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 MACHINERY AND MILLWORK. 
 
 BY 
 
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 CIVIL ENGINEEB; LL.D. TKIN. COLL. DUB. ; F.B.S.S. LOUD. AND BDIN. ; F.E.S.S.A. ; 
 LATE EEGIUS PBOFESSOK OF CIYIL KKGISEEKISQ AND MECHANICS IK TEE UNIVEESITT OF GLASGOW. 
 
 With numerous 
 
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 BT 
 
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 SECRETARY, INST. ESGISEBES AHD 3HIPBDILDBBS IK SCOTLAND. 
 
 LONDON: 
 CHARLES GRIFFIN AND COMPANY, 
 
 EXETER STBEET, STBAND. 
 1887. 
 
 [The Sight of Translation Reserved.]
 
 TJ 
 PRESAGE. 7 
 
 THIS book is divided into three parts: the first treats of the 
 Geometry of Machinery; the second of the Dynamics of Ma- 
 chinery; and the third of the Materials, Strength, and Con- 
 struction of Machinery. 
 
 Under the head of the Geometry of Machinery, machines are 
 considered with reference to the comparative motions only of 
 their moving parts; and rules are given for designing and arrang- 
 ing those parts so as to produce any given comparative motion. 
 
 Considering that the object of such rules is to adjust the dimen- 
 sions of the parts of machines by processes of practical geometry, 
 I have thought it advisable to solve every question by drawing, 
 rather than by calculation, except in a few special cases where 
 calculation is indispensable. 
 
 Many of the graphic rules thus obtained are made more easy 
 and accurate, and some, indeed, are made possible which were not 
 so before, by the aid of new methods of measuring and laying off 
 the lengths of curved lines. 
 
 Two chapters of the first part are devoted to the detailed con- 
 sideration of the movements of single pieces in machines. The 
 remainder of the part relates to Pure Mechanism, as defined and 
 reduced to a system by Professor WILLIS. The order in which 
 the various combinations in mechanism are treated of is different 
 from that adopted by him; but the principles are the same. 
 
 Several problems in mechanism are solved by methods which, 
 so far as I know, have not hitherto been published; and which 
 possess advantages in point of ease or of accuracy. I may specify, 
 in particular, the drawing of rolling curves, and of some kinds of 
 cams; the construction of the figures of teeth of skew-bevel wheels, 
 and of threads of gearing screws, by the help of the normal section; 
 and some improvements in the details of processes for designing 
 intermittent gear, link-motions, and parallel motions. 
 
 Under the head of the Dynamics of Machinery are considered
 
 VI PREFACE. 
 
 the forces exerted and the work done in machines; the means of 
 measuring those quantities by indicators and dynamometers, of 
 determining and balancing . the reactions of moving masses in 
 machines, and of regulating work and speed ; and the efficiency, or 
 proportion in which the useful work is less than the total work, in 
 the different sorts of moving pieces, and in their various combina- 
 tions. 
 
 Considering that a convenient single word is wanted to denote 
 the proportion in which the total work in a machine is greater 
 than the useful work, I have ventured to propose the word COUNTER- 
 EFFICIENCY for that purpose. 
 
 Under the head of the Materials, Strength, and Construction of 
 Machinery are considered, first, the properties of various materials, 
 as affecting their treatment and use in the construction of machines; 
 secondly, the general principles of the strength of materials; thirdly, 
 the special application of those principles to questions relating to 
 the strength and the construction of various parts of machines; 
 and fourthly, the principles of the action of cutting tools. 
 
 Great care has been taken to ascertain the values of the factor 
 of safety and of the working stress in successful examples of actual 
 machinery; and some of the problems respecting the strength of 
 special parts of machines have not been published previously except 
 in scientific journals and in lectures. 
 
 Authorities for facts and information are cited where it is 
 necessary to do so. The following works are so frequently referred 
 to, that it may be desirable to mention them here specially : 
 
 WILLIS On Mechanism, first edition, 1841; second edition, 1870. 
 
 FAIRBAIRN On MiUwork. 
 
 HOLTZAPFFEL On Mechanical Manipulation. 
 
 BUCHANAN On Millwork; edited by TREDGOLD and G. RENNIE, 
 
 with an Essay on Tools by NASMYTH. 
 
 W. J. M. R 
 
 The Sixth Edition has been revised, and additional matter 
 relating to recent improvements added to Appendix. 
 
 W. J. M. 
 GLASGOW, May, 1887.
 
 CONTENTS. 
 
 INTRODUCTION. 
 
 Art Page 
 
 1. Nature and Use of Machinery in 
 
 General, 1 
 
 2. Distinction between the Geometry 
 
 and the Dynamics of Machinery, 1 
 
 Art. 
 
 3. Strength of Machinery, 
 
 4. Construction of Machinery, 
 
 5. Division of the Subject, 
 
 Page 
 
 . 2 
 
 . 2 
 
 2 
 
 PART I. GEOMETRY OF MACHINERY. 
 CHAPTER I. ELEMENTARY RULES IN DESCRIPTIVE GEOMETRY. 
 
 SECTION I. General Explanations: 
 Projection of Points and Lines. 
 
 6. Descriptive Geometry, 3 
 
 7. Projection of a Point, . 3 
 
 8. Position of a Point, . 3 
 
 9. Axis of Projection, . . 4 
 
 10. Rabatment, ... 4 
 
 11. Projections of Lines, . . 4 
 
 12. Drawings of a Machine, . 5 
 
 SECTION II. Traces of Lines and 
 Surfaces. 
 
 13. Trace, .... 5 
 
 14. Traces of a Straight Line, . . 5 
 
 15. Traces of a Plane, ... 6 
 
 SECTION III. Rules Relat ing to Straight 
 Lines. 
 
 16. General Explanations, . . 6 
 
 17. Given, the Traces of a Straight 
 
 Line; to draw its Projections, 6 
 
 18. Given, the Projections of a 
 
 Straight Line; to find its 
 Traces, 7 
 
 19. Given, the Projections of Two 
 
 Points; to measure the distance 
 between them, ... 7 
 
 20. Given, the Projections of a Point, 
 
 and the Projections of a Straight 
 Line through that Point; to lay 
 off a given distance from the 
 Point along the Line, . . 7 
 
 21. Given, the ProjectionsofaStraight 
 
 Line ; to find the Angle which 
 it makes with one of the Planes 
 of Projection, ... 8 
 
 22. Given, the Projections of a Pair 
 
 of Straight Lines which Inter- 
 sect each other; to find the 
 Angle between those Lines, . 8 
 
 22A. Given, the Projections of a 
 Straight Line, and One Trace 
 of a Plane traversing that Line ; 
 to find the Projections of a 
 Straight Line which shall, at 
 a given Point, make a given 
 Angle in the given Plane with 
 the given Straight Line, 
 
 9 
 
 SECTION TV.Jtdles Relating to Planes. 
 
 23. Given, the Projections of Three 
 
 Points; to draw the Traces of 
 a Plane passing through them, 9 
 23A. Given, the Projections of Two 
 Points and of a Straight Line; 
 to draw the Traces of a Plane 
 traversing the Points and 
 parallel to the Line, . . 9 
 
 24. Given, the Traces of a Plane ; to 
 
 find the Angle which it makes 
 with one of the Planes of Pro- 
 jection, 9 
 
 25. Given, the Traces of a Plane; to 
 
 find the Angle which it makes 
 with the Axis of Projection, . 9 
 
 26. Given, the Traces of a Plane; to 
 
 draw the Traces of another 
 Plane which shall be parallel 
 to the given Plane, and at a 
 given perpendicular distance 
 from it in either direction, . 10 
 
 27. Given, the Traces of Two Planes; 
 
 to draw the Projections of 
 their Line of Intersection, . 10 
 
 28. To find the Projections of the 
 
 Point where a Straight Line 
 intersects a Plane, . . .11 
 
 29. Given, the Traces of Two Planes; 
 
 to find the Angle between them, 12 
 
 30. Given, the Traces of a Plane, 
 
 also the Traces of a Straight
 
 CONTENTS. 
 
 Art Page 
 
 Line in that Plane; to draw 
 the Traces of a Plane which 
 shall cut the given Plane in 
 that Line at a given Angle, . 12 
 
 81. Given, the Traces of a Plane and 
 
 the Projections of a Point; to 
 draw the Traces of a Plane 
 Traversing the given Point, 
 and parallel to the given Plane, 13 
 
 82. Given, the Traces of a Plane, and 
 
 one Projection of a Point in 
 that Plane; to find the other 
 Projection of that Point, . 13 
 
 83. Given, the Traces of a Plane, and 
 
 the Projections of a Point ; to 
 draw the Projections of a Per- 
 pendicular let fall from the 
 Point on the Plane, . . 13 
 
 Art. Page 
 
 34. Given, the Projections of a Point, 
 
 and those of a Straight Line ; 
 to draw the Traces of a Plane 
 which shall Traverse the Point, 
 and be perpendicular to the 
 Line, 14 
 
 35. Given, the Projections of a Point 
 
 and of a Straight Line; to 
 draw the Projections of a Per- 
 pendicular let fall from the 
 Point upon the Straight 
 Line, 14 
 
 36. Given, the Projections of Two 
 
 Straight Lines that are neither 
 parallel nor intersecting; to 
 find the Projections of their 
 Common Perpendicular, . 15 
 36A. Projections of a Circle, . . 15 
 
 CHAPTER II. OF THE MOTIONS OF PRIMARY MOVING PIECES IN MACHINES. 
 
 SECTION I. General Explanations. 
 87. Frame; Moving Pieces, Primary 
 
 and Secondary, . . .17 
 38. Bearings, . ... 17 
 
 89. Motions of Primary Moving Pieces, 18 
 
 SECTION II. Straight Motion of 
 Primary Pieces. 
 
 40. Straight Translation, ,. . 18 
 
 41. Resolution and Composition of 
 
 Motions, . . . .18 
 
 42. Relative Motions of Two Moving 
 
 Pieces, 21 
 
 43. Comparative Motion, . . .22 
 
 44. Driving Point and Working Point, 23 
 
 SECTION III. Rotation of Primary 
 Pieces. 
 
 45. Rotation of a Primary Piece, . 24 
 
 46. Speed of Rotation, . . 24 
 
 47. Rotation is Common to all parts 
 
 of the Turning Body, . . 25 
 
 48. Right and Left-handed Rotation, 25 
 
 49. Translation of a Point in a Ro- 
 
 tating Piece, . . . .26 
 
 50. Motion of a Part of a Rotating 
 
 Piece, 27 
 
 51. Rules as to Lengths of Circular 
 
 Arcs, . . 27 
 
 52. Relative Translation of a Pair of 
 
 Points in a Rotating Piece, . 30 
 
 53. Comparative Motion of Points in 
 
 a Rotating Piece, . . .31 
 
 54. Relative and Comparative Trans- 
 
 lation of a Pair of Rigidly 
 Connected Points, . . .32 
 
 55. Components of Velocity of a 
 
 Point in a Rotating Piece 
 Periodical Motion, . . .33 
 
 56. Comparative Motion of Two 
 
 Rotating Pieces, and of Points 
 in them, . . . .35 
 
 SECTION IV. Screw-like Motion of 
 Primary Pieces. 
 
 57. Helical or Screw-like Motion, . 
 
 58. General Figure of a Screw Pitch, 
 
 59. Right-handed and Left-handed 
 
 Screws, 
 
 60. Comparative Motion of a Point 
 
 in a Screw, .... 
 
 61. Path of a Point in a Screw 
 
 Linear Screw or Helix, . 
 
 62. Projection of a Linear Screw, 
 
 63. Development of a Linear Screw, 
 
 64. Radius of Curvature, . . 
 
 65. Normal Pitch, . 
 
 66. Divided Pitch, . 
 
 36 
 36 
 
 37 
 
 38 
 38 
 40 
 41 
 41 
 42 
 
 CHAPTER III. OF THE MOTIONS OF SECONDARY MOVING PIECES. 
 
 67. General Principles, . . . 43 I 72. 
 
 68. Translation of Secondary Moving 
 
 Pieces, 44 73. 
 
 69. Rotation parallel to a Fixed 
 
 Plane Temporary Axis In- 74. 
 
 stantaneous Axis, . . .45 
 
 70. Rotation about a Fixed Point, . 48 75. 
 
 71. Unrestricted Motion of a Rigid Body, 50 i 
 
 Instantaneous Axis of a Rolling 
 
 Body, 51 
 
 Composition of Rotation with 
 
 Translation, . . . .52 
 Rolling of a Cylinder on a Plane 
 
 Trochoid Cycloid, . . 53 
 Rolling of a Plane on a Cylinder; 
 
 Involutes Spirals, . . 53
 
 CONTENTS. 
 
 Art. Page 
 
 76. Composition of Rotations about 
 
 Parallel Axes, . . .54 
 
 77. Rolling of a Cylinder on a 
 
 Cylinder Epitrochoids Epi- 
 cycloids, . . . .56 
 
 78. Curvature of Involutes of Circles, 
 
 Epicycloids, and Cycloids, . 56 
 
 79. To draw Rolled Curves, . . 58 
 
 80. Resolution of Rotation in Gene- 
 
 ral, 63 
 
 Art. 
 
 81. Rotations about Intersecting 
 
 Axes Compounded, 
 
 82. Rolling Cones, ... 
 
 83. Resolution of Helical Motion, 
 
 84. Rolling Hyperboloids, . 
 
 85. Cylinders Rolling Obliquely, 
 
 86. Cones Rolling Obliquely, . 
 
 87. BandsorFlexibleSecondaryPiece 
 
 Cords Belts Chains, 
 
 88. Fluid Secondary Pieces, 
 
 Page 
 
 66 
 68 
 68 
 70 
 73 
 73 
 
 74 
 
 75 
 
 CHAPTER IV. OF ELEMENTARY COMBINATIONS IN MECHANISM. 
 
 SECTION I. Definitions, General Prin- 
 ciples, and Classification. 
 
 89. Elementary Combinations De- 
 
 fined, 77 
 
 90. Line of Connection, . . 77 
 
 91. Comparative Motions of Con 
 
 nected Points and Pieces, 78 
 
 92. Adjustments of Speed, . 80 
 
 93. Train of Mechanism, . 80 
 
 94. Elementary Combinations Classed, 80 
 
 SECTION II. Of Rolling Contact and 
 Pitch Surfaces. 
 
 95. Pitch Surfaces, . . .81 
 
 96. Toothless Wheels, Rollers, 
 
 Toothless Racks, . . .81 
 
 97. Ideal Pitch Surfaces, . . 82 
 
 98. Pitch Line, . . . . 82 
 
 99. General Conditions of Perfect 
 
 Rolling Contact, . . .82 
 
 100. Wheels with Parallel Axes, . 83 
 
 101. Wheel and Rack, ... 84 
 
 102. Circular Wheels in General, . 84 
 
 103. Circular Spur Wheels, . . 85 
 
 104. Circular Wheel and Straight 
 
 Rack, 85 
 
 105. Circular Bevel Wheels, . . 86 
 
 106. Circular Skew Bevel Wheels, . 87 
 
 107. Non-Circular Wheels in General, 92 
 
 108. Elliptic Wheels, ... 95 
 
 109. Lobed Wheels, ... 97 
 
 110. Logarithmic Spiral Sectors or 
 
 Rolling Cams, . . .99 
 
 111. Frictional Gearing, . . .102 
 
 SECTION III. Of the Pitch and Number 
 of the Teeth of Wheels. 
 
 112. Relation between Teeth and 
 
 Pitch Surfaces Nature of the 
 Subject, . 103 
 
 113. Pitch Defined, . 
 
 114. General Principles, 
 
 115. Frequency of Contact Hunt- 
 
 ing-Cog, 
 
 116. Smallest Pinion, 
 
 117. Arithmetical Rules, 
 
 118. Train of Wheelwork, 
 
 103 
 104 
 
 104 
 105 
 105 
 108 
 
 119. Diametral and Radial Pitch, . Ill 
 
 120. Relative Positions of Parallel 
 
 Axes in Wheelwork, . .113 
 
 121. Laying-off Pitch, and Sub- 
 
 division of Pitch-lines, . . 113 
 
 SECTION TV. Sliding Contact; Teeth, 
 Screw-Gearing, and Cams. 
 
 122. General Principle of Sliding 
 
 Contact, . . . .114 
 
 123. Teeth of Wheels and Racks; 
 
 General Principle, . . 115 
 
 124. Teeth Definition of their Parts, 115 
 
 125. Customary Dimensions of Teeth, 116 
 
 126. Teeth for Inside Gearing, . . 117 
 
 127. Common Velocity and Relative 
 
 Velocity of Teeth Approach 
 and Recess Path of Contact, 117 
 
 128. Arc of Contact, . . .119 
 
 129. Obliquity of Action,. . . 119 
 
 130. Teeth of Spur-Wheels and 
 
 Racks, 120 
 
 131. Involute Teeth for Circular 
 
 Wheels, 120 
 
 132. Involute Teeth for Racks, . 125 
 
 133. Peculiar Properties of Involute 
 
 Teeth, 125 
 
 134. Teeth for a given Path of 
 
 Contact, .... 128 
 
 135. Teeth Traced by Rolling Curves, 129 
 
 136. Epicycloidal Teeth in General, . 130 
 
 137. Tracing Epicycloidal Teeth by 
 
 Templets, . . . .131 
 
 138. Straight Flanked Epicycloidal 
 
 Teeth, 133 
 
 139. Epicycloidal Teeth Traced by an 
 
 Uniform Describing Circle^ . 134 
 
 140. Approximate Drawing of Epi- 
 
 cycloidal Teeth, . . .134 
 
 141. Teeth Gearing with Round 
 
 Staves Trundles, Pin- Wheels, 
 and Pin-Racks, . . . 137 
 
 142. Intermittent Gearing, . . 139 
 
 143. Teeth of Non-Circular Wheels, 141 
 
 144. Teeth of Bevel-Wheels, . . 143 
 
 145. Teeth of Skew-Bevel Wheels- 
 
 General Conditions, . . 146
 
 CONTBNTa 
 
 Art Page 
 
 146. Skew-Bevel Teeth Rules, . 147 
 
 147. Transverse Obliquity of Skew- 
 
 Bevel Teeth 152 
 
 148. Skew-Bevel Wheels in Douhle 
 
 Pairs, .... 152 
 
 149. Teeth with Sloping Backs, 152 
 
 150. Stepped Teeth, . . 155 
 
 151. Helical Teeth, ... 156 
 
 152. Screw and Nuk . . 157 
 
 153. Screw Wheel-Work in General, 157 
 
 154. Screw Wheel- Work -Rules for 
 
 Drawing, .... 159 
 
 155. Figures of Threads found by 
 
 Means of Normal Screw-Lines, 163 
 
 156. Figures of Threads designed on 
 
 a Plane Normal to One Axis, 163 
 
 157. Close- Fitting Tangent Screws, 165 
 
 158. Oldham's Coupling, . . .166 
 
 159. Pin and Straight Slot . .167 
 
 160. Cams and Wipers in General, . 170 
 
 161. Cam with Groove and Pin, . 170 
 
 162. Drawing a Cam by Circular Arcs, 173 
 
 163. Many-Coiled Cams; Spiral and 
 
 Conoidal Cams, . . . 174 
 
 164. Wipers and Pallets Escape- 
 
 ments, 175 
 
 SECTION V. Connection by Sands. 
 
