Mo. GIFT Of Daughters of Frederick Slate ftfefl. 'i / /^ \& A MANUAL OF APPLIED MECHANICS. to, -'I-/ In One Volume, Royal Boo, with Portrait on Sleel, Plates, and Diagrams. A SELECTION FROM THE MISCELLANEOUS SCIENTIFIC PAPERS OF W. J. MACQUORN RANKINE, C.E, LL.D., F.R.S., From the Transactions and Proceedings of the Royal and other Scientific and Philosophical Societies, and the Scientific Journals. WITH AN INTRODUCTORY MEMOIR OF THE AUTHOR, Br P. G. TA1T, M.A., Prof, of Natural Philosophy in Vie University of Edinburgh. EDITED BY W. J. MILLAE, C E., Sec. to the Inst. of Engineers and Shipbuilders in Scotfand. CHARLES GRIFFIN AND COMPANY, LONDON. A MANUAL I OF APPLIED MECHANICS BY WILLIAM JOHN MACQUORN RANKINE, CIVIL ENGINEER ; LL.D. TKIN. COLL. DUB. ; F.E BS. LOSD. AND EDIN. ; F.R.S.S.A. ; LATE REGIUS PBOFESSOR OF CIVIL ENGINEERING AND MECHANICS IN THE UNIVERSITY OS GLASGOW. With Numerous IHagrams. NINTH EDITION, THOROUGHLY REVISED, BY EDWARD FISHER BAMBER, C.E. LONDON: CHARLES GRIFFIN AND COMPANY, STATIONERS' HALL COURT. 1877. [The Right of Translation Reserved.} _. \) V fe^aJsAJs. e. v . v^V ^ V tX /v \ c 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 Lemons sur la TJieorie maihematique de fjElasticite des Corps solides. 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 Memoir es des Savans etr angers. 994578 iv PBEFACE. 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 and 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. K. GLASGOW UNIVERSITY, May, 1858. ADVERTISEMENT TO THE NINTH EDITION. This Ninth Edition has been carefully revised, and contains n^ alterations and additions. E. F. B. LONDON, October, 1877. CONTENTS. PRELIMINARY DISSERTATION ON THE HARMONY OF THEORY AND PRACTICE Page INTRODUCTION, Definition of General TCJ Article Page ms, and Division of the Subjec(, 8. Rest, . . . . . 13 14 14 14 15 15 15 13 15 t,16 9. Motion, 10. Fixed Point, 11. Cinematics, 12. Force, 13. Equilibrium or Balance, 14. Statics and Dynamics, 15. Structures and Machines, 16. General Arrangement of the i Jubjec 2. Applied Mechanics, ... 13 3. Matter, 13 4. Bodies, Solid, Liquid, Gaseous, 13 5. Material or Physical Volume, . 13 6. Material or Physical Surface, . 13 7. Line, Point, Physical Point, Mea- sures of Length, . . .13 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, . 18 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 CF PARALLEL FORCES. SECTION 3. On Parallel Forces. 38. Balanced Parallel Forces in general, 25 39. Equilibrium of three Parallel Forces in one Plane Principle of Lever, 26 40. Resultant of two Parallel Forces, SECTION 1. On Couples with the Same Axis. 25. Couples, 21 26. Force of a Couple Arm or Lever- age Moment, . . .21 27. Tendency of a Couple Plane and Axis of a Couple Right-handed and Left-handed Couples, . 21 28. Equivalent Couples of equal Force and Leverage, . . .21 29. Moment of a Couple, ... 22 30. Addition of Couples of equal Force, 22 31. Equivalent Couples of equal Mo- ment, 22 32. Resultant of Couples with the same Axis 23 33. Equilibrium of Couples having the same Axis, . . . .23 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 41. Resultant of a Couple, and a single Force in Parallel Planes, . 42. Moment of a Force with respect to an Axis, .... 43. Equilibrium of any system of Par- allel Forces in one Plane, 44. Resultant of any number of Paral- lel Forces in one Plane, . 45. Moments of a Force with respect to a pair of Rectangular Axes, 46. Equilibrium of any system of Par- allel Forces, .... 47. Resultant of any number of Paral- lel Forces, .... SECTION 4. On Centres of Parallel Forces. 48. Centre of a pair of Parallel Forces, 49. Centre of any system of Parallel Forces, 50. Co-ordinates of Centre of Parallel Forces, 26 27 27 23 28 29 N :]) 1 32 32 CHAPTER III BALANCE OP INCLINED FORCES. 52. Equilibrium of three Forces acting through one Point in one Plane, 36 53. Equilibrium of any system of 35 Forces acting through one Point, 3G SECTION 1. Inclined Forces applied at One Point. 51. Parallelogram of Forces, VI CONTENTS. Page 54. Parallelepiped of Forces, . . 37 55. Resolution of a Force into two Components, . . . .37 56. Resolution of a Force into three . . .37 SECTION 2. Inclined Forces applied to a System of 'Paints. 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 CHAPTER IV ON PARALLEL PROJECTIONS IN STATICS. 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 Components, . . 57. Rectangular Components, 61. Parallel Projection of a Figure de- nned, 62. Geometrical Properties of Parallel Projections, . 63. Application to Parallel Forces, . 38 CHAPTER V. ON DISTRIBUTED FORCES. 67. Restriction of the subject to Par- allel Distributed Forces, . . 48 68. Intensity of a Distributed Force, 48 SECTION 1. 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 73. System of Symmetrical Bodies, . 50 74. Homogeneous Body of any Figure, 51 75. Centre of Gravity found by Addi- tion, . . . . .53 76. Centre of Gravity found by Sub- traction, 53 77. Centre of Gravity altered byTrans- 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, .... 68 SECTION 2. Of Stress, and its Resultants and Centres. 86. Stress its Nature and Intensity 87. Classes of Stress, 88. Resultant of Stress, its Magni- tude, ..... 89. Centre of Stress, or of Pressure, 90. Centre of Uniform Stress, . 91. Moment of Uniformly -Varying Stress, Neutral Axis, . 92. Moment of Bending Stress, 93. Moment of Twisting Stress, 94. Centre of Uniformly - Varying Stress, Conjugate Axis, . 76 95. Moments of Inertia of a Surface, 77 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, . . 90 106. Transformation of Stress, . . 92 107. Principal Axes of Stress, . . 93 108. Stress Parallel to one Plane, . 95 109. Principal Axes of Stress Parallel to one Plane, . . . .98 110. Equal Principal Stresses Fluid Pressure, . . . .99 111. Opposite Principal Stresses com- posing Shear, .... 101 112. Ellipse of Stress, Problems, . 101 113. Combined Stresses in one Plane, 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 InternalEqui- 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 Solid, . . 126 126. On the Parallel Projection of Stress and Weight, . . .127 CONTENTS. vu CHAPTER VI ON STABLE AND UNSTABLE EQUILIBRIUM. Page ! 127 Stable and Unstable Equilibrium 128. Stability of a Fixed Body, . 128 of a Free Body,' . . . 128 ' PART II. THEORY OF STRUCTURES. CHAPTER I DEFINITIONS AND GENERAL PRINCIPLES.' 129. Structures Pieces Joints, . 129 131. Conditions of Equilibriur of a 130. Supports Foundations, 129 Structure, 129 132. Stability, Strength, and Stiffiiess, 130 CHAPTER II. STABILITY. 133. Resultant Gross Load, . . 131 I 172. Suspension Bridge with Sloping 134. Centre of Resistance of a Joint, 131 Rods, . . . . - 17J 135. Line of Resistance, . . . 131 I 173. Extrados and Intrados, . .173 136. Joints Classed, . . . 131 I 174. Cord with Horizontal Extrados, 175 SECTION 1. Equilibrium and Stability . \1 b -' Catenary^ .^ . _. TO - . ti - 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 j 145. Frame of Two Bars Equilibrium, 136 ! 146. Frame of Two Bars Stability, 136 I 147. Treatment of Distributed Loads, 137 148. Triangular Frame, . . .137 149. Triangular Frame under Parallel Forces, 138 150. Polygonal Frame Equilibrium, 139 151. Open Polygonal Frame, . . 140 152. Polygonal Frame Stability. . 140 153. Polygonal Frame under Parallel Forces, 141 154. 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, . 162 176. Centre of , 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 Frictional 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,21] 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 pwnWeight-,216 198. Pressure of Earth against a Ver- tical Plane, -.- 218 SECTION 2. Equilibrium of Chains, Cords, Eibs, 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, . . . .169 219 220 221 199. Supporting Power of Earth Foundations, . 200. Abutting Power of Earth, 201. Table of Examples, . 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 Vlll CONTENTS. SECTION 4. On the Stability of Abut- ments and Vaults. 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, 233 211. Moment of Stability, 233 212. Abutments classed, . 235 213. Buttresses in general, 235 214. Rectangular Buttress, 238 215. Towers and Chimneys, 240 216. Dams or Reservoir- Walls, 243 *217. Retaining Walls in general, 249 218. Rectangular Retaining Walls. 252 219. Trapezoidal Walls, . . ' 254 220. 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 23.5. Strength of Abutments and V.iults,268 235A. Transformation of Structures in Masonry, . . . .268 CHAPTER III. STRENGTH AND STIFFNESS. . 270 . 270 . 270 . 271 . 271 . 272 SECTION 1. Summary of General Principles. 236. Theory of Elasticity, . 270 237. Elasticity defined, " . 238. Elastic Force or Stress, 239. Fluid Elasticity, 240. Liquid Elasticity, 241. Rigidity or Stiffness, . 242. Strain and Fracture, >J43. Perfect and Imperfect Elasticity Plasticity, - . . .272 244. Strength, Ultimate and Proof 1 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, . 254. Co-efficients of Pliability, . 255. Axes of Elasticity, . 256. Isotropic Solid, ... 257. Modulus of Elasticity, 258. Examples of Co-efficients, . 259. General Problem of Internal Equilibrium of an Elastic Solid, 280 SECTION 2. On Relations between Strain and Stress. 260. Ellipse of Strain, 261. Ellipsoid of Strain, . 262. Transverse Elasticity of an Iso tropic Substance, . . 284 263. Cubic Elasticity, . . 285 264. Fluid Elasticity, . . 285 277 277 278 278 279 279 280 283 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, .... 28 270. Strength of Rivetted Joints, 288 271. Thin Hollow Cylinders Boiler 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,' . . .302 289 290 290 294 293 296 CONTENTS. Page 2E4. Explanation of Table of the Re- sistance of Materials to Crash- ing by a Direct Thrust (see Appendix), . . . . 304 285. 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. '283. 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 ot 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 of 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, . . . i 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 IX Pago 347 348 348 348 349 353 315. Limiting Length of Beam, 316. Sloping Beam, . 317. Originally Curved Beam, . 318. Expansion and Contraction of Long Beams, . 319. Elastic Curve, . 319A. Hydrostatic Arch, . 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 Ci-ushing 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, . . . .363 351. Mr. Hodgkinson's Formula for the Ultimate Strength of Cast Iron Pillars, . . . .363 332. Wrought Iron Framework, . 364 333. Wrought Iron Cells, . . . 3M= 834. Sides cf 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 or POINTS. SECTION 1 Uniform Motion of a Pair of Points. 348. Fixed and Nearly Fixed Directions, 379 349. Motion of a Pair of Points, . 380 350. Fixed Point and Moving Point, 381 351. Component andResultant Motions, 381 3.52. Measurement of Time, . . 381 353. Uniform Velocity, 354. Uniform Motion, 382 382 SECTION 2. Uniform Motion of several Points. 355. Motion of Three Points, . . 383 356. Motions of a Series of Points, . 383 CONTENTS. 357. Parallelepiped of Motions, 358. Comparative Motion, Page . 38-1 . 38A SECTION 3. Varied Motion of Points. 359. Velocity and Direction of Varied Motion, ..... 385 300. Components of Varied Motion, . 386 361. Uniformly-Varied Velocity, . 3 86 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. 868. Rigid Bodies how understood, 390 369. Translation or Shifting, . .390 SECTION 2. Simple Hotation. 370. Rotation defined Centre of Rotation, . 390 371. Axis of Rotation, 390 372. Plane of Rotation Angle of Ro tation, . 391 373. Angular Velocity, 891 374. Uniform Rotation, 391 375. Rotation common to all parts of body, 391 376. Right and Left-handed Rotation, 391 877. 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, . 393 380. 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 Intersecting Axes 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 OF PLIABLE BODIES, AND OF FLUIDS. 399. Division of the Subject, 408 SECTION 1. Motions of Flexible Cords. 413. General Differential Equations of 400. General Principles, . 408 Steady Motion, . 414 402. Cord guided by Surfaces of Revo- lution, 409 Unsteady Motion, . . . 415 415. Equations of Displacement, . 415 A -I ft Wntro 41 R SECTION 2. Motions ofFl' Constant Density. 403. Velocity and Flow, . 404. Principle of Continuity, 405. Flow in a Stream, . 406. Pipes, Channels, Currents Jets, lids of . 410 . 411 . 411 and . 411 417. Oscillation, . . . .416 SECTION 3. Motions of Fluids of Varying Density. 418. Flow of Volume and Flow of Mass, 417 410. Principle of Continuity, . . 417 420 Stream . ... 418 407. Radiating Current, . 408. Vortex, Eddy, or Whirl, 409. Steady Motion, . 410. Unsteady Motion, 411. Motion of Piston, 412. General Differential Equat Continuity, . 412 . 412 . 412 . 413 . 413 ons of . 413 421. Steady Motion, . . . .419 422. Pistons and Cylinders, . . 419 423. General Differential Equa- tions, . . 419 424. Motions of Connected Bodies, . 420 CONTENTS. XI PART IV. THEORY OF MECHANISM. 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 PrimaiyMoving Pieces, 423 430. Motions of secondaryMovingPieces, 423 AND GENERAL PRINCIPLES. Page 431. Elementary Combinations in Me- chanism, . . . 423 432. Line of Connection, . . f . 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; SlidingofTeeth,440 456. Inside Gearing, . . . .441 457. Involute Teeth for CircularWheels, 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, . 449 449 449 450 451 45CL . 452 470. Cam, or Wiper, 471. Screws Pitch, . 472. Normal and Circular Pitch, 473. Screw Gearing, 474. Hooke's Gearing, 475. Wheel and Screw, 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 9f 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 Universal Joint, . 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, . . . 406 j 506. Link Motion, 504. Differential Windlass, 505. Compound Screws, . .466 5C7. Parallel Motions, . 467 I 508. Epicyclic Trains, .468 . 469 . 473 Xll CONTENTS. PART V. PRINCIPLES OF DYNAMICS. 609. Division of the Subject, Pape . 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 1 Definitions. 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, . . . .483 628. Relations between Impulse, Energy, and Work, . . 484 SECTION 2. Law of Varied Translation. 529. Second Law of Motion, . . 484 530. General Equations of Dynamics, 484 631. Mass in terms of Weight, . . 485 532. Absolute Unit of Force, . .486 533. Motion of a Falling Body, . . 486 584. Projectile, Unresisted, . . 487 535. Motion along an Inclined Path, . 489 536. Uniform Effort, or Resistance, . 490 537. Deviating Forces, . . .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, . . .494 643. Elliptical Oscillations, or Revo- lutions, 495 544. Simple Oscillating Pendulum, . 496 545. Cycloidal Pendulum, . . 497 546. Residua,! Forces, . . . 498 SECTION 3. Transformation of Energy. 547. Actual Energy denned, Vis-Viva, 499 548. Components of Actual Energy, . 499 549. Energy of Varied Motion, . ' .490 550. 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 4. Varied Translation of a System of Bodies. 558. Conservation of Momentum, . 505 559. Motion of Centre of Gravity, . 505 560. Angular Momentum Defined, . 50. r > 561. Angular Impulse Defined, . 506 562. Relations of Angular Impulse and Angular Momentum, . . 50G 563. Conservation of Angular Momen- tum, 506 564. Actual Energy of a System of Bodies, 507 565. Conservation of Internal Energy, 500 566. Collision, . . . . 5Gb 1 567. 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 OF RIGID BODIES. 570. Motion of a Rigid Body in General, 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,.5l 1 574. Radius of Gyration, . . . 515 575. Components" of Moments of In- ertia,, 515 513 576. 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, . 517 579. Moments of Inertia found by Division and Subtraction, . 51 9 580. Moments of Inertia found by Transformation, . . .519 581. Centre of Percussion Centre of Gyration, . . . .520 CONTEXTS. Xlll Page 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 Pajre 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 IV dv with a Fixed Axis, . .543 SECTION 4. Varied notation 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, . . 47 607. Ballistic Pendulum, . . . 54& CHAPTER IV. MOTIONS OF PLIABLE BODIES. 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 CHAPTER V. MOTIONS OF FLUIDS HI-DRODYNAMICS. 617. Division of the Subject, . . .566 608. 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 SECTION 1. Motion of Liquids without Friction. 618. General Equations, . . .567 619. Dynamic Head, ... 568 620. General Dynamic Equations in terms of Dynamic Head, . 568 621. Law of Dynamic Head for Steady Motion, ... . 568 622. Total Energy, . .569 623. Free Surface, . . .570 624. Surface of Equal Pressure, . 570 625. Motion in Plane Layers, . 570 626. Contracted Vein, . . 572 627. Vertical Orifices, . . 572 628. Surfaces of Equal Head, . 573 629. Radiating Current, . . 574 630. Free Circular Vortex, . 574 631. Free Spiral Vortex, . . 576 632. Forced Vortex, . . .576 633. Combined Vortex, . . 576 634. Vertical Revolution, . . 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 C37. Flow of Gas from an Orifice, . 581 637 A. Maximum Flow of Gas, . . 582 SECTION 3. Motions of Liquids with Friction. 638. General Laws of Fluid Friction, 584 639. Internal Fluid Friction, . . 585 640. Friction in an Uniform Stream Hydraulic Mean Depth, . 586 641. Varying Stream, . . .587 642. Friction hi 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 witfi 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, . 598 653. 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 G04 XIV CONTENTS. PART VI. THEORY OF MACHINES. C.;7. Nature and Division of the Subject, Page . 609 CHAPTER I. WORK OF MACHINES WITH UNIFORM OR PERIODIC MOTION. Page SECTION 1. General Principles. 658. Useful and Lost Work, . . 610 (.59. Useful and Prejudicial Resistance, 610 660. Efficiency, . . . .610 661. Power and Effect : Horse Power, 610 662. Driving Point : Train : Work- ing Point, .... 663. Points of Resistance, 664. Efficiencies of Pieces of a Train, 665. Mean Efforts and Resistances, . 666. General Equations, . 667. Equations in terms of Compara- tive Motion. 610 610 610 611 611 612 UVG JJlUUMUf .... Uii* 668. Reduction of Forces and Couples, 612 SECTION 2. On the Frictionof Machines. 669. Co-efficients of Friction, . . 612 670. Unguents, . . . .613 671. Limit of Pressure between Rub- bing Surfaces, . . .613 672. Friction of a Sliding Piece, . 614 673. Moment of Friction, . 614 674. Friction of an Axle, 614 675. Friction of a Pivot, .616 676. Friction of a Collar, , 616 677. Friction of Teeth, .617 678. Friction of a Band, . 617 679. Frictional Gearing, . 618 680. Friction Couplings, . 618 681. Stiffness of Ropes, . 619 682. Rolling Resistance of Smooth Surfaces, . . . .619 683. Resistance of Carriages on Roads, 619 684. Resistance of Railway Trains, . 620 685. Heat of Friction, . . .620 CHAPTER II. VARIED MOTIONS OF MACHINES. 689. Fluctuations of Speed, . . 622 690. Fly-Wheel, . . . .623 691. Starting and Stopping Brakes, 624 686. Centrifugal Forces and Couples, 621 687. Actual Energy of a Machine, . 621 688. Reduced Inertia, . . .621 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, 625 625 625 625 626 627 628 699. Overshot and Breast Wheels, . 628 700. Undershot Wheels, 701. Turbines, . 702. Windmills, 703. Efficiency of Heat Engines in General, 704. Steam Engines, 705. Electrodynamic Engines: Science of Energetics, .... 628 629 629 629 630 APPENDIX. I. Tables 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. 377, 631 . 633 . 633 . 634 , 636 . 637 . 640 ADDENDUM. XV ADDENDUM (referred to in Article 634, page 579). of Water in Wares. I. Rolling Waves. In waves which are not accompanied by permanent translation of the particles of water, it is known by observation that those particles revolve in orbits situated in vertical plantfC which are perpendicular to the ridges and furrows of the waves, and parallel to their direction of advance ; also, that each revolving particle moves forward while on the crest of a wave, downward when on the back slope, backward when in the trough, and upward when on the front slope. The length of a wave is the distance, in the direction of advance, from crest to crest ; the height is equal to the vertical diameter of the orbit of a surface particle. Each particle makes one revolution while the wave advances through a wave-length ; the interval of time thus occupied is called the period. Let L denote the wave-length, T the period, a the velocity of advance; then a = ; and also, mean velocity of revolution of a particle = circumference of orbit T. The orbits of the particles are approximately elliptic, with the longer axis horizontal. In going from the surface towards the bottom, the dimensions of the orbits are found to diminish, the vertical axis diminishing faster than the horizontal axis, as shown at A, B, C, in fig. A. At the bottom the particles move back and forward in a straight line, as --- --. atD. The deeper the water is, as jj v : ^^^%^%%^^$%%%%^^ are the two axes of the orbit of Fig. A. a surface particle ; and in water whose depth is half a wave-length and upwards, those axes are sensibly equal, and the orbit of a surface particle sensibly circular. II. Relation between Figure of Surface and Velocity of Advance. In fig. 252, page 578, let C be the centre, and C B the radius of the circular orbit of a particle. Lay off C A vertically upwards, of a length equal to that of the equivalent pendulum (that is, the pendulum whose period is T) viz., c \ = .? Ta - T (seconds) 4ir2 0-815 foot nearly"" Then we have gravity : centrifugal force: : A C : C B ; and A B represents (as in Article 634, page 578) the resultant of gravity and centrifugal force ; so that a surface of uniform pressure traversing B is normal to A B. The upper surface of the wave is such a surface ; and in order to fulfil that condition its profile must be a trochoid traced by the point B while a circle of the radius C A rolls on the under side of a horizontal straight line traversing A. The length of such a wave, and its velocity of advance, are given by the following equations : TT = (in feet) 5 '12 T 2 ; .................. (2.) a = = = (in feet per second) 512 T. ... ....(3.) JL 2i "TT When the orbits of the surface particles are elliptic, let m be the ratio in which the vertical axis is less than the horizontal axis. Then it is evident that in order that the surface of the wave may still be everywhere normal to the resultant of gravity and re-action, we must have XVI ADDENDUM. (4.) a- ^~ = (in feet per second) 512 mT (5.) III. Relation between Velocity of Advance and Depth of Uniform Disturbance. Let h be the height of a wave ; that is, the vertical diameter of the orbit of a surface particle. Then, in an indefinitely short interval of time, the front slope of the wave advances through the distance a d t, and the volume of water con- tained between the original and new positions of the front slope, per unit of breadth, is h a d t. In the same interval of time there passes into the space vertically below the front slope, per unit of breadth, the volume of water 2 u c d t, where u is the forward velocity of a surface particle at the crest, -u the equal backward velocity of a surface particle in the trough, and c a depth which may be called the depth of uniform disturbance, because it is equal to the mean depfli of a canal in which the volume of water displaced per second would be equal to that displaced per second in the actual wave, if the horizontal velocity of disturbance were the same from surface to bottom. Equating the two volumes just given, we have h a = 2 u c; but u can be shown to be = g h -r- 2 a; there- fore c = a 2 -r- fr. Hence the velocity of advance of a wave of any figure in which the volume displaced horizontally per second is equivalent to that due to a hori- zontal velocity of disturbance equal to the surface velocity down to the depth (, is given by the equation For waves rolling in deep water, without interference by external forces, it can be shown that the diameters of the orbits of particles at different depths vary proportionally to e c ; where z is the depth of the centre of the orbit of the particle in question below the centre of the orbit of a surface particle. In water of the depth k, let L -r- 2 TT = 6; then it can be shown that at the ( ~ -*^ ( * ~~\ surface, m \e b e ^ ) -r- \e h + e b ) ; that c = m 6; and that the hori- fr_z ik zontal and vertical diameters of an orbit vary respectively as c 6 + e 6 , and k t * J. as e b e b In very deep water, m sensibly = 1, and c, b. In very shallow water the horizontal disturbance is sensibly uniform from the surface to the bottom, so that c represents the actual depth ; and the vertical disturbance is sensibly proportional to the height above the bottom. IV. Waves of Translation are those which are accompanied by a permanent travelling of the particles of water, and are said to be positive or negative accord- ing as that travelling is forward or backward. Their motions may be expressed by taking two different quantities, u' and u', to denote respectively the for- ward velocity of a particle at the crest of a wave, and the backward velocity of a particle in the trough; when the velocity of advance will be given by the formula a + KV! - u") (7.) V. Authorities on Waves. Weber's Wellenlehre; Scott Eussell, in Reports of the British Association, 1844; Airy, On Tides and Waves; Stokes, Cambridge Transactions, 1842, 1850; Earnshaw, Ib., 1845; Froude, Trans, of the Institution of Naval Architects, 1862; Rankine, Philos. Trans., 1863; Do., Philos. Mag., November, 1864; Do., Proceedings of the Jloyal Society, 1868; Watts, Kankine, Napier, and Barnes, On Shipbuilding; Thomas Stevenson, On Harbours, Caligny, Liouvillc's Journal, June and July, 1866; Cialdi, Sul Moto Ondoso dd Mare. PRELIMINARY DISSERTATION . . ON TDK HARMONY OF THEORY AND PRACTICE IN MECHANICS.' THE words, theory 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 e very-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 tmd 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 Scientiarum 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, 1856. 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 cuiiayatea only for the sake of gain or some other material fc l ", -advantage. ^"A'Kchytas of Tarentum might illustrate the truths of t s geometry .by .mechanical contrivances; his methods were regarded ' \ *\ ^ ^4 S pup4 JP? a V t 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 dry 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 Buffon, burn fleets with the concentrated sunbeams j 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 Ttoger 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 j 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 tbe 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 h away from his official duties."* VoL w., 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. 5 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 sh6uld 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. 6 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 iense, 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. 9 to qualify him to become a scientific discoverer. In this branch of study exactness is an essential feature; and mathematical difficulties must not be shrunk from when the nature of the subject le^-ds 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 What 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 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 j 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 ^ihis 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 ; tor 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. r i 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. natter (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, and possibly also between liquid and gaseous. 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. lane, 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 the two ends of 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 are very commonly 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 has been introduced of expressing fractions of an inch in decimals. The French unit of length is the me"tre, being about 4ooo 1 ooo<> 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. inroilou 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 wITat 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 causes, is tho subject of a branch of' geometry called " Cinematics. 1 ' 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 general mechanics. 15. structures and machines. The works of human art to whicli 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 OP STATICS. IL THEORY OP STRUCTURES. III. FIRST PRINCIPLES OF CINEMATICS. IV. THEORY OP MECHANISM. V. FIRST PRINCIPLES OP DYNAMICS. VI- THEORY OF MACHINES. K~ rf\ I q ft iAL ,, /\7" L vTp PART I. PKINCIPLES 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 booties 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. 18. 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 rea.l 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. 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 as The grain = r^nnr of a pound avoirdupois. The troy pound = 5,760 grains = 0-82285714 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 0> 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 IJiic. 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. 19 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 " sum " 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 Lines. 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. Flg< j. For example, in fig. 1, the fact that the body B B B B is acted upon at the point O a by a given force, may be expressed by drawing from O l 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^ is balanced by a force applied either at the same point, or at another point O 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 O 1 F a , 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 L L, applied at points O t O 2 , &c., and re- presented by lines O^FJ, CVF& &c. j *h en eit ner direction in the line 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 g 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 " pres- sure" that the context shall show in what sense it is used. CHAPTER IL THEORY OF COUPLES AND OF THE BALANCE OF PABALLEL FORCES. SECTION 1. On Couples with tJie 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 Bight-handed and ^eft-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 Oj F 1? O 2 F 2 , whose leverage is L, L 2 , form a right- handed couple; and the equal and opposite forces 3 F 3 , O* F 4 , form a left-handed couple. 28. Equivalent Couples of Equal 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. Rg. 8. 22 PRINCIPLES OF STATICS. 3 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 J> moment in. foot-pounds, and similarly for other measures. ' XV^SO. Addition of Couples of Equal Force. LEMMA. Two COUples of \ r 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 Fig- 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 6, 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 6, which form a couple whose forco is F A = F B , and its arm ~ab = 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 k 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 F A L A = F B L B . If the forces and arms of the two couples be commensurable, so that REPRESENTATION OF COUPLES. F A : F B : : L B : L A : : m : n (m and n being two whole numbers), let -.* and Z = * = *. m n Then the couple A is equivalent to m n couples of the moment f I ; 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 j 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 same axis is equiva- lent to a couple whose moment is the algebraical sum of the moments of the combined couples. 33. Equilibrium of Couples baring 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. ie resultant moment is nothing. Representation of Couples by Lines. The nature and amount the tendency of a couple to turn a body are completely known len 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 O towards M 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 the 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 ca = L A , cb = L B , and join ab. 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 + 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. _JNow from any point O draw O M A perpendicular to ac, 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 b c, M c M B , the following pro- portions will exist : "OM A :CTM B : OM C , :: L A : L B : L c ; and consequently OM C will also represent the moment of the re- Bultant couples. Q. E. D. PARALLEL FORCES. 25 36. Equilibrium of Three Couples with Different Axes in the Same Plane COROLLARY. A couple equal and opposite to tliat ^represented by tJte diagonal OM C balances the couples represented by thersides QM A , OM B . In other words, three couples represented by tJie three Asides of a triangle balance each other. ^i 37. Equilibrium of any Number of Couples. COROLLARY. If a \number of couples acting on tJie 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 MX is represented by O 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 CT), from D towards C. M 5 IK), from towards D. Then by the theorem of Article 35, the resultant of Mj and M 2 is O B j the resultant of this and M 3 is O C ; the resultant of this and M 4 is 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 1 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. I >\ X *V Ic 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. Priii- ciple of the I/ever. 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, FA L A = F B L B , 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 + 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 L B ; and consequently F A :F B :F C : :C~B: A~(T: AB; 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 them, 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 Coheir 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. M denote the moment of a couple applied to a body (fig. 9) ; id 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 O , . M , , being =: , and of course par- F Fig. 9. allel to the plane of the couple M. Then the forces at O balance each other, and F. applied at A is the resultant of the single force F applied at O, and the couple M ; 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 distance O A = ; to the left if M is right- 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 lie the common perpendicular of those two lines. At B conceive a pair of equal and directly op- ] osed forces to be applied in a line of action varallel to F, viz.: F=F, and-F = -F. The t imposed application of such a pair of balanced F - 10 I u-ces does not alter the statical condition of the 1'ody. Then the original single force F, applied in a line tra- 28 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 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 BDE being = J F AB, represents one-half of the moment of F relatively to 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 OX,?/ 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 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. 29 action of that resultant from the assumed axis X to which the positions of forces are referred : then F r =2- F; 2-yF petel r""'45. / In 'force In some cases, the forces may have no single resultant, 2 p being = ; and then, unless the forces balance each other com- pletely, their resultant is a couple of the moment 2 . y F. moments of a Force with respect to a Pair of Rectangular Axes fig. 11, let F be any single force; O an arbitrarily-assumed pointjCalled 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 O Y. x and y are the "rectan- gular co-ordinates" of the line of action of F relatively to the axes - X + 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 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', 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 = F applied at B, combined with the couple constituted by F and F 7 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 0, conceive a pair of equal and opposite forces, F" and F", to be applied, F" being equal and parallel to F and F 7 , and in the same direction. Then the single force F / is equivalent to the single force F" = F 7 = F applied at O, combined with the couple constituted by F' and F" with the arm OB = x } whose moment is Fig. 11. to be considered as : f :t \ V u 30 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 y, 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 O 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 : 2-F = 0; S-y P = 0; 2 x F = 0; for by the last Article, each force F is equivalent to an equal and parallel force F" applied directly to O, combined with two couples, y F with the axis OX, and -a? 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, 2 ccF 2 In some cases, the forces may have no single resultant, 2 J? CENTRE OP 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 O X and O 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^ }; and if Q be the angle which its axis makes with O X, cos=g;. / /" 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- 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 m f b , in any other angular position, and if those new forces bear to each other the same proportion with the original forces, viz. : / a :/ 6 ::F A :F B ::BC:AC; 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 B C : AC. 49. Centre of any System of Parallel Forces. Let parallel forces, F , F 1? be applied at the points A A l (fig. 13.), / x Draw the straight line AQ A 1} in which * A take C,, so that F : F, : : C, A, : C, A ; then will C^ be the centre of a pair of Fig. 13. N ' V * P ara ll e l forces applied at A and A lt and having the proportion F : Fj. At a third point, A 2 , let a third parallel force, F 2 , be applied. Then, because the forces F , Fj, are together equivalent to a parallel force, F + F lf applied at C lt draw the straight line Cj A 2 , in which take C 2 , so that F + F! : F 2 : : C 2 A 2 : C 2 Cj ; then will C 2 be the centre of three parallel forces applied at A , A 19 A 2 , and having the proportions F : ^ : F 2 . At a fourth point, A 3 , let a fourth parallel force, F 3 , be applied. Then, because the forces F , F x , F 2 , 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 Cg, so that F 4- F 1 + F 2 : F 3 : : C 3 A 3 : C 3 C 2 ; then will C 3 be the centre of four parallel forces applied at A , A,, A 2 , A 3 , and having the proportion F : Fj : 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 tJie resultant of every system of parallel forces having the given mutual ratios and applied to the given system of points, whatsoever may be the absolute magnitudes of those forces, and tlie angular position of their lines of action. 50. Co-ordinates of Centre of Parallel Forces. The method of iding centres of parallel forces described in the preceding Article, i 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, Y, OZ, three axes of co-ordinates at right angles to each other. CENTRE OF PARALLEL FORCES. S3 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 YZ, y parallel to 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 -*F; 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 Firr. 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 2-yF; and consequently the distance of their resultant, and also of the centre of parallel forces from that plane is given by the equation 2-F ' 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 and consequently the distance of their resultant, and also of the centie of parallel forces from that plane is given by the equation z r = Thus are found x r) y r) z r , the three rectangular co-ordinates of i> 34 PRINCIPLES OP 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 die system from that plane. 35 CHAPTER HI BALANCE OP 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 ly the diagonal. First Demonstration. Through the point (fig. 15), let two forces act, represented in direction and magnitude by OA and OB. .x'^ 1 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 A and O B, is the sum of the moments of O A and O B relatively to the same axis. "x. Now, first, the force represented in/ direction and magnitude by the dia- Fig. 15. gonal C of the parallelogram A B \ fulfils this condition. For let P be any point in the plane of A and O B, and let an axis perpendicular to that plane traverse P. Join P A, P B, P C, P Q. Then,_as_already shown in Art. 42, the moments of the forces O 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, PE. Then A__P D_= AJP_O A, and A POE = A POB ; but be- cause OD + OE" = OC, .-. A POC = AIM3D + A POE = A P A + APOBj and the moment_of_ O C relatively to P is equal to the sum of the moments of O A and OB; 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 A and O B. JoinPQ, Then AOPQ = APOC,-and.-. CQ||PO; so that 36 PRINCIPLES OF STATICS. O Q fulfils the required condition for those axes only which are situated in a line OP || C Q, and not for any other axis. Therefore the diagonal O C of the parallelogram A B represents tta resultant, and the only resultant, of the forces represented by OA. and OB. Q. E. D. Second Demonstration. Suppose a perpendicular to be erected to the plane A B at the point O, of any length whatsoever ; call the other extremity of that perpendicular B, ; and at K conceive two forces to be applied, respectively equal, parallel, and opposite to O A and O B. Then B. is the arm common to two couples whose axes and moments are represented (in the manner described in Art. 34) by lines perpendicular and proportional respectively to O A and O B. On the lines so representing the couples, construct a paral- lelogram ; then, as shown in Art. 35, the diagonal of that parallelo- gram represents the resultant couple constituted by the resultant of OA and O B acting at O, and an equal and opposite force at B ; and as the parallelogram of couples has its sides perpendicular and proportional to O A and O B, its diagonal must be perpendicular and proportional to O C, which consequently represents the result- ant of OA and OB. Q. E. D. [There are various other modes of demonstrating the theorem of the parallelogram of forces, all of which may be studied with ad- vantage : especially those given by Dr. Whewell in his Elementary Treatise on Mechanics', and by Mr. Moseley in his Mechanics of En- gineering and Architecture.] 52. Equilibrium of Three Forces acting through One Point in One Plane. To balance the forces O~A and OB, a force is required equal and directly opposed to their resultant O~C. This may be otherwise expressed by saying, that if the directions and mag- nitudes ofjhree Jorcesbe^ represented by the three sides of a triangle, / (such as O A, A C, C O), then those three forces, acting through one point, balance each other. -^/ 53. Equilibrium of any System of Forces acting through One Point. /COROLLARY. If a number of f orces acting through the same point be Represented by lines equal and parallel to the sides of a closed polygon, those forces balance each other. To fix the ideas, let there be five forces acting through the point (fig. 16), and re- presented in direction and magnitude by the lines F 15 F 2 , F 8 , F 4 , F 5 , which are equal and parallel to the sides of the closed polygon A B C D O \ viz. : RESOLUTION OF A FORCE. 37 F, = and || A; F, = and || A B ; F 3 = and || B C ; F 4 = and||CD; F 5 = and || D 0. Then by the theorem of Art. 52, the resultant of F x and F 2 is B; the resultant of F 15 F , and F, is C; the resultant of F,, F 2 , F 8 , and F 4 is D, equal and opposite to F 5 , so that the final resultant is nothing. The closed polygon may be either plane or gauche. 54. Parallelepiped of Forces. The simplest gauche polygon ia one of four sides. Let O A B C E F G H (fig. 1 7), be a parallelepiped whose diagonal is OH. Then any three successive edges so placed as to begin at O and end at H, form, together with the dia- gonal H O, a closed quadrilateral ; conse- quently if three forces F 1? F 2 , F 3 , acting through Q, be represented by the three edges CTA, O B, CTC", of a parallelepiped, the diagonal O H represents their resultant, and a fourth force F 4 equal and opposite to OH balances them. Fi g- 17. ^55. Resolution of a Force into Two Components. From the theo- rem of Art. 51, it is evident that in order that a given single force may be resolvable into two components acting in given lines inclined to each other, it is necessary, first, that the lines of action of those components should intersect the line of action of the given force in one point ; and secondly, that those three lines of action should be in one plane. Returning, then, to fig. 15, let C represent the given force, which it is required to resolve into two component forces, acting in the lines OX, Y, which lie in one plane with C, and intersect it in one point O. Through C draw C A || Y, cutting O X in A, and C B || X, cutting O Y in B. Then will O A and O B represent the com- ponent forces required. Two forces respectively equal to and directly opposed to O A and CTB will balance O C. 56. Resolution of a Force into Three Components. In Order that a given single force may be resolvable into three components acting in given lines inclined to each other, it is only necessary that the lines of action of the components should intersect the line of action of the given force in one point. 38 PRINCIPLES OF STATICS. Keturning to fig. 17, let O H represent the given force which it is required to resolve into three component forces, acting in the lines O X, Y, O Z, which intersect O H in one point O. Through H draw three planes parallel respectively to the planes Y OZ, Z X, X Y, andcutting respectively O X in A, O Y in B, Z in C. Then will O A, (XB, O~C, represent the component forces required. __Three forces respectively equal to, and directly opposed to (5.) 3 = 0; F 2 =Ff + Fl- j In using these equations, the rule respecting the positive and negative signs of cosines is to be observed ; and it is also to be borne in mind, that the angle is reckoned from O X in the direction towards Y, and the angle ft from Y in the reverse direction, that is, towards X, and that f to 180 ) f positive. the sines of angles from j 18QO fo 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 3 = 0;2 F 8 = ............. (6.) SECTION 2. Inclined Forces Applied to a System of Points. f 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 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 R both have magnitude, then the line of action of the resultant R is at the perpendicular distance from 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 as origin of co-ordinates, let any two axes be assumed, 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 ct, be the angle which its line of action makes to the right of 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 cos *; F 2 = F sin a; then the rectangular components of the resultant are respectively parallel to X, 2 (F cos ) = R : , ) n . paraUel to O Y, 2 (F- sin ) = R 2 , / ' ' its magnitude is given by the equation and the angle r which it makes to the right of X is found by the equations 1 . ' _ 2 /Q \ T "o ^ ** ~D "\ - / ASY SYSTEM OF FORCES. 41 The quadrant in which the direction of R lies is indicated by the algebraical signs of R] and R^, as already stated in Art. 57. The perpendicular distance from O of the line of action of any force F is L = x sin y cos 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 O Z is 2-FL = 2-F (x sin y cos a) = *(x-F, yV l ) ........................ (4.) whence it follows, that the perpendicular distance of the resultant force from 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 ovR* y r Ri = RL r ) which is equivalent to V ............... (6.) x r sin r y r cos x r = L,. j As in Art. 58, if 2-F L = 0, the resultant acts through the origin O; if 2-FL 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 FL = 0; ) V .... (7. Y,y'F l ) = 0.) or in other symbols V .... (7.) ~T7i A . _ . ~GI " * " ~ 2 i = U, 2' 2 = 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 2 , which is right-handed when x and F 2 are both positive, and of y ~F lt which is left-handed when y and Fj aro 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 ) C Y towards Z } Y > towards 0, rotation from < Z towards X V Zj (X towards YJ 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 , /8, 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 a. along OX,) F 2 = F cos ft along O Y, V (1.) F 3 = F cos y along 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 fy relatively to O X, ) z F! x F 3 = F (z cos a x cos y) relatively to Y, > (2.) x F 2 y Fj = F (x cos ft 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 1 = 2 F cos , O Y ; R,= 2 - F cos /3, \ (3.) OZ; K 3 = 2- F cosy, Couples. X round O X ; MI = 2 F (y cos y z cos /3) F (z cos a x cos y) ..(4.) O Z ; M 8 = 2[F (x cos ft - #cos The three forces "R lf R 2 , R 3 , are equivalent to a single force B= ^(RJ + EJU-R,-), ...................... (5.) acting through O in a line which makes with the axes the angles given by the equations "R ~R "R r=j cos/3 '-=- f cos 7r=r " ............ ( 6 -) The three couples M,, M 2 , M s , according to Article 37, are equi- valent to one couple, whose magnitude is given by the equation M= ^(Mf + MI + Ml) ..................... (7.) ANY SYSTEM OF FORCES. 43 and whose axis makes with the axes of co-ordinates the angles given by the equations ' L r f I denote respectively the angles I r\ ^r I in which 1 f. V ^ de by ^ e axi / of M ih 1 O Y V. \ V J \ ) 1 J The Conditions of Equilibrium of the system of forces may be ex- pressed in either of the two following forms : = ; R s = ; R 3 = : M x = ; M 2 = ; M 3 = 0. . .(9.) or R = 0j 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 O. Case ii. When the axis of M. is at right angles to tfie direction of R, a case expressed by either of the two following equations: COS r COS X 4 COS /3 r COS {A + COS y r COS V = j ) /-. -. ^ 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 (12)- Case in. WJien R = 0, there is no single resultant; and the only resultant is the couple M. Case iv. When the axis o/*M is parallel to tlie line of action o/*R> that is, when either A = a r ; p = ft f - v = y r , .................. (13). or x = * r ; p = p,; v = Vr ; ............ (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. Wlien the axis of M is oblique to the direction of R, making with it the angle given by the equation cos & = cos x cos ct r + cos (*> cos /3 r + cos v cos y r ,....(15). the couple M is to be resolved into two rectangular components, viz. : I 44 PRINCIPLES OF STATICS. M sin & round an axis perpendicular to R, and in "1 the plane containing the direction of R and of I (16.) the axis of M; M cos 6 round an axis parallel to R. The force R and the couple M sin 6 are equivalent, as in Cast II., to a single force equal and parallel to R, whose line of action is in a plane perpendicular to that containing E, and the axis of M, and whose perpendicular distance from is (17.) The couple M cos &, whose axis is parallel to the line of action of R, is incapable of further combination. Hence it appears finally, that every system of forces which is not self-balanced, is equivalent either, (A) ; to a single force, as in Cases I. and II. (B) ; to a couple, as in Case III. (C) ; to a force, com- bined with a couple whose axis is parallel to the line of action of the force, as in Cases IY. and Y. This can occur with inclined forces only, it having been shown in Article 47, that the resultant of any number of parallel forces is either a single force or a couple. 45 CHAPTER IT. ON PARALLEL PROJECTIONS IN STATICS. 61. Parallel Projection of a Figure defined. If two figures be BO related, that for each point in one there is a corresponding point in the other, and that to each pair of equal and parallel lines in the one there corresponds a pair of equal and parallel lines in the other, those figures are said to be PARALLEL PROJECTIONS of each other. The relation between such a pair of figures may be otherwise expressed as follows : Let any figure be referred to axes of co- ordinates, whether rectangular or oblique ; let x, y, z, denote the co-ordinates of any point in it, which may be denoted by A : let a second figure be constructed from a second set of axes of co-ordinates, either agreeing with, or differing from, the first set as to rectan- gularity or obliquity ; let x', T/, z', be the co-ordinates in the second figure, of the point A' which corresponds to any point A in the first figure, and let those co-ordinates be so related to the co-ordi- nates of A, that for each pair of corresponding points, A, A', in the two figures, the three pairs of corresponding co-ordinates shall bear to each other three constant ratios, such as x y z then are these two figures parallel projections of each other. 62. Geometrical Properties of Parallel Projections. The following are the geometrical properties of parallel projections which are of most importance in statics. Being purely geometrical propositions, they are not here demonstrated. I. A parallel projection of a system of three points, lying in one straight line and dividing it in a given proportion, is also a system of three points, lying in one straight line and dividing it in the same proportion. II. A parallel projection of a system of parallel lines whose lengths bear given ratios to each other, is also a system of parallel lines whose lengths bear the same ratios to each other. III. A parallel projection of a closed polygon is a closed polygon. 46 PRINCIPLES OF STATICS. IV. A parallel projection of a parallelogram is a parallel- ogram. V. A parallel projection of a parallelepiped is a parallelepiped. VI. A parallel projection of a pair of parallel plane surfaces, whose areas are in a given ratio, is also a pair of parallel plane surfaces, whose areas are in the same ratio. VII. A parallel projection of a pair of volumes having a given ratio, is a pair of volumes having the same ratio. 63. Application to Parallel Forces. It has been shown in Chap. II., Sect. 3, that the equilibrium of any system of parallel forces depends on the mutual proportions of the forces and on those of the distances of their lines of action from given planes. By considering this in connection with the principles I. and II. of Article 62, it is evident, that if a balanced system of parallel forces be represented by a system of lines, then any system of lines which is a parallel projection of the first system, will also represent a balanced system of parallel, forces ; and also, that if there be two systems of parallel forces represented by systems of lines which are parallel projections of each other, then are the respective resultants of those systems of forces, whether single forces or couples, represented by lines which are parallel projections of each other related in the same manner with the other pairs of corresponding lines in the two systems. In applying this principle to couples, it is to be observed, that they are not to be represented by single lines, as in Art. 34, but by pairs of equal and opposite lines, as in the previous articles, or by areas, as in Articles 42 and 51. 64. Application to Centres of Parallel Forces. If two Systems of points be parallel projections of each other ; and if to each of those systems there be applied a system of parallel forces bearing to each other the same system of ratios, then, by considering the principles I. and II. of Article 62 in conjunction with those of Chap. II., Sect. 4, it is evident that the centres of parallel forces for those two systems of points will be parallel projections of each other, mutually related in the same manner with the other pairs of corresponding points in the two systems. 65. Application to Inclined Forces acting through One Point. From principles III., IV., and V., of Article 62, taken in conjunc- tion with the principles of Chap. III., Sect. l,it follows, that if a given system of lines represents a balanced system of forces acting through one point, then will any parallel projection of that system of lines also represent a balanced system of forces acting through one point ; and also, that if two systems of forces, each acting through one point, be represented by two systems of lines which are parallel projections of each other, then will the respective resultants of those two systems of forces be represented by a pair of lines which are PARALLEL PROJECTION OF FORCES. 47 parallel projections of each other, mutually related in the same manner with other pairs of corresponding lines. 66. Application to anj System of Forces. As every system of forces applied to any system of points' can be reduced, as in Art. 60, to a system of 'forces acting through one point, and certain systems of parallel forces, it follows that if a balanced system of forces acting through any system of points be represented by a system of lines, then will any parallel projection of that system of lines represent a balanced system of forces ; and that if any two systems of forces be represented by lines which are parallel projections of each other, the lines, or sets of lines, representing their resultants, will be cor- responding parallel projections of each other : it being still ob- served, as in Article 63, that couples are to be represented by pairs of lines, as pairs of opposite forces, or by areas, and not by single lines along their axes. 43 CHAPTER Y. ON DISTRIBUTED FORCES. 67. Restriction of the Subject. In Article 18 it has already been explained, that the action of every real force is distributed through- out some volume, or over some surface. It is always possible, however, to find either a single resultant, or a resultant couple, or a combination of a single force with a couple (like that described in Art 60), to which a given distributed force is equivalent, so far as it affects the equilibrium of the body, or part of a body, to which it is applied. In the application of Mechanics to Astronomy, Electricity, and Magnetism, it is often necessary to find the resultant of a distri- buted attraction or repulsion, whose direction is sensibly different at different points of the body to which it is applied ; and problems thus arise of great difiiculty and complexity. But in the applica- tion of Mechanics to Structures and Machines, the only force dis- tributed throughout the volume of a body which it is necessary to consider, is its weight, or attraction towards the earth ; and the bodies considered are in every instance so small as compared with the earth, that this attraction may, without appreciable error, be held to act in parallel directions at each point in each body. More- over, the forces distributed over surfaces, which have to be consi- dered in applied mechanics, are either parallel at each point of their surfaces of application, or capable of being resolved into sets of parallel forces. Hence, in applied mechanics, parallel distributed forces have alone to be considered ; every such force is statically equivalent either to a single resultant, or to a resultant couple ; and the problem of finding such resultant is comparatively simple. 68. The Intensity of a Distributed Force is the ratio which the magnitude of that force, expressed in units of force, bears to the space over which it is distributed, expressed in units of volume, or in units of surface^ as the case may be. An unit of Intensity is an unit of force distributed over an unit of volume or of surface, as the case may be ; so that there are two kinds of units of intensity. For example, one pound per cubic foot is an unit of intensity for a force distributed throughout a volume, such as weight ; and one CENTRE OF GRAVITY OF SYMMETRICAL BODT ; 49 pound per square foot is an unit of intensity for a force distributed over a surface, such as pressure or friction. The intensity of a force acting at a single point would be infinite, if such a force were possible. SECTION 1. Of Weight, and Centres of Gravity. 69. The Specific Grarity of a body is a number proportional to the weight of an unit of its volume; for example, the weight in pounds, of a cubic foot of the volume of the body. The pound per cubic foot is the most convenient unit of specific gravity for practi- cal purposes ; but in tables of specific gravity, a special unit is usu- ally employed, viz., the weight, at a fixed temperature, of unity of volume of water. In Britain, that fixed temperature is usually 62 Fahrenheit; in France, and on the continent of Europe generally, it is the temperature at which water is most dense, viz,, 3-95 centigrade, or 39-l Fahrenheit. In a table at the end of this volume are given the specific gravities of such materials as most commonly occur in structures and machines. So far as this and similar tables relate to solid materials, they must be regarded as approximate only; for the specific gravity of the same solid substance varies not only in different specimens, but frequently even in different parts of the same specimen ; still the approximate values are sufficiently near the truth for practical purposes in the art of construction. 70. The Centre of Grarity of a body, or of a system of bodies, is the point always traversed by the resultant of the weight of the body or system of bodies, in other words, the centre of parallel forces for the weight of the body or system of bodies. To support a body, that is, to balance its weight, the resultant of the supporting force must act through the centre of gravity. 71. Centre of Grarity of a Homogeneous Body baring a Centre of Figure. Let a body be homogeneous, or of equal specific gravity throughout ; let it also be so far symmetrical, as to have a centre of figure; that is, a point within the body, which bisects every diameter of the body drawn through it; then it is self-evident, that the centre of figure of the body must also be its centre of gravity. Amongst the bodies which answer this description are, the sphere, the ellipsoid, the circular cylinder, the elliptic cylinder, prisms whose bases have centres of figure, and parallelepipeds, whether right or oblique. ^VjjU \ r"~72. Bodies baring Planes or Axes of Symmetry. If a homogene- \ lous body be of a figure which is symmetrical on either side of a I given plane, the centre of gravity must be in that plane. If two or more such planes of symmetry intersect in one line, or axis of 50 PRINCIPLES OF STATICS. symmetry, the centre of gravity must be in that axis. If three or more planes of symmetry intersect each other in a point, that point must be the centre of gravity. The following are examples : I. In fig. 18, let AB C be an equilateral triangle, the base of a rigfa equilateral triangular prism. This prism has one plane of symmetry parallel to its bases at the middle of its length. It has also three planes of symmetry, A. a, B 6, C c, each traversing one edge of the prism and bisecting the opposite side, and those three planes intersect in an axis Gy whose perpendicular distance from any edge is two-thirds of the distance from that edge to the opposite side, that is, GO GA GB 2 The centre of gravity of the prism is at the middle of this axis. Fig. 18. Fig. 19. II. In fig. 19, let A B C D be a regular tetraedron, or triangular pyramid, bounded by four equilateral triangles. Bisect any edge D C in E j then the plane ABE drawn through the point of bisec- tion and the opposite edge is a plane of symmetry. There are six such planes, and they intersect each other in one point G, which is therefore the centre of gravity of the tetraedron. It may be shown by geometry, that the point G can be found in the following manner. Prom any summit, such as B, draw B E, bisecting one of the opposite edges, such as D C. In B E take 2 - 3 BF = BE. Join AF, in which take AG = AF : then O 4: is G the centre of gravity sought. 