PREFACE. THE object of tliis 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 Legons sur la Tkeorie mathematique de I'jElasticite 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 flius hitherto done so ki an exact manner, is M. Yvon-Yillarceaux, in the Mf moires des PREFACE. t)f;$ie ^transformation of structures and its appli- cations" hate* liiftierlx)- 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_, arc indicated in their places. In the arrangement of this treatise an effort has been made to adhere as rigidly as possible to a methodical classification, of its subjects; and, in particular, care has been taken to keep in view the distinction between the comparison of motions with each other, and the relations between motions and forces, which was first pointed out by Monge and Ampere, and which Mr. Willis has so successfully applied to the subject of mechanism. The observing of that distinction is highly conducive to the correct understanding and ready application of the principles of Mechanics. W. J. M. R. GLASGOW UNIVEESITY, May, 1858. ADVERTISEMENT TO THE TENTH EDITION. The Tenth Edition has been carefully revised, and new matter, bearing on subjects treated of in the text, has been added to the Appendix. The Index has also been enlarged, and rendered more suitable for reference. For various notes and suggestions, the Editor begs to thank, amongst others, Prof. Eddy, of Cincinnati University, anu Mr. Arthur W. Thompson, B.Sc., late of the Imperial Collegw of Engineering, Japan. W. J. M. GLASGOW, Jan., 1882. CONTENTS. PRELIMINARY DISSERTATION ON THE HARMONY OF THEORY AND PRACTICE IN MECHANICS, 1 INTRODUCTION, Definition of General Terms, and Division of the Subject, . 13 Article Page 1. Mechanics, 13 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 8. Rest, 14 9. Motion, 14 10. Fixed Point, . . 14 11. Cinematics, 12. Force, 13. Equilibrium or Balance, 14. Statics and Dynamics, 15. Structures and Machines, 15 15 15 15 15 16. General Arrangement of the Subject, 16 PART I. PRINCIPLES OF STATICS. CHAPTER I. BALANCE AND MEASUREMENT OF FORCES ACTING IN ONE STRAIGHT LINE. 17. Forces, how Determined, 18. Place of Application Point of Application, . 19. Supposition of Perfect Rigidity, . 20. Direction Line pf Action, . 17 21. Magnitude Units of Force, British and French, .... 18 22. Resultant of Forces acting in one Line,18 23. Representation of Forces by Lines, 19 24. Pressure, 20 CHAPTER II. THEORY OF COUPLES AND OF THE BALANCE OF PARALLEL FORCES. SECTION 3. On Parallel Forces. 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 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, . 26 41. Resultant of a Couple, and a single Force in Parallel Planes, . . 27 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, . . . . SEQTION 4. On Centres of Parallel Forces. 48. Centre of a pair of Parallel Forces, 31 49. Centre of any system of Parallel Forces, ..... 32 50. Co-ordinates of Centre of Parallel Forces, ..... 32 CHAPTER III BALANCE OP INCLINED FORCES. SECTION 1. Inclined Forces applied at One Point. 51. Parallelogram of Forces, . . 35 52. Equilibrium of three Forces acting through one Point in one Plane, 36 53. Equilibrium of any system of Forces acting through one Point, 36 VI CONTENTS. 54. Parallelepiped of Forces, 55. Resolution of a Force into two Components, .... 56. Resolution of a Force into three Components, .... 57. Rectangular Components, . sc | SECTION 2. Inclined Forces applied to a 37 System of Points. _ I 58> Forces acting in one Plane _ 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. 45 01. Parallel Projection of a Figure de- fined, ..... 62. Geometrical Properties of Parallel Projections, . . . . 45 63. Application to Parallel Forces, . 46 4G 67. Restriction of the subject to Par- allel Distributed Forces, . . ' 68. Intensity of a Distributed Force, SECTION 1. Of Weight, and Centres of Gravity. 69. Specific Gravity, . . . 70. Centre of Gravity, . . . 71. Centre of Gravity of a Homoge- neous Body having a Centre of Figure, . . . . . 72. Bodies having Planes or Axes of Symmetry, . 73. System of Symmetrical Bodies, . . 74. Homogeneous Body of any Figure, . 75. Centre of Gravity "found by Addi- tion, . . . . i 76. Centre of Gravity found by Sub- traction, ...... 77. Centre of Gravity altered by Trans- position, . . , 78. Centres of Gravity of Prisms and Flat Plates, . 79. Body with similar Cross-Sections, 80. Curved Rod, .... 81. Approximate Computation of In- tegrals, ..... 82. Centre of Gravity found by Pro- jection, 83. Examples of Centres of Gravity, 84. Heterogeneous Body, . 85. Centre of Gravity found experi- mentally, .... 64. Application to Centres of Parallel Forces, ..... 65. Application to Inclined Forces acting through one Point, . 46 66. Application to any system of Forces, 47 CHAPTER V. ON DISTRIBUTED FORCES. 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. P*ir 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 54 5G 57 08 61 68 G8 68 SECTION 2. Of Stress, and its Resultants and Centres. 86. Stress its Nature and Intensity, 68 87. Classes of Stress, ... 69 88. Resultant of Stress, its Magni- tude, 70 89. Centre of Stress, or of Pressure, . 71 90. Centre of Uniform Stress, . . 73 91. Moment of Uniformly -Varying Stress, Neutral Axis, . . 73 92. Moment of Bending Stress, . 75 U.S. Moment of Twisting Stress, . 76 94. Centre of Uniformly - Varying Stress, Conjugate Axis, . 76 95. Moments of Inertia of a Surface, 77 105 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 Internal Equi- librium, ... . 113 117. Equilibrium of Fluids, . 116 118. Equilibrium of a Liquid, . 118 119. Equilibrium of different Fluids in contact with each other, . . 118 120. Equilibrium of a Floating Body, 120 121. Pressure on an Immersed Body, 122 122. Apparent Weights, . . .123 123. Relative Specific Gravities, . 124 124. Pressure on an Immersed I 3 lane, 125 125. Pressure in an Indefinite Uni- formly Sloping Solid, . . 126 126. On the Parallel Projection of Stress and Weight, . . .127 CONTEXTS. Vll CHAPTER VI ON STABLE AND UNSTABLE EQUILIBRIUM. Page ! 127. Stable and Unstable Equilibrium I 128. Stability of a Fixed Body, of a Free Body, . . . 128 ' PART II. THEORY OF STRUCTURES. CHAPTER I DEFINITIONS AND GENERAL PRINCIPLES. 129. Structures Pieces Joints, . 129 131. Conditions of Equilibrium of a Page , 128 130. Supports Foundations, 129 Structure, .... 129 132. Stability, Strength, and Stiffness, 130 133. Resultant Gross Load, . . 131 | 134. Centre of Resistance of a Joint, 131 j 135. Line of Resistance, . . . 131 ! 136. Joints Classed, . . . . 131 I SECTION 1. Equilibrium and Stability , of Frames. 137. Frame, 132 138. Tie, . . . . . .132 139. Strut, 133 140. Treatment of the Weight of a Bar, 133 141. Beam under Parallel Forces, . 133 142. Beam under Inclined Forces, . 134 143. Load supported by three Parallel Forces, . . . ... 135 144. Load supported by three Inclined Forces, 135 145. Frame of Two Bars Equilibrium, 136 146. Frame of Two Bars Stability, 136 j 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 1 56 164. Lattice Girder, any Load, . . 160 165. Lattice Girder, Uniform Load, . 161 166. Transformation of Frames, . 162 SECTION 2. Equilibrium of Chains, Cords, Ribs, 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 163 CHAPTER II. STABILITY. 172. Suspension Bridge with Sloping Rods, . . . . . 171 173. Extrados and Intrados, , .173 174. Cord with Horizontal Extrados, 175 175. Catenary, 177 176. Centre of Gravity of a Flexible Structure, . . . .180 177. Transformation of Cords and Chains, 180 178. Linear Arches or Ribs, . . 182 179. Circular Arch for Uniform Fluid Pressure, . . . .183 180. Elliptical Arches for Uniform Pressures, . . . .184 181. Distorted Elliptic Arch, . 186 182. Arches for Normal Pressure in general, . . . . 189 183. Hydrostatic Arch (see also p. 353) 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,211 194. Frictional Stability of Earth, . 212 195. Mass of Earth with Plane Surface,214 196. Principle of Least Resistance, . 215 197. Earth Loaded with its ownWeight,216 198. Pressure of Earth against a Ver- tical Plane, . . . .218 199. Supporting Power of Earth- Foundations, . . . .219 200. Abutting Power of Earth, . 220 201. Table of Examples, . . .221 202. Frictional Tenacity or Bond of Masonry and Brickwork, . 222 203. Friction of Screws, Keys, and VVedges, . . . .226 204. Friction of Rest and Friction of Motion, ..... 226 Vlll COXTE.NTS. SECTION 4. On the Stability of Abut- ments and Vaults. Page 205. Stability at a Plane Joint, . 226 206. Stability of a series of Blocks Line of Resistance Line of Pressures, . . . .230 207. Analogy of Blockwork and Framework, . . . .231 208. Transformation of Blockwork Structures, . . . .232 209. Frictional Stability of a Trans- formed Structure, . . .233 210. Structure not laterally pressed, 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, . 25 i Paga 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, . . . . 229. Groined Vaults, . . . 230. Clustered Arches, . . . 231. Piers of Arches, . . . 232. Open and Hollow Piers and Abutments, . . . . 233. Tunnels, ..... 264 234. Domes, ..... 265 235. Strength of Abutments and Vaults,268 235A. Transformation of Structures in Masonry, . . . .268 261 262 263 263 263 CHAPTER III. STRENGTH AND STIFFNESS. SECTION 1. Summary of General Principles. 236. Theory of Elasticity, . 270 237. Elasticity defined, . . 270 238. Elastic Force or Stress, . 270 239. Fluid Elasticity, . . 270 240. Liquid Elasticity, . . 271 241. Rigidity or Stiffness, . . 271 242. Strain and Fracture, . 272 243. Perfect and Imperfect Elasticity Plasticity, . . . .272 244. Strength, Ultimate and Proof- Toughness Stiffness Spring, 273 245. Determination of Proof Strength, 274 246. Working Stress, . . .274 247. Factors of Safety, . . .274 248. Divisions of Mathematical Theory of Elasticity, . . . .275 249. Resolution and Composition of Strains, 275 250. Displacements, .... 276 251. Analogy of Stresses and Strains, 276 252. Potential Energy of Elasticity. 253. Co-efficients of Elasticity, . 254. Co-efficients of Pliability, . 255. Axes of Elasticity, . 256. Isotropic Solid, 257. Modulus of Elasticity, 277 277 277 278 278 279 279 258. 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, . 280 261. Ellipsoid of Strain, 262. Transverse Elasticity of an Iso- tropic Substance, 263. Cubic Elasticity, 264. Fluid Elasticity, 283 284 285 285 SECTION 3. On Resistance to Stretching and Tearing. 265. Stiffness and Strength of a Tie-bar, 286 266. Resilience, or Spring of a Tie-Bar Modulus of Resilience , ; 287 267. Sudden Pull, . . . .287 268. Explanation of Table of the Re- sistance of Materials to Stretch- ing and Tearing. (see page 646), . . . . .288 269. Additional data Welded Joint Iron Wire Ropes Hempen Cables Leather Belts Chain Cables, 288 270. Strength of Rivetted Joints, . 288 271. Thin Hollow Cylinders Boilers 289 290 290 294 295 296 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 p. 648), 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 CONTENTS. Page 284. Explanation of Table of the Re- sistance of Materials to Crush- ing by a Direct Thrust (see p. 648), . . . .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 nd only, . 310 290. Beams Supported at both ends, 310 291. Moments of Flexure in terms of Load and Length, . . .311 292. Uniform Moment of Flexure Railway Carriage Axles, . . 312. 293. Resistance of Flexure, . . 312 294. Transverse Strength of Beams in General, . . . .315 295. Transverse Strength in Terms of ' Breadth and Depth, . . 316 296. Explanation of Table of Moduli of Rupture. (see p. 649), 317 297. Modulus of Rupture of Cast Iron Beams, ..... 318 298. Section of Equal Strength for Cast Iron Beams, . . . 319 299. Beams of Uniform Strength, . 320 300. Proof Deflection of Beams, . 322 301. Deflection found by Graphic Con- struction, . . . .326 302. Proportion of the greatest Depth of a Beam to the Span, . . 327 303. Slope and Deflection of a Beam under any load, . . . 328 304. Deflection with Uniform Moment, 330 305. Resilience or Spring of a Beam, 330 306. Suddenlyapplied Transverse Load, 332 307. Beam Fixed at both ends, . . 332 308. Beam Fixed at one end, and Sup- ported at both, . . . 338 309. Shearing Stress in Beams, . 338 310. Lines of Principal Stress in Beams, 341 311. Direct Vertical Stress, . . 342 312. Small effect of Shearing Stress upon Deflection, . . . 342 513. Partially-loaded Beam, . . 344 514. Allowance for Weight of Beam, 346 IX Pago 347 348 348 315. Limiting Length of Beam, 316. Sloping Beam, . 317. Originally Curved Beam, 318. Expansion and Contraction of Long Beams, . . . 348 319. Elastic Curve, . . . 349 319A. Hydrostatic Arch, . . 353 SECTION 7 On Resistance to Twisting and Wrenching. 320. Twisting Moment, . . . 353 321. Strength of a Cylindrical Axle. 353 322. Angle of Torsion of a Cylindrical Axle, 356 323. Resilience of a Cylindrical Axle, 357 324. Axles not Circular in Section, . 358 325. Bending and Twisting combined ; Crank and Axle, . . .358 326. Teeth of Wheels, . . .359 SECTION 8. On Crushing by Bending. 327. Introductory Remarks, . . 360 328. Strength of Iron Pillars and Struts, 361 329. Connecting Rods, Piston Rods, . 363 330. Comparison of Cast and Wrought Iron Pillars, . . . .363 331. Mr. Hodgkinson's Formula? for the Ultimate Strength of Cast Iron Pillars, . . . .363 332. Wrought Iron Framework, . 364= 333. Wrought Iron Cells, ... .364= 334. Sides of Plate Iron Girders, . 365 335. Timber Posts and Struts, . . 365 SECTION 9. On Compound Girders, Frames, and Bridges. 336. Compound Girders in General, . 366 337. Plate Iron Girders, . . . 366 338. Half-Lattice and Lattice Beams, 369 339. Bowstring Girder, . . .369 340. Stiffened Suspension Bridges, . 370 341. Ribbed Arches, . . .376 SECTION 10. Miscellaneous Remarks on Strength and Stiffness 342. Effects of Temperature, . 376 343. Effects of Repeated Meltings on Cast Iron, . . . 376 344. Effects of Ductility, . . 376 345. Internal Friction, . . 377 346. Concluding remarks on Strength and Stiffness, .... 377 PART III. PRINCIPLES OF CINEMATICS, OR THE COMPARISON a>17 TV . , OF MOTIONS. 347. Division of the Subject, ........ . .379. CHAPTER I. MOTIONS OF POINTS. SECTION 1 Uniform Motion of a Pair of Points. 548. Fixed andNearly Fixed Directions, 379 349. Motion of a Pair of Points, . 380 [60. Fixed Point and Moving Point, 381 151, Component andResultant Motions, 381 552. Measurement of Time, . 381 353. Uniform Velocity, . . . 382- 354. Uniform Motion, . . .382 SECTION 2. Uniform Motion of several Points. 355. Motion of Three Points, . . 383 356. Motions of a Series of Points, . 383 CONTENTS. Pase . 384 . 384 357. Parallelepiped of Motions, 358. Comparative Motion, SECTION 3. Varied Motion of Points. 359. Velocity and Direction of Varied Motion, 385 360. Components of Varied Motion, . 386 361. Uniformly- Varied Velocity, . 386 Paga 362. Varied Rate of Variation of Velo- city, 387 363. Uniform Deviation, . 364. Varying Deviation, . 365. Resultant Rate of Variation, 366. Rates of Variation of Component Velocities, .... 367. Comparison of Varied Motions, 387 388 388 388 389 CHAPTER II MOTIONS OF RIGID BODIES. SECTION 1 Rigid Bodies, and their Translation. 368. Rigid Bodies how understood, 390 369. Translation or Shifting, . . 390 SECTION 2. Simple Rotation. 370. Rotation defined Centre of Rotation, . .390 371. Axis of Rotation, . . 390 372. Plane of Rotation Angle of Ro- tation, ... .391 373. Angular Velocity, . . 391 374. Uniform Rotation, . .391 375. Rotation common to all parts of body, 391 376. Right and Left-handed Rotation, 391 377. Relative Motion of a pair of Points in a Rotating Body, . . 392 378. Cylindrical Surface of equal Ve- locities, 392 379. Comparative Motions of two Points relatively to an Axis, . 393 389. Components of Velocity of a Point in a Rotating Body, . . 393 SECTION 3. Combined Rotations and Translations. 381. Property of all Motions of Rigid Bodies, 394 382. Helical Motion. . . .394 383. To find the Motion of a Rigid Body from the Motions of three of its Points, . . . .395 384. Special cases, . . . .396 385. Rotation combined with Transla- tion in the same Plane, . . 397 386. Rolling Cylinder Trochoids, . 398 387. Plane rolling on Cylinder Spiral Paths, ; . . . .398 388. Combined Parallel Rotations, . 399 389. Cylinder rolling on Cylinder Epitrochoids, .... 400 390. Curvature of Epitrochoids, . 401 391. Equal and Opposite Parallel Ro- tations combined, . . . 404 392. Rotations about IntersectingAxes combined, .... 404 393. Rolling Cones, . . . .405 394. Analogy of Rotations and Single Forces, 405 395. Comparative Motions in Com- pound Rotations, . . . 406 SECTION 4. Varied Rotation. 396. Variation of Angular Velocity, . 406 397. Change of the Axis of Rotation, 407 398. Components of Varied Rotation, 407 CHAPTER III. MOTIONS OF PLIABLE BODIES, AND OF FLUIDS. 399. Division of the Subject, SECTION 1. Motions of Fkxible Cords. 400. General Principles, . . .408 401. Motions classed, . . .409 402. Cord guided by Surfaces of Revo- lution, 409 and SECTION 2. Motions of Fluids of Constant Density. 403. Velocity and Flow, " . .410 404. Principle of Continuity, 405. Flow in a Stream, 406. Pipes, Channels, Currents Jets, 407. Radiating Current, . 408. Vortex, Eddy, or Whirl, 409. Steady Motion, . 410. Unsteady Motion, 411. Motion of Piston, 412. General Differential Equations of Continuity, .... 413 411 411 411 412 412 412 413 413 408 413. General Differential Equations of Steady Motion, . . .414 414. General Differential Equations of Unsteady Motion, . . . 415 415. Equations of Displacement, . 415 416. Wave, 416 417. Oscillation, . . . .416 SECTION 3. Motions of Fluids of Varying Density. 418. Flow of Volume and Flow of Mass, 417 410. Principle of Continuity, . . 417 420. Stream, 418 421. Steady Motion 419 422. Pistons and Cylinders, . . 419 423. General Differential Equa- tions, 419 424. Motions of Connected Bodies, . 420 CONTEXTS. PART IV. THEORY OF MECHANISM. XI CHAPTER I. DEFINITIONS Page 425. Theoryof Pure Mechanism defined,421 426. General Problem of Mechanism stated, 422 427. Frame Moving Pieces Con- nectors, 422 428. Bearings, 422 429. Motions of'PrimaryMoving Pieces, 423 430. Motions of secondarjMovingPieces,423 AND GENERAL PRINCIPLES. Page 431. Elementary Combinations in Me- chanism, ... . . 423 432. Line of Connection, . . . 424 433. Principle of Connection, . . 424 434. Adjustments of Speed, . . 424= 435. Train of Mechanism, . . 425 436. Aggregate Combinations in Me- . 425 , Aggregate Combinatio chanism, .. CHAPTER II. ON ELEMENTARY COMBINATIONS AND TRAINS OF 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 CircularWheei; 427 442. Bevel Wheels, 443. Non-Circular Wheels, . 428 . 428 SECTION 2. Sliding Contact. 444. Skew-Bevel Wheels, . . 430 445. Grooved Wheels, or Frictional Gearing, .... 431 446. Teeth of Wheels Definitions and General Principles, . . 432 447. Pitch and Number of Teeth, . 433 448. Hunting Cog, . . . .434 449. Trains of Wheelwork, . . 434 450. Principle of Sliding Contact. . 436 451. Teeth of Spur-Wheels and Racks General Principle, . . 438 452. Teeth described by rolling curves, 438 453. Sliding of a pair of Teeth (see also 455, 458, 462A), . . 439 454. Arc of Contact on Pitch Lines, . 440 455. Length of a Tooth; Slidingof Teeth,440 456. Inside Gearing, .... 441 457. Involute Teeth for 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, . 449 470. Cam, or Wiper, . . .449 471. Screws Pitch, . . . .440 472. Normal and Circular Pitch, . 450 473. Screw Gearing, . . . 451 474. Hooke's Gearing, . . . 451 475. Wheel and Screw, . . .452 476. Relative Slidingof Pair of Screws, 453 477. Oldham's Coupling, . . . 453 SECTION 3. Connection 'by Bands. 478. Bands classed: Belts,Cords,Chains,454 479. Principle of Connection by Bands, 454: 480. Pitch Surface of u 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, 465 503. General Principles, , 504. Differential Windlass, 505. Compound Screws, , CHAPTER III. ON AGGREGATE COMBINATIONS. 506. Link Motion, 5C7. Parallel Motions, 466 466 . 467 508. Epicyclic Trains, . 4G8 . 469 . 473 Xll CONTENTS. PART V. PRINCIPLES OF DYNAMICS. -509. Division of the Subject, Page . 475 CHAPTER I UNIFORM MOTION UNDER BALANCED FORCES. Page 510. First Law of Motion, . . 476 511. Effort, Resistance, Lateral Force, 476 512. Conditions of Uniform Motion, . 477 513. Work, 477 514. Energy, . . . . .477 515. Energy and Work of Varying Forces, ..... 477 516. Dynamometer, or Indicator, . 478 517. Energy and Work of Fluid Pres- sure, 478 518. Conservation of Energy, . . 478 519. Principle of Virtual Velocities, . 479 520. Energy of Component Forces and Motions, .... 480 CHAPTER II. ON THE VARIED TRANSLATION OF POINTS AND RIGID BODIES. SECTION 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, . . . .483 528. Relations between Impulse, Energy, and Work, . . 484 SECTION 2. Law of Varied Translation. 529. Second Law of Motion, . . 484 530. General Equations of Dynamics, 484 531. Mass in terms of Weight, . . 485 532. Absolute Unit of Force, . . 486 533. Motion of a Falling Body, . . 486 534. Projectile, Unresisted, . . 487 535. Motion along an Inclined Path, . 489 536. Uniform Effort, or Resistance, . 490 537. Deviating Force, . . .491 538. Centrifugal Force, . . .491 539. Revolving Simple Pendulum, . 492 540. Deviating Force in terms of An- gular Velocity, . . .492 541. Rectangular Components of De- viating Force, . . .493 542. Straight Oscillation, . . . 494 543. Elliptical Oscillations, or Revo- lutions, 495 544. Simple Oscillating Pendulum, . 496 545. Cycloidal Pendulum, . . 497 546. Residual Forces, . . .498 SECTION 3. Transformation of Energy. 547. Actual Energy defined, Vis- Viva, 499 548. Components of Actual Energy, 499 549. Energy of Varied Motion, . 499 550. Energy Stored and Restored, 501 551. Transformation of Energy, 501 552. Conservation of Energy in Variec 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, . 505 561. Angular Impulse Defined, . 506 562. Relations of Angular Impulse and Angular Momentum, . . 506 563. Conservation of Angular Momen- tum, 506 564. Actual Energy of a System of Bodies, ... . . .507 565. Conservation of Internal Energy, 508 566. Collision, . . . .508 567. Action of Unbalanced External Forces on a System. General Equations, . . . .510 568. Determination of Internal Forces D'Alembert's Principle, . 511 569. Residual External Forces in a System of Bodies, . . . 511 CHAPTER III. ROTATIONS OF RIGID BODIES. 570. Motion of a Rigid Body in General, . . .513 SECTION 1 . On Moments of Inertia, Radii of Gyration, Moments of Devia- tion, and Centres of Percussion. 571. Moment of Inertia Defined, . 514 572. Moment of Inertia of a System of Physical Points, . . . 514 573. Moment of Inertia of a Rigid Body,514 574. Radius of Gyration, . . .515 575. Components of Moments of In- ertia 515 576. Moments of Inertia round Pa- rallel Axes compared, . ^ .516 577. Combined Moments of Inertia, . 517 578. Examples of Moments of Iner- tia and Radii of Gyration, . 517 579. Moments of Inertia found by Division and Subtraction, . 519 580. Moments of Inertia found by Transformation, . . .519 581. Centre of Percussion Centre of Gyration, . . . .520 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 ^ Xlll Page 595. Varied Rotation about a fixedAxis,540 596. Analogy of Varied Rotation and Varied Translation, . . 541 597. Uniform Variation. . . .541 598. Gyration, or Angular Oscillation, 542 599. Single Force applied to a Body with a Fixed Axis, . . 543 SECTION 4. Varied Rotation and Trans- lation Combined. 600. General Principles, . . .543 601. Properties of the Centre of Per- cussion, ..... 544 602. Fixed Axis, . . . .545 603. Deviating Force, . . .545 604. Compound Oscillating Pendulum Centre of Oscillation, . . 546 605. Compound Revolving Pendulum, 547 606. Rotating Pendulum, . . 547 607. Ballistic Pendulum, . . .548 CHAPTER IV. MOTIONS OF PLIABLE BODIES. 608. Nature of the Subject Vibration, 552 609. Isochronous Vibration. Condi- tion of Isochronism, . . 553 610. Vibrations of a Mass held by a light Spring, .... 554 611. Superposition of Small Motions, 555 612. Vibrations not Isochronous, . 557 613. Vibrations of an Elastic Body in General, .... 557 614. Waves of Vibration, . . .562 615. Velocity of- Sound, . . .563 616. Impact and Pressure; Pile-driving,564 622. 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 Contimu'ty for a Steady Stream of Gas, . .581 637. Flow of Gas from an Orifice, . 581 637A. Maximum Flow of Gas, . . 582 SECTION 3. Motions of Liquids with Friction. 638. General Laws of Fluid Friction, 584 566 639. Internal Fluid Friction, . . 585 640. Friction in an Uniform Stream CHAPTER V. MOTIONS OF FLUIDS HYDRODYNAMICS. 17. Division of the Subject, SECTION 1. Motion of Liquids without Friction. 18. General Equations, . . .567 619. Dynamic Head, . . . 568 20. General Dynamic Equations in terms of Dynamic Head, . 568 621. Law of Dynamic Head for Steady Motion, 568 Hydraulic Mean Depth, 641. Varying Stream, 642. Friction in a Pipe running full, 643. Resistance of Mouthpieces, 644. Resistance of Curves and Knees 645. Sudden Enlargement of Chan nel, .... 646. General Problem, 586 587 588 589 589 589 590 SECTION 4. Flow of Gases with Friction. 647. General Law, . . . .590 SECTION 5. Mutual Impulse of Fluids and Solids. 648. Pressure of a Jet against a fixed Surface, 591 649. Pressure of a J 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, . , .- . .604 XIV CONTENTS. PART VI. THEORY OF MACHINES. 657. 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, . . 610 659. Useful and Prejudicial Resistance, 610 660. Efficiency, . . . .610 661. Power and Effect : Horse Power, 610 662. Driving Point : Train : Work- 610 610 610 611 611 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, . . . .612 668. Reduction of Forces and Couples, 612 SECTION 2. On the Friction of 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. Factional Gearing, . . 618 680. Friction Couplings, . . 618 681. Stiffness of Ropes, . . 619 682. Rolling Resistance of Smooth Surfaces, . . ... 619 683. Resistance of Carnages on Roads, 619 684. Resistance of Railway Trains, . 620 685. Heat of Friction, . . . 620 CHAPTER II. VARIED MOTIONS OF MACHINES. 686. Centrifugal Forces and Couples, 621 687. Actual Energy of a Machine, . 621 688. Keduced Inertia, . . .621 689. Fluctuations of Speed, . . 622 690. Fly-Wheel, . . . .623 691. Starting and Stopping Brakes, 624 CHAPTER III. ON PRIME MOVERS. 692. Prime Mover denned, . 625 693. Regulators Governors, . 625 694. Prime Movers Classed, . 625 695. Muscular Strength, . . 625 696. Water- Pressure Engines, . 626 697. Water Wheels in General, . 627 698. Classes of Water Wheels, . 628 699. Overshot and Breast Wheels, . 628 700. Undershot Wheels, . . .628 701. Turbines 629 702. Windmills, .... 629 703. Efficiency of Heat Engines in General, . . . .629 704. Steam Engines, . . .630 705. Electrodynamic Engines: Science of Energetics, . 630 APPENDIX. Motion of Water in Waves, 631 Continuous Girders, ............ 634 Reservoir Walls, 638 American Bridges, ............ 639 Wind Pressure, , 639 Continuous Brakes, 640 Steel Boilers, . 640 Pulsometer Pump, ............ 641 Strength of Steel, 641 Ship Resistance, ............ 641 Hydraulic Rivetting, &c., 642 Safety Valves, 642 Coefficient of Contraction, . . 642 Gas Engines, ............. 643 Railway Resistance, ............ 643 Compound Engines, ............ 643 Dimensions of Great Chimneys, 644 I. Tables of the Resistance" of Materials to Stretching and Tearing, . . .646 II. Table of the Resistance of Materials to Shearing and Distortion, . . .648 IK. Table of the Resistance of Materials to Crushing, 648 IV. Table of the Resistance of Materials to Breaking Across, .... 649 V. Comparative Tables of British and French Measures, 651 VI. Table of Specific Gravities of Materials, 652 r PRELIMINARY DISSERTATION HARMONY OF THEORY AND PRACTICE IN MECHANICS/ THE words, theory and practice, are of Greek origin : they cany 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 j and great are our obligations to the ideas and the models which they have transmitted to our times. But in physics and in mechanics their notions were very generally pervaded by a great fallacy, which attained its complete and most mischievous development amongst the mediaeval school- men, and the remains of whose influence can be traced even at the present day the fallacy of a double system of natural laws; one theoretical, geometrical, rational, discoverable by contemplation, applicable to celestial, setherial, indestructible bodies, and being an object of the noble and liberal arts ; the other practical, mechanical, empirical, discoverable by experience, applicable to terrestrial, gross, destructible bodies, and being an object of what were once called the vulgar and sordid arts. The so-called physical theories of most of those whose under- standings were under the influence of that fallacy, being empty dreams, with but a trace of truth here and there, and at variance with the results of every- day observation on the surface of the planet we inhabit, were calculated to perpetuate the fallacy. The stars were celestial, incorruptible bodies ; their orbits were circular and their motions perpetual ; such orbits and motions being charac- teristic of perfection. Objects on the earth's surface were terrestrial * This Dissertation contains the substance of a discourse, " De ConcordiS, inter Scientiarum Machinalium Coutemplationem 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 'moTJicttis being characteristic of imperfection, were in mixed straight and curved lines, and of limited duration. Rational and practical mechanics (as Newton observes in his preface to the Principia) were considered as in a measure opposed to each other, the latter being an inferior branch of study, to be cultivated only for the sake of gain or some other material advantage. Archytas of Tarentum might illustrate the truths of geometry by mechanical contrivances ; his methods were regarded by his pupil Plato as a lowering of the dignity of science. Archi- medes, to the character of the first geometer and arithmetician of his day, might add that of the first mechanician and physicist, he might, by his unaided strength acting through suitable machinery, move a loaded ship on 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 ; 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 011 the part of the architects. But the very names of those architects, with few and doubtful exceptions, were suffered to be forgotten. ; and the prin- ciples which guided their work remain unrecorded, and were left to be re-discovered in our own day ; for the scholars of those times, despising practice and observation, were occupied in developing and magnifying the numerous errors, and in perverting and obscur- PRELIMINARY DISSERTATION. 3 ing the much more numerous truths, which are to be found in the writings of Aristotle ; and those few men who, like Roger Bacon, combined scientific with practical knowledge, were objects of fear and persecution, as supposed allies of the powers of darkness. At length, during the great revival of learning and reformation of science in the fifteenth, sixteenth, and seventeenth centuries, the system falsely styled Aristotelian was overthrown : so also was the fallacy of a double system of natural laws ; and the truth began to be duly appreciated, that sound theory in physical science con- sists simply of facts, and the deductions of common sense from them, reduced to a systematic form. The science of motion was founded by Galileo, and perfected by Newton. Then it was estab- lished that celestial and terrestrial mechanics are branches of one science ; that they depend on one and the same system of clear and simple first principles; that those very laws which regulate the motion and the stability of bodies on earth, govern also the revolutions of the stars, and extend their dominion throughout the immensity of space. Then it" came to be acknowledged, that no material object, however small, no force, however feeble, no phenomenon, however familiar, is insignificant, or beneath the attention of the philosopher ; that the processes of the workshop, the labours of the artizan, are full of instruction to the man of science ; that the scientific study of practical mechanics is well worthy of the atten- tion of the most accomplished mathematician. Then the notion, that scientific men are unfit for business, began to disappear. It was not court favour, not high connection, not Parliamentary in- fluence, which caused Newton to be appointed Warden, and after- wards Master, of the Mint j it was none of these ; but it was the knowledge possessed by a wise minister of the fact, that Newton's skill, both theoretical and practical, in those branches of knowledge which that* office required, rendered him the fittest man in all Britain to direct the execution of a great reform of the coinage. Of the manner in which Newton performed the business entrusted to him, we have the following account in the words of Lord Macaulay, an author who cannot be accused of undue partiality to speculative science or its cultivators : " The ability, the industry, and the strict uprightness of the great philo- sopher, speedily produced a complete revolution throughout the depart- ment which was under his direction. He devoted himself to the task with an activity which left him no time to spare for those pursuits in which he had surpassed Archimedes and Galileo. Till the great work was com- pletely done, he resisted firmly, and almost angrily, every attempt that was made by men of science, here or on the Continent, to draw him away from his official duties."* * Vol. iv., p. 703. 4: PRELIMINARY DISSERTATION. Then the historian proceeds to detail the results of Newton's exertions, and shows, that within a short time after his appoint- ment, the weekly amount of the coinage of silver was increased to eightfold of that which had been looked upon as the utmost practi- cable amount by his predecessors. The extension of experimental methods of investigation, has caused even manual skill in practical mechanics, when scientifically exercised, to be duly honoured, and not (as in ancient times) to be regarded as beneath the dignity of science. As a systematically avowed doctrine, there can be 110 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. O the engineer who has to lift a great mass of real granite by the instru- mentality of real timber and real hemp, should absolutely rely on the pro- positions which he finds in treatises on Dynamics, and should make no allowance for the imperfection of his materials, his whole apparatus of beams, wheels, and ropes, would soon come down in ruin, and with all his geometrical skill, he would be found a far inferior builder to those painted barbarians who, though they never heard of the parallelogram of forces, managed to pile up Stonehenge."* It is impossible to read this passage without feeling admiration for the force and clearness (and I may add, for the brilliancy and wit) of the language in which it is expressed; and those very qualities of force and clearness, as well as the author's eminence, render it one of the best examples that can be found to illustrate the lurking influence of the fallacy of a double set of mechanical laws, rational and practical. In fact, the mathematical theoiy 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 theoiy 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 theoiy 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 j 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 j 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. ? are destitute of that knowledge of scientific principles which is requisite to prevent their being misled by their own ingenuity. Such men too often spend their money, waste their lives, and it may be lose their reason, in the vain pursuit of visionary inventions, of which a moderate amount of theoretical knowledge would be sufficient to demonstrate the fallacy; and for want of such know- ledge, many a man who might have been a useful and happy member of society, becomes a being than whom it would be hard to find anything more miserable. The number of those unhappy persons to judge from the patent- lists, and from some of the mechanical journals must be much greater than is generally believed. The most absurd of all their delusions, that commonly called the perpetual motion, or to speak more accurately, the inexhaustible source of power, is, in various forms, the subject of several patents in each year. The ill success of the projects of misdirected ingenuity has very naturally the effect of driving those men of practical skill who, though without scientific knowledge, possess prudence and common sense, to the opposite extreme of caution, and of inducing them to avoid all experiments, and to confine themselves to the careful copying of successful existing structures and machines : a course which, although it avoids risk, would, if generally followed, stop the progress of all improvement. A similar course has sometimes, indeed, been adopted by men possessed of scientific as well as practical skill : such men having, in certain cases, from deference to popular prejudice, or from a dread of being reputed as theorists, considered it advisable to adopt the worse and customary design for a work in preference to a better but unusual design. Some of the evils which are caused by the fallacy of an incom- patibility between theory and practice having been described, it must now be admitted, that at the present time those evils show a decided tendency to decline. The extent of intercourse, and of mutual assistance, between men of science and men of practice, the practical knowledge of scientific men, and the scientific knowledge of practical men, have been for some time steadily increasing ; and that combination and harmony of theoretical and practical knowledge that skill in the application of scientific principles to practical purposes, which in former times was confined to a- few remarkable individuals, now tends to become more generally diffused. With a view to promote the diffusion of that kind of skill, Chairs were instituted at periods of from fifteen to ten years ago, in the two Colleges of the University of London, in the University of Dublin, in the three Queen's Colleges of Belfast, Cork, and Galway, and in this University of Glasgow. For the sake of a parallel, it may here be worth while to refer 8 PRELIMINARY DISSERTATION. to another branch of practical science that of Medicine. From the time of the first establishment of Medical Schools in Universities, there have existed, not only Chairs for the teaching of the purely scientific departments of Medical Science, such as Anatomy and Physiology,, but also Chairs for instruction in the art of applying scientific principles to practice, such as those of Surgery, the Practice of Physic, and others. The institution of a Chair of Mechanics and Engineering in a University where there have long existed Chairs of Mathematics and Natural Philosophy, is an endeavour to place Mechanical Science on the same footing with that of Medicine. Another parallel may be found in an Institution, which, though not a University, and though established as much for the advance- ment as for the diffusion of knowledge, has had a most beneficial effect in promoting the appreciation of science by the public, I mean the British Association. When that body was first instituted, both the theoretical advancement and the practical application of Mechanics, and the several branches of Physics, were allotted to a single section, called Section A. The business before that Section soon became so excessive in amount, and so multifarious in its character, that it was found necessary to institute Section G, for the purpose of considering the practical application of those branches of science to whose theoretical advancement Section A was now devoted; and notwithstanding this separation, those two Sections work harmoniously together for the promotion of kindred objects ; and the same men are, in many instances, leading members of both. What Section G is to Section A in the British Association, this class of Engineering and Mechanics is to those of Physics and Mathematics in the University. It being admitted, that Theoretical and Practical Mechanics are in harmony with each other, and depend on the same first prin- ciples, and that they differ only in the purposes to which those principles are applied, it now remains to be considered, in what manner that difference affects the mode of instruction to be followed in communicating those branches of science. Mechanical knowledge may obviously be distinguished into three kinds : purely scientific knowledge, purely practical knowledge and that intermediate knowledge which relates to the application of scientific principles to practical purposes, and which arises from understanding the harmony of theory and practice. The objects of instruction in purely scientific mechanics and physics are, first, to produce in the student that improvement of the understanding which results from the cultivation of natural knowledge, and that elevation of mind which flows from the con- templation of the order of the universe; and secondly, if possible, PRELIMINARY DISSERTATION. 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 leads to them. The ascertainment and illustration of truth are the objects; and structures and machines are looked upon merely as natural bodies are : namely, as furnishing experimental data for the ascer- taining of principles, and examples for their illustration. Instruction in purely practical knowledge is that which the student acquires by his own experience and observation of the transaction of business. It enables him to judge of the quality of materials and workmanship, and of questions of convenience and commercial profit, to direct the operations of workmen, to imitate existing structures and machines, to follow established practical rules, and to transact the commercial business which is connected with mechanical pursuits. The third and intermediate kind of instruction, which connects the first two, and for the promotion of which this Chair was estab- lished, relates to the application of scientific principles to practical purposes. It qualifies the student to plan a structure or a machine for a given purpose, without the necessity of copying some existing example, and to adapt his designs to situations to which no existing example aifords 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; and if existing data are insuflicient 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. Eobison of PRELIMINARY DISSERTATION. 11 Edinburgh, after stating that the principles of Carpentry depend on two branches of the science of Statics, remarks "It is thi& which makes Carpentry a liberal art." So also is Masonry a liberal art, so is the art of working in Iron, so is every art, when guided by scientific principles. Every structure or machine, whose design evinces the guidance of science, is to be regarded not merely as an instrument for promoting con- venience and profit, but as a monument and testimony that those who planned and made it had studied the laws of nature; and this renders it an object of interest and value, how small soever its bulk, how common soever its material. For a century there has stood, in a room in this College, a small, rude, and plain model, of appearance so uncouth, that when an artist lately introduced its likeness into a historical painting, those who saw the likeness, and knew nothing of the original, wondered what the artist meant by painting an object so unattractive. But the artist was right ; for ninety-one years ago a man took that model, applied to it his knowledge of natural laws, and made it into the first of those steam engines that now cover the land and the sea; and ever since, in Reason's eye, that small and uncouth mass of wood and metal shines with imperishable beauty, as the earliest embodiment of the genius of James Watt. Thus it is that the commonest objects are by science rendered precious; and in like manner the engineer or the mechanic, who plans and works with understanding of the natural laws that regulate the results of his operations, rises to the dignity of a Sage. INTRODUCTION. DEFINITION OF GENERAL TERMS AND DIVISION OF THE SUBJECT. ART. 1. Mechanics is the science of rest, motion, and force. The laws, or first principles of mechanics, are the same for all bodies, celestial and terrestrial, natural and artificial. The methods of applying the principles of mechanics to particular cases are more or less different, according to the circumstances of the case. Hence arise branches in the science of mechanics. 2. Applied Mechanics. The branch to which the term " APPLIED MECHANICS" has been restricted by custom, consists of those ' consequences of the laws of mechanics which relate to works of human art. A treatise on applied mechanics must commence by setting forth those first principles which are common to all branches of mechanics ; but it must contain only such consequences of those principles as are applicable to purposes of art. 3. Matter (considered mechanically) is that which fills space. 4. Bodies are limited portions of matter. Bodies exist in three conditions the solid, the liquid, and the gaseous. Solid bodies tend to preserve a definite size and shape. Liquid bodies tend to preserve a definite size only. Gaseous bodies tend to expand inde- finitely. Bodies also exist in conditions intermediate between the solid and liquid, 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. Line, Point, Physical Point, Measure of Length. In mechanics, as in geometry, a LINE is the boundary of a surface, or between two 14 INTRODUCTION. parts of a surface ; and a POINT is the boundary of a line, or be- tween two parts of a line j 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 j" 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 metre, being about ^ooooooo of the earth's circumference, measured round the poles. (See table at the end of the volume.) 8. Rest is the relation between two points, when the straight line joining them does not change in length nor in direction. A body is at rest relatively to a point, when every point in the body is at rest relatively to the first mentioned point. 9. Motion is the relation between two points when the straight line joining them changes in length, or in direction, or in both. A body moves relatively to a point when any point in the body moves relatively to the first mentioned point. 10. Fixed Point. When a single point is spoken of as having motion or rest, some other point, either actual or ideal, is always either expressed or understood, relatively to which the motion or rest of the first point takes place. Such a point is called a fixed point. So far as the phenomena of motion alone indicate, the choice of a fixed point with which to compare the positions of other points appears to be arbitrary, and a matter of convenience alone ; but when the laws of force, as affecting motion, come to be considered, STRUCTURES AND MACHINES. 15 it will be seen that there are reasons for calling certain points fixed, in preference to others. In the mechanics of the solar system, the fixed point is what is called the common centre of gravity of the bodies composing that system. In applied mechanics, the fixed point is either a point which is at rest relatively to the earth, or (if the structure or machine under consideration be moveable from place to place on the earth), a point which is at rest relatively to the structure, or to the frame of the machine, as the case may be. Points, lines, surfaces, and volumes, which are at rest relatively to a fixed point, are fixed. 11. Cinematics. The comparison of motions with each other, without reference to their causes, is the subject of a branch of geometry called "Cinematics" ' 12. Force is an action between two bodies, either causing or tending to cause change in their relative rest or motion. The notion of force is first obtained directly by sensation; for the forces exerted by the voluntary Trmsjcjps 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 aud 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 ami Machines. The works of human art to which the science of applied mechanics relates, are divided into two classes, according as the parts of which they consist are intended to rest or to move relatively to each other. In the former case they are called Structures; in the latter, Machines. Structures are sub- jects of Statics alone; Machines, when the motions of their parts are considered alone, are subjects of Cinematics; when the forces acting on and between their parts are also considered, machines are subjects of Statics and Dynamics. 16 INTRODUCTION. 16. General Arrangement of the Subject. The subject of the pre- sent treatise will be arranged as follows : I. FIRST PRINCIPLES OP STATICS. II. THEORY OF STRUCTURES. III. FIRST PRINCIPLES OF CINEMATICS. IV. THEORY OF MECHANISM. "V. FIRST PRINCIPLES OF DYNAMICS. VI. THEORY OF MACHINES. PART I. PRINCIPLES OF STATICS. -$4? CHAPTER I. BALANCE AND MEASUREMENT OF FORCES ACTING IN ONE STRAIGHT LINE. 17. Forces how Determined. Although, every force (as has been stated in Art. 12) is an action between two bodies, still it is con- ducive to simplicity to consider in the first place the condition of one of those two bodies alone. The nature of a force, as respects one of the two bodies between which it acts, is determined, or made known, when the following three things are known respecting it : first, the place, or part of the body to which it is applied; secondly, the direction of its action; thirdly, its magnitude. 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 t]je 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, uither a volume or a surface ; and a force concentrated at a single point has no real existence. Nevertheless it is necessary, in treating of the principles of statics, to begin by demonstrating the properties of such ideal forces, conceived to be concentrated at single points. It will afterwards be shown how the conclusions so arrived at re- specting single forces (as they may be called), are made applicable to the distributed forces which really act in nature. In illustrating the principles of statics experimentally, a force concentrated at a single point may be represented with any required degree of accuracy "by a force distributed over a very small space, if that space be made small enough. c 18 PRINCIPLES OP 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 Yict., 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 = TTO of a pound avoirdupois. The troy pound = 5,760 grains = 0*822857 14 pound avoirdupois. The hundredweight = 112 pounds avoirdupois. The ton = 2,240 pounds avoirdupois. The French standard unit of force is the gramme, which is the weight, in the latitude of Paris, of a cubic centimetre of pure water, measured at the temperature at which the density of water is greatest, viz., 4'l centigrade, or 39-4 Fahrenheit, and under the pressure which supports a barometric column of 760 millimetres of mercury. A comparison of French and British measures of force and of size is given in a table at the end of this volume. 22. Resultant of Forces Acting in One Straight Line. The RE- SULTANT of any number of given forces applied to one body, is a single force capable of balancing that single force which balances the given forces ; that is to say, the resultant of the given forces is equal and directly opposed to the force which balances the given forces ; and is equivalent to the given forces so far as the balance of REPRESENTATION OF FORCES. 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 For example, in fig. 1, the fact that the body B B B B is acted upon at the point G! by a given force, may be expressed by drawing from O l a straight Hne G! F! in the direction of the force, and of a length representing the magnitude of the force. If the force represented by G^Fj is balanced by a force applied either at the same point, or at another point G 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 G 2 F 2 , opposite in direc- tion, and equal in length to G! F 1? and lying in the same line of action L L. If the body B B B B (fig. 2), be balanced by several forces acting in the same straight line LL, applied at points G x O 2 , &c., and re- presented by lines G x F 1? 2 F 2 , &c. ; then either 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 Fi 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 II. THEOKY OP COUPLES AND OF THE BALANCE OF PARALLEL FORCES. SECTION 1. On Couples with the Same Axis. 25. Couples. Two forces of equal magnitude applied to the same body in parallel and opposite directions, but not in the same line of action, constitute what is called a " couple" 26. Force of a Couple Arm or leverage. The force of a COUple is the common magnitude of the two equal forces ; the arm or leverage of a couple is the perpendicular distance between the lines of action of the two equal forces. 27. Tendency of a Couple Plane of a Couple Right-handed and Left-handed Couples. The tendency of a couple is to turn the body to which it is applied in the plane of the couple that is, the plane which contains the lines of action of the two forces. (The plane in which a body turns, is any plane parallel to those planes in the body whose position is not altered by the turning). T]ig^*.r?'. : LP i and consequently O M c will also represent the moment of the re- sultant 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 that represented by the diagonal OM C balances the couples represented by the sides OM A , O M B . In other words, three couples represented by the three sides of a triangle balance each other. 37. Equilibrium of any Number of Couples. COROLLARY. If a number of couples acting on tlie 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 MI, M 2 , M 3 , M 4 , M 5 ; and let them be represented by the sides of the polygon in fig. 7 in such a manner that M! is represented by O A, and seems right-handed looking from A towards 0. M 2 AB, from B towards A. M 3 BC, from C towards B. M 4 CD, from D towards C. M 5 D O, from towards D. Then by the theorem of Article 35, the resultant of M x and M a is O B ; the resultant of this and M 3 is O~C j 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 by the excepted side. SECTION 3. On Parallel Forces. 38. Balanced Parallel Forces in General. A balanced system of parallel forces consists either of pairs of directly opposed equal forces, or of couples of equal forces, or of combinations of such pairs and couples. Hence the following propositions as to the relations amongst the magnitudes of systems of parallel forces are obvious : I. In a balanced system of parallel forces, the sums of the forces acting in opposite directions are equal ; in other words, the alge- PRINCIPLES OF STATICS. braical sum of the magnitudes of all the forces taken with their proper signs is nothing. II. The magnitude of the resultant of any combination of par- allel forces is the algebraical sum of the magnitudes of the forces. The relations amongst the positions of the lines of action of balanced parallel forces remain to be investigated; and in this inquiry, all pairs of directly opposed equal forces may be left out of consideration j for each such pair is independently balanced what- soever its position may be ; so that the question in each case is to be solved by means of the theory of couples. 39. Equilibrium of Three Parallel Forces in One Plane. Prin- ciple of the Lever. 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 the 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 tJie lines oj action of the other two. Let a body (fig. 8) be maintained 8 * in equilibrio by two opposite couples having the same axis, and of equal moments, F A 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 + FA, + 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 ~ AB = L A + L B ; and consequently F A :F B :F ::C~B:A~C~: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 FOKCE. 27 and in the same plane j if they act in the same direction, then their resultant is their sum, acts in the same direction, and lies between them ; if they act in opposite directions, their resultant is their difference, acts in the direction of, and lies beyond, the prepon- derating force ; and the distance between the lines of action of any two of those three forces the resultant and its two components is proportional to the third force. In order that two opposite parallel forces may have a single resultant, it is necessary that they should be unequal, the resultant being their difference. Should they be equal, they constitute a "V couple, which has no single resultant. ; ^ 41. Resultant of a Couple and a Single Force in Parallel Planes. Let M denote the moment of a couple applied to a body (fig. 9); and at a point O let a single force F be applied, in a plane parallel to that of the couple. For the given couple substitute an equivalent couple, consisting of a force F equal and directly opposed to F at O, and a force F applied at A, the arm. A O , . M , . being = - , and 01 course par- Jj allel to the plane of the couple M. Then the forces at balance each other, and F applied at A is the resultant of the single force F applied at 0, 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 M distance O A = -=^-j to the left if M is right- _b 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 X be any straight line perpendicular in direction to the line of action of the force, and not intersecting it, and let A B be the common perpendicular of those two lines. At B conceive a pair of equal and directly op- posed forces to be applied in a line of action parallel to F, viz. : F = F, and - F = - F. The supposed application of such a pair of balanced forces does not alter the statical condition of the Fig. 10. body. Then the original single force F, applied in a line tra- 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 O X, and sometimes also, the moment of the force F relatively to the plane which contains 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 O X. 43. Equilibrium of any System of Parallel Forces in One Plane. In order that any system of parallel forces whose lines of action are in one plane may balance each other, it is necessary and suffi- cient that the following conditions should be fulfilled : I. (As already stated in Art. 38) that the algebraical sum of the forces shall be nothing : II. That the algebraical sum of the moments of the forces rela- tively to any axis perpendicular to the plane in which they act shall be nothing : two conditions which are expressed symbolically as follows : - let F denote any one of the forces, considered as positive or nega- tive, according to the direction in which it acts ; let y be the per j pendicular distance of the line of action of this force from an arbitrarily assumed axis O X, y also being considered as positive or negative, according to its direction ; then, Sum of forces, 2 F = ; Sum of moments, 2 y F = 0. For, by the last Article, each force F is equivalent to an equal and parallel force F applied directly to O X, combined with a couple y F ; and the system of forces F', and the system of couples y F, must each be in equilibrio, because when combined they are equiva- lent to the balanced system of forces F. In summing moments, right-handed couples are usually considered as positive, and left-handed couples as negative. 44. Resultant of any Number of Parallel Forces in One Plane. The resultant of any number of parallel forces in cne 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-F In some cases, the forces may have no single resultant, 2 J? being = ; and then, unless the forces balance each other com- pletely, their resultant is a couple of the moment 2 . y F. 4:5. Moments of a Force with respect to a Pair of Rectangular Axes In fig. 11, let F be any single force; O an arbitrarily-assumed point, called the "originof co-ordin- ates;" -X + X, - Y O + Y, a pair of axes traversing O, 'at right angles to each other and to the line of action of F. Let A B = y, be the common perpen- dicular of F and OX ; let AC = x, be the common perpendicular of F and OY. x and y are the "rectan- gular co-ordinates" of the line of action of F relatively to the axes - X + X, - Y + Y, re- spectively. According to the ar- rangement of the axes in the figure, x is to be considered as positive to the right, and nega- tive to the left, of YO + Y; and y is to be considered as positive to the left, and negative to the right, of-XO + X ; right and left referring to the spectator's right and left hand. In the particular case represented, x and y are both positive. Forces, in the figure, are considered as positive upwards, and negative downwards ; and in the particular case represented, F is positive. At B conceive a pair of equal and opposite forces, F' and F', 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' 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', and in the same direction. Then the single force F' is equivalent to the single force F' f = F 7 = F applied at O, combined with the couple constituted by F' and - F" with the arm OB = x } whose moment is 11. 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 : S-F = 0; S-yF = 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 In some cases, the forces may have no single resultant, 2 J 1 CENTRE OF PARALLEL FORCES. 31 being = ; and then, unless the forces balance each other com- pletely, their resultant is a couple, whose axis, direction, and moment are found as follows : Let M, = 2 . y F ; M y = - 2 . x F ; be the moments of the pair of partial resultant couples relatively to the axes O X and Y respectively. From O, along those axes, set off two lines representing respectively M, and M y according to the rule of Art. 34 ; that is to say, proportional to those moments in length, and pointing in the direction from which those couples must respectively be viewed in order that they may appear right- handed. Complete the rectangle whose sides are those lines ; 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 ; + Mj and if 6 be the angle which its axis makes with O X, 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 f a y & C depends solely on the proportionate mag- F j g 12 . nitude of the parallel forces F A and F B , and not on their absolute magnitude, nor on the angular position of their lines of action; so that if for those forces there be substituted another pair of parallel forces, f a , f b , in any other angular position, and if those new forces bear to each other the same proportion with the original forces, viz. : / a :/ 6 ::F A :F B : : BC" the point C where the resultant cuts A B will still be the same, This point is called the Centre of Parallel Forces, for a pair of 32 PRINCIPLES OF STATICS. forces applied at A and B respectively, and having the given ratio BC : AC. 49. Centre of any System of Parallel Forces. Let parallel forces, F , Fj, be applied at the points A A! (fig. 13.), Draw the straight line AQ A 1? in which take Cj, so that F :F i: then will C x be the centre of a pair of Fig 13 ^'^ P ara ll e l forces applied at A and A lt and having the proportion F : F x . At a third point, A 2 , let a third parallel force, F 2 , be applied. Then, because the forces F , F 15 are together equivalent to a parallel force, F + F x , applied at 0^ draw the straight line Cj A 2 , in which take C 2 , so that then will C 2 be the centre of three parallel forces applied at A , A 1? A 2 , and having the proportions F : Fj : F 2 . At a fourth point, A 3 , let a fourth parallel force, F 3 , be applied. Then, because the forces F , F 15 F 2 , are together equivalent to a parallel force, F + j + F 2 , applied at C 2 , draw the straight line C 2 , A 3 , in which take C 3 , so that F 2 : F 8 : : 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 : F! : 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, wliatsoever may be the absolute magnitudes of those forces, and tJie angular position of their lines of action. 50. Co-ordinates of Centre of Parallel Forces. The method of finding centres of parallel forces described in the preceding Article, though suitable for the demonstration of the theorem just stated, is tedious and inconvenient when the number of forces is great, in which case the best method is to find the rectangular co-ordinates of that point relatively to three fixed axes, as follows : Let O be any convenient point, taken as the origin of co-ordi- nates, and OX, Y, Z, 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 O Y, and perpendicular to the plane Z X, and z parallel to O Z, and perpendicular to the plane X Y. x, y, and z are the rectangu- lar co-ordinates of A, which, being known, the position of A is deter- mined. Let F denote either the magnitude of the force applied at A, or any magnitude proportional to that magnitude, x, y, z, and F are supposed to be known for every point of the given system of points. Then first, conceive all the parallel forces to act in lines parallel to the plane Y Z. Then the sum of their moments relatively to an axis in that plane is 2- #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 2 -aF Ffc. 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 2'2/F; and consequently the distance of their resultant, and also of the centre of parallel forces from that plane is given by the equation - S> ^ F ~ 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 centime of parallel forces from that plane is given by the equation 2 - F ' Thus are found x r , y r) z f , the three rectangular co-ordinates of 34 PRINCIPLES OF STATICS. the centre of parallel forces, for a system of forces applied to any given system of points, and having any given mutual ratios. If the parallel forces applied to a system of points are all equal, then it is obvious that the distance of the centre of parallel forces from any given plane is simply the mean of the distances of the points of ie system from that plane. 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 by the diagonal. First Demonstration. Through the point O (fig. 15), let two forces act, represented in direction and magnitude by OA and OB. x -- x 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. Now, first, the force represented in direction and magnitude by the dia- Fig. 15. gonal O C of the parallelogram A B fulfils this condition. For let P be any point in the plane of O A and B, and let an axis perpendicular to that plane traverse P. Join P A, P B, P C, P O. Then, as already shown in Art. 42, the moments of the forces O A, OB, OC, relatively to the axis P, are represented respectively by the doubles of the triangles POA, POB, POO. Draw AD || BE || OP, and join PD, P E. Then A_PO D_= AJM) A, and A POE = A P OB ; but be- cause OD+OE = OC, .-. APOC = A POD + A P O E = A P O A + A P B ; and the moment j O~C relatively to P is equal to the sum of the moments of O A and B ; and that whatsoever the position of P may be. Secondly. The force represented by O is the only force which fulfils this condition. For let O~Q represent a force whose moment relatively to P is equal to the sum of the moments of O A and O B. Join P Q. Then A O P Q = A P C, and .-. C Q || P 0; so that 36 PRINCIPLES OF STATICS. O Q fulfils the required condition for those axes only which are situated in a line O P || C Q, and not for any other axis. Therefore the diagonal O C of the parallelogram A B represents the resultant, and the only resultant, of the forces represented by OA and OR Q. E. D. Second Demonstration. Suppose a perpendicular to be erected to the plane O A B at the point O, of any length whatsoever ; call the other extremity of that perpendicular R ; and at II conceive two forces to be applied, respectively equal, parallel, and opposite to O A and B. Then R 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 O A and O B acting at O, and an equal and opposite force at R ; 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 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 OA and OB, a force is required equal and directly opposed to their resultant C. This may be otherwise expressed by saying, that if the directions and mag- nitudes of Jhree JbrcesJ^represented by the three sides of a triangle, (such as O A, AC, 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 forces 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 O (fig. 16), and re- presented in direction and magnitude -* ~'^"~ are equal and parallel to the sides of the closed polygon O A B C D viz. : RESOLUTION OF A FORCE. 37 Fi = and || A ; F 2 = and || A B ; F 5 = and || D 0. Then by the theorem of Art. 52, the resultant of Fj and F 2 is B ; the resultant of F 1; F , and F 3 is O C; the resultant of F,, F 2 , F 3 , 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. Paraiieiopiped of Forces. The simplest gauche polygon is one of four sides. Let O A B C E F G H (fig. 17), 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 0, a closed quadrilateral ; conse- quently if three forces F a , F 2 , F 3 , acting through O, be represented by the three edges OA, Q~B, OCf, of a parallelepiped, the diagonal H represents their resultant, and a fourth force F 4 equal and opposite to Oil balances them. Fi S- !7- 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, O Y, which lie in one plane with C, and intersect it in one point 0. 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 B represent the com- ponent forces required. Two forces respectively equal to and directly opposed to OA and O~B will balance UC. 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. OS PRINCIPLES OF STATICS. Returning 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, O Y, O Z, which intersect O H in one point 0. 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, O Z in C. Then will O A, OB, C, represent the component forces required. Three forces respectively equal to, and directly opposed to O A, O B, and dC", will balance 0~H. 57. Rectangular Components. The rectangular components of a force are those into which it is resolved when the directions of their lines of action are at right angles to each other. For example, in fig. 17, suppose O X, O Y, Z, to be three axes of co-ordinates at right angles to each other. Then O H is resolved into three rectangular components simply by letting fall from H perpendiculars on O X, O Y, O Z, cutting them at A, B, C, respectively. To express this case algebraically, let F = O H denote the force to be resolved. Let be the angles which its line of action makes with the three rect- angular axes. Then, as is well known, those three angles are con- nected by the equation cos 2 * + cos 2 /3 4- cos 2 y = l, .................. (1.) Let be the three rectangular components of F ; then In order to distinguish properly the direction of the resultant F as compared with the directions of the axes, it is to be borne in mind that f acute ) i ( positive. the cosme of an j obtuge j angle * | J From a well known property of right-angled triangles (also em- bodied in equation 1), it follows that F 2 =F? + FS + F 2 ........................ (3.) To express algebraically the case in which a force is resolved into FORCES ACTING IN ONE PLANE. OV two rectangular components in one plane with it, let the plane in question be that of X and O Y. Then the angles are subject to the following equations '- >a ^ y = a right angle ; * + /3 = a right angle; cosy 0; cos /3 = sin ; cos sin /3. ......... (4.) and consequently the equations 2 and 3 are reduced to the following : F, = F- cos * = F- sin ;... ........... ) F 2 = F-sin * = F- cos/3; 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 X in the direction towards Y, and the angle /3 from Y in the reverse direction, that is, towards X, and that the sines of angles from { } are If a system of forces acting through one point balance each other, their resultant is nothing ; and therefore the rectangular components of their resultant, which are the resultants of their parallel systems of rectangular components, are each equal to nothing; a case re- presented as follows : 2-F 1 = 0; 2-F 2 = 0; 2 F 3 = ............. (6.) SECTION 2. Inclined Forces Applied to a System of Points. 58. Forces acting in One Plane. Graphic Solution. Let any system of forces whose lines of action are in one plane, act together on a rigid body, and let it be required to find their resultant. Assume an axis perpendicular to the plane of action of the forces at any point, and let it be called O Z. According to the principle of Art. 42, let each force be resolved into an equal and parallel force acting through O, and a couple tending to produce rotation about O Z; so that if a force F be applied along a line whose per- pendicular distance from is L, that force shall be resolved into F = and || F acting through 0, and a couple whose moment is M = LF, and which is right or left-handed according as 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 E 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 E =0, then the couples and forces balance completely, and there is no resultant. If 2-M = 0, while E has magnitude, then the resultant acts through 0. If 2 M and E, both have magnitude, then the line of action of the resultant E 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 E, = 0, while 2-M has magnitude, the only resultant of the given system of forces is the couple 2-M. 59. Forces acting in One Plane. Solution by Rectangular Co-or- dinates. Through the point O as origin of co-ordinates, let any two axes be assumed, O X and Y, perpendicular to each other and to O Z, and in the plane of action of the forces ; and in looking from Z towards 0, let Y lie to the right of X, so that rotation from X towards Y shall be right-handed. Let F, as before, denote any one of the forces ; let * be the angle which its line of action makes to the right of O X ; and let x and y be the co-ordinates of its point of application, or of any point in its line of action, relatively to the assumed origin and axes. Eesolve each force F into its rectangular components as in Art. 57, Fj =F cos ,', Fo = F sin ; then the rectangular components of the resultant are respectively parallel to O X, 2 (F cos ) = E 1? ) ,, v parallel to Y, 2 (F- sin ) = E 2 , j ' its magnitude is given by the equation E 2 = E* + EI; ......................... (2.) and the angle T which it makes to the right of X is found by the equations E/i . E<2 /o \ cos * r = sm ce r = -=g- ................... (3.) ANY SYSTEM OF FORCES. 41 The quadrant in which the direction of R lies is indicated by the algebraical signs of R } and R 2 , a s already stated in Art. 57. The perpendicular distance from of the line of action of any force F is L = x 'sin ee. y cos * which is positive or negative according as 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 a. y cos a) = ?(xF 2 yF,) ........................ (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^ ~ which is equivalent to V ............... (6.) x r sin ct r y r cos a r = L r As in Art. 58, if 2-F L = 0, the resultant acts through the origin 0; if 2-FL has magnitude, and R = (in which case Rj = 0, R 2 = 0) the resultant is a couple. The conditions of equili- brium of the system of forces are or in other syj-nbols V .... (7.) 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 1? which is left-handed when y and F! 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 V towards O, 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 *, /3, y, the angles which its direction makes with the axes respectively. Then the three rectangular components of F being as in Art. 57, F, = F cos * along OX,) F 2 = F-cos l 3 along OY, V (1.) F 3 = F cos y along Z, ] 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 0) relatively to O X, \ % F! x F 3 = F (z cos * x cos y) relatively to O Y, > (2.) x F 2 y F! = F (x cos /3 y cos a) relatively to O Z ; j so that the force F is equivalent to the three forces of the formulae 1 acting through 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. X j Ej = 2 F cos , OY; E a =: s-Fcos/3, V (3.) OZ ; E 3 = s-Fcosy, Couples. round OX; M, = OZ -M 3 = F (y cos y - z cos /3) F (2 cos ex, x cos y) F (x cos /3 2/ cos a) The three forces Ej, E 2 , E 3 , are equivalent to a single force E|+E), (5.) acting through O in a line which makes with the axes the angles given by the equations cos =5i- cos/3 =^2. cos -5l '" r ~~E ' CDS r ~ E j C S rr ~~ E ' ' The three couples M,, M 2 , M 3 , according to Article 37, are equi- valent to one couple, whose magnitude is given by the equation M= ^(Mf + Mi + MS) (7.) ANY SYSTEM OF POECES. 43 and whose axis makes with the axes of co-ordinates the angles given by the equations M, M 2 M 3 * I denote respectively the angles I r> v t ^ j made by the axis of M with J A j " The Conditions of Equilibrium of the system of forces may be ex- pressed in either of the two following forms : R 1 = 0; R 2 = 0; R 3 = 0: M 1 = 0; M 2 = 0; M 3 = 0...(9.) or R = 0; M = ......................... (10.) When the system is not balanced, its resultant may fall under one or other of the following cases : Case I. When M = 0, the resultant is the single force R acting through 0. Case II. When the axis o/*M is at right angles to tfie direction of R, a case expressed by either of the two following equations : COS ct r COS X -f COS /3 r COS ft, + COS y r COS V = j or R 1 M 1 + R a M 2 + E 3 M 3 = 0; the resultant of M and R is a single force equal and parallel to R, acting in a plane perpendicular to the axis of M, and at a perpen- dicular distance from O given by the equation Case in. When R, = 0, there is no single resultant; and the only resultant is the couple M. Case iv. When the axis ofM. is parallel to the line of action o/R, that is, when either * = *r-, t* = P r ; = w "V" #o> 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 AX, &y, AS. Let a;, y, , be the co-ordinates of the centre of one of these mole- cules. Then its volume is its weight w A x A y A , 52 PRINCIPLES OF STATICS. and its moments relatively to the three co-ordinate planes re- spectively, xw AX Ay AZ; yw AX Ay AZ' } zw AX Ay AZ. 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 AZ nearly; 'W = w'V w'2'AXAyAZ nearly; to V x = w -2,'x A # Ay AZ nearly; therefore omitting the common and constant factor w, and similar approximate formulae for y Q and z . a-) 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 formulse continually approach nearer and nearer, as the dimensions A x, Ay, A z, are diminished. Such limits are found by the process called integration* and are expressed in the following manner : volume weight moments V= f f f dxdydz; = w V = w f j j dx dy dz ; = w / / / xdxdydz-j = wJJJydxdydz- J = wjjjzdxdydz; (2.) .(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 ii the sequel. CENTRE OP GRAVITY. ^J . 53 co-ordinates of the centre of gravity .(4.) / / / xdxdydz XQ , - ~~ y ill dxdydz I ydxdydz y ~ = r r r * \ / / dxdydz II zdxdydz II dxdydz Such are the general formulae for finding the centre of gravity of a 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. 7G. 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, G, its known centre of gravity, Wi its weight. Let A B E be the smaller figure, whose centre of gravity G 2 is known, W 2 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 8 = W, W fl . Join G! G 2 ; G 3 will be in the prolongation of that straight line be- yond G!. In the same straight line produced, take any point as origin of co-ordinates, and an axis at perpendicular to O G 2 Gj as axis of moments. Make OG^ = Xt, j OG 2 = x a , O G 8 (the unknown quantity) = x a . Then the moment of W 3 relatively to the axis at is and therefore _x l W l x *~ - E PRINCIPLES OF STATICS. 77. Centre of Gravity Altered by Transposition. In fig. 21, let A B C D 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 D H, so that the new figure of the body is A B H E. Let Gj 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 F - 21. amount "Wj G! G 2 ; and the centre of gravity of the whole body will be shifted to G s , in a direction G G 3 parallel to GI G 2 , and through a distance given by the formula w /N /"^ /"N />* 1 ' 78. Centres of Gravity of Prisms and Flat Plates. The general for- mulae of Article 74 are intended not so much for direct use in finding centres of gravity, as for the deduction of formulae of a more 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 midwayjbetween 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 a be the distance of the middle line ol one of these bands from Y, anc Fig. 22. yi, 2/ 2 , the distances of the two extremities of that middle line fron O X. Then the band is approximately equal to a rectangular bane of the length y z y lt and breadth A x, the co-ordinates of whose centre are x, and ^-~ Consequently, if z be the uniform thick PEISMS AND FLAT PLATES. 55 ness of the plate, and w its specific gravity, we have for a single band, area = (y z -y^x nearly ; volume = (2/2 - 2/0 A x nearly ; weight =w z (2/2-2/0 Ax nearly; moment relatively to Y, = w z x (y z - 2/0 A a? nearly ; moment relatively to X, = wz J - - AX nearly; 2t and for the whole plate area = 2 ' (2/2 2/0 A ^ nearly ; volume Y = z 2 (y 2 2/0 A ^ nearly ; weight W = w z 2 (y 2 y^ A x nearly ; moment relatively to Y, x W = wz ' 2 x (2/g 2/0 A ^ moment relatively to X, y^T = wz -2^^ A xnearly; I (1.) consequently, the co-ordinates of the centre of gravity of the plate (omitting the common factors w z), are x _ 2>a; (2/2-2/0 A **"> (2/2-2/0 A nearly. 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 = z I (y z 2/1) dx; weight w Y = w z f (y 2 y^) d x ; PKINCIPLES OP STATICS. moments x W = wzJ x (y*-yi)dx-, .(3.) _/ a(y-yi) J /2~2/i) da .(4.) co-ordinates of the centre of gravity The foregoing process is what is usually called by writers on mechanics, "finding the centre of gravity of a plane surface ; " but this phrase ought always to be understood to signify "finding the centre of gravity of a Iwmogeneous plate of uniform thickness, the faces of which are plane surf aces 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 x y\ 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 e 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\ t 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 ? dz; '* dz; .(2.) x ~W = wa x^s 2 - dz^ y ~W = wajyisldz; (3.) CUEVED HOD. co-ordinates of centre of gravity / Xi d z jfdz Z = *. dz [zt dz .(4) When the centres of all the cross-sections lie in one straight line, as in pyramids, cones, conoids, and solids of revolution generally, the centre of gravity lies in that line, which may be taken as the axis of z, making x = 0, y = ; so that z is the only co-ordinate which requires to be determined. 80. Curved Rod. In fig. 23, let K B, represent a curved rod so slender, that its diameter may, without sensible error, be neglected in comparison with its radius of curva- ture at any point ; let a denote its sectional area, uniform throughout, and w, as usual, its specific gravity ; so that the weight of an unit of length of the rod is wa. Let X, Y, O Z be rect- angular axes of co-ordinates. Suppose the rod to be divided into arcs, so short as to be nearly straight ; let the length of any one of these arcs be denoted by A s ; let S S represent it in the figure, and let M be the middle of its length. Then M is nearly the centre of gravity of A Let Fig. 23. M P = x be the perpendicular distance from M to the plane of y z. Then for the short arc S S we have, weight wa A s; moment with respect to an axis in the plane yz, and for the entire rod, = w a, x A W = w a 2 A S ') moment x W = w a 2 x A s nearly; dinate of } 2 x A s of gravity JT ** ' = 2^^ nearly, co-ordinate of centre a-) PRINCIPLES OF STATICS. and similar equations for y and z . Proceeding by the method of limits as before, we obtain as the exact formulae .(2.) and similar equations for y and z . The foregoing process is what is often called by writers on Mechanics, "folding the centre of gravity of a curved line /" but what ought more properly to be called, "finding the centre of gravity of a slender curved rod of uniform thickness." 81. Approximate Computation of integrals. Frequent reference having been made to the process of integration, as being essential to the solution of most problems connected with distributed force, the present article is intended to afford to those who have not made that branch of mathematics a special study, some elementary information respecting it. The meaning of the symbol of an integral, viz. : is of the following kind: / u d x t In fig. 24, let AC D B be a plane area, of which one boundary, AB, is a portion of an axis of abscissae OX, the opposite boundary, C D, a curve of any figure, and the remaining boundaries A C, B D, ordinates perpendicular to O X, whose respective abscissae, or distances from the origin O, are OA OB = b. Let E F = u be any ordinate whatsoever of the curve C D, and O E = x the corresponding abscissa. Then the integral denoted by the symbol, l\udx, means, the area of tJte figure A C D B. The abscissae a and b which are the least and greatest values of x, and which indicate COMPUTATION OF INTEGRALS. 59 the longitudinal extent of the area, are called the limits of in- tegration; but when the extent of the area is otherwise indicated, the symbols of those limits are sometimes omitted, as in the pre- ceding Articles. When the relation between u and x is expressed by any ordinary algebraical equation, the value of the integral for a given pair of values of its limits can generally be found by means of formulae which are contained in works on the Integral Calculus, or by means of mathematical tables. Cases may arise, however, in which u cannot be so expressed in terms of x; and then approximate methods must be employed. Those approximate methods, of which two are here described, are founded upon the division of the area to be measured into bands by parallel and equi-distant ordinates, the approximate computation of the areas of those bands, and the adding of them together j and the more minute that division is, the more near is the result to the truth. First Approximation. Divide the area A C D B, as in fig. 25, into any convenient number of bands by parallel or- dinates, whose uniform distance apart is A x; so that if n be the number of bands, n -f 1 will be the number of ordinates, and . 