 165. Bands and Pulleys Classed, . 179 
 
 166. Principles of Connection by 
 
 Bands, 180 
 
 167. Pulleys with Equal Angular 
 
 Velocities, .... 182 
 
 168. Bands and Pulleys for a Con- 
 
 stant Velocity-Ratio, . . 182 
 
 169. Length of an Endless Band, . 183 
 
 170. Pulleys with Flat Belts, . . 184 
 
 171. Speed Cones, . . . .185 
 
 172. Pulleys for Ropes and Cords, . 187 
 
 173. Guide Pulleys, . . . .188 
 
 174. Straining Pulleys, . . .188 
 
 175. Eccentric and Non - Circular 
 
 Pulleys, . . . .188 
 
 176. Chain Pulleys and Gearing 
 
 Chains, 190 
 
 177. Suspended Pulleys, . . .191 
 
 SECTION VI. Connection by Linkwork. 
 
 178. Definitions, . . .192 
 
 179. Principles of Connection, . 193 
 
 180. Dead Points, . . .193 
 
 181. Coupled Parallel Shafts, . 194 
 
 182. Drag-Link, . . .194 
 
 183. Link for Contrary Rotations, . 196 
 
 184. Linkwork with Reciprocating 
 
 Motion Crank and Beam 
 Crank and Piston-Rod, . . 196 
 
 185. Eccentric . . . .197 
 
 186. Length of Stroke, . . .197 
 
 187. Mean Velocity-Ratio, . . 199 
 
 Art. 
 
 Page 
 199 
 
 188. Extreme Velocity-Ratios, . 
 
 189. Doubling of Oscillations by 
 
 Linkwork 201 
 
 190. Slow Motion by Linkwork, . 202 
 
 191. Hooke's Coupling or Universal 
 
 Joint, 203 
 
 192. Double Hooke's Joint, . . 205 
 
 193. Hooke-and-Oldham Coupling, . 206 
 
 194. Intermittent Linkwork Click 
 
 and Ratchet, . . 206 
 
 195. Silent Click, 
 
 196. Double-acting Click 
 
 197. Fractional Catch, 
 
 198. Slotted Link, . 
 
 199. Band Links, . 
 
 208 
 209 
 211 
 213 
 213 
 
 SECTION VII. Connection by Plies of 
 Cord, or by Reduplication. 
 
 200. General Explanations, . . 214 
 
 201. Velocity- Ratios, . . .215 
 
 202. Ordinary form of Pulley- Blocks, 216 
 
 203. Whites Pulleys, . . .216 
 
 204. Compound Purchases, . . 217 
 
 205. Rope and Space required for a 
 
 Purchase, . . . .218 
 
 206. Obliquely-acting Tackle, . . 218 
 206A. Tiller-Ropes, . . 219 
 
 SECTION VIII. Hydraulic Connection. 
 
 207. General Nature of the Combina- 
 
 tions, 221 
 
 208. Cylinders, Pistons, and Plungers, 221 
 
 209. Comparative Velocitiesof Pistons, 223 
 
 210. Comparative Velocities of Fluid 
 
 Particles, . . . .223 
 
 211. Use of Valves Intermittent 
 
 Hydraulic Connection, . . 224 
 21lA. Flexible Cylinders and Pistons, 226 
 
 SECTION IX. Miscellaneous Principles 
 respecting Trains. 
 
 212. Converging Trains, . . .227 
 
 213. Diverging Trains, . . .227 
 
 214. Train for diminishing Fluctua- 
 
 tions of Speed, . . .228 
 
 SECTION X. Beferencesto Combinations 
 Arranged in Classes. 
 
 215. Object of this Section, . . 229 
 
 216. Directional- Relation Constant 
 
 Velocity-Ratio Constant, . 229 
 
 217. Directional-Relation Constant 
 
 Velocity-Ratio Variable, . 230 
 
 218. Directional-Relation Variable, . 231 
 
 219. Intermittent Connection, . 231 
 
 SECTION XI. Comparative Motion in 
 the "Mechanical Powers." 
 
 220. Classification of the Mechanical 
 
 Powers 231
 
 CONTENTS. 
 
 XI 
 
 Art Page 
 
 221. Lever Wheel and Axle, . . 232 
 
 222. Inclined Plane Wedge, . . 233 
 
 Art. 
 
 223. Screw, .... 
 
 224. Pulley, 234 
 
 CHAPTER V. OF AGGREGATE COMBINATIONS IN MECHANISM. 
 
 SECTION I. General Explanations. 
 
 225. Aggregate Combinations De- 
 
 nned, 235 
 
 226. General Principle ef their 
 
 Action, 235 
 
 227. Aggregate Combinations Ter- 
 
 minating in a Primary Piece, 235 
 
 228. Shifting Trains, . . .235 
 
 229. Methods of Treating Problems 
 
 Respecting Aggregate Com- 
 binations, .... 238 
 
 230. Aggregate Combinations Classed 
 
 according to their Purposes 
 Aggregate Velocities Aggre- 
 gate Paths, . . . .239 
 
 231. Converging Aggregate Com- 
 
 binations, .... 240 
 
 SECTION II. Production of Uniform 
 Aggregate Velocity-Ratios. 
 
 232. Differential Pulley and Windlass, 240 
 
 233. Compound Screws, . . .242 
 
 234. Epicyclic Trains with Uniform 
 
 Action, 243 
 
 SECTION III. Production oj Varying 
 Aggregate Velocity-Ratios. 
 
 235. Reciprocating Endless Screw, . 246 
 
 236. Epicyclic Trains with Periodic 
 
 Action, 246 
 
 236A. Sun-and- Planet Motion, . 246 
 
 237. Eccentric Gearing, . . .247 
 
 ! 238. Aggregate Linkwork in General, 248 
 
 239. Harmonic Motion in Aggregate 
 
 Linkwork, .... 250 
 
 240. Link-Motions for Slide-Valves, 253 
 
 241. Differential Harmonic Motions, 260 
 
 SECTION IV. Production of Curved 
 Aggregate Paths. 
 
 242. Circular Aggregate Paths, . 261 
 
 243. Epitrochoidal Paths, . . 262 
 
 244. Rolled Paths in General, . . 265 
 
 245. Elliptic Paths Traced by 
 
 Rolling, . . . .266 
 
 246. Trammel, . . . .267 
 
 247. Feathering Paddle-Wheels, . 270 
 
 248. Spherical Epitrochoidal Paths 
 
 Z-Crank, . . . .272 
 
 SECTION V. Parallel Motions. 
 
 249. Parallel Motions in General, . 274 
 
 250. Exact Parallel Motions, . . 274 
 
 251. Approximate Grasshopper Par- 
 
 allel Motion, . . . .275 
 
 252. Watt's Parallel Motion General 
 
 Description,/. . . . 275 
 
 253. Watt's Parallel Motion Rules 
 
 for Designing, . . .277 
 
 254. Watt's Parallel Motion Extent 
 
 of Deviation,. . . .281 
 
 255. Tracing Approximate Circular 
 
 Arcs by the Parallel Motion, . 283 
 
 256. Roberts's Parallel Motion, . 285 
 
 ADDENDA TO CHAPTERS IV. AND V. 
 
 Article 142, page 141. 
 Intermittent Gearing Counter- 
 Wheels Geneva Stop, . . 286 
 Article 154, page 161. 
 
 Racks in Screw Gearing, . . 289 
 Article 243, page 265, and Article 
 
 245, page 267. 
 Epitrochoidal Paths, . . .290 
 
 Article 244, page 265. 
 Aggregate Paths traced by Cam- 
 Motions, 291 
 
 Article 251, page 275. 
 Approximate Grasshopper Par- 
 allel Motion, . . . .292 
 Article 143, page 143. 
 Involute Teeth for Elliptic Wheels, 292 
 
 CHAPTER VI. OF ADJUSTMENTS. 
 
 257. Adjustments Defined and Classed, 2 93 
 
 258. Traversing - Gear and Feed- 
 
 Motions in General, . . 293 
 
 SECTION I. Of Engaging, Disengaging, 
 and Reversing-Gear. 
 
 259. General Explanations, . . 294 
 
 260. Clutch, 295 
 
 261. Friction - Clutch Friction - 
 
 Cones Friction - Sectors 
 Friction-Discs, . . .296 
 
 262. Disengagements acting by Roll- 
 
 ing Contact, . . . .297 
 
 263. Disengagements and Reversing- 
 
 Gear acting by Sliding Con- 
 tact, 298 
 
 264. Disengagements and Reversing- 
 
 Gear by Bands, . . . 299 
 
 265. Disengagements and Reversing- 
 
 Gear acting by Linkwork, . 299 
 
 266. Disengagements acting by Hy- 
 
 draulic Connection Valves, . 301 
 
 267. Principles of the Action of 
 
 Valves, . . . .301 
 
 268. Periodical Motion of Slide- 
 
 Valves. . 306
 
 Xll 
 
 CONTENTS. 
 
 SECTION II. Of Adjustments for 
 
 Changing Speed and Stroke. 
 Art Page 
 
 269. General Explanations, . . 310 
 
 270. Changing Speed by Friction 
 
 Wheels, . . . .310 
 
 271. Changing Speed by Toothed 
 
 Wheels, . . . .311 
 
 Art Page 
 
 272. Changing Speed by Bands and 
 
 Pulleys, . . . .312 
 
 273. Changing Stroke in Linkwork, 312 
 
 274. Changing Speed with Hydraulic 
 
 connection, .... 313 
 Addendum to Article 267, page 306. 
 Slide Valves, . . . .314 
 
 PART II. DYNAMICS OF MACHINERY. 
 CHAPTER I. SUMMARY OF GENERAL PRINCIPLES. 
 
 275. Nature and Division of the Sub- 
 
 ject, 315 
 
 276. Forces Action and Re-action, 316 
 
 277. Forces, how Determined and 
 
 Expressed, .... 317 
 277A. Measures of Force and Mass, 318 
 
 278. Resultant and Component 
 
 Forces their Magnitude, . 319 
 
 279. Couples, . . . .321 
 
 280. Parallel Forces, . . .322 
 
 281. Specific Gravity Heaviness, 
 
 Density, Bulkiness, . . 325 
 
 282. Centre of Gravity Moment of 
 
 Weight, . . . .328 
 
 283. Centre of Pressure, . . 329 
 
 284. Centre of Buoyancy, . . 329 
 
 285. Resultant of a Distributed 
 
 Force, 329 
 
 286. Intensity of Pressure, . . 329 
 
 287. Principles Relating to Varied 
 
 Motion, .... 329 
 
 288. Deviated Motion and Centri- 
 
 fugal Force, .... 330 
 288A. Falling Bodies, . . .330 
 
 Supplement to Chapter I. 
 Rules for the Mensuration of Figures and finding of Centres of Magnitude. 
 
 289. To Measure any Plane Area, . 331 
 
 290. To Measure the Volume of any 
 
 Solid, 333 
 
 291. To Measure the Length of any 
 
 Curve, 333 
 
 292. Centre of Magnitude, . . 334 
 
 293. Centre of a Plane Area, . 
 
 294. Centre of a Volume, 
 
 295. Centre of Magnitude of 
 
 Curved Line, 
 
 296. Special Figures, 
 
 334 
 336 
 
 336 
 
 CHAPTER II. OF THE PERFORMANCE OF WORK BY MACHINES. 
 
 SECTION I. Of Resistance and Work. 
 
 297. Action of a Machine, . . 338 
 
 298. Work, 338 
 
 299. Rate of Work, . . .339 
 
 300. Velocity, . . . .339 
 
 301. Work in Terms of Angular 
 
 Motion, . . . .340 
 
 302. Work in Terms of Pressure and 
 
 Volume, . . . .341 
 
 303. Algebraical Expressions for 
 
 Work 342 
 
 804. Work against an Oblique 
 
 Force, 343 
 
 305. Summation of Quantities of 
 
 Work, 343 
 
 306. Representation of Work by an 
 
 Area, . . . . .345 
 
 307. Work against varying Resist- 
 
 ance. 346 
 
 308. Useful Work and Lost Work, . 347 
 
 309. Friction, . . . .348 
 
 310. Unguents, .... 350 
 310A. Friction of a Band, . . 351 
 
 311. Work Performed against Fric- 
 
 tion, 353 
 
 312. Work of Acceleration, . . 354 
 
 313. Summation of Work of Accele- 
 
 rationMoment of Inertia, . 357 
 
 314. Centre of Percussion Equiva- 
 
 lent Simple Pendulum Equi- 
 valent Fly-wheel, . . .361 
 
 315. Reduced Inertia, . . .362 
 
 316. Summary of Various Kinds of 
 
 Work, 362 
 
 SECTION II. Of Deviating and 
 Centrifugal Force. 
 
 317. Deviating Force of a Single 
 
 Body, 363 
 
 318. Centrifugal Force, . . .364 
 
 319. Revolving Pendulum, . . 364 
 
 320. Deviating Force in Terms of 
 
 Angular Velocity, . . . 364 
 
 321. Resultant Centrifugal Force, . 365 
 
 322. Centrifugal Couple Permanent 
 
 Axis, 365
 
 CONTENTS. 
 
 Xlll 
 
 SECTION III. Of Effort, Energy, 
 
 Power, and Efficiency. 
 Art Page 
 
 323. Effort, 368 
 
 324. Condition of Uniform Speed, . 369 
 
 325. Energy Potential Energy, . 370 
 
 326. Equality of Energy Exerted and 
 
 Work Performed, . . .370 
 
 327. Various Factors of Energy, . 371 
 
 828. Energy Exerted in producing 
 
 Acceleration, .... 371 
 
 829. Accelerating Effort, . . .371 
 
 830. Work during Retardation 
 
 Energy Stored and Restored, . 373 
 
 331. Actual Energy, . . . 373 
 
 332. Reciprocating Force, . . 374 
 
 333. Periodical Motion, . . . 375 
 
 334. Starting and Stopping, . . 376 
 
 335. Efficiency, . . . .376 
 
 Art. Page 
 
 336. Power and Effect Horse-power, 378 
 
 337. General Equation, . . . 378 
 
 338. Principle of Virtual Velocities, . 378 
 
 339. Forces in the Mechanical Powers, 
 
 neglecting Friction Purchase, 379 
 
 SECTION IV. Of Dynamometers. 
 
 340. Dynamometers, . . . 382 
 
 341. Prony's Friction Dynamometer, 383 
 
 342. Morin's Traction Dynamometer, 383 
 
 343. Morin's Rotatory Dynamometer, 386 
 
 344. Torsion Dynamometer, . . 387 
 
 345. Elasticity of Spiral Springs, . 389 
 
 346. Steam-Engine Indicator, . . 390 
 
 347. Integrating Dynamometers, . 394 
 
 348. Measurement of Friction, . . 395 
 Addendum to Article 309, page 348. 
 
 Friction of Pistons and Plungers, 399 
 
 CHAPTER III. OF REGULATING APPARATUS. 
 
 849. Regulating Apparatus Classed 
 
 Brake Fly Governor, . 400 
 
 SECTION I. Of Brakes. 
 
 850. Brakes Defined and Classed, . 400 
 
 351. Action of Brakes in General, . 400 
 
 352. Block-Brakes,. . . 401 
 
 353. Brakes of Carriages, . 402 
 
 354. Flexible Brakes, . . 402 
 
 355. Pump-Brakes, . . 404 
 
 356. Fan-Brakes, . .406 
 
 SECTION II. Of Fly-Wheels. 
 
 357. Periodical Fluctuations of Speed, 407 
 
 358. Fly-Wheels, . . . .409 
 
 CHAPTER IV. OF THE EFFICIENCY 
 COMBINATIONS, AND 
 870. Nature and Division of the 
 
 Subject, . . . .422 
 
 SECTION I. Efficiency and Counter- 
 Efficiency of Primary Pieces. 
 371. Efficiency of Primary Pieces in 
 
 General, .... 423 
 871 A. Conditions Assumed to be 
 
 Fulfilled, . . . .424 
 872. Efficiency of a Straight-Sliding 
 
 Piece, 426 
 
 373. Efficiency of an Axle, . . 427 
 
 374. Axles of Pulleys connected by 
 
 Bands, 431 
 
 375. Efficiency of a Screw, . . 433 
 
 376. Efficiency of Long Lines of 
 
 Horizontal Shafting, . . 433 
 
 SECTION II. Efficiency and Counter- 
 Efficiency of Modes of Connection in 
 Mechanism. 
 
 877. Efficiency of Modes of Con- 
 nection in General, . . 436 
 
 SECTION III. Of Governors. 
 
 359. Regulator, . . . .410 
 
 360. Pendulum-Governors, . . 411 
 
 361. Loaded Pendulum-Governor, . 413 
 
 362. Parabolic Pendulum - Gover- 
 
 nors, 414 
 
 363. Isochronous Gravity-Governor, 415 
 
 364. Fluctuations^ of Isochronous 
 
 Governors, .... 418 
 
 365. Balanced, or Spring Governors, 418 
 
 366. Disengagement Governors, . 419 
 
 367. Differentia] Governors, . . 420 
 
 368. Pump-Governors, . . .421 
 
 369. Fan-Governors, . . . 421 
 
 AND COUNTER-EFFICIENCY OF PIECES, 
 TRAINS IN MECHANISM. 
 