73. System of Symmetrical Bodies. Let a connected system of bodies whose absolute or proportional weights are known, and whose centres of gravity are also known by reason of the symmetry BODY OF ANY FIGURE. 51 and homogeneity of each body, be arranged in any manner ; then the common centre of gravity of the whole system of bodies is the same with the centre of parallel forces for a system of forces equal or proportional- to the weights of the bodies, and acting through their respective centres of gravity. Consequently, applying to this case the principles of Chap. II., Section 4, Article 50, the centre of gravity is found in the following manner. Let yz denote any fixed plane, x the perpendicular distance of the centre of gravity of any one of the bodies from that plane, and W the weight of that body, so that Wx is the moment of the weight of the body in question with respect to any axis in the plane y z. Let XQ denote the perpendicular distance of the common centre of gravity from the plane y z. Then we have, total moment of the system relatively to any axis in the plane yz, XQ -2 W = and consequently, By proceeding in a similar manner, the distances of the common centre of gravity of the system of bodies from two other fixed planes, either perpendicular or oblique to ~y~z and to each other, are /found so as to determine its position completely. The same process is applicable to any body whose figure is capable ' of being divided into symmetrical figures. \t-{ ^,74 Homogeneous Body of any Figure. Let W be the Specific ^gravity of a homogeneous body of any figure, Y its volume, and W = -M>Y its weight. Conceive three fixed co-ordinate planes, yz,zx, and x y, perpendicular to each other, and let XQ, y^ z^ be the co-ordinates of the centre of gravity, which it is required to find ; so that w Y XQ, w Y y , w Y ZQ, are the moments of the body relatively to the three co-ordinate planes respectively. Conceive the space in and near the body to be divided by three series of equi- distant planes parallel to the co-ordinate planes respectively, into equal and similar small rectangular molecules, whose dimensions, parallel to x, y, and z, respectively, are Let x, y, z, be the co-ordinates of the centre of one of these mole- cules. Then its volume is A a; Ay AZ; its weight w A x A y A z t (i.) 52 PRINCIPLES OF STATICS. and its moments relatively to the three co-ordinate planes re- spectively, XWAXAy&Z' } yW&XAyAZ', ZWAXAyAZ. "Whatsoever may be the figure of the body whose centre of gravity is sought, a figure approximating to it may be built by putting together a proper number of suitably arranged rectangular mole- cules ; so that V = 2 AX Ay AS nearly; "W = ; V w * 2 * AX &y AZ nearly; w~V XQ W-S-XAX Ay AZ nearly; therefore omitting the common and constant factor w, 2 ' X AX Ay AZ X Q = nearly; 2 AX Ay AZ and similar approximate formulae for y Q and ZQ. Now, it is evident, that the smaller the dimensions AX, Ay, AZ, of each rectangular molecule, or in other words, the more minute the subdivision of the space in and near the body into small rectangles, the more nearly will the approximate figure, built up of rectangular molecules, agree with the exact figure of the body, and, consequently, the more nearly will the results of the approximate formulae (1.) agree with the true results ; which, therefore, are the limits towards which the results of these formulae continually approach nearer and nearer, as the dimensions A x, Ay, Az, are diminished. Such limits are found by the process called integration* and are expressed in the following manner : volume 'V=llldxdydz' ~| weight W = w>V = w f f f dxdydz' t J Wxo = w J \\xdxdydz; moments \ Wyo = wfjjydxdydz' Wz = w I I I zdxdydz; ...(3.) * For further elucidation of the meaning of symbols of integration, and for explana- tions of processes of approximately computing the values of integrals, see Art. 81 in the sequel. CENTRE OF GRAVITY. 53 co-ordinates of the centre of gravity / / / xdxdydz 3 dxdydz 06 = r~ j 1 1 ydxdydz III dxdydz III zdxdydz III dxdydz Lsuch are the general formulae for finding the centre of gravity of homogeneous body, of any form whatsoever. 75. Centre of Gravity found by Addition. When the figure of a body consists of parts, whose respective centres of gravity are known, the centre of gravity of the whole is to be found as in Article 73. 76. Centre of Gravity found by Subtraction. When the figure of a homogeneous body, whose centre of gravity is sought, can be made by taking away a figure whose centre of gravity is known from a larger figure whose centre of gravity is known also, the following method may be used. Let A C D be the larger figure, GI its known centre of gravity, W! its weight. Let A B E be the smaller figure, whose centre of gravity G 2 is known, W a its weight. Let E B C D be the figure whose centre of gravity G 3 is sought, made by taking away ABE from A C D, so that its weight is W 3 = W, W* Join G! G 2 ; G 3 will be in the prolongation of that straight line be- yond GI. In the same straight line produced, take any point O as origin of co-ordinates, and an axis at O perpendicular to O G 2 Gj as axis of moments. Make OGJ = x^ j OG 2 = a^, Q G 8 (the unknown quantity) = x* Then the moment of W 3 relatively to the axis at O is and therefore Fig. 20. /> " ** / 3 ~ -W-B-T- PRINCIPLES OF STATICS. 77. Centre of Gravity Altered by Transposition. In fig. 21, let ABCD be a body of the weight W , whose centre of gravity G is known. Let the figure of this body be altered, by trans- posing a part whose weight is Wj, from the position E C F to the position F B H, so that the new figure of the body is A B H E. Let G! be the original, and G 2 the new- position of the centre of gravity of the transposed part. Then the moment of the body relatively to any axis in a plane per- pendicular to Gj G 2 will be altered by the amount "Wj Fig. 21. GO G 3 = G! G 2 == T! G 2 ; and the centre of gravity of the whole body will be shifted to G 3 , in a direction G G 3 parallel to GI G 2 , and through a distance given by the formula W, W &O * 78. Centres of Gravity of Prisms and Flat Plates. The general for- \N^ Jmulse of Article 74 are intended not so much for direct use in if finding centres of gravity, as for the deduction of formulae of a more I simple form adapted to particular classes of cases. Of such the fol- lowing is an example. The centre of gravity of a right prism with parallel ends lies in a plane midway between its ends ; that of a flat plate of uniform thickness, which in fact is a short prism, in a plane midway between its faces. Let such middle plane be taken for that of x y ; any point in it (fig. 22), for the origin, and two rectangular axes in it, OX and O Y, for axes of co-ordinates, to which A B, the transverse section of the plate, is referred. Conceive the figure A B to be divided into narrow bands, by equi-distant lines parallel to one of the axes of co-ordinates O Y, and at the distance A x apart. Let x be the distance of the middle line of one of these bands from O Y, and 3ft, 2/3, the distances of the two extremities of that middle line from O X Then the band is approximately equal to a rectangular band of the length y z - y^ and breadth A x, the co-ordinates of whose centre are x } and ~ 2 ~-> Consequently, if % be the uniform thick- Yig. 22. PRISMS AND FLAT PLATES. 55 ness of the plate, and w its specific gravity, we have for a single band, area = (y z - Vi) A x nearly ; volume = z (y z - 2/,) A x nearly ; -weight ==w z (y 2 2A) *x nearly; moment relatively to O Y, = wzx(y 2 y?) *x nearly ; moment relatively to X, WZ AX and for the whole plate area = 2 (y. 2 y^ A x nearly ; volume Y = z 2 (y 2 - y^) A x nearly ; weight "W = w z 2 (y a y?)*x nearly ; moment relatively to O Y, x ~W = w z 2 x (2/32/1) A x nearly; moment relatively to X, y^W = w z 2 -^2 A x nearly; consequently, the co-ordinates of the centre of gravity of the plate (omitting the common factors w z\ are x n = nearly ; .(1.) 2 2 '(2/3-2/1) A* The more minutely the cross-section AB is subdivided into bands, the more nearly do these . approximate formulae agree with the truth; so that the true results are the limits to which the results of the approximate formulae (1.) approach continually as A x becomes smaller ; that is to say, in the notation of the integral calculus, area = volume Y = z \ (y 2 y\) dx; weight w Y = w z \ (y 3 y^) d x ; (2-) PRINCIPLES OF STATICS. moments x W = w z j x z x .(3.) .(4.) co-ordinates of the centre of gravity The foregoing process is what is usually called by writers on mechanics, "finding tJie centre of gravity of a plane surface ; " but this phrase ought always to be understood to signify "finding the centre of gravity of a homogeneous plate of uniform thickness, the faces of which are plane surfaces of a given figure." 79. Body with Similar Cross-sections. Let all the cross-sections of /a body made by planes parallel to a given plane (say that of xy), be similar figures, but of different sizes. The areas of the different cross-sections are to each other as the squares of their corresponding linear dimensions. Let s denote some definite linear dimension of a cross-section whose distance from the plane x y is z, so that its area shall be a being a constant. Let x h y^ z, be the co-ordinates of the centre of gravity of a flat plate having its middle plane coincident with the given cross-section. Then, by reasoning similar to that of Articles 74 and 78, we find the following results for the whole body : volume weight moments Y= dz; '* dz- .(2.) x "W = wa \ a*j s? dz' } y BG sin. ^ O D C II. Polygon. Divide it into triangles ; find the centre of gravity of each; then find their f common centre of gravity as in Art. 75. -""III. Trapezoid. ABCE. (Fig. 29.) Greatest breadth, A B = B. Least C E = b. Bisect A B in O, C E in D join O D. 1 B I AY = w O D \ A B + I Fig. 29. OT n ^ O T) TC 2 64 PRINCIPLES OF STATICS. IY. Trapezoid. (Second solution.) (Fig. 30.) O, point where inclined sides meet. Let F = a?i, O D = x* O~G = x . x = J_ ^LTZ^ W = tu . ^=^ - sin 2 ^ F B. Fig. 30. (cotan ^ A B + cotan ^ B A). W = sui" OFB. (cotan ^: A B + cotan ^ B A). *\; ^ Y. Parabolic Half - Segment. (OAB, fig. 31.) Q, vertex of diameter OX; OA= ,; AB = y w ordinate || tangent O C Y. 3 '= -7 ^ ' i 2/1 * sin ^^: X O Y. Fig. 31. VI. Parabolic Spandril (0 B C, fig. 31.) G', centre of gravity, = -- ^ ^ 2A sin ^: X Y. - (OAC, fig. 32.) Let OX bisect the angle AOCj OY-LQX. AC Radius O A = r Half-arc, to radius unity, tTVVv = 2 sin Fig. 32. CENTRES OF GRAVITY. TJ 1 1. Circular Half-Segment. (A B X, fig. 32.) V 2 sin" jj _, y* _ . 3 6 sin 6 cos 6 4 sin 2 sin 2 6 cos ^ = r 3 (* - cos 6 sin 0) f IX. Circular Spandril. (ADX, fig. 32.) _!_ sinM ^ ~~ 3 r *2 sin Q - sin cos B - 6' 3 sin 9 6 - 2 sin 2 6 cos - 4 sin 8 - y= "3 2 sin & sin cos 6 4 /I 0\ W = tw 2 (sin sin 6 cos 6 ~\. \ ^ J/ fetor of Ring. (A C F E, fig. 32.) OA = r ; OE = r'. 2 r 3 - r* sin 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 ledge, and by a cylindrical or prismatic surface (cylindrical, as before, being used in the most general sense). """XII. General Formula for Wedges. (Fig. 33.) All wedges may 1 be divided into parts such as the figure here represented. O 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 ty /> = #,; X A = z v Then z = -; = W 'L I #! J v Fig. 33. 66 PRINCIPLES OF STATICS. ) ofydx I xy dx I = J xydx = -* . (This last equation denoting that G is in the plane which traverses O Y and bisects AX.) In a symmetrical wedge, if be taken at the middle of the edge, y = O. Such is the case in the following examples, in each of which, length of edge = 2 y lt ^dail. Rectangular Wedge. (= Triangular Prism.) (Fig. 34.) ; Fig. 34. * = 3 *. XIV. Triangular Wedge. (= Triangular Pyramid.) , 2 ^ O w - Kg. 35. XQ = *~~ XV. Semicircular Wedge. (Fig. 36.) Radius OX = OY = r. y= JJ*^?. ^, Fig. 36, = 3 U16 CENTRES OF GRAVITY. XVI. Annular, or Hollow Semicircular Wedge. (Fig. 37.) External radius, r' } internal, /. Fig. 37. C. CONES AND PYRAMIDS. Let 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 &. XVIL Complete Cone or Pyramid. Let the height OX = h; ' 3 h. I * : ' y W = \ 10 A A am A o ru XVIII. Truncated Cone or Pyramid. Height of portion trun- \*| < cated = h'. sin D. PORTIONS OP A SPHERE. XIX. Zone or Eing of a Spherical Shell, bounded by two conical j surfaces having their common apex ! at the centre O of the sphere (fig. 38). IT ' OX, axis of cones and zone. r. external radius } /, , . , i -,. > ot s r , internal radius } 1 ^ XO A. = a, half-angle of less ) greater / cone. Fig. 38. 68 PRINCIPLES OP STATICS. cos cos A cos /3 - cos I. \ XX. tfecfor of a Hemisplierical SMI (C X D, fig. 39.) OY Bisects angle DOC; i r 4 - r'* sin y c Fig. 39. 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 twa different positions, and finding the single point in the body wh in both positions is intersected by the axis of the cord. For tH resistance of the cord is equivalent sensibly to a single force actiij 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. 69 in a table annexed to this volume. The following table shows a comparison between different British units of intensity of stress : Founds on the Pounds on tho square foot. square inch. One pound on the square inch, 144 One pound on the square foot, 1 rir 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*2021 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 parts 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. SJiear, or Tangential Stress, is the force which acts between "0 contiguous bodies or parts of a body, when each draws the other Ideways, in a direction parallel to their surface of contact, and hich 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, 1 ' 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 PRINCIPLES OF STATICS. / t| ~l 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 I 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 jp, the amount or magnitude of its resultant will be T= P S .............................. (1.) If the intensity or 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, O X, Y. Conceive that plane to be divided into small rectangles by a network of lines parallel to O X and Y respectively, and let A #, A y } be the dimensions of any one Fig, 40. o these rectangles, such as that marked a in the figure. Conceive a figure approximating to that of the given plane surf ace to be composed of several of these small rectangles, so that S = 2 A x A 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 Then the amount of the resultant stress is given approximately by , the equation P = 2 p AX &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, ff , , r < 4 ') Jr = / I p 'dxdy. \ The mean intensity of the stress is given by the following equation : p \\pdxdy I I dxdy CENTRE OF 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, O Y, the two axes of co-ordinates in its plane; 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, Fig. 41. 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 = W .(6.) The volume of this ideal solid will be V =// z 'dxdy (7.) So that if it be conceived to consist of a material whose specific gravity is w 9 the amount of the stress will be equal to the weight of the solid, that is to say, P = w 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 j -teand in this case, the two parts into which the solid A B A B is divided by the plane X Y, are to be regarded as having opposite signs, and V is to be held to represent the difference of their volumes. The Tneoffi stress of equation 5 is evidently Po = WZ Q (9.) in which 2 is the height of a parallel-ended prism or cylinder standing on the base A A A. and of volume equal to the solid AB. 9. 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 Fig. 42. 72 PRINCIPLES OP 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 at, /3 denote the angles which the direction of the stress makes with X, Y respectively. Then the moments, relative to the co-ordinate planes, Z X, Z Y, of the components parallel to those planes of the stress on A x A y, are given by the approxi- mate equations. Moment relatively toZOX, y p AX &y sin /3 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 /3 = sin /3 / / yp dx dy ) (\\ X Q P ' sin * = sin a. j J xp ' dx dy \ Consequently the co-ordinates of the centre of stress are I I xp 'dxdy \ I I p-dxdy rt } (2-) _ J J y p-dxdy p 'dxdy which are evidently the same with the co-ordinates, parallel to OX and 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, \ j pdxdy = 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 XQ and y Q , 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. Hloment 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 O 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= [ I p-dxdy = a I \ x-dxdy (2.) which, if the axis O Y 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 A A A 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); -YOY the neutral axis of the stress applied; -XOX perpendi- cular to -YOY, and in the plane of AAA; ZOZ perpendicular to that plane. Conceive a plane BB inclined to AAA to traverse the neutral axis, and to form, with the plane A A A, a pair Fig, 43. of wedges bounded by a cylindrical or prismatic surface parallel to ZOZ. The ordinate z, drawn from any point of AAA to BB, will be proportional to the intensity of the stress at that point of AAA, 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 , /3, y, be the angles which the direction of the stress makes with OX, OY, OZ, respectively. Let AceA?/ denote, as before, the area of a small rectangular portion of the surface, x, 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 X; A P y cos y ; j, O Y ; A P x cos y j O Z ; A P (x cos /3 y cos a). Summing and integrating those moments, the following are found to be the total moments : round OX; M { =a 'cosy f f xy 'dxdy OY;M 2 z= a cosy f [tf'dxdy .(3.) OZj M 3 = a jcos/3 f f x 2 -dxdy- cos * \ (xydxdyl For the sake of brevity, let J J J J then, as in equation 7 of Article 60, we find, for the moment of the resultant couple, M = = a - J {(I 2 + K 2 ) cos 2 y + P cos 2 ft + K 2 cos 2 - 2 I K cos # cos /3. } = a J (I 2 sin 2 a + K 2 sin 2 /3 2 I K cos a cos /3) ;...(4.) and for the angles A, ^, *, made by the axis of that couple with the axes of co-ordinates, we find the angles whose cosines are as follows: M! M 2 M 3 ~W COSf *~~W' COSi/ ~lvf ( 5 *) MOMENT OF BENDING STRESS. 75 The following equation is easily verified : COS a- COS A + COS (3 COS (A + 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, Xi, from the neutral axis; then 92. Moment of Bending Siress. If the uniformly varying be normal to the surface at which it acts ; that is to say in symbols, cos* = 0; cos/3 = 0; cosy = l; (1.) then it is evident that M 5 = 0; cos * = 0; (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 : (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 M! = a K ; M, = al, M = a- N /(I 2 fK 2 ); cos A = sin i * = K JVTW' cos^ = sin x= I VrTK 2 ' .. tan *-fs and the same equation may be fulfilled also for certain unsymme- trical figures. In this case we have M!=OJ M = M 2 = al- t* = 0', (4.) 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 oe. = sin & ', cos /3 = since ; M = M 3 = a (I sin K cos ) ; cos A ; cos p = j cos v = 1. 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. 4. Centre of Uniformly Varying Stress. When the amount of 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 figure, O, of the surface of action. As in Article 88, equation 5, this mean intensity may be represented by P total stress Q- S area Secondly, an uniformly- varying stress, whose neutral axis tra- verses 0, whose amount is 0, and whose intensity, p', 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 P = PQ + p = PQ + a X (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 L = ^ (3.) to the opposite side to that whose name designates the tendency of the couple M ; 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 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 to the neutral axis lar) is f* along the neutral axis j J xp'-dxdy a\\x"dxdy al P " P = T J jyp'-dxdy aj J xy 'dxdy aK P; The angle 6 which the line joining and the centre of stress makes with the neutral axis OY, is that whose cotangent is cotan I =-= ~ x 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. ~ I 95. moments of Inertia of a Surface. The integral I = I f gp V 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, its centre of gravity, YOY, XOX, 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 = j j x*'dxdy; let the moment of inertia re- latively to XOX as a neutral axis be r r f( *) J = J J tf 'dady, and let K = j J xydndy- Fig. 44 Now let Y'OY', X'OX', be a new pair of rectangular axes, in any position making the angle Y Y' = X X' = ,3 78 PRINCIPLES OF STATICS. with the original pair of axes; and let I' = f f = I jy'*'dx'dy'', ' = f fjj (2.) 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 f = x cos & y sin /3; \ y = csin/3 + ycos/3;V .................. (3.) x' z + y' 2 = x 2 + y\ J (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-ordiuates are placed.) From the equations (3), the following relations are easily deduced between the original integrals I, J, K, and the new integrals I',J',K':- i I' = I cos 2 /3 + J sin 2 /3 2 K cos /3 sin /3; ) 4 J' = I sin 2 /3 + J cos 2 /3 + 2 K ' cos /3 sin /3; >,..(4:.) J K' = (I J) cos /3 sin/3 + K (cos 2 /3 sin 2 A) j Also, the following functions of those integrals are found to be isotropic; I + J = r + J' = 1 1 ( x * + y*) dxdy ...... (5.) (called the polar moment of inertia) I J -- 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 "elatively 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' + 3' is constant, I' must be a maximum and J' a minimum for that position of the rectangular axes which makes the difference I ; J' a maximum. And because (r JO 2 = (P + J') 2 r J ; , 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, THEOKEM 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 other less, than for any other neutral axis. These axes are called Principal Axes. Let I,, J,, be the maximum and minimum moments of inertia relatively to them, and let fa be the angle which their position makes with the originally-assumed axes ; then because E^ = 0, we have, from the third of the equa- tions (4) tan<>a - 2co3 ^i sin ^i - 2K (7 \ ^-cos^-sin 2 ^"^^" and because I t + J l = I + J, and I t J x = IJ K 2 , we have, by the solution of a quadratic equation, Tl = The position of the principal axes, and the values of Ij, J,, being once known, the integrals I', J', K', for any pair of axes which make the, angle /3' with the principal axes, are given by the equations I' = Ii cos 8 /3 f + J l sin 8 /3' ; } J' = I 1 sin 2 /3' + JiCos 2 /3'; > (9.) J K' = ft, J,) cos /3' sin '. J -If I x = Jj, then I' = J' = I,, 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 Y makes with OY cotan & = -=-. 1 \ For the principal axes, K = 0, cotan & = 0, and 6 is a right \anglej 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 YOY, has been determined, and that its position is Y' Y', so that 80 PRINCIPLES OF STATICS. Let tliis conjugate axis be assumed as a new neutral axis. Then the integrals I', J', K', belonging to it are determined by substituting 6 for /3 in the equation 4 ; that is, by substituting for cos ft and sin /3, the values of cos & and sin Q in terms of K and I, viz. : cos 6 = K sin o = VP + K* .yi' + K 3 which substitution having been made, we find .(10.) Now let it be required to find the angle 0, which the new con- jugate axis makes with the new neutral axis Y'O Y'. This angle is given by the equation cotan ff = -r= = cotan t, whence t = -, (no \ or in words, ' THEOREM IT. If the axis conjugate to a given neutral axis be 1 taken as a new neutral axis, the original neutral axis will be the new L^fonjugate axis. > <*The following mode of graphically representing the preceding ^ theorems and relations depends on well known properties of the ellipse. In fig. 45, let X : Y, perpendicular to each other, represent the principal axes of a surface. With the semi-axes, = OY- = VJTj describe an ellipse, so that the square of each semi-axis shall represent the moment of inertia round the other. Let the semidiameter OY 7 be drawn in the direction of any assumed neutral axis, and let ^ Y X O Y 7 = ft'. Draw 00, the MOMENTS OF INERTIA OF A SURFACE. 81 semidiameter conjugate to OY', so that the tangent CT shall be parallel to O Y'. Let CT = t, and let the normal OT = n. Then it is well known that v? = a 9 cos 2 & + l>" sin 2 &; \ and that V (13.) n t = (a 2 6) cos & sin p -, J consequently, comparing this equation with the equation 9, we find, K' =nt; ~T7" j. 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 Q the distance of the centre of gravity of the given figure from the axis O 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 /*/ /* __ ff from O'Y'. The required moment of inertia is but x* therefore, I^s^B + Sxo ff af-dxdy+ f f x'*-dxdy; and because O'Y' traverses the centre of gravity of S, x'-dxdy=Q' t 82 PRINCIPLES OF STATICS. so that the middle term of the expression for I vanishes, leaving I == x 2 S + j j x'*'dxdy 3 (15.) or in words, THEOREM Vi The moment of inertia of a surface relatively to an axis not traversing its centre of gravity is greater than the moment of j inertia round a parallel axis traversing its centre of gravity, by the ' product of the area of the surface into tlie square of the distance between I 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 : -p. Maximum T! Minimum J (neutral axis Y). (neutral axis X). I. RECTANGLE. Length along X, ) h s b hb s h', "breadth along Y, b j 12" 12" II. SQUARE. Side = h,. -A*- ^ ITI. ELLIPSE. Longer axis, h ) h s b -jrhW Shorter axis, b J Q4- $4. IV. CIRCLE. Diameter, h , - _- 54 64 V. Hollow symmetrical figures ; sub- tract I or J for inner figure, from I or J for outer figure. VI. Symmetrical assemblage of rec- 1 j^^ ^^s tangles ; dimensions of any one I 2 y^- 3 -y^- h || x, b || j/; distance of its centre j fromOY, X Q ; from OX, y J +^'hbxl +?'hbyl 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 j 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. fr- r""~ 97. Simple Stress and its Normal Intensity. A simple stress is a \ pull or a thrust. In the following investigations a pull will be I treated as positive, and a thrust as negative. \ In fig. 46, 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 O X, of the amount P, and of the intensity T \ B m Fig. 46. 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 j let O 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 = ^:XON=^TOA. 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 0X5 but the area over which that pull is distributed is now 84: PRINCIPLES OF STATICS. consequently, the intensity of the stress, as reduced to the oblique plane of section, is Pcos* 99. Resolution of Oblique Stress into Normal and Tangential Component*. 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- \ -p 1 s~\ TL-T- /* JL COS U * long ON, J Tangential component ) r> /i . i r\rr\ f * Sin 8 , along OT, J and the intensities of these components are, Normal ; p n =p r cos 6 p m cos 2 6 ', ) ,-,^ Tangential ; p t =p r sin 6 =p x ' cos & sin } ^ *' Suppose another oblique plane of section to cut the body at right angles to B B, so that its obliquity is ^ = 90 0; and let the intensity of the stress on the new plane be denoted by accented letters ; then P' n =P,' cos 2 tf =p m sin 2 6 ) (2 . p' t = p t ; p n +p' n =p x ; J" 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. Pa"' of Conjugate Stresses. THEOREM. If the stress OH a given plane in a body be in a given direction, ttie 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 ABCD, bounded by a pair of planes AB, 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 XOX, 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 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 fourtli 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, z x, 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 xy ; A A A v w, w u, uv, the inclinations of those perpendiculars to each other ; A A A ux, vy, wz, the respective obliquities of the stresses. Then those nine angles are related as follows : T 4- i 2 A A A A A A .Let 1 cos y z cos z x cos 3 x y + 2 cos y z cos z x cos x y = C; ....................... (1.) Then (2.) sn vw = - ; cos vw = sn z x ' sn xy sin z x sin x y sn x y sn y z sin x y sin y z A A A ^ A smuv= A / r\ A J ^ A cos y z cos z x cos xy = - -* - -- - - A sn y z sin z x sin y z * sin z x = fSr ( 3 -) 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 \ 3 > (4) A A = 1- cos 2 y z - cos 2 z xj ) PLANES OF EQUAL SHEAR. A sin vw = A sinwu = A JC RJ n ?/ fl? ^ = JC sin 20; Vc A cos y z on 5 ? flj in j V/V/O V */ ^ , sin zx A cos z x /\ ^ wo w * A ' sin y z sin ^ z A A A cos y z cos 2 x sin yz * sin zx smyz'sinzx A /C A vy = -^^ ; cos wz = sin # sin ' A > cos /2 ,(.) RESTRICTED CASE II. Suppose one of the stresses (such as z) to be perpendicular to the other two, which are oblique to each other, Then ..(7.) (8-) tr n I V A A cos y z = 0; cos z x = ; - A i - A i sin y ^ = 1 ; sin z x = 1 ; C = S in* A y. sin v w = 1; cos vw = ; (or v w = 90); sinwm=z=l; cost0w = 0; (or^; = 90 ); A A A A sin u v = sin x y; cos wv = cos x y; (or, uv + xy = 180). A A A A A cos u x = sin a; y; cos v y = sin a; y; cos wz = A A A A or wee = t?2/ = "0 xy } wz = 0; 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. V\ 103. Planes of Equal Shear, or Tangential Stress. THEOREM. If tJie stresses on a given pair of planes be tangential to tJiose 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. tiaJ, 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 B C D, and bounded by a pair of parallel planes, AB, CP, and a pair of parallel planes, AD, C B. 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. Fig. 48. The forceg exerte( j j^ tne 0^,, par ts o f 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, jVare&AB'EF. The forces exerted by the other parts of the body on the pair o 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 p't' 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 bt oblique stress on three planes at right angles to each other, the tangential components of the stress on any two of tJwse planes in direction^ 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, CB, to the plane z 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 X^N", 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, YTZ-, whose resul- tants act through the axis of z, and the compo- nents Ytf, which form a couple acting round that axis, which, by the conditions of equilibrium of the rectangular solid ABCD, must be equal and opposite to the former couple; and by reasoning similar to that of Article 103, 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 ys 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. yz ... intensities. -- .rjwe jryy - ry* xy Then, in virtue of the Theorems of Articles 101 and 102, we have the normal stresses, p It ,Pyy, P> conjugate and independent; and PRINCIPLES OF STATICS. in virtue of the theorem of this Article, there are three pairs of tangential stresses of equal intensify, [The reader who wishes to confine his attention to the more simple class of problems may pass at once to Article 108, page 95.] _~ ^ 105. Tctraedron 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 oftlie stress on a fourth plane, traversing tlw same body in any direction. In fig. 50, let Y O Z, Z O X, X Y, be the three planes, on which act conjugate stresses in the directions X, Y, O Z, of the intensities Pv Py) Pz- 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 50 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 GAB. On O X, O Y, O Z, respectively take OD = total stress on O B C = p, area O B C, O E = total stress onQCA.=p 9 ' area OCA, O E = total stress on O A B = p* area O A B. Complete the parallelepiped O D E E B, ; 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 - L . Q. 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 oftlie 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 Y, in fig. 50, represent the rectangular co-ordinate planes, so that OX, O Y, O Z, are now at right angles to each other (instead of being, as in Problem I., in TETRAEDROX 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 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, C A, and O A B. Therefore, on O X, O Y, Z, respectively, take OD = p tx area B C + p^ area OCA + p t , area O A B, O E =p fy * area B C + p n area OCA + p yt area O A B, =p zg ' area O B C + p yi area OCA + p tg area O A B ; Complete the rectangle D E F R ; then will its diagonal Oil re- present the direction and amount of the total stress on an area of the fourth plane equal to ABC, and the intensity of that stress will be - - . Q.E.L area ABC A A A To express this algebraically, let x n, yn, zn, denote the angles which a normal to the fourth plane makes with the three rectangu- lar axes respectively ; x r, yr, z r, the angles which the direction of the stress on that plane makes with the three rectangular axes respectively; and p r the intensity of that stress. Then, it is well known that A area O B C = area ABC' cos x n, area O C A = area ABC* cos y n, area A B = area ABC' cos z n m y so that the rectangular components of the intensity p r are ........ (i.) A A A p M = p xt cos x n + p^ cos y n + p M cos z n A A A p ny = p^ ' cos x n + p n ' cos y n + p yz ' cos z n A A A p ng = p n cos x n + p yil cos y n + p zg ' cos z n The resultant intensity of the stress required is given by the equation ....................... (2.) and its direction by the equation Pn, A Pny A p n - ; cosyr= ; coszr= J 92 PRINCIPLES OF STATICS. Hence it appears, that if the rectangular components of the stress on three rectangular planes in a body be given, the stress on any fourth plane may be determined ; from which it follows, That every possible system of stresses which can co-exist in a body, is capable of being resolved into, or expressed by means of, the three normal stresses, and the six pairs of tangential stresses, on three rectangular co-ordinate 106. Transformation of stress. For the direction of the normal to the new plane of action, ABC, which direction is denoted by n in Problem II. of Article 105, let there be successively assumed the directions of three new rectangular axes x 1 , tf, z', and let it be required to express the rectangular components, p x ' x ', &c., of a given compound stress relatively to those new axes, in terms of the rectangular components, p ix , &c., of the same compound stress relatively to the original rectangular axes, x, y, z. To solve this question, let n be taken to denote any one of the three new axes. The three components, parallel to the original axes, of the stress on the plane normal to n, are given by equation 1 of Article 105. Each of these components being further resolved into its components parallel to the new axes, and the nine com- ponents so found collected into three sums of intensities parallel to the new axes, the following results are obtained : A A A, p M ' = p M ' COS X X' + p ny ' COS yx' + Pm ' COS Z X \ A A A Pny =p n * ' cos x y t- #,, cos y tf + p M cos z y ; A A A Pz Pn* ' cosa z 1 + p ny -cosyz + p nt coszz. For n are now to be substituted successively, both in p na ', &c., and in the values ofp nx , &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. p,Y p M cos 8 x x' -f- p yy cos 2 y x' -\-p u cos 2 z x' A A , A A A A -f- ^ 2 } yz cosy x cos # + 2 p., x cos z x cos x x' -f- 2p xy cos xx' cosy x' ', Pyy = P** cos 2 x y' + p yy cos 2 ytf+p^ cos 2 z y' A A A A A A -f- 2p yt cosy i/ cos zy f + 2 p M cos z y 1 cos x y -f 2 p xy cos x y cos y y' ', A o A A p/; = p M cos j x z' + p y!> cos" y z -{->,, cos 2 z z r -}- 2 p yt cos y z' cos z + 2 p ts cos z z cos x z + 2 p, y cos x z cos y sf $ PRINCIPAL AXES OF STRESS. 93 TANGENTIAL STRESSES. A A A A A A ftV = P*. cos * y' cos x z -f p n cos y y cos y z +p lt cos z y' cos ^ A. A AA AA AA -f p^. (cossycosyz' + cosyy'coszz') +p f . t (cosxy f coszz + coscy'cosajs) A A A A 4- ^ (cos y y' cos x z 1 4- cos a; y' cos y ^) ; A A A A A A, j,V =/? cos a: s cos x x' -Tp n cos y z' cos y a;' + p a cos z z' cos z x A, A AA AA AA f p, g (coszz cosyx' + cosyz coszx') + p zf (cosxz coszx + coszz'cosxx') A A A A -f /? jy (cos y * cos a; x 1 + cos a; s' cos y a/) 5 A A A A A A f>,' f = jt? M cos x x' cos xtf +Pyy cos y x' cos y y + p a cos z x' cos z y' A A A A A A, A, A -f j9 yr (cossa/cosyy'+cosya/coszy') + ^,(cosa^c'cos^y'+ cosa/cos#y') A A A A + jo xy (cos y a/ cos x y + cos a: x cos y y). The two systems of component stresses, p xx) &c., relative to the axes x, y, z, and_p,V, &c., relative to the axes a/, y, s', which con- stitute the same compound stress, are said to be equivalent to each other. 107. Principal Axes of stress. THEOREM. For every state stress in a body, there is a system of three planes perpendicular to each other, on each of which the stress is wholly 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 Pnx A A p n A cos x r = = cos x n : cos yr = = cos y n ; Pr Pr A , A coszr = - = coszn (1.1 Pr Introducing these values into the equation 1 of Article 105, we obtain the following : \ A A A n (fix* Pr) cos x n + p^ cos yn + p fx cos z n = \ A =0;' r (2.) A A A p, x cos x n -f > cos y n + (p f , - p r ) cos z n \ =o. J 94 PRINCIPLES OF STATICS. From these equations, by elimination of the three cosines, is obtained the following cubic equation ; Then tf-Ap' r + Ep r -C = ..................... (4.) The solution of this cubic equation gives three roots, or values of the stress p r , which satisfy the condition of being normal to their planes of action; and according to the properties of conjugate stresses stated in Article 102, the directions of those three normal stresses must be perpendicular to each other. Q. E. D. The three conjugate normal stresses are called principal stresses, and their directions, principal axes of stress. If p r denote the intensity of one of those principal stresses, the angles which it makes with the originally assumed axes of x, y, z, are found by means of the following equations, deduced by elimination from the equation 2 of this Article : cos x n \p fl p ly + (p r - p xx )p y ,} = cosyn {p xy p yz + (p r = coszn\p yz p zl +(p r ~p zz )p fy ] ............ (5.) Let pi, p 2 , p s , denote the three values of p r , which satisfy equation 4. Then, from the well known properties of equations, it follows that the co-efficients of that equation have the following values : A = (6.) G=p l p 2 p 3 . Hence it appears, that for a given state of stress, the three functions denoted by A, B, C, in the equations 3 and 6, are the same for all positions of the set of rectangular axes of x, y, z, or are isotropic, in the sense already explained in Article 95. Let the principal axes of stress now be taken for axes of rectan- gular co-ordinates, and denoted by x,y,z' } and let it be required to find the direction and the intensity p, of the stress on a plane whose 111 A A A normal makes the angles xn, yn, zn, with those axes. For this purpose the equations 1, 2, and 3, of Article 105, are to be modified by making P** =Pi ; P n =P 2 ; p u =p 3 ', p yz =p tf =p fy = 0. STRESS PARALLEL TO ONE PLANE. 95 Thus we obtain A A A A p cos x p = p l cos x n j p cos y p = p 2 cos y n ; A A p cos zp = 2? 3 eos zn ..................... (7.) p = ^/ < p? cos 2 sen +_pj cos 2 7/72 +j?J cos 8 z n 8 z n \ ...(8.) The equations 7 are easily transformed into the following : A A A A A A cosa;rc_cosa;j? m cosyn_cosyp. coszn_coszp P Pi P P 2 P Ps 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 o A 1 -=^)cos a xp cos-yp cos-zpl ...HO) 'Jl o ~*~ o "t" - n - * I \ J p ( fi pi pl j the well known equation of an ellipsoid, in which p lt 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 + cos z n cos zp ( A A A ) == p J cos 2 x p .cos 2 y p .cos 3 z p ( ( Pi P* Ps ) 1 A A = - (pi cos 2 x n -rp a cos 2 y n+ p 3 cos 3 z n) ........ (11.) 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 their 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 OX 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 g- . to X and O Y respectively. The force exerted by the other parts of the body on the face A of the prism, will be proportional to p/OA; on 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 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 OD R E ; its diagonal OR will represent the direction and amount of the stress on A B, and the intensity of that stress will be OR QB a + pi O"A 2 + Zp.p, ' OB OB'+OA 2 -2OB-O~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,. agrees in kind PRISM OF STRESS. 97 52 ' 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 find the intensity 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 O X 5 let Pyy denote the intensity of the normal stress on" the plane X, which stress is parallel to O Y. In virtue of the Theorem of Article 103, the tangential stresses on those two planes must be of equal intensity; and they may therefore be denoted by one symbol, p^, which sym- bol may be read as meaning the intensity of ( x ) on a plane ( y } the stress along ( y J normal to ( x J 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 A JLy, O B JL x, A B JL O K The areas of the faces of that prism have the following proportions : _ _ /\ _ _ A O B = AB cos xn ; OA = AB sin x n. The forces exerted on the faces O A and O B, in a direction parallel to x, consist of the normal stress on O B, and the tangential stress on O A ; that is to say, p xx 6B-f# E2 ,-OA = AB' < p xx - cos x n + Pxy' sin xn \ Let this be represented by O D. The forces exerted on the faces A and O B, in a direction paral lei to y, consist of the normal stress on OA, and the tangential stress on B ; that is to say, Pxy ' O B -f- pyy * O A = A B ' \ pxy cos xn-{- p Let this be represented by E. sin x n 98 PRINCIPLES OF STATICS. Complete the rectangle O D E 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 f A \H Ut f A A , ?V = = \M fe 2 ' cos2 x n ~T~Pyy 2 * sm 2 a; n -{ p xy + 2p X y(p xx +p m )cosxn'sinx A n\ ...(1J j^)V _ #ry fe Pssr |- jxiJ/lTrom R draw R P perpendicular to the normal N; then the, formal and the tangential components of the total stress on A B will represented respectively by OP = OD cos xn + OE sin xn', PR = OD sin x-n - O E cos x n', f and the intensities of these components by O~P A A A A p n = -=r = p xx cos 2 xn -rpyy ' sin 2 xn + 2$^ cos #?z * sin sew; ] _A_ Jb> A . A A A Pt = ~^=~ = (pxx~Pw) cos xn * sin xn<\- p xy (sin xn cos" am). ^ AB A The obliquity, ^ NOR = n r, of the stress on A B is given by the equation A Pt /<* \ tsmnr -^ \ o< / 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 xxj p yy , p,^, of the stress on two rectangular planes, 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 -, A 2 A cos" xn-sm 2 xn 9 or, what is the same thing, tan 2 xn = -^>- ; (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 ON 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 r//v*/3ie/30 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. ^ --- * ^ =4.10. 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. I In fig. 53, let OX, OY, be the direc- x tions of the given principal stresses, and p x , p y , their intensities. By the condi- tions of the question, those intensities are equal, or P x =Py 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 Y be unity. Then_the total stresses Fig. 53. _ ;on the faces OB and OA will be respectively p x ' CKB and p y - OA. On X and Y respectively, take OD to represent p f * O"B, and E to represent p y - O A; complete the rectangle O D B, E; then its diagonal O B, will represent the amount and direction of the stress on the face A B of the prism, and the intensity of that stress will be OR AB = P " 100 PRINCIPLES OF STATICS. Now, because p x - p y , we have OD OE OK CHI OA AB' and consequently p r =--P*=P y '> and because of the similarity of the triangles A B, E K, R 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. D. In this case it is obvious, that every direction in the plane X 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 denned 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 str ess _ exerted amongst and by them at every point and in every 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 appli 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 %air of prin- cipal stresses be of equal intensities, but of opposite *MK&, 'il<& stress on any plane perpendicular to the plane of the dw&titjfyiS oj the principal stresses is of the same intensity, and fa angles wtyfh its direction makes with the normal to its plane are Us^ied by the axe. of 2Mincipal stress. In fig. 53, let the stresses acting along the rectangular axes OX, OY, 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 OR is normal. In this case OD is to be taken as before, to represent p, OB, the total stress on the face OB of the triangular prism O AB; 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 OR 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 of principal st of any intensities, and of the same or opposite kinds, being given, it required to find the direction and intensity of the stress on a plane i any position at rigJit angles to the plane parallel to which the two principal stresses act. Let O X and O 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. 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 x 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 y is held to be implied in that symbol. Consider the two equations P.= From these it appears, that the pair of stresses, p x 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 , and a pair of stresses of equal intensity, but 2 opposite kinds, whose values are + -. a JSTow 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 O N = xn. ELLIPSE OF STKESS. 103 On fc N" take O M = ; this will represent a normal stress J 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 L, will be the effect upon the plane AB, of the pair of stresses **" ?*. 2i 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, MR = P " ~ P \ 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 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: I Intensity, ORorp r = J {pl'cos*xn + p a y 'sm 2 xn} ..... (1.) I an equation which might have been obtained by making p^ = in equation 1 of Article 108, Problem II. Obliquity, ^ N O R or n r. = arc sin (sin 2 A -fi^fc) ............... (2.) '\ This obliquity is always towards the axis of greatest stress. Injig. 54, p x and p y are represented as being of the same kind; and MR is consequently less than OM, so that OR fells on the same side of OX with OlST, that is to say, n r ^L x n. _ In fig. 55, p x and|? y are of opposite kinds, MR is greater than OM, and OR A A falls on the opposite side of OX to OM; that is to say, nr^xn. The locus of the point M is obviously a circle of the radius *** a P "; and that of the point R, an ellipse whose semi-axes are p, and p y , and which may be called the ELLIPSE OF STKESS, because its semidiameter in any direction represents the intensity of the stress in that direction. 104 PRINCIPLES OF 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 E upon O N, and are as follows: Direct, p n = p x cos*nx 4- p y ' sin 2 ajw; (3.) * = (P* ~ p y )wsxn'smxn', (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 the planes on which the shear, or tangential component of the stress, is most intense, and tJie intensity of that shear. It is evident that the shear is greatest when M E is perpendicular to O M ; and then M E 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 on which the obliquity of the stress is greatest, the intensity of that stress, and the angle of its obliquity. CASE 1. When the pricinpal stresses are of the same kind. (Fig. 54.) In this case M E ^ M O, and it is evident that the angle of obliquity, ^ M E = n r is greatest, when M E is perpendicular to E, and that its value is given by the equation A ME maximum n r arc sin - . OM P* * Pi To find the position of the normal N to the plane A B, we have to consider that, A= i-^rPMN; ELLIPSE OP STRESS PROBLEMS. 105 but^r PMN = ^ MRO + ^ MOB consequently in this case, 90 + max. nr\ 90 + max. n r (an obtuse angle). And for the position of the plane AB itself, we have = 90-*; = (an acute angle). These equations apply to a pair of planes, making equal angles at opposite sides of X. The intensity of the most oblique stress is obviously =>{ MR 2 ) (p.pl 4 or a weow proportional between the principal stresses. This is otherwise evident from the consideration, that when OR -L PRQ, then OB = J (PR RQ), and that RQ = p^ PR = p r 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 O R lies in the plane AB. In this case it is evident, that the triangle O M R becomes right-angled at 0, and conse- quently, that the intensity of the stress is given by the equation = */(-P.ft), ......................... (10-) 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 1ST is found by considering, that and that ^ P M N = ^ M O R + ^ MRO = 90 + arc- sin P * + Py 106 PEINCIPLES OP STATICS. consequently, (an obtuse angle); A x n -if 90 + arc sin (11.) = 90*- xn = I J90- arc -sin -&& 1 21 p*-p 9 ) (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 f + p y means the difference, and p x p y the sum, of the arithmetical values of the principal stresses. .V "^PROBLEM TV. The intensities, kinds, and obliquities, of any two stresses whose planes of action are perpendicular to the plane of their directions, being given, it is required to find the principal stresses and axes of stress. CASE 1. When tJie given stresses are of t/te same kind, and unequal. In fig. 56, let A B, A'B', represent the given planes, O JST, O N', their normals, O R, O R', the stresses upon them. Let the intensities be denoted algebraically p = OR; p = OR', and the obliquities by Fig. 56. In fig. 57, take N to represent at once the normals to both planes. Make ^ 1ST O E, = nr; ^zN O H' = rir; OR = p- } OE' = p'. Join HE,', bisect it in S, from which draw SM -L R.B,', cutting OMinM. Join MR, MR', which lines are evidently 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 OM = 2LILS; MK = MR' = y -^q^; Z '2t 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. E. 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, f either x n + x n = + nn= < A ^ A (or x n xn ^ 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 formula? of trigonometry : P 2 ~P r2 - . ...ns.\ =OM = 2 (p cos n r p' cos n' r) A A 2 p cos nrp x p v cos 2 x n = c ; A n A , 2 ' cos n' r' p. p v cos 2 x n = -- 1 f 2 .(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 In this case, equation 15 becomes A > 2 cos n r j^'"' -PI/} = equation 16 becomes ^=-a cos nr equations 17 are modified only by the equality of ri r' to nr. CASE 4. When the planes of action of the two given stresses are perpendicular to each other j 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 n r =p r sin n r 1 =p? Let the normal components of the given stresses be denoted by A , A p n =p cos nr ; p' n =p' cos n r'. Then equation 15 becomes equation 16 becomes The equations 17 become cos 2 x n = cos 2 x n' = or, what is equivalent, tan 2 A = - tan 2 V = being the same with equation 1 of Article 109. ELLIPSE OF STRESS - PROBLEMS. 109 PROBLEM V. The stress in every direction being a thrust, and tlte greatest obliquity being given, it is required to find the ratio of two conjugate thrusts whose common obliquity is given. Let 0- denote the given greatest obliquity. Then according to Problem III., Let n r, which must not exceed the less, then Pr - P,_ 1 , f / A "~2~ : 2" v 1 V 2 'P^os 2np) -i A 1 2 n sin 2 n r> nx = -% arc-tan-^- ^ .............. (3.) 2 ->cos2 rap The equation 2 is capable of being expressed in another form, as follows. Let a, a! be any two angles. Then cos a cos a' + sin a sin a' = cos (a a'). Now the quantity under the sign J, in equation 2, consists of the following classes of terms : 1. All the squares > 2 cos 3 2 np ; 2. All the products 2 pp cos 2 w p cos 2 w^/ ; where p, p', are OTM/ jpcwV of the given stresses ; 3. AH the squares p 2 sin 3 2 np j 4. All the products 2 p p' sin 2 wjp sin 2 w^X. The first and third of these classes being added together, make 2 (p 2 ); the second and fourth make 2 2 (p p ' cos Spp 1 ) j pp' being the angle between p and p'. Equation 2 thus becomes ...... (4.) Prom the equations (1) and (4) it appears that the intensifies 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 A equations are the several angles pj/ t which the given stresses make L 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 2 p p exceeds 90, we have = cos (l80 %pp')> 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 uniform 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 ; 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 three 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 the subject of the strength of materials, which will be treated of iii 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 10 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 formulse are simplified by assuming one axis to be vertical; let 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 Fig. 58. Let the dimensions of the molecule A be A a; parallel to OX, Ay OY, A* OZ. Then its weight is represented by w AX Ay AS. 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 xx , pyy, p a - Tangential, p yz , Farthest from 0. Nearest to O. -f Ay A - Ay AZ + AZ AX AZ AX 4- AX Ay ) (the upper.) J AX Ay \ (the lower.) J As for the signs of normal stress, let pull be positive and thrust I 114 PRINCIPLES OF STATICS. negative. As for the signs of tangential stress, let those stresses be considered as j ^ ^tive | which tend to make the pair of cor- ners of the molecule which are nearest and farthest from O ( sharper ) \ natter kind In the first place, let the rate of variation of the stress, of what ind 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 A y be p, then at the face + A x A y, whose distance from A x A y is A % 9 let the mean intensity of the same component be d f) in which -= is a constant co-efficient or factor, meaning " the rate 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 tlie weight of the molecule a principle expressed by the three following equa- tions : = W ' A X A y A Z. NTERNAL EQUILIBRIUM. 115 Eacn 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, A x & y A z dividing by this, they are reduced to the followin : s. 4. + * = dx dy dz dp* dp ys dp a ~d~x~ ~d^j ~d^ 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 instead 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 , 10 cos /3, w cos y. The equations of this Article are not in general sufficient of 116 PRINCIPLES 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. Equilibrium of Fluids. It has already been explained in ""), that in a fluid the only stress to be considered in a thrust or pressure, normal and of equal intensity in all directions. This is expressed symbolically in the following manner : means of the /117. Equil \L^I Article 110, ^practice is a OA 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 ' 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 OP FLUIDS. 117 Those principles may also be proved directly. Let fig. 59 re- present a vertical section of a fluid; Y O Y any horizontal plane, O X a -- vertical axis. Let BB be a hori- zontal plane at the depth x below O; _ A C C another horizontal plane at the - I L depth x + A#. Let A be a small rectangular molecule contained be- tween those two horizontal planes; Fi g. 59. and let A y and A % 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 p , then, be the uniform pressure at the horizontal plane YO Y, p, that at the plane B B, and^> + - *x that at the plane C C, cL x cl x 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 The difference between those forces, viz. : dp '*x-*y*s, 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 obtaiiL the first of the equations 2 of this Article. The pressure p 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^ I (3.) 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 118 PRINCIPLES OP 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 20000, or 2 5 A o o 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 + wxj (1.) that is to say : let p Q be the pressure at the upper surface, Y 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 PQ 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 different 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 U (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 p Q 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 r* * ........ ~ ..... (*) P =p] o w"dx', which equations, when the fluids are liquids, and wf, w", constants, become p = p Q - wx; p" = p - 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 x 1 , x", 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 n =.p, and consequently, for fluids in general, 120 PRINCIPLES OF STATICS. * j w'dx= I w"dx, (4.) o Jo 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 / w 1 dx w" x" ; (6.) and on this last formula is founded the method of determining differences of level by barometric observations of the atmospheric J 120. Equilibrium of a Floating Body. THEOREM. A Solid body _ loating on the surface of a liquid is balanced, wJien it displaces a volume of liquid whose weight is equal to the weight of the floating body, and when the centre of gravity of the floating body, and that of the 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- mersed part of the solid body. The liquid will exert against S a normal pressure, whose amount will be ex- pressed by Fig. 60. Sp = Swx, 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 * 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 Ssin. 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 S/> cos = S w x cos , being equal to the product of the intensity p by the horizontal projection of S. But S x cos 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 S 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 Y the volume of liquid displaced by it. Then the conditions of equilibrium of the floating body are ob- viously the following : First : W = w Y; or its weight must be equal to the weight of the volume of liquid displaced by it; 122 PRINCIPLES OF STATICS, Secondly: 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 EJocly. 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, Y Y whether liquid or gaseous. . Conceive a small vertical prism STJ 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 prism of which S is an oblique section, and TJY a horizontal prism of which U is an 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 Yj 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 TJ is downward, and equal to the weight of the prismatic portion of the fluid which stands vertically above TJ. 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. I In practical questions relative to the equilibrium of ships, the I 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. L-^ 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 dry 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 WQ. 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, 493-2 ,, x ^ = ^7Tl6r^ ; W and if w, /, be the weights of a given volume of air, under the respective pressures p,p', and at the temperatures t, t', of Fahrenheit's scale, then to' __p_ t + 461-2 (2 v w ~~ ' t -f-461-2'" 124 PRINCIPLES OF STATICS. Let Wi denote the true weight of a body, Yi its volume, Wi its weight per unit of volume, w the weight of unity of volume of air. Then and the apparent weight of the same body in air is W' = (w, - 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 _ w l so that the proportion between the real weights of the bodies is "W ww. w.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 and consequently Let the first weighing take place in air and the second in the liquid, and let W be the apparent weight in air ; then -^w, and consequently W" w t so that if is known, may be found by the equation IMMERSED PLANE. 125 W'-W"- w* W W" ' (3.) "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, 62-425 pounds avoirdupois. For any other temperature t on Fahrenheit's scale, the weight of a cubic foot of pure water is 62-425 (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. t?=10-l (t 39-1) 2 0-0369 (t 39 -I) 3 -5-10,000,000. (5.) The relative specific gravities of two liquids are determined by weighing the same solid body immersed in them successively Comparing its apparent losses of weight. 124. Pressure on an Immersed Plane. If a horizontal plane SUT-j I face of any figure be immersed in I a fluid, the pressure on that sur- ( 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 ri s- oz - usually termed, the centre of gravity of the plane surface. If an inclined or vertical plane surface be immersed in a liquid, let O Y (fig. 62), represent a section of the horizontal plane at which the pressure is nothing, and BF a vertical section of the 126 rKLNCIPLES OF STATICS. immersed plane. Let X L = BE be the depth, to which the lower edge of this plane is immersed below OY. From B draw BD = BE, and -L BE; produce the plane BE till it cuts the horizontal plane of no pressure, OY, in the line represented in section by 0; through O and I) draw a plane H D, and conceive the prism B D H E to stand normally upon the base B E and to be bounded above by the plane D H. The pressure on the plane BE will be normal; its amount will be equal to the weight of fluid contained in the volume B D H E ; that is to say, let X Q denote the depth of the centre of gravity of the plane BE below O Y, and w the weight of unity of the volume of liquid; then the mean intensity of the pressure on B E is p = WXQ,... ............. . ............ (1.) and the amount of the pressure P = MOV area 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 BE 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 BE, 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 = w sin a. ............................ (3.) where a, is the angle of inclination of the plane B E 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 & with a horizontal plane. In fig. 63, let Y O Y represent a ver- tical section of that upper sloping surface along its direction of greater declivity, and O X a vertical plane Fig. 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 L 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 = wxcost) (1.) From the above theorem, combined with the principle of conjugate stresses of Article 101, there follows THEOREM II. T/ie 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 I the planes B B, C C ; whence follows THEOREM III. The state of stress, at a given uniform, depth belmv 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 their 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 Avill appear in the Second Part. CHAPTER VI. ON STABLE AND UNSTABLE EQUILIBRIUM, 127. Slable and Unstable Equilibrium of a Free Bodf. 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, its 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 mafy 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, j^econd part of this treatise. THEORY OF STRUCTURES. CHAPTER I. DEFINITIONS AND GENERAL 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 ; and the deter- mination of the forces by which it is supported requires the con- sideration of dynamical as well as of statical principles j 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 : * 1. That the forces exerted on the whole structure by external bodies sludl balance each other. 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 ; (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 THEOEY OP STRUCTURES. II. That the forces exerted on each piece of tlie structure shall 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 fche 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 shall 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 j 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 Jirst 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- J&ary, 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 H. STABILITY, 133. Resultant Gros* 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, whet. 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 Line 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. s are such as occur in carpentry, in frames of 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 tin's 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. BlockworJc 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 j 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 Jg ' ' 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. The 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 w.65. 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 they 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 direction of the load ; and let the load act between the points of support, as in fig. 66 ; *K where P represents the resultant of the gross |fp j 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, Ej, E 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, E 2 and P = E! + (1.) (2.) Fig. 67. 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 w 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 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^Ei - E 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 E 1? E 2 , be now inclined to that of the resultant of the load, P, as in fig. 68. This case is that of the equili- \ brium of 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 Fig. 68. fog i n one plane. II. They must intersect in one point (C, fig. 68). IIL 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 oftlie supporting forces, Ej, and the centre of resistance of the other, S 2 ; required the line of action of ilie 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 j this will be the line of action of R^; 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 t,, %, 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 t : R 2 : : sin (ii + *s) : sin ^ : 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 63 six straight lines so as to form four triangles; each force will bepro- poriionoL 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 in the points L, Sj, S 2 , S 3 ; let P, Rj, R 2 , R 3 , 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 Rg, whose magnitude is Fig. 69. R 2 -f- Rg, must traverse Bj ; and because the resultant of that resultant and R! is equal and opposite to P, we must have the following proportion : P:R t : : S^ :LB, : : A 8^83 : AS 2 L S 3 ; and applying the same reasoning to the forces R 2 , Rg, we find the proportions, P:R 1 :R 2 :R^-::AS 1 S 2 S 3 :AS,LS 3 :AS 3 LS 1 :AS 1 LS 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 thrt^ props, exerting parallel supporting forces. 144. JLoad 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, and 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, Sj, S 3 , 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, R l5 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,, S 2 , are equal and directly opposed. Q. E. I. The symbolical expression of this solution is as follows : let i lt 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 : R! : R 2 : : sin (% + i 2 ) : sin ^ : sin %. 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. 71 that of two struts (such as a pair of rafters abutting against two walls) ; fig. 72 that of a strut, L S^ 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 x 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 S! 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 bars, 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. IY. 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 t be applied at the joint 1 in any given direction ; let supporting forces, P 2 , P 3 , be applied at the joints -p. 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 , B~C ... P s , Then by the conditions of equilibrium of a frame of two bars (Article 145), the external force Pj 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 0, 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 tJie 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 oftlie 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 froni O * 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 A, O B, O C ; then if the_load 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, OC, represent the stresses on the bars A, B, C, respectively. From O let fall OH 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 CA, as is usually the case, is vertical, OH 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 Fi 74 POLYGONAL FRA1IE. 135 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, &, 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 GA = OIL -(tan c =: tan a); "] Supporting j AB = OH -(tana:qp tan b) ; [ (1.) Forces j BC = O~H (tan 6 tan c) ; j ( + ) is to be used when the two ) opposite directions ( J inclinations are in j the same direction. (2.) r~, / 'l50. Polygonal Frame Equilibrium. The Theorem ^of Article '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 P, 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, OE, 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 ABCPEA; then it is evident that AB 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 O A on A and B on B ; that B C is the external ' force which being applied at the joint 2 of B and 0, will produce the stress OB on B (already mentioned) and O(J on 0; 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 of the 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 0*\e bars whose lines of resistance are parallel to the pair of radiating nes that enclose the side of the polygon of forces, representing the force i question. Also, the lengths of * the radiating lines will represent the 'resses along the bars to whose lines of resistance they are respectively iralleL 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 E, D E, E A. Then the polygon of external forces becomes A B C D A 3 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 ar*e 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 unstayed 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. oS. Polygonal Frame under Parallel Forces. J7JL^J^ , en 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 Fig. 75**. 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. AC represents the resultant of the load applied between the bars A. and C; and similarly for any other pair of bars. From O draw OH 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 "horizontal thrust" of the frame, and, as in Article 149, might more correctly be called liori- 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 OH be denoted simply by H. Let i, i', denote the inclinations to O H of the lines of resistance of any two bars, contiguous or not. Let R, R, 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 Pw = H sec i ; R/ = H sec i' ; P = H (tan i zr tan i'). m, f 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 D E and E A of the line A D, but by the inclined lines D O and O A, equal and directly opposed respectively to the stresses along the extreme bars of the frame, D and A. Let i d and i a denote the angles of inclination of those bars. Let ~R d = O D and E a = O 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, H = taiii d + ta,ni a _ = 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, ar 2/D k e * ne 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 on the part of the frame to the right of the section are ^ = 0; T? 9 = P M= P^. Then, observing that for the strut A C, n = 0, and that for the tie A A, n = y l9 we have, by the equations 1 of this Article HALF-LATTICE GIEDER. 153 E cos i + H = F, = ; ini = P whence we obtain, from the kst equation, (4.) from the first, or from the second -pi- Pi, = -- . = P cosec i. cos i 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 0, and P at A, have to be taken into account j so that from the first of which equations we obtain H + T = F, = 0, and T= -H= -Pcotani ................... (5.) 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 Half-Lattice Girder, sometimes called a "Warren Girder,'* 1 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 L A A j! A A A A jfrV-/ between the upper and lower iWViV VVVVg bars into a series of triangles. In the example to be consi- -p. 9 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 OP 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 j 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 sj 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 j shearing force, F y = P A - s P ; a force which is { P^ ive r _ u P ward i according as the plane of section ( negative or downward } lies < ~ n fh' rei f f 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 o^ s 'P#j 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, K 2 that 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 : Hj -f E 2 + R 3 cos i = ; or R x + R 3 cos i = E^...(a.) that is, the stress along the upper bar, and the horizontal component HALF-LATTICE GIEDEE. 155 of the stress along the diagonal, are equal and opposite to the stress along the lower bar j Eg sin i = F, = P A ^ P; (b.) that is, the vertical component of the stress along the diagonal, balances the shearing force ; R2 y = Us h = M = P A x l *c " P j ( c -) that is, the couple formed by the equal and opposite horizontal stresses of equation (a), acting at the ends of the arm h, balances the bending couple. Finally, from the equations (a), (b), (c), are deduced the following values of the stresses : Pull on lower bar. Stress on diagonal, ES= cosec i (P A 2j -P); Thrust on upper bar, Rj = RS E 3 cos ^ (I-) 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 and also cosec i = ; cotan i ' ; (2.) which substitutions having been made, give h (3.) 156 THEORY OF STRUCTURES. in which x' is taken tp 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 ^\^ | the direction of the shearing force Fj, 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 < J??f > from the middle towards the ends are < > . By these, principles the kind of ( struts j 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. N Between and either pier there are -- 1 loaded joints ; be- 2t 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 F y = P A - 20P = (-l) P; ............. (2.) So that it increases at an uniform rate from the middle towards the ends. /N \ The distance of the n ih joint from the pier is x { = ( 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 /N \ that of the farthest ( - 1 n\ I, so that the mean leverage is 1 /N \ A ( -- 1 n] I ; which being multiplied by the load 2 P as 2 \2 / c found above, gives for the moment hence the bending couple is =1(|-I-,) (-)" duct of the segme th of the girder, greatest at the middle, where it is P I. that is to say, it is proportional to the product of the segments into which the plane of section divides tJie length of the girder, and is 158 THEORY OF STRUCTURES. The uniform inclination of the diagonals, in one direction or the other, being denoted by i, we have s cosec * = -7- = h h and hence the amounts of the stresses are, Along the diagonal, B,' = F, cosec i=.ln_ p ; Along the horizontal bar. M /N" E / N = T=U ") si- .(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, for which n=\ 9 and R' = . Its greatest value is for the dia- 2i /i N" (N-l)sP . f , gonals next the piers, for which n - -^, and R, = i - j in fact, 2i ^ ill 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 E (g-i)pi 2A The greatest value of E, is at the division of one of the horizontal N 3 P bars opposite the middle joint, for which n = 0, and R = 5-= . 8 h 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 ("!N" \ -- 1 J 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, 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 The upward moment of the supporting force is at the joint rc,P A ^:= ( _ -) (- -w atthejointn- l,P A ( a?1 +Q ._ 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 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 ( -~- J P, gives for the moment, The corresponding moment for the joint n 1 is Hence the bending couples are At the loaded joint n, - r (7 -KOG-)"- ?-*)"> At the unloaded joint n 1, i 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 = ^pi . *- ............... (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, M 1 /N 2 A P I Along the bar opposite the unloaded joint n 1, 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 ttirdcr Any Load. In a lattice girder, as in a half- lattice girder, there are a hori- zontal upper and lower bar; but whereas a half-lattice girder contains but one zig-zag set of xxxxxxxx I J ^ I J diagonal bars, a lattice girder I | J | contains two or more sets, cross- Yig. 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 m 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 | t ^ t | for bars which slope | a |^ 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 165. Lattice carder 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 Article 164. 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 the new frame, will be the corresponding parallel projections of the lines representing tlie 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- Fig 84 ' 3C 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 L The resultant of the load between two given points in a balanced cord acts through the point of intersection of the tangents to tlie cord at tJwse points; and that resultant, and the pulls along the cord at tlie 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 a, 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 O 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 C, 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 OB; 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 draum, such 1G4 THEORY OF STRUCTURES. Cord tJiat 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 y 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 the 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, OC, O D. A loaded cord, hanging freely, is obviously stable, but capable of oscillation. der Parallel T,oad*. 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 AP is horizontal. Then in fig. 84**, O A will be horizontal, and perpendicular to C D. Let H = O A = horizontal tension along the cord at A; B, OB = pull along the cord at B ; Fig. 84** p __ AB __ load on the cord between A and B; i = ^ X P B (fig. 84) = -^ A B (fig. 84**) = inclination of cord at B ; then, * , ,-r^n -*--nt --- . f-i \ * (*) 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, AX = ce, XB = ?/j then dy tan i = -', dx' whence we obtain a differential equation which enables the form assumed by the cord to be determined when the distribution of the load is known. 169. Cord under Uniform Vertical i,oad. 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 P = px- (1.) where p is a constant quantity, denoting the intensity of the load in units 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 AX; 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 parabola. Also, the proportions of the load, and the horizontal and oblique tensions are as follows : P:H:R::BX:XP: : \px : : px ' Second Solution. In the present case equation 2 of Article 16& becomes which being integrated with due regard to the condition that when x = 0, y = 0, gives 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, a? H ,-. 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 2y tan ^ = -7 = ~ = : dx 2m x' (6.) whence we have the proportions as before. The following are the solutions of some useful problems respecting uniformly loaded cords. PROBLEM I. Given the elevations, y ly 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, Xi, x 2 , of the lowest point from the two points of support; also the modulus m. In a parabola, therefore, also TTAeTi the points of support are at the same level, PROBLEM II. Given the same data, to Jmd the inclinations i 2 , of the cord at the points of support. By equations 6, we have, a when yj 2/ 2 , tan ^ = tan i 8 = -^ ............... (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 R^ Rg, at the points of support. By equation 5, we find, and by the proportional equation 7, R, = H sec i, = H ,/l + ............... (14.) When y^ = y Z) those equations become PROBLEM IV. Given the same data as in Problem I., to find the length of the cord. The following are two well known formulae 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. a? 2 = m|tan i sec i + hyp. log. (tan i + sec i) }...(! 6.) The length of the cord is s, + s 2) where ^ is found by putting x l and y l in the first of the above formula, or ^ in the second, and s 2 by putting x 2 and y 2 in the first formula, or i z 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 ; s = x + - nearly; .......... .....(17.) o x which gives for the total length of the cord 168 THEORY OF STRUCTURES. ffc. (18.) and when y l = y 2 , this becomes 2s, = a + 4- ' - nearly, (19.) o (jj PROBLEM Y. Given the same data, to find, approximately, the small elongation of the cord d (sj + s 2 ) required to produce a given small depression djofthe lowest point A, and conversely. Differentiating equation 18, we find which serves to compute the elongation from the depression ; and conversely, y\ i_ 2/2 which serves to compute the depression of the lowest point from the elongation of the cord, "When y : = y 2 , those formulae become, (22.) j 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 Verlieal 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 fig. 85, represent the vertical axis of a pier, and C 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 ; E the common value of the two tensions ; then the vertical pressure on the pier is V = 2Bsini = 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 D A C, 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 E, E', their tensions, then E = H sec t ; R' = H sec i'; V = E sin i + E f 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, ADC, 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, the 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 ; 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; be P + ; be 2i = P + W. From b draw b parallel to the strut A B ; that is, horizontal ; from e draw e parallel to C A, cutting b in ; 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 EF in D. Then will AF 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 O c, O e, O/, and O b, 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 , K,., be the tensions of the tie at A, D, C, respectively, and H the horizontal thrust; then 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 term of equation 19, Article 169, AC for a, and that is to say, 172. Suspension Bridge with Sloping Bods. Let the uniformlv- 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 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, e:jKjf::CH:OE :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 V: P: Q ::CH: CE:E~H::1 : sec j: tan/ ..... (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. BP; and the ratio of the oblique load P, the horizontal tension. H along the chain at A, and the tension R, along the chain at B, is that of the sides of the triangle B X P ; that is to say, P : H : B, :: BX :"XP = ~ : BP. ........... (3.) a 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 : ^= .......................... <*> where m, as in Article 169, denotes the modulus of the parabola, given by the equation 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 formulse 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 Q = qx = vx tan./; ,.... o-P px* 2pm H = = = = 2vm ' sec J ........... (a = _=. ..... (9.) 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 anclwred or made fast afc 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 tan./) ................ (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 j (11.) i = arc 'cos I =- -cos.; 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 4-hyp. log. fa-nt + sec^) ^^ tan.;-|-sec.;j In most cases which occur in practice, however, it is sufficient to use the following approximate formula : 2 arc AB = x + y ' sin j + - ^J nearly. ..... (13.) 3 aH- 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. Extra do* 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 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 / 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, (1.) the integral representing the area between the axis of y, the given ordinate, the extrados and the intrados. 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. TT 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 9 Then that line is called the pwrameter of the intrados, or curve in which the cord hangs. When the vertical load is of uniform intensity, as in Article 169, 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 O A be the axis of y. Denote the length of O A, which is the least ordinate of the intrados, by y& Let BX = y be any other ordinate, at the end Fis ' 87 A> of the abscissa OX =, x. Let the area O A B X be denoted by u. Then equations 1 and 2 of Article 172 become the following : dy dx d x 2 The general integral of the latter of these equations is u= Ae B in which A and B are constants, which are determined by the special conditions of the problem in the following manner. When -fL * ic= 0, e a =e T = 1; but at the same time u = 0, therefore A = B, and equation (a.), may be put in the form, u = A\e* e~~ ) ..................... (b.) This gives for the ordinate, y= A ( which, for x = 0, becomes y Q = - and therefore (d.) \ ' 176 THEORY OF STRUCTURES. which value being introduced into the various preceding equations, gives the following results, as to the geometrical properties of the intrados : ay Q ( *. --M Area, u = ^- V.e e u ) ', Ordinate, y = ( * + e ~ " ) ; Deviation , -^ = %,= /^ (e *" + e ~ "^ ). (2.) The relations amongst the forces which act on the cord are given by the equations a x R (tension at B) = j?* + H 2 = H 1 + (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 amoujtts 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 y = ,(4.) hyperbolic cosine of -, as this function is called. Supposing a table of hyperbolic cosines /v to be at hand, - is found by its being the number whose hyper- y bolic cosine is ; so that number to hyp. cos. 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 a = 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 alos.e. 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_ d7 . . dy / dx* int= -y- = A/ 1 --TT; ds V dsr' dy -~ dx ds 2 dx (1.) 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 1G8, we deduce the following : .(3.) ds 178 THEORY OP STRUCTURES. which, by a few reductions, is brought to the following form : dx = _J**_ />n d s J n the integral of which (paying due regard to the conditions that when s = 0, x = 0) is known to be m - ......... (5.) This equation gives the abscissa x of the extremity of an arc A B =r 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 fds* ~ s 1 / _* x d^ = Vdxl- 1 =^ == 2( em - e m )> ....... ( 7 '> which gives '-- ;.... (8.) 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; (9.) so that the tension at any point is equal to the weight of a piece oj 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 y' 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 = a = 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 lt y 1? 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^ 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 Xi + x. 2 = h (an unknown quantity). Then we have _h + k h k fii\ Putting these values of a; in the equations 6 and 8, we find l== __ / JL L\ / JL __*A " j Ui>. / JL _JL N / JL JL\ .---.(.. + , .-).(...-..)] Square those two equations and take the difference of the squares ; then, (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 i- l + v * =TTv and consequently , , , I + v h m- hyp. log. (14.) I/ v Either or both of the abscissae x t and x 2 , 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^ y^ 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 s = - h = (15.) _ k In this case equation 13 becomes / *. *\ l = m (e*> e-\ (16.) from which m is to be found by successive approximations. Then the computation of y v = 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 = m-sec 2 i (17.) 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 ra. IY. 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 which 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 ly 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 OP 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 y Q 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; 7 T> let p = be its intensity per horizontal unit of length; let H be f ' -'.' 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 (1.) but for each horizontal co-ordinate x, let there be substituted an oblique co-ordinate a/, inclined at the angle j to the horizon, and altered in length by the constant ratio - = a. Then for the hori- zontal tension H, there will be substituted an oblique tension H' , parallel to x f ) and altered in the same proportion with that co- ordinate; that is to say, x' = ax ; H' = a H ...................... (2.) The original tension at B is the resultant of the vertical load P and the horizontal tension H. Let E, be its amount, and i its in- clination to H; then ___ _ B = J P J + H 2 ; ........................ (3.) 182 THEORY OF STRUCTURES. and the ratios of those three forces are expressed by the proportion P : H : B : : tan i : 1 : sec i : ; sin i : cos i : 1 ........ (4) Let B' 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 11'= N /(P + EP l ==2FH'sin t ; < ) ............ (5.) P':H': B': : sn^: cos(i / zt t /) : cos./ ............ (6.) The alternative signs rr are to be used according as i' and 3 in direction. ( differ j The intensity of the load in the transformed structure per unit of oblique length measured along dot, is but if the intensity of the load be estimated per unit of horizontal length, it becomes p' secj = (8.) a- cos,; 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 formulae 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 & 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 j 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 farther 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, Fig. 88. 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 OF STHUCTURES. 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, f , be the intensities of the vertical and horizontal pressure respectively, then p t =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 is p y 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. 0. 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" r 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 LINEAR ARCH. 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 x f = 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, the total horizontal pressure, on one quadrant of the circle A B. Then Let P', be the total vertical pressure, and P'y the total horizontal pressure, on one quadrant of the ellipse, as A' K, or A" B" ; and let T, be the vertical thrust on the rib at B' or B ', and T', the hori- zontal thrust at A' or A". Then, by the principle of transformation, rrv __ TV _ ~p _ FTI __ . or, the total thrusts are as tlie axes to which, they cure parallel. Further, let P' = T" be the total pressure, parallel to any semi- imeter of the ellipse (as O' D' or 0" D") on the quadrant D' C 7 or C", which force is also the thrust of the rib at C' or C", the ex- 3mity of the diameter conjugate to O' D' or O" D"; and let O'D' )rO r 'D" = r- then (3.) or, the total thrusts are as the diameters to which tliey are parallel. Next, let pt, p' y , be the intensities of the conjugate horizontal and vertical pressures on the elliptic arch ; that is, of the " principal tresses" (Articles 109, 112). Each of those intensities being found }y dividing the corresponding total pressure by the area of the plane to which it is normal, they are given by the following equa- on : cr c P' y = -= c P> ,(4.) 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 'the principal pressures ; that is, ,' <*> The external pressure on any point, D' x>r 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 (0' C' or O" C"), is given by the following equation, in which / denotes the semidiameter (O' D' or O" D") parallel to the pressure in question, and r" the con- jugate semidiameter (0' C' or 0" 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" Tbe the intensity of the external pressure in the direction of the semidiameter r*'. Then it is evident that p :p" ::r'* : r "* ', (7.) that is, the intensities of a pair of conjugate pressures are to each otJier as t/ie squares of the conjugate diameters of the elliptic rib to which tliey 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, DISTORTED ELLIPTIC ARCH. 187 E D = y. Let B' A' CT 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' tf = x' = x (I.) Then for the vertical semidiameter of the circle OA = r, will bo substituted the equal vertical semidiameter of the ellipse O' A' = r ; and for the horizontal diameter of the circle C B = 2 r, will be substituted the inclined diameter of the ellipse C / B'=2cr,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' x denote the total vertical pressure, and P y the total inclined pressure, on either of the elliptic quadrants, (J A, A B' j T' y the inclined thrust of the arch at A, T', the vertical thrust at B' or (T. Then that is to say, those forces are, as before, proportional to the dia- meters to which they are parallel. Let pf 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. P o J (3.) so that, as before, the intensities of the conjugate 2wessures are as the squares of the diameters to which they are parallel. The thrust of the arch at any point D' is as before, proportional to the diameter conjugate to O' JD'. 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 >' x -sec?' = 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 0' a -J- and = 0' A ; join B' a, which bisect in m ; in B'a produced both ways take mp = mq = O'm ; 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 the semiaxes are respectively equal to the segments of the line p q, viz., B'^> = aq, T& q a p. The following is the algebraical expression of this solution. Let denote the major and B the minor semi-axis of the ellipse. whence we have for the lengths of the semi-axes, B (5.) ARCHES FOR NORMAL PRESSURE. 189 The angle ^ B' O'p = k, which the nearest axis makes with the diameter C' B', is found by the equation 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 aTlinear 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 tJie product of the radius of curvature by the intensity of the pressure; that is, denoting the radius of curvature by f, 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, cV At the ends of r : e, = = (2) r 3 r ' At the ends of c r : e, = = - , cr c J Introducing these values into the equations of Article 180, and into equation 1 of this Article, we find, T 7 * = P'y ?y = cp - = pr SLS before j c ny T' y = p' x e x - - dyf = dx; d y = cd v> c we find the following relations between the total vertical and horizontal pressures in a given arc of the hydrostatic arch, and the total vertical and conjugate pressures on the corresponding arc of the transformed arch, P', = P.; F y = cP y ; ..................... (8.) being the same with the relations which, according to equation 5, exist between the co-ordinates respectively parallel to the pressures in question. Therefore the transformed arch is a parallel projection of the original arch under forces represented by lines which are the corresponding parallel projections of the lines representing the forces acting on the original arch: therefore it is in equilibria The conclusions of the preceding investigation may be summed up in the following THEOREM. A geostatic arch, transformed from a hydrostatic arch by preserving the vertical co-ordinates, and substituting for the hori- zontal co-ordinates, conjugate co-ordinates, either horizontal or inclined, and altered in a given ratio, sustains vertical and conjugate pressures, the ratio of the intensity of the conjugate pressure to that of the vertical pressure being the square of the ratio of the conjugate co-ordinates to \e original horizontal co-ordinates. transformation is exactly analogous to that of a circular arch into an elliptic arch, in Articles 180, 181. Let T be the thrust, horizontal or inclined as the case may be, at the crown of a geostatic arch, and T, the vertical thrust at the J9S THEORY OF STRUCTURES. points where the arch is vertical, which in this, as in other cases, is the vertical load of the semi-arch; then To = cT, (10.) All the equations relative to the co-ordinates of a hydrostatic arch, given in Article 183, are made applicable to a geostatic arch, by substituting x' for oc, and for y. This principle, however, is appli- cable to co-ordinates only, and not to angles of inclination, radii of curvature, nor lengths of arcs. The modulus #, and amplitude > W dy* 1 For rectangular co-ordinates - r - : . = 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^ 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, = Ho - f"p,dx, (5.) J TO the conjugate component of the thrust of the arch at the point of springing; also, let x a be the depth of the resultant conjugate thrust below the conjugate co-ordinate plane; then - (6-) H TT H Example I. Circular arch under 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 TT __^ fTI __ /*? \ Let the crown be taken for origin of co-ordinates, so that XQ = 0. CASE 1. Semicircle. Here x { = r; Hj = 0; and I UOO THEORY OP STRUCTURES. CASE 2. Segment. Inclination at springing, v Here x^ = r (1 cos i)j HJ p r cos i' } and _ J p x\ + p r x { cos i p r _ r O. (1 cos i) 2 + cos i (I cos i)) !1 sin 3 i...(9.) 43 Example II. Semi-elliptic arch, under conjugate uniform vertical and horizontal pressures (Art. 180). Let a = a?, be the rise, or vertical semi-axis; cathe horizontal semi-axis, or Iialf-span; and let the origin of co-ordinates be at the crown. Then p y = c*p x ; and we have H = T = a p y = c 2 a p. == c P x ; x n = | .... (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 Jialf-span, being denoted by a and c a respectively, the equations 10 apply to this case also. Example IY. 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 .^T^ ; H! = 0; and * = !V/>^f = !. *=* (11 .) Xlfl " # ^0 Example Y. Gfeostatic arch, with horizontal or inclined extrados (Art. 184). Here p, = w x - cosj; p y = c*p x = c 2 wx- cosj; H = T = c P, = c 2 w cosj - ' ; and consequently as in the last example. Example YI. Semicircular arch with Jwrizontal extrados. In this case the angle of rupture i is to be determined by means of eqxiation 13 of Article 185; and thence, by equation 12 of the same Article, is to be found H . The springing being vertical, we have i, = 90; H, = 0. Let the crown of the arch be taken as origin ; then x = r (1 cos i), d x = r sin t d i, and equation 6 of the present Article becomes APPROXIMATE HYDROSTATIC AND GEOSTATIC ARCHES. 207 X H = -- f^sin^l cosi)'di' } ......... (13.) H J to Example VII. Circular segmental arch with horizontal extrados. Let i { be the inclination of the arch at the springing, P x the total vertical load; then Hj = P! cotan ^ ....................... (14.) Let 2 be determined as in the last example. CASE 1. IQ > or = t,. In this case H = H 1? and the conjugate thrust is simply the single horizontal force HI at the point of spring- ing. CASE 2. i Q < i^ Find HO as in the last example, and let the origin of co-ordinates be at the crown; then Xi = r (1 cos tj); and we have H _ i 188. Approximate Hydrostatic and Oeostatic 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 XQ, a?!, as in Article 183, be the depth of load of a hydrostatic arch at the crown and springing respectively; r , r u its radii of curvature at those points; a = x l XQ, 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 extrenvities of the semi-axes, we have (1.) 208 THEORY OP 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 TB- and A, and is about uV on an average. Hence we may take, as a first approximation whose utmost error in practice is about TO, and whose average error is about TS, the following formula, giving the lialf-span in terms of the depths of 'load at the crown and springing : 19 . Suppose the rise a and half-span yi 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, 19 a and because x l x = a } we have i _1 (^i 1 .(3.) A closer approximation is given by the equations -b- ' l> = yi + -5^-; 60 a I 9 3 /y /y _ m f U/j U, , o ' " o , V .(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 m the depths of load at its crown and springing, X Q , x l} and the vertical load at the springing, P x . Determine, by equation 2 or equation 4, the span y t of a hydro- static arch for the depths of load X Q , x l} and let ?!=,.. ..(5.) 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- its vertical ordinates and load, and altering its conjugate >rdinates 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^P! (6.) SECTION 3. On Frictional Stabili 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 acfliesion. 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. i,aw 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 j 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 210 THEORY OP STRUCTURES. they are pressed together by 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 = 191. Angle of Repose. Let A A, in fig. 93, represent any solid body, BB 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 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 C 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 or P C N be denoted by 0, so that ) / " T = P -sin* = N -tan A 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 jfH, is equivalent to the condition, that ^-, or tan 6, 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 wJwse tangent is the co-efficient of friction. This angle is called the angle of repose, and is denoted by 48 3'23 1*23 to o'9 193. Frictional Stability of Plane Joints. In a structure COm- posed of a number of pieces connected only by touching each other at plane surfaces (as is the case in masonry and brickwork), it is necessary to stability that the obliquity of the pressure should at no joint exceed the angle of repose. 212 THEORY OF STRUCTURES. In structures of masonry, this condition can almost always be complied with by suitably placing the joints. Both this and other principles depending on the effect of friction in promoting the stability of masonry, will be considered in subse- quent sections. s\ 194. Friciionni stability of Earth.* A structure of earth, whether ' produced by excavation or by embankment, preserves its figure at first partly by means of the friction between its grains, and partly by means of their mutual cohesion or tenacity; which latter force is considerable in some kinds of earth, such as clay, especially when moist. It is by its tenacity that a bank of earth is enabled to stand with a vertical face, or even an overhanging face, for a few feet below its upper edge; whereas friction alone, as will afterwards appear, would make it assume an uniform slope. But the tenacity of earth is gradually destroyed by the action of air and moisture, and of the changes of the weather; so that its friction is the only force which can be relied upon to produce permanent stability. In the present investigation, therefore, the stability of a mass of earth, or of shingle or gravel, or of any other material consisting of separate grains, will be treated as arising wholly from the mutual friction of those grains, and not from any adhesion amongst them. Previous researches on this subject are based (so far as I am acquainted with them) on some mathematical artifice or assumption, suchas Coulomb's "Wedge of Least Resistance." Researches so based, although leading to true solutions of many special problems, are both limited in the application of their results, and unsatisfactory in a scientific point of view. I propose, therefore, to investigate the mathematical theory of the frictional stability of a granular mass, without the aid of any artifice or assumption, and from the following sole PRINCIPLE. The resistance to displacement by sliding along a given plane in a loose granular mass, is equal to the normal pressure exerted between the parts of the mass on either side of that plane y multiplied by specific constant. The specific constant is the co-efficient of friction of the mass, and is the tangent of the angle of repose. Let p n denote the normal pressure per unit of area of the plane in question ; q the resistance to sliding (per unit of area also) ; (p the angle of repose ; then the symbolical expression of the above principle is as follows : - == tan

) /0 x p 9 = w x cos 6 . ^7 n ,'* ( 2 cos 6 + J (cos 2 6 cos 2 /0 x p v = w x - : \\ (3.) 1 + sin V ' " Natural slope," 6 = ?, p y = p f = w x ' cos p., (4.) The third pressure p, is found in the following manner. Being perpendicular to the plane of p x and p y , it must be a principal pres- sure (Arts. 107, 109). Being a passive force, it must have the least intensity consistent with stability, and must therefore be equal to the least pressure in the plane of p x and p y . The greatest and least stresses, or principal pressures, in that plane, are to be found by means of Problem III. of Article 112, case 3, from the pair of conjugate pressures^,, p y , whose obliquity is e. Let pi be the greatest, and p a the least principal pressure ; then u equations 19 and 20 of Art. 112, for A p, p', n r, p t) p y , we are to substitute respectively, Px, Pv *, Pi> P* giving the following results : +* STABILITY OP EARTH. P* ~^~ Py W X ' COS t 217 ...(5.) 2 2 cos & cos 6 + J (cos 2 * cos 2 ) + 7? y ) 2 ) w a; cos o ' sin p 2 J ( 4< and consequently, Greatest pressure, Least pressure, p 2 w x cos 4 * (1 + sin ) /7\ cos 6 -\- ,J (cos 2 4 cos 2 $)' w x ' cos 6 (1 sin , is given by the follow- ing formula, deduced from equation 17 of Article 112, by making the proper substitutions : cos = Pi p* from which is easily deduced, In using this formula, the arc sin - is to be taken as greater sin

3 = j>, = w x (1 sin 'i33 157 0-165 REBIAEK. The column headed o is applicable to liquids. '222 THEORY OF STRUCTURES. 202. Frictional Tenacity or Bond of Masonry and Brickwork. The overlapping or breaking of the joints, commonly called the bond, in masonry and brickwork, has three objects first, to dis- tribute the vertical load which rests on each stone or brick over two or three of the stones or bricks of the course next below, and so to produce a more nearly uniform distribution of the load than would otherwise take place; secondly, to enable the structure to resist forces tending to break it by shearing, or sliding of one part on another, in a vertical plane; and thirdly, to enable it to resist forces tending to tear it asunder horizontally. For masonry and brickwork laid either dry, or in common mor- tar which has not had time to acquire practically appreciable tenacity, the resistance to horizontal tension mentioned above as the third object of the bond, 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 0-74. Let fig. 94 A represent a portion of a wall with a horizontal top A ; and let it be required to determine i i L. , ' i .l"!" the frictional tenacity at a horizontal joint B, whose depth below A is x, the intensity of that tenacity per unit of 'J I ( I i!JIC area of a vertical plane at B, and the III ;~T~ aggregate tenacity of the wall from A i ii i ' i- down to B, with which it is capable of -p. Q4 A 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 wall ; 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, if floe the co-efficient of friction, the amount of frictional tenacity for the joint B is fwbtx (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 TT BOND OF MASONEY 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 .(2.) Let n be the number of courses from A down to B. Then the value of x for the uppermost course is = h, and for the lowest course, = n h ; and the mean value of a; is ^ ' h ; so that the mean tenacity per course is and the mean intensity, -f 1 Hence the amount of the aggregate frictional tenacity of the wall, from A down to B, is , qx (3.) From the equations 2 and 3 it is obvious that the frictional tenacity of masonry and brickwork is increased by increasing the ratio T 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 T O depth, of a brick. The value of T is therefore j ; but to allow for irregularities of figure and of laying in the bricks, it is safe to make it n - in the formulae. Substituting this in equations 2 and 3, and 224 THEORY OF STRUCTURES. Q making /= -, we find for the intensity of the frictional tenacity, where one-lialf of the face of the wall consists of ends of headers, and for the amount from the top of the wall down to the depth x, wt(x*-\-hx) x r v 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. fi 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 war, ................................. (6.) and its amount from the top of the wall down to the depth x, for a length of wall denoted by I, In a brick wall consisting entirely of stretchers, as in fig. 94 B, -i 1 , the longitudinal tenacity is double of I I .' I I ~T tliat of tn e wal1 in % 94 A, where I I I I I one-half of the face consists of ends of I III"" 1 " headers. But that increased longitu- ^ ^"^ ' 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 headers, and T-^ of sides of stretchers. - 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 1 W X 2 T -r wx oo 1 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 |-y, and the transverse tenacity as the proportion of ends of S ~T~ 1 headers; whence we have the following formula for the intensi- ties : Consequently, for the aggregate tenacities down to a given depth x y when the length of the wall is I, and its thickness t, we have Longitudinal, . . wt (x 2 -f hx)j ......... (10.) 4 (s -t- l; Transverse, . . w I (x 2 + h ) ........... (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 (12.) In round factory chimneys, it is usual to make s = 4 ; and then we have L= -wx; T = *0* ............... (13.) v 226 THEORY OP 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 11 may also be used to find the transverse tenacity of a rubble wall, if be taken to represent the propor- 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. lAA. 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 SOine Sub- 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 JODsT. 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 tKe. 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 94), 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 Q 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 Q + a x, so that if x { be the greatest positive distance of a point at the edge of the joint from the neutral axis, the maximum pressure will be Pi = Po~^ ax i- Now, by the condition stated above, p L = 2p Q , and, consequently, ...................... . XL XL X L S If the diameter of the joint is bisected by the centre of figure, and if X Q (as in Article 94) be the distance of the centre of pressure from the neutral axis, we shall have -ft> 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 al I 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 j so that if h be the diameter in that direction, XL = - - m STABILITY AT A PLANE JOINT. FIGUKE OF BASE. I. Rectangle Length K) I tfb S 1 Breadth,... ,...&f T^~ hb 6 II. Square Side, h 1 III. Ellipse 12 *hb 6 1 fi4. A. 8 IV. Circle Diameter, h 1 V. Hollow rectangle Outside dimensions,... , J) 04 tfb-WV 4 Ji A />' A' 8 Inside dimensions,... A', b') VI. Hollow square Outside dimensions, h\ 12 A 4 -A'* no ii o k a h' 3 A 2 + A' 2 Inside dimensions, h') VII. Circular ring Diameter, Ostside, k\ 12 6A a tf + 7i' a Do. Inside, h') 64 4 8 A 3 "When the solid parts of the hollow square and of the circular ring are very thin, the expressions for q in Examples VI. and VIL become approximately equal to the following : VIII. Hollow square, ......................................... q =~o> ~o> IX. Circular ring, ........................................... q = -r' } 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, e 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 of the pressure must not exceed the angle of repose, that is to say, 230 THEORY OF STRUCTURES. ^ (3.) II. The ratio which the deviation of the centre of pressure from the centre of figure of the joint bears to tJie length of the diameter of tJie joint traversing those two centres, must not exceed a certain fraction, ivhose value varies, according to circumstances, from one-eighth to three-eighths, that is to say, \ ^E C~E =^^q (4.) eE The first of these conditions is called that of stability offriction t the second, that of stability of position. LI" 206. Stability of a Series of Blocks; JLiiie of Resistance ; Line of 1 ^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 G! 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 Gj. 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 joirt v divides the structure, precisely as in the "method of sections" alreaJy 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 K, K, in fig. 95, has received from Mr. Moseley the name of the " line of resistance ;" and that author has also shown ANALOGY OP 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 titan a certain fraction (q) of the diameter of the 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 Block work 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 j 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.) Determ'^e and lay down on a drawing of the structure the line of action \Jnd 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 0, 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- U^Jion of the load. 2 OS. Transformation of Block work Structures. THEOREM. If a structure composed of blocks have stability of position when acted on Til ty forces represented by a given system oflines y then will a structure 111 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 'es 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 1IOMENT 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. 299. Frictional Stability of a Transformed Structure. The ques- tion, whether the new structure obtained by transformation will possess stability offridion, 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 pressure as originally determined. / ^10. Structure not Laterally Pressed. If fig. 96 represents ft 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 r> 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 ^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 of forces 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. pres stru coui 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 j - 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 of the weight of the structure into the horizontal dis- tance of a point in this line from a vertical line traversing the centre of gravity of the 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 iJie 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 qt be the distance of the given limiting centre of resistance from the middle point of that diameter, and ( 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 = n-whbt, 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 : (3.) This quantity is divided by points into three factors, viz. : (1.) n (q rr q') cosj, a numerical factor, depending on the figure of the structure, the obliquities of its co-ordinates, and the direction in which the applied force tends to overturn it. (2.) w, the specific gravity of the material. (3.) h b t 2 , a geometrical factor, depending on the dimensions of the structure, Now the first factor is the same in all structures having figures of the same class, with co-ordinates of equal obliquity, and exposed to similarly applied external forces; that is say, to all structures whose figures, together with the lines of action of the applied forces, are parallel projections of each other, with co-ordinates of equal obli- quity ; hence for any set of structures which fulfil that condition, the moments of stability are proportional to I. The specific gravity of the material ; II. The height; III. The breadth ; IV. The square of the thickness ; that is, of the dimension of the base which is parallel to the vertical plane of the applied force. 2T2. Abutments classed. In the title of the present section, the word " abutment " is used in an extended sense, to denote every structure, which by its stability of position and of friction, sustains some pressure which abuts or acts laterally against it. The structures comprehended under this definition may be classed as follows : I. Buttresses, which sustain the thrust of a frame or a rib, at one or more definite points. II. Towers and chimneys, which sustain the lateral pressure of the wind, uniformly or almost uniformly distributed, and liable to act in every horizontal direction. III. Dams for sustaining the lateral pressure of water, and retaining walls for sustaining that of earth the intensity of the pressure being proportional to the depth beneath the surface. IY. Arch abutments, which resemble both buttresses and retain- ing walls, and whose properties will be treated of after those of stone and brick arches shall have first been considered with refer* ^^.eace to the stability at their joints. * J fl ;213. Buttresses in General. Let fig. 97 represent a vertical sec- ion of a buttress, against which a strut, rib, or piece of frame- work abuts at C, exerting a given force P in a given direction OA. In order that the buttress may be stable, it must fulfil 236 THEORY OF STRUCTURES. the conditions of stability at each of its bed-joints. Let D E be one of those joints. i j Should several pressures abut against the buttress, p the force P acting in the line CA may be held to . represent the resultant of all the forces which are applied above the particular joint DE under con- sideration. Let G be the centre of gravity of that part of the buttress which is above the joint D E, and let W denote the weight of the same part. Through G draw the vertical line A G B, cutting the direction of the lateral thrust in A, and the joint D E in B ; make AW = W, AT? = P ; complete the parallelo- gram A P R W ; then A R will represent the result- ant of all the forces which act on the part of the buttress above the joint D E, and to which the resultant of the resistance at that joint must be equal and directly opposed. A R being produced, cuts D E in F, the centre of resistance of that joint, which must not fall beyond a certain prescribed limit, that the condition of stability of position may be fulfilled. In order that the condition of stabi- lity of friction may be fulfilled, the angle A F B must not be less than the complement of the angle of repose. The most convenient mode of expressing this problem algebrai- cally depends on the circumstances of the particular case. The following example is that which is most frequent and useful in practice ; viz., when the inner face C D of the buttress is vertical, and the joint D E horizontal. In this case, let the point of application of the lateral force, C, be taken for the origin of co-ordinates. Let i denote the angle of inclination of the applied lateral pressure to the horizon ; x = G I), the depth of the joint in question below C ; 2/ = B D, the horizontal distance of the centre of gravity of the part of the buttress above that joint from the inner face ; y = D F, the horizontal distance of the centre of resistance of the joint from its inner edge. The resultant resistance, which acts through F in the direction F A, may be resolved into two components, respectively parallel, equal, and opposite to the weight W and applied force P. The couple of forces W is right-handed, and has the arm F B = y y ff The couple of forces P is left-handed, and has for its arm the per- pendicular distance of F from the line of action C A of the applied force, viz. : x cos i y sin i. BUTTRESS. 237 The former of those couples tends to maintain the stability of the buttress : the latter tends to overturn it. Equating their magni- tudes, we obtain for the expression of the condition of stability of position the following : w (2/~2/o) = P(cosi-y sint) .............. (1.) From this fundamental equation the solutions of various pro- blems may be deduced, of which the following are examples : I. The buttress and the lateral force being given, to find the centre of resistance at a given joint. ,^ W + Psinz - This is the equation of the " line of resistance." The condition of stability is expressed in terms of y thus . e relation between the weight and the dimensions of the part of the buttress under consideration being given as in equations 2 and 3 of Article 211, it is required to find the least thickness at the joint D E consistent with stability. For this purpose we must substitute for "W (y 2/ ) in equation 1 of this Article its limit ; that is to say, the moment of stability, as expressed in equation 3 of Article 211 ; and for y we must substi- tute its limiting value in terms of the thickness, as given by equa- tion 3 of this Article. Thus we obtain the following equation : sini) ...... (4.) To simplify the form of this quadratic equation, make (^ 4- -z\ P sin * Pa? cos V r 2j _ . ' " n(q + q')whb then equation 4 becomes p- 4 ^ J^ t - - .J] t 2 = A 2Et, the solution of which is r 4 /J t = ^/A + B 3 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 \ tan ^ FAB = w ~. -I " (6.) W + P sm ^ 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" defined 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 cosi ^ r , p . . -^ tan

