from the point B there be drawn two straight lines B D and B E, to the extremities of the line F representing the force, the area of the triangle B D E being = J F AB, represents one-half of the moment of F relatively to O X. 43. Equilibrium of any System of Parallel Forces in One Plane. In order that any system of parallel forces whose lines of action are in one plane may balance each other, it is necessary and suffi- cient that the following conditions should be fulfilled : I. (As already stated in Art. 38) that the algebraical sum of the forces shall be nothing : II. That the algebraical sum of the moments of the forces rela- tively to any axis perpendicular to the plane in which they act shall be nothing : two conditions which are expressed symbolically as follows : let F denote any one of the forces, considered as positive or nega- tive, according to the direction in which it acts ; let y be the per- pendicular distance of the line of action of this force from an arbitrarily assumed axis O X, y also being considered as positive or negative, according to its direction ; then, Sum of forces, 2 F - ; Sum of moments, 2 y F = 0. For, by the last Article, each force F is equivalent to an equal and parallel force F' applied directly to O X, combined with a couple y F ; and the system of forces F', and the system of couples y F, must each be in equilibrio, because when combined they are equiva- lent to the balanced system of forces F. In summing moments, right-handed couples are usually considered as positive, and left-handed couples as negative. 44. Resultant of any Number of Parallel Forces in One Plane. The resultant of any number of parallel forces in one plane is a force in the same plane, whose magnitude is the algebraical sum of the magnitudes of the component forces, and whose position is such, that its moment relatively to any axis perpendicular to the plane in which it acts is the algebraical sum of the moments of the com- ponent forces. Hence let F r denote the resultant of any number of parallel forces in one plane, and y r the distance of the line of MOMENTS OF A FORCE. action of that resultant from the assumed axis O X to which the positions of forces are referred : then F r = 2 F ; v - 2 ^- F y ' - - ' In some cases, the forces may have no single resultant, 2 F being = ; and then, unless the forces balance each other com- pletely, their resultant is a couple of the moment 2 . y F. 45. Moments of a Force wish respect to a Pair of Rectangular Axe*. In fig. 11, let F be any single force; O an arbitrarily-assumed point, called the " originof co-ordin- ates;" -X O + X, - Y O + Y, a pair of axes traversing O, at right angles to each other and to the line of action of F. Let A B = y, be the common perpen- dicular of F and OX ; let AC = x, be the common perpendicular of F and OY. x and y are the "rectan- gular co-ordinates" of the line of action of F relatively to the axes - X O + X, - Y O + Y, re- spectively. According to the ar- rangement of the axes in the figure, x is to be considered as positive to the right, and nega- tive to the left, of - YO + Y; and y is to be considered as positive to the left, and negative to the right, of XO + X ; right and left referring to the spectator's right and left hand. In the particular case represented, x and y are both positive. Forces, in the figure, are considered as positive upwards, and negative downwards ; and in the particular case represented, F is positive. At B conceive a pair of equal and opposite forces, F' and F 7 , to be applied ; F' being equal and parallel to F, and in the same direction. Then, as in Article 42, F is equivalent to the single force F 1 = F applied at B, combined with the couple constituted by F and F' with the arm y, whose moment is y F ; being positive In the case represented, because the couple is right-handed. Next, at the origin O, conceive a pair of equal and opposite forces, F" and F", to be applied, F" being equal and parallel to F and F', and in the same direction. Then the single force F is equivalent to the single force F" = F = F applied at O, combined with the couple constituted by F' and F" with the arm B = aj, whose moment is Fig. 11. 3U PRINCIPLES OF STATICS. x F ; being negative in the case represented, because the couple is left-handed. Hence it appears finally, that a force F acting in a line whose co-ordinates with respect to a pair of rectangular axes perpendicular to that line are x and ?/, is equivalent to an equal and parallel force acting through the origin, combined with two couples whose moments are, y F relatively to the axis X, and x F relatively to the axis O Y ; right-handed couples being considered positive ; and + Y lying to the left of + X, as viewed by a spectator looking from + X towards O, with his head in the direction of positive forces. 46. Equilibrium of any System of Parallel Forces. In order that any system of parallel forces, whether in one plane or not, may balance each other, it is necessary and sufficient that the three following conditions should be fulfilled : I. (As already stated in Art. 38), that the algebraical sum of the forces shall be nothing : II. and III. That the algebraical sums of the moments of the forces, relatively to a pair of axes at right angles to each other, and to the lines of action of the forces, shall each be nothing : conditions which are expressed symbolically as follows : F = 0; 2'7/F - 0; 2 F =0; for by the last Article, each force F is equivalent to an equal and parallel force F" applied directly to 0, combined with two couples, y F with the axis O X, and - x F with the axis O Y ; and the system of forces F", and the two systems of couples y F and x F, must each be in equilibrio, because when combined they are equi- valent to the balanced system of forces F. 47. Resultant of any Number of Parallel Forces. The resultant of any number of parallel forces, whether in one plane or not, is a force whose magnitude is the algebraical sum of the magnitudes of the component forces, and whose moments relatively to a pair of axes perpendicular to each other and to the lines of action of the forces, are respectively equal to the algebraical sums of the moments of the component forces relatively to the same axes. Hence let F r denote the resultant, and x r and y r the co-ordinates of its line of action, then F r = 2 - F, *' = f^J _ 2-yF Vr ~ I~~~T In some cases, the forces may have no single resultant, 2 j 1 CENTRE OF PARALLEL FORCES. 31 being = ; and then, unless the forces balance each other com- pletely, their resultant is a couple, whose axis, direction, and moment are found as follows : Let M. = 2 . y F ; M y = - 2 . x F ; be the moments of the pair of partial resultant couples relatively to the axes X and Y respectively. From O, along those axes, set off two lines representing respectively M, and M y according to the rule of Art. 34 ; that is to say, proportional to those moments in length, and pointing in the direction from which those couples must respectively be viewed in order that they may appear right- handed. Complete the rectangle whose sides are those lines j its diagonal (as shown in Art. 35) will represent the axis, direction, and moment of the final resultant couple. Let M r be the moment of this couple ; then M, + M y and if 6 be the angle which its axis makes with X, M. COS & = -^f ' Mr SECTION 4. On Centres of Parallel Forces. 48. Centre of a Pair of Parallel Forces. In fig. 12, let A and B represent a pair of points, to which a pair of parallel forces, F A and F B , of any given magnitudes, are applied. In the straight line joining A and B take the point C such, that its distances from A and B respec- tively shall be inversely proportional to the forces applied at those points. Then from the principle of Art. 40 it is obvious that the resultant of F A and F B traverses C. It is also obvious that the position of the point C depends solely on the proportionate mag- Fio . 12 nitude of the parallel forces F A and F B , and not on their absolute magnitude, nor on the angular position of their lines of action; so that if for those forces there be substituted another pair of parallel forces, f a ,f b , in any other angular position, and if those new forces bear to each other the same proportion with the original forces, viz. : fa :/ : :F A :F B : : BC the point C where the resultant cuts A B will still be the same. This point is called the Centre of Parallel Forces, for a pair of 32 PRINCIPLES OF STATICS. forces applied at A and B respectively, and having the given ratio BC : AC). 49. Centre of any System of Parallel Forces. Let parallel forces, F , F b be applied at the points A Aj (fig. 13.), Draw the straight line A A 1} in which take Cj so that F : F, : then will C x be the centre of a pair of Fig. 13. NMh * parallel forces applied at AO and A,, and having the proportion F : F,. At a third point, A 2 , let a third parallel force, F 2 , be applied. Then, because the forces F , F,, are together equivalent to a parallel force, F + F,, applied at C b draw the straight line C, A 2 , in which take C 2 , so that F + F, : F 2 : : C^A, : C7C7; then will C 2 be the centre of three parallel forces applied at A , A,, A 2 , and having the proportions F : F l : F 2 . At a fourth point, A 3 , let a fourth parallel force, F 3 , be applied. Then, because the forces F , F,, F 8 , are together equivalent to a parallel force, F + Fj + F 2 , applied at C 2 , draw the straight line C 2 A 3 , in which take C 3 , so that F + F 1 + F 2 :F 3 : : C^A a : C~C^ then will C 3 be the centre of four parallel forces applied at A , A,, A.,, A 3 , and having the proportion F : F x : F 2 : F 3 . By continuing this process the centre of any system of parallel forces, how nume- rous soever, may be found ; and hence results the following THEOREM. If there be given a system of points, and the mutual ratios of a system of parallel forces applied to those points, then there is one point, and one only, which is traversed by the line of action of the resultant of every system of parallel forces having the given mutual ratios and applied to tlie given system of points, whatsoever may be the absolute magnitudes of those forces, and the angular position of their lines of action. 50. Co-ordinates of Centre of Parallel Forces. The method of finding centres of parallel forces described in the preceding Article, though suitable for the demonstration of the theorem just stated, is tedious and inconvenient when the number of forces is great, in which case the best method is to find the rectangular co-ordinates of that point relatively to three fixed axes, as follows : Let O be any convenient point, taken as the origin of co-ordi- nates, and OX, O Y, OZ, three axes of co-ordinates at right angles to each other. CENTRE OF PARALLEL FORCES. 33 Let A be any one of the points to which the system of parallel forces in question are applied. From A draw x parallel to OX, and perpendicular to the plane Y Z, y parallel to O Y, and perpendicular to the plane Z X, and z parallel to O Z, and perpendicular to the plane X Y. x, y, and z are the rectangu- lar co-ordinates of A, which, being known, the position of A is deter- mined. Let F denote either the magnitude of the force applied at A, or any magnitude proportional to that magnitude, x, y, z, and F are supposed to be known for every point of the given system of points. Then first, conceive all the parallel forces to act in lines parallel to the plane Y Z. Then the sum of their moments relatively to an axis in that plane is 2'tfFj and consequently the distance of their resultant, and also of the centre of parallel forces from that plane is given (as in Articles 44 and 47), by the equation 2 -#F Fig. 14. Secondly, conceive all the parallel forces to act in lines parallel to the plane Z X. Then the sum of their moments relatively to an axis in that plane becomes 3-yF; and consequently the distance of their resultant, and also of the centre of parallel forces from that plane is given by the equation Thirdly, conceive all the parallel forces to act in lines parallel to the plane X Y. Then the sum of their moments relatively to an axis in that plane becomes 2 -*F; and consequently the distance of their resultant, and also of the centre of parallel forces from that plane is given by the equation Thus are found x r , y r , z r , the three rectangular co-ordinates of D 34 PRINCIPLES OF STATICS. the centre of parallel forces, for a system of forces applied to any given system of points, and having any given mutual ratios. If the parallel forces applied to a system of points are all equal, then it is obvious that the distance of the centre of parallel forces from any given plane is simply the mean of the distances of the points of the system from that plane. 35 CHAPTER III. BALANCE OF INCLINED FORCES. SECTION 1. Inclined Forces applied at One Point. 51. Parallelogram of Forces. THEOREM. If two forces whose lines of action traverse one point be represented in direction and magnitude by the sides of a parallelogram, their resultant is represented by the diagonal. First Demonstration. Through the point (fig. 15), let two forces act, represented in direction and magnitude by O A and O B. ^-^ The resultant or equivalent single ^ ,*'*' force of those two forces must be a force such, that its moment relatively to any axis whatsoever perpendicu- lar to the plane of O A and O B, is the sum of the moments of A and O B relatively to the same axis. Now, first, the force represented in direction and magnitude by the dia- Fig. 15. gonal O C of the parallelogram A B fulfils this condition. For let P be any point in the plane of O A and O B, and let an axis perpendicular to that plane traverse P. Join PA, P B, PC. Then, as already shown in Art. 42, the moments of the forces A, OB, O C, relatively to the axis P, are represented respectively by the doubles of the triangles POA, POB, POC. Draw AD || BE || OP, and join PD, P E. Then A P O D = A PQ A, and A P OE = A P O B ; but be- cause OI) + OE" - OC~ .-. A P O C - A POD + A P O E - A POA + A P B ; and the moment_o O C relatively to P is equal to the sum of the moments of O A and B ; and that whatsoever the position of P may be. Secondly. The force represented by O C is the only force which fulfils this condition. For let O Q represent a force whose moment relatively to P is equal to the sum of the moments of O A and B. Join P Q. Then A O P Q = A P C, and .-. C Q II P O ; so that 36 PRINCIPLES OF STATICS. O Q fulfils the required condition for those axes only which are situated in a line O P || C Q, and not for any other axis. Therefore the diagonal O C of the parallelogram A B represents the resultant, and the only resultant, of the forces represented by is reckoned from O X in the direction towards Y, and the angle /3 from O Y in the reverse direction, that is, towards X, and that ,-, - f T f ( to 180 ) f positive. the sines of angles from j lg()0 to 36QO j are j ^^ If a system of forces acting through one point balance each other, their resultant is nothing ; and therefore the rectangular components of their resultant, which are the resultants of their parallel systems of rectangular components, are each equal to nothing ; a case re- presented as follows : 2-F 1 = 0; 2-F 2 = 0; 2 - F 3 = ............. (6.) SECTION 2. Inclined Forces Applied to a System of Points. 58. Forces acting in One Plane Graphic Solution. Let any system of forces whose lines of action are in one plane, act together on a rigid body, and let it be required to find their resultant. Assume an axis perpendicular to the plane of action of the forces at any point, and let it be called O Z. According to the principle of Art. 42, let each force be resolved into an equal and parallel force acting through O, and a couple tending to produce rotation about O Z ; so that if a force F be applied along a line whose per- pendicular distance from O is L, that force shall be resolved into F - and || F acting through O, and a couple whose moment is M = LF, and which is right or left-handed according as O lies to the right or left of the direction of F. 40 PRINCIPLES OF STATICS. The magnitude and direction of the resultant are to be found by forming a polygon with lines equal and parallel to those representing the forces, as in Art. 53, when, if the polygon is closed, the forces have no single resultant; but if not, then the resultant is equal, parallel, and opposite to that represented by the line which is required in order to close the polygon. Let R be its magnitude if any. The position of the line of action of the resultant is found as follows : Let 2 M be the resultant of the moments of all the couples M, distinguishing right-handed from left-handed, as in Arts. 27 and 32. If 2-M = 0, and also R =0, then the couples and forces balance completely, and there is no resultant. If 2-M = 0, while R has magnitude, then the resultant acts through O. If 2 M and E, both have magnitude, then the line of action of the resultant II is at the perpendicular distance from O given by the equation 2-M and the direction of that perpendicular is indicated by the sign of 2 M. If R 0, while 2 M has magnitude, the only resultant of the given system of forces is the couple 2-M. 59. Forces acting in One Plane. Solution by Rectangular Co-or- dinates. Through the point O as origin of co-ordinates, let any two axes be assumed, O X and Y, perpendicular to each other and to Z, and in the plane of action of the forces ; and in looking from Z towards O, let Y lie to the right of X, so that rotation from X towards Y shall be right-handed. Let F, as before, denote any one of the forces ; let * be the angle which its line of action makes to the right of O X ; and let x and y be the co-ordinates of its point of application, or of any point in its line of action, relatively to the assumed origin and axes. Resolve each force F into its rectangular components as in Art. 57, F! -F-cosa; F 2 = F-sin *; then the rectangular components of the resultant are respectively parallel to OX, 2(F cos *) = R l3 ) n v parallel to O Y, 2 (F- sin ) - R 2 , / " its magnitude is given by the equation R 2 = R? + R1; ......................... (2.) and the angle r which it makes to the right of O X is found by the equations -p T> cos r=- ', sin * r =- ................... (3.) ANY SYSTEM OF FORCES. 41 The quadrant in which the direction of K lies is indicated by the algebraical signs of Rj and R 2 , as already stated in Art. 57. The perpendicular distance from O of the line of action of any force F is L = x ' sin oe, y cos a, which is positive or negative according as O lies to the right or to the left of that line of action ; and hence the resultant moment of the system of forces relatively to the axis Z is 2-FL = 2'F (x sin a. y cos ) = s( aJ F,-yF 1 ) ........................ (4.) whence it follows, that the perpendicular distance of the resultant force O is Let x r and y r be the co-ordinates of any point in the line of action of the resultant; then the equation of that line is which is equivalent to V ............... (6.) x r sin ce, r y r cos a, = L r As in Art. 58, if 2 F L = 0, the resultant acts through the origin O ; if 2 F L has magnitude, and R = (in which case R! = 0, R 2 = 0) the resultant is a couple. The conditions of equili- brium of the system of forces are or in other symbols > . . . . (7. ) 2-^ = 0; 2-F 2 = 0; *(x F 2 y F>) = 0. The moment of the resultant relatively to the axis O Z can also be arrived at by considering the moment F L of each force as the resultant of x F 8 , which is right-handed when x and F 2 are both positive, and of y F 1? which is left-handed when y and F t are both positive. 60. Any System of Forces. To find the resultant and the con- ditions of equilibrium of any system of forces acting through any system of points, the forces and points are to be referred to three rectangular axes of co-ordinates. As in Art. 57, let O denote the origin of co-ordinates, and OX, OY, OZ, the three rectangular axes; and let them be arranged (as in fig. 17), so that in looking from X | ( Y towards Z \ Y > towards O, rotation from < Z towards X > Z j (X towards Y j shall appear right-handed. 42 PRINCIPLES OF STATICS. Let F denote any one of the forces ; x, y, z, the co-ordinates of a point in its line of action ; and a, , /3, y, the angles which its direction makes with the axis respectively. Then the three rectangular components of F being as in Art. 57, F, = F cos along OX,) F 2 = F -cos ^ along OY, V ................. (1.) F 3 = F cos y along O Z, j it can be shown by reasoning similar to that of Art. 59, that the total moments of these components relatively to the three axes are respectively y F 3 z F 2 F (y cos y z cos /3) relatively to O X, ) z F! x F 3 = F (z cos et x cos y) relatively to O Y, V (2.) x F 2 y F! = F (x cos /3 y cos ) relatively to O Z ; j so that the force F is equivalent to the three forces of the formulae 1 acting through O along the three axes, and the three couples of the formulae 2 acting round the three axes. Taking the algebraical sums of all the forces which act along the same axes, and of all the couples which act round the same axes, the six following quantities are found, which compose the resultant of the given system of forces ; Forces. along OX; R : = 2 F cos a, ) OY; R 2 = 2-Fcos/8, I .............. (3.) OZ ; R 3 = 2 -Fcosy, j Couples. round OX; M t = 2 F (y cos y z cos /3) O Y; M 2 =s[F (z cos - acosy) O Z ; M 3 = 2 [F (x cos /3 - y cos a)} The three forces "R lt R 2 , R 3 , are equivalent to a single force R= ^(BJ + RS+EJ), ...................... (5.) , j. ........ (4.) > I acting through O in a line which makes with the axes the angles given by the equations R! RZ RS /r> \ COS r =z: ; COS/3,. = ; COS 7f = ............ (6.) The three couples M, 1 , M 2 , M 3 , according to Article 37, are equi- valent to one couple, whose magnitude is given by the equation ^) ..................... (7.) ANY SYSTEM OF FORCES. 43 and whose axis makes with the axes of co-ordinates the angles given by the equations M 2 M 3 ' , . T ! denote respectively the angles I ^ ^ T I m which | ,, j; made by * he ^^ M ^ JO Y j. The Conditions of Equilibrium of the system of forces may be ex- pressed in either of the two following forms : Ri = 0; R 2 = 0; R 3 = 0: M, = ; M 2 = 0; M, = 0...(9.) or R = 0; M = ......................... (10.) When the system is not balanced, its resultant may fall under one or other of the following cases : Case I. When M = 0, the resultant is the single force R acting through 0. Case II. When the axis of ]& is at right angles to the direction of R, a case expressed by either of the two following equations : COS r COS * + COS /3 r COS ft, + COS y f COS V j ) /-,-,>, or R 1 M, + R 2 M 2 + R 3 M 3 = Oj /' the resultant of M and R, is a single force equal and parallel to R, acting in a plane perpendicular to the axis of M, and at a perpen- dicular distance from O given by the equation Case in. When R = 0, there is no single resultant ; and the only resultant is the couple M. Case iv. When the axis ofM. is parallel to the line of action of ~R, that is, when either A = a r ; p = ft r ; v = yr , .................. (13). or * = r ; f* = p r ; v = v r ; ............ (14). there is no single resultant; and the system of forces is equiva- lent to the force R and the couple M, being incapable of being farther simplified. Case v. When the axis of M. is oblique to the direction of R, making with it the angle given by the equation cos 6 = cos A cos , (dotted))-^, ................... (2.) It is obvious, that if the values of the ordinates u required in these computations can be calculated, it is unnecessary to draw the figure to a scale, although a sketch of it maybe useful to assist the memory. When the symbol of integration is repeated, so as to make a double integral, such as f r u dxdy, II or a triple integral, such as r - dxdy dz, it is to be understood as follows :- Let III" = / u ' d be the value of this single integral for a given value of y. Con- struct a curve whose abscissae are the various values of y within the prescribed limits, and its ordinates the corresponding values of v. Then the area of that curve is denoted by / v ' d y = I I u d xdy. PROJECTION OF CENTRE OF GRAVITY. 61 Next, let t=fvdy be the value of this double integral for a given value of z. Con- struct a curve whose abscissae are the various values of z within the prescribed limits, and its ordinates the corresponding values of t, Then the area of that curve is denoted by / t . dz = I I v ' dydz / / / u ' dxdy dz; and so on for any number of successive integrations. 82. Centre of Gravity found by Projection -- -According to the geo- metrical properties of parallel projections, as stated in Chap. IV., Article 62, a parallel projection of a pair of volumes having a given ratio is a pair of volumes having the same ratio ; and hence, if a body of any figure be divided by a system of plane or other sur- faces into parts or molecules, either equal, or bearing any given system of proportions to each other, and if a second body, whose figure is a parallel projection of that of the first body, be divided in the same manner by a system of plane or other surfaces which are the corresponding projections of the first system of plane or other surfaces, the parts or molecules of the second body will bear to each other the same system of ratios, of equality or otherwise, which the parts of the first body do. Also, the centres of gravity of the parts of the second body will be the parallel projections of the centres of gravity of the parts of the first body. And hence it follows (according to Article 64), that if the figures of two bodies are parallel projections of each other, the centres of gravity of these two bodies are corresponding points in these parallel projections. To express this symbolically, as in Article 61, let x, y, z, be the co-ordinates, rectangular or oblique, of any point in the figure of the first body ; x', y', z, those of the corresponding point in the second body ; x 0) y , z , the co-ordinates of the centre of gravity of the first body ; x' , y f 0) sf m those of the centre of gravity of the second body ; then This theorem facilitates much the finding of the centres of gravity of figures which are parallel projections of more simple or more sym- metrical figures. For example : it appears, from symmetry, as in Art. 72, that the centre of gravity of an equilateral triangular prism is at the 62 PKINCIPLES OF STATICS. point of intersection of the lines joining the three angles of the middle section of the prism with the middle points of the opposite sides of that section. But all triangular prisms are parallel pro- jections of each other ; hence the above described point of inter- section is the centre of gravity of any triangular prism. Also, as in Art. 72, the centre of gravity of a regular tetraedron is at the point of intersection of the planes joining each of the edges with the middle point of the opposite edge. But all tetrae- drons are parallel projections of each other ; hence that point of intersection is the centre of gravity in any tetraedron. As a third example, let it be supposed that a formula is known (which will be given in the sequel) for finding the centre of gravity of a sector of a circular disc, and let it be required to find the centre of gravity of a sector of an elliptic disc. In fig. 27, let A B' A B' be the ellipse, A O A = 2 a, and B' O B' = 2 b, its axes, and C' O D' the sector whose centre of gravity is required. One of the parallel pro- jections of the ellipse is a circle, ABAB, whose radius is the semi-axis major a. The ellipse and the circle being both referred to rectangular co-ordinates, with their centre as origin, x and y denoting the co- Fig. 27. ordinates parallel to O A and O B respectively of a point in the circle, and x' and y' those of the corresponding point in the ellipse, those co-ordinates are thus related : y Through C' and D' respectively draw E C' C and FD'D, parallel to O B, and cutting the circle in C and D respectively ; the cir- cular sector C O D is the parallel projection of the elliptic sector C' O D'. Let G be the centre of gravity of the sector of the circular disc, its co-ordinates being Then the co-ordinates of the centre of gravity G' of the sector or the elliptic disc are "OH = x" = xl', EXAMPLES OF CENTRES OF GRAVITY. 63 Further examples of the results of this process will be found in the next Article. 83. Examples of Centres of Oravity. The following examples consist of formulae for the weight, the moment with respect to some specified axis, and the position of the centre of gravity, of homo- geneous bodies of those forms which most commonly occur in practice. In each case, as in the formulae of the preceding Articles, w denotes the specific gravity of the body, W, its weight, and x , &c., the co-ordinates of its centre of gravity, which in the diagrams is marked G, the origin of co-ordinates being marked O. A. PRISMS AND CYLINDERS WITH PARALLEL BASES. The word cylinder is here to be taken in its most general meaning, as comprehending all solids traced by the motion of a plane curvi- linear figure parallel to itself. The examples here given apply, of course, to flat plates of uni- form thickness. In the formulae for weights and moments, the length or thickness is supposed to be unity. The centre of gravity, in each case, is at the middle of the length (or thickness) ; and the formulae give its situation in the plane figure which represents the cross-section of the prism or cylinder, and which is specified at the commencement of each example. I. Triangle. (Fig. 28) O, any angle. Bisect .o opposite side B C in D. Join A D. x = 0~G = - OlX O W = w OT) ITC sin. ^ O D C ~2~ c II. Polygon. Divide it into triangles ; find the centre of gravity of each; then find their Fig. 28. common centre of gravity as in Art. 75. III. TrapessM. (Fig. 29.) AB||CE. Greatest breadth, A B = B. Least C E = 6. Bisect AB in 0, CE in D; join O D. OD B W = w -OD sin Fig. 29. ODE. 64 PRINCIPLES OF STATICS. IV. Trapezoid. (Second solution.) (Fig. 30.) O, point where inclined sides meet. Let O F 2 aj a- 2 sin 2 ^ O F B. (cotan ^ O A B + cotan ^L B A). Xo W = w Xl ~~ x * sins ^ O F B. o (cotan ^ O A B + cotan ^ B A). Y. Parabolic Half- Segment. (0 A B, fig. 31.) Q, vertex of diameter OX; A = #, ; A B = y lt ordinate || tangent O Y. 3 W.4-- XOY. Fig. 31. VI. Parabolic Spandril. (0 B C, fig. 31.) G', centre of gravity, _3_ ^3_ 10 4 W = w - - sin ^ X O Y. VII. Circular Sector. (O A C, fig. 32.) Let O X bisect the angle AOC; OY-i-OX. AC Fig. 32. Radius O A = r Half-arc, to radius unity, n \ r\ :== 2 sintf x = -y- r -y-j T/ O = 0. W = w r 2 & CENTRES OF GRAVITY. VIII. Circular Half-Segment. (A B X, fig. 32.; 2 sin 3 6 /v .-... ^ /n 3 S - sin cos / 2/o = 4 sin 2 sin 2 cos 6 3 4 cos sin 1 W = - wr* (& cos d sin 0). 4H IX. Circular Spandril. (A D X, fig. 32.) _1_ sin 3 6 ~~ 3 2 sin & - sin & cos 6 - ti , 3 sin 2 6 - 2 sin 2 cos d - 4 sin 2 ^ "3 2 sin - sin 6 cos - ^ ' W sin 6 - n sin & cos 6 -2)' X. Sector of Ring. (A C F E, fig. 32.) O A = r ; OE = /. 2 r 3 - r* sin tf W == w (r 2 - r' 2 ) 6. XL Elliptic Sector, Half-Segment, or Spandril. Centre of gravity to be found by projection from that of corresponding circular figure, as in Article 82. B. WEDGES. A Wedge is a solid bounded by two planes which meet in an edge, and by a cylindrical or prismatic surface (cylindrical, as before, being used in the most general sense). XII. General Formulae for Wedges. (Fig. 33.) All wedges may be divided into parts such as the figure here represented. A Y, OXY, planes meeting in the edge O Y; AX Y, cylindrical (or pris- matic) surface perpendicular to the plane OXYj OX A, plane triangle perpendicular to the edge OY; OZ, axis perpendicular to XO Y. Let OX =*=*,; X A =s Then z = ; x l :=w '~f x y d vO| J Fig. 33. PRINCIPLES OF STATICS. / x z ydx I xy dx J xy z 'dx xydx z\o (This last equation denoting that G is in the plane which traverses O Y and bisects AX.) In a symmetrical wedge, if O be taken at the middle of the edge, 2/ = O. Such is the case in the following examples, in each of which, length of edge = 2 y v XIII. Rectangular Wedge. (= Triangular Prism.) (Fig. 34.) W = w Fig. 34. XIV. Triangular Wedge. (= Triangular Pyramid.) w=I w ^Jx Fig. 35. XQ XV. Semicircular Wedge. (Fig. 36.) Radius OX = OY = r. Fig. 36. X Q --- 16 (^ = 3 1416 nearly). CENTRES OF GRAVITY. 67 XVI. Annular, or Hollow Semicircular Wedge. (Fig. 37.) External radius, r; internal, r. 2 XQ = /* 16 r 3 r* C. CONES AND PYRAMIDS. Fig. 37. Let O denote the apex of the cone or pyramid, taken as the origin, and X the centre of gravity of a supposed prism whose middle section coincides with the base of the cone, or pyramid. The centre of gravity will lie in the axis OX. Denote the area of the base by A, and the angle which it makes with the axis by 6. XVII. Complete Cone or Pyramid. Let the height OX h; 3 h *o = T "W = -5 w ' A h sin Q. o XVIII. Truncated Cone or Pyramid. Height of portion trun- cated = h'. * 0= I 'A'-**'. W - I 3 D. PORTIONS OF A SPHERE. XIX. Zone or Ring of a Spherical Shell, bounded by two conical surfaces having their common apex at the centre O of the sphere (fig. 38). O X, axis of cones and zone. r, external radius ) f , 1T J . . i T . > oi shell. r , internal radius j XO A = x, half-angle of less ) greater j cone. Fig. 38. 68 PRINCIPLES OF STATICS. W 3 r 4 - r' 4 cos a. + cos ft 4 r 3 r' 3 2 = ^ ^^ (f 3 - f*J . (cos ft - COS aj. XX. tfecfor o/ a Hemispherical Shell. (C X D, fig. 39.) O Y bisects angle DOC; D C = * r 4 - r 4 _ r '< sn 84. Heterogeneous Body. If a- body consists of parts of definite figure and extent, whose specific gravities are different, although each individual part is homogeneous, the centres of gravity of the parts are to be found as in Article 74 and the subsequent Articles, and the common centre of gravity of the whole as in Article 73. 85. Centre of Gravity found Experimentally/ The centre of gravity of a body of moderate size may be found approximately by experiment, by hanging it up successively by a single cord in two different positions, and finding the single point in the body which in both positions is intersected by the axis of the cord. For the resistance of the cord is equivalent sensibly to a single force acting along its axis ; and as that force balances the weight of the body when hung by the cord, its line of action must, in all positions of the body, traverse the centre of gravity of the body. SECTION 2. Of Stress, and its Resultants and Centres. 86. Stress, its Nature and Intensity. The word STRESS has been adopted as a general term to comprehend various forces which are exerted between contiguous bodies, or parts of bodies, and which are distributed over the surface of contact of the masses between which they act. The INTENSITY of a stress is its amount in units of force, divided by the extent of the surface over which it acts, in units of area. The French and British units of intensity of stress are compared CLASSES OF STRESS. 6 ( J in a table annexed to this volume. The following table shows a comparison between different British units of intensity of stress : Pounds on the Pounds on the square foot. square inch. One pound on the square inch, 144 1 One pound on the square foot, 1 T^T One inch of mercury (that is, weight of a column of mercury at 32 Fahr., one inch high), 70*73 0-4912 One foot of water (at 39-4 Fahr.), 62-425 0-4335 One inch of water, 5-2011 0-036125 One atmosphere, of 29 '922 inches of mercury, 2116*4 14-7 87. Classes of Stress. Stress may be classed as follows : I. Thrust, or Pressure, is the force which acts between two con- tiguous bodies, or parbs of a body, when each pushes the other from itself, and which tends to compress or shorten each body on which it acts, in the direction of its action. It is the kind of force which is exerted by a fluid tending to expand, against the bodies which surround it. Thrust may be either normal or oblique, relative to the surface at which it acts. II. Pull, or Tension, is the force which acts between two con- tiguous bodies, or parts of a body, when each draws the other towards itself, and which tends to lengthen each body on which it acts, in the direction of its action. Pull, like thrust, may be either normal or oblique, relatively to the surface at which it acts. III. Shear, or Tangential Stress, is the force which acts between two contiguous bodies or parts of a body, when each draws the other sideways, in a direction parallel to their surface of contact, and which tends to distort each body on which it acts. In expressing a Thrust and a Pull in parallel directions algebrai- cally, if one is treated as positive, the other must be treated as negative. The choice of the positive or negative sign for either is a matter of convenience. In treating of the general theory of stress, the more usual system is to call a pull positive, and a thrust negative : thus, let p denote the intensity of a stress, and n a certain number of pounds per square foot ; p = n will denote a pull, and p = n a thrust of the same intensity. But in treating of certain special applications of the theory, to cases in which thrust is the only or the predominant stress, it becomes more convenient to reverse this system, calling thrust positive, and pull negative. The word " Pressure," although, strictly speaking, equivalent to " thrust," is sometimes applied to stress in general; and when this is the case, it is to be understood that thrust is treated as positive. 70 PKINCIPLES OF STATICS. 88. Resultant of Stress : its Magnitude. If to a plane surface of any figure, whose area is S, there be applied a stress, either normal, oblique, or tangential, and parallel in direction at all points of the surface (according to the restriction stated in Art. 67), then if the intensity of the stress be uniform over all the surface, and denoted by p, the amount or magnitude of its resultant will be ?=p$ .............................. (1.) If the intensity of the stress is not uniform, that amount is to be found by integration. For example, in fig. 40, let A A A be the plane surface, and let it be referred to rectangular axes of co-ordinates in its own plane, OX, O Y. Conceive that plane to be divided into small rectangles by a network of lines parallel to X and O Y respectively, and let A x, y, be the dimensions of any one - 40 - O f these rectangles, such as that marked a in the figure. Conceive a figure approximating to that of the given plane surface to be composed of several of these small rectangles, so that S = 2 AX y nearly ; ....................... (2.) let p be the intensity of the stress at the centre of any particular rectangle, so that the stress on that rectangle is p A x A y nearly. Then the amount of the resultant stress is given approximately by the equation P = 2 p &x &y nearly ...................... (3.) Then passing, as in previous examples, to the integrals, or limits towards which the sums in the equations 2 and 3 approach as the minuteness of the subdivision into rectangles is indefinitely in- creased, we find, for the exact equations, The mean intensity of the stress is given by the following equation : p \\pdxdy P = -g- = 7 r . , .................. ( 5 -) I I dxdy CENTRE OP STRESS. 71 A convenient mode of representing to the mind the foregoing process is as follows: In fig. 41, let A A be the given plane surface ; O X, Y, the two axes of co-ordinates in its plane ; O Z, a third axis perpendicular to that plane. Conceive a solid to exist, bounded at one end by the given plane surface A A, laterally by a cylindrical or prismatic surface generated by the motion of a straight line par- allel to O Z round the outline of A A, and at the other end by a surface B B, of such a figure, that its ordinate z at any point shall be proportional to the intensity of the stress at the point of the surface A A from which that ordinate proceeds, as shown by the equation .(6.) The volume of this ideal solid will be V =J j z 'dxdy (7.) So that if it be conceived to consist of a material whose specific gravity is w, the amount of the stress will be equal to the weight of the solid, that is to say, P = V... (8.) If the stress be of opposite signs at different points of the plane surface A A, the surface B B and the solid which it terminates will be partly at one side of A A and partly at the opposite side, as in fig. 42 ; and in this case, the two parts into which the solid A B A B is divided by the plane X O Y, are to be regarded as having opposite signs, and V is to be held to represent the difference of their volumes. Fig. 42. The mean stress of equation 5 is evidently (9.) in which z is the height of a parallel-ended prism or cylinder standing on the base A A A, and of volume equal to the solid A BAB. 89. The Centre of Stress, or of Pressure, in any surface, is the point traversed by the resultant of the whole stress, or in other words, the Centre of Parallel Forces for the whole stress. From the principles already proved in Chap. II., Section 4, it follows, that 72 PRINCIPLES OF STATICS. the position of this point does not depend upon the direction of the stress, nor upon its absolute magnitude ; but solely on the form of the surface at which the stress acts, and on the proportions between the intensities of the stress at different points. As in Article 88, conceive a figure approximating to that of the given plane surface A A A (fig. 40), to be composed of several small rectangles ; let /3 denote the angles which the direction of the stress makes with X, O Y respectively. Then the moments, relative to the co-ordinate planes, Z O X, Z O Y, of the components parallel to those planes of the stress on A x A y t are given by the approxi- mate equations. Moment relatively to Z X, yp A x A y ' sin $\ mar i Summing all such moments, and passing to the integral or limit of the sum, as in former examples, we find the following expressions, in which x and y Q denote the co-ordinates of the centre of stress ; 2/o P sin ft = sin ft / / yp - dx dy ) f \, n \ X P sin ex, = sin a. I I xp 'dxdy I Consequently the co-ordinates of the centre of stress are / / xp 'dxdy I I p dxdy Y f I (2-) J J yp-dxdy I 1 p-dxdy which are evidently the same with the co-ordinates, parallel to O X and O Y, of the centre of gravity of the ideal solid of fig. 41, whose ordinates z are proportional to the intensity of the pressure at the points on which they stand. When the intensity of the stress is positive and negative at different points of the surface A A A, cases occur in which the positive and negative parts of the stress balance each other, so that the total stress is nothing, that is to say, p d x d y = 0. In such cases, the resultant of the stress (if any) is a couple, and there is no centre of stress. This case will be further considered in the sequel. UNIFORMLY VARYING STRESS. 73 90. Centre of Uniform Stress. If the intensity of the stress be uniform, the factor p in equation 2 of Article 89 becomes constant, and may be removed from both numerator and denominator of the expressions for x and y^ which then become simply the co- ordinates of the centre of gravity of a flat plate of the figure A A A. This also appears from the consideration, that the surface B B in fig. 41 becomes a plane parallel to A A, and the solid ABAB, a parallel-ended prism or cylinder. 91. Moment of Uniformly Varying Stress. By an Uniformly varying stress is understood a stress whose intensity, at a given point of the surface to which it is applied, is proportional to the distance of that point from a given straight line. For example, let the given straight line be taken as the axis Y ; then the following equation p = ax, (1.) a being a constant, represents the law of variation of the intensity of an uniformly varying stress. The amount of an uniformly varying stress is given by the equa- tion P = ( ! p-dxdy = a f I x- dxdy (2.) which, if the axis OY traverses the centre of gravity of a plate of the figure of the surface of action A A A, becomes equal to nothing, the positive and negative values of p balancing each other. In this case, OY is called a NEUTRAL AXIS of the surface A A A. In fig. 43, let AAA represent the plane surface of action of a stress ; let O be its centre of gravity (that is, the centre of gravity of a flat plate of which AAA is the figure) ; - Y O Y the neutral axis of the stress applied ; X O X perpendi- cular to YOY, and in the plane of AAA; Z Z perpendicular to that plane. Conceive a plane B B inclined to AAA to traverse the neutral axis, and to form, with the plane AAA, a pair Fig. 43. of wedges bounded by a cylindrical or prismatic surface parallel to Z Z. The ordinate z, drawn from any point of AAA to BB, will be proportional to the intensity of the stress at that point of A A A, and will indicate by its upward or downward direction whether that stress is positive or negative ; and the nullity of the total stress will be indicated by 74 PRINCIPLES OF STATICS. the equality of the positive wedge above A A A, and the negative wedge below A A A. The resultant of the whole stress is a couple, whose moment, and the position of its axis, are found in the following manner, by the application of the process of Chap. III., Sect. 2, Article 60. Let , /8, y, be the angles which the direction of the stress makes with OX, OY, OZ, respectively. Let Ax&y denote, as before, the area of a small rectangular portion of the surface, re, y, the co- ordinates of its centre (for which z = 0), and p = ax, the intensity of the stress on it, so that is the force acting on this rectangle. The moments of this force relatively to the three axes of co-ordi- nates, are found to be as follows, by making the proper substitutions in equation 2 of Article 60 : round OX; A P * y cos y ; O Y ; A P x cos y ; OZ; AP(#cos/3 2/cosa). Summing and integrating those moments, the following are found to be the total moments : round OX ; M, = a cos y \\xydxdy O Y ; M 2 = a cos y / / x 2 d x d y OZ; M 3 = jcos/3 f fa?-dxdy-cosei f fxydxdyl For the sake of brevity, let / / x 2 dxdy = I' } I I xy 'dxdy = K; (^A.) J J J J then, as in equation 7 of Article 60, we find, for the moment of the resultant couple, -_ a . j {(P + K 2 ) cos 2 y + P cos 2 /3 + K a cos 2 * - 2 I K cos - cos /3. } = a J(I 2 ' sin 2 /3 + K 2 sin 2 a 2 I K cos * cos );... (4.) and for the angles A, p., v, made by the axis of that couple with the axes of co-ordinates, we find the angles whose cosines are as follows: M 2 MS ; cosv = (5.) MOMENT OF BENDING STRESS. 75 The following equation is easily verified : cos a cos A + cos /3 cos p, + cos y cos v (^ A )- This indicates what is of itself obvious ; that the axis of the resul- tant couple M is perpendicular to the direction of the stress. The following form is often the most convenient for the constant a. Let p l be the intensity of the stress at some fixed distance, x^ from the neutral axis; then (6.) #1 92. Moment of Bending stress. If the uniformly varying stress be normal to the surface at which it acts ; that is to say in symbols, if cos a = 0; cos /3 =r ; cosy 1; (1.) then it is evident that (2.) or in words, that the axis of the resultant couple is in the plane of the surface A A A. Such a stress as this is called a bending stress, for reasons which will be explained in treating of the strength of materials. The equations of Article 91, when applied to this case, become as follows : cos A = sin (* =. K ..(3.) If the figure AAA is symmetrical on either side of the axis OX, then for every point at which y has a given positive value, there is a corresponding point for which y has a negative value of equal amount ; so that for such a figure = / / =//*' and the same equation may be fulfilled also for certain unsymme- trical figures. In this case we have so that the axis of the couple coincides with the neutral axis. 76 PRINCIPLES OF STATICS. 93. Moment of Twisting stress. If the stress be tangential, its tendency is obviously to twist the surface AAA about the axis O Z. In this case we have cos y = ; cos a. = sin ft ; cos ft = sin ; ] M i== 0;M 2 ^0; M = M 3 = a (I sin *-K cos ) ; cos A = ; cos ^ = ; cos v = 1. J In the cases referred to in Article 92, for which K = 0, we find M = alsin ; ........................ (2.) so that in these cases it is only the component of the stress parallel to the neutral axis which produces the twisting couple. 94. Centre of Uniformly Varying Stress. When the amount of an uniformly varying stress has magnitude, that stress may be con- sidered as made up of two parts, viz. : First, an uniform stress, whose intensity is the mean intensity of the entire stress, and whose centre is the centre of gravity, O, of the surface of action. As in Article 88, equation 5, this mean intensity may be represented by P total stress Secondly, an uniformly-varying stress, whose neutral axis tra- verses O, whose amount is = 0, and whose intensity, p' 9 at a given point, is the deviation of the intensity at that point from the mean ; so that the intensity of the entire stress is given by the equation (2.) Let M be the moment of this second part of the stress; its effect, as has been already shown in Article 60, case 2, is to shift the resultant P parallel to itself through a distance to the opposite side to that whose name designates the tendency of the couple Mj and the direction of the line L is perpendicular at once to that of the stress, and to that of the axis of the couple M. The co-ordinates relatively to the point O of the centre of stress as thus shifted, being the point where the line of action of the shifted resultant cuts the plane of AAA, are most easily found by adapting the equation 2 of Art. 89 to the present case, as follows: MOMENTS OF INERTIA OF A SURFACE. 77 perpendicular) {{xri'dxdy a((y?'dxdy al to the \ XQ L2 __ UL^ _ = . neutral axis j along the ) ^yp'-dxdy a j jxydxdy aK neutral axis j 2/o = p p : p The angle 6 which the line joining O and the centre of stress makes with the neutral axis OY, is that whose cotangent is cotan 6 = = XQ I (5.) This line will be called the axis conjugate to the neutral axis -YOY. When K = 0, it is perpendicular to the neutral axis. 95. moments of Inertia of a Surface. The integral I = I f x 2 dxdy is sometimes called the moment of inertia of the surface AAA relatively to the neutral axis YOY. This is a term adopted from the science of Dynamics for reasons which will after- wards appear. The present Article is intended to point out certain relations which exist amongst the moments of inertia of a plane surface of a given figure relatively to different neutral axes ; a knowledge of which relations is useful in the determination of the moment of a bending or twisting stress. Let A A in fig. 44 represent a plane surface of any figure, O its centre of gravity, YOY, X O X, a pair of rectangular axes crossing each other at O, in any position. Taking YOY as a neutral axis, let the moment of inertia relatively to it be I =1 $ J-dxdy; let the moment of inertia re- latively to XOX as a neutral axis be J = I j if dxdy; and let K = / / xy d xdy. Now let Y'OY', X'OX', be a new pair of rectangular axes, in any position making the angle 78 PRINCIPLES OF STATICS. with the original pair of axes ; and let I' = J fx' 2 'dx'dy'>, J' = f ftf'-dafdtf; K' = The following relations exist between the original co-ordinates, x, y, of a given point, and the new co-ordinates of, y', of the same point ; x' = x cos /3 y sin /3; \ 2/ = x sin /3 + y cos /3; > .................. (3.) (This last quantity, which is the square of the distance of the given point from 0, is what is called an Isotropic Function of the co-ordinates ; being of equal magnitude in whatsoever position the rectangular co-ordinates are placed.) From the equations (3), the following relations are easily deduced between the original integrals I, J, K, and the new integrals cos 2 ft + J sin 2 /3 2 K cos /3 sin p-,\ J' = I sin 2 /3 + J cos 2 /3 + 2 K cos /3 sin /8; V ...(4.) K' = (I J) cos /3 sin /3 + K (cos /3 sin 2 /3.) j Also, the following functions of those integrals are found to be isotropic; I + J - T - J' = (x 2 + y^'dxdy ...... (5.) (called the polar moment of inertia); IJ K 2 = I' J' -- K /2 ................... (6.) Equation 5 may be thus expressed in words : THEOREM I. The sum of the moments of inertia of a surface relatively to a pair of rectangular neutral axes is isotropic. Equations 5 and 6 in conjunction lead to the following conse- quences. Because the sum I' + J' is constant, I' must be a maximum and 3' a minimum for that position of the rectangular axes which makes the difference I' J' a maximum. And because (I' _ J') 2 = (!' + J') 2 4 F J 7 , I' J' must be a maximum for that position of the axis which makes I' J' a minimum. But by equation 6, 1' J' K' 2 is constant CONJUGATE AXES. 79 for all positions of the axes; therefore when K' 0, I'J' is a minimum, I' J' a maximum, I' a maximum, and J' a minimum. Hence follows, in the first place, THEOREM II. In every plane surface there is a pair of rect- angular neutral axes for one of which the moment of inertia is greater, and for the oilier less, than for any other neutral axis. These axes are called Principal Axes. Let I 15 J 15 be the maximum and minimum moments of inertia relatively to them, and let ft 1 be the angle which their position makes with the originally-assumed axes ; then because Kj = 0, we have, from the third of the equa- tions (4) 2 cos ft sin ft 2 K tan 2 ft = - r-j- = T T" ff.) cos 2 ft-- sin 2 ft I J and because I, + J x = I + J, and ^ Jj = IJ K 2 , we have, by the solution of a quadratic equation, Il = li. V {!l=il!.K.j I ,._* y{tj2<,} The position of the principal axes, and the values of I w J,, being once known, the integrals I', J', K', for any pair of axes which make the angle ft' with the principal axes, are given by the equations I' = T! cos 2 /3' + J, sin 8 /3'; ) J' = Ij sin 2 /3' + J, cos 2 /3'; V (9.) K' = (I, J,) cos ft' sin ft'. } If I : = J 1? then I' = J' = Ij, and K' = 0, for all axes whatso- ever ; and the given figure may be said to have its moment of inertia completely isotropic. Next, as to Conjugate Axes. By equation 5, Article 94, we have for the angle which the axis conjugate to O Y makes with OY TT cotan 6 = . For the principal axes, K = 0, cotan & 0, and 6 is a right angle ; from which follows THEOREM III. The principal axes are conjugate to each other : that is, if either of them be taken for neutral axis, the other will be the conjugate axis. Returning to equation 4 of the present Article, let us suppose, that the axis conjugate to the originally assumed neutral axis YO Y, has been determined, and that its position is Y' O Y', so that 80 PRINCIPLES OF STATICS. Let this conjugate axis be assumed as a new neutral axis. Then the integrals I', J', K', belonging to it are determined by substituting Q for /3 in the equation 4 ; that is, by substituting for cos ft and sin /3, the values of cos & and sin in terms of K and I, viz. : COS 8 = K P + K 2 sin Q = which substitution having been made, we find T< _-* I 8 + K 2 .(10.) F + K J Now let it be required to find the angle 0, which the new con- jugate axis makes with the new neutral axis Y'OY'. This angle is given by the equation 1C K cotan ff = -r -- ~ = cotan whence or in words, THEOREM IV. If the axis conjugate to a given neutral axis be taken as a new neutral axis, the original neutral axis will be the new conjugate axis. The following mode of graphically representing the preceding theorems and relations depends on well known properties of the ellipse. x> In fig. 45, let O Xx O YI perpendicular to each other, represent the principal axes of a surface. With the semi-axes, Fig. 45. = 0^ = describe an ellipse, so that the square of each semi-axis shall represent the moment of inertia round the other. Let the semidiameter OY' be drawn in the direction of any assumed neutral axis, and let ^ YiO Y x - '. Draw O C, the MOMENTS OF INERTIA OF A SURFACE, 81 semidiameter conjugate to OY', so that the tangent CT shall be parallel to Y'. Let CT = t, and let the normal OT = n. Then it is well known that n 2 = a*' cos 2 /3' + 6 2 sin 2 /3'; ) and that V ............ (13.) n t = (a 2 b*) ' cos ff sin /3'; J consequently, comparing this equation with the equation 9, we find, r = n'; 1 = ?'' I ...... (14.) cotan 6 = = - == cotan Y' O C; I n } so that the square of the normal O T represents the moment of inertia for the neutral axis O Y', and the semidiameter O C con- jugate to OY' is also the conjugate axis of the neutral axis OY', and vice versd. In finding the moment of inertia of a surface of complex figure, it may sometimes be desirable to divide it into parts, each of more simple figure, find the moment of inertia of each, and add the results together. In a case of this kind, the neutral axis of the whole surface will not necessarily traverse the centre of gravity of each of its parts, and it becomes necessary to use formulae for finding the moment of inertia of a figure relatively to an axis not traversing its centre of gravity. Let O Y denote such an axis, x the distance of any point of the given figure from it, and x the distance of the centre of gravity of the given figure from the axis Y. Through that centre of gravity conceive an axis O' Y' to be drawn parallel to O Y ; the point which is at the distance x from O Y, is at the distance X f = X - XQ fromO'Y'. The required moment of inertia is I = I I xdxdy; but therefore x'-dxdy+ and because O' Y' traverses the centre of gravity of S, / / x ' dxdy = ; 82 PRINCIPLES OF STATICS. so that the middle term of the expression for I vanishes, leaving (15.) or in words, THEOREM Y. The moment of inertia of a surface relatively to an axis not traversing its centre of gravity is greater than the moment of inertia round a parallel axis traversing its centre of gravity, by the product oftlie area of the surface into the square of the distance between those two axes. The following is a table of the principal (or maxima and minima) moments of inertia of surfaces-of-action of stress of those figures which most commonly occur in practice : !?;_,,.,, Maximum Ij Minimum 3 1 (neutral axis Y). (neutral axis X). I. RECTANGLE. Length along O X, ) h s b hb 3 h' } breadth along O Y, b .......... } "i2~ T2 II. SQUARE. Side = h ................. 12 12 III. ELLIPSE. Longer axis, h ...... ) ^h s b vkb s Shorter axis, b ...... / ~64~ . ~64~ IY. CIRCLE. Diameter, h 64 64 Y. Hollow symmetrical figures; sub- tract I or J for inner figure, from I or J for outer figure. YI. Symmetrical assemblage of rec- 1 ^s j j b fis tangles; dimensions of any one 1 2 h || x, b || y, distance of its centre j from O Y, x ; from O X, y J + SECTION 3. Of Internal Stress, its Composition and Resolution. 96. Internal Stress in General. If a body be conceived to be divided into two parts by an ideal plane traversing it in any direction, the force exerted between those two parts at the plane of division is an internal stress. The finding of the resultant, and of the centre of stress, for an internal stress, depend upon the principles relating to stress in general, which have been explained in the last section. The present section refers to a different class of problems, viz., the relations between the different stresses which can exist together in one body at one point. SIMPLE STRESS. 83 A body may be divided into two parts by a plane traversing a given point, in an indefinite number of ways, by varying the angular position of the plane ; and the stress which acts between the two parts may vary in direction, or intensity, or in both, as the position of the plane varies. The object of the present section is to show the laws of such variation ; and also the effect of applying different stresses simultaneously to one body. The investigations in this section relate strictly to stress of uniform intensity ; but their results are made applicable to stress of variable intensity to any required degree of accuracy, by sufficiently contracting the space under consideration, so that the variations of the stress within its limits shall not exceed the assigned limits of deviation from uniformity. 97. Simple Stress and its Normal Intensity. A simple stress IS a pull or a thrust. In the following investigations a pull will be treated as positive, and a thrust as negative. In fig. 47, let a prismatic solid body, or part of a solid body, whose sides are parallel to the axis O X, be kept in equilibrio by a pull applied in opposite directions to its two ends, of uniform intensity, and of the amount P. Let an ideal plane A A, perpendicular to O X, be conceived to divide the body into two parts, and let the area of that plane of section be S. That each of these parts may be in equilibrio, it is necessary that they should act upon each other, at the plane of section A A, with a pull in the direction Fig. 46. O X, of the amount P, and of the intensity This, which is the intensity of the stress as distributed over a plane normal to its direction, may be called its normal intensity. 98. Reduction of Simple Stress to an Oblique Plane. Next, let the plane of section be conceived to have the position B B, oblique to O X ; let N be a line normal to B B, and O T a line at the intersection of the planes B B and X O N. Let the obliquity of the plane of section be denoted by The two parts into which B B divides the body must exert on each other, as in the former case, a pull of the amount P, and in the direction O X ; but the area over which that pull is distributed is now 84 PRINCIPLES OF STATICS. areaBB = - - COS0 consequently, the intensity of the stress, as reduced to the oblique plane of section, is Pcostf Pr a =Px ' COS 6, to 99. Resolution of Oblique Stress into Normal and Tangential Components. The oblique stress P on the plane of section B B may be resolved by the principles of Articles 55, 57, into two compo- nents, viz. : Normal component a- ) longON,!. } Pcos ' ; Tangential component ) -D /j along OT, / and the intensities of these components are, Normal ; p n =p r cos 6 = p x cos 2 6 j Tangential ; p t = p r sin 6=p x - cos 6 sin Suppose another oblique plane of section to cut the body at right angles to B B, so that its obliquity is and let the intensity of the stress on the new plane be denoted by accented letters ; then p' n =p x ' cos 2 & = p,. ' sin 2 6 so that we obtain the following THEOREM. On a pair of planes of section whose obliquities are together equal to a right angle, the tangential components of a simple stress are of equal intensity, and the intensities of the normal com- ponents are together equal to the normal intensity of the stress. 100. Compound Stress is that internal condition of a body which is made by the combined action of two or more simple stresses in different directions. A compound stress is known when the direc- tions and the intensities, relatively to given planes, of the simple stresses composing it are known. The same compound stress may be analyzed (as the ensuing Articles will show) into groups of simple stresses, in different ways ; such groups of simple stresses are said to be equivalent to each other. The problems of finding of a group of stresses equivalent to another, and of determining the relations which must exist between co-existing stresses, are solved by con- sidering the conditions of equilibrium of some internal part of the solid, of prismatic or pyramidal figure, bounded by ideal planes. THREE CONJUGATE STRESSES. 85 101. Pair of Conjugate Stresses. THEOREM. If the stress OH a given plane in a body be in a given direction, the stress on any plane parallel to that direction must be in a direction parallel to the first-mentioned plane. In fig. 47, let YOY represent, in section, a given plane tra- versing a body, and let the stress on that plane be in the direction X O X. Consider the condition of a prismatic portion of the body represented in sec- tion by A B C D, bounded by a pair of planes A B, D C, parallel to the given plane, and a pair of planes A D, B C, parallel to each other and to the given direction X O X, and having for its axis a line in the plane YOY, cutting ' Fig. 47. XOX in O. The equal resultant forces exerted by the other parts of the body on the faces AB and D C of this prism are directly opposed, their common line of action traversing the axis O ; and they are there- fore independently balanced. Therefore the forces exerted by the other parts of the body on the faces A D and B C of the prism must be independently balanced, and have their resultants directly opposed ; which cannot be unless their direction is parallel to the plane YOY. Therefore, &c. Q. E. D. A pair of stresses, each acting on a plane parallel to the direction of the other, are said to be conjugate. In a rigid body, it is evident that their intensities are independent of each other, and that they may be of the same, or of opposite kinds : a pair of pulls, a pair of thrusts, or a pull and a thrust. In those cases (of frequent occurrence in practice) in which the planes of action of a pair of conjugate stresses are both perpendi- cular to the plane which contains their two directions, their obli- quity is the same, being the complement of the angle which they make with each other. 102. Three Conjugate Stresses may act together in one body, the direction of each being parallel to the line of intersection of the planes of action of the other two ; and in a rigid body, the kinds and intensities of those stresses are independent of each other. Thus, in fig. 47, if X O X and YOY represent the directions of two stresses, each acting on a plane which traverses the direction of the other, the intersection of those planes (which may make any angle with XOX and Y O Y), will give a third direction, being that of a third stress of either kind and of any intensity, which may act on the plane X O Y, and will be conjugate to each of the other two. 86 PRINCIPLES OF STATICS. Three is the greatest number of a group of conjugate stresses ; for it is evidently impossible to introduce a fourth stress which shall be conjugate at once to each of the other three. The relations between the three angles which the directions of three conjugate stresses make with each other, the three obliquities of those stresses (being the angles which they make with the per- pendiculars to their respective planes of action), and the three angles which those perpendiculars make with each other, as found by the ordinary rules of spherical trigonometry, are given by the following formulae. GENERAL CASE. Let x, y, z, denote the directions of the three conjugate stresses: A A A y z, zxj xy, their inclinations to each other; u, v, w, the directions of the perpendiculars to their planes of action, so that u -L plane y z, v -L plane z x, w -L plane x y ; A A A vw f w u } uv, the inclinations of those perpendiculars to each other ; u x, vy, wz, the respective obliquities of the stresses. Then those nine angles are related as follows : , A -A A A A A Let 1 cos 2 y z cos 2 z x cos 2 x y + 2 cos y z cos z x cos x y = 0; (1.) Then A A A A / C A cos z x cos x y cos y z sin v w = ^ ^-; cos v w = ^ " ^ ; sin z x ' sin x y sin z x sin x y A A A .A JQ ^ A cos x y ' cos y z cos z x ^ A . A } A A > sin x y ' sin y z sin x y sin y z A A A A J C f A cos y z ' cos z x cosa?2/ sin y z ' sin z x sin y z sin z x -(2.) A JC A v cos v y = A j cos w z = ^- (3.) sin y z sin z x sin xy RESTRICTED CASE I. Suppose two of the stresses, for example, those parallel to x and y, to be perpendicular to each other, and oblique to the third. Then A A cosxy= 0; sin xy = 1 ; ) / 4 v C = 1 cos 2 yz cos 2 z x' } ) " PLANES OF EQUAL SHEAR. 87 A JO . /\ A cos y z A sin wu =. A v/C Sill Id V A > sin zx JO . sin ?/2 sin 2ic A A COS 2 X sin 2/2 A A ^ A^ COS 2/ 2 ' COS 2 iC sin ?/2 sin , A , cos saj sin y z sin 2 # (5.) cos u x = * A ; cos v ? sin y z = * A ; cos w z = N / C . . . (6.) sin 2 re RESTRICTED CASE II. Suppose one of the stresses (such as 2) to be perpendicular to the other two, which are oblique to each other. Then A A 1 cos y z = ; cos 2 # = j .A A sin 2/2^1; sm z x = 1 ', C = sin 2 re 2/ sin v w = 1 ; cos v w = ; (or v w = 90) ; A A A sm w u = 1 ; cos w u = ; (or w u = 90 ' ) : A A A sin wv = sin x y; cos u v = cos a? 2/; (or, w V + xy = 180). A A A A A cos u x sin cc 2/; cos v y sm x y ; cos w 2 = A A A A or^ic = vy = 90 xy\ wz 0; .(7.) (8.) ....(9.) results identical with those given at the end of Article 101. RESTRICTED CASE III. All three stresses perpendicular to each other. In this case the normals to the three planes of action are perpendicular to each other, and coincide with the directions of the stresses. 103. Planes of Equal Shear, or Tangential Stress. THEOREM. If tJie stresses on a given pair of planes be tangential to those planes, and parallel to a third plane which is perpendicular to the pair of planes, those stresses must be of equal intensity. Let the third plane be represented by the plane of the paper in fig. 48, and let the pair of planes on which the stresses are tangen- 88 PRINCIPLES OF STATICS. tial, and parallel to the plane of the paper, be parallel respectively to AB and AD. Consider the condition of a right prism of any length, represented in section by A BCD, and bounded by a pair of parallel planes, AB, CD, and a pair of parallel planes, AD, CB. Let p t denote the intensity of the shear or tangential stress on AB, CD, and planes parallel to them, and p' t the intensity of the shear, or tangential ^. stress on AD, CB, and planes parallel to them. The forces exerted by the other parts of the body on the pair of faces AB, CD, form a couple (right-handed in the figure), of which the arm is the perpendicular distance EF, between AB and CD, and the moment, 2VareaAB-EF. The forces exerted by the other parts of the body on the pair of faces AD, CB, form a couple (left-handed in the drawing), of which the arm is the perpendicular distance GH between AD and CB, and the moment pv area AD GH. The equilibrium of the prism requires that these opposite moments shall be equal. But the products, area AB EF, and area AD GH are equal, each of them being the volume of the prism; there- fore the intensities of the tangential stresses Pt=P't are equal. Q. E. D. The above demonstration shows that a shear upon a given plane cannot exist alone as a solitary or simple stress, but must be com- bined with a shear of equal intensity on a different plane. The tendency of the action of the pair of shearing stresses represented in the figure on the prism A B C D is obviously to distort it, by lengthening the diagonal DB, and shortening the diagonal AC, so as to sharpen the angles D and B, and flatten the angles A and C. 104. Stress on Three Rectangular Planes. THEOREM. If there be oblique stress on three planes at right angles to each other, the tangential components of the stress on any two of those planes in directions parallel to the third plane must be of equal intensity. Let yz,zx,xy, denote the three rectangular planes whose intersec- tions are the rectangular axes of x, y, and z. Consider the condition of a rectangular portion of the body, having its three pairs of faces parallel respectively to the three planes, and its centre at the point of intersection of the three axes. Let ABCD (fig. 49), represent the section of that rectangular solid by the plane of xy, the faces STRESS ON THREE RECTANGULAR PLANES. 89 AB, CD being parallel to the plane yz, and the faces AD, C B, to the plane * x. Let the equal and parallel lines XR represent the intensities of the forces exerted by the other parts of the body on the pair of faces AB, CD; resolve each of these forces into a component XN, parallel to the plane z x, and a tangential component, XT, parallel to the axis of y; the resultants of the components X N will act through the axis of z, and will produce no couple round that axis; the com- ponents XT will form a couple acting round that axis. In the same manner the intensities of the forces exerted on the faces AD, CB, being re- presented by the equal and parallel lines, Yr, are resolved into the components, Y%, whose resul- tants act through the axis of , and the compo- nents Y, which form a couple acting round that axis, which, by the conditions of equilibrium of the rectangular solid ABCD, mast be equal and opposite to the former couple; and by reasoning similar to that of Article 104, it is shown that the intensities of the tangential stresses constituting these couples, must be equal ; and similar demonstrations apply to the other planes and stresses. To represent this symbolically: let p, as before, denote the intensity of a stress ; and let small letters affixed below p be used, the first small letter to denote the direction perpendicular to the plane on which the stress acts, and the second to denote the direc- tion of the stress itself: for example, let p yg denote the intensity of the stress on the plane normal to y (that is, the plane zx), in the direction of z. Then resolving the stress on each of the three rectangular planes into three rectangular components, we have the following notation : PLANE. DIRECTION. x y z y P p*v P I zx p yx p yy p ys \ intensities. Then, in virtue of the Theorems of Articles 101 and 102, we have the normal stresses, p xj: , p yy , p zz , conjugate and independent ; and 90 PRINCIPLES OF STATICS. in virtue of the theorem of this Article, there are three pairs of tangential stresses of equal intensity, [The reader who wishes to confine his attention to the more simple class of problems may pass at once to Article 108, page 95.] 1 05. Tetraedron of stress. PROBLEM I. The intensities of three conjugate stresses on three planes traversing a body being given, it is required to find the direction and intensity of the stress on a fourth plane, traversing the same body in any direction. In fig. 50, let Y O Z, Z X, X O Y, be the three planes, on which act conjugate stresses in the directions OX, O Y, O Z, of the intensities p s , p y , p g . Draw a plane parallel to the fourth plane, cutting the three conjugate planes in the triangle ABC, so as to form with them the tri- angular pyramid or tetraedron O A B C. Then must the stresses on the four triangular faces of Fig. 50. *kat tetraedron balance each other ; and the total stress on A B C will be equal and opposite to the resultant of the total stresses on O B C, O C A, and O A B. On O X, O Y, O Z, respectively take Ol) = total stress on O B C = p x area B C, OE = total stress on C A = p, area OCA, O F = total stress on A B = p* area O A B. Complete the parallelepiped O D E F R ; then will its diagonal OR represent the direction and amount of the total stress on an area of the fourth plane equal to that of A B C ; and the intensity of that stress will be QR O. E. I. area ABC Hence it appears, that if the stresses on three conjugate planes in a body be given, the stress on any other plane may be deter- mined; from which it follows, That every possible system of stresses which can co-exist in a body, is capable of being resolved into, or ex- pressed by means of, a system of three conjugate stresses. PROBLEM II. The directions and intensities of the stresses on three rectangular co-ordinate planes being given, it is required to find the direction and intensity of the stress on a fourth plane in any posi- tion. Let the planes Y Z, Z O X, X O Y, in fig. 50, represent the rectangular co-ordinate planes, so that OX, Y, O Z, are now at right angles to each other (instead of being, as in Problem I., in TETRAEDRON OF STRESS. 91 any directions). Reduce the three given stresses, as in Article 104, to rectangular components, with the notation already explained. Let A B C, as in Problem I., be a triangle parallel to the fourth plane, enclosing, with three triangles in the co-ordinate planes, the tetraedron O A B C. The total stress on A B C will be equal and opposite to the resultant of all the rectangular components of the total stresses on O B C, O C A, and O A B. Therefore, on X, O Y, O Z, respectively, take O D p xa ' area O B C + p xy area OCA + p tf m area A B, O E = p xy ' area O B C + p yy area OCA + p yz area O A B, + p ny -co$yy' + p nt - cos z y 1 ; A A, A, p M r =p nx cos xzf + p ny ' cos y z 4- p nz cos z z. For n are now to be substituted successively, both in p^ x , &c., and in the values of p M , &c., according to equation 1 of Article 105, the symbols x', y', z' ; and thus are obtained finally the following equations of transformation : NORMAL STRESSES. jp,V =p fa . cos 2 xx' -\r p yy cos 2 y x' -f- p u cos 2 z x' A A A A A A + 2 p yz cos y x 1 cos z x + 2 p zx cos z x' cos x x' -}- 2 p xy cos x x' cos y x' \ A A A Pyy = P** cos x tf + Pyy co ^ V !/ + P** co ^ z V A A A A A A + 2 p yz cos y if cos z y' -f- 2 p, x cos z y' cos x y + 2 p xy cos x y' cos y y' ; A A A P*'*' = P,, cos 2 xz'+ pyy cos 2 y z 1 +p u cos 2 z z' A A A A, A A -f- 2p yz cos y z cos zz -}-2p za! cos z z cos x z + %p xy cos x z' cos y z 1 ; PRINCIPAL AXES OF STRESS. 93 TANGENTIAL STRESSES. A A A A A A p 9 r g ' = p xai - cos x y' cos x z' -\- p m cos y y - cos y z -\-p zz cos z y cos z z . AA AA AA A . A + p yz (cos 2 2/ cos 2/2' + cos yy'coszz') +p zx (cosxy'coszz + coszy'cosxz) A A A A + pgy (cos y y' cos x z 1 + cos x y' cos y z') ; A A A A A A, pi.' =PS X cos x z cos x x 1 -\-pyy cos y z' cos y x' + p z , cos z z' cos # A A A A A, A, A A -f pj, z (coscos2/a;' + cos/2'cosa/) + p^cosxz'coszx + coszz'cosxx 1 ) A A A A -f- p xy (cos ?/ ^ cos x x' + cos ic z* cos ?/ &') ; A A A A A A j^' f = jt^ cos cc a/ cos xy* + pyy cos ?/ ic' cos y y + p zz cos x' cos * y' A A , A A A A A, A + p yg (cos z x" cos y y -{- cos y x' cos ?/') + jo^ (cos cccc' cos zy -t-coszx cos a; y') A A A A + P^y ( cos 2/ ^' cos x y + cos ^ ^ cos 2/ 2/')- The two systems of component stresses, p xx , &c., relative to the axes 05, ?/, , and ^,V, &c., relative to the axes a/, y , ^', which con- stitute the sarae compound stress, are said to be equivalent to each other. 107. Principal Axes of Stress. THEOREM. For every state of stress in a body, there is a system of three planes perpendicular to each ot/ier, on each of which the stress is ivholly normal. Referring to the equation 3 of Article 105, it is evident that the condition, that the direction of stress on a plane shall coincide with the normal to that plane, is expressed by the equations A P nx A A P*y A cos x r = cos x n : cos y r = = cos y n : Pr Pr A P* A /i ^ cos z r = = coszn (1.) Pr Introducing these values into the equation 1 of Article 105, we obtain the following : A A A (Pxx p r ) cos x n + p xy cos y n + p zx cos z n = ; A A A p xy cos x n + (p yy Which equations being squared and added, and the square root of the sum extracted, give the following value for the reciprocal of the intensity required : 1 ( A A A ) _ /Jcos #j0 cos y p coszp\ (10") P ( ~?T * j' the well known equation of an ellipsoid, in which p l} p 2 , p s , denote the three semi-axes, and p the semidiameter in any given direction. The cosine of the obliquity of the stress p is given by the equation A A A A A A A cos n p = cos x n cos x p -f- cos y n cos yp -f~ cos z n cos zp ( 2 A 2 A A ) __ p ) cos j x p. cos^ y p , cos* z p I { PI ~~:P*~ PS ) 1 A A _(^jcos 2 # n-\-p 2 cos 2 yn+ /> 3 cos 2 zn) ; (H-) P and this cosine, by being positive } indicates ( a pull \ nothing > that the < a shear > negative J stress p is ( a thrust j 108. Stress Parallel to One Plane. In most practical questions respecting the stress in structures, the directions of the stresses chiefly to be considered are parallel to one plane, to which their planes of action are perpendicular, the remaining stress, if any, being a principal stress, and perpendicular to the plane to which the others are parallel. The problems concerning the relations amongst stresses parallel to one plane, might be solved by considering them as particular cases of the more general problems respecting stresses in any direc- 96 PRINCIPLES OF STATICS. tion, which have been treated of in Articles 105, 106, and 107 ; but the complexity of the investigations and results in those Articles, makes it preferable to demonstrate the principles relating to stresses parallel to one plane, independently. PROBLEM I. The intensities and directions of a pair of conjugate stresses, parallel to a plane which is perpendicular to tlieir planes of action, being given, it is required to find the direction and intensity of the stress on a fourth plane, perpendicular also to the first mentioned plane. In fig. 51, let the plane of the paper represent the plane to which the stresses are parallel ; let X and O Y represent the directions of the pair of conjugate stresses, whose intensities are p x and p y ; and let AB be the plane, the stress on which is sought. Consider the condition of a prism, O A B, bounded by the plane A B, and by planes parallel 51 - to O X and O Y respectively. The force exerted by the other parts of the body on the face O A of the prism, will be proportional to on O Y take OE to represent that force. The force exerted by the other parts of the body on the face O B of the prism, will be pro- portional to jtvOB; on O X take O D to represent this force. The force exerted by the other parts of the body on the face A B of the prism, must balance the forces exerted on O A and A B ; therefore complete the paral- lelogram O D R E ; its diagonal O R will represent the direction and amount of the stress on A B, and the intensity of that stress will be OE, pi - Q^ a + pi - O~A 2 + 2p x p y QHB (JA cos^XO Y ) OB 2 +OA 2 -2OB-0~Acos^XOY. J The parallelogram marked in the figure with the capital letters R, E, corresponds to the case in which p x and p y are of the same kind, both pulls, or both thrusts, in which case p r is of the same kind also ; the parallelogram marked with the small letters, r, e, corresponds to the case in which p x and p y are of opposite kinds, one being a pull and the other a thrust ; in which case p r agrees in kind PRISM OF STRESS. 97 with that one of the given conjugate stresses whose direction falls to the same side of A B with it. When O r is parallel to A B, p r is a shear, or tangential stress. PROBLEM II. The intensities and directions of the stresses on a pair of planes perpendicular to each other and to a plane to which the stresses are parallel, being given, it is required to jind the inte r iisity and direction of the stress on a plane in any position perpendicular to that plane to which the stresses are parallel. In fig. 52, let the plane of the paper represent the plane to which the stresses are parallel, and OX, O Y, the pair of rectangular planes on which the stresses are given. Let those stresses be resolved, as in Article 99, into rectangular normal and tangential components. Let p xx de- note the intensity of the normal stress on the plane O Y, which stress is parallel to X ; let p yy denote the intensity of the normal stress on ~ the plane O X, which stress is parallel to O Y. In virtue of the Theorem of Article 103, the Fig. 52. me (y\ to \x] tangential stresses on those two planes must be of equal intensity; and they may therefore be denoted by one symbol, p xy , which sym- bol may be read as meaning the intensity of ( x ) on a plane the stress along \y] normal Let O N be a line normal to the plane the stress on which is sought, making with O X the angle X O N = x n. Consider the condition of a prism O A B, of the length unity, bounded by the planes O A J_y, O B J_ x, A B _L O N. The areas of the faces of that prism have the following proportions : A A OB = AB cos xn; OA = AB sin xn. The forces exerted on the faces O A and B, in a direction parallel to x, consist of the normal stress on O B, and the tangential stress on A ; that is to say, p xx O B + p xy - O A = A B < pxx ' cos x n + p xy ' sin x n > Let this be represented by O D. The forces exerted on the faces O A and O B, in a direction paral- lel to y, consist of the normal stress on OA, and the tangential stress on O B ; that is to say, p xy O B -f pyy O A = A B < pxy cos x n + p yy - sin x n > Let this be represented by O E. 98 PRINCIPLES OF STATICS. Complete the rectangle D R, E ; the amount and direction of the stress on A B will be represented by its diagonal, and the intensity of that stress by OR- ( A A Pr = j= = */ | Pxx* ' cos 2 x n +p 2 sin 2 x n 4~ %pxy (Pxx ~\~ Pyy) COS x ^ ' sin X n > ................. (1.) From K draw E, P perpendicular to the normal O N ; then the normal and the tangential components of the total stress on A B will be represented respectively by OP = OD cos xn -f- OE sin xn; _ _ A - A PR = OE cos xn - ODsimew; and the intensities of these components by OP A A A A x p n = == = p xx cos 2 xn -rp yy ' sm 2 xn -f 2p^y cos xn sin xn- } -Ci- .D _ [(2.) PR A A A A r ' p t = =r = (pyy-p xx ) cos xn ' sin xn + pxy (cos 2 xn - sin 2 xn). ) A The obliquity, ^ N O B, = n r, of the stress on A B is given by the equation 109. Principal Axes of Stress Parallel to One Plane. THEOREM. For every condition of stress parallel to one plane, there are two planes perpendicular to each other, on which there is no tangential stress. As in Article 108, let the three rectangular components, p xx , p yy , p xy , of the stress on two rectangular planes, O Y, OX, be given. The condition, that there shall be no tangential stress on a plane normal to O N, is expressed by making p t in the second of the equations 2 of that Article ; and in order that this may be fulfilled, we must have A A cos x n ' sin x n p^y cos 2 xn sin 2 x n ^ xx . *W or, what is the same thing, tan 2 xn = 2pxy ; ........................ (1.) FLUID PRESSURE. 99 Now for two values of x n, differing by a right angle, the values of tan 2 xn are equal ; hence there are two directions of the normal O N perpendicular to each other, which fulfil the condition of having no tangential stress. Those two directions are called principal axes of stress, and the stresses along them (which are conjugate to each other) principal There may be a third principal stress, conjugate and at right angles to the first two ; but as, with one exception, the ensuing in- vestigations of this section relate to stresses upon planes parallel to the direction of this third principal stress, which does not affect such planes, it may be left out of consideration. The most simple mode of expressing the relations amongst inter- nal stresses parallel to a plane is obtained by taking the two prin- cipal axes of stress in that plane for axes of co-ordinates ; and this is done in the ensuing Articles. 110. Equal Principal Stresses Fluid Pressure. THEOREM I. If a pair of principal stresses be of the same kind and of equal intensity, every stress parallel to the same plane is of the same kind, of equal in- tensity, and normal to its plane of action. In fig. 53, let OX, OY, be the direc- x tions of the given principal stresses, and PX) p y) their intensities. By the condi- tions of the question, those intensities are equal, or Let it be required to find the direction and intensity of the stress on any plane A B. As in Article 108, consider the condition of the triangular prism O A B ; and let the length of that prism, in a direction perpendicular to the plane X O Y be unity. Then the total stresses Fig. 53, on the faces OB and OA will be respectively p x O~B and p y - OA. On O X and Y respectively, take d~I) to represent p x O~B, and O E to represent p y - O~A; complete the rectangle O D R E; then its diagonal O R will represent the amount and direction of the stress on the face AB of the prism, and the intensity of that stress will be OR 100 PRINCIPLES OF STATICS. Now, because p x = p y , we have OP (JE OR OB OA AB' and consequently Pr=P*=p y ', and because of the similarity of the triangles A O B, O E R, O~Il is perpendicular to A B. Therefore, the stress on each plane per- pendicular to X O Y is normal, and of equal intensity in all direc- tions. Q. E. P. In this case it is obvious, that eveiy direction in the plane X O Y has the properties of an axis of stress. COROLLARY. If the stress in all directions parallel to a given plane be normal, it must be of equal intensity in all those directions. THEOREM II. In a perfect fluid, the pressure at a given point is normal and of equal intensity in all directions. Fluid is a term opposed to solid, and comprehending the liquid and gaseous conditions of bodies, which have been defined in Article 4. The property common to the liquid and the gaseous conditions is that of not tending to preserve a definite shape, and the possession of this property by a body in perfection throughout all its parts, con- stitutes that body a perfect fluid. The parts of a body resisting alteration of shape must exert tangential stress ; a perfect fluid does not resist alteration of shape ; therefore the parts of a perfect fluid cannot exert tangential stress ; therefore the stress exerted amongst and by them at every point and in eveiy direction is normal ; there- fore at a given point, it is of equal intensity in every direction. Q. E. D. This theorem, and its consequences, form the branch of statics called Hydrostatics, which is sometimes treated of separately, but which, in this treatise, it has been considered more convenient to include in the subject of the statics of distributed forces in general. Gaseous fluids always tend to expand, so that the stress in them is always a pressure. Liquid fluids are capable of exerting to a slight extent tension, or resistance to dilatation, as well as pressure ; but in all cases of practical importance in applied mechanics, the only kind of stress in liquids which is of sufficient magnitude to be considered, is pressure. The term, fluid pressure is used to denote a thrust which is normal and equally intense in all directions round a point. The idea of perfect fluidity is not absolutely realized by actual liquids, they having all more or less a tendency in their parts to resist distortion, which is called viscosity, and which constitutes an approach to the solid condition ; nevertheless, in problems of applied ELLIPSE OF STRESS. 101 hydrostatics, the assumption of perfect fluidity gives results near enough to the truth for practical purposes. 111. Opposite Principal Stresses. THEOREM. If a pair of prin- cipal stresses be o f equal intensities, but of opposite kinds, the stress on any plane perpendicular to the plane of the directions of the principal stresses is of the same intensity, and the angles which its direction makes with the normal to its plane are bisected by tlie axes of principal stress. In fig. 53, let the stresses acting along the rectangular axes OX, O Y, be as before, of equal intensity; but let them now be, not as before, of the same kind, but of opposite kinds, one being a thrust and the other a pull : a condition expressed by the equation and let it be required to find the direction and intensity of the stress on the plane A B, to which O R is normal. In this case OD is to be taken as before, to represent^. OB, the total stress on the_face OB of the triangular prism O A B ; but instead of taking OE in the direction from O towards B, to represent the total stress on O A, viz., p y OA, we are now to take Oe of equal length, but in the contrary direction. Complete the rectangle ODre; then the diagonal Or will represent the total stress on AB. The intensity of this stress is the same as before, viz., Pr=P*', but its direction Or, instead of being perpendicular to AB, makes an angle XOr on one side of the axis OX, equal to the angle XOR which the normal OE- makes on the other side of that axis; and O X bisects the angle of obliquity R Or. Q. E. D. The stress p r agrees in kind with that one of the principal stresses to which its direction is nearest ; and when it makes angles of 45 with each of the axes, it is shearing or tangential; so that a pull and a thrust of equal intensity, on a pair of planes at right angles to each other, are equivalent to a pair of shearing stresses of the same intensity on a pair of planes at right angles to each other, and making angles of 45 with the first pair. 112. Ellipse of stress. PROBLEM I. A pair o f principal stresses of any intensities, and oftlie same or opposite kinds, being given, it is required to find tJie direction and intensity of the stress on a plane in any position at right angles to the plane parallel to which the two principal stresses act. Let O X and Y (figs. 54 and 55), be the directions of the two principal stresses; OX being the direction of the greater stress. 102 PKINCIPLES OF STATICS. Let p. be the intensity "of the greater stress ; and p y that of the less. Q Y Fig. 54. Fig. 55. The kind of stress to which each of these belongs, pull or thrust, is to be distinguished by means of the algebraical signs. If a pull is considered as positive, a thrust is to be considered as negative, and vice versd. It is in general convenient to consider that kind of stress as positive to which the greater principal stress belongs. Fig. 54 represents the case in which p a and p y are of the same kind ; fig. 55 the case in which they are of opposite kinds. In all the following equations, the sign of p v is held to be implied in that symbol. Consider the two equations Py = From these it appears, that the pair of stresses, p m and p y , may be considered as made up of two pairs of stresses, viz. : a pair of stresses of equal intensity and of the same kind, whose common value is H~~> and a pair of stresses of equal intensity, but f) ~ j) opposite kinds, whose values are + ** ry . L/ Now let AB be the plane on which it is required to ascertain the direction and intensity of the stress, and ON a normal to that plane, making with the axis of greatest stress the angle ^ X N = A ELLIPSE OF STRESS. 103 On N take M = ^ y ; this will represent a normal stress 2 on A B of the same kind with the greater principal stress, and of an intensity which is a mean between the intensities of the two principal stresses ; and this, according to Article 110, Theorem I., will be the effect upon the plane AB, of the pair of stresses - 2t Through M draw PMQ, making with the axis of stress the same angles which ON makes, but in the opposite direction; that is to say, take MP = MTQ = MO. On the line thus found set off from M towards the axis of greatest stress, M R = ** 9 . This, ac- cording to Article 111, will represent the direction and the intensity of the oblique stress on AB, which is the effect of the pair of stresses P.P, 2 Join OR. Then will that line represent the resultant of the forces represented by OM and MR; that is to say, the direction and intensity of the entire stress on AB. Q. E. I. The algebraical expression of this solution is easily obtained by means of the formulae of plane trigonometry, and consists of the two following equations: Intensity, O R or p r = J {pi' cos 2 x n + pi' sin 2 x n] ..... (1.) an equation which might have been obtained by making p^ = in equation 1 of Article 108, Problem II. Obliquity, ^ N R or n r. (sin 2xn- P *~~ P*}... ..(2.) V 2p r / This obliquity is always towards the axis of greatest stress. In fig. 54, p x and p y are represented as being of the same kind ; and MR is consequently less than OM, so that OR falls on the same side of OX with ON, that is to say, n r ^ x n. In fig. 55, p x andp,, are of opposite kinds, MR is greater than OM, and OR A A = arc sin falls on the opposite side of OX to OM; that is to say, The locus of the point M is obviously a circle of the radius P X + P y , and that of the point R, an ellipse whose semi-axes are 2i p x and p y , and which may be called the ELLIPSE OF STRESS, because its semidiameter in any direction represents the intensity of the stress in that direction. 104 PRINCIPLES OP STATICS. The principal stresses, being represented by the semi-axes of this ellipse, are respectively the greatest and least of the stresses parallel to the plane XOY. The direct and shearing, or normal and tangential components of O R = p r are found by letting fall a perpendicular from R upon O N, and are as follows: Direct, p = p x ' cos 2 n x + p y ' sin 2 x nj (3.) A A 1>P* = (P* - Py) cos an- sinajw; (4.) equations which might have been deduced from the equations 2 of Article 108, Problem II. From equation 3 it is obvious, that the sum of the normal stresses on a pair of planes at right angles to each other is equal to the sum of the principal stresses ; and from equation 4 follows the principle, already demonstrated otherwise in Article 104, of the equality of the shearing stress on a pair of planes perpendicular to each other. PROBLEM II. A pair of principal stresses being given, it is required to find the positions of tJie planes on which the shear, or tangential component of the stress, is most intense, and the intensity of that shear. It is evident that the shear is greatest when M R is perpendicular to O M; and then M R itself represents the intensity of the shear; that is to say, maximum p t = - (5.) 2 In this case, A B is either of the two planes which make angles of 45 with the axes of stress. PROBLEM III. To find the planes mi ivhich the obliquity of the stress is greatest, the intensity of that stress, and tJte angle of its obliquity. CASE 1. WJien tlie pricinpal stresses are of the same kind. (Fig. 54.) In this case M R ^ M O, and it is evident that the angle of obliquity, ^: M O R = n r is greatest, when M R is perpendicular to O R, and that its value is given by the equation maximum n r arc sin . OM = arc sin ? (6.) P X + Py To find the position of the normal N to the plane A B, we have to consider that, A - -^ PMN; a ELLIPSE OF STRESS - PROBLEMS. 105 A = 90 + max. n r; consequently in this case, A 90 + max. n r 2 .......... ' ............ ' (an obtuse angle). And for the position of the plane AB itself, we have vn A 90 max. wr /Q , ^XOA = 90 xn=- ......... (8.) A (an acute angle). These equations apply to a pair of planes, making equal angles at opposite sides of O X. The intensity of the most oblique stress is obviously MR" 2 ) . P ,) ....... (9.) or a mean proportional between the principal stresses. This is otherwise evident from the consideration, that when O R -L PRQ, then OR - J (PR RQ), and that RQ - p xy PR = p y . CASE 2. When the principal stresses are of opposite kinds (fig. 55), it is evident, that the most oblique stress possible is a tangential stress, and that the problem amounts to finding the circumstances under which R lies in the plane AB. In this case it is evident, that the triangle O M R becomes right-angled at O, and conse- quently, that the intensity of the stress is given by the equation , = N /MK<_ o N being, as before, a mean proportional between the principal stresses. The product p x p y is a positive quantity, notwithstanding its negative sign, because p y in this case is implicitly negative. The position of the normal O N" is found by considering, that xn= -i and that ^PMISr=-^MOR + ^ MRO - 90 + arc sin L&. 106 PKINCIPLES OF STATICS. consequently, A x n ~ 2 i 90 + arc - sin (11.) (an obtuse angle) ; ^ XOA = 90 xn = ~ j 90 arc -sin 2i \ (an acute angle). In these, as in the other formulae applicable to the case in which p x and p y are of opposite kinds, it is to be borne in mind that p y is implicitly negative, and that consequently p,. + p y means the difference, and p x p y the sum, of the arithmetical values of the principal stresses. PROBLEM IY. The intensities, kinds, and obliquities, of any two stresses whose planes of action are perpendicular to the plane of their x/ directions, being given, it is required to find the principal stresses and axes of stress. CASE 1. When the given stresses are of tlie same kind, and unequal. In fig. 56, let A B, A B', represent the given planes, N, O N', their normals, O B, O B', the stresses upon them. Let the intensities be denoted algebraically p = OB; p = OK, Fig. 56. and the obliquities by NOB = n r B' - n'r. In fig. 57, take O N to represent at once the normals to both planes. ^ N B = n r ; Make O B' = rir; OB = p; OB' = p'. Join BB', bisect it in S, from which draw SM -L BB', cutting OMinM. Join MB, MB', which lines are evidently M - ~~" equal. Then from a com- parison of the construction of this figure with the gene- ration of the ellipse of stress, as described under Problem Fig. 57. I., is evident, that ELLIPSE OF STRESS PROBLEMS. 107 and consequently that the principal stresses are and it is also evident, that the angles made by the axis of greatest stress, with the two normals respectively, are which data are sufficient to determine the position of the axes. Q. K I. CASE 2. When the given stresses are of opposite kinds, the con- struction is the same in every respect, except that the lesser of the given stresses must be represented in fig. 57 by a line in the pro- longation of its direction beyond O, making an obtuse angle with O N, equal to the supplement of its obliquity. In either of the two cases that have been stated, the angle between the normals to the two given planes must have one or other of the two following values : A, ( either x ri + ^ra^^NMS) nn=l A A V ......... (14.) (or x ri ajw=r^:RMS j according as the two normals are at opposite sides, or at the same side of the axis of greatest stress. The solution of cases 1 and 2 is expressed algebraically by the following equations, which are deduced from the geometrical solution by means of well known formulae of trigonometry : 2 (p cos n r p' cos rv r J MR = Mil rt A , 2 cos 2 x n = (17.) 108 PRINCIPLES OF STATICS. In using these equations, it is to be observed that the cosine of an obtuse angle is negative. Simplified Forms of Cases 1 and 2. CASE 3. When the two given stresses are conjugate, they are of equal obliquity; and the points O, R', S, R, in fig. 57, are in one straight line, to which M S is perpendicular ; the angle between the two normals being ^NMS = w A 7i' = 90 + 7fr ..................... (18.) In this case, equation 15 becomes A 2 cos n r equation 16 becomes J** -*-*-&} (20.) A cos 2 wr I equations 17 are modified only by the equality of n' r 7 to nr. CASE 4. When the planes of action of the two given stresses are perpendicular to each other, M S is perpendicular and R R' parallel to O N, in fig. 57, so that we have, for the tangential component of each stress, MS = p sin nr=p' sin n' r" =p t . Let the normal components of the given stresses be denoted by A A p n =p cos n r ; p n p' cos n r. Then equation 15 becomes equation 16 becomes : ( The equations 17 become A A v\ ' ~\ cos 2 x n = - cos 2 x'n' = P n ~~P ; or, what is equivalent, [ (23. ) tan 2 x n = - tan 2 x 1 n f = J being the same with equation 1 of Article 109. ELLIPSE OP STRESS PROBLEMS. 109 PROBLEM V. The stress in every direction being a thrust, and the greatest obliquity being given, it is required to find the ratio of two conjugate thrusts whose common obliquity is given. Let

-fp') 3 cos 2 wr Hence it follows that the ratio of the conjugate stresses, p, p', is that of the two roots of a quadratic equation. tf 2coswr "w-|-cos 2