 378. Efficiency of Rolling Contact, . 436 
 
 379. Efficiency of Sliding Contact in 
 
 General, ... 437 
 
 380. Efficiency of Teeth, . . 438 
 
 381. Efficiency of Bands, . 440 
 
 382. Efficiency of Linkwork, . 442 
 
 443 
 
 383. Efficiency of Blocks and Tackle 
 
 384. Efficiency of Connection by 
 
 Means of a Fluid, . . .444 
 Addendum to Article 343, page 
 
 386. 
 Rotatory Dynamometers Epi- 
 
 cyclic-Train Dynamometers, . 446 
 Addenda to Article 381, page 440. 
 I. Use of Relieving Rollers between 
 
 Pulleys, . . . .447 
 II. Efficiency of Telody nan iio Trans- 
 mission, .... 447 
 Addendum to Article 382, page 
 
 442. 
 
 Effect of Obliquity of a Con- 
 necting-Rod on Friction, . 449
 
 XIV 
 
 CONTENTS. 
 
 PAET III. MATERIALS, CONSTRUCTION, AND STRENGTH OF 
 MACHINERY. 
 
 CHAPTER L OF MATERIALS USED IN MACHINERY. 
 
 Art. 
 
 385. General Explanations, 
 
 Page 
 . 450 
 
 SECTION I. Of Iron and Steel. 
 
 450 
 451 
 451 
 453 
 453 
 455 
 457 
 
 -886. Kinds of Iron and Steel, 
 
 387. Impurities of Iron, . 
 
 388. Cast Iron, 
 
 389. Strength of Cast Iron, 
 
 390. Castings for Machinery, 
 
 391. Wrought or Malleable Iron, 
 
 392. Steel and Steely Iron, . 
 
 393. Strength of Wrought Iron and 
 
 Steei 459 
 
 394. Preservation of Iron, . . 460 
 
 SECTION II. Of Various Metals and 
 Alloys. 
 
 395. Zinc Tin Lead Copper, . 461 
 
 396. Bronze and Brass, . . . 462 
 
 397. Other Alloys, . . . .463 
 
 SECTION IIL Of some Stony Materials. 
 
 398. Stone Bearings for Shafts, . 464 
 
 SECTION IV. Of Wood and other 
 
 Organic Materials. 
 Art Page 
 
 399. Structure of Wood, . . . 464 
 
 400. Classification of Wood, . . 466 
 
 401. Appearance of Good Timber, . 467 
 
 402. Examples of Pine Wood, . . 468 
 
 403. Examples of Leaf Wood with 
 
 Large Rays, . . . .468 
 
 404. Examples of Leaf Wood without 
 
 Large Rays, .... 469 
 
 405. Examples of Leaf Wood without 
 
 Large Rays continued, . 469 
 
 406. Seasoning, .... 470 
 
 407. Durability, Decay, and Preserva- 
 
 tion of Wood, . . . 471 
 
 408. Strength of Timber, . . 471 
 
 409. Use of Wood in Machinery, . 472 
 409 A. Pasteboard, . . . .474 
 
 410. Organic Materials for Bands 
 
 Leather, Gutta-Percha, Indian 
 Rubber, Cotton, Flax, &c., . 474 
 Addendum to Note, Article 309, . 476 
 
 GENERAL TABLES OF THE STRENGTH OF MATERIALS. 
 L Table of the Resistance of 
 
 Materials to Stretching and 
 Tearing by a Direct Pull, . 477 
 II. Table of the Resistance of 
 Materials to Shearing and 
 Distortion, . . . 479 
 III. Table of the Resistance of 
 Materials to Crushing by a 
 Direct Thrust, . . .479 
 
 IV. Table of the Resistance of 
 
 Materials to Breaking across, 480 
 V. Miscellaneous Supplementary 
 
 Table, . . . .482 
 VI. Supplementary Table for 
 
 Wrought Iron and Steel, . 482 
 VII. Resilience of Iron and Steel, . 485 
 VIII. Supplementary Table for Cast 
 
 Iron, . 486 
 
 CHAPTER IL PRINCIPLES AND RULES RELATING TO STRENGTH OF MATERIALS. 
 
 411. Object of this Chapter, . . 487 
 
 SECTION L Of Strength and Stiffness 
 in General. 
 
 412. Load, Stress, Strain, Strength, 487 
 
 413. Co-efficients or Moduli of 
 
 Strength, ... 488 
 
 414. Factors of Safety, . . 488 
 
 415. Proof or Testing, . . 490 
 
 416. Stiffness or Rigidity, Pliability, 
 
 their Moduli or Co-efficients, 491 
 
 417. Elasticity of a Solid, . 492 
 
 418. Resilience or Spring, . 492 
 
 419. Heights or Lengths of Moduli 
 
 of Stiffness and Strength, . 492 
 
 SECTION II. Of Resistance to Direct 
 Tension. 
 
 420. Strength, Stiffness, and Re- 
 
 silience of a Tie, . . . 493 
 
 421. Thin Cylindrical and Spherical 
 
 Shells, 494 
 
 422. Thick Hollow Cylinders and 
 
 Spheres, .... 495 
 
 SECTION IIL Of Resistance to Distor- 
 tion and Shearing. 
 
 423. Distortion and Shearing Stress 
 
 in General, .... 496 
 
 424. Strength of Fastenings and 
 
 Joint-Pins, .... 497
 
 CONTENTS. 
 
 XV 
 
 Art Page 
 
 425. Rivets, 498 
 
 426. Pins, Keys, Wedges, Gibs, and 
 
 Cottars, . . . .499 
 
 427. Bolts and Screws, . . .499 
 
 SECTION IV. Of Resistance to Twist- 
 ing and Wrenching. 
 
 428. Twisting or Torsion in General, 500 
 
 429. Strength of a Cylindrical Shaft, 500 
 
 430. Angle of Torsion, . . .502 
 
 431. Resilience of a Cylindrical Shaft, 504 
 
 432. Shafts not Circular in Section, . 504 
 
 SECTION V. Of Resistance to Bending 
 and Cross-Breaking. 
 
 433. Resistance to Bending in 
 
 General, . . . .504 
 
 434. Calculation of Shearing Loads 
 
 and Bending Moments, . . 505 
 
 Art. Page 
 
 435. Examples, . . . .509 
 
 436. Bending Moments produced by 
 
 Longitudinal and Oblique 
 Forces, . . . 510 
 
 437. Moment of Stress Transverse 
 
 Strength, . . . .510 
 
 438. Longitudinal Sections of Uni- 
 
 form Strength, . . . 517 
 
 439. Deflection of Beams, . . 517 
 
 440. Beam fixed at the Ends, . . 521 
 
 441. Resilience of a Beam, . . 521 
 
 SECTION VI. Of Resistance to Thrust 
 or Pressure. 
 
 442. Resistance to Compression and 
 
 Direct Crushing, . .522 
 
 443. Crushing by Cross-Breaking, . 524 
 
 444. Collapsing of Tubes, . . 525 
 
 CHAPTER III.- 
 
 -C>F SPECIAL PRINCIPLES RELATING TO STRENGTH AND 
 STIFFNESS IN MACHINES. 
 
 445. Subjects of this Chapter, .527 
 
 SECTION I. Summary of Principles. 
 
 446. Load in Machines, . . . 527 
 
 447. Straining Actions computed from 
 
 Power, . . 527 
 
 448. Alternate Strains, . 529 
 
 449. Straining Effects of Re- ction 529 
 
 450. Framework, . . 530 
 
 451. Stiffness and Pliability, 531 
 
 452. Compound Stress, . 532 
 
 SECTION II. Special Rules as to Bands, 
 Rods, and Links. 
 
 453. Belts and Cords at Moderate 
 
 Speeds, . . . .532 
 
 454. Allowance for Centrifugal Ten- 
 
 sion, 532 
 
 455. Wire Ropes, . . . .533 
 
 456. Deflection and Length of Bands, 534 
 
 457. Chains, 535 
 
 458. Rods or Links for Tension, . 535 
 
 459. Rods or Links for Reciprocating 
 
 Stress, 537 
 
 SECTION III. Special Rules relating 
 to Axles and Shafts. 
 
 460. General Explanations as to 
 
 Shafts, Axles, and Journals, 540 
 
 461. Gudgeons or End-Journals, 541 
 
 462. Bearing Axles, . . 543 
 
 463. Neck-Journals, . . 544 
 
 464. Shafts under Torsion, . 544 
 
 465. Span between Bearings of Shafts, 545 
 
 466. Shafts under Combined Bending 
 
 and Twisting Actions, . . 547 
 
 467. Centrifugal Whirling of Shafts, 549 
 
 468. Dimensions of Couplings, . 552 
 
 469. Bushes and Plumber-Blocks, 552 
 
 SECTION IV. Special Rules relating to 
 Pulleys, Wheels, Teeth, and Levers. 
 
 470. Teeth and Rims of Wheels, and 
 
 Dimensions depending on them, 553 
 
 471. Boss and Arms of a Wheel or 
 
 Pulley, 554 " 
 
 472. Centrifugal Tension in Wheels 
 
 and Pulleys, .... 556 
 
 473. Arms of Vertical Water- Wheels, 556 
 
 474. Braced Wheels, . . .557 
 
 475. Levers, Beams, and Cranks, . 557 
 
 CHAPTER IV. ON THE PRINCIPLES OF THE ACTION OF CUTTING TOOLS. 
 
 476. General Explanations, . . 559 
 
 477. Characteristics of Cutting Tools 
 
 in General, .... 559 
 
 478. Classification of Cutting Tools 
 
 Shearing Paring Scrap- 
 ing, 560 
 
 479. Shearing and Punching Tools, . 560 
 
 480. Paring and Scraping Tools in 
 
 General, .... 562 
 
 481. Cutting Angles of Tools, . . 566 
 
 482. Speed of Cutting Tools. . . 567 
 
 483. Resistance and Work of Paring 
 
 Tools, 567 
 
 484. Combinations of Cutting Tools, 568
 
 XVI 
 
 CONTENTS. 
 
 Art. Page 
 
 485. Motions of Machine Tools in 
 
 General, . . . .568 
 
 486. Making Ruled Surfaces Plan- 
 
 ing Slotting Shaping, . 569 
 
 487. Scraping Plane Surfaces, . . 571 
 
 Art. Page 
 
 488. Making Surfaces of Revolution 
 
 Turning Drilling Bor- 
 ing, 572 
 
 489. Screw-Cutting, . . ,574 
 
 490. Wheel-Cutting, . . .575 
 
 APPENDIX. 
 
 To Article 263, page 298 (Clasp-Nut), 576 
 
 To Table V., page 482 (Strength and Elasticity of Silk and Flax), ... 576 
 
 To Article 465, page 547, and Article 466, page 549 (Braced Shaft), . . 576 
 
 Additional Authorities, 576 
 
 To Article 262, page 297 (Disengagements), 577 
 
 To Section VIIL page 221 (Accumulator and Hydraulic Ram), . . . 577 
 
 To Article 353, page 402 (Continuous Brakes), 578 
 
 To Article 392, page 457 (Steel), 578 
 
 To Article 397, page 463 (Phosphor Bronze), 578 
 
 Manganese Bronze, &*& 
 
 Coefficients of Friction in Railway Wheels, 578 
 
 Marine Engine Governors, 579 
 
 Feed Water Heaters, . 679 
 
 Fuel Feeders, 579 
 
 Preservation of Timber, 680 
 
 Ventilating Fans for Mines, 580 
 
 Hydraulic Machinery, . 680 
 
 Differential Screw-Nut, 581 
 
 Strength of Crank Shafts, 681 
 
 Valve Gear Motions, 582 
 
 Cable Tramways, 382 
 
 Strength of Wire Ropes, 584 
 
 Strength of Corrugated Flues, 584 
 
 Strength of Shafts for Screw Steamers, 584 
 
 Fly- Wheels, 585 
 
 Strength of Leather Belts, 585 
 
 Strength of Wires, 585 
 
 Strength of Lead Pipes, 585 
 
 Comparative Table of British and French Weights and Measures, . . . 586 
 
 IKDEX, 589
 
 A MAOTAL OF MACHINEBY AND MILLWOM. 
 
 INTKODUCTION 
 
 ART. 1. Nature and Use of machinery *n General. The use of 
 
 machinery is to transmit and modify motion and force. The parts of 
 which it consists may be distinguished into two principal divisions, 
 the Mechanism, or moving parts; and the Frame, being the structure 
 which supports the pieces of the mechanism, and to a certain extent 
 determines the nature of their motions. In the action of a machine 
 the following three things take place : First, Sonfe natural source 
 of energy communicates motion and force to a part of the mechan- 
 ism called the Prime Mover; Secondly, The motion and force are 
 transmitted from the prime mover through the train of mechanism 
 to the working piece; and during that transmission the motion and 
 force are modified in amount and in direction, so as to be rendered 
 suitable for the purpose to which they are to be applied; and, 
 Thirdly, The working piece, by means of its motion, or of its motion 
 and force combined, accomplishes some useful purpose. 
 
 2. Distinction between the Geometry and the Dynamics of ma- 
 chinery. The modification of motion in machinery depends on the 
 figures and arrangement of the moving pieces, and the way in 
 which they are connected with the frame and with each other; and 
 almost all questions respecting it can be solved by the application 
 of geometrical principles alone. The modification offeree depends 
 on the modification of motion; and those two phenomena always 
 take place together; but in solving questions relating to the modi- 
 fication of force, the principles of dynamics have to be applied in 
 addition to those of geometry. Hence, in treating of the art of 
 designing machinery, arises a division into two departments, the 
 "Geometry of Machinery " or "Science of Pwre Mechanism" (to use 
 a term introduced by Professor Willis), which shows how the figure, 
 arrangement, and mode of connection of the pieces of a machine 
 are to be adapted to the modification of motion which they are to 
 
 B
 
 2 INTRODUCTION. 
 
 produce; and the "Dynamics of Machinery" which shows what 
 modifications of force accompany given modifications of motion, 
 and what modifications of motion are required in order to produce 
 given modifications of force. 
 
 3. Strength of Machinery. In order that a machine may be fit 
 for use, every part, both of the machinery and of the framework, 
 must be capable of bearing the utmost straining action which can 
 be exerted upon it during the working of the machine, without any 
 risk of being broken or overstrained ; and the dimensions required 
 for that purpose are to be determined by the proper application of 
 the principles of the strength of materials. 
 
 4. The Art of the Construction of Machinery consists of three de- 
 partments, the selecting and obtaining of suitable materials for the 
 parts of the mechanism and framework; the shaping of those parts 
 to the proper figures and dimensions by means of suitable tools; 
 and the fitting-up of the machine, by putting its parts together. 
 
 5. Division of the Subject. For the reasons explained in the 
 preceding Articles, the subjects of this work are treated of under 
 four principal heads, Geometry of Machinery, or Pure Mechanism; 
 Dynamics of Machinery; Materials, Construction, and Strength of 
 Machinery.
 
 PAET I. 
 
 GEOMETKY OF MACHINERY. 
 
 CHAPTER L 
 
 ELEMENTARY RULES IN DESCRIPTIVE GEOMETRY. 
 
 SECTION I. General Explanations Projection of Points and Lines. 
 
 6. Descriptive Geometry is the art of representing solid figures 
 upon a plane surface. In the present chapter are given some 
 general elementary rules in that art, whose application is of 
 very frequent occurrence in designing mechanism. The more 
 special and complex rules will be given in the ensuing chapters, in 
 treating of the particular kinds of mechanism to which those rules 
 belong.* 
 
 7. By the Projection of a Point upon a given plane is meant 
 the foot of a perpendicular 
 
 let fall from the point on 
 the plane. For example, 
 in fig. 1, X Z Z X repre- 
 sents a plane (called a 
 plane of projection), A a 
 point, and A B a perpen- 
 dicular let fall from the 
 point on the plane; the 
 foot, B, of that perpen- 
 dicular is the projection 
 of the point A on the 
 plane X Z Z X. 
 
 
 Fig. 1. 
 
 8. The Position of a 
 
 Point is completely deter- 
 mined when its projections upon two planes not parallel to each 
 other are known. In descriptive geometry a pair of planes of 
 projection at right angles to each other are used; and in general 
 one of these is vertical and the other horizontal. Thus, in fig. 1, 
 
 * For complete information on the subject of descriptive geometry, refer- 
 ence may be made to the works of Monge and Hachette in French, and of 
 Dr. Woolley in English.
 
 GEOMETRY OP MACHINERY. 
 
 X Z Z X is the vertical plane of projection, and X Y Y X the 
 horizontal plane of projection; B is the vertical projection, and C 
 the horizontal projection of the point A; and those two projections 
 eompletely determine the position of the point A; for no other 
 point can have the same pair of projections. 
 
 9. The Axis of Projection is the line X X, in which the two 
 planes of projection cut each other. 
 
 10. Rabatment. When the two projections of an object are 
 shown in one drawing, it is convenient to represent to the mind 
 that the following process has been performed : Suppose that the 
 vertical plane of projection is hinged to the horizontal plane at the 
 axis X X, and that after the projection of the object on the vertical 
 plane has been made, that plane is turned about that axis until it 
 lies flat in the position X z z X, so as to be continuous with the 
 horizontal plane : thus bringing down the projection B to 6. This 
 process is called the rabatment of the vertical plane upon the 
 horizontal plane (to use a term borrowed from the French 
 "rabattement" by Dr. Woolley). The two points C and b are in 
 one straight line perpendicular to X X. The process of rabatment 
 may be conceived also to be performed upon a plane in any position 
 when a figure contained in that plane is shown in its true dimen- 
 sions on one of the planes of projection. 
 
 11. Projections of Lines. The projection of a line is a line con- 
 taining the projections of all the points of the projected line. The 
 projection of a straight line perpendicular to the plane of projection 
 is a point; for example, the projection on the vertical plane, 
 X Z Z X (fig. 1 ), of the straight line A B, perpendicular to that 
 plane, is the point B. The projection of a straight line in any 
 other position relatively to the plane of projection is a straight 
 line. If the projected line is parallel to the plane of projection, 
 its projection is parallel and equal to the projected line itself; thus 
 the projection on the horizontal plane, X Y Y X, of the horizontal 
 straight line A B, is the parallel and equal line C D. If the pro- 
 jected line is oblique to the plane of projection, the projection 
 is shorter than the original line. 
 