25 . b a = n A a?, the length of the figure. Let u, u", denote the two ordinates which bound one of the bands ; then the area of that band is u' + u" A X, and consequently, adding together the approximate areas of all the bands, denoting the extreme ordinates as follows, AC = Um ; BD = u b - and the intermediate ordinates by u { , we find for the approximate value of the integral 4 GO Second Approximation. Divide the area A C D B, as in fig. 26, into an even number of bands, by parallel ordinates, whose D uniform distance apart is A x. The ordinates are marked alternately by plain lines and by dotted lines, so as to arrange the bands in pairs. Con- sidering any one pair of bands, such. as E F H G, and assuming that the curve F H is nearly a parabola, it appears, from the properties of that curve, that the area of that pair of bands is M + 4 u" -f u'") A j in which u' and u" r denote the plain ordinates E F and G H, and u" the intermediate dotted ordinate ; and consequently, adding together the approximate areas of all the pairs of bands, we find, for the approximate value of the integral u d x = ( u a -p u b -}- 2 2 u t (plain) ^ + 4 2 -u, (dotted)) *, ................... (2.) It is obvious, that if the values of the ordinates u required in these computations can be calculated, it is unnecessary to draw the figure to a scale, although a sketch of it maybe useful to assist the memory. When the symbol of integration is repeated, so as to make a double integral, such as / / u'dxdy, or a triple integral, such as u-dxdydz, it is to be understood as follows : Let ..,Y.* be the value of this single integral for a given value of y. Con- struct a curve whose abscissae are the various values of y within the prescribed limits, and its ordinates the corresponding values of v. Then the area of that curve is denoted by / v ' dy = / / u dxdy. PROJECTION OF CENTRE OF GRAVITY. 61 Next, let t= j v . dy be the value of this double integral for a given value of z. Con- struct a curve whose abscissae are the various values of z within the prescribed limits, and its ordinates the corresponding values of t. Then the area of that curve is denoted by / t.dz = l v'dydz = l I I u'dxdydz; and so on for any number of successive integrations. 82. Centre of Gravity found by Projection* 'According to the geo- metrical properties of parallel projections, as stated in Chap. IV., Article 62, a parallel projection of a pair of volumes having a given ratio is a pair of volumes having the same ratio j and hence, if a body of any figure be divided by a system of plane or other sur- faces into parts or molecules, either equal, or bearing any given system of proportions to each other, and if a second body, whose figure is a parallel projection of that of the first body, be divided in the same manner by a system of plane or other surfaces which are the corresponding projections of the first system of plane or other surfaces, the parts or molecules of the second body will bear to each other the same system of ratios, of equality or otherwise, which the parts of the first body do. Also, the centres of gravity of the parts of the second body will be the parallel projections of the centres of gravity of the parts of the first body. And hence it follows (according to Article 64), that if the figures of two bodies are parallel projections of each other, the centres of gravity of these two bodies are corresponding points in these parallel projections. To express this symbolically, as in Article 61, let x, y, z, be the co-ordinates, rectangular or oblique, of any point in the figure of the first body ; x f , y', z, those of the corresponding point in the second body ; x , y 0) z 0) the co-ordinates of the centre of gravity of the first body ; d , tf , d^ those of the centre of gravity of the second body; then -._ . . . _ . . _ * ' y ~ y ' z ' ' z' This theorem facilitates much the finding of the centres of gravity of figures which are parallel projections of more simple c* more sym- metrical figures. For example : it appears, from symmetry, as in Art. 72, that the centre of gravity of an equilateral triangular prism is at the 62 PEINCIPLES OP STATICS. point of intersection of the lines joining the three angles of the middle section of the prism with the middle points of the opposite sides of that section. But all triangular prisms are parallel pro- jections of each other ; hence the above described point of inter- section is the centre of gravity of any triangular prism. Also, as in Art. 72, the centre of gravity of a regular tetraedron is at the point of intersection of the planes joining each of the edges with the middle point of the opposite edge. But all tetrae- drons are parallel projections of each other ; hence that point of intersection is the centre of gravity in any tetraedron. As a third example, let it be supposed that a formula is known (which will be given in the sequel) for finding the centre of gravity of a sector of a circular disc, and let it be required to find the centre of gravity of a sector of an elliptic disc. In fig. 27, let A B' A B' be the ellipse, A 6 A = 2 a, and . B' B' = 2 b, its axes, and C' O D' the sector whose centre of gravity is required. One of the parallel pro- jections of the ellipse is a circle, ABAB, whose radius is the semi-axis major a. The ellipse and the circle being both referred to rectangular co-ordinates, with their centre as Fig. 27. origin, x and y denoting the co- ordinates parallel to O A and O B respectively of a point in the circle, and x' and ?/ those of the corresponding point in the ellipse, those co-ordinates are thus related : x y Through C' and D' respectively draw EC'C and FD'D, parallel to O B, and cutting the circle in C and D respectively ; the cir- cular sector C D is the parallel projection of the elliptic sector C' D'. Let G- be the centre of gravity of the sector of the circular disc, its co-ordinates being Then the co-ordinates of the centre of gravity G-' of the sector of the elliptic disc are "OH = af = oc ; HG'= /0= a EXAMPLES OF CENTRES OF GRAVITY. 63 Further examples of the results of this process will be found in the next Article. 83. Examples of Centres of Gravity. The following examples consist of formulae for the weight, the moment with respect to some specified axis, and the position of the centre of gravity, of homo- geneous bodies of those forms which most commonly occur in practice. In each case, as in the formulae of the preceding Articles, w denotes the specific gravity of the body, "W, its weight, and x , &c., the co-ordinates of its centre of gravity, which in the diagrams is marked G, the origin of co-ordinates being marked 0. A. PRISMS AND CYLINDERS WITH PARALLEL BASES. The word cylinder is here to be taken in its most general meaning, as comprehending all solids traced by the motion of a plane curvi- linear figure parallel to itself. The examples here given apply, of course, to flat plates of uni- form thickness. In the formulae for weights and moments, the length or thickness is supposed to be unity. The centre of gravity, in each case, is at the middle of the length (or thickness) ; and the formulae give its situation in the plane figure which represents the cross-section of the prism or cylinder, and which is specified at the commencement of each example. I. Triangle. (Fig. 28) O, any angle. Bisect .o C in D. Join O D. opposite side B W= II. Polygon. Divide it into triangles; find the centre of gravity of each; then find their common centre of gravity as in Art. 75. III. Trapezoid. fRScr 99\ 7 *C,=*j AB.IICE. Fig. 28. 29.) Greatest breadth, A B = B. f f Least C E = b. Bisect A B in 0, C E in W OD/ IB- b x =OG=--- I i __-_- 2 \ W = w - -y< ij^^ZfxO ~V*3.';x ct * ee^ c.^*?- 1 * 3 rf- k; i 5 ..' PRINCIPLES OF STATICS. 7 IV. Trapezoid. (Second solution.) (Fig. 30.) O, point where inclined sides meet. Let O F = x lf OD = x z) 3 A 3 - sin 6 cos * 4 sin 2 sin 2 # cos & 2 2/0 ==r 3 (6 - cos sin B) ' "W = - wr* (Q - cos 6 sin 6). 2i IX. Circular Spandril. (ADX, fig. 32.) _1 X ~~ sn "W 2 sin 6 - sin cos 0-6' 3 sin 2 6 - 2 sin 2 tfcos 6 - 4 sin 8 ^ ^ 2 sin 6 - sin cos 6 d sin Q cos / . 1 = ttr 2 f sin & ~ si X. Sector of Ring. (A C F E, fig. 32.) O A = r ; OE = r. 2 r 3 - r' 3 sin W = w(r*-r'*)0. XL Elliptic Sector, Half-Segment, or Spandril. Centre of gravity to be found by projection from that of corresponding circular figure, as in Article 82. B. WEDGES. A Wedge is a solid bounded by two planes which meet in an edge, and by a cylindrical or prismatic surface (cylindrical, as before, being used in the most general sense). XII. General Formulae for Wedges. (Fig. 33.) All wedges may be divided into parts such as the figure here represented. O A Y, OXY, planes meeting in the edge OYj AXY, cylindrical (or pris- matic) surface perpendicular to the plane OXY; OX A, plane triangle perpendicular to the edge OY; OZ, axis perpendicular to XO Y. Let OX = i; XA = Then z = j % Fig. 33. 66 PRINCIPLES OF STATICS. I ofydx = -r -- "I \ xy dx dx 2 I dx (This iasfc equation denoting that G is in the plane which traverses Y and bisects AX.) In a symmetrical wedge, if be taken at the middle of the edge, y Q = O. Such is the case in the following examples, in each of which, length of edge = 2 y v $ A XIII. Rectangular Wedge. (= Triangular Prism.) (Fig. 34.) W Fig. 34. XIY. Triangular Wedge. (= Triangular Pyramid.) Fig. 35. XQ = XV. Semicircular Wedge. (Fig. 36.) Kadius OX = OY = r. Fig. 36. = -5 w f* t o ( = 3 1416 CENTRES OF GRAVITY. 67 XVI. Annular, or Hollow Semicircular Wedge. (Fig. 37.) External radius, r j internal, /. 16" r 3 r' 3 ' C. CONES AND PYRAMIDS. Fig. 37. 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 &. XVII. Complete Cone or Pyramid. Let the height OX = fa; 3 h. 4 "W = - w A h sin & o XVIII. Truncated Cone or Pyramid. Height of portion trun- cated = h'. W- sin e. hi D. POKTIONS OP A SPHEKE. XIX. Zone or Ring of a Spherical Shell, bounded by two conical surfaces having their common aj at the centre O of the sphere (fig. OX, axis of cones and zone. r, external radius ) /, internal radius } fslie11 XO A. = a, half-angle of less ) cone ' greater / *#-, Fig. 38. 68 PRINCIPLES OF STATICS. r 4 r' 4 cos . + cos 2 _ cos XX. Sector of a Hemispherical Shell. bisects angle DOC; \ D C = 4. (0 X D, fig. 39.) OY 3 r 4 - 3 r 4 - r' 4 ^ sin 16" * r _ / * ~T~* 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 Crravity found [Experimentally. The centre of gravity of a body of moderate size may be found approximately by experiment, by hanging it up successively by a single cord in two different positions, and finding the single point in the body which in both positions is intersected by the axis of the cord. For the resistance of the cord is equivalent sensibly to a single force acting along its axis ; and as that force balances the weight of the body when hung by the cord, its line of action must, in all positions of the body, traverse the centre of gravity of the body. SECTION 2. Of Stress, and its Resultants and Centres. 86. Stress, its Nature and Intensity. The word STRESS has been adopted as a general term to comprehend various forces which are exerted between contiguous bodies, or parts of 'bodies, and which are distributed over the surface of contact of the masses between which they act. The INTENSITY of a stress is its amount in units of force, divided by the extent of the surface over which it acts, in units of area. The French and British units of intensity of stress are compared CLASSES OF STRESS. 69 in a table annexed to this volume. The following table shows a comparison between different British units of intensity of stress : Pounds on the Pounds on the square foot. square inch. One pound on the square inch, 144 1 One pound on the square foot, 1 ^ T One inch of mercury (that is, weight of a column of mercury at 32 Falir., one inch high), 70-73 0-4912 One foot of water (at 39-4 Fahr.), G2-425 0-4335 One inch of water, 5*2021 0-036125 One atmosphere, of 29 -922 inches of mercury, 211C-4 14-7 87. Classes of stress. Stress may be classed as follows : I. Thrust, or Pressure, is tjhe 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. Sliear, or Tangential Stress, is the force which acts between two contiguous bodies or parts of a body, when each draws the other sidewaj^s, in a direction parallel to their surface of contact, and which tends to distort each body on which it acts. In expressing a Thrust and a Pull in parallel directions algebrai- cally, if one is treated as positive, the other must be treated as negative. The choice of the positive or negative sign for either is a matter of convenience. In treating of the general theory of stress, the more usual ^system is to call a pull positive, and a thrust negative : thus, let p denote the intensity of a stress, and n a certain number of pounds per square foot ; p = n will denote a pull, and p = n a thrust of the same intensity. But in treating of certain special applications of the theory, to cases in which thrust, is the only or the predominant stress, it becomes more convenient to reverse this system, calling thrust positive, and pull negative. The word " Pressure," although, strictly speaking, equivalent to " thrust," is sometimes applied to stress in general ; and when this is the case, it is to be understood that thrust is treated as positive. 70 PRINCIPLES OF STATICS. 88. Resultant of Stress : its Magnitude. If to a plane surface of any figure, whose area is S, there be applied a stress, either normal, oblique, or tangential, and parallel in direction at all points of the surface (according to the restriction stated in Art. 67), then if the intensity of the stress be uniform over all the surface, and denoted , the amount or magnitude of its resultant will be P=.PS .............................. (1.) If the intensity of the stress is not uniform, that amount is to be found by integration. For example, in x fig. 40, let A A A be the plane surface, and let it be referred to rectangular axes of co-ordinates in its own plane, OX, O Y, Conceive that plane to be divided into small rectangles by a network of lines parallel to O X and O Y respectively, and let A x, *y } be the dimensions of any one Fig. 49. o f ^ ege rectangles, such as that marked a in the figure. Conceive a figure approximating to that of the given plane surface to be composed of several of these small rectangles, so that g = s-AajA^ nearly; ..................... ..(2.) let p be the intensity of the stress at the centre of any particular rectangle, so that the stress on that rectangle is p A x A y nearly. Then the amount of the resultant stress is given approximately by the equation P = 2 p&x&y nearly , (3.) Then passing, as in previous examples, to the integrals, or limits towards which the sums in the equations 2 and 3 approach as the minuteness of the subdivision into rectangles is indefinitely in- creased, we find, for the exact equations, .(4.) The mecm intensity of the stress is given by the following equation : P / jpdxdy = -- = ~T~~ (5>) dxdy CENTRE OP STKESS. 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; O Z, a third axis perpendicular to that plane. Conceive a solid to exist, bounded at one end by the given plane surface A A, laterally by a cylindrical or prismatic surface generated by the motion of a straight line par- allel to Z round the outline of A A, and at * the other end by a surface B B, of such a figure, that its ordinate z at any point shall be proportional to the intensity of the stress at the point of the surface A A from which that ordinate proceeds, as shown by the equation S =P .................... . ..... (6.) w The volume of this ideal solid will be =ff z -dxdy ........................ (7.) So that if it be conceived to consist of a material whose specific gravity is w, the amount of the stress will be equal to the weight of the solid, that is to say, P = wV ............................. (8.) If the stress be of opposite signs at different points of the plane surface A A, the surface B B and the solid which it terminates will be partly at one side of A A and partly at the opposite side, as in fig. 42; and in this case, the two parts into which the solid A B A B is divided by .the plane X O Y, are to be regarded as having opposite signs, and Y is to be held to represent the difference of their volumes. Fig. 42. The mean stress of equation 5 is evidently Po = in which Z Q is the height of a parallel-ended prism or cylinder standing on the base A A A, and of volume equal to the solid ABAB. 89. The Centre of Stress, or of Pressure, in any surface, IS the point traversed by the resultant of the whole stress, or in other words, the Centre of Parallel Forces for the whole stress. From the principles already proved in Chap. II., Section 4, it follows, that 72 PRINCIPLES OF STATICS. the position of this point does not depend upon the direction of the stress, nor upon its absolute magnitude ; but solely on the form of the surface at which the stress acts, and on the proportions between the intensities of the stress at different points. As in Article 88, conceive a figure approximating to that of the given plane surface A A A (fig. 40), to be composed of several small rectangles ; let ex. /3 denote the angles which the direction of the stress makes with O X, O Y respectively. Then the moments, relative to the co-ordinate planes, Z O X, Z O Y, of the components parallel to those planes of the stress on A x A y, are given by the approxi- mate equations. Moment relatively to Z X. yp&x&y sin /3\ 7 71 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 Q and y denote the co-ordinates of the centre of stress ; y Q P sin /3 = sin /3 / / yp dx dy } >. (1 ) XQ P ' sin t = sin a. I I xp dx dy j Consequently the co-ordinates of the centre of stress are - r J j p-dxdy which are evidently the same with the co-ordinates, parallel to OX and Y, of the centre of yravity 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, / I 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 inteiisit}^ of the stress be uniform, the factor jp in equation 2 of Article 89 becomes constant, and may be removed from, both numerator and denominator of the expressions for OC Q and y , which then become simply the co- ordinates of the centre of gravity of a flat plate of the figure A A A. This also appears from the consideration, that the surface B B in fig. 41 becomes a plane parallel to A A, and the solid AB AB, a parallel-ended prism or cylinder. 91. Moment of Uniformly Varying Stress. By an uniformly varying stress is understood a stress whose intensity, at a given point of the surface to which it is applied, is proportional to the distance of that point from a given straight line. For example, let the given straight line be taken as the axis 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 i p-dxdy = a I I x'dxdy (2.) which, if the axis OY traverses the centre of gravity of a plate of the figure of the surface of action AAA, 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 be its centre of gravity (that is, the centre of gravity of a flat plate of which A A A is the figure); -YOY the neutral axis of the stress applied ; - X X perpendi- cular to -YOY, and in the plane of A A A ; ZOZ perpendicular to that plane. Conceive a plane BB inclined to AAA to traverse the neutral axis, and to form, with the plane AAA, a pair Fig. 43. of wedges bounded by a cylindrical or prismatic surface parallel to 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 Ax&y denote, as before, the area of a small rectangular portion of the surface, x } y, the co- ordinates of its centre (for which z = Q), and p~ax, the intensity of the stress on it, so that A P P AX&y = Cl/XAXAy 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 ;i P-2, cos ; e/"\ ~\.7~ T> U x ; A r ' x cos y : ' * OZ; AP(#eos/3 ycosot). Summing and integrating those moments, the following are found to be the total moments : round OX ; M t = a cos y f ( xy 'dxdy OY;M 2 = acosy f j x* -dxdy OZ; M 3 = a jcos/3 f fx 2 'dxdy-coset f (xydxdyl . For the sake of brevity, let J Jx 2 ' dxdy = 1, f f ay dxdy = K-, (3A.) then, as in equation 7 of Article 60, we find, for the moment of the resultant couple, = a ,y {(I 2 + K 2 )cos 2 y + F -cos 2 /3 + K 2 cos 2 a -2IK-cos*-cos/3.} = a J (I 2 sin 2 + K 2 sin 2 p 2 I K cos * cos /3);. ..(4.) and for the angles A, <", v, 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, _; cos^ = w ; cos,^ (5.) MOMENT OP BENDING STRESS. 75 The following equation is easily verified : cos os cos x + cos /3 cos p + cos y cos v = ......... (5 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, ccj, from, the neutral axis; then (6.) 92. Moment of Bending Stress. If the uniformly varying stress be normal to the surface at which it acts ; that is to say in symbols, = 0j cos/3 = 0; cosy = (1.) then it is evident that M 3 = Oj cos 9 = j ........................ (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 : Mi = aK; M 2 = al, cos A = sin ^ = K J r + K 3 cos = 0; n , M = M 3 = a (I sin - K cos *) ; cos A = ; cos ^ = j cos = 1. J In the cases referred to in Article 92, for which K = 0, we find so that in these cases it is only the component of the stress parallel to the neutral axis which prod aces the twisting couple. 94. Centre of Uniformly Varying Stress. When the amount of an uniformly varying stress has magnitude, that stress may be con- sidered as made up of two parts, viz. : First, an uniform stress, whose intensity is the mean intensity of the entire stress, and whose centre is the centre of figure, O, of the surface of action. As in Article 88, equation 5, this mean intensity may be represented by P total stress Secondly, an uniformly-varying stress, whose neutral axis tra- verses O, whose amount is = 0, and whose intensity, p', 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=Po + P'=Po + a* .................. (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 tl resultant P parallel to itself through a distance L = ............................. (3.) to the opposite side to that whose name designates the tendency o the couple M ; and the direction of the line L is perpendicular a once to that of the stress, and to that of the axis of the couple M. The co-ordinates relatively to the point O of the centre of stres as thus shifted, being the point where the line of action of th shifted resultant cuts the plane of AAA, are most easily found b 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 along the neutral axis / xp'-dxdy a I x*' dxdy al '=- p " p = T ) / lyp''dxdy a I Ixydxdy aK |2/o= p- ~p =-p~ K*-) The angle 6 which the line joining and the centre of stress makes with the neutral axis O Y, is that whose cotangent is (5.) XQ I This line will be called the axis conjugate to the neutral axis YOY. When K = 0, it is perpendicular to the neutral axis. 95. Moments of Inertia of a Surface. The integral I = / / x 3 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 urface of a given figure relatively to different neutral axes ; a knowledge of which relations is useful in the determination of the moment of a bending or twisting stress. Let A A in fig. 44 represent a plane surface of any figure, O its entre of gravity, YOY, XOX, a pair of rectangular axes crossing ach other at O, in any position. Taking YOY as a neutral axis, let he moment of inertia relatively to it be I = J / x 3 ' dxdy; et the moment of inertia re- ativelyto XOX as a neutral be 'dxdy; ndlet = j j xydxdy. ,(1.) Now let Y'OY', X'OX', be a new pair of rectangular axes, in Jiy position making the angle YOY' = XOX' = ft f 8 PRINCIPLES OF STATICS. with the original pair of axes; and let J' = fftf'-dofdi/', K' =ffo!y f -dafdy'. .(2.) The following relations exist between the original co-ordinates, x, y, of a given point, and the new co-ordinates a/, ?/', of the same point; a/ = x cos y sin /3 (3.) (This last quantity, which is the square of the distance of the given point from O, is what is called an Isotropic Function of the co-ordinates j being of equal magnitude in whatsoever position the rectangular co-ordinates are placed.) From the equations (3), the following relations are easily deduced between the original integrals I, J, K, and the new integrals I',J',K':- I r = I cos 2 ft + J sin 2 /3 2 K cos /3 sin /3; ] J' = I sin 2 /3 + J cos 2 /3 + 2 K * cos /3 sin /3; V ...(4.) K' = (I J) cos /3 sin ft + K (cos s /3 sin 2 /3.) j Also, the following functions of those integrals are found to b< isotropic; I + J = T + J' = (x 2 + f) ' dxdy ...... (5.) (called the polar moment of inertia) I J K 2 = I 1 J' K' 2 .................. (6.) Equation 5 may be thus expressed in words : THEOREM I. The sum of the moments of inertia of '" a surfai relatively to a pair of rectangular neutral axes is isotropic. Equations 5 and 6 in conjunction lead to the following cons( quences. Because the sum I' + J' is constant, I' must be maximum and J ; a minimum for that position of the rectangulr axes which makes the difference I' J' a maximum. And becaus (T J') 2 = (F + J') 2 41' J', I' J' must be a maximum for that position of the axis whk makes I' J' a minimum. But by equation 6, 1' J' K/ 2 is constaj CONJUGATE AXES. 79 for all positions of the axes; therefore when K' == 0, I' J' is a minimum, I' J' a maximum, I' a maximum, and J' a minimum. Hence follows, in the first place, THEOREM II. In every plane surface there is a pair of rect- angular neutral axes for one of which the moment of inertia is greater, and for the other less, than for any other neutral axis. These axes are called Principal Axes. Let I 1? J 15 be the maximum and minimum moments of inertia relatively to them, and let ft be the angle which their position makes with the originally-assumed axes j then because K, = 0, we have, from the third of the equa- tions (4) and because I> + J, = I + J, and I, J = IJ __ K 2 , we have, by the solution of a quadratic equation, The position of the principal axes, and the values of I,, J u being once known, the integrals I', J', K', for any pair of axes which make le angle /3 with the principal axes, are given by the equations r = I,co80 + J lS in'/3'j ) J = Ij sm 2 p + J, cos 2 p; }. , (9.) K' = (I, JJ cos p sin - ( If I a = J,, 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 ie angle which the axis conjugate to Y makes with O Y TT cotan 6 = - -. For the principal axes, K = 0, cotan 6 = 0, and 6 is a right angle; from which follows THEOREM III. The principal axes are conjugate to each other: that is, if either of them be taken for neutral axis, the other will be the conjugate axis. Returning to_ equation 4 of the present Article, let us suppose, that the axis conjugate to the originally assumed neutral axis YO Y has been determined, and that its position is Y'O Y', so that 80 PRINCIPLES OF STATICS. Let this conjugate axis be assumed as a new neutral axis. Then the integrals I', J', K', belonging to it are determined by substituting 6 for /3 in the equation 4 ; that is, by substituting for cos /3 and sin /3, the values of cos 6 and sin 6 in terms of K and I, viz. : K I cos d = r== - ; sin & = J~F + E? ,yi a + & 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 TC' K. cotan ff = -^j- = ,- = cotan 6, whence .(11.) or in words, THEOREM IT. If the axis conjugate to a given neutral axis be taken as a new neutral axis, the original neutral axis will be the new conjugate axis. The following mode of graphically representing the preceding theorems and relations depends on well known properties of the ellipse. *> In fig. 45, let X : Y, perpendicular to each other, represent the principal axes of a surface. With the semi-axes, Fig. 45. describe an ellipse, so that the square of each semi-axis shall represent the moment of inertia round the other. Let the semidiameter OY' be drawn in the direction of any assumed neutral axis, and let ^L YjO Y' = /3'. Draw 00, the MOMENTS OF INERTIA OF A SURFACE. 81 semidiameter conjugate to OY', so that the tangent CT shall be parallel to Y'. Let CT = t, and let the normal OT = n. Then it is well known that n 2 = a 3 cos 2 p + V sin 2 /3'; ) and that I ............ (13.) n t = (a 2 5 8 ) cos ' sin /3'; J consequently, comparing this equation with the equation 9, we find, I' = n'; cotan 6 = -=- = = cotaii Y' C ; In } so that the square of the normal O T represents the moment of inertia for the neutral axis O Y', and the semidiameter O con- jugate to OY' is also the conjugate axis of the neutral axis OY'. and vice versd. 5, & 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 land, 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 I conceive an axis 0' Y' to be drawn parallel to O Y ; the point which is at the distance x from O Y, is at the distance O/ = X - XQ I from O'Y'. The required moment of inertia is therefore, I = o?JS + 2fl3 / / x' 'dxdy + I I x' 3 'dxdy ; and because O'Y' traverses the centre of gravity of S, x' 'dxdy = ; o 82 PRINCIPLES OF STATICS. so that the middle term of the expression for I vanishes, leaving I = ic 2 S + j j x'*'dxdy } ............... (15.) or in words, THEOREM "V* The moment of inertia of a surface relatively to an axis not traversing its centre of gravity is greater than the moment of inertia round a parallel axis traversing its centre of gravity, by the product of the area of the surface into the square of the distance between those two axes. The following is a table of the principal (or maxima and minima) moments of inertia of surfaces-of-action of stress of those figures which most commonly occur in practice : -p. Maximum 1^ Minimum J t (neutral axis Y). (neutral axis X). I. RECTANGLE. Length along X, ) h s b hb s h' } breadth along O Y, b .......... J Ti2~ "12" II. SQUARE. Side = h,. ......... ...... ** ^ u u III. ELLIPSE. Longer axis, h ...... ) nl^b vhb s Shorter axis, b ...... j ~~64~ 64 IV. CIRCLE. Diameter, h .............. 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 tangles ; dimensions of any one I h\\x, b || w; distance of its centre j from Y, a? ; from OX, y ...... j SECTION 3. Of Internal Stress, its Composition and Resolution. 96. internal stress in General. If a body be conceived to be divided into two parts by ar ideal plane traversing it in any direction, the force exerted between those two parts at the plane of division is an internal stress. The finding of the resultant, and of the centre of stress, for an internal stress, depend upon the principles relating to stress in general, which have been explained in the last section. The present section refers to a different class of problems, viz., the relations between the different stresses which can exist together in one body at one point. SIMPLE STRESS. 83 A body may be divided into two parts by a plane traversing a. given point, in an indefinite number of ways, by varying the angular position of the plane ; and the stress which acts between the two parts may vary in direction, or intensity, or in both, as the position of the plane varies. The object of the present section is to show the laws of such variation ; and also the effect of applying different stresses simultaneously to one body. The investigations' in this section relate strictly to stress of uniform intensity ; but their results are made applicable to stress of variable intensity to any required degree of accuracy, by sufficiently contracting the space under consideration, so that the variations of the stress within its limits shall not exceed the assigned limits of deviation from uniformity. 97. Simple Stress and its Normal Intensity. A simple stress is a pull or a thrust. In the following investigations a pull will be treated as positive, and a thrust as negative. In fig. 46, let a prismatic solid body, or part of a solid body, whose sides are parallel to the axis 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 Xj of the amount P, and of the intensity This, which is the intensity of the stress as distributed over a plane normal to its direction, may be called its normal intensity. 98. Reduction of Simple Stress to an Oblique Plane. Next, let the plane of section be conceived to have the position B B, oblique to O X ; let N be a line normal to B B, and O T a line at the intersection of the planes B B an \ X O N. Let the obliquity of the plane of section be denoted b^ = ^: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 OX} but the area over which that pull is distributed is now 84 PRINCIPLES OP STATICS. . cos 4' consequently, the intensity of the stress, as reduced to the oblique plane of section, is .= g - 99. Resolution of Oblique Stress into Normal and Tangential Components. The oblique stress P on the plane of section B B may be resolved by the principles of Articles 55, 57, into two compo- nents, viz. : Normal component a- ) T , a i S\-KT r -t COS 1 long ON, ............ j Tangential component ) T> /i . i r\ m r Jt Sin along OT, ............ J and the intensities of these components are, Normal ; p n = p r cos &=p x ' cos 2 6 ', Tangential ; p t =p r sin d=p x ' cos 6 sin Suppose another oblique plane of section to cut the body at right angles to B B, so that its obliquity is ff = 90 0; and let the intensity of the stress on the new plane be denoted by accented letters ; then (2 } = p t ',pn+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 rigJit 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. JPai* of Conjugate Stresses. THEOREM. If tfte stress OH a given plane in a body be in a given direction, the stress on any plane parallel to that direction must be in a direction parallel to the first-mentioned plane. In fig. 47, let YOY represent, in section, a given plane tra- versing a body, and let the stress on that plane be in the direction X O X. Consider the condition of a prismatic portion of the body represented in sec- tion by A B C D, bounded by a pan: 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 7 Fig. 47. XOX iii 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 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 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 X X and Y 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 fourth stress which shall be conjugate at once to each of the other three. The relations between the three angles which the directions of three conjugate stresses make with each other, the three obliquities of those stresses (being the angles which they make with the per- pendiculars to their respective planes of action), and the three angles which those perpendiculars make with each other, as found by the ordinary rules of spherical trigonometry, are given by the following formulae. GENERAL CASE. Let x, y, z, denote the directions of the three conjugate stresses; 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, vw, wu, uv, the inclinations of those perpendiculars to each other ; A A A u x, vy, wz, the respective obliquities of the stresses. Then those nine angles are related as follows : T4.1 2 A 2 A * A A A A .Let 1 cos* y z cos z x cos* x y + 2 cos y z cos z x cos x y = 0; (1:) Then A J C A cos z x cos xy tt nil *y . f,f\0 / nit " (2.) 3mvw , = ___ f _ _ ;cos ^ = _ ^ _ sin zx ' sin xy sin z x * sin x y A A A A / C A cos x y ' cos y z cos z x 8LVWU = ^ ^GOSWU= -* ^ ^ : sin x y ' sin y z sin x y ' sin y z A A A A /C A cosy z' cos zx cosxy SWUV= t A . A jCOSMP= K - - sin y z sin z x sin y z sin z x A JO A /C A /C cos u x = A ; cos v y = A ; cos wz = -^^ (3.) sin y z sin z x sin xy RESTRICTED CASE I. Suppose two of the stresses, for example, those parallel to x and y, to be perpendicular to each other, and oblique to the third. Then A A \ cos x y = ; sin x y 1 ; ( C = 1 cos 2 y z - cos 2 z Xy ) ...(4.) PLANES OF EQUAL SHEAR. 87 A sin v w = A _ JO sin z x A A COS 11 Z fo^ ^" w "" A ' sin y z sin y z A A A cos y z cos z x sin yz- sin rt/-\o ni n> . _ V A > C03WV A A ^ a; sin y sin a; . A ^C A . rr><3 -71 it 31 . rrk<3 tii * 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 A A cos y z = ; cos z x = ; .A . A sin y z = 1 ; sin z x = 1 ; C = sin 2 x y. sin v w = 1 ; cos v w = ; (or v w = 90); Ednt0t*=l; cosw?w = 0j (or wu = 90); A A A A sin u v = sin x y; cos uv = cos x y\ (m } uv + xy = 180). .(7.) A A A A cos u x = sin x y } cos v y = sin # y ; A A A A or wa? = -yy = 90 xy\ wz = 0; cos w z = 1 ; .(8.) ....(9.) results identical with those given at the end of Article 101. RESTRICTED CASE III. All three stresses perpendicular to each other. In this case the normals to the three planes of action are perpendicular to each other, and coincide with the directions of the stresses. 103. Planes of Equal Shear, or Tangential Stress. THEOREM. If tJie stresses on a given pair of planes be tangential to those planes, and parallel to a third plane which is perpendicular to the pair of planes, those stresses must be of equal intensity. Let the third plane be represented by the plane of the paper in fig. 48, and let the pair of planes on which the stresses are tangen- 88 PRINCIPLES OF STATICS. tial, and parallel to the plane of the paper, be parallel respectively to AB and AD. Consider the condition of a right prism of any length, represented in section by ABCD, and bounded by a pair of parallel planes, AB, CD, 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. The- forces exerted by the other parts of the body on the pair of faces AB, CD, form a couple (right-handed in the figure), of which the arm is the perpendicular distance EF, between AB and CD, and the moment, 2vareaAB-EF. The forces exerted by the other parts of the body on the pair 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 ABCD is obviouily 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 bf oblique stress on tlvree planes at riglit angles to each other, tJie tangential components of the stress on any two of those planes in direction* parallel to the third plane must be of equal intensity. Let yz, zXj 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 X R represent the intensities of the forces exerted by the other parts of the body on the pair of faces AB, CD; resolve each of these forces into a component XN, parallel to the plane z x, and a tangential component, XT, parallel to the axis of y\ the resultants of the components X N will act through the axis of z, and will produce no couple round that axis; the com- ponents XT will form a couple acting round that axis. In the same manner the intensities of the forces exerted on the faces AD, CB, being re- presented by the equal and parallel lines, "Yr, are resolved into the components, Yn, 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 ABC D, 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, XT==Y, 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 ex), in the direction of z. Then resolving the stress on each of the three rectangular planes into three rectangular components, we have the following notation : PLANE. DIRECTION. x y Z yz p xx p xy P,, \ . zx P 3 x Pn p yz j- intensities. xy Pzx P*, A J Then, in virtue of the Theorems of Articles 101 and 102, we have the normal stresses, p^pyy, Put conjugate and independent; and i)0 PRINCIPLES OF STATICS. in virtue of the theorem of this Article, there are three pairs of tangential stresses of equal intensity, [The reader who wishes to confine his attention to the more simple class of problems may pass at once to Article 108, page 95.] (5 105. Tetraedron of stress. PROBLEM I. The intensities of three conjugate stresses on three planes traversing a body being given, it is required to find the direction and intensity oftJie stress, on a fourth plane, traversing tlie same body in any direction. In fig. 50, let Y O Z, Z O X, X O Y, be the three planes, on which act conjugate stresses in the directions OX, O Y, O Z, of the intensities p x , py p.,. Draw a plane parallel to the fourth plane, cutting the three conjugate planes in the triangle ABC, so as to form with them the tri- angular pyramid or tetraedron O A B C. Then must the stresses on the four triangular faces of Fig 50 *kat tetraedron balance each other; and the total stress on A B C will be equal and opposite to the resultant of the total stresses on B C, OCA, and A B. On X, O Y, O Z, respectively take Ol) = total stress on O B C = p K area O B C, O E = total stress on O C A = p y ' area OCA, OF = total stress on A B = p* ' area A B. Complete the parallelepiped O D E F E. ; then will its diagonal OK 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 OR _ Q. E. I. area A B =p gx ' area B C + p KV area OCA + p t , - area O A B, O E = p g9 ' area O B C + p yy ' area OCA + p yg - area O A B, OF =p ix ' area B C + p yt area OCA + p a - area O A B; Complete the rectangle D E F R then will its diagonal O R 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 be . Q. E. I. area ABC A A A To express this algebraically, let x n, y n, 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 t area A B = area ABC* cos z n; so that the rectangular components of the intensity p r are A A A " p na =. p xie cos x n + p xy cos yn-v p fa . cos z n A A A ft \ p ny = p x ,j cos x n + Pyy ' cos y n + p yg cos z n r" ........ \ Lt ) A A A p ng = p zx cos x n + p yz ' cos y n + p zis cos z n . The resultant intensity of the stress required is given by the equation and its direction by the equation A p ne A p ny A p nz cosa;r = ; cos2/r= -; cosr= Pr Pr Pr 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 planes. 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', y f , 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 fl , &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 nx f =p nie 'cosxxf + p ny - cos y x' + p^ cos z x ; A A A / -f > A A A, p M f =p M cosxz' + p ny cos yz + p M coszz. For n are now to be substituted successively, both in p n ., &c., and in the values of p 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. pJJ =p, x cos s x x' + p yy cos 2 y x' -f p u cos 2 z x f + 2 p y , cos y x' cos s x + 2p fx cosz x' cos xx' -f- 2p jfy cosxx' cosy x 1 ', P y ' y '=P** cos 2 x y' +p yy cos 2 y if +p zt cos 2 z y A A A A A A 4- 2p y ,cosyy'coszy' + 2 p u cos z y f cos x y -f 2p fy cosxycosyy' ; PS* = Px c s 2 xz'-\- p y}> cos 2 y z' -\-p zt cos 2 z z f + 2p yt cos y z cos z z -f 2 p tf cos z z cos xz'-\- 2p, y cos xzcosyz*", PRINCIPAL AXES OF STRESS. 93 TANGENTIAL STRESSES. A A A A A A p 9 ' t r p xx cos x y' cos x z -\- p yy cos y y cos y z -\-p zz cos z y' cos z z 1 AA AA AAAA + p vz (coszy'cosyz' + cosyy'coszz') +p sx (cosxy'coszz + coszy'cosxz) A A A A + p xy (cos y y' cos xz' + cos x y' cos y z') ; A A A A A A, pit =p xa; cos x z cos x x' -rp yy cos y z f cos y x 1 -f p^ cos cos z x AA AA A A, A A + p yz (coszz' cosy x' + cosy z coszx) + p ix (cosxz coszx + coszz'cosxx') A A A A x f ^ (cos y z 1 cos # a/ + cos x z 1 cos y x 1 ) ; A A A A A A Pxy =Pxx COS X X ' COS X 'U' ^~Pyy COS 2/ ^ COS 2/ 2/ + P*t COS ^ ^ COS Z V' A A , A A A A, A, A + p y;s (coszx'cosyy -{- cosy x' coszy') + p. x (cosxx'coszy + coszx cosxy') A A , A A -j-^j, (cos y x' cos x y -f cos cc x' cos y y'\ The two systems of component stresses, p xx , &c., relative to the axes x, y, z, and jp,V, &c., relative to the axes x' } y, z 1 , which con- stitute the same compound stress, are said to be equivalent to each other. 107. Principal Axes of Stress. THEOREM. For every state of stress in a body, there is a system of three planes perpendicular to each oilier, 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 Pny A cos x r = = cos x n ; cos y r = = cos y n ; A P~ A /i \ coszr = = coszn (I.) Introducing these values into the equation 1 of Article 105, we obtain the following : A A A (P p r ) cosxn + p.^ cos y n + p tx cos z n = ; p^ cos x n + (pyy - p r ) cos y n + p ye cos z n = ; j- (2.) A A A p a cos x n JTP^ cos y n+(p fz - p r ) cos z n f = 0. J 94 PRINCIPLES OF STATICS. From these equations, by elimination of the three cosines, is obtained the following cubic equation ; Let P*,+P n +P = A-; ] yy -pi -p*l -P' y = B ; \ (3.) **Pl -PyyP?*-P**P*y = C > J Then pi- Ajp2 + B#.-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 gx p xy + (p r - p xx )p y ,} = cos y n {p xy p yz + (p f -p w )P**} = coszn\p yz p zx + (Pr-p*,)p xy ] ............ (&) Let p l} p 2 , p 3 , 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 =Pi+P* +P m , 'B=p*p 3 +p 3 pi+pip 2 ; > ........... ...(6.) G=p 1 p a p s . 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 normal makes the angles x n, 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 ; Pvy =P* ' P** =Ps ', Pyz =P** =P = - STRESS PARALLEL TO ONE PLANE. 95 Thus we obtain A A A A p cos x p = pi cos x n ', p cos yp = p 2 cosyn; A A p cos zp = p s cos zn.. ....... (7.) p = ,J < p\ ' cos 2 x n +p 2 2 cos 2 y n -\-pl cos 3 z n I . . .(8.) The equations 7 are easily transformed into the following : A A A A A A cos x n cosxp, cosyn eo&yp coszn coszp , . P Pi ~ P P*~ 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-./J P ( * A A 2 A ") / J cos 2 ^ cos- yp co^zp { ...HO.) /v \ 1 -j- a / \ / ( P* V* PI j the well known equation of an ellipsoid, in which p l9 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 np = cos x n cos x p {- cos y n cos yp + cos z n cos zp == p A cos 2 2/ ft 4- j p 3 and this cosine, by being positive ) indicates ( a pull ^ nothing V that the < a shear negative j stress ^? 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 tlie stress on a fourth plane, perpendicular also to the first mentioned plane. In fig. 51, let the plane of the paper represent the plane to which the stresses are parallel ; let X and O Y represent the directions of the pair of conjugate stresses, whose intensities are p x and p y ; and let AB be the plane, the stress on which is sought. Consider the condition of a prism, O A B, bounded o R >j y by the plane A B, and by planes parallel Fig- 51. to X and Y respectively. The force exerted by the other parts of the body on the face A of the prism, will be proportional to on Y take O E 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 J.'OBj 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 n AB pi (7B 2 + pi O"A 2 + 2p fP , OB OXcos ^ XO Y \ OB 2 +OA 2 -2OB-0~Acos^XOY. J The parallelogram marked in the figure with the capital letters K, E, corresponds to the case in which p x and p y are of the same kind, both pulls, or both thrusts, in which case p r is of the same kind also ; the parallelogram marked with the small letters, r, e, corresponds to the case in which p x and p y are of opposite kinds, one being a pull and the other a thrust ; in which case p r agrees in kind PRISM OF STRESS. 97 with that one of the given conjugate stresses whose direction falls to the same side of A B with it. When Or 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 tliat 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 Y, which stress is parallel to O X , 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 52 - tangential stresses on those two planes must be of equal intensity; and they may therefore be denoted by one symbol, p xy , which sym- bol may be read as meaning the intensity of ( x \ on a plane ( 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 X the angle X N = x n. Consider the condition of a prism A B, of the length unity, bounded by the planes O A _L#, O B JL x, A B _L O K The areas of the faces of that prism have the following proportions : _ _ A _ _ A OB = AB -cosxn; OA = AB sin xn. The forces exerted on the faces O A and O B, in a direction parallel to x, consist of the normal stress on B, and the tangential stress on O A ; that is to say, p xx ' O B -f p xy O A = A B < p xx cos x n + p xy * sin x n > Let this be represented by O D. The forces exerted on the faces O A and O B, in a direction paral lei to y, consist of the normal stress on OA, and the tangential stress on O B ; that is to say, Pxy B -|- pyy OA = AB ' | p xy ' cos x n + p yy sin x n i Let this be represented by OE. 98 PRINCIPLES OF STATICS. Complete the rectangle O D R E ; the amount and direction of the stress on A B will be represented by its diagonal, and the intensity of that stress by _ -^.jL^^t.* ~- OB, f A A + %Pxy (Pxx ~}-pyy) cos xn ' sin x n V ................. (1.) From K draw BP perpendicular to the normal O Nj then the normal and the tangential components of the total stress on A B will be represented respectively by OP = OD cos xn + OE sin xn; - _ , A __ A "fe-'dZ'iT- PB = O D sin x n - O E cos xn; and the intensities of these components by OP A ; A , A Ax p n = ^= = p xx cos 2 xn -\~pyy ' sin 2 xn -f 2p xy - cos xn sin am; -X _D A A . A A f . w-z* ryy/ cos xn * s i n xn -\-Pxy ( sm " ^' ft- -cos x n). { AB The obliquity, ^ NOB = in, r, of the stress on A B is given by the equation ismnr -L (3-) P 109. Principal Axes of Stress Parallel to One Plane. THEOREM. For every condition of stress parallel to one plane, there are two planes perpendicular to each other, on which there is no tangential stress. As in Article 108, let the three rectangular components, p xx) p yy , p^, of the stress on two rectangular planes, O Y, OX, be"given. The condition, that there shall be no tangential stress on a plane normal to 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 xy , A . A : " n xx -v m 9 GQ^ xn sm'xn or, what is the same thing, A %Pxy n < : z IT* ' PXX Pyy FLUID PRESSURE. 99 Now for two values of x n, differing by a right angle, the values of tan 2xn 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 There may be a third principal stress, conjugate and at right angles to the first two ; but as, with one exception, the ensuing in- vestigations of this section relate to stresses upon planes parallel to the direction of this third principal stress, which does not affect such planes, it may be left out of consideration. The most simple mode of expressing the relations amongst inter- nal stresses parallel to a plane is obtained by taking the two prin- cipal axes of stress in that plane for axes of co-ordinates; and this ' is done in the ensuing Articles. 110. Equal Principal Stresses Fluid Pres -THEOREM I. If O, pair of principal stresses be of the same kind and of equal intensity, every stress parallel to the same plane is of the same kind, of equal in- tensity, and normal to its plane of action. In fig. 53, let OX, OY, be the direc- x tions of the given principal stresses, and p*> Py) their intensities. By the condi- tions of the question, those intensities are equal, or Let it be required to find the direction and intensity of the stress on any plane ~ T " 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 A will be respectively p x ' OB and p y OA. On X and Y respectively, take OT> to represent p, B, and O E to represent p y - O~A; complete the rectangle O I> R E; then its diagonal R 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 OB 100 PRINCIPLES OF STATICS. Now, because p x = p y , we have or> OJE OR OR OA AB' and consequently and because of the similarity of the triangles A B, E R, R is perpendicular to A B. Therefore, the stress on each plane per- pendicular to X Y is normal, and of equal intensity in all direc- tions. Q. E. D. In this case it is obvious, that eveiy direction in the plane X O Y has the properties of an axis of stress. COROLLARY. If the stress in all directions parallel to a given plane be normal, it must be of equal intensity in all those directions. THEOREM II. In a perfect fluid, the pressure at a given point is normal and of equal intensity in all directions. Fluid is a term opposed to solid, and comprehending the liquid and gaseous conditions of bodies, which have been defined in Article 4. The property common to the liquid and the gaseous conditions is that of not tending to preserve a definite shape, and the possession of this property by a body in perfection throughout all its parts, con- stitutes that body a perfect fluid. The parts of a body resisting alteration of shape must exert tangential stress; a perfect fluid does not resist alteration of shape ; therefore the parts of a perfect fluid cannot exert tangential stress ; therefore the stress exerted amongst and by them at every point and in 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 applied ELLIPSE OF STRESS. 101 hydrostatics, the assumption of perfect fluidity gives results near enough to the truth for practical purposes. ; 111. Opposite Principal Stresses. THEOREM. -'If a", pair of prin- cipal stresses be of equal intensities, but of, opposite kinds, the stress on any plane perpendicular to the plane yf, tfie 'directions bflt&e principal stresses is of tlie same intensity, and the angles which its direction makes with the normal to its plane are bisected by the axes of principal 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 O E is normal. In this case OD is to be taken as before, to represent^ OB, the total stress on the face (JB of the triangular prism A B ; but instead of taking OE in the direction from O towards B, to represent the total stress on OA, viz.,^> 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., 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 XOE which the normal OE makes on the other side of that axis; and X bisects the angle of obliquity E 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 stresses- of any intensities, and of the same or opposite kinds, being given, it is required to find the direction and intensity of the stress on a plane in any position at right angles to the plane parallel to which the two principal stresses act. Let X and Y (figs. 54 and 55), be the directions of the two principal stresses; OX being the direction of the greater stress. 102 PRINCIPLES OF STATICS. Let p m be the intensity of the greater stress ; . and p.. that- of the less* * - * ^ - % v2) i / * 9 - A, .< ^ /^-^ l 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 m and j0 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 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 + - > J Now let AB be the plane on which it is required to ascertain the direction and intensity of the stress, and ON a normal to that plane, making with the axis of greatest stress the angle ^ XON = xn. ELLIPSE OF STRESS. 103 On O N take M = ^* P "' } this -will represent a normal stress 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 ^ 2 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 = M~Q = MO. On the line thus found set off from M towards the axis of greatest stress, ME, = ^* 9 . This, ac- cording to Article 111, will represent the direction and the intensity of the oblique stress on AB, which is the effect of the pair of stresses P.P, 1 Join OK Then will that line represent the resultant of the forces represented by OM and ME,; that is to say, the direction and intensity of the entire stress on AB. Q. E. I. "" The algebraical expression of this solution is easily obtained by ^ means of the formulae of plane trigonometry, and consists of the two following equations: ' Intensity, B, or p r = J [pi' cos 2 xn + pi' sin 2 x n\ (1.) an equation which might have been obtained by making p^ = in equation 1 of Article 108, Problem II. A Obliquity, ^ N O B orn r. = arc sin (sin 2 x n ^ ) (2.) This obliquity is always towards the axis of greatest stress. k^J%- ^> p x &n & p y are represented as being of the same kind; and ME, is consequently less than OM, so that OR falls on the same side of OX with ON, that is to say, n r ^ x n. In fig. 55, p x and^>y are of opposite kinds, ME is greater than OM, and OE 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 * n > an ^ that of the point E, an ellipse whose semi-axes are p x and py, and which may be called the ELLIPSE OF STRESS, because its semidiameter in any direction represents the intensity of the stress in that direction. 104 PRINCIPLES 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 H upon O N, and are as follows: Direct, p n = p x cos 2 nee + p y * sin 2 xw; (3.) A A Shearing, p t = (p x - p y ) cos x n ' sin x n; ...(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 tJie normal stresses on a pair of planes at right angles to each other is equal to the sum of the principal stresses j 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 the intensity of that shear. It is evident that the shear is greatest when M R is perpendicular to OM; and then M JR, itself represents the intensity of the shear; that is to say, maximum p t = x ~ (5.) 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!. W/ien the principal stresses are of the same kind. CFig. 54.) In this case M R, ^ M O, and it is evident that the angle of obliquity, ^ M O II = nris greatest, when M R, is perpendicular to R, and that its value is given by the equation A ME, maximum n r arc sin _ : OM To find the position of the normal O N" to the plane A B, we have to consider that, A= \^L PMN; ELLIPSE OF STRESS PROBLEMS. 105 but^: PMN = ^L MRO + ^ MOR consequently in this case, = 90 + max. r (an obtuse angle). And for the position of the plane AB itself, we have ^ X A = 90= - xn = 9- ...... - -(&) 2 (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 p p = ^off ME or a wean proportional between the principal stresses. This is otherwise evident from the consideration, that when OR -L PRQ, then OR - J (PR * RQ), and that RQ = p x , PR = p y . CASE 2. When the principal stresses are of opposite kinds (fig. 55), it is evident, that the most oblique stress possible is a tangential stress, and that the problem amounts to finding the circumstances under which R lies in the plane AB. In this case it is evident, that the triangle M R becomes right-angled at 0, and conse- quently, that the intensity of the stress is given by the equation MK? - OW) = 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 N is found by considering, that A 1 and that ^ PMN = ^ MOR -f ^L MRO = 90 + arc -sin^ + Py '; P* ~ Pi 106 consequently, (an obtuse angle) ; PRINCIPLES OF STATICS. x n = - < 90 + arc sin XOA = 90 xn = ^ ( 90 arc sin 2 I (11.) P*~P (an acute angle). In these, as in the other formulas 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 a + p y means the difference, and p x p y the sum, of the arithmetical values of the principal stresses. PROBLEM IV. The intensities, "kinds, and obliquities, of any two stresses wJwse planes of action are perpendicular to the plane of their directions, being given, it is required tofindtlie principal stresses and axes of stress. CASE 1. When the given stresses are oftJie same kind, K and unequal. In fig. 56, let A B, A'B', represent the given planes, N, O N", their normals, O B, O B', the stresses upon them. Let the intensities be denoted algebraically p = B; p = B', and the obliquities by \m ifr: ^. N'OB'= fy. Fig. 56. In fig. 57, take O N to represent at once the normals to both planes. Make ^ N B = n r ; ^ N B' = n' r'; OB = p; OB' =p'. Join BB', bisect it in S, from which draw SM -L BB', cutting ONinM. Join MB, MB', 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 = &&; MS = MR' = ^=&; and consequently that the principal stresses are MR; ............ (12.) and it is also evident, that the angles made by the axis of greatest stress, with the two normals respectively, are A-^^NMB,; x A ri = i- ^N MR' j ......... (13.) which data are sufficient to determine the position of the axes. E. I. CASE 2. WJien tJie 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 0, 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 ri + x n = ^iNMS ) nn = \ A A V ......... (14.) or xri- aw^^RMS 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 formulse of trigonometry : - p2 " P/2 - A A .> 2 (p cos n r p cos w r J = MR = ME? 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 tJie two given stresses are conjugate, they are of equal obliquity; and the points 0, R', S, R, in fig. 57, are in one straight line, to which M S is perpendicular ; the angle between the two normals being ^NMS = A' = 90^/fr ................... .,(18.) In this case, equation 15 becomes 3 ~v~ ^ & ^ ' A >-< 2 cos n r equation 16 becomes _JU- . f^v J > //J / 1 -- .................... (20.) equations 17 are modified only by the equality of n' Y* to n r. CASE 4. TFAe7& the planes of action of the two given stresses are perpendicular to each otfier, M S is perpendicular and II R/ parallel to N, in fig. 57, so that we have, for the tangential component of each stress, A A MS =p sin nr=p' sin n' r 1 =p t . Let the normal components of the given stresses be denoted by A , ^ p n =p cos n r \ p n =p' cos n r f . Then equation 15 becomes 2 - 9 . ' () equation 16 becomes ~2 The equations 17 become cos 2 x n = - cos 2 x'ri = P~P - } or, what is equivalent, |> ..... .. .(23.) tan 2 x n = - tan 2 a' n' = 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 the greatest obliquity being given, it is required to find the ratio of two conjugate thrusts whose common obliquity is given. Let (p denote the given greatest obliquity. Then according to Problem III., Let n r, which must not exceed ^ 90 C nr y\ the angle between those planes themselves. Let p be the intensity; of the greater, and p that of the less, of those conjugate thrusts whose ratio is sought ; then dividing equation 20 of this Article by * equation 19, and squaring the result, we find A (24.) or transposing (p+p')- = cos 2 n r (25) 4pp " cos 2

> i2 A (3.) 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' -f sin a sin d = cos (a a'). Now the quantity under the sign J, in equation 2, consists of the following classes of terms : 1. All the squares p 2 cos 2 2 np ; 2. All the products 2 p p cos 2 n p cos 2 wp' j where p, p', are OMM/ ^air of the given stresses ; 3. All the squares p 2 sin 2 2 np ; 4. All the products 2 p p f sin 2 wp sin 2np'. The first and third of these classes being added together, make A A 2 (p s ); the second and fourth make 2 2 (p p' ' cos 2^>') ; pp' being the angle between p and /. Equation 2 thus becomes From the equations (1) and (4) it appears that the intensities of the principal stresses p x and ^> y can be computed without assuming planes of reduction ; for the only angles involved in this pair of equations are the several angles pp', which the given stresses make 112 PRINCIPLES OF STATICS. with each other when compared by pairs in every possible com- bination. To find the directions, however, of those principal stresses, planes of reduction must be assumed. In using the equation (4), it is to be remembered that when 2pp exceeds 90 P we have A / A = cos (lSO 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 W denote the weight per unit of volume of a body, or part of a body, and let it be required to determine what modes of variation of internal stress are consistent with that specific gravity. Consider the condition of a rectangular molecule A (fig. 58), bounded by ideal planes, whose edges are parallel to three rectangular axes, OX, OY, OZ. The position of this set of axes is immaterial to the result; but the algebraic formulae are simplified by assuming one axis to be vertical; let O Z, then, be vertical, and let distances along it be positive upwards. Then weight must be treated as a nega- tive force ; and the weight of a portion of the body of the volume V will be denoted by Fig. 58. Let the dimensions of the molecule A be AX parallel to OX, A* OZ. Then its weight is represented by w ' AX Ay AZ. The six faces will be designated as follows : The pair parallel to Y Z zox XOY1 (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 , p yy , p zz - Tangential, p yzj p zx , p xy . As for the signs of normal stress, let pull be positive and thrust I Farthest from 0. Nearest to 0. + A y A z Ay AZ + AZ AX - AZ AX + AX Ay} AX Ay (the upper.) j (the lower.) 114 PKINCIPLES OF STATICS. negative. As for the signs of tangential stress, let those stresses be considered as < * ^^7 e I which tend to make the pair of cor- ( negative J ners of the molecule which are nearest and farthest from O ( sharper ) \ natter J ' In the first place, let the rate of variation of the stress, of what kind soever, from point to point, be uniform; that is to say, for example, if the mean intensity of any one of the components of the stress at the face A x A y be p, then at the face + A x A y, whose distance from A x A y is A z, let the mean intensity of the same component be in which -~ is a constant co-efficient or factor, meaning " the rate cu z of variation of p along z," which is positive or negative, according as the variation of p is of the same or of the contrary kind to that of z. Rates of variation are also known by the name of differential co-efficients. As there are six components in the stress, and three axes of co-ordinates, there are eighteen possible differential co- efficients of the stress with respect to the co-ordinates j 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, tha 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 : OJ-A^AZ + -^-AyAZAX + -^ A Z ' A X A y = j X'AyAZ+^A x dy dz AX" AV A Z + -- A1J-AZAX + - AZ'AXAy dy dz = W' A X Ay AZ. INTERNAL EQUILIBRIUM. 116 Eacii 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 A y A z ; dividing by this, they are reduced to the following : , d Pxy , dp zx dx ~d dz . dx dy dz d Pzx , d dp yz .(2.) , yz , zz dx dy dz In this second form, the equations are applicable to rates of varia- tion which are not uniform as well as to those which are uniform. For as the' rectangular molecule, from the conditions of whose equilibrium these equations are deduced, is of arbitrary size, it may be supposed as small as we please j 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 j so that for 0, 0, w, in the first, second, and third equations respectively, are to be substituted w cos a, w 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. 117. Equilibrium of Fluids. It has already been explained in Article 110, that in a fluid the only stress to be considered in practice is a thrust or pressure, normal and of equal intensity in all directions. This is expressed symbolically in the following manner : p yt = ', p gje = ; p xy = Ptx = Pyy Pzz P ') 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;- 1**'$* = ; dy dz (2.) The first of these equations expresses the fact, that in a balanced fluid, the pressure increases with the vertical depth, at a rate expressed by the weight of the fluid per unit of volume; and the second and third express the fact, that in a balanced fluid, the pressure has no variation in any horizontal direction ; in other words, that the pressure is equal at all points in the same level surface. [The exact figure of a level surface is spheroidal ; but for pur- poses of applied mechanics it may be treated as a plane, without sensible error.] EQUILIBRIUM OF FLUIDS. 117 Those principles may also be proved directly. Let fig. 59 re- present a vertical section of a fluid; Y O Y any horizontal plane, X a -~ ' vertical axis. Let BB be a hori- zontal plane at the depth x below 0; * C C another horizontal plane at the Z_ O- depth x + AX. Let A be a small rectangular molecule contained be- tween those two horizontal planes; Fi S- 59. and let A y and A z be its horizontal dimensions, so that its weight is W A X A y A Z. The pressure exerted by the other portions of the fluid against the vertical faces of this molecule are horizontal, and must balance each other; therefore there can be no variation of pressure horizontally. Let PQ, then, be the uniform pressure at the horizontal plane YO Y, p, that at the plane B B, and p + --- A x that at the plane C 0, -= (L x , d 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 ~- ' A x A y A s, dx has to be balanced by the weight of the molecule; equating it fo which, and dividing by the common factor A x A y A z t we obtain the first of the equations 2 of this Article. The pressure p Q at the surface Y Y being given, the pressure p at any given depth x below Y Y is found by means of the integral, dp (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 unify, 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 25/000 for each pound on the square inch. If, then, the specific gravity w be treated as a constant in equation 3 of Article 117, it becomes as follows: p = p Q + wx-j (1.) that is to say : let p be the pressure at the upper surface, Y O Y, (fig. 59) of a mass of liquid; then the pressure p at any given depth x below that surface is greater than the superficial pressure p 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 insh. 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 TJ (and called an "inverted siphon"), or if one of the two fluids be contained in a vertical tube open below, and the other in the space surrounding that tube ; or, generally, if the two fluids be partially separated from each other by a vertical or nearly vertical partition, below which there is a com- munication between the spaces on either side of it; the horizontal surface of contact of the fluids will be at that side of the partition at which the lighter fluid is found, so that it may be above, and the heavier fluid below, that surface of contact. Let PQ denote the common pressure of the two fluids at their sur- face of contact, and let any ordinate measured from that surface upwards, be denoted by x. Let w' denote the specific gravity, and p' the pressure, of the lighter fluid; w" the specific gravity, and p" the pressure, of the heavier fluid. Then at any given elevation a? above the surface of contact which equations, when the fluids are liquids, and u/ t w", constants, become p' = PQ - wx- } p" - p Q - 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 j consequently (3.) Let a/, 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; ihen p" = $/, and consequently, for fluids in general, 120 PRINCIPLES OF STATICS. 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' dx w" x"', (6.) and on this last formula is founded the method of determining differences of level by barometric observations of the atmospheric pressure. / 120. Equilibrium of a Floating Body. THEOREM. A solid body floating on tlie 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- Fig. 60. &p = 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 Iwrizontal projection of the area S, and its value is S cos . Conceive a horizontal prism ST to have its axis in the vertical plane which is perpendicular to S, and to have the area S for an FLOATING BODY. 121 oblique section ; the vertical transverse section of this prism is what is called the vertical projection of the area S, and its value is S sin . 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 = 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 $p 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 ex, 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 HS; 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"V; 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 Body. THEOREM. If a Solid body l>e 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, whether liquid or gaseous. Conceive a small vertical prism S U 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 TJV 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 V; so that the horizontal component of the resultant of the pressure of the fluid on the entire body is nothing, and that resultant is vertical. The vertical component of the pressure on S is upward, and equal to the weight of the prismatic portion of the fluid which would stand vertically above S if a part of it were not displaced by the solid body. The vertical component of the pressure on U is downward, and equal to the weight of the prismatic portion of the fluid which stands vertically above U. The vertical force arising from the pressures on S and on U together is upward, and equal to the difference between those two weights; that is, it is equal and directly opposed to the weight of the portion of the fluid dis- placed by the prismatic portion S U of the immersed body. Hence the resultant of the pressure of the fluid over the entire surface of the immersed body is equal and directly opposed to the weight of the portion of fluid displaced by that body. Q. E. D. The centre of gravity C, of the portion of fluid which would occupy the position of the body if it were not immersed, is called, as before, the centre of buoyancy, and is traversed by the vertical line of action of the resultant of the pressure of the fluid, which is APPARENT WEIGHTS IN AIR. 123 itself called the buoyancy of the immersed body, and sometimes the apparent loss of weight. To maintain an immersed body in equilibrio, there must "be applied to it a force or couple, as the case may be, equal and directly op- posed to the resultant, if any, of its downward weight and upward buoyancy; that resultant being determined according to the principles of Articles 39 and 40. "When a body floats in a heavier fluid (as water) having its upper portion surrounded by a lighter fluid (as air), its total buoyancy is equal and opposite to the resultant of the weights of the two portions of the respective fluids which it displaces. In practical questions relative to the equilibrium of ships, the buoyancy arising from the displacement of air is too small as com- pared with that arising from the displacement of water, to require to be taken into account in calculation. 122. Apparent Weights. The only method of testing the equality of the weights of two bodies which is sufficiently delicate for exact scientific purposes, is that of hanging them from the opposite ends of a lever with equal arms. If this process were performed in a vacuum, the balancing of the bodies would prove their weights to be equal ; but as it must be performed in air, the balancing only proves the equality of the apparent weights of the bodies in air, that is, of the respective ex- cesses of their weights above the weights of the volumes of air which they displace. The real weights of the bodies, therefore, are not equal unless their volumes are equal also. If their volumes are unequal, the real weight of the larger body must be the greater by an amount equal to the weight of the difference between the volumes of air which they displace. The weight of a cubic foot of pure dry air, under the pressure of one atmosphere (14 -7 Ibs. on the square inch), and at the temperature of melting ice (32 Fahrenheit) is 0' 080728 pound avoirdupois. Let this be denoted by W Q . 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 , . and if w, W, be the weights of a given volume of air, under the respective pressures >,>', and at the temperatures t, t', of Fahrenheit's scale, then w' _!/ t + 461-2 (2 . w "'tf-f 461-2'" '"* '' 124: PRINCIPLES OP STATICS. Let Wi denote the true weight of a body, Vj 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 """ ' A/) * ' 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 a the weight per unit of volume, of the second body, we have !J - w SL^W, (4) Wi W z so that the proportion between the real weights of the bodies is Wa = WjW.2-W.2W 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 Wj, as in equation 3 of Article 122, denote the true weight of the solid body, w^ 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 and consequently g^Wi-w (1) Let the first weighing take place in air and the second in the liquid, and let W be the apparent weight in air ; then Wi Wi and consequently so that if is known, may be found by the equation IMMERSED PLANE. 125 W a (3.) w, - W~ W" When the object of weighing of this kind is to determine the specific gravities of solids, the liquid usually employed is pure water; and the results obtained are the ratios of the specific gravities of solid bodies to that of pure water. If these ratios, or relative spe- cific gravities, be multiplied by the weight of a cubic foot of pure water, the weight of a cubic foot of the solid is obtained. The weight of a cubic foot of pure water at the temperature of its maximum density (being, according to Playfair and Joule, 39'l Fahrenheit) is, according to the best existing data, 62 '425 pounds avoirdupois. For any other temperature t on Fahrenheit's scale, the weight of a cubic foot of pure water is 62425 /A \ where v denotes the volume to which a mass of water measuring one cubic foot at 39-l expands at t Q ; 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. = 10-1 (t 39-1) 2 0-0369 ( 39 -I) 3 -10,000,000. (5.) The relative specific gravities of two liquids are determined by weighing the same solid body immersed in them successively and comparing its apparent losses of weight. 124. Pressure on an Immersed Plane. If a horizontal plane SUr- face of any figure be immersed in a fluid, the pressure on that sur- face is vertical, and uniformly distributed; its amount is the D 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 Fig. 62. usually termed, the centre of gravity of the plane surface. If an inclined or vertical plane surface be immersed in a liquid, let OY (fig. 62), represent a section of the horizontal plane at which the pressure is nothing, and B F a vertical section of the 126 PRINCIPLES OF STATICS, immersed plane. Let x v = BE be the depth to which the lower edge of this plane is immersed below OY. From B draw BD = BE, and -L BF; produce the plane BF till it cuts the horizontal plane of no pressure, O Y, in the line represented in section by O ; through and D draw a plane O H D, and conceive the prism B D H F to stand normally upon the base B F and to be bounded above by the plane D H. The pressure on the plane BF will be normal; its amount will be equal to the weight of fluid contained in the volume B D H F ; that is to say, let x denote the depth of the centre of gravity of the plane BF below O Y, and w the weight of unity of the volume of liquid; then the mean intensity of the pressure on B F is Po = wxo, (1.) and the amount of the pressure P = wo; -areaBF (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 CP let fall on it from C. As the intensity of the pressure on any point of BF is propor- tional to its depth below OY, and consequently to its distance from O, this is a case of uniformly varying stress, and the formulae of Article 94 are applicable to it. In the application of those formulae it is to be observed, that the ordinates y are to be measured hori- zontally in the plane BF, 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 = wsma. (3.) where a is the angle of inclination of the plane B F to a horizontal plane. 120. Pressure in an Indefinite Uniformly Sloping Solid. Conceive a mass of homogeneous solid mate- rial to be indefinitely extended laterally and downwards, and to be bounded above by a plane sur- face, making a given angle of de- clivity 6 with a horizontal plane. In fig. 63, let Y O Y represent a ver- tical section of that upper sloping surface along its direction of greatest declivity, and 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 I. In an indefinite homogeneous solid bounded above by a sloping plane, the pressure on any plane parallel to that sloping surface is vertical, and of an uniform, intensity equal to the weight of the vertical prism which stands on unity of area of the given plane. The area of the horizontal section of that prism is cos 6, conse- quently, the intensity of the vertical pressure on the plane B B at the depth x is p, = wxcosQ (1.) From the above theorem, combined with the principle of conjugate stresses of Article 101, there follows THEOREM II. The stress, if any, on any vertical plane 'is parallel to the sloping surface, and conjugate to the stress on a plane parallel to that surface. Consider now the condition of a prismatic molecule A, bounded above and below by planes B B, C C, parallel to the sloping surface Y Y, and laterally by two pairs of parallel vertical planes. Let the common area of the upper and lower surfaces of this prism be unity, and its height A x ; then its volume is A x cos d, and its weight w A x ' cos 6, which is equal and opposite to, and balanced by the excess of the vertical pressure on its lower face above the vertical pressure on its upper face. Therefore, the pressures paral- lel to the sloping surface, on the vertical faces of the prism, must balance each other independently ; therefore they must be of equal mean intensity throughout the whole extent of the layer between the planes B B, CO; whence follows THEOREM III. The state of stress, at a given uniform depth below the sloping surface, is uniform. 126. On the Parallel Projection of Stress and Wei -lit. 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 will appear in the Second Part. 12* CHAPTER VI ON STABLE AND UNSTABLE EQUILIBRIUM. 127. Stable and Unstable Equilibrium of a Free Bodf. Siq> 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 may be by the deviation, in amount, in position, or in both. If this resultant acts towards the same direc- tion with the deviation, the equilibrium is unstable if towards the opposition direction, stable and if the resultant is still nothing, the equilibrium is indifferent. The disturbance of a free body from a position of stable equi- librium causes it to oscillate about that position. 128. stability of a Fixed Body. The term "stability," as ap- plied to the condition of a body forming part of a structure, has, in most cases, a meaning different from that explained in the last Article, viz., the property of remaining in equilibria, without sen- sible deviation of position, notwithstanding certain deviations of the load, or externally applied force, from its mean amount or posi- tion. Stability, in this sense, forms one of the principal subjects of the second part of this treatise. PART II. THEORY OF STRUCTURES. CHAPTER I. i% DEFINITIONS AND GENEKAL PRINCIPLES. 129. Structures Pieces Joints. Structures have already, in Article 15, been distinguished from machines. A structure con- sists of two or more solid bodies, called its pieces, which touch each other, and are connected at portions of their surfaces called joints. 130. Supports Foundations. Although the pieces of a structure are fixed relatively to each other, the structure as a whole may be either fixed or moveable relatively to the earth. A fixed structure is supported on a part of the solid material of the earth, called the foundation of the structure j 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 ; 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 shall 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 THEORY OF STRUCTURES. II. That the forces exerted on each piece of the 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 the pieces in contact with it. III. That the forces exerted on each of the parts into which the pieces of the structure can be conceived to be divided 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; the forces which act on the part so considered are (1.) its weight, and (2.) (if it is at the external surface of the piece) the external stress applied to it, if any, making together its gross load; (3.) the stress exerted at the ideal surface of division, between the part in ques- tion and the other parts of the piece. 132. stability, strength, and stiffness. It is necessary to the per- manence of a structure, that the three foregoing conditions of equilibrium should be fulfilled, not only under one amount and one mode of distribution of load, but under all the variations of the load as to amount and mode of distribution which can occur in the use of the structure. Stability consists in the fulfilment of the first and second condi- tions of equilibrium of a structure under all variations of load within given limits. A structure which is deficient in stability gives way by the displacement of its pieces from their proper posi- tions. Strength consists in the fulfilment of the third condition of equi- librium of a structure for all loads not exceeding prescribed limits ; that is to say, the greatest internal stress produced in any part of any piece of the structure, by the prescribed greatest load, must be such as the material can bear, not merely without immediate break- ing, but without such injury to its texture as might endanger its breaking in the course of time. A piece of a structure may be rendered unfit for its purpose not merely by being broken, but by being stretched, compressed, bent, twisted, or otherwise strained out of its proper shape. It is neces- sary, therefore, that each piece of a structure should be of such dimensions that its alteration of figure under the greatest load applied to it shall not exceed given limits. This property is called stiffness, and is so connected with strength that it is necessary to consider them together. From the foregoing considerations, it is evident that the theory of structures may be divided into two divisions, relating, the first to STABILITY, or the property of resisting displacement of the pieces, and the second to STRENGTH and STIFFNESS, or the power of each piece to resist fracture and disfigurement. 131 CHAPTER H, STABILITY. 133. Resultant Gross Load. The mode of distribution of the intensity of the load upon a given piece of r a~structure affects the strength and s^jfoss only. So far as stability alone is concerned, it is sufficient to know the magnitude and position of the resultant of that load, which is to be found by means of the principles ex- plained in the First Part of this work, and may then be treated as a single force. 134. Centre of Resistance of a Joint. In like manner, when stability only is in question, it is sufficient to consider the position and magnitude of the resultant of the resistance or stress exerted between two" pieces of a structure at the joint where they meet, and to treat that resultant as a single force. The point where its line of action traverses the joint is called the centre of^resistance of that joint. 135. A 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. I. Framework joints are such as occur in carpentry, in frames of metal bars, and in structures of ropes and chains, fixing the ends of two or more pieces together, but offering little or no resistance to change in the relative angular positions of those pieces. In a joint of this class, the centre of resistance is at the middle of the joint, and does not admit of any variation of position consistently with security, II. Blockwork joints are such as occur in masonry and brickwork, being plane or curved surfaces of contact, of considerable extent as compared with the dimensions of the pieces which they connect, capable of resisting a thrust more or less oblique, according to laws to be afterwards explained, but not of resisting a pull of suf- 132 THEORY OF STRUCTURES. ficient intensity to be taken into account in practice. In such joints the position of the centre of resistance may be varied within certain limits. III. Fastened joints, at which, by means of some strong cement, or of bolts, rivets, or other fastenings, two pieces are so connected that the joint fixes their relative angular position, and is capable of resisting a pull as well as a thrust. In this case, the centre of resistance may be at any distance from the centre of the joint ; and there may even be no centre of resistance, when the resultant of the stress at the joint is a couple, as explained in Articles 91, 92, and 93. It is obvious that the effect of a joint thus cemented or fastened is to make the two pieces which it connects act as one piece, and that the resistance which it is capable of exerting is a question not of stability but of strength. SECTION 1. Equilibrium and Stability of Frames. 