, p', are any pair of the given stresses ; 3. All the squares p 2 sin 2 2 n p ; 4. All the products 2 p p' sin 2 n p sin 2 71^3'. The first and third of these classes being added together, make 2 (p 2 ); the second and fourth make 2 2 (p p' cos 2pp') pp' being the angle between p and /. Equation 2 thus becomes ^T*' -g- V {2 (/) + 2 > (pp cos 3/p-)} ...... (4.) From the equations (1) and (4) it appears that the intensities of the principal stresses p x and p y can be computed without assuming planes of reduction ; for the only angles involved in this pair of equations are the several angles pp', which the given stresses make 112 PRINCIPLES OF STATICS. with each other when compared by pairs in every possible com- bination. To find the directions, however, of those principal stresses, planes of reduction must be assumed. In using the equation (4), it is to be remembered that when 2pp' exceeds 90, we have cos 2pp> = cos (l80 2 p"p'\ SECTION 4. Of the Internal Equilibrium of Stress and Weight, and the Principles of Hydrostatics. 114. Varying internal stress. The investigations of the preced- ing section have been conducted as if the internal stress, whether simple or compound, were iiniform at all points in the body under consideration ; but their results are nevertheless correctly applicable to internal stress which varies from point to point of the body ; for those results are arrived at by considering the conditions of equilibrium of a pyramidal or prismatic portion of the body con- taining the point at which the relations amongst the components of the stress are to be determined ; and when the stress varies from point to point, then by supposing the pyramid or prism to be small enough, its condition of stress may be made to deviate from uni- formity to an extent less than any assigned limit of deviation ; but the truth of the propositions of the preceding section for an uniform stress is independent of the size of the prism or pyramid ; therefore they can be proved to deviate from the truth for a vary- ing stress by less than any assignable error j therefore they must be true for a varying as well as for an uniform stress. 115. Causes of Varying stress. The internal stress exerted amongst the parts of a body, may vary from point to point, from tliree classes of causes, viz. : I. Mutual attractions and repulsions between the parts of the body; II. Attractions and repulsions exerted between the parts of the body in question and external bodies ; III. Stress exerted between the body in question and external bodies at their surfaces of contact. I. The first of these classes of causes may be left out of considera- tion in the present treatise ', because the mutual attractions and repulsions of the parts of an artificial structure are too small to be of practical importance in the art of construction. II. Of the second class of causes, the only force which is of sufficient magnitude to be considered in the art of construction, is weight. III. The consideration of the third class of causes belongs to INTERNAL EQUILIBRIUM. 113 Fig. 58. the subject of the strength of materials, which will be treated of in the sequel. The subject of the present section, therefore, is the relation be- tween the weight of the parts of a body, and the variation of its condition of stress from point to point. 116. General Problem of Internal Equilibrium. Let W denote the weight per unit of volume of a body, or part of a body, and let it be required to determine what modes of variation of internal stress are consistent with that specific gravity. Consider the condition of a rectangular molecule A (fig. 58), bounded by ideal planes, whose edges are parallel to three rectangular axes, OX, OY, OZ. The position of this set of axes is immaterial to the result ; but the algebraic formulae are simplified by assuming one axis to be vertical; let O Z, then, be vertical, and let distances along it be positive upwards. Then weight must be treated as a nega- tive force ; and the weight of a portion of the body of the volume Y will be denoted by - wV. Let the dimensions of the molecule A be A x parallel to OX, A2/ OY, AZ OZ. Then its weight is represented by w ' AX Ay A z. The six faces will be designated as follows : The pair parallel to Y O Z zox . . XOY) (That is, the horizontal pair.) J Let the six intensities of the components of the stress be denoted as in Article 104, viz. : Normal, p xxt p yy , p zz ' } Tangential, p yz , p zx , p xy . As for the signs of normal stress, let pull be positive and thrust i Farthest from 0. + A y A z + AS! AX 4- AX Ay} (the upper.) j Nearest to O. - Ay AZ AZ AX (the lower.) j 114 PRINCIPLES OF STATICS. negative. As for the signs of tangential stress, let those stresses be considered as < ^ ... > which tend to make the pair of cor- [ negative j ners of the molecule which are nearest and farthest from O { sharper ) \ natter ( In the first place, let the rate of variation of the stress, of what kind soever, from point to point, be uniform; that is to say, for example, if the mean intensity of any one of the components of the stress at the face A x & y be p, then at the face + A x A y, whose distance from A x A y is A z, let the mean intensity of the same component be in which - is a constant co- efficient or factor, meaning " tJie rate a z of variation of p along z," which is positive or negative, according as the variation of p is of the same or of the contrary kind to that of z. Rates of variation are also known by the name of differential co-efficients. As there are six components in the stress, and three axes of co-ordinates, there are eighteen possible differential co- efficients of the stress with respect to the co-ordinates ; but it will presently appear that nine only of those co-efficients are concerned in the solution of the present problem. The relations amongst the weight of the molecule A, and the variations of the intensities of the component stresses on its differ- ent faces, depend on this principle, that the force arising from the variations of stress must balance the weight of the molecule; that is to say, the resultant force parallel to each of the horizontal axes, which arises from the variation of stress, must be nothing, and the resultant force parallel to the vertical axis, which arises from the variation of stress, must be upward, and equal to the weight of the molecule a principle expressed by the three following equa- tions : yy vz AX' Ay AZ 4- -~ A y A Z A X + -2p5 AZ'AXAy-Q; ax ay dz dp zr dp,,, dp zz -4-= A X ' A y A Z + *- A V A Z A Z H = A Z ' A X A y dx dy dz w ' A x A y A c. (1.) INTERNAL EQUILIBRIUM. 115 Each of the nine terms which compose the left sides of the above equations is the product of four factors ; the first being the rate of variation of a stress, the second the distance between two faces on which that stress acts, and the third and fourth the dimensions of those faces, whose product is their common area. Each term of those three equations contains as a common factor the volume of the molecule, AX & y A z ; dividing by this, they are reduced to the following : .(2.) dpxx i dp xy , dp zx Q dx dy dz dpxy + dpyy + d = Q dx dy dz dpzx ^02 J_ ^jZ2 j ~\~ j ~j~ j W. dx dy dz In this second form, the equations are applicable to rates of varia- tion which are not uniform as well as to those which are uniform. For as the rectangular molecule, from the conditions of whose equilibrium these equations are deduced, is of arbitrary size, it may be supposed as small as we please ; and when the rates of variation of the stress are not uniform, we can always, by supposing the molecule small enough, make the rates of variation of the stresses throughout its bulk deviate from uniform rates to an extent less than any given limit of error. The equations 2 can easily be modified so as to adapt them to any different arrangement of the axes of co-ordinates. Thus, if z be made positive downwards io stead of upwards, w is to be put for w in the third equation. If x or y, instead of z, be made the vertical axis, w is to be substituted for in the first or the second equation, as the case may be, and for w in the third equation. If the axes of x, y, and z make respectively the angles , /3, and y, with a line pointing vertically upwards, the force of gravity is to be resolved into three rectangular components, each of which must be separately balanced by variations of stress ; so that for 0, 0, w, in the first, second, and third equations respectively, are to be substituted w cos *, w cos P, w cos y. The equations of this Article are not in general sufficient of 116 PEINCIPLES OF STATICS. themselves to determine the mode of variation of the intensity of the stress in a solid body, because of their number not being so great as that of the number of unknown quantities to be determined. They have therefore to be combined with other equations, deduced from the relations which are found by experiment to exist between the alterations of figure, which the parts of a solid body undergo when a load acts on it, and the stresses which at the same time act amongst the disfigured parts. These relations belong to the sub- ject of elasticity and of the strength of materials, and not to that of the principles of statics. The remainder of the present section will relate to those more simple problems which can be solved by means of the equations 2 alone. 117. Equilibrium of Fluids. It has already been explained in Article 110, that in a fluid the only stress to be considered in practice is a thrust or pressure, normal and of equal intensity in all directions. This is expressed symbolically in the following manner : 0'A J" the single symbol p being used, for the sake of convenience and brevity, to denote the intensity of the fluid pressure at any given point in the fluid. In adapting the equations 2 of Article 116 to this case, it is con- venient to take x to denote vertical co-ordinates, and to make it positive downwards. Then, bearing in mind that p is now a thrust, being positive (and not a pull when positive and a thrust when negative, as in the general problem), we obtain the following equations : dp ~T-~ W '> dx -- -- dy~ ' dz~ The first of these equations expresses the fact, that in a balanced fluid, the pressure increases with the vertical depth, at a rate expressed by the weight of the fluid per unit of volume; and the second and third express the fact, that in a balanced fluid, the pressure has no variation in any horizontal direction ; in other words, that the pressure is equal at all points in the same level surface. [The exact figure of a level surface is spheroidal ; but for pur- poses of applied mechanics it may be treated as a plane, without sensible error.] EQUILIBRIUM OF FLUIDS. 117 Those principles may also be proved directly. Let fig. 59 re- present a vertical section of a fluid; Y Y any horizontal plane, O X a ' ' vertical axis. Let B B be a hori- zontal plane at the depth x below O ; B A C C another horizontal plane at the 1^, depth x + A SB. Let A be a small rectangular molecule contained be- x tween those two horizontal planes; Fi g- 59 - and let A y and A z be its horizontal dimensions, so that its weight is W A X A y A Z. The pressure exerted by the other portions of the fluid against the vertical faces of this molecule are horizontal, and must balance each other; therefore there can be no variation of pressure horizontally. Let PQ, then, be the uniform pressure at the horizontal plane YO Y, p, that at the plane B B, and^> + - A x that at the plane C C, ~- being the rate of increase of pressure with depth. The molecule is pressed downwards by the pressure whose amount is p A y A z, and upwards by the pressure whose amount is (p + ? AX ]*y A . dx / The difference between those forces, viz. : dp jjj-A*;**** has to be balanced by the weight of the molecule ; equating it to which, and dividing by the common factor A x A y A z, we obtain the first of the equations 2 of this Article. The pressure p Q at the surface Y Y being given, the pressure p at any given depth x below Y Y is found by means of the integral, (3.) P ~** ' ' 9dX = p Q + I wdx; that is to say, it is equal to the pressure at the plane Y Y, added to the weight of a vertical column of the fluid whose area of base is unity, and which extends from the plane YY down to the given depth x below that plane. It is obviously necessary to the equilibrium of a fluid, that the 1 1 S PRINCIPLES OF STATICS. specific gravity, as well as the pressure, should be the same at all points in the same level surface. The preceding principles are the base of the science of Hydro- statics. 118. Equilibrium of a Liquid. A liquid is a fluid whose parts tend to preserve a definite size ; that is to say, a portion of a liquid of a given weight tends to occupy a certain definite volume ; and to make it occupy a greater or a less volume, tension or pressure, as the case may be, must be applied to it. The volume occupied by an unit of weight is the reciprocal of the weight of an unit of volume ; so that the preceding principle might otherwise be stated by say- ing, that a liquid tends to preserve a definite specific gravity, which may be increased by pressure, or diminished by tension. The volume which a given weight of a liquid tends to occupy depends on its temperature according to laws which belong to the science of Heat. The alterations of the specific gravity of liquids produced by any pressures which occur in practice, are so small, that in most pro- blems respecting the equilibrium of liquids, the specific gravity w may be treated without sensible error as a constant quantity, inde- pendent of the pressure p. In the case of water, for example, the compression of volume, and increase of specific gravity, produced by a pressure of one atmosphere, or 14*7 pounds per square inch, is about TOTHT, or ^s/ooo for each pound on the square inch. If, then, the specific gravity w be treated as a constant in equation 3 of Article 117, it becomes as follows : p = p Q + wx; (1.) that is to say : let p be the pressure at the upper surface, Y O Y, (fig. 59) of a mass of liquid; then the pressure p at any given depth x below that surface is greater than the superficial pressure p Q by an amount found by multiplying that depth by the weight of an unit of volume of the liquid. When the mass of liquid is in the open air, the superficial pres- sure p Q is that arising from the weight of the earth's atmosphere of air, and at places near the level of the sea, is estimated on an average at 14 -7 pounds on the square inch. In a close vessel, the superficial pressure may be greater or less than that of the atmosphere. 119. Equilibrium of different Fluids in contact with each other. If two different fluids exist in the same space, they may unite so that each of them shall be distributed throughout the whole space, either by chemical combination or by diffusion ; but in such cases they form, in fact, but one fluid, which is a compound or mixture, as the case may be. The present Article has reference to the case EQUILIBRIUM OF DIFFERENT FLUIDS. 119 when fluids of different kinds remain in contact, uncombined and unmixed. In this case, the condition of equilibrium is, that the pressures of two fluids at each point of their surface of contact shall be equal to each other, a condition which, when the two fluids are of difierent specific gravities, can only be fulfilled when the surface of contact is horizontal. If, then, two or more fluids of difierent specific gravities, which do not combine nor mix with each other, be contained in one vessel uninterrupted by partitions, they will arrange themselves in hori- zontal strata, the heavier fluids being below the lighter. If two fluids of different specific gravities be contained in the two legs of a tube shaped like the letter TJ (and called an "inverted siphon"), or if one of the two fluids be contained in a vertical tube open below, and the other in the space surrounding that tube ; or, generally, if the two fluids be partially separated from each other by a vertical or nearly vertical partition, below which there is a com- munication between the spaces on either side of it; the horizontal surface of contact of the fluids will be at that side of the partition at which the lighter fluid is found, so that it may be above, and the heavier fluid below, that surface of contact. Let PQ denote the common pressure of the two fluids at their sur- face of contact, and let any ordinate measured from that surface upwards, be denoted by x. Let w denote the specific gravity, and p the pressure, of the lighter fluid; w" the specific gravity, and p" the pressure, of the heavier fluid. Then at any given elevation x above the surface of contact _ \ L -) = W dx which equations, when the fluids are liquids, and W, w", constants, become P' = Po ~ wx'> p" = Po - w"x ................ (2.) As in the case of the barometer, and the mercurial pressure gauge, the height at which a liquid stands in a tube, closed and empty at the upper end, above its surface of contact with another fluid, may be used to determine the pressure exerted by that other fluid at the surface of contact. In this case, p" = 0, or nearly so ; consequently (3.) Let a/, a?", be two heights above the surface of contact at which the respective pressures of the lighter and the heavier fluid are either equal to each other, or both equal to nothing ; then p" = p', and consequently, for fluids in general, 120 PRINCIPLES OF STATICS. f^w'dx = f*Jw"dx, (4.) If the fluids be both liquids, this becomes, w' x = w" x", (5.) or, the heights are inversely as the specific gravities. If the heavier fluid be a liquid (such as the mercury in the baro- meter) and the lighter a gas (such as the atmosphere) the equation becomes (6.) and on this last formula is founded the method of determining differences of level by barometric observations of the atmospheric pressure. 120. Equilibrium of a Floating Body. THEOREM. A solid body floating on the surface of a liquid is balanced) wJien it displaces a volume of liquid whose weight is equal to the weight of ttie floating body, and when the centre of gravity of the floating body, and that of tJie volume from which the liquid is displaced, are in the same vertical line. Let fig. 60 represent a solid body (such as a ship), floating in a liquid, whose horizontal upper surface is Y Y. Suppose, in the first place, that there is no pressure on the surface YY. Consider a small portion S of the surface of the im- JT mersed part of the solid body. The liquid will exert against S a normal pressure, whose amount will be ex- Fig. 60. Sp = S wx } where S is the area of the small portion of the immersed surface, x the depth of immersion of its centre below the level surface YY, and w the weight of unity of volume of the liquid. Let a denote the angle of inclination of the area S to a horizontal plane, or, what is the same thing, the angle of inclination of the pressure on S to the vertical. Conceive a vertical prism H S to stand on the area S ; the area of the horizontal transverse section of this prism is what is called the horizontal projection of the area S, and its value is S cos *>. Conceive a horizontal prism ST to have its axis in the vertical plane which is perpendicular to S, and to have the area S for an FLOATING BODY. 121 oblique section ; the vertical transverse section of this prism is what is called the vertical projection of the area S, and its value is S sin a.. This horizontal prism cuts the immersed surface in another small area T, whose projection on a vertical plane perpendicular to the axis of the prism S T is equal to that of S, and which is immersed to the same depth, and sustains pressure of the same intensity. Resolve the total pressure on S into a horizontal component and a vertical component. The horizontal component is S p ' sin a. = S w x ' sin , being equal to the product of the intensity p by the vertical projection of S ; but this component is balanced by an equal and opposite com- ponent of the total pressure on T; and the same is the case for every portion such as S into which the immersed surface can be divided ; therefore the resultant of all the horizontal components of the pressure exerted by the liquid against the solid is nothing. The vertical component of the pressure on S is Sp cos = S w x cos ee, being equal to the product of the intensity p by the Jwrizontal projection of S. But S x cos a. is the volume of the vertical prism H S, standing upon the small area S, and bounded above by the horizontal surface YY, and w is the weight of unity of volume of the liquid ; therefore^ w x cos is the weight of liquid which the prism H S would contain ; so that the vertical component of the pressure on S is an upward force, equal and opposite to the weight of the liquid displaced by the prismatic portion of the solid body which stands vertically above S. Then if the whole of the immersed surface be divided into small areas such as S, the resultant of the pressure of the liquid against that entire surface is the sum of all the vertical components of the pressures on the small areas ; that is, a force equal and opposite to the sum of the weights of liquid displaced by all the prisms such as H S ; that is, a sum equal and opposite to the weight of the whole volume of liquid displaced by the floating body ; and the line of action of that resultant traverses the centre of gravity of the volume of liquid so displaced. Let C denote that centre of gravity, which is also called the Centre of Buoyancy. Let G- denote the centre of gravity of the floating body. Let W denote the weight of the floating body, and V the volume of liquid displaced by it. Then the conditions of equilibrium of the floating body are ob- viously the following : First : W = w V ; or its weight must be equal to the weight of the volume of liquid displaced by it j 122 PRINCIPLES OP STATICS. j: its centre of gravity G, and the centre of buoyancy C, must be in the same vertical line. Q. E. D. The preceding demonstration has reference to the case in which the pressure on the horizontal surface Y Y is nothing. In the case of bodies floating on water, that surface, as well as the non-immersed part of the surface of the floating body, have to sustain the pressure of the air. To what extent this fact modifies the conclusions arrived at will appear in the next Article. 121. Pressure on an Immersed Body. THEOREM. If a solid body be wholly immersed in a fluid, the resultant of the pressure of the fluid on the solid body is a vertical force, equal and directly opposed to the weight of the portion of the fluid which the solid body displaces. Let fig. 61 represent a solid body totally immersed in a fluid, T Y whether liquid or gaseous. Conceive a small vertical prism SU to extend from a portion S of the lower surface of the body, to the portion U of the upper surface which is ver- tically above S. Also let S T be a horizontal H _ prism of which S is an oblique section, and y~^ ^ UV a horizontal prism of which U is an Fi 61 oblique section, as in Article 120. Then, as in Article 120, it may be proved that the horizontal component of the pressure on S is balanced by an equal and opposite component of the pressure on T, and the horizontal component of the pressure on U by an equal and opposite component of the pressure on V; so that the horizontal component of the resultant of the pressure of the fluid on the entire body is nothing, and that resultant is vertical. The vertical component of the pressure on S is upward, and equal to the weight of the prismatic portion of the fluid which would stand vertically above S if a part of it were not displaced by the solid body. The vertical component of the pressure on U is downward, and equal to the weight of the prismatic portion of the fluid which stands vertically above U. The vertical force arising from the pressures on S and on U together is upward, and equal to the difference between those two weights; that is, it is equal and directly opposed to the weight of the portion of the fluid dis- placed by the prismatic portion S U of the immersed body. Hence the resultant of the pressure of the fluid over the entire surface of the immersed body is equal and directly opposed to the weight of the portion of fluid displaced by that body. Q. E. D. The centre of gravity C, of the portion of fluid which would occupy the position of the body if it were not immersed, is called, as before, the centre of buoyancy, and is traversed by the vertical line of action of the resultant of the pressure of the fluid, which is APPARENT WEIGHTS IN AIR. 123 itself called the buoyancy of the immersed body, and sometimes the apparent loss of weight. To maintain an immersed body in equilibrio, there must be applied to it a force or couple, as the case may be, equal and directly op- posed to the resultant, if any, of its downward weight and upward buoyancy; that resultant being determined according to the principles of Articles 39 and 40. When a body floats in a heavier fluid (as water) having its upper portion surrounded by a lighter fluid (as air), its total buoyancy is equal and opposite to the resultant of the weights of the two portions of the respective fluids which it displaces. In practical questions relative to the equilibrium of ships, the buoyancy arising from the displacement of air is too small as com- pared with that arising from the displacement of water, to require to be taken into account in calculation. 122. Apparent Weights. The only method of testing the equality of the weights of two bodies which is sufficiently delicate for exact scientific purposes, is that of hanging them from the opposite ends of a lever with equal arms. If this process were performed in a vacuum, the balancing of the bodies would prove their weights to be equal ; but as it must be performed in air, the balancing only proves the equality of the apparent weights of the bodies in air, that is, of the respective ex- cesses of their weights above the weights of the volumes of air which they displace. The real weights of the bodies, therefore, are not equal unless their volumes are equal also. If their volumes are unequal, the real weight of the larger body must be the greater by an amount equal to the weight of the difference between the volumes of air which they displace. The weight of a cubic foot of pure diy air, under the pressure of one atmosphere (147 Ibs. on the square inch), and at the temperature of melting ice (32 Fahrenheit) is 0-080728 pound avoirdupois. Let this be denoted by w . Then the weight of a cubic foot of air under any other pressure of p atmospheres, and at the temperature t of Fahrenheit's scale, is given with a degree of accuracy sufficient for most purposes, by the formula, 4930-2 /i x "' and if w, W, be the weights of a given volume of air, under the respective pressures /?,>', and at the temperatures t, t', of Fahrenheit's scale, then t^_ t + 461-2 w ~ p' tf + 461-2'" 124 PRINCIPLES OF STATICS. Let W l denote the true weight of a body, Vj its volume, w^ its weight per unit of volume, w the weight of unity of volume of air. Then W< ='* and the apparent weight of the same body in air is W = to - w) V, = -^ W, (3.) Let this body now be balanced against another body in an accurate pair of scales, and let their apparent weights be equal. Then, if W 2 denote the true weight, and w 2 the weight per unit of volume, of the second body, we have Wi w ~, w a w ^, T ,. . so that the proportion between the real weights of the bodies is W^ 2 Wi w 2 w 2 w / K , = (5.) Vv i Wi w 2 Wi w 123. Relative Specific Gravities. If the true weight of a solid body be known, and that body be next weighed while immersed in a liquid, the proportion of the specific gravities of the solid body and of the liquid can be deduced from the apparent loss of weight, which is the weight of the volume of liquid displaced by the body. Let W,, as in equation 3 of Article 122, denote the true weight of the solid body, w l its weight per unit of volume, w 2 the weight of an unit of volume of the liquid in which its apparent weight is found, and W" the apparent weight; then by the equation already referred to W" = ?1 ~ 2 Wi = (l - and consequently 111 TIT ITT" Wn yy W Let the first weighing take place in air and the second in the liquid, and let W be the apparent weight in air ; then and consequently W" _ w, w 2 m , 9 v W ~ w, w '' so that if is known, may be found by the equation w 2 w 2 J IMMERSED PLANE. 125 W" w a (3.) w 2 ~ W W" When the object of weighing of this kind is to determine the specific gravities of solids, the liquid usually employed is pure water; and the results obtained are the ratios of the specific gravities of solid bodies to that of pure water. If these ratios, or relative spe- cific gravities, be multiplied by the weight of a cubic foot of pure water, the weight of a cubic foot of the solid is obtained. The weight of a cubic foot of pure water at the temperature of its maximum density (being, according to Playfair and Joule, 39-l Fahrenheit) is, according to the best existing data, 62425 pounds avoirdupois. For any other temperature t on Fahrenheit's scale, the weight of a cubic foot of pure water is 62425 (4) v where v denotes the volume to which a mass of water measuring one cubic foot at 39-l expands at ; a volume which may be computed for temperatures from 32 to 77 Fahrenheit, by means of the follow- ing empirical formula, extracted from Prof. W. H. Miller's paper on the Standard Pound in the Philosophical Transactions for 1856 : log. v = 10/1 (t 39-1) 2 0-0369 (t 39-1) 3 ; (5.) The relative specific gravities of two liquids are determined by weighing the same solid body immersed in them successively and comparing its apparent losses of weight. 124. Pressure on an Immersed Plane. If a horizontal plane SUr- face of any figure be immersed in a fluid, the pressure on that sur- Y face is vertical, and uniformly distributed ; its amount is the product of the intensity of the pressure at the depth to which the plane is immersed by the area of the plane; and the centre of pressure (as already shown in Art. 90) is the centre of gravity of a flat plate of the figure of the plane surface, or, as it is Fi S- 62 - usually termed, the centre of gravity of the plane surface. If an inclined or vertical plane surface be immersed in a liquid, let Y (fig. 62), represent a section of the horizontal plane at which the pressure is nothing, and B F a vertical section of the 126 PRINCIPLES OF STATICS. immersed plane. Let Xi = BE be the depth to which the lower edge of this plane is immersed below OY. From B draw BD BE, and -L BF; produce the plane BF till it cuts the horizontal plane of no pressure, O Y, in the line represented in section by O ; through O and D draw a plane O H D, and conceive the prism B D H F to stand normally upon the base B F and to be bounded above by the plane D H. The pressure on the plane B F will be normal ; its amount will be equal to the weight of fluid contained in the volume B D H F ; that is to say, let x denote the depth of the centre of gravity of the plane BF below O Y, and w the weight of unity of the volume of liquid; then the mean intensity of the pressure on B F is and the amount of the pressure P = BF (2.) Let C be the centre of gravity of the volume B D H F; then the centre of pressure of the surface B F is the point where it is cut by the perpendicular C P let fall on it from C. As the intensity of the pressure on any point of BF is propor- tional to its depth below OY, and consequently to its distance from O, this is a case of uniformly varying stress, and the formulae of Article 94 are applicable to it. In the application of those formulae it is to be observed, that the ordinates y are to be measured hori- zontally in the plane B F, whose centre of gravity is to be taken as the origin; that the co-ordinates x are to be measured in the same plane, along the direction of steepest declivity, and reckoned positive downwards ; and that the value of the constant a in the equations of Article 94 is given by the formula a = wsina. ....................... ,....(3.) where is the angle of inclination of the plane B F to a horizontal plane. 125. Pressure in an Indefinite Uniformly Sloping Solid. Conceive a mass of homogeneous solid mate- rial to be indefinitely extended laterally and downwards, and to be bounded above by a plane sur- face, making a given angle of de- clivity 6 with a horizontal plane. In fig. 63, let Y Y represent a ver- tical section of that upper sloping surface along its direction of greatest declivity, and O X a vertical plane 63. perpendicular to the plane of vertical PARALLEL PROJECTION OF STRESS AND WEIGHT. 127 section which is represented by the paper. Let w be the uniform weight of unity of volume of the substance. Let B B be any plane parallel to, and at a vertical depth x below the plane Y Y. If the substance is exposed to no external force except its own weight, the only pressure which any portion of the plane B B can have to sustain is the weight of the material directly above it. Hence follows THEOREM I. In an indefinite homogeneous solid bounded above by a sloping plane, the pressure on any plane parallel to that sloping surface is vertical, and of an uniform intensity equal to the weight of the vertical prism which stands on unity of area of the given plane. The area of the horizontal section of that prism is cos 6, conse- quently, the intensity of the vertical pressure on the plane B B at the depth x is p x wxcosff (1.) From the above theorem, combined with the principle of conjugate stresses of Article 101, there follows THEOREM II. The stress, if any, on any vertical plane is parallel to the sloping surface, and conjugate to the stress on a plane parallel to that surface. Consider now the condition of a prismatic molecule A, bounded above and below by planes B B, C C, parallel to the sloping surface Y Y, and laterally by two pairs of parallel vertical planes. Let the common area of the upper and lower surfaces of this prism be unity, and its height A x ', then its volume is A x ' cos 6, and its weight w A x ' cos 0, which is equal and opposite to, and balanced by the excess of the vertical pressure on its lower face above the vertical pressure on its upper face. Therefore, the pressures paral- lel to the sloping surface, on the vertical faces of the prism, must balance each other independently ; therefore they must be of equal mean intensity throughout the whole extent of the layer between the planes B B, CO; whence follows THEOREM III. TJie state of stress, at a given uniform depth below the sloping surface, is uniform. 126. On the Parallel Projection of Stress and Weight. In apply- ing the principles of parallel projection to distributed forces, it is to be borne in mind that those principles, as stated in Chapter IV., are applicable to lines representing the amounts or resultants of distributed forces, and not tJieir intensities. The relations amongst the intensities of a system of distributed forces, whose resultants have been obtained by the method of projection, are to be arrived at by a subsequent process of dividing each projected resultant by the projected space over which it is distributed. Examples of the application of processes of this kind to practical questions will appear in the Second Part. 128 CHAPTER YI. ON STABLE AND UNSTABLE EQUILIBRIUM. 127. Stable and Unstable Equilibrium of a Free Body. Sup- pose a body, which is in equilibrio under a balanced system of forces, to be free to move, and to be caused to deviate to a small extent from its position of equilibrium. Then if the body tends to deviate further from its original position, its equilibrium is said to be un- stable; and if it tends to return to its original position, its equi- librium is said to be stable. Cases occur in which the equilibrium of the same body is stable for one kind or direction of deviation, and unstable for another. When the body neither tends to deviate further, nor to recover its original position, is equilibrium is said to be indifferent. The solution of the question, whether the equilibrium of a given body under given forces is stable, unstable, or indifferent, for a given kind of deviation of position, is effected by supposing the deviation made, and finding the resultant of the forces which act on the body, altered as they may be by the deviation, in amount, in position, or in both. If this resultant acts towards the same direc- tion with the deviation, the equilibrium is unstable if towards the opposition direction, stable and if the resultant is still nothing, the equilibrium is indifferent. The disturbance of a free body from a position of stable equi- librium causes it to oscillate about that position. 128. stability of a Fixed Body. The term "stability," as ap- plied to the condition of a body forming part of a structure, has, in most cases, a meaning different from that explained in the last Article, viz., the property of remaining in equilibrio, without sen- sible deviation of position, notwithstanding certain deviations of the load, or externally applied force, from its mean amount or posi- tion. Stability, in this sense, forms one of the principal subjects of the second part of this treatise. PART II. THEORY OF STRUCTURES. CHAPTER I. DEFINITIONS AND GENEKAL PRINCIPLES. 129. structures Pieces joints. Structures have already, in Article 15, been distinguished from machines. A structure con- sists of two or more solid bodies, called its pieces, which touch each other and are connected at portions of their surfaces called joints. 130. Supports Foundations. Although the pieces of a structure are fixed relatively to each other, the structure as a whole may be either fixed or moveable relatively to the earth. A fixed structure is supported on a part of the solid material of the earth, called the foundation of the structure; the pressures by which the structure is supported, being the resistances of the various parts of the foundation, may be more or less oblique. A moveable structure may be supported, as a ship, by floating in water, or as a carriage, by resting on the solid ground through wheels. When such a structure is actually in motion, it partakes to a certain extent of the properties of a machine j and the deter- mination of the forces by which it is supported, requires the con- sideration of dynamical as well as of statical principles ; but when it is not in actual motion, though capable of being moved, the pres- sures which support it are determined by the principles of statics ; and it is obvious that they must be wholly vertical and have their resultant equal and directly opposed to the weight of the structure. 131. The Conditions of Equilibrium of a Structure are the three following : I. That the forces exerted on the whole structure by external bodies shall balance each otJier. The forces to be considered under this head are (1.) the Attraction of the Earth, that is, the weight of the structure ; (2.) the External Load, arising from the pressures exerted against the structure by bodies not forming part of it nor of its foundation j (these two kinds of forces constitute the gross or -total load) ; (3.) the Supporting Pressures, or resistance of the founda- tion. Those three classes of forces will be spoken of together as the External Forces. 130 THEORY OF STRUCTURES. II. That the forces exerted on each piece of the structure slwll balance each other. These consist of (1.) the Weight of the piece, and (2.) the External Load on it, making together the Gross Load; and (3.) the Resistances, or stresses exerted at the joints, between the piece under consideration and the pieces in contact with it. III. That the forces exerted on each of the parts into which the pieces of the structure can be conceived to be divided Sliall balance each other. Suppose an ideal surface to divide any part of any one of the pieces of the structure from the remainder of the piece ; the forces which act on the part so considered are (1.) its weight, and (2.) (if it is at the external surface of the piece) the external stress applied to it, if any, making together its gross load; (3.) the stress exerted at the ideal surface of division, between the part in ques- tion and the other parts of the piece. 132. stability, Strength, and Stiffness. It is necessary to the per- manence of a structure, that the three foregoing conditions of equilibrium should be fulfilled, not only under one amount and one mode of distribution of load, but under all the variations of the load as to amount and mode of distribution which can occur in the use of the structure. Stability consists in the fulfilment of the first and second condi- tions of equilibrium of a structure under all variations of load within given limits. A structure which is deficient in stability gives way by the displacement of its pieces from their proper posi- tions. Strength consists in the fulfilment of the third condition of equi- librium of a structure for all loads not exceeding prescribed limits ; that is to say, the greatest internal stress produced in any part of any piece of the structure, by the prescribed greatest load, must be such as the material can bear, not merely without immediate break- ing, but without such injury to its texture as might endanger its breaking in the course of time. A piece of a structure may be rendered unfit for its purpose not merely by being broken, but by being stretched, compressed, bent, twisted, or otherwise strained out of its proper shape. It is neces- sary, therefore, that each piece of a structure should be of such dimensions that its alteration of figure under the greatest load applied to it shall not exceed given limits. This property is called stiffness, and is so connected with strength that it is necessary to consider them together. From the foregoing considerations, it is evident that the theory of structures may be divided into two divisions, relating, the first to STABILITY, or the property of resisting displacement of the pieces, and the second to STRENGTH and STIFFNESS, or the power of each piece to resist fracture and disfigurement. 131 CHAPTER II. STABILITY. 133. Resultant dross Load. The mode of distribution of the intensity of the load upon a given piece of a structure affects the strength and stiffness only. So far as stability alone is concerned, it is sufficient to know the magnitude and position of the resultant of that load, which is to be found by means of the principles ex- plained in the First Part of this work, and may then be treated as a single force. 134. Centre of Resistance of a Joint. In like manner, when stability only is in question, it is sufficient to consider the position and magnitude of the resultant of the resistance or stress exerted between two pieces of a structure at the joint where they meet, and to treat that resultant as a single force. The point where its line of action traverses the joint is called the centre of resistance of that joint. 135. A tine of Resistance is a line, straight, angular, or curved, traversing the centres of resistance of the joints of a structure. It is to be borne in mind, that the direction of this line at any given joint does not necessarily coincide with the direction of the resist- ance at that joint, although it may so coincide in certain cases. 136. Joints classed. Joints, and the structures in which they occur, may be divided into three classes, according to the limits of the variation of position of which their centres of resistance are capable. I. Framework joints are such as occur in carpentry, in frames of metal bars, and in structures of ropes and chains, fixing the ends of two or more pieces together, but offering little or no resistance to change in the relative angular positions of those pieces. In a joint of this class, the centre of resistance is at the middle of the joint, and does not admit of any variation of position consistently with security, II. Blockwork joints are such as occur in masonry and brickwork, being plane or curved surfaces of contact, of considerable extent as compared with the dimensions of the pieces which they connect, capable of resisting a thrust more or less oblique, according to laws to be afterwards explained, but not of resisting a pull of suf- 132 THEORY OF STRUCTURES. ficient intensity to be taken into account in practice. In such joints the position of the centre of resistance may be varied within certain limits. III. Fastened joints, at which, by means of some strong cement, or of bolts, rivets, or other fastenings, two pieces are so connected that the joint fixes their relative angular position, and is capable of resisting a pull as well as a thrust. In this case, the centre of resistance may be at any distance from the centre of the joint ; and there may even be no centre of resistance, when the resultant of the stress at the joint is a couple, as explained in Articles 91, 92, and 93. It is obvious that the effect of a joint thus cemented or fastened is to make the two pieces which it connects act as one piece, and that the resistance which it is capable of exerting is a question not of stability but of strength. SECTION 1. Equilibrium and Stability of Frames. 137. Frame is here used to denote a structure composed of bars, rods, links, or cords, attached together or supported by joints of the first class described in the last Article, the centre of resistance being at the middle of each joint, and the line of resistance, con- sequently, a polygon whose angles are at the centres of the joints. The condition of a single bar will be considered first, then that of a combination of two bars, then of three bars, and then of any number. 138. Tie. Let fig. 64 represent a single bar of a frame, L the centre of resistance where the load is ap- plied, and S the centre of resistance where the support- ing force is applied ; so that the straight line L S is the " line of resistance." The bar is represented as being straight itself, that being the figure which connects the points L and S, and gives adequate stiffness and strength, with the least ex- penditure of material. But the bar may, consistently with the principles of this Article, be of any other figure connecting those two points, provided it is sufficiently strong and stiff to prevent their distance from altering to an extent inconsistent with the purposes of the structure. The condition of the bar is the same with that of the solid in Article 23; and it is obvious that the load P, and the supporting resistance R, must be equal and directly opposed, and must both act along the line of resistance L S. In the present case those forces are supposed to be directed out- ward, or from each other. The bar between L and S is in a state of tension, and the stress exerted between any two divisions of it is a pull, equal and opposite to the loading and supporting forces. A BEAM UNDER PARALLEL FORCES. 133 bar in this condition is called a tie. It is obvious that a rope or chain will answer the purpose of a tie. Tlie equilibrium of a tie is stable; for if its angular position be deviated, the equal forces P and R, which originally were directly opposed, now constitute a couple tending to restore the tie to its original position. 139. strut. If the equal and opposite forces applied to the two ends, L and S, of the line of resistance of a bar be direct- ed (as in fig. 65) inwards, or towards each other, the bar, be- tween L and S, is in a state of compression, and the stress exerted between any two divisions of it is a thrust equal and opposite to the loading and supporting forces. It is obvious that a flexible body will not answer the purpose of a strut. The equilibrium of a moveable strut is unstable; for if its angular position be deviated, the equal forces P and R, which originally were directly opposed, now constitute a j>j gg couple tending to make it deviate still farther from its original position. In order that a strut may have stability, its ends must be pre- vented from deviating laterally. Pieces connected with the ends of a strut for this purpose are called stays. 140. Treatment of the Weight of a Bar. In the two preceding Articles, the weight of the bar itself has not been taken into ac- count. But the principles of those Articles, so far as tliey relate to the equilibrium of the bar as a whole, continue to be applicable when the weight of the bar is treated in the following manner. Resolve that weight, by the principles of Articles 39 and 40, into two paral- lel components, acting through L and S respectively. Let P now represent not merely the external load, but the resultant of that load, and of the component of the weight which acts through L. Let R represent not merely the supporting force, but the resultant of that force and of the component of the weight which acts through S. Then P and R, as before, must be equal and directly opposed. In many cases, the weight of a strut or tie is too small as com- pared with the load applied to it to require to be specially con- sidered in practice. 141. Beam under Parallel Forces. A bar supported at two points, and loaded in a direction perpendicular or oblique to its length is called a beam. In the first place, let the supporting pressures be parallel to each other and to the K* direction of the load ; and let the load act ^ ** between the points of support, as in fig. 66 ; -||A T where P represents the resultant of the gross |p \, load, including the weight of the beam itself, L, the point where the line of action of that Fi S- 66 - 134 THEORY OF STRUCTURES. resultant intersects the axis of the beam, R R 2 , the two sup- porting pressures or resistances of the props parallel to, and in the same plane with P, and acting through the points S 1; S 2 , in the axis of the beam. Then, according to the Theorem of Article 39, each of those three forces is proportional to the distance between the lines of action of the other two ; and the load is equal to the sum of the two supporting pressures ; that is to say, P : R, : R 2 :: S^ : TW 2 : H^ ; (1.) and P = R! -f R 2 ..* (2.) Next, let the load act beyond the points of support, as in fig. 67, which represents a canti- lever or projecting beam, held up by a wall or other prop at S,, held down by a notch in a mass of masonry or otherwise at S 2 , and loaded so that P is the resultant of the load, including the Fig. 67. weight of the beam. Then the proportional equation (1) remains exactly as before ; but the load is equal to the difference of the supporting pressures ; that is to say, P = BI- R 2 (3.) In these examples the- beam is represented as horizontal; but the same principles would hold if it were inclined ; for the proportions amongst the distances between parallel lines in the same plane are the same, whether they be measured in a direction perpendicular or oblique to those lines. 142. Beam under Inclined Forces. Let the directions of the supporting forces R 1? R 2 , be now inclined to that of the resultant of the load, P, as in figi^8. This case is that of the equili- brium-oi three forces treated of in Articles 51 and 52; and consequently the following principles apply to it. I. The lines of action of the supporting forces and of the resultant of the load must Fi S- 68 - be in one plane. II. They must intersect in one point (C, fig. 68). III. Those three forces must be proportional to the three sides of triangle A, respectively parallel to their directions; or in other words, to the sides and diagonal of a parallelogram. PROBLEM. Given the resultant of the load in magnitude and position, P, the line of action of one of the supporting forces, R 1? and the centre of resistance of the other, S 2 ; required the line of action of the second supporting force, and the magnitudes of both. LOAD SUPPORTED BY THREE FORCES. 135 Produce the line of action of R till it cuts the line of action of P at the point C ; join C S 2 ; this will be the line of action of R 2 ; construct a triangle A with its sides respectively parallel to those three lines of action ; the ratios of the sides of that triangle will give the ratios of the forces. Q. E. I. To express this algebraically, let * i 2 , be the angles made by the lines of action of the supporting forces with that of the resultant of the load ; then because each side of a triangle is proportional to the sine of the angle between the other two, P : R! : R 2 : : sin (^ + i 2 ) : sin i 2 : sin ^. 143. Load supported by Three Parallel Forces THEOREM. If four parallel forces balance each other, let their lines of action be inter- sected by a plane, and let the four points of intersection be joined by six straight lines so as to form four triangles; each force mill be pro- portional to the area of the triangle whose angles are in the lines of action of the other three. In fig. 69, let the plane of the paper represent the plane which is cut by the lines of action of the four forces s in the points L, Sj, S 2 , S 3 ; let P, R l5 R 2 , B-s, denote the four parallel forces. Join the four points by six lines as in the figure, and pro- duce each of the three lines S L till it cuts the opposite line S S in one of the points B. Because the forces balance each other, the resultant of R 2 and R 3 , whose magnitude is Fig. 69. R 2 + R 3 , must traverse B x ; and because the resultant of that resultant and R x is equal and opposite to P, we must have the following proportion : P : R, : : Si B! : L B, : : A S t S 2 S 3 : A S 2 L S 3 ; and applying the same reasoning to the forces R 2 , R 3 , we find the proportions, P : R! : R 2 : R 3 : : A S x S, S 3 : A S 2 L S 3 : A S 3 L Si : A Sj L S 2 . Q. E. D. By the aid of this Theorem may be determined the proportion in which the load of a given body is distributed amongst three props, exerting parallel supporting forces. 144. Load supported by Three Inclined Forces. The case of a load supported by three inclined forces is that considered in Articles 54 and 56. The lines of action of the three supporting forces must intersect that of the load in one point ; and the magnitudes of the three supporting forces are represented by the three edges of a parallelepiped, whose diagonal represents the load. 136 THEORY OF STRUCTURES. 145. Frame of Two Bars Equilibrium. PROBLEM. Figures 70, 71, arid 72 represent three cases in which a frame consisting of two Fig. 70. Fig. 71. Fig. 72. bars, jointed to each other at the point L, is loaded at that point with a given force, P, and is supported by the connection of the bars at their farther extremities, S 1? S 2 , with fixed bodies. It is required to find the stress on each bar, and the supporting forces at S x and S 2 . Resolve the load P (as in Article 55) into two components, E^, R 2 , acting along the respective lines of resistance of the two bars. Those components are the loads borne by the two bars respectively ; to which loads the supporting forces at S b S 2 , are equal and directly opposed. Q. E. I. The symbolical expression of this solution is as follows : let i lf i 2 , be the respective angles made by the lines of resistance of the bars with the line of action of the load ; then P : Rj : R 2 : : sin (^ + * 2 ) : sin i z : sin i^. The inward or outward direction of the forces acting along each bar indicates that the stress is a thrust or a pull, and the bar a strut or a tie, as the case may be. Fig. 70 represents the case of two ties ; fig. 7 1 that of two struts (such as a pair of rafters abutting against two walls); fig. 72 that of a strut, L Sj, and a tie, L S 2 (such as the gib and the tie-rod of a crane). 146. Frame of Two Bars Stability. A frame of two bars is stable as regards deviations in the plane of its lines of resistance. With respect to lateral deviations of angular position, in a direction perpendicular to that plane, a frame of two ties is stable ; so also is a frame consisting of a strut and a tie, when the direction of the load inclines from the line S, S 2 , joining the points of support. A frame consisting of a strut and a tie, when the direction of the load inclines towards the line Sj S 2 , and a frame of two struts in all cases, are unstable laterally, unless provided with lateral stays. These principles are true of any pair of adjacent bars whose farther cerdres of resistance are fixed; whether forming a frame by them- selves, or a part of a more complex frame. 147. Treatment of Distributed Loads. Before applying the prin- ciples of Article 145, or those of the following Articles, to frames in which the load, whether external or arising from the weight of TRIANGULAR FRAME. 137 the batons, is distributed over their length, it is necessary to reduce that distributed load to an equivalent load, or series of loads, applied at the centres of resistance. The steps in this process are as follows : I. Find the resultant load on each single bar. II. Resolve that load, as in Article 141, into two parallel compo- nents acting through the centres of resistance at the two ends of the bar. III. At each centre of resistance where two bars meet, combine the component loads due to the loads on the two bars into one resultant, which is to be considered as the total load acting through that centre of resistance. IV. When a centre of resistance is also a point of support, the component load acting through it, as found by step II. of the pro- cess, is to be left out of consideration until the supporting force required by the system of loads at the other joints has been deter- mined; with this supporting force is to be compounded a force equal and opposite to the component load acting directly through the point of support, and the resultant will be the total supporting force. In the following Articles of this section, all the frames will be supposed to be loaded only at those centres of resistance which are not points of support ; and therefore, in those cases in which components of the load act directly through the points of support also, forces equal and opposite to such components must be com- bined with the supporting forces as determined in the following Articles, in order to complete the solution. 148. Triangular Frame. Let fig. 73 represent a triangular frame, consisting of the three bars A, B, C, con- / nected at the three joints 1, 2, 3, viz. : C and A at 1, A and B at 2, B and C at 3. Let a load P l be applied at the joint 1 in any given direction ; let supporting forces, P 2 , P 3 , be applied at the joints FJ 73 2, 3 the lines of action of those two forces must be in the same plane with that of Pj, and must either be parallel to it or intersect it in one point. The latter case is taken first, because its solution comprehends that of the former. The three external forces, in virtue of Article 131, condition I., balance each other, and are therefore proportional to the three sides of a tri- angle respectively parallel to their directions. In fig. 73* let A B C be such a triangle, in which CA represents P,, AB ... P 2 , BG ... P 3 , Fig. 73*. Then by the conditions of equilibrium of a frame of two bars (Article 145), the external force P 1 applied at the joint 1, and the 138 THEORY OF STRUCTURES. resistances or stresses along the bars C and A which meet at that joint, are represented in magnitude by the sides of a triangle re- spectively parallel to their directions. Therefore, in fig. 73*, draw C O parallel to the bar C, and A O parallel to the bar A, meeting in the point O, and those two lines will represent the stresses on the bars C and A respectively. In the same manner it is proved, that B O represents the stress on the bar B. The three lines C O, A O, BO, meet in one point O, because the components along the line of direction of a given bar, of the external forces applied at its two extremities, are equal and directly opposed. Hence follows the following THEOREM. If three forces be represented by the three sides of a triangle, and if three straight lines radiating from one point be drawn to the three angles of that triangle, then a triangular frame whose lines of resistance are parallel to the three radiating lines will be in equilibria under the three given forces, each force being applied to the joint where the two lines of resistance meet, which are parallel to the radiating lines contiguous to that side of the original triangle which represents the force in question. Also, the lengths of the three radiating lines will represent the stresses on the bars to which they are respectively parallel. 149. Triangular Frame under Parallel Forces. When the three external forces are parallel to each other, the triangle of forces A B C of fig. 73* becomes a straight line C A, as in fig. 74*, divided into two segments by the point B. Let straight lines radiate from O to A, B, C ; and let fig. 74 represent a triangular frame whose sides 1 2 or A, 2 3 or B, 3 1 or C, are respectively parallel to O A, O B, O C ; then if the joad CA be applied at 1 (fig. 74), AB applied at 2, and B C applied at 3, are the supporting forces required to balance it ; and the radiating lines O A, OB, O C, represent the stresses on the bars A, B, C, respectively. From O let fall O H perpendicular to C A, the com- ^ mon direction of the external forces. Then that line will represent a component of the stress, which is of equal amount in each bar. When C A, as is usually the case, is vertical, O H is horizontal ', and the force represented by it is called the " horizontal thrust " of the frame. Horizontal Stress or Resistance would be a more precise term ; because the force in question is a pull in some parts of the frame, and a thrust in others. In fig. 74, A and C are struts, and B a tie. If the frame were POLYGONAL FRAME. 139 exactly inverted, all the forces would bear the same proportions to each other ; but A and C would be ties, and B a strut. The trigonometrical expression of the relations amongst the forces acting in a triangular frame, under parallel forces, is as follows : Let a, b, c, denote the respective angles of inclination of the bars A, B, C, to the line O H (that is, in general, to a horizontal line). Then, Load CA = Oil (tan c :z tan a) ; 1 Supporting f AB = 001 (tan a =p tan b) ; \- (1.) Forces j BC = OH (tan b == tan c) ; j Th * f + ) is to be used when the two ) opposite directions I J inclinations are in J the same direction. O A = O H Stresses sec a sec b .(2.) O C = O H sec c GA ~ tan c t tan a ' ' 150. Polygonal Frame Equilibrium. The Theorem of A.rticle 148 is the simplest case of a general theorem respecting polygonal frames consisting of any number of bars, which is arrived at in the fol- lowing manner. In fig. 75, let A, B, C, D, E, be the lines of resistance of the bars of a polygonal frame, connected together at the joints, whose centres of resistance are, 1 between A and B, 2 between B and C, 3 between C and D, 4 between D and E, and 5 between E and A. In the figure, the frame consists of five bars; but the demonstra- tion is applicable to any number. From a point O, in fig. 75* (which may be called the Diagram of Forces), draw radiating lines OA, OB, OC, OD, O~E, parallel respectively to the lines of resistance of the bars; and on those radiating lines take any lengths whatsoever, to represent the stresses on the several bars, which may have any magnitudes within the limits of strength of the material. Join the points thus found by straight lines, so as to form a closed polygon A B C D E A ; then E it is evident that A B is the ex- Fig- 75. Fig. 75* 140 THEORY OF STRUCTURES. ternal force, which being applied at the joint 1 of A and B, will produce the stress A on A and OB on B ; that BO is the external force which being applied at the joint 2 of B and C, will produce the stress O B on B (already mentioned) and on ; and so on for all the sides of the polygon of forces A B C D E A. Hence follows this THEOREM. If lines radiating from a point be drawn parallel to the lines of resistance oftlie bars of a polygonal frame, then the sides of any polygon whose angles lie in those radiating lines will represent a system of forces, which, being applied to the joints of the frame, will balance each other ; each such force being applied to the joint between the bars whose lines of resistance are parallel to the pair of radiating lines that enclose the side of the polygon of forces, representing the force in question. A Iso, the lengths of the radiating lines will represent the stresses along the bars to whose lines of resistance they are respectively parallel. 151. Open Polygonal Frame. When the polygonal frame, instead of being closed, as in fig. 75, is converted into an OPEN frame, by the omission of one bar, such as E, the corresponding modification is made in the diagram of forces by omitting the lines O E, D E, E A. Then the polygon of external forces becomes A B D A ; and D O and O A represent the supporting forces respectively, equal and directly opposed to the stresses along the extreme bars of the frame, D and A, which must be exerted by the foundations (called in this case abutments), at the points 4 and 5, against the ends of those bars, in order to maintain the equilibrium. 152. Polygonal Frame Stability. The stability or instability of a polygonal frame depends on the principles already stated in Articles 138 and 139, viz., that if a bar be free to change its angular position, then if it is a tie it is stable, and if a strut, unstable ; and that a strut may be rendered stable by fixing its ends. For example, in the frame of fig. 75, E is a tie, and stable ; A, B, C, and D, are struts, free to change their angular position, and therefore unstable. But these struts may be rendered stable in the plane of the frame by means of stays ; for example, let two stay-bars connect the joints 1 with 4, and 3 with 5 ; then the points 1, 2, and 3, are all fixed, so that none of the struts can change their angular positions. The same effect might be produced by two stay-bars connecting the joint 2 with 5 and 4. The frame, as a whole, is unstable, as being liable to overturn laterally, unless provided with lateral stays, connecting its joints with fixed points. POLYGONAL FRAME UNDER PARALLEL FORCES. 141 Now, suppose the frame to be exactly inverted, the loads at 1, 2, and 3, and the supporting forces at 4 and 5, being the same as before. Then E becomes a strut ; but it is stable, because its ends are fixed in position ; and A, B, C, and D become ties, and are stable without being stayed. An open polygon consisting of ties, such as is formed by A, B, C, and D when inverted, is called by mathematicians a funicular poly- gon, because it may be made of ropes. It is to be observed, that the stability of an unstated polygon of ties is of the kind described in Article 127, and admits of oscillation to and fro about the position of equilibrium. This oscillation may be injurious in practice, and stays may be required to prevent it. 153. Polygonal Frame under Parallel Forces. When the external forces are parallel to each other, the polygon of forces of fig. 75* becomes a straight line A D, as in fig. 75**, divided into segments by the radiating lines ; and each segment represents the external force which acts at the joint of the bars whose lines of resistance are parallel to the radiating lines that bound the segment. Moreover, the seg- ment of the straight AD which is intercepted be- tween the radiating lines parallel to the lines of resistance of any two bars whether contiguous or not, represents the resultant of the external forces which F] S- 7 act at points between the bars. Thus, AD represents the total load, consisting of the three por- tions AB, B C, CD, applied at 1, 2, 3 respectively. DA represents the total supporting force, equal and opposite to the load, consist- ing of the two portions D E, E A, applied at 4 and 5 respectively. A C represents the resultant of the load applied between the bars A and C ; and similarly for any other pair of bars. From draw O H perpendicular to A D ; then that line re- presents a component of the stress, whose amount is the same in each bar of the frame. When the load, as is usually the case, is vertical, that component is called the "Jiorizontal thrust" of the frame, and, as in Article 149, might more correctly be called Jiori- zontal stress or resistance, seeing that it is a pull in some of the bars and a thrust in others. The trigonometrical expression of these principles is as follows: Let the force O H be denoted simply by H. Let i, i f , denote the inclinations to O H of the lines of resistance of any two bars, contiguous or not. Let E, B,', be the respective stresses which act along those bars. 142 THEORY OF STRUCTURES. Let P be the resultant of the external forces acting through the joint or joints between those two bars, Then B, = H sec i ; R' = H sec H ; P = H (tan i *~ tan i"). m, r sum ) of the tangents of the inclinations is ( opposite ) \ difference J to be used according as they are [ similar j ' 154. Open Polygonal Frame under Parallel Forces. When the frame becomes an open polygon by the omission of the bar E, the diagram of forces 75** is modified by omitting the line O E. Then the supporting forces exerted by the abutments at 4 and 5, are no longer represented by the segments I) E and E A of the line A D, but by the inclined lines D O and OA, equal and directly opposed respectively to the stresses along the extreme bars of the frame, D and A. Let i d andj a Denote the angles of inclination of those bars. Let R,,, = P and R a = A be the stresses along them. Let 2 P = A D denote the total load on the frame. Then by the equations of Article 153, tani/ K d = H sec i d ; R a = H sec i a . 155. Bracing of Frames. A brace is a stay-bar on which there is a permanent stress. When the external forces applied to a poly- gonal frame, although balancing each other as an entire system, are distributed in a manner not consistent with the equilibrium of each bar separately, then by connecting two or more joints together by means of braces, which may be either struts or ties, the resistances of those braces may be made to supply, at the joints which they connect, the forces wanting to produce equilibrium of each bar. The resistance of a brace introduces a pair of equal and opposite forces, acting along the line of resistance of the brace, upon the pair of joints which it connects. It therefore does not alter the resultant of the forces applied to that pair of joints in amount nor in position ; but only the distribution of the components of that resultant on the pair of joints. The same remark applies to any number of joints connected by a system of braces. To exemplify the use of braces and the mode of determining the stresses on them, let fig. 76 represent a frame such as frequently occurs in iron roofs, consisting of two struts or rafters, A and E, and three tie-bars, B, C, and D, forming a polygon of five sides, jointed at 1, 2, 3, 4, 5, loaded vertically at 1, and supported by the vertical resistance of a pair of walls at 2 and 5. The joints 3 and BRACING OF FRAMES. 143 4, having no loads applied to them, are connected with 1 by the those Fig. 76. braces 1 4 and 1 3. It is required to find the stresses on braces, and on the other pieces of the frame. To make the diagram of forces (fig. 76*), draw as before from a point O, radiating lines O E, O D, O C, OB, O A, parallel to the bars forming the external polygon of the frame ; and across those lines draw the vertical line E A, as in Article 153, to represent the direction of the load and of the supporting forces. The two extreme segments of this line A7B and TTE, are to be taken to represent the supporting forces at 2 and 5 ; and their sum Hi i) + B A, will represent the actual load at 1. E A represents what the load at 1 ought to be, to fulfil the conditions of equilibrium of the bars A and E ; therefore D B represents the deficiency of the actual load at 1, or the additional downward force, to be supplied at 1 by means of the bracing. Also B D, equal and opposite to D B, represents the resultant deficiency of supporting pressure at the joints 3 and 4, where the actual external force applied is nothing ; and this has to be supplied by means of the bracing. In the diagram of forces, draw D C parallel to the brace 1 4, and G B parallel to the brace 1 3, intersecting the radiating line O C in C. Then will D C represent the pull along the brace 1 4, acting obliquely upwards at 4, and obliquely downwards at 1 ; and CB will represent the pull along the brace 1 3, acting obliquely upwards at 3, and obliquely downwards at 1. The resultant of the additional forces thus applied at 1 is D B, a vertical downward force, equal to the deficiency before-mentioned ; and the oblique upward forces at 3 and 4 complete a polygon of forces A B C D E A, whose angles are on the radiating lines parallel to the bars of the frame, and which therefore fulfils the conditions of the Theorem of Article 150. The stresses on the external bars of the frame are represented as before by the lengths of the radiating lines in fig. 76*. The method of arranging the positions of braces, and determining the stresses along them, of which an example has been given, may be thus described in general terms. 144 THEORY OF STRUCTURES. If the distribution of the loads on the joints of a polygonal frame, though consistent with its equilibrium as a whole, be not consistent with the equilibrium of each bar, then, in the diagram of forces, upon attempting to construct a polygon of external forces, having its angles on the radiating lines, gaps will be left in the outline of that polygon. The lines necessary to fill up those gaps will indicate the forces to be supplied by means of the resistance of braces. 156. Rigidity of a Truss. The word truss is applied in carpentry and iron framing to a triangular frame, and to a polygonal frame to which rigidity is given by staying and bracing, so that its figure shall be incapable of alteration by turning of the bars about their joints. If each joint were absolutely of the kind described as the first class in Article 136, that is, like a hinge, incapable of offering any resistance to alteration of the relative angular position of the bars connected by it, it would be necessary, in order to fulfil the condition of rigidity, that every polygonal frame should be divided by the lines of resistance of stays and braces into triangles and other polygons so arranged, that every polygon of four or more sides should be surrounded by triangles on all but two sides and the included angle at farthest. For every unstayed polygon of four sides or more, with flexible joints, is flexible, unless all the angles except one be fixed by being connected with triangles. Sometimes, however, a certain amount of stiffness in the joints of a frame, and sometimes the resistance of its bars to bending, is relied upon to give rigidity to the frame, when the load upon it is sub- ject to small variations only in its mode of distribution. For example, in the truss of fig. 81 (for which see Article 161, farther on), the tie-beam A A is made in one piece, or in two or more pieces, so connected together as to act like one piece ; and part of its weight is suspended from the joints C, C, by the rods C B, C B. These rods also serve to make the resistance of the tie-beam C C to being bent, act so as to prevent the struts AC, C C, C A, from deviating from their proper angular positions, by turning on the joints A, C, C, A. If A B, B B, and B A, were three distinct pieces, with flexible joints at B, B, it is evident that the frame might be disfigured by distortion of the quadrangle B C C B. 157. Variations of Load on Truss The object of stiffening a truss by braces is to enable it to sustain loads variously distributed ; for were the load always distributed in one way, a frame might be designed of a figure exactly suited to that load, so that there should be no need of bracing. The variations of load produce variations of stress on all the pieces of the frame, but especially on the braces ; and each piece must be suited to withstand the greatest stress to which it is liable. Some pieces, and especially braces, may have to act sometimes as SECONDARY TRUSSING. 145 struts and sometimes as ties, according to the mode of distribution of the load. 158. Bar common to several Frames. When the same bar forms at the same time part of two or more different frames, the stress along it is determined by the aid of the following THEOREM. The stress on a bar common to two or more frames, is the resultant of the different stresses to which it is subject, in virtue of its position in the different frames. Illustrations of this will be found in the following Articles. 159. Secondary Trussing. A secondary truss is a truss which is supported by another truss. When a load is distributed over a great number of centres of resistance, it may be advantageous, instead of connecting all those centres by one polygonal frame, to sustain them by means of several small trusses, which are supported by larger trusses, and so on, the whole structure of secondary trusses resting finally on one large truss, which may be called the primary truss. In such a combina- tion, the same piece may often form part of different trusses ; and then the stress upon it is to be determined according to the Theorem of Article 158. Example I. Fig. 77 represents a kind of secondary trussing com- mon in the framework of iron roofs. Fig. 77. The entire frame is supported by pillars at 2 and 3, each of which sustains in all, half the weight. 1 2 3 is the primary truss, consisting of two rafters 1 3, 1 2, and a tie-rod 2 3. The weight of a division of the roof is distributed over the rafters. The middle point of each rafter is supported by a secondary truss; one of those is marked 143; it consists of a strut, 1 3 (the rafter itself), two ties 4 1, 4 3, and a strut-brace, 5 4, for transmitting the load, applied at 5, to the point where the ties meet. Each of the two larger secondary trusses just described supports two smaller secondary trusses of similar form and construction to itself; two of those are marked 1 7 5, 5 6 3 ; and the subdivision of the load might be carried still farther. In determining the stresses on the pieces of this structure, it is indifferent, so far as mathematical accuracy is concerned, whether we 146 THEORY OF STRUCTURES. commence with the primary truss or with the secondary trusses ; but by commencing with the primary truss, the process is rendered more simple. (1.) Primary Truss 123. Let W denote the weight of the roof; then i W is distributed over each rafter, the resultants acting through the middle points of the rafters. Divide each of those resultants into two equal and parallel components, each equal to ^ W, acting through the ends of the rafter ; then ^ W is to be considered as directly supported at 3, ^ W at 2, and J W + -J W = i W at 1 ; therefore the load at the joint 1 is P = 1W. Let i be the inclination of the rafters to the horizon ; then by the equations of Article 149 2 tan i 4 tan i' This is the pull upon the horizontal tie-rod of the primary truss, 2 3 ; and the thrust on each of the rafters 1 3, 1 2, is given by the equation . W cosed R = H sec ^ = j (2.) (2.) Secondary Truss 1435. The rafter 1 3 has the load W distributed over it; and reasoning as before, we are to leave two quarters of this out of the calculation, as being directly supported at 1 and 4, and to consider one-half, or \ W, as being the vertical load at the point 5. The truss is to be considered as consisting of a polygon of four pieces, 5 1, 1 4, 4 3, 3 5, two of which happen to be in the same straight line, and of the strut-brace, 5 4, which exerts obliquely upwards against 5, and obliquely downwards against 4, a thrust equal to the component perpendicular to the rafter of the load \ W; which thrust is given by the equation Then we easily obtain the following values of the stresses on the rafter and ties, in which each stress is distinguished by having affixed to the letter R the numbers denoting the two joints between which it acts. Pulls on ties v ( R.. 1_ 1_ W Thrusts [B" = 5h+ Sprint = 1 W cosec ^ On 1 I R 6 SECONDARY TRUSSING. 147 The difference between the thrusts on the two divisions of the rafter, is the component along the rafter of the load at the point 5. (3.) Smaller Secondary Trusses, 175, 56 3. These trusses are similar in every respect to the larger secondary trusses, except that the load on each point is one-half, and consequently each of the stresses is reduced to one-half of the corresponding stress in the equations 3 and 4. (4.) Resultant Stresses. The pull on the middle division of the great tie-rod 2 3 is simply that due to the primary truss, 123. The pull on the tie 4 7 is simply that due to the secondary truss 143. The pulls on the ties 57, 56, are simply those due to the smaller secondary trusses, 157,563. But agreeably to the Theorem of Art. 158, the pull on the tie 1 7 is the sum of those due to the larger secondary truss 143, and the smaller secondary truss 175. The pull on 6 4 is the sum of those due to the primary truss 123 and to the larger secondary truss 143. The pull on 6 3 is the sum of those due to the primary truss 1 2 3, to the larger secondary truss 143, and to the smaller secondary truss 563. The thrust on each of the four divisions of the rafter 1 3, is the sum of three thrusts, due re- spectively to the primary truss, the larger secondary truss, and one or other of the smaller secondary trusses. Example II. Fig. 78 represents another form of truss common in roofs. Let W be the weight of the roof, as before, distributed over the rafters 1 2, 1 3. 2 3 is the great tie-rod ; 1 7, 6 5, 8 9, suspension- rods; 7 6, 7 8, 5 4, 9 10, struts. (1.) Primary Truss 123. The load at 1, as before, is to be taken as - i W. (2.) Secondary Trusses 7 6 3, 7 8 2. The load at 6 is to be held to consist of one-half of the load between 6 and 1 , and one-half of the load between 6 and 3 ; that is, one-half of the load between 1 and 3, or i W. The trusses are triangular, each consisting of two struts and a tie, and the stresses are to be found as in Article 149. The suspension-rod 1 7 supports two- thirds of the load on 7 6 3, and two-thirds of the load on 7 8 2 ; that is, I I W = W; and 148 THEORY OF STRUCTURES. this, together with W which rests directly on 1, makes up the load of I W, already mentioned. (3.) Smaller Secondary Trusses 3 4 5, 9 102. Each of the points 4 and 10 sustains a load of \ W, from which the stresses on the bars of those smaller trusses can be determined. One-half of the load on 4, that is TI W, hangs by the suspension- rod 65; and this, together with -g- W, which rests directly on 6, makes up the load of T W on that point, formerly mentioned. The same remarks apply to the suspension-rod 8 9. (4.) Resultant Stresses. The pull between 5 and 9 is the sum of those due to the primary and larger secondary trusses ; that between 5 and 3, and between 9 and 2, is the sum of the pulls due to the primary, larger secondary, and smaller secondary trusses. The thrust on 1 6 is due to the primary truss alone ; that on 6 4 to the primary and larger secondary truss ; that on 4 3 to the primary, larger secondary, and smaller secondary trusses ; and similarly for the divisions of the other rafter. Example III. Suppose that instead of only three divisions, there are n divisions in each of the rafters 1 3, 1 2, of fig. 78 ; so that be- sides the middle suspension-rod 1 7, there are n 2 suspension-rods under each rafter, or 2 n 4 in all ; and n 1 sloping struts under each rafter, or 2 n 2 in all. There will thus be 2 n 1 centres of resistance; that is, the ridge-joint 1, and n 1 on each rafter ; and the load directly supported on each of these W points will be ^. 2n W The total load on the ridge-joint, 1, will be as before, ^ ; that "W W / 1\ is to say, directly supported, and -- ( 1 ) hung by the middle suspension-rod. The total load on the upper joint of any secondary truss, distant from the ridge-joint by m divisions of the rafter, will be, - W; that is to say, directly supported, and ~ ^ nun g by a suspension-rod. The stresses on the struts and tie of each truss, primary and secondary, being determined as in Article 149, are to be combined as in the preceding examples. 160. Compound Trusses. Several frames, without being distin- guishable into primary and secondary, may be combined and con- COMPOUND TRUSSES. 149 nected in such a manner, that certain pieces are common to two or more of them, and require to have their stresses determined by the Theorem of Article 158. Example I. In fig. 79, 8 9 represents part of the horizontal plat- form of a suspension bridge, supported and balanced by being hung from the top of a central pier, 1, by pairs of equally inclined rods or ropes, viz. : 1 8 and 1 9 ; 1 6 and 1 7 j 1 4 and 1 5 1 2 and 1 3. Fig. 79. Here 8 1 9 is to be considered as a distinct triangular frame, consisting of a strut 8 9, and two ties 1 8 and 1 9, loaded with equal weights at 8 and 9, and supported at 1. Let x denote the height of the point of suspension 1 above the level of the loaded points, 2/ 8 = 2/9, the distance of those points on either side of the middle of the pier, P the load at each point, R 8 = B^ the pull on each of the ties, 1 8, 1 9, T 89 the thrust between 8 and 9 along the platform. Then we have and similar equations for each of the other distinct frames 617, 4 1 5, 2 1 3. Then using a similar notation in each case, the thrust along the platform between 8 and 6 | . T ^ 7 and 9 / lfc l89 + 67 6 and 4 ) . T T T 5 and 7 f ls9 + 67 + ^ B) and so on for as many pairs of divisions as the platform consists of. Example II. Fig. 80 represents the framework for supporting one side of a timber bridge, resting on two piers at 1 and 4. consists of four distinct trusses, viz., It 150 THEORY OF STRUCTURES. 1234 loaded at 2 and 3, 1564 5 6, 1784 7 8, 194 9 but all those trusses have the same tie-beam, 1 4 ; and the pull along that tie-beam is the sum of the pulls due to the four trusses. 161. Resistance of Frame at a Section. THEOREM. If a frame be acted upon by any system of external forces, and if that frame be conceived to be completely divided into two parts by an ideal surface, the stresses along the bars which are intersected by that surface, balance the external forces which act on each of the two parts of the frame. This theorem, which requires no demonstration, furnishes in some cases the most convenient method of determining the stresses along the pieces of a frame. The following consideration shows to what extent its use is limited. CASE 1. When the lines of resistance of the bars, and the lines of action of the external forces, are all in one plane, let the frame be supposed to be intersected anywhere by a plane at right angles to its own plane. Take the line of intersection of these two planes for an axis of co-ordinates ; say for the axis of y t and any convenient point in it for the origin O ; let the axis of x be perpendicular to this, and in the plane of the frame, and the axis of z perpendicular to both, and in the plane of section. The external forces applied to the part of the frame at one side of the plane of section (either may be chosen) being treated as in Article 59, give three data, viz., the total force along x = F,; the total force along y = F y , and the moment of the couple acting round z = M \ and the bars which are cut by the plane of section must exert resistances capable of balancing those two forces and that couple. If not more than three bars are cut by the plane of section, there are not more than three unknown quantities, and three relations between them and given quantities, so that the problem is determinate ; if more than three bars are cut by the plane of section, the problem is or may be indeterminate. The formulae to which this reasoning leads are as follows : Let x be positive in a direction from the plane of section towards the part of the structure which is considered in determining F,, F y , and M ; let -j- y lie to the right of -j- x when looking from z ; let angles measured from x towards -f- y, that is, towards the right, be positive; and let the lines of 'resistance of the three bars cut by the plane of section make the angles i,, i 2 , i s , with x. Let n l} n 2 , n s , be the perpendicular distances of those three lines of resistance from O, distances towards the "ffl* } of x bein S considered { negative } METHOD OF SECTIONS. 151 Let R b E^, R 3 , be the resistances, or total stresses, along the three bars, pulls being positive, and thrusts negative. Then we have the following three equations : F, = R! cos it + R 2 cos i 2 + R 3 cos 4 ; F y = R! sin ^ -f R 2 sin i. 2 + R 3 sin i 3 ; (1.) - M = R! n + R2 ^2 + RS MS', from which the three quantities sought, R t , R 2 , Rg, can be found. Speaking with reference to the given plane of section, F^. may be called the normal stress, F y the shearing stress, and M the moment of flexure or bending stress; for it tends to bend the frame at the section under consideration. CASE 2. When the bars of the frame, and the forces applied to them, act in any direction, the forces applied to one of the two divisions of the frame are to be reduced to rectangular components ; and the three resultant forces along these rectangular axes, F.,., F y , F,, and the three resultant couples round these three axes, M x , M y , M,, are to be found as in Article 60. Those forces and couples must be equal and opposite to the corresponding forces and couples arising from the stresses along the bars cut by the section ; and thus are obtained six equations between those stresses and known quantities ; so that if the section cuts not more than six bars, the problem is determinate ; if more, it is or may be indeterminate. The equations are obtained as follows : Let R denote the stress along any one of the bars, pull being positive and thrust negative. Let a, /3, y, be the inclinations of the line of resistance of that bar to the axes of x, y, z. Let n be its perpendicular distance from O. Conceive a plane to pass through O and through the line of resistance of the bar, and a normal to be drawn to that plane in such a direc- tion, that looking from the end of that normal towards O, the bar is seen to lie to the right of O, and let A, ^, v, be the angles of inclination of that normal to the three axes. Let s denote the summation of six corresponding quantities for the six bars. Then the six equations are, F^ = 2 R cos * ; F y = 2 R cos ft ; F, = 2 R cos y ; " - M, = 2 R n cos X ; M y = 2 R n cos ^ ; (2). M z = 2 R n cos v from which the six stresses sought can be computed by elimination. The plane of y z being as before, that of the section, F x is the total direct stress on it; F y and F, are the total shearing stresses ; M y and M. are bending couples, and M, a turisting couple. REMARKS. Every problem respecting the equilibrium of frames which can be solved by the method of sections explained in this 152 THEORY OF STRUCTURES. Article, can also be solved by the method of polygons explained in the previous Articles; and the choice between the two methods is a question of convenience and simplicity in each particular case. The following is one of the simplest examples of the solution of a problem in both ways. Fig. 81 represents a truss of a form very c c common in carpentry (already referred to in Article 156), and consisting of three struts, A C, C C, C A, a tie- beam A A, and two suspension-rods, '{jj- C B, C B, which serve to suspend part of the weight of the tie-beam from the joints C C, and also to stiffen the truss in the manner mentioned in Article 156. Let i denote the equal and opposite inclinations of the rafters A C, C A, to the horizontal tie-beam A A ; and leaving out of consideration the portions of the load directly supported at A A, let P, P, denote equal vertical loads applied at C C, and P, - P, equal upward vertical supporting forces applied at A A, by the resistance of the props. Let H denote the pull on the tie- beam, R the thrust on each of the sloping rafters, and T the thrust on the horizontal strut C C. Proceeding by the method of polygons, as in Article 153, we find at once, H = T = P cotan i; ) \ ................ (3.) E, = P cosec i. ) (Thrusts being considered as negative.) To solve the same question by the method of sections, suppose a vertical section to be made by a plane traversing the centre of the right hand joint C ; take that centre for the origin of co-ordinates ; let x be positive towards the right, and y positive downwards ; let a?!, y lt be the co-ordinates of the centre of resistance at the right hand point of support A. When the plane of section traverses the centre of resistance of a joint, we are at liberty to suppose either of the two bars which meet at that joint on opposite sides of the plane of section to be cut by it at an insensible distance from the joint. First, consider the plane of section as cutting C A. The forces and couple acting 011 the part of the frame to the right of the section are Then, observing that for the strut A C, n = 0, and that for the tie A A, n 2/ b we have, by the equations 1 of this Article HALF-LATTICE GIRDER. 153 R, cos i + H = F, = ; R sin i = P ; Hyi = M= whence we obtain, from the last equation, from the first, or from the second TT R = -- : = P cosec i. cos ^ (*) Next, conceive the section to cut C C at an insensible distance to the left of C. Then the equal and opposite applied forces + P at C, and P at A, have to be taken into account ; so that from the first of which equations we obtain H + T = F, = 0, and T = - H - -Pcotani ................... (o.) In the example just given, the method of sections is tedious and complex as compared with the method of polygons, and is intro- duced for the sake of illustration only ; but in the problems which are to follow, the reverse is the case, the solution by the method of sections being by far the more simple. 162. A Haif-JLauice Girder, sometimes called a "Warren Girder," is represented in fig. 82. It consists essentially of a horizontal upper bar, a horizontal lower bar, and a series of diagonal bars sloping alternately in opposite direc- tions, and dividing the space between the upper and lower bars into a series of triangles. In the example to be consi- -p. dered, the girder is supposed to be supported by the vertical resistance of piers at its ends A and B, and loaded with weights acting at or through the joints at the angles of the several triangles. This girder might be treated as a case of secondary trussing, by considering the upper and lower and endmost diagonal bars as forming a polygonal truss like fig. 81, but inverted, supporting a smaller erect truss of the same kind, which supports a still smaller inverted truss, which supports a still smaller erect truss, and so on to the smallest truss, which is the middle triangle. But it is more 154 THEORY OF STRUCTURES. simple to proceed by the method of sections, which must be applied successively to each division of the girder. The load at each joint being known, the two supporting forces at A and B, are to be determined by the principles of the equili- brium of parallel forces in one plane (Articles 43, 44). Let P A , P B , denote those supporting forces, upward forces being treated as positive, and downward as negative and let P denote the load at any joint, which may be a constant or a varying quantity for different joints. Suppose now that it is required to find the stress along any one of the diagonals, such as C E, along the top bar immediately to the right of C, and along the bottom bar immediately to the left of E. Conceive the girder to be divided by a vertical plane of section C D, at an insensibly small distance to the right of C ; take the intersection of this plane with the line of resistance of the top bar for the origin of co-ordinates, which sensibly coincides with C. Let x denote the distance of any one of the joints to the left of the plane of section, from that plane. Let x l be the distance of the point of support A to the left of the same plane. Let y be positive upwards ; so that for the joints of the upper bar, y = 0, and for those of the lower bar, y = h, h denoting the vertical depth between the lines of resistance of the upper and lower bars. Let i be the inclination of the diagonal C E to the horizontal axis of x. In the present instance this is positive ; but had C E sloped the other way, it would have been negative. Let the symbol - s P denote the sum of the loads acting at the joints between the plane of section and the point of support A, the load at the joint C being included. Then for the total forces and couple acting on the division of the girder to the left of the plane of section, we have, direct force, F, 0, because the applied forces are all vertical ; shearing force, F y = P A s P a force which is I n ^tt7or r d l oward } accordin g as the P lane of section lies < ~ , , > the point of support A, than a plane which divides the load into two portions equal respectively to the support- ing pressures; bending couple M P A x l 2^ -Px; which is upward, and right-handed with respect to the axis of z. Now let R! denote the stress along the upper bar at C, R 2 * na * along the lower bar at D, and R 3 that along the diagonal C E ; then the equations 1 of Article 161 become the following : R! + R 3 + R 3 cos i = ; or Rj + R 3 cos i - R 2 ...(a.) that is, the stress along the upper bar, and the horizontal component HALF-LATTICE GIRDER. 155 of the stress along the diagonal, are equal and opposite to the stress along the lower bar ; that is, the vertical component of the stress along the diagonal, balances the shearing force j R, 2 y = R-2 h = M P A x l s P x \ (c.) that is, the couple formed by the equal and opposite horizontal stresses of equation (a), acting at the ends of the arm li, balances the bending couple. Finally, from the equations (a), (b), (c), are deduced the following values of the stresses : Pull on lower bar, j Stress on diagonal, (i-) R 3 = cosec i (P A 2 P) ; Thrust on upper bar, R! = R 2 R 3 cos i = -i (P A x l 2* P x) cotan i (P A 2* P). Another, and sometimes a more convenient form, can be found for the second and third of those expressions. Let s denote the length of the diagonal C E, and a?/ the horizontal distance of its lower end E from the point of support A ; then 8 = V (A 2 + K - Xtf), and also g x' x cosec i = ; cotan i = - ' ' ; (2.) which substitutions having been made, give R, = - 1 | P A!BI ij Px + (x,' x,) (P A i -P) j. (3.) 156 THEORY OF STRUCTURES. in which x' is taken to denote the horizontal distance of any joint to the left of a vertical plane traversing E. The last expression for R! is exactly what would have been obtained by supposing the plane of section to traverse E instead of C. "Any given diagonal is j *^ j according as it slopes j ^^ j the direction of the shearing force F y acting on a plane of section traversing it. 163. Half-Lattice Girder Uniform Load CASE 1. Every joint loaded. When the joints of a half-lattice girder are at equal dis- tances apart horizontally, and loaded with equal weights, the equations take the following form : Let N" denote the even number of divisions into which vertical lines drawn through the joints divide the total length or span between the points of support. Let I be the length of one of these divisions, so that N" I is the total span. The total number of loaded joints is N 1 ; this must be an odd number, and there must be a middle joint dividing the girder into two halves, sym- metrical to each other in every respect, figure, load, support, and stress, so that it is sufficient to consider one half only; let the left hand half be chosen. Let the middle joint be denoted by O, and the other joints by numbers in the order of their distances from the middle joint, so that the joint numbered n shall be at the distance n I from O. The even numbers denote joints on the same horizontal bar with O ; the odd numbers those on the other. The total load on the girder is -(N-l)P, of which one-half is supported on each pier ; that is to say, The stress on the upper bar is everywhere a thrust ;- that on the lower bar a pull. Diagonals which < ,, > from the middle towards the ends are -j , , V . By these principles the kind of stress on each piece is determined ; it remains only to compute the amount. Let n be the number of any joint; it is required to find the stress along the diagonal which runs from that joint towards the middle of the girder, and the stress along that part of either of the hori- zontal bars which is opposite the joint. Suppose a vertical section to be made at an insensible distance HALF-LATTICE GIRDER. 157 from the joint, intersecting the diagonal in question and the hori- zontal bars. 1ST Between O and either pier there are -- 1 loaded joints ; be- Z tween O and the plane of section in question, there are n 1 joints ; hence between the plane of section and the pier there are n joints. Consequently and the shearing force is So that it increases at an uniform rate from the middle towards the ends. The distance of the n th joint from the pier is x l = ( -- n) I. Hence the upward moment of the supporting force is The downward moment of the load at the joints between the plane of section and the pier is found from the consideration, that the leverage of the nearest portion of that load is nothing, and that of the farthest f 1 nj I, so that the mean leverage is 1 /N \ A ( 1 n] I ; which being multiplied by the load 2 P as 2 \ 2i / Q found above, gives for the moment hence the bending couple is M.p.-.-.'p.-i (!+.)(-) - that is to say, it is proportional to the product of t/ie segments into which the plane of section divides the length of the girder, and is N 2 greatest at the middle, where it is P L 158 THEORY OF STRUCTURES. The uniform inclination of the diagonals, in one direction or the other, being denoted by i, we have cosec ^ = = and hence the amounts of the stresses are, Along the diagonal, Along the horizontal bar, M /N 3 \ PI (4. ) These stresses are stated irrespective of their signs, which are to be determined by the rules laid down after equation 1. The least value of R' is for the diagonals next the middle point, sP for which n=I, and R' = -_ . Its greatest value is for the dia- 2 h N CN \}sp gonals next the piers, for which n = , and R' = - , in fact, '2 2/i these diagonals sustain the entire load. The least value of the horizontal stress R is at the divisions of N one of the horizontal bars next the piers, for which n = l, and (N-l)Pj 2h ' The greatest value of R is at the division of one of the horizontal N 2 PI bars opposite the middle joint, for which n = Q, and R = -,. 8 n CASE 2. Every alternate joint loaded. Suppose those joints only to be loaded which are distant by an even number of divisions from N the piers. The total number of loaded joints is -- 1, the load 2 /N \ on the girder - I - -- 1 ) P, and the supporting pressures Let n be the number of any loaded joint, n 1 that of the unloaded joint nearest to it on the side next the middle of the girder, 0. If a plane of section traverse the girder at an insensible HALF-LATTICE GIRDER. 159 distance from either of those joints on the side next O, the shearing force is the same, being the excess of the supporting pressure, P A (equation 5) above the load on n y and the other loaded joints between it and A, whose number is one-half of what it was in N n case 1, that is Hence we find F, = !L-J.P ......................... (6.) The upward moment of the supporting force is at the joint n, P A ^ = ( _ - J ( - n j -PI; at the joint n - 1, P A (^+ I) = (^ - I ) (^ -n + l) PL The downward moment of the load from the joint n inclusive to the pier, relatively to the plane of section near that joint, is found by considering that the leverage of the nearest portion of that load is /N" \ is nothing, and that of the farthest ( 2 n\ I; so that the mean leverage is f 2 n\ I, which being multiplied by the load ( - \ P, gives for the moment, The corresponding moment for the joint n 1 is Hence the bending couples are At the loaded joint n, 1 /N PI, / 4: \ 4: / At the unloaded joint n 1, 160 THEORY OF STRUCTURES. Using these data, we obtain for the stress along the diagonal con- necting the joints n and n 1, R' = F y cosec i = n ~ . S ~ (8.) (The stress along the diagonal connecting the joints n 1 and n 2 is of equal amount and opposite kind). Along the bar opposite the loaded joint n, R - 1 *L = 'L('?- n*\ ...(9.) Along the bar opposite the unloaded joint n 1, El M, !{N:_ }^. h -i ( 4 ) fi The last two stresses are of opposite kinds ; and the kind of each stress is to be determined, as before, by the rule given after equa- tion 1 of this Article. 164. Lattice Girder Any r,oad. In a lattice girder, as in a half- lattice girder, there are a hori- zontal upper and lower bar; N/N/N/N/N/N/^V^N/I ^ u * wnereas a half-lattice girder /\/V/\/\/\X\/\/\l B contains but one zig-zag set of Hjj {| ) diagonal bars, a lattice girder j \ contains two or more sets, cross- Fig. 83. ing each other, usually at equal inclinations to the horizon. Fig. 83 represents the simplest form of a lattice girder, in which there are two sets of diagonals, crossing each other midway between the upper and lower horizontal bars. The load is supposed to be applied at the joints. Suppose the girder to be cut by a vertical plane of section C D, traversing one of the joints where the diagonals cross. The shearing force and bending couple at this plane of section are to be deter- mined exactly in the same manner as for a half-lattice girder, in Article 162. In the present case, because the plane of section C D cuts four bars, the problem, in a strict mathematical sense, is indeterminate, according to the principles stated in. Article 161 ; but it is solved by taking for granted what is the fact in well-constructed lattice girders, that each of the two diagonals which cross each other at the section C D bears one-half of the shearing force ; and in like manner, when several pairs of diagonals cross each other at the LATTICE GIRDER UNIFORM LOAD. 161 same cross section, it is assumed that the resistance to the shearing force is equally distributed amongst them. To fulfil this condition where a pair of diagonals, as in fig. 83, cross each other, with equal and opposite inclinations, the stresses along them must be equal, and of opposite kinds. Then let R' and - R' be the stresses along the pair of diagonals, and i and i their inclinations to the horizon, we shall have for the vertical component of the force sustained by them, F y = R' sin i R' sin ( i)= 2 R' sin i', ........ (1.) and for the horizontal component, R' cos i R' cos ( i) = ; so that the horizontal components of the stresses along the two diagonals at the plane of section balance each other. Let 2 ra be the number of diagonal bars which cross each other at a given vertical section, the amount of the stress along each bar is which is a | f .g^ t j for bars which slope | ft J^ j the shearing force. The pull along the lower bar, and the thrust along the upper bar, at the given vertical section, must constitute a couple which balances the bending couple M ; hence their common amount is (3.) 1 65. Lattice Girder Uniform Load. If N denote the even num- ber of equal divisions into which the length of a lattice girder is divided by vertical lines traversing all the joints, whether of meeting of diagonal and horizontal bars, or of crossing of diagonal bars, and I the length of one of those divisions, so that N I, as before, is the span of the girder, then the effect of a load equally distributed amongst all those vertical lines, or amongst the alternate lines, may be found by means of the formulae for a half-lattice girder, Article 163, as follows : I. When the load is distributed over all the vertical lines, the formulae for case 1, equations 1, 2, 3, 4, are to be applied to vertical sections, such as C D, traversing the joints of crossing of diagonals ; observing only, that the resistance to the shearing force is distributed amongst the diagonals as shown by equation 2 of this Article, M 162 THEORY OP STRUCTURES. II. When the load is distributed over those vertical lines only which traverse joints of meeting of diagonal and horizontal bars, the formulae of case 2, equations 5, 6, 7, 8, 9, so far as they relate to sections made at unloaded joints, are to be applied to vertical sections, such as C D, traversing the joints of crossing of diagonals; attending as before to the distribution of the stress amongst the diagonals by equation 2 of this Article. 166. Transformation of Frames. The principle explained in Article 66, of the transformation of a set of lines representing one balanced system of forces into another set of lines representing another system of forces which is also balanced, by means of what is called " PARALLEL PROJECTION," being applied to the theory of frames, takes obviously the following form : THEOREM. If a frame whose lines of resistance constitute a given figure, be balanced under a system of external forces represented by a given system of lines, then will a frame whose lines of resistance con- stitute a figure which is a parallel projection of the original figure, be balanced under a system of forces represented by the corresponding parallel projection of the given system of lines; and the lines repre- senting the stresses along the bars of tJie new frame, will be tlie corresponding parallel projections of the lines representing the stresses along the bars of the original frame. This Theorem is called the " Principle of the Transformation of Frames." It enables the conditions of equilibrium of any unsym- metrical frame which happens to be a parallel projection of a symmetrical frame (for example, a sloping lattice girder), to be deduced from the conditions of equilibrium of the symmetrical frame, a process which is often much more easy and simple than that of finding the conditions of equilibrium of the unsymmetrical frame directly. SECTION 2. Equilibrium of Chains, Cords, Ribs, and Linear Arches. 167. Equilibrium of a Cord. Let D A C in fig. 84 represent a flexible cord supported at the points C and D, and loaded by forces in any direction, constant or vary- ing, distributed over its whole length with con- v ^t ^ stant or varying intensity. Let A and B be any two points in this cord ; from those points draw tangents to the cord, A P and B P, meeting in P. The load acting on the cord between the points A and B is balanced by the pulls along the EQUILIBRIUM OP A CORD. 163 cord at those two points respectively ; those pulls must respectively act along the tangents A P, B P ; hence follows THEOREM I. The resultant of the load between two given points in a balanced cord acts through the point of intersection of the tangents to the cord at those points; and that resultant, and the pulls along the cord at the two given points, are proportional to the sides of a triangle which are respectively parallel to their directions. The more the number of loaded points in & funicular polygon (as denned in Article 150) is increased, or, in other words, the more the number of sides in the polygon is multiplied, the more nearly does it approximate to the condition of a cord continuously loaded ; while at the same time, the number of lines radiating from the point O in the diagram of forces (exemplified in fig. 75*) increases with the number of sides of the funicular polygon, and the polygon of external forces of fig. 75* approximates to a continuous line, curved or straight. A diagram of forces for a continuously loaded cord may be con- structed in the following manner (fig. 84*). Let radiating lines be drawn from the point O parallel to the tangents of the cord at any points which may be under consideration : for example, let C, O D, be parallel to the tangents at the points of support, and O A, O B, parallel to the tangents at the points A and B of fig. 84 re- spectively. Let the lengths of those radiating lines represent the pulls along the cord at the points to whose tangents they are parallel ; and let a line D A B 0, curved or straight, as the case may be, be drawn so as to pass through the extremities of all the radiating lines which represent the pulls along the cord at different points. Then from Theorem I. it appears, that a straight line drawn from B to A in fig. 84*, will represent in magnitude and direction the resultant of the load on the cord between A and B (fig. 84). Now, suppose the point marked A in fig. 84 to be taken gradually nearer and nearer to B; then will O A in fig. 84* approach gradually nearer and nearer to O B ; and while the direction of the straight line drawn from B to A gradually approaches nearer and nearer to the direction of the tangent at the point B to the line C B A D in fig. 84% the resultant load between B and A represented by that straight line gradually approaches nearer and nearer in direction to the direction of the load at the point B in fig. 84 ; therefore, the direction of the load at any point B of the cord (fig. 84), is represented by the direction of a tangent at B (fig. 84*), to the line C B A D. Hence follows THEOREM II. If a line (called a line of loads) be drawn, such 164 THEORY OF STRUCTURES. that while its radius-vector from a given point is parallel to a tangent to a loaded cord at a given point, its own tangent is parallel to the direction of the load at the point in the cord; then will the length of a radius-vector of the line of loads represent the pull at the corre- sponding point of the cord; and a straight line drawn between any two points in tfie line of loads will represent in magnitude and direction the resultant load between the two corresponding points in the cord. The supporting forces required at the points C and D (fig. 84), are obviously represented in magnitude and direction by the ex- treme radiating lines, O C, O D. A loaded cord, hanging freely, is obviously stable, but capable of oscillation. 168. Cord under Parallel Loads. If the direction of the load be everywhere parallel and vertical, the line of loads be- comes a vertical straight line, as C B A D (fig. 84**). To express this case algebraically, let A in fig. 84 be the lowest point of the cord, so that the tangent A P is horizontal. Then in fig. 84**, O A will be horizontal, and perpendicular to C D. Let H = OA = horizontal tension along the cord at A; R = O B = pull along the cord at B ; Fig. 84**. p __ AB = load on the cord between A and B ; i = ^L X P B (fig. 84) = ^. A O B (fig. 84**) = inclination of cord at B ; P = Htani; H = J (P 2 +[H 2 ) = Hseci (1.) To deduce from these formulae an equation by which the form of the curve assumed by the cord can be determined when the distri- bution of the load is known, let that curve be referred to rectangular horizontal and vertical co-ordinates, measured from the lowest point A, the co-ordinates of B being, A X = #, X B = ?/; then . dy tan& = -7; dx whence we obtain y ( e i\ 55~B ; ' a differential equation which enables the form assumed by the cord to be determined when the distribution of the load is known. 1 69. Cord under Uniform Vertical Load. By an uniform vertical load is here meant a vertical load uniformly distributed along a CORD UNDER UNIFORM VERTICAL LOAD. 165 horizontal straight line ; so that if A (fig. 85), be the lowest point of the rope or cord, the load suspended between A and B shall be Fig. 85. proportional to AX = x, the horizontal distance between those points, and capable of being expressed by the equation (i.) where p is a constant quantity, denoting the intensity of the load in tinits of weight per unit of horizontal length : in pounds per lineal foot, for example. It is required to find the form of the curve D A B C, and the relations amongst the load P, the horizontal pull at A (H), the pull at B (R), and the co-ordinates AX = x, BX = y. First Solution. Because the load between A and B is uniformly distributed, its resultant bisects A X therefore, the tangent B P bisects A X : this is a property characteristic of a PARABOLA whose vertex is at A ; therefore, the curve assumed by the cord is such a Also, the proportions of the load, and the horizontal and oblique tensions are as follows : ................. (2.) Second Solution. In the present case equation 2 of Article 168 becomes - dx ~ H which being integrated with due regard to the condition that when x =-. 0, y = 0, gives p x* y = 2H the equation of a parabola whose focal distance (or modulus, to use the term adopted in Dr. Booth's paper on the "Trigonometry of the Parabola," Reports of the British Association, 1856), is, 166 THEORY OF STRUCTURES. For a parabola we have also the inclination i to the horizon re- lated to the co-ordinates by the following equations : dy x 2 y tan i -r- = s - - = -^; ax 2m x whence we have the proportions P : H : K : : tan i : 1 : sec i : : ^ : 1 : J (l + */ \ as before. The following are the solutions of some useful problems respecting uniformly loaded cords. PROBLEM I. Given the elevations, y,, y 2 , of the two points of support of the cord above its lowest point, and also the horizontal distance, or span a, between those points of support; it is required to find the horizontal distances, x a , x 2 , of the loivest point from the two points of support; also the modulus m. In a parabola, yi:y 2 : :^:#1; therefore, x l = a- , J* ,-; ^ a .-_^L_; ...... (8.) TyT+V^ Jfr+Jy* also _ ^i _ ^2 _ ffi H~ ^2 _ _ ^ _ ^ /g \ " ~ ~ "" "' the points of support are at the same level, a a? PROBLEM II. Given the same data, to find the inclinations i M i 2 , of the cord at the points of support. By equations 6, we have, i when ^ = 2/2? tan^ =: tan i s = (12.) PARABOLIC CORD. 167 PROBLEM III. Given the same data, and the load per unit of length; required the horizontal tension H, and the tensions HI, K 2 , at the points of support. By equation 5, we find, H = 2pm = - -cr^r = ; .......... (13.) 2 y, + 2 y, + and by the proportional equation 7, t = HsectJ = H ^ l + rj R 2 = H sec *, ..................... (U.) When i/j = y^ those equations become fl =:; B, = E, = H sec t t = H V l + PROBLEM IY. Given the same data as in Problem I., to find the length of the cord. The following are two well known formulse for the length of a parabolic arc, commencing at the vertex, one being in terms of the co-ordinates x and y of the farther extremity of the arc, and the other in terms of the modulus m, and the inclination i of the farther extremity of the arc to a tangent at the vertex. y + * - m-jtan i sec i + hyp. log. (tan i + sec t)} The length of the cord is 5, + s 2 , where s l is found by putting x l and y l in the first of the above formula, or i in the second, and s a by putting x z and y z in the first formula, or i 2 in the second. The following approximate formula for the length of a parabolic arc is in many cases sufficiently near the truth for practical purposes ; * = x + f nearly; (17.) which gives for the total length of the cord 168 THEORY OF STRUCTURES. Sl +s. 2 = a ++ nearly ........... (18.) O \Cj X% / and when y^ = y 2 , this becomes 2s, = a + A V l nearly; ................ (19.) O a PROBLEM V. Given the same data, to find, approximately, tJie small elongation of the cord d (sj + s 2 ) required to produce a given small depression dyofthe lowest point A, and conversely. Differentiating equation 18, we find which serves to compute the elongation from the depression ; and conversely, 4 ^1-4- zL which serves to compute the depression of the lowest point from the elongation of the cord. When y = y 2 , those formulas become, 2 .(22.) The preceding formulae serve to compute the depression which the middle point of a suspension bridge undergoes in consequence of a given elongation of the cable or chain, whether caused by heat or by tension. 170. Suspension Bridge with Vertical Rods. In a Suspension bridge the load is not continuous, the platform being hung by rods from a certain number of points in each cable or chain : neither is it uniformly distributed ; for although the weight of the platform per unit of length is uniform or sensibly so, the load arising from the weight of the cables or chains and of the suspending rods is more intense near the piers. Nevertheless, in most cases which occur in practice, the condition of each cable or chain approaches sufficiently near to that of a cord continuously and uniformly loaded to enable the formulae of Article 169 to be applied without material error. SUSPENSION BRIDGE FLEXIBLE TIE. 169 When the piers of a suspension bridge are slender and vertical (as is usually the case), the resultant pressure of the chain or cable on the top of the pier ought to be vertical also. Thus, let C E, in tig. 85, represent the vertical axis of a pier, and G the portion of the chain or cable behind the pier, which either supports another division of the platform, or is made fast to a mass of rock, or of masonry, or otherwise. If the chain or cable passes over a curved plate on the top of the pier called a saddle, on which it is free to slide, the tensions of the portions of the chain or cable on either side of the saddle will be equal; and in order that those tensions may compose a vertical pressure on the pier, their inclinations must be equal and opposite. Let i be the common value of those inclina- tions ; R the common value of the two tensions ; then the vertical pressure on the pier is V = 2R sini = 2Htani = 2px; (1.) that is, twice the weight of the portion of the bridge between the pier and the lowest point, A, of the curve C B A D. But if the two divisions of the chain or cable DAG, C G, which meet at C, be made fast to a sort of truck, which is supported by rollers on a horizontal cast iron platform on the top of the pier, then the pressure on the pier will be vertical, whether the inclina- tions of the two divisions of the chain or cable be equal or unequal; and it is only necessary that the horizontal components of their ten- sion should be equal ; that is to say, let i, i', be the inclinations of the two divisions of the chain or cable in opposite directions at C, and R, R', their tensions, then R = H sec i; R' = H sec i' ; V = R sin i + R' sin i' = H (tan i -\- tan i') (2.) 171. Flexible Tie. Let a vertical load, P, be applied at A, fig. 86, Fig. 86. Fig. 86*. and sustained by means of a horizontal strut, A B, abutting against a fixed body at B, and a sloping rope or chain, or other flexible tie, A D C, fixed at C. The weight of the strut, A B, is supposed to be divided into two components, one of which is supported at B, while the other is included in the load P. The weight, W, of the 170 THEORY OF STRUCTURES. flexible tie, A D C, is supposed to be known, and to be considered separately ; and with these data there is proposed the following PROBLEM. W being small compared with P, to find approximately the vertical depression E D of the flexible tie below the straight line A C, tJw pulls along it at A, D, and C, and the horizontal thrust along A B. Because W is small compared with P, the curvature of the tie will be small, and the distribution of its weight along a hori- zontal line may be taken as approximately uniform j therefore its figure will be nearly a parabola ; the tangent at D will be sensibly parallel to A C, and the tangents at A and C will meet in a point which will be near the vertical line E D F, which line bisects A C, and is bisected in D. Hence we have the following construction : Draw the diagram of forces, fig. 86*, in the following manner. W On the vertical line of loads b c, take bf = P ; 6 e = P + ; be 2i = P + W. From b draw b O parallel to the strut A B ; that is, horizontal from e draw e parallel to C A, cutting b O in O ; join c O, /O. In fig. 86, bisect AC in E, through which draw a vertical line ; through A and C respectively draw A F || O/, C F || O c, cutting that vertical line in F ; bisect E F in D. Then will A F and C F be tangents to the flexible tie at A and C, D will be its most depressed point, and D E its greatest depression ; and the pulls along the tie at C, D, and A, and the thrust along the strut A B, will, in virtue of the principle of Article 168, be represented by the radiating lines c, O e, Of, and 6, in fig. 86*. This solution is in general sufficiently near the truth for practi- cal purposes. To express it algebraically, let R a , R d , R,., be the tensions of the tie at A, D, C, respectively, and H the horizontal thrust; then (i.) SUSPENSION BRIDGE WITH SLOPING RODS. 171 The difference of length between the curve ADC and the straight line A E C is found very nearly, by substituting, in the second AB-DE term of equation 19, Article 169, AC for a, and === * or V\\ that is to say, ADC-AEC = ?-^H = 1.^J5-1 172. Suspension Bridge with Sloping Rods. Let the uniformly- loaded platform of a suspension bridge be hung from the chains by parallel sloping rods, making an uniform angle j with the vertical. The condition of a chain thus loaded is the same with that of a chain loaded vertically, except in the direction of the load ; and the form assumed by the chain is a parabola, having its axis paral- lel to the direction of the suspension rods. In fig. 87, let C A represent a chain, or portion of a chain, sup- ported or fixed at C, and horizontal at A, its lowest point. Let AH be a horizontal tangent at A, representing the platform of the bridge ; and let the suspension rods be all parallel to C E, which makes the angle ^ E C H J = j with the vertical. Let B X re- present any rod, and suppose a vertical load v to be supported at the point X. Then, by the principles of the equilibrium of a frame of two bars (Article 145), this load will produce a putt, p, on the rod X B, and a thrust, q, on the platform between X and H ; and the three forces v, p, q, will be proportional to the sides of a triangle parallel to their directions, such as the triangle C E H ; that is to say, v:p:q:: CH : CE: EH: : 1 : sec,/ : tan.;. (1.) Next, instead of considering the load on one rod B X, consider the entire vertical load Y between A and X. This being the sum of the loads supported by the rods between A and X, it is evident that the proportional equation (1) may be applied to it; and that if P represent the amount of the pull acting on the rods between A and X, and Q the total thrust on the platform at the point X, we shall have Y:P:Q::CH:CE:E~H::1 : sec./ : tan j. (2.) The oblique load P = Y sec j is what hangs from the chain between A and B. Being uniformly distributed, its resultant bisects A X in P, which is also the point of intersection of the tangents A P, 172 THEORY OF STRUCTURES. B P ; and the ratio of the oblique load P, the horizontal tension H along the chain at A, and the tension K along the chain at B, is that of the sides of the triangle B X P ; that is to say, P :H :R ::BX : XP = ^- : BP ............ (3.) Comparing this with the case of Article 169 and fig. 85, it is evi- dent that the form of the chain in fig. 87 must be similar to that of the chain in fig. 85, with the^exception that the ordinate X B = y is oblique to the abscissa AX = x, instead of perpendicular ; that is to say, C B A is a parabola, having its axis parallel to the inclined suspension rods. The equation of such a parabola, referred to its oblique co-ordi- nates, with the origin at A, is as follows : X 2 ' COS 2 J = -- ........................... < 4 -) where m, as in Article 169, denotes the modulus of the parabola, given by the equation x 2 ' cos 2 / m=- J - ........................... (5.) y x and y being the co-ordinates of any known point in the curve. The length of the tangent B P = t is given by the following equa- tion : (6.) Hence are deduced the following formulae for the relations amongst the forces which act in a suspension bridge with inclined rods. Let v now be taken to denote the intensity of the vertical load per unit of length of horizontal platform per foot, for ex- ample ; p the intensity of the oblique load ; q the rate at which the thrust along the platform increases from A towards H. Then Y = v x ; P = p x = v x sec j ; Q = q x = v x tan j ; xP px* 2pm H = = __ = -J- 2vm sec 3 ? (8.) O,>. f)ai nrkC J ft * \ / K = -== COS'J [ ptx y x y EXTRADOS AND INTRADOS. 173 The horizontal pull H at the point A may be sustained in three different ways, viz. : I. The chain may be anchored or made fast at A to a mass of rock or masonry. II. It may be attached at A to another equal and similar chain, similarly loaded by means of oblique rods, sloping at an equal angle in the direction opposite to that of the rods B X, &c., so that A may be in the middle of the span of the bridge. III. The chain may be made fast at A to the horizontal platform A H, so that the pull at A shall be balanced by an equal and op- posite thrust along the platform, which must be strong enough and stiff enough to sustain that thrust. In this case, the total thrust at any point, X, of the platform is no longer simply Q = q x, but x tanj) ................ (10.) The length of the parabolic arc, A B, is given exactly by the following formulae. Let i denote the inclination of the parabola at the point B to a line perpendicular to its axis. Then i = arc cos ( = cosjj (H-) which, when B coincides with A, becomes simply i = j. Then from the known formulae for the lengths of parabolic arcs, we have parabolic arc A B = m < tan i sec i tan j sec j + h ! tani-Heoi) ..... (12) tan j -f sec j j In most cases which occur in practice, however, it is sufficient to use the following approximate formula : 2 y 2 arc A B = * + y sin j + - , nearly ...... (13.) The formulae of this Article are applicable to Mr. Dredge's sus- pension bridges, in which the suspending rods are inclined, and although not exactly parallel, are nearly so. 173. Kxirado* and intrados. When a cord is loaded with parallel vertical forces, and ordinates are drawn downwards from the cord, of lengths proportional to the intensity of the vertical load at the points of the cord from which they are drawn, a line, straight or 174 THEORY OF STRUCTURES. curved as the case may be, which traverses the lower ends of all these ordinates, is called the extrados of the given load. The curve formed by the cord itself is called the intrados. The load suspended between any two points of the cord is proportional to the vertical plane area, bounded laterally by the vertical ordinates at those two points, above by the cord or intrados, and below by the extrados ; and may be regarded as equal to the weight of a flexible sheet of some heavy substance, of uniform thickness, bounded above by the intrados, and below by the extrados. The following is the alge- braical expression of the relations between the extrados and the intrados. Assume the horizontal axis of x to be taken at or below the level of the lowest point of the extrados ; and let the vertical axis of y, as in Articles 168, 169, and 170, traverse the point where the intrados is lowest. For a given abscissa x, let y' be the ordinate of the extrados, and y that of the intrados, so that y y' is the length of the vertical ordinate intercepted between those two lines, to which .the intensity of the load is proportional. Let w be the weight of unity of area of the vertical sheet by which the load is considered to be represented. Then we have for the load between the axis of y and a given ordinate at the distance x from that axis, 'y -tf)dx; (1.) the integral representing the area between the axis of y, the given ordinate, the extrados and the infcrados. Combining this equation with equation 2 of Article 168, we obtain the following equation : an equation which affords the means of determining, by an indirect process, the equation of the intrados, when the horizontal tension H, and the equations of the extrados are given, and also, by a some- what more indirect process, the equation of the intrados and the horizontal tension, when the equation of the extrados and one of the points of the intrados are given. Both these processes are in general of considerable algebraical intricacy. H obviously represents the area of a portion of the sheet above mentioned, whose weight is equal to the horizontal tension. Let that area be the square of a certain line, a ; that is, let CORD WITH HORIZONTAL EXTRADOS. 175 Then that line is called the parameter of the intrados, or curve in which the cord hangs. When the vertical load is of uniform intensity, as in Article 1 69, so that the intrados is a parabola, it is obvious that the extrados is an equal and similar parabola, situated at an uniform depth below the intrados. [The reader who has not studied the properties of exponential functions may pass at once to Article 176.] 174. Cord with Horizontal Extrados. If the extrados be a horizontal straight line, that line may itself be taken for the axis of x. Thus, in fig. 87 A, let OX be the straight horizontal extrados, A the lowest point of the intrados, and let the vertical line A be the axis of y. Denote the length of O A, which is the least ordinate of the intrados, by y Q . Let o x. FX = y be any other ordinate, at the end Fig * 87 A> of the abscissa O X =. x. Let the area O A B X be denoted by u. Then equations 1 and 2 of Article 172 become the following : P w u = w I ydx; J o dx dx 2 H a 2 The general integral of the latter of these equations is u=Ae^ Ee~^ W in which A and B are constants, which are determined by the special conditions of the problem in the following manner. When x= 0, e a =e a rr 1 ; but at the same time u = 0, therefore A = B, and equation (a.), may be put in the form, u = A\e~ e~ " J (b.) This gives for the ordinate, A ( =., --M y= U- +e * ) (c.) (hich, for x = 0, becomes y Q = - - ; and therefore A , 176 THEORY OF " STRUCTURES. which value being introduced into the various preceding equations, gives ib,e 'following results, as to the geometrical properties of the intradds : a ?/ rt ( ~ \ , u = ^ \e a e a ) ; Area Ordinate, y = | (e^ -f e~ ) dy u y ( *. - :?. "\ 3%>e, tan i = -y^ = ~ =- \ e a e a ) : d x or 2 a Deviation, -s-~ = ~ 9 = d x* a* (2.) The relations amongst the forces which act on the cord are given by the equations H = ! _ R (tension at B) = ^/P 2 + H 2 = H \/ 1 + j (3.) In the course of the application of these principles, the following problem may occur : given, the extrados O X, the vertex A of the intrados, and a point of support B ; it is required to complete the figure of the intrados. For this purpose it is necessary and sufficient to find the parameter a; so that the problem in fact amounts to this ; given the least ordinate y Q , and the ordinate y corresponding to one given value of the abscissa x, it is required to find a, so as to fulfil the equation ........ (4.) = hyperbolic cosine of -, as this function is called. Supposing a table of hyperbolic cosines to be at hand, - is found by its being the number whose hyper- a y bolic cosine is ; so that a = y number to hyp. cos. - (5.) CATENARY. 177 but such a table is rarely to be met with ; and in its absence a is found as follows : The value of x is given in terms of y by the equation and hence 175. Catenary is the name given to the curve in which a cord or chain of uniform material and sectional area (so that the weight of any part is proportional to its length) hangs when loaded with its own weight alone. Let fig. 87 A, serve to represent this curve; but let A be taken as the origin of co-ordinates, so that the axis of x is a horizontal tangent at A. Let s denote the length of any given arc A B. Then if p be the weight of an unit of length of the cord or chain, the load suspended between A and B is P = p s. The inclination i of the curve at B to a horizontal line is expressed by the equations cos* = dx ---^_A /1__^ ds ~ V ds*' i = * = + /l-** dx V ds> tan (!) dx ds Let the horizontal tension be equal to the weight of a certain length of chain, m, so that (2.) From these equations, and from the general equation 2 of Article 168, we deduce the following : tani = dx* dx "H" .(3.) 178 THEORY OF STRUCTURES. which, by a few reductions, is brought to the following form : gg = __ m * _ _ (4i) the integral of which (paying due regard to the conditions that when s - 0, x = 0) is known to be a = m -hyp. log. (_L + A /i + ^\ ......... (5.) \ m V m 2 / This equation gives the abscissa x of the extremity of an arc A B = s, when the parameter of the catenary (as m is called) is known. Transforming the equation so as to have s in terms of x, we obtain The ordinate y is found in terms of x by integrating the equation dy which eri gves m the term . 2 being introduced in order that when x - 0, y may be also = 0. This is the equation of the catenary, so far as its form is concerned. The mechanical condition is given by the equations H =pm-, P =ps; = 3~j^(e +e m \ =p(y + m); (9-) so that the tension at any point is equal to the weight of a piece of the chain, whose length is the ordinate added to the parameter. Suppose the axis of #, instead of being a tangent at the vertex of the curve, to be situated at a depth A O = m below the vertex, and let i/ denote any ordinate measured from this lowered axis ; then ; (10.) which, being compared with the expression for the ordinate amongst equations 2, Article 173, shows, that the intradosfor a horizontal ex- CATENARY. 179 trados when the least ordinate is equal to the parameter (y = a), becomes identical with a catenary, having the same parameter (m = SL = y ). PROBLEM. Given, two points in a catenary, and the length of chain between them; required the remainder of the curve. Let k be the horizontal distance between the two points, v their difference of level, I the length of chain between them. Those three quantities are the data. The unknown quantities may be expressed in the following manner. Let x { , y lt be the co-ordinates of the higher given point, and Si the arc terminating at it, all measured from the yet unknown vertex of the catenary, and x 2 , y 2 , s 2 , the corresponding quantities for the lower given point. (The particular case when the points are at the same level will be afterwards considered). Also let x i H~ X 2 ^ ( an unknown quantity). Then we have _ X Putting these values of x in the equations 6 and 8, we find (A _ -A_\ / * * Q^n. + e 2^\f e ^e~~^ -2/i y,- [e j.^e J ^ Square those two equations and take the difference of the squares ; then, / _*^ * \ v 2 = m ie m_ e ~smV (13.) In this equation the only unknown quantity is the parameter m, which is to be determined by a series of approximations. Next, divide the sum of the equations (12) by their difference. This gives - l + v '-r^, and consequently I ~f v li m- hyp. log. _ ^ (14.) Either or both of the abscissae x^ and x a , being computed by the equations 11, we find the position of the vertical axis. Then com- puting by equation 8, either or both of the ordinates, y lf y 2 , we find 180 THEORY OF STRUCTURES. the vertex of the catenary, which, together with the parameter, being known, completely determines the curve. Q. E. I. When the given points are at the same level, that is, when v - 0, the vertical axis must be midway between them, so that x, = x 2 = - h = (15.) In this case equation 13 becomes ^-~^) (16.) from which m is to be found by successive approximations. Then the computation of y i = y 2 by means of equation 8 determines the vertex of the curve, and completes the solution. The following are some of the geometrical properties of the catenary : I. The radius of curvature at the vertex is equal to the para- meter, and at any other point is given by the equation r = t . 8 eci = - = . 8 eoS (17.) dx 2 dx> d x II. The length of a normal to the catenary, at any point, cut off by a horizontal line at the depth m below the vertex, is equal to the radius of curvature at that point. III. The involute of a catenary commencing at its vertex, is the tractory of the horizontal line before mentioned, with the constant tangent m. IV. If a parabola be rolled on a straight line, the focus of the parabola traces a catenary whose parameter is equal to the focal distance of the parabola. 176. Centre of Gravity of a Flexible Structure. In every case in Avhich a perfectly flexible structure, such as a cord, a chain, or a funicular polygon, is loaded with weights only, the figure of stable equilibrium in the structure is that which corresponds to the lowest possible position of the centre of gravity of the entire load. This principle enables all problems respecting the equilibrium of ver- tically loaded flexible structures to be solved by means of the " Calculus of Variations." 177. Transformation of Cords and Chains. The principle of Transformation by Parallel Projection is applicable to continuously loaded cords as well as to polygonal frames : it being always borne in mind, that in order that forces may be correctly transformed by parallel projection, their magnitudes must be represented by the lengths of straight lines parallel to their directions, so that if in any case TRANSFORMATION OF CORDS. 181 the magnitude of a force is represented by an area (as in Articles 173 and 174) or by the length of a curve (as in Article 175), we must, in transforming that force by projection, first consider what length and position a straight line should have in order to represent it. Some of the cases already given might have been treated as ex- amples of transformation by parallel projection. For instance, the bridge-chain with sloping rods of Article 172 might be treated as a parallel projection of a bridge-chain with vertical rods, made by substituting oblique for rectangular co-ordinates ; and the intrados for a horizontal extrados of Article 174, where the least ordinate ?/ and parameter a have any ratio, might be treated as a parallel projection deduced, by altering the proportions of the rectangular co-ordinates, from the corresponding curve in which the least co- ordinate is equal to the parameter; that is, from the catenary. The algebraical expressions for the alterations made by parallel projection in the co-ordinates of a loaded chain or cord, and in the forces applied to it, are as follows : In the original figure, let y be the vertical co-ordinate of any point, and x the horizontal co-ordinate. Let P be the vertical load applied between any point B of the chain and its lowest point A ; let p = be its intensity per horizontal unit of length; let H be the horizontal component of the tension; let R be the tension at the point B. Suppose that in the transformed figure, the vertical ordinate y', and the vertical load P', which is represented by a vertical line, are unchanged in length and direction, so that we have y' = y; P = P; (1.) but for each horizontal co-ordinate x, let there be substituted an oblique co-ordinate x f , inclined at the angle j to the horizon, and gj altered in length by the constant ratio - =a. Then for the hori- x zontal tension H, there will be substituted an oblique tension H' , parallel to a/, and altered in the same proportion with that co- ordinate; that is to say, x' = ax- H' =aIL (2.) The original tension at B is the resultant of the vertical load P and the horizontal tension H. Let R be its amount, and i its in- clination to H ; then R = V^ + H 2 ; (3.) 182 THEORY OF STRUCTURES. and the ratios of those three forces are expressed by the proportion P : H : R : : tan i : 1 : seci : :sini icosi : 1 ........ (4.) Let R/ be the amount of the tension at the point B in the new structure, corresponding to B, and let i' be its inclination to the oblique co-ordinate x' , then R' = ,y (F 8 + H* =*= 2P'H' sin./) ............ (5.) P' : H' : R/ : : sini' : cos (i' =+= j) : cosj ............ (6.) The alternative signs == are to be used according as i' and j in direction. | differ j The intensity of the load in the transformed structure per unit of oblique length measured along dot, is (7.) but if the intensity of the load be estimated per unit of horizontal length, it becomes p r secj = (8.) a -cosj 178. linear Arches or Ribs. Conceive a cord or chain to be exactly inverted, so that the load applied to it, unchanged in direc- tion, amount, and distribution, shall act inwards instead of out- wards; suppose, further, that the cord or chain is in some manner stayed or stiffened, so as to enable it to preserve its figure and to resist a thrust ; it then becomes a linear arch, or equilibrated rib ; and for the pull at each point of the original cord is now substi- tuted an exactly equal thrust along the rib at the corresponding point. Linear arches do not actually exist ; but the propositions respect- ing them are applicable to the lines of resistance of real arches and arched ribs, in those cases in which the direction of the thrust at each joint is that of a tangent to the line of resistance, or curve connecting the centres of pressure at the joints. All the propositions and equations of the preceding Articles, respecting cords or chains, are applicable to linear arches, substi- tuting only a thrust for a pull, as the stress along the line of resist- ance. The principles of Article 167 are applicable to linear arches in general, with external forces applied in any direction. The principles of Article 168 are applicable to linear arches under parallel loads; and in such arches, the quantity denoted by CIRCULAR LINEAR ARCH. 183 H in the formula represents a constant thrust, in a direction per- pendicular to that of the load. The form of equilibrium for a linear arch under an uniform load is a, parabola, similar to that described in Article 169. In the case of a linear arch under a vertical load, intrados denotes the figure of the arch itself, and extrados a line traversing the upper ends of ordinates, drawn upwards from the intrados, of lengths pro- portional to the intensities of the load ; and the principles of Article 173 are applicable to relations between the intrados and the extrados. The curve of Article 174 is the figure of equilibrium for a linear arch with a horizontal extrados ; and from Article 175 it appears, that the figures of all such arches may be deduced from that of a catenary, by inverting it and altering its horizontal and vertical co-ordinates in given constant proportions for each case. The principles of Article 177, relative to the transformation of cords and chains, are applicable also to linear arches or ribs. This subject will be further considered in the sequel. The preceding Articles of this section contain propositions which, though applicable both to cords and to linear arches, are of impor- tance in practice chiefly in relation to cords or chains. The follow- ing Articles contain propositions which, though applicable also to cords as well as linear arches, are of importance in practice chiefly in relation to linear arches. 179. Circular Arch for Uniform Fluid Pressure. It is evident that a linear arch, to resist an uniform normal pressure from with- out, should be circular ; because, as the force to which it is sub- jected is similar all round, its figure ought to be similar to itself all round a property possessed by the circle alone. In fig. 88, let A B A B be a circular linear arch, rib, or ring, whose centre is O, pressed upon from without by a normal pressure of uniform intensity. In order that the intensity of that pressure may be conveniently expressed in units of force per unit of area, conceive the ring in 184 THEORY OP STRUCTURES. question to represent a vertical section of a cylindrical shell, whose length, in a direction perpendicular to the plane of the figure, is unity. Let p denote the intensity of the external pressure, in units of force per unit of area ; r the radius of the ring ; T the thrust exerted round it, which, because its length is unity, is a thrust per unit of length. The uniform normal pressure p, if not actually caused by the thrust of a fluid, is similar to fluid pressure ; and, according to Article 110, it is equivalent to a pair of conjugate pressures in any two directions at right angles to each other, of equal intensity. For example, let x be vertical, y horizontal, and let p x) p y , be the intensities of the vertical and horizontal pressure respectively, then P*=P y =P'> (1.) and the same is true for any pair of rectangular pressures. To find the thrust of the ring, conceive it to be divided into two parts by any diametral plane, such as C C. The thrust of the ring at the two ends of this diameter, of the amount 2 T, must balance the component, in a direction perpendicular to the diameter, of the pressure on the ring; the normal intensity of that component isj9, as already shown ; and the area on which it acts, projected on the plane, C C, which is normal to its direction, is 2r ; hence we have the equation 2T = 2pr; or T =pr (2.) for the thrust all round the ring ; which is expressed in words by this THEOREM. The thrust round a circular ring under an uniform normal pressure is. the product of the pressure on an unit of circum- ference by the radius. 180. Elliptical Arches for Uniform Pressures. If a linear arch has to sustain the pressure of a mass in which the pair of conjugate thrusts at each point are uniform in amount and direction, but not equal to each other, all the forces acting parallel to any given direc- tion will be altered from those which act in a fluid mass, by a given constant ratio ; so that they may be represented by parallel projec- tions of the lines which represent the forces that act in a fluid mass. Hence the figure of a linear arch which sustains such a system of pressures as that now considered, must be a parallel projection of a circle ; that is, an ellipse. To investigate the relations which must exist amongst the dimensions of an elliptic linear arch under a pair of conjugate pressures of uniform intensity, let A' B' A' B', B" A" B", in fig. 88, represent elliptic ribs, transformed from the circular rib A B A B by parallel projection, the vertical dimensions being un- changed, and the horizontal dimensions either expanded (as B" B"), ELLIPTIC LINEAK AKCH. 185 or contracted (as B' B'), in a given uniform ratio denoted by c ; so that r shall be the vertical and c r the horizontal semi-axis of the ellipse ; and if x, y, be respectively the vertical and horizontal co- ordinates of any point in the circle, and x' y', those of the corre- sponding point in the ellipse, we shall have xf = x; y' = cy (1.) If C C, D D, be any pair of diameters of the circle at right angles to each other, their projections will be a pair of conjugate diameters of the ellipse, as C' C', D' D'. Let P,,. be the total vertical pressure, and P y the total horizontal pressure, on one quadrant of the circle A B. Then Let P'j, be the total vertical pressure, and P' y the total horizontal pressure, on one quadrant of the ellipse, as A B', or A" B" ; and let T' x be the vertical thrust on the rib at B' or B", and T' y the hori- zontal thrust at A' or A". Then, by the principle of transformation, T { IT Y y Z c P^ = TT = C P r;}" (2 ') or, the total thrusts are as the axes to which they are parallel. Further, let P' T' be the total pressure, parallel to any semi- diameter of the ellipse (as O' D' or O" D") on the quadrant D' C' or D" C", which force is also the thrust of the rib at C' or C", the ex- tremity of the diameter conjugate to O' D' or O" D" ; and let O' D' or O" D" = r' ; then (3.) or, the total thrusts are as the diameters to which tliey are parallel. Next, let p'j,, p' y) be the intensities of the conjugate horizontal and vertical pressures on the elliptic arch ; that is, of the " principal stresses" (Articles 109, 112). Each of those intensities being found by dividing the corresponding total pressure by the area of the plane to which it is normal, they are given by the following equa- tion : , P, P = =' 186 THEORY OF STRUCTURES. so that the intensities of the principal pressures are as the squares of the axes of the elliptic arch to which they are parallel. Hence the " ellipse of stress " of Article 112 is an ellipse whose axes are proportional to the squares of the axes of the elliptic arch ; and to adapt an elliptic arch to uniform vertical and horizontal pressures, the ratio of the axes of the arch must be the square root of the ratio of the intensities of tJie principal pressures ; that is, .(5.) The external pressure on any point, D' or D", of the elliptic arch, is directed towards the centre, O' or O", and its intensity, per unit of area of the plane to which it is conjugate (O' C' or O" C"), is given by the following equation, in which r 1 denotes the semidiameter (0' D' or 0" D") parallel to the pressure in question, and r" the con- jugate semidiameter (O' C' or O" C") : that is, the intensity of the pressure in the direction of a given dia- meter is directly as that diameter and inversely as the conjugate dia- meter. Let p" be the intensity of the external pressure in the direction of the semidiameter r". Then it is evident that p' :p":: f :,".;.., (7.) that is, the intensities of a pair of conjugate pressures are to each other as the squares of the conjugate diameters of the elliptic rib to which they are respectively parallel. These results might also have been arrived at by means of the principles relative to the ellipse of stress, which have been explained in Article 112. 181. Distorted Elliptic Arch. To adapt an elliptic linear arch to the sustaining of the pressure of a mass in which, while the state of stress is uniform, the pressure conjugate to a vertical pressure is not horizontal, but inclined at a given angle j, the figure of the ellipse must be derived from that of a circle by the substitution of inclined for horizontal co-ordinates. In fig. 89, let BAG be a semicircular arch on which the ex- ternal pressures are normal and uniform, and of the intensity p } as before ; the radius being r, and the thrust round the arch, and load on a quadrant, being as before, P = T = p r. Let D be any point in the circle, whose co-ordinates are, vertical, O E = x, horizontal, DISTOKTED ELLIPTIC ARCH. 187 E D = y. Let B' A' C' be a semi-elliptic arch, in which the verti- cal ordinates are the same with those of the circle, while for each horizontal ordinate is substituted an ordinate inclined to the hori- zon by the constant angle j, and bearing to the corresponding hori- zontal ordinate of the circle the constant ratio c ; that is to say, let O' E' = x' = x 00 Then for the vertical semidiameter of the circle OA = r, will be substituted the equal vertical semidiameter of the ellipse O' A' = r j and for the horizontal diameter of the circle C B = 2 r, will be substituted the inclined diameter of the ellipse C' B' = 2 c r, which is conjugate to the vertical semidiameter. The forces applied to the elliptic arch are to be resolved into vertical and inclined components, parallel to O' A and C' B', instead of vertical and horizontal components. Let P'.,. denote the total vertical pressure, and P y the total inclined pressure, on either of the elliptic quadrants, C' A, A B' ; T' y the inclined thrust of the arch at A', T', the vertical thrust at B' or C'. Then T. = P', = T = P = pr- \ T, = P, = cT = cP = cpr; f that is to say, those forces are, as before, proportional to the dia- meters to which they are parallel. Let p x be the intensity of the vertical pressure on the elliptic arch per unit of area of the inclined plane to which it is conjugate, C' B' ; let p' y be the intensity of the inclined pressure per unit of area of the vertical plane to which it is conjugate ; then 188 THEORY OF STRUCTURES. cr F p' y = = so that, as before, tlie intensities of the conjugate pressures are as the squares of the diameters to which tJiey are parallel. The thrust of the arch at any point D' is as before, proportional to the diameter conjugate to O' D'. It is sometimes convenient to express the intensity of the verti- cal pressure per unit of area of the horizontal projection of the space over which it is distributed ; this is given by the equation f) p' x secj = - . ..................... (4.) c cos j It is to be borne in mind, that this is not the pressure on unity of area of a horizontal plane (which pressure is inversely as the horizontal diameter of the ellipse and directly as the diameter con- jugate to that diameter, to which latter diameter it is parallel), but the pressure on that area of a plane inclined at the angle j, whose horizontal projection is unity. The following geometrical construction serves to determine the major and minor axes of the ellipse B' A' C'. Draw O' a -L and = O' A' ; join B' a, which bisect in m in B'& produced both ways take mp = mq = O'ra ; join O'p, O' q these lines, which are perpendicular to each other, are the directions of the axes of the ellipse, and the lengths of those axes are respectively equal to the segments of the line p q, viz., J$'p = aq,l$'q a p. The following is the algebraical expression of this solution. Let A denote the major and B the minor semi-axis of the ellipse. Then c 2 + 2c -cosj); whence we have for the lengths of the semi-axes, ARCHES FOR NORMAL PRESSURE. 189 The angle ^L B' O'p = k, which the nearest axis makes with the diameter C' B', is found by the equation B /A'- r according as that axis is the longer ; - the shorter. The axes of the elliptic arch are parallel to, and proportional to the square roots of, the axes of the ellipse of stress in the pressing mass ; so that they might be found by the aid of case 3 of Problem IV., Article 112. 182. Arches for Normal Pressure in General. The condition of a linear arch of any figure at any point where the pressure is nor- mal, is similar to that of a circular arch of the same curvature under a pressure of the same intensity; and hence modifying the Theorem of Article 179 to suit this case, we have the following : THEOREM I. The thrust at any normally pressed point of a linear arch is the product of the radius of curvature by the intensity of the pressure; that is, denoting the radius of curvature by j, the normal pressure per unit of length of curve by p, and the thrust (1.) Example. This Theorem is verified by the vertically and hori- zontally pressed elliptic arches of Article 180 ; for the radii of curvature of an ellipse at the ends of its two axes, r and c r, are respectively, c 2 r a At the ends of r j f x = = cV ; r At the ends of c r ; ? = = - : cr c j Introducing these values into the equations of Article 180, and into equation 1 of this Article, we find, .(3.) T'* = p' y e y = cp ' - = p r as before ; T) T' y = p' x f x - - c?r = cpr as before; C It is further evident, that if the pressure be normal at every point of the arch (which it is not in the cases cited), the thrust must be constant at every point ; for it can vary only by the application of a tangential pressure to the arch ; and hence follows 190 THEORY OF STRUCTURES. THEOREM II. In a linear arch sustaining a pressure which is everywhere normal, the thrust is uniform, and the radius of curva- ture is inversely as the pressure a theorem expressed symbolically thus : T pf constant (4.) The only arch of this class which has hitherto been considered is the circular arch under uniform normal pressure. Another instance will be given in the following Article. 183. The Hydrostatic Arch is a linear arch suited for sustaining normal pressure at each point proportional, like that of a liquid in repose, to the depth below a given horizontal plane ; and is some- times called " the arch of Yvon-Villarceaux," from the name of the mathematician who first thoroughly investigated the properties of its figure by the aid of elliptic functions. The radius of curvature at a given point in the hydrostatic arch being, in virtue of Theorem II. of the last Article, inversely propor- tional to the intensity of the pressure, is also inversely proportional to the depth below the horizontal plane at which vertical ordinates representing that intensity commence. In fig. 90, let Y O Y represent the level surface from which the Fig. 90. pressure increases at an uniform rate downwards, so as to be similar to the pressure of a liquid having its upper surface at Y Y. Let A be the crown of the hydrostatic arch, being the point where it is nearest the level surface, and consequently horizontal. Let co-ordi- nates be measured from the point O in the level surface, directly above the crown of the arch ; sojfchat CTX = Y C = x shall be the vertical ordinate, and O Y = X C = y the horizontal ordinate, of any point, C, in the arch. Let O A, the least depth of the arch below the level surface, be denoted by x , the radius of curvature at the crown by r Q , and the radius of curvature at any point C by r. Let w be the weight of an unit of volume of the liquid, to whose pressure the load on the arch is equivalent. Then the intensities of the external normal pressure at the crown A, and at any point C, are expressed respectively by HYDKOSTATIC ARCH. 191 Po = wx *'> P = WX The thrust of the arch, which, in virtue of the principles of Article 1 82, is a constant quantity, is given by the equation T = Po r o = wx o r <> = P r = from which follows the following geometrical equation, being that which characterizes the figure of the arch : .......................... (3.) When x n and r are given, the property of having the radius of curvature inversely proportional to the vertical ordinate from a given horizontal axis enables the curve to be drawn approximately, by the junction of a number of short circular arcs. It is found to present some resemblance to a trochoid (with which, however, it is by no means identical). At a certain point, B, it becomes vertical, beyond which it continues to turn, until at D it becomes horizontal; at this point its depth below the level surface is greatest, and its radius of curvature least. Then ascending, it forms a loop, crosses its former course, and proceeds towards E to form a second arch similar to B A B. Its coils, consisting of alternate arches and loops, all similar, follow each other in an endless series. It is obvious that only one coil or division of this curve, viz., from one of the lowest points, D, through a vertex, A, to a second I>oint, D, is available for the figure of an arch ; and that the por- tion BAB, above the points where the curve is vertical, is alone available for supporting a load. Let a?!, y lt be the co-ordinates of the point B. The vertical load above the semi-arch A B is represented by v: I 'xdy; J o and this being sustained by the thrust T of the arch at B, must obviously be equal to that thrust j whence follows the equation (4.) That is to say, the area of tlie figure between the shortest vertical ordinate, and the vertical tangent ordinate, is equal to the constant jyroduct oftlie vertical ordinate and radius of curvature. The vertical load above any point, C, is w 192 THEOKY OF STRUCTUKES. and this is sustained by and equal to the vertical component of the thrust of the arch at C, which is T sin i (i being the inclination of the arch to the horizon). Hence follows the equation That is to say, the area of the figure between the shortest vertical ordinate and any vertical ordinate, varies as the sine of the angle of inclination to the horizon of the curve at the latter ordinate. The horizontal external pressure on the semi-arch from B to A is the same with that on a vertical plane, A F, immersed in a liquid of the specific gravity w with its upper edge at the depth x below the surface (see Article 124); so that its amount is and this is balanced by the thrust of the arch, T, at the crown. Hence follows the equation r*2 /y.2 M\ *O0 , n \ xr = x Q r = ...................... (6.) That is to say, halftJie difference of the squares of the least vertical ordinate and of the tangent vertical ordinate is equal to the constant product of the vertical ordinate and radius of curvature. Equation 6 gives for the value of the vertical tangent ordinate, The horizontal external pressure between B and any point, C, is equal to the pressure of a liquid of the specific gravity w on a ver- tical plane X F with its upper edge immersed to the depth x, so that its amount is f" , x*-x* w pdx = w-L. .-_; J X Zl and this is balanced by the horizontal component T cos i of the thrust of the arch at C ; whence follows the equation x\ x 2 . , . 2 = % r Q cos i; (8.) which gives for the value of any vertical ordinate, HYDROSTATIC ARCH. 193 x x xr-cos = x + 2x Q r Q (l cosi) Let jc, #', be any two vertical ordinates. Then from equation 8 it follows that x' 2 x 2 = 2x <) r Q (cos i cos i') .............. (10-) or, tlie difference of the squares of two ordinates varies as the difference of the cosines of the respective inclinations of the arc at their lower ends. From equation 9 is deduced the following expression of the in- clination in terms of the vertical ordinate : 1 The various properties of the figure of the hydrostatic arch ex- pressed by the preceding equations are thus summed up in one formula : f yi 7 * o *^i ***o ^i x n r fl = xr = xdy = . = s = T J V smi 2 2 To obtain expressions for the horizontal co-ordinate y, whose maximum value is the half-span ?/ and also for the lengths of arcs of the curve, it is necessaiy to use elliptic functions. [The reader who has not studied elliptic functions may here pass at once to Article 184.] In the use of elliptic functions the notation employed will be that of Legendre ; and the classes of functions referred to will be those called by that author the first and second kind respectively, and tabulated by him in the second volume of his treatise. Let 6 denote a constant angle, called the modulus of the func- tions; = 45, and at A, (f> = 90. Let the vertical ordinate and radius of curvature at the point D be denoted respectively by X, R ; then r x = r ) for the modulus & take an angle such that #** v .__ Then equation 9, the expression for the vertical ordinate, becomes x = A/ ( xl + 4 # r sin 2 %\ = X A/ f 1 sin 2 ^ sin The values of this for the points B and A are respectively r= X ' COS 6 ......................... (1* A.) Introducing the above value of x into equation 5, we obtain for the area between O A and any other vertical ordinate, f x d y = X Q TQ sin i = 2 X R cos

2 2 sm ^ J wr : . . sin i a i ( (i cos i sin i\ l+m cosi . . ) (10.) 2 sin 3 ^ y The value of the horizontal pressure itself is given by introducing the values of T and P x from equations 8 and 9 into equation 3, and is as follows : J / /I _L \ ' _L COS2 * ,*' COS i ) Py = wr \ m (1 -f-m) cosz -h ^ r ^~. . t ...til.) ( ' 2 2smi) v ' The horizontal component of the thrust of the arch at C is given by the equation When i 0, that is, for the crown of the arch, p y takes the fol- lowing value : wr 202 THEORY OF STRUCTURES. so that for every circular linear arch in which the depth of load at the crown, m r, is less than one-third of the radius, p y has negative values at and near the crown, showing that outward horizontal pressure or tension is required to preserve equilibrium. In such cases, there is a certain value of the angle i for which p y = 0. At the point where this takes place, P y consequently attains a negative maximum, and the horizontal component T cos i of the thrust along the arch attains a positive maximum, greater than T , because of P y being negative. Let this point be called C , and let the in- clination of the arch at it be denoted by i Q . This angle must satisfy the transcendental equation sin L - = 0, ......... (13.) 2 sin 3 ? and can therefore be found by approximation only. As a first approximation, may be taken 3m+l and then by successive substitutions, nearer and nearer approxi- mations may be found. Supposing i Q to have been thus determined to a sufficient degree of accuracy, its substitution for i in the equation 12 will give the maximum value of the horizontal component of the thrust of the arch. By expanding or contracting the horizontal dimensions of a cir- cular arch, it can be transformed into an elliptic arch, which will be balanced under forces deduced from those applied to the circular arch according to the principles explained in Articles 180, 184. In adapting the equations from 7 to 13 inclusive to an elliptic arch, it is to be observed that i represents not the inclination of the elliptic arch itself at a given point, but that of the circular arch from which the elliptic arch is derived at the corresponding point. CASE 2. When the direction of the conjugate pressure is inclined. This case is represented in the lower diagram of fig. 91. The in- clined axis of co-ordinates, Y' O' Y', is taken parallel to the direc- tion of the conjugate pressure, and touching the arch at the point O', which is now its crown. Each double ordinate of the arch, C' X' C' = 2 2/', is bisected by the vertical axis, on either side of which the vertical load is symmetrically distributed. Let j denote the inclination of the conjugate pressure to the horizon. Construct a parallel projection of the given arch, like the upper diagram of the figure, having its vertical ordinates equal to those of the distorted arch, and its horizontal ordinates less in the TOTAL THRUST OF LINEAR ARCHES. 203 ratio cos j : 1 ; conceive it to be under a vertical load, of equal amount to that on the distorted arch, and similarly distributed ; determine the horizontal pressures required to keep it in equilibrio ; then will the proper projection of those pressures keep the dis- torted arch in equilibrio. The relations amongst the co-ordinates of the two arches, and the amounts and magnitudes of the vertical and conjugate pres- sures, are as follows, quantities relating to the distorted arch being distinguished by accented letters : x = x ; y = y sec j ; F, = P, ; T' = T sec j ; F, P, sec j ; I ... (14.) p'. = p x cos j ; p' y = p 9 sec j. Let H' denote the conjugate component of the thrust of the dis- torted arch at any point C' ; then we have H' T' - F, = (T - P,) sec j ; (15.) and if T' be the thrust along the distorted arch at C', then the positive or negative sign being used according as the point C' is at the depressed or the elevated side of the arch. 186. Pointed Arches If a linear arch, as in fig. 92, consists of two arcs, B C, C B, meeting in a point at C, it is necessary to equilibrium that there should be con- centrated at the point C a load equal to that which would have been distributed over the two arcs AC, C A, extending from the point C to the respective crowns, A, A, of the curves of which two portions form the pointed arch. 187. Total Conjugate Thrust of Linear Arches. The total con- jugate thrust of an arch is the conjugate component, horizontal or inclined, as the case may be, of the entire pressure exerted between one semi-arch and its abutment, whether directly, at the point from which the arch springs, or above that point, through the material of the spandril. When a linear arch is of such a figure as to be balanced under a load of which the pressure is wholly vertical (as in the case de- scribed in Article 174), that is to say, when its figure is that in which a cord would hang, loaded with the same weight distributed in the same manner, its conjugate thrust is exerted simply at the point from which it springs, and is equal to the conjugate com- ponent of the thrust along the arch, which is a constant quantity throughout its whole extent. 204 THEORY OF STRUCTURES. When an arch, springs vertically from its abutments, the point of springing sustains the vertical load of the semi-arch only ; and the conjugate thrust is exerted wholly through the spandril. In other cases, the conjugate thrust is exerted partly at the point of springing and partly through the spandril. THEOREM. The amount oftJie conjugate thrust is equal to the con- jugate component of the thrust along the arch at the point where that component is a maximum; for at that point, as appears from the reasoning of Article 185, the intensity of the conjugate pressure between the arch and its spandril is nothing : it is, therefore, en- tirely below that point that the conjugate thrust, whether through the spandril or at the point of springing, is exerted; and conse- quently the amount of that thrust must be equal to the maximum conjugate component of the thrust along the arch, which is balanced by it. The point of the arch where the conjugate component of the thrust along it is a maximum, is called the point of rupture, for reasons which will afterwards appear. It may be at the crown ; or it may be in a lower position, to be determined by solving the equa- tion formed by making the intensity of the conjugate pressure between the arch and spandril, as found by the method of Article 1 85, equal to nothing : that is, d, d ( dy\ =lU = - d~x ( P * dx) == ............ This equation having been solved so as to give the position of the point of rupture, the corresponding value of P.,,, being the vertical load supported at that point, is to be computed; and then the conju- gate thrust is given by the equation H = max. value of P - .................. 2. (Where the conjugate pressures, as is generally the case, are hori- zontal, -j = cotan i ; and the value of i } the inclination of the arch, cL sc which fulfils equation 1, is called the angle of rupture). When the point of rupture is the crown of the arch (as in hydro- static and geostatic arches), equation 2 gives no result, because of d ?y P. vanishing and -~ increasing indefinitely; but it has already Ct 30 been shown bv other methods that in this case, where the conjugate pressures are horizontal E =T = Po r ; (3.) p^ being the intensity of the vertical load, and r the radius of cur- TOTAL THRUST OF LINEAR ARCHES. 205 vature ; but in order to form an equation which shall be applicable whether the conjugate pressures and co-ordinates are horizontal or inclined, the above equation must be converted into one expressed in terms of the co-ordinates ; that is to say, H = T = - (for y = 0) = fL. (for y = 0)...(4.) u/ u/ ju a x dy' dy dy* For rectangular co-ordinates -rr, = at the crown of the arch, so dy* r that equation 4 is converted into equation 3. Thus far as to finding the amount of the conjugate thrust. To find the position of its resultant, that is to say, the depth of its line of action below the conjugate co-ordinate plane, we must conceive it to act against a vertical plane, extending from the depth of the point of rupture below the conjugate co-ordinate plane, down to the depth of the point of springing below that plane, and find, by the methods of Article 89, the vertical co-ordinate of the centre of pressure of the plane so acted upon. That is to say, let X Q denote the depth of the point of rupture, and X Y that of the point of spring- ing below the conjugate co-ordinate plane ; p y the intensity of the conjugate pressure between the arch and spandril at any point between those points, and H = - r P ,dx, .................... (o.) J xo the conjugate component of the thrust of the arch at the point of springing; also, let X R be the depth of the resultant conjugate thrust below the conjugate co-ordinate plane; then Example I. Circular arch wider uniform normal pressure of intensity, p. 183 (Art. 179). Here p x = p y = p and the point of rupture is at the crown, the horizontal thrust is H = T=pr .......................... (7.) Let the crown be taken for origin of co-ordinates, so that X Q =. 0. CASE 1. Semicircle. Here x { = r, Hj = 0; and 20G THEORY OF STRUCTURES. CASE 2. Segment. Inclination at springing, i^ Here (1 cos i)\ H, p r ' cos i', and i p x\ + p r x { ' cosi r (i (1 cos i) 2 + cos i (I cos i)^ I sin 2 i...(9.) a Example II. Semi-elliptic arch, under conjugate uniform vertical and horizontal pressures (Art. 180). Let a = x be the rise, or vertical semi-axis; cathe horizontal semi-axis, or half -span; and let the origin of co-ordinates be at the crown. Then p y = c 2 p x ; and we have H = T = ap y = #ap x = c P,; X H =| .... (10.) Example III. Semi-elliptic distorted arch, with conjugate uniform vertical and oblique pressures (Art. 181). The vertical and conju- gate semidiameters, or rise and inclined half-span, being denoted by a and c a respectively, the equations 10 apply to this case also. Example IV. Hydrostatic arch (Art. 183). The origin of co- ordinates being taken, as in the article referred to, at the point of the extrados vertically above the crown, we have p y = p x = w x, H = T = w-.!LyJ$j H> = 0; and /Xl x 2 d x 2 v 3 T 3 _ Q _ _ _ . ai a? ............ (11.) H ~3 x*-xl Example Y. Geostatic arch, with horizontal or inclined extrados (Art. 184). Here p x = w x ' cosj; p y = (?p x = c 2 w x cosj ; H or = i^. In this case H = Hj, and the conjugate thrust is simply the single horizontal force H t at the point of spring- ing. CASE 2. i Q < i t . Find H as in the last example, and let the origin of co-ordinates be at the crown ; then Xi = r (I cos ij); and we have 1 ( f tl ) x n = - < r 2 1 p y smi(l cos i) - d i + r Hj (1 cos ^) \ (15.) H ( J JO 188. Approximate Hydrostatic and Cteostatic Arches. The Subject of elliptic functions is so seldom studied, and complete tables of them are so scarce, that it is useful to possess a method of finding the proper proportions of hydrostatic and geostatic arches (Articles 183, 184) to a degree of approximation sufficient for practical pur- poses, using algebraic functions alone. Such a method is founded on the fact that a hydrostatic arch approaches nearly to the figure of a semi-elliptic arch of the same height, and having its maximum and minimum radii of curvature in the same proportion. Let x , x lt as in Article 183, be the depth of load of a hydrostatic arch at the crown and springing respectively; r , r lf its radii of curvature at those points ; a = #, x , its rise ; y l its half-span, given in Article 183 by means of elliptic functions. Suppose a semi-elliptic arch to be drawn, having the same rise, a, with the hydrostatic arch; let r' , r\, be its radii of curvature at the crown and springing, whose proportion to each other is the same with that of the radii of the hydrostatic arch; that is to say, Let b be the half-span of this semi-ellipse. Then because the cubes of the semi-axes of an ellipse are to each other inversely as the radii of curvature at the respective extremities of the semi-axes, we have - ..(l.) 208 THEORY OF STRUCTURES. A rough approximation to the half-span of the hydrostatic arch is found by making y\ = b ; but this, in the cases which occur in practice, is too great by an excess which varies between T V and -rnr, and is about ^V 011 an average. Hence we may take, as a first approximation whose utmost error in practice is about TTO, and whose average error is about T ao, the following formula, giving the half -span in terms of the depths of load -ah the crown and springing : 19 , yi = OA fa ~ ^ Suppose the rise a and half-span y of a proposed hydrostatic arch to be given, and that it is required to find the depths of load ; equa- tion 2 gives us, as an approximation, 19a/' and because x\ x a, we have /202/,V U9a/ /9f) ,, \ 3 /^J/l\ I f ^ ^/1\ V19^/ \19^7 A closer approximation is given by the equations (3.) = a ' .(4.) A semicircular or semi-elliptic arch may have its conjugate thrust approximately determined, by considering it as an approximate geo- static arch, as follows : Let there be given, the half-span of the arch in question, horizontal or inclined, as the case may be, y n the depths of load at its crown and springing, x , x^ and the vertical load at the springing, P v Determine, by equation 2 or equation 4, the span y i of a hydro- static arch for the depths of load # , a^, and let <'> FRICTIONAL STABILITY. 209 be the ratio of the half-span of the actual arch to that of the hydro- static arch. The actual arch may now be conceived as an approximation to a geostatic arch, transformed from the hydrostatic arch by pre- serving its vertical or din at es and load, and altering its conjugate ordinates and thrust in the ratio c. The conjugate thrust of a hydrostatic arch being equal to the load, we have, as an approxi- mation to the conjugate thrust of the given semi-elliptic or semi- circular arch, HO^C?! (6.) SECTION 3. On Frictional Stability. 189. Friction is that force which acts between two bodies at their surface of contact, and in the direction of a tangent to that surface, so as to resist their sliding on each other, and which depends on the force with which the bodies are pressed together. There is also a kind of resistance to the sliding of two bodies upon each other, which is independent of the force with which they are pressed together, and which is analogous to that kind of strength which resists the division of a solid body by shearing, that is, by the sliding of one part upon another. This kind of resistance is called adhesion. It will not be considered in the present section. Friction may act either as a means of giving stability to struc- tures, as a means of transmitting motion in machines, or as a cause of loss of power in machines. In the present section it is to be considered in the first of those three capacities only. 190. Law of Solid Friction. The following law respecting the friction of solid bodies has been ascertained by experiment : The friction which a given pair of solid bodies, with their surfaces in a given condition, are capable of exerting, is simply proportional to the force with which they are pressed together. If the bodies be acted upon by a lateral force tending to make them slide on each other, then so long as the lateral force is not greater than the amount fixed by this law, the friction will be equal and opposite to it, and will balance it. There is a limit to the exactness of the above law, when the pressure becomes so intense as to crush or grind the parts of the bodies at and near their surface of contact. At and beyond that limit the friction increases more rapidly than the pressure ; but that limit ought never to be attained in a structure. From the law of friction it follows, that the friction between two bodies may be computed by multiplying the force with which p 210 THEORY OF STRUCTURES. they are pressed together t>y a constant co-efficient which is to be determined by experiment, and which depends on the nature of the bodies and the condition of their surfaces : that is to say, let N denote the pressure, f the co-efficient of friction, and F the force of friction, then F =/N. 191. Angle of Repose. Let A A, in fig. 93, represent any solid body, B B a portion of the surface of another, body, with which A A is in contact throughout the plane surface of contact e E. Let P C re- present the amount, direction, and position of the resultant of a force by which A A is urged obliquely towards B B, so that C is the centre of pressure of the surface of contact e E. (Art. _ 89.) Let P be resolved into two rectangular components : one, N C, normal to the plane of contact, and pressing the bodies to- gether: the other, TO, tangential to the plane of contact, and tending to make the bodies slide on each other. Let the total force P C, be denoted by P, its normal component by N, and its tangential component by T ; and let the angle of obliquity T P C or P C N be denoted by 6, so that T = P-sin0 = N -tan ) I Then so long as the tangential force T is not greater than fN, it will be balanced by the friction, which will be equal and opposite to it ; but the friction cannot exceed f N; so that if T be greater than this limit, it will be no longer balanced by the friction, but will make the bodies slide on each other. Now the condition, that T T shall not exceed /IST, is equivalent to the condition, that , or tan 0, shall not exceed/! Hence it follows, that the greatest angle of obliquity of pressure between two planes which is consistent with stability, is the angle whose tangent is the co-efficient of friction. This angle is called the angle of repose, and is denoted by ) p 9 = wx cos 6 . TLA =-/; (2.) cos & + fj (cos 2 6 cos 2

Py) ^ Pi > P-'> giving the following results : P.+P' STABILITY OF EARTH. P X ~\~ p y W X ' COS 4 217 C5 ^ 2 2 cos 6 cos * + J (cos 2 * sin 2 ) - Py) z } wx ' cos 6 sin 4 ... \y. ) ) 2 " { 4 c< and consequently, Greatest pressure, ; OS 2 * P*Pyf cog -{- ^/ (COS 2 f si] w # cos / (1 T- sin (9.} sin a> ( \ / sm In using this formula, the arc sin - - is to be taken as greater than a right angle. The following are the results of the equations 7, 8, 9, for the extreme cases : Horizontal surface, 6 ; p l = w x =. p x , 1 sin ^ =90, or the axis of greatest pressure is vertical. Natural Slope, 6 =

'' p wx sn? 6(p' + wx)~ 3(1 +sin 2 ?)'" 200. Abutting Power of Earth. If a vertical plane surface of some body which is pressed horizontally, such as a buttress, or a retaining wall, abuts or presses horizontally against a horizontal layer of earth, of the depth x, the limit of the resistance which that layer is capable of opposing to the horizontal thrust of the vertical plane is determined by the greatest horizontal pressure consistent with the stability of the earth. Hence the amount of STABILITY OF EARTH. 221 that horizontal resistance, per unit of horizontal breadth of the vertical abutting plane, is given by the equation = w x 2 1 + sin sn The centre of resistance is at below the surface of the earth. o 201. Table of Examples of the results of the formulae in Articles 197, 198, 199, and 200. 90 2 /=tan 45 -, = cotan

T sin

\ : i sin

ond, is due to the mutual friction of the overlapping portions of the beds or horizontal faces of the stones or bricks, and may be called "frictional tenacity.'" The amount of the frictional tenacity at any horizontal joint is the product of the ver- tical load upon the portion of that joint where two blocks of stone or brick overlap each other, into the co-efficient of friction, which, as stated in the table of Article 192, is about 074. Let fig. 94 A represent a portion of a wall with a horizontal top A ; and let it be required to determine 1,1,1,1 I I ^ ne frictional tenacity at a horizontal I ' ' ( ' ' [ ' ' p ' | ' ' |-*- joint B, whose depth below A is oc, the I 1 1 1 I | I I ]~l I I I I j I intensity of that tenacity per unit of "' i i i ' i i i ] ii i T~ area of a vertical plane at B, and the i i ' i ii I 1 1 I ' I \ ' i aggregate tenacity of the wall from A ' i"-*-! r^ r^r r * i r^ i- down to B, with which it is capable of 9 , . resisting a force tending to tear it into two parts by separation at the serrated dark line which extends from A to B in the figure. Let w be the weight of an unit of volume of the material of the Avail ; b the length of the overlap at each joint; t the thickness of the wall. Then wbtx is the vertical pressure on the overlapping portions of the stones or bricks at B, and consequently, ify*be the co-efficient of friction, the amount of frictional tenacity for the joint B is fivbtx (1.) The intensity of that tenacity per unit of area of a vertical plane is found by dividing its amount by the area of a vertical section of one course of stones or bricks. Let h be the depth of a BOND OF MASONRY AND BRICKWORK. 223 course ; then h t is the area of its vertical section ; and the intensity of the frictional tenacity of the joint immediately below is Let n be the number of courses from A down to B. Then the value of x for the uppermost course is = k, and for the lowest course, = n h ; and the mean value of x is ^ h ; so that the mean tenacity per course is d the mean intensity, n ~Y- 'f. Hence the amount of the aggregate frictional tenacity of the wall, from A down to B, is + * -fbtl = /^*i(^M cy / o 7, * \ i *j* j From the equations 2 and 3 it is obvious that the frictional tenacity of masonry and brickwork is increased by increasing the ratio - which the length of the overlap bears to the depth of a course. This may be effected either by increasing the length of the stones or bricks (to which the overlap bears a definite proportion, depending on the style of bond adopted), or by diminishing their depth ; but to both those expedients there is a limit fixed by the liability of stones and bricks to break across when the length exceeds the depth in more than a certain ratio, which for brick and stone of ordinary strength is about 3. For English bond (as in fig. 94 A), consisting of a course "of stretchers (or bricks laid lengthwise), and a course of headers (or bricks laid crosswise), alternately, and also for Flemish bond, in which each course consists of alternate headers and stretchers, the overlap b is one-fourth of the length, or about three-fourths of the 7 O depth, of a brick. The value of j is therefore - ; but to allow for irregularities of figure and of laying in the bricks, it is safe to make it ^ in the formulae. Substituting this in equations 2 and 3, and 224 THEORY OF STRUCTURES. o making f -, we find for the intensity of the frictional tenacity, wJiere one-half of tJie face of the wall consists of ends of f waders, wx and for the amount from the top of the wall down to the depth x, wt(x 9J rhx) -4- The tenacity of the wall in the direction of its thickness, which resists the separation of its front and back portions by splitting, is often as important as its longitudinal tenacity, and sometimes more so. Where one-half of the face, as in fig. 94 A, consists of ends of headers, the overlap of each course in the direction of the thickness is generally one-half of the length of a brick instead of one quarter ; so that - is to be made = - instead of two-thirds. n * o Hence in this case, the transverse frictional tenacity (as it may be called) is double of the longitudinal frictional tenacity, its intensity at the depth x being wx, ................................. (6.) and its amount from the top of the wall down to the depth x, for a length of wall denoted by I, wl(x* + hx\ 2 '"^ ' In a brick wall consisting entirely of stretchers, as in fig. 94 B, , -- - -- the longitudinal tenacity is double of | | | | [~ that of the wall in fig. 94 A, where I I ! one-half of the face consists of ends of i 1 > 2 1 | | | - headers. But that increased longitu- Fig 94 B dinal tenacity is attained by a total sacrifice of transverse tenacity, when the wall is more than half a brick thick. In brickwork, therefore, in which the longitudinal is of more importance than the transverse tenacity (as is the case in furnace chimneys), a sufficient amount of transverse tenacity is to be preserved by having courses of headers at intervals. The effects of this arrangement are computed as follows : Let s be the number of courses of stretchers for each course of BOND OF MASONRY. 225 headers : so that : =- of the face of the wall consists of ends of s + 1 headers, and -r-j of sides of stretchers. S ~T~ 1 Let L denote the intensity of the longitudinal frictional tenacity, and T that of the transverse frictional tenacity, at the depth x. The following table represents the values of those intensities in the extreme cases : 1 s t + 1 8+1 JU 1 1 w a? 2" '2 2 1 w x W X Now, in intermediate cases, the longitudinal tenacity will vary nearly as the proportion of sides of stretchers in the face of the wall , , and the transverse tenacity as the proportion of ends of headers; whence we have the following formulae for the intensi- ties : Consequently, for the aggregate tenacities down to a given depth x, when the length of the wall is I, and its thickness t, we have //. n ' w * (*" + 7ia; )^ ( la ) Transverse, ^ > wl (x 2 + hx) ........... (11.) To make the longitudinal and transverse frictional tenacities of equal intensity, we should have s = 2, or two courses of stretchers for one course of headers. This makes In round factory chimneys, it is usual to make s 4 ; and then we have L= i-waj; T = ^-wx ............... (13.) 226 THEORY OF STRUCTURES. The preceding formulae are applicable not only to brickwork, but to ashler masonry in which the proportions of the dimensions of the stones are on an average nearly the same with those of bricks. The formulae 9 and 1 1 may also be used to find the transverse tenacity of a rubble wall, if be taken to represent the propor- s ~T~ 1 tion of the face of the wall which consists of the ends of squared headers or bond stones, connecting the front and back of the wall together. The principles of the present Article may be relied on as a means of comparing one piece of masonry or brickwork with another, so far as their security depends on the horizontal tenacity produced by the friction of the courses. But inasmuch as the absolute numerical results have been arrived at by an indirect process, from the tangent of the angle of repose of masonry and brickwork laid with damp mortar, these results are to be considered as uncertain, and as requiring direct experiments for their verification or correc- tion. No such experiments have yet been made. 203. Friction of Screws, Keys, and Wedges. The pieces of structures in timber and metal are often attached together by the aid of keys or wedges, or of screws. The stability of those fasten- ings arises from friction, and requires for its maintenance that the obliquity of the pressure between the wedge or key and its seat, or between the thread of the screw and that of its nut, shall not exceed the smallest value of the angle of repose of the materials. 204. Friction of Rest and Friction of Motion. For some Sllb- stances, especially those whose surfaces are sensibly indented by a moderate pressure, such as timber, the friction between a pair of surfaces which have remained for some time at rest, relatively to each other, is somewhat greater than that between the same pair of surfaces when sliding on each other. This excess, however, of the friction of rest over the friction of motion, is instantly destroyed by a slight vibration ; so that the friction of motion is alone to be relied on as giving stability to a structure. In Article 192, accordingly, the co-efficients of friction and angles of repose in the table relate to the friction of motion, where there is any sensible difference between it and the friction of rest. SECTION 4. On the Stability of Abutments and Vaults. 205. stability at a Plane Joint. The present section relates to the stability of structures composed of blocks, such as stones or bricks, touching each other at joints, which are plane surfaces, capable of exerting pressure and friction, but not tension. The conclusions of the present section are applicable to structures STABILITY AT A PLANE JOINT. 227 of masonry or brickwork, uncemented, or laid in ordinary mortar ; for although ordinary mortar sometimes attains in the course of years a tenacity equal to that of limestone, yet, when fresh, its tenacity is too small to be relied on in practice as a means of resisting tension at the joints of the structure; so that a structure of masonry or brick- work, requiring, as it does, to possess stability while the mortar is fresh, ought to be designed on the supposition, that the joints have no appreciable tenacity. The mortar adds somewhat to the frictional stability, as has already been stated in the table of Article 192, and thus contributes indirectly to the frictional tenacity described in Article 202. There are kinds of cement whose tenacity becomes at once equal to that of brick, or even to that of stone. So far as the joints are cemented with such kinds of cement, a structure is to be considered as one piece, and its safety is a question of strength. A plane joint which has no tenacity is incapable of resisting any force, except a pressure, whose centre of stress falls within the joint, and whose obliquity does not exceed the angle of repose. If the resistance of the material of the blocks which meet at the joint to a crushing force were infinitely great, it would be suffi- cient for stability that the centre of pressure should fall anywhere within the joint, how close soever to the edge ; but for the actual materials of construction, it is necessary that the centre of pressure should not be so near the nearest edge of the joint as to produce a pressure at that edge sufficiently intense to injure the material. Hence it appears that the exact determination of the limiting posi- tion of the centre of pressure at a plane joint is, strictly speaking, a question relating to the strength of materials. Nevertheless, an approximation to that position can be deduced from an examina- tion of the examples which occur in practice, without having recourse to an investigation founded on the theory of the strength of materials. Some of the most useful results of such an examina- tion are expressed as follows : Let q denote the ratio which the distance of the centre of pressure of a given plane joint from its centre of figure bears to the diameter or breadth of the same joint, measured along the straight line which traverses its centre of pressure and centre of figure ; so that if t be that diameter, q t shall be the distance of the centre of pres- sure from the centre of figure. Then the ratio q is found in prac- tice to have the following values : 3 In retaining walls designed by British engineers,..., or 0-375. 3 In retaining walls designed by French engineers,..., or 0-3. 228 THEORY OF STRUCTURES. In the abutments of arches, in piers and detached buttresses, and in towers and chimneys exposed to the pressure of the wind, it has been found by experience to be advisable so to limit the deviation of the centre of pressure from the centre of figure, that the maxi- mum intensity of the pressure, supposing it to be an uniformly varying pressure (see Article 9^), shall not exceed the double of the mean intensity. As in Article 94, let P be the total pressure ; S p the area of the joint ; let -^- = p be the mean intensity of the pres- sure, which is also the intensity at the centre of figure of the joint, and at each point in a neutral axis traversing that centre of figure ; let x be the perpendicular distance of any point from that axis, and let the pressure at that point be p = p -f- a x, so that if x t be the greatest positive distance of a point at the edge of the joint from the neutral axis, the maximum pressure will be Now, by the condition stated above, p = 2p Q , and, consequently, = Rz. = a = P ...................... (1.) X 1 #; X l S If the diameter of the joint is bisected by the centre of figure, and if x (as in Article 94) be the distance of the centre of pressure from the neutral axis, we shall have and by inserting in this equation the value of # , as given by equa- tion 4 of Article 94, and having regard to the value of a, as given by equation 1 of this Article, we find /o\ an expression whose value depends wholly on the figure of the joint that is, of the transverse section of the abutment, pier, buttress, tower, or chimney. Referring to the table at the end of Article 95 for the values of the moment of inertia I, the following results are obtained for joints of different figures. In each case in which there is any difference in the values of q for different directions, the deviation of the centre of pressure is supposed to take place in that direction in which the greatest deviation is admissible that is to say, at right angles to the neutral axis for which I is a maximum ; so that if h be the diameter in that direction, x l -. STABILITY A PLANE JOINT. 229 FIGURE OF BASE. I. Rectangle ft) i S 1 Breadth . ... 1} To" hb fi II. Square Side I A* h* 1 III. Ellipse- Longer axis, 12 6 1 Shorter axis, .ik 64. A. 8 IV. Circle Diameter, I 1 V. Hollow rectangle Outside dimensions,... h ,6) 64 4 Z. Z, V JJ 8 Inside dimensions, . . . h' t VI. Hollow square Outside dimensions, ... V{ .h\. 12 6 h 2 (Ji b h' b Inside dimensions, VII. Circular ring Diameter, Outside,.... .h'f Ji\ 12 .g-n 6 A 2 h 2 + h 12 When the solid parts of the hollow square and of the circular ring are very thin, the expressions for q in Examples VI. and VII. become approximately equal to the following : VIII. Hollow square, q = ; o IX. Circular ring, q -; which values are sufficiently accurate for practical purposes when applied to square and round factory chimneys. The conditions of stability of a block supported upon another block at a plane joint may be thus summed up : Referring to fig. 93, Article 191, let A A represent the upper block, B B part of the lower block, c E the joint, C its centre of pressure, P C the resultant of the whole pressure distributed over the joint, whether arising from the weight of the upper block, or from forces applied to it from without. Then the conditions of sta- bility are the following : I. The obliquity oftJie pressure must not exceed the angle of repose, that is to say, 230 THEORY OP STRUCTURES. (3.) II. The ratio which the deviation of the centre of pressure from, the centre of figure of the joint bears to the length of the diameter of the joint traversing those two centres, must not exceed a certain fraction, whose value varies, according to circumstances, from one-eighth to three-eighths, that is to say, TE CE eE The first of these conditions is called that of stability of friction, the second, that of stability of position. 206. Stability of a Series of Blocks; Une of Resistance 5 Line of Pressures. In a structure composed of a series of blocks, or of a series of courses so bonded that each may be considered as one block, which blocks or courses press against each other at plane joints, the two conditions of sta- bility must be fulfilled at each joint. Let fig. 95 represent part of such a structure, 1, 1, 2, 2, 3, 3, 4, 4, being some of its plane joints. Suppose the centre of pressure C^ of the Fig. 95. joint 1, 1, to be known, and also the amount and direction of the pressure, as indicated by the arrow traversing Cj. With that pressure combine the weight of the block 1, 2, 2, 1, together with any other external force which may act on that block ; the resultant will be the total pressure to be resisted at the joint 2, 2, will be given in magnitude, direction, and position, and will intersect that joint in the centre of pressure C 2 . By continu- ing this process there are found the centres of pressure C 3 , C 4 , &c., of any number of successive joints, and the directions and magni- tudes of the resultant pressures acting at those joints. The magnitude and position of the resultant pressure at any joint whatsoever, and consequently the centre of pressure at that joint, may also be found simply by taking the resultant of all the forces which act on one of the parts into which that joint divides the structure, precisely as in the "method of sections" already described in its application to framework, Article 161. The centres of pressure at the joints are sometimes called centres of resistance. A line traversing all those centres of resistance, such as the dotted line R, R, in fig. 95, has received from Mr. Moseley the name of the " line of resistance ;" and that author has also shown ANALOGY OF BLOCKWORK AND FRAMEWORK. 231 how in many cases the equation which expresses the form of that line may be determined, and applied to the solution of useful problems. The straight lines representing the resultant pressures may be all parallel, or may all lie in the same straight line, or may all intersect in one point. The more common case, however, is that in which those straight lines intersect each other in a series of points, so as to form a polygon. A curve, such as P, P, in fig. 95, touching all the sides of that polygon, is called by Mr. Moseley the " line of pressures." The properties which the line of resistance and line of pressures must have, in order that the conditions of stability may be fulfilled, are the following : To insure stability of position, the line of resistance must not deviate from the centre of figure of any joint by more than a certain fraction (q) of tJie diameter oftJie joint, measured in the direction of deviation. To insure stability of friction, the normal to each joint must not make an angle greater than the angle of repose with a tangent to the line of pressures drawn through the centre of resistance of that joint. 207. Analogy of Blockwork and Framework. The point of in- tersection of the straight lines representing the resultant pressures at any two joints of a structure, whether composed of blocks or of bars, must be situated in the line of action of the resultant of the entire load of the part of the structure which lies between the two joints; and those three resultants must be proportional to the three sides of a triangle parallel to their directions. Hence the polygon formed by the intersections of the lines repre- senting the pressures at the successive joints in fig. 95, is analogous to a polygonal frame ; for the sides of that polygon represent the directions of resistances, which sustain loads acting through its angles, as in the instances of framework described in Articles 150, 151, 153, and 154, and represented in fig. 75. A structure of blocks is especially analogous to an open polygonal frame, like those in Articles 151 and 154, represented by fig. 75, with the piece E omitted because of the absence of ties. The question of the stability of a structure composed of blocks with plane joints may therefore be solved in the following manner : (1.) Determine and lay down on a drawing of the structure the line of action and the magnitude of the resultant of the external forces applied to each block, including its own weight. Either one or two of those resultants, as the case may be, will be the support- ing force or forces. (2.) Draw a polygon of external forces, like that in fig. 75* or 75**. Two contiguous sides of that polygon will represent the external forces 232 THEORY OF STRUCTURES. acting on the two extreme blocks of the series, of which one may be a supporting pressure and the other a load, or both may be supporting pressures. In either case their intersection gives the point O, from which radiating lines are to be drawn to the angles of the polygon of external forces, to represent the directions and magnitudes of the resistances of the several joints. (3.) Draw a polygon having its angles on the lines of action of the external forces, as laid down in step (1.) of the process, and its sides parallel to the radiating lines of step (2). This polygon will represent the equivalent polygonal frame of the given structure, and will have a side corresponding to each joint; and each side of the polygon (produced if necessary) will cut the corresponding plane joint in its centre of pressure, and will show the direction of the resultant pressure at the joint. Then if each centre of pressure falls within the proper limits of position, and the direction of each resultant pressure within the proper limits of obliquity, as prescribed in Article 205, the structure will be balanced ; and the conditions of stability will be fulfilled under variations of the distribution of the load, which will be the greater, the greater is the diameter of each joint; for every increase in the diameters of the joints increases the limits within which the figure of the equivalent polygonal frame may vary, and every variation of that figure corresponds to a variation in the distribu- tion of the load. 208. Transformation of Block work Structures. THEOREM. If a structure composed of Hocks have stability of position when acted on by forces represented by a given system of lines, then will a structure whose figure is a parallel projection of the original structure have stability of position when acted on by forces represented by the corre- sponding parallel projection of the original system of lines; also, the centres of pressure and the lines representing the resultant pressures at the joints of the new structure will be the corresponding projections of the centres of pressure and the lines representing the resultant pressures at the joints of the original structure. For the relative volumes, and consequently the relative weights, of the several blocks of which the structure is composed, are not altered by the transformation; and if those weights in the new structure be represented by lines, parallel projections of the lines representing the original lines, and if the other forces applied externally to the pieces of the new structure be represented by the corresponding parallel projections of the lines representing the corresponding forces applied to the pieces of the original structure, then will each external force acting on the new structure be the parallel projection of a force acting on the corresponding point of the original structure; therefore the resultant pressures at the MOMENT OF STABILITY. 233 joints of the new structure, which balance the external forces, will be represented by the parallel projections of the lines representing the resultant pressures at the corresponding joints in the original structure ; therefore (Article 62, Proposition I.), the centres of pressure, where those resultants cut the joints, will divide the diameters of the joints in the same ratios in the new and in the original structures ; therefore if the original structure have stability of position, the new structure will also have stability of position. This is the extension, to a structure composed of blocks, of the principle of the transformation of structures, already proved for frames in Article 166, and for cords and linear arches in Article 177. 209. Frictional Stability of a Transformed Structure. The ques- tion, whether the new structure obtained by transformation will possess stability of friction, is an independent problem, to be solved by determining the obliquity of each of the transformed pressures relatively to the joint at which it acts. Should the pressure at any joint in the transformed structure prove to be too oblique, frictional stability can in most cases be secured, without appreciably affecting the stability of position, by altering the angular position of the joint, without shifting its centre of figure, until its plane lies sufficiently near to a normal to the pressure as originally determined. 210. Structure not Laterally Pressed. If fig. 96 represents a structure consisting of a single series of blocks, or courses, separated by plane joints, and has no lateral pressure applied to it from without, then the centre of resistance at any one of those joints, such as D E, is simply the point C where that joint is intersected by a vertical let fall from the centre of gravity G of the part of the structure ABED which lies above that joint ; and the conditions of stability are, that no joint shall be inclined to the horizon at an angle steeper than the angle of repose, and that the point C shall not at Fi S- 96 - any joint approach the edge of the joint within a distance bearing a certain proportion to the diameter of the joint. 211. The Moment of Stability of a body or structure supported at a given plane joint is the moment of the couple offerees which must be applied in a given vertical plane to that body or structure in addition to its own weight, in order to transfer the centre of resistance of the joint to the limiting position consistent with stability. The applied couple usually consists of the thrust of a frame, or an arch, or the pressure of a fluid, or of a mass of earth, against the structure, together with the equal, opposite, and parallel, but not directly opposed, resistance of the joint to that lateral force. 234 THEORY OF STRUCTURES. The moment of stability may be different according to the position of the axis of the applied couple. The moment of that couple is determined in the following manner : Conceive a line to pass through all the limiting positions of the centre of resistance of the joint, so as to enclose a space beyond which that centre must not be found. The product oftlie weight of the structure into the horizontal dis- tance of a point in this line from a vertical line traversing the centre of gravity oftlie structure is the MOMENT OF STABILITY of the struc- ture, when the applied thrust acts in a vertical plane parallel to that horizontal distance, and tends to overturn the structure in tlie direc- tion of the given point in the line limiting the position of the centre of resistance; for that, according to Article 41, is the moment of the couple, which, being combined with a single force equal to the weight of the structure, transfers the line of action of that force parallel to itself through a distance equal to the given horizontal distance of the centre of resistance from the centre of gravity of the structure. To express this symbolically, let t be the length of the diameter of the joint where it is cut by the vertical plane traversing the centre of gravity of the structure and parallel to the applied thrust ; let j be the inclination of that diameter to the horizon ; let q t be the distance of the given limiting centre of resistance from the middle point of that diameter, and c[ t the distance from the same middle point to the point where the diameter is cut by the vertical line through the centre of gravity of the structure, and let W be the weight of the structure. Then the moment of stability is W (q rr q') t cosj; (1.) the sign { "" i being used according as the centre of resistance, and the vertical line through the centre of gravity, lie towards j opposite sides ) f ^ ^^ of ^ diametei , ( the same side j Let h denote the height of the structure above the middle of the plane joint which is its base, b the breadth of that joint in a direc- tion perpendicular or conjugate to the diameter t, and w the weight of an unit of volume of the material. Then we shall have W = n -whbt, (2.) where n is a numerical factor depending on the figure of the structure, and on the angles which the dimensions, h } b, t } make with each other; that is, the angles of obliquity of the co-ordinates BUTTRESSES IN GENERAL. 235 to which the figure of the structure is referred. Introducing this value of the weight of the structure into the formula 1, we find the following value for the moment of stability : n (q . ' O . /-. I l\ IT. ~~ > then equation 4 becomes t 2 = A 2 B t, the solution of which is * = x/A + B 2 B (5.) In detached buttresses, it is in general desirable to give q the value assigned by equation 2 of Article 205, for the reason there stated. 238 THEORY OF STRUCTURES. III. To find the obliquity of the pressure at the joint D E, we have the equation W + P sin .(6.) As the resultant of the resistance at each joint must act in a line traversing the point A, the locus of that point is the " line of pres- sures" denned in Article 206. The greatest obliquity of pressure occurs at that joint which is immediately below the point of abutment C. Let W , therefore, denote the weight of material above that joint, and the condition of stability of friction will be given by the equation P cos i tan (p .................... 7. W ft + P sin i 214. Rectangular Buttress. In a rectangular buttress, the breadth b and thickness t are constant ; and if h^ be taken to denote the height of the top of the buttress above the point C, h h -j- x will be its height above a given joint. Also, because the centre of gravity of the portion above any bed-joint is vertically above the centre of the joint, q' = 0, and y Q = t; and because * These values being substituted in equations 2, 4, 5, and 7 of Article 213, give the following results : Equation of the line of resistance - w (k Q + x) b & + P x cos i _ ^ _ _ /i \ The least thickness compatible with stability (x being the depth of the base of the wall below C) is found by making __- P si n i _ P x 1 cos i _ V ~ 2 q w (h + a?,) b ' whence follows RECTANGULAR BUTTRESS PINNACLE. 239 P sin* (2.) The least volume of material above the level of the point C which is compatible with stability of friction, is given by making P cosi 7 T .~j_ p . . = tan / T\ ) (5 A.) x v \/ \ 6 n (q ~r q)wj The vertical and horizontal components of the pressure of the water are respectively Vertical, P sin^* = ~g tanj, a Horizontal, P cosj = ^ ; Consequently the condition of stability of friction at the joint D E is given by the equation w'x* RESERVOIR- WALLS. 247 If the ratio - has been determined by means of equation 5, then we have W = nwxt = nw a? '- x so that by cancelling the common factor a? 8 , equation 6 is brought to the following form : w' -^tan* ( 8 -) 2 n w + w' x Eocample I. Rectangular Wall. In this case n=\; , = W T -- . - -, = V, 1 + sm

) \ \ -7: 7^ ; > ( A w (1 -f- sin \ 2 1 - sin g> ~ 1 - sin 8 ~ \1 - sin J 1 + sin

16 f ) ,~ , j V 219. Trapezoidal Walls. In fig. 102, let E Q represent the vertical face of a rectangular wall, suited to sustain the thrust of a given bank, and let F be the centre of resistance of the base. Make QN = 3 EF = 3 ( q) t ; then the centre of gravity g of the triangular prism of masoniy E Q N will be vertically above the centre of resistance F; therefore if that prism be removed, so as to reduce the cross section of the wall to a trapezoid with a sloping face E ]ST, the position of the centre of resistance F will not be altered, and the wall will still fulfil the condition of stability of position, the thickness t being determined q as for a rectangular wall. If q - , the thickness of the wall at the o K summit will be -5 of the thickness at the base. The face of the wall o Q N is said to batter: the rate of the batter being the ratio -=- = EQ G-.)-:- As the vertical component of the pressure on the base of the wall is diminished by this change, the obliquity of that pressure will be increased; and it may in some cases be necessary to make the base slope backwards, as in fig. 101. 220. Battering Walls of Uniform Thickness. When a wall for supporting a horizontal-topped bank is of uniform thickness, and has a sloping or curved face, as in figs. 103 and 104, its mo- ment of stability may be deter- mined with a degree of accuracy sufficient for practical purposes, in the following manner : Let E Q in each figure repre- sent the vertical face of a rec- tangular wall of the same height Fig."103. Fig. 104. centre of gravity of that rectangular wall. Then x and thickness t with the pro- posed wall, and let g be the COUNTERFORTS. W ' q t = q W X t 2 255 will be its moment of stability per unit of length. Divide the area E Q N included between the vertical face E Q and the face of the proposed wall, E N, by the height x. Then will be the distance of the centre of gravity G of the sloping or curved wall from that of the rectangular wall ; and the change of figure will increase the stability in the ratio q + q' : q; that is to say, the moment of stability will now be W (q + q') t = (q + q') w x t 2 If E N is a straight line (fig. 103), .(3.) If E N is a parabolic arc, 2QN. .(4.) a formula which is also sensibly correct when EN is an arc of a circle. Walls with a " curved batter " are usually built as shown in fig. 105, with the bed-joints perpendicular to the face of the wall. This diminishes the obliquity of the pressure on the base. 221. Foundation Courses of Retaining Walls have their width increased beyond the thick- ness of the wall by a series of steps in front, as shown in figs. 102 and 105. The objects of this are, at once to distribute the pressure over a greater area than that of any bed-joint in the body of the wall, and to diffuse that pressure more equally, by bringing the centre of resistance nearer to the middle of the base Fl S- 105> than it is in the body of the wall. The power of earth to support foundations has already been considered in Article 199. 222. Counterforts are projections from the inner face of a retain- ing wall. A wall and its counterforts, if the bond of the masonry is well preserved, constitute a wall having successive divisions of its length alternately of greater and of less thickness. The moment of stability of a wall with counterforts, per unit of length, 256 THEORY OF STRUCTURES. when the wall is well bonded, may be found, with sufficient accuracy for practical purposes, by adding together the moments of stability of one of the parts between two counterforts, and of one of the parts whose thickness is augmented by the addition of a counterfort, and dividing the sum by the joint length of those two parts. For example, let fig. 106 represent a portion of the plan, or hori- zontal section, of a vertical rectangular retaining wall whose height is h, with a row of rectangular counterforts of the same height with the wall. ' c Let t = FE be the thickness of a part of the wall between two counterforts, and b = E D its length ; let T = A B be the thickness of a coun- terforted part of the wall, including the counter- fort, and c = B C its length. The moment of stability of the first part is q w h b t s and that of the second part, q w h c T 2 . Fig. 106. Adding together those moments, and dividing their sum by the total length b + c = A F, the mean moment of stability per unit of length is found to be This is the same with the moment of stability per unit of length of a wall of the uniform thickness, (2.) which may be called the equivalent uniform wall. The quantity of masonry in the counterforted wall is to the quantity in the equivalent uniform wall in the ratio bt + cT cK, which is always less than unity; so that there is a saving of masonry (though in general but a small one) by the use of counter- forts. 223. Arches of masonry. An arch of masonry consists of a ring of wedge-formed stones, called arch-stones or voussoirs, pressing against each other at surfaces called bed-joints, which are, or ought LINE OF PRESSURES IN AN ARCH. 257 to be, perpendicular or nearly perpendicular to the soffit, or internal concave surface of the arch. The outer or convex surface of the ring of arch-stones, which may be either a curved surface parallel to the soffit, or, what is better, a series of steps, sustains the vertical pressure of that part of the load which arises from the weight of materials other than the arch-stones themselves ; and that outer surface also exerts in many cases a horizontal or inclined thrust against the spandrils and abutments. The abutments sus- tain also the thrust of the lowest voussoirs, vertical or inclined, as the case may be. Sometimes an arch springs at once from the ground, so that its abutments are its foundations. A wall standing on an arch, in the plane of the arch-ring, is called a spandril wall. The arch of a bridge requires a pair of external spandril walls, one over each face of the arch ; the space between them is filled up to a certain level with solid masonry, and above that level it is sometimes filled with earth or rubbish, and sometimes occupied by a series of internal spandril walls parallel to the external spandril walls, and having vacant spaces between them a mode of construction favourable both to stability and to lightness. In order to form a continuous platform for the road- way, the spaces between the internal spandril walls are sometimes covered with flags of some strong stone (such as slate), and some- times arched over with small transverse arches. The external spandril walls are the abutments of those arches, and must have stability sufficient to sustain their thrust : when the spandrils are filled with earth or rubbish, the external spandril walls must have stability sufficient to withstand the pressure of the filling material. In determining the conditions of stability of an arch, it is con- venient to consider only a rib, or vertical layer, of arch, abutment, and spandril, of the thickness unity (e. g., one foot). When there are spandril walls with vacant spaces between, an ideal specific gravity is to be adopted for the material of the spandrils, found by supposing the weight of the material of the spandril walls to be uniformly distributed, so as to fill the vacuities ; that is to say, let w be the weight of an unit of volume of the material of the walls, 2 T the sum of the thicknesses of all the walls, and 2 * S the sum of the widths of the spaces between them ; then in computations respecting the stability of the arch, the spandrils may be supposed to be completely filled with a material whose weight per unit of volume is 224. Line of Pressures in an Arch; Condition of Stability. According to the principles explained in Articles 206 and 207, if a s 258 THEORY OP STRUCTURES. straight line be drawn through each bed-joint of the arch-ring representing the position and direction of the resultant of the pres- sure at that joint, the straight lines so drawn form a polygon, and each of the angles of that polygon is situated in the line of action of the resultant external force acting on the arch-stone, which lies between the pair of joints to which the contiguous sides of the polygon correspond ; so that the polygon is similar to a poly- gonal frame, loaded at its angles with the forces which act on the arch-stones (their own weight included). A curve inscribed in that polygon, so as to touch all its sides, is the line of pressures of the arch. The smaller and the more numerous the arch-stones into which the arch-ring is subdivided, the more nearly does the poly- gon coincide with the curve \ and the curve or line of pressures represents an ideal linear arch, which would be balanced under the continuously-distributed forces which act on the real arch under consideration. From the near approach of this linear arch to the polygon whose sides traverse the centres of resistance of the bed- joints, the points where the linear arch cuts those joints may be taken without sensible error for the centres of resistance. Now in order that the stability of the arch may be secure, it is necessary that no joint should tend to open either at its outer or at its inner edge ; and in order that this may be the case, the centre of resistance of each joint should not deviate from the centre of the joint by more than one-sixth of the depth of the joint ; that is to say, the centre of resistance should lie within the middle third of the depth of the joint ; whence follows this THEOREM. The stability of an arch is secure, if a linear arch, balanced under the forces which act on the real arch, can be drawn within the middle third of the depth of the arch-ring. It has already been stated that the tenacity of fresh mortar is not sufficiently great to be taken into account in determining the stabi- lity of masonry ; and hence, where cement is not used, all horizon- tal or oblique conjugate forces which maintain the equilibrium of the arch-ring must be pressures, acting on the arch from without inwards. The linear arch, therefore, is limited in such cases to those forms which are balanced under pressures from, without alone; that is to say, that the intensity of the horizontal or conjugate pressure, denoted by p y in Article 185, equation 4, must not at any point be negative. It is true that arches have stood, and still stand, in which the centres of resistance of joints fall beyond the middle third of the depth of the arch-ring ; but the stability of such arches is either now precarious, or must have been precarious while the mortar was fresh. When tenacity to resist horizontal or oblique tension is given to ANGLE, JOINT, AND POINT OF EUPTURE. 259 the spandrils of an arch, and to the joints between them and the arch-stones, by means of cement, hoop-iron bond, iron cramps, or otherwise, the conjugate tension denoted by p y must not at any point exceed a safe proportion of that tenacity ; that is to say., about one-eighth. By this means stability may be given to arches of seemingly anomalous figures ; but such structures are safe on a small scale only. 225. Angle, Joint, and Point of Rupture. The first step towards determining whether a proposed arch will be stable, is to assume a linear arch parallel to the intrados or soffit of the proposed arch, and loaded vertically with the same weight, distributed in the same manner. The size of this assumed linear arch is a matter of indifference, provided each point in it is considered as subjected to the same forces which act at the corresponding joint in the real arch ; that is, the joint at which the inclination of the real arch to the horizon is the same with that of the assumed arch at the given point. The assumed arch is next to be treated as a stereostatic arch, according to the method of Article 185; and by equation 4 of that Article is to be determined, either a general expression or a series of values of the intensity p y of the conjugate pressure, horizontal or oblique, as the case may be, required to keep the arch in equilibrio under the given vertical load. If that pressure is nowhere negative, a curve similar to the assumed arch, drawn through the middle of the arch-ring, will be either exactly or very nearly the line of pres- sures of the proposed arch ; p y will represent, either exactly or very nearly, the intensity of the lateral pressure which the real arch, tending to spread outwards under its load, will exert at each point against its spandril and abutments ; and the thrust along the linear arch at each point will be the thrust of the real arch at the corre- sponding joint. On the other hand, if p y has some negative values for the assumed linear arch, there must be a pair of points in that arch where that quantity changes from positive to negative, and is equal to nothing. The angle of inclination i Q at that point, called the angle of rupture, is to be determined by solving equation 1 of Article 187. The corresponding joints in the real arch are called the joints of rupture; and it is below those joints only that conjugate pressure from with- out is required to sustain the arch. In fig. 107, let BC A represent one-half of a symmetrical arch, O Y a horizontal axis of co-ordinates in or above the spandril, K L D E an abutment, and C the joint of rupture, found by the method already described. The point of rupture, which is the centre of resistance of the joint of rupture, is somewhere within the middle third of the depth of that joint; and from that point 260 THEORY OF STRUCTURES. down to the springing joint B, the line of pressures is a curve similar to the assumed linear arch, and parallel to the intrados, being kept in equilibrio by the lateral pres- sure between the arch and its spandril and abutment. From the joint of rupture C to the crown A, the fact that the assumed linear arch would require lateral tension to keep it in equilibrio, shows that the true line of pressures must be a flatter curve than the assumed linear arch ; the figure of the true line of pressures being determined by the condition, that it shall be a linear arch balanced under vertical forces only; that is to say, that the horizontal com- ponent of the thrust along it at each point is a constant quantity, and equal to the horizontal component of the thrust along the arch at the joint of rupture. That horizontal thrust, denoted by H , is found by means of equa- tion 2 of Article 187, and is the horizontal thrust of the entire arch. [If the arch is distorted, conjugate thrust is to be read instead of "horizontal thrust" wherever that phrase occurs.] The only point in the line of pressures above the joints of rup- ture which it is important to determine, is that which is at the crown of the arch, A ; and it is found in the following manner : Find the centre of gravity of the load between the joint of rup- ture C and the crown A ; and draw through that centre of gravity a vertical line. Then if it be possible, from one point in that vertical line, to draw a pair of lines, one parallel to a tangent to the soffit at the joint of rupture, and the other parallel to a tangent to the soffit at the crown, so that the former of those lines shall cut the joint of rup- ture, and the latter the keystone, in a pair of points which are both within the middle third of the depth of the arch-ring, the stability of the arch will be secure ; and if the first point be the point of rupture, the second will be the centre of resistance at the crown of the arch, and the crown of the true line of pressures. When the pair of points related as above do not fall at opposite limits of the middle third of the arch-ring, their exact positions are to a small extent uncertain ; but that uncertainty is of no conse- quence in practice. Their most probable positions are equi-distant from the middle line of the arch-ring. Should the pair of points fall beyond the middle third of the arch-ring, the depth of the arch-stones must be increased. 226. Thrust of an Arch of Masonry. The line of pressures, or equivalent linear arch, of an arch of masonry, with its point of rup- ABUTMENTS SKEW ARCHES. 261 ture and total thrust, having been determined by the methods described in the two preceding Articles, the distribution of that thrust, and the line of action of its resultant, are to be found by the methods of Article 187. 227. Abutments of Arches. The abutment of an arch, when it is not simply a foundation, is a buttress, or a wall with or without counterforts, which is bounded, or may be considered as bounded by a vertical face L D (fig. 107) towards the arch. Two external forces are applied to the abutment of an arch besides its own weight, viz., the vertical load of the half-arch, P, whose resultant acts through B, the centre of resistance of the springing joint, and the thrust H, found in amount and position by methods already referred to, which acts through B also if the angle of rupture is equal to or greater than the inclination of the arch at B; and which, if there is either no joint of rupture, or a joint of rupture above B, is distributed between B and A, or B and C, as the case may be. The resultant of the vertical load and conjugate thrust being taken as the entire pressure applied to the abutment, its conditions of stability and requisite dimensions are to be found by the methods described in Articles 213, 214, and 222. For the abutment of an arch, as for the arch-ring, the centre of resistance should fall within the middle third of the base, so that the proper value of q is one-sixth. If the figure of an arch be transformed by parallel projection, the proper figures for the abutments of the new arch are the corre- sponding parallel projections of the original abutments. 228. Skew Arches are of figures derived from those of symmetri- cal arches by distortion in a horizontal plane. The eleva- tion of the face of a skew arch, and every vertical section par- allel to its face, being similar to the corresponding elevation and vertical section of a sym- metrical arch, the forces which act in a vertical layer or rib of a skew arch with its abut- ments, are the same with those which act in an equally thick vertical layer of a symmetrical arch with its abutments, of the same dimensions and figure, and similarly and equally loaded. Fig. 108. Fig. 109. Fig. 108 represents a plan of a skew arch, with count erforted abutments. The angle of skew, or obliquity, is the angle which the 262 THEORY OF STRUCTURES. axis of the archway, A A, makes with a perpendicular to the face of the arch, B C A B. The span of the archway, " on the square," as it is called (that is, the perpendicular distance between the abut- ments), is less than the span on the skew, or parallel to the face of the arch, in the ratio of the cosine of the obliquity to unity. It is the span on the skew which is equal to that of the corresponding symmetrical arch. The best position for the bed-joints of the arch-stones is perpen- dicular to the thrust along the arch. If, therefore, there be drawn on the soffit of a skew arch, a series of parallel curves, made by the intersections of the soffit with vertical planes parallel to the face of the arch, the best forms for the bed-joints will be a series of curves drawn on the soffit of the arch so as to cut the whole of the former series of curves at right angles, such as C C in figs. 108 and 109. Joints of the best form being difficult to execute/ spiral joints are used in practice as an approximation. 229. Groined Vaults. A groined vault, represented in plan, looking upwards, by fig. 110, is formed by the intersection of two archways. The ribs at the edges where the soffits of the archways intersect and interrupt each other, are called the groins. The portions of the arches which form the groined vault, properly speaking, abut against the groins ; the groins themselves, and the four inde- pendent portions of the archways, abut against four buttresses at the corners of the vault. The crown of the vault is the point where the groins meet. The line marked B' is the length from the crown to the face of one of the arch- ways; and B is the breadth of the por- Fig. 110. tion of one of the buttresses against which that archway abuts, whether directly or through the groin. The thrust due to the length of archway B' is concentrated upon the breadth of abut- B' ment B its intensity is therefore increased in the ratio ; and if t be the thickness which an abutment requires to withstand the thrust of the plain archway, the thickness D required for the but- tress, in a direction perpendicular to B, will be At the left-hand side of the figure, the buttresses are compound and rectangular : at the right-hand side, a single diagonal buttress PIEKS AND ABUTMENTS. 263 is opposed to the thrust of each groin, and to the combined thrusts of the two archways which abut against it. The breadth of the dia- gonal buttress being the resultant of the breadths of the compound buttresses, its thickness is simply equal to theirs. 230. clustered Arches are arched ribs, of which several spring from one buttress, as is shown in plan in fig. 111. The thrust against the buttress is the resultant of the thrusts of the ribs ; the vertical pressure is the sum of their loads. 231. Piers of Arches. A pier is a pillar against which two or more arches abut, in such a manner s ' that their horizontal thrusts balance each other, so that the pier has only to sustain the vertical pressure of the half-arches which rest on it. The piers of a bridge or viaduct are usually oblong walls, of a length equal to that of the soffits of the arches, two of which spring from the opposite sides of each pier. It is customary to make the thickness of a pier, at the springing of the arches, from one-sixth to one-ninth of the span of the arches which it sustains. Mr. Hosking, in his Treatise on Bridges, has pointed out, that this thickness is usually greater than is necessary ; and that there is in general no reason that the thickness of the pier should be more than is just sufficient to support the rings of arch-stones that spring from it. If one of two arches which abut against the same pier falls, the other arch, having its thrust unbalanced, usually overthrows the pier, and consequently falls also ; so that if a viaduct consists of a series of arches with piers between, the fall of a single arch causes the destruction of the whole viaduct. To lessen the damage caused by accidents of this kind, it is customary in long viaducts, to introduce at intervals what are called abutment piers, which have stability sufficient to resist the thrust of a single archj so that when an arch falls, the destruction is limited to the division of the viaduct between the two nearest abutment piers. In some important bridges over large rivers, where it has been considered advisable to spare no expense in order to render the structure durable, each pier is an abutment pier. 232. Open and Hollow Piers and Abutments. In Some cases the piers and abutments of bridges, in order to save materials, and to diminish the pressure on the foundations, are made with arched openings through them, or with rectangular hollows in their in- terior. The bottoms of such openings or hollows should be closed, when they are small by courses of large stones, and when they are large by inverted arches, in order that the area of the foundation, over which the pressure is distributed, may be as large as if the building were solid. The moment of stability of an abutment, with arched openings 264 THEORY OF STRUCTURES. through it, or hollows in its interior, is less than that of a solid abutment of the same external dimensions, very nearly in the same ratio in which the moment of inertia of the horizontal section of the abutment is diminished by means of the vacuities. (See Article 95.) 233. Tunnels. If the depth of a tunnel beneath the surface of the ground is great compared with the height of its archway, the proper form for the line of pressures, which must lie within the middle third of the thickness of its arch, is the elliptic linear arch of Article 180, in which the ratio of the horizontal to the vertical semi-axis is the square root of the ratio of the horizontal to the vertical pressure of the earth, as already shown in Article 180, equation 5, and Article 197, equation 3; that is to say, horizontal semi-axis _ /p y / /I sin ?, , be the projections, parallel to x, y, z, respectively, of the displacement of a particle in a strained solid from its position when the solid is free, expressed as functions of x, y, z. Then _ = d-y + ~dz> '"'-dl + dTx'' dv dl y _ _ I _ ~~ dx dy 251. Analogy of Stresses and Strains. It has been shown in Article 104, that the elastic forces exerted on and by an originally cubical particle, which constitute the state of stress of the solid at the point where that particle is situated, may be resolved into six elementary stresses, viz.: three normal stresses, perpendicular re- spectively to the three pairs of faces, and tending directly to alter the three linear dimensions of the particle and three pairs of tangential stresses acting along the double pairs of faces to which they are applied, and tending directly to alter the angles made by such double pairs of faces. To reduce the state of stress at a given point expressed by a system of six elementary stresses referred to one system of rectangular co-ordinates to an equivalent system of elementary stresses referred to a new system of rectangular co-ordi- nates, equations have been given in Articles 105, 106, 107, 108, 109, and 112. The whole of those equations are virtually compre- hended under the following theorem : Let p xx , p yy , p zz , be the CO-EFFICIENTS OF ELASTICITY AND PLIABILITY. 277 three normal stresses, and p yz , p zx , p xy , the three tangential stresses ; conceive the surface whose equation is Transform this equation so as to refer the same surface to the new set of axes ; the six co-efficients of the transformed equation will be the six elementary stresses referred to the new axes. For the complete investigation of this subject, see M. Lame's Legons sur la Theorie mathematique de I* Elasticite des Corps solides, Paris, 1852. The above equation is tranformed into the equation of Article 249 by substituting respectively , /3, y, A, ^, V) for p xx , p yy , p zi , 2p yz , 2 p^, 2 p xy ', and by making corresponding substitutions in all the equations of Articles 105, 106, 107, 108, 109, and 112, they are made applicable to strains instead of stresses. 252. The Potential Energy of Elasticity of an originally cubic particle in a given state of strain is the work which it is capable of performing in returning from that state of strain to the free state ; and is the product of the volume of the particle by the following function : U = 327> ...... 2,159,100. j- ...................... 15,121,000 ...... 5,643,800. - ..................... 14,300,000 ...... 5,746,000. a a ..................... 0-0000000699 ...... 0-0000001740. i ..................... 0-0000000239 ...... 0-0000000575. C ..................... 0-0000001877 ...... 0-0000004631. fc ..................... 0-0000000661 ...... 0-0000001772. 280 THEORY OF STRUCTURES. 259. The General Problem of the Internal Equilibrium of an Elas- tic Solid is this : Given the free form of a solid, the values of its co-efficients of elasticity, the attractions acting on its particles, and the stresses applied to its surface : to find its change of form, and the strains of all its particles. This problem is to be solved, in general, by the aid of an ideal division of the solid (as already described) into molecules rectangular in their free state, and re- ferred to rectangular co-ordinates. For isotropic solids, some par- ticular cases are most readily solved by means of spherical, cylin- drical, or otherwise curved co-ordinates. The general equation of internal equilibrium in a solid acted on by its own weight, has already been given in Article 116, equation 2. If, in that equa- tion, the values of the stresses in terms of the strains, expressed, as in Article 250, in terms of the displacements of the particles, be introduced, equations are obtained, which being integrated, give the displacements, and consequently the strains and stresses. The general problem is of extreme complexity ; but the cases which occur in practice, and to which the remainder of this chapter re- lates, can generally be solved with sufficient accuracy by compara- tively simple approximate methods. Most of those approximate methods are analogous to the " method of sections " described in its application to framework in Article 161. The body under consideration is conceived to be divided into two parts by an ideal plane of section ', the forces and couples acting on one of those two parts are computed, and they must be equal and opposite to the forces and couples resulting from the entire stress at the ideal sectional plane, which is so found. Then as to the distribution of that stress, direct and shearing, some law is assumed, which if not exactly true, is known either by experiment or by theory, or by both combined, to be a sufficiently close approximation to the truth. Except in a few comparatively simple cases, the strict method of investigation, by means of the equations of internal equilibrium, has hitherto been used only as a means of determining whether the ordinary approximative methods are sufficiently close. SECTION 2. On Relations between Strain and Stress. 