 The projections, on the same plane, of parallel and equal straight 
 lines are parallel and equal. The projections, on the same plane, 
 of parallel lines bearing given proportions to each other are 
 parallel lines bearing the same proportions to each other. When 
 the plane of a plane curved line is perpendicular to a plane of 
 projection, the projection of the curve on this plane is a straight 
 line, being the intersection of the plane of the curve with the plane 
 of projection. When the plane of the projected curve is parallel 
 to a plane of projection, the projection of the curve on this plane 
 is similar and equal to the original curve. In all other cases, it 
 follows from the preservation of the proportions of a set of parallel
 
 PROJECTIONS AND TRACES OF FIGURES. 5 
 
 ordinates amongst their pi-ejections, that the projections of a plane 
 curve of a given algebraical order are curves of the same algebraical 
 order. The projections of a circle are ellipses; the projections of a 
 parabola of a given order are parabolas of the same order. The 
 projections of a straight tangent to a plane curve are straight 
 tangents to the projections of that curve. The projections of a 
 point of contrary flexure in a plane curve are points of contrary 
 flexure in its projections. 
 
 12. Drawings of a machine. A third plane of projection, per- 
 pendicular to the first two, is often employed, not as being 
 mathematically necessary, but as being more convenient for the 
 representation of certain lines. Thus, for example, the drawings 
 of a machine usually consist of three projections on three planes at 
 right angles to each other; one horizontal (the plan), and the other 
 two vertical (the elevations). Any two of those projections are 
 mathematically sufficient to show the whole dimensions and figure 
 of the machine ; and from any two the third can be constructed ; 
 but it is convenient, for purposes of measurement, calculation, and 
 construction, to have the whole three projections. 
 
 In the application of the rules about to be stated in the sequel 
 of this Section, the two planes of projection may be held to repre- 
 sent any two of the three views of a machine ; 4nd the axis of 
 projection will then have the directions stated in the following 
 table : 
 
 Views Represented by the Planes of Direction of the Aria 
 
 Projection. of Projection. 
 
 Longitudinal Elevation and Plan, Longitudinal 
 
 Longitudinal and Transverse Elevations,.... Vertical. 
 Plan and Transverse Elevation, Transverse. 
 
 Projections of figures upon planes oblique to the principal planes 
 of projection may be used for special purposes. 
 
 SECTION II. Traces of Lines and Surfaces. 
 
 13. By a Trace is meant the intersection of a line with a sur- 
 face, or of one surface with another. The trace of a line upon 
 a surface is a point; the trace of one surface upon another is a 
 line. 
 
 In descriptive geometry the term traces is specially employed, 
 when not otherwise specified, to denote the intersections of a line 
 or surface with the planes of projection. 
 
 14. Trace* of a straight Line. The position of a straight line is 
 completely determined when its traces are known. For example, 
 *,he straight line A C, in fig. 2, has its position completely deter- 
 mined by its traces, A and 0, being the points where it cuts the
 
 GEOMETRY OF MACHINERY. 
 
 two planes of projection. The rabatment of the trace C is repre- 
 sented by c. 
 
 A straight line parallel to one of the planes of projection has 
 
 only one trace, being the 
 point where it cuts the 
 other plane of projec- 
 tion. 
 
 A straight line paral- 
 lel to the axis of projec- 
 tion has no traces. 
 
 15. The Traces of a 
 Plane are straight lines 
 which (unless they are 
 both parallel to the axis 
 of projection) meet that 
 axis in one point. The 
 position of a plane is 
 completely determined 
 when its traces are 
 
 known. For example, the plane A B C, in fig. 2, has its position 
 completely determined by its traces, B A and B C. 
 
 A plane perpendicular to one of the planes of projection has its 
 trace on the other plane of projection perpendicular to the axis of 
 projection. A plane perpendicular to both planes of projection 
 has for its traces two lines perpendicular to the axis. Thus, in 
 fig. 1, page 3, the traces of the plane A B C D are D C and D B, 
 both perpendicular to X X. 
 
 A plane parallel to one of the planes of projection has a trace on 
 the other plane of projection only, being a straight line parallel 
 to X X. 
 
 If a plane traverses a straight line, the traces of the plane 
 traverse the traces of the line. 
 
 Fig. 2. 
 
 SECTION III. Rules Relating to Straight Lines. 
 
 16. General Explanations. In each of the figures illustrating 
 the following rules the axis of projection is represented by X X ; 
 and in general the part of the figure above that line represents 
 the rabatment of the vertical plane of projection, and the part 
 below, the horizontal plane of projection. The projections of 
 points on the horizontal plane are in general marked with 
 capital letters, and the projections on the vertical plane with small 
 letters. 
 
 17. Given (in fig. 3), the Traces, A, b, of a Straight Line, to 
 
 Draw its Projections. From A and b let fall A a and b B perpen- 
 dicular to X X. Then a will be the vertical projection of the
 
 RULES RELATING TO STRAIGHT LINES. 
 
 Fig. 3. 
 
 trace A, and B the horizontal projection of the trace b. Join a b, 
 A B; these will be the projections required. 
 
 (It may here be remarked, that a A and a b are the traces of a 
 plane traversing the given line, and perpendicular to the vertical 
 plane of projection, and that B A and B b are the traces of a plane 
 traversing the given line, and perpendicular to the horizontal plane 
 of projection.) 
 
 18. Given (in fig. 3), the Projections, A B, a b, of a Straight 
 
 Line, to Find its Traces. From a and B, where the given projections 
 meet the axis, draw a A and 
 B b perpendicular to X X, cut- 
 ting the given projections in A 
 and b respectively. These points 
 will be the required traces. 
 
 19. Given, the Projections of 
 two Points, A, a, B, b (fig. 4), 
 to measure the Distance between 
 them. Join a b, A B; these will 
 be the projections of the straight 
 line to be measured. Through 
 either end of either of those 
 
 projections (as 6) draw d b e parallel to X X ; through the other end, 
 a, of the same projection, draw a d perpendicular to X X, cutting 
 d b e in d' } make d e = the other projection, A B; join a e; this 
 will be the length required. 
 
 The same operation may be performed on the other plane of 
 projection. 
 
 20. Given (in fig. 4), the Projections, A, a, of a Point, and the 
 Projections, A B, a b, of a Straight 
 
 Line through that Point, to Lay off 
 a given Distance from the Point 
 
 along the Line. In any con- 
 venient position, draw a straight 
 line, B b, perpendicular to X X, 
 meeting the projections of the 
 given straight line in two points, 
 B, b, which are the projections of 
 one point; then perform the con- 
 struction described in Article 19, 
 so as to find a e. From the point 
 a, in the line a e, lay off the given 
 distance, a f. Through f draw 
 f h parallel to X X, cutting 
 a b in g; a g will be one of 
 the projections of the given dis- 
 tance. Then draw g G perpen- 
 
 d
 
 O GEOMETRY OF MACHINERY. 
 
 dicular to X X, cutting A B in G; A G will be the other projection 
 of the given distance. 
 
 Another method of finding G is to lay off A G = hf. 
 
 21. Given (in fig. 4), the Projections, a b, A B, of a Straight Line, 
 to Find the Angle which it makes with One of the Planes of Pro- 
 
 jection (for example, the horizontal plane). Perform the construc- 
 tion described in Article 19; then d e a is the angle made by the 
 given line with the horizontal plane. The same construction 
 performed in the horizontal plane of projection will give the angle 
 made by the given line with the vertical plane of projection, 
 
 22. Given (in fig. 5), the Projections, a b and A B, a C and A C, 
 of a Pair of Straight Lines which Intersect each other in the Point 
 whose Projections are a, A, to find the Angle between those Lines. 
 
 In either of the planes of projection (for example, the vertical 
 
 Fig. 5. 
 
 plane) find the points, d, e, where the projections of the given line 
 cut the axis X X; these will be also the vertical projections of the 
 horizontal traces of the lines. Through e and d draw e E, d D, 
 perpendicular to X X, cutting A C and A B in E and D respec- 
 tively; these points will be the horizontal traces of the lines. Join 
 D E (which will be the horizontal trace of the plane containing 
 the lines), and on it let fall the perpendicular F A. Join A a 
 (which of course is perpendicular to X X) ; let it cut XX in G. 
 Make G/= A F, and join af. In F A produced, take F H = a/; 
 join H E, H D ; E H D will be the angle required. 
 
 REMARK. The triangle E H D is the rabatment upon the 
 horizontal plane of the triangle whose projections are E A D 
 and e a d.
 
 EULES BELATING TO PLANES. 
 
 22 A. Given (in fig. 5), the Projections, a b and A B, of a Straight 
 Line, and One Trace (say D E) of a Plane Traversing that Line, to 
 Find the Projections of a. Straight Line which shall, at a given Point, 
 a A make a given Angle in the given Plane with the given Straight 
 
 Line. Join A a, which will be perpendicular to X X. On D E 
 let fall the perpendicular A F. In X X take G/= A F; join 
 af. In F A produced take F H = af. Join H D; and draw 
 H E, making D H E = the given angle, and cutting D E in E. 
 From E let fall Ee perpendicular to X X; join A E, ae; these 
 will be the projections of the line required. 
 
 SECTION IV. Rules Relating to Planes. 
 
 23. Given, the Projections of Three Points, to draw the Traces 
 of a plane Passing through them. Draw straight lines from one 
 of the points to the two others ; find, by Article 18, the traces of 
 those straight lines; through those traces, on the two planes of 
 projection respectively, draw two straight lines; these will be the 
 traces required. 
 
 23 A. Given, the Projections of Two Points and of a Straight Line, 
 to Draw the Traces of a Plane Traversing the Points and Parallel 
 
 to the Line. Through the projections of either of the given points 
 draw straight lines parallel respectively to the corresponding pro- 
 jections of the given line; these will be the projections of a straight 
 line through the given point, parallel to the given straight line; 
 then, by Article 23, find the traces of a plane traversing the new 
 straight line and the other given point. 
 
 24 Given (in fig. 6), the Traces of a Plane, B A, B C, to Find 
 the Angle which it makes with one of the Planes of Projection 
 
 (for example, the vertical plane). 
 From any convenient point, 
 A, in the horizontal trace let fall 
 A D perpendicular to X X. 
 From D let fall D e perpen- 
 dicular to B C. In D B lay off 
 D/ = D e. Join /A (this will 
 represent the perpendicular dis- 
 tance from B C of the point ^ 
 
 whose projections are D and A). 
 A/ D will be the angle re- 
 quired. 
 
 25. Given (in fig. 7), the Trace* 
 
 of a Plane, B A, B C, to Find ^>v 
 
 the Angle which it makes with 
 the Axis of Projection, X X. 
 
 In either of the two traces (for 
 
 Fig. 6.
 
 10 
 
 GEOMETRY OP MACHINERY. 
 
 example, B A) take any convenient point, A, from which let fall 
 A D perpendicular to X X; and on B D as a diameter describe a 
 circle. From D let fall perpendiculars, D e, D F, on the two given 
 
 traces. From the point 
 e, thus found on the op- 
 posite trace to that on 
 which the point A was 
 assumed, let fall e E per- 
 pendicular to X X; join 
 E A, cutting D F in G. 
 From G draw Gr H per- 
 pendicular to X X, cut- 
 ting the circle in H; 
 D B H will be the re- 
 quired angle. 
 
 26. Given (in fig. 7), 
 the Traces of a Plane, 
 B A, B C, to Draw the 
 Traces of another Plane 
 which shall be Parallel to 
 the given Plane, and at n 
 given Perpendicular Dis- 
 tance from it in either 
 
 Direction. Complete the 
 Fig. 7. construction described in 
 
 Article 25. Join D H 
 
 (this represents the perpendicular distance of the point D in 
 the axis from the given plane); then from H, along H D (or 
 along D H produced, according to the direction in which the 
 new plane is to lie), lay off the given perpendicular distance 
 between the planes, H K. From K draw K M parallel to H B, 
 cutting X X in M. From M draw M N parallel to B C, and 
 M L parallel to B A; these will be the traces of the plane 
 required. 
 
 Or otherwise : Complete the construction described in Article 
 24 (see fig. 8). A.f is the rabatment of the intersection of the 
 given plane with a plane, A. D e, perpendicular to the ver- 
 tical trace B C. Through A draw A M perpendicular to Af, 
 and make A M equal to the given distance between the planes; 
 draw M N parallel to A/, cutting X X in N In D e produced 
 take D O equal to D N". O is a point in the trace of the plane 
 required. Through draw O P parallel to B C, cutting X X in 
 P ; and through P draw P Q parallel to B A. O P Q is the plane 
 required. 
 
 27. CJiren (in fig. 9), the Traces of Two Planes, C A C? and C B d, 
 to Draw the Projections of their JLine of Intersection. The traces of
 
 RULES RELATING TO PLANES. 
 
 11 
 
 the required line are and d, where the traces of the given 
 planes intersect Prom those points respectively let fall C c and 
 
 Fig. 8. 
 
 d D perpendicular to X X; join C D, c d; these will be the 
 projections required. ^ 
 
 28. To Find the Projections of the 
 Point where a Straight Ijine Intersects 
 
 a Plane (the traces of the line and of 
 the plane being given), it is only 
 necessary to draw the traces of two 
 planes traversing the given line in 
 convenient directions, and find the 
 projections of the lines in which those 
 two planes cut the given plane; the 
 intersections of those projections will 
 be the projections of the point required. 
 
 Fig. 9. 
 
 29. GircH (in fig. 10), the Traces of Two Planes, A d, C B d,
 
 12 
 
 GEOMETRY OF MACHINERY. 
 
 to Find the Angle between them. From either of the intersections 
 of the traces (say d) let fall d D perpendicular to X X ; draw D 0, 
 joining D with the other intersection of the traces. Through any 
 convenient point, I, in D 0, draw G I H pei-pendicular to D C, 
 
 Fig. 10. 
 
 cutting A C in G and B C in H. Along X X lay off D E = D C, 
 and Di = D I; join d E (this will be the length of the line of 
 intersection of the planes). From i let fall i k perpendicular to 
 d E; in I C take I K = i k; join K G, KH; G K H will be the 
 angle required. 
 
 When the traces of the two given planes are inconveniently 
 placed for the completion of the figure, we may substitute for either 
 pair of traces another pair of traces parallel to them, and more 
 conveniently placed. 
 
 30. CHveto (in fig. 10), the Traces, A d and AC, of a Plane; also 
 the Trace*, d and C, of a Straight Line in that Plane; to Draw the 
 Traces of a Plane which shall Cut the given Plane in that I^ine at 
 
 a given Angle. From either of the traces of the sti-aight line, as d, 
 let fall d D perpendicular to X X ; draw the straight line D C, 
 joining D with the other trace, C, of the straight line. Through 
 any convenient point, I, in D C, draw I G perpendicular to D C, 
 cutting C A in G. In X X lay off D E - D C and D i = D I; 
 join d E, and on it let fall the perpendicular i k. In I C take 
 I K = i k; join K G. Then draw K H, making G K H = the 
 given angle, and cutting G I, produced if necessaiy, in H. Draw 
 C H, cutting X X in B, and join B d' } these will be the traces of 
 the plane required.
 
 RULES RELATING TO PLANES. 
 
 31. Given (in fig. 11), the Traces of a Plane, ABC, and the 
 Projections of a Point, G, y, to Draw the Traces of a Plane Tra- 
 versing the given Point, and Parallel to the given Plane. Through 
 
 either of the projections of the given point (say G) draw G H 
 parallel to the corresponding trace of the given plane, and cutting 
 X X in H. (This will be one of the projections of a line through 
 the given point, parallel to the trace A B of the given plane.) 
 
 Fig. 11. 
 
 Through H draw H D perpendicular to X X ; and through g draw 
 g D parallel to X X, cutting H D in D (g D will be the projection 
 and D one of the traces of the line before mentioned). Through 
 D draw D E parallel to C B, cutting X X in E; and through E 
 draw E F parallel to B A; D E F will be the traces of the 
 required plane. 
 
 32. Given, the Traces of a Plane, E F, E D (in fig. 11), and One 
 Projection of a Point in that Plane, to Find the other Projection 
 
 of that Point. Suppose g, the vertical projection of the point, to 
 be given. Draw g D parallel to X X, cutting E D in D. From 
 D let fall D H perpendicular to X X. From g draw g G perpen- 
 dicular to X X, and from H draw H G parallel to E F; the 
 intersection of those lines, G, will be the required horizontal pro- 
 jection of the given point. 
 
 33. Given (in fig. 12), the Traces, A B C, of a Plane, and the 
 Projections, D, d, of a Point, to Draw the Projections of a Perpen- 
 dicular let Fall from the Point on the Plane. From One of the 
 
 projections of the given point (say D) draw D E F perpendicular 
 to the corresponding trace, B A, of the given plane, and cutting 
 B A in E, and X X in F. From E let fall E e perpendicular to
 
 14 
 
 GEOMETRY OP MACHINERY. 
 
 XX; from F draw F./" perpendicular to X X, cutting the trace 
 B C in fi join ,/e; from d draw d g perpendicular to B C, cutting 
 
 f e in g; and from g draw g G 
 perpendicular to X X, cutting 
 D F in G. D G and d g will 
 be the projections of the per- 
 pendicular required. 
 
 34. Given (in fig. 13), the Pro 
 jections of a Point, D, d, and 
 those of a Straight J,im, A B, 
 a b, to Draw the Traces of a 
 Plane which shall Traverse the 
 Point, and be Perpendicular to 
 
 the Line. Through one of the 
 projections of the given point 
 (say D) draw D G perpendicular 
 to A B (the corresponding pro- 
 jection of the given line), cutting 
 X X in G. Through G draw 
 G g perpendicular to X X ; 
 through d, the other projection 
 
 Fig. 12. 
 
 of the point, draw dg parallel to X X, cutting G g in g- through g 
 draw E C perpendicular to a b, cutting X X in C ; and through C 
 
 draw C F perpendicular 
 to A B. EOF will 
 be the traces of the re- 
 quired plane. 
 
 35. Given, the Pro- 
 jections of a Point and 
 of a Straight Line, to 
 Draw the Projections 
 of a Perpendicular let 
 Fall from the Point up- 
 on the Straight Line. 
 
 Find by the preceding 
 rule the traces of a 
 plane traversing the 
 given point, and per- 
 pendicular to the given 
 line; then, by Article 
 23, find the traces of 
 Fig. 13. a plane traversing the 
 
 given point and line; 
 
 and finally, by Article 27, find the projection of the line of 
 
 intersection of those two planes. 
 
 36. Given, the Projections of Two Straight Lines that are neither
 
 COMMON PERPENDICULAR CIRCLE. 
 
 15 
 
 Parallel nor Intersecting, to Find the Projections of their Common. 
 
 Perpendicular. By Article 23 A, find the traces of a plane travers- 
 ing one of the lines and parallel to the other. Then, by Article 33, 
 find the projections of a perpendicular let fall on that plane from 
 any convenient point in the second line. Then through the pro- 
 jections of the foot of that perpendicular draw the projections of 
 a straight line parallel to the second straight line; these will cut 
 the projections of the first straight line at one end of the common 
 perpendicular, whose projections will be parallel and equal to those 
 of the perpendicular already found. 
 