, and 7 of Article 213, give the following results : Equation of the line of resistance The least thickness compatible with stability (x l being the depth of the base of the wall below C) is found by making -n P x l cos ^ _ ~ qw(k Q -{-x 1 )b } " 2 q w (h Q + x,) b* whence follows RECTANGULAR BUTTRESS PINNACLE. 239 The least volume of material above the level of the point which is compatible with stability of friction, is gixgnJ)Y. making Pcos^ _ 10 A 6* + Print" ' that is to say, sin The equation 1 of the line of resistance is that of a rectangular hyperbola traversing the point A (which is in this case invariable), and having a vertical asymptote, whose distance from the inner face of the buttress is P cos * wbt being the limit which y continually approaches, but never attains, as the de^fch x increases without limit. As the depth x increases without limit, the thickness required for the wall approaches the following limit : -VGSr) ** qwb which depends on the horizontal component of the lateral force alone. Supposing this value to be adopted for the thickness of the but- tress, in order that it may be stable, how deep soever the base may be below the point C, then to insure stability of friction, the height of the top above C must have the following value : cos (9 + i) ko = qt sin* cos i- ..................... < 6 -> Instead of the rectangular mass A b t, there may be substituted a pin'/utde of the same volume, and of any figure. 240 THEORY OF STRUCTURES. 215. Towers and Chimneys are exposed to the lateral pressure of the wind, which, without sensible error in practice, may be assumed to be horizontal, and of uniform intensity at all heights above the ground. The surface exposed to the pressure of the wind by such struc- tures is usually either flat, or cylindrical, or conical, and differing very little from the cylindrical form. Octagonal chimneys, which are occasionally erected, may be treated as sensibly circular in plan. The inclination of the surface of a tower or chimney to the vertical is seldom sufficient to be worth taking into account in determining the pressure of the wind against it. The greatest intensity of the pressure of the wind against a flat surface directly opposed to it hitherto observed in Britain, has been 55 Ibs. per square foot ; and this result, obtained by observations with anemometers, has been verified by the effects of certain vio- lent storms in destroying factory chimneys and other structures. In any other climate, before designing a structure intended to resist the lateral pressure of wind, the greatest intensity of that pressure should be ascertained, either by direct experiment, or by observation of the effects of the wind on previous structures. The total pressure of the wind against the side of a cylinder is about one-half of the total pressure against a diametral plane of that cylinder. Let fig. 98 represent a chimney, square or circular, and let it be required to determine the conditions of stability of a given bed-joint D E. Let S denote the area of a diametral vertical section of the part of the chimney above the given joint, and p the greatest intensity of pres- sure of the wind against a flat surface. Then the total pressure of the wind against the chim- ney will be sensibly P = p S for a square chimney ; ) -D S, , , . >...(!) r = p - ior a round chimney ; j p. g& and its resultant may, without appreciable error, be assumed to act in a horizontal line through the centre of gravity of the vertical diametral section, C. Let H denote the height of that centre above the joint 1) E ; then the moment of the pressure is \v* H P = H p S for a square chimney ; H P = - for a round chimney ; ' * CHIMNEYS. 241 and to this the least moment of stability of the portion of the chim- ney above the joint D E, as determined by the methods of Article 211, should be equal For a chimney whose axis is vertical, the moment of stability is the same in all directions. But few chimneys have their axes exactly vertical \ and the least moment of stability is obviously that which opposes g, lateral pressure acting in that direction to- ward which the chimney leans. Let G be the centre of gravity of the part of the chimney which is above the joint D E, and B a point in the joint D E vertically below it ; and let the line D E = t represent the diameter of that joint which traverses the point B. Let q', as in former examples, represent the ratio which the deviation of 5 from the middle of the diameter D E bears to the length t of that diameter. Let F be the limiting position of the centre of resistance of the joint D E, nearest the % edge of that joint towards which the axis of the chimney leans, and let q, as before, denote the ratio which the deviation of that centre from the middle of the diameter D E bears to the length t of that diameter. Then, as in equation 3 of Article 211, the least moment of stability is denoted by W BF = < VTt The value of the coefficient q is determined by considering the manner in which chimneys are observed to give way to the pressure of the wind. This is generally observed to commence by the opening of one of the bed-joints, such as D E, at the windward side of the chimney. A crack thus begins, which extends itself in a zig-zag form diagonally downwards along both sides of the chimney, tending to separate it into two parts, an upper leeward part, and a lower wind- ward part, divided from each other by a fissure extending obliquely downwards from windward to leeward. The final destruction of the chimney takes place, either by the horizontal shifting of the upper division until it loses its support from below, or by the crushing of a portion of the brickwork at the leeward side, from the too great concentration of pressure on it, or by both those causes combined ; and in either case the upper portion of the structure falls in a shower of fragments, partly into the interior of the portion left standing, and partly on the ground beside its base. It is obvious that in order that the stability of a chimney may be secure, no bed-joint ought to tend to open at its windward edge ; that is to say, there ought to be some pressure at every point of each bed-joint, except the extreme windward edge, where the in- tensity may diminish to nothing ; and this condition is fulfilled 242 THEORY OF STRUCTURES. 41 with sufficient accuracy for practical purposes, by assuming the pressure to be an uniformly varying pressure, and so limiting the position of the centre of pressure F, that the intensity at the lee- ward edge E shall be double of the mean intensity. It has already been shown, in Article 205, what values this con- dition assigns to the co-efficient q for different forms of the bed-joints. Chimneys in general consist of a hollow shell of brickwork, whose thickness is small as compared with its diameter ; and in that case it is sufficiently accurate for practical purposes to give to q the fol- lowing values : For square chimneys, q = ; For round chimneys, q = The following general equation, between the moment of stability and the moment of the external pressure, expresses the condition of stability of a chimney : This becomes, when applied to square chimneys, Hi>S=Q and when applied to round chimneys, The following approximate formulae, deduced from these equations, are useful in practice : Let B be the mean thickness of brickwork above the joint D E under consideration, and b the thickness to which that brickwork would be reduced, if it were spread out flat upon an area equal to the external area of the chimney. That reduced thickness is given with sufficient accuracy by the formula but in most cases the difference between b and B may be neglected. Let w be the weight of an unit of volume of brickwork; being, on an average, about 112 Ibs. per cubic foot, or, if the bricks are KESEEVOIK-WALLS. 243 dense, and laid very closely, with thin layers of mortar in the joints, from 115 to 120 Ibs. per cubic foot. Then we have, very nearly, for square chimneys, ~W = 4 w 6 S; ) /o\ for round chimneys, W = 3 14 w 6 S j / " which values being substituted in the equation 6, give the following formulae : /4 For square chimneys, Up = ( 4 For round chimneys, TLp = 1 1 57 These formulae serve two purposes j first, when the greatest in- tensity of the pressure of the wind, p, and the external form and dimensions of a proposed chimney are given, to find the mean re- duced thickness of brickwork, b, required above each bed-joint, in order to insure stability ; and secondly, when the dimensions and form and the thickness of the brickwork of a chimney are given, to find the greatest intensity of pressure of wind which it will sustain with safety. The shell of a chimney consists of a series of divisions, one above another, the thickness being uniform in each division, but diminish- ing upwards from division to division. The lreUjoints between the divisions, where the thickness of brickwork changes (including the bed-joint at the base of the chimney), have obviously less stability than the intermediate bed-joints j hence it is only to the former set of joints that it is necessary to apply the formulae. To illustrate the application of the formulae, a table is given in the Appendix, showing the dimensions and figure, and the stability against the wind, of the great chimney of the works of Messrs. Tennant and Company, at St. Kollox, near Glasgow, which was erected from the designs of Messrs. Gordon and Hrll, and is, with the exception of the spire of .Strasburg, the Great Pyramid, and the spire of St. Stephen's at Vienna, the most lofty building in the world. 216. Dams or Reservoir- Walls of masonry are intended to resist the direct pressure of water. A dam, when a current of water falls o\er its upper edge, becomes a weir, and requires protection for its base against the undermining action of the falling stream. Such structures are not considered in the present Article, which is confined to walls for resisting the pressure of water only. In fig. 99, let E D represent a horizontal bed-joint of a reservoir- wall, which wall has a plane surface O D exposed to the pressure 7 244 THEORY OF STRUCTURES. of the contained water, whose upper surface is the horizontal plane O Y. Consider a vertical layer of the wall of the length unity, sustaining the pressure of a ver- tical layer of water of the length unity also. Then from Articles 89 and 124 it appears, that the total pressure exerted against that layer of the wall is equal to the weight of the triangular prism of water O D K, right angled at D, whose thickness is unity, and whose side D K is " equal to the depth of the joint DE beneath the surface Y; and it also appears, that the resultant of that pressure acts in the line H C, being a perpendicular upon O D from the centre of gravity H of the prism of water; so that C D = - -. Let G be the centre of gravity of the vertical layer o of masonry above D E, and G B "W a vertical line drawn through it ; produce H 0, cutting that vertical line in A ; take AW to represent the weight of the layer of masonry, and AP to represent the pressure _p the layer of water ; complete the parallelogram A P R, W j A B, will represent the total pressure on the joint D E for each unit of length of the wall, and F, where that line cuts D E, will be the centre of resistance of that joint, which must fall within the limits consistent with stability of position, while at the same time the angle A F D must not be less than the complement of the angle of repose. To treat this case algebraically, let x denote the depth of D beneath the surface of the water, w' the weight of an unit of volume of water, and j the inclination of D to the vertical. Then the pressure of the vertical layer of water is (i.) 2 its centre C being at the depth - x. O This force, together with the equal and ^opposite oblique com- ponent of the resistance of the joint D E at*F, constitute a couple tending to overturn the wall, whose arm is the perpendicular dis- tance of F from C P ; that is to say, CB-FD -sin./. RESERVOIR- WALLS. 245 Xow C~D = x ' f C ^, and if, as before, we make ED = t, FD = o (^ "J~ 9) ^ consequently we have for the arm. of the couple in question, x - which being multiplied by the pressure, gives the moment of the overturning couple ; and this being made equal to moment of stability of the wall, we obtain the following equation : /W-fB-Wfe fit^-sec'j-w'x't (f + i) tan/....(a) When the inner face of the wall is vertical, secj= 1, and tanj = 0;' and the above equation becomes (2 A.) To obtain a convenient general formula for comparing walls of similar figures but different dimensions, let n, as in Article 211, denote the ratio of the area of the vertical section of the wall to that of the circumscribed rectangle, so that if w be the weight of an unit of volume of masonry, the weight of the vertical layer of masonry under consideration is "W = nwht, 9 where h is the depth of the joint D E below the top of the wall,. Then equations 2 and 2 A take the following forms : ; ...... (3.) (3 A.) equations analogous to equation 4 of Article 213. To obtain a formula suitable for computing the requisite thickness of wall , let W tf - sec 2 .; A; 6 n (q + q i* -i/ - B 2 n (q +. $') w h 246 THEORY OF STRUCTURES. then which quadratic equation being solved, gives t = N/A + tf-B j ........................ (4.) or for a wall with a vertical inner face, for which B = 0, t=jA.. ........................... (4 A.) most cases which occur in practice, the surface of the watei Y either is, or may occasionally be, at or near the level of the of the wall, so that h may be made =? x. In such cases, let w' sec 2 ^* I and 1 = 6, x zn\q^q)W and we have \ which being solved, gives = j^b'-b; ........................ (5.) X and for a wall with a vertical inner face, ............ (5 A '> The vertical and horizontal components of the pressure of the water are respectively Vertical, P sinj = -g tan j, Horizontal, P cos,/ = -g j Consequently the condition of stability of friction at the joint D B is given by the equation P COS J W' X 3 , v L ^ .......... RESERVOIR-WALLS. 247 If the ratio - has been determined by means of equation 5, then oc we have W = nwxt = nwy? - :... ...(7.} x so that by cancelling the common factor ce 2 , equation 6 is brought to the following form : ^- ^tan

' being the angle of repose of the foundation of the wall. The object of giving the base of the wall an inclined position is to diminish the obliquity of the pressure on it, and so to enable the condition of frictional stability to be fulfilled. 3 3 The values adopted for q in practice vary from to -. ^^ ** 10 o i ifl "Is 18. Rectangular Retaining Walls. In a vertical rectangular 7 r\ all, n = 1, q' = 0, i = j so that, in equations 3 and 4 of Article V' w\ cos 6 d 6 q w 3 h(t+,Ts).* (I-) .%- ,/ sm b = q w RECTANGULAR RETAINING WALLS. 253 Example I. When the surface of the bank is horizontal, so that 6 = 0, then and Also 1 sin

J / triangular prism of masonry 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 N, 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 as for a rectangular wall. If q = , the thickness of the wall at the o summit will be -5 of the thickness at the base. The face of the wall o is said to batter; the rate of the batter being the ratio = G-)-:- EQ 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 slope backwards, as in fig. 101. 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 x and thickness t with the pro- posed wall, and le.t g be tho Fig. it4. ntre of gravity of that rectangular" wall. Then r COUNTERFORTS. W ' q t = q w x ? 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 .(2.) If E N is a straight line (fig. 103), E N is a parabolic arc, .(3.) .(4) a formula which is also sensibly correct when E K 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. foundation Courses of Retaining Wall* 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 Fi S- 105> than it is in the body of the wall. The power of earth to suppo jions has already been considered in Article 199. 2. 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 p wall whose height is h, with a row of rectangular counterforts of the same height with the wall. 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 qwhbt*-, W~'tfw and that of the second part, 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, *, = 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 : (b + c)^, 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 commutations 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 258 THEORY OF 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 tlie 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 OP RUPTURE. 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 if seemingly anomalous figures; but such structures are safe on a p&ll scale only. Angle, Joint, and Point of Rupture. The first step towards Letermining whether a proposed arch will be stable, is to assume a inear arch parallel to the intrados or soffit of the proposed arch, ind 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 t/ie 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 inteosity 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 BOA 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 2 GO THEOKY 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 J A 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 yfoWer 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. 2C1 ture and total thrust, Laving 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 B (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 G, 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- n 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 counterforted abutments. The anale 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- TV ment B ; its intensity is therefore increased in the ratio ;, and B 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 PIERS AKD 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 ** 111 ' 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 j ust 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 arch; 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 2G4 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 2G6 THEORY OF STRUCTURES. It has been sliown in Article 179, that if there be an inward radiating pressure upon a ring, of a given intensity per unit of arc, there is a thrust exerted all round that ring, whose amount is the product of that intensity into the radius of the ring. The same proposition is true, substituting an outward for an inward radiating pressure, and a tension all round the ring for a thrust. If, there- fore, the horizontal radiating pressure of the dome at the joint C C be resisted by the tenacity of a hoop, the tension at each point of that hoop, being denoted by P y , is given by the equation _> P* cotan i ( . ? = ypy= 2v ..................... (2.) !Now conceive the hoop to be removed to the circular joint D D, distant by the arc d s from C C, and let its tension in this new position be The difference, d P y , when the tension of the hoop at C C is the greater, represents a thrust which must be exerted all round the ring of brickwork C C D D, and whose intensity per unit of length of the arc CD is Every ring of brickwork for which p x is either nothing, or positive, is stable, independently of the tenacity of cement ; for in each such ring there is no tension in any direction. When p z becomes negative, that is, when P y has passed its maxi- mum, and begins to diminish, there is tension horizontally round each ring of brickwork, which, in order to secure the stability of the dome, must be resisted by the tenacity of cement, or of external hoops, or by the resistance of abutments. Such is the condition of stability of a dome. The inclination to the horizon of the surface of the dome at the joint where p t = 0, and below which that quantity becomes negative, is the angle of rupture of the dome ; and the horizontal component of its thrust at that joint, is its total horizontal thrust against the abutment, hoop, or hoops, by which it is prevented from spreading. A^dome may have a circular opening in its crown. Oval arched openings may also be made at lower points, provided at such points there is no tension ; and the ratio of the horizontal to the inclined axis of any such opening should be fixed by the equation horiz. axis = A / P, * inclined axis V p y sec i *\ 'I DOMES, SPHERICAL AND CONICAL. 267 Example I. Spherical Dome. Uniform thickness, t ; weight of material per unit of volume, w ; radius, T. ds = rdi. _ Ps cos i ^ to t r* cos i sin i cos i } y ~ -p cos ^ cos 2 i 4- cos i 1 1 + cos i The angle of rupture, for which p s = 0, is (5.) = arc cos ~ 1 = 51 49' ;. .(6.) and from this angle we obtain, for the horizontal thrust of the dome, per unit of periphery at the joint of rupture, p y = 0-382 wtr; and for the tension on a hoop to resist that thrust, y Example II. Truncated Conical Dome (fig. 113). Apex, O. Depth of top of dome below apex, x ; of base of dome, ajj ; i, uni- form inclination ; t, uniform thickness ; y = x cotan i. Then at the base of the dome, w t cos i _ y ~ 2 sin 3 ; (Xl z = wtx l ' cotan 2 i. ...(8.) Fig. 113. p z being everywhere positive, there is in this dome no joint of rupture. Example III. Truncated Conical Dome, supporting on its summit a turret or "lantern" of the weigU L. 268 THEORY OF STRUCTURES. w t cos i ( xl . -' ] 2* a?/ , (90 P^^Lf^.^ + L cotan 2 i. 235. Strength of Abutments and Vaults. The dimensions required in an abutment, arch, or dome, to insure stability, are in most cases sufficient to insure strength also ; but instances occur, in which the condition of sufficient strength requires to be indepen- dently considered, and it may be convenient here so far to antici- pate the subject of strength as to state that condition, viz., that the intensity of the thrust in the materials shall at no point exceed a certain limit, found by dividing the resistance of the material to crushing by a number called the factor of safety. The factor of safety in existing bridges ranges from 3 or 4 to 50 and upwards. In tunnels it is about 4. Tredgold considers, that in bridges the best value for the factor of safety is about 8 (Treatise on Masonry). The resistance of some of the most important materials of masonry to crushing is stated in a table at the end of this volume ; but a prudent engineer, who contemplates a great work in masonry, will not trust to tables alone, but will ascertain the strength of the materials at his command by direct experiment. 235 A. Transformation of Structures in Masonry. The principle already stated in Article 126, that to determine the intensity of a force in a transformed structure, the projected line representing the amount of the force must be divided by the projected area over which it is distributed, requires special attention in considering the strength of transformed structures of masonry. To exemplify the application of that principle, conceive a rec- tangular prism whose dimensions are x, y, z, x being vertical : its volume is Y = x y z. Let w be the weight of unity of volume of the material of which it is composed ; and let the weight of the prism be represented by a line parallel to x, of the length W; then "W = wxyz. , (1.) The amount of an upward vertical pressure on the base of this prism, which balances W, will be represented by a line equal and opposite to W : that is P - AY- (2 \ . * ) \ / TRANSFORMATION OF STRUCTURES. 269 and the intensity of that pressure will be p p = = -wx ......................... (3.) yz "Now let there be a parallel projection of this prism, whose dimen- sions, x = a x, y = b y, z = c z, are oblique to each other. The weight of the new prism will be represented by a line parallel to a/, of the length W = aW ............................ (4.) Let C = 1 cos 7 y' z' cos 2 z' x' cos &' y' + 2 cos y' z cosz x* ' cosx' y .................. (-5.) Then the volume of the new prism is V = 0/2/2' /C"= V-abc J~C; ............ (6.) consequently the intensity of its weight is W , = W = qW_ w V abc JG'V lc VC " The area of the lower surface of the new prism is 2/ z' ' sin y 1 z' = y z b c sin i/ z' } ............... (8.) The amount of the stress on that area is -W = l?=aP= apyz being represented by a line P 7 , which is the projection of P, and parallel to x'. The intensity of this new stress is A yf z f sin y z 1 b c ' sin y f z' and if we consider the relation between stress and weight, F = - W, that is, p'y 1 z smi?z = - w'x'y' z 1 JU. ............ (11.) we find sin 270 CHAPTER III. STRENGTH AND STIFFNESS. SECTION 1. Summary of the Theory of Elasticity as applied to Strength and Stiffness, 236. The Theory of Elasticity relates to the laws which connect the stresses, or pressures and tensions, which act at the surface and in the interior of a body, with the alterations of dimensions and figure which the body and its parts simultaneously undergo. That theory, therefore, is the foundation of the principles of the strength and stiffness of materials of construction. The theory of elasticity has many other applications, to crystallography, to light, to sound, to heat, and to other branches of physics. Its full discussion would of itself require a voluminous work; in the present section, its principles are to be briefly summed in so far as they are appli- cable to the strength and stiffness of structures. 237. Elasticity is the property which bodies possess of occupying, and tending to occupy, portions of space of determinate volume and figure, at given pressures and temperatures, and which, in a homo- geneous body, manifests itself equally in every part of appreciable magnitude. 238. An Elastic Force is a force exerted between two bodies at their surface of contact, or between two parts into which a body either is divided or is capable of being divided at the surface of actual or ideal separation between those parts. The intensity of an elastic force is stated in units of weight per unit of area of the surface at which it acts. That kind of force is in fact identical with stress, the statical laws of which have already been explained in Part I., Chapter V., Sections 2, 3, and 4, Articles 86 to 126. 239. Fluid Elasticity. The elasticity of a perfect fluid is such that its parts resist change of volume only, and not change of figure j whence it follows, that the pressure exerted by a perfectly fluid mass is wholly perpendicular to its surface at every point : principles which form the basis of hydrostatics and hydrodynamics. Fluids are either gaseous or liquid. A gaseous fluid is one whose parts (so far as is known by experiment) exert a pressure against LIQUID ELASTICITY RIGIDITY. 271 each other and against the vessel containing them, how great soever the volume to which they are expanded. See Arts. 110, and 117 to 124 240. Liquid Elasticity. The elasticity of a perfect liquid resists change of volume only, and differs from that of a gaseous fluid chiefly in this : that the greatest variations of the pressure which it is possible to apply to a liquid mass produce very small variations of its volume. The compression undergone by a liquid mass in consequence of the application of a given pressure over its surface, is measured by the ratio of the diminution of volume produced by the given pres- sure to the entire volume of the mass : a ratio which is always a very small fraction. The compressibility of a given liquid is the compression produced by a unit of elastic pressure ; in other words, the ratio of a compression to the pressure producing it. The modulus or co-efficient of elasticity of a liquid is the ratio of a pressure applied to and exerted by the liquid, to the accompanying compres- sion, and is therefore the reciprocal of the compressibility. The following empirical formula for the compressibility of pure water at any temperature between 32 and 128 Fahrenheit has been deduced from the experiments of M. Grassi (Comptes Rendus, XIX. / Phttos. Mag., June, 1851). Compressibility per Atmosphere, 40 (T + 461) D T, temperature in degrees of Fahrenheit. D, density of water at that temperature under one atmosphere, the maximum density of water under the pressure of one atmosphere being taken as unity. See Art. 123, equation 5. At the temperature of maximum density, 39 ! Fah., the compressibility of water per atmosphere is 0-00005, and its modulus of elasticity, 20,000 atmospheres, or 294,000 Ibs. per square inch. Compressibilities of some Liquids, per Atmosphere, from M. Grasses experiments. Saturated aqueous solution of nitrate of potash, o '0000306565 Saturated aqueous solution of carbonate of potash,... .0*0000303294 Artificial sea water, 0*0000445029 Saturated aqueous solution of chloride of calcium,.... 0*00002 09830 ./Ether, 0*00011137 to 0*00013073 Alcohol, o -00008 245 to o -00008587 The compressibility of aether and alcohol increases with the pressure. 241. Rigidity or Stiffness. A solid body, besides resisting change of volume like a liquid, possesses also rigidity, or the property of 272 THEORY OF STRUCTURES. resisting change of figure. As in the case of liquids, the utmost alteration of volume of which a solid body is capable by any pressure which can be applied to it, is always a very small fraction of its entire volume. The stresses at the surface of a solid body or particle are not necessarily normal, but may have any direction, from normal to tangential. 242. strain and Fracture. In popular language the words strain and stress are applied indifferently to denote either the system of forces at the surface of a solid body whereby its volume and figure are altered, or the alteration of volume and figure of the body and its parts thereby produced. For the sake of clearness in scientific language, certain authors have recently endeavoured to appropriate the word strain to the alterations, of what nature soever, in the volume and figure of a solid body and of its parts, produced by forces applied to it, and the word stress as formerly defined. This nomenclature will be used in the present treatise. Fracture of a solid occurs when a strain is carried so far as to cause actual division of the solid into parts. The strains and fractures to which a solid, considered as a whole, is subject, may be classified according to the following table. To each kind of strain there corresponds a kind of stress ; being the external force which produces that strain, and the equal and opposite force wherewith the solid resists that strain : Strain. Fracture. ^ ( Compression Crushing and Cleaving. [ Distortion Shearing. Transverse < Torsion Wrenching. ( Bending Breaking across. 243. Perfect and Imperfect Elasticity. Plasticity. A body is Said to be perfectly elastic, which, if strained at a constant temperature by the application of a stress, recovers its original volume, or volume and figure, when such stress is withdrawn. Deviations from this property constitute imperfect elasticity. Gases, and liquids perfectly free from viscosity, are perfectly elastic. The elasticity of every solid is sensibly perfect when the strain does not exceed a certain limit. This has been proved to be the case even for solids so plastic as moistened clay. For every solid there are limits, which if a strain exceed, set, or permanent altera- tion of volume or figure, is produced , and such limits of elasticity are less, and often considerably less, than the strains required to produce fracture. It has been proved by Mr. Hodgkinson that these limits depend on the duration of the strain, being less for a long-continued strain than for a brief strain. The elasticity of volume , STRENGTH TOUGHNESS STIFFNESS RESILIENCE. 273 in solids is in general much more nearly perfect than the elasticity of figure. It is true that the density of many metals is perma- nently increased by hammering, rolling, and wiredrawing, and that of some other materials by intense pressure (Fairbairn ; Report oj tlie British Association, 1854); but the stresses which operate during these processes are very great. A body which is capable of undergoing great alterations of figure, and whose elasticity of figure is very imperfect, is a plastic solid. The gradations are insensible between plastic solids and viscous liquids, in which there is a resist- ance to change of figure, but no tendency to recover any particulai figure. Rise of temperature, so far as we yet know, increases elasticity of volume in all substances, and at the same time diminishes the amount and the perfection of elasticity of figure, so as to make solids more plastic and liquids less viscous. 244. The Ultimate Strength of a solid is the stress required to produce fracture in some specified way. The Proof strength is the stress required to produce the greatest strain of a specific kind consistent with safety j that is, with the retention of the strength of the material unimpaired. A stress exceeding the proof strength of the material, although it may not produce instant fracture, pro- duces fracture eventually by long-continued application and fre- quent repetition. Strength, whether ultimate or proof, is the product of two quantities, which may be called Toughness and stiffness. Toughness, ultimate or proof, is here used to denote the greatest strain which the body will bear without fracture or with- out injury, as the case may be : stiffness, which might also be called hardness, is used to denote the ratio borne to that strain by the stress required to produce it, being, in fact, a modulus of elasticity of some specified kind. Malleable and ductile solids have ultimate toughness greatly exceeding their proof toughness. Brittle solids have their ultimate and proof toughness equal or nearly equal. Resilience or Spring is the quantity of mechanical work required to produce the proof strain, and is equal to the product of that strain, by the mean stress in its own direction which takes place during the production of that strain, such stress being either exactly or nearly equal to one-half of the stress corresponding to the proof strain. Hence the resilience of a solid is exactly or nearly one-half of the product of its proof toughness by its proof strength ; in other words, one-half of the product of the square of its proof toughness by its stiffness. Each solid has as many different kinds of stiffness, toughness, strength, and resilience as there are different ways of straining it, as the following table shows. In that table pliability is used as a general term to denote the inverse of stiffness : T 274 THEORY OP STRUCTURES. Stress. Strain. Stiffness. Pliability. Fracture. Strength. Pull. Stretching or Extension. ... Extensibi- lity. Tearing. Tenacity. Thrust. Squeezing or Compres- sion. ... Compressibi- lity. Crushing. Shearing. Distortion. ... ... Shearing. ... j Twisting. Twisting or Torsion. ... ... Wrenching. Bending. Bending. Transverse Stiffness. Flexibility. Breaking Across. ... Those kinds of stiffness and strength which have no single word to designate them, are called resistance to the kind of strain or frac- ture to which they are opposed. 245. Determination of Proof Strength. It was formerly Supposed that the proof strength of any material was the utmost stress con- sistent with perfect elasticity ; that is, the utmost stress which does not produce a set, as denned in Article 243. Mr. Hodgkinson, however, has proved that a set is produced in many cases by a stress perfectly consistent with safety. The determination of proof strength by experiment is now, therefore, a matter of some obscu- rity ; but it may be considered that the best test known is, the not producing an INCREASING SET by repeated application. 246. The Working stress on the material of a structure is made less than the proof strength in a certain ratio determined "by prac- tical experience, in order to provide for unforeseen contingencies. 247. Factors of Safety are of three kinds, viz. : the ratio in which the ultimate strength exceeds the proof strength, the ratio in which the ultimate strength exceeds the working stress, and the ratio in which the proof strength exceeds the working stress. The following table gives examples of the values of those factors which occur in practice : Ult. Strength. Ult. Strength. Proof Strength. Proof Strength. Working Stress. Working Stress. Strongest steel, 1A Ordinary steel and wr. iron, steady load, " moving load, Wrought iron boilers, . 2 2 3 4 to 6 8 i'i 2 to 3 4 2 to 3 3 to 4 about 1 ^ " moving load, .. 6 to 8 2 to 3~ Timber; average 3 10 34- Stone and brick,.., about 2 4 to 10, av.abt.8 av. about 4 11ESOLTJTION AND COMPOSITION OF STRAINS. 275 248. Divisions of the mathematical Theory of Elasticity. The theory of the elasticity of solids has been reduced to a body of mathematical principles applicable to those cases in which the strains of the particles of the body are so small, that quantities in the stresses depending on the squares, products, and higher powers of the strains may be neglected without appreciable error, and that, consequently, Hooke's Law " ut tensio sic vis " is sen- sibly true for all relations between strains and stresses. This con- dition is fulfilled in nearly all cases in which the stresses are within the limits of proof strength the exceptions being a few substances, very pliable, and at the same time very tough, such as caoutchouc. The mathematical theory, as thus limited, consists of three parts, viz., the resolution and composition of stresses, the resolution and composition of strains, and the relations between strains and stresses. The resolution and composition of stresses has already been fully discussed in Part I., Chapter V., Section 3. 249. Resolution and Composition of Strains. Let a Solid of any figure be conceived to be ideally divided into a number of inde- finitely small cubes by three series of planes parallel respectively to three co-ordinate planes. Each such elementary cube is dis- tinguished by means of the distances, x, y, z, of its centre from the three co-ordinate planes. If the solid be strained in any manner, each of the elementary cubical particles will have its dimensions and figure changed, and will become a parallelepiped, which may be right or oblique its size being conceived to be so small, that the curvature of its faces is inappreciable. The simple or elementary strains of which a particle, cubical in its free state, is susceptible, are six in number, viz. : three longitudinal or direct strains, being the three proportional variations of its linear dimensions, which are elongations when positive, and compressions when negative ; and three transverse strains, being the three distortions, or variations of the angles between its faces from right angles, which are considered as positive or negative according to some arbitrary but fixed rule, and are expressed by the proportions of the arcs subtending them to radius. When the values of those six strains for every particle are expressed by functions of the co-ordinates, x, y, z, the state of strain of the solid is completely expressed mathematically. The six elementary strains, in the cases to which the theory is limited, are very small fractions. The method of reducing the state of strain of the solid at a given point, as expressed by a system of six elementary strains relatively to one system of rectangular axes, to an equivalent system of six elementary strains relatively to a new system of rectangular axes, is founded on the following theorem. Let at, ft, y, be the longitu- dinal strains of the dimensions of a given particle along x } y, z. 276 TIIEOUY OF STRUCTURES. A, ft, v, the distortions of its angles in the planes y z, z x, x y. Con- ceive the surface of the second order whose equation is ctx* + /3 2/ 3 + yz? + *yz + pzx + vxy = 1. Transform this equation so as to refer the same surface to the new axes of co-ordinates ; the six co-efficients of the transformed equa- tion will be the elementary strains referred to the new axes. Other ways of resolving strains have been pointed out by Professor W. Thomson, Cambridge and Dublin MatJiematical Journal, May, 1855. The sum of the direct strains ct + P + y represents the cubic dila- tation of a particle when positive, and the cubic compression when negative. The state of strain of a transparent body may be ascer- tained experimentally by its action on polarized light. On this subject experiments have been made by Fresnel, Sir D. Brewstcr, M. Wertheim, and Mr. Clerk Maxwell. 250. Displacements. Let , 37, f, 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% d* d? M _ _ * . a _ _ . *. _ _ 5. ~dx' ft ~d' ~dz' = + . ,, = . dy dz } dz dx' _dv d_Z ~ dx dy 251. Analogy of Stresses and Strains. It has been shown ill 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, f , p yy , p.,, be the CO-EFFICIENTS OF ELASTICITY AND PLIABILITY. 277 three normal stresses, and p yz , p tx , p^,, 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 matMinatique de rElastidte des Corps solides, Paris, 1852. The above equation is transformed into the equation of Article 249 by substituting respectively , /3, y, x, p, t>, for p xx , p^ p zt , 2p y! , 2p ZJC , 2p 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 : This function was first employed by Mr. Green, Cambridge Trans- actions, vol. vii. 253. Co-efficients of Elasticity. According to Hooke's Law, each of the six elementary stresses may, without sensible error, be treated as a linear function of the six elementary strains, each multiplied by a particular co-efficient or modulus of elasticity. By expressing all the stresses in terms of the strains, the potential energy U is transformed into a homogeneous quadratic function of the six elementary strains, which must have twenty-one terms, and consequently twenty-one co-efficients, multiplying respectively the six half-squares and the fifteen binary products of the six ele- mentary strains. The co-efficient of - 2 in U is that of in f) xx ; the co-efficient of a /3 in U is that of in p n and also that of . /3 in p xx ; and so on. 254. Co-efficients of Pliability. According to Hooke's Law also, each of the six elementary strains may be treated, without sensible error, as a linear function of the six elementary stresses, so as to transform U to a homogeneous quadratic function of the elemen- tary stresses p xt , &c., having twenty-one terms, and twenty-one co- efficients expressing different kinds of pliability. The word " plia- bility " is here used in an extended sense, to include liability to 278 THEORY OF STRUCTURES. alteration of figure of every kind, whether by elongation, linear compression, or distortion. Co-efficients, whether of elasticity or of pliability, may be thus classified : Direct, or longitudinal, when they express relations between longitudinal strains, and normal stresses in the same direction; lateral, when they express relations between longitu- dinal strains, and normal stresses in directions at right angles to the strains j transverse, when they express relations between dis- tortions, and tangential stresses in the same direction ; oblique, when they express any other relations between strains and stresses. 255. An Axis of Elasticity is any direction in a solid body, with respect to which some kind of symmetry exists in the relations between strains and stresses. An axis of direct elasticity is a direc- tion in a solid body, such that a longitudinal strain in that direc- tion produces a normal stress, and no tangential stress on a plane normal to that direction. Every such axis is a direction of maxi- mum or minimum direct elasticity relatively to the directions adjacent. By the aid of the calculus of forms, and of an improvement in the geometry of oblique co-ordinates, it has been shown that every homogeneous solid must have at least three axes of direct elasticity, which may be rectangular or oblique with respect to each other, that the number of such axes increases as the symmetry of the action of elastic forces becomes greater, and that their various possible arrangements correspond exactly with those of the normals to the faces and edges of the various primitive crystalline forms (Phil. Tram., 1856-7). 256. In an isotropic or Amorphous Solid the action of elastic forces is alike in all directions. Every direction is an axis of elas- ticity. The co-efficients of oblique elasticity and oblique pliability are all null. The number of different co-efficients of elasticity, and of different co-efficients of pliability, is three. The following nota- tion and equations show their relations to each other : Elasticities. a-* Direct, A = Lateral, B = Transverse, ........................... C= ~ ; A Elasticity of volume, - - - MODULUS OF ELASTICITY CO-EFFICIENTS. 279 Pliabilities. TV A + B Direct, a = + AB-2B 3 ' (otherwise, the extensibility.) B Lateral, ft = A 2 + AB-2B 2 ' Transverse, ........................ \j Cubic compressibility, ........... fo = 3 a 6fo. 257. modulus of Elasticity. The quantity to which the term " modulus of elasticity " was first applied by Dr. Young, is the reciprocal of the extensibility, or longitudinal pliability; that is to say, 1 2B This quantity expresses the ratio of the normal stress on the trans- verse section of a bar of an isotropic solid to the longitudinal strain, only when the bar is perfectly free to vary in its transverse dimensions, but not under other circumstances. The values of Young's modulus have been determined experimentally for almost every solid substance of importance, and a table of them is given at the end of the volume. 258. Examples of Co-efficients. The only complete sets of co- efficients of elasticity and pliability which have yet been computed are those for brass and crystal, deduced from the experiments of M. Wertheim (Annales de Chimie, 3d series, vol. xxiii.), and are as follows the unit of pressure being one pound on the square inch : Brass. Crystal. A ..................... 22,224,000 ...... 8,522,600. B ...................... 11,570,000 ...... 4,204,400. C ...................... 5,327,000 ...... 2,159,100. jj ...................... 15,121,000 ...... 5,643,800. 1 ~ ..................... 14,300,000 c ..... 5,746,000. B a ..................... 0-0000000699 ...... 0-0000001740. fc ..................... 0*0000000239 ...... 0*0000000575. C ..................... 0-0000001877 ...... 0-0000004631. fc ..................... 0-0000000661 ...... o'ooooooi772. 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 a 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 5 and similar to each other notwithstand- ing their alteration of dimensions and figure. Round 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 Ob its semi-axes, the body being so placed in its strained condition that the central par- ticle 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 andf. In the direction O A, the body has undergone the extension Aa = at.-, and in the direction O B, at right angles to O A, the extension Fig. 115. 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 , ft, and the directions of the pair of rectangular axes of strain O A, B, 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- noting them would have been negative, to express diminution of size. 282 THEORY OF STRUCTURES. A square whose sides are unity, and parallel to A and O 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 + and 1 + /3, and still parallel to O 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, C and O 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 O C = 90 - A O F = e. Then x = cos 6 =. y' y sin 6 x'. Also, let x + I = YD, y + n = OY + DC, J>e_the co-ordinates of c,jfche new position of C j and let a/ + i' = Y'G, 2/ + if = O Y' + G/j be the co-ordinates off, the new position of F. Then because of the uniformity of the strain, the component displacements %, v, ', a/, have the following values : I = CD = a. x = a. cos 6 ', *? = D c = ft y = ft sin & ; ' = F G- = a. x' = a, y = at, sin 6 ; a t/ = ft cos 6. 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 from./ let fall /N perpendicular to F N". Then ' = M is the extension of O C ; ft' = F N is the extension of O F ; and ' = c M + f N 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 9 with the principal axes O A, B, in terms of the principal elementary stresses, *, ft, are as follows : ELLIPSOID OF STRAIN. 283 ....(2.) et r = cos 6 + " sin 6 = 0. cos 2 + /3 sin 2 /? = ' sin 6 ' cos 4 = sin 2 * + ft cos 2 V = i sin 6 >j cos 6 -\- % cos * + tf sin 4 = 2 (a /3) cos * sin *. Those three equations are exactly analogous to the equations 3 and 4 of Article 112, from which they may be formed by substituting a. for p x , and ft for p y in both equations; and then, in the first place, a.' for p n , and 6 for x n ; in the second place, & for p n) and (90 *) for a; n, and in the third place, v for _p w and 8 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 = 6, 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 and ft is to be referred Draw C c, ~Ff, parallel to o 6, cutting the ellipse in c andy, from which points respectively draw c m -L o C, and/N -L o F. Then om = ' } on = &, 2 cm = 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, and 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, ft . 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 : ' = 0; 0=0; '=2*' ; (1.) that is to say, an extension, and an equal compression, along a pair of rectangular axes, are equivalent to a simple distortion relatively to Oj pair of axes making angles of 45 with the original axes; and the amount of the distortion is double that 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 , /8 = oc, will be accompanied by a pair of principal stresses along O A and OB respectively, given by the following equations : along A, p x = A * + B /3 (A - B) * ; OB,^ y ^B^ + A/3=:(B-A) a =- p,', (2.) that is to say, there will be a pull along 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^ to represent the intensity of the shearing stress on each of a pair of planes normal to the new pair of axes, # = p, = (A-B); (3.) but if C be the co-efficient of transverse elasticity of the substance, we have also fl = C.; (4.) and consequently, for an isotropic substance, CUBIC AND FLUID ELASTICITY. 285 or the transverse elasticity is Iwlfthe difference of the direct and latei'al elasticities. This is the. demonstration of a principle already stated in Article 256. The corresponding principle for pliabilities, viz. : that the transverse pliability is twice the sum of the direct and lateral extensi- bilities, is demonstrated by a similar process, of which the steps may be briefly summed as follows : = %P* %p 9 = /3 = %p y bp x = ( a . Q. E.D ................... (G.) 263. Cubic Elasticity. If the three rectangular dimensions of a body or particle are changed in the respective proportions 1 + *, 1 -f- /8, 1 -f- y, its volume is altered in the proportion and when the elementary strains a, /3, y, are very small fractions this is sensibly equal to l-t- Consequently, as in Article 249, may be called the cubic strain, or alteration of volume. In an iso tropic substance, the three rectangular direct stresses which accompany those three strains are (1.) The third part of the sum of those stresses, which may be called the mean direct stress, has the following value : 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 xi = p yy =p 2Z = B (* + /3 + 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 ,5- . 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 " = aP= ......................... (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 + a) 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-BAE 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 f 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 the length, as before, the elongation of the bar under the proof load is f x *==%; the force which acts through this space has for its least value 0, for ^S its greatest value P =fS, and for its mean value ^- ; so that the work performed in stretching the bar to the proof strain is /S fx__ S* , n '2 ' E ~ = E ' 2 f 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 OP RESI- LIENCE. /"S 267. Sudden Pull.- A pull of *^-, or one-half of the proof load, being suddenly applied to the bar, will produce the entire proof strain of ^ which is produced by the gradual application of the Jii proof load itself ; for the work performed by the action of the con- /*S stant force ^p through a given space, is the same with the work 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 288 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 the 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 Rivcttcd Joints of iron plates is given in the table, in Ibs. per square inch of section of the 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 116. row of rivet-holes. The strength of a single-rivetted 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. Hi vetted 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 \\ Jl if we consider a ring, being a portion of the cylin- \v^ .,// der of the length unity, the tension on that ring ^==^ will be Fig. 117. ? = 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 K 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 = ^- 2 ....................... (3.) a being an arbitrary constant, and r' any value of the radius, from r to R inclusive ; for this reduces both sides of equation 1 to CASE 3. General solution. By combining the two partial solu- tions of equations 2 and 3 together^ we find Radial pressure, q = n m = ^ raj Hoop-tension, p = n + m = ^ + ra (*) To determine the constants a and ra we have the equations a a whence we obtain by elimination THICK HOLLOW CYLINDER. 293 giving, finally, for the maximum lioop-ten&ion, The mean hoop-tension is which is exceeded by the maximum in the proportion r 2 ) 2 gl R 8 "" 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 po may be =f, the bursting, proof, or working tension, as the case may be : In most cases which occur in practice, the external fluid pressure /I \ -?I R C 1 -) From symmetry it appears, that the axes of stress at any particle must be, one in the direction of a radius, with the pressure q along it, and the other two in any two directions perpendicular to the first and to each other, with equal tensions p along them. Two partial solutions are obtained in the following manner : Let 2p q Q *=> so that p = n + m; q = 2n m. CASE 1. w = 0, p = ^ = m ; being the case of & fluid tension, equal in all directions. In this case, equation 1 is solved by making p = q = m = constant, .................. (2.) which reduces both sides of that equation to m (R 2 _r 2 ) CASE 2. m = 0, p=jj-=n-, being the case of a pair of circumfer- * % ential tensions, each equal to half of the radial pressure. In this case, equation 1 is solved by making which reduces both sides of that equation to 296 THEORY OF STRUCTURES. CASE 3. General solution. 2a q=2n m = m, \ p = n-\-m = 3 + m, ) The constants a and m, deduced from the equations 2a 2a q Q = -^m' J q l== m , are found by elimination to have the following values _(_*d.*'> B Ii B. 3 r 3 giving finally, for the maximum tension, ? 2(R 3 -r 3 ) ........ > 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 Q may be = f, the bursting, proof, or working ten- sion, as the case may be : B '// 2 (/ + ?) I F V (2/- ?0 + 3 ?1 r This equation shows, that if 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 T which are fitted to resist the pressure from within ooo jj3oj by b e i n g connected together across the water-space oooo or steam-space between them by tie-bars, called o stays when long, bolts when short. For example, fig. 120 represents part of the flat side of a loco- oooo 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 f 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- meter 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 x , B C, whose weight per unit of volume is w, and let it be \_ / 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 Sdx from the weight of the rod for a height x above B j and this must be equal to the pull /S. Hence f $dx=fS; .................. (1.) Fig.121. which being solved, gives for the cross section of the rod, "W 1PX S = ./j ...... . .................... (2.) 298 THEORY OP STRUCTURES. and for its weight, for a height x above B, fa- W = W (eT- 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 pi of that stress is represented simply by -^ ; F being the shearing force, and S the sectional area which resists it. 279. A Table of the Resistance of Materials to Shearing and Dis- tortion, in Ibs. avoirdupois per square inch, is given at the end of the volume. It is of small extent, because of the small number of substances whose resistances to shearing and distortion haverl/ /* been ascertained by satisfactory experiments. The resistance of / \ M timber to shearing is in each case that which acts between conti-v guous layers of fibres. ""1280. 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; f, 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' ; or |-' = (1.) For wrought iron rivetted plates, taking the value of/' from the table (as determined by the experiments of Mr. Doyne), we have i = I nearly, and .-. S'= S (2.) For wrought iron bars connected by bolts or rivets, we have ( = 1 nearly, and.-. S' = | S (3.) jo o 300 THEORY OF STRUCTURES. Example I. Plate-joint overlapped, single-rivetted. Fig. 122. A, front view ; B, side view. Let t = thickness of plate, diameter of rivet. distance from centre to centre of rivets. Fig. 122. Then H \\ t = tn: 1 f d=di S A I II* c = dis Sectional area of one rivet S Sectional area of plate between two holes _ 0-7854 d\ ~ t(cd) ' so that, d and t being given, and c required, we have 0-7854 d* (5.) d in practice is usually from 2 1 to lt ; and the overlap from c II to 1-nr C. 0000 coo .A. Ffe. 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 1-5708 d 2 t (cd) } 1-5708 d 2 t d f / lap in practice = 1| 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 0000 - v Fig. 124. 2 x Sectional area of rivet Sectional area of plate between two holes t (c d) '"'^ 1-5708 c (9.) Length of each covering plate = 2 x overlap = from 2 c to 2J c. RIVETS TIMBE3 TIES. 301 Example 1Y. Plate Butt-joint, with a pair of covering plates, double- rivetted. Fig. 125. S' 4 x Sectional area of rivet S Sectional area of plate between two holes in one row 3-1416 d' z t(c d) >' .(10.) 3-1416 d 2 t (11.) o o Length of each covering-plate = 2 x overlap = from 3J to 31 c. Flg> 125, NOTE. The length of a rivet, before being clenched, measuring from the head, is about 4J t for overlapped-joints, and 5| t for butt-joints with covering-plates. Example Y. Suspension bridge chain-joint. The chain of a sus- pension bridge consists of long and short links alternately. Each long linTc 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 lias 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 07854 tf = 1-5708 n d 2 ; and because S' should be = - S, we have -V (1 1-309 n 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, S' 2nd / /ox . (3.) In determining the number w, 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 nearly parallel to the direction of the crushing force, is characteristic CRUSHING. 303 of hard homogeneous substances 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 o 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 J 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. Hodgkinsoii, 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 THBUST. 305 maximum intensity; that is to say, in a ratio which may be denoted by mean intensity __;? maximum intensity p L ' That ratio may be found with a precision sufficient for practical purposes, by considering the pressure at any 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 x Q r sin e. 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 0} the greatest intensity of pressure will be . ,., ................ (1.) in which a = = Bb^ - - ; P being the total pressure, and S the area of the section of the pillar. Consequently the ratio required is Po _ * / 2 \ ft , iBo^S"' "* V ' I 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. -j-. I. Eectangle, h b; b, neutral axis, ) 6 / ................. II. Square, h 2 , II. Ellipse : n IV. Circle : diameter, h, III. Ellipse : neutral axis, b ; other axis, h j ) j ....... x 306 THEORY OP STRUCTURES. V. Hollow ollow rectangle : outside dimensions, h, bj ) Qk(kb h'b') inside dimensions, h', V ; neutral axis, &,.../ ]$ h' s V ' Qh VI. Hollow square, h 2 h' 2 , .................... ....... O 7 VII. Circular ring : diameter, outside, h; inside, h', , 2 , , , 2 . 286. Limitations of the Preceding Formulae. The formulas 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. Crashing 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 B, and r denote respectively the outer and inner radii of the cylinder, qi 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 When the pressure from within is null or insensible, this becomes and supposing the 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-BREAKING. 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 thick hollow cylinders. But where the thickness is small (as in the internal flues of boilers), the cylinder gives way, not by direct crushing, but by COLLAPSING, which, as it consists in an alteration of figure, is analogous to crushing by bending. According to Mr. Fairbairn's experiments, published in the Philosophical Transactions for 1858, the intensity of the pressure from without which makes a thin wrought iron tube collapse is in- versely as the length, inversely as the radius, and directly as the power of the thickness whose index is 2 -19. In most calculations for practical purposes, the square of the thickness may be used in- stead of that power. For plate iron flues, let I be the length, d the diameter, t the thickness, all in the same units of measure, and let q be the collapsing pressure in Ibs. on the square inch ; then ? = 9,672,000 nearly ..................... (4.) ' cC Mr. Fairbairn strengthens long flues by means of rings of T-iron ; in which case I is the distance between two adjacent rings. SECTION 6. On Resistance to Bending and Cross-Breaking. 288. Shearing Force and Bending moment in General. It has already been shown, in Articles 14J. 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 ajone 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 & 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 tocases 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 ( positive ) T -,. , j f - / upward ) { negative / '> let vertlcal ^stances and forces in an | downward J direction, be considered as j J S ^g [ ; and let tlie moments of according as they are Let F denote the resultant of all 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 M = V(af-x) (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 * 1 ^ 6 1 , it 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 formulae as folr lows : CASE 1. The load applied at detached 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 Xi and x a 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 - 2-x"W = 0; from which follow the equations x 2 2 W - 2 x" W .(2.) . W _ v ?r" W P, To show how the shearing force and moment of flexure at any cross section are found, let W be applied to the left of the origin, and let the plane of section, whose distance from the origin is x, lie between P x and P 2 ; then the force acting on the beam to the left of x will be and the moment of flexure ........... (3.) M = jfo-a?)?! - ^'(x"-x i the symbol 2*1 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 cordinuwisly 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 I 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 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 : EXAMPLE. SHEABING FOKCE 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 x)W cW w(c x) we w(cx? w cW -M Right of 0, W Y W " 2 2 2 V. Single load, W, ap- plied at a/' Left of x" (c+*)W (c + /)W (c +0 ( M )W Rieht of a/' .. 2c _(c-a/')W 2c 2c (c a?")(c+a;)W (Cr x ) VV 2c M" nt V 2c 2c 2c VI. Uniform load of in- WJX 2 T =M 291. Moments of Flexure in Terms of Load 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 Q = mWl, (1.) where m is a numerical factor. For beams fixed at one end, I ~ c \ 312 THEORY OP STRUCTURES. for beams supported at both ends, I = 2 c = the span ; for an uniform load, W = w I. 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, ........ ^. IV. Beam supported at both ends, loaded in the ) 1 middle, ............................................. J 4' V. Beam supported at both ends, loaded at x" ) 1 / 4# f>2 \ from the middle, ................................. J 4 V ~ I- )' VI. Beam supported at both ends, uniformly loaded, ^. 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 Z 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 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 Fi - 13 - 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 0' be the surface origi- nally plane, now curved, at which that layer is situated ; this is called the neutral surface 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 O. Therefore the direct stress at each section, such as A B, whose moment constitutes the resistance to bending, is an uniforjnly-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 ilw cross section A B. 314 THEORY OF STRUCTURES. The moment of a lending stress has already been given in Article 92, equations 3 and 4 j 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- w\ stant ratio - as positive when it is such as to give resistance to an upward moment of flexure ; that is, when p is a thrust for positive values of y, and a pull for negative values j consequently, p is to be considered as { } according as it is a { "* } 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, TT- P = 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 G, 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 = ~ ........................ (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 j and then equa- tion 1 becomes M=*I ................... .