137. Frame is here used to denote a structure composed of bars, rods, links, or cords, attached together or supported by joints of the first class described in the last Article, the centre of resistance being at the middle of each joint, and the line of resistance, con- sequently, a polygon whose angles are at the centres of the joints. The condition of a single bar will be considered first, then that of a combination of two bars, then of three bars, and then of any number. 138. Tie. Let fig. 64 represent a single bar of a frame, L the centre of resistance where the load is ap- plied, and S the centre of resistance where the support- ing force is applied ; so that the straight line L S is the "line of resistance." The bar is represented as being straight itself, that being the figure which connects the points L and S, and gives adequate stiffness and strength, with the least ex- v G4 P en diture of material. But the bar may, consistently with the principles of this Article, be of any other figure connecting those two points, provided it is sufficiently strong and stiff to prevent their distance from altering to an extent inconsistent with the purposes of the structure. The condition of the bar is the same with that of the solid in Article 23; and it is obvious that the load P, and the supporting resistance R, must be equal and directly opposed, and must both act along the line of resistance L S. In the present case those forces are supposed to be directed out- ward, or from each other. The bar between L and S is in a state of tension, and the stress exerted between any two divisions of it is a pull, equal and opposite to the loading and supporting forces. A BEAM UNDER PARALLEL FORCES. 133 bar in this condition is called a tie. It is obvious that a rope or chain will answer the purpose of a tie. 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. I 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 Yig.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 tJiey 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 xg Ri direction of the load ; and let the load act ^ * between the points of support, as in fig. 66 ; J^*|k T w"/~ where P represents the resultant of the gross '''-Q. E. D. By the aid of this Theorem may be determined the proportion in which the load of a given body is distributed amongst three props, exerting parallel supporting forces. 144. Load supported by Three Inclined Forces. The Case of a load supported by three inclined forces is that considered in Articles 54: and 56. The lines of action of the three supporting forces must intersect that of the load in one point ; and the magnitudes of the three supporting forces are represented by the three edges of a parallelepiped, whose diagonal represents the load. 136 THEORY OF STRUCTURES. 145. Frame of Two Bars Equilibrium. PROBLEM. Figures 70, 71, and 72 represent three cases in which a frame consisting of two p Fig. 70. Fig. 71. Fig. 72. bars, jointed to each other at the point L, is loaded at that point with a given force, P, and is supported by the connection of the bars at their farther extremities, S b S 2 , with fixed bodies. It is required to find the stress on each bar, and the supporting forces at S A and S 2 . Resolve the load P (as in Article 55) into two components, R b 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 2 , be the respective angles made by the lines of resistance of the bars with the line of action of the load ; then P : Rj : R 2 : : sin (^ + i 2 ) : sin % : sin ij. 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 Sj, and a tie, L S 2 (such as the gib and the tie-rod of a crane). 146. Frame of Two Bars Stability. A frame of two bars is stable as regards deviations in the plane of its lines of resistance. With respect to lateral deviations of angular position, in a direction perpendicular to that plane, a frame of two ties is stable ; so also is a frame consisting of a strut and a tie, when the direction of the load inclines from the line Sj S 2 , joining the points of support. A frame consisting of a strut and a tie, when the direction of the load inclines towards the line Sj S 2 , and a frame of two struts in all cases, are unstable laterally, unless provided with lateral stays. These principles are true of any pair of adjacent bars whose farther centres 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 I, 2, 3, viz. : C and A at 1, A and B at 2, B and C at 3. Let a load P! be applied at the joint 1 in any given direction ; let supporting forces, P 2 , P 3 , be applied at the joints ^ 73 2, 3 ; the lines of action of those two forces must be in the same plane with that of P 1? and musb 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 , BC ... 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 OP 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 CO parallel to the bar C, and AO parallel to the bar A, meeting in the point O, and those two lines will represent the stresses on the bars C and A respectively. In the same manner it is proved, that B O represents the stress on the bar B. The three lines C O, A O, BO, meet in one point O, because the components along the line of direction of a given bar, of the external forces applied at its two extremities, are equal and directly opposed. Hence follows the following THEOREM. If three forces be represented by 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 equilibrio under the three given forces, each force being applied to the joint wJiere the two lines of resistance meet, which are parallel to the radiating lines contiguous to that side of the original triangle which represents the force in question. Also, tJie lengths of the three radiating lines will represent the stresses on the bars to which they are respectively parallel. 149. Triangular Frame under Parallel Forces. When the three external forces are parallel to each other, the triangle of forces A B C of fig. 73* becomes a straight line C A, as in fig. 74*, divided into two segments by the point B. Let straight lines radiate from to A, B, C; and let fig. 74 represent a triangular frame whose sides 1 2 or A, 2 3 or B, 3 1 or C, are respectively parallel to O A, OB, C; then if thejload 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 TT M~y P.lort Ma.YWpl1 in 144 THEORY OF STRUCTURES. with the equilibrium of each bar, then, in the diagram of forces, when converging lines respectively parallel to the lines of resistance are drawn from the angles of the polygon of external forces, those converging lines, instead of meeting in one point, will be found to have gaps between them. The lines necessary to fill up those gaps will indicate the forces to be supplied by means of the resistance of braces. 156. Rigidity of a Truss. The word trus is applied in carpentry and iron framing to a triangular frame, and to a polygonal frame to which rigidity is given by staying and bracing, so that its figure shall be incapable of alteration by turning of the bars about their joints. If each joint were absolutely of the kind described as the first class in Article 136, that is, like a hinge, incapable of offering any resistance to alteration of the relative angular position of the bars connected by it, it would be necessary, in order to fulfil the condition of rigidity, that every polygonal frame should be divided by the lines of resistance of stays and braces into triangles and other polygons so arranged, that every polygon of four or more sides should be surrounded by triangles on all but two sides and the included angle at farthest. For every unstayed polygon of four sides or more, with flexible joints, is flexible, unless all the angles except one be fixed by being connected with triangles. Sometimes, however, a certain amount of stiffness in the joints of a frame, and sometimes the resistance of its bars to bending, is relied upon to give rigidity to the frame, when the load upon it is sub- ject to small variations only in its mode of distribution. For example, in the truss of fig. 81 (for which see Article 161, farther on), the tie-beam A A is made in one piece, or in two or more pieces, so connected together as to act like one piece ; and part of its weight is suspended from the joints C, C, by the rods C B, C B. These rods also serve to make the resistance of the tie-beam C C to being bent, act so as to prevent the struts AC, C C, C A, from deviating from their proper angular positions, by turning on the joints A, C, C, A. If A B, B B, and B A, were three distinct pieces, with flexible joints at B, B, it is evident that the frame might be disfigured by distortion of the quadrangle B C C B. 157- Variations of J,ou G and 4 5 and 7 }isT 8 -67, and so on for as many pairs of divisions as the platform consists of. Example II. Fig. 80 represents the framework for supporting Fig. 80. one side of a timber bridge, resting on two piers at 1 and 4. consists of four distinct trusses, viz., It 150 THEORY OF STRUCTURES. 1234 loaded at 2 and 3, 15G4 5 6, 1784 7 8, 194 9 ; but all those trusses have the same tie-beam, 1 4 ; and the pull along that tie-beam is the sum of the pulls due to the four trusses. 161. Resistance of Frame at a Section. THEOREM. If a frame acted upon by any system of external forces, and if that frame be conceived to be completely divided into two parts by an ideal surface, the stresses along the bars which are intersected by that surface, balance the external forces which act on each of the two parts of the frame. This theorem, which requires no demonstration, furnishes in some cases the most convenient method of determining the stresses along the pieces of a frame. The following consideration shows to what extent its use is limited. CASE 1. When the lines of resistance of the bars, and the lines of action of the external forces, are all in one plane, let the frame be supposed to be intersected anywhere by a plane at right angles to its own plane. Take the line of intersection of these two planes for an axis of co-ordinates ; say for the axis of y, and any convenient point in it for the origin O ; let the axis of x be perpendicular to this, and in the plane of the frame, and the axis of z perpendicular to both, and in the plane of section. The external forces applied to the part of the frame at one side of the plane of section (either may be chosen) being treated as in Article 59, give three data, viz., the total force along x = T? x ; the total force along y = F y , and the moment of the couple acting round z = M ; and the bars which are cut by the plane of section must exert resistances capable of balancing those two forces and that couple. If not more than three bars are cut by the plane of section, there are not more than three unknown quantities, and three relations between them and given quantities, so that the problem is determinate ; if more than three bars are Cut by the plane of section, the problem is or may be indeterminate. The formulae to which this reasoning leads are as follows : Let x be positive in a direction from the plane of section towards the part of the structure which is considered in determining F.,, F y , and M; let + y lie to the right of + x when looking from z j let angles measured from x towards -f- y, that is, towards the right, be positive ; and let the lines of resistance of the three bars cut by the plane of section make the angles i l} i?, is, with x. Let n l} n 2 , n s , be the perpendicular distances of those three lines of resistance from O, distances towards the . being considered METHOD OF SECTIONS. 151 Let EJ, E 2 , RS> t> e the resistances, or total stresses, along the three bars, pulls being positive, and thrusts negative. Then we have the following three equations : F x = E! cos % + E 2 cos ia + E 3 cos t' 3 ; F y E! sin ii + E 2 sin i 2 + E 3 sin i s ; ., (1.) - M E! % + E 2 n 2 + E 3 w 3 ; from which the three quantities sought, E b E 2 , E 3 , can be found. Speaking with reference to the given plane of section, F.,. may be called the normal stress, F y the shearing stress, and M the moment of flexure or lending stress; for it tends to bend the frame at the section under consideration. CASE 2. When the bars of the frame, and the forces applied to them, act in any direction, the forces applied to one of the two divisions of ne frame are to be reduced to rectangular components; a.nd the three resultant forces along these rectangular axes, F,, F y , F,, and the three resultant couples round these three axes, M,, M y , M,, are to be found as in Article 60. Those forces and couples must be equal and opposite to the corresponding forces and couples arising from the stresses along the bars cut by the section; and thus are obtained six equations between those stresses and known quantities ; so that if the section cuts not more than six bars, the problem is determinate ; if more, it is or may be indeterminate. The equations are obtained as follows : Let E denote the stress along any one of the bars, pull being positive and thrust negative. Let a, '/3, y, be the inclinations of the line of resistance of that bar to the axes of x, y, z. Let n be its perpendicular distance from O. Conceive a plane to pass through O and through the line of resistance of the bar, and a normal to be drawn to that plane in such a direc- tion, that looking from the end of that normal towards O, the bar is seen to lie to the right of 0, and let A, ^, , be the angles of inclination of that normal to the three axes. Let s denote the summation of six corresponding quantities for the six bars. Then the six equations are, F x = 2 E cos , Fy = s E cos /3; F r = 2 E cos y ; M, = 2 E n cos A ; M y = 2 E n cos ^ ; j- (2). M, = 2 E n cos " ; from which the six stresses sought can be computed by elimination. The plane of T/ z being as before, that of the section, F a is the total direct stress on it; F y and F, are the total shearing stresses ; M y and M z are lending couples, and M^. a twisting couple. EEMARKS. Every problem respecting the equilibrium of frames which can be solved by the method of sections explained in this 152 THEORY OF STRUCTURES. Article, can also be solved by the method of polygons explained in the previous Articles; and the choice between the two methods is a question of convenience and simplicity in each particular case. The following is one of the simplest examples of the solution of a problem in both ways. Fig. 81 represents a truss of a form very c c common in carpentry (already referred to in Article 156), and consisting of three struts, AC, CO, A, a tie- beam A A, and two suspension-rods, C B, C B, which serve to suspend part of the weight of the tie-beam from Flg " 81 * the joints G C, and also to stiffen the truss in the manner mentioned in Article 156. Let i denote the equal and opposite inclinations of the rafters AC, C A, to the horizontal tie-beam A A ; and leaving out of consideration the portions of the load directly supported at A A, let P, P, denote equal vertical loads applied at C C, and P, P, equal upward vertical supporting forces applied at A A, by the resistance of the props. Let H denote the pull on the tie- beam, R, the thrust on each of the sloping rafters, and T the thrust on the horizontal strut C C. Proceeding by the method of polygons, as in Article 153, we find at once, x H = T = P cotan i; } \ ................ (3.) K- = P cosec i. (Thrusts being considered as negative.) To solve the same question by the method of sections, suppose a vertical section to be made by a plane traversing the centre of the right hand joint C ; take that centre for the origin of co-ordinates; let x be positive towards the right, and y positive downwards ; let as,, y lt be the co-ordinates of the centre of resistance at the right hand point of support A. When the plane of section traverses the centre of resistance of a joint, we are at liberty to suppose either of the two bars which meet at that joint on opposite sides of the plane of section to be cut by it at an insensible distance from the joint. First, consider the plane of section as cutting C A. The forces and couple acting on the part of the frame to the right of the section are Then, observing that for the strut A C, n = 0, and that for the tie A A, n y l} we have, by the equations 1 of this Article HALF- LATTICE GIRDER, 153 B, cos i + H = F, = ; K sin i = P ; HTA^ M= + whence we obtain, from the last equation, Pa- H = 2/i from the first, or from the second (4.) -r -H- T j> _. . _ j-> 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 C, and P at A, have to be taken into account ; so that F. = 0; F y = 0; M = -P* i; 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," is represented in fig. 82. It consists essentially of a horizontal upper bar, a horizontal lower bar, and a series of diagonal bars sloping alternately in opposite direc- tions, and dividing the space between the upper and lower |fl\/\/\/!\/\/ \/\/\/ m bars into a series of triangles. In the example to be consi- -P. 00 i t -i -i -i -^ '{? &" 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 THEOEY 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 ; 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 0, 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 1 be the distance of the point of support A to the left of the same plane. Let y be positive upwards ; so that for the joints of the upper bar, y = 0, and for those of the lower bar, y = h, h denoting the vertical depth between the lines of resistance of the upper and lower bars. Let i be the inclination of the diagonal C E to the horizontal axis of x. In the present instance this is positive ; but had C E sloped the other way, it would have been negative. Let the symbol s P denote the sum of the loads acting at the joints between the plane of section and the point of support A, the load at the joint C being included. Then for the total forces and couple acting on the division of the girder to the left of the plane of section, we have, direct force, F.,. = 0, because the applied forces are all vertical j shearing force, F y = P A s P a force lies < ^ *!rt? 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 x t s -Pa;; which is upward, and right-handed with respect to the axis of z. Now let R! denote the stress along the upper bar at C, R 2 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 : H! + R 2 + E 3 cos t = ; or K! + R 3 cos i = "R...(a.) that is, the stress along the upper bar, and the horizontal component HALF-LATTICE GIRDER. 155 of the stress along the diagonal, are equal and opposite to the stress along the lower bar ; K 3 sin i = F y = P A ^ P j ............ ,.. .(&.) that is, the vertical component of the stress along the diagonal, balances the shearing force j R 2 y = E 2 h = M = P' A x l ^ - P# ; ......... (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), (6), (c), are deduced the following values of the stresses : Pull on lower bar, -i K 2 = -(P A o ;j z Stress on diagonal, KK, = cosec i (P A 2 P); , Thrust on upper bar, R! = R 2 RS cos * = |(P A aj 1 sj P aj) cotan i (P A _2j -P). Another, and sometimes a more convenient form, can be found for the second and third of those expressions. Let s denote the length of the diagonal C E, and x{ the horizontal distance of its lower end E from the point of support A; then rr - (*) 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 A X ; therefore, the tangent B P bisects A X : this is a property characteristic of a PARABOLA whose vertex is at A , therefore, the curve assumed by the cord is such a parabola. Also, the proportions of the load, and the horizontal and oblique tensions are as follows : P : H : R : : BIT: XP : PB : : v : ? : Second Solution. In the present case equation 2 of Article 168 becomes ^ = ^ (a) 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 ,_ m = = 5. 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 ^ = - = -5 = j dx 2m x' (6.) whence we have the proportions P : H : K : : tan i : 1 : sec i : : ^ : 1 : ^ (l + if 2 fly \ 32 +' .............. < 7 "> as before. The following are the solutions of some useful problems respecting uniformly loaded cords. PROBLEM I. Given the elevations, y b y 2 , of the two points of support of the cord above its lowest point, and also the horizontal distance, or span a, between those points of support; it is required to find the horizontal distances, x a , x 2 , of the lowest point from the two points of support; also the modulus m. In a parabola, '- :o:a therefore, ; ...... (8.) *yi-r4ya-f-8,/ft When the points of support are at the same level, a a? PROBLEM II. Given the same data, to find the inclinations i v i 2 , of the cord at the points of support. By equations 6, we have, x 2 when 2/i = y a , tant, = tan i s = -? >(12.) a PARABOLIC COED. 167 PROBLEM III. Given the same data, and the load per unit of length ; required the horizontal tension H, and the tensions B^, B^ at the points of support. By equation 5, we find, H = 2p m = , ) PO \_ . ; .......... (13.) 2^/1 -f * and by the proportional equation 7, ............... (H.) When yj = y^ those equations become H = f^- 8 j E, = B 2 = H sec i, = H V f 1 + r PROBLEM IY. 6rwm 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. = w-jtan i sec i + hyp. log. (tan i + sec *)}... (16.) The length of the cord is s { + s 2 , where 81 is found by putting Xi and 2/i in the first of the above formula, or % in the second, and s 2 by putting x 2 and y z in the first formula, or i 2 in the second. The following approximate formula for the length of a parabolic arc is in many cases sufficiently near the truth for practical purposes ; if" nea/rly ' which gives for the total length of the cord 168 THEORY OF STRUCTURES. s, + s. 2 = a + + nearly. .......... (18.) O \Cj iCg ' and when y l = y 2 , this becomes 2s, = a + |- ' - nearly, ................ (19.) PROBLEM V. Given the same data, to find, approximately, the small elongation of the coi*d d (sj + s 2 ) required to produce a given small depression d y of the lowest point A, and conversely. Differentiating equation 18, we find <* (, v^ = 4 (- +-)<*? -( 2 -) O \Xi Xn / which serves to compute the elongation from the depression ; and conversely, dy = ' ( Sl ' s -' '; (21.) 4 V, . i/a ' which serves to compute the depression of the lowest point from the elongation of the cord. When y^ = y 2 , those formulae become, .(22.) The preceding formulae serve to compute the depression which the middle point of a suspension bridge undergoes in consequence of a given elongation of the cable or chain, whether caused by heat or by tension. 170. Suspension Bridge with Vertical Rods. In a Suspension bridge the load is not continuous, the platform being hung by rods from a certain number of points in each cable or chain : neither is it uniformly distributed ; for although the weight of the platform per unit of length is uniform or sensibly so, the load arising from the weight of the cables or chains and of the suspending rods is more intense near the piers. Nevertheless, in most cases which occur in practice, the condition of each cable or chain approaches sufficiently near to that of a cord continuously and uniformly loaded to enable the formulae of Article 169 to be applied without material error. SUSPENSION BRIDGE FLEXIBLE TIE. 169 When the piers of a suspension bridge are slender and vertical (as is usually the case), the resultant pressure of the chain or cable on the top of the pier ought to be vertical also. Thus, let 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 masomy, 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 ; B, the common value of the two tensions ; then the vertical pressure on the pier is Y = 2Rsin*=: 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 R, R', their tensions, then V = R sin i + E' sin i' = H (tan i + tan i'} ...... (2.) 171. Flexible Tie. Let a vertical load, P, be applied at A, fig. 86, t 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 j therefore its figure will be nearly a parabola . the tangent at D will be sensibly parallel to A C, and the tangents at A and C will meet in a point which will be near the vertical line E D F, which line bisects A C, and is bisected in D. Hence we have the following construction : Draw the diagram of forces, fig. 86% in the following manner. ~\y On the vertical line of loads b c, take &y =: P ; be = P -\- ; be Z = P + W. From b draw b parallel to the strut A B ; that is, horizontal j from e draw e parallel to C A, cutting b in ; join c 0, /O. In fig. 86, bisect A C in E, through which draw a vertical line ; through A and C respectively draw A F || O/, C F || c, cutting that vertical line in F ; bisect E F in D. Then will A F and C F be tangents to the flexible tie at A and C, D will be its most depressed point, and D E its greatest depression ; and the pulls along the tie at C, D, and A, and the thrust along the strut A B, will, in virtue of the principle of Article 168, be represented by the radiating lines O c, O e, O/, and b, in fig. 86*. This solution is in general sufficiently near the truth for practi- cal purposes. To express it algebraically, let E a , R d , B,,., be the tensions of the tie at A, D, C, respectively, and H the horizontal thrust ', then (I-) SUSPENSION BRIDGE WITH SLOPING RODS. 171 The difference of length between the curve ADC and the straight line A E C is found very nearly, by substituting, in the second AB-UE term of equation 19, Article 169, AC for a, and === >r 2AJ A. \j that is to say, ABO AW S.^'PE 8 1 A&-BC' W = 3 T&- =24 172. Suspension Bridge with Sloping Bods. Let the uniformly- loaded platform of a suspension bridge be hung from the chains by parallel sloping rods, making an uniform angle j with the vertical. The condition of a chain thus loaded is the same with that of a chain loaded vertically, except in the direction of the load; and the form assumed by the chain is a parabola, having its axis paral-' lei 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 ^ ECH = j with the vertical. Let B X re- m & 87 present any rod, and suppose a vertical load v to be supported at the point X. Then, by the principles of the equilibrium of a frame of two bars (Article 145), this load will produce a putt, p, on the rod X B, and a thrust, q, on the platform between X and H ; and the three forces v, p, q, will be proportional to the sides of a triangle parallel to their directions, such as the triangle C E H ; that is to say, v:p:q:: CH: CE :EH:: 1: sec,/: tan,/. (1.) Next, instead of considering the load on one rod B X, consider the entire vertical load Y between A and X. This being the sum of the loads supported by the rods between A and X, it is evident that the proportional equation (1) may be applied to it; and that if P represent the amount of the pull acting on the rods between A and X, and Q the total thrust on the platform at the point X, we shall have Y: P: Q ::CH: CE:E~H::1 : sec j : tan,/. (2.) The oblique load P = V 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 K along the chain at B, is that of the sides of the triangle B X P ; that is to say, (3.) Comparing this with the case of Article 169 and fig. 85, it is evi- dent that the form of the chain in fig. 87 must be similar to that of the chain in fig. 85, with the exception that the ordinate X B = y is oblique to the abscissa AX = x, instead of perpendicular ; that is to say, C B A is a parabola, having its axis parallel to the inclined suspension rods. The equation of such a parabola, referred to its oblique co-ordi- nates, with the origin at A, is as follows : where m, as in Article 169, denotes the modulus of the parabola, given by the equation x* ' cos 2 j m= - = (5.) 4y 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 : t = Hence are deduced the following formulae for the relations amongst the forces which act in a suspension bridge with inclined rods. Let v now be taken to denote the intensity of the vertical load per unit of length of horizontal platform per foot, for ex- ample ; p the intensity of the oblique load ; q the rate at which the thrust along the platform increases from A towards H. Then V = vx' } P =px = vx ' sec j , } , . Q = q x v x tan j ; J x P p x 9 2pm H = =V- = 2 = 2vm -sec 3 ? (8.) 2y 2y cos 2 ,; J EXTRADOS AND INTRADOS, 173 The horizontal pull H at the point A may be sustained in three different ways, viz. : I. The chain may be anchored or made fast 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 -aj-tanj) (10.) The length of the parabolic arc, A B, is given exactly by the following formulae. Let i denote the inclination of the parabola at the point B to a line perpendicular to its axis. Then i arc* cos fg-j -cosjj (11.) 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 secj + hyp. log .fc^HL^.l ...(12.) tan^ -f secj j In most cases which occur in practice, however, it is sufficient to use the following approximate formula : arc AB = x + y ' sin J + g ^~^ nearly. ..... (13.) The formulae of this Article are applicable to Mr. Dredge's sus- pension bridges, in which the suspending rods are inclined, and although not exactly parallel, are nearly so. 173. Extra tins and inirndos. When a cord is loaded with parallel vertical forces, and ordinates are drawn downwards from the cord, of lengths proportional to the intensity of the vertical load at the points of the cord from which they are drawn, a line, straight or 174 THEORY OF STRUCTURES. curved as the case may be, which traverses the lower ends of all these ordinates, is called the extrados of the given load. The curve formed by the cord itself is called the intrados. The load suspended between any two points of the cord is proportional to the vertical plane area, bounded laterally by the vertical ordinates at those two points, above by the cord or intrados, and below by the extrados ; and may be regarded as equal to the weight of a flexible sheet of some heavy substance, of uniform thickness, bounded above by the intrados, and below by the extrados. The following is the alge- braical expression of the relations between the extrados and the intrados. Assume the horizontal axis of x to be taken at or below the level of the lowest point of the extrados; and let the vertical axis of y, as in Articles 168, 169, and 170, traverse the point where the intrados is lowest. For a given abscissa x, let y' be the ordinate of the extrados, and y that of the intrados, so that y ?/ is the length of the vertical ordinate intercepted between those two lines, to which the intensity of the load is proportional. Let w be the weight of unity of area of the vertical sheet by which the load is considered to be represented. Then we have for the load between the axis of y and a given ordinate at the distance x from that axis, (y - y'}dx; ....(i.) 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. - 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 j that is, let CORD WITH HORIZONTAL EXTRADOS. 175 * Then that line is called the parameter of the intrados, or curve in which the cord hangs. When the vertical load is of uniform intensity, as in Article 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 87 A, let O X 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 Q . Let B~X = y be any other ordinate, at the end Fig ' 87 A> of the abscissa O X x. Let the area O A B X be denoted by . Then equations 1 and 2 of Article 172 become the following : P = w u = w I ydx; J o == __ == dx~ dx*~ H ~a 3 J The general integral of the latter of these equations is u= A e^ B e in which A and B are constants, which are determined by the special conditions of the problem in the following manner. When X __ * ' & = 0, e* =e a = 1 ; but at the same time u 0, therefore A = B, and equation (a.\ may be put in the form, This gives for the ordinate, * - - (c.) A ( *- - -\ \e+e a ) which, for x = 0, becomes y Q = - ; and therefore & 176 THEORY OP STRUCTURES. which, value being introduced into the various preceding equations, gives the following results, as to the geometrical properties of the intradoa : (2.) V ( ^ Ordinate, y =~ \e a -{- e " J ; Slop,, tan; = ==t a T^ ' . tt .y ,v 2/o | Deviation, -j-~ = <> ^-^ \e a 4- e a y. d x a" 2 a' The relations amongst the forces which act on the cord are given by the equations a x B, (tension at B) = ^/P 3 + H 2 = H \f 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 amounts to this ; given the least ordinate y , 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 hyperbolic cosine of -, as this function is called. Supposing a table of hyperbolic cosines or to be at hand, - is found by its being the number whose hyper- 9 bolic cosine is ; so that a = number to hyp. cos. ,(5.) CATENARY. 177 but such a table is rarely to be met with; and in its absence a is found as follows : The value of x is given in terms of y by the equation and hence 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 alone. Let fig. 87 A, serve to represent this curve; but let A be taken as the origin of co-ordinates, so that the axis of & 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 . . dy / dx 2 in * = -7- = \/ I -- -~z : ds V ds z) dy /~ ~dtf = -y^- = A / 1 -- dx V ds* tan * ..(1.) dx ds Let the horizontal tension be equal to the weight of a c&rtain length of chain, m } so that (2.) From these equations, and from the general equation 2 of Article 168, we deduce the following : tani = dx ^ IT m .(3.) 178 THEORY OP STRUCTURES. which, bj a few reductions, is brought to the following form : gg- m (4) ** Jni' + s 2 ' the integral of which (paying due regard to the conditions that when s = 0, x = 0) is known to be + A / 1 +4) ......... (& m V m / This equation gives the abscissa x of the extremity of an arc A B = s, when the parameter of the catenary (as m is called) is known. Transforming the equation so as to have s in terms of x, we obtain =f (e e~~\. ..................... (6.) The ordinate y is found in terms of x by integrating the equation dy / ds 2 s 1 / - _-\ ^=-V ^- 1= ^=2( e "- e -)> ....... ( 7 -> which gives y = ^ (e^+e-* ^ = J t? + m 2 m ;.... (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 = pm; P =ps; ' 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 cc, 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 174, shows, that the intradosfor a horizontal ex- CATENARY. 17 9 trados when the least ordinate is equal to the parameter (y = a), becomes identical with a catenary, having the same parameter (m = a = y c ). PROBLEM. Given, two points in a catenary, and the length of chain between them; required the remainder of the curve. Let k be the horizontal distance between the two points, v their difference of level, I the length of chain between them. Those three quantities are the data. The unknown quantities may be expressed in the following manner. Let x^ y^ be the co-ordinates of the higher given point, and $! the arc terminating at it, all measured from the yet unknown vertex of the catenary, and x 2 , y 2 , s 2 , the corresponding quantities for the lower given point. (The particular case when the points are at the same level will be afterwards considered). Also let x l + x s = h (an unknown quantity). Then we have Putting these values of a; in the equations 6 and 8, we find -\ / _*- - \ = 8 1 - S 9 =mte*m+e 2J.Je2m - e 2m\ . ...(12.) *.\ / JL -ix. =}(,*-,-.-) j Square those two equations and take the difference of the squares ; then, *^'~ K )i < 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 6 "=rr^ and consequently h = m- hyp. log. ? -2 (14.) Either or both of the abscissae Xj, 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 l9 y 2 , we find 180 THEORY OF STRUCTURES. the vertex of the catenary, which, together with the parameter, being known, completely determines the curve. Q. E. I. When the given points are at the same level, that is, when v - 0, the vertical axis must be midway between them, so that x, = - x, = |; h = .................. (15.) 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^ = y a 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 (17.) II. The length of a normal to the catenaiy, at any point, cut off by a horizontal line at the depth m below the vertex, is equal to the radius of curvature at that point. III. The involute of a catenary commencing at its vertex, is the tractory of the horizontal line before mentioned, with the constant tangent m. 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 by Parallel Projection is applicable to continuously loaded cords as well as to polygonal frames : it being always borne in mind, that in order that forces may be correctly transformed by parallel projection, their magnitudes must be represented by the lengths of straight lines parallel to their directions, so that if in any case TRANSFORMATION OF CORDS. 181 the magnitude of a force is represented by an area (as in Articles 173 and 174) or by the length of a curve (as in Article 175), we must, in transforming that force by projection, first consider what length and position a straight line should have in order to represent it. Some of the cases already given might have been treated as ex- amples of transformation by parallel projection. For instance, the bridge-chain with sloping rods of Article 172 might be treated as a parallel projection of a bridge-chain with vertical rods, made by substituting oblique for rectangular co-ordinates; and the intrados for a horizontal extrados of Article 174, where the least ordinate 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; let p = be its intensity per horizontal unit of length; let H be CL 00 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 ?/, and the vertical load P', which is represented by a vertical line, are unchanged in length and direction, so that we have tf = y,P = V; (1.) but for each horizontal co-ordinate x, let there be substituted an oblique co-ordinate x', 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', and altered in the same proportion with that co ordinate ; that is to say, x' = a x ; H' == a II.. (2.) The original tension at B is the resultant of the vertical load P and the horizontal tension H. Let R be its amount, and i its in- clination to H ; then R = VP- + H';........' ....(3.) 182 THEORY OF STRUCTURES. and the ratios of those three forces are expressed by the proportion P:H:B: ttani :1 :seci: :sini:cosi: 1 (4.) Let B' be the amount of the tension at the point B in the new structure, corresponding to B, and let i f be its inclination to the oblique co-ordinate x' ; then B' = ^(F 2 + H' 2 =i 2 P'H' sin.;) (5.) B' : H' : B' : : sin : cos (t' =*=.;) : cos./ (6.) The alternative signs are to be used according as i' and 3 {dMfer} indirection - The intensity of the load in the transformed structure per unit of oblique length measured along dx 1 , is but if the intensity of the load be estimated per unit of horizontal length, it becomes p' sec ; = (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 Tinder 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 upp&r ends of ordinates, drawn upwards from the intrados, of lengths pro- portional to the intensities of the load ; and the principles of Article 173 are applicable to relations between the intrados and the extrados. The curve of Article 174 is the figure of equilibrium for a linear arch with a horizontal extrados ; and from Article 175 it appears, *that the figures of all such arches may be deduced from that of a catenary, by inverting it and altering its horizontal and vertical co-ordinates in given constant proportions for each case. The principles of Article 177, relative to the transformation of cords and chains, are applicable also to linear arches or ribs. This subject will be further considered in the sequel. The preceding Articles of this section contain propositions which, though applicable both to cords and to linear arches, are of impor- tance in practice chiefly in relation to cords or chains. The follow- ing Articles contain propositions which, though applicable also to cords as well as linear arches, are of importance in practice chiefly in relation to linear arches. 179. Circular Arch ibr 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 STRUCTURES. question to represent a vertical section of a cylindrical shell, whose length, in a direction perpendicular to the plane of the figure, is wtity. Let p denote the intensity of the external pressure, in units of force per unit of area ; r the radius of the ring ; T the thrust exerted round it, which, because its length is unity, is a thrust per unit of length. The uniform normal pressure p, if not actually caused by the thrust of a fluid, is similar to fluid pressure ; and, according to Article 110, it is equivalent to a pair of conjugate pressures in any two directions at right angles to each other, of equal intensity. For example, let x be vertical, y horizontal, and let p x , p y , be the intensities of the vertical and horizontal pressure respectively, then P*=p y =p; ........................... (1.) and the same is true for any pair of rectangular pressures. To find the thrust of the ring, conceive it to be divided into two parts by any diametral plane, such as C C. The thrust of the ring at the two ends of this diameter, of the amount 2 T, must balance the component, in a direction perpendicular to the diameter, of the pressure on the ring; the normal intensity of that component is p, as already shown ; and the area on which it acts, projected 011 the plane, C C, which is normal to its direction, is 2r ; hence we have the equation or T=pr ................... (2.) for the thrust all round the ring ; which is expressed in words by this THEOREM. The thrust round a circular ring under an uniform normal pressure is the product of the pressure on an unit of circum- ference by the radius. 180. Elliptical Arches for Uniform Pressures. If a linear arch has to sustain the pressure of a mass in which the pair of conjugate thrusts at each point are uniform in amount and direction, but not equal to each other, all the forces acting parallel to any given direc- tion will be altered from those which act in a fluid mass, by a given constant ratio ; so that they may be represented by parallel projec- tions of the lines which represent the forces that act in a fluid mass. Hence the figure of a linear arch which sustains such a system of pressures as that now considered, must be a parallel projection of a circle ; that is, an ellipse. To investigate the relations which must exist amongst the dimensions of an elliptic linear arch under a pair of conjugate pressures of uniform intensity, let A' B' A' B', B" A" B", in fig. 88, represent elliptic ribs, transformed from the circular rib A B A B by parallel projection, the vertical dimensions being un- changed, and the horizontal dimensions either expanded (as B" B"), ELLIPTIC 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 1 y', those of the corre- sponding point in the ellipse, we shall have x' = 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 CO', D'D'. Let P.,. be the total vertical pressure, and P y the total horizontal pressure, on one quadrant of the circle A B. Then Let P', be the total vertical pressure, and P',, the total horizontal pressure, on one quadrant of the ellipse, as A' B', or A" B" ; and let T'j. be the vertical thrust on the rib at B' or B", and T' y the hori- zontal thrust at A' or A". Then, by the principle of transformation, . . . T y = F, = cP,= C T = c P r;|" or, the total thrusts are as ttie axes to which they are parallel. Further, let P' = T be the total pressure, parallel to any semi- diameter of the ellipse (as 0' D' or 0" D") on the quadrant D' C' or D" C", which force is also the thrust of the rib at C' or C", the ex- tremity of the diameter conjugate to O' D' or O" D"; and let 0' D' or 0" D" = r' ; then (3.) or, the total thrusts are as the diameters to which they are parallel. Next, let p' JC , p' y , be the intensities of the conjugate horizontal and vertical pressures on the elliptic arch ; that is, of the " principal stresses" (Articles 109, 112). Each of those intensities being found by dividing the corresponding total pressure by the area of the plane to which it is normal, they are given by the following equa- tion : cr c P ' y f=^ . V 4. / A / /V 186 THEORY OP STRUCTURES. so that the intensities of the principal pressures are as the squa/res 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 ofilie axes of the arch must be the square root of ike ratio of the intensities of 'the principal pressures ; that is, The external pressure on any point, D' or D", of the elliptic arch, is directed towards the centre, O' or 0", and its intensity, per unit of area of the plane to which it is conjugate (0' C' or = a q, B'^ = a p. The following is the algebraical expression of this solution. Let A denote the major and B the minor semi-axis of the ellipse. Then whence we have for the lengths of the semi-axes, 9 f J(l + c 2 + 2c'coBJ)-J(l+(f-2c' cos./) 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 a linear arch of any figure at any point where the pressure is nor- mal, is similar to that of a circular arch of the same curvature under a pressure of the same intensity; and hence modifying the Therein of Article 179 to suit this case, we have the following : THEOREM I. The thrust at any normally pressed point of a linear arch is the product of the radius of curvature ~by the intensity of the pressure; that is, denoting the radius of curvature by , the normal pressure per unit of length of curve by 7?, and the thrust byT, T=Pt (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, At the ends of r ; e* = = c 2 r ' T *\> *TK~ r* r At the ends of c r ; e == = - ; cr c Introducing these values into the equations of Article 180, and into equation 1 of this Article, we find, Y T' x = p' y ? y = cp - = pr as before ; : .(3.) ' = p' x p x - c 2 r = cpr as before ; c It is further evident, that if the pressure be normal at every point of the arch (which it is not in the cases cited), the thrust must be constant at every point ; for it can vary only by the application of a tangential pressure to the arch ; and hence follows 190 THEORY OF STRUCTURES. THEOREM II. In a linear arch sustaining a pressure which is everywhere normal, the thrust is uniform, and the radius of curva- ture is inversely as the pressure & theorem expressed symbolically thus : T = pe = constant (4.) The only arch of this class which has hitherto been considered is the circular arch under uniform normal pressure. Another instance will be given in the following Article. 183. The Hydrostatic Arch is a linear arch suited for sustaining normal pressure at each point proportional, like that of a liquid in repose, to the depth below a given horizontal plane j and is some- times called " the arch of Yvon-Villarceaux," from the name of the mathematician who first thoroughly investigated the properties of its figure by the aid of elliptic functions. The radius of curvature at a given point in the hydrostatic arch being, in virtue of Theorem II. of the last Article, inversely propor- tional to the intensity of the pressure, is also inversely proportional to the depth below the horizontal plane at which vertical ordinates representing that intensity commence. In fig. 90, let Y O Y represent the level surface from which the Fig. 90. pressure increases at an uniform rate downwards, so as to be similar to the pressure of a liquid having its upper surface at Y O Y. Let A be the crown of the hydrostatic arch, being the point where it is nearest the level surface, and consequently horizontal. Let co-ordi- nates be measured from the point O in the level surface, directlj above the crown of the jirch ; so_that (JX = Y C = x shall be the vertical ordinate, and O Y X C y the horizontal ordinate, of any point, C, in the arch. Let O A, the least depth of the arch below the level surface, be denoted by X Q , the radius of curvature at the crown by r , and the radius of curvature at any point C by r. Let w be the weight of an unit of volume of the liquid, to whose pressure the load on the arch is equivalent. Then the intensities of the external normal pressure at the crown A, and at any point C, are expressed respectively by HYDROSTATIC ARCH. 19i p Q = wx d ; p wx. t (1.) The thrust of the arch, which, in virtue of the principles of Article 182, is a constant quantity, is given by the equation = #> r o = wx o r o = pr = wxr-. from which follows the following geometrical equation, being that which characterizes the figure of the arch : xr _ x ^ (3 ) "When x n and r are given, the property of having the radius of curvature inversely proportional to the vertical ordinate from a given horizontal axis enables the curve to be drawn approximately, by the junction of a number of short circular arcs. It is found to present some resemblance to a trochoid (with which, however, it is by no means identical). At a certain point, B, it becomes vertical, beyond which it continues to turn, until at D it becomes horizontal; at this point its depth below the level surface is greatest, and its radius of curvature least. Then ascending, it forms a loop, crosses its former course, and proceeds towards E to form a second arch similar to BAB. Its coils, consisting of alternate arches and loops, all similar, follow each other in an endless series. It is obvious that only one coil or division of this curve, viz., from one of the lowest points, D, through a vertex, A, to a second point, D, is available for the figure of an arch ; and that the por- tion BAB, above the points where the curve is vertical, is alone available for supporting a load. Let ajj, y lt be the co-ordinates of the point B. The vertical load above the semi-arch A B is represented by w and this being sustained by the thrust T of the arch.at B, must obviously be equal to that thrust ; whence follows the equation xr = x r = \ x-dy (4.) That is to say, the area of the figure between the slwrtest vertical ordinate, and the vertical tangent ordinate, is equal to the constant product of the vertical ordinate and radius of curvature. The vertical load above any point, C, is /: 192 THEORY OF STRUCTURES. and this is sustained by and equal to the vertical component of the thrust of the arch at C, which is T sin i (i being the inclination of the arch to the horizon). Hence follows the equation j ' xdy = x That is to say, the area of the figure between the shortest vertical ordinate and any vertical ordinate, varies as the sine of the angle of inclination to tJie horizon of tJie curve at the latter ordinate. The horizontal external pressure on the semi-arch from B to A is the same with that on a vertical plane, AF, immersed in a liquid of the specific gravity w with its upper edge at the depth x below the surface (see Article 124), so that its amount is /'*i 'V 2 _'y 2 T w\ *>VA xdx = w ' -- SJ x 2 and this is balanced by the thrust of the arch, T, at the crown, Hence follows the equation / r> \ (6.) That is to say, half the difference of the squares of the least vertical ordinate and of the tangent vertical ordinate is equal to the constant product of the vertical ordinate and radius of curvature. Equation 6 gives for the value of the vertical tangent ordinate, ...(7.) The horizontal external pressure between B and any point, C, is equal to the pressure of a liquid of the specific gravity w on a ver- tical plane X F with its upper edge immersed to the depth x, so that its amount is ft wl xdx=z and this is balanced by the horizontal component T cos i of the thrust of the arch at C ; whence follows the equation x\a? . /QX j~ = o r o cos t ; (8.) which gives for the value of any vertical ordinate, HYDROSTATIC ARCH. 193 Let x, x', be any two vertical ordinates. Then from equation 8 it follows that x' 2 or = 2# r (cos i cos i') .............. (10.) or, the difference of the squares of two ordinates varies as the difference of the cosines of the respective inclinations of the arc at tlieir lower ends. From equation 9 is deduced the following expression of the in- clination in terms of the vertical ordinate : 1 2 sin 2 'I = 1 cos i = 1 A /, , dx* = ^=^....(11.) V l ^djf 2 Vo The various properties of the figure of the hydrostatic arch ex- pressed by the preceding equations are thus summed up in one formula : To obtain expressions for the horizontal co-ordinate y, whose maximum value is the half-span y lt and also for the lengths of arcs of the curve, it is necessary to use elliptic functions. [The reader who has not studied elliptic functions may here pass at once to Article 184.] In the use of elliptic functions the notation employed will be that of Legendre; and the classes of functions referred to will be those called by that author the first and second kind respectively, and tabulated by him in the second volume of his treatise. Let & denote a constant angle, called the modulus of the func- tions ; Such are the formulae expressing the geometrical properties of the hydrostatic arch. Numerical results can easily be computed from them by the aid of Legendre's tables of the functions F andE. The relation between the thrust of the arch, the specific gravity of the load, and the modulus is given by the equation 196 THEORY OF STRUCTURES. 184. Ocosaatic Arches. It is proposed, by the term " Geostatic Arch" to denote a linear arch of a figure suited to sustain a pressure similar to that "of earth, which (as will be shown 'in Section 3 of this Chapter) .consists, in a given vertical plane, of a pair of con- jugate pressures, one vertical, as in Article 125 of Part I., and proportional to the depth below a given. plane, horizontal or sloping, and the other parallel to the horizontal or sloping plane, and bearing .to the vertical pressure a certain 'constant ratio, depending on the nature of the material, and other circumstances to be explained in the sequel. In what follows, the horizontal or sloping plane will be called the conjugate plane, and ordinates parallel to its line of steepest declivity,! when it slopes, or to : any line in it, when it is horizontal, conjugate ordinates. The intensity "of the vertical pres- sure will be estimated per unit' of area" of the conjugate plane; aiid the pressure parallel to the line of steepest declivity of that plane, when it slopes, or to any line in it, when it is horizontal, will be called the conjugate pressure, and its intensity will be estimated per unit of area of a vertical plane. Let the origin of co-ordinates be taken at a point in the conju- gate plane vertically above the crown of the proposed arch ; let x denote the vertical co-ordinate of any point, and y' the conjugate co-ordinate. Letj be the angle of inclination of the conjugate plane to the horizon. Let w' be the weight of unity of volume of the material to which the pressure is due, and whose upper surface is at the conjugate, plane. Then the intensity of the f vertical pressure at a given depth x', according to Theorem I. of Article 125, is p' x = w' x -cos,/;... .............. ;......(!.} and that of the conjugate pressure Py ' = 2 P'X = c *. w'x'-cosj-,..: (2.) c 2 being a constant ratio, expressed in the form of a square, for a reason which will afterwards appear. Conceive a hydrostatic arch, whose vertical and horizontal co- ordinates are x and y, and which is subjected to the pressure of a material whose weight per cubic foot is w = c w'cosj. (3.) Then at any given point in that hydrostatic arch, whose depth below the surface is x = x f , we shall have for the intensities of the vertical and horizontal pressures p x = p y = w x = c w' x' "cosj = cp' x = (4.) Now let the figure of an arch be transformed from that of the hydrostatic arch "by parallel projection, in such a manner that the GEOSTATIC ARCH. 197 vertical co-ordinate '. of saiiy point in the. new arch shall be the same with that of the corresponding point in the hydrostatic arch, and that the conjugate co-ordinate . of any point in the new arch shall bear to the horizontal co-ordinate of the corresponding point in the hydrostatic arch the constant ratio c; that is to say, let .V ; =>jV = cy.. ....... ..... ... ....... (5.) The total vertical and horizontal pressures on the arc between two given points in. the hydrostatic arch are respectively y dx.......... ...... (6.) ' The total vertical and conjugate pressures on the arc between the two corresponding points in : the new arch are respectively and if into these two expressions we introduce the values ofp' x ,p' y , dxf, aiid dy', deduced from equations 4 and 5, viz. : v ./* = ^ '>'&,-= G Py> dx ''= dx', dy* = cdy; i o . 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, 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 equilibrio. The conclusions of the preceding investigation may be summed up in the following THEOREM. A geostatic arch, transformed from a hydrostatic arch by preserving tJie 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 the original horizontal co-ordinates. This 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, arid Tf the vertical thrust at the 198 THEORY OF STRUCTUEES. points where the arch is vertical, which in this, as in other cases, is the vertical load of the semi-arch ; then T = 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 ocf for x, 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 6, and amplitude q>, are therefore to be considered as functions, not of inclinations, nor of radii of curvature, but of vertical ordinates ; that is to say, let XQ be the least vertical ordinate at the crown, x l the vertical tangent ordinate, and X the greatest vertical ordinate at the loop (which are the same in both kinds of arch), then 6 arc cos ^ = arc cos .X. = arc sm X sin (11.) and is the same function of 6 and

+ x) d y = w r 2 1 (m -f- 1 cos i) cos i d i ... cos i sin i i } ( 1 -f m) sin t -- - -- - ; j ........ (9.) = w r which value being introduced into equation 4, gives for the inten- sity of the horizontal pressure _ dP,-_ d(Pcotant)_ 1 . d (P. cotan t) d x d x r sin i d i wr d ( ' . cos 2 i i cos i \ = -- : . .<(! + m) cos i -- ff. . > sin ^ d i ( ^ 2 2 sin t J cos * sin i\ ........ (10.) The value of the horizontal pressure itself is given by introducing the values of T and P.,, from equations 8 and 9 into equation 3, and is as follows : or = i t . In this case H = H l5 and the conjugate thrust is simply the single horizontal force Hj at the point of spring- ing. CASE 2. i < v Find H as in the last example, and let the origin of co-ordinates be at the crown; then Xi = r (1 cos *j); and we have XB = {^/%,sin*(l- coBfy'di + r'H. l (l cos^)l (15.) H ( J o ) 188. Approximate Hydrostatic and Gcoslatic Arches. The Subject of elliptic functions is so seldom studied, and complete tables of them are so scarce, that it is useful to possess a method of finding the proper proportions of hydrostatic and geostatic arches (Articles 183, 184) to a degree of approximation sufficient for practical pur- poses, using algebraic functions alone. Such a method is founded on the fact that a hydrostatic arch approaches nearly to the figure of a semi-elliptic arch of the same height, and having its maximum and minimum radii of curvature in the same proportion. Let x , x lt as in Article 183, be the depth of load of a hydrostatic arch at the crown and springing respectively; r , r lt its radii of curvature at those points; a = Xi X Q , 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 6 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 extren^ities of the semi-axes, we have ............ a-) 208 THEORY OF STRUCTURES. A rough approximation to the half-span of the hydrostatic arch is found by making y { = b ; but this, in the cases which occur in practice, is too great by an excess which varies between TS and -gV, and is about -^V 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 T is, the following formula, giving the half-span in terms of the deptJis of load &t the crown and springing : Suppose the rise a and half-span y l 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, xj_ _ /20j/A 3 x ~ \l9aj' and because x\ X Q = a, we have /aoyA' ~ U/ \19a/ A closer approximation is given by the equations 7) 2 2/i = & 30 a' (4.) a 3 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 K the depths of load at its crown and springing, a? , x^ and the vertical load at the springing, P,. Determine, by equation 2 or equation 4, the span y^ of a hydro- static arch for the depths of load X Q , x l} and let *=* (5.) ft FRICTIONAL STABILITY. 209 be the ratio of the half-span of the actual arch to that of the hydro- static arch. The actual arch may now be conceived as an approximation to a geostatic arch, transformed from the hydrostatic arch by pre- serving its vertical ordinates and load, and altering its conjugate ordinates and thrust in the ratio c. The conjugate thrust of a hydrostatic arch being equal to the load, we have, as an approxi- mation to the conjugate thrust of the given semi-elliptic or semi- circular arch, Hi-eP, .............................. (6.) SECTION 3. On Frictional Stability. 189. Friction is that force which acts between two bodies at their surface of contact, and in the direction of a tangent to that surface, so as to resist their sliding on each other, and which depends on the force with which the bodies are pressed together. There is also a kind of resistance to the sliding of two bodies upon each other, which is independent of the force with which they are pressed together, and which is analogous to that kind of strength which resists the division of a solid body by shearing, that is, by the sliding of one part upon another. This kind of resistance is called adhesion. It will not be considered in the present section. Friction may act either as a means of giving stability to struc- tures, as a means of transmitting motion in machines, or as a cause of loss of power in machines. In the present section it is to be considered in the first of those three capacities only. 190. r,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 ', but that limit ought never to be attained in a structure. From the law of friction i; follows, that the friction between. two bodies may be computed by multiplying the force with which 210 THEORY OF 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 191. Angle of Repose. Let A A, in fig. 93, represent any solid body, B B a portion of the surface of another body, with which A A is in contact throughout the plane surface of contact e E. Let P C re- present the amount, direction, and position of the resultant of a force by which A A is urged obliquely towards B B, so that C is the centre of I?- no B pressure of the surface of contact eE. (Art. 89.) Let P G be resolved into two rectangular components : one, N C, normal to the plane of contact, and pressing the bodies to- gether: the other, TO, tangential to the plane of contact, and tending to make the bodies slide on each other. Let the total force P C, be denoted by P, its normal component by N", and its tangential component by T ; and let the angle of obliquity T P C or P C N be denoted by 6, so that T = P -sin0 - N -tan ) j " 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 fN, is equivalent to the condition, that r-, or tan &, shall not exceed f. Hence it follows, that the greatest angle of obliquity of pressure between two planes which is consistent with stability, is the angle whose tangent is the co-efficient of friction. This angle is called the angle of repose, and is denoted by (p. It is the steepest inclination of a plane to the horizon, at which a block of a given substance will remain in equilibrio upon it ; for if P represents the weight of the body A A, so that P C is vertical, and & = ; then p, as before, being the greater conjugate pressure, and p' the less, we obtain the following proposition : THEOREM III. The following is the expression oftJie condition of the stability of a mass of earth, in terms of the ratio of a pair of con- jugate pressures in the plane of greatest and least pressures : p _ cos 6 -f- J (cos 2 6 cos 2 4]) .. cos 6 + J (cos 2 cos 2 0) p f cos 6 ,J (cos 2 6 cos 2 61) cos 6 V (cos 2 6 cos 2 = p x V*'/ For all values of greater than Q, equation 2 becomes impossible; which shows what is otherwise evident, that the angle of repose is the steepest possible slope. There is a third pressure which may be denoted byj^,in a direction perpendicular to the first two, .>.,. and p y ; that is, horizontal, and perpendicular to the vertical plane in which the declivity is steepest; but the intensity of that third pressure will be considered in a subsequent Article. It is of secondary importance in practice, seeing that walls for the support of sloping banks of earth are gene- rally placed so as to resist the pressure of the earth in the direction of steepest declivity. With the exception of equation 4, the equations of the present Article give only the limits of the intensity of the conjugate pressure parallel to the steepest declivity. To find the exact intensity of that pressure, it is necessary to have recourse to a statical principle, first discovered by Mr. Moseley, which is stated in the following Article. 196. Principle of Least Resistance. THEOREM. If the forces which balance each other in or upon a given body or structure be distinguished into two systems, called respectively active and passive, which stand to each other in the relation of cause and effect, then will the passive forces be the least which are capable of balancing the active forces, consistently with the physical condition of the body or structure. For the passive forces being caused by the application of the ^active forces to the body or structure, will not increase after the )tive forces have been balanced by them ; and will therefore not vacti 216 THEORY OF STRUCTURES. increase beyond the least amount capable of balancing the active forces. Q. E. D. 197. Earth Loaded with its own Weight. In a mass of earth loaded with its own weight only, the gravitation of the earth causes the vertical pressure, the vertical pressure causes a tendency to spread laterally, and the tendency to spread causes the conjugate pressure; therefore the vertical and conjugate pressures stand to each other in the relation of cause and effect, or active and passive respectively; therefore the intensity of the conjugate pressure is the least which is consistent with the conditions of stability given in Articles 194 and 195. Applying this principle to the equations of Article 195, relative to a bank with a plane upper surface, they become the following : as before), jo, = wx cos 6 ........................ (1.) Conjugate pressure parallel to steepest declivity : General case, . cos 6 J (cos 2 & cos 9 "Natural slope," 6 = 9, p y =p ae = w x - cos p., ................... (4.) The third pressure p 2 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 IV. of Article 112, case 3, from the pair of conjugate pressures^,, p y , whose obliquity is 6. Let p l be the greatest, and p a the least principal pressure ; then u equations 19 and 20 of Art. 112, for A P,P, nr,p x ,p v , we are to substitute respectively, P*, P y > 0, Piy P giving the following results : 1 STABILITY OP EARTJ. W X ' COS 6 217 & *, Pi ft- / / Gv+/?)l 2 ~ * ( 4 cos COR 6 COS 6 -}- ^ (COS 2 COS 2 J (cos- 6 cos a .(8.) The axis of greatest pressure lios in the acute angle between the direction of greatest declivity and tjie vertical ; and its inclination j to the horizon, which may be denoted by ^, is given by the follow- ing formula, deduced from equation IT of Article ll2, by making the proper substitutions : cos 2 ilt = Pi Ps from which is easily deduced, ...-(9.) In using this formula, the arc sin - - is to be ^aken as greater smp than a right angle. The following are the results of the equations 7, 8, 9, for the extreme cases : Horizontal surface, 6 ; 1 sin tp 1 + sin

down to X, at a depth OX = x beneath the surface, and the/direction and position of the resultant of that pressure. The direction of that resultant i?J already known to be parallel to the declivity Y O Y. Let B B be a plane traversing X, parallel to Y O Y. In that plane take a point D, at srach a distance X D from X, that the weight of a prism of earth of the length X D and having an oblique base of the area unity in the plane O X, shall represent the inten- sity of the conjugate pressure per unit of area of a vertical plane at the depth X. Draw the straight line O D ; then will the ordinate, parallel +< O Y, drawn from X to D at any depth, be the ;in oblique prism, whose weight, per unit of area of its oe, will be the intensity of the conjugate pressure at that jt O D X be a triangular prism of earth of the thickness j weight of that prism will be the amount of the conju- gate pressure sought, and a line parallel to Y, traversing its centre of gravity, and cutting X in the centre of pressure C, will be the position of the resultant of that pressure. The depth O C of that centre of pressure beneath the surface is evidently two- thirds of the total depth OX. To express this symbolically, make ^-f- p y p y cos 6 /s/(cos 2 6 cos 2 I 7 2 0-072 i 1-700 3-000 5-826 13*924 i 0-966 0-866 0-707 0-500 i 0-933 0-750 0-500 0-250. i 0-346 o-iit 0-0295 0-0052 3(1 9-000 33-94 193-8 i i '945 S'ooo 17-47 97-4 o 0-081 0-133 0>I 57 0-165 KEJIARK. 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, arid may be called "frictional tenacity" The amount of the frictional tenacity at any horizontal joint is the product of the ver- tical load upon the portion of that joint where two blocks of stone or brick overlap each other, into the co-efficient of friction, which, as stated in the table of Article 192, is about 074. Let fig. 94 A represent a portion of a wall with a horizontal top ^ A ; and let it be required to determine ! 1 i I i i the frictional tenacity at a horizontal I | 1 1 1 1 | i ' 1 1 i ' ' | i ' ' I ' ' joint B, whose depth below A is x, the j I ' | I | \ ' ' j-j ' | ' I t | intensity of that tenacity per unit of ' 11 'I i 1 11 ' i area of a vertical plane at B, and the . ii i i II II i i . aggregate tenacity of the wall from A !i i J i i ' i i * i i ' r r~ down to B, with which it is capable of -p. g4 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 w b t x is the vertical pressure on the overlapping portions of the stones or bricks at B, and consequently, if/* be 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 BOND OF MASONRY AND BRICKWORK. 223 course ; then h t is the area of its vertical section ; and the intensity of the frictional tenacity of the joint immediately below is Let n be the number of courses from A down to B. Then the value of x for the uppermost course is = A, and for the lowest course, = n h ; and the mean value of x is = h ; so that the mean tenacity per course is Y~ fwbth; and the mean intensity, "+ 1 s i -j-:-/* . Hence the amount of the aggregate frictional tenacity of the wall, from A down to B, is (3.) From the equations 2 and 3 it is obvious that the frictional tenacity of masonry and brickwork is increased by increasing the ratio 7 which the length of the overlap bears to the depth of a n/ course. This may be effected either by increasing the length of the stones or bricks (to which the overlap bears a definite proportion, depending on the style of bond adopted), or by diminishing their depth ; but to both those expedients there is a limit fixed by the liability of stones and bricks to break across when the length exceeds the depth in more than a certain ratio, which for brick and stone of ordinary strength is about 3. For English bond (as in fig. 94 A), consisting of a course of stretchers (or bricks laid lengthwise), and a course of headers (or bricks laid crosswise), alternately, and also for Flemish bond, in which each course consists of alternate headers and stretchers, the overlap b is one-fourth of the length, or about three-fourths of the 7 O depth, of a brick. The value of -7- is therefore j ; but to allow for Ib 4: irregularities of figure and of laying in the bricks, it is safe to make it o - in the formulae. Substituting this in equations 2 and 3, and 224 TIIEORY OF STRUCTURES. o making /= -, we find for the intensity of the frictional tenacity, wJiere one-half of tJie 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) ^ , 5 v The tenacity of the wall in the direction of its thickness, which resists the separation of its front and "back portions by splitting, i 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. 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 wXj ................................. (6.) and its amount from the top of the wall down to the depth x, for a length of wall denoted by 1 3 w I (a? + h x) 2 In a brick wall consisting entirely of stretchers, as in fig. 94 B, I - 1 - the longitudinal tenacity is double of [ | i -- 1 -- [- that of the wall in fig. 94 A, where I I I I I one-half of the face consists of ends of -| -- 1 ' | I - headers. But that increased longitu- ^ ^j~^j " 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 Leaders j so that of the face of the wall consists of ends of o headers, and = - of sides of stretchers. s + 1 Let L denote the intensity of the longitudinal frictional tenacity, and T that of the transverse frictional tenacity, at the depth x. The following table represents the values of those intensities in the extreme cases : .-TT rh L .' 1 1 wx wx oo I wx Now, in intermediate cases, the longitudinal tenacity will vary nearly as the proportion of sides of stretchers in the face of the wall -j_-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 : (8.) Consequently, for the aggregate tenacities down to a given depth x, when the length of the wall is I, and its thickness t, we have Longitudinal, ' w t (or + hx)' } ......... (10.) 4 ( s ' -v Transverse, . wl (x* + hx) ........... (11.) To make the longitudinal and transverse frictional tenacities of equal intensity, we should have s = 2, or two courses of stretchers for one course of headers. This makes (130 In round factory chimneys, it is usual to make s = 4 ; and then we have L= **** = ,, (13.) Q 22 G THEORY OF STRUCTURES. The preceding formulae are applicable not only to brickwork, but to ashler masonry in which the proportions of the dimensions of the stones are on an average nearly the same with those of bricks. The formulae 9 and 11 may also be used to find the transverse tenacity of a rubble wall, if be taken to represent the propor- S ~TT 1 tion of tJie face of the wall which consists of tlie ends of squared headers or bond stones, connecting the front and back of the wall together. The principles of the present Article may be relied on as a means of comparing one piece of masonry or brickwork with another, so far as their security depends on the horizontal tenacity produced by the friction of the courses. But inasmuch as the absolute numerical results have been arrived at by an indirect process, from the tangent of the angle of repose of masonry and brickwork laid with damp mortar, these results are to be considered as uncertain, and as requiring direct experiments for their verification or correc- tion. No such experiments have yet been made. 203. Friction of Screws, Keys, and Wedges. The pieces of structures in timber and metal are often attached together by tho 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 oi Rest and Friction of Motion. For some 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 JOINT. 227 of masonry or brickwork, uncemented, or laid in ordinary mortar ; for although ordinary mortar sometimes attains in the course of years a tenacity equal to that of limestone, yet, when fresh, its tenacity is too small to be relied on in practice as a means of resisting tension at the joints of the structure; so that a structure of masonry or brick- work, requiring, as it does, to possess stability while the mortar is fresh, ought to be designed on the supposition, that the joints have no appreciable tenacity. The mortar adds somewhat to the frictional stability, as has already been stated in the table of Article 192, and thus contributes indirectly to the frictional tenacity described in Article 202. There are kinds of cement whose tenacity becomes at once equal to that of brick, or even to that of stone. So far as the joints are cemented with such kinds of cement, a structure is to be considered as one piece, and its safety is a question of strength. A plane joint which has no tenacity is incapable of resisting any force, except a pressure, whose centre of stress falls within the joint, and whose obliquity does not exceed the angle of repose. If the resistance of the material of the blocks which meet at the joint to a crushing force were infinitely great, it would be suffi- cient for stability that the centre of pressure should fall anywhere within the joint, how close soever to the edge ; but for the actual materials of construction, it is necessary that the centre of pressure should not be so near the nearest edge of the joint as to produce a pressure at that edge sufficiently intense to injure the material. Hence it appears that the exact determination of the limiting posi- tion of the centre of pressure at a plane joint is, strictly speaking, a question relating to the strength of materials. Nevertheless, an approximation to that position can be deduced from an examina- tion of the examples which occur in practice, without having recourse to an investigation founded on the theory of the strength of materials. Some of the most useful results of such an examina- tion are expressed as follows : Let q denote the ratio which the distance of the centre of pressure of a given plane joint from its centre of figure bears to the diameter or breadth of the same joint, measured along the straight line which traverses its centre of pressure and centre of figure ; so that if t be that diameter, q t shall be the distance of the centre of pres- sure from the centre of figure. Then the ratio q is found in prac- tice to have the following values : o In retaining walls designed by British engineers,..., or 0-375. o 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 + a x, so that if 05, be the greatest positive distance of a point at the edge of the joint from the neutral axis, the maximum pressure will be p l =p -^-ax l . Now, by the condition stated above, p^ = 2p Q , and, consequently, = ..................... . XL Xi XL S If the diameter of the joint is bisected by the centre of figure, and if x (as in Article 94) be the distance of the centre of pressure from the neutral axis, we shall have < 2X 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. Eeferring to the table at the end of Article 95 for the values of the moment of inertia I, the following results are obtained for joints of different figures. In each case in which there is any difference in the values of q for different directions, the deviation of the centre of pressure is supposed to take place in that direction in which the greatest deviation is admissible that is to say, at right angles to the neutral axis for which I is a maximum ; so that if h be the diameter in that direction, XL = ^. STABILITY j FIGURE OF BASE. I. Rectangle Length, K\ VT A PLAN1 I tfb : JOINT. S Breadth, 6) 12 ho II. Square Side, h III. Ellipse Longer axis, Ti\ 12 *#5 ,hb 64 4 IV. Circle V. Hollow -rectangle Outside dimensions,...^, 6) 64 4 7) 7) ft H Inside dimensions,... h', 6'j" VI. Hollow square Outside dimensions, h\ 12 I 2 7/ 3 Inside dimensions, h') VII. Circular ring Diameter, Outside, /t\ 12 Do. Inside, h') 64 4 229 rr Qh 2 (hb-h'b') AM-^ 2 ~~61iT 8 A 2 When the solid parts of the hollow square and of the circular ring are very thin, the expressions for q in Examples VI. and VII. become approximately equal to the following : VIII. Hollow square, q = -$ o IX. Circular ring,. 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 THEOKY OF STRUCTURES. (3.) II. The ratio which tJie deviation of the centre of pressure from the centre of figure of the joint bears to the length of the diameter of the joint traversing those two centres, must not exceed a certain fraction, ivhose value varies, according to circumstances, from one-eighth to three-eighths, that is to say, _ . eE The first of these conditions is called that of stability offriction t the second, that of stability of position. 206. Stability of a Series of Blocks; Line of Resistance ; lane of Pressures. In a structure composed of a series of blocks, or of a series of courses so bonded that each may be considered as one block, which blocks or courses press against each other at plane joints, the two conditions of sta- bility must be fulfilled at each joint. Let fig. 95 represent part of such a structure, 1, 1, 2, 2, 3, 3, 4, 4, being some of its plane joints. Suppose the centre of pressure Cj of the Fig. 95. j oint 1 , 1 , to be known, and also the amount and direction of the pressure, as indicated by the arrow traversing Cj. With that pressure combine the weight of the block 1, 2, 2, 1, together with any other external force which may act on that block ; the resultant will be the total pressure to be resisted at the joint 2, 2, will be given in magnitude, direction, and position, and will intersect that joint in the centre of pressure C 2 . By continu- ing this process there are found the centres of pressure C 3 , C 4 , &c., of any number of successive joints, and the directions and magni- tudes of the resultant pressures acting at those joints. The magnitude and position of the resultant pressure at any joint whatsoever, and consequently the centre of pressure at that joint, may also be found simply by taking the resultant of all the forces which act on one of the parts into which that joint divides the structure, precisely as in the " method of sections" already described in its application to framework, Article 161. The centres of pressure at the joints are sometimes called centres of resistance. A line traversing all those centres of resistance, such. as the dotted line E,, R, in fig. 95, has received from Mr. Moseley the name of the " line of resistance ;" and that author has also shown ANALOGY OF BLOCKWORK AND FRAMEWORK. 231 how in many cases the equation which expresses the form of that line may be determined, and applied to the solution of useful problems. The straight lines representing the resultant pressures may be all parallel, or may all lie in the same straight line, or may all intersect in one point. The more common case, however, is that in which those straight lines intersect each other in a series of points, so as to form a polygon. A curve, such as P, P, in fig. 95, touching all the sides of that polygon, is called by Mr. Moseley the " line of pressures" The properties which the line of resistance and line of pressures must have, in order that the conditions of stability may be fulfilled, are the following : To insure stability of position, the line of resistance must not deviate from the centre of figure of any joint by more than a certain fraction (q) of 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 ; for the sides of that polygon represent the directions of resistances, which sustain loads acting through its angles, as in the instances of framework described in Articles 150, . 151, 153, and 154, and represented in fig. 75. A structure of blocks is especially analogous to an open polygonal frame, like those in Articles 151 and 154, represented by fig. 75, with the piece E omitted because of the absence of ties. The question of the stability of a structure composed of blocks with plane joints may therefore be solved in the following manner : (1.) Determine and lay down on a drawing of the structure the line of action and the magnitude of the resultant of the external forces applied to each block, including its own weight. Either one or two of those resultants, as the case may be, will be the support- ing force or forces. (2.) Draw a polygon of external forces, like that in fig. 75* or 75**. Two contiguous sides of that polygon will represent the external for 1 f 232 THEORY OF STRUCTURES. acting on the two extreme blocks of the series, of which one may be a supporting pressure and the other a load, or both may be supporting pressures. In either case their intersection gives the point O, from which radiating lines are to be drawn to the angles of the polygon of external forces, to represent the directions and magnitudes of the resistances of the several joints. (3.) Draw a polygon having its angles on the lines of action of the external forces, as laid down in step (1.) of the process, and its sides parallel to the radiating lines of step (2). This polygon will represent the equivalent polygonal frame of the given structure, and will have a side corresponding to each joint; and each side of the polygon (produced if necessary) will cut the corresponding plane joint in its centre of pressure, and will show the direction of the resultant pressure at the joint. Then if each centre of pressure falls within the proper limits of position, and the direction of each resultant pressure within the proper limits of obliquity, as prescribed in Article 205, the structure will be balanced ; and the conditions of stability will be fulfilled under variations of the distribution of the load, which will be the greater, the greater is the diameter of each joint; for every increase in the diameters of the joints increases the limits within which the figure of the equivalent polygonal frame may vary, and every variation of that figure corresponds to a variation in the distribu- tion of the load. 208. Transformation of Block work Structures. THEOREM. If a structure composed of blocks have stability of position when, acted on by forces represented by a given system of lines, then will a structure whose figure is a parallel projection of the original structure have stability of position when acted on by forces represented by the corre- sponding parallel projection of the original system of lines; also, the centres of pressure and the lines representing the resultant pressures at the joints of the new structure will be the corresponding projections of the centres of pressure and the lines representing the resultant pressures at the joints of the original structure. For the relative volumes, and consequently the relative weights, of the several blocks of which the structure is composed, are not altered by the transformation; and if those weights in the new structure be represented by lines, parallel projections of the lines representing the original lines, and if the other forces applied externally to the pieces of the new structure be represented by the corresponding parallel projections of the lines representing the corresponding forces applied to the pieces of the original structure, then will each external force acting on the new structure be the parallel projection of a force acting on the corresponding point of the original structure; therefore the resultant pressures at the MOMENT OF STABILITY. 233 joints of the new structure, which balance the external forces, will be represented by the parallel projections of the lines representing the resultant pressures at the corresponding joints in the original structure j therefore (Article 62, Proposition I.), the centres of pressure, where those resultants cut the joints, will divide the diameters of the joints in the same ratios in the new and in the original structures ; therefore if the original structure have stability of position, the new structure will also have stability of position. This is the extension, to a structure composed of blocks, of the principle of the transformation of structures, already proved for frames in Article 166, and for cords and linear arches in Article 177. 209. Frictional Stability of a Transformed Structure. The ques- tion, whether the new structure obtained by transformation will possess stability of friction, is an independent problem, to be solved by determining the obliquity of each of the transformed pressures relatively to the joint at which it acts. Should the pressure at any joint in the transformed structure prove to be too oblique, frictional stability can in most cases be secured, without appreciably affecting the stability of position, by altering the angular position of the joint, without shifting its centre of figure, until its plane lies sufficiently near to a normal to the pressure as originally determined. I 210. Structure not Laterally Pressed. If fig. 96 represents a structure consisting of a single series of blocks, or courses, separated by plane joints, and has no lateral pressure applied to it from without, then the centre of resistance at any one of those joints, such as D E, is simply the point C where that joint is intersected by a vertical let fall from the centre of gravity G of the part of the structure ABED which lies above that joint ; and the conditions of stability are, that no joint shall be inclined to the horizon at an angle steeper than the angle of repose, and that the point C shall not at 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 ^pf a fluid, or of a mass of earth, against the structure, together with 'lie equal, opposite, and parallel, but not directly opposed, resistance of the joint to that lateral force. 234 THEORY OF STRUCTURES. The moment of stability may be different according to tlie position of the axis of the applied couple. The moment of that couple is determined in the following manner : Conceive a line to pass through all the limiting positions of the centre of resistance of the joint, so as to enclose a space beyond which that centre must not be found. TJie 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 Jwrizontal distance, and tends to overturn the structure in t/ie direc- tion of the given point in the line limiting the position of the centre of resistance; for that, according to Article 41, is the moment of the couple, which, being combined with a single force equal to the weight of the structure, transfers the line of action of that force parallel to itself through a distance equal to the given horizontal distance of the centre of resistance from the centre of gravity of the structure. To express this symbolically, let t be the length of the diameter of the joint where it is cut by the vertical plane traversing the centre of gravity of the structure and parallel to the applied thrust; let j be the inclination of that diameter to the horizon; let q t be the distance of the given limiting centre of resistance from the middle point of that diameter, and being used according as the centre of resistance, and the vertical line through the centre of gravity, lie towards {r, } f ae ***** f the **- Let h denote the height of the structure above the middle of the plane joint which is its base, b the breadth of that joint in a direc- tion perpendicular or conjugate to the diameter t, and w the weight of an unit of volume of the material. Then we shall have W = n -whbt, (2.) where n is a numerical factor depending on the figure of the structure, and on the angles which the dimensions, h, b, t, make with each other; that is, the angles of obliquity of the co-ordinates BUTTRESSES IN GENERAL. 235 to which the figure of the structure is referred. Introducing this value of the weight of the structure into the formula 1, we find the following value for the moment of stability : n (q=zq') cosj-whbt 2 (3.) This quantity is divided by points into three factors, viz. : (1.) n (q ~ + " cos t v ' Instead of the rectangular mass h b t, there may be substituted a pinnacle of the same volume, and of any figure. w \tan