 36A. Projections of a Circle. When an instrument which draws 
 ellipses accurately is at hand, it may be used for the purpose of 
 drawing the projections of a circle of a given radius, described 
 about a given point in a given plane, and may thus facilitate 
 much the solution of various problems. The following is the 
 process for obtaining the projections of a circle : 
 
 Given (in fig. 14), the Traces of a Plane, ABC, and the Pro. 
 jections of a Point in that Plane, D, d, to Draw the Projections of 
 a Circle of a given Radius, described in the given Plane and about 
 
 the given Point. For the vertical projection, describe about d a 
 circle of the given radius, df = d e, and draw the diameter ej 
 parallel to the trace 
 C B; ef will itself be 
 the vertical projection 
 of one diameter of the 
 circle. Draw dg per- 
 pendicular to ef. Find, 
 by Article 24, the angle 
 which the given plane 
 makes with the vertical 
 plane of projection, and 
 lay off g d h equal to 
 the angle so found. 
 From h, in the circle, 
 draw h k parallel tofe, 
 and cutting d g in k' } 
 then d k will be the 
 vertical projection of a 
 radius of the circle perpendicular to ef. Then on the major axis, 
 ef, and minor semi-axis, d k, describe an ellipse; that ellipse will 
 be the required vertical projection of the circle. 
 
 The horizontal projection is obtained by a precisely similar 
 process, the rule of Article 24 being now used to find the angle 
 which the given plane makes with the horizontal plane of pro- 
 jection. 
 
 The two ellipses are both touched by a pair of tangents, M m f 
 
 Fig. 14.
 
 16 GEOMETRY OP MACHINERY, 
 
 "L I, perpendicular to X X; and the diameters, I m, L M, are the 
 projections of one diameter of the circle viz., that diameter 
 in which the plane A B C is cut at right angles by a plane parallel 
 to X X. The perpendicular distance, N n, between the two 
 tangents is equal to the diameter of the circle multiplied by the 
 cosine of the angle which the given plane makes with X X, and 
 is bisected by the line D d.
 
 17 
 
 CHAPTER II. 
 
 OF THE MOTIONS OP PRIMARY MOVING PIECES IN MACHINES. 
 
 SECTION I. General Explanations. 
 
 37. Frame; Moving Pieces, Primary and Secondary, (A. M., 427.) 
 The 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 mathemati- 
 cally, 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 dynamics of machinery, the motion of the frame 
 relatively to the earth does not require to be considered. 
 
 The moving pieces may be distinguished into primary and 
 secondary; the former being those which are directly carried by 
 the frame, and have their motion wholly guided by their con- 
 nection with the frame ; and the latter, those which are carried by 
 other moving pieces, or which have their motion not wholly guided 
 by their connection with the frame. For example, the crank-shaft 
 and the piston-rod of a steam engine are primary moving pieces; 
 the wheels of a locomotive ere primary moving pieces; the 
 connecting-rod of a steam engine is a secondary moving piece 
 
 Connectors are those secondary moving pieces, such as links, belts, 
 cords, and chains, which transmit motion from one moving piece 
 to another, when that transmission is not effected by immediate 
 contact. 
 
 38. Bearings (A. M., 428,) 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 
 translation in a straight line must have plane or cylindrical* 
 
 * The word "cylindrical" is here used in the comprehensive sense, whicji 
 denotes any surface generated by the motion of a straight line parallel to 
 itself. 
 
 G
 
 18 GEOMETRY OF MACHINERY. 
 
 surfaces, exactly straigM in the direction of motion. The bearings 
 of rotating pieces must have surfaces accurately turned tojigures 
 of revolution, such as circular cylinders, spheres, cones, conoids, and 
 flat discs. The bearing of a piece whose motion is helical, must be 
 an exact screw. Those parts of moving pieces which touch the 
 bearings should have sm'faces 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. 
 . The accurate formation and fitting of bearing surfaces is of 
 primary importance to the correct and efficient working of 
 machines. 
 
 39. The Motions of Primary Moving Pieces (.'1 M., 429,) are 
 Limited by the fact, that in order that different portions of a pair 
 of bearing surfaces may accurately fit each other during their 
 relative motion, those surfaces must be either straight, circular, or 
 helical ; from which it follows, that the motions in question can be 
 of three kinds only, viz. : 
 
 I. Straight translation, or shifting, which is necessarily of limited 
 extent, and which, if the motion of the machine is of indefinite 
 duration, must be reciprocating; that is to say, must take place 
 alternately in opposite directions : for example, the piston-rod of a 
 steam engine. 
 
 II. Simple rotation, or turning about a fixed axis, which motion 
 may be either continuous or reciprocating, being called in the 
 latter case swinging, rocking, or oscillation. Continuous rotation 
 is exemplified by the shaft of a steam engine ; reciprocating rotation 
 by various beams or levers. 
 
 IIL Helical, or screw-like motion, compounded of rotation about 
 a fixed axis, and translation along that axis. 
 
 SECTION II. Straight Motion of Primary Pieces. 
 
 40. Straight Translation is the motion of a primary piece sliding 
 along a straight guiding surface. All the particles of the piece 
 move through equal distances in a given time, along parallel straight 
 lines ; and the line joining any two particles remains unaltered in 
 length and in direction. 
 
 41. Resolution and Composition of Motions The resultant of 
 
 two or more component motions is the motion which results from 
 putting them together. If the component motions are represented 
 by straight lines, their resultant is found geometrically by joining 
 together, end to end, a series of straight lines respectively equal 
 and parallel to the given straight lines, and pointing in the same 
 directions, and then drawing a straight line from the starting point 
 to the further end of the series. For example :
 
 COMPONENT AND RESULTANT MOTIONS. 
 
 19 
 
 ..--X 
 
 Fig. 15. 
 
 I. (See fig. 15.) To find the resultant of two component motions, 
 A B and A C. Let the paper represent the plane of those motions. 
 From B draw B D parallel and equal to A C, and pointing in the 
 same direction; join A D; this will be 
 
 the required resultant motion; or, in other 
 words, complete the parallelogram, A B 
 D C; its diagonal, A D, will be the re- 
 quired resultant. 
 
 A motion may, if required, be resolved 
 into components. The following are the 
 cases most useful in mechanism : 
 
 II. (Fig. 15.) To resolve a given motion, 
 A D, into components in two given direc- 
 tions in the same plane, A X and A Y. 
 
 Through D draw D C parallel to X A, cutting A Y in C, and D B 
 parallel to Y A, cutting AX in B; A B and A C will be the 
 required components. 
 
 III. (Fig. 16.) To resolve a given motion, A D, into one component 
 parallel and another component perpendicular to a given direction. 
 Through A, parallel to the given direction, , 
 
 draw A X, upon which let fall the perpen- 
 dicular D B; then A B will be the first of 
 the required components, and A C parallel 
 and equal to B D will be the second. 
 
 IV. Given, the traces of a plane (Article 
 15, page 6) and the projections of a straight 
 line representing a motion (Article 11, page 
 4), to find the projections of two component 
 
 Fig. 16. 
 
 motions, one perpendicular and the other parallel to the plane. By 
 the rule of Article 31, page 13, draw the traces of a second plane 
 parallel to the given plane, and traversing the point which repre- 
 sents one end of the given motion. Then by the rule of Ai-ticle 33, 
 page 13, find the projections of the perpendicular let fall on the 
 second plane from the point representing the other end of the 
 given motion. That perpendicular will be one of the required 
 components ; and the straight line from the first-mentioned point 
 to the foot of the perpendicular will be the other. The lengths of 
 the lines representing the component motions may be found, if 
 required, by Article 1 9, page 7. 
 
 The component of the motion parallel to the given plane is 
 obviously its projection on that plane. It is sometimes called the 
 tangential component, and the component perpendicular to the given, 
 plane the normal component of the given motion. 
 
 V. Given, a resultant motion and one of two component motions, 
 to find the other component motion. Combine the given resultant 
 motion with a motion equal and opposite to the given component
 
 20 
 
 GEOMETRY OF MACHINERY. 
 
 motion; the resultant of these two will be the required other com- 
 ponent motion. For example, in fig. 15, let A D be the given 
 resultant motion, and A B the given component; draw D C equal 
 and parallel to A B, and pointing the opposite way; join A C; 
 this will be the required other component ; or otherwise, join B D 
 aad draw A C equal and parallel to it. 
 
 VI. (Fig. 17.) Given, the vertical projection, A B, and the hori- 
 zontal projection, A' B', of a straight line representing a motion, to 
 resolve that motion into three rectangular components parallel and 
 perpendicular to the planes of projection. Let O X be the axis of 
 projection (Article 9, page 4). Draw the straight lines A A', B B', 
 
 \ 
 
 B' 
 
 Fig. 17. 
 
 cutting the axis of projection (of course at right angles) in C and D. 
 Then through any convenient point, 0, in the axis of projection, 
 draw the straight line Z O Y' at right angles to that axis; and 
 take O Y' to represent a transverse horizontal axis, and O Z to 
 represent a vertical axis. (The point O is called the origin.} Then 
 parallel to X draw A' E' and B' F' to meet O Y', and A G and 
 B H to meet O Z. The three components required will be repre- 
 sented by C D, E' F', and G H. 
 
 VII. Given (in fig. 17), the vertical projection, A B, and the 
 horizontal projection, A' B', of a straight line representing a motion, 
 to draw a third projection of the same straight line on a vertical 
 transverse plane of projection perpendicular to the first two planes of 
 projection. Construct fig. 17 as described in the preceding Rule. 
 O Z and Y' will be the traces of the third plane of projection. 
 Produce X towards Y"; then Y" will represent the rabatnient
 
 EELATIVE MOTION OF TWO PIECES. 21 
 
 of Y', and Z O Y" the rabatment of the vertical transverse 
 plane upon the vertical longitudinal plane of projection. In O Y" 
 take O E" = O E', and F" = O F; draw E" A" and F'B" parallel 
 to O Z, to meet A G and B H produced in A" and B" respectively; 
 join A" B"; this will be the projection required. 
 
 According to the rule already stated in Article 19, page 7, the 
 motion of which A B and A' B' are the projections is to be found 
 by making K L = A' B', and joining L B, which line will repre- 
 sent the extent of the resultant motion. 
 
 The following are the relations between a resultant motion and 
 its components as expressed by calculation. In fig. 15, 
 
 sin C A B : sin C A D : sin D A B : : A D : A B : A C; 
 also, A D 2 = A B 2 + A C 2 + 2 A B A C cos C A B. 
 In fig. 16, 
 
 AB = AD-cosB AD; AC = AD-sinB AD; 
 A D 2 = A B 2 + A C 2 . 
 
 In fig. 17, 
 
 L B 2 = C D 2 + E' F 2 + G H 2 . 
 
 42. Relative motion of Two Moving Pieces. All motion is 
 
 relative : that is to say, every conceivable motion consists in a 
 change of the relative position of two or more points. In speaking 
 of the motions of the moving pieces of machines, motions relatively 
 to the frame ai-e always to be understood, unless it is otherwise 
 specified. It is often requisite, however, to express the motion of 
 a point in a moving piece relatively to a point in the same or in 
 another moving piece. 
 
 In the case considered in the present section, where the relative 
 position of two points in the same moving piece remains unaltered, 
 not only as to distance but as to direction, the relative motion of 
 such a pair of points is nothing. The motion of one moving piece 
 relatively to another is determined by the following principle : Let 
 P, Q, and R denote any three points; then the motion of R 
 relatively to P is the resultant of the motion of R relatively to Q, 
 combined with the motion of Q relatively to P; so that if the 
 motions of Q relatively to P, and of R relatively to P are given, 
 the motion of R relatively to Q is to be found according to Rule 
 V. of the preceding Article, by compounding with the motion of R 
 relatively to P a motion equal and opposite to that of Q relatively 
 to P. For example, let P stand for the frame of a machine, and 
 Q and R for two moving pieces which slide along straight guides;
 
 22 . GEOMETRY OF MACHINERY. 
 
 and in a given interval of time let A B, in fig. 15, page 19, repre- 
 sent the motion of Q relatively to P, and A D the motion of R 
 relatively to P; then A C, found by Rule V. of Article 41, will 
 , represent the motion of R relatively to Q. 
 
 In all cases whatsoever of relative motion of two bodies, the 
 motion of one relatively to the other is exactly equal and 
 contrary to that of the second relatively to the first. For example, 
 let P and Q be two points; and when P is treated as fixed, let Q 
 move through a given distance in a given direction relatively to P j 
 then if Q is treated as fixed, P moves through the same distance 
 in the contrary direction relatively to Q. 
 
 43. Comparative motion (A. M., 358,) is the relation borne to each 
 other by the simultaneous motions of two points, either in the same 
 body or in different bodies, relatively to one and the same fixed point 
 or body. It consists of two elements : the velocity-ratio, which is the 
 proportion borne to each other by the distances moved through by 
 the two points in the same interval of time; and the directional 
 relation, which is the relation between the directions in which the 
 two points are moving at the same instant. 
 
 In the case of two points in a primary piece whose motion is 
 one of translation, the velocity-ratio is that of equality, and the 
 directional relation that of identity ; for all points in such a piece 
 are moving with equal speed in parallel directions at the same 
 instant. 
 
 When two points in two different pieces are compared, the 
 comparison may give a different result. For example, let P, as 
 before, stand for the frame of a machine, and Q and R for two 
 moving pieces; and while Q performs relatively to P the motion 
 represented by A B (fig. 15, page 19), let R perform relatively to 
 P the motion represented by A D. Then the comparative motion 
 of R and Q consists of the following elements : 
 
 A D 
 the velocity-ratio, -T-O j 
 
 -\ _L> 
 
 and the directional relation, represented by the angle BAD. 
 
 In most of the cases which occur in mechanism the motion of each 
 point is limited to two directions forward or backward in a fixed 
 path; so that the directional relation of two points may often be 
 sufficiently expressed by prefixing the sign + or to their velocity- 
 ratio, according as their motions are similar or contrary; that is, 
 the sign + denotes that those motions are both forward or both 
 backward; and the sign that one is forward and the other 
 backward. 
 
 We may compare together the different components of the 
 motion of one point, and the resultant motion. For example, in
 
 COMPARATIVE MOTION DRIVING AND WORKING POINTS. 23 
 
 figs. 15 and 16, page 19, the velocity-ratios of two component 
 motions, as compared with, their resultant, are expressed by 
 
 AB , A.C 
 AD 
 
 and in fig. 17, page 20, the velocity-ratios of three rectangular 
 component motions, as compared with their resultant, are ex- 
 pressed by 
 
 C D E' F G H 
 
 ITS' LB' ai d I7B' 
 
 Strictly speaking, the principles of the geometry of machines, or 
 of pure mechanism, are concerned with comparative motions only, 
 and not with absolute velocities : or, in other words, those principles 
 relate to the motions which different moving points perform in the 
 course of the same interval of time, but not to the length of the 
 interval of time in which such motions are performed. For 
 example, in the case of a direct-acting steam engine, the principles 
 of pure mechanism show that the piston makes one double stroke 
 for each revolution of the crank; that the directional relation of 
 the piston and crank-pin varies periodically, the piston moving to 
 and fro, while the crank-pin moves continuously round in a circle; 
 and that in particular positions of those pieces their velocity-ratio 
 takes particular values; but the question of what interval of time 
 is occupied by a revolution, or of how many revolutions are per- 
 formed in a minute, belongs not to the geometry, but to the 
 dynamics of machines. Further, in the case of a pair of spur 
 wheels gearing into each other, the principles of pure mechanism 
 show that in any given interval of time the numbers of revolutions 
 performed by those wheels respectively are inversely as their 
 numbers of teeth, and that the directions in which they turn are 
 contrary; but those principles do not inform us how many revolur 
 tions either wheel makes in a minute. 
 
 44. Driving Point and Working Point. The term driving point 
 is used to denote that point, either in a whole machine or in a 
 given moving piece of a machine, where the force is applied that 
 -causes the motion; and the term working point is used to denote 
 the point where the useful work is done. These explanations 
 contain references to the dynamics of machines; but it is to be 
 understood that in the geometry of machines, or pure mechanism, 
 it is the comparative motion only of the driving point and working 
 point that is taken into consideration. It is to be observed, too, 
 that the word "point" is here taken in an extended meaning; for 
 the exertion of force or communication of motion at a mathematical 
 point, of no sensible magnitude, is purely ideal ; and when the word 
 point is used with reference to the driving or the work of machines*
 
 24 GEOMETRY OF MACHINERY. 
 
 it is to be held to moan the place where the action that drives or 
 that resists a machine is exerted, of what magnitude soever that 
 place may be, whether a surface or a volume. Thus, the driving 
 point in a steam engine comprehends the whole surface of the 
 piston that is pressed upon by the steam which drives the engine; 
 and the working point, where friction is overcome, comprehends 
 the whole of the rubbing surface, and where a heavy body is lifted, 
 the whole volume of that body. Nevertheless, for the sake of con- 
 venience in mathematical investigation, such places of the action of 
 driving or resisting forces are often treated on the supposition that 
 they may be represented by single points ; for when such points ar 
 properly chosen, no error is incurred by making that supposition. 
 
 SECTION III. Rotation of Primary Pieces. 
 
 45. Rotation of a Primary Piece. (A. M., 370-372.) Eolation Or 
 
 Turning is the motion of a rigid body when lines in it change their 
 directions; and it is the only kind of motion involving change of the 
 relative positions of the particles of a body that is possible consistently 
 with rigidity; that is to say, with the maintenance of the distance 
 between every pair of particles in the body unchanged. An axis of 
 rotation is a line in a rigid body whose direction is unchanged by the 
 rotation ; and a fixed axis of rotation is a line whose position, as well as 
 its direction, is unchanged by the rotation. Every line in a rotating 
 body which is parallel to the axis has its direction unchanged by 
 the rotation. The rotation of a primary piece in a machine always 
 takes place about an axis that is fixed relatively to the frame of the 
 machine; that axis being the geometrical axis, or centre line, of a 
 bearing surface (such as that of the journals or gudgeons of a shaft), 
 whose form is that either of a circular cylinder or of some other 
 surface of revolution The plane of rotation is any plane perpen- 
 dicular to the axis. Every such plane in a rotating body has its 
 position unchanged by the rotation ; and straight lines in such a 
 plane that is, straight lines perpendicular to the axis of rotation 
 change their directions more rapidly than any other straight lines 
 in the same body. 
 