(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 Transrerse 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 < concave (. s ide of the beam, according as the mate- Convex / j essure . | rial gives way most readily to j ^ nsioiL j 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 h is that depth, h y= ~2 and the limiting value of pis 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 j and let the limiting stresses be p =f a for pressure j and = f b for tension ; then the beam will give way by < J. ^ > according as ^? is This point having been determined, the equation from which the strength of the beam may be found is 316 THEORY OF STRUCTURES. M =mWZ=^ ...................... (2.) "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. Transverse 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 = n'bh s ........................... (1.) n' 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' = ^. Thus it appears, that the resistances of flexure of similar cross sections are as their breadths and as tlie 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 m! section of the beam, and m 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, 7i', m', n, for some of the more usual forms of cross section : .-- 1 '=f- I tn FORM OF CROSS SECTIONS. -IF h ~ybh*' I. Rectangle b h, ) (including square) J i 12 I 2 1 6 II. Ellipse- Vertical axis h, ^ Horizontal axis 6, ... > (including circle) J 1 64 "20-4 = 0-0491 1 2 1 32 ~ 10-2 = 0-0982 III. Hollow rectangle, 5 h b' h'j also I-formed section, where H is the - sum of the breadths of the lateral hollows,... I/ b'h'\ 5V 1 ny 1 2 l (l--\ 6V 6/iV IY. Hollow square | tf-W J 1 d- h '*\ T~9 \ Til 1 9 1 / 1 7i' 4 \ 6 ( ~h*J LA \ n / Zi D \ /t / V Hollow ellipse 1 / V*^ 1 1 / VI* 20-4 V bh s J 2 10-2V 6AV VI. Hollow circle, 1 d_**\ 1 1 (l- h "} 20-4V AV 2 10-2V AV 1 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 /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. 29 6. 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 Us 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 1855, 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 : /=/.+/?, a.) where f is the modulus of rupture of the beam in question; f 0f the direct tenacity of the iron of which it is made ; /', a co-efficient TT determined empirically; and , the ratio which the depth of solid tit 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, / = 18,750 Ibs. per square inch ; /' = 23,000 Ibs. per square inch; = H/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 skin, and diminished towards the interior ; that the tenacity found by directly tearing a bar asunder,^, is the tenacity of the interior; that the modulus of rupture of a solid rectangular beam, f Q + f', 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 Equal Strength for Cast Iron Beams was first proposed by Mr. Hodgkinson, in consequence 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 +H . 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 Fig< 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 : 320 THEORY OF STRUCTURES. Areas. Depths. Upper flange, A lf h v Lower flange, A 2 , h 2 . Vertical web, A 3 , h. Totals,... Aj_ + A 2 + A 3 = A, 7^ -f A 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 _h (k + WA-H*! 1 >---- A - h A, ,(1.) 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 : . 1 f ; , 2 4A 1 A i A 2V i i + /i 2 + J/i s) ; ......... (2.) .(3.) and the strength of the beam is expressed by the equation 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 error being on the safe side. Let h' be the depth from the middle of the upper flange to the middle of the lower flange ; then (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 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 Fig. 135. Fig. 1JG. BEAMS OF UNIFORM STRENGTH. 821 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. c Ffe. 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 b constant; Figure of Vertical Longitudinal Section. I. (Figs. 133, 134). Fixed at A, load- ed at 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 THEOEY OP STRUCTURES. The formulae and figures for a constant depth are applicable to the breadths of the flanges of the j^-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. Reverting to fig. 130, it is evident that if * represents the proportionate elongation of the layer C C', whose distance from the neutral surface O 0' is y, and if r be the radius of curvature of the neutral surface, we must have 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 * = ^ , and consequently, the cur- Jcj vature has the following values : -.-_ r Ey El 7 " 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 MQ 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 7/ 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 greate at any other section, st stress, = 1 31 / = Mo .' MI E^o / T EI~ E yc IMo' J .(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. Fig. 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). 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 a.xis 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 = ^ 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 dv i =: arc tan -7; a 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. 1 _ di di r d s 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 ds is sensibly equal to its horizontal dx projection dx \ so that the following equations may be used without sensible error : Slope, i = ^; 1 di d 2 v Curvature, = -7 = -7 r, . r dx dx' 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 : ai C x dx f r*MI , felope, i = / = =v . / T-Tiv- d x J o r E y J o I M Ordinate, v = \ idx = ~l~ . / / T-TV- d x\ Jo Ey joJolM The greatest slope ^ that is, the slope at D and the deflection or greatest ordinate v^ 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 xeferred to Article 81 for explanations of the nature of the process of integration.] MI In both the integrals of the formulae 4, the quantity 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 ' P^.^ = V; ..... (5.) o J I M . IM ' J o J I M where m" and %" 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. 323 For beams of similar figures, and similarly loaded and supported, 2/o is as the depth, 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 dejlection is directly as the square of the length, and inversely as the 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 -- being simply equal to , depends on the 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, 1-5 C 1 2 1 3 Ill Uniformly loaded .. (\ ^ 1 1 ( c) 3 4 SUPPORTED AT BOTH ENDS. IV. Loaded in the middle, 1-- 1 1 C o 3 "V Uniformly loaded 1 X ~ 2 5 e 3 12 For a beam of uniform strength and uniform depth, the quantity t_r- 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, M I ho I Mo (7-) 326 THEORY OF STRUCTURES. V 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 1Yl' iflf I M VI. Uniform strength) and uniform depth, .... j 1 1 I 2 VII. Uniform strength,! uniform breadth ; fixed 1 \/ c 2 2 at one end, loaded at | the other I V c-x 3 VIII. Uniform strength,! uniform breadth ; sup- 1 ported at both ends, F loaded in the middle,.. J v^ 2 2 3 IX. Uniform strength, ") uniform breadth ; fixed I at one end, uniformly f c c x Infinite. 1 X. Uniform strength,! uniform breadth ; sup- 1 ported at both ends, | uniformly loaded, J c Jc^? = 1-5708 ^-1 = 0-5708 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 T am aware, by Mr. C. H. Wild, enables a diagram to be drawn, which represents, with a near approach to BATIO 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 Proportion 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, appeal's to be For the working load, = from to For the proof load, ... = from m to i. 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 Now 2/0 = m' ho, m being a numerical factor, which for symmetri- cal sections is -^ ; and consequently the required ratio is given by A the equation _Ao _ ffo n"fc ri'f _2jc ,- . 2 c~2m'*~2'~' ' ' n" an expression consisting of three factors : a factor, depending on the distribution of the load and the figure of the beam ; a factor, 2c , being the prescribed ratio of the span to the deflection ; and a Vi f factor, -4=r> being the proof strain, or the working strain, of the & 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 ratio of the span to the working deflection. Let the material be wrought iron, for which ^ is a safe value for the working strain C Then 7*o _ j^ 1000 _ 5 I Tc ~ 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 : dx I f M , da? ^ '' To integrate these equations, it is only necessary to substitute for the constant factor , in the equations 4, 5, 6, Article 300, its M' y equivalent -2, M' being now not the proof moment of flexure, but M) the actual moment of flexure at the point where the beam is hori- zontal ; that is to say, m"M' c ~ ,iv , ,. . Greatest slope i, = ^ ; deflection ^ = Jii IG m" and n" 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., MO = m W I ; and the value of IQ, 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 s ; hence the above equations 2 become .. 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, REFLECTION OP BEAMS. 329 m!" = m ff m; ri" = n"m; and for beams supported at both ends, m" 1 = 2m"m', ri" = 2ri'm; the equations 3 become Whence it appears, that the deflections of similar beams under equal loads are as the cubes of their lengths, and inversely as their breadths and the cubes of their depths. The values of n' = ~~ S1 for the ordinary forms of cross section, are given in the table of Article 295. The following table gives the values of m'" and ri" for different modes of loading and support- ing, for beams of uniform cross section, and for beams of uniform strength : m'" ri" Factor for Factor for A. UNIFORM CROSS SECTION. slope. Deflection. I. Fixed at one end, loaded at the other, ....... - ...... -. II. Fixed at one end, loaded uniformly, ......... ~ ...... E. o III. Supported at both ends, loaded in the middle, r ...... -. 1 5 IV. Supported at both ends, uniformly loaded, ,. ^ ...... . B. UNIFORM STRENGTH AND UNIFORM DEPTH. V. Fixed at one end, loaded at the other, ....... 1 ...... - VI. Fixed at one end, loaded uniformly, ......... - ...... T . ^ * VII. Supported at both ends, loaded in the middle, - ...... -. VIII. Supported at both ends, loaded uniformly, .. - ...... -. 330 THEORY OF STRUCTURES. m"' n"' C. UNIFORM STRENGTH AND UNIFORM Factor for Factor for BREADTH. Slope. Deflection. 2 IX. Fixed at one end, loaded at the other, 2 ^. 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 w" = - ; 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 2 V l = 0~TTT JDJ 1 A Jli 1 305. 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, P. 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 j then for beams fixed at one end, u = v ; and for beams supported at both ends, } ....................... (2.) u = v l v. 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 / uwdx ........................... (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 m = 1, 1 = c), (n being given by the table of Article 295), and by Article 300, equation 6, _ n"fJ _ ri'fc* Vi ~ Ey ~ m'E/i' (n" being given by the table of Article 300, and m by that of Article 295). Consequently, .,. l nn ~ ,, f . . Resilience = - ~ =. - f .<= -cbh ............ (4.) 2 40flt Jil It will be observed that this expression consists of three factors, viz.: (1.) The volume of the prism circumscribed about the beam, cbh. f~ (2.) A Modulus of Resilience, ^=, of the kind already mentioned JJi in Article 266. (3.) A numerical factor, ^ , ; in which n and m' (Article 295) 2t m 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 f , . , 1,1 , , f nn" n" section, for which n = -z, m = -^. and therefore ^ > = -^ : 6* I. Uniform breadth and depth, ................................... . II. Uniform strength, uniform depth, ........................... -. 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 l 2c, 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 : n 6' IV. Uniform breadth and depth,..., -. */ V. Uniform strength, uniform depth, -. VI. Uniform strength, uniform breadth, 306. A Suddenly- Applied Transverse Load, like the Suddenly- applied pull of Article 267, produces at first double the maximum stress, and double the strain, which the application of a load gradually increasing from nothing to the amount of the given load would produce. It is unnecessary to demonstrate this in detail, the reasoning being the same with that employed in Article 267. The contingency of the sudden application. of a moving load is provided for by the factor of safety, which expresses the ratio of the proof load to the working load (Article 247). The action of the rolling load to which a railway bridge is sub- jected is intermediate between that of an absolutely sudden load and a perfectly gradual load. It has been investigated mathemati- cally by Mr. Stokes, and experimentally by Captain Galton, and the results are given in the Report of the Commissioners on the Application of Iron to Railway Structures. The practical con- clusion to be drawn from them is, that a moving load requires a larger factor of safety than a steady load. 307 Beam Fixed at Both Ends. A beam is fixed, as well as supported, at both ends, when a pair of equal and opposite couples are made to act on the Fi g 143 ' ' vertical sectional planes at its points of support, of magnitude sufficient to maintain its longitudinal axis horizontal there, and so to diminish the deflection, slope, and curvature of its middle por- EEA5I FIXED AT BOTH ENDS. 333 tion. This is generally accomplished by making the beam form part of one continuous girder with several points of support, or by making it project on either side beyond its points of support, and so fastening or loading the projecting portions, that their loads, or the resistance of their fastenings, shall give the required pair of couples. In fig. 143, let C B A B C represent a beam supported at the points C, C, loaded along its intervening portion, and so fixed or loaded beyond these points that at them its longitudinal axis is horizontal, instead of having the slope i lt which it would have if the beam were simply supported at C, C, and not fixed. At each of the vertical sections above the points of support, C, C, there is an uniformly-varying horizontal stress, being a pull above and a thrust below the neutral axis; and the moment of that pair of stresses is that of the pair of equal and opposite couples which maintain the beam horizontal at the points of support. It is re- quired to find, in the first place, that resisting moment at the vertical planes of support (from which the stress on ^fche material there may at once be found); and secondly, the effect of that moment on the curvature, slope, deflection, and strength of the beam. The general method of solution of this question is as follows : Compute, by equation 3 of Article 303, lt the slope which the neutral surface of the beam would have at the points C, C, if it were simply supported there, and not fixed. Then, by Article 304, find the uniform moment of flexure, which, if it acted on the beam in such a manner as to make it become convex upwards, would produce a slope at the points C, 0, equal and contrary to i'i. This will be the required moment of resistance at the vertical sections C, C, from which the greatest stress on the material at those sections can be found by equation 4 of Article 293. It will afterwards appear that this is the greatest stress on the beam ; so that by putting it instead of M = m "VV I in equations 2 of Article 294, and 3 of Article 295, the conditions of strength of the beam are determined. Denote this moment by M 1; the negative sign denoting that it tends to produce convexity upwards, while the load on the beam tends to produce convexity downwards. Let M be what the moment of flexure at any point of the beam u'ould be, if it were simply supported at C, C. Then the actual moment of flexure is and by substituting this for M in the equations of Articles 300 and 303, the curvature, slope, and deflection, with the proof load, or with any load, are found. 334 THEORY OF STRUCTURES. Where M is the greater, as at A, the beam is convex down- wards. Where MI is the greater, as at C, the beam is convex up- wards. There are a pair of points, B, B, at which M = M M so that the moment of flexure, and consequently the curvature, vanish, and the beam is subjected to a shearing force alone; these are called the points of contrary flexure; and they divide the middle part of the beam, which is convex downwards, from the two end- most parts, which are convex upwards. In expressing the solution of this problem by formulae, four cases will be taken into consideration, viz. : 1. The case of an uniform beam, with a symmetrical load in general 2. Beam of uniform section, loaded in the middle. 3. Beam of uniform section, loaded uniformly. 4. Beam of uniform strength and uniform depth, uniformly loaded. CASE 1. Symmetrical load on a beam of uniform section. By Article 303, equation 3, observing that I 2 c, we have ., _ 2m" m We 2 %1 - ~^ and by Article 304, .... Bit', M > = = T consequently, M! = 2m"mWc = ra" -mWl = m" M , ......... (1.) M being what the moment of flexure at A would have been, had the beam been simply supported. The values of m" are given in Article 300. Let M' be the actual moment of flexure at A. Then M' =(l m") M ....................... (2.) The greatest moment of flexure must be either at A or C, or at both, if the moments at these sections be equal and opposite. But for beams of uniform section, m" is never less than ^ ; therefore the greatest moment of flexure is at C, or both at C and A, and never at A alone. The strength of the beam is expressed by the following formula, obtained by putting M! instead of m W l } in equation 3 of Article 295: W = r .............. (3.) m ml BEAM FIXED AT BOTH ENDS. 335 / being the limit of proof or working stress, as the case may be, and n a factor suitable to the form of section of the beam, as given by the table of Article 295. Hence it appears, that by fixing the ends of an uniform beam, so that they shall be horizontal, its strength is increased in the ratio 1 : m". The deflection is found, by subtracting that due to the uniform moment Mj from that which the load would produce if the beam were simply supported at C and C. The former of these quan- tities, according to Article 304, is 2EI~ 2EI ' and the latter, according to Article 303, equation 2, is E " m"EI so that the deflection, their difference, is 1\ M,c 2 E ...... - From the last of those expressions, it appears that by fixing the ends horizontal, an uniform beam is made stiffer under a given load in the ratio If, in the first expression for the deflection, it be considered that Mj is the moment of resistance corresponding to the proof or limit- ing stress at the section C, we may make so as to obtain the following expression for the deflection under the proof load : being less than the proof deflection of a beam simply supported, as given by equation 6, Article 300, in the ratio The points of contrary flexure are to be found in each particular case by solving the equation M-M, = (6.) 336 THEORY OF STRUCTURES. CASE 2. Uniform section, loaded in tJie middle. - 1 " - 1 _ 1 - ...(7.) The points of contrary flexure are midway between A and C. CASE 3. Uniform section, uniformly loaded. W = 2 c to 1= M O = W?= W c (8.) The points of contrary flexure are thus found. By the table of Article 300, case 5, so that in order to have M = M,, we must make 1 - = *; or* = -j = 0-577 c; ............. (9.) c 3 N / 3 which equation gives the distance of each of the points of contrary flexure B, from A, the middle of the beam. CASE 4. Uniform strength, uniform depth, uniform load. In this case the uniformity of strength is attained by making the breadth at each point proportional to the moment of flexure, as c shown in the plan, fig. ] 44, preserving, at the points of contrary flexure B, B, a sufficient thickness only to l44 - 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, CB and BA, 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 = -, J c supported, but not fixed at B, B; that is to say, and therefore, the moment of flexure at C is 61 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 6, 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 STK.UCTUB.ES. intersecting each other in the points of contrary flexure, B, B, for which x = zr -. 2t The breadth which must be left at B, to resist shearing, will appear from the next Article. 308. A Beam Fixed at One Eutl and Supported at JBoth 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 + x : 2 c, where X Q 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 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 * See Article 303A, p. C41. SHEARING STRESS IN BEAMS. 339 equilibrium of that solid, the amount of shearing stress on the plane F 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; 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 imiform cross section; let O B = x, CTfl = c, BE = y, B G = y,, iTD = EF (sensibly) = dx; let the breadth of the beam at any point E be denoted by z, and at the neutral surface by ZQ, Let p be the intensity of the direct horizontal stress at E, q that of 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 ir . y z ' d y. 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 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 l ; 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 = f zdy 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 : <7 () S S 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 3 I. Eectangle, z d = b, ............................ 2' II. Ellipse, III . Hollow Rectangle 1 S = bh-b'h*z = b-V. 1 3 This includes I-shaped sec- f 2 ' (b- b') (6 A 8 - V h") tions, ........................... J IV. Hollow square, &' - X>, .................... | (l + /? ^L). 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. Lines of Principal Stress in Beams. 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 n , the normal com- ponent of the stress on a vertical plane, is to be put p ; for p n , the normal component of the stress on a horizontal plane, is to be put 0; and for p t , 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 p, 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-2 _ P . ~~ ' equation 22 becomes 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 tan 2 i, = -i .................... .....(2.) or in another form tan in = 342 THEORY OF STRUCTURES. If i a be the angle which the axis of least stress makes with the horizon, then, because i l -i a = 90, we have 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 i n g. 146, may be 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 FE (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 j 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 01 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. Small Effect of Shearing Stress upon Deflection A shearing stress of the intensity q produces a distortion represented by ~ 9 L C being the transverse elasticity, as already explained in Article 262. The slope of any given originally horizontal layer of the SMALL DEFLECTION DUE TO SHEAR. 343 beam at a given point will be increased by this distortion to the extent denoted by 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 _dF I fy " 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.) Observing that / F d x = M^ the above equation becomes, for uniform beams, Supposing the beam to be under the proof load, we may put for -=- its value , making the equation 1 y\ (5 -> 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 y l = ^, z = b, \* l yz d=. Then 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 OP STRUCTURES. 3 E tf __ 20 ' C' For wrought iron (for example) = about 3. Suppose - = C c t which is an ordinary proportion in practice ; then = ^^ = Vj you iu/ 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 wlwle learn 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 3L 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 ; so that the removal PARTIALLY-LOADED BEAM. 345 of the load from any portion of the learn diminishes the moment of flexure at each point. Q. K 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 R 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, diminislies 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 R 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 846 THEORY 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 imiform 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 (2.) At the ends of the beam this excess vanishes. At the middle, it consists of the whole shearing force !P = j 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 b' denote the breadth as computed by considering the ex- ternal load alone, W. Compute the weight of the beam from that TV provisional breadth, and let it be denoted by B'. Then = is the proportion which the weight of the beam must bear to the entire or W 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, t _. i'W nv ~ 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 and the true gross load will be W' 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, 6, 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 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 348 THEORY OF STRUCTURES. weight (treated as uniformly distributed) is its working load, is T> given by the condition = 1 ; that is, _ = kw r 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 :l - L-l : :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, w e = 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 that/= 10,000 Ibs. per square inch ; and let - = . Then 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 the 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 JLoug 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 Pahr., 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 3\[r. Adie.) Brick, common, '355 fire, -0005 Cement, -0014 Glass, average of different kinds, -0009 Granite, -0008 to -0009 Marble,. -00065 to "ooii Sandstone, -0009 to -0012 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 Carre, in the widest sense of the term, is the figure assumed by the longitudinal axis of an originally straight 350 THEORY OP 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 Arlicle 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 implied, and A B their common line of action. The figures from Fig. 146 a. Fig. 146 b. Fig. 146 c. Fig. 146 d. Fig. 146 e. 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 A D, B E, 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 G in the elastic curve from the line of action A B. Then the mo- ment of flexure at that point is obviously M = *P; ............................. (2.) and consequently the radius of curvature at that point is given by the equation that is to say, the radius of curvature is inversely proportional to the perpendicular distance from the line of action of the forces. At each of the points in figs. 146 a, b, 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 the 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 'Sd'2 THEORY OF STRUCTURES. and opposite forces of the magnitude P are applied, directed to- wards each other ; the spring forming a single arc A C B, of the length I. 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 &t 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 : which value being inserted in equation 3, gives _^4 = Z. a; . ( 4.) The integral of this equation in y x a ' sin -, C I (5.) /EI where c = \ / -=r. In order that x may be = at the points A and B, it is necessary that when y = I, - should be = n K, n being any whole number ; c and consequently that c = ^ (6.) Now of all the possible values of n, that which gives the least value of P is n = 1 ; whence we find I ^ ^ ,EI = T = -; and P = 2 ; and this jfowte 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. l^Qfj 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 t 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 ^foment, 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 Arfe. 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 L ^ any intermediate cross section such as S, and also the angular displace- Fl S- 147< 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 ; 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 S, 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 di 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 working resistance if the working moment of torsion is to be determined. Let r be the external radius of the axle. Then / is the value of q at the distance ^ from the axis ; and at any other distance r, the intensity of the shearing stress is Conceive the cross section S 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 dr ; 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 OP 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 S, gives for the moment of torsion, and of resistance to torsion, 2 = 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^ j 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 A, be the external diameter of the axle, and h$ its internal diameter, if hollow; then For a solid axle, M - ^-* - ^ j For a hollow axle, M= 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 = _-, an a f or the moment of torsion, ^r = ^. Hence we have this useful principle, that for equal values of the limiting stress f, the resistance of a cylinder, solid or hollow, 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 STIIUCTUKES. table, viz., 27,700, 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 24,000, and others a value as high as 30,000. With respect to the working values of the limiting stress f, 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, ..... ./=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 = 07 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 = -^ > and therefore, x dx I. Let the moment of torsion be the working moment, for which 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 C" 1,000'" " II. Let the moment of torsion 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 7 for f in the numerators and r for ?*, in the denominators ; that is to say, q 2M For solid axles. == 7 ', and r * 1 1 ~Cr~ For Jwllow axles. - 7. ,-r : and r *-(rf-r?,V q*_ Cr~*C(r?-r<) 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 : = . ' for a solid shaft or ^ = W-*p for a hollow shaft. 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 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 j so that formulae founded on the assumption of its being constant are erroneous. The mathematical investigations of M. de St. Yenant 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 5 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 leip 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 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 j 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 p. 148 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 CRANK AND AXLE TEETH OF WHEELS. 359 A bending couple P N S = M cos j ; and A twisting couple P N P = M sinj. 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 ptanj = ~ ~~ consequently, by the equations 1 of this Article, the resultant greatest stress at S, and its inclination to the axis of the shaft, are 5*1 M Pi = (*) and by making p l =f t 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 S 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. x section of the rim of the wheel A of fig. 149, and let EHKP 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- Fiff 150 * tersect the tooth from the side EP to the crest PK, and let PG be perpendicular to that plane* Let h be the thickness of the tooth, and let B EF = 6, FG = I Then the moment of flexure at the section EF is P?, and the greatest stress produced by that moment of flexure at that section is GPJ P = 6 A 2 which is a maximum when then the value, PEF = 45, and 6 = 21, having Consequently, the proper thickness for the tooth is given by the equation h = 3P 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 A 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- p. 151 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 -fi 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. Hodgkinsou'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 : P J>'- . 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, P* * " 6V 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~ 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 oj 'the propor- tion in which the length of the pillar exceeds tfie 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 t 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 : (2.) 1+'J 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. par inch. a. Wrought iron, solid rectangular section, 36,000 Cast iron, hollow cylinder, 80,000 solid .. 80,000 3,000' 1 800' 1 400* 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- U| | I fore be expected to be the same; a Li ^ ' ' ^ ' conclusion verified by the experiments of Mr. Hodgkinson. Hence, for such pillars, (3.) Fig. 152. Fig. 153. Fig. 154. Mr. Hodgkinson found the strength of a pillar, jfec? at one end and rounded at the other (fig. 153), to be a mean between the strengths of two pillars of 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 IROX PILLARS AND STRUTS. I. 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 is computed by means of equation 2 of Article 328, it appears that for the smaller pro- portions 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^d : 1 : : 26J : 1 nearly, those materials in the shape of solid pillars, rectangular for wrought iron, cylindrical for cast, are equally strong, and beyond that pro- portion wrought iron is the stronger. This result was pointed out by Mr. Gordon. The following table illustrates it : I T O 2O ">6* A 00 AO k ^\j 4 3 4 Breaking load, ("Wrought, Ibs. per square p solid rect- angular, 34,840 31,765 29,230 27,700 23,480 inch, = -;r. Cast solid S' cylindrical, 64,OOO 40,000 29,230 24,620 16,000 331. 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 incites, and L the length in feet, (i.) For hollow cylindrical pillars, h Y being the external, and internal diameter, in inches, and L the length in feet, the = A .(2.) The values of the co-efficient A are as follows ; 364 THEORY OF STRUCTURES. (1 . (2. 3. Tons. For solid pillars with rounded ends, ............... 14-9 flat ends, ...................... 44*i6 For hollow pillars with rounded ends, ............ 1 3 'o 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 tho 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- v i - - -n- icf T- f -, T* ne * iron), ID i (a cross- Fig, loo. Fig. 15G. Jig. T57. F iff. 158. i , '' . v , . shaped section, used in half-lattice girders), and 158 (T-iron). In some large lattice girders, the struts are composed of a pair of parallel T-iron bars, such as fig. 158, with their middle ribs turned towards each other, and connected together by a lattice work of small diagonal bars. In applying to wrought-iron struts the formulae of Article 328 I 2 72 Q pages 361, 362, for ^ there is to be substituted ^; J bein^ the 1 *2i t) least moment of inertia of the section (Article 95, pages 77-82). 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 Fig 159 one-thirtieth of the dia?neter of the cell, as determined by Mr. Fairbairn and Mr. Hodgkinson, is 27,000 Ibs. per square inch section of iron j CELLS SIDES OF GIRDERS TIMBER POSTS. 365 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- | j 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.) F!g . m 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 (2.) 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. The above formulae are for posts and struts fixed at both ends. For those which are freely jointed at both ends, the strength is reduced to one fourth. Weisbach applies to timber posts and struts a formula identical with equation 2 of page 362, with the following values of the con- stants : f 7,200 Ibs. on the square inch. 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 formulae 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 web or 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- Fig. 163. beam" long employed in the platforms of blast furnaces, and first used in a railway bridge by Andrew Thom- son about 1832, on the Pollok and Govan Kailway. 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 j the upper and lower pairs of angle irons are rivetted to the tipper 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. 1 65 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. 164. Smaller cellular girders are sometimes used, in I I I 1 I which the top alone consists of one or two lines _, of cells, the girder in other respects being similar to fig. 164, 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 and 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. Let 8j be the sectional area of the compressed rib, yj its resistance to crushing per square inch, S 2 the sectional area of the stretched rib, j^ its resistance to tearing per square inch; then P^M^P M '"/ f\h' *~ /a ~ f,h The values of the tenacity^ 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, f lt 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 empirical formula, of the kind already ex- plained in section 8, Article 328 : / "l +'" wherey= 36,000, a = , h' = the breadth of the compressed o,UOU lib, and I' = the 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 z be its resistance per unit of section to the shear- ing force, F F J 3 ; fsh 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 9 should be the resistance of the vertical web to crushing, as determined by equation 2 of Article 328, page 362, in which, for y is to be substituted -^ h being the depth of the web, as before, li ii 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, when first loaded (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 about two-thirds of the value for a continuous bar, so that the deflection is about one-half greater. But the part of that deflection due to the yielding of the joints is permanent; so that after the joints have "come to their bearing" the modulus of elasticity becomes the same as for a continuous bar. 33S. For IBalf-I.attice 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 313. The ultimate tenacity of the ties may be taken at / 2 = from 50,000 to 60,000 Ibs. per square inch. The resistance 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. The remarks made in the last Article on abutting joints and on deflection are equally applicable in the present case. In designing those joints which are connected by means of bolts, rivets, or keys, the principles of Article 280 should be observed. 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 unilbimly- 2 B 370 THEORY OF 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 w eight 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 s/tearing 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. 1G7. 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- 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 the centre of the 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 I), 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 fixed horizontal at A, and loaded uniformly between A and D ; and hence (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, A r\ = CD = =^=0-577 x AC; ............ (1.) V 3 and consequently, making A C, the distance between the lowest and highest points, = (/, 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 ;, 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 : in r' 2 r' W M, = ^- =-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 Gy and the second v. 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 figs. 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 -f x, its gross load is W = w(c-\-x)', and the intensity of the force exerted between the girder and chain, This is the intensity of the upward load on 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 n/(c-a;) w(c-}-x)(c-xf . Mc = ~~~ ~~ The amount of the upward force exerted between the chain and BDis W = w' (c-x) = |; ................. (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, 374 THEORY OF STRUCTURES. ' W w(c*-x*) - 2 - 4c is tlie 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 ~^= 27~~' ....................... (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 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 o the 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. : 2 we* max. M A = max. M = ; .............. (7.) 27 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 we 2 therefore the transverse strength of the stiffening girder should be four twenty-seventh parts of that of a simple girder of tlie same span of th suited to bear an uniform load o 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 .(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 4 5 f../*. m \ -^c 9 12 Ey 108 E^" GY five-ninths of the proof deflection of an uniformly loaded beam. '"" * In the preceding solution of Case 2, winch-appeared in the first edition of this work the effect of the resistance of the chain to disfigurement upon the figure of the auxilian- girder is neglected ; and hence the result is in almost every case an approximation only ; but it can be shown that the error is always on the safe side, four twenty-sevenths o;' the strength of a simple girder being somewhat more than sufficient for the strength of the stiffening girder. In order to make the solution exact, the extensibility of the chain should be so great as to make its proof central depression nearly equal to the proof deflection of the stiffening girder ; but in practice the proof depression of the chain is always much less. The first solution in which the action of the chain just referred to is taken into account appeared in an editorial article of the Civil Engineer and Architects Journal for November and December, 1860 ; and this is done by introducing into the conditions of the problem an equation, expressing that under all the alterations of the figure of the chain produced by the bending of the stiffening girder, the span continues constant. Having applied the principle just stated to the problem of Case 2, the author of this work has arrived at the following results, supposing the chain to be inextensible. The greatest bending moment of the stress on the stiffening girder takes place when 0-417, or about five-twelfths, of the span of the bridge are loaded, and 0-583, or about seven-twelfths, unloaded. That moment is 0-138 of the bending moment which would be produced by an uniform load of the same intensity on a girder supported at the ends only. Hence it appears that if the chain be supposed inextensible, the proportion borne by the strength of the stiffening girder to that of a simple girder of the same span, suited to bear an uniform load of the same intensity with the travelling load, ought to be .............................................. 0-138:1; while if the chain is supposed very extensible, as in the approximate solu- tion, that proportion is found to be 4:27, or ................................... 0148:1; so that in the intermediate cases that occur in practice no material error will be committed if that proportion be made 1 : 7, or ............................ 0-143: L 376 THEORY OF STRUCTURES. 341. Ribbed Archc*. 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- 171. 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 ;i 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 Sir 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. CONDENSED SUMMARY OF EXPERIMENTS BY MESSRS. ROBERT NAPIER AND SONS ON THE TENACITY OF IRON AND STEEL. (For details, see Transactions of the Institution r>f Engineers in Scotland, 1858-59.) STEEL BARS. Cast Steel Tenacity in Ibs per square inch. Strongest Weakest Quality. Quality. ..132.909 92.015 STKBL PLATES. Cast Steel Tenacity in Ibs. per square inch. Strongest Weakest Quality. Quality. 95,299 72,338 96,715 72.994 93,979 72,366 56,735 49,338 43,979 64,12S 45,584 51,349 41,743 55,937 41,386 Blistered Steel (one quality only) . - Homogeneous Metal, .... Puddled Steel, Bessemer's (do.) Homogeneous Metal, . Puddled Steel 111,400 ... 90,647 89.724 71 486 62 769 IRON PLATES. Yorkshire, IRON BARS. ...62,231 56,715 ...64,795 56,655 Durham (one quality only] Staffordshire, Lancashire, Swedi e h ... 60,110 53,775 ...48,232 47,855 Lanarkshire, IRON STRAP?, &c. ...56,805 49,564 55 87S 53.420 Cut out of large forged crank. . 47,582 41,758 The strength ol each quality is the mean of at least four experiments, and sometimes of eight. 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, 11; 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. Ill Pliable Bodies and Fluids. IY. . 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 lino 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 the 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. Motion of a Pair of Points. In fig. 172, let A l B, 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, Fig. 172. from the length and direc- Fig- 1 . 174. tion represented by A l 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, fig. 173, a pair of lines, AB 1? AB 2 , equal and parallel to A l Bj, A 2 B 2 , of fig. 172 ; then A represents one of the pair of points whose relative motion is under consideration, and Bj, BJJ, represent the two successive positions of the other point B relatively to A j and the line B7B 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, BAj, BAo, equal and parallel to A^, A 2 B,, of fig. 172; then Aj, A 2 , represent the two successive positions of A relatively to B; and the line A l A M equal and parallel to Bj Ba of fig. 173, but pointing in the contrary direction, represents the motion of A relatively to B. COMPONENT AND RESULTANT MOTIONS TIME. 381 350. Fixed Point and Hloring 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 nnd 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 0, such as OX, OY, OZ. Through A and B draw straight lines AC, BD, parallel to the plane of O Y and Z, and cutting the axis O X in C and D. Then CD is said to be the com- ponent of the motion of the second point relatively to O, along or in tlie direction o/*the axis O X ; and by a similar process are found the components of the motion AB along O Y and O 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 be the angle made by the direction of the motion A B with O X, CTD = AB cos *. / The relations between resultant and component motions are f 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 86164-09 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 bo 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, thero 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 minute. per hour. i = 1-46 =88 = 5280- 0-6818 = i 60 = 3600 o-oiiSd^ = 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, O Y, O Z, fig. 176, be axes traversing it in fixed directions. Let A 1 and Bj be the positions of A and B relatively Fig ' 176 * 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 l A 2 and Bj B 2 are the respective motions of A and B relatively to O. Complete the parallelogram A t B : b A 2 ; then because A a 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 2 is the motion of B relatively to A. Then in the triangle B, b B 2 , B! b = Aj A 3 is the motion of A relatively to O, b B 2 is the motion of B relatively to A, Bj B, 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, the motion of the third relatively to the first is the resultant of the motion of tlie third relatively to the second, and of the motion of the second relatively to the first; the word "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 of each point deter- mined relatively to that which precedes it in the series, and if the relative motion of the last point and the first point be also determined, then will those motions be represented by tlie sides of a closed polygon. Let O be the first point, A, B, C, \ / 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 % -^=n> + ^; (2.) VARIATION OF VELOCITY DEVIATION. 387 and the distance described is = V + ^ ............................ (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 1? A s 2 , described in two equal intervals of time, each equal to At, 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 362. Varied Kate 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 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 " \ 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 t, let A! and A 2 be the positions of the moving point. Then. the arc A x A 2 = v A t ; and chord Fig. 179. the chord A, A 2 = v A t arc The velocities at A) and A 2 are represented by the equal lines 388 PRINCIPLES OF CINEMATICS. = v, touching the circle at A l and A 2 respec- tively. From A 2 draw A s v equal and parallel to A, V,, and join VT-y. Then the velocity A g Y a may be considered as compounded of ~A~v and v V a ; so that v V 3 is the deviation of the motion dur- ing the interval &t', and because the isosceles triangles A 2 vV 8 , C A! A 3 , are similar : -=. _ ATV 2 -A^ 2 _ tf -A* chord CA r ^c ' 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. 0* Now the factor remains unaltered by that diminution ; and the 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 Delation. 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 : 366. The Rates of Variation of the Component Velocities of a point parallel to three rectangular axes, are represented as follows : COMPARISON OF VARIED MOTIONS. 389 ^f. y. d l^. n\ dt de 3 dt*>~ " w and if a rectangular parallelepiped be constructed, of which the edges represent these quantities, its diagonal, wkose length is will represent the resultant rate of variation, already given in another form in equation 1 of Article 365. 67. 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 IL 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 j 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 change of position of a rigid body, relatively to a fixed centre, there is a line 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 1? and after the turning, A 2 , B 2 . Join A, A 2 , Bj B 2 , forming the isosceles trian- gles O A! A 2, O Bj 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 O E, then EAjAa, and E B, 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 Telocity of a turning body is the ratio of the angle :>f rotation, expressed in terms of radius, to the number of units cf 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- a *. a = 2 *-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 soeve*r, 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-Handed Rotation. The direction of rotation round a given axis is distinguished in an arbitrary manner into 392 PRINCIPLES OF CINEMATICS. right-Jianded and left-Jianded. 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 fiom the other end. 377. Relative Motion of a Pair of Points in a Rotating Body. Let O and A denote any two points in a rotating body ; and consider- ing O 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 traverang O is one of revolution, or translation in a circular path of the 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 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 POIXTS IX A KOTATING BODF. o93 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 tlie directional relation consists in tlie angle between their directions of motion being tlie same with that between their radii-vectores. Or symbolically : Let r lf r 2 , be the per- pendicular distances of A and B from the axis traversing O, and v l and v z their velocities ; then v 3 r a A A = j and VtV a = r^* v \ *i 3SO. 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 O, 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 TJ = 6 be the angle made by that line with A Y From Y 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 t (1.) From O let fall O B perpendicular to B A. Then ^ A O B = ^ Y A TJ = t ; and the right-angled triangles O B A and A U V are similar j so that AY : AU: : OA : OB = r cos t (2.) Now the entire velocity of B relatively to the axis O is ar cos 6 = u, , (3.) so that the component, along a given straight line in tlie 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 line. 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. Ttie 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 otlierwise, 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 Vi 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 w t ap p = ~~~> whence v i = ( L ) Let r, as before, be the radius- vector of any point in the body, and let * = <" (2.) denote its velocity of revolution, or velocity relatively to the axis, due to the rotation alone. Then the resultant velocity of that point is ' ~ 2 + r2 } ( 3 -) The inclination of the helix described by that point to the plane of rotation is given by the equation i = arc tan = arc tan ^ . ...C4.\ MOTION OF A RIGID BODY. 395 the tangent of that angle being the ratio of the pitch to the circum- ference of the circle described by the point relatively to the axis of rotation. 383. PROBLEM. To Find the Motion of a Rigid Body from the motions of Three of its Points. Let A, B, C, fig. 182, be three points in a rigid body, and at a given instant let them have mo- tions relatively to a point indepen- dent of the body, which motions are represented in velocity and direction by the three lines A Y n , B Y 6 , C Y,. It is required to find the motion of the entire rigid body relatively to the same fixed point. Through any point o, fig. 183, draw three lines oa,ob,oc, equal and parallel to the three lines AV a , WT b , C V,. Through a, b, and c, draw a plane a be, on which let fall a perpendicular o n 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 Fiff. 182. points in the translation. body; that is, it is a velocity of Fig. 183. From the points Y a , Y 6 , Y c , draw lines Y a TJ, Y 6 TJ 6 , Y e TJ e , equal and parallel to o n, but opposite in direction to it ; and join A U a , BU 6 , CU C , which will all be parallel to the same plane ; that is, to the plane a be. 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 ab 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 TJ,., will cut the other two planes in the same axis : the three revolving component velocities AU a ,BU 6 ,CU, 396 PRINCIPLES OP CINEMATICS. will be respectively proportional to the perpendicular distances, or radii-vectores, AD, Bl), OF, 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 Telocity represented by o n, constitutes a Jielical 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 the 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 0, 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 0, 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 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 , V 6 , V e , 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 O. 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 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 G 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 ' = c -CE; but CD : CTE : : siu^AOC : sin ^COT; and therefore sin^COT :sin^:AOC : : a :c; and, combining this proportion with that given in equation 1, we obtain the following proportional equation : sin^COT rsin^AOT .-si : : _a_ : b_ : c__ > ..... (2.) : : Oa : Ob : Oc } that is to say, the angular velocities of the component and resultant rotations are each proportional to the sine of the angle between the axes of the other two ; and the diagonal of the parallelogram O b c a repre- sents both the direction oft/te instantaneous axis and the angular velo- city about that axis. 393. Boiling Cones. All the lines which successively come into the position of instantaneous a.xis are situated in the surface of a cone described by the revolution of O T about C ; and all the positions of the instantaneous axis lie in the surface of a cone described by the revolution of OT about O A. 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 j 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 PK1NCIPLES 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. Tariation 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 A a the interval of time, , converges, as the length of the interval is indefinitely diminished ; being represented by da ~dt' and found by the operation of differentiation. 397. Change of the Axis of Rotation has been already 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 with 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 v 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 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 afc once the resultant angular velocity, and the direction of the instan- taneous axis. For example, let a t , a y , a,, 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 z towards x, and about z from x towards y, being considered as positive ; then a=V(i + o; + a3 (1.) is the resultant angular velocity, and cos = ; cos/3 = -*: cos / ; (2.) a a a 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 Volume. 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 offerees 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- The length between each pair of points in the cord is constant. II. That length is the shortest line which can be drawn between its extremities over the surfaces by which the cord is guided. MOTIONS OF FLEXIBLE CORDS. 409 401. motions Classed. 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 Gnided by Surfaces of Rcvolntiou. Let a COrd in some portions of its course be straight, and in others guided by the sur- iaces 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 upper signs being employed when the cord crosses, and the lower when it does not cross the line 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 -(COS./+COB/); (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 ; and v = 2ntt (6.) X 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 = wA (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 the^ow?, or rate of tfow, through A. When the particles of fluid move obliquely to A, let 4 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 6 (2.) When the velocity normal to A varies at different points, either from the variation of v, or of 6 9 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 OP FLUIDS OF CONSTANT DENSITY. 411 Q = [ udA. = ( v cos 6 c?A; ............... (3.) and if we now distinguish the mean normal velocity from the velocity at any particular point by the symbol u , we have, 404. Principle of Continuity. AXIOM. When the motion of 'a fluid of constant density is considered relatively to an enclosed space of invariable volume which is always filled with the fluid, the flow 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 self-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.) consequently, * = ; .............................. (2.) u A or, the normal velocities at a given instant at two fixed cross sections are inversely as the areas oftliese 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 OP CINEMATICS. nels 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. r 409. steady motion of a fluid relatively to a given space considered las fixed is that in which the velocity and direction of the motion of I 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 ?, , represent the co-ordinates of an individual particle ; then the three components of the velocity of that particle have the values d d* d? ... u = - : v = -j- - : w - /" ; (1.) a t ' dt' dt } v ' 416 PRINCIPLES OF CINEMATICS. and the three components of the rate of variation of its motion, as denned in Article 366, are 5?i _ d-u . cPv _ d-v d^ _ d-w ~d?~~dt' } d?~'dt''dt 2 ~ dt > ......... W the values of 7 , ~~TT> an( ^ 7. ? being taken from Article 413 for 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 If 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 of, 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 ittass. In the case of a fluid / lof varying density, the volume, which in an unit of time flows rthrough a given area A, with a normal velocity u, is still repre- sented, as for a fluid of constant density, by (i.) but the absolute quantity, or mass 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 maybe expressed, either in units of weight per unit of volume, or in arbitrary units suited to the particular case. Let j be the density; then the flow ofjnass may be thus expressed : 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 Y, 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 ft to ^ Then the mass of fluid which at the beginning of the interval occupied the volume Y, occupies at the end of the interval the volume ; and the difference of those volumes is the volume which flows through the surface bounding the space, outward if ? 2 is less than ft, inward if ? is greater then ft. Let t a ( be the length of the interval of time ; then the rate of flow of volume is expressed as follows : v(S-i) Q= 2'- tl < h > 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 of flow 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 \ 2i 418 PRINCIPLES OP 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, g 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 % 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 ]} s 2 , be the positions of two cross sections of the stream whose distance apart along the axis is s 2 s^, then the volume of the space between those cross sections is = f'*A.da (1.) J s i Let Q! be the rate of flow of volume through the first cross section ; Qo that through the second; u v u 2) the corresponding mean velo- cities normal to the respective cross sections ; g the mean density of the fluid in the space Y ; ? L 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 g 2 -Q,ei *,-*! converges as the distance s 2 ^ is 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 .A. STEADY MOTION PISTONS GENERAL EQUATIONS. 419 d-Au 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 that is to say, the flaw 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 ; / the density of the fluid in contact with it; Y the whole volume of fluid enclosed; e its mean density. Then equation 4 becomes the last expression being introduced because ? V = the mass en- closed, is constant. If the density is uniform, then S ' AM= W> ....................... (1 A -> 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 ^", then the flow of mass through that opening, A" u" e", is to be included in the sum- r ___ Lmation at the left side of equation 1. :ss> 423. General Differential Aquations. 