&> 2 its cenifce C being at the depth - x. 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, RESERVOIR-WALLS. 245 Now C D = ^ -, and if, as before, we make E D = t, F D = ( 9 ~J~ 9 ) t\ 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 : u< , When the inner face of the wall is vertical, secj= 1, and tan^ 0; and the above equation becomes w/ , .. w'x* ^fe + sO* = g-. (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, 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 : + -J tanjj ...... (3.) (3 A.) equations analogous to equation 4 of Article 213. To obtain a formula suitable for computing the requisite thickness of wall t, let w' x* ' sec 2 / 6 n (q + q) w h w' x' (- + - J tan j 246 THEORY OF STRUCTURES. then t* - A-2B*; which quadratic equation being solved, gives or for a wall with a vertical inner face, for which B = 0, t = JA.. (4 A.) In most cases which occur in practice, the surface of the watet O Y either is, or may occasionally be, at or near the level of the top of the wall, so that h may be made = x. In such cases, let A w' sec 2 ,/ x 2n(q+_q')w and we have =a-2& x 2 x 9 which being solved, gives ; () and for a wall with a vertical inner face, The vertical and horizontal components of the pressure of the water are respectively w'x* Vertical, P sin,/ = - tan,/, Horizontal, P cos j = ~ ', Consequently the condition of stability of friction at the joint D E is given by the equation P cos./ w' a? W + PsinJ = SW + ti/artan/ ^^ W RESERVOIR-WALLS. 247 If the ratio - has been determined by means of equation 5, then we have W = nwxt = nwx 3 - :... ...(7.} x so that by cancelling the common factor x s , equation 6 is brought to the following form : 2 n w - + w tan / Example I. Rectangular Watt. In this case n=l;q = ',j=Q ', consequently, equation 5 A becomes and equation 8, wf = \ / 5_i^.' -^ tan

i tanj = ^-= \/ 4 = 0707; and 4 .(15.) so that the base of the wall is to its height as the diagonal to the side of a square. Equation 8 in this case becomes w tan (p .(10.) This condition is always fulfilled so far as the frictional stability of one course of masonry on another is concerned. As the object, however, of giving the wall the figure now in question, is to dis- tribute the pressure uniformly over a soft foundation, let it be supposed that its base rests on a material for which tan

i^#i (6.) / \1 + sin ^ S *'^ = P '' C SeC ^ and the total horizontal component of that radiating thrust is Let Py denote the intensity of that horizontal radiating thrust per unit of periphery of the joint C C ; then because the periphery of that joint is 2 T y ( = 6-2832 y\ we have _ P., cotan i 266 THEORY OF STRUCTURES. It has been shown 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,,, is given by the equation 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 p,-n r The difference, d Pj,, 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 C D is Every ring of brickwork for wliwh p s is either nothing, or positive, is stable, independently of the tenacity of cement j for in each such ring there is no tension in any direction. When p, 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 g = 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 j and the ratio of the horizontal to the inclined axis of any such opening should be fixed by the equation horiz. inclined axis DOMES, SPHERICAL AND CONICAL. 267 Example I. Splierical Dome. Uniform thickness, t ; weight of material per unit of volume, w j radius, r. x = r(l cos i) j y=.r sin i', ds = rdi. wtr cos i wtr 2 cos i sin i P >=l+cosi> T >= 1+cosi p. = r? = w t r r d^ cos 2 i 4- cos i 1 1 + cos i <&) The angle of rupture, for which p s = 0, is = arc cos J&~~ 1 = 51 49' j. 2 .(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, ,(7.) Example II. Truncated Conical Dome (fig. 113). Apex, O. Depth of top of dome below apex, a? ; of base of dome, a^ ; i, uni- form inclination ; t } uniform thickness ; y = x cotan i. Then at the base of the dome, ...(8.) w t cos i 2 sin 2 * W t COS 3 t - cotan 2 i. Fig. 113. Pz 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" oftfte weight L. 268 THEORY OF STRUCTURES. P. = ^ w t - . (x\ -xl) 4- L ; sin i wtcosi ( r| ^ (9-) 2 sin 2 w? cos 2 * , . L cotan i p s = w t x 1 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 V = 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 =r wxy z. ........................... (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= -W; ............................ (2.) TRANSFORMATION OF STRUCTURES. 269 arid the intensity of that pressure will be p = = wx ......................... (3.) yz Now let there be a pai-allel projection of this prism, whose dimen- sions, x' = ax, y = by, z = cz, are oblique to each other. The weight of the new prism will be represented by a line parallel to x' t of the length W = aW ............................ (4.) Let C = 1 cos 2 y' z' cos 2 z' x' cos 2 ct' y' A A A + 2 cos y z ' cos z x' cos x' y . . ................ (o.) Then the volume of the new prism is Y' = x' y f z' /C"= V-abc ,/C; ............ (6.) consequently the intensity of its weight is ' aW w a b c ,/CT- Y ~ b c The area of the lower surface of the new prism is y' z' sin y' z' = y z ' b c sin y' z' ; ............... (8.) The amount of the stress on that area is _ W = F = a P = ap y z .................. (9.) being represented by a line F, which is the projection of P, and parallel to x'. The intensity of this new stress is y ' ~ ~ "' (10-) , A y z f - sin y z" b c ' sin y' z' and if we consider the relation between stress and weight, F = - W, that is, p f y' z sin /' = - yf x y' z 1 JC. ............ (11.) we find sin J70 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. laqnid 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. ; Philos. May., June, 1851). Compressibility per Atmosphere, 1 ~ 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, 39-1 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, 0-0000306565 Saturated aqueous solution of carbonate of potash,... .0*0000303294 Artificial sea water, 0*0000445029 Saturated aqueous solution of chloride of calcium,... .0*0000209830 ^Ether, 0*00011137 to 0*00013073 Alcohol, 0*00008245 to 0*00008587 The compressibility of Esther 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. T ., ,. -, /Extension Tearing. al | 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 ; Eeport of the 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 ; 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- ... Compressibi- lity. Crushing. sion. Shearing. Distortion. ... ... Shearing. 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 bhe 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 Stresa U Ordinary steel and wr. iron, steady load, " " moving load, 2* 2 3 4 to 6 8 H 2 to 3 4 Cast iron, steady load, 2 to 3 3 to 4 about 1^ 6 to 8 2 to 3 Timber; average 3 10 3i Stone and brick,.. about 2 4tolO,av.abt.8 av. about 4 RESOLUTION 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 #, /3, y, be the longitu- dinal strains of the dimensions of a given particle along x, y t #, 276 THEORY 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 a-x* + /3?/ 2 + y s? + A ?/ ' z + (*zx + v xy = 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 Mathematical Journal, May, 1855. The sum of the direct strains a, + /3 + 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. Brewster, M. Wertheim, and Mr. Clerk Maxwell. 250. Displacements. Let , w, , 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 A _ + . - + . ~~ dy dz } dz dx* d YI d% V = - + - . dx dy 251. Analogy of Stresses and Strains. It has been shown in Article 104, that the elastic forces exerted on and by an originally cubical particle, which constitute the state of stress of the solid at the point where that particle is situated, may be resolved into six elementary stresses, viz.: three normal stresses, perpendicular re- spectively to the three pairs of faces, and tending directly to alter the three linear dimensions of the particle and three pairs of tangential stresses acting along the double pairs of faces to which they are applied, and tending directly to alter the angles made by such double pairs of faces. To reduce the state of stress at a given point expressed by a system of six elementary stresses referred to one system of rectangular co-ordinates to an equivalent system of elementary stresses referred to a new system of rectangular co-ordi- nates, equations have been given in Articles 105, 106, 107, 108, 109, and 112. The whole of those equations are virtually compre- hended under the following theorem : Let p xx , p yy) p^ be the CO-EFFICIENTS OF ELASTICITY AND PLIABILITY. 277 three normal stresses, and p yz , p lx , 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 mathematique de T Elasticite des Corps solides, Paris, 1852. The above equation is transformed into the equation of Article 249 by substituting respectively *, /3, /, A, ^, v, for p xx , p yy , p*,, %P yt , 2 PS*, 2 p xy ; and by making corresponding substitutions in all the equations of Articles 105, 106, 107, 108, 109, and 112, they are made applicable to strains instead of stresses. 252. The Potential Energy of Elasticity of an originally cubic particle in a given state of strain is the work which it is capable of performing in returning from that state of strain to the free state ; and is the product of the volume of the particle by the following function : TJ = -(# + Pp yy + yp et + *p yz + pp zx + v p xy \ This function was first employed by Mr. Green, Cambridge Trans- actions, vol. vii. 25JB. 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 TJ 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 - a 2 in U is that of in p xx the co-efficient of ft in TJ is that of a. in p yy 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 TJ to a homogeneous quadratic function of the elemen- tary stresses p lx , &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 STRUCTUEES. 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 ; 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 Trans., 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-b Direct, . A = Lateral, B = -5 - Transverse, C = s ; A Elasticity of volume, - = . MODULUS OP ELASTICITY CO-EFFICIENTS. 279 Pliabilities. TV A+B Direct, a = + AB-2B 2> (otherwise, the extensibility.) B Lateral, fc = + AB-2B 3 ' Transverse, ........................ C = - = 2(a + fo); Cubic compressibility, ........... tl = 3 a - 6 fo. 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, 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-ciiicicms. 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. ...................... 15,121,000 ...... 5,643,800. 1 - ..................... 14,300,000 . ..... 5,746,000. tl a ..................... 0-0000000699 ...... 0-0000001740. fo ..................... 0-0000000239 ...... 0-0000000575. C ..................... 0-0000001877 ...... 0-0000004631. fo ..................... 0-0000000661 ...... 0-0000001772. 280 THEORY OF STRUCTURES. 259. The Oeneral 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 j 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 principled respecting the relations amongst strains, and the analogies and other relations between strain and stress, are stated without a detailed demonstration. In the present section the more simple cases of those principles, to which there will be occasion to refer in the sequel, are to be demonstrated. ELLIPSE OF STRAIN. 281 Let a solid body be supposed to undergo a strain, or small alteration of dimensions and figure, of such a nature that all the displacements of its particles from their original positions are parallel to one plane ; and let that plane be repre- sented by the plane of the paper in fig. 114. In the first instance, let the state of strain of the body be uniform throughout ; that is, let all parts of the body which originally were equal and similar to each other, continue equal and similar to each other notwithstand- ing their alteration of dimensions and 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 a and Ob its semi-axes, the body being so placed in its strained condition that the central par- ticle O may remain unchanged in position, in order that the circle and ellipse may be the more easily compared. Then the particle which was at A is displaced to a, and the particle which was at B is displaced to b ; and particles which were at points in the circle, such as C and F, are displaced to corresponding points in the ellipse, such as c andy! In the direction A, the body has undergone the extension A a = ; and in the direction O B, at right angles to O A, the extension 116 * 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 a, /3, and the directions of the pair of rectangular axes of strain O A, OB, are known. One or both of the elementary strains might have been compres- sive, instead of tensile, in which case one or both of the quantities de- noting them would have been negative, to express diminution of size. 282 THEORY OF STRUCTURES. A square whose sides are unity, and parallel to O A and 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 4~ a and 1 + /3, and still parallel to A and O B. Let it now be required to express the state of strain of the body with reference to two new rectangular axes, O C and F, that is to say, to find the alterations of dimensions and figure produced by the strains on a figure originally square, described on O C and O F. Let x = O X, y = O Y, be the original co-ordinates of C, and x r = OX', y' = OY', those of F; and let the angle A O C = 90 - A O F = 6. Then x = cos 6 = 2/ y = sin 6 x f . Also, let x + | = Yl), y + * = OY + Dc,^be_the co-ordinates of c^the new position of ; and let of + % = Y'G, y' + tf = ()Y' + Gf } be the co-ordinates of f, the new position of F. Then because of the uniformity of the strain, the component displacements %, n, % t tf, have the following values : .(i.) = ao; = a cos 6 ] = /3 y = /3 sin 6 aaj' = 2/ otr sin 6 ; ' = G~f= 13 y' = - /3COS0. O c and Of are the sides of the oblique parallelogram into which the square on 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 C M; and from./ let fall /N perpendicular to F N. Then a' = M is the extension of O C; P = FN is the extension of OF; and v' = c M +/2ST is the distortion or deviation from rectan- gularity; and the values of those three new elementary strains, relatively to the pair of axes which make the angle 6 with the principal axes O A, O B, in terms of the principal elementary stresses, , /3, are as follows : ELLIPSOID OP STRAIN. 283 *' = i cos 6 -\- y sin 6 = at, cos 2 6 + /3 sin 2 d ; iff = ' sin ^ n' cos 4 = sin 2 ^ + /3 cos 2 ; *' = sin 4 cos 4 -f- 1' cos 6 + / sin 4 = 2 (ec /3) cos 6 sin ....(2.) Those three equations are exactly analogous to the equations 3 and 4 of Article 112, from which they may be formed by substituting for p x , and ft for p y in both equations; and then, in the first place, of for p n , and 6 for x n ; in the second place, ft' for jt? n , and (90 6) for ic ?fc, and in the third place, ' for p ( , and 4 for x n. 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 = a, o b ft, and draw the ellipse b c af, and the circumscribing circle C a F. Let ^ ao C = 0, and let o F be perpendicular to o C, so that those lines represent the direction of the new rectangular axes, to which the strain composed of a and ft is to be referred. Draw C c, F/j parallel to o b, cutting the ellipse in c and/, from which points respectively draw c m -L o C, and/Ti -L o F. Then o m = #', o T& = /3', 2 c m = 2/w = *', 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, a-, let the strain along O B be a compression, ft = ot. The ellipse will fall beyond the circle at A, and as much within it at B, and will cut it at an intermediate point near the middle of each quadrant. Take a pair of new axes bisecting the right angles between the original axes ; that is, let 6 = 45; then the equations 2 of Article 260 give the following result : '=0j #=0; *' = 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 a pair of axes making angles of 45 with the original axes; and the amount of the distortion is double tJiat of either of the two direct strains tvhich 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 a,fi= , will be accompanied by a pair of principal stresses along O A and OB respectively, given by the following equations : along O A, p x = A + B /3 = (A - B) * ; OB,j?, = B + A|S = (B-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 p t to represent the intensity of the shearing stress on each of a pair of planes normal to the new pair of axes, but if C be the co-efficient of transverse elasticity of the substance, we have also (3.) tanc (4.) and consequently, for an isotropic substance, CUBIC AND FLUID ELASTICITY. 285 or the transverse elasticity is half the difference of the direct and lateral 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 : up y %p x = (a . E. D ................... (6.) 263. Cubic Elasticity. If the three rectangular dimensions of a body or particle are changed in the respective proportions 1 + , 1 + /3, 1 + - . JD 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 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 j that is to say, the extension ex. increases faster than the intensity of the stretching force p, until the bar is torn asunder. The ultimate strength of the bar, or the total pull required to tear it instantly asunder the proof strength, or the greatest pull TIE-BAR SUDDEN PULL. 287 of which it can safely bear the long-continued or repeated applica- tion and the working load are computed by means of the formula p=/,orP=/S, (3.) where 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 ~~ = ! (*> where f 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 fx ~%j the force which acts through this space has for its least value 0, for its greatest value P =/S, and for its mean value ~- ; so that the 2i work performed in stretching the bar to the proof strain is /s fx_r s* a 2 * E ~" E ' 2 ' "' v ; / 2 The co-efficient ~-, by which one-half of the volume of the bar is IS multiplied in the above formula, is called the MODULUS OF RESI- LIENCE. /*S 267. Sadden Pull. A pull of ^~j or one-half of the proof load, Z 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 ^- 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 tlie fibres, or " grain." Both the ten- acity and the elasticity of timber against forces acting across the grain are much smaller than against forces acting along the grain, and are also of uncertain amount, the results of experiments being few and contradictory. 269. Additional Data. The following are a few experimental results in addition to those given in the table : Welded joint of a wrought iron retort. Ultimate tena- city, by a single experiment, in Ibs. per square inch,... 30750' Iron wire-ropes. Strength in Ibs., for each Ib. weight per fathom, Ultimate, 4480- Proof,.... 2240- "Working load J 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 Riretted 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. Rivetted joints will be further considered in the sequel. 271. Thin Hollow Cylinders; Boilers; Pipes. Let q denote the uniform intensity of the pressure exerted by a fluid which is confined within a hollow cylin- der of the radius r, and of a thickness, t, which is small as compared with that radius. The demonstration in Article 179 shows, that \\ 11 if we consider a ring} being a portion of the cylin- \v Jr der of the length unity, the tension on that ring ^==^ will be Pig. 117. P=?r, (1.) being the force per unit of length with which the internal pressure tends to split the cylinder from end to end. The sectional area of the ring under consideration is t. Then assuming, what is very nearly correct, that the tension is uniformly distributed, the intensity of that tension is The ratio of thickness to radius, which a thin hollow cylinder requires, to fit it for a given intensity of bursting pressure, proof pressure } or ivorking pressure, is given by the formula /being the ultimate tenacity, the proof tension, or the working ten- #ion, as the case may be. It is considered prudent, in STEAM-BOILERS, to make the working tension only one-eighth of the ultimate tenacity. The joints of plate iron boilers are single-rivetted ; but from the manner in which the plates break joint, analogous to the bond in masonry, the tenacity of such boilers is considered to approach more nearly to that of a double-rivetted joint than that of a sin gle-ri vetted joint. Mr. Fairbairn estimates it at 34,000 Ibs. per square inch ; so that the values of /for wrought iron boilers may be thus stated : Bursting tension, ................ 34,000 Proof tension, ..................... 17,000 Working tension, ............... 4,250 For CAST IRON WATER PIPES, the working tension may be made one-sixth of the bursting tension, which for cast iron, on an average, is 16,500 Ibs. per square inch ; that is to say, the values of /are Bursting tension, ............... 1 6,500 Proof tension (one-third), ...... 5,500 Working tension, ............... 2,750 U 290 THEORY OF STRUCTURES. For steam-pipes, as for steam-boilers, the factor of safety should be eight. 272. Thin Hollow Spheres. Let fig. 117 now be conceived to represent a diametral section of a thin hollow sphere, filled with a fluid which presses from within with the intensity q. The area of the fluid cut by the section is *r5 hence the whole force to be resisted by the tenacity of the section of the spherical shell is ? = vqr\ (1.) The area of the section of the spherical shell, supposing the thick- ness t to be small as compared with the radius r, is very nearly S = 2vrt' } (2.) hence assuming, what is very nearly correct, that the tension is uniform, its intensity is or, one-half Q the tension round a cylindrical shell having the same internal pressure, and the same proportion of thickness to radius ; so that, in these circumstances, the sphere is twice as strong as the cylinder. Equation 3 gives also the longitudinal tension in a thin hollow cylinder, which, being only one-half of the circumferential tension round the cylinder, does not require to be considered in practice. The proper ratio of thickness to radius in a thin hollow sphere is given by the formula *._!.. (4) r-2f" " W f being the bursting, proof, or working tension, according as q is the bursting, proof, or working pressure. 273. Thick Hollow Cylinder. The assumption that the circum- ferential tension, or hoop-tension as it may be called, in a hollow cylinder is uniformly distributed, is approxi- mately true only when the thickness is small as compared with the radius ; for if a ring of the cylinder be conceived to be divided into several concentric hoops, one within another, the tension of the innermost hoop balances part of the radial pressure of the confined fluid, so that a dimin- Fig. 118. ished radial pressure is transmitted to the second hoop, which has therefore a less tension than the first hoop, and so on. THICK HOLLOW CYLINDER. 291 Equation 2 of Article 271 gives the inean hoop- tension in a thick as well as in a thin cylinder ; but it is not the mean, but the greatest hoop-tension (that is, the tension round the inner surface of the cylinder), which is limited by the strength of the material. The object of the present investigation is to show what law the variation of hoop-tension follows, and thence, what relation the maximum tension bears to the fluid pressure. To make the solution perfectly general, it will be supposed that the cylinder is pressed from without as well as from within. Let fig. 118 represent a cross section of the cylinder; let R, denote its external and r its internal radius. Let q Q denote the fluid pressure from within, and g^ that from without ; p the hoop- tension at the inner surface of the cylinder, and p l the hoop-tension at the outer surface. Consider, as before, a ring whose length, parallel to the axis of the cylinder, is unity. The radial section of that ring, from r to R in fig. 118, has to sustain the difference between the total pressures from within and without, in a direction perpendicular to the radius O r R, on a quadrant bounded by that radius. That difference is Conceive the ring to be divided into an indefinite number of con- centric iioops, each of the thickness d r, and exerting a tension of the intensity^; then the total hoop-tension will be (1.) From the symmetry of the ring and f the forces acting on it in all directions round the centre 0, it is obvious that the axes of stress of any particle of metal must be respectively in the direction of a radius, and perpendicular to that direction. The principal stresses at any particle are a radial pressure, q (which for each particle at the inner surface is q 0) and for each particle at the outer surface, q } ) and a hoop-tension p. As in the case of the ellipse of stress, Article 112, we may con- ceive this pair of principal stresses to be made up of two component pairs, viz. : A pair of equal stresses of the same kind, constituting a Jiuid pressure or tension, whose common intensity, stated so as to be a tension when positive, a pressure when negative, is and a pair of equal stresses of contrary kinds, whose common intensity is 292 THEORY OF STRUCTURES. - 2 - =n. 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 time for all values of E, 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 and the solution of equation 1 is p = q = n = -^ ....................... (3.) a being an arbitrary constant, and r' any value of the radius, from r to E, inclusive ; for this reduces both sides of equation 1 to /I 1\ a [ ) . \r R/ CASE 3. General solution. By combining the two partial solu- tions of equations 2 and 3 together, we find Radial pressure, q = n m m; Hoop-tension, p = n + m = + m To determine the constants a and m we have the equations a a m = q ; 2 m = q l ; whence we obtain by elimination THICK HOLLOW CYLINDER. 293 r" (5.) r p2 ' "" R2 r ? (") The mean hoop-tension is giving, finally, for the maximum Jioop-tension, go r gi R R, r ' which is exceeded by the maximum in the proportion n /T?2 1 it*-\ 9 ft T?2 f. () 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 p may be =f, the bursting, proof, or working tension, as the case may be : T / / J* I \ (9.) / In most cases which occur in practice, the external fluid pressure q { is so small compared with the internal, that it may be neglected. One important consequence of equation 9 is, that if the internal pressure q is equal to or greater than the sum f + 2 q, of the co- efficient of strength and twice the external pressure, no thickness, how great soever, will enable the cylinder to resist the pressure. The following is a geometrical representation of the A foregoing solution. In fig. 119, let O represent the centre of the cylinder ; O r its internal, and O R its external radius. To represent the value of n= , draw two ordinates r A, R B, at right angles to the direction of those radii, such that o r~A : EJB : : R 2 : r\ Then A and B will be points in a hyperbola of the Fig ' 119 ' second order, A B, which has the property that 294 THEORY OF STRUCTURES. area r A B R = r x fA~- R x RB; so tliat it represents case 2. Draw CD || O r R, cutting off from the ordinates the parts C A, D B, which bear to each other the proportions CA. : DB : : q : q v Then r C = R D will represent m, the solution of ^ase 1 . Draw E F || r R at the same distance r E = r C on the opposite side. Then if any ordinate be drawn across the two straight lines E F ^and C D, and the curve A B, at a given distance r' from O, the segment of that ordinate between C D and A B will represent the radial pressure q, and the entire ordinate from E F to A B will represent the hoop-tension p, at that distance from O ; and in par- ticular E A will represent the maximum hoop-tension p& The formulse of this Article are the same with those given by M. Lame* in his Traite de TElasticite; but they are arrived at in a different manner. 274. Cylinder of Strained Rings. To obviate, in whole or in part, the unequal distribution of the hoop-tension in thick hollow cylinders for withstanding great pressures, it has been proposed to construct such cylinders of concentric hoops or rings built together, the outer hoops being "shrunk" on to the inner hoops, in such a manner, that before any internal pressure is applied, the hoops within a certain distance of the centre may be in a state of circum- ferential compression, and those beyond that distance in a state of circumferential tension. If the stress thus produced by the mutual action of the concentric hoops could be adjusted with such accuracy, as to be at each point exactly equal and opposite to the difference between the actual hoop tension at the same point due to the internal pressure, as given by equations 4, 5, and 6, of Article 273, and the mean hoop-tension as given by equation 7, then upon applying the proper internal pressure, there would result simply an uniform tension equal to the mean, and the formulse of Article 271 would become applicable to thick as well as to thin cylinders. Even although it may be impracticable to adjust the previous stress with the accuracy above described, any approach to its proper distribution must increase the strength of the cylinder. This method of construction has been carried into effect in Captain Blakely's gun, Mr. Mallet's mortar, and some other pieces of artillery. The only equation which the stress of the concentric hoops will of itself fulfil is THICK HOLLOW SPHERE. 295 275. Thick Hollow Sphere. Let fig. 118 now represent a diame- tral section of a hollow sphere, the fluid pressures within and without being q and q lt as before. The pressure to be resisted at the section is ^fer 2 ?iR 2 ); and if the section of the metal be conceived to be divided into an indefinite number of concentric rings, the breadth of one of these rings being dr, its radius r', and the tension at it p, it appears that the total resistance of the section will be 2v f^pr' dr; J r* and hence the equation to be fulfilled, for all values of q , q lt r, and R, is 2 pr'dr = q r- ?iK 2 .................. (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 BO that p = n-\-m' } q = 2n ra. CASE 1. n = Q, p = q=m', being the case of a 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=-=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 -'-8- 296 THEORY OF STRUCTURES. CASE 3. General solution. 2a q=2n m = ^ m, \ \ The constants a and m, deduced from the equations 2a 2a ?o = -^f m; 2 1== m , are found by elimination to have the following values : 2(K S r 3 ) R 3 r 3 giving finally, for the maximum tension, (5.) A transformation of this equation gives the following value of ratio of the external to the internal radius of the sphere, required in order that p may be = f, the bursting, proof, or working ten- sion, as the case may be : 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 siays. The sides of locomotive fire-boxes, the ends of cylindrical boilers, and the sides of boilers of irregular figures like those of marine steam engines, are often made of flat plates, r , which are fitted to resist the pressure from within ooo [^ ooj by b e i n g connected together across the water- space oooo or steam-space between them by tie-bars, called stays when long, bolts when short. For example, fig. 120 represents part of the flat side of a loco- 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 /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 , , 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 \ Sdx from the weight of the rod for a height x above B; and this must be equal to the pull /S. Hence ^f"" W + w f" o &dx=fS; (1.) Fig.121. which being solved, gives for the cross section of the rod, Ty -* S = .e7; (2.) 298 THEORY OF STRUCTURES. and for its weight, for a height x above B, /S W = W (e? 1) (3.) The most useful application of this is to the determination of the dimensions of the pump-rods of deep mines. They are not made with the section varying continuously, according to the formula "2, but in a series of divisions, each of uniform scantling ; neverthe- less that formula will serve to show approximately the law which tlie 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 F of that stress is represented simply by -g ; 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 have been ascertained by satisfactory experiments. The resistance of timber to shearing is in each case that which acts between conti- guous layers of fibres. 280. Economy of Material in Bolts and Rivets. There are many structures, such as boilers, wrought iron bridges, and frames of tim- ber or iron, in which the principal pieces, such as plates, links, or bars, being themselves subjected to a direct pull, are connected with each other at their joints by fastenings, such as rivets, bolts, pins, or keys, which are under the action of a shearing force. It is in every such case important, that the pieces connected and their fastenings should be of equal strength ; for if the fastenings be the weaker, either the whole structure is insufficiently strong, or the material which gives the additional strength to the plates or bars is wasted : and if the fastenings be the stronger, the plates and bars are weak- ened more than is necessary by the holes or sockets j 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 j 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' ;<*!' = (1.) For wrought iron rivetted plates, taking the value off from the table (as determined by the experiments of Mr. Doyne), we have -,= 1 nearly, and.-. S'=S (2.) For wrought iron bars connected by bolts or rivets, we have = I nearly, and. . S' = | S (3.) 300 THEORY OF STRUCTURES. O O O O O Example I. Plate-joint overlapped, single-rivetted. Fig. 122. A, front view ; B, side view. Let g _ thickness of plate. 7 ). /. . , d= diameter of nvet. c = distance from centre to centre of rivets. Fig. 122. Then 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 2 C ~~~ t (5.) d in practice is usually from 2 1 to l^t ; and the overlap from c ii to ITS- c. 6 "6 5 " o coo 1 Example II. Plate-joint overlapped, double- 3 rivetted. Fig. 123. Sectional area of two rivets S Sectional area of plate between two holes in same line A Fig. 123. (c d) 3 1-5708 d 2 (6.) (7.) Overlap in practice = Ifcto Ifc. Example III. Plate Butt-joint, with a pair of covering plates, single-rivetted. Pig. 124. 5 Here each rivet can give way only by being } sheared across in two places at once ; there- fore o 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<2 2 , *. c (9.) V Length of each covering plate = 2 x overlap = from 2 c to 2| c. o o o o o O O O o c RIVETS TIMBEH TIES. 301 Example IV. Plate Butt-joint, with a pair of covering plates, double- rivetted. Fig. 125. S f 4 x Sectional area of rivet Sectional area of plate between two holes in one row ..8-U16J. , m -t(c d)'- -^ > .:c = 3 -l^+d (11.) Length of each covering-plate = 2 x overlap = from 3J to 3J c. -pig. 125. NOTE. The length of a rivet, before being clenched, measuring from the head, is about 4^ t for overlapped-joints, and 5 t for butt-joints with covering-plates. Example V. Suspension bridge chain-joint. The chain of a sus- pension bridge consists of long and short links alternately. Each long link consists of one or more, say of n, parallel flat bars, of a shape resembling fig. 64, Article 138, placed side by side; each bar has a round eye at each end. Each short link consists of n + 1 parallel flat bars, with round eyes at their ends, which are placed between and outside of the ends of the parallel bars of the long links ; so that the end of each long bar is between the ends of a pair of short bars. The eyes of the long and short bars at each joint form one continuous cylindrical hole or socket, into which a bolt or pin is fitted, to connect the links together. To break the chain at a joint, by the giving way of the bolt, that bolt must be sheared across at 2 n places at once. Hence, let S denote the total sectional area of the bars in a link, and d the diameter of the bolt; then S' == 2 n x 07854 d 2 = 1-5708 n d 2 ; and because S' should be = - S, we have 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 OP 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 B d f and for bolt-holes, if n be their number, ' .................. ( } In determining the number n, it is to be observed, that if two or more bolts pierce the same layer of fibres, the resistance to the shearing out of the part of that layer between the end of the tie and the most distant of the bolts is nearly the same as if that bolt existed alone ; so that the most distant only of such a set of bolts is to be reckoned in using equation 3. In general, the piercing of the same layer of fibres by more than one bolt is unfavourable to economy. SECTION 5. On Resistance to Direct Compression and Crushing. 282. Resistance to Compression, when the limit of proof stress is not exceeded, is sensibly equal to the resistance to extension, and is expressed by the same " modulm 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. 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 IY. 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. 2S4. A Table of the Resistance of Materials to Crushing by a Direct Thrust, in pounds avoirdupois per square inch, is given at the end of the volume. So far as that table relates to the strength of brick and stone, reference has already been made to it in Article 235. It is condensed from the experimental data given by various authorities, especially by Tredgold, Mr. Fairbairn, Mr. Hodgkinson, and Captain Fowke. 285. Unequal Distribution of the Pressure On a pillar arises from the line of action of the resultant of the load not coinciding with the axis of figure of the pillar, so that the centre of pressure of a cross section of the pillar does not coincide with its centre of figure, but deviates from it in a certain direction by a certain distance, which may be denoted by TQ. In this case the strength of the pillar is diminished in the same ratio in which the mean intensity of the pressure is less than the UNEQUAL THRUST. 305 maximum intensity; that is to say, in a ratio which may be denoted by mean intensity _ /? 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 j and thence determine the neutral axis conjugate to the direction of the devia- tion r . Let d 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 6. 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 x\ be the greatest perpendicular distance of the edge of the cross section from the neutral axis in the same direction with X Q) the greatest intensity of pressure will be in which a = *j = XQpo . S . j ................ (I-) P being the total pressure, and S the area of the section of the pillar. Consequently the ratio required is Pl 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 OF CROSS SECTION. -j-. I. Rectangle, h b; b, neutral axis, ) 6 II. Square, A 2 , ....... , .................. f ................. ^ III. Ellipse : neutral axis, b ', other axis, hj\ 8 IY. Circle : diameter, h, ........................ J ....... " x 306 THEOKY OP STRUCTURES. V. Hollow rectangle : outside dimensions, 7i, b', ) 6 h (h b tib') inside dimensions, h', b' ; neutral axis, &,... j 7^5 #35* / 7 VI. Hollow square, h 2 h' 2 , .* , 2 . , , a . Q T VII. Circular ring : diameter, outside, h ; inside, h', . 286. Limitations of the Preceding Formulae. The formulae of the preceding Article of this section have reference to direct crush- ing only, and are therefore limited in their application to those cases in which the pillars, blocks, or struts along which the pres- sure acts are not so long in proportion to their diameter as to have a sensible tendency to be crushed by bending. Those cases com- prehend Stone and brick pillars, and blocks of ordinary proportions ; Pillars and struts of cast iron, in which the length is not more than five times the diameter, approximately ; Pillars and struts of wrought iron, in which the length is not more than ten times the diameter, approximately ; Pillars and struts of dry timber, in which the length is not more than about twenty times the diameter. 287. Crushing and Collapsing of Tubes. When a hollow cylin- der is exposed to a pressure from without, there is a circumferen- tial thrust round it, whose greatest intensity takes place at the inner surface of the cylinder, and may be computed by suitably modifying the formulae of Article 273. That is to say, let E and r denote respectively the outer and inner radii of the cylinder, qi the intensity of the radial pressure from without, <^ 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 K ^ being tlie -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 .(4.) 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 141 and 142, how to determine the proportions between the resultant of the gross load of a beam and the two forces which support it, whether those three forces are perpendicular or oblique to the beam, and whether they are par- allel or inclined to each other. In the present section those cases alone will be considered in which the loading and supporting forces are perpendicular to the beam, and parallel to each other, and in one plane ; for such forces alone tend simply to bend the beam, and if sufficiently great, to break it across. In Article 161 it has been shown how to determine the resist- ances exerted by the pieces of a frame which are cut by an ideal sectional plane, in terms of the forces and couples which act on one of the portions into which that plane of section divides the frame j and in Articles 162, 163, 164, and 165, that metliod of sections, as it is called, has been applied to the determination of the stresses 308 THEORY OF STRUCTURES. acting along the bars of half-lattice or Warren girders and of lattice girders. The method followed in determining the effect of a transverse load on a continuous beam is similar ; except that the resistance at the plane section, which is to be determined, does not consist of a finite number of forces acting along the axes of certain bars, but of a distributed stress, acting with various intensities, and, it may be, in various directions, at different points of the section of the beam. In what follows, the load of the beam will be conceived to con- sist of weights acting vertically downwards, and the supporting forces will also be conceived to be vertical. The longitudinal axis of the beam being perpendicular to the applied forces, will accord- ingly be horizontal. The conclusions arrived at will be applicable to cases in which the axis of the beam and the direction of the applied forces are inclined, so long as they are perpendicular to each other. Let any point in the longitudinal axis of the beam be taken as the origin of co-ordinates ; and at a given horizontal distance x from that origin, conceive a vertical section perpendicular to the longitudinal axis to divide the beam into two parts. To fix the ideas, let horizontal distances to the < . e , , > be considered as { ^tive } *> let vertical Dances and forces in an j d ^^d } direction, be considered as < negative I > an( * * e * *^ e moments ^ couples be ( P osi f e 1 according as they are I ^; h , and , ed , 1 . } negative / ( right-handed / 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 = F(or-oj) (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 osi *i ye I 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 fol- 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 x l and x 2 be the horizontal distance of the points of support from the origin, and let Pj, P 2 , be the supporting forces ; then to determine those forces we have the conditions of equilibrium Pi-HP,- 2- .(2.) from which follow the equations x., 2 W - 2 x " W 1 = x -x ~ j Xi 2 ' W - 2 ' X" W 2 = 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 j then the force acting on the beam to the left of x will be and the moment of flexure M = fa-x) (3.) 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 continuously distributed. On any indefinitely short division of the beam whose length is d x t and distance from 310 THEORY OP 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 FOECE F BENDING MOMENT M Anywhere. F Greatest FO Anywhere. M Greatest. M I. Loaded at extreme end with W, II. Uniform load of in- W W -( C -*)W cW w(c x) we w(c a;) 2 2 we 2 2 Ill. Uniform load of in- tensity w, and ad- ditional load at extreme end W, W' w(c x) WWC _ W '(c-oO- w(c x? -* 2 290. In Beams Supported at Both Ends, and loaded on the inter- mediate portion, like those represented in fig. 66 of Article 141, and in figs. 138 and 140 of a subsequent Article, it is most conve- nient to take the middle of the beam as the origin of co-ordinates. Then let c denote the half-span of the beam, so that 2 c is the span, or distance between the points of support ; the positions of those points will be expressed by = c', x, = -c; x l -x z = 2c (I-) which substitutions convert equation 2 of Article 288 into the following : P.= (2.) MOMENTS OF FLEXURE. If the load is symmetrically distributed, 2 a?" W = 0, ^-Vvir ^ Jin 311 and The equations 3 of Article 288 also become F = P l - 2 -W; M = (c-x)? l -? e x >(x"-x)W and for a symmetrically distributed load, F = s-Wj M = (c-)2S'W-si- .(2 A.) ,...(3.) ,(3 A.) The results in the cases most important in practice are given in the following table : SHEAEIN G FOECE F BENDING I M lOMENT Anywhere. F Greatest F! or.F 2 Anywhere. M Greatest M or M". IV. Single load, W, in middle- Left of 0, , W 2 W 2 (ca;)^ cW_ M Eight of 0, W ~T W " 2 2 2 ~ V. Single load, W, ap- plied at x" Left of x", (c + *")W (c + ^)W (o + *'2 \ from the middle, ................................. j 4 \ l ~~l r ~)' VI. Beam supported at both ends, uniformly loaded, j% 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 2i 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. KESISTANCE OF FLEXUKE. 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 BB', which in the free condition of the beam is straight, must be bent into an arc of a circle ; and that any surface originally plane and longitudinal, and perpendicular to the plane in which the curva- ture takes place, such as the upper surface A A', or the lower surface B B', must be bent into a cylin- drical form ; and the cylindrical surfaces so produced will have a common axis. Any two transverse sectional planes, such as A B and A B', which in the free state of the beam are parallel to each other, will have, in the curved state of the beam, positions radiating from the axis of curvature. Therefore, if the portion of the beam between the transverse planes A B, A' B', be conceived to be divided into layers, such as CO', originally plane, parallel, and of equal length, these layers, in the bent condition of the beam, must have lengths proportional to their distances from the axis of curvature. The layers near the concave side of the beam, A A', are shortened by the bending, and the layers near the convex side, B B', lengthened ; and there must be some intermediate layer which is neither lengthened nor short- ened, but preserves its free length. Let O 0' be the surface origi- nally plane, now curved, at which that layer is situated ; this is called the neutral surf ace of the beam, and the line O 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. Therefore the direct stress at each section, such as A B, whose moment constitutes the resistance to bending, is an uniformly-vary- ing stress, as defined in Article 91 ; and in order that the longi- tudinal resultant of that stress may be null, the neutral axis (as shown in that Article) must traverse the centre of gravity of the cross section A B. 314 THEORY OF STRUCTURES. The moment of a bending stress has already been given in Article 92, equations 3 and 4 ; and the methods of determining the inte- grals I and K, which occur in those equations, have been explained and illustrated in Article 95. To apply the equations of those Articles to the present purpose, let p be the intensity of the direct stress at a layer of the beam whose distance from the neutral axis is y : height above the neutral axis being considered as positive, and depth below it as negative. Then because a moment of flexure tending to make the beam con- cave upwards has been treated as positive, it is convenient, in order to avoid the unnecessary use of negative signs, to consider the con- stant ratio - as positive when it is such as to give resistance to an upward moment of flexure ; that is, when p is a thrust for positive values of y, and a pull for negative values ; consequently, p is to be considered { P^ } according a S 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, T7- V- = - arc tan j .................... (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 &, the angle made by the neutral axis with its conju- gate aods, in which the plane of the bending forces cuts the plane of section A B, viz. : cotan 6 = 5 ........................ (3.) In almost every case which occurs in practice, the plane of the bending forces cuts each cross section of the beam in one or other of its principal axes, for which K = 0, ^ = 0, = 90; and then equa- tion 1 becomes M=** ............................ (4.) y In beams whose transverse sections and moments of flexure are not uniform, no error appreciable in practice is produced by applying equation 4 to each cross section, and to the moment of flexure which TRANSVERSE STRENGTH. 315 acts upon it, as if the given section and moment belonged to an uniform beam with an uniform moment of flexure. 294. The Transverse strength of a beam, ultimate, proof, or work- ing, as the case may be, is the load required to break it across, or to produce the proof stress or the working stress, as the case may be. It is found by equating the greatest moment of flexure, ex- pressed in terms of the load and length, as in Article 291, to the moment of resistance at the cross section where that moment of flexure acts : such moment of resistance being found from the equa- tions of Article 293, by putting for p the ultimate, proof, or working direct stress of the material, as the case may be, and for y the distance from the neutral axis to the point in the given cross section where the limiting stress p is first attained. That point will be at the < c ncave I side of the beam, according as the mate- ( convex J , reassure ) rial gives way most readily to | J engion> ' 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- v"" A __ presents a beam giving way by "~~ 4 " 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, and the limiting value of p is the resistance to pressure or to ten- sion, whichever is least. For other forms of section, let y = y a for the concave side j and = y b for the convex side ; and let the limiting stresses be p =f a for pressure j and = f b for tension ; then the beam will give way by \ + us . m ^> I according as ^ is ( tearing J y b .............................. a.) This point having been determined, the equation from which the strength of the beam may be found is 316 THEORY OF STRUCTURES. M. =mWl = ^ (2.) t/ "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 = ribh* (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 y = m'h (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 breadtJis and as the squares of tJieir 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 : o M = m W I = 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, n', m', n, for some of the more usual forms of cross section : n'- 1 '=? I IY) FORM OF CROSS SECTIONS. ~bh* h ybh*' I. Rectangle b A, ) (including square) J I 12 I 2 I 6 II. Ellipse Vertical axis h ^ 1 1 1 Horizontal axis b, ... > (including circle) j 64 "20-4 = 0-0491 2 32 10-2 = 0-0982 III. Hollow rectangle, b h b'h'; also I-formed section, where b' is the sum of the breadths of the lateral hollows, . . . ldjv*\ i2V ny 1 2 l( l VK*\ 6V TW IY. Hollow square ) A 2 -A' 2 J 1/1-^ i o V Ti 1 1 1 (i-^t ft V 7)4 / LA \ fa*/ O \ /i*/ "V Hollow ellipse 1 (l VK*\ 1 1 / VK* 20-4\ TV) 2 10-2\ bh s ) ~VT. Hollow circle, 1 ^ **. 1 1 /j 7'H 20-4 \ ~&) 2 10-2\ AV 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. 296. A Table of the Resistance of Materials to Breaking Across is given at the end of the volume. It gives values of the modulus of rupture, being that for which the co-efficient /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 : /=/+/?, (i.) where/ is the modulus of rupture of the beam in question; f , 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 metal H in the cross section of the beam bears to the total depth of section h. The following were the values of the constants for the cast iron experimented on : Direct tenacity, f = 18,750 Ibs. per square inch ; \ f = 23,000 Ibs. per square inch ; V (2.) = l%fo nearly. Mr. Barlow has since made further experiments on cast iron CAST IKON 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, f , is the tenacity of the interior; that the modulus of rupture of a solid rectangular beam, / + /', 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 A of his discovery of the fact, that the resistance of cast iron to direct crushing is more than six times its resistance to tearing. It consists, as in fig. 132, of a lower flange B, an upper flange A, and a vertical 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 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 : O T-r- 320 THEORY OF STRUCTURES. Areas. Depths. Upper flange, A 1 , Lower flange, A 2 , Vertical web, A 3 , Totals,... AJ + A 2 -f- A 3 = A, 7^ -j- Ji 2 4. h 3 = < "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 Ji _ (7?, 3 + ^) A 2 h - 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 : ; ......... (2.) and the strength of the beam is expressed by the equation (3.) ifb 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 2 ..................... (4.) 299. Beams of Uniform strength are those in which the dimen- sions of the cross section are varied in such a man- ner, that its ultimate or proof resistance bears at each point of the beam the same proportion to the Fig. 133. Fig. 134. 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. 136. BEAMS OF UNIFORM STRENGTH. 321 law of variation depends upon the mode of variation of the moment of flexure of the beam from point to point, and this depends on the Fig. 137. Fig. 138. r> c Fig. 139. Fig. 140. distribution of the load and of the supporting forces, in a way which has been exemplified in Articles 289 and 290. When the depth of the beam is made uniform, and the breadth varied, the vertical longitudinal section is rectangular, and the plan is of a figure depending on the mode of variation of the breadth. When the breadth of the beam is made uniform, and the depth varied, the plan is rectangular, and the vertical longitudinal section is of a figure depending on the mode of variation of the depth. The following table gives examples of the results of those principles : Mode of Loading and Supporting. 6 2 proportional to Depth h constant; Figure of Plan. Breadth 6 constant ; Figure of Vertical Longitudinal Section. I. (Figs. 133, 134). Fixed at A, load- ed atB, 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 OF STRUCTURES. The fornmlse 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 ex. represents the proportionate elongation of the layer C C', whose distance from the neutral surface O O' 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 1 T ~ y ' Let p be the direct stress at the layer C C', and E the modulus of elasticity of the material; then a = ^ , and consequently, the cur- iii vature has the following values : 1 ' M 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 7/0 the distance from the neutral axis of that section to the layer where the limiting intensity /of the stress is attained. Then the curvature will be, PROOF DEFLECTION OF BEAMS. 323 at the section of greatest stress, = -^ = r -&y at any other section, 1 = ^ 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 axis of abscissae. Let A C, the horizontal distance from the origin to one end of the beam, be denoted by c, which, as in Articles 289 and 290, is the length of the projecting portion of a beam fixed at one end, and the half-span of a beam supported at both ends and symmetrically loaded. Let AX, 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, whieh 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 j being the difference of level between that point and the origin A. Let C D = v l 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 ^ = arc tan -7 : dx and the curvature, being the rate of variation of the inclination in a given length of the curve, is expressed by 324 THEORY OF STRUCTURES. di ~d~S di dx 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 d x projection dx ', so that the following equations may be used without sensible error : Slope, Curvature, dv 1 _ di r ~~ d x d*v d x 2 ' .(3.) Therefore, when the curvature at each point is given by equation 2, the slope and the ordinate are to be found by two successive integrations, as shown by the following equations : r*dx f /"*MIo , Slope, i = I = . ^r^.dx; J o r E y ./ o I M Ordinate, v = I idx = ~ . I I ^-~ d x 2 . Jo E y J o J o I M The greatest slope i v that is, the slope at D and the deflection or greatest ordinate v l} are found by performing the complete inte- grations between the limits x = and x = c. [Readers who are not familiar with the integral calculus are referred to Article 81 for explanations of the nature of the process of integration.] MI In both the integrals of the formulae 4, the quantity -j-^-is a 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 where m" and n" are two numerical factors depending on the dis- tribution of the forces and the figure of the beam ; so that the greatest slope and the deflection are given by the equations PROOF DEFLECTION OF BEAMS. 325 m"fc ri'fc* .(6.) For beams of similar figures, and similarly loaded and supported, 2/ 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 deflection is directly as the square of the length, and inversely as the depth. The following table gives the values of the factors m r ' and n" for some of the more ordinary cases of beams of uniform section, in which the ratio ML M being simply equal to =-p 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 III. Uniformly loaded, fi--Y 1 1 \ c) 3 4 SUPPORTED AT BOTH ENDS. IV Loaded in the middle, 1-- 1 1 c 2 3 V Uniformly loaded, I-* 3 2 5 c 8 3 12 For a beam of uniform strength and uniform depth, the quantity T\T Y 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, MI h IM f h (70 326 THEORY OF STRUCTURES. ^ 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 m f/ I M VI. Uniform strength ) and uniform depth, . . . . J 1 1 1 2 VII. Uniform strength, 1 uniform breadth ; fixed 1 \/ C 2 2 at one end, loaded at f the other, J V c - x 3 VIII. Uniform strength,! uniform breadth ; sup- 1 ported at both ends, j loaded in the middle,.. J VS, 2 2 3" IX. Uniform strength, ^| uniform breadth ; fixed 1 at one end, uniformly j loaded 1 c C X Infinite. 1 X. Uniform strength, uniform breadth j sup- c 7F ported at both ends, uniformly loaded, N/=? s=l-5708 Zi -1 = 0-5708 J 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 /", the factor for the maximum slope. For the factor for the deflection, n", the error is inconsiderable, as experiment has shown. 301. Deflection found by Graphic Construction. The great length of the radii of curvature, which are the reciprocals of the curva- tures given by equation 2 of Article 300, and the smallness of the ordinates of the curve of the neutral surface, in all cases which occur in practice, render it neither practicable nor useful to draw the figure of that curve in its natural proportions. But the following process, invented, so far as I am aware, by Mr. C. H. Wild, enables a diagram to be drawn, which represents, with a near approach to RATIO OF DEPTH TO SPAN. 327 accuracy, that curve, with its vertical dimensions exaggerated, so as to show conspicuously the slopes and ordinates Compute, by equation 2 of Article 300, the radii of curvature for a series of equi-distant points in the beam. Diminish all those radii in any proportion which may be convenient, and draw a curve composed of small circular arcs with the diminished radii. Then in the same ratio that the radii, as compared with the horizontal scale of the drawing, are diminished, will the vertical scale of the draw- ing, according to which the ordinates are shown, be exaggerated. 302. The 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, appears to be .. 1 For the working load, -^- = from - tor 2 c ~ 600 u " 1200* For the proof load, ... = from A to _L. The determination of the proportion, 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 B A, at each side of the beam, are two arcs of circles of equal radii, horizontal at A and C, and touching each other at B; therefore those arcs are of equal length ; therefore each point of contrary flexure B is midway between the middle of the beam A and the point of sup- port C. It is evident also, that the proof deflection of the beam must be double of that of an uniformly curved beam of half the span, sup- ported at the ends without being fixed ; that is to say, one-half of that of an uniformly curved beam of the same span, supported but not fixed; or symbolically The actual moment of flexure at A must be the same as in an W uniformly loaded beam, with the same intensity of load w = , supported, but not fixed at B, B; that is to say, ____ " "" ~~ 6 "" 32 ~~ and therefore, the moment of flexure at C is ,-r 70 TIT TIT TIT' 3M 3Wc 3WI - &! 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 STRUCTURES. intersecting each other in the points of contrary flexure, B, B, for which x :r . 2t The breadth which must be left at B, to resist shearing, will appear from the next Article. 308. A Beam Fixed at One End and Supported at Both is sensibly in the same condition with the part C B A B of the beam in fig. 143, extending from one of the fixed points C to the farther point of contrary flexure, which now represents a, point supported, but not fixed. Hence if a continuous girder be supported on a series of piers, the span of each of the endmost bays should be to the span of each intermediate bay, in the ratio c-+-XQ:2c, 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 rig. 145. , , * ' , 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 30SA, p. G41. SHEAKING 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 uniform cross section; let O B = x, O C = c, B~E = y, B G = y lt B D = E F (sensibly) = dx; let the breadth of the beam at any point E be denoted by z, and at the neutral surface by # . 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 p= j-y and the amount of the direct stress on the sectional plane between G and E is M /* - J ^ y -dy. 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 arrwunt 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 pi The same results are in eveiy 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 = / % d y be the area of the cross section of the beam. Then the mean intensity of the shearing stress is I S' and the maximum intensity exceeds the mean in the following ratio : a ratio depending wholly on the figure of the cross section of the beam. The following table gives some of its values : FIGURE OP CROSS SECTION. L Rectangle, z = 6, IL Ellipse, III. Hollow Rectangle This includes I-shaped sec- 3 (6 h -V h')' (bh*-b'h'*) 2' (b-b')'(bh 3 -b'It,'>) tions, IV. Hollow square, V - V, |(l + J^). Y. VI. Hollow ellipse and hollow circle; the numerical factor ; 9 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 M the normal com- ponent of the stress on a vertical plane, is to be put p ; for p' M the normal component of the stress on a horizontal plane, is to be put 0; and for p tt 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^ and p. 2 may be taken to represent the greatest and least principal stresses instead of p, and p st 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 2 " 2> 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 "**-2i (2.) or in another form 342 THEORY OF STRUCTURES. If i a be the angle which the axis of least stress makes with the horizon, then, because % i 2 = 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 fig. l4.Q } mav 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 ; 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 Ojc 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 ^, O 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 di" dY I for uniform beams, and to nearly the same amount for other beams j " and there will be an additional deflection of the layer under con- sideration, of the amount "dx , (3.) Observing that \^dx = M , the above equation becomes, for uniform ' Supposing the beam to be under the proof load, we may put for - its value , making the equation 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, f* 1 yz dy = _-. Then 4C For the same beam, according to equation 6 of Article 300, we have the proof deflection due to flexure proper, so that the ratio of those two parts of the deflection is 344 THEORY OF STRUCTURES. For wrought iron (for example) ~ = about 3. Suppose --, v" 9 which is an ordinary proportion in practice; then 1 = -r^ nearly, V-, 4 Ju a quantity practically inappreciable. It appears, then, that the distortion produced by the shearing stress in beams, even at the neutral surface, where it is greatest, produces a deflection which is very small compared with that due to the bending action of the load ; and that the alteration of the external figure of the beam must be smaller still ; from which it may be concluded, that in ordinary practical cases there is no occa- sion to compute the additional deflection due to the shearing stress. 313. Partially-Loaded Beam. In designing beams for the sup- port of roads and railways, or for any other situation in which one part of a beam may be loaded and another unloaded, it is necessary to consider whether a partial load may or may not produce, at any point of the beam, a more intense stress than an uniform load over the whole beam. The case of this kind, which is most important in practice, is that in which a beam supported at both ends is uniformly loaded throughout a certain portion of its length and unloaded throughout the remainder ; and its solution depends on two theorems. THEOREM I. For a given intensity of load per unit of length, an uniform load over the whole beam produces a greater moment of flexure at each cross section than any partial load. Let the two ends of the beam be called C and D, and aoy 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 imiform load between that end and H 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 j therefore the moment of flexure is dinmrished. 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 tJie removal PARTIALLY-LOADED BEAM. 345 of the load from any portion of tlie beam diminisJies tJie moment of flexure at each point. Q. E. D. Hence it follows, that if a beam be strong enough to bear an uni- form load of a given intensity, it will bear any partial load of the same intensity. THEOREM II. For a given intensity of load per unit of length, the greatest shearing force at any given cross section of a beam takes place when the longer of the two parts into which that section divides the beam is loaded and the shorter unloaded. Let the ends of the beam, as before, be called C and D, and the given cross section E ; and let C E be the longer part, and E D the shorter part of the beam. In the first place, let C E be loaded and E D unloaded. Then the shearing force at E is equal to the support- ing force at D, and consists in a tendency of E D to slide upwards relatively to C E. The load may be altered, either by putting weight between D and E, or by removing weight between C and E. If any weight be put between D and E, a force equal to part of that weight is added to the supporting force at D, and therefore to the shearing force at E ; but a force equal to the whole of that weight is taken away from that shearing force ; therefore the shear- ing force at E is diminished by the alteration of the load. If weight be removed from the load between C and E, the shearing force at E is diminished also, because of the diminution of the supporting force at D. Therefore any alteration from tliat distri- bution of the load in which the longer segment C E is loaded., and the shorter segment E D unloaded, diminishes the shearing force at E. Q. E. D. In designing beams where the shearing force is borne by a thin vertical web, or by lattice work (as in plate, lattice, and other compound girders, to be considered more fully in a subsequent sec- tion), it is necessary to attend to this Theorem, and to provide strength, at each cross section, sufficient to bear the shearing force which may arise from the longer segment of the beam being loaded and the shorter unloaded. To find a formula for computing that force, let c be the half-span of the beam, x the distance of the given cross section, E, from the middle of the beam, and w the uniform load per unit of length on the loaded part of the beam C E. The length of that part is and the amount of the load upon it, w (c + x). The centre of gravity of that load lies at a distance from the end, C, of the beam which is represented by 346 THEORY OF STRUCTURES. and therefore the upward supporting force at the other end of the beam, D, which is also the shearing force at E, is given by the equation It has already been shown, in Article 290, that the shearing force at a given cross section with an uniform load is F = w x ; hence the excess of the greatest shearing force at a given cross section with a partial load, above the shearing force at the same cross section with an uniform load of the same intensity, is (2.) At the ends of the beam this excess vanishes. At the middle, it consists of the whole shearing force 1? jwc, 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 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, b ' w ' WEIGHT OP 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 B' W B =ra; ........................... < 2 -> and the true gross load will be W 72 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 j 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 : _ 8 nfb W . l >- >"^') I, 5, 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 r 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 __. W~~ Snfh'"' ...... W 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 tlie condition = 1 ; that is, 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, Z, according to the fol- lowing proportional equation : L :l :L-J : :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' = 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 thaty = 10,000 Ibs. per square inch ; and let - = . Then I 15 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 r.n 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 Long Beams, which EXPANSION AND CONTRACTION OP BEAMS. 349 arise from the changes of atmospheric temperature, are usually pro- vided for by supporting one end of each beam on rollers of steel or hardened cast iron. The following table shows the proportion in which the length of a bar of certain materials is increased by an elevation of temperature from the melting point of ice (32 Fahr., or Centigrade) to the boiling point of water under the mean atmospheric pressure (212 Fahr., or 100 Centigrade); that is, by an elevation of 180 Fahr., or 100 Centigrade I- 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, '002 94 EARTHY MATERIALS. (The expansibilities of stone from the experiments of Mr. Adie.) Brick, common, "00355 ^e, -0005 Cement, -0014 Glass, average of different kinds, '0009 Granite, '0008 to '0009 Marble, '00065 to "oon 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 OF STRUCTURES. bar under any system of bending forces. All the examples of the curvature, slope, and deflection of beams in Article 300 and the subsequent Articles, are cases in which the elastic curve has been determined with a degree of approximation sufficiently close under the circumstances; that is, when the deflection is a very small fraction of the length. The present Article relates to the figure of the elastic curve for a slender fiat 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.: 1 M I being the uniform moment of inertia of the section of the spring, E the modulus of elasticity, M the moment of flexure at a given point, and r the radius of curvature at that point. When a spring is under the action of a pair of equal and opposite couples applied to its two ends, then, as in Article 304, M is constant, r is constant, and the elastic curve is a circular arc of the radius r. When a spring is under the action of a pair of equal and opposite forces, let A and B denote the two points to which those forces are applied, and A B their common line of action. The figures from Fig. 146 a. tftP- Fig. 146 c. Fig. 146 b. Fig. 146/ 146 a to 146^ 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 A3), BE, to which the spring is fixed at D and E : the spring forming one or more looped coils, lying altogether at one side of the line of action A B. Let P be the common magnitude of the equal and opposite forces applied at A and B, and x the perpendicular distance of any point C in the elastic curve from the line of action A B. Then the mo- ment of flexure at that point is obviously M = o;P; ............................. (2.) and consequently the radius of curvature at that point is given by the equation EI EI 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 tJie ends of whose neutral surface a pair of forces are applied ', will not be bent {f 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 352 THEORY OF STRUCTURES. and opposite forces of the magnitude P are applied, directed to- wards each other ; the spring forming a single arc ACE, of the length 1. x being, as before, the ordinate of any point C, let y be the distance of that ordinate from A. The smaller the force P, the more nearly will the arc A C B approach to the straight line A B ; and in order to find the small- est value of P Avhich is compatible with any bending of the spring, that force must be computed on the supposition that the ordinate x at each point is insensibly small compared with the length of the spring, and consequently, that the length of the arc A C does not sensibly differ from that of its abscissa y. This being the case, the curvature at any point C is to be taken as sensibly given by the following equation : 1 d~x which value being inserted in equation 3, gives d*x P The integral of this equation is y x = a ' sin - / E * where c = \ / -=-. In order that x may be = at the points A and B, it is necessary that when y = I, - should be = n v } n being any whole number ; c and consequently that c= (6.) UK Now of all the possible values of n, that which gives the least value of P is n = 1 ; whence we find "El and P r= A /* l =v -F=V> and this finite quantity is the smallest force which loill bend the given spring in the manner proposed. Q. E. I). 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 ininate, 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. 146y"; for its radius of curvature at any point is inversely pro- portional to the perpendicular distance of that point from a given straight line. In order to transform all the equations given in that Article for the hydrostatic arch into the corresponding equa- tions for the coiled and looped elastic curve of fig. 146 f, it is only necessary to put for the constant product of the ordinate and radius of curvature the following value : El xr=-. An instrument consisting of an uniform spring attached to a pair of levers, might be used for tracing the figures of hydrostatic arches on paper. This property of the coiled and looped elastic curve is analogous to that discovered by James Bernouilli in the simple bow of fig. 14G a, viz., that it is the figure assumed by the vertical longitu- dinal section of an indefinitely broad sheet, containing a liquid mass whose upper horizontal surface is represented by A B. SECTION 7. On Resistance to Twisting and Wrenching. 320. The Twisting moment, or moment of torsion, applied to a bar, is the moment of a pair of equal and opposite couples applied to two cross sections of the bar, in planes perpendicular to the axis of the bar, and tending to make the portion of the bar between those cross sections rotate in opposite directions about that axis. In the following Articles, twisting moments are supposed to be expressed in inch-pounds. 321. Strength of a Cylindrical Axle. A Cylindrical axle, A B, fig. 147, being subjected to the twisting moment of a pair of equal and oppo- site couples applied to the cross sec- tions A and B, it is required to find the condition of stress and strain at any intermediate cross section such as S, and also the angular displace- ment of any cross section relatively to any other. From the uniformity of the figure of the bar, and the uniformity of the twisting moment, it is evident that the condition of stress and strain of all cross sections is the same ; also, because of the 354 THEORY OP 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 f is the value of q at the distance r 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 radium of one of these rings. Then its area is 2 rdr; 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 STKENGTH OF AN AXLE. 355 moment of the shearing stress of the ring in question, being the product of those three quantities,, is which being integrated for all the rings from the centre to the circumference of the cross section S, gives for the moment of torsion, and of resistance to torsion, 1-5708 V (I- If the axle is hollow, r being the radius of the hollow, the integral is to be taken from r = r to r = r l ' } and the moment of torsion becomes It is in general more convenient to express the strength of an axle in terms of the diameter than in terms of the radius. Let h t be the external diameter of the axle, and h its internal diameter, if hollow; then For a solid axle, For a hollow axle, M = (6.) 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 ~ = ^-^ an d for the moment of torsion, ~ = ^. 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 35 G THEORY OF STRUCTURES. table, viz., 27,700, is adopted on the authority of Mr. Hodgkin- soii'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/, 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, ./= 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 clx = C7 "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 = -y-> and therefore, = <>> I. Let the moment of torsion be the working moment, for which lJ. r r l TORSION OF AN AXLE RESILIENCE. 357 Then the angle of torsion is and is the same whether the axle is solid or hollow. A value of C, the co-efficient of transverse elasticity for cast iron, is given in the table ; but it is uncertain, as experiments are dis- cordant. For wrought iron, that constant has been found with more precision, its mean value being about 9,000,000 Ibs. per square inch. Hence, for the working torsion of wrought iron shafts, we may make (3.) 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 r from the equations 4 and 5 of Article 321, by substituting q for f in the numerators and r for r l in the denominators j that is to say, For solid axles. - = ,- ; and 7* CT* */* ._qx 2Ma_32Mcc_ Ma; For hollow axles, - = ^ - r . j and r vr\-r .___ 32Mo; -~ 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 : Mi / 2 h\ x _ = for a solid shaft ; or i O'L \j M . '., , } a-) jv/i n tin h } 1(* -Tr ^^-i-f^-j-/ for a hollow shaft. 2i o ' 1 (_/ til 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 ; 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 /7* 8 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 j let p^ be the intensity of the greatest result- ant stress, and i the angle which its direction makes with the axis of the shaft. Then (I-) tan 2 i = ; P One of the most important examples of this is illustrated in fig. 148, which represents a shaft having a crank at one end. At the centre of the crank-pin, P, is applied the pressure of the connecting rod ; and at the bearing, S, acts the equal and opposite resistance of that bearing. Represent- ing the common magnitude of those forces by P, they form a couple whose moment is M = P -SP. . Draw P N perpendicular to S 1ST, 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 sin j. Equal and opposite couples act on the farther end of the shaft. Let h be its diameter. By the formulae of Article 295, the greatest stress produced at S by the bending couple is and that produced by the twisting couple, according to Article 321, is 5-1 Msinj _ ptsmj m ( . ~~ ~~ consequently, by the equations 1 of this Article, the resultant greatest stress at S, and its inclination to the axis of the shaft, are Pi = | (sec j + 1) =rr~ (1 + cos j) ; and by making p l =/*, 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 . (5.) 