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MECHANICAL TEXTBOOK; 
 
 INTRGDlTCTION TO THE STUDY OF MECHANICS 
 AND ENGINEERING. 
 
 WILLIAM JOHN MACQUORN RANKINE, 
 
 CIVIL engikeek; ll.d. tein. coll. due.; f.hbs. lond. ahd'edin. ; p.b.b.s.a. ; 
 
 LATE BEGIUe PROFESSOR OF CIVIL KNGIMEERING AND MECHANICS IN THE DHIVERSITY OF GLASGOW; 
 
 EDWARD FISHER BAMBER, C.E. 
 
 With Numerous diagrams. 
 
 LONDON: 
 
 CHARLES GRirPIN AND COMPANY, 
 
 stationers' hall court. 
 
 1873. 
 
Cornell University 
 Library 
 
 The original of tliis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31 924031 21 51 67 
 
PEEFACE. 
 
 This book is designed as an Introduction to more abstruse works 
 on Engineering and Mechanics, and in particular to those of 
 the late Professoi!»Eankine. 
 
 Its study demands only a previous acquaintance with the 
 ordinary Rules of Arithmetic, and with the Elementary Alge- 
 braical Notation. A few pages have been devoted to the 
 Differential and Integral Calculus, as these have been used in 
 different parts of the book, their application having been in every 
 instance explained. 
 
 Professor Eankine's Manual of Applied Mechanics has been 
 taken as tlie model for this work, the only alteration being the 
 treating of the Theory of Motion before that of Force, as more 
 in harmony with modern practice, and as proposed by himself for 
 the present purpose. 
 
 The general design of the work having been indicated, it onl3' 
 remains for me to explain briefly how my name has been con- 
 nected with that of Professor Eankine on the Title-page, and 
 also in what condition it was left at the time of his recent 
 lamented death. 
 
 I was Professor Eankine's Assistant, and lectured for Iiim 
 during his illness, and it was whilst on a visit which his death 
 suddenly terminated, that the arrangement was made whicli 
 connected me with him in the task. My duty was simply to 
 assist him in its preparation. On my mentioning to hiift that 
 the amount of labour I should have to do hardly justified my 
 
IV PREFACE. 
 
 name appearing with his as joint-author, he replied, that, owing 
 to his state of health, more of the work might devolve upon 
 me than I expected. The issue has proved the correctness of 
 his surmise. 
 
 As to the state of the MS. at the time of his death, two 
 hundred pages had heen already completed, and the general scope 
 and plan of the work decided upon. I need hardly say that his 
 wishes have been implicitly carried out in every respect, so far 
 as lay in my power. The work has been completed at the request 
 of Professor Rankine's Executrix, and at that of the Publishers, 
 at whose desire also I have undertaken the superintendence of 
 New Editions of his other Scientific Manuals, some of which 
 have already been submitted to the Public. 
 
 E. F. B. 
 
 Glasgow, October, 187$. 
 
CONTENTS. 
 
 MATHEMATICAL INTRODUCTION. 
 
 Article 
 
 Akithmetical Rules. 
 
 Page 
 
 1. To find the Prime Factors of a 
 
 Given Number, ... 1 
 
 2. To find the GreaAist Common 
 
 Measure (othervrise called the 
 Greatest Comtnon Divisor) of 
 Two Numbers, ... 1 
 
 3. To Reduce the Ratio oi'Two Num- 
 
 bers to its Least Terms, . 2 
 
 4. ToExpresstheEatioofTwoNum- 
 
 bers in the Form of a Continued 
 Fraction, .... 2 
 
 5. To forrii a series of Approxima- 
 
 tions to a Given Ratio, . . 2 
 
 6. Logarithms — Definitions, . . 4 
 6-16. Logarithms, . . . 4, 5 
 17. Antilogarithms, . . .6 
 
 Trigonometeical Rules. 
 
 18. Trigonometrical Functions De- 
 
 iined, 6 
 
 19. Relations amongst the Trigono- 
 
 metrical Functions of One 
 Angle, A, and of its Supple- 
 ment 7 
 
 20. The Circular Measure of an 
 
 Angle, 8 
 
 21. Trigonometrical Functions of 
 
 Two Angles, ... 8 
 
 22. Formulae for the Solution of 
 
 Plane Triangles, ... 8 
 
 23. To Solve a Right-angled Triangle, 9 
 
 24. To Express the Area of a Plane 
 
 Triangle in terms of its Sides 
 and .£igle3, . . . .10 
 
 Rules of the Diffekkntial and 
 Integbal Calculus. 
 
 25. Definitions 10 
 
 26. Rules for finding Differential Co- • 
 
 efficients, . . . .11 
 
 27. Illustration of the Difierential 
 
 Calculus to Geometry, . . 12 
 
 28. The Integral Calculus the Inverse 
 
 ofthelDifierential, . . 13 
 
 29. Approximate Computation of 
 
 Integrals, . . . .13 
 
 Rules for the . Mensuration of 
 Figures and Findiboof Centres 
 OF Magnitude. 
 
 Section 1.— Areas of Plane Surfaces. 
 Article Page 
 
 30. Parallelogram, . . .16 
 
 31. Trapezoid, . . .16 
 
 32. Triangle, ..... 16 
 
 33. Parabolic Figures of the Third 
 
 Degree 16 
 
 34. Any Plane Area, . . .17 
 
 35. Circle 21 
 
 36. Area of a Circular Sector, . . 22 
 Section 2,— Volumes of Solid Figures. 
 
 37. To Measure the Volume of any 
 
 SoUd, 22 
 
 Section 3. — Lengths of Curved Lines. 
 
 38. To Calculate the Lengths of Cir- 
 
 cular Arcs, . . . .23 
 
 39. To Measure the Length of any 
 
 Curve 25 
 
 Section 4. Geometrical Centres and 
 Moments. 
 
 40. Centre of Magnitude— General 
 
 Principles, . . . .25 
 
 41. Centre of a Plane Area, . . 26 
 
 42. Centre of a Volume, . . 27 
 
 43. Centre of Magnitude of a Curved 
 
 Line, 27 
 
 44. Special Figures, . . .28 
 Elementary Mechanical Notions. 
 
 Definition of General Terms and 
 Division of the Subject. 
 
 45. Mechanics 30 
 
 46. Matter, 30 
 
 47. Bodies 30 
 
 48. A Material or Physical Volume, 30 
 
 49. A Material or Physical Surface, 30 
 
 50. Line, Point, Physical Point, 
 Measure of Length, 
 
 51. Rest, 
 
 52. Motion, 
 
 53. Fixed Point, 
 
 54. Cmematics, 
 
 55. Force, 
 
 56. EquUibrinm or Balance, 
 67. Dynamics — Statics and Kinetics. 
 S8. Structures and Machines, 
 
 30 
 31 
 31 
 31 
 31 
 31 
 31 
 32 
 32 
 
CONTEajTS. 
 
 PART I.— PRINCIPLES OF CINEMATICS, OR THE COMPARISON 
 
 OF MOTIONS. 
 Article ^ Page 
 
 89. Division of the Subject 33 
 
 Chapter I. — Motions of Points. 
 
 Seotioh 1. — Motions of a Fair of 
 Points. 
 
 60. Fixed and nearly Fixed Direc- 
 
 tions 31 
 
 61. Motion of a Pair of Points, . 34 
 
 62. Fixed Point and Moving Point, 35 
 
 63. Component and Resultant Mo- 
 
 tions, 35 
 
 64. The Measurement of Time, . 35 
 
 65. Velocity, 36 
 
 66. Uniform Motion, . . .37 
 
 Sectioit 2. — Vniform Motion, of 
 Several Points. 
 
 67. Motion of Three Points, . . 37 
 
 68. Motions of a Series of Points, . 37 
 
 69. The Parallelopiped of Motions, . 38 
 
 70. Comparative Motion, . . 38 
 
 Section 3. — Varied Motion of Points. 
 
 71. Velocity and Direction of Varied 
 
 Motion 39 
 
 72. Components of Varied Motion, . 40 
 
 73. Uniformly Varied Velocity, • 4] 
 
 74. Graphical Representation of Mo- 
 
 tions, . . . - . 
 
 75. Varied Rate of Variation of 
 
 Chaptek II.— Motions op Rigid Bodies. 
 
 Section 1. — Rigid Bodies, and their 
 Translation, 
 
 82. The Term Rigid Body, . . 47 
 
 83. Translation or Shifting, . . 47 
 
 Section 2i Simple Rotation. 
 
 84. Rotation or Turning, . . 47 
 
 85. Axis of Rotation, . . .47 
 
 86. Plane of Rotation, ... 48 
 
 87. Angular Velocity, . . . 48 
 
 88. Uniform Rotation, . . .48 
 
 89. Rotation Common to all Parts of 
 
 Body, 49 
 
 90 Right and Lefl-Handed Rotation, 49 
 
 91. Relative Motion of a Pair of 
 
 Points in a Rotating Bbdy, . 49 
 
 92. Cylindrical Surface of Ec[ual 
 
 Velocities 50 
 
 93. Comparative Motions of Two 
 
 Points relatively to an Axis, . 
 
 94. Components of Velocity of a 
 
 Point in a Rotating Body, 
 Section 3. — Comhined Rotations and 
 Translations. 
 
 95. Property of all Motions of Rigid 
 
 Bodies, 51 
 
 50 
 
 SO 
 
 42 
 
 Velocity 
 
 76. Combination of Uniform andUni- 
 
 formly Accelerated Motion, . 
 
 77. Uniform Deviation, . 
 
 78. Varying Deviation, . 
 
 79. The Resultant Rate of Variation, 
 
 80. The Rates of Variation of the 
 
 Component Velocities, . 
 
 81. The Comparison of the Varied 
 
 Motions 
 
 96. Helical Motion, 
 
 97. To find the Motion of a Rigid 
 
 Body from the Motions of 
 Three of its Points, 
 
 98. Special Cases, .... 
 
 99. Rotation Combined with Trans- 
 
 lation in the same Plane, 
 
 100. Rolling Cylmder ; Trochoid, . 
 
 101. Plane Rolling on Cylinder; 
 
 Spiral Paths, 
 
 102. Combined Parallel Rotations, . 
 
 103. Cylinder Rolling on Cyhnder; 
 
 Epitrochoids, 
 
 104. Equal and Opposite Parallel 
 
 Rotations Comoined, 
 
 105. Rotations about Intersecting 
 
 Axes Combined, . 
 
 106. Rolling Cones, 
 
 107. Comparative Motions in Com- 
 
 pound Rotations, . 
 
 Section i.— Varied Rotation. 
 
 108. Variation of Angular Velocity, 63 
 
 109. Components of Varied Rotation, 64 
 
 43 
 
 43 
 
 4-i 
 45 
 45 
 
 45 
 
 45 
 
 51 
 
 52 
 63 
 
 54 
 55 
 
 55 
 56 
 
 58 
 
 62 
 
 63 
 
 Chapter III.— Motions of Pliable Bodies, and of Fluids. 
 
 110. Division of the Subject 65 
 
 Section 1. — Motions of Flexible Cm-ds. 
 
 tit. General Principles^ 
 
 65 
 
 112. Motions Classed, . . .65 
 
 113. Cord Guided by Surfaces of 
 
 Revolution 66 
 
CONTENTS. 
 
 Vll 
 
 Section 2. — Motions of Fluids of 
 
 Constant Density. 
 
 Article Page 
 
 114. Velocity and Flow, . . .66 
 
 115. Principle of Continuity, . . 67 
 
 116. Flow in a Stream, . . .67 
 
 117. Pipes, Channels, Currents, and 
 
 Jets, 67 
 
 Article Page 
 
 118. Steady Motion, . . . 68 
 
 119. Motion of Pistons, . . . 68 
 Section 3. — Motions of Fluids of 
 
 Varying Density. 
 
 120. Flow of Volume and Flow of 
 
 Mass, 69 
 
 121. The Principle of Continuity, . 69 
 
 PART II.— THEORY OF MECHANISM. 
 Chapter I. — Definitiohs and Genebai, Pkinciples. 
 
 122. Theory of Pure Mechanism De- 
 
 fined, 
 
 123. The General Problem, . 
 
 124. Frame; Moving Bisces; Con- 
 
 nectors; Bearm^s, 
 
 125. The Motions of Primary -Moving 
 
 Pieces, 
 
 71 
 
 126. The Motions of Secondary Mov- 
 
 inePieccs, . 
 
 127. An Elementary Combination, 
 
 128. Line of Connection, . 
 
 129. Principle of Connection, . 
 
 130. Adjustments of Speed, 
 
 131. A Train of Mechanism, . 
 
 132. Aggregate Combinations, 
 
 72 
 72 
 73 
 73 
 73 
 73 
 78 
 
 Chapter II.— On Elementary Combinations and Trains of Mechanism. 
 
 Section \.— boiling Contact. 
 
 133. Pitch Surfaces, . . . 74 
 
 134. SmoothWheets, Bolters, Smooth 
 
 Racks, . ... 74 
 
 135. General Conditions of Rolling 
 
 Contact, .... 74 
 
 136. Circular Cylindrical Wheels, . 75 
 
 137. A Straight Rack and Curcular 
 
 Wheel 75 
 
 138. Bevel Wheels, .... 76 
 
 139. Non:Ciroular Wheels, . . 77 
 Section 2. — Sliding Contact. 
 
 140. Skew-Bevel Wheels, . . 77 
 
 141. Principle of Sliding Contact, . 79 
 
 142. Teeth of Wheels, . . .81 
 
 143. Pitch and Number of Teeth, . 81 
 
 144. Hunting Gog, . . . .83 
 
 145. A Train of Wheelwork, . . 83 
 
 146. Teeth of Spur -Wheels and 
 
 Racks. General Principle, . 86 
 
 147. Teeth Described by KoUing 
 
 Curves, . . . .86 
 
 148. The Sliding of a Pair of Teeth 
 
 on eabh other, . . .87 
 
 149. The A;-c of Contact on the 
 
 Pitch Lines, .... 88 
 
 150. The Length of a Tooth, . . 88 
 
 151. Involute Teeth for Circular 
 
 Wheels 88 
 
 152. The Smallest Pinion with Invo- 
 
 lute Teeth, .... 89 
 
 153. Epicydoidal Teeth, ... 89 
 
 154. Teeth of Wheel and Trundle, . 90 
 
 155. Dimensions of Teeth, . . 91 
 
 156. The Teeth of a Bevel Wheel, . 91 
 167. The Teelh of Non- Circular 
 
 ■ Wheels, .... 92 
 
 158. A Cam or Wiper, ... 92 
 
 159. Screws. Pitch, . . . 92 
 
 160. Normal and Circular Pitch, . 93 
 
 161. Screw Gearing, . . .94 
 
 162. The Wheel and Screw, . . 95 
 
 163. The Relative Sliding of a Pan: 
 
 of Screws, . . . .95 
 
 164. Oldham's Coupling, . . 96 
 
 Section 3. — Connection by Bands. 
 
 166. Bands Classed, 
 
 166. Prindple of Connection by 
 
 Bands, 
 
 167. The Pitch Surface of a Pulley 
 
 or Drum, .... 
 
 168. Circular Pulleys and Droms, . 
 
 169. The Length of an Endless Belt, 
 
 170. Speed Cones 
 
 Section 4. — Linkwork. 
 
 171. Definitions, .... 
 
 172. Principles of Connection, . 
 
 173. Dead Points, . . . . 
 
 174. Coupling of Parallel Axes, 
 
 175. The Comparative Motion of the 
 
 Connected Points, 
 
 176. An Eccentric, .... 
 
 177. The Length of Stroke, . 
 
 178. Hooke's UniversalJoint, . 
 
 179. The Double Hooke's Joint, 
 
 180. A Click, . . . . 
 
 97 
 
 97 
 
 98 
 
 98 
 
 99 
 
 100 
 
 101 
 101 
 101 
 101 
 
 102 
 103 
 104 
 104 
 105 
 105 
 
 Section 5. — Seduplication of Cords. 
 
 181. Definitions, . . . .105 
 
 182. Velocity Ratio, . . .106 
 182a. The Velocity of any Ply, .106 
 
 183. White's Tackle, . . .106 
 
VUl 
 
 CONTENTS. 
 
 Section 6. — Cmnparatwe Motion in 
 
 the "Mechanical Powers." 
 
 Article Page 
 
 184. Classification of the Mechanical 
 
 Powers," . . . .107 
 
 Section 7. — Hydraulic Connection. 
 
 185. The General Principle, . . 110 
 
 Article Page 
 
 186. Valves, 110 
 
 187. The Hydraalic Press, . . 110 
 
 188. The Hydraulic Hoist, . . Ill 
 
 Section 8.— Trajus of Mechanism. 
 
 189. Trains of Elementary Combina- 
 
 tions, Ill 
 
 190. The General Principles, 
 1^1. Differential Windlass, 
 
 Chaptee III, — On Aogbeoatb Combinations. 
 192. Compound Screws, 
 
 112 
 112 
 
 113 
 
 PART III.— PEINCIPLES OF STATICS. 
 
 Chapter I.— Sdmmart op General Principles. — Nature and Division 
 OP THE Subject. 
 
 193. Forces — Action and Ee-action, 115 
 
 194. Forces, how Determined and 
 
 Expressed, .... 115 
 
 195. Measures of Force and Mass, 
 
 196. Eepresentation of Forces by 
 
 Lmes, . 
 
 197. Eesultant and Component 
 
 Forces — Their Magnitude, 
 
 198. Equilibrium or Balance, . 
 
 199. Parallel Forces, 
 
 117 
 
 117 
 
 118 
 118 
 118 
 
 200. Couples, . . . .118 
 
 201. The Centre of Parallel Forces, 119 
 
 202. Distributed Forces in General, 119 
 
 203. Specific Gravity — Heaviness— 
 
 TDensity — Bulkiness, . . 120 
 
 204. The Centre of Gravity, . 121 
 
 205. The Centre of Pressure, . . 121 
 
 206. The Centre of Buoyancy, , . 121 
 
 207. The Intensity of Pressure, .121 
 
 Chapter II.— Composition, Eesolution, and Balance of Forces. 
 
 Section 1. — Forces Acting through 
 One Point. 
 
 208. Resultant of Forces Acting in 
 
 One Straight Line, . . 122 
 
 209. Kesnltant and Balance of In- 
 
 clined Forces — Parallelogram 
 of Forces, . . . .122 
 
 210. Triangle of Forces, . . . 122 
 
 211. Polygon of Forces, . . .123 
 
 212. Principles of the Parallelopiped 
 
 of Forces, . . . .123 
 
 213. Eesolution of a Force into Two 
 
 Components, . . . 123 
 
 214. Eesolution of a Force into Three 
 
 Components, . . . 124 
 
 215. Eesolution of a Force. Eect- 
 
 angular Components, . . 124 
 
 216. Eesultant and Balance of any 
 
 number of Inclined Forces 
 acting through One Point, . 125 
 
 Section 2. — Eesultant and Balance of 
 Couples. 
 
 217. Equivalent Couples, . . 125 
 
 218. Eesultant of Couples, . . 125 
 
 219. Equilibrium of Couples with 
 
 same Axis, .... 126 
 
 220. Parallelogram of Couples, . 1-26 
 
 221. Polygon of Couples, . . 126 
 
 222. Eesultant ofa Couple and Single 
 
 Force in Parallel Planes, . 126 
 
 223. Moment of Force with respect 
 
 to an Axis, .... 127 
 
 Section 3. — Resultant and Balance of 
 Parallel Forces. 
 
 224. Magnitude of Eesultant of Par- 
 
 allel Forces 127 
 
 225. Direction of Resultant of Parallel 
 
 Forces— Principle of the Lever, 123 
 
 226. To find the Resultant of Two 
 
 Parallel Forces, . . .128 
 
 227. To find the Relative Proportions 
 
 of Three Parallel Forces which 
 Balance each other, Acting in 
 One Plane: their Lines of 
 Action being given, . . 129 
 
 228. To find the Relative Proportions 
 
 of Four Parallel Forces which 
 Balance each other, not Acting 
 in One Plane : their Lines of 
 Action being given, . . 129 
 
 229. Moments of a Force with re- 
 
 spect to a Pair of Rectangular 
 Axes 130 
 
 230. Balauoeof any System of Parallel 
 
 Forces in One Plane, . . 131 
 
 231. Resultant of any System of 
 
 Parallel Forces in One Plane, 131 
 
CONTENTS. 
 
 Article Page 
 
 232. Balance of any System of Par- 
 
 allel Forces 132 
 
 233. Besoltant of any System of 
 
 Parallel Forces, . . .132 
 23i. To find the Centre of Parallel 
 
 Forces, , . . .133 
 
 Suction 4.-0/ any System of Forces. 
 Article Page 
 
 235. Resultant and Balance of any 
 
 System of Forces in One Plane, 134 
 
 236. Resultant and Balance of any 
 
 Systemof Forces, . . . 135 
 
 237. Parallel Projections or Trans- 
 
 formations in Statics, . . 13S 
 
 Ohaptee III. — Distributed Forces. 
 
 SECTioif 1.— Centres of Gravity. 
 
 238. Centre of Gravity of a Symme- 
 
 trical Homogeneous Body, . 140 
 
 239. The Common Centre of Gravity 
 
 of a Set of Bodies, . - 140 
 
 240. Planes of Symmetey— Axes of 
 
 Symmetry, .... 140 
 
 241. To find the Centre of Gravity 
 
 of a Homogeneous Body of 
 any Figure 140 
 
 242. If the Specific Gravity of the 
 Body Vanes 141 
 
 243. Centre of Gravity found by Ad- 
 
 dition 141 
 
 244. Centre of Gravity found by Sub- 
 
 traction, 141 
 
 245. Centre of Gravity Altered by 
 
 Transposition, . . , 142 
 
 246. Centre of Gravity found by Pro- 
 
 jection or Transformation, . 142 
 
 247. Centre of Gravity found Expe- 
 
 rimentally, .... 143 
 SECTioif 2.— Of Stress. 
 218. Stress — its Intensity, . . 143 
 
 , 144 
 145 
 
 145 
 146 
 
 249. Classes of Stress, . 
 
 250. Stress of Uniform Intensity, 
 
 251. Stress of Varying Intensity, 
 
 but of One Sign, . 
 
 252. Stress of Contrary Signs, 
 
 Section 3. — Principles of ffydrostatica 
 and Internal Stress of Solids. 
 
 253. Pressure and Balance of Fluids: 
 
 Principles of Hydrostatics, . 147 
 
 254. Compound Internal Stress of 
 
 Solids, 149 
 
 256. Conjugate Stresses — Principal 
 
 . 150 
 150 
 
 150 
 
 150 
 
 152 
 
 153 
 153 
 
 256. The Shearing Stress, 
 
 257. A Psii of Equal and Opposite 
 
 Principal Stresses, 
 268. Combination of any Two Prin- 
 cipal Stresses, 
 
 259. Deviation of Principal Stresses 
 
 by a Shearing Stress, . 
 
 260. Parallel Projection of Distri- 
 
 buted Forces, 
 
 261. Friction, .... 
 
 PART IV.— THEORY OP STRUCTURES. 
 Chapter 1.— Shmmaey op Prisciples op Stability and Strength. 
 
 Section 1. — Of Structures in General. 
 
 262. A Structure, . . . .156 
 
 263. Pieces — Joints — Supports — 
 
 Foundations, . . . 166 
 
 264. The Conditions of Equilibrium 
 
 of a Structure, . . . 157 
 
 265. Stability, Strength, and Stiff- 
 
 ness 157 
 
 Section 2. — Balance and Stability of 
 Frames, Chains, and Blocks. 
 
 266. A Frame, . , . .158 
 
 267. A Single Bar 158 
 
 268. Distributed Loads, . . .160 
 
 269. Frames of Two Bars, . . 161 
 
 270. Triangular Frames, . . 162 
 
 271. Polygonal Frame, . . .163 
 
 272. Open Polygonal Frame . . 165 
 
 273. Polygonal Frame— Stability, . 165 
 
 274. Braomg of Frames, . . 166 
 
 275. Rigidity of a Truss, . 168 
 
 276. Secondary and Compound 
 
 Trussing 169 
 
 277. Resistance of a Frame at a 
 
 Section, .... 171 
 
 278. Balance of a Chain or Cord, . 174 
 
 279. Stability of Blocks, . ,175 
 
 280. Transformation of Blockwork 
 
 Structures, .... 178 
 
 Chapter II. — Principles and Rules Relating to Strength and Stiffness, 
 
 Moduli of 
 
 281. The Object of this Chapter, . 179 
 Section 1. — Of Strength and Stiffness 
 
 in General. 
 
 282. Load, Stress, Strain, Strength, 179 
 
 283. Coefficients 
 
 Strength, . . . .180 
 
 284. Factors of Safety, . . .180 
 286. The Proof or Testing, . . 182 
 
CONTENTS. 
 
 Article Page 
 
 286. Stiffness or Rigidity, Pliability, 
 
 their-Modnll or Coefficients, 183 
 
 287. Tlie Elasticity of a Solid,. . 183 
 
 288. Resilience or Spring, . . 18t 
 
 289. Heights or Lengths of Moduli 
 
 of Stifiiiesa and Strength, .184 
 
 Section 2. — Of Besktance to Direct 
 Tension. 
 
 290. Strength, Stiffness, and Resili- 
 
 ence of a Tie, . - . 184 
 
 291. Thin Cylindrical and Spherical 
 
 Shells, 186 
 
 Section 3. — Of Resistanceto Distortion 
 and Shearing. 
 
 292. Distortion and Shearing Stress 
 
 in General 186 
 
 Section 4. — Of Resistance to Ttmting 
 and Wrenching. 
 
 293. Twisting or Torsion in General, 187 
 
 Article Pa«o 
 
 294. Strength of a Cylindrical Shaft, 187 
 
 SecTIOh 5.— Of Resistance to Bending 
 and Cross-Breaking. 
 
 295. Resistance to Bending in General, 189 
 
 296. Calculation of Shearing Loads 
 
 and Bending Moments, - . 190 
 
 297. Examples 194 
 
 298. Bending Moments produced by 
 
 Longitudinal and Oblique 
 Forces, . . . .196 
 
 299. Moment of Stress — ^Transverse 
 
 Strength, . ^ . . .196 
 
 300. Allowance for Weight of Beam, 
 
 — Limiting Len^h of Beam, 200 
 
 Sbotion 6. — Of Resistance to Thrust 
 or Pressure. 
 
 301. Resistance to Compression and 
 
 Direct Crushing, . . .202 
 
 PART V. -PRINCIPLES OF KINETICS. 
 
 Chapter I.— Summaky of General Principles. — Nature and Division 
 
 OF THE Subject. 
 
 302. 
 
 804. 
 305. 
 
 807 
 
 309. 
 310. 
 311. 
 312. 
 
 Effort — Resistance — Lateral 
 
 Forc^ 205 
 
 The Conditions of Uniform 
 
 Motion 205 
 
 Work, . . . . 206 
 
 Energy 206 
 
 The Conservation of Energy, . 206 
 The Principle of Virtual Velo- 
 cities, 206 
 
 The Mass, or Inertia, . . 206 
 The Centre of Mass, . - 207 
 The Momentum, . . .207 
 The Resultant Momentum, . 207 
 Variations and Deviations of 
 Momentum, . . , 207 
 
 313. Impulse, . . . '. .207 
 
 314. Impulse, Accelerating, Retard- 
 
 ing, Deflecting, . , .207 
 
 glS. A Deviating Force, . . .207 
 
 316. Centrifugal Force, . . .207 
 
 317. The Actual Energy, . . 207 
 
 318. Energy Stored and Restored, . 208 
 
 319. The Transformation of Energy, 208 
 
 320. Periodical Motion, . . .208 
 
 321. A Reciprocating Force, . . 208 
 
 322. Colhsion, . . . .208 
 
 323. The Moment of Inertia, . . 208 
 
 324. The Radius of Gyration, . . 208 
 
 325. The Centre of Percussion, . 208 
 
 326. Principles of Kinetics, , , 209 
 
 Chapter 2.— On Uniform Motion under Balanced 'Forces. 
 27. First Law of Motibn, 
 
 210 
 
 Chapter III. — 0» the Varied Translation of Points and Rigid Bodies. 
 
 Section 1. — Law of VariedTranslation. 
 
 328. Second Law of Motion, . . 211 
 
 329. General Equations of Kinetics, 211 
 
 330. Mass in Terms of Weight, . 212 
 
 331. An Absolute Unit of Force, . 213 
 
 332. The Motion of a Falling Body, 213 
 
 833. An Unresisted Projectile, . 214 
 
 834. An Uniform Effort or Resist- 
 
 ance, 215 
 
 335. Deviating Force of a Single 
 
 Body, 216 
 
 336. A Revolving Simple Pendlilum, 217 
 
 337. Deviating Force in Terms of 
 
 Angular Velocity, . . 217 
 
 S38. A Simple Oscillating Pendulum, 218 
 
 Section 2. — Va/ried Translation of a 
 System, of Bodies. 
 
 339. Conservation of Momentum, . 219 
 
 340. Motion of Centre of Gravity, . 219 
 
 341. The Angular Momentum, . 219 
 
 342. Angular Impulse, . . .220 
 
 343. Rdations or Angular Impulse 
 
 and Angular Momentum, . 220 
 
 344. ConeervatiSh ' of Angular Mo- 
 
 mentum, .... 220 
 
 345. CoUisSon, . . . .221 
 
CONTENTS. 
 
 CB4PIBB IV. — Rotations op Rigid Bodies. 
 
 Article 
 
 8-16. The Motion ofaRi^ Body, . 222, 
 
 Section 1. — On Momania of Inertia,, 
 
 Radii of Gyration, and Cmtres of 
 
 Percussion. 
 
 347. The Moment of Inertia, . . 222 
 
 348. The Moment of Inertia of a 
 
 System of Physical Points, . 223 
 
 349. The Moment of Inertia of a 
 
 RiridBody, . . . .223 
 360. The Radius of Gyration, . . 223 
 351. Components of Sloment of In- 
 ertia 224 
 
 Article Page 
 
 3o2. Moments of Inertia Gonnd 
 
 Parallel Axes Compared, . 224 
 
 353. Combined Moments of Inertia, 225 
 
 354. Examples of Moments of Inertia 
 
 and Radii of Gyration, , , 235 
 
 355. The Centre of Percussion, , 227 
 
 Section 2. — On Uniform Rotation. 
 
 356. The Momentum, . . .228 
 
 357. The Angular Momentum, . 228 
 
 358. The Actual Energy of Rotation, 229 
 
 CHAPTEn V. — Motions op Fluids. 
 
 359. Division of the Sabjeot, 
 
 230 
 
 Section 1. — Jffotions of Liquids 
 without Friction. 
 
 860. Dynamic Head, . . .230 
 
 361. Law of Rynamio Head for 
 
 .Steady Motion, . . .231 
 
 362. The Total Energy, . . .231 
 
 363. The Free Surface, . . .231 
 
 364. A Surface of Equal Pressure, . 231 
 865. Motion in Plain Layers, . . 232 
 366. The Contracted Vein, . . 233 
 
 Section 2. — Mofifms of Liquids with 
 FrMion. 
 
 367. General Laws of Fluid Friction, 233 
 
 368. Internal Fluid Friction, . . 235 
 
 369. Friction in an Uniform Stream, 235 
 
 370. Varying Stream, . . .236 
 
 371. The Fnotion in a Pipe Running 
 
 Full 237 
 
 372. Resistance of Mouthpieces, . 238 
 
 373. The Resistance of Curves and 
 
 Knees, . . . .238 
 
 374. A Sudden Enlargement, . . 238 
 
 375. The General Problem, . . 239 
 
 PART VI.— THEORY OF MACHINES. 
 Chapter I. — Definitions and Gesekal Pkinciples. 
 
 376. Nature and Division of the 
 
 Subject, 
 
 377. A Prime Mover, 
 
 378. The Regulator, 
 
 379. A Governor, . 
 
 380. Fluctuations of Speed, 
 
 381. A Fly-Wheel, 
 
 382. A Brake, 
 
 383. Useful and Lost Work, 
 
 240 
 240 
 241 
 241 
 241 
 241 
 241 
 241 
 
 384. Useful and Prejudicial Resist- 
 
 ance 241 
 
 385. The Efficiency, . . .241 
 
 386. Power and Effect; Horse Power, 241 
 
 387. Driving Point: Train: Working 
 
 Point, . . . . .242 
 
 388. Points of Resistance, . . 242 
 
 389. Efficiencies of Pieces of a Train, 242 
 
 Chaptek IL— Op the Peefokmanoe of Work by Machines. ' 
 
 Section 1. — Of Work. 
 
 390. The Action of a Machine, .243 
 
 391. Work, 243 
 
 392. The Rate of Work, . . .243 
 
 393. Velocity 244 
 
 3S4. Work in Terms of Angular Mo- 
 tion .244 
 
 395. Work in Terms of Pressure and 
 
 Volume 245 
 
 396. Algebraical Expressions for 
 
 Work, 246 
 
 397. Work against an Oblique Force, 246 
 
 398. Summation of Quantities of 
 
 Work, 247 
 
 399. Representation of Work by an 
 
 Area, . . . . .249 
 
 400. Work agdnst Varying Resist- 
 
 ance, 249 
 
 40t Useful Work and Lost Work, . 251 
 
 402. The Work Performed against 
 
 Friction 251 
 
 403. Work of Acceleration, . . 252 
 
 404. Summation of Work of Accel- 
 
 eration, .... 256 
 
xu 
 
 COKTENTS. 
 
 Article Page 
 
 405. Reduced Inertia, . . . ,257 
 
 406. Summary of Various Kinds of 
 
 Work, 258 
 
 Section 2.— Of Energy, Power, and 
 Efficiency. 
 
 407. Condition of Uniform Speed, . 2.58 
 
 408. Energy— Potential Energy, . 259 
 
 409. EquaUty of Energy Exerted 
 
 and Work Performed, or the 
 Conservation of Energy, . 260 
 
 410. Various Factors of Energy, . 260 
 
 411. The Energy Exerted in Pro- 
 
 ducing Acceleration, . . 260 
 
 412. The Accelerating Effort, . . 260 
 
 413. Work during Retardation — 
 
 Energy Stored aad Restored, 262 
 
 Article Paee 
 
 414. The Actual Energy, , . 262 
 
 415. A Reciprocating Force, . . 263 
 
 416. Periodical Motion, . . . 264 
 
 417. Starting and Stopping, . . 265 
 
 418. The Efficiency, . . .265 
 
 419. Power and Effect; Horse Power, 266 
 
 420. General Equation, . . .267 
 
 421. The Principle of Virtual Velo- 
 
 cities, 267 
 
 422. Forces in the Mechanical Powers 
 
 Neglecting Friction — Purchase 268 
 
 Sectioit 3. — Of Dynamometers. 
 
 423. Dynamometers . . . 271 
 424 Steam Engine Indicator, . . 271 
 
 Chapter III. — Of Eegulatinu Apparatus. 
 
 425. Begnlating Apparatus Classed 
 
 — Brake — Fly — Governor, . 276 
 Section 1. — Of Brakes. 
 
 426. Brakes DeSned and Classed, . 276 
 
 427. Action of Brakes in General, . 276 
 
 428. Block Brakes, . . 277 
 
 429. The Brakes of Carriages, . 278 
 
 Section 2.— Of Fly-Wheels. 
 
 430. Periodical Fluctuations of Speed, 278 
 
 431. Fly-Wheels 280 
 
 Section 3. — Of Governors. 
 
 432. The Regulator, . . .282 
 
 433. Pendulum-Governors, . . 283 
 
 434. Loaded Pendulum-Governor, . 285 
 
 Chapter IV.— Of the Efficiency and Counter-Efficiency of Pieces, 
 Combinations, and Trains m Mechanism. 
 
 435. Nature and Division of the , 
 
 Subject, .... 286 
 
 Section 1. — Efficiency and Counter- 
 Efficiency of Primary Pieces. 
 
 436. Efficiency of Primary Pieces in 
 
 General, . . .287 
 
 437. Efficiency of a Straight-sliding 
 
 Piece, 288 
 
 438. Efficiency of an Axle, . . 289 
 
 439. Efficiency of a Screw, . .293 
 
 Section 2. — Effiaency and Counter- 
 Efficiency of Modes of Connection in 
 Mechanism,. 
 
 440. Efficiency of Modes of Connec- 
 
 tion in General, . . . 293 
 
 441. Efficiency of Rolling Contact, 294 
 
 442. Efficiency of Sliding Contact 
 
 in General, .... 295 
 
 443. Efficiency of Teeth, . . 296 
 
 444. Efficiency of Bands. . . 297 
 446. Efficiency of Linkwork, . . 299 
 
 446. Efficiency of Blocks and Tackle, 300 
 
 447. Efficiency of Connection by 
 
 means of a Fluid, . . 301 
 
MATHEMATICAL INTEODUCTION. 
 
 ARITHMETICAL EXILES. 
 
 For convenience sake the following Arithmetical Rules are here 
 given : they will be referred to hereafter in the designing of toothed 
 gearing, under The Theory of Pure Mechanism, Part II. 
 
 Definition. — A prime number is one which is only divisible by 
 the number 1. 
 
 1. To find the Prime Factors of a Given Number Try the prime 
 
 numbers, 2, 3, 5, 7, .11, &c., as divisors in succession, until a prime 
 number has been found to divide the given number without a 
 remainder; then try whether and how many times over the quotient 
 is again divisible by the same j)rime number, so as to obtain a 
 quotient not divisible again by the same prime numberj then try 
 the division of that quotient by the next greater prime number; and 
 so on until a quotient is obtained which is itself a prime number; 
 that is, a number not divisible by any other number except 1. This 
 final quotient and the series of divisors will be the prime factors of 
 the given number. To test the accuracy of the process, multiply 
 all the prime factors together; the product should be the given 
 number. 
 
 2. To find the Greatest Common Measure (otherwise called the 
 greatest common divisor) of Two Numbers. — Divide the greater 
 number by the less, so as to obtain a quotient, and a remainder less 
 than the divisor; divide the divisor by the remainder as a new 
 divisor; that new divisor by the new remainder; and so on, until a 
 remainder is obtained which divides the previous divisor without 
 a remainder. That last remainder will be the required greatest 
 common measure. 
 
 If the last remainder is 1, the two numbers are said to be " prime 
 to each other." 
 
 Example. — Required, the greatest common measure of 1420 and 
 1808. 
 
Zi MATHEMATICAL INTRODUCTION. 
 
 Divisor, 1420) 1808 (1, Quotient. 
 
 1420 
 Eemainder, 388) 1420 (3, Quotient. 
 1164 
 Eemainder, 256) 388 (1, Quotient. 
 256 
 Eemainder, 132)256(1, Quotient. 
 132 
 Eemainder, 124) 132 (1, Quotient. 
 124 
 Eemainder, 8) 124(15, Quotient. 
 120 
 Eemainder, 4 ) 8 ( 2, Quotient. 
 
 The last remainder, 4, is the required gi-eatest common measure. 
 Definition. — Eatio is the mutual relation of two quantities in 
 respect of magnitude. 
 
 3. To reduce the Eatio of Two Numbers to its Least Teims, 
 divide both numbers by their greatest common measure. 
 
 ■J, , 1808-4 452 
 
 For example, ^^^^^^=3gg. 
 
 4. To express the Eatio of Two Numbers in the form of a Con- 
 tinned Fraction. — Let A be the lesser of the two numbers, and B 
 the greater; and let a, b, c, d, &c., be the quotients obtained during 
 the process of finding the greatest common measure of A and B. 
 Then, in the equation 
 
 B=a+1 
 A 6 + 1 
 
 c + 1 
 
 d + &o., 
 
 the right-hand side is the continued fraction required. 
 
 To save space in printing, a continued fraction is often arranged 
 as follows : — 
 
 ^ 1 1 1 i 
 
 a + 1 5 — oso. 
 
 6+ c+ d + 
 
 The ratio of two incommensurable quantities is expressed by an 
 
 endless continued fraction. For example, the ratio of the diagonal 
 
 1111 
 to the side of a square is expressed by 1 + „ — jj — s — -s — &c., 
 
 without end. 
 
 5. To form a series of Approximations to a Given Ratio. — Express 
 
APPROXIMATIONS TO A GIVEN RATIO. 3 
 
 the ratio in the form of a continued fraction. Then -write the 
 quotients in their order; and in a line below them write =- to the 
 
 left of the first quotient, and ^ directly under the first quotient. 
 
 Then calculate a series of fractions by the following rule : — Multiply 
 the first quotient by the numerator of the fraction that is below 
 it, and add the numerator of the fraction next to the left; the 
 sum will be the numerator of a new fraction : multiply the first 
 quotient by the denominator of the fraction that is below it, and 
 add the denominator of the fraction that is next to the left; the 
 sum will be the denttminator of the new fraction ; then write that 
 new fraction under the second quotient, and treat the second 
 quotient, the fraction below it, and the fraction next to the left, as 
 before, to find a fraction which is to be written under the third 
 quotient, and so on. For example : 
 
 Quotients, a, h, c, d, &c. 
 
 _, ,. 1 n n' n" 
 Fractions, =-, t, — , — , — ;,; 
 1 m ml ml 
 
 n _0 + a _a n' _1 + b n ^ n" _ n + c n' , 
 m l + 0~ I' mi'~0 + bm,' m,"~ m + cm" 
 
 452 
 To take a particular case; let the given ratio be as before, -w^., 
 
 then we have the following series : — 
 
 Quotients, 1 3 1 1 1 15 2 
 
 ^ ^. 1 1 4 5 9 14 219 452 
 
 ^'*°*'°"'' 1 1 3 4 7 n 172 355 
 
 Less or greater than )-L Q l G L G L G 
 given ratio, / 
 
 The fractions in a series formed in the manner just described are 
 called converging fractions, and they have the following properties : — 
 First, each of them is in its least terms; secondly, the difierence 
 between any pair of consecutive converging fractions is equal to 
 unity divided by the product of their denominators; for example, 
 9 S 36-35 1 9 14 99-98 1 ,,, „ ^, 
 
 f-4 = Tin- = 2-8^ 7-ri= 7-7ir=T7' ^''^^y' ^^'^ 
 
 are alternately less and greater than the given ratio towards which 
 they approximate, as indicated by the letters L and G in the 
 example; and, fowrthly, the difference between any one of them 
 and the given ratio is less than the difierence between that one and 
 the next fraction of the series. 
 
 Fractions intermediate between the converging fractions may be 
 
4 MATHEMATICAL INTRODUCTION. 
 
 o .-, I. 1 hn + kn' , w , w' 
 
 found by means of tne formula -r = — ^; where — and - > are 
 
 •' hm + km ni m 
 
 any two of the converging fractions, and h and k are any two whole 
 
 numbers, positive or negative, that are prime to each other. 
 
 6. Logarithius. Deflnitions. — The power of a number is the 
 product of itself multiplied a certain number of times. The index 
 or exponent of the power is the small figure placed above the right- 
 hand comer, which denotes the number of times the multiplication 
 takes place. The Logarithm of a number to a given base is the 
 index of the power to which the base must be raised to be equal 
 to the given number. That number of which the indices of the 
 powers are the logarithms, is called the base of the system. A suffix 
 denotes the base of the logarithm ; if a* = w, a; is the logarithm of 
 the number n to the base a, or log„ n = x. 
 
 Logarithms to the base 10 are called common logarithms. 
 
 7. The logarithm of 1 is 0. 
 
 8. The common logarithm of 10 is 1, and that of any power of 
 10 is the index of that power; in other words, it is equal to the 
 number of noughts in the power; thus the common logarithm of 
 100 is 2; that of 1000, 3; and so on. 
 
 9. The common logarithm of '1 is — 1, and that of any power of 
 •1 is the index of that power with the negative sign; that is, it is 
 equal to one more than the number of noughts between the decimal 
 point and the figure 1, with the negative sign; for example, the 
 common logarithm of "01 is — 2 ; that of "001, — 3; and so on. 
 
 10. The logarithms given in tables, are merely the fractional 
 parts of the logarithms, correct to a certain number of places of 
 decimals, without the integral parts or indices; which are supplied 
 in each case according to the following rules ; — 
 
 The index of the common logarithm of a number not less than 
 1 is one less than the number of integer places of figures in that 
 number; that is to say, for numbers less than 10 and not less than 
 1, the index is 0; for numbers less than 100 and not less than 10, 
 the index is 1 ; for numbers less than 1000, and not less than 100, 
 the index is 2; and so on. 
 
 The index of the common logarithm of a decimal fraction less 
 than 1 is negative, and is one more than the number of noughts 
 between the decimal point and the significant figures; and the 
 negative sign is usually written above instead of before the index ; 
 that is to say, for numbers less than 1 and not less than -1, the 
 index is 1^; for numbers less than -1 and not less than '01, the 
 index is 2; and so on. 
 
 The fractional part of a common logarithm is always positive, 
 and depends solely upon the series of figures of which the number 
 consists, and not upon the place of the decimal point amongst 
 them. 
 
logarithms — definitions. 6 
 
 Examples. 
 
 Numlier. Logarithms. 
 
 377000 5-57634 
 
 37700 4-57634 
 
 3770 3-57634 
 
 377 2-57634 
 
 37-7 1-57634 
 
 3-77 0-57634 
 
 •377 1-57634 
 
 •0377 2-57634 
 
 . -00377 3-57634 
 
 and so on. 
 
 11. The logaritlim of a product is the sum of the logarithms of 
 its factors. 
 
 12. The logarithm of a power is equal to the logarithm of the 
 root multiplied by the index of the power. 
 
 13. The logarithm of a quotient is found by subtracting the 
 logarithm of the divisor from the logarithm of the dividend. 
 
 14. The logarithm of a root is found by dividing the logarithm 
 of one of its powers by the index of that power. 
 
 Note. — In applying these principles to logarithms of numbers 
 less than I, it is to be observed that negative indices are to be 
 subtracted instead of being added, and added instead of being 
 subtracted. 
 
 15. To avoid the inconvenience which attends the use of nega- 
 tive indices to logarithms, it is a very common practice to put, 
 instead of a negative index to the logarithm of a fraction, the 
 aiynvplement (as it is called) of that index _to 10; that is to say, 
 9 instead of 1, 8 instead of 2, 7 instead of 3, and so on. In such 
 cases, it is always to be understood that each such complementary 
 index has - 10 combined with it; and to prevent mistakes, it is 
 useful to prefix - 10 + to it; for example, 
 
 >f umber. 
 
 Logarithm wlfh 
 Negative Index. 
 
 logarithm with 
 Complementary Index. 
 
 -377 
 
 T-57634 
 
 -10 + 9-57634 
 
 ■0377 
 
 2-57634 
 
 -10 + 8-57634 
 
 •00377 
 
 3-57634 
 
 -10 + 7-57634 
 
 16. To find the fractional part of the common logarithm of a 
 number of five places of figures; take from the table the logarithm 
 corresponding to the first three figures, and the difference between 
 that logarithm and the next greater logarithm in the table; mul- 
 tiply that difference by the two remaining figures of the given 
 number, and divide by 100; the quotient will be a correction, to 
 be added to the Ipgarithm already found. 
 
6 MATHEMATICAL INTEODUCTIOIT. 
 
 Example. — Find the common logarithm of 37725. 
 
 Log. 377, 57634 
 
 Log. 378, 57749 
 
 Difference, 115 
 
 X 25 -=-100 
 
 Correction — 29 
 
 Add log. 377, 57634 
 
 Log. 37725, 57663 Answer. 
 
 17. To find the natural number, or cmtilogariihm, coiTesponding 
 to a common logarithm of five places of decimals, which is not in 
 the table; find the next less, and the next greater logarithm in the 
 table, and take their difference. Opposite the next less logarithm 
 ■will be the first three figures of the antUogarithm. Subtract the 
 next less logarithm from the given logarithm ; annex two noughts 
 to the remainder, and divide by the before-mentioned difference ; 
 the quotient will give two additional figures of the required anti- 
 logarithm. (The first of those figures may be a nought.) 
 
 Exomiple. — Find the antUogarithm of the common logarithm 
 •57663. 
 
 Next less log. in table, 57634 
 
 Next greater, 57749 
 
 Difference, 115 
 
 Given logarithm, 57663 
 
 Subtract log. 377, 57634 
 
 Divide by difference, 115)2900 
 
 Two additional figures, 25 
 
 so. that the answer is 37725. 
 
 Note. — ^The last two rules refer particularly to the tables in 
 Eankine's Usefvil Rules and Tables, but are equally applicable to 
 other tables. For instance, where the logarithm of a number of 
 5 figures is given in the tables; in these last two rules, for 3 read 5, 
 and for 5 read 7. 
 
 TEIGONOMETEICAL EULES. 
 
 The following is a summary of the Principles and Chief Eules of 
 Trigonometry : — 
 
 Definition. — Every expression which in any way contains a 
 number, or depends for its value upon the value of the number, 
 is said to be a, function of that number, as 2x, x^, log. x, tan x are 
 all functions of x. 
 
 18. Trigonometrical Functions Defined.— Suppose that A, B, C 
 
TEIGONOMETBICAL FUNCTIONS. 7 
 
 stand for tie three angles of a right-angled triangle, C being the 
 right angle, and that a, h, c stand for the sides respectively opposite 
 to those angles, c being the hypothenusej then the various names 
 of trigonometrical functions of the angle A have the following 
 meanings : — 
 
 smA=-: cosA=-: 
 c c 
 
 , c-b . , c - a 
 
 versin A = : coversm A = : 
 
 c ' c 
 
 tau A = -y : cotan A = - : 
 a 
 
 A " A <= 
 
 secA=7: cosecA = -. 
 a 
 
 The complement of A means the angle B, such that A+B = a 
 right angle ; and the sine of each of those angles is the cosine of 
 the other, and so of the other functions by pairs. 
 
 19. Belations amongst the Trigonometrical Functions of One 
 Angle, A, and of its Supplement, 180°- A: — 
 
 • A Ti m. tan A 1 _ 
 
 sin A;= Jl — cos^ A = r = r> 
 
 ^ sec A cosec A 
 
 A #7 ^-TT cotan A 1 . 
 
 cos A.= Jl- sin'' A = 7- = T- ; 
 
 ^ cosec A sec A 
 
 versin A = 1 - cos A ; 
 
 coversia A = 1 - sin A; 
 
 tan A= r = ^ i-=sin A-sec A= Jsec^ A-l; 
 
 cos A cotan A 
 
 cotan A = -: — r = i a- = cos A'coseo A= ^/cosec^A-l; 
 
 sm A tan A v * 
 
 1 
 
 sec A = 7- = ^1 + tan^ A ; 
 
 cos A ^ 
 
 cosec A = -^ — r = \/l + cotan^ A. 
 sm A ^ 
 
 sin(180°-A) = sinAj 
 
 cos (180°- A) = -cos A; 
 
 versin (180° - A) = 1 + cos A =2 - versin A ; 
 
 coversin (180° - A) = coversin A ; 
 
 tan(180°-A)=-tanAj 
 
 cotan (180° - A) = - cotan A ; 
 
 sec (180° - A) = -sec A; 
 
 cosec (180° - A) = cosec A 
 
8 MATHEMATICAL INTRODUCTION. 
 
 20. The Circular Measure of an Angle. — If a right line as radius 
 hy revolution about a fixed point at its extremity as centre, traces 
 out an angle from a fixed position, the angle may be measured by 
 the ratio of the arc to the radius; this mode of measurement is called 
 circular measure. The unit of circular measure is the angle Vhose 
 arc is equal to the radius, that is, 360° -=- Sa- = (57° 17' 45" = 206265"). 
 
 To compute sines, &c., approximately by series ; reduce the 
 angle to circular measure — that is, to radius-lengths and fractions 
 of a radius-length let it be denoted by A. Then 
 
 A* A^ A^ 
 
 sinA = A-^ + ^^3^g- ^3^^g_g^ +&o. 
 
 A^ A* A* 
 
 cosA = 1-^ + 2J:^-2-3^j^+&c. 
 
 21. Trigonometrical Functions of Two Angles: — 
 
 sin (A ± B) = sin A cos B ± cos A sin B ; 
 
 cos (A ± B) = cos A cos B + sin A sin B ; 
 
 / A J. T.\ tS'n A ± tan B 
 
 tan (A ± B) =.j -r— — ^. 
 
 ^ 1 + tan A tan B 
 
 22. Formulse for the Solution of Plane Triangles. — Let A, B, C 
 be the angles, and a, b, c the sides respectively opposite them. 
 
 I. Eelations amongst the Angles — 
 
 A + B + C=180"; 
 or if A and B are given, 
 
 C = 180° - A - B. 
 
 II. When the Angles and One Side are given, let a be the given 
 side; then the other two sides are 
 
 , sin B sin C 
 
 o = a •—. — J- : c = a'—. — r. 
 
 sm A ' sin A 
 
 III. When Two Sides and the Included Angle are given, let a, h 
 be the given sides, C the given included angle ; then 
 
 To find the third side. First Method : 
 
 c= J(a^ + b^-2abcosC); 
 
 n 
 
 then 
 
 Second Method : 
 
 Make sin D-^^/"'*- cos 5 
 a + b ^ 
 
 
 c = {a + b) cos D. 
 
 Third Method. ■ 
 
 Make tan E-^^/"*- sin ^ 
 a-b 2 
 
 
 c = {a-b) sec E. 
 
 then 
 
s = — 3 — ; then 
 
 TO SOLVE A EIGHT-ANGLED TRIANGLE. 9 
 
 J 
 
 Tojmd the remaming Angles, A and B. 
 If the third side has been computed, 
 
 sin A = --sin C; sin B = --sin C. 
 c c 
 
 If the third side has not been computed, 
 
 tan .^ = cotan |; tanV = — J -tan ?; 
 •a J 2 a + b 2 
 
 ._A+B A-B ^ A+B A-B 
 
 ^— 2-+-2-' ^=-r — 2— 
 
 IV. When the Three Sides are given, to find any one of the 
 Angles, such as — 
 
 cosC = — ■ ; 
 
 or otherwise, let 
 
 a + b + e 
 
 „«= C! /s(s-c) . C ^ /(s-a)(7-6) 
 
 cotan I = V ;^--';) , • tan ^ = . /Sip; 
 2 V {s-a){s-b)' 2 V «(s-c) ' 
 
 sin C = 2\/^(«-")(g-^)(^-o) 
 
 N'ote. — In all trigonometrical problems, it is to be borne in 
 mind, that small acute angles, and large obtuse angles, are most 
 accurately determined by means of their sines, tangents, and 
 coseoants; and angles approaching a right angle by their cosines, 
 cotangents, and secants. 
 
 23. To Solve a Eight-angled Triangle Let C denote the right 
 
 angle; c the hypothenuse; A and B the two oblique angles; a and 
 6 the sides respectively opposite them. 
 
 Given, the right angle, another angle B, the hypothenuse c. 
 Then 
 
 A = 90°-B; a = c-cosBj 6 = c-sinB. 
 
 Given, the right angle, another angle B, a side a, 
 A = 90''-B; 6 = a-tanB; c = a-secB. 
 
 Given, the right angle, and the sides a, h. 
 
 a 
 
 tan A = 7: tan B = -: c= Ja^ + l 
 a ^ 
 
10 MATHEMATICAL INTKODUCTION. 
 
 Given, the right angle, the hypothenuse c; a side a, 
 
 sin A = cos B = - : h= J c^ — a\ 
 c ^ 
 
 Given the three sides, a, b, e, which fulfilling the equation 
 
 c^ = a^ + b% the triangle is known to be right-angled at 0, 
 
 ■ A * ■ -D * 
 
 sm A = - : sm ±5 = -. 
 c c 
 
 24. To Express the Area of a Plane Triangle in terms of its 
 
 Sides and Angles. 
 
 Given, one side, c, and the angles. 
 
 A _ ''^. ^^"^ -^ ^^ -^ 
 2 Sin 
 Given, two sides, 6, c, and the included angle A. 
 
 h • sin A 
 
 Area = 
 
 2 
 
 Given, the three sides a, b, e. Let ^ =s; then 
 
 Area = ^ <, s{s-a)(s-b)s - c)> . 
 
 RULES OF THE DIFFERENTIAL AND INTEGRAL 
 CALCULUS. 
 
 25. Definitions. — A. function has already been defined. When a 
 function of one quantity is assumed equal to another quantity, 
 both quantities are called variables, the one upon whose assumed 
 value the other depends being called the independent variable, 
 while the other, whose value depends upon it, is called the de- 
 pendent variable. The expression y = <i>x for instance denotes that 
 the dependent variable y, depends for its value upon the independent 
 variable x, the expression denoting that y is a, function of x. 
 
 A quantity, x, may be assumed to be made up of an infinite 
 number of infinitesimal parts, dx, this expression meaning simply 
 one of the small infinitesimal difierences of which x is made up, 
 i.e., x = n-dx, where n is assumed to increase without limit, and 
 dx to diminish without limit: this process of considering a quantity 
 to be diminished without limit is called differentiation. 
 
 The quotient, if it has a limit formed by taking the difierence of 
 the function of a quantity, and the function of that quantity with 
 a small increment, and dividing by the increment, is termed the 
 differential coefficient of the function, with regard to the quantity 
 
 ^^ — -r-^ — — is the difierential coeflacient oi <i>x with respect to 
 
 X, this is generally written <t>' x; or otherwise the small increment 
 
DIFFERENTIAL AND INTEGRAl CALCULUS. H 
 
 or decrement of the dependent variable divided by that of ihe 
 independent variable, the former being a function of the latter, is 
 
 called the differential coefficient, thus ■/ is the differential co- 
 
 efficient of y -with respect to x, it being always borne in mind that 
 
 -^ is one quantity, which cannot be divided into a numerator dy, 
 
 and a denominator dx. 
 26. Eules for finding differential coefficients,— 
 
 If y = C (a constant) ; -—- = 0. 
 
 (L SC 
 
 The Differential Coefficient of the sum of functions is equal 
 to the sum of the differential coefficients of the functions, or if 
 'v = w + y + z -where all of these quantities are functions of x, then 
 dv _ dw dy dz 
 dx dx dx dx' 
 
 In the same way to find the differential coefficient of the differ- 
 ence, product, and quotient of functions of quantities. 
 
 If « = « - », then -^=^- ^, where v, y, z, are functions of x. 
 " ' dx dx dx ' 
 
 dv dw dy dz , 
 
 If« = «,2/«,then^ = ^-2,-« + J-«;-«+^-«'-y, where 
 
 V, w, y, z, are functions of x. • 
 
 dy dz 
 z ■ —^ — y — 
 
 liv = —, then ^ = :; , where v, y, z, are functions of x. 
 
 z dx z- 
 
 dy 
 If 4ix = nx, <P'x = n; or otherwise let <t>x = y=nx, then ^ = »»> 
 
 thus if <t>x = 7x, 4>'x = 7. 
 
 If ,px = a", <t>'x = wa;" - S thus if <l>x = x^, 'P'x = 7x^. 
 
 If fx =log,a!, f'x = ^j^, thus if *«; = log^o », *'« = ^j^^o = 
 1 _ -43429 
 2-30258a;~ x 
 
 l{<px = log,x,4,'x=-. 
 
 If <l>x = a", tji'x = a" logE a. 
 If <t>x = s^, <i>'x = i''. 
 If ^x = sin X, <p'x = cos x. 
 If 'i>x = cos X, 'P'x = - sin x. 
 
 li<t>x = tan X, 4>'x = -^^^. 
 
 Definition. — By sine"'a!_is meant the angle whose sine is 
 X, thus if 03 = sine y,y = sine ' x. 
 
12 
 
 MATHEMATICAL INTRODUCTION. 
 
 It fx = sin "' X, <f'X = 
 
 1 
 
 pa; = COS 
 
 1 
 
 X, f' x= - 
 
 J{l-x^) 
 
 ipx = tan "'a;, 9'x = ^ ». 
 
 1 + a;^ 
 
 l.iv = <Ptc, and ^v = xx, then x' x = -^ v <^' x, or otherwise; 
 
 ,, rfw ^y i^'' 
 It v = ip X, y = xx, y = '^v then -^ =---. -y-. 
 
 Jt fx = log. sin ar, then p' a; = 
 
 1 
 
 <ia; dv ' dx' 
 • cos X ; here ^oy. siw x is first 
 
 difierentiated with respect to sin x, and then sin x with respect to ar. 
 
 Definition. — The differential coefficient of the differential coeffi- 
 cient is called the second differential coefficient; the differential 
 coefficient of the second is called the third; and so on : — 
 
 Thus let fx = x", then ^'a; = ma;""', ip" x = n {n - l)x''~\ 
 f'" a; = Ji (ra - 1) (n- 2) x"~' ; where f" x stands for the second 
 differential coefficient, ?'" x for the third differential coefficient. 
 So let ? a; = sin x, then ip' x = cos x, <p" x= — sin x, f'" x= — cos x. 
 This process is called successive differentiation. 
 
 Another mode of representing this is the following : — -^ is the 
 differential coefficient of y with respect to x ; the second differ- 
 ential coefficient dxis represented by -5-^, where -j-^ is a quan- 
 
 dx 
 tity which cannot be divided into a numerator d^y and deno- 
 
 minator d x\ as already stated of the quantity 
 
 dy 
 dx' 
 
 27. The following is an illustration of the application of the differ- 
 ential calculus to Geometry. In fig. 1, 
 let O X, O Y be two axes of co-ordi- 
 nates at right angles to each other. 
 Let A B be a curve, whose equation 
 is represented hj y = fx, i.e., ' each 
 point of the curve has ordinates and 
 abscissK bearing to each other the 
 ratio represented by the equation, and 
 all the points in the curve are known 
 when their ordinates and abscissae 
 (co-ordinates) are known. Let O C 
 = OS be a particular value of a;, and A 
 Kg. 1. =b = (pa be a particular value of y; 
 
APPROXIMATE COMPUTATION OF INTEGRALS. 13 
 
 also let A E be a small increment of a; = Ax, and E B a correspond- 
 ing increment of y = Ay. The trigonometrical tangent of the 
 
 angle E A B = — . Let there be a line, A T, lying between A B 
 and AE, whose tangent is -^ = (('x. Then as B approaches 
 
 CL OS 
 
 towards A, the line A B will ultimately be the line A T; that is, 
 its limiting value as soon as A and B coincide, will be AT. 
 Hence if x be an abscissa of a curve, and j a; = 3/ be an ordinate, 
 
 the differential coefficient ■— =f'x is the trigonometrical tangent 
 
 of the angle which the geometrical tangent of the curve makes 
 with the axis of x, at the point where the abscissa is x. 
 
 The differential calculus will occasionally be applied in an ele- 
 mentary manner to portions of the following work, as, for example, 
 in treating of the Varied Motion of Points (Part I., Section 3). 
 
 28. The integral calculus is the inverse of the differential; it 
 determines the whole magnitude of a quantity of which the dif- 
 ferentials are given. 
 
 If a number of points be taken in a curve, and chords drawn 
 joining the points, and also tangents drawn through the points 
 intersecting each other, the sum of the one will be less than, and that 
 of the other (intersected portions) will be greater than the length 
 of the curve ; if the chords, and tangents are increased in number, 
 they will approximate to the length of the curve. The integral 
 calculus is used for finding the exact length of the curve. A 
 mechanical illustration is the computation of the space passed over 
 by a point having varied motion. 
 
 29. Approxiiuate Computation of lutegrals.-r-The present article 
 is intended to afford to those who have not made that branch of 
 mathematics which treats of the process of integration a special 
 study, some elementary information respecting it. 
 
 The meaning of the symbol of an integral, viz.: — 
 
 /■ 
 
 u d X, 
 
 is of the following kind : — 
 
 In fig. 2, let A C B D be a plane area, of which one boundary, 
 A B, is a portion of an axis of 
 abscissae O X, — the opposite 
 boundary, C D, a curve of any 
 figure, — and the remaining 
 boundaries A 0, B D, ordinates 
 perpendicular to X, whose 
 
 respective abscissse, or dis- Kg. 2. 
 
 tances from the origin O, are 
 
14 
 
 MATHEMATICAL INTRODUCTION. 
 
 O A =a; OB = o. 
 
 Let E F = M be any ordinate whatsoever of the curve C T), and 
 O E = a; the corresponding abscissa. Then the integral denoted 
 by the symbol, 
 
 
 means, the area of the figure A C B D. The abscissEe a and 6 
 ■which are the least and greatest values of x, and which indicate 
 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. 
 
 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. 
 
 Oases may arise, however, in which w 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 computa- 
 tion of the areas of those bands, and the adding of them together; 
 and the more minute that division is, the more near is the result 
 to the truth. 
 
 First Approtdmation. 
 
 Di-vide the area A D B, as in fig. 3, into any convenient 
 
 number of bands by parallel or- 
 -"'> dinates, whose uniform distance 
 apart is' A as ; so that if n be the 
 number of bauds, n+\ will be the 
 number of ordinates, and 
 
 Fig. 3. 
 
 b - a = n l\x, 
 the length of the figure. 
 Let u', 1/1', denote the two ordinates which bound one of the 
 bands ; then the area of that band is 
 
 W + M" 
 
 — ;r — • A «, newrly; 
 
 and consequently, adding together the approximate areas of all 
 the bands, — denoting the extreme ordinates as follows, — 
 
 AC = M„: BD = M. 
 
APPROXIMATE COMPUTATION OP INTEGRALS. 
 
 15 
 
 Fig. 4. 
 
 and the intermediate ordinates by «« we find for the approximate 
 value of the integral — 
 
 f\dx=(^ + '^ + li-u^A<»,-: (!•) 
 
 Second Approximation. 
 
 Divide the area A C D B, as iu fig. 4, into an even number of 
 bands, by parallel ordinates, whose 
 uniform distance apart is A a^- 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 theo 
 curve F H is nearly a parabola, it 
 appears from the properties of that curve, that the area of that 
 pair of bands is 
 
 (u' + 4 M" + u"') A X 
 5 ^-^=^- , nearly; 
 
 in which u' and u'" 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 — 
 
 / udx = (u^ + Mj + 2 2 ■ Mj (plain) 
 
 + 42 -M. (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 may be useful to assist the memory. 
 
 When the symbol of integration is, repeated, so as to make a 
 doiible integral, such as 
 
 / \u-dxdy, 
 
 or a triple integral, such as 
 
 I J j wdxdydz, 
 
 it is to be understood as follows : — 
 
 J^e* v^Ju-dx 
 
 be the value of this single integral for a given value of y. Con- 
 
16 MATHEMATICAL INTEODUCTION. 
 
 struct a curve whose abscissse 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 
 
 I V dy= \ I u- dxdy. 
 
 N«^*'l«* t=fvdy 
 
 be the value of this double integral for a given value of z. Con- 
 struct a curve whose abscissse 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 
 
 1 1- dz= j I vdydz= I I I u-dxdydz; 
 and so on for any number of successive integrations. 
 
 EXILES FOR THE MENSURATION OF FIGURES AND 
 FINDING OF CENTRES OF MAGNITUDE. 
 
 Section 1. — Areas of Plane Surfaces. 
 
 30. Parallelogram. Rule A. — Multiply the length of one of the 
 sides by the perpendicular distance between that side and the 
 opposite side. 
 
 Htde B. — Multiply together the lengths of two adjacent sides 
 and the sine of the angle which they make with each other. 
 (When the parallelogram is right-angled, that sine is = 1.) 
 
 31. Trapezoid (or four-sided figure bounded by a pair of parallel 
 straight lines, and a pair of straight lines not parallel). Multiply 
 the half sum of the two parallel sides by the perpendicular distance 
 between them. 
 
 32. Triangle. Bule A. — Multiply the length of any one of the 
 sides by one-half of its perpendicular distance from the opposite 
 angle. 
 
 Bule B. — -Multiply one-half of the product of any two of the 
 sides by the sine of the angle between them. 
 
 Rule G. — Multiply together the following four quantities : the 
 half sum of the three sides, and the three remainders left after 
 subtracting each of the three sides from that half sum ; extract the 
 square root of the quotient ; that root will be the area required. 
 
 Note. — Any Polygon may be measured by dividing it into tri- 
 angles, measuring those triangles, and adding their areas together. 
 
 33. Parabolic Figures of the Third Degree.— The parabolic 
 
ANY PLANE AREA. 
 
 ir 
 
 Fi2. 5. 
 
 Fig. 6. 
 
 figures to which the following rules apply are of the following kind 
 (see figs. 5 and 6.) One boundary is a straight line, A X, called the 
 base or aods; two other boundaries are 
 either points in that line, or straight 
 lines at right angles to it, such as 
 A B and X C, called ordinates; and 
 the fourth boundary is a curve, B C, 
 o^the parabolic class, and of the third 
 degree; that is, a curve whose ordinate 
 (or perpendicular distance from the base A X) at any point is 
 expressed by what is called an algebraical function of the third 
 degree of the abscis^ (or distance of that ordinate from a fixed 
 point in the base). An algebraical function of the third degree of a 
 quantity consists of terms not exceeding four in number, of which 
 one may be constant, and the rest must be proportional to powers 
 of that quantity not higher than the cube. 
 
 Eule A. — Divide the base, as in fig. 5, into two equal parts or 
 intervals; measure the endmost ordinates, A B and X 0, and the 
 middle oi'dinate (which is dotted in the figure) at the point of 
 divison; add together the endmost ordinates and four times the 
 middle ordinate, and divide the sum by six ; the quotient will be 
 the mean breadth of the figure, which, being multiplied by the 
 length of the base, A X, will give the area. 
 
 Bule B. — Divide the base, as in fig. 6, into three equal intervals; 
 measure the endmost ordinates, A B and X C, and the two inter- 
 mediate ordinates (which are dotted) at the points of division ; add 
 together the endmost ordinates and three times each of the inter- 
 mediate ordinates; divide the sum hj eight; the quotient will be 
 the mean breadth of the figure, which, being multiplied by the 
 length of the base, A X, will give the area. 
 
 In applying either of those rules to figures whose curved 
 boundaries meet the base at one or both ends, the ordinate at each 
 such point of meeting is to be made = 0. 
 
 34. Any Plane Area. — Drawanaxisor base-line, AX, 
 in a convenient position. The most convenient position 
 is usually parallel to the greatest length of the area to 
 be measured. Divide the length of the figure into a 
 convenient number of equal intervals, and measure 
 breadths in a direction perpendicular to the axis at 
 the two ends of that length, and at the points of 
 division, which breadths will, of course, be one more 
 in number than the intervals. (For example, in fig. 
 7, the length of the figure is divided into ten equal 
 intervals, and eleven breadths are measured at &g, &j, 
 &c.) Then the following rules are exact, if the sides 
 of the figures are bounded by straight lines, and by 
 
IS MATHEMATICAL INTRODUCTION. 
 
 parabolic curves not exceeding the third degree, and are approxi- 
 mate for boundaries of any other figures. 
 
 Rule A. — (" Simpson's First Ride," to be used when the number 
 of intervals is even.) — Add together the two endmost breadths, 
 twice every second intermediate breadth, and/ow?- times each of the 
 remaining intermediate breadths; multiply the sum by the common 
 interval between the breadths, and divide by 3j the result will be 
 the area required. 
 
 For two intervals the multipliers for the breadths are 1, 4, 1 
 (as in Eule A of the preceding Article); for four intervals, 
 1, 4, 2, 4, 1 ; for six intervals, 1, 4, 2, 4, 2, 4, 1 ; and so on. These 
 are called " Simpson's Multipliers." 
 
 Example. — Length, 120 feet, divided into six intervals of 20 
 feet each. 
 
 Breadths in Feet Simpson's Prodncts. 
 
 and Decimals. Multipliers. 
 
 17-28 1 17-28 
 
 16-40 4 65-60 
 
 14-08 2 28-16 
 
 10-80 4 43-20 
 
 7-04 2 14-08 
 
 3-28 4 13-12 
 
 1 0-00 
 
 Sum, 181-44 
 X Common interval, 20 feet. 
 
 -=- 3)3628-8 
 
 Area required, 1209-6 square feet. 
 
 Eule B. — ("Simpson's Second Rule" to be used when the 
 number of intervals is a multiple of 3.) — Add together the two 
 endmost breadths, twice every third intermediate breadth, and 
 thrice each of the remaining intermediate breadths; multiply 
 the sum by the common interval between the breadths, and 
 by 3; divide the product by 8; the result will be the area 
 required. 
 
 " Simpson's Multipliers " in this case are, for three intervals, 
 1, 3, 3, 1; for six intervals, 1, 3, 3, 2, 3, 3, 1; for nine intervals, 
 1, 3, 3, 2, 3, 3, 2, 3, 3, 1 ; and so on. 
 
 Example. — Length, 120 feet, divided into six intervals of 20 
 feet each. 
 
TBAPEZOIDAL RULE. 19 
 
 Breadths in Feet Simpson's _ , , 
 
 and Decimals. Multipliers. Products. 
 
 17-28 1 17-28 
 
 16-40 3...' 49-20 
 
 14-08 3 42-24 
 
 10.80 2 21-60 
 
 7-04 3 21-12 
 
 3-28 3 9-84 
 
 1 0-00 
 
 Sum, 161-28 
 X Common interval, 20 feet. 
 
 3225-6 
 X 3 
 
 ■i- 8)9676-8 
 
 Area required, 1209-6 square feet. 
 
 Remarks. — The preceding examples are taken from a parabolic 
 figure of the third degree, for which both. Simpson's Rules are 
 exact; and the results of using them agree together precisely. For 
 other figures, for which the rules are approximate only, the first 
 rule is in general somewhat more accurate than the second, and is 
 therefore to be used unless there is some special reason for pre- 
 ferring the second. 
 
 The probable extent of error in applying Simpson's First Rule 
 to a given figure is, in most cases, nearly proportional to the fourth 
 power of the length of an interval. 
 
 The errors are greatest where the boundaries of the figure at-e 
 most curved, and where they are nearly perpendicular to the axis. 
 In such positions of a figure the errors may be diminished by sub- 
 dividing the axis into smaller intervals. 
 
 Bule G. — (" Merrifield's Trapezoidal Rule," for calculating sepa- 
 rately the areas of the parts into which a figure is subdivided by 
 its equidistant ordinates or breadths.) — Write down the breadths 
 in their order. Then take the differences of the successive breadths, 
 distinguishing them into positive and negative, according as the 
 breadths are increasing or diminishing, and write them oppo- 
 site the intervals between the breadths. Then take the dif- 
 ferences of those difierences, or second differences, and write them 
 opposite the intervals between the first difierences, distinguishing 
 them into positive and negative, according to the following 
 principles : — 
 
20 MATHEMATICAL INTRODUCTION. 
 
 First Differences. Second Difference. 
 
 Positive increasing, or 1 ^^^^^.^^_ 
 
 JNegative dimmisning, J 
 
 Negative increasing, or) j^-egative. 
 
 Positive diminisnmg, j ° 
 
 In the column of second differences there will now be two blanks 
 opposite the two endmost breadths; those blanks are to be filled 
 up with numbers each forming an arithmetical progression with 
 the two adjoining second differences, if these are unequal, or equal 
 to them, if they are equal. 
 
 Divide each second difference by 12; this gives a correction, 
 which is to be subtracted from the breadth opposite it if the second 
 difference is positive, and added to that breadth if the second 
 difference is negative. 
 
 Then to find the area of the division of the figure contained 
 between a given pair of ordinates or breadths ; multiply the half 
 sum of the corrected breadths by the interval betvieen them. 
 
 The area of the whole figure may be formed either by adding 
 together the areas of all its divisions, or by adding together the 
 halves of the endmost corrected breadths, and the whole of the 
 intermediate breadths, and multiplying the sum by the commou 
 interval. 
 
 Examfiple. — Length, 120 feet, divided into six intervals of 20 feet 
 each. 
 
 Breadths in 
 Feet and 
 
 First 
 Differences. 
 
 Second 
 Diflerenoes. 
 
 17-28 
 
 -0-88 
 
 (-1-92) 
 
 1640 
 
 -2-33 
 
 -1-44 
 
 14-08 
 
 -3-28 
 
 -0-96 
 
 10-80 
 
 -3-76 
 
 -0-48 
 
 7-04 
 
 -3-76 
 
 
 
 3-28 
 
 -3.28 
 
 + 0-48 
 
 
 
 
 ( + 0-96) 
 
 Corrections. 
 
 Corrected Areaa of 
 Breadths. DiTisions. 
 
 Feet. Sq. Feet. 
 
 + 0-16 17-44. 
 