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 e d y d z, and the flow of mass out at the second face is (u t + -j^- dx)dydz;ite resultant of which pair of flows is - * . d x d y d z. dx Taking the corresponding resultant for the other two pairs of faces, addino- the three quantities thus found together, observing that 'V = dxdydz, and dividing by that common factor, the equation 4 of Article 419, which expresses the principle of continuity, becomes the following : ~dx dy dz ~ d~t r which is the equation of continuity for a fluid of varying density. This equation may be otherwise expressed as follows : or dividing by e, dx dy dz \ dx dy dz 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 : dl which conditions apply to a fixed 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 . sinj sin t, = sin to = J (a\ 4- ajj + 2 i a 2 cosj) ' &, sinj ! a t cos j) ' J 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 variable velocity-ratio between a pair of parallel axes. In fig. 191, let C 1} C s , 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 Uj, U 2 , another pair of points, which touch each other at another instant of the motion; and let the four points, T,, fr, * NON-CIRCULAR WHEELS. 429 TV, TJ,, U 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 : . arcT 1 U 1 = arcT 3 U 2 ; " 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 t , r. be the radii vector -es of a pair of points which touch each other; ds^ ds. a a pair of elementary arcs of the cross sections T! Uj, T 2 TJ 2 , of the pitch surfaces, and c the line of centres or distance between the axes. Then r, + r a = cj \ d*>_ _d^ ........................ (2.) dr, dr t ' } 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 n on- 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 and unity. excentricity 1 430 THEORY OF MECHANISM. Ill 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 Maxwells paper on Rolling Curves, Trans. Roy. Soc. Edin., vol. xvi., and Professor Willis's work on Mechanism.) SECTION 2. Sliding Contact. 444. Skew-Bevel Wheels are employed to transmit an uniform velocity-ratio between two axes which are neither parallel nor Fig. 192. Fig. 194. 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 j 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. 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 : a l : a 2 : : hp : 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. From p let fall p m perpendicular to h e. Then divide the common perpendicular G K in the ratio given by the proportional equation h~e:e~m: ~mh : : GK : G H : ]O[ ; then the two segments thus found will be the least distances of the line of contact from the axes. 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 5 ; the second, by the rotation of the same* line about the axis C D with the radius vector H K = r f 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 J_ 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 (GIF + JTW 2 ). Then will Y be a point in the hyperbola. 445. Grooved 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 VVvV -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 the edges and 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 hyperboloidal 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 piano 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 tJiat the pitch sJwuld 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 c?, a num- ber less than n, so that n = m d, N = M d, then a = wN = Mw = Mmc?; 6 = M; c = m. ........ (2.) CASE 3. If N and n be prime to each other, 2F 434 THEORY OF MECHANISM. a = Nn; b = N ; 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 4 s 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. 435 motion to a wheel on the following axis. Such a wheel is called an idle wheel : 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, NU N 2 , &c N m _i; and let the numbers of teeth in the driven or following wheels be denoted by n's, each with the number of its axis affixed ; thus, ?^, n 3) &c n m . Then the ratio of the angular velocity a m of the m^ axis to the angular velocity a^ of the first axis is the product of the m 1 velocity-ratios of the succes- sive elementary combinations, viz. : a m _ N! . ]ST 2 . &c K ro _ t > a t ~ n a ' nz &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 ; 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, t, 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 toted 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 C minimum number of teeth (which may be called t) } 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 iricoii- T> veniently large, and prime, then instead of the exact ratio , some C ratio approximating to that ratio, and capable of resolution into con- venient factors, is to be found by the method of continued fractions. T> Should be greater than 6, the best number of elementary O combinations, m 1, will lie between log B - log C log B - log ~P~ Iog3 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. C So far as the resultant velocity-ratio is concerned, the order of the drivers N 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 Cj, C 2 , represent the axes of rotation of the two pieces; AI, A 2 , two portions of their respective surfaces; and T 1? Tz, a pair of points in those surfaces, which, at the instant under consideration, are in contact with each other. Let P 1 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, P,, 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 P, P 2 of the velocities of the points Pj, P 2 , are equal. Let i t , ij, be the angles which that line makes with the direc- tions of the axes respectively. Let i, a 2 , be the respective angular velocities of the moving pieces; then a t Cj P! * sin i t = a a C 2 P 2 sin i 2 ; consequently, Oy G! P, sin ij f i C 2 P 2 sin t 2 which is the principle stated above. When the line of action is perpendicular in direction to both axes, then sin ij = sin i% = 1 ; and equation 1 becomes a-) .(1 A.) WJien the axes are parallel, ij = i,. Let I be the point where the line of action cuts the plane of the two axes ; then the triangles P! Ci I, P ? C 2 I, are similar; so that equation 1 A is equivalent to the following : a* TC, 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 (2.) "WTien the line of action is perpendicular in direction to the axis of the rotating piece, sin i = 1 ; and ~ec- t / = a-rCj .............. (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 the 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 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 C 19 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 tJie 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 Carres. 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 T! in fig. 197), and the corresponding position of the pitch point I in the pitch line I B! of the wheel to which that tooth belongs, are so related, that the line I. Tj which joins them is normal to the outline of the tooth Aj T t 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 Bj, its tracing-point Tj 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 Tr^Tr;; (i.) a-nd this condition is fulfilled, if the outlines of the two teeth be traced by the motion of t/ie 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 th.e 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 fa^es 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 P t towards P s ; 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 flatter 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 = ^i T I C), and the radius of curvature of the curve traced by the point T (j), 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, f 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 GJ with the angular velocity a lf and the wheel 2 to rotate round C 2 as before with the angular velocity a 2 relatively to the line of centres (\ 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 388 ; that is, a rotation about the instantaneous axis I with the angular velocity a^ + a? Hence the velocity of sliding is that due to this rotation about I, with the radius IT = r' } that is to say, its value is r (a l + %); (1.) so that it is greater, the farther the point of contact is from the It is usual to make the arcs of ap 4:55. The Length of a Tooth that of the face and that of the wheel, the length of the flank depi of the face, on the arc of recess ; the length of the flank depends c face, on the arc of approach. Let <7 t be the arc of approach, the flank, ^ the length of the fac< Let T-J be the radius of curvature of curve, r the radius vector of the t angular velocity of the rolling cui dq m fl Tt LENGTH OF 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 and consequently a.) 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 f, 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 M C 2 , be the centres of two circular wheels v whose pitch circles are Bi, B 2 . Through the pitch point I draw the intended line of action P : P 2 , making the angle I P = 6 with the line of centres. From C w C 2 , draw ' 0, Pi = I O z sin 6, .(1.) Fig. 198. C 2 P 2 = I C 2 sin 0, perpendicular to P! P 2 , with which two perpendiculars as radii, describe circles (called base circles) D u D^ 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.J. 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 x I P 2 . That point will trace, on a plane rotating alon 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 x 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 Pj 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^ times the pitch ; and with this length of path and the value of which is usual in practice, viz., 75, the addendum is about $> 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 Q T> r = q'-=-=q-sin6; ..................... (1.) which reduces equation 3 of Article 455 to the following : This distance may also be expressed in terms of the extreme dis- tances of the point of contact from the pitch point. Let these be denoted by t v t 2 ; then (1 1 - + - For inside gearing, the difference of the reciprocals of the radii of the wheels is to be taken instead of their sum. The preceding formulae, which are exact for involute teeth, are approximately correct for all teeth, if 6 be taken to represent the mean value of the angle C I P between the line of centres and the line of action. 31 The usual value of 6 being 75^, sin 6 = -^ nearly. 459. The Addendum of involute Teeth, that is, their projection beyond the pitch circle, is found by considering, that for one of the wheels in fig. 198, such as the wheel 1, the real radius, or radius of the addendum circle, is the hypothenuse of a right-angled tri- angle, of which one side is the radius of the base circle C P, and the other is P I + the portion of the path of contact beyond I. Now Cl? = r l - sin 6 ; P I = r l . cos 6. Let t 2 be the portion of the path of contact above mentioned ( = q 2 sin 0), and d, the addendum of the wheel 1 ; then (r l + di) 2 = rl -sin 2 6 + fa cos 6 + tjf ; ............. (1.) and for the wheel 2. the suffixes 1 and 2 are to be interchanged. 31 1 The usual value of sin 6 is about , and that of cos 6 about -. 62 4 The same formulae apply to teeth of any figure, if 6 be taken to represent the extreme value of the angle C I P. 460. The Smallest Pinion with Involute Teeth of a given pitch p, has its size fixed by the consideration that the path of contact of the flanks of its teeth, which must not be less than p sin 0, cannot 444 THEORY OF MECHANISM. be greater than the distance along the line of action from the pitch point to the base circle, I P = r cos 6. Hence the least radius is r = ptan0; (1.) which, for 6 75 J, gives for the radius' r 3'867p, and for the circumference of the pitch circle, p x 3'867 x 2 * 24-3 p\ to which the next greater integer multiple of p is 25 pj and therefore twenty-five, as formerly stated, in Article 447, is the least number of involute teeth to be employed in a piDion. 461. Epicycloidal Teeth. For tracing the figures of teeth, the most convenient rolling curve is the circle. The path of contact which a point in its circumference traces is identical with the circle itself; the flanks of the teeth are internal, and their faces external epicycloids, for wheels; and both flanks and faces are cycloids for a rack. Wheels of the same pitch, with epicycloidal teeth traced by the same rolling circle, all work correctly with each other, whatsoever may be the numbers of their teeth ; and they are said to belong to the same set. For a pitch circle of twice the radius of the rolling or describing circle (as it is called), the internal epicycloid is a straight line, being in fact a diameter of the pitch circle ; so that the flanks of the teeth for such a pitch circle are planes radiating from the axis. For a smaller pitch circle, the flanks would be convex, and incurved or under-cut, which would be inconvenient ; therefore the smallest wheel of a set should have its pitch circle of twice the radius of the describing circle, so that the flanks may be either straight or concave. In fig. 199, let B be part of the pitch circle of a wheel, C C the line of centres, I the pitch-point, E- the internal, and E,' the equal external describing circles, so placed as to touch the pitch circle and each other at I; let DID' be the path of contact, consisting of the path of approach D I, and the path of re- cess ID'. In order that there may always be at least two pairs of teeth in action, each of those arcs should be equal to the pitch. The angle 6, on passing the line of centres, is 90; the least value of that angle is & = ^ C I D = ^ C' I D'. It appears from experience that the least value of & should be about therefore the arcs D I = I D' should each be one-sixth of a cir- EPICYCLOIDAL TEETH. 445 conference; therefore the circumference of the describing circle should be six times tlie pitch. It follows that the smallest pinion of a set, in which pinion the flanks are straight, should have twelve teeth, as has already been stated in Article 447. 462. The Addendum for Epicycloidal Teeth is found from the formula already given in Article 459, equation 1, by putting for 6 the angle C I D, and for t a the chord I D' = 2 r Q cos d, r being the radius of the rolling circle. Hence (n-Mi) 2 = 1 sin 2 * + (r x + 2 T^ -cos 2 4 .......... (1.) 3 1 For the usual value of 6, 60, sin 2 6 = -, and cos 8 & = - whence (2.) 462 A. The Sliding of Epicycloidal Teeih is deduced from equation 3 of Article 455, by observing, that the radius vector of the point of contact is and that the extreme values of q are the arcs of approach and recess (2.) whence we have r 2 r + i) ; ................ (3.) l r 2' which, for 6 = 60, has the value 463. Approximate Epicycloidal 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, be the radius of the pitch circle, r that of the rolling circle, e the radius of curvature required; then -^;^; ...... (1.) *! nz 2 T Q 3==n the sign + applying to an external epicycloid, that is, to the /ace 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 d\ that is to say, - r = 2r cos* The value to be assumed for & is its mean value, that is, 75 J; and cos = - nearly : r Q is nearly equal to the pitch, p- } and if n be the number of teeth in the wheel, 6 : n : : r : r,. Therefore, for the proportions approved of by Mr. Wil l is, equation 2 becomes ' =1 idhj' .................. < 3 -> + being used for the face, and - for the flank ; also r = - nearly .......................... (4.) Hence the following con- struction. In fig. 200, let B be part of the pitch circle, A the point where a tooth is to cross it. Set off AB = AC = |. Draw radii , ' 20A< of the pitch circle, D B, E C. JJraw X B, O G, making angles of 751 with those radii, in which take TRUNDLE DIMENSIONS OF TEETH. 447 Bound 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 tmndk, 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 3 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, thickness of tooth = pitch, 44:8 THEORY OF MECHANISM. /> width of space = pitch. The difference of 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. air. San sj'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-Avheels, 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 O 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 FJ of that cone thus employed will approach sufficiently near to a zone of the sphere described about O with the radius 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 kept always in the position of the generating line of a hyperboloi'dal surface similar to the pitch-surface (see Article 444, pages 430, 431). 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 and female, or external and internal; a short internal screw is called a nut; and when a screw is not otherwise specified, external is 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 ^.bout 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. 2o 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 f the advance takes place. Fig. 204 re- in presents a right-handed screw, and fig. i 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 TT 9n4 IT- OAK 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 Jielix, 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 n 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 aodal 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 tfce normal helix to the the normal n p a the circular pitch ; Then .. . PC = Pa' cotan * = p n cosec ^ = 2 sr r ' tan i Pa =p n 'seci=p e 'tem = - - - ; 2 v r ' sin i p n =p e - sm i=p a -cosi= - . 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. Hooke'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 *" F . 2 o6 ' 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 n 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 : BjBj, SLIDING OF SCREWS OLDHAll's COUPLING. 453 in fig. 208, for example, being the pitch circle of the wheel, and B-, 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 l9 r 2 , be the radii of their pitch cylin- ders, and ii, i% 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 so that u = Oj TI ' sn j = a 2 r' sn % ; ^ . .............. j = - : - ; 2 = - = - ; J r^ ' sin tj r z ' sin ij and v = cijTi' cos ?! + 2 r 2 cos i 2 = u (cotan ? a -f- cotan i%) ..... (2.) When the screws are contrary-handed, the difference instead of the sum of the terms in equation 2 is to be taken. 477. oidham's Coupling. A coupling is a mode of connecting a pair of shafts so that they shall rotate in E j>/ 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. C 1 , C 2 , are the axes of the two parallel shafts ; D v D 2 , two cross- heads, facing each other, fixed on the ends of the two shafts re- spectively; E x , Ej, a bar, sliding in a diametral groove in the face of 454 THEORY OP MECHANISM. D x ; E 2 , E 2 , 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 shafts 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 C x C 2 , as a diameter, twice for each turn of the shafts and cross ; the instan- taneous axis of rotation of the cross, at any instant, is at I, the point in the circle C x 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. JBeltSj 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 ; 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 l} r z , be the perpendiculars let fall from their axes on the centre line of the band, a^, a 3 , their angular velocities, and i^ 4 the angles which the centre line of the BAXDS 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 I T = r a a? sin ^ ; whence the angular velocity-ratio is .(1.) Oj r a sin ? 2 When the axes are parallel (which is almost always the case), ^ = ^ and ?=5 P.) 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 j for then sin ^ = sin i, = 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.) sin cos./ .(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 Policy 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. constant velocity-ratio. Fig. 212. In each of them, the length denoted by 456 THEORY OF MECHANISil. 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 Beit, connecting a pair of pulleys whose effective radii are C^ T 4 = r lt C 2 T 2 = r 2 , with parallel axes whose distance apart is C x C 2 = c, is given by formulae founded on equation 1 of Article 402, viz., L = s s + 2 r i. Each of the two equal straight parts of the belt is evidently of the length s = Jc 2 (?*! + r 2 ) 2 for a crossed belt ; ( s = ^/c* (r x r a ) 2 for an open belt ; r t being the greater radius, and r 2 the less. Let ^ be the arc to radius unity of the greater pulley, and ?' 3 that of the less pulley, with which the belt is in contact ; then for a crossed belt O fl + IT + 2 arc sin -J / ^ l = ^ a = ( I \ c and for an open belt, j, (2.) i,= ^ + 2 arc . sin TI ~ r *\ ; i- 2 = (*-2 arc sin r * ~ r * \, and the introduction of those values into equation 1 of Article 402 gives the following results : For a crossed belt, L = 2 ^/c 2 - (r x + r 2 ) 2 + (r l + r a )' (^ + 2 arc sin and for an open belt, j- (3.) (r l + r 2 ) + 2 (r x - r a ) ' arc sin ^ ^ 8 . j 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 2 is small compared with c, J (2.) a\ ~~ d C 3 , there is introduced a short interme- diate shaft C 2 , making equal angles with d and C 3 , connected with each 221 - of them by a Hooke's joint, and having both its own forks in the same plane. Let i be the angle of inclination of d an d C 2 , and also that of C 2 and C 3 . Let

The motion of the centre of the pulley is the same with that of a point in a rope wound on a barrel of the radius ^^. The use of 2i the contrivance is to obtain a slow motion of the pulley without using a small, and therefore a weak, barrel. 505. Compound Screws. (Fig. 225.) On the same axis let there be two screws S 4 S u and S 2 S 2 , of the respective pitches Fig. 225. p l and pz, p t being the greater, and let the screws in the first in- stance be both right-handed or both left-handed. Let N t and ET 2 be two nuts, fitted on the two screws respectively. When the compound screw rotates with the angular velocity a, the nuts ap- proach towards or recede from each other with the relative velocity, .(I.) 468 THEORY OF MECHANISM. being that due to a screw whose pitch is the difference of the two pitches of the compound screw. (See Article 382, equation 1.) The object of this contrivance is to obtain the slow advance due to a fine pitch, together with the strength of large threads. Fig. 226 represents a compound screw in which the two screws are contrary-handed, and the relative velocity of the nuts N M N 2 , is that due to the sum of the two pitches ; or, as these are usually equal, to double the pitch of each screw. This combination is used in coupling railway carriages. 506. i^ink Motion. Let C be the axis of the shaft of a steam engine, CT the crank, /the connected point (see Article 489) of the 3805^ forward eccentric (which is suited to move the slide valve when the engine moves s forwards), b the connected point of the backward eccen- tric (which is suited to move the slide valve when the engine is reversed), fF the 997. forward and b B the back- ward eccentric rods, F B a piece called the link, jointed to those two rods at F and B, S a slider, which is capable of being slid to and fixed at different positions in the link, and to which the slide valve rod is jointed. Let the arrow represent the direction of forward rotation of the shaft, and at the instant represented in the figure, let the piston be at one end of its stroke. Let L L be a line showing the position in which the crank arm of an eccentric should stand, in order that the middle of the stroke of the slide valve should be at the same instant with the extremity of the stroke of the piston. The angle ^L L Cf is the angular lead or advance of the forward eccentric, and the angle ^ L C b (usually equal to the former) the angular lead or advance of the backward eccentric. "When S is at F, the engine is in full forward gear, the motion of the slide valve being governed by the forward eccentric alone. The stroke or throw of the slide valve is 2 Cf, and its lead corre- sponds to the angle ^ L Cf. When S is at B, the engine is in full backward gear, the motion of the slide valve being governed by the backward eccentric alone. The stroke or throw of the slide valve is 2 C b (usually = 2 Cf), and its lead corresponds to the angle ^ L C b (usually = ^ L Cf). When S is at A, the engine is in mid gear, the velocity of the valve rod at each instant being a mean between those which it would receive from either eccentric separately. LINK MOTION PARALLEL MOTION. 469 The lead corresponds to 90, or a quarter of a revolution. The throw is nearly, though not exactly, = 2 C a, a being the middle of the straight line fb. To find exactly the motions of the slide valve for different posi- tions of the slider S, it is best to draw a diagram to a scale, repre- senting the positions of the eccentrics, rods, and link, for a series of angular positions of the crank (usually dividing a revolution into 24 equal angles) ; and the corresponding series of positions of S when fixed at various points in the link. Several examples of this process are given in Mr. D. K. Clark's treatise on Railway Machinery. A useful approximation to the motions of the valve, when the rods are long compared with the link, is got by dividing the line fb at s in the same proportion in which S divides F B, and con- sidering the motion of the valve as produced by the crank C 8 ; so that the throw is approximately 2 C s, and the lead approxi- mately ^1 L C s. 507. Parallel motions are jointed combinations of linkwork, designed to guide the motion of a reciprocating piece, such as the piston rod of a steam engine, either exactly or approximately in a straight line, in order to avoid the friction which attends the use of straight guides. Four kinds of parallel motion will now be described : I. An Exact Parallel motion, believed to have been first proposed by Mr. Scott Russell, is represented in fig. 228. The same parts of the mechanism are marked with the same letters, and different successive positions are indicated by numerals afiixed. The lever CT turns about the fixed centre C, and carries, jointed to its other_end^the bar or link P T Q, in which PT = ~T~Q = C~T. The point Q is jointed to a slider which slides in guides along the straight line C Q. From Q draw Q D J- CQ, cutting CT Fig. 228. produced in D; then by Article 488, D is the instantaneous axis of the link ; and because D P || C Q, the motion of P, which is -L D P, is always JL C Q ; that is to say, the point P moves in the straight line Pj C P 3 , -i- C Q. In a steam engine, a pair of the combinations here shown are used, one at each side of the cylinder ; and the pair of bars P Q are jointed at their extremities P to the head of the piston rod. The distance through which Q slides at each single stroke of the piston, of the length P x P 3 = S, is given by the equation 470 .THEORY OF MECHANISM. - (1) and is small compared with, the length of stroke of the piston. II. An Approximate Parallel Motion, somewhat resembling the preceding, is obtained by guiding the link P Q entirely by means of oscillating levers, instead of by a lever and a slide. To find the length and the position of the axis of one of those levers, c t, select any convenient point, t, in the link P Q, and lay down on a drawing the extreme and middle positions, t 1} t 2 , t s , of that point, corre- sponding to the extreme and middle positions of the link P Q. The centre c of a circle traversing those three points will be the required axis of the lever, and c t will be its length; and if the link P Q is guided by two such levers, the extreme and middle positions of P will be in one straight line, and the other positions of that point very nearly in one straight line. III. Watt's Approximate Parallel Motion. In fig. 229, let C T, c t, be a pair of levers, connected by a link T t, and oscillating about Fig. 229. the axes C, c, between the positions marked 1 and 3. Let the middle positions of the levers, C T 2 , c t 2 , be parallel to each other. It is required to find a point P in the link T t, such, that its middle position P.,, and its extreme positions P,, P 3 , shall be in the same WATT'S PARALLEL MOTIONS. 471 straight line perpendicular to C T 2 , c t 2 , and so to place the axes C, c, on the lines C T 2 , c t 2 , that the path of P, between the positions Pj, P 2 , P 3 , shall be as near as possible to a straight line. The axes C, c, are to be so placed, that the middle M of the versed sine Y T 2 , and the middle m of the versed sine v t 2 , of the respective arcs whose equal chords Tj T 3 = ^ t s represent the stroke, may each be in the line of stroke M m. Then Tj and T 3 will be as far to one side of that line as T 2 is to the other, and t and t s will be as far to the latter side of the same line as 4 is to the former; consequently, the two extreme positions of the link, TX t l) T 3 ts, are parallel to each other, and inclined to M m at the same angle in one direction that the middle position of the link T 2 3 is inclined to that line in the other direction; and the three intersections P! P 2 P 3 , are at the same point on the link. The position of the point P on the link is found by the following proportional equation : Yt : PT : P7 : tv i cm :C"M The positions of the point P in the link, intermediate between its middle and extreme positions, are near enough to a straight line for practical purposes. When there are given, the axes C, c, the line of stroke P x P 2 Pg, the length of stroke > 1 P 3 = S, and the per- pendicular distance M m between the middle positions of the two levers, the following equations serve to compute the lengths of the levers and link ; Yersed sines, ~TY = -JL- - tv = -4L ; 8CM' 8cm' Levers, C~T = CM + 5Z; ti = c~m~+^-' \ (3.) Link, lY. Watt's Parallel Motion modified by having the guided point P in the prolongation of the link T t beyond its connected points, instead of between those points, is represented by fig. 230. In this case, the centres of the two levers are at the same side of the link, instead of at opposite sides, the shorter lever being the farther from the guided point P j and the equations 2 and 3 are modified as follows : 472 THEORY OP MECHANISM. Segments of the link, Versed sines, Levers, Link, T* : PT : P t : :t^TV : TY \Tv : : C M cm -.cm: CM. . S2 Q2 ^ T >5 .(4.) cm (5.) This parallel motion is used in some marine engines, in a position inverted with respect to that in the figure, P being the upper, and t the lower end of the link. Fig. 231. When Watt's parallel motion (III.) is applied to steam engines with beams, it is more usual to guide the air pump rod than the piston rod directly by means of the point P. The head of the piston rod is guided by being con- nected with that point by means of a parallelogram of bars, shown in fig. 231. c is the axis of motion of the beam of the engine, c t A one arm of that beam, C T a lever called the radius bar or bridle rod, T t a link called the back link. C T, c t, and T t, form the com- bination already described (IIL)> and shown in fig. 229 ; and the point P, found as already shown, is guided in a verticaUine, almost exactly straight. The total length of the beam arm, c A, is fixed by the proportion Fig. 230. PARALLELOGRAM EPICYCLIC TRAINS. 473 T~t :Tt : :CTt : CA; (6.) that is, t A is very nearly a third proportional to C T and c t. Draw A B || T t, and c P B intersecting it; then from the proportion 6 it follows that AB = T t. A B is the main link, by the lower end of which, B, the head of the piston rod is guided. B T = and || t A is the parallel bar, by which the main and back links are connected. P moves sensibly in a straight line; = = - is a constant ratio; c r ct therefore B moves sensibly in a straight line parallel to that in which P moves. A parallelogram analogous to A B T t may also be combined with the parallel motion IV. 508. Epicyclic Trains. The term epicyclic train is used by Mr. Willis to denote a train of wheels carried by an arm, and having certain rotations relatively to that arm, which itself rotates. The arm may either be driven by the wheels, or assist in driving them. The comparative motions of the wheels and of the arm relatively to each other and to the frame, and the aggregate paths traced by points in the wheels, are determined by the principles of the com- position of rotations, already explained in Articles 385 to 395. y- PAIT T. PRINCIPLES OF BYSTAMICS. 519. Division of the Subject. The science of Bynamics, which treats of the relations between the motions of bodies and the forces acting amongst them, may be divided into two primary divisions, according as it has reference to balanced forces and uniform motions, or to unbalanced forces and varying motions. A secondary mode of dividing the subject is founded on the distinction between ques- tions respecting the motions of masses which are either insensibly small, or which, being of sensible magnitude, have motions of trans- lation only, questions respecting the motions of rigid bodies and rigidly connected systems which rotate, and questions respecting the motions of pliable bodies and of fluids. The dynamics of fluids has received the special name of hydrodynamics. It is a branch of mechanics so extensive in its applications, and depending so much in its details upon special experiments, as to require a separate work for its full exposition ; nevertheless, in the present treatise its fundamental principles will be set forth in their proper place. The dynamical principles of the motions of rotating rigid bodies, of pliable bodies, and of fluids, are deduced from those of the motions of rigid bodies having motions of simple translation, by conceiving the bodies under consideration to be divided into indefinitely small molecules or particles, so that the laws of the motion of each mole- cule shall differ from those of a body having a motion of simple translation to an extent less than any given difference. It is to such indefinitely small molecules that the term physical point, already mentioned in Article 7, is applied. Hence it appears that the laws of the relations between the motions of a so-called physical point, and the forces acting on it, are the foundation of the science of dynamics ; and the same laws are applicable to a rigid body in which every point moves in the same manner at the same instant ; that is to say, which has a motion of translation, as defined in Article 369. 476 PRINCIPLES OF DYNAMICS. The subjects to which the principles of dynamics relate will therefore be classed in the following manner : I. Uniform Motion. II. Varied Translation of Points and Kigid Bodies. III. Kotations of Rigid Bodies. IY. Motions of Pliable Bodies. Y. Motions of Fluids. CHAPTER I. ON UNIFORM MOTION UNDER BALANCED FORCES. 510. First taw of motion. A body under the action of no force, or of balanced forces, is either at rest, or moves uniformly. (Uniform motion has been defined in Article 354.) Such is the first law of motion as usually stated ; but in that statement is implied something more than the literal meaning 01 the words ; for it is understood, that the rest or motion of the body to which the law refers, is its rest or motion relatively to another body which is also under the action of no force, or of balanced forces. Unless this implied condition be fulfilled, the law is not true. Therefore the complete and explicit statement of the first law of motion is as follows : If a pair of bodies be each under the action of no force, or of balanced forces, the motion of each of those bodies relatively to the other is either none or uniform. The first law of motion has been learned by experience and observation : not directly, for the circumstances supposed in it never occur ; but indirectly, from the fact that its consequences, when it is taken in conjunction with other laws, are in accordance with all the phenomena of the motions of bodies. The first law of motion may be regarded as a consequence of the definitions of force and of balance (Articles 12, 13) : at the same time it is to be observed, that the framing of those definitions has been guided by experimental knowledge. 511. .Effort; Resistance; Lateral Force. Let F denote a force applied to a moving point, and 6 the angle made by the direction of that force with the direction of the motion of the point. Then, by the principles of Article 57, the force F may be resolved into two rectangular components, one along, and the other across, the direction of motion of the point, viz. : UNIFORM MOTION WORK POTENTIAL ENERGY. 477 The direct force, F cos 6. The lateral force, F sin e. A direct force is further distinguished, according as it acts with or against the motion of the point (that is, according as 6 is acute or obtuse), by the name of effort, or of resistance, as the case may be. Hence each force applied to a moving point may be thus decom- posed : Effort, P = F cos 6, if 6 is acute ; } Resistance, K = F cos ( 0) if 6 is obtuse ; > (1.) Lateral force, Q = F sin 6. 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 slwll balance each otfier, and that t/ie 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 X I ?ds (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 EF, perpendicular to the zero line OX 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, / P d s, 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 the 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 B, * As)=p A &s=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, (1.) CASE 2. For tlie 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-pA S: =2-RA S ....................... (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 the 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 HI ; the efforts acting between the points of the system and external bodies by Po, and the resistances by Rj. Then by case 2, but by the condition of rigidity, 2-?! As = 0j 2-R 1 &s = (); therefore, 2'P 2 AS=2-It, A s ...................... (3.) 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 OF DYNAMICS. velocity, or any number proportional to the velocity, of the point to which the external force F is applied, and 6 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 If is s-Fvcos0 = Oj (1.) attention being paid to the principle, that cos 6 is < ^ o , . V when 6 is < a ^ e [ . 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 H is applied; then 2'Pv = 2'Kw (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 IY. 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, , /3, 7, with three rectangular axes ; then A S ' COS a, 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 ', ', y', so that its rectangular components are F cos *', F cos /3', F cos */'. 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. 461 F * A S ' COS COS '; F A s cos ft cos ft; F A * cos y cos y'; and the sum of those three quantities of energy is the whole energy exerted. Now it is well known, that cos a, cos ' H- cos /3 cos # -f- cos y cos y' = cos 0, 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 formulas 1 gives for the whole energy exerted, simply F A s cos 6, as in former examples; and similar remarks apply to work per- formed. 482 CHAPTER IL 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 eartKs surface. This fact has been learned by experiment j 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. Whewell's 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 compooent 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 9 v^ w^ the rectangular components of the velocity of the centre of gravity of that system of bodies ; then u 2 m = 2 mu; v 2 m = 2 m v j w Q ' 2 m = 2 m w. COROLLARY. The resultant momentum of a system of bodies rela- tively to t/Leir common centre of gravity is nothing ; that is to say, /2 \ " 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 dt an interval of time during which it acts unbalanced, ~Fdtis 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 6, 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 dt is accelerating impulse if 6 is acute ; \ K dt = F cos (x - 6) dt is retarding impulse if 6 is obtuse ; > (1.) Q d t = F sin 6 d t is deflecting impulse. j 528. Relations between Impulse, Energy, and Work. If V be the mean velocity of a moving body during the interval dt of the action of the unbalanced force F, then ds = vdt is the distance described by that body ; and according as 6 is acute or obtuse, there is either energy exerted on the body by the accelerating impulse to the amount Pc?s = Fv cos 6 ' dt; (1.) or work performed by the body against the retarding impulse to the amount R,ds = Fv cos (*-') * 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 me- chanics, the motion of any part of the earth's surface may be treated as uniform without sensible error in practice. The units of mass and of force may be so adapted to each other as to make change of momentum equal to the impulse producing it. (See Articles 531, 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, m being the mass of the body, v its velocity, and T the radius of curvature of its path, it follows from Articles 361 and 362 that the rate of direct variation of its momentum is ^y 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> m. r Equating these respectively to the direct and lateral impulse per unit of time, exerted by an unbalanced force F, making an angle * with the direction of the body's motion, we find the two following equations : Por -R = Fcos* = m--r- = w-j-g j ............. (!) Q = Fsin* = ............................ (2.) The radius of curvature r is in the direction of the deviating force Q. Second metlwd. As in Article 366, let the velocity of the body be resolved into three rectangular components, -^, - ^, -^- so that tit (.it dtt the three component rates of variation of its momentum are drx Also let the unbalanced force F, making the angles , /6, y, with the axes of co-ordinates, and its impulse per unit of time, be resolved into three components, F,, F f , F,. Then, we obtain , = 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 A denote the latitude of the place, h its elevation above the mean level of the sea, g 1 = 32-1695 feet, or 9-8051 metres, per second; being the value of g for * = 45 3 and h = 0, and K = 20900000 feet, or 6370000 metres, nearly, 486 PRINCIPLES OF DYNAMICS. being the earth's mean radius; then g = ffl -(1-0 -00284 cos 2 A) l-~ ............ (1.) For latitudes exceeding 45, it is to be borne in mind that cos 2 x 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 g = 32-2 feet, or 9'81 metres, 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 whence W 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 standard avoirdupois pound. The weight of an unit of mass, in any given locality, has for its value, in absolute units of force, the co-efficient g. When the unit of weight is employed as the unit of force, instead of the absolute unit, the corresponding unit of mass becomes g times the unit just mentioned: that is to say, in British measures, the mass of 32'2 bs. ; or in French measures, the mass of 9 '81 kilogrammes. 533. The motion of a Falling Body, under the unbalanced action f its own weight, a sensibly uniform force, is a case of the uni- brmly varied velocity described in Article 361. In the equations of that Article, for the rate of variation of velocity a, is to be sub- stituted the co-efficient g, mentioned in the last Article. Then if VQ be the velocity of the body at the beginning of an interval of time t } its velocity at the end of that time is . FALLING BODY UNEESISTED PROJECTILE. 87 V = V + gtj ~ (1.) the mean velocity during that time is PO + , 9 * , 9 \ ~2~ = v + T' W and the vertical height fallen through is * = *! + (3.) The preceding equations give the final velocity of the body, and the height fallen through, each in terms of the initial velocity and the time. To obtain the height in terms of the initial and final velo- cities, or vice versa, equation 2 is to be multiplied by v v = g t, the acceleration, and compared with equation 3 ; giving the follow- ing results : When the body falls from a state of rest, v is to be made = ; so that the following equations are obtained : The height h in the last equation is called the lieight or fall due to the velocity v; and that velocity is called tlie velocity due to the height or fall h. Should the body be at first projected vertically upwards, the initial velocity v is to be made negative. To find the height to which it will rise before reversing its motion and beginning to fall, v is to be made = in the last of the equations 4; then '" being a rise equal to the fall due to the initial velocity V& \fy^ * ^T 534. An Unresisted Projectile, or a projectile to whose motion ( (jy! there is no sensible resistance, has a motion compounded of the \ vertical motion of a falling body, and of the horizontal motion due to the horizontal component of its velocity of projection. In fig. 232, let O represent the point from which the projectile is originally 488 PRINCIPLES OP DYNAMICS. projected in the direction O A, making the angle X A = 6 with a horizontal line O X in the same vertical plane with O A. Let horizontal distances parallel to O X be denoted by x, and verti- cal ordinates. parallel to Z by z, positive upwards, and negative downwards. In the equations of vertical motion, the symbol h of Fig. 232. the equations of Article 533 is to be replaced by z, because of h and z being measured in opposite directions. Let v be the velocity of projection. Then at the instant of pro- jection, the components of that velocity are, horizontal, - r - = v cos &' } vertical, = V Q it, t (.It sin and after the lapse of a given time t, those components have become dx -7-7- = V Q cos 8 = constant; dz . . sin I g t. Hence the co-ordinates of the body at the end of the time t are horizontal, x = v cos t t; (2.) vertical, and because t z = v sin 6 t ~- : M . those co-ordinates are thus related, f g - 2 ' ........ (3.) v ' V Q COS z = x - tan 6 2 v* cos 2 6 an equation which shows the path B C of the projectile to be a parabola with a vertical axis, touching O A in 0. 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 from the last form of which is obtained the equation MOTION ALONG AN INCLINED PATH. 489 (5 -> which, being compared with equation 4 of Article 033, 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 tfte 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 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, a 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 6, which is a resistance if 6 is positive, and an effort if 6 is negative; therefore Tt = ~9^ 6 (1.) 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 & 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 fhe body is the following : vl v 2 = 2 g sin 6 %' 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 z r , positive upwards. But z' sin 6 = 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, dv . .- . 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 CENTRIFUGAL FORCE. 491 Consequently the two weights move according to the same law with " a falling body, but slower in the ratio of f to 1. NQ W 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 -f- g its mass, & 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 ._ : ^~ ~~~ '" ( ' 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 eacn 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 I 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 winch 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 C 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 B C weight _ g r . radius AB ~~ centrifugal force v' ' ' where r = A B. Let n be the number of turns per second of the pendulum; then v = 2 -3- n r ; and therefore, making B G = h, h - 9.?- - g - - , - , . -T , , = (in the latitude of London) 4 ^2 n 2 0-8154 foot 9-7848 inches 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 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 x n, where n is the number of revolutions per second, so that W .(3.) from which equation the height of a revolving pendulum might be deduced with the same result as in the last Article. ||N l. Rectangular Components of Deviating Force. First Demon- > stratum. 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 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 ) /i \ AB = y =- r sin 6, J ' Fig. 234. be the rectangular co-ordinates of the revolving body at any in- stant. Let Q x , Q y , be the components of the deviating force, parallel to X and O Y respectively. Then from the obvious proportion between the magnitudes of those components, Q:Q*:Q, :: r : x : y, (2.) combined with the equation 2 of Article 540, follow the values of those components, Q, = - ; Q, = - .(3.) 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 & = a t, and the values of tfce 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 PRINCIPLES OP DYNAMICS. dx . dy /KX . = ar sin at, -^- ar cos at, ........... (o.) d t at the velocity parallel to each co-ordinate being proportional to the other. The components of the variation of motion are d* x ~ = a?r cos a t = a" 1 x ; * 72 a V -~ = a? r sin at a*y cL t" W which being multiplied by the mass , reproduce the components ^ c/ of the deviating force as before given in equation 3. --W)42. straight Oscillation is the motion performed by a body ~ V^Which moves to and fro in a straight line, alternately to one side atnd 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 || O Y at the distance x = r cos a t = r cos 6 ..................... (2.) from O : the velocity of B being at each instant equal to the par- allel component of the velocity of A ; that is to say, dx = arsmat = arsintf; .............. (3.) a t 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 ,(4.) where r is the semi-amplitude of the oscillation, E = O G, Q is the corresponding greatest magnitude of the force urging the body towards 0, 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 Q= being similar to equation 3 of Article 540. b Revolution in a circle may be regarded as compounded of two dilations of equal amplitude, in directions at right angles to each her. 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 ............... <> 5 being a constant. Then the rectangular com- ponents of that force are ;...(2.) 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 viz.: (3.) according to equation 4 of Article 542. Hence let a^ = O E = O G be the semi-amplitude of the former straight oscillation, and y t = 496 PRINCIPLES OF DYNAMICS. O F = H that of the latter ; then at any instant the co-ordinates of the body will be x = Xi cos bt; y 2A sin 1 1; (4.) which equations being respectively divided by x l and y it 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 h y The components of the velocity of the body at any instant are dx _ (6.) rfrl C\f TTif\ f\f TIT- 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 ^ D C A = W EG CA' acting along C A, and balanced by the tension of the rod or cord, and Fig. 236. 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. : ivx- where r denotes the semi-amplitude, and Q the maximum value of W ==^. But if the length of the pendulum, CA, be made = I, C A we have Q AB r = max. -=- = 7, nearly ; W C A 6 whence, approximately, for small arcs of oscillation, 1 = 2*- A/-; n V g' and .(i.) which being compared with equation 2 of Article 539, shows, that tJie length of a simple oscillating pendulum, making a given number of small double oscillations in a second, is sensibly equal to the Jieight of a revolving pendulum, making the 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. Cydoidal 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 PETNCIPLES 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 = C D = the length of each of the semi- cycloids OF, C G. Then as the pendulum C A swings between the p. 237 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 earth's 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 height due to its velocity; that is to say, it is represented by ~T -27" The product m v 2 , the double of the actual energy of a body, was formerly called its vis-viva. Actual energy, being the product of a weight into a height, is expressed, like potential energy and work, iu foot-pounds (Article -5 13, 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, dx dy dz T-, -7-7, -j-, parallel to three rectangular axes; then the quantities of actual energy due to those three components respectively are W da? W dy 1 "W d 2~jj' df> 2j"d' } 2^'d^' But the square of the resultant velocity is the sum of the squares of its three components, or ~~ d t 2 dt 2 dt 2 ' therefore the actual energy of the body is simply the sum of the actual energies due to the rectangular components of its velocity. _J149. Energy of Varied Motion. THEOREM I. A deviating force '''produces no change in a bodys 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 the potential energy exerted. This theorem is a consequence of the second law of motion, deduced as follows : > 500 PRINCIPLES OF DYNAMICS. Let m = W -J- g, be the mass of a moving body acted upon by an effort P, and a resistance R, 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 lt and its velocity at the end of that interval v 2 , the increase of the body's momentum is W -(*,-,) = (P - R) A * .................. GO Because of the uniformity of the acceleration of the body, its mean velocity is ^ -, and the distance traversed by it is a V l + V 2 A 8 = ' A t. a 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* the change in the square of the velocity in that distance ; then Wvdv - or if s 15 s 2 , denote the two extremities of a finite portion of the body's path, ........... (3,) THEOREM III. TJie diminution of actual energy produced by an the resistance. This is a consequence of the second law of motion, demonstrated by considering R, to be greater than P in the equa- tions of the preceding theorem ; so that equation 1 becomes ( Vl Va ) (R P) &t; (4.) equation 2 becomes TRANSFOHMATIOX OP ENERGY. 501 and equation 3 and 3 A become 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 motion, the potential energy exerted, added to the energy restored, is equal to the energy stored added to tJie work 'per- formed. This principle, expressed in the form of a differential equation, is as follows : which includes equations 3 and 6 of Article 549 ; and in the form of an integral equation, f *,-- -flld. = Q ............ (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 518). 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, X 502 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 : p__W dv W vdv ~ g dt g ds 1 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 & with the direction of the force, the product F cos 6 'ds represents the energy exerted, if 6 is acute, or the work performed, if 6 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 (^ 6) = R) ? the component of the force along the direction of motion, into d s, the motion ; or as the product of F, the entire force, into cos & 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 surfaca, as the case may be ; then s ............................. (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 v z , be the velocities of the body, and z v 2 , 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 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. 503 for in that case, z denotes the elevation of the body above a given level, F = - W (because it tends to diminish z), and therefore .(3.) 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 Iff, %& 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 1. If a body whose weight is W be at a height z 1 above the ground, and be moving with the velocity VL in any direction, its initial total energy relatively to the ground is W (*' + ^)i O-) V s of which W Zi 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 ? 2 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. #04 PRINCIPLES 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 t the greatest value of the moving force, so that -- l is the value 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 *rd,= a;, Jo 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. : v being the maximum velocity. At any intermediate point, the total energy, partly actual and partly potential, is still the same, being where, as before, a=2irn; 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 OP BODIES ANGULAR MOMENTUM. 505 SECTION 4. Varied Translation of a System of Bodies. 558. Conservation of Jloinentum. THEOREM. TJiC mutual OCtWUS of a system of bodies cannot change their resultant momentum. (Re- sultant momentum has been defined 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 bodies, 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 of the centre of gravity of a system of bodies are wholly produced by forces exerted by bodies extenwl 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. The term angular momentum was introduced by Mr. Hay ward. 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 Flat is angular impulse. Angular impulses are compounded and resolved like the moments of couples. 562. Relations of Angular Impulse and Angular Momentum. THEOREM. The variation, in a given time, of the angular momentum of a body, is equal to the angular impulse producing tJiat 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 J?ldt. At the same time, the body m acquires a variation of momentum in the direction of the force applied to it, of the amount so that relatively to the fixed point, the variation of the body's angular momentum is rnldv = ldt;.... a ................... (1.) being equal to the angular impulse, and having the same axis. Q. E. D. 563. Conservation of Angular momentum. THEOREM. The 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 ACTUAL ENERGY OF A SYSTEM. 507 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 a,xis 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. ^/L-7 564. Actual Energy of a System of Bodies. THEOREM. Theoctuafv ty energy of a system of bodies relatively to a point external to the system, is the sum oftJie 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 whole 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 , W Q , 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 the centre of gravity rela- tively to the external point, and that which it has relatively to t/te centre of gravity. The component velocities of the first part are and those of the second part u UQ = u'; v v = v'j w W Q = w'', so that the components of the whole motion of the body may be represented by u = U Q + u' ; v = v + i/' } w = w -{- w 1 . Then the actual energy of the system relatively to the external point is ^ 2 m [(u + u')* + (V Q + v') 2 -f- (WQ -f- vTf j ; 508 PRINCIPLES OF DYNAMICS. which, being developed, and common factors removed outside the sign of summation, gives i (ul -}- v% + w*) 2 m + U Q 2 m u' + v 2 m v' + w Q ' 2 m w' + i 2 m (u' 2 + iP + 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 m w' = ; 2'mv' = Qj 2 mw' = ; so that the above expression for the actual energy of the system becomes simply i (ul + vl + wl) ' sm + ls -m(u' 2 + v' a + 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 oftlie 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 to be necessary to the permanent existence of the universe as actually constituted. 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 disappears as visible motion, and appears as vibration and heat. If the bodies are equal, similar, and perfectly elastic, that fraction is nothing. Let mi, m a , be the masses of the two bodies, and ^ u a , 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 ............................ (2.) The internal energy of the system of two bodies is >\ (MI - *4)) 2 ,nh(u t - u7S. Examples of moments 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 mass of the body; that headed I gives the moment of inertia; that headed e|J, the square of the radius of gyration. The mass of an unit of volume is in each case denoted by w. BODY. Axis. W la e 2 o I. Sphere of radius r, II. Spheroid of revolution polar semi-axis a, equa- 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*W 15 2r 2 3 4-rwabc III. Ellipsoid semi-axes, a, b c 15 5 ~5 2(r 5 r' 5 ) IV. Spherical shell external radius r, internal /,.... V. Spherical shell, insensibly thin radius r, thick- ness dr . . 3 4*w(r 3 r' 3 ) 15 3 2irwabc 2ai0a(r 2 r^ 4vwardr 2trwabc 4*ioardr Swdbc 4wdbc 4wabc 15 5(r'-r' 3 ) 2r 2 VI. Circular cylinder length 3 *war* 3 r 2 2 VII. Elliptic cylinder length 2a, transverse semi- axes b, c,... VIII. Hollow circular cylinder lengttrlJ'a, external ra- dius r, internal /, IX. Hollow circular cylinder, insensibly thin length 2 a, radius r, thickness dr, X. Circular cylinder length 2 4 2 r 2 r 2 a 2 4^3 r^+r 12 a 2 XI. Elliptic cylinder length 2a, transverse semi-axes 6, c, 6 XII. Hollow circular cylinder- length 2o, external ra- 6 XIII. Hollow circular cylinder, insensibly thin radius r, thickness dr, XIV. Rectangular prism di- mensions 2a, 26, 2c, XV. Rhombic prism length 2a, diagonals 26, 2c,.... XVI. Rhombic prism, as above, 6 v +4a 2 (r 2 -r' 2 )| 3 4 ' 3 ~3 3 3 6 c 2 a 2 3 MOMENTS OF INERTIA. 519 579. Moments of Inertia found br 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, ? 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 B 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 BA~B = 26, <3A~C = 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 EG in D, and join AT), in which take r Q = AE = | AD = _ J 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- tenuse of a right-angled triangle whose sides are, n parallel to y y and parallel to z, and are given by the equations .(1.) and the radius of gyration J of the transformed body will be the hypotenuse of a new right-angled triangle whose sides are m n and n ; that is to say/ p = m 'f + n *? (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 (i6., case 1). ' r I "581. The Centre of Percussion of a body, for a given axis, is a ' Jpoint 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 Fig. 239. * ne whole mass, concentrated partly at C, and partly at the centre of percus- 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 : : BO : 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 (2.) where r Q = 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 Q -=- 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 = $ The last equation may also be expressed in the form GB = BC r = ...................... (4.) e!; ........................ (5.) 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 (S z *) S (S x y) 2 . 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 af - z yj + S (z x' - x zj + S (a; y'-yxj', x, y, z, x 1 , y f , 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" + x'y" z + x" y z' x y" z' x" y z x 1 y zj; 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 yf, S z\\ 526 PRINCIPLES OP 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 Xi : cos y x l : cos z x l zx-Sxy' (Szl Sy 2 )Szx+Sxy'Syz 1 (6.) "(S xl S z*} S x y + Sy z Szx Similar equations, substituting y l and z l successively for x l} 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 orf, &c. : 1 = S R 2 as before ; ' (7.) and hence it appears, that the functions of the six integrals S #*, &c., denoted by A, B, and C, in the equations 5, are ieotropic; that is, are the same in magnitude for all directions of the rectangular axes of x, y, and z. 585. Ellipsoid of inertia. Let the principal axes of a body, tra- versing a given point, be now taken for axes of co-ordinates; and the moments of inertia about them, called the principal moments of inertia, being given, and denoted by I 3 , I 2 , I 3 , let it be required to determine the moment of inertia, I, about any axis traversing the same point, and making with the principal axes the angle , /3, y. Let co-ordinates along this new axis be denoted by x } and along the principal axes by x {) y l} z ]} as before. It has already been shown that S x 2 = cos 2 * S xl + cos 2 ft S y\ + cos 2 v S zf,...(l.) and that S* 2 ; ........................ (2.) and from these equations the following is easily deduced : I = Ij -cos 2 a + I 2 ' cos 2 /3 + I 3 - cos 2 y ......... (3.) ELLIPSOID OP INERTIA. 527 Let a, b, GJ be the three semi-axes of an ellipsoid, and s its semi~ diameter in any direction which makes the angles , /3, y, with those semi-axes. Then it is well known that 1 COS 2 O, COS 3 /3 COS 2 y . . . = ~~ ~~ ~~ ~*' 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 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 centred ellipsoid. If I,, I 2 , I 3 , be ranged in their order of magnitude, it is evident that the greatest of them, Ij, 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, L, 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 Ij. 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 Ij are equal. In this case equation 3 becomes I = Ii cos 3 + L 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 528 PRINCIPLES OF DYNAMICS. gravity of the body, whose weight is "W, and that I 0l , I 02 , 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 , ft, 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 y = I 02 -f W rl sin V; j- (8.) I. = I 03 + W rl sin 2 y and there are at the same time moments of deviation represented by S yz = W 7-5 ' cos /3 cos y; Szo; = "W r 2 , cos y cos *j ) /q \ S # ?/ = "W rl ' cos a cos ; J 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 iUoment 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 A A A A S y z = cos y x t cos z x^ S x\ -f- cos y y l cos z y^ S y\ -J- cos y Zi ' cos zz l S z\, (1.) and similar equations for S z x, and S x y, mutatis mutandis. Sub- stituting for S xl, &c., their values, S E, 8 Ii, &c., and observing that A A A A A A cos y Xi cos z Xj_ + cos y y^ cos z y^-{- cos y z^ cos z z l = 0, those equations become _ A A A A b y z = lj cos y x cos z x^ 1 2 * cos y y l cos #y, A A 1 3 -cos yzi "cos zzij (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 K x = J ]Il cos 3 x x i + If, cos 2 x y l + If cos 2 x z l (Ii cos 2 x x l + 1 2 cos* x y^ + I 3 cos 8 x z^f\ ; = J [ll cos 8 x Xi + 1| cos 2 x 2/j + II cos 2 xz l - 1 2 } This equation, expressed in terms of the axes of the ellipsoid of inertia, becomes as follows : , , A . A , A )s 2 a;c 1 , cos 2 ;*;^ , cos^xz 1 _ - | Z~^ I "J * " but the positive part of this expression is well known to be the value of - 2 , where n represents the normal let fall from the centre 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 s*n in which it is to be observed, that Js?-ri* 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^n 3 : n = tan 6, 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 OP DYNAMICS. angular velocity of rotation. Then the velocity of any particle W, whose radius vector is r = J if + z*, is and the angular momentum of that particle, relatively to the axis of rotation, is being the product of Us 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 z' } the distance of that line from the particle is x, and the angular momentum of the particle relatively to that line is W W . , . , ar x a x J y* -f- z , y j and this may be resolved into two components ; one relatively to the axis of y, W azx and the other relatively to the axis of z, Waxy ~~ 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, _ (S y- + S ) 9 and relatively to the transverse axes, .(i.) 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 wvtk 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, y. Resolve the angular velocity a about that axis into three components about the axes of inertia a cos ; a cos /8 ; a cos y ; then the angular momenta due to those three components are respectively a T a - a T - Ij cos * ; - I 2 cos ft ', - 1 3 cos y ; y y y the resultant angular momentum is A '^/JIfcos 8 * + If cos a /3 + Icos 2 y}; ............ (2.) and the axis of angular momentum makes with the axes of inertia the angles whose cosines are a Ii cos at, aI 2 cos & al s cos y 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 : 532 PRINCIPLES OF DYNAMICS. I, COS *> I 2 cos /3 I 3 COS y 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 oftJie normal let fall from the centre of the ellipsoid of inertia upon a plane touching that ellipsoid at the end of that diameter v&ich is the axis of rotation; and the angular momentum itself is directly as the 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 6 in Article 586, whose value is given by the equation n I 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 R be the axis of rotation, whether permanent or instanta- neous, O B, being the semidiameter of the ellipsoid of inertia. Let B, T be part of a plane touching the ellipsoid at B, and O N a normal upon that plane from 0. Then the moment of inertia, the moment of deviation, and their resultant, the total moment, have the following proportions : I :K + K* BN 1 ,(8.) "OBr 'OB 2 -ON 'OB-ON' the direction of the axis of angular momentum is O ET j and its amount is proportional to = ON 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* WaV 57 : ~^~ j ................... ;*- (L) 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 ike 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 ->~ ' ' 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 , a cos /3 a cos y j then the quantities of actual energy due to those three component rotations are a 2 Ij cos 2 a, o? I 2 cos 2 /3 a~ I 3 cos 2 y ,. which being added together, reproduce the amount of actual energy given in formula 2; showing that the actual energy of rotation about a given axis is the sum of the actual energies due to the component* of that rotation about the three axes of inertia. yvv/590. Free Rotation is that of a body turning about its centre of gravity under no force. The principles of the conservation of angular momentum (Article 563), and of the conservation of in- ternal energy (Article 565), oeing applied to free rotation, show that it is governed by the following laws ; 534: PRINCIPLES OF DYNAMICS. I. The direction oftJie axis of angular momentum is fixed. II. The angular momentum, is constant. III. The actual energy is constant. The first law shows, that the direction of the normal O N, fig. 241, is fixed; and consequently, that unless that normal coincides with the axis of rotation O R, which takes place for axes of inertia only, the axis of rotation is not a fixed direction, and is therefore an instantaneous axis only (Articles 385 to 393). Hence the axes of inertia are sometimes called "permanent axes of rotation." The second and third laws are expressed by the following equa- tions : A = ^/(P + K) = constant; ..(i.) constant. To find how these laws regulate the changes of direction of the instantaneous axis, eliminate the angular velocity as follows : # A 2 = I 2 + K 2 _ If cos 2 a + Ij cos 2 /3 + I^cos'y 2 E I I x cos 2 a + I 2 cos 2 /3 + I 3 cos' 2 y = constant (2.) Now, referring to fig. 241, and to equation 8 of Article 588, it appears that I 2 + K 2 oc 1 -i- OR 2 O~N 2 , and that I oc 1 -- ; and analogous equations for a t \ ) <7M y and<7M,; .....(3.) while the equations 2 become 540 PRINCIPLES OP DYNAMICS. 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