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 or D, fig. 149. In fig. 150, let the shaded part represent a portion of a cross 360 THEORY OF STRUCTURES. section of the rim of the wheel A of fig. 149, and let 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- tersect the tooth from the x Fig. 149. Fig. 150. side EP to the crest PK, and let PG be perpendicular to that plane. Let h be the thickness of the tooth, and let EF = 6, PG = 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 _6JPj P= ~ bh 2 ' which is a maximum when ^L P E F = 45, and b = 21, having then the value, Consequently, the proper thickness for the tooth is given by ths equation This formula is Tredgold's ; according to whom the proper value for the greatest working stress /is 4,500 Ibs. per square inch, when the teeth are of cast iron. SECTION 8. On Crushing by Bending. 327. introductory Remarks. Pillars and struts whose lengths exceed their diameters in considerable proportions (as is almost always the case with those of timber and metal), give way not by direct crushing, but by bending sideways and breaking across, being crushed at one side, as at A, fig. 151, and torn asunder at the other, as at B. There does not yet exist any complete theory of this phenomenon. The formulae which have been provision- F . 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 IRON PILLARS AND STRUTS. 361 revived by Mr. Lewis Gordon, who determined the values of the constants contained in them by a comparison of them with Mr. Hodgkinson's experiments. Then will be given Mr. Hodgkinson's own empirical formulae for the ultimate strength of cast iron pillars. 328. Strength of Iron Pillars and Struts. Let P be the load which acts on a long pillar or strut, and S its sectional area. Then one part of the intensity of the greatest stress on the material is simply the intensity due to the uniform distribution of the load over the section, and may be represented thus : **". 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 j 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, **" But the greatest deflection consistent with safety is directly as the square of the length, and inversely as the thickness (Article 300) ; that is, also, the product b h 2 is proportional to the sectional area S and to the thickness L Consequently we have the proportional equation that is, the additional stress due to bending is to the stress due to direct pressure, in a ratio which increases as the square of the propor- tion in which the length of the pillar exceeds the least diameter. The whole intensity of the greatest stress on the material of the pillar, being made equal to a co-efficient of strength /, is expressed by the following equation : a.) 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 : F = /S (20 The following are the values of f and a for the ultimate strength, as computed by Mr. Gordon from Mr. Hodgkinson's experiments on pillars FIXED AT THE ENDS, by having fiat capitals and bases, as in fig. 152 : /, Ibs. per inch. a. Wrought iron, solid rectangular section, 36,000 Cast iron, hollow cylinder, 80,000 solid .. 80,000 3,000' 1 I 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- Ul I fore be expected to be the same; a ' (\ ' (\ conclusion verified by the experiments of Mr. Hodgkinson. Hence, for such pillars, Mr. Hodgkinson found the strength of a pillar^icec? at one end and rounded Fig. 152. Fig. 153. Fig. 154. 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 IRON PILLARS AND STRUTS. 363 In using the formulae 2 and 3, the ratio -is generally fixed before- hand, to a degree of approximation sufficient for the purposes of the calculation. 329. Connecting Rods of engines are to be considered as in the condition of struts rounded at both ends ; Piston Bods, 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~695 : 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 TO 2O h Breaking load, Ibs. per square P (-Wrought, solid rect- angular, 34,840 ^u 4 29,230 3 27,700 4 23,480 inch, = -p:, Cast, solid S cylindrical, 64,000 40,000 29,230 24,620 16,000 ^ 331. Mr, Oodgkinson's Formulae for the Ultimate Strength of Cast iron Pillars, as deduced by that author from his own experiments, are as follows : I. When the length is not less than thirty times the diameter. For solid cylindrical pillars, h being the diameter, in inches, and L the length in feet, P = A ........................... (i.) For hollow cylindrical pillars, h v being the external, and h the internal diameter, in inches, and L the length in feet, P = A ',8-6 L 1 ' 7 .(2.) The values of the co-efficient A are as follows 364 THEORY OF STRUCTURES. Tons. S (3. For solid pillars with rounded ends, 14*9 flat ends, 44'* For hollow pillars with rounded ends, 13-0 flat ends, 44'3 II. When the length is less than thirty times the diameter. Let b denote the breaking load of the pillar, as computed by the preceding formulae. Let c denote the crushing load of a short block of the same sectional area S, as computed by the formula c = 49 tons x S in square inches (3.) Then the correct crushing load of the pillar is be /I \ 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- nel iron), 157 (a cross- Fig. 156. Fig. 157. Fig. 158. s]mped s ' ectioii ; us ed 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, 72 72 Q pages 361, 362, for ^ there is to be substituted npp J being the 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 one-thirtieth of the diameter of the cell, as determined by Mr. Fairbairn and Mr. Hodgkinson, is 27,000 Ibs. per square inch section of iron ; 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. J" 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 : . P = A^S; ..................... (1.) Fig . 160 . { S being the sectional area in square inches, h : I the ratio of the least diameter to the length, and A = 3,000,000 Ibs. per square inch. The factor of safety for the working load of timber being 10, A is to be made = 300,000 only, if P is the working load. For square posts and struts, the formula becomes If the strength of a timber post be computed both by this formula and by the formula for direct crushing, viz. : P=/S, ............................. (3.) the lesser value should be adopted as the true strength. 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 : /= 7,200 Ibs. on the square inch. 366 THEORY OF 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 j 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 vertical web is double : this is the " box- beam" long employed in the platforms of Fig. 164. Fig. 163. blast furnaces, and first used in a railway bridge by Andrew Thom- son about 1832, on the Pollok and Govan Railway. In fig. 164, the upper and lower ribs are each built of several layers of narrow plates or flat bars, rivetted to each other and to a pair of angle PLATE IRON GIRDERS. 367 irons j the upper and lower pairs of angle irons are rivetted to the upper and lower edges of the vertical web, and the plates of the vertical web are connected and stiffened at each of their vertical joints by a pair of T irons, in the manner of which a horizontal section has been already given in fig. 160, Article 334. The object of building the larger sizes of horizontal ribs in layers, instead of making them in one piece, is to make them of those sizes of iron which can easily be rolled of good quality, and which are usually found in the market. Beams resembling fig. 164 are sometimes made with a double vertical web, for the sake of lateral stiffness. Fig. 165 represents the general form of the cross section of great tubular or cellular girders, characterized by Mr. Stephenson's principle, of carrying the railway through the interior of the beam, and by Mr. Fairbairn's principle, of giving stiffness by means of cells, already described in Article 333. The joints of the cells are connected and stiffened by covering plates outside as well as angle irons inside ] and the plates of the two sides, which form a double vertical web, are stiffened and connected by T-irons, like those of fig. 164. Smaller cellular girders are sometimes used, in which the top alone consists of one or two lines ~ of cells, the girder in other respects being similar to fig. 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 Si be tlie sectional area of the compressed rib,/J its resistance to crushing per square inch, S 2 the sectional area of the stretched rib, /., its resistance to tearing per square inch ; then P _ M P M 1= 7 " 71' 2= 7" = ~ /I Jl /l J2 The values of the tenacity f 2 have already been considered in sec- tion 3. For plate beams with double-rivetted covering plates, its ultimate value may be taken at about 45,000 Ibs. per square inch of section of rib. The ultimate resistance to crushing, / may be taken at its full value of 3G,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 : where/" = 36,000, a = .. m , ti = the breadth of the compressed 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 3 be its resistance per unit of section to the shear- ing force, 7**~;aiid = .^; ...................... (4.) J 3 /$ 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 a 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 li 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. 3G9 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. 338. For Half-lattice Beams and Lattice Beams, the methods of determining the total stresses have been fully considered in Articles 162, 163, 164, and 165; and it has only to be added here, that the shearing force should be computed for a partial 'load, as in Article 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 uniformly* 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 fornmke 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 shearing force due to a partial load above that due to an uniform load, as given by the formulae of Article 313. 340. Stiffened Suspension Bridges. The suspension bridge is that which requires the least quantity of material to support a given load. But when it consists, as in Article 169, solely of cables or chains, suspending rods, and platform, it alters its figure with every alteration of the distribution of the load ; so that a moving load causes it to oscillate in a manner which, if the load is heavy and the speed great, or even if the application of a small load takes place by repeated shocks, may endanger the bridge. To diminish this evil, it has long been the practice partially to stiffen suspension bridges by means of framework at the sides resembling a lattice girder. It was formerly supposed that, to make a suspension bridge as stiff as a girder bridge, we should use lattice girders sufficiently strong to bear the load of themselves, and that, such being the case, there would be no use for the suspending chains. But Mr. P. W. Barlow, having made some experiments upon models, finds that very light girders, in comparison with what were supposed to be necessary, are sufficient to stiffen a suspension bridge. If mathe- maticians had directed their attention to the subject, they might have anticipated this result. The present is believed to be the first investigation of its theory which has appeared in print. The weight of the chain itself, being always distributed in the same manner, resists alteration of the figure of the bridge. By leaving it out of account, therefore, an error will be made on the safe side as to the stiffness of the bridge, and the calculation will be simplified. Let fig. 167 represent one side of a suspension bridge, in which a STIFFENED SUSPENSION BRIDGE. 371 girder is used to stiffen the bridge. In order that it may do so effectually, any partial or concentrated load on the platform must, by Fig. 167. means of the girder, be trans- mitted to the chain in such a manner as to be uniformly distributed on the chain. The girder must have its ends so fixed to the piers as to be incapable of rising or falling. Then the forces which act upon it may be thus classed : downward, the load as applied ; down- Fig. 170. ward or upward, the resistances of the fastenings of the ends to their vertical displacement; upward, the uniformly distributed tension, acting through the suspension rods, between the girder and the chain. The girder will be supposed to be of uniform section throughout its length. Two cases will be considered : first, that in which a given load is concentrated in the middle of the girder; and secondly, that in which a given portion of the length of that girder is uniformly loaded, and the remainder unloaded, like the partially loaded beam of Article 313. The second case is the most important in practice. In each case, the half-span of the bridge will be denoted by c, and the horizontal distance of any point from the middle of the bridge by x. CASE I. A single load W, applied at 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 D, D, the points of attachment to the piers. Now this curved figure is the effect of three downward forces, applied at D, A, D, respectively, and of an uniformly distributed upward force, acting on the whole length of the girder. Each half 372 THEORY OF STRUCTURES. of the girder, therefore, is in the condition of the beam described in Article 308, inverted; that is to say, the half-girder from A to D, if inverted, becomes a beam supported at D, supported and 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, BC = CD = ^.= 0-577 x ITC; ............ (1.) N/3 and consequently, making A C, the distance between the lowest and highest points, = c', we have In order t(3 determine the greatest moment of flexure, and the deflection, of the stiffening girder, A C = c' is to be taken as the half-span of a girder like that considered in Article 307, case 3, fixed at both ends, and loaded with an uniform load of the intensity W _ W ~ 27~ F268^ ....................... ( ' 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 : 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 c , and the second V D . Now by Article 307, case 3, we have The points of support D are at the same level with the points of contrary flexure B, being, in fact, points of no curvature them- selves ; and from this it is easily found that STIFFENED SUSPENSION BRIDGE. 373 CASE 2. The girder partially loaded. Let E B, in either of the 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 j 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 ETB = c + x, its gross load is W = w(c + x); and the intensity of the force exerted between the girder and chain 00 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 MC __ c-* = - ..... The amount of the upward force exerted between the chain and BD is 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. is the value at once of the supporting force exerted by the pier against the girder at E, of the shearing force between the two divisions of the girder at B, and of the downward force by which the end D of the girder is held at its point of attachment to the pier. The intensity of the net load on E B is w(c-x) . . ~ ~ and the length of that segment being c -f- x, its greatest moment of flexure, at A, according to Articles 290 and 291, is w By the usual process of finding maxima and minima, it is easily ascertained, that the greatest moment of flexure of the loaded 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 s\ unloaded division of the girder occurs when x = -, or when 6 two-thirds of the beam are unloaded ; and further, that those two greatest moments are of equal magnitude though opposite in direction, viz. : max. M A = -max. M c = -^- ; .............. (7.) and the stiffening girder must be made sufficiently strong to bear this bending moment safely in either direction. Now, the greatest moment of flexure which would arise from an uniform load of the given intensity w over the whole beam unsupported by the chain is we 2 therefore the transverse strength of the stiffening girder should be four twenty-seventh parts of that of a simple girder of the same span suited to bear an uniform load of tlie same intensity. The greatest value of the shearing force F in equation 4 occurs when one-half of the girder is loaded, or x = 0, and its amount is STIFFENED SUSPENSION" BRIDGE. 375 (8.) When two-tLirds 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 ./oj 5. /I 2 . ~9 12 E?~27 E'"' V ' or four-ninths of the proof deflection of a beam of the same figure, uniformly loaded, of the span 2 c, unsupported by a chain. At the same time, the elevation of C above a straight line joining B and Dis . 12 E 4y ~9 12 Es/~108 Bf The proof depression of the lowest point of the beam, A, below the highest, C, is given by the equation _ 5. A ./*_-_?*./*: V* v c g 12 Ey 108 Ey'" QY five-nintlis of the proof deflection of an uniformly loaded beam. * * In the preceding solution of Case 2, whiclrappeared in the first edition of this work, the effect of the resistance of the chain to disfigurement upon the figure of the auxiliary 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 oa the safe side, four twenty-sevenths of 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 Architect's 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 ................................... 0*148:1; so that in the intermediate cases that occur in practice no material error will be committed if that proportion be made 1: 7, or ............. . .............. 0-143:1. 37G THEORY OF STRUCTURES. 341. Bibbed Arches. Bridges are frequently constructed whose arches consist of iron or timber ribs springing from stone abutments, as in fig. 171. In such cases it ought to be considered, that each rib fulfils at once the functions of an equi- librated arch, sustain- ing an uniform load of a certain intensity, and having a certain thrust along it, to be computed by the principles of Articles 169 and 178, and those of a stiffening girder, suited to produce an uniform distribution of a partial load, according to the principles of Article 340. Therefore, in designing the cross section of a rib for such a bridge, a provisional cross section ought first to be designed, suitable to bear a bending moment, upward or down- ward, of four twenty-sevenths of that which an uniform load of the given intensity would produce on a straight girder of the same span; and in the second place, it should be determined in what proportion the thrust along the rib, considered as an equilibrated arch, will increase the intensity of the greatest stress on the pro- visional section already designed, and the breadths of that section should be increased in that proportion, to obtain the final cross. section. SECTION 10. Miscellaneous Remarks on Strength and Stiffness. 342. Effects of Temperature. At a temperature of 600 Fahren- heit, the tenacity of iron was found by Mr. Fairbairn not to be diminished. That of copper and brass, at the same temperature, is reduced to about two-thirds of its ordinary magnitude. Sudden cooling from a high temperature tends to make most substances hard, stiff, and brittle ; gradual cooling tends to make them soft and tough ; and if often repeated or performed slowly from a very high temperature, to weaken them. Various effects of temperature on the elasticity of solids have been ascertained by Dr. Joule, Dr. Thomson, and Professor Kupfer ; but they are more important to the science of molecular physics than to the art of construction. 343. The Effects of Repeated Meltings 011 Cast Iron have been ascertained by Mr. Fairbairn. Up to and beyond the fourteenth melting the resistance to crushing increases ; but the resistance to cross-breaking reaches its maximum about the twelfth melting, and afterwards diminishes, from the metal becoming brittle and crys- talline. 344. The Effects of Ductility on strength form the subject of a DUCTILITY INTERNAL FRICTION. 377 paper by Professor James Thomson in the Cambridge and Dublin Mathematical Journal. That author shows, that a bent bar or a twisted rod of a ductile material, by being slowly and gradually strained, may be brought into such a condition as to have nearly the whole of its cross section in the condition of proof or limiting stress instead of the outer layers only, and may thus have its strength increased much beyond that given by the ordinary formulae. 345. internal Friction is a term which may be used until a better shall be devised to express a phenomenon recently observed by 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 NAPIEB AND SONS ON THE TENACITY OF IRON AND STEEL. (For details, see Transactions of the Institution of Engineers in /Scotland, 1858-59.) nacity in Ibs. per square inch. Strongest Weakest STEEL BARS- Quality. Quality. Cast Steel 132,909 92,015 Blistered Steel (one quality only) 104,298 Bessemer's (do.) .. 111,460 Homogeneous Metal, ....90,647 89.724 Puddled Steel, 71,486 62,761) IRON BARS. Yorkshire 66,392 60,075 Staffordshire, 62,231 66,715 Lanarkshire, 64,795 6,655 Lancashire, 60,110 63,775 Swedish, 48,232 47,855 Russian 56,805 49,564 Hammered Scrap 55,878 53,420 Cut out of lurjje forged crank, 47,582 44,758 Tenacity in Ibs. per square inch. Strongest Weakest STEEL PLATES. Quality. Quality. CastSteel, 95.299 72,338 Homogeneous Metal 96,715 72.91)4 Puddled Steel, 93,979 72,366 IKON PLATES. Yorkshire, 56,735 49,338 Durham (one quality only) 48,979 Staffordshire 64,) 28 45,584 Lanarkshire, 61,349 41,743 IRON STRAP.?, Ac. Various districts, 55,937 41,386 The strength ot each quality is the mean of at least four experiments, and sometimes of eight. X PART III. PRINCIPLES OF CINEMATICS, OR THE COMPARISON OF MOTIONS. 347. Division of the Subject. The science of cinematics, and the fundamental notions of rest and motion to which it relates, having already been defined in the Introduction, Articles 8, 9, 10, 1 1 ; it remains to be stated, that the principles of cinematics, or the comparison of motions, will be divided and arranged in the present part of this treatise in the following manner : I. Motions of Points. II Rigid Bodies or Systems. 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 line must be compared, at the beginning and end of the motion con- sidered, with some fixed or standard length, and with at least two fixed directions. Standard lengths have already been considered in Article 7. An absolutely fixed direction may be ascertained by means whose principles cannot be demonstrated until the subject of dynamics is considered. For the present it is sufficient to state, that when a solid body rotates free from the influence of any external force tending to change its rotation, there is an absolutely fixed direction called that of the axis of angular momentum, which bears certain relations to the successive positions of the body. A nearly fixed direction is that of a straight line joining a pair 380 PRINCIPLES OF CINEMATICS. of points in two bodies whose distance from each other is very- great, such as the earth and a fixed star. A line fixed relatively to tlie earth changes its absolute direction (unless parallel to the earth's axis) in a manner depending on the earth's rotation, and returns periodically to its original absolute direction at the end of each sidereal day of 86,164 seconds. This rate of change of direction is so slow compared with that which takes place in almost all pieces of mechanism to which cinematical and dynamical principles are applied, that in almost all questions of applied mechanics, directions fixed relatively to the earth may be treated as sufficiently nearly fixed for practical purposes. When the motions of pieces of mechanism relatively to each other, or to the frame by which they are carried, are under consi- deration, directions fixed relatively to the frame, or to one of the pieces of the machine, may be considered provisionally as fixed for the purposes of the particular question. 349. Motion of a Pair of Points In fig. 172, let A a 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, from the length and direc- tion represented by AJ 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,, AB 2 , equal and parallel to A^Bj, A 2 B 2 , of fig. 172 ; then A represents one of the pair of points whose relative motion is under consideration, and B,, B.,, represent the two successive positions of the other point B relatively to A ; and the line Bj B 2 represents the motion of B rela- tively to A. Or otherwise, as in fig. 174, from a single point B let there be drawn a pair of lines, BA,, BA 2 , equal and parallel to AiB,, A 2 B 2 , of fig. 172; then A l5 A 2 , represent the two successive positions of A relatively to B; and the line Aj A,, equal and parallel to Bj Ba of fig. 173, Outpointing in the contrary direction, represents the motion of A relatively to B. COMPONENT AND RESULTANT MOTIONS TIME, 381 350. Fixed Point and moving Point In fig. 173, A is treated as tlie fixed point, and B as the moving point ; and in fig. 174, B is treated as the fixed point, and A as the moving point ; and these are simply two different methods of representing to the mind the same relation between the points A and B (see Article 10). 351. Component and Resultant Motions. Let O be a point assumed as fixed, and A and B two suc- cessive positions of a second point rela- tively to O. In order to express mathe- matically the amount and direction of AB, the motion of the second point .relatively to O, that line may be com- pared with three axes, or lines in fixed directions, traversing the fixed point O, such as OX, OY, OZ. Through A and B draw straight lines AC, B D, 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- Fi S 175 - ponent of the motion of the second point relatively to 0, along or in tJie 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 j and if be the angle made by the direction of the motion A B with X, CD = AB cos . The relations between resultant and component motions are exactly analogous to those between the lines representing resultant and component couples, which have already been explained in Articles 32, 33, 34, 35, 36, and 37. 352. The Measurement of Time is effected by comparing the events, and especially the motions, which take place in intervals of time. Equal times are the times occupied by the same body, or by equal and similar bodies, under precisely similar circumstances, in per- forming equal and similar motions. The standard unit of time is the period of the earth's rotation, or sidereal day, which has been proved by Laplace, from the records of celestial phenomena, not to have changed by so much as one eight-millionth part of its length in the course of the last two thousand years. 382 PKINCIPLES 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 -00273791 of a sidereal day ; so that a sidereal day is 86164-09 seconds. The length of a solar day is variable j but the mean solar day, being the exact mean of all its different lengths, is the period already mentioned of 1-00273791 of a sidereal day, or 86,400 seconds. The divisions of the mean solar day into 24 hours, of each hour into 60 minutes, and of each minute into 60 seconds, are familiar to all. Fractions of a second are measured by the oscillations of small pendulums, or of springs, or by the rotations of bodies so contrived as to rotate through equal angles in equal times. 353. Velocity is the ratio of the number of units of length described by a point in its motion relatively to another point, to the number of units of time in the interval occupied in describing the length in question ; and if that ratio is the same, whether it be computed for a longer or a shorter, an earlier or a later, part of the motion, the velocity is said to be UNIFOKM. Telocity is expressed in units of distance per unit of time. For different purposes, there are employed various units of velocity, some of which, together with their proportions to each other, are given in the following table : Comparison of Different Measures of Velocity. Miles Feet Feet Feet per hour. per second. per minute. per hour. i 1-46 =88 = 5280- 0-6818 . = 1 . ss 60 == 3600 0-01136^ = 0-016 = i 60 0*0001893 = 0-00027 = 01016 = 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 0, A, B, denote the three points. Any one of them may be taken as a fixed point; let be so chosen; and let OX, O Y, O Z, fig. 176, be axes traversing it in fixed directions. Let Aj and B x be the positions of A and B relatively to O at the beginning of the given interval of time, and A 2 and B 2 their positions at the end of that interval. Then A x A 2 and Bj B> are the respective motions of A and B relatively to O. Complete the parallelogram A! B : b A 2 ; then because A a b is parallel and equal to A a B 13 b is the position which B would have at the end of the interval, if it had no motion relatively to A ; but B 2 is the actual position of B at the end of the interval ; therefore, b B 2 is the motion of B relatively to A. Then in the triangle Bj b B 2 , B! b = A! A 2 is the motion of A relatively to 0, b B 2 is the motion of B relatively to A, B! B 2 is the motion of B relatively to O ; so that those three motions are represented by the three sides of a triangle. Q. E. D. This Theorem might be otherwise expressed by saying, that if three moving points be considered in any order, the motion oftJie third relatively to the first is the resultant of the motion of the 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 the sides of a closed polygon. Let be the first point, A, B, C, &c., successive points following it, M the last point but one, and N" the last point; and, for brevity's sake, let the relative motion of two points, such as B and C, be denoted thus (B, C). Then by the Theorem of Article 355 (0, A), (A, B), and (0, B) are the three sides of a triangle ; also (0, B), (B, C), and (0, C), are the three sides of a triangle; therefore 384 PRINCIPLES OF CINEMATICS. (O, A), (A, B), (B, C), and (O, C), are the four sides of a quadri- lateral ; and by continuing the same process, it is shown, that how great soever the number of points, (O, N), is the closing side of a polygon, of which (O, A), (A, B), (B, C), (C, D), &c, (M, N) are the other sides. Q. E. D. In other words, the motion of the last point relatively to the first is the resultant of the motions of each point of the series relatively to that preceding it. This proposition is exactly analogous to that of the " polygon of couples," Article 37. 357. The Puraiieiopiped of Motions is a case of the polygon of motions, analogous to the parallelepiped of forces in Article 54. In fig. 177, let there be four points, O, A, B, C, of which one, O, is assumed as fixed, and is traversed by three axes in fixed direc- tions, O X, O Y, O Z. In a given interval of time, let A have the motion Aj A 2 along or parallel to OX ; let B have, in the same interval, the motion b B 3 parallel to O Y, and relatively to A; then B x B,,, the diagonal of the parallelogram whose sides Fig. 177. are prj __ AjAjand 6B~ is the motion of B relatively to O. Let C have, relatively to B, the motion c C 2 parallel to O Z; then C t C 2 , the diagonal of the parallelopiped whose edges are A, A 2 , b B L >, and c C 2 , is the motion of C relatively to O, being the resultant of the motions represented by those three edges. This is a mechanical explanation, of the composition of motions, leading to results corresponding with the geometrical explanation of Article 351. 358. Comparative Motion is the relation which exists between the simultaneous motions of two points relatively to a third, which is assumed as fixed. The comparative motion of two points is ex- pressed, in the most general case, by means of four quantities, viz. : (1.) The velocity ratio* or the proportion which their velocities bear to each other. (2.) (3.) (4.) The directional relation* which requires, for its com- plete expression, three angles. Those three angles may be measured in different ways, and one of those ways is the following : (2.) The angle made by the directions of the compared motions with each other. (3.) The angle made by a plane parallel to those two directions with a fixed plane. * These terms are adopted from Mr. Willis's work on Mechanism. COMPARISON OF VARIED MOTIONS. 38 (4.) The angle made by the intersection of those two planes with a fixed direction in the fixed plane. Thus, the comparative motion of two points relatively to a third, is expressed by means of one of those groups of four elements which Sir William Rowan Hamilton has called "quaternions" In most of the practical applications of cinematics, the motions to be compared are limited by conditions which render the comparison more simple than it is in the general case just described. In machines, for example, the motion of each point is limited to two directions, forward or backward in a fixed path; so that the comparative motion of two points is sufficiently expressed by means of the velocity ratio, together with a directional relation expressed by + or , according as the motions at the instant in question are similar or contrary. SECTION 3. Varied Motion of Points. 359. Velocity and Direction of" Varied Motion. The motion of one point relatively to another may be varied, either by change of velocity, or by change of direction, or by both combined, which last case will now be considered, as being the most general. In fig. 178, let O represent a point assumed as fixed, O X, O Y, O Z, fixed directions, and A B part of the path or orbit traced by a second point in its varied motion relatively to O. At the Fig " 178 ' instant when the second point reaches a given position, such as P, in its path, the direction of its motion is obviously that of P T, a tangent to the path at P. To find the velocity at the instant of passing P, let A t denote an interval of time which includes that instant, and A s the distance traced in that interval. Then is an approximation to the velocity at the instant in question, which will approach, continually nearer and nearer to the exact velocity as the interval A t and the distance A s are made shorter and shorter ; and the limit towards which converges, as A s and A t are inde- A t finitely diminished, and which is denoted by ds f , \ v = dT ............................... (L) 2o 386 PRINCIPLES OP CINEMATICS. is the exact velocity at the instant of passing P. This is the process called "differentiation.'" Should the velocity at each instant of time be known, then the distance s l s , described during an interval of time ^ t Q , is found by integration (see Article 81), as follows : SI -S Q = / vdt (2.) 360. Components of Varied Motion. All the propositions of the two preceding sections, respecting the composition and resolution of motions, are applicable to the velocities of varied motions at a given instant, each such velocity being represented by a line, such as P T, in the direction of the tangent to the path of the point which moves with that velocity, at the instant in question. For example, if the axes OX, O Y, O Z, are at right angles to each other, and if the tangent P T makes with their directions respec- tively the angles , /3, y, then the three rectangular components of the velocity of the point parallel to^fchose three axes are v cos a.', v cos /3; v cos y. Let x, y, z } be the co-ordinates of any point, such as P, in the path A P B, as referred to the three given axes. Then it is well known that d x d y d z cos , = j cos /3 = cos y = ; ds ds ds and consequently the three components of the velocity v are d x d y d z vcos a = : v cos/3 = z. v cosy = ; (3.} dt dt dt 3 and these are related to their resultant by the equation 361. Uniformly-varied Velocity. Let the velocity of a point either increase or diminish at an uniform rate; so that if t repre- sents the time elapsed from a fixed instant when the velocity was v , the velocity at the end of that time shall be v = v + at; ............................ (1.) a being a constant quantity, which is the rate of variation of the velocity, and is called acceleration when positive, and retardation when negative. Then the mean velocity during the time t is VARIATION OF VELOCITY DEVIATION. 387 and the distance described is = *< + ............................ (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 l9 A s 2 , described in two equal intervals of time, each equal to A, before and after the instant in question, be observed. Then the velocity at the instant between those inter- vals is . ....... and its rate of variation is ~ At 362. Varied Rate of Variation of Velocity. When the velocity of a point is neither constant nor uniformly- varied, its rate of variation may still be found by applying to the velocity the sam^, 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 2 s m dt dt 2 ' and is the limit to which the quantity obtained by means of the formula 5 of Article 361 continually approximates, as the interval denoted by A t is indefinitely diminished. 363. uniform Deviation is the change of motion of a point which moves with uniform velocity in a circular path. The rate at which uniform deviation takes place is determined in the following manner. Let C, fig. 179, be the centre of the cir- cular path described by a point A with an uniform velocity v, and let the radius C A be denoted by r. At the beginning and end of an interval of time A t, let A, and A 2 be the positions of the moving point. Then the arc A! A 2 = v A t ; and , chord the chord A t A* = v A t arc The velocities at AI and A 2 are represented by the equal lines 388 PRINCIPLES OF CINEMATICS. | = A 2 V 2 = v, toucliing the circle at A! and A 2 respec- tively. From A 3 draw A 2 v equal and parallel to A, V 17 and join V 2 v. Then the velocity A 2 Y 2 may be considered as compounded of A* v and v V 2 ; so that v V 2 is the deviation of the motion dur- ing the interval A; and because the isosceles triangles A,vV 2J C A! A, are similar : A7T 2 A, A 2 v*'At chord ~-'~- and the approximate rate of that deviation is v 2 chord r * arc J but the deviation does not take place by instantaneous changes of velocity, but by insensible degrees ; so that the true rate of deviation is to be found by finding the limit to which the approximate rate continually approaches as the interval At is diminished indefinitely. V* Now the factor remains unaltered by that diminution ; and the r ratio of the chord to the arc approximates continually to equality ; so that the limit in question, or true rate of deviation, is expressed by r- ................................. <> 364. Varying Deviation. When a point moves with a varying velocity, or in a curve not circular, or has both these variations of motion combined, the rate of deviation at a given instant is still represented by equation 1 of Article 363, provided v be taken to denote the velocity, and r the radius of curvature of the path, of the' point at the instant in question. 365. The Resultant Rate of Variation of the motion of a point is found by considering the rate of variation of velocity and the rate of deviation as represented by two lines, the former in the direction of a tangent to the path of the point, and the latter in the direction of the radius of curvature at the instant in question, and taking the diagonal of the rectangle of which those two lines are the sides, which has the following value : ^ff (-<> 366. The Ratc of Variation of the Component Velocities of a point parallel to three rectangular axes, are represented as follows : COMPARISON OF VARIED MOTIONS. 389 d z x d"~y d*z ' and if a rectangular parallelepiped be constructed, of which the edges represent these quantities, its diagonal, whose length is (S)'} ............ will represent the resultant rate of variation, already given in another form in equation 1 of Article 365. 367. The Comparison of the Varied Motions of a pair of points relatively to a third point assumed as fixed, is made by finding the ratio of their velocities, and the directional relation of the tangents of their paths, at the same instant, in the manner already described in Article 358 as applied to uniform motions. It is evident that the comparative motions of a pair of points may be so regulated as to be constant, although the motion of each point is varied, pro- vided the variations take place for both points at the same instant, and at rates proportional to their velocities. 390 CHAPTER IL MOTIONS OF KIGID BODIES. SECTION 1. Rigid Bodies, and their Translation. 368. The term Rigid Body is to be understood to denote a body, or an assemblage of bodies, or a system of points, whose figure undergoes no alteration during the motion which is under con- sideration. 369. Translation or Shifting is the motion of a rigid body rela- tively to a fixed point, when the points of the rigid body have no motion relatively to each other ; that is to say, when they all move with the same velocity and in the same direction at the same instant, so that no line in the rigid body changes its direction. It is obvious that if three points in the rigid body, not in the same straight line, move in parallel directions with equal velocities at each instant, the body must have a motion of translation. The paths of the different points of the body, provided they are all equal and similar, and at each instant parallel, may have any figure whatsoever. SECTION 2. Simple Rotation. 370. Rotation or Turning is the motion of a rigid body when lines in it change their direction. Any point in or rigidly attached to the body may be assumed as a fixed point to which to refer the motions of the other points. Such a point is called centre of rotation. 371. Axis of Rotation. THEOREM. In every possible 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?, B 2 . Join Aj A 2 , Bj B 2 , forming the isosceles trian- gles O Aj A 2, Bj B,. Bisect the bases of those triangles in C and SIMPLE ROTATION. 391 D respectively, and through the points of bisection draw two planes perpendicular to the respective bases, intersecting each other in the straight line O E, which must traverse O. Let E be any point in the line OE; then EAiAj, and EB : B 2 , are isosceles triangles; and E is at the same distance from O, A, and B, before and after the turning; therefore E is one and the same point in the body, whose place is unchanged by the turning; and this demonstration applies to every point in the straight line O E; therefore that line is unchanged in direction. Q. E. D. COROLLARY. It is evident that every line in the body, parallel to the axis, has its direction unchanged. 372. The Plane of Rotation is any plane perpendicular to the axis. The Angle of Rotation, or angular motion, is the angle made by the two directions, before and after the turning, of a line per- pendicular to the axis. 373. The Angular Velocity of a turning body is the ratio of the angle of rotation, expressed in terms of radius, to the number of units of time in the interval of time occupied by the angular motion. Speed of turning is sometimes expressed also by the number of turns or fractions of a turn in a given time. The relation between these two modes of expression is the following : Let a be the angular velocity, as above defined, and T the turns in the same unit Gf time; then (2 * = 6-2831852). 374. Uniform Rotation consists in uniformity of the angular velo- city of the turning body, and constancy of the direction of its axis of rotation. 375. Rotation common to all Parts of Body. Since the angular motion of rotation consists in the change of direction of a line in a plane of rotation, and since that change of direction is the same how short soever the line may be, it is evident that the condition of rotation, like that of translation, is common to every particle, how small soever, of the turning rigid body, and that the angular velocity of turning of each particle, how small soever, is the same with that of the entire body This is otherwise evident by con- sidering, that each part into which a rigid body can be divided turns completely about in the same time with every other part, and with the entire body. 376. Right and Lcft-Handed Rotation. The direction of rotation round a given axis is distinguished in an arbitrary manner into 392 PRINCIPLES OF CINEMATICS. right-handed and left-handed. One end of the axis is chosen, as that from which an observer is supposed to look along the direction of the axis towards the rotating body. Then if the body seems to the observer to turn in the same direction in which the sun seems to revolve to an observer north of the tropics, the rotation is said to be right-handed ; if in the contrary direction, left-handed : and it is usual to consider the angular velocity of right-handed rotation to be positive, and that of left-handed rotation to be negative ; but this is a matter of convenience. It is obvious that the same rotation which seems right-handed when looked at from one end of the axis, seems left-handed when looked at from the other end. 377. Relative Motion of a Pair of Points in a Rotating Body. Let 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 traversing O is one of revolution, or translation in a circular path of the radius r j 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 Oia v = ar ; (1.) and the direction of that velocity is at each instant perpendicular to the plane drawn through A and the axis. The rate of deviation of A in its motion relatively to the given axis is ? = *'; (2.) in which the first expression is that already found in Article 363, and the second is deduced from the first by the aid of equation 1 of this Article. It is evident that for a given rotation the motion of O relatively to an axis of rotation traversing A is exactly the same with that of A relatively to a parallel axis traversing O ; for it depends solely on the angular velocity a, the perpendicular distance r of the moving point from the axis, and the direction of the axis ; all which are the same in either case. r is called the radius-vector of the moving point. 378. Cylindrical Surface of Equal Velocities. If a Cylindrical surface of circular cross section be described about an axis of rota- tion, all the points in that surface have equal velocities relatively to the axis, and the direction of motion of each point in the cylin- MOTIONS OF POINTS IN A ROTATING BODf. 393 drical surface relatively to tlie 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 be considered as fixed, and let an axis of rotation be drawn through it. Then the comparative motions of A and B relatively to that axis are expressed as follows : the velocity-ratio is that oftlie radii- vectores of the points, and the directional relation consists in the angle between their directions of motion being the same ivith that between their radii-vector es. Or symbolically : Let r }) r%, be the per- pendicular distances of A and B from the axis traversing O, and v l and v% their velocities ; then v 2 r 2 A A ' 2 2 = ', and vi n 380. Components of Velocity of a Point in a Rotating Body. The component parallel to an axis of rotation, of the velocity of a point in a rotating body relatively to that axis, is null. That velocity may be resolved into components in the plane of rotation. Thus let 0, in fig. 181, represent an axis of rotation of a body whose plane of rota- tion is that of the figure ; and let A be any point in the body whose radius- vector is O A = r. The velocity of that point being v = a r, let that velocity be repre- sented by the line A V perpendicular to O A. Let B A be any direction in the plane of rotation, along which it is desired to find the component of the velocity of A ; and let ^ Y A U = * be the angle made by that line with A Y From Y let fall Y U perpendicular to B A ; then A U represents the component in question , and denoting it by u, ^(, = v ' cos 6 = ar cos $ ..................... (1.) From O let fall B perpendicular to B A. Then ^L A O B = ^ Y A U = 6 ; and the right-angled triangles B A and A U Y are similar ; so that AY : ALT : : OA : towards T. The following limited cases : I. When the tracing-point is the surface of the rolling cylinder, r = 2 r a cos 6 ; and therefore, CURVATURE OF EPITROCHOIDS. 403 which is the radius of curvature of an epicycloid. II. When a cylinder rolls on a plane, r^ becomes infinitely great as compared with r 2 , and thus reduces equation 3 to p = - r - - ......................... (5.) r 2 cos 6 r winch is the radius of curvature of a trochoid. III. When a cylinder rolls on a plane, and the tracing-point is in the surface of the cylinder, r = 2 r z cos 0, and p = 2r = 4r 2 cos 6, ..................... (6.) which is the radius of curvature of a cycloid. IY. When a plane rolls on a cylinder, r 2 becomes infinitely great as compared with ^ and r; and equation 3 becomes which is the radius of curvature of a spiral of the class mentioned in Article 387. V. When a plane rolls on a cylinder, and the tracing-point is in tJie plane, cos 6 = and equation 7 becomes which is the radius of curvature of the involute of a circle. "VI. When a plane rolls on a cylinder, and the tracing-point is at the distance r r from the plane on the side next the cylinder, cos 6 = -- - j and equation 7 takes the following form : < 9 -> which is the radius of curvature of an Archimedean spiral. Let B, be the distance of a point in that spiral from the fixed axis O ; then and (9A>) As to rolling curves in general, see Professor Clerk Maxwell's paper in the Transactions of the Eoyal Society of Edinburgh, vol. xvi. 404 PRINCIPLES OF CINEMATICS. 391. Equal and Opposite Parallel Rotations Combined. Let a plane C rotate with an angular velocity a about an axis O con- tained in the plane, and let a rigid body rotate about the axis C in that plane parallel to O, with an angular velocity a, equal and opposite to that of the plane. Then the angular velocity of the rigid body is nothing ; that is, its motion is one of translation only, all its points moving in equal circles of the radius O C, with the velocity a O C. This case is not capable of being represented by a rolling action. 392. Rotations about Intersecting Axes Combined. In fig. 190, let O A be an axis assumed as fixed ; and about it let the plane A O C rotate with the angular velocity a. Let C be an axis in the rotating plane ; and about that axis let a rigid body rotate with the angular velocity b re- latively to the rotating plane. Because the point O in the rigid body is fixed, the instantaneous axis must traverse that point. The direction of that axis is determined, as before, by considering that each point which arrives at that line must have, in virtue of the rotation about O C, a velocity relatively to the rotating plane, equal and directly opposed to that which the coincident point of the rotating plane has. Hence it follows, that the ratio of the per- pendicular distances of each point in the instantaneous axis from the fixed and moving axes respectively that is, the ratio of the sines of the angles which the instantaneous axis makes with the fixed and moving axes must be the reciprocal of the ratio of the component angular velocities about those axes ; or symbolically, if O T be the instantaneous axis, Fig. 190. sin A O T : sin C T : : b : a ,(1.) This determines the direction of the instantaneous axis, which may also be found by graphic construction as follows : On O A take O a proportional to a ; and on O C take O b proportional to b. Let those lines be taken in such directions, that to an observer looking from their extremities towards O, the component rotations seem both right-handed. Complete the parallelogram O b c a ; the dia- gonal c will be the instantaneous axis. The resultant angular velocity about this instantaneous axis is found by considering, that if C be any point in the Jnoving axis, the linear velocity of that point must be the same, whether com- puted from the angular velocity a of the rotating plane about the ROLLING CONES ANALOGY OP ROTATIONS AND FORCES. 40^ fixed axis A, or from the resultant angular velocity c of the rigid body about the instantaneous axis. That is to say, let CD, C E, be perpendiculars from C upon O A, T, respectively ; then = c -CE; but CD : CTE : : sin ^i AOC : sin ^C T 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 : sin^rAOT :sin^:AOC} : : _^ : 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 otJier two ; and the diagonal oftJie parallelogram b c a repre- sents both the direction of the instantaneous axis and the angular velo- city about that axis. 393. Roiling c one*. All the lines which successively come into the position of instantaneous axis are situated in the surface of a cone described by the revolution of O T about C ; and all the positions of th instantaneous axis lie in the surface of a cone described by the revolution of OT about OA. Therefore the motion of the rigid body is such as would be produced by the roll- ing of the former of those cones upon the latter. It is to be understood, that either of the cones may become a flat disc, or may be hollow, and touched internally by the other. For example, should ^ A O T become a right angle, the fixed cone would become a flat disc j and should ^ A T become obtuse, that cone would be hollow, and would be touched internally by the rolling cone ; and similar changes may be made in the rolling cone. The path described by a point in or attached to the rolling cone is a spherical epitrochoid; but for the purposes of the present trea- tise, it is unnecessary to enter into details respecting the properties of that class of curves. 394. Analogy of Rotations and Single Forces. If the proportional equation 3 of Article 388, which shows the relations between the component angular velocities of rotation about a pair of parallel axes, the resultant angular velocity, and the position of the instan- taneous axis, be compared with the proportional equation of Article 39, by means of which, as explained in Article 40, the magnitude and position of the resultant of a pair of parallel forces are found, it will be evident that those equations are exactly analogous. The result of the combination of a rotation with a translation in 406 PRINCIPLES OF CINEMATICS. the same plane, in producing a rotation of equal angular velocity about an instantaneous axis at a certain distance to one side of the moving axis, as explained in Article 385, is exactly analogous to the result of the combination of a single force with a couple in pro- ducing an equal single force transferred laterally to a certain dis- tance, as explained in Article 41. The result of the combination of two equal and opposite rotations about parallel axes, in producing a translation with a velocity which is the product of the angular velocity into the distance between the axes, as explained in Article 391, is exactly analogous to the production of a couple by means of a pair of equal and oppo- site forces, as explained in Article 25. The result of the combination of two rotations about intersecting axes, as explained .in Article 392, is exactly analogous to the result of the combination of a pair of inclined forces acting through one point, as explained in Article 51. The combination of a rotation about a given axis with a transla- tion parallel to the same axis, as explained in Article 382, is exactly analogous to the combination of a force acting in a given line with a couple whose axis is parallel to the same line, as explained in Article 60, cases 4 and 5. It thus appears, that just as the composition and resolution of translations are exactly analogous to the composition and resolution of couples, so the composition and resolution of rotations are exactly analogous to the composition and resolution of single forces; that is to say, if lines be taken, representing in direction axes of rotation, and in length the angular velocities of rotation about such axes, all mathematical theorems which are true of lines representing single forces are true of such lines representing rotations : and if with this be combined the principle, that all mathematical theorems which are true of lines representing in direction the axes and in length the moments of couples are true also of lines representing the velocities and directions of translations, all problems of the resolution and composition of motions may be solved by referring to the solutions of analogous problems of statics. 395. Comparative motions in Compound Rotation. The velocity - ratio of two points in a rotating rigid body at any instant is that of their perpendicular distances from its instantaneous axis ; and the angle between the directions of motion of the two points is equal to that between the two planes which traverse the points and the instantaneous axis. SECTION 4. Varied Rotation. 396. variation of Angular Telocity 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 (L the interval of time, , converges, as the length of the interval is indefinitely diminished ; being represented by da 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 Sit the lines of contact of rolling cylinders and cones with bases not circular ; and that r v r z , in each case, represent the values of the variable radii of curvature of non-circular cylinders at their lines of contact, and ^ A O T, ^ COT, the variable angles of obliquity of the osculating circular cones of non-circular cones. 398. Components of Varied Rotation. The most convenient way, in most cases, of expressing the mode of variation of a rotatory motion, is to resolve the angular velocity at each instant into three component angular velocities about three rectangular axes fixed in direction. The values of those components, at any instant, show at once the resultant angular velocity, and the direction of the instan- taneous axis. For example, let a,., a y ,*a f , 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(<4+aJ + <*;) ...................... (1.) is the resultant angular velocity, and cos=2?j cos /3 = 2r; cos y = -'; ............... (2.) a a a are the cosines of the angles which the instantaneous axis makes with the axes of x, y and z, respectively. 403 CHAPTER III. MOTIONS OF PLIABLE BODIES, AND OP FLUIDS. 399. Division of the Subject. The subject of the present chapter, so far as it comprehends the relative motions of the points of pliable solids, has been already treated of in those portions of the Third Chapter of Part II. which relate to strains. There remain now to be considered the following branches : I. The Motions of Flexible Cords. II. The Motions of Fluids not altering in Yolume. III. The Motions of Fluids altering in Yolume. 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- ing principles : I. The length between each pair of points in the cord is constant. II. That length is itie 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 IY. of this treatise. Next in point of frequency in practice are the problems to be considered in the ensuing Article. 402. Cord Guided by Surfaces of Revolution. Let a COrd in some portions of its course be straight, and in others guided by the sur- faces of circular drums or pulleys, over each of which its track is a circular arc in a plane perpendicular to the axis of the guiding surface. Let r be the radius of any one of the guiding surfaces, i the angle of inclination which the two straight portions of the cord contiguous to that surface make with each other, expressed in length of arc to radius unity. Then the length of the portion of the cord which lies on that surface is r i ; and if s be the length of any straight portion of the cord, the total length between two given points fixed in the cord may be expressed thus : L = 2 + s 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 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 / + cos/) ; (3.) and consequently, if u represent the velocity of translation of the 410 PRINCIPLES OF CINEMATICS. guiding surface in the given direction, and v the longitudinal velo- city of the point B in the cord, and if any number of guiding surfaces between A and B be trans- lated, each in its own direction, V = 2 ' U (COS^+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 j = cos/ = 1 ; and v = 2nu .............................. (6.) SECTION 2. Motions of Fluids of Constant Density. 403. Velocity and Flow. The density of a moving fluid mass may be either exactly invariable, from the constancy or the adjust- ment of its temperature and pressure, or sensibly invariable, from the smallness of the alterations of volume which the actual altera- tions of pressure and temperature are capable of producing. The latter is the case in most problems of practical mechanics affecting liquids. Conceive an ideal surface of any figure, and of the area A, to be situated within a fluid mass, the parts of which have motion rela- tively to that surface ; and let u denote, as the case may be, the uni- form velocity, or flie mean value of the varying velocity, resolved in a direction perpendicular to A, with which the particles of the fluid pass A. Then Q = uA ............................. (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 \heflow, or rate of flow, through A. When the particles of fluid move obliquely to A, let ^ denote the angle which the direction of motion of any particle passing A makes with a normal to A, and v the velocity of that particle ; then u = v ' cos & ............................ (2.) When the velocity normal to A varies at different points, either from the variation of v, or of 6, or of both, the flow may also be expressed as follows : Let A be divided into indefinitely small elements, each of which is represented by d A ; then MOTIONS OF FLUIDS OF CONSTANT DENSITY. 411 Q = [ udA. = I v cos 6 dAj (3.) and if we now distinguish, the mean normal velocity from the velocity at any particular point by the symbol u , we have, (4.) 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 2-Q = (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 of these sections. 406. Pipes, Channels, Currents, and Jets. When a stream of fluid completely fills a pipe or tube, the area of each cross section is given by the figure and dimensions of the pipe, and for similar forms of section varies as the square of the diameter. Hence the mean normal velocities of a stream flowing in a full pipe, at differ- ent cross sections of the pipe, are inversely as the squares of the diameters of those sections. A channel partially encloses the stream flowing in it, leaving the upper surface free ; and this description applies not only to chan- 412 PRINCIPLES OF CINEMATICS. 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. 409. steady Motion of a fluid relatively to a given space considered as fixed is that in which the velocity and direction of the motion of the fluid at each fixed point is uniform at every instant of the time under consideration ; so that although the velocity and direction of the motion of a given particle of the fluid may vary while it is transferred from one point to another, that particle assumes, at each fixed point at which it arrives, a certain definite velocity and direction depending on the position of that point alone ; which velocity and direction are successively assumed by ^ach particle which successively arrives at the same fixed point. The steady motion of a stream is expressed by the two conditions, that the area of each fixed cross section is constant, and that the flow through each cross section is constant ; that is to say, dA dQ If u represents the normal velocity of a fluid moving steadily, at a given fixed point, then du /0 \ MOTION OF PISTONS DIFFERENTIAL EQUATIONS. 413 expresses the condition of steady motion. Next, let u represent the normal velocity, not at a given fixed point, but of a given identical particle of fluid; then the variation undergone by u in an indefi- nitely small interval of time, d t, is that arising from its being transferred from one cross section to another, whose distance down the stream from the former is d s = u - d t. Hence, denoting by - d s, the indefinitely small variation of velocity which takes d s place from this cause, and by -~rr, the rate at which that variation takes place, we have d ' u _ du d s _ d u ~dT~ ~Ts'Tt =U ' ~d~s .................. W Most of the problems respecting streams which occur in practice have reference to steady motion. 410. In Unsteady motion, the velocity at each fixed point varies, at a rate denoted by ; and the total rate of variation of the cu t velocity of an individual particle in a stream, being found lay adding together the rates of variation due to lapse of time and to change of position, is expressed by d'u _ du du ds _ du _. du , . ~dt = ~dl~^~ds ' ~dt~~dJ^ 'ITs ...... 1 L ' 411. Motion of Pistons. Let a mass of fluid of invariable volume be enclosed in a vessel, two portions of the boundary of which (called pistons) are moveable inwards and outwards, the rest of the boundary being fixed. Then, if motion be transmitted between the pistons by moving one inwards and the other outwards, it follows, from the invariability of the volume of the enclosed fluid, that the velocities of the two pistons at each instant will be to each other in the inverse ratio of the areas of the respective projections of the pistons on planes normal to their directions of motion. This is the principle of the transmission of motion in the hydraulic press and hydraulic crane. The flow produced by a piston whose velocity is u, and the area of whose projection on a plane perpendicular to the direction of its motion is A, is given, as in other cases, by the equation Q = uA .............................. (1.) 412. General Differential Equations of Continuity. When the motions of a fluid of invariable density are considered in the most 414 PRINCIPLES OF ClSTEMATICS. general way, the principle of continuity stated in Article 404 is expressed symbolically in the following manner. The space as- sumed as fixed, to which the motion of the fluid is referred, is con- ceived to be divided into indefinitely small rectangular elementary spaces, each having for its linear dimensions, d x, dy, dz, and for the areas of its three pair* of faces, dy dz, dzdx, dxdy. Let x, x + dx, be the co-ordinates of the pair of faces, dy dz ; y,V + dy, ,> dzdx; z,z + dz, dxdy. Let the velocity of the particles of water at any point be resolved into three rectangular components, u, v, w, parallel respectively to x, y, z } with proper algebraical signs. Let outward flow be posi- tive, and inward flow negative. The values of the flow for the six faces are as follows : Through the first face dy dz, u'dydz', second face dydz, (u + -7 dx)dydz; first face d z d x, vdzdx; second face dzdx, (v + dy) dzdx; ay first face d x d y, wdxdy, second face dxdy, (w + -y- dz) dxdy. " ' ' '' dx dy dz ' dt which is -the equation of continuity for a fluid of varying density. This equation may be otherwise expressed as follows : fdu . dv . dw\ . ( d . d . d . d\ f ( -= + -y- + -j ) +(^T -- H V -,"- + W -j- + ) a ~ j (2.) \dx dy dz) \ dx dy dz dt} * or dividing by e, du dv dw id 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 imotion are the following : -0- -0-^^-0- ^ e n- m dt~"> dt~ ' dt > Tt = () > ............ ( 3 ') 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 du and similar equations for r , = , and - dt' dt ' dt 424. The Motions of Connected Bodies form the subject of the Theory of Mechanism, to which the Fourth Part of this treatise relates. PART IV. THEORY OF MECHANISM. CHAPTER I. DEFINITIONS AND GENERAL PRINCIPLES. 425. Theory of Pure Mechanism Defined. Machines are bodies, or assemblages of bodies, which transmit and modify motion and force. The word " machine," in its widest sense, may be applied to every material substance and system, and to the material uni- verse itself ; but it is usually restricted to works of human art, and in that restricted sense it is employed in this treatise. A machine transmits and modifies motion when it is the means of making one motion cause another ; as when the mechanism of a clock is the means of making the descent of the weight cause the rotation of the hands. A machine transmits and modifies force when it is the means of making a given kind of physical energy perform a given kind of work ; as when the furnace, boiler, water, and mechanism of a marine steam engine are the means of making the energy of the chemical combination of fuel with oxygen perform the work of overcoming the resistance of water to the motion of a ship. The acts of transmitting and modifying motion, and of transmitting and modifying force, take place together, and are connected by a cer- tain law ; and until lately, they were always considered together in treatises on mechanics ; but recently great advantage in point of clearness has been gained by first considering separately the act of transmitting and modifying motion. The principles which re- gulate this function of machines constitute a branch of Cinematics, called the theory of pure mechanism. The principles of the theory of pure mechanism having been first established and understood, those of the theory of the work of machines, which regulate the act of transmitting and modifying force, are much more readily de- monstrated and apprehended than when the two departments of the theory of machines are mingled. The establishment of the theory of pure mechanism as an independent subject has been mainly accomplished by the labours of Mr. "Willis, whose no- menclature and methods are, to a great extent, followed in this treatise. 422 THEORY OF MECHANISM. 426. The General Problem of the theory of pure mechanism may be stated as follows : Given t/ie mode of connection of two or more moveable points or bodies with each oilier, and with certain fixed bodies ; required iJie comparative motions of the moveable points or bodies : and conversely, when the comparative motions of two or more moveable points are given, to find their proper mode of connec- tion. The term "comparative motion" is to be understood as in Articles 358, 367, 379, and 395. In those Articles, the compara- tive motions of points belonging to one body have already been considered. In order to constitute mechanism, two or more bodies must be so connected that their motions depend on each other through cinematical principles alone. 427. Frame; Moving Pieces; Connectors. The frame of a ma- chine is a structure which supports the moving pieces, and regulates the path or kind of motion of most of them directly. In consider- ing the movements of machines mathematically, the frame is con- sidered as fixed, and the motions of the moving pieces are referred to it. The frame itself may have (as in the case of a ship or of a locomotive engine) a motion relatively to the earth, and in that case the motions of the moving pieces relatively to the earth are the resultants of their motions relatively to the frame, and of the motion of the frame relatively to the earth ; but in all problems of pure mechanism, and in many problems of the work of machines, the motion of the frame relatively to the earth does not require to be considered. The moving pieces may be distinguished into primary and second- ary; the former being those which are directly carried by the frame, and the latter those which are carried by other moving pieces. The motion of a secondary moving piece relatively to the frame is the resultant of its motion relatively to the primary piece which carries it, and of the motion of that primary piece relatively to the frame. Connectors are those secondary moving pieces, such as links, belts, cords, and chains, which transmit motion from one moving piece to another, when that transmission is not effected by immediate contact. 428. Bearings are the surfaces of contact of primary moving pieces with the frame, and of secondary moving pieces with the pieces which carry them. Bearings guide the motions of the pieces which they support, and their figures depend on the nature of those motions. The bearings of a piece which has a motion of transla- tion in a straight line, must have plane or cylindrical surfaces, exactly straight in the direction of motion. The bearings of rotat- ing pieces must have surfaces accurately turned to figures of revolu- BEARINGS MOVING PIECES ELEMENTARY COMBINATION. 423 tion, such as cylinders, spheres, conoids, and flat discs. The bearing of a piece whose motion is helical, must be an exact screw, of a pitch equal to that of the helical motion (Article 382). Those parts of moving pieces which touch the bearings, should have surfaces accurately fitting those of the bearings. They may bo distinguished into slides, for pieces which move in straight lines, gudgeons, journals, bushes, and pivots, for those which rotate, and screws for those which move helically. The accurate formation and fitting of bearing surfaces is of primary importance to the correct and efficient working of machines. Sur- faces of revolution are the most easy to form accurately, screws are more difficult, and planes the most difficult of all. The success of Mr. Whitworth in making true planes, is regarded as one of the greatest achievements in the construction of machinery. 429. The Motions of Primary Moving Pieces are limited by the fact, that in order that different portions of a pair of bearing sur- faces may accurately fit each other during their relative motion, those surfaces must be either straight, circular, or helical; from which it follows, that the motions in question can be of three kinds only, viz : I. Straight translation, or shifting, which is necessarily of limited extent, and which, if the motion of the machine is of indefinite duration, must be reciprocating ; that is to say, must take place alternately in opposite directions. (See Part III., Chapter II., Section 1.) II. Simple rotation, or turning about a fixed axis, which motion may be either continuous or reciprocating, being called in the latter case oscillation. (See Part III., Chapter II., Section 2.) III. Helical or screw-like motion, to which the same remarks apply as to straight translation. (See Part III., Chapter II., Section 3, Article 382.) 430. The Motions of Secondary Moving Pieces relatively to the pieces which carry them, are limited by the same principles which apply to the motions of primary pieces relatively to the frame. But the motions of secondary moving pieces relatively to the frame may be any motions which can be compounded of straight translations and simple rotations according to the principles already explained in Part III., Chapter II., Section 3. 431. An Elementary Combination in mechanism consists of a pair of primary moving pieces, so connected that one transmits motion to the other. The piece whose motion is the cause is called the driver ; that whose motion is the effect, the follower. The connection between the driver and the follower may be I. By rolling contact of their surfaces, as in toothless wheels. 424 THEORY OF MECHANISM. II. By sliding contact of their surfaces, as in toothed wlieels, screws, wedges, cams, and escapements. III. By bands or wrapping connectors, such as belts, cords, and gearing-ctiains. IY. By link-work, such as connecting rods, universal joints, and clicks. V. By reduplication of cords, as in the case of ropes and pulleys. VI. By an intervening fluid, transmitting motion between two pistons. The various cases of the transmission of motion from a driver to a follower are further classified, according as the relation between their directions of motion is constant or changeable, and according as the ratio of their velocities is constant or variable. This latter principle of classification is employed by Mr. Willis as the founda- tion of a primary division of the subject of elementary combinations in mechanism into classes, which are subdivided according to the mode of connection of the pieces. In the present treatise, elemen- tary combinations will be classed primarily according to the mode of connection. 432. Une of Connection. In every class of elementary combina- tions,- except those in which the connection is made by reduplica- tion of cords, or by an intervening fluid, there is at each instant a certain straight line, called the line of connection, or line of mutual action of the driver and follower. In the case of rolling contact, this is any straight line whatsoever traversing the point of contact of the surfaces of the pieces ; in the case of sliding contact, it is a line perpendicular to those surfaces at their point of contact ; in the case of wrapping connectors, it is the centre line of that part of the connector by whose tension the motion is transmitted ; in the case of link- work, it is the straight line passing through the points of attachment of the link to the driver and follower. 433. Principle of Connection. The line of connection of the driver and follower at any instant being known, their comparative velocities are determined by the following principle : The respec- tive linear velocities of a point in t/ie driver, a'/id a point in the fol- lower, each situated anywhere in the line of connection, are to each otJier inversely as the cosines of the respective angles made by tJie paths of the points with the line of connection. This principle might be otherwise stated as follows : The components, along the line of con- nection, of tJie velocities of any two points situated in that line, are equal. 434. Adjustments of Speed. The velocity-ratio of a driver and its follower is sometimes made capable of being changed at will, by means of apparatus for varying the position of their line of connec- tion; as when a pair of rotating cones are embraced by a belt TRAIN AGGREGATE COMBINATIONS. 425 which can be shifted so as to connect portions of their surfaces of different diameters. 435. A Train of mechanism consists of a series of moving pieces, each of which is follower to that which drives it, and driver to that which follows it. 436. Aggregate Combinations in mechanism are those by which compound motions are given to secondary pieces. 426 CHAPTER IL ON ELEMENTAEY COMBINATIONS AND TRAINS OP MECHANISM. SECTION 1. Rolling Contact. 437. Pitch Surfaces are those surfaces of a pair of moving pieces, which touch each other when motion is communicated by rolling contact. The LINE OP CONTACT is that line which at each instant traverses all the pairs of points of the pair of pitch surfaces which are in contact. 438. Smooth Wheels, Rollers, Smooth Backs. Of a pair of pri- mary moving pieces in rolling contact, both may rotate, or one may rotate and the other have a motion of sliding, or straight translation. A rotating piece, in rolling contact, is called a smooth wheel, and sometimes a roller ; a sliding piece may be called a smooth rack. 439. General Conditions of Rolling Contact. The whole of the principles which regulate the motions of a pair of pieces in rolling contact follow from the single principle, that eachpair of points in the pitch surfaces, which are in contact at a given instant, must at that instant be moving in the same direction with the same velocity. The direction of motion of a point in a rotating body being per- pendicular to a plane passing through its axis, the condition, that each pair of points in contact with each other must move in the same direction leads to the following consequences : I. That when both pieces rotate, their axes, and all their points of contact, lie in the same plane. II. That when one piece rotates and the other slides, the axis of the rotating piece, and all the points of contact, lie in a plane per- pendicular to the direction of motion of the sliding piece. The condition, that the velocities of each pair of points of con- tact must be equal, leads to the following consequences : III. That the angular velocities of a pair of wheels, in rolling contact, must be inversely as the perpendicular distances of any pair of points of contact from the respective axes. IY. That the linear velocity of a smooth rack in rolling contact with a wheel, is equal to the product of the angular velocity of the wheel by the perpendicular distance from its axis to a pair of points of contact. CIRCULAR WHEELS STRAIGHT RACK. 427 Respecting the line of contact, the above principles III. and IV. lead to the following conclusions : V. That for a pair of wheels with parallel axes, and for a wheel and rack, the line of contact is straight, and parallel to the axes or axis ; and hence that the pitch surfaces are either plane or cylin- drical (the term " cylindrical" including all surfaces generated by the motion of a straight line parallel to itself). YI. That for a pair of wheels, with intersecting axes, the line of contact is also straight, and traverses the point of intersection of the axes ; and hence that the rolling surfaces are conical, with a common apex (the term " conical" including all surfaces generated by the motion of a straight line which traverses a fixed point). 440. Circular Cylindrical Wheels are employed when an uniform velocity-ratio is to be communicated between parallel axes. Figs. 187, 188, and 189, of Article 388, may be taken to represent pairs of such wheels ; C and O, in each figure, being the parallel axes of the wheels, and T a point in their line of contact. In fig. 187, both pitch surfaces are convex, the wheels are said to be in outside gearing, and their directions of rotation are contrary. In figs. 188 and 189, the pitch surface of the larger wheel is concave, and that of the smaller convex ; they are said to be in inside gearing, and their directions of rotation are the same. To represent the comparative motions of such pairs of wheels symbolically, let be their radii : let O = c be the line of centres, or perpendicular distance between the axes, so that for y Let a v a 2 , be the angular velocities of the wheels, and v the common linear velocity of their pitch surfaces ; then the sign applying to gearing. 441. A Straight Rack and Circular Wheel, which are used when an uniform velocity-ratio is to be communicated between a sliding piece and a turning piece, may be represented by fig. 185 of Article 385, C being the axis of the wheel, P T P the plane surface of the rack, and T a point in their line of contact. Let r be the radius of the wheel, a its angular velocity, and v the linear velocity of the rack ; then v = r a. 428 THEORY OF MECHANISM. 442. Bevel Wheels, whose pitch surfaces are frustra of regular cones, are used to transmit an uniform angular velocity-ratio between a pair of axes which intersect each other. Pig. 190 of Article 392 will serve to illustrate this case; O A and O C being the pair of axes, intersecting each other in O, O T the line of con- tact, and the cones described by the revolution of O T about O A and O C respectively being the pitch surfaces, of which narrow zones or frustra are used in practice. Let ,, a 2j be the angular velocities about the two axes respec- tively; and let ^ = ^ A O T, i 2 = ^_ C O T, be the angles made by those axes respectively with the line of contact ; then from the principle III. of Article 439 it follows, that the angular velocity- ratio is 2 _ sin i, ~ r ....... 11. ) CTJ sin i. 2 Which equation serves to find the angular velocity-ratio when the axes and the line of contact are given. Conversely, let the angle between the axes, ^ A O C = % -f i 2 =j, CL be given, and also the ratio ; then the position of the line of a i contact is given by either of the two following equations : a. 2 sinj fw f*f\c* n\ .(2.) sin t 1 = sin ^o = J (a! -f ai + 2 a { a. 2 cosj) } ai sin 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 b C 3 , 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,, NOX-CIRCULAR WHEELS. 429 TV, IT,, Uo, 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 IJ 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 { , r 2 , be the radii vectores of a pair of points which touch each other; ds^ ds. z , a pair of elementary arcs of the cross sections T! Uu T 2 TJ 2 , of the pitch surfaces, and c the line of centres or distance between the axes. Then (2.) If one of the wheels be fixed and the other be rolled upon it, a point in the axis of the rolling wheel describes a circle of the radius c round the axis of the fixed wheel. The equations 1 and 2 are made applicable to inside gearing by putting instead of + and + instead of . The angular velocity-ratio at a given instant has the value (3.) As examples of non-circular wheels, the following may be mentioned : I. An ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity-ratio varying from 1 excentricity 1 + excentricity 1 + excentricity 1 excentricity* II. A hyperbola rotating about its farther focus, rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hyperbolas, and the velocity- ratio varying between excentricity + 1 , . .. = and unity. excentricity 1 * 430 THEORY OF MECHANISM. III. Two logarithmic spirals of equal obliquity rotate in rolling contact with each other through an indefinite angle. (For further examples of non-circular wheels, see Professor Clerk Maxwell's paper on Rolling Curves, Trans. Roy. Soc. Edin., vol. xvi., and Professor "Willis's work on Mechanism.) SECTION 2. Sliding Contact. 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- ,f ing about axes A B, C D. In fig. 193 are shown a pair of narrow zones of the same figures, such as are employed in practice. A hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight line round an axis from which it is at a constant distance, and to which it is inclined at a constant angle. If two such hyperboloids, equal or unequal, be placed in the closest possible contact, as in fig. 192, they will touch each other along one of the generating straight lines of each, which will form their line of contact, and will be inclined to the axes A B, C D, in opposite directions. The axes will neither be parallel, nor will they intersect each other. The motion of two such hyperboloids, rotating in contact with each other, has sometimes been classed amongst cases of rolling contact; but that classification is not strictly correct; for although the component velocities of a pair of points of contact in a direction at right angles to the line of contact are equal, still, as the axes are neither parallel to each other nor to the line of contact, the velocities of a pair of points of contact have components along the line of SKEW-BEVEL WHEELS GROOVED WHEELS. 431 contact, which are unequal, and their difference constitutes a lateral sliding. The directions and positions of the axes being given, and the required angular velocity-ratio, , it is required to find the obli- a\ quities of the generating line to the two axes, and its radii vectores, or least perpendicular distances from these axes. In fig. 194, let A B, C D, be the two axes, and G K their common perpendicular. 