260. Ellipse of strain. In Articles 249, 251, 252, 253, 254, 256, and 257, of the preceding section, certain general principles respecting the relations amongst strains, and the analogies and other relations between strain and stress, are stated without a detailed demonstration. In the present section the more simple cases of those principles, to which there will be occasion to refer in the sequel, are to be demonstrated. ELLIPSE OF STRAIN. 281 Let a solid body be supposed to undergo a strain, or small alteration of dimensions and figure, of such a nature that all the displacements of its particles from their original positions are parallel to one plane ; and let that plane be repre- sented by the plane of the paper in fig. 114. In the first instance, let the state of strain of the body be uniform throughout ; that is, let all parts of the body which originally were equal and similar to each other, continue equal and similar to each other notwithstand- ing their alteration of dimensions and Eound any centre O, with the radius unity, let a circle be traced amongst the particles of the body, B C A F. Because of the uniformity of the strain, this circle will be changed into a parallel projection of a circle; that is, into an ellipse. Let b c af be that ellipse, and O a and b its semi-axes, the body being so placed in its strained condition that the central par- ticle O may remain unchanged in position, in order that the circle and ellipse may be the more easily compared. Then the particle which was at A is displaced to a, and the particle which was at B is displaced to b ; and particles which were at points in the circle, such as C and F, are displaced to corresponding points in the ellipse, such as c and/! Fig. 114. Fig. 115. In the direction O A, the body has undergone the extension A a = and in the direction O B, at right angles to A, the extension and the combination of those two extensions or elementary direct strains, in rectangular directions, constitutes the state of strain of the body parallel to the given plane; that state of strain being completely known, when , /8, and the directions of the pair of rectangular axes of strain O A, OB, are known. One or both of the elementary strains might have been compres- sive, instead of tensile, in which case one or both of the quantities de- "~ting them would have been negative, to express diminution of size. 282 THEORY OF STRUCTURES. A square whose sides are unity, and parallel to O A and B, being traced amongst the particles of the body in the free state, is converted by the strain into a rectangle whose sides are 1 -f- * and 1 + ft, and still parallel to A and O B. Let it now be required to express the state of strain of the body with reference to two new rectangular axes, O C and F , that is to say, to find the alterations of dimensions and figure produced by the strains on a figure originally square, described on O C and O F. Let x = O X, y = O Y, be the original co-ordinates of C, and x' = OX', y = OY', those of F; and let the angle A C = 90 - A O F = 6. Then x = cos 6 = y' y = sin 6 x f . Also, let x + = Y D, y + = O Y -f- DC, be the co-ordinates of c,_the new position of C ; and let x 1 -f- % = Y 7 , y' + u' = OY' + Gf } be the co-ordinates off, the new position of F. Then because of the uniformity of the strain, the component displacements f, u, ', n', have the following values : g = C D ot. x a, cos 6 ; = FG = a x ' = * y = sin 6 ; it = G = pi = - /3cos &. O c and O/ are the sides of the oblique parallelogram into which the square on O C and O F has been transformed by the strain. The relations between the new and the original figure are distin- guished into two direct strains and a distortion, in the following manner : From c let fall c M perpendicular to O C M; and fromy let fall /N perpendicular to F N. Then a' = C M is the extension of O C \ ft' = F~N is the extension of Olf; and V = c M +y*]Sr is the distortion or deviation from rectan- gularity ; and the values of those three new elementary strains, relatively to the pair of axes which make the angle 6 with the principal axes A, O B, in terms of the principal elementary stresses, , ft, are as follows : ELLIPSOID OF STRAIN. 283 cos 6 + >? sin & = a. cos 2 d + ft sin 2 ; I' sin r! cos 4 = a sin 2 ^ + /3 cos 2 6 ; = I sin 6 >j cos 4 + ' cos ^ + >/ sin 4 = 2 (oe /3) cos ^ sin 6. Those three equations are exactly analogous to the equations 3 and 4 of Article 1 1 2, from which they may be formed by substituting <* for p x , and ft for p y in both equations ; and then, in the first place, a' for p M and 6 for x n; in the second place, ft' for jt? n , and (90 ^) for ic n, and in the third place, v for jt? and 6 for a? T&. This illustrates the general principle of analogy of stresses and strains stated in Article 251. That principle is further illustrated by the following geometrical construction of the preceding problem. In fig. 115, make o a = , o b = ft, and draw the ellipse b c af, and the circumscribing circle C a F. Let ^ a o C = 0, and let o F be perpendicular to o C, so that those lines represent the direction of the new rectangular axes, to which the strain composed of et, and ft is to be referred. Draw C c, F/, parallel to o b, cutting the elli] in c and/, from which points respectively draw c m -J- o C, ai -L o F. Then o m = *', o n = ft', 2 c m = 2fn = *', are the components of the strain, referred to the new axes ; and the ellipse of strain b c af is analogous to the ellipse of stress of Article 112. The results of the preceding investigation are applicable not only to an uniform state of strain, but to a state of strain varying from point to point of the body, provided the variation is continuous, so that it shall be possible, by diminishing the space under considera- tion, to make the strain within that space deviate from uniformity by less than any given deviation. 261. Ellipsoid of strain. A strain by which the size and figure of a body are altered in three dimensions may be represented in a manner analogous to that of the preceding Article, by conceiving a sphere of the radius unity to be transformed by the strain into an ellipsoid, and considering the displacement of various particles, from their original places in the sphere, to their new places in the ellipsoid. The three axes of the ellipsoid are the principal axes of strain, arid their extensions or compressions, as compared with the coincident diameters of the sphere, are the three principal elementary strains which compose the entire strain. It is by this method, which it is unnecessary here to give in detail, that the general principles stated in Articles 249 and 251 are arrived at. 284 THEORY OF STRUCTURES. 262. Transverse Elasticity of an Isotropic Substance. Let the two principal elementary strains in one plane be of equal magnitude, but opposite kinds ; that is, supposing the strain in fig. 114 along O A to be an extension, , let the strain along O B be a compression, /3 = a. The ellipse will fall beyond the circle at A, and as much within it at B, and will cut it at an intermediate point near the middle of each quadrant. Take a pair of new axes bisecting the right angles between the original axes ; that is, let 6 = 45 ; then the equations 2 of Article 260 give the following result : = 0j 0=0; = that is to say, an extension, and an equal compression, along a pair of rectangular axes, are equivalent to a simple distortion relatively to a pair of axes making angles of 45 with the original axes; and the amount of tfie distortion is double tJiat of either of the two direct strains which compose it ; a proposition which is otherwise evident, by con- sidering that a distortion of a square is equivalent to an elongation of one diagonal, and a shortening of the other, in equal proportions. The body being isotropic, or equally elastic in all directions, let A be its direct and B its lateral elasticity ', then the pair of principal strains , /3 = , will be accompanied by a pair of principal stresses along A and OB respectively, given by the following equations : along O A, p x = A + B ft = (A - B) ; OB,^ = B* + A/3 = (B - A) = -p,; (2.) that is to say, there will be a pull along O A, and an equal thrust along O B. It has already been proved, in Article 111, that such a pair of principal stresses, of equal intensities and opposite kinds, are equivalent to a pair of shearing stresses of the same intensity on a pair of planes making angles of 45 with the axes of principal stress; or taking p t to represent the intensity of the shearing stress on each of a pair of planes normal to the new pair of axes, p t =p, = (A-H.); (3.) but if C be the co-efficient of transverse elasticity of the substance, we have also P, = C,; (4.) and consequently, for an isotropic substance, C= ^,.. ...(*.) CUBIC AND FLUID ELASTICITY. 285 or the transverse elasticity is half the difference of tlie direct and lateral This is the demonstration of a principle already stated in Article 256. The corresponding principle for pliabilities, viz. : that the transverse pliability is twice tlie sum of the direct and lateral extensi- bilities, is demonstrated by a similar process, of which the steps may be briefly summed as follows : . Q. E. D ................... (6.) 263. Cubic Elasticity. If the three rectangular dimensions of a body or particle are changed in the respective proportions 1 + , 1 + 0, 1 -f- y, its volume is altered in the proportion and when the elementary strains , ft, y, are very small fractions ris sensibly equal to l+ Consequently, as in Article 249, may be called the cubic strain, or alteration of volume. In an isotropic substance, the three rectangular direct stresses which accompany those three strains are (1.) Tlie third part of the sum of those stresses, which may be called the mean direct stress, has the following value : P +Pvy + J> = (A + 3 B) ^ + ^ + y) . ........... (2>) o o The co-efficient contained in this expression, being the ratio of the mean direct stress to the cubic strain, is the cubic elasticity, or elasticity of volume, already mentioned in Article 256, its reciprocal being the cubic compressibility. 264. Fluid Elasticity. The distinction between solids and fluids is well illustrated by applying to fluids the equations of Articles 262 and 263. Fluids offer no resistance to distortion, that is, they have no transverse elasticity; therefore for them 286 THEORY OF STRUCTURES. Introducing this into the equations 1 and 2 of Article 263, we find P** =P yy =P** = B ( + ft + y), and the cubic elasticity The equality of the pressures in all directions at a given point in a fluid has already been proved by another process in Article 110. The equations of Article 256 show the pliabilities of a perfect fluid to be infinite, with the exception of the cubic compressibility, which is ^c- . SECTION 3. On Resistance to Stretching and Tearing. 265. Stiffness and Strength of a Tie-Bar. If a cylindrical or prismatic bar, whose cross section is S (as in Article 97, fig. 46), be subjected to a pull whose resultant acts along the axis of figure of the bar, and whose amount is P, the intensity of the pull will be uniform on each cross section of the bar, and will have the value This direct stress will produce a strain, whose principal element will be a longitudinal extension of each unit of length of the bar, of the value (2.) where a denotes the direct extensibility, and E its reciprocal, the modulus of elasticity, or co-efficient of resistance to stretching, as explained in Articles 256 and 257. Let x denote the length of the bar, or of any portion of it, in the free or unloaded state; that length, under the tension p, becomes (1 + ) x. The co-efficient is nearly constant until p passes the limit of the proof stress; but after that limit has been passed, that co-efficient diminishes ; that is to say, the extension * increases faster than the intensity of the stretching force p, until the bar is torn asunder. The ultimate strength of the bar, or the total pull required to tear it instantly asunder the proof strength, or the greatest pull TIE-BAR SUDDEN PULL. 287 of which it can safely bear the long-continued or repeated applica- tion and the working load are computed by means of the formula p=/,orP=/S, (3.) where /represents the ultimate tenacity, the proof tenacity, or the working stress, as the case may be. The toughness of the bar, or the extension corresponding to the proof load, is given by the formula =! < 4 -> where /is the proof tenacity. 266. The Resilience, or spring of the bar, or the work performed in stretching it to the limit of proof strain, is computed as follows : x being thfe length, as before, the elongation of the bar under the proof load ig fx --v'> the force which acts through this space has for its least value 0, for its greatest value P =/S, and for its mean value ^~ ; so that the 2t work performed in stretching the bar to the proof strain is / l x - Z! ?*f n\ 2 ' E " E ' 2 The co-efficient ~, by which one-half of the volume of the bar is Hi multiplied in the above formula, is called the MODULUS OF RESI- LIENCE. /*S 267. Sudden Pull. A pull of , or one-half of the proof load, 2] being suddenly applied to the bar, will produce the entire proof f strain of ", which is produced by the gradual application of the Hi proof load itself ; for the work performed by the action of the con- /*S stant force ~r- through a given space, is the same with the work a performed by the action, through the same space, of a force increas- ing at an uniform rate from up to/S. Hence a bar, to resist with safety the sudden application of a given pull, requires to have twice the strength that is necessary to resist the gradual applica- tion and steady action of the same pull. The principle here applied belongs to the subject of dynamics, and is stated by anticipation, on account of its importance as THEORY OF STRUCTURES. respects the strength of materials. It is the chief reason for mak- ing the factor of safety for a moving load considerably greater than for a steady load (see Article 247). 268. A Table of the Resistance of Materials to Stretching and Tearing, by a direct pull, in pounds per square inch, is given at the end of the volume. The tenacity, or resistance to tearing, given in that table, is in each case the ultimate tenacity, being the quantity as to which experimental data are most abundant and precise. The proof ten- acity and working tension, when required, are to be found by dividing the ultimate tenacity by the proper factors, according to Article 247. The modulus of elasticity in each case is given from experiments made within the limits of proof strain. Both co-efficients, for fibrous substances, have reference to the effects of tension acting along tlie fibres, or "grain." Both the ten- acity and the elasticity of timber against forces acting across the grain are much smaller than against forces acting along the grain, and are also of uncertain amount, the results of experiments being few and contradictory. 269. Additional Data. The following are a few experimental results in addition to those given in the table : Welded joint of a wrought iron retort. Ultimate tena- city, by a single experiment, in Ibs. per square inch,... 30750- Iron wire-ropes. Strength in Ibs., for each Ib. weight per fathom, Ultimate, 4480- Proof,.... 2240- Working load of ultimate, or J of proof strength. Hempen cables. Ultimate strength = (girth in inches) 2 x 448 Ib. Leathern belts. Working tension in Ibs. per square inch, according to General Morin 285- Chain cables, when the tendency of each link to collapse is resisted by means of a cross-bar, as shown in fig. 116, have a strength per square inch of cross section of the link equal to that of the iron of which they are made, when it is in the form of bars. 270. The strength of Rivetted Joints of iron plates is given in the table, in Ibs. per square inch of section of 'tfie plate, from the experiments of Mr. Fairbairn. The strength of a double-rivetted joint is seven-tenths of that of the iron plate, simply because of three-tenths of the breadth of the plate being punched out in each Pig. 116. row of rivet-holes. The strength of a single-ri vetted joint is diminished not merely by the removal of the iron at the CYLINDERS BOILERS PIPES. 289 rivet-holes, but by the unequal distribution of the stress. Rivetted joints will be further considered in the sequel. 271. Thin Hollow Cylinders; Boilers; Pipes. Let q denote the uniform intensity of the pressure exerted by a fluid which is confined within a hollow cylin- der of the radius r, and of a thickness, t, which is small as compared with that radius. The demonstration in Article 179 shows, that if we consider a ring, being a portion of the cylin- der of the length unity, the tension on that ring will be Fig. 117. P = *', ............................ (I-) being the force per unit of length with which the internal pressure tends to split the cylinder from end to end. The sectional area of the ring under consideration is t. Then assuming, what is very nearly correct, that the tension is uniformly distributed, the intensity of that tension is The ratio of thickness to radius, which a thin hollow cylinder requires, to fit it for a given intensity of bursting pressure, proof pressure, or working pressure, is given by the formula /being the ultimate tenacity, the proof tension, or the working ten- sion, as the case may be. It is considered prudent, in STEAM-BOILERS, to make the working tension only one-eighth of the ultimate tenacity. The joints of plate iron boilers are single-ri vetted ; but from the mariner in which the plates break joint, analogous to the bond in masonry, the tenacity of such boilers is considered to approach more nearly to that of a double-ri vetted joint than that of a single-ri vetted joint. Mr. Fairbairn estimates it at 34,000 Ibs. per square inch ; so that the values of /for wrought iron boilers may be thus stated : Bursting tension, ................ 34,000 Proof tension, ..................... 17,000 Working tension, ............... 4,250 For CAST IRON WATER PIPES, the working tension may be made one-sixth of the bursting tension, which for cast iron, on an average, is 16,500 Ibs. per square inch ; that is to say, the values of /are Bursting tension, ............... 16,500 Proof tension (one-third), ...... 5,500 Working tension, ............... 2,750 290 THEORY OF STRUCTURES. For steam-pipes, as for steam-boilers, the factor of safety should be 272. Thin Hollow Spheres. Let fig. 117 now be conceived to represent a diametral section of a thin hollow sphere, filled with a fluid which presses from within with the intensity q. The area of the fluid cut by the section is vr 2 ; hence the whole force to be resisted by the tenacity of the section of the spherical shell is P = irqr' i ........................... (1.) The area of the section of the spherical shell, supposing the thick- ness t to be small as compared with the radius r, is very nearly S = 2 7 rrt; ........................... (2.) hence assuming, what is very nearly correct, that the tension is uniform, its intensity is or, one-half of the tension round a cylindrical shell having the same internal pressure, and the same proportion of thickness to radius ; so that, in these circumstances, the sphere is twice as strong as the cylinder. Equation 3 gives also the longitudinal tension in a thin hollow cylinder, which, being only one-half of the circumferential tension round the cylinder, does not require to be considered in practice. The proper ratio of thickness to radius in a thin hollow sphere is given by the formula f being the bursting, proof, or working tension, according as q is the bursting, proof, or working pressure. 273. Thick Hollow Cylinder. The assumption that the circum- ferential tension, or hoop-tension as it may be called, in a hollow cylinder is uniformly distributed, is approxi- mately true only when the thickness is small as compared with the radius ; for if a ring of the cylinder be conceived to be divided into several concentric hoops, one within another, the tension of the innermost hoop balances part of the radial pressure of the confined fluid, so that a dimin- Fig. 118. ished radial pressure is transmitted to the second hoop, which has therefore a less tension than the first hoop, and HO on. THICK HOLLOW CYLINDER. 291 Equation 2 of Article 271 gives the mean hoop-tension in a ick as well as in a thin cylinder ; but it is not the mean, but the greatest hoop-tension (that is, the tension round the inner surface of the cylinder), which is limited by the strength of the material. The object of the present investigation is to show what law the variation of hoop-tension follows, and thence, what relation the maximum tension bears to the fluid pressure. To make the solution perfectly general, it will be supposed that the cylinder is pressed from without as well as from within. Let fig. 118 represent a cross section of the cylinder; let R denote its external and r its internal radius. Let q denote the fluid pressure from within, and q l that from without ; p the hoop-tension at the inner surface of the cylinder, and p^ the hoop-tension at the outer surface. Consider, as before, a ring whose length, parallel to the axis of the cylinder, is unity. The radial section of that ring, from r to R in fig. 118, has to sustain the difference between the total pressures from within and without, in a direction perpendicular to the radius O r R, on a quadrant bounded by that radius. That difference is Conceive the ring to be divided into an indefinite number of con- centric hoops, each of the thickness d r, and exerting a tension of the intensity p ; then the total hoop-tension will be /R f pdr = q Q r q l R (1.) From the symmetry of the ring and f the forces acting on it in all directions round the centre O, it is obvious that the axes of stress of any particle of metal must be respectively in the direction of a radius, and perpendicular to that direction. The principal stresses at any particle are a radial pressure, q (which for each particle at the inner surface is q , and for each particle at the outer surface, q } ) and a hoop-tension p. As in the case of the ellipse of stress, Article 112, we may con- ceive this pair of principal stresses to be made up of two component pairs, viz. : A pair of equal stresses of the same kind, constituting a fluid pressure or tension, whose common intensity, stated so as to be a tension when positive, a pressure when negative, is P 9 _ 2 and a pair of equal stresses of contrary kinds, whose common intensity is '292 THEORY OF STRUCTURES. Thus we have p = n -^- m, q -=.n m ; and the problem is to be solved by first supposing m to act alone, then supposing n to act alone, and lastly combining their effects ; observing, that the only solutions of equation 1 which are admissible, are those which are true for all values of R and r. CASE 1. Equal and similar stresses, or n = 0. In this case p = q = m, showing, that instead of a radial pressure, there is a radial tension equal to the hoop-tension, and constituting along with it simply a fluid tension of the intensity m at each point. Equation 1 is ful- filled by making p = q = m = constant, .................. (2.) which reduces both sides of equation 1 to m (R _ r ). CASE 2. Equal and contrary stresses, or m = 0. In this case p q n, and the solution of equation 1 is p = q = n = -^ ....................... (3.) a being an arbitrary constant, and r 1 any value of the radius, from r to II inclusive ; for this reduces both sides of equation 1 to G-i)- CASE 3. General solution. By combining the two partial solu- tions of equations 2 and 3 together, we find a Radial pressure, q = n m = m; ..(*) Hoop- tension, p=zn + m = ^ + m To determine the constants a and m we have the equations a a ?- =? ; E- =" whence we obtain by elimination THICK HOLLOW CYLINDER. 293 _ R 2 r 2 R 8 r 2 giving, finally, for the maximum Iwop-tension, (5 The mean hoop-tension is q,r q^, R r which is exceeded by the maximum in the proportion a proportion which tends towards equality, as R and r become more nearly equal. A transposition of equation 6 gives the following value of the ratio of the external to the internal radius, required in order that PQ may be =/, the bursting, proof, or working tension, as the case may be : In most cases which occur in practice, the external fluid pressure second order, A B, which has the property that 294 THEORY OF STRUCTURES. area r A B K = r x 7A - R x RB; so that it represents case 2. Draw CD || O r R, cutting off from the ordinates the parts C A, D B, which bear to each other the proportions CA : D~B : . q Q : qi . Then r C = R D will represent m, the solution of case 1 . Draw E F || O r R at the same distance r E = r C on the opposite side. Then if any ordinate be drawn across the two straight lines E F and C D, and the curve A B, at a given distance r' from O, the segment of that ordinate between C D and A B will represent the radial pressure q, and the entire ordinate from E F to A B will represent the hoop-tension p, at that distance from ; and in par- ticular E A will represent the maximum hoop- tension p . The formulae of this Article are the same with those given by M. Lame in his Traite de VElasticite; but they are arrived at in a different manner. 274. Cylinder of Strained Rings. To obviate, in whole or in part, the unequal distribution of the hoop-tension in thick hollow cylinders for withstanding great pressures, it has been proposed to construct such cylinders of concentric hoops or rings built together, the outer hoops being " shrunk " on to the inner hoops, in such a manner, that before any internal pressure is applied, the hoops within a certain distance of the centre may be in a state of circum- ferential compression, and those beyond that distance in a state of circumferential tension. If the stress thus produced by the mutual action of the concentric hoops could be adjusted with such accuracy, as to be at each point exactly equal and opposite to the difference between the actual hoop tension at the same point due to the internal pressure, as given by equations 4, 5, and 6, of Article 273, and the mean hoop-tension as given* by equation 7, then upon applying the proper internal pressure, there would result simply an uniform tension equal to the mean, and the formulse of Article 27 1 would become applicable to thick as well as to thin cylinders. Even although it may be impracticable to adjust the previous stress with the accuracy above described, any approach to its proper distribution must increase the strength of the cylinder. This method of construction has been carried into effect in the thirty- six inch mortar of Mr. Mallet. The only equation which the stress of the concentric hoops will of itself fulfil is THICK HOLLOW SPHERE. 295 275. Thick Hollow Sphere. Let fig. 118 now represent a diame- tral section of a hollow sphere, the fluid pressures within and without being q Q and < & > m=- R.- r . - J giving finally, for the maximum tension, (6 ' } ,* 2(R'-r) A transformation of this equation gives the following value of ratio of the external to the internal radius of the sphere, required in order that p may be = f, the bursting, proof, or working ten- sion, as the case may be : 2 + This equation shows, that if 2o= or no thickness will be sufficient to enable the sphere to withstand the pressure. The formulae of this Article agree with those given by M. Lame, though arrived at by a different process. 276. Boiler Stays. The sides of locomotive fire-boxes, the ends of cylindrical boilers, and the sides of boilers of irregular figures like those of marine steam engines, are often made of flat plates, r ., which are fitted to resist the pressure from within !_ oa j by being connected together across the water-space oooo or steam-space between them by tie-bars, called stays when long, bolts when short. For example, fig. 120 represents part of the flat side of a loco- o o o o motive fire-box, and shows the arrangement of the Fig. 120. bolts by which it is tied to the flat plate at the other side of the water-space. BOILER STAYS ROD OF UNIFORM STRENGTH. 297 Each of these bolts or stays sustains the pressure of the steam against a certain area of the plate to which it is attached. Thus, in fig. 120, the bolt a resists the pressure of the steam on the square area which surrounds it, and whose side is equal to the distance from centre to centre of the bolts. Let a be the sectional area of a stay ; A, that of the portion of flat plate which it holds ; q, the bursting, proof, or working pres- sure, and /the ultimate, proof, or working tension of the material of the stay. Then fa = q A. The proper factor of safety is eight, as for other parts of boilers. Experience has shown, that the plate, if its material is as strong as that of the stay, should have its thickness equal to half the dia- 7tieter of the stay. If the plate be of a weaker material than the stay, its thickness should be proportionally increased. The flat ends of cylindrical boilers are sometimes stayed to the cylindrical sides by means of triangular plates of iron called " gus- sets." These plates are placed in planes radiating from the axis of the boiler, and have one edge fixed to the flat end, and the other to the cylindrical body. Each gusset sustains the pressure of the steam against a sector of the flat circular end. Considering that the resultant tension of a gusset must be concentrated near one edge, it appears advisable that its sectional area should be three or four times that of a stay-bar suited for sustaining the pressure on the same area. The best experimental data respecting the strength of boilers are due to the researches of Mr. Fairbairn, especially those recorded in his work called Useful Information for Engineers. 277. Suspension Rod of Uniform Strength. In fig. 121, let W be a weight hung from the lower end of a vertical rod v B C, whose weight per unit of volume is w, and let it be \_J required to find how the transverse section S of the rod must vary with the height x above B, in order that the tension may be everywhere of equal intensity/ The total load at any point is, W from the weight hung at B, w I S dx from the weight of the rod for a height x above Bj and this must be equal to the pull /S. Hence $dx=fS', .................. (1.) Kg. 121. which being solved, gives for the cross section of the rod, 8 = .7; ........................... (2.) f 298 THEORY OF STRUCTURES. and for its weight, for a height x above B, /S W = W(e/ 1) (3.) The most useful application of this is to the determination of the dimensions of the pump-rods of deep mines. They are not made with the section varying continuously, according to the formula "2, but in a series of divisions, each of uniform scantling ; neverthe- less that formula will serve to show approximately the law which the dimensions of those divisions should follow. SECTION 4. On Resistance to Shearing. 278. Condition of Uniform intensity. The present section refers to those cases only in which the shearing stress on a body is uni- form in direction and in intensity. The effects of shearing stress varying in intensity will be considered under the head of Resist- ance to Bending, which is in general accompanied by such a stress ; and the effects of shearing stress varying in direction as well as in intensity under the head of Resistance to Torsion. It has been shown in Article 103 that shearing stresses can only exist in pairs, every shearing stress on a given plane being neces- sarily accompanied by a shearing stress of equal intensity on another plane. In Article 112, Problem II., it is shown that for any combination of stress parallel to a given plane, the planes rela- tively to which the shearing stress is greatest are at right angles to each other, and make angles of 45 with the axes of principal stress. When equal forces are applied to the opposite sides of a wedge, bolt, rivet, or other body, in such a manner as to tend to shear it into two parts at a particular transverse plane of section, then at any given point in that transverse sectional plane the shearing stress is of equal intensity relatively to that plane itself, and to a longitudinal plane traversing the same point, perpendicular to the direction of the externally-applied shearing forces. If the wedge, bolt, or rivet is loose in its hole or socket at and near the plane of shearing, there can be no shearing stress on those free parts of its external surface which are at right angles to the direction of the external shearing force ; and hence the intensity of the shearing stress at the plane of shearing, how great soever it may be in the internal parts of the body, must diminish to nothing at certain parts of the external edges of that sectional plane, and must be unequally distributed ; so that the most intense shearing stress must be greater than the intensity of a stress of equal amount uni- formly distributed. To insure uniform distribution of the stress, it is necessary that the rivet or other fastening should fit so tight in its hole or socket, BOLTS AND RIVETS. 299 that the friction at its surface may be at least of equal intensity to the shearing stress. When this condition is fulfilled, the intensity TT of that stress is represented simply by ^-; F being the shearing fe force, and S the sectional area which resists it. 279. A Table of the Resistance of Materials to Shearing and Sis- tortion, in Ibs. avoirdupois per square inch, is given at the end of the volume. It is of sma]l extent, because of the small number of substances whose resistances to shearing and distortion have been ascertained by satisfactory experiments. The resistance of timber to shearing is in each case that which acts between conti- guous layers of fibres. 280. Economy of Material in Bolts and Rivets. There are many structures, such as boilers, wrought iron bridges, and frames of tim- ber or iron, in which the principal pieces, such as plates, links, or bars, being themselves subjected to a direct pull, are connected with each other at their joints by fastenings, such as rivets, bolts, pins, or keys, which are under the action of a shearing force. It is in every such case important, that the pieces connected and their fastenings should be of equal strength ; for if the fastenings be the weaker, either the whole structure is insufficiently strong, or the material which gives the additional strength to the plates or bars is wasted : and if the fastenings be the stronger, the plates and bars are weak- ened more than is necessary by the holes or sockets ; and as before, either the structure is too weak, or material is wasted. Let f denote the resistance per square inch of the material of the principal pieces to tearing ; S, the total sectional area, whether of one piece or of two or more parallel pieces, which must be torn asunder in order that the structure may be destroyed ; /', the resistance per square inch of the material of the fastenings to shear- ing; S', the total sectional area of fastenings at one joint, which must be sheared across in order that the structure may be destroyed ; then, if the conditions of uniform distribution of stress are fulfilled, the principal pieces and their fastenings ought to be so propor- tioned, that /S=/S' ;<*!' = (1.) For wrought iron rivetted plates, taking the value off' from the table (as determined by the experiments of Mr. Doyne), we have C=l nearly, and . . S'= S (2.) For wrought iron bars connected by bolts or rivets, we have f ft * ft j., = ^ nearly, and. '- S' = j=- S (3.) 300 THEOKY OF STRUCTURES. Example I. Plate-joint overlapped, single-rivetted. Fig. 122. A, i O O O O O overlapped, front view ; B, side view. Let t = thickness of plate. 7 ! r- a = diameter 01 rivet. Fig. 122. c = distance from centre to centre of rivets. Then Sectional area of one rivet S Sectional area of plate between two holes 07854 d 2 ~ t (cd) ' so that, d and t being given, and c required, we have 0-7854 C = (5.) d in practice is usually from 2 1 to \t ; and the overlap from c II to ITS- c. 0000 coo A Fig. 123. Example II. Plate-joint overlapped, double- rivetted. Fig. 123. Sectional area of two rivets S Sectional area of plate between two holes in same line ..(6.) t(cd) ' 1-5708 d + d Overlap in practice = If c to If c. Example III. Plate Butt-joint, with a pair of covering plates, single-rivetted. Fig. 124. Here each rivet can give way only by being sheared across in two places at once ; there- fore o o o o o -V Fig. 124. - S 2 x Sectional area of rivet ~~ Sectional area of plate between two holes t (c d) ' ' ... 0= lJW8 + rf (9 .) Length of each covering plate = 2 x overlap = from 2 c to 2^ c. RIVETS TIMBER TIES. 301 Example IV. Plate Butt-joint, with a pair of covering plates, double- rivetted. Fig. 125. 4 x Sectional area of rivet Sectional area of plate between two holes in one row 3-1416 d* PV , ( 10 ') -,J^+ d (11.) t Length of each covering-plate = 2 x overlap = from 3J to 3^ c. Fig. 125. NOTE. The length of a rivet, before being clenched, measuring from the head, is about 4^ t for overlapped-joints, and 5^ t for butt-joints with covering-plates. Example V. Suspension bridge chain-joint. The chain of a sus- pension bridge consists of long and short links alternately. Each long link consists of one or more, say of n, parallel flat bars, of a shape resembling fig. 64, Article 138, placed side by side; each bar has a round eye at each end. Each short link consists of n + 1 parallel flat bars, with round eyes at their ends, which are placed between and outside of the ends of the parallel bars of the long links ; so that the end of each long bar is between the ends of a pair of short bars. The eyes of the long and short bars at each joint form one continuous cylindrical hole or socket, into which a bolt or pin is fitted, to connect the links together. To break the chain at a joint, by the giving way of the bolt, that bolt must be sheared across at 2 n places at once. Hence, let S denote the total sectional area of the bars in a link, and d the diameter of the bolt ; then S' = 2 n x 0-7854 d? = 1-5708 n d 2 ; and because S' should / be = - S, we have 5 281. Fastenings of Timber Ties. In timber framing, a tie may be connected with the adjoining pieces of the frame either by having their ends abutting against notches cut in the tie (as shown at A, A, fig. 81, Article 161), or by means of bolts or pins. In either case, the tie may yield to the stress in two ways, by being torn asunder at the place where its transverse section is least (that is, where it is notched or pierced, as the case may be), or by having the part beyond the notch, or beyond the bolt-hole, sheared off or sheared 302 THEORY OF STRUCTURES. out, as the case may be. In order that the material may be econo- mically used, equation 1 of Article 280 should be fulfilled, viz. : (1.) This condition serves to determine the distance of the notch, or of the bolt-hole, or of the nearest bolt-hole where there are more than one, from the end of the tie, in the following manner : Let h be the effective depth of the tie, left after deducting the depth of the notch, or the diameters of bolt-holes, and d the distance of the notch, or of the nearest bolt-hole, from the end of the tie; then for a notch and for bolt-holes, if n be their number, In determining the number n, it is to be observed, that if two or more bolts pierce the same layer of fibres, the resistance to the shearing out of the part of that layer between the end of the tie and the most distant of the bolts is nearly the same as if that bolt existed alone ; so that the most distant only of such a set of bolts is to be reckoned in using equation 3. In general, the piercing of the same layer of fibres by more than one bolt is unfavourable to economy. SECTION 5. On Resistance to Direct Compression and Crushing. 282. Resistance to Compression, when the limit of proof stress is not exceeded, is sensibly equal to the resistance to extension, and is expressed by the same " modulus of elasticity" already mentioned and explained in Articles 257, 265, 266, and 268. When that limit is exceeded, the irregular alterations undergone by the figure of the substance render the precise determination of the resistance to compression difficult, if not impossible. 283. Modes of Crushing. Splitting, Shearing, Bulging, Buckling, Cross-breaking. Crushing, or breaking by compression, is not a simple phenomenon like tearing asunder, but is more or less complex and varied, according to the texture of the substance. The modes in which it takes place may be classed as follows : I. Crushing by splitting (fig. 126) into a number of prismatic fragments, separated by smooth surfaces whose general direction is lei to the direction of the crushing force, is characteristic CRUSHING. 303 of hard homogeneous siibstances of a glassy texture, such as vitrified bricks. X Fig. 126. Fig. 127. Fig. 128. Fig. 129. II. Crushing by shearing or sliding of portions of the block along oblique surfaces of separation is characteristic of substances of a granular texture, like cast iron, and most kinds of stone and brick. Sometimes the sliding takes place at a single plane surface, like A B in fig. 127; sometimes two cones or pyramids are formed, like c, c, in fig. 128, which are forced towards each other, and split or drive outwards a number of wedges surrounding them, like w, w, in the same figure. Sometimes the block splits into four wedges, as in fig. 129. The surfaces of shearing make an angle with the direction of the crushing force, which Mr. Hodgkinson (who first fully investigated those phenomena) found to have values depending on the kind and quality of material. For different qualities of cast iron, for example, that angle ranges from 42 to 32. The greatest intensity of shearing stress is on a plane making an angle of 45 with the direction of the crushing force ; and the deviation of the plane of shearing from that angle shows that the resistance to shearing is not purely a cohesive force, independent of the normal pressure at the plane of shearing, but consists partly of a force analogous to friction, increasing with the intensity of the normal pressure. Mr. Hodgkinson considers that in order to determine the true resistance of substances to direct crushing, experiments should be made on blocks in which the proportion of length to diameter is not less than that of 3 to 2, in order that the material may be free to divide itself by shearing. When a block which is shorter in pro- portion to its diameter is crushed, the friction of the flat surfaces between which it is crushed has a perceptible effect in holding its parts together, so as to resist their separation by shearing; and thus the apparent strength of the substance is increased beyond its real strength. In all substances which are crushed by splitting and by shearing, the resistance to crushing considerably exceeds the tenacity, as an examination of the tables will show. The resistance of cast iron to crushing, for example, was found by Mr. Hodgkinson to be somewhat more than six times its tenacity. 304 THEORY OF STRUCTURES. III. Crushing by bulging, or lateral swelling and spreading of the block which is crushed, is characteristic of ductile and tough materials, such as wrought iron. Owing to the gradual manner in which materials of this nature give way to a crushing force, it is difficult to determine their resistance to that force exactly; that resistance is in general less, and sometimes considerably less, than the tenacity. In wrought iron, the resistance to the direct crush- ing of short blocks, as nearly as it can be ascertained, is from 2 4 - to - of the tenacity. o 5 IV. Crushing by buckling or crippling is characteristic of fibrous substances, under the action of a thrust along the fibres. It consists in a lateral bending and wrinkling of the fibres, sometimes accom- panied by a splitting of them asunder. It takes place in timber, in plates of wrought iron, and in bars longer than those which give way by bulging. The resistance of fibrous substances to crushing is in general considerably less than their tenacity, especially where the lateral adhesion of the fibres to each other is weak compared with their tenacity. The resistance of most kinds of timber to 1 2 crushing, when dry, is from - to - of the tenacity. Moisture in the A o timber weakens the lateral adhesion of the fibres, and reduces the resistance to crushing to about one-half of its amount in the dry state. V. Crushing by cross-breaking is the mode of fracture of columns and struts in which the length greatly exceeds the diameter. Under the breaking load, they yield sideways, and are broken across like beams under a transverse load. This mode of crushing will be con- sidered after the subject of resistance to bending. 284. A Table of the Resistance of Materials to Crushing by a Direct Thrust, in pounds avoirdupois per square inch, is given at the end of the volume. So far as that table relates to the strength of brick and stone, reference has already been made to it in Article 235. It is condensed from the experimental data given by various authorities, especially by Tredgold, Mr. Fairbairn, Mr. Hodgkinson, and Captain Fowke. 285. Unequal Distribution of the Pressure on a pillar arises from the line of action of the resultant of the load not coinciding with the axis of figure of the pillar, so that the centre of pressure of a cross section of the pillar does not coincide with its centre of figure, but deviates from it in a certain direction by a certain distance, which may be denoted by r . In this case the strength of the pillar is diminished in the same ratio in which the mean intensity of the pressure is less than the UNEQUAL THRUST. 305 maximum intensity; that is to say, in a ratio which may be denoted by mean intensity p Q maximum intensity - p l ' That ratio may be found with a precision sufficient for practical purposes, by considering the pressure at &ny cross section of the pillar as an uniformly varying stress, as denned in Article 94. Consequently the following is the process to be pursued : Find, by the methods of Article 95, the principal axes and moments of inertia of the cross section of the pillar; and thence determine the neutral axis conjugate to the direction of the devia- tion r . Let 6 be the angle made by that axis with the direction of the deviation r ; then the perpendicular distance of the centre of pressure from the neutral axis will be Find the moment of inertia of the cross section relatively to the neutral axis, and denote it by I ; then from equations 1, 2, and 4 of Article 94, it appears that if Xi be the greatest perpendicular distance of the edge of the cross section from the neutral axis in the same direction with x , the greatest intensity of pressure will be , . , #o -P ^ r (*) m which a = = X Q p Q - ; j P being the total pressure, and S the area of the section of the pillar. Consequently the ratio required is P? = __J__ / 2 \ p, , asoOJiS'" ~T~ Values of S, for certain symmetrical figures, and of I for the principal axes of these figures, have already been given in the table of Article 205, from which are computed the following values of the factor - in the denominator of the preceding formula : FIGURE OP CROSS SECTION. -4-. I. Rectangle, h b ; 6, neutral axis, ) 6 II. Square, A 2 , , J h' III. Ellipse : neutral axis, b ; other axis, h ; IV. Circle: diameter, A, x 306 THEORY OF STRUCTURES. ollow rectangle : outside dimensions, inside dimensions, h' } U ; neutral axis, b V. Hollow rectangle : outside dimensions, h, b ; ) 6 h (h b tib') ,... j 7^5 7^5' VI. Hollow square, A 2 A' 2 , Q 7 VII. Circular ring : diameter, outside, h ; inside, h', 286. UmiiationM of the Preceding Formulae. The formulae of the preceding Article of this section have reference to direct crush- ing only, and are therefore limited in their application to those cases in which the pillars, blocks, or struts along which the pres- sure acts are not so long in proportion to their diameter as to have a sensible tendency to be crushed by bending. Those cases com- prehend Stone and brick pillars, and blocks of ordinary proportions ; Pillars and struts of cast iron, in which the length is not more than five times the diameter, approximately ; Pillars and struts of wrought iron, in which the length is not more than ten times the diameter, approximately ; Pillars and struts of dry timber, in which the length is not more than about twenty times the diameter. . 287. Crushing and Collapsing of Tubes When a hollow cylin- der is exposed to a pressure from without, there is a circumferen- tial thrust round it, whose greatest intensity takes place at the inner surface of the cylinder, and may be computed by suitably modifying the formulae of Article 273. That is to say, let R and r denote respectively the outer and inner radii of the cylinder, q l the intensity of the radial pressure from without, q Q that of the radial pressure from within, and let p now denote, not a tension, but a thrust, viz., the maximum circumferential thrust which acts round the inner surface of the cylinder. Then reversing the signs of the second side of equation 6 of Article 273, we obtain _ Po ~ R, 2 -?- 8 When the pressure from within is null or insensible, this becomes and supposing tJie material to give way by direct crushing, the proper ratio of the internal to the external radius is given by the equation BENDING AND CROSS-BREAKINQ. 307 q l being the working, proof, or crushing external pressure, and f the working, proof, or crushing thrust of the material, as the case may be. This formula gives correct results for hollow cylinders, very accurately formed, and of considerable thickness as compared with the radius. But where the figure is not very accurately cylindri- cal, and where the thickness is small compared with the radius (as in the internal flues of boilers), the cylinder gives way, or tends to give way, not by direct crushing, but by collapsing, which, as it consists in an alteration of figure, is analogous to crushing ~by bend- ing. The laws of the resistance of tubes to collapsing are not yet completely known; but experiments by Mr. Fairbaim on that subject are now in progress, from which, when they are complete, these laws may be ascertained. According to an interim report addressed to the British Association, the intensity of the pressure from without which makes a thin tube collapse is inversely as the length, inversely as the radius, and directly as some function of the thickness which is nearly proportional to the square. The practical rule at present followed with respect to plate iron tubes exposed to pressure from without, is to make the thickness double of what it would require to be if the pressure were from within. SECTION 6. On Resistance to Bending and Cross-Breaking. 288. Shearing Force and Bending Moment in General. It has already been shown, in Articles 141 and 142, how to determine the proportions between the resultant of the gross load of a beam and the two forces which support it, whether those three forces are perpendicular or oblique to the beam, and whether they are par- allel or inclined to each other. In the present section those cases alone will be considered in which the loading and supporting forces are perpendicular to the beam, and parallel to each other, and in one plane ; for such forces alone tend simply to bend the beam, and if sufficiently great, to break it across. In Article 161 it has been shown how to determine the resist- ances exerted by the pieces of a frame which are cut by an ideal sectional plane, in terms of the forces and couples which act on one of the portions into which that plane of section divides the frame ; and in Articles 162, 163, 164, and 165, that method of sections, as it is called, has been applied to the determination of the stresses 308 THEORY OF STRUCTURES. acting along the bars of half-lattice or Warren girders and of lattice girders. The method followed in determining the effect of a transverse load on a continuous beam is similar ; except that the resistance at the plane section, which is to be determined, does not consist of a finite number of forces acting along the axes of certain bars, But of a distributed stress, acting with various intensities, and, it may be, in various directions, at different points of the section of the beam. In what follows, the load of the beam will be conceived to con- sist of weights acting vertically downwards, and the supporting forces will also be conceived to be vertical. The longitudinal axis of the beam being perpendicular to the applied forces, will accord- ingly be horizontal. The conclusions arrived at will be applicable to cases in which the axis of the beam and the direction of the applied forces are inclined, so long as they are perpendicular to each other. Let any point in the longitudinal axis of the beam be taken as the origin of co-ordinates ; and at a given horizontal distance x from that origin, conceive a vertical section perpendicular to the longitudinal axis to divide the beam into two parts. To fix the ideas, let horizontal distances to the \ '. > be considered as 1 right J { negative } ' let vertical d*** 8 and forces in an { downward } direction, be considered as \ * .. > : and let the moments of I negative j couples be I P osit /7 e 1 according as they are / I^VT^ \ \ negative j ( right-handed J Let F denote the resultant of ail the vertical forces, whether loading or supporting, which act on the part of the beam to the left of the vertical plane of section, and let x' be the horizontal distance of the line of action of that resultant from the origin. If the beam is strong enough to sustain the forces applied to it, there will be a shearing stress whose amount is equal to F, distri- buted (in what manner will afterwards appear) over the given vertical section ; and that shearing stress, or vertical resistance, will constitute, along with the applied force F, a couple whose moment is a:) .......................... (1.) This is called the bending moment or moment of flexure of the beam at the vertical section in question ; and it is resisted by the normal stress at that section, in a manner to be explained in the sequel. If the bending moment is / P 081 *^ 6 1 j t tends to make the ( negative J ' SHEARING FORCE AND BENDING MOMENT. 309 originally straight longitudinal axis of the beam become concave ( upwards ) | downwards j * The determination of the magnitude and position of the resultant F consists simply in finding the resultant of a number of parallel forces in one plane, as explained in Article 44, the supporting forces having first been found by the principles of Articles 39 and 141. These processes are expressed by general formulse as fol- lows : CASE 1. The load applied at detacfad points. Let W denote one of the weights of which the load consists ; x" its horizontal distance from the origin ; then 2 W is the total load, made negative as acting downwards ; and 2 x" W is its moment relatively to the origin. Let x^ and x 2 be the horizontal distance of the points of support from the origin, and let P 1? P 2 , be the supporting forces ; then to determine those forces we have the conditions of equilibrium ^H-P, - s-W = 0; x l P, + x, P 2 - 2 - x" W = ; from which follow the equations P,* X, 2 - W - 2 X " W .(2.) _ Xi 2 ' W - 2 ' X" W X, - X 2 To show how the shearing force and moment of flexure at any cross section are found, let Pj be applied to the left of the origin, and let the plane of section, whose distance from the origin is x, lie between P l and P 2 ; then the force acting on the beam to the left of x will be and the moment of flexure > ........... (3. ) the symbol s* denoting in each case, that the summation extends to that part of the beam only which lies between the given plane of vertical section and the point of support (if any) to the left of that plane. CASE 2. The load continuously distributed. On any indefinitely short division of the beam whose length is d x, and distance from 310 THEORY OF STRUCTURES. the origin x" , let the intensity of the load per unit of length be w. Then in the equations 2 and 3, given above, it is only necessary to substitute w d x for W, and the sign / for the sign 2. 289. In Beams Fixed at One End Only, and loaded on the pro- jecting portion, as in fig. 67 of Article 141, and figs. 133 to 136 of a subsequent Article, the shearing force and moment of flexure can be N determined for any vertical section of the projecting part of the beam, without considering the supporting pressures. Let the plane at which the beam is fixed be taken as the origin ; let c be the length of the projecting part of the beam. The results in the cases most important in practice are given in the following table : i EXAMPLE. 1 SHEARING FORCE F BENDING MOMENT M Anywhere. F Greatest. F. Anywhere. M Greatest. M I. Loaded at extreme end with W, II. Uniform load of in- W W -(c-*)W cW w(c-xf we 2 2 2 Ill. Uniform load of in- tensity w, and ad- ditional load at extreme end W, W w(c x) W we W(c a;) w (cxf -We-* 2 290. In Beams Supported at Both Ends, and loaded on the inter- mediate portion, like those represented in fig. 66 of Article 141, and in figs. 138 and 140 of a subsequent Article, it is most conve- nient to take the middle of the beam as the origin of co-ordinates. Then let c denote the half-span of the beam, so that 2 c is the span, or distance between the points of support ; the positions of those points will be expressed by xj, = c; x., = -c; x^x z = 2c; (1.) which substitutions convert equation 2 of Article 288 into the following : -L ' X 2 W 2 ' X" W (2.) MOMENTS OF FLEXURE. If the load is symmetrically distributed, *-x"W - 0, P^P/.'^W^ 311 and .(2 A.) The equations 3 of Article 288 also become F = Pt-a;-W; M - (c-x)P l -^ x '(x"- and for a symmetrically distributed load, F = aJ'W-; M (c-x) aj W-aJ- (x"-x) W....(3 A.) The results in the cases most important in practice are given in the following table : SHEABING FORCE BENDING MOMENT F M Anywhere. Greatest. Anywhere. Greatest F F! or F 2 M M or M". IV. Single load, W, in middle Left of 0, W 2 W ~2 (o-)W Right of 0, W W 1, 2 2 - M(> ! V. Single load, W, ap- plied at x" Left of x", (c + o;")W (c+a/OW 2c (c+y)(c s)W ( #-jw 2c 2c Right of a/', (c-a/')W (c a/')W (c aO(c+a;)W = M" at a/' 2c 2c 2c VI. Uniform load of in- tensity, w, wx we W ( C 2_ ar 2 ) 2 2 M 291. Moments of Flexure in Terms of JLoad and Length. For practical purposes, it is often convenient to express the greatest bending moment of a beam in terms of the total load, W, and un- supported length, I, of a beam, by means of a formula of this kind, M = mWZ, (1.) where m is a numerical factor. For beams fixed at one end, I = c ; 312 THEORY OF STRUCTURES. for beams supported at both ends, I 2 c the span ; for an uniform load, W = wl. Hence, comparing equation 1 with Examples I., II., IV., V., and VI. of Articles 289 and 290, we find the follow- ing values of the factor m : m I. Beam fixed at one end, loaded at the other, ...... 1 . II. Beam fixed at one end, loaded uniformly, ........ ^. a IV. Beam supported at both ends, loaded in the ) 1 middle, ............................................. j 4' V. Beam supported at both ends, loaded at x" ) 1 / 4 x n2 \ from the middle, ................................. J J \ ~ I 2 / VI. Beam supported at both ends, uniformly loaded, -. o 292. Uniform Moment of Flexure. If a pair of equal and oppo- site couples, acting in the same longitudinal plane, be applied at or near the ends of a beam, the part of the beam intermediate between the portions to which the couples are applied is under the influence of an uniform moment of flexure, and of no shearing force. An illustration of this is the condition of that part of the axle of a railway carriage which lies between the pair of wheels, if the bearings are outside of the wheels, or between the bearings if the bearings are inside of the wheels. Let W be the weight which W rests on one pair of wheels ; then is the weight resting on each 2 wheel, and on each bearing. Let I be the distance from the centre of each wheel to the middle of the adjoining bearing. Then a pair of equal and opposite couples, each of the moment, are applied to the two ends of the axle ; and this is the uniform moment of flexure of the portion of the axle lying between the portions acted upon by the forces which constitute the couples ; and the shearing force on the same portion is null. 293. Resistance of Flexure means, the moment of the resistance which a beam opposes to being bent or broken across ; and if the beam is strong enough, that moment, at each cross section of the beam, is equal and opposite to the moment of the bending forces at the same cross section. RESISTANCE OF FLEXURE, 313 13 * Let fig. 130 represent a side view of part of a beam which is of uniform cross section, and which is sub- jected to an uniform moment of flexure; and let fig. 130* represent the cross sec- tion of the same beam. It is self-evident that the curvature produced in the part of the beam in question must be uniform ; that is to say, that any longitudinal line in the beam, such as its upper edge A A', or its lower edge B B', which in the free condition of the beam is straight, must be bent into an arc of a circle ; and that any surface originally plane and longitudinal, and perpendicular to the plane in which the curva- ture takes place, such as the upper surface A A', or the lower surface B B', must be bent into a cylin- drical form ; and the cylindrical surfaces so produced will have a common axis. Any two transverse sectional planes, such as A B and A' B', which in the free state of the beam are parallel to each other, will have, in the curved state of the beam, positions radiating from the axis of curvature. Therefore, if the portion of the beam between the transverse planes A B, A' B', be conceived to be divided into layers, such as CO', originally plane, parallel, and of equal length, these layers, in the bent condition of the beam, must have lengths proportional to their distances from the axis of curvature. The layers near the concave side of the beam, A A', are shortened by the bending, and the layers near the convex side, B B', lengthened ; and there must be some intermediate layer which is neither lengthened nor short- ened, but preserves its free length. Let O 0' be the surface origi- nally plane, now curved, at which that layer is situated ; this is called the neutral surf ace of the beam, and the line 0, fig. 130*, in which it intersects a given cross section, is called the neutral axis of that section. The direct strains, or proportionate elongations and compressions, of the layers of the beam are proportional to their distances below and above the neutral surface; and hence, within the limits of proof stress, the direct stresses, or tensions and pressures, at the different points of the cross section AB, fig. 130*, have intensities sensibly proportional to their distances from the neutral axis O 0. Therefore the direct stress at each section, such as A B, whose moment constitutes the resistance to bending, is an uniformly-vary- ing stress, as defined in Article 91 ; and in order that the longi- tudinal resultant of that stress may be null, the neutral axis (as shown in that Article) must traverse the centre of gravity of the cross section A B. .314 THEORY OF STRUCTURES. The moment of a bending stress has already been given in Article 92, equations 3 and 4 ; and the methods of determining the inte- grals I and K, which occur in those equations, have been explained and illustrated in Article 95. To apply the equations of those Articles to the present purpose, let p be the intensity of the direct stress at a layer of the beam whose distance from the neutral axis is y : height above the neutral axis being considered as positive, and depth below it as negative. Then because a moment of flexure tending to make the beam con- cave upwards has been treated as positive, it is convenient, in order to avoid the unnecessary use of negative signs, to consider the con- stant ratio - as positive when it is such as to give resistance to an 7 upward moment of flexure ; that is, when p is a thrust for positive values of y, and a pull for negative values ; consequently, p is to i i f positive ) i . ... ( thrust. ) be considered as | %&&* } accordin g as * 1S a { pu n. } This being understood, we have, for the moment of the resistance opposed by the beam to bending, ................... (1.) and for the angle made by the neutral axis with the direction of the axes of the bending couples, wr (*=. arc tan ; .................... (2.) I and K being found by the methods of Article 95. In some cases, a more convenient form of equation 2 is that which gives 0, the angle made by the neutral axis with its conju- gate axis, in which the plane of the bending forces cuts the plane of section A B, viz. : cotan 9= y ........................ (3.) In almost every case which occurs in practice, the plane of the bending forces cuts each cross section of the beam in one or other of its principal axes, for which K = 0, ^ 0, = 90; and then equa- tion 1 becomes M=! ............................ (4.) y In beams whose transverse sections and moments of flexure are not uniform, no error appreciable in practice is produced by applying equation 4 to each cross section, and to the moment of flexure which TRANSVERSE STRENGTH. 315 acts upon it, as if the given section and moment belonged to an uniform beam with an uniform moment of flexure. 294. The Transverse Strength of a beam, ultimate, proof, or work- ing, as the case may be, is the load required to break it across, or to produce the proof stress or the working stress, as the case may be. It is found by equating the greatest moment of flexure, ex- pressed in terms of the load and length, as in Article 291, to the moment of resistance at the cross section where that moment of flexure acts : such moment of resistance being found from the equa- tions of Article 293, by putting for p the ultimate, proof, or working direct stress of the material, as the case may be, and for y the distance from the neutral axis to the point in the given cross section where the limiting stress p is first attained. That point will be at the < > side of the beam, according as the mate- i convex i / \ rial gives way most readily to j p 1 ^-^ 6 ' f In fig. 131, A represents a beam of a granular material, like cast iron, giving way by the crushing of the concave side, out of which a sort of wedge is forced. B re- presents a beam giving way by the tearing asunder of the con- Fig. 131. vex side. In a beam symmetrical above and below, or otherwise of such a form that the neutral axis is at the middle of the depth of the cross section, if A is that depth, h and the limiting value of p is the resistance to pressure or to ten- sion, whichever is least. For other forms of section, let y = y a for the concave side ; and = y b for the convex side ; and let the limiting stresses be p fa for pressure ; and = f b for tension ; then the beam will give way by j ^ us . mg 1 according as #? is 2/& <7. W This point having been determined, the equation from which the strength of the beam may be found is 316 THEORY OF STRUCTURES. M Q =mWl=^ ..................... (2.) y When the breaking load is in question, the co-efficient/ is what is called the modulus of rupture of the material. It does not always agree with the resistance of the same material to direct crushing or direct tearing, but has a special value, which can be found by experiments on cross-breaking only. One of the causes of this phenomenon is probably the fact, already stated in Article 257, that the resistance of a material to a direct stress is increased by preventing or diminishing the alteration of its transverse dimen- sions ; and another cause may be the fact, that the strength of masses of metal, especially when cast, is greater in the external layer, or skin, than in the interior of the mass. When a bar is directly torn asunder, the strength indicated is that of the weakest part of the mass, which is in the centre ; when it is broken across, the strength indicated is that either of the skin, which is the strongest part, or of some part near the skin (See the Article 296). When the proof load or working load is in question, the co-effi- cient /is the modulus of rupture divided by a suitable factor of safety, as to which see Article 247. 295. Transvcre Strength in Terms of Breadth and Depth. From the principles explained in Article 95, it is obvious that the moments of inertia, I, of similar sections are to each other as the breadths, and as the cubes of the depths. If, therefore, b be the breadth, and h the depth, of the rectangle circumscribing the cross section of a given beam at the point where the moment of flexure is greatest, we may put I = ribh 5 ........................... (1.) ri being a numerical factor depending on the form of the section. It is also evident, that for similar figures, the values of y are as the depths ; so that we may put (2.) m' being another numerical factor depending on the form of section. If the section is symmetrical above and below, m 1 = ^. Thus it appears, that the resistances of flexure of similar cross sections are as their breadths and as the squares of their depths, and that equation 2 of Article 294, which expresses equality between the greatest moment of flexure, as stated in terms of "the load and length, and the resistance of the cross section where that moment acts, is equi- valent to the following : M = mWl = nfbh* ..................... (3.) TRANSVERSE STRENGTH. 317 where n = , is a numerical factor Depending on the form of cross section of the beam, and ra is the numerical factor depending on the mode of distribution of the loading and supporting forces, of which examples have been given in Article 291. The following table gives examples of the values of the three factors, n' y ra', n, for some of the more usual forms of cross section : FORM OF CROSS SECTIONS. I. Rectangle bh, (including square) II. Ellipse- Vertical axis h, Horizontal axis b, ... (including circle) III. Hollow rectangle, b h b' li ; also I-formed section, where b' is the sum of the breadths of the lateral hollows, . . . IV. Hollow square Y. Hollow ellipse, VI. Hollow circle, . I n '=blf- . h ra=-. y I . ~ybh- } A 1 ~2 1 6" * *- l 1 * 1 ( 64 204 j a 0-0491 1 2 32~I(F2 = 0-0982 l (l b ' k ' Z ] 1 1 A b'h'\ 12 \ &*V 2 HV~T) 1 -1 (l-J^ j 12\ tt) 1 2" J(--) 1 / b>h'*\ 20-4 V bh*J 1 2 1 (l-b'h'\ 1.0:8 \ bh,*) __I(l-H 20-4 V A*/ 1 2" d-^\ 10-2 V & 4 ) In using the equation 3 for any of the purposes to which it may be applied such as computing the strength of a beam of which the dimensions and figure are given, or fixing the transverse dimen- sions of a beam of which the strength, length, and figure are given care is to be taken to use the same unit of measure throughout the calculation ; that is to say, when the transverse dimensions, as is usually the case, are stated in inches, and the co-efficient of strength f in pounds on the square inch, the length I should be stated in inches also. This caution is necessary on account of that diversity of units which is characteristic of British measures. 296. A Table of the Resistance of materials to Breaking Across is given at the end of the volume. It gives values of the modulus of rupture, being that for which the co-efficient f stands in Article 318 THEORY OF STRUCTURES. 294, equation 2, and in Article 295, equation 3, when m W I is the breaking moment. It will be observed, that this modulus is, for most materials, intermediate between the tenacity and the resistance to direct crushing. 297. Cast iron Beams. The values of the modulus of rupture for cast iron require special remark. It had for some time been known, that while the direct tenacity of cast iron (as determined by Mr. Hodgkinson) is on an average 16,500 Ibs. per square inch, the modulus of rupture of rectangular cast iron beams is on an average about 40,000 Ibs. per square inch, or two and a-half times as great. This was supposed to be accounted for by the assumption, that the stress on a cross section of a cast iron beam is not an uniformly varying stress, and that the neutral axis does not traverse the centre of gravity of the section. But in 1 855, Mr. William Henry Barlow, by experiments of which an account is published in the Philosophical Transactions for that year, showed, in the first place, that the stress is an uniformly varying stress, and that the neutral axis, in symmetrical sections at all events, traverses the centre of gravity of the section, and in the second place, that the modulus of rupture has various values, ranging from the mere direct tenacity of the iron up to about two and a-third times that tenacity, accord- ing to the figure of the cross section of the beam. The beams on which the experiments of Mr. Barlow, now referred to, were made, were in some cases of a solid rectangular section, and in other cases of an open-work rectangular section, consisting of equal rectangular upper and lower horizontal bars, with alternate open spaces and vertical connecting bars between. As far as those experiments went, they were in accordance with the following empirical formula : /=/.+/ -5, (i.) where f is the modulus of rupture of the beam in question; f 0) the direct tenacity of the iron of which it is made : /'', a co-eificient TT determined empirically; and , the ratio which the depth of solid li metal H in the cross section of the beam bears to the total depth of section h. The following were the values of the constants for the cast iron experimented on : Direct tenacity, f Q = 18,750 Ibs. per square inch ; f' = 23,000 Ibs. per square inch ; = l^/o nearly. Mr. Barlow has since made further experiments on cast iron CAST IRON BEAMS. 319 beams of various forms of section, and also experiments on wrought iron beams, showing, though not so conclusively, variations in the modulus of rupture of wrought iron analogous to those which have been proved to exist in the case of cast iron ; but as those further experiments, though communicated to the Royal Society, have not yet been published in detail, it would be premature to make remarks on them here. Mr. Barlow has proposed a theory of those phenomena, to the effect that the curvature of the layers of the beam produces a peculiar kind of resistance to bending, distinct from that which arises from the direct elasticity \ and he adduces in support of that theory the fact that the additional strength represented by the second term of equation 1 increases with the ultimate curvature of the beam ; that is, its curvature just before breaking. Another conceivable theory has already been mentioned in Article 294, viz., that the strength of a metal bar, and in particular of a cast iron bar, is greatest at the sldn, and diminished towards the interior ; that the tenacity found by directly tearing a bar asunder, f , is the tenacity of the interior; that the modulus of rupture of a solid rectangular beam,/* + f 1 , is the tenacity of the skin, and that the modulus of rupture of an open-work beam is the tenacity at a distance from the skin depending on the form of section. But until conclusive experimental data shall have been obtained, all theories on the subject must be considered as provisional only. 298. The Section of I<]qual Strength for Cast Iron Beams was first proposed by Mr. Hodgkinson, in consequence A of his discovery of the fact, that the resistance of cast iron to direct crushing is more than six times its resistance to tearing. It consists, as in fig. 132, of a lower flange B, an upper flange A, and a vertical ^.4. web connecting them. The sectional area of the lower flange, which is subjected to tension, is nearly six times that of the upper flange, which is subjected Fi * 132- to thrust. In order that the beam, when cast, may not be liable to crack from unequal cooling, the vertical web has a thickness at its lower side equal to that of the lower flange, and at its upper side equal to that of the upper flange. The tendency of beams of this class to break by tearing of the lower flange is slightly greater than the tendency to break by crushing of the upper flange ; and their modulus of rupture is equal, or nearly equal, to the direct tenacity of the iron of which they are made, being, on an average of different kinds of iron, 16,500 Ibs. per square inch. Let the areas and depths of the parts of which the section in fig. 132 consists be denoted as follows : THEORY OF STRUCTURES. Areas. Depths. Upper flange, ..., Aj, h v Lower flange, A 2 , h y Vertical web, A 3 , h 3 . Totals,. . .Aj + A 2 + A 3 = A, h^ + h 2 -f h 3 = h. No appreciable error will arise from treating the section of the vertical web as rectangular instead of trapezoidal. The height of the neutral axis above the lower side of this section is -i i ("'a ~~ 'h) A 3 /t x ~ 2 2A Then by applying the formula of Article 95, Example VI., to this case, the moment of inertia of the section is found to be as follows : 12 + A, A s (h, + h s ? + A 2 A 3 (h, + A 3 ) 2 } ; (2.) and the strength of the beam is expressed by the equation (3.) 2/6 It is seldom necessary, however, to use the formulae 1 and 2 in all their complexity; the following approximate formula being usually sufficiently near the truth for practical purposes, and its eiTor being on the safe side. Let h 1 be the depth from the middle of the upper flange to the middle of the lower flange ; then M.Q = m W l=f b ti A 2 (4.) 299. Beams of Uniform Strength are those in which the dimen- sions of the cross section are varied in such a man- ner, that its ultimate or proof resistance bears at each point of the beam the same proportion to the Fig. 134. Fig. 133. Fig. 135. Fig. 136. moment of flexure. That resistance, for figures of the same kind, being pro- portional to the breadth and to the square of the depth, can be varied either by varying the breadth, the depth, or .both. The BEAMS OF UNIFORM STRENGTH. 321 law of variation depends upon the mode of variation of the moment of flexure of the beam from point to point, and this depends on the Fig. 137. Fig. 138. Fig. 139. Fig. 140. distribution of the load and of the supporting forces, in a way which has been exemplified in Articles 289 and 290. "When the depth of the beam is made uniform, and the breadth varied, the vertical longitudinal section is rectangular, and the plan is of a figure depending on the mode of variation of the breadth. When the breadth of the beam is made uniform, and the depth varied, the plan is rectangular, and the vertical longitudinal section is of a figure depending on the mode of variation of the depth. The following table gives examples of the results of those principles : Mode of Loading and Supporting. 6A2, proportional to Depth h constant; Figure of Plan. Breadth 6 constant ; Figure of Vertical Longitudinal Section. I. (Figs. 133, 134). Fixed at A, load- ed a t B Distance from B. Triangle, apex at B, fig. 133. Parabola, vertex at B, fig. 134. II. (Figs. 135, 136). Fixed at A, uni- formly loaded,... Square of distance from B. Pair of parabolas, vertices touching each other at B, fig. 135. Triangle, apex at B, fig. 136. III. (Figs. 137, 138). Supported at A and B, loaded at C. .. Distance from adjacent point of support. Pair of triangles, common base at C, apices at A and B, fig. 137. Pair of parabolas, vertices at A and B, meeting at C, fig. 138. IV. (Figs. 139, 140). Supported at A and B, uniformly loaded Product of dis- tances from points of support. Pair of parabolas, vertices at C, C, in middle of beam ; common base A B, Ellipse A D B, fig. 140. fig. 139. 322 THEOKY OF STRUCTURES. The formulae and figures for a constant depth are applicable to the breadths of the flanges of the ^-shaped girders described in Article 298. In applying the principles of this Article, it is to be borne in mind, that the shearing force has not yet been taken into account ; and that, consequently, the figures described in the above table require, at and near the places where they taper to edges, some additional material to enable them to withstand that force. In figs. 137 and 139, such additional material is shown, disposed in the form of projections or palms at the points of support, which serve both to resist the shearing force, and to give lateral steadiness to the beams. 300. Proof Deflection of Beams. Eeverting to fig. 130, it is evident that if * represents the proportionate elongation of the layer C C', whose distance from the neutral surface 0' is y, and if r be the radius of curvature of the neutral surface, we must have 1 : 1 + * : : r : r + y y and consequently, the radius of curvature is and the curvature, which is the reciprocal of the radius of curvature, is expressed by the equation r ~~ y Let p be the direct stress at the layer C C', and E the modulus of elasticity of the material; then a, = *- , and consequently, the cur- Ji< vature has the following values : El the second value being deduced from the first by means of equation 4 of Article 293. When the quantity - varies for different points of the beam, / the curvature varies also. Suppose now that the beam is under its proof load, and let MO denote the greatest moment of flexure arising from that load, I the moment of inertia of the cross section at which that moment acts, and 2/0 the distance from the neutral axis of that section to the layer where the limiting intensity /of the stress is attained. Then the curvature will be, PROOF DEFLECTION OF BEAMS. 323 at the section of greatest stress. == -=r~ = ^rr > at any other section, / .(2.) The exact integration of this equation for slender springs, in certain cases, will be considered in a subsequent Article. For beams it is integrated approximately in the following manner : Let the middle of the neutral axis of the section of greatest stress be taken as the origin of co-ordinates, and represented by A in figs. r. 141. Fig. 142. 141 and 142. For a beam supported at both ends and symme- trically loaded, A is in the middle of the beam (fig. 141)i For a beam fixed at one end and projecting, A is at the fixed end (fig. 142). Let the beam be so fixed or supported that at this point its neutral surface shall be horizontal, and let a horizontal tangent, A X C, to that surface at that point be taken as the axis of abscissae. Let A C, the horizontal distance from the origin to one end of the beam, be denoted by c, which, as in Articles 289 and 290, is the length of the projecting portion of a beam fixed at one end, and the half-span of a beam supported at both ends and symmetrically loaded. Let A X, the abscissa of any other point in the beam = x. Let A B D be the curved form assumed by the neutral surface when the beam is bent, which form, in a beam supported at both ends, is concave upwards, as in fig. 141, and in a beam fixed at one end concave downwards, as in fig. 142. Let X B = v be the ordinate of any point B in the curve A B D ; being the difference of level between that point and the origin A. Let C D = Vi be the greatest ordinate : this is what is termed the deflection. The inclination of the beam at any point B, is expressed by the equation d v ^ = arc tan : d x and the curvature, being the rate of variation of the inclination in a given length of the curve, is expressed by 324 THEORY OF STRUCTURES. di dx /i , d ^ V 1 + ^ But in cases which occur in practice, the curvature of the beam is so slight, that the arc i is sensibly equal to its tangent, the slope and the elementary arc d s is sensibly equal to its horizontal d x projection d x ; so that the following equations may be used without sensible error : Slope, Curvature, d v ^d^' 1 d i d a v r dx dot?' (3.) Therefore, when the curvature at each point is given by equation 2, the slope and the ordinate are to be found by two successive integrations, as shown by the following equations : Slope, Ordinate, . P^=-_/_ TM 1 ' J o r E 2/0 ' J o I M ' / . , / "/ r* ^ v = I i d x = ==r ' / I v Jo E 2/ J o J o I T. , M , ., a xr. The greatest slope i^ that is, the slope at D and the deflection or greatest ordinate Vj, are found by performing the complete inte- grations between the limits x = and x = c. [Readers who are not familiar with the integral calculus are referred to Article 81 for explanations of the nature of the process of integration.] MI In both the integrals of the formulae 4, the quantity ^ivr-is a 1 M numerical ratio depending on the mode of distribution of the load- ing and supporting forces, and the mode of variation of the section of the beam. Hence it is evident that we must have the complete integrals MI " re / M I !K = mc '> Jo] TM ' (lx -= nc ' ..... <") where m" and n" are two numerical factors depending on the dis- tribution of the forces and the figure of the beam ; so that the greatest slope and the deflection are given by the equations PROOF DEFLECTION OF BEAMS. 325 . _m"fc _ ' - ~~ ; Vl -' ri'fc 2 .(6.) For beams of similar figures, and similarly loaded and supported, ?/ is as the depjbh, and c as the length ; hence, for such beams, the greatest slope under the proof load is directly as the length, and inversely as the depth ; and the proof deflection is directly as the square of the length, and inversely as tlie depth. The following table gives the values of the factors m" and n" for some of the more ordinary cases of beams of uniform section, in which the ratio ^W^, being simply equal to ^-, depends on the 1 M M distribution of the load alone, and may be found by the aid of the tables of Articles 289 and 290. M M m" n" I. Constant moment of flexure, 1 1 1 2 FIXED AT ONE END. II. Loaded at extreme end, i-s c 1 2 1 3 III. Uniformly loaded, d -V 1 . 1 \ *) 3 4 SUPPORTED AT BOTH ENDS. I~V Loaded in the middle .... 1 x 1 1 c 2 3 "V" Uniformly loaded, 1 i 2 2 5 c 2 3 12 For a beam of uniform strength and uniform depth, the quantity t=- is constant ; hence in every such beam, in what manner soever it may be supported and loaded, the curvature is uniform, as in the case of Example I. of the above table. For a beam of uniform strength and uniform breadth, the quantity is constant ; and therefore in such beams, IM~h } 326 THEORY OF STRUCTURES. A being the depth at the section of greatest bending moment, and h the depth at any other section. The following table shows some of the consequences of these principles : MI I M m" n" VI. Uniform strength ) and uniform depth, / 1 I 1 2 VII. Uniform strength, 1 uniform breadth ; fixed 1 \ / C 2 2 at one end, loaded at f the other I V c - x 3 VIII. Uniform strength, ~| uniform breadth ; sup- 1 ported at both ends, | loaded in the middle,.. J V c=* 2 2 3 IX. Uniform strength, ] uniform breadth ; fixed 1 at one end, uniformly j c C X Infinite. 1 X. Uniform strength, 1 uniform breadth j sup- ! c 7T tf ported at both ends, j uniformly loaded, J Jtt 5 =1-5708 J I -1-0-5708 2i It is to be borne in mind, that the values of m" and n" for beams of uniform strength, as given in the above table, are somewhat less than those which occur in practice, because, in computing the table, no account has been taken of the additional material which is placed at the ends of such beams, in order to give sufficient resistance to shearing. The error thus arising applies chiefly to m", the factor for the maximum slope. For the factor for the deflection, n", the error is inconsiderable, as experiment has shown. 301. Deflection found by Graphic Construction. The great length of the radii of curvature, which are the reciprocals of the curva- tures given by equation 2 of Article 300, and the smallness of the ordinates of the curve of the neutral surface, in all cases which occur in practice, render it neither practicable nor useful to draw the figure of that curve in its natural proportions. But the following process, invented, so far as I am aware, by Mr. C. H. Wild, enables a diagram to be drawn, which represents, with a near approach to RATIO OF DEPTH TO SPAN. 327 accuracy, that curve, with its vertical dimensions exaggerated, so as to show conspicuously the slopes and ordinates Compute, by equation 2 of Article 300, the radii of curvature for a series of equi-distant points in the beam. Diminish all those radii in any proportion which may be convenient, and draw a curve composed of small circular arcs with the diminished radii. Then in the same ratio that the radii, as compared with the horizontal scale of the drawing, are diminished, will the vertical scale of the draw- ing, according to which the ordinates are shown, be exaggerated. 302. The 1> i-o portion of the Greatest Depth of a Beam to the Span is so regulated, that its greatest deflection shall not exceed a cer- tain proportion of the span which experience has shown to be con- sistent with convenience. That proportion, from various examples, appears to be For the working load, = from JL to J . For the proof load, ... = from J- to g -l. The determination of the proportion, ^-, of the greatest depth of the beam to the span, so as to give the required stiffness, is effected by the aid of equation 6 of Article 300, from which we obtain _^i_ __ ri'fc 2~c~ ' Now 2/0 = m' 7i , m' being a numerical factor, which for symmetri- cal sections is -^ ; and consequently the required ratio is given by the equation _^o __ %_ n"fc ri'f 2_c 2c 2m'c~2m'Etf 1 4m'E' v, } n" an expression consisting of three factors : a factor, - -, depending on the distribution of the load and the figure of the beam ; a factor, , being the prescribed ratio of the span to the deflection ; and a *>i / factor, -4r, being the proof strain, or the working strain, of the _fcj material, as the case may be. To illustrate this, let the beam be under its working load, uni- formly distributed, and let it be of uniform section, alike above and 328 THEORY OP STRUCTURES. below. Then n" , m' = . Let = 1000 be the prescribed 2t v l ratio of the span to the working deflection. Let the material be wrought iron, for which ^ is a safe value for the working strain 1- Then k 5 1000 5 _ _ 2~c~24* 3000 72 ~ 144 ' which is very nearly the average proportion of depth to span adopted for wrought iron girders in practice. 303. The Slope and Deflection of a Beam under any Load are given by the following formulae : do? 1 r rM To integrate these equations, it is only necessary to substitute for the constant factor , in the equations 4, 5, 6, Article 300, its ivr y equivalent -y?, M' being now not the proof moment of flexure, but J-o the actual moment of flexure at the point where the beam is hori- zontal j that is to say, m"M' c , , "M' c 2 Greatest slope * l = ; deflection v l = -^-y - Ji 1 L 1 m" and w" being factors depending on the distribution of the load, and having the values given in the table of Article 300. Now the value of the moment of flexure is given in terms of the load and length by equation 1 of Article 291, and the ensuing table, viz., M = m "W I ; and the value of I , in terms of the dimensions of the rectangle circumscribing the cross section, is given by equation 1 of Article 295, and the ensuing table, viz., I = n' b h 3 ; hence the above equations 2 become ., _ m"mWlc , _ n" mW I c , . w'E6A' ; n'Ebh* ..... Moreover, I = c, or = 2 c, according as the beam is fixed at one end only, or supported at both ; so that if m'", ri", be a pair of numeri- cal factors, whose values are, for beams fixed at one end only, DEFLECTION OF BEAMS. 329 m m m ' } n = and for beams supported at both ends, in,"( 2m"m; ri" - the equations 3 become ni m W IV. Supported at both ends, uniformly loaded, . ^ ...... . B. UNIFORM STRENGTH AND UNIFORM DEPTH. Y. Fixed at one end, loaded at the other, . . ..... 1 ...... -. VI. Fixed at one end, loaded uniformly, ......... ^ ...... -. VII. Supported at both ends, loaded in the middle, - ...... -. VIII. Supported at both ends, loaded uniformly,.. - 330 THEORY OF STRUCTURES. C. UNIFORM STRENGTH AND UNIFORM Factor for Factor for BREADTH. Slope. Deflection. 2 IX. Fixed at one end, loaded at the other, 2 -5. X. Fixed at one end, uniformly loaded, infinite -. XL Supported at both ends, loaded in the middle, 1 o XII. Supported at both ends, uniformly loaded, 0-3927 ... 0-1427. 304. Deflection with Uniform Moment. In Article 292 the case has already been described, in which a beam or bar of uniform section has a pair of equal and opposite couples in the same plane applied to its ends, and the same case is the first given in the table of Article 300. In this case, M and I are constants, m" 1, and n" = - ; and accordingly, if c be the length of the part of the beam under consideration, and i\ the slope, and v\ the deflection, of one end relatively to a tangent at the other, ., _ Me , Me* *' ~ E I ; Vl ~ 2EI 305. The Resilience or Spring of a Beam is the work performed in bending it to the proof deflection. This, if the load is concen- trated at or near one point, is the product of half the proof load into the proof deflection ; that is to say, If the load is distributed, the length of the beam is to be divided into a number of small elements, and half the proof load on each element multiplied by the distance through which that element is moved during the proof deflection of the beam. Let u be that dis- tance ; then for beams fixed at one end, u ' = v ; and for beams supported at both ends, u = v l v. .(2.) Let d x be the length of an element of the beam ; w the intensity of the load on it, per unit of length ; then the resilience is RESILIENCE OF BEAMS. 331 wdx (3.) The cases in which the determination of resilience is most useful in practice are those in which the load is applied at one point. Let the beam be fixed at one end and loaded at the other, c being the length of its projecting part. Then by Article 295, equation 3 (observing that in 1, 1 = c), (n being given by the table of Article 295), and by Article 300, equation 6, _ ri'fc* _ ri'fc 2 ~E ~y~ Q m' E h' (n" being given by the table of Article 300, and in' by that of Article 295). Consequently, .,. Wv, nn" f 2 . . Reeilunce-- -y- --. _.--<;& A (4.) It will be observed that this expression consists of three factors, viz. : (1.) The volume of the prism circumscribed about the beam, c b h. f z (2.) A Modulus of Resilience, , of the kind already mentioned in Article 266. 7? / YL (3.) A numerical factor, ^ , ; in which n and m' (Article 295) 27/1 depend on the form of cross section of the beam, and n" (Article 300) on the form of longitudinal section and of plan. The follow- ing are values of this compound factor for a rectangular cross section, for which n = -z,m' = -, and therefore ^ , = ~Q ' n" IT I. Uniform breadth and depth, . II. Uniform strength, uniform depth, -x. III. Uniform strength, uniform breadth, -^. 332 THEORY OF STRUCTURES. If a beam be supported at both ends and loaded in the middle, its length being 1 = 2 c, its proof deflection is the same with that of a beam of the same transverse dimensions and of the length c, fixed at one end and loaded at the other ; and its proof load is double of that of the latter beam ; therefore its resilience is double of that of the latter beam. Consequently, for rectangular beams of the half-span c, supported at both ends and loaded in the middle, we have the following values for the numerical factor of the resilience : IV. Uniform breadth and depth, .................................... ' v \. preserving, at the points of ?Sl contrary flexure B, B, a Flg * 144- sufficient thickness only to resist the shearing force* BEAM FIXED AT BOTH ENDS. 337 As shown in Article 300, case 6, the curvature of the beam is uniform in amount, changing in direction only at the points of contrary flexure. Therefore, in fig. 143, C B and B A, at each side of the beam, are two arcs of circles of equal radii, horizontal at A and C, and touching each other at B; therefore those arcs are of equal length ; therefore each point of contrary flexure B is midway between the middle of the beam A and the point of sup- port C. It is evident also, that the proof deflection of the beam must be double of that of an uniformly curved beam of half the span, sup- ported at the ends without being fixed ; that is to say, one-half of that of an uniformly curved beam of the same span, supported but not fixed; or symbolically The actual moment of flexure at A must be the same as in an W uniformly loaded beam, with the same intensity of load w = , supported, but not fixed at B, B ; that is to say, = _ 16 ' 32 " " 4 " and therefore, the moment of flexure at C is 3 M 3 W c 3 W I nfb 1 k 2 = M l = ^L Q W = - = -- = 6 X being the breadth of the beam at C, which is three times the breadth b at A. To find the breadth at any other point, it is to be observed, that the moment of flexure at the distance x from A is and that consequently the breadth b, which is proportional to the moment of flexure, is given by the equation In using this equation, the positive or negative sign of the result merely indicates the direction of the curvature. According to equation 14, the figure of the beam in plan (fig. 144) consists of two parabolas, having their vertices at A, and 338 THEORY OF STRUCTURES. intersecting each other in the points of contraiy flexure, B, B, for which x = rr ^ . 2i The breadth which must be left at B, to resist shearing, will appear from the next Article. 308. A Beam Fixed at One End and Supported at Both is sensibly in the same condition with the part C B A B of the beam in fig. 143, extending from one of the fixed points C to the farther point of contrary flexure, which now represents a point supported, but not fixed. Hence if a continuous girder be supported on a series of piers, the span of each of the endmost bays should be to the span of each intermediate bay, in the ratio c -j- a? : 2 c, where XQ is the distance A B from the lowest point to a point of contrary flexure. 309. Shearing Stress in Beams. It has already been shown, in Article 288, how to find the amount F of the shearing force at a given vertical cross section of a beam ; and examples of that force in particular cases have been given in Articles 289 and 290. The object of the present Article is to show the manner in which the stress which resists that force is distributed. In Article 104 it has been shown, that the intensities of the tan- gential stresses at a given point, on a pair of planes at right angles to each other and to the plane parallel to which the stresses act , are necessarily equal. Hence, in order to determine the intensity of the vertical shearing stress at a given point in a vertical section of a beam, such as the point E in the vertical section G E B of the beam repre- sented in fig. 145, it is sufficient to find the equal Fig 145 intensity of the horizontal shearing stress >at the same point E in the horizontal plane E F. The existence of that hori- zontal shearing stress is familiarly known by the fact, that if a beam, instead of being one continuous mass, be divided into separate horizontal layers, those layers will slide on each other like the layers of a coach spring. The intensity of that stress is found as follows : Let H F D be another vertical section near to G E B. If the moment of flexure at H F D differs from that at G E B, there must be a corresponding difference in the amount of the direct stress on two corresponding parts of the planes of section, such as G E and H F. (In the case shown in the figure, that direct stress is a thrust, and is greatest at G E). That difference constitutes a horizontal force acting on the solid H F E G and in order to maintain the SHEAKING STRESS IN BEAMS. 339 equilibrium of that solid, the amount of shearing stress on the plane P E must be equal and opposite to that horizontal force. That amount being divided by the area of the plane F E, gives the intensity of the shearing stress. Q. E. I. From the foregoing solution it is obvious, that the shearing stress is nothing at the upper and lower surfaces of the beam j because the entire direct stress on each cross section is nothing. This might also be proved by reasoning like that of Article 278. It is also obvious that the shearing stress in the vertical layer between the two planes of section is greatest at D B, where they cut the neutral surface O C, at which the direct horizontal stress changes from thrust to pull ; for at that surface the horizontal force to be balanced by the shearing stress reaches its maximum. To express this solution symbolically in the case of a beam of uniform cross section ; let B = x, O C = c, B E = y, B G = ?/ B D = E F (sensibly) = d x ; let the breadth of the beam at any point E be denoted by , and at the neutral surface by . Let p be the intensity of the direct horizontal stress at E, q that ef the shearing stress at E, and q Q that of the maximum shearing stress at B. Then by equation 4 of Article 293, M and the amount of the direct stress on the sectional plane between G and E is M j The horizontal force by which the solid H F E G is pressed from O towards C, is the excess of the value of the above quantity for G E above its value for H F ; which excess arises from the excess of the moment of flexure M at G E B above the moment of flexure at H F D, farther from the middle of the beam by the distance d x. That difference of the moments of flexure is obviously equal to Fdx. F being the amount of the shearing force at the vertical layer in question; consequently, the horizontal force, which the shearing stress on the plane F E is to balance, is ~Fdx fvi j _ I ^ yz-dy. Dividing this by the area of the plane F E, which is z d x, the required intensity of the shearing stress is found to be 340 THEORY OF STRUCTURES. and the maximum value of that intensity, for the given vertical layer, which acts at D B in the neutral surface, is The same results are in every case obtained, whether the upper or the lower surface of the beam be taken as the limit of integration indicated by y^ ; the complete integral / y z d y, for the whole cross section of the beam, being =. 0, because of y being measured from the neutral axis, which traverses the centre of gravity of that section. Let S = I z d y be the area of the cross section of the beam. Then the mean intensity of the shearing stress is F S ' and the maximum intensity exceeds the mean in the following ratio : a ratio depending wholly on the figure of the cross section of the beam. The following table gives some of its values : FIGURE OF CROSS SECTION. -. r o I. Rectangle, z = b, ............................ ^. II Ellipse, .......................................... i. III. Hollow Rectangle 3 This includes I-shaped sec- f 2 ' (b- b') '(bh s - b 1 A") tions, ........................... J IV. Hollow square, tf - A' 2 , .................... |/1 +^ra)' 4 ' V. VI. Hollow ellipse and hollow circle ; the numerical factor ; o the symbolical factor, the same as for the hollow rectangle and hollow square respectively. LINES OF PRINCIPAL STRESS. 341 For beams of variable cross section, the preceding results, though not absolutely correct, are near enough to the truth for practical purposes. When a beam consists of strong upper and lower flanges or horizontal bars, connected by a thin vertical web or webs, like the wrought iron plate girders to be treated of in a subsequent section, the shearing force is to be treated as if it were entirely borne by the vertical web or webs, and uniformly distributed. 310. Hiinc* of Principal Stress in Beam*. Let p be the intensity of the direct horizontal stress, and q that of the shearing stress, at any point, such as E, fig. 145, in a beam. Then the axes of principal stress at that point, and the intensities of the pair of principal stresses, may be found by Article 112, Problem IV., case 4. In the equa- tions 21, 22, 23, which solve that problem, for p M the normal com- ponent of the stress on a vertical plane, is to be put p ; for p' M the normal component of the stress on a horizontal plane, is to be put 0; and for p tj the common tangential component, is to be put q. x and y having already been taken to denote the horizontal and vertical co-ordinates of the point E, p l and p. 2 may be taken to represent the greatest and least principal stresses instead of ^ and p y , and i, the angle which the axis of greatest stress makes with the horizon, instead of x n. Then equation 21 of Article 112 becomes Pi + P* _ P . 2 ~ 2 } equation 22 becomes Pi ~P* = from which we have These equations show, that the greatest principal stress is of the same kind with the direct horizontal stress, and the least principal stress of the contrary kind. Further, equation 23 becomes 2 a tan2*' 1 = -i (2.) or in another form THEORY OP STRUCTURES. If i a be the angle which the axis of least stress makes with the horizon, then, because i^ i? = 90, we have tan f j V 4 (f 2 q q " Equations 3 and 4 show that the axes of greatest and least stress are inclined opposite ways to the horizon (as indeed they must be, being perpendicular to each other), the inclination of the axis of least stress being the steeper. If those inclinations be computed for a number of different points in the vertical section of a beam, and the directions of the axes of stress at those points laid down on a drawing, a network of lines, con- sisting of two series of lines inter- secting each other at right angles, Fig. 146. as j n fig 146^ mav fc e drawn, so that each line shall touch the axes of stress traversing a series of points, and so that the tangents to the pair of lines which cross at any given point shall be the axes of stress at that point. These lines may be called the lines of principal stress. For a beam supported at the ends, the lines convex upwards are lines of thrust, and those convex downwards lines of tension. They all intersect the neutral surface at angles of 45. The stress along each of those lines is greatest where it is horizontal, and gradually diminishes to nothing at the two ends of the line, where it meets the surface of the beam in a vertical direction. 311. Direct Vertical Stress. It is to be observed, that no account has yet been taken of the direct vertical stress upon such planes as F E (fig. 145) in a loaded beam, that stress having been treated in the last Article as if it were null. The reasons for this are first, That the direct vertical stress is in most practical cases of small intensity compared with the other elements of stress ; secondly, That the mode of its distribution can be modified in an indefinite variety of ways by the modes of placing the load on or attaching it to the beam, so that formulae applicable to one of those modes would not be applicable to another (in fact, by a certain mode of loading, it can even be reduced to nothing) ; and thirdly, That its introduction would complicate the formulae without adding mate- rially to their accuracy. 312. SmaU Effect of Shearing Stress upon Deflection. A shearing stress of the intensity q produces a distortion represented by ~, C being the transverse elasticity, as already explained in Article 262. The slope of any given originally horizontal layer of the beam at a given point will be increased by this distortion to the extent denoted by SMALL DEFLECTION DUE TO SHEAR. 343 which additional slope is to be added to the slope due to the bend- ing stress, in order to find the total slope. The curvature of the layer will also be increased by the amount TV/ ,71? 1 cii __ a r 1 dx dx ' C 1 for uniform beams, and to nearly the same amount for other beams ; and there will be an additional deflection of the layer under con- sideration, of the amount dx ........................... (3.) /c F d x = M , the above equation becomes, for uniform beams, M Supposing the beam to be under the proof load, we may put for -=- its value , making the equation y i The greatest value of this is that for the neutral surface, for which the limits of integration are and y v To compare this additional deflection due to distortion with that due to flexure proper, let us take the case of a rectangular beam, in which 3/1 = ~, 2 &> d y = . Then ' j 1 For the same beam, according to equation 6 of Article 300, we have the proof deflection due to flexure proper, so that the ratio of those two parts of the deflection is 344 THEORY OF STRUCTURES. t>, 20 ' C ' ~TX 7 -j For wrought iron (for example) - = about 3. Suppose - = - v" 9 1 which is an ordinaiy proportion in practice ; then - = - = i}-^ t/ou j.v/y nearly, a quantity practically inappreciable. It appears, then, that the distortion produced by the shearing stress in beams, even at the neutral surface, where it is greatest, produces a deflection which is very small compared with that due to the bending action of the load ; and that the alteration of the external figure of the beam must be smaller still ; from which it may be concluded, that in ordinary practical cases there is no occa- sion to compute the additional deflection due to the shearing stress. 313. Partially-Loaded Beam. In designing beams for the sup- port of roads and railways, or for any other situation in which one part of a beam may be loaded and another unloaded, it is necessary to consider whether a partial load may or may not produce, at any point of the beam, a more intense stress than an uniform load over the whole beam. The case of this kind, which is most important in practice, is that in which a beam supported at both ends is uniformly loaded throughout a certain portion of its length and unloaded throughout the remainder ; and its solution depends on two theorems. THEOREM I. For a given intensity of load per unit of length, an uniform load over the whole beam produces a greater moment of flexure at each cross section than any partial load. Let the two ends of the beam be called C and D, and any inter- mediate cross section E. Then for an uniform load, the moment of flexure at E is an upward moment, being equal to the upward moment of the supporting force at either of the ends relatively to E, minus the downward moment of the uniform load between that end and E. A partial load is produced by removing the uniform load from part of the beam, situated either between E and C, be- tween E and D, or at both sides of E. First, let the load be removed from any part of the beam between E and C. Then the downward moment, relatively to E, of the load between E and D is unaltered ; and the upward moment, relatively to E, of the support- ing force at D is diminished, in consequence of the diminution of that force ; therefore the moment of flexure is diminished. A similar demonstration applies to the case in which the load is removed from a part of the beam between E and D ; and the combined effect of those two operations takes place when the load is removed from portions of the beam lying at both sides of E j so that the removal PARTIALLY-LOADED BEAM. 345 of the load from any portion of the beam diminishes the moment of flexure at each point. Q. E. D. Hence it follows, that if a beam be strong enough to bear an uni- form load of a given intensity, it will bear any partial load of the same intensity. THEOREM II. For a given intensity of load per unit of length, the greatest shearing force at any given cross section of a beam takes place when the longer of the two parts into which that section divides the beam is loaded and the shorter unloaded. Let the ends of the beam, as before, be called C and D, and the given cross section E ; and let C E be the longer part, and E D the shorter part of the beam. In the first place, let C E be loaded and E D unloaded. Then the shearing force at E is equal to the support- ing force at D, and consists in a tendency of E D to slide upwards relatively to C E. The load may be altered, either by putting weight between D and E, or by removing weight between C and E. If any weight be put between D and E, a force equal to part of that weight is added to the supporting force at D, and therefore to the shearing force at E ; but a force equal to the whole of that weight is taken away from that shearing force ; therefore the shear- ing force at E is diminished by the alteration of the load. If weight be removed from the load between C and E, the shearing force at E is diminished also, because of the diminution of the supporting force at D. Therefore any alteration from that distri- bution of the load in which the longer segment C E is loaded, and the shorter segment E D unloaded, diminishes the shearing force at E. Q. E. D. In designing beams where the shearing force is borne by a thin vertical web, or by lattice work (as in plate, lattice, and other compound girders, to be considered more fully in a subsequent sec- tion), it is necessary to attend to this Theorem, and to provide strength, at each cross section, sufficient to bear the shearing force which may arise from the longer segment of the beam being loaded and the shorter unloaded. To find a formula for computing that force, let c be the half-span of the beam, x the distance of the given cross section, E, from the middle of the beam, and w the uniform load per unit of length on the loaded part of the beam C E. The length of that part is and the amount of the load upon it, w (c + x). The centre of gravity of that load lies at a distance from the end, C, of the beam which is represented by 346 THEOEY OF STRUCTURES. c-\-x ~2~ ' and therefore the upward supporting force at the other end of the beam, D, which is also the shearing force at E, is given by the equation It has already been shown, in Article 290, that the shearing force at a given cross section with an uniform load is F = w x ; hence the excess of the greatest shearing force 'at a given cross section with a partial load, above the shearing force at the same cross section with an uniform load of the same intensity, is 4c At the ends of the beam this excess vanishes. At the middle, it consists of the whole shearing force F' = - w c, or one quarter of the shearing force at the ends ; that is, one-eighth of the amount of an uniform load. 314. Allowance for Weight of Beam. When a beam is of great span, its own weight may bear a proportion to the load which it has to carry, sufficiently great to require to be taken into account in determining the dimensions of the beam. Before the weight of the beam can be known, however, its dimensions must have been de- termined, so that to allow for that weight, an indirect process must be employed. As already explained in Article 302, the depth of a beam is de- termined by the deflection which it is desired to allow ; and the breadth remains to be fixed by conditions of strength, the strength being simply proportional to the breadth. Let H denote the breadth as computed by considering the ex- ternal load alone, W. Compute the weight of the beam from that B' provisional breadth, and let it be denoted by B'. Then =- f is the proportion which the weight of the beam must bear to the entire or W 7 gross load which it is calculated to support; and === ^ is the proportion in which the gross load exceeds the external load. Consequently, if for the provisional breadth b' there be substituted the exact breadth, , _ b'W W'-B" WEIGHT OF BEAM LIMITING LENGTH. 