 I 339-6 
 
 + 0-12 16-52 J 
 
 306-8 
 
 14-161 
 
 + 0-08 
 
 + 0-04 10-84 
 
 I 250-0 
 84. J 
 
 
 
 ■0-04 
 
 7-04 ( 
 
 \ 102-i 
 3-24 
 
 178-8 
 8 
 
 -0-08 -008 
 
 i 31-6 
 
 Total area, square feet, 1209-6 
 
 . The second differences enclosed in parentheses at the top and 
 bottom of the column are those filled in by making them form au 
 arithmetical progression with the second differences adjoining them. 
 
CIRCLE. 21 
 
 Tlie last corrected breadth in the present example is negative, 
 and is therefore subtracted instead of added in the ensuing com- 
 putation. 
 
 Jiule D. — (" Common Trapezoidal Bule," to be used when a 
 rough approximation is sufficient.) Add together the halves of 
 the endmost breadths, and the whole of the intermediate breadths, 
 and multiply the sum by the common interval. 
 
 Example. — The same as before. 
 
 Half breadth at one end, 17-28 -f- 2 = 
 
 Feet. 
 
 8-64: 
 
 
 f 16-40 
 
 
 14-08 
 
 intermediate breadths, ■ 
 
 10-80 
 
 
 7-04 
 
 
 3'28 
 
 Half breadth at the other end, . 
 
 
 
 
 60-24 
 
 X Common interval, 
 
 20 
 
 Approximate area, .... 
 
 1204-8 square feet. 
 
 True area as before computed, . 
 
 1209-6 
 
 Error, 
 
 — 4-8 square feet. 
 
 35. Circle. — The area of a circle is equal to its circumference 
 miiltiplied by one-fourth of its diameter, and therefore to the square 
 of the diameter multiplied by one-fourth of the ratio of the circum- 
 ference to the diameter. The ratio of the area of a circle to the 
 
 square of its diameter f which ratio is denoted by the symbol -rj 
 
 is incommensurahle; that is, not expressible exactly in figures; but 
 it can be found approximately, to any required degree of precision. 
 Its value has been computed to 250 places of decimals; but the 
 following approximations are clo.se enough for most purposes,, 
 scientific or practical : — 
 
 , ^ , „ tr Errors in Fractions of the 
 
 Appromnate Values of j. ^^^^^^ ^^^^^ 
 
 •7853981 634 - -t- one-300,000,000,000th. 
 
 •785398 + -one-5,000,000th. 
 
 •7854 - + one-400,000th. 
 
 -P-^^ - -i-one-13,000,000th. 
 
 4 X 113 
 
 =-r - 4 one-2,500th. 
 
 14 
 
 The diameter of a circle equal in area to a given square is very 
 nearly 1-12838 x the side of the square. The following table gives 
 examples of this : — 
 
22 
 
 MATHEMATICAL INTRODUCTION. 
 
 Table — Multipliers for Converting 
 
 
 Sides of Squares into 
 Diameters of 
 Equal Ciroles. 
 
 Diameters of Circles 
 
 into Bides of 
 
 Equal Squares. 
 
 
 1 
 
 1-12838 
 
 0-88623 
 
 1 
 
 2 
 
 2-25676 
 
 1-77245 
 
 2 
 
 3 
 
 3-38514 
 
 2-65868 
 
 3 
 
 4 
 
 4-51352 
 
 3-54491 
 
 4 
 
 5 
 
 5-64190 
 
 4-43113 
 
 5 
 
 6 
 
 6-77028 
 
 5-31736 
 
 6 
 
 7 
 
 7-89866 
 
 6-20359 
 
 7 
 
 8 
 
 9-02704 
 
 7-08981 
 
 8 
 
 9 
 
 10-15542 
 
 7-97604 
 
 9 
 
 10 
 
 11-28380 
 
 8-86227 
 
 10 
 
 36. The area of a Circular Sector (0 A B, fig. 8) is the same 
 fraction of the whole circle that the 
 angle A O B of the sector is of a -whole 
 revolution. In other -words, multiply 
 holf the squa/re of the radius, or one-eighth 
 of the square of the diameter, by the 
 circular measure (to radius unity) of the 
 angle A Bj the product will be the 
 
 Fig. 8. 
 area of the sector. 
 
 Section 2. — Volumes of Solid Figures. 
 
 37. To Measure the Volume of any Solid. — Method I. By 
 layers. — Choose a straight axis in any convenient position. (The 
 most convenient is usually parallel to the greatest length of the 
 solid.) Divide the whole length of the solid, as marked on the 
 axis, into a convenient number of equal intervals, and measure 
 the sectional area of the solid upon a series of planes crossing the 
 axis at right angles at the two ends and at the points of division. 
 Then treat those areas as if they were the breadths of a plane 
 figure, applying to them Rule A, B, or C of Article 34, page 17; 
 and the result of the calculation will be the volume required. If 
 Eule is used, the volume will be obtained in separate layers. 
 
 Method II.. By pi-isms or columns (" Wooley's Eule"). — ^Assume 
 a plane in a convenient position as a base, divide it into a network 
 of equal rectangular divisions, and conceive the solid to be built of 
 a set of rectangular prismatic columns, having those rectangular 
 divisions for their sectional areas. Measure the thickness of the 
 solid at the centre and at the middle of each of the sides of each of 
 those rectangular columns; add together the doubles of all the 
 thicknesses before-mentioned, which are in the interior of the solid, 
 and the simple thicknesses which are at its boundaries ; divide the 
 sum by six, and multiply by the area of one rectangular division 
 of the base. 
 
to calculate the lengths op circular arcs. 23 
 
 Section 3. — LIengths op Curved Lines. 
 
 38. To Calculate the Lengths of Circular Arcs. — ^When the 
 
 proportion of the arc to an entire circumference is given, the 
 
 length of the arc, in terms of the radius, is to be calculated by 
 
 multiplying that proportion by the well-known approximate value 
 
 of the ratio of the circumference of a circle to its radius : viz., 
 
 circumference 710 , oooqiqc i j-t, u i- • 
 — j: = YY=^nearly, = 6'283185 nearly: the above ratio is 
 
 commonly denoted by the symbol 2 ff; the reciprocal of the above 
 
 113 
 ratio is very nearly tr.-js =0-159155 nearly; but it is often much 
 
 more convenient in practice to proceed by drawing ; and then the 
 following rules are the most accurate yet known : — * 
 
 L (Pig. 9). To drojw a straight line approximately equal to a 
 given circular arc, A B. Draw the straight 
 chord B A; produce A to C, making 
 A C = |- B A j about C, with the radius 
 C B = |- B A, draw a circle; then draw the 
 straight line A D, touching the given arc b^ 
 in A, and meeting the last-mentioned circle Fig. 9. 
 
 in D ; A D will be the straight line required. 
 
 The error of this rule consists in the straight line being a little 
 shorter than the arc : in fractions of the length of the arc, it is 
 about xwff ^°^ ^^ ^"^^ equal in length to its own radius ; and it 
 varies as the fourth power of the angle subtended by the arc ; so 
 that it may be diminished to any required extent by subdividing 
 the arc to be measured by means of bisections. For example, in 
 drawing a straight line approximately equal to an arc subtending 
 •60°, the error is about -g^ of the length of the arc ; divide the arc 
 into two arcs, each subtending 30° ; draw a straight line approxi- 
 mately equal to one of these, and double it; the error will be 
 reduced to one-sixteenth of its former amount ; 
 that is, to about -j ^Tirff °^ *^^ length of the arc. 
 The greatest angular extent of the arcs to : 
 which the rule is applied should be limited 
 in each case according to the degree of pre- 
 cision required in the drawing. 
 
 11. (Fig. 10). To- draw a straight line ap- 
 prooeimatdy equal to a given circular o/rc, A B. 
 (Another Method.) Let be the centre of 
 the arc. Bisect the arc A B in D, and the Pig. 10. 
 
 arc A D in Ej draw the straight secant 
 
 • These rules are extracted from Papers read to the Bri&h Association 
 in 1867, and published tin the Philosophioal Magaziaelor Sq>tember 'and 
 October of that year. 
 
24 MATHEMATICAL INTRODtTCTION. 
 
 C E F, and the straight tangent A F, meeting each other in F ; 
 draw the straight line F B ; then a straight line of the length A F 
 + F B will be approximately equal in length to the arc A B. 
 
 The error of this rule, in fractions of the length of the arc, is 
 just one-fourth of the error of Rule I., but in the contrary direction ; 
 and it varies as the fourth power of the angle subtended by the arc. 
 
 III. To lay off upon a given circle an a/rc approximately equal in 
 length to a given stra/ight line. In fig. 11, let A D be part of the 
 
 circumference of the given circle, A one end of 
 
 the required arc, and A B a straight line of the 
 
 given length, drawn so as to touch the circle at 
 
 the point A. In A B take A = ;| A B, and 
 
 about C, with the radius B = f A B dfaw a 
 
 Pig. 11. circular arc B D, meeting the given circle in D. 
 
 A D will be the arc required. 
 
 The error of this rule, in fractions of the given length, is the 
 
 same as that of Rule I., and follows the same law. 
 
 IV. (Fig. 11.) To draw a circular arc which shall -he approxi- 
 Tfiatdy equal in length to Hie straight line A B, sliall with one of its 
 ends touch that straight line at A, and shall subtend a given angle. 
 In A B take A.G = \ A.'B; and about C, with the radius C B 
 = J A B, draw a circle, B D. Draw the straight line A D, 
 
 making the angle B A D = one-half of the given angle, and meeting 
 the circle B D in E. Then D will be the other end of the required 
 arc, which may be drawn by well-known rules. 
 
 The error of this rule, in fractions of the given length, is the 
 same with that of Rules I. and III., and follows the same law. 
 
 V. To' divide a circular arc, approodmately, into any required 
 number of equal parts. By Rule I. or II., draw a straight line 
 approximately equal in length to the given arc; divide that straight 
 line into the required number of equal parts, and then lay off upon, 
 the given arc, by Rule III., an arc approximately equal in length 
 to one of the parts of the straight line. 
 
 Rule V. becomes unnecessary when the number of parts is 2, 4, 
 8, or any other power of 2 ; for then the required division can 
 be performed exactly by plane geometry. 
 
 VI. To divide the whole circumference of a circle approximately 
 into any required number of equal arcs. When the required 
 number of equal arcs is any one of the following numbers, the 
 division can be made exactly by plane geometry, and the present 
 rule is not needed : — any power of 2 ; 3 ; 3 x any power of 2 ; 5; 
 5 X any power of 2 ; 15 ; 15 x any power of 2v* In other cases 
 
 * It may be convenient here to state the methods of subdividing arcs and 
 whole circles by plane geometry. (1.) To bisect any circular arc On the 
 chord of the arc as a base, construct any convenient isosceles trianc;le, with 
 the summit pointing away from the centre of the arc ; a straight line from 
 
CENTEE OF MAGNITUDE. 25 
 
 proceed as follows : — Divide the circumference exactly, by jjlane 
 geometry, into such a number of equal arcs as may be required, in 
 order to give sufficient precision to the approximative part of the 
 process. Let the number of equal arcs in that preliminary division 
 be called n. Divide one of them, by means of Eule V., into the 
 required number of equal parts ; n times one of those parts will 
 be one of the required equal arcs into which the whole circumfer- 
 ence is to be divided. 
 
 Rules I., III., and V., are applicable to arcs of other curves 
 besides the circle, provided the changes of curvature in such arcs 
 are small and gradual. 
 
 39. To Measure thj Length of any Curve. — Divide it into short 
 arcs, and measure each of them by Eule I. of Article 38, page 23. 
 
 Section 4.— Geometrical Centres and Moments. 
 
 40. Centre of Magnitude — General Principles.— By the magni- 
 tude of a figure is to be understood its length, area, or volume, 
 according as it is a line, a surface, or a solid. 
 
 The centre of magnitude of a figure is a point such that, if the 
 figure be divided in any way into equal parts, the distance of the 
 centre of magnitude of the whole figure from any given plane is 
 the mean of the distances of the centres of magnitude of the several 
 equal parts from that plane. 
 
 The geometrical moment of any figure relatively to a given plane 
 is the product of its" magnitude into the perpendicular distance of 
 its centre from that plane. 
 
 I. Symmetrical figure. — If a plane divides a figure into two 
 symmetrical halves, the centre of magnitude of the figure is in that 
 plane; if the figure is symmetrically divided in the like manner 
 by two planes, the centre of magnitude is in the line where those 
 planes cut each other; if the figure is symmetrically divided by 
 three planes, the centre of magnitude is their point of intersection; 
 and if a figure has a centre of figure (for example,, a circle, a sphere, 
 
 the centre of the arc to that summit will bisect the arc. (2.) To marh 
 
 the sixth part of the circumference of a circle. Lay off 
 
 a chord equal to the radius. (3.) To marh the tenth 
 
 part of the circumference of a drcle. In fig. 12, 
 
 draw the straight line AB=the radius of the circle; 
 
 and perpendicular to A B, draw B C=4 A B. Join 
 
 A C, and from it cut ofif C D = C B. AD will be the 
 
 chord of one-tenth part of the circumference of the 
 
 circle, (i.) For the fifteenth part, take ths dJiSevencQ Fig. 12. 
 
 between one-sixth and one-tenth. It may be added 
 
 that Gauss discovered a method of dividing the circumference of a circle by 
 
 geometry exactly, when the number of equal parts is any prime number that 
 
 IS equal tol + a power of 2; such as l-f2' = 17; l-f2"=257, &c.; but the 
 
 method is too laborious for use in designing mechanism. 
 
26 MATHEMATICAL INTEODUCTION. 
 
 an ellipse, an ellipsoid, a parallelogram, &c.), that point is its centre 
 of magnitude. 
 
 II. Compound figure. — To find the perpendicular distance from 
 a given plane of the centre of a compound figure made up of parts 
 ■whose centres are known. Multiply the magnitude of each part 
 by the perpendicular distance of its centre from the given plane; 
 distinguish the products (or geometrical moments) into positive or 
 negative, according as the centres of the parts lie to one side or to the 
 other of the plane ; add together, separately, the positive moments 
 and the negative moments : take the difference of the two sums, 
 and call it positive or negative according as the positive or negative 
 sum is the greater; this is the resultant moment of the compound 
 figure relatively to the given plane ; and its being positive or nega- 
 tive shews at which side of the plane the required centres lies. 
 Divide the resultant moment by the magnitude of the compound 
 figure; the quotient will be the distance required. 
 
 The centre of a figure in three dimensions is determined by find- 
 ing its distances from three planes that are not parallel to each 
 other. The best position for those planes is perpendicular to each 
 other; for example, one horizontal, and the other two cutting each 
 other at right angles in a vertical line. To determine the centre 
 of a plane figure, its distances from two planes perpendicular to the 
 plane of the figure are sufiicient. 
 
 41. Centre of a Plane Area. — To find, approximately, the centre 
 of any plane area. 
 
 Rule A. — Let the plane area be that represented in fig. 7 (of 
 Article 34, page 17). Draw an axis, A X, in a convenient posi- 
 tion, divide it into equal intervals, measure breadths at the ends 
 and at the points of division, and calculate the area^ as in Article 
 34. 
 
 Then multiply each breadth by its distance from one end of the 
 axis (as A) ; consider the pi-oducts as if they were the breadths of 
 a new figure, and proceed by the rules of Article 34 to calculate 
 the area of that new figure.^ The result of the operation will be 
 the geometrical moment of the original figure relatively to a plane 
 perpendicular to A X at the point A. 
 
 Divide the moment by the area of the original figure; the 
 quotient will be the distance of the centre required from the plane 
 perpendicular to A X at A. 
 
 Draw a second axis intersecting A X (the most convenient posi- 
 tion being in general perpendicular to A X), and by a similar pro- 
 cess find the distance of the centre from a plane perpendicular to 
 the second axis at one of its ends; the centre will then be completely 
 determined. 
 
 Mule B.—li convenient, the distance of the required centre from 
 a plane cutting an axis at one of the intermediate points of divi- 
 
CENTEE OF MAGNITUDE OF A CURVED LINE. 27 
 
 sion, instead of at one of its ends, may be computed as follows : — 
 Take separately the moments of the two parts into which that 
 plane divides the figure ; the required centre will lie in the part 
 which has the greater moment. Subtract the less moment from 
 the greater; the remainder will be the resultcmt moment of the 
 whole figure, which being divided by the whole area-, the quotient 
 will be the distance of the required centre from the plane of 
 division. 
 
 Rema/rk. — When the resultant moment is = 0, the centre is in 
 the plane of division* 
 
 Bute G. — To find the perpendicular distance of the centre from 
 the axis A X. Multiply each breadth by the distance of the 
 middle point of that breadth from the axis, and by the proper 
 "Simpson's Multiplier," Article 34, page 18; distinguish the pro- 
 ducts into right-Tianded and left-handed, accordiog as the middle 
 points of the breadths lie to the right or left of the axis; take 
 separately the sum of the right-handed products and the sum of 
 the left-handed products; the required centre will lie to that side 
 of the axis for which the sum is the greater; subtract the less sum 
 from the greater, and multiply the remainder by \ of the common 
 interval if Simpson's first rule is used, or by |- of the common 
 interval if Simpson's second rule is used ; the product will be the 
 resultant moment relatively to the axis A X, which being divided 
 by the area, the quotient will be the required distance of the centre 
 from that axis.* 
 
 42. Centre of a Volume. — To find the perpendicular distance of 
 the centre of magnitude of any solid figure from a plane perpen- 
 dicular to a given axis at" a given point, proceed as in Rule A of 
 the preceding Article to find the moment relatively to the plane, 
 substituting sectional areas for breadths; then divide the moment 
 by the volume (as found by Article 37) ; the quotient will be the 
 required distance. 
 
 To determine the centre completely," find its distances from three 
 planes, no two of which are parallel. In general it is best that 
 those planes should be perpendicular to each other. 
 
 43. Centre of Magnitude of a Curved lAvLQ.—Rule A.— To find 
 approximately the centre of magni- 
 tude of a very fiat curved line. — ^ 15 — 
 
 In fig. .13, let ADB be the arc. a .^^""'^ T 
 Draw the straight chord A B, which '' 
 
 bisect in C; draw C D (the deflec- Fig. 13. 
 
 Hon of the arc) perpendicular to 
 
 AB; from D lay off DE = ^ CD; E will be very nearly the 
 
 centre required. 
 
 * The rules of this Article are expressed in symbols, as follows : — Let x and 
 y be the perpendicular distances of any point in the plane area from two 
 
:28 
 
 MATHEMATICAL INTRODUCTION. 
 
 Fi". 14. 
 
 This process is exact for a cydoidal arc wLose chord, A B, is 
 parallel to the base of the cycloid. For other curves it is approxi- 
 mate. For example, in the case of a circular arc, it gives D E too 
 small; the error, for an arc subtending 60°, being about ^^ of the 
 deflection, and its proportion to the deflection varying nearly as 
 the square of the angular extent of the arc. 
 
 Rule B. — When the curved line is not very fiat, divide it into 
 very flat arcs; find their several centres of magnitude by Eule A, 
 and measure their lengths; then treat the whole curve as a com- 
 pound figure, agreeably to Rule II. of Article 40, page 26. 
 
 44. Special Figures. — I. Triangle (fig. 14). — From any two of 
 the angles draw straight lines to the middle 
 points of the opposite sides; these lines will 
 cut each other in the centre required; — or 
 otherwise, — from any one of the angles draw 
 a straight line to the middle of the opposite 
 side, and cut off one-third part from that line 
 commencing at the side. 
 II. Quadrilateral (fig. 15). — Draw the two diagonals A and 
 B D, cutting each other in E. If the quadrilateral is a parallelo- 
 gram, E will divide each diagonal into two equal parts, and will 
 itself be the centre. If not, one or both of the diagonals will be 
 divided into unequal parts by the point E. Let B D be a diagonal 
 that is unequally divided. From D lay off D F in that diagonal 
 = B E. Then the centre of the triangle F A C, found as in the 
 preceding rule, will be the centre required. 
 
 III. Plane "polygon. — Divide it into tri- 
 angles ; find their centres, auct measure 
 their areas ; then treat the polygon as a 
 compound figure made up of the triangles, 
 by Rule II. of Article 40, page 26. 
 
 IV. Prism or cylinder with pla/ne par- 
 allel ends.' — Find the centres of the ends ; 
 
 •pj jg a straight line joining them will be the axis 
 
 of the prism or cylinder, and the middle 
 point of that line will be the centre required. 
 
 'B '---i 
 
 planes perpendicular to the area and to each other, and x„ and 2^,, the per- 
 pendicular distances of tha centre of magnitude of the area from the same 
 planes; then 
 
 /fxdxdy _ _ /fydxdy 
 " Jfdxdy ' ^"^ f/dxdy ' 
 
 See Article 29, page 16. 
 
SPECIAL FIGUEES. 
 
 29 
 
 16). — Bisect 
 
 V. Tetrahedron, or triangular pyramid ( fig. 
 any two opposite edges, as A D and 
 B C, in E and F ; join E F, and bisect 
 it in G j this point will be the centre 
 required. 
 
 VI. Any pyramid or cone with a 
 plane base. — Find the centre of the 
 base, from •which draw a straight line 
 to the summit ; this will be the axis of 
 the pyramid or cone. From the axis 
 cut off one-fourth of its length, begin- 
 ning at the base ; th^g will give the 
 centre required. 
 
 VII. Any 'polyhedron or plane-faced solid. — Divide it into 
 pyramids; find their centres and measure their volumes; then 
 treat the whole solid as a compound figure by Paxle II. of Article 
 22. 
 
 VIII. Gircidar arc. — In fig. 17, let A B be the arc, and C the 
 the centre of the circle of which it is part. 
 
 Bisect the arc in D, and join D and* A B. 
 Multiply the radius C D by the chord A B, 
 and divide by the length of the arc A D B ; 
 lay off the quotient C E upon C D ; E will be 
 the centre of magnitude of the arc. 
 
 IX. Circula/r sector, C A D B, fig. 17. — 
 Find C E as in the preceding rule, and 
 make C F = f C E ; F will be the centre re- 
 quired. 
 
 Ji.. Sector of aflat ring. — Let r be the external and r' the 
 internal radius of the ring. Draw a circular arc of the same 
 
 2 r^ - r'^ 
 angular extent with the sector, and of the radius ^ ■ -^ r^, 
 
 and find its centre of magnitude by Rule VIII. 
 
30 MECHANICS. 
 
 ELEMENTARY MECHANICAL NOTIONS. 
 Definitioit of Geneeal Terms and Division of the Subject. 
 
 45. 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. 
 
 46. Matter (considered mechanically) is that which fills space. 
 
 47. 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. Liqiiid 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 the liquid and the 
 gaseous. 
 
 48. A Material or Physical Volume is the space occupied by a 
 body or by a part of a body. 
 
 49. A Material or Physical Surface is the boundary of a body, 
 or between two parts of a body. 
 
 50. Line, Point, Physical Point, Measure of Length. — In 
 mechanics, as in geometry, a Line is the boundary of a surface, or 
 between two parts of a surface ; and a Point is the boundary of a 
 line, or between two parts of a line; but the term "Physical 
 Point" is sometimes used by mechanical writers to denote an 
 inmieasurdbly small body — a sense inconsistent with the strict 
 meaning of the word " point ;" but still not leading to erroi-, so 
 long as it is rightly understood. 
 
 In measuring the dimensions of bodies, the standard British unit 
 of length is the ya/rd, being the length at the temperature of 62° 
 Eahrenheit, and at the mean atmospheric pressure, between the 
 two ends of a certain bar' which is kept in the office of the Ex- 
 chequer, 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/oot, 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- 
 
STRUCTURES AND MACHINES. 31 
 
 teenths, and thirty-second parts; but according to a resolution of 
 the Institution of Mechanical Engineei-s, 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 mStre, being about ^innnnnro "^ 
 the earth's circumference, measured round the poles. 
 
 51. 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. 
 
 52. 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. 
 
 53. 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, relativdy to which the motion or 
 rest of the first point takes place. Such a point is called & 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 afiecting motion, come to be considered, 
 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 mass 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 movable 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. 
 
 54. Cinematics. — The comparision of motions with each other, 
 without reference to their causes, is the subject of a branch of 
 geometry called " Cinematics." 
 
 55. Force is an action between two bodies, either causing or 
 tending to cause change in their relative rest or motion. 
 
 The notion of force is first obtained directly by sensation; for 
 the forces exerted by the voluntary muscles can be felt. The 
 existence of forces other than muscular tension is inferred from 
 their efiects. 
 
 56. 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. 
 
32 
 
 MECHANICS. 
 
 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. 
 
 57. Dynamics — Statics and Kinetics. — Forces may take effect, 
 either by balancing other forces, or by producing change of motion. 
 The former of those effects is the subject oi Statics; the latter that 
 of Kinetics, and the Science which treats of both is by modern 
 practice entitled Dyna/mics ; these, together with Cinematics, 
 already defined, form the three great divisions of pure, abstract, or 
 general mechanics. 
 
 58. Structures and Machines. — The works of human art to 
 which the science of applied mechaaics 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 Stnictures ; in the latter, Machine. 
 Structures are subjects of Statics alone j Machines, when the 
 motions of their parts are considered alone, are subjects of Cine- 
 matics; when the forces acting on and between their parts are 
 also considered, machines are subjects of Dynamics. 
 
PART I. 
 
 PRINCIPLES OF CINEMATICS, OR THE COMPARISON 
 OF MOTIONS. 
 
 59. Division of the*Subject.— The Science of Cinematics, and 
 the fundamental notions of rest and motion to which it relates, 
 having already been defined among the Elementary Mechanical 
 Notions, Articles 51, 52, 53, 54, 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. 
 TI. „ Rigid Bodies or Systems. 
 
 III. „ Pliable Bodies and Fluids. 
 
 CHAPTER I. 
 
 MOTIONS OF POINTS. 
 Section 1. — Motions of a Paie of Points. 
 
 60. Fixed and Nearly Fixed Directions. — From the definition 
 of motion given in Article 52, it follows, that in order to deter- 
 mine 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 considered, with some fixed or standard length, and with 
 at least two fixed directions. Standard lengths have already been 
 considered in Article 50. 
 
 An absolutely fixed direction may be ascertained by means whose 
 principles cannot be demonstrated until the subject of kinetics 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. 
 
 D 
 
34 PEINCIPLES OF CINEMATICS. 
 
 A. nearly fixed direction is \h&t of a straight line joining a pair 
 of points in two bodies whose distance from each other is very 
 great, such as the earth and a fixed star. 
 
 A line fixed relatively to the earth changes its absolute direction 
 (unless parallel to the earth's axis) in a manner depending on the 
 earth's rotation, and returns periodically to its original absolute 
 direction at the end of each sidereal day of 86,164 seconds. This 
 rate of change of direction is so slow compared with that which 
 takes place in almost all pieces of mechanism to which cinematical 
 and kinetic 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 con- 
 sideration, 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. 
 
 Postulate. — Let it be granted that a line may represent a 
 motion, where the term motion is employed to represent the path 
 of motion, the direction and the velocity or length of motion. 
 This is a self-evidently possible problem, for a line may be drawn 
 to represent any path, in any direction to represent any direction 
 of motion, and of any length to represent any length of motion, or 
 velocity, limited always by the space within which motions can 
 take place or lines be drawn. 
 
 61. Motion of a Pair of Points.— In fig. 18, let Aj B^ repre- 
 „ sent the relative situation 
 
 ^2 ^ of a pair of points at one 
 
 instant, and Ag Bg 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- 
 Pig. 19. Fig. 20, tion represented by Aj Bj 
 
 to the length and direction 
 represented by Ag Bj, 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. 19 , a pair of lines, ABj^, A Bg, equal and 
 parallel to A^ B^, Ag Bj, of fig. 18 ; then. A represents one of the 
 pair of points whose relative motion is under consideration, and 
 Bj, B2, represent the two successive positions of the other point B 
 
THE MEASUREMENT OF TIME. 35 
 
 relatively to A ; and the line B^ B^ represents the motion of B 
 relatively to A which, for the purposes of the representation, is 
 assumed to be fixed. 
 
 Or otherwise, as in_fig^ 20, from a single point B let th ere be 
 drawn a pair of lines, BA^, B A^, equal and parallel to A^, A^ 
 of fig. 18 ; then Aj^ Ag, represe nt the two successive positions of 
 A relatively to B j and the line Aj^ Aj, equal and parallel to B^^ 
 of fig. 19, but pointing in the contrary direction, represents the 
 motion of A relatively to B. 
 
 62. Fixed Point and Moving Point— In fig. 19, A is treated 
 as the fixed point, and B as the moving point; and in fig. 20, B 
 is treated as the fixed ptiint, and A as the moving point; and these 
 are simply two different methods of representing to the mind the 
 same relation between the poiats A and B (see Article 53). 
 
 63. Component and Resultant Motions.— Let be a point 
 assumed as fixed, and A and B two 
 successive positions of a second point 
 relatively to O. In order to express 
 ma them atically the amount and direction 
 of A B, 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 X, O Y, O Z. 
 
 Through A and B, draw straight lines 
 A 0, B D, parallel to the plane of O Y 
 and Z, and cutting the axis X in C 
 and D. Then CD is said to be the com- ^'S- 21. 
 
 ponent of the motion of the second point relatively to 0, along, o 
 in the direction o/the axis O X; and by a similar process are found 
 the components of the motion A B along 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 parallelopiped of which the components 
 are the sides. 
 
 The three axis are usually taken at right angles to each other ; 
 in which case AC and B D are perpendiculars let fall from A and 
 B upon OX; and if » be the angle made by the direction of the 
 motion A B with O X, 
 
 CD = AB ■ cos cc. 
 
 64. The Measurement of Time is effected by comparing the 
 events, and especially the motions, which take place in intervals 
 of time. 
 
 Eqmd times are the times occupied by the same body, or by 
 equal and similar bodies, under precisely similar circumstances, in 
 
36 PRINCIPLES OF CINEMATICS. 
 
 performing 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. 
 
 A subordinate unit is the second, being the time of one swing of 
 a pendulum, so adjusted as to make 86,400 oscillations in 1-0027379 1 
 of a sidereal day; so that a sidereal day is 86164-09 seconds. 
 
 The length of a solar day is variable; but the mean solar day, 
 being the exact mean of all its different lengths, is the period 
 already mentioned of 1-00273791 of a sidereal day, or 86,400 
 seconds. The divisions of the mean solar day into 24 hours, of 
 each hour into 60 minutes, and of each minute into 60 seconds, 
 are familiar to all. 
 
 Fractions of a second are measured by the oscillations of small 
 pendulums, or of springs, or by the rotations of bodies so contrived 
 as to rotate through equal angles in equal times. 
 
 65. Velocity is the ratio of the number of units of length 
 described by a point in its motion relatively to another point, to 
 the number of units of time in the interval occupied in describing 
 the length in question; and if that ratio is the same, whether it be 
 computed for a longer or a shorter, an earlier or a later, part of the 
 motion, the velocity is said to be uniform. Velocity is expressed 
 in units of distance per unit of time. For different purposes, there 
 are employed various units of velocity, some of which, together 
 with their proportions to each other, are given in the following 
 table : — 
 
 Comparison of Different Measures of Velocity. 
 
 MUes 
 per houi". 
 
 Feet 
 per second. 
 
 Feet Feet 
 per minute, per hour. 
 
 1 
 
 = 1-46 
 
 =88 = 5280 
 
 0-68i8 
 
 = 1 
 
 =60 = 3600 
 
 001136 
 
 = 0016 
 
 = 1 =60 
 
 0-0001893 
 
 = 0-00027 
 
 = 0'016 = 1 
 
 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 propor- 
 tions 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 
 
MOTIONS OP A SERIES OF POINTS. 37 
 
 way -with those motions, which have already been treated of in 
 Article 63. 
 
 66. Uniform Motion consists in the combination of uniform 
 velocity with uniform direction; that is, with motion along a 
 straight line whose direction is fixed. 
 
 Section 2. — Uniform Motion op Several Points. 
 
 67. Motion of Three Points Theorem. Tlie relative motions 
 
 of tliree points in a given interval of time j^^ 5 
 
 are represented in direction and magni- ' 
 
 tude by the three sides o^a triangle. Let 
 
 O, A, B, denote the three points. Any^ 
 
 one of them may be taken as a fixed 
 
 point; let O be so chosen; and let X, 
 
 O Y, O Z, fig. 22, be axes traversing 
 
 it in fixed directions. Let A^ and Bj 
 
 be the positions of A and B relatively 
 
 to at the beginning of the given injterval of time, and Ag and Bj 
 
 their positions at the end of that interval. Then Aj Ag and Bj Bg 
 
 are the respective motions of A and B relatively to 0. Complete 
 
 the parallelogram Aj Bj J Ag ; then because Ag h is parallel and 
 
 equal to Aj Bj, h is the position which B would have at the end of 
 
 tlie interval, if it had no motion relatively to A; but Bg is the 
 
 actual position of B at the end of the interval ; therefore, h Bg is 
 
 the motion of B relatively to A. Then in the triangle Bj^ 6 Bg, 
 
 B]^ 6 = Aj Aj is the motion of A relatively to O, 
 h Bg is the motion of B relatively to A, 
 
 B^ Bg 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, tliat if 
 three moving points he considered in any order, the motion of the 
 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 ex- 
 plained in Article 63. 
 
 68. Motions of a Series of Points. — Corollary. If a series of 
 points be considered in any order, and the motion of each point 
 determined relatively to that which precedes it in the series, and if 
 the relative motion of the last point and the first point be also deter- 
 mined, then wiU those motions be represented by the sides of a closed 
 
38 PRINCIPLES OF CINEMATICS. 
 
 polygon. Let O be the first point, A, B, C, &o.; 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 0, be denoted thus (B, C). Then by the Theorem of Article 
 67, (0, A), (A, B), and (0, B) are the three sides of a triangle; 
 also (0, B), (B, 0), and (0, C), are the three sides of a triangle ; 
 therefore (O, A), (A, B), (B, C), and (O, C), are the four sides of a 
 quadrilateral ; and by continuing the same process, it is shewn, 
 that how, great soever the number of points, (O, N), is the closing 
 side of a polygon, of which (0, A), (A, B), (B, C), (C, D), &c., 
 (M, N) are the other sides.— Q. E. D. In other words, the motion 
 oftM last point relatively to the first is the resultant of the motions 
 of each point of the series relatively to that preceding it. 
 
 69. The Parallelopiped of Motions.— In fig. 23, 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 directions, O X, Y, 
 O Z. In a given interval of time, let A 
 have the motion Aj Ag along or parallel 
 to X; let B have, in the same interval, 
 the motion 6 B^ parallel to O Y, and rela- 
 tively to A; then Bj^ B^, the diag onal of 
 the parallelogram whose sides are B^ 6 = 
 
 Pig. 23. ^^ ^^ ^j^^ 6B2, is the motio n of B rela- 
 
 tively to O. Let C have, relatively to B, the motion c Og parallel 
 to Z ; then Oi C2, the diagonal of the parallelopiped whose edges 
 are Aj Aj, b Bg, and c Cj, 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 63. 
 
 70. 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 expressed, 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, that is, the proportion borne to each other by 
 the distances moved through by the two points in the same interval 
 of time. 
 
 (2.) (3.) (4.) The directional relation* which is the relation be- 
 tween the directions in which the two points are moving at the 
 same instant, and which requires, for its complete expression, three 
 
 * These terms are adopted from Prof. Willis's work on Mechanism. 
 
VELOCITY AND DIRECTION OF VARIED MOTION. 50 
 
 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 plaice. 
 
 (L) 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 " qiiatemions." In most 
 of the practical applications of cinematics, the motions to be com- 
 pared are limited by coftditions which render the comparision 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 sufiiciently expressed by means of the velo- 
 city 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 op Points. 
 
 71. 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. 24, let O represent a point 
 assumed as fixed, O X, O Y, Z, fixed 
 directions, and A B part of the path or 
 orbit traced by a second point in its '' -p- 24 
 
 varied motion relatively to 0. At the 
 instant when the second point reaches a given position, suc h 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 f denote 
 an interval of time which includes that instant, and A s the dis- 
 tance traced in that interval. Then 
 
 AjS 
 At 
 
 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 
 
40 PRINCIPLES OF CINEMATICS. 
 
 and sliortei"; and the limit towards which —^ converges, as A * 
 and A t are indefinitely diminished, and which is denoted by 
 
 is the exact velocity at the instant of passing P. In the language 
 of the differential calculus, the apace is a function of the time and 
 the velocity is the differential coefficient of the space with respect 
 
 ds 
 to the time, thus s = ^t and -7-'= ^' t — v. It will be seen here- 
 after that, the velocity {v) itself is a function of the time (<). This 
 is the process called " differentiation." 
 
 Should the velocity at each instant of time be known, then the 
 distance Sj — Sq, described during an interval of time i^ - ?q, is found 
 by integration (see Article 29), as follows: — 
 
 J '0 
 
 vdt (2.) 
 
 72. 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 O X, 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 those three axes are 
 
 V cos a; V cos /3; V cos y. 
 
 Let X, y, e, be the co-ordinates of any point, such as P, in the path 
 A P B, as referred to the three given axes. If a point p be 
 assumed indefinitely near to the point P, its co-ordinates will be 
 x + dx, y + dy, z + dz, and \i ds have the already assumed value, 
 dx, dy, dz, will be its projections on the three axes; that is, the 
 lengths bounded by perpendiculars let fall from the extremities of 
 ds on the three respective axes. Then it is well known that 
 
 dx , dy dz 
 
 cos «= 3—: cos /3=-^: cos 5' = -r-; 
 ds ds ds 
 
 and consequently the three components of the velocity v I = -j- I are 
 
 dx „ dy dz 
 « cos « = ^^; « cos /3 = -;,, cosy = j^; (3.) 
 
UNIFORMLY TAEIED VELOCITY. 41 
 
 uow by the Geometry of three dimensions 
 
 cos^ a + cos^ /3 + cos^ y = 1. 
 and hence these are related to their resultant by the equation 
 
 m'^m^m-- w 
 
 73. 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 
 Vq, 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 retaardatiov, 
 when negative. Theii the mean velocity during the time * is 
 
 ^^0 + ^ V0 + V0 + at at 
 
 ~^^- 2 ^''o + T (^-^ 
 
 and the distance described is 
 
 s = Vo« + -2- ■ (3.) 
 
 If there be no initial velocity, that is, if the body start from a 
 state of rest, then v=at and s = -=-, and these equations are illus- 
 trations of the use of the differential calculus; for first differentiate 
 s with respect to t in the equation s = -^, and there is obtained 
 
 —.(=«)) = — ^ — = at, which is the first equation, then differentiate 
 
 v = at, and there is obtained t^ = «- 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, Asj, As2> 
 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 intervals is 
 
 Afi + M 
 2A< ■■■■ 
 
 and its rate of variation is 
 
 _ A" _ Ag2 - Asi 
 *"A« {Atf • 
 
 .(4.) 
 .(5.) 
 
42 
 
 PRINCIPLES OF CINEMATICS. 
 
 /\'8 — /\8 
 
 where the variation of velocity = — ?- 1, and the rate of varia- 
 tion being either acceleration or retardation, as the velocity of the 
 point is being increased or diminished, is that quantity divided 
 by A *. 
 
 74. Graphical Representation of Motions. — Since in uniform 
 motion the space is equal to the product of the velocity and time, 
 and since in geometry a rectangiflar area is the product of a base 
 line and perpendicular, an uniform motion may be represented by 
 a rectangular area, as in fig. 25, where A B represents a certain 
 number of units of time, and A C a certain 
 number of units of velocity per unit of 
 time. It will be noticed that in uniform 
 motion, the velocity or number of units of 
 velocity at each unit of time is the same, 
 as at A, B, E. Varied motion and uni- 
 formly varied motion may also be graphi- 
 cally represented: in the first, the line 
 CD will be a curve; and in the second, 
 the line C D will form a constant angle with A B; hence in 
 varied motion any ordinate, E F, depends upon the abscissa A E, 
 and the mean velocity is the mean ordinate of a figure so formed, 
 
 or is the quotient of the area 
 (space) divided by the base (time), 
 whereas in uniformly-varied mo- 
 tion, the space described depends 
 iipon the initial and final velo- 
 cities alone, and not upon the 
 intermediate velocities. Fig. 26 
 represents varied motion where 
 the velocity at each point is re- 
 presented by the ordinate at that point, and the mean velocity is 
 
 equal to the area of the figure divided 
 by the base A B. Fig. 27 represents 
 uniformly- varied motion, and it is evi- 
 dent that, in order to estimate the area 
 of the figure ABOD, that is, the space, 
 it is only necessary to consider the 
 initial and final velocities. In these 
 figures, if the velocity be null at any 
 point, there will be no ordinate at 
 that point : if the direction of motion 
 change, this wiU be represented by a change of sign of the ordinate 
 or velocity. 
 
 There is another method of graphically representing the motion 
 of a point: in this the abscissae represent the time, and the ordinates 
 
 3 
 
 A. 
 
 Fig. 27. 
 
ACCELEEATED MOTION. 43 
 
 at each point the space passed over in the corresponding number 
 of units of time, or the distance of the point from a certain datum 
 point. In this case the space described in any number of units of 
 time is equal to the difference of the lengths of the ordinates at 
 the corresponding intervals, and the velocity is proportional to the 
 quotient of the difference of the ordinates divided by the difference 
 of the abscissDe. 
 
 75. Varied Kate of Variation of Velocity. — When the velocity 
 of a point is neither constant nor uniformly-varied, its rate of 
 variation may still be found by applying to the velocity the same 
 operation of differentiation, which, in Article 73, was applied to 
 the distance described i^ order to find the velocity. The result of 
 this operation is expressed by the symbols, 
 
 dv_ dt_d^s 
 "'"dt Tt d¥' 
 
 and is the limit to which the quantity obtained by means of the 
 formula 5 of Article 73 continually approximates, as the interval 
 
 denoted hy At is indefinitely diminished. In the fraction dt 
 
 "IT' 
 ds is the limit of the difference of either of the spaces As in equa- 
 tion (5), Article 73, and d • ds, is the limit of the difference of that 
 difference, viz., Asg ~ AsjJ that is, d in this fraction is represented 
 by the minus sign ( — ) in the other, and dshy the limit of either 
 of the quantities Asj, A^- Here in the language of the differential 
 calculus, the velocity (*i) is a function of the time (t), and the 
 acceleration (a) is the differential coeflS.cient of the velocity with 
 
 respect to the time, thus v = <t>t and a = 4"}, or=--r^. Also the 
 
 velocity, v, being the differential coefficient of the space with respect 
 to the time, see Article 71j the acceleration a is the 2nd differ- 
 ential coefficient of the space with respect to the time, or v being 
 ^'i, a = yp"t. 
 
 76. Combination of Uniform and Uniformly Accelerated Motion. 
 
 Assume a pair of rectangular axes of co-ordinates. Let the 
 
 uniform motion be represented by abscissae along X, and the 
 uniformly accelerated motion by ordinates parallel to Y; let 
 OB ( = x) = vt, represent the space described in the time t with 
 
 a t^ 
 the -vjelocity v, and let (=y)=—^, represent the space de- 
 
44 
 
 PRINCIPLES OP CINEMATICS. 
 
 scribed with a uniform rate of acceleration, a, in the same time t, 
 
 see Article 73, then x'^ = v^t^ and 
 
 y = -^-, .: x^ = y-^, where the 
 
 square of any abscissa bears a con- 
 stant ratio to the corresponding ordi- 
 nate, and the path of the point is 
 known by Conic Sections to be a 
 Pig- 2S. Parabola. The same follows for any 
 
 axes of co-ordinates; but if the direction of the uniformly accelerated 
 motion be that of the uniform motion or directly opposed to it, 
 the resultant direction will be the same as that of either motion, 
 or will be that of the greater component. 
 
 77. 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. 29, 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 i, let Aj and Ag be the 
 positions of the moving point. Then 
 
 the arc Aj A.2 = v^t; 
 the chord A-^^ A^ = v^t 
 
 chord 
 
 The velocities at Ai and Ag are represented by the equal lines 
 Ai Vi = A2 Y^ = v, touching the circle at Aj and Ag respec- 
 tively. From Ag draw A ^v eq ual and parallel to A^ Vj, and join 
 Y^v- Then the velocity A^Vg may be considered as compounded 
 of Ag'y and vY^; so that vY^ is the deviation of the motion dur- 
 ing the interval A <; and because the isosceles triangles As i; Vj, 
 C Aj Aj, are similar : — 
 
 vY^ = 
 
 Aj Vs • Ai Aa 11^ • At chord 
 
 CA 
 
 arc 
 
 deduced by substituting the value of Aj Aj already found; and the 
 approximate rate of that deviation being the deviation divided by 
 the interval of time in which it occurs, is 
 
 v^ chord 
 
THE COMPARISON OP THE VARIED MOTIONS. 45 
 
 but the deviation does not take place by instantaneous changes of 
 velocity, but by insensible degrees ; so that the true rate of devia- 
 tion is to be found by finding the limit to which the approximate 
 rate continually approaches as the interval A* is diminished 
 
 indefinitely. Now the factor — remains unaltered by that diminu- 
 
 r 
 
 tion ; and the ratio of the chord to the arc approximates continually 
 
 to equality; so that the limit in question, or true rale of deviation, 
 
 is expressed by 
 
 ? (1.) 
 
 78. 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 77, 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. 
 
 79. 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 : — 
 
 the first term of the quantity under the first radical is the square of 
 
 dv v^ 
 
 -J- in Article 73, and the second the square of - , Equation (1), 
 
 Article 77. 
 
 80. The Rates of Variation of the Component Velocities of a 
 point parallel to three rectangular axes, are represented as follows: — 
 
 d'^x d^y ^z .^ . 
 
 arid if a rectangular parallelepiped be constructed, of which the 
 edges represent these quantities, its diagonal, whose length is 
 
 vm'*mr<iiy} ■■« 
 
 will represent the resultant rate of variation, already given in 
 another form in Equation 1 of Article 79. 
 
 81. The Comparison of the Varied Motions of a pair of points 
 
46 , PKINCIPLES OF CINEMATICS. 
 
 relatively to a tbird 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 70, 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 motiMi of each point is varied, pro- 
 vided the variations take place for ooth points at the same instant, 
 and at rates proportional to their velocities. 
 
47 
 
 CHAPTER II. 
 
 MOTIONS OF KIGID BpDIES. 
 
 Section 1. — Rigid Bodies, and their Translation. 
 
 82. The term Rigid Body is to be understood to denote a body, 
 or an assemblage of booties, or a system of points, whose figure 
 undergoes no alteration during the motion which is under con- 
 sideration. 
 
 83. 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. 
 
 84. 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 a centre of 
 rotation. 
 
 85. Axis of Rotation. — Theorem. In every possible -change of 
 position of a rigid hody, relatively to a fixed centre, there is a line 
 traversing that centre whose direc- 
 tion is not changed.) In fig. 30, 
 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, Aj, Bj, and a fter the 
 turning, Aj, Bj. Join Aj A^. 
 Bi Bj, forming the isosceles tri- 
 angles O Ai Aj, Bi Bj. Bisect the bases of those triangles in C 
 
48 PRINCIPLES OF CINEMATICS. 
 
 and D respectively, and through the points of bisection draw two 
 planes perpendicular to the respective bases, intersecting each 
 other in the straight line O SS, which must traverse O. Let E be 
 any point in the line O B; then E Aj A^, and E Bj Bj, are isosceles 
 triangles ; and E is at the same distance from O, A, and B, before 
 and after the turning; therefore jE 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. 
 
 In fig. 31, the same construction and reasoning being appliedj 
 the point E being supposed vertically 
 above or below the point O, it is evident 
 that the planes through O D, and C 
 intersect, and the axis will be represented 
 by a straight line perpendicular to the 
 plane of the paper through O and E. 
 Corollary. It is evident that every 
 Fig. 31. line in the body, parallel to the axis, has 
 
 its direction unchanged. 
 
 86. The Plane of notation is any plane perpendicular to the 
 axis, such as any plane parallel to the plane of the paper, in 
 fig. 31.^ The Angle of Rotation, or angular motion, is the angle 
 made by the two directions, before and after the turning, of a line 
 perpendicular to the axis, as Aj As, or Bj O Bj, in fig. 31. 
 
 87. The Angular Velocity of a turning body is the ratio of the 
 I angle of rotation, expressed jn 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 rela- 
 tion 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 of time ; then 
 
 (710\* 
 2r= 6-2831852 = |j^.j 
 
 88. Uniform Botation consists in uniformity of the angular 
 
 * The value of tr may he easily remembered by taking the first three odd 
 numbers twice each, and placing the six in a row, using the first three as 
 the denominator, and the last three as the numerator of a fraction : we thus 
 
 355 
 obtain 113 | 355=:pp=; this is a nearer approximation than 314159, and 
 
 is generally much more easily employed in calculation. 
 
ROTATING BODY. 49 
 
 ^'elocity of the turning body, and constancy of the direction of its 
 axis of rotation. 
 
 89. Rotation common to all Parts of Body.— Since the angu- 
 lar 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 considering, 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. 
 
 90. Right and Left-Handed Rotation. — The direction of rota- 
 tion round a given axis is distinguished in an arbitrary manner 
 into rigiht-hcmded 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, or in 
 that in which the hands of a watch or clock revolve, the rotation 
 is said to be right-hamded; if in the contrary direction, hft-hamded: 
 and it is usual to consider the angular velocity of right-handed 
 lotation 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. 
 
 91. Relative Motion of a Pair of Points in a Rotating Body. — 
 Let O and A denote any two points in a rotating body; and con- 
 sidering 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 
 perpendicular. 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; 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, that is, the velocity of a point situate 
 at the distance unity from the axis of rotation, then the velocity of 
 A relatively to the axis traversing is 
 
 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 (^deviation 
 of A in its motion relatively to the given axis is 
 
 ^ = <^'r,- (2.) 
 
50 PRINCIPLES OF CINEMATICS. 
 
 in ■which the first expression is that already found in Article 77, 
 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. 
 
 92. 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- 
 drical surface relatively to the axis is a tangent to the surface in a 
 plane perpendicular to the axis. 
 
 93. Comparative Motions of Two Points relatively to an Asds. 
 — Let O, A, B, denote three points in a rotating rigid body; let O 
 be considered as fixed, and let an axis of rotation be drawn through 
 it. Then the comparaiiwe motions of A and B relatively to that 
 axis are expressed as follows: — The velocity-ratio is that of the radii,- 
 vectores of the points, and the directional relation consists in the 
 amgle between their directions of motian, being the same with that 
 between their radii-vectores. Or symbolically : Let r^, r^, be the per- 
 pendicular distances of A and B from the axis traversing 0, and 
 ■Pj and ■^2 their velocities; then 
 
 «2 r„ A A 
 
 94. 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 re- 
 solved into components in the plane of 
 rotation. Thus let O, in fig. 32, represent 
 an axis of rotation of a body whose plane 
 of rotation 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 = ar (a representing the 
 '-'^S- 32. velocity of a point situated at the distance 
 
 unity from the axis of rotation), let that velocity be represented 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 com- 
 ponent of the velocity of A; and let ^ V A XJ = e be the angle 
 made by that line with A V. Prom V let fall V TJ perpendicular 
 
HELICAL MOTION. 51 
 
 to B A; then A TJ represents the component in question; and de- 
 noting it by u, 
 
 u = v cos 6 = ar- cos 6 (1.) 
 
 From let fall O B perpendicular to B A. Then Z A O B = 
 Z V A TJ = tf ; and the right-angled triangles O B A and A U V 
 are similar; so that 
 
 ATY : AAJ ■ : OX: OB = r cos 6 (2.) 
 
 Now the entire velocity of B relatively to the axis is 
 
 ar cos e = u, (3.) 
 
 « 
 so that tTie component, along a given, straight line in the plane oj 
 rotation, of the velocity of any point in that line, is egtud to t/ie velo- 
 city of the point where a perpendicular from the axis meets that line. 
 
 Sectiou 3. — Combined Eotations and Translations. 
 
 95. Property of all Motions of Rigid Bodies.— The foregoing 
 proposition may be regarded as a particular case of the following, 
 which is true of all motions of a rigid body. 
 
 The components, along a given straight line in a rigid body, of the 
 velocities of the points in that line relatively to any point, whether in 
 or attached to the body or otherwise, are all equal to each other; for 
 otherwise, the distances between points in the given straight line 
 must alter, which is inconsistent with the idea of rigidity. 
 
 96. Helical Motion. — Rotation is the only movement which a 
 rigid body as a whole can have relatively to a point belonging to 
 it or attached to it. But if the motion of the body be determined 
 relatively to a point not attached to it, a translation may be com- 
 bined with the rotation. When that translation takes place in 
 the direction of the axis of rotation, the motion of the rigid body 
 is said to be helical, or screw-like, because each point in the rigid 
 body describes a helix or screw, or a part of a helix or screw. 
 
 Let Vi denote the velocity of translation, parallel to the axis of 
 rotation, which is common to all points of the body; this is called 
 the velocity of advance. The advance during one complete turn of 
 the rotating body is the pitch of each of the helical or screw-like 
 paths described by its particles; that is, the distance, in a direc- 
 tion parallel to the axis, between one turn of each such helix and 
 
 2 ■IT . 
 
 the next; and a being the angular velocity, so that — is the time 
 
 a 
 
 of one turn (2 «■ being the space traversed in one turn by a point 
 
 at the distance unity from the axis), the value of the pitch (or the 
 
 space passed over, which is equal to the product of the velocity 
 
 and time) is 
 
52 
 
 PKINCIPLES OP CINEMATICS. 
 
 2 B-O), , 
 
 p =i '; whence v^ = 
 
 a p 
 
 '2^ 
 
 •(!•) 
 
 Let r, as before, be tie radius-vector of any point in the body, and 
 let 
 
 V2 = ar (2.) 
 
 denote its velocity of revolution, or velocity relatively to the axis, 
 due to the rotation alone. Then the resultant velocity of that 
 point is 
 
 v=MTW=<^-'y/ {^2+'-'} (3-) 
 
 The inclination of the helix described by that point to the plane of 
 rotation is given by the equation 
 
 i = arc • tan • — = arc • tan ' 
 
 P 
 
 .(4.) 
 
 that is, an angle whose tangent is equal to % divided by v^, or to 
 p divided by 2 w r, the tangent of that angle being the ratio of the 
 pitch to the circumference of the circle described by the point rela- 
 tively to the axis of rotation. 
 97. Problem. — To find the Motion of a Rigid Body from the 
 
 Motions of Three of its Points.^- 
 Let A, B, C, fig. 33, be three 
 points in a rigid body, and at a 
 given instant let them have mo- 
 "^ tions relatively to a point indepen- 
 dent of the body, which motions 
 are represented in velocity and 
 direction by the three lines A V„, 
 B Vj, C Vj. It is required to find 
 the motion of the entire rigid 
 body relatively to the same fixed 
 point. 
 
 Through any point o, fig. 34, 
 _,. „„ draw three lines oa, oh, oc, equal 
 
 ^^' \ and parallel to the three lines 
 
 , A V„, B Vj, C V„. Through a, h, and c, draw a 
 plane a 6 c, on which let fall a perpendicular o n 
 from 0. Then o n represents a component, which 
 is common to the velocities of all the three points 
 A, B, C, and must therefore be common to all the 
 Fig. 34. points in the body ; that is, it is a velocity of 
 
 translation. 
 
 From the points V„, V„ Y„ draw lines V^ Ij;, yyUl, VJJ« 
 equal and parallel to o n, but opposite in direction to it; and join 
 
SPECIAL CASES. 53 
 
 A U„, B XJj, C U,,, which will all be parallel to the same plane; 
 that is, to the plane ab c. The last three lines -will represent the 
 component velocities which, along with the common velocity of 
 translation parallel to o n, make up the resultant velocities of the 
 three points. Through the point A draw a plane perpendicular to 
 t he co mponent of its motion, which is parallel to a & c; that is, to 
 A XJ„, and through B draw a plane perpendicular to B Uj. These 
 two planes will intersect each other in a line ODE, which will 
 be parallel to o n. The perpendicular distances of that line from 
 the points A B being unchanged by the motion, it represents one 
 and the same line in or attached to the rigid body, and it is there- 
 fore the axis of rota tion. * A plane drawn through the third point, 
 C, perpendicular to C U„, will out the other two planes in the same 
 axis : the three revolving component velocities 
 
 AIJ~ BU; CU:, 
 will be respectively proportional to the perpendicular distances, or 
 radU-vectores, A D, B E, C P, 
 
 of the three points from that axis ; and the angular velocity will be 
 equal to each of the three quotients made by dividing the revolving 
 component velocities of the points by their respective radii-vectores. 
 This rotation, combined with a translation parallel to the axis, 
 with a velocity represented by o n, constitutes a helical or screw-like 
 motion, being the required motion of the rigid body. — Q. E. I. 
 
 98. Special Cases of the preceding problem occur, in which 
 either a more simple method of solution is sufficient, or the general 
 method fails, and a special method has to be employed. 
 
 I. When the motions of the points of 
 the body are knovm to be aM parallel to 
 one plane, it is sufficient to know the 
 motions of two points, such as A, B, fig. 
 35. Let A O, BO, be two planes tra- 
 versing A and B, and perpendicular to 
 the respective directions of the simul- 
 taneous velocities of those points ; if those 
 planes cut each other, the entire motion 
 is a rotation ; the line of intersection of pj„ 35^ 
 the planes O, being the axis of rotation, 
 
 and the angular velocity, are found as in the last Article. If the 
 two planes are parallel, the motion is a translation. 
 
 II. If three points in the same plane have parallel motions oblique 
 to the plane, the motion is a translation. 
 
 III. If three points in the sa/me plane move perpendicularly to the 
 plame, as A B C, fig. 35 a, then if their velocities are equal, the 
 motion is a translation; and if their velocities are unequal, the 
 
54 
 
 PRINCIPLES OP CINEMATICS. 
 
 motion is a rotation about the axis wHch is the intersection of the 
 plane of the three points with the plane drawn through the^extre- 
 
 Fig. 36. 
 
 Fig. 35 a. -^ T^g.?,oc. 
 
 mities of the three lines which represent their velocities viz., 
 through the points, V„, V„ V„; the angular velocity being found 
 
 If the plane of rotation is known, then the simultaneous veloci- 
 ties of two points, as A and B in figs. 35 h and 35 c, are sufficient to 
 determine the axis O. 
 ^99. Eotation Combined with Translation in the Same Plane.— 
 Let a body rotate about an axis C (fig. 36), fixed 
 relatively to the body, with an angular velocity 
 a, and at the same time let that axis have a 
 motion of translation in a straight path perpen- 
 dicular to the direction of the axis, with the 
 velocity u, represented by the line C U. It is 
 required to find the velocity and direction of 
 motion of any point in the body. From the 
 moving axis draw a straight line C T perpendi- 
 cular to that axis and to U, and in that direction into whichthe 
 rotation (as represented by the feathered aiTow) tends to turn C U, 
 and make CT"=- (1.) 
 
 Then the point T has, in virtue of translation along with the axis 
 C, a forward motion with the velocity u; and in virtue of rotaiioii' 
 about that axis, it has a backward motion with the velocity 
 
 a ■ CT= M, 
 equal and opposite to the former; and its resultant velocity is 0. 
 Hence every point in the body, which comes in succession into the 
 
 position T, situated at the distance - from the axis C in the direo- 
 
 '■ a 
 
 tion above described, is at rest at the instant of its arriving at that 
 position; that is, it has just ceased to move in one direction, and 
 is about to move in another direction; and this is true of every 
 
SPIRAL PATHS. 55 
 
 point which arrives at a line trwversmg T parallel to 0. Conse- 
 quently the resultant motion of the body, at any given instant, is 
 the same as if it were rotating about the line which at the insfcmi 
 in queBiion occupies the position T, parallel to C, at the distance 
 
 -• ; and that line is called the instantaneous axis. To find the 
 a 
 
 motion of any point A in the body at a given instant, let fall the 
 
 perpendicular A T from that point on the instantaneous axis; then 
 
 the motion of A is in the direction AV perpendicular to the plane 
 
 of the instantaneous axis and of the instantaneous %adiii8-vector 
 
 A T, and the velocity of that motion is 
 
 v = a-XT (2.) 
 
 100. Rolling Cylinder; Trochoid. — Every straight line parallel 
 to the moving axis C, in a cylindrical surface described about O 
 
 •with the radius -, becomes in turn the instantaneous axis. Hence 
 a 
 
 the motion of the body is the same with that produced by the roll- 
 ing of such a cylindrical surface on a plane P T P parallel to C and 
 
 to C U, at the distance -. 
 a 
 
 The path described by any pomt in the body, such as A, which 
 is not in the moving axis C, is a curve well known by the name of 
 trochoid. The particular form of trochoid called the cyehid, is 
 described by each of the points in the rolling cylindrical surface; 
 being such a curve as is described by a nail in the tyre of a revolv- 
 ing wheel. 
 
 101. Plane Boiling on Cylinder; Spiral Paths. — ^Another mode 
 of representing the combination of rotation with translation in the 
 same plane as follows :— Let O, fig. 37, be an axis assumed as fixed, 
 about which let the plane O C (containing the axis 0) rotate (right- 
 handedly, in the figure), with the angular 
 velocity a. Let a rigid body have, rela- 
 tively to the rotating plane, and, in a direc- 
 tion perpendicular to it, a translation 
 with the velocity u. In the plane C, 
 and at right angles to the axis O, take 
 
 O T = -, in such a direction that the 
 a 
 
 velocity 
 
 u-a-Wr, 
 
 which the point T in the rotating plane 
 has at a given instant, shall be in the 
 contrary direction to the equal velocity 
 of translation u, which the rigid body has relatively to the rotating 
 
56 PRINCIPLES OF CINEMATICS. 
 
 plane. Then each point in tlie rigid body which arrives at the 
 position T, or at anjr position in a line traversing T parallel to the 
 fixed axis O, is at rest at theAnstant of its occupying that position; 
 therefore the line traversing T parallel to the fixed axis O is the 
 instantaneous axis; the motion at a given instant of any point in 
 the rigid body, such as A, is at right angles to the radius-vector 
 A T drawn perpendicular to the instantaneous axisj and the 
 velocity of that motion is given by the equation 
 
 ' v = a-AT. 
 
 All the lines in the rigid body which successively occupy the 
 position of instantaneous axis are situated in a plane of that body, 
 P T P, perpendicular to O C; and all the positions of the instan- 
 taneous axis are situated in a cylinder described about O with the 
 radius O T ; so that the motion of the rigid body is such as is pro- 
 duced by the rolling of the plane P P ow the cylinder whose radius is 
 
 O T = — . ' Each point in the rigid body, such as A, describes a 
 
 plane spiral about the fixed axis O. For each point in the rolling 
 plane, P P, that spiral is the involute of the circle whose radius is 
 O T. The simplest method of understanding the nature of this 
 curve, is to wrap a cord round ,the perimeter of a cylinder, placed 
 on a sheet of paper, to attach a tracing point to any point in the 
 cord in juxtaposition with the cylinder, and then to unwrap the 
 cord from the cylinder, keeping the cord always in the same plane 
 parallel to the plane of the paper; the tracing point will trace the 
 involute of a circle on the sheet of paper. For each point whose 
 path of motion traverses the fixed axis ; that is, for each point 
 in a plane of the rigid body traversing parallel to P P, the spiral 
 is Archime4ean, having a radius-vector increasing by the length 
 u for each angle a through which it rotates; this spiral is traced 
 by a point moving uniformly from the centre along the radius, 
 while the radius itself revolves. 
 
 102. Combined Parallel Rotations.— In figs. 38, 39, and 40, let 
 O be an axis assumed as fixed, and O C a plane traversing that 
 axis, and rotating about it with the angular velocity a. Let C be 
 an axis in that plane, parallel to the fixed axis 0; and about the 
 moving axis let a rigid body rotate with the angular velocity 6 
 relatively to the plane O C ; and let the directions of the rotations 
 a and h be distinguished by positive and negative signs. The body 
 is said to have the rotations about the parallel axes O and C com- 
 bined ov fiompownded, and it is required to find the result of that 
 combination of parallel rotations. 
 
 Fig. 38 respresents the case in which a and 6 are similar ia 
 direction; fig. 39, that in which a and b are in opposite directions, 
 
COMBINED PARALLEL ROTATIONS. 
 
 57 
 
 and b 18 the greater; and fig. 40, that in which a and 6 are in 
 opposite directions, and a is the greater. 
 
 Kg. 39. Fig. 40. 
 
 Let a common perpendicular O C to the fixed and moving axes 
 lie intersected in T by a straight line parallel to both those axes, in 
 such a manner that the distances of T from the fixed and moving 
 axes respectively shall be inversely proportional to the angular 
 velocities of the component rotations about them, as is expressed 
 by the following proportion : — 
 
 a lb: :CT:OT (1.) 
 
 When a and b are similar in direction, let T fall between O and C, 
 as in fig. 38 ; when they are contrary, beyond, as in figs. 39 and 
 40. Then the velocity of the line T of the plane C is a • OT; 
 and the velocity of the line T of the rigid body, relatively to the 
 plane O C, is 6 • C T, equal in amount and contrary in direction to 
 the former; therefore each line of the rigid body which arrives at 
 the position T is at rest at the instant of its occupying that position, 
 and is then the instantaneous axis. The resultant angular velocity 
 is given by the equation 
 
 c = a + b- (2.) 
 
 regard being had to the directions or signs of a and b ; that is to 
 say, if we now take a and 6 to represent arithmetical magnitudes, 
 and affix explicit signs to denote their directions, the direction of 
 c will be the same with that of the greater ; the case of fig. 3S 
 will be represented by Equation 2, already given ; and those of 
 figs. 39 and 40 respectively by 
 
 c = b-a; c = a — b (2 A.) 
 
 The relative proportions of a, b, and c, and of the distances 
 between the fixed, moving, and instantaneous axes, are given by 
 
 the equation 
 
 a:b:o: :CT:OT:OC (3.) 
 
 The motion of any point, such as A, in the rigid body, is at each 
 
58 PKINOIPLES OF CINEMATICS. 
 
 I 
 
 instant at right angles to the radius-vector A T drawn from the 
 point perpendioxilar to the instantaneous axis; and the velocity of 
 that motion is 
 
 v = c-A?I..... (4.) 
 
 103. Cylinder Eolling on Cylinder; Epitrochoids. — All the lines 
 in the rigid body which successively occupy the position of instan- 
 taneous axis are situated in a cylindrical surface described about 
 with the radius CT; and all the positions of the instantaneous 
 axis are contained in a cylindrical surface described about O with 
 the radius O T ; therefore the resultant motion of the rigid body is 
 that which is produced by rolling the former cylinder, attached to 
 the body, on the latter cylinder, considered as fixed. 
 
 In fig. 38, a convex cylinder rolls on a convex cylinder; in fig. 
 39, a smaller convex cylinder rolls in a larger concave cylinder; in 
 fig. 40, a larger concave cylinder rolls on a smaller convex cylinder. 
 
 Each point in the rolling rigid body traces, relatively to the 
 fixed axis, a curve of the kind called epitrochoids. The epitrochoid 
 traced by a point in the surface of the rolling cylinder is an 
 epicycloid. 
 
 In certain cases, the epitrochoids become curves of a more 
 simple class. For example, each point in the moving axis G traces 
 a circle. 
 
 When a cylinder, as in fig. 39, rolls within a concave cylinder 
 of double its radius, each point in the surface of the rolling cylinder 
 moves backwards and forwards in a straight line, being a diameter 
 of the fixed cylinder; each point in the axis of the rolling cylinder 
 traces a circle of the same radius with that cylinder, and each other 
 point in or attached to the rolling cylinder traces an ellipse of 
 greater or less eccentricity, having its centre in the fixed axis 0. 
 
 In the examples shewn in figs. 41, 42, and 43 the ratio of the 
 
 roUing-circle to the base-circle* is -x, so that the epitrochoids are 
 
 three-lobed. Each figure shews an external and an internal epitro- 
 choid, traced by rolling the rolling-circle outside and inside the 
 base-circle respectively. The centres of the base-circles are marked 
 A; those of the external rolling-circles, B; those of the internal 
 rolling-circles, 6; and the tracing points of the external and in- 
 ternal rolling-circles are marked C and c respectively. 
 
 In fig. 41 the tracing-points are in the circumferences of the 
 rolling-circles; and the curves traced are epicycloids, distinguished 
 by having cusps at the points where the tracing-point coincides 
 with the base-circle. In fig. 42 the tracing points are inside the 
 rolling-circles; and the curves traced are prolate epitrochoids, dis- 
 tinguished by their wave-like form. In fig. 43 the tracing-points 
 ' The fixed circle is called a 'base-cirole. 
 
EPrrEOCHOID& 
 
 59 
 
 Kg, 41. 
 
 Fi.2. 42. 
 
GO PRINCIPLES OF CINEMATICS. 
 
 are outside the rolUng-circles ; and the curves traced are curtate 
 
 emProchoids, distinguished by their looped form. . ^, , ^ 
 
 An important property of curves traced by rolling is that at 
 
 Fig. 43. 
 
 every instant the straight line joining the tracing-point and the 
 pitch-point, or point of contact of the rolling-curve and base-curve, 
 is normal to the traced curve at the tracing point. 
 
 The distance B C or be may in each case be called the tracing- 
 arm. 
 
 In mechanism for the tracing of epitrochoids (used chiefly in 
 ornamental turning), the rolling and base-circles are the pitch- 
 circles of a pair of spur-wheels, made with great accuracy. 
 
 Elliptic paths traced by rolling form a particular case of internal 
 epitrochoids. In fig. 44 is represented a rolling-circle, which rolls 
 inside a base-circle of exactly twice its radius. Then (considering 
 a quarter of a revolution at a time), while the centre of the rolling- 
 circle traces a quadrant, B 6, of an equal circle about A, a point 
 D in the circumference of the rolling-circle traces a straight line 
 traversing A, and a point C, inside the rolling-circle, traces a 
 quadrant, c, of an ellipse whose semiaxes are AC = AB + BC, 
 and Ac = OI) = AB-BC; also a point C' outside the rolling- 
 
EPITHOCHOIDS. 
 
 61 
 
 circle, but rigidly attaclied to it, traces a quadrant, C c', of an 
 ellipse whose semiaxes are A C = B C + A B, and A c' = C D = 
 B C' - A B. The former may be called an interned, and the latter 
 an external, ellipse. The proportions of the axes of either of them 
 
 Kg. 44. 
 
 may be indefinitely varied by adjusting the position of the tracing- 
 pointj but in every internal ellipse the sum, and in every external 
 ellipse the difference, of the semiaxes is equal to the diameter of 
 the rolling-circle J that is, to the radius of the base-circle. 
 
 This is the principle of the mechanism commonly used for 
 turning ellipses. 
 
 It is evident that by having a number of tracing-points carried 
 by one rolling-circle, several ellipses differently proportioned and 
 in different positions may be traced at the same time. 
 
02 PRIKCIPLES OP CINEMATICS. 
 
 104. Equal and Opposite Parallel Rotations Combined. — Let a 
 plane O C rotate with an angular velocity a about an axis con- 
 tained in the plane, and let a rigid body rotate about the axis C 
 in that plane parallel to O, -with an angular velocity - a, equal and 
 opposite to that of the plane. Then the angular velocity of the 
 rigid body is nothing; that is, its motion is one of translation only, 
 all its points moving in equal circles of the radius O C, with the 
 velocity a ■ O 0. This case is not capable of being represented by 
 a rolling action. 
 
 105. Rotations about Intersecting Axes Combined. — In fig. 45, 
 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 h re- 
 latively to the rotating plane. Pig. 45. 
 
 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 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 
 perpendicular 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 T be the instantaneous axis, 
 
 sin AOT:sinCOT : :6:a (1.) 
 
 This determines the direction of the instantaneous axis, which may 
 also be found by graphic construction as follows : — On A take 
 O a proportional to a; and on O C take O 6. proportional to 6. 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 6ca; the dia- 
 gonal c will represent the direction of the instantaneous axis. 
 
 The resultant angular velocity about this instantaneous axis is 
 found by considering, that if C be any point in the moving axis, 
 the linear velocity of that point must be the same, whether com- 
 puted from the angular velocity a of the rotating plane about the 
 fixed axis O A, or from the resultant angular velocity c of the rigid 
 
VARIATION OP ANGUBAE VELOCITY. 63 
 
 body about the instantaneous axis. That is so say, let C D, C E, 
 be perpendiculars from C upon A, O T, respectively; then 
 
 a-CD = c-CEj 
 but CD : CE : : sinZ A O C : sinZC O Tj and therefore 
 
 sinZC0T:sinZAOC: :a:c; 
 and, combining this proportion \7ith that given in Equation 1, we 
 obtain the following proportional equation: — 
 
 sinZCOT :sinZAOT 
 : a : b 
 
 : Oa :» OJ 
 
 sinZAOC 
 
 ^ } (2.) 
 
 Oc 
 
 that is to say, the cmgula/r velocities of the component and resultant 
 rotations are each proportional to the sine of the angle between the 
 axes of the other two; and the diagonal of the paraMdogram O b c a 
 represents both the direction of the instantaneous axis and the gmgu- 
 lar velocity about that axis. 
 
 106. Boiling Cones. — All the lines which successively come into 
 the position of instantaneous axis are situated in the surface of a 
 cone described by the revolution of O T about O C ; and all the 
 positions of the instantaneous axis lie in the surface of a cone 
 described by the revolution of O T about O A. Therefore the 
 motion of the rigid body is such as would be produced by the 
 rolling 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. 
 Por example, should Z A O T become a right angle, the fixed cone 
 would become a flat disc j and should Z A O T become obtuse, 
 that cone would be hollow, and would be touched internally by the 
 rolling cone; and similar changes may be made in the rolling cone. 
 
 The path described by a point in or attached to the rolling cone 
 is a spherical epitrochoid; but for the purposes of the present trea- 
 tise, i«t is unnecessary to enter into details respecting the properties 
 of that class of curves. 
 
 107. Comparative Motions in Compound Rotations. — The velo- 
 city 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. 
 
 108. Variation of Angular Velocity is measured like variation 
 of linear velocity, by comparing the change which takes place in 
 
64 PRINCIPLES OF CINEMATICS. 
 
 the 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 ofva/riatum is the value towards which the ratio of the change 
 
 of angular velocity to the interval of time, -j—z, converges, as the 
 
 length of the interval is indefinitely diminished; being represented 
 
 by -J-, and found by the operation of differentiation. 
 
 Qt t 
 
 109. 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 these components, at any instant shew 
 at once the resultant angular velocity and the direction of the 
 instantaneous axis. For example, let a„ «„, a^, be the rectangular 
 components of the angular velocity of a rigid body at a given 
 instant, — 
 
 rotation about x from 3/ towards «, 
 
 about y from z towards x, 
 
 and about z from x towards y, 
 
 being considered as positive; then 
 
 a= J(«/ + o/ + a,2) (1.) 
 
 is the resultant angular velocity, and 
 
 ax .ay as ,„. 
 
 cos a = — ; cos/3=— !i; cosy = — ; (2.) 
 
 a ' a ' a ' ^ ' 
 
 are the cosines of the angles which the instantaneous axis makes 
 with the axis of as, y, and z respectively. 
 
G5 
 
 CHAPTER III. 
 MOTIONS OF PLIABLE BODIES, AND OF FLUIDS. 
 
 110. Division of the Subject The subject of the present 
 
 chapter wiU be considered under the following branches : — 
 
 I. The Motions of Flexible Cords. 
 II. The Motions of Fluids not altering in Volume. 
 
 111. The Motions of Fluids altering in Volume. 
 
 SECTioif 1. — Motions op Flexible Cords. 
 
 111. General Principles. — As those relative motions of the 
 points of a cord which may arise from its extensibility, belong to 
 the subject 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 
 j)oints fixed in the cord, is invariable, or sensibly invariable. 
 
 In order that the figure and motions of a flexible cord may be 
 determined from cinematical considerations alone, independently of 
 the magnitude and distribution of forces acting on the cord, its 
 weight must be insensible compared with the tension on it, and it 
 must everywhere be tight; and when that is the case, each part of 
 the cord which is not straight is maintained in a curved figure by 
 passing 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 or tangential plane at each point 
 of such a line is perpendicular 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. Tliat length is the shortest line which can be drawn between its 
 extremities over the surfaces hy which the cord is guided. 
 
 112. Motions Classed. — The motions of a cord are of two 
 kinds — 
 
 I. Travelling of a cord along a ti-ack of invariable form; in 
 which case the velocities of all points of the cord are equal. 
 
 F 
 
66 PRINCIPLES OF CINEMATICS. 
 
 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 
 motion between two pieces in a train of mechanism. Such pro- 
 blems will be considered in Part II. of this treatise. 
 