 46. speed of Rotation. (A. M., 373.) Although in the case of 
 rotation, as well as in that of translation, the principles of pure 
 mechanism are concerned with comparative velocities only, still 
 it is desirable here to state, that the speed with which a rotating 
 body turns is expressed in two different ways. For most practical 
 purposes it is usually stated in turns and fractions of a turn in some 
 convenient unit of time; such as a second, or (more commonly) a 
 minute. For scientific purposes, and for some practical purposes 
 also, it is expressed in angular velocity; which means, the angle 
 swept through in a second by a line perpendicular to the axis of
 
 ROTATION ANGULAR VELOCITY. 25 
 
 rotation: that angle being stated in circular measure; which 
 means the ratio of the length of the arc subtended by an angle to 
 the radius of that arc. The following are examples of the values 
 of angles in circular measure : 
 
 One degree, ................. 0-0174533 nearly; 
 
 A right angle, or quarter ) , K - AO , 
 revolutioS,.. ..... . ....... } 1 ' 5708 nearl 
 
 6 ' 2832 -* veiy nearly - 
 
 Hence, to convert turns per second into angular velocity, multiply by 
 
 710 
 
 :pj-^ = 6 '2832 nearly; and to convert angular velocity into turns per 
 
 I It) 
 
 113 
 
 second, multiply by =y^ = 0-159155 nearly. The time of revolution 
 
 in seconds is the reciprocal of the speed expressed in turns per 
 second. The comparative speed or angular velocity-ratio of two 
 rotating pieces is independent of the kind of unit in which their 
 absolute speeds may be expressed; it is the reciprocal of the ratio 
 of their times Df revolution. 
 
 47. Rotation is Common to all Parto of the Turning Body. 
 (A. M., 375.) Since the angular motion of rotation consists in the 
 change of direction of a line in a plane of rotation, and since that 
 change of direction is the same how short soever the line may be, 
 it is evident that the condition of rotation, like that of translation, 
 is common to every particle, how small soever, of the turning rigid 
 body, and that the angular velocity of turning of each particle, how 
 small soever, is the same with that of the entire body. This is 
 otherwise evident, by 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, and makes the same 
 number of turns in a second, or a minute, or any other interval 
 of time. 
 
 48. Bight and Left-Banded Rotation. (A. M., 376.) The direction 
 
 of rotation round a given axis is distinguished in an arbitrary 
 manner into right-handed and left-handed. One end of the axis is 
 chosen as that from which an observer is supposed to look along 
 the direction of the axis towards the rotating body. Then if the 
 body seems to the observer to turn in the same direction in which 
 the sun seems to revolve to an observer north of the tropics, the 
 rotation is said to be right-handed; if in the contrary direction, 
 left-handed; and it is usual to consider the angular velocity of 
 right-handed rotation to be positive, and that of left-handed rota-
 
 26 
 
 GEOMETRY OF MACHINERY. 
 
 tion 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. In fig 18, the arrow R, represents right- 
 handed rotation, and the arrow L 
 left-handed rotation. When a 
 body oscillates about an axis its 
 
 rotation is alternately right-handed 1 i* \ \ B. 
 
 and left-handed. 
 
 18. 
 
 49. Translation of a Point in a 
 Rotating Piece. (A. M., 377.) 
 
 Each point in a rotating piece 
 
 (except those situated in the axis) 
 
 lias a motion of revolution that is, translation in a circular path, 
 
 round the axis of rotation ; and the velocity of that translation is 
 
 the product of the perpendicular distance of the point from, the 
 
 axis that is, the radius of the circular path, into the angular 
 
 velocity of rotation (Article 46, page 24). Thus, in fig. 19, let the 
 
 Fig. 21. 
 
 Fig. 19. 
 
 surface of the paper represent a plane of rotation ; let O be at once 
 the trace and the projection of the axis of rotation on that plane, 
 and A the projection of a point in the rotating piece under con- 
 sideration. Then the motion of that point (and of its projection 
 A), takes place in a circle of the radius O A; and if A A' be the 
 arc described in a second, then 
 
 also, 
 
 A A' = O A x angular velocity; 
 
 YYS 
 llo 
 
 = O A x S x number of turns per second; 
 
 angular velocity = 
 
 A A'
 
 ROTATING PIECE CIRCULAR ARCS. 27 
 
 The velocity at a given instant of a point which moves in a curve, 
 as distinguished from the arc traced in a second by that point, is 
 represented by a straight line equal in length to that arc, and 
 pointing in the direction in which the point is moving at the given 
 instant; that is to say, being a tangent to the path of the point at 
 that instant. Therefore, to represent by a straight line the velocity 
 of the point now in question at the instant when its projection is 
 at A, draw A a perpendicular to O A, and equal in length to A A' 
 ( <= O A x angular velocity). 
 
 50. motion of a Part of a Rotating Piece. When what has just 
 
 been explained is considered together with the statement in Article 
 47, it is easily seen, that if the centre of any part of a moving 
 piece rotating about a fixed axis is situated in that axis, then the 
 motion of that part is simply a rotation similar and equal to that 
 of the whole piece ; but if the centre of the part is situated at a 
 distance, O A, from that axis, the motion of that part consists of a 
 translation of its centre, with the velocity O A x the angular 
 velocity, in a circle described about the fixed axis with the radius 
 O A, combined with a rotation similar and equal to that of the 
 whole piece, about a moving axis traversing A, and parallel to the 
 fixed axis which traverses O. Consider, for example, that rotating 
 piece in a steam engine which consists of the shaft, crank, and 
 crank-pin, and which turns about the axis of the shaft, as a fixed 
 axis of rotation, to which the axis of the crank-pin is parallel. 
 Then the motion of the shaft consists simply in a rotation about 
 its own axis; while the motion of the crank-pin consists in a 
 translation of its centre, and of each point in its axis, in a circular 
 path described about the axis of the shaft, combined with a rota- 
 tion about its own moving axis similar and equal to that of the 
 shaft. As an additional illustration, suppose one end of a cord to 
 be held still, and the other to be attached to a hook which is fixed 
 at the centre of a rotating wheel, and which therefore rotates 
 along with and as part of the wheel. The cord will undergo one 
 twist for each turn of the wheel. Now let the hook be reinoved 
 from the centre, and fixed at any point in an arm of the wheel, or 
 in its rim ; the cord will still undergo one twist, neither more nor 
 less, for each turn of the wheel; thus showing, as before, the effect 
 of the rotation of the hook along with the wheel; and the only 
 difference in the motion will be that the end of the cord attached 
 to the hook will be carried round in a circle, at the same time that 
 the whole cord is twisted. A secondary piece in a machine may 
 be so contrived as to have translation in a circle or some other 
 curved path without rotation; this will be considered in a later 
 chapter. 
 
 51. Rules as to Lengths of Circular Arcs. In connection with 
 
 the motion of points in rotating pieces, and with various other
 
 28 GEOMETRY OF MACHINERY. 
 
 questions in mechanism, there is frequent occasion to measure the 
 lengths of circular arcs, and to lay off circular arcs of given lengths. 
 These processes may be performed by the help of calculation, and 
 of the well-known approximate values of the ratio which the 
 radius and the circumference of a circle bear to each other, viz. : 
 
 circumference 710 . /.OOOIOR- i 
 
 r . = VTT nearly = 6-283185 nearly: 
 
 radius 113 
 
 radius 113 A _ __, ,.,.. . 
 
 = jrvr: nearly = - 159155 nearly: 
 
 circumference 710 
 
 but it is often much more convenient in practice to proceed by 
 drawing; and then the following rules are the most accurate yet 
 known : * 
 
 I. (Fig. 22.) To draw 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 ; about C, with the radius 
 C B = f B A, draw a circle ; then draw the 
 straight line A D, kmching the given arc 
 Fig. 22. in A, and meeting the last-mentioned circle 
 
 in D ; AD 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 T3^ ff for an arc 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 ai'c 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 ^a 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 T?4Tjrr f the 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 
 precision required in the drawing. 
 
 II. (Fig. 23.) To draw a straight line 
 approximately equal to a given circular arc, 
 A B. (Another Method.) Let C be the 
 Fig. 23. centre of the arc. Bisect the arc A B in D, 
 
 and the arc AD in E; draw the straight 
 
 * These rules are extracted from Papers read to the British Association 
 in 1867, and published in the Philosophical Magazine for September and 
 October of that year.
 
 LENGTHS OP ARCS. 29 
 
 secant 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 ; 
 aud it varies as the fourth power of the angle subtended by the arc. 
 
 III. To lay off upon a given circle an arc approximately equal in 
 length to a given straight line. In fig. 24, 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 C = ^ A B, and 
 about C, with the radius C B = f A B, draw a 
 circular ai'C B D, meeting the given circle in D. Fig. 24. 
 
 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. 
 
 IY. (Fig. 24.) To draw a circular arc which shall be approxi- 
 mately equal in length to the straight line A B, shall with one of its 
 ends touch that straight line at A, and shall subtend a given angle. 
 In A B take A C = A. B; and about C, with the radius C B 
 = | 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 meet- 
 ing the circle B D in D. 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, approximately, 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 Y. 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. 
 
 Yl. To divide the whole circumference of a circle approxi- 
 mately 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 2.* In other 
 
 * It may be convenient here to state the methods of subdividing area 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 triangle,
 
 30 GEOMETRY OF MACHINERY. 
 
 cases proceed as follows: Divide the circumference exactly, by 
 plane geometry, into such a number of equal arcs as may be re- 
 quired 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 Rule 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 
 circumference 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. 
 
 52. Relative Tr.-uiwlnt ion of a Pair of Point* in a Rotating Piece. 
 
 In fig. 19, page 26 (where O, as already explained, is at once the 
 projection and the trace of a fixed axis of rotation on a plane 
 perpendicular to it, and A the projection of a point in the rotating 
 piece), let B be the projection of another point in the rotating piece, 
 and A B the projection of the straight line connecting those two 
 points. The point B describes a circle of the radius O B about the 
 fixed axis; and the radii O A and O B sweep round with the 
 angular velocity common to all parts of the rotating piece, so that 
 by the time that A has moved to the position A', B has moved to 
 the position B', such that the angles A O A and BOB' are equal. 
 In order to determine the motion of one of those moving points 
 (as A) relatively to the other (as B), it is to be considered that, 
 owing to the rigidity of the body, the length of A B is invariable, 
 and that the change of direction of that line (as projected on the 
 plane of rotation), consists in turning in a given time through an 
 angle equal to that through which the whole piece turns. In fig. 
 20, take B to represent at once the trace and the projection, on a 
 plane of rotation, of an axis parallel to the fixed axis, and tra versing 
 the point B. Draw B A in fig. 20 parallel and equal to B A in 
 fig. 19; and B A' in fig. 20 parallel and equal to B' A' in fig. 19. 
 Then A and A in fig. 20 represent two successive positions of A 
 
 with the summit pointing away from the centre of the arc; a straight line 
 from the centre of the arc to that summit will bisect the arc. (2. ) To mark 
 the siyth part of the circumference of a circle. Lay off 
 a chord equal to the radius. (3. ) To mark the tenth 
 part of the circumference of a circle. In fig. 24 A, 
 draw the straight line A B = the radius of the circle ; 
 and perpendicular to A B, draw B C = 4 A B. Join, 
 A C, and from it cut off C D = C B. AD will be the 
 chord of one-tenth part of the circumference of the 
 Fig. 24 A. circle. (4) For theffteenth part, take the difference 
 
 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 to 1 + a power of 2 ; such asl+2 4 =17;l+2 8 = 257, &c, ; but 
 the method is too laborious for use in designing mechanism,
 
 ROTATING PIECE. 31 
 
 relatively to the axis traversing B, at the beginning and end 
 respectively of the interval of time in which the rotating piece 
 turns through the angle A A' (fig. 19) = A B A (fig. 20). The 
 translation of A relatively to this new axis consists in revolution, 
 in a circle of the radius B A, in the same direction with the rota- 
 tion (that is, in the present example, right-handed); and the 
 velocity of that relative translation is B A x the angular velocity 
 of rotation. Fig. 21 shows how, by a similar construction, the 
 motion of B relatively to an axis traversing A is represented. 
 Take A in fig. 21 to represent at once the trace and the projec- 
 tion, on a plane of rotation, of an axis parallel to the fixed axis, and 
 traversing A. Draw A B and A B' in fig. 21 parallel and equal 
 respectively to A B and A' B' in fig. 19. Then B and B' in fig. 
 21 represent two successive positions of B relatively to the axis 
 traversing A, at the beginning and end respectively of the interval 
 of time in which the rotating piece turns through the angle A A' 
 (fig. 19) = B A B' (fig. 21); the translation of B relatively to this 
 new axis consists in revolution in a circle of the radius A B, in 
 the same direction with the rotation (that is, in the present ex- 
 ample, right-handed) ; and the velocity of that relative translation 
 is A B x the angular velocity, and is at each instant equal, parallel, 
 and contrary to the velocity of translation of A relatively to B, 
 agreeably to the general principle stated at the end of Article 42, 
 page 21. 
 
 53. Comparative Motion of Points in a Rotating Piece. In fig. 
 19, page 26, as before, let A and B be the projections at a given 
 instant, on a plane of rotation, of two points whose motions are to 
 be compared. The directions of motion of those points at that 
 instant are represented by the straight lines A a, B 6, tangents to 
 the circles in which the points revolve about the axis O ; and the 
 directional relation of the points is expressed by the fact, that the 
 angle between those directions of motion is equal to the angle 
 A O B, between the perpendiculars let fall from the two points on 
 the axis O; or, in other words, the angle between the planes 
 traversing that axis and the two points respectively; of which 
 planes O A and O B are the traces upon the plane of rotation ; for 
 the directions of motion, A a, B b, are respectively perpendicular 
 to those two planes. 
 
 The velocity-ratio of the two points is equal to the ratio B : O A 
 borne to each other by the radii of their circular paths. In other 
 words, if A a = A A' be taken, as before, to represent the velocity 
 of A, and B b = B B' to represent the velocity of B, then 
 
 O A : O B : : A a : B 6; 
 
 and if the velocities of any number of points in a rotating piece 
 are compared together, they are all proportional respectively to
 
 32 
 
 GEOMETRY OF MACHINERY. 
 
 the perpendicular distances of those points from the axis of 
 rotation. 
 
 It is obvious that all points in a circular cylindrical surface 
 described about the axis of rotation have equal velocities. The dotted 
 circles in fig. 19, page 26, represent the traces of two such surfaces. 
 
 The relative motions of any two pairs of points in a rotating 
 piece may be compared together. For example, let it be pro- 
 posed to compare the motion of A relatively to B with the motion of 
 B relatively to O. Then, because the velocity of the motion of A 
 relatively to B is proportional to B A, and its direction perpen- 
 dicular to the plane whose trace is B A, while the velocity of the 
 motion of B relatively to O is proportional to O B, and its 
 direction perpendicular to the plane of which O B is the trace, the 
 directional relation is expressed by the angle made by those planes 
 with each other, and the velocity-ratio by the ratio B A : B 
 borne to each other by the projections on the plane of rotation of 
 the two lines of connection of the two pairs of points. 
 
 54. Relative and Comparative Translation of a Pair of Rigidly 
 
 Connected Points. The following proposition is applicable to all 
 motions whatsoever of a pair of points so connected that the 
 distance between them is invariable. It forms the basis of nearly 
 the whole theory of combinations in mechanism, and many of its 
 consequences will be explained in the ensuing chapters of this 
 Part. At present it is introduced with a view to its application 
 to pairs of points in a rotating piece. 
 
 Fig. 25. 
 
 Fig. 26. 
 
 THEOREM. If two points are so connected tJiat their distance apart 
 is invariable, tJie components oftJieir velocities along the straight line 
 which traverses them both must be equal; for if those component 
 velocities are unequal, the distance between the points must 
 necessarily change. 
 
 The straight line which traverses the points is called their Line 
 of Connection. 
 
 For example, in fig. 25, let A and B represent two points in the 
 plane of the paper, whose distance apart, A B, is invariable. At a 
 given instant let the velocities of those points be represented by 
 straight lines, which may be in the same plane, or in different planes,
 
 BOTATING PIECE PERIODICAL MOTION. 33 
 
 according to circumstances; and let A a and B6 be the projections 
 of those lines. From a and b let fall a c and b c perpendicular to 
 the line of connection, A B ; these will be the traces of two planes 
 perpendicular to the line of connection, and traversing respectively 
 the points of which a and b are the projections; the parts Ac and 
 Be', cut off by those planes from the line of connection (produced 
 where necessary), will be the components along that line of the 
 velocities of A and B respectively; and those components must 
 necessarily be equal that is, B c' ~ A c. The component velo- 
 cities transverse to the line of connection are represented by the 
 lines whose projections are c a and c' b, and may bear to each other 
 any proportion whatsoever. 
 
 The same principle is illustrated in fig. 19, page 26. In that 
 figure A a and B 6 represent the velocities of two points, A and 
 B, whose line of connection is A B, and is of invariable length ; 
 c and be' are perpendiculars let fall from a and b upon A B, 
 produced where necessary; and Ac and Be' represent the com- 
 ponent velocities of A and B along the line of connection, which 
 are equal to each other. 
 
 RULE. Given (in Jig. 25), a pair of rigidly connected points, A 
 and B, and the directions of the projections A a and B b upon a 
 plane traversing A B, of their velocities at a given instant, to find 
 the ratio of those projections or component velocities to each other. In 
 fig. 26, draw O c of any convenient length parallel to A B, and a c b 
 perpendicular to it; through O draw O a in fig. 26 parallel to 
 A a in fig. 25, and b in fig. 26 parallel to B 6 in fig. 25 ; then 
 the required ratio is 
 
 B^ = Ob 
 A a ~~ a* 
 
 55. Components of Velocity of a Point in a Rotating Piece 
 
 Periodical Motion. (A. M., 380.) The component parallel to an 
 axis of rotation, of the velocity of a point 
 in a rotating body relatively to that axis, 
 is nothing. That velocity may be resolved 
 into rectangular components parallel to the 
 plane of rotation. Thus let O in fig. 27 
 represent the projection and trace of the 
 axis of rotation of a body whose plane of 
 rotation is that of the figure; and let A be 
 the projection of a point in the body, the Fig"27. 
 
 radius of whose circular path is O A. The 
 velocity of that point being = O A x angular velocity, let it be 
 represented by the straight line A Y perpendicular to O A. Let 
 B A be any direction in the plane of rotation parallel to which it 
 is desired to find the component of the velocity of A. From V 
 
 D
 
 GEOMETRY OF MACHINERY. 
 
 let fall V TJ perpendicular to B A; then A U represents the com- 
 ponent in question. Sometimes the more convenient way of 
 finding that component is the following : 
 
 From O let fall O B perpendicular to B A. Then A and B 
 represent a pair of rigidly connected points; therefore, according 
 to Article 24, the component velocities of A and B along A B are 
 equal. But B A, being perpendicular to O B, is the direction of 
 the whole velocity of B; therefore the component, along a given 
 straight line in the plane of rotation, of the velocity of any point 
 whose projection is in thai line, is equal to the whole velocity of the 
 point where a perpendicular from the axis meets that line. 
 