347 the beam will now be strong enough to bear both the proposed external load W, and its own weight, which will now be TV W B =ra; ........................... w and the true gross load will be In the preceding formulae, both the external load and the weight of the beam are treated as if uniformly distributed a supposition which is sometimes exact, and always sufficiently near the truth for the purposes of the present Article. 315. limiting Length of Beam. The gross load of beams of similar figures and proportions, varying as the breadth and square of the depth directly, and inversely as the length, is proportional to the square of a given linear dimension. The weights of such beams are proportional to the cubes of corresponding linear dimen- sions. Hence the weight increases at a faster rate than the gross load ; and for each particular figure of a beam of a given material and proportion of its dimensions, there must be a certain size at which the beam will bear its own weight only, without any addi- tional load. To reduce this to calculation, let the gross working uniformly- distributed load of a beam of a given figure, as in Article 295, be expressed as follows : (i.) I, b, and h being the length, breadth, and depth of the beam, f the limit of working stress, and n a factor depending on the form of cross section. The weight of the beam will be expressed by (2.) w' being the weight of an unit of volume of the material, and k a factor depending on the figure of the beam. Then the ratio of the weight of the beam to the gross load is B Jew' I 9 which increases in the simple ratio of the length, if the proportion - is fixed. When this is the case, the length L of a beam, whose n 348 THEORY OF STRUCTURES. weight (treated as uniformly distributed) is its working load, is T> given by the condition = 1 ; that is, L - Sn f k - Wl kw'l ~ B " ^ * This limiting length having once been determined for a given class of beams, may be used to compute the ratios of the gross load, weight of the beam, and external load to each other, for a beam of the given class, and of any smaller length, I, according to the fol- lowing proportional equation : L : I : L-Z : : W : B : W-B (5.) To illustrate this by a numerical example, let the beams in ques- tion be plain rectangular cast iron beams, so that n = , k = 1, 6 w' = 0-257 Ib. per cubic inch ; let 40,000 Ibs. per square inch be taken as the modulus of rupture, and 4 as the factor of safety, so ih'd,tf= 10,000 Ibs. per square inch ; and let -= = . Then I lo L = 3,459 inches = 288 feet, nearly. 316. A Sloping Beam, like that represented in fig. 68, Article 142, is to be treated like a horizontal beam, so far as the bending stress produced by that component of .^he load which is normal to the beam, is concerned. The component of the load which acts along the beam, is to be considered as producing a direct thrust along the beam, which is to be combined with the stress due to the bending component of the load. 317. An Originally Curved Beam, at any given cross section made at right angles to its neutral surface, so far as the bending stress is concerned, is in the same condition with an originally straight beam at a similar and equal cross section to which the same moment of flexure is applied. Beams are sometimes made with a slight convexity upwards, called a camber, equal and opposite to the curvature which the intended working load would produce in an originally straight beam. The effect of this is to make the beam become straight under the working load, instead of curved, and to diminish the additional stress due to rapid motion of the load, which additional stress arises partly from the curvature of the beam. 318. The Expansion and Contraction of I^ong Beams, which EXPANSION AND CONTRACTION OF BEAMS. 349 arise from the changes of atmospheric temperature, are usually pro- vided for by supporting one end of each beam on rollers of steel or hardened cast iron. The following table shows the proportion in which the length of a bar of certain materials is increased by an elevation of temperature from the melting point of ice (32 Fahr., or Centigrade) to the boiling point of water under the mean atmospheric pressure (212 Fahr., or 100 Centigrade) ; that is, by an elevation of 180 Fahr., or 100 Centigrade : METALS. Brass, -00216 Bronze, -00181 Copper, -00184 Gold, '0015 Cast iron, -ooin Wrought iron and steel, -00114 to -00125 Lead, -0029 Platinum, -0009 Silver, -002 Tin, -002 to -0025 Zinc, -00294 EARTHY MATERIALS. (The expansibilities of stone from the experiments of Mr. Adie.) Brick, common, '355 ^e, -0005 Cement, -0014 Glass, average of different kinds, -0009 Granite, -0008 to -0009 Marble, -00065 * ' OO1 1 Sandstone, -0009 to -OOT 2 Slate, -00104 TIMBER. (Expansion along the grain, when dry, according to Mr. Joule, Proceed. Roy. Soc., Nov. 5, 1857.) Baywood, -000461 to -000566 Deal, -000428 to -000438 Mr. Joule found that moisture diminishes, annuls, and even re- verses, the expansibility of timber by heat, and that tension in- creases it. 319. The Elastic Curve, in the widest sense of the term, is the figure assumed by the longitudinal axis of an originally straight 600 THEORY OF STRUCTURES. bar under any system of bending forces. All the examples of the curvature, slope, and deflection of beams in Article 300 and the subsequent Articles, are cases in which the elastic curve has been determined with a degree of approximation sufficiently close under the circumstances ; that is, when the deflection is a very small fraction of the length. The present Article relates to the figure of the 'elastic curve for a slender flat spring of uniform section, when acted upon either by a pair of equal and opposite couples, or by a pair of equal and opposite forces. The general equation of Article 300 applies to this case, viz. : I being the uniform moment of inertia of the section of the spring, E the modulus of elasticity, M the moment of flexure at a given point, and r the radius of curvature at that point. When a spring is under the action of a pair of equal and opposite couples applied to its two ends, then, as in Article 304, M is constant, r is constant, and the elastic curve is a circular arc of the radius r. When a spring is under the action of a pair of equal and opposite forces, let A and B denote the two points to which those forces are applied, and A B their common line of action. The figures from c Fig. 146 6. Fig. 146 c. Fig. 146 d. Fig. 146/ 146 a to 146 f, inclusive, represent various forms which the spring may assume, viz. : I. When the forces are directed towards each other ELASTIC CURVE. 351 a. A simple arc, like a bow, meeting A B at the points A and B only. b, c. An undulating figure, crossing A B at any number of inter- mediate points. d. The points A and B coinciding, which may give, with an endless spring, a figure of 8. II. When the forces are directed from each other e. One or more loops, with the ends and intermediate portions meeting or crossing A B. f. The forces acting from each other at the points A, B, in two rigid levers AD, BE, to which the spring is fixed at D and E : the spring forming one or more looped coils, lying altogether at one side of the line of action A B. Let P be the common magnitude of the equal and opposite forces applied at A and B, and x the perpendicular distance of any point C in the elastic curve from the line of action A B. Then the mo- ment of flexure at that point is obviously M = aP; (2.) and consequently the radius of curvature at that point is given by the equation El El -w = ^> that is to say, the radius of curvature is inversely proportional to the perpendicular distance from the line of action of tJie forces. At each of the points in figs. 146 a, 6, c, d, and e, where the curve meets or crosses A B, the radius of curvature is infinite ; that is, there is a point of contrary flexure. The above geometrical property is common to all the varieties of curves formed by an uniform spring bent by a pair of forces, and is sufficient to enable any one of them to be drawn approximately, by means of a series of short circular arcs. It is sufficient, also, to establish all their other geometrical properties, such as the rela- tions between their rectangular co-ordinates, and the lengths of their arcs. These are expressed by means of elliptic functions ; and it is unnecessary to give them in detail in this treatise, except in one case, which will be mentioned in the next Article, 319 A. There is one important proposition, however, which it is here necessary to prove ; and that is the following THEOREM. That a spring of a given length and section, to tlie ends of whose neutral surface a pair of forces are applied, will not be bent if those forces are less than a certain finite magnitude. Let A and B in fig. 146 a be the two ends of the spring, to which two equal 3 5 '2 THEORY OF STRUCTURES. and opposite forces of the magnitude P are applied, directed to- Avards each other ; the spring forming a single arc A C B, of the length 1. x being, as before, the ordinate of any point C, let y be the distance of that ordinate from A. The smaller the force P, the more nearly will the arc A C B approach to the straight line A B ; and in order to find the small- est value of P which is compatible with any bending of the spring, that force must be computed on the supposition that the ordinate x at each point is insensibly small compared with the length of the spring, and consequently, that the length of the arc A C does not sensibly differ from that of its abscissa y. This being the case, the curvature at any point C is to be taken as sensibly given by the following equation : r d y* ' which value being inserted in equation 3, gives d*x_ P_ ~^~El' The integral of this equation is x == a ' sin -, where c = A / -r=r- In order that x may be = at the points A and B, it is necessary AJ that when y = I, - sh c and consequently that qj that when y = I, - should be = n K, n being any whole number ; C (6.) Now of all the possible values of n, that which gives the least value of P is n = 1 ; whence we find and this^mfe quantity is the smallest force which will bend the given spring in the manner proposed. Q. E. D. This investigation proves the Theorem in question, and gives the least bending force ; but as it leaves the constant a indeter- HYDROSTATIC ARCH TWISTING AND WRENCHING. 353 minate, it does not give the figure assumed by the spring, which cannot be found exactly except by the use of elliptic functions. 319 A. The Hydrostatic Arch, described in Article 183, is of the same figure with the coiled and looped elastic curve represented in fig. 146/; for its radius of curvature at any point is inversely pro- portional to the perpendicular distance of that point from a given straight line. In order to transform all the equations given in that Article for the hydrostatic arch into the corresponding equa- tions for the coiled and looped elastic curve of fig. 146 f, it is only necessary to put for the constant product of the ordinate and radius of curvature the following value : El xr=-. An instrument consisting of an uniform spring attached to a pair of levers, might be used for tracing the figures of hydrostatic arches on paper. This property of the coiled and looped elastic curve is analogous to that discovered by James Bernouilli in the simple bow of fig. 146 a, viz., that it is the figure assumed by the vertical longitu- dinal section of an indefinitely broad sheet, containing a liquid mass whose upper horizontal surface is represented by A B. SECTION 7. On Resistance to Twisting and Wrenching. 320. The Twisting Moment, or moment of torsion, applied to a bar, is the moment of a pair of equal and opposite couples applied to two cross sections of the bar, in planes perpendicular to the axis of the bar, and tending to make the portion of the bar between those cross sections rotate in opposite directions about that axis. In the following Articles, twisting moments are supposed to be expressed in inch-pounds. 321. strength of a Cylindrical Axle. A cylindrical axle, A B, fig. 147, being subjected to the twisting moment of a pair of equal and oppo- site couples applied to the cross sec- tions A and B, it is required to find the condition of stress and strain at any intermediate cross section such as C, and also the angular displace- ment of any cross section relatively to any other. From the uniformity of the figure of the bar, and the uniformity of the twisting moment, it is evident that the condition of stress and strain of all cross sections is the same j also, because of the 2A 354 THEORY OF .STRUCTURES. circular figure of each cross section, the condition of stress and strain of all particles at the same distance from the axis of the cylinder must be alike. Suppose a circular layer to be included between the cross section C, and another cross section at the distance dx from it. The twisting moment causes one of those cross sections to rotate rela- tively to .the other, about the axis of the cylinder, through an angle which may be denoted by d i. Then if there be two points at the same distance r from the axis of the cylinder, one in the one cross section, and the other in the other, which points were origi- nally opposite to each other, in a line parallel to the axis, the twisting moment shifts one of those points laterally, relatively to the other, through the distance rdi. Consequently the part of the layer which lies between those points is in a condition of distortion, in a plane perpendicular to the radius r ; and the dis- tortion is expressed by the ratio which varies proportionally to the distance from the axis. There is therefore a shearing stress at each point of the cross section C, whose direction is perpendicular to the radius drawn from the axis to that point, and whose intensity is proportional to that radius, being represented by The STRENGTH of the axle is determined in the following manner : Let f be the limit of the shearing stress to which the material is to be exposed, being the ultimate resistance to wrenching if it is to be broken, the proof resistance if it is to be tested, and the ivorkiny resistance if the working moment of torsion is to be determined. Let ^ be the external radius of the axle. Then f is the value of q at the distance r l from the axis ; and at any other distance r, the intensity of the shearing stress is Conceive the cross section C to be divided into narrow concentric rings, each of the breadth dr. Let r be the mean radius of one of these rings. Then its area is 2 * r d r ; the intensity of the shear- ing stress on it is that given by equation 3, and the leverage of that stress relatively to the axis of the cylinder is r; consequently, the STRENGTH OF AN AXLE. 355 moment of the shearing stress of the ring in question, being the product of those three quantities, is which being integrated for all the rings from the centre to the circumference of the cross section C, gives for the moment of torsion, and of resistance to torsion, M = ^-. 'rdr = -^ (*) (i ""=1-5708 If the axle is hollow, r being the radius of the hollow, the integral is to be taken from r = r to r = r x ; and the moment of torsion becomes It is in general more convenient to express the strength of an axle in terms of the diameter than in terms of the radius. Let h v be the external diameter of the axle, and h Q its internal diameter, if hollow ; then For a solid axle, M = ^Y^- = -^ ; W For a hollow axle, M = Vj \ n ^ Jt " 16 A! If these formulae be compared with those applicable to solid and hollow cylindrical beams in Article 295, it will be seen that they differ only in the numerical factor, which, for the moment of flexure, is JL = _ , and for the moment of torsion, Y6 = 5T Hence we have this useful principle, that for equal values of the limiting stress f, the resistance of a cylinder, solid or holloiv, to wrenching, is double of its resistance to breaking across. Values of the co-efficient of ultimate resistance to shearing for cast and wrought iron, are given in a table which has already been referred to. The co-efficient for cast iron is somewhat doubtful, because the experiments give varying results. That given in the 356 THEORY OF STRUCTURES. table, viz., 32,500, is adopted on the authority of Mr. Hodgkin- son's work on cast iron, as the mean of the experiments considered by him the most trustworthy ; but some experiments give a value as low as 24,300, and others a value as high as 42,000. With respect to the working values of the limiting stress/) the following are those adopted by Tredgold in his practical rules : For cast iron, ................ 7, 650 Ibs. per square inch. For wrought iron, .......... 8,570 This amounts to allowing a factor of safety of about 4 for cast iron and 6 for wrought. Practical experience of the strength of wrought iron axles confirms the co-efficient given above for wrought iron very closely, it having been found that such axles bear a work- ing stress of 9,000 Ibs. per square inch for any length of time, if well manufactured of good material. The co-efficient for cast iron appears to leave too small a factor of safety for any motion except one that is very smooth and steady, and it may be considered that 5,000 Ibs. per square inch is a safer co-efficient for general use. Hence we may put, as the limit of working stress in shafts, For cast iron, ........... .f= 5,000 Ibs. per square inch. For wrought iron, ..... .f= 9,000 322. Angle of Torsion of a Cylindrical Axle -- Suppose a pair of diameters, originally parallel, to be drawn across the two circular ends, A and B, of a cylindrical axle, solid or hollow ; it is proposed to find the angle which the directions of those lines make with each other when the axle is twisted, either by the working moment of torsion, or by any other moment. This question is solved by means of equation 2 of Article 321, which gives for the angle of torsion per unit of length, di q dx = Cr' The condition of the axle being uniform at all points of its length, the above quantity is constant ; and if x be the length of the axle, and i the angle of torsion sought, expressed in length of arc to radius 1, we have - = -5-, and therefore, x dx I Let the moment of torsion be the working moment, for which r r, TORSION OF AN AXLE RESILIENCE. 357 Then the angle of torsion is and is the same whether the axle is solid or hollow. A value of C, the co-efficient of transverse elasticity for cast iron, is given in the table ; but it is uncertain, as experiments are dis- cordant. For wrought iron, that constant has been found with more precision, its mean value being about 9,000,000 Ibs. per square inch. Hence, for the working torsion of wrought iron shafts, we may make .(3.) / i C" 1,000' " II. Let the moment of flexure have any amount M consistent with safety. Then for , we have to put the equal ratio deduced from the equations 4 and 5 of Article 321, by substituting q for f in the numerators and r for r l in the denominators ; that is to say, q 2M For solid axles. - 7 ', and r irr\ _ ~ For hollow axles, - = . - -^ ; and r *-* ~Cr" 323. The Resilience of a Cylindrical Axle is the product of one- half of the greatest moment of torsion into the corresponding angle of torsion ; and it is given by the following equation : 1VT " f^ J)*^ IT - = ' for a solid shaft ; or 2 O'L C f K r\ 12 ^ or a n Uow shaft. Q ' I O n>i 358 THEORY OF STRUCTURES. 324. Axles not Circular in Section. When the cross section of a shaft is not circular, it is certain that the ratio - of the shearing T stress at a given point to the distance of that point from the axis of the shaft, is not a constant quantity at different points of the cross section, and that in many cases it is not even approximately constant ; so that formulae founded on the assumption of its being constant are erroneous. The mathematical investigations of M. de St. Tenant have shown how the intensity of the shearing stress is distributed in certain cases. The most important case in practice to which M. de St. Tenant's method has been applied is that of a square shaft ; and it appears that its moment of torsion is given by the formula M = 0-281 fh? nearly. 325. Bending and Twisting combined; Crank and Axle. A shaft is often acted upon by a bending load and a pair of twisting couples at the same time. In that case, the greatest direct stress due to the bending load, and the greatest shearing stress due to the moment of torsion, are to be combined in the manner already illustrated for beams, in Article 310. That is to say, let p be the greatest stress due to bending, and q that due to twisting ; \etp l be the intensity of the greatest result- ant stress, and i the angle which its direction makes with the axis of the shaft. Then 2(7 tan2t = ^ ; P .(I.) One of the most important examples of this is illustrated in fig. 148, which represents a shaft having a crank at one end. At the centre of the crank-pin, P, is applied the pressure of the connecting .rod ; and at the bearing, S, acts the equal and opposite resistance of that bearing. Represent- ing the common magnitude of those forces by P, they form a couple whose moment is M = P -SP. Draw P N perpendicular to S N, the axis of the shaft ; and let the angle P S N =j. Then the couple M may be resolved into Fig. 148. CRANK AND AXLE TEETH OF WHEELS. 359 A bending couple P N S = M cos j ; and A twisting couple P N P = M sin j. Equal and opposite couples act on the farther end of the shaft. Let h be its diameter. By the formulae of Article 295, the greatest stress produced at S by the bending couple is and that produced by the twisting couple, according to Article 321, is 5-1 M sinj p tan^ . Cf Tg ' ^ j \ V consequently, by the equations 1 of this Article, the resultant greatest stress at S, and its inclination to the axis of the shaft, are p. 5-1 M,. Pi=~(^cj + 1) = 75 (1 + 008 j) <-J' and by making p l =f, the proper diameter can be determined. These results may be represented graphically as follows : Draw S Q bisecting the angle N S P, and P Q perpendicular to S Q. S Q will be the direction of the resultant greatest stress at S, and the intensity of that stress will be the same as if it were caused by the bending action of a force equal to P and applied at Q, on an oblique section of the shaft perpendicular to P Q ; and also the same as the greatest intensity of the stress which would be produced at S by the direct bending action of a force equal to P applied at M in the axis of the shaft, with the leverage 326. The Teeth of wheels are made sufficiently strong, to provide against an action analogous to combined twisting and bending, which may arise from the whole force transmitted by a pair of wheels happening to act on one corner of one tooth, such as C or D, fig. 149. In fig. 150, let the shaded part represent a portion of a cross 360 THEORY OF STRUCTURES. section of the rim of the wheel A of fig. 149, and let E H K P be the face of a tooth, on one corner of which, P, acts the force represented by that letter. Conceive any sectional plane E F to in- tersect the tooth from the side EP to the crest PK, Fig. 150. and let PG be perpendicular to that plane. Let k be the thickness of the tooth, and let EF = 6, PG = I Then the moment of flexure at the section EF is PI, and the greatest stress produced by that moment of flexure at that section is 6 PZ Fig. 149. b A 2 ' which is a maximum when then the value, PE F = 45, and b = 2i, having 3P Consequently, the proper thickness for the tooth is given by the equation This formula is Tredgold's ; according to whom the proper value for the greatest working stress /is 4,500 Ibs. per square inch, when the teeth are of cast iron. SECTION 8. On Crushing by Bending. 327. Introductory Remarks. Pillars and struts whose lengths exceed their diameters in considerable proportions (as is almost always the case with those of timber and metal), give way not by direct crushing, but by bending sideways and breaking across, being crushed at one side, as at A, fig. 151, and torn asunder at the other, as at B. There does not yet exist any complete theory of this phenomenon. The formulae which have been provision- ally adopted are founded on a mode of investigation partly theoretical and partly empirical. Those which will first be explained are of a form proposed by Tredgold on theo- retical grounds. Having fallen for a time into disuse, they were Fig. 151. IRON PILLARS AND STRUTS. 361 revived by Mr. Lewis Gordon, who determined the values of the constants contained in them by a comparison of them with Mr. Hodgkinson's experiments. Then will be given Mr. Hodgkinson's own empirical formulae for the ultimate strength of cast iron pillars. 328. Strength of Iron Pillars and Struts. Let P be the load which acts on a long pillar or strut, and S its sectional area. Then one part of the intensity of the greatest stress on the material is simply the intensity due to the uniform distribution of the load over the section, and may be represented thus : ,4 Another part of the greatest stress is that which arises from the lateral bending, which will take place in that direction in which the pillar is most flexible ; that is, in the direction of its least dia- meter, if the diameters are unequal. Let h be that diameter, and b the diameter perpendicular to it let I be the length of the pillar, and let v be the greatest deflection of the axis of the pillar from its original straight position. Then, as in the case of a spring, Article 319, the greatest moment of flexure is P v ; and the greatest stress produced by that moment (which will be denoted by p") is directly as the moment, and inversely as the breadth and square of the thickness of the pillar (Article 295) ; that is, Pv * "bh* But the greatest deflection consistent with safety is directly as the square of the length, and inversely as the thickness (Article 300) ; that is, also, the product b h 2 is proportional to the sectional area S and to the thickness h. Consequently we have the proportional equation that is, the additional stress due to bending is to the stress due to direct pressure, in a ratio which increases as the square of the propor- tion in which the length of the pillar exceeds the least diameter. The whole intensity of the greatest stress on the material of the pillar, being made equal to a co-efficient of strength f, is expressed by the following equation : 362 THEORY OF STRUCTURES. in which, a is a constant co-efficient, to be determined by experi- ment. Hence the following is the strength of a long pillar : p _ 1 + (2.) The following are the values of /and a for the ultimate strength, as computed by Mr. Gordon from Mr. Hodgkinson's experiments on pillars FIXED AT THE ENDS, by having flat capitals and bases, as in fig. 152 : /, Ibs. per inch. a. Wrought iron, 36,000 oTw^n- 0,000 Cast iron, 80,000 A pillar ROUNDED AT BOTH ENDS, as in fig. 154, is as flexible as a pillar of the same diameter, fixed at both ends, and of double the length; and its strength might there- fore be expected to be the same; a \ 7* r\ f\ conclusion verified by the experiments of Mr. Hodgkinson. Hence, for such pillars, P= /S rf (3.) Mr. Hodgkinson found the strength of a pillar ,/jrec? at one end and rounded ,-,. lK/< at the other (fig. 153), to be a mean Fig. 152. Fig. 153. Fig. 154. V /> ... _ between the strengths of two pillars 01 the same length and diameter, one fixed at both ends, and the other rounded at both ends. Taking the proof load as one-half of the breaking load for wrought iron, and one-third for cast iron, and the working load as from one- fourth to one-sixth of the breaking load for both materials, the following are the values to be assigned to the limit of stress /under different circumstances : LOAD Breaking. Proof. Working. Wrought iron, 36,000 18,000 6,000 to 9,000 Cast iron, 80,000 26,700 13,300 to 20,000 IRON PILLARS AND STRUTS. 363 In using the formulae 2 and 3, the ratio -is generally fixed before- hand, to a degree of approximation sufficient for the purposes of the calculation. 329. Connecting Rods of engines are to be considered as in the condition of struts rounded at both ends ; Piston Rods, as in the condition of struts fixed at one end and rounded at the other. 330. Comparison of Cast and Wrought Iron. When the ultimate strength per square inch of section of pillars of cast and wrought iron respectively, and having various proportions of length to diameter, is computed by means of equation 2 of Article 328, it appears that for the smaller proportions of length to diameter, cast iron is the stronger material ; but that its strength diminishes as the proportion of length to diameter increases, faster than that of wrought iron ; so that for the proportion I : h : : J~Q95 : 1 : : 26J : 1 nearly, those materials are equally strong, and beyond that proportion wrought iron is the stronger. This result was first pointed out by Mr. Gordon. The following table illustrates it : I 10 20 26*.! on AO h * -w T- O u 4 Breaking load, ' Wrought, 34,840 31,765 29,230 27,700 23,480 Ibs. per square inch, = ,... Cast, 64,000 40,000 29,230 24,620 T 6,000 331. Mr, Hodgkinson's Formulae for the Ultimate Strength of Cast iron Pillars, as deduced by that author from his own experiments, are as follows : I. When the length is not less than thirty times the diameter. For solid cylindrical pillars, h being the diameter, in inches, and L the length in feet, P A _ - ^ - .,(1.) For hollow cylindrical pillars, 7i t being the external, and A the internal diameter, in inches, and L the length in feet, - A .(2.) The values of the co-efficient A are as follows : 364 THEORY OF STRUCTURES. Tons. For solid pillars with rounded ends, ............... 14-9 flat ends, ...................... 44-16 For hollow pillars with rounded ends, ............ 13-0 flat ends, .................. 44-3 II. When the length is less than thirty times the diameter. Let b denote the breaking load of the pillar, as computed by the preceding formulae. Let c denote the crushing load of a short block of the same sectional area S, as computed by the formula c = 49 tons x S in square inches .............. (3.) Then the correct crushing load of the pillar is 332. In Wrought iron Framework, the bars which act as struts, in order that they may have sufficient stiffness, are made of various figures in cross section, of which some examples are given in figs. 155 (angle iron), 156 (chan- *. *!* *7. *>* sBSLfe&si half-lattice girders), and 158 (T-iron). In each of these figures, the line which is to be considered as represented by h in the formula- of Article 328 is marked A H. In some large lattice girders, the struts are composed of a pair of parallel T-iron bars, such as fig. 158, with their flanges or ribs A turned towards each other, and connected together by a lattice work of small diagonal bars. In this case h may be taken to represent the diameter of the compound strut in that direction in which it is most flexible. 333. Wrought iron Cells are rectangular tubes (generally square) composed of four plate iron sides, rivetted to. angle iron bars at the corners, as shown in the section, fig. 159. This mode of construction was designed by Mr. Fair- bairn, to resist a thrust along the axis of the tube. The ultimate resistance of a single square cell to crushing by the buckling or bending of its sides, when the thickness of the ^plates is not less than one-thirtieth of the diameter of the cell, as determined by Mr. Fairbairn and Mr. Hodgkinson, is 27,000 Ibs. per square inch section of iron ; Fig. 159. CELLS SIDES OF GIRDEKS TIMBER POSTS. 36-5 but when a number of cells exist side by side in one girder, their stiffness is increased, and their ultimate resistance to a thrust may be taken at 33,000 to 36,000 Ibs. per square inch section of iron. The latter co-efficients apply also to cylindrical cells. 334. The sides of Plate iron Girders are subjected to a diagonal thrust arising from the shearing stress, and are usually stiffened by means of T-iron ribs, in the manner shown in fig. 160. The entire depth across the ribs may be taken to represent h in the formulae of Article 328. 335. Timber Posts and Struts. The following for- mula is given on the authority of Mr. Hodgkinson's experiments, for the ultimate resistance of posts of oak and red pine to crushing by bending : (1.) Fig. 160. S being the sectional area in square inches, h : I the ratio of the least diameter to the length, and A = 3,000,000 Ibs. per square inch. The factor of safety for the working load of timber being 10, A is to be made = 300,000 only, if P is the working load. For square posts and struts, the formula becomes If the strength of a timber post be computed both by this formula and by the formula for direct crushing, viz. : P=/S, ............................. (3.) the lesser value should be adopted as the true strength. Thus the ultimate strength per square inch for direct crushing is For oak, ......................... ./= 10,000 Ibs. = A . For red pine, .................... 6,000 Ibs. =-; oOO so that equation 1 or equation 3 should be used according as - is greater or less than a limit, which is, for oak, J 300 = 17 -32; for red pine, ^"500 = 22*36. 366 THEORY OP STRUCTURES. The resistance of timber to crushing, while green, is about one- half of its resistance after having been dried. SECTION 9. On Compound Girders, Frames, and Bridges. 336. Compound Girders in General. A compound girder is a structure which, as a whole, acts as a beam, resisting bending and breaking by a transverse load ; but whose parts are subjected to a variety of stresses of different kinds, requiring to be separately considered; such as the Warren girder of Articles 162 and 163, and the Lattice girder of Articles 164 and 165. In Part II., Chapter II., Section 1, it has already been shown how to determine the total stresses which act on the several pieces of a frame ; in section 6 of the present chapter, it has been shown how the stress is distributed in a continuous beam ; and in that and other sections, the resistance of materials to the various kinds of stress has been considered. The principal object of the present section is to indicate, by referring back to previous Articles, where the data and forinulse for determining the strength of the different parts of certain compound structures are to be found. A girder consists of three principal parts : a lower rib, to resist tension ; an upper rib, to resist thrust \ and a vertical iveb QIC frame, to resist shearing force. 337. Plate iron Girders are treated of in this section rather than in section 6, because the slender proportions of the parts subjected to a thrust sometimes render it necessary to compute their strength according to the laws of resistance to crushing by bending, explained in Ar- ticle 328. Some of the forms of cross sec- tion employed in such beams are shown in figs. 161, 162, 163, 164, and 165. Fig. 161 is a plain I-shaped beam, rolled in one piece. In fig. 162, the upper and lower ribs consist each of a flat bar or narrow plate rivetted to a pair of angle irons, the two pairs of angle irons being rivetted to the upper and lower edges of the vertical web. In fig. 163 the con- struction is the same, except that the Fig. 164- vertical web is double : this is the " box- long employed in the platforms of Fig. 163. blast furnaces, and first used in a railway bridge by Andrew Thom- son about 1832, on the Pollok and Govan Railway. In fig. 164, the upper and lower ribs are each built of several layers of narrow plates or flat bars, rivetted to each other and to a pair of angle PLATE IRON GIRDERS. 367 irons ; the upper and lower pairs of angle irons are rivetted to the upper and lower edges of the vertical web, and the plates of the vertical web are connected and stiffened at each of their vertical joints by a pair of T-irons, in the manner of which a horizontal section has been already given in fig. 160, Article 334. The object of building the larger sizes of horizontal ribs in layers, instead of making them in one piece, is to make them of those sizes of iron which can easily be rolled of good quality, and which are usually found in the market. Beams resembling fig. 164 are sometimes made with a double vertical web, for the sake of lateral stiffness. Fig. 165 represents the general form of the cross section of great, tubular or cellular girders, characterized by Mr. Stephenson's principle, of carrying the railway through the interior of the beam, and by Mr. Fairbairn's principle, of giving stiffness by means of cells, already described in Article 333. The joints of the cells are connected and stiffened by covering plates outside as well as angle irons inside ; and the plates of the two sides, which form a double vertical web, are stiffened and connected by T-irons, like those of fig. 1 64. Smaller cellular girders are sometimes used, in which the top alone consists of one or two lines ._. of cells, the girder in other respects being similar to fig. 1 64, with either a single or a double vertical web. In all plate iron girders, the joints exposed to tension should have covering plates, double rivetted if the stress is great enough to require it, which is almost always the case in the lower rib (see Article 280). The joints exposed to thrust should be exactly plane, exactly perpendicular to the direction of the thrust, accurately fitted, and perfectly close, that the surfaces may abut equally over their whole extent. Should open or irregular abutting joints be discovered after the girder has been put together, they should be filed out, and a flat plate of steel driven tight into each opening. The plates or bars of which built ribs are composed should break joint in a manner similar to the bond of brickwork. In plate iron girders generally, it is sufficiently accurate for prac- tical purposes to consider the whole bending moment M at any vertical section as borne by the upper and lower ribs, and the whole shearing stress F by the vertical web ; and also to consider the resistance of each of the horizontal ribs as concentrated at the centre of gravity of its section. Let h be the vertical depth between the centres of gravity of the sections of the upper arid lower ribs ; then the common value of the thrust along the compressed rib, and the tension along the stretched rib, is 368 THEORY OF STRUCTURES. M Let S t be the sectional area of the compressed rib,/! its resistance to crushing per square inch, S 2 the sectional area of the stretched rib, ft its resistance to tearing per square inch; then P M P M The values of the tenacity f 2 have already been considered in sec- tion 3. For plate beams with double-rivetted covering plates, its ultimate value may be taken at about 45,000 Ibs. per square inch of section of rib. The ultimate resistance to crushing, _/J, may be taken at its full value of 36,000 Ibs. per square inch in great tubular girders ; but when the compressed rib is narrow as compared with its length, the tendency to lateral bending may be allowed for by means of the following formula, of the kind already explained in section 8, Article 328 : where f= 36,000, a = , h' = the breadth of the compressed rib, and I' = the half span of the girder,. if it is not laterally stiffened by framing. In cases in which parallel beams are stiffened by hori- zontal diagonal braces, I' may be taken to denote the distance along the rib between a pair of the points to which braces are attached. Let t be the thickness of the vertical web if single, or the sum of the thicknesses if double. Then its sectional area is h t nearly ; consequently, if f s be its resistance per unit of section to the shear- ing force, F F ^ = -;and* = ; ...... . ............... (4.) Js Js /l and as the shearing stress is equivalent to a pull and a thrust in directions perpendicular to each other, and at angles of 45 to the horizon, f s should be the resistance of the vertical web to crushing, as determined by equation 3, in which, for j t is to be substituted r-77, h being the depth of the web, as before, and h" the width across the flanges of the stiffening ribs. The shearing force F at each cross section is to be computed as for a partial load, extending over the greater of the two segments GIRDERS HALF-LATTICE LATTICE BOWSTRING. 369 into which the section divides the beam, as explained in Article 313. The weight of the beam itself may be allowed for, either by the method of Article 314, or by the approximate method of Article 315. Owing probably to the yielding of the joints, it is found that in computing the deflection of plate girders (Articles 300 to 303), a smaller modulus of elasticity ought to be taken than for continuous iron bars. Its value in Ibs. per square inch is E = from 16,000,000 to 18,750,000. . 338. For Half-Lattice Beams and Lattice Beams, the methods of determining the total stresses have been fully considered in Articles 162, 163, 164, and 165 ; and it has only to be added here, that the shearing force should be computed for a partial load, as in Article 315. The ultimate tenacity of the ties, if the iron is of good qua- lity, may be taken at^ = 60,000 Ibs. per square inch. The resist- ance of the struts is to be computed as in Article 328. The figure of the strut diagonals has been considered in Article 332. The compressed rib may be a T-bar in small beams, and in larger beams a built rib or a cell. It has been sometimes made of cast iron. The remarks made in the last Article on abutting joints are equally applicable in the present case. In designing those joints which are connected by means of bolts, rivets, or keys, the prin- ciples of Article 280 should be observed. The value of the modulus of elasticity E for lattice girders, as computed from the deflection of Mr. Barton's viaduct at Drogheda, with/= about 10,000 Ibs. per square inch, is 16,000,000 Ibs. per square inch nearly. 339. A Bowstring Girder consists of an arched rib resisting thrust ; a horizontal tie resisting tension, and holding together the ends of the arched rib; a series of vertical suspending bars, by Fig. 166. which the platform is hung from the arched rib, and a series of diagonal braces between the suspending bars. Such girders are executed in timber and in iron; sometimes the arched rib is made of cast iron, as being stronger against crushing than wrought iron, and the remainder of the structure of wrought iron. The arched rib may be treated as uniformly loaded. Accord- ing to Article 178, its condition is like that of an uniformly - 2B 370 THEORY OP STRUCTURES. loaded chain inverted, and its proper form a parabola; and the thrust along it at each point is to be found by the formulae of Article 169. The tension along the horizontal tie is equal to the uniform horizontal component of the thrust along the arched rib. The tension on each vertical suspending bar is the weight of those portions of the platform and of the tie rod which hang from it. To give lateral stability to the girder, the suspending bars are usually made of considerable breadth, and of a form of horizontal section resembling figs. 160 and 161, and are firmly bolted to the cross beams of timber or of wrought iron which carry the roadway. When the beam is uniformly loaded, the arched rib is equilibrated, and there is no stress on the diagonals. The strength of the two diagonals which cross each other at a given plane of section S S ; , is to be adapted to sustain the excess of the greater shearing force due to a partial load above that due to an uniform load, as given by the formulae of Article 313. 340. Stiffened Suspension Bridges. The suspension bridge is that which requires the least quantity of material to support a given load. But when it consists, as in Article 169, solely of cables or chains, suspending rods, and platform, it alters its figure with every alteration of the distribution of the load ; so that a moving load causes it to oscillate in a manner which, if the load is heavy and the speed great, or even if the application of a small load takes place by repeated shocks, may endanger the bridge. To diminish this evil, it has long been the practice partially to stiffen suspension bridges by means of framework at the sides resembling a lattice girder. It was formerly supposed that, to make a suspension bridge as stiff as a girder bridge, we should use lattice girders sufficiently strong to bear the load of themselves, and that, such being the case, there would be no use for the suspending chains. But Mr. P. W. Barlow, having made some experiments upon models, finds that very light girders, in comparison with what were supposed to be necessary, are sufficient to stiffen a suspension bridge. If mathe- maticians had directed their attention to the subject, they might have anticipated this result. The present is believed to be the first investigation of its theory which has appeared in print. The weight of the chain itself, being always distributed in the same manner, resists alteration of the figure of the bridge. By leaving it out of account, therefore, an error will be made on the safe side as to the stiffness of the bridge, and the calculation will be simplified. Let fig. 167 represent one side of a suspension bridge, in which a STIFFENED SUSPENSION BRIDGE. 371 girder is used to stiffen the bridge. In order that it may do so effectually, any partial or concentrated load on the platform must, by Fig. 167. means of the girder, be trans- mitted to the chain in such a manner as to be uniformly distributed on the chain. The girder must have its ends so fixed to the piers as to be incapable of rising or falling. Then the forces which act upon it may be thus classed : downward, the load as applied ; down- Fig. 168. Fig. 169. Fig. 170. ward or upward, the resistances of the fastenings of the ends to their vertical displacement; upward, the uniformly distributed tension, acting through the suspension rods, between the girder and the chain. The girder will be supposed to be of uniform section throughout its length. Two cases will be considered : first, that in which a given load is concentrated in the middle of the girder; and secondly, that in which a given portion of the length of that girder is uniformly loaded, and the remainder unloaded, like the partially loaded beam of Article 313. The second case is the most important in practice. In each case, the half-span of the bridge will be denoted by c, and the horizontal distance of any point from the middle of the bridge by x. CASE I. A single load W, applied at t/te centre of tlie girder, tends to depress the chain in the middle, and consequently to raise it at the sides, and along with it to raise the beam near the ends; but the beam being, by its attachment to the piers, prevented from rising at the ends, takes a form like that represented by fig. 168 : depressed in the middle at A, and concave upwards; elevated, and convex upwards at C, C ; having points of contrary flexure at B, B ; and again depressed at D, D, the points of attachment to the piers. Now this curved figure is the effect of three downward forces, applied at D, A, D, respectively, and of an uniformly distributed upward force, acting on the whole length of the girder. Each half 372 THEORY OF STRUCTURES. of the girder, therefore, is in the condition of the beam described in Article 308, inverted; that is to say, the half-girder from A to D, if inverted, becomes a beam supported at D, supported and fioced horizontal at A, and loaded uniformly between A and D ; and lience (referring to the formulae of Article 307, case 3, and of Article 308) we have the following proportions amongst the lengths of the parts into which the half-girder is divided by the highest point C, and the point of contrary flexure B, BC ! = C~D = ^ = 0-577 x ATC; ............ (1.) x/3 and consequently, making A C, the distance between the lowest and highest points, = c f , we have In order to determine the greatest moment of flexure, and the deflection, of the stiffening girder, A C = c' is to be taken as the half-span of a girder like that considered in Article 307, case 3, fixed at both ends, and loaded with an uniform load of the intensity W W The greatest moment of flexure, as thus determined by the for- mulae of Article 307, case 3, is at the point A, and has the following value : .. r >2 J TXT M, = ^- = c -f- = 0-1057 c W; ............ (4.) and to that moment of flexure must the strength of the stiffening girder be adapted. The proof deflection may be measured in two ways : either between the highest and lowest points, C and A, or between the ends and the lowest point, D and A. The first may be called v c , and the second V D . Now by Article 307, case 3, we have The points of support D are at the same level with the points of contrary flexure B, being, in fact, points of no curvature them- selves ; and from this it is easily found that STIFFENED SUSPENSION BRIDGE. 373 CASE 2. The girder partially loaded. Let E B, in either of the tigs. 169, 170, represent the length of the loaded part of the stiffening girder, and B D that of the unloaded part ; let w be the uniform intensity of the load, and x the distance of the point where the load terminates from the middle of the beam ; x being considered as a positive quantity when the loaded part is the longer, as in fig. 169, and as a negative quantity when the loaded part is the shorter, as in fig. 170. The ends E and D of the beam being fastened so as to be in- capable of vertical displacement, the loaded segment E B is convex downwards, and the unloaded segment B D convex upwards : the loaded segment is in the condition of a beam supported at E and B, and uniformly loaded with the excess of the weight sustained above the force exerted between the girder and the chain ; and the unloaded segment is in the condition of a beam held down at B and D, and loaded with an uniformly distributed upward force, being that exerted between the girder and chain. The greatest moment of flexure of each segment is at its middle point, being A for the loaded part, and C for the unloaded part. The length of the loaded segment being E B = c + x, its gross load is W = w(c + x); and the intensity of the force exerted between the girder and chain, * = ) ........................... a-) This is the intensity of the upward load 011 the segment B D, whose length is B D = c x ; and consequently, according to Articles 290 and 291, the greatest moment of flexure of that seg- ment, at C, is xf The amount of the upward force exerted between the chain and BDis # : *4W-49.i ................. (3.) and this also is the amount of the net load on E B, being the excess of the gross load above the part borne by the chain. The half of this quantity, ">74 THEORY OF STRUCTURES. is the value at- once of the supporting force exerted by the pier against the girder at E, of the shearing force between the two divisions of the girder at B, and of the downward force by which the end D of the girder is held at its point of attachment to the pier. The intensity of the net load on E B is w(c-x) w-W = ^ 2c ' ; ....................... (5.) and the length of that segment being c + x, its greatest moment of flexure, at A, according to Articles 290 and 291, is _ (w-w)(c + x? _ w(c + xf'(c-x) A " 8 16 c By the usual process of finding maxima and minima, it is easily ascertained, that the greatest moment of flexure of the loaded s* division of the girder occurs when x = ^ ', or when two-thirds of tlie beam are loaded; and that the greatest moment of flexure of the unloaded division of the girder occurs when x = -, or when o two-thirds of the beam are unloaded; and further, that those two greatest moments are of equal magnitude though opposite in direction, viz. : max. M A = -max. M c = ; .............. (7.) and the stiffening girder must be made sufficiently strong to bear this bending moment safely in either direction. Now, the greatest moment of flexure which would arise from an uniform load of the given intensity w over the whole beam unsupported by the chain is therefore the transverse strength of the stiffening girder should be four twenty-seventh parts of that of a simple girder of the same span suited to bear an uniform load of the same intensity. The greatest value of the shearing force F in equation 4 occurs when one-half of the girder is loaded, or x = 0, and its amount is STIFFENED SUSPENSION BRIDGE. 375 max. F = - ........................... (8.) When two-thirds of the beam are loaded, the proof deflection of A below a straight line joining E and B, according to Article 300, is 4y ~9 12 E2/~27 Ey'"'' or four-ninths of the proof deflection of a beam of the same figure, uniformly loaded, of the span 2 c, unsupported by a chain. At the same time, the elevation of C above a straight line joining B and D is _ ..-. -__ % 13 E 4*, ~ 9 12 E2/~108 Ey'" v ' The proof depression of the lowest point of the beam, A, below the highest, C, is given by the equation _5 _5_ yv _25 yv v A -t-^ c 9 12 E2/~108 ES/'" QT foe-ninths of the proof deflection of an uniformly loaded beam. To find the depression of A below and the elevation of C above the level of the ends E and D, there must be added to V A and sub- tracted from V Q one-half of the depression of B below that level, which is found as follows : The greatest slope of E B under the proof load, relatively to a tangent at A, is and *./'. * ~3 E that of B D relatively to t - 2 ' / - c -f- x 4: fc 2y ' ~ 9 Ey' a tangent at C is c x 2 fc * D ~ 3 E Zy 9 E?/' j .(12.) C and the depression of B is easily seen to be _ (' * \ ./ . /I Q \ which gives finally, for the proof depression of A and elevation of C, <, B _17./V. V B _ 7 Jc\ V *^T 81 Vy' VG ~J 324 %'- the former being almost exactly one-half of the proof deflection of an uniformly-loaded beam of the same dimensions. 376 THEORY OF STRUCTURES. 341. Ribbed Arches. Bridges are frequently constructed whose arches consist of iron or timber ribs springing from stone abutments, as in fig. 171. In such cases it ought to be considered, that each rib fulfils at once the functions of an equi- librated arch, sustain- ing an uniform load of a certain intensity, and having a certain thrust along it, to be computed by the principles of Articles 169 and 178, and those of a stiffening girder, suited to produce an uniform distribution of a partial load, according to the principles of Article 340. Therefore, in designing the cross section of a rib for such a bridge, a provisional cross section ought first to be designed, suitable to bear a bending moment, upward or down- ward, of four twenty-sevenths of that which an uniform load of the given intensity would produce on a straight girder of the same span ; and in the second place, it should be determined in what proportion the thrust along the rib, considered as an equilibrated arch, will increase the intensity of the greatest stress on the pro- visional section already designed, and the breadths of that section should be increased in that proportion, to obtain the final cross section. SECTION 10. Miscellaneous Remarks on Strength and Stiffness. 342. Effects of Temperature. At a temperature of 600 Fahren- heit, the tenacity of iron was found by Mr. Fairbairn not to be diminished. That of copper and brass, at the same temperature, is reduced to about two-thirds of its ordinary magnitude. Sudden cooling from a high temperature tends to make most substances hard, stiff, and brittle ; gradual cooling tends to make them soft and tough ; and if often repeated or performed slowly from, a very high temperature, to weaken them. Various effects of temperature on the elasticity of solids have been ascertained by Dr. Joule, Dr. Thomson, and Professor Kupfer ; but they are more important to the science of molecular physics than to the art of construction. 343. The Effects of Repeated meltings on Cast Iron have been ascertained by Mr. Fairbairn. Up to and beyond the fourteenth melting the resistance to crushing increases ; but the resistance to cross-breaking reaches its maximum about the twelfth melting, and afterwards diminishes, from the metal becoming brittle and crys- talline. 344. The Effects of Ductility on strength form the subject of a DUCTILITY INTERNAL FRICTION. 377 paper by Professor James Thomson in the Cambridge and Dublin Mathematical Journal. That author shows, that a bent bar or a twisted rod of a ductile material, by being slowly and gradually strained, may be brought into such a condition as to have nearly the whole of its cross section in the condition of proof or limiting stress instead of the outer layers only, and may thus have its strength increased much beyond that given by the ordinary formulae. 345. internal Friction is a term which may be used until a better shall be devised to express a phenomenon recently observed by Dr. William Thomson in the extension of copper wire by a direct pull. The tension of the wire is increased, step by step, by successive augmentations of the load within the limits of permanent elasticity, and the elongation is observed at each step. Then by successive diminutions of the load, the tension is diminished by the same series of steps in the reverse order, and the elongation observed. When the load is completely removed, the wire recovers its original length without " set " or permanent elongation, but for each degree of tension the elongation is greater during the shortening of the wire than during the lengthening ; as if there were some molecular force analogous to friction, in so far as it impedes motion both ways, making the elongation less than it would otherwise be while the wire is being elongated, and greater than it would otherwise be while the wire is returning to its original length. It appears also that the force in question must depend in some way on the stress, from its disappearing when the tension is removed. 346. It must be obvious that much of the subject of strength and stiffness is in a provisional state, both as to mathematical theory and as to experimental data. Considerable improvement in both these respects may be anticipated from researches now in progress. PART III. PRINCIPLES OF CINEMATICS, OR THE COMPARISON OF MOTIONS. 347. Division of the Subject. The science of cinematics, and the fundamental notions of rest and motion to which it relates, having already been defined in the Introduction, Articles 8, 9, 10, 1 1 ; it remains to be stated, that the principles of cinematics, or the comparison of motions, will be divided and arranged in the present part of this treatise in the following manner : I. Motions of Points. II Rigid Bodies or Systems. III. Pliable Bodies and Fluids. IV. . Connected Bodies. CHAPTER I. MOTIONS OF POINTS. SECTION 1. Motion of a Pair of Points. 348. Fixed and Nearly Fixed Directions. From the definition of motion given in Article 9, it follows, that in order to determine the relative motion of a pair of points, which consists in the change of length and direction of the straight line joining them, that line must be compared, at the beginning and end of the motion con- sidered, with some fixed or standard length, and with at least two fixed directions. Standard lengths have already been considered in Article 7. An absolutely fixed direction may be ascertained by means whose principles cannot be demonstrated until the subject of dynamics is considered. For the present it is sufficient to state, that when a solid body rotates free from the influence of any external force tending to change its rotation, there is an absolutely fixed direction called that of the axis of angular momentum, which bears certain relations to the successive positions of the body. A nearly fixed direction is that of a straight line joining a pair 380 PRINCIPLES OF CINEMATICS. of points in two bodies whose distance from each other is very great, such as the earth and a fixed star. A line fixed relatively to tlie earth changes its absolute direction (unless parallel to the earth's axis) in a manner depending on the earth's rotation, and ' returns periodically to its original absolute direction at the end of each sidereal day of 86,164 seconds. This rate of change of direction is so slow compared with that which takes place in almost all pieces of mechanism to which cinematical and dynamical principles are applied, that in almost all questions of applied mechanics, directions fixed relatively to the earth may be treated as sufficiently nearly fixed for practical purposes. When the motions of pieces of mechanism relatively to each other, or to the frame by which they are carried, are under consi- deration, directions fixed relatively to the frame, or to one of the pieces of the machine, may be considered provisionally as fixed for the purposes of the particular question. 349. rotation of a Pair of Points 111 fig. 172, let Aj BI repre- sent the relative situation of a pair of points at one instant, and A 2 B 2 the relative situation of the same pair of points at a later instant. Then the change of the straight line A B between those points, from the length and direc- Fig. 172. Fig. 173. Fig. 17-i. tion represented by A! Bi to the length and direction represented by A 2 B 2 , constitutes the relative motion of the pair of points A, B, during the interval between the two instants of time considered. To represent that relative motion by one line, let there be drawn, from one point A,Jig. 173, a pair of lines, AB,, AB 2 , equal and parallel to A, B 1? A 2 B 2 , of fig. 172 ; then A represents one of the pair of points whose relative motion is under consideration, and Bj, B 2 , represent the two successive positions of the other point B relatively to A ; and the line Bi B 2 represents the motion of B rela- tively to A. Or otherwise, as in fig. 174, from a single point B let there be drawn a pair of lines, B A,, B A 2 , equal and parallel to A^, A 2 B 2 , of fig. 172; then Aj, A s , represent the two successive positions of A relatively to B ; and the line A l A 8 , equal and parallel to B! Bg of fig. 173, "but pointing in the contrary direction, represents the motion of A relatively to B. COMPONENT AND KESULTANT MOTIONS TIME. 381 350. Fixed Point and Uloring Point. In fig. 173, A is treated as the fixed point, and B as the moving point ; and in fig. 174, B is treated as the fixed point, and A as the moving point ; and these are simply two different methods of representing to the mind the same relation between the points A and B (see Article 10). 351. Component and Resultant motions. Let O be a point assumed as fixed, and A and B two suc- cessive positions of a second point rela- tively to O. In order to express mathe- matically the amount and direction of AB, the motion of the second point relatively to O, that line may be com- pared with three axes, or lines in fixed directions, traversing the fixed point O, such as O X, Y, O Z. Through A and B draw straight lines AC, B D, parallel to the plane of O Y and O Z, and cutting the axis X in C and D. Then CD is said to be the com- Fi S- 175 - ponent of the motion of the second point relatively to O, along or in tlie direction of the axis O X ; and by a similar process are found the components of the motion AB along O Y and Z. The entire motion A B is said to be the resultant of these components, and is evidently the diagonal of a parallelepiped of which the components are the sides. The three axes are usually taken at right angles to each other ; in which case A C and B D are perpendiculars let fall from A and B upon O X ; and if at, be the angle made by the direction of the motion A B with O X, CT3 = AB cos *. The relations between resultant and component motions are exactly analogous to those between the lines representing resultant and component couples, which have already been explained, in Articles 32, 33, 34, 35, 36, and 37. 352. The measurement of Time is effected by comparing the events, and especially the motions, which take place in intervals of time. Equal times are the times occupied by the same body, or by equal and similar bodies, under precisely similar circumstances,, in per- forming equal and similar motions. The standard unit of time is the period of the earth's rotation, or sidereal day, which has been proved by Laplace, from the records of celestial phenomena, not to have changed by so much as one eight-millionth part of its length in the course of the last two thousand years. 382 PRINCIPLES OF CINEMATICS. A subordinate unit is the second, being the time of one swing of a pendulum, so adjusted as to make 86,400 oscillations in 1 -0027379 1 of a sidereal day ; so that a sidereal day is 86 164 '09 seconds. The length of a solar day is variable ; but the mean solar day, being the exact mean of all its different lengths, is the period already mentioned of 1-00273791 of a sidereal day, or 86,400 seconds. The divisions of the mean solar day into 24 hours, of each hour into 60 minutes, and of each minute into 60 seconds, are familiar to all. Fractions of a second are measured by the oscillations of small pendulums, or of springs, or by the rotations of bodies so contrived as to rotate through equal angles in equal times. 353. Velocity is the ratio of the number of units of length described by a point in its motion relatively to another point, to the number of units of time in the interval occupied in describing the length in question ; and if that ratio is the same, whether it be computed for a longer or a shorter, an earlier or a later, part of the motion, the velocity is said to be UNIFORM. Velocity is expressed in units of distance per unit of time. For different purposes, there are employed various units of velocity, some of which, together with their proportions to each other, are given in the following table : Comparison of Different Measures of Velocity. Miles Feet Feet Feet per hour. per second. per n.inute. per hour. i =1-46 =88 = 5280- 0-6818 = i 60 = 3600 o > oii36 > =. 0*016 = i 60 0-0001893 = 0-00027 = 0-016 = i 1 nautical mile ) per hour, or >= 1*1507 = 1-6877 101-262 = 6075-74 "knot," j In treating of the general principles of mechanics, the foot per second is the unit of velocity commonly employed in Britain. The units of time being the same in all civilized countries, the proportions amongst their units of velocity are the same with those amongst their linear measures. Component and resultant velocities are the velocities of component and resultant motions, and are related to each other in the same way with those motions, which have already been treated of in Article 351. 354. Uniform Motion consists in the combination of uniform velocity with uniform direction \ that is, with motion along a straight line whose direction is fixed. MOTIONS OF POINTS. 383 SECTION 2. Uniform Motion of Several Points. 355. Motion of Three Points. THEOREM. The relative motions of three points in a given interval of time are represented in direction and magni- tude by the three sides of a triangle. Let O, A, B, denote the three points. Any one of them may be taken as a fixed point; let O be so chosen; and let OX, OY, O Z, fig. 176, be axes traversing it in fixed directions. Let A x and B! be the positions of A and B relatively to O at the beginning of the given interval of time, and A 2 and B 2 their positions at the end of that interval. Then A t A 2 and IVB 2 are the respective motions of A and B relatively to 0. Complete the parallelogram A! B x b A 2 ; then because A 3 b is parallel and equal to A! Bj, b is the position which B would have at the end of the interval, if it had no motion relatively to A ; but B 2 is the actual position of B at the end of the interval ; therefore, b B 3 is the motion of B relatively to A. Then in the triangle B, b B 2 , Bj b = A! A 3 is the motion of A relatively to O, b B 2 is the motion of B relatively to A, Bj B 2 is the motion of B relatively to O ; so that those three motions are represented by the three sides of a triangle. Q. E. D. This Theorem might be otherwise expressed by saying, that if three moving points be considered in any order, tlie motion oft/te third relatively to the first is the resultant of the motion oftfie third relatively to the second, and of the motion of the second relatively to tlie first; the word li resultant" being understood as already explained in Article 351. 356. Motions of a Series of Points. COROLLARY. If a series of points be considered in any order, and the motion o f each point deter- mined relatively to that which precedes it in tlie series, and if the relative motion of the last point and tlie first point be also determined., then will those motions be represented by the sides of a closed polygon. Let O be the first point, A, B, C, is found by integration (see Article 81), as follows : dt ........................ (2.) 360. Components of Varied motion. All the propositions of the two preceding sections, respecting the composition and resolution of motions, are applicable to the velocities of varied motions at a given instant, each such velocity being represented by a line, such as P T, in the direction of the tangent to the path of the point which moves with that velocity, at the instant in question. For example, if the axes OX, O Y, O Z, are at right angles to each other, and if the tangent P T makes with their directions respec- tively the angles *, ft, y, then the three rectangular components of the velocity of the point parallel to those three axes are v cos ; v cos /3; v cos y. Let x, y, z, be the co-ordinates of any point, such as P, in the path A P B, as referred to the three given axes. Then it is well known that d x dy dz cos a = ; cos /3 = ^ ; cos y = ; d s ds ds and consequently the three components of the velocity v are d x d y d z vcosa = ^j- -; v cos/3 = 2 ; v cos y = ; ...... (3.) at at at and these are related to their resultant by the equation 361. Uniformly-Varied Velocity. Let the velocity of a point either increase or diminish at an uniform rate ; so that if t repre- sents the time elapsed from a fixed instant when the velocity was V Q , the velocity at the end of that time shall be (1.) a being a constant quantity, which is the rate of variation of the velocity, and is called acceleration when positive, and retardation when negative. Then the mean velocity during the time t is VARIATION OF VELOCITY DEVIATION. 387 and the distance described is s = vt + a - ............................ (3.) To find the velocity of a point, whose velocity is uniformly varied, at a given instant, and the rate of variation of that velocity, let the distances, A s l} A s 2 , described in two equal intervals of time, each equal to A , before and after the instant in question, be observed. Then the velocity at the instant between those inter- vals is and its rate of variation is A 362. Varied Rate of Variation of Velocity. When the velocity of a point is neither constant nor uniformly-varied, its rate of variation may still be found by applying to the velocity the same operation of differentiation, which, in Article 359, was applied to the distance described in order to find the velocity. The result of this operation is expressed by the symbols, dv d*s ~dt = dt 2 ' and is the limit to which the quantity obtained by means of the formula 5 of Article 361 continually approximates, as the interval denoted by A t is indefinitely diminished. 363. Uniform Deviation is the change of motion of a point which moves with uniform velocity in a circular path. The rate at which uniform deviation takes place is determined in the following manner. Let C, fig. 179, be the centre of the cir- cular path described by a point A with an uniform velocity v, and let the radius C A be denoted by r. At the beginning and end of an interval of time A, let Aj and A 2 be the positions of the moving point. Then the arc A l A 2 = v A t ; and ch the chord Aj A 2 = v A t a The velocities at A, and A 2 are represented by the equal lines chord Fig. 179. the chord Aj A 2 = v A t arc 388 PRINCIPLES OF CINEMATICS. A, YI = A 2 Y 2 = v, touching .the circle at A! and A 2 respec- tively. From A 2 draw A 2 v equal and parallel to A, Y 1? and join ~V^v. Then the velocity A 2 Y 3 may be considered as compounded of ~A^v and v Y 2 ; so that v Y 2 is the deviation of the motion dur- ing the interval At; and because the isosceles triangles A 2 vV,, A! A 2 , are similar : -=. ATVg'A, A 2 _ v^_*t chord C~A r arc ' and the approximate rate of that deviation is v 2 chord r ' arc ' but the deviation does not take place by instantaneous changes of velocity, but by insensible degrees ] so that the true rate of deviation is to be found by finding the limit to which the approximate rate continually approaches as the interval At is diminished indefinitely. v 2 Now the factor remains unaltered by that diminution ; and the r ratio of the chord to the arc approximates continually to equality ; so that the limit in question, or true rate of deviation, is expressed by 364. Varying Deviation. When a point moves with a varying velocity, or in a curve not circular, or has both these variations of motion combined, the rate of deviation at a given instant is still represented by equation 1 of Article 363, provided v be taken to denote the velocity, and r the radius of curvature of the path, of the point at the instant in question. 365. The Resultant Rate of Variation of the motion of a point is found by considering the rate of variation of velocity and the rate of deviation as represented by two lines, the former in the direction of a tangent to the path of the point, and the latter in the direction of the radius of curvature at the instant in question, and taking the diagonal of the rectangle of which those two lines are the sides, which has the following value : I (S)'+i ()'}<> 366. The Rates of Variation of the Component Velocities of a point, parallel to three rectangular axes, are represented as follows : COMPARISON OP VARIED MOTIONS. 389 d a x d^y d a z ,.. * ~df'~dP' dt*'~ and if a rectangular parallelepiped be constructed, of which the edges represent these quantities, its diagonal, whose length is A / / (**}*+ (w + /**y i (2 .) V I \dt 2 ) \dt) \dt?) J will represent the resultant rate of variation, already given in another form in equation 1 of Article 365. 367. The Comparison of the Varied Motions of a pair of points relatively to a third point assumed as fixed, is made by finding the ratio of their velocities, and the directional relation of the tangents of their paths, at the same instant, in the manner already described in Article 358 as applied to uniform motions. It is evident that the comparative motions of a pair of points may be so regulated as to be constant, although the motion of each point is varied, pro- vided the variations take place for both points at the same instant, and at rates proportional to their velocities. 390 CHAPTER II. MOTIONS OF RIGID BODIES. SECTION 1. Rigid Bodies, and their Translation. 368. The term Rigid Body is to be understood to denote a body, or an assemblage of bodies, or a system of points, whose figure undergoes no alteration during the motion which is under con- sideration. 369. Translation or Shifting is the motion of a rigid body rela- tively to a fixed point, when the points of the rigid body have no motion relatively to each other ; that is to say, when they all move with the same velocity and in the same direction at the same instant, so that no line in the rigid body changes its direction. It is obvious that if three points in the rigid body, not in the same straight line, move in parallel directions with equal velocities at each instant, the body must have a motion of translation. The paths of the different points of the body, provided they are all equal and similar, and at each instant parallel, may have any figure whatsoever. SECTION 2. Simple Rotation. 370. Rotation or Turning is the motion of a rigid body when lines in it change their direction. Any point in or rigidly attached to the body may be assumed as a fixed point to which to refer the motions of the other points. Such a point is called centre of rotation. 371. Axis of Rotation. THEOREM. In every possible cJiange of position of a rigid body, relatively to a fixed centre, there is a line A 2 traversing that centre whose direc- tion is not changed. In fig. 180, let O be the centre of rotation, and let A and B denote any two other points in the body, whose situa- tions relatively to O are, before the turning, A,, B J? and after the 12 turning, A 2 , B 2 . Join A, A 8 , B! B 2 , forming the isosceles trian- gles O A! A 2, O B! Bo. Bisect the bases of those triangles in C and SIMPLE ROTATION. 391 D respectively, and through the points of bisection draw two planes perpendicular to the respective bases, intersecting each other in the straight line O E, which must traverse O. Let E be any point in the line OE; then EA^, and E Bj B 2 , are isosceles triangles; and E is at the same distance from O, A, and B, before and after the turning; therefore E is one and the same point in the body, whose place is unchanged by the turning; and this demonstration applies to every point in the straight line O E ; therefore that line is unchanged in direction. Q. E. D. COROLLARY. It is evident that every line in the body, parallel to the axis, has its direction unchanged. 372. The Plane of Rotation is any plane perpendicular to the axis. The Angle of Rotation, or angular motion, is the angle made by the two directions, before and after the turning, of a line per- pendicular to the axis. 373. The Angular Velocity of a turning body is the ratio of the angle of rotation, expressed in terms of radius, to the number of units of time in the interval of time occupied by the angular motion. Speed of turning is sometimes expressed also by the number of turns or fractions of a turn in a given time. The relation between these two modes of expression is the following : Let a be the angular velocity, as above denned, and T the turns in the same unit of time; then T- - -> (2 >* = 6-2831852). 374. Uniform Rotation consists in uniformity of the angular velo- city of the turning body, and constancy of the direction of its axis of rotation. 375. Rotation common to all Parts of Body. Since the angular motion of rotation consists in the change of direction of a line in a plane of rotation, and since that change of direction is the same how short soever the line may be, it is evident that the condition of rotation, like that of translation, is common to every particle, how small soever, of the turning rigid body, and that the angular velocity of turning of each particle, how small soever, is the same with that of the entire body. This is otherwise evident by con- sidering, that each part into which a rigid body can be divided turns completely about in the same time with every other part, and with the entire body. 376. Right and Left-Handcd Rotation. The direction of rotation round a given axis is distinguished in an arbitrary manner into 392 PRINCIPLES OF CINEMATICS. right-handed and left-handed. One end of the axis is chosen, as that from which an observer is supposed to look along the direction of the axis towards the rotating body. Then if the body seems to the observer to turn in the same direction in which the sun seems to revolve to an observer north of the tropics, the rotation is said to be right-handed ; if in the contrary direction, left-handed : and it is usual to consider the angular velocity of right-handed rotation to be positive, and that of left-handed rotation to be negative ; but this is a matter of convenience. It is obvious that the same rotation which seems right-handed when looked at from one end of the axis, seems left-handed when looked at from the other end. 377. Relative Motion of a Pair of Points in a Rotating Body. Let and A denote any two points in a rotating body ; and consider- ing as fixed, let it be required to determine the motion of A relatively to an axis of rotation drawn through O. On that axis let fall a perpendicular from A ; let r be the length of that perpen- dicular. Then the motion of A relatively to the axis traversing O is one of revolution, or translation in a circular path of tJie radius r ; the centre of that circular path being at the point where the perpendicular from A meets the axis. If a be the angular velocity of the body, then the velocity of A relatively to the axis traversing Ois v = ar; (1.) and the direction of that velocity is at each instant perpendicular to the plane drawn through A and the axis. The rate of deviation of A in its motion relatively to the given axis is ? = ^ ; ., ...(2.) in which the first expression is that already found in Article 363, and the second is deduced from the first by the aid of equation 1 of this Article. It is evident that for a given rotation the motion of O relatively to an axis of rotation traversing A is exactly the same with that of A relatively to a parallel axis traversing O ; for it depends solely on the angular velocity a, the perpendicular distance r of the moving point from the axis, and the direction of the axis ; all which are the same in either case. r is called the radius-vector of the moving point. 378. Cylindrical Surface of Equal Velocities. If a Cylindrical surface of circular cross section be described about an axis of rota- tion, all the points in that surface have equal velocities relatively to the axis, and the direction of motion of each point in the cylin- MOTIONS OF POINTS IN A ROTATING BODY. 393 drical surface relatively to the axis is a tangent to the surface in a plane perpendicular to the axis. 379. Comparative Motions of Two Points relatively to an Axis. Let O, A, B, denote three points in a rotating rigid body ; let O be considered as fixed, and let an axis of rotation be drawn through it. Then the comparative motions of A and B relatively to that axis are expressed as follows : the velocity-ratio is that oftlie radii- vectores of the points, and the directional relation consists in tJie angle between their directions of motion being the same with that between their radii-vector es. Or symbolically : Let r b r 2 , be the per- pendicular distances of A and B from the axis traversing O, and v 1 and v 2 their velocities ; then v 2 r 2 A A - ; and v^v z = 380. Components of Telocity of a Point in a Rotating Body. The component parallel to an axis of rotation, of the velocity of a point in a rotating body relatively to that axis, is null. That velocity may be resolved into components in the plane of rotation. Thus let 0, in fig. 181, represent an axis of rotation of a body whose plane of rota- tion is that of the figure ; and let A be any point in the body whose radius- vector is O A = r. The velocity of that point being v = a r, let that velocity be repre- sented by the line A V perpendicular to O A. Let B A be any direction in the plane of rotation, along which it is desired to find the component of the velocity of A ; and let ^ Y A IT = 6 be the angle made by that line with A V. From V let fall Y U perpendicular to B A ; then AU represents the component in question ; and denoting it by u, u = v ' cos 6 = ar cos 6 ..................... (1.) From let fall O B perpendicular to B A. Then ^ A O B = ^L Y A U = *; and the right-angled triangles B A and A U V are similar ; so that AY : ATI : : OA : OB = r cos t ............. (2.) Now the entire velocity of B relatively to the axis is ar cos 6 = u, ......... , ................ (3.) so that the component, along a given straight line in the plane of rotation, of the velocity of any point in that line, is equal to the velo- city of the point where a perpendicular from the axis meets that 394 PRINCIPLES OF CINEMATICS. SECTION 3. Combined Rotations and Translations. 381. Property of all motions of Rigid Bodies. The foregoing pro- position may be regarded as a particular case of the following, which is true of all motions of a rigid body. The components, along a given straight line in a rigid body, of the velocities of the points in that line relatively to any point, whether in or attached to the, body or otJierwise, are all equal to each other; for otherwise, the distances between points in the given straight line must alter, which is inconsistent with the idea of rigidity. 382. Helical Motion. Rotation is the only movement which a rigid body as a whole can have relatively to a point belonging to it or attached to it. But if the motion of the body be determined relatively to a point not attached to it, a translation may be com- bined with the rotation. When that translation takes place in the direction of the axis of rotation, the motion of the rigid body is said to be helical, or screw-like, because each point in the rigid body describes a helix or screw, or a part of a helix or screw. Let v l denote the velocity of translation, parallel to the axis of rotation, which is common to all points of the body ; this is called the velocity of advance. The advance during one complete turn of the rotating body is the pitch of each of the helical or screw-like paths described by its particles ; that is, the distance, in a direc- tion parallel to the axis, between one turn of each such helix and the next; and a being the angular velocity, so that is the time of one turn, the value of the pitch is 2, C V fl . Through a, b, and c, draw a plane a b c, on which let fall a perpendicular on from o. Then o n represents a component, which is common to the velocities of all the three points A, B, C, and must therefore be common to all the points in the body; that is, it is a velocity of translation. Fig. 183. From the points V a , V 6 , V w draw lines V a U a , V 6 U 6 , V c TJ e , equal and parallel to o n, but opposite in direction to it ; and join A U a , B TJ 6 , C I7 W which will all be parallel to the same plane ; that is, to the plane a b c. The last three lines will represent the component velocities which, along with the common velocity of translation parallel to o n, make up the resultant velocities of the three points. Through any two of the points A, B, draw planes perpendicular to the respective components of their motions which are parallel to a b c. These two planes will intersect each other in a line ODE, which will be parallel to o n. The perpendicular distances of that line from the points A, B, being unchanged by the motion, it represents one and the same line in or attached to the rigid body, and it is therefore the axis of rotation. A plane drawn through the third point C, perpendicular to C U e , will cut the other two planes in the same axis : the three revolving component velocities AU a ,BU 6 ,CU, 396 PRINCIPLES OF CINEMATICS. will be respectively proportional to the perpendicular distances, or radii-vectores, A~D, BE, CF, of the three points from that axis ; and the angular velocity will be equal to each of the three quotients made by dividing the revolving component velocities of the points by their respective radii-vectores. This rotation, combined with a translation parallel to the axis, with a velocity represented by o n, constitutes a lielical motion, being the required motion of the rigid body. Q. E. I. 384. Special Cases of the preceding problem occur, in which either a more simple method of solution is sufficient, or the general method fails, and a special method has to be employed. I. When the motions of tJie points of the body are known to be all parallel to one plane, it is sufficient to know the motions of two points, such as A, B, fig. 184. Let A O, B O, be two planes tra- versing A and B, and perpendicular to the respective directions of the simul- taneous velocities of those points ; if those planes cut each other, the entire motion is a rotation ; the line of intersection of the planes O, being the axis of rotation, and the angular velocity, are found as in the last Article. If the two planes are parallel, the motion is a translation. II. If three points, not in the same plane, have parallel motions, or if three points in the same plane have parallel motions oblique to the plane, the motion is a translation. III. If three points in the same plane move perpendicularly to the plane, as A B C, fig. 184 a, then if their velocities are equal, the Fig. 184. Fig. 184 c. motion is a translation; and if their velocities are unequal, the motion is a rotation about the axis which is the intersection of the ROTATION INSTANTANEOUS AXIS. 397 plane of the three points with the plane drawn through the extre- mities V a , Y fr , V c , of the three lines which represent their veloci- ties ; the angular velocity being found as in Article 383. If the plane of rotation is known, then the simultaneous veloci- ties of two points, as A and B in figs. 184 b and 184 c, are sufficient to determine the axis 0. 385. Rotation Combined with Translation in the Same Plane. Let a body rotate about an axis C (fig. 185), fixed relatively to the body, with an angular velocity a, and at ^the same time let that axis have a motion of translation in a straight path perpendicular to the direction of the axis, with the velocity u, represented by the line C U. It is required to find the velocity and direction of motion of any point in the body. From the F moving axis draw a straight line C T perpendi- cular to that axis and to CTJ, and in that direction into which the rotation (as represented by the feathered arrow) tends to turn C U, and make CT = - * (1.) a Then the point T has, in virtue of translation along with the axis C, a forward motion with the velocity u ; and in virtue of rota- tion about that axis, it has a backward motion with the velocity a CT = u, equal and opposite to the former and its resultant velocity is 0. Hence every point in the body, which comes in succession into the nt position T, situated at the distance from the axis C in the direc- tion above described, is at rest at the instant of its arriving at that position; that is, it has just ceased to move in one direction, and is about to move in another direction ; and this is true of every point which arrives at a line traversing T parallel to C. Conse- quently the resultant motion of the body, at any given instant, is the same as if it were rotating about the line which at the instant in question occupies the position T, parallel to C, at the distance - ; and that line is called THE INSTANTANEOUS AXIS. To find the a motion of any point A in the body at a given instant, let fall the perpendicular A T from that point on the instantaneous axis ; then the motion of A is in the direction A V perpendicular to the plane 398 PRINCIPLES OF CINEMATICS. of the instantaneous axis and of the instantaneous radius-vector A T, and the velocity of that motion is v = a .(2.) 386. Rolling Cylinder; Trochoid. Every straight line parallel to the moving axis C, in a cylindrical surface described about C with the radius -, becomes in turn the instantaneous axis. Hence a the motion of the body is the same with that produced by the roll- ing of such a cylindrical surface on a plane FTP parallel to C and to C U, at the distance -. a The path described by any point in the body, such as A, which is not in the moving axis C, is a curve well known by the name of trochoid. The particular form of trochoid called the cycloid, is described by each of the points in the rolling cylindrical surface. 387. Plane Rolling on Cylinder; Spiral Paths. Another mode of representing the combination of rota- tion with translation in the same plane is as follows : Let O be an axis assumed as fixed, about which let the plane O C (containing the axis O) rotate (right- handedly, in the figure), with the angu- lar velocity a. Let a rigid body have, relatively to the rotating plane, and in a direction perpendicular to it, a transla- tion with the velocity u. In the plane O C, and at right angles to the axis 0, Fig. 186. take T = -, in such a direction that a the velocity u = a - OT, which the point T in the rotating plane has at a given instant, shall be in the contrary direction to the equal velocity of translation u t which the rigid body has relatively to the rotating plane. Then each point in tJie rigid body which arrives at the position T, or at any position in a line traversing T parallel to the fixed axis O, is at rest at the instant of its occupying that position ; therefore the line traversing T parallel to the fixed axis O is the instantaneous axis; the motion at a given instant of any point in the rigid body, such as A, is at right angles to the radius- vector A T drawn per- COMBINED PARALLEL ROTATIONS. 399 pendicular to the instantaneous axis; and the velocity of that motion is given by the equation v = a- AT. All the lines in the rigid body which successively occupy the position of instantaneous axis are situated in a plane of that body, FTP, perpendicular to O C ; and all the positions of the instan- taneous_axis are situated in a cylinder described about O with the radius O T ; so that the motion of the rigid body is such as is pro- duced by the rolling of the plane P P on tlie cylinder whose radius is O T = -. Each point in the rigid body, such as A, describes a plane spiral about the fixed axis 0. For each point in the rolling plane, P P, that spiral is the involute of the circle whose radius is O T. For each point whose path of motion traverses the fixed axis O, that is, for each point in a plane of the rigid body traversing O parallel to P P, the spiral is Archimedean, having a radius- vector increasing by the length u for each angle a through which it rotates. 388. Combined Parallel Rotations In figs. 187, 188, and 189, Fig. 188. Fig. 189. let O be an axis assumed as fixed, and O C a plane traversing that axis, and rotating about it with the angular velocity a. Let C be an axis in that plane, parallel to the fixed axis O ; and about the moving axis C let a rigid body rotate with the angular velocity b relatively to the plane O C ; and let the directions of the rotations a and b be distinguished by positive and negative signs. The body is said to have the rotations about the parallel axes O and C combined or compounded, and it is required to find the result of that com- bination of parallel rotations. Fig. 187 represents the case in which a and b are similar in direction; fig. 188, that in which a and b are in opposite direc- 400 PRINCIPLES OF CINEMATICS. tions, and b is the greater ; and fig. 189, that in which a and b are in opposite directions, and a is the greater. Let a common perpendicular O C to the fixed and moving axes be intersected in T by a straight line parallel to both those axes, in such a manner that the distances of T from the fixed and moving axes respectively shall be inversely proportional to the angular velocities of the component rotations about them, as is expressed by the folloAving proportion : a :b : : CT : OT (1.) When a and b are similar in direction, let T fall between and C, as in fig. 187 ; when they are contrary, beyond, as in figs. 188 and 189. Then the velocity of the line T of the plane O C is a O T ; and the velocity of the line T of the rigid body, relatively to the plane C, is b ' C T, equal in amount and contrary in direction to the former ; therefore each line of the rigid body which arrives at the position T is at rest at the instant of its occupying that posi- tion, and is then the instantaneous axis. The resultant angular velocity is given by the equation c = a + b- (2.) regard being had to the directions or signs of a and b ; that is to say, if we now take a and b to represent arithmetical magnitudes, and affix explicit signs to denote their directions, the direction of c will be the same with that of the greater; the case of fig. 187 will be represented by the equation 2, already given ; and those of figs. 188 and 189 respectively by c = b-a; c = a-b (2 A.) The relative proportions of a, b, and c, and of the distances between the fixed, moving, and instantaneous axes, are given by the equation a :b :c :: CT : OT :OC (3.) The motion of any point, such as A, in the rigid body, is at each instant at right angles to the radius- vector A T drawn from the point perpendicular to the instantaneous axis j and the velocity of that motion is v = c A^T (4.) 389. Cylinder Boiling on Cylinder; Epifrochoids. All the lines in the rigid body which successively occupy the position of instan- taneous axis are situated in a cylindrical surface described about C with the radius C T ; and all the positions of the instantaneous ROLLING CYLINDERS CURVATURE OF EPITROCHOIDS. 401 axis are contained in a cylindrical surface described about O with the radius O T ; therefore the resultant motion of the rigid body is that which is produced by rolling the former cylinder, attached to the body, on the latter cylinder, considered as fixed. In fig. 187, a convex cylinder rolls on a convex cylinder ; in fig. 188, a smaller convex cylinder rolls in a larger concave cylinder; in fig. 189, a larger concave cylinder rolls on a smaller convex cylinder. Each point in the rolling rigid body traces, relatively to the fixed axis, a curve of the kind called epitrochoids. The epitrochoid traced by a * point in the surface of the rolling cylinder is an epi- cycloid. In certain cases, the epitrochoids become curves of a more simple class. For example, each point in the moving axis C traces a circle. When a cylinder, as in fig. 188, rolls within a concave cylinder of double its radius, each point in the surface of the rolling cylinder moves backwards and forwards in a straight line, being a diameter' of the fixed cylinder ; each point in the axis of the rolling cylinder traces a circle of the same radius with that cylinder, and each other point in or attached to the rolling cylinder traces an ellipse of greater or less eccentricity, having its centre in the fixed axis O. This principle has been made available in instruments for drawing and turning ellipses. 390. Curvature of Epitrochoids. The following being given : the radius of the fixed cylinder, O T = ^ ; the radius of the rolling cylinder, C T = r 2 ; the instantaneous radius- vector of a tracing-point A, A T = T ; the angle made by that radius- vector with the rotating plane, as 6 is it is required to find the radius of curvature, p, of the pa.th of the tracing-point A, at the instant under consideration. The radius of a convex cylinder is to be considered as positive, and that of a concave cylinder as negative ; and regard is to be paid to the principle, that cos d is < P ** 1 * ve I according ( negative j ( acute ) { obtuse j ' Let d t be an indefinitely short interval of time ; then during that interval the tracing-point A moves through the distance crdt. Let the direction of the radius- vector r, which is perpendicular t( > the path traced by A, alter in the same time by the angle di. Then the radius of curvature of the path of A is 2D 402 PRINCIPLES OF CINEMATICS. cr dt To determine the angular motion di of the radius- vector, it has to be considered, that the absolute angular velocity of the rolling cylinder is c, which gives that cylinder an angular motion, c d t, in the given interval ; and also that, in the course of the same inter- val, a new line comes to occupy the position of instantaneous axis, distant from the original line by the length br s dt, in a direction opposite to that of the rotation of the rolling cylinder. The effect of this shifting of the instantaneous axis is, to turn the angular position of the radius-vector r, in a negative direction relatively to the rolling cylinder, through the angle b r a cos 6 ' d t which being combined with the angular motion of the cylinder, c d t, gives as the resultant angular motion of the radius- vector, / b r a cos d i> z^z 1 c v f which being substituted in equation 1, gives for the radius of cur- vature of the path traced by A, cr ~ b r a cos 6 n b r a cos 6 c -- - - 1 -- r cr Now, b r, /2 \ c - n + V (attention being paid to the implicit signs of r^ and r a ) ; and con- sequently, r i ~^~ r 2 C\ \ P = r " r.r.coe< () *+- The sign of this result, when j ^ative f ' shows that the curve traced by A is j conca ^ I towards T. The following are some limited cases : I. When the tracing-point is the surface of the rolling cylinder, r = 2 r 2 cos 6 ; and therefore, CURVATURE OF EPITROCHOIDS. 403 -j .................... (4.) which is the radius of curvature of an epicycloid. II. When a cylinder rolls on a plane, r^ becomes infinitely great as compared with r 2 , and thus reduces equation 3 to P = r 2 cos 6 J. which is the radius of curvature of a trochoid. III. When a cylinder rolls on a plane, and the tracing-point is in the surface of the cylinder, r = 2 r 2 cos 6, and p = 2r = 4r 2 cos 6, (6.) which is the radius of curvature of a cycloid. IV. When a plane rolls on a cylinder, r z becomes infinitely great as compared with r and r ; and equation 3 becomes i (7.) 1 0-1 cos* which is the radius of curvature of a spiral of the class mentioned in Article 387. Y. When a plane rolls on a cylinder, and the tracing-point is in tlie plane, cos 6 = ; and equation 7 becomes p = r, .............................. (8.) which is the radius of curvature of the involute of a circle. VI. When a plane rolls on a cylinder, and the tracing-point is at the distance r from the plane on the side next the cylinder, cos 6 -- - ; and equation 7 takes the following form : which is the radius of curvature of an Archimedean spiral. Let R be the distance of a point in that spiral from the fixed axis ; then r 2 = R 2 + r\, and (9A.) As to rolling curves in general, see Professor Clerk Maxwell's paper in the Transactions of the Royal Society of Edinburgh, vol. xvi. PRINCIPLES OF CINEMATICS. 391. Equal and Opposite Parallel Rotations Combined. Let a plane O C rotate with aii angular velocity a about an axis con- tained in the plane, and let a rigid body rotate about the axis C in that plane parallel to O, with an angular velocity a, equal and opposite to that of the plane. Then the angular velocity of the rigid body is nothing ; that is, its motion is one of translation only, all its points moving in equal circles of the radius O C, with the velocity a O C. This case is not capable of being represented by a rolling action. 392. Rotations about Intersecting Axes Combined. In fig. 190, let O A be an axis assumed as fixed ; and about it let the plane A O C rotate with the angular velocity a. Let O C be an axis in the rotating plane ; and about that axis let a rigid body rotate __ with the angular velocity b re- F - 190 latively to the rotating plane. Because the point O in the rigid body is fixed, the instantaneous axis must traverse that point. The direction of that axis is determined, as before, by considering that each point which arrives at that line must have, in virtue of the rotation about O C, a velocity relatively to the rotating plane, equal and directly opposed to that which the coincident point of the rotating plane has. Hence it follows, that the ratio of the per- pendicular distances of each point in the instantaneous axis from the fixed and moving axes respectively that is, the ratio of the sines of the angles which the instantaneous axis makes with the fixed and moving axes must be the reciprocal of the ratio of the component angular velocities about those axes ; or symbolically, if O T be the instantaneous axis, : ^COT : : b : a .................. (1.) This determines the direction of the instantaneous axis, which may also be found by graphic construction as follows : On O A take O a proportional to a ; and on O C take O b proportional to b. Let those lines be taken in such directions, that to an observer looking from their extremities towards O, the component rotations seem both right-handed. Complete the parallelogram b c a ; the dia- gonal 6 c will be the instantaneous axis. The resultant angular velocity about this instantaneous axis is found by considering, that if C be any point in the moving axis, the linear velocity of that point must be the same, whether com- puted from the angular velocity a of the rotating plane about the ROLLING CONES ANALOGY OF ROTATIONS AND FORCES. 405 fixed axis O A, or from the resultant angular velocity c of the rigid body about the instantaneous axis. That is to say, let CD, C E, be perpendiculars from C upon O A, T, respectively ; then a CD = c ' C E ; but CD : CE : :sin^AOC : sin ^:COT; and therefore sin ^ C O T : sin ^ A O C : : a : c ; and, combining this proportion with that given in equation 1, we obtain the following proportional equation : sin^COT : sin^AOT isin^ _ __ ___ : : Oa : Ob : Oc that is to say, the angular velocities of the component and resultant rotations are each proportional to the sine of t/ie angle between the axes of the otlier two ; and the diagonal of the parallelogram b c a repre- sents both the direction of the instantaneous axis and the angular velo- city about that axis. 393. Rolling Cones. All the lines which successively come into the position of instantaneous axis are situated in the surface of a cone described by the revolution of O T about O C ; and all the positions of the instantaneous axis lie in the surface of a cone described by the revolution of OT about OA. Therefore the motion of the rigid body is such as would be produced by the roll- ing of the former of those cones upon the latter. It is to be understood, that either of the cones may become a flat disc, or may be hollow, and touched internally by the other. For example, should ^ A O T become a right angle, the fixed cone would become a flat disc ; and should ^ A O T become obtuse, that cone would be hollow, and would be touched internally by the rolling cone ; and similar changes may be made in the rolling cone. The path described by a point in or attached to the rolling cone is a spherical epitrochoid; but for the purposes of the present trea- tise, it is unnecessary to enter into details respecting the properties of that class of curves. 394. Analogy of Rotations and Single Forces. If the proportional equation 3 of Article 388, which shows the relations between the component angular velocities of rotation about a pair of parallel axes, the resultant angular velocity, and the position of the instan- taneous axis, be compared with the proportional equation of Article 39, by means of which, as explained in Article 40, the magnitude and position of the resultant of a pair of parallel forces are found, it will be evident that those equations are exactly analogous. The result of the combination of a rotation with a translation in 406 PRINCIPLES OF CINEMATICS. the same plane, in producing a rotation of equal angular velocity about an instantaneous axis at a certain distance to one side of the moving axis, as explained in Article 385, is exactly analogous to the result of the combination of a single force with a couple in pro- ducing an equal single force transferred laterally to a certain dis- tance, as explained in Article 41. The result of the combination of two equal and opposite rotations about parallel axes, in producing a translation with a velocity which is the product of the angular velocity into the distance between the axes, as explained in Article 391, is exactly analogous to the production of a couple by means of a pair of equal and oppo- site forces, as explained in Article 25. The result of the combination of two rotations about intersecting axes, as explained in Article 392, is exactly analogous to the result of the combination of a pair of inclined forces acting through one point, as explained in Article 51. The combination of a rotation about a given axis with a transla- tion parallel to the same axis, as explained in Article 382, is exactly analogous to the combination of a force acting in a given line with a couple whose axis is parallel to the same line, as explained in Article 60, cases 4 and 5. It thus appears, that just as the composition and resolution of translations are exactly analogous to the composition and resolution of couples, so the composition and resolution of rotations are exactly analogous to the composition and resolution of single forces ; that is to say, if lines be taken, representing in direction axes of rotation, and in length the angular velocities of rotation about such axes, all mathematical theorems which are true of lines representing single forces are true of such lines representing rotations : and if with this be combined the principle, that all mathematical theorems which are true of lines representing in direction the axes and in length the moments of couples are true also of lines representing the velocities and directions of translations, all problems of the resolution and composition of motions may be solved by referring to the solutions of analogous problems of statics. 395. Comparative motions in Compound Rotation. The velocity- ratio of two points in a rotating rigid body at any instant is that of their perpendicular distances from its instantaneous axis; and the angle between the directions of motion of the two points is equal to that between the two planes which traverse the points and the instantaneous axis. SECTION 4. Varied Rotation. 396. Variation of Angular Velocity is measured like variation of linear velocity, by comparing the change which takes place in the VARIED ROTATION. 407 angular velocity of a rotating body, A a, during a given interval of time, with the length of that interval, A t, and the rate of variation is the value towards which the ratio of the change of angular velocity to the interval of time, , converges, as the length of the interval is A t indefinitely diminished ; being represented by da dl' and found by the operation of differentiation. 397. Change of the Axis of Rotation has already been considered, so far as it is consistent with uniform angular velocity, in the pre- ceding section. All the propositions of that section are applicable also to cases in which the angular velocity is varied, so long as the ratio of each pair of component angular velocities, such as a : b, is constant. When that ratio varies, the propositions are true also, provided it be understood, that the rolling cylinders and cones ynth circular bases, spoken of in section 3, are simply the osculating cylinders and cones at the lines of contact of rolling cylinders and cones with bases not circular; and that r lf r 2 , in each case, represent the values of the variable radii of curvature of non-circular cylinders at their lines of contact, and ^ A O T, ^ COT, the variable angles of obliquity of the osculating circular cones of non-circular cones. 398. Components of Varied Rotation. The most convenient way, in most cases, of expressing the mode of variation of a rotatory motion, is to resolve the angular velocity at each instant into three component angular velocities about three rectangular axes fixed in direction. The values of those components, at any instant, show at once the resultant angular velocity, and the direction of the instan- taneous axis. For example, let a x , a y , a z , be the rectangular com- ponents of the angular velocity of a rigid body at a given instant, rotation about x from y towards z, about y from towards x, and about z from x towards y, being considered as positive; then is the resultant angular velocity, and cos * == ; cos/3 = -i; cO8y= ; ............ (2.) are the cosines of the angles which the instantaneous axis makes with the axes of x, y and z, respectively. 408 CHAPTER III. MOTIONS OF PLIABLE BODIES, AND OF FLUIDS. 399. Division of the Subject. The subject of the present chapter, so far as it comprehends the relative motions of the points of pliable solids, has been already treated of in those portions of the Third Chapter of Part II. which relate to strains. There remain now to be considered the following branches : I. The Motions of Flexible Cords. II. The Motions of Fluids not altering in Yolume. III. The Motions of Fluids altering in Volume. SECTION 1. Motions of Flexible Cords. 400. General Principles. As those relative motions of the points of a cord which may arise from its extensibility, belong to the sub- ject of resistance to tension, which is a branch of that of strength and stiffness, the present section is confined to those motions of which a flexible cord is capable when the length, not merely of the whole cord, but of each part lying between two points fixed in the cord, is invariable, or sensibly invariable. In order that the figure and motions of a flexible cord may be determined from cinematical considerations alone, independently of the magnitude and distribution of forces acting on the cord, its weight must be insensible compared with the tension on it, and it must everywhere be tight; and when that is the case, each part of the cord which is not straight is maintained in a curved figure by pass- ing over a convex surface. The line in which a tight cord lies on a convex surface is the shortest line which it is possible to draw on that surface between each pair of points in the course of the cord. (It is a well known principle of the geometry of curved surfaces, that the osculating plane at each point of such a line is perpendi- cular to the curved surface.) Hence it appears, that the motions of a tight flexible cord of invariable length and insensible weight are regulated by the follow- ing principles : I. The length between each pair of points in the cord is constant. II. That length is the shortest line which can be drawn between it$ extremities over the surfaces by which tfie cord is guided. MOTIONS OF FLEXIBLE CORDS. 409 401. Motions Clawed. The motions of a cord are of two kinds I. Travelling of a cord along a track of invariable form ; in which case the velocities of all points of the cord are equal. II. Alteration of the figure of the track by the motion of the guiding Surfaces. Those two kinds of motion may be combined. The most usual problems in practice respecting the motions of cords are those in which cords are the means of transmitting mo- tion between two pieces in a train of mechanism. Such problems will be considered in Part IV. of this treatise. Next in point of frequency in practice are the problems to be considered in the ensuing Article. 402. Cord Guided by Surfaces of Revolution. Let a cord in some portions of its course be straight, and in others guided by the sur- faces of circular drums or pulleys, over each of which its track is a circular arc in a plane perpendicular to the axis of the guiding surface. Let r be the radius of any one of the guiding surfaces, i the angle of inclination which the two straight portions of the cord contiguous to that surface make with each other, expressed in length of arc to radius unity. Then the length of the portion of the cord which lies on that surface is r i; and if s be the length of any straight portion of the cord, the total length between two given points fixed in the cord may be expressed thus : L = 2 s + 2 ri ........................... (1.) Let c be the distance between the centres of a given adjacent pair of guiding surfaces, s the length of the straight portion of cord which lies between them, and r, r', their respective radii ; then evidently (2.) the < ( jg> eren > of the radii being employed, according as the cord . ( crosses ) , , , . ,, i does not cross } tlie lme of centres c ' Now let a given point in the cord, A, be considered as fixed, and let L be the constant length of cord between A and another point in the cord, B. Let one of the guiding surfaces between A and B be moved through an indefinitely short distance, dx, in a direction which makes angles,^',/, with the two contiguous straight divisions of the cord respectively. Then, in order to keep the cord tight, B must be drawn longitudinally through the distance, dx (cosj + cos/); ..................... (3.) and consequently, if u represent the velocity of translation of the 410 PRINCIPLES OF CINEMATICS. guiding surface in the given direction, and v the longitudinal velo- city of the point B in the cord, v = u (cos J + cos/) ; (4.) and if any number of guiding surfaces between A and B be trans- lated, each in its own direction, v = 2 u (cos j + cos /) (5.) The case most common in practice is that in which the plies, or straight parts of the cord, are all parallel to each other ; so that i = 180 in each case, while a certain number, n, of the guiding bodies or pulleys all move simultaneously in a direction parallel to the plies of the cord with the same velocity, u. Then cos,; = cos/ = 1 j and v = nu (6.) SECTION 2. Motions of Fluids of Constant Density. 