 Next in point, of frequency in practice are the problems to be 
 considered in the ensuing Article. 
 
 113. Cord Guided by Surfaces of Revolution. — Let a cord in 
 some portions of its course be straight, and in others guided by the 
 surfaces 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 + 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 
 
 «= ^ o^-{r±r'f (3.) 
 
 the ■) j-fT \ oi the radii being employed, according as the cord 
 
 < J „„ , > the line of centres c. 
 
 [ does not cross j 
 
 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 
 
 4 = 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; where u represents 
 
 the velocity of translation of the guiding surfaces, and v the 
 
 longitudinal velocity of any point in the cord 
 
 v = 2nu (3.) 
 
 Section 2. — Motions op Fluids op Constant Density. 
 
 114. 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- 
 
PIPES, CHANNELS, CURRENTS, JETS. 67 
 
 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 
 uniform velocity, or the mean value of the varying velocity, 
 resolved in a direction perpendicular to A, with which the particles 
 of the fluid pass A. Then 
 
 Q = wA (1.) , 
 
 is the volume of fluid which passes from one side to the other of 
 the surface A in an unit oPtime, and is called the flow, or rate of 
 flow, through A. 
 
 When the particles of fluid move obliquely to A, let S denote 
 the angle which the direction of motion of any particle passing A 
 makes with a normal to A, and v the velocity of that particle; 
 then 
 
 u = v cos 6. (3.) 
 
 115. 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 amd the flow out of it, in any one given interval of time, 
 must he equal — a principle expressed symbolically by 
 
 2-Q = (3.) 
 
 The precedihg 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. 
 
 .116. Flow in a Stream. — A stream is a moving fluid mass, 
 indefinitely 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, con- 
 sidered 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 continuity may be fulfilled, those rates of flow must be 
 equal; that is, 
 
 M A = m' A' = Q = Q' = constant for all cross 
 
 sections of the channel at the given instant; (1.) 
 
 consequently, 
 
 11' A 
 
 l=X'-' (2-) 
 
 or, the normal velocities at a given instant at two fixed cross sections 
 are inversely as the areas of these sections. 
 117. Pipes, Channels, Currents, and Jets.— When a stream of 
 
68 PRINCIPLES OF CINEMATICS. 
 
 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 
 different 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 
 npper surface free; and this description applies not only to channels 
 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 elevation 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. 
 
 118. Steady Motion of a fluid relatively to a given space con- 
 sidered 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 
 ^■elocity and direction depending on the position of that point 
 alone; which velocity and direction are successively assumed by 
 each particle which successively arrives at the same fixed point. 
 
 The steady motion of a stream is expressed by the two conditions, 
 that the area of each fixed cross section is constant, and that the 
 flow through each cross section is constant ; that is to say, 
 
 dt ' dt^ ^^■' 
 
 If u represents the normal velocity of a fluid moving steadily, 
 at a given fixed point, then the differential coefficient of a constant 
 being equal to (see Article 26, page 1 1), 
 
 a-:-»^ w 
 
 expresses the condition of steady motion. 
 
 119. Motion of Eistons.— Let a mass of fluid of invariable 
 volume be enclosed in a vessel, two portions of the boundary of 
 which (called\pi«tows) are movable 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 out- 
 wards, it follows, from the invariability of the volume of the 
 enclosed fluid, that the velocities of the two pistons at each instant 
 
TUB PRINCIPLE OF CONTINUITY. 69 
 
 will be to each other in the inverse ratio of the areas of the respec- 
 tive 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 Jlow 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 = mA (1.) 
 
 Section 3. — Motions op Fluids op Varying Density. 
 
 * 
 
 120. Flow of Volume and Flow of Mass.— In the case of a fluid 
 of varying density, the volume, which in an unit of time flows 
 through a given area A, with a normal velocity m, is still repre- 
 sented, as for a fluid of constant density, by 
 
 Q = Am; (1.) 
 
 but the absolute quantity, or mass of fluid which so flows, bears no 
 longer a constant proportion to that volume, but is proportional 
 to the volume multiplied by the density. The density may be 
 expressed, either in units of weight per unit of volume, or in 
 arbitrary units suited to the particular case. Let e be the density; 
 then the flow of mass may be thus expressed: — 
 
 jQ = eAM (2.) 
 
 121. The Principle of Continuity, as applied to fluids of varying 
 density, takes the following form : — the flow into or out of any 
 fixed space of constant volume is that due to the variation of density 
 alone. 
 
 To express this symbolically, let there be a fixed space of the 
 
 constant volume V, and in a given interval of time let the density 
 
 of the fluid in it, which in the first place may be supposed uniform 
 
 at each instant, change from Si to Si- Then the mass of fluid which 
 
 at the beginning of the interval occupied the volume V, occupies 
 
 Y p, 
 at the end of the interval the volume — ^ : and the difierence of 
 
 those volumes is the volume which flows through the surface 
 bounding the space, outward if {^ is less than d, inward if fa is 
 greater than ft. Let t^-tihe the length of the interval of time ; 
 then the rate of flow of volume is expressed as follows : — 
 
 V 
 
 Q = -)V^ (1-) 
 
 2 ^ *1 
 
70 
 
 PAET II. 
 
 THEOEY OF MECHANISM. 
 
 'CHAPTER I. 
 
 DEFINITIONS AND GBNEEAL PRINCIPLES. 
 
 122. Theory of Pure Mechanism Defined.— i/ac^mes 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 roaking 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 meolianism. The principles of the theory 
 of pure mechanism having been first established and understood, 
 those of the theory of the work of rrMchines, which wUl form the 
 subject of Part VI. of this work, which regulate the act of trans- 
 mitting and modifying force, are much more readily demonstrated 
 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 ac- 
 
MOViKG pieces; connectoksj bearings. 71 
 
 complished by the labours of Professor Willis, -whose nomenclature 
 and methods are, to a great extent, followed in this treatise. 
 
 123. The General Problem of the theory of pure mechanism 
 may be stated as follows : — Given the mode of connection of two 
 or more movable points or bodies with each other, and with certain 
 fixed bodies; required the comparative motions of the movable 
 points or bodies : and conversely, vihen the comparative motions of 
 two or more movable points are given, to find their proper mode of 
 connection. 
 
 The term "comparative motion" is to be understood as in 
 Articles 70, 81, 93, and 107. In those Articles, the comparative 
 motions of points belonging to one body have already been consid- 
 ered. In order to constitifte mechanism, two or more bodies must 
 be so connected that their motions depend on each other through 
 cinematical principles alone. 
 
 124. Frame; Moving Pieces; Connectors; Bearings Th6 frame 
 
 of a machine is a structure which supports the moving pieces, and 
 regulates the path or kind of motion of most of them directly. In 
 considering the movements of machines mathematically, the frame 
 is considered 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 maybe distinguished vnto primary &uA. 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 
 I'rame 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 imme- 
 diate contact. 
 
 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- 
 
72 THEORY OP MECHANISM. 
 
 ing pieces must have surfaces accurately turned to figures of revolu- 
 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 96). Those 
 parts of moving pieces which touch the bearings, should have 
 surfaces accurately fitting those of the bearings. They may be 
 distinguished into slides, for pieces which move in straight lines, 
 gudgeons, journals, bushes, and pivots, for those which rotate, and 
 screws for those which move helically. 
 
 125. 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 iu opposite directions. (See Part I., 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 I., Chapter II., Section 2.) 
 
 III. Helical or screw-like motion, to which the same remarks 
 apply as to straight translation. (See Part I., Chapter II., Section 
 3, Article 96.) 
 
 126. 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 I., Chapter IT., Section 3. 
 
 127. 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 JoUower. The connection between 
 the driver and the follower may be — 
 
 I. By rolling contact of their surfaces, as in toothless wheels. 
 
 II. By sliding contact of their surfaces, as in toothed wheels, 
 screws, wedges, cams, and escapements. 
 
 III. By hands or wrapping connectors, such as belts, cords, and 
 gewring-chains. 
 
 _IV- By link-work, such as connecting rods, universal joints, and 
 clicks. 
 
AGGREGATE COMBINATIONS. 73 
 
 Y. 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 was employed by Professor Willis, in the 
 first edition of his Principles of Mechanism, as the foundation 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; which is the classification employed by Professor 
 Willis in the Edition of 1870. 
 
 128. Line 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 li7ie 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 sUding 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. 
 
 129. 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 the driver, and a point in the fol- 
 lower, each situated anywhere in the line of connection, are to each 
 other inversely as the cosines of the respective angles made by the paths 
 qfthepoints with the line of connection. This principle might be other- 
 wise stated as follows : — The components, along the line of connec- 
 tion, oftlie velocities of any two points situated in that line, are equal. 
 
 130. 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 con- 
 nection, as when a pair of rotating cones are embraced by a belt 
 which can be shifted so as to connect portions of their surfaces of 
 different diameters. 
 
 131. 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. 
 
 132. Agregate Combinations in mechanism are those by which 
 compound motions are given to secondary pieces. 
 
74 
 
 CHAPTER II. 
 
 ON ELEMEITTAIIY COMBINATIONS AND TRAINS OP 
 MECHANISM. 
 
 Section 1. — ^Eolling Contact. 
 
 133. Pitch Surfaces are those surfaces of a pair of moving pieces, 
 wMcli touoli eacla other when motion is communicated by rolling 
 contact. The line of contact is that line which at each instant 
 traverses all the pairs of points of the pair of pitch surfaces which 
 are in contact. 
 
 134. Smooth Wheels, Rollers, Smooth Racks Of a pair of pri- 
 mary moving pieces in rolling contact, both may rotate, or one may 
 rotate and the other have a motion of sliding, or straight transla- 
 tion. A rotating piece, in rolling contact, is called a smooth wheel, 
 and sometimes a roller; a sliding piece may be called a smooth 
 rack. 
 
 135. General Conditions of Rolling Contact The whole of the 
 
 principles which regulate the motions of a pair of pieces in rolling 
 contact follow from the single principle, that each pair of points in 
 the pitch swrfaces, which are in contact at a given instant, must at 
 tliat instant he moving in the sa/me direction with the same velocity ; 
 that this must be the case is evident from the rigidity of the bodies, 
 for did the pair of points vary in velocity, it would follow that 
 there was motion among the particles, or in a particle at least, of 
 the body, which is contrary to the hypothesis of rigidity. 
 
 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. 
 
 IV. 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 
 
A STRAIGHT RACK AND CIRCULAR WHEEL. 75 
 
 ■wlieel by the perpendicular distance from its axis to a pair of points 
 of contact. 
 
 Eespecting the line of contact, the above principles III. and lY. 
 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). 
 
 VI. That for a pair of wheels, with intersecting axes, the line of 
 contact is also straight, and traverses the point of intersection of 
 the axes; and hence that>the rolling surfaces are conical, with a 
 common apex (the term " conical" including all surfaces generated 
 by the motion of a straight line which traverses a fixed point). 
 
 136. Circular Cylindrical Wheels are employed when an uniform 
 velocity-ratio is to be communicated between parallel axes. Figs. 
 38, 39, and 40, of Article 102, may be taken to represent pairs 
 of such wheels ; and 0, in each figure, being the parallel axes of 
 the wheels, and T a point in their line of contact. In fig. 38, 
 both pitch surfaces are convex, the wheels are said to be in outside 
 gearing, and their directions of rotation are contrary. In figs. 39 
 and 40, the pitch surface of the larger wheel is concave, and that 
 of the smaller convex j 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 
 
 OT = ri, CT = r2. 
 
 be their radii : let O C = c be the line of centres, or perpendicular 
 distance between the axes, so that for 
 
 -^f [gearing, o = ^±.. (1.) 
 
 Let Oi, Wj, be the angular velocities of the wheels, and v the common 
 linear velocity of their pitch surfaces; then 
 
 
 .(2.) 
 
 the sign ± applying to | °^^l^^^j gearing. 
 
 137. A Straight Eack 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. 36 of Article 
 99, C being the axis of the wheel, FTP the plane surface of the 
 rack, and T a point in their line of contact. Let r be the radius 
 
76 THEORY OF MECHANISM. 
 
 of fcbe wheel, a its angular velocity, and v the linear velocity of the 
 rack; then 
 
 v==r a. 
 
 138. Bevel Wheels, whose pitch surfaces are frustra of regular 
 cones are used to transmit an uniform angular velocity-ratio 
 between a pair of axes which intersect each other. Fig. 45 of 
 Article 105 will serve to illustrate this case; O A and O 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 A 
 and C respectively being the pitch surfaces, of which narrow 
 zones or frustra ai'e used in practice. 
 
 Let Oj, ttj, be the angular velocities about the two axes respec- 
 tively; and let ii = ZAOT, i2 = ZC0T, be the angles made 
 by those axes respectively with the line of contact ; then from 
 the principle III. of, Article 135 it follows, that the angular velocity- 
 ratio is 
 
 a^ sin ij _ ,, . 
 
 »! sin i^' 
 
 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, 
 
 Z A O C = ii + 4 =j, 
 
 be given, and also the ratio — ; then the position of the line of 
 contact is given by either of the two following equations : — 
 
 sin i "^^'"J . ] 
 
 . . r V'-) 
 
 bin lo ^ J-T n n c\ ' • * J 
 
 which are formed from equation (1) by substituting for ii its value 
 = (j - ij), and for % its value = (_;' - ii). 
 
 As this is the first instance of the use of Trigonometrical 
 analysis, the method of formation of these equations will be ex- 
 plained : — 
 
 Prom Equation (1) it follows that — 
 
 sin j'l • % = sin 4 • a^ 
 
 = sin 0" - ^i) ■ «2 
 
 = sin^" • cos ij^-a^-cosj ■ sin ij • a^ 
 
 = sin j ■ ,^y(l - sin2 i^) • a.^ - cosj ■ sin I'l • a^. 
 (See Trigonometrical Rules, Sections 19 and 21.) 
 
 Squaring both sides, and tran.^posing 
 
SKEW-BEVEL WHEELS. 
 
 77 
 
 shi^j^ • sin2 ij • aj + (sin ij • Oj + cos^ ■ sin i^ • a^f = aiu^j ■ a\ 
 
 SVD?il 
 
 - coa^j ■ sin^ i^ • o| + sin^ ii " a? + cos^^' • sin^ij • al 
 
 + 2 sin if • «! • cos J • a^ - sin^y • a| 
 sin^ ij • Oa + sin^ ij • a? + 2 sin ^ ■ Oi • cos j' • «» = sin^ J • a\ 
 
 sin2ii= - 
 
 sin^j • tta 
 
 , sin ^l = 
 
 a? + a| + 2 «! • tta' cos j 
 
 fflj • siny 
 ;^(^flSi + aj + 2 Oi • Oa ■ cos^)' 
 
 Graphically, the same problem is solved as follows : — On the two 
 axes respectively, take lengths to represent the angular vejooities 
 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 106, one of a pair of bevel wheels 
 may be a flat disc, or a concave cone. 
 
 139. Non-Circular Wheels are used to transmit 
 a variable velocity-ratio between a pair of parallel 
 axes. In fig, 46, let Ci, Cj, represent the axes of 
 such a pair of wheels ; Tj, T^, 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 TJi, Uj, another 
 pair of points, which touch each other at another 
 instant of the motion ; and let the four points, Tj, 
 To, Ui, Uj, 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 : — 
 
 Fig. 46. 
 
 = Ci Ca 
 
 arcTiUi = arcT2Ua 
 
 and those equations shew the geometrical relations which must 
 exist between a pair of rotating surfaces in order that they may 
 move in rolling contact round fixed axes. 
 
 Section 2. — Sliding Contact. 
 140. Skew-Bevel Wheels are employed to transmit an uniform 
 
 Fig. 47. 
 
 Fig. 48. 
 
78 THEOKY OF MECHANISM. 
 
 velocity-ratio between two axes whicli are neither parallel nor 
 
 intersecting. The pitch surface of a 
 skew-bevel wheel is a frustrum or 
 zone of a hyp&rholoid of revolution. 
 In fig. 47, a pair of large portions of 
 such hyperboloids are shewn, rotat- 
 ing about axes A B, C D. In fig. 48 
 are shewn a pair of narrow zones of 
 the same figures, such as are employed 
 in practice. 
 Fig. 49.. -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. 49, 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 velo- 
 cities of a pair Of points of contact have components along the line 
 of 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 olli- 
 
 quities of the generating line to the two axes, and its radii vectores, 
 or least perpendicular distances from these axes. 
 
 In fig. 49, let A B, D, be the two axes, and G K their common 
 perpendicular. 
 
 On any plane normal to the common perpendicular G- K A, draw 
 a & II A B, c (Z II C D, in which take, lengths in the following pro- 
 portions ; — 
 
 Oi : flSa : -.hp :hq; 
 
 complete the parallelogram hp e q, and draw its diagonal e hf; the 
 line of contact E H P 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 
 
PRINCIPLE OP SLIDING CONTACT. 
 
 7J 
 
 Pig. 50. 
 -Draw X "W Y parallel to 
 
 he.em:mh: -.GK-.GrH.-.KB.; 
 
 then the two segments thus found wUl be the least distances of 
 
 the Une of contact from the axes. 
 
 The first pitch surface is generated by the rotation of the line 
 
 E H P about the axis A B with the radius vector 011 = ^ ; the 
 
 second, by the rotation of the same line about the axis D with 
 
 the radius vector J£K = r^. 
 
 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. 50, re- 
 present the axis, G H i. A,G B, the 
 
 radius vector of the generating line, 
 
 and let the straight line E G JF 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 
 
 the asymptotes,t of the required hyper- 
 bola. To find any number of points 
 in that hyperbola, proceed as follows :- 
 
 G H, cutting G E in W, and make X7T= ^(GW+ XlP). 
 Then will Y be a point in the hyperbola. 
 
 141. 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 j and the principle of their comparative motion is, that the 
 components, along that perpendicular, of the velocities of any two 
 points tra/oersed hy 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 3. 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, ^ach 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. 51, let Ci, Os, represent the axes of rotation of the two 
 pieces; A,, Aj, two portions of their respective surfaces; and Tj, 
 Tj, a pair of points in those surfaces, which, at the instant under 
 consideration, are in contact with each other. Let Pj Pa be the 
 common perpendicular of the surfaces at the pair of points Tj, T^; 
 
 • The Hyperbola is the curve traced out by a point which moves in such 
 a manner that its distance from a given, fixed point (I), continually bears the 
 same ratio greater than unity to its distance from a given fixed lino (A B). 
 
 f An Asymptote is a straight line whose distance from a curve diminishes 
 as the curve extends away from the origin. 
 
80 
 
 THEORY OF MECHANISM. 
 
 that is, the line of action; and let Cj Pj, C2 Pj, be the common per- 
 pendiculars of the line of action and of the two axes respectively. 
 Then at the given instant, the components 
 along the line Pi Pj of the velocities of the 
 points Pi, Pj, are equal. Let ii, ij, be the 
 angles which that line makes with the direc- 
 tions of the axes respectively. Let a^, a^, he 
 the respective angular velocities of the moving 
 pieces; then 
 
 tti' 
 
 ■ Ci Pi • sin I'l : 
 consequently, 
 
 «2 ■ Cj Pj • sint'j; 
 
 a' _ Oi Pjjinii 
 
 Oi Ca Pj sin ii ' 
 
 ■(!•) 
 
 which is the principle stated above. 
 
 When the line of action is perpendicular in direction to both 
 axes, then sin «i = sin ij = 1 ; and Equation 1 becomes 
 
 «2. 
 
 C^i 
 
 .(lA.) 
 
 When the axes are parallel, ii = i^. Let I be the point where 
 the line of action cuts the plane of the two axes; then the triangles 
 Pi Ci I, P2 Ca I, are similar; so that Equation 1 a is equivalent to 
 the following : — 
 
 ICi 
 
 .(IB.) 
 
 «! I Ca 
 
 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; » the linear velocity of the sliding 
 piece; ^ the angle which its direction of motion makes with the 
 line of action ; then 
 
 « = » • C P • sini • sec j (2.) 
 
 When the line of action is perpendicular in direction to the axis 
 of the rotating piece, sin i = 1 ; and 
 
 v = a-G'P-aec j = a -10; (2a.) 
 
 where I C denotes the distance from the axis of the rotating piece 
 to the point where the line of action cuts a perpendicular from that 
 axis on the direction of motion of the shifting piece. 
 
PITCH AND NUMBER OP TEETH. 81 
 
 142. Teeth of Whjeels.— The most usual method of communi- 
 cating motion between a pair of wheels, or a wheel and a rack, 
 and the only method which, by preventing the possibility of the 
 rotation of one wheel unless accompanied by the other, insures the 
 preservation of a given velocity-ratio exactly, is by means of the 
 projections called teeth. 
 
 The pitch surface of a wheel is an ideal smooth surface, inter- 
 mediate between the crests of the teeth and the bottoms of the 
 spaces between them, which, by rolling contact with the pitch sur- 
 face of another wheel, would communicate the same velocity-ratio 
 that the teeth communicate by their sliding contact. In designing 
 wheels, the forms of the ideal pitch surfaces are first determined, 
 and from them are deduced'the forms of the teeth. 
 
 Wheels with cylindrical pitch surfaces are called spur wheels; 
 those with conical pitch surface.s, 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; 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 /ace; that which lies within the pitch 
 surface, the flank. 
 
 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 difierence between those radii, 
 the addendum,. 
 
 143. 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 t/iat the pitch should be an aliquot 
 part of the circumference. 
 
 In wheels which reciprocate without performing a complete 
 revolution, this condition is not necessary. Such wheels are called 
 sectors. 
 
82 THEORY OF MECHANISM. 
 
 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 
 shovM he the same in each. 
 
 III. Hence, in any pair of circular wheels which work together, 
 the numbers of teeth in a complete circumference are directly as 
 the radii, and inversely as the angular velocities. 
 
 IV. Hence also, in any pair of circular wheels which rotate 
 continuously for one revolution or more, the ratio of the numbers 
 of teeth, and its reciprocal, the angular velocity-ratio, must be 
 expressible in whole numbers. 
 
 V. Let n, N, be the respective numbers of teeth in a pair of 
 wheels, N being the greater. Let t, T, be a pair of teeth in the 
 smaller and larger wheel respectively, which at a particular instant 
 work together. It is required to find, first, how many pairs of 
 teeth must pass the line of contact of the pitch surfaces before t and 
 T work together again (let this number be called a); secondly, 
 with how many different teeth of the larger wheel the tooth t will 
 work at different times (let this number be called h) ; 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 w is a divisor of N, 
 
 a = N; 6 = —; c=l (1.) 
 
 Case 2. If the greatest common divisor of N and nhe d a, number 
 less than n, so that n = md, N = M c?, then 
 
 a = m'S = 'M.n = 'M.md; 5 = M; c = m (2.) 
 
 Case 3. If N and n be prime to each other, 
 
 ra = Nji; 6 = Nj c = n (3.) 
 
 It is considered desirable by millwrights, with a view to the 
 preservation of the uniformity of shape of the teeth of a pair of 
 wheels, that each given tooth in one wheel should work with as 
 many different teeth in the other wheel as possible. They, there- 
 fore, study to make the numbers of teeth in each pair of wheels 
 which work together such as to be either prime to each other, or to 
 have their greatest common divisor as small as is possible con- 
 sistently with the pui-poses 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 
 
A "train of wheelwork. 83 
 
 teeth which can be usually employed in pinions having teeth of the 
 three classes of figures named below, whose properties will be 
 explained in the sequel : — 
 
 I. Involute teeth, 25. 
 
 II. Epicycloidal teeth, 12. 
 
 III. Cylindrical teeth, or staves, 6. 
 
 144. Hunting Cog. — When the ratio of the angular velocties of 
 two wheels, being reducedj^ 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 th? wheels, millwrights and engine-makers 
 avoid the evil of frequent contact between the same pairs of teeth, 
 by giving one additional tooth, called a hunting cog, to the larger 
 of the two wheels. This expedient causes the velocity-ratio to be 
 not exactly but only approximately equal to that which was at 
 first contemplated; and therefore it cannot be used where the 
 exactness of certain velocity-ratios amongst the wheels is of impor- 
 tance as in clockwork. 
 
 145. A Train of Wheelwork 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 ai'e 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 answei's the purpose both 
 of receiving motion from a wheel on the preceding axis and giving 
 motion to a wheel on the following axis. Such a wheel is called 
 an idle wheel: it afiects the direction of rotation only, and not the 
 velocity-ratio. 
 
 Let the series of axes be distinguished by numbers 1, 2, 3, 
 
 &o m ; let the numbers of teeth in the driving wheels be 
 
 denoted by N's, each with the number of its axis affixed; thus, 
 
 Nj, Nj, &c N"m_iJ and let the numbers of teeth in the «?ri'uen 
 
 or following wheels be denoted by n's, each with the number of its 
 
 axis affixed; thus, n^, n^, &c n^. Then, the ratio of the 
 
 angular velocity a„ of the m"' axis to the angular velocity a^ of the 
 first axis is the product of the m—\ velocity-ratios of the succes- 
 sive elementary combinations, viz. : — 
 
 a„_ -E^-^.-&o N„_i . 
 
 «i n^-n^-kc. . . . . n,„ ' *- ■' 
 
 that is to say, the velocity-ratio of the last and first axes ia the 
 ratio of the product of the numbers of teeth in the drivers to the 
 product of the numTsers of teeth in the followers; and it is obvious 
 that so long as the same drivers and followers constitute the train, 
 
84 THEORY OP MECHANISM. 
 
 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, »i - 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 shewn 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. 
 
 Let j^ be the velocity-ratio required, reduced to its least terms, 
 
 and let B be greater than C. 
 
 If pf is not greater than 6, and C lies between the prescribed 
 
 minimum number of teeth (which may be called (), and its double 
 2 1, then one pair of wheels will answer the purpose, and B and C 
 will themselves be the numbsrs required. Should B and C be 
 inconveniently large, they are if possible to be resolved into factors, 
 and those factors, or if they are too small multiples of them, used 
 for the numbers of teeth. Should B or 0, or both, be at once 
 inconveniently large, and prime, then instead of the exact ratio 
 
 ^, some ratio approximating to that ratio, and capable of resolu- 
 tion into convenient factors, is to be found by the method of 
 continued fractions. See Mathematical Inteoduction, page % 
 Article 4. 
 
 Should p be greater than 6, the best number of elementary 
 
 combinations is found by dividing by 6 again and again till a 
 quotient is obtained less than unity, when the number of divisions 
 will be the required number of combinations, in—\. 
 
 Then, if possible, B and C themselves are to be resolved each 
 into TO - 1 factors, which factors, or multiples of them, shall be not 
 less than t, nor greater than 6i; or if B and C contain incon- 
 veniently large prime factors, an approximate velocity-ratio, found 
 
 by the method of continued fractions, is to be substituted for p^, as 
 
 before. "When the prime factors of either B or C are fewer in 
 
A. TRAIN OP 'WHEELWORK. S5 
 
 number than wi - 1, tlie required number of factors is to be made 
 up by inserting 1 as often as may be necessary. In multiplying 
 factors that are too small to serve for numbers of teeth, prime 
 numbers differing from those already amongst the factors are to be 
 preferred as multipliers; and in general, where two or more factors 
 require to be multiplied, different prime numbers should be used 
 for the different factors. 
 
 So far as the resultant velocity-ratio is concerned, the wd&r of 
 the drivers N, and of the followers n, is immatei-ial ; but to secure 
 equable wear of the teeth, as explained in Article 143, 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 ; and if the preceding rules have been 
 observed in the choice of multipliers, this will be insured by so 
 placing each driving wheel that it shall work with a following 
 wheel whose number of teeth does not contain any of the same 
 multipliers; for the original numbers B and C contain no common 
 factor except 1. 
 
 The following is an example of a case requiring the use of 
 additional multipliersj — Let the required velocity-ratio, in its least 
 terms, be 
 
 B 360 
 
 To get a quotient less than 1, this ratio must be divided by 6 
 three times, therefore m - 1 = 3. The prime factors of 360 are 
 2'2:2*3'3"5; these may be combined so as to make three 
 factors in various different ways; and the preference is to be given 
 to that which makes these factors least vinequal, viz., 5 • 8 • 9. 
 Hence, resolving numerator and denominator into three factors, 
 each, we have 
 
 B_ 5-8-9 
 
 C~l -1 •?• 
 
 It is next necessary to multiply the factors of the numerator and 
 denominator by a set of three multipliers. Suppose that the wheels, 
 to be used are of such a class that the smallest pinion has 12 teeth, 
 then those multipliers must be such that none of their products by 
 the existing factors shall be less than 12; and for reasons already- 
 given, it is advisable that they should be different prime numbers- 
 Take the prime numbers, 2, 13, 17 (2 being taken to multiply 7);. 
 then the numbers of teeth in the followers will be 
 
 13x1 = 13; 17x1 = 17; 2x7 = 14. 
 
 In distributing the multipliers amongst the factors of the num- 
 erator, let the smallest multiplier be combined with the largest 
 factor, and so on; then we have 
 
88 THEORY OF MECHANISM. 
 
 17x5 = 85; 13x8 = 104; 2x9 = 18. 
 
 Finally, in combining the drivers -with the followers, those 
 
 numbers are to be combined which have no common factor; the 
 
 result being the following train of wheels : — 
 
 85^18 104 _ 360 
 
 14*13' 17 " 7 ■ 
 
 146. Teeth of Spur- Wheels and Racks. General Principle.— The 
 figures of the teeth of wheels are regulated by the principle, that 
 tJie teeth of a pair of wheels shall give the same velocity-ratio hy their 
 sliding contact, which the ideal smooth pitch surfaces would give hy 
 their rolling contact. Let Bi, Bj, in fig. 51, 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 Oj, 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 141, Case 2, Equation 1 b). Hence the condition of 
 correct working for the teeth of wheels with parallel axes is, that 
 the line of action of the teeth shall at every instant traverse the line 
 of contact of tJie pitch surf aces ; 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 worlcs. 
 
 147. 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 T^ in the cross section of the acting surface 
 of a tooth (such as the line Aj Tj in fig. 51), and the corresponding 
 position of the pitch point I in the pitch line I Bj of the wheel to 
 which that tooth belongs, are so related, that the line I Ti which 
 joins them is normal to the outline of the tooth Aj Tj at the point 
 Tj. Now, this is the relation which exists between the tradng- 
 point Ti, and the instantaneous axis or line of contact I, in a rolling 
 curve of such a figure, that being rolled upon the pitch surface Bj, 
 its tracing-point Tj traces the outline of the tooth. (As to rolling 
 curves, see Articles 100, 101, 103, and 106). 
 
 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 expresse'd by the equation 
 
 TT, = n\; (1.) 
 
 and this condition is fulfilled if the outlives of the two teeth be traced 
 by the motion of the same tracing-point, in rolling the same rolling 
 curve on the same side of the pitch surfaces of the respective wheels. 
 
THE SLIDING OF A PAIR OF TEETH ON EACH OTHER. 87 
 
 Thejlank of a tootli is traced ■while the rolling curve rolls inside 
 of the pitch line ; the face, while it rolls outside. Hence it is 
 evident that the flcmks 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. 51, supposing 
 the motion to be from Pj towards Pa ; 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 
 irito two parts, called the path of approach and the path of recess; 
 and the lengths of those psPths must be so adjusted, that two pairs 
 of teeth at least shall be in action at each instant. 
 
 It is evidently necessary that the surfaees of contact of a pair 
 of teeth should either be both convex, or that if one is convex 
 and the other concave, the concave surface should have the flatter 
 curvature. 
 
 148. 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 Cj to rotate backwards round Ci with the angular 
 velocity Oj, and the wheel 2 to rotate round Cj as before with the 
 angular velocity a^ relatively to the line of centres Cj Cj, 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 102; that is, a rotation 
 about the instantaneous axis I with the angular velocity flj + «j. 
 Hence the velocity of sliding is that due to this rotation about I, 
 with the radius IT-r; that is to say, its value is 
 
 r(ai + Oj); (1.) 
 
 so that it is greater, the farther the point of contact is from the 
 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 roll- 
 ing contact. 
 
 The roots of the teeth slide towards each other during the ap- 
 jiroach, and from each other during the recess. To find the amount 
 or total distance through which the sliding takes place, let i, be the 
 time occupied by the approach, and t^ that occupied by the recess; 
 then the distanpe of sliding is 
 
 s= / V(ai+fflo)<^i+ / 'rioi + a^dt; (2.) 
 
 •'0 •' 
 
 or in another form, if d i denote an element of the change of angu- 
 
88 
 
 THEORY OF MECHANISM. 
 
 lar position of one wheel relatively to tlie other, i^ the amount of 
 that change during the approach, and % during the recess, then 
 (ai + tti) dt = di ; and 
 
 ! - l^r d i + I 
 
 r di . 
 
 .(3.) 
 
 ^rdi + 
 ■'0 
 
 149. The Arc of Contact on the Pitch Lines 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 eqtial to the length of the required arc of contact. 
 It is usual to make the arcs of approach and recess equal. 
 
 150. 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. 
 
 151. Involute Teeth for Circular Wheels.— In fig. 52, let Cj, Cj, 
 be the centres of two circular wheels, whose pitch circles are Bj, Bj. 
 Through the pitch point I draw the intended line of action Pj, Pj, 
 making the angle C I P = ^ with the line of centres. From Cj, Cj, draw 
 
 •(!■) 
 
 CiPi = ICi-sin 6, 
 
 perpendicular to Pj P,, with which two perpendiculars as radii, 
 
 describe circles (called base circles) D^, Dj. 
 
 Suppose the base circles to be a pair of 
 circular pulleys, connected by means of a 
 cord whose course from pulley to pulley is 
 Pi I Pj. 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 be fixed to the cord, so as to be carried 
 along the path of contact Pj I Pj. That 
 point will trace, on a plane rotating along 
 with the wheel 1, part of the involute of 
 the base circle Dj, and on a plane rotating 
 along with the wheel 2, part of the involute 
 Fig. 52. of the base circle D^, and the two curves so 
 
EPICYCLOIDAL TEETH. 89 
 
 traced will always touch each other in the required point of contact 
 T, and will therefore fulfil the condition required by Article 14G. 
 
 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 Pj, is less 
 than the pitch in the ratio sin 6 : 1, for the former is proportional 
 to r • sin 6, and the latter to r • 6, and consequently that if dis- 
 tances not less than the pitch x sin 6 be marked off either way from 
 I towards Pj and Pa respectively, as the extremities of the path of 
 contact, and if the addendum circles be described through the 
 points so found, there will al\jjpys 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 6 which is usual in practice, viz., 
 75^°, the addendum is about -^^ of the pitch. 
 
 The teeth of a rack, to work correctly with wheels having invo- 
 lute teeth, should have plane surfaces, perpendicular to the line of 
 connection, and consequently making, with the direction of motion 
 of the rack, angles equal to the before-mentioned angle 6. 
 
 152. 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 the teetli, which must not be less than p • sin 6, 
 cannot be greater than the distance along the line of action from 
 
 IP 
 
 the pitch point to the base circle, !? = »■• cos 6. Then r = ^ 
 
 and substituting for I P its least possible value p ■ sin 6, hence the 
 leobst radius is 
 
 r=p tan 6; (1.) 
 
 which, for ^ = 75J°, gives for the radius r = 3-867 p, and for the 
 circumference of the pitch circle, p x 3-867 x 2 s- = 24-3 p; to 
 which the next greater integer multiple of ^is 25 jo; and therefore 
 twenty-five, as formerly stated, in Article 143, is the least number 
 of involute teeth to be employed in a pinion. 
 
 153. Epicyoloidal 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 J 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 
 
90 
 
 THEORY OF MECHANISM. 
 
 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 ivr 
 curved or under-cut, wbich 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. 53, let B be part of the pitch circle of a wheel, C C the 
 
 line of centres, I the pitch-point, 
 E. the internal, and E,' the equal 
 external describing circles, so placed 
 as to touch the pitch circle and each 
 other at I ; let D I D' be the path 
 of contact, consisting of the path of 
 - B 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 angle is tf = ZCID = Z 
 C I D'. 
 
 It appears from experience that 
 the least value of ^ should be about 60°; therefore the arcs D 1 = 
 I D' should each be one-sixth of a circumference ; 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 been already 
 stated in Article 143. 
 
 154. Teeth of Wheel and Trundle.— A trundle, as in fig. 54, 
 has cylindrical pins called staves for teeth. The face of the teeth 
 
 Kg. 53. 
 
 Kar. 55. 
 
 of a wheel suitable for driving it, in outside gearing, are described 
 by first tracing external epicycloids by rolling the pitch circle Bj of 
 
DIMENSIONS OF TEETH. 91 
 
 the trundle on the pitch circle Bj of the driving wheel, -witli the 
 centre of a stave for a tracing point, as shewn 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 gea/ring, the outlines of the teeth of 
 the wheel should be curves parallel to internal epicycloids. A 
 peculiar case of this is represented in fig. 55, 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 Billing 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. 
 
 155. Dimensions of Teeth. — Toothed wheels being in general 
 intended to rotate either way, the hacks of the teeth are made 
 similar to the fronts. The space between two teeth, measured on 
 the pitch circle, is made about one-fifth part wider than the thick- 
 ness of the tooth on the pitch circle ; that is to say, 
 
 5 
 thickness of tooth = =-t pitch, 
 
 width of space = — pitch. 
 
 The difierence of ^^ of the pitch is called the hack-lash. 
 
 The clearance allowed between the points of teeth and the bot- 
 toms 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 of 
 strength; and the breadth is so adjusted, that when multiplied by 
 pitch, the product shall contain one square inch for each 160 lbs. of 
 force transmitted by the teeth. 
 
 156. 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 
 
92 THEORY OF ilECHANISlI. 
 
 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 0, fig. 56, 
 be the apex, and O C the axis of the pitch • 
 cone of a bevel- wheel; and let the largest 
 pitch circle be that whose radius is C B. 
 Perpendicular to O B draw B A cutting 
 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 
 ■^'S' ^^- sufficiently near to a zone of the sphere 
 
 described about O with the radius O B, to be used in its stead. On 
 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. 
 
 167. The Teeth of Non-Circular Wheels are described by rolling 
 circles or other curves on the pitch surfaces, like the teeth of cir- 
 cular wheels; and when they are small compared with the wheels 
 to which they belong, each tooth is nearly similar to the tooth of a 
 circular wheel, having the same radius of curvature with the pitch 
 surface of the actual wheel at the point where the tooth is situated. 
 
 158. 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 141, 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. 
 
 159. 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 femde, 
 or external and internal; a short internal screw is called a nvi; 
 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 
 
NORMAL AND CIRCULAR PITCH. 93 
 
 helical guide, has already been demonstrated in Article 96, 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. 
 
 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 yroni whom 
 the advance takes place. Kg. 57 re- ; 
 presents a right-handed screw, and fig. p\ 
 58 a left-handed screw. I 
 
 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 
 helical motion of that screw, as ex- 
 plained in Article 96, and is the dis- ^'S- ^7- ^ig 58. 
 tance (marked^ in figs. 57 and 58) measured parallel to the axis 
 of the screw, between the corresponding points in two consecutive 
 turns of the same thread. 
 
 In a screw of two or more threads, the distance measured parallel 
 to the axis, between the corresponding points in two adjacent 
 threads, may be called the divided pitch. 
 
 160. Normal and Circular Pitch. — When the pitch of a screw is 
 not otherwise specified, it is always understood to be measured 
 parallel to the axis. But it is sometimes convenient for particular 
 purposes to measure it in other directions; and for that purpose a 
 cylindrical pitch surface is to be conceived as described about the 
 axis of the screw, intermediate between the crests of the threads 
 and the bottoms of the grooves between them. 
 
 If a helix be now described upon the pitch cylinder, so as to 
 cross each turn of each thread at right angles, the distance between 
 two corresponding points on two successive turns of the same 
 thread, measured along this normal helix, may be called the normal 
 pitch; and when the screw has more than one thread, the normal 
 pitch from thread to thread may be called the normal divided pitch. 
 
 The distance from thread to thread measured on a circle described 
 on the pitch cylinder, and called the pitch circle, may be called the 
 circular pitch ; for a screw of one thread it is one circumference; 
 for a screw of n threads 
 
 one circumference 
 
 n 
 
 The following set of formulae shew the relations amongst the differ- 
 
94 THEORY OF MECHANISM. 
 
 ent modes of measuring the pitch of a screw. The piteh, properly 
 speaking, as originally defined, is distinguished as the axial jnMi, . 
 and is the same for all parts of the same screw : the normal and 
 circular pitch depend on the radius of the pitch cylinder. 
 
 Let r denote the radius of the pitch cylinder; 
 
 n, the number of threads; 
 
 i, the obliquity of the threads to the pitch circles, and of the 
 normal helix to the axis; 
 
 V " I +h ■ 1 / pi*"^ ' 
 
 —f =p^C I divided pitch; 
 
 P_ > the normal < 5- • / i •. i 
 
 —2 = OT I [ divided pitch; 
 
 j>a the circular pitch ; 
 Then 
 
 . 3 vr 
 Pc=pa'oota,m=pn' coseGi= ; ■ 
 
 . 2 5r ?• • tan i 
 Pa =Vn ■seci=p,- tan i = ; 
 
 . . . 2 ^ r • sin i 
 Pn=Pc- sm I =p, • cos * = ^ . 
 
 Fig. 59 will make these formulae clear, in which the several 
 lines are lettered to represent the pitches : the hypoteause 
 of the larger triangle is the linear development on the plane 
 of the paper of one coil of the screw which, it will be remarked, 
 = JiPa+Pe^)') Pn the normal pitch is normal to this: it is also 
 evident from the figure that with a constant axial pitch, the normal 
 and radial or circumferential pitch, as well as the angle of obliquity 
 of the threads to the pitch cylinders, vary with the radii of those 
 cylinders. 
 
 161. 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 diflference of their obliquities. 
 
 II. The normal pitch, for a screw of one thread, and the normal 
 
THE RELATIVE SLIDING OF A PAIR OF SCREWS. 
 
 95 
 
 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. 
 
 162. 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 combination is to produce a change of angular velocity in a 
 ratio greater than can be obtained by any single pair of ordinary 
 wheels, one of the screws is commonly wheel-like, being of large 
 diameter and many-threaded, while the other is short and of few 
 threads; and the angular velocities are inversely as the number 
 of threads. • 
 
 Fig. 60. 
 
 Kg. 61. 
 
 Kg. 60, represents a side view of this combination, and fig. 61 
 a cross section at right angles to the axis of the smaller screw. It 
 has been shewn by Prof. 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 lai-ger and smaller screw 
 should be those of the teeth of a wheel and rack respectively : Bi Bj, 
 in fig. 60 for example, being the pitch circle of the wheel, and 
 Ba Ba the pitch line of the rack. 
 
 The periphery and teeth of the wheel are usually hollowed to 
 fit the screw, as shewn at T, fig. 61. 
 
 To make the teeth or threads of a pair of screws fit correctly apd 
 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. 
 
 163. The Relative Sliding of a Pair of Screws at their point of 
 contact is found thus : — Let Ti, r^, be the radii of their pitch cylin- 
 ders, and ii, i^, the obliquities of their threads to their pitch circles, 
 one of which is to be considered as negative if the screws are con- 
 trary-handed. Let u be the common component of the velocities 
 of a pair of points of contact along a line touching the pitch sur- 
 
93 
 
 THEORY OP MECHANISM. 
 
 faces and perpendicular to' the threads at the pitch point, and v 
 the velocity of sliding of the threads over each other, where v may 
 he considered to be made up of the algebraical sura of two quantities, 
 Vi and v^, which act perpendicularly to u, and whose values are 
 ■Ui = Oi ri cos ii, and v^ = % r^ cos 4 the sum or difference being taken 
 as the screws are similar or cont»ary-handed. Then 
 
 so that 
 
 and 
 
 V, = flSj ri ■ sin «j = a^ r^ ■ sin ij; 
 
 «! = 
 
 a. 
 
 ri • sin tj " rj • sm i^ 
 
 ■(I.) 
 
 « = Oil rj • cos ii + flij ''2 ■ cos i^ = u (cotan ^l + cotan 4) (2.) 
 
 "When the screws are contrary-handed, the difference instead of the 
 sum of the terms in Equation 2 is to be taken. 
 
 164. Oldham's Coupling. — A coupling is a mode of connecting a 
 pair of shafts so that they shall rotate in 
 the same direction, with the same mean 
 angular velocity. If the axes of the shafts 
 are in the same straight line, the coupling 
 consists in so connecting their contiguous 
 ends that they shall rotate as one piece; 
 but if the axes are not in the same straight 
 line, combinations of mechanism are re- 
 quired. A coupling for parallel shafts 
 Ficr 62. which acts by sliding contact was invented 
 
 °' ' by Oldham, and is represented in fig. 62. 
 
 are the axes of the two parallel shafts; Dj, Dj, two cross- 
 heads, facing each other, fixed on the ends of the two shafts 
 i-espectively ; Ej, Ej, a bar, sliding in a diametral groove in the 
 face of Di; Ej, Ej, a bar, sliding in a diametral groove in the 
 face of Dj; 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 hne 
 of 'centres Cj, Cj, as a diameter, twice for each turn of the shafts 
 and cross ; the instantaneous axis of rotation of the cross, at any 
 instant, is at I, the point in the circle Ci Cj, diametrically oppo- 
 site 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 practicability 
 or permanency of their exact continuity. 
 
 '\,j, O2, 
 
PRINCIPLE OF CONNECTION BY BANDS. 97 
 
 Section 3. — Connection by Bands. 
 
 165. 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. Bdts, 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 puUey, 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 
 neai'ly 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. 
 
 166. 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 129 being applied to this case, 
 leads to the following consequences : — 
 
 I. For a pair of rotating pieces, let rj, r^, be the perpendiculars 
 let fall from their axes on the centre line of the band, osi, a^, their 
 singular velocities, and ij, 4, the angles which the centre line of the 
 band makes with the two axes respectively. Then the longitudinal 
 velocity of the band, that is, its component velocity in the direction 
 of its own centre line, is 
 
 u = ri Oi sin ii = r^a2 sin i^; (1.) 
 
 whence the angular velocity-ratio is 
 
 CTa^ riSin% .g. 
 
 Oj ra sin ij ^ '' 
 
 When the axes are parallel (which is almost always the case), 4 = 4j 
 and 
 
 ^ = ^ (3.) 
 
yo THEORY OF MECHANISM. 
 
 The same equation holds when both axes, whether parallel or not, 
 are perpendicular in direction to that part of the band which 
 transmits the motion; for then sin % = sin 4 = 1- 
 
 II. F<yr a rotaUng piece cmd a sliding piece, let r be the perpen- 
 cular from the axis of the rotating piepe on the centre line of the 
 band, a the angular velocity, i the angle between the directions of 
 the band and axis, u the longitudinal velocity of the band, j the 
 angle between the direction of the centre line of the band and that 
 of the motion of the sliding piece, and v the velocity of the sliding 
 piece; then 
 
 u = ra sin i = v cosj; (4.) 
 
 for r sin * is the projection on the plane of motion of r, and u the 
 longitudinal velocity of the band must necessarily be equal to 
 V cos j, the longitudinal velocity of the sliding piece owing to the 
 rigidity of the band; and 
 
 cos_7 
 
 .(5.) 
 
 When the centre line of the band is parallel to the direction of 
 motion of the sliding piece, and perpendicular to the direction of 
 the axis of the rotating piece, sin i (90°) = cos j (0°) = 1, and 
 
 v = u = ra (6.) 
 
 167. 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. 
 
 168. Circular Pulleys and Drums are used to commuicate a 
 
 constant velocity-ratio. In each of them, the length denoted by f 
 in the equations of Article 166 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 puUeys, as in fig. 63, 
 reverses the direction of rotation; an open belt, as in fig. 64, pre- 
 serves that direction. 
 
THE LENGTH OP AN ENDLESS BELT. 99 
 
 169. The length of an E ndles s Bel t, con necting a pair of puUeys 
 ■whose effective radii are CiTi'=ri, C^T^^r^, with parallel axes 
 ■whose distance apart is Cj Ca = c, is given by formulae founded on 
 Equation 1 of Article 113, viz., L = 2's + 2*ri. Each of the two 
 equal straight parts of the belt is evidently of the length 
 
 8 = Ti Ta = Jc^ — (r, + r,)^ iov a crossed belt;^ 
 
 y (1.) 
 
 s = Ti Ta = Jc^ - (r^ _ ra)2 for an open belt; j 
 
 Ti being the greater radius, and r^ the less. Let i^ be the arc to 
 Radius unity of the greater pulley, and i^ that of the less pulley, 
 ■with ■which the belt is in contact; then for a crossed belt 
 
 «• + 2 arc • sm — i] ; 
 
 (2.) 
 
 for the angle Vj Cj Wj at the centre is double of the angle at 
 the circumference Cj Tj Wj, and this is equal to the angle 
 Si Ca Qi as they both differ from a right angle by the same 
 angle Tj Oi Vj; and for an open belt, 
 
 / « . '•i - »'a\ • / n ■ »*l - ''aN 
 
 ij = I w + 2 arc • sm — -) ; *a = I «■ - 2 arc • sm -j ; 
 
 and the introduction of those values into Equation 1 of Article 113 
 gives the following results : — 
 For a crossed belt 
 
 L = 2 ^c2-(ri + ra)2+ (r^ + ra) • (x + 2 arc -sin -^J-^IJ; 
 
 and if similar reasoning be applied, it may be she^wn that 
 for an open belt, 
 
 L = 2 J<y^ - (ri - j-a)^ + w(»*i + r^) + 2{ri - r^ • arc • sin • — 
 
 r,-}-. 
 
 (3.) 
 
 As the last of these equations would be troublesome to employ in 
 a practical application to be mentioned in the next Article, an 
 approximation to it, suflSciently close for practical purposes, is 
 obtained by considering, that if rj - j-a is small compared with c, 
 
 ^e2-(r^-r^)2 = e- ^'''~J°^^ nearly, and arc • sin • !iZii = ?iZ^» 
 
 nearly; whence, for an open belt, 
 
 L nearly = 2c + ;r(»-i + r,) + hzllt (3a.) 
 
100 THEORY OF MECHANISM. 
 
 170. Speed-Cones (figs. 65, 66, 67, 68) are a contrivance for 
 
 Fig. 65. 
 
 Fig. 66. 
 
 Fig. 67. 
 
 Fig. 68. 
 
 varying and adjusting the velocity-ratio communicated between a 
 pair of parallel shafts by means of a belt, and may be either conti- 
 nuous cones or conoids, as in figs. 65, 66, -whose velocity-ratio can 
 be varied gradually while they are in motion by shifting the belt; 
 or sets of pulleys whose radii vary by steps, as in figs. 67, 68, in 
 which case the velocity-ratio can be changed by shifting the belt 
 from one pair of pulleys to another. 
 
 In order that the belt may be equally tight in every possible 
 position on a pair of speed-cones, the quantity L in the equations 
 of Article 169 must be constant. 
 
 For a crossed belt, as in figs. 65 and 66, L depends solely on 
 c and on rj -t- rj. Now c is constant, because the axes are parallel, 
 therefore the sum of the radii of the pitch circles connected in every 
 position of the belt is to be constant. That condition is fulfilled 
 by a pair of continuous cones generated by the revolution of two 
 straight lines inclined opposite ways to their respective axes at 
 equal angles, and by a set of pairs of pulleys in which the sum of 
 the radii is the same for each pair. 
 
 For an open belt, the following practical rule is deduced from the 
 approximate Equation 3a of Article 169 : — 
 
 Let the speed-cones be equal and similar conoids, as in fig. 66, 
 but with their large and small ends turned opposite ways. Let rj 
 be the radius of the large end of each, r^ that of the small end, r, 
 that of the middle ; and let y be the sagitta, measured perpendi- 
 cular to the axis, of the arc by whose revolution each of the conoids 
 is generated, or, in other, words, the bulging of the conoids in the 
 middle of their length; then 
 
 y 
 
 = r '^i + '^^A^^-^^f 
 
 2s-c 
 
 •(!•) 
 
 where the second value is obtained from the first by considering 
 that in Equation 3a, 2 ^ j-q = t (r^ + r^) + yIzIA- 2 n- = 6-2832; but 
 6 may be used in most practical cases without sensible error. 
 
COUPLING OP PARALLEL AXES. 101 
 
 The radii at the middle and ends being thus determined, make 
 the generating curve an arc either of a circle or of a parabola. 
 
 For a pair of stepped cones, as in fig. 68, let a series of differ- 
 ences of the radii, or values of r^—r^, be assumed; then for each 
 pair of pulleys, the sum of the radii is to be computed from the 
 difference by the formula 
 
 r, + r, = 2r,-(^i^'; (2.) 
 
 2 j-o being that sum when the radii are equal. 
 
 Section 4. — Linkwork. 
 
 171. Definitions. — The pieces which are connected by linkwork, 
 if they rotate or oscillate, are usually called cranks, beams, and 
 levers. The link by which they are connected is a rigid bar, which 
 may be straight or of any other figure; the straight figure being 
 the most favourable to strength, is used when there is no special 
 reason to the contrary. The link is known by various names under 
 various circumstances, such as coupling rod, connecting rod, crcmk 
 rod, eccentric rod, &c. It is attached to the pieces which it con- 
 nects by two pins, about which it is free to turn. The effect of the 
 link is to maintain the distance between the centres of those pins 
 invariable; hence the line joining the centres of the pins is the line 
 of connection; and those centres may be called the connected points. 
 In a turning piece, the perpendicular let fall from its connected 
 point upon its axis of rotation is the a/rm or crank arm. 
 
 172. Principles of Connection. — The whole of the equations 
 already given in Article 166 for bands, are applicable to linkwork. 
 The axes of rotation of a pair of turning pieces connected by a link 
 are almost always parallel, and perpendicular to the line of connec- 
 tion; in which case the angular velocity-ratio at any instant is the 
 reciprocal of the ratio of the common perpendiculars let fall from 
 the line of connection upon the respective axes of rotation (Article 
 166, Equation 3)_. 
 
 173. Dead Points. — If at any instant the direction of one of the 
 crank arms coincides with the line of connection, the common 
 perpendicular of the line of connection and the axis of that crank 
 arm vanishes, and the directional relation of the motions becomes 
 indeterminate. The position of the connected point of the crank 
 arm in question at such an instant is called a dead point. The 
 velocity of the other connected point at such an instant is null, 
 unless 'it also reaches a dead point at the same instant, so that the 
 line of connection is in the plane of the two axes of rotation, in 
 which case the velocity-ratio is indeterminate. 
 
 174. Coupling of Parallel Axes. — The only case in which an uni- 
 
103 THEOEy OP MECHANISM. 
 
 form angular velocity-ratio (being ttat of equality) is communicated 
 by linkwork, is that in which two or more parallel shafts (such as 
 those of the driving wheels of a locomotive engine) are made to 
 rotate with constantly equal angular velocities, by having equal 
 cranks, which are maintained parallel by a coupling rod of such a 
 length that the line of connection is equal to the distance between 
 the axes. The cranks pass their dead points simultaneously. To 
 obviate the unsteadiness of motion which this tends to cause, the 
 shafts are provided with a second set of cranks at right angles to 
 the first, connected by means of a similar coupling rod, so that one 
 set of cranks pass their dead points at the instant when the other 
 set are farthest from theirs. 
 
 175. The Comparative Motion of the Connected Points in a piece 
 of linkwork at a given instant is capable of determination by the 
 method explained in Article 98; that is, by finding the instantan- 
 eous axis of the link ; for the two connected points move in the 
 same manner with two points in the link, considered as a rigid 
 body. 
 
 If a connected, point belongs to a turning piece, the direction of 
 its motion at a given instant is perpendicular to the plane contain- 
 ing the axis and crank arm of the piece. If a connected point 
 belongs to a shifting piece, the direction of its motion at any 
 instant is given, and a plane can be drawn perpendicular to that 
 direction. 
 
 The line of intersection of the planes perpendicular to the paths 
 of the two connected points at a given instant, is the instantcme(Ms 
 axis of the link at that instant j and the velocities of the connected 
 points are directly as their distcmces from that aods. 
 
 In drawing on a plane surface, the two planes perpendicular to 
 the paths of the connected points are represented by two lines 
 (being their sections by a plane normal to them), and the instanta- 
 neous axis by a point; and should the length of the two lines 
 render it impracticable to produce them nntil they actually inter- 
 sect, the velocity-ratio of the connected points may be found by 
 the principle, that it is equal to the ratio of the segments which a 
 line parallel to the Hne of connection cuts off from any two lines 
 drawn from a given point, perpendicular respectively to the paths 
 of the connected points. 
 
 Example I. Two Rotating Pieces with Parallel Axes (fig. 69) — 
 Let Ci, Ca, be the parallel axes of the pieces; Tj, Ts, their con- 
 nected points; Ci Tj, CTT,, their crank arms ; Tj Tj, the link. At 
 a given instant, let vi be the velocity of %; v^ that of T^. 
 
 To find the ratio of those velocities, produce Cj Tj, C^ Tj, till 
 they intersect in K ; K is the instantaneous axis of the link or 
 connecting rod, and the velocity ratio is 
 
 v.:v ::irTt;^T; (1.) 
 
AN ECCENTRIC. 
 
 103 
 
 Should K be inconveniently far off, draw any triangle with its 
 sides respectively parallel to Ci Tj, Ca Tj, and Ti T,; the ratio of 
 the two sides first mentioned will be the velocity-ratio required. 
 For example, draw Cj A parallel to Oi T„ cutting Ti T^ in A, then 
 
 Fig. 70. 
 
 Kg. 69. 
 
 Exawjple II. Rotating piece and sliding piece (fig. 70). — Let 
 Cj be the axis of a rotating piece, and Ti E, the straight line along 
 which a sliding piece moves. Let Tj, Tj, be the connected points, 
 Ca Tj the crank arm of the rotating piece, and Tj Ta the link or 
 connecting rod. The point Tj, %, and the line Ti R, are supposed 
 to be in one plane, perpendicular to the axis C. Draw Ti K per- 
 pendicular to T, R, intersecting Ca Ta in K j K is the instantaneous 
 axis of the link; and 
 
 ^•, : «j : : K Ti : K T, 
 
 Or otherwise draw from a point C,, Ca A perpendicular to T, R the 
 "direction of motion of the sliding piece, Ca Ta perpendicular to the 
 direction of motion of the rotating piece, then the line Ti Ta, or a 
 line parallel thereto cuts off the segments Ca A, Cj Tj, or segments 
 , proportional thereto, and the velocity-ratio of the rotating piece 
 to the sliding piece is as C, Tj to Ca A. 
 
 176. An Eccentric (fig. 71) being a circular 
 shaft, with whose axis its centre does not coin- 
 cide, and used to give a reciprocating motion to 
 a rod, is equivalent to a crank whose con- 
 nected point is T, the centre of the eccentric 
 disc, and whose crank arm is C T, the distance 
 of that point from the axis of the shaft, called 
 the eccentricity. 
 
 on a 
 
104 
 
 THEORY OP MECHANISM. 
 
 177. The Length of Stroke of a point in a reciprocating piece is 
 the distance between the two ends of the path in which that point 
 moves. When it is connected by a link with 
 a point in a continuously rotating piece, the 
 ends of the stroke of the reciprocating point 
 correspond with the dead points of the continu- 
 ously revolving piece (Article 173). 
 
 Let S = B C be the length of stroke of 
 the reciprocating piece, L = E C = D B the 
 length of the line of connection, and E. = A E = 
 A D the crank arm of the continuously turning 
 piece. Then if the two ends of the stroke be 
 in one straight line with the axis of the 
 crank, 
 
 S = 2Ej (1.) 
 
 and if their ends be not in one straight line 
 with that axis, then S, L - R, and L + E, are 
 the three sides of a triangle, having the angle 
 opposite S at that axis; so that if ) be the 
 supplement of the arc between the dead points, 
 D and E, 
 
 S2 = (L-E)2 + (L + E)2-2(L-E) 
 (L + R) cos « ; "4 
 
 2 {Jj + E2) - 2 (L2 - R») cos 6 J ' ,„ . 
 2L2 + 2E2 S2 r-W 
 
 Kg. 72. 
 
 S2 = 
 
 cos 6 = '- 
 
 2(L2-E2) 
 
 178. Hooke's Universal Joint (fig. 73) is a contrivance for 
 coupling shafts whose axes intersect each other in a point. 
 
 Let O be the point of intersection of the axes C,, Cj, and i 
 
 their angle of inclination to each 
 other. The pair of shafts C„ Cj, 
 terminate in a pair of forks, Fj, Fj, 
 
 in bearings at the extremities of 
 
 which turn the gudgeons at 'the 
 ends of the arms of a rectangular 
 cross, having its centre at 0. This 
 cross is the link; the connected 
 points are the centres of the bear- 
 ings Fi, Fj. At each instant each 
 of those points moves at right angles 
 to the central plane of its shaft and fork, therefore the line of 
 intersection of the central planes of the two forks, at any instant, 
 is the instantaneous axis of the cross, and the velocity-ratio of the 
 
 Kg. 73. 
 
DEFINITIONS. 105 
 
 points F„ Fj (which, as the forks are equal, is also the angular 
 velocity-ratio of the shafts), is equal to the ratio of the distances of 
 those points from that instantaneous axis. The mean value of that 
 velocity-ratio is that of equality; for each successive quarter turn 
 is made by both shafts in the same time; but its actual value 
 fluctuates between the limits, 
 
 d^ r \ '\ 
 
 '^' = : — ?rw^o — ^ = ■ when Fi is in the plane ! 
 
 «! r • sm (90° - 1) cos t ' 
 
 of the axis; l (1.) 
 
 — = cos i when Fj is in th^J; plane. 
 
 179. The Double Hooka's Joint (fig. 74) is used to obviate the 
 vibratory and unsteady motion caused by the fluctuation of the 
 velocity-ratio indicated in the equa- o/x ,„ 
 tion of Article 178. Between the "^^ /if* 
 two shafts to be connected, Cj, C,, ^'^-"^ <!**^-5s^^ 
 there is introduced a short interme- /( ^g^^*— ^§»'U 
 diate shaft Cj, making equal angles ^j^^^^^'^^t^^^iij) 
 with Gj. and C3, connected with each ^ . *» 
 
 of them by a Hooke's joint, and ■'^^S- '^^■ 
 
 having both its own forks in the same plane. 
 
 By this arrangement the angular velocities of the first and third 
 shafts are equal to each other at every instant. 
 
 180. A GUck, being a reciprocating bar, acting upon a rachefc 
 wheel or rack, which it pushes or pulls through a certain arc at 
 each forward stroke, and leaves at rest at each backward stroke, is 
 an example of intermittent linkwork. During the forward stroke^ 
 the action of the click is governed by the principles of linkwork ; 
 during the backward stroke, that action ceases. A. caich or 'pall, 
 turning on a fixed axis, prasrents the ratchet wheel or rack from 
 
 Section 5. — Eeduplication op Coeds. 
 
 181. Definitions — The combination of pieces connected by the 
 several plies of a cord or rope consists of a pair of cases or frames 
 called blocks, each containing one or more pulleys called sheaves. 
 One of the blocks called the fall-hloch, Bi, is fixed; the other, or 
 rurming-hlock, Bj, is movable to or from the fall-block, with which 
 
IOC 
 
 THEORY OF MECHANISM. 
 
 it is connected 
 
 Fig. 75. 
 
 by means of a rope of whicli one end is attached 
 either to the fall-block or to 
 the running-block, -while the 
 other end, Ti, called the faU, 
 or tacUe-foM, is free; while 
 the intermediate portion o£ 
 the rope passes alternately 
 round the pulleys in the fall- 
 block and running-block. The 
 whole combination is called a 
 tackle or pwrchase. 
 
 182. The Velocity-Ratio chief- 
 ly considered in a tackle is that 
 between the velocities of the 
 running-block, u, and of the 
 tackle-faU, v. That ratio is 
 given by Equation 3 of Article 
 113 (which see), viz.: — 
 
 Fig. 75a. 
 
 v = nu; (1.) 
 
 where n is the number of plies of rope by which the running-block 
 is connected with the fall-block. Thus, in fig. 75 n = 7 ; and in 
 fig. 75am = 6. 
 
 182a. The Velocity of any Ply of the rope is found in the follow- 
 ing manner : — 
 
 I. For a ply on the side of the fall-block next the tackle-fall, 
 such as 2, 4, 6, fig. 75, and 3, 5, fig. 75a, it is, to be considered 
 what would be the velocity of that ply if it were itself the t&ckle- 
 fall. Let that velocity be denoted by v', and let n' be the number 
 of plies between the ply in question and the point of attachment by 
 which the first ply (marked 1 in the figures) is fixed to one or 
 other block. Then 
 
 V' = n' u. 
 
 •(!•) 
 
 II. For a ply on the side of the fall-block farthest from the 
 tackle-fall, the velocity is equal and contrary to that of the next 
 succeeding ply, with which it is directly connected over one of the 
 sheaves of the fall-block. 
 
 III. If the first ply, as in fig. 75a, is attached to the fall-block, 
 its velocity is nothing; if to the running-block, its velocity is equal 
 to that of the block. 
 
 183. White's Tackle. — The sheaves in a block are usually made 
 all of the same diameter, and turn on a fixed pin; and they have, 
 consequently, different angular velocities. But by making the 
 diameter of each sheaf proportional to the velocity, relatively to the 
 
CLASSIFICATION OF THE MECHANICAL POWERS. 107 
 
 bhck, of the ply of rope -wliich it is to carry, the angular velocities 
 of the sheaves in one block may be rendered equal, so that the 
 sheaves may be made all in one piece, and may have journals 
 turning in fixed bearings. This is called White's Tackle, from the 
 inventor, and is represented in figs. 75 and 75a. 
 
 Section 6. — Compaeative Motion in the " Mechanical 
 Powers." 
 
 184. Classification of the Mechanical Powers. — "Mechanical 
 Powers" is a name given to certain simple or elementary machines, 
 all of which, with the sin^ exception of the pulley, are more 
 simple than even an elementary combination of a driver and fol- 
 lower j for, with that exception, a mechanical power consists 
 essentially of only one primary moving piece; and the comparative 
 motion taken into consideration is simply the velocity-ratio either 
 of a pair of points in that piece, or of two components of the 
 velocity of one point. There are two established classifications of 
 the mechanical powers; an older classification, which enumerates 
 six; and a newer classification, which ranges the six mechanical 
 powers of the older system under three heads. The following table 
 shews both these classifications : — 
 
 KBWBa CiAssinoATioH. Older OiAssmOAiioir. 
 
 The T.PVii'P i ^^^ ^^^^'■' 
 
 J.HE L,EVEE, I ^j^g ^jjggj ^^^ ^^jg_ 
 
 ( The Inclined Plane. 
 The Inclined Plane, J The Wedge. 
 
 ( The Screw. 
 The Pullet...... The Pulley. 
 
 In the present section the comparative motions in the mechanical 
 powers are considered alone. The relations amongst the forces 
 which act in those machines will be treated of in the kinetic 
 division of this Treatise. 
 
 In the lever and the wheel and axle of the older classification, 
 which are both comprehended under the lever of the newer classi- 
 fication, the primary moving piece turns about a fixed axis; and 
 the comparative motion taken into consideration is the velocity- 
 ratio of two points in that piece, which may be called respectively 
 the driving point and the following point. The principle upon 
 which that velocity-ratio depends has already been s^ted in Article 
 93, page 50 — viz., that the velocity of each point is proportional 
 to the radius of the circular path which it describes; that is, to its 
 perpendicular distance from the axis of motion. 
 
 The distinction between the lever and the wheel and axle is 
 
108 
 
 THEORY OF MECHANISM. 
 
 this: that in the lever, the driving point, D, and the following 
 
 point, F, are a pair of determinate 
 points in the moving piece, as in 
 figs. 76a to 76d; whereas in the 
 wheel and axle they may be any 
 pair of points which are situated 
 respectively in a pair of cylindrical 
 pitch-surfaces, D and F, described 
 about the axis A, fig. 76. 
 
 In each of these figures the plane 
 of projection is normal to the axis, 
 and A is the trace of the axis. In 
 fig. 76, D and F are the traces of 
 two cylindrical pitch-surfaces. In 
 figs. 76a to 76d, D and F are the 
 S- '"• projections of the driving and 
 
 following points respectively. 
 
 The axis of a lever is often called the fulcrum. 
 
 A lever is said to be straight, when the driving point, D, and 
 
 following point, F, are in one plane traversing the axis A, as in 
 
 figs. 76a, 76b, and 76c. In other cases the lever is said to be bent, 
 
 as in fig. 76d. 
 
 A 
 
 Fig. 76a. 
 
 Fig 76b. 
 
 Fig. 76c. 
 
 The straight lever is Siiid to be of one or other of three kinds, 
 according to the following classification : — 
 In a lever of the first kind, fig. 76a, the 
 driving and following points are at opposite 
 sides of the fulcrum A. 
 
 In a lever of the second hind, fig. 76b, the 
 
 driving and following points are at the same 
 
 side of the fulcrum, and the driying point is 
 
 the further from the fulcrum. 
 
 In a lever of the third hind, fig. 76c, the driving and following 
 
 points are at the same side of the fulcrum, and the following point 
 
 is the further from the fulcrum. 
 
 In the inclined plane, and in the wedge, the comparative motion 
 considered is the velocity-ratio of the entire motion of a straight- 
 sliding primary piece and one of the components of that motion ; 
 the principles of which velocity-ratio have been stated in Article 
 70, pages 38, 39. 
 
THK INCLINED PLANE. 
 
 109 
 
 In the inclined plane, fig. 76b, A A is the trace of a fixed plane; 
 B, a block sliding on that 
 plane in the direction BC; 
 the plane of projection being 
 perpendicular to the plane 
 A A, and parallel to the 
 direction of motion of B. 
 B D is some direction oblique 
 to B C. From any convenient 
 point, 0, in B C, let fall C D 
 perpendicular to B D; then 
 B D .^ B C is the ratio of ^'g- 76b. 
 
 the component velocity in the'direction B D to the entire velocity 
 of B. 
 
 In fig. 76p, a a is the trace of a fixed plane; BCD, the trace 
 of a wedge which slides on that plane. While the wedge advances 
 through the distance C c, its oblique face advances from the posi- 
 tion D to the position c d; and if C e be drawn normal to the 
 plane C D, the ratio borne by the component velocity of the wedge 
 
 Fig, 76r. 
 
 in a direction normal to its oblique face to its entire velocity will 
 be expressed by C e : C c. 
 
 In the screw the comparative motion considered is the ratio 
 borne by the entire velocity of some point in, or rigidly connected 
 with, the screw, to the velocity of advance of the screw. 
 
 The helical path of motion of a point in, or rigidly attached to, 
 a screw may be developed (as has been already explained in Article 
 160, page 94) into a straight line : being the hypotenuse of a 
 right angled triangle whose height is equal to the pitch of the 
 screw, and its base to the circumference of a circle whose radius 
 is the distance of the given point from the axis of the screw. Then 
 if B D in fig. 76e be taken to represent the pitch of the screw, and 
 D 0, perpendicular to B D, the circumference of the circle described 
 by the point in question about the axis, B C wiU be the develop- 
 ment of one turn of the screw-line described by that point as it 
 revolves and advances along with the screw; and B C -^ B D will 
 be the ratio of its entire velocity to the velocity of advance; just 
 as in the case of a body sliding on an inclined plane, A A, parallel 
 to B 0. This shews why the screw is comprehended under the 
 
110 THEORY OF MECHANISM. 
 
 general head of the inclined plane, in the newer classification of 
 the mechanical powers. 
 
 The term pulley, in treating of the mechanical powers, means 
 any ptirchase or tackle of the class already described in Section 
 5 of this Chapter, pages 105 to 107. 
 
 Section 7. — Hydraulic Connection. 
 
 185. The General Principle of the communication of motion 
 between two pistons by means of an intervening fluid of constant 
 density has already been stated in Article 119, viz., that the velo- 
 cities of the pistons are inversely as their areas, measured on planes 
 normal to their directions of motion. 
 
 Should the density of the fluid vary, the problem is no longer 
 one of pure mechanism; because in that case, besides the communi- 
 cation of motion from one piston to the other, there is an additional 
 motion of one or other, or both pistons, due to the change of volume 
 of the fluid. 
 
 186. Valves are used to regulate the communication of motion 
 through a fluid, by opening and shutting passages through which 
 the fluid flows; for example, a cylinder may be provided with 
 valves which shall cause the fluid to flow in through one passage, 
 and out through another. Of this use of valves, two cases may be 
 distinguished. 
 
 I. When the piston moves the fluid, the valves may be what is 
 called self-acting; that is, moved by the fluid. If there be two 
 passages into the ■ cylinder, one provided with a valve opening 
 inwards, and the other with a valve opening outwards; then 
 during the outward stroke of the piston the former valve is opened 
 and the latter shut by the inward pressure oi, the fluid, which 
 flows in through the former passage; and during the inward stroke 
 of the piston, the former valve is shut and the latter opened by the 
 outward pressure of the fluid, which flows out . through the latter 
 passage. This combination of cylinder, piston, and valves, consti- 
 tutes a pwmp. 
 
 II. When the fluid moves the piston, the valves must be opened 
 and shut by mechanism, or by hand. In this case the cylinder is 
 a worTdng cylinder. 
 
 187. In the Hydraulic Press, the rapid motion of a small piston 
 in a pump causes the slow motion of a large piston in a working 
 cylinder. The pump draws water from a reservoir, and forces it 
 into the working cylinder : during the outward stroke of the pump 
 piston, the piston of the working cylinder stands still; during the 
 inward stroke of the pump piston, the piston of the working 
 cylinder moves outward with a velocity as much less than that of 
 the pump piston as its area is greater. When the piston of the 
 
TRAINS OF ELEMENTAKT COMBINATIONS. Ill 
 
 ■working cylinder has finished its outward stroke, which may be of 
 any length, it is permitted to be moved inwards again by opening 
 a valve by hand and allowing the water to escape. 
 
 188. In the Hydraulic Hoist, the slow inward motion of a large 
 piston drives water from a large cylinder into a smaller cylinder, 
 and causes a more rapid outward motion of the piston of the 
 smaller cylinder. When the latter piston is to be moved inward, 
 a valve between the two cylinders is closed, and the valve of an 
 outlet from the smaller cylinder opened, by hand, so as to allow 
 the water to escape from the smaller cylinder. The larger cylinder 
 is filled and its piston moved outward, when required, by means of 
 a pump, in a manner resembli^ the action of a hydraulic press. 
 
 Section 8. — Trains or Mechanism. 
 
 189. Trains of Elementary Combinations have been defined in 
 Article 131, and illustrated in the case of wheelwork, in Article 145, 
 and in the case of a double Hooke's joint, in Article 179. The 
 general principle of their action is that the comparative motion of 
 the first driver and last follower is expressed by a ratio, which is 
 found by multiplying together the several velocity-ratios of the 
 series of elementary combinations of which the train consists, each 
 with the sign denoting the directional relation. 
 
 Two or more trains of mechanism may converge into one; as 
 when the two pistons of a pair of steam engines, each through its 
 own connecting rod, act upon one crank shaft. One train of 
 mechanism may diverge into two or more; as when a single shaft, 
 driven by a prime mover, carries several pulleys, each of which 
 drives a different machine. The principles of comparative motion 
 in such converging and diverging trains are the same as in simple 
 trains. 
 
112 
 
 CHAPTER III. 
 
 ON AGGEEGATE COMBINATIOKS. 
 
 190. The General Principles of aggregate combinations have 
 already been given in Part I., Chapter II., Section 3. The pro- 
 blems to •which those principles are to be applied may be divided 
 into two classes. 
 
 I. Where a secondary moving piece is connected at three, or at 
 two points, as the case may be, with three or with two other pieces 
 whose motions are given; so that the problem is, from the motions 
 of three or of two points in the secondary piece, to find its motion as 
 a whole, and the motion of any point in it. The solution of this 
 problem is given in Articles 97 and 98. 
 
 IT. Where a secondary piece, C, is carried by another piece, B; 
 and denoting the frame of the machine by A, there are given two 
 out of the three motions of A, B, and C, relatively to each other, 
 and the third is required. The motion of C relatively to A is the 
 resultant of the motion of C relatively to B, and of B relatively to 
 A; and the problem is solved by the methods already explained iu 
 Articles 99 to 107, inclusive. 
 
 Professor Willis distinguishes the effects of aggregate combina- 
 tions into aggregate velocities, whether linear or angular, produced 
 in secondary pieces by the combined action of different drivers, and 
 aggregate paths, being the curves, such 
 as cycloids and trochoids, epicycloids and 
 epitrochoids, described by given points in 
 such secondary pieces. 
 