 The whole velocity of B is = O B x the angular velocity; and 
 the velocity-ratio of B to A, or, in other ~words, the ratio of the 
 component velocity of A along B A to the whole velocity of A, is 
 OB :OA. 
 
 The velocity of a point such as A in a rotating piece may be 
 resolved into components, oblique (see fig. 19) or rectangular (see 
 fig. 27) as the case may 
 be, by regarding the 
 velocity of A relatively 
 to O as the resultant 
 of the velocity of A 
 relatively to B, and of 
 that of B relatively to 
 O. The directions of 
 that resultant velocity 
 and its two components 
 are respectively per- 
 pendicular to A, 
 B A, and O B, and 
 their ratios to each 
 other are equal to 
 those of the lengths of 
 the same three lines. 
 This is a particular 
 case of a more general 
 
 Fig. 28. 
 
 proposition, viz., that the velocities of three points rdatively to each 
 other are proportional to the three sides of a triangle which make 
 with each other the same angles that the directions of those three 
 relative velocities do (A. M., 355). 
 
 In fig. 28, let O be the trace of the axis on a plane of rotation, 
 and A a point in the rotating piece, revolving in the circle O A, 
 so as to assume successively a series of positions such as 1, 2, 3, 4, 
 5, 6, 7, 8; and in each position of A, let the component velocity 
 A U, parallel to a fixed plane whose trace is the diameter 8 O 4, 
 be compared with the whole velocity of revolution, A Y.
 
 ROTATING PIECES COMPARATIVE MOTION. 35 
 
 Let 2 6 be a diameter perpendicular to 8 O 4 ; and through 
 A draw A Y parallel to 4 O 8 and A X parallel to 2 O 6. Then, 
 according to the principles already explained, the value of the 
 velocity ratio in question is, 
 
 AU _ XA _ QY 
 
 AV " OA ~ OA ; 
 
 and it is evident that this ratio is equal to nothing when A is at 
 the points 4 and 8, and to unity when A is at the points 2 and 6. 
 Further, if the velocity of revolution be considered as always 
 positive, and if the component velocity A U be considered as 
 positive when from left to right, and negative when from right to 
 
 A U 
 
 left, the ratio v is, 
 -A. V 
 
 in the quadrant 812, positive and increasing; 
 in the quadrant 234, positive and diminishing ; 
 in the quadrant 456, negative and increasing; 
 in the quadrant 678, negative and diminishing. 
 
 It thus undergoes a series of periodical variations. All this is 
 expressed in symbols by the formula, 
 AU 
 
 where t denotes the angle that the radius A makes at any 
 instant with the radius 8. 
 
 Other and more complex ways of resolving the motions of points 
 in rotating bodies into components will be considered in the next 
 chapter. 
 
 56. Comparative Motion of Two Rotating Pieces, and of Points 
 
 in them. In comparing together the rotations of two rotating 
 pieces without reference to the translations of points in them, their 
 comparative speed is expressed (as already stated in Article 46) 
 by the angular-velocity ratio, or ratio of the numbers of turns in a 
 given time ; which is also the reciprocal of the ratio of the periodic 
 times of revolution. When the axes are parallel, or nearly parallel, 
 the directional relation may be expressed simply by prefixing 4- 
 or to the velocity-ratio, according as the directions of rotation 
 are similar or contrary; but there are cases to be considered 
 further on, where the relative angular positions of the axes have to 
 be considered with precision. 
 
 When the translations of two points in two different rotating 
 pieces are compared, the directional relation is determined by 
 the fact, that each point moves in a direction normal to a plane 
 traversing itself and the axis about which it revolves; and that 
 the velocity of each point is proportional to its perpendicular 
 distance from that axis, and the speed of* rotation about that axis,
 
 36 GEOMETRY OF MACHINERY. 
 
 jointly. Hence, let a and a! denote the angular velocities of two 
 rotating pieces, or a pair of numbers proportional to those angular 
 velocities ; r and r 1 , the perpendicular distances of a pair of points 
 in those two pieces from their respective axes, or a pair of numbers 
 proportional to those distances ; and v and v , the respective velocities 
 of those two points, or a pair of numbers proportional to those 
 velocities; then the velocity-ratio of the points is, 
 
 v a' r' 
 
 v ar' 
 In order that a pair of points in a pair of rotating pieces may have 
 
 v' 
 equal velocities that is, in order that -\rnay be = 1, we must 
 
 make the radii inversely proportional to the angular velocities 
 
 XT. j. i r' a 
 
 that is, a r = a r, or - = .. 
 r a 
 
 SECTION IV. Screw-like Motion of Primary Pieces. 
 
 57. Helical or Screw-like Motion (A. M., 382,) is compounded 
 of rotation about a fixed axis, and of translation along that axis : 
 the advance (as the translation in a given time is called) bearing a 
 constant proportion to the rotation in the same time; in other 
 words, the moving piece advances along the axis of rotation 
 through an uniform length during each turn. 
 
 The subject of the resolution of screw-like motion into com- 
 ponents in other and more complex ways will be considered in 
 the next chapter. 
 
 58. General Figure of n Screw Pitch. (A. M., 471.) In order 
 that a primary moving piece may have screw-like motion, its 
 figure ought to be that of a true screw ; and it ought to turn in 
 a bearing of the same figure, fitting it accurately. The figure 
 of a screw may be described in genei-al terms as consisting of a 
 projection of uniform cross-section called the thread, winding in 
 successive coils round a circular cylinder. The best form of 
 section for the thread of a screw that is to be used as a primary 
 moving piece for producing helical motion only, and not as a 
 fastening, nor in " screw gearing," is rectangular. The forms 
 suited for other purposes will be considered later. There are two 
 sorts of screws, convex, or external, and concave, or internal; in 
 the former the thread winds round the outside of a cylindrical 
 spindle; in the latter it winds round the inside of a hollow cylinder. 
 When the word " screw " is used without qualification, an external 
 screw is usually meant; an internal screw is called a "nut." 
 When a primary moving piece is an external screw, its bearing is 
 an internal screw ; when the primary moving piece is an internal 
 screw, the bearing is an external screw. The truth or accuracy of
 
 SCREW COMPARATIVE MOTION. 
 
 37 
 
 figure of a screw depends mainly on the perfect uniformity of the 
 pitch; that is to say, the distance, measured parallel to the axis, 
 from any point in one coil of the thread to the corresponding 
 point in the next coil. For example, the pitch of the screw E, in 
 fig. 29, so long as it is measured parallel to the axis, may be 
 measured either from D to F, from E to G, from F to H, or from 
 G to K, or between any pair of corresponding points in two 
 successive coils; and it ought to be exactly the same wheresoever 
 it is measured. The pitch is also the uniform distance through 
 which the screw advances at each turn. 
 
 59. Right-Handed and Left-Handed Screws. A screw is said to 
 
 ^ be right-handed or left-handed 
 c) according as right-handed or 
 left-handed rotation is required 
 in order to make it advance ; 
 and this is a permanent dis- 
 tinction, and not dependent 
 on the position of the spec- 
 tator, as the distinction be- 
 tween right-handed and left- 
 handed rotation is (Article 48, 
 page 25). For example, in fig. 
 29, L is a left-handed screw, 
 and II a right-handed screw. 
 
 Fig. 29. 
 
 Most screws used in the arts are right-handed ; left-handed screws 
 are made for special purposes only. 
 
 60. Comparative Motion of a Point in a Screw. The principles 
 
 of the present Article apply not only to any point in the thread or 
 in the spindle of a screw, but to any point in a body that is rigidly 
 attached to the screw, so as to move along with the screw as one 
 piece ; such as a wheel or a lever fixed to and turning with the 
 screw. In fig. 29, let A B be the axis of the screw, and C a point 
 rigidly attached to it at the perpendicular distance C A from the 
 axis. Then, while the screw makes one turn the motion of the 
 point C is the resultant of two components at right angles to each 
 other: an advance, along with the whole screw, in a direction 
 parallel to the axis, through a distance equal to the pitch of the screw ; 
 and a revolution, round a circle described about the axis with the 
 radius A C, and having, therefore, the circumference 6'2832 A C. 
 In most questions of comparative motion connected with screws, 
 the quantity of most importance is the velocity-ratio of those two 
 components of the motion of a given point, and it is expressed as 
 follows : 
 
 velocity of revolution circumference 6-2832 A 
 
 velocity of advance 
 
 pitch 
 
 DF
 
 30 GEOMETRY OF MACHINERY. 
 
 61. Path of a Point in a Screw I,imnr Screw or Ilclix. A 
 
 point in, or rigidly attached to, a screw, traces a path which may 
 be called a screw-shaped line or linear screw. By mathematicians 
 it is called a helix. A. helix winds round in successive similar coils 
 upon a cylindrical surface described about the axis of rotation with 
 a radius equal to the perpendicular distance of the tracing point 
 from the axis. The distance, measured parallel to the axis, between 
 any two successive coils is everywhere the same, and is identical 
 with the pitch of the screw; and the angle of inclination of the 
 linear screw to the axis is everywhere the same. 
 
 Points in, or rigidly attached to, a screw, at equal distances from 
 the axis, trace by their motion equal and similar linear screws on 
 one and the same cylindrical surface. Points at unequal distances 
 from the axis trace different linear screws, inclined to the axis at 
 different angles, and situated on cylindrical surfaces of unequal 
 radii ; but the pitch of all those linear screws is the same. All the 
 edges, whether projecting or re-entering, of a screw-thread are 
 linear screws. 
 
 A linear screw may be traced on a cylindrical surface by any 
 mechanical contrivance which ensures that, while the cylinder 
 rotates, the tracing point shall advance along a line parallel to the 
 axis at a rate bearing a constant proportion to the rate of rotation. 
 This will be further considered in that part of this treatise which 
 relates to the construction of machinery. 
 
 A linear screw is the shortest line on the surface of a cylinder 
 between two points that are neither in one plane traversing the 
 axis nor in one plane perpendicular to the axis; and a cord or a 
 flexible wire stretched on a cylindrical surface between two such 
 points tends to assume of itself the figure of a linear screw. 
 
 62. Projection of a Linear Screw. The most useful projection 
 of a linear screw is that upon a plane traversing the axis, and is 
 drawn as follows : In fig. 30, let A B represent the axis of the screw. 
 Draw DAG perpendicular to A B, making A C = A D = the 
 radius of the cylindrical surface in which the helix is to be situated. 
 Draw D I and C F parallel to A B; those two lines will be the 
 traces of the cylindrical surface. About A, with the radius A C, 
 draw the semicircle C K D ; this represents the trace of one-half 
 of the cylindrical surface on a plane perpendicular to its axis, 
 "rabatted" upon the plane of projection. Divide the semicircle 
 into any convenient number of equal arcs (Article 51, page 27); 
 the greater the number of those divisions, the greater will be the 
 accuracy of the projection. In fig. 29 the semicircle is divided into 
 six equal arcs only; in practice a greater number will in general be 
 required. 
 
 On C F, or any other line parallel to the axis, lay off C E = the 
 intended pitch, of the screw, and divide it into twice as many
 
 HELIX OR LINEAR SCREW. 
 
 equal parts as there are equal divisions in the semicircle 
 CfKD. 
 
 Through the points of division, of the semicircle draw straight 
 
 s 
 
 lines parallel to A B; and through the points of division of the 
 pitch draw straight lines perpendicular to A B; the points of 
 intersection of successive pairs of those two sets of lines will be 
 points in the required projection of the linear screw, C G L E,
 
 40 GEOMETRY OF MACHINERY. 
 
 which can then be drawn through those points. This is the pro- 
 jection of one coil ; and as many successive coils as may be required 
 may be projected by simply repeating the same curve. In tig. 30 
 the projection, E H M F, of a second coil is shown; and it has 
 been constructed by laying off an uniform distance parallel to the 
 axis and equal to the pitch from each projected point of the first 
 coil; for example, G H, L M, E F. 
 
 The half coils on the nearer side of the cylinder, viz., G L E 
 and H M F, are drawn in thicker lines than the half coils, C G, 
 E H, on the farther side of the cylinder. The screw is right- 
 handed. Had it been left-handed, C G and EH would have been, 
 on the nearer side, and G E and H F on the farther side of the 
 cylinder. 
 
 63. Development of a Linear Screw. The development of any 
 figure drawn on a curved surface consists in supposing the sm-face 
 to be a flexible sheet, and drawing the appearance which the figure 
 would present if that sheet were spread out flat on a plane. Some 
 curved surfaces only are capable of being developed, such as a 
 cylinder, and a cone ; others, such as a sphere, are not. To draw 
 the development of the cylindrical surface in fig. 30, as bounded by 
 the two circles whose projections are C D and F I, draw C c per- 
 pendicular to A B, and equal in length to the circumference of the 
 cylinder (see Article 51, page 27), and complete the rectangular 
 parallelogram C c/F; this will be the required development of the 
 cylindrical surface. 
 
 To draw the development of one coil, C G E, of the linear screw, 
 take ce = the pitch C E; draw the straight line C e; this will be 
 the required development. To draw the development of the 
 second coil, E H F, take ef the pitch, and draw the straight 
 line Ey"; and so on for any required number of coils. 
 
 The uniform angle of inclination of the linear screw to the axis 
 is E C e = F E/. 
 
 One method of drawing a screw on a cylindrical surface is first 
 to draw its development on a sheet of some flexible substance, and 
 then to roll that sheet round a cylinder of the proper radius. 
 
 The several points in the development marked with small letters 
 are the respective developments of the points marked with the 
 corresponding capital letters in the projection. 
 
 To draw the development of the series of lines parallel to the 
 axis which pass through the points of division of the circumference, 
 divide C c into twice as many equal parts as the semicircle C D 
 is divided into, and draw straight lines parallel to C F through the 
 points of division, such as d i, q w, <fec. 
 
 The length of one coil of the screw is 
 
 C e - >J (circumference 2 + pitch 2 ).
 
 LINEAR SCREWS CURVATURE NORMAL PITCH. 41 
 
 64. The Radius of Curvature of a linear screw is found by the 
 following construction : In fig. 31, let A C be the radius 
 of the cylindrical surface in which the screw is situated. 
 Draw A Y, making the angle CAY equal to the angle 
 which the screw makes with a plane perpendicular to its 
 axis. Draw C Y perpendicular to A C, cutting A Y in 
 Y, and Y Z perpendicular to A Y, cutting A C produced 
 in Z. A Z will be the required radius of curvature. Its 
 length may also be found by calculation, as follows : 
 
 AZ = AC (1 + 
 
 pitch 2 \ 
 
 circumference 2 /' 
 
 65. Normal Pitch. (A.M. ,472.) By the normal pitch 
 of a linear screw is to be understood the distance from 
 one coil to the next, measured, not parallel to the axis, but along 
 the shortest line on the cylindrical surface between the two coils ; 
 that is to say, along another helix or linear screw which cuts all 
 the coils of the original linear screw at right angles. The normal 
 pitch is to be determined from the development of the screw, as 
 follows : In fig. 30, from c let fall c n perpendicular to C e; en will 
 be the normal pitch. The straight line c n is part of the develop- 
 ment of the normal helix, as it may be called. When produced, it 
 cuts E/j the development of the next coil, in o, and n o = c n is also 
 the normal pitch. By finding the intersections, such as p, of the 
 development of the normal helix with the series of straight lines 
 parallel to the axis, any number of points, such as P, in the pro- 
 jection of the normal helix, C N" O P, may be found if required. 
 The normal pitch may be calculated by the following formula : 
 
 C c ce circumference x pitch 
 
 iy> __ . i 
 
 C e length of one coil 
 
 The pitch of the normal helix, if required, may be found by pro- 
 ducing c p in fig. 30 till it cuts G F produced, and measuring the 
 distance of the point of intersection from C; and then its radius 
 of curvature may be found by a construction like that in fig. 31 ; 
 but in general it is more convenient to find these quantities by 
 calculation, as follows : 
 
 ... . , i ,. circumference 2 
 
 pitch of normal helix = -r. 7: . . ..- 
 pitch of original helix 
 
 radius of curvature of normal helix = A C ( 1 + : -. - : . .. ) . 
 
 \ pitch of ong. helix 2 / 
 
 The sum of the reciprocals of the radii of curvature of the 
 original helix and the normal helix is equal to the reciprocal of 
 
 the radius of the cylinder ; that is, to
 
 42 GEOMETRY OF MACHINERY. 
 
 The pitch of a screw as measured parallel to the axis may be 
 called the axial pitch, in order to distinguish it from the normal 
 pitch ; but when the word " pitch " is used without qualification, 
 axial pitch is always to be understood. 
 
 The several linear screws which exist in the figure of an actual 
 solid screw, or which are described by points in it or rigidly 
 attached to it, have all the same axial pitch ; but they have not 
 the same normal pitch except when they are situated on the same 
 cylindrical surface. 
 
 66. Divided Pilch. A screw sometimes has more than one 
 thread, in which case the distance between any coil of any one 
 thread and the next coil of the same thread is divided by the other 
 threads into as many parts as the total number of threads. In. 
 that case the distance from a point in one thread to the correspond- 
 ing point in the next thread may be called the divided pitch, to dis- 
 tinguish it from the distance between two successive coils of the same 
 thread, or pitch proper, which may, when required, be designated 
 as the total pitch. The advance of the screw at each revolution 
 depends on the total pitch only, in the manner already explained, 
 and is wholly independent of the number of threads and of the 
 divided pitch; so that division of the pitch does not affect the 
 motion of a screw as a primary piece. Its use in combinations of 
 pieces will be afterwards explained. 
 