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 Une 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 : K~H ; 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 : ; the second, by the rotation of the same* line about the axis C D with the radius vector H K = r 2 . To draw the hyperbola which is the longitudinal section of a skew-bevel wheel whose generating line has a given radius vector and obliquity, let A G B, fig. 195, re- present the axis, G H _L A G B, the radius vector of the generating line, and let the straight line E G F make with the axis an angle equal to the obliquity of the generating line. H will be the vertex, and E G F one of Fi S- 195 - the asymptotes, of the required hyperbola. To find any number of points in that hyperbola, proceed as follows : Draw X W Y parallel to G H, cutting G E in W, and make XY = J (G~H 2 + XW~ 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 432 THEORY OF MECHANISM. section illustrating the kind of frictional gearing invented by Mr. Robertson. The comparative motion of a pair of wheels* thus I ridged and grooved is nearly the same with that I of a pair of smooth wheels in rolling contact, V A A A / having cylindrical or conical pitch surfaces lying V V V V midway between the tops of the ridges and bottoms of the grooves. Fig 196 ^ ie re l a ti ve 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 ihe 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 plane perpendicular to the axis j 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, 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 bs an aliquot part of the circumference. In wheels which reciprocate without performing a complete re- volution, this condition is not necessary. Such wheels are called sectors. II. In order that a pair of wheels, or a wheel and a rack, may work correctly together, it is in all cases essential that the pitch should be the same in each. III. Hence, in any pair of circular wheels which work together, the numbers of teeth in a complete circumference are directly as the radii, and inversely as the angular velocities. IY. 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. Y. 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 IS" and n be d } a num- ber less than n } so that n = m d, N = M d, then a = mN = Mrc = M.md ; 6 = M; c = m ......... (2.) CASE 3. If N and n be prime to each other, 2F 434 THEORY OF MECHANISM. a = Nnj b = N"; c = % (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 ** 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 OP 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 j let the numbers of teeth in the driving wheels be denoted by N's, each with the number of its axis affixed ; thus, Nj, N 2 , &c ..... N_! ; 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 2 , n s , &c ..... n m . Then the ratio of the angular velocity a m of the m th axis to the angular velocity a of the first axis is the product of the m l velocity-ratios of the succes- sive elementary combinations, viz. : &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, m, it is evident that if m is odd, the direction of rotation is preserved, and if even, reversed. It is often a question of importance to determine the numbers of teeth in a train of wheels best suited for giving a determinate velocity-ratio to two axes. It was shown by Young, that to do this with the least total number of teeth, the velocity-ratio of each elementary combination should approximate as nearly as possible 3*59. This would in many cases give too many axes; and as a useful practical rule it may be laid down, that from 3 to 6 ought to be the limit of the velocity-ratio of an elementary combination in wheelwork. T> Let be the velocity-ratio required, reduced to its least terms, O and let B be greater than C. T> If is not greater than 6, and C lies between the prescribed O mininmm 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 will themselves be the numbers required. Should B and C be inconveniently large, they are if possible to be resolved into factors, 436 THEORY OF MECHANISM. and those factors, or if they are too small, multiples of them, used for the numbers of teeth. Should B or C, or both, be at once incon- T> veniently large, and prime, then instead of the exact ratio , some 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, ra 1, will lie between and - ...... log 6 log 3 Then, if possible, B and C themselves are to be resolved each into w 1 factors (counting 1 as a factor), which factors, or multiples of them, shall be not less than t, nor greater than 6 t; 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. \j 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 C^ C 2 , represent the axes of rotation of the two pieces; A,, A 2 , two portions of their respective surfaces; and T^ Tz, a pair of points in those surfaces, which, at the instant under consideration, are in contact with each other. Let P! P 2 be the common perpendicular of the surfaces at the pair of points Tj, T 8 ; 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 P,, P 2 , are equal. Let t,, i 2 , be the angles which that line makes with the direc- tions of the axes respectively. Let a*, a 2 , be the respective angular velocities of the moving pieces ; then sin i x = a a C 2 P 2 sin i 2 ; consequently, a 2 Cj PI sin tj m (1.) i G 2 P 2 sini 2 which is the principle stated above. When the line of action is perpendicular in direction to both axes, then sin ^ = sin i 2 = 1 ; and equation 1 becomes G.P! C 2 P 2 * .(1 A.) When the axes are parallel, i, = i. 2 . Let I be the point where the line of action cuts the plane of the two axes ; then the triangles P! GI I, P 2 C 2 1, are similar; so that equation 1 A is equivalent to the following : ,(1 B.) <*i 1G 2 CASE 3. A rotating piece and a shifting piece, in sliding contact, have their comparative motion regulated by the following prin- ciple : Let C P denote the perpendicular distance from the axis of the rotating piece to the line of action ; i the angle which the direc- tion of the line of action makes with that axis; a the angular velocity of the rotating piece; v the linear velocity of the sliding piece ; j the angle which its direction of motion makes with the line of action ; then v = a G P 'sini -secj (2.) When the line of action is perpendicular in direction to the axis of the rotating piece, sin i = 1 ; and = a -CP (2 A.) where I C denotes the distance from the axis of the rotating piece 438 THEOllY 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 Backs. Oeneral 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 tlieir 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 Cj, I C 2 , of the line of centres; and this also is the proportion of the angular velocities given by a pair of surfaces in sliding contact whose line of action traverses the point I (Article 450, case 2, equation 1 B). Hence the condition of correct working for the teeth of wheels with parallel axes is, that tlie line of action of the teeth sJiall at every instant traverse the line of contact of the pitch surfaces; and the same condition obviously applies to a rack sliding in a direction perpendicular to that of the axis of the wheel with which it works. 452. Teeth Described by Rolling Curves. From the principle of the preceding Article it follows, that at every instant, the position of the point of contact Tj in the cross section of the acting surface of a tooth (such as the line A! TJ in fig. 197), and the corresponding position of the pitch point I in the pitch line I B x 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, at the point Tj. Now this is the relation which exists between the tracing- point Tj, and the instantaneous axis or line of contact I, in a rolling curve of such a figure, that being rolled upon the pitch surface B J? its tracing-point T! 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,; ........................... (i.) a-nd this condition is fulfilled, iftlie outlines of the two teeth be traced by the motion of the same tracing-point, in rolling the same rolling curve on the same side of the pitch surfaces of the respective wheels. Thejlank of a tooth is traced while the rolling curve rolls inside of the pitch line; the face, while it rolls outside. Hence it is TEETH DESCRIBED SLIDING OF TEETH. 439 evident that the flanks of the teeth of the driving wheel drive the faces of the teeth of the driven wheel; and that the faces of the teeth of the driving wheel drive the flanks of the teeth of the driven wheel. The former takes place while the point of contact of the teeth is approaching the pitch point, as in fig. 197, supposing the motion to be from P x towards P a ; the latter, after the point of contact has passed, and while it is receding from, the pitch point. The pitch point divides the path of the point of contact of the teeth into two parts, called the path of approach and the path of recess; and the lengths of those paths must be so adjusted, that two pairs of teeth at least shall be in action at each instant. It is evidently necessary that the surfaces of contact of a pair of teeth should either be both convex, or that if one is convex and the other concave, the concave surface should have the natter curvature. The equations of Article 390 give the relations which exist between the radius of curvature of a pitch line at the pitch point (rj, the radius of curvature of the rolling curve at the same point (r 2 ), the radius vector of the tracing-point (r\ = I T), the angle made by that line with the line of centres of the fixed and rolling curves (0 = ^L T I C), and the radius of curvature of the curve traced by the point T (^), all at a given instant. When a pair of tooth surfaces are both convex absolutely, that which is a face is concave, and that which is a flank is convex, towards the pitch point; and this is indicated by the values of having contrary signs for the two teeth, being positive for the face and negative for the flank. The face of a tooth is always convex absolutely, and concave towards the pitch point, ^ being positive; so that if it works with a concave flank, the value of 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 Cj C 2 to rotate backwards round Cj with the angular velocity a 1} 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 2 . Hence the velocity of sliding is that due to this rotation about I, with the radius I T = r; that is to say, its value is r (ai + a 2 ); (1.) so that it is greater, the farther the point of contact is from the 440 THEORY OF MECHANISM. line of centres ; and at the instant when that point, passing the line of centres, coincides with the pitch point, the velocity of sliding is null, and the action of the teeth is, for the instant, that of rolling contact. The roots of the teeth slide towards each other during the ap- proach, and from each other during the recess. To find the amount or total distance through which the sliding takes place, let t { be the time occupied by the approach, and t 2 that occupied by the recess ; then the distance of sliding is j (2.) or in another form, if di denote an element of the change of angu- lar position of one wheel relatively to the other, i^ the amount of that change during the approach, and i 2 during the recess, then (a l -f- a 2 ) d t = d i ; and s= f'rdi + f rdi (3.) JO J (See also Article 455.) 454. The Arc of Contact on the Pitch Unes is the length of that portion of the pitch lines which passes the pitch point during the action of one pair of teeth ; and in order that two pairs of teeth at least may be in action at each instant, its length should be at least double of the pitch. It is divided into two parts, the arc of ap- proach and the arc of recess. In order that the teeth may be of length sufficient to give the required duration of contact, the dis- tance moved over by the point I upon the pitch line during the rolling of a rolling curve to describe the face and flank of a tooth, must be in all equal to the length of the required arc of contact. It is usual to make the arcs of approach and recess equal. 455. The Length of a Tooth may be divided into two parts, that of the face and that of the flank. For teeth in the driving wheel, the length of the flank depends on the arc of approach, that of the face, on the arc of recess ; for those in the following wheel, the length of the flank depends on the arc of recess, that of the face, on the arc of approach. Let q l be the arc of approach, q a that of recess ; , the length of the flank, l\ the length of the face of a tooth in the driving wheel. Let n be the radius of curvature of the pitch line, r that of the rolling curve, r the radius vector of the tracing-point, at any instant. The angular velocity of the rolling curve relatively to the wheel is li dt 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 For the following wheel, q and q 3 have to be interchanged, so that, if r s be the radius of that wheel, 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 4, 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 w C 3> be the centres of two circular wheels, whose pitch circles are Bi, B 2 . Through the pitch point I draw the intended line of action P : P 2 , making the angle C I P = 6 with the line of centres. From C u C 2 , draw Cj P L = I C, sin 6, (VP 3 = rC 2 sin 6, .(1.) perpendicular to P! P 2 , with which two perpendiculars as radii, describe circles (called base circles) "D u Dg. Suppose the base circles to be a pair of circular pulleys, connected by means of a cord whose course from pulley to pulley is P! I P 2 . As the line of connection of those pulleys is the same with that of the proposed teeth, they will rotate with the required velocity-ratio. Now suppose a tracing-point T to be fixed to the cord, so as to be carried along the path of contact P 1 I P 2 . That point will trace, on a plane rotating along with the wheel 1, part of the involute of the base circle D t , 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 Pj I P 2j is less than the pitch in the ratio sin 0:1, and consequently that if dis- tances not less than the pitch x sin 6 be marked off either way from I towards P! and P 2 respectively, as the extremities of the path of contact, and if the addendum circles be described through the points so found, there will always be at least two pairs of teeth in action at once. In practice, it is usual to make the path of contact somewhat longer, viz., about 2J times the pitch ; and with this length of path and the value of & which is usual in practice, viz., 75, the addendum is about T % 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 6. INVOLUTE TEETH. 443 458. Sliding of involute Teeth. The distance through which a pair of involute teeth slide on each other, is found by observing that the distance from the point of. contact of the teeth to the pitch point is given by the equation r = V * 7=7" = ? ' sin 6 ..................... (1.) L- 1 which reduces equation 3 of Article 455 to the following : n* .................. (2.) This distance may also be expressed in terms of the extreme dis- tances of the point of contact from the pitch point. Let the'se be denoted by t v t 2 ; then t t = q l sin 6 ; t 2 = q z sin 6 j and s = ( - + - ) ' ' . 2 ..(2 A.) \^*j ^*2' S1H v For inside gearing, the difference of the reciprocals of the radii of the wheels is to be taken instead of their sum. The preceding formulse, which are exact for involute teeth, are approximately correct for all teeth, if & 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 (TP = r v ' sin 6 ; P I = r x . 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 1 + d l ) 2 = r 2 l sin 2 + (r 1 cos 9 + t 2 ) 2 ', ............. (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 & about -. 6Z 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 d, 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 = p tan 6' } (!) which, for 6 751, gives for the radius 1 r = 3 -867;?, and for the circumference of the pitch circle, p x 3*867 x 2 K = 24-3 p' } to which the next greater integer multiple of p is 25 p; and therefore twenty-five, as formerly stated, in Article 447, is the least number of involute teeth to be employed in a pinion. 461. Epicycioidal 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 R' 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 I D'. 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 Je is 6 =^ C I D = ^ C' I D'. It appears from experience that Fi g- 199- the least value of 6 should be about 60 ; therefore the arcs D I = I D' should each be one-sixth of a cir- EPICYCLOIDAL TEETH. 445 eumference; therefore the circumference of the describing circle should be six times the 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 2 the chord I D' = 2 r - cos ^, r being the radius of the rolling circle. Hence ( Tl + dtf = r\ sin 8 6 + (r^r 2 rtf - cos 3 6 .......... (1.) 3 1 For the usual value of 6, 60, sin 2 d = -, and cos 8 6 = - whence ................... (2.) 462 A. The Sliding of Epicycloidal Teeth is deduced from equation 3 of Article 455, by observing, that the radius vector of the point of contact is (1.) and that the extreme values of q are the arcs of approach and recess (2.) whence we have = 8 (I - sin 6) r* (I + I) ; ................ (3.) which, for 6 = 60, has the value (3A -> 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 t be the radius of the pitch circle, r that of the rolling circle, e the radius of curvature required; then (1.) the sign + applying to an external epicycloid, that is, to the face of a tooth, and the sign to an internal epicycloid, that is, to the flank of a tooth. To find the distances of the centres of curvature of the given point in an epicycloid from the point of contact I of the pitch circle and rolling circle, there is to be subtracted from the radius of cur- vature, the instantaneous radius vector, v 2 r Q cos 0; that is to say, ^z^ * r (2.) The value to be assumed for & is its mean value, that is, 75 J; and cos 6 - nearly : r is nearly equal to the pitch, pj and if n be the number of teeth in the wheel, 6 : n : : r Q : r t . Therefore, for the proportions approved of by Mr. Willis, equation 2 becomes being used for the face, and for the flank ; also r =| nearly (4.) Hence the following con- struction. In fig. 200, let B C be part of the pitch circle, A the point where a tooth is to cross it. Set oif = AC = . Drawradii of the pitch circle, D B, E C. Draw F B, C G, making angles of 75^ with those radii, in which take TRUNDLE DIMENSIONS OP TEETH. 447 Round F, with the radius FA, draw the circular arc AH; this will be the face of the tooth. Round G, with the radius G A, draw the circular arc G K ; this will be the flank of the tooth. To facilitate the application of this rule, Mr. Willis has published tables of the values of e r, and invented an instrument called the " odontograph" 464. Teeth of Wheel and Trundle. A trundle, as in fig. 201, has cylindrical pins called staves for teeth. The face of the teeth of a wheel suitable for driving it, in outside gearing, are described by first tracing external epicycloids by rolling the pitch circle B 2 of the trundle on the pitch circle BI 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. Tootned wheels being in general intended to rotate either way, the backs of the teeth are made similar t6 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, 448 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 1 60 Ibs. of force transmitted by the teeth. 466. Mr. Sans;' 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. Sang's work. 467. The Teeth of a Bevel- Wheel have acting surfaces of the conical kind, generated by the motion of a line traversing the apex of the conical pitch surface, while a point in it is carried round the outlines of the cross section of the teeth made by a sphere described about that apex. The operations of describing the exact figures of the teeth of bevel- wheels, whether by involutes or by rolling curves, are in every respect analogous to those for describing the figures of the teeth of spur-wheels, except that in the case of bevel-wheels, all those operations are to be performed on the surface of a sphere described about the apex, instead of on a plane, substituting poles for centres, and great circles for straight lines. In consideration of the practical difficulty, especially in the case of large wheels, of obtaining an accurate spherical surface, and of drawing upon it when obtained, the following approximate method, proposed originally by Tredgold, is generally used : Let O, fig. 203, be the apex, and O C the axis of the pitch cone of a bevel- wheel; and let the largest pitch circle be that whose radius is CB. 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 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 Wheel* 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 w r heels 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 about an axis fixed longitudinally relatively to the frame- work, and the advance of a nut in which that screw rotates, the nut being free to shift longitudinally, but not to turn. The advance of the nut in the latter case is in the direction opposite to that of the advance of the screw in the former case. 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 the advance takes place. Fig. 204 re- presents a right-handed screw, and fig. 205 a left-handed screw. The pitch of a screw of one thread, and the total pitch of a screw of any number of threads, is the pitch of the 9 the axial ^> - v . -, , ., , = p a j ( divided pitch j *u p.) 1-4 the normal p a the circular pitch ; Then IV 2 TC r tan i p a = p n ' sec i = p e ' tan ^ = ; 2 ir r ' sin i *p n j) e ' sin i = jO a * cos i . n 473. Screw Gearing. A pair of convex screws, each rotating about its axis, are used as an elementary combination, to transmit motion by the sliding contact of their threads. Such screws are commonly called endless screws. At the point of contact of the screws, their threads must be parallel \ and their line of connection is the common perpendicular to the acting surfaces of the threads at their point of contact. Hence the following principles : I. If the screws are both right-handed or both left-handed, the angle between the directions of their axes is the sum of their obli- quities : if one is right-handed and the other left-handed, that angle is the difference of their obliquities. II. The normal pitch, for a screw of one thread, and the normal divided pitch, for a screw of more than one thread, must be the same in each screw. III. The angular velocities of the screws are inversely as their number of threads. 474. Mooke'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 . 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. 20 /. circular pitch, and n the number of steps. Then the arc of contact, the addendum, and the extent of sliding, are those due to the smaller pitch , while the strength of the teeth is that due to the thickness corresponding to the entire pitch p ; so that the smooth action of small teeth and the strength of large teeth are combined. Stepped teeth being more expensive and difficult to execute than common teeth, are used for special pur- poses only. 475. The Wheel and .Screw is an elementary combination of two screws, whose axes are at right angles to each other, both being right-handed or both left-handed. As the usual object of this com- bination is to produce a change of angular velocity in a ratio greater than can be obtained by any single pair of ordinary wheels, t 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 : BiBj, SLIDING OF SCREWS OLDHAM'S COUPLING, 453 in fig. 208, for example, being the pitch circle of the wheel, and B 2 B 2 the pitch line of the rack. The periphery and teeth of the wheel are usually hollowed to fit the screw, as shown at T, fig. 209. To make the teeth or threads of a pair of screws fit correctly and work smoothly, a hardened steel screw is made of the figure of the smaller screw, with its thread or threads notched so as to form a cutting tool ; the larger screw, or wheel, is cast approximately of the required figure ; the larger screw and the steel screw are fitted up in their proper relative position, and made to rotate in contact with each other by turning the steel screw, which cuts the threads of the larger screw to their true figure. 476. The Relative Sliding of a Pair of Screws at their point of contact is found thus : Let r{, r 2 , be the radii of their pitch cylin- ders, and ii, i 2 , the obliquities of their threads to their pitch circles, one of which is to be considered as negative if the screws are con- trary-handed. Let u be the common component of the velocities of a pair of points of contact along a line touching the pitch sur- faces and perpendicular to the threads, at the 'pitch point, and i> the velocity of sliding of the threads over each other. Then u = a,! TI sn i = a 2 r 2 sn j so that /i \ u u " " v '* O> 2 = T! sin ^ 1 and v = aft- cos tj + a z r 2 cos i 2 = u (cotan ^ + cotan i 2 ) ..... (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 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. CJL, 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 ; Ep E 1? a bar, sliding in a diametral groove in the face of 454 THEORY OF MECHAiaSM. T) l ; 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 Cj C 9 , 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 Cj C 2 , diametrically opposite to A. Oldham's coupling may be used with advantage where the axes of the shafts are intended to be as nearly in the same straight line as is possible, but where there is some doubt as to the practica- bility or permanency of their exact continuity. SECTION 3. Connection by Bands. 478. Bands Classed. Bands, or wrapping connectors, for com- municating motion between pulleys or drums rotating about fixed axes, or between rotating pulleys and drums and shifting pieces, may be thus classed : I. JBelts, 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 lt r a , be the perpendiculars let fall from their axes on the centre line of the band, %, a t , their angular velocities, and i v , 8 , the angles which the centre line of the BANDS PULLEYS DRUMS. 455 band makes with the two axes respectively. Then the longitudi^ nal velocity of the band, that is, its component velocity in the direction of its own centre line, is ................... (1.) whence the angular velocity-ratio is % r z sn ? When the axes are parallel (which is almost always the case), ^ = % and The same equation holds when both axes, whether parallel or not, are perpendicular in direction to that part of the band which trans- mits the motion for then sin ^ = sin ? 3 = 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 cos,/ j and ..................... (4.) rasini /er v v = -- r- .............................. (5.) cos,; When the centre line of the band is parallel to the direction of motion of the sliding piece, and perpendicular to the direction of the axis of the rotating piece, sin i = cos j = 1, and v = u = ra ............................ (6.) 480. The Pitch Surface of a Pulley or Drum is a Surface to which the line of connection is always a tangent ; that is to say, it is a surface parallel to the acting surface of the pulley or drum, and distant from it by half the thickness of the band. 481. circular Pulleys and Drums are used to communicate a Fig. 211. Fig. 212. constant velocity-ratio. In each of them, the length denoted by r 456 THEORY OF MECHANISM. in the equations of Article 479 is constant, and is called the effec- tive radius, being equal to the real radius of the pulley or drum added to half the thickness of the band. A crossed belt connecting a pair of circular pulleys, as in fig. 211, reverses the direction of rotation; an open belt, as in fig. 212, pre- serves that direction. 482. The Length of an Endless Belt, connecting a pair of pulleys whose effective radii are Cj Tj = r^ C 2 T 2 = r 2 , with parallel axes whose distance apart is C^ C 2 = c, is given by formulae founded on equation 1 of Article 402, viz., L = % s + 2-ri. Each of the two equal straight parts of the belt is evidently of the length 8 = Jc 2 fa + r 2 ) 2 for a crossed belt ; | _ r ........ \^') s = Jo 3 fa r 2 ) a for an open belt ; j r l being the greater radius, and r 2 the less. Let ^ be the arc to radius unity of the greater pulley, and ? 2 that of the less pulley, with which the belt is in contact ; then for a crossed belt ^>(cr + 2arc-sin?l:p); and for an open belt, (r r\ / r r\ v + 2 arc . sin _! 2 J : i 2 = 1^ 2 arc sin _? : ) ; c / \ c J (2.) and the introduction of those values into equation 1 of Article 402 gives the following results : For a crossed belt, L = 2 J (2.) a>\ ~~ dV^ tan

(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 shall balance each other, and that the 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 4:78 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 Yds. 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 j and the area between tivo 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 R- 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 tJie 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 OF ENERGY VIRTUAL VELOCITIES. 479 self-evident; for as the effort applied to the point balances the resistance, the products of these forces into the distance traversed by the point in any interval must be equal; that is, P-A S = R-A S ......................... (1.) CASE 2. For the forces acting on any system of balanced points, the principle must be true, because it is true for those acting on each single point of the system. This is expressed as follows : 2'PA 5 =:2-BA 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 Rj ; the efforts acting between the points of the system and external bodies by P 2 , and the resistances by R 2 - Then by case 2, but by the condition of rigidity, 2'Pj AS = 0; 2-]^ AS= rO; therefore, 2'P 2 AS = 2'R, 2 AS ...................... (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 F is = 0; ........................ (1.) attention being paid to the principle, that cos 6 is < ^ , . > when & is < i , > . 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 E, is applied ; then 2-Pv = s-E,M ......................... (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, y } 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 <*', /3', 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. 481 3? A 5 * cos at, cos '; 'I F AS cos /3 cos /3'j L (1.) F A s cos y cos y'; J and the sum of those three quantities of energy is the whole energy exerted. Now it is well known, that cos a cos ' + cos ft cos /3' + cos y cos y' cos Q, & 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 fornmlse 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. I 1 I 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 earth's 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 tJie sys- tem. Conceive the velocity of each of the bodies to be resolved into three rectangular components. Consider all the component velocities parallel to one of the rectangular directions. These are the rates of variation of the perpendicular distances of the bodies from a certain plane. If the mass of each of the bodies be multi- plied by its distance from a certain plane, the products added, and the sum divided by the sum of the masses, the result is the distance of the centre of gravity of the whole system from that plane ; there- fore, if the component velocity of each of the bodies in a direction perpendicular to that plane be multiplied by the mass of the body, the sum of such products for all the bodies of the system will be the product of the entire mass of the system into the velocity of its centre of gravity in a direction perpendicular to the plane in ques- tion; so that this product is one of the three rectangular com- ponents of the resultant momentum of the system of bodies ; and the same may be proved for the other rectangular components. Expressed symbolically, let u, v, w, be the three rectangular com- ponents of the velocity of any mass, m, belonging to a system of bodies, and U Q , V Q , w , the rectangular components of the velocity of the centre of gravity of that system of bodies ; then U Q ' 2 m V Q ' im = 2 mv j ................... (1.) W Q ' 2 m = s mw. j COROLLARY. The resultant momentum of a system of bodies rela- tively to their common centre of gravity is nothing ; that is to say, 2 m (u - w c ) = ; 2 m (v - v ) = ; ) /^ \ " 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 L, Section 3. 526. impulse is the product of an unbalanced force into the time during which it acts unbalanced, and can be resolved and com- pounded exactly like force. If F be a force, and d t an interval of time during which it acts unbalanced, 3?dt is the impulse exerted by the force during that time. The impulse of an unbalanced force in an unit of time is the magnitude of the force itself. 527. Impulse, Accelerating, Retarding, Deflecting. Correspond- ing to the resolution of a force applied to a moving body into effort or resistance, as the case may be, and lateral stress, as explained in 484 PRINCIPLES OF DYNAMICS. Article 511, there is a resolution of impulse into accelerating or retarding impulse, which acts with or against the body's motion, and deflecting impulse, which acts across the direction of the body's motion. Thus if t, as before, be the angle which the unbalanced force F makes with the body's path during an indefinitely short interval, dt, P d t = F cos e ' dt is accelerating impulse if 6 is acute ; \ R d t = F cos ( *) ' d t is retarding impulse if 6 is obtuse ; V (1.) Qdt = ~F sin #dt 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 = v dt 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 Yds = Fv cos 6 ' dtj ................... (1.) or work performed by the body against the retarding impulse to the amount R,ds =Fv cos (~-0) ' dt .................. (2.) SECTION 2. Law of Varied Translation. 529. Second l^avr 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, 532.^ 530. General Equations of Dynamics. To express the SCCOnd 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 ot the body, v its velocity, and r the radius of curvature of its path, it follows from Articles 361 and 362 that the rate of direct variation of its momentum is dv d 2 s EQUATIONS OF DYNAMICS GRAVITY. 485 and from Articles 363 and 364, that the rate of deviation of its momentum is v 2 m. r Equating these respectively to the direct and lateral impulse per unit of time, exerted by an unbalanced force F, making an angle 6 with the direction of the body's motion, we find the two following equations : P or -R = Fcos^ = m -=- = m ^j ............. (!) (2.) The radius of curvature r is in the direction of the deviating force Q. Second method. As in Article 366, let the velocity of the body be resolved into three rectangular components, -y , - , ; so that (Ji t Ci t U t the three component rates of variation of its momentum are Also let the unbalanced force F, making the angles e, /3, y, with the axes of co-ordinates, and its impulse per unit of time, be resolved into three components, F,, F v , F,. Then, we obtain F, = F cos m ' -- j FcoM^m^, d?z COS '/ = m -y-;r j (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 x denote the latitude of the place, h its elevation above the mean level of the sea, g^ 32*1695 feet, or 9 - 8051 metres, per second; being the value of g for a 4o D and h = 0, and 11 = 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 A is negative, and the terms containing it as a factor have their signs reversed. For practical purposes connected with ordinary machines, it is sufficiently accurate to assume 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 m 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 Ibs. ; or in French measures, the mass of 9*81 kilogrammes. 533. The motion of a Falling Body, under the unbalanced action of its own weight, a sensibly uniform force, is a case of the uni- formly 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 y its velocity at the end of that time is FALLING BODY UNRESISTED PROJECTILE. 487 v = v + gt, ............................ (1.) the mean velocity during that time is 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 : .(4.) When the body falls from a state of rest, v is to be made = ; so that the following equations are obtained : =">*== < 5 -> The height h in the last equation is called the height or fall due to the velocity v; and that velocity is called the 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 * i? < 6 -> being a rise equal to the fall due to the initial velocity VQ. 534. An Unresisted Projectile, or a projectile to whose motion 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 represent the point from which the projectile is originally 488 PRINCIPLES OF DYNAMICS. projected in the direction O A, making the angle X A = t with a horizontal line 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 O Z by z } positive upwards, and negative -^ downwards. In the equations of \ vertical motion, the symbol h of the equations of Article 533 is to be replaced by z, because of h Fig. 232. an( j z ^^g pleasured in opposite directions. Let v be the velocity of projection. Then at the instant of pro- jection, the components of that velocity are, , dx , dz . A horizontal, - 7 - = v cos 6- } vertical, - = V Q sm 0; a t u t and after the lapse of a given time t, those components have become (1.) dx -T = v cos 6 = constant; dz Hence the co-ordinates of the body at the end of the time t are horizontal, x V Q cos 6 tj \ vertical, z ^ VQ sin 9 * c *~~ ~~^ | 2t ) x and because t = , those co-ordinates are thus related, Vn COS 6 'JQ COS z = x tan 6 g 2 vl cos 3 6 or .(3.) an equation which shows the path B C of the projectile to be a parabola with a vertical axis, touching O A in O. The total velocity of the projectile at a given instant, being the resultant of the components given by equation 1, has for the value of its square from the last form of which is obtained the equation MOTION ALONG AN INCLINED PATH. 489 which, being compared with equation 4 of Article 533, shows that the relation between the variation of vertical elevation, and the varia- tion of tlie square of the resultant velocity, is the same, whether the velocity is in a vertical, inclined, or horizontal direction. This is a particular case of a more general principle, to be explained in the sequel. The resistance of the air prevents any actual projectile near the earth's surface from moving exactly as an unresisted projectile. The approximation of the motion of an actual projectile to that of an unresisted projectile is the closer, the slower is the motion, and the heavier the body, because of the resistance of the air increasing with the velocity, and because of its proportion to the body's weight being dependent upon that of the body's surface to its weight. 535. The motion of a Body Along an Inclined Path, under the force of gravity alone, is regulated by the principle, that the varia- tion of momentum in each interval of time is equal to the impulse exerted in that interval, by that component of the body's weight which acts along the direction of motion. If the path is straight, the other rectangular component of the body's weight is balanced by the resistance of the surface or other guiding body which causes the inclined path to be described; if the path is curved, the difference between those two forces which act across it is employed in deviat- ing the direction of motion of the body. Let v be the velocity of the body at any instant, , as before, du t the rate of variation of that velocity, ^ 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** ........................ (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 9 for the inclined path than for the vertical path. The motion of a body in any path on an INCLINED PLANE being resolved into two rectangular components, one horizontal, and the other in the direction of steepest declivity, the horizontal com- ponent (in the absence of friction) is uniform, and the inclined 490 PRINCIPLES OF DYNAMICS. component takes place according to the law expressed by equation 1 of this Article. Consequently, the resultant motion of the body is that of an unresisted projectile, as described in Article 534, except that g sin 6 is to be substituted for g. The motions of bodies on inclined planes being slower, and there- fore more easily observed than their vertical motions, were used by Galileo to ascertain the laws of dynamics, which he discovered. For a body sliding on an inclined plane without friction, the equation connecting the velocity directly with the position of the body is the following : f vl v s 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*, 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 ., . &=**' < L > where yis 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 E-, 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, P-R -2. - _JL_ . x_ /. -. - m - r" "" [/'+* DEVIATING ANtf^ENTRIFUGAL FORCE. 491 ffy *?* Consequently the two weights move according to the same law with a falling body, but slower in the ratio of fto 1. 537. A Deviating Force, which acts unbalanced in a direction perpendicular to that of a body's motion, and changes that direc- tion without changing the velocity of the body, is equal to the rate of deviation of the body's momentum per unit of time, as the fol- lowing equation expresses : Q being the. deviating force, W the weight of the body, W -f- g its mass, v 2 its velocity, and r the radius of curvature of its path. In the case of an unresisted projectile, already mentioned in Article 534, the deviating force at any instant is that component of the body's weight which acts perpendicular to its direction 9f. ,;< 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 . g g Consequently Q r = - , which agrees with equation 1. In the case of projectiles, just described, and of the heavenly bodies, deviating force is supplied by that component of the mutual attraction of two masses which acts perpendicular to the direction of their relative motion. In machines, deviating force is supplied by the strength or rigidity of some body, which guides the deviating mass, making it move in a curve. A pair of free bodies attracting each other have both deviated motions, the attraction of each guiding the other; and their devia- tions of momentum are equal in equal times; that is, their devia- tions of motion are inversely as their masses. In a machine, each revolving body tends to press or draw the body which guides it away from its position, in a direction from the centre of curvature of the path of the revolving body; aad that tendency is resisted by the strength and stiffness of the guiding body, and of the frame with which it is connected. 538. Centrifugal Force is the force with which a revolving body reacts on the body that guides it, and is equal and opposite to the 492 PRINCIPLES OF DYNAMICS. deviating force with which ths guiding body acts on the revolving body. In fact, as has been stated in Article 12, eveiy 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 a^ heightj^C _ weight _ gr '"' radius A B ~~ centrifugal force v*'" " '' where r = A B. Let n be the number of turns per second of the pendulum; then v 2 K n r; and therefore, making B C = 7i, = (in the latitude of London) 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. I>< vim in- 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, v a = -. r Let ar be substituted for v in the value of deviating force of Article 537, and that value becomes Q = 5^T... ...(,) 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 IF n, where n is the number of revolutions per second, so that Q from which equation the height of a revolving pendulum might be deduced with the same result as in the last Article. 541. Rectangular Components of Deviating Force. First Demon- stration. Let O in fig. 234 be the centre of the circular path E F G H of a body revolving in a circle with an uniform velocity, through which centre draw rectangular axes, X and O Y, in the plane of revolution. Let the angle ^ X O A, which at any instant the - 3 radius vector of the revolving body makes with the axis of x, be denoted by 6. Let A D x = r cos 6. and ) /, x . . Ml.) Fig. 234. A B = y - r sin 0, J be the rectangular co-ordinates of the revolving body at any in- stant. Let Q*, Qj,, be the components of the deviating force, parallel to O X and O Y respectively. Then from the obvious proportion between the magnitudes of those components, Q:Q,:Q y :: r : x : y, ..................... (2.) combined with the equation 2 of Article 540, follow the values of those component!- 1 ., ; 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 the co-ordinates x and y, in terms of the time, are x = r cos at', y = rsina ................. (4.) The components of the velocity of the body are, 494 PRINCIPLES OP DYNAMICS. dx . dy t = ar sin at; -~ = ar cos at, ........... (5.) at at the velocity parallel to each co-ordinate being proportional to the other. The components of the variation of niotion are = a 2 r cos a t = a? as; * \ ............. (6.) d y ^ = O?T sin a t a 2 y d? W which being multiplied by the mass , reproduce the components of the deviating force as before given in equation 3. 542. Straight Oscillation is the motion performed by a body which moves to and fro in a straight line, alternately to one side and to the other of a central point; and in order that this motion may take place, the body must be urged at each instant towards the central point. In most cases, the force so acting on the oscillating body is either exactly or very nearly proportional to its displacement, or distance from the central point of equilibrium ; that is to say, that force follows the law of one of the rectangular components of the deviat- ing force of a body revolving uniformly in a circle once for each double oscillation of the oscillating body. In fig. 234, let a body B, equal in weight to the body A, start at the same instant from E, and oscillate to and fro along the dia- meter E G-, while A revolves in the circle E F G H. Then if B is urged towards the centre O with a force at each instant propor- tional to its distance from that point, and given by the equation a.) 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 & ..................... (2.) from : the velocity of B being at each instant equal to the par- allel component of the velocity of A j that is to say, dx - arsmat = arsm0; .............. (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 ^-^-* * 3 sh-S* ' 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 O, being the same with the entire deviating force of A, and n is the number of double oscillations in a second. (The angle = a t is called the PHASE of the oscillation.) The greatest value Q of the force which must act on B to pro- duce n double oscillations of the semi-amplitude r in a second, is given by the equation Wa*r _4:v 2 Wri 3 r being similar to equation 3 of Article 540. Revolution in a circle may be regarded as compounded of two oscillations of equal amplitude, in directions at right angles fco each other. 543. Elliptical Oscillations or Revolutions compounded of two straight oscillations of equal periods, but un- equal amplitudes, may be performed by a body urged towards a central point by a force pro- portional to its distance from that point. In fig. 235, let A be the position of the body at any instant ; let O A = ? , and let the force urging the body towards O be a.) b being a constant. Then the rectangular com- ponents of that force are W6> 9 Fig. 235. the former force being suited to maintain a straight oscillation parallel to O X, and the latter, a straight oscillation parallel to O Y, the period of a double oscillation in either case being the same, viz. : 1 2* (3.) according to equation 4 of Article 542. Hence let x l = E = G be the semi-amplitude of the former straight oscillation, and y l = 496 PRINCIPLES OF DYNAMICS. O F = O H that of the latter ; then at any instant the co-ordinates of the body will be x = Xi cos bt; y y^ sin bt ; (4.) which equations being respectively divided by ^ and y lt 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 t , y lf The components of the velocity of the body at any instant are dx - = b Xi sin b t = b dt ' 544. A Simple Oscillating Pendulum consists of an indefinitely small weight A, fig. 236, hung by a cord or rod of in- sensible weight A C from a point C, and swinging in a vertical plane to and fro on either side of a central point D vertically below C. The path of the weight or bob is a circular arc, A D E. The weight W of the bob, acting vertically, may be resolved at any instant into two components, viz. : W cos ^ D C A = W BCJ CA' acting along C A, and balanced by the tension of the Fig. 23G. rod or cord, and sn C A = W 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 veiy nearly equal to the distance of the bob from D, the central point of its oscillations. Hence the bob is very nearly in the condition of straight oscillation described in Article 542 ; and the time which it occupies in making a double oscillation is there- OSCILLATING PENDULUM. 497 fore found approximately by means of equation 4 of that Article, viz.:- where r denotes the semi-amplitude, and Q the maximum value of W - -. But if the length of the pendulum, C A, be made = I, wehl^ Q AB r ^ = max. -= = 7 , nearly ; YY \j JEL I whence, approximately, for small arcs of oscillation, -(!-) -; and n v g ff which being compared with equation 2 of Article 539, shows, that the length of a simple oscillating pendulum, making a given number of small double oscillations in a second, is sensibly equal to the height of a revolving pendulum, making 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. Cycloidai 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 >-o = J (1.) 2K 498 PRINCIPLES OF DYNAMICS. It is well known that a cycloid is the involute of an equal and similar cycloid. Hence, in fig. 237, let C F, C G, be a pair of cycloidal cheeks, described by rolling a circle of the radius r Q on a horizontal line -traversing C ; let C A be a flex- ible line, fixed at C, and having a bob at A, its length being I = 4 r = C D = the length of each of the semi- cycloids OF, C G-. Then as the _ pendulum C A swings between the Fj . 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-Jialf of the square of its velocity, or, as Article 533 shows, the product of the weight of the body into the Iieight due to its velocity ; that is to say, it is represented by mv 2 Wv 2 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, in foot-pounds (Article 513, 514). 548. Components of Actual Energy. The actual energy of a body (unlike its momentum) is essentially positive, and irrespective of direction. Let the velocity v be resolved into three components, -^ , -~, -=-, parallel to three rectangular axes; then the quantities cl t at Co t of actual energy due to those three components respectively are da? W d W dz 2 But the square of the resultant velocity is the sum of the squares of its three components, or _ ~ dt 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. 549. Energy of Varied Motion. THEOREM I. A deviating force produces no change in a body's actual energy, because such force produces change of direction only, and not of velocity; and actual energy is irrespective of direction, and depends on velocity only. THEOREM II. The increase of actual energy produced by an un- balanced effort is equal to tlie potential energy exerted. This theorem is a consequence of the second law of motion, deduced as follows : 500 PRINCIPLES OF DYNAMICS. Let m = W H- 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 R ; 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 momentuip, is W -^ ' ' -(v 2 -V 1 ) = (P-K)At .................. (1.) y Because of the uniformity of the acceleration of the body, its mean velocity is -^= -, and the distance traversed by it is 2 _ Vi + V* . *>*,. -~2~ 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 bodys actual energy, and the second is the potential energy exerted by the un- balanced effort; and those two quantities are equal. Q. E. D. When the unbalanced effort varies, let d s be taken to denote a distance in which it varies less than in any given proportion, and d v 9 the change in the square of the velocity in that distance ; then ............. (3.) or if 8u s 2 , denote the two extremities of a finite portion of the body's path, W'fo! fi THEOREM III. The diminution of actual energy produced by an unbalanced resistance is equal to the work performed in moving against the resistance. This is a consequence of the second law of motion, demonstrated by considering R to be greater than P in the equa- tions of the preceding theorem ; so that equation 1 becomes \ m 9 equation 2 becomes TRANSFORMATION OP ENERGY. 501 and equation 3 and 3 A become w_dry 2? 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 the work per- formed. This principle, expressed in ^he form of a differential equation, is as follows : (1.) ' 9 which includes equations 3 and 6 of Article 549 ; and in the form of an integral equation, (2.) 553. Periodical Motion. If a body moves in such a manner that it periodically returns to its original velocity, then at the end of each period, the entire variation of its actual energy is nothing; and in each such period the whole potential energy exerted is equal to the whole work performed, exactly as in the case of a body moving uniformly (Article 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, 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 : ' g dt cj ds ' and each is a necessary consequence of the other ; yet in former times a fallacy prevailed that they were inconsistent and contra- dictory, and a bitter controversy long raged between their respec- tive partizans. 555. Energy due to Oblique Force. It has already been stated in Chapter I. of this Part, and especially in Article 520, that if an unbalanced force F acts on a body while it moves through the dis- tance ds, making the angle & with the direction of the force, the product F cos 6 'ds represents the energy exerted, if d 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 (K 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 surface . as the case may be ; then dz = cos0 'ds ............................. (1.) is the component, of the motion of the body in the direction of z. The force F is to be treated as positive or negative according as it tends to increase or diminish z. Then if v v v 2 , be the velocities of the body, and z l9 # 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 *-* m -~2 = i * .. (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 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^ above the ground, and be moving with the velocity v t in any direction, its initial total energy relatively to the ground is of which W z l is potential and W - actual. Supposing the body ^ to have moved without any resistance except such as may arise from a component of its own weight, which is a reciprocating force, to a different height z 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. 504 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 w **+ = w *>+-*<** ........ < 3 -> Example III. Let a body oscillate (as in Article 54:2) 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 x t be the semi- amplitude of oscillation, x the displacement at any instant, Q t the greatest value of the moving force, so that ---- 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 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. : ( > VQ 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 = 2 n ; n being the number of double oscilla- tions in a second. For the elliptic oscillations of Article 543, the total energy of the body is at each instant the sum of the quanti- ties of energy due to the two straight oscillations of which the elliptic oscillation is compounded ; and for a body revolving in a circle, and urged towards the centre by a deviating force propor- tional to the radius vector, the total energy relatively to the centre is one-half actual and one-half potential, viz. : SYSTEM OF BODIES ANGULAR MOMENTUM. 505 SECTION 4. Varied Translation of a System ofJBodies. 558. Conservation of Momentum. THEOREM. The mutual actions 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 external to the system; for the motion of the centre of gravity is that which, being multiplied by the total mass of the system, gives the resultant momentum, and this can be varied by external forces only. It follows that in all dynamical questions in which the mutual actions of a certain system of bodies are alone considered, the centre of gravity of that system of bodies may be correctly treated as a point whose motion is none or uniform ; because its motion cannot be changed by the forces under consideration. 560. The Angular Momentum, relatively to a fixed point, of a body having a motion of translation, is the product of the momen- tum of the body into the perpendicular distance of the fixed point from the line of direction of the motion of the body's centre of gravity at the instant in question ; and is obviously equal to the product of the mass of the body into double the area swept by the radius vector drawn from the given point to its centre of gravity in an unit of time. Let m be the mass of the body, v its velocity, I the length of the before-mentioned perpendicular ; then Wvl m v i = g 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. Hayward. 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 tJie angular impulse producing that variation, and has tlie same axis. This is a consequence which is deduced from the second law of motion in the following manner : Conceive an unbalanced force F to be applied to a body m, and an equal, opposite, and parallel force, to a fixed point, during the interval d t and let I be the perpendicular distance from the fixed point to the line of action of the first force. Then the couple in question exerts the angular impulse Fldt. At the same time, the body m acquires a variation of momentum in the direction of the force applied to it, of the amount mdv = F dt-, so that relatively to the fixed point, the variation of the body's angular momentum is rnldv = Fldt;.... 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 axis of angular momentum is called by some writers the invariable plane. The nearest approach to an absolutely fixed direction yet known is the invariable axis of the discovered bodies of the solar system. 564. Actual Energy of a System of Bodies. THEOREM. The actual energy of a system of bodies relatively to a point external to the system, is the sum oft/ie 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 wJwle 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 , 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 tlie centre of gravity rela- tively to the external point, and that which it has relatively to the centre of gravity. The component velocities of the first part are U , V , W ', and those of the second part U UQ = U'j V VQ = V* ; W WQ = w' j so that the components of the whole motion of the body may be represented by U = UQ + U'j V = VQ + V'' } W = WQ + W* . Then the actual energy of the system relatively to the external point is 1 2 m {(u + uj -f (v + vj + (w + wj] ; 508 PRINCIPLES OF DYNAMICS. which being developed, and common factors removed outside the sign of summation, gives i (! + ^ + ^5) 2 rti -f- UQ ' 2 * m U 1 -f- V ' 2 ' m V' + WQ ' 2 W w' + i 2 m (u 12 + v' 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 ww' = ; 2 ' mvf ; 2 m w' = ; so that the above expression for the actual energy of the system becomes simply 2 -m(u' 2 + v' 2J [-w f2 ); (1.) of which the first term is the actual energy of the ivhole 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 oft/ie bodies relatively to their common centre of gravity. Q. E. I). 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 m l} m 2 , be the masses of the two bodies, and u lt u 2 , their velocities before the collision, whose directions should be indicated by their signs. Then the velocity of their common centre of gra- vity is and this is not altered by the collision; neither is the external energy, whose amount is ............................ (2.) The internal energy of the system of two bodies is m l (u, - ntf _j_ m 2 (u 2 - u$ ~2~ ~2~~ ................... (3>) When the bodies strike together, this actual internal energy is expended in altering the figures of the bodies at and near their surface of contact, in opposition to their elastic force. So soon as the relative motion of the bodies has been thus stopped, the elastic force begins to restore their figures, and drive them asunder; and if they were equal, similar, and perfectly elastic, it would reproduce all the energy of relative motion given by the formula 3, so that the bodies would separate with velocities relatively to their common centre of gravity, equal and opposite to their original velocities relatively to that point ; that is to say, with the velocities UQ-UH U -U 2 , relatively to the common centre of gravity, and the velocities ^ = 2^0-^, v 2 = 2u u. 2) .................. (4.) relatively to the earth. But as a certain proportion, which may be denoted by 1 & 2 , of the internal actual energy takes the forms of internal vibration and of heat, the internal actual energy due to visible motion after the collision is tfm 2 (u 2 -u )~ ~2 j the velocities of the bodies, relatively to their common centre of gravity, after the collision, are 510 PRINCIPLES OF DYNAMICS. JC(UQ-U^), k (uo-u a ); and their velocities relatively to the earth are V! = (L + &) Uo-kUi'y v 2 = (l +k)u Q Jcu z ............ (6.) Should the bodies be perfectly soft, or inelastic, k = ; in which case Vl = v a = u Q ; ............................ (7.) that is, the bodies do not fly asunder, but proceed together with the velocity of their common centre of gravity. (See Addendum, p. 512.) 567. The Action of Unbalanced External Forces on a system of bodies, considered as a whole, is to vary the resultant momentum and the resultant angular momentum. It has been shown in Article 60, that every system of forces can be reduced to a single force and a couple. The system of forces applied to a system of bodies is to be reduced to a single force acting through the centre of gravity of the system, and a couple, as shown in equations 5, 6, 7, 8, of Article 60 ; then in a given interval of time, the variation of resultant momentum of the system is equal to and in the direc- tion of the impulse of the single resultant force, and the variation of angular momentum is equal to the angular impulse, and about the axis, of the resultant couple. To express this by general equations, let the components of the momentum of any mass m belonging to the system, whose rectan- dx dv dz gular co-ordinates are x, y, z, be m -=, m -~, m -=-. Then the u t ut Cut rates of variation of these components are j . r Aj~ 2U d z x d 2 d*z Also, the rectangular components of the angular momentum of that mass are about *, m (._) ; about ,, . (_. JJ) ; whose rates of variation are , - - , (&) "ysf-'s?} PRINCIPLE OF D'ALEMBERT. 511 Let F,, F y , F z , be the components of the force externally applied to a point whose co-ordinates are x, y, z. Then by the equality of the resultant impulse to the variation of resultant momentum, $ {*-$& '- *"" - 1 ^^ 2 I *~ ar } and by the equality of the resultant angular impulse to the varia- tion of the resultant angular momentum. <*> The use of those equations is to determine the effect of a given system of external forces on a system of bodies when the relations amongst the motions of those bodies are known, without taking into consideration the internal forces acting between the bodies, which latter forces it is sometimes difficult or impossible to determine until the effects of the external forces have first been found. 568. Determination of the Internal Forces. When the relations which exist between the motion of the system as a whole, that is, its resultant momentum and angular momentum, and the motions of the several bodies of which it consists, are fixed by cinematical principles, then the motion of each body can be determined when the externally applied forces are known. Then if, from the force externally applied to each body at each instant, there is taken away the force required to produce the change of motion of the body which takes place at that instant, the remainder must be balanced by, and equal and opposite to, the internal force acting on the body in ques- tion ; and this, which is the PRINCIPLE OF D'ALEMBERT, serves to determine the internal forces. Using the notation of the last Article, the components of the internal force applied to a given body of the system are 569. Residual External Forces. If the resultant external force acting through the centre of gravity of a system of bodies be sup- 512 PRINCIPLES OF DYNAMICS. posed to be divided into parallel components, each applied to one of the bodies and proportional to the mass of the body to which it is applied, such will be the system of external forces required to make all the bodies of the system have equal and parallel motions at each instant in common with their centre of gravity. Then if the forces so determined be taken away from the forces actually applied to the several bodies, the residual external forces, being combined with the internal forces, will constitute those forces which regulate the motions of the bodies relatively to their com- mon centre of gravity considered as a fixed point. ADDENDUM TO ARTICLE 566, PAGE 510. Collision. It was formerly supposed that the disappearance of energy after collision was wholly due to imperfect elasticity, and that any two perfectly elastic bodies would fly asunder after col- lision with a relative velocity equal to their relative velocity of approach before collision. But M. de St. Yenant showed that, except when the bodies are similar and equal, a certain quantity of energy disappears, even in perfectly elastic bodies, in producing internal vibrations of each body. The value of the co-efficient k, being the ratio of the relative velocity of the recoil to that of the approach, in the case of a pair of perfectly elastic prismatic bars, striking each other endwise, is given as follows : let a L and 2 be the lengths of the bars ; p l and p 2 their weights per unit of length ; s 1 and s 2 the velocities of the transmission of sound (that is, of longitudinal vibrations) along them; let - 1 ^ -^ ; and also let s i s z s, p, ^ s 9 p<> : in other words, let ^L and ^" l : then As to the velocity of sound, see Article 615, page 563. The paper of M. de St. Venant is published in full in the Journal des Mathematiques pures et appliquees, 1867 ; and an abstract in English of the more simple of its results in The Engineer for the 15th Feb- ruary, 1867. 513 CHAPTER IIL ROTATIONS OF RIGID BODIES. 570. The motion of a Rigid Body, or of a body which sensibly preserves the same figure, has already been shown in Part III., Chapter II., to be always capable of being resolved at each instant into a translation and a rotation ; and by the aid of the principles explained in Section 3 of that chapter, the component rotation can always be conceived to take place about an axis traversing the centre of gravity of the body, and to be combined, if necessary, with a translation of the whole body in a curved or straight path along with its centre of gravity. The variations of the momen- tum of the translation, whether in amount or in direction, are due to the resultant force acting through the centre of gravity of the body, and are exactly the same with those of the momen- tum of the entire mass if it were concentrated at that centre; the variations of the angular momentum of the rotation are due to the resultant couple which is combined with that re- sultant force. The variations of actual energy are due to both causes. When the translation of the centre of gravity of a rotating body, and its rotation about an axis traversing that centre, are known, the motion of every point in the body is determined by cinematical principles, which have been explained in Part IIL, Chapter II., Section 3 ; so that by the aid of D'Alembert's principle (Article 568) the internal forces acting amongst the parts of the body can be completely determined. In the investigations of questions respecting the motions of rigid bodies, there are certain quantities, lines, and points, de- pending on the figures of the bodies, the mode of distribution of their masses, and the way in which their motions are guided, whose use facilitates the understanding of the subject and the computation of results, and which are related to each other by geometrical principles. These are, moments of inertia, radii of gyration, moments of deviation, and centres of percussion. Their geometrical relations are considered in the following sec- tion. 2L 514 PBINCIPLES OF DYNAMICS. SECTION 1. On Moments of Inertia, Radii of Gyration, Moments of Deviation, and Centres of Percussion. 571. The Moment of inertia of an indefinitely small body, or " physical point," relatively to a given axis, is the product of the mass of the body, or of some quantity proportional to the mass, such as the weight, into the square of its perpendicular distance from the axis : thus in the following equation': I Wr 2 - = r- = , (1.) r is the perpendicular distance of the mass m, whose weight is W, from a given axis; and the moment of inertia, according to the unit employed, is either I, or I -- g ; the former, when the unit is the moment of inertia of an unit of weight at the end of an arm whose length is unity; and the latter, when the unit is the moment of inertia of an unit of mass at the end of the same arm. For the purposes of applied mechanics, the former is the more convenient unit, and will be employed in this treatise. By an extension of the term " moment of inertia," it is applied to the product of any quantity, such as a volume, or an area, into the square of the distance of the point to which that quantity relates from a given axis, as has already been exemplified in Article 95, and in the theory of resistance to bending; but in the remainder of this treatise the term will be used in its strict sense, and accord- ing to the unit of measure already specified; that is, in British measures, moment of inertia will be expressed by the product of a certain number of pounds avoirdupois into the square of a certain number of feet. The geometrical relations amongst moments of inertia, to which the present section refers, are independent of the unit of measure. 572. The Moment of Inertia of a System of Physical Points, rela- tively to a given axis, is the sum of the moments of inertia of the several points , that is, 1 = 2 -Wr 2 (1.) 573. The Moment of Inertia of a Rigid Body is the Slim of the moments of inertia of all its parts, and is found by integration; that is, by conceiving the body to be divided into small parts of regular figure, multiplying the mass of each of those parts into the square of the distance of its centre of gravity from the axis, adding the products together, and finding the value towards which their sum converges when the size of the small parts is indefinitely diminished. For example, let the body be conceived to be built up of rectangular MOMENT OF INERTIA. ol5 molecules, whose dimensions are d x, d y, and d z, the volume of each dxdy dz, and the mass of unity of volume w. Then 1= J j J r*wdxdydz (1.) Hence follows the general principle which will afterwards be illustrated in special cases, that propositions relative to the geome- trical relations amongst the moments of inertia of systems of points are made applicable to continuous bodies by substituting integration for ordinary summation; that is, for example, by putting iff for 2, and w ' d x d y d z for "W. 574. The Radius of Gyration of a body about a given axis is that length whose square is the mean of all the squares of the distances of the indefinitely small equal particles of the body from the axis, and is found by dividing the moment of inertia by the mass, thus, When symbols of integration are used, this becomes / / / r^w'dxdydz e 2 = r-fr~ ................. (2.) / / / w ' dx dydz 575. Components of foment of Inertia. Let the positions of the particles of a body be referred to three rectangular axes, one of which, O X, is that about which the moment of inertia is to be taken. Then the square of the radius vector of any particle is so that the moment of inertia round the axis of x is I. = 2 W y 2 + 2 W z 2 ; .................. (1.) that is to say, the moment of inertia of a body round a given axis may be found by adding together the sumoftlie products of the masses of the particles, each multiplied by the square of each of its distances from a pair of planes cutting each other at right angles in the given axis. In the same manner it may be shown that the moments of inertia of the same body round the other two axes are given by the equations I, = 2 W z 2 + 2 W or 3 ; T f =s-Waj' + s-W ^...(2.) 016 PRINCIPLES OF DYNAMICS. 576. Moments of Inertia Round Parallel Axes Compared. THEOREM. The moment of inertia of a body about any given axis is equal to its moment of inertia about an axis traversing its centre of gravity parallel to the given axis, added to tJie moment of inertia about the given axis due to the whole mass of the body concentrated at its centre of gravity. Take the given axis for the axis of x, and any two planes tra- versing it at right angles to each other as the planes of x y and z Xy then, as in the preceding Article, I, = 2- Wy 2 + 2 -AY's 2 . Let y , 2 , oe the perpendicular distances of the centre of gravity of the body from the two co-ordinate planes before mentioned ; con- ceive a new axis to traverse that centre of gravity, parallel to the given axis; let two co-ordinate planes parallel to the original co-ordinate planes traverse that new axis; and let ?/, z', be the perpendicular distances of a given particle from those new co- ordinate planes. Then and introducing those values of the original co-ordinates into the value of I t , we find I, = , - W (y. + yj + 2 - W (z. + zj + 2 2/ 2 W y 1 + 2 z 2 W z + 2 W (y 12 but because y' and z' are the distances of a particle from planes traversing the centre of gravity of the body, 2 'Wy=0; 2 -W2' = 0; J& and the preceding equation is reduced to the following : i. = (2/0 + *5) 2 w + 2 . w (^ + o ........... .(i.) which expresses the theorem to be proved. This theorem may be more briefly expressed as follows: Let I be the moment of inertia of a body about an axis traversing its centre of gravity in any given direction, and I the moment of inertia of the same body about an axis parallel to the former at the perpendicular distance r ; then I = rJ-2W + I ........................ (2.) An analogous proposition for surfaces has been demonstrated in Article 95, Theorem Y. COROLLARY I. The radius of gyration (e) of a body about any MOMENTS OF INERTIA. 517 axis is equal to the hypotenuse of a right-angled triangle, of which the two legs are respectively equal to the radius of gyration of the body about an axis traversing the centre of gravity parallel to the given axis (e n ), and to the perpendicular distance between these two axes (r ). That is to say, e 2 = ^ + e? ............................ (3.) COROLLARY II. The moment of inertia of a body about an axis traversing its centre of gravity in a given direction, is less than the moment of inertia of the same body about any other axis parallel to the first. COROLLARY III. The moments of inertia of a body about all axes parallel to each other, which lie at equal distances from its centre of gravity, are equal. 577. Combined Moments of Inertia. THEOREM. Tlw combined moment of inertia of a rigidly connected system of bodies about a given axis, is equal to the combined moment of inertia which the sys- tem would have about tlie given axis, if each body were concentrated at its own centre of gravity, added to the sum oftJie several moments of inertia of the bodies, about axes traversing tJieir respective centres of gravity, parallel to the given axis. Let W now denote the mass of one of the bodies, I its moment of inertia about an axis traversing its own centre of gravity parallel to the given common axis, and r Q the distance of its centre of gravity from that common axis. Then the moment of inertia of that body about the common axis, according to Article 576, equation 2, is Consequently, the combined moment of inertia of the system of bodies is E D I = -Wr;+il,; ..................... (1.) 578. 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 ej, "the square of the radius of gyration. The mass of an unit of volume is in each case denoted by w. BOOT. AXIS. W Io tl 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, 25 Transverse diameter Transverse diameter Axis, 2a Axis, 2a Diagonal, 26 4*^ 15 2r 2 5 2r 2 5 3 4-rwar* III. Ellipsoid semi-axes, a. 3 15 IV. Spherical shell external radius r, internal />.... V. Spherical shell, insensibly thin radius r, thick- 3 15 8*w(r 5 r' 5 ) 5 2(r 5 r' 5 ) 3 2*001* 2gewabc 2-rwa(r 2 r' 2 ) 4>rwardr kxwardr Swdbc 4wabc 15 5(r 3 -r' 3 ) 2r 2 VI. Circular cylinder length 3 3 r- 2 VII. Elliptic cylinder length 2a, transverse semi-axes 5. c, VIII. Hollow circular cylinder- length 2a, external ra- dius ?*, internal r', 2 4 IX. Hollow circular cylinder, insensibly thin length 2a, radius r, thickness dr, X. Circular cylinder length 2 r 2 r 2 + a 2 c 2 a 2 XI. Elliptic cylinder length 2a, transverse semi-axes J c, ... 6 XII. Hollow circular cylinder- length 2a, external ra- dius r internal r' 6 -<3(r* r'*) 6 (. + 4a 2 (r 2 -r' 2 )} 4 3 8wabc(b z + c 2 ) 43 r 2 +r '2 tt 2 XIII. Hollow circular cylinder, insensibly thin radius 4 '3 2" + 3~ XIV. Rectangular prism di- mensions 2a, 25, 2c, XV. Rhombic prism length 2a, diagonals 25, 2c, .... XVI. Rhombic prism, as above, 3 3 6 c 2 a 2 6 3 3 3 MOMENTS OF INEKTIA. 519 579. Moments of Inertia found by Division and Subtraction. - Each of the solids mentioned in the table of the preceding Article can be divided into two equal and symmetrical halves by a plane perpendicular to the axis. The radius of gyration of each of those halves is the same with that of the original solid. Each of the solids can also be divided into four equal and symmetrical wedges or sectors by planes traversing the axis j 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 Q 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 5 2 + c 2 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 B A B = 2 6, CAC = 2 c, its two diagonals. Divide the prism into four equal right- angled triangular prisms by two planes traversing the diagonals and the longitudinal axis ; the radius of gyration of each of those prisms about that axis is the same with that of the original prism. Bisect B C in D, and join AD, in which take r Q = AE = f A D = Y+S- 1 B C = - } then E is the extremity of a longitudinal axis 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 OP 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, >j parallel to y, and f parallel to z, and are given by the equations U_ . tt _ __ > a TIT .(1.) and the radius of gyration % 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, '? (2.) This method may be exemplified by deducing the radius of gyration of an ellipsoid about any one of its axes (Article 578, case 3) from that of a sphere (ib., case 1). 581. The Centre of Percussion of a body, for a given axis, is a point so situated, that if part of the mass of the body were concen- trated at that point, and the remainder at the point directly oppo- site in the given axis, the statical moment of the weight so distri- buted (Article 42), and its moment of inertia about the given axis, would be the same as those of the actual body in every position of the body. In fig. 239 let XX be the given axis, and G the centre of gravity of the body. It is evident, in the first place, that the centre of percussion must be somewhere in the perpenfli- 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. the 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 : : BC : GB : GC ................. (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 = G C, and ^ is the radius of gyration of the body about an axis parallel to X X and traversing G ; and substituting for B its value from equation 1, viz., B = Wr -^ B C, we find, for the dis- tance of the centre of percussion from the axis, 50 = = -jf+r,j ........................ (3.) r o r o and for its distance from the centre of gravity, Q GB = BC r = ^- 5 ...................... (4.) TQ The last equation may also be expressed in the form GB-GC = eS; ........................ (5.) which preserves the same value when GB and GO are inter- changed 5 thus showing, that if a new axis parallel to the original axis XX be made to traverse the original centre of percussion, the new centre of percussion is the point C in the original axis. The proportion in which the mass of the body is to be considered as distributed between B and C takes the following form, when each of the last three terms of the proportion 1 is multiplied by r = GC : 44** ^ L ' :^ .................. (6.) The preceding solution is represented by the following geometrical construction : Draw G D _L C G and ^ ; join C D, perpendicu- lar to which draw D B cutting C G produced in B ; this point is the centre of percussion. Also, C D = e, the radius of gyration about X X ; and I) B is the radius of gyration about an axis traversing B parallel to XX. 522 PRINCIPLES OP DYNAMICS. If C E be taken = C D, E is sometimes called tlie CENTRE OF GYRATION of the body for the axis XX.* 582. No Centre of percussion exists when the axis traverses the centre of gravity of the body. In that case, the statical moment of the body is nothing ; and an equal mass, concentrated and uni- formly distributed round the circle BBB, whose radius is g , the radius of gyration, or at a set of symmetrically arranged points in that circle, has the same moment of inertia with the actual body. 583. Moments of Inertia about Inclined Axes. The object of the present Article and the remaining Articles of this section is to show the relations which exist amongst the moments of inertia of a body about axes traversing a fixed point in it in different direc- tions. The mathematical processes which it is necessary to employ for that purpose, though not very abstruse, are somewhat complex ; and the reader who wishes to study the more simple parts of the subject only, may take the conclusions for granted. It has already been shown in Article 575 that the moment of inertia of a body about a given axis denoted by x, is given by the equation (1.) in which, for the sake of brevity, 2 "W has been replaced by the single symbol S. The fixed point being the origin of co-ordinates, let S R 3 be the sum of the products of the weight of each particle into the square of its distance from that point ; a sum which is independent of the directions of the axis. Then because R 2 = a; 2 + y* -f 2?, the moments of inertia of the body relatively to three rec- tangular axes may be expressed as follows : Further, let the three sums of the weights of the particles of the body, each multiplied by the product of a pair of its co-ordinates, be thus expressed : Sy#; Szx; Sxy (3.) These will be called moments of deviation. Now, let three new rectangular axes of co-ordinates, denoted by x', y', zf, traverse the same fixed point in the body; let the angles which they make with the original axes be denoted by * As to the centres of percussion and gyration, and other remarkable points in a rigid body, see a memoir by M. Poinsot in Liouvillds Journal for 1857. MOMENTS OF INERTIA. 523 A , A , A , xx, xy, xz, y , zx, z^,zz f Then for any given particle, the new co-ordinates are thus expressed in terms of the original co-ordinates : x 1 = x ' cos xx' + y ' cosyos' + z cos zx', ............ (5.) and analogous equations for y* and z' ; and the original co-ordinates are thus expressed in terms of the new co-ordinates : x = x' ' cos x x' + y' cos x j/ + z' ' cos x z'', &c ....... (6.) The nine angles of equation 4 are connected by the relations : that the sum of the squares of the cosines of any three angles in one line, or in one column, is unity ; for example, COS 2 XX' + COS 2 Xtf + COS 2 C^ = lj ................. (7.) and that the sum of the three products of the pairs of cosines of the angles in a pair of lines, or a pair of columns, is nothing ; for example, A A A A A A cos y x' ' cos z x 1 + cos y y' ' cos z y 4~ cos y z' ' cos z z' = 0. ... (8.) A relation deduced from the preceding is this, that the cosine of each angle is equal to the difference between the binary products of the cosines of the four angles, which are neither in the same line nor in the same column with the first, these binary products being taken diagonally ; for example, A A A A A cos xx 1 = cosyy'-cos zz 1 - cos 2/^-cos zy'. ......... (9.) and similarly for the other cosines. Now, if for the new co-ordinates x 1 , tf, z', in the six integrals, S< Sy' 2 , S* Sy'z', Sz'V, Stf^, there are substituted their values in terms of the original co-ordi- nates, as given by equation 5 for x', and analogous equations for y' and zf, there are obtained the six expressions for those integrals relatively to the new axes, in terms of the integrals relatively to the original axes, and of the cosines of the nine angles between the 524 PRINCIPLES OP DYNAMICS. new and the original axes ; but it is unnecessary here to write those equations at length, for they are precisely similar to the equations of transformation in Article 106 (pages 92, 93), substituting only Sx*, Sy 8 , Sz 2 , Syz, Szx, for p, f , p yy , p ft , p yt , p f ,, p ty , and making the like substitutions in the symbols referring to the new co-ordinates. 