 The following Articles give examples 
 of two simple aggregate combinations. 
 
 191. Differential Windlass.— In fig. 77, 
 the axis Aj carries two barrels of different 
 radii, r-^ being the greater, and r^ the less. 
 A running block containing a single 
 pulley is hung by a rope which passes 
 below the pulley, and has one end wound 
 round the larger barrel, and the other 
 wound the contrary way round the 
 smaller barrel. When the two barrels 
 rotate together with the common angular velocity a, the division 
 of the rope which hangs from the larger barrel moves with the 
 
COMPOUSD SCREWS. 
 
 113 
 
 velocity a rj, and the division -which hangs from the smaller barrel 
 moves in the contraty direction with the velocity -avi (whose 
 direction is denoted by the negative sign). These are also the 
 velocities of the two points at opposite extremities of a diameter of 
 the pulley, where it is touched by the two vertical divisions of the 
 rope. The velocity of the centre of the pulley is a mean between 
 those two velocities; that is, their half-diflFerence, because their 
 signs are opposite ; or denoting it by v, 
 
 a(ri-_r^) 
 
 .(1.) 
 
 The insfantaneotts axis of tjjje pulley may be found by the method 
 of Article 98, as follows : — In fig. 35c, let A and B be the two 
 ends of the horizontal diameter of the pulley, and let A V„ = a rj, 
 and B Vj = ar2 represent their velocities; join V^ Vj cutting AB 
 in O j this is the instantaneous axis. Now 
 
 A0-OB = AC + C0-OB = BC + O0-OB = 20C, 
 AO + OB : A0-0B::AV„ + BV5:AV„-BVs, 
 AB: 200:: a(ri + r2): air^-r^); 
 
 and hence the distance of the instantaneous axis from the centre 
 or moving axis of the pulley is obviously 
 
 AB' 
 
 .(2.) 
 
 2{r, + r,) 
 
 The motion of the centre of the pulley is the same with that of a 
 
 point in -a rope wound on a barrel of the radius — = — . The use of 
 
 the contrivance is to obtain a slow motion of the pulley without 
 using a small, and therefore a weak, barrel. 
 
 192. Compound Screws. — (Fig. 78). On the same axis let 
 there be two screws Sj Sy and SaSj, of the respective pitches 
 
 o 
 
 N/ 
 
 Kg. 78. 
 
 Pi and Pi, Pi being the greater, and let the screws in the first 
 instance be both right-handed or both left-handed. Let Nj and Nj 
 be two nuts, fitted on the two screws respectively. "When the com- 
 pound screw rotates with the angular velocity a, the nuts ap roach 
 towards or recede from each other with the relative velocity 
 
 I 
 
114 THEOET OF MECHANISM. 
 
 -"-%i^^ (1-) 
 
 teing that due to a screw whose pitch is the difference of the two 
 pitches of the compound screw. (See Article 96, Equation 1.) 
 The object of this contrivance is to obtain the slow advance due to 
 a fine pitch, together with the strength of large threads. 
 
 Fig. 79 represents a compound screw in which the two screws 
 are contrary-handed, and the relative velocity of the nuts Nj Nj is 
 that due to the sum of the two 'pitches ; or as they are usually 
 equal, to double the pitch of each screw. This combination is 
 used in coupling railway-carriages. 
 
PAET III. 
 
 PRINCIPLES OF STATICS. 
 
 CHAPTER I. 
 
 SUMMARY OF'feENERAL PRINCIPLES. 
 
 Natueb and Division of the Subject.' 
 
 The present Chapter contains a summary of the Principles of 
 Statics. 
 
 193. Forces — Action and Re-action. — Every force is an action 
 exerted between a pair of bodies, tending to alter their condition 
 as to relative rest and motion; it is exerted equally, and in con- 
 trary directions, upon each body of the pair. That is to say, if A 
 
 I and B be a pair of bodies acting mechanically on each other, the 
 force exerted by A upon B is equal in magnitude and contrary in 
 direction to the force exerted by B upon A. This principle is 
 sometimes called the equality of action and re-action. It is ana- 
 logous to that of relative motion, explained in Article 61, page 34. 
 
 194. Forces, how Determined and Expressed. — A force, as 
 respects one of the two bodies between which it acts, is deter- 
 mined, 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. 
 
 The Place of the application of a force to a body may be the 
 ■whole of its volume, as in the case of gravity; or the surface at 
 which two bodies touch each other, or the bounding surface 
 between two parts of the same body, as in the case of pressure, 
 tension, shearing stress, and friction. 
 
 Thus every force has its action distributed over a certain space, 
 either a volume or a surface; and a force concentrated at a single 
 point has no real existence. Nevertheless, in investigations respect- 
 ing the action of a distributed force upon the position and move- 
 ments, as a whole, of a rigid body, or of a body which without 
 error may be treated as rigid, like the solid parts of a machine, 
 fixed or moving, that force may be treated as if it were concen- 
 trated at a point or points, determined by suitable processes ; and 
 
116 PRINCIPLES OF STATICS. 
 
 such is the use of those numerous propositions in statics which 
 relate to forces concentrated at points; or single forces, az they 
 are called. 
 
 The Direction of a force is that of the motion which it tends 
 to produce. A straight line drawn through the points of applica- 
 tion of a single force, and along its direction, is the line op action 
 of that force. 
 
 The Magnitudes of two forces are equal when, being applied to 
 the same body in opposite directions 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 standcurd of force, which, 
 for practical purposes, 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 
 standwrd pound a/voirdupois ; which is the weight, in the latitude 
 of London, of a certain piece of platinum kept in a public office. 
 
 For the sake of convenience, or of compliance with custom, other 
 units of weight are occasionally employed in Britain, bearing certain 
 ratios to the sttodard pound; such as — 
 
 The grain = T-jnrir °^ ^ pound avoirdupois. 
 
 The troiy pound = 5,760 grains = 0-82285714 pound avoirdupois. 
 
 The hundredweight = 112 pounds avoirdupois. 
 
 The ton = 2,240 pounds avoirdupois. 
 
 The French standard of -v^eight is the Mhgramme, which is the 
 weight, in the latitude of Paris, of a certain piece of platinum kept 
 in a public office. It was originally intended to be the weight of 
 a cubic decimetre of pure water, measured at the temperature at 
 which the density of water is greatest — viz., 4°-l Cent., or 39°4 
 Fahr., and under the pressure which supports a barometric column 
 of 760 millimetres of mercury; but it is in reality a little heavier. 
 
 A kilogramme is 2'20462125 lbs. avoirdupois. 
 A pound avoirdupois is 04535926525 of a kilogramme. 
 
 For scientific purposes, forces are sometimes expressed in 
 Absolute Units. The "Absolute Unit of Force" is a term used to 
 denote the force which, acting on an unit of mass for an unit of 
 time, produces an unit of velocity. 
 
 The unit of time employed is always a second. 
 
 The unity of velocity is in Britain one foot per second ; in 
 
 France one mfetre per second. 
 The unit of mass is the mass of so much matter as weighs one 
 
REPRESENTATION OF FORCES BY LINES. 117 
 
 unit of -weight near the level of the sea, and in some 
 definite latitude. 
 
 In Britain the latitude chosen is that of London ; in France, 
 that of Paris. 
 
 In Britain the unit of weight chosen is sometimes a grain, 
 sometimes a pound avoirdupois; and it is equal to 32-187 
 of the corresponding absolute units of force. In France the 
 unit of weight chosen is either a gramme or a kilogramme, 
 and it is equal to 9-8087 of the corresponding absolute 
 units of force. Each of those coefficients is denoted by the 
 letter g. 
 
 195. Measures of Force ^and Mass. — If by the unit of force 
 is understood the weight of a certain standayd, such as the 
 avoirdupois pound, then the mass of that standard is l^g'j and 
 the unit of mass is g times the mass of the standard; and this 
 is the most convenient system for calculations connected with 
 mechanical engineering, and is therefore followed in the present 
 work. 
 
 But if we take for the unit of mass, the mass of the standard 
 itself, then the unit of force is the absolute unit; and the weight of 
 the standard in such units is expressed by g; iov g is the velocity 
 which a body's own weight, acting unbalanced, impresses on it 
 in a second. This wiU be specially treated of in Part V. This is 
 the system employed in many scientific writings, and in particular, 
 in Thomson and Tait's Natural Philosophy. It has great advan- 
 tages in a scientific point of view; but its use in calculations for 
 practical purposes would be inconvenient, because of the prevailing 
 custom of expressing forces in terms of the standard of weight. 
 
 196. Representation of Forces by Lines. — A single force may be 
 represented in a drawing by a straight line; an extremity of the 
 line indicating the point of 
 application of the force, — the 
 direction of the line, the direc- 
 tion of the force, — and the length 
 of the line, the magnitude of the 
 force, according to an arbitrary 
 scale. j-ig, 80. 
 
 For example, in fig. 80, the 
 fact that the body B B B B is acted upon at the point Oj by a 
 given force, may be expressed by drawing from Oj a straight line 
 Ox Fj in the direction of the force, and of a length representing the 
 magnitude of the force. 
 
 If the force represented by Oj Fj is balanced by a force applied 
 either at the same point, or at another point Og (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 Oa Fa, opposite in diree- 
 
118 PRINCIPLES OF STATICS. 
 
 tion, and equal in length, to Oj Fj, and lying in the same line of 
 
 action L L. 
 
 If the body B B B B (fig. 81), be balanced by several forces acting 
 
 in the same straigh t lin e LL, applied at points Oi Oj, &c., and 
 
 represented by lines Oj F^ Oa Fa, &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 
 jijg gj_ positively be equal to the 
 
 sum of all those which point 
 
 negatively, the algebraical sum of all the forces is nothing, and the 
 
 body is balanced. 
 
 197. Resultant and Component Forces — Their Magnitude.— 
 The Resultant of any combination of forces applied to one body 
 is a single force capable of balancing that single force which 
 balances the combined forces; that is to say, the resultant of the 
 combined forces is equal and directly opposed to the force which 
 balances the combined forces, and is equivalent to the combined 
 forces so far as the balance of the body is concerned. The com- 
 bined forces are called components of their resultant. 
 
 The resultant of a set of mutually balanced forces is nothing. 
 
 The magnitudes and directions of a resultant force and of its 
 components are related to each other exactly in the same manner 
 with the velocities and directions of resultant and component 
 motions. 
 
 As to the position of the resultant, if the components act through 
 one point, the resultant acts through that point also; but if the 
 components do not act through one point, the position of the re- 
 sultant is to be found by methods which will be stated further on. 
 
 198. 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, and that each force 
 merely tends to cause such change, without actually causing it. 
 
 In treatises on statics, the word pressure is often used to denote 
 any balanced force; although in the popular sense that word is 
 used to denote a force, of the nature of a thrust or push, distributed 
 over a surface. 
 
 199. Parallel Forces are forces whose directions of motion are 
 parallel, excepting couples and directly opposed forces. 
 
 200. Couples. — Two forces of equal magnitude applied to the 
 same body in parallel and opposite directions, but not in the same 
 
DISTRIBUTED FORCES IN GENERAL. 119 
 
 line of action (such as T, F, in fig. 82), constitute what is called a 
 " couple." 
 
 The CM-m or leverage of a couple (L, fig. 82) is the perpendicular 
 distance between the lines of action of the two equal forces. 
 
 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 con- 
 tains the lines of action of the two 
 forces. (The plane in which a body 
 turns is any plane parallel to those 
 planes in the body whose position is not 
 altered by the turning). The turning 
 of a body is said to be *ight-h(mded 
 when it appears to a spectator to take 
 place in the same direction with that of ^'S- 82. 
 
 the hands of a watch, and left-handed when in the opposite direc- 
 tion; and couples are designated as right-handed or left-handed 
 according to the direction of the turning which they tend to pro- 
 duce. The couple represented in fig. 82 appears right-handed to 
 the reader. 
 
 The Moment of a couple means the product of the magnitude of 
 its force by the length of its arm (F L) ; and may be represented 
 by the area of a rectangle whose sides are F and L. If the force 
 be a certain number of pounds, and the arm a certain number of 
 feet, the product of those two numbers is called the moment in 
 foot-poimds, and similarly for other measures. The moment of a 
 couple may also be represented by a single line on paper, by setting 
 off upon its aocis (that is, upon any line perpendicular to the plane 
 of the couple) a length proportional to that moment (O M, fig. 62) 
 in such a direction, that to an observer looking from O towards M 
 the couple shall seem right-handed. 
 
 201. The Centre of Parallel Forces is the single point 
 referred to in the following principle. The forces to which that 
 principle is applied are in general either weights or pressures; 
 and the point in question is then called the Centre of OramJty or 
 the Centre of Pressure, as the case may be. 
 
 Jf there he given a system, of points, and the mutual ratios of a 
 system of pa/rallel forces applied to those points, which forces hwve a 
 single resultant, then there is one point, and one only, which is tra- 
 versed by the line of action of the resultant of every system of pa/rcdlel 
 forces having the given mutual ratios and applied to the given system, 
 of points, whatsoever may be the absolute magnitudes of those forces 
 a/iid the angukur position of their limes of action. 
 
 202. Distributed Forces in General. — In Article 194, page 115, 
 it has already been explained, that the action of every real force is 
 distributed throughout some volume, or over some surface. It is 
 always possible, however, to find either a single resultant, or a 
 
120 PEINCIPLES OP STATICS. 
 
 resultant couple, or a..comMnation of a single force with a couple, to 
 •which a given distributed force is equivalent, so far as it affects 
 the equilibrium of the body, or part of a body, to which it is 
 applied. 
 
 In the application of Mechanics to Structures, the only force dis- 
 tributed throughout the volume of a body which it is necessary to 
 consider, is its weight, or attraction towards the earth ; and the 
 bodies considered are in every instance so small as compared with 
 the earth, that this attraction may, without appreciable error, be 
 held to act in parallel directions at each point in each body. More- 
 over, the forces distributed over surfaces are either parallel at each 
 point of their surfaces of application, or capable of being resolved 
 into sets of parallel forces; hence, parallel distributed forces have 
 alone to be considered; and every such force is statically equivalent 
 either to a single resultant, or to a resultant couple. 
 
 The intensity of a distrilmted force is the ratio which the magni- 
 tude of that force, expressed in units of weight, bears to the space 
 over which it is distributed, expressed in units of volume, or in 
 units of surface, as the case may be. An unit of intensity is an 
 unit of force distributed over an unit of volume or of surface, as 
 the case may be; so that there are two kinds of units of intensity. 
 For example, one pound per cubic foot is an unit of intensity for a 
 force distributed throughout a volume, such as weight; and one 
 pound per square foot is an unit of intensity for a force distributed 
 over a surface, such as pressure or friction. 
 
 203. Specific Gravity — Heaviness — Density — Bulkiness.— I. 
 Specific Gravity is the ratio of the weight of a given bulk of a 
 given substance to the weight of the same bulk of pure water at 
 a standard temperature. In Britain the standard temperature 
 is 62° Fahr. = 16°-67 Cent. In France it is the temperature of 
 the maximum density of water = 3°'94 Cent. = 39°-l Fahr. 
 
 In rising from 39°'l Fahr. to 62° Fahr., pure water expands in 
 the ratio of 1 '0011 18 to 1 ; but that difference is of no consequence 
 in calculations of specific gravity for engineering purposes. 
 
 II. The heaviness of any substance is the weight of an unit of 
 volume of it in units of weight. In British measures heaviness is 
 most conveniently expressed in lbs. avoirdupois to the cubic foot ; 
 in French measures, in Jdlogrammes to the cubic decimetre (or to 
 the litre). The values of the heaviness of water at 39°-l Fahr., 
 and at 62° Fahr., are respectively 62425 and 62-355 lbs. to the 
 cubic foot. 
 
 III. The density of a substance is either the number of units of 
 mass in an unit of volume, in which case it is equal to the heavi- 
 ness, — or the ratio of the mass of a given volume of the substance 
 
 ito the mass of an equal volume of water, in which case it is equal 
 to the specific gravity. In its application to gases, the term 
 
THE INTENSITY Or PRESSURE. 121 
 
 "Density" is often used to denote the ratio of the heaviness of a 
 given gas to that of air, at the same temperature and pressure. 
 
 IV. The bulkiness of a substance is the number of units of 
 volume which an unit of weight fills; and is the reciprocal of t/ie 
 heaviness. In British measures bulkiness is most conveniently- 
 expressed in cubic feet to the lb. avoirdupois; in French measures^ 
 in cubic decimetres (or in litres) to the hilogramme. Rise of temper- 
 ature produces (with certain exceptions) increase of bulkiness. 
 The linear expansion of a solid body is one-third of its expansion 
 in bulk. 
 
 204. The Centre of Gravity of a body or of a system of bodies, 
 is the point always traversed by the resultant of the weight of the 
 body or system of bodies, — itf other words, the centn-e of parallet 
 forces for the weight of the body or system of bodies. 
 
 To support a body, that is, to balance its weight, the resultant of 
 the supporting force must act through the centre of gravity. 
 
 When the centre of gravity of a geometrical figure is spoken of, 
 it is to be iraderstood to mean the point where the centre of 
 gravity would be, if the figure were formed of a substance of 
 uniform heaviness. 
 
 205. The Centre of Pressure in a plane surface is the point 
 traversed by the resultant of a pressure that is exerted at that 
 surface. When the intensity is uniform, the centre of pressure is- 
 at the centre of magnitude of the pressed surface. 
 
 206. The Centre of Buoyancy of a solid wholly or partly im- 
 mersed in a liquid is the centre of gravity of the mass of liquid 
 displaced. The resultant pressure of the liquid on the solid is. 
 equal to the weight of liquid displaced, and is exerted vertically 
 upwards through the centre of buoyancy. 
 
 207. The Intensity of Pressure is expressed in units of weight 
 on the unit of area; as pounds on the square inch, or kilogrammes 
 on the square mfetre ; or by the height of a column of some fluid ;. 
 or in atmospheres, the unit in this case being the average pressure 
 of the atmosphere at the level of the sea. 
 
123 
 
 CHAPTER II. 
 
 COMPOSITION, RESOLUTION, AND BALANCE OE EOECES. 
 Section 1. — Forces Acting Theough One Point. 
 
 208. Resultant of Forces Acting in One Straight Line. — The 
 resultant of any number of forces acting on one body in tlie same 
 straight line of action, acts along that line, and is equal in magni- 
 tude to the sum of the component forces; it being understood, 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. 
 
 When a system of forces acting along one straight hne are 
 balanced, the sum of the forces acting in one direction is equal to 
 the sum of the forces acting in the opposite direction. 
 
 209. Resultant and Balance of Inclined Forces — Parallelogram 
 of Forces. — The smallest number of inclined forces which can 
 balance each other is three. Those three forces must act through 
 one point, and in one plane. Their relation to each other depends 
 on the following theorem, called the " Parahelogeam of Foecbs," 
 from which the whole science of statics may be deduced. 
 
 If tiop forces whose lines of action traverse one point be repre- 
 sented in direction and magnitude by the sides of a pa/raUdogram, 
 their resultant is represented hy the diagonal. 
 
 For example, through the point O (fig. 83) let two forces act, 
 represented in direction and magnitude by O A and B. The re- 
 sultant or equivalent single force of 
 those two forces is represented in 
 direction and magnitude by the 
 diagonal of the parallelogram 
 O A C B. Its magnitude is given 
 algebraically by the equation.* 
 
 0C=\/ |oA2-i-OB2 
 + 20A-OBcosAOb] 
 
 (!•) 
 
 Kg. 83. 
 
 210. Triangle of Forces. — To balance the forces A and B, 
 a force is required equal and directly opposed to their resultant 
 O 0. This may be expressed by saying, that if the directions and 
 
RESOLUTION OP A FORCE INTO TWO COMPONENTS. 
 
 123 
 
 mngnitudes of three forces he represented hy the three sides of a 
 tricmgle, taken in the same ordeir (such as A, A 0, C O), then those 
 three forces, acting through one point, balance each other, or ia other 
 words, that three forces in the same plane balance each other at 
 one point, when each is proportional to the sine of the angle 
 between the other two. 
 
 211. Polygon of Forces. — If a number of forces acting through 
 the same point he represented hy limes 
 equal and parallel to the sides of a 
 closed polygon, taken in tlie same order, 
 those forces balance each other. To fix 
 the ideas, let there be fiye forces 
 acting through the point O (fig. 84), 
 and represented in direction and 
 
 magnitude by the lines F^, Fj, Fg, 
 
 Fig. 84 
 
 F^, Fg, which are equal and parallel to the sides of the closed poly- 
 gon OAB CD O; viz.:— 
 
 Fi = and || O A; F2 = andil AB; F3 = andll BC; 
 
 F^ = and 
 
 CD; F5 = andilD0. 
 
 Then, by the principle of the parallelogram of forces, the resultant 
 of Fi and Fg is OB; the, resultant of F^, F^, and Fg is O C ; the 
 resultant of F^, Fg, Fg, and F^ is O D, equal and opposite to Fj, 
 so that the final resultant is nothing. 
 
 The closed polygon may be either plane or " gauche" — that is, 
 not in one plane. 
 
 212. Principles of the Parallelepiped of Forces.— The simplest 
 gauche polygon is one of four sides. Let AOBCEFGH (fig. 
 85), be a parallelopiped whose diagonal is O H. 
 Then any three successive edges so placed as 
 to begin at and end at H, form, together 
 with the diagonal H O, a closed quadrilateral; 
 consequently, if three forces Fj, Fg, Fg, acting 
 through 0, be represented by the three edges 
 AO, OB, O'O]^ of a parallelopiped, the dia- 
 gonal OH represents their resultant, and a 
 fourth force F^ equal and opposite to 
 OH balances them. /y- 
 
 213. Resolution of a Force into two / 
 Components. — In order that a given ^^ 
 single force may be resolvable into two 
 components acting in given lines in- 
 clined 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. 
 
 Fig. 85. 
 
124 PRINCIPLES OF STATICS. 
 
 Returning then to fig. 83, let O represent the given force, 
 ■which it is required to resolve into two component forces, acting in 
 the lines O X, Y, which lie in one plane with C, and intersect 
 it in one point 0. 
 
 Though C draw C A II O T, cutting X in A, and C B ii X, 
 cutting O Y in B. Then will O A and O B represent the com- 
 ponent forces required. 
 
 Two forces respectively equal to and directly opposed to A 
 and B will balance O G. 
 
 The magnitudes of the forces are in the following proportions; — 
 
 OC:OA:OB 
 ::sinAOB:sinBOC:sinAOC (1.) 
 
 214. 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 necessary that the 
 lines of action of the components should intersect the line of action 
 of the given force in one point. 
 
 Returning to fig. 85, 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 H in one point 0. 
 
 Through H draw three planes parallel respectively to the planes 
 Y Z, Z Y, X Y, and cutting respectively O X in A, Y ia 
 B, O Z in C. Then will O A, OB, O C, represent the component 
 forces required. 
 
 Three forces respectively equal to, and directly opposed to A, 
 OB, and 0, will balance OH. 
 
 215. Resolution of a Force. 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. 85, suppose X, Y, Z, to be three 
 axes of co-ordinates at right angles to each other. Then O H is 
 resolved into three rectangular components, A O, B, O 0, simply 
 by letting fall from H perpendiculars on O X, Y, Z, cutting 
 them at A, B, C, respectively. 
 
 Let the three rectangular components be denoted respectively by 
 X, Y, Z, the resultant by E, and the angles which it makes with 
 the components by a, /3, y, respectively ; then the relations between 
 the three rectangular components and their resultant are expressed 
 by the following equations : — 
 
 X = Rcos«; Y = Rcos/3; Z = Rcosy; (2.) 
 
 R2 = X2-hY2-l-Z2 (3.) 
 
RESULTANT OP COUPLES. 125 
 
 "When the resultant is in the same plane with two of its com- 
 ponents (as X and Y), the third component is nuU, and the 
 Equations 2 and 3 take the following form : — 
 
 X = Il cos « = E sin /3; Y = E cos /3 = E sin «; Z = 0j...(4.) 
 E2 = X2 + Y2 (5.) 
 
 In using Equations 2, 3, 4, and 5, it is to be remembered that 
 cosines of obtuse angles are negative. 
 
 216. Resultant and Balance of any number of inclined Forces 
 acting through one Point. — To find this resultant by calculation, 
 assume any three directions at right angles to each other as axes; 
 resolve each force into threwi components (X, Y, Z) along those 
 axes, and consider the components along a given axis which act in 
 one direction as positive, and those which act in the opposite direc- 
 tion as negative; take the algebraical sums of the components 
 along the three axes respectively (S • X, 2 • Y, 2 • Z); these will be 
 the rectangular components of the resultant of all the forces; and its 
 magnitude and direction will be given by the following equations : — 
 
 E2 = (2-X)2 + (2-Y)2-i-(2-Z)2; (1.) 
 
 2-X «2-Y 2-Z 
 
 cos et = „ ; C0S/3=^5— ; COS y = -^5- (2.) 
 
 If the forces all act in one plane, two rectangular axes in that 
 plane are sufficient, and the terms containing Z disappear from the 
 equations. 
 
 If the forces balance each other, the components parallel to each 
 axis balance each other independently; that is to say, the three 
 following conditions are fulfilled : — 
 
 2-X = 0; 2-Y = 0; 2-Z = (3.) 
 
 If the forces all act in one plane, these conditions of equilibrium 
 are reduced to two. 
 
 Section 2. — Eesultant and Balance of Couples. , 
 
 217. Equivalent Couples. — If the moments of two couples acting 
 in the same direction and in the same or parallel plcmes are equal, 
 those couples are equivalent : that is, their tendencies to turn the 
 body to which they are applied are the same. 
 
 The following propositions are the chief consequences of the 
 principle just stated : — 
 
 218. Resultant of Couples. — The resultant of any number of 
 couples acting in the same or parallel planes is equivalent to a 
 couple whose moment is the algebraical sum of the moments of the 
 combined couples. 
 
126 PKINCIPLES OF STATICS. 
 
 219. Equilibrium of Couples with same Axis. — Two opposite 
 couples of equal moment in the same or parallel planes balance 
 each other. Any number of couples in the same or parallel planes 
 balance each other when the moments of the right-handed couples 
 
 vare together equal to the moments of the left-handed couples; in 
 other words, when the resultant moment is nothing — a condition 
 expressed algebraically by 
 
 2-FL = (1.) 
 
 220. Parallelogram of Couples. — If the two sides of a parallelo- 
 gram represent the axes and moments of two couples acting on 
 the same body in planes inclined to each other, the diagonal of the 
 parallelogram will represent the axis and moment of the resultant 
 couple, which is equivalent to those two. 
 
 In other words, three couples represented by the three sides of a 
 triangle, taken in the same order, balance each other. 
 
 221. Polygon of Couples. — If any number of couples acting on 
 the same body be represented by a series of lines joined end to end, 
 and taken in the same order so as to form sides of a polygon, and 
 if the polygon is closed, those couples balance each other. 
 
 These propositions are analogous to corresponding propositions 
 relating to single forces; and couples, like single forces, can he 
 resolved into components acting about two or three given axes. 
 
 222. Besultant of a Couple and Single Force in Parallel Planes. 
 — ^Let M denote the moment of a couple applied to a body (fig. 86); 
 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 equi- 
 valent couple, consisting of a 
 force - F equal and directly op- 
 posed to F at 0, and a force F 
 acting through the point A, the 
 arm A O perpendicular to F 
 
 M 
 
 being = ^, and parallel to the 
 
 Pig. 86. plane of the couple M. Then 
 
 the forces at O balance each 
 other, and F acting through A is the resultant of the single force 
 F applied at 0, and the couple M; that is to say, that if with a 
 single force F there be combined a couple M whose plane is parallel 
 to the force, the effect of that combination is to shift the line of 
 
 M 
 action of the force parallel to itself through a distance A = •=; j — 
 
 to the left if M is right-handed — to the right if M is left-handed. 
 
MAGNITUDE OP RESULTANT OP PARALLEL FORCES. 127 
 
 223. Moment of Force with respect to an Axis.— In fig. 87, 
 let the straight line F represent a force. Let 
 O X be any straight line perpendicular in direc- 
 tion to the line of action of the force, and not 
 intersecting it, and let A B be the common per- 
 pendicular of those two lines. At B conceive a 
 pair of equal and directly opposed 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 body. Then the original 
 single force F, applied in a line traversing A, is 
 equivalent to the force F' appllbd in a line travers- 
 ing B, the point in O X which is nearest to A, 
 conibined with the couple composed of F and - F', 
 whose moment is F • A. B. This is called the moment of the force 
 F relatively to the axis X, and sometimes also, the moment of the 
 force F relatively to the plane traversing O X, parallel to the line 
 of action of the force. 
 
 If from the point B there be drawn two straight lines B D and 
 B E, to the extremities of the line F representing the force, the 
 area of the triangle B D E being = J F ■ A B, represents one-half of 
 the moment of F relatively to X. 
 
 Section 3. — Kesultant and Balance op Parallel Forces. 
 
 224. Magnitude of Resultant of Parallel Forces. — 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. 
 
 I. In a balanced system of parallel forces the sums of the forces 
 acting in opposite directions are equal; in other words, the alge- 
 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 parallel 
 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 shewn j and in this inquiry 
 all pairs of directly opposed equal forces may be left out of con- 
 sideration' for each such pair is independently balanced whatso- 
 ever its position may be; so that the question in each case is to be 
 solved by means of the theory of couples. 
 
 The following is the simplest case : — 
 
128 
 
 PRINCIPLES OF STATICS. 
 
 225. Direction of Resultant of Parallel Forces— Principle of the 
 
 Lever. — If three parcdld forces 
 applied to one body balcmce each 
 other, they must he m one plome; 
 the two extreme forces must act 
 in the somie direction; the middle 
 force must act in the opposite 
 direction; and the magnitude of 
 each force must be proportional 
 to the distance between the lines 
 of action of the oilier two. Let 
 a body (fig. 88) be maintained 
 
 in equilibrio by two opposite couples acting in the same plane, and 
 of equal moments, 
 
 FiL^=F„LB, 
 
 and let those couples be so applied to the body that the lines of 
 action of two of those forces, - F^ - Fu, which act in the same 
 direction, shall coincide. Then those two forces are equivalent to 
 the single middle force F,, = - (F^ + F^), equal and opposite to the 
 sum of the extreme forces + F^, + Fj, 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 
 
 AU=L^j CB = L3; AB = L^ + L3; 
 
 and consequently 
 
 F^ : F3 : F„ : :^B : AC : AB; (L) 
 
 This proposition holds also when the straight line A C B cresses 
 the lines of action of the three forces obliquely. 
 
 226. To find the Resultant of .Two Parallel Forces.— The 
 resultant is in the same plane with, and parallel to, the com- 
 ponents. It is their sum or difference, according as they act in 
 the same or contrary directions; and in the latter case its direction 
 JB that of the greater component. To find its line of action by 
 construction, proceed as follows : — Fig. 89 representing the case 
 in which the components act in the same direction, fig. '90 that in 
 which they act in contrary directions. Let A D and B E be the 
 components. Join A E and B D, cutting each other in F. In 
 BD (produced in fig. 90) take BG = DF. Through G draw a 
 line par^lel to the components; this will be the line of action of 
 the resuftant. To find its magnitude ,by, construction : parallel to 
 A E, draw B C and D H, cutting the line of action of the result- 
 ant in C and H; H will represent the resultant required; and a 
 force equal and opposite to H will balance A D and B ,E. 
 
 To find the line of action of the resultant by calculation; make 
 either 
 
riELATIVE PROPORTIONS OF FOUR PARALLEL FORCES. 129 
 
 Kg. 89. 
 
 BG: 
 
 AD-DB 
 
 orDG = 
 
 Fig. 90. 
 BEDB 
 
 Fig. 91. 
 
 OH ^ "' " " C H ■ 
 
 When the two given, parallel forces are opposite and equal, they 
 form a couple, and have no single resultant. 
 
 227. To find the Relative Proportions of 
 Three Parallel Forces which Balance each 
 other, Acting in One Plane: their Lines of 
 Action being given. — Across the three lines 
 of action, in any convenient position, draw 
 a straight line A C B, fig. 91, and measure 
 the distances between the points where it 
 cuts the lines of action. Then each force 
 will be proportional to the distance between 
 the lines of action of the other two. The 
 direction of the middle force, C, is contrary 
 to that of the other two forces, A and B. 
 
 In symbols, let A, B, and C be the forces; then, 
 
 A + B + C = 0; AB:BC:CA::C:A:B. 
 Each of the three forces is equal and opposite to the resultant of 
 the other two; and each pair of forces are 
 equal and opposite to the components of the 
 third. Hence this rule serves to resolve a 
 given force into two parallel components 
 acting in given lines in the same plane. 
 
 228. To find the Relative Proportions of 
 Four Parallel Forces which Balance each 
 other, not Acting in One Plane: their Lines 
 of Action being given. — Conceive a plane to 
 cross the lines of action in any convenient 
 position; and in fig. 92 or fig. 93, let A, B, 
 
 Fig. 92. 
 
 C, D repre- 
 
130 
 
 PRINCIPLES OF STATICS. 
 
 sent the points where the four lines of action cut the plane. 
 Draw the six straight lines joining those four 
 points by pairs. Then the force which acts through 
 each point will be proportional to the area of the 
 triangle formed by the other three points. 
 
 In fig. 92 the directions of the forces at A, B, 
 and C are the same, and are contrary to that of 
 the force at D. In fig. 93 the forces at A and D 
 act in one direction, and those at B and C in the 
 
 ■■4^- 
 
 x> 
 
 93. 
 
 Fig. 
 
 contrary direction. 
 In symbols, A 
 
 BOD 
 
 t- B + C + D = 
 CD A:D AB 
 
 0; 
 
 :ABC 
 : D. 
 
 Each of the four forces is equal and opposite to the resultant 
 of the other three ; and each set of three forces are equal and oppo- 
 site to the components of the fourth. Hence the rule serves to resolve 
 a force into three parallel components not acting in one plane. 
 
 229. Moments of a Force with respect to a Pair of Rectangular 
 Axes. — In fig. 94, let F be any single force; O an arbitrarily- 
 assumed point, called the " origin 
 of co-ordinates ; " - X -I- X, 
 - Y O -I- Y, a pair of axes travers- 
 ing O, at right angles to each 
 other and to the line of action 
 of F. Let A B = j^, be the com- 
 mon perpendicular of F and 
 O X; let A C = X, be the common 
 perpendicular of F and Y. x 
 and y are the "rectangular co- 
 ordinates" of the line of action 
 of F relatively to the axes 
 -XO + X, -YO + Y, respec- 
 tively. According to the arrange- 
 ment of the axes in the figure, 
 X is to be considered as positive 
 to the right, and negative to the 
 - Y O -H Y; and y is to be considered as positive to the left, 
 and negative to the right, of - X O -t- 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 223, F is eqvdvalent to the single force 
 
 Pig. 94. 
 
 left, of 
 
KESULTAUT OF ANT SYSTEM OF PARALLEL FORCES IN ONE FLANE. 131 
 
 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 O, conceive a pair of equal and opposite forces, F" and - F", 
 to be applied, F" being equal and parallel to F aaid F, and in the 
 same direction. Then the single force F' is equivalent to the 
 single force F" = F' = F applied at O, combined with the couple 
 constituted by F' and - F" with the arm OB = x, whose moment is 
 
 - a; 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 - a: F relatively to the axis 
 OY right-handed couples being considered positive; and + Y lying 
 to the left of + X, as viewed by a spectator looking from + X 
 towards 0, with his head in the direction of positive forces. 
 
 230. Balance 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 
 sufficient that the following conditions should be fulfilled : — 
 
 Firgt — (As already stated) that the algebraical sum of the forces 
 shall be nothing. 
 
 Secondly — That the algebraical sum of the moments of the forces 
 relatively 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 
 negative, according to the direction in which it acts; let y be the 
 perpendicular 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, 
 
 2-F=0j 2-yF = 0. 
 
 In summing moments, right-handed couples are usually con- 
 sidered as positive, and left-handed couples as negative. 
 
 231. Let R denote the Resultant of any System of Parallel 
 Forces in one Plane, and y„ the distance of the line of action of 
 that resultant from the assumed axis O X to which the positions 
 of forces are referred; then, 
 
 E = 2-F; 
 y*- - 2 ■ F ' 
 
132 PRINCIPLES OP STATICS. 
 
 In some cases the forces may have no single. resultant, S-P 
 being =0; and then, unless the forces' balance each other com- 
 pletely, their resultant is a couple of the moment 2 • y F. 
 
 232. Balance 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 shall be fulfilled : — 
 
 First — (As already stated) that the algebraical sum of the forces 
 shall be nothing. 
 
 Secondly amd Thirdly — 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 for'ces, shall each be nothing, 
 
 two conditions which are expressed symbolically as follows : — 
 Let O X and O Y denote the pair of axes; let F be the magnitude 
 of any one of the forces; y its perpendicular distance from X, 
 and X its perpendicular distance from O Y ; then, 
 
 2-F = 0; 2-yF = 0; 2-a!F = 0; 
 
 233. Let R denote the Resultant of any System of Parallel 
 Forces, and x^ and y^ the distances of its line of action from two 
 rectangular axes; then, 
 
 ■ _ „ _ 2-a!F 2-wF 
 
 R = 2-F;ar, = ^^-;2/,= ^^. 
 
 In some cases the forces may have no single resultant, 2 • F 
 being = ; and then, unless the forces balance each other com- 
 pletely, their resultant is a couple, whose axis, direction, and 
 moment, are found as follows : — 
 
 Let M^ = 2-2/F; Mj, = -2-a;F; 
 
 be the moments of the pair of partial resultant couples about the 
 axes O X and Y respectively. From O, along those axes, set off 
 two lines representing respectively M,. and Mj, ; that is to say, pro- 
 portional 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 will represent the axis, direction, 
 and moment of the final resultant couple. Let M,. be the moment 
 of this couple; then 
 
 M,= /y^|Ml-t-M^| 
 and if e be the angle which its axis makes \vith X, 
 
 00.6 = ^. 
 
TO FIND THE CENTRE OP PARALLEL FORCES. 
 
 133 
 
 234. To find the Centre of Parallel Forces.— Let O in fig. 95 
 be any convenient point, taken as the origin of co-ordinates, and 
 X, O Y, Z, three axes of co-ordinates at right angles to each 
 other. 
 
 Let A be any one of the points to which the system of parallel 
 forces in question is applied. From A draw x parallel to O X, 
 and perpendicular to the plane Y Z, 
 y parallel to O Y, and perpendicular 
 to the plane Z X, and z parallel to 
 O Z, and perpendicular to the plane 
 X Y. X, y, and z are the rectangu- 
 lar co-ordinates of A, wbich, 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. 
 
 The position of the centre of parallel forces depends solely on 
 the ipropoTtionate magnitudes of the parallel forces, not on their 
 absolute magnitudes, nor on the angular positions of their lines of 
 actions; so that for any system of parallel forces another may be 
 substituted in any angular position : this is the statement of the 
 principle of the centre of parallel forces given at Article 201, 
 page 119. This is evident since, in considering the relations of 
 parallel forces, they are not considered with reference to any parti- 
 cular plane, and hence these relations must hold for any plane. 
 
 First, conceive all the parallel forces to act in lines parallel to 
 the plane Y Z. Then the distance of their resultant, and of the 
 centre of parallel forces from that plane is 
 
 Kg. 95. 
 
 ;F 
 
 2-F 
 
 •(1) 
 
 Secondly, conceive all the parallel forces to act in lines parallel 
 to the plane Z X. Then the distance of their resultant, and of the 
 centre of parallel forces from that plane is 
 
 Vr 
 
 2-yF 
 2-F" 
 
 .(2.) 
 
 Thirdly, conceive all the parallel forces to act in lines parallel to 
 the plane X Y. Then the distance of their resultant, and of the 
 centre of parallel forces from that plane is 
 
 S-«F 
 
 «r = 
 
 2-F* 
 
 .(3.) 
 
134 PRINCIPLES OP STATICS. 
 
 If the forces have no single resultant, so that S • F = 0, there is 
 no centre of parallel forces. This may be the case with pressures, 
 but not with weights. 
 
 If the parallel forces applied to a system, of points are all equal 
 and in the same direction, it is obvious that the distance of the 
 centre of parallel forces from any given plane is simply the mean 
 of the distances of the points of the system from that plane. 
 
 Section 4. — Op ant System op Forces. 
 
 235. Eesultant and Balance of any System of Forces in One 
 Plane. — Let the plane be that of the axes O X and O Y in fig. 95; 
 and in looking from ^ towards 0, let Y lie to the right of X, so 
 that rotation from X towards Y shall be right-handed. Let a; and 
 y be the co-ordinates of the point of application of one of the 
 forces, or of any point in its line of action, relatively to the assumed 
 origin and axes. Resolve each force into two rectangular com- 
 ponents X and Y, as in Article 215, page 125; then the rectangular 
 components of the resultant are S • X and 2 • Y ; its magnitude is 
 given by the equation 
 
 E2 = (2;X)«+(S-Y)2, (1.) 
 
 and the angle «, which it makes with O X is found by the equations 
 
 S-X . 2-Y ,,, 
 
 cos «,. = — ^—; sin«,= -^ (2.) 
 
 This angle is acute or obtuse according as 2 • X is positive or nega- 
 tive ; and it lies to the right or left of O X according as 2 • Y is 
 positive or negative. 
 
 The resultant moment of the system of forces about the axis 
 OZis* 
 
 M = 2(a;Y-yX), (3.) 
 
 and is right or left-handed according as M is positive or negative. 
 The perpendicular distance of the resultant force E from is 
 
 T ^ u\ 
 
 ^ = K- : (*•) 
 
 Let x^ and y, be the co-ordinates of any point in the line of 
 action of that resultant; then the equation of that line is* 
 
 a!,2-Y-y,S-X = M. (5.) 
 
 The method of obtaining this result by Co-ordinate Geometry is the 
 
RESULTANT AND BALANCE Or ANY SYSTEM OP FORCES. 135 
 
 If M = the resultant acts through, the origin 0; if M has 
 magnitude, and B, = (in which case S • X = 0, S • T = 0) the 
 resultant is a couple. The conditions of equilibrium of the system 
 of forces are 
 
 S-X = 0; 2-Y = 0; M = 0. 
 
 ..(6.) 
 
 236. Eesultant and Balance of any System of Forces To 
 
 find the resultant and the conditions 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 before, let O in fig. 95, p. 133, denote the origin of co-or- 
 dinates, and O X, O Y, Z, the three rectangular axes: and let 
 them be arranged so that in looking from 
 
 X) 
 
 Y > towards 
 
 Z) 
 
 0, rotation from 
 
 C Y towards Z ) 
 ^ Z towards XV' 
 ( X towards Y j 
 
 shall appear right-handed. 
 
 Let X, Y, Z, denote the rectangular components of any one 
 of the forces; x, y, z, the co-ordinates of a point in its line of 
 action. 
 
 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, 
 
 foUowin? :— Let C = L, AB=E, ZXAy=«r; and let ECi=Xr and 
 
 OG=yr'be the co-ordinates of the 
 
 point E. Then by Trigonometry 
 
 sin «r=ainOAO = oosOOA=siu 
 
 DOG=oos DGO = smEGI' and 
 
 -cos a,= oos OAC=sin CO A 
 
 = 003 DOG. 
 
 L=DC-^OD=FE-^OD 
 
 =EG •sinEGF + 
 
 G ■ cos D G 
 = K, • sin a, - y, ■ cos «, 
 
 multiplying by K 
 
 L • E,=M = Xr •E. -sin «,- 
 
 2/, ■ R ■ cos ar 
 
 =Xr-sY-i/r-2X Pig. 96. 
 
 by substituting the values in Equation 2 supi-a. Equation 3 was thus 
 obtained. 
 
 ^ 
 
 : 
 
 \ 
 
 
 
 
 
 
 \o 
 
 
 
 / 
 
 d/ y 
 
 
 \ 
 
 
 
 
 yx G 
 
 : 
 
 ^\ 
 
136 PRINCIPLES OF STATICS. 
 
 the six following quantities are found, which compose the resultant 
 of the given system of forces : — 
 
 Forces. 
 
 S-X;2-Y; S'Z; (1-) 
 
 Couples. 
 
 about OX; Mi = 2 (yZ-a Y); ) 
 
 „ OY; M2 = S(^X-a!Z);V (2.) 
 
 found as already explained in the footnote to Article 235. 
 The three forces are equivalent to a single force 
 
 = Y^ {{2-X)2 + (2-Y)2 + (2-Z)2| (3.) 
 
 acting through in a line which makes with the axes the angles 
 given by the equations 
 
 2-X ^ 2-Y 2-Z ... 
 
 cos «= -^-i cos^=-^;cosy = -^g- (4.) 
 
 The three couples, Mj, Mj, Mj, are equivalent to one couple, 
 whose magnitude is given by the equation 
 
 M= J{M.l + Ml + MD, (5.) 
 
 and whose axis makes with the axes of co-ordinates the angles 
 given by the equations 
 
 , M, M, Mg .,. 
 
 E 
 
 in which 
 
 . , J I denote respectively the angles j (-> v t 
 ^"'^ "J '' f made by the axis of M with [q Z) 
 
 The conditions of equilibrium of the system of forces may be 
 expressed in either of the two following foi-ms : — 
 
 2-X = 0; 2-Y = 0; 2-Z = 0; Mi = Oj Mj, = 0; M3 = 0; (7.) 
 or 
 
 K = 0; M = (8.) 
 
 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 E. acting 
 through O. 
 
RESULTANT AND BALANCE OF ANT SYSTEM OP FORCES. 137 
 
 Case II. — When the axis of M. is at right angles to the direction of 
 E, — a case expressed by the following equation: — 
 
 cos « cos 'i + cos /3 cos ft + cos y cos •' = j (9.) 
 
 (an equation of Co-ordinate Geometry) 
 
 the resultant of M and K is a single force equal and parallel to E, 
 acting in a plane perpendicular to the axis of M, and at a perpen- 
 dicular distance from given by the equation 
 
 M 
 
 ^=S w 
 
 Case III. When E = Of there, is no single resultant; and the 
 only resultant is the couple M. 
 
 Case IV. When the axis of M. is parallel to the line of action of 
 E, that is, when either 
 
 A = «j fi = li; » = y, (11.) 
 
 or 
 
 ?i= -a; /*= -/3j »= -y; (12.) 
 
 there is no single resultant; and the system of forces is equivalent 
 to the force E and the couple M, being incapable of being farther 
 simplified. 
 
 Case V. — When the aads of M is oblique to the direction of E, 
 making with it the angle given by the equation 
 
 cos 6 = cos A. cos a + cos It cos iS + cos 1/ cos y,....(13.) 
 
 the couple M is to be resolved into two rectangular components. 
 
 M sin 6 round an axis perpendicular to E, and in 
 the plane containing the direction of E and of 
 the axis of M; 
 
 M cos 6 round an axis parallel to E. 
 
 (14.) 
 
 The force E and the couple M sin 6 are equivalent, as in Case 
 II., to a single force equal and parallel to E, whose line of action 
 is in a plane perpendicular to that containing E and axis of M, 
 and whose perpendicular distaflce from is 
 
 L = ^|^ (15.) 
 
 The couple M cos 6, whose axis is parallel to the line of action of 
 E, is incapable of further combination. 
 
 Hence it appears finally, that every system of forces which is not 
 self-balanced, is equivalent either, (A); to a single force, as in 
 
138 PIIINCIPLE8 OP STATICS. 
 
 Cases I. and II. (B); to a couple, as in Case III. (C); to a 
 force, combined with a couple whose axis is parallel to the line of 
 action of the force, as in Cases IV. and V. This can occur with 
 inclined forces only; for the resultant of any number of parallel 
 forces is either a single force or a couple. 
 
 237. Parallel Projections or Transformations in Statics. — If two 
 figures be so related, that for each point in one there is a corre- 
 sponding point in the other, and that to each pair of equal and 
 parallel lines in the one, there corresponds a pair of equal and 
 parallel lines in the other, those figures are said to be parallel 
 PKOJECTiONS of each other. 
 
 The relations between such a pair of figures is expressed alge- 
 braically as follows : — Let any figure be referred to axes of co- 
 ordinates, whether rectangular or oblique; let as, y, z, denote the 
 co-ordinates of any point in it, which may be denoted by A : let a 
 second figure be constructed from a second set of axes of co-ordinates, 
 either agreeing with, or difiering from, the first set as to rectangu- 
 larity or obliquity; let x', y', z', be the co-ordinates in the second 
 figure, of the point A' which corresponds to any point A in the 
 first figure, and let those co-ordinates be so related to the co-ordi- 
 nates of A, that for each pair of corresponding points. A, A', in 
 the two figures, the three pairs of corresponding co-or'dinates shall 
 bear to each other three constant ratios, such as 
 
 ar' y' , «' 
 
 - =a; — —b; - =c; 
 X y z 
 
 then are those two figures parallel projections of each other. 
 
 For example, all circles and ellipses are parallel projections of 
 each other; so are all spheres, spheroids, and ellipsoids; so are all 
 triangles; so are all triangular pyramids; so are all cylinders; so 
 are all cones. 
 
 The following are the geometrical properties of parallel projec- 
 tions which are of most importance in statics : — 
 
 I. A parallel projection of a system of three points, lying in one 
 straight line and dividing it in a given proportion, is also a system 
 of three points, lying in one straight line and dividing it in the 
 same proportion. 
 
 II. A parallel projection of a system of parallel lines, whose 
 lengths bear given ratios to each other, is also a system of parallel 
 lines whose lengths bear the same ratios to each other. 
 
 III. A parallel projection of a closed polygon is a closed 
 polygon. 
 
 IV. A parallel projection of a parallelogram is a parallelogram. 
 
 V. A parallel projection of a parallelopiped is a paraUelopiped. 
 
 ' VI. A parallel projection of a pair of parallel plane surfaces, 
 
PARALLEL PEOJECTIONS OR TRANSFORMATIONS IN STATICS. 139 
 
 ■whose areas are in. a given ratio, is also a pair of parallel plane 
 surfaces, whose areas are in the same ratio. 
 
 "VII. A parallel projection of a pair of volumes having a given 
 ratio, is a pair of volumes having the same ratio. 
 
 The following are the mechanical properties of parallel projec- 
 tions in connection with the principles set forth in this section : — 
 
 "VIII. If two systems of points be parallel projections of each 
 other; and if to each of those systems there be applied a system of 
 parallel forces bearing to each other the same system of ratios, then 
 the centres ofpa/rallel forces for those two systems of points will be 
 parallel projections of each other, mutually related in the same 
 manner with the other ^irs of corresponding points in the two 
 systems. 
 
 IX. If a balanced system of forces acting through any system of 
 points be represented by a system of lines, then wiU any parallel 
 projection of that system of lines represent a balanced system of 
 forces; and if any two systems of forces be represfented by lines 
 which are parallel projections of each other, the lines, or sets of 
 lines, representing their resultants, axe corresponding parallel pro- 
 jections of each other, — it being observed that couples are to be 
 represented by pairs of lines, as pairs of opposite forces, or by areas, 
 and not by single lines along their axes. 
 
140 
 
 CHAPTEE III. 
 
 DISTRIBUTED FORCES. 
 
 Section 1. — Centres op Gravity. 
 
 238. Centre of Gravity of a Symmetrical Homogeneous Body. 
 — If a body is homogeneous, or of equal specific gravity throughout, 
 and so far symmetrical as to have a centre of figure ; that is, a 
 point within the body, which bisects every diameter of the body 
 drawn through it, that point is also the centre of gravity of the 
 body. 
 
 Amongst the bodies which answer this description, are the 
 sphere, the ellipsoid, the circular cylinder, the elliptic cylinder, 
 prisms whose bases have centres of figure, and parallelepipeds, 
 whether right or oblique. 
 
 239. The Common Centre of Gravity of a Set of Bodies whose 
 several centres of gravity are known, is the centre of parallel forces 
 for the weights of the several bodies, each considered as acting 
 through its centre of gravity. (See Article 234, p. 133.) 
 
 240. Planes of Symmetry — Axes of Symmetry. — If a homogeneous 
 body be of a figure which is si/mmetrical on either side of a given 
 plane, the centre of gravity is in that plane. If two or more such 
 places of symmetry intersect in one line, or axis of symmetry, the 
 centre of gravity is in that axis. If three or more planes of 
 symmetry intersect each other in a point, that point is the centre 
 of gravity. 
 
 241. To find the Centre of Gravity of a Homogeneous Body of 
 any Figure, assume three rectangular co-ordinate planes in any 
 convenient position, as in fig. 95, p. 133. 
 
 To find the distance of the centre of gravity of the body from 
 one of those planes (for example, that of Y Z), conceive the body 
 to be divided into indefinitely thin plane layers parallel to that 
 plane. Let s denote the area of any one of those layers, and d x 
 its thickness, so that sdx is the volume of the layer, and 
 
 ' — j sdx, 
 
 the volume of the whole body, being the sum of the volumes of 
 
CENTRE OP GRAVITY FOUND BY SUBTRACTION. 
 
 141 
 
 the layers. Let x be the perpendicular distance of the centre of 
 the layer sdx from the plane of Y Z. Then the perpendicular 
 distance a;„ of the centre of gravity of the body from that plane is 
 given by the equation 
 
 x,=h^. (1.) 
 
 rind, by a similar process, the distances y„ z„, of the centre of 
 gravity from the other two co-ordinate planes, and its position will 
 be completely determined. 
 
 If the centre of gravity is previously known to be in a particular 
 plane, it is sufScient to finjj by the above process its distances from 
 two planes perpendicular to that plane and to each other. 
 
 If the centre of gravity is previously known to be in a particular 
 line, it is sufficient to find its distance from one plane, perpendicular 
 to that line. 
 
 242. If the Specific Gravity of the Body Varies, let w be the 
 mean heaviness of the layer s dsc, so that 
 
 W = j w sdx, 
 is the weight of the body. Then 
 
 J xwsdx 
 
 '' w ■ 
 
 .(2.) 
 
 243. Centre of Gravity found by Addition. — When the figure of 
 a body consists of parts, whose res])eotive centres of gravity are 
 known, the centre of gravity of the whole is to be found as in 
 Article 239. 
 
 244. 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, Gj its 
 known centre of gravity, Wj its weight. 
 Let A B E be the smaller figure, whose 
 centre of gravity Ga is known, Wj its 
 weight. Let E B C D be the figure whose 
 centre of gravity Gg is sought, made by taking away ABE from 
 A C D, so that its weight is 
 
 W3 = Wi-W2. 
 
 Fig. 97. 
 
142 
 
 PRINCIPLES OF STATICS. 
 
 Join Gi Graj Gg ''^ill ^^ i^ t^^ prolongation of that straight line 
 beyond Gj. In the same straight line produced, take any point O 
 as origin of co-ordinates. Make OGi = a;i; O G^ = X2, O Gj (the 
 ■unknown quantity) = 1X3. ■ 
 Then 
 
 _ Xi Wi — aja W 2 
 
 ''^- Wi-W, ^"^-^ 
 
 245. Centre of Gravity Altered by Transposition.— In fig. 98, 
 let A B C D be a body of the weight W„, 
 whose centre of gravity Go 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 V D H, 
 so that the new figure of the body is A B 
 H E. Let Gj be the original, and G^ 
 the new position of the centre of gravity 
 of the transposed part. Then the centre 
 of gravity of the whole body will be shifted 
 3- to G3, in a direction G^ Gg parallel to 
 Fig. 98. G2 Gi, and through a distance given by 
 
 the formula. 
 
 w 
 
 .(4.) 
 
 246. Centre of Gravity found by Projection or Transformation. 
 — If the figures of two homogeneous bodies are parallel projections 
 of each other, the centres of gravity of those two bodies are corres- 
 ponding points in those parallel projections. 
 
 To express this symbolically, — as in Article 237, let x, y, z, be 
 the co-ordinates, rectangular or oblique, of any point in the figure 
 of the first bodyj x', y, z', those of the corresponding point in the 
 second body; x„ y:„, «„, the co-ordinates of the centre of gravity of 
 the first body; x'„ y „ z'„ those of the centre of gravity of the 
 second body, then 
 
 »» x' y„ y' «o «' 
 
 .(5.) 
 
 This theorem facilitates much the finding of the centres of gravity 
 of figures which are "parallel projections of more simple or more 
 symmetrical figures. 
 
STRESS — ITS INTENSITY. 
 
 143 
 
 For example, let it be supposed that the centre of gravity of a 
 sector of a circular disc has been 
 found (Case IX. Article 44), and let 
 it be required to find the centre of 
 gravity of a sector of an elliptic 
 disc. In fig. 99, let A B' A B' be 
 the ellipse, A O A = 2 a, and 
 B' O B' = 2 6, its axes, and C D' 
 the sector whose centre of gravity 
 is required. About the centre of 
 the ellipse, O, describe the circle, 
 A B A B, whose radius is <Jie semi- 
 axis major a. Through (7 and D' 
 respectively draw E C C and FD'D, 
 parallel to O B, and cutting the 
 circle in C and I) respectively; the circular sector C O 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 
 
 GH = a!,; HG = y, 
 
 Then the co-ordinates of the centre of gravity G' of the sector of 
 the elliptic disc are 
 
 
 ^ 
 
 -H'^ 
 
 Y 
 
 c 
 
 A 
 
 fi 
 
 
 \-] 
 
 &i ^ 
 
 \ 
 
 F 
 
 ^ 
 
 s 
 
 A 
 
 
 ^ 
 
 E HP 
 
 " "^ 
 
 \ 
 
 V 
 
 ■v. 
 
 -^^ 
 
 B' 
 
 7 
 
 Fig. 99. 
 
 OH = a!'„ = a!„; \ 
 
 .(6.) 
 
 247. Centre of Gravity found Experimentally.— The centre of 
 gravity of a body of moderate size may be found approximately by 
 experiment, by hanging it up successively by a single cord in two 
 different positions, and finding the single point in the body which 
 in both positions is intersected by the axes of the cord. 
 
 Section 2. — Or Stress. 
 
 248. Stress — ^its 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 dis- 
 tributed over the surface of contact of the masses between which 
 they act. 
 
 The Intensity of a stress is its amount in units of weight, 
 divided by the extent of the surface over which it acts, in units 
 of area. 
 
144 PRINCIPLES OF STATICS. 
 
 The following table gives a comparison of various units in whicli 
 the intensity of stress is expressed : — 
 
 Founds on the Fonnda on the 
 
 square loot. square inch. 
 
 One pound on the square inch,.... 144 1 
 
 One pound on the square foot 1 y^ 
 
 One inch of mercury(that is, weight 
 
 of a column of mercury at 32° 
 
 Fahr., one inch high), 70-.73 0-4912 
 
 One foot of water (at 39°-lFahr.), 62-425 0-4335 
 
 One inch of water (at 39°-l Fahr.), 5-2021 0-036125 
 
 One foot of -water (at 62° Fahr.),... 62355 0-43302 
 
 One inch of water (at 62° Fahr.),... 5-19625 0036085 
 
 One atmosphere, of 29-922 inches 
 
 of mercury, or 760 millimetres, 2116-4 14-7 
 
 One foot of air, at 32° Fahr., and 
 
 under the pressure of one atmo- 
 sphere, 0-080728 0-0005606 
 
 One kilogramme on the square 
 
 mfetre, 0-20481 0-00142228 
 
 One kilogramme on the square 
 
 miUimltre, 204810 1422-28 
 
 One millimetre of mercury, 2-7847 0-01934 
 
 249. Classes of Stress. — The various kinds of stress may be thttS 
 classed : — 
 
 I. Thrust, or Pressure, is the force which acts between two con- 
 tiguous bodies, or parts of a body, when each pushes the other from 
 itself. 
 
 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. 
 
 Pressure and tension may be either normal or oblique, relatively 
 to the surface at which they act. 
 
 III. Shewr, or Tcmgential Stress, is the force which acts between 
 two contiguous bodies, or parts of a body, when each draws the 
 other sideways, in a direction parallel to their surface of contact. 
 
 In expressing a Thrust and a Pull in parallel directions alge- 
 braically, 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. 
 
 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. 
 
 The following are the processes for finding the magnitude of the 
 resultant of a stress distributed over a plane surface, and the centre 
 
IN STIIESS OF VARYING INTENSITY. 145 
 
 of stress; that is, the point where the line of action of that resultant 
 cuts the plane surface : — 
 
 250. In Stress of Unifonn Intensity, the magnitude of the re- 
 sultant is the product of that intensity and the area of the surface; 
 and the centre of stress is at the centre of magnitude of the surface. 
 Or in symbols, let S be the area of the surface, p the intensity of 
 the stress, P its resultant, then — 
 
 P=^S. 
 
 251. In Stress of Varying Intensity, but of One Sign, there is 
 all tension, or all pressure, or all shear in one direction. 
 
 In fig. 100, let A A be the given plane surface at which the stress 
 acts; X, O Y, two rectangular 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 parallel to Z 
 
 round the outline of A A, and at the other 
 
 ' w' 1 nn ^^^ ^y ^ surface B B, of such a figure, that its 
 
 Jiig. lUO. ordinate « at any point shall be proportional to 
 
 the intensity of the stress at the point a of the surface A A from 
 which that ordinate proceeds, as shewn by the equation 
 
 « = ^ (1.) 
 
 w 
 
 where p represents the intensity of the stress and w the heaviness, 
 or weight per unit. 
 
 Conceive the surface A A to be divided into an indefinite number 
 of small rectangular areas, each denoted hy dxdy, and so small 
 that the stress on each is sensibly uniform; the entiire area being 
 
 S = / I dxdy. 
 
 The volume of the ideal solid will be 
 
 Y=j je-dxdy (2.) 
 
 So that if it be conceived to consist of a material whose heaviness 
 
 is w = -, the amount of the stress will be equal to the weight of the 
 
 z 
 solid; that is to say, 
 
 '= / I pdxdy = wY (3.) 
 
146 PRINCIPLES OF STATICS. 
 
 TJhe centre of stress is the point on the surface A A perpendicu- 
 larly opposite the centre of gravity of the ideal solid. 
 
 The simplest, and at the same time the commonest, case of this 
 kind is where the stress is uniformly-varying; that is, where its 
 intensity at a given point is simply proportional to the per- 
 pendicular distance of that point from a given straight line in 
 the plane of the surface A A. To express this symboUically, 
 take the straight line in question for the axis O Y; conceive 
 the substance to be divided into bands by lines parallel to Y ; 
 let y denote the length of one of these bands, and d x its breadth, 
 
 so that y dx is its area, and S = I ydx the area of the whole 
 
 surface. Let x be the perpendicular distance of the centre of 
 a band from the line of no stress O Y, and let the intensity of the 
 stress there be 
 
 P = ax; (4.) 
 
 a being a constant coefficient; then the amount or' resultant of 
 the stress is 
 
 F= J pydx = a fssydx; (5.) 
 
 and the perpendicular distance of the centre of stress from Y is 
 j pxydx \ x^y dx; 
 
 j pydx- 
 
 .(6.) 
 
 252. In Stress of Contrary Signs, for example, pressure at 
 one part of the surface and tension at another, the resultants 
 and centres of stress of the pressure and tension are to be 
 found separately. Those partial resultants are then to be treated 
 as a pair of parallel forces acting through the two respective 
 centres of stress; their final resultant will be equal to their 
 difference, if any, acting through a point found as in Article 226, 
 page 128. 
 
 If the total pressure and total tension are equal to each other, 
 they have no single resultant and no single centre of stress : their 
 resultant being a couple, whose moment is equal to the total 
 stress of either kind multiplied by the perpendicular distance 
 between the resultant of the pressure and the resultant of the 
 tension. 
 
principles of hydrostatics. 147 
 
 Section 3. — Principles of Hydrostatics and Internal 
 Stress of Solids. 
 
 253. Pressure and Balance of Fluids: Principles of Hydro- 
 statics. — Fluid, is a term opposed to solid, and comprehending the 
 liquid and gaseous conditions of bodies. The property common to 
 the liquid and the gaseous conditions is that oinot tending to preserve 
 a definite shape, and the possession of this property by a body in 
 perfection throughout all its parts, constitutes that body a perfect 
 fluid. 
 
 A necessary consequence of that property is the following prin- 
 ciple, -which is the foundation of the whole science of hydro- 
 statics: — • 
 
 I. In a perfect fluid, when still, the pressure exerted at a given 
 point is normal to the surface on which it acts, and of equal intensity 
 for all positions of that surface. 
 
 The following are some of the most useful consequences of that 
 principle : — 
 
 II. A surface of equal pressure in a still fluid mass is everywhere 
 perpendicular to the direction of gravity; that is, horizontal through- 
 out. In other words, the pressure at all points at the same level is 
 of equal intensity. 
 
 III. The intensity of the pressure at the lower of tv)o points in a 
 still fluid mass is greater than the intensity at the higher point, by 
 an amount equal to the weight of a vertical column of the fluid whose 
 height is the difference of elevation of the points, and base an unit of 
 area. 
 
 To express this symbolically, let p^ denote the intensity of the 
 pressure at the higher of two points in a fluid mass, and jt>j the 
 intensity at a point whose vertical depth below the former point is 
 X. Let w be the mean heaviness of the layer of fluid between those 
 two points; then 
 
 p^=^p^ + wx (1.) 
 
 In a gas, such as air, w varies, being nearly proportional to p ; but 
 in a liquid, such as water, the variations of w are too small to be 
 considered in practical cases. 
 
 For example, let the upper of the two points be the surface of a 
 mass of water where it is exposed to the air; then p„ is the atmos- 
 pheric pressure; let the depth x of the second point below the 
 surface be given in feet, and let the temperature be 39°-l; then 
 
 p-y in lbs. on the square foot=jOo+ 62-425 x (2.) 
 
 In many questions relating to engineering, the pressure of the 
 atmosphere may be left out of consideration, as it acts with sensibly 
 equal intensity on all sides of the bodies exposed to it, and so 
 balances its own action. The pressure calculated, in such cases, is 
 
148 PEINCIPLES OF STATICS. 
 
 the excess of the pressure of the water above the atmospheric 
 pressure, which may be thus expressed, — 
 
 p =Pi-po = G2'4:25 a; nearly (3.) 
 
 IV. The pressure of a liquid on a floating or immersed body, is 
 equal to the weight of the volume of fluid displaced by that body; 
 and the resultant of that pressure acts vertically upwards through 
 the centre of gravity of that volume; which centre of gravity is 
 called the " centre of hvayancy." 
 
 V. The pressure of a liquid" against a plane surface im/mersed in 
 it is perpendicular to that surface in direction i its magnitude is 
 equal to the weight of a volume of the liquid, found by multiplying 
 the area of the surface by the depth to which its centre of gravity 
 is immersed. 
 
 VI. The centre of pressure on such a surface, if the surface is 
 horizontal, coincides with its centre of gravity; if the surface is 
 vertical or sloping, the centre of pressure is always below the centre 
 of gravity of the surface, and is found by considering that the 
 pressure is an uniformly-varying stress, whose intensity at a given 
 point varies as the distance of that point from the line where the 
 given plane surface (produced if necessary) intersects the upper 
 surface of the liquid. 
 
 To express the last two principles by symbols in the case in 
 which the pressed surface is vertical or sloping, let the line where 
 the plane of that surface cuts the upper surface of the liquid be 
 taken as the axis O Y. Let S denote the angle of inclination of 
 the pressed surface to the horizon. Conceive that surface to be 
 divided by pai'allel horizontal lines into an indefinite number of 
 narrow bands. Let y be the length of any one of those bands, d x 
 its breadth, x the distance of its centre from Y; then ydxia its 
 area, x sin 6 the depth at which it is immersed; and if w be the 
 weight of unity of volume of the fluid, the intensity of the pressure 
 on that band is 
 
 p = wx&mS (4.) 
 
 The whole area of the pressed surface, being the sum of the areas 
 
 of all the bands, is S = ydx; the whole pressure upon it is 
 
 P= I py d x — wsm sixydx; (5.) 
 
 the mean intensity of the pressure is 
 
 P 
 
 j pydx j xydx 
 
 = wsin $ ; (6.) 
 
 I y dx I ydx 
 
«'o = p =— (7.) 
 
 COMPOUND INTERNAL STRESS OP SOLIDS. 149 
 
 and the distance of the centre of pressure from Y is 
 Ixpydx ix^ydx 
 I xydx 
 
 For example, let the sloping pressed surface be rectangular, like 
 a sluice, or the back of a reservoir-wall j and in the first instance, 
 let it extend from the surface of a mass of water down to a distance 
 x^, measured along the slope, so that its lower edge is immersed to 
 the depth x-^ sin 6. Then its centre of gravity is immersed to the 
 depth T-^^ sin 6-^i, and £he mean intensity of the pressure in lbs. on 
 the square foot, is 
 
 P _ &2-i:X^s\n.6 
 
 S~ 2 ^ ' 
 
 The breadth y is constant; so that the area of the surface is 
 S = »! y; and the total pressure is 
 
 ^J2-ia^y.\n6 ^^ ^ 
 
 The distance of the centre of pressure from the upper edge is 
 
 2 
 a"o = 3 ^1- (10.) 
 
 Next, let the upper edge, instead of being at the surface of the 
 water, be at the distance x^ from it, so as to be immersed to the 
 depth Kg sin 6. Then the centre of gravity of the pressed surface 
 is immersed to the depth {x^ + x^ sin 6^'i, and the mean intensity 
 of the pressure upon it, in lbs. on the square foot, is 
 
 P _ 624 (a!i + x^ sin 6 .... 
 
 S= 2 ' *■''••' 
 
 the area of the surface is (a^ - x^ y, and the total pressure on it 
 
 p^ 62-4(aj-a|)ysia<l ,j2 . 
 
 The distance of the centre of pressure from the line Y is 
 
 -o = |-S-^ (13-) 
 
 254. Compound Internal Stress of Solids.— If a body be con- 
 ceived to be divided into two parts by an ideal plane traversing it 
 in any direction, the force exerted between those two parts at the 
 plane of division is an internal stress. 
 
150 PEINCIPLES OF STATICS. 
 
 According to the principles stated in the preceding article, the 
 internal stress at a given point in a fluid is normal and of equal 
 intensity for all positions of the ideal plane of division. In a solid 
 body, on the other hand, the stress may be either normal, oblique, 
 or shearing; and it may vary in direction and intensity, as the 
 position of the ideal plane of division varies. 
 
 255. Conjugate Stresses — Principal Stresses. — If two planes 
 traverse a point in a body, and the direction of the stress on the 
 first plane is parallel to the second plane, then the direction of the 
 stress on the second plane is parallel to the first plane. Such a 
 pair of stresses are said to be conjugate; and if they are both 
 normal to their planes of application (and consequently perpendi- 
 cular to each other) they are called principal stresses. Three con- 
 jugate stresses, or three principal stresses, may act through one 
 point; but in the present treatise it is sufiicient to consider two. 
 
 Fig. 101 represents a pair of conjugate oblique tensions acting 
 in the direction X X and Y Y through 
 a prismatic particle A B D. 
 
 The rectangular directions in which 
 principal stresses — that is, direct pulls . 
 and thrusts — act, through a given point 
 in a solid, are called axes of stress, 
 
 la a fluid, the stress at a given point 
 being of equal intensity in all directions, 
 every direction has the property of an 
 'fib. 101. ^^^^ °^ stress. A solid may be in the 
 
 same condition with a fluid as to stress; 
 but it may also have the principal stresses at a given point of 
 different intensities. In a mass of loose grains, the ratio of those 
 intensities has a limit depending on friction : — in a firm continuous 
 solid, the principal stresses at a point may bear any ratio to each 
 other, and may be either of the same or of opposite kinds. 
 
 256. The Shearing Stress, on two planes traversing a point in a 
 solid at right angles to each other, is of equal intensity. 
 
 257. A Pair of Equal and Opposite Principal Stresses ; that is, 
 a pull and a thrust of equal intensity acting through a particle of a 
 solid in directions 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 of planes. 
 
 258. Combination of any Two Principal Stresses. 
 
 Pboblem. — A pair of principal stresses of any intensities, aild 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. 
 
 Xi 
 
COMBINATION OP ANY TWO PRINCIPAL STRESSES. 
 
 151 
 
 Let O X and Y (figs. 102 and 103) be the directions of the 
 two principal stresses; OX being the direction of the greater 
 stress. 
 
 Let p^ be the intensity of the greater stress; 
 
 and p^ that of the less. 
 
 Fig. 102. 
 
 The kind of stress to which each of these belongs, puU 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. 102 represents the case in which p^ and P2 are of the same 
 kind; fig. 103 the case in which they are of opposite kinds. In all 
 the following equations, the sign oi p2 is held to be implied in that 
 symbol; that is to say, when p^ is of the contrary kind to p^, the 
 sign applied to its arithmetical value, in computing by means of 
 the equations, is to be reversed. 
 
 Let A B be the plane on which it is required to ascertain the 
 direction and intensity of the stress, and N a normal to that 
 plane, making with the axis of greatest stress the angle 
 
 ZXO]Sr = a;m. 
 
 On O N take OM = „^' ; 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. 
 
 Through M draw P M Q, making with the axes of stress the 
 same angles which N" makes, but in the opposite direction ; that 
 
'■^^ PRINCIPLES OF STATICS. 
 
 is to say, take ITP =irQ = M~0. On the line thus found set ofi 
 from M tow, take axis of greatest stress, ITE^P^ZP". 
 
 Join O jK. Then -will that line represent the direction and 
 intensity of the stress on A B. 
 
 In fig. 102, pi and p^ are represented as being of the same kind ; 
 and M K is consequently less than O M, so that OK falls on the 
 
 same side of O X with N; that is to say, nr<xn. In fig. 103, 
 Pi and Pi are of opposite kinds, ME, is greater than O M, and O R 
 
 fells on the opposite side of X to M ; that is to say, nr>xn. 
 
 The locus of the point M is a circle of the radius ^-i^» ^^^ 
 
 Z 
 that of the point E, an ellipse whose semi-axes are p^ and p^, and 
 which may be ca,lled the Ellipse op Steess, because its semi- 
 diameter in any direction represents the intensity of the stress in 
 that direction. 
 
 259. Deviation of Principal Stresses by a Shearing Stress. — 
 Problem. Let p^ and p^ denote the original intensities of a pair ' 
 of principal stresses acting at right angles to each other through 
 one particle of a solid. Suppose that with these there is combined 
 a shearing stress of the intensity q, acting in the same plane with 
 the original pulls or thrusts; it is required to find the new inten- 
 sities and new directions of the principal stresses. 
 
 To assist the conception of this problem, the original stresses 
 referred to are represented in fig. 104, as acting through a particle 
 of the form of a square prism. The principal 
 stresses, both original and new, are represented 
 as tensions, although any or all of them might 
 be pressures. In the formulae annexed, tensions 
 are considered positive, pressures negative; 
 angles lying to the right of A A are considered 
 as positive, to the left as negative ; and a shear- 
 ing stress is considered as positive or negative 
 according as it tends to make the upper right- 
 hand and lower left-hand corner of the square 
 particle acute or obtuse. 
 
 The arrows A A represent the greater original 
 Fig- 104. tension p^; the arrows B B, the less original 
 
 tension p„; C, C, D, D, represent the positive shear of the inten- 
 sity q, as acting at the four faces of the particle. The combination 
 of this shear with the original tensions is equivalent to a new pair 
 of principal tensions, oblique to the original pair. The greater new 
 
.(3.) 
 
 FRICTION. 133 
 
 principal tension, p^, is represented by the arrows E, E ; it deviates 
 to the right of p^ through an angle which will be denoted by t. 
 The less new principal tension p^ is represented by the arrows F, F ; 
 it deviates through the same angle to the right of p,. 
 
 Then the intensities of the new principal stresses are given by 
 the equations, 
 
 and the double of the angle of deviation by either of the following, 
 tan2i'= -^-^■, or cotan 2 (I =^V^- (4.) 
 
 The greatest value of i is 45°, when p^ — p„. 
 The new principal stresses are to be conceived as acting normally 
 on the faces of a new square prism. 
 
 260. Parallel Projection of Distributed Forces. — In applying 
 the principles of parallel projection to distributed forces, it is to be 
 borne in mind that those principles, as stated in Article 237, are 
 applicable to lines representing the amounts or resultants of distri- 
 buted 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. 
 
 261. Friction is that force which acts between two bodies at 
 their surface of contact so as to resist their sliding on each other, 
 and which depends on the force with which the bodies are pressed 
 together. It is a kind of shearing stress. 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 ofeocerting, is simply proportional 
 to the force with which they are pressed together. 
 