403. Velocity and Flow. The density of a moving fluid mass may be either exactly invariable, from the constancy or the adjust- ment of its temperature and pressure, or sensibly invariable, from the smallness of the alterations of volume which the actual altera- tions of pressure and temperature are capable of producing. The latter is the case in most problems of practical mechanics affecting liquids. Conceive an ideal surface of any figure, and of the area A, to be situated within a fluid mass, the parts of which have motion rela- tively to that surface ; and let u denote, as the case may be, the uni- form velocity, or the mean value of the varying velocity, resolved in a direction perpendicular to A, with which the particles of the fluid pass A. Then Q-^A (1.) is the volume of fluid which passes from one side to the other of the surface A in an unit of time, and is called iheflow, or rate of flow, through A. When the particles of fluid move obliquely to A, let t denote the angle which the direction of motion of any particle passing A makes with a normal to A, and v the velocity of that particle ; then u = v cos & (2.) When the velocity normal to A varies at different points, either from the variation of v, or of 6, or of both, the flow may also be expressed as follows : Let A be divided into indefinitely small elements, each of which is represented by d A ; then MOTIONS OF FLUIDS OF CONSTANT DENSITY. 411 Q= [ udA = I vcoBd-dA', .......... .....(3.) and if we now distinguish the mean normal velocity from the velocity at any particular point by the symbol u , we have, O / udA. 404. Principle of Continuity. AXIOM. When tlie motion of a fluid of constant density is considered relatively to an enclosed space of invariable volume which is always filled with the fluid, tJieflow into the space and the flow out of it, in any one given interval of time, must be equal a principle expressed symbolically by (5.) The preceding sejf-evident principle regulates all the motions of fluids of constant density, when considered in a purely cinematical manner. The ensuing articles of this section contain its most usual applications. 405. Flow in a Stream. A stream is a moving fluid mass, in- definitely extended in length, and limited transversely, and having a continuous longitudinal motion. At any given instant, let A, A', be the areas of any two of its transverse sections, considered as fixed ; u, u', the mean normal velocities through them ; Q, Q', the rates of flow through them ; then in order that the principle of con- tinuity may be fulfilled, those rates of flow must be equal ; that is, u A = u' A' = Q = Q' == constant for all cross sections of the channel at the given instant ; ..................... (1.) consequentlv, ' A. u = T' .............................. (2 '> or, the normal velocities at a given instant at two fixed cross sections are inversely as the areas of these sections. 406. Pipes* Channels, Currents, and Jets. When a stream of fluid completely fills a pipe or tube, the area of each cross section is given by the figure and dimensions of the pipe, and for similar forms of section varies as the square of the diameter. Hence the mean normal velocities of a stream flowing in a full pipe, at differ- ent cross sections of the pipe, are inversely as the squares of the diameters of those sections. A channel partially encloses the stream flowing in it, leaving the upper surface free ; and this description applies not only to chan- 412 PRINCIPLES OF CINEMATICS. iiels commonly so called, but to pipes partially filled. In this case the area of a cross section of the stream depends not only on the figure and dimensions of the channel, but on the figure and eleva- tion of the free upper surface of the stream. A current is a stream bounded by other portions of fluid whose motions are different. A jet is a stream whose surface is either free all round, or is touched by a solid body in a small portion of its extent only. 407. A Radiating Current is a part of a stream which moves towards or from an axis. It is evident that such a stream cannot extend to the axis itself, but must turn aside into a different course at some finite distance from the axis. Conceive a radiating cur- rent to be cut by a cylindrical surface of the radius r described about the axis, and let h be the depth, parallel to the axis, of the portion of that surface which is traversed by the current ; then the mean radial component, u, of the velocity of the current at that surface has the value, 408. A Vortex, Eddy, or Whirl, is a stream which either returns into itself, or moves in a spiral course towards or from an axis. In the latter case two or more successive turns of the same vortex may touch each other laterally without the intervention of any solid partition. 409. Steady Motion of a fluid relatively to a given space considered as fixed is that in which the velocity and direction of the motion of the fluid at each fixed point is uniform at every instant of the time under consideration ; so that although the velocity and direction of the motion of a given particle of the fluid may vary while it is transferred from one point to another, that particle assumes, at each fixed point at which it arrives, a certain definite velocity and direction depending on the position of that point alone ; which velocity and direction are successively assumed by each particle which successively arrives at the same fixed point. The steady motion of a stream is expressed by the two conditions, that the area of each xed cross section is constant, and that the flow through each cross section is constant; that is to say, If u represents the normal velocity of a fluid moving steadily, at a given fixed point, then = ' ............................. < 2 -> MOTION OF PISTONS DIFFERENTIAL EQUATIONS. 413 expresses the condition of steady motion. Next, let u represent the normal velocity, not at a given fixed point, but of a given identical particle of fluid; then the variation undergone by u in an indefi- nitely small interval of time, d t, is that arising from its being transferred from one cross section to another, whose distance down the stream from the former is d s = u ' d t. Hence, denoting by d s, the indefinitely small variation of velocity which takes Cv S place from this cause, and by ^ , the rate at which that variation d t takes place, we have d-udu d s du Most of the problems respecting streams which occur in practice have reference to steady motion. 410. In Unsteady motion, the velocity at each, fixed point varies, (J 9f at a rate denoted by ; and the total rate of variation of the u t velocity of an individual particle in a stream, being found by adding together the rates of variation due to lapse of time and to change of position, is expressed by d'u _ du du ds _ du . du 7 , l" ~T * J . - ' 7 , \ W dt ~ dt ' ds ' dt~ dt ' ds'" 411. Motion of Pistons. Let a mass of fluid of invariable volume be enclosed in a vessel, two portions of the boundary of which (called pistons) are moveable inwards and outwards, the rest of the boundary being fixed. Then, if motion be transmitted between the pistons by moving one inwards and the other outwards, it follows, from the invariability of the volume of the enclosed fluid, that the velocities of the two pistons at each instant will be to each other in the inverse ratio of the areas of the respective projections of the pistons on planes normal to their directions of motion. This is the principle of the transmission of motion in the hydraulic press and hydraulic crane. The flow produced by a piston whose velocity is u, and the area of whose projection on a plane perpendicular to the direction of its motion is A, is given, as in other cases, by the equation Q - uA (1.) 412. General Differential Equations of Continuity. When the motions of a fluid of invariable density are considered in the most 414 PRINCIPLES OF CINEMATICS. general way, the principle of continuity stated in Article 404 is expressed symbolically in the following manner. The space as- sumed as fixed, to which the motion of the fluid is referred, is con- ceived to be divided into indefinitely small rectangular elementary spaces, each having for its linear dimensions, d x, dy, dz, and for the areas of its three pairs of faces, dy dz, dzdx, dxdy. Let x, x + d x, be the co-ordinates of the pair of faces, dy dz; y,y + dy, dzdx; z,z + dz, dxdy. Let the velocity of the particles of water at any point be resolved into three rectangular components, u, v, w, parallel respectively to x, y, z, with proper algebraical signs. Let outward flow be posi- tive, and inward flow negative. The values of the flow for the six faces are as follows : Through the first face dy dz, u-dydz; second face dy dz, (u + -7 d x) d y d z ; (JL OG first face dzdx, vdzdx; second face dzdx, (v + -= d y) d z d x ; first face d x d y, wdxdy; second face dxdy, (w + -y- dz) dxdy. dz Adding those six parts of the flow together, and equating the result, in virtue of the principle of continuity, to nothing, we find the following equation : (du dv d d x + dj + and, striking out the common factor, du dv dw This is the general differential equation of continuity in a fluid of invariable volume. 413. General Differential Equations of Steady Motion -- If each particle which arrives successively at a given point assumes a velo- city and direction of motion depending on the position of the point DIFFERENTIAL EQUATIONS OF LIQUID MOTION. 415 alone, and not on the lapse of time, that state of steady motion is represented by the equations, where u, v, w, are the component velocities at a fixed point. Next, instead of the velocities at a fixed point, let u, v, w, be the compo- nent velocities of an individual particle ; then in the indefinitely short interval dt, the co-ordinates of that particle alter by the lengths dx = udt, dy=.vdt, dz wdt', and it assumes the component velocities proper to its new position, differing from its original velocities by quantities, which, being divided by dt, give the rates of variation of the component velocities of an individual particle, viz. : d ' u du . du . du = = u -j \- v -j -\-w-j ; dt ax dy dz d ' v d v , d v , d v 5 =u -j \-v-j--\-w-j ; dt dx dy dz d 'w dw . dw . dw = u-- T - -\r (2.) dt ~ dx ' dy ' dz' } 414. General Differential Equations of Unsteady iVIotion. When the motion is not steady, each of the three rates of variation in the equations 2 of Article 413 requires the addition of a term represent- ing the rate of variation of velocity due to lapse of time indep&n- dently of change of position, as follows : d-u du . du . du . du -7- = T7 + u 5--r v ^-+ w T-J (1.) dt dt dx dy dz' and similar equations for - and : the presence of the dot cL t at denoting that the velocities are those of an individual particle, and its absence, that they are those at a fixed point. 415. Equations of Displacement. In all the preceding Articles, x, y, and z, denote the co-ordinates of a real or ideal fixed point in the space to which the motions of the fluid are referred ; and the differentials , &c., refer to the differences amongst the condi- (Ju 00 tions of the fluid at different points in that space. Let , u, , represent the co-ordinates of an individual particle ; then the three components of the velocity of that particle have the values d% d* d? 416 PKINCIPLES OF CINEMATICS. and the three components of the rate of variation of its motion, as denned in Article 366, are d 2 1 d-u d 2 n d-v d 2 d'w the values of -^ , 5, and , being taken from Article 413 for du t c/u t at steady motion, and from Article 414 for unsteady motion. 416. A Wave is a state of unsteady motion of a mass, whether solid or fluid, such, that the state of motion which at a given instant of time takes place amongst the particles occupying a certain space, is transmitted to other particles occupying a certain other space, along a continuous course, it may be unchanged, or it may be with modifications which still leave a certain similarity between the motions of the particles originally affected, and of those affected in succession. For example, let a given fixed point O be taken as the origin, and let the particle which is at that point, at an instant of time denoted by 0, have a certain velocity and direction of motion. After the lapse of the time t, let another particle which is at a point A, distant from O by the length x, have either the same velocity and direction of motion, or a velocity and direction bearing a definite relation to those of the original particle; the motion so communicated having been transmitted in succession to all the particles between O and A. The velocity of transmission or propagation of a wave, when con- stant, is the ratio, - , of the distance between two points to the time t which elapses between the instants when the motions at those points are similar. Let a denote that velocity ; then the condition of motion at any point whose distance from the origin is x, at the instant t, depends upon, or is a function o^ a t x] which quantity, or a quantity bearing some definite proportion to it, is called the phase of the wave motion. Wave motion in fluids of invariable density is regulated by the principle of continuity already stated. 417. Oscillation in a fluid, is a motion in which each individual particle of the fluid returns over and over again to the same posi- tion, and repeats over and over again the same motions. The period of an oscillation is the interval of time which elapses between the commencement of a series of movements, and the commencement of the repetition of the same movements. The most usual kind of oscillation in a fluid is that of a series of oscil- latory waves, in which a certain state of motion is transmitted onward from particle to particle, that motion being oscillatory. FLOW OF FLUIDS OF VARYING DENSITY. 417 SECTION 3. Motions of Fluids of Varying Density. 418. Flow of Volume and Flow of ittas*. In the case of a fluid of varying density, the volume, which in an unit of time flows through a given area A, with a normal velocity u, is still repre- sented, as for a fluid of constant density, by Q = A^; (1.) but the absolute quantity, or inass of fluid which so flows, bears no longer a constant proportion to that volume, but is proportional to the volume multiplied by the density. The density may be expressed, either in units of weight per unit of volume, or in arbitrary units suited to the particular case. Let be the density ; then iheflow of mass may be thus expressed : eQ = e Au (2.) 419. The Principle of Continuity* as applied to fluids of varying density, takes the following form : the flow into or out of any fixed space of constant volume is that due to the variation of density alone. To express this symbolically, let there be a fixed space of the constant volume V, and in a given interval of time let the density of the fluid in it, which in the first place may be supposed uniform at each instant, change from ^ to 2 . Then the mass of fluid which at the beginning of the interval occupied the volume V, occupies at the end of the interval the volume - ; and the difference of e those volumes is the volume which flows through the surface bounding the space, outward if & is less than ft, inward if ^ 2 is greater then ft. Let t z t { be the length of the interval of time ; then the rate of flow of volume is expressed as follows : V( -_ t (1.) If the rate of flow is variable during the instant in question, the above equation gives its mean value ; and in that case the exact rate oiflow of volume at a given instant is the value towards which the result of equation 1 converges as the interval of time is inde- finitely diminished, viz. : of mass at the same instant is 2E 418 PRINCIPLES OF CINEMATICS. Next let it be supposed that the density of the fluid varies at different points of the space. Then on the right-hand side of equation 3, e is to be held to represent the mean density throughout tJie space at the given instant; while on the left-hand side, ? must be held to represent the mean density at the surface through which the flow takes place. Let that surface be divided into parts, over each of which the density is uniform at a given instant ; let Q' represent the part of the flow of volume which takes place through one of those parts of the surface, and e' the density of the fluid so flowing, so that Q' $ is the part of the flow of mass which takes place through the part of the surface in question ; then for equation 3 is to be substituted 420. stream. To apply the preceding principles to a stream of fluid of varying density, let the axis of the stream be a line, straight or curved, which traverses the centres of gravity of all the cross sections of the stream made at right angles to that axis, and let distances from a fixed point in that axis, measured down-stream, be denoted by s, and the area of any cross section by A. Let s b s 2 , be the positions of two cross sections of the stream whose distance apart along the axis is s 2 Si ; then the volume of the space between those cross sections is Y=: [ s *Ads .......................... (1.) J *i Let Q, be the rate of flow of volume through the first cross section ; Q 2 that through the second; u } , u 2 , the corresponding mean velo- cities normal to the respective cross sections ; e the mean density of the fluid in the space V ; 1 the mean density at the first cross section, and ? 2 that at the second. Then equation 4 of Article 419 becomes The rate at which the flow of mass varies, in passing from one cross section of the stream to another, is the limit to which the ratio Q 2 e 2 - Qi gi *.-*! converges as the distance s 2 5 i i s indefinitely diminished ; that is to say, The mean normal velocity at a given cross section of a stream having the value u , is subject to the equation STEADY MOTION PISTONS GENEEAL EQUATIONS. 419 dt ' 421. Steady Motion. In the case of steady motion in a fluid of varying density, the density, velocity, and direction of motion at each fixed point of the space to which the motion is referred, are constant, and are assumed successively by each particle which arrives at the given point. Hence in this case, equation 4 of Article 419 becomes 2-QY=0 (1.) The case of a stream is expressed by the forms assumed by equations 3 and 4 of Article 420, viz. : ds ds that is to say, the flow of mass is uniform for all cross sections of the stream; and being also constant for all instants of time, is therefore absolutely constant. 422. Pistons and Cylinders. Let a mass of fluid of variable density be enclosed in a space whose volume is capable of being varied by the motion of one or more pistons. Let A be the area of the projection of a piston on a plane perpendicular to its direction of motion; u its normal velocity, positive if outward, negative if inward ; * ne density of the fluid in contact with it ; V the whole volume of fluid enclosed; e i* s niean density. Then equation 4 becomes the last expression being introduced because ^ Y = the mass en- closed, is constant. If the density is uniform, then as is otherwise evident. If the space is not completely enclosed, but has an opening whose cross section is A", and at which the mean normal velocity of the stream is u" (positive outward), and the density e ", then the flow of mass through that opening, A" u" e", is to be included in the sum- mation at the left side of equation 1. 423. General Differential Equations. As in Article 412 and the subsequent Articles, let u, v, and w, be the rectangular components of the velocity of the fluid at any given fixed point in the space to which the motion is referred, and dx,dy,dz, the dimensions of an indefinitely small fixed rectangular portion of that space. Then considering the pair of faces of that space whose common area is 420 PRINCIPLES OF CINEMATICS. d y d z, the flow of mass in at the first face is u % ' d y d z, and the flow of mass out at the second face is (u g -f - d x) d y d z ; the resultant of which pair of flows is d u Taking the corresponding resultant for the other two pairs of faces, adding the three quantities thus found together, observing that Y = dxdy dz, and dividing by that common factor, the equation 4 of Article 419, which expresses the principle of continuity, becomes the following : d'u? d-v? ,d-w e _ _ df ( dx " dy ' dz dt'" which is the equation of continuity for a fluid of varying density. This equation may be otherwise expressed as follows : . dv . dw\ d . d d . d or dividing by ?, du dv dw ( d d d d\, . . . -r + -; I- -t + I u -= + v -= h w -= K-r: 1 hyp. log. P = 0. ( 2 A. ) dx dy dz \ dx dy dz dt) * The first three terms of the last equation are identical with the three terms of the equation of continuity for a fluid of uniform density. The conditions of steady motion are the following : ^A.^-O-^- o-^-o- m 17-^- > 57- ' Tt~ which conditions apply to a jixed point in space, and not to an individual particle of fluid. The rates of variation of the component velocities and of the density of an individual particle of fluid are expressed as follows : d u du . du . du . du -TT-=-TT +W-J +V~ -f W-T- j ............ (4.) dt dt d x dy dz d-v d-w -.d'e and similar equations for r , ^ , and -= dt dt dt 424. The motions of Connected Bodies form the subject of the Theory of Mechanism, to which the Fourth Part of this treatise relates. PART IV. THEORY OF MECHANISM. CHAPTER I. DEFINITIONS AND GENERAL PRINCIPLES. 425. Theory of Pure Mechanism Defined. Machines are bodies, or assemblages of bodies, which transmit and modify motion and force. The word " machine," in its widest sense, may be applied to every material substance and system, and to the material uni- verse itself ; but it is usually restricted to works of human art, and in that restricted sense it is employed in this treatise. A machine transmits and modifies motion when it is the means of making one motion cause another ; as when the mechanism of a clock is the means of making the descent of the weight cause the rotation of the hands. A machine transmits and modifies force when it is the means of making a given kind of physical energy perform a given kind of work ; as when the furnace, boiler, water, and mechanism of a marine steam engine are the means of making the energy of the chemical combination of fuel with oxygen perform the work of overcoming the resistance of water to the motion of a ship. The acts of transmitting and modifying motion, and of transmitting and modifying force, take place together, and are connected by a cer- tain law ; and until lately, they were always considered together in treatises on mechanics ; but recently great advantage in point of clearness has been gained by first considering separately the act of transmitting and modifying motion. The principles which re- gulate this function of machines constitute a branch of Cinematics, called the theory of pure mechanism. The principles of the theory of pure mechanism having been first established and understood, those of the theory of the work of machines, which regulate the act of transmitting and modifying force, are much more readily de- monstrated and apprehended than when the two departments of the theory of machines are mingled. The establishment of the theory of pure mechanism as an independent subject has been mainly accomplished by the labours of Mr. Willis, whose no- menclature and methods are, to a great extent, followed in this treatise. 422 THEORY OF MECHANISM. 426. The General Problem of the theory of pure mechanism may be staled as follows : Given the mode of connection of two or 'more moveable points or bodies with each other, and with certain fixed bodies; required the comparative motions of the moveable points or bodies : and conversely, when the comparative motions of two or more moveable points are given, to find their proper mode of connec- tion. The term "comparative motion" is to be understood as in Articles 358, 367, 379, and 395. In those Articles, the compara- tive motions of points belonging to one body have already been considered. In order to constitute mechanism, two or more bodies must be so connected that their motions depend on each other through cinematical principles alone. 427. Frame; Moving Pieces; Connectors. The frame of a ma- chine is a structure which supports the moving pieces, and regulates the path or kind of motion of most of them directly. In consider- ing the movements of machines mathematically, the frame is con- sidered as fixed, and the motions of the moving pieces are referred to it. The frame itself may have (as in the case of a ship or of a locomotive engine) a motion relatively to the earth, and in that case the motions of the moving pieces relatively to the earth are the resultants of their motions relatively to the frame, and of the motion of the frame relatively to the earth ; but in all problems of pure mechanism, and in many problems of the work of machines, the motion of the frame relatively to the earth does not require to be considered. The moving pieces may be distinguished into primary and second- ary; the former being those which are directly carried by the frame, and the latter those which are carried by other moving pieces. The motion of a secondary moving piece relatively to the frame is the resultant of its motion relatively to the primary piece which carries it, and of the motion of that primary piece relatively to the frame. Connectors are those secondary moving pieces, such as links, belts, cords, and chains, which transmit motion from one moving piece to another, when that transmission is not effected by immediate contact. 428. Bearings are the surfaces of contact of primary moving pieces with the frame, and of secondary moving pieces with the pieces which carry them. Bearings guide the motions of the pieces which they support, and their figures depend on the nature of those motions. The bearings of a piece which has a motion of transla- tion in a straight line, must have plane or cylindrical surfaces, exactly straight in the direction of motion. The bearings of rotat- ing pieces must have surfaces accurately turned to figures of revolu- BEARINGS MOVING PIECES ELEMENTARY COMBINATION. 423 tion, such as cylinders, spheres, conoids, and flat discs. The bearing of a piece whose motion is helical, must be an exact screw, of a pitch equal to that of the helical motion (Article 382). Those parts of moving pieces which touch the bearings, should have surfaces accurately fitting those of the bearings. They may be distinguished into slides, for pieces which move in straight lines, gudgeons, journals, busJies, and pivots, for those which rotate, and screws for those which move helically. The accurate formation and fitting of bearing surfaces is of primary importance to the correct and efficient working of machines. Sur- faces of revolution are the most easy to form accurately, screws are more difficult, and planes the most difficult of all. The success of Mr. Whitworth in making true planes, is regarded as one of the greatest achievements in the construction of machinery. 429. The Motions of Primary Moving Pieces are limited by the fact, that in order that different portions of a pair of bearing sur- faces may accurately fit each other during their relative motion, those surfaces must be either straight, circular, or helical ; from which it follows, that the motions in question can be of three kinds only, viz. : I. Straight translation, or shifting, which is necessarily of limited extent, and which, if the motion of the machine is of indefinite duration, must be reciprocating ; that is to say, must take place alternately in opposite directions. (See Part III., Chapter II., Section 1.) II. Simple rotation, or turning about a fixed axis, which motion may be either continuous or reciprocating, being called in the latter case osculation. (See Part III., Chapter II., Section 2.) III. Helical or screw-like motion, to which the same remarks apply as to straight translation. (See Part III., Chapter II., Section 3, Article 382.) 430. The Motions of Secondary Moving Pieces relatively to the pieces which carry them, are limited by the same principles which apply to the motions of primary pieces relatively to the frame. But the motions of secondary moving pieces relatively to the frame may be any motions which can be compounded of straight translations and simple rotations according to the principles already explained in Part III., Chapter II., Section 3. 431. An Elementary Combination in mechanism consists of a pair of primary moving pieces, so connected, that one transmits motion to the other. The piece whose motion is the cause is called the driver ; that whose motion is the effect, the follower. The connection between the driver and the follower may be I. By rolling contact of their surfaces, as in toothless wJieels. 424 THEORY OF MECHANISM. II. By sliding contact of their surfaces, as in toothed wheels, screws, -wedges, cams, and escapements. III. By bands or wrapping connectors, such as belts, cords, and gearing-chains. IV. By link-wor/c, such as connecting rods, universal joints, and dicks. V. By reduplication of cords, as in the case of ropes and pulleys. VI. By an intervening fluid, transmitting motion between two pistons. The various cases of the transmission of motion from a driver to a follower are further classified, according as the relation between their directions of motion is constant or changeable, and according as the ratio of their velocities is constant or variable. This latter principle of classification is employed by Mr. Willis as the founda- tion of a primary division of the subject of elementary combinations in mechanism into classes, which are subdivided according to the mode of connection of the pieces. In the present treatise, elemen- tary combinations will be classed primarily according to the mode of connection. 432. rane of Connection. In every class of elementary combina- tions, except those in which the connection is made by reduplica- tion of cords, or by an intervening fluid, there is at each instant a certain straight line, called the line of connection, or line of mutual action of the driver and follower. In the case of rolling contact, this is any straight line whatsoever traversing the point of contact of the surfaces of the pieces ', in the case of sliding contact, it is a line perpendicular to those surfaces at their point of contact ; in the case of wrapping connectors, it is the centre line of that part of the connector by whose tension the motion is transmitted ; in the case of link- work, it is the straight line passing through the points of attachment of the link to the driver and follower. 433. Principle of Connection. The line of connection of the driver and follower at any instant being known, their comparative velocities are determined by the following principle : The respec- tive linear velocities of a point in tJte driver, and a point in the fol- lower, each situated anywhere in the line of connection, are to each other inversely as the cosines of the respective angles made by the paths of the points with the line of connection. This principle might be otherwise stated as follows : The components, along the line of con- nection, of the velocities of any two points situated in that line, are equal. 434. Adjustments of Speed. The velocity-ratio of a driver and its follower is sometimes made capable of being changed at will, by means of apparatus for varying the position of their line of connec- tion; as when a pair of rotating cones are embraced by a belt TRAIN AGGREGATE COMBINATIONS. 425 which can be shifted so as to connect portions of their surfaces of different diameters. 435. A Train of Mechanism consists of a series of moving pieces, each of which is follower to that which drives it, and driver to that which follows it. 436. Aggregate Combinations in mechanism are those by which compound motions are given to secondary pieces. 426 CHAPTER II. ON ELEMENTARY COMBINATIONS AND TRAINS OF MECHANISM. SECTION 1. Rolling Contact. 437. Pitch Surfaces are those surfaces of a pair of moving pieces, which touch each other when motion is communicated by rolling contact. The LINE OF CONTACT is that line which at each instant traverses all the pairs of points of the pair of pitch surfaces which are in contact. 438. Smooth Wheels, Rollers, Smooth Racks. Of a pair of pri- mary moving pieces in rolling contact, both may rotate, or one may rotate and the other have a motion of sliding, or straight translation. A rotating piece, in rolling contact, is called a smooth wheel, and sometimes a roller ; a sliding piece may be called a smooth rack. 439. General Conditions of Rolling Contact. The whole of the principles which regulate the motions of a pair of pieces in rolling contact follow from the single principle, that each pair of points in the pitch surfaces, which are in contact at a given instant, must at that instant be moving in the same direction with the same velocity. The direction of motion of a point in a rotating body being per- pendicular to a plane passing through its axis, the condition, that each pair of points in contact with each other must move in the same direction leads to the following consequences : I. That when both pieces rotate, their axes, and all their points of contact, lie in the same plane. II. That when one piece rotates and the other slides, the axis of the rotating piece, and all the points of contact, lie in a plane per- pendicular to the direction of motion of the sliding piece. The condition, that the velocities of each pair of points of con- tact must be equal, leads to the following consequences : III. That- the angular velocities of a pair of wheels, in rolling contact, must be inversely as the perpendicular distances of any pair of points of contact from the respective axes. IY. That the linear velocity of a smooth rack in rolling contact with a wheel, is equal to the product of the angular velocity of the wheel by the perpendicular distance from its axis to a pair of points of contact. CIRCULAR WHEELS STRAIGHT RACK. 427 Respecting the line of contact, the above principles III. and IY. lead to the following conclusions : Y. That for a pair of wheels with parallel axes, and for a wheel and rack, the line of contact is straight, and parallel to the axes or axis ; and hence that the pitch surfaces are either plane or cylin- drical (the term " cylindrical " including all surfaces generated by the motion of a straight line parallel to itself). VI. That for a pair of wheels, with intersecting axes, the line of contact is also straight, and traverses the point of intersection of the axes ; and hence that the rolling surfaces are conical, with a common apex (the term " conical " including all surfaces generated by the motion of a straight line which traverses a fixed point). 440. Circular Cylindrical Wheels are employed when an uniform velocity-ratio is to be communicated between parallel axes. Figs. 187, 188, and 189, of Article 388, may be taken to represent pairs of such wheels ; C and O, in each figure, being the parallel axes of the wheels, and T a point in their line of contact. In fig. 187, both pitch surfaces are convex, the wheels are said to be in outside gearing, and their directions of rotation are contrary. In figs. 188 and 189, the pitch surface of the larger wheel is concave, and that of the smaller convex; they are said to be in inside gearing, and their directions of rotation are the same. To represent the comparative motions of such pairs of wheels symbolically, let be their radii : let O C = c be the line of centres, or perpendicular distance between the axes, so that for outside ) /, v inside }geanng,c = r 1 r 2 (I.) Let Oi t %., be the angular velocities of the wheels, and v the common linear velocity of their pitch surfaces ; then c \ TI '. 7*2 : : &2 "~*~ G&I : a% ; a l ; J the sign =t applying to ^^ gearing. 441. A Straight Rack and Circular Wheel, which are used when an uniform velocity-ratio is to be communicated between a sliding piece and a turning piece, may be represented by fig. 185 of Article 385, C being the axis of the wheel, FTP the plane surface of the rack, and T a point in their line of contact. Let r be the radius of the wheel, a its angular velocity, and v the linear velocity of the rack; then 428 THEORY OF MECHANISM. 442. Bevel wheels, whose pitch surfaces are frustra of regular cones, are used to transmit an uniform angular velocity-ratio between a pair of axes which intersect each other. Fig. 190 of Article 392 will serve to illustrate this case; O A and O C being the pair of axes, intersecting each other in O, O T the line of con- tact, and the cones described by the revolution of O T about O A and O C respectively being the pitch surfaces, of which narrow zones or frustra are used in practice. Let a 1} a 2 , be the angular velocities about the two axes respec- tively; and let t\ = ^ A O T, i 2 = ^ C O T, be the angles made by those axes respectively with the line of contact ; then from the principle III. of Article 439 it follows, that the angular velocity- ratio is a 2 sin i t = r ; ........................... (i.) a, sin i. 2 Which equation serves to find the angular velocity-ratio when the axes and the line of contact are given. Conversely, let the angle between the axes, ^ A O C = i, -f i, =j, be given, and also the ratio ; then the position of the line of a i contact is given by either of the two following equations : . . a. 2 sinj 11/ ~ . . . _ a, sn lt= Graphically, the same problem is solved as follows : On the two axes respectively, take lengths to represent the angular velocities of their respective wheels. Complete the parallelogram of which those lengths are the sides, and its diagonal will be the line of contact. As in the case of the rolling cones of Article 393, one of a pair of bevel wheels may be a flat disc, or a concave cone. 443. Non-Circular wheel* are used to transmit a ^s\-L- 2 i variable velocity-ratio between a pair of parallel -"/ / axes. In fig. 191, let C l7 C 2 , represent the axes of / / such a pair of wheels; T,, T 2 , a pair of points which at a given instant touch each other in the line of contact (which line is parallel to the axes and in the same plane with them) ; and ~U l} U 8 , another 191 P a * r f Pi n * s > which touch each other at another instant of the motion; and let the four points, T,, NON-CIRCULAR WHEELS. 429 T 2 , U|, TJ 2 , be in one plane perpendicular to the two axes, and to the line of contact. Then for every such set of four points, the two following equations must be fulfilled : C,;) J " and those equations show the geometrical relations which must exist between a pair of rotating surfaces in order that they may move in rolling contact round fixed axes. The same conditions are expressed differentially in the following manner : Let r lt r 2 , be the radii vectores of a pair of points which touch each other; ds l} ds 2 , a pair of elementary arcs of the cross sections T t TJj, T 2 U 2 , of the pitch surfaces, and c the line of centres or distance between the axes. Then n + r a = c j \ ds, _dsz V ........................ (2.) dri dr 2 ' J If one of the wheels be fixed and the other be rolled upon it, a point in the axis of the rolling wheel describes a circle of the radius c round the axis of the fixed wheel. The equations 1 and 2 are made applicable to inside gearing by putting instead of + and + instead of . The angular velocity-ratio at a given instant has the value (3.) As examples of non-circular wheels, the following may be mentioned : I. An ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity-ratio varying from 1 excentricity 1 + excentricity 1 + excentricity 1 excentricity' II. A hyperbola rotating about its farther focus, rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hyperbolas, and the velocity- ratio varying between excentricity + 1 r~ - T and unity. excentricity 1 430 THEOKY OF MECHANISM. III. Two logarithmic spirals of equal obliquity rotate in rolling contact with each other through an indefinite angle. (For further examples of non-circular wheels, see Professor Clerk Maxwell's paper on Rolling Curves, Trans. Roy. Soc. Edin., vol. xvi., and Professor Willis's work on Mechanism.) SECTION 2. Sliding Contact. i 444. skew-Bevel wheel* are employed to transmit an uniform velocity-ratio between two axes which are neither parallel nor Fig. 192. Fig. 193. intersecting. The pitch surface of a skew-bevel wheel is a frustrum or zone of a hyperboloid of revolution. In fig. 192, a pair of large portions of such hyperboloids are shown, rotat- / ing about axes A B, C D. In fig. 193 are shown a pair of narrow zones of the same figures, such as are employed in practice. A hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to which it is inclined at a constant angle. If two such hyperboloids, equal or unequal, be placed in the closest possible contact, as in fig. 192, they will touch each other along one of the generating straight lines of each, which will form their line of contact, and will be inclined to the axes A B, C D, in opposite directions. The axes will neither be parallel, nor will they intersect each other. The motion of two such hyperboloids, rotating in contact with each other, has sometimes been classed amongst cases of rolling contact ; but that classification is not strictly correct \ for although the component velocities of a pair of points of contact in a direction at right angles to the line of contact are equal, still, as the axes are neither parallel to each other nor to the line of contact, the velocities of a pair of points of contact have components along the line of SKEW-BEVEL WHEELS GROOVED WHEELS. 431 contact, which are unequal, and their difference constitutes a lateral sliding. The directions and positions of the axes being given, and the required angular velocity-ratio, , it is required to find the obli- a\ quities of the generating line to the two axes, and its radii vectores, or least perpendicular distances from these axes. In fig. 194, let A B, C D, be the two axes, and G K their common perpendicular. Divide that perpendicular at H in the ratio given by the proportional equation r l :KH = r 2 ; ...... (1.) then the two segments thus found, whose lengths are inversely as the angular velocities, will be the least distances of the line of con- tact from the axes. On any plane normal to the common perpendicular G K h, draw a b || A B, c d \\ C D, in which take lengths in the following pro- portions : i : a 2 : : h p : h q ; complete the parallelogram hpeq, and draw its diagonal e hf; the line of contact E H F will be parallel to that diagonal. The first pitch surface is generated by the rotation of the line E H F about the axis A B with the radius vector G H = r, ; the second, by the rotation of the same line about the axis C D with the radius vector H K = r 2 . To draw the hyperbola which is the longitudinal section of a skew-bevel wheel whose generating line has a given radius vector and obliquity, let A G B, fig. 195, re- present the axis, G H _L A G B, the radius vector of the generating line, and let the straight line E G F make with the axis an angle equal to the obliquity of the generating line. H will be the vertex, and E G F one of Fi S- 195 - the asymptotes, of the required hyperbola. To find any number of points in that hyperbola, proceed as follows : Draw X W Y parallel to G H, cutting G E in W, and make XY = J (Gil 3 + XW" 2 ). Then will Y be a point in the hyperbola. 445. Orooved wheels. To increase the friction or adhesion between a pair of wheels, which is the means of transmitting force and motion from one to the other, their surfaces of contact are sometimes formed into alternate circular ridges and grooves, con- stituting what is called frictional gearing. Fig. 196 is a cross 432 THEORY OF MECHANISM. section illustrating the kind of frictional gearing invented by Mr. Robertson. The comparative motion of a pair of wheels thus ridged and grooved is nearly the same with that of a pair of smooth wheels in rolling contact, having cylindrical or conical pitch surfaces lying midway between the tops of the ridges and bottoms of the grooves. The relative motion of the faces of contact of Fig. 196. ^e ec [g es an( j grooves is a rotatory sliding, about the line of contact of the ideal pitch surfaces as an instantaneous axis. The angle between the sides of each groove is about 40 ; and it is stated that the mutual friction of the wheels is about once and a-half the force with which their axes are pressed towards each other. 446. Teeth of Wheels. The most usual method of communi- cating motion between a pair of wheels, or a wheel and a rack, and the only method which, by preventing the possibility of the rotation of one wheel unless accompanied by the other, insures the preservation of a given velocity-ratio exactly, is by means of the projections called teeth. The pitch surface of a wheel is an ideal smooth surface, inter- mediate between the crests of the teeth and the bottoms of the spaces between them, which, by rolling contact with the pitch sur- face of another wheel, would communicate the same velocity-ratio that the teeth communicate by their sliding contact. In designing wheels, the forms of the ideal pitch surfaces are first determined, and from them are deduced the forms of the teeth. Wheels with cylindrical pitch surfaces are called spur wheels ; those with conical pitch surfaces, bevel wheels; and those with h^perboloidal pitch surfaces, skew-bevel wheels, The pitch line of a wheel, or, in circular wheels, the pitch circle, is a transverse section of the pitch surface made by a surface per- pendicular to it and to the axis ; that is, in spur wheels, by a plane perpendicular to the axis ; in bevel wheels, by a sphere described about the apex of the conical pitch surface ; and in skew-bevel wheels, by any oblate spheroid generated by the rotation of an ellipse whose foci are the same with those of the hyperbola that generates the pitch surface. The pitch point of a pair of wheels is the point of contact of their pitch lines ; that is, the transverse section of the line of contact of the pitch surfaces. Similar terms are applied to racks. That part of the acting surface of a tooth which projects beyond the pitch surface is called the face ; that which lies within the pitch surface, the flank. PITCH AND NUMBER OF TEETH. 433 The radius of the pitch circle of a circular wheel is called the geometrical radius; that of a circle touching the crests of the teeth is called the real radius; and the difference between those radii, the addendum. 447. Pitch and Number of Teeth. The distance, measured along the pitch line, from the face of one tooth to the face of the next, is called the PITCH. The pitch, and the number of teeth in circular wheels, are regu- lated by the following principles : I. In wheels which rotate continuously for one revolution or more, it is obviously necessary that the pitch should be an aliquot part of the circumference. In wheels which reciprocate without performing a complete re- volution, this condition is not necessary. Such wheels are called sectors. II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essential that the pitch should be the same in each. III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii, and inversely as the angular velocities. IV. Hence also, in any pair of circular wheels which rotate continuously for one revolution or more, the ratio of the numbers of teeth, and its reciprocal, the angular velocity-ratio, must be ex- pressible in whole numbers. V. Let n, N, be the respective numbers of teeth in a pair of wheels, N being the greater. Let t, T, be a pair of teeth in the smaller and larger wheel respectively, which at a particular instant work together. It is required to find, first, how many pairs of teeth must pass the line of contact of the pitch surfaces before t and T work together again (let this number be called a) ; secondly, with how many different teeth of the larger wheel the tooth t will work at different times (let this number be called b) ; and thirdly, with how many different teeth of the smaller wheel the tooth T will work at different times (let this be called c). CASE 1. If n is a divisor of N, (1.) CASE 2. If the greatest common divisor of N and n be d, a num- ber less than n, so that n = md, N M d, then a = mN = M.n = Mme?; b = M j c = m ....... (2.) CASE 3. If N and n be prime to each other, 434 THEORY OF MECHANISM. a = Nw; b = 1ST; c = n (3.) It is considered desirable by millwrights, with a view to the preservation of the uniformity of shape of the teeth of a pair of wheels, that each given tooth in one wheel should work with as many different teeth in the other wheel as possible. They, there- fore, study to make the numbers of teeth in each pair of wheels which work together such as to be either prime to each other, or to have their greatest common divisor as small as is possible con- sistently with the purposes of the machine. VI. The smallest number of teeth which it is practicable to give to a pinion (that is, a small wheel), is regulated by the principle, that in order that the communication of motion from one wheel to another may be continuous, at least one pair of teeth should always be in action ; and that in order to provide for the contingency of a tooth breaking, a second pair, at least, should be in action also. For reasons which will appear when the forms of teeth are con- sidered, this principle gives the following as the least numbers of teeth which can be usually employed in pinions having teeth of the three classes of figures named below, whose properties will be ex- plained in the sequel : I. Involute teeth, 25. II. Epicycloidal teeth, 12. III. Cylindrical teeth, or staves, 6. 448. Hunting Cog. When the ratio of the angular velocities of two wheels, being reduced to its least terms, is expressed by small numbers, less than those which can be given to wheels in practice, and it becomes necessary to employ multiples of those numbers by a common multiplier, which becomes a common divisor of the numbers of teeth in the wheels, millwrights and engine-makers avoid the evil of frequent contact between the same pairs of teeth, by giving one additional tooth, called a hunting cog, to the larger of the two wheels. This expedient causes the velocity-ratio to be not exactly but only approximately equal to that which was at first contemplated ; and therefore it cannot be used where the exactness of certain velocity-ratios amongst the wheels is of importance, as in clockwork. 449. A Train of Wheeiwork consists of a series of axes, each having upon it two wheels, one of which is driven by a wheel on the preceding axis, while the other drives a wheel on the following axis. If the wheels are all in outside gearing, the direction of rotation of each axis is contrary to that of the adjoining axes. In some cases, a single wheel upon one axis answers the purpose both of receiving motion from a wheel on the preceding axis and giving TRAINS OF WHEELWORK. 4.V) motion to a wheel on the following axis. Such a wheel is called an idle wJieel : it affects the direction of rotation only, and not the velocity-ratio. Let the series of axes be distinguished by numbers 1, 2, 3, &c m ] let the numbers of teeth in the driving wheels be denoted by N's, each with the number of its axis affixed ; thus, N!, N 2 , &c N m _! ; and let the numbers of teeth in the driven or following wheels be denoted by w's, each with the number of its axis affixed ; thus, n 2 , n 3) &c n m . Then the ratio of the angular velocity a m of the m th axis to the angular velocity a^ of the first axis is the product of the in 1 velocity-ratios of the succes- sive elementary combinations, viz. : 61 = N! N. &c y,.^ a\ n a ' n 3 &c n m that is to say, the velocity-ratio of the last and first axes is the ratio of the product of the numbers of teeth in the drivers to the product of the numbers of teeth in the followers j and it is obvious, that so long as the same drivers and followers constitute the train, the order in which they succeed each other does not affect the resultant velocity-ratio. Supposing all the wheels to be in outside gearing, then as each elementary combination reverses the direction of rotation, and as the number of elementary combinations, m 1, is one less than the number of axes, m, it is evident that if m is odd, the direction of rotation is preserved, and if even, reversed. It is often a question of importance to determine the numbers of teeth in a train of wheels best suited for giving a determinate velocity-ratio to two axes. It was shown by Young, that to do this with the least total number of teeth, the velocity-ratio of each elementary combination should approximate as nearly as possible 3-59. This would in many cases give too many axes; and as\a useful practical rule it may be laid down, that from 3 to 6 ought to be the limit of the velocity-ratio of an elementary combination in wheelwork. -p Let be the velocity-ratio required, reduced to its least terms, C and let B be greater than C. T> If is not greater than 6, and C lies between the prescribed O minimum number of teeth (which may be called ), and its double 2 t, then one pair of wheels will answer the purpose, and B and C will themselves be the numbers required. Should B and C be inconveniently large, they are if possible to be resolved into factors, 436 THEORY OF MECHANISM. and those factors, or if they are too small, multiples of them, used for the numbers of teeth. Should B or C, or both, be at once incon- T> veniently large, and prime, then instead of the exact ratio , some \j ratio approximating to that ratio, and capable of resolution into con- venient factors, is to be found by the method of continued fractious. T> Should be greater than 6, the best number of elementary C combinations, m 1, will lie between and g-;g ...... log 6 log 3 Then, if possible, B and C themselves are to be resolved each into m 1 factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t, nor greater than 6 1 ; or if B and C contain inconveniently large prime factors, an approxi- mate velocity-ratio, found by the method of continued fractions, is T> to be substituted for as before. L- So far as the resultant velocity-ratio is concerned, the order of the drivers K and of the followers n is immaterial ; but to secure equable wear of the teeth, as explained in Article 447, Principle V., the wheels ought to be so arranged that for each elementary com- bination the greatest common divisor of N and n shall be either 1, or as small as possible. 450. Principle of Sliding Contact. The line of action, or of con- nection, in the case of sliding contact of two moving pieces, is the common perpendicular to their surfaces at the point where they touch ; and the principle of their comparative motion is, that the components, along that perpendicular, of the velocities of any two points traversed by it, are equal. CASE 1. Two shifting pieces, in sliding contact, have linear velo- cities proportional to the secants of the angles which their directions of motion make with their line of action. CASE 2. Two rotating pieces, in sliding contact, have angular velocities inversely proportional to the perpendicular distances from their axes of rotation to their line of action, each multiplied by the sine of the angle which the line of action makes with the particular axis on which the perpendicular is let fall. In fig. 197, let C 1? C 2 , represent the axes of rotation of the two pieces; Ai, A 2 , two portions of their respective surfaces; and T 1? T, a pair of points in those surfaces, which, at the instant under consideration, are in contact with each other. Let Pj P 2 be the common perpendicular of the surfaces at the pair of points T u T 2 ; PRINCIPLE OF SLIDING CONTACT. 437 that is, the line of action; and let C, Pj, C 2 P 2 , be the common per- pendiculars of the line of action and of the two axes respectively. Then at the given instant, the components along the line PI P 2 of the velocities of the points P], P 2 , are equal. Let i lt i. 2 , be the angles which that line makes with the direc- tions of the axes respectively. Let Oi, a 2 , be the respective angular velocities of the moving pieces; then ! sin ^ = a a C 2 P 2 sin i z ; consequently, g 2 _ G! PI sin ii t ~ v ~ v< Ca Fig. 197. which is the principle stated above. When the line of action is perpendicular in direction to both axes, then sin i x = sin i z = 1 ; and equation 1 becomes CVB, C 2 P 2 ' .(1 A.) When the axes are parallel, i^ = i. 2 . Let I be the point where the line of action cuts the plane of the two axes ; then the triangles PI G! I, P 2 C 2 1, are similar; so that equation 1 A is equivalent to the following : CASE 3. A rotating piece and a shifting piece, in sliding contact, have their comparative motion regulated by the following prin- ciple : Let C P denote the perpendicular distance from the axis of the rotating piece to the line of action ; i the angle which the direc- tion of the line of action makes with that axis; a the angular velocity of the rotating piece ; v the linear velocity of the sliding piece ; j the angle which its direction of motion makes with the line of action ; then v = a ' C P sini -secj ..................... (2.) When the line of action is perpendicular in direction to the axis of the rotating piece, sin i = 1 ; and v = a-CP-sec'j = a'TC; .............. (2 A.) where I C denotes the distance from the axis of the rotating piece 438 THEORY OF MECHANISM. to the point where the line of action cuts a perpendicular from that axis on the direction of motion of the shifting piece. 451. Teeth of Spur-Wheels and Racks. General Principle. The figures of the teeth of wheels are regulated by the principle, that tlie teeth of a pair of wheels shall give the same velocity-ratio by their sliding contact, which the ideal smooth pitch surfaces would give by their rolling contact. Let B b B 2 , in fig. 197, be parts of the pitch lines (that is, of cross sections of the pitch surfaces) of a pair of wheels with parallel axes, and I the pitch point (that is, a section of the line of contact). Then the angular velocities which would be given to the wheels by the rolling contact of those pitch lines are inversely as the segments I Cj, I C 2 , of the line of centres; and this also is the proportion of the angular velocities given by a pair of surfaces in sliding contact whose line of action traverses the point I (Article 450, case 2, equation 1 B). Hence the condition of correct working for the teeth of wheels with parallel axes is, that the line of action of the teeth shall at every instant traverse the line of contact of the pitch surfaces; and the same condition obviously applies to a rack sliding in a direction perpendicular to that of the axis of the wheel with which it works. 452. Teeth Described by Rolling Curves. From the principle of the preceding Article it follows, that at every instant, the position of the point of contact Tj in the cross section of the acting surface of a tooth (such as the line A x Tj in fig. 197), and the corresponding position of the pitch point I in the pitch line I Bj of the wheel to which that tooth belongs, are so related, that the line I T a which joins them is normal to the outline of the tooth A l T l at the point Tj. Now this is the relation which exists between the tracing- point Tj, and the instantaneous axis or 'line of contact I, in a rolling curve of such a figure, that being rolled upon the pitch surface B 15 its tracing-point 5\ traces the outline of the tooth. (As to rolling curves, see Articles 386, 387, 389, 390, 393, 396, 397, and Professor Clerk Maxwell's paper there referred to). In order that a pair of teeth may work correctly together, it is necessary and sufficient that the instantaneous radii vectores from the pitch point to the points of contact of the two teeth should coincide at each instant, as expressed by the equation (i.) and this condition is fulfilled, if the outlines of the two teeth be traced by the motion of the same tracing-point, in rolling the same rolling curve on the same side of the pitch surfaces of the respective wheels. The flank of a tooth is traced while the rolling curve rolls inside of the pitch line; the face, while it rolls outside. Hence it is TEETH DESCRIBED SLIDING OF TEETH. 439 evident that the flanks of the teeth of the driving wheel drive the faces of the teeth of the driven wheel ; and that the faces of the teeth of the driving wheel drive the flanks of the teeth of the driven wheel. The former takes place while the point of contact of the teeth is approaching the pitch point, as in fig. 197, supposing the motion to be from Pj towards P a ; the latter, after the point of contact has passed, and while it is receding from, the pitch point. The pitch point divides the path of the point of contact of the teeth into two parts, called the path of approach and the path of recess; and the lengths of those paths must be so adjusted, that two pairs of teeth at least shall be in action at each instant. It is evidently necessary that the surfaces of contact of a pair of teeth should either be both convex, or that if one is convex and the other concave, the concave surface should have the natter curvature. The equations of Article 390 give the relations which exist between the radius of curvature of a pitch line at the pitch point (rj), the radius of curvature of the rolling curve at the same point (r 2 ), the radius vector of the tracing-point (r, = I T), the angle made by that line with the line of centres of the fixed and rolling curves (6 - ^ T I C), and the radius of curvature of the curve traced by the point T ({), all at a given instant. When a pair of tooth surfaces are both convex absolutely, that which is a face is concave, and that which is a flank is convex, towards the pitch point; and this is indicated by the values of having contrary signs for the two teeth, being positive for the face and negative for the flank. The face of a tooth is always convex absolutely, and concave towards the pitch point, g being positive; so that if it works with a concave flank, the value of e for that flank is positive also, and greater than for the face with which it works. 453. The Sliding of a Pair of Teeth on Each Other, that is, their relative motion in a direction perpendicular to their line of action, is found by supposing one of the wheels, such as 1, to be fixed, the line of centres C a C 2 to rotate backwards round C x with the angular velocity a lt and the wheel 2 to rotate round C 2 as before with the angular velocity a 2 relatively to the line of centres C x C 2 , so as to have the same motion as if its pitch surface rolled on the pitch surface of the first wheel. Thus the relative motion of the wheels is unchanged ; but 1 is considered as fixed, and 2 has the resultant motion given by the principles of Article 389 ; that is, a rotation about the instantaneous axis I with the angular velocity a l + a 2 . Hence the velocity of sliding is that due to this rotation about I, with the radius I T = r; that is to say, its value is r(a l + a a ); (1.) so that it is greater, the farther the point of contact is from the 440 THEORY OF MECHANISM. line of centres ; and at the instant when that point, passing the line of centres, coincides with the pitch point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact. The roots of the teeth slide towards each other during the ap- proach, and from each other during the recess. To find the amount or total distance through which the sliding takes place, let , be the time occupied by the approach, and t 2 that occupied by the recess ; then the distance of sliding is f J o (2.) or in another form, if di denote an element of the change of angu- lar position of one wheel relatively to the other, -i v the amount of that change during the approach, and i 2 during the recess, then (a l + a 2 ) d t = d i and s= Crdi + Trdi .................... (3.) Jo Jo (See also Article 455.) 454. The Arc of Contact on the Pitch lanes is the length of that portion of the pitch lines which passes the pitch-point during the action of one pair of teeth ; and in order that two pairs of teeth at least may be in action at each instant, its length should be at least double of the pitch. It is divided into two parts, the arc of ap- proach and the arc of recess. In order that the teeth may be of length sufficient to give the required duration of contact, the dis- tance moved over by the point I upon the pitch line during the rolling of a rolling curve to describe the face and flank of a tooth, must be in all equal to the length of the required arc of contact. It is usual to make the arcs of approach and recess equal. 455. The Length of a Tooth may be divided into two parts, that of the face and that of the flank. For teeth in the driving wheel, the length of the flank depends on the arc of approach, that of the face, on the arc of recess ; for those in the following wheel, the length of the flank depends on the arc of recess, that of the face, on the arc of approach. Let (?! be the arc of approach, q 2 that of recess ; ^ the length of the flank, l\ the length of the face of a tooth in the driving wheel. Let TI be the radius of curvature of the pitch line, r Q that of the rolling curve, r the radius vector of the tracing-point, at any instant. The angular velocity of the rolling curve relatively to the wheel is LENGTH OP TEETH INSIDE GEARING INVOLUTE TEETH. 441 the positive sign applying to rolling outside, or describing the face, and the negative sign to rolling inside, or describing the flank. Hence the velocity of the tracing-point at a given instant is dt r and consequently =/: e For the following wheel, q l and q 2 have to be interchanged, so that, if r a be the radius of that wheel, .(2.) The equations 2 and 3 evidently give the means of finding the dis- tance of sliding between a pair of teeth, in a different form from that given in Article 453 ; for that distance is 456. To inside Gearing all the preceding principles apply, ob- serving that the radius of the greater, or concave pitch surface, is to be considered as negative, and that in Article 453, the difference of the angular velocities is to be taken instead of their sum. 457. Involute Teeth for Circular Wheels, being the first of the three kinds mentioned in Article 447, are of the form of the in- volute of a circle, of a radius less than the pitch circle in a ratio which may be expressed by the sine of a certain angle I, and may be traced by the pole of a logarithmic spiral rolling on the pitch circle, the angle made by that spiral at each point with its own radius vector being the complement of the given angle 6, But this mode of describing involutes of circles, being more com- plex than the ordinary method, is mentioned merely to show that they fall under the general description of curves described by rolling. 442 THEORY OF MECHANISM. In fig. 198, let C,, C 2 , be the centres of two circular wheels, whose pitch circles are B, B 2 . Through the pitch point I draw the intended line of action P! P 2 , making the angle C I P = 6 with the line of centres. From C w C 2 , draw j P! = I C, sin 6, C 2 P 2 = I C 3 sin 9, .(1.) perpendicular to P t P 2 , with which two perpendiculars as radii, describe circles (called base circles) Dj, D 2 . Suppose the base circles to be a pair of circular pulleys, connected by means of a cord whose course from pulley to pulley is P! I P 2 . As the line of connection of those pulleys is the same with that of the proposed teeth, they will rotate with the required velocity-ratio. Now suppose a tracing-point T to be fixed to the cord, so as to be carried along the path of contact P 1 I P 2 . That point will trace, on a plane rotating along with the wheel 1, part of the involute of the base circle Dj, and on a plane rotating along with the wheel 2, part of the involute of the base circle D 2 , and the two curves so traced will always touch each other in the required point of contact T, and will therefore fulfil the condition required by Article 451. All involute teeth of the same pitch work smoothly together. To find the length of the path of contact on either side of the pitch point I, it is to be observed that the distance between the fronts of two successive teeth as measured along P l I P 2 , is less than the pitch in the ratio sin 0:1, and consequently that if dis- tances not less than the pitch x sin 6 be marked off either way from I towards P! and P 2 respectively, as the extremities of the path of contact, and if the addendum circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice, it is usual to make the path of contact somewhat longer, viz., about 2 T 4 times the pitch; and with this length of path and the value of which is usual in practice, viz., 75|, the addendum is about T 3 o of the pitch. The teeth of a rack, to work correctly with wheels having invo- lute teeth, should have plane surfaces, perpendicular to the line of connection, and consequently making, with the direction of motion of the rack, angles equal to the before-mentioned angle 0. INVOLUTE TEETH. 443 458. Sliding of involute Teeth. The distance through which a pair of involute teeth slide on each other, is found by observing that the distance from the point of contact of the teeth to the pitch point is given by the equation r = q '-fTt = 9 'sin 463. Approximate Epicycloidnl Teeth. Mr. Willis has shown how to approximate to the figure of an epicycloidal tooth by means of two circular arcs, one concave, for the flank, the other convex, for the face, and each having for its radius, the mean radius of curva- ture of the epicycloidal arc. Mr. Willis's formulae are deduced in his own work from certain propositions respecting the transmission of motion by linkwork. In the present treatise they will be deduced from the values already given for the radii of curvature of 446 THEORY OF MECHANISM. epicycloids in Article 390, case 1, equation 4 : viz., let r l be the radius of the pitch circle, r that of the rolling circle, g the radius of curvature required ; then = 2 r Q cos 6 the sign -f applying to an external epicycloid, that is, to the face of a tooth, and the sign to an internal epicycloid, that is, to the flank of a tooth. To find the distances of the centres of curvature of the given point in an epicycloid from the point of contact I of the pitch circle and rolling circle, there is to be subtracted from the radius of cur- vature, the instantaneous radius vector, r 2 r cos 6; that is to say, P - r = 2r ft eos4- ?--... ...(2.} The value to be assumed for 6 is its mean value, that is, 7 5^ 5 and cos 6 - nearly : r is nearly equal to the pitch, p- and if n be the number of teeth in the wheel, Therefore, for the proportions approved of by Mr. Willis, equation 2 becomes ti r 2' (3.) + being used for the face, and - for the flank ; also r = ^- nearly. 2 .(4.) Hence the following con- struction. In fig. 200, let B C be part of the pitch circle, A the point where a tooth is to cross it. Set off Fig. 200. = AC=. Drawradii Z of the pitch circle, D B, E C. Draw F B, C G, making angles of 75^ with those radii, in which take n P TRUNDLE DIMENSIONS OF TEETH. 447 Round F, with the radius F A, draw the circular arc A H ; this will be the face of the tooth. Round G, with the radius G A, draw the circular arc G K ; this will be the flank of the tooth. To facilitate the application of this rule, Mr. Willis has published tables of the values of e r, and invented an instrument called the " odontograph" 464. Teeth of Wheel and Trundle. A trundle, as in fig. 201, has cylindrical pins called staves for teeth. The face of the teeth of a wheel suitable for driving it, in outside gearing, are described by first tracing external epicycloids by rolling the pitch circle B 2 of the trundle on the pitch circle Bj of the driving wheel, with the Fig. 201. Fig. 202. centre of a stave for a tracing-point, as shown by the dotted lines, and then drawing curves parallel to and within the epicycloids, at a distance from them equal to the radius of a stave. Trundles having only six staves will work with large wheels. To drive a trundle in inside gearing, the outlines of the teeth of the wheel should be curves parallel to internal epicycloids. A peculiar case of this is represented in fig. 202, where the radius of the pitch circle of the trundle is exactly one-half of that of the pitch circle of the wheel ; the trundle has three equi-distant staves ; and the internal epicycloids described by their centres while the pitch circle of the trundle is rolling within that of the wheel, are three straight lines, diameters of the wheel, making angles of 60 with each other. Hence the surfaces of the teeth of the wheel form three straight grooves intersecting each other at the centre, each being of a breadth equal to the diameter of a stave of the trundle. 465. Dimensions of Teeth. Toothed wheels being in general intended to rotate either way, the backs of the teeth are made similar to the fronts. The space between two teeth, measured on the pitch circle, is made about one-fifth part wider than the thick- ness of the tooth on the pitch circle; that is to say, Q thickness of tooth = pitch, 4 8 THEORY OF MECHANISM. / width of space = pitch. The difference of 7 of the pitch is called the back-lash. The clearance allowed between the points of teeth and the bottoms of the spaces between the teeth of the other wheel, is about one- tenth of the pitch. The thickness of a tooth is fixed according to the principles already stated in Article 326; and the breadth is so adjusted, that when multiplied by the pitch, the product shall contain one square inch for each 160 Ibs. of force transmitted by the teeth. 466. Mr. Sang' s Process. Mr. Sang has published an elaborate work on the teeth of wheels, in which a process is followed differing in some respects from any of those before described. A form is selected for the path of the point of contact of the teeth, and from that form the figures of the teeth are deduced. For details, the reader is referred to Mr. Bang's work. 467. The Teeth of a Bevel-Wheel have acting surfaces of the conical kind, generated by the motion of a line traversing the apex of the conical pitch surface, while a point in it is carried round the outlines of the cross section of the teeth made by a sphere described about that apex. The operations of describing the exact figures of the teeth of bevel- wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel- wheels, all those operations are to be performed on the surface of a sphere described about the apex, instead of on a plane, substituting poles for centres, and great circles for straight lines. In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by Tredgold, is generally used : Let O, fig. 203, be the apex, and O C the axis of the pitch cone of a bevel- wheel; and let the largest pitch circle be that whose radius is C B. Perpendicular to B draw B A cut- ting the axis produced in A, let the outer rim of the pattern and of the wheel be made a portion of the surface of the cone whose apex is A and side A B. The narrow zone of that cone thus employed will approach lg ' ' sufficiently near to a zone of the sphere described about O with the radius O B, to be used in its stead. On TEETH CAMS SCREWS PITCH. 449 a plane surface, with the radius A B, draw a circular arc B D ; a sector of that circle will represent a portion of the surface of the cone ABC developed, or spread out flat. Describe the figures of teeth of the required pitch, suited to the pitch circle B D, as if it were that of a spur-wheel of the radius A B ; those figures will be the required cross sections of the teeth of the bevel-wheel, made by the conical zone whose apex is A. 468. Teeth of Skew-Bevel Wheels. The cross sections of the teeth of a skew-bevel wheel at a given pitch circle are similar to those of a bevel wheel whose pitch surface is a cone touching the hyperbo- loidal pitch surface of the skew-bevel wheel at the given pitch circle; and the surfaces of the teeth of the skew-bevel wheel are generated by a straight line which moves round the outlines of the cross section and at the same time is maintained at the constant perpendicular distance GH (fig. 195, Article 444) from the axis. 469. The Teeth of Non-Circular Wheels are described by rolling circles or other curves on the pitch surfaces, like the teeth of cir- cular wheels; and when they are small compared with the wheels to which they belong, each tooth is nearly similar to the tooth of a circular wheel having the same radius of curvature with the pitch surface of the actual wheel at the point where the tooth is situated. 470. A Cam or Wiper is a single tooth, either rotating continu- ously or oscillating, and driving a sliding or turning piece, either constantly or at intervals. All the principles which have been stated in Article 450, as being applicable to sliding contact, are applicable to cams ; but in designing cams, it is not usual to deter- mine or take into consideration the form of the ideal pitch surface which would give the same comparative motion by rolling contact that the cam gives by sliding contact. 471. Screws. Pitch. The figure of a screw is that of a convex or concave cylinder with one or more helical projections called threads winding round it. Convex and concave screws are dis- tinguished technically by the respective names of male said female; a short concave screw is called a nut; and when a screw is spoken of without qualification, a convex screw is usually understood. The relation between the advance and the rotation, which com- pose the motion of a screw working in contact with a fixed nut or helical guide, has already been demonstrated in Article 382, equa- tion 1 ; and the same relation exists between the rotation of a screw about an axis fixed longitudinally relatively to the frame- work, and the advance of a nut in which that screw rotates, the nut being free to shift longitudinally, but not to turn. The advance of the nut in the latter case is in the direction opposite to that of the advance of the screw in the former case. 450 THEORY OF MECHANISM. A screw is called right-handed or left-handed, according as its advance in a fixed nut is accompanied by right-handed or left-handed rotation, when viewed by an observer from whom if the advance takes place. Fig. 204 re- la presents a right-handed screw, and fig. 205 a left-handed screw. * The pitch of a screw of one thread, and the total pitch of a screw of any number of threads, is the pitch of the Fig 204 Fte 205 helical motion of that screw, as ex- plained in Article 382, and is the dis- tance (marked p in figs. 204 and 205) measured parallel to the axis of the screw, between the corresponding points in two consecutive turns of the same thread. In a screw of two or more threads, the distance measured parallel to the axis, between the corresponding points in two adjacent threads, may be called the divided pitch. 472. Normal and Circular Pitch. When the pitch of a screw is not otherwise specified, it is always understood to be measured parallel to the axis. But it is sometimes convenient for particular purposes to measure it in other directions ; and for that purpose a cylindrical pitch surface is to be conceived as described about the axis of the screw, intermediate between the crests of the threads and the bottoms of the grooves between them. If a helix be now described upon the pitch cylinder, so as to cross each turn of each thread at right angles, the distance between two corresponding points on two successive turns of the same thread, measured along this normal lielix, may be called the normal pitch; and when the screw has more than one thread, the normal pitch from thread to thread may be called the normal divided pitch. The distance from thread to thread measured on a circle described on the pitch cylinder, and called the pitch circle, may be called the circular pitch; for a screw of one thread it is one circumference ; for a screw of n threads one circumference The following set of formulae show the relations amongst the differ- ent modes of measuring the pitch of a screw. The pitch, properly speaking, as originally defined, is distinguished as the axial pitch, and is the same for all parts of the same screw : the normal and circular pitch depend on the radius of the pitch cylinder. Let r denote the radius of the pitch cylinder ; n, the number of threads ; SCREW GEARING HOOKE'S GEARING. 451 i, the obliquity of the threads to the pitch circles, and of the normal helix to the axis ; P. P " the normal divided pitch ; ( pitch ; ( divided pitch ; p e , the circular pitch ; Then P Pa' cotan i = p n cosec i = ; 2 ^ r tan i Pa = Pn ' sec i = p e ' tan i = - - ; /(/ 2 T r sin i Pn PC' sm i = Pa.' cos z = . n 473. Screw Gearing. A pair of convex screws, each rotating about its axis, are used as an elementary combination, to transmit motion by the sliding contact of their threads. Such screws are commonly called endless screws. At the point of contact of the screws, their threads must be parallel ; and their line of connection is the common perpendicular to the acting surfaces of the threads at their point of contact. Hence the following principles : I. If the screws are both right-handed or both left-handed, the angle between the directions of their axes is the sum of their obli- quities : if one is right-handed and the other left-handed, that angle is the difference of their obliquities. II. The normal pitch, for a screw of one thread, and the normal divided pitch, for a screw of more than one thread, must be the same in each screw. III. The angular velocities of the screws are inversely as their number of threads. 474. iiooke's Gearing is a case of screw gearing, in which the axes of the screws are parallel, one screw being right-handed and the other left-handed, and in which, from the shortness and great diameter of the screws, and their large num- ber of threads, they are in fact wheels, with teeth whose crests, instead of being parallel to the line of contact of the pitch cylinders, cross it obliquely, so as to be of a screw-like " . or helical form. In wheelwork of this kind, the contact of each pair of teeth commences at the foremost end of 452 THEORY OF MECHANISM. the helical front and terminates at the aftermost end ; and the helix is of such a pitch that the contact of one pair of teeth does not terminate until that of the next pair has commenced. The object of this is to increase the smoothness of motion. "With the same object, Dr. Hooke invented the making of the fronts of teeth in a series of steps. A wheel thus formed resembles in shape a series of equal and similar toothed discs placed side by side, with the teeth of each a little behind those of the preced- ing disc. In such a wheel, let p be the Fig. 207. circular pitch, and n the number of steps. Then the arc of contact, the addendum, and the extent of sliding, are those due to the smaller pitch , while the strength of the teeth is that due to the thickness corresponding to the entire pitch p ; so that the smooth action of small teeth and the strength of large teeth are combined. Stepped teeth being more expensive and difficult to execute than common teeth, are used for special pur- poses only. 475. The wheel and Screw is an elementary combination of two screws, whose axes are at right angles to each other, both being right-handed or both left-handed. As the usual object of this com- bination is to produce a change of angular velocity in a ratio greater than can be obtained by any single pair of ordinary wheels, one of the screws is commonly wheel-like, being of large diameter and many-threaded, while the other is short and of few threads ; and the angular velocities are inversely as the number of threads. Fig. 208. Fig. 209. Fig. 208 represents a side view of this combination, and fig. 209 a cross section at right angles to the axis of the smaller screw. It has been shown by Mr. Willis, that if each section of both screws be made by a plane perpendicular to the axis of the large screw or wheel, the outlines of the threads of the larger and smaller screw should be those of the teeth of a wheel and rack respectively : B x B 1? SLIDING OF SCREWS - OLDHAM'S COUPLING. 453 in fig. 208, for example, being the pitch circle of the wheel, and B 2 B 2 the pitch line of the rack. 'The periphery and teeth of the wheel are usually hollowed to fit the screw, as shown at T, fig. 209. To make the teeth or threads of a pair of screws fit correctly and work smoothly, a hardened steel screw is made of the figure of the smaller screw, with its thread or threads notched so as to form a cutting tool ; the larger screw, or wheel, is cast approximately of the required figure ; the larger screw and the steel screw are fitted up in their proper relative position, and made to rotate in contact with each other by turning the steel screw, which cuts the threads of the larger screw to their true figure. 476. The Relative Sliding of a Pair of Screws at their point of contact is found thus : Let r l7 r 2) be the radii of their pitch cylin- ders, and i}, i 2 , the obliquities of their threads to their pitch circles, one of which is to be considered as negative if the screws are con- trary-handed. Let u be the common component of the velocities of a pair of points of contact along a line touching the pitch sur- faces and perpendicular to the threads, at the 'pitch point, and v the velocity of sliding of the threads over each other. Then u = ! rj ' sin ^ = a 2 r z - sin i. z ; so that n.) u u " _ _ i - ; - ~ * ^ ~ ; " T! ' sm % r z ' sm h and v = a l r l - cos tj -\-a 2 r 2 - cos % = u (tan ^ + tan 4) ..... (2.) When the screws are contrary-handed, the difference instead of the sum of the terms in equation 2 is to be taken. 477. Oldham's Coupling. A coupling is a mode of connecting a pair of shafts so that they shall rotate in the same direction, with the same mean angular velocity. If the axes of the shafts are in the same straight line, the coupling consists in so connecting their contiguous ends that they shall rotate as one piece; but if the axes are not in the same straight line, combinations of mechanism are re- quired. A coupling for parallel shafts which acts by sliding contact was invented by Oldham, and is represented in fig. 210. Cj, C 2 , are the axes of the two parallel shafts; D x , D 2 , two cross- heads, facing each other, fixed on the ends of the two shafts re- spectively; Ej, Ej, a bar, sliding in a diametral groove in the face of 454 THEORY OF MECHANISM. Dj ; E 2 , E^, a bar, sliding in a diametral groove in the face of D 2 ; those bars are fixed together at A, so as to form a rigid cross. The angular velocities of the two discs and of the cross are all equal at every instant. The middle point of the cross, at A, revolves in the dotted circle described upon the line of centres Cj, C 2 , as a dia- meter, twice for each turn of the discs and cross ; the instantaneous axis of rotation of the cross, at any instant, is at I, the point in the circle Cj, C 2 , diametrically opposite to A. Oldham's coupling may be used with advantage where the axes of the shafts are intended to be as nearly in the same straight line as is possible, but where there is some doubt as to the practica- bility or permanency of their exact continuity. SECTION 3. Connection by Bands. 478. Bands Classed. Bands, or wrapping connectors, for com- municating motion between pulleys or drums rotating about fixed axes, or between rotating pulleys and drums and shifting pieces, may be thus classed : I. Belts, which are made of leather or of gutta percha, are flat and thin, and require nearly cylindrical pulleys. A belt tends to move towards that part of a pulley whose radius is greatest j pulleys for belts, therefore, are slightly swelled in the middle, in order that the belt may remain on the pulley unless forcibly shifted. A belt when in motion is shifted off a pulley, or from one pulley on to another of equal size alongside of it, by pressing against that part of the belt which is moving towards the pulley. II. Cords, made of catgut, hempen or other fibres, or wire, are nearly cylindrical in section, and require either drums with ledges, or grooved pulleys. III. Chains, which are composed of links or bars jointed together, require pulleys or drums, grooved, notched, and toothed, so as to fit the links of the chains. Bands for communicating continuous motion, are endless. Bands for communicating reciprocating motion have usually their ends made fast to the pulleys or drums which they connect, and which in this case may be sectors. 479. Principle of Connection by Bands. The line of connection of a pair of pulleys or drums connected by means of a band, is the central line or axis of that part of the band whose tension transmits the motion. The principle of Article 433 being applied to this case, leads to the following consequences : I. For a pair of rotating pieces, let r lt r 2 , be the perpendiculars let fall from their axes on the centre line of the band, Oj, a 2 , their angular velocities, and i^ ?' a , the angles which the centre line of the BANDS PULLEYS DRUMS. 455 band makes with the two axes respectively. Then the longitudi- nal velocity of the band, that is, its component velocity in the direction of its own centre line, is u = r 1 a l sin ^! = r a & 2 sin i 2 ; (1.) whence the angular velocity-ratio is a 2 r x sin *! /0 , sin JTAew and axes are parallel (which is almost always the case), ?\ = .(3.) The same equation holds when both axes, whether parallel or not, are perpendicular in direction to that part of the band which trans- mits the motion ; for then sin % = sin i 2 = 1. II. For a rotating piece and a sliding piece, let r be the perpendi- cular from the axis of the rotating piece on the centre line of the band, a the angular velocity, i the angle between the directions of the band and axis, u the longitudinal velocity of the band, j the angle between the direction of the centre line of the band and that of the motion of the sliding piece, and v the velocity of the sliding piece ; then u = ra sin i v cosj ; and (4.) ra sin ^ cosj .(5.) When the centre line of the band is parallel to the direction of motion of the sliding piece, and perpendicular to the direction of the axis of the rotating piece, sin i = cos j = 1 , and v = u = ra (6.) 480. The Pitch Surface of a Pulley or Drum is a Surface to which the line of connection is always a tangent ; that is to say, it is a surface parallel to the acting surface of the pulley or drum, and distant from it by half the thickness of the band. 481. Circular Pulleys and Drums are used to communicate a Fig. 211. Fig. 212. constant velocity-ratio. In each of them, the length denoted by r 456 THEORY OF MECHANISM. in the equations of Article 479 is constant, and is called the effec- tive radius, being equal to the real radius of the pulley or drum added to half the thickness of the band. A crossed belt connecting a pair of circular pulleys, as in fig. 211, reverses the direction of rotation ; an open belt, as in fig. 212, pre- serves that direction. 482. The Length of an Endless Belt, connecting a pair of pulleys whose effective radii are Cj T^ = r^ C 2 T 2 = r 2 , with parallel axes whose distance apart is Cj C 2 == c, is given by formulae founded on equation 1 of Article 402, viz., L = 2 8 + 2 r i. Each of the two equal straight parts of the belt is evidently of the length s = Jc? - fa + r 2 ) 2 for a crossed belt ; | f (1.) s = Jc* - (T! - r 2 )* for an open belt ; j i\ being the greater radius, and r 2 the less. Let i T be the arc to radius unity of the greater pulley, and i z that of the less pulley, with which the belt is in contact ; then for a crossed belt + 2 arc -sin TI + r *\ 1 and for an open belt, ij = (K + 2 arc . sin r ' ~ r * j ; i 2 = ( v - 2 arc -sin TI ~ r * j ; and the introduction of those values into equation 1 of Article 402 gives the following results : For a crossed belt, sn and for an open belt, L - 2 J<*-( ri -r a y + <*(r 1 + r a ) + 2 (^ - r,) arc sin r -Ll^ (3.) As the last of these equations would be troublesome to employ in a practical application to be mentioned in the next Article, an approximation to it, sufficiently close for practical purposes, is obtained by considering, that if r r z is small compared with c, ^-(rj-rji = c - ^ 1 ~ r ^ nearly, and arc sin T -^ = r -l^-- a (s C C nearly ; whence, for an open belt, L nearly = 2 c + v (r l + r z ) + ^^ (3 A.) SPEED-CONES. 457 483. Speed-cone* (figs. 213, 214, 215, 216) are a contrivance for Fig. 213. Fig. 214. Fig. 215. Fig. 216. varying and adjusting the velocity-ratio communicated between a pair of parallel shafts by means of a belt, and may be either conti- nuous cones or conoids, as in figs. 213, 214, whose velocity-ratio can be varied gradually while they are in motion by shifting the belt ; or sets of pulleys whose radii vary by steps, as in figs. 215, 216, in which case the velocity-ratio can be changed by shifting the belt from one pair of pulleys to another. In order that the belt may be equally tight in every possible position on a pair of speed-cones, the quantity L in the equations of Article 482 must be constant. For a crossed belt, as in figs. 213 and 215, L depends solely on c and on r^ + r 2 . Now c is constant, because the axes are parallel, therefore the sum of the radii of the pitch circles connected in every position of the belt is to be constant. That condition is fulfilled by a pair of continuous cones generated by the revolution of two straight lines inclined opposite ways to their respective axes at equal angles, and by a set of pairs of pulleys in which the sum of the radii is the same for each pair. For an open belt, the following practical rule is deduced from the approximate equation 3 A of Article 482 : Let the speed-cones be equal and similar conoids, as in fig. 214, but with their large and small ends turned opposite ways. Let r x be the radius of the large end of each, r 2 that of the small end, r that of the middle ; and let y be the sagitta, measured perpendi- cular to the axis, of the arc by whose revolution each of the conoids is generated, or, in other words, the bulging of the conoids in the middle of their length j then 2 -TT = 6 -2832 ; but 6 may be used in most practical cases without sensible error. 458 THEORY OF MECHANISM. The radii at the middle and ends being thus determined, make the generating curve an arc either of a circle or of a parabola. For a pair of stepped cones, as in fig. 216, let a series of differ- ences of the radii, or values of r t r 2 , be assumed ; then for each pair of pulleys, the sum of the radii is to be computed from the difference by the formula , 1 + , 2 = 2, -fe; ................... (2.) 2 r Q being that sum when the radii are equal. SECTION 4. Linkwork. 484. Definitions. The pieces which are connected by linkwork, if they rotate or oscillate, are usually called cranks, beams, and levers. The link by which they are connected is a rigid bar, which may be straight or of any other figure ; the straight figure being the most favourable to strength, is used when there is no special reason to the contrary. The link is known by various names under various circumstances, such as coupling rod, connecting rod, crank rod, eccentric rod, &c. It is attached to the pieces which it connects by two pins, about which it is free to turn. The effect of the link is to maintain the distance between the centres of those pins in- variable ; hence the line joining the centres of the pins is the line of connection; and those centres may be called the connected points. In a turning piece, the perpendicular let fall from its connected point upon its axis of rotation is the arm or crank arm. 485. Principles of Connection. The whole of the equations already given in Article 479 for bands, are applicable to linkwork. The axes of rotation of a pair of turning pieces connected by a link are almost always parallel, and perpendicular to the line of connec- tion j in which case the angular velocity-ratio at any instant is the reciprocal of the ratio of the common perpendiculars let fall from the line of connection upon the respective axes of rotation (Article 479, equation 3). 486. Dead Points. If at any instant the direction of one of the crank arms coincides with the line of connection, the common perpendicular of the line of connection and the axis of that crank arm vanishes, and the directional relation of the motions becomes indeterminate. The position of the connected point of the crank arm in question at such an instant is called a dead point. The velocity of the other connected point at such an instant is null, unless it also reaches a dead point at the same instant, so that the line of connection is in the plane of the two axes of rotation, in which case the velocity-ratio is indeterminate. LINKWORK. 459 487. Coupling of Parallel Axes. The only case in which an uni- form angular velocity-ratio (being that of equality) is communicated by linkwork, is that in which two or more parallel shafts (such as those of the driving wheels of a locomotive engine) are made to rotate with constantly equal angular velocities, by having equal cranks, which are maintained parallel by a coupling rod of such a length that the line of connection is equal to the distance between the axes. The cranks pass their dead points simultaneously. To obviate the unsteadiness of motion which this tends to cause, the shafts are provided with a second set of cranks at right angles to the first, connected by means of a similar coupling rod, so that one set of cranks pass their dead points at the instant when the other set are farthest from theirs. 488. The Comparative Motion of the Connected Points in a piece of linkwork at a given instant is capable of determination by the method explained in Article 384 ; that is, by finding the instanta- neous axis of the link ; for the two connected points move in .the same manner with two points in the link, considered as a rigid body. If a connected point belongs to a turning piece, the direction of its motion at a given instant is perpendicular to the plane contain- ing the axis and crank arm of the piece. If a connected point belongs to a shifting piece, the direction of its motion at any instant is given, and a plane can be drawn perpendicular to that direction. The line of intersection of the planes perpendicular to the paths of the two connected points at a given instant, is the instantaneous axis of the link at that instant; and the velocities of the connected points are directly as their distances from that axis. In drawing on a plane surface, the two planes perpendicular to the paths of the connected points are represented by two lines (being their sections by a plane normal to them), and the instanta- neous axis by a point; and should the length of the two lines render it impracticable to produce them until they actually inter- sect, the velocity-ratio of the connected points may be found by the principle, that it is equal to the ratio of the segments which a line parallel to the line of connection cuts off from any two lines drawn from a given point, perpendicular respectively to the paths of the connected points. Example I. Two Rotating Pieces with Parallel Axes (fig. 217). Let C 1? C 2 , be the parallel axes of the pieces ; T 1? T 2 , their con- nected points ; GI T 1? C 2 T 2 , their crank arms ; TI T 2 , the link. At a given instant, let v^ be the velocity of T! ; v z that of T 2 . To find the ratio of those velocities, produce Cj TI, C 2 T 2 , till they intersect in K ; K is the instantaneous axis of the link or connecting rod, and the velocity-ratio is 460 THEORY OF MECHANISM. v l : v 2 : : K T, : K T, .(1.) Should K be inconveniently far off, draw any triangle with its sides respectively parallel to Cj T,, C 2 T 2 , and T t T 8 ; the ratio of the two sides first mentioned will be the velocity-ratio required. For example, draw C 2 A parallel to Cj T u cutting T l T 2 in A; then C 2 A : C 2 T 2 .(2.) Fig. 217. Example II. Rotating Piece and Sliding Piece (fig. 218). Let C 2 be the axis of a rotating piece, and T\ R the straight line along which a sliding piece moves. Let T 1? T 2 , be the connected points, C7T 2 the crank arm of the rotating piece, and T! T 2 the link or connecting rod. The points T\, T 2 , and the line T l R, are supposed to be in one plane, perpendicular to the axis C. Draw Tj K per- pendicular to T! R, intersecting C 2 T 2 in K ; K is the instantaneous axis of the link ; and the rest of the solution is the same as in Example I. 489. An Eccentric (fig. 219) being a circular disc keyed on a shaft, with whose axis its centre does not co- incide, and used to give a reciprocating motion to a rod, is equivalent to a crank whose con- nected point is T, the centre of the eccentric disc, and whose crank arm is C T, the distance of that point from the axis of the shaft, called the eccentricity. An eccentric may be made capable of having its eccentricity altered by means of an adjusting screw, so as to vary the extent of the reciprocating motion which it communicates, and which is called the throw, or travel, or length of stroke. 490. The JLength of Stroke of a point in a reciprocating piece is the distance between the two ends of the path in which that point moves. When it is connected by a link with a point in a con- Fig. 219. LENGTH OP STROKE HOOKE'S JOINT. 461 tinuously rotating piece, the ends of the stroke of the reciprocating point correspond with the dead points of the continuously revolving piece (Article 486). Let S be the length of stroke of the reciprocating piece, L the length of the line of connection, and K the crank arm of the con- tinuously turning piece. Then if the two ends of the stroke be in one straight line with the axis of the crank, S = 2R,; (1.) and if their ends be not in one straight line with that axis, then S, L R, and L -f- R, are the three sides of a triangle, having the angle opposite S at that axis ; so that if & be the supplement of the arc between the dead points, S 2 =: 2 (L 3 + R 2 )-2 (L 2 -R 2 ) cos 6- COS0 = 2(L 3 -R a ) 491. Hooke's Universal Joint (fig. 220) is a contrivance for coup- ling shafts whose axes intersect each other in a point. Let O be the point of intersection of the axes G lt O C 2 , and i their angle of inclination to each other, The pair of shafts d, C 2 , terminate in a pair of forks F J? F 2 , in bearings at the extremities of which turn the gudgeons at the ends of the arms of a rectangular cross having its centre at O. This cross is the link ; the connected points are the centres of the bearings F 1? F 2 . At each instant each of those points moves at right angles to the central plane of its shaft and fork, therefore the line of intersection of the central planes of the two forks, at any instant, is the instantaneous axis of the cross, and the velocity-ratio of the points F 1? F 2 (which, as the forks are equal, is also the angular velocity-ratio of the shafts), is equal to the ratio of the distances of those points from that instan- taneous axis. The mean value of that velocity-ratio is that of equality ; for each successive quarter turn is made by both shafts in the same time ; but its actual value fluctuates between the limits, 6J2 1 = -. when Fi is in the plane of the axes : \ \ ...a) = cos i when F 2 is in that plane. 462 THEORY OF MECHANISM. Its value at intermediate instants, as well as the relation between the positions of the shafts, are given by the following equations : Let (2.) i ~ d (ft ~~ tan