584. Principal Axes of Inertia. THEOREM. At each point in a body there is a system of three rectangular axes, for which ilie moments of deviation are each equal to nothing. Supposing such a set of axes to exist, let co-ordinates parallel to them be denoted by x l} y ly z v Then the property which they are required to have is expressed by the equations Sy 1 2r 1 = 0; S 1 a? 1 = 0; Saj iyi = ............... (1.) Co-ordinates parallel to a set of axes, for which the integrals So 2 , \ S ~ " 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 111 . then will the reciprocal of the square of the semidiameter of that ellipsoid in any direction represent the moment of inertia about an axis traversing the origin in that direction, as expressed by the equation Such an ellipsoid, when described about the centre of gravity of the body as a centre, is called by M. Poinsot the central ellipsoid. If Ij, I 2 , I 3 , be ranged in their order of magnitude, it is evident that the greatest of them, T lt is the greatest moment of inertia of the body about any axis traversing the fixed point ; that the least, I 3 , is the least moment of inertia about any such axis j 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 I x are equal. In this case equation 3 becomes I = Ii cos 2 * + 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^, 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 c , in a direction making the angles , /3, y, with the principal axes at the centre of gravity. Then with respect to a set of rectangular axes traversing the new point parallel to the original axes, the new moments of inertia are I, = I 02 4- W rl sin 2 /3; . ................ (8.) I, = I 08 + W rl sin and there are afc the same time moments of deviation represented by S y z = W rl cos /3 cos y ; S z x = "W r% ' cos y cos * j 1 /n \ S x y = W rjj cos a, cos (& ; ) so that the principal axes at the new point are not parallel to those at the centre of gravity, unless two at least of the direction cosines of r are null ; that is to say, unless the new point is in one of the original principal axes, when all the moments of deviation vanish, and the new axes are parallel to the original axes. 586. The Resultant Moment of ]>< riatiou about a given axis is represented by the diagonal of a rectangular parallelogram of which the sides represent the moments of deviation relatively to two rectangular co-ordinate planes traversing the given axis. Let the principal axes and moments of inertia at a given point be known, and let three new axes of moments, denoted by x, y, z, be taken in any three rectangular directions making angles with the original axes denoted as in the equations of Article 583. Then the moments of deviation in the new co-ordinate planes are S y z = cos y x^ * cos z x^ S x\ + cos y y cos z y { S y\ -j- cos y Zi ' cos z %i S z\, .................... (1.) and similar equations for S z x, and S x y, mutatis mutandis. Sub- stituting for S x\, &c., their values, S R 2 L, &c., and observing that A A A A A A cos y #! cos z x l + cos y y l cos z y + cos y z l cos z z l = 0, those equations become A A A A b y z = Li ' cos y x-, cos z x l I 2 * cos y y cos c y, 1 3 -cos yzi -cos zzi, ..................... (2.) MOMENT OF DEVIATION UNIFORM ROTATION. 529 and similar equations, mutatis mutandis, for Szo;, Secy; 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: K x = J [ll cos 2 xx^ + Il cos 2 x y l + II cos 2 x z l (Ii cos 2 x Xi + 1 2 cos 2 x yi + I 3 cos 2 x 2-j) 2 j ; = ij {ll cos3 35*1 + 11 ' COS 2 X 2/j 4- Ijj COS 2 # 2?j I This equation, expressed in terms of the axes of the ellipsoid of inertia, becomes as follows : A A A TT _ //COS 2 030?!, COS 2 ^?/! , COS* 35^ _ 1 x ~ ~~ ~~ ~~ ~ but the positive part of this expression is well known to be the value of t 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 in which it is to be observed, that Js 2 n 2 represents the length of the tangent to the ellipsoid, from the point of contact to the foot of the normal. Also, let 6 be the angle between the normal n and the semidiameter 5; then Js s n s : w = tan 4, and K, = I,tan* ........................... (6.) SECTION 2. On Uniform flotation. 587. The Momentum of a body rotating about its centre of gravity is nothing, according to the principle of Article 524. As every motion of a rigid body can be resolved into a translation, and a rotation about its centre of gravity, the rotation will be supposed to take place about the centre of gravity of the body throughout this section. 588. The Angular Momentum is found in the following manner : Let x denote the axis of rotation, and y and z any two axes fixed in the body, perpendicular to it and to each other. Let a be the 530 PRINCIPLES OF DYNAMICS. angular velocity of rotation. Then the velocity of any particle AY, whose radius vector is r = J y s + z?, is ar = and the angular momentum of that particle, relatively to the axis of rotation, is War* Wa, =_ (s . +y); being the product of its moment of inertia into its angular velocity, divided by g, because of the weights of the particles having been used in computing the moment of inertia. Now let a line, parallel to the radius vector of the particle, be drawn in the plane of y and *j 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-= ax J y 2 + *?; and this may be resolved into two components; one relatively to the axis of y, W azx 9 ' and the other relatively to the axis of , W a xy ~~ 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, 9 and relatively to the transverse axes, and if lines proportional to those three components be set off upon the three axes, the diagonal of the rectangle described upon them ANGULAR MOMENTUM. 531 will represent in direction the axis, and in length the magnitude, of the resultant angular momentum. It follows that the axis of angular momentum of a rotating body does not coincide with the axis of rotation, unless that axis is an axis of inertia; in which case the moments of deviation are each equal to nothing, and the resultant angular momentum is simply the product of the moment of inertia about the axis of rotation into the angular velocity, divided by g. Now let the axes of inertia be taken for axes of co-ordinates, and let the axis of rotation make with them the angles , /3, y. Resolve the angular velocity a about that axis into three components about the axes of inertia a cos ; a cos /3 ; a cos y; then the angular momenta due to those three components are respectively cb T a T a T - li cos :- \. z cos /3 ; - 1 3 cos / ; c7 . c7 U the resultant angular momentum is A = -' J{I* COS 8 * + I 2 2 COS 2 ft + I COS 2 y] j ..... ....... (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 & a I 3 cos / ~ ~~ 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 maybe 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 OP DYNAMICS. I, cos I 8 cos ft I s cos are the direction cosines of the normal of the ellipsoid of inertia. Hence the axis of angular momentum at any instant is in the direc- tion of tlw normal let fall from the centre of the ellipsoid of inertia upon a plane touching that ellipsoid at the end of that diameter which is tJie axis of rotation; and the angular momentum itself is directly as tlte 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 By the following geometrical construction, the preceding prin- ciples are represented to the eye : In fig. 241, let 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 l l .-.(7.) Let B 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 O. Then the moment of inertia, the moment of deviation, and their resultant, the total moment, have the following proportions : I :K : 1 OB 3 OB 2 -ON 'OB (8.) the direction of the axis of angular momentum is ON; and its amount is proportional to >B -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 Wtt 2 m WaV t being the moment of inertia of the particle multiplied by . Hence for the whole body the actual energy of rotation is that is to say, actual energy bears the same relation to angular velo- city and moment of inertia that it does to linear velocity and weight. Referring again to fig. 241, it appears that the actual energy of rotation is proportional to <*_ ,0V 2g -OR 2 '" " V '' 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 ; then the quantities of actual energy due to those three component rotations are a 2 T! cos 2 * a? I 8 cos 2 /3 a 2 I 3 cos s 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 tJie actual energies due to tJie components of that rotation about the three axes of inertia. 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 of the 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 K, 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 2 ) = constant; (i.) constant. To find how these laws regulate the changes of direction of the instantaneous axis, eliminate the angular velocity as follows : g A 2 r + K 2 I? cos 2 + I* cos 2 /3 + I* cos 2 y 2 E I I t cos 2 + I 2 cos 2 /3 + I a 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 + OK 2 ON 2 , and that I oc 1 ^ OB? ; whence J2 Tjr2 1 That is to say, the normal N is constant in length as well as fixed in direction; and therefore a body rotating freely moves in such a manner, that its ellipsoid of inertia always touches a fixed plane (viz., the plane T N K), the instantaneous axis traversing the point of contact. The second of the equations (1.) further shows, that the angular velocity, being given by the equation is at each instant proportional to the semidiameter O K. If the instantaneous axis O E. and the position of the body are known at any instant of the rotation, the invariable plane FREE ROTATION FIXED AXIS DEVIATING COUPLE. 535 and the length and direction of the fixed normal O N, are also known. Conceive a curve to be drawn on the ellipsoid of inertia through all the points whose tangent planes are at the same perpendicular distance O N" from the centre then the instantaneous axis O R will always traverse that curve, and will always be found in the surface of a cone of the second order fixed relatively to the axes of inertia, whose equation is Let this be called the rolling cone. Then the motion of the body will be such as would be produced by the rolling of the rolling cone upon a fixed cone generated by the motion of O R relatively to ON. As free rotation is of unusual occurrence in practical mechanics, I shall refrain from applying its principles to special examples here, and shall refer the reader to the work of M. Poinsot on Rotation, and to a paper by Professor Clerk Maxwell in The Transactions of ike Royal Society of Edinburgh, vol. xxi. 591. Uniform Rotation about a Fixed Axis. When a body ro- tates about a fixed axis traversing its centre of gravity, with an uniform angular velocity, its actual energy is still represented, as in the case of free rotation, by E = - = constant; , ..................... (!) and its angular momentum by A-- /v /(I 2 + K 2 ) = constant; ................. (2.) y but unless the axis of rotation is an axis of inertia, the axis of angu- lar momentum N is no longer fixed, but revolves about the fixed axis of rotation O R, with the angular velocity a. In order to produce that continual change in the direction of the axis of angu- lar momentum, a continual angular impulse, or continuously acting couple, must be applied to the body ; and unless that couple be applied, the axis of rotation will not remain fixed. 592. The Deviating Couple, as the couple required for the above purpose is called, must have its axis always perpendicular to the axis of angular momentum, otherwise it would alter the amount of the angular momentum, contrary to the condition of uniform rota- tion. The axis of the deviating couple must also be always per- 536 PRINCIPLES OF DYNAMICS. pendicular to the axis of rotation, because, in order that it may not alter the actual energy of the body (contrary to the condition of uniform rotation), the pair of equal and opposite forces composing it must act through points having no motion ; that is, through points in the axis of rotation. (In machines, the forces constitut- ing the deviating couples are supplied by the pressures of the bear- ings against the axles.) It appears, therefore, that the axis of the deviating couple must always be perpendicular to the plane O K N, which contains the axes of rotation and of angular momentum ; and that the pair of forces constituting it must always act in that plane, changing their direction as the body rotates, with an angular velo- city equal to that of the body. The direction of the deviating couple must be such as would of itself tend to turn ON towards OR. To determine the amount of the deviating couple, let tf, as before, denote the angle R IsT. Then in the indefinitely short interval of time d t, the direction of the axis of angular momentum is shifted through the indefinitely small angle a d t sin 0, and the result differs to an indefinitely small extent from that which would be produced by combining with the actual angular momentum A, an angular momentum about the axis of the deviat- ing couple represented by -VI' + K 2 ' sin 6 dt\ y and this is the angular impulse to be supplied in the interval d t by the deviating couple ; therefore the deviating couple is M = A a-sin 6 = - ^/I + K 8 sin FT but sin d = ; therefore (1) and if Q be the magnitude of each of the forces constituting this couple, and I the length of the arm on which they act (being the distance between their points of application to the axis), so that M = QJ, then which being compared with the expression for deviating force in Article 537, shows that the force of a deviating couple bears the DEVIATING AND CENTRIFUGAL COUPLES ENERGY OP COUPLES. 537 same relation to the angular velocity #, the moment of deviation K, and the arm I, which a simple deviating force bears to the linear velocity v, the weight W, and the radius vector r. To represent these principles graphically, it is to be observed that in fig. 241, the ratio of moment of deviation to the moment of inertia is K:I ::BN :ON; ....................... (3.) and that this also expresses the ratio of the deviating couple to double the actual energy, viz. : The reaction of the axis of the rotating body on its bearings, equal and opposite to the deviating couple, that is, tending to turn the axis of those bearings towards the axis of angular momentum O N, ** is called the CENTRIFUGAL COUPLE. It is balanced, in machines, by the strength and rigidity of the framework. The amount and direction of the deviating couple might have been determined by finding the resultant couple of the deviating forces required to make each particle of the body revolve in a circle about O .R with the common angular velocity; and the result would have been exactly the same. 593. Energy and Work ol Couples. The energy exerted by a couple is the product of the common magnitude of its pair of forces into the sum of the distances through which their points of appli- cation move in the interval of time under consideration; and as that sum is the product of the length of the arm of the couple into the angle through which it rotates about its axis in that time, the energy exerted may be expressed by Vldi = M.di = M.adt, .................. (1.) d i being the angle of rotation about the axis of the couple in the interval d t, with the angular velocity a. When the couple acts against the direction of rotation, the above expression becomes negative, and represents work performed. If a couple be applied to a rotating body whose axis of rotation makes an angle

; also, let the unbalanced couple which acts on the body be resolved into three rectangular components denoted by then LAW OF VARIED ROTATION. 539 M,; M y ; M,; Those three equations express the relations between the vinbalanced couple and the rate of change of the angular momentum. Those relations may otherwise be expressed as follows : let $ be the angle made by the axis of the unbalanced couple with the axis of angular momentum; then the couple may be resolved into two components, M cos *]/ and M sin -$/, of which the former produces variation in the amount of angular momentum, and the latter, deviation of the axis of angular momen- tum, according to the following laws : = M cos *; A = M sin +; ............. (2.) in the latter of which equations, d % denotes the angle through which the axis of angular momentum deviates in the indefinitely small interval dt, in the plane which contains that axis and the axis of the couple M, and in a direction towards the latter axis. This equation of deviation of angular momentum has in fact been already employed in Article 592, to find the deviating couple required in order to fix the axis of rotation, when that differs from the axis of angular momentum. The equations 1, or their equivalents 2, are not of themselves sufficient to determine the variations of motion of a body rotating without a fixed axis; for in such a body, the angular momentum may change by a change of the direction of its axis relatively to the body, as well as by a variation of amount, or a deviation of its axis in absolute direction. This is expressed by putting for the angular momentum its value in terms of the moments of inertia and devia- tion relatively to the instantaneous axis, viz., A = - j when the equations 1 take the following form : g3 x = < a cos A Jl? + K 2 I ; and analogous equations for dt ( ) <7M y and^M,;. ........................ (3.) while the equations 2 become 540 PRINCIPLES OF 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

and by dividing this equation by a, and observing that adt = di, where di is any indefinitely small angle of rotation, it is made to assume the following forms : VARIED ROTATION ABOUT A FIXED AXIS. 541 ._-. Ida I d s i lada M- ot0 =7- = - '-779 = -JT-; ............... (2.) - showing that the direct couple is equal at once to the variation of angular momentum about the fixed axis divided by the time, and to the variation of actual energy divided by tJie angular motion. 596. Analogy of Varied Rotation and Varied Translation. When the equation of Article 554 is compared with equation 2 of Article 595, it appears that those equations are exactly analogous to each other, and that the former is transformed into the latter, when for P, W, s, v, there are respectively substituted M cos 0, I, i, a ; that is to say, a direct couple for a direct force, moment of inertia for weight, angular motion for linear motion, and angular velocity for linear velocity. Consequently, by making those substitutions, any equation relat- ing to the varied translation produced by a direct force, may be transformed into a corresponding equation respecting the varied rotation of a body about a fixed axis traversing its centre of gravity produced by a direct couple. Examples of this principle are given in the two following Articles. 597. Uniform Variation of angular velocity is produced by a con- stant couple, and is analogous to the vertical motion of a heavy body, as given in Article 533. In that Article, g is the proportion of the moving force to the mass of the body. Let M be the couple, and let (p = ; that is, let the couple be altogether about the axis of rotation- Then for g is to be substituted which is to be considered positive when in the direction of the initial angular velocity a ; and for h is to be substituted i. Then equations 1 and 3 of Article 533, being transformed, give for the angular velocity and total angular motion at the end of a given time t t the expressions Equation 4 gives 54:2 PRINCIPLES OP DYNAMICS. which is also the result of applying to the present case the law of the conservation of energy; the right hand side of the equation being the potential energy exerted, and the left hand side the actual energy stored. To find through what angle a body will turn before stopping against a constant resistance, its initial angular velocity being a , it is to be considered that if II is the resistance, and I its perpendi- cular distance from the fixed axis, the resisting couple is and tha,t a is to be made = ; whence equation 2 gives 598. Gyration about a fixed axis, or Angular Oscillation, is alter- nate rotation to one side and to the other of a middle position. Let a straight line be conceived to be drawn perpendicular to the axis of the gyrating body, to serve as an index ; let its middle posi- tion be denoted by 0, and its angular displacement from that posi- tion by i, positive or negative according as it is to one side or to the other ; and let ^ be the semi-amplitude of gyration, or extreme displacement. To produce gyration, the body must be acted upon by a couple directed towards the middle position ; that is, contrary to the displacement i. In most cases which occur, the couple is either exactly or nearly proportional to the displacement. Suppos- ing it to be exactly proportional, let Mj be its extreme magnitude irrespective of sign ; then the negative sign showing that the couple is contrary to the dis- placement, tending to restore the body to its middle position. It is obvious from this equation, that gyration is analogous to straight oscillation, explained in Article 542 j and that the equa- tions of that Article are to be transformed by substituting respec- tively for f. Wa 2 _ dx r, x, Q, _, Q x , _, fl , . . ., ! __ , ^ ^, M 15 * M, a, Vy 1 . t-j l/i -L For brevity's sake, let the substitute for a 2 be thus expressed : *; ........................... (2-) GYRATION SINGLE FORCE - ROTATION WITH TRANSLATION. 543 then by transforming equation 4 of Article 542, it appears that the number of double gyrations per second is which is independent of the semi-amplitude ^ so long as MI is pro- portional to tj, and I is constant. This constitutes isochronism, and is the property aimed at in the balance wheels of watches, where I is the moment of inertia of the wheel, and the couple is derived from the elasticity of the balance spring. The equations 2 and 3 being transformed, give for the angle and angular velocity of displacement at any instant, i ii cos k t ; di I ............... (4.) a = = -k^ l $mkt, u t and the maximum couple M^ in terms of the number of double oscillations per second n, is given by the equation M. = ***=*-< * (5) g g 599. A Single Force applied to a body with a fixed axis causes the bearings of the axis to exert a pressure equal, opposite, and parallel ; so that if the line of action of the force traverses the fixed axis, it is balanced ; and if not, a couple is formed whose moment is the product of the force into its perpendicular distance from the axis, and whose effects are such as have been already described. SECTION 4. Varwd Rotation and Translation Combined. 600. General Principles. All rotation of a body about an axis, fixed or instantaneous, which does not traverse the centre of gravity of the body, is to be considered as compounded of rotation about a parallel axis traversing the centre of gravity, and translation of the centre of gravity with a velocity equal to the product of the angu- lar velocity into the distance of the centre of gravity from the actual axis of rotation. Consequently, every variation of the motion of a body, which consists in a variation of the angular velocity about an axis, fixed or instantaneous, and not traversing the centre of gravity, is to be considered as producing a change of the momentum, which is the product of the mass of the entire body into the velocity of its centre of gravity, and a simultaneous change of the angular momentum due to the rotation of the body with the given angular velocity 544 PRINCIPLES OF DYNAMICS. about an axis traversing its centre of gravity parallel to the actual axis of rotation ; and the force required to produce the given varia- tion of motion will be the resultant of the force required to produce the change of momentum, applied at the centre of gravity, and the couple required to produce the change of angular momentum. 601. Properties of the Cenlre of Percussion. In fig. 239, Article 581, page 520, let G be the centre of gravity of a rigid body whose weight is W, XX the axis about which, in the interval dt, a change of angular velocity denoted by d a takes place, and G C =. r , the perpendicular distance of the centre of gravity from that axis. Then the force, in a direction perpendicular to the plane of X X and G 0, required at G to produce the change of momentum, is and the couple required to produce the change of angular momen- tum due to the change of angular velocity d a about the axis GDXX is and the resultant of that force and couple (according 'to Article 41) is a force acting in the same plane with them, parallel and equal to F, and in the same direction, but acting through a point whose distance from G, in a direction opposite to G C, is M L el that is, the resultant of the force and couple is a single force F act- ing through the centre of percussion B corresponding to the given axis. (See Article 581, equation 4.) Now suppose, as in Article 581, that the weight of the body is distributed in two rigidly connected masses, one concentrated at C and the other at B, and having their common centre of gravity still at G. Then in producing the same change of angular velo- city d a about the axis X C X, the momentum of C is unchanged, while that of B undergoes the change 9 9 being the exact change of momentum already given in equation 1 ; a consequence, indeed, of the fact, that the centre of gravity is not changed by the concentration of the masses at B and C; and to CENTRE OF PERCUSSION FIXED AXIS DEVIATING FORCE. 545 produce this change of momentum in the interval d t, there is re- quired the same force F applied at B, which has already been found; which proves the following THEOREM I. If the mass of a body be conceived to be concentrated at two rigidly connected points, one at a given axis, and the otlier at the corresponding centre of percussion, so as not to alter the position of the centre of gravity of the body, the force required to produce a given change of angular velocity in the body about the given axis is tJie same, in magnitude, direction, and line of action, with that required to produce the corresponding change of motion in that part of tlie mass which is conceived to be concentrated at tJie centre of percussion. This proposition might also have been arrived at by considering THEOREM II. If a body rotates about a given axis not traversing its centre of gravity, and the mass of that body be conceived to be con- centrated at the axis of rotation and centre of percussion so as not to alter the centre of gravity, tJie momentum, the angular momentum, and the actual energy of the body are not changed by that concentra- tion of mass. For the centre of gravity being unchanged, the momentum ia unchanged ; and because (by the definition of the centre of percus- sion) the moment of inertia about the axis of rotation is unchanged, the angular momentum and actual energy are unchanged. Q. E. D. COROLLARY. From Theorem I., and from equation 5 of Article 581, it follows, that the action of an impulse upon a free body at either of the points B or C, produces a rotation about an axis tra- versing the other point. 602. Fixed Axis. When the axis of rotation X X is fixed, an impulse applied to the centre of percussion B, in a direction per- pendicular to the plane B X X, simply alters the angular velocity according to the principles explained in the last Article, without causing any additional pressure between the axis and its bearings. But should the force giving the impulse not traverse the centre of percussion, or traverse it in a different direction, it is to be resolved by the principles of statics into two components, one traversing the centre of percussion in the required direction, and the other tra- versing the axis of rotation; when the former will produce change of motion, and the latter will be balanced by the resistance of the bearings of the axis. 603. The Deviating Force of a body rotating about a fixed axis not traversing its centre of gravity is the resultant of the deviating force due to the revolution of the whole mass conceived as concen- trated at its centre of gravity, found as in Article 540, combined with the deviating couple due to the rotation of the body with the same angular velocity about a parallel axis traversing the centre of gravity, found as in Article 592. This resultant deviating force is 546 PRINCIPLES OP DYNAMICS. supplied by the resistance of the bearings of the axis, and an equal and opposite CENTRIFUGAL FORCE is exerted by the axis against the bearings. 604. A Compound Oscillating Pendulum is a body supported by a horizontal fixed axis, about which it is free to swing under the action of its own weight, its centre of gravity not being in the axis. Now, by Article 601, Theorem II., the momentum and angular momentum of the body are at every instant the same as if its mass were concentrated at the axis and at the centre of oscillation in the proportions given by Article 58 Inequations 1 and 6 ; and by the definition of the centre of oscillation, the statical moment of the weight of the body with respect to the axis, being the couple which causes the motion, is in every position the same as if the mass were concentrated in these proportions ; therefore, the motion of the body is exactly the same as if it were so concentrated ; that is to say, it oscillates in the same time and according to the^ same laws, with a simple oscillating pendulum as defined in Article 544, whose length is the distance from the axis X C X to the centre of oscillation B, as given by equation 3 of Article 581, viz. : (1.) Such a simple pendulum is called the equivalent simple pendulum. It is obvious that, for a given body swinging about all possible axes parallel to a given direction in the body, the shortest equiva- lent simple pendulum is that whose length is the minimum value of 3JC as given by the above equation. That minimum length corresponds to the condition, P '"' 7* whence, \ ....................... (2.) min. B C = 2, ft :j that is to say, the least period of oscillation of a pendulous body takes place when the distance of its centre of gravity from its axis is equal to the radius of gyration about a parallel axis traversing the centre of gravity; and the length of the equivalent simple pen- dulum is double of that radius of gyration. If for a given direction of axis, a pair of points be so related that each is the centre of percussion for an axis in the given direction traversing the other (as shown by Article 581, equation 5), then the period of oscillation about either axis is the same. From the properties of the centre of percussion explained in this Article, it is sometimes called the CENTRE OF OSCILLATION. COMPOUND PENDULUMS GOVERNORS. 547 ; 605. Compound Revolving Pendulum. To avoid unnecessary complexity in the theory of a compound revolving pendulum, let the body of which it consists be of such a figure and so suspended, that the straight line C G B (fig. 239), traversing the point of sus- pension C and the centre of gravity G, shall be one of the axes of inertia, and that the moments of inertia about the other two axes shall be equal. Then for every axis traversing the centre of gravity at right angles to C G B, the radius of gyration is the same ; and consequently, for every axis traversing the point of suspension C at right angles to C G B, the centre of percussion B is the same ; and the body moves exactly like a simple revolving pendulum of the length C B, and height C B cos 6, if 6 is the angle which it makes with the vertical. It is to be borne in mind, that in order that a pendulum may revolve according to the above law, it must have no rotation about its longitudinal axis B G C, but must swing as if hung by a double universal joint at C (Article 492). 606. A Rotating Pendulum (fig. 242) is a body B C G suspended by a point C not in the centre of N 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 I t be its moment :x of inertia about G E, and I 2 its moment of inertia about G C, and ^, ft, the corresponding radii of gyration. Let the angle X C G = *; let C G = r ; and let the weight of the pendulum be W. Then, a being the angular velocity of rotation about the vertical axis, it appears from Articles 592 and 586 that the deviating couple due to rotation about a vertical axis traversing Gis (I, L) cos a sin a. = (g* el) cos a. sin <7 V g v 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 Q cos a, being the height of C above G j that is to say, Wa 2 rl cos . sm : 9 making for the entire deviating couple Wo, 2 548 PRINCIPLES OF DYNAMICS. and this couple has to be supplied by means of the weight of the pendulum acting with the leverage r sin at; that is, it must be equal to W r sin . Dividing by this quantity, we find and putting for a 2 its value, 4 w 2 T 2 , where T is the number of turns per second, this leads to the equation A being the height of the equivalent simple revolving pendulum, as given in Article 539, equation 2. When t t , the radius of gyration about C G, is insensibly small compared with g u 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 & = ?,, 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 5, and to move with the velocity v in a line of flight whose perpen- dicular distance from the axis of suspension is r'. Then the angular momentum of the ball relatively to the axis of suspension is <> and because the ball lodges in the pendulum, this angular momen- tum is wholly communicated to the joint mass consisting of the ball and the pendulum, which swings forward, carrying with it 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 B = ~, ............................ (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, / versin i; that is to say, and the corresponding angular momentum of the combined mass is B ~V I ; which, being equated to the angular momentum of the ball / before the collision (1), gives the equation (4.) giving for the velocity, momentum, and actual energy of the ball, respectively, BVZ (5.) BV* The energy of the combined mass after the collision being , ? and less than that of the ball before the collision in the proportion of 6 r' 2 : B I 2 , shows, that an amount of energy denoted by b disappears in producing heat and molecular changes in the ball and in the soft mass in which it is lodged. To measure the impulse produced by the explosion of gunpowder, the gun to be experimented on is to be fixed to and form part of the pendulum, and a ball is to be fired from it. The gas produced by the explosion exerts equal pressures during the same time, that is, equal impulses, forwards against the ball, and backwards against the gun, and the pendulum swings back through a certain angle, which is registered by an index as before, and from which the 550 PRINCIPLES OF DYNAMICS. maximum velocity of the centre of percussion of the pendulum can be calculated as before by equation 3. Let r' now denote the distance from the axis of suspension to the axis of the gun, and P the pressure exerted by the explosive gas at any instant ; the total impulse exerted by the gas is / P d tj and the angular impulse / / P d t' } which being equated to the angular momentum pro- duced in the pendulum, gives BVJ 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 &v 2 :BY 2 ::BZ 2 :6r' 2 .................... (11.) 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. : 1) w , B w and + 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, i - -in? and instead of 6, The equation 10, hi 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. 552 CHAPTER IV. MOTIONS OF PLIABLE BODIES. COS. Nature of the Subject; Vibration. The motion of each par- ticle of a pliable body may always be resolved into three components : that which it has in common with the centre of gravity of the body, being the motion due to translation of the whole body; that which it has about the centre of gravity of the body, being the motion due to rotation of the whole body; and a third component, being the motion due to alterations of the volume and figure of the body and of its parts. This third component is alone to be considered in the present chapter. The cinematical branch of the present subject, that is to say, the branch which comprehends the relations amongst the displace- ments of the particles in a strained solid from their free positions, and the strains or disfigurements of its parts accompanying such displacements, has already been treated of generally in Articles 248, 249, 250, 260, and 261 ; with reference to bending, in part of 293, part of 300, 301, part of 303, part of 304, part of 307, part of 309, part of 312, and part of 319; with reference to twisting, in part of 321 and part of 322; and again with reference to bending, in part of Article 340. The dynamical branch of the subject has been, to a certain extent anticipated in Article 244, where resilience is defined ; in Article 252, where potential energy of elasticity is defined ;* in Articles 266 and 269, which relate to the resilience of a stretched bar and the effect of a sudden pull; in Article 305, which relates to the resilience of a beam; in Article 306, which relates to the effect of a suddenly applied transverse load ; and in Article 323, which relates to the resilience of an axle. The motions due to strains amongst the particles of pliable bodies being all of limited extent, and consisting in changes of the dis- placement of each particle from the position which it would occupy in a state of equilibrium, which displacement is limited and gene- rally small, are of the kind called VIBRATIONS, and are more or less * In Article 252, the first employment of this function is correctly ascribed to Mr. Green; but it is right also to mention, that its use was independently discovered by M. Clapeyron. VIBRATION CONDITION OF ISOCHROXISJI. 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 tlie displacements at each instant. This is the CONDITION OF ISOCHRONISM, and has already been illustrated in Articles 542, 543, 544, 545, and 557, Example III., for the case of a single par- ticle acted on by a single force, and in Article 598 for the analogous case of a gyrating rigid body, where angular is substituted for linear displacement, and a couple for a force. To express that condition by an equation suited to the present class of questions, let "W -f- g be the mass of a particle, 5 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 2 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 a = A 7T and the period of a double oscillation by L (2.) 1 2* 554: PRINCIPLES OF DYNAMICS. All the equations of Article 542 and Article 557, Example III., are made applicable to the present case, by substituting respectively for QorQ,, Q,, rorajz, x, F,, F, & S, respectively, where Fj represents the maximum force, corresponding to Sj, the maximum displacement, or semi-amplitude; consequently, if in order to make the formulae more general we represent by t any instant of time at which the particle reaches the extremity of an oscillation, we have 8 = \ cos a (t t ) ; "j - = - a \ sin a (t - 1 ). ( ' ' CL t J 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 8 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 tJie 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 Ught 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, &, of the point for which deflection? 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"'y are two arbitrary constants. Thus the numbers in a second of the two series of component vibrations, viz., are proportional to the square roots of the stiffnesses of the spring in the directions of the two rectangular axes; that is, they are proportional to its thicknesses in these two directions respectively. If n x and n f are commensurable, the path of the vibrating mass is a closed curve ; for example, to take the simplest case, if n f = 2 n y) that path is such a curve as is represented in fig. 243. If n f VIBRATIONS IN GENERAL. 557 V) and n y are incommensurable, the path is of indefinite length ; but in every case it is wholly inscribed within the rectangle whose sides are the amplitudes 2 &, 2 v lt of the component vibrations. 612. Vibrations not Isochronous can only x be expressed mathematically by conceiving them to be compounded of a number of superposed vibrations, each isochronous in itself, but not isochronous with each other, as in the last example of the preceding Article; and the forces under which such vibrations take place are in like manner to be conceived to be resolved into component F . 243 forces, each proportional to a parallel com- ponent of the displacement. The art of resolving displacements of any kind whatsoever into components, each of which separately satisfies the conditions of isochronism, is a mathematical process which it will not be necessary to exemplify in this treatise. 613. Vibrations of an Elastic Body in General The general equations of the vibration of an elastic body are found by the aid of D'Alembert's principle (Article 568), by conceiving the body to be divided into indefinitely small rectangular or other regularly shaped molecules, and equating the components of the rate of varia- tion of momentum of each molecule to the corresponding com- ponents of the restoring force caused by the internal stresses, which restoring force, for each molecule, is at each instant equal and opposite to the share belonging to that molecule, of a distributed ex- ternal load that would, in a state of equilibrium, produce the actual state of disfigurement of the body at the instant. The condition of isochronism is expressed by making each restoring force propor- tional and opposite to the displacement of the molecule to which it is applied ; and the displacements, velocities, and forces for vibra- tions not isochronous are expressed by sums of series of corre- sponding quantities for isochronous vibrations. By the application of D'Alembert's principle as stated above, every equation concerning the equilibrium of an elastic body under external forces distributed amongst its molecules can be converted into a corresponding equation concerning its vibration. Example I. General Differential Equations. In Article 116, illustrated by fig. 58, are given the equations of internal equili- brium (2.) of an elastic solid for a rectangular molecule dx dy dz, expressing the three components of the external force per unit of volume of that molecule, in terms of the equal and opposite com- ponents of the internal forces arising from the variations of the six elementary stresses, pulls being considered as positive, and thrusts as negative. Those equations are converted into general 558 PRINCIPLES OF DYNAMICS. equations of vibration of the same molecule by substituting, at the right-hand sides of the three equations respectively, for 0, w d* w w, w . d*( ~g'~di* (I-) W where is the mass per unit of volume, and |, >?, , are the three components of the displacement of the molecule from its position of equilibrium. To make use of the three equations thus obtained, each of the six elementary stresses is to be expressed in terms of the six ele- mentary strains multiplied by the proper co-efficients of elasticity of the substance (Article 253); then each of the six elementary strains is to be expressed as in Article 250, by means of the differential co-efficients of the three component displacements g, *j, , and thus the three original equations are converted into three linear differen- tial, equations of the second order in i, >?, and , by the integration of which, with due regard to the circumstances of each particular problem, all questions respecting vibration are solved. It is un- necessary here to enter into details respecting those integrations. The most complete compendium of the processes which they in- volve and the results to which they lead, is contained in M. Lame's Legons sur VElasticite des Corps solides. Example II. Case of an Axis of Vibration. In figs. 244 and 245, Fig. 244. Fig. 245. S S ard the lines parallel to it represent a series of planes VIBRATIONS OF ISOTROPIC SUBSTANCE. 559- parallel to each other, and let the mode of vibration of the particles of the body be such, that all the particles in any one of those planes have equal displacements in parallel directions at the same instant. A straight line O X, perpendicular to all those surfaces, may be called an axis of vibration. Let the displacement of each particle, denoted at any instant by o, take place in a direction making an angle & with O X, in the plane of x y ; so that its component dis- placements are In the condition of equilibrium, conceive a square prism to extend along the axis X, as in fig. 244, and to be divided into cubical molecules, each of the volume dxdy dz, and mass d x d y d z. \y At a given instant in the state of vibration, let those molecules be displaced in the manner shown in fig. 245, the displacement of each point in each molecule depending, according to some law yet to be determined, upon the lapse of time and upon the distance, when in a state of equilibrium, of the plane of equal displacement containing it from 0, which distance is denoted by #; that is, let I = function of (t, x) ........................ (3.) Then it is evident, that each molecule, originally cubical, becomes directly strained and distorted; the direct strain along x (an elonga- tion if positive) being represented at any instant by ...................... (4.) dx dx and the distortion, in the plane of x y, by v = = sin 4 (5.) c?c dx The vibrating substance will be supposed to be isotropic as to elasticity, according to the definition given in Article 256, A being its direct and C its transverse elasticity. Then at a given plane of equal displacement, and at a given instant, there is a direct stress (tension being positive) of the intensity and a tangential stress of the intensity ~ ~ d W 560 PRINCIPLES OF DYNAMICS. and applying to these data the reasoning of the preceding example, we find that the components of the moving force, per unit of volume, acting on a given molecule, at a given instant, are as follows : d* Longitudinal, Q, = A -T-T, = - transverse, Q y = C -=-. - t = C -= a sin 6- } so that if we make g A gC (8.) 2 ' ^ - (\ w , z^: C .........It/.) W W we find for the equations of vibration, longitudinal, -; = a? -jAi (10.) d*m -d 3 * n . . transverse. ~v- 5 r=c 2 - T (11.) a r dx- The general integral of those two equations is given by the pair of equations, where (p, ^, x, , represent any functions whatsoever. But to obtain definite results, which can be used in calculation, the conditions of isochronism are to be applied ; and they lead to the following con- sequences : First, in order that vibrations may be isochronous, the restoring force must act along the direction of vibration ; that is, we must have Q, : Q y : : cos I : sin t- 3 (13.) and because for every known substance, A and C are unequal, this condition can only be fulfilled when either cos t or sin 6 is nothing ; that is to say, in an isotropic substance, isochronous vibrations are eitJier wholly longitudinal, or wholly transverse. Secondly, the moving force acting on a particle must be propor- tional and opposite to its displacement ; a condition expressed for longitudinal and transverse vibrations respectively, by VIBRATIONS. 561 where b* and b' s are two arbitrary positive constants. The most convenient way of expressing those constants, for reasons which will afterwards appear, is the following : b = b' = r A and A' being arbitrary lengths. Then it is easily seen, that to satisfy the equations 14 and 15, the displacements must be expressed as follows : i^-cos-^-cos^-^; (16.) Q ,_ O _ ., Yl = ^ COS ^- (# #' ) COS - (t t' Q ) (17.) i> ^u x x '> BO* ^'0? ^o> an d ^o being arbitrary constants, having values depending on the circumstances of each particular problem. These constants have the following meanings : ^! and >?! are the maximum semi- amplitudes of vibration. and , are the periodic times of a complete oscillation. 1 K a J!i ir c A and A' 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- A A' ment, x x , or x a/ , being an odd multiple of or . Their 4 4 A A' distance apart is or - (A, C, and E, in the figures). Ventral planes are those of maximum displacement, x # , or A A' x X'Q, 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% A A A (18.) d% 2* , . 2*, x 2, . ' direct strain, = ?! sin (x X Q ) cos (t 1 ). For transverse vibrations, 2o 562 PRINCIPLES OF DYNAMICS. velocity of ) dn 2 KG ST. . 2*c ,. .,. aparticlejj-^- "'- 003 ^^-^)- 8111 -^ ('-'); \ (19.) d w 2 w . 2 T , . x 2 T c , Distortion, -=- = r vi ' sin (a; x ) cos (t t ). a# 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. Wares of Vibration consist in the transmission of a vibra- tory state from particle to particle through a body. Let OX 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 t is expressed by a functipn 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 where a and c are the velocities of transmission of longitudinal and transverse vibrations respectively. Now the equations 1 have already been shown in Article 613 to be forms of the integrals of the general equations of vibratory motion, a and c having the values there given, viz. : (2.) which accordingly are the respective velocities of transmission of waves of longitudinal and transverse vibration in a medium whose weight per unit of volume is w, and its direct and transverse elas- ticities A and C. In a fluid, for which C = 0, the transmission of waves of transverse vibration is impossible. It may here be observed, that it is essential to the exactness of WAVES OF VIBRATION VELOCITY OF SOUND. 563 the values given above for the velocities of the transmission of waves, that the surfaces of simultaneous displacement (called some- times wave-surfaces) should also be surfaces of equal amplitude of vibration. If the amplitude varies at different points of the same wave-surface, the velocity of transmission becomes less than that given by the equations 2, according to a law which it is unnecessary here to explain in detail. 615. Telocity of Sound.' Longitudinal vibrations, being those which can be transmitted through all substances, solid -and fluid, are the ordinary means of transmitting sound ; so that the velocity of sound in a given medium is the co-efficient a in the equations 2 of Article 614 j being the velocity which a body would acquire in falling from the height A -^ 2 w ; that is, a height equal to half the length of a prism of the substance of the base unity, whose weight is equal to the co-efficient of longitudinal elasticity. The velocity of sound, as determined by experiment, is, In water, at 61 Fahr... .. 4,708 feet per second ; In dry air, at 32 Fahr. ... 1,092 In air and other gases, the velocity of sound depends on the pres- sure, density, and temperature in the following manner : When a nearly perfect gas has its density changed, and is kept at a constant temperature, the pressure varies nearly in proportion to the density simply. But with every change of density which takes place under circumstances such that the gas cannot gain- or lose heat by con- duction, a variation of temperature occurs depending on the change of density in such a manner, that the pressure, instead of varying simply as the density, varies as a power of the density higher than the first. Let y denote the index of that power, p the pressure, and w the density of the gas ; then PKW, (1.) so that the co-efficient of elasticity A has the following value : A dp - yp (9\ A- = -= (2.1 aw w The value of the index y for air is all expended. The mean value of P is -5 . The distance through which it is overcome in compressing the pile is the compression due T? T to its maximum value, viz., ^TET? where E is the modulus of elasti- jit O city of the pile, and L the length of a post, which, if uniformly compressed throughout its length, would be as much shortened as the pile. Considering that the pile is held in a great measure by friction against its sides, L may be made equal to half its length. Then the work performed in compressing the pile is ~ ; and the work performed in driving it deeper is E, x, where x is the depth through which it is driven by a blow ; and equating these to the energy of the blow, we find fm + * x ..................... W When x has been ascertained by observation, K is found by solving a quadratic equation, viz., Piles are in general driven till R amounts to between 2,000 and 3,000 Ibs. per square inch of the area of head S, and are loaded with from 200 to 1,000 Ibs. per square inch ; so that the factor of safety is from 10 to 3. The overcoming of any resistance by blows is analogous to the example here given, which is extracted, and somewhat modified, from a section by Mr. Airy in Dr. Whewell's treatise on Mechanics. 566 CHAPTER V. MOTIONS OF FLUIDS. 617. Division of the Subject. The principles of dynamics, as applied to fluids, so far as small and rapid changes of dnsity 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 cineniatical 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 exerted between masses of fluid and solid surfaces require to be considered separately. The following is the division of the subject of this chapter : I. Motions of Liquids under Gravity and Pressure alone. II. Motions of Gases under Gravity and Pressure alone. III. Motions of Liquids affected by Friction. IV. Motions of Gases affected by Friction. V. Mutual Impulses of Fluid Masses and Solid Surfaces. GENERAL EQUATIONS OF HYDRODYNAMICS. 567 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 d x d y d z denote the volume of a rectangular molecule, and p the intensity of the pressure of the liquid at a point whose co- ordinates are x, y, z. Let z be vertical, and positive downwards. w being used to denote one of the components of the velocity at a point, the symbol g will now be employed to denote the weight of an unit of volume. Then the forces by which the molecule is acted upon are along x, -~ .dxdydz'y along y, -^- ' dxdy dz', / dp along*, e - Let the rates of variation of the components of the momentum of the molecule be found by multiplying the three rates of variation of the components of the velocity in Article 415, equation 2, each by * ; then equating these respectively to the three forces in g equation 1 above, dividing by d x d y d z, so as to reduce the equa- tion to the unit of volume, and then by f, so as to reduce them to the unit of weight, the following results are obtained : dp 1 d 2 % 1 ( du , du . du . du\ ~^t 'die ~dy ~dz] 3 dp 1 d 2 YI 1 f d v , d v , dv . dv\^ ?dy~ g dt z ~ g\ dt dx dy d z ) dp 1 d 2 1 ( dw , dw . dw . dw Combining with those three equations of motion the equation of continuity, viz. : - 568 PRINCIPLES OF DYNAMICS. du dv dw_ d^+d^+d^- 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 du dv dw as in Article 413. 619. Dynamic Head. The quotient is what is called the height, or head, due to the pressure; that is, the height of a column of the liquid, of the uniform specific gravity , 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, ez is the weight of a column of liquid per unit of base extending down from that plane to a particle under consideration ; p g 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 *-* = * .............................. (I-) is the height or head due to that difference of intensity, being what will be termed the dynamic Jtead. When z is measured positively upwards from a datum horizontal plane, its sign is to be changed j so that the expression for the dynamic head in that case becomes (2.) 620. General Dynamic Equations in Terms of Dynamic Head. If instead of the rates of variation of the pressure in the equations 2 of Article 618, there are substituted their values in terms of the dynamic head, those equations take the following forms : dh I d 2 % I ( du . du . du . du\ = \ -f- u r V r~ W r ' dx~ g dt 2 g\dt^ dx^ d ij dz)> dh 1 d 2 YI 1 ( dv dv d v d v\ dy~ g dt 1 g\dt dx dy dz j ' dh 1 d 2 \ ( dw dw . dw dw 621. Law of Dynnmic Ilend for Steady Motion. From these equations is deduced the following consequence, in the case of DYNAMIC HEAD - TOTAL ENERGY. 569 steady motion, in which there is no variation of the dynamic head at a particle, except that arising from the change of position of the particle. _ Let V be the velocity of a given particle. Its value, in terms of its rectangular components, is given by the equation 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 !+?".*j}+i(.*|W, dt dt* dt at 2 / dh d % . d h d n , dh a 2g (t \dt ' (2.) This principle might otherwise be stated thus : In steady motion, the sum of the height due to the velocity of a particle and of its dynamic head is constant, or symbolically V 2 jj -- \- h = constant ....................... (3.) This equation applies to the particles which successively occupy the same fixed point, as well as to each individual particle. 622. The Total Energy of a particle of a moving liquid without friction is expressed by multiplying the expression in equation 3 of the last Article by the weight of the particle W, thus : W V 2 iii which - is the actual energy of the particle, and W h is its potential energy ; because, from the last Article it appears, that by W V 2 the diminution of W h, may be increased by an equal amount, A 9 and vice versa; so that the dynamic head of a particle is its potential energy per unit of weight. In the case of steady motion, the total energy of each particle is constant ; and the total energy of each of the equal particles which successively occupy the same position is the same. In the case of unsteady motion of a liquid mass, the total internal energy of the entire mass is constant; that is, if the centre of gravity of the mass, or a point either fixed or moving uniformly, 570 PRINCIPLES OF DYNAMICS. with respect to that centre of gravity, is takeii as the fixed point to which the motions of all the particles are referred, the following equation is fulfilled : 2-W l. + h\oT J J jQ~ dxdydz = constant. ..(2.) 623. The Free Surface of a moving liquid mass, being that which is in contact with the air only, is characterized by the pressure being uniform all over it, and equal to that of the atmosphere. Let pi be the atmospheric pressure, z l the vertical ordinate, mea- sured positively upwards from a given horizontal plane, of any point in the free surface of the liquid, and hi the dynamic head at the same point; then it appears from Article 619, equation 2, that for that surface, hi Zi = == constant .(1.) e 624. A Surface of Equal Pressure is characterized by an analo- gous equation, /vy 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. 625. Motion iu 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 UST 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- pendicular to the plane, and equal in velocity. It is illustrated by the three figures 246, 247, and 248, each of which represents a reservoir containing liquid up to the elevation OZ 1 = z l above a given datum, and discharging the liquid from an orifice A at the smaller elevation O Z F . 94g = z . The liquid moves exactly or nearly in plane layers at the upper surface A x and at the orifice A . Let these symbols denote the areas of the upper surface and of the issuing stream respectively. Let Q denote the rate of flow per second, ^ 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 -Q; The pressures at the upper surface and at the orifice respectively are each equal to the atmospheric pressure ; hence the difference of dynamic head is simply the difference of elevation ; that is to say, therefore, according to Article 621, equations 2 and 3, vl-vl \ This gives for the velocity of outflow, v =V jnrM from which can be computed the rate of flow or discharge by means of equation 1. The general equation of motion, for every part of the vessel or channel at which the motion takes place in plane layers, is, accord- ing to Article 621, equation 3, 572 PRINCIPLES OF DYNAMICS. The motion may be considered to take place in plane layers at any part of the channel whose sides are nearly straight and parallel, such as A 2 in fig. 246, whose elevation above the datum is z r To find the dynamic head, and thence the pressure, at this intermediate section of the channel, the velocity through it is to be computed by the formula whence the dynamic head relatively to the datum O is obtained by the equation (6.) and thence the pressure by the formula ft = e(/*2-^ ............................ (7.) When a large vessel discharges liquid through a small orifice, the ratio 5 is often so small a fraction, that it may be neglected in AJ equations 2 and 3. 626. The Contracted Veiu 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 O7, 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 '..(I.) 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 were -' ' /ox (2.) and the formulae of Article 625 applied accordingly. For a rectan- gular orifice for which y is constant, this gives o O and if it is a TiotaA, or a rectangular orifice extending to the upper surface, so that z" z 1} (4.) 628. Surfaces of Equal Head, which for steady motion are also SURFACES OF EQUAL VELOCITY, are ideal surfaces traversing a fluid mass, at each of which the dynamic head is uniform. Their posi- tions are related to the direction, velocity, curvature, and variation of velocity of the fluid motion in the following manner : In fig. 249, let H^ H w H 2 H 2 , represent a pair of such surfaces, very near each other; their normal distance apart being dn, measured forwards from H t 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 K * Fifr 249 curvature r, its thickness dr, and the varia- tion of its velocity d Y ; 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, YdY Y 2 dh', or - -\-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 OF DYNAMICS. _ d r cos nr dn (M dn A cos n r. .(2.) 629. In a Radiating Current, flowing towards or from an axis, as described in Article 407, the surfaces of equal dyna- mic head and equal velocity are cylinders described about the axis. The equation of continuity, 1 of Article 407, putting b instead of h to denote the depth, parallel to the axis, of the cylindrical space in which the current flows, gives for the velocity the formula Fig. 250. v = Q _ 00 . f . - r > ...... \ ' where r is the radius of the cylindrical surface R , fig. 250, at which the radiating part of the current begins or ends, according as it flows outwards or inwards. The radiating current may ex- tend indefinitely in all directions beyond this surface, the velocity being at any point inversely as the distance from the axis O. Let h Q be the dynamic head at R ; then at any other cylindrical sur- face of the radius O E- = r, we have the dynamic head, A 2? = A.+ Let h L be the limit towards which the dynamic head approxi- mates as the distance from the axis is indefinitely increased ; then (3.) 630. Free circular Vortex. In the cylindrical space of fig. 250, lying outside of the surface E, > 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 h + ~ = Aj = constant .................. (1.) h^ 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 which is equal and opposite to the deviating force tdh; that is to say, - m dr gr ............................ W v 2 But by the condition of freedom in equation 1, we have - g h), which being substituted in equation 2, gives dr whence < 3 -) or, the velocity is inversely as the distance from the axis, exactly as in a radiating current. Then let v be the velocity of revolution, and h the dynamic head, at the inner boundary R of the vortex ; we have for the general equations amongst the dynamic heads and velocities at all points, 2g ~~ 2g ~ 2g y*' 2 2 j- (4.) * = * ~ 2# = h ~ 2~g ?' 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, eo 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 0, is analogous to the motion of a radiating current, and that whose velocity is v sin 6 is analogous to the motion of a free circular vortex. 632. A Forced Vortex is one in which the velocity of revolution of the particles follows any law different from that of a free vor- tex ; but the kind of forced vortex which it is most useful to con- sider, is one in which the particles revolve with equal angular velocities of revolution, as if they belonged to a rotating solid body ; so that if r be the radius of the outer boundary of the vor- tex, where the velocity is v , v = ^ .............................. (1.) The equation of deviating force, 2 of Article 630, is applicable to all vortices, forced as well as free. Introducing into it the value of v from equation 1, above, we find, d h vl r f9 . z?'T*" - 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 Itead 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 head A, of the free vortex, we have this principle : In a combined vortex, the maximum dynamic liead is double of the dyna- mic Jiead at the surface of junction, each being measured relatively to the axis of the vortex; or symbolically, To illustrate this geometrically, let a combined vortex revolve about a vertical axis, O Z Z,, fig. 251, the upper surface of the liquid being free, and represented in section by D B O B D. Let A B, A B, be the cylindrical sur- face of junction between the free and the forced vortices. Let A O A be a horizontal plane, Fig. 251. touching the upper surface at its lowest point, which is at the axis, and let vertical ordinates be measured from this plane. The pressure of the atmosphere being equal at all points, may be left out of consideration ; so that if z be the height of any point in the surface of the vortex above A O A, we shall have simply Then for the forced vortex, z = .(3.) so that B B is a paraboloid of revolution with its vertex at 0. Make AC = 2AB = 2^ ; this will represent h l} the maximum dynamic head j and for the free vortex, Z = II (4.) 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 7^ z, measured downwards from C Z l C, is inversely as the square of the distance from the axis. The two surfaces have a common tangent at B B, where they join. The velocity of any particle in the free vortex is that due to its depth below C C ; that of any particle in the forced vortex is that due to its height above A A ; and B, where those velocities are equal, is midway between C C and A A 578 PRINCIPLES OF DYNAMICS. The theory of the combined vortex was made, by Professor James Thomson of Belfast, the principle of the action of his tur- bine or vortex water- wheel. 634. Tertical Revolution. When a mass of liquid revolves in a vertical plane about a horizontal axis (like the water in a bucket of an overshot wheel), its upper surface is not horizontal, but assumes a figure depending on the deviating force required by its revo- lution. In fig. 252, let C represent a horizontal axis, and B a bucket of liquid revolving round it in a vertical circle of the radius B C, with the angular velocity of revolution a. Let "W" be the weight of liquid in the bucket. Then the deviating force required is given by the formula Take the radius B C itself to represent the devi- ating force, and C A vertically upwards from the axis to represent the weight ; the height C A is given by the proportion CA : BC : : W : BC, that is, (I-) where n is the number of revolutions per second. Now A C representing the weight, and C B the centrifugal force, equal and opposite to the deviating force, the internal condition of the liquid in the bucket, according to D'Alembert's principle, is the same as if it were under a force represented by A B, the resultant of these two forces ; therefore the surface of the liquid is perpendi- cular to A B. Now it appears from equation 1, that the height of A above is independent of the radius of the wheel, and of every circumstance 3xcept 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, SECTION 2. Motions of Gases without Friction. 635. Dynamic Head in Oases. The dynamical equations of motion of a gas are the same with those already given in Article 618, equation 2; and in their integration, it has to be observed that , the density, is no longer constant, but depends on the pres- sure. The equations of continuity have been given in Articles 419 to 423. In finding the DYNAMIC HEAD for a particle of a gas, instead of there is to be taken / , as is evident from the general equa- 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, i a-) 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 j 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, ^oc e', ............................... (2.) 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, in which, for air, 580 PRINCIPLES OF DYNAMICS. y 1408 Let T = T -f 461-2 Fahr. (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 (6.) be the absolute temperature of melting ice. Then for gases sen sibly perfect, p - P T ' (7\ ~ ~^T ; l*v from which we have the following value of the integral in terms of the temperature : o p 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 " 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.) The following are the values of and c' for certain gases and vapours : ^ feet. C '. po Air, ................................... 26,214 ...... 0-238 Oxygen, .............................. 23,710 Hydrogen, ........................... 37 8 > 8 ip Steam, ................................ 42,141* ...... 0-480 -^Ether vapour, ...................... 10,110 ...... 0-481 Bisulphuret of carbon vapour, ... 9, 902 ...... " L 57S Carbonic acid, if a perfect gas, ... 17,264 Do., actually, ........... 17,145 ...... 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 cc 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 _i_ i p oc p* cc T y - T oc (h-s) y ~ l ................. (2.) the exponents having, for air, the values * = 0-71 ; ^ = 2451 ................... (3.) Hence the equation of continuity, in terms of the pressure, of the absolute temperature, and of the dynamic head respectively, takes the followin forms : = A vp* = constant ; .................. (4.) i_ _j_ Qr 3 1 = A-vr*- 1 = constant; ............... (5.) i i Q (h - z)*~ l = A v (h - zy~ l = constant ; ......... (6.) 637. Flow of Gas from an Orifice. Let the pressure of a gas within a receiver be p lt and without, p 2 let A be the effective area of an orifice with thin edges ; that is, the product of the actual area. by a co-efficient of contraction, whose value is 0-6, nearly. Let the receiver be so large that the velocity within it is insensible. Let the absolute temperature and density of the gas within the receiver be T W f 1? and those of the issuing jet r 2 , ^ 2 . The latter are 582 PRINCIPLES OF DYNAMICS. not the same with those of the still gas outside, for reasons to be stated afterwards. Then y-l 1 ft and by equation 8 of Article 635, and equation 3 of Article 621, we have for the height due to the velocity of outflow, ~ :^= /I, fl% =^ :r * 2g y I e, i _(P 1 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 i i L y On. T n r). giving the following results : es Q = v A e2 For small differences of pressure, such that ^ is nearly = 1, the JPi following approximate formula may be used where great accuracy is not required : Pl 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, TI. 637 A. maximum Flow of Gas. "When is indefinitely dimin- Pi ished, the velocity of outflow given by equation 2 of Article 637 increases towards the limit // 2ygp r l 1 V t(y-l), r h MAXIMUM FLOW OF GAS. 583 V~~ 2 ~" f : 1 whose value for air is 2-21, giving for the limiting velocity of flow of air 2,413 feet per second x A/ (2.) V T o The flow of weight, however, as given by equation 4 of Article 637, does not continuously increase as is indefinitely diminished, but reaches a maximum for the value corresponding to " 1 Pl V/ + 1- The values of these ratios for air are (3.) ^ = 0-527 ; ? = 0-6345 ; - 2 = 0-8306 ...... (4.) Pi d and the corresponding velocity of flow is being less than the velocity of sound in the ratio \/ - -r-y : 1, whose value for air is 0-912 ; 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/ 7 ............ (&) It is often convenient to express the flow of weight in the following manner : ,,Q = ^-A, l5 ..: ..................... (7.) in which is what is called the reduced velocity, being the velo^ ft 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 y+l ^ = velocity of sound x (-^TJ) ' ' .......... (8.) whose value for air is velocity of sound x 0-579 = 632 feet per sec. x \ / -...(9. V T 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 1408. 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 ' TQ 4:90"^ 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 2 : p lt 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 i, an* 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 the 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, ? the weight of an unit of volume of the fluid, and f a factor called the co-efficient of friction; then is the amount of the friction at the surface S. 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/; then to compute the value of / 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, or for velocities that are not very small, / = 0-005 Iron pipes, value of/ for first approximation, 0*0064 Beds of rivers (Weisbach),.../ = a + -; a = 0-0074. b = 0-00023 foot- Beds of rivers, value of / for first ) ^ r . . > 0*0076. 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 V Eau dans les Tuyaux. 639. internal Fluid Friction. Although the particles of fluids have no transverse elasticity that is, no tendencv 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 iii 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 Y is the greatest velocity in feet per second : 10-25 + 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 E, we find or dividing by common factors, and by the area of section A, we find for the value of the fall in terms of the velocity STREAMS HYDRAULIC MEAN DEPTH. 587 Let s be what is called the wetted perimeter of the cross-section of the stream; that is, the cross-section of the rubbing surface of the stream and channel ; then and dividing both sides of equation 3 by I, we find for the relation between the rate of declivity and the velocity, (4.) 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 S- 253 - wards. Consider any indefinitely short portion of the stream whose horizontal length is dx; in practice this may almost always be con- sidered as equal to the actual length. The fall in that portion of 588 PRINCIPLES OF DYNAMICS. the stream is d z, and the acceleration d v, because of v being opposite to x. Then modifying the expression for the loss of head due to friction in equation 3 of Article 640 to meet the present case, and adding the loss of head due to acceleration, we find dz . _^ g +y "A 20" 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 a?, 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 t or of z and (the Cv CC 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 j dp vdv , sdl v 2 -dh = -dz --- - = - - +/ '-sr- ......... (1.) 9 A 2g dh The ratio is called the virtual or hydraulic declivity, being the 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, -=-=. cL L "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 L If Q is unknown, the values of h for two particular values of I, or of h and for one FLOW IN PIPES SUDDEN ENLARGEMENT. 589 particular value of I, are required for the solution, which compre- hends the determination of Q. 643. Resistance of Mouthpieces. A mouthpiece is the part of a channel or pipe immediately adjoining a reservoir. The internal friction of the fluid on entering a mouthpiece causes a loss of head equal to the height due to the velocity multiplied by a constant depending on the figure of the mouthpiece, whose values for certain figures have been found empirically ; that is to say, let A h be the loss of head ; then f being a constant. For the mouthpiece of a cylindrical pipe, issuing from the flat side of a reservoir, and making the angle i with a normal to the side of the reservoir, according to Weisbach, / = 0-505 + 0-303 sin ^ + 0-226 sin'i .......... (2.) 644. The Resistance of Curves and Knees in pipes Causes a loss of head equal to the height due to the velocity multiplied by a co- efficient, whose values, according to Weisbach, are given by the following formulae : For curves, let i be the arc to radius unity, r the radius of curvature of the centre line of the pipe, and d its diameter. Then for a circular pipe, .(i.) and for a rectangular pipe, 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 f = 0-9457 sin 2 1 + 2-047 sin 4 I (2.) 2i 2i 645. A Sudden Enlargement of the channel ill 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 due to the velocity of the narrow stream relatively to the wide stream, that is, to the difference v l (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 \ + ~, and h 2 + ^-, be the total heads Ag & 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 Law of the friction of gases is the same with that of the friction of liquids as expressed by equation 1, Article 638, the value of the co-efficient /being 0-006, nearly, for friction against the sides of the pipe or channel. 3?or a cylin- drical mouthpiece, the co-efficient of resistance is 0-83 ; for a conical mouthpiece diminishing from the reservoir, 0-38. When the pressures at the beginning and end of a stream of gas do not differ by more than of their mean amount, problems respecting its flow may be solved approximately by means of the above data, treating it as if it were a liquid of the density due to the lesser pressure, as in the approximate equation of Article 637. In seeking the exact solution of the flow of a gas with friction, it is necessary to take into account the effect of 'the friction in pro- ducing heat, and so raising the temperature of the gas above what it would be if there were no friction, as supposed in Section 2. In the flow of a perfect gas with friction, if the heat produced by the friction is not lost by conduction, the friction causes no loss of total FRICTION OF GASES PRESSURE OF A JET. 591 head ; so that if at the beginning and end of a stream the velocities of a perfect gas are the same, its temperatures must also be the same. In an imperfect gas, there is a small depression of tempera- ture, which has been employed by Dr. Joule and Dr. Thomson as a means of determining or verifying the laws of the deviation of different gases from the condition of perfect gas. SECTION 5. Mutual Impulse of Fluids and Solids. 648. Pressure of a Jet against a Fixed Surface. A jet of fluid A, fig. 254, striking a smooth surface, is deflected so as to glide Fig. 254. Fig. 255. Fig. 256. along the surface in that path B E which makes the smallest angle with its original direction of motion A B, and at length glances off at the edge E in a direction tangent to the surface. To simplify the question, the surface is sup- posed to be curved in such a manner as to guide the jet to glance off it in one definite direction. The fric- tion between the jet and the surface is supposed insensible. This being the case, as the particles of fluid in contact with the sur- face move along it, and the only sensible force exerted between them and the surface is perpendicular to their direction of motion, that force cannot accelerate or retard the motion of the particles, but can only deviate it. Let v, then, be the velocity of the par- ticles of fluid, Q the volume discharged per second, p the density, and /3 the angle by which the direction of motion is deflected; then is the momentum of the quantity of fluid whose motion is deflected per second. Also conceive an isosceles triangle whose legs are each 592 PRINCIPLES OF DYNAMICS. equal to the velocity v, and make with each other the angle /3 ; then the base of that triangle, whose value is ft 2 v sin -, 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 : 2 sin -^ : 1 -cos/3 :sin/3 ........... (2.) J whence the components have the values, F, = l(l-cos,3);F y = lsm,3 ......... (3.) If the surface struck by the jet is of a symmetrical figure about the original direction of the jet as an axis, the quantity of fluid Q which strikes the surface in each second spreads and glides off in various directions distributed symmetrically round the axis, and making equal angles /3 with it ; so that the forces exerted perpen- dicular to the axis by the different parts of the spread and diverted jet balance each other, and nothing remains but the sum of the components parallel to the axis, whose value is F,, as given in the first of the equations 3. By substituting A v for Q, the forces may be expressed in terms of the sectional area of the jet. As a particular case, let the surface be a plane, as in fig. 255. The jet, on striking the surface, spreads and glances off in all direc- tions at right angles to its original direction, so that /3 = 90, cos ft = 0, and .-t2-f ..................... (4.) PRESSURE BETWEEN A JET AND A SURFACE. 593 being equal to the weight of a column of fluid whose base is the sectional area of the jet, and its height double of the height due to the velocity. This result is confirmed by experiment. As another case, let the surface be a hollow hemisphere (fig. 256), so that the jet in spreading is turned directly backwards. Then ft 180, -cos/3 = +1, and 2 P Qv _ 2 P Av* JP, - j ,.....(o.) 9 9 being equal to the weight of a column of fluid whose base is the sectional area of the jet, and its height four times the height due to the velocity. 649. The Pressure of a Jet against a Moving Surface is found by substituting in the equations of the preceding Article, the motion of the jet relatively to the surface for its motion relatively to the earth. In this case there is energy transmitted from the jet to the solid surface or from the solid surface to the jet; and the deter- mination of the amount of energy so transmitted per second forms an important part of the problem. CASE 1. When the surface has a motion of translation parallel to tlie 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 . which, if u is positive, is transmitted from the fluid to the solid, and if u is negative, from the solid to the fluid. The energy thus transmitted per second is equal to the difference of the actual energies of the volume Q of fluid before and after acting on the solid. Let v 2 be the velocity of the fluid after the collision ; this being the resultant of u, and of v l u in tlie devi- ated direction, its square is given by the equation v\ = u* 4- (v r - uf + 2 u (v, - u) ' cos ft = J-2w(t; 1 -w)(l-cos/8); ................. (3.) 2o 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 F* = g ^ ' (1 cos /3) ; F, u = g */ 1 (1 cos ft). If ft = 90, as in fig. 255, the maximum energy transmitted is e Q v* -r- 4 gr, or half of the original actual energy of the fluid. If ft = 180, as in fig. 256, the maximum energy transmitted is C Q v\ -T- 2 gr, or /te M?Aofe of the original actual energy of the fluid, which, after the collision, is left at rest. CASE 2. W/ien 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 v, of the jet. Then D C represents the direction and velocity of the original motion of the jet relatively to the surface. Draw E F = D C tangent to the surface at E, where the jet glances off; this represents the relative velocity and direction with which the jet leaves the surface. Draw F G || and = B D, and join E G ; this last line represents the direction and velocity relatively to the earth, with which the jet leaves the surface, being the resultant of E F and F G. The total force exerted between the fluid and the surface might be determined by finding the change of the momentum of the volume of fluid Q, due either to the change of direction and velo- city relatively to the earth, viz., from BC to EG; or to that relatively to the surface, viz., from DC to EF. But the force which it is most important to determine is that to which the trans- mission of energy is due, viz., the force parallel to B D, which will be denoted by F.,.. This force is equal to the change in one second of the component momentum of the fluid in the direction B D. Let = ^_ D B C, denote the angle between the direction of the jet and that of the body's translation; then the component, in the direction B D > of the original velocity of the jet is V) COS ' . \: ' PRESSURE OF A VORTEX ON A WHEEL. 595 Let w = D C be the velocity of the jet relatively to the surface ; then w 2 = w 2 -f- vl2uvi cos et .................. (6.) Let y = supplement of ^ E F G, denote the angle which a tan- gent to the surface at the edge where the fluid leaves it makes with the direction of translation. Then the component, in the direction B D, of the new velocity of the jet is u + w cos y ] and the change of momentum in that direction in one second is F, = - (vj cos etuw ' cos y) ............... (7.) which gives for the energy transferred per second, F, M = - U (Vj COS et U W ' COS y) ............ (8.) y Let v 2 be the resultant velocity of the fluid after the collision ; then vl = u 2 + w 2 + 2uwcosy .................. (9.) and it is easily verified that P.. = '-^3 ...................... (1 o.) 650. Pressure of a Forced Vortex Against a Wheel. In a free vortex (Article 630, 631), because the velocity of each particle is inversely as its distance from the axis, the angular momentum of every particle of equal weight is the same ; and a particle in mov- ing nearer to or farther from the axis of the vortex, preserving its angular momentum, requires no external force to be applied to it in order to make it assume the motion proper to each part of the vortex at which it arrives. If, in a forced vortex, there is at the same time a radiating current by which the fluid moves towards or from the axis, then by means of solid surfaces, such as those of the vanes of a wheel, there must be applied to the fluid in the vortex a couple sufficient in each second to produce the requisite change of angular momentum in the quantity of fluid which flows radially through the vortex in a second, and the fluid will react upon the wheel with an equal and opposite couple. Symbolically, let r Q , r lt be the radii of the cylindrical surfaces at which a forced vortex begins and ends ; v , v u the velocities of fehe 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 (1.) A vortex- wheel, or turbine, when working in the most favourable manner, receives the fluid at ends of its vanes which have a velocity of revolution equal to that of the particles of fluid in contact with them ; so that relatively to the wheel, the motion of the fluid is at first radial. The fluid glances off from the vanes at their other ends, which are of such a figure and position that they leave the fluid behind them with only a radial motion relatively to the earth ; so that the whole of the energy due to the revolution of the fluid is transmitted to the wheel. That is to say, let a be the angular velocity of the wheel ; then we must have 9 99 The last quantity, M a, is the energy transmitted in each second from the fluid to the wheel, which, in the case supposed, is the whole energy due to the motion of revolution and centrifugal pressure of the weight e Q of fluid in a rotating forced vortex, as already shown in Article 632. The ends of the vanes which receive the fluid should be radial, because the motion of the fluid relatively to them is radial. The ends of the vanes where the fluid glances off should be inclined backwards so as to make with the radii intersecting them, an angle given by the following equation : Let u = - - - be the velocity J *7C TI of the radial current at the ends of the vanes now in question; then (3-) b being the depth of the wheel in a direction parallel to the axis. Fig. 257 represents 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 6 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 K = 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 Q = V Q 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, .