 If a body be acted upon by a force tending to make it slide on 
 another, then so long as that force does not exceed 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 indent the parts of the 
 bodies at and near their surface of contact. At and loeyond that 
 limit the friction increases more rapidly than the pressurej but 
 
154 PJJINCIPLES OF STATICS. 
 
 tliat limit ought never to be attained in any structure. For some 
 substances, 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. That excess, how- 
 ever, 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 taken into account, as contributing to the stability of 
 a structure. 
 
 The friction between a pair of surfaces is calculated by multiply- 
 ing the force with which they are directly pressed together, by a 
 factor called the coefficient of friction, which has a special value 
 depending on the nature of the materials and the state of the 
 surfaces. Let F denote the friction between a pair of sur- 
 faces; N, the force, in a direction perpendicular to the surfaces, 
 with which they are pressed together; and / the coefficient of 
 friction; then 
 
 F=/N (1.) 
 
 The coefficient of friction of a given pair of surfaces is the 
 tangent of an angle called the angle of repose, being the greatest 
 angle which an oblique pressure between the surfaces can make 
 with a perpendicular to them, without making them slide on aceh 
 other. 
 
 Let P denote the amount of an oblique pressure between two 
 plane surfaces, inclined to their common normal at the angle of 
 repose ?; then 
 
 F=/lsr = N tan ?i = Psin (P= ■- ,d=, (2.) 
 
 V 1 +/^ 
 
 The angle of repose is the steepest inclination of a plane to the 
 horizon, at which a block of a given substance will remain bal- 
 anced on it without sliding down. 
 
 The intensity of the friction between two surfaces bears the same 
 proportion to the intensity of the pressure that the whole friction 
 bears to the whole pressure. 
 
 The following is a table of the angle of repose ip, the coefficient 
 of friction /= tan p, and its reciprocal 1 : /, for various materials — 
 condensed from the tables of General Morin, and other sources, and 
 arranged in a few comprehensive classes. The values of those 
 constants which are given in the table have reference to the friction 
 of motion. 
 
FKICTION. 
 
 155 
 
 Surfaces. 
 
 / 
 
 1 
 / 
 
 Dry masonry and brickwork, 
 
 Masonry and brickwork with wet 
 
 mortar, ' 
 
 Masonry and brickwork, with 
 
 slightly damp mortar, 
 
 Wood on stone 
 
 Iron on atone, 
 
 Masonry on dry clay, 
 
 ,, on moist clay>k 
 
 Earth on earth, 
 
 ,, „ dry sand, clay, 
 
 and mixed earth 
 
 Earth on earth, damp clay, 
 
 „ wet clay, 
 
 ,, ,; shingle and gravel, 
 
 Wood on wood, dry, 
 
 „ „ soaped, 
 
 Metals on oak, dry, .,o>,.. 
 
 » „ wet, 
 
 „ „ soapy, 
 
 Metals on elm, dry, 
 
 Bronze on lignum vitse, constantly ) 
 
 wet,..., \ 
 
 Hemp on oak, dry, 
 
 » „ wet, 
 
 Leather on oak, 
 
 Leather on metals, dry, 
 
 „ „ wet, , 
 
 „ „ greasy 
 
 oily, 
 
 Metals on metals, dry 
 
 ,, ,, wet and clean, . . 
 
 „ ,, damp and sliiny. 
 
 Smooth surfaces, occasionally ) 
 
 Smfloth surfaces, continually ) 
 
 Smoothest and best greased surfaces, 
 
 31° to 35° 
 
 36J° 
 
 22° 
 35° to 16J° 
 
 27° 
 
 18i° 
 14° to 45° 
 
 21° to 37° 
 
 45° 
 
 17° 
 35° to 48° 
 14° to 264o 
 llio to 2" 
 26*0 to 31° 
 13*° to 14J° 
 
 114" 
 Hi" to 14 
 
 3°? 
 
 28° 
 184° 
 15° to 194° 
 294° 
 20° 
 13° 
 
 84° to 114° 
 
 164° 
 
 8° 
 
 4° to 4J° 
 
 3° 
 
 1-1° to 2o 
 
 0-6 to 0-7 
 0-47 
 
 074 
 
 about 0'4 
 0-7 to 0-3 
 
 0-51 
 
 0-33 
 0-25 to 10 
 
 0-38 to 0-75 
 
 10 ■ 
 
 0-31 
 
 0-7 to I-H 
 
 •25 to -5 
 
 •2 to -04 
 
 •5 to -6 
 
 ■24 to -26 
 
 •2 
 .2 to -25 
 
 •05? 
 
 ■53 
 •33 
 
 •27 to ^38 
 
 ■56 
 
 ■36 
 
 •23 
 
 ■15 
 •15 to -2 
 
 •3 
 
 •14 
 
 •07 to ^08 
 
 •05 
 •03 to ^036 
 
 1^67 to 1^43 
 2-1 
 
 1^35 
 
 2*5 
 
 1-43 to 3^33 
 
 1-96 
 
 3 
 
 4 to 1 
 
 2-63 to 1-33 
 
 1 
 
 3-23 
 
 1^43 to •a 
 
 4 to 2 
 
 5 to 25 
 2 to 1^67 
 
 4'17 to 3-85 
 
 5 
 
 5to4 
 
 20? 
 
 I •SO 
 
 3 
 
 3-7 to 2-86 
 
 r79 
 
 2^78 
 
 4-35 
 
 6-67 
 
 6^67 to 5 
 
 3-33 
 
 7^14 
 
 14-3 to 12-5 
 
 20 
 33 3 to 27^6 
 
PAET lY. 
 
 THEORY OF STEUCTUEES. 
 
 CHAPTEE I. 
 
 STJMMAHY OF PRINCIPLES OF STABILITY AND STRENGTH. 
 
 Section 1. — Of Structures in General. 
 
 262. A Structure consists of portions of solid materials, put 
 together so as to preserve a definite form and arrangement of parts, 
 and to withstand external forces tending to disturb such form and 
 arrangement. As the parts of a structure are intended to remain 
 at rest relatively to each other, the forces which act on the whole 
 structui-e, and on each of its parts, should be balanced, so that the 
 mechanical principles on which the permanence and eflSciency of 
 structures depend for the most part belong to Statics, or the 
 science of balanced forces. 
 
 The materials of a structure may be more or less stiff, like stone, 
 timber, and metals, or loose, like earth. 
 
 In the present chapter are given a summary of mechanical 
 principles applicable to structures. 
 
 263. Pieces — Joints — Supports — Foundations. — A structure 
 consists of two or more solid bodies, called its pieces, which touch 
 each other and are connected at portions of their surfaces, called 
 joints. This statement may appear to be applicable to structures 
 of stiff materials only; but, nevertheless, it comprehends masses 
 of earth also, if they are considered as consisting of a very great 
 number of very small pieces, touching each other at innumerable 
 joints. 
 
 Although the pieces of a structure are fixed relatively to each 
 other, the structure as a whole may be either fixed or movable 
 relatively to the earth. 
 
 A fixed structure is supported on a part of the solid material of 
 the earth, called the foundation of the structure; the pressures by 
 
STABILITY, STRENGTH, AND STIFFNESS. 157 
 
 ■which the structure is supported, being the resistances of the 
 various parts of the foundation, may be more or less oblique. 
 
 A movable 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 kinetic as -well as of statical principles; but when 
 it is not in actual motion, though capable of being moved, the 
 pressures ■which support it are determined by the principles of 
 statics ; and it is obvious that they have their resultant equal and 
 directly opposed to the ■weight of the structure. 
 
 264. The Conditiolis of Equilibrium of a Structure are the three 
 following : — 
 
 I. Tlmt the forces exerted on the whole structure hy external bodies 
 shall halcmce each other. — The forces to be considered under thi's 
 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 
 foundation. Those three classes of forces will be spoken of together 
 as the External Forces. 
 
 II. That the forces exerted on each piece of the structure sludl 
 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 forces 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 each 
 piece 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 force 
 applied to it, if any, making together its gross load; (3.) the stress, 
 or force, exerted at the ideal surface of di-vision, between the part 
 in question and the other parts of the piece. 
 
 265. Stability, Strength, and Stiffness. — It is necessaiy to the 
 permanence 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 aU 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 i^e first and second condi- 
 tions of equilibrium of a structure under all variations of the load 
 within given limits. A structure which is deficient in stability 
 
158 THEORY OF STRUCTURES. 
 
 gives way by the displacement of its pieces from their proper posi- 
 tions. 
 
 When a structure, or one of its parts, is flexible, like the chain 
 of a suspension bridge, or in any other way free to move, its 
 stability consists in a tendency to recover its original jSgure and 
 position after having been disturbed. 
 
 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 
 breaking, 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 binder 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. 
 
 Section 2. — Balance and Stability of Frames, Chains, 
 AND Blocks. 
 
 266. A Frame is a structure composed of bars, rods, links, or 
 cords, attached together or supported by joints, such as occur in 
 carpentry, in irames 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, or 
 point through which the resultant of the resistance to displacement 
 of the pieces connected at the joint acts, is at or near the middle of 
 the joint, and does not admit of any variation of position consis- 
 tently with security. 
 
 The line of resistance of a frame is a line traversing the centres 
 of resistance of the joints, and is in general a polygon, having its 
 angles at these centres. 
 
 267. A Single Bar in a frame may act as a Tie, a Strut, or a 
 Beam. 
 
 I. A tie has equal and directly opposite forces applied to its two 
 ends, acting outwards, or from each other. The bar is in a state 
 of tension, and the stress exerted between any two divisions of it 
 is a pull, equal and opposite to the applied forces. A rope or 
 chain will answer the purpose of a tie. 
 
 17ie equilibrium of a movable tie is stable; for if its angular posi- 
 
A SINGLE BAB. 159 
 
 tion be devoted, the forces applied to its ends, wbich originally 
 •were directly opposed, now constitute a couple tending to restore 
 the tie to its original position. 
 
 II. A strut has equal and directly opposite forces applied to its 
 two ends, acting inwards, or towards each other. The bar is in a 
 state of compression, and the stress exerted between any two divi- 
 sions of it is a thrust equal and opposite to the applied forces. It 
 is obvious that a flexible body will not answer the purpose of a 
 strut. 
 
 The equilibrium of a movable strut is unstable; for if its angular 
 position be deviated, the forces applied to its ends, which originally 
 were directly opposed, now constitute a couple tending to make it 
 deviate stUl farther f!bin 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. 
 
 III. A beam is a bar supported at two points, and loaded in a 
 direction perpendicular or oblique to its length. 
 
 Case I. — Let the supporting pressures be parallel to each 
 other and to the direction of the load; and let the load act between, 
 the points of support, as in fig. 105; where P u.^ 
 represents the resultant of the gross load, in- ^ I ^ 1 ^ 
 
 Wf 
 
 eluding the weight of the beam itself; L, the ypr 
 
 point where the line of action of that resultant & 
 
 intersects the axis of the beam; Ej, R2> ^^ ''"'' 
 
 two supporting pressures or resistances of the -^S- 105. 
 
 props parallel to, and in the same plane with P, and acting through 
 
 the points Sj, Sg, in the axis of the beam. 
 
 Then, according to the principle of the lever. Article 225, 
 page 128, each of those tliree forces is proportional to the distance 
 between' the lines of action of the other two; and the load is equal 
 to the sum of the two supporting pressures; that is to say, 
 
 P : Rj : Eg : : S^ Sj : L Sj : L S^; (1.) 
 
 andP = R^ + E2 (2.) 
 
 Case II. — Let the load act beyond the points of support, as in 
 fig. 106, which represents a cantilever or project- 
 ing beam, held up by a wall or other prop at S^, 
 held down by a notch in a mass of masonry or 
 otherwise at Sj, and loaded so that P is the re- 
 sultant of the load, including the weight of the 
 .beam. Then the proportional Equation 1. re- "^^j^e* 106 
 mains exactly as before; but the load is equal to 
 the difference of the supporting pressures; that is to say, 
 
 P = El - E^. 
 
160 
 
 THEORY OP STRUCTURES. 
 
 In these examples the beam is represented as horizontal; but the 
 same principles would hold if it were inclined. 
 
 Case III. — Let the directions of the supporting forces Ei, Itj, 
 be now inclined to that of the resultant of 
 the load, P, as in fig. 107. This case is that 
 of the equilibrium of three forces treated of 
 in Article 209, page 122, and consequently 
 the following principles apply to it : — 
 
 The lines of action of the supporting 
 forces and of the resultant of the load must 
 be in one plane. 
 
 They must intersect in one point (0, 
 fig. 107). 
 
 Those three forces must be proportional to the three sides of a 
 triangle A, respectively parallel to their directions. 
 
 Problem. — Given, the resultant of the load in magnitude and 
 position, P, the line of action of one of the supporting forces, Rj, 
 and the centre of resistance of the other, Sj; required, the line 
 ■of action of the second supporting force, and the magnitudes of 
 both. 
 
 Produce the line of action of Ji^, till it cuts the line of action of 
 P at the point C; join C Sjj this will be the line of action of 'R^; 
 construct a triangle A with its sides respectively parallel to those 
 three lines of action; the ratios of the sides of that triangle will 
 give the ratios of the forces. 
 
 To express this algebraically, let ii, %, be the angles made by the 
 lines of action of the supporting forces with that of the resultant 
 of the load; then 
 
 P : Hi : Ej • • sin (i^ + ij) : sin i^ : sin i^. 
 
 .(4.) 
 
 The same piece in a frame may act at once as a beam and tie, or 
 as a beam and strut; or it may act alternately as a strut and as a 
 tie, as the action of the load varies. 
 
 The load tends to break a tie by tearing it asunder, a strut by 
 crushing it, and a beam by breaking it across. The power of 
 materials to resist those tendencies will be considered in a later 
 section. 
 
 268. Distributed Loads. — Before applying the principles of 
 the present section to frames in which the load, whether external 
 or arising from the weight of the bars, is distributed over their 
 length, it is necessary to reduce that distributed load to an equiva- 
 lent 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. 
 
 IL Eesolve that load, as in Article 267, Equation 1, page 159, 
 
FRAMES OP TWO BARS. 
 
 161 
 
 into two parallel components acting through the centres of resist- 
 ance at the two ends of the bar. 
 
 III. At each centre of resistance where two bars meet, combine 
 the component loads due to the loads on the two bars into one 
 resultant, which is to be considered as the total load acting through 
 that centre of resistance. 
 
 IV. When a centre of resistance is also a point of support, the 
 component load acting through it, as found by step II. of the pro- 
 cess, is to be left out of consideration until the supporting force 
 required by the system of loads at the other joints has been deter- 
 :nined; with this supporting force is to be compounded a force 
 equal and opposite tj^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 siipport 
 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. 
 
 269. Frames of Two Bars.— Figures 108, 109, and 110, repre- 
 sent cases in which a frame of two bars, jointed to each at the 
 point L, is loaded at that point with a given force, P, and is sup- 
 
 Fig. 109. 
 
 Fig. 110. 
 
 ported by the connection of the bars at their farther extremities, 
 Si, Sa, with fixed bodies. It is required to find the stress on each 
 bar, and the supporting forces at Sj and Sj. 
 
 Eesolve the load P (as in Article 213, page 123) into two com- 
 ponents, Ri, Eai 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 Si, Sj, 
 are equal and directly opposed. 
 
 The symbolical expression of this solution is as follows: — Let 
 ii, ij, be the respective angles made by the lines of resistance of the 
 bars with the line of action of the load; then 
 
 P : El : Rj : : sin (ii + ij) : sin ij : sin ij. 
 
 M 
 
162 THEORT OP STRtJCTUKES. 
 
 The inward or out-word direction of the forces acting along each 
 har indicates that the stress is a thrust or a pull, and the bar a strut 
 or a tie, as the case may be. Fig. 108 represents the case of two 
 ties; fig. 109 that of two struts (such as a pair of rafters abutting 
 against two walls).; fig. 110 of a strut, L Sj, and a tie, L S^ (such 
 as the jib and the tie-rod of a crane). 
 
 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 Sj, joining the points of sup- 
 port. 
 
 A frame consisting of a strut and a tie, when the direction of 
 the load inclines towards the line Sj Sj, and a frame of two struts 
 in all cases, are unstable laterally, unless provided with lateral 
 
 These principles are true ot any pair of adjacent ha/rs whose farther 
 centres of resistance are fixed; whether forming a frame by them- 
 selves, or a part of a more complex frame. 
 270. Triangular Frames. — Let fig. Ill represent a frame, con- 
 sisting of three bars. A, B, 0, connected at the 
 three joints 1, 2, 3, — viz., and A at 1, A and B 
 at 2, B and at 3. Let a load Pj^ be applied at 
 _ the joint 1 in any given direction; let supporting 
 ■p.", J, ^ forces, Pg, Pg, be applied at the joints 2, 3; the 
 lines of action of those two forces must be in the 
 same plane with that of Pj, and must either be parallel to it or 
 intersect it in one point. The latter case is taken first, because its 
 solution comprehends that of the former. 
 
 p The three external forces balance each other, 
 
 and are therefore proportional to the three sides 
 of a triangle respectively parallel to their direc- 
 tions. In fig. 112, let A B be such a triangle, 
 in which 
 
 C A: 
 
 represents 
 
 Pi. 
 
 AB 
 
 3J 
 
 P2. 
 
 B C 
 
 » 
 
 Pa, 
 
 Draw C O parallel to the bar C, and A O parallel to the bar A, 
 meeting in the point O, and join B 0, which will be parallel to B. 
 
 The lengths of the three lines radiating from O will represent 
 the stresses on the bars to which they are respectively parallel. 
 
 When the three external forces are parallel to each other, the 
 triangle of forces A B C of fig. 112, becomes a straight line C A, as 
 
POLYGONAL FRAME. 163 
 
 in fig. 113, divided into two segments by the point B. Let straight 
 lines radiate from O to A, B, C, respectively parallel 
 to the bars of the frame; then if the load C A be 
 applied at 1 (fig. Ill), A B applied at 2, and B C 
 applied at 3, are the supporting forces required to 
 balance it ; and the radiating lines O A, O B, O C, 
 represent the stresses on the bars A, B, C, respec- 
 tively, as before. 
 
 From O let fall O H perpendicular to C A, the 
 common direction of the external forces. Then that ™ iis^ 
 line will represent a component of the stress, which is 
 of equal amount in eaph bar. When A, as is usually the case, is 
 vertical, O H is horizontal; and the force represented by it is called 
 the " horizontal thrust" of the frame. Horizontal Stress or Resist- 
 ance would be a more precise term; because the force in question 
 is a pull in some parts of the frame, and a thrust in others. 
 
 In fig. Ill, A and C are struts, and B a tie. If the frame were 
 exactly inverted, all the forces would bear the same proportions to 
 each other; but A and C would be ties, and B a strut. 
 
 The trigonometrical expression of the relations amongst the forces 
 acting in a triangular frame, under parallel forces, is as follows : — 
 
 Let a, h, c, denote the respective angles of inclination of the bars 
 A, B, C, to the line O H (that is, in general, to a horizontal line) ; 
 viz., the angles A O H, B H, C O H of fig. 113, then 
 
 Horizontal Stress O H = 7 :;— (1.) 
 
 tan c ± tan a ^ 
 
 Supporting f A B = O H • (tan a + tan i);\ /o \ 
 
 Forces | B C = H • (tan 6 ± tan c); / ^ > 
 
 rp, . f + I is to be used when the two \ opposite directions, 
 ° I - J inclinations are in J the same direction. 
 
 fO A = OH-seca) 
 
 Stresses^ O B = H-sec6 V (3.) 
 
 (O C = OH-secc j 
 
 271. Polygonal Frame.— In fig. 114, let A, B, C, D, E, be the 
 lines of resistance of the bars of 
 a frame connected together at 
 the joints, whose centres of re- 
 sistance are, 1 between A and B, 
 2 between B and 0, 3 between 
 C and D, 4 between D and E, 
 and 5 between E and A. In the Pig. 114. 
 
164 
 
 THEORY OP STRUCTURES. 
 
 Fig. 115. 
 
 figure, the frame consists of fi.ve bars; but the principle is appli- 
 cable to any number. From a point O, in fig. 116, 
 (which may be called the Biagra/m of Forces), draw 
 radiating lines O A, O B, C, O D, O E, parallel 
 respectively to the lines of resistance of the bars; 
 and on those radiating lines take any lengths 
 whatsoever, to represent the stresses on the 
 several bars, which may have any magnitudes 
 within the limits of strength of the material. 
 Join the points thus found by straight lines, so as 
 to form a closed polygon A B C D E A; then the 
 sides of the polygon will represent a system offerees, 
 which, being applied to the joints of the frame, will 
 balance each other ; each such force being applied to the joint 
 between the bars whose lines of resistance are parallel to the pair 
 of radiating lines that enclose the side of the polygon of forces 
 representing the force in question. 
 
 When the external forces are parallel to each other, the polygon 
 of forces of fig. 115 becomes a straight line AD, 
 as in fig. 116, divided into segments by the radiating 
 lines; and each segment represents the external force 
 which acts at the joint of the bars whose lines of 
 resistance are parallel to the radiating lines that 
 bound the segment. Moreover, the segment of the 
 line A D which is intercepted between the radiating 
 lines parallel to the lines of resistance of any two 
 bars whether contiguous or not, represents the re- 
 sultant of the external forces which act at points 
 between the bars. 
 
 Thus, A D represents the total load, consisting 
 of the three portions A B, B C, CD, applied at 
 1, 2, 3, respectively. DA represents the total supporting force, 
 equal and opposite to the load, consisting of the two portions D E, 
 E A, applied at 4 and 5 respectively. A represents the resultant 
 of the load applied between the bars A and ; and similarly for 
 any other pair of bars. 
 
 From O draw H perpendicular to AD; then that line repre- 
 sents a component of the stress, whose amount is the same in each 
 bar of the frame. When the load, as is usually the case, is verti- 
 cal, that component is called the "horizontal thrust" of the frame, 
 and, as in Article 270, might more correctly be called horizontal 
 stress or resistance, seeing that it is a pull in some of the bars and 
 a thrust jn others. 
 
 The trigonometrical expression of those principles is as follows : — 
 Let the force O H be denoted simply by H. 
 Let i, i', denote the inclinations to H of the lines of resistance 
 of any two bars, contiguous or not. 
 
POLYGONAL FRAME — STABIHTT. 165 
 
 Let E, E,', be the respective stresses which act along those bars. 
 Let P be the resultant of the external forces acting through the 
 joint or joints between those two bars. 
 Then 
 
 P = H (tan i ± tan i'); (1.) 
 
 E = II-seci; E' = H-seci' (2.) 
 
 The S j-fp , [ of the tangents of the inclinations is to bo 
 used, according as the inclinations are < • -if, f 
 
 272. Open Polygflhal Frame. — When the frame, instead of being 
 closed, as in fig. 114, is converted into an open frame, by the omis- 
 sion of one bar, such as E, the corresponding modification is made 
 in the diagram of inclined forces, fig. 115, by omitting the lines 
 E, D E, E A, and in the diagram of parallel forces, fig. 116, by 
 omitting the line E. Then, in both diagrams, D O and A. 
 represent the supporting forces respectively, equal and directly 
 opposed to the stresses along the extreme bars of the frame, D and 
 A, which must be exerted by the supports (called in this case 
 abutments), at the points 4 and 5, against the ends of those bars, 
 in order to maintain the equilibrium. 
 
 In the case of parallel loads, the following formulae give the 
 horizontal stress and supporting pressures. 
 
 Let ia and, 4 denote the angles of inclination of the bars D and A 
 respectively. 
 
 Let E^ = D and E„ = A be the stresses along them. 
 Let 2 • P = A D denote the total load on the frame; then, 
 I 
 
 jj[_ 2 • P ... 
 
 tan i^ + tantf' ^ '' 
 
 Ei = H -sec i^; Ea = H -seci^ (2.) 
 
 273. Polygonal Frame — Stability. — The stability or instability 
 of a polygonal frame depends on the principles stated in Article 
 267, page 159, viz., that il'a bar be free to change its angular position, 
 then if it is a tie it is stable, and if a strut, unstable; and that a 
 strut may be rendered stable by fixing its ends. 
 
 For example, in the frame of fig. 1 14, E is atie, and stable ; A, B, 
 C, and D, are struts, free to change their angular position, and 
 therefore unstable. 
 
 But these struts may be rendered stable in the plane of the 
 frame by means of stays; for example, let two stay-bars connect the 
 joints 1 with 4, and 3 with 5 ; then the points 1, 2, and 3, are all 
 fixed, so that none of the struts can change their angular posi- 
 
166 THEORY OF STRTJCTUEES. 
 
 tions. The same effect might be produced by two stay -bars con- 
 necting the joint 2 with 5 and 4. 
 
 The frame, as a whole, is unstable, as being liable to overturn 
 laterally, unless provided with lateral stays, connecting its joints 
 with fixed points. 
 
 Now, suppose the frame to be exactly inverted, the loads at 1,2, 
 and 3, and the supporting forces at 4 and 5, being the same as 
 before. Then E becomes a strut; but it is stable, because its ends 
 are fixed in position ; and A, B, C, and D becomes ties, and are 
 stable without being stayed. 
 
 An open polygon consisting of ties, such as is formed by A, B, 0, 
 and D, when inverted, is called by mathematicians, a funicular 
 polygon, because it may be made of ropes. 
 
 It is to be observed, that the stability of an unstayed polygon of 
 ties is of the kind which admits of oscillation to and fro about the 
 position of equilibrium. That oscillation may be injurious in 
 practice, and stays may be required to prevent it. 
 
 274. Bracing of Frames. — A Irace is a stay-bar on which there 
 is a permanent stress. If the distribution of the loads on the 
 joints of a polygonal frame, though consistent with its equilibrium 
 as a whole, be not consistent with the equilibrium of each bai-, 
 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 meet- 
 ing 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.* 
 
 The resistance of a brace introduces a pair of equal and opposite 
 forces, acting along the line of resistance of the brace, upon the 
 pair of joints which it connects. It therefore does not alter the 
 resultant of the forces applied to that pair of joints in amount nor 
 in position, but only the distribution of the components of that 
 resultant on the pair of joints. 
 
 To exemplify the use of braces, and the mode of determining the 
 stresses on them, let fig. 117 represent a frame such as frequently 
 
 * Thiia method of treating traced frames contaiiis an improvement sug- 
 gested by Prof. Clerk Maxwell iu 1867. 
 
BRACING OF FRAMES. 
 
 167 
 
 occurs in iron roofs, consisting of two struts or rafters, A and E, 
 and three tie-bars, B, C, and D, form- 
 ing a polygon of five sides, jointed at , 
 1, 2, 3, 4, 5, loaded vertically at 1, and ^ 
 
 supported by the vertical resistance of c> — ^^ — ^^H'RC.D 
 
 a pair of walls at 3 and 5. The joints 
 3 and 4 having no loads applied to 
 them, are connected with 1 by the 
 braces 1 4 and 1 3. 
 
 To make the diagram of forces (fig. 118), draw the vertical line 
 E, A, as in Article 271, to represent the direction of the load and of 
 the supporting forces. 
 
 The two segment^of that line, A B and D E, are to be taken to 
 represent the supporting forces at 2 and 5 ; and the whole line E A 
 will represent the load at 1. Erom the ends, and from the point 
 of division of the scale of external forces, E A, draw straight lines 
 parallel respectively to the lines of resistance of the frame, each 
 line being drawn from the point in E A that is marked with the 
 corresponding letter. Then A a and B 6, meeting at a, b, will 
 represent the stresses along A and B respectively ; and E e and 
 D d, meeting in D e, will represent the stresses along D and E 
 respectively ; but those four lines, instead of meeting each other 
 and c parallel to C in one point, leave gaps, which are to be filled 
 up by drawing straight lines parallel to the braces: that is to say, 
 from a, b, to c, parallel to 1 3; and from d, e, to c parallel to 4 1. 
 Then those straight lines will represent the stresses along the braces 
 to which they are respectively parallel; and C c will represent the 
 tension along C. To each joint in the frame, fig. 117, there corre- 
 sponds, in fig. 118, a triangle, or other closed polygon, having its 
 sides respectively parallel, and therefore proportional, to the forces 
 that act at that joint. Eor example, 
 
 Joints, 1, 2, 3, 4, 5, 
 
 Polygons, EAaceE; AB6A; Bc6B; BdcB; D E eD. 
 The order of the letters indicates the directions in which the forces 
 act relatively to the joints. 
 
 Another method of treating simple cases of bracing is illustrated 
 by fig. 119. A and B are two struts, forming the two halves of 
 
 Fig. X19. 
 
 Fig. 120. 
 
168 THEORY OF STRUCTURES. 
 
 one straight bar ; C and D are two equal tie-rods ; E, a strut brace. 
 A vertical rod P is applied at the joint 1, between A and B; two 
 vertical supporting pressures, each denoted by E, = P -h 2, act at 
 the joints 4 and 2. The joiat 3 has no external load. 
 
 Pig. 120 is the diagram of forces, constructed as follows: — 
 Through a point O draV O B A parallel to A and B, O C parallel 
 to 0, and OD parallel to D. Make OD = 0; join CD; this 
 line will be parallel to the brace E, and perpendicular to A. 
 
 Through D and draw vertical lines D B, C A ; these, b^ing 
 equal to each other, are to be taken to represent the two sup- 
 porting pi-essures E ; and their sum D B -i- A C will repi-esent 
 the load P. The equal tensions on C and D will be represented 
 by C and O D, and the thrusts along A, B, and E, by O A, B, 
 and C D. 
 
 The polygon of external forces in this case is the crossed quad- 
 rilateral A C D B, in -which C A and B D represent (as already 
 stated) the supporting pressures, and D C and A B the components 
 of the load P respectively parallel and perpendicular to the brace 
 E. When A and B are horizontal, and E vertical, A B in fig. 120 
 vanishes, and B D and C A coincide with the two halves of C D. 
 
 275. Rigidity of a Truss. — The word truss is applied in car- 
 pentry 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 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 u^on it is 
 
 subject to small variations only in its mode of distribution. For 
 
 ,. c example, in the truss of fig. 121, the 
 
 y X ~"|nN. tie-beam A A is made in one piece, ov 
 
 k/y ^sXa. i*^ *^° '^^ more pieces so connected 
 
 ■■/^M 1 "^^wk// ' *og6*^6' ^s *° ^^^ l''^® °^^ piece; and 
 
 ^B ^P P^'"* °f i*^ weight is suspended from 
 
 ■^ . r^ the joints C, C, by the rods B, C B. 
 
 Fig. 121. These rods also serve to make the re- 
 
 sistance of the tie-beam A A to being bent act so as to prevent the 
 
SECONDARY AND COMPOUND TRUSSING. 169 
 
 struts A C, 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. 
 
 The object of stiffening a truss by braces is to enable it to sustain 
 loads variously distributed ; for were the load always distributed in 
 one way, a frame might be designed of a figure exactly suited to 
 that load, so that there should be no need of bracing. 
 
 The variations of load produce variations of stress on all the 
 pieces of the frame, but especially on the braces; and each piece 
 must be suited to withstand the greatest stress to which it is liable. 
 
 Some pieces, and especially braces, may have to act sometimes as 
 struts and sometimes as ties, according to the mode of distribution 
 of the load. 
 
 276. Secondary and Compound Trussing. — A secondary truss 
 is a truss which is supported by another truss. 
 
 When a load is distributed over a great number of centres of 
 resistance, it may be advantageous, instead of connecting all those 
 centres by one polygonal frame, to sustain them by means of several 
 small trusses, which are supported by larger trusses, and so on, the 
 whole structure of secondary trusses resting finally on one large 
 truss, which may be called the primary truss. In such a combina- 
 tion the same piece may often form part of different trusses ; and 
 then the stress upon it is to be determined according to the follow- 
 ing principle : — 
 
 When the same bar forms at the same time part of two or more 
 different frames, the stress on it is tlie resultcmt of the several stresses 
 to which it is svhject by reason of its position in the several frames. 
 
 In a Compound Truss, several frames, without being distinguish- 
 able into primary and secondary, are combined and connected in 
 such a manner that certain pieces are common to two or more of 
 them, andtrequire to have their stresses determined by the principle 
 above stated. 
 
 Eommple. — Fig. 122, represents a kind of secondary trussing 
 common ill the framework of iron roofs. 
 
 Fig. 122. 
 
 The entire frame is supported by pillars at 2 and 3, each of which 
 sustains in all, half the weight. 
 
170 THEOEY OF STEUCTUEES. 
 
 1 2 3 is the primary truss, consisting of two rafters 1 3, 1 2, and 
 a tie-rod 2 3. 
 
 The weight of a division of the roof is distributed over th? 
 rafters. 
 
 The middle point of each rafter is supported by a seconda/ry truss; 
 one of those is marked 14 3; it consists of a strut, 1 3 (the rafter 
 itselfj, two ties 4 1, 4 3, and a strut-brace, 5 4, for transmitting the 
 load, applied at 5, to the point where the ties meet. 
 
 Each of the two larger secondary trusses just described supports 
 two smaller secondary trusses of similar form and construction to 
 itself; two of those are marked 1 7 5, 5 6 3 ; and the subdivision of 
 the load might be carried still farther. 
 
 In determining the stresses on the pieces of this structure, it is 
 indifferent, so far as mathematical accuracy is concerned, whether 
 we commence with the primary truss or with the secondary trusses; 
 but by commencing with the primary truss, the process is rendered 
 more simple. 
 
 (1.) Primary Truss 12 3. Let W denote the weight of the roof; 
 then J W is distributed over each rafter, the resultants acting 
 through the middle points of the rafters. Divide each of those 
 resultants i^ito two equal and parallel components, each equal to 
 ^ W, acting through the ends of the rafter; then ^ W is to be 
 considered as directly supported at 3, ^ W at 2, and ^ W -f- J W 
 = ^ W at 1 ; therefore the load at the joint 1 is 
 
 Let i be the inclination of the rafters to the horizon ; then by the 
 equations of Article 270. 
 
 H= ^ = -— • (1.) 
 
 2 tan i 4 tan i' 
 
 This is the pull upon the horizontal tie- rod of the primary truss, 
 
 2 3 ; and the thrust on eagh of the rafters 1 3, 1 2, is given by the 
 
 equation 
 
 . W cosec i ' /o ^ 
 Il = H sec t = J (2.) 
 
 (2.) Secondary Truss 14 3 5. The rafter 1 3 has the load J W 
 dsstributed over it; and reasoning as before, we are to leave two 
 quarters of this out of the calculation, as being directly supported 
 at 1 and 3, and to consider one-half, or \ W, as being the vertical 
 load at the point 5. The truss is to be considered as consisting of 
 a polygon of four pieces, 5 1, 1 4, 4 3, 3 5, two of which happen to 
 be in the same straight line, and of the strut-brace, 5 4, which 
 exerts obliquely upwards against 5, and obliquely downwards 
 
EESISTAKCE OF A PKAME AT A SECTION. 171 
 
 against 4, a thrust equal to the component perpendicular to the 
 rafter of the load ^ W; which thrust is given by the equation 
 
 ^6i = l W cos *' (3-) 
 
 Then we easily obtain the following values of the stresses on the 
 rafter and ties, in which each stress is distinguished by having affixed 
 to the letter E, the numbers denoting the two joints between which 
 it acts. 
 
 Pulls / E.. 1 
 
 on ties 1 1^43 = 1^41 = 2^ = 8 "^ ''°**°- *^ 
 
 Thrusts 
 
 on 
 
 rafter 
 
 '' E 1 1 
 
 - ot °* ■ - 5 W sin i = 5 ■ 
 2 tan i 8 8 
 
 ^51 = 91:^-5 W sin i = gW(coseci-2sint); 
 
 The difference between the thrusts on the two divisions of the 
 rafter, 
 
 ia the component along the rafter of the load at the point 5. 
 
 (3.) Smaller Secondary Trusses, 1 7 5, 5 6 3. — These trusses are 
 similar in every respect to the larger secondary trusses, except 
 that the load on each point is one-half, and consequently each of 
 the stresses is reduced to one-half of the corresponding stress in the 
 Equations 3 and i. 
 
 (4.) Resultani Stresses. The pull on the middle division- of the 
 great tie-rod 2 3 is simply that due to the primary truss, 12 3. The 
 pull on the tie 4 7 is simply that due to the secondary truss 14 3. 
 The pulls on the ties 5 7, 5 6, are simply those due to the smaller 
 secondary trusses, 1 5 7, 5 6 3. But agreeably to the Theorem stated 
 at the commencement of this article, the pull on the tie 1 7 is the 
 sum of those due to the larger secondary truss 14 3, and the smaller 
 secondary truss 17 5. The pull on 6 4 is the sum of those due to 
 the primary truss 12 3, and to the larger secondary truss 14 3. The 
 pull on 6 3 is the sum of those due to the primary truss 1 2 3, to the 
 larger secondary truss 14 3, and to the smaller secondary truss 5 6 3. 
 The thrust on each of the four divisions of the rafter 1 3, is the sum 
 of three thrusts, due respectively to the primary truss, the larger 
 secondary truss, and one or other of the smaller secondary trusses. 
 
 277. Resistance of a Frame at a Section The labour of calcu- 
 lating the stress on the bars of a frame may sometimes be abridged 
 by the application of the following principle : — 
 
 If a frame be acted upon hy any system of esctemal forces, and if 
 that frame he concewed to he coTnpletely divided into two pa/rts hy an 
 ideal surface, the stresses along the hars which are intersected hy that' 
 
172 THEORY or STRUCTURES. 
 
 surface, balance tlie external forces which act on each of the two parts 
 of the frame. 
 
 In most cases whicli occur in practice, the lines of resistance of 
 the bars, and the lines of action of the external forces, are all in one 
 vertical plane, and the external forces are verticaL In such cases 
 the most convenient position for an assumed plane of section is 
 vertical, and perpendicular to the plane of the frame. Take the 
 vertical 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 0; let the axis of x be horizontal, and in the 
 plane of the frame, and the axis of z horizontal, 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 combined, as 
 in Article 235, page 134, give three data — viz., the totsJ force along 
 a; = 2 • X : the total force along y = '2, • Y ; and the moment of 
 the couple acting round s = M; and the bars which are cut by 
 the plane of section miist 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 thvee 
 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 formuliE 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 2 ■ X, S • Y, 
 and M; let +2/ be measured upwards; let angles measured from 
 Ox towards + 2/, that is, upwards, be positive; and let the lines of 
 resistance of the three bars cut by the plane of section make the 
 angles ij, i^, %, with x. Let n^, n^, n^, be the perpendicular dis- 
 tances of those three lines of resistance from O, distances lying 
 
 {upwards 1 » n, ^ • -j j f positive 1 
 
 1 ^ , > from O X being considered as ■{ ^ , • >■ 
 
 downwards J I, negative. J 
 
 Let Rj, Eg; -^3! ^s ^^ resistances, or total stresses, along the 
 three bars, pulls being positive, and thrusts negative. Then we 
 have the following three equations : — 
 
 S • X = Rj cos i^ + Rj cos i^ + R3 cos ig , 
 
 2 • Y = Rj sin ii + Rj sin i2 + ^^ sin i^; V (1.) 
 
 - M = Rj jij -t- Rg ^2 + -E^s '^a j 
 
 :;}. 
 
 from which the three quantities sought, R^, Rj, Rg can be found. 
 
 Sjieaking with reference to the given plane of section, S • X may 
 be called the normal stress, 2 ■ Y, the s/iea/rijig stress, and M, the 
 
EESISTANCE OF A FRAME AT A SECTION. 173 
 
 moment offlexwre, or bending stress; for it tends to bend the frame 
 at the section under consideration. M is to be considered as 
 
 {" , . > according as it tends to make the frame become con- 
 negative J ° 
 
 cave 
 
 {upwards 
 downwards. 
 
 .} 
 
 The following is one of the simplest examples of the solution of 
 a problem by the method of polygon^s, and the method of sections. 
 Fig. 121 represents a truss of a form very common in carpentry 
 (already referred to in Article 275), and consisting of three struts, 
 A C, C 0, C A, a tie-ljeam A A, and two suspension-rods, C B, C B, 
 which serve to suspend part of the weight of the tie-beam from 
 the joints C, and also to stiffen the truss in the manner men- 
 tioned in Article 275. 
 
 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, 
 E, the thrust on each of the sloping rafters, and T the thrust on 
 the horizontal strut C C. 
 
 Proceeding by the method of polygoTis, as in Article 271, we find 
 at once, 
 
 H = - T = P cotan i ; \ 
 
 ( (3.) 
 
 E = - P cosec i. ) 
 
 (Thrusts being considered as negative.) 
 
 To solve the same question by the m,eihod 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 origiu of co-ordinates; 
 let X be positive towards the right, and y positive downwards; let 
 ajj, yy 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 ar^ 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 
 
 F, = 0;r,= -P 
 M = - Vxy 
 
174 THEOKY OF STRUCTUEBS. 
 
 Then, observing that for the strut A C, to =0, and that for the tie 
 A A, M = 2/1, ■we have, by the equations 1 of this Article 
 
 R cos i + H = F^ = j 
 
 E. sin. i = — P ; 
 H2/i= -M= +Pa;i; 
 
 ■whence we obtain, from the last equation, 
 
 H = ^ = Pcotan* ] 
 2/1 I 
 
 from the first, or from the second J- (3.) 
 
 H 
 
 E = -. = - P cosec i 
 
 cos I 
 
 J 
 
 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; P, = 0; M=-Pari; 
 from the first of which equations we obtain 
 H + T = F^ = 0, and 
 T= -H= -Pcotani (4.) 
 
 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. 
 
 278. Balance of a Chain or Cord. — A loaded chain may be looked 
 tipon as a polygonal frame whose pieces and joints are so numerous 
 that its figure may without sensible error be treated as a continuous 
 curve. The following are the principles respecting the equilibrium 
 of loaded chains and cords which are of most importance in practice. 
 I. Balance of a Chain in general. — Let D A C, in fig. 123, repre- 
 sent a flexible cord or chain supported at the points C and D, and 
 
 c loaded by forces in any 
 direction, constant or vary- 
 ing, distributed over its 
 whole length with constant 
 or varying intensity. 
 Let A and B be any 
 _ two points in this chain ; 
 
 J^ig. 123. from those points draw 
 
 tangents to the chain, A P and B P, meeting in P, The load acting 
 on the chain between the points A and B is balanced by the pulls 
 along the chain at those two points respectively; those pulls must 
 respectively act along the tangents A P, B P; hence the resultant 
 of the load between A and B acts through the point of intersection 
 of the tangents at A and B; and that load, and the tensions on the 
 
STABILITY OP BLOCKS. 175 
 
 chain at A and B, are respectively proportional to the sides of a 
 triangle parallel to their directions. 
 
 II. Ghmi under Vertical Load. — Curve of Equilibrium. — If the 
 direction of the load be everywhere parallel and vertical, draw a 
 vertical straight line, C D, fig. 124, to represent the total load, amd 
 from its ends draw C O and D O, parallel to two tangents at the 
 points of support of the chain, and meeting in O; 
 those lines will represent the tensions on the chain 
 at its points of support. 
 
 Let &., in fig. 123, be the lowest point of the 
 chain. In fig. 124, draw the horizontal line O A; 
 this will represent the horizontal component of the 
 tension of the chain at every point, and if O B be 
 parallel to a tangent to the chain at B (fig. 123), 
 A B will represent the portion of the load sup- 
 ported between A and B, and O B the tension at B. 
 
 To express this algebraically, let Kg. 124. 
 
 H = A = horizontal tension along the chain at A; 
 
 11 = B = pull along the chain at B j 
 
 P = A B = load on the chain between A and B; 
 
 i= ZX P B (fig. 123) = Z A O B fig. 124) = inclination of 
 chain at Bj 
 then, 
 
 P = Htani;E= J (P^ + H^) = H sec i (1.) 
 
 To deduce from these formulae an equation by which the form of 
 the curve assumed by the chain can be determined when the dis- 
 tribution of the load is known, let that curve be referred to rect- 
 angular, horizontal, and vertical co-ordinates, measured from the 
 lowest point A, fig. 123, the co-ordinates of B being, AX = a;, 
 
 X B = y, then tan i = 3^ = =., a differential equation, which enables 
 ax a. 
 
 the form assumed by the eord (or " curve of equilibrium") to be 
 
 determined when the distribution of the load is known. 
 
 279. Stability of Blocks. — The conditions of stability of a single 
 block supported upon another body at a plane joint may be thus 
 summed up : — 
 
 In fig. 125, let A A represent the upper block, 
 B B part of the supporting body, e E the joint, 
 C its centre of pressure, P O the resultant of 
 the whole pressure distributed over the joint, 
 NO, TO, its components perpendicular and 
 parallel to the joints respectively. Then the 
 conditions of stability are the following : — 
 
 I. In order that the hhch mouy not slide, tJie obliquity of ffie 
 
176 THEORY OF STEUCTUEES. 
 
 pressure must not exceed the angle of repose (Article 261, page 154), 
 that is to say, 
 
 Z P C N^ «> (1.) 
 
 II. In order that the block may he in no dang6r of overturning, the 
 ratio which the deviation of the centre of pressure from, the centre of 
 figure of the joint hears to the length of the diaarneter of the joint 
 traversing those two centres, must not exceed a certain fraction. The 
 value of that fraction varies, according to circumstances, from one- 
 eighth to three-eighths. 
 
 The first of these conditions is called that of stability of friction, 
 the second, that of stability of position. 
 
 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 
 
 '^^f^^ ^* plane joints, the two conditions of 
 stability must be fulfilled at each joint. 
 
 Let fig. 126 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 joint 1, 1, to be known, and also the 
 amount and direction of the pressure, as 
 Fig. 126. 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 lalock; 
 the resultant will be the total pressure to be resisted at the joint 
 2, 2, which will be given in magnitude, direction, and position, and 
 will intersect that joint in the centre of pressure C^. By continu- 
 ing this process there ai'e found the centres of pressure Cg, C^, &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. 
 
 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. 126, has received from Mr. Moseley 
 the name of the "line of resistance;" and that author has also 
 shewn how in many cases the equatioti 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 
 
STABILITY OF BLOCKS. H7 
 
 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. 126 
 touching all the sides of that polygon, is called by Mr. Moseley 
 the " line of presni/res." 
 
 The properties which the line of resistance and line of pressures 
 must have, in order that the conditions of stability may be fulfilled, 
 are, as already stated, the following : — 
 
 To inswre stability of position, the line of resistance must not 
 deviate from the centre of figure of any joint by more them a certain 
 fraction of the diameter of the joint, measured in the direction qf 
 deviation. 
 
 To insure stabilitygof friction, the normal to each joint must not 
 make an angle greater iham the angle of repose with a tangent to 
 tlie line of pressures drawn through the centre of resistance ofthatjoint. 
 
 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. 
 
 Tlie product of the weight of the structure into the horizontal dis- 
 tance qf a point in this line from a vertical line traversing tlie centre 
 of gramiy of the structure is the moment of stability of the struc- 
 ture, when the applied thrust acts in a vertical plane parallel to that 
 horizontal distance, and tends to overturn the structure in the direc- 
 tion of the given point in the line limiting the position qf the centre of 
 resistance; for that, according to Article 222, 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. The applied couple usually consists of the thrust of 
 a frame, or an arch, or the pressure of a fluid, or of a mass of earth, 
 against the structure, together with the equal, opposite, and parallel, 
 but not directly opposed, resistance of the joint to that lateral 
 force. 
 
 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; 
 lety be the inclination of that diameter to the horizon; let q the 
 the distance of the given limiting centre of resistance from the 
 middle point of that diameter, and q' t the distance from the same 
 middle point to th« point where the diameter is cut by the vertical 
 line through the centre of gravity of the structure, and let W be 
 the weight of the structure. Then the moment of stability is 
 
 W{q± q') t cos/; (1.) 
 
 the sign < > being used according as the centre of resistance, 
 
 N 
 
178 THEOKY OF STRUCTURES. 
 
 and the vertical line through the centre of gravity, lie towards 
 
 ( opposite sides ) ^^ ^^^ ^^^^ ^^ ^j^^ diameter. 
 
 ( the same side J 
 
 Let h denote the height of the structure above the middle of the 
 plane joint which is its base, b the breadth of that'joint in a direc- 
 tion perpendicular or conjugate to the diameter t, and w the weight 
 of an unit of volume of the material. Then we shall have 
 
 W = n-w7ibt (2.) 
 
 where Ji 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 
 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 
 followiog value for the moment of stability : — 
 
 n{q + q') cosj-w • hb t^ (3.) 
 
 This quantity is divided by points into three factors, viz. : — 
 
 (1.) n{(i±. g'') cos_7, a numerical factor, depeuding on %he figure 
 of the structure, the obliquities of its co-ordinates, and the direction 
 in which the applied force tends to overturn it. 
 
 (2.) w, the specific gravity of the material. 
 
 (3.) hbt^, a, geometrical factor, depending on the dimensions of 
 the structure. 
 
 Now the first factor is the same in all structures having figures 
 of the same class, with co-ordinates of equal obliquity, and exposed 
 to similarly applied external forces; that is say, to all structures 
 whose figures, together with the lines of action of the applied forces, 
 are parallel projections of each other, with co-ordinates of equal obli- 
 quity; hence for any set of structures which fulfil that condition, 
 the moments of stability are proportional to — 
 I. The specific gravity of the material; 
 II. The height; 
 
 III. The breadth; 
 
 TV. The squa/re of the thickness; that is, of the dimension of 
 the base which is parallel to the vertical plane of the applied force. 
 
 280. TransformatiQn of Blockwork Structures. — 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 
 corresponding parallel projection of the original system of lines; 
 also, the centres of pressure in the new ' structure will be the 
 corresponding projections of the centres of pressure in the original 
 structure. 
 
 The question, whether the new structure obtained by transfor- 
 mation will possess stability of friction is an independent problem. 
 
179 
 
 CHAPTEE II. 
 
 PRINCIPLES AXD RULES RELATING TO STRENGTH AND 
 STIFFNESS. 
 
 281. The Object of this Chapter is to give a summary of the 
 principles, and of the general rules of calculation, which are 
 applicable to problems of strength and stiffness, ■whatsoever the 
 particular material may be. 
 
 • 
 Section I. — Or Strength and Stiffness in General. 
 
 282. Load, Stress, Strain, Strength. — The load, or combination 
 of external forces, which is applied to any piece, moving or fixed, 
 in a structure or machine, produces stress amongst the particles 
 of that piece, being the combination of forces which they exert in 
 resisting the tendency of the load to disfigure and break the 
 piece, accompahied by si/rain, or alteration of the volumes and 
 figures of the whole piece, and of each of its particles. 
 
 If the load is continually increased, it at length produces either 
 fraciure or (if the material is very tough and ductile) such a 
 disfigurement as is practically equivalent to fracture, by rendering 
 the piece useless. 
 
 The Ultimate Strength of a body is the load required to produce 
 fracture in some specified way. The Proof Strength is the load 
 required to produce the greatest strain of a specific kind con- 
 sistent with safety; that is, with the retention of the strength of 
 the material unimpaired. A load exceeding the proof strength of 
 the body, although it may not produce instant fracture, produces 
 fracture eventually by long-continued application and frequent 
 repetition. 
 
 The Working Load on each piece of a machine is made less than 
 the ultimate strength, and less than the proof strength, in certain 
 ratios determined partly by experiment and partly by practical 
 experience, in order to provide for unforeseen contingencies. 
 
 Each solid has as many difierent kinds of strength as there are 
 different ways in which it can be strained or broken, as shewn in 
 the following classification : — 
 
 strain. Fracture. 
 
 ■pi, , /Extension Tearing. 
 
 ^ \ Compression Crushing. 
 
 ( Distortion Shearing. 
 
 Compound < Twisting Wreucl^ing. 
 
 ( Bending Breaking across 
 
180 TUEOEY OF STEUCTUEES. 
 
 283. Coefficients or Moduli of Strength are quantities expressing 
 the intensity of the stress under which a piece of a given material 
 gives way when strained in a given manner; such intensity being 
 expressed in units of weight for each unit of sectional area of the 
 layer of particles at which the body first begins to yield. In 
 Britain, the ordinary unit of intensity employed in expressing the 
 strength of materials is the pound a/ooirdupois on the squa/re inch. 
 
 Coefiiciepts of strength are of as many different kinds as there 
 are different ways of breaking a body. Their use will be explained 
 in the sequel. 
 
 Coefiicients of strength, when of the same kind, may still vary 
 according to the direction in which the stress is applied to the 
 body. Thus the tenacity, or resistance to tearing, of most kinds of 
 wood is much greater against tension exerted along than across 
 the grain. 
 
 284. Factors of Safety. — -A factor of safety, in the ordinary sense, 
 is the ratio in which the load that is just sufficient to overcome 
 instantly the strength of a piece of material is greater than the 
 greatest safe ordinary working load. 
 
 The proper value for the factor of safety depends on the nature 
 of the material; it also depends upon how the load is applied. 
 The load upon any piece in a structure or in a machine is distin- 
 guished into dead load and live load. A dead load is a load which 
 is put on by imperceptible degrees, and which remains steady; siich 
 as the weight of a structure, or of the fixed framing in a machine. 
 A live load is one that is or may be put on suddenly, or accom- 
 panied with vibration; like a swift train travelling over a railway 
 bridge; or like most of the forces exerted by and upon the moving 
 pieces in a machine. 
 
 It can be shewn that in most cases which occur in practice a 
 live load produces, or is liable to produce, tvnce, or very nearly 
 twice, the effect, in the shape of stress and strain, which an equal 
 dead load would produce. The mean intensity of the stress pro- 
 duced by a suddenly applied load is no greater than that produced 
 by the same load acting steadily; but in the case of the suddenly 
 applied load, the stress begins by being insensible, increases to 
 double its mean intensity, and then goes through a series of 
 fluctuations, alternately below and above the mean, accompanied 
 by vibration of the strained body. Hence the ordinary practice is 
 to make the factor of safety for a live load double of the factor 
 of safety for a dead load. 
 
 A distinction is to be drawn between real and apparent factors 
 of safety. A real factor of safety is the ratio in which the ultimate 
 or breaking stress is greater than the real working stress at the 
 time when the straining action of the load is greatest. The 
 ajjparent factor of safety has to be made greater than the real 
 
FACTORS OP SAFETY. 181 
 
 factor of safety in those cases in -wbich the calculation of strength 
 is based, not upon the greatest straining action of the load, but 
 upon a mean straining action, which is exceeded by the greatest 
 straining action in a certain proportion. In such cases the apparent 
 factor of safety is the product obtained by multiplying the real 
 factor of safety by the ratio in which the greatest straining action 
 exceeds the mean. 
 
 Another class of cases in which- the apparent exceeds the real 
 factor of safety is when there are additional straining actions 
 besides that due to the transmission of motive power, and when 
 those additional actions, instead of being taken into account in 
 detail, are allowed for in a rough way by means of an increase of 
 the factor of safety. A third class of cases is when there is a 
 possibility of an increased load coming by accident to act upon the 
 piece under consideration. For example, a steam engine may 
 drive two lines of shafting, exerting half its power on each; one 
 may suddenly break down, or be thrown out of gear, and the 
 engine may for a short time exert its whole power on the other. 
 
 The following table shews the ordinary values of real factors of 
 safety : — 
 
 Beal Factors op Sapbtt. 
 Dead Load. LItb Load. 
 
 Perfect materials and workmanship, .... 2 4 
 
 Ordinary materials and workmanship — 
 
 Metals, .' 3 6 
 
 Wood, Hempen Ropes, from 3 to 5 10 
 
 Masonry and Brickwork, 4 8 
 
 The following are examples of apparent factors of safety : 
 
 Batio in which 
 Greatest Effort 
 
 Eeal Factor of Safety, 6 exceeds Mean s^ixL 
 
 Effort, nearly. ^*'^'^- 
 
 Apparent 
 Tactor of 
 Safety. 
 
 Steam engines acting against a constant 
 resistance — 
 
 Single engine, '. TS 9'6 
 
 Pair of engines driving cranks at right 
 
 angles, 
 
 Three engines driving equiangular 
 
 cranks, 
 
 Ordinary cases of varying effort and 
 
 resistance, 
 
 Linesof shafting inmillwork; apparent 
 
 factor of safety for twisting stress 
 
 due to motive power, to cover allow- 
 ances for bending actions, accidental 
 
 extra load, &c., 
 
 11 6-6 
 
 1-05 6-3 
 
 2-0 120 
 
 y from 18 to 36 
 
182 THEORY OF STEUCTUfiES. 
 
 Almost all tlie experiments hitherto ms^de on the strength of 
 materials give coefficients or moduli of ultimate strength; that is, 
 coefficients expressing the intensity of the stress exerted by the 
 most severely strained particles of the material just before it gives 
 way. In calculations for the purpose of designing framework or 
 machinery to bear a given working load, there are two ways of 
 using the factor of safety, — one is, to multiply the working load 
 by the factor of safety, so as to determine the breaking load, and 
 use this load in the calculation, along with the modulus of ultimate 
 strength : the other is, to divide the modulus of ultimate strength 
 by the factor of safety, and thus to find a modulus or coefficient 
 of working stress, which is to be used in the calculation, along 
 with the worTdng load. It is obvious that the two methods are 
 mathematically equivalent, and must lead to the same result; 
 but the latter is on the whole the more convenient in designing 
 machines. 
 
 285. The Proof or Testing by experiment of the strength of a 
 piece of material is conducted in two different ways, according to 
 the object in view. 
 
 I. If the piece is to be afterwa/rds used, the testing load must be 
 so limited that there shall be no possibility of its impairing the 
 strength of the piece; that is, it must not exceed the proof strength, 
 being from one-third to one-half of the ultimate strength. About 
 double or treble of the working load is in general sufficient. Care 
 should be taken to avoid vibrations and shocks when the testing 
 load approaches near to the proof strength. 
 
 II. If the piece is to We sacrificed for the sake of ascertaining the 
 strength of the material, the load is to be increased by degrees until 
 the piece breaks, care being taken, especially when the breaking 
 point is approached, to increase the load by small quantities at a 
 time, so as to get a sufficiently precise result. 
 
 The proof strength requires much more time and trouble for its 
 determination than the ultimate strength. One mode of approxi- 
 mating to the proof strength of a piece is to apply a moderate load 
 and remove it, apply the same load again and remove it^ two or 
 three times in succession, observing at each time of application of 
 the load the strain or Alteration of figure of the piece when loaded, 
 by stretching, compression, bending, distortion, or twisting, as the 
 case may be. If that alteration does not sensibly increase by re- 
 peated applications of the same load, the load is within the limit 
 of proof strength. The effects of a greater and a greater load being 
 successively tested in the same way, a load will at length be reached 
 whose successive applications produce increasing disfigurements of 
 the piece; and this load will be greater than the proof strength, 
 which will lie between the last load and the last load but one in 
 the series of experiments. 
 
ELASTICITY OP A SOLID. 183 
 
 It was formerly supposed that the production of a set — that is, a 
 disfigurement which continues after the removal of the load — was 
 a test of the proof strength being exceeded; but Mr. Hodgkinson 
 shewed that supposition to be erroneous, by proving that in most 
 materials a set is produced by almost any load, how small soever. 
 
 The strength of bars and beams to resist breaking across, and of 
 axles to resist twisting, can be tested by the application of known 
 weights either directly or through a lever. 
 
 To test the tenacity of rods, chains, and ropes, and the resist- 
 ance of pillars to crushing, more powerful and complex mechanism 
 is required. The apparatus most commonly employed is the 
 hydraulic press. In computing the stress which it produces, no 
 reliance ought to be placed on the load on the safety valve, or on 
 a weight hung to the pump handle, as indicating the intensity of 
 the pressure, which should be ascertained by means of a pressure 
 gauge. This remark applies also to the proving of boilers by water 
 pressure. From experiments by Messrs. Hick and Liithy it appears 
 that, in calculating the stress produced on a bar by means of a 
 hydraulic press, the friction of the collar may be allowed for by 
 deducting a force equivalent to the pressure of the water upon an ■ 
 area of a length equal to the circumference of the collar, and one- 
 eightieth of an inch broad. 
 
 For the exact determination of general laws, although the load 
 may be applied at one end of the piece to be tested by means of a 
 hydraulic press, it ought to be resisted and measured at the other 
 end by means of a combination of levers. 
 
 286. Stiffness or Rigidity, Pliability, their Moduli or CoeflScients. 
 — Rigidity or stiffness is the property which a solid body possesses 
 of resisting forces tending to change its figure. It may be expressed 
 as a quantity, called a modulus or coefficient of stiffness, by taking 
 the ratio of the intensity of a given stress of a given kind to the 
 strain, or alteration of figure, with which that stress is accom- 
 panied — that strain being expressed as a quantity by dividing the 
 alteration of some dimension of the body by the original length of 
 that dimension. In most materials -vtrhich are used in machinery, 
 the moduli of stiffness, though not exactly constant, are nearly 
 constant for stresses not exceeding the proof strength. 
 
 The reciprocal of a modulus of stiffness may be called a " modulus 
 of pliabiliti/ ;" that is to say. 
 
 Modulus of Stiffness = =n^-: ; 
 
 iStrain 
 
 Modulus of Pliability = z;r— — rr — 5-5- . 
 
 ■' Intensity of Stress 
 
 287. The Elasticity of a Solid consists of stiffness, or resistance 
 to change of figure, combined with the nower of recovering the 
 
184 THEORY OP STRUCTURES. 
 
 original figure when the straining force is withdrawn. If that 
 recovery is complete and immediate, the body is 'perfectly elastic; 
 if there is a set, or permanent change of figure, after the removal 
 of the straining force, the body is imperfectly elastic. The elasticity 
 of no solid substance is absolutely perfect, but that of many sub- 
 stances is nearly perfect when the stress does not exceed the proof 
 strength, and may be made sensibly perfect by restricting the stress 
 ■within small enough limits. 
 
 Moduli or Goeffiidents of Elasticity are the values of moduli of 
 stiffness when the stress is so limited that the value of each of 
 those moduli is sensibly constant, and the elasticity of the body 
 sensibly perfect. 
 
 288. Resilience or Spring is the quantity of mechanical work* 
 required to produce the proof stress on a given piece of material, 
 and is equal to the product of the proof strain, or alteration of 
 figure, into the mean load which acts during the production of 
 that strain ; that is to say, in general, very nearly one-half of the 
 proof load. 
 
 289. Heights or Lengths of Moduli of Stiflfness and Strengths— 
 The term height or length, as applied to a modulus or coefficient of 
 strength or of stiffness, means the length of an imaginary vertical 
 column of the material to which the modulus belongs, whose 
 weight would cause a pressure on its base equal in intensity to 
 the stress expressed by the given modulus. Hence 
 
 Height of a modulus in feet 
 
 Modulus in lbs. on the square foot 
 
 Heaviness of material in lbs. to the cubic foot' 
 
 Modulus in lbs. on the square inch 
 "Weight of 12 cubic i)iohes of the material' 
 
 Height of a modulus in inches 
 
 Modulus in lbs. on the square inch 
 "~ Heaviness of material in lbs. to the cubic inch' 
 
 Height of a modulus in metres 
 
 Modulus in kilogrammes on the square metre 
 "Heaviness of material in kilogrammes to the cubic metre* 
 
 Section 2. — Or Eesistance to Direct Tension. 
 
 290. Strength, Stiffness, and Resilience of a Tie. — The word tie 
 is here used to denote any piece in framing or in mechanism, such 
 
 * Mechanical Work, which will be fully treated of in Part VI. , may be 
 defined as tlie product of a, force into the space through which it acts. 
 
RESILIENCE OR SPRING. 185 
 
 as a rod, bar, band, cord, or chain, which is under the action of a 
 pair of equal and opposite longitudinal forces tending to stretch 
 it, and to tear it asunder. The common magnitude of those two 
 forces is the load ; and it is equal to the product of the sectional 
 area of the piece into the intensity of the tensile stress. The 
 values of that intensity, corresponding to the immediate breaking 
 load, the proof load, and the working load, are called respectively 
 the moduli or coefficients of ultimate tenacity, of proof tension, and 
 of working tension. 
 
 In symbols, let P be the load, S the sectional area, and p the 
 intensity of the tensile stress; then 
 
 . T^-P^ (1) 
 
 If the sectional area varies at different points, the lea^t area is to 
 be taken into account in calculations of strength. 
 
 The elongation of a tie produced by any load, P, not exceeding 
 the proof load, is found as follows, provided the sectional area is 
 uniform : — 
 
 Let X denote the original length of the tie, A x the elongation, 
 
 A A" 
 and a = the extension ; that is, the proportion which that 
 
 elongation bears to the original length of the bar, being the 
 numerical measure of the strain. 
 
 Let E denote the modulus of direct elasticity, or resistance to 
 stretching. Then 
 
 « = ^; A« = «« = |a' (2-) 
 
 Lety denote the proof tension of the material, so that/' S is the 
 proof load of the tie ; then the proof extension isf -=- E. 
 
 The Resilience or Spring of the tie, or the work done in stretch- 
 ing it to the limit of proof strain, is computed as follows. The 
 length, as before, being x, the elongation of the tie produced by the 
 proof load is / a; -=- E. The force which acts through this space has 
 for its least value 0, for its greatest value P =/' S, and for its mean 
 value /■ S -=- 2 ; so that the work done in stretching the tie to the 
 proof strain, that is, its resUierwe or spring, is 
 
 flfx.nsx 
 
 2 E ~ E 2 ^ ' 
 
 The coefficient f^ 4- E, by which one-half of the volume of the 
 tie is multiplied in the above formula, is called the Modulus op 
 Resilience. 
 
 A sudden pull of /" S -f- 2, or one-half of the proof load, being 
 applied to the bar, will produce the entire proof strain of /'h-E, 
 which is produced by the gradual applica^on of the proof load 
 itself; for the work performed bv the action of the constant force 
 
186 THEORY OP STRUCTUEES. 
 
 / S -f- 2, through a given space, is the same with the work per- 
 formed by the action, through the same space, of a force increasing 
 at an uniform rate from up to /' S. Hence a tie, to resist with' 
 safety the sudden application of a given pull, requires to have twice 
 the strength that is necessary to resist the gradual application and 
 steady action of the same pull. This is an illustration of the 
 principle, that the factor of safety for a live load is twice that for a 
 dead load. 
 
 291. Thin Cylindrical and Spherical Shells. — Let r denote the 
 radius of a thin hollow cylinder, such as the shell of a high-pressure 
 boiler ; 
 
 t, the thickness of the shell; 
 
 /, the idtimate tenacity of the material, in pounds per square 
 inch j 
 
 •p, the intensity of the pressure, in pounds per square inch, re- 
 quired to burst the shell. This ought to be taken at six times the 
 effective working pressure — effective pressure meaning the excess of 
 the pressure from within above the pressure from without, which 
 last is usually the atmospheric pressure, of 14-7 lbs. on the square 
 inch or thereabouts. 
 
 Then 
 
 .4*^ « 
 
 and the proper proportion of thickness to radius is given by the 
 formula, — 
 
 ^=■1 (2.) 
 
 Thin spherical shells are twice as strong as cylindrical shells of 
 the same radius and thickness. 
 
 The tenacity of good wrought-iron boiler-plates is about 50,000 lbs. 
 
 Section 3. — Of Resistance to Distortion and Shearing. 
 
 292. Distortion and Shearing Stress in General. — In framework 
 and mechanism many cases occur in which the principal pieces, such 
 as plates, links, bars, or beams, being themselves subjected to ten- 
 sion, pressure, ^twisting, or bending, are connected with each other 
 at their joints by rivets, bolts, pins, keys, or screws, which are 
 under the action of a shearing force, tending to make them give 
 way by the sliding of one part over another. 
 
 Every shearing stress is equivalent to a pair of direct stresses of 
 the same intensity, one tensile and the other compressive, exerted 
 
STRENGTH OP A CYLINDRICAL SHAFT. 187 
 
 in directions making angles of 45° with the shearing stress. Hence 
 it follows that a body may give way to a shearing stress either by 
 actual shearing, at a plane parallel to the direction of the shearing 
 force, or by tearing, in a direction making an angle of 45° with that 
 force. The manner of breaking depends on the structure of the 
 material, hard and brittle materials giving way by tension, and soft 
 and tough materials by shearing. 
 
 When a shearing force does not exceed the limit within which 
 moduli of stiffness are sensibly constant, it produces distortion of 
 the body on which it acts. Let q denote the intensity of shearing 
 stress applied to the four lateral faces of an originally square 
 ]jrismatic particle, so ^ to distort it; and let » be the distortion, 
 expressed by the tangent of the difference between each of the distorted 
 angles of the prism and a right angle; then 
 
 9' 
 
 = 0, (1.) 
 
 is the modulus of transverse elasticity, or resistance to distortion. 
 
 One mode of expressing the distortion of an originally square 
 prism is as follows : — Let » denote the proportionate elongation of 
 one of the diagonals of its end, and - « the proportionate shorten- 
 ing of the other; then the distortion is 
 
 (- = 2«. 
 
 C 
 
 The ratio = of the modulus of transverse elasticity to the modulus 
 
 of direct elasticity defined in Article 287, page 184, has different 
 
 values for different materials, ranging from to ^r. For wrought- 
 
 1 ^ 
 
 iron and steel it is about 5. 
 o 
 
 Section 4. — Of Eesistance to Twisting and Wkenching. 
 
 293. Twisting or Torsion in General Torsion is the condition 
 
 of strain into which a cylindrical or prismatic body is put when a 
 pair of couples of equal and opposite moment, tending to make it 
 rotate about its axis in contrary directions, are applied to its two 
 ends. Such is the condition of shafts which transmit motive power. 
 The moment is called the tioisting moment, and at each cross- 
 section of the bar it is resisted by an equal and opposite moment of 
 stress. Each particle of the shaft is in a state of distortion, and 
 exerts shearing stress. 
 
 In British measures, twisting moments are expressed in inch-lbs. 
 
 294. Strength of a Cylindrical Shaft.— A cylindrical shaft, A B, 
 
188 THEORY OP STKUCTURES. 
 
 fig. 127, being subjected to the twisting moment of a pair of eqnal 
 
 and opposite couples applied to the 
 cross-sections, A and B, it is required 
 to find the condition of stress and 
 strain at any intermediate cross-sec- 
 tion, such as S, and also the angular 
 displacement of any cross-section rela- 
 Fig. 127. tively 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 
 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 longitudinal distance d x from 
 it. The twisting moment causes one of those cross-sections to 
 rotate relatively to the other, about the axis of the cylinder, through 
 an angle which may be denoted by d 6. 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 originally in one straight line parallel to the axis of the 
 cylinder, the twisting moment shifts one of those points laterally, 
 relatively to the other, through the distance r d 6. Consequently, 
 the part of the layer which lies between those points is in a con- 
 dition of distortion, in a plane perpendicular to the radius r; and 
 the distortion is expressed by the ratio 
 
 "^'---T,' (!•) 
 
 ■which varies proportionally to the distance from the axis. There 
 is therefore a shearing stress at each point of the cross-section, 
 ■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 
 
 ^=^" = ^'■■1! (2.) 
 
 The STEENGTH of the shaft is determined in the following man- 
 ner: — Let ji be the limit of the shearing stress to which the 
 material is to be exposed, being the ultimate resistance to wrench- 
 ing 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 
 
IlESISTANCE TO BENDING IN 'GENERAL. 189 
 
 yi is the value of q at the distance j-i from the axis ; and at any 
 other distance, r, the intensity of the shearing stress is 
 
 5==^ (3.) 
 
 Conceive the cross-section to be divided into narrow concentric 
 rings, each of the breadth d r. Let r be the mean radius of one of 
 these rings. Then its area is 2vrdr; 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 moment of the shearing stress of the ring in question, being 
 the product of the three quantities, 
 
 iJ— , r, and 2 a-rc^r is -^^^ • r^ d 
 
 ■which being integrated for all the rings from the centre to the 
 circumference of the cross-section, gives for the moment of torsion, 
 and of resistance to torsion, 
 
 M = ^9'i»1 = JyiA!; (4.) 
 
 if A = 2 rj^ be the diameter of the shaft, 
 
 (^=1-5708 ; ~ = 0-196 nearly). 
 
 If the axle is hollow, \ being the diameter of the hollow, the 
 moment of torsion becomes 
 
 ^ = 1^-^^^" c^-) 
 
 The following forraulse serve to calculate the diameters of shafts 
 when the twisting moment and stress are given ; solid shafts : — 
 
 hollow shafts — 
 
 51 M 
 
 '^-(^W' <«•> 
 
 
 .(7.) 
 
 Section 5. — Or Resistance to Bending and Uboss-Beeaking. 
 
 295. Resistance to Bending in General.— In explaining the prin- 
 ciples of the resistance which bodies oppose to bending and cross- 
 breaking, it is convenient to use the word beam, as a general term 
 
190 THEORY OP STRUCTURES. 
 
 to denote the body under consideration ; but those principles are 
 applicable, not only to beams for supporting weights, but to levers, 
 cross-heads, cross-tails, shafts, journals, cranks, and all pieces in 
 machinery or framework to which forces are applied tending to 
 bend them and to break them across ; that is to say, forces trans- 
 verse to the axis of the piece. 
 
 Conceive a beam which is acted tipon by a combination of 
 parallel transverse forces that balance each other, to be divided 
 into two parts by an imaginary transverse section ; and consider 
 separately the conditions of equilibrium of one of those parts. The 
 external transverse forces which act on that part, and constitute 
 the load on it, do not necessarily balance each other. Their result- 
 ant may be found by the rule of Article 233, page 132. That 
 resultant is called the Shea/ring Load at the cross-section under con- 
 sideration, and it is balanced by the Sliecmng Stress exerted by the 
 particles which that cross-section traverses. The resultant moment 
 of the same set of forces, relatively to the same cross-section, may 
 be found by the same rule ; it is called the Bending Moment at that 
 cross-section, and it is balanced (if the beam is strong enough) by 
 the Moment of Stress exerted by the particles which the cross-section 
 traverses, called also the Moment of Resistance. That moment of 
 stress is due wholly to longitudinal stress, and it is exerted in the 
 following way: — The bending of the beam causes the originally 
 straight layers of particles to become curved ; those near the 
 concave side of the beam become shortened ; those near the convex 
 side, lengthened ; the shortened layers exert longitudinal thrust ; 
 the lengthened layers, longitudinal tension j the resultant thrust and 
 the resultant tension are equal and opposite, and compose a couple, 
 whose moment is the moment of stress, equal and opposite to the 
 bending moment. 
 
 In the solution of problems respecting the transverse strength of 
 beams, it is necessary to determine the shearing load and bending 
 moment produced by the transverse external forces at different 
 cross-sections, and especially at those cross-sections at which they 
 act most intensely, and the relations between the dimensions and 
 figure of a cross-section of the beam, and the moment of stress 
 which that cross-section is capable of exerting, so that each cross- 
 section, and especially that at which the bending moment is 
 greatest, may have sufficient strength. 
 
 296. Calculation of Shearing Loads and Bending Moments. — 
 In the formulse which follow, the shearing load at a given cross- 
 section will be denoted by F, and the bending moment by M. In 
 British measures it is most convenient to express the bending 
 moment in inch-lbs., because of the transverse dimensions of pieces 
 in machines being expressed in inches. 
 
 The mathematical process for finding !P and M at any given 
 
CALCULATION OF SHEARING LOADS. 
 
 191 
 
 cross-section of a beam, though always the same in principle, may- 
 be varied considerably in detail. The following is on the whole 
 the most convenient way of conducting it : — 
 
 Fig. 128 represents a beam swppmied at both ends, and loaded 
 between them. Fig. 129 represents a bracket; that is, a beam 
 supported erndfieed at one end, and loaded on a projecting portion. 
 P, Q, represent in each case the supporting forces; in fig. 128, Wi, 
 
 Fig. 128. Fig. 129. 
 
 Wj, Wg, &c., represent portions of the load: in fig. 129, Wq re- 
 presents the endmost portion of the load, and Wp Wg, Wg, other 
 portions; in both figures, Axi, Aa52i Aaig, &o., denote the lengths of 
 the intervals into which the lines of action of the portions of the 
 load divide the longitudinal axis of the beam. The forces marked 
 W may be the weights of parts of the beam itself, or of bodies 
 carried by it ; or they may be forces exerted by moving pieces in a 
 machine on each other; or, in short, they may be any external 
 transverse forces. If the body called the beam is a shaft, P and 
 Q will be the bearing pressures. 
 