 Division of the pitch of a linear screw is illustrated in fig. 30, 
 where two additional linear screw threads, marked by dotted lines, 
 are shown dividing the pitch of the original screw into three equal 
 parts. To avoid confusion, one only of the additional screw-lines 
 is lettered, viz., that marked QHSTUYW in the projection, 
 and qr, ~E,stu, ~Uvw, in the development. The other is un- 
 lettered. The divided axial pitch is C B, = ^ C E, and the divided 
 normal pitch c x = ^ en. 
 
 The several linear screw threads in a case of this kind are all 
 parallel and similar to each other; and in the development they are 
 represented by parallel straight lines. They divide the circum- 
 ference into as many equal parts as there are threads; and the 
 length of one of those parts may be called the circular or 
 circumferential pitch. In fig. 30, the circular pitch is represented 
 by the arc C Q', and by its development c q.
 
 43 
 
 CHAPTER III. 
 
 OF THE MOTIONS OF SECONDARY MOVING PIECES. 
 
 67. enerai Principles. (A. M., 383, 384,503.) In the present 
 chapter the general principles only of the motions of secondary 
 moving pieces in machines can be given, many of their most 
 important applications being reserved for that chapter which will 
 treat of " Aggregate Combinations in Mechanism," and some for 
 the chapter on " Elementary Combinations." The mechanism for 
 producing the motions of secondary moving pieces belongs wholly 
 to those later chapters. 
 
 Secondary moving pieces have already been defined (in Article 
 37, page 17) as those which are carried by other moving pieces, 
 or which have their motions not wholly guided by their con- 
 nection with the frame. Their motions, therefore, are not 
 restricted, like those of primary pieces, to translation in a straight 
 line, rotation about a fixed axis, and that combination of those 
 two motions which constitutes the motion of a screw with a fixed 
 axis; they comprehend translations along curved lines of various 
 figures, rotations about shifting axes, and various combinations of 
 translations and rotations. The paths of points, too, in secondary 
 pieces are not restricted to three forms the straight line, the circle, 
 and the helix ; they comprehend a great variety of curved lines, 
 both plane and of double curvature. The comparative motions 
 of any two points in a primary piece are constant. The com- 
 parative motions of two points in a secondary piece very often, 
 vary from instant to instant as the piece changes its position. 
 
 In many cases the motions of secondary pieces are partially 
 guided or restricted. For example, a secondary piece may be 
 so guided that all its movements take place parallel to a fixed 
 plane; in which case its motions are restricted to translations 
 parallel to the fixed plane, and rotations about axes perpendicular 
 to it; and the paths of its points are restricted to lines, straight or 
 curved, in or parallel to that plane; and this restricted case is by 
 far the most common in mechanism. Another kind of restriction 
 on the movements of a secondary piece is when it turns about a 
 ball and socket joint, or some equivalent contrivance, so that one 
 point at the centre of the joint is kept fixed: in this case its 
 motions are restricted to rotations about axes traversing that 
 fixed point; and the motions of points in it are restricted to
 
 44 
 
 GEOMETRY OP MACHINERY. 
 
 curves situated in spherical surfaces described about the fixed point. 
 Cases in which the movements of secondary moving pieces are not 
 restricted in one or other of those ways are comparatively rare. 
 
 The geometrical problems relating to the motions of secondary 
 moving pieces may be divided into the two following classes: 
 
 I. When the motions, in most cases, of two, and at furthest of 
 three, points in a secondary moving piece are given, and it is 
 required to find the motion of any other point in the piece, or of 
 the piece as a whole. All problems of this class depend for their 
 solution on the principle of Article 54, page 32. 
 
 II. When there are two moving pieces or moving points, C and 
 B, the frame of the machine being denoted by A, and two out of 
 the three motions of A, B, and C relatively to each other being 
 given, it is required to find the third of those motions. All 
 problems of this class depend for their solution on the principle 
 (already stated in Article 42, page 21) that the motion of C rela- 
 tively to A is the resultant of the motions of B relatively to A, 
 and of C relatively to B. 
 
 68. Translation of Secondary IWovIng Piece*. (A. M. t 369.) If, 
 
 in a moving piece whose movements are not restricted, the direc- 
 tions of motion of three points not in the same straight line are 
 parallel to each other and oblique to the plane of the three points ; 
 or if, in a moving piece restricted to movements parallel to one 
 plane, the motions of two points are parallel to each other and 
 oblique to the line of connection of the points ; then the motion of 
 the whole piece is a translation. All the points in the piece describe 
 equal and similar paths, straight or curved; and all, at a given 
 instant, move with equal velocities in parallel directions. The 
 motion of any pair of points in the moving piece relatively to 
 each other is nothing; and their comparative motion consists 
 in the directional relation of parallelism and the velocity-ratio 
 of equality. 
 
 To exemplify the translation of all the points of a moving piece 
 in equal and similar curved paths, we may take the case of a 
 coupling-rod (fig. 32) which connects together a pair of equal 
 
 Fig. 32.
 
 TEMPORARY AND INSTANTANEOUS AXES. 45 
 
 cranks, AC, B D, and has its effective length, C D, equal to the 
 perpendicular distance, A B, between the axes of rotation of the two 
 cranks. The motion of that coupling-rod is one of translation, in 
 which all the particles describe with equal speed equal and similar 
 circles of the radius AC = B D, in planes perpendicular to the 
 axes A and B. The same is the case with any particle rigidly 
 attached to the coupling-rod; such as F, which revolves in a circle 
 of the radius E F = A C ; so that, for example, the points C, D, 
 and F move simultaneously through the equal and similar arcs 
 C C', D D', F F'. 
 
 69. Rotation Parallel to a Fixed Plane Temporary Axis 
 
 instantaneous Axis. The cases next in order as to complexity are 
 those in which all the movements of the piece are parallel to a 
 fixed plane; and the following are the problems which present 
 themselves : 
 
 I. Given, tlie paths of two points in a moving piece, the distance 
 between their projections on the plane of motion, and two successive 
 positions of one of them, to find the temporary axis of motion of t/ie 
 piece. 
 
 In fig. 33, let the plane of projection and of motion be that of 
 
 the paper, and let the partly dotted lines A A' and B B' be the 
 projections of the paths of the two points, which may be straight 
 lines or plane curves of any figure, subject only to the limitation 
 that the distance between the points is invariable. Let A and A' 
 be the given two successive positions of one of the points. About 
 A and A' respectively, with the projection of the line of connection 
 as a radius, draw circular arcs cutting the projected path of the 
 other point in B and B'; these will be the projections of the two 
 successive positions of the second point; and the straight lines 
 A B and A' B' will be the projections of the line of connection in 
 the two successive positions of the moving piece. Draw the
 
 46 
 
 GEOMETRY OF MACHINERY. 
 
 Fig. 35. 
 
 straight lines A A' and B B' ; bisect them in C and D, through 
 which points draw C I perpendicular to A A' and D I perpen- 
 dicular to B B', meeting each other in I. Then, because A I = 
 
 A I and B' I = B I, I represents 
 the same point in the two positions 
 of the piece; and therefore I is the 
 projection and the trace of a line per- 
 pendicular to the plane of motion, 
 whose position is the same after the 
 motion of A to A' and of B to B' that 
 it was before. That line may be called 
 the Temporary Axis of Motion of the 
 moving piece, x because the change of 
 position of the piece is the same as if 
 it had been turned through an angle 
 A I A' = B I B' about that line. 
 
 Let E be the point of intersection 
 of A B and A' B'. Then the straight 
 line E I bisecting the angle A E B' 
 traverses the temporary axis I; and 
 this affords a means of finding that axis when C I and D I cut each 
 other at an angle so oblique as to make it difficult to determine 
 precisely their point of intersection. 
 
 When B B' is parallel to A A', as in figs. 34 and 35, C I and 
 D I become parts of one straight line, and have no intersection ; 
 and then the point I is determined by its coinciding with E. In 
 most cases of this kind it is necessary that the two successive 
 positions of B should be given as well as those of A. 
 
 II. Given (in fig. 36), t/ie projections A and B, at a given instant, 
 
 of two j)oints in a moving piece 
 on the plane of motion, and the 
 simultaneous directions of mo- 
 tion of those points, A a and Bb, 
 to find the instantaneous axis 
 of the moving piece ; and thence 
 r to deduce the comparative mo- 
 tions, at the given instant, of the 
 given points, and of any other 
 points in the moving piece. 
 
 If the simultaneous direc- 
 tions of motion of the given 
 points are perpendicular to 
 their line of connection, the 
 problem requires additional 
 data for its solution, which will be stated in Eule III. If those 
 directions are parallel to each other, and not perpendicular to the 
 
 Fig. 36.
 
 INSTANTANEOUS AXIS. 47- 
 
 line of connection, the motion of the piece is one of translation, 
 like that referred to in Article 68, page 44. The present rule com- 
 prehends all cases in which the given directions are not parallel to 
 each other. 
 
 Through A and B draw A I and B I perpendicular respectively 
 to A a and B b, and cutting each other in I. Then I will be the 
 projection and the trace on the plane of motion of the required 
 INSTANTANEOUS AXIS : that is to say, of a line such that the motion 
 of the piece at the instant in question is one of rotation about that 
 axis. 
 
 An instantaneous axis is so called because it is an imaginary 
 line which is continually changing its position, both relatively to 
 the frame of the machine and relatively to the secondary piece 
 to which it belongs; so that it occupies any particular position, 
 whether relatively to the frame or relatively to the secondary 
 piece, at a particular instant only. 
 
 The comparative motions at the given instant of points in the 
 secondary piece are deduced from the principle that the velocities 
 of those points are proportional in magnitude and perpendicular in 
 direction to the perpendiculars let fall from the points upon the 
 instantaneous axis. For example, let A. a, B b, c, D d, E e, 
 represent the directions and velocities of the points whose projec- 
 tions are A, B, C, D, E; then 
 
 Aa : B6 : Cc : D d : Ee 
 are respectively proportional and perpendicular to 
 
 From I let fall I D perpendicular to the projection, A B, of the 
 line of connection of the given points. Then all points whose 
 projections are at D are at the given instant in the act of moving 
 parallel to A B ; and all points whose projections are in A B, or 
 in A B produced, such as A, B, and E, have for their component 
 velocities along A B velocities equal to the velocity of D ; that 
 is to say, 
 
 D d = A d' = B d" = E d'" 
 
 a consequence which follows also from the principle of Article 53, 
 page 31. 
 
 The components perpendicular to A B of the velocities of points 
 whose projections are in that line, such as A, B, and E, are 
 proportional to the distances of those projections from D; that is 
 to say, if A/", B g, and E h represent those transverse component 
 velocities, we have the proportions, 
 
 DA :DB :DE 
 
 and the points /, h, D, g are in one straight line.
 
 48 GEOMETRY OF MACHINERY. 
 
 Hence, when A I and B I form an angle with each other so 
 oblique as to make it difficult to determine precisely their point of 
 intersection, we may proceed as follows to increase the precision of 
 that determination: Lay off any convenient equal distances, A d' 
 = Bo?", along AB from A and B respectively, to represent the 
 longitudinal component of their velocities. Then complete the 
 rectangular parallelograms Ad'af, T$d"bg; draw the straight 
 line fg, cutting A B in D. Then from D perpendicular to A B 
 draw I) I ; this line will traverse the instantaneous axis, and will 
 increase the precision with which it is determined. 
 
 This last way of considering the motion of the piece is equivalent 
 to regarding that motion as compounded of a rotation about an axis 
 at D and a translation of that axis, and of the whole body along 
 with it, with the velocity represented by D d. 
 
 III. Given (in fig. 37 or fig. 38), the projections A and B, at a 
 given instant, of two points in a moving piece on tlie plane of motion, 
 and the ratio of t/teir velocities, which are both perpendicular to the 
 
 Fig. 37. Fig. 38. 
 
 projection, A B, of their line of connection, to find tlie instantaneous 
 axis of motion of the piece. Perpendicular to A B draw the straight 
 lines A a, B6, bearing to each other the given proportion of the 
 velocities of the two points : draw the straight line a b ; the point 
 of intersection, I, of A B and a b (produced if necessary) will be 
 the projection and trace on the plane of motion of the required 
 instantaneous axis. 
 
 That axis may then be used as in the preceding Kule to determine 
 the comparative motions of any set of points in the moving piece. 
 
 70. Rotation about a Fixed 1'oint. Every possible motion of 
 a rigid body relatively to a point in the body is reducible to 
 rotation about an axis, permanent, temporary, or instantaneous, 
 as the case may be, which traverses that point. This is proved 
 by showing that the following problem is always capable of 
 solution : 
 
 I. Given, at any instant, the directions of motion of any two 
 points, B, C (fig. 39), in a rigid body relatively to a point, A, in the
 
 INSTANTANEOUS AXIS. 
 
 body, to find the instantaneous axis of the motion of the whole body 
 relatively to A. In the first place, it is to be observed that when 
 the motions of three points in a rigid body are determined, the 
 
 A 
 
 Fig. 40. 
 
 Fig. 39. 
 
 motion of the whole body is determined ; for the distances of any 
 fourth point in the body from those three points being invariable, 
 the position of that fourth point at every instant is completely 
 determined by the positions of the three points. 
 
 In order that the solution may be put in the simplest possible 
 form, let the plane of the three points themselves, or a plane 
 parallel to it, be taken for one plane of projection; and in fig. 39 
 let A, B, C be the projections of the three points on that plane. 
 For a second plane of projection, take a plane perpendicular to the 
 first plane, and traversing B C, and let A', B', and C' (which are 
 in one straight line) be the projections of the three points on that 
 second plane; so that B' C' is parallel to B C, and A A', B B', and 
 C C ' are perpendicular to B C. 
 
 Because the instantaneous axis must traverse A, it is obvious 
 that A B and A C are the traces on the first plane of projection of 
 two planes traversing the instantaneous axis and the points B and! 
 C respectively; and also, that if B 6 and C c are the projections' 
 on the first plane of projection of the directions of motion of B" 
 and C at the given instant, those projections must be perpendicular 
 to A B and A C. Let B' 6' and C 7 c represent the projections of 
 the directions of motion of B and C on the second plane of 
 projection. Draw B' I' and C' I' perpendicular respectively to 
 B' b' and C' c', and meeting each other in I'; then B' I' and C' I* 
 are the traces, on the second plane of projection, of two planes 
 perpendicular respectively to the instantaneous directions of motion
 
 60 GEOMETRY OP MACHINERY. 
 
 of B and C ; that is to say, of the two planes already mentioned, 
 which traverse the instantaneous axis and the points B and C 
 respectively ; and I' is tJie trace of the instantaneous axis on the 
 second plane of projection. From I' let fall I' I perpendicular to 
 B C; then I is the projection of I' on the first plane of projection. 
 Draw the straight lines A I, A' I': those are the projections of 
 the instantaneous axis. 
 
 II. To draw the projections of the points B and C on a plane 
 perpendicular to the instantaneous axis, and to find the comparative 
 motion of those points. In BC. fig. 39, take IF = I' A'; draw 
 A G parallel and F G perpendicular to B C, cutting each other in 
 G; join I G : this line will be the rabatment of I A. From B' 
 and C' let fall B' H and C' EL perpendicular to I' A' (produced if 
 required). In B C take I L = I' H, and I M = I' K; then G, L, 
 and M will represent the respective projections of A, B, and C 
 upon a plane which traverses the instantaneous axis, and is per- 
 pendicular to the second plane of projection. From L and M let 
 fall L N and M P perpendicular to I G. Then, in fig. 40, let 
 the paper represent a plane of projection perpendicular to the in- 
 stantaneous axis : let A be the trace and projection of that axis, and 
 A I the trace of the plane already mentioned as being perpendicular 
 to the second plane of projection in fig. 39. Make A I in fig. 40 
 = N L in fig. 39, and A m in fig. 40 = P M in fig. 39. Draw 
 I B in fig. 40 perpendicular to A I in fig. 40 and = H B 1 in fig. 39 ; 
 also m C in fig. 40 perpendicular to A in fig. 40 and K C' in 
 fig. 39. Join A B, A C. Then B and C in fig. 40 will be tlie 
 projections required; and the velocities of B and C relatively to 
 A will be perpendicular in direction and proportional in magnitude 
 to A B and A C respectively. 
 
 Another mode of finding the comparative motion of A and B is 
 the following : According to the principle of Article 54, page 32, 
 the component velocities of B and C along their line of connection, 
 B 0, are equal. Therefore, in fig. 39, lay off along B C and B' C' 
 tbe equal distances Be?, C e, B' d', C' e\ to represent that com- 
 ponent ; then draw d' b' db, c' e ce perpendicular to B C, cutting 
 B b in 6, B b' in b', C c in c, and C' c' in c'; then B b and B' b' will 
 be the projections of the velocity of B relatively to A; and C e and 
 C' e' will be the projections of the velocity of C relatively to A 
 Then, by the rule of Article 19, page 7, find the lengths of the 
 lines of which B b and B' b', C c and C' c' are the projections; the 
 ratio of those lengths to each other will be the velocity-ratio of the 
 two points. 
 
 71. Unrestricted Motion of a Rigid Body How complicated 
 
 soever the motion of a rigid body may be, it may always be 
 regarded as made up of a change of position of the body as a 
 whole that is, a translation of the body, and a change of position of
 
 ROLLING BODY. 5? 
 
 the body relatively to some point in it; that is to say, as has been, 
 shown in the preceding Article, a rotation about an axis traversing 
 that point, which axis may be either permanent, temporary, or 
 instantaneous. It will be shown further on that a rotation and a 
 translation parallel to the plane of rotation, when compounded, are 
 equivalent to a motion of rotation about an instantaneous axis 
 perpendicular to that plane. Hence it follows, that if a rigid 
 body has any translation combined with any rotation, the transla- 
 tion is to be resolved into two components, one parallel and one 
 normal to the plane of rotation ; when the parallel component of 
 the translation, being combined with the rotation, will be equivalent 
 to a rotation about a new instantaneous axis perpendicular to the 
 plane of rotation; and the whole motion of the body will be 
 equivalent to this new rotation combined with the normal com- 
 ponent of the translation, the direction of which component is 
 parallel to the axis. In short, how complex soever the motion of 
 a rigid body may be, it is at each instant equivalent to a helical 
 or screw-like motion about an instantaneous axis. The most 
 comprehensive problem that can occur respecting the unrestricted 
 motion of a rigid body at a given instant is this: given the 
 simultaneous velocities and directions of motion of three points (say 
 A, B, and C), to find the motion of the whole body at that instant. 
 This is to be solved by choosing one of the points (say A), and 
 regarding it for the time as fixed, and determining the motions of 
 B and C