....................... (2.) min. B C = 2 e ; ) 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 0, if Q 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 m C G suspended by a point C not in the centre of gravity G, and rotating about a vertical axis C 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 . x of inertia about G E, and I 2 its moment of inertia about G C, and f lt &, the corresponding radii of gyration. Let the angle X C G = j let C G = r ; and let the weight of the pendulum be \V. 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 Gis a a /T _, Wa (A! 1 2 ) cos a, sin a. = ( & J) cos * sin ; 9 9 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 a, being the height of C above G ; that is to say, 'Ty cos sin ec- making for the entire deviating couple Wa 2 ~~ (ff* & + r%) cos at, sin * ; 9 548 PRINCIPLES OP DYNAMICS. and this couple has to be supplied by means of the weight of the pendulum acting with the leverage r sin #- f that is, it must be equal to W r sin . Dividing by this quantity, we find !. .................. (1.) and putting for a 2 its value, 4 ^ 2 T 2 , where T is the number of turns per second, this leads to the equation li being the height of the equivalent simple revolving pendulum, as given in Article 539, equation 2. "When fr, the radius of gyration about C G, is insensibly small compared with r lt 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 ea = ?i> 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 g (1.) 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 .(2.) 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, -V = J(2gl' versin i) = 2 sin ^ - jjl; ......... (3.) and the corresponding angular momentum of the combined mass is B YZ ; which, being equated to the angular momentum of the ball c/ before the collision (1), gives the equation giving for the velocity, momentum, and actual energy of the ball, respectively, BYJ -T7 6 v B Y I bv 3 g g ft ' 2 Q (5.) BY 2 The energy of the combined mass after the collision being , 1 and less than that of the ball before the collision in the proportion of b r' 2 : B Z 2 , shows, that an amount of energy denoted by 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 PEINCIPLES 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 1 " I P d tj 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 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. : bw , B w and , . .^ ; 6 + B 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 instead of B, and instead of 6, bw Bto 2 (b + B) .(12.) 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 Robins, celebrated for his investigations on gunnery. 52 CHAPTER IV. MOTIONS OP PLIABLE BODIES. 608. ivatnre 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 OP ISOCHRONISiL 553 analogous to the oscillations already treated of in Articles 542 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 the displacements at each instant. This is the CONDITION OP 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 -r- g be the mass of a particle, 3 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 9 a numerical con- stant, expressed as a square for reasons which will presently appear; then the condition of isochronism is expressed as follows : 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 and the period of a double oscillation by j. (2.) i n a 554 PRINCIPLES OP 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,, r or x h x, F,, F, 3,, 3, respectively, where F x represents the maximum force, corresponding to S ]? 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 3 = &j cos a (t 1 ) ; =- ., sin a (*-*). 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 3 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 the mass of the particle, and directly as the ratio of the restoring force to the dis- placement. 610. Vibrations of a Mass held by a XJght 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, 5, 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, n'Ebh 3 ri"?!, A, x', x Q , x' 0) t , and <' being arbitrary constants, having values depending on the circumstances of each particular problem. Theso constants have the following meanings : |j and n l are the maximum semi-amplitudes of vibration. - - and -^ - , are the periodic times of a complete oscillation. 2 v a 2*c x and x' are the distances (for the longitudinal and transverse vibrations respectively) between a pair of planes in which the particles are in the same phase of vibration at the same instant ; such as the planes A and E in figs. 244, 245. Nodal planes are planes in which the particles have no displace- x x' ment, x x , or x #' , being an odd multiple of or . Their x x' distance apart is or - (A, C, and E, in the figures). '2t *2t Ventral planes are those of maximum displacement, x x , or x x' x o/o, being a multiple of -- or - (B and D in the figures). They '- ' lie midway between the nodal planes. The following quantities for isochronous vibrations are deduced from equations 16 and 17 : For longitudinal vibrations, velocity of ) d % 2-a a particle, j ~dt~ A a? z^ . ^ direct strain, -7 = - t sin - (a; ar ) * cos ax x x For transverse vibrations, 2o 562 PRINCIPLES OP DYNAMICS. velocity of | dn 2-rc t 2?r. , . . d* 2* 2* aJ ' (19>> Distortion, -j- = - 7 >?r sin - (x - x' ) -cos 7- (t - tf ). ax A A A Vibrations may exist in which the displacements, strains, velocities, and forces, are the resultants of combinations of isochronous vibra- tions, having any number of different sets of arbitrary constants, and having only in common the co-efficients a and c. The results of the preceding investigation, so far as they relate to longitudinal vibrations, are applicable to fluids as well as to solids. Transverse vibrations are impossible in fluids, because in them there is no transverse elasticity. 614. Waves of Vibration consist in the transmission of a vibra- tory state from particle to particle through a body. Let X denote the direction in which the vibratory state is transmitted, being, as in the last Article and its figures, an axis of vibration, or line per- pendicular to a series of surfaces of simultaneous and equal displace- ment, which surfaces do not now remain stationary, but advance from particle to particle with a velocity called the velocity of trans- mission or of propagation. With respect to wave motion in general, it has already been explained in Article 416, that the condition of motion of any particle, whose distance from the origin is x, is expressed by a function of at x } where t is the time elapsed from a given instant, and a the velocity of transmission. Applying this to the displacements in longitudinal and transverse vibrations re- spectively; we find the equations = all expended. The mean value of P is -. The distance through which it is overcome in compressing the pile is the compression due ~p T to its maximum value, viz., -=TO> where E is the modulus of elasti- Ji* fe 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 halfiis length. T>2 T Then the work performed in compressing the pile is ; and the work performed in driving it deeper is R x y 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 2ESWA E 2 S 2 x 2 ) ESa; J^iles are in general driven till E- 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 ciuematical 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 those of gases. 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 exei^d 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. IY. Motions of Gases affected by Friction. V. Mutual Impulses of Fluid Masses and Solid Surfaces. GENERAL EQUATIONS OF HYDRODYNAMICS. 5G7 SECTION 1. Motions of Liquids without 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 dxdy dz 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 Then the forces by which the molecule is acted an unit of volume. upon are along x, ~- . d x d y d z along y, -~ 'dxdy dz; ' along z, ) dxdydz. 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 \)jdxdydz, 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 : dp 1 d 2 % 1 ( du , du , du I d 2 * 1 ( dv , dv , dv du _ dz dv dp dw . dw . dw . dw Combining with those three equations of motion the equation of continuity, viz. : 568 PRINCIPLES OP DYNAMICS. dudvdw 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 ... ................ ( } as in Article 413. 19. Dynamic Head. The quotient is what is called the height, or heady due to the pressure; that is, the height of a column of the liquid, of the uniform specific gravity ^ , 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, e z is the weight of a column of liquid per unit of base extending down from that plane to a particle under consideration ; p e 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 is the height or head due to tJiat difference of intensity, being what will be termed the dynamic Itead. When z is measured positively upwards from a datum horizontal plane, its sign is to be changed 5 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"% 1 / du . du . du , du dy g dt 1 g ( dt dx dy dz dh_l d^_l(dw du> . ^dw dw :ri = ' g ' lTt*~ '~g \~d~i~* U dx dy W dz 621. JLaw 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 Y be the velocity of a given particle. Its value, in terms of its rectangular components, is given by the equation 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 dt dfdt dtdt dt* ^(dh^.dh^dvdh^df^ _ \dx dt dy dt dz dt) This principle might otherwise be stated thus : In steady motion^ the sum of the fieight due to the velocity of a particle and of its dynamic head is constant, or symbolically Y 2 + h = constant ........ ...* ........... (3.) ~* J This equation applies to the particles which successively occupy the same fixed point, as well as to each individual particle. 622. The Toial Energy of a particle of a moving liquid without friction is expressed by multiplying the expression in equation 3 of the last Article by the weight of the particle W, thus : (i.) W Y 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" Y s the diminution of \Y h } may be increased by an equal amount, ^9 and vice versa; so that tlie 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 taken as the fixed point to which the motions of all the particles are referred, the following equation is fulfilled : ("V 2 \ r c c f~V \ + 7t) or / / / ( ^ H7n e 'dxdydz = constant... (2.) A g J J J J \A g / 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, V) 7ij 2j = = constant (1.) e 624. A Surface of Equal Pressure is characterized by an analo- gous equation, P h z = = constant; (1.) and all surfaces of equal pressure fulfil the differential equation, dh = dz; (2.) which, for steady motion, becomes - } . ..(3.) expressing that the variations of actual energy are those due to the variations of level simply. ^>25. 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 LAYERS. 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- z 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 z up to the elevation O Z l = z l above a given datum, and discharging the liquid from _- an orifice A at the smaller elevation O Z = z . The liquid moves exactly or nearly in plane layers at the upper surface 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 and at the orifice A -.- , . 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, = constant = - + + ....... (4.) 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 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 Q VnA 0a /i\ *>a T- = I- > ........................ (-J-) A 2 A 3 whence the dynamic head relatively to the datum O is obtained by the equation and thence the pressure by the formula P* = e(?h-Zt) ............................ (7.) a large vessel discharges liquid through a small orifice, the A 2 atio ^ is often so small a fraction, that it may be neglected in AI quations 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' the elevation of the lower, and z" that of the upper edge of the orifice, so that f 2 " AQ = C / 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 t wero 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" = , 9 i*^ v/zi *o=3 - VM -' (4.) r 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- jfcions are related to the direction, velocity, curvature, and variation iof velocity of the fluid motion in the following manner : In fig. 249, let H t H 1? H 2 H 2 , represent a pair of such surfaces, very near each other; their normal distant apart being d n, measured forwards from E^ 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 H * F . 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, V A V V* - = dh; oT--t-h = constant; (1.) 9 9 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 OP DYNAMICS. .(2.) - dr 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 Q Fig. 250, 2 TT b r r where r is the radius of the cylindrical surface K , 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 O. Let h Q be the dynamic head at K ; then at any other cylindrical sur- face of the radius O ft = r, we have the dynamic head, Let ^ be the limit towards which the dynamic head approxi- mates as the distance from the axis is indefinitely increased \ then (3.) h = h, - 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 + n- = h t = constant. ................. (1.) 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* odr which is equal and opposite to the deviating force t dh; that is to say, rf * - *- (9\ dr ~ gr" " W But by the condition of freedom in equation 1, we have - = g - h), which being substituted in equation 2, gives whence 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 o the dynamic head, at the inner boundary E- of the vortex ; we have for the general equations amongst the dynamic heads and velocities at all points, A 7 7i 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 the velocity is inversely as the 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 6 the con- stant inclination of the current to the radius vector ; then the com- ponent of the motion whose velocity is v cos 6, 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 ; so that if r be the radius of the outer boundary of the vor- tex, where the velocity is t? , 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, .(2.) which being integrated, with the understanding that the dynamic head is to be reckoned relatively to the axis of the vortex, gives _ _ --' - from which it appears, that in a rotating vortex, the dynamic Jiead at any point is the height due to the velocity, and the energy of any particle is half 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 R 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. 577 the sum of those heights for the surface of junction is equal to the maximum /lead 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, .(I.) To illustrate this geometrically, let a combined vortex revolve about a vertical axis, O Z Z l} fig. 251, the upper surface of the liquid being free, and represented in section by DBOBD. 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 z = h (2.) Then for the forced vortex, .(3.) so that B O B is a paraboloid of revolution with its vertex at 0. Make AC = 2 AB = 2 z ; this will represent h l} the maximum dynamic head ; and for the free vortex, 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 Zj 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 th#t due to its depth below C C ; that of any particle in the forced vortex is thai due to its height above A A ; and B, where those velocities are equal, is midway between C C and A A. PRINCIPLES OP 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 I 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 0, 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 g ire. 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 that is, ~W /y 2 CA:BC::W: -Bl, 9 rr\ _ 9 9 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 tJie 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 sxcept 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 may be deduced from the above proposition. For a brief sketch of that theory, see Addendum, tv. 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 2j and in their integration, it has to be observed that j, 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- s 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, 7 f p dp , ,- . "=/of + ^ < L > 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, the exponent y having the values therein stated, of which the most important is 1 -408 for air. This gives for the value of the integral in equation 1, h z f'dP y .P. /ox - LT ~*-i e' ( } in which, for air, 580 PRINCIPLES OF DYNAMICS. Let T = T + 461 -2Fahr. ...................... (5.) denote the absolute temperature of the gas, T being its temperature on the ordinary Fahrenheit's scale ; and let T O== 493-2 Fahr. ........................ (G.) be the absolute temperature of melting ice. Then for gases seri sibly perfect, P __ P T . /7 \ ~*-^." from which we have the following value of the integral in terms of the temperature : *-* = /' *2V*"?5 ............... (8.) J o P y-1 p 9 T/ so that it is simply proportional to the absolute temperature. It is known by the science of thermodynamics, that the above expression is equivalent to Jc'r; ................................ (9.) where c' is the specific heat of the gas at constant pressure, and J is 41 Joule's equivalent" or the height from which a given weight must fall, in order to produce by friction as much heat as will raise the temperature of an equal weight of water by one degree. For Fahrenheit's scale, * J = 772 feet ........................... (10.) *D The following are the values of and c' for certain gases and Po vapours : ^ feet. c '. e Air, ................................... 26,214 ...... 0-238 Oxygen, .............................. 23,710 Hydrogen, ........................... 378,819 Steam, ................................ 42,141* ...... 0-480 JEther 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>i45 ...... 0-217 * This is an ideal result, arrived at not by direct experiment, but by calculation from the chemical composition of steam. FLOW 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 r oc p*~ l oc 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 = A v p = constant, (1.) we have to consider that, by the equations of the last Article, we have i JL i p oc pr oc T 5 - 1 oc (h-zy~ l (2.) the exponents having, for air, the values \ = 0-71 ;-L=2451 (3.) Hence the equation of continuity, in terms of the pressure, of the absolute temperature, and of the dynamic head respectively, takes the following forms : i^ i^ Qp'' = A vp*' = constant; (4.) i _i Qr 3 - 1 = Avi*- } = constant; (5.) j_ j__ Q(h-zy~* = Av(h-2)~'- 1 = constant; (6.) 637. Flow of Gas from an Orifice. Let the pressure of a gas within a receiver be p lt and without, p a ; 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 r u ? v and those of the issuing jet 7 a , & The latter are 582 PRINCIPLES OF DYNAMICS. not the same with those of the still gas outside, for reasons to be stated afterwards. Then y-l 1 and by equation 8 of Article 635, and equation 3 of Article 621, we have for the height due to the velocity of outflow, y_ ,PO . TI (2.) .Po . T! - - 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 go TO giving the following results : & Q = v A ess -1 For 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, n. 637 A. maximum Flow of Gas. When is indefinitely dimin- fh ished, the velocity of outflow given by equation 2 of Article 637 increases towards the limit A /J Zygpo^ } V t( y -i), oTo h MAXIMUM FLOW OF GAS. 583 being greater than the velocity of sound in the ratio A/ : 1, whose value for air is 2-21, giving for the limiting velocity of flow of air 2,413 feet per second x A/-^- f .--(2.) The flow of weight, however, as given by equation 4 of Article 637, does not continuously increase as is indefinitely diminished, Pi but reaches a maximum for the value s-fcfcy corresponding to (3.) The values of these ratios for air are to = 0-527 ; & = 0-6345; -* = 0-8306 (4) Pi ei T > and the corresponding velocity of flow is / 2 being less than the velocity of sound in the ratio A / - -r-y : 1, whose value for air is 0-912 j giving for the velocity of flow of air corresponding to the greatest flow of weight through a given orifice from a receiver where the pressure and temperature are given, v = 997 feet per second x A/ ~ ............ (6.) It is often convenient to express the flow of weight in the following manner : (7.) in which is what is called the rediiced velocity, being the velo- Pi city of a current of a density equal to that of the gas in the receiver, whose flow of weight would be equal to that of the actual current. 584 PRINCIPLES OF DYNAMICS. The maximum reduced velocity corresponds to the maximum flow; and its value is jL v pj / 2 \ 2 (y-0 - = velocity of sound x [ =-^1 ...(8.} Pi \V + V whose value for air is velocity of sound x 0-579 = 632 feet per sec. x A/ -... (9. The investigations in this and the preceding Article are substan- tially the same with those originally communicated to the Royal Society in May, 1856, by Dr. Joule and Dr. Thomson; and the results differ by small quantities arising mainly from those gentle- men having taken y 1 41, instead of 1 408. Messrs. Joule and Thomson tested the theoretical result as to the maximum reduced velocity given in equation 9, by experiments on the flow of air through orifices in plates of copper of 0-029, 0-053, and 0-084 of an inch in diameter, at the temperature of 57 Fahrenheit, for which = , and the calculated maximum T O 4yo*j reduced velocity is 647 feet per second. The maximum reduced velocity found by experiment was 550 feet per second, or 0-84 of that found by theory; but in calculating the velocity from the experiments, the actual area of the orifice was employed ; so that the difference probably arises from contraction. The corresponding value of the ratio p z : pi, as found by experiment, was 0-375 instead of 0-527; a difference produced by friction. SECTION 3. Motions of Liquids with Friction. 638. General Laws of Fluid Friction. It is known by experi- ment, that between a fluid, and a solid surface over which it glides, there is exerted a resistance to their relative motion which is pro- portional to their surface of contact, and to the density of the fluid, and is approximately proportional to the square of the velocity of the relative motion ; that is, the resistance is approximately pro- portional to tJie weight of a prism of the fluid, whose base is the surface of contact, and its height the Jieight due to the relative velocity. Let S be the surface of contact, v the velocity, g the weight of an unit of volume of the fluid, and f& factor called the co-efficient of friction; then is the amount of the friction at the surface FLUID FRICTION. 585 The co-efficient /is not absolutely constant at different velocities. The mode of calculation employed in practice, where the velocity is one of the unknown quantities to be determined, is to find an approximate value of the velocity from the mean value of f; then to compute the value of f corresponding to that approximate velocity, and use it to compute the velocity more exactly. The following are some of the values of the co-efficients of friction, according to different authorities, for streams of WATER, gliding over various surfaces; v being the mean velocity of the stream, in feet per second : Iron pipes (Darcy). Let d = diameter of pipe in feet; then, /..^(..J or for velocities that are not very small, Iron pipes, value off for first approximation, 0-0064 Beds of rivers (Weisbach), . . .f = a -f- -; a = 0*0074. b = 0*00023 foot, Beds of rivers, value of f for first ) , approximation, ........................... J * " A collection of numerous formulae for fluid friction, proposed by different authors, together with tables of the results of the best formulae, is contained in Mr. Neville's work on hydraulics. The formulae of many 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 formulae given above, on the authority of Darcy, for iron pipes, are based on his experiments as recorded in his treatise du Mouvement de VEau dans les Tuyaux. 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 PRINCIPLES 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, 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 V is the greatest velocity in feet per second : V ~ "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 z. 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 K, 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 ,. s v* i = y=/-T- ' 5-. 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- Fi - 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 PEIXCIPLES OP DYNAMICS. the stream is d z, and the acceleration d v t 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 v d v - s d x v 2 . . In applying this differential equation to the solution of any parti- cular problem, for v is to be put Q ^- 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 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 77 7 dp vdv , sdl if ,, . -dh=-d a ~^-- +/_ ._ ......... (1.) The ratio -=-= is called the virtual or hydraulic declivity, being the d I 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, --j. 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 -=- 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 -yy for one ii i FLOW IN PIPES SUDDEN ENLARGEMENT. 589 particular value of ?, are required for the solution, which, compre- hends the determination of Q. 643. Resistance of .ii on th pieces. 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 h be the loss of head ; then - = i .......................... - 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 + 0-22G sin'i ......... (2.) 644. The Resistance of Carres 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, and for a rectangular pipe, *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 1 (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 number 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 in due to the velocity of the narrow stream relatively to the wide stream, that is, to the difference v^ (1 -- J, 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 646. The General Problem of the flow of a stream with friction is thus expressed : Let 7^ + -, and Jh + ^-, be the total heads ^9 Ay at the beginning and end of the stream respectively ; then the loss of total head is represented by 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 G38, 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 j^ 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. la j 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 FRICTION OP GASES PRESSURE OF A JET. 591 head j 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 ^urface, is deflected so as to glide Fig. 256. Fig. 254. 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 i\* Let v, then, be the velocity of the par- ticles of fluid, Q the volume discharged per second, p the density, and /8 the angle by which the direction of motion is deflected; then pQv 9 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 ft ; then the base of that triangle, whose value is 2 v sin , J represents the change of velocity undergone by each particle of fluid ; so that the change of momentum per second is ('> 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, F : F x : F y :: 2 sin ~ : I -cos ft -.sin ft ........... (2.) a whence the components have the values, F. = 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 Uloring 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 has 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 ..(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 Fxtt = f*,- (1 _ c03/8) . ............... (2.) y 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 a 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 9 4- (i\ uf + 2 u (Vi u) cos ft -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.) V. = tgi(l-cos,3); F x = ^(l-cos/3). j If ft = 90, as in fig. 255, the maximum energy transmitted is e Q $ * 4 g, or half of the original actual energy of the fluid. If ft = 180, as in fig. 256, the maximum energy transmitted is Q v? -j- 2 #, or tfAe wAofe of the original actual energy of the fluid, which, after the collision, is left at rest. CASE 2. When the 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 Vt 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 EG; 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 B C to E G; 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 x . This force is equal to the change in one second of the component momentum of the fluid in the direction B D. Let = ^ 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 l cos et. PRESSURE OP A VORTEX ON A WHEEL. Let w = IVC be the velocity of the jet relatively to the surface ; then v? =. y? -f- -y? %uv l cos ............ ...... (6.) Let y supplement of ^L 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 T>, of the new velocity of the jet is u + w cos y ; and the change of momentum in that direction in one second is F, = i-Z (vjCOSa U W'COSy) ............... (7.) which gives for the energy transferred per second, H? X U = - Ufa COSet U W 'COS 7) ............ (8.) y Let v 2 be the resultant velocity of the fluid after the collision; then ^ = ^ 2 + ^+ Sun-cosy .................. (9.) and it is easily verified that 650. Pressure ef 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 lt be the radii of the cylindrical surfaces at which a forced vortex begins and ends ; V Q , v ly the velocities of the 596 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 (i.) 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 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 ^ 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 6 given by the following equation : Let u = - - - be the velocity 2l 7T 7*1 of the radial current at the ends of the vanes now in question; then b being the depth of the wheel in a direction parallel to the axis. Fig. 257 represents part of 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, C, 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 & above described. The guide VORTEX-WHEEL CENTRIFUGAL PUMP FAN. 597 blades are moveable about pivots at A, 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 E- = r , fig. 258. 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 = 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, .p.) 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, = 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 j 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+ ,jTd = 0, 1, 2, 3. &= 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 OP FLUIDS PROJECTILES SHIPS. 593 ! ' ; B = *A (i.) 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-f- >Jl)~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, k for a cylinder, moving sideways, about 0-77 , for a sphere, ................ 0*51; for a thin hollow hemisphere moving with the hollow foremost, .............................. about 2*0; for a prism with wedge-formed ends = k for same prism with flat ends, x (1 cos /3), where ft = | uiigle of wedge (doubtful). The following are results deduced from Mr. Bashforth's experi- ments on elongated projectiles at velocities of from 1,300 to 1,500 feet per second (see Proceedings of the Royal Society, Feb., 1868): where A is in square feet, and v in feet per second; and c has the following values, according to the shape of the head of the projectile, hemispherical, 0-0000245; oval and pointed, from 0-0000191 to 0-0000204. 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. The opinion that the resistance to the motion of ships which are not very bluff consists almost wholly of friction, has been confirmed by subsequent experiments. The co-efficient of the friction between water and the bottom of an iron ship is nearly the same with that of water in iron pipes. The friction varies nearly as the square of the velocity (TyO PRINCIPLES OF DYNAMICS. of rubbing between the water and the ship's bottom. That velocity is different at different points of the ship's bottom, and bears to the speed of the ship a ratio at each point depending on the ship's figure and on the position of the point in question. The average velocity of rubbing exceeds the speed of the ship; and the excess is the greater the bluffer her shape. Thus, though a long and sharp vessel presents a greater rubbing surface than a short and bluff vessel of the same size, the average velocity of rubbing is less in the longer vessel at the same speed; so that there is a certain degree of sharpness which gives the least resistance for a given size and speed. What that degree of sharpness is cannot yet be fixed with any great precision; but in general it does not greatly differ from that which is given by making the sum of the lengths of the bow and stern equal to about seven times the greatest breadth. The following formula has been found to agree well with experi- ments on the resistance of ships : Let Gr be the mean immersed girth ; L, the length on the water line ; s 2 , the mean of the squares of the sines of the angles of obliquity of the stream lines, or lines which the particles of water follow in gliding over the ship's bottom; let v be the velocity of the ship in feet per second, and/ a co-efficient, whose value for a clean painted iron bottom is about 0004; then the resistance is nearly K =L G The factor, L G (1 + 4 s 2 + s 4 ), is called the "augmented surface." See Civil Engineer and Architect's Journal, October, 1861; Phil. Trans., 1862, 1863; Trans, of the Institution of Naval Architects, 1864; also Shipbuilding, Theoretical and Practical, by "Watts, Rankine, Napier, and Barnes. Mr. Scott Russell has proved that, when the length of a ship bears less than a certain proportion to that of the wave which naturally travels with the same speed, there is a rapidly increasing additional resistance. The least proper length in feet suitable for a given speed is about fifteen-sixteenths of the square of the speed in knots. (As to "Waves, see page xv.) 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- orium must cause a deviation of the centre of buoyancy, relatively to a STABILITY OF SHIPS METACEXTRE. 601 vertical line traversing the centre of gravity, in the direction towards which the floating body heels; 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 heel 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 moment is known. Let W be the weight of the ship, F . and S the volume of water displaced by her, so that W = e S (e 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 6', consequently, the moment of the righting couple, equal and opposite to the moment of the heeling couple, is G M sin 6. .(1.) The comparative stability of a ship is proportional to the arm of the righting couple for the same angle of heel ; and that arm is propor- tional to G M, which length thus becomes a measure of the stability of the ship. The point M, when determined for an indefinitely small angle of heel, is called the METACENTRE; it may be the same, or it may be different for finite angles. When the position of M is variable, the angle of heel 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 602 PRINCIPLES OF DYNAMICS. 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, e its density, 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 le s I s , then is D the new centre of buoyancy. The angle which C D makes with the horizon is in general either & B exactly or very nearly = 5 ; so that C D = M C 2 sin , approxi- mately. Also, the volume s is in general either exactly or nearly A proportional to 2 sin 9 ; so that if c be a constant volume depend- ing on the figure of the water line, s = c ' 2 sin 5, approximately. Consequently, to find the height M C of the point M above the centre of buoyancy, and its height M G above the centre of gravity, we have the approximate formulae, MO=CD - 2 sin =; The sign z+= denotes that G C is to be subtracted or added according as G is above or below C. The product I c is found approximately in the following manner, for those cases in which 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 x is the double of the statical moment of one of the wedges relatively to the line H, supposing the density equal to unity. Let distances measured lengthways on the line H be denoted by x; let the perpendicular distance of any point in a water line plane bisecting the angle A H E from STABILITY AND OSCILLATIONS OF SHIPS. 603 the line H be denoted by y, and let the thickness of the wedge at the point whose co-ordinates are x and y be z = y 2 sin . Then we have s = 2 sing ' J J ydy dx;c = J J y -dy d x; ls = 4:sm^ and therefore lc=*2 I j 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 x, measured length- ways from an assumed origin. Then / y 8 d y = ^; and Ic = ^ J b* - d x. ......... (5.) 2 As to the moments of inertia of different plane figures, see Article 95. Thus, equation 3 becomes fb s 'dx MG = .L -^ G The theory of the stability of ships was first investigated by Bossut, and was further developed by Atwood. The most impor- tant contributions to that theory, of later date, have been, the memoir of Dupin, Sur la Stabilite des Corps Flottans, a paper by Canon Moseley in the Philosophical Transactions for 1850, and various papers by Rawson, Froude, Merrifield, Barnes, and others. 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 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. If that time is too long, the ship is deficient in stability; if too short, her movements are abrupt, and tend to overstrain her. 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 604 PRINCIPLES OF DYNAMICS. gravity does not alter its level, then her rolling gyrations are per- formed about a permanent longitudinal axis traversing her centre of gravity, and are not accompanied by vertical oscillations, and her moment of inertia is constant while she rolls. That condition is fulfilled if all the water line planes, such as AB and E F, are tangents 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 point M is sensibly constant. According to Article 654, equation 1, the righting couple for a given angle of heel 6 is W GM sin 0; but in an approximate solution we may substitute 6 for sin ff. Let I be the moment of inertia of the ship about her axis of rolling; then equations 2 and 3 of Article 598 give the following value for the time of a double gyration : 2 v / ( I \ 2 ~* : "V ( 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 R 2 -r- G~M. The researches of Mr. William Fronde, first described to the British Association in July, 1860, and afterwards laid more fully before the Institute of Naval Architects, have shown, first, that the same forces which tend to keep a ship upright in still water tend to place her perpendicular to the surface of the water amongst waves, and thus to increase rolling; secondly, that the chief cause of excessive rolling is too near a coincidence between the periodic time of the vessel's rolling and that of her being acted upon by successive waves; and thirdly, that the most efficient method of preventing excessive rolling is to adjust the moment of inertia and the stability of a vessel, so that her periodic time of rolling shall be longer than the period of any waves she is likely to en- counter, taking care at the same time to leave sufficient stability to prevent the risk of upsetting, or of heeling too far over with a side wind. See Trans, of the Institution of Naval Architects, passim; also Shipbuilding, by Watts, Rankine. Napier, and Barnes. (As to Waves, see page xv.) 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. 605 cylinder provided with a piston, any such space being denoted by s -, and let ordinates measured parallel to the line P, perpendi- Fig. 260. Fig. 261. cular to 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^ p?) d s (1.) Secondly, for any given pressure O F = p, let F E = 81 and F D = s 2 be the greater and the less of the spaces occupied ; then energy transferred = / (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 E = s x at the pressure B = p lt is introduced into a cylinder and made to drive a piston, is theii 606 PRINCIPLES 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 areaABC D = [ l sdp = W (* ; ......... (3.) J p 2 J P2 e 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 Jiead. 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 s 2 (p 2 p s ) ............................ (5.) If the fluid which acts 011 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 r the corresponding boiling point of its liquid, in degrees reckoned from the absolute zero, 274 Centigrade or 4 9 3 -2 Fahrenheit below the melting point of ice. Then B Lo gJP = A - _ B !-. /JA-logy BM . T" V I C r 4C 2 J 20 (See Edin. Philos. Jour., July, 1849 ; Edin. Transac., xx; PMlos. Mag., Dec., 1854; Nicholas Cyclopaedia, art. "Heat, Mechanical Action of.") The following are the values of some of the constants in the above formulse, 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 OP STEAM. 607 A LogB LogC A |1 Water,... 8-2591 3-43642 5*59873 0-003441 0-00001184 ^Ether,...7'5732 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 T, 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 T ; and J the equivalent in foot pounds of a British thermal unit, or 772 ; then .(7.) (hyp. log. 10 = 2-3026); H=H -J(c b)(rr ) (for water, c b = 07) ; D = L -f- H. (for water at the temperature of melting ice, H = 842872.) J c denotes the value in foot pounds of the specific heat of the liquid, which for water is 772, and for aether, 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 expansion. Then ratio of expansion, - =^-< + J c Di hyp log f ; ...... (8.) s l -Ml V T l T 2 ) energy transferred, U = / ' s d p + # 2 (p a -_g? 3 ) J j> = =* fr + J C Dl ( Tl - Ts )} .................... (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 608 PRINCIPLES OP DYNAMICS. following formulae give approximate results for steam, where the pressure of its admission p^ is from one to twelve atmospheres : Sft energy transferred, U = | * s dp -f s 2 (p 2 -p 3 ) J Pa - Pl Sl ' 10 { 1 -(^5 } + ** (ft I. V>/ 7 The expenditure of heat in foot pounds may be computed roughly to about -, when the feed water is supplied to the boiler at about 100 Fahrenheit, by the formula TL=\ n sdp + np 2 s 2 ' } .................. (13.) -' pi where n is a co-efficient whose value is, for condensing engines, 1 6 ; for non-condensing engines, 15. Equations 11 and 12 are applicable to non-conducting cylinders without steam-jackets. For cylinders with steam-jackets, acting so as to keep the steam dry, it is more accurate to substitute 16 for 9, 17 for 10, and if, *|, and iV, respectively, for A, V, and j, throughout the equations 11 and 12. For the exact theory of this case, see A Manual of the Steam Engine and other Prime Movers; also, Philosophical Transactions, 1859, Part I. The following are the ordinary formulae, which give a good approximation when the steam is slightly moist : ,(14.) p l l"2. = Pi s 1 hyp. log. + s 2 (p 2 P$) (15.) i The approximate formula (13) is applicable in all cases. PART VI. THEORY OF MACH 657. Wature 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 j and therefore a large portion of the present part of this treatise will consist of references back to Part I Y. and Part Y. 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. WORK OF MACHINES WITH UNIFORM OR PERIODIC MOTION. SECTION 1. General Principles. 658. Useful and i^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; tvurking 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 oftlie series MEAN EFFORTS AND RESISTANCES GENERAL EQUATIONS. 611 of moving pieces which transmit energy from the 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 c? s for the energy exerted by a varying effort whose magnitude at any instant is P ; and a corresponding expression f^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 8 be the space described by the point of application of the effort or resist- ance in the interval in question j then / P ds -=- s or I Rds -f- 8 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 j and therefore, in the remainder of the present chapter, P and E, 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 E, 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 : Energy "1 f Force in pounds x distance in feet ; or Couple in foot pounds x angular motion to work - = - radius unity; or in Pressure in pounds per square foot x space foot pounds \ [ described by a piston in cubic feet. 666. 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, R! the mean useful resist- ance, s t the space through which it is overcome in the same inter- val, R 2 any one of the prejudicial resistances, s 2 the space through which it is overcome j then Ps = R l5l + 2-R>5 2 (1.) The efficiency of the machine is expressed by 612 THEORY OF MACHINES. Ps ,o x 667. Equations in terms of Comparative Motions. - Let 5j : S = n w s 2 : s = ?i 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 : Przr^Ei + 2-ra 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. Reduction 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, 191, and 192. The following is a table of the angle of repose , the co-efficient of friction f = tan 79. in Frictionai Gearing, described in Article 445, it appears that when the angle of the grooves is 40, and when their surfaces are smooth, clean, and dry, the tangential force transmitted between the wheels is once and a-half the force with which their axes are pressed together. This proportion is much greater than that due to ordinary friction, and must arise partly from adhesion. 680. Friction Couplings are used to communicate rotation be- tween pieces having the same axis, where sudden changes of force or of velocity take place; being so adjusted as to limit the force transmitted within the bounds of safety. Contrivances of this kind STIFFNESS OF ROPES ROLLING FRICTION CARRIAGES. 619 are very numerous; one of the most common and most useful is that called a pair of friction cones. The angle made by the sides of the cones with the axis should not be less than the angle of repose. 681. stiffness of Ropes. Ropes offer a resistance to being bent, and when bent to being straightened again, which arises from the mutual friction of their fibres. It increases with the sectional area of the rope, and is inversely proportional to the radius of the curve into which it is bent. The work lost in pulling a given length of rope over a pulley, is found by multiplying the length of the rope in feet, by its stiffness in pounds ; that stiffness being the excess of the tension at the leading side of the rope above that at the following side, which is necessary to bend it into a curve fitting the pulley, and then to straighten it again. The following empirical formula3 for the stiffness of hempen ropes have been deduced by General Morin from the experiments of Coulomb : Let K be the stiffness in pounds avoirdupois ; dj the diameter of the rope, in inches ; n = 4=8 d 2 for white ropes, 35 d z for tarred ropes ; r, the effective radius of the pulley, in inches ; T, the tension, in pounds ; then, For white ropes, E, = -(0-0012 + 0-001026 n + 0-0012 T); . (I-) For tarred ropes, E, = - (0-006 + 0001392 w + 000168 T). J 682. Rolling Resistance of Smooth Surfaces. By the rolling of two surfaces over each other without sliding, a resistance is caused, which is called rolling friction. It is of the nature of a couple resisting rotation ; its moment is found by multiplying the normal pressure between the rolling surfaces by an arm whose length depends on the nature of the rolling surfaces ; and the work lost in an unit of time in overcoming it is the product of its moment by the angular velocity of the rolling surfaces relatively to each other. The following are approximate values of the arm in decimals of afoot : Oak upon oak, 0-006 (Coulomb). Lignum-vitse on oak, 0-004 Cast iron on cast iron, 0-002 (Tredgold). 683. The Resistance of Carriages on Roads consists of a constant part, and a part increasing with the velocity. According to Gene- ral Morin, it is given approximately by the following formula : 620 THEORY OF MACHINES. (1.) where Q is the gross load, r the radius of the wheels in indies, v the velocity in feet per second, and a and b two constants, whose values are a b For good broken stone roads, ............ '4 to -55 -024 to '026 For paved roads, ........................... '27 -0684 For the pavement of Paris, ............... '39 '03 On gravel roads the resistance is about double, and on sandy and gravelly soft ground, five times the resistance on good broken stone roads. 684. Resistance of Railway Trains. In the following formula, which are all empirical E denotes the weight of the engine ; T the gross load drawn by it ; V the velocity, in miles an hour; r the radius of curvature of the line, in miles; B ,, the resistance in pounds ; f a co-efficient of friction ; c a co-efficient for resistance due to curvature. Then for single carriages with cylindrical wheels, at velocities up to 12 miles an hour, according to the experiments of Lieutenant David Bankine and the Author, (1.) where f= 0-002; and c = 0'3. (See Experimental Inquiry on the Use of Cylindrical Wheels on Railways, 1842.) For an engine and train, the following is an empirical formula deduced from the experiments of various authors : (2.) where/ ranges from -0027 to -004, according to the state of the line and carriages, and c from 0'3 to 0-1. (See Bankine' s Manual of Civil Engineering.) 685. Heat of Friction. The work lost in friction produces heat in the proportion of one British thermal unit, being so much heat as raises the temperature of a pound of water one degree of Fahrenheit, for every 772 foot pounds of lost work. Excessive heating is prevented by a constant and copious supply of a goojL unguent. 621 CHAPTER IL VARIED MOTIONS OF MACHINES. 686. The Centrifugal Forces and Couples exerted by the various rotating pieces of a machine against the bearings of their axles are to be determined by the principles of Articles 540, 592, and 603, and taken into account in determining the lateral pressures which cause friction, and the strength of the axles and framework. As those centrifugal forces and couples cause increased friction and stress, and sometimes also, by reason of their continual change of direction, produce detrimental or dangerous vibration, it is de- sirable to reduce them to the smallest possible amount ; and for that purpose, unless there is some special reason to the contrary,^?! the axis of rotation of every piece which rotates rapidly ought to^/H^ traverse its centre of gravity, that the resultant centrifugal force f may be nothing, and ought to be an axis of inertia, that the centri- fugal couple may be nothing. As to axes of inertia, see Article 584. ^rr-687. Actual Energy of a Machine. To determine the entire actual energy of a machine at a given instant, it is necessary to I know (1.) The weight of each of its sliding pieces : let any one of those weights be denoted by W; (2.) The velocity of translation of each of those pieces at the given instant : let v denote any one of these velocities ; (3.) The moment of inertia of each of its rotating pieces : let any one of these moments be denoted by I ; (4.) The angular velocity of each of those pieces at the given instant ; let a be any one of these angular velocities. These quantities being given, the actual energy of the machine is (1.) and if the moment of inertia of each rotating piece be expressed in the form I = W e 2 , "W being its weight and e its radius of gyra- tion, the above expression may be put in the form, E = -fr-Wtf + 2-WYa 2 ) .............. (2.) 88. Reduced inertia, The figures, sizes, and connection of the 622 THEORY OF MACHINES. pieces of a machine being known, the principles of the Theory of Mechanism (Part IV.), enable the comparative motions of all its points to be determined, and in particular, the several ratios of their velocities to that of the driving point at any instant. Let Y be the velocity of the driving point, and for any given piece of the machine whose weight is W, let n denote the ratio v : V if it is a sliding piece, and the ratio f a : V if it is a turning piece. Then the sum (1.) expresses tfie weight which, if concentrated at the driving point, would have the same actual energy with the entire machine. This quantity may be called the inertia reduced to the driving point. By Mr. Moseley, who first introduced its consideration into mechanics, it is called the " co-efficient of steadiness." The actual energy of the machine at any instant may now be expressed by Another mode of expressing the reduced inertia is with reference to the driving axis. Let A represent the angular velocity, at any instant, of the axis of the piece which first receives the motive power ', for any shifting piece let v : A = I ; and for any rotating piece let a : A = n. Then the reduced moment of inertia is 2- WZ 3 + 2 -In 2 ; ...................... (3.) and the actual energy at any instant, E =- 689. Fluctuations of Speed in a machine are caused by the alter- nate excess of the energy received above the work performed, and of the work performed above the energy received, which produce an alternate increase and diminution of actual energy, according to the law of the conservation of energy explained in Article 552. G. K To determine the greatest fluctuations of speed in a machine moving periodically, take ABC, in fig. 265, to represent the motion of the driving point during one period j let the effort P of the prime mover at each instant be represented by the ordinate of the Fig. 265. Clirve B G E I F ; and let the sum of the resistances, reduced to the driving point, as in Article 668, at each instant, be denoted by R, and represented by the ordinate of the G FLUCTUATION OP SPEED FLY-WHEEL. 623 curve D H E K F, which cuts the former curve at the ordinates A. D, B E, F. Then the integral being taken for any part of the motion, gives, as in Article 549, the excess or deficiency of energy, according as it is positive or negative. For the entire period ABC this integral is nothing. For A B, it denotes an excess of energy received, represented by the area D G E H ; and for B C, an equal excess of work performed, repre- sented by the equal area E K F I. Let those equal quantities be each represented by A E. Then the actual energy of the machine attains a maximum value at B, and a minimum value at A and C, and A E is the difference of these values. Now let V be the mean velocity, Yj the greatest velocity, and Y 2 the least velocity of the driving point ; then V 2 _ V 2 28 AE; .................. (1.) which, being divided by twice the mean actual energy gives a ratio which may be called the co-efficient of fluctuation of speed. The ratio of the periodical excess and deficiency of energy A E to the whole energy exerted in one period or revolution, fl? ds, has been determined by General Morin for steam engines under various circumstances, and found to be from to j, for single cylinder engines. For a pair of engines driving the same shaft, with cranks at right angles to each other, the value of this ratio is about one-fourth of its value for single cylinder engines. 