 The figures represent the load as Applied at detached points ; 
 but when it is continuously distributed, the length of any inde- 
 finitely short portion of the beam may be denoted byt^ar, the 
 intensity of the load upon it per unit of length by w, and the 
 amount of the load upon it by w d x. 
 
 The process to be gone through will then consist of the foUowr 
 ing steps : — 
 
 Step I. To find the Supporting Forces or Searing Pressures, P 
 
 and Q. — Assume any convenient point in the longitudinal axis as 
 
 origin of co-ordinates, and find the distance x^ of the resultant of 
 
 the load from it, by the rule of Article 233, page 132; that is 
 
 to say, 
 
 -2-xW 
 Xn= ^ -.T7- i or 
 
 ^0- 2-W 
 
 a^n = 
 
 J 
 
 xwdx 
 
 I 
 
 wdx 
 
 .(2.) 
 
192 THEOBY OF STRUCTURES. 
 
 Then, by the rule of Article 227, page 129, find the two sup- 
 porting forces or bearing pressures, P and Q ; that is to say, let R 
 be the resultant load, and P E and B Q its distances from the 
 points of support; and make 
 
 PQ:PIl:QR 
 ; : R : Q : P 
 
 ^} (3.) 
 
 Step II. To find, the, shea/ring loads at a series of sections. — In 
 ■what position soever the origin of co-ordinates may have been 
 during the previous step, assume it now, in a beam supported at 
 both ends, to be at one of the points of support (as A, fig. 128), and 
 in a bracket to be at the loaded point farthest from the fixed end 
 (as A, fig. 129). Consider P as positive and W as negative. 
 
 Then the shearing load in any given interval of the length of 
 the beam is the resultant of all the forces acting on the beam from 
 the origin to that interval; so that it has the series of values, 
 
 In Fig. 128. 
 
 Foi = P; 
 
 r23 = P-Wi-W,; 
 
 F3, = P-Wi-W,-W3; &c.; 
 
 and generally, 
 
 F = P-S-W; (4.) 
 
 In Fig. 129. 
 
 -Foi = W„; 
 
 -F^ = Wo + Wi + W,; 
 
 - F3, = Wo + Wi + W, + W3,&c. ; 
 
 and generally, 
 
 -F = 2-W; (5.) 
 
 so that the shearing loads which act in a series of intervals of the 
 length of the beam can be computed by successive subtractions or 
 successive additions, as the case may be. 
 
 For a continuously distributed load, these equations become 
 respectively, 
 
 In a beam supported at both ends, F = P- 1 wdx;...I.Q.) 
 
 
 
 In a bracket, - F = I wdx; (7.) 
 
 in wliich expressions, x' denotes the distance from the origin. A, to 
 the plane of section under consideration. 
 
 The positive and negative signs distinguish the two contrary 
 •directions of the distortion which the shearing load tends to 
 produce. 
 
 The Greatest Shearing Load acts in a beam supported at both 
 «nds, close to one or other of the points of support, and its value 
 is either P or Q. In a bracket, the greatest shearing load on the 
 projecting part acts close to the puter point of support, and its 
 ^alue is equal to the entire load. 
 
 In a beam supported at both ends the Shearing Load vanishes, 
 or changes from positive to negative at some intermediate section, 
 
GREATEST SHEAEING LOAD. 193 
 
 ■whose position may be found from Equation 4 or Equation 6, 
 by making F = 0. At the second point of support, F = - Q. 
 
 Step III. Tojmd the bending moments at a series of sections. — 
 At the origin A there is no bending moment. Multiply the 
 length of each of the intervals A « of the longitudinal axis of the 
 beam by the shearing load F, which acts throughout that interval; 
 the first of the products so obtained is the bending moment at 
 the inner end of the first interval ; and by adding to it the other 
 products successively, there are obtained the bending moments at 
 the inner ends of the other intervals in succession.* 
 
 That is to say,: — bending moment 
 
 at the origin A ; * Mg = ; 
 
 at the line of action of Wj^; Mj = Foi ■ A«i; 
 
 „ „ „ ■ Wg; Mg = F„i • A«i + Fi2 A x^^ 
 
 „ „ „ Wg; M3 = F„i-Aa;i + -Fi2Aa;2 + F23-Aa;3; 
 
 &c. &c. 
 
 and generally, M = 2 • F Aa;. (8.) 
 
 If the divisions A x are of equal length, this becomes 
 
 M = Aa!-2F; (9.) 
 
 and for a continuously distributed load, 
 
 M= {"'Edx (10.) 
 
 The three preceding Equations 8, 9, and 10, are applicable to 
 beams -whether supported at both ends or fixed at one end. By 
 substituting for F in Equation 10 its values as given by Equations 
 6 and 7 respectively, we obtain the following results : — 
 
 For a beam supported at both ends, 
 
 M = ?!»;'- r f'wdx^ 
 
 = Pia;'- [' {oii'-x)wdx; (11.) 
 
 •J 
 
 For a beam fixed at one end, 
 
 -M=/ f'wdx^= {' (x'~x)wdx; (12:) 
 
 in the latter of which equations the symbols —M denotes that the 
 bending moment acts downwards. 
 
 • This process is substantially the same with that employed by Mr. 
 Herbert Latham, in his work On Iron Bridges, to compute the stress iu a 
 half-lattice girder. 
 
 O 
 
194 THBOKT OP STRUCTURES. 
 
 The Greatest Bending Moment acts, in a bracket, at the outer 
 point of support; and in a beam supported at both ends, at the 
 section where the shearing load vanishes; found, as abeady stated 
 in Step II., from the Equation F = 0. 
 
 When the transverse forces applied to a beam supported at both 
 ends are symmetrically distributed relatively to its middle section, 
 the Greatest Bending Moment acts at that section; and it is some- 
 times convenient to assume a point in that section as the origin of 
 co-ordinates. 
 
 Step IV. To deduce the shearing load and hendvng moment in 
 one beam from those in another beam, similarly supported and 
 haded. — This is done by the aid of the following principle : — 
 
 When bea/ms differing in length and in the amounts of the loads 
 upon them a/re similarly supported, and have theH/r loads similwrly 
 distributed, the shearing loads at corresponding sections in them vary 
 as the total loads, and the bending moments as the products of the 
 loads and lengths. 
 
 This principle may be expressed by symbols in either of the two 
 following ways : — 
 
 First, Let I, I', denote the lengths of two beams, similarly sup- 
 ported; let W, W', denote their total loads, similarly distributed; 
 let F, F', be the shearing forces, and M, M', the bending moments, 
 at sections similarly situated in the two beams ; then 
 
 W: W : :F :F; (13.) 
 
 IW -.I'W ■.-.M.: M'...., (14.) 
 
 Secondly, Let k and m be two numerical factors, depending on 
 the way in which a beam is supported, the mode of distribution of 
 its load, and the position of the cross-section under consideration; 
 then 
 
 F = AW; (15.) 
 
 M = mWl (16.) 
 
 The length between the points of support of a beam supported 
 at the ends, as in fig. 128, is Often called the span. 
 
 297. Examples. — In the following formulae, which are examples 
 of the application of the principles of the preceding Article to the 
 cases which occur most frequently in practice, W denotes the total 
 load ; 
 
 w, when the load is distributed, the load per unit of length of 
 the beam ; 
 
 c, in brackets, the length of the free part of the bracket ; 
 
 c, in beams either loaded or supported at both ends, the half 
 span, between the extreme points of load or support and the 
 middle ; 
 
 M, the greatest bending moment. 
 
EXAMPLES. 195 
 
 I. Bracket fixed at one end and loaded 1 M = c"W (1.) 
 
 at the other, J 
 
 II. Bracket fixed at one end and uni- ) j^ ^ cW ^ wc^ .^ > 
 formerly loaded, / 2 2 ""^ "'' 
 
 III. Beam supported at both ends and "J 
 
 loaded at an intermediate point, f-]yj._(c2-ayW^ ,g . 
 whose distance from the middle of | 2 c •■■■\ •/ 
 
 the span is a;, ) 
 
 IV. Beam supported at both ends and ) ,, _ cW .. , 
 
 loaded in the middle, / ^^~ 2 ^ ' 
 
 V. Beam supported at both ends and \-%t_<:'^_wc^ ,„, 
 uniformly loaded, J^^^ — i T ^ ' 
 
 In Example III. the greatest force exetted is -^ — W, and the 
 
 leverage with which it acts is c + x; and Examples IV. and V. 
 follow from it by making x = o. 
 
 VI. If a beam has equal and opposite couples applied to its two 
 ends J for example, if the beam in fig. 130 has the couple of equal 
 and opposite forces Pj applied at A and B, and the couple of 
 equal and opposite forces P3 at C and D, 
 and if the opposite moments Pj • A B 
 = Pa • C D = M are equal, then each of 
 the endmost di-^isions, A B and C D, is ^ 
 in the condition of a bracket fixed at one 
 end and loaded at the other (Example 1.); ^^'' v- ion " 
 and the middle division B C is acted upon '^" 
 
 by the uniform bending moment M, and by no shearing load. 
 
 VII. Let a beam of the half span 6 be loaded with an uniformly 
 distributed load of w units of weight per unit of span; and at a 
 point whose distance from the middle of the span is a, let there 
 be applied an additional load W. It is required to find x, the dis- 
 tance from the middle of the span at which the greatest bending 
 moment is exerted, and M, that greatest moment. 
 
 Make 
 
 W 
 2 CW 
 
 then the solutions are as follows : — 
 
 Cask I. — When - = or;^, '; x = m(G-a) ; and 
 
 c l+m 
 
 1 
 
 1 
 
196 
 
 THEOKY OF STRUCTURES. 
 W I 
 
 
 .(6.) 
 
 Case 2. — ^When - = or .i^, 
 
 c 1 + m 
 
 ; 33 = a ; and 
 
 M=!i^^l.2.)(l-g (7.) 
 
 In the following case both sets of formulse give the same result; 
 a = m {c — a); and 
 
 wc^ (1 + 2. njA2 
 
 , a m 
 
 ■when - = = ; 
 
 c 1 +m 
 
 M 
 
 298. Bending Moments produced by Longitudinal and Oblique 
 Forces. — ^When a bar is acted upon at a given cross-section by any 
 external force, whose line of action, whether transverse, oblique, or 
 parallel to the axis of the bar, does not traverse the centre of 
 magnitude of that cross-section, that force exerts a moment upon 
 that, cross-section equal to the product of the force into the perpen- 
 dicular distance of its line of action from the centre of the cross- 
 section, and that moment is to be balanced by the moment of 
 longitudinal stress at the cross-section. 
 
 The external force may be resolved into a longitudinal and a 
 transverse component. The longitudinal component is balanced 
 by an imiform longitudinal tension or pressure, as the case may be, 
 exerted at the cross-section, and combined with the stress which 
 resists the bending moment; and the transverse component is re- 
 sisted by shearing stress. 
 
 299. Moment of Stress — Transverse Strength. — The bending 
 moment at each cross-section of a beam bends the beam so as to make 
 any originally plane longitudinal layer of the beam perpendicular 
 to the plane in which the load acts, become concave in the direction 
 towards which the moment acts, and convex in the opposite 
 direction. Thus, fig. 131 represents a side view of a short portion 
 
 of a bent beam; C C is a layer, origin- 
 ally plane, which is now bent so as to 
 become concave at one side and convex 
 at the other. 
 
 The layers at and near the concave 
 side of the beam, A A', are shortened, 
 and the layers near the convex side, 
 B B' lengthened, by the bending action 
 
 Fig. 131. 
 
MOMENT OF STRESS. 197 
 
 of the load. There is one intermediate surface, 0', which is 
 neither lengthened nor shortened ; it is called the " neutral surface." 
 The particles at that surface are not necessarily, however, in a state 
 devoid of strain; for, iu common with the other particles of the 
 beam, they are compressed and extended in a pair of diagonal 
 directions, making angles of 45° with the neutral surface, by the 
 shearing action of the load, when such action exists. 
 
 The condition of the particles of a beam, produced by the com- 
 bined bending and shearing actions of the load, is illustrated by fig. 
 132, which represents a vertical longitudinal section of a rectangular 
 beam, supported at the ends, and loaded at intermediate points. 
 It is covered with a netwjrk consist- 
 ing of two sets of curves cutting each 
 other at right angles. The curves 
 convex upwards are lines of direct 
 
 thrust; those convex downwards are Fig. 132. 
 
 lines of direct tension. A pair of 
 
 tangents to the pair of curves which traverse any particle are the 
 axes of stress of that particle. The neutral surface is cut by both 
 sets of curves at angles of 45°. At that vertical section of the 
 beam where the shearing load vanishes, and the bending moment 
 is greatest, both sets of curves become parallel to the neutral 
 surface. 
 
 When a beam breaks under the bending action of its load, it 
 gives way, either by tlie crushing of the compressed side, A A', or 
 by the tearing of the stretched side, B B'. 
 
 In fig. 133, A represents a 
 beam of a granular material, like 
 
 cast iron, giving way by the 
 
 crushing of the compressed side, lis i q9 ° 
 
 out of which a sort of wedge is ^' 
 
 forced. B represents a beam giving way by the tearing asunder of 
 the stretched side. 
 
 The resistance of a beam to bending and cross-breaking at any 
 given cross-section is the moment of a couple, consisting of the 
 thrust along the longitudinally-compressed layers, and the equal 
 and opposite tension along the longitudinally-stretched layers. 
 
 It has been found by experiment, that in most cases which occur 
 in practice, the longitudinal stress of the layers of a beam may, 
 without material error, be assumed to be uniformly va/rying, its 
 intensity being simply proportional to the distance of the layer 
 from the neutral surface. 
 
 Let fig. 134 represent a cross-section of a beam (such as that 
 represented in fig. 131), A the compressed side, B the extended 
 side, C any layer, and the iwutral axis of the section, being 
 the line in which it is cut by the neutral surface. Let p denote 
 
198 THEORY OF STRUCTURES. 
 
 the intensity of the stress along the layer C, and y the distance 
 of that layer from the neutral axis. Because the 
 stress is uniformly varying, p ~ y is a constant 
 quantity. Let that constant be denoted for the 
 present by a. 
 
 Let z be the breadth of the layer 0, and ci! y its 
 thickness; 
 Fia ^134. Then the amount of stress along it is 
 
 pzdy = ayzdy; 
 
 the amount of the stress along all the layers at the given cross- 
 section is 
 
 ajyzdy; 
 
 and this amount must be nothing, — in other words, the total thrust 
 and total tension at the cross-section must be equal, — because the 
 forces applied to the beam are wholly transverse; from which it 
 follows that 
 
 j yzdy = 0, (1.) 
 
 and the neutral aocis traverses the centre of 'magnitude of the cross- 
 section. This principle enables the neutral axis to be found by the 
 aid oTthe methods explained in Section 1, Chapter IIL, Part IIL 
 
 To find the greatest value of the constant p ■- y consistent with 
 the strength of the beam at the given cross-section, let y^ be the 
 distance of the compressed side, and y^ that of the extended side 
 from the neutral axis; f^ the greatest thrust, and /j the greatest 
 tension which the material can bear in the form of a beam ; com- 
 pute/, -i- y„, and/j-=-2/j, and adopt the less of those two quantities 
 as the value oi p -i- y, which may now be denoted hyf^y^i 
 / being _/ji or/j, and y^ being y^ or y^, according as the beam is 
 liable to- give way by crushing or by tearing. 
 
 For the best economy of joaterial, the two quotients ought to be 
 equal; that is to say, 
 
 (lA.) 
 
 and this gives what is called a cross-section of equal strength. 
 
 The moment ifelsttively to the neutral axis, of the stress exerted 
 along any given layer of the cross-section, is 
 
 f 
 ypzdy^J-y^zdy; 
 
 f 
 
 /» 
 
 /. 
 
 _/» 
 
 +/. . 
 
 2/1 
 
 Ua 
 
 Vi 
 
 
 h ' 
 
MOMENT OP SXEESS. 199 
 
 and tlie sum of all such moments, being the moment of stress, or 
 MOMENT OF RESISTANCE of the given cross-section of the beam to 
 breaking across, is given by the formula, 
 
 M = 
 
 making j y^zdy = l, 
 
 jpytidy=—jy^zdy; (2.) 
 
 Vv 
 
 M=^ (2 a.) 
 
 When the hreaking load is in question, the coefficient / is what 
 is called the modulus of kuptuee of the material. 
 
 When the proof loaa or worhmg load is in question, the co- 
 efficient / is the modulus of rupture divided by a suitable factor 
 of safety, which, for the working stress in parts of machinery that 
 are made of metal, is usually 6, and for the parts made of wood, 10. 
 Thus, the working modmlus f is usually 9,000 lbs. on the square 
 inch for wrought iron, 4,500 for cast iron, and from 1,000 to 1,200 
 for wood. 
 
 The factor denoted by I in the preceding equation is what is 
 called the "geometrical moment of inertia" of the cross-section of 
 the beam. For sections whose figures are similar, or are parallel 
 projections of each other, the moments of inertia are to each other 
 as the breadths, and as the cubes of the depths of the sections; and 
 the values of y-i are as 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 stress is greatest, 
 we may put 
 
 l=n'h¥, (3.) 
 
 yx = m!h, (4.) 
 
 n and m! being numerical factors depending on the form of section, 
 and making n' -i-m' = n, the moment of resistance may be thus 
 expressed, — 
 
 M = w/6A2 (5.) 
 
 Hence it appears that the resistances of simila/r cross-sections to 
 cross-breaking are as their breadtlis and as the squares of their depths. 
 
 The relation between the load and the dimensions of a beam is 
 found by equating the value of the greatest bending moment in 
 terms of the load and span of the beam, as given in Article 296, 
 Equations 10, 11, 12, 16, to the value of the moment of resist- 
 ance of the beam, at the cross-section where that greatest bending 
 moment acts, as given in Equation 5 of this Article. 
 
 The depth A is usually fixed by considerations of stiffness, and 
 then the unknown quantity is the breadth, b. Sometimes, as when 
 
200 
 
 THEORY OP STRUCTURES. 
 
 the cross-section is circular or square, we have b = h; and then we 
 lave h^, instead of b h? in Equation 5, which is solved so as to 
 give h by extraction of the cube root. The following are the 
 formulse for these calculations : — 
 
 6 = 
 
 M 
 
 and when h = h, 
 
 nfW-' ^^^ 
 
 Examples of the Numerical Eactors in Equations 3, 4, 
 
 5 AND 6. 
 
 Form of Cross-Sections. 
 
 , I 
 
 '»-! 
 
 M 
 "^fbh^- 
 
 I. Bectangle&A, 
 
 (including square) 
 
 IL Ellipse- 
 Vertical axis Ti '\ 
 
 Horizontal axis 2>, > 
 
 (including circle) ) 
 
 IIL Hollow rectangle, 5 A — V K; . 
 also I -formed section, J 
 where b' is the sum of the > 
 breadths of the lateral \ 
 hollows, ' 
 
 rv. Hollow square — ) 
 h'-h^, \ 
 
 V Hollow ellinae 
 
 1 
 
 12 
 
 1 
 2 
 
 1 
 
 6 
 
 nr 1 
 64 20-4 
 = 0-0491 
 
 1 
 2 
 
 •n- 1 
 32 10-2 
 = 0-09S2 
 
 1 / l^K«\ 
 liV bh'J 
 
 1 
 2 
 
 1/ I-K'S- 
 6\ bh'J 
 
 Ao-a 
 
 1 
 2 
 
 6V h'J 
 
 64: V bh'J 
 
 1 
 2 
 
 32 \ bh'J 
 
 
 64 \ AV 
 
 1 
 2 
 
 ki^-S) 
 
 VII. Isosceles triangle; tase &, ) 
 heiglit h; y^ measured 
 
 1 
 36 
 
 2 
 3 
 
 1 
 
 24 
 
 
 
 
 300. Allowance for Weight of Beam — Limiting Length of 
 Beam. — When a beam is of great span, its own weight may bear 
 a proportion to the load which it has to carry, sufl5ciently great to 
 require to be taken into account in determining the dimensions 
 of the beam. The following is the process to be performed for 
 that purpose, when the load is uniformly distributed, and the 
 
ALLOWANCE FOE WEIGHT OF BEAM. 201 
 
 beam of uniform cross-section. Let W' be the external working 
 
 load, Si its factor of safety, Sj a factor of safety suited to a steady 
 
 load, like the weight of the beam. 
 
 Let b' denote the breadth of any part of the beam, as computed 
 
 by considering the external breaking load alone, 8-^ W'- Compute 
 
 the weight of the beam from that provisional breadth, and let it be 
 
 s W 
 denoted by B.' Then — =i =7 is the proportion in which the 
 
 gross breaking load exceeds the external part of that load. Conse- 
 quently, if for the promsional breadth b' there be substituted the 
 exact breadth, 
 
 ^s,W'-s,-B" (^-^ 
 
 the beam will now be strong enough to bear both the proposed 
 external load W, and its own weight, which will now be 
 
 ^-s,W'-s,B" (^•> 
 
 and the true gross breaking load will be 
 
 ^-^^'+*^^=i:w^- (3-) 
 
 As the factor of safety for a steady load is in general one-half of 
 that for a moving load, Sj may be made = 2 Sa; in which case the 
 preceding formulee become 
 
 . 25'W' . 
 
 ■"2W'-B'' ^ '' 
 
 2B''W' 
 ^"2W'-B'^ (^-^ 
 
 2W'-B" ^ •' 
 
 In all these formulse, 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. 
 
 The gross load of beams of similar figures and proportions, vary- 
 ing 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 dimensions. Hence the weight 
 increases at a faster rate than the gross load ; and for each parti- 
 
202 THEORY OF STKUCTUEES. 
 
 cular 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 additional load. 
 
 To reduce this to calculation, let the uniformly distributed gross 
 breaking load of a beam of a given figure be expressed as follows:-— 
 
 W = «,W' + .,B=^ = -^; , (7.) 
 
 the value of m for an uniformly distributed load and rectangular 
 
 cross-section being ^; and nfhA being = ray 6 A^, Equation 5, 
 
 Article 299; I, A and A being the length, depth, and sectional 
 area of the beam, / the modulus of rupture, and n a factor depend- 
 ing on the form of cross-section. The weight of the beam will be 
 expressed by 
 
 B-^kw'lA; (8.) 
 
 W being the weight of an unit of volume of the material, and k a 
 factor depending on the figure of the beam. Then the ratio of 
 the weight of the beam multiplied by its proper factor of safety to 
 the gross breaking load is 
 
 Sj B _ Ss ^ w)' P ,Q . 
 
 'W~ 8nfh' ^ ' 
 
 which increases in the simple ratio of the length, if the proportion 
 Ixh is fixed. When this is the case, the length L of a beam, 
 whose weight (treated as uniformly distributed) is its working load, 
 is given by the condition Sa B = W ; that is, 
 
 L = i^; (10.) 
 
 This limUing length having once been determined for a given class 
 of beams, may be used to compute the ratios of the gross breaking 
 load, weight of the beam, and external working load to each other, 
 for a beam of the given «lass, and of any smaller length, I, according 
 to the following proportional equation;' — 
 
 L :- ■}^: : W : B : W ; (11.) 
 
 Section 6. — Of Resistance to Thrust or Pressure. 
 
 301. Resistance to Compression and Direct Cresliing Eesist- 
 
 ance to longitudinal compression, when the proof stress is not 
 
RESISTANCE TO COMPRESSION. 
 
 203 
 
 exceeded, is sensibly equal to the resistance to stretching, and is 
 expressed by the same modulus of elasticity, denoted by E. When 
 that limit is exceeded, it becomes irregular. 
 
 The present Article has reference to direct and simple crushing 
 only, and is limited to those cases in -which the pillars, blocks, 
 struts, or rods along which the thrust acts are not so long in pro- 
 portion to their diameter as to have a sensible tendency to give way 
 by bending sideways. Those cases comprehend — 
 
 Stone and brick pillars and blocks of ordinary proportions ; 
 
 Pillars, rods, and struts of cast iron, in which the length is not 
 more than five times the diameter, approximately; 
 
 Pillars, rods, and struts of wrought iron, in which the length is 
 not more than ten times the diameter, approximately; 
 
 Pillars, rods, and struts of dry timber, in which the length is not 
 more than about five times the diameter. 
 
 In such cases the rules for the strength of ties (Article 290) are 
 approximately applicable, substituting thrust for tendon, and using 
 the proper modulus of resistance to direct crushing instead of the 
 tenacity. 
 
 Blocks whose lengths are less than about once-and-a-half their 
 diameters ofier greater resistance to crushing than that given by the 
 rules; but in what proportion is uncertain. 
 
 The modulus of resistance to direct crushing often difiers con- 
 siderably from the tenacity. The nature and amount of those 
 differences depend mainly on the modes in which the crushing 
 takes place. These may be classed as follows : — 
 
 I. Crushing hy splitting (fig. 135) into a number of nearly pris- 
 matic fragments, separated by smooth surfaces whose general 
 direction is nearly parallel to the direction of the load, is character- 
 istic of very hard homogeneous substances, in which the. resistance 
 to direct crushing is greater than the tenacity ; being in many 
 examples about double. 
 
 Pig. 136. 
 
 Kg. 137. 
 
 II. Crushing hy shearing or sliding of portions of the block along 
 obliqtie 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. 136; sometimes two cones or pyramids are formed, like 
 c, c in fig, 137, which are forced towards each other, and split or 
 drive outwards a number of wedges surrounding them, like w, w, 
 
204 THEORY or STEUCTURES. 
 
 in the same figure. Sometimes tlie block splits into four wedges, 
 as in fig. 138. In substances whicli are crushed by shearing, 
 the resistance to crushing is always much greater than the tenacity j 
 for example, in cast iron it is from four times to six times. 
 
 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 crashing load, it is 
 difficult to determine their resistance to that load 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 pillars or struts of moderate length, as nearly as it can be 
 
 2 4 
 ascertained, is from to ■= of the tenacity. 
 o o 
 
 TV. Crushing by budding or crippling is charalcteristic of fibrous 
 subtances, such as wood, under the action of a thrust along the 
 fibres. ■ It consists in a lateral bending and wrinkling of the fibres, 
 sometimes accompanied by a splitting of them asunder, 
 
 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. 
 
PAKT V. 
 
 PRINCIPLES OF KINETICS. 
 
 CHAPTER I. 
 
 SUMMABY OF GENERAL PRINCIPLES. 
 
 NATURE AND DIVISION OF THE SUBJECT. 
 
 The present Chapter contains a summary of the Principles of 
 Kinetics. 
 
 302. Effort ; Resistance ; Lateral Force. — Let F denote a force 
 applied to a moving point, and 6 the angle made by the direction 
 of that force with the direction of the motion of the point. Then, 
 by the principles of Article 215, the force F may be resolved into 
 two rectangular components, one along, and the other across, the 
 direction of motion of the point, viz : — 
 
 The direct force, F cos 6. 
 The lateral force, F sin 0. 
 
 A direct force is fm-ther distinguished, according as its acts vnth or 
 against the motion of the point (that is, according as i is acute or 
 obtuse), by the name of effort, or of resistance, as the case may be. 
 Hence, each force applied to a moving point may be thus decom- 
 posed : — 
 
 Effort, P = F cos 6, if 6 is acute ; 
 
 Resista/nce, R = F cos (a- - e) if tf is obtuse ; 
 
 Lat&ral Force, Q = F sin i. 
 
 303. 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 lat&ral forces applied to each point shaU 
 lalance each other, and that the efforts applied to each point shall 
 balance the resistances. 
 
 The direction of a force being, as stated in Article 194, 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. 
 
206 PRINCIPLES OP KINETICS. 
 
 304. Work consists in moving against resistance. The work is 
 said to be 'performed, and the resistance overcome. Work is mear 
 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 afoot-pound. 
 
 305. Energy means capacity for performing work. The energy 
 of an effort, ot 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. 
 
 306. The Conservation of Energy, in the case of uniform motion, 
 means the fact, that the energy exerted is equal to the work per- 
 formed. 
 
 307. 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 ajsplied to any connected system of points. 
 
 308. The Mass, or Inertia, of a body, is a quantity proportional 
 to the unbalanced force which is required in order to produce a 
 given definite change in the motion of the body in a given interval 
 of time. 
 
 It is known that the weight of a body, that is, the attraction 
 between it and the earth, at a fixed locality on the earth's surface, 
 acting unbalanced on the body for a fixed interval of time (e. g., 
 for a second), produces a change in the body's motion, which is the 
 same for all bodies whatsoever. Hence it follows, that the masses 
 of all bodies are proportional to their weights at a given locality on 
 the earth's surface. 
 
 This fact has been learned by experiment; but it can also be 
 shewn 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.* 
 • See the Eev. Dr. Whewell's demonstration " that all matter gravitates.'' 
 
THE ACTUAL ENERGY. 
 
 207 
 
 309. The Centre of Mass of a body is its centre of gravity, found 
 in the manner explained in Part III., Chapter III., Section 1. 
 
 310. 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 I., Chapter I. 
 
 311. The Resultant Momentum of a system of bodies is the 
 resultant of their separate momenta, compounded as if they were 
 motions or statical couples. 
 
 312. Variations and Deviations of Momentum are the products 
 of the mass of a body intojihe rates of variation of its velocity 
 and deviation of its direction, found as explained in Part I., 
 Chapter I., Section 3. 
 
 313. Impulse is the product of an unbalanced force into the Ume 
 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, Fdtia 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. 
 
 314. Impulse, Accelerating, Retarding, Deflecting. — Correspond- 
 ing to the resolution of a force applied to a moving body into eiFort 
 or resistance, as the case may be, and lateral stress, as explained 
 in Article 302, there is a resolution of impulse into accelerating 
 or retarding impulse, which acts with or against the body's motion, 
 and deflecting impulse, which acts across the direction of the body's 
 motion. ) Thus, if 6, as before, be the angle which the unbalanced 
 force F makes with the body's path during an indefinitely short 
 interval, d t. 
 
 7 dt = F cos i • dtia accelerating impulse if i is acute; 
 Hd t = ¥ cos (■TT - 6) ■ d t is retarding impulse if 6 is obtuse ; 
 Qdt = 'F sin 6 • d t is deflecting impulse. 
 
 315. A Deviating Force is one which acts unbalanced in a direc- 
 tion perpendicular to that of a body's motion, and changes that 
 direction without changing the velocity of the body. 
 
 316. Centrifugal Force is the force with which a revolving body 
 reacts on the body that guides it, and is equal and opposite to the 
 deviating force with which the guiding, body acts on the revolving 
 body. 
 
 In fact, as has been stated in Article 193, every force is an action 
 between two bodies; and deviating force and ceritn/tcgal 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. 
 
 317. The Actual Energy of a moving body relatively to a fixed ' 
 
208 PRINCIPLES or KINETICS. 
 
 point is the product of the mass of the body into one-haifoi the 
 square of its velocity, that is to say, it is represented by 
 
 2 2g 
 
 The product m t^, 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^oo^22M™(li'^''ticles 304, 305.) y 
 
 318. Energy Stored and Restored. — A body alternately acceler- 
 ated and retarded, so as to be brought back to its original speed, 
 performs 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 
 acceleration, and restored during the retardation. 
 
 319. 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. 
 
 320. 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 (Ai-ticle 306.) 
 
 321. 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. The work which a body performs 
 in moving against a reciprocating force is employed in increasing 
 its own potential energy, and is not lost by the body. 
 
 322. Collision is a pi'essure of inappreciably short duration be- 
 tween two bodies. 
 
 323. 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. 
 
 324. The Radius of Gyration of a body about a given axis is 
 that length whose square is the meam, of all the squares of the dis- 
 tances of the indefinitely small equal particles of the body from the 
 axis, and is found by dividing the moment of inertia by the mass. 
 
 325. 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 con- 
 centrated at that point, and the remainder at the point directly 
 opposite in the given axis, the statical moment of the weight so 
 distributed, and its moment of inei'tia about the given axis, would 
 
THE CENTRE OP PERCUSSION. 209 
 
 be the same as those of the actual body in every position of the 
 body. 
 
 326. The subjects to which the principles of kinetics relate will 
 be classed in the following manner : — 
 
 I. Uniform Motion. 
 II. Varied Translation of Points and Rigid Bodies. 
 
 III. notations of Bigid Bodies. 
 
 IV. Motions of Fluids. 
 
210 
 
 CHAPTER n. 
 
 ON UNIFORM MOTION UNDER BALANCED FORCES. 
 
 327. First Law of Motion. — A body under the action of no force, 
 or of balanced forces, is either at rest, or moves uniformly. (Uni- 
 form motion has been defined in Article 66.) 
 
 Such is the first law of motion as usually stated; but in that 
 statement is implied something more than the literal meaning of 
 the words; for it is understood, that the rest or Tnotion of the body 
 to which the law refers, is its rest or motion relatively to another 
 body which is also under the action of no force or of balanced forces. 
 TJnless this implied condition be fulfilled, the law is not true. 
 Therefore the complete and explicit statement of the first law of 
 motion is as follows : — 
 
 If a pair of bodies be each under the action of no^ force, or of 
 bala/nced forces, the motion of each of those bodies relatively to the 
 other is either none or uniform. 
 
 The first law of motion has been learned by experience and 
 observation: not directly, for the circumstances supposed in it 
 never occur; but indirectly, from the fact that its consequences, 
 when it is taken in conjunction with other laws, are in accordance 
 with all the phenomena of the motions of bodies. 
 
 The first law of motion may be regarded as a consequence of the 
 definitions of /orce and oi balance (Articles 55, 56); at the same 
 time it is to be observed, that the framing of those definitions has 
 been guided by Experimental knowledge. 
 
211 
 
 CHAPTER III. 
 
 ON THE VARIED TEANSLATION OF POINTS AND RIGID 
 
 BODIES. 
 
 Section, 1. — Law of Varied Translation. 
 
 328. Second Law of Motion. — Chcmge of momentvim is propor- 
 tional to the impulse proiMcing it. In this statement, as in that of 
 the first law of motion, Article 327, 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 330, 331.) 
 
 329. General Equations of Dynamics. — To express the second 
 law of motion algebraically, two- methods may be followed : the 
 first method being to resolve the change of momentum into direct 
 variation and xleviation, and the impulse into direct and deflecting 
 impulse; and the second method being to resolve both the change 
 of momentum and the impulse into components parallel to three 
 rectangular axes. 
 
 First method, m being the mass of the body, v its velocity, and 
 r the radius of curvature of its path, it follows from Articles 73 
 and 75 that the rate of direct va/riation of its momentum is 
 dv d^s 
 
 '^d-t='"'-dfi' 
 and from Articles 77 and 78, that the rate of deviation of its 
 momentum is 
 
 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 : — 
 
 -n T, -r, A dv (Ps ,_, 
 
 Por -E = Pcos ^ = »»*T4 = ''*j^j V--I 
 
 Q = P sm e = (2.) 
 
212 PRINCIPLES OP KINETICS. 
 
 The radius of curvature r is in the direction of the deviating 
 
 force Q. 
 
 Second method. As in Article 80, let the velocity of the body 
 
 , . , , , , , dx dy dz 
 
 be resolved into three rectangular components, -r-, -f-, -j— ; so that 
 
 the three component rates of variation of its momentum are 
 
 d^ X d^y d^z 
 a<^ dv dt'' 
 
 Also let the unbalanced force F, making the angles «, /3, y, with 
 the axes of co-ordinates, and its impulse per unit of time, be 
 resolved into three components, 'S^, F^, F^. Then we obtain 
 
 .(3.) 
 
 F^ = Fcos« = ?n-^; 
 
 F„ = Fco3/3= m-^; 
 
 d^ z 
 F, = Fcosy=m^; 
 
 three equations, which are substantially identical with the Equa- 
 tions 1 and 2. 
 
 330. Mass in Terms of Weight. — A body's own weight, acting 
 unbalanced on the body, produces velocity towards the earth, 
 increasing at a rate per second denoted by the symbol g, whose 
 numerical value is as follows; — ^Let ». denote the latitude of the 
 place, h its elevation above the mean level of the sea, 
 
 g'i = 32-1695 feet^ or 9-8051 metres, per second; 
 being the value of g for X = 45° and h = 0, and 
 
 R = 20900000 feet, or 6370000 mfetres, nearly, 
 being the earth's mean radius; then 
 
 ?=?i ■(! -0-00284 cos 2 '^) • (l -^) (1-) 
 
 For latitudes exceeding 45°, it is to be borne in mind that cos 2 x 
 is negative, and the terms containing it as a factor have their signs 
 reversed. 
 
 For practical purposes connected with ordinary macliines, it is 
 sufficiently accurate to assume 
 
 ^ = 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 
 
THE MOTION OF A FALLING BODY. 213 
 
 force F, tlie change of velocity in the direction of F produced in a 
 second will be 
 
 F_F^ 
 
 m~W' 
 ■whence 
 
 m = — (o.) 
 
 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 diflFerent elevations and in 
 diflferent latitudes. 
 
 331. 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 coefficient g. When the imit 
 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 
 lbs.; or in French measures, the mass of 9*81 kilogrammes. 
 
 332. 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 73. In the equations 
 of that Article, for the rate of variation of velocity a, is to be sub- 
 stituted the coefficient g, mentioned in the last Article. Thejji if 
 Vq be the velocity of the body at th* beginning of an interval of 
 time t, its velocity at the end of that time is 
 
 v = VQ + gt, (1.) 
 
 the mean velocity during that time is 
 
 — 2~-^o + -^> W'> 
 
 and the vertical height fallen through is 
 
 9 
 
 <2 
 
 ^ = «o< + -2- (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 
 tirne. To obtain the height in terms of the initial and final velo- 
 cities, or vice versa, Equation 2 is to be multiplied hj v-Vf,=gt, 
 
214: PRINCIPLES OF KINETICS. 
 
 the acceleration, and compared with Equation 3; giving the follow- 
 ing results : — 
 
 2 ^ I (4.) 
 
 ^9 ' J 
 
 When the body falls from a state of rest, v^ is to be made =0; so 
 that the following equations are obtained : — 
 
 v=gt;h^\ = ^^ (5.) 
 
 The height h in the last equation is called the Imght 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, 
 « is to be made = in the last of the Equations 4; then 
 
 ''=-r/ « 
 
 being a rise equal to the fall due to the initial velocity v^. 
 
 333. 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. 
 139, let O represent the point from which the projectile is originally 
 projected in the direction A, making the angle X O A = e with 
 a horizontal line O X in the same vertical plane with O A. Let 
 
 horizontal distances parallel to 
 X be denoted by x, and verti- 
 cal ordinates parallel to Z by z, 
 positive upwards, and negative 
 downwards. In the equations of 
 vertical motion, the symbol h of 
 the equations of Article 332 is to , 
 be replaced by - a, because of h 
 Kg. 139. and z being measured in opposite 
 
 directions. 
 Let «fl be the velocity of projection. Then at the instant of pro- 
 jection, the components of that velocity are, 
 
 horizontal, -^ = V()Cos 6; vertical, -r- =Vq sin 6; 
 
AN UNIFORM EFFORT OR RESISTANCE. 215 
 
 and after the lapse of a given time t, those components have become 
 ;t- = Vo cos tf = constant j 
 
 •(!•) 
 
 Hence the co-ordinates of the body at the end of the time t are 
 horizontal, x = VqCos S -t; \ 
 
 vertical, is = VQm.n S 't-^ ; I ^ ' 
 
 the Equations 2 being tliose of which the differential coefficients 
 
 are Equations 1, and because t= z, those co-ordinates are 
 
 ^ Vq cos 6 
 
 thus related, 
 
 '-'•'^^''-M^y'^' ('■> 
 
 an equation which shews the path O B C of the projectile to be a 
 parabola with a vertical axis, touching 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 (remembering that sin^ 6 + cos^ 8=1), 
 
 ^ = ^ + % = ^'-^^o^i^e-gt + gH^ = vl-2gz; (4.) 
 
 from the last form of which is obtained the equation 
 
 "^■' (=•) 
 
 which, being compared with Equation. 4 of Article 332, shews that 
 the relation between the va/riation of vertical elevation, and ilie vaca- 
 tion of the squa/re of the resultant velocity, is the same, whether the 
 velocity is in a vertical, inclined, or horizontal direction. 
 
 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. 
 
 334. An Uniform Effort or Resistance, unbalanced, causes the 
 velocity of a body to vary according to the law expressed by this 
 equation, 
 
 dt=^3; (1.) 
 
216 PEINCIPLES OP KINETICS. 
 
 where ^f is the constant ratio which the unbalanced force bears to 
 the weight of the moving body, positive or negative according to 
 the direction of the force; so that by substituting / g- for g in the 
 equations of Article 332, 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, Attwood, 
 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. Two weights, P and K, 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 + E, and the moving force P - R, being less than the 
 weight in the ratio, 
 
 P-R 
 ■'"P + ll' 
 consequently the two weights move according to the same law 
 with a falling body, but more slowly in the ratio of/ to 1. 
 
 335. Deviating Force of a Single Body. — It is part of the first 
 law of motion, that if a body moves under no force, or balanced 
 forces, it moves in a straight line. 
 
 It is one consequence of the second law of motion, that in order 
 that a body may move in a curved path, it must be continually 
 acted upon by an unbalanced force at right angles to the direction 
 of its motion, the direction of the force being that towards which 
 the path of the body is curved, and its magnitude bearing the same 
 ratio to the weight of the body that the height due to the body's 
 velocity bears to half the radius of curvature of its path. 
 
 This principle is expressed symbolically as follows : — 
 
 Half radius of HoigM due Body's Deviating 
 
 curvature. to velocity. weight, force. 
 
 or otherwise that the acceleration produced by gravity^ bears 
 the same ratio, to the rate of deviation, that the weight beafs to 
 the magnitude of the deviating force, which may be symbolically 
 
 expressed g ■- : : W : Q = . 
 
 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. 
 
DEVIATING FOKCE IN TERMS OF ANGULAR VELOCITY. 217 
 
 A pair of free bodies attracting each other have both deviated 
 motions, the attraction of each guiding the other; and their devia- 
 tions of momentum are equal in equal times; that is, their devia- 
 tions of motion are inversely as their masses. 
 
 In a machine, each revolving body tends to press or draw the 
 body which guides it away from its position, in a direction from 
 the centre of curvature of the path of the revolving body; and that 
 tendency is resisted by the strength and stiffness of the guiding 
 body, and of the frame with which it is connected. 
 
 336. 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 therefoi-e the rod 
 will assume such an inclination that Fig. 140. 
 
 height B C _ weight _gr .^ . ' 
 
 radius AB centrifugal force i>^ ^ "^ 
 
 where r = A B. Let n be the number of turns per second of the 
 pendulum; then 
 
 v = 2 irnr; 
 
 and therefore, making B C = A, 
 
 7. = ^- ^ 
 
 1)2 4 w^ n^ 
 
 ,. , , . , -T , ,0-8154 foot 9-7848 inches ,„ , 
 = (m the latitude of London) j 2 ••••(2.) 
 
 When the speed of revolution varies, the inclination of the pendu- 
 lum varies so as to adjust the height to the varying speed. 
 
 337. Deviating Force in Terms of Angular Velocity.— If the 
 radius of curvature of the path of a revolving body be regarded as 
 a sort of arm of constant or variable length at the end of which 
 the body is carried, the angular velocity of that arm is given by 
 the expression, 
 
 •(!•) 
 
 Let ar be substituted for v in the valije of deviating force of 
 Article 335, and that value becomes 
 
 Q=:^^ (2.) 
 
218 
 
 PEINCIPLES OF KINETICS. 
 
 In the case of a body revolving with uniform velocity in a circle, 
 like the bob A of the revolving pendulum of Article 336, a = 2 a- «, 
 ■where n is the number of revolutions per second, so that 
 
 Q = 
 
 4 w^W w^ r 
 
 .(3.) 
 
 from which equation the height of a revolving pendulum might be 
 deduced with the same result as in the last Article. 
 
 338. A Simple Oscillating Pendulum consists of an indefimtely 
 small weight A, fig. 141, 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 hob 
 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-cosZDCA = W-=, 
 CA 
 
 acting along C A, and balanced by the tension of the 
 Fig. 141. rod or cord, and 
 
 W-sinZDCA = W-=, 
 A 
 
 acting in the direction of a tangent to the arc, towards D, and un- 
 balanced. The motion of A depends on the latter force. 
 
 When the arc A D E is small compared with the length of the 
 pendulum A C, it very nearly coincides with the chord ABE; and 
 the horizontal distance A B, to which the moving force is propor- 
 tional, is very nearly equal to the distance of the bob from D, the 
 central point of its oscillations. Then if the length of the pendu- 
 lum, A, be denoted by I, we have approximately, for small arcs 
 of oscillation, 
 
 — = 2 ■k\/ -\ and 
 n V 9 
 
 b = 
 
 4^2 ^2^ 
 
 .(1.) 
 
 and the following statement shews the connection between a simple 
 oscillating and revolving pendulum, viz., thai the length of a simple 
 oscillating pendulvmi, making a given number of small double oscUlor 
 tions in a second, is sensibly equal to the hdght of a revolving pendu- 
 lum, making tlie same number of revolutions in a second. 
 
THE ANGULAE MOMENTUM. 219 
 
 Section 2. — ^Yaeied Teanslation of' a System op Bodies. 
 
 339. Conservation of Momentum. — Theoeem. The mutual 
 actions of a system of bodies cannot change their resultant momentum. 
 (Resultant momentum has been defined in Article 311.) 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 opposite impulses on those bodies, and produces equal and 
 opposite momenta. Therefore the momenta produced in a system 
 of bodies by their muti:(|p,l actions neutralize each other, and have 
 no resultant, and cannot change the resultant momentum of the 
 system. 
 
 340. Motion of Centre of Gravity. — Corollaey. The variations 
 of the motion of the centre of gravity of a system of bodies are wholly 
 produced by forces exerted hy 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. 
 
 341. The Angular Momentum, relatively to a fixed point, of a 
 body ha'fring 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. Let m be the mass of 'the body, 
 V its velocity, I the length of the before-mentioned perpendicular; 
 then 
 
 , "Wvl 
 
 mv 1 = 
 
 9 
 
 is the angular momentum relatively to the given point. 
 
 Angular momenta are compounded and resolved like forces, 
 each angular momentum being represented by a line whose length 
 is proportional to the magnitude of the angular momentum, and 
 whose direction is perpendicular to the plane of the motion of the 
 body and of the fixed point, and such, that when the motion of the 
 body is viewed from the extremity of the line, the radius vector of 
 the body seems to have right-handed rotation. The direction of 
 such a line is called the axis of the angular momentum which it 
 represents. The resultant angular momentum of a system of bodies 
 is the resultant of all their angular momenta relatively to their 
 common centre of gravity; and the axis of that resultant angular 
 
220 PRINCIPLES OF KINETICS. 
 
 momentum is called the axis of anguh/r momentum of the system. 
 The term angulcM" momentum was introduced by Mr. Hayward. 
 
 342. Angular Impulse is the product of the moment of a couple 
 of forces (Article 200) into the time during which it acts. Let F 
 be the force of a couple, I its leverage, and dt the time during 
 which it acts, then 
 
 Yldt 
 
 is the angular impulse. Angular impulses are compounded and 
 resolved like the moments of couples. 3ee a.Aja<x~vrt.n.vv,A,AwW) 
 - 343. Eelations of Angular Impulse and Angular Momentum. — 
 Theorem. The variation, in a given tim^, of the angular momentum 
 of a body, is equal to the anguh/r impulse producing that vaHation, 
 and has the same axis. This is a consequence which is deduced 
 from the second law of motion in the following manner : — Conceive 
 an unbalanced force F to be applied to a body m, and an equal, 
 opposite, and parallel force, to a fixed point, during the interval dt; 
 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 
 
 Vldt. 
 
 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^lP dt; 
 
 so that relatively to the fixed point, the variation of the body's 
 angular momentum is 
 
 m ldv=¥ Idt; 
 
 being equal to the angular impulse, and having the same axis. — 
 Q. E. D. 
 
 344. Conservation of Angular Momentum. — Theorem. The 
 resultant angular momentum of a system of bodies cannot be chamged 
 in magnitude, nor in the direction of its axis, by the mviual 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 efiect the equilibrium or motion of the system. Also, the 
 resultant of all the couples thus formed is nothing; therefore, the 
 
COLLISION. 221 
 
 resultant of their angular impulses is nothing; therefore, the 
 1 resultant of the several variations of angular momentum produced 
 by those angular impulses is nothing; therefore, the resultant 
 jangular momentum of the system is invariable in amount and in 
 the direction of its axis. — Q. E. D. 
 
 346. Collision. — The most useful 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 
 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 m^ion, and appears as vibration and heat. 
 If the bodies are equal, similar, and perfectly elastic, that fraction 
 is nothing. 
 
 Let Ml, m,, be the masses of the two bodies, and Uj, U2, their 
 velocities before the collision, whose directions should be indicated 
 by their signs. Then the velocity of their common centre of 
 gravity is 
 
 Mo = i (l.) 
 
 and this is not altered by the collision. 
 
222 
 
 CHAPTER IV. 
 ROTATIONS OF EIGID BODIES. 
 
 346. The Motion of a Rigid Body, or of a body which sensibly 
 preserves the same figure, has already been shewn in Part I., 
 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 moTnentwrn, 
 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 momentum of the 
 entire mass if it were Ooncentrated at that centre; the variations 
 of the angular momentum of the rotation are due to the resultant 
 couple which is combined with that resultant force. The varia- 
 tions 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 I., Chapter II., 
 Section 3. 
 
 Section 1. — On Moments op Inertia, Eadii of Gyration, 
 'AND Centres of Percussion. 
 
 347. 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 : — 
 
 ~ = mr^ = , (1.) 
 
 9 9 
 
 r is the perpendicular distance of the mass m, whose weight is Wj 
 from a given axis; and the moment of inertia, according to the 
 unit employed, is either I, or I-H.9; 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 
 
THE KADIUS OF GYEATIOK 223 
 
 of inertia of an unit of mass at the end of the same arm. 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; but in the remainder of this treatise the 
 term will be used in its strict sense, and according 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 avoirdi^ois into the square of a certain number oi/eet. 
 
 The geometrical relati<^s amongst moments of inertia, to which 
 the present section refers, are independent of the imit of measure. 
 
 348. The Moment of Inertia of a System of Physical Points, 
 relatively to a given axis, is the sum of the moments of inertia 
 of the several points; that is, 
 
 I=S-W5-^ (1.) 
 
 349. The Moment of Inertia of a Rigid Body is the sum of the 
 moments of inertia of all its parts, and is found by integration; that 
 is, by conceiving the body to be divided into small parts of regular 
 figure, multiplying the 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 
 molecules, whose dimensions are d x, d y, and d z, the volume of 
 each d X d y d z, and the mass of unity of volume w. Then 
 
 1= / / j r'w dxdy dz (1.) 
 
 Hence follows the general principle that propositions relative to 
 the geometrical relations amongst the moments of inertia of systems 
 of points are made applicable to continuous bodies by substituting 
 integration for ordinary summation; that is, for example, by putting 
 
 // / for 2, and wd x dy d miorW. 
 
 350. The Radius of Gyration of a body about a given axis is that 
 length whose square is the mecm 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, 
 
 ..=^.1^^ 
 
224 PRINCIPLES OP KINETICS. 
 
 When symbols of iutegratioa are used, tliis^becomes 
 / I It^w'dxdydz 
 
 e r f f W 
 
 I / j w -dx dydz 
 
 351. Components of Moment of Inertia. — Let the positions of 
 the particles of a body be referred to three rectangular axes, one of 
 •which, X, is that about which the moment of inertia is to be 
 taken. Then the square of the radius vector of any particle is 
 
 80 that the moment of inei-tia round the axis of x is 
 
 I, = 2-W 2/^ + 2 •Wa'j (1.) 
 
 that is to say, the moment of inertia of a body round a given axis 
 may be found by adding together the sum of the products of the 
 masses of the particles, each multiplied by the squxure of each of its 
 distances from a pair of planes euiting each other at right angles in 
 the given axis. 
 
 In the same manner it may be shewn that the moments of 
 inertia of the same body round the other two axes are given by 
 the equations 
 
 I, = 2-Wa» + 2-War'j 1, = '2-W a^ + ^Z-W y^ (2.) 
 
 352. Moments of Inertia Bound Parallel Axes Compared.— 
 Theorem. The mmnerU of inertia of a body about any given axis 
 is equal to its moment of inertia about am axis traversing its centre 
 of gravity parallel to the given axis, added to the moment of inertia 
 (ibout the given asais due to the wJtole mass of the body concentrated 
 at its centre of gravity. 
 
 This theorem may be 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 = ,^-2W + Io (1.) 
 
 Corollary I. The radius of gyration ({) of a body about any 
 axis is equal to the hypotenuse of a right-angled triangle, of which 
 the two sides are respectively equal to the radius of gyration of the 
 body about an axis traversing the centre of gravity parallel to the 
 
MOMENTS. OP INERTIA AND RADII OF GYRATION. 225 
 
 given axis (»„), and to the perpendicular distance between these 
 axes (ro). That is to say, 
 
 i' = ri + <:l (2.) 
 
 CoROLLAfiT 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. 
 
 353. Combined Moments of Inertia. — Theorem. TIw combined 
 moment of inertia of a rigidly connected system of bodies abovi a 
 given axis, is equal to the combined moment of inertia which the sys- 
 tem, would have about the given axis, if each body were concentrated 
 at its own centre of gravity, added to the sum of the several moments 
 of inertia of the bodies, about axes traversing their respective centres 
 of gravity, parallel to the given aads. 
 
 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^ 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 352, Equation 1, is 
 
 I = Wrl + J,. 
 
 Consequently, the combined moment of inertia of the system of 
 bodies is 
 
 2I = S-Wr? + 2Io; (1). 
 
 — Q. E. D. 
 
 354. 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 352 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. 
 
 The column headed W gives the mass of the body ; that headed 
 Id gives the moment of inertia j that headed d, the square of the 
 radius of gyration. The mass of an unit of volume is in each case 
 denoted by w. 
 
226 
 
 PRINCIPLES OP KINETICS. 
 
 Body. 
 
 W 
 
 eS 
 
 I. Sphere of radius r, 
 
 II. Spheroid of revolution — 
 polar semi-axis a, equa- 
 torial radius r,...,; 
 
 III. Ellipsoid — semi-axes, a, 
 T>,e, 
 
 IV. Spherical shell — external 
 radius r, internal /,.... 
 
 V. Spherical shell, insensibly 
 thin — radius r, thick- 
 ness dr, 
 
 VI. Circular cylinder — ^length 
 2a, radius r, 
 
 VII. Elliptic cylinder — ^length 
 2o, transverse semi-axes 
 *|C, 
 
 VIII. Hollow circular cylinder- 
 length '2a, external ra- 
 dius r, internal r', 
 
 IX. Hollow circular cylinder, 
 insensibly thin — length 
 2a, radius r, thickness dr, 
 
 X Circular cylinder — ^length 
 2a, radius r, 
 
 XI. Elliptic cylinder — length 
 2a, transverse semi-axes 
 *i Ci 
 
 XII. Hollow circular cylinder- 
 length 2a, external ra- 
 dius r, internal i^, 
 
 XIII. Hollow circular cylinder, 
 insensibly thin — radius 
 r, thickness dr, 
 
 XIV. Rectangular prism — di- 
 mensions 2a, 2b, 2c, 
 
 XVi Bhombic prism — length 
 2a, diagonals 2i, 2c,.... 
 
 XVI. Khombic prism, as above. 
 
 Diameter 
 
 Polar axis 
 Axis, 2a 
 Diameter 
 
 Diameter 
 
 Lon^tudinal 
 axis, 2a 
 
 Longitudinal 
 axis, 2a 
 
 Longitudinal 
 axis, 2a 
 
 Longitudinal 
 axis, 2a 
 
 Transverse 
 diameter 
 
 Transverse 
 axis, 25 
 
 Transverse 
 diameter 
 
 Transverse 
 diameter 
 
 Axis, 2a 
 
 4ir!/>o5e 
 
 3 
 
 irtiyr'dr 
 
 29WII3^ 
 
 2<rmdbc 
 
 •iiwaff—r^ 
 
 itmardr 
 
 2ifwabc 
 
 ivwardr 
 iwdbc 
 
 Axis, 2a iwabo 
 
 Diagonal, 2i iteabc 
 
 15 
 15 
 
 3 
 
 iwafy* — /*) 
 inwai^dr 
 
 2»f 
 5 
 
 2)f 
 5 
 
 5 
 
 6 
 
 »raioic(3c'+4a') 
 
 6 
 
 +4oX»^-r^\ 
 
 ■ 4 
 
 !rM;o(2r' + goV)tfr 
 3 
 
 Swaicffl + e') 
 3 
 
 2wiac(fi+c^ 
 3 
 
 ■c^ + 2on 
 
 2(r'— r") 
 
 V + <? 
 i 
 
 2 
 
 i^ a' 
 4"'"3 
 
 1 + ^ 
 4 3 
 
 r'+t^' y 
 
 4 'S- 
 
 2 3 
 i' + c' 
 
 3 
 i'+c» 
 
 6 
 
 6 ■^S 
 
CENTRE OF PERCUSSION. 
 
 227 
 
 355. 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 (Ai-ticle 223), 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. 142 let XX be the given 
 axis, and G the centre of gravity of 
 the body. It is evident, in the first 
 place, that the centre of percussion 
 must be somewhere in the perpendi- 
 cular C G B let fall from the centre of 
 gravity on the given axis. Secondly, 
 in order that the statical moment of 
 the whole mass, concentrated partly at _. 
 
 C, and partly at the centre of percus- ^^' 
 
 sion B (still unknown), may be the same with that of the actual 
 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:G:B: :BC:GB:GC. 
 
 ■(!•) 
 
 Lastly, in order that the moment of inertia of the mass as supposed 
 to be concentrated at B and C, about the axis X X, may be the 
 same with that of the actual body, we must have 
 
 B ■ BC^ = W(t^ = W iil + rl) (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^ h- BC, we find, for the dis- 
 tance of the centre of percussion from the axis, 
 
 BG = '~=^+r, 
 
 Or 
 
 .(3.) 
 
 and for its distance from the centre of gravity. 
 
 
 .(1) 
 
228 PRINCIPLES OP KINETICS. 
 
 The last equation may also be expressed in the form. 
 
 GB-GC = e§; (5.) 
 
 which preserves the same value when G B and G C are inter- 
 changed ; thus shewing, that if a new axis parallel to the original 
 axis X X 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:- 
 
 W:C:B::fS + r?:e?:»^ (6.) 
 
 The preceding solution is represented by the following geometrical 
 construction: — Draw GD perpendicular to C G and = fg; join 
 C D, perpendicular to which draw D B cutting C G produced in 
 B; this point is the centre of percussion. 
 
 Also, D = s, the radius of gyration about X X ; and D B is the 
 radius of gyration about an axis traversing B parallel to X X. 
 
 If CE be taken = CD, E is sometimes called the Centre of Gyra- 
 tion of the body for the axis X X. ' 
 
 Section 2. — On Uniform Rotation. 
 
 356. The Momentum of a body rotating about its centre of 
 gravity is nothing, according to the principle of Article 344. 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. 
 
 357. The Angular Momentum is found in the following manner : 
 — Let a: 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 
 angular velocity of rotation. Then the velocity of any particle W, 
 whose radius vector is *• = Jy^ + ;?, is 
 
 ar = a ^y^ + z^, 
 
 and the angular momentum of that particle, relatively to the aocis of 
 rotation, is 
 
 9 9. ' 
 
 being the product of its moment of inertia into its angula/r velocity 
 
THE ACTUAL ENERGY OF ROTATION. 229 
 
 divided by g, because of the weights of the particles having been 
 used in computing the moment of inertia. 
 
 358. The Actual Energy of Rotation of a body rotating about its 
 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 
 
 '-2^='-2r' ^ ^^ 
 
 being the moment of inertia of the particle multiplied by „— . Hence 
 for the whole body the actual energy of rotation is 
 
 ^ = ^^ (2-) 
 
 that is to say, actual energy bears the same relation to angular velo- 
 city and momeni of inertia that it does to linear velocity and weight. 
 
230 
 
 CHAPTER V. 
 MOTIONS OF FLUIDS. 
 
 359. Division of the Subject. — The mode of division that will 
 be employed in this chapter in treating of the motions of fluids is 
 founded on the distinction between motions not sensibly afiected 
 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 conse- 
 quences, 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 following is the division of the subject of this chapter : — 
 
 I. Motions of Liquids under Gravity and Pressure alone. 
 II. Motions of Liquids affected by Friction. 
 
 Section 1. — Motions op Liquids without Friction. 
 
 360. Dynamic Head. — Let p denote the intensity of the pressure 
 of the liquid at a given point and i the weight of an unit of volume; 
 
 then 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 f, whose weight per uiit of base would be 
 equal to the pressure p. Now, let a vertical ordinate z be measured 
 positively downwards from a datum horizontal plane, g is is the 
 weight of a column of liquid per unit of base extending down from 
 that plane to a pai'ticle under consideration; p — js is the difference 
 between the intensity of the actual pressure at that particle and 
 the pressure due to its depth below the datum horizontal plane ; 
 and 
 
 ^-z = h (1.) 
 
 e 
 
 is the height or head due to that difference of intensity, being what 
 will be termed the dynamic head. When z is measured positively 
 
A SURFACE OF EQUAL PRESSURE. 231 
 
 upwards from a datum horizontal plane, its sign is to be clianged; 
 so that the expression for the dynamic head in that case becomes 
 
 P 
 
 + z = h (2.) 
 
 361. Law of Dynamic Head for Steady Motion. — This principle 
 may 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 
 
 Y2 
 
 ir— + h = constant. 
 ^9 
 
 This equation applies to the particles ■which successively occupy the 
 same fixed point, as well as to each individual particle. 
 
 362. The Total Energy of a particle of a moving liquid without 
 friction is expressed by multiplying the expression in the previods 
 equation by the weight of the particle W, thus : — 
 
 in which — = — is the actual energy of the particle, and W h is its 
 
 potential energy; because, from the last Article it appears, that by 
 
 the diminution otWh, —^ — may be increased by an equal amount, 
 
 and vice versa; so that the dynamic Jiead 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. 
 
 363. 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, % the vertical ordinate, mea- 
 sured positively upwards from a given horizontal plane, of any point 
 in the free surface of the liquid, and h^ the dynamic head at the 
 same point; then it appears from Article 360, Equation 2, that for 
 that surface, 
 
 hT-z,= — = constant (1.) 
 
 e 
 
 364. A Surface of Equal Pressure is characterized by an ana- 
 logous equation, 
 
 A-« = — = constant; (1.) 
 
232 PEINCIPLES OF KINETICS. 
 
 and all surfaces of equal pressure fulfil the differential equation, 
 
 dh = dz; (2.) 
 
 for the differential coefficient of a constant being equal to dh 
 - dz = Equation 1, and .: d h = dz ■which, for steady motion, 
 becomes 
 
 dz = dh= -d-^; (3.) 
 
 found by differentiating the equation of Article 361, expressing 
 that the vai-iations of actual energy are those due to the variations 
 of level simply. 
 
 365. Motion in Plane Layers is a state which is either exactly 
 or approximately realized in many ordinary cases of liquid motion ; 
 
 1 -A/ 
 
 
 
 
 
 < 
 
 
 1 
 
 
 
 
 
 
 
 Fig. 143. 
 
 Fig. 144. 
 
 and the assumption of which is often used as a first approximation 
 
 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 143, 144, and 145, each of which 
 represents a reservoir containing liquid 
 up to the elevation OZi = Zi above a given 
 datum, and discharging the liquid from 
 an orifice A,, at the smaller elevation O Zo 
 = Zq. The liquid moves exactly or nearly 
 
 in plane layers at the upper surfuce Ai 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, Vi the velocity of descent 
 
 of the liquid at the upper surface, v^ its velocity of outflow from the 
 
 Fig. 145. 
 
GENERAL LAWS OF FLUID FRICTIOK. 233 
 
 orifice; then, according to Article 116, the equation of continuity is 
 
 •(!•) 
 
 Q Q 
 
 The pressures at the upper surface and at the orifice respectively 
 are each equal to the atmospheric pressure; hence the difierence of 
 dynamic head is simply the difierence of elevation ; that is to say, 
 
 hi — nQ = Zi~ ZqI 
 
 therefore, according to Article 361 and Article 364, Equations 2 
 and 3, • 
 
 This gives for the velocity of outflow, 
 
 / r 2gr(gi-a:o) '\ 
 
 -Vj i-| I- <^) 
 
 from which can be computed the rate of flow or discharge by means 
 of Equation 1. 
 
 366. The Contracted Vein is the name given to a portion of a 
 jet of fluid at a short distance from an orifice in a plate, which is 
 smaller in diameter and in area than the orifice, owing to a spon- 
 taneous 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 
 Ao in the equations of the last Article. 
 
 The ratio of the area of the contracted vein, or efiective orifice, 
 to that of the actual orifice, is called the coefficient of contraction. 
 For sharp edged orifices in thin plates, it has difierent values for 
 difierent figures and proportions of the orifice, ranging from about 
 0'58 to 0'7, and being on an average about ^. It diminishes some- 
 what for great pressures, and for dynamic heads of six feet and 
 upwards may be taken at about 0'6. The most elaborate table of 
 those coefiicients is that of Poncelet and Lesbros. 
 
 For orifices with edges that are not sharp and thin, the discharge 
 is modified sensibly by friction. 
 
 Section 2. — Motions of Liquids with Friction. 
 
 367. General Laws of Fluid Friction. — It is known by experi- 
 ment, that between a fluid, and a solid surface over which it glides. 
 
234 PRINCIPLES OP KINETICS. 
 
 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- 
 jjortional to tlie weight of a prism of the fluid, whose hose is the sur- 
 face of contact, aiid its height the height due to the relative velocity. 
 
 Let S be the surface of contact, v the velocity, e the weight of an 
 unit of volume of the fluid, and / a factor called the comment of 
 friction; then 
 
 ^=/eS|J, (1.) 
 
 is the amount of the friction at the surface S. 
 
 The coefficient / is not absolutely constant at difierent 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 coefficients of 
 friction, according to difierent 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 (i = diameter of pipe in feet; then, 
 
 or for velocities that are not very small, 
 
 /=0-0°<1^12j- 
 
 Iron pipes, value of/ for first approximation, 0'0064 
 
 Beds of rivers (Weisbaoh), / = « + -; a = 0'0074. 
 
 6 = 0-00023 foot. 
 Beds of rivers, value of / for first 1 0-0076 
 
 approximation, J 
 
 A collection of numerous formulse for fiuid friction, proposed by 
 difierent authors, together with tables of the results of the best 
 formulse, is contained in Mr. Neville's work on hydraulics. The 
 formulae of many authors, though difiering 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 difier. The two 
 formulse given above, on the authority of Darcy, for iron pipes, 
 
FEICTION IN AN UNIFORM STREAM. 
 
 235 
 
 are based on his experiments as recorded in his treatise dv, Mouve- 
 ment de VEau dans les TuyoMX. 
 
 368. Internal Fluid Friction. — Although the particles of fluids 
 liave no transverse elasticity — that is, no tendency to recover a 
 certain figure after having been distorted — it is certain that they 
 resist being made to slide over each other, and that there is a 
 lateral communication of motion amongst them; that is, that there 
 is a tendency of particles which move side by side in parallel lines 
 to assume the same velocity. The laws of this lateral communica- 
 tion of motion, or internal friction of fluids, are not known exactly; 
 but its efiects 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 perpen- 
 dicular to the axis of the channel (Article 365). 
 
 369. Friction in an Uniform Stream. — It is this last fact which 
 renders possible the existence of an open stream of uniform section, 
 velocity, and declivity. In hydraulic calculations respecting the 
 resistance of this, or any other stream, the value given to the 
 velocity is its mean value throughout a given cross-section of the 
 stream A, 
 
 "I <■•) 
 
 The greatest velocity in each cross-section of a stream takes place 
 at the point most distant from the rubbing surface of the channel. 
 Its ratio to the mean velocity is given by the following empirical 
 formula of Prony, where V is the greatest velocity in feet per 
 second : — 
 
 ^ _ 7-71 +V 
 
 V 10-25 + V ^ ' 
 
 In an uniform stream, the dynamic head which would otherwise 
 have been expended in producing increase of actual energy, is 
 wholly expended in overcoming friction. Consider a portion of 
 the stream whose length-is I, and fall z. The loss of head is equal 
 to the fall of the surface of the stream, according to Article 363 ; 
 and the expenditure of potential energy in a second is accordingly 
 
 »eQ = «e«A. 
 Equating this to the work performed in a second in overcoming 
 iriction, viz., v E, Equation 1, Article 367, we find 
 
236 
 
 PRINCIPLE^ OF KINETICS. 
 
 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 
 
 
 .(3.) 
 
 A 2^ 
 
 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, 
 
 A. 
 
 ,1* J, o u 
 
 sim = j = j — ^— • 
 
 .(4.) 
 
 is what is called the " hydkaulic mean depth" of the stream; 
 
 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 shewn 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. 
 
 370. Varying Stream. — In a stream whose area of cross-section 
 varies, and in which, consequently, the mean velocity varies at 
 difierent 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 sym- 
 bolically is the most convenient 
 for practical purposes. In fig. 
 146, let the origin of co-oi-- 
 dinates be taken at a point 
 completely below the part of the 
 stream to be considered ; let ho- 
 rizontal abscissse x be measured 
 against the direction of flow, 
 and vertical ordinates to the 
 surface of the stream, z, up- 
 wards. Consider any indefinitely short portion of the stream whose 
 
THE FRICTION OF A PIPE RUNNING FULL. 237 
 
 horizontal length ia dx; in practice this may almost always be con- 
 sidered as equal to the actual length. The fall in that portion of 
 the stream is d z, and the acceleration -dv, because of v being 
 opposite to X. Then modifying the expression for the loss of head 
 due to friction in Equation 3 of Article 369 to meet the present 
 case, and adding the loss of head due to acceleration, we find 
 
 . sdx v^ vdv ., . 
 
 '^^ = ^-A-'2^— T ^-^ 
 
 In applying this differential equation to the solution of any parti- 
 cular problem, for v is to be put Q -r- A, and for A and s are to be 
 put their values in terijs of x and z. Thus is obtained a differential 
 equation between z and x, and the constant quantity Q, the flow 
 per second. If Q is known, then it is sufficient to know the value 
 of z for one particular value of x, in order to be able to determine 
 the integral equation between z and x If Q is unknown, the 
 
 dz 
 values of z for two particular values of x, or of z and -j— (the 
 
 Of X 
 
 declivity), for one particular value of x, are required for the solu- 
 tion, which comprehends the determination of the value of O. 
 
 371. The Friction in a Pipe Running Pull 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 he the length of an indefinitely short 
 portion of a pipe measured in the direction of flow, « its internal 
 circumference, 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 
 
 ,, , dp vdv . sdl v^ ... 
 
 -dh= -dz i- = +/•— ^ ■ -— (1.) 
 
 The ratio -r^ is called the virtual or hydraulio 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 
 
 dz 
 actual declivity of the pipe, -=-,. 
 
 When. the pipe is of uniform section, dv = 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 I. If Q is given, « = Q-r A is also a given function of I; and 
 to solve the equation completely, there is only required in addition 
 
238 PRINCIPLES OP KINETICS. 
 
 the value of h for one particular value of I. If Q is unknown, the 
 
 fl h 
 
 values of h for two particular values of I, or of h and -j-v for one 
 
 particular value of I, are required for the solution, which compre- 
 hends the determination of Q. 
 
 372. Resistance of Mouthpieces. — A mouthpiece is the part of 
 a channel or pipe immediately adjoining a reservoir. The internal 
 friction of the fluid on entering a mouthpiece causes a loss of head 
 equal to the height due to the velocity multiplied by a constant 
 depending on the figure of the mouthpiece, whose values for 
 certain figures have been found empirically; that is to say, let 
 - A A he the loss of head; then 
 
 f v^ 
 
 -^^=T7' (^-^ 
 
 /' 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) 
 
 373. The Kesistance of Curves and Knees in pipes causes a loss 
 of head equal to the height due to the velocity multiplied by a 
 Goefficient, 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{0.13I.1.847(A/}, 
 
 and for a rectangular pipe, \ (!•) 
 
 /" = !{ 0-124+ 3-m(^AJ}; 
 
 for Ttnees, or sudden bends, let i be the angle made by the two por- 
 tions of the pipe at either side of the knee with each other; then 
 
 /" = 0-9457 sin2 I + 2-047 sin* 5 (2.) 
 
 374. A Sudden Enlargement of the channel in which a stream 
 of liquid flows, causes a sudden diminution of the mean velocity in 
 the same proportion as that in which the area of section is in- 
 
THE GENERAL PROBLEM. 239 
 
 creased. Thus, let v^ 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 
 
 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 Vj^ ( 1 — • 1, is expended in over- 
 coming the internal fluid friction of eddies, and so producing heat; 
 so that there is a loaa of total head, represented by 
 
 ^(i-iV (1.) 
 
 2gy 
 
 375. The General Problem of the flow of a stream with friction 
 
 1^ v^ 
 
 is thus expressed : — Let hi + -^--, and h^ + =^, be the total heads at 
 
 the beginning and end of the stream respectively; then the loss of 
 total head is represented by 
 
 ^-^^^V=^"^^ (!■) 
 
 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. 
 
240 
 
 PAET YI. 
 
 THEORY OP MACHINES. 
 
 CHAPTER I. 
 DEFINITIONS AND GENERAL PRINCIPLES. 
 
 376. Nature and Division of the Subject.— In the present Part 
 of this work, machines are to be considered not merely as modify- 
 ing motion, but also as modifying force, and transmitting energy 
 from one body to another. The theory of machines consists 
 chiefly in the application of the principles of dynamics to trains 
 of mechanism; and therefore much of the present Part of this 
 treatise will consist of references back to Parts II. and V. 
 
 There are two fundamentally different ways of considering a 
 machine, each of which must be employed in succession, in order 
 to obtain a complete knowledge of its working. 
 
 I. In the first place is considered the action of the machine 
 during a certain period of time, with a view to the determination 
 of its EFFICIENCY ; that is, the ratio which the -useful part of its 
 work bears to the whole expenditure of energy. The motion of 
 every ordinary machine is either uniform or periodical; and there- 
 fore the principle of the equality of energy and work 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. 
 
 377. 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 em- 
 ployed, and to transform energy from the various forms in which 
 
POWER AND EFFECT J HORSE POWER. 241 
 
 it may occur, such as chemical aflBnity, 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 availahle power of a prime mover is its useful work 
 in some given unit of time, such as a second, a minute, an hour, 
 or a day. 
 
 378. 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. 
 
 379. A Governor is a self-acting adjusting apparatus, usually 
 consisting of a pair of rotating pendulums, whose angle of devia- 
 tion from their axis depends upon the speed. 
 
 380. Fluctuations of Speed in a machine are caused by the 
 alternate excess of the energy received above the work performed, 
 and of the work performed above the energy received, which pro- 
 duce an alternate increase and diminution of actual energy. 
 
 381. A Ply-Wheel is a wheel with a heavy rim, whose great 
 moment of inertia reduces the coefficient of fluctuation of speed 
 to a certain fixed amount. 
 
 382. A Brake is employed to stop a machine in a shorter time 
 than can be done by simply suspending the effort of the prime 
 mover. 
 
 383. Useful and Lost 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. 
 
 384. Useful and Prejudicial Resistance are overcome in per- 
 forming useful work and lost work respectively. 
 
 385. 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. 
 
 386. 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 minutCj or 
 1,980,000 foot pounds per hour. The effect is equal to the power 
 multiplied by the efficiency. 
 
242 THEORY OF MACHINES. 
 
 387. Driving Point J Train; WorMng 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. 
 
 388. Points of Besistance are points in the train of mechanism 
 through which the resultants of prejudicial resistances act. 
 
 389. 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 
 of moving pieces which transmit energy from the driving point to the 
 worldng point. The same principles apply to a train of suocessive 
 machines, each driving that which follows it. 
 
2-13 
 
 CHAPTER II. 
 
 OF THE PERFOEMANCE OF WORK BY MACHINES. 
 
 Section 1. — Of Woek. 
 
 390. The Action of a Machine is to produce motion against 
 Resistance. For exam^e, if the machine is one for lifting solid 
 bodies, such as a crane, or fluid bodies, such as a pump, its action 
 is to produce upward motion of the lifted body against the resist- 
 ance arising from gravity ; that is, against its own weight : if 
 the machine is one for propulsion, such as a locomotive engine, its 
 action is to produce horizontal or inclined motion of a load against 
 the resistance arising from friction, or from friction and gravity 
 combined : if it is one for shaping materials, such as a planing 
 machine, its action is to produce relative motion of the tool and of 
 the piece of material shaped by it, against the resistance which that 
 material ofiers to having part of its surface removed; and so of 
 other machines. 
 