(1.) where z is the elevation of the point of delivery, V the velocity in the discharge pipe, and 2 -f 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 Q 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 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-r- J~bd = 0, 1, 2, 3. &= 1-864, 1-477, 1-347, 1-328. The value headed is applicable to very thin plates. 653. The Resistance of Fluids to the motion of bodies floating or immersed in them is subject to the same remarks which have been made respecting the pressure of currents against solid bodies. It is also capable in many cases of being approximately represented by the formula RESISTANCE OF FLUIDS PRO JECTILES SHIPS. 599 The co-efficient k is less for a solid moving in a fluid, than for a fluid moving past the same solid. The following values are given chiefly on the authority of Duchemin. For prisms and cylinders, moving in the direction of their axes, the symbols having the same meaning as in the last Article : (1 + '!i 7 V 96^+ = 0, 1, 2, 3; average above 3. k = I -254, 1 -282, 1 -306, 1 -330 ; 1 4. These results are also given by the empirical formula, & =1-254 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 = ^ angle 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): c A v\ -7-' 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 dthout 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 600 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 - 004; then the resistance is nearly G(1 + 4* 2 + s 4 ) ................. (3.) 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 631.) 654. stability of Floating Bodies In Article 120 it has been shown, that in order that a body floating in a liquid may be in equilibrio, the weight of liquid displaced must be equal to the weight of the floating body, and the centre of buoyancy must be in the same vertical line with the centre of gravity of the floating body. In order that the equilibrium of a floating body may be stable, every angular displacement of the body from the position of equili- brium must cause a deviation of the centre of buoyancy, relatively to a STABILITY OF SHIPS METACENTRE. 601 vertical line traversing the centre of gravity, in the direction towards which tlie 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, arid 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, and S the volume of water displaced by her, so that W = . e S (? 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 Q; consequently, the moment of the righting couple, equal and opposite to the moment of the heeling couple, is W -G^M-sintf (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 G D parallel to the line joining those two centres of gravity; and, according to Article 77, make then is D the new centre of buoyancy. The angle which C D makes with the horizon is in general either exactly or very nearly = K > 8O tna * C D = M C 2 sin , approxi- mately. Also, the volume s is in general either exactly or nearly proportional to 2 sin g ; so that if c be a constant volume depend- ing on the figure of the water line, s = c 2 sin , 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, The sign =+=. 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. Q The product I s = I c 2 sin h is the double of the statical J 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 II 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 o. Then we have s = 2 sin 2' j j y'dy dx;c = J J ydy dx\ s = 4 sin A / / y 2 ' dy d x' } and therefore I c 2 I I yt-dy dx; 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 2 1 2/ 2 d y = ^; and I c = ~ J 6 3 dx. (5.) As to the moments of inertia of different plane figures, see Article 95. Thus, equation 3 becomes b s 'dx -^r^-^^C (6.) The theory of the stability of ships was first investigated by Bossut, and was further developed by At wood. The most impor- tant contributions to that theory, of later date, have been, the memoir of Dupin, Sur la Stabilise 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 A B 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 6 ; but in an approximate solution we may substitute 6 for sin 6. 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 : = 2xA /( - V \g W 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 -- GH. The researches of Mr. William Froude, 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 631.) 656. The Action between n Fluid and a Piitton, 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 2* 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 O 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 ATB = p. 2 be the greater and the less intensities of the pressure ; then energy transferred = / (p^ p 2 )ds (1.) Secondly, for any given pressure O F = p, let F E = s t and FT) = Sj be the greater and the less of the spaces occupied ; then energy transferred = / (^ s 2 ) dp (2.) which is another expression for the same quantity. Fig. 261 represents the case in which a given weight of an elastic substance occupying the space O E = s t at the pressure B = p lt is introduced into a cylinder and made to drive a piston, is then 606 PRINCIPLES OF DYNAMICS. allowed to expand, its volume increasing to O F = s 2 , and its pres- sure falling to FD = 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 D= [ Pl sdp = W P ; (3.) J J>2 ^ J>2 6 area ABC being, in fact, as the last expression shows, equal to the weight of the elastic substance employed, W, multiplied by its loss of dyna- mic head. The same equation gives the energy transferred from the piston to the elastic substance, when the latter is introduced into the cylinder at the lower pressure and expelled at the higher. For a perfect gas (Article 635) this expression becomes If the fluid is discharged from the cylinder under a pressure p 3 less than that at which the expansion terminates, there is to be added to the preceding formula the term ** (P*P3) ............................ (5.) If the fluid which acts on the piston is introduced in the state of saturated vapour, it is discharged as a mixture of saturated vapour at a lower pressure with more or less of liquid. In this case, the following equations belonging to the science of thermo- dynamics are to be used. Let p be the pressure of saturation of a vapour, and t the corresponding boiling point of its liquid, in degrees reckoned from the absolute zero, "274 Centigrade or 493 0< 2 Fahrenheit below the melting point of ice. Then B C 1 Lo gJP = A ; ---; (6.) C 40 20 (See Edin. Philos. Jour., July, 1849 ; Edin. Transac., xx; Pliuos. Mag., Dec., 1854; NichoVs Cyclopaedia, art. "Heat, Mechanical Action of.") The following are the values of some of the constants in the above formulae, selected from a table in the Philosophical Magazine for Dec., 1854, p being in Ibs. per square foot, and * in degrees of Fahrenheit: HEAT AND WORK OF STEAM. 607 Tl T>J A LogB LogC -x r-^ fi\j * \j Water,... 8-2591 3-43642 5'59 8 73 '344i 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 mere in the state of vapour than it does in the state of liquid ; B 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 -J( C -6)(r-r ) (for water, c b = 0-7) j (for water at the temperature of melting ice, H = 842872.) 3 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 ; s } 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^ ( r \ T 2 ) energy transferred, U = / * s dp + * 2 (p 2 -p s ) ( ~ T2 (9.) (10.) These formulas 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 OF DYNAMICS. following formulae give approximate results for steam, where the pressure of its admission pi is from one to twelve atmospheres : (11.) P* energy transferred, TJ = f s dp -f s 2 (p 2 - p & ) ...(12.) The expenditure of heat in foot pounds may be computed roughly to about Y^ -, when the feed water is supplied to the boiler at about 100 Fahrenheit, by the formula Tl H=/ sdp + npoS^, (13.) J P2 where nis a co-efficient whose value is, for condensing engines, 16; 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, 1, and iV, respectively, for iV, V, and j, throughout the equations 11 and 12. For the exact theory of this case, see A Manual of the Steam Engine and oilwr Prime Movers; also, Philosophical Transactions, 1859, Part I. The following are the ordinary formula?, which give a good approximation when the steam is slightly moist : V =p l s 1 hyp. log.-? + s 2 (p 2 -p s ) ...(15.) s i The approximate formula (13) is applicable in all cases. PART VI. THEORY OF MACHINES. 657. rvatnrc and Division of the Subject. In the present Part of this work, machines are to be considered not merely as modify- ing motion, but also as modifying force, and transmitting energy from one body to another. The theory of machines consists chiefly in the application of the principles of dynamics to trains of me- chanism ; and therefore a large portion of the present part of this treatise will consist of references back to Part IV. and Part 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 tost Work The whole work performed by a machine is distinguished into useful work, being that performed in producing the effect for which the machine is designed, and lost work, being that performed in producing other effects. 659. Useful and Prejudicial Resistance are overcome in perform- ing useful work and lost work respectively. 660. The Efficiency of a machine is a fraction expressing the ratio of the useful work to the whole work performed, which is equal to the energy expended. The limit to the efficiency of a machine is unity, denoting the efficiency of a perfect machine in which no work is lost. The object of improvements in machines is to bring their efficiency as near to unity as possible. 661. Power and Effect; Horse Power. The power of a machine is the energy exerted, and the effect, the useful work performed, in some interval of time of definite length. The unit of power called conventionally a horse power, is 550 foot pounds per second, or 33,000 foot pounds per minute, or 1,980,000 foot pounds per hour. The effect is equal to the power multiplied by the efficiency. 662. Driving Point; Train; Working Point. The driving point is that through which the resultant effort of the prime mover acts. The train is the series of pieces which transmit motion and force from the driving point to the working point, through which acts the resultant of the resistance of the useful work. 663. Points of Resistance are points in the train of mechanism through which the resultants of prejudicial resistances act. 664. Efficiencies of Pieces of a Train. The useful Work of an intermediate piece in a train of mechanism consists in driving the piece which follows it, and is less than the energy exerted upon it by the amount of the work lost in overcoming its own friction, Hence the efficiency of such an intermediate piece is the ratio of the work performed by it in driving the following piece, to the energy exerted on it by the preceding piece ; and it is evident that the efficiency of a machine is the product of the efficiencies of the series MEAN EFFORTS AND RESISTANCES GENERAL EQUATIONS. 61 1 of moving pieces which transmit energy from tJie driving point to the working point. The same principle applies to a train of successive machines, each driving that which follows it. 665. Mean Efforts and Resistances. In Article 515 is given the expression / P d s for the energy exerted by a vaiying effort whose magnitude at any instant is P ; and a corresponding expression / T\,ds denotes the work performed in overcoming a variable re- sistance. In a machine moving uniformly, let these expressions have reference to any interval of time, and in a machine moving periodically, to one or any whole number of periods ; let s be the space described by the point of application of the effort or resist- ance in the interval in question j then / P d s -f- s or /Re?$ -=- s is the mean effort or mean resistance as the case may be. The fluc- tuations of the efforts and resistances above and below their mean values concern only the variations of velocity in a machine ; and therefore, in the remainder of the present chapter, P and K 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 * and II 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 J [ 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, RJ the mean useful resist- ance, s { the space through which it is overcome in the same inter- val, R> any one of the prejudicial resistances, s. 2 the space through which it is overcome ; then .(1.) The efficiency of the machine is expressed by 612 THEORY OF MACHINES. PS 667. Equations in terms of Comparative Motions. Let 5 t : S = n if 8. 2 : s =. n 2 , &c. , be the ratios of the spaces described in a whole num- ber of periods by the working point and the several points of resistance, to the space described, in the same interval of time, by the driving point ; then equation 1 of Article 666 takes the follow- ing form, which expresses the " Principle of Virtual Velocities " (Article 519) as applied to machines : P = rc 1 R I + s-rc 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 tJie 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 / I'/ 1. 14to2G^ 25 to -5 4 to 2 2. soapy, 2 5 3 Metals on oak drv . 26lto31 '5 to - 6 2 to 1-67 4. ,, wet, mtol4i 24 to 26 4-17 to 3 85 5. 114 2 5 6 111 to 14 2 to -25 5 to 4 7. Hemp on oak, drv, " 28 53 1-89 18 wet 18j 33 3 9 15 to 19i 27 to -38 3-7 to 2-86 ,10 Leather on metals, dry, 29^ 56 1-79 n wet 20 36 2-78 12 13 23 4 35 13. oily, 8i 15 667 14 Metals on metals dry 81 to H J - 15 to -2 6-67 to 5 15 '" 164 J 3 333 16. 17. 18. Smooth surfaces, occasionally greased, continually greased, best results, 4 to"4i 3 1| to 2 07 to -08 05 03 to -036 14-3 to 12-5 20 33-3 to 27-6 670. Unguents. The last three results in the preceding table, Nos. 16, 17, and 18, have reference to smooth firm surfaces of any kind, greased or lubricated to such an extent that the friction depends chiefly on the continual supply of unguent, and not sensibly on the nature of the solid surfaces ; and this ought almost always to be the case in machinery. Unguents should be thick for heavy pres- sures, that they may resist being forced out, and thin for light pres- sures, that their viscidity may not add to the resistance. 671. JAmit of Pressure between Rubbing Surfaces. The law of the simple proportionality of friction to pressure (Article 190) is only true for dry surfaces, when the pressure is not sufficiently intense to indent or grind the surfaces ; and for greased surfaces, when the pressure is not sufficiently intense to force out the unguent from between the surfaces, where it is held by capillary attraction. If the proper limit of intensity of pressure be exceeded, the friction increases more rapidly than in the simple ratio of the pressure. That limit diminishes as the velocity of rubbing increases, according to some law not yet exactly determined. The following are some of its values deduced from experience : Railway Carriage Axles. Velocity of rubbing 1 foot per second, :t %k j> ?> & > Timber ways for launching ships, about Limit of Pressure, Ib. per square inch. 39 2 224 140 50 614 THEORY OP MACHINES. The inclination given to these ways varies from about 1 in 10 for the smallest vessels, to about 1 in 20 for the largest. The co-efficient of friction, when the ways are well lubricated with tallow or soft soap, is probably between -03 and -04. 672 Friction of a Sliding Piece. In fig. 262, let A represent a sliding piece, which moves uniformly along the straight guide B B in the direction indi- cated by the arrow, under two forces which may be direct or oblique, but which are re- presented as oblique, to make the solution general. The force F 2 opposed to the motion, is the resultant of the useful resistance or Fig. 262 . force which A exerts on the next piece In the train, and of the weight of A itself, and will be called the given force. Let the angle which it makes with the guide B B be denoted by i r The force Fj is that which drives the piece ; the angle i t which its direction makes with the guide B B is supposed to be known ; but its magnitude remains to be determined, as well as the friction, which it has to overcome in addition to the useful resistance. Let Q denote the normal pressure of A against B B, so that f Q is the friction. Then we have the two equations of equilibrium : Q = Ft sin i, + Fo sin i 2 ; 1 F! cos i, = F 2 cos i, + /Q I .......... (1.) = Fi/sin i L + F 2 (cos i 2 + /sin i 2 ) ; J from which are easily deduced the following equations, solving the problem : /sin (i, + i 2 ) ' ! 2 - ; - -f. - ~ - -- : . .... cos ^ 1 f sin ^ 1 ' cos i x / sm z l The moment of Friction of a rotating piece is the statical moment of the friction relatively to the axis of rotation of the piece, and is the moment of a couple consisting of the friction, and of an equal and opposite component of the pressure exerted by the bearings of the piece against its axle. The moment of friction, being multiplied by the angular motion in a given time, gives the work lost in friction in that time. 674. Friction of an Axle. After a cylindrical axle has run for some time in contact with its bearing, the bearing becomes slightly larger than the axle, so that the point of most intense pressure, which is also the point of resistance, traversed by the resultant of the friction, adapts its position to the direction of the lateral pressure. FRICTION OF AN AXLE. 615 Iii fig. 263, let A A A be a transverse section of the cylindrical axle of a rotating piece, and C its axis of rotation; let B represent the direction and magnitude of what will be called the given force', being the resultant of the useful resistance, and of the weight of the piece under consideration. Let P represent the effort required to drive the piece, whose line of action is known, but its magnitude remains to be determined. Let D be the point where the directions of P and B intersect, and D Q the line of action of their resultant, which resultant is equal and opposite to Q, the pressure exerted by the bearing against the axle, and is there- fore inclined to the radius C Q by an angle CQD = $, being the angle of repose, in such Fi &- 263< ' a manner as to resist the rotation, whose direction is indicated by the arrow. Then to find the line of pressure D Q, it is obviously sufficient to describe about the centre C a circle B B whose radius is fr (1.) r = C Q being the radius of the axle, and to draw from the known point D a line D T Q touching that circle in T, which point of contact is at that side of the circle which makes a force acting from Q towards T oppose the rotation. From T draw T B -L B, and T P -L P. Then the magnitude of the effort P is given by the equation P = B-Tll + TP ..................... (2.) and that of the pressure Q by the equation Q 2 = P 2 + B 2 + 2 P B cos ^ P D B .......... (3.) (the last term of which becomes negative when ^ P D B is obtuse) ; while the friction is and its moment Q T sin $ = Q CT ...................... (5.) When P and B are parallel to each other, Q is their difference or their sum, according as they act at the same or at opposite sides of the axle, and Q T is to be drawn parallel to them both, so that sfr. 616 THEORY OF MACHINES. R T, T P, and C T, lie in one straight line, when equations 2, 4, and 5 will still hold. In order to diminish the lateral pressure Q, and the friction arising from it, to the least possible amount, the mechanism should be so arranged as to make P and R act parallel to each other at the same side of the axle. _ In most actual cases, sin =/ : J 1 +/ 2 differs from tan $ =f in a proportion too small to be of any practical importance. The bearings of axles should be made of materials which, though hard enough to resist the rubbing without abrasion, are not so hard as the axle. Hence for wrought iron axles, bronze bearings are commonly used. Bearings of cast iron, millboard, and hardwood, such as elm, with the grain set radially, have also been used with advantage. 675. Friction of a Pivot. A pivot is the termination of an axle, which presses endways against a bearing called a step, or footstep. Pivots require great hardness, and are usually made of steel. A fiat pivot is a short cylinder of steel, having a plane circular end for a rubbing surface. If the pressure Q be equally distributed over that surface whose radius is r, the moment of friction is easily found by integration to be . .............................. In flat pivots, the intensity of the pressure, which is given by the equation is usually limited to 2,240 Ibs. per square inch. In the cup and ball pivot, the end of the shaft, and the step, present two recesses facing each other, into which are fitted two shallow cups of steel or hard bronze. Between the concave spherical surfaces of those cups is placed a steel ball, being either a complete sphere, or a lens having convex surfaces of a somewhat less radius than the concave surfaces of the cups. The moment of friction of this pivot is at first almost inappreciable, from the extreme small- ness of the radius of the circles of contact of the ball and cups ; but as they wear, that radius and the moment of friction increase. 676. Friction of a. Collar. When it is impracticable or incon- venient to sustain the pressure which acts along a shaft by means of a pivot at its end, that pressure is borne by means of one or more collars, or rings projecting from the shaft, and pressing against corresponding ring-shaped bearings, for which, in the case of shafts of screw propellers, hardwood set with the grain endways has been FRICTION OF COLLARS - OF TEETH - OF BANDS. 617 found a good material amongst others. Let r be the external, and r the internal radius of a collar ; its moment of friction for the pressure Q is given by the formula 677. Friction of Teeth. When a pair of wheels work together, let P be the pressure exerted between each pair of their teeth which comes into action, s the distance through which each pair of teeth slide over each other, as found in Articles 453, 455, 458, and 462 A, and n the number of pairs of teeth which pass the line of centres in a given interval of time. Then in that interval, the work lost by the friction of the teeth is fnsP .............................. (1.) 678. Friction of a Band. A flexible band, such as a cord, rope, belt, or strap, may be used either to exert an effort or a resistance upon a drum or pulley round which it wraps. In either case, the tangential force, whether effort or resistance, exerted between the band and the pulley, is their mutual friction, caused by and pro- portional to the normal pressure between them. In fig. 264, let C be the axis of a pulley A B, round an arc of which there is wrapped a band, T t A B T 2 ; let the outer arrow represent the direction in which the band slides, or tends to slide, relatively to the pulley, and the inner arrow the direction in which the pulley slides, or tends to slide, relatively to the band. Let T, be the tension of the free part of the band at that side towards which it tends to draw the pulley, or from which the pulley tends to draw it ; T 2 the tension of the free part at the other side; T the tension of the band at any intermediate point of its arc of contact with the pulley; 6 the ratio of the length of that arc to the radius of the pulley ; d 6 the ratio of an indefinitely small element of that arc to the radius ; E, = T, T 2 , the total friction between the band and the pulley ; d R the elementary portion of that friction due to the elementary arc d6] /the co-efficient of friction between the materials of the band and pulley. Then according to a principle proved in Articles 179 and 271, it is known that the normal pressure at the elementary arc d 6 is 618 THEORY OP MACHINES. T being the mean tension of the band at that elementary arc; con- sequently, the friction on that arc is Now that friction is also the difference between the tensions of the band at the two ends of the elementary arc ; or which equation being integrated throughout the entire arc of contact, gives the following formulse : ' (1.) When a belt connecting a pair of pulleys has the tensions of its two sides originally equal, the pulleys being at rest ; and when the pulleys are set in motion, so that one of them drives the other by means of the belt ; it is found that the advancing side of the belt is exactly as much tightened as the returning side is slackened, so that the mean tension remains unchanged. Its value is given by this formula : JL-^U^-^*^ T 1 + T 2 _ e- f +l _. TBT ~ 2 (/-!)' ^ which is useful in determining the original tension required to enable a belt to transmit a given force between two pulleys. If the arc of contact between the band and pulley, expressed in turns and fractions of a turn, be denoted by n, 6=2 *n; e/=10 8 /..- (3.) When the band is used to resist the motion of the pulley, it constitutes a kind of brake called & friction strap. In this case the rubbing surfaces of the band and pulley may either be both of iron, or may be protected by a covering made of pieces of wood, which is renewed from time to time as it wears out. 679. 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 formulae 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 ; d, the diameter of the rope, in inches ; n = 48 d' z for white ropes, 35 d 2 for tarred ropes ; r, the effective radius of the pulley, in inches ; T, the tension, in pounds ; then, For white ropes, B = -(0-0012 + 0-001026 rc + 0-0012 T); , (I-) For tarred ropes, R = - (0-006 + 0-001392 rc, + 0-00168 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. R = Q[ a + b(v - 3-28)3; 0-) where Q is the gross load, r the radius of the wheels in inclies, 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 formulae, which are all empirical E denotes the weight of the engine; the gross load drawn by it; the velocity, in miles an hour; r R the radius of curvature of the line, in miles; the resistance in pounds; a co-efficient of friction ; 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 Rankine and the Author, where /= 0-002; and c = 0-3. (See Experimental Inquiry on the Use of Cylindrical Wheels on Railways, 1842.) For an engine arid train, the following is an empirical formula deduced from the experiments of various authors : where /ranges from -0027 to -004, according to the state of the line and carriages, and c from 0'3 to O'l. (See Rankine' 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 tht 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 good unguent. 621 CHAPTER II. VARIED MOTIONS OP 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 traverse its centre of gravity, that the resultant centrifugal force 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. G87. Actual Energy of a Machine. To determine the entire actual energy of a machine at a given instant, it is necessary to 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 .....? ......... (i.) 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 = /i'-.W* + 2-WYa 2 ) .............. (2.) *9 688. 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 V 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 2 -Ww 2 .............................. (1.) expresses the 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 -4- 2-lTj, 2 - ...................... (3.) and the actual energy at any instant, E = - s-WZ 2 + s-Irc 2 ................ (4.) 689. Fluctuation* of Speed in a machine are caused by the alter- nate excess of the energy received above the work performed, and oi 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. To determine the greatest fluctuations of speed in a machine moving periodically, take ABC, in fig. 265, to represent the motior of the driving point during one period; let the effort P of the prime mover at each instant be represented by the ordinate of the Fig. 265. curve D G E I F ; and let the sum of the resistances, reduced to the driving point, as in Article 6G8, at each instant, be denoted by 11, and represented by the ordinate of the FLUCTUATION OF SPEED FLY-WHEEL. 623 curve D H E K F, which cuts the former curve at the ordinates A D, B E, C 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 5 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 Y be the mean velocity, Y x the greatest velocity, and Y 2 the least velocity of the driving point ; then V 2 _ V 2 li=.Il'.,.W*= AE; .................. (1.) which, being divided by twice the mean actual energy ' V i _y 2 _ A E E gives V i V " 2E a ratio which may be called the co-efficient of fluctuation of speed. The ratio of the periodical excess and deficiency of energy A to the whole energy exerted in one period or revolution, \ 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 -^r o2i 00 oU 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 : the second of which equations gives the requisite moment of inertia of the fly-wheel. 691. starting mid Stopping Brakes. The starting of a maqhine 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 t be the ordinary resistance of the machine, reduced to tlm 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 = E- (R, + R 2 ); (1.) and because the mean velocity of rubbing during the operation of stopping is v - 2, t = = 2 E + r (B, + R 2 ) (2.) 625 CHAPTER IIL ON PRIME MOVERS. 692. A Prime Mover 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 supplying 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 mover* may be classed according to the forms in which the energy is first obtained. These are I. Muscular Strength. II. The Motion of Fluids. III. Heat. IV. Electricity and Magnetism. 695. Muscular Strength. The daily effect exerted by the muscu- lar strergth 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 OF 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. fit . 3,600 = hrs. p. day. RV ft. Ib. p. sec. RVT ft. Ib. p. day. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Raising his own weight up stair 143 0-5 0-75 0-55 0-13 1-3 0-075 2-0 5-0 2-5 14-4 25 ? 14| 3-6 8 10 6 6 6 10 10 8 ? 8 (2 mins.) 10 8? 4 8 72-5 30 24-2 18-5 7-8 9-9 53 62-5 45 288 33 ? 447 432 2,088,000 2,616,000 648,000 522,720 399,600 280,800 366,400 1,526,400 1,296,000 1,188,000 480,000 6,444,000 12,441,600 (Tread- wheel, see 1.) Hauling up weight with rope, Lifting weights by hand . . 40 44 143 6 132 26-5 {12-5 18-0 20-0 13-2 15 (min. 22) -jmeanSO^V (max. 50) 120 Carrying weights up stairs, Shovelling up earth to a height of 5 feet 3 inches, Wheeling earth in barrow up slope of 1 in 12, horiz. veloc. 0-9 ft. per sec. (return, empty), Pushing or pulling horizontally Ccapstan or oar').. Turning a crank or winch, HORSE. Cantering and trotting, draw- ing a light railway carriage (thoroughbred), .... . Horse drawing cart or boat, walking (draught horse), 696. A Water Pressure Engine consists essentially of a working cylinder, in which water moves a piston in the manner stated in Article 499, case 2. Let h be the virtual fall, that is, the excess of the dynamic head of the water entering the cylinder above that of the water leaving the cylinder; Q the volume of water supplied per second; ? its weight per unit of volume; 1 k the efficiency of the engine; then WATER-WHEELS. 627 is its effect per second. In well constructed water pressure en- gines, 1 k varies from '66 to '8. 697. Water-wheels in General. Water may act on a wheel either by its weight and pressure, or by its velocity; that is, either by its potential, or by its actual energy. See Article 622. Let Q denote the weight of water, in pounds, supplied to the wheel in a second; h the difference of dynamic head, in feet, of the water before and after its action on the wheel ; Vi the velocity of the water, in feet per second, just before it begins to press on the wheel, or supply-velocity; v 2 the velocity of the water just after it has ceased to act on the wheel, or discharge-velocity. Then the total energy of the water, as in Article 622, is ( Q ( h + jp- J foot pounds per second; the energy of the water when discharged, v* e Q ~j foot pounds per second; *y the total power of the wheel, e Q \h+ *o~ y f ot P oun d s P er second; ........ (1.) the maximum theoretical efficiency, the quantity may be called the theoretical fall or head. The available efficiency of a water-wheel falls short of the maximum theoretical efficiency principally from the following causes : 1. The resistance of the channel and orifices by which the water is supplied, which causes the actual height from which the water must descend in order to acquire the supply- velocity v to be greater than w? : 2 g. The effect of such resistance is expressed by putting for the actual fall, (4.) 2 '/being the co-efficient of resistance of the channel and orifices of supply, determined according to the principles of Articles 638 to 646. 2. The escape of part of the water before it has completed its action on the wheel. 3. The agitation and mutual friction of the 628 THEORY OF MACHINES. particles of water acting on the wheel j and, 4. The friction of the wheel. The effects of the last three causes are expressed by multi- plying the total power and the theoretical efficiency of the wheel by an empirically determined fractional co-efficient & ; so that the effect or available power is denoted by (1 &) e QV and the available efficiency by (1 *).Ai -(5.) H 698. Classes of Water-Wheels. "Water-wheels may be classed as follows'. Ovetr.shot-wheels and breast-wheels, undershot-wheels and 699. Overshot and Breast-Wheels. The water is Supplied to this class of wheels at or below the summit, and acts wholly, or partly by its weight, as it descends in the buckets. (See Article 634). Formerly the buckets used to be closed at their inner sides, but now they are made with openings for the escape and re-entrance of air : an invention of Mr. Fairbairn. A breast- wheel differs from an overshot- wheel chiefly in having the water poured into the buckets at a somewhat lower elevation as compared with the summit of the wheel, and in being provided with a casing or trough, called a breast, of the form of an arc of a circle, extend- ing from the regulating sluice to the commencement of the tail- race, and nearly fitting the periphery of the wheel, which revolves within it. The effect of the breast is to prevent the overflow of water from the lips of the buckets until they are over the tail-race. The usual velocity of the periphery .of overshot and high breast- wheels is from three to six feet per second ; and their available efficiency, when well designed and constructed, is from 0'7 to 0-8. 700. Undershot- Wheels are driven by the impulse of water, dis- charged from an opening at the bottom of the reservoir with the velocity produced by the fall, against floats or boards, as to which see Article 649. Every such wheel has a certain velocity of maodmum efficiency, which does not in any case differ much from half the velocity of the water striking it. In undershot- wheels of the old construction, the floats are flat boards in the direction of radii of the wheel ; and the maximum theoretical efficiency is ^. The available efficiency is about 0-3. This class of wheels was much improved by Poncelet, who curved the floats with a con- cavity backwards, adjusting their position and figure so that the water should be supplied to them without shock, and should drop from them into the tail-race without any horizontal velocity. The available efficiency of such wheels is about 0'6. TURBINES WINDMILLS HEAT ENGINES. 629 701. A Turbine is a horizontal water-wheel with a vertical axis, receiving and discharging water in all directions round that axis : that is, driven by a vortex; its efficiency ranges from -6 to '8 (see Article 650). 702. windmills are driven by the impulse of the air against oblique surfaces called sails, rotating in a plane perpendicular to the direction of the wind. The best figure and proportions for windmill sails, as determined experimentally by Smeaton, are given by the following formulae, in which the whip means, the length of an arm, or the distance of the tip of a sail from the axis : length of sail, - whip : breadth at end nearest axis, = whip : at tip, - whip : angles made by the surface O o of the sail with the plane of rotation ab the end nearest the axis, 1 8 : at the tip, 7. The efficiency of a good windmill is about 0-29. (See Smeaton on Windmills, in Tredgold's Hydraulic Tracts.) 703. The Efficiency of neat Engines is the subject of a peculiar branch of science, Thermodynamics ; and an outline only of the principles on which it depends can here be given. If the number of British Fahrenheit units of heat produced by the combustion of one pound of a given kind of fuel, be multiplied by Joule's equivalent, 772 foot pounds, the result is the total lieat of combustion of the fuel in question, expressed in foot pounds. For different kinds of coal, it varies from 6,000,000 to 12,000,000 foot rounds. This total heat is expended, in any given engine, in pro- ducing the following effects, whose sum is equal to the heat so expended t 1. The waste Jieat of the furnace, being from 015 to 0-6 of the total heat, according to the construction of the furnace, and the skill with which the combustion is regulated. 2. The necessarily rejected heat of the engine, being = - x the heat n received by the elastic fluid : ^ being the upper, and t z the lower limits of absolute temperature, which is measured from the absolute zero, 493-2 Fahrenheit below the melting point of ice. 3. The heat wasted by the engine, whether by conduction, or by non-fulfilment of the conditions of maximum efficiency. 4. The useless work of the engine, employed in overcoming friction and other prejudicial resistances. 5 The useful work. The efficiency of a thermodynamic engine is improved by diminishing as far as possible the first four of these effects, so as to increase the fifth. The efficiency of a heat engine is the product of three factors; viz. : the efficiency of the furnace, being the ratio of the heat 630 THEORY OF MACHINES. transferred to the elastic fluid to the total heat of combustion; the efficiency of the fluid, being the fraction of the heat received by it which is transformed into mechanical energy ; and the efficiency of the mechanism, being the fraction of that energy which is available for driving machines. The maximum efficiency of the fluid between given limits of absolute temperature is expressed by As to the mechanical action of an elastic fluid on a piston, see Article 656. 704. steam Engines. Formulae for the mechanical action of steam on a piston, both exact and approximate, have been given in Article 656, equations 6 to 13. The efficiency of the steam lies between the limits '02 and *2 in extreme cases, and -04 and -1 in ordinary cases. The details of the construction and working of steam engines can be explained in a special treatise only. The duty of an engine is the work performed by a given quantity of fuel, such as one pound. The duty of a pound of coal varies in difierent classes of engines from about 100,000 to 1,900,000 foot pounds. These are extreme results, as respects wastefulness on the one hand, and economy on the other. In good ordinary engines, the duty varies from 200,000 to 700,000. 705. Eiectrodyuaniic Engines, though capable of higher efficiency than heat engines, are not so economical commercially, on account of the greater cost of the materials consumed in them. Their theo- retical efficiency, according to a law demonstrated by Mr. Joule, is given by the formula "> where yi is the strength which the electric current would have if the machine performed no mechanical work, and y 2 is * ne actual strength of the current. This law, and the law of the maximum efficiency of heat engines, are particular cases of a general law which regulates all transforma- tions of energy, and is the basis of the Science of Energetics.* * Edinburgh Philosophical Journal, July, 1855; Proceedings of ike Philotophical Society of Glasgow, 1853-5. APPENDIX. ARTICLE 634, p, 579. Motion of water in Waves. 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 planes which are perpen- dicular 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 = j 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 to- wards 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 bacl? f p and forward in a straight line, as at D. F i g . A. The deeper the water is, as compared with the length of a wave, the more nearly equal are the two axes of the orbit of 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 up- wards, of a length equal to that of the equivalent pendulum (that is, the pendulum whose period is T) viz., T 2 T 2 (seconds) (n 4 * 2 ~ 0-815 foot nearly " 632 APPENDIX. 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 : L = 2 * C A- = (in feet) 5-12 T 2 ; ............ (2.) 2i1F a = ^ = |? = (in feet per second) 5-12 T ....... ........ (3.) 1 ^ ar 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 (in feet) 5-12 w T 2 ; ............ ...... (4.) 2 =(m feet per second) 5-12 m T ............ (5.) III. Delation between Velocity of Advance and Depth of Uniform, Disturbance. Let h be the height of a wave j that is, the vertical diameter of the orbit of a surface particle. Then, in an inde- finitely short interval of time, the front slope of the wave ad- vances through the distance a d t, and the volume of water contained 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 depth of a canal in which the volume of water displaced per second would be equal to that dis- placed 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 b< APPENDIX. 633 shown to be = g h -f- 2 a ; therefore c = a 2 -f- r M, =d f ^ n + 9 - - j ' ' r , + : yr ....... (2.) C, _ i _ a T * i^*_ i + V The values of the quantities c, d are o - ^ I -I then . M r M,.^! ffor the right hand shear at the rth r " ~~^~ a t support, .... M r + ! M r . J for the left hand shear at the r + 1th r + ! r . = ~~ support, _, M w M n+ ! f for the right hand shear at all supports w== J^~ ~ \ except r f . _ M w - M w _! f for the left hand shear at all supports- " - J ~ ; n _ 1 ( except r + 1, ^ . then the reaction at any support is Let M and F denote the moment and shearing force at any section; then APPENDIX. 637 M = M r - F r x + W (x - a) for a section between W and "j the following support : > (6.) M = M M F n x for any other section, j F = W - F r for a section between W and the r + 1th ^ support, V (7.) F = F M for any other section, Equations (6) and (7) refer to a concentrated load ; for an uniform load for W substitute / w d a. CASE 2. One end free and the other fixed horizontally. In the application of the above formulae {!) to (7) to this case, make I, = 0, and let s 1 be the number of spans. CASE 3. Both ends fixed horizontally. In the above formulae make ^ = and l t = 0, and let 8 2 be the number of spans. 638 APPENDIX. The diagram annexed shows the form and proportions adopted by Professor Rankine for reservoir-walls of great height. tJRDINATES TO OUTER FACE iT 1"-40' FEET FUST. ^ *.. W| - .. 1 00.154.FT ^it80 \_ 12.128 \_ rT - xi For detailed description see The Engineer for January 5, 1872 (a reprint of this paper is embodied in the Rankine Memorial Volume of Selected Papers). The Unes of resistance lie within, or near to, the middle third of the fjiickness of the wall. The outer and inner faces are logarithmic curves. It is desirable to give such walls a curva- ture in plan convex towards the reservoir, to counteract the tendency of the wall to being bent by the pressure of the water into a curved shape, concave towards the water. APPENDIX. 639 The following notes on American Bridge Practice are taken from The Transactions of the American Society of Civil Engineers, Vol. VIII. : "American bridges are generally built up from the following individual members, most, if not all the mechanical work upon them being done in the shop. 1st. Chord and web eye-bars j round, square, or flat bars, with a head at each end, formed by some process of forging. These are tension members. 2nd. Lateral, diagonal, and counter rods. 3rd. Floor-beam hangers. 4th. Pins. 5th. Lateral struts. 6th. Posts. 7th. Top chord sections. The last three being columns formed by rivetting together various rolled forms ; plates, angles, channels, I beams, Ac. Some are square-ended, others pin-connected. These are compression members. 8th. Floor-beams and stringers. These consist either of rolled beams, rivetted plate girders^ or occa- sionally of latticed or trussed girders. The proportion of depth to span in American bridges is from one-fifth to one-seventh. " In top chords, posts^ and struts the strains are calculated by a modification of Rankine's formula, as follows : 8000 p = * or square-end compression members. + 400007 2 8000 p ^ for compression members with one pm and one square 1 + 30000,* end ' 8000 p for compression members with pin bearings. + 20000 r2 where p = the allowed compression per square inch of cross section. I = the length of compression member, in inches. r = the least radius of gyration of the section, in inches." ARTICLE 240, p. 215. The correctness of the value for wind pressure, as adopted by Professor Rankine, has been lately proved in the severe storms which have visited this country, a recent committee of enquiry having fixed this pressure on bridge surfaces at 56 Ibs. per square foot. 640 APPENDIX. CONTINUOUS BRAKES FOR RAILWAY TRAINS. The use of brake power is now considerably extended in railway traffic, and instead of the brakes being only applied on tender and guard's van, the application has been extended to the carriages composing the train. Yery considerable resistance is thus obtained, and consequent cessation of motion at a much earlier period. Various forms of continuous brake have been tried recently, and the results of the experiments are familiar to engineers. Some of the various forms are the screw-brake, chain- brake, vacuum-brake, hydraulic-brake, and compressed-air-brake, in all of which, by means of mechanism extending below the carriages and actuated by the engine-driver or guard, the whole or part of the wheels of the train can be braked. In the first two methods, rigid or flexible bodies are employed to transmit the power required, whilst, in the others, the same object is accomplished through the medium of fluids. In the hydraulic- brake, water at a high pressure from a pump on the engine is conveyed by a pipe ; in the vacuum-brake the air is removed, and in the air-brake the air is forced under pressure to the points required. In the automatic arrangements, whether of air or vacuum, there are reservoirs. It has been found desirable to adopt reservoirs or vessels having pistons immediately in connec- tion with the brake blocks, the object in the automatic arrange- ments being to keep up a certain condition in the chambers, whether of pressure or vacuum, by which, if destroyed either intentionally or accidentally (as through the breakage of a pipe), the braking action may at once take place. In some cases 1J seconds is sufficient to apply the brakes, and fast trains can be stopped in about 300 yards. ARTICLE 271, p. 289. Boilers are now largely made of steel, and when iron is used in the shell, the flues and ends are sometimes made of steel. Iron rivets, however, appear to be still in favour, as the shearing strength of mild steel appears less in proportion to tensile strength than in the case of iron. The tensile strength of steel boiler plates is about 29 tons per square inch, and the elastic limit appears to be about half of that. Punching out the rivet holes weakens the metal about 30 per cent. This loss, however, may be restored by annealing. Drilling the rivet holes does not affect the strength, and is the usual method adopted. Some advantage should arise, so far as steaming pro- perties are concerned through the use of steel, as lighter sections can be used for the plates. APPENDIX. 641 The Pulsometer Pump, recently introduced for engineering work, is a steam pump with few or no moving parts, the water is forced to the height of delivery by the direct action of the steam, the supply being raised by the condensation of the same steam. The action is regulated by a ball valve at the upper part, and clack valves at the lower part, of the working chamber. ARTICLE 346, p. 377. Steel is now largely used for rails, boiler and ship plates, &c., and combines great strength with ductility, the ultimate tensile strength varying from 27 to 33 tons per square inch, with an elongation of about 20 per cent. The limit of elasticity is about one-half of the ultimate or breaking strength. The Board of Trade allows 6 tons per square inch as the safe working strength for bridge structures. It appears that by the use of steel in the construction of ships a saving of weight of about 16 per cent, is obtained. Some peculiarities exist as to the behaviour of steel, and care must be taken both in the working of it from the ingot into plates, and in the workshop or yard, one special point being, that it should not be worked at a " black heat," or about 550 F. The steel referred to is known as " Mild Steel" or " Ingot Metal," and safety in working lies above or below this temperature. The question as to the comparative wear of iron and steel by corrosion seems still undecided, but so far no practical difference has been observed. Steel rails and tires are now used for railway traffic, and articles of cast steel can now be manufactured possessing considerable strength and ductility. ARTICLE 653, p. 598. Of late years much experimental information has been obtained as to the resistance of both vessels and their models, Mr. Froude, of Torquay, having determined the relations which exist between the ship and her model. Again, trials of the same vessel, at different speeds, have shown clearly that the law of resistance varies for different speeds. This subject is one of much im- portance, especially in connection with new and untried forms of vessels, and has already received much advancement from ex- perimental research.* * For authorities on this subject, see Transactions of the Institution oj Engineers and Shipbuilders in Scotland, and Transactions of the Institute oj Naval Architects. 2T 642 APPENDIX SECTION 6, p. 464. Hydraulic pressure is now largely used for engineering work, such as rivetting, flanging, punching, shearing, &c., and for the motive power for moving heavy loads, such as swing bridges, cranes, &c. By the introduction of flexible pipes and portable rivetting machines, hydraulic rivetting can be performed on bridges and ships. To accomplish this a. high pressure is used, varying from 700 to- 1,500 Ibs. per square inch, and this is obtained by means of an " accumulator," or cylinder containing a loaded piston, against which water is pumped in by a steam engine. This water under pressure being connected with the die of, say, a rivetting machine, is free to exert its pressure on the opening of the connecting valve, and a fall of the loaded plunger taking place at same time the rivet is closed up. This powerful closing pressure is very serviceable when a number of plates are to be bound together, as is the case for some of the keels of our large steam vessels. ARTICLE 637, p. 581. In a Report on Safety Valves,* drawn up by a Committee, of which Professor Rankine was a member, but whose decease happened prior to the completion of the experiments which were being carried out, it is stated that the weight of steam in pounds discharged per minute per square inch of opening, with square-edged entrance, corresponds very nearly with three- fourths of the absolute pressure in the boiler, the range being from 25-37 Ibs. to 100 Ibs. per square inch. And the requisite area for a safety valve is shown to be 4 x square feet of fire-grate a = . Pi where a = area of orifice in square inches, and p l = the absolute pressure. ARTICLE 626, p. 572, The Editor appends the following investigation by the late Professor Rankine of the value of the theoretical co-efficient ot contraction in a jet of water issuing from a large cistern with a pipe going into it. The investigation was laid before the Pro- fessor's Class of Civil Engineering and Mechanics in Glasgow University, Session 1866-67. * See Transactions of the Institution of Engineers and Shipbuilders in Scot- land, vol. xviii. APPENDIX. 643 Let a, s= area of orifice in feet. v = velocity of outflow in feet per second. c a v = number of cubic feet per second. D = weight of a cubic foot of water, then, D c a v = weight of flow per second. Now, the reaction or backward pressure exerted against the . Dcav 2 . . Dv 2 reservoir = - ; the pressure in the reservoir = -^ ; multi- y ^y plying the latter expression by a, and equating, we have Gas Engines have now been introduced with success, espe- cially for the lighter classes of work. These engines have the great advantage of being easily managed, as the starting and stopping of them only entails the opening or shutting of the tap regulating the gas supply. The action in the cylinder is an explosive one, the explosion taking place at certain intervals of the revolutions, a heavy fly-wheel being added to store up the excess of energy over work done when the explosion takes place. ARTICLE 684, p. 620. From recent experiments made with trains brought to rest simply by their own resistance, it appears that the latter amounts to about 9 J Ibs. per ton of weight, or J per cent. The effect of handbrakes as applied to tender and guards' vans was found to be about 2J per cent, of the weight. A train going about 45 miles per hour could be stopped in 1,000 yds., or at 60 miles per hour in about 1 mile. With the continuous brake system, as much as 10 per cent, of the train weight can be obtained as a retarding force, and a train travelling 60 miles per hour can be stopped in 400 yards. Brake-blocks are of cast- iron, steel, or wood. Some difference of opinion appears as to the effect of skidding the wheels ; no doubt, greater wear and tear is caused when the wheels are skidded. ARTICLE 704, p. 630. In Compound Engines as now used for marine and also land purposes, the consumpt of coal is less than 2 Ibs. per indicated horse power, or probably about If lb., being a duty of over 1,000,000 foot-pounds. 644 APPENDIX. The economy of the marine engine is largely due to high pres- sure steam, about 90 Ibs. per square inch being now often carried, to surface condensation, and the large ratio of expansion obtained by the compound system where the steam passes from one cylinder into one or two others, before reaching the con- denser; our best engines, however, are only yielding an efficiency of from about % to about ^. This appears to be made up more or less as follows : Efficiency of furnace and boiler, ^ ; efficiency of the steam, -f$ ; or total efficiency -^ x -$ = about | ; again, if we take the efficiency of the propeller as -3^, we shall have about Jg- as the final efficiency. DIMENSIONS AND STABILITY OF THE OUTER SHELL OF THE GREAT CHIMNEY .OF ST. ROLLOX. Greatest pres- Divisions of Heights above Chimney. Ground. External Diameters. TK?/.!^. sure of Wind Thicknesses. ^^ntvith Security. Feet Feet Inches. Feet Inches. lb. per square foot V. 435i 13 6 > I 2 IV. 35<>i 16 9 m. aioj 24 o If I lOf' 114^ 30 6 57 IL 2 3 54 35 o ~~ i 6S i. o 40 o 71 Fo-mdMion. "fi^T External Diameter. Thicknesses. Concrete. Brick. Feet Feet Feet Inches. Feet L 50 5 o 3 8 50 4 8 3 II. 14 5 12 III. 20 50 25 o O Total height from base of foundation to top of chimney, 455 feet TOWNSEND'S CHIMNEY, GLASGOW. BUILT, 1857-59. Total height 468 feet. Height from surface of ground to cope, * Joint of least stability. APPENDIX. 645 454 feet. Extra height of 20 feet of ornamental iron work since added, and connected with the lightning conductor. Outside diameter at foundations, 50 feet ; outside diameter at surface of ground, 32 feet ; outside diameter at top of cope, 12 feet 8 inches. The sides have a straight batter. The thickness varies from 7 bricks at base to 1 J brick at top. 646 APPENDIX. TABLE OP THE RESISTANCE OP MATERIALS TO STRETCHING AND TEARING BY A DIRECT PULL, in pounds avoirdupois per square inch. Tenacity, or Resistance to Tearing. Modulus of Elasticity, or Resistance to Stretching. MATERIALS. STONES, NATURAL AND ARTIFICIAL : Ceteit,} 28oto 3co Glass, 9,400 8,000,000 gl ate ./ 9, 6 o 13,000,000 I to 12,800 to 16,000,000 Mortar, ordinary, 50 METALS : Brass, cast, 18,000 9,170,000 wire, 49,o. 00 14,230,000 Bronze or Gun Metal (Copper 8, ) , rj^ j\ ( 3^,000 9,900,000 Copper, cast, 19,000 sheet, 30,000 bolts, 36,000 wire, 60,000 17,000,000 Iron, cast, various qualities, ( J *%& average, 16,500 17,000,000 Iron, wrought, plates, 5 1 ,000 joints, double rivetted, 35,7 single rivetted, fc 8,600 ^ bars and bolts, < ^ >7o'ooo I 2 9,ooOj O00 hoop, best-best, 64,000 1 25,300,000 wire-ropes, ^.. 90,000 15,000,000 Lead, sheet, 3,300 720,000 Steel bars, j IOO > 000 29,000,000 ( to 130,000 to 42,000,000 Steel plates, average, 80,000 Tin, cast, , 4,600 Zinc, 7,000 to 8,000 APPENDIX. 647 MATERIALS. TIMBER AND OTHER ORGANIC FIBRE: Acacia, false. See " Locust." Ash (Fraxinus excelsior), Bamboo (Bambusa arundinacea), Beech (Fagus sylvatica), Birch (Betula alba), Box (Buxus semper vir ens), Cedar of Lebanon (CedrusLibani), Chestnut (Castanea Vesca), Tenacity, or Resistance to Tearing. I7,OOO 6,300 11,500 15,000 20,000 11,400 {10,000 ) to 13,000] ( Ulmus campestris), Fir : Bed Pine (Pinus sylvestris), Spruce (A bies excelsa), Larch (Larix Europcea), Hoxen Yam, about Hazel (Corylus Avellana), Hempen Bopes, from 12,000 to 16,000 Hide, Ox, undressed, Hornbeam (Carpinus Betulus), . . . Lance wood (Guatteria virgata),... Leather, Ox, ' Lignum- Yitse (Guaiacum offici- \ nale), / Locust (Robinia Pseudo-Acacia), Mahogany (Swietenia Mahagoni), Maple (A cer campestris), Oak, European (Quercus sessili- id Quercus pedunculata), I Bed (Quercus ) 14,000 12,000 to 14,000 12,400 9,000 to 10,000 25,000 18,000 6,300 20,000 23,400 4,200 1 1, 80O l6,OOO f 8,000 ) ( to 21,800 J 10,600 ( 10,000 ( to 19,800 Modulus of Elasticity, or Resistance to Stretching. I,6oo,OOO 1,350,000 1,645,000 486,000 1,140,000 7OO,000 to 1,340,000 1,460,000 to 1,900,000 1,400,000 to 1,800,000 900,000 to 1,360,000 American rubra), , Silk Fibre, Sycamore(J. cerPseudo-Platanus), Teak, Indian (Tectona grandis), African, (?) Whalebone, Yew (Taxus baccata), 10,250 52,000 13,000 15,000 21,000 7,700 8,OOO 24,300 1,255,000 1,200,000 to 1,750,000 2,150,000 1,300,000 1,040,000 2,400,000 2,300,000 648 APPENDIX. II. TABLE OF THE RESISTANCE OP MATERIALS TO SHEARING AND DISTORTION, in pounds avoirdupois per square inch. _ . , Transverse Resistance Elasticity, MATERIALS. to or Res i st ance to METALS: Shearing. Distortion. Brass, wire-drawn, ....................... 5j33>o Copper, ................................... 6,200,000 Iron, cast, ................................. 27,700 2,850,000 TIMBER : Fir: Red Pine, ......................... 5 to 800 Spruoe, ............................. 600 Larch, .............................. 970 to 1,700 Oak, ....................................... 2,300 82,000 Ash and Elm, ........................... 1,400 76,000 III. TABLE OF THE RESISTANCE OF MATERIALS TO CRUSHING BY A DIRECT THRUST, in pounds avoirdupois per square inch. MATERIALS. to Crushing. STONES, NATURAL AND ARTIFICIAL: Brick, weak red, 5 50 to 800 strong red, 1,100 fire, 1,700 Chalk, 330 Granite, 5,5oo to 11,000 Limestone, marble, 5^5 granular, 4,000 to 4,500 Sandstone, strong, 5>5 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 APPENDIX. 649 Resistance MATERIALS. to Crashing. 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 (Achras Sideroxylori), ............... 14,000 Cabacalli, ........................................... 9,900 Cedar of Lebanon, ................................. 5,86o Ebony, West Indian (Brya Ebenus), .......... 19,000 Elm, .................................................. 10,300 Fir: Red Pine, ............................... ..... 5,375 to 6,200 Ameiic&nYellowJ?iii.e(Pinusvariabilis), Hornbeam, ......... . ............................... 7>3oo Lignum- Yitse, .............. ........................ 9>9oo Mahogany, .......................................... 8,200 Mora (Mora excelsa), .......................... .... 9>9oo Oak, British, ....................................... 10,000 Dantzic, ....................................... 7,700 American Red, .............................. 6",ooo Teak, Indian, .................................... ... 12,000 Water-Gum (Tristania nerifolia), ............. 1 1,000 IV, TABLE OF THE RESISTANCE OP MATERIALS TO BREAKING ACROSS, in pounds avoirdupois per square inch. MATERIALS. or Modulus of Rupture.* STONES: Sandstone, .. 1,100 to 2,360 Slate, .. ...,. 5,000 * The resistances stated are for .dry timber. Green timber is much weaker, having sometimes only half the strength of dry timber against crushing. f The modulus of rupture is eighteen times the load which is required to break a bar of one inch square, supported at two points one foot apart, and loaded in the middle between the points of support. 650 APPENDIX. Resistance to breaking, MATERIALS. or Modulus of Rupture. METALS: Iron, cast, open-work beams, average, 17,000 solid rectangular bars, var. qualities, 33,000 to 43,500 average, 40,000 wrought, plate beams, 42,000 TIMBER : 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 (JDammara australis), 11,000 Ebony, "West Indian, 27,000 Elm, 6,000 to 9,700 Fir: Red Pine, 7,100 to 9,540 Spruce, 9,900 to 12,300 Larch, 5,000 to 10,000 Greenheart (Nectandra Rodicei), 16,500 to 27,500 Lancewood, *7,35o Lignum- Yitse, 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 I* 00 , 11 , 13,3 baul, , 16,300 to 20,700 Sycamore, , 9,600 Teak, Indian, 12,000 to 19,000 African, 14,980 Tonka (Dipt&ryx odorata), 22,000 Water-Gum, 17,460 Willow (Salix, various species), 6,600 APPENDIX. 651 ONOOOVOO VO "* -"t-VO OO VO ON O OVO N O 10 vo CO J>. CO N vo *>. ^ ^- o ^ IM O IM M IN IM O I N M-VO ON o 1000 CO COOO 00 l:^ IO Jr- -^- co O ON H-I co ON 00 CO CO IO XO CO w oo "* ON M ON ON co M co ON to 10 f- O ' ' g3 a 03 4$ .3 * c3 * a| a-9.9.9gg.5*& ligi'iailllil ofE3^HSS^^ 5^fS 652 APPENDIX. VI. TABLE OF SPECIFIC GRAVITIES OF MATERIALS. Weight of a cubic GASES, at 32 Fahr., and under the pressure of one foot in atmosphere, of 2116-4 H>. on tne square foot: lb - avoirdupois Air., 0-080728 Carbonic Acid, 0-12344 Hydrogen, 0-005592 Oxygen, .. 0-089256 Nitrogen, ....... 0-078596 Steam (ideal), 0*05022 - 169 27 Granite, 164 to 172 2-631x5276 Gypsum, 143-6 2-3 Limestone (including marble),.. 169 to 175 27 to 2-8 magnesian, 178 2*86 Marl, 100 to 119 1-6 to 1-9 Masonry,. 116 to 144 1-85 to 2-3 Mortar, 109 175 Mud, 102 1-63 Quartz, 165 2-65 Sand (damp), 118 1-9 (dry), 88-6^ 1-42 Sandstone, average, 1 44 2 -3 various kinds, 130^0157 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, Ii86toi224 19 to 19-6 Iron, cast, various, 434 to 456 6-95^7-3 average, 444 7-n Iron, wrought, various, 474 to 487 7-6 to 7-8 average, 480 7-69 kead, 712 n- 4 Platinum, 1311 to 1373 21 to 22 Silver, 655 10-5 487 to 493 7-8 to 7-9 456 to 468 7'3to7'5 424 to 449 6-8 to 7-2 654 APPENDIX. TIMBER:* Ash, Bamboo, Beech, Birch, Blue-Gum, Box, Bullet-tree, Cabacalli, Cedar of Lebanon, Chestnut, Cowrie, Ebony, West Indian, Elm, Fir: Red Pine, Spruce, American Yellow Pine,. Larch, Greenheart, Hawthorn, , Hazel, Holly, Hornbeam, Laburnum, , Lancewood, , Larch. See "Fir." Lignum- Vitse, Locust, Mahogany, Honduras, Spanish, , Maple, Mora, Oak, European, American Bed, Poon, Saul, Sycamore, Teak, Indian, African, Tonka, Water-Gum, Willow, Yew, Weight of a cubic foot in Ib. avoirdupois. 47 25 43 44*4 52'5 60 65-3 56-2 30*4 33'4 36-2 74*5 34 30 to 44 30 to 44 29 31 t 35 62-5 57 54 47 47 57 42 to 63 41 to 83 44 35 53 49 57 43 to 62 37 41 to 55 61 62 to 66 62-5 25 Specific gravity, pure water = 1. 0753 0-4 0-69 0711 0-843 0-96 1-046 0-9 0-486 o-579 i-i 9 3 o'544 0*48 to 07 0-48 to 07 0-46 0-5 to 0-56 I '00 1 0-91 0-86 076 076 0-92 0*675 to I'OI 0-65 to 1-33 071 0-56 0-85 079 0-92 0-69 to 0-99 0-87 0-58 0*96 0-66 to 0-88 0-98 0-99 to i -06 i-ooi 0-4 0-8 * The Timber in every case is supposed to be dry. INDEX. ABSOLUTE Unit of Force, 486. Abutments of Arches, 261. Open and Hollow, 263. Stability of, 226, 235. Strength of, 268. Accelerating Effect of Gravity, 485. Force, 490. Impulse, 483. Acceleration, 386. Accumulator, 641. Actual Energy, 499, 507. of a Machine, 621. of Rotation, 532. Adhesion, 209. Aggregate Combinations, 466. Air Apparent Weight of Bodies in, 123. Expansion of, 123. Velocity of Sound in, 563. Weight of, 123, 652. Angle of Repose, 210. of Rotation, 391. of Rupture, 204, 259. of Torsion, 356. Angular Impulse, 506. ,, Momentum, 505, 529. Velocity, 391, 492. Arch Abutments of, 261. , Angle, Joint, and Point of Rupture of, 259. Circular Linear, 183, 200. Clustered, 263. Distorted, 202. Distorted Elliptic Linear, 186. -Elliptic Linear, 184. Geostatic approximate, 196, 207. Hydrostatic, 190, 207, 353. Iron-ribbed, 376. Line of Pressures in, 257. Linear or Equilibrated Rib, 162, 175, 182. Linear for Normal Pressure, 189. Piers of, 263. Pointed, 203. -Skew, 261. Stability of, 226, 257. Stereostatic, 198. -Strength of, 268. Arch-Total Thrust of, 203, 260. Areas Conservation of, 507. Measurement of, 58. Atmospheric Pressure, 69. Authorities on Waves, 633. Axes Conjugate, 77, 79. of Elasticity, 278. of Inertia, 524, 526. of Stress, 93, 98. Axis Fixed, 545. Instantaneous, 397. of Angular Momentum, 505, 529. of Rotation, 390. Axle Friction of, 614. Resilience of, 357. Strength of, 353. 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. of Forces 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 Friction of, 617. ,, in Mechanism, 454. Bars Strength of Iron and Steel, 377. Beams, 133. Allowance for Weight of, 346. Cast-iron, 318. Deflection under any Load, 328. Direct Vertical Stress in, 342. Expansion and Contraction of, 348. ' fixed at both ends 332. Limiting Length of, 347. Lines of Principal Stress in, 341. j, of Uniform Strength, 320. originally Curved, 348. Partially loaded, 344. Proof Deflection of, 322. 656 INDEX. Beams Proportion of Depth to Span of, 327. Resilience of, 330. Shearing Stress in, 338, 342. -Sloping, 348. Strength of, 307, 315, 318, 634. Bearings, 422. Belts, 288, 454. Bending Moment of, 308. ,, Resistance to, 307. Bevel-wheels, 428, 448. Blocks Stability of a series of, 230. and Tackle, 462. Bodies, 13. Boiler Stays, 296. Boilers Strength of, 289, 296, 299, 306, 640. Bond in Brickwork, 222. ,, Masonry, 222. Bowstring Girder, 369. Bracing of Frames, 142. Brakes, 242, 624, 640, 643. Breaking across Resistance to, 307. Breast-wheel, 628. Brickwork, 222, 226, 242. Bridges, 149, 153, 263, 344, 639. Suspension, 149, 171, 286, 370. Buckling, 302. Bulging, 302. Buoyancy, 120. Centre of, 121, 601. Buttresses^ 228, 235. CABLES Strength of, 288. Cam, 449. Cast-iron Beams, 318. Strength of, 362, 646. Catenary, 177. Cells Strength of, 364. Centre of Buoyancy, 121, 601. of Gravity, 49, 180, 505. of Mass, 482. of Oscillation, 546. of Parallel Forces, 31. of Percussion, 520, 544. of Pressure, 71, 76, 125, 227. of Resistance, 131. Centrifugal Couple, 537, 621. Force, 387, 491, 546. Pump, 597, Chains Equilibrium of, 162. for Pulleys, 454. Channel Flow in, 411. Chimneys-Stability of, 228, 235, 240, Cinematics, 15, 421. <* ,, Principles of, 379. Click, 462. Coefficient of Contraction, 572, 642. Coefficient of Elasticity, 277, 279. of Friction, 210, 612. of Pliability, 277, 279. Collapsing Resistance to, 306. Collar Friction of, 616. Collision, 508. Columns Strength of, 360, 638. Comparative Motion, 384, 389. Components, 19, 37, 381. Composition of Couples, Forces, Motions, 23, 381. Compound Engines, 642, 643. Screws, 467. Compressibility of Liquids, 271. Compression Resistance to, 302. Cones Speed, 457. Conjugate Axes, 79. Stress, 85. Connected Bodies Motions of, 420, 421. Connecting Rods Strength of, 363. Conservation of Angular Momentum, or of Areas, 506. of Energy, 478, 501, 508. ,, of Momentum, 505. Continuity Equations of in Gases, 417. Equations of in Liquids, 411, 413. Continuous Brakes, 640, 643. Girders, 338, 634. Contracted Vein, 572. Contraction Coefficient of, 572, 642. Cord Equilibrium of, 162. Motion of, 408. Cords, 454. Reduplication of, 462. Counterforts, 255. Couples Centrifugal, 537. Deviating, 535. Energy and Work of, 537. Polygon of, 25. Reduction of, 612. Statical Theory of, 21, with Inclined Axes, 24. with Parallel Axes, 21. Coupling Friction, 618. Hooke's, 461. Oldham's, 453. of Parallel Axes, 459. Crank and Axle Motion of, 458. ,, Strength of, 358. Cross-breaking Resistance to, 314. Crushing by Bending Resistance to, 360. Direct Resistance to, 302, Table of, 648. Cup and Ball Pivot, 616. Current, 412. ,, Pressure of on a Solid Body, 598. Radiating, 412, 574. Cycloid, 398. INDEX. 657 Cjcloidal Pendulum, 497. Cylinders Strength of, 289, 294. DAMS Stability of, 235, 243. Day Mean Solar, 382. Sidereal, 380, 381. Dead Points, 458. Deflection of Beam, 312, 322, 328. Deviating Couple, 535. Force, 491, 492, 545. Deviation (of Motion) Moment of, 528. Uniform, 387. Varying, 388. Differential Windlass, 466. Differentiation, 386. Direction Fixed and nearly fixed, 379. Distributed Forces, 48. Domes Stability of, 265. Driving Point, 610. Drums in Mechanism, 455. Ductility, 273, 376. Duty of Engines, 630. Dynamic Head, 568, 579. Dynamics, 15. General Equations of, 484. Principles of, 475. Dynamometer, 478. EARTH Friction of, 211. ,, Foundations, 219. Pressure of, 218, 249, 277. Stability of, 212. Table of Examples of; 221. Eccentric Motion of, 460. Eddy, 412. Effect of a Machine, 610. Efficiency, 609, 610. of Heat Engine, 629, 644. ofWaterwheels, 628. of Windmills, 629. Effort, 476, 611. Elastic Curve, 349. Force, 270. Elasticity Coefficients of, 277. Fluid, 270. Liquid, 271. Modulus of, 279, 646. Potential Energy of, 277. Theory of, 270, 275. Electro-dynamic Engine Efficiency of, 630. Elementary Combinations, 465. Ellipsoid of Inertia, 526. Energy, 477. Actual, 499, 507. ,, Actual of a Rotating Body, 532. Components of, 480, 499. Conservation of Motion being Uniform, 478. Energy Conservation of -in Varied Motion, 501, 508. Initial, 503. of Couples, 537. Potential, 477. Total, 503, 569. Transformation of, 499. Engines, 625, 643. Epicycloid, 401. Epicycloid al Teeth, 444. Epitrochoid, 401. Equilibrated Arch, 162, 175, 182. Equilibrium of forces, 25, 43. of Structures, 129. Stable and Unstable, 128. Expansion of Air, 123, 606. of Metals, Stones, Briok, Glass, Timber, 349. of Steam, 606. of Water, 125. Extrados of Arch, 173. FACTORS of Safety, 274, 362, 365. Fall, or Head, 627. Falling Body, 485, 486. Fan Blowing, 598. First Law of Motion, 476. Fixed Direction, 379. Point, 14, 381. Flexure Moment of, 311. ,, Resistance of, 312. Floating Bodies, 120, 600. Oscillation of, 603, Floats of Waterwheels, 628. Flow of Fluids, 417. of Liquid, 410. Flues Strength of, 306. Fluid, 100. Elasticity of, 285. Equilibrium of, 116. Flow of, 417. Impulse of on a Solid Surface, 591. M Motion of, 410, 566. ,, Pressure of, 99. , Resistance, 598. Fly-wheel, 623. Foot-pound, 477. Force, 15, 17. Absolute Unit of, 486. .,, Centrifugal, 387, 491. Deviating 491, 492, 595. ,, Distributed, 48. Reciprocating, 503. Representation of, 19. ,, -T-Unbalanced Measures of, 001. Forces Action of on a System of Bodiei, 510. Parallelogram of, 35. -Parallelepiped of, 37. 2U 658 INDEX. Forces Polygon of, 86. Reduction of, 612. Residual, 498, 61L Resolution of, 37. Foundation, 129. 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. Free Rotation, 533. Surface, 570. Friction, 209. Coefficient of, 210, 612. Couplings, 618. Heat of 620. Internal, 377. Moment of, 614. of Gas, 590. of Liquids, 584. of Machines, 612, 614. of Solid Bodies Law of, 209. Strap, 618. Tables of, 211, 613. Frictional Gearing, 618. Stability, 209. Tenacity, 222. Furnace Waste Heat of, 629. GAS, 13. Action of on a Piston, 604. Dynamic Head in, 579. Engine, 642. Equation of Continuity in, 581. Flow of from an Orifice, 581, 642. Flow of with Friction, 590. Motion of, 566. Motion of without Friction, 579. Geostatic Arch, 196. 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, 153. Governor, 548, 625. Gravity Accelerating Effect of, 485. Centre of, 15, 49, 61, 180. Motion under, 485, 486. Specific, 49, 124. Specific Table of, 652. Grease, 613. Groined Vaults, 262. Gyration, 515, 542. Radius of, 515. Table of Radii of, 518. HEAD Dynamic of Gas, 579. ,, of Liquid, 568, 627. ,, Equal Surfaces of, 573. Headers in Masonry, 223. Heat Engine Efficiency of, 629. Heat of Friction, 620. of Steam, 607. Specific of Gases at Constant Pressure, 580. Height due to Velocity, 487. Helical Motion, 394. Hool-3's Double Joint, 462. Gearing, 451. Law, 275. Universal Joint, 461. Hoop-tension, 290. Horse-power, 610. Horse Work of, 626. Hunting Cog, 434. Hydraulic Hoist, 465. Mean Depth, 587. Press, 462. Rivetting, 642. Hydraulics, 585. Hydrodynamics, 475, 566. Hydrostatics Principles of, 100, 112. 117. Hydrostatic Arch, 190, 206, 353. IMMERSED Body Pressure on, 122. ,, Plane Pressure on, 125. Impact, 564. Impulse, 483. Angular, 506. between Solids and Fluids, 591. ., and Momentum Law of, 484. Inclined Plane, 489. Indicator, 478. Inertia, or Mass, 482. Axes of, 524. Ellipsoid of, 526, 532. Moment of, 77, 514, 518. Reduced, 621. Inside Gearing, 441. Instantaneous Axis, 397, 404, 467. Integrals Approximate Computation of, 58, 386. Intensity of Distributed Force, 48. of Pressure, 69. of Stress, 68. Internal Equilibrium of Stress and Weight, 112. Internal Stress, 280. INDEX. 659 Intrados of Arch, 173. Isochronous Vibration, 553. Isotropic Solid, 278. JET Impul'se of, 591. Joints of a Structure, 129, 131. ,, of Masonry, 211. ,, of Rupture, 259. KEYS Friction of, 226. LATERAL Force, 476. Lattice Girder, 153, 160. Least Resistance Principle of, 215. Leather Strength of, 288. Length Measure of, 13, 14. Lever, 26. Line, 13. Linear Arch, 162, 182, 189, 203, 258. Link Motion, 468. Linkwork in Mechanism, 424, 458. Liquid, 13. Dynamic Head of, 568. Equilibrium of, 118. Flow of from an Orifice, 570. Flow of in a Pipe, 411, 588. Flow of in a Stream, 586. Free Surface of, 570. Motion of, 410. Motion of in Plane Layers, 570. Jf Motion of with Friction, 584. Surface of Equal Pressure in, 570. without Friction Motion of, 567. MACHINES, 5, 15, 421. Actual Energy of, 621. Pieces of, 422. Reduced Inertia of, 621. Theory of, 609. n Varied Motion of, 621. J} Work of with Uniform or Periodic Motion, 610. Man Work of, 626. Masonry and Brickwork Bond of, 222. -Frictionof,2ll, 222. Stability of, 230. Mass, 482, 484, 485. Centre of, 482. Matter, 13. Measures Comparative Table of British and French, 651. of Length, 13, 14. of Stress, 69. of Time, 381. of Velocity, 382. of Weight, 18. Mechanical Equivalent of Heat, 620, Mechanics Applied, 13. Dissertation on, 1. Mechanism Aggregate Combinations in, 425, 466, Elementary Combinations in. 423, 426. Principle of Connection in, 424. Theory of, 421. Mercury Weight of, 69. Metacentre, 601. Modulus of Elasticity, 279, 646. of Kesilience, 287. of Rupture, 316, 649. Moment Bending, 307. of a Couple, 22. ,, of Deviation, 528. of Flexure, 311. ,, ot Friction, 614. ,, of Inertia, 514. ,, of Inertia of a Surface, 77. , , of Inertia-Tables of, 82, 229, 518. of Stability, 233. of Stress, 73. of Tomon, 353. Statical, 27, 29. Momentum, 482. ,, Angular, 505, 529. ,, Conservation of, 505. and Impulse Law of, 484. of a Rotating Body, 529. Motion, 14. Comparative, 384, 389. Component and Resultant, 381, 383. First Law of, 476. Friction of, 226, 612. of a System of Bodies, 505. of Fluids Dynamics of 475, 566. of Gases, 417. n of Liquids, 410, 566. of Pliable Bodies and Fluids, 408. of Pliable Bodies Dynamics of, 552. of Points, 379. of Points Varied, 385. of Rigid Bodies, 390. Second Law of, 484. Uniform Dynamical Princi- ples of, 476. VariedDynamical Principles of, 482. Muscular Strength Work of, 625. NEUTRAL axis, 73. Notch Flow through, 573. OIL, 613. Oldham's Coupling, 453. Orifice Flow through, 571. 660 INDEX. Oscillation, 416. ^-Angular, 542. Centre of, 546. Elliptical, 495. Straight, 494. Oscillating Pendulum, 496, 546. Overshot Wheel, 628. PARABOLA Formulae relating to, 165. Parallel Forces, 25. Motion, 469. Projection, 45, 61, 127. Pendulum Ballistic, 548. Compound Oscillating, 546. ,, Compound Revolving, 547. Cycloidal, 497. Rotating, 547. Simple Oscillating, 496. Simple Revolving, 492. Percussion Centre of, 520, 644. Periodical Motion of Machines, 501. Pieces of a Structure, 129. Piers Stability of, 228. of Arches, 263. Open and Hollow, 263. Pile-driving, 564. Pillars Strength of Long, 360, 639. Strength of Short, 302. Pinion, 434, 443. Pinnacle on a Buttress, 239. Pipes Flow in, 411, 588. Friction in, 585, 588. Resistance caused by Sudden En- largement in, 589. Resistance of Curves and Knees in, 589. Resistance of Mouthpieces of, 589. Strength of, 289. Piston, 413. Action of a Fluid upon, 604. Piston Rods Strength of, 363. Pitch, 394, 433, 449. Surface, 426, 454. Pivot Friction of, 616. Plasticity, 272. Plate-iron Girders, 365, 366. Joints, 299. Plates Strength of Iron and Steel, 377, 641. Pliability, 273. Coefficients of, 277. Point -Fixed. 14, 381. Motions of, 379. Physical, 13, 475. Pointed Arch, 203. Posts Timber Strength of, 365. Potential Energy, 477. of Elasticity, S77. Pound Standard, 18. Power, 610. Preliminary Dissertation, 1. Press Hydraulic, 464. Strength of, 290. Pressure, 20, 69, 564. ,, between Rubbing Surfaces, 615. in a Sloping Solid Mass, 126. Internal, 289. of Earth, 218, 249, 277. ,, of Fluids, 99. Prime Movers, 609, 625. Principle of D'Alembert, 511. Projectiles, 487, 491, 599. Projection Parallel, 45, 61, 127. Proof Strength, 273, 274. Pulsometer Pump, 641. Pull, 69. Pulleys and Belts, 454. and Cords, 462. Speed, 457. Pump Centrifugal, 597. Pulsometer, 641. Rods Strength of, 297. RACK Motion of, 427. Teeth of, 438. Radiating Current, 574. Radius of Gyration, 515. Vector, 392. Railways Resistance on, 620. Reciprocating Force, 503. Reduced Inertia, 621. Reduction of Forces and Couples in Ma- chines to the Driving Point, 612. Regulator of a Prime Mover, 625. Repose Angle of, 210. Reservoir Walls Stability of, 243, 638. 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. of Materials, 273, 646. of Railway Trains and En- gines, 620, 643. of Rolling, 619. of Ships, .598, 641. Point of, 610. Resolution of Forces, 37. of Internal Stress, 82. Rest, 14. Friction of, 226, 612. Resultant, 18. Momentum, 482. of any System of Forces, 41. INDEX. 661 Reiultant of Couples, 23, 24. of Inclined Forces, 35. of Motions, 381. of Parallel Forces, 26, 28, 30. of Stress, 70. of Weight, 49. Retaining Walls, 227, 249, 638. Retardation, 386. Revetements, 227, 249. Revolving Simple Pendulum, 492. Rib Arch, 182. Ribbed Arches, 376. Rigid Body Action of a Single Force on, 543. Motion of, 390, 394, 513. Rigidity or Stiffness, 271. of a Truss, 144. Supposition of Perfect, 18. Rivets Strength of, 299. Rivetted Joints Strength of, 289, 299. Rivetting Hydraulic, 642. Roads Resistance of, 619. * Rolling Cones, 405, 535. Contact in Mechanism, 426. Load, 332. of Cylinder on Cylinder, 400. of Cylinder on Plane, 398. of Plane on Cylinder, 398. of Ships, 604. Resistance, 619. Roof, 142, 145. Ropes Stiffness of, 619. Strength of, 288. Rotating Body Comparative Motion of Points in, 393. Relative Motion of a Pair of Points in, 392. Rotation, 390. ,, Actual Energy of, 532. and Force Analogy of, 405. Angular Velocity of, 391. Axis of, 390. ,, Combined with Translation, 394. ,, Comparative Motionsin Com- pound, 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, 649. Angle of, 204, 259. Point of, 204, 259. SAFETY Factors of, 274, 641. Valvw, 642. Screw-like Motion, 894. ,, Gearing, 451. Screws Compound, 467. Friction of, 226. in Mechanism, 449. Second Law of Motion, 484. Sections Method of Applied to Frame- work, 150. Set. 271. Shafts and Axles Strength of, 353. Shear, 69, 87. Shearing Force in Beams, 307. ,, Resistance to, 298; Stress in Beams, 338. Table of, 648. Shifting or Translation, 390. Ship Resistance, 599, 641. Shrunk Rings, 294. Skew Arches, 261. Bevel Wheels, 430, 449. Sliding Contact in Mechanism, 436. Solid, 13. Sound Velocity of, 563. Spandril Wall, '257. Specific Gravity, 42, 124, 652. Speed-cones, 457. Speed Fluctuations of, 622. Spheres Strength of, 290. Spiral, 398. Stability, 128. Frictional, 230. of Floating Bodies, 600. of Structures, 130, 131. Standard Measure of Length, 14. ,, Measure of Weiglit, 18. Starting of Machines, 624. Statics, 15. Principles of, 17. Stays, 133, 136. Steady Motion of a Gas, 419. of a Liquid, 412, 414. Steam Action of, 606. Boilers, 289. Engine Efficiency of, 629, 630, 643. Steel Strength of, 377, 631, 641, 646. Stereostatic Arch, 198. Stiffness, 130, 270, 273. of Beams, 322. Stopping of Machines, 624. Strain, 272. and Stress Relations between, 280. -Ellipse of, 280. Resolution and Composition of. 275. Stream Friction of, 586. Hydraulic Mean Depth of, 587. Lines, 600. of Gas, 417. of Liquid, 411, 586. 662 INDEX. Stream Varying, 587. Strength, 130, 270. of Abutments and Vaults, 268. of Axles, 353, 358. of Beams, 30 /, 315. of Boilers, Pipes, and Cylinders, '289, 299, 306. of Bolts, Pins, Keys, and Rivets, 299. of Iron and Steel, 377,640, 646. of Iron Effects of Repeated Melting on, 376. of Leathern Belts, 288. of Long Pillars and Struts, 360, 639. of Masonry and Brickwork, 268, 302. of Pump-rods, 298. of Ropes and Cables, 288. of Short Pillars, 304. of Spheres. 290, 295. of Teeth, 359. of Tie -bar, 286. of Tubes and Flues, 306. Proof, 273. Tables of, 377, 646. Transver se, 315. Ultimate, 273. Stress, 68. and Strain Relations between, 280. Internal, 82. Stretchers in Masonry, 223. Stretching Resistance to, 286. Stroke Length of in Mechanism, 460. Structures, 15. ,, Theory of, 129. Transformation of, 129. Struts, 133. Strength of, 302, 360, 365. VVrought-iron, 364. Superposition of Small Motions, 555. Surface, 13. Suspension Bridge, 149, 165, 168, 171. Stiffened, 370. M Strength of, 288, 301. ,, with Sloping Rods, 171. with Vertical Rods, 168. System of Bodies Motion of, 505. TABLES of Bending Moments, 310, 311. of Coefficient of Friction. 211, 613. ,, of Compressibilty of Liquids, of Expansion by Heat, 349. of Factors of Safety, 274, 356. of Figures of Beams, 321. ,, of French and British Measures, 651, Tables of Measures of Velocity, 382. of Moments of Inertia, 82, 229. 518. of Shearing Forces, 310, 311. of Specific Gravities, 652. of Stability of Earth, 221. of Strength of Iron Pillars, 363. ,, of Materials, 274, 288, 646-650. of Work of Men and Horses, 626. Tangential Stress, 69, 87. Tearing Resistance to, 286. Tables of Resistance to, 288, 289, 377, C46. Teeth of Wheels, 432. ,, 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. Temperature Effects of, 376. Tenacity, 2S6. Tension, 69. Testing Strength, 273. Theory and Practice in Mechanics Har- mony of, 1, 10. Thrust, 69. Tie, 132. Flexible, 169. ,, Strength of, 286. Timber Struts, 365. Ties, 301. Time Measure of, 881. Torsion Moment of, 353. Toughness, 273. Towers Stability of, 240. Trains of Mechanism, 465. ,, Efficiency of, 610. Epicyclic, 473. of Wheels, 434. Transformation, 66, 127. ,, of Cords and Chains, 180. of Energy, 501. of Frames, 162. of Stress, 92. of Structures in Masonry, 332, 268. Translation or Shifting, 390. Varied, 482. Transverse Strength, 315. Table of, 649. Trochoid, 398. Trundle, 447. Truss, 144. Compound, 148. Trussing Secondary, 146. INDEX. 663 Tunnels, 264. Turbine, 595, 628, 629. Turning, 390. Twisting and Bending, 358. Moment, 353. UNBALANCED Force Measures of, 501. Undershot Wheel, 628. Unguents, 613. Uniform Deviation, 387. Effort or Resistance Effect of, 490. Motion, 382. Motion under Balanced Forces, 476. Velocity, 382. Unit of Force, 18, 486. of Length, 14. of Specific Gravity, 49. of Time, 381. Universal Joint, 461. Double, 462. Unsteady Motion of Fluid, 413, 415. VANES Impulse of Liquid on, 593. Varied Rotation, 538. Vaults Groined, 262. Stability of, 226. Velocity, 382. ' Angular, 391. of Sound. 563. Ratio, 463. Uniform, 382. Uniformly-varied, 388. Jf Varied, 385. Varied Rate of Variation of, 387. Velocities Virtual, 479. Vibration, 552. ,, Isochronous, 553. ,, Not Isochronous, 557. of Elastic Body, 557. Virtual Velocities, 479. Viscous Liquid, 273. Vis-viva, 499. Volume, 13. Vortex, 412, 574. Action of on Wheel, 595, 629. Combined, 576. Forced, 576. ., Free Circular, 574. Free Spiral, 576. Vortex Wheel, 596. WALLS Retaining, 227, 249. Stability of, 226, 638. Warren Girder, 153. Water Apparent Weight of Bodies im- mersed in, 125. Expansion of, 125. Flow of, 585. Pipes, 289, 585. Pressure Engine, 626. Velocity ot Sound in, 563. Weight of, 125. Water-wheel, 578, 627. A ction of Vortex on, 595, 629. Efficiency of, 627. Impulse of Water on Floats of, 593. Waves Motion in, 416, 579, 631. of Vibration, 562. Wedges Friction of, 226. Weight, 49, 485. Apparent of Body immersed in Fluid, 123. Measures of, 18, 651. Table of; 652. Weir, 243. Wheel and Screw, 452. Wheels Bevel, 428, 448. Grooved, 431. Motion of, 426. Non-circular, 428, 449. Skew-bevel, 430, 449. Teeth of, 432. Train of, 434. White's Tackle, 463. Wind Action of on Towers and Chim- neys, 240. Pressure of, 240, 639. Windlass Differential, 466. Windmills, 629. Wiper or Cam, 449. Work, 477. of Machines, 610. Useful and Lost, 610. Working Point, 611. ,, Stress, 274. Wrenching Resistance to, 353. Wrought -iron Strength of, 362, 377, 646. YARD Standard, 14, BILL AND BAIN, PRJDJTEBS, GLASGOW, UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. ? 5 1C.:0' LD 21-100m-9,'48(B399 8 16)476 THE UNIVERSITY OF CALIFORNIA LIBRARY