(1.) Lateral force, Q = F sin Q. 512. The Conditions of Uniform Motion of a pair of points are, that the forces applied to each of them shall balance each other ; that is to say, that the lateral forces applied to each point shall balance each other, and that tlie efforts applied to each point shall balance the resistances. The direction of a force being, as stated in Article 20, that of the motion which it tends to produce, it is evident that the balance of lateral forces is the condition of uniformity of direction of motion, that is, of motion in a straight line ; and that the balance of efforts and resistances is the condition of uniformity of velocity. 513. Work consists in moving against resistance. The work is said to be performed, and the resistance overcome. Work is mea- sured by the product of the resistance into the distance through which its point of application is moved. The unit of work com- monly used in Britain is a resistance of one pound overcome through a distance of one foot, and is called a, foot-pound. 514. Energy means capacity for performing work. The energy of an effort, or potential energy, is measured by the product of the effort into the distance through which its point of application is capable of being moved. The unit of energy is the same with the unit of work. When the point of application of an effort has been moved through a given distance, energy is said to have been exerted to an amount expressed by the product of the effort into the distance through which its point of application has been moved. 515. Energy and Work of Varying Forces. If an effort has dif- ferent magnitudes during different portions of the motion of its point of application through a given distance, let each different magnitude of the effort P be multiplied by the length A s of the corresponding portion of the path of the point of application ; the sum 2- PAS (1.) is the whole energy exerted. If the effort varies by insensible 478 PRINCIPLES OF DYNAMICS. degrees, the energy exerted is the integral or limit towards which that sum approaches continually, as the divisions of the path are made smaller and more numerous, and is expressed by Pds (2.) Similar processes are applicable to the finding of the work per- formed in overcoming a varying resistance. As to integration in general, see Article 81. 516. A Dynamometer or Indicator is an instrument which mea- sures and records the energy exerted by an effort. It usually con- sists essentially, first, of a piece of paper moving with a velocity proportional to that of the point of application of the effort, and having a straight line marked on it parallel to its direction of motion, called the zero line ; and secondly, of a spring, acted upon and bent by the effort, and carrying a pencil whose perpendicular distance from the zero line, as regulated by the bending of the spring, is proportional to the effort. The pencil traces on the piece of paper a line like that in fig. 24 of Article 81, such that its ordi- nate E F, perpendicular to the zero line O X at a given point, represents the effort P for the corresponding point in the path of the point of application of the effort ; and the area between two ordinates, such as A C D B, represents the energy exerted, T*ds, for the corresponding portion, A B, of the path of the point of application of the effort. 517. The Energy and Work of Fluid Pressure may be expressed as follows : Let A denote the projection on a plane perpendicular to tJie direction of motion of the moving body, of that portion of the body's surface to which the pressure is applied, p the intensity of the pressure in units of force per unit of area (Article 86), and A s the distance through which the body is moved in a given interval of time ', then during that interval, the energy exerted by, or work performed against, the fluid pressure, according as it acts with or against the motion, is given by the formula P AS (or II &s)=p A As=p AY; (1.) where A V is the volume of the space swept through by the portion of the body's surface which is pressed upon, during the given interval of time. 518. The Conservation of Energy, in the case of uniform motion, means the fact, that the energy exerted is equal to the work performed; and is a consequence of the first law of motion, as is shown by the consideration of the following cases : CASE 1. For the forces acting on a single point, the principle is CONSERVATION OP ENERGY VIRTUAL VELOCITIES. 479 self-evident; for as the effort applied to the point balances the resistance, the products of these forces into the distance traversed by the point in any interval must be equal ; that is, P-AS^K-AS ......................... (1.) CASE 2. For the forces acting on any system of balanced points, the principle must be true, because it is true for those acting on each single point of the system. This is expressed as follows : (2.) CASE 3. When a system of points are rigidly connected, so that their relative positions do not alter, there is neither energy exerted nor work performed by the forces which act amongst the points of tJie system themselves; and therefore, from case 2 it follows, that the principle of the conservation of energy is true of the forces acting between the points of the system and external bodies. Symbolically, let the efforts acting amongst the points of the system be denoted by P,, the resistances by R.J ; the efforts acting between the points of the system and external bodies by P, and the resistances by R 2 . Then by case 2, but by the condition of rigidity, 2-P t As = therefore, 2 'P 2 AS =2 -R A S CASE 4. The same principle is demonstrable in the same manner, for the forces acting between external bodies and the points of a system so connected, that though not absolutely rigid, they do not vary their relative positions in the directions in which the internal forces of the system act. Such is the ideal condition in which a train of mechanism would be, if no resistance arose from the mode of connection of the pieces. 519. The Principle of Virtual Velocities is the name given to the application of the principle of the conservation of energy to the determination of the conditions of equilibrium amongst the forces externally applied to any connected system of points. That appli- cation is effected in the following manner : Let F be any one of the externally applied forces in question. The conditions of equili- brium are those of uniform motion. Conceive the points of the system to be moving with uniform velocities in any manner which is consistent with the absence of all exertion of energy and perfor- mance of work by their mutual or internal forces. Let v be the 480 PRINCIPLES OP DYNAMICS. velocity, or any number proportional to the velocity, of the point to which the external force F is applied, and & the angle between the direction of that force and the direction of motion of its point of application. Then from cases 3 and 4 of the principle of the conservation of energy, it follows that the condition of equilibrium amongst the forces F is s-Fcos* = 0; (1.) attention being paid to the principle, that cos 6 is < ^ 'V^ 6 > (^ negative I when 6 is < \^ > The same principle may be otherwise ex- pressed thus : let v be the virtual velocity of any point to which an effort P is applied, u the virtual velocity of any point to which a resistance II is applied ; then *-P = -R (2.) The principle thus expressed is called that of virtual velocities, because the velocities denoted by v are merely velocities which the points of the system might have. As the proportions of the several velocities v are all that are required in using this principle, it enables the conditions of equili- brium of the forces applied to any body or machine to be found, so soon as the comparative velocities of the points of application of those forces have been determined by means of the principles of cinematics, and of the theory of mechanism ; and every proposition which has been proved in Parts III. and IV. of this treatise, respecting the comparative velocities of points in a body or in a train of mechanism, can at once be converted into a proposition respecting the equilibrium of forces applied to those points in given directions. 520. Energy of Component Forces and Motions. Let the motion A s of a point in a given interval of time make angles, , j8, y, with three rectangular axes ; then A S ' COS M, A S ' COS /3, A S ' COS y, are the three components of that motion. To that point let there be applied a force F, making with the same axes the angles a', /3', y', so that its rectangular components are F cos a!, F cos /3', F cos y'. Then multiplying each component of the motion by the component of the force in its own direction, there are found the three quantities of energy exerted, COMPONENTS OF ENERGY AND WORK. 481 F AS cos cos *'; 1 F AS cos ft cos /3'; I (1.) F A s ' cos y cos y' ; J and the sum of those three quantities of energy is the whole energy exerted. Now it is well known, that cos at, cos a! -j- cos |3 cos /3' -j- cos y cos y' cos 0, 6 being the angle between the directions of the force and of the motion; so that the addition of the three quantities of energy in the formulae 1 gives for the whole energy exerted, simply F * s cos &, as in former examples; and similar remarks apply to work per- formed. 482 CHAPTEE II. ON THE VARIED TRANSLATION OF POINTS AND RIGID BODIES. SECTION 1. Definitions. 521. The Mass, or inertia, of a body, is a quantity proportional to the unbalanced force which is required in order to produce a given definite change in the motion of the body in a given interval of time. It is known that the weight of a body, that is, the attraction between it and the earth, at a fixed locality on the earth's surface, acting unbalanced on the body for a fixed interval of time (e. g., for a second), produces a change in the body's motion, which is the same for all bodies whatsoever. Hence it follows, that the masses of all bodies are proportional to their weights at a given locality on the earth's surface. This fact has been learned by experiment ; but it can also be shown that it is necessary to the permanent existence of the uni- verse ; for if the gravity of all bodies whatsoever were not propor- tional to their respective masses, it would not produce similar and equal changes of motion in all bodies which arrive at similar posi- tions with respect to other bodies, and the different parts which make up stars and systems would not accompany each other in their motions, never departing beyond certain limits, but would be dis- persed and reduced to chaos. Neither an imponderable body, nor a body whose gravity, as compared with its mass, differs in the slightest conceivable degree from that of other bodies, can belong to the system of the universe.* 522. The Centre of Mass of a body is its centre of gravity, found in the manner explained in Part I., Chapter V., Section 1. 523. The Momentum of a body means, the product of its mass into its velocity relatively to some point assumed as fixed. The momentum of a body, like its velocity, can be resolved into com- ponents, rectangular or otherwise, in the manner already explained for motions in Part III., Chapter I. 524. The Resultant Momentum of a system of bodies is the re- sultant of their separate momenta, compounded as if they were motions or statical couples. * See the Rev. Dr. WhewelTs demonstration " that all matter gravitates." MOMENTUM IMPULSE. 483 THEOREM. The momentum of a system of bodies is the same as if all their masses were concentrated at the centre of gravity of the sys- tem. Conceive the velocity of each of the bodies to be resolved into three rectangular components. Consider all the component velocities parallel to one of the rectangular directions. These are the rates of variation of the perpendicular distances of the bodies from a certain plane. If the mass of each of the bodies be multi- plied by its distance from a certain plane, the products added, and the sum divided by the sum of the masses, the result is the distance of the centre of gravity of the whole system from that plane ; there- fore, if the component velocity of each of the bodies in a direction perpendicular to that plane be multiplied by the mass of the body, the sum of such products for all the bodies of the system will be the product of the entire mass of the system into the velocity of its centre of gravity in a direction perpendicular to the plane in ques- tion; so that this product is one of the three rectangular com- ponents of the resultant momentum of the system of bodies ; and the same may be proved for the other rectangular components. Expressed symbolically, let u, v, w, be the three rectangular com- ponents of the velocity of any mass, m, belonging to a system of bodies, and u^ V Q , w , the rectangular components of the velocity of the centre of gravity of that system of bodies ; then .(i.) 2 m = 2 m v ; ' 2 m = 2 ' m w. COROLLARY. The resultant momentum of a system of bodies rela- tively to tJieir common centre of gravity is nothing ; that is to say, 525. Variations and Deviations of Momentum are the products of the mass of a body into the rates of variation of its velocity and deviation of its direction, found as explained in Part III., Chapter I., Section 3. 526. impulse is the product of an unbalanced force into the time during which it acts unbalanced, and can be resolved and com- pounded exactly like force. If F be a force, and d t an interval of time during which it acts unbalanced, F d t is the impulse exerted by the force during that time. The impulse of an unbalanced force in an unit of time is the magnitude of the force itself. 527. Impulse, Accelerating, Retarding, Deflecting. Correspond- ing to the resolution of a force applied to a moving body into effort or resistance, as the case may be, and lateral stress, as explained in 484 PRINCIPLES OF DYNAMICS. Article 511, there is a resolution of impulse into accelerating or retarding impulse, which acts with or against the body's motion, and deflecting impulse, which acts across the direction of the body's motion. Thus if t, as before, be the angle which the unbalanced force F makes with the body's path during an indefinitely short interval, dt, P d t = F cos 6 - d t is accelerating impulse if 6 is acute ; \ R d t = F cos (v 0) d t is retarding impulse if 6 is obtuse ; > (1.) Q dt F sin t dt is deflecting impulse. 528. Relations between Impulse, Energy, and Work. If v be the mean velocity of a moving body during the interval d t of the action of the unbalanced force F, then ds = v dt is the distance described by that body ; and according as & is acute or obtuse, there is either energy exerted on the body by the accelerating impulse to the amount ds = l?vcost' dt; (1.) or work performed by the body against the retarding impulse to the amount ~Rds =Fv cos (v-6) dt (2.) SECTION 2. Law of Varied Translation. 529. Second Law of Motion. Change of momentum is propor- tional to the impulse producing it. In this statement, as in that of the first law of motion, Article 510, it is implied that the motion of the moving body under consideration is referred to a fixed point or body whose motion is uniform. In questions of applied mechanics, the motion of any part of the earth's surface approaches sufficiently near to uniformity to be treated as uniform without sensible error in practice. The unit of mass being arbitrary, may be so assumed as to make change of momentum equal to the impulse producing it. 530. General Equations of Dynamics. To express the second law of motion algebraically, two methods may be followed : the first method being to resolve the change of momentum into direct variation and deviation, and the impulse into direct and deflecting impulse ; and the second method being to resolve both the change of momentum and the impulse into components parallel to three rectangular axes. First method, in being the mass of the body, v its velocity, and r the radius of curvature of its path, it follows from Articles 361 and 362 that the rate of direct variation of its momentum is dv d 2 s EQUATIONS OF DYNAMICS GRAVITY. 485 and from Articles 363 and 364, that the rate of deviation of its momentum is v 2 in. r Equating these respectively to the direct and lateral impulse per unit of time, exerted by an unbalanced force F, making an angle 6 with the direction of the body's motion, we find the two following equations : Por -R, = Fcos4 = w-j- = ^-r^; ............. (1.) 2 Q = Fsin* = ^- ............................ (2.) The radius of curvature r is in the direction of the deviating force Q. Second method. As in Article 366, let the velocity of the body be resolved into three rectangular components, , -~^ - } so that d t d t dt the three component rates of variation of its momentum are Also let the unbalanced force F, making the angles , ft, y, with the axes of co-ordinates, and its impulse per unit of time, be resolved into three components, F,, F y) F r Then we obtain d*x F = F cos * = w -r-? g = F cos y m (3.) three equations, which are substantially identical with the equa- tions 1 and 2. 531. Mass in Terms of Weight. A body's own weight, acting unbalanced on the body, produces velocity towards the earth, increasing at a rate per second denoted by the symbol g, whose numerical value is as follows : Let * denote the latitude of the place, h its elevation above the mean level of the sea, g l = 32-169545 feet per second ; being the value of g for A = 45 and h 0, and R = 20887540 feet (1 + 0-00164 cos 2 x), 486 PRINCIPLES OF DYNAMICS. the earth's radius at the locality of observation ; then g = 9l ' (1-0-00284 cos 2 A) (l -^) ............ (1.) For latitudes exceeding 45, it is to be borne in mind that cos 2 A is negative, and the terms containing it as a factor have their signs reversed. For practical purposes connected with ordinary machines, it is sufficiently accurate to assume # = 32-2 feet per second nearly .................... (2.) If, then, a body of the weight W be acted upon by an unbalanced force F, the change of velocity in the direction of F produced in a second will be m W whence m = .............................. (3.) g is the expression for the mass of a body in terms of its weight, suited to make a change of momentum equal to the impulse pro- ducing it. m being absolutely constant for the same body, g and W vary in the same proportion at different elevations and in different latitudes. 532. An Absolute Unit of Force is the force which, acting during an unit of time on an arbitrary unit of mass, produces an unit of velocity. In Britain, the unit of time being a second (as it is else- where), and the unit of velocity one foot per second, the unit of mass employed is the mass whose weight in vacuo at London and at the level of the sea is a grain, being WQ.. ...(2.) vertical, z = v sin 6 t *-- ; and because t = , those co-ordinates are thus related, v cos w z = x ' tan* 9 2 -x 2 - (3.) 2 v 2 cos 2 an equation which shows the path B C of the projectile to be a parabola with a vertical axis, touching O A in O. The total velocity of the projectile at a given instant, being the resultant of the components given by equation 1, has for the value of its square ^ _ d&_ , d _ 3 _ from the last form of which is obtained the equation MOTION ALONG AN INCLINED PATH. 489 which, being compared with equation 4 of Article 533, shows that the relation between the variation of vertical elevation, and the varia- tion of the square of the resultant velocity, is the same, whether the velocity is in a vertical, inclined, or horizontal direction. This is a particular case of a more general principle, to be explained in the sequel. The resistance of the air prevents any actual projectile near the earth's surface from moving exactly as an unresisted projectile. The approximation of the motion of an actual projectile to that of an unresisted projectile is the closer, the slower is the motion, and the heavier the body, because of the resistance of the air increasing with the velocity, and because of its proportion to the body's weight being dependent upon that of the body's surface to its weight. 535. The Motion of a Body Along an Inclined Path, under the force of gravity alone, is regulated by the principle, that the varia- tion of momentum in each interval of time is equal to the impulse exerted in that interval, by that component of the body's weight which acts along the direction of motion. If the path is straight, the other rectangular component of the body's weight is balanced by the resistance of the surface or other guiding body which causes the inclined path to be described; if the path is curved, the difference between those two forces which act across it is employed in deviat- ing the direction of motion of the body. Let v be the velocity of the body at any instant, , as before, u t the rate of variation of that velocity, 6 the inclination of the body's path to the horizon, positive upwards, and negative downwards. Then the body is acted upon in a direction along its path by a force equal to its weight multiplied by sin t t which is a resistance if 6 is positive, and an effort if 6 is negative; therefore When the inclination of the path is uniform, this rate of varia- tion of velocity is constant, and the body moves in the same manner with an unresisted body moving vertically, except that each change of velocity occupies an interval of time longer in the ratio of 1 : sin 6 for the inclined path than for the vertical path. The motion of a body in any path on an INCLINED PLANE being resolved into two rectangular components, one horizontal, and the other in the direction of steepest declivity, the horizontal com- ponent (in the absence of friction) is uniform, and the inclined 490 PRINCIPLES OF DYNAMICS. component takes place according to the law expressed by equation 1 of this Article. Consequently, the resultant motion of the body is that of an unresisted projectile, as described in Article 534, except that g . sin 6 is to be substituted for g. The motions of bodies on inclined planes being slower, and there- fore more easily observed than their vertical motions, were used by Galileo to ascertain the laws of dynamics, which he discovered. For a body sliding on an inclined plane without friction, the equation connecting the velocity directly with the position of the body is the following : vl - v 2 = 2 g sin 6 - z' ; where v is the velocity at the origin of the motion, and v the velocity which the body has when it reaches a position whose inclined co-ordinate relatively to the origin of the motion is ^, positive upwards. But 2' sin t = z, the difference of vertical eleva- tion of the two positions of the body; so that the variation of the square of the velocity bears the same relation to the difference of vertical elevation in the present case as in the case of an unresisted projectile, or a free body moving vertically. 536. An Uniform Effort or Resistance, unbalanced, causes the velocity of a body to vary according to the law expressed by this equation, where /is the constant ratio which the unbalanced force bears to the weight of the moving body, positive or negative according to the direction of the force; so that by substituting f ' g for g in the equations of Article 533, those equations are transformed into the equations of motion of the body in question, h being taken to represent the distance traversed by it in a positive direction. In the apparatus known by the name of its inventor, Atwood, for illustrating the effect of uniform moving forces, this principle is applied in order to produce motions following the same law with those of falling bodies, but slower, by a method less liable to errors caused by friction than that of Galileo. Two weights, P and R, of which P is the greater, are hung to the opposite ends of a cord passing over a finely constructed pulley. Considering the masses of the cord and pulley to be insensible, the weight of the mass to be moved is P + R, and the moving force P R, being less than the weight in the ratio, DEVIATING AND CENTKIFUGAL FORCE. 491 Consequently the two weights move according to the same law with a falling body, but slower in the ratio of /to 1. 537. A Deviating Force, which acts unbalanced in a direction perpendicular to that of a body's motion, and changes that direc- tion without changing the velocity of the body, is equal to the rate of deviation of the body's momentum per unit of time, as the fol- lowing equation expresses : Q being the deviating force, W the weight of the body, W -r g its mass, v 2 its velocity, and r the radius of curvature of its path. In the case of an unresisted projectile, already mentioned in Article 534, the deviating force at any instant is that component of the body's weight which acts perpendicular to its direction of motion ; that is to say The well known expression for the radius of curvature of any curve whose co-ordinates are x and z is / dvy^ ' \ dx*J ' dx* r = \ dz "W v Consequently Qr = , which agrees with equation 1. In the case of projectiles, just described, and of the heavenly bodies, deviating force is supplied by that component of the mutual attraction of two masses which acts perpendicular to the direction of their relative motion. In machines, deviating force is supplied by the strength or rigidity of some body, which guides the deviating mass, making it move in a curve. A pair of free bodies attracting each other have both deviated motions, the attraction of each guiding the other; and their devia- tions of momentum are equal in equal times ; that is, their devia- tions of motion are inversely as their masses. In a machine, each revolving body tends to press or draw the body which guides it away from its position, in a direction from the centre of curvature of the path of the revolving body; and that tendency is resisted by the strength and stiffness of the guiding body, and of the frame with which it is connected. 538. Centrifugal Force is the force with which a revolving body reacts on the body that guides it, and is equal and opposite to the 492 PRINCIPLES OF DYNAMICS. deviating force with which the guiding body acts on the revolving body. In fact, as has been stated in Article 12, every force is an action between two bodies ; and deviating force and centrifugal force are but two different names for the same force, applied to it according as its action on the revolving body or on the guiding body is under consideration at the time. 539. A Revolving Simple Pendulum consists of a small mass A, suspended from a point C by a rod or cord C A of insensibly small weight as compared with the mass A, and revolving in a circle about a vertical axis A B. The tension of the rod is the resultant of the weight of the mass A, acting vertically, and of its centrifugal force, acting horizontally ; and therefore the rod Fig. 233. will assume such an inclination that height BC weight gr . radius AB centrifugal force v* ' ' where r = A B. Let T be the number of turns per second of the pendulum; then v = 2srTr; and therefore, making B C = h, h- 9 - g ' ~ .0-8154 foot 9-7848 inches = (in the latitude of London) -- - = - =^ - (2.) When the speed of revolution varies, the inclination of the pendu- lum varies, so as to adjust the height to the varying speed. 540. Deviating Force in Terms of Angular Velocity. If the radius of curvature of the path of a revolving body be regarded as a sort of arm of constant or variable length at the end of which the body is carried, the angular velocity of that arm is given by the expres- sion, = Let ar be substituted for v in the value of deviating force of Article 537, and that value becomes Q = DEVIATING FORCE. 493 In the case of a body revolving with uniform velocity in a circle, like the bob A of the revolving pendulum of Article 539, a - 2 T, where T is the number of revolutions per second, so that (3.) from which equation the height of a revolving pendulum might be deduced with the same result as in the last Article. 541. Rectangular Components of Deviating Force. First Demon- stration. Let O in fig. 234 be the centre of the circular path E F G H of a body revolving in a circle with an uniform velocity, through which centre draw rectangular axes, O X and O Y, in the plane of revolution. Let the angle ^ X O A, which at any instant the radius vector of the revolving body makes with the axis of x, be denoted by 6. Let A D x = r ' cos 0, and ) ^ ^ be the rectangular co-ordinates of the revolving body at any in- stant. Let Q. r , Qj,, be the components of the deviating force, parallel to O X and O Y respectively. Then from the obvious proportion between the magnitudes of those components, Q:Q,:Q y :: r : x : y, ..................... (2.) combined with the equation 2 of Article 540, follow the values of those components, y Those two components have the negative sign affixed, because they represent forces tending to diminish the co-ordinates x and y, to which they are proportional. Second Demonstration. The same result may be obtained, though less simply, by the second method described in Article 530, as fol- lows : Let intervals of time, t, be reckoned from an instant when the revolving body is at E. Then 6 = a t, and the values of the co-ordinates x and y, in terms of the time, are x = r cos at; y = rsinat ................. (4.) The components of the velocity of the body are, 494 PEINCIPLES OF DYNAMICS. dx . dy . = ar sin at', = ar cos at, ........... (5.) (t t U t the velocity parallel to each co-ordinate being proportional to the other. The components of the variation of motion are .(6.) = ar r cos a t a^x; dt - | - a 2 r sin a t a 2 y; d t" w which being multiplied by the mass , reproduce the components of the deviating force as before given in equation 3. 542. Straight Oscillation is the motion performed by a body which moves to and fro in a straight line, alternately to one side and to the other of a central point; and in order that this motion may take place, the body must be urged at each instant towards the central point. In most cases, the force so acting on the oscillating body is either exactly or very nearly proportional to its displacement, or distance from the central point of equilibrium ; that is to say, that force follows the law of one of the rectangular components of the deviat- ing force of a body revolving uniformly in a circle once for each double oscillation of the oscillating body. In fig. 234, let a body B, equal in weight to the body A, start at the same instant from E, and oscillate to and fro along the dia- meter E G, while A revolves in the circle E F G H. Then if B is ' urged towards the centre O with a force at each instant propor- tional to its distance from that point, and given by the equation being equal to the parallel component of the deviating force of A, B will accompany A in its motion parallel to O X ; both those bodies being at each instant in the same straight line B A || Y at the distance x T cos a t = r cos & ................ ..... (2.) from O : the velocity of B being at each instant equal to the par- allel component of the velocity of A j that is to say, arsmat arsmO; .............. (3.) and each double oscillation of B, from E to G and back again to E, OSCILLATION. 495 being performed in the same time with a revolution of A ; that is in the time "W where r is the semi-amplitude of the oscillation, O E = O Gr, Q is the corresponding greatest magnitude of the force urging the body towards O, being the same with the entire deviating force of A, and n is the number of double oscillations in a second. (The angle 6 = a t is called the PHASE of the oscillation.) The greatest value Q of the force which must act on B to pro- duce n double oscillations of the semi-amplitude r in a second, is given by the equation Wa*r being similar to equation 3 of Article 540. Revolution in a circle may be regarded as compounded of two oscillations of equal amplitude, in directions at right angles to each other. 543. Elliptical Oscillations or Revolutions Compounded of two straight oscillations of equal periods, but un- equal amplitudes, may be performed by a body urged towards a central point by a force pro- portional to its distance from that point. In fig. 235, let A be the position of the body at any instant ; let O A = $, and let the force urging the body towards O be F = b being a constant. Then the rectangular com- ponents of that force are F.= - *= WVy. . ;...(*) Fig. 235. the former force being suited to maintain a straight oscillation parallel to O X, and the latter, a straight oscillation parallel to O Y, the period of a double oscillation in either case being the same, viz. : j-v- according to equation 4 of Article 542. Hence let x l = O E = O G be the semi-amplitude of the former straight oscillation, and y l = 496 PRINCIPLES OF DYNAMICS. O F = O H that of the latter ; then at any instant the co-ordinates of the body will be ; y = ^ sin b t\ ................ (4.) = x cos which equations being respectively divided by x l and y 1} the results squared, and the squares added together, give -* + ;;: =!; ( 5 -) the well known equation of an ellipse described about O as a centre with the semi-axes x i} y v The components of the velocity of the body at any instant are dx dt=- b dy d t Xi sin 6 1 = b y: (6.) 544. A Simple Oscillating Pendulum consists of an indefinitely small weight A, fig. 236, hung by a cord or rod of in- sensible weight A C from a point C, and swinging in a vertical plane to and fro on either side of a central point D vertically below C. The path of the weight or bob is a circular arc, A D E. The weight W of the bob, acting vertically, may be resolved at any instant into two components, viz. : W cos ^z D C A = W EG CA' acting along C A, and balanced by the tension of the Fig. 236. rod or cord, and W sin ^ D C A = W =, O A. acting in the direction of a tangent to the arc, towards D, and un- balanced. The motion of A depends on the latter force. When the arc A D E is small compared with the length of the pendulum A C, it very nearly coincides with the chord ABE; and the horizontal distance A B, to which the moving force is propor- tional, is very nearly equal to the distance of the bob from D, the central point of its oscillations. Hence the bob is very nearly in the condition of straight oscillation described in Article 542 ; and the time which it occupies in making a double oscillation is there- OSCILLATING PENDULUM. 497 fore found approximately by means of equation 4 of that Article, viz. : where r denotes the semi-amplitude, and Q the maximum value of W = But if the length of the pendulum, C A, be made = I, \~/ -A. we have Q AB r =.. = max. -== y , nearly ; W \j A. v whence, approximately, for small arcs of oscillation, -j and 9 ' (i.) which being compared with equation 2 of Article 539, shows, i/ie length of a simple oscillating pendulum, making a given number of small double oscillations in a second, is sensibly equal to the height of a revolving pendulum, making tJie same number of revolutions in a second. When the amplitude of oscillation becomes of considerable mag- nitude, the period of oscillation is no longer sensibly independent of the length of the arc, but becomes longer for greater amplitudes, according to a law which can be expressed by an elliptic function, but which it is unnecessary to explain in this treatise. (See Le- gendre, Traite des Fonctions elliptiques, vol. i., chap, viii.) 545. Cycloidal Pendulum. In order that the oscillations of a simple pendulum may be exactly isochronous (or of equal duration) for all amplitudes, the bob must oscillate in a curve, the lengths of whose arcs, measured from its lowest point, are proportional to the sines of their angles of declivity at their upper ends, to which sines the moving forces at those upper ends are proportional. That this may be the case, the radius of curvature at each point of the curve must be proportional to the cosine of the declivity : the greatest radius of curvature, at the lowest point of the curve, being equal to I, as given by equation 1 of Article 544 ; and from Article 390, case 3, equation 6, it appears that such a curve is a cycloid, traced by a rolling circle whose radius is -.= 2K 498 PRINCIPLES OF DYNAMICS. It is well known that a cycloid is the involute of an equal and similar cycloid. Hence, in fig. 237, let C F, C G, be a pair of cycloidal cheeks, described by rolling a circle of the radius r Q on a horizontal line traversing C ; let C A be a flex- ible line, fixed at C, and having a bob at A, its length being I = 4 r Q = C D = the length of each of the semi- cycloids C F, C G. Then as the pendulum C A swings between the cycloidal cheeks, the bob oscillates in an arc of the cycloid F D G ; its double oscillations, for all amplitudes, have exactly the periodic time given by equation 1 of Article 544, being that of a revo- lution of a revolving pendulum of the height C D ; and the motion of the bob in its cycloidal path follows the law of straight oscillations described in Article 542. 546. Residual Forces. If two bodies be acted upon at every instant by unbalanced forces which are parallel in direction, and proportional to the masses of the bodies in magnitude, the varia- tions of the motions of those two bodies, relatively to a fixed body, whether by change of velocity or by deviation, are simultaneous and equal; so that their motion, relatively to each other, is the same with that of a pair of bodies acted upon by no force or by balanced forces ; that is, according to the first law of motion, Article 510, that motion is none or uniform. If two bodies, A and B, be acted upon by any unbalanced forces whatsoever, and if from the force acting on B there be taken away a force parallel to that acting on A, and proportional to the mass of B (in other words, if with the actual force acting on B there be combined a force equal and opposite to that which would make the motion of B change in the same manner with that of A), then the resultant or residual unbalanced force acting on B is that corre- sponding to the variations of the motion, o/*B relatively to A. This is the exact statement of the case of a body near the earth's surface. From the total attraction between the body and the earth is to be taken away the deviating force necessary to make the body accompany the earth's surface in its motion, by revolving in a circle round the earth's axis once in a sidereal day (Article 352). The residual force is the weight of the body, W = g m, which regulates its motions relatively to the earths surface. Thus the variations of the co-efficient g in different localities of the earth's surface, at different elevations, expressed by the formulae of Article 531, are due partly to variations of attraction, and partly to variations of deviating force. ACTUAL ENERGY. 499 When bodies are carried in a ship or vehicle, and are free to move with respect to it, then when the ship or vehicle varies its motion, the bodies in question perform motions relatively to the ship or vehicle, such as would, in the case of the uniform motion of the ship or vehicle, be produced by the application to the bodies of forces equal and contrary to those which would make them accom- pany the ship or vehicle in the variations of its motion. SECTION 3. Transformation of Energy. 547. The Actual Energy of a moving body relatively to a fixed point is the product of the mass of the body into one-half of the square of its velocity, or, as Article 533 shows, the product of the weight of the body into the Iwight due to its velocity ; that is to say, it is represented by m v 2 _ W v 2 T- ---Tg" The product m v*, the double of the actual energy of a body, is sometimes called its vis-viva. Actual energy, being the product of a weight, into a height, is expressed, like potential energy and work, m foot pounds (Article 513, 514). 548. Components of Actual Energy. The actual energy of a body (unlike its momentum) is essentially positive, and irrespective of direction. Let the velocity v be resolved into three components, d x dy d z -T-, M , -j- , parallel to three rectangular axes ; then the quantities (JL t> (JL u CL v of actual energy due to those three components respectively are _ _ 2g ' df' 27* 1? '' 2^ ' dff' But the square of the resultant velocity is the sum of the squares of its three components, or 2 _ _i_ i -~dt*^~~dt*^~dt*'> therefore the actual energy of the body is simply the sum of the actual energies due to the rectangular components of its velocity. 549. Energy of Varied motion. THEOREM I. A deviating force 'produces no change in a body's actual energy, because such force produces change of direction only, and not of velocity; and actual energy is irrespective of direction, and depends on velocity only. THEOREM II. The increase of actual energy produced by an un- balanced effort is equal to tJie potential energy exerted. This theorem is a consequence of the second law of motion, deduced as follows : 500 PRINCIPLES OF DYNAMICS. Let m W -4- g, be the mass of a moving body acted upon by an effort P, and a resistance H, the effort being the greater, so that there is an unbalanced effort P B, ; and in the first place let that unbalanced effort be constant. Then the body is uniformly acce- lerated ; and if its velocity at the beginning of a given interval of time A t is v l} and its velocity at the end of that interval v. 2 , the increase of the body's momentum is ^(.-,) = (P-B)Al .................. (1.) y Because of the uniformity of the acceleration of the body, its mean velocity is -^ l , and the distance traversed by it is Let both sides of equation 1 be multiplied by that mean velocity ; the following equation is obtained : now the first side of this equation is the increase of the body's actual energy, and the second is the potential energy exerted by the un- balanced effort; and those two quantities are equal. Q. E. D. When the unbalanced effort varies, let d s be taken to denote a distance in which it varies less than in any given proportion, and d ' v 9 the change in the square of the velocity in that distance ; then Wvdv^ ~~ or if * w $ 2 > denote the two extremities of a finite portion of the body's path, w (if _ i?} r*2 J|u_y = ^(P_B)l. ............ (3 A.) THEOREM III. The diminution of actual energy produced by an unbalanced resistance is equal to the work performed in moving against the resistance. This is a consequence of the second law of motion, demonstrated by considering H to be greater than P in the equa- tions of the preceding theorem ; so that equation 1 becomes ^( I _^) = (R-P)^; ................. (4.) equation 2 becomes TRANSFORMATION OF ENERGY. 501 "VV (^i vl) and equation 3 and 3 A become Wd'tf Wvdv (6.) 9 ,(6 A.) 550. Energy stored and Restored. A body alternately accelerated and retarded, so as to be brought back to its original speed, per- forms work by means of its retardation exactly equal in amount to the potential energy exerted in producing its acceleration ; and that amount of energy may be considered as stored during the accelera- tion, and restored during the retardation. 551. The Transformation of Energy is a term applied to such processes as the expenditure of potential energy in the production of an equal amount of actual energy, and vice versa. 552. The Conservation of Energy in Varied motion is a fact or principle expressed by combining the Theorems II. and III. of Article 549 with the definition of stored and restored energy of Article 550, and may be stated as follows : in any interval of time during a body's '/notion, the potential energy exerted, added to tlte energy restored, is equal to the energy stored added to tlie work per- formed. This principle, expressed in the form of a differential equation, is as follows : T> -, V? vdv /1X P ds~ -- Rds = Q; .................. (1.) y which includes equations 3 and 6 of Article 549; and in the form of an integral equation, (2.) 553. Periodical Motion. If a body moves in such a manner that it periodically returns to its original velocity, then at the end of each period, the entire variation of its actual energy is nothing ; and in each such period the whole potential energy exerted is equal to the whole work performed, exactly as in the case of a body moving uniformly (Article 517). 554. Measures of Unbalanced Force. From Articles 530 and 531, and from Article 549, it appears that the magnitude of an un- balanced force may be computed in two ways, either from the change of momentum which it produces by acting for a given time, PRINCIPLES OF DYNAMICS. or by the change of energy which it produces by acting along a given distance. Both those ways of computing are expressed in the following equation : g ' dt g ds ' and each is a necessary consequence of the other; yet in former times a fallacy prevailed that they were inconsistent and contra- dictory, and a" bitter controversy long raged between their respec- tive partizans. 555. Energy due to Oblique Force. It has already been stated in Chapter I. of this Part, and especially in Article 520, that if an unbalanced force F acts on a body while it moves through the dis- tance d s, making the angle 6 with the direction of the force, the product F cos f - ds represents the energy exerted, if 6 is acute, or the work performed, if is obtuse, during that motion. Now that product may be treated mathematically in two ways : either as the product of F cos 6 - P (or, as the case may be, F cos ( - ff) = K), the component of the force along the direction of motion, into ds, the motion ; or as the product of F, the entire force, into cos 6 - ds, the component of the motion in the direction of the force. The former method is that pursued in the preceding Articles ; but occasionally the latter may be the more convenient. For example, when the force F is either directed towards or from a central point, or is always per- pendicular to a given surface ; let z denote the distance of the body at any instant from the central point, or its normal distance from the given surface, as the case may be ; then dz = cos 6 ds ........................... (1.) is the component of the motion of the body in the direction of z. The force F is to be treated as positive or negative according as it tends to increase or diminish z. Then if v^ v 2 , be the velocities of the body, and e l9 g , its distances from the given point or surface at the beginning and end of a given interval, the change of its actual energy in that interval is (2.) and if F is either constant, or a function of z only, the velocity of v varies with z alone. This principle, as applied to the force of gravity near the earth's surface, has already been illustrated in Articles 533, 534, and 535 ; RECIPROCATING FORCE TOTAL ENERGY. for in that case, z denotes the elevation of the body above a given level, F W (because it tends to diminish #), and therefore ^~^ = z l -z tt . ...(3.) 4( as was formerly proved by another process. 556. A Reciprocating Force is a force which acts alternately as an effort and as an equal and opposite resistance, according to the direction of motion of the body. Such a force is the weight of a body which alternately rises and falls ; or the attraction of a body towards a point from which its distance periodically changes. Such a force is the force F in the last Article, when it is constant, or a function of z only ; and such is the elasticity of a perfectly elastic- body. The work which a body performs in moving against a reci- procating force is employed in increasing its own potential energy, and is not lost by the body. 557. The Total Energy of a body is the sum of its potential and actual energies. It is evident, that if at each point of the course of a moving body its total energy, or capacity for performing work, be added to the work which it has already performed, the sum must be a constant quantity, and equal to the INITIAL ENERGY which the body possessed before beginning to perform work. If. a body performs no work, its total energy is constant ; and the same is the case if its work consists only in moving itself to a place where its potential energy is greater, that is, moving against a reciprocating force ; and the increase of potential energy so obtained being taken into account, balances the work performed in obtaining it. Example I. If a body whose weight is W be at a height z^ above the ground, and be moving with the velocity v l in any direction, its initial total energy relatively to the ground is w (*. + ); (i-) v 2 of which W z l is potential and "W - actual. Supposing the body / to have moved without any resistance except such as may arise from a component of its own weight, which is a reciprocating force, to a different height z t above the ground, its total energy relatively to the ground is now being the same in amount as before, but differently divided between the actual and potential forms. 504 PKTNCIPLES OF DYNAMICS. Example II. Should the motion of the body be opposed by a resistance such as friction, which is not a reciprocating force, then the total energy in the second position of the body is diminished to Example III. Let a body oscillate (as in Article 542) in a straight line traversing a central point towards which the body is urged by a force varying as the distance from the point ; let #, be the semi- amplitude of oscillation, x the displacement at any instant, Q, the greatest value of the moving force, so that ---- ' is the value 00^ for the displacement x. Then when the body is at its extreme displacement, its actual energy is nothing ; and its total energy, which is all potential, is (4.) When the body is in the act of passing the central point, its poten- tial energy is nothing, and its total energy, which is now all actual, is in amount the same as before, viz. : 2g y being the maximum velocity. At any intermediate point, the total energy, partly actual and partly potential, is still the same, being Wv Qx Wv* . 2 ,Q,a;i Q^ ' ' ' where, as before, a = 2 ir n ; n being the number of double oscilla- tions in a second. For the elliptic oscillations of Article 543, the total energy of the body is at each instant the sum of the quanti- ties of energy due to the two straight oscillations of which the elliptic oscillation is compounded ; and for a body revolving in a circle, and urged towards the centre by a deviating force propor- tional to the radius vector, the total energy relatively to the centre is one-half actual and one-half potential, viz. : SYSTEM OF BODIES ANGULAR MOMENTUM. 505 SECTION 4. Varied Translation of a System of Bodies. 558. Conservation of Momentum. THEOREM. The mutual actions of a system of bodies cannot change tJieir resultant momentum. (Re- sultant momentum has been denned in Article 524.) Every force is a pair of equal and opposite actions between a pair of bodies ; in any given interval of time it constitutes a pair of equal and oppo- site impulses on those bodiesj and produces equal and opposite momenta. Therefore the momenta produced in a system of bodies by their mutual actions neutralize each other, and have no result- ant, and cannot change the resultant momentum of the system. 559. motion of Centre of Gravity. COROLLARY. The variations of the motion oftJie centre of gravity of a system of bodies are wlwlly produced by forces exerted by bodies external to the system ; for the motion of the centre of gravity is that which, being multiplied by the total mass of the system, gives the resultant momentum, and this can be varied by external forces only. It follows that in all dynamical questions in which the mutual actions of a certain system of bodies are alone considered, the centre of gravity of that system of bodies may be correctly treated as a point whose motion is none or uniform ; because its motion cannot be changed by the forces under consideration. 560. The Angular momentum, relatively to a fixed point, of a body having a motion of translation, is the product of the momen- tum of the body into the perpendicular distance of the fixed point from the line of direction of the motion of the body's centre of gravity at the instant in question ; and is obviously equal to the product of the mass of the body into double the area swept by the radius vector drawn from the given point to its centre of gravity in an unit of time. Let m be the mass of the body, v its velocity, I the length of the before-mentioned perpendicular ; then Wvl mvl = 9 is the angular momentum relatively to the given point. Angular momenta are compounded and resolved like forces, each angular momentum being represented by a line whose length is proportional to the magnitude of the angular momentum, and whose direction is perpendicular to the plane of the motion of the body and of the fixed point, and such, that when the motion of the body is viewed from the extremity of the line, the radius vector of the body seems to have right-handed rotation. The direction of such a line is called the axis of the angular momentum which it represents. The resultant angular momentum of a system of bodies is the resultant of all their angular momenta relatively to their 506 PRINCIPLES OF DYNAMICS. common centre of gravity ; and the axis of that resultant angular momentum is called the axis of angular momentum of the system. 561. Angular impulse is the product of the moment of a couple of forces (Article 29) into the time during which it acts. Let F be the force of a couple, I its leverage, and d t the time during which it acts, then is angular impulse. Angular impulses are compounded and resolved like the moments of couples. 562. Relations of Angular Impulse and Angular Momentum. THEOREM. TJie variation, in a given time, of the angular momentum of a body, is equal to the angular impulse producing that variation, and has the same axis. This is a consequence which is deduced from the second law of motion in the following manner : Conceive an unbalanced force F to be applied to a body m, and an equal, opposite, and parallel force, to a fixed point, during the interval d t ; and let I be the perpendicular distance from the fixed point to the line of action of the first force. Then the couple in question exerts the angular impulse Fldt. At the same time, the body m acquires a variation of momentum in the direction of the force applied to it, of the amount m dv = F dt' } so that relatively to the fixed point, the variation of the body's angular momentum is mldv = Fldt- ........................ (1.) being equal to the angular impulse, and having the same axis. Q. E. D. 563. Conservation of Angular momentum. THEOREM. TJie result- ant angular momentum of a system of bodies cannot be changed in magnitude, nor in the direction of its axis, by the mutual actions of the bodies. Considering the common centre of gravity of the system of bodies as a fixed point, conceive that for each force with which one of the bodies of the system is urged in virtue of the combined action of all the other bodies upon it, there is an equal, opposite, and parallel force applied to the common centre of gravity, so as to form a couple. The forces with which the bodies act on each other are equal and opposite in pairs, and their resultant is nothing ; there- fore, the resultant of the ideal forces conceived to act at the common centre of gravity is nothing, and the supposition of these forces does not effect the equilibrium or motion of the system. Also, the resultant of all the couples thus formed is nothing; therefore, the ;he ;he ACTUAL ENERGY OF A SYSTEM. OUI resultant of their angular impulses is nothing ; therefore, the result- ant of the several variations of angular momentum produced by those angular impulses is nothing ; therefore, the resultant angular momentum of the system is invariable in amount and in the direc- tion of its axis. Q. E. D. This theorem is sometimes called the principle of the conservation of areas. When applied to a system consisting of two bodies only, it forms one of the laws discovered by Kepler, by observation of the motions of the planets. In considering the relative motions of a system of bodies as depending on their mutual actions only, the axis of angular momen- tum may be treated as & fixed direction, as already stated in Article 348. A plane perpendicular to the axis of angular momentum is called by some writers the invariable plane. The nearest approach to an absolutely fixed direction yet known is the invariable axis of the discovered bodies of the solar system. 564. Actual Energy of a System of Bodies. THEOREM. The actual energy of a system of bodies relatively to a point external to the system, is the sum of the actual energies of the bodies relatively to their common centre of gravity, added to the actual energy due to the motion of the mass of the w/wle system with a velocity equal to that which its centre of gravity has relatively to the external point. Let the motion of each of the bodies, and of their common centre of gravity, relatively to the external point, be resolved into three rectangular components. Let m be any one of the masses, and u, v, w, the components of its velocity relatively to the external point ; let 2 m be the mass of the whole system, and U Q , V Q , w , the com- ponents of the velocity of its centre of gravity relatively to the external point. Conceive the motion of each of the bodies to be resolved into two parts; that which it has in common with tJie centre of gravity rela- tively to the external point, and that which it has relatively to the centre of gravity. The component velocities of the first part are U Q , V Q , W Q ; and those of the second part U UQ = U' ; V VQ = V' ', W WQ = W' \ so that the components of the whole motion of the body may be represented by u = u + u' ; v = v Q + v' ; w = w + w'. Then the actual energy of the system relatively to the external point is 1 2 m[(u + uj+ (t? + t/)'+ fa + wj] 508 PRINCIPLES OF DYNAMICS. which being developed, and common factors removed outside the sign of summation, gives I (ul + vl + w$) ' 2 m + U 2 ' 171 U 1 -f- V ' 2 ' m v' + W n 2 m W' + 1 2 m (u 12 + */ 2 + w' 2 ). But in Article 524 it has been shown, that the resultant momentum of a system of bodies relatively to their common centre of gravity is nothing; that is to say, 2' mu' = Q; 2 "niv 1 ; 2 mw' = ; so that the above expression for the actual energy of the system becomes simply | (ul + vl +wl) am + ia w(w' 2 + v' 2 + w/ 2 ); (1.) of which the first term is the actual energy of the whole mass of the system due to the motion of the centre of gravity relatively to the external point, and the second term is the sum of the actual energies of tJie bodies relatively to their common centre of gravity. Q. E. D. Those two parts of the actual energy of a system may be distin- guished as the external and internal actual energy. COROLLARY. The mutual actions of a system of bodies change their internal actual energy alone. 565. Conservation of Internal Energy. LAW. The total internal energy, actual and potential, of a system of bodies, cannot be changed by their mutual actions. This is a proposition made known partly by reasoning and partly by experiment. The total internal energy of a system is the sum of the total energies of the bodies of which it consists relatively to their common centre of gravity. It has been shown in Articles 549 to 557, that the total energy of a single body can be diminished only by performing work against a resist- ance which is not a reciprocating force ; in other words, against an irreversible or passive resistance. Now it has been proved by experiment, that all work performed against passive resistances is accompanied by the production of an equal amount of energy in a different form (as when friction pro- duces heat); therefore the total internal energy of a system of bodies cannot be changed by their mutual actions. Q. E. D. Although this law has become known in the first instance by experiment and observation, it can be shown <%, priori to be necessary to the stability of the universe. 566. Collision is a pressure of inappreciably short duration be- tween two bodies. The most usual problem in cases of collision is, when two bodies whose masses are given move before the collision in one straight line with given velocities, and it is required to find COLLISION. 509 their velocities after the collision. The two bodies form a system whose resultant momentum and internal energy are each unaltered by the collision ; but a certain fraction of the internal energy, ascertained by experiment, disappears as motion, and appears as heat. If the bodies were perfectly elastic, this fraction would be nothing ; but no actual body is perfectly elastic. Let mi, m 2) be the masses of the two bodies, and u l} u 2) their velocities before the collision, whose directions should be indicated by their signs. Then the velocity of their common centre of gra- vity is and this is not altered by the collision ; neither is the external energy, whose amount is The internal energy of the system of two bodies is m, (u, - U Q )* + m 2 (u s - u )* ^ 2 2 When the bodies strike together, this actual internal energy is expended in altering the figures of the bodies at and near their surface of contact, in opposition to their elastic force. So soon as the relative motion of the bodies has been thus stopped, the elastic force begins to restore their figures, and to cause them to fly asunder ; and if they were perfectly elastic, this would reproduce all the actual energy of relative motion given by the formula 3, so that the bodies would separate with velocities relatively to their common centre of gravity, equal and opposite to their original velocities relatively to that point ; that is to say, with the velocities relatively to the common centre of gravity, and the velocities v l = 2u -u 1 , V 2 =2u -u 2 , .................. (4.) relatively to the earth. But as a certain proportion, which may be denoted by 1 & 2 , of the internal actual energy takes the form of heat, owing to the imperfect elasticity of the bodies, the internal actual energy due to visible motion after the collision is k 2 n^ (% - u$ . k* m 2 (u 2 - u$ JF I -- ; ............... (5<) the velocities of the bodies, relatively to their common centre of gravity, after the collision, are "HO PRINCIPLES OF DYNAMICS. k (u Q - MJ), k (u Q - u 2 ) y and their velocities relatively to the earth are Vi = (l +b) UQ kui v 2 = (l +k)u ku z ............ (6.) Should the bodies be perfectly soft, or inelastic, k ; in which case Vl = v 2 = u - ............................ (7.) that is, the bodies do not fly asunder after the collision, but proceed together with the velocity of their common centre of gravity. 567. The Action of Unbalanced External Forces on a System of bodies, considered as a whole, is to vary the resultant momentum and the resultant angular momentum. It has been shown in Article 60, that every system of forces can be reduced to a single force and a couple. The system of forces applied to a system of bodies is to be reduced to a single force acting through the centre of gravity of the system, and a couple, as shown in equations 5, 6, 7, 8, of Article 60 ; then in a given interval of time, the variation of resultant momentum of the system is equal to and in the direc- tion of the impulse of the single resultant force, and the variation of angular momentum is equal to the angular impulse, and about the axis, of the resultant couple. To express this by general equations, let the components of the momentum of any mass m belonging to the system, whose rectan- gular co-ordinates are x, y, z, be m -= , m -=-- , m -j. Then the (JL t Ct/ L Cu it rates of variation of these components are d"x d 2 d 2 z Also, the rectangular components of the angular momentum of that mass are about , i ,_ _, JD ; about ,, ( _. Jf) j about *,-; .................. (2.) whose rates of variation are ('- PRINCIPLE OF D'ALEMBERT. 511 Let F,., F y , F,, be the components of the force externally applied to a point whose co-ordinates are x, y, z. Then by the equality of the resultant impulse to the variation of resultant momentum, , and by the equality of the resultant angular impulse to the varia- tion of the resultant angular momentum, (5.) The use of those equations is to determine the effect of a given system of external forces on a system of bodies when the relations amongst the motions of those bodies are known, without taking into consideration the internal forces acting between the bodies, which latter forces it is sometimes difficult or impossible to determine until the effects of the external forces have first been found. 568. Determination of the Internal Forces. When the relations which exist between the motion of the system as a whole, that is, its resultant momentum and angular momentum, and the motions of the several bodies of which it consists, are fixed by cinematical principles, then the motion of each body can be determined when the externally applied forces are known. T/ien if, from the force externally applied to each body at each instant, there is taken away the force required to produce the change of motion of the body which takes place at that instant, the remainder must be balanced by, and equal and opposite to, tJie internal force acting on the body in ques- tion ; and this, which is the PRINCIPLE OF D'ALEMBERT, serves to determine the internal forces. Using the notation of the last Article, the components of the internal force applied to a given body of the system are ~-F a ; m^|-F y ; m^-F,. 569. Residnai External Forces. If the resultant external force acting through the centre of gravity of a system of bodies be sup- 512 PKINCIPLES OF DYNAMICS. posed to be divided into parallel components, each applied to one of the bodies and proportional to the mass of the body to which it is applied, such will be the system of external forces required to make all the bodies of the system have equal and parallel motions at each instant in common with their centre of gravity. Then if the forces so determined be taken away from the forces actually applied to the several bodies, the residual external forces, being combined with the internal forces, will constitute those forces which regulate the motions of the bodies relatively to their com- mon centre of gravity considered as a fixed point. 513 CHAPTER III. ROTATIONS OF RIGID BODIES. 570. The Motion of a Rigid Body, or of a body which sensibly preserves the same figure, has already been shown in Part III., Chapter II., to be always capable of being resolved at each instant into a translation and a rotation ; and by the aid of the principles explained in Section 3 of that chapter, the component rotation can always be conceived to take place about an axis traversing the centre of gravity of the body, and to be combined, if necessary, with a translation of the whole body in a curved or straight path along with its centre of gravity. The variations of the momen- tum of the translation, whether in amount or in direction, are due to the resultant force acting through the centre of gravity of the body, and are exactly the same with those of the momen- tum of the entire mass if it were concentrated at that centre; the variations of the angular momentum of the rotation are due to the resultant couple which is combined with that re- sultant force. The variations of actual energy are due to both causes. When the translation of the centre of gravity of a rotating body, and its rotation about an axis traversing that centre, are known, the motion of every point in the body is determined by cinematical principles, which have been explained in Part III., Chapter II., Section 3 : so that by the aid of D'Alembert's principle (Article 568) the internal forces acting amongst the parts of the body can be completely determined. In the investigations of questions respecting the motions of rigid bodies, there are certain quantities, lines, and points, de- pending on the figures of the bodies, the mode of distribution of their masses, and the way in which their motions are guided, whose use facilitates the understanding of the subject and the computation of results, and which are related to each other by geometrical principles. These are, moments of inertia, radii of gyration, moments of deviation, and centres of percussion. Their geometrical relations are considered in the following sec- tion. 2L 514 PRINCIPLES OF DYNAMICS. SECTION 1. On Moments of Inertia, Radii of Gyration, Moments of Deviation, and Centres of Percussion. 571. The Moment of inertia of an indefinitely small body, or " physical point," relatively to a given axis, is the product of the mass of the body, or of some quantity proportional to the mass, such as the weight, into the square of its perpendicular distance from the axis : thus in the following equation : a.) 9 9 r is the perpendicular distance of the mass m, whose weight is "W, from a given axis; and the moment of inertia, according to the unit employed, is either I, or I -f- g ; the former, when the unit is the moment of inertia of an unit of weight at the end of an arm whose length is unity ; and the latter, when the unit is the moment of inertia of an unit of mass at the end of the same arm. For the purposes of applied mechanics, the former is the more convenient unit, and will be employed in this treatise. By an extension of the term " moment of inertia," it is applied to the product of any quantity, such as a volume, or an area, into the square of the distance of the point to which that quantity relates from a given axis, as has already been exemplified in Article 95, and in the theory of resistance to bending; but in the remainder of this treatise the term will be used in its strict sense, and accord- ing to the unit of measure already specified; that is, in British measures, moment of inertia will be expressed by the product of a certain number of pounds avoirdupois into the square of a certain number of feet. The geometrical relations amongst moments of inertia, to which the present section refers, are independent of the unit of measure. 572. The Moment of Inertia of a System of Physical Points, rela- tively to a given axis, is the sum of the moments of inertia of the several points ; that is, 1 = 2 - Wr 2 ........................... (1.) 573. The Moment of Inertia of a Rigid Body is the sum of the moments of inertia of all its parts, and is found by integration ; that is, by conceiving the body to be divided into small parts of regular figure, multiplying the weight of each of those parts into the square of the distance of its centre of gravity from the axis, adding the products together, and finding the value towards which their sum converges when the size of the small parts is indefinitely diminished. For example, let the body be conceived to be built up of rectangular MOMENT OF INERTIA. 515 molecules, whose dimensions are d x, d y, and d z, the volume of each dxdydz, and the weight of unity of volume w. Then r 2 w d x d y d z (1.) Hence follows the general principle which will afterwards be illustrated in special cases, that propositions relative to the geome- trical relations amongst the moments of inertia of systems of points are made applicable to continuous bodies by substituting integration for ordinary summation; that is, for example, by putting / / / for 2, and w 'dxdydz for W. 574. The Radius of Gyration of a body about a given axis is that length whose square is the mean of all the squares of the distances of the indefinitely small equal particles of the body from the axis, and is found by dividing the moment of inertia by the weight, thus, ~2 - W~ 2 - W * '" ' When symbols of integration are used, this becomes / / / r*w-dxdydz ~ (2.) r r r / / / w d x d y d z 575. Components of moment of Inertia. Let the positions of the particles of a body be referred to three rectangular axes, one of which, X, is that about which the moment of inertia is to be taken. Then the square of the radius vector of any particle is so that the moment of inertia round the axis of x is ! = * W^+s-W* 2 ; .................. (1.) that is to say, the moment of inertia of a body round a given axis may be found by adding together the sum of the products of the weights of the particles, each multiplied by the square of each of its distances from a pair of planes cutting each other at right angles in the given axis. In the same manner it may be shown that the moments of inertia of the same body round the other two axes are given by the equations I, = 2 W z* + 2 W x 2 - I, = 2 W x 2 -f 2 W 2/ 2 ...(2.) 516 PRINCIPLES OF DYNAMICS. 576. moments of Inertia Round Parallel Axes Compared.- THEOREM. The moment of inertia of a ~body about any given axis is equal to its moment of inertia about an axis traversing its centre of gravity parallel to the given axis, added to tJie moment of inertia about the given axis due to the whole mass of the body concentrated at its centre of gravity. Take the given axis for the axis of x, and any two planes tra- versing it at right angles to each other as the planes ofxy and z x; then, as in the preceding Article, I, = 2 W if + 2 W V. Let ?/ , ft , be the perpendicular distances of the centre of gravity of the body from the two co-ordinate planes before mentioned ; con- ceive a new axis to traverse that centre of gravity, parallel to the given axis; let two co-ordinate planes parallel to the original co-ordinate planes traverse that new axis; and let ?/, z', be the perpendicular distances of a given particle from those new co- ordinate planes. Then y = yo + y'', Z = Z Q + z'- } and introducing those values of the original co-ordinates into the value of I,., we find I, = 2 W (2/ + yj + 2 W (z + zj = +4) 2 W + 2 2/ 2 W y 1 + 2 Z Q 2 W z + 2 W (y'* + z' 2 ); but because y' and z' are the distances of a particle from planes traversing the centre of gravity of the body, and the preceding equation is reduced to the following : I, = + *5) 2 W + 2 - W (y' 2 4 O ............ (1.) which expresses the theorem to be proved. This theorem may be more briefly expressed as follows: Let I be the moment of inertia of a body about an axis traversing its centre of gravity in any given direction, and I the moment of inertia of the same body about an axis parallel to the former at the perpendicular distance r ; then I ........................ (2.) An analogous proposition for surfaces has been demonstrated in Article 95, Theorem Y. COROLLARY I. The radius of gyration (?) of a body about any MOMENTS OF INERTIA. 517 axis is equal to the hypothenuse of a right-angled triangle, of which the two legs are respectively equal to the radius of gyration of the body about an axis traversing the centre of gravity parallel to the given axis (? ), and to the perpendicular distance between these two axes (r ). That is to say, f = rl + tl ............................ (3.) COROLLARY II. The moment of inertia of a body about an axis traversing its centre of gravity in a given direction, is less than the moment of inertia of the same body about any other axis parallel to the first. COROLLARY III. The moments of inertia of a body about all axes parallel to each other, which lie at equal distances from its centre of gravity, are equal. 577. Combined Moments of Inertia. THEOREM. TJw combined moment of inertia of a rigidly connected system of bodies about a given axis, is equal to the combined moment of inertia which tlie sys- tem would have about the given axis, if each body were concentrated at its own centre of gravity, added to the sum of the several moments of inertia of the bodies, about axes traversing their respective centres of gravity, parallel to tJie given axis. Let W now denote the weight of one of the bodies, I its moment of inertia about an axis traversing its own centre of gravity parallel to the given common axis, and r the distance of its centre of gravity from that common axis. Then the moment of inertia of that body about the common axis, according to Article 576, equation 2, is Consequently, the combined moment of inertia of the system of bodies is 2l = 2 .Wr;+ 2 I ; ..................... (1.) Q. E. D. 57 8. Examples of foments of Inertia and Radii of Gyration of homogeneous bodies of some of the more simple and ordinary figures, are given in the following tables. In each case, the axis is supposed to traverse the centre of gravity of the body; for the principles of Article 576 enable any other case to be easily solved. The axes are also supposed, in each case, to be axes of symmetry of the figure of the body. In subsequent Articles, it will be shown what relations exist between the moments of inertia of the same body about axes traversing it in different directions. The column headed W gives the weight of the body; that headed I gives the moment of inertia; that headed ^, the square of the radius of gyration. The weight of an unit of volume is in each case denoted by w. BODY. Axis. W Io e 2 o Diameter Polar axis Axis, 2a Diameter Diameter Longitudinal axis, 2a Longitudinal axis, 2a Longitudinal axis, 2a Longitudinal axis, 2a Transverse diameter Transverse axis, 26 Transverse diameter Transverse diameter Axis, 2a Axis, 2a Diagonal, 26 4-rzmr 2 B*ur* 2r 2 II. Spheroid of revolution polar semi-axis a, equa- 5 2r 2 5 b^ + c- 5 2(r 5 r' 5 ) 2r 2 r 2 ^ III. Ellipsoid semi-axes, a. 3 15 IV. Spherical shell external radius r, internal/,.... V. Spherical shell, insensibly thin radius r, thick- 3 15 3 2wa(r 2 r' 2 ) 15 VI. Circular cylinder length 3 VII. Elliptic cylinder length 2a, transverse semi- axes J, c, VIII. Hollow circular cylinder- length 2a, external ra- dius r, internal /, IX. Hollow circular cylinder, insensibly thin length 2a, radius r, thickness c?r, X. Circular cylinder length 2a, radius r, 2 4 2-rwar 2 2irwabc 2wzt'a(r 2 / 2 ) 4-rwardr Swabc 4wabc 2 r 2 r 2 a 2 c 2 a 2 4^3 XI. Elliptic cylinder length 2a, transverse semi-axes 6, c , 6 XII. Hollow circular cylinder- length 2a, external ra- dius T internal / 6 XIII. Hollow circular cylinder, insensibly thin radius r, thickness dr, 6 I + 4a 2 (r 2 -r' 2 )| 3 8w?a6c(6 2 + c 2 ) 4 h 3 ? + 3~ XIV. Rectangular prism di- mensions 2a, 26, 2c, XV. Rhombic prism length 2a, diagonals 26, 2c,.... XVI. Rhombic prism, as above, 3 3 3 6 c 2 a 2 3 6 + 3 MOMENTS OF INERTIA. 519 579. Moments of Inertia found by Division and Subtraction. Each of the solids mentioned in the table of the preceding Article can be divided into two equal and symmetrical halves by a plane perpendicular to the axis. The radius of gyration of each of those halves is the same with that of the original solid. Each of the solids can also be divided into four equal and symmetrical wedges or sectors by planes traversing the axis ; and those which are solids of revolution can be divided into an unlimited number of such wedges or sectors. The radius of gyration of each such sector about the original axis, which forms its edge, is the same with that of the original solid. To find the radius of gyration of any such sector about an axis parallel to its edge, the original axis, and traversing the centre of gravity of the sector, let r be the distance of that centre of gravity from the original axis, e the radius of gyration of the original solid, and e' the radius of gyration of the sector about the new axis in question; then from Article 576, equation 3, it follows that Example. In case 15 of Article 578, the square of the radius of gyration of a rhombic prism about its & 3 + c longitudinal axis is found to be , b and c being the two semi-diagonals. Let fig. 238 represent such a prism, and let A be one end of its longitu- dinal axis, and BAB = 2b, OAC = 2 c, its two diagonals. Divide the prism into four equal right- angled triangular prisms by two planes traversing the diagonals and the longitudinal axis ; the radius of gyration of each of those prisms about that axis is the same with that of the original prism. Bisect B C in D, and join AD, in which take r = 1 B C == ^ - then E is the extremity of a longitudinal axis o traversing the centre of gravity of the triangular prism ABC, and the radius of gyration of that prism about that new axis is given by the equation 580. Moments of Inertia found by Transformation. The moment of inertia and radius of gyration of a body about a given axis are not changed by any transformation of its figure which can be effected by shifting its particles parallel to the given axis ; and the 520 PRINCIPLES OF DYNAMICS. radius of gyration is not altered by altering the dimensions of the body parallel to the axis in a constant ratio ; for example, in cases 1 and 2 of Article 578, the radius of gyration of a spheroid about its polar axis is the same with that of a sphere of the same equa- torial radius. If the dimensions of a body in all directions transverse to the axis are altered in a constant ratio, the radius of gyration is altered in the same ratio. If the dimensions of a body transverse to its axis, in two direc- tions perpendicular to each other, are altered in different ratios ; for example, if the dimensions denoted by y are altered in the ratio m, and the dimensions denoted by z in the ratio n, then the radius of gyration e of the original body is to be conceived as the hypo- thenuse of a right-angled triangle whose sides are, parallel to y, and parallel to z, and are given by the equations 2 ' * a-Ws* aW .(1.) and the radius of gyration j of the transformed body will be the hypothenuse of a new right-angled triangle whose sides are m n and n g ; that is to say, '2 (2-) This method may be exemplified by deducing the radius of gyration of an ellipsoid about any one of its axes (Article 578, case 3) from that of a sphere (ib., case 1). 581. The Centre of Percussion of a body, for a given axis, is a point so situated, that if part of the mass of the body were concen- trated at that point, and the remainder at the point directly oppo- site in the given axis, the statical moment of the weight so distri- buted (Article 42), and its moment of inertia about the given axis, would be the same as those of the actual body in every position of the body. In fig. 239 let XX be the given axis, and G the centre of gravity of the body. It is evident, in the first place, that the centre of percussion must be somewhere in the perpendi- cular C G B let fall from the centre of gravity on the given axis. Secondly, in order that the statical moment of the whole mass, concentrated partly at C, and partly at the centre of percus- Fig. 239. sion B (still unknown), may be the same with that of the actual CENTRE OF PERCUSSION. 521 body, the centre of gravity must be unaltered by that concen- tration of mass ; that is to say, the masses concentrated at B and C must be inversely as the distances of those points from G. Hence denoting the weights of those masses by the letters B and C respectively, and the weight of the whole body by W, we have the proportion W : C : B : : BC : GB : GO ................. (1.) Lastly, in order that the moment of inertia of the mass as supposed to be concentrated at B and C, about the axis X X, may be the same with that of the actual body, we must have B BC 3 = We = WfeJ + rS) ................. (2.) where r = G C, and { is the radius of gyration of the body about an axis parallel to X X and traversing G ; and substituting for B its value from equation 1, viz., B = Wr + B C, we find, for the dis- tance of the centre of percussion from the axis, and for its distance from the centre of gravity, GB = BC r = ^ ...................... (4.) r The last equation may also be expressed in the form which preserves the same value when GB and GC are inter- changed ; thus showing, that if a new axis parallel to the original axis XX be made to traverse the original centre of percussion, the new centre of percussion is the point C in the original axis. The proportion in which the mass of the body is to be considered as distributed between B and C takes the following form, when each of the last three terms of the proportion 1 is multiplied by r = GO : W:C:B: : & + i* : & : 1$ .................. (6.) The preceding solution is represented by the following geometrical construction : Draw G D J_ C G and - & ; join C D, perpendicu- lar to which draw D B cutting C G produced in B ; this point is the centre of percussion. Also, C D = e, the radius of gyration about X X ; and D B is the radius of gyration about an axis traversing B parallel to X X. 522 PRINCIPLES OF DYNAMICS. If C E be taken = C D, E is sometimes called the CENTRE OF GYRATION of the body for the axis X X.* 582. No Centre of Percussion exists when the axis traverses the centre of gravity of the body. In that case, the statical moment of the body is nothing ; and an equal mass, concentrated and uni- formly distributed round the circle BBB, whose radius is ^ , the radius of gyration, or at a set of symmetrically arranged points in p.j 240 that circle, has the same moment of inertia with the actual body. 583. moments of Inertia about Inclined Axes. The object of the present Article and the remaining Articles of this section is to show the relations which exist amongst the moments of inertia of a body about axes traversing a fixed point in it in different direc- tions. The mathematical processes which it is necessary to employ for that purpose, though not very abstruse, are somewhat complex ; and the reader who wishes to study the more simple parts of the subject only, may take the conclusions for granted. It has already been shown in Article 575 that the moment of inertia of a body about a given axis denoted by #, is given by the equation ; .......................... (1.) in which, for the sake of brevity, 2 W has been replaced by the single symbol S. The fixed point being the origin of co-ordinates, let S R 2 be the sum of the products of the weight of each particle into the square of its distance from that point ; a sum which is independent of the directions of the axis. Then because R 2 = x 2 + f/ 2 + z 2 , the moments of inertia of the body relatively to three rec- tangular axes may be expressed as follows : I, = SR 8 -Sa 2 ; I y = SR 2 -S2/ 2 ; I^SR'-S* 2 ........ (2.) Further, let the three sums of the weights of the particles of the body, each multiplied by the product of a pair of its co-ordinates, be thus expressed : (3.) These will be called moments of deviation. Now, let three new rectangular axes of co-ordinates, denoted by x', 2/, zf, traverse the same fixed point in the body; let the angles which they make with the original axes be denoted by * As to the centres of percussion and gyration, and other remarkable points in a rigid body, see a memoir by M. Poinsot in Liouvillds Journal for 1857. MOMENTS OF INERTIA. 523 A , A , A , xx,xy, xz', A , A , A , zx, zy,zz' ; Then for any given particle, the new co-ordinates are thus expressed in terms of the original co-ordinates : A A A x = x cos xx + y - cosyx' -}- z ' cos zx', ............ (5.) and analogous equations for y' and *' ; and the original co-ordinates are thus expressed in terms of the new co-ordinates : A A A x = x cos x x + y cos x y' + z 1 ' cos x z'- } &c ....... (6.) The nine angles of equation 4 are connected by the relations : that the sum of the squares of the cosines of any three angles in one line, or in one column, is unity ; for example, cos 2 xx r -\- cos 2 xi/ + cos 2 xz' = 1; ................. (7.) and that the sum of the three products of the pairs of cosines of the angles in a pair of lines, or a pair of columns, is nothing ; for example, A A A A A A , cos y x' ' cos z x' -f- cos y y cos z y ~\- y z' ' cos z x = 0....(8.) A relation deduced from the preceding is this, that the cosine of each angle is equal to the difference between the binary products of the cosines of the four angles, which are neither in the same line nor in the same column with the first, these binary products being taken diagonally ; for example, A A A A A cos xx 1 = cos yy' -cos zz 1 cos yz 1 'cos zy .......... (9.) and similarly for the other cosines. Now, if for the new co-ordinates x", y', z*, in the six integrals, S^ 2 , S7/ 2 , S**, SyV, S*V, Srfy', there are substituted their values in terms of the original co-ordi- nates, as given by equation 5 for x', and analogous equations for y' and #', there are obtained the six expressions for those integrals relatively to the new axes, in terms of the integrals relatively to the original axes, and of the cosines of the nine angles between the 524 PRINCIPLES OP DYNAMICS. new and the original axes ; but it is unnecessary here to write those equations at length, for they are precisely similar to the equations of transformation in Article 106 (pages 92, 93), substituting only Sa 2 , Sy, S;s 2 , Syz, Szx, Secy, for p M , p w p zt) p y! , p zx , p ty , and making the like substitutions in the symbols referring to the new co-ordinates. 584. Principal Axes of Inertia. THEOREM. At each point in a body there is a system of three rectangular axes, for which the moments of deviation are each equal to nothing. Supposing such a set of axes to exist, let co-ordinates parallel to them be denoted by x lt y l9 z^. Then the property which they are required to have is expressed by the equations 8^ = 0; 8*^ = 0; Sa? l y 1 = (1.) Co-ordinates parallel to a set of axes, for which the integrals See 2 , &c., have been determined, being denoted by x, y, z, we have for each particle, A A .A x = x l cos x Xi + 2/i cos x y l -f- z l cos x Zi ; A A A x l x =. xl cos x Xi -f Xi yi cos x y l + x t z l cos x z { ; and consequently, S x l x = cos x Xi ' S x\ + cos x y l S a^ y^ + cos xz^ S z l x l ; but because of the conditions expressed by the equations 1, this is reduced to Sajj x = cos xx l So? 9 ; (2.) and by similar reasoning it is shown that Sx l y = cosyx l -$x 2 l ; ) ^ ...(2 A.) S Xi z = cos z x l ' S x\. ) Now, from the equation A A A #1 = x cos x a-, + y cos yx l -\-z cos z x^ are deduced the following values of the integrals in the equations 2, 2 A : AXES OF INERTIA. 525 Sx l x = cos x HI . 80^ + cos y x ' S x y + cos zx^ ' S z x ; $x l y = cos x x l * S x y + cos y x l S ^ + cos z #1 S y z S Xi z cos ic ^ S a; + cos y a^ S y z + cos a^ S z 2 . Subtracting the equations 2 and 2 A from these, we find the fol- lowing equations : cos x x l (S x 2 - S as?) + cos y x l S # y + cos z x l ' S z x = ; 1 cos a? ! S a? i/ + cos y x l (S y 2 - S ajj) + cos z x l S y z = ; } (3.) cos a; O5i S z x + cos 2/0^ S y z + cos z x\ (S ^ 2 S xi) = 0. J The elimination of the three cosines from these three equations leads to the following cubic equation : (Sa;?) s -A(S^) 2 -{-B-Sa?-C = 0; ............ (4.) in which the co-efficients have the following values : B = S y 2 S z 2 + S - S x* + S y? S y z S x 2 (S y zf S f - (S z xf S a 8 (S a y) 2 . (5.) It is evident that A is always positive. By considering the terms of which B is composed, it can be shown that it is equivalent to S (y * - * yj + S(*x'-x O 2 + S (x y' - y aTf ; x t y, z, x', y', z', being the co-ordinates of a pair of different particles, and the particles being taken in pairs in every possible way ; and by considering the terms of which C is made up, it can be shown to be equivalent to S (x y' z" + tf y" z + x" y z x y" z' x" y' z x' y z") 2 ; in which the letters without accents, with one accent, and with two accents, denote the co-ordinates of a set of three different particles, and the particles are taken in triplets in every possible way. Hence B and C, being both sums of squares, are positive, as well as A; and the cubic equation 4 has three real positive roots, corresponding to the three rectangular axes which satisfy the con- ditions of equation 1. These roots are the values of S x\, S ?/?, S z\ ; 526 PRINCIPLES OF DYNAMICS. and their existence proves the existence of the three rectangular PRINCIPAL AXES OF INERTIA. - Q. E. D. The angles which any one of the principal axes makes with the three original axes are given by the following equations, which are deduced from the equations 3 : A A A cos x a?i : cos y x l : cos z Xi (S a* S^Sajy Similar equations, substituting y l and z l successively for x lt give the ratios of the other two sets of cosines. From the properties of the roots of equations, it follows, that the co-efficients of the cubic equation 4 have the following values in terms of the integrals S x\, &c. : S^ 2 =SR 2 as before; and hence it appears, that the functions of the six integrals S # 3 , /, with those semi-axes. Then it is well known that 1 COS 2 ot . COS 2 /3 . COS 2 f a 2 b 2 c 2 (4.) and by comparing this with equation 3 it is made evident, that if an ellipsoid be constructed whose semi-axes are in direction the principal axes of the body at a given point, and represent in magni- tude the reciprocals of the square roots of the moments of inertia about those axes respectively, as shown by the equations 1 1 1 = 1 = < - .............. (5 - } then will the reciprocal of the square of the semidiameter of that ellipsoid in any direction represent the moment of inertia about an axis traversing the origin in that direction, as expressed by the equation Such an ellipsoid, when described about the centre of gravity of the body as a centre, is called by M. Poinsot the central ellipsoid. If I 15 I 2 , I 3 , be ranged in their order of magnitude, it is evident that the greatest of them, I lt is the greatest moment of inertia of the body about any axis traversing the fixed point ; that the least, I 3 , is the least moment of inertia about any such axis; and that the intermediate principal moment of inertia, I a , is the least moment of inertia about any axis traversing the fixed point perpendicular to the axis of I 3 , and the greatest moment of inertia about any axis traversing the fixed point perpendicular to the axis of I x . Should two of the principal moments of inertia be equal, as I 2 = I 3 , the ellipsoid becomes a spheroid of revolution : all the mo- ments of inertia about axes traversing the fixed point in the plane of the axes of I 2 and I 3 are equal; and the moments of inertia about all axes traversing the fixed point and equally inclined to the axis of T! are equal. In this case equation 3 becomes 1 = 1, cos 2 + I 2 sin 2 * .................... (7.) If all three principal moments of inertia are equal, the ellipsoid becomes a sphere, and the moments of inertia are equal about all axes traversing the fixed point. Suppose the fixed point in the first place to be the centre of 02b PRINCIPLES OF DYNAMICS. gravity of the body, whose weight is W, and that I 0l , I^, I 03 , are the principal moments of inertia about rectangular axes traversing it. Let a new fixed point be taken whose distance from the centre of gravity is r , in a direction making the angles a, /3, y, with the principal axes at the centre of gravity. Then with respect to a set of rectangular axes traversing the new point parallel to the original axes, the new moments of inertia are I, - Io l2 + W r* - (1 -cos 2 /3); (8.) and there are at the same time moments of deviation represented by S y z = W r\ ' cos /3 cos y ', S z x = W rl cos y cos *j/q\ r cos cos $ ; so that the principal axes at the new point are not parallel to those at the centre of gravity, unless two at least of the direction cosines of r are null; that is to say, unless the new point is in one of the original principal axes, when all the moments of deviation vanish, and the new axes are parallel to the original axes. 586. The Resultant Moment of Deviation about a given axis is represented by the diagonal of a rectangular parallelogram of which the sides represent the moments of deviation relatively to two rectangular co-ordinate planes traversing the given axis. Let the principal axes and moments of inertia at a given point be known, and let three new axes of moments, denoted by x, y, z, be taken in any three rectangular directions making angles with the original axes denoted as in the equations of Article 583. Then the moments of deviation in the new co-ordinate planes are S y z = cos y x t cos z x^ S #f -j- cos y y l cos z yi S y\ A A + cos y Zi cos *i 8 2& .................... (!) and similar equations for S z x, and S x y, mutatis mutandis. Sub- stituting for S #?, &c., their values, SB, 2 Ij, &c., and observing that A A A A A A cos y x 1 ' cos z x l -{- cos y y ' cos z y -f~ cos y z v cos z z 1 = 0, those equations become A A A A &yz= Ij cos y x l ' cos z x l 1 2 * cos y y-^ cos z y l -I 3 cosyzi -cos zzi, ..................... (2.) MOMENT OP DEVIATION UNIFORM ROTATION. 529 and similar equations, mutatis mutandis, for Szx, Sxy; from which, by the aid of relations amongst the direction cosines already stated in Article 583, the following value is found for the resultant moment of deviation about one of the new axes, such as x: c A A A I K, J [If cos 2 x x { + II cos 2 x ^ + If cos 2 x z l (Ij cos 2 xx l + 1 2 cos 2 xyi + I 3 cos 2 xz^f\ ; f (3.) A A A = ,J \Il cos 2 x #! + II ' cos 2 xy^ + Il cos 2 x z^ I* This equation, expressed in terms of the axes of the ellipsoid of inertia, becomes as follows : , eosxy 1 , -- but the positive part of this expression is well known to be the value of --5 . where n represents the normal let fall from the centre s 2 n 2 of the ellipsoid of inertia upon a plane which touches the ellipsoid at the point where it is cut by the new axis x. Hence in which it is to be observed, that Js^-n 2 represents the length of the tangent to the ellipsoid, from the point of contact to the foot of the normal. Also, let 6 be the angle between the normal n and the semidiameter s ; then Js 2 n 2 :n = tan 4, and (6.) SECTION 2. On Uniform Rotation. 587. The momentum of a body rotating about its centre of gravity is nothing, according to the principle of Article 524. As every motion of a rigid body can be resolved into a translation, and a rotation about its centre of gravity, the rotation will be supposed to take place about the centre of gravity of the body throughout this section. 588. The Angular Momentum is found in the following manner : Let x denote the axis of rotation, and y and z any two axes fixed in the body, perpendicular to it and to each other. Let a be the 530 PRINCIPLES OF DYNAMICS. angular velocity of rotation. Then the velocity of any particle W, whose radius vector is r = J y* + ?, is a r = a J y* + *?', and the angular momentum of that particle, relatively to the axis of rotation, is W a r> Wa. _ being the product of its moment of inertia into its angular velocity, divided by g, because of the weights of the particles having been used in computing the moment of inertia. Now let a line, parallel to the radius vector of the particle, be drawn in the plane of y and * ; the distance of that line from the particle is a?, and the angular momentum of the particle relatively to that line is W W a r x = a 9 9 and this may be resolved into two components ; one relatively to the axis of y, W a z x and the other relatively to the axis of , W a x y and these are equal respectively to the angular velocity divided by the acceleration produced by gravity in a second, multiplied by the moments of deviation of the particle in the co-ordinate planes of z x and x y. Hence it appears that the resultant angular momentum of the whole body consists of three components, viz. : Relatively to the axis of rotation, a (S y~ + S z 2 ) = - I, and relatively to the transverse axes, a a _ - S z xj - - S x y; 9 9 and if lines proportional to those three components be set off upon the three axes, the diagonal of the rectangle described upon them ANGULAR MOMENTUM. 531 will represent in direction the axis, and in length the magnitude, of the resultant angular momentum. It follows that the axis of angular momentum of a rotating body does not coincide with the axis of rotation, unless that axis is an axis of inertia; in which case the moments of deviation are each equal to nothing, and the resultant angular momentum is simply the product of the moment of inertia about the axis of rotation into the angular velocity, divided by g. Now let the axes of inertia be taken for axes of co-ordinates, and let the axis of rotation make with them the angles a, /3 3 y. Resolve the angular velocity a about that axis into three components about the axes of inertia a cos ; a cos ft ; a cos y ; then the angular momenta due to those three components are respectively a _ a T a T - Ii cos a; - I 8 cos /3; - I 8 COS yj 9 y y the resultant angular momentum is A = -- J[I* cos 8 * + IJcos'0 + II cos 8 y}; ............ (2.) and the axis of angular momentum makes with the axes- of inertia the angles whose cosines are a Ii cos a, a I B cos a I 3 cos y ~7X~ ~iA~ ~TA- Now, as already shown in Article 586, the quantity whose square root is extracted in equation 2 is the reciprocal of the product of the squares of the semidiameter and normal of the ellipsoid of inertia ; and by inspecting the equations of Article 586, it is evident, that the square root itself, in equation 2 of this Article, is the resultant of the moment of inertia and moment of deviation proper to the axis of rotation ; so that equation 2 may be expressed in the following form : n being, as before, the normal, and s the semidiameter of the ellipsoid of inertia at the point cut by the axis of rotation; for which the moments of inertia and of deviation are I and K. Further, the direction cosines of the axis of angular momentum, in the formula 3, which may otherwise be expressed as follows : PRINCIPLES OF DYNAMICS. I 3 COS I 2 COS ft I s COS y _ , > ^(P + Ky ^(P + Ky are the direction cosines of the normal of the ellipsoid of inertia. Hence the axis of angular momentum at any instant is in the direc- tion of the normal let fall from the centre of the ellipsoid of inertia upon a plane touching that ellipsoid at the end of that diameter which is tJie axis of rotation; and the angular momentum itself is directly as tlw angular velocity of rotation, and inversely as the product of the normal and semidiameter. The angle between the axes of rotation and of angular momentum is the angle already denoted by & in Article 586, whose value is given by the equation cos 6 = - = x/ Fig. 241. ^A :OB : By the following geometrical construction, the preceding prin- ciples are represented to the eye : In fig. 241, let O be the point about which the body rotates, and A B C A B C its ellipsoid of inertia, whose semi-axes have the proportions 1 1 1 Let O R be the axis of rotation, whether permanent or instanta- neous, O R being the semidiameter of the ellipsoid of inertia. Let R, T be part of a plane touching the ellipsoid at R, and O N a normal upon that plane from O. Then the moment of inertia, the moment of deviation, and their resultant, the total moment, have the following proportions : I :K + K 2 RN 1 (8.) OR 2 OR 2 -OISr OR-ON the direction of the axis of angular momentum is O N ; and its amount is proportional to - . 589. The Actual Energy of Rotation of a body rotating about its ACTUAL ENERGY FREE ROTATION. 533 centre of gravity, being the sum of the masses of its particles, each multiplied into one-half of the square of its velocity, is found as follows: a being the angular velocity of rotation, the linear velo city of any particle whose distance from the axis of rotation is r, is v = ar\ and the actual energy of that particle, its weight being W, is Wv 2 WaV being the moment of inertia of the particle multiplied by -. Hence for the whole body the actual energy of rotation is that is to say, actual energy bears the same relation to angular velo- city and moment of inertia that it does to linear velocity and weight. Referring again to fig. 241, it appears that the actual energy of rotation is proportional to .(3.) 2g -OR 2 Conceive, as in the last Article, the angular velocity a to be re- solved into three components about the three axes of inertia respectively, viz. : a cos at, a cos /3, a cos y ; then the quantities of actual energy due to those three component rotations are a 2 Ii cos 2 oc, a z I 2 cos 2 /3 a? I 3 cos 2 y ~w ~w ~~z dt, (2.) which represents energy exerted or work performed, according as the couple acts with or against the rotation. When the direct couples applied to a rotating body are balanced, the actual energy of the body remains constant, the potential energy exerted in any interval of time is equal to the work performed ; that is 2- Mcos//, of which the former produces variation in tJte amount of angular momentum, and the latter, deviation of the axis of angular momen- tum, according to the following laws : = M cos *; A = M sin *; ............. (2.) d t d t in the latter of which equations, d & denotes the angle through which the axis of angular momentum deviates in the indefinitely small interval d t } in the plane which contains that axis and the axis of the couple M, and in a direction towards the latter axis. This equation of deviation of angular momentum has in fact been already employed in Article 592, to find the deviating couple required in order to fix the axis of rotation, when that differs from the axis of angular momentum. The equations 1, or their equivalents 2, are not of themselves sufficient to determine the variations of motion of a body rotating without a fixed axis; for in such a body, the angular momentum may change by a change of the direction of its aods relatively to the body, as well as by a variation of amount, or a deviation of its axis in absolute direction. This is expressed by putting for the angular momentum its value in terms of the moments of inertia and devia- / 1 2 I K 2 tion relatively to the instantaneous axis, viz., A = - - ; when the equations 1 take the following form : g M x = < a cos A ^/P + K a I ; and analogous equations for dt ( ......................... (3.) while the equations 2 become 540 PRINCIPLES OF DYNAMICS. .(4.) It is therefore necessary to have an additional equation to complete the data for the solution of the problem ; and this is afforded by the law of the conservation of energy, in virtue of which the actual energy stored or restored by the rotating body is equal to the energy exerted or consumed by the unbalanced couple, according as it acts with or against the rotation, as the following equation expresses, where

, M, a, >. For brevity's sake, let the substitute for a 2 be thus expressed : GYRATION SINGLE FORCE ROTATION WITH TRANSLATION. 543 then by transforming equation 4 of Article 542, it appears that the number of double gyrations per second is .(3.) which is independent of the semi-amplitude i^ so long as MI is pro- portional to i^, and I is constant. This constitutes isochronism, and is the property aimed at in the balance wheels of watches, where I is the moment of inertia of the wheel, and the couple is derived from the elasticity of the balance spring. The equations 2 and 3 being transformed, give for the angle and angular velocity of displacement at any instant, i = i-L cos k t ', di ................ (4.) a = = -ki l sin kt; a t and the maximum couple Mj, in terms of the number of double oscillations per second n, is given by the equation 9 9 599. A single Force applied to a body with a fix*ed axis causes the bearings of the axis to exert a pressure equal, opposite, and parallel ; so that if the line of action of the force traverses the fixed axis, it is balanced ; and if not, a couple is formed whose moment is the product of the force into its perpendicular distance from the axis, and whose effects are such as have been already described. SECTION 4. Varied Rotation and Translation Combined. 600. General Principles. All rotation of a body about an axis, fixed or instantaneous, which does not traverse the centre of gravity of the body, is to be considered as compounded of rotation about a parallel axis traversing the centre of gravity, and translation of the centre of gravity with a velocity equal to the product of the angu- lar velocity into the distance of the centre of gravity from the actual axis of rotation. Consequently, every variation of the motion of a body, which consists in a variation of the angular velocity about an axis, fixed or instantaneous, and not traversing the centre of gravity, is to be considered as producing a change of the momentum, which is the product of the mass of the entire body into the velocity of its centre of gravity, and a simultaneous change of the angular momentum due to the rotation of the body with the given angular velocity 544 PRINCIPLES OF DYNAMICS. about an axis traversing its centre of gravity parallel to the actual axis of rotation ; and the force required to produce the given varia- tion of motion will be the resultant of the force required to produce the change of momentum, applied at the centre of gravity, and the couple required to produce the change of angular momentum. 601. Properties of the Centre of Percussion. In tig. 239, Article 581, page 520, let G be the centre of gravity of a rigid body whose weight is W, X X the axis about which, in the interval' dt, a change of angular velocity denoted by d a takes place, and G C = r , the perpendicular distance of the centre of gravity from that axis. Then the force, in a direction perpendicular to the plane of X X and G C, required at G to produce the change of momentum, is Wr^ gdt and the couple required to produce the change of angular momen- tum due to the change of angular velocity d a about the axis G D X X is and the resultant of that force and couple (according to Article 41) is a force acting in the same plane with them, parallel and equal to F, and in the same direction, but acting through a point whose distance from G, in a direction opposite to G C, is that is, the resultant of the force and couple is a single force F act- ing through the centre of percussion B corresponding to the given axis. (See Article 581, equation 4.) Now suppose, as in Article 581, that the weight of the body is distributed in two rigidly connected masses, one concentrated at C and the other at B, and having their common centre of gravity still at G. Then in producing the same change of angular velo- city d a about the axis X C X, the momentum of C is unchanged, while that of B undergoes the change 9 9 being the exact change of momentum already given in equation 1 ; a consequence, indeed, of the fact, that the centre of gravity is not changed by the concentration of the masses at B and C ; and to CENTKE OF PERCUSSION FIXED AXIS DEVIATING FORCE. 545 produce this change of momentum in the interval d t, there is re- quired the same force F applied at B, which has already been found; which proves the following THEOREM I. If the mass of a body be conceived to be concentrated at two rigidly connected points, one at a given axis, and the other at tJie corresponding centre of percussion, so as not to alter the position of the centre of gravity of the body, the force required to produce a given change of angular velocity in the body about tJie given axis is the same, in magnitude, direction, and line of action, with that required to produce the corresponding change of motion in that part of the mass which is conceived to be concentrated at the centre of percussion. This proposition might also have been arrived at by considering THEOREM II. If a body rotates about a given axis not traversing its centre of gravity, and the mass of that body be conceived to be con- centrated at the axis of rotation and centre of percussion so as not to alter the centre of gravity, the momentum, the angular momentum, and tJie actual energy of the body are not changed by that concentra- tion of mass. For the centre of gravity being unchanged, the momentum is unchanged; and because (by the definition of the centre of percus- sion) the moment of inertia about the axis of rotation is unchanged, the angular momentum and actual energy are unchanged. Q. E. D. COROLLARY. From Theorem I., and from equation 5 of Article 581, it follows, that the action of an impulse upon a free body at either of the points B or C, produces a rotation about an axis tra- versing the other point. 602. Fixed Axis. When the axis of rotation X X is fixed, an impulse applied to the centre of percussion B, in a direction per- pendicular to the plane B X X, simply alters the angular velocity according to the principles explained in the last Article, without causing any additional pressure between the axis and its bearings. But should the force giving the impulse not traverse the centre of percussion, or traverse it in a different direction, it is to be resolved by the principles of statics into two components, one traversing the centre of percussion in the required direction, and the other tra- versing the axis of rotation; when the former will produce change of motion, and the latter will be balanced by the resistance of the bearings of the axis. 603. The Deviating Force of a body rotating about a fixed axis not traversing its centre of gravity is the resultant of the deviating force due to the revolution of the whole mass conceived as concen- trated at its centre of gravity, found as in Article 540, combined with the deviating couple due to the rotation of the body with the same angular velocity about a parallel axis traversing the centre of gravity, found as in Article 592. This resultant deviating force is 546 PRINCIPLES OF DYNAMICS. supplied by the resistance of the bearings of the axis, and an equal and opposite CENTRIFUGAL FORCE is exerted by the axis against the bearings. 604. A Compound Oscillating Pendulum is a body supported by a horizontal fixed axis, about which it is free to swing under the action of its own weight, its centre of gravity not being in the axis. Now, by Article 601, Theorem II., the momentum and angular momentum of the body are at every instant the same as if its mass were concentrated at the axis and at the centre of oscillation in the proportions given by Article 581, equations 1 and 6 ; and by the definition of the centre of oscillation, the statical moment of the weight of the body with respect to the axis, being the couple which causes the motion, is in every position the same as if the mass were concentrated in these proportions ; therefore, the motion of the body is exactly the same as if it were so concentrated ; that is to say, it oscillates in the same time and according to the same laws, with a simple oscillating pendulum as defined in Article 544, whose length is the distance from the axis X C X to the centre of oscillation B, as given by equation 3 of Article 581, viz. : ^H-r .......................... (1.) Such a simple pendulum is called the equivalent simple pendulum. It is obvious that, for a given body swinging about all possible axes parallel to a given direction in the body, the shortest equiva- lent simple pendulum is that whose length is the minimum value of BC as given by the above equation. That minimum length corresponds to the condition, *o = r o > } whence, > ....................... (2.) min. B~C = 2 eo J that is to say, the least period of oscillation of a pendulous body takes place when the distance of its centre of gravity from its axis is equal to the radius of gyration about a parallel axis traversing the centre of gravity; and the length of the equivalent simple pen- dulum is double of that radius of gyration. If for a given direction of axis, a pair of points be so related that each is the centre of percussion for an axis in the given direction traversing the other (as shown by Article 581, equation 5), then the period of oscillation about either axis is the same. From the properties of the centre of percussion explained in this Article, it is sometimes called the CENTRE OF OSCILLATION. COMPOUND PENDULUMS GOVERNORS. 547 605. Compound Revolving Pendulum. To avoid unnecessary complexity in the theory of a compound revolving pendulum, let the body of which it Consists be of such a figure and so suspended, that the straight line C G- B (fig. 239), traversing the point of sus- pension C and the centre of gravity G, shall be one of the axes of inertia, and that the moments of inertia about the other two axes shall be equal. Then for every axis traversing the centre of gravity at right angles to C G B, the radius of gyration is the same ; and consequently, for every axis traversing the point of suspension C at right angles to C G B, the centre of percussion B is the same ; and the body moves exactly like a simple revolving pendulum of the length C B, and height C B cos 6, if 6 is the angle which it makes with the vertical. It is to be borne in mind, that in order that a pendulum may revolve according to the above law, it must have no rotation about its longitudinal axis B G C, but must swing as if hung by a double universal joint at C (Article 492). 606. A Rotating Pendulum (fig. 242) is a body E C G suspended by a point C not in the centre of N gravity G, and rotating about a vertical axis G X traversing the point of suspension. To avoid needless complexity, as before, let C G, and E G perpendicular to it in the vertical plane of C G and C X, be two of the axes of inertia of the pendulum. Let Ij be its moment of inertia about G E, and I 2 its moment of inertia about G C, and ft, &> the corresponding radii of gyration. Let the angle X C G = * ; let C G = r Q and let the weight of the pendulum be W. Then, a being the angular velocity of rotation about the vertical axis, it appears from Articles 592 and 586 that the deviating couple due to rotation about a vertical axis traversing G is a? /T . , (Ij 1 2 ) cos a sin = (f\ fi) cos et sm * ; y y to which has to be added, the couple due to the deviating force of W revolving along with the centre of gravity G, and to the leverage r cos , being the height of C above G ; that is to say, Wa 2 - 7*0 cos tx, sin * ; y making for the entire deviating couple W a -(? d + o)cossin; y .548 PRINCIPLES OF DYNAMICS. and this couple has to be supplied by means of the weight of the pendulum acting with the leverage r sin a; that is, it must be equal to W r sin . Dividing by this quantity, we find and putting for a 2 its value, 4 ^r 2 T 2 , where T is the number of turns per second, this leads to the equation h being the height of the equivalent simple revolving pendulum, as given in Article 539, equation 2. When f 2 , the radius of gyration about C G, is insensibly small compared with r the radius of gyration about G E, h becomes equal to the height of the simple pendulum equivalent to the pen- dulum in the figure, when made to revolve without rotation about C G, as in the last Article. When &, = ?1 , the height becomes simply r cos #, being the same as if the whole mass were concen- trated at the centre of gravity. This is very nearly the case in the rotating pendulums used as GOVERNORS for prime movers, which are in general large heavy spheres hung by slender rods. 607. The Ballistic Pendulum is used to measure the momentum of projectiles, and the impulse of the explosion of gunpowder. To measure the momentum of a projectile, such as a rifle ball, the pendulum must consist of a mass of material in which the ball can lodge, such as a block of wood, or a box full of moist clay, hung by rods from a horizontal axis. Suppose the ball to be of the weight b, and to move with the velocity v in a line of flight whose perpen- dicular distance from the axis of suspension is r'. Then the angular momentum of the ball relatively to the axis of suspension is and because the ball lodges in the pendulum, this angular momen- tum is wholly communicated to the joint mass consisting of the ball and the pendulum, which swings forward, carrying with an index that remains, and points out on a scale the extreme angular displacement. Let this be denoted by i. Let I denote the length of the simple pendulum equivalent to that mass, which can be found by means of Article 544, equation 1, from the number of BALLISTIC PENDULUM. 549 oscillations in a given time; let W be the joint weight of the pen- dulum and ball, and r the distance of their common centre of gravity from the axis ; then is the portion of the joint weight to be treated as if concentrated at the centre of oscillation. Let V be the velocity of the centre of oscillation at the lowest point of its arc of motion ; this is the velocity due to the height, I ' versin i ; that is to say, Y = J (2gl-versini) = 2 sin 3 >J~g~l; (3.) and the corresponding angular momentum of the combined mass is B V I ; which, being equated to the angular momentum of the ball */ before the collision (1), gives the equation giving for the velocity, momentum, and actual energy of the ball, respectively, ^_BY } I (5.) g ~ g r' ' 2 g 2 g b r' 2 BV* The energy of the combined mass after the collision being , and less than that of the ball before the collision in the proportion of b r' 2 : B %, shows, that an amount of energy denoted by br' disappears in producing heat and molecular changes in the ball and in the soft mass in which it is lodged. To measure the impulse produced by the explosion of gunpowder, the gun to be experimented on is to be fixed to and form part of the pendulum, and a ball is to be fired from it. The gas produced by the explosion exerts equal pressures during the same time, that is, equal impulses, forwards against the ball, and backwards against the gun, and the pendulum swings back through a certain angle, which is registered by an index as before, and from which the 550 PRINCIPLES OF DYNAMICS. maximum velocity of the centre of percussion of the pendulum can be calculated as before by equation 3. Let r' now denote the distance from the axis of suspension to the axis of the gun, and P the pressure exerted by the explosive gas at any instant ; the total impulse exerted by the gas is / P d t ; and the angular impulse r' - I P d t; which being equated to the angular momentum pro- duced in the pendulum, gives in which it is to be observed, that B does not now include the weight of the ball. The impulse exerted by the powder is therefore and the velocity of the ball b on leaving the gun is consequently The energy exerted by the exploding powder is of which the portions communicated to the ball and to the pendulum are indicated by the two terms, being in the ratio J In the preceding calculations, the momentum and energy pro- duced in the explosive gases themselves are not considered ; but it is very doubtful whether any attempt to take them into account, hypothetical as it must be, adds to the practical correctness of the result. As a probable approximation, the following may be em- ployed : Let w be the weight of powder used. Divide this into two parts proportional to b and B, viz. : b w 1$ w and consider the smaller part to move with half the velocity of B, and the larger with half the velocity of b ; that is to say, in equations 7, 8, and 9, put, BALLISTIC PENDULUM. 551 ,(12.) instead of B, and instead of b, The equation 10, in its original form, will still show the actual energies of the pendulum and of the ball, and their sum ; but that sum will be exclusive of the energy exerted in giving motion to the explosive gases themselves. The ballistic pendulum was invented by Eobins, celebrated for his investigations on gunnery. 552 CHAPTER IV. MOTIONS OF PLIABLE BODIES. 608. Nature of the Subject ; Vibration. The motion of each par- ticle of a pliable body may always be resolved into three components : that which it has in common with the centre of gravity of the body, being the motion due to translation of the whole body ; that which it has about the centre of gravity of the body, being the motion due to rotation of the whole body; and a third component, being the motion due to alterations of the volume and figure of the body and of its parts. This third component is alone to be considered in the present chapter. The cinematical branch of the present subject, that is to say, the branch which comprehends the relations amongst the displace- ments of the particles in a strained solid from their free positions, and the strains or disfigurements of its parts accompanying such displacements, has already been treated of generally in Articles 248, 249, 250, 260, and 261 ; with reference to bending, in part of 293, part of 300, 301, part of 303, part of 304, part of 307, part of 309, part of 312, and part of 319; with reference to twisting, in part of 321 and part of 322; and again with reference to bending, in part of Article 340. The dynamical branch of the subject has been, to a certain extent, anticipated in Article 244, where resilience is defined ; in Article 252, where potential energy of elasticity is defined ;* in Articles 266 and 269, which relate to the resilience of a stretched bar and the effect of a sudden pull; in Article 305, which relates to the resilience of a beam; in Article 306, which relates to the effect of a suddenly applied transverse load; and in Article 323, which relates to the resilience of an axle. The motions due to strains amongst the particles of pliable bodies being all of limited extent, and consisting in changes of the dis- placement of each particle from the position which it would occupy in a state of equilibrium, which displacement is limited and gene- rally small, are of the kind called VIBRATIONS, and are more or less * In Article 252, the first employment of this function is correctly ascribed to Mr. Green; but it is right also to mention, that its use was independently discovered by M. Clapeyron. VIBRATION CONDITION OF ISOCHRONISM. 553 analogous to the oscillations already treated of in Articles 512 and 543. The complete theory of vibration embraces all the phenomena of the production and transmission of sound, and all those of the pro- pagation of light, as well as those of the visible and tangible vibra- tions of bodies. Many of its branches are foreign to the objects of this treatise and therefore in the present chapter there will be given only an outline of the general principles of the theory of vibration, and an explanation of such of its applications as are of importance in practical mechanics. 609. isochronous vibrations of an elastic body are those in which each particle of the body performs a complete oscillation in the same period of time, so that all the particles return to the same relative situations at the end of each equal period of time, and that whether the oscillations are of greater or of less amplitude. Iso- chronous vibrations being communicated to the ear produce the sensation of a sound of uniform pitch, or musical tone. In order that oscillations of different amplitudes may be performed by equal masses in the same time, it is evidently necessary that the forces under which they are performed should be proportional, and directly opposed, to tJie displacements at each instant. This is the CONDITION OF ISOCHRONISM, and has already been illustrated in Articles 542, 543, 544, 545, and 557, Example III., for the case of a single par- ticle acted on by a single force, and in Article 598 for the analogous case of a gyrating rigid body, where angular is substituted for linear displacement, and a couple for a force. To express that condition by an equation suited to the present class of questions, let W -4- g be the mass of a particle, its displacement from its position of equilibrium at any given instant, F an unbalanced force by which it is urged directly towards that position, and a 8 a numerical con- stant, expressed as a square for reasons which will presently appear ; then the condition of isochronism is expressed as follows : do an equation identical with equation 1 of Article 542 ; while from equation 4 of the same Article it appears that the number of double oscillations per second is expressed by a n = , 2* and the period of a double oscillation by 1 2* ..(2.) 