 391. Work. — The action of a machine is measured, or expressed 
 as a definite quantity, by multiplying the motion which it pro- 
 duces into the resistance, or force directly opposed to that motion, 
 which it overcomes; the product resulting from that multiplication 
 being called work. 
 
 In Britain, the distances moved through by pieces of mechanism 
 are usually expressed in feet ; the resistances overcome, in pounds 
 avoirdupois; and quantities of work, found by multiplying dis- 
 tances in feet by resistances in pounds, are said to consist of so 
 ■ma.nj foot-pounds. Thus the work done in lifting a weight of one 
 pound, through a height of one foot, is one foot-pound; the work 
 done in lifting a weight of twenty pounds, through a height of one 
 hundred feet, is 20 x 100 = 2,000 foot-pounds. 
 
 In France, distances are expressed in metres, resistances over- 
 come in kilogrammes, and quantities of work in what are called 
 kUogrammetres, one kilogrammStre being the work performed in 
 lifting a weight of one killogramme through a height of one 
 mfetre. 
 
 392. The Rate of Work of a machine means, the quantity of work 
 which it performs in some given interval of time, such as a second, 
 a minute, or an hour. It may be expressed in units of work (such 
 as foot-pounds) per second, per minute, or per hour, as the case 
 
244 ' THEORY OP MACHINES. 
 
 may be ; but there is a peculiar unit of power appropriated to its 
 expression, called a hoese-powee, -which is, in Britain, 
 
 550 foot-pounds per second, 
 or 33,000 foot-pounds per minute, 
 or 1,980,000 foot-pounds per hour. 
 
 In France, the term Force de Cheval is applied to the following 
 rate of work : — 
 
 Foot-ponnda. 
 
 75 kilogrammfetres per second = 542J 
 
 or 4,500 kilogrammfetres per minute = 32,549 
 
 or 270,000 kilogrammStres per hour = 1,952,932 
 
 being about one-seventieth part less than the British horse-power. 
 
 393. Velocity. — If the velocity of the motion which a machine 
 causes to be performed against a given resistance be given, then the 
 product of that velocity into the resistance obviously gives the rate 
 of work, or effective power. If the velocity is given in feet per 
 second, and the resistance in pounds, then their product is the rate 
 of work in foot-pounds per second, and so of minutes, or hours, or 
 other units of time. 
 
 It is usually most convenient, for purposes of calculation, to 
 _ express the velocities of the parts of machines either in feet i)er 
 second or in feet per minute. For certain kinetic calculations 
 the second is the more convenient unit of time • in stating the 
 performance of machines for practical purposes, the minute is the 
 unit most commonly employed. 
 
 394. Work in Terms of Angular Motion. — When a resisting 
 force opposes the motion of a part of a machine which moves round 
 a fixed axis, such as a wheel, an axis, or a crank, the. product of 
 the amount of that resistance into its leverage (that is, the perpen- 
 dicular distance of the line along which it acts from the fixed axis) 
 is called the moment, or statical mximent, of the resistance. If the 
 resistance is expressed in pounds, and its leverage in feet, then its 
 moment is expressed in terms of a measure which may be called 
 & foot-pound, but which, nevertheless, is a quantity of an entirely 
 different kind from a foot-pound of work. 
 
 Suppose now that the body to whose motion the resistance is 
 opposed turns through any number of revolutions, or parts of a 
 revolution ; and let T denote the angle through which it turns, 
 expressed in revolutions, and parts of a revolution , also, let 
 
 710 
 2 !r = 6-2832 = {{-^ 
 
 denote, as is customary, the ratio of the circumference of a circle to 
 
WORK IN TERMS OF PRESSURE AND VOLUME 245 
 
 its radius. Then the distance through which the given resistance 
 is overcome is expressed by 
 
 the leverage x 2 s- x Tj 
 
 that is, by the product of the circumference of a circle whose radius 
 is the leverage, into the number of turns and fractions of a turn 
 made by the rotating body. 
 
 The distance thus found being multiplied by the resistance over- 
 come, gives the work performed ; that is to say. 
 
 The work performed 
 =the resistance x the leverage x 2 t x T; 
 
 But the product of the resistance into the leverage is what is called 
 the moment of the resistance, and the product 2 «■ T is called the 
 angular motion of the rotating body ; consequently. 
 
 The work performed 
 = the moment of the resistance x the angvla/r motion. 
 
 The mode of computing the work indicated by this last equation 
 is often more convenient tlRln the direct mode already explained in 
 Article 391. 
 
 The angular motion 2 t T of a body during some definite unit of 
 time, as a second or a minute, is called its angula/r velocity; that is 
 to say, angula/r velocity is the 'product of the turns and fractions of a 
 turn made in an unit of time into the ratio of the circv/mference 
 of a circle to its radius. Hence it appears that 
 
 The rate of work 
 = the moment of the resistance x the angular velocity. 
 
 395. Work in Terms of Pressure and Volume. — If the resistance 
 overcome be a pressure uniformly distributed over an area, as when 
 a piston drives a fluid before it, then the amount of that resistance 
 is equal to the intensity of the pressure, expressed in units of force 
 on each unit of area (for example, in pounds on the square inch, 
 or pounds on the square foot) multiplied by the area of the sur- 
 face at which the pressure acts, if that area is perpendicular to 
 the direction of the motion ; or, if not, then by the projection of 
 that area on a plane perpendicular to the direction of motion. In 
 practice, when the a/rea of a piston is spoken of, it is always 
 understood to mean the projection above mentioned. 
 
 Now, when a plane area is multiplied into the distance through 
 which it moves in a direction perpendicular to itself, if its motion 
 is straight, or into the distance through which its centre of gravity 
 moves, if its motion is curved, the product is the volume of the 
 space traversed by the piston. 
 
 Hence the work performed by a piston in driving a fluid before 
 
246 THEORY OP MACHINES. 
 
 it, or by a fluid in driving a piston before it, may be expressed in 
 either of. the following -ways : — 
 
 Resistance x distance traversed 
 
 = intensity of pressure x (vrea x distance traversed; 
 
 — intensity of pressure x volume traversed. 
 
 In order to compute the work in foot-pounds, if the pressure is 
 stated in pounds on the square foot, the area should be stated in 
 square feet, and the volume in cubic feet; if the pressure is stated in 
 pounds on the square inch, the area should be stated in square inches, 
 and the volume in units, each of which is a prism, of one foot in 
 
 length, and one square inch in area ; that is, of ^77 of a cubic foot 
 
 in volume. 
 
 396. Algebraical Expressions for Work. — To express the results 
 of the preceding articles in algebraical symbols, let 
 
 s denote the distance in feet through which a resistance is over- 
 come in a given time ; 
 
 E, the amount of the resistance overcome in pounds. 
 Also, supposing the resistance to be overcome by a piece which 
 turns about an axis, let 
 
 T be the number of turns and fractions of a turn made in the 
 given time, and i = 2 b- T = 6-2833 T the angular motion in the 
 given time ; and let 
 
 I be the leverage of the resistance ; that is, the perpendicular 
 distance of the line along which it acts from the axis of motion ; 
 so that s = il, and R i is the statical moment of the resistance. 
 Supposing the resistance to be a pressure, exerted between a piston 
 and a fluid, let A be the area or projected area of a piston, a.ndp 
 the intensity of the pressure in pounds per unit of area. 
 
 Then the following expressions aU give quantities of work in the 
 given time in foot-pounds : — 
 
 Rs; iJS.1; p As ; ipA.1. 
 
 The last of these expressions is applicable to a piston turning on 
 an axis, for which I denotes the distance from the axis to the centre 
 of gravity of the area A. 
 
 397. Work against an Oblique Force. — The resistance directly 
 due to a force which acts against a moving body in a direction 
 oblique to that in which the body moves, is found by resolving 
 that force into two components, one at right angles to the direction 
 of motion, which may be called a lateral force, and which must be 
 balanced by an equal and opposite lateral force, unless it takes 
 effect by altering the direction of the body's motion, and the other 
 component directly opposed to the body's motion, which is the 
 
SUMMATION OF QUANTITIES OF WORK. 247 
 
 resistance required. Tbat resolution is effected by means of the 
 well-known principle of the parallelogram of forces as follows : — 
 
 In fig. 147, let A represent the point at which a resistance is over- 
 come, A. B the direction in which 
 that point is moving, and let A F 
 be a line whose direction and 
 length represent the direction and 
 magnitude of a force obliquely 
 opposed to the motion of A. 
 From F upon B A produced, let fall the perpendicular F R; the 
 length of that perpendicular will represent the magnitude of the 
 lateral component of »the oblique force, and the length A R will 
 represent the direct component or reaistoice. 
 
 The work done against an oblique resisting force may also 
 be calculated by resolving the motion into a direct component 
 in the line of action of the force, and a transverse component, 
 and multiplying the whole force by the direct component of the 
 motion. 
 
 398. Smnmation of Quantities of Work. — In every machine, 
 resistances are overcome during the same interval of time, by 
 different moving pieces, and at different points in the same moving 
 piece ; and the whole work performed during the given interval is 
 found by adding together the several products of the resistances 
 into the respective distances through which they are simultaneously 
 overcome. It is convenient, in algebraical symbols, to denote the 
 result of that summation by the symbol — 
 
 S-Es; (1.) 
 
 in which 2 denotes the operation of taking the sum of a set of 
 quantities of the kind denoted by the symbols to which it is pre- 
 fixed. 
 
 When the resistances are overcome by pieces turning upon axes, 
 the above sum may be expressed in the form — 
 
 -2'iRl; (2.) 
 
 and so of other modes of expressing quantities of work. 
 
 The following are particular cases of the summation of quantities 
 of work performed at different points : — 
 
 I. In a shifting piece, or one which has the kind of movement 
 called ^•anslation only, th« velocities of every point at a given 
 instant are equal and parallel ; hence, in a given interval of time, 
 the motions of all the points are equal ; and the -work praformed 
 is to be found by multiplying the sum of the resistances into the 
 motion as a common factor ; an operaiaon expressed algelbmieally 
 thus — 
 
 sSE; (3.) 
 
248 THEORY OF MACHINES. 
 
 IT. For a turning piece, the angular motions of all the points 
 during a given interval of time are equal; and the work performed 
 is to be found by multiplying the sum of the moments of the resist- 
 ances relatively to the aocis into the angular motion as a common 
 factor — an operation expressed algebraically thus — 
 
 iS-EZ; (4.) 
 
 The sum denoted by 2 • E / is the total moment of resistance of the 
 piece in question. 
 
 III. In every t/rain of mechanism, the proportions amongst the 
 motions performed during a given interval of time by the several 
 moving pieces, can be determined from the mode of connection of 
 those pieces, independently of the absolute magnitudes of those 
 motions, by the aid of the Theory of Pure Mechanism, Part II. 
 This enables a calculation to be performed which is called 
 reducing the resistances to the driving point; that is to say, 
 determining the resistances, which, if they acted directly at the 
 point where the motive power is applied to the machine, would 
 require the same quantity of work to overcome them with the 
 actual resistances. 
 
 Suppose, for example, that by the principles of pure mechanism 
 it is found, that a certain point in a machine, where a resistance R 
 is to be overcome, moves with a velocity bearing the ratio »i : 1 to 
 the velocity of the driving point. Then the work performed in 
 overcoming that resistance will be the same as if a resistance n R 
 were overcome directly at the driving point. If a similar calcula- 
 tion be made for each point in the machine where resistance is 
 overcome, and the results added together, as the following symbol 
 denotes : — 
 
 2-nB, (5.) 
 
 that sum is the equivalent resistcmce at the driving poini ; and if in 
 a given interval of time the driving point moves through the dis- 
 tance s, then the work performed in that time is — 
 
 s2-nR (6.) 
 
 The process above described is often applied to the steam engine, 
 by reducing all the resistances overcome to equivalent resistances 
 acting directly against the motion of the piston. 
 
 A similar method may be applied to the moments of resistances 
 overcome by rotating pieces, so as to reduce them to equivalent 
 moments at the, driving axle. Thus, let a resistance R, with the 
 leverage I, be overcome by a piece whose angular velocity of rota- 
 tion bears the ratio n:\ to that of the driving axle. Then the 
 equivalent moment of resistance at the driving axle is Ji R Z ; and 
 if a similar calculation be made for each rotating piece in the 
 machine which overcomes resistance, and the results added to- 
 gether, the sum — 
 
 2-wEZ (7.) 
 
WORK AGAIKST VARYING RESISTANCE, 
 
 249 
 
 is the total equivalent moment of resistance at the driving axle; and 
 if in a given interval of time the driving axle turns through the 
 arc i to radius unity, the work performed in that time is — 
 
 i 2 • w R ;. 
 
 .(8.) 
 
 IV. Centre of gravity. — The work performed in lifting a body 
 is the product of tihe weight of the body into the height through which 
 its centre of gravity is Ufted. 
 
 If a n\aohine lifts the centres of gravity of several bodies at once 
 to heights either the same or different, the whole quantity of work 
 performed in so doing is the sum of the several products of the 
 weights and heights ; but that quantity can also be computed by 
 multiplying the sum oJ*aU the 
 weights into the height through 
 which their common centre of 
 gravity is lifted. 
 
 399. Representation of Work 
 by an Area. — As a quantity of 
 work is the product of two 
 quantities, a force and a motion, 
 it may be represented by the *^'S- ^™- 
 area of a plane figure, which is the product of two dimensions. 
 Let the base of the rectangle A, fig. 148, represent one foot of 
 motion, and its height one pound of resistance ; then wUl its area 
 represent one foot-pound of work. 
 
 In the larger rectangle, let the base O S represent a certain 
 motion s, on the same scale with the base of the unit-area A; and 
 let the height (J U represent a certain resistance R, on the same 
 scale with the height of the unit^area A; then will the number of 
 times that the rectangle 0^ • R contains the unit-rectangle A, 
 express the number of foot-pounds in the quantity of work R s, 
 which is performed in overcoming the resistance R through the 
 distance s. 
 
 400. Work against Varying Resistance.— In fig. 149, let dis- 
 tances, as before, be represented y 
 by lengths measured along the 
 base line X of the figure ; and 
 let the magnitudes of the resist- 
 ance overcome at each instant be 
 represented by the lengths of 
 ordinates drawn perpendicular to , 
 X, and parallel to O Y :— For 
 example, when the working body 
 
 Fig. 
 
 A. H 
 
 149. 
 
 has moved through the distance re presen ted by O S, let the resist- 
 ance be represented by the ordinate S R. 
 
250 THEORY OP MACHINES. 
 
 If the resistance were constant, the summits of those ordinates 
 -would lie in a straight line parallel to O X, like E, B in fig. 148; 
 but if the resistance varies continuously as the motion goes on, the 
 summits of the ordinates will lie in a line, straight or curved, such 
 as that marked E R Gr, fig. 149, which is not parallel to X. 
 
 The values of the resistance at each instant being represented by 
 the ordinates of a given line ERG, let it now be required to deter- 
 mine the work performed against that resistance during a motion 
 represented by O F = s. 
 
 Suppose the area O E G F to be divided into bands by a series of 
 parallel ordinates, such as A C and B D, and between the upper 
 ends of those ordinates let a series of short lines, such as C D, be 
 drawn parallel to O X, so as to form a stepped or serrated outline, 
 consisting of lines parallel to O X and O Y alternately, and approxi- 
 mating to the given continuous line E G. 
 
 Now conceive the resistance, instead of varying continuously, to 
 remain constant during each of the series of divisions into which 
 the motion is divided by the parallel ordinates, and to change 
 abruptly at the instants between those divisions, being represented 
 for each division by the height of the rectangle which stands on 
 that division : for example, during the division of the motion 
 represented by A B, let the resistance be represented by A G, and 
 so for other divisions. 
 
 Then the work performed during the division of the motion re- 
 presented by A B, on the supposition of alternate constancy and 
 abrupt variation of the resistance, is represented by the rectangle 
 A B • A 0; and the whole work performed, on the same supposition 
 during the whole motion O F, is represented by the sum of all the 
 rectangles lying between the parallel ordinates; and inasmuch as 
 the supposed mode of variation of the resistance represented by the 
 stepped outline of those rectangles is an approximation to the real 
 mode of variation represented by the continuous line E G, and is a 
 closer approximation the closer and the more numerous the parallel 
 ordinates are, so the sum of the rectangles is an approximation to 
 the exact representation of the work performed against the conti- 
 nuously varying resistance, and is a closer approximation the closer 
 and more numerous the ordinates are, and by making the ordinates 
 numerous and close enough, can be made to differ from the exact 
 representation by an amount less than any given difference. 
 
 But the sum of those rectangles is also an approximation to the 
 area O E G F, bounded above by the continuous line E G, and is 
 a closer approximation the closer and the more numerous the ordi- 
 nates are, and by making the ordinates numerous and close enough, 
 can be made to diflfer from the area O E G F by an amount less 
 than any given difference. 
 
 Therefore the area O E G F, bounded ly the straight line F, which 
 
THE WORK PERFORMED AGAINST FRICTION. 251 
 
 represents the motion, by the line E G, whose ordinates represent the 
 values of the resistance, and by the two ordinates O E emd F G, repre- 
 sents exactly the work performed. (See Article 34, page 17). 
 
 The MEAN RESISTANCE during the motion is found by dividing 
 the area O E G F by the motion O F. 
 
 401. Useful Work and Lost Work.— The useful work of a ma- 
 chine is that which is performed in effecting the purpose for which 
 the machine is designed. The lost work is that which is performed 
 in producing effects foreign to that purpose. The resistances over- 
 come in performing those two kinds of work are called respectively 
 useful resistance and prejudicial resistance. 
 
 The useful work and the lost work of a machine together make 
 up its total or gross worTt. 
 
 In a pumping engine, for example, the useful work in a given 
 time is the product of the weight of water lifted in that time into 
 the height to which it is lifted : the lost work is that performed in 
 overcoming the, friction of the water in the pumps and pipes, the 
 friction of the plungers, pistons, valves, and mechanism, and the 
 resistance of the air pump and other parts of the engine. 
 
 For example, the useful work of a marine steam engine in a 
 given time is the product of the resistance opposed by the water to 
 the motion of the ship, into the distance through which she 
 moves : the lost work is that performed in overcoming the resist- 
 ance of the water to the motion of the propeller through it, the 
 friction of the mechanism, and the other resistances of the engine, 
 and in raising the temperature of the condensation water, of the 
 gases which escape by the chimney, and of adjoining bodies. 
 
 There are some cases, such as those of muscular power and of 
 windmills, in which the useful work of a prime mover can be 
 determined, but not the lost work. 
 
 402. The Work Performed against Friction in a given time, 
 between a pair of rubbing surfaces, is the product of that friction 
 into the distance through which one surface slides over the other. 
 
 When the motion of one surface relatively to the other consists 
 in rotation about an axis, the work performed may also be cal- 
 culated by multiplying the relative angular motion of the surfaces 
 to radius unity into the moment of friction; that is, the product of 
 the friction into its leverage, which is the mean distance of the 
 rubbing surfaces from the axis. 
 
 For a cylindrical journal, the leverage of the friction its simply 
 the radius of the journal. 
 
 For a flat pivot, the leverage is two-thirds of the radius of the 
 pivot. 
 
 For a collar, let r and r be the inner and outer radii; then the 
 leverage of the friction is 
 
 2 T^-r'^ 
 
 3 • r2-r'2 (^-^ 
 
252 
 
 THEORY OF MACHINES. 
 
 In the cwp amd hall pivot, the enA of the shaft, and the step on 
 which it presses, 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 some- 
 what less radius than the concave surfaces of the cups. The 
 moment of friction of this pivot is at first almost inappreciable, 
 from the extreme smallness 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. 
 
 By the rolling of two surfaces over each other without sliding, a 
 resistance is caused, which is called sometimes " rolling friction," 
 but more correctly rolling resistance. 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 angula/r velocity of the rolling surfaces relatively to each 
 other. The following are approximate values of the arm in decimals 
 of a foot: — 
 
 Oat upon oak, 0'006 (Coulomb). 
 
 Lignum-vitse on oak, 0-004 „ 
 
 Cast-iron on cast-iron, 0002 (Tredgold). 
 
 The work lost in friction produces heat in the proportion of one 
 British thermal unit, being so much heat as raises the temperature 
 of a pound of water 1° of Fahr., for every 772 foot-pounds of 
 lost work. 
 
 The heat produced by friction, when moderate in amount, is 
 useful in softening and liquefying unguents; but when excessive 
 it is prejudicial by decomposing the unguents, and sometimes even 
 by softening the metal of the bearings, and raising their tempera- 
 ture so high as to set fire to neighbouring combustible matters. 
 
 Excessive heating is prevented by a constant and copious supply 
 of a good unguent. When the velocity of rubbing is about four 
 or five feet per second, the elevation of temperature is found to 
 be, with good fatty and soapy unguents, 40° to 50° Fahr., with 
 good mineral unguents, about 30°. The efiect of friction upon the 
 efiiciency of machines will be considered at the end of this Part. 
 
 403. Work of Acceleration. — In order that the velocity of a body's 
 motion may be changed, it must be acted upon by some other body 
 with a force in the direction of the change of velocity, which force 
 is proportional directly to the change of velocity, and to the mass 
 of the body acted upon, and inversely to the time occupied in pro- 
 ducing the change. If the change is an acceleration or increase of 
 velocity, let the first body be called the driven body, and the second 
 
WOKK OF ACCELERATION. 253 
 
 the driving body. Then the force must act upon the driven body 
 in the direction of its motion. Every force being a pair of equal 
 and opposite actions between a pair of bodies, the same force which 
 accelerates the driven body is a resistance as respects the driving 
 body. 
 
 For example, during the commencement of the stroke of the 
 piston of a steam engine, the velocity of the piston and of its rod is 
 accelerated J and that acceleration is produced by a certain part of 
 the pressure between the steam and the piston, being the excess of 
 that pressure above the whole resistance which the piston has to 
 overcome. The piston and its rod constitute the driven body; the 
 steam is the driving body ; and the same part of the pressure which 
 accelerates the piston, afcts as a resistance to the motion of the 
 steam, in addition to the resistance which would have to be over- 
 come if the velocity of the piston were uniform. 
 
 The resistance due to acceleration is computed in the following 
 manner : — ^It is known by experiment, that if a body near the 
 earth's surface is accelerated by the attraction of the earth, — that 
 ia, by its own weight, or by a force equal to its own weight, its 
 velocity goes on continually increasing very nearly at the rate of 
 32-2 feet per second of additional veloaity, for each second during 
 which the force acts. This quantity varies in different latitudes, 
 and at different elevations, but the value just given is near enough 
 to the truth for purposes of mechanical engineering. For brevity's 
 sake, it is usually denoted by the symbol g; so that, if at a given 
 instant the velocity of a body is % feet per second, and if its own 
 weight, or an equal force, acts freely on it in the direction of its 
 motion for t seconds, its velocity at the end of that time will have 
 increased to 
 
 Vi = v^ + gt (1.) 
 
 If the acceleration be at any different rate per second, tlm force 
 necessary to produce that acceleration, being the resistance on the 
 driving body dice to the acceleration of the driven body, bears the 
 same proportion to the driven body's weight which the actual rate of 
 acceleration bears to the rate of acceleration produced by gravity 
 acting freely. (In metres per second, gr = 9-81 nearly.) 
 
 To express this by symbols, let the weight of the driven body be 
 denoted by W. Let its velocity at a given instant be Vi feet per 
 second; and let that velocity increase at an uniform rate, so that 
 at an instant t seconds later, it is v^ feet per second. 
 
 Let /denote the rate of acceleration ; then 
 
 f=^; (2.) 
 
254 THEORY OF MACHINES. 
 
 and the force E necessary to produce it will be given by the pro- 
 portion, 
 
 g:f::W:B.; 
 
 that is to say, 
 
 ^^/W^W^v-0 ,, (3, 
 
 W 
 
 The factor — , in the above expression, is called the mass of the 
 
 driven body; and being the same for the same body, in what place 
 soever it may be, is held to represent the quantity of matter in the 
 body. (See Article 195, page 117,) 
 
 The product of the mass of a body into its velocity at any 
 
 instant, is called its momentum ; so that the resistance due to a 
 given acceleration is equal to the increase of momerdum divided hy 
 the time which that increase occupies. 
 
 If the product of a force by which a body is accelerated, equal 
 and opposite to the resistance due to acceleration, into the time 
 during which it acts, be called impulse, the same principle may be 
 otherwise stated by saying, that the increase of momentum is equal 
 to tliie impulse by which it is caused. 
 
 If the rate of acceleration is not constant, but variable, the force 
 E varies along with it. In this case, the value, at a given instant 
 
 of the rate of acceleration, is represented by /= j^, and the cr- 
 
 responding value of the force is 
 
 ^J_W^Wd. 
 
 g g dt ^ ' 
 
 The WOKK performed in accelerating a body is the product of 
 the resistance due to the rate of acceleration into the distance 
 moved through by the driven body while the acceleration is going 
 on. The resistance is eqiial to the mass of the body, multiplied by 
 the increase of velocity, and divided by the time which that 
 increase occupies. The distance moved through is the product of 
 the mean velocity into the same time. Therefore, the work per- 
 formed is equal to the mass of the body multiplied by the increase 
 of the velocity, and by the mean velocity; that is, to the mass of 
 the body, muMplied by the increase of the half-square of its vdocity. 
 
 To express this by symbols, in the case of an uniform rate of 
 acceleration, let s denote the distance moved through by the driven 
 body during the acceleration ; then 
 
 -'^'i (5.) 
 
WORK OF ACCELERATION. 255 
 
 •wtich being multiplied by Equation 3, gives for the work of accele- 
 ration, 
 
 g t 2 9 2 ^ '' 
 
 In the case of a variable rate of acceleration, let v denote the 
 mean velocity, and ds the distance moved through, in an interval 
 of time dt so short that the increase of velocity dv ia indefinitely 
 small compared with the mean velocity. Then 
 
 ds=^vdt; (7.) 
 
 which being multiplied by Equation 4, gives for the work of accele- 
 ration during the intervsrt d t, 
 
 „ , W dv ,, 
 &as = — • -=— • V dt 
 9 dt 
 
 W 
 
 =— • vdv; (8.) 
 
 9 ^ ' 
 
 and the integration of this expression (see Article 29) gives for 
 the work of acceleration during a finite interval, 
 
 jRds = ~Jvdv=--^^ (9.) 
 
 being the same with the result already arrived at in Equation 6. 
 
 From Equation 9 it appears that the work performed in producing 
 a given acceleration depends on the initial and final velocities, Vj and 
 Vj, and not on the intermediate changes of velocity. 
 
 If a body falls freely under the action of gravity from a state of 
 rest through a height h, so that its initial velocity is 0, and its final 
 velocity v, the work of acceleration performed by the earth on the 
 body is simply the product W A of the weight of the body into the 
 height of fall. Comparing this with Equation 6, we find — 
 
 ^-1^ (10-) 
 
 This quantity is called the height, or fall, due to the velocity v ; 
 and from Equations 6 and 9 it appears that the work performed in 
 producing a given acceleration is the same with that performed in 
 lifting the driven body through the ^fference of the heights due to Us 
 initial <md final velocities. 
 
 If work of acceleration is performed by a prime mover upon 
 bodies which neither form part of the prime mover itself, nor of the 
 machines which it is intended to drive, that work is lostj as when 
 a marine engine performs work of acceleration on the water that is 
 struck by the propeller. 
 
256 THEORY OP MACHINES. 
 
 Work of acceleration performed on the moving pieces of the 
 prime mover itself, or of the machinery driven by it, is not neces- 
 sarily lost, as will afterwards appear. (Article 413.) 
 
 404. Summation of Work of Acceleration. — If several pieces of 
 a machine have their velocities increased at the same time, the 
 work performed in accelerating them is the sum of the several 
 quantities of work due to the acceleration of the respective pieces; 
 a result expressed in symbols by 
 
 ^V^'} (4 
 
 The process of finding that sum is facilitated and abridged in 
 certain cases by special methods. 
 
 I. Accelerated Rotation. — Let a denote the angular velocity of a 
 solid body rotating about a fixed axis; — that is, as explained in 
 Article 87, the velocity of a point in the body whose radius- 
 vector, or distance from the axis, is unity. 
 
 Then the velocity of a particle whose distance from the axis 
 is r is 
 
 D = a r; (2.) 
 
 and if in a given interval of time the angular velocity is accelerated 
 from the value a^, to the value a^, the increase of the velocity of the 
 particle in question is 
 
 ''2-^i = »'(«2-«i) (3-) 
 
 Let w denote the weight, and — the mass of the particle in ques- 
 tion. Then the work performed in accelerating it, being equal to 
 the product of its mass into the increase of the half-square of its 
 velocity, is also equal to the product of its mass into the square of its 
 radius-vector, and into the increase of the half-square of the angvla/r 
 velocity; that is to say, in symbols, 
 
 w v\-v\ wr^ al- a\ .. , 
 
 g'-i^-f'^ir ^*-^ 
 
 To find the work of acceleration for the whole body, it is to be con- 
 ceived to be divided into small particles, whose velocities at any 
 given instant, and also their accelerations, are proportional to their 
 distances from the axis; then the work of acceleration is to be found 
 for each particle, and the results added together. But in the sum 
 so obtained, the increase of the half-square of the angular velocity 
 is a common factor, having the same value for each particle of the 
 body; and the rate of acceleration produced by gravity, g = 32'2 is 
 a common divisor. It is therefore sufficient to add together the 
 
EEDUCED INERTIA. 
 
 257 
 
 products of the weight of each particle (w) into the squa/re of its 
 radius-vector (r^), and to multiply the sum so obtained (2 • w r^) by 
 
 the increase of the half-square of the angular velocity I n(^l " ^i) )> 
 
 and divide hy the rate of acceleration due to graviti) (g). The 
 result, viz. : — 
 
 J«,.<-j^1 «|-^.2^,2 ^5 
 
 (g 2 j 2g 
 
 is the •work of acceleration sought. In fact, the sum S w r^ is the 
 weight of a body, which, if concentrated at the distance unity from 
 Uie adds of rotation, would require the same work to produce a given 
 increase of angular velo(j^ty which the actual body requires. 
 
 405. Reduced Inertia. — If in a certain machine, a moving piece 
 whose weight is W has a velocity always bearing the ratio n : 1 to 
 the velocity of the driving point, it is evident that when the driving 
 point undergoes a given acceleration, the work pei'formed in pro- 
 ducing the corresponding acceleration in the piece in question is 
 the same with that which would have been required if a weight 
 n^ W had been concentratfti at the driving point, the work per- 
 formed in producing the acceleration depending on the square of 
 the velocity. 
 
 If a similar calculation be performed for each moving piece in the 
 machine, and the results added together, the sum 
 
 S-n^W (1.) 
 
 gives the weight which, being concentrated at the driving point, 
 would require the same work for a given acceleration of the driving 
 point that the actual machine requires; so that if v^ is the initial, 
 and ^2 the final velocity of the driving point, the work of accelera- 
 tion of the whole machine is 
 
 '^ti^-'Z-n^W (2.) 
 
 This operation may be called the reduction of the' inertia to the 
 driving point. Mr. Moseley, by whom it was first introduced into 
 the theory of machines, calls the expression (1.) the "coefficient of 
 steadiness." 
 
 In finding the reduced inertia of a machine, the mass of each 
 rotating piece is to be treated as if concentrated at a distance from 
 its axis equal to its radius of gyration {j so that if v represents the 
 velocity of the driving point at any instant, and a the corresponding 
 angular velocity of the rotating piece in question, we are to make 
 
 «^ = ¥ (^•) 
 
 in performing the calculation expressed by the formula (1.) 
 
 s 
 
258 THEOKT OP MACHINES. 
 
 406. Summary of Various Kinds of Work. — In order to present 
 at one view the symbolical expression of the various modes of per- 
 forming work described in the preceding articles, let it be supposed 
 that in a certain interval of time d t the driving point of a machine 
 moves through the distance ds; that during the same time its 
 centre of gravity is elevated through the height dh; that resist- 
 ances, any one of which is represented by E., are overcome at 
 points, the respective ratios of whose velocities to that of the 
 driving point arfe denoted by n ; that the weight of any piece of 
 the mechanism is W, and that n' denotes the ratio of its velocity 
 (or if it rotates, the ratio of the velooity'of the end of its radius of 
 gyration) to the velocity of the driving point ; and that the driving 
 
 ds 
 point, whose mean velocity is « = -j~, undergoes the acceleration 
 
 Clr t 
 
 d V. Then the whole work performed during the interval in ques- 
 tion is • 
 
 The vnean, total resistance, reduced to the driving point, may be 
 computed by dividing the above expression by the motion of the 
 driving point d8 = vdt, giving the following result : — 
 
 J^-SW + S«E + 4^-S«'2W (2.) 
 
 d s gdt ^ ' 
 
 Section 2. — Or Eneegt, Power, and Efficiency. 
 
 407. Condition of Uniform Speed. — According to the first law 
 of motion, in order that a body may move uniformly, the foi'ces 
 applied to it, if any, must balance each other; and the same 
 principle holds for a machine consisting of any number of bodies. 
 
 When the direction of a body's motion varies, but not the velocity, 
 the lateral force required to produce the change of direction depends 
 on the principles set forth in Article 335; but the condition of balance 
 still holds for the forces which act along the direction of the body's 
 motion, that is, for the efforts and resistances ; so that, whether for 
 a single body or for a machine, the condition of uniform velocity is, 
 that the efforts shall balance the resistances. 
 
 In a machine, this condition must be fulfilled for each of the 
 single moving pieces of which it consists. 
 
 It also follows, from the principles of statics, that in any 
 body, system, or machine, that condition is fulfilled when the sum, 
 of the products of the efforts into the velocities of their respective 
 points of action is equal to the sum, of the produ^cts of the resistances 
 into the velocities of tlie points where they a/re overcome. 
 
ENERGY — POTENTIAL ENERGT. 259 
 
 Thus, let v be the velocity of a driving pomt, or point where an 
 effort P is applied; v' the velocity of a working point, or point where 
 a resistance E. is overcome ; the condition of uniform, velocity for 
 any body, system, or machine is 
 
 2 • Pi; = 2 • R< (1.) 
 
 If there be only one driving point, or if the velocities of all the 
 driving points be alike, then. P being the total effort, the single 
 product P V may be put in in place of the sum 2 • P « ; reducing 
 the above equation to 
 
 P« = 2-E,«' (2.) 
 
 Referring now to Article 398, let the machine be one in which 
 the comipa/rative or proportionate velocities of all the points at a 
 given instant are known independently of their absolute velocities, 
 from the construction of the machine ; so that, for example, the 
 velocity of the point where the resistance E is overcome bears to 
 that of the driving point the ratio 
 
 v 
 -=n; 
 
 V 
 
 then the condition of uniform speed may be thus expressed : — 
 
 P = 2»iE; (3.) 
 
 that is, the total effort is equal to the sum of the resistances reduced to 
 the driving point. 
 
 408 Energy — Potential Energy. — Energy means capacity for 
 performing work, and is expressed, like work, by the product of a 
 force into a space. 
 
 The energy of an effort, sometimes called "potential energy '' (to 
 distinguish it from another form of enetgy to be referred to in Article 
 414), is the product of the effort into the distance through which it is 
 capable of acting. Thus, if a weight of 100 pounds be placed at an 
 elevation of 20 feet above the ground, or above the lowest plane 
 to which the circumstances of the case admit of its descending, 
 that weight is said to possess potential energy to the amount of 
 \QQ x. 2{i = 2i,QQQ foot-pounds ; which means, that in descending 
 from its actual elevation to the lowest point of its course, the 
 weight is capable of performing worh to that amount. 
 
 To take another example, let there be a reservoir containing 
 10,000,000 gallons of water, in such a position that the centre of 
 gravity of the mass of water in the reservoir is 100 feet above the 
 lowest point to which it can be made to descend while overcoming 
 resistance. Then as a gallon of water weighs 10 lbs., the weight of 
 the store of water is 100,000,000 lbs., which being multiplied' by 
 the height through v/hich that weight is capable of acting, 100 feet, 
 gives 10,000,000,000 foot-pounds for the potential energy of the 
 weight of the store of water. 
 
260 
 
 THEORY OF MACHINES. 
 
 409. Equality of Energy Exerted and Work Performed, or the 
 Conservation of Energy. — When an effort actually does drive 
 its point of application through a certain distance, energy to the 
 amount of the product of the effort into that distance is said to 
 be exerted ; and the potential energy, or energy which remains 
 capable of being exerted, is to that amount diminished. 
 
 When the energy is exerted in driving a machine at an uniform 
 speed, it is equal to the work performed. 
 
 To express this algebraically, let t denote the time during -which 
 the energy is exerted, v the velocity of a driving point at which an 
 effort P is applied, s the distance through which it is driven, v' the 
 velocity of any working point at which a resistance R is overcome, 
 s' the distance through which it is driven; then 
 
 s = vtj s' = v' t ; 
 
 and multiplying Equation 1 of Article 407 by the time t, we obtain 
 the following equation : — 
 
 2'Pv« = 2-E«'i! = 2-P« = 2-E«'j (1.) 
 
 which expresses the equality of energy exerted, and work per- 
 forined, for constant efforts and resistances. 
 
 When the efforts and resistances vary, it is sufficient to refer to 
 Articles 400 and 29, to shew that the same principle is expressed 
 as follows : — 
 
 ifVds = 2 JRds'; (2.) 
 
 where the symbol f expresses the operation of finding the work 
 
 performed against a varying resistance, or the energy exerted by a 
 varying effort, as the case may be; and the symbol 2 expresses the 
 operation of adding together the quantities of energy exerted, or work 
 performed, as the case may be, at different points of the machine. 
 
 410. Various Factors of Energy. — A quantity of energy, like a 
 quantity of work, may be computed by multiplying either a force 
 into a distance, or a statical moment into an angular motion^ or 
 the intensity of a pressure into a volume. These processes have 
 already been explained in detail in Articles 394 and 395, pages 
 244 to 246. 
 
 411. The Energy Exerted in Producing Acceleration is equal to 
 the work of acceleration, whose amount has been investigated in 
 Articles 403 and 404, pages 252 to 257. 
 
 412. The Accelerating Effort by which a given increase of 
 velocity in a given mass is produced, and which is exerted by the 
 driving body against the driven body, is equal and opposite to the 
 resistance due to acceleration which the driven body exerts against 
 the driving body, and whose amount has been given in Articles 
 
THE ACCKLERATING EFFORT. 261 
 
 403 and 404. Referring, therefore, to Equations 4 and 8 of Article 
 403, we find the two following expressions, the first of whiqh gives 
 the accelerating effort required to produce a given acceleration d v 
 in a body whose weight is W, when the time dt in which that 
 acceleration is to be produced is given, and the second, the same 
 accelerating effort, when the distance ds = vdt in which the ac- 
 celeration is to be produced is given : — 
 
 P=^-§^, (1.) 
 
 g d t 
 
 ^W ^vdv^W _d{v') .^. 
 
 ^g ds g ids 
 
 Referring next to Article 404, page 257, we find, from Equation 5, 
 that the work of acceleration corresponding to an increase da in the 
 lingular velocity of a rotating body whose moment of inertia is I, is 
 
 I • d (a^) _1 a d a 
 
 Let d t hs the time, and di = adt the angular motion in which 
 that acceleration is to be produced ; let P be the accelerating effort, 
 and I its leverage, or the perpendicular distance of its line of action 
 from the axis ; then, according as the time dt, or the angle di,'-'is 
 given, we have the two following expressions for the accelerating 
 couple: — 
 
 Fi=^ •^,:. : ;.; (3.) 
 
 g dt ^ ' 
 
 ^I ,ada^l Jja^)- ,^ , 
 
 g di- g 2di \'^' 
 
 Lastly, referring to Article 405, page 257, Equation 2, we find, 
 
 that if a train of mechanism consists of various parts, and if W be 
 
 the weight of any one of those parts, whose velocity v' bears to that 
 
 v' 
 of the driving point v the ratio — = n, then the accelerating effort 
 
 which must be applied to the driving point, in order that, during 
 the interval d t, in which the driving point moves through the 
 distance ds = v dt, that point may undergo the acceleration d v, 
 and each weight W the corresponding acceleration ndv, is given 
 by one or other of the two formulse — 
 
 Sw^w dv ' 
 
 ^-^^-'dt (^-^ 
 
 _ l,n^W vdv _ 'Zn^W d{i^) 
 
 ~ q ' ds g 2ds ^ '' 
 
262 THEOKT OF MACHINES. 
 
 both of wliioli are derived from, the equation "P ds ="£ v • dt = 
 
 V dv „ 3 
 
 9 
 
 413. Work During Eetardation — Energy Stored and Eestored. — 
 In order to cause a givei;i retardation, or diminution of the velocity 
 of a given body, in. a given time, or whUe it traverses a given dis- 
 tance, resistance must be opposed to its motion equal to the effort 
 •which would be required to produce in the same time, or in the 
 same distance, an acceleration equal to the retardation. 
 
 A moving body, therefore, while being retarded, overcomes re- 
 sistance and performs work; and that work is equal to the energy 
 exerted in producing an acceleration of the same body eqiial to the 
 retardation. 
 
 It is for this reason that it has been stated, in Article 403, that 
 the work performed in accelerating the speed of the moving pieces 
 of a machine is not necessarily lost ; for those moving pieces, by 
 returning to their original speed, are capable of performing an 
 equal amount of work in overcoming resistance; so that the per- 
 formance of such work is not prevented, but only deferred. Hence 
 energy exerted in acceleration is said to be stored; and when by a 
 subsequent and equal retardation an equal amount of work is per- 
 formed, that energy is said to be restored. 
 
 The algebraical expressions for the relations between a retarding 
 resistance, and the retardation which it produces in a given body 
 by acting during a given time or through a given space, are ob- 
 tained from the equations of Article 412 simply by putting K, the 
 symbol for a resistance, instead of P, the symbol for an effort, and 
 — dv, the symbol for a retardation, instead of d v, the symbol for 
 an acceleration. 
 
 414. The Actual Energy of a moving body is the work which 
 it is capable of performing against a retarding resistance before 
 being brought to rest, and is equal to the energy which must be 
 exerted on the body to bring it from a state of rest to its actual 
 velocity. The value of that quantity is the product of the weight 
 of the body into the height from which it must fall to acquire its 
 actual velocity ; that is to say, 
 
 Z^ (1.) 
 
 The total actual energy of a system of bodies, each moving with 
 its own velocity, is denoted by 
 
 ~^v ^^^ 
 
 and when those bodies are the pieces of a machine, whose velocities 
 
A RECIPROCATING FORCE. 263 
 
 bear definite ratios (any one of which is denoted by n) to the velo- 
 city of the driving point v, their total actual energy is 
 
 |^-2«^W, (3.) 
 
 being the product of the reduced inertia (or coefficient of steadiness, 
 as Mr. Moseley calls it) into the height due to the velocity of the 
 driving point. 
 
 The actual energy of a rotating body whose angular velocity is a, 
 and moment of inertia 2 W r^ = I, is 
 
 that is, the product of the moment ofvnertia into the height due to the 
 velocity, a, of a point, whose distance from the axis of rotation is 
 
 When a given amount of energy is alternately stored and restored 
 by alternate increase and diminution in the speed of a machine, 
 the actual energy of the machine is alternately increased and 
 diminished by that amount. 
 
 Actual energy, like motion, is relative only. That is to say, in. 
 computing the actual energy of a body, which is the capacity it 
 possesses of performing work upon certain other bodies by reason 
 of its motion, it is the motion relatively to those other bodies that is 
 to be taken into account. 
 
 For example, if it be wished to determine how many turns a 
 wheel of a locomotive engine, rotating with a given velocity, would 
 make, before being stopped by the friction of its bearings only, sup- 
 posing it lifted out of contact with the rails, — the actual energy of 
 that wheel is to be taken relatively to the frame of the engine to 
 which those bearings are fixed, and is simply the actual energy due 
 to the rotation. But if the wheel be supposed to be detached from 
 the engiis*, and it is inquired how high it unll ascend up a perfectly 
 smooth inclined plane before being stopped by the attraction of the 
 earth, then its actual energy is to be taken relatively to the earth; 
 that is to say, to the energy of rotation already mentioned, is to be 
 added the energy due to the translation or forward motion of the 
 wheel along with its axis. 
 
 415. A Beciprocating Force is a force which acts alternately as 
 an efibrt and as an equal and opposite resistance, according to the 
 direction of motion of the body. Such a force is the weight of a 
 moving piece whose centre of gravity alternately rises and falls; 
 and such is the elasticity of a perfectly elastic body. The work 
 which a body performs in moving against a reciprocating force is 
 employed iu increasing its own potential energy, and is not lost by 
 
264 THEORY OP MACHINES. 
 
 the body; so that by the motion of a body alternately against and 
 with a reciprocating force, energy is stored and restored, as well as' 
 by alternate acceleration and retardation. 
 
 Let 2 W denote the weight of the whole of the moving pieces of 
 any machine, and h a height through which the common centre of 
 gravity of them all is alternately raised and lowered. Then the 
 quantity of energy — 
 
 is stored while the centre of gravity is rising, and restored whUe it 
 is falling. 
 
 These principles are illustrated by the action of the plungers 
 of a single-acting pumping steam engine. The weight of those 
 plungers acts as a resistance while they are being lifted by the 
 pressure of the steam on the piston; and the same weight acts as 
 effort when the plungers descend and drive before them the water 
 ■with which the pump barrels have been filled. Thus the energy 
 exerted by the steam on the piston is stored during the up-stroke 
 of the plungers; and during their down-stroke the same amount of 
 energy is restored, and employed in performing the work of raising 
 water and overcoming its friction. 
 
 416. 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 if, during any part of the period of motion, energy has been 
 stored by acceleration of the body, the same quantity of energy 
 exactly must have been during another part of the period restored 
 by retardation of the body. 
 
 If the body also returns in the course of the same period to the 
 same position relatively to all bodies which exert reciprocating 
 forces on it — for example, if it returns periodically to the same 
 elevation relatively to the earth's surface — any quantity of energy 
 which has been stored during one part of the period by moving 
 against reciprocating forces must have been exactly restored during 
 another part of the period. 
 
 Hence at the end of each period, the equality of energy and work; 
 and the balance of mean effort and mean resistance, holds with 
 respect to the driving effort amd the resistances, exactly as if the speed 
 were uniform and the reciprocating forces null; and all the equa- 
 tions of Articles 407 and 409 are applicable to periodic motion, pro- 
 vided that in the equations of Article 407, and Equation 1 of 
 Article 409, P, E, and v are held to denote the mea/n values of the 
 efforts, resistances, and velocities, — that s and s' are held to denote 
 spaces moved through in one or more efniire periods, — and that in 
 
 Equation 2 of Article 409, the . integrations denoted by ' /, be held 
 
 to extend to one or more entire periods. 
 
THE EFFICIENCY OF A MACHINE. 265 
 
 These principles are illustrated by the steam engine. The velo- 
 cities of its moving parts are continually vaiying, and those of 
 some of them, such as the piston, are periodically reversed in direc- 
 tion. But at the end of each period, called a revolution, or double- 
 stroke, every part returns to its original position and velocity; so 
 that the equality of energy cmd worh, and the equality of the mean, 
 effort to the mean resistance reduced to the driving point, — that is, 
 the equality of the mean effective pressure of the steam on the 
 piston to the mean total resistance reduced to the piston — hold for 
 one or any whole number of complete revolutions, exactly as for 
 uniform speed. 
 
 It thus appears that (as stated at the commencement of this 
 Part) there are two fundamentally different ways of considering a 
 periodically moving 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 one or more whole periods, with a view to the determination 
 of the relation between the mean resistances and mean efforts, and 
 of the 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. 
 
 " II. In the second place is to be considered the action of the 
 machine during intervals of time less than its period, in order to 
 determine the law of the periodic changes in the motions of the 
 pieces of which the machine consists, and of the periodic or recip- 
 rocating forces by which such changes are produced." 
 
 417. Starting and Stopping. — The starting of a machine consists 
 in setting it in motion from a state of rest, and bringing it up to 
 its proper mean velocity. This operation requires the exertion, 
 besides the energy required to overcome the mean resistance, 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 414, page 262. 
 
 If, in order to stop a machine, the effort of the prime mover is 
 simjfly suspended, the machine will continue to go until work has 
 been performed in overcoming resistances equal to the actual energy 
 due to the speed of the machine at the time of suspending the 
 effort of the prime mover. 
 
 In order to diminish the time required by this operation, the 
 resistance may be increased by means of the friction of a brake. 
 Brakes will be further described in the sequel. 
 
 418. The Efficiency of a machine is a fraction expressing tho 
 ratio of the useful work to the whole work, which is equal to the 
 energy expended. The Counter-efficiency is the reciprocal of 
 the efficiency, and is the ratio in which the energy expended is 
 greater than the useful work. The object of improvements in 
 
266 THEOKT OF MACHINES. 
 
 machines is to bring their efficiency and counter-efficiency as near 
 to unity as possible. 
 
 As to useful and lost work, see Article 401. The algebraical 
 expression of the efficiency of a machine having uniform or perio- 
 dical motion, is obtained by introducing the distinction between 
 useful and lost work into the equations of the conservation of energy. 
 Article 409. Thus, let P denote the mean eSoit at the driving point; 
 s, the space described by it in a given interval of time, being a 
 whole number of periods of revolutions; 'R^, the mean useful resist- 
 ance; Sj, the space through which it is overcome in the same 
 interval; Eg, any one of the wasteful resistances; «£> ^^^ space 
 through which it is overcome; then 
 
 Ps = E,Si + S • E2S2; (1.) 
 
 and the efficiency of the machine is expressed by 
 
 Ps EiSi + S-E2«2 ^ ' 
 
 In many cases the lost work of a machine, Eg Sg, consists of a con- 
 stant part, and of a part bearing to the useful work a proportion 
 depending in some definite manner on the sizes, figures, arrange- 
 ment, and connection of the pieces of the train, on which also 
 depends the constant part of the lost work. In such cases the 
 whole energy expended and the efficiency of the machine are 
 expressed by the equations 
 
 Ps = (l-i-A)EiSi-hB; 1 
 Ej_Sj__ 
 
 .(3.) 
 
 Ps , . B 
 l+A+= — 
 
 and the first of these is the mathematical expression of what Mr. 
 Moseley calls the " modulus" of a machine. 
 
 The iiseful 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 over- 
 coming its own friction. Hence the efficiency of such an inter- 
 mediate 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 aTnachine is the product 
 of the efficiencies of the series ofmcming pieces which t/ransmit energy 
 from the driving point to the worMng point. The same principle 
 applies to a train of successive machines, each driving that whioh 
 follows it ; and to counter-efficiency as well as to efficiency. , 
 
 419. Power and Effect — Horse Power. — The power of a machine 
 
THE PRINCIPLE OP VIRTUAL VELOCITIES. 267 
 
 is the energy exerted, and the effect, the useful -work performed, in 
 some interval of time of definite length, such as a second, a minute, 
 an hour, or a day. 
 
 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; and the power is equal to the effect 
 multiplied by the counter-efficiency. The loss of power is the dif- 
 ference between the effect and the power. As to the French 
 " Force de Cheval," see Article 392, page 244. It is equal to 
 0-9863. of a British horse-power; and a British horse-power is 
 1'0139 of a French force de cheval'. 
 
 420. General Equationt — The following general equation pre- 
 sents at one view the principles of the action of machines, whether 
 moving uniformly, periodically, or otherwise : — 
 
 J7ds = ^j-Rds'±h^W + ^-^Mz^- 
 
 where W is the weight of any moving piece of the machine; 
 
 h, when positive, the elevation, and when negative, the depres- 
 sion, which the common centre of gravity of all the moving pieces 
 undergoes in the interval of time under consideration; v^ the 
 velocity at the beginning, and v^ the velocity at the end, of the 
 interval in question, with which a given particle of the machine of 
 the weight "W is moving; 
 
 g, the acceleration which gravity causes in a second, or 32-2 feet 
 per second, or 9 '81 metres per second. 
 
 fRds', the work performed in overcoming any resistance during 
 
 the interval in question; 
 
 I P ds, the energy exerted during the interval in question. 
 
 The second and third terms of the right-hand side, when positive, 
 are energy stored; when negative, energy restored. 
 
 The principle represented by the equation is expressed in words 
 as follows : — 
 
 The energy exerted, added to the energy restored, is equal to the 
 energy stored added to the work performed. 
 
 421. The Principle of Virtual Velocities, when applied to the 
 uniform motion of a machine, is expressed by Equation 3 of Article 
 407, already given in page 259; or in words as follows : — Tfie effort 
 is equal to the sum of the resistances reduced to tlie driving point ; 
 that is, each multiplied by the ratio of the velocity of its working 
 point to the velocity of the driving point. The same principle, 
 when applied to reciprocating forces and to re-actions due to 
 varying speed, as well as to pafesive resistances, is expressed by 
 
268 THEORY OF MACHINES. 
 
 means of a modified forjn of the general equation of Article 420, 
 
 obtained in tlie following manner : — Let n denote either the ratio 
 
 borne at a given instant by the velocity of a given working point, 
 
 ■where the resistance R is overcome, to the velocity of the driving 
 
 point, or the mean value of that ratio during a given interval of 
 
 time; let n" denote the corresponding ratio for the vertical ascent 
 
 or descent (according as it is positive or negative) of a moving 
 
 piece whose weighb is W; let n' denote the corresponding ratio 
 
 for the mean velocity of a mass whose weight is W, undergoing 
 
 dv' 
 acceleration or retardation, and — =— either the rate of acceleration 
 
 at 
 
 of that mass, if the calculation relates to an instant, or the mean 
 
 value of that rate, if to a finite interval of time. Then the efibrt 
 
 at the instant, or the mean efibrt during the given interval, as the 
 
 case may be, is given by the following equation : — 
 
 gdt 
 
 If the ratio n', which the velocity of the mass W bears to that of 
 the driving point, is constant, we may put -7- = , , where -=-- 
 denotes the rate of acceleration of the driving point ; and then the 
 third term of the foregoing expression becomes —j-. 2 • n'^ W, as 
 
 in formula 2 of Article 406, page 258. 
 
 422. Forces in the Mechanical Powers, Neglecting Friction- 
 Purchase. — The mechanical powers, considered as means of modi- 
 fying motion only, have been considered in Section 6, Part II., 
 pages 107 to 110. "When friction is neglected, any one of the 
 mechanical powers may be regarded as an uniformly-moving simple 
 machine, in which one effort balances one resistance; and in which, 
 consequently, according to the principle of virtual velocities, or 
 of the equality of energy exerted and work done, the effort and 
 resistance are to each other inversely as the velocities along their lines 
 of action of the points where they are applied. 
 
 In the older writings on mechanics, the effort is called the 
 power, and the resistance the- weight; but it is desirable to avoid 
 the use of the word "power" in this sense, because of its being 
 very commonly used in a different sense — viz., the rate at which 
 energy is exerted by a prime mover ; and the substitution ot 
 " resistance " for " weight " is made in order to express the fact, 
 that the principle just stated applies to the overcoming of all sorts 
 of resistance, and not to the lifting of weights only. 
 
 The weight of the moving piece itself in a mechanical power 
 may either be wholly supported at the bearing, if the piece is 
 
FORCES IN THE MECHANICAL POWERS. 263 
 
 balanced ; or if not, it is to be regarded as divided into two 
 parallel components, one supported directly at the bearing, and 
 the other being included in the effort or in the resistance, as the 
 Case may be. 
 
 The relation between the effort and the resistance in any 
 mechanical power may be deduced from the pi'inciples of statics; 
 viz.: — In the case of the lever (including the wliecl and aode), 
 from the balance of couples of equal and opposite moments ; in the 
 case of the inclined plane (including the wedge and the sarew), 
 from the parallelogram of forces; and in the case of the pulley, 
 from the composition of parallel forces. The principle of virtual 
 velocities, however, is more convenient in calculation. 
 
 The total load in a me^ianical power is the resultant of the 
 effort, the resistance, the lateral components of the forces acting at 
 the driving and working points, and the weight directly carried at 
 the bearings; and it is equal and directly opposed to the re-action 
 of the bearings or supports of the machine. 
 
 By the purchase of a mechanical power is to be understood the 
 ratio borne by the resistance to the effort, which is equal to the 
 ratio borne by the velocity of the driving point to that of the 
 working point. This term has already been employed in connec- 
 tion with the pulley. 
 
 The following are the results of the principle of virtual velocities, 
 as applied to determine the purchase in the several mechanical 
 powers : — 
 
 I. Levee. — The effort and resistance are to each other in the 
 inverse ratio of the perpendicular distances of their lines of action 
 from the axis of rotation or fulcrum ; so that the purchase is the 
 ratio which the perpendicular distance of the effort from the axis 
 bears to the perpendicular distance of the resistance from the axis. 
 
 Under the head of the lever may be comprehended all turning 
 or rocking primary pieces in mechanism which are connected with 
 their drivers and followers by linkwork. 
 
 II. Wheel and Axle. — The purchase is the same as in the case 
 of the lever; and the perpendicular distances of the lines of action 
 of the effort and of the resistance from the axis are the radii of the 
 pitch-circles of the wheel and of the axle respectively. 
 
 Under the head of the wheel and axle may be comprehended 
 all turning or rocking primary pieces in mechanism which are 
 connected with their drivers and followers by means of rolling 
 contact, of teeth, or of bands. By the " wheel " is to be understood 
 the pitch-cylinder of that part of the piece which is driven ; and by 
 the " axle," the pitch-cylinder of that part of the piece which drives. 
 
 III. Inclined Plane, and lY. Wedge. — Herethe purchase, or 
 ratio of the resistance to the effort, is the ratio borne by the whole 
 velocity of the sliding body (represented by B in fig. 76b, 
 
270 THEOET OP MACHINES. 
 
 and C c in fig. 76f, page 109) to that component of the 
 velocity (represented by B D in fig. 76e, and C e in fig. 76f, 
 page 109) which is directly opposed to the resistance : it being 
 understood that the efibrt is exerted in the direction of motion 
 of the sliding body. 
 
 The term inclined plane may be used when the resistance to 
 the motion of a body that slides along a guiding surface consists 
 of its own weight, or of a force applied to a point in it by means 
 of a lint; and the term wedge, when that resistance consists of a 
 pressure applied to a plane surface of the moving body, oblique 
 to its direction of motion. 
 
 V. ScKEW. Let the resistance (R) to the motion of a screw 
 be a force acting along its axis, and directly opposed to its advance; 
 and let the efibrt (P) which drives the screw be applied to a pouit 
 rigidly attached to the screw, and at the distance r from the axis, 
 and be exerted in the direction of motion of that point. Then, 
 while the screw makes one revolution, the working point advances 
 against the resistance through a distance equal to the pitch (p); 
 and at the same time the driving point moves in its helical path 
 through the distance J (4 ifl'fi-vp^'); therefore the purchase of 
 the screw, neglecting friction, is expressed as follows : — 
 
 B _ 7 4 ir2 r2 + j?8 
 
 P- p 
 
 _ length of one coil of path of driving point 
 ~ pitch 
 
 VI. Pullet. — In the pulley without friction, the purchase is 
 the ratio borne by the resistance which opposes the advance of 
 the running block to the efibrt exerted on the hauling part of 
 the rope ; and it is expressed by the number of plies of rope by 
 which the running block is connected with the fixed block. 
 
 VII. The Hydraulic Peess, when friction is neglected, may 
 be included amongst the mechanical powers, agreeably to the 
 definition of them given at the beginning of this Article. By the 
 resistance is to be understood the force which opposes the outward 
 motion of the press-plunger; and by the efibrt, the force which 
 drives inward the pump-plunger. The intensity of the pressure 
 exerted between each of the two plungers and the fiuid is the 
 same; therefore the amount of the pressure exerted- between 
 each plunger and the fluid is proportional to the area of that 
 plunger; so that the purchase of the hydraulic press is expressed 
 as follows : — 
 
 B. _ transverse area of press-plunger _ 
 P ~ transverse area of pump-plunger' 
 
STEAM ENGINE INDICATOR. 271 
 
 and this is the reciprocal of the ratio of the velocities of those 
 plungers, as already shewn in Article 185, page 110. 
 
 The purchase of a train of mechanical powers is the product of 
 the purchases of the several elementary parts of that train. 
 
 The object of producing a purchase expressed by a number 
 greater than unity is, to enable a resistance to be overcome by 
 means of an effort smaller than itself, but acting through a greater 
 distance; and the use of such a purchase is found chiefly in 
 machines driven by muscular power, because of the efibrt being 
 limited in amount. 
 
 Section 3. — Of Dynamometers. 
 
 423. Dynamometers are instruments for measuring and record- 
 ing the energy exerted and work performed by machines. They 
 may be classed as follows : — 
 
 I. Instruments which merely indicate the force exerted between 
 a driving body and a driven body, leaving the dista/nce through 
 which that force is exerted to be observed independently. 
 
 II. Instruments which record at once the force, motion, and 
 work of a machine, by drawing a line, straight or curved, as the 
 case may be, whose abscissse represent the distances moved through, 
 its ordinates the resistances overcome, and its area the work per- 
 formed (as in fig. 149, page 249). 
 
 A dynamometer of this class consists essentially of two principle 
 parts : a spring whose deflection indicates the force exerted between 
 a driving body and a driven body ; and a band of paper, or a card, 
 moving at right angles to the direction of deflection of the spring 
 with a velocity bearing a known constant proportion to the velo- 
 city with which the resistance is overcome. JOie spring carries a 
 pen or pencil, which marks on the paper or card the required 
 line. The Steam Engine Indicator is an example of this class of 
 instruments. 
 
 III. Instruments called Integrating Dynamometers, which re- 
 cord the work performed, but not the resistance and motion 
 separately. 
 
 434. SteCim Engine Indicator. — This instrument was invented 
 by Watt, and has been improved by other inventors, especially 
 M'Naught and Richards. Its object is to record, by means of a 
 diagram, the intensity of the pressure exerted by steam against one 
 of the faces of a piston at each point of the piston's motion, and so 
 to aflTord the means of computing, according to the principles of 
 Articles 395 and 400, first, the energy exerted by the steam in 
 driving the piston during the forward stroke; secondly, the work 
 lost by the piston in expelling the steam from the cylinder during 
 the return stroke; and thirdly, the difierence of those quantities, 
 
272 
 
 THEORY OF MACHINES. 
 
 ■which is the available or effective energy exei-ted by the steam 'on 
 the piston, and which, being multiplied by the number of strokes 
 per minute and divided by 33,000 foot-pounds, gives the indicated 
 
 HORSE-POWER. 
 
 The indicator in a common form is represented by fig. 150. A B 
 is a cylindrical case. Its lower end, A, contains a smaller cylinder, 
 fitted with a piston, which cylinder, by means of the screwed 
 nozzle at its lower end, can be fixed in any convenient position 
 on a tube communicating with that end of the engine-cylinder 
 where the work of the steam is determined. The communication 
 between the engine-cylinder and the indicator-cylinder can be 
 opened and shut at will by means of the cock K. 
 When it is open, the intensity of the pressure of 
 the steam on the engine-piston and on the indi- 
 cator-piston is the same, or nearly the same. 
 
 The upper end, B, of the cylindrical case con- 
 tains a spiral spring, one end of which is at- 
 tached to the piston, or to its rod, and the other 
 to the top of the easing. The indicator-piston 
 is pressed from below by the steam, and from 
 above by the atmosphere. When the pressure 
 of the steam is equal to that of the atmosphere, 
 the spring retains its unstrained length, and the 
 piston its original position. When the pressure 
 of the steam exceeds that of the atmosphere, 
 the piston is driven outwards, and the spring 
 compressed ; when the pressure of the steam is 
 less than that of the atmosphere, the piston is 
 driven inwards, and the spring extended. The 
 compression or extension of the spring indicates 
 the difference, upward or downward, between the pressure of the 
 steam and that of the atmosphere. 
 
 A short arm, C, projecting from the indicator piston-rod carries at 
 one side a pointer, D, which shews the pressure on a scale whose 
 zero denotes the pressure of the atmosphere, and which is graduated 
 into pounds on the square inch both upwards and downwards 
 from that zero. At the other side the short arm has a longer arm 
 jointed to it, carrying a pencil, E. 
 
 r is a brass drum, which rotates backward and forward about a 
 vertical axis, and which, when about to be used, is covered with a 
 piece of paper called a " card." It is alternately puUed round in 
 one direction by the cord H, which wraps on the pulley G, and 
 pulled back to its original position by a spring contained within 
 itself. The cord H is to be connected with the mechanism of the 
 steam engine in any convenient manner which shall ensure that 
 the velocity of rotation of the drum shall at every instant bear a 
 
 Fig. 150. 
 
STEAM ENGINE INDICATOR. 
 
 273 
 
 1 
 
 H 
 
 
 
 B _-£. 
 
 ,±:^— ' 
 
 B 
 
 constant ratio to that of the steam engine piston : the back and 
 forward motion of the surface of the drum representing that of the 
 steam engine piston on a reduced scale. This having been done, 
 and before opening the cock K, the pencil is to be placed in con- 
 tact with the drum during a few strokes, when it will mark on the 
 card a line which, when the card is afterwards spread out flat, 
 becomes a straight line. This line, whose position indicates the 
 pressure of the atmosphere, is called the atmospheric line. In fig. 
 151 it is represented by A A. 
 
 The cock K is opened, and the pencil, moving up and down 
 with the variations of the pressure 
 of the steam, traces on the card 
 during each complete or double 
 stroke a curve such as B^ DEB. 
 The ordinates drawn to that curve 
 from any point in the atmospheric 
 line, such as U K and H G, indi- 
 cate the difierences between the 
 pressure of the steam and the at- 
 mospheric pressure at the corre- pj- J51 
 spending point of the motion of the 
 
 piston. The ordinates of the part BODE represent the pres- 
 sures of the steam during the forward stroke, when it is driving 
 the piston; those of the part EB represent the pressures of the 
 steam when the piston is expelling it from the cylinder. 
 
 To found exact investigations on the indicator-diagrams of steam 
 engines, the atmospheric pressure at the time of the experiment 
 ought to be ascertained by means of a barometer; but this is 
 generally omitted; in which case the atmospheric pressure may be 
 assumed at its mean value, being 14-7 lbs. on the square inch, or 
 2116-3 lbs. on the square foot, at and near the level of the sea. 
 
 Let AO = HF be ordinates representing the pressure of the 
 atmosphere. Then F V parallel to A A is the absolute or true 
 zero line of the diagram, corresponding to no pressure; and ordi- 
 nates drawn to the curve from that line represent the absolute 
 intensities of the pressure of steam. Let B and L E be ordi- 
 nates touching the ends of the diagram ; then 
 
 OX represents the volume traversed by the piston at each single 
 stroke ( = s A, where s is the length of the stroke and A the area 
 of the piston) ; 
 
 The area O B C D E L O represents the energy exerted by the 
 steam on the piston during the forward stroke; 
 
 The area B E L O represents the work lost in expelling the 
 steam during the return stroke; 
 
 The area B C D E B, being the difference of the above areas, 
 
 T 
 
274 TmsojKi: un MAvms^tLS. 
 
 represents the effective work of the steam ou the piston during the 
 complete stroke. 
 
 Those areas can be found by the rules of Article 34, page 
 17; and the common trapezoidal rule, D, page 21, is in general 
 sufficiently accurate. The number of intervals is usually ten, and 
 of ordinates eleven. 
 
 The mean forward pressure, the mean hack pressitre, and the mean 
 effective pressure; are found by dividing those three areas respec- 
 tively by the volume s A, which is represented by O L. 
 
 Those mean pressures, however, can be found by a direct process, 
 ■without first measuring the areas, viz. : — having multiplied each 
 ordinate, or breadth, of the area under consideration by the proper 
 multiplier, divide the sum of the products by the sum of the 
 multipliers, which process, when the common trapezoidal rule is 
 used, takes the following form: add together the halves of the 
 endmost ordinates, and the whole of the other ordinates, and 
 divide by the number of intervals. That is, let Sg be the first, b„ 
 the last, and b^, b^, (fee, the intermediate breadths; then let n be 
 the number of intervals, and 6„ the mean breadth; then 
 
 ^'n = ~J^-^ + h^ + h + &^); (1.) 
 
 and this represents the mean forward pressure, mean back .pressure, 
 or mean efiective pressure, as the case may be. Let p, be the 
 mean efiective pressure; then the efiective energy exerted by the 
 steam on the piston during each double stroke is the product of 
 the mean efiective pressure, the area of the piston, and the length 
 of stroke, or 
 
 AAs;..... (2.) 
 
 and if N be the number of double strokes in a minute, the indicated, 
 power in foot-pounds per minute, in a single-acting engine, is 
 
 aANs; (3.) 
 
 from which the indicated horse-power is found by dividing by 33,000. 
 In a double-acting engine the steam acts alternately on either 
 side of the piston ; and to measure the power accurately, two indi- 
 cators should be used at the same time, communicating respectively 
 with the two ends of the cylinder. Thus a pair of diagrams will 
 be obtained, one representing the action of the steam on each face 
 of the piston. The mean efiective pressure is to be found as above 
 for each diagram separately, and then, if the areas of the two faces 
 of the piston are sensibly equal, the meam of those two results is to 
 be taken as the general mean effective pressure; which being multi- 
 plied by the area of the piston, the length of stroke, and twice the 
 
STEAM ENGINE INDICATOR. 275 
 
 number of double strokes or revolutions in a minute, gives the 
 indicated power per minute; that is to say, if y denotes the general 
 mean effective pressure, the indicated power per minute is 
 
 y A-aiSTsj (4.) 
 
 If the two faces of the piston are sensibly of unequal areas 
 (as in " trunk engines"), the indicated power is to be computed 
 separately for each face, and the results added together. 
 
 If there are two or more cylinders, the quantities of power 
 indicated by their respective diagrams are to be added together. 
 
 The reactions of the moving parts of the indicator, combined 
 with the elasticity of the spriDg, cause oscillations of its piston. 
 In order that the errors thus produced in the indicated pressures 
 at particular instants may be as small as possible, and may 
 neutralize each other's effects on the whole indicated power, the 
 moving masses ought to be as small as practicable, and the spring 
 as stiff as is consistent with shewing the pressures on a visible 
 scale. In Richard's indicator this is effected by the help of a train 
 of very light linkwork, which causes the pencil to shew the move- 
 ments of the spring on a magnified scale. 
 
 The friction of the moving parts of the indicator tends on 
 the whole to make the. indicated power and indicated mean 
 effective pressure less than the truth, but to what extent is un- 
 certain. 
 
 Every indicator should have the accuracy of the graduation of its 
 scale of pressures frequently tested by comparison with a standard 
 pressure gauge. 
 
 The indicator may obviously be used for measuring the energy 
 exerted by any fluid, whether liquid or gaseous, in driving a 
 piston; or the work performed by a pump, in lifting, propelling, 
 or compressing any fluid. 
 
zia 
 
 CHAPTER III. 
 OF REGULATING APPARATUS. 
 
 425. Regulating Apparatus Classed — Brake — Fly — Governor.— 
 The effect of all regulating apparatus is to control the speed of 
 machinery. A regulating instrument may. act simply by con- 
 suming energy, so as to prevent acceleration, or produce re- 
 tardation, or stop the machine if required; it is then called a 
 brake; or it. may act by storing surplus energy at one time, and 
 giving it out at another time when energy is deficient: in this 
 case it is called a, fly; or it may act by adjusting the power of the 
 prime mover to the work to be done, when it is called a ffovernor. 
 The use of a brake involves waste of power. A fly and a governor, 
 on the other hand, promote economy of power and economy of 
 strength. 
 
 Section 1. — Op Brakes. 
 
 426. Brakes Defined and Classed. — The contrivances here com- 
 prehended under the general title of Brakes are those by means of 
 which friction, whether exerted amongst solid or fluid particles, 
 is purposely opposed to the motion of a machine, in order either to 
 stop it, to retard it, or to employ superfluous energy during uniform 
 motion. The use of a brake involves waste of energy, which is in 
 itself an evil, and is not to be incurred unless it is necessary to 
 convenience or safety. 
 
 Brakes may be classed as follows : — 
 
 I. Block-brakes, in which one solid body is simply pressed against 
 another, on which it rubs.. 
 
 II. Flexible brakes, which embrace the periphery of a drum or 
 pulley. 
 
 III. Pump-brakes, in which the resistance employed is the 
 friction amongst the particles of a fluid forced through a narrow 
 
 IV. Fan-brakes, in which the resistance employed is that of a 
 fluid to a fan rotating in it. 
 
 427. Action of Brakes in General. — The work disposed of by a 
 brake in a given time is the product of the resistance which it pro- 
 duces into the distance through which that resistance is overcome 
 in a given time. 
 
BLOCK -BRAKES. 277 
 
 To Stop a macliine, the brake must employ work to the amount 
 of the whole actual energy of the machine, as already stated in 
 Article 417. To retard a machine, the brake must employ work 
 to an amount equal to the difference between the actual energies 
 of the machine at the greater and less velocities respectively. 
 
 To dispose of surplus energy, the brake must employ work equal 
 to that energy ; that is, the resistance caused by the brake must 
 balance the surplus effort to which the surplus energy is due; so 
 that if n is the ratio which the velocity of rubbing of the brake 
 bears to the velocity of the driving point, P, the surplus effort at 
 the driving point, and R the resistance of the brake, we ought to 
 have — 
 
 • Il = - (1.) 
 
 It is obviously better, when practicable, to store surplus energy, 
 or to prevent its exertion, than to dispose of it by means of a 
 brake. 
 
 When the action of a brake composed of solid material is long- 
 continued, a stream of water must be supplied to the rubbing 
 surfaces, to abstract the heat that is produced by the friction, 
 according to the law stated in Article 402, page 252. 
 
 428. Block-Brakes. — When the motion of a machine is to be 
 controlled by pressing a block of solid material against the rim of 
 a rotating drum, it is advisable, inasmuch as it is easier to renew 
 the rubbing surface of the block than that of the drum, that the 
 drum should be of the harder, and the block of the softer material 
 — ^the drum, for example, being of iron, and the block of wood. 
 The best kinds of wood for this purpose are those which have con- 
 siderable strength to resist crushing, such as elm, oak, and beech. 
 The wood forms a facing to a frame of iron, and can be renewed 
 when' worn. 
 
 When the brake is pressed against the rotating drum, the direc- 
 tion of the pressure between them is obliquely opposed to the 
 motion of the drum, so as to make an angle with the radius of the 
 drum equal to the angle of repose of the rubbing surfaces (denoted 
 by ?; see page 154). The component of that oblique pressure in 
 the direction of a tangent to the rim of the drum is the friction 
 (E) ; the component perpendicular to the rim of the drum is the 
 normal pressure (N) required in order to produce that friction, and 
 is given by the equation 
 
 N = 5; (1.) 
 
 / being the coefficient of friction, and the proper value of R being 
 determined by the principles stated in Article 427. 
 
278 THEORY OF MACHINES. 
 
 It is in general desirable that the brake should be capable of 
 effecting its purpose when pressed against the drum by means of 
 the strength of one man, pulling or pushing a handle with one 
 hand or one foot. As the required normal pressure N is usually 
 considerably greater than the force which one man can exert, a 
 lever, or screw, or a train of levers, screws, or other convenient 
 mechanism, must be interposed between the brake block and the 
 handle, so that when the block is moved towards the drum, the 
 handle shall move at least through a distance as many times greater 
 than the distance by which the block directly approaches the drum, 
 as the required normal pressure is greater than the force which 
 the man can exert. 
 
 Although a man may be able occasionally to exert with one 
 hand a force of 100 lbs., or 150 lbs., for a short time, it is desirable 
 that, in working a brake, he should not be required to exert a force 
 greater than he can keep up for a considerable time, and exert re- 
 peatedly in the course of a day, without fatigue — ^that is to say, 
 about 20 lbs. or 25 lbs. 
 
 429. The Brakes of Carriages are usually of the class just de- 
 scribed, and are applied either to the wheels themselves or to 
 drums rotating along with the wheels. Their effect is to stop or to 
 retard the rotation of the wheels, and make them slip, instead of 
 rolling on the road or railway. The resistance to the motion of a 
 carriage which is caused by its brake may be less, but cannot be 
 greater, than the friction of the stopped or retarded wheels on the 
 road or rails under the load which rests on those wheels. The 
 distance which a carriage or train of carriages wUl riin on a level 
 line during the action of the brakes before stopping, is found by 
 dividing the actual energy of the moving mass before the brakes 
 are applied, by the sum of the ordinary resistance and of the addi- 
 tional resistance caused by the brakes; in other words, that dis- 
 tance is as many times greater than the height due to the speed as 
 the weight of the moving mass is greater than the total resistance. 
 