690. A Fly- wheel is a wheel with a heavy rim, whose great moment of inertia reduces the co-efficient of fluctuation of speed to a certain fixed amount, being about in ordinary machinery, and or o2i 00 oO in machinery for fine purposes. Let be the intended value of the co-efficient of fluctuation of m speed, and A E, as before, the fluctuation of energy; then if this is 624: THEORY OF MACHINES. to be provided for by the moment of inertia I of the fly-wheel alone, let a be its mean angular velocity; then equation 2 of Article 689 is equivalent to the following : 7)1 a% I cto the second of which equations gives the requisite moment of / inertia of the fly-wheel. - 691. Starting and Stopping Brakes. The starting of a machine consists in setting it in motion from a state of rest, and bringing it up to its proper mean velocity. This operation requires the ex- penditure, besides the energy required to overcome the resistance of the machine, of an additional quantity of energy equal to the actual energy of the machine when moving with its mean velocity, as found according to the principles of Article 687. If, in order to stop a machine, the effort of the prime mover is simply suspended, the machine will continue to go until work has been performed in overcoming its resistances equal to the actual energy due to its speed at the time of suspending the effort of the prime mover. In order to stop the machine in less time than this operation would require, the resistance may be artificially increased by means of a brake, which may be a friction-strap, as described in Article 678, or a block pressed against the rim of a wheel, or a grooved sector pressed against a wheel grooved as for fractional gearing ^Articles 445, 679). Let R! be the ordinary resistance of the machine, reduced to the rubbing surface (Article 668), R 2 the friction produced by the brake, v the velocity of the surface on which it acts at the time when it is first applied, s the distance through which rubbing must take place in order to stop the machine, t the time required for the same effect, E the actual energy of the machine when the brake begins to act. Then s = E + (R! + R 2 ) ; (1.) and because the mean velocity of rubbing during the operation of stopping is v -f- 2, 625 CHAPTER IIL ON PRIME MOVERS. 692. A Prime Merer is an engine, or combination of moving pieces, which serves to transfer energy from those bodies which naturally develop it, to those by means of which it is to be employed, and to transform energy from the various forms in which it may occur, such as chemical affinity, heat, or electricity, into the form of mechanical energy, or energy of force and motion. The mechanism of a prime mover comprehends all those parts by means of which it regulates its own operations. The useful work of a prime mover is the energy which it trans- mits to any machine driven by it; and its efficiency is the ratio of that useful work to the whole energy received by it from a natural source of energy. The effect or available power of a prime mover is its useful work in some given unit of time, such as a second, a minute, an hour, a day. 693. The Regulator of a prime mover is some piece of apparatus by which the rate at which it receives energy from the source of energy can be varied; such as the sluice or valve which adjusts the size of the orifice for siipplying water to a water-wheel, the appara- tus for varying the surface exposed to the wind by windmill-sails, the throttle-valve of a steam engine. In prime movers, whose speed and power have to be varied at will, such as locomotive engines, and winding engines for mines, the regulator is adjusted by hand. In other cases it is adjusted by a self-acting apparatus called a Governor usually consisting of a pair of rotating pen- dulums, whose angle of deviation from their axis depends upon the speed. (Article 606). 694. Prime Morera may be classed according to the forms in which the energy is first obtained. These are I. Muscular Strength. II. The Motion of Fluids. IIL Heat. IV. Electricity and Magnetism. 695. Muscular strength. The daily effect exerted by the muscu- lar strength of a man or of a beast is the product of three quan- tities; the useful resistance, the velocity with which that resistance 2s 626 THEORY OP MACHINES. is overcome, and the number of units of time per day during which work is continued. It is known that for each individual man or animal there is a certain set of values of those three quanti- ties which makes their product a maximum, and is therefore the best for economy of power; and that any departure from that set of values diminishes the daily effect. The following table of the effects of the strength of men and horses employed in various ways, is compiled from the works of Poncelet and General Morin, and some other sources : MAN. R Ib. V ft. p. sec. T"~ 3,600 = hrs. p. day. RV ft. Ib. p. sec. RVT ft. Ib. p. day. 1. Kaising his own weight up stair 143 0*5 8 72-5 2,088,000 2. 3. 4. 5. 6. 7. Do. do. do., (Tread-wheel, see 1.) Hauling up weight with rope, Lifting weights by hand, Carrying weights up stairs, Shovelling up earth to a height of 5 feet 3 inches, 40 44 143 Q 0-75 0-55 0-13 1-3 10 6 6 6 10 30 24-2 18-5 7-8 2,616,000 648,000 522,720 399,600 280,800 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 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 25 ? 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) 000 9,900,000 Copper, cast, 19,000 sheet, 30,000 bolts, 36,000 wire, ; 60,000 17,000,000 Iron, cast, various qualities, {^gg to average, 16,500 17,000,000 Iron, wrought, plates, 5 1,000 joints, double rivetted, 35>7oo single rivetted, 28,600 bars and bolts, j ^ * | 29,000,000 hoop, best-best, 64,000 e > {to^ooo} 2 S,3o,ooo wire-ropes, *.. 90,000 15,000,000 Lead, sheet, 3,300 720,000 Steel bars, j IOO > OO 2 9,ooo,ooo \ to 130,000 to 42,000,000 Steel plates, average, 80,000 Tin, cast, 4,600 Zinc, 7,000 to 8,000 63:2 APPENDIX. Tenacity, MATERIALS. or Resistance to Modulus of Elasticity, or Resistance to i earing. Stretching. TIMBER AND OTHER ORGANIC FIBRE: Acacia, false. See " Locust." . Ash (Fraxinus excelsior), 17,000 I,6oo,OOO Bamboo (Bamhusa 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), < 10,000 ) to 13,000 j 1,140,000 Elm ( Ulmus campestris), 14,000 \ 7OO,000 to 1,340,000 Fir : Red Pine (Pinus sylvestris), < 12,000 to 14,000 1,460,000 to 1,900,000 Spruce (A bies excelsa), 12,400 \ 1,400,000 to 1,800,000 Larch (Larix Europoea), < 9,000 to 10,000 900,000 to 1,360,000 Hoxen Yarn, about 25,000 Hazel (Corylus Avellana), 18,000 Hempen Ropes, from 12,000 to 16,000 Hide, Ox, undressed, 6,300 Hornbeam (Carpinus Betulus), . . . 20,000 Lance wood (Guatteria virgata),... 23,400 Leather, Ox, 4,200 24,300 Lignum- Vita3 (Guaiacum qffici- ) II 800 nale), J " s Locust (Robinia Pseudo-Acacia), 16,000 Mahogany (Swietenia Hahagoni), < 8,000 ) to 21,800 J 1,255,000 Maple (A cer campestris), 10,600 Oak, European (Quercus sessili- ( 10,000 1,200,000 flora and Quercus pedunculata), \ to 19,800 to 1,750,000 American Red (Quercus ) 10,250 2,150,000 Silk Fibre, 52,000 1,300,000 Sycamore^! cerPseudo-Platanus\ Teak, Indian (Tectona grandis), 13,000 15,000 1,040,000 2,400,000 African, (?) 21,000 2,300,000 Whalebone, 7,700 Yew (Taxus laccata), 8,000 APPENDIX. G33 II. TABLE OP .THE RESISTANCE OP MATERIALS TO SHEARING AND DISTORTION, in pounds avoirdupois per square inch. to METALS I Shearing. Distortion. Brass, wire-drawn, ....................... 5>33jo Copper, ................................... 6,200,000 Iron, cast, ................................. 27,700 2,850,000 TIMBER : f , Soto 800 { to t % Spruoe, ............................. 600 ...... Larch, .............................. 970101,700 ...... Oak, ....................................... 2,300 82,000 Ash and Elm, ........................... 1,400 76,000 III. TABLE OP 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, ...................................... 550 to 800 strong red, ..................................... 1,100 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, ..................... 82,000 to 145,000 average, ................................ 112,000 wrought, .............................. about 36,000 to 40,000 634 APPENDIX. Resistance MATERIALS. to Crushing* TIMBER,* Dry, crushed along the grain : Ash, 9,000 Beech, 9,360 Birch, 6,400 Blue-Gum (Eucalyptus Globulus), 8,800 Box, 10,300 Bullet-tree (A chras Sideroxylori), 1 4,000 Cabacalli, 9,900 Cedar of Lebanon, 5,86o Ebony, "West Indian (Brya Ebenus), 1 9,000 Elm, 10,300 Fir: Ked Pine, 5,375 to 6,200 American ~Yenowl?ine(Pinusvariabilis), 5, 400 Larch, 5,570 Hornbeam, 7, 300 Lignum- Yitse, 9,900 Mahogany, 8,200 Mora (Mora excelsa), 9,9o Oak, British, 10,000 Dantzic, 7>7oo American Ked, 6,000 Teak, Indian, 12,000 "Water-Gum (Tristania nerifolia), 1 1,000 IV. TABLE OF THE KESISTANCE OF MATERIALS TO BREAKING ACROSS, in pounds avoirdupois per square inch. Resistance to Breaking, MATERIALS. or Modulus of Rupture.f Sandstone, 1,100 to 2,360 Slate, : 5,ooo * The resistances stated are for dry timber. Green timber is much weaker, havirg 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 Rupture. 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 TIJIBEB : Ash, 12,000 to 14,000 Beech, 9,000 to 12,000 Birch, 11,700 Blue-Gum, 16,000 to 20,000 Bullet-tree, 15,900 to 22,000 Cabacalli, 15,000 to 16,000 Cedar of Lebanon, 7, 400 Chestnut, 10,660 Cowrie (Dammara austrcdis), 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 (Nedamdra Rodioei), 16,500 to 27,500 Lancewood, I 7)35 Lignum- VitaB, 12,000 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, , 13>3 Saul, 16,300 to 20,700 Sycamore, 9,600 Teak, Indian, 12,000 to 19,000 African, 14,980 Tonka (Dipteryxodorata), 22,000 Water-Gum, 17,460 Willow (Salix, various species), 6,600 636 MEASURES. "Sfl-T O 2 $ * slli!il CO M "* ON O M CO M ON ON O ON vo O -^ covp w ^ 00 co 7* ON O 00 M vo vo ^ ^ 5 M O N VO O vo CO ON VO O | |M H M IN co *>. 10 O VO VO |W CO CO vo o N N ^VO ON O VOOO CO COOO OO J>. VO J>- ^- co O ON M co ON OO CO CO 1C vo CO M si - O |w O fed I'M M O t^. CO vo vo T- vo N -rf vo CO ON CO CO CO O -t> ON -t** M ON M- vo c* M M VOOO OO CO ICO M O IH O vo oo ON VO b VO N VO CO CO J>- M covo O ON ^*- M Tj- N VO O COVO ^ CO O -4-OO J T+- O o o CO VO 00 CO vo o vo M co ON vo co co M vo CO ri - d ^ a d ^ '% g *-M g^ a P4 PH PQ APPENDIX. VI. TABLE OF SPECIFIC GRAVITIES OF MATERIALS. 637 Weightofacubio GASES, at 32 Fahr., and under the pressure of one foot in atmosphere, of 2116-4 tt>- on the square foot: lb - avoirdupois Air, 0-080728 Carbonic Acid, 0-12344 Hydrogen, 0-005592 Oxygen, 0-089256 Nitrogen, 0-078596 Steam (ideal), 0-05022 ^Ether vapour (ideal), 0-2093 Bisulphuret-of-carbon 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 3 9 -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 JEther, 44'7o 0-716 Mercury, 848-75 13*596 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, t 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, 62-43 to 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. avoirdupois. pure water = 1. SOLID MINERAL SUBSTANCES continued. Glass, crown, average, 156 2*5 flint, 187 3-0 green, 169 27 plate, 169 27 Granite, 164 to 172 2-63 to 276 Gypsum, 143-6 2-3 Limestone (including marble),.. 169 to 175 27 to 2-8 magnesian, 178 2-86 Marl, iootoii9 i -6 to 1*9 Masonry, 116 to 144 1*85 to 2-3 Mortar, 109 175 Mud, c 102 i'6~3 Quartz, 165 2*65 Sand (damp), 118 1-9 (dry), 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, 474 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 8 Specific TIMBER:* -.. foot in Ib. avoirdupois. gravity, pure water = 1. Ash, 47 0753 Bamboo, 25 0-4 Beech, 43 0-69 Birch, 44'4 0*711 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 0-579 Ebony, "West Indian, 74-5 1-193 Elm, 34 0-544 Fir: Red Pine, 30 to 44 0-48 to 0-7 Spruce, 30 to 44 0-48 to 0-7 American Yellow Pine,... 29 0-46 Larch, 31 ^ 35 0-5 to 0-56 Greenheart, 02-5 I '00 1 Hawthorn, 57 0-91 Hazel, 54 0-86 Holly, 47 0-76 Hornbeam, 47 0-76 Laburnum, 57 0-92 Lancewood, 42 to 63 0-675 to I ' I Larch. See "Fir." Lignum- Yitae, 41 to 83 0-65 to 1-33 Locust, 44 0-71 Mahogany, Honduras, 35 0-56 Spanish, 53 0-85 Maple, 49 0-79 Mora, .: 57 0-92 Oak, European, 43 to 62 0-69 to 0-99 American Red, 54 0-87 Poon, 36 0-58 Saul, 60 0*96 Sycamore, 37 0-59 Teak, Indian, 41 to 55 0-66 to 0-88 African, 61 0-98 Tonka, 62 to 66 0-99 to i -06 "Water-Gum, 62-5 I -00 1 Willow, 25 0-4 Yew, 0-8 * The Timber in every case is supposed to be dry. 640 APPENDIX. YIL DIMENSIONS AND STABILITY OF THE OUTER SHELL OP THE GREAT CHIMNEY OF ST. KOLLOX. Greatest pres Divisions of Chimney. Heights above Ground. External Diameters. Thicknesses. Security. Feet. Feet. Inches. Feet. Inches. Ib. per square foe V. 1 435i 13 6 } r i IV. j 35oi 16 9 24 o . ; ;I^ S5 III. { "* r 30 6 57 II. 2 3 54i 35 o 63 I. ' 74 o 40 o 7 1 Foundation. Depth below Ground. External Diameter. Thick Concrete. iesses. Brick. Feet. Feet Feet. Inches. Feet. I. / 50 rt o -* 3 8 50 4 8 3 II. i 14 50 \ 12 III. { ~ 25 o O Total height from base of foundation to top of chimney, 455^ feet * Joint of least stability. APPENDIX. 641 308 A.* Continuous Girders. The fundamental principle of the theory of continuous girders, with the load distributed in any manner, is the "Theorem of Three Moments," due originally to Clapeyron and Bresse and improved by Heppel (See Bresse, Mecanique Appliquee, Part III., and the Proceedings of the Royal Society for 1869). The subject is treated of in Art. 178 of the tenth edition of Civil Engineering. The following demonstration, with deduced formulae, is abstracted from a paper communicated by Mr. Mans- field Merriman, C.E., to the Philosophical Magazine, September, 1875. The elastic curve (Art. 319) has the following equation: d*y M E1* The equation for any particular case is obtained by sub- stituting the values of M and I (constant) in terms of x, and integrating twice. Let I be length of first span, I' of second. Let M , M 1? M 2 , be the moments at three points of support. Let "W be a single con- centrated load at a distance a from support 0, and W' at a distance a 1 from 1. Since equilibrium prevails, we have for a section between "W and 1, the equation M -F# + W(#-a)-M = 0, ............... (I.) making x I, M becomes M 1? and F = M L -M 1 + W-a) = M 2 -M 1 + W(1 _ /l) ....... i L k being any fraction. Insert now the value of M in the differ- ential equation of the curve, and integrate it twice : t the tangent of the angle which the curve at the origin makes with the axis of abscissa is the constant in the first integration, and zero in the second. The required equation is y =t x + 3M x*-F*? + W(x-a)s ........ (II.) Substituting the value of F in terms of M , Mj, and W, making = l } and a = k I \ y = 0, and we obtain 6 El t= -2 M Z-M! 1 + W P (2 k- 3 ^ + ^) (.) * See Article 308, p. 338. 2T 642 APPENDIX. Now in the value of -^ make x = 1; ^| becomes ^, the tangent at 1, and we obtain 6 El ^ = M 1+2 M x I- W Z 2 (&->) ........... (III.) If the origin be taken at 1, we obtain an equation analagous to . where k denotes -=,-, and is not necessarily the same in the two t expressions. "We thus obtain the Theorem of Three Moments for concentrated loads For many loads 2 has to be prefixed to the terms involving W and W. For uniformly distributed loads w and w' per unit of length, we place 2 W = J w d (&Z)and 2 W = j w d(k l'\ integrating between the required limits. If the loads extend over the whole span, the first integral is taken between kl = Q and k 1=1, the second between k I' = 0, and k I' = I'. "Then M 1+ 2 M! (I + + M 2 r = J w P + 1 to' P, ....... (V.) which is the theorem as first deduced by Clapeyron. The following are the formulae deduced by Mr. Merriman CASE 1. Ends resting freely upon abutments. Let the girder consist of any number of unequal spans, the rth only being loaded. Let s = the number of spans, and ^, / 3 ,| &c., their lengths; 1 being the first ands+1 the last support, the index n will refer to any support. A single load in the rth span is called "W, and its distance from the rth support k l r or a. Referring to (IV.) it is seen there are two functions of W and Jc l r of frequent occurrence. Denoting these by A and B for the supports r and r + 1. for a single load in the rth span. A = ( | 2 w II (2 Tc - 3 k 2 + &) d k 1 for an uniform load whose ends yv 1 j- are distant ^ l r and 2 l r from B = J Ik w % & ~ ^ d ^ the su PP ort r ' APPENDIX. 643 From the equations of moments, and the solution of these equations by the method of indeterminate co-efficients, the two following equations are derived. When n r The values of the quantities c, d are -2 ~3 7 -la C - 2 * - 2 - 3 * - 3 (3.) Let the shearing stress in the span l^ at a point infinitely near to the rth support, be denoted by F r , and to the r + 1th support by F' r ; then M r M r+1 ("for the right hand shear at the rth l r a \ support, a M r , J for the lef l r \ support, M r + a M r , J for the left hand shear at the r + 1th M n M n + 1 f for the right hand shear at all supports '~l~ n 1 except r, . . M n M n _j ( for the left hand shear at all supports ^ w - 1= Y n ]. I except r+1, .... then the reaction at any support is Rn=F / -l + ^;Rr=F' r -l+F r ,&C., ............. (5.) Let M and F denote the moment and shearing force at any section; then 641 APPENDIX. M - M r F r x + W (x - a) for a section between W and \ the followi M = M n F M x for any other section, the following support : f(6- j F = W-F r for a section between W and the r+ 1th } support, s (7.) F = F M for any other section, Equations (6) and (7) refer to a concentrated load ; for an uniform load for W substitute I w d a. CASE 2. One end free and the other fixed horizontally. In the application of the above formulae (1) to (7) to this case, make l s = 0, and let s - 1 be the number of spans. CASE 3. Both ends fixed horizontally. In the above formulae make l = and l t 0, and let s - 2 be the number of spans. INDEX. ABUTMENTS, stability of, 226. of arches, 261, open and hollow, 263 (see also 82). strength of, 268. Accelerating effect of gravity, 485. impulse, 483. force, 490. Acceleration, 386. Air, apparent weight of bodies in, 123. expansion of, 123. velocity of sound in, 563. weight of, 123. (see Gas). Angle of repose, 210. of rotation, 391. of rupture, 204, 259. of torsion, 356. Angular impulse, 506. momentum, 505, 529. velocity, 391. Arch, abutments of, 261. angle, joint, and point of rapture of,259. circular linear, 183, 201. clustered, 263. distorted, 202. distorted elliptic linear, 186. elliptic linear, 184. geostatic approximate, 209. geostatic (to sustain earth), 196. groined (see Vaults). hydrostatic approximate, 207. hydrostatic (to sustain fluid pressure), 190, 353. iron- ribbed, 376. line of pressures in, 257. linear, for normal pressure, 189. linear, or equilibrated rib, 162, 175, 182. of masonry or brick- work, stability of, 226, 257. piers of, 263. pointed, 203. skew, 261. stereostatic (with rigid load), 198. strength of, 268. total thrust of, 203, 260. Areas, measurement of, 58. conservation of (see Conservation). Atmospheric pressure, 69. Axes of inertia, 524. of elasticity, 278. of stress, 9*3, 98. Axis of rotation, 390. fixed, 545. instantaneous, 397. of angular momentum, 505, 529. Axle, strength of, 353. friction of, 614. resilience of, 357. torsion of, 356. with crank, strength of, 358. BALANCE, 15. of any system of forces, 41. of couples, 21. of floating bodies, 120. of fluids, 116. offerees in one line, 19. of inclined forces, 35. of parallel forces, 21, 25. of stress and weight, 112. of structures, 129. Balanced forces, motion under, 476. Ballistic pendulum, 548. Bands in mechanism, 454. friction of, 617. Bars, strength of iron and steel, 377 Beam, 133. allowance for weight of, 346. cast iron, 318. deflection under any load, 328. direct vertical stress in, 342. expansion and contraction of, C48. fixed at both ends, 332. limiting length of, 347. lines of principal stress in, 341. of uniform strength, 320. originally curved, 348. partially loaded, 344. proof deflection of, 322. proportion of depth to span of, 327. 646 INDEX. Beam, resilience of, 330. shearing stress in, 338, 342. sloping, 348. strength of, 307, 315, 634. (see also Girder). Belts, strength of, 288 (see also Bands). Bending, resistance to, 307. moment of, 307. (see also Beam). Bevel-wheels, 428, 448. Blocks, stability of a series of, 230. and tackle, 462. Bodies, 13. Boilers, strength of, 289, 296, 299, 306. Boiling point, 606. Bracing of frames, 142. Brake, 624. Breaking across, resistance to, 307 (see also Beam). Brickwork (see Masonry). Bridge (see Arch, Beam, Girder). suspension (see Suspension Bridge). Buoyancy, 121. Buttresses, 228, 235. CABLES, strength of, 288. Cam, 449. Catenary, 177. Cells, strength of, 364. Centre of buoyancy, 121. of gravity, 49, i80. of mass, *482. of oscillation or percussion, 520, 544. of parallel forces, 31. of pressure, 71, 76, 125. of resistance, 131. Centrifugal force, 491, 546 (see also De- viating Force). couple, 537. pump, 597. Chains, equilibrium of, 162 (see also Sus- pension Bridge). Channel, flow in, 411. Chimneys, stability of, 228, 240, 640. Cinematics, 15. principles of, 379. Click, 462. Collapsing, resistance to, 306. Collar, friction of, 616. Collision, 508. Columns, strength of (see Pillars). Comparative motion, 384, 389. Components, 19, 381. Composition of couples, forces, motions, &c. (see Resultant). Compressibility of liquids, 271. Compression, resistance to, 302. Cones, speed, 457. Connected bodies, motions of, 420, 421. Connecting rods, strength of, 363. Conservation of energy, 478, 501. of angular momentum, or of areas, 506. of momentum, 505. Continuity, equations of, in liquids, 411, 413. equations of, in gases, 417. Contracted vein, 572. Contraction, co-efficient of, 572. Cord, equilibrium of, 162. motion of, 408. Counterforts, 255. Couples, deviating, 535. centrifugal, 537. energy and work of, 537. polygon of, 25. statical, theory of, 21. with inclined axes, 24. with parallel axes, 21. Coupling, Oldham's, 453. friction, 618. Hooke's, 461. of parallel axles, 459. Crank and axle, strength of, 358. motion of, 458. Cross- breaking, resistance to (see Beam). Crushing, direct resistance to, 302, table, 633. by bending, resistance to, 360. Current, 412. pressure of, on a solid body, 598. radiating, 412, 574. Cycloid, 398. Cylinders, strength of, 289, 294. DAMS, stability of, 243. Day, sidereal, 380, 381. mean solar, 382. Deflection (see Beam). Deviating force, 491, 492, 545. couple, 535. Deviation (of motion), uniform, 387. moment of, 528. varying, 388. Direction, fixed and nearly fixed, 379. Distributed forces, 48. Dome, stability of, 265. Drums, in mechanism, 455. INDEX. 647 Dynamics, 15. general equations of, 484. principles of, 475. Dynamometer, 478. EARTH, friction of, 211. foundations, 219. pressure of, 218 (see also Retaining Walls). stability of, 212. table of examples, 221. Eccentric, motion of, 460. Eddy (see Vortex). Effect of a machine, 610. Effort, 476. Efficiency, 609, 610. Elastic curve, 349. Elasticity, theory of, 270, 275. co-efficients of, 277. modulus of, 279, 631. potential energy of, 277. Electro-dynamic engine, efficiency of, 630. Energy, 477. actual, 499, 507. actual, of a rotating body, 532. components of, 480, 499. conservation of, in varied motion, 501, 508. conservation of, motion being uniform, 478. initial, 503. of couples, 537. potential, 477. total, 503. transformation of, 499. Epicycloid, 401. Epicycloidal teeth (see Teeth). Epitrochoid, 401. Equilibrated Arch (see Arch). Equilibrium (see Balance). stable and unstable, 128. Expansion of air, 123, 606. of metals, stones, brick, glass, timber, 349. of steam, 606. of water, 125. Extrados, 173. FALLING BODY (see Gravity). Fan, 598. Fixed direction, 379 point, 14, 381. Flexure, moment of, 311. resistance of, 312. Floating bodies, 120, GOO. Flow of liquid, 410. of gas, 417. (see Liquid, Gas). Flues, strength of, 306. Fluid, 100. elasticity of, 285. equilibrium of, 116. impulse of, on a solid surface, 59L motion of, 410. pressure of, 99. (see Liquid, Gas). Fly-wheel, 623. Foot-pound, 477. Force, 15, 17. absolute unit of, 486. centrifugal (see Deviating Force). deviating (see Deviating Force). distributed, 48. reciprocating, 503. representation of, 19. unbalanced, measures of, 501. Forces, action of, on a system of bodies, 510. parallelogram of, 35. parallelepiped of, 37. polygon of, 36. residual, 498, 511. resolution of, 37. Foundations, earth, 219, 255. Fracture, 272. Frames, bracing of, 142. equilibrium and stability of, 132. of two bars, 136. polygonal, 139. resistance of, at a section, 150 triangular, 137. Friction, 209. coupling, 618. heat of, 620. internal, 377. moment of, 614. of gas, 590. of liquid, 584. of machines, 612. of solid bodies, law of, 209. strap, 618. tables of, 211, 613. Frictional stability, 209. gearing, 618. GrAS, 13. action of, on a piston, 604. dynamic head in, 579. 648 INDEX. Gas, equation of continuity in, 581. flow of, from an orifice, 581. flow of, with friction, 590. motion of, 417. motion of, without friction, 579. Girder, bowstring, 369. cellular, 367. compound, 366. half-lattice, 153, 369. lattice, 160, 369. plate, 366. stiffening, for suspension bridges, 370. tubular, 366, 367. Warren's, 153, 369. Governor (see Pendulum, revolving). also 625. Gravity, accelerating effect of, 485. centre of, 49, 180. motion under, 485, 4 80. specific, 49, 124. specific, table of, 637. Grease, 613. Groined vaults, 262. Gyration, 542. radius of, 515. table of radii of, 518. HEAD, dynamic, of liquid, 568. dynamic, of gas, 579. equal, surfaces of, 573. Heat of friction, 620. engine, efficiency of, 629. of steam, 607. specific, of gases at constant pressure, 580. Height due to Velocity, 487. Horse-power, effective, 610. Horse, work of, 626. Hydraulic hoist, 465. mean depth, 587. press, 464. Hydraulics (see Hydrodynamics). Hydrodynamics, 566. Hydrostatic arch, 190, 353. Hydrostatics, principles of, 100, 112, 117. IMMERSED BODY, 122. plane, 125. Impact (see Collision). and pressure, 564. Impulse, 483. and momentum, law of, 484. angular, 506. between solids and fluids, 591. Indicator, 478. Inertia, or mass, 482. ellipsoid of, 526, 532. moment of (see Moment). reduced, 621. Inside gearing, 441. Integrals, approximate computation of, 58. Intensity of distributed force, 48. of pressure, 69. of stress, 68. Internal equilibrium of stress and weight, 112. Internal stress (see Stress). Intrados, 173. Isochronous vibration, 553. JET, impulse of, 591. Joints of a structure, 129, 131. of masonry, 211. KEYS, friction of, 226. LATERAL FORCE, 476. Leather, strength of, 288. Length, measure of, 13, 14. Lever, 26. Line, 13. Link motion, 468 Linkwork in mechanism, 458. Liquid, 13. dynamic head of, 568. equilibrium of, 118. flow of, from an orifice, 570. flow of, in a pipe (see Pipe). flow of, in a stream (see Stream). free surface of, 570. motion of, 410. motion of, in plane layers, 570. motion of, with friction, 584. surface of equal pressure in, 570. without friction, motion of, 567. MACHINES, 15. actual energy 'of, 621. pieces of, 422. reduced inertia of, 621. theory of, 609. varied motion of, 621. work of, with uniform or periodic mo- tion, 610. Man, work of, 625. Masonry and brickwork, bond of, 222 ; friction of, 211, 222. stability of, 230. INDEX. 649 Mass, 482, 484, 485. centre of, 482. Matter, 13. Measures, comparative table of British and French, 636. of length, 13, 14. of stress, 69. of time, 381. of velocity, 382. of weight, 18. Mechanics, 13. applied, 13. Mechanism, theory of, 421. aggregate combinations in, 425, 466. elementary combinations in, 423, 426. principle of connection in, 424. Mercury, weight of, 69. Modulus of elasticity, 279, 631. of rupture, 316, 634. Moment, bending, 307. of a couple, 22. of deviation, 528. of flexure, 311. of friction, 614. of inertia, 514. of inertia of a surface, 77. of inertia, table of, 518. of stability, 233. of stress, 73. of torsion, 353. statical (see Moment of a Couple), also 27, 29. Momentum, 482. and impulse, law of, 484. angular (see Angular Momentum). conservation of, 505. of a rotating body, 529. Motion, 14. comparative, 384, 389. component and resultant, 381, 383. deviated (see Deviation). first law of, 476. of a system of bodies, 505. of fluids, dynamics of (see Hydrody- namics. of gases (see Gas). of liquids (see Hydrodynamics and Liquid). of points, 379. of points, varied, 385. of pliable bodies and fluids, 408. of pliable bodies, dynamics of, 552. of rigid bodies, 390". eecond law of, 484. Motion, uniform, dynamical principles of, 4*C. varied, dynamical principles of, 482. Muscular strength, work of, 625, NOTCH, flow through, 573. OIL, 613. Orifice, flow through, 571. Oscillation, 416. angular (see Gyration). centre of (see Centre). elliptical, 495. straight, 494. PARABOLA, formulae relating to, 165. Parallel Forces, 25. motion, 469. projection (see Projection, Parallel). Pendulum, ballistic, 548. compound oscillating, 546. compound revolving, 547. cycloidal, 497. rotating, 547. simple oscillating, 496. simple revolving, 492. Percussion, centre of (see Centre). Periodical motion of machines, 611. Pieces of a structure, 129. Piers, stability of, 228. of arches, 2*63. open and hollow, 263. Pile driving, 564. Pillars, strength of short, 302. strength of long, 360. Pinion (see Wheel). Pinnacle on a buttress, 239. Pipes, friction in, 588. flow in, 411, 588. resistance caused by sudden enlarge- ment in, 589. resistance of curves and knees' in, 589. resistance of mouthpieces of, 589. strength of, 289. Piston, 413, 419. action of a fluid upon, 604. Piston rods, strength of, 363. Pivot, friction of, 616. Plasticity, 272. Plate-iron girders (see Beam, Girder). Plates, strength of iron and steel, 377. Pliability, 273. co-eflicients of, 277. Point, 13. 650 INDEX. Point, fixed, 14, 381. motions of, 379. physical, 13. Posts, timber, strength of, 365. Pound, standard, 18. Power, 610. Press, Hydraulic, 464. strength of, 290. Pressure, 20, 69. in a sloping solid mass, 126. internal (see Stress). of earth (see Earth). of fluids (see Fluid). Prime movers, 625. Projection, parallel, 45, 61, 127. Proof strength, 273, 274. Pull (see Tension). Pulleys and belts, 454. and cords, 462. speed, 457. Pump, centrifugal, 597. Pump rods, strength of, 297. RACK, motion of, 427. Railways, resistance on, 620. Reciprocating force, 503. Reduced inertia, 621. Reduction of forces and conples in ma- chines to the driving point, 612. Regulator of a prime mover, 625. Repose, angle of (see Angle). Reservoir walls, stability of, 243. Resilience, 273. of axle, 357. of beam, 330. of tie-bar, 287. Resistance, 476. centre of, 131. line of, 131. of carriages on roads, 619. of fluids, 598, of machines, 610 (see Friction). of materials (see Strength). of railway trains and engines, 620. of rolling, 619. Resolutionof forces, 37. of internal stress, 82. Rest, 14. Resultant, 18. momentum, 482. of any system of forces, 41. of couples, 23, 24. of inclined forces, 35. of motions, 381. Resultant of parallel forces, 26, 28, 30. of stress, 70. of weight, 49. Retaining walls, 227. Revetements (see Retaining Walls). Rib (see Arch, Linear). Rigid body, motion of, 390, 394, 513, (see Rotation). action of a single force on, 543- Rigidity or stiffness, 271. of a truss, 144. supposition of perfect, 18. Rivets, strength of, 299. Ri vetted joints, strength of, 289, 299. Roads, resistance of, 619. Rolling of cylinder on plane, 398. cones, 405, 535. contact in mechanism, 426. of cylinder on cylinder, 400. of plane on cylinder, 398. resistance, 619. Roof (see Frame, also Truss). Ropes, strength of, 288. stiffness of, 619. Rotating body, comparative motion of points in, 393. relative motion of a pair of points in, 392. Rotation, 390. actual energy of, 532. alternate (see Gyration). and force, analogy of, 405. angular velocity of, 391. axis of, 390. combined with translation, 394. comparative motions in compound, 406. compound, 399. dynamical principles of, 513. free, 533. instantaneous axis of, 397, 543. uniform, 535. varied, 406, 538. varied, combined with translation, 543. Rupture, modulus of, 316, 634. SAFETY, factors of, 274. Screw-like motion, 394. Screws, friction of, 226. compound, 467. in mechanism, 449. Sections, method of, applied to frame- work, 150. Set, 272. Shaft, strength of (see Axle). INDEX. 651 Shear, 69, 87. Shearing, resistance to, 298; table, 633. force in beams, 307. stress in beams, 338. Shifting or translation, 390. Sliding contact in mechanism, 436. Solid, 13. Specific gravity (see Gravity, Specific). Speed-cones, 457. Speed, fluctuations of, 622. Spheres, strength of, 290. Spiral, 398. Stability, 128. frictional (see Frictional). of structures, 130, 131. Standard measure of length, 14. measure of weight, 18. Starting machines, 624. Statics, 15. principles of, 17. Stays, 133, 136. Steady motion of a liquid, 412, 414. of a gas, 419. Steam, action of, 606. engine, efficiency of, 629, 630. Stiffness, 130, 270", 273. of beams (see Beam, deflection of). Stopping machines, 624. Strain, 272. and stress, relations between, 280. ellipse of, 280. resolution and composition of, 275. Stream of liquid, 411, 586. friction of, 586. of gas, 417. hydraulic mean depth of, 587. varying, 587. Strength, 130, 270. of abutments and vaults, 268. of axles, 353, 358. of beams, 307, 315 (see Beam). of boilers, pipes, and cylinders, 289, 299, 306. of bolts, pins, keys, and rivets, 299. of iron and steel, 377. of iron, effects of repeated melting on, 376. of leathern belts, 288. of long pillars and struts, 360. of masonry and brickwork, 268, 302. of pump-rods, 297. of ropes and cables, 288. of short pillars, 304. of spheres, 290, 295. Strength of teeth, 359. of tie-bar, 286. of tubes and flues, 306. proof, 273. tables of, 377, 631. transverse, 315. ultimate, 273. Stress, 68. and strain, relations between, 280. internal, 82. Stretching, resistance to, 286. Stroke, length of, in mechanism, 46(^ Structures, 15. theory of, 129. transformation of, 129. Struts, 133. strength of (see Pillars). wrought-iron, strength of, 364. Superposition of small motions, 555. Surface, 13. Suspension bridge, 149, 165. stiffened, 370. strength of, 286, 288, 301. with sloping rods, 171. with vertical rods, 168. System of bodies, motion of, 505. TEARING, resistance to, 286. tables of resistance to, 288, 289, 377, 631. Teeth of wheels, dimensions of, 447. epicycloidal, 444. friction of, 617. form of, 438. involute, 441. of bevel wheels, 448. of wheel and trundle, 447. pitch and number of, 432. strength of, 359. Tenacity, 286 (see Tearing, Resistance to). Tension, 69 (see Stretching). Testing strength (see Proof). Theory and practice hi mechanics, har- mony of, 1. Thrust, 69. Tie, 132. flexible, 169. strength of, 286. Time, measure of, 381. Torsion (see Wrenching). Toughness, 273. Towers, stability of, 240. Trains of mechanism, 465. epicyclic, 473. 652 INDEX. Trains of Wheels, 434. Transformation (see Projection). of cords and chains, 180. of frames, 162. of stress, 92. of structures in masonry, 232, 268. Translation or shifting, 390. varied, 482. Transverse strength, 315 ; table, 634. Trochoid, 398. Trundle, 447. Truss, 144. compound, 148. Trussing, secondary, 145. Turbine, 595, 629* Turning (see Rotation). Twisting (see Wrenching). UNBALANCED FORCE, measures of, 501. Unguents, 613. Uniform motion, 382. deviation, 387. effort or resistance, effect of, 490. motion under balanced forces, 476. velocity, 382. Universal joint, 461. double, 462. Unsteady motion of fluid, 413, 415. VANES, impulse of liquid on, 593. Vaults, stability of, 226 (see Arch). groined, 262. Velocities, virtual, 479. 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In short, no labour or expense has been spared to sustain the well-known reputation of this Work, as " the Representative Book of the Medical Science and Practice of the day." Opinions of the Press. "The work is an admirable one, and adapted to the requirements of the Student, Professor, and Practitioner of Medicine Malignant Cholera is very fully dis- cussed, and the reader will find a large amount of information not to be met with in other books, epitomised for him in this The part on Medical Geography forms an admirable feature of the volumes We know of no work that contains so much, or such full and varied, information on all subjects connected with the Science and Practice of Medicine." Lancet. " The extraordinary merit of Dr. Aitken's work The author has unquestionably performed a service to the profession of the most valuable kind. The article on Cholera undoubtedly offers the most clear and satisfactory summary of our knowledge respecting that disease which has yet appeared." 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By ALEXANDER WYNTER BLYTH, M.R.C.S., F.C.S., etc., Analyst for the County of Devon, and Medical Officer of Health for the North Devon Combination of Sanitary Authorities. Medium 8vo, 672 pp., cloth bevelled, with Map, Diagram, and 140 Illustrations, Price 28/- GENERAL CONTENTS. The Work comprises over Seven Hundred Articles, embracing the following subjects : I. SANITARY CHEMISTRY: the Composition and Dietetic Value of Foods, with the latest Processes for the Detection of Adul- terations. II. SANITARY ENGINEERING : Sewage, Drainage, Storage of Water, Ventilation, Warming, etc. III. SANITARY LEGISLATION : the whole of the PUBLIC HEALTH ACT, 1875, together with sections and portions of other Sanitary Statutes, (without alteration or abridgment, save in a few unimportant instances) in a form admitting of easy and rapid reference. IV. EPIDEMIC AND EPIZOOTIC DISEASES : their History and Pro- pagation, with the Measures for Disinfection. V. HYGIENE MILITARY, NAVAL, PRIVATE, PUBLIC, SCHOOL. 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" The articles on Food and its Adulterations are good, the most recent methods of examination being given, and the chemical processes well described." Lancet. " A very important Treatise ... an examination of its contents satisfies us that it is a work which should be highly appreciated." Medico-Chirurgical Review. " A work that must have entailed a vast amount of labour and research. . . . Will be found of extreme value to all who are specially interested in Sanitation. It is more than probable that it will become a STANDARD WORK IN HYGIENE AND PUBLIC HEALTH." Medical Times and Gazette. " Mr. Blyth has ably filled a void in British Sanitary literature. . . . This STANDARD WORK . . . indispensable for all who are interested in Public-Health matters, and for all Public Libraries." Public Health. " Contains a great mass of information of easy reference ... a compilation carefully made from the best sources. Many of the articles are very good." Sani- tary Record. 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VOL. 8. MATHEMATICAL SCIENCE. Philosophy of Arithmetic; Algebra and its Solutions ; Plane Geometry ; Logarithms ; Plane and Spherical Trigonometry; Mensuration and Practical Geometry, with use of Instruments, by Prof. YOUNG, Rev. J. F. TWISDEN, M.A., Sand- hurst College, and ALEXANDER JARDINE, C.E. VOL. 9. MECHANICAL PHILOSOPHY. The Properties of Matter, Elementary Statics; Dynamics; Hydrostatics; Hydrodynamics; Pneu- matics ; Practical Mechanics ; and the Steam Engine, by the Rev. WALTER MITCHELL, M.A., J. R. YOUNG, and JOHN IMRAY. 12 CHARLES GRIFFIN & COMPANY'S THE CIRCLE OF THE SCIENCES, In Separate Treatises. Cloth. 5. d. 1. ANSTED'S Geology and Physical Geography . . .26 2. BREEM'S Practical Astronomy 26 3. BRONNER and SCOFFERN'S Chemistry of Food and Diet . i 6 4. BUSHNAN'S Physiology of Animal and Vegetable Life . i 6 5. GORE'S Theory and Practice of Electro-Deposition . i 6 6. IMRAY'S Practical Mechanics i 6 7. JARDINE'S Practical Geometry i 6 8. 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S., F.L.S.): A POPULAR HISTORY OF THE ANIMAL CREATION : being a Systematic and Popular Description of the Habits, Structure and Classification of Animals. With coloured Frontispiece and many hundred Illustrations. Crown 8vo. T.loth, 8/6. New Edition. SCIENTIFIC PUBLICATIONS. 13 DOUGLAS'S TELEGRAPH CONSTRUCTION. Published with the Approval of the Director-General of Telegraphs in India. A MANUAL OF TELEGRAPH COiNSTRUC- TION : The Mechanical Elements of Electric Telegraph Engineering. For the use of Telegraph Engineers and others. By JOHN CHRISTIE DOUGLAS, Society of Telegraphic Engineers, East India Government Telegraph Department, &c. With numerous Diagrams. Crown 8vo. Cloth, bevelled, i5/- Second Edition, with Appendices and Copious Index, now ready. GENERAL CONTENTS. PART I. GENERAL PRINCIPLES OF STRENGTH AND STABILITY, comprising the Strength of Materials; the Dis- tribution of Load and Stress in Telegraph Structures, such as Poles simple, strutted, tied, stayed, coupled, and trussed; the Catenary, with application of its Formulas to the cases of Wires and Cables; Theory of the Submersion of Cables, &c. PART II. PROPERTIES AND APPLICATIONS OF MATE- RIALS, OPERATIONS, AND MANIPULATION, including the Prin- ciples and Practice of, and Numerical Data for, designing Simple Structures, such as Poles of Iron and Wood ; Iron and Wooden Masts simple and compound; Specifications for Wire, &c. ; Soldering; Surveying; the Raising of Heavy Masts ; Insulating Materials and their Applications, &c. PART III. TELEGRAPH CONSTRUCTION, MAINTENANCE AND ORGANISATION, treating of the Application of the In- formation conveyed in Parts I and II. to the case of Combined Structures, including the Construction of Overground, Subter- ranean, and Subaqueous Lines ; Office Fittings ; Estimating; Organisation, &c. " Mr. Douglas deserves the thanks of Telegraphic Engineers for the excellent ' Manual ' now before us .... he has ably supplied an existing want the subject is treated with great clearness and judgment .... good practical information given in a clear, terse style." Engineering. "Mr. Douglas's work is, we believe, the first of its kind. . . . The author is evidently a practical Telegraphic Engineer The amount of information given is such as to render this volume a most useful guide to any one who may be engaged in any branch of Electric-Telegraph Engineering." Athenccum. " The book is calculated to be of great service to Telegraphic Engineers. . . . the arrangement is so judicious that with the aid of the full table of contents, reference to any special point should be easy." Iron, GRIFFIN (John Joseph, F.R.S.) : CHEMICAL RECREATIONS: A Popular Manual of Experi- mental Chemistry. With 540 Engravings of Apparatus. Crown 4to. Cloth. Tenth Edition. Part I. Elementary Chemistry, price 2/- Part II. The Chemistry of the Non-Metallic Elements, including a Comprehensive Course of Class Experiments, price xo/6. Or, complete in one volume, cloth, gilt top, 12/6 CHARLES GRIFFIN & COMPANY'S LEAKED (Arthur, M.D., F.R.C.P., Senior Phy- sician to the Great Northern Hospital) : IMPERFECT DIGESTION : Its Causes and Treatment. Post Svo. Cloth, 4/6. Sixth Edition. " It now constitutes about the best work on the subject." Lancet. " Dr. Leared has treated a most important subject in a practical spirit and popular manner." Medical Times and Gazette. " A useful manual of the subject upon which it treats, and we welcome it as an addition to our Medical Literature." Dublin Quarterly Journal of Medical Science. MOFFITT (Staff-Assistant-Surgeon A., of the Royal Victoria Hospital, Netley) : A MANUAL OF INSTRUCTION FOR ATTENDANTS ON THE SICK AND WOUNDED IN WAR. Published under the sanction of the National Society for Aid to the Sick and Wounded in War. With numerous Illustrations. Post Svo. Cloth, 5/- " A work by a practical and experienced author. After an explicit chapter on the Anatomy of the Human Body, directions are given concerning bandaging, dressing of sores, wounds, &c., assistance to wounded on field of action, stretchers, mule litters, ambulance, transport, &c. All Dr. Moffitt's instructions are assisted by well executed illustrations." Public Opinion. " A well written volume. Technical language has been avoided as much as possible, and ample explanations are afforded on all matters on the uses and management of the Field Hospital Equipment of the British Army." Standard. NAPIER (James, F.R.S.E., F.C.S.): A MANUAL OF ELECTRO-METALLURGY. With numerous Illustrations. Crown Svo, cloth, 7/6. Fifth Edition, revised and enlarged. GENERAL CONTENTS. I. HISTORY of the ART. II. DESCRIPTION of GALVANIC BAT- TERIES and their RESPECTIVE PECULIARITIES. III. ELECTROTYPE PROCESSES. IV. BRONZING. V. MISCELLANEOUS APPLICATIONS of the Process of COATING with COPPER. VI. DEPOSITION of METALS upon one another. VII. ELECTRO-PLATING. VIII. ELECTRO-GILDING. IX. RESULTS of EXPERIMENTS on the DEPOSITION of other METALS as COATINGS. X. THEORETICAL OBSERVATIONS. " A work that has become an established authority on Electro-Metallurgy, an art which has been of immense use to the Manufacturer in economising the quantity cf the precious metals absotbed, and in extending the sale of Art Manufactures We can heartily commend the work as a valuable handbook on the subject on which it treats." Journal of Applied Science. " The fact of Mr. Napier's Treatise having reached a FIFTH EDITION is good evidence of an appreciation of the Author's mode of treating his subject A very useful and practical little Manual." Iron. " The Fifth Edition has all the advantages of a new work, and of a proved and tried friend. Mr. Napier is well-known for the carefulness and accuracy with which he writes . . . there is a thoroughness in the handling of the subject which is far from general in these days . . . The work is one of those which, besides supplying first-class information, are calculated to inspire invention." Jeweller and Watchmaker. SCIENTIFIC PUBLICATIONS. 15 NAPIER (James, F.R.S.E., F.C.S.) : A MANUAL OF THE ART OF DYEING AND DYEING RE- CEIPTS. Illustrated by Diagrams and Numerous Specimens of Dyed Cotton, Silk, and Woollen Fabrics. Demy 8vo., cloth, 2i/-. Third Edition, thoroughly revised and greatly enlarged. GENERAL CONTENTS: PART!. HEAT AND LIGHT: Their effects upon Colours, and the changes they produce in many Dyeing Operations. PART II. A CONCISE SYSTEM OF CHEMISTRY, with special reference to Dyeing : Elements of Matter, their physical and chemical properties, producing in their combination the different Acids, Salts, &c., in use in the Dye-House. PART III. MORDANTS AND ALTERANTS : Their composition, properties, and action in fixing Colours within the Fibre. PART IV. VEGETABLE MATTERS in use in the Dye-House: ist, those containing Tannin, Indigo, &c. ; 2ndly, the various Dyewoods and Roots, as Logwood, Madder, Bark, &c. PART V. ANIMAL DYES : Cochineal, Kerms, Lac, &c. PART VI. COAL-TAR COLOURS : Their Discovery, Manufacture, and Introduction to the Dyeing- Art, from the discovery of MAUVE to ALIZARIN. APPENDIX. RECEIPTS FOR MANIPULATION : Bleaching; Removing Stains and Dyes; Dyeing of different Colours upon Woollen, Silk, and Cotton Materials, with Patterns. " The numerous Dyeing Receipts and the Chemical Information furnished will be exceedingly valuable to the Practical Dyer a Manual of necessary reference to all those who wish to master their trade, and keep pace with the scientific discoveries of the time." Journal of Applied Science. "In this work Mr. Napier has done good service being a Practical Dyer himself, he knows the wants of his confreres the Article on Water is a very valuable one to the Practical Dyer, enabling him readily to detect impurities, and correct their action The Article on Indigo is very exhaustive the Dyein^ Receipts are very numerous, and well illustrated." Textile Manufacturer. PHILLIPS (John, M.A., F.R.S., F.G.S.,late Pro- fessor of Geology at the University of Oxford). A MANUAL OF GEOLOGY : Practical and Theoretical. Revised and Edited by ROBERT ETHERIDGE, F.R.S., F.G.S., of the Museum of Practical Geology. (In Preparation). 16 CHARLES GRIFFIN &> COMPANY'S PHILLIPS (J.Arthur,M.Inst.C.E.,F.C.S.,F.G.S., Ancien Eleve de 1'Ecole des Mines, Paris) : ELEMENTS OF METALLURGY: A Practical Treatise on the Art of Extracting Metals from their Ores. With over two hundred Il- lustrations, many of which have been reduced from Working Drawings. Royal 8vo, 764 pages, cloth, 34/- GENERAL CONTENTS: I. A TREATISE on FUELS and REFRACTORY MATERIALS. II. A Description of the principal METALLIFEROUS MINERALS, with their DISTRIBUTION. III. STATISTICS of the amount of each METAL annually produced throughout the World, obtained from official sources, or, where this has not been practicable, from authentic private information. IV. The METHODS of ASSAYING the different ORES, together with the PROCESSES of METALLURGICAL TREATMENT, comprising : IRON, COBALT, NICKEL, ALUMINIUM, COPPER, TIN, ANTIMONY, ARSENIC, ZINC, MERCURY, BISMUTH, LEAD, SILVER, GOLD and PLATINUM. "In this most useful and handsome volume Mr. Phillips has condensed a large amount of valuable practical knowledge Wehave not only the results of scientific inquiry most cautiously set forth, but the experiences of a thoroughly practical man, very clearly given." AthencEum. " For twenty years the learned author, who might well have retired with honour on account of his acknowledged success and high character as an authority in Metal- lurgy, has been making notes, both as a Mining Engineer and a practical Metallurgist, and devoting the most valuable portion of his time to the accumulation of materials for this, his Masterpiece. There can be no possible doubt that ' Elements of Metal- lurgy' will be eagerly sought for by Students in Science and Art, as well as by Practi- cal Workers in Metals Two hundred and fifty pages are devoted exclusively to the Metallurgy of Iron, in which every process of manufacture is treated, and the latest improvements accurately detailed." Colliery Guardian. "The value of this work is almost inestimable. There can be no question that the amount of time and labour bestowed on it is enormous There is certainly no Metallurgical Treatise in the language calculated to prove of such general utility to the Student really seeking sound practical information upon the subject, and none which gives greater evidence of the extensive metallurgical knowledge of its author." Mining Journal. PORTER : (Surgeon-Major J. H., Assistant- Professor of Military Surgery in the Army Medical School, Hon. Assoc. of the Order of St. John of Jerusalem) : THE SURGEON'S POCKET-BOOK: An Essay on the Best Treat- ment of the Wounded in War ; for which a Prize was awarded by Her Majesty the Empress of Germany. Specially adapted to the PUBLIC MEDICAL SERVICES. With numerous Illustrations, i6mo, roan, 7/6. "Just such a work as has long been wanted, in which men placed in a novel position, can find out quickly what is best to be done. We strongly recommend it to every officer in the Public Medical Services." Practitioner. "A complete vade mecum to guide the military surgeon in the field." British Medical Journal. "A capital little book . . . of the greatest practical value. . . . A surgeon with this Manual in his pocket becomes a man of resource at once." Westminster Review. SCIENTIFIC PUBLICATIONS. 17 SCIENTIFIC MANUALS BY W. J. MACQUOEN EANKINE, O.E., LL.D., F.E.S., Late Regius Professor of Civil Engineering in the University of Glasgow. I. RANKINE (Prof.): APPLIED MECHANICS (A Manual of) ; comprising the Principles of Statics and Cinematics, and Theory of Structures, Mechanism and Machines. With numerous Diagrams. Revised by E. F. B AMBER, C.E. Crown 8vo. Cloth, 12/6. Ninth Edition. " Cannot fail to be adopted as a text-book The whole of the information so admirably arranged that there is every facility for reference." Mining Journal. II. RANKINE (Prof.): CIVIL ENGINEERING (A Manual of) ; comprising Engineering Surveys, Earthwork, Founda- tions, Masonry, Carpentry, Metal-work, Roads, Railways, Canals, Rivers, Water- works, Harbours, &c. With numerous Tables and Illus- trations. Revised by E. F. BAMBER, C.E. Crown 8vo. Cloth, id/- Twelfth Edition. " Far surpasses in merit every existing work of the kind. As a Manual for the hands of the professional Civil Engineer it is sufficient and unrivalled, and even when we say this we fall short of that high appreciation of Dr. Rankine's labours which we should like to express." The Engineer. III. RANKINE (Prof.): MACHINERY AND MILL WORK (A Manual of); comprising the Geometry, Motions, Work, Strength, Construction, and Objects of Machines, &c. Illus- trated with nearly 300 Woodcuts. Revised by E. F. BAMBER, C.E. Crown 8vo. Cloth, 12/6. Third Edition. "Professor Rankine's 'Manual of Machinery and Millwork' fully maintains the high reputation which he enjoys as a scientific author ; higher praise it is difficult to award to any book. It cannot fail to be a lantern to the feet of every engineer." The Engineer. IV. RANKINE (Prof.): The STEAM ENGINE and OTHER PRIME MOVERS (A Manual of). With Diagram of the Mechanical Properties of Steam, numerous Tables and Illustrations. Revised by E. F. BAMBER, C.E. Crown 8vo. Cloth, 12/6. Eighth Edition. V. RANKINE (Prof.): USEFUL RULES and TABLES. For Architects, Builders, Carpenters, Coachbuilders, En- gravers, Engineers, Founders, Mechanics, Shipbuilders, Surveyors, Wheelwrights, &c. Crown 8vo. Cloth, g/- Fifth Edition. 44 Undoubtedly the most useful collection of engineering data hitherto produced." Mining Journal. VI. RANKINE (Prof.): A MECHANICAL TEXT-BOOK. By Professor M ACQUORN RANKINE & E. F. BAMBER, C.E. With numerous Illustrations. Crown 8vo. Cloth, g/- Second Edition. " The work, as a whole, is very complete, and likely to prove invaluable for furnish- ing a useful and reliable outline of the subjects treated of." Mining Journal. %* The MECHANICAL TEXT-BOOK forms a simple Introduction to PROF. RANKINE'S SERIES Of MANUALS On ENGINEERING and MECHANICS i8 CHARLES GRIFFIN & COMPANY'S SHELTON (W. Vincent, Foreman to the Imperial Ottoman Gun-Factories, Constantinople). THE MECHANIC'S GUIDE: A Hand-book for Engineers and Artizans. With Copious Tables and Valuable Recipes for Practical Use. Illustrated. Crown 8vo, cloth, 7/6. GENERAL CONTENTS: PART I. Arithmetic. PART II. Geometry. PART III. Mensuration. jects PART IV. Velocities in Boring and Wheel-Gearing. PART V. 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" ' Cobbett's French Grammar ' comes out with perennial freshness. There are few grammars equal to it for those who are learning, or desirous of learning, French without a teacher. The work is excellently arranged, and in the present edition we note certain careful and wise revisions of the text." School Board Chronicle. " Business men commencing the study of French will find this treatise one of the best aids It is largely used on the Continent." Midland Counties Herald. COBBETT (James Paul): A LATIN GRAM- MAR. Fcap. 8vo. Cloth, 2/- COLERIDGE (Samuel Taylor) : A DISSERTA- TWN ON THE SCIENCE OF METHOD. (Encyclopedia Metro, politana.) With a Synopsis. Crown 8vo. Cloth, 2/- Ninth Edition* 20 CHARLES GRIFFIN & COMPANY'S CRAIK'S ENGLISH LITERATURE. A COMPENDIOUS HISTORY OF ENGLISH LITERATURE AND OF THE ENGLISH LANGUAGE from the Norman Conquest. With numerous specimens. By GEORGE LILLIE CRAIK, LL.D., late Professor of History and English Literature, Queen's College, Belfast. In two vols. 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COBBIN'S MANGNALL: MANGNALL'S HISTORICAL AND MISCEL- LANEOUS QUESTIONS, for the Use of Young People. By RICHMAL MANGNALL. Greatly enlarged and corrected, and continued to the pre- sent time. By INGRAM COBBIN, M.A. i2mo. Cloth 4/- Forty-eighth Thousand. New Illustrated Edition. MENTAL SCIENCE: SAMUEL TAYLOR COLERIDGE'S CELEBRATED ESSAY ON METHOD; Arch- bishop WHATELY'S Treatises on Logic and Rhetoric. Crown 8vo. Cloth, 5/- Tenth Edition. WORKS BY WILLIAM RAMSAY, M.A., Trinity College, Cambridge, late Professor of Humanity in the University of Glasgow. A MANUAL OF ROMAN ANTIQUITIES. For the use of Advanced Students. With Map, 130 Engravings, and very copious Index. Revised and enlarged, with an additional Chapter on Roman Agriculture. Crown 8vo. Cloth, 8/6. Tenth Edi- tion. GENERAL CONTENTS. I. The Typography of Rome. II. The Origin of the 'Roman People ; their Political and Social Organization ; Religion ; Kalendar ; and Private Life. III. 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Cloth, 2/- EDUCATIONAL PUBLICATIONS. 23 THE SCHOOL BOARD READERS: A NEW SERIES OF STANDARD READING BOOKS. Edited by a former H.M. INSPECTOR of SCHOOLS. RECOMMENDED BY THE LONDON SCHOOL BOARD. And adopted by many School Boards throughout the Country. *** AGGREGATE SALE, igO,OOO COPIES. ELEMENTARY READING BOOK, PART I. Containing Lessons s. d. in all the Short Vowel Sounds. Demy i8mo., 16 pages. In stiff wrapper or ELEMENTARY READING BOOK, PART II. Containing the Long Vowel Sounds and other Monosyllables. Demy i8mo, 48 pages. In stiff wrapper . . . . .02 STANDARD I. Containing Reading, Dictation, and Arith- metic. Demy i8mo, 96 pages. Neat cloth . . .04 STANDARD II. Containing Reading, Dictation and Arith- metic. Demy i8mo, 128 pages. Neat cloth . .06 STANDARD III. Containing Reading, Dictation and Arith- metic. Fcap. 8vo, 160 pages. Neat cloth . . .09 STANDARD IV. Containing Reading, Dictation and Arith- metic. Fcap. 8vo, 192 pages. Neat cloth . . .10 STANDARD V. 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" It is difficult to determine which of these volumes is the most attractive. Will be found equally enjoyable on a railway journey, or by the fireside." Mining Journal. " These additions to the Library, produced by Mr. Timbs 1 industry and ability, are useful, and in his pages many a hint and suggestion, and many a fact of importance, is stored up that would otherwise have been lost to the public." Builder. " Capital little books of about a hundred pages each, wherein the indefatigable Author is seen at his best." Mechanics' Magazine. " Extremely interesting volumes." Evening Standard. "Amusing, instructive, and interesting As food for thought and pleasant reading, we can heartily recommend the 'Shilling Manuals. 1 "Birmingham Daily (razette. TIMBS (John, F.S.A.): PLEASANT HALF- HOURS FOR THE FAMILY CIRCLE. Containing Popular Sci- ence, Thoughts for Times and Seasons, Oddities of History, Charac- teristics of Great Men, and Curiosities of Animal and Vegetable Life. Fcap. 8vo. 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Presentation Edition, Cloth and Gold ... 12s. 6d. each volume. Library Edition, Half Bound, Boxburghe ... 14s. Do., Morocco Antique 21s. M Each Series is complete in itself t and sold separately. "MAmr THOUGHTS," &c., are evidently the pro- dace of years of research, we look up any subject under the sun, and are pretty sure to find something that has been said generally well said upon it." Examiner. " Many beautiful examples of thought and style are to be found among the selections." Leader. "There can be little doubt that it is destined to take a high place among books of this class. "Note* and Queries. 11 A treasure to every reader who may be fortunate enough to possess it. Its perusal is like inhaling essences ; we have the cream only of the great authors quoted. Here all are seeds or gems." English Journal of Education. "Mr. Southgate's reading will be found to extend over nearly the whole known field of literature, ancient and modern." Gentleman's Magazine. "Here is matter suited to all tastes, and illustrative Of all opinions; morals, politics, philosophy, and solid Information. We have no hesitation in pronouncing it one of the most important books of the season. Credit is due to the publishers for the elegance with which the work is got up, and for the extreme beauty and correctness of the typography." Morning Chronicle. " Of the numerous volumes of the kind, we do not remember having met with one in which the selection vas more judicious, or the accumulation of treasures so truly wonderful." Morning Herald. "Mr. Southgate appears to have ransacked every nook and corner for gems of thought." Allen's Indian Mail. " The selection of the extracts has been made with taste, judgment, and critical nicety." Morning Post. "This is a wondrous book, and contains a great many gems of thought." Daily News. ' As a work of reference, it will be an acquisition to any man's library. "-Publisher's Circular. "This volume contains more gems of thought, re- fined sentiments, noble axioms, and extractable sentences, than have ever before been brought together iu our language." The Field. " Will be found to be worth its weight in gold by literary men." The Builder. " All that the poet has described of the beautiful in nature and art ; all the wit that has flashed from I-regnant minds ; all the axioms of experience, the collected wisdom of philosopher and sage, ar garnered into one heap of useful and well-arranged instruction fid amusement." The Era. " The mind of almost all nations and ages Of th world is recorded here." John Bull. " This is not a law-book ; but, departing from 01 usual practice, we notice it because it is likely ' be very useful to lawyers." Law Times. " The collection will prove a mine, rich and inffl haustible, to those in search of a quotation/'-.^ Journal. " There is not, as we have reason to know, a sing trashy sentence in this volume. Open where we ma; every page is laden with the wealth of profounde thought, and all aglow with the loftiest inspirational genius. To take this book into our hands is like a'ttir down to a grand conversazione with the greate thinkers of all ages." Star. " The work of Mr. Southgate far outstrips all other; of its kind. To the clergyman, the author, the artis and the essayist, 'Many Thoughts of Many Mind cannot fail to render almost incalculable service." Edinburgh Mercury. " We have no hesitation whatever in describing M Sonthgate's as the very best book of the class. The: is ^positively nothing of the kind in the language thi will bear a moment's comparison with it." ManchesH Weekly Advertiser. " There is no mood in which we can take it n without deriving from it instruction, consolation, an amusement. We heartily thank Mr. Southgate for book which we shall regard as one of our best frien< and companions." Cambridge Chronicle. " This work possesses the merit of being a magn ficent gift-book, appropriate to all times and seasons a book calculated to be of use to the scholar, tb divine, or the public man." Freemason's Magazine. "It is not so much a book as a library of quote tions." Patriot. " The quotations abound in that thought which i themainspring of mental exercise. "LiverpoolCouriet " For purposes of apposite quotation, it cannot b surpassed." Bristol Times. "It is impossible to pick out a single passage ii the work which does not, upon the face of it, just if; its selection by its intrinsic merit." Dorset Chroniclt " We are not surprised that a Second Series of thi work should have been called for. Mr. Southgat has the catholic tastes desirable in a good Editor Preachers and public speakers will find that it ha special uses for them." Edinburgh Daily Review. "The SKCOND.SERIES fully sustains the deserv* reputation of the First." John Bull. London : CHARLES GRIFFIN & COMPANY. o t r - 7 r - ~ >M^- V . 1 *; V v