554 PKINCIPLES OF DYNAMICS. All the equations of Article 542 and Article 557, Example III., are made applicable to the present case, by substituting respectively for Q or Q,, Q x , r or x h x, F,, F, d,, S, respectively, where F t represents the maximum force, corresponding to 81, the maximum displacement, or semi-amplitude ; consequently, if in order to make the formulae more general we represent by t Q any instant of time at which the particle reaches the extremity of an oscillation, we have - = - a \ sin a (t - t ). ( ................. ' '' Ci t ) When the restoring force corresponding to a given displacement is known, the constant a 2 is computed by the formula in which the negative sign denotes, that although F being contrary to d in direction, their quotient is implicitly negative, it is to have that negativity reversed and to be treated as positive. The equations 2 and 4 show, that the square of the number of oscillations made by a particle in a second, is inversely as tJie mass of tJie particle, and directly as tJie ratio of tlie restoring force to the dis- placement. 610. Vibrations of a Mass held by a Light Spring. The deflection of a straight spring or elastic beam under any load is given by the equations of Article 303 for those cases in which it is sensibly pro- portional to the load. The position of equilibrium of the spring, if not affected by a lateral transverse load (for example, if it is placed vertically), may be straight ; or if there be a permanent transverse load, that posi- tion may be more or less deflected. In either case, the production of an independent deflection, S, of the point for which deflections are computed by the formulae, to one side or to the other of the position of equilibrium, provided the limits of perfect elasticity are not exceeded, causes the spring to exert a restoring force F, whose value is found by applying to this case equation 4 of Article 303 ; that is to say, ri"~Fc* n'Ebh* ' w'V ' [ ........ (1.) = ft for brevity's sake ; in which /may be called the stiffness of the spring. LOADED SPRING SUPERPOSITION OF MOTIONS. 555 Now suppose that there is attached to the point of the spring for which is calculated, a mass W -f- g, in comparison with which the mass of the spring is inappreciably small. Then if that mass be drawn to one side or to the other of the position of equilibrium, and left free to vibrate, the spring will make it vibrate according to the law already explained in Article 609 ; putting for the constant a the value " = Vw < 2 -> If the mass gyrates about a fixed axis traversing its centre of gravity, let I denote the distance from that axis to the point upon which the spring acts ; then in the equations of motion, substitu- tions are to be made according to the principles of Article 598, when the above equation becomes k = (3.) If the mass oscillates about a fixed axis not traversing its centre of gravity, the above equation is still applicable, when the proper value is put for the moment of inertia I. The restoring couple F I for a gyrating body may be supplied by the resistance of a rod or wire to torsion ; in which case f I is to be taken to represent the ratio of the moment of torsion to the angle of torsion, which, for a cylindrical rod or wire, is given in Article 322, case 2, equation 4, viz. : x being the length, and h the diameter of the rod or wire, and C the co-efficient of transverse elasticity of the material. By the aid of the principles here explained, experiments on the numbers of vibrations per second made by springs and wires loaded with masses great in proportion to the masses of the springs and wires, may be used to determine the co-efficients of elasticity E andC. 611. Superposition of Small motions. If the restoring force of a particle for vibrations in a given direction be opposite and pro- portional to the displacement, and if the same be the case for one or more other directions of vibration, then for a displacement which is the resultant of two or more displacements in the given directions, the force acting on the particle will evidently be the resultant of the separate forces corresponding to the component displacements, and the velocity the resultant of the component velocities. 556 PEINCIPLES OF DYNAMICS. This is called the principle of the SUPERPOSITION OF SMALL MOTIONS. If the co-efficient a of Article 609 is the same for the different directions of the component displacements, the component vibra- tions will not ooly be isochronous in themselves, but isochronous with each other, or simultaneous, and so also will be the resultant vibration. This has already been sufficiently illustrated in Articles 542 and 543, where circular and elliptic oscillations are treated as compounded of a pair of straight oscillations in directions perpen- dicular to each other. Such, for example, is the oscillation of a mass placed at the end of a spring whose stiffness is the same for all directions of deflection. If the co-efficient a has different values for the different direc- tions of the component vibrations, they will no longer be isochronous with each other; the resultant restoring force will not at every instant act directly towards the position of equilibrium, and the resultant vibration will take place in a complex curve which may have a great variety of figures. For example, let a mass W -f- g be fixed at the end of a spring whose cross section is a rectangle of unequal dimensions, so that its stiffness is different for displace- ments in the directions of two rectangular axes, denoted by x and y. Let f x , f y , be the two values of the stiffness of the spring for those two directions of displacement ; and let % and w denote com- ponent displacements in those two directions respectively, and ^ and ! their maximum values or semi-amplitudes. Then the equa- tions of motion of the mass are the following : = u x cos a y (t - f where * - and t 0iK and t 0 x/ ) ^o> ^05 ^o> an( i ^o being arbitrary constants, having values depending on the circumstances of each particular problem. These constants have the following meanings : ij and ^ are the maximum semi-amplitudes of vibration. and -^ - , are the periodic times of a complete oscillation. 2 * a 2v all expended. The mean value of P is . The distance through which it is overcome in compressing the pile is the compression due T? T to its maximum value, viz., -_ , where E is the modulus of elasti- JL o city of the pile, and L the length of a post, which, if uniformly compressed throughout its length, would be as much shortened as the pile. Considering that the pile is held in a great measure by friction against its sides, L may be made equal to half its length. T> 2 J^ Then the work performed in compressing the pile is ; and the work performed in driving it deeper is R x, where x is the depth through which it is driven by a blow ; and equating these to the energy of the blow, we find When x has been ascertained by observation, R is found by solving a quadratic equation, viz., / V Piles are in general driven till R amounts to between 2,000 and 3,000 Ibs. per square inch of the area of head S, and are loaded with from 200 to 1,000 Ibs. per square inch ; so that the factor of safety is from 10 to 3. The overcoming of any resistance by blows is analogous to the example here given, which is extracted, and somewhat modified, from a section by Mr. Airy in Dr. Whewell's treatise on Mechanics. 566 CHAPTER V. MOTIONS OF FLUIDS. 617. Division of the Subject. The principles of dynamics, as applied to fluids, so far as small and rapid changes of density are concerned, have already been discussed under the head of vibratory motions. Now the only changes of density which occur during the motions of liquids are small and rapid ; so that in the present chapter those motions of liquids are alone to be considered in which the density is constant, and whose cinematical principles have been treated of in Part III., Chapter III., Section 2. In the motions of gases, great and continuous changes of density occur, such as those whose cinematical principles have been treated of in section 3 of the chapter already referred to ; and the dynamical laws of motions affected by such changes have still to be considered. One mode of division, therefore, of hydrodynamics, is founded on the distinction between the motions of liquids, regarded as of constant density, and Another mode of division is founded on the distinction between motions not sensibly affected by friction, and those which are so affected. The motions of fluids not sensibly affected by friction, and therefore governed by pressure and weight only, take place according to laws which are exactly known ; so that any difficulty which exists in tracing their consequences, in particular cases, arises from mathematical intricacy alone. The laws of the friction of fluids, on the other hand, are only known approximately and empirically ; and the mode of operation of that force amongst the particles of a fluid is not yet thoroughly understood; so that the solution of a particular problem has often to be deduced, not from first principles representing the condensed results of all experience, but from experiments of a special class, suited to the problem under consideration. The laws of the mutual impulses exerted between masses of fluid and solid surfaces require to be considered separately. The following is the division of the subject of this chapter : I. Motions of Liquids under Gravity and Pressure alone. II. Motions of Gases under Gravity and Pressure alone. III. Motions of Liquids affected by Friction. IV. Motions of Gases affected by Friction. Y. Mutual Impulses of Fluid Masses and Solid Surfaces. GENERAL EQUATIONS OF HYDRODYNAMICS. 567 SECTION 1. Motions of Liquids wiifiout Friction. 618. General Equations. In Articles 414 and 415 have been given the three general equations, by which the rates of variation of the components of the velocity of an individual particle of liquid are expressed in terms of those of the velocity at a point given in position; and in Article 412 has been given the equation of con- tinuity which connects the components of the latter velocity with each other. To obtain the general dynamical equations of the motion of a liquid, the first three equations are to be converted into expressions for the rates of variation of the components of the mo- mentum of a particle, and the results equated to the unbalanced forces which act upon it. Let d x d y d z denote the volume of a rectangular molecule, and p the intensity of the pressure of the liquid at a point whose co- ordinates are x, y, z. Let z be vertical, and positive downwards. w being used to denote one of the components of the velocity at a point, the symbol ^ will now be employed to denote the weight of an unit of volume. Then the forces by which the molecule is acted upon are along x, -f-t dxdy dz; along y, -, dxdy dz; y (1 along z, ( f j- \ dxdy dz. Let the rates of variation of the components of the momentum of the molecule be found by multiplying the three rates of variation of the components of the velocity in Article 415, equation 2, each by ; then equating these respectively to the three forces in g equation 1 above, dividing by d x d y d z, so as to reduce the equa- tion to the unit of volume, and then by , so as to reduce them to the unit of weight, the following results are obtained : (2.) dp 1 d 2 % 1 ( du . du . du . du dp 1 d 2 YI 1 ( d v . d v . d v . d v} tdy~ g dt 2 ~ g\ dt dx dy d z S dp 1 d 2 Z 1 ( dw . dw . dw . dw} 1 f- = - ' -5-3 =* i ; r u-z r v -y -\-w f . %dz g dt* g ( dt ax ay a z ) Combining with those three equations of motion the equation of continuity, viz. : 568 PRINCIPLES OF DYNAMICS. du dv dw ^ + ^+^ = we have the data for solving all dynamical questions as to liquids without friction. These equations are adapted to the case of steady motion by making dt dt dt~ as in Article 413. rv\ 619. Dynamic Head. The quotient - is what is called the height, or head, due to the pressure; that is, the height of a column of the liquid, of the uniform specific gravity e, whose weight per unit of base would be equal to the pressure p. Now as the vertical ordinate z is measured positively downwards from a datum horizontal plane, z is the weight of a column of liquid per unit of base extending down from that plane to a particle under consideration ; p z is the difference between the intensity of the actual pressure at that particle and the pressure due to its depth below the datum hori- zontal plane ; and (1.) is the height or head due to that difference of intensity, being what will be termed the dynamic head. When z is measured positively upwards from a datum horizontal plane, its sign is to be changed ; so that the expression for the dynamic head in that case becomes (2.) 620. General Dynamic Equations in Terms of Dynamic Head. If instead of the rates of variation of the pressure in the equations 2 of Article 618, there are substituted their values in terms of the dynamic head, those equations take the following forms : dh I d 2 % 1 ( du . du , d u . d dt d x dy d ^ v I . dzl' dh_l d 2 _l(dw dw dw dw^ -'~-~^ U ^ V ^ W 621. i,aw of Dynamic Head for Steady Motion. From these equations is deduced the following consequence, in the case of DYNAMIC HEAD TOTAL ENERGY. 569 steady motion, in which there is no variation of the dynamic head at a particle, except that arising from the change of position of the particle. Let V be the velocity of a given particle. Its value, in terms of its rectangular components, is given by the equation dt which, being divided by 2 g, gives the height due to the velocity ; so that the variation of that height, in a given indefinitely short interval of time, is (2.) 2g g \dt dt*dt dedt at 2 dh dl.dh d* dh d (d \d , x d This principle might otherwise be stated thus : In steady motion, the sum of the height due to the velocity of a particle and of its dynamic head is constant, or symbolically V 2 2~ -}-h = constant ....................... (3.) This equation applies to the particles which successively occupy the same fixed point, as well as to each individual particle. 622. The Total Energy of a particle of a moving liquid without friction is expressed by multiplying the expression in equation 5 of the last Article by the weight of the particle W, thus : (1.) W V 2 in which - is the actual energy of the particle, and W h is its potential energy; because, from the last Article it appears, that by W V a the diminution of W h, may be increased by an equal amount, and vice versa; so that the dynamic head of a particle is its potential energy per unit of weight. In the case of steady motion, the total energy of each particle is constant ; and the total energy of each of the equal particles which successively occupy the same position is the same. In the case of unsteady motion of a liquid mass, the total internal energy of the entire mass is constant; that is, if the centre of gravity of the mass, or a point either fixed or moving uniformly, 570 PRINCIPLES OF DYNAMICS. with respect to that centre of gravity, is takeii as the fixed point to which the motions of all the particles are referred, the following equation is fulfilled : 2 W Q^ + h) or j j I Q- -f /A e . d xdydz = constant .. .(2.) 623. The Free Surface of a moving liquid mass, being that which is in contact with the air only, is characterized by the pressure being uniform all over it, and equal to that of the atmosphere. Let pi be the atmospheric pressure, z l the vertical ordinate, mea- sured positively upwards from a given horizontal plane, of any point in the free surface of the liquid, and 7^ the dynamic head at the same point ; then it appears from Article 619, equation 2, that for that surface, fy\ h 1 z 1 = = constant ..................... (1.) 624. A Surface of Equal Pressure is characterized by an analo- gous equation, fv\ h-z = constant; ..................... (1.) and all surfaces of equal pressure fulfil the differential equation, dh = dz; .............................. (2.) which, for steady motion, becomes dz=zdh = -d ' ; ..................... (3.) %9 expressing that the variations of actual energy are those due to the variations of level simply. 625. motion in Plane Layers is a state which is either exactly or approximately realized in many ordinary cases of liquid motion ; Fig. 246. Fig. 247. and the assumption of which is often used as a first approximation MOTION OF LIQUID IN PLANE LAYEKS. 571 to the solution of various questions in hydraulics. It consists in the motions of all the particles in one plane being parallel to each other, per- pendicular to the plane, and equal in velocity. It is illustrated by the three figures 246, 247, and 248, each of which represents a reservoir containing liquid up to the elevation OZ 1 = z l above a given datum, and discharging the liquid from an orifice A at the smaller elevation O Z Fig. 248. = Z Q . The liquid moves exactly or nearly in plane layers at the upper surface A^ and at the orifice A^ Let these symbols denote the areas of the upper surface and of the issuing stream respectively. Let Q denote the rate of flow per second, v l the velocity of descent of the liquid at the upper surface, v its velocity of outflow from the orifice ; then, according to Article 405, the equation of continuity is v l A l = v A Q = Q; The pressures at the upper surface and at the orifice respectively are each equal to the atmospheric pressure ; hence the difference of dynamic head is simply the difference of elevation ; that is to say, therefore, according to Article 621, equations 2 and 3, This gives for the velocity of outflow, from which can be computed the rate of flow or discharge by means of equation 1. The general equation of motion, for every part of the vessel or channel at which the motion takes place in plane layers, is, accord- ing to Article 621, equation 3, 572 PRINCIPLES OF DYNAMICS. The motion may be considered to take place in plane layers at any part of the channel whose sides are nearly straight and parallel, such as A 2 in fig. 246, whose elevation above the datum is z 2 . To find the dynamic head, and thence the pressure, at this intermediate section of the channel, the velocity through it is to be computed by the formula ........................ <*> whence the dynamic head relatively to the datum is obtained by the equation and thence the pressure by the formula When a large vessel discharges liquid through a small orifice, the AO ratio ^ is often so small a fraction, that it may be neglected in A? equations 2 and 3. 626. The Contracted Vein is the name given to a portion of a jet of fluid at a short distance from an orifice in a plate, which is smaller in diameter and in area than the orifice, owing to a sponta- neous contraction which the jet undergoes after leaving the orifice. The area of the narrowed part of the contracted vein is in every case to be considered as the virtual or effective outlet, and used for A in the equations of the last Article. The ratio of the area of the contracted vein, or effective orifice, to that of the actual orifice, is called the co-efficient of contraction. For sharp edged orifices in thin plates, it has different values for different figures and proportions of the orifice, ranging from about 0-58 to 0'7, and being on an average about |. It diminishes some- what for great pressures, and for dynamic heads of six feet and upwards may be taken at about 0-6. The most elaborate table of those co-efficients is that of Poncelet and Lesbros. For orifices with edges that are not sharp and thin, the discharge is modified sensibly by friction. 627. Vertical Orifices of discharge, whose vertical dimensions are not small in comparison with their depths below the upper surface of the reservoir, are treated as having a mean velocity of discharge through their contracted veins due to the mean value of the square root of the dynamic head for the several parts of the orifice. . For example, let y be the horizontal breadth of an orifice at any given VERTICAL ORIFICES SURFACES OF EQUAL HEAD. 573 elevation z above the datum, z 1 the elevation of the lower, and z" that of the upper edge of the orifice, so that ,f = C J z t ydz ........................ (1.) is its effective area, c being the co-efficient of contraction. Then that orifice is to be treated as if its depth below the upper surface A! were .__ ' ~ and the formulae of Article 625 applied accordingly. For a rectan- gular orifice for which y is constant, this gives and if it is a notch, or a rectangular orifice extending to the upper surface, so that z" = 628. Surfaces of Equal Head, which for steady motion are also SURFACES OF EQUAL VELOCITY, are ideal surfaces traversing a fluid mass, at each of which the dynamic head is uniform. Their posi- tions are related to the direction, velocity, curvature, and variation of velocity of the fluid motion in the following manner : In fig. 249, let Hj H w H 2 H 2 , represent a pair of such surfaces, very near each other ; their normal distance apart being dn, measured forwards from Hj towards H 2 , and the difference of dynamic head at them being d h. Let A B be part of the moving fluid, forming an elementary stream whose velocity is V, its radius of "* curvature r, its thickness dr, and the varia- tion of its velocity d V; velocities from A towards B being posi- tive, and curvature concave towards H 2 being positive. Then the equations 2 and 3 of Article 621 give, as before, y d y v 2 = dhj or - -\-h = constant; (1.) and in order that the variation of head may supply the deviating force necessary to produce the curvature of the stream A B, the radius of curvature must be in a plane perpendicular to the surfaces of equal head, and the following equation must be fulfilled : 574 PRINCIPLES OF DYNAMICS. .(2.) l , d r cos nr ; dn dn 629. ID a Radiating Current, flowing towards or from an axis, as described in Article 407, the surfaces of equal dyna- mic head and equal velocity are cylinders described about the axis. The equation of continuity, 1 of Article 407, putting b instead of h to denote the depth, parallel to the axis, of the cylindrical space in which the current flows, gives for the velocity the formula Fig. 250. 2 sr b r r where r Q is the radius of the cylindrical surface R , fig. 250, at which the radiating part of the current begins or ends, according as it flows outwards or inwards. The radiating current may ex- tend indefinitely in all directions beyond this surface, the velocity being at any point inversely as the distance from the axis 0. Let h Q be the dynamic head at R ; then at any other cylindrical sur- face of the radius O R = r, we have the dynamic head, 7 7 h = h + r g - _ r (2.) Let ^ be the limit towards which the dynamic head approxi- mates as the distance from the axis is indefinitely increased ; then h o + ^ = (3.) 630. Free Circular Vortex. In the cylindrical space of fig. 250, lying outside of the surface R , let the particles of the fluid revolve in a circular current round the axis O ; and let the velocity of each circular current be such, that if, owing to a slow radial movement, particles should find their way from one circular current to another, they would assume freely the velocities proper to the several cur- VORTEX. 575 rents entered by them, without the action of any force but weight and fluid pressure. This last condition is what constitutes a free, vortex, and is a condition towards which every vortex not acted on by external forces tends, because of the tendency to the inter- mixture of the particles of adjoining circular currents. It is ex- pressed mathematically by v 2 h + ~- ^ = constant .................. (1.) *9 A! will be called the maximum head. Conceive a portion of a thin circular current of the mean radius r, contained between two cylindrical surfaces at the indefinitely small distance apart d r, and of the area unity, the current having the velocity v. Then the centrifugal force of that portion of the current is v* dr gr ' which is equal and opposite to the deviating force that is to say, <^=^ (2) dr gr V 2 But by the condition of freedom in equation 1, we have - 2 (A t - h), which being substituted in equation 2, gives dh _ ^(h,-h) dr r ' whence *-*- ....................... (3 -> or, the velocity is inversely as the distance from the axis, exactly as in a radiating current. Then let v be the velocity of revolution, and h the dynamic head, at the inner boundary H of the vortex ; we have for the general equations amongst the dynamic heads and velocities at all points, tl = , ___ , , ! , J _ h = * - " = * ~ 576 PRINCIPLES OF DYNAMICS. 631. Free Spiral Vortex. As the equations of the motion of a free circular vortex are exactly the same with those of a radiating current, it follows that they also apply to a vortex in which the motion is compounded of those two motions in any proportions, so long as tfie velocity is inversely as ilie distance from the axis. To fulfil this condition, the currents of liquid must have a form that is at every point equally inclined to the radius drawn from the axis ; a property of the logarithmic spiral. Let v be the velocity of the current in a free spiral vortex at any point, and & the con- stant inclination of the current to the radius vector ; then the com- ponent of the motion whose velocity is v cos 0, is analogous to the motion of a radiating current, and that whose velocity is v sin 6 is analogous to the motion of a free circular vortex. 632. A Forced Vortex is one in which the velocity of revolution of the particles follows any law different from that of a free vor- tex ; but the kind of forced vortex which it is most useful to con- sider, is one in which the particles revolve with equal angular- velocities of revolution, as if they belonged to a rotating solid body j so that if r be the radius of the outer boundary of the vor- tex, where the velocity is v , v r v = - .............................. (1.) * The equation of deviating force, 2 of Article 630, is applicable to all vortices, forced as well as free. Introducing into it the value of v from equation 1, above, we find, dh vlr d-r = j* ..... which being integrated, with the understanding that the dynamic head is to be reckoned relatively to the axis of the vortex, gives _ _ -27? -2^ A -V- from which it appears, that in a rotating vortex, the dynamic head at any point is the height due to the velocity, and the energy of any particle is Jialf actual and half potential. 633. A Combined Vortex consists of a free vortex without and a forced vortex within a given cylindrical surface, such as K in fig. 250. In order that such a combined vortex may exist, the velo- city v and the dynamic head h at the surface of junction must be the same for the two vortices ; consequently, as the dynamic head of the forced vortex is equal to the height due to its velocity, and COMBINED VORTEX. the sum of those heights for the surface of junction is equal to the maximum liead hi of the free vortex, we have this principle : In a combined vortex, the maximum dynamic head is double of the dyna- mic head at the surface of junction, each being measured relatively to the axis of the vortex ; or symbolically, To illustrate this geometrically, let a combined vortex revolve about a vertical axis, Z Z b fig. 251, the upper surface of the liquid being free, and represented in section by D B B D. Let A B, A B, be the cylindrical sur- face of junction between the free and the forced vortices. Let A O A be a horizontal plane, Fig. 251. touching the upper surface at its lowest point, which is at the axis, and let vertical ordinates be measured from this plane. The pressure of the atmosphere being equal at all points, may be left out of consideration ; so that if z be the height of any point in the surface of the vortex above A O A, we shall have simply = h. .(2.) Then for the forced vortex, z = .(3.) so that B O B is a paraboloid of revolution with its vertex at 0. Make AC = 2AB=:22 j this will represent A,, the maximum dynamic head ; and for the free vortex, z = h L _j, (*) and D B, D B, is a hyperboloid of the second order, described by the rotation round the vertical axis of a hyperbola of the second order, whose ordinate hi z, measured downwards from C Z t C, is inversely as the square of the distance from the axis. The two surfaces have a common tangent at B B, where they join. The velocity of any particle in the free vortex is that due to its depth below C C ; that of any particle in the forced vortex is that due to its height above A A ; and B, where those velocities are equal, is midway between C C and A A. 578 PRINCIPLES OF DYNAMICS, The theory of the combined vortex was made, by Professor James Thomson of Belfast, the principle of the action of his tur- bine or vortex water-wheel. 634. Vertical Revolution. When a mass of liquid revolves in a vertical plane about a horizontal axis (like the water in a bucket of an overshot wheel), its upper surface is not horizontal, but assumes a figure depending on the deviating force required by its revo- lution. In fig. 252, let C represent a horizontal axis, and B a bucket of liquid revolving round it in a vertical circle of the radius B C, with the angular velocity of revolution a. Let W be the weight of liquid in the bucket. Then the deviating force required is given by the formula Wa 2 BC. Fig. 252. Take the radius B C itself to represent the devi- ating force, and C A vertically upwards from the axis to represent the weight ; the height C A is given by the proportion CA : BC : : W : - B C, that is, (1.) where n is the number of revolutions per second. Now A C representing the weight, and C B the centrifugal force, equal and opposite to the deviating force, the internal condition of the liquid in the bucket, according to D'Alembert's principle, is the same as if it were under a force represented by A B, the resultant of these two forces ; therefore the surface of the liquid is perpendi- cular to A B. Now it appears from equation 1, that the height of A above C is independent of the radius of the wheel, and of every circumstance except the time of revolution; being, in fact, the height of a revolv- ing pendulum which revolves in the same time with the wheel. (See Article 539.) Therefore the point A is the same for all buckets carried by the same wheel with the same angular velocity, and for all points in the surface of the liquid in the same bucket, whether nearer to or farther from the axis C ; therefore the upper VERTICAL REVOLUTION DYNAMIC HEAD IN GASES. 579 surface of the liquid in each bucket is part of a cylinder described about a horizontal axis passing through A and parallel to C. The theory of rolling waves might be deduced from the above proposition; but it is a subject foreign to the purposes of this treatise. SECTION 2. Motions of Gases without Friction. 635. Dynamic Head in Oases. The dynamical equations of motion of a gas are the same with those already given in Article 618, equation 2; and in their integration, it has to be observed that , the density, is no longer constant, but depends on the pres- sure. The equations of continuity have been given in Articles 419 to 423. In finding the DYNAMIC HEAD for a particle of a gas, instead of there is to be taken / , as is evident from the general equa- e J e tions of fluid motion already referred to. Consequently, the dyna- mic head for a gaseous particle, at a given elevation z above a fixed horizontal plane, is, relatively to that plane, and the putting of this value for h in all the dynamical equations relating to liquids, transforms them into the corresponding equa- tions for gases. In most practical problems respecting the flow of gases, the dif- ferences of level of different points of the gaseous mass have little or no sensible effect on the motion ; so that z may often be omitted from the preceding formula. In determining the value of the integral in that formula, it is to be observed that almost all changes of velocity of gases take place so rapidly, that the particles have no time to receive or to emit heat to any sensible amount ; and therefore the pressure and den- sity of each particle are related to each other according to the law already explained in treating of the velocity of sound; that is to say, p I 4 I * '475 .^Ether vapour, 10,110 0-481 Bisulphuret of carbon vapour, ... 9, 902 o 1 57 5 Carbonic acid, if a perfect gas, ... 17,264 Do., actually, i7>!45 0-217 * This is an ideal result, arrived at not by direct experiment, but by calculation from the chemical composition of steam. PLOW OF A GAS. 581 The variations of pressure, volume, and absolute temperature of a gas during rapid changes of motion, are connected by the propor- tional equation >-i *=! r oc p a p * .. .................... (11.) The equations in this Article are all adapted to perfect gases. Actual gases deviate from the perfectly gaseous condition more or less ; but in most practical questions of hydrodynamics the equa- tions for perfect gases may be applied to them without material error. 636. The Equation of Continuity for a Steady Stream of Gas takes the following form, when the laws stated in the last Article are taken into account. The original equation, as given in Article 421, being equivalent to Q p = Av p = constant, .................... (1.) we have to consider that, by the equations of the last Article, we have 1 i i p oc pY oc r*- 1 oc (h-zf- 1 ................. (2.) the exponents having, for air, the values Hence the equation of continuity, in terms of the pressure, of the absolute temperature, and of the dynamic head respectively, takes the following forms : * = constant; .................. (4.) i i Qr 3 '- 1 = Avr*- 1 = constant ; ............... (5.) 1 i Q (h - z)*- 1 Av(h- zy~ l = constant ; ......... (6.) 637. Flow of Gas from an Orifice. Let the pressure of a gas within a receiver be p lt and without, p 2 ; let A be the effective area of an orifice with thin edges ; that is, the product of the actual area by a co-efficient of contraction, whose value is 0-6, nearly. Let the receiver be so large that the velocity within it is insensible. Let the absolute temperature and density of the gas within the receiver be T U &> and those of the issuing jet r st ^ 2 . The latter are 582 PRINCIPLES OF DYNAMICS. not the same with those of the still gas outside, for reasons to be stated afterwards. Then and by equation 8 of Article 635, and equation 3 of Article 621, we have for the height due to the velocity of outflow, J^!_ - fa _ 7 y ^? TI r 2 2g~ 2 y Ig T o to from which the velocity itself, and the flow of volume Q = v A at the contracted vein, are easily computed. To find the flow of weight, the last quantity is to be multiplied by giving the following results : g a Q = V A i --( Eor small differences of pressure, such that is nearly = 1, the following approximate formula may be used where great accuracy is not required : When the motion of the jet is finally extinguished by friction, heat is reproduced sufficient to raise the absolute temperature nearly to its original value, TV 637 A. Maximum Flow of Gas. When is indefinitely dimin- Pi ished, the velocity of outflow given by equation 2 of Article 637 increases towards the limit A// syg^M V t(7-l), rj' MAXIMUM FLOW OF GAS. 583 being greater than the velocity of sound in the ratio \/ -r- : 1 , V JV K b = 0-0043. Iron pipes, average value of /for first ) . ^ approximation, .......................... J a = 0-007 1 b 0-001363 foot. ; b = 0-000227 Beds of rivers (Eytelwein),. . ./ + -; a = 0-007 164. Beds of rivers ( Weisbach),... /= a+ - ; a = 0-007409. Beds of rivers, average value of / for ) ~ ' , ' > O' first approximation, .................... J A collection of numerous formulae for fluid friction, proposed by different authors, together with tables of the results of the best for- mulae, is contained in Mr. Neville's work on Hydraulics. The formulae of different authors, though differing in appearance, are founded on the same, or nearly the same, experimental data, being chiefly those of Du Buat, with additions by subsequent inquirers ; and their practical results do not materially differ. The two for- mulae given above, on the authority of Weisbach, may on the whole be considered the best. 639. internal Fluid Friction. Although the particles of fluids have no transverse elasticity, that is, no tendency to recover a certain figure after having been distorted, it is certain that they resist being made to slide over each other, and that there is a lateral communication of motion amongst them ; that is, that there is a tendency of particles which move side by side in parallel lines to 586 PPtlNCIPLES OF DYNAMICS. assume the same velocity. The laws of this lateral communication of motion, or internal friction, of fluids, are not known exactly; but its effects are known thus far : that the energy due to differ- ences of velocity, which it causes to disappear, is replaced by heat in the proportion of one thermal unit of Fahrenheit's scale for 772 foot pounds of energy, and that it causes the friction of a stream against its channel to take effect, not merely in retarding the film of fluid which is immediately in contact with the sides of the channel, but in retarding the whole stream, so as to reduce its motion to one approximating to a motion in plane layers perpendicular to the axis of the channel (Article 625). 640. Friction in an Uniform Stream. It is this last fact which renders possible the existence of an open stream of uniform section, velocity, and declivity. In hydraulic calculations respecting the resistance of this, or any other stream, the value given to the velocity is its mean value throughout a given cross-section of the stream A, * = 5 (1) A -w The greatest velocity in each cross-section of a stream takes place at the point most distant from the rubbing surface of the channel. Its ratio to the mean velocity is given by the following empirical formula of Prony, where Y is the greatest velocity in feet per second : Y ~ 10-25 + V" " ' In an uniform stream, the dynamic head which would otherwise have been expended in producing increase of actual energy, is wholly expended in overcoming friction. Consider a portion of the stream whose length is I, and fall *. The loss of head is equal to the fall of the surface of the stream, according to Article 623 ; and the expenditure of potential energy in a second is accordingly Equating this to the work performed in a second in overcoming friction, viz., v E, we find or dividing by common factors, and by the area of section A, we find for the value of the fall in terms of the velocity STREAMS HYDRAULIC MEAN DEPTH. 587 Let s be what is called the wetted perimeter of the cross-section of the stream; that is, the cross-section of the rubbing surface of the stream and channel ; then and dividing both sides of equation 3 by I, we find for the relation between the rate of declivity and the velocity, . . Z j. S V* sin ^ = y =/-r- . I A. 2g (*) is what is called the " HYDRAULIC MEAN DEPTH " of the stream ; s and as the friction is inversely proportional to it, it is evident that the figure of cross-section of channel which gives the least friction is that whose hydraulic mean depth is greatest, viz., a semicircle. When the stability of the material limits the side-slope of the channel to a certain angle, Mr. Neville has shown that the figure of least friction consists of a pair of straight side-slopes of the given inclination connected at the bottom by an arc of a circle whose radius is the depth of liquid in the middle of the channel; or, if a flat bottom be necessary, by a horizontal line touching that arc. For such a channel, the hydraulic mean depth is half of the depth of liquid in the middle of the channel. 641. Varying stream. In a stream whose area of cross-section varies, and in which, consequently, the mean velocity varies at different cross-sections, the loss of dynamic head is the sum of that expended in overcoming friction, and of that expended in producing increased velocity, when the velocity increases, or the difference of those two quantities when the velocity diminishes, which difference may be positive or negative, and may represent either a loss or a gain of head. The following method of representing this principle symbolically is the most con- venient for practical purposes. In fig. 253, let the origin of co- ordinates be taken at a point O completely below the part of the stream to be considered ; let ho- rizontal abscissae x be measured against the direction of flow, and vertical ordinates to the surface of the stream, z, up- Fig. 253. wards. Consider any indefinitely short portion of the stream whose horizontal length is dx; in practice this may almost always be con- sidered as equal to the actual length. The fall in that portion of 588 PRINCIPLES OF DYNAMICS. the stream is d z, and the acceleration d v, because of v being opposite to x. Then modifying the expression for the loss of head due to friction in equation 3 of Article 640 to meet the present case, and adding the loss of head due to acceleration, we find ................. (1.) In applying this differential equation to the solution of any parti- cular problem, for v is to be put Q -f- A, and for A and s are to be put their values in terms of x and z. Thus is obtained a differential equation between x and z, and the constant quantity Q, the flow per second. If Q is known, then it is sufficient to know the value of z for one particular value of x } in order to be able to determine the integral equation between z and x. If Q is unknown, the values of z for two particular values of x, or of z and -= (the d x declivity), for one particular value of x } are required for the solu- tion, which comprehends the determination of the value of Q. 642. The Friction in a Pipe Running Full produces loss of dynamic head according to the same law with the friction in a channel, except that the dynamic head is now the sum of the ele- vation of the pipe above a given level, and of the height due to the pressure within it. The differential equation which expresses this is as follows : Let d I be the length of an indefinitely short portion of a pipe measured in the direction of flow, s its internal circumfer- ence, A its area of section, z its elevation above a given level, p the pressure within it, h the dynamic head. Then the loss of head is 7 dp vdv .. sdl V* /n x ^_^ = _^__? = _ +/ ._._ ........ (1 .) The ratio is called the virtual or hydraulic declivity, being the cL v rate of declivity of an open channel of the same flow, area, and hydraulic mean depth. This may differ to any extent from the actual declivity of the pipe, -=-=. d I When the pipe is of uniform section, d v = 0, and the first term of the right-hand side of equation 1 vanishes. When the section of the pipe varies, s and A are given functions of 1. If Q is given, v Q -H A is also a given function of I \ and to solve the equation completely, there is only required in addition the value of h for one particular value of 1. If Q is unknown, the values of h for two particular values of I, or of h and -=j for one ct> L FLOW IN PIPES SUDDEN ENLABGEMENT. 589 particular value of I, are required for the solution, which compre- hends the determination of Q. 643. Resistance of mouthpieces. A mouthpiece is the part of a channel or pipe immediately adjoining a reservoir. The internal friction of the fluid on entering a mouthpiece causes a loss of head equal to the height due to the velocity multiplied by a constant depending on the figure of the mouthpiece, whose values for certain figures have been found empirically; that is to say, let A A be the loss of head ; then f being a constant. For the mouthpiece of a cylindrical pipe, issuing from the flat side of a reservoir, and making the angle i with a normal to the side of the reservoir, according to "Weisbach, / = 0-505 + 0-303 sini + 0-226 sin 2 i .......... (2.) 644. The Resistance of Curves and Knees in pipes causes a loss of head equal to the height due to the velocity multiplied by a co- efficient, whose values, according to Weisbach, are given by the following formulae : For curves, let i be the arc to radius unity, r the radius of curvature of the centre line of the pipe, and d its diameter. Then for a circular pipe, . 0-1.1 + 1*47 ( )'}; and for a rectangular pipe, /' = * { 0-124 + 3-104 (f- 7T ( V For knees, or sudden bends, let i be the angle made by the two por- tions of the pipe at either side of the knee with each other; then /' = 0-9457 sin 2 ^ + 2-047 sin 4 | (2.) 645. A Sudden Enlargement of the channel in which a stream of liquid flows, causes a sudden diminution of the mean velocity in the same proportion as that in which the area of section is in- creased. Thus, let v l be the velocity in the narrower portion of the channel, and let m be the mimber expressing the ratio in which the channel is suddenly enlarged: the velocity in the enlarged part 590 PRINCIPLES OF DYNAMICS. is . Now it appears from experiment, that the actual energy 7)1 due to the velocity of the narrow stream relatively to the wide stream, that is, to the difference v 1 (l V is expended in over- coming the internal fluid friction of eddies, and so producing heat ; so that there is a loss of total head, represented by IK)' (i.) 646. The Oeneral Problem of the flow of a stream with friction is thus expressed : Let ^ + , and h 2 + ~-^ be the total heads at the beginning and end of the stream respectively ; then the loss of total head is represented by fc-a. + ^.F* .................. (i.) where the right-hand side of the equation represents the sum of all the losses of head due to the friction in various parts of the channel. SECTION 4. Flow of Gases with Friction. 647. The General l^aw of the friction of gases is the same with that of the friction of liquids as expressed by equation 1, Article 638, the value of the co-efficient/ being 0-006, nearly, for friction against the sides of the pipe or channel. For a cylin- drical mouthpiece, the co-efficient of resistance is 0-83 ; for a conical mouthpiece diminishing from the reservoir, 0'38. When the pressures at the beginning and end of a stream of gas do not differ by more than ^ of their mean amount, problems respecting its flow may be solved approximately by means of the above data, treating it as if it were a liquid of the density due to the lesser pressure, as in the approximate equation of Article 637. In seeking the exact solution of the flow of a gas with friction, it is necessary to take into account the effect of the friction in pro- ducing heat, and so raising the temperature of the gas above what it would be if there were no friction, as supposed in Section 2. In the flow of a perfect gas with friction, if the heat produced by the friction is not lost by conduction, the friction causes no loss of total FEICTION OF GASES PRESSURE OF A JET. 591 head ; so that if at the beginning and end of a stream the velocities of a perfect gas are the same, its temperatures must also be the same. In an imperfect gas, there is a small depression of tempera- ture, which has been employed by Dr. Joule and Dr. Thomson as a means of determining or verifying the laws of the deviation of different gases from the condition of perfect gas. SECTION 5. Mutual Impulse of Fluids and Solids. 648. Pressure of a Jet against a Fixed Surface. A jet of fluid A, fig. 254, striking a smooth surface, is deflected so as to glide Fig. 254. Fig. 255. Fig. 256. along the surface in that path B E which makes the smallest angle with its original direction of motion A B, and at length glances off at the edge E in a direction tangent to the surface. To simplify the question, the surface is sup- posed to be curved in such a manner as to guide the jet to glance off it in one definite direction. The fric- tion between the jet and the surface is supposed insensible. This being the case, as the particles of fluid in contact with the sur- face move along it, and the only sensible force exerted between them and the surface is perpendicular to their direction of motion, that force cannot accelerate or retard the motion of the particles, but can only deviate it. Let v, then, be the velocity of the par- ticles of fluid, Q the volume discharged per second, p the density, and ft the angle by which the direction of motion is deflected ; then is the momentum of the quantity of fluid whose motion is deflected per second. Also conceive an isosceles triangle whose legs are each 592 PRINCIPLES OF DYNAMICS. equal to the velocity v, and make with each other the angle /3 ; then the base of that triangle, whose value is 2 v sin -, represents the change of velocity undergone by each particle of fluid ; so that the change of momentum per second is F-; ...................... a-) and this also is the amount of the total pressure acting between the fluid and the surface, in the direction of a line which is parallel to the base of the isosceles triangle before mentioned ; that is, which makes equal angles in opposite directions with the original and new directions of motion of the jet. The force represented by F may be resolved into two compo- nents, F, and F y , respectively parallel and perpendicular to the original direction of the jet. The values of the resultant and its two components evidently bear to each other the proportions, /3 F : F x : F y : : 2 sin ^ : l-cos/8 : sin /3 ........... (2.) Z whence the components have the values, If the surface struck by the jet is of a symmetrical figure about the original direction of the jet as an axis, the quantity of fluid Q which strikes the surface in each second spreads and glides off in various directions distributed symmetrically round the axis, and making equal angles /3 with it ; so that the forces exerted perpen- dicular to the axis by the different parts of the spread and diverted jet balance each other, and nothing remains but the sum of the components parallel to the axis, whose value is F^., as given in the first of the equations 3. By substituting A v for Q, the forces may be expressed in terms of the sectional area of the jet. As a particular case, let the surface be a plane, as in fig. 255. The jet, on striking the surface, spreads and glances off in all direc- tions at right angles to its original direction, so that ft = 90, cos /Si = 0, and (*) PRESSURE BETWEEN A JET AND A SURFACE. 593 being equal to the weight of a column of fluid whose base is the sectional area of the jet, and its height double of the height due to the velocity. This result is confirmed by experiment. As another case, let the surface be a hollow hemisphere (fig. 256), so that the jet in spreading is turned directly backwards. Then ft = 180, -cos/3 = +1, and _ 9 9 ' being equal to the weight of a column of fluid whose base is the sectional area of the jet, and its height four times the height due to the velocity. 649. The Pressure of a Jet against a Moving Surface is found by substituting in the equations of the preceding Article, the motion of the jet relatively to the surface for its motion relatively to the earth. In this case there is energy transmitted from the jet to the solid surface or from the solid surface to the jet; and the deter- mination of the amount of energy so transmitted per second forms an important part of the problem. CASE 1. When the surface lias a motion of translation parallel to the original direction of the jet, let u be the velocity of that motion, positive if it is along with the motion of the jet, and negative if against it ; let v l be the original velocity of the jet ; then v l u is the velocity of the jet relatively to the surface. Consequently, the component force acting between the fluid and the solid surface, in the direction of motion of the latter, is P. = (l-cos,3); ................. (1.) representing also the equal and opposite force which must be ap- plied to the solid to make its motion uniform ; and the energy transmitted per second is which, if u is positive, is transmitted from the fluid to the solid, and if u is negative, from the solid to the fluid. The energy thus transmitted per second is equal to the difference of the actual energies of the volume Q of fluid before and after acting on the solid. Let v 2 be the velocity of the fluid after the collision ; this being the resultant of u, and of v l u in the devi- ated direction, its square is given by the equation vl = u z + (^ - uf + 2 u (! - u) - cos ft = v*-2u(v l -u)(l-cosft); ................ (3.) 594 PRINCIPLES OF DYNAMICS. by comparing which with equation 2 it is evident that as has been stated. The maximum transmission of energy from the fluid to the solid, for a given velocity of jet, is obviously given by the velocity, which gives j* ...(5.) If ft = 90, as in fig. 255, the maximum energy transmitted is Q v\ -^ 4 ff) r half of the original actual energy of the fluid. If ft 180, as in fig. 256, the maximum energy transmitted is e Q v \ -*- 2 g, or the whole of the original actual energy of the fluid, which, after the collision, is left at rest. CASE 2. When t/ie surface has a motion of translation in any direction, with the velocity u, let B D, fig. 254, represent that direction and velocity, and B C the direction and original velocity Vi of the jet. Then D C represents the direction and velocity of the original motion of the jet relatively to the surface. Draw E F = D C tangent to the surface at E, where the jet glances off; this represents the relative velocity and direction with which the jet leaves the surface. Draw F G || and = B D, and join E G ; this last line represents the direction and velocity relatively to the earth, with which the jet leaves the surface, being the resultant of E F and F G. The total force exerted between the fluid and the surface might be determined by finding the change of the momentum of the volume of fluid Q, due either to the change of direction and velo- city relatively to the earth, viz., from BC to EG; or to that relatively to the surface, viz., from DC to E F. But the force which it is most important to determine is that to which the trans- mission of energy is due, viz., the force parallel to B D, which will be denoted by F^. This force is equal to the change in one second of the component momentum of the fluid in the direction B D. Let et = ^- D B C, denote the angle between the direction of the jet and that of the body's translation; then the component, in the direction B D, of the original velocity of the jet is v, cos . PRESSURE OF A VORTEX ON A WHEEL. 595 Let w = D C be the velocity of the jet relatively to the surface ; then w 2 = it? + v\ 2 uv l cos oc, (6.) Let y = supplement of ^ E F G, denote the angle which a tan- gent to the surface at the edge where the fluid leaves it makes with the direction of translation. Then the component, in the direction B D, of the new velocity of the jet is u -f- w cos 7 ; and the change of momentum in that direction in one second is Fj, = (0j cos ocuw cos 7) (7.) J which gives for the energy transferred per second, F^ = - ufa cos a u io cos y) (8.) 7 Let v 2 be the resultant velocity of the fluid after the collision ; then v\ = u 9 -f- w 2 + 2 uw -cos y (9.) and it is easily verified that *. = '-^ do.) 650. Pressure of a Forced Vortex Against a Wheel. In a free vortex (Article 630, 631), because the velocity of each particle is inversely as its distance from the axis, the angular momentum of every particle of equal weight is the same ; and a particle in mov- ing nearer to or farther from the axis of the vortex, preserving its angular momentum, requires no external force to be applied to it in order to make it assume the motion proper to each part of the vortex at which it arrives. If, in a forced vortex, there is at the same time a radiating current by which the fluid moves towards or from the axis, then by means of solid surfaces, such as those of the vanes of a wheel, there must be applied to the fluid in the vortex a couple sufficient in each second to produce the requisite change of angular momentum in the quantity of fluid which flows radially through the vortex in a second, and the fluid will react upon the wheel with an equal and opposite couple. Symbolically, let r , r i} be the radii of the cylindrical surfaces at which a forced vortex begins and ends ; v , v lt the velocities of the PRINCIPLES OF DYNAMICS. revolving motion at these two surfaces; Q, the flow of the radial current ; then the moment of the couple exerted between the vortex and the wheel is (1.) A vortex- wheel, or turbine, when working in the most favourable manner, receives the fluid at ends of its vanes which have a velocity of revolution equal to that of the particles of fluid in contact with them ; so that relatively to the wheel, the motion of the fluid is at first radial. The fluid glances off from the vanes at their other ends, which are of such a figure and position that they leave the fluid behind them with only a radial motion relatively to the earth ; so that the whole of the energy due to the revolution of the fluid is transmitted to the wheel. That is to say, let a be the angular velocity of the wheel ; then we must have * (2-) The last quantity, M a, is the energy transmitted in each second from the fluid to the wheel, which, in the case supposed, is the whole energy due to the motion of revolution and centrifugal pressure of the weight e Q of fluid in a rotating forced vortex, as already shown in Article 632. The ends of the vanes which receive the fluid should be radial, because the motion of the fluid relatively to them is radial. The ends of the vanes where the fluid glances off should be inclined backwards so as to make with the radii intersecting them, an angle Q d given by the following equation : Let u = - ' - be the velocity of the radial current at the ends of the vanes now in question ; then . (3.) b being the depth of the wheel in a direction parallel to the axis. Fig. 257 represents Mr. James Thomson's vortex water-wheel, designed on these principles. The water is supplied to the wheel from a large external casing, in which it forms a free spiral vortex ; it is directed by guide blades, B, against the outer circumference of the wheel, where the vanes are radial, and is discharged at the central orifice of the wheel, the inner ends of the vanes being directed backwards at the angle 6 above described. The guide VORTEX- WHEEL CENTRIFUGAL PUMP FAN. 597 blades are moveable about pivots at C, in order to adjust the angle of obliquity of the external free spiral vortex at pleasure, and so to adapt the flow Q of the radial current to the work to be performed. Fig. 257. Fig. 258. A vortex-wheel has been applied to steam by Mr. William Gorman of Glasgow. 651. A Centrifugal Pump consists mainly of a vortex- wheel which communicates motion to the water, so as to make it form a forced vortex of the radius C K = r , fig. 257. The water is supplied by a radiating current proceeding outwards from the central orifice towards the circumference. The inner ends of the vanes should make with the radii traversing them the angle already denoted by 0, Article 650, equation 3, that they may cleave the fluid as it moves radially outwards, without striking it, which would cause agitation, and waste of energy in friction. The outer ends of the vanes should be radial. Beyond the wheel, the water forms a free spiral vortex in a casing, from which it is discharged at A through a pipe. The surface velocity a r Q = v of the wheel is regulated by the total head required, consisting of the elevation at which the water is to be delivered, the height due to its velocity of delivery, and the head lost in overcoming friction ; that is to say, according to the prin- ciples of Article 630 to 633, -* l h = z+?P (1 + 2 /) (1.) where z is the elevation of the point of delivery, V the velocity in the discharge pipe, and 2 -/the sum of the various quantities by which the height due to that velocity is to be multiplied to find the 598 PRINCIPLES OF DYNAMICS. loss of head from various causes of friction. The ratio of C A to C R ?* is regulated by the law that in a free vortex the velocity is inversely as the radius; that is to say, (2.) Guide blades in the free vortex are here unnecessary. A blowing fan is a centrifugal pump applied to air. 652. The Pressure of a Current upon a solid body floating or immersed in it would be equal in opposite directions, and have nothing for its resultant, if fluids moved without friction. But because of the energy of the diverted streams which glance from the body being to a greater or less extent expended in fluid friction, the pressure on the back of the solid body becomes less intense than the pressure on the front ; and to the resultant pressure in the direction of the current thus arising, has to be added the resultant of the direct friction of the fluid against the surface of the solid body. Our knowledge of the laws of the force exerted by a current against a solid body is almost wholly empirical. It is known that that force can be approximately represented by a formula of this kind : being the product of the height due to the velocity of the current, the area A of the greatest cross-section of the solid body; the weight e of an unit of volume of the fluid, and a co-efficient k depending on the figure of the body. The values of this co-efficient have been found experimentally for a few figures. The following, according to Duchemin, are some of its values for rectangular prisms and cylinders, placed with their axes along the current : Let L be the length of the prism or cylinder, A its transverse area, b and d its transverse dimensions, if a rectangular prism, or its axes, if a cylinder. Then for L- J~bd= 0, 1, 2, 3. k= 1-864, 1477, -1-347, 1-328. The value headed is applicable to very thin plates. 653. The Resistance of Fluids to the motion of bodies floating or immersed in them is subject to the same remarks which have been made respecting the pressure of currents against solid bodies. It is also capable in many cases of being approximately represented by the formula RESISTANCE OF FLUIDS - SHIPS. 599 The co-efficient k is less for a solid moving in a fluid, than for a fluid moving past the same solid. The following values are given chiefly on the authority of Duchemin. For prisms and cylinders, moving in the direction of their axes, the symbols having the same meaning as in the last Article : L -r fj b d 0, 1, 2, 3; average above 3. k = 1 -254, 1 -282, 1 -306, 1 -330 ; 1 -4. These results are also given by the empirical formula, 9 J b d + L (2.) k for a cylinder, moving sideways, about 0*77 k for a sphere, 0'51 k for a thin hollow hemisphere moving with the hollow foremost, about 2*0 k for a prism with wedge-formed ends = k for same prism with flat ends, x (1 cos /3), where /3 = ^ angle of wedge (doubtful). For bodies moving through air at high velocities, Duchemin found that the resistance increased faster than the square of the velocity, according to a law which is very nearly expressed by multiplying the resistance found by the formulae already given by 1+ 1^0' v being in feet per second. From the results of observations of the engine power required to propel various steam vessels of different sizes and figures at different velocities, there is reason to think it probable, that when ships are built of such figures that the water glides round their surfaces without forming surge or large eddies, the principal part, if not the only appreciable part, of the resistance, is due to the direct friction between the water and the bottom of the ship ; and if this opinion shoiild be confirmed by further investigation, it will show, that the advantage of Mr. Scott Russell's " wave line " figures, and of any other figures which may be successful in combining speed with economy of power, consists in their causing the water to glide past the ship with the least possible agitation and friction; and also, 600 PRINCIPLES OF DYNAMICS. that a certain definite proportion of length to breadth is the best, and that not only excessive bluffhess, but excessive sharpness also, is unfavourable to speed combined with economy of power. 654. stability of Floating Bodies. In Article 120 it has been shown, that in order that a body floating in a liquid may be in equilibrio, the weight of liquid displaced must be equal to the weight of the floating body, and the centre of buoyancy must be in the same vertical line with the centre of gravity of the floating body. In order that the equilibrium of a floating body may be stable, every angular displacement of the body from the position of equili- brium must cause a deviation of the centre of buoyancy, relatively to a vertical line traversing the centre of gravity, in the direction towards which the floating body leans; so that the weight of the body acting through its centre of gravity, and the equal and opposite pressure of the liquid acting through the centre of buoyancy, may constitute a restoring or righting couple, tending to bring the body back to the position of equilibrium. Should the relative deviation of the centre of buoyancy take place in the opposite direction, a couple is pro- duced tending to upset the body, which is accordingly unstable; should the centre of buoyancy continue to be in the same vertical line with the centre of gravity, the body continues to be in equili- brio in its new position, and its equilibrium is indifferent. Let fig. 259 represent a cross-section of a ship, G her centre of . gravity, A B the water line, / and C the centre of buoyancy in the position of equilibrium. Let the ship roll through an angle 6, and let E F be the new water line, and D the new centre of buoyancy; and let the ship be kept in this position by a couple whose mo- ment is known. Let W be the weight of the ship, and S the volume of water displaced by her, so that W = ^ S (i> being the weight of a cubic foot of water). Through D draw a vertical line D M, cutting the line C G, which was originally vertical, in M. The force of the righting couple is W, and its arm is the horizontal distance from G to the line D M ; that is, G M sin 0; consequently, the moment oftlie righting couple, equal and opposite to the moment of the rolling couple, is W GM - sin 6 (1.) The comparative stability of different ships is considered as propor- STABILITY OF SHIPS METACENTRE. 601 tional to the arms of their righting couples for the same angle of deviation ; and those arms are proportional to G M, which length thus becomes a measure of the stability of a ship. The point M is called the METACENTRE ; it may be the same, or it may be different for different angles ; the variation of its position is in general small. When the position of the metacentre in the ship is variable, the angle of deviation to be adopted in finding it should be the greatest which under ordinary circumstances is likely to occur \ for different ships this varies from 6 to 20. If the metacentre is above the centre of gravity, the equilibrium is stable if it coincides with the centre of gravity, the equilibrium is indifferent ; if it is below the centre of gravity, the equilibrium is unstable. Let H be the line of intersection of the planes of the two water lines A B, E F. The deviation C D of the centre of buoyancy is the same with the deviation of the centre of gravity of the mass of water displaced, which would arise from removing the wedge A H E into the position F H B. Let s be the volume of that wedge, its weight, and let I denote the distance between the centres of gravity of its two positions, A H E and F H B. Draw C D parallel to the line joining those two centres of gravity; and according to Article 77, make then is D the new centre of buoyancy. The angle which C D makes with the horizon is in general either 6 & exactly or very nearly = ; so that CD = M C 2 sin ^, approxi- J -j mately. Also, the volume s is in general either exactly or nearly I proportional to 2 sin ; so that if c be a constant volume depend- m ing on the figure of the water line, s = c 2 sin - , approximately. Z Consequently, to find the height M of the metacentre above the centre of buoyancy, and its lieight M G above the centre of gravity, we have the approximate formulae, SF8 ^ C D + 2 sin = M G = ^ z^ G C. The sign qp denotes that G C is to be subtracted or added according .(3.) 602 PRINCIPLES OF DYNAMICS. as Gr is above or below C. The product I c is found approximately in the following manner, for those cases in which, either from the smallness of the deviation, or from the figure of the vessel, or from both those causes, the water lines A B and E F are sensibly equal and similar figures, so that the line H, where their planes intersect, traverses the centre of gravity of each of those figures, and the wedges A H E, F H B, are similar as well as equal. $ The product I s = I c 2 sin - is the double of the statical mo- ment of one of the wedges relatively to the line H, supposing its density equal to unity. As in Article 83, case 12, let distances, measured lengthways, along the line H be denoted by y ; let the perpendicular distance of any point in a water line plane bisecting the angle A H E from the line H be denoted by x, and let the thickness of the wedge at the point whose co-ordinates are x and y be z = x - 2 sin - . Then by the formulae of the Article and case referred to, 6 r r r r s = 2 t sin g / / x d y d x' } c = / / x dy dx' } I s = 2 s X Q = 4 sin ^ * / \ x* ' dy d x; and therefore being the moment of inertia of the water line plane about the axis H. To express this in a convenient form, let b be the breadth of the ship at the water line, at a given distance y, measured length- ways from an assumed origin. Then and (5.) As to the moments of inertia of different plane figures, see Article 95. Thus, equation 3 becomes g zqpGC .................. (6.) The theory of the stability of ships was first investigated by Bossut. PITCHING AND ROLLING OF SHIPS. 603 The most important contributions to that theory of later date have been, the memoir of Mr. Dupin, Sur la Stabilize des Corps flottans, a paper by Mr. Moseley in the Philosophical Transactions for 1 850, and the experiments of Mr. Rawson. 655. Oscillations of Floating Bodies. The theory of the oscilla- tions of ships was investigated in an approximate manner by Bossut and other mathematicians, and was first brought into a complete state by Mr. Moseley, in the paper already referred to. Its details are of much complexity ; and an outline of its leading principles, and of their results in the most simple cases, is all that needs be given in this treatise. The oscillation of a ship may be resolved into rolling, or gyration about a longitudinal axis, pitching, or gyration about a transverse axis, and vertical oscillation, consisting in an alternate rising above and sinking below the position of equilibrium. The point of chief importance in practice is the time occupied by a rolling oscillation. One quarter of the time of a double rolling oscillation is the time the ship takes to right herself, and it is important that this should not be too long. If a ship is of such a figure that, when she rolls into a new posi- tion of equilibrium under the action of a couple, her centre of gravity does not alter its level, then her rolling gyrations will be performed about a permanent longitudinal axis traversing her centre of gravity, and will not be accompanied by vertical oscilla- tions, and her moment of inertia will be constant while she rolls. But if rolling into a new position of equilibrium alters the level of her centre of gravity, then her instantaneous axis of gyration and her moment of inertia must vary as she rolls, and her rolling must necessarily be accompanied by vertical oscillations. Such oscilla- tions tend to strain and damage the ship and her contents ; and it is therefore desirable that her figure should be such that her centre of gravity shall not alter its level when she rolls a condition ful- filled if all the water line planes, such as A B and E F, are tan- gents to one sphere described about G. In what follows it will be supposed that this condition is fulfilled, and also that the position in the ship of the metacentre M is sen- sibly constant. There are two methods of approximating to the time of gyration of a ship. By the first method, the inertia of the displaced mass of water, which has to be shifted from side to side as the vessel rolls, is neglected ; by the second, that inertia is considered as equivalent to the inertia of an equal mass, moving from side to side with a velocity at each instant equal to that of the centre of buoyancy ; that is, to the horizontal velocity of the metacentre, which is verti- cally above the centre of buoyancy. 604 PRINCIPLES OF DYNAMICS. According to Article 654, equation 1, the righting couple for a given angle of roll 6 is W ' GM sin 0; but in an approximate solution we may substitute & for sin &. Let I be the moment of inertia of the ship about her axis of rolling ; then, according to the first supposition, equations 2 and 3 of Article 598 give the following value for the time of a double gyration : , : ^-- where R is the radius of gyration of the ship. This is the same with the time of a double oscillation of a simple pendulum whose length is B, 2 -^ G M ; and to make it a minimum, the radius of gyration should be as small and the height of the metacentre as great as possible. But, according to the second mpposition, the time of oscillation of the ship is the same as if she formed a compound pendulum, hung from the metacentre M ; so that, by Article 604, the length of the equivalent simple pendulum is and the time of a double oscillation, J- (2. ) 2*- 9 * and to make this a minimum, as in Article 604, equation 2, we should have GM = R; L 2R (3.) It is probable that the truth lies between those two suppositions ; but itf either case, the dynwniical stability of a ship, which consists in her righting herself rapidly, is favoured by making her moment of inertia as small as is practicable. It is easily seen how the same principles may be applied to pitching, or gyration about a transverse axis. 656. The Action between a Fluid and a Piston* consisting in the transmission of energy from the one to the other, has already been considered in a general way in Article 517. In the present Article it will be treated more in detail. In figs. 260 and 261, let abscissae measured parallel to the line O S represent the spaces successively occupied by a fluid in a ACTION BETWEEN FLUID AND PISTON. (505 cylinder provided with a piston, any such space being denoted by s ; and let ordinates measured parallel to the line O P, perpendi- Fig. 260. E Fig. 261. cular to O S, represent the intensities of the pressure exerted by the fluid against the piston, any such intensity being denoted by p. Let a given weight of a gaseous substance go through a succes- sion of arbitrary changes of pressure and volume, so as to return in the end to the condition from which it set out. Such a succes- sion of changes is called a cycle of changes ; it is represented by a closed curve, such as D C E B in fig. 260, and the area of that curve represents the energy transferred during the cycle of changes. If the changes take place in the order D C E B, that is, if greater pressures are exerted during the expansion of the substance than during its compression, energy is transferred from the gas to the piston ; if the changes take place in the order D B E C, that is, if greater pressures are exerted by the substance during its compres- sion than during its expansion, energy is transferred from the pis- ton to the gas. The amount of energy transferred may be expressed in two ways. First, for any given volume O A = s, let A C = p l and A B = p 2 be the greater and the less intensities of the pressure ; then energy transferred = / (p l p 2 ) ds ( 1 . ) J ' V Secondly, for any given pressure O F = p, let F E = ^ and FD = s 2 be the greater and the less of the spaces occupied ; then energy transferred == f (s l s 2 ) dp (2.) which is another expression for the same quantity. Fig. 261 represents the case in which a given weight of an elastic substance occupying the space O E = s l at the pressure O B = p }) is introduced into a cylinder and made to drive a piston, is then 606 PEINCIPLES OF DYNAMICS. allowed to expand, its volume increasing to O F = s 2 , and its pres- sure falling to F D = p 2 , according to a law represented by the curve C D, and is lastly expelled from the cylinder at the final pressure. In this case the energy transferred from the elastic sub- stance to the piston is represented by areaABCD = /"* sdp = W ^ ^; '...(3.) being, in fact, as the last expression shows, equal to the weight of the elastic substance employed, W, multiplied by its loss of dyna- mic head. The same equation gives the energy transferred from the piston to the elastic substance, when the latter is introduced into the cylinder at the lower pressure and expelled at the higher. For a perfect gas (Article 635) this expression becomes If the fluid is discharged from the cylinder under a pressure p s less than that at which the expansion terminates, there is to be added to the preceding formula the term If the fluid which acts on the piston is introduced in the state of saturated vapour, it is discharged as a mixture of saturated vapour at a lower pressure with more or less of liquid. In this case, the following equations belonging to the science of thermo- dynamics are to be used. Let p be the pressure of saturation of a vapour, and T the corresponding boiling point of its liquid, in degrees reckoned from the absolute zero, 274 Centigrade or 493' 2 Fahrenheit below the melting point of ice. Then B__C i ~' r? '.v.ii <> 4C 2 J 2C J (See Edin. Philos. Jour., July, 1849 ; Edin. Transac., xx; Philos. Mag., Dec., 1854; Nicholas Cyclopcedia, art. "Heat, Mechanical Action of.") The following are the values of some of the constants in the above formulae, selected from a table in the Philosophical Magazine for Dec., 1854, p being in Ibs. per square foot, and r in degrees of Fahrenheit : HEAT AND WORK OF STEAM. 607 B LogB LogC B 2 Water,... 8 -25 9 1 3-43642 5-59873 0-003441 0-00001184 3-31492 5-21706 0-006264 0-00003924 Let L be the value, in foot pounds of energy, of the latent heat of evaporation, at the absolute temperature r, of so much fluid as fills a cubic foot more in the state of vapour than it does in the state of liquid ; D the weight of that fluid ; H the value, in foot pounds of energy, of the latent heat of evaporation of one pound of the fluid at the absolute temperature r ; and J the equivalent in foot pounds of a British thermal unit, or 772 ; then dp (hyp. log. 10 = 2-3026) ; H = H -J(c-6)(r-r ) (for water, c b = 0-7) ; D == L - H. J c denotes the value in foot pounds of the specific heat of the liquid, which for water is 772, and for sether, 399. Let the suffixes 1, 2, and 3, denote the pressures and tempera- tures respectively, of the introduction of the vapour, the end of its expansion, and its final discharge, and quantities corresponding to them; &\ and s 2 being, as before, the spaces filled by it at the begin- ning and end of its expansions. Then ratio of expansion, - = ~ \ - -f J c DI hyp log f ; ...... (8.) $1 -^2 I T l T 2 ) energy transferred. / ' s dp -f s a (p a -p 3 ) ^>+ Jc D, (r, - r 2 (l +hyplog 1' T l N T 8 heat expended ) in foot pounds, } f > [ L i + J (10.) These formulae are demonstrated in a paper on Thermodynamics in the Philosophical Transactions for 1854. The complexity of the preceding formulae renders their use incon- venient, except with the aid of tables of the quantities p, L, and D, for different boiling points. In the absence of such tables, the 008 PRINCIPLES OF DYNAMICS. following formulae give approximate results for steam, where the pressure of its admission pi is from one to twelve atmospheres : (11.) energy transferred from ) r p i steam to piston, / J ^ sd P + ' (*' ~ 10 ...(12.) The expenditure of heat in foot pounds may be computed roughly to about ^r, when the feed water is supplied to the boiler at about oU 100 Fahrenheit, by the formula * (13^ + 4,000) ..................... (13.) Those approximate formulae, 11, 12, and 13, are now published for the first time. PART VI. THEORY OF MACHINES. 657. Nature and Division of the Subject. In the present Part of this work, machines are to be considered not merely as modify- ing motion, but also as modifying force, and transmitting energy from one body to another. The theory of machines consists chiefly in the application of the principles of dynamics to trains of me- chanism ; and therefore a large portion of the present part of this treatise will consist of references back to Part IV. and Part V. There are two fundamentally different ways of considering a machine, each of which must be employed in succession, in order to obtain a complete knowledge of its working. I. In the first place is considered the action of the machine during a certain period of time, with a view to the determination of its EFFICIENCY ; that is, the ratio which the useful part of its work bears to the whole expenditure of energy. The motion of every ordinary machine is either uniform or periodical. Hence, as has been shown in Article 553, the principle of the equality of energy and work, as expressed in Article 518, is fulfilled either constantly or periodically at the end of each period or cycle of changes in the motion of the machine. II. In the second place is to be considered the action of the machine during intervals of time less than its period or cycle, if its motion is periodic, in order to determine the law of the periodic changes in the motions of the pieces of which the machine con- sists, and of the periodic or reciprocating forces by which such changes are produced (Article 556). The first chapter of the present Part relates to the work of machines moving uniformly or periodically, and the second chapter to variations of motion and force in machines. In a third chapter will be stated briefly the general principles of the action of the more important prime movers. With respect to those machines, it is impossible to enter fully into details within the limits of such a treatise as the present, especially as the most important of them all, the steam engine, depends on the laws of the phenomena of heat, which could not be completely explained except in a special treatise. 610 CHAPTER I. WOEK OF MACHINES WITH UNIFORM OR PERIODIC MOTION. SECTION 1. General Principles. 658. Useful and l^ost Work. The whole work performed by a machine is distinguished into useful work, being that performed in producing the effect for which the machine is designed, and lost work, being that performed in producing other effects. 659. Useful and Prejudicial Resistance are overcome in perform- ing useful work and lost work respectively. 660. The Efficiency of a machine is a fraction expressing the ratio of the useful work to the whole work performed, which is equal to the energy expended. The limit to the efficiency of a machine is unity, denoting the efficiency of a perfect machine in which no work is lost. The object of improvements in machines is to bring their efficiency as near to unity as possible. 661. Power and Effect; Horse Power. The power of a machine is the energy exerted, and the effect, the useful work performed, in some interval of time of definite length. The unit of power called conventionally a horse power, is 550 foot pounds per second, or 33,000 foot pounds per minute, or 1,980,000 foot pounds per hour. The effect is equal to the power multiplied by the efficiency. 662. Driving Point; Train; Working Point. The driving point is that through which the resultant effort of the prime mover acts. The train is the series of pieces which transmit motion and force from the driving point to the working point, through which acts the resultant of the resistance of the useful work. 663. Points of Resistance are points in the train of mechanism through which the resultants of prejudicial resistances act. 664. Efficiencies of Pieces of a Train. The useful work of an intermediate piece in a train of mechanism consists in driving the piece which follows it, and is less than the energy exerted upon it by the amount of the work lost in overcoming its own friction. Hence the efficiency of such an intermediate piece is the ratio of the work performed by it in driving the following piece, to the energy exerted on it by the preceding piece ; and it is evident that the efficiency of a machine is the product of the efficiencies of the series MEAN EFFORTS AND RESISTANCES GENERAL EQUATIONS. 611 of moving pieces which transmit energy from tfte driving point to the working point. The same principle applies to a train of successive machines, each driving that which follows it. 665. Mean Efforts and Resistances In Article 515 is given the expression / P d s for the energy exerted by a varying effort whose magnitude at any instant is P ; and a corresponding expression / ~Rds denotes the work performed in overcoming a variable re- sistance. In a machine moving uniformly, let these expressions have reference to any interval of time, and in a machine moving periodically, to one or any whole number of periods ; let s be the space described by the point of application of the effort or resist- ance in the interval in question ; then / P ds -=- s or I ~Rds -=- s is the mean effort or mean resistance as the case may be. The fluc- tuations of the efforts and resistances above and below their mean values concern only the variations of velocity in a machine ; and therefore, in the remainder of the present chapter, P and R will be used to denote such mean values only; so that energy exerted and work performed, whether the forces are constant or varying, will be respectively denoted by P s and R s. By referring to Articles 517 and 593, it appears, that besides a force and a length, as expressed above, the two factors of a quantity of energy may be a stress and a cubic space, or a couple and an angle, as shown in the following table : Force in pounds x distance in feet ; Couple in foot pounds x angular motion to radius unity; or Pressure in pounds per square foot x space described by a piston in cubic feet. ). The General Equations of the uniform or periodical working of a machine are obtained by introducing the distinction between useful and lost work into the equations of the conservation of energy. Thus, let P denote the mean effort at the driving point, s the space described by it in a given interval of time, being a whole number of periods or revolutions, Rj the mean useful resist- ance, s l the space through which it is overcome in the same inter- val, R 2 an y one of the prejudicial resistances, s 2 the space through which it is overcome ; then Ps - R lSl + 2 -R 2 s 2 (1.) The efficiency of the machine is expressed by .(2.) 612 THEORY OF MACHINES. R! S l Ps R^ + 2 -R 2 s 2 667. Equations in term* of Comparative Motions. Let S 1 : S = n lt s. 2 : s =. n 2 , &c., be the ratios of the spaces described in a whole num- ber of periods by the working point and the several points of resistance, to the space described, in the same interval of time, by the driving point ; then equation 1 of Article 666 takes the follow- ing form, which expresses the " Principle of Virtual Velocities " (Article 519) as applied to machines : P = TO! R, + 2- 2 R 2 , (1.) Thus the mean effort at the driving point is expressed in terms of the several mean resistances, and of the comparative motions alone, which last set of quantities are deduced from the construction of the machine by the principles of the theory of mechanism ; so that every proposition in Part IV., respecting the comparative motions of the points of a machine, can at once be converted into a proposi- tion respecting the relation between the mean effort and resistances \ and the mean effort required to drive the machine can be deter- mined if the resistances are known. 668. Redaction of Forces and Couples. In calculation it is often convenient to substitute for a force applied to a given point, or a couple applied to a given piece, the equivalent force or couple applied to some other point or piece ; that is to say, the force or couple, which, if applied to the other point or piece, would exert equal energy, or employ equal work. The principles of this reduction are, that the ratio of the given to the equivalent force is the reciprocal of the ratio of the velocities of their points of appli- cation ; and the ratio of the given to the equivalent couple is the reciprocal of the ratio of the angular velocities of the pieces to which they are applied. SECTION 2. On the Friction of Machines. 669. Co-efficients of Friction. The nature and laws of the fric- tion of solid surfaces, and the meanings of co-efficients of friction and angles of repose, have been explained in Articles 189, 190, 11)1, and 192. The following is a table of the angle of repose Of the co-efficient of friction f= tan (p, and its reciprocal 1 :f, for the materials of mechanism, condensed from the tables of General Morin, and other sources, and arranged in a few com- prehensive classes. The values of those constants which are given in the table have reference to the friction of motion. As to the difference between that and the friction of rest, see Article 304. FRICTION UNGUENTS. 613 No. SURFACES. / I:/ 1. 14to26A 25 to -5 4 to 2 I ni 2 5 3. Metals on oak, dry, 26to31 5 to -6 2 to 1-67 4. wet 13Atol4i 24 to 26 4'17 to 3 85 5, 1U ' 2 5 6 lHtol4 2 to -25 5 to 4 7. Hemp on oak, dry, " 28 53 1-89 8. g wet, Leather on oak 18 15 to 19i 33 27 to '38 3 3-7 to 2-86 10 29i 56 1-79 11. 20 36 2-78 12 13 23 4-35 13. 8 15 6-67 14, 8i to ll/, 15 to -2 6-67 to 5 15. wet, 16A " 3 3-33 16. 17. 18. Smooth surfaces, occasionally greased, continually greased, best results, 4 to 4i 3 lf to 2 07 to -08 05 03 to -036 14-3 to 12-5 20 33-3 to 27-6 670. Unguents. The last three results in the preceding table, Nos. 16, 17, and 18, have reference to smooth firm surfaces of any kind, greased or lubricated to such an extent that the friction depends chiefly on the continual supply of unguent, and not sensibly on the nature of the solid surfaces ; and this ought almost always to be the case in machinery. Unguents should be thick for heavy pres- sures, that they may resist being forced out, and thin for light pres- sures, that their viscidity may not add to the resistance. 671. Limit of Pressure between Rubbing Surfaces. The law of the simple proportionality of friction to pressure (Article 190) is only true for dry surfaces, when the pressure is not sufficiently intense to indent or grind the surfaces ; and for greased surfaces, when the pressure is not sufficiently intense to force out the unguent from between the surfaces, where it is held by capillary attraction. If the proper limit of intensity of pressure be exceeded, the friction increases more rapidly than in the simple ratio of the pressure. That limit diminishes as the velocity of rubbing increases, according to some law not yet exactly determined. The following are some of its values deduced from experience : Railway Carriage Axles. Velocity of rubbing 1 foot per second, j) ?? ^2 " jj ?) jj Timber ways for launching ships, about Limit of Pressure, Ib. per square inch. 392 224 140 50 614 THEORY OF MACHINES. The inclination given to these ways varies from about 1 in 10 for the smallest vessels, to about 1 in 20 for the largest. The co-efficient of friction, when the ways are well lubricated with tallow or soft soap, is probably between -03 and -04. 672 Friction of a Sliding Piece. In fig. 262, let A represent a sliding piece, which moves uniformly along the straight guide B B in the direction indi- r cated by the arrow, under two forces which "~~~ -p=CA\^^-B may be direct or oblique, but which are re- \^^ presented as oblique, to make the solution general. The force F 2 opposed to the motion, is the resultant of the useful resistance or Fig. 262 . force which A exerts on the next piece in the train, and of the weight of A itself, and will be called the given force. Let the angle which it makes with the guide B B be denoted by 4- The force Fj is that which drives the piece ; the angle ^ which its direction makes with the guide B B is supposed to be known j but its magnitude remains to be determined, as well as the friction, which it has to overcome in addition to the useful resistance. Let Q denote the normal pressure of A against B B, so that f Q is the friction. Then we have the two equations of equilibrium : Q = F, sin ij + F 2 sin i 2 ; Fj cos i x = F 2 cos ig + /Q = Fj/sin ii + F 2 (cos i 2 + /sin i 2 ) ; from which are easily deduced the following equations, solving the problem : ooB-fns .n+y cos ii .'/ sin i l ' 9 cos ii j sin ^ 1 673. The Moment of Friction of a rotating piece is the statical moment of the friction relatively to the axis of rotation of the piece, and is the moment of a couple consisting of the friction, and of an equal and opposite component of the pressure exerted by the bearings of the piece against its axle. The moment of friction, being multiplied by the angular motion in a given time, gives the work lost in fricion in that time. 674. Friction of an Axle. After a cylindrical axle has run for some time in contact with its bearing, the bearing becomes slightly larger than the axle, so that the point of most intense pressure, which is also the point of resistance, traversed by the resultant of the friction, adapts its position to the direction of the lateral pressure. FRICTION OF AN AXLE. 615 In fig. 263, let A A A be a transverse section of the cylindrical axle of a rotating piece, and C its axis of rotation ; let R represent the direction and magnitude of what will be called the given force, being the resultant of the useful resistance, and of the weight of the piece under consideration. Let P represent the effort required to drive the piece, whose line of action is known, but its magnitude remains to be determined. Let D be the point where the directions of P and R intersect, and D Q the line of action of their resultant, which resultant is equal and opposite to Q, the pressure exerted by the bearing against the axle, and is there- fore inclined to the radius C Q by an angle C QD = 9, being the angle of repose, in such a manner as to resist the rotation, whose direction is indicated by the arrow. Then to find the line of pressure D Q, it is obviously sufficient to describe about the centre C a circle B B whose radius is /~i 7~fs> \ ) r C Q being the radius of the axle, and to draw from the known point D a line D T Q touching that circle in T, which point of contact is at that side of the circle which makes a force acting from Q towards T oppose the rotation. From T draw T R JL R, and T P J- P. Then the magnitude of the effort P is given by the equation P = R TR -=- TP ..................... (2.) and that of the pressure Q by the equation Q 2 = P 2 + R 2 + 2 P R cos ^ P D R .......... (3.) (the last term of which becomes negative when ^ P D R is obtuse) ; while the friction is and its moment Q r sin

. p. sec. RVT ft. Ib. p. day. 1. Raising his own weight up stair 143 0-5 8 72-5 2,088,000 2. Do. do. do., 10 2,616,000 3. 4. 5. 6. 7. (Tread-wheel, see 1.) Hauling up weight with rope, Lifting weights by hand, Carrying weights up stairs, Shovelling up earth to a height 40 44 143 6 0-75 0-55 0-13 1-3 6 6 6 10 30 24-2 18-5 7-8 648,000 522,720 399,600 280,800 8. 9. Wheeling earth in barrow up slope of 1 in 12, horiz. veloc. 0'9 ft. per sec. (return, empty), Pushing or pulling horizontally fcaDstan or oar} . 132 26-5 0-075 2-0 10 8 9*9 53 356,400 1,526 400 10. 11. Turning a crank or winch, {12-5 18-0 20-0 13'2 5-0 2-5 14-4 2-5 9 8 (2 mins.) ' 10 ' 62-5 45 288 33 1,296,000 1,188,000 12. 15 ? 8? ? 480,000 13. HORSE. Cantering and trotting, draw- ing a light railway carriage (thoroughbred) (min. 22) -J mean3 o|y ( max 50 j Wf 4 447 6,444,000 14. Horse drawing cart or boat, walking (draught horse), 120 3-6 8 432 12,441,600 696. A -Water Pressure Engine consists essentially of a working cylinder, in which water moves a piston in the manner stated in Article 499, case 2. Let h be the virtual fall, that is, the excess of the dynamic head of the water entering the cylinder above that of the water leaving the cylinder ; Q the volume of water supplied per second; ^ its weight per unit of volume; 1 k the efficiency of the engine; then (1 fy WATER-WHEELS. 627 is its effect per second. In well constructed water pressure en- gines, 1 k varies from '8 to '85. 697. Water-wheels in General. Water may act on a wheel either by its weight and pressure, or by its velocity; that is, either by its potential, or by its actual energy. See Article 622. Let Q denote the weight of water, in pounds, supplied to the wheel in a second ; h the difference of dynamic head, in feet, of the water before and after its action on the wheel ; v l the velocity of the water, in feet per second, just before it begins to press on the wheel, or supply-velocity ; v 2 the velocity of the water just after it has ceased to act on the wheel, or discharge-velocity. Then the total energy of the water, as in Article 622, is e Q (j l H~ ep- ) ft pounds per second ; the energy of the water when discharged, v* { Q ~ t foot pounds per second; *g the total power of the wheel, e Q (/i + -^~- - J foot pounds per second; ........ (1.) the maximum theoretical efficiency, the quantity, may be called the theoretical fall or head. The available efficiency of a water-wheel falls short of the maximum theoretical efficiency principally from the following causes : 1. The resistance of the channel and orifices by which the water is supplied, which causes the actual height from which the water must descend in order to acquire the supply-velocity v to be greater than v\ : 2 g. The effect of such resistance is expressed by putting for the actual fall, 2 f being the co-efficient of resistance of the channel and orifices of supply, determined according to the principles of Articles 638 to 646. 2. The escape of part of the water before it has completed its action on the wheel. 3. The agitation and mutual friction of the 628 THEORY OF MACHINES. particles of water acting on the wheel ; and, 4. The friction of the wheel. The effects of the last three causes are expressed by multi- plying the total power and the theoretical efficiency of the wheel by an empirically determined fractional co-efficient k ; so that the effect or available power is denoted by (1 k) e Q h, and the available efficiency by a 7 \ 7 \ . / ) &i H 698. Classes of Water- Wheels. Water-wheels may be classed as follows: Overshot-wheels and breast-wheels, undershot-wheels and turbines. 699. Overshot and Breast-Wheels. The water is supplied to this class of wheels at or below the summit, and acts wholly, or partly by its weight, as it descends in the buckets. (See Article 634). Formerly the buckets used to be closed at their inner sides, but now they are made with openings for the escape and re-entrance of air : an invention of Mr. Fairbairn. A breast- wheel differs from an overshot- wheel chiefly in having the water poured into the buckets at a somewhat lower elevation as compared with the summit of the wheel, and in being provided with a casing or trough, called a breast, of the form of an arc of a circle, extend- ing from the regulating sluice to the commencement of the tail- race, and nearly fitting the periphery of the wheel, which revolves within it. The effect of the breast is to prevent the overflow of water from the lips of the buckets until they are over the tail-race. The usual velocity of the periphery of overshot and high breast- wheels is from three to six feet per second ; and their available efficiency, when well designed and constructed, is from 075 to 0*85. 700. Undershot-Wheels are driven by the impulse of water, dis- charged from an opening at the bottom of the reservoir with the velocity produced by the fall, against floats or boards, as to which see Article 649. Every such wheel has a certain velocity of maximum efficiency, which does not in any case differ much from lialf the velocity of the water striking it. In undershot- wheels of the old construction, the floats are flat boards in the direction of radii of the wheel; and the maximum theoretical efficiency is t j. The available efficiency is about 0-3. This class of wheels was much improved by Poncelet, who curved the floats with a con- cavity backwards, adjusting their position and figure so that the water should be supplied to them without shock, and should drop from them into the tail-race without any horizontal velocity. The available efficiency of such wheels is about 0'6. TURBINES WINDMILLS HEAT ENGINES. 629 701. A Turbine is a horizontal water-wheel with a vertical axis, receiving and discharging water in all directions round that axis : that is, driven by a vortex (see Article 650). 702. 'Windmills are driven by the impulse of the air against oblique surfaces called sails, rotating in a plane perpendicular to the direction of the wind. The best figure and proportions for windmill sails, as determined experimentally by Smeaton, are given by the following formulae, in which the whip means, the length of an arm, or the distance of the f tip of a sail from the axis : length of sail, - whip : breadth at end 6 nearest axis, whip : at tip, - whip : angles made by the surface o o of the sail with the plane of rotation ab the end nearest the axis, 18 : at the tip, 7. (See Smeaton on Windmills, in Tredgold's Hydraulic Tracts.} 703. The Efficiency of Heat Engines is the subject of a peculiar branch of science, Thermodynamics ; and an outline only of the principles on which it depends can here be given. If the number of British Fahrenheit units of heat produced by the combustion of one pound of a given kind of fuel, be multiplied by Joule's equivalent, 772 foot pounds, the result is the total Iteat of combustion of the fuel in question, expressed in foot pounds. For different kinds of coal, it varies from 5,000,000 to 11,000,000 foot pounds. This total heat is expended, in any given engine, in pro- ducing the following effects, whose sum is equal to the heat so expended :- 1. The waste heat of the furnace, being from 0'15 to 0-6 of the total heat, according to the construction of the furnace, and the skill with which the combustion is regulated. 2. The necessarily rejected heat of the engine, being = x the heat *i received by the elastic fluid : ^ being the upper, and t a the lower limits of absolute temperature, which is measured from the absolute zero, 493 0< 2 Fahrenheit below the melting point of ice. 3. The heat wasted by the engine, whether by conduction, or by non-fulfilment of the conditions of maximum efficiency. 4. The useless work of the engine, employed in overcoming friction and other prejudicial resistances. 5. The useful work. The efficiency of a thermodynamic engine is improved by diminishing as far as possible the first four of these effects, so as to increase the fifth. The efficiency of a heat engine is the product of three factors; viz. : the efficiency of the furnace, being the ratio of the heat 630 THEORY OF MACHINES. transferred to the elastic fluid to the total heat of combustion ; the efficiency of the fluid, being the fraction of the heat received by it which is transformed into mechanical energy ; and the efficiency of the mechanism, being the fraction of that energy which is available for driving machines. The maximum efficiency of the fluid between given limits of absolute temperature is expressed by (i.) As to the mechanical action of an elastic fluid 011 a piston, see Article 656. 704. steam Engines. Formulae for the mechanical action of steam on a piston, both exact and approximate, have been given in Article 704, equations 6 to 13. The efficiency of the steam lies between the limits '02 and -15 in extreme cases, and -04 and '1 in ordinary cases. The details of the construction and working of steam engines can be explained in a special treatise only. The duty of an engine is the work performed by a given quantity of fuel, such as one pound. The duty of a pound of coal varies in different classes of engines from about 100,000 to 1,200,000 foot pounds. These are extreme results, as respects wastefulness on the one hand, and economy on the other. In good ordinary engines, the duty varies from 200,000 to 500,000. 705. JElectrodynamic Engines, though capable of higher efficiency than heat engines, are not so economical commercially, on account of the greater cost of the materials consumed in them. Their theo- retical efficiency, according to a law demonstrated by Mr. J[oule, is given by the formula where yi is the strength which the electric current would have if the machine performed no mechanical work, and y 2 is the actual strength of the current. This law, and the law of the maximum efficiency of heat engines, are particular cases of a general law which regulates all transforma- tions of energy, and is the basis of the Science of Energetics.* * Edinburgh Philosophical Journal, July, 1855; Proceedings of the Philosophical Society of Glasgow, 1853-5. APPENDIX. TABLE OF THE RESISTANCE OF MATERIALS TO STRETCHING AND TEARING BY A DIRECT PULL, in pounds avoirdupois per square inch. Modulus of Elasticity, or Res i stan i e to MATERIALS. Tenacity, or Resistance to Tearing. STONES, NATURAL AND ARTIFICIAL : Glass, Mortar, ordinary, METALS : Brass, cast, Bronze or Gun Metal (Copper 8, Tini), Copper, cast, sheet, bolts,... wire, Iron, cast, various qualities, average, Iron, wrought, plates, joints, double rivetted, single rivetted, bars and bolts,, hoop, best-best,. wire-ropes, Lead, sheet, Steel, Tin, cast, Zinc, 9)4 18,000 49,000 36,000 19,000 30,000 36,000 60,000 13,400 to 29,000 16,500 51,000 28,600 60,000 to 70,000 64,000 70,000 to 100,000 90,000 3,300 100,000 to 130,000 4,600 7,000 to 8,000 Stretching. 8,000,000 13,000,000 to 16,000,000 9,170,000 14,230,000 9,900,000 17,000,000 14,000,000 to 22,900,000 17,000,000 29,000,000 V/4J-j\-/WW I 70,000 ) ( to 100,000 J { 25,300,000 15,000,000 720,000 29,000,000 to 42,000,000 632 APPENDIX. Tenacity. Modulus of MATERIALS. or Resistance to Elasticity, Tearing. Stretching. TIMBER AND OTHER ORGANIC FIBRE: Acacia, false. See " Locust." Ash (Fraxinus excelsior), 17,000 1,600,000 Bamboo (Bambusa arundinacea), 6,300 Beech (Fagus sylvatica), 11,500 1,350,000 Birch (Betula alba), 15,000 1,645,000 Box (Buxus sempervirens), 2O,OOO Cedar of Lebanon (CedrusLibani), 11,400 486,000 Chestnut (Castanea Vesca), i to 13^000 / 1,140,000 Elm ( Ulmus campestris), 14,000 \ to 1,340,000 Fir : Red Pine (Pinus sylvestris), fc$. to i;i3E Larch (Larix Europcea), | 9,000 ( to 10,000 900,000 to 1,360,000 tt&wtlLOYn(CratcegusOxyacantha,), 10,500 Hazel (Corylus A vellana), 18,000 Hempen Cables, 5,600 Holly (Ilex A quifolium), 16,000 Hornbeam (Carpinus Betulus),.. . 20,000 Laburnum (Cytisus Laburnum), 10,500 Lancewood (Guatteria virgata), 23,400 Lignum- Yits3 (Guaiacum offici- ) TVOLLQ i 1 1, 800 Locust (Robinia Pseudo-Acacia), 16,000 ( 8 nnr> ) Mahogany (Swietenia Mahagoni), O-UUU f } to 21,800 J 1,255,000 Maple (A cer campestris), IO,6oo Oak, European (Quercus sessili- / IO,OOO 1,200,000 Jlora and Quercus pedunculata), \ to 19,800 to 1,750,000 American Red (Quercus ) rubra), J 10,250 2,150,000 Poplar (Populus alba), Sycamore(2 cerPseudo-Platanus), 7,200 13,000 1,130,000 1,040,000 Teak, Indian (Tectona grandis), 15,000 2,400,000 African, (?) 21,000 2,3OO,OOO Whalebone, 7,700 Yew (Taxus toccata), 8,OOO APPENDIX. 633 II. TABLE OF THE RESISTANCE OF MATERIALS TO SHEARING DISTORTION, in pounds avoirdupois per square inch. Transverse Resistance Elasticity, MATERIALS. to^ or Resistance to METALS : Distortion. Brass, wire-drawn, ....................... 5j33? oo Copper, ................................... 6,200,000 Iron, cast, ................................. 32,500 2,850,000 8.S * TIMBER : Fir: Red Pine, ......................... 5ooto 800 Spruce, ............................. 600 Larch, ............................... 970 to 1,700 Oak, British, .............................. 2,300 Poplar, .................................... i, 800 III. TABLE OF THE RESISTANCE OF MATERIALS TO CRUSHING BY A DIRECT THRUST, in pounds avoirdupois per square inch. Resistance MATERIALS. to Crushing. STONES, NATURAL AND ARTIFICIAL: Brick, weak red, 5 50 to 800 strong red, 1,100 fire, 1,700 Chalk, 330 Granite, 5,5oo to 11,000 Limestone, marble, 5?5oo granular, 4,000 to 4,500 Sandstone, strong, 5,5oo ordinary, 3,300 to 4,400 weak, 2,200 Rubble masonry, about four- tenths of cut stone. METALS : Brass, cast, 10,300 Iron, cast, various qualities, 8 2,000 to 1 45,000 average, 112,000 wrought, about 36,000 to 40,000 634 APPENDIX. Resistance MATERIALS. to Crushing. TIMBER,* Dry, crushed along the grain : Ash, .................................................. 9> Beech, ................................................ 9>3 6 Birch, ...................... .' ......................... 6,400 Blue-Gum (Eucalyptus Gflobulus), .............. 8,800 Box, ..................... . ............................ 10,300 Bullet- tree (A ckras Sideroxylon), ............... 1 4,000 Cabacalli, ........................................... 9,900 Cedar of Lebanon, ................................. 5>^6o Ebony, West Indian (Brya Ebenus), .......... 1 9,000 Elm, .................................................. 10,300 Fir: Red Pine, .................. . ........ 5,375 to 6,200 Larch, ........................................ 5,570 Hornbeam, ......................................... 7,3 Lignum- Vitae, ...................................... 9>9o Mahogany, .......................................... 8,200 Mora (Mora excelsa), .............................. 9,9 Oak, British, ....................................... 10,000 Dantzic, ....................................... 7,7 American Red, .............................. 6,000 Teak, Indian, ....................................... 12,000 Water-Gum (Tristania nerifolia), ............. 1 1,000 IV. TABLE OF THE RESISTANCE OP MATERIALS TO BREAKING ACROSS, in pounds avoirdupois per square inch. Resistance to Breaking, MATERIALS. or Modulus of Ruptnre.f STONES : Sandstone, 1,100 to 2,360 Slate, 5,ooo * The resistances stated are for dry timber. Green timber is much weaker, having sometimes only half the strength of dry timber against crushing. f The modulus of rupture is eighteen times the load which is required to break a bar of one inch square, supported at two points one foot apart, and loaded in the middle between the points of support. APPENDIX. 635 Resistance to Breaking, MATERIALS. or Modulus of Kupture. METALS : Iron, cast, open-work beams, average, 1 7,000 solid rectangular bars, var. qualities, 33,000 to 43,500 average, 40,000 wrought, plate beams, 42,000 TIMBER : Ash, 1 2,000 to 1 4,000 Beech, 9,000 to 1 2,000 Birch, 1 1,700 Blue-Gum, 1 6,000 to 20,000 Bullet-tree, 15,900 to 22,000 Cabacalli, 15,000 to 16,000 Cedar of Lebanon, 7,4o Chestnut, 10,660 Cowrie (Dammara australis), 1 1 ,000 Ebony, West Indian, 27,000 Elm, 6,000 to 9,700 Fir: Red Pine, 7,100 to 9,540 Spruce, 9,900 to 12,300 Larch, 5,000 to 10,000 Greenheart (Nectandra Rodicei\ 16,500 to 27,500 Lancewood, 1 7,350 Lignum- Vitse, 1 2,000 Locust, 11,200 Mahogany, Honduras, 11,500 Spanish, 7,600 Mora, 22,000 Oak, British and Russian, ' 10,000 to 13,600 Dantzic, 8,700 American Red,.. 10,600 Poon, i3>3 Poplar, Abele, 10,260 Sycamore, 9, 600 Teak, Indian, I4,77o African, 14,980 Tonka (Dipteryx odorata), 22,000 Water-Gum, 17,460 Willow (Salix, various species), 6,600 636 APPENDIX. ON co 10 ON ON Os O ^ iO CO ON ON ON ON co M ON ON O 00 iO O O O O M ON O N 00 VO T t> IO IH Tj- VO VO M O N "* M ON O 00 vo O O *$ TVO 00 IO O 00 O O vo Op vp O 7!- T*- N ON O |H st* |C4 |M M O co M ON oo M CO 00 M N J>. IO IO O VO VO VO N O N OO IO rh *^ OO 7$- M M vp IN |M b b . $ ^ O O O ON M O 00 N rf vo N T ON 10 co co oo ONJ>.IOOO * co O ON M co ON 00 CO CO IO 10 CO M OO T ON M ON ON CO MCOONlOlO.t O M M O N O |N |M M |CO M oo Tf co & J>. ON M 00 VO M ON CO j>. ON o 1000 N b 00 IO 00 ON M IO 7953553 IO VO vo 00 ON 10 ^ IO N vo N ON CO CO covo O ONJT^OO '* N -* N ON O covo O CO O TOO t^ M j^ 10 TNOOOCONX IO " ONOO ONVO 2 M b ^P b 8 O I TO N VO T J^- vo M 10 M co ON co co M ^ W vo" COJ^ ^P O ffl APPENDIX. 637 VI. TABLE OP SPECIFIC GRAVITIES OF MATERIALS. Weight of a cubic G-ASRB, at 32 Fahr., and under the pressure of one foot in atmosphere, of 2116-4 U>. on the square foot: lb - avoirdupois. Air, 0-0807 28 Carbonic Acid, 0-12344 Hydrogen, 0-005592 Oxygen, 0-089256 Nitrogen, 0*078596 Steam (ideal), 0-05022 ^Bther vapour (ideal), 0-2093 Bisulphuret-of-earbon vapour (ideal), 0-2137 Olefiant gas, 0-0795 Weight of a cubic Specific foot in gravity, lb. avoirdupois. pure water = 1. LIQUIDS at 32 Fahr. (except Water, which is taken at 39'4 Fahr.): Water, pure, at 39"4, 62-425 i-ooo sea, ordinary, 64-05 1-026 Alcohol, pure, 49*38 0791 proof spirit, 57' J 8 0-916 447o 0716 Mercury, 848-75 I3'59 6 Naphtha, 52*94 0-848 Oil, linseed, 58*68 0-940 -, olive, 57*12 0-915 whale, 57*62 0*923 of turpentine, 54'S 1 0-870 Petroleum, 54'8i 0-878 SOLID MINERAL SUBSTANCES, non-metallic : Basalt, 187*3 3' Brick, 125 to 135 2 to 2-167 Brickwork, 112 1*8 Chalk, 117 to 174 1-87 to 2*78 Clay, 120 1*92 Coal, anthracite, 100 1-602 bituminous, 77-4 to 89-9 1*24 to 1*44 Coke, 6V43 * 103-6 I'oo to 1-66 Felspar, 162-3 2-6 Flint,.., 164-2 2-63 638 APPENDIX. Weight of a cubic Specific foot in gravity, Ib. a\oirdupois. pure water = 1. SOLID MINERAL SUBSTANCES continued. Glass, crown, average, 156 2-5 flint, 187 3-0 green, 169 27 P^te, 169 27 Granite, 164 to 172 2-63 to 2-76 Gypsum, I43' 6 2 '3 Limestone (including marble),.. 169 to 175 2-7 to 2-8 magnesian, 178 2-86 Marl, iootoii9 1-6 to 1*9 Masonry, 116 to 144 1-85 to 2-3 Mortar, 109 175 Mud, 102 1-63 Quartz, 165 2-65 Sand (damp), 1 18 1-9 (^7), 88-6 1-42 Sandstone, average, 144 2-3 various kinds, 130 to 157 2 '08 to 2-52 Shale, 162 2-6 Slate, 175 to 181 2-8 to 2-9 Trap, 170 272 METALS, solid : Brass, cast, 487 to 524-4 7 -8 to 8-4 wire, 533 8-54 Bronze, 524 8-4 Copper, cast, 537 8-6 sheet, ' 549 8-8 hammered, 556 8-9 Gold, n86toi224 19 to 19*6 Iron, cast, various, 434 to 456 6 -95 to 7 -3 average, ; 444 7-11 Iron, wrought, various, 47 4 to 487 7 *6 to 7 '8 average, 480 7-69 Lead, 712 11-4 Platinum, 1311 to 1373 21 to 22 Silver, 655 10-5 Steel,.. 487 to 493 7-8 to 7-9 Tin, 456 to 468 7-3 to 7-5 Zinc, 424 to 449 6-8 to 7-2 APPENDIX. 639 Weight of a cubi c Specific TIMBER :* foot in Ib. avoirdupois. gravity, pure water = 1. Ash, 47 0753 Bamboo, 25 0-4 Beech, 43 0-69 Birch, 44'4 0711 Blue-Gum, 52-5 0-843 Box, 60 0-96 Bullet-tree, 65-3 1-046 Cabacalli, 56-2 0-9 Cedar of Lebanon, 30-4 0-486 Chestnut, 33'4 0*535 Cowrie, 36-2 Q'579 Ebony, West Indian, 74'5 1-193 Elm, 34 0-544 Fir: Red Pine, 30 to 44 0-48 to 07 Spruce, 30 to 44 0-48 to 0-7 American Yellow Pine,... 29 0-46 Larch, 3i t 35 0-5 to 0-56 Greenheart, 62-5 I'OOI Hawthorn, 57 0-91 Hazel, 54 0-86 Holly, 47 076 Hornbeam, 47 076 Laburnum, 57 0-92 Lance wood, 42 to 63 0-675 * I ' I Larch. See " Fir." Lignum- Vitse, 41 to 83 0-65 to 1-33 Locust, 44 071 Mahogany, Honduras, 35 0-56 Spanish, 53 0-85 Maple, 49 079 Mora, 57 0-92 Oak, European, 43 to 62 0-69 to 0-99 American Red, 54 0-87 Poon, , 36 0-58 Poplar (Abele), 32 0-51 Sycamore, , 37 0-59 Teak, Indian, 41 0-66 African, 61 0-98 Tonka, 62 to 66 0-99 to i -06 Water-Gum, 62-5 i-ooi Willow, 25 0-4 Yew, : 5o 0-8 * The Timber in every case is supposed to be dry. 640 APPENDIX. VII. DIMENSIONS AND STABILITY OF THE OUTER SHELL OF THE GREAT CHIMNEY OF ST. ROLLOX. Greatest pres- Divisions of Heights above External Chimney. Ground. Diameters. Security. Feet. Feet. Inches. Feet Inches. lb. per square foot. 435i 13 6 \ I 2 35i 16 9 77 IV. - I 6 2I0 i 24 o \ 55* III. \ I loi 114^ 30 6 57 II. V 2 3 f 54i 35 o 63 I. > 2 74 40 o 71 Depth below External Thicknesses. Foundation. Ground. Diameter. Concrete. Brick. Feet. Feet. Feet. Inches. Feet. I. O 50 5 O Q 8 50 4 8 3 II. : 14 50 12 III. 20 50 } 25 Total height from base of foundation to top of chimney, 455| feet. * Joint of least stability. ERRATA. 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