 The skid, or slipper-drag, being placed under a wheel of a carriage, 
 causes a resistance due to the friction of the skid upon the road or 
 rail under the load that rests on the wheel. 
 
 Section 2. — Of Fly-Wheels. 
 
 430. Periodical Fluctuations of Speed in a machine are caused 
 by the alternate excess and deficiency of the energy exerted above 
 the work performed in overcoming resisting forces, which produce 
 an alternate increase and diminution of actual energy, according to 
 the law explained in Article 413, page 263. 
 
PEKIODICAI/ FLUCTUATIONS OF SPEED. 270 
 
 To determine the greatest fluctuation of speed in a machine 
 moving periodically, take A B C, in fig. 152, 
 to represent the motion of the driving point 
 
 (CIO- 
 
 during one period; let the effort P of the prime 
 
 mover at each instant be represented by the 
 
 ordinate of the curve D G E I F ; and let the 
 
 sum of the resistances, reduced to the driving 
 
 point as in Article 398, at each instant, be jig. 152. 
 
 denoted by R, and represented, by the ordinate 
 
 of the curve D H E K F, which cuts the former curve at the 
 
 ordinates A D, B E, F. Then the integral, 
 
 f (P-E)ds, 
 
 being taken for any part of the motion, gives the excess or defi- 
 ciency of energy, according as it is positive or negative. For the 
 entire period ABC, this integral is nothing. For A B, it denotes 
 an excess of energy received, represented by the area D G E H ; and 
 for B C, an equal excess of work performed, repi'esented 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 0, and A E is the 
 difference of those values. 
 
 Now let Vq be the mean velocity, v^ the greatest velocity, v^ the 
 least velocity of the driving point, and S • n^ W the reduced inertia 
 of the machine (see Article 405, page 257); then 
 
 ^'~^^^ ■Z-n^W = A^; ,....(1.) 
 
 2g 
 
 which, being divided by the mean actual energy, 
 
 gives 
 
 vl-i^ AE 
 
 Eo 
 
 .(2.) 
 
 .(3.) 
 
 and observing that v^ = {v^ + Vj)-;- 2, we find 
 
 vi - «2 _A-E_ yAE 
 
 «„ ~ 2 1Eo~vtl^-n^W' 
 
 a ratio which may be called the coefficient of fiuctuatium of speed 
 or of unsteadin/ess. 
 
 The ratio of the periodical excess and deficiency of energy A E 
 
280 TUEOKY OP MACHINES. 
 
 to the whole energy exerted in one period or revolution, \ "P d s, 
 
 has been determined by General Morin for steam engines under 
 
 various circumstances, and found to be from -r^ to - for sin^le- 
 
 10 4 " 
 
 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, and for three engines with cranks at 120° 
 one-twelfth of its value for single-cylinder engines. 
 
 The following table of the ratio, A E H- j P d s, for one revolution 
 
 of steam engines of different kinds is extracted and condensed from 
 General Morin's works : — 
 
 Non-Expansive Engines. 
 
 Length of connecting rod 
 
 Length of crank ~ 
 
 AE-^l^ds = -105 -118 -125 -132 
 
 Expansive Condensing Engines. 
 Connecting rod = crank x 5. 
 
 Fraction of Stroke at) 11 1111 
 
 which steam is cut off, J 3 4 5 6 7 8 
 
 AE-v-Jpc^s = -163 -173 -ITS' -184 -189 -191 
 
 Expansive Non-Condensing Engines. 
 
 Steam cut off at x s 7 5: 
 
 2 3 4 
 
 AE-/] 
 
 -p d s= -160 -186 -209 -232 
 
 For double-cylinder expansive engines, the value of the ratio 
 A E -=- I P d s may be taken as equal to that for single-cylinder 
 
 non-expansive engines. 
 
 For tools working at intervals, such as punching, slotting, and 
 plate-cutting machines, coining presses, &c., A E is nearly equal to 
 the whole work performed at each operation. 
 
 431. Fly-Wheels. — A fly-wheel is a wheel with a heavy rim, 
 whose great moment of inertia being comprehended in the 
 
FLY-WHEELS. 281 
 
 reduced moment of inertia of a machine, reduces the coefficient 
 of fluctuation of speed to a certain fixed amount, being about .5^ for 
 
 ordinary machinery, and -^ or ^ for machinery for fine purposes. 
 
 Let — be the intended value of the coefficient or fluctuation of 
 m 
 
 speed, and A E, aa before, the fluctuation of energy. If this is to 
 
 be provided for by the moment of inertia, I, of the fly-wheel alone, 
 
 let ap be its mean angular velocity ; then Equation 3 of Article 
 
 430 is equivalent to the following : — 
 
 m all ' • ^^-f 
 
 j^rn^AE. ^2.) 
 
 the second of which equations gives the requisite moment of inertia 
 of the fly-wheel. 
 
 The fluctuation of energy may arise either from variations in the 
 efibrt exerted by the prime mover, or from variations in the resist- 
 ance, or from both those causes combined. When but one fly- 
 wheel is used, it should be placed in as direct connexion as 
 possible with that part of the mechanism where the greatest 
 amount of the fluctuation originates; but when it originates at 
 two or more points, it is best to have a fly-wheel in connection 
 with each of those points. 
 
 For example, let there be a steam engine which drives a shaft 
 that traverses a workshop, having at intervals upon it pulleys for 
 driving various machine-tools. The steam engine should have a 
 fly-wheel of its own, as near as practicable to its crank, adapted to 
 that value of A E which is due to the fluctuations of the effort 
 applied to the crank-pin above and below the mean value of that 
 effort, and which may be computed by the aid of General Morin's 
 tables, quoted in Article 430 ; and each machine tool should also 
 have a fly-wheel, adapted to a value of A E equal to the whole 
 work performed by the tool at one operation. 
 
 As the rim of a fly-wheel is usually heavy in comparison with 
 the arms, it is often sufficiently accurate for practical purposes to 
 take the moment of inertia as simply equal to the weight of the 
 rim multiplied by the square of the mean between its outside aud 
 inside radii — a calculation which may be expressed thus : — 
 
 I = Wr2; (3.) 
 
 whence the weight of the rim is given by the formula — 
 
 ^^rng^^rng^ 
 
 if v' be the velocity of the rim of the fly-wheel. 
 
282 THEORY OF MACHINES. 
 
 In millwork the ordinary values of the product mg, the unit 
 of time being the second, lie between 1,000 and 2,000 feet, or 
 approximately between 300 and 600 metres. In pumping- 
 machinery it is sometimes only about 300 feet, or 90 mfetres. 
 
 The rim of the fly-wheel of a factory steam engine is very often 
 provided with teeth, or with a belt, in order that it may directly 
 drive the machinery of the factory. 
 
 Section 3. — Of Goveenoes. 
 
 432. 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 
 apparatus for varying the surface exposed to the wind by windmill 
 sails, the throttle- valve which adjusts the opening of the steam pipe 
 of a steam engine, the damper which controls the supply of air to 
 its furnace, and the expansion gear which regulates the volume of 
 steam admitted into the cylinder at each stroke of the piston. 
 
 In prime movers whose speed and power have to be frequently 
 and rapidly varied at will, such as locomotives and winding 
 engines for mines, the regulator is adjusted by hand. In other 
 cases the regulator is adjusted by means of a self-acting instrument 
 driven by the prime mover to be regulated, and called a Goveenoe. 
 
 The special construction of the different kinds of regulators is a 
 subject for a treatise on prime movers. In the present treatise it 
 is sufficient to state that in every governor there is a moving piece 
 ■which acts on the regulator through a suitable train of mechanism, 
 and which is itself made to move in one direction or in another 
 according as the prime mover is moving too fast or too slow. 
 
 The object of a governor, properly so called, is to preserve a 
 certain uniform speed, either exactly or approximately; and such 
 is always the case in millwork. There are other cases, as in 
 marine steam engines, where it may be considered sufficient to 
 prevent sudden variations of speed, without preserving an uniform 
 speed; and in those cases an apparatus may be used possessing 
 only in part the properties of a governor : this may be called a 
 fly-governor, to ,distingui^ it from a governor proper. 
 
 Governors proper may be .distinguished into position-governors, 
 disengagement-governors, and differential governors: a position-gov- 
 ernor being one in which the moving piece that acts on the regu- 
 lator assumes positions depending on the speed of motion, as in 
 the common steam engine governor, which consists of a pair of 
 revolving pendulums acting directly on a train of mechanism which 
 adjusts the throttle-valve : a disengaging-governor being one which, 
 when the speed deviates above or below its proper value, throws 
 
PENDULUM-GOVERNOES. 283 
 
 the regulator into gear with one or other of two trains of 
 mechanism which move it in contrary directions so as to diminish 
 or increase the speed, as the case may require, as in water-mill 
 governors; and a dififerential-governor being one which, by means 
 of an aggregate combination, moves the regulator in one direction 
 or in another with a speed proportional to the difierence between 
 the actual speed and the proper speed of the engine. 
 
 In almost all governors the action depends on the centrifugal 
 force exerted by two or more masses which revolve round an axis. 
 By another classification, different from that which has already 
 been described, governors may be distinguished into gravity- 
 governors, in which gravity is the force that opposes the centrifugal 
 force; and balanced-governor^ in which the actions of gravity on 
 the various moving parts of the governor are mutually balanced, 
 and the centrifugal force is opposed by the elasticity of a spring. 
 
 Governors may be further distinguished into those which are 
 truly isochronous — that is to say, which remain without action on 
 the regulator at one speed only ; and those which are nearly 
 isochronous — that is to say, which admit of some variation of the 
 permanent or steady speed when the resistance overcome by the 
 engine varies; and lastly, governors may be distinguished into 
 those which are specially adapted to one speed, and those which 
 can be adjusted at will to different speeds. 
 
 433. Pendulum-Governors.— ;-A pendulum-governor is the simplest 
 kind of gravity-governor. It has a vertical spindle, driven by the 
 engine to be regulated; and from that spindle there hang, at 
 opposite sides, a pair of revolving pendulums, which, by the posi- 
 tions that they assume at different speeds, act on the regulator. 
 
 The relation between the height of a simple revolving pendulum 
 and the number of turns which it makes per second has already 
 been stated in Article 336; but for the sake of convenience it 
 may here be repeated : — Let h denote the height or altitude of the 
 pendulum ( = O H in fig. 153), and T the number of turns per 
 second; then 
 
 y_ g _ -815 foot _ 9- 78 inches 0-248 mfetre , . 
 4^2X2- ^2 ~ 'i'2 ~ Y- ' ' 
 
 If the rods of the revolving pendulums are jointed, as in fig. 
 154, not to a point in the vertical axis, but to a pair of points, 
 such as C, c, in arms projecting from that axis, the height is to be 
 measured to the point O, where the lines of tension of the rods cut 
 the axis. 
 
 In most cases which occur in practice, the balk are so heavy, as 
 compared with the rods, that the height may be measured without 
 sensible error from the level of the centres of the balls to the point 
 O, where the lines of suspension cut the axis. This amounts to 
 
284 
 
 THEORY OP MACHINES. 
 
 neglecting the effects both, of the weight and of the centrifugal 
 force of the rods. 
 
 The ordinary steam engine governor invented by Watt, which 
 is represented in fig. 153, is a position-governor, and acts on the' 
 
 Fig. 154. 
 
 regulator by means of the variation of its altitude, through a train 
 of levers and linkwork. That train may be very much varied in 
 detail. In the example shewn in the figure, the lever O forms 
 one piece with the ball-rod B, and the lever O c with the ball- 
 rod O 6; so that when the speed falls too low, the balls B, b, by 
 approaching the spindle, cause the point E to rise; and when the 
 speed rises too high, the balls, by receding from the spiadle, cause 
 the point E to falL At the point E there is a collar, held in the 
 forked end of the lever E F, which communicates motion to the 
 regulator. 
 
 The ordinary pendulum-governor is not truly isochronous ; for 
 when, in order to adapt the opening of the regulator to different 
 loads, it rotates with its revolving pendulums at different angles 
 to the vertical axis, the altitude h assumes different values, corre- 
 sponding to different speeds. 
 
 As in Article 431, let the utmost extent of fluctuation of the 
 speed of the engine between its highest and lowest limits be the 
 
 fraction — of the mean speed : let h be the altitude of the governor 
 m -^ 
 
 corresponding to the mean speed; and let R be the utmost extent 
 
 of variation of the altitude between its smaller limit, when the 
 
 regulator is shut, and its greater limit, when the regulator is full 
 
 open. Then we have the following proportion : — 
 
LOADED PENDULUM-GOVEKNOK. 285 
 
 and consequently 
 
 |=i (3.) 
 
 434. Loaded Pendulum-Governor. — From the balls of the com- 
 mon governor, whose collective weight is (say) A, let there be 
 hung by a pair of links of lengths equal to the ball-rods, a load 
 B, capable of sliding up and down the spindle, and having its 
 centre of gravity in the axis of rotation. Then the centrifugal 
 force is that due to A alone; and the effect of gravity is that due 
 to A + 2 B ; for when the ball-rods shift their position, the load 
 B moves through twice the vertical distance that the balls move 
 through, and is therefore equivalent, to a double load, 2 B, acting 
 directly on the balls. Consequently the altitude for a given speed 
 is greater than that of a simple revolving pendulum, in the ratio 
 
 ■I + —r- ; a given absolMte variation of altitude in moving the 
 
 regulator produces a proportionate variation of speed smaller than 
 
 A 
 m the common governor, in the ratio -j — ^-^ ; and the governor 
 
 is said to be more sensitive than a common governor, in the ratio of 
 A : A H- 2 B. Such is the construction of Porter's governor. 
 
 The links by which the load B is hung may be attached, not 
 to the balls themselves, but to any convenient pair of points in 
 the ball-rods; the links, and the parts of the ball-rods to which 
 they are jointed, always forming a rhombus, or equilateral par- 
 allelogram. Let q be the ratio borne by each of the sides of that 
 rhombus to the length on the ball-rods from the centre of a ball 
 to the point where the line of suspension cuts the axis ; then in 
 the preceding expressions 2 g B is to be substituted for 2 B. 
 
 In the one cabe 2 B, and in the other 2 g B, is the weight, 
 applied directly at A, which would be statically equivalent to the 
 load B, applied where it is. 
 
38G 
 
 CHAPTER IV. 
 
 OF THE EFFICIENCY AND COUNTER-EFFICIENCY OF PIECES, 
 COMBINATIONS, AND TRAINS IN MECHANISM. 
 
 435. Nature and Division of the Subject. — The terms Effieiency 
 and Counter-efficiency have already been explained in Article 418, 
 page 265; and the laws of friction, the most important of the 
 wasteful resistances which cause the efficiency of a machine to 
 be less than unity, have been stated in Articles 261 and 402, pages 
 153 and 251. In the present chapter are to be set forth the effects 
 of wasteful resistance, and especially of friction, on the efiSciency 
 and counter-efficiency of single pieces, and of combinations and trains 
 of pieces, in mechanism. In practical calculations the counter- 
 efficiency is in general the quantity best adapted for use; because 
 the useful work to be done in an unit of time, or effective power, is 
 in general given ; and from th^t quantity, by multiplying it by the 
 counter-efficiency, of the machine — that is, by the continued product 
 of the counter-efficiencies of all the successive pieces and combina- 
 tions by means of which motion is communicated from the driving- 
 point to the useful working-point — is to be deduced the value of the 
 expenditure of energy in an unit of time, or total power, required 
 to drive the machine. In symbols, let TJ be the useful work to be 
 done per second; c, c', c", he, the counter-efficiencies of the several 
 parts of the train ; T, the total energy to be expended per second ; 
 then 
 
 T = c-c'-c"-<fec....U (1.) 
 
 When the mean effijrt required at the driving-point can con- 
 veniently be computed by reducing each resistance to the driving- 
 point, and adding together the reduced resistances (as in Ajticle 
 407, page 253, and Article 421, page 267), the ratio in which the 
 actual effort required at the driving-point is greater than what the 
 required effort would be, in the absence of wasteful resistance, is 
 expressed by the continued product of the counter-efficiencies of 
 the parts of the train, as follows : let Pq be the effort required, in 
 the absence of wasteful resistance; P, the actual effort required; 
 then 
 
 P = c-c'-c"-&c....Po; (2.) 
 
 and in determining the efficiency or the counter-efficiency of a 
 single piece, the most convenient method of proceeding often con- 
 
EFFICIENCY OP PRIMARY PIECES IN GENERAL. 287 
 
 sists in comparing together the efforts required to drive that piece, 
 with and without friction, and thus finding the ratios 
 
 p'' = efficiency; p- = counter-efficiency. (3.) 
 
 In the ensuing sections of this chapter, the efficiency of single 
 primary pieces is first treated of, and then that of the various 
 modes of connexion employed in elementary combinations. 
 
 Section 1. — Efficiency and Countee-efpiciency op Primaey 
 
 Pieces. 
 
 436. Efficiency of Primary Pieces in General. — A primary piece 
 in mechanism, moving with an uniform velocity, is balanced under 
 the action of four forces, viz. : — 
 
 I. The re-action of the piece which it drives : this may be called 
 the Useful Besistance, and denoted by E,; 
 
 II. The weight of the piece itself: this may be denoted by W; 
 
 III. The effort by which the piece is driven : this may be 
 denoted by P; and its values with and without friction by P„ and 
 Pj respectively; 
 
 IV. The resultant pressure at the bearings, or bearing-pressure, 
 which may be denoted by Q; and which of course is equal and 
 directly opposed to the resultant of the first three forces. 
 
 In the absence of friction, the bearing-pressure would be normal 
 to the bearing surface. The effect of friction is, that the line of 
 action of the bearing-pressure becomes oblique to the bearing- 
 surface, making with the normal to that surface the angle of 
 repose ( j), whose tangent (/= tan f)' is the coefficient of friction 
 (see Article 261, page 154); and the amount of the friction is 
 expressed by Q sin ?, or very nearly by / Q, when the coefficient 
 of friction is small. 
 
 In the class of problems to which this chapter relates, the first 
 two forces — that is, the useful resistance R, and the weight W — 
 are given in magnitude, position, and direction ; and in most cases 
 it is convenient to find their resultant, in magnitude, position, and 
 direction, by the rules of statics: that is to say, if the line of 
 action of E is vertical, by Article 226, page 128; and if inclined, 
 by the rules given or referred to in Article 209, page 122. In 
 what follows, the resultant of the useful resistance and weight 
 will be called the given force, and denoted by R'. 
 
 The third force — that is, the effort required in order to drive 
 the piece at an uniform speed — is given in position and direction ; 
 for its line of action is the line of connection of the piece under 
 consideration with the piece that drives it. The magnitude of the 
 effort is one of the quantities to be found. 
 
 The fourth force — that is, the bearing-pressure — has to be found 
 
288 
 
 THEOEr OF MACHINES. 
 
 in position, direction, and magnitude. The general principles 
 according to -which it is determined are the following : — 
 
 First, That if the lines of action of the given force and the effort 
 are parallel to each other, the line of action of the resultant 
 bearing-pressure must be parallel to them both; and that if they 
 are inclined to each other, the line of action of the resultant 
 bearing-pressure must traverse their point of intersection. 
 
 Secondly, That at the centre of pressure, where the line of action 
 of the resultant bearing-pressure cuts the bearing surfaces, it makes 
 an angle with, the common normal of those surfaces equal to their 
 angle of repose, and in such a direction that its tangential 
 component (being the friction) is directly opposed to the relative 
 sliding motion of that pair of surfaces over each other. 
 
 Thirdly, That the given force, the effort, and the bearing- 
 pressure, form a system of three forces that balance each other; 
 and are therefore proportional to the three sides of a triangle 
 parallel respectively to their directions. 
 
 437. Efficiency of a Straight-sliding Piece. — In fig. 155, let A A 
 be a straight guiding-surface, upon which there slides, in the direc- 
 
 Fig. 155. 
 
 tion marked by the feathered arrow, the moving piece B. Let 
 C D represent the given force, being the resultant of the useful 
 resistance and of the weight of the piece B. (The figure shews 
 the motion of B as horizontal ; but it may be in any direction.) 
 Let C J be the line of action of the effort by which the piece B is 
 driven. 
 
 Praw ON perpendicular to A A; and C F making the angle 
 NC r = the angle of repose. Through D, parallel to C J, draw 
 the straight line D H Q, cutting ON in H, and C F in Q; and 
 
EFFICIENCY OF AN AXLE. 289 
 
 through H and Q, and parallel to D C, draw H K,, and Q Kj, 
 cutting C J in Ko and Kj respectively. Produce H C to H', and 
 Q C to Q', making C H' = H C, and C Q' = Q C. 
 
 Then, in the absence of friction, H' will represent the resultant 
 bearing-pressure exerted upon B by A A; and CKo = DHwill 
 represent the force in the given direction J required to drive B 
 at an uniform speed; and when friction is taken into account, C Q' 
 will represent the resultant bearing-pressure, and C K^ the actual 
 driving force required ; and we shall have 
 
 C K OK 
 
 the efl5ciency = ^^p=r£; and the counter-efficiency = ppj^^ 
 
 If from D, K,,, and Kj ther^ be let fall upon A A the perpen- 
 diculars D E, Ko Po, and Kj Pj, E, will represent the direct 
 resistance to the advance of B; C P,, the direct effort in the 
 absence of friction; and C Pj, the direct effort taking friction into 
 account; so that the distance P,, Pi will represent the friction 
 itself; which is also represented by Q N perpendicular to C N. 
 
 To express these results by symbols, let C D = E' (the given 
 force) ; let the acute angle A C D be denoted by «, and the acute 
 angle A J by /3; and let (p denote the angle of repose N Q. 
 
 Then, in the triangle C D H, we have ZDCH=- .<, and 
 C H D = ^ - /3; and in the triangle C Q D, we have Z D C Q 
 = '^ » + <p, and z C Q D =^ - /3 - ?; consequently 
 DH=E'''°^";DQ = E'-°-Hife); 
 
 cos i8 COS (/3 -1- ?) ' 
 
 v^hence it follows that the efficiency and counter-efficiency are 
 given by the following equations : — 
 
 _„ . Po D H cos « • cos (/3 -1- «)) 1 -/tanfl ., . 
 
 Efficiency = p^ = pQ = cos ^ • cos (- - ?) - l+/tan« ^^'^ 
 
 Counter-efficiency = ^= ^ -/tan/3 ' ^'^ 
 
 It is to be remarked, that the efficiency diminishes to nothing 
 when cotan /3=/; that is to say, when /3 is the complement of the 
 angle of repose, ?>. In other words, if the oblique effort is applied 
 in the direction Q, no force, how great soever, will be sufficient 
 to keep the piece B in motion. 
 
 438. EflBciency of an Axle. — In fig. 156, let the circle AAA 
 represent the trace of the bearing-surface of an axle on a plane 
 perpendicular to its axis of rotation, — in other words, the trans- 
 
 II 
 
290 
 
 THEORY OP MACHINES. 
 
 verse section of that surface. Let the arrow near the letter N 
 represent the direction of rotation. Let D be the given force ; 
 that is, as before, the resultant of the weight of the whole piece 
 that rotates with the axle, and of the useful resistance or re-action 
 exerted on that piece by the piece which it drives ; C J, the line of 
 action of the effort by which the rotating piece is driven. 
 
 Fig. 156. 
 
 Let r denote the radius of the bearing-surface. 
 
 About O describe the small circle B B, with a radius = 
 r sin (p=fr, very nearly. Draw the line of action, T Q, of the 
 resultant bearing-pressure, touching the small circle at that side 
 which will make the bearing-pressure resist the rotation. In the 
 case in which C I) and C J intersect each other in a point, 0, as 
 shewn in the figure, C T Q will traverse that point also; and in 
 the case in which the lines of action of the given force and the 
 effort are parallel to each other, G T Q will be parallel to both. 
 The centre, of bearing-pressure is at Q; and O QT = f, the angle 
 of repose. 
 
 In the former case the efficiency may be found by parallelo- 
 grams of forces, as follows : — Draw the straight line CON; this 
 would be the line of action of the resultant bearing-pressure in the 
 absence of friction, and N would be the centre of bearing-pressure. 
 Through D, parallel to C J, draw D H E, cutting C O N" in H, 
 and C T Q in E. Through H and E, parallel to D C, draw H P, 
 
EFFICIENCY OF AN AXLE. 291 
 
 and E Pi- Then, in the absence of friction, H C would represent 
 the bearing-pressure, and C Po = D H the effort; the actusi bear- 
 ing-pressure is represented by B C, and the actual effort by C Pi = 
 D E. Hence the eflaciency and counter-eflBciency are as follows : — 
 
 Po_D_H Pi_DE 
 
 Pi DE' Po"DH ^ ' 
 
 Another method, applicable whether the forces are inclined or 
 parallel, is as follows : — From the axis of rotation O, let fall L^ 
 and Mq perpendicular respectively to the lines of action of the 
 given force and of the effort. Then, by the balance of moments, 
 the effort in the absenc^ of friction is 
 
 Erom a convenient point in the actual line of action, C Q, of 
 the bearing-pressure (such, for example, as T, where it touches the 
 small circle B B), let fall T Li and T Mj perpendicular respec- 
 tively to the same pair of lines of action; then the actual effort 
 will be 
 
 Hence the efficiency and the counter-efficiency have the following 
 value : — 
 
 Pq OLq -TMi . 
 Pi OMj-TLi' 
 
 PiPM^j^I^ 
 Po"0"Lo-TMi^ J 
 
 )■ (2.) 
 
 The same results are expressed, to a degree of approximation 
 sufficient for practical purposes, by the following trigonometrical 
 formulae :— Let O Iif, = l; O Mo = »i; ZCO Lo = a; ZCO M^= . 
 Then we have, very nearly, 
 
 .(3.) 
 
 Pq I m —fr sm /8 _ m 
 
 'Pi~m I +fr sin a~ , fr . 
 ^ ■' 1 +•-=- • sm a 
 
 V 
 
 In making use of the preceding formula, it is to be observed 
 that the conPraay algebraicai signs of sin « and sin (3 apply to those 
 cases in which the two angles « and /3 lie at contrary sides of C. 
 In the cases in which those angles Ue at the same side of C, their 
 alwebraioal signs are the same; and in the formula they are to be 
 
292 THEORY OF MACHINES. 
 
 made both positive or both negain/ee, according as /3 is less or greaier 
 than « ; so that the efficiency may be always expressed by a frac- 
 tion less than unity. That is to say, 
 
 fr 
 p 1 - — sin /3 
 
 if^>»,|o. ;. . ■' (3--) 
 
 1 --y- sin » 
 L 
 
 , fr . 
 ■p 1 + — sin /3 
 
 ^'^<-4;=^^ — ■' (3b.) 
 
 1 + -, sin « 
 c 
 
 When the lines of action intersect, let O be denoted by c; 
 then I = c cos «, and to = c cos /3; and consequently the three 
 preceding equations take the following form ; — 
 
 j8 and « of contrary signs; p^ = j 
 
 ' 1 + - 
 ^ and a of the same sign; 
 
 fr 
 1 -•'— tan /3 
 
 ^'-^v—' c^) 
 
 - tana 
 c 
 
 p 1 --^ tan iS 
 
 "^"^pr—^^ — ^ (^^•> 
 
 1 tan » 
 
 c 
 
 p 1+— taniS 
 
 ''<»^p-=— ^^ ; (4 b.) 
 
 ^ 1 + — tan a 
 c 
 
 When the lines of action of the forces are parallel, we have sin fi 
 and sin «= +1 or-1, as the case may be; and the formulas 
 take the following shape : — 
 
 When I and m lie at contrary sides of 0, the piece is a " lever ' 
 of the first kind; " and 
 
 i_/r 
 ^-'^ "■' 
 
 "When I and m lie at the same side of O; 
 
EFFICIENCY OP MODES OF CONNECTION IN GENERAL. 293 
 
 Iim>l, the piece is a « lever of the second kind; " and 
 
 prTjr (^^-^ 
 
 I 
 
 Um<l, the piece is a " lever of the third kindj" and 
 P ^-^- 
 
 Px=T7v (^"> 
 
 ■^ I 
 
 (As to levers of the first, second, and third kinds, see Article 
 184, page 108.) 
 
 The following method is applicable whether the forces are inclined 
 or parallel; in the former case it is approximate, in the latter 
 exact. Throug:h O, perpendicular to O C, draw U O V, cutting 
 the lines of action of the given force and of the eflfort in U and V 
 respectively. The point where this transverse line cuts the small 
 circle B B coincides exactly with T when the forces are parallel, 
 and is very near T when they are inclined; and in either case the 
 letter T will be used to denote that point. Then 
 
 Po_OU.Ty 
 
 Pa OV TTJ ^^■' 
 
 It is evident that with a given radius and a given coefficient of 
 friction, the efficiency of an axle is the greater the more nearly 
 the effijrt and the given force are brought into direct opposition to 
 each other, and also the more distant their lines of action are from 
 the axis of rotation. 
 
 439. Efficiency of a Screw. — The efficiency of a screw acting as 
 a primary piece is nearly the same with that of a block sliding on 
 a straight guide, which represents the development of a helix situated 
 midway between the outer and inner edges of the screw-thread ; 
 the block being acted upon by forces making the same angles with 
 the straight guide that the actual forces do with that helix. As to 
 the development of a helix, see Article 160, page 94 ; and as to the 
 efficiency of a piece sliding along a straight guide, see Article 437, 
 page 288. 
 
 Section 2. — Efficiency and Countee-efficiency op Modes op 
 Connection in Mechanism. 
 
 440. Efficiency of Modes of Connection in General. — In an ele- 
 mentary combination consisting of two pieces, a driver and a 
 
294 THKOKir OF MACHIKIBB. 
 
 follower, there is always some work lost in overcoming wasteful 
 resistance occasioned by the mode of connection ; the result being 
 that the work done by the driver at its working-point is greater 
 than the work done upon the follower at its driving-point, in a 
 proportion which is the counter-efficiency of the connection ; and the 
 reciprocal of that proportion is the effidency of the connection. In 
 calculating the efficiency or the counter-efficiency of a train of 
 mechanism, therefore, the factors to be multiplied togother comprise 
 not only the efficiencies, or the counter-efficiencies, of the several 
 primary pieces considered separately, but also those of the several 
 modes of connection by which they communicate motion to each 
 other. 
 
 441. Efficiency of Eolling Contact. — The work lost when one 
 primary piece drives another by rolling contact is expended in 
 overcoming the rolling redstcmce of the pitch-surfaces, a kind of 
 resistance whose mode of action has been explained in Article 402, 
 page 251 ; and the value of that work in units of work per second 
 is given by the expression ahlS ; in which N is the normal pressure 
 exerted by the pitch-surfaces on each other; h, a constant arm, of a 
 length depending on the nature of the surfaces (for example 0'002 
 of a foot = 0'6 millimetre for cast iron on cast iron, see page 252); 
 and a the relative angular velocity of the surfaces. 
 
 The useful work per second is expressed by m/N, in which y is 
 the coefficient of friction of the surfaces, and u the common velocity 
 of the pitch lines. Hence the counter-efficimey is 
 
 c = l-H— 7. (1.) 
 
 uf ^ ' 
 
 Let pi and p^ be the lengths of two perpendiculars let faii from 
 the two axes of rotation on the common tangent of the two pitch- 
 lines; if the pieces are circular wheels, those perpendiculars will 
 be the radii. Then the absolute angular velocities of the pieces 
 
 are respectively - and ; and their relative angular velocity is 
 
 therefore 
 
 a = u\— -V — ); 
 
 which value being substituted in Equation 1, gives for the counter- 
 efficiency the following value : — 
 
 o=l + l.(l+l\ (2.) 
 
 It is assumed that the normal pressure is not greater than is 
 
EFFICIENOr OF SLIDING CONTACT IN GENERAL. 
 
 295 
 
 necessary in <Drder to give sufficient friction to communicate tlie 
 motion. 
 
 It is evident, from the smallness of h, that the lost work in this 
 case must be almost always a very small fraction of the whole. 
 
 442. Efficiency of Sliding Contact in General,— In fig. 157, let T 
 be the point of contact of a pair of , 
 
 moving pieces connected by sliding 
 contact. Let the plane of the figure 
 be that containing the directions of 
 motion of the two particles which 
 touch each other at the point T ; and 
 let T V be the velocity of the driving- 
 particle, and T W the velocity of the 
 following particle ; wltence V W will 
 represent the velocity of sliding, and 
 T TJ, perpendicular to V W, the 
 common component of the velocities 
 of the two particles along their line of 
 connection E. T P. C T C, parallel to 
 V W, and perpendicular to E, T P, is 
 a common tangent to the two acting 
 surfaces at the point T j the arrow A 
 represents the direction in which the 
 driver slides relatively to the follower; 
 and the arrow B, the direction in 
 which the follower slides relatively to 
 the driver. 
 
 Along the line of connection, that is, normal to the acting sur- 
 faces at T, lay off T P to represent the effort exerted by the driver 
 on the follower, and T E ( = - T P) to represent the equal and 
 opposite useful resistance exerted by the follower against the driver. 
 Draw S T Q, making with R T P an angle equal to the angle of 
 repose of the rubbing surfaces, (see Article 261, page 154), and 
 inclined in the proper direction to represent forces opposing the 
 sliding motion; draw P Q and E, S parallel to C C. Then T Q 
 will represent the resultant pressure exerted by the driver on the 
 follower, and T S (= — T Q), the equal and opposite resultant 
 pressure exerted by the follower against the diiver, and P Q = - R S 
 will represent the friction which is overcome, through the dis- 
 tance V W, in each second ; while the useful resistance, T E, 
 is overcome through the distance T U. Hence the useful work 
 per second is T U • T E; the lost work is V W • E S; and the 
 counter-efficiency is 
 
 Kg. 157. 
 
 c = 1 + 
 
 YWE S 
 T U • T E" 
 
 .(1.) 
 
296 THEORY OP MACHINES. 
 
 Let the angle TJ T V = «, the angle TJ T "W = /3, and let/ be the 
 coefficient of friction. Then -w^e have — 
 
 -^^ =tan a + tan /3 ; i^=f; 
 
 and consequently 
 
 c = l +/(tan « + tan fi) (2.) 
 
 443. EflSciency of Teeth. — It has already been shewn, in Article 
 148, page 87, that the relative velocity of sliding of a pair of 
 teeth in outside gearing is expressed at a given instant by 
 
 (Oi + ffla) t; 
 
 ■where t denotes the distance at that instant of the point of contact 
 from the pitch-point. (In inside gearing the angular velocity of 
 the greater wheel is to be taken with the negative sign.) 
 
 The distance t is continually varying from a maximum at the 
 beginning and end of the contact, to nothing at the instant of 
 passing the pitch-point. Its mean value may be assumed, with 
 sufficient accuracy for practical purposes, to be sensibly equal 
 to one-half of its greatest value ; and in the formulae , which 
 follow, the symbol t stands for that mean value. 
 
 Let P be the mutual pressure exerted by the teeth; /, the 
 coefficient of friction j then the work lost per second through 
 the friction of the teeth is 
 
 (oi + a^) t/F. 
 
 Let M be the common velocity of the two pitch-circles ; 6 , the 
 meam obliquity of the line of connection to the common tangent of 
 the pitch-circles ; then u cos 6 is the mean value of the common 
 component of the velocities of the acting surfaces of the teeth along 
 the line of connection ; and the useful work done per second is 
 expressed by 
 
 P u cos e. 
 
 so that the counter-efficiency is 
 
 ,^l^ («i + «2y n.) 
 
 U COS '' ^ ' 
 
 Let Ti and r^ be the radii of the two pitch circles; then we have 
 M u 
 
 and consequently 
 
 c.l+/isece{i + i} (2.) 
 
EFFICIENCY OP BANDS. 297 
 
 If two pairs of teeth at least are to be in action at each instant 
 (as in the case of involute teeth, and of some epicycloidal teeth), 
 
 and if the pitch be denoted hyp, we have < see e=^; and therefore 
 
 where Wj. and n^ are the number of teeth in the two wheels. 
 
 In many examples of epicycloidal teeth, especially where small 
 
 2 3 
 pinions are used, the duration of the contact is only ^ or j of that 
 
 assumed in Equation 3 j and the work lost in friction is less in the 
 same proportion. * 
 
 444. EfiBciency of Bands. — ^A band, such as a leather belt or a 
 hempen rope, which is not perfectly elastic, requires the expenditure 
 of a certain quantity of work — first to bend it to the curvature of 
 a pulley, and then to straighten it again; and the quantity of work 
 so lost has been found by experiment to be nearly the same as 
 would be required in order to overcome an additional resistance, 
 varying directly as the sectional area of the band, directly as its 
 tension, and inversely as the radius of the pulley. In the follow- 
 ing formulae for leather belts, .the stiffness is given as estimated by 
 Beuleaux (fionsti-uctionslekre/iir Maschmenbau, § 307). 
 
 Let T be the mean tension of the beltj S, its sectional area; 
 r, the radius of the pulley; 6, a constant divisor determined by 
 experiment; E.', the resistance due to stiffness; then 
 
 E' = |^ (1.) 
 
 or ^ ' 
 
 h (for leather) = 3-4 inch = 87 millimetres. 
 
 To apply this to an endless belt' connecting a pair of pulleys of 
 
 the respective radii r^ and r^, let Tj and Ta be the tensions of the 
 
 two .sides of the belt. Then the useful resistance is Tj - T^ . the 
 
 T + T 
 mean tension is -^ — ^j and the additional resistance due to 
 
 stiffness is 
 
 T. + T, SJ1 1 ) . 
 2 hir^'^rj' 
 
 consequently the counter-efficiency is 
 
 T, + T, Sfl ,_11. 
 
 - 1 , I^ + 1 -S f 1 , M . ^ ^'^ 
 
298 THEORY OF MACHINES. 
 
 T 
 
 N denoting ^r- The sectional area, S, of a leather belt is given 
 
 by the formula 
 
 T 
 S = -^; (3.) 
 
 where p denotes the safe working tension of leather belts, in units 
 of weight per unit of area; its value being, according to Morin, 
 
 0'2 kilogramme on the square millimetre, or 
 285 lbs. on the square inch. 
 
 The ordinary thickness of the leather of which belts are made is 
 about 0*16 of an inch, or 4 millimetres; and from this and from 
 the area the breadth may be calculated. A double belt is of double 
 thickness, and gives the same area with half the breadth of a single 
 belt. 
 
 When a band runs at a high velocity, the centrifugal tension, 
 or tension produced by centrifugal force, must be added to the 
 tension required for producing friction on the pulleys, in order to 
 find the total tension at either side of the band, with a view to 
 determining its sectional area and its stiffness. The centrifugal 
 tension is given by the following expression : — 
 
 M> S »2 ,, , 
 
 -T-' ^'-^ 
 
 in which w is the heaviness (being, for leather belts, nearly equal to 
 that of water) ; S, the sectional area; v, the velocity; and g, gravity 
 ( = 32-2 feet, or 9-81 metres per second). 
 
 When centrifugal force is taken into consideration, the following 
 formula is to be used for calculating the sectional area; Ti being 
 the tension at the driving-side of the belt, exclusive of centrifugal 
 tension : — 
 
 s = _A^ 
 
 .(5.) 
 
 wv'i 
 
 and the following foiTnuIa for the counter-efficiency : — 
 T +T 1 ^"'^' 
 
 "'^^ 2(t.-tJ TU'-tA (^-^ 
 
 'For calculating the efficiency of hempen ropes used as bands, it 
 is unnecessary in such questions as that of the present article to 
 use a more complex formula than that of Eytelwein — viz., 
 
 T)2T 
 
 I^' = W' (7-) 
 
EFFICIENCY OF LINKWORK. 
 
 299 
 
 where D is the diameter of the rope, and V = 54 millimetres = 
 2-125 inches. 
 
 D2 g 
 
 In all the formulae, -rr is to be substituted for -r. The proper 
 value of D^ is given by the formula 
 
 D2 = 
 
 
 .(8.) 
 
 ■where p' = 1,000 for measures in inches and lbs. ; and 
 
 p'= 0*7 for measures in millimetres and killogrammes. 
 445. Efficiency of Linkwork.— In fig. 158, let 0^ Tj, CgT^ be two 
 levers, turning about ptirallel axes at Cj and Ca, and connected with 
 each other by the link Tj. Tgj Ti and Tg being the connected points. 
 
 Fig. 158. 
 
 The pins, which are connected with each other by means of the 
 link, are exaggerated in diameter, for the sake of distinctness. Let 
 C^ Ti be the driver, and Cg Tj the follower, the motion being as 
 shewn by the arrows. From the axes let fall the perpendiculars 
 Ci Pi, Ca Pj, upon the line of connection. Then the angular 
 velocities of the driver and follower are inversely as those perpen- 
 diculars; and, in the absence of friction, the driving moment of the 
 first lever and the working moment of the second are directly as 
 those perpendiculars; the driving pressure being exerted along the 
 line of connection Tj Tj. Let M^ be the working moment; and 
 let Mq be the driving moment in the absence of friction; then we 
 have 
 
 M, 
 
 M,-CiP, 
 
 Cj Pg 
 
300 THEOET OF MACHINES. 
 
 To allow for the friction of the pins, multiply the radius of each 
 ^in by the sine of the angle of repose; that is, very nearly by the 
 coefficient of friction; and with the small radii thus computed, 
 Ti Ai and Tj Aj, draw small circles about the connected points. 
 Then draw a straight line, Qi A, Bj Q^ A^ Bj, touching both the 
 small circles, and in such a position as to represent the line of 
 action of a force that resists the motion of both pins in the eyes of 
 the link. This will be the line of action of the resultant force 
 exerted through the link. Let fall upon it the perpendiculars 
 Ci Qi, Ca Q2; these will be proportional to the actual driving 
 moment and working moment respectively; that is to say, let Mj 
 be the driving moment, including friction; then 
 
 Comparing this with the value of the driving moment without 
 friction, we find for the counter-efficiency 
 
 ^ Ml CiQi-0,P„ 
 
 ,.(1.) 
 .(2.) 
 
 — M.-C.Q.-CF,' 
 
 and for the efficiency 
 
 1 Mq C,Q,-CiPi 
 
 c Ml CiQi-c^p; 
 
 446. Efficiency of Blocks and Tackle.— (See Articles 181, 182, 
 pages 105 and 106.) — In a tackle composed of a fixed and a running 
 block containing sheaves connected together by means of a rope, let 
 the number of plies of rope by which the blocks are connected with 
 each other be n. This is also the collective number of sheaves in 
 the two blocks taken together, and is the number expressing the 
 purchase, when friction is neglected. 
 
 Let c denote the counter-efficiency of a single sheave, as depend- 
 ing on its friction on the pin, according to the principles of Article 
 373, page 290. Let c' denote the counter-efficiency of the rope, 
 
 when passing over a single sheave, determined by the principles 
 
 p 
 of Article 444, the tension being takep as nearly equal to — ; 
 
 where E is the useful load, or resistance opposed to the motion of 
 the running block. R -^ «. is also the effijrt to be exerted on the 
 hauling part of the rope, in the absence of friction. Then the 
 counter-efficiency of the tackle will be expressed approximately by 
 
 (oc)"; (1.) 
 
 so that the actual or effective purchase, instead of being expressed 
 by n, will be expressed by 
 
 nice')-" (2.) 
 
EFFICIENCY OF CONNECTION BT MEANS OP A FLUID. 301 
 
 447. Eflaciency of Connection by means of a Fluid. — ^Whea 
 motion is communicated from one piston to another by means of an 
 intervening mass of fluid, as described in Articles 185 to 188, 
 pages 110 and 111, the eflBciencies and counter-efficiencies of the two 
 pistons have in the first place to be taken into account; that is 
 to say, -with ordinary workmanship and packing, the efficiency of 
 each piston may be taken at 0-9 nearly j while with a carefully 
 made cupped leather collar the counter-efficiency of a plunger may 
 be taken at the following value : — 
 
 ■-¥^ a-) 
 
 in which d is the diameter of the plunger; and 6 a constant, whose 
 
 value is from 0-01 to 0-015 of an inch, or from 0-25 to 038 of a 
 
 millimetre. For if c be the circumference of the plunger, and p 
 
 the effective pressure of the liquid, the whole amount of the^pres- 
 
 • V c d 
 sure on the plunger is ^-r— ; and the pressure required to overcome 
 
 the friction is ^ c 6. 
 
 The efficiency and counter-efficiency of the intervening mass of 
 fluid remain to be considered ; and if that fluid is a liquid, and 
 may therefore be regarded as sensibly incompressible, these quan- 
 tities depend on the work which is lost in overcoming the resist- 
 ance of the passage which the liquid has to traverse. 
 
 To prevent unnecessary loss of work, that passage should 
 be as wide as possible, and as nearly as possible of uniform 
 transverse section; and it should be free from sudden enlarge- 
 ments and contractions, and from sharp bends, all necessary 
 enlargements and contractions which may be required being made 
 by means of gradually tapering conoidal parts of the passage, 
 and all bends by means of gentle curves. When those conditions 
 are fulfilled, let Q be the volume of liquid which is forced through 
 the passage in a second; S, the sectional area of the passage; 
 then, 
 
 -I' (^•> 
 
 is the velocity of the stream of fluid. Let 6 denote the wetted 
 border or circumference of the passage ; then. 
 
 S 
 
 .(3.) 
 
 is what is called the hyikaulio mean depth of the passage. In a 
 cylindiical pipe, m = \ diameter. Let I be the length of the 
 
302 THEORY OP MACHINES. 
 
 passage, and w the heaviness of the liquid. Then the loss of pres- 
 sure in overcoming the friction of the passage is 
 
 , fl wv^ ,, . 
 
 P-mTg-' (^•> 
 
 in which g denotes gravity, and f a coefficient of friction whose 
 value, for water in cylindrical cast-iron pipes, according to the 
 experiments of Darcy, is 
 
 /=0005(l+^^);* (5.) 
 
 d being the diameter of the pipe in feet. 
 
 Let p be the pressure on the driven or following piston ; then 
 the pressure on the driving piston is p + p", and the counter- 
 efficiency ofthejhbid is 
 
 '-i' (^•) 
 
 which, being multiplied by the product of the. counter-efficiencies 
 of the two pistons, gives the counter-efficiency of the intervening 
 liquid. 
 
 When the intervening fluid is air, there is a loss of wort 
 through friction of the passage, depending on principles similar to 
 those of the friction of liquids ; and there is a further loss thi-ough 
 the escape by conduction of the heat produced by the compression 
 of the air. 
 
 The friction which has to be overcome by the air, and which 
 causes a certain loss of pressure between the compressing pumps 
 and the working machinery, consists of two parts, one occasioned 
 by the resistance of the valves, and the other by the friction along 
 the internal surface of pipes. 
 
 To overcome the resistance of valves, about five per cent, of the 
 effective pressure may be allowed. 
 
 The friction in the pipes depends on their length and diameter, 
 and on the velocity of the current of air through them. It is 
 nearly proportional to the square of the velocity of the air. 
 
 A velocity of about forty feet per second for the air in its com- 
 pressed state has been found to answer in practice. The diameter 
 of pipe required in order to give that velocity can easily be com- 
 puted, when the dimensions of the cylinders of the machinery to be 
 driven, and the number of strokes per minute, are given. 
 
 When the diameter of a pipe is so adjusted that the velocity of 
 the air is 40 feet per second, the pressure expended in overcoming 
 its friction may be estimated at one per cent, of the total or absolute 
 
 * When the diameter is expressed in millimetres, for^;--, substitute — t-' 
 
 12 d d 
 
EFFICIENCY OP CONNECTION BY MEANS OP A FLUID. 303 
 
 pressv/re of the aw, far every five hundred diameters of the pipe that 
 its length contains. ' 
 
 Although the abstraction from the air of the heat produced by 
 the compression involves a certain sacrifice of motive power (say 
 from 30 to 35 per cent.) still the effects of the heated air are so 
 inconvenient in practice, that it is desirable to cool it to a certain 
 extent during or immediately after the compression. This may be 
 effected by injecting water in the form of spray into the com- 
 pressing pumps; and for that purpose a small forcing pump of 
 about yJtt*'* °f ^^^ capacity of the compressing pumps has been 
 found to answer in practice. The air may thus be cooled down to 
 about 104" Fahr. or 40° Cent. 
 
 The factor in the counter-efficiency due to the loss of heat 
 expresses the ratio in which the volume of air as discharged from 
 the compressing pump at a high temperature is greater than the 
 volume of the same air when it reaches the working machinery at 
 a reduced temperature; which ratio may be calculated approxi- 
 mately by taking two-sevenths of the logwrithm of the absolute 
 working pressure of the compressed air in atmospheres, and finding 
 the corresponding natural number. That is to say, let pf, denote 
 one atmosphere ( = at the level of the sea 14-7 lbs. on the square 
 inch, or 10,333 kilogrammes on the square m6tre); let^i be the 
 absolute working pressure of the air, so that pi -p^ is the effective 
 pressure ; then the counter-efficiency due to the escape of heat is, 
 
 "(gr- 
 
 .(7.) 
 
 From examples of the practical working of compressed air, 
 Vhen used to transmit motive power to long distances, it appears 
 that in order to provide for leakage and various other imperfec- 
 tions in working, the capacity of the compressing pumps should be 
 very nearly double of the net volume of uncompressed air required; 
 and it has also been found necessary, in working the compressing 
 pumps, to provide from three to four times the power of the 
 machinery driven by the compressed air. 
 
INDEX. 
 
 ■ Absolute unit of force, 213. 
 
 - Acceleration, work of, 252. wJ^ ^^ 
 Accelerating effect of gravity, 213. 
 
 force, 213. 
 
 - impulse, 207. 
 
 " Action and re-action, 113. 
 ' Actual energy, 207. , 
 
 Addendum of a tooth, 81. 
 
 Aggregate combinations, 73, 112. 
 "Angle of repose, 154. 
 .of rotation, 48. 
 
 Angular impulse, 220. 
 
 - momentum, 219, 228. 
 momentum, conservation of, 220. 
 momentum and angular impulse 
 
 - relation of, 220. 
 
 - velocity, 48. 
 
 - velocity, variation of, 63. 
 
 - Arch, line-of-^ressures-in, VnJxr.M 
 Arcs, measurement of, 23, 24. 
 Areas, centre of, 26. 
 
 mensuration of, 16, 17. 
 Axis, instantaneous, 55. ^■'^^- '-' 
 
 of rotation, 47, 48. 
 Axle, strength of, 187. 
 
 torsion of, 187. 
 Axles and shafts, efficiency of, 289. 'T7 
 
 friction of (see Efficiency). ^-O"'- 
 
 Balance, 31, 118. 
 
 of any system of forces, 135, 136, 
 137. 
 
 of any system of forces in one 
 plane, 134. 
 
 of chain or cord, 174. 
 
 of couples, 126. 
 
 of forces in one line, 118. 
 
 of inclined forces, 122. 
 
 of parallel forces, 131, 132. 
 
 of structures, 157. 
 Balanced forces, motion under, 210. 
 Bands, classed, 97. 
 
 connection by, 72, 97, 98. 
 
 efficiency of, 297. 
 
 length of, 99. 
 
 motion of, 97. 
 
 principle of connection by, 97. 
 Bar. 158. 
 
 Beam, 158. 
 
 allowance for weight of, 200. 
 
 limiting length o^ 200. 
 
 in linkwork, 101. 
 Bearings, 71. 
 
 friction of, 251. 
 Belt, with speed cones, 100. 
 Bending moment, at a series of sec- 
 tions, 193. 
 Bending moment, greatest, 194. 
 Bending moments, calculation of, 
 
 190. -^ 
 Bending, resistance to, 189. 
 
 moment of, 190. 
 Bevel-wheels (see Wheels). 
 Blocks and tackle, 105. 
 
 efficiency of, 300. 
 Blocks, stability of a series of, 158, 
 
 175. 
 Bodies, 30. 
 
 rigid, 47. 
 Bracing of frames, 166, 167, 168. 
 Brake, 241. 
 Brakes, 276. 
 
 block, 277. 
 Bulkiness, 121. 
 Buoyancy, centre o^ 121. 
 
 Cam or Wiper, 92. 
 Centre of area, 26. 
 
 of a curved Une, 27. 
 
 of a piano area, 26. 
 
 of buoyancy, 121. 
 
 of gravity, 121, 140. 
 
 of magnitude, 25, 26, 27, 28, 29. 
 
 of mass, 207. 
 
 of oscillation or percussion, 208, 
 227. 
 
 of parallel forces, 119, 133. 
 
 of pressure, 121. 
 
 of resistance, 176. 
 
 of special figures, 28. 
 
 of volume, 27. 
 Centrifugal force, 207 (see also De- 
 
 viatmg Force). 
 Chains, equilibrium of, 158, 174. 
 Channel, 68. 
 Cinematics, 31. 
 
3o6 
 
 INDEX. 
 
 Cinematics, principles of, 33. 
 Circle, involute of (see Involute). 
 
 area of, 21. 
 Circular arcs, measurement of, 23. 
 Circular measure, 8. 
 
 sector, area of, 22. 
 
 arcs, length of, 23, 24. 
 Click, 105. 
 Coefficient of stiffness, 183. 
 
 of elasticity, 184. 
 
 of pliability, 183. 
 Cog, hunting, 83. 
 Collar, friction of, 231. 
 ColHaion, 208, 221. 
 Combinations, aggregate (see Aggre- 
 gate). 
 
 elementary (see Elementary). 
 Comparative motion, 38, 45, 50, 63. 
 Components, 123. 
 
 of motion, 35, 
 
 of varied motion, 40. 
 Compression, resistance to, 202. 
 Cones, pitch (see also Wheels, bevel). 
 
 rolling, 63. 
 
 speed, 100. 
 Connected points, motion of, 102. 
 Connecting-rod, 101 (see Linkwork). 
 Connection, line of, 73. 
 
 principle of, 73. 
 Connectors, 71. 
 Conservation of energy, 206, 260. 
 
 of angular momentum, 220. 
 
 of momentum, 219. 
 Continued fractions, 2. 
 Continuity, equations of, in liquids, 
 
 67, 69. 
 Contracted vein, 233. 
 Contraction, coefficient of, 233- 
 Cord, equilibrium of, 158-, 174. 
 
 guided by surfaces of revolution, 
 66. 
 
 motion of, 65. 
 Counter-efficiency, (see Efficiency). 
 Coupled parallel shafts, 101. 
 Couples, 118, 119. 
 
 equivalent, 125. 
 
 parallelogram of, 126. 
 
 polygon of, 126. 
 
 resultant of, 125.' 
 
 with parallel axes, 126. 
 Coupling, double, Hooke's, 105. 
 
 Hooke's, 104. 
 Oldham's, 96. 
 Coupling-rod, 101 (see Linkwork). 
 Crank-rod, 101 (see Linkwork). 
 Cross-breaking; resistance to. 189. 
 
 Crushing, direct resistance to, 202. 
 Curved fines, measurement of, 23. 
 Curves, measurement of the length 
 
 of, 23, 24, 25. 
 Cycloid, 55. 
 Cylinders, strength of, 186, 187. 
 
 Dead points in linkwork, 101. 
 
 Dead load, 180. 
 
 Density, 120. 
 
 Deviating force, 207, 216. 
 
 in terms of angular velocity, 217. 
 Deviation (of motion), uniform, 44. 
 
 varying, 45. 
 Differential and integral calculus, 10. 
 
 coefficients, 11, 12. 
 
 calculus, geometrical illustration 
 of; 12. 
 Direction, fixed and nearly fixed, 
 
 33. 
 Directional relation, 38. 
 Distributed forces, 119, 120, 140. 
 
 loads, 160. 
 Driving-point, 242. 
 Dynamics, 32. 
 
 general equations of, 211. 
 Dynamometer, 271. 
 
 Eccentric, 103. 
 
 rod, 101. 
 Effect and power, 241, 266. 
 Efficiency and counter-efficiency, 
 241, 265, 286. 
 of a machine, 265, 266. 
 of a shaft or axle, 289. 
 of a sliding piece, 288. 
 of modes of connection in mechan- 
 ism, 293. 
 of primary pieces, 287. 
 of bands, 297. 
 of linkwork, 299. 
 of blocks and tackle, 300. 
 of fluid connection, 301. 
 of a screw, 293. 
 of rolling contact, 294. 
 of sliding contact, 295. 
 of teeth, 296. 
 Effort, 205. 
 
 accelerating, 260. 
 when speed is uniform, balances 
 resistances, 215. 
 Elasticity, l83. 
 coefficients of, 184. 
 modulus of, 184. 
 Elementary combjnjations, 72. 
 classed generally, 72. 
 
IKDEX. 
 
 307 
 
 Energy, 206, 259. 
 
 actual (or kinetic), 207, 262. 
 
 and work, general equation of, 267. 
 
 exerted and work done, equality 
 of, 260. 
 
 potential, 259. 
 
 stored and restored, 208, 262. 
 
 conservation of, 206, 260. 
 
 transformation of, 208. 
 Epicycloid, 58. 
 Epicyoloidal teeth, 89, 90. 
 Epitroohoid, 58. 
 
 curtate, 60. 
 
 prolate, 59. 
 Equilibrium (see Balance^ 
 
 Pace of a tooth, 81. 
 Factors of safety, 180. 
 
 prime, of a number, 1. 
 Falling body (see Gravity). 
 Fixed direction, 33. 
 
 point, 31. 
 Flank of a tooth, 81. 
 Flow of liquid, 66, 67. 
 
 in a stream, 67. 
 Fluctuations of speed, 241. 
 Fluid, motion of, 66, 68, 69, 230. 
 
 pressure of, 147. 
 
 steady motion of, 68. 
 
 velocity and flow of, 66. 
 Fluids, flow of volume of, 69. 
 
 balance of, 147. 
 
 flow of mass of, 69. 
 Fly-wheels, 241, 278, 280. 
 Foot-pound, 243. 
 Force, 31, 
 
 absolute unit of, 116, 213. 
 
 centrifugal (see Deviating Force). 
 
 deviating (see Deviating Force). 
 
 direction of, 116. 
 
 distributed, 119, 120, 140. 
 
 magnitude of, 116. 
 
 measure of, 117. 
 
 moments of, 127, 130; 
 
 rectangular components of, 124. 
 
 representation of, 115, 116. 
 
 reciprocating, 208, 263. 
 Forces, action and reaction, 115. 
 
 how determined and expressed, 
 115. 
 
 inclined, resultant and balance 
 of, 122, 125. 
 
 parallel, 118. 
 
 parallel, magnitude of resultant of, 
 127. 
 
 direction of, 128. 
 
 Forces, parallelogram o^ 122. 
 
 paridlelopiped of, 123. 
 
 polygon of, 123. 
 
 representation of by line, 117. 
 
 resolution of, 122, 123, 124. 
 
 resultaat and component of, 118. 
 
 triangle of, 122. 
 Fractions, continued, 2. 
 Frames, 71. 
 
 bracing of, 166. 
 
 equilibrium and stability of, 158. 
 
 of two bars, 161. 
 
 polygonal, 163, 164^ 165. 
 
 resistance of, at a section, 171. 
 
 triangular, 162, 163. 
 Friction, 153, 154. 
 
 coefficient of, 154, 
 
 moment of, 251. 
 
 of liquid, 235. 
 
 of solid bodies, law of, 153. 
 
 tables of, 155. 
 
 work done against, 251. 
 Frictional stabmty, 176. 
 Function, 6. 
 
 Governors, 241, 282. 
 
 pendulum, 283. 
 
 loaded, 285. 
 Gravity, accelerating effect of, 213. 
 
 centre of, 121, 140. 
 
 motion under, 213. 
 
 specific, 120. 
 Greatest common measure, 1. 
 Gyration, radius of, 208, 223. 
 
 table of radii of, 226. 
 
 Head, dynamic, of liquid, 230. 
 Heat of friction, 252. 
 Heaviness, 120. 
 Helical motion, 51, 52. 
 Helix (see Screw-line). 
 
 normal, 93. 
 Horse-power, 241, 266. 
 Hunting-cog, 83. 
 Hydraulic connection, 110. 
 
 efficiency o^ 301. 
 
 hoist. 111. 
 Hydraulic press, 110. 
 Hydrostatics, principles of, 147, 148, 
 149. 
 
 Impulse, 207. 
 
 and momentum, law of, 254. 
 Inclined plane, 107. 
 Indicator, 271. 
 
308 
 
 INDEX, 
 
 Indicator diagram, 273. 
 Inertia, or mass, 206. 
 
 moment of (see Moment). 
 
 reduced, 257. 
 Integrals, approximate computation 
 
 of, 13, 14, 15. 
 Intensity of distributed force, 120. 
 
 of pressure, 121. 
 
 of stress, 143. 
 Intervening fluid, connection by, 73. 
 Involute, 56. 
 
 Joints, of a structure, 156. 
 Journal, friction of, 251. 
 
 Kinetics, 32, 205. 
 
 general equations of, 211. 
 
 Lateral poece, 205. 
 
 Lengtb, measure of, 30, 31. 
 
 Lever, 101, 107, 128. 
 
 Line, 30. 
 
 Link, 101. 
 
 Linkwork, connection by, 72, 101. 
 
 comparative motion of the cou 
 nected points in, 102. 
 
 efficiency o^ 299. 
 Liquid, dynamic head of, 230. 
 
 equilibrium of, 147. 
 
 free surface of, 231. 
 
 motion of, 230, 233. 
 
 motion of, in plane layers, 232. 
 
 motion o£ with friction, 233. 
 
 surface of equal pressure in, 231. 
 
 ■without friction, motion of, 230. 
 Live load, 180. 
 Load, 179. 
 
 dead, 180. 
 
 live, 180. 
 
 working, 179. 
 Logarithms, common, 4, 5, 6. 
 
 Machine, efficiency of (see Effi- 
 ciency), 
 action of, 243. 
 general equation of the action of, 
 
 267. 
 moving pieces in, primary and 
 secondary, 72. 
 Machines, 82. 
 
 theory of, 240. 
 Magnitude, centre of, 25. 
 Mass, 206. 
 centre of, 207. 
 in terms of weight, 212. 
 measure of, 117. 
 
 Matter, 30. 
 
 Measure, greatest common, 1. 
 
 Measures offeree and mass, 117. 
 
 of length, 30, 
 Mechamcal powers, comparative mo- 
 tion in, 107. 
 
 forces in, 268. 
 Mechanics, 30. 
 Mechanism, theory of, 70. 
 
 aggregate combinations in, 73. 
 
 elementary combinations in, 72. 
 
 principle of connection in, 73. 
 Mensuration of areas, 17. 
 
 of curved lines, 23. 
 
 of geometrical moments, 25 
 
 of volumes, 22. 
 Merrifield's trapezoidal rule, 19, 20. 
 Modulus of elasticity, 184. 
 
 height or length of, 184. 
 
 of pliability, 183. 
 
 of resilience, 185. 
 
 of stifiness, 183. 
 
 of transverse elasticity, 187. 
 Moment, bending, 190. 
 
 geometrical, 25. 
 
 geometrical, of inertia, 199. 
 
 greatest, 194. 
 
 of a couple, 127. 
 
 of a force, 127, 130. 
 
 of inertia, 208, 222. 
 
 of inertia, table of, 226. 
 
 of stability, 177. 
 
 of stress, 196. 
 Momentum, 207. 
 
 and impulse, law of, 254. 
 
 angular (see Angular Momentum). 
 
 conservation of, 219. 
 
 of a rotating body, 228. 
 
 resultant, 207. 
 
 variation and deviation of, 207. 
 Motion, 31. 
 
 combination of uniform, and uni- 
 formly accelerated, 43. 
 
 comparative, 38, 39, 50, 63. 
 
 component and resultant, 35. 
 
 first law of, 210. 
 
 graphical representation o^ 42. 
 
 of a falling body, 213. 
 
 of fluid of constant density, 66. 
 
 of pistons, 68. 
 
 of points, 34, 37. 
 
 of jjoints, varied, 39, 40. 
 
 of pliable bodies and fluids, 65. 
 
 of rigid bodies, 47. 
 
 of varying density, 69. 
 
 periodical, 208, 264, 278. 
 
INDEX. 
 
 309 
 
 Motion, second law of, 211. 
 uniform, 37, 205. 
 uniform, dynamical principles of, 
 210. 
 
 Neuteai suepace, 197. 
 
 Pakabolio curves, 16, 17. 
 Parallel forces, 118, 127. 
 
 centre of, 119, 133. 
 
 forces, resultant of, 127, 128, 129, 
 131, 132. 
 
 projection (see Projection,Parallel). 
 Parallelogram, area of, 16. 
 Parallelepiped of motions, 38. 
 Pendulum, rotating, 217. 
 
 simple oscillating, 218. * 
 
 simple revolving, 217. 
 Percussion, centre of (see Centre). 
 Periodic motion, 208, 264, 278. 
 Periodical mbtion of madiines, 208. 
 Pieces, moving, 71. 
 
 of a structure, 156. 
 Pinion, smallest, with involute teeth, 
 
 89. 
 Pipes, friction in, 237. 
 
 resistance caused by sudden en- 
 largement in, 238. 
 
 resistance pf curves' and knees in, 
 238. 
 
 resistance of mouthpieces of, 238. 
 Piston, 110. 
 
 action of a fluid upon, 110. 
 
 motion of, 68. 
 Piston-rod, 101. 
 Pitch of a screw, axial, 94. 
 
 divided, 93. 
 
 normal, 93. 
 
 of teeth, 81 (see Teeth). 
 Pitch-circles, 81. 
 Pitch-lines, 81. 
 Pitch-point, 81. 
 
 Pitch-surfaces, 74, 81 (see Wheels). 
 Pivot, friction of, 251. 
 Plane of rotation, 48. 
 PliabiUty, 183. 
 
 coefficients of, 183. 
 Point, 30. 
 
 fixed, 31, 35. 
 
 motions of, 34. 
 
 moving, 35. 
 
 physical, 30. 
 Power, 241. 
 
 and effect, 241, 26.6. 
 
 horse, 241, 266. 
 Powers, mechanical (see Mechanical 
 powers). 
 
 Press, hydraulic (see Hydraulic 
 
 press). 
 Pressure, 144. 
 
 centre of, 121. 
 
 intensity of, 121. 
 Primary moving pieces, efficiency of, 
 287. 
 
 motions of, 72. 
 Prime factors, 1. 
 Prime movers, 240. 
 Projection, parallel, 138, 153, 178. 
 Projectile, unresisted, 214. 
 Proof strength, 182, 183. 
 Pull (see Tension). 
 Pulley-blocks (see Tackle). 
 PuUey (mechanical power), 107. 
 
 Racks, toothless, 74. 
 
 smooth, 74. 
 
 straight and circular wheels, 75. 
 Padius, geometrical, 81. 
 
 of gyration, 208. 
 
 real, 81. 
 Batio, 2. 
 
 approximation to, 2. 
 Reaction and action, 115. 
 Reciprocating force, 208, 263. 
 Reduced inertia, 257. 
 Reduction of forces and couples in 
 machines to the driving point, 
 257. 
 Reduplication (see Tackle). 
 Regulating apparatus, 276. 
 Regulator of a prime mover, 241. 
 Repose, angle of (see Angle). 
 Resilience, 184. 
 Resistance, 205. 
 
 centre o^ 176. 
 
 line of, 176. 
 
 points of, 242. 
 
 of curves and knees, 238. 
 
 of mouthpieces, 238. 
 
 of rolling, 252. 
 
 useful and prejudicial, 241. 
 Resolution offerees, 122. 
 Rest, 31. 
 Resultant, 118. 
 
 momentum, 207. 
 
 of any system of forces, 135. 
 
 of any system of forces in one 
 plane, 134. 
 
 of couples, 125. 
 
 of inclined forces, 125. 
 
 motions, 35. 
 
 of parallel forces, 127, 128, 129, 
 131, 132. 
 
310 
 
 INDEX, 
 
 JBigid body, motion of, 47, 222 (see 
 
 Rotation), 
 iligidity or stiffness, 183. 
 
 coefficients of, 183. 
 Eod (see Crank-rod, Coupling-rod, 
 Connecting-rod, EoCentric-rod, 
 Link, Piston-rod). 
 Kolled curves (see Cyelttid, Epicy- 
 cloid, Epitrochoid, Involute, 
 Spiral, Trochoid). 
 EoUers, 74. 
 
 KoUing contact, connection by, 72. 
 cones, 63. 
 efficiency of, 294. 
 general conditions of, 74. 
 of cylinder on plane, 55. 
 of cylinder on cylinder, 58. 
 of plane on cylinder, 55. 
 resistance, 252. 
 Rotating body; comparative motion 
 of points in, 50. 
 components of velocity of a point 
 
 in, 50. 
 relative motion of a pair of points 
 in, 49. 
 Rotation, 47. 
 actual energy of, 229. 
 angle of, 48. 
 angular velocity of, 48. 
 axis of, 47, 48. 
 
 combined with translation, 51, 54. 
 combined parallel, 56, 57, 62. 
 components o^ varied, 64. 
 instantaneous axis of, 55. 
 plane of, 48. 
 
 right and left handed, 49. 
 uniform, 48, 228. 
 varied, 63, 64. 
 Rotations about intersecting axes 
 combined, 62. 
 
 Safety, factors of, 180. 
 Screw, 92. 
 
 circular, pitch of, 93. 
 
 efficiency of, 293. 
 
 mechanical power, 107. 
 
 pitch of, 92, 93. 
 Screw-gearing, 94. . . 
 
 axial pitch of, 94. 
 • development of, 94. 
 
 divided pitch of, 93. 
 Screw -like or helical motion, 51, 52. 
 Screw-Hne, normal pitch of, 93. 
 Screws, compound, 113. 
 
 relative sliding of a pair of, 95. 
 
 right and left handed, 93. 
 
 Secondary moving pieces, 72. 
 
 efficiency of, 289. 
 Sections, method of, applied to frame- 
 work, 171. 
 Shaft, strength of (see Axle). 
 Shear, 144. 
 Shearing load, greatest, 192. 
 
 at a series of sections, 192. 
 Shearing loads, calculation of, 190. 
 Shearing, resistance to, 186. 
 Sheaves, 105. 
 
 Shifting, or translation, 47, 
 Simpson's Rules, 18, 19. 
 Skew-bevel wheels (see Wheels). 
 Sliding contact, connection by, 72. 
 
 efficiency of, 295. 
 
 principle of, 79, 80. 
 Sliding piece, efficiency of, 288. 
 Solid, 30. 
 
 Solids, mensuration of, S2. 
 Specific gravity (see Gravity, 
 
 Specific). 
 Speed (see Velocity). 
 Speed, adjustments o^ 73. 
 
 cones, 100. 
 
 fluctuations of, 241. 
 
 periodic fluctuations of (see 
 Periodic motion). 
 
 uniform, condition of, 258. 
 Spheres, strength 6f, 186. 
 Spiral, 55, 56. 
 Spring, 184. 
 Stability, 156. 
 
 frictional, 176. 
 
 of position, 176. 
 Standard measure of length, 30. 
 
 measure of weight, 116. 
 Starting a machine, 265. 
 Statics, 32. 
 
 principles of, 115. 
 Stifihess, 157, 179. 
 Stopping a machine, 265. 
 Strain, 179. 
 Stream of liquid, friction of, 235. 
 
 hydraulic, mean depth o^ 236. 
 
 varying, 236. 
 Strength, 156, 179, 
 
 coefficients or moduli of, 180. 
 
 proof, 179. 
 
 transverse, 196. 
 
 ultimate, 179. 
 Stress, 143, 179. 
 
 classes of, 144. 
 
 compound internal, 149. 
 
 intensity of, 143. 
 
 internal, 147. 
 
INDEX. 
 
 311 
 
 Stress, moment of, 196. 
 
 shearing, ISO. 
 
 tangential, 144. 
 
 uniiorm, 145. 
 
 varying, 145. 
 Stresses, conjugate, principal, 150. 
 Stretching, resistance to, 184. 
 Structures, 32. 
 
 equilibrium of, 157. 
 
 theory of, 156. 
 Stroke, length of, in linkwork, 104. 
 Struts, 158. 
 Supports, 156. 
 Surface, 30. 
 System of parallel forces, 1^1. 
 
 Tackle, 105. 
 
 connection by, 73, 105. 
 
 efficiency of, 300. 
 Tearing, resistance to, 184. 
 Teeth, arc of contact of, 88. 
 
 dimensions of, 91. , 
 
 efficiency of, 296. 
 
 epicycloidal, 89. - 
 
 involute, for circular wheels, 88, 
 89. 
 
 of mitre or bevel-wheels, 91, 92. 
 
 of non-circular wheels, 92. 
 
 of spur wheels and racks, 86. 
 
 of wheels, 81. 
 
 of wheel and trundle, 90. 
 
 pitch and number of, 81. 
 
 sliding of, 87. 
 
 traced by roUine; curves, 86. 
 Tension, 144, 184. 
 Testing, 182. 
 Thrust, 144. 
 Tie, 158. 
 
 strength of, 184. 
 Time, measure of, 35. 
 Tooth, face of, 81. 
 
 flank of, 81. 
 Torsion (see Wrenching). 
 Trains of mechanism, 73, 111. 
 
 ofwteelwork, 83, 84, 85. 
 Transformation (see Projection). 
 Transformation of energy, 208. 
 Translation or shifting, 47. 
 
 varied, 211, 219. 
 Transverse strength, 196. 
 
 table, 200. 
 Trapezoid, area of, 16. 
 Trapezoidal rule, Merrifleld's, 19, 20. 
 
 common, 21. 
 Triangles, area of, 10, 16. 
 
 solution of plane, 8, 9. 
 
 Trigonometrical rules, 6. 
 
 functions of one angle, 7. 
 
 functions of two angles, 8. 
 Trochoid, 55. 
 Trundle, 90. 
 Truss, 168. 
 
 compound, 169. 
 Trussing, secondary-, 169, 170, 171. 
 Turning (see Botation). 
 Twisting (see Wrenching). 
 
 Unotjents, 252. 
 Uniform motion, 37, 205. 
 
 deviation, 44. 
 
 effort or resistance, effect of, 215. 
 
 motion under balanced forces, 210. 
 
 rotation, 48. 
 
 stress, 145. 
 
 velocity, 36. 
 Universal joint, 104. 
 
 double, 105. 
 
 Valves, 110. 
 
 VeJocities, virtual, 206, 267. 
 
 Velocity, 36, 244. 
 
 angular, 48. 
 
 angular, variation of, 63. 
 
 ratio, 38. 
 
 uniform, 36. 
 
 uniformly-varied, 41. 
 
 varied, 39. 
 
 varied rate of variation of, 43. 
 Virtual velocities, 206, 267. 
 Volume, 30. 
 Volumes, measurement of, 22. 
 
 Wedge (mechanical power), 107. 
 Weight, 116. 
 
 mass in terms of, 212. 
 Wheel and axle, 107. 
 
 and rack, 75. 
 
 and screw, 95. 
 Wheels, bevel, 76, 81. 
 
 circular, in general, 75. 
 
 non-oircular, 77. 
 
 pitch-surfaces, pitch-lines, pitch- 
 points of, 81. 
 
 skew-bevel, 77, 78, 81. 
 
 spur, 81. 
 Wheelwork, train o^ 83. 
 White's tackle, 106. 
 Windlass, differential, 112. 
 Wooley's rule, 22. 
 Work, 206; 243. 
 
 against an oblique force, 246. 
 
 against friction, 251. 
 
312 
 
 INDEX 
 
 Work,against varying resistance, 249, 
 
 250. 
 algebraical expressions for, 246. 
 and energy, general equation o^ 
 
 267. 
 done, and energy exerted, equality 
 
 of, 260. 
 done during retardation, 262. 
 in terms of angular motion, 244. 
 in terms of pressure and volume, 
 
 245. 
 measures of, 243. 
 of acceleration, 252. 
 
 Work, of acceleration, summation of, 
 256. 
 
 of machines, 243. 
 
 rate of, 243. 
 
 represented by an area, 249. 
 
 summary of various kinds of, 258. 
 
 summation of, 247, 248, 256. 
 
 useful and lost, 241, 251. 
 Working point, 242. 
 Working stress, working load, 179. 
 Wrenchmg, resistance to, 187. 
 
 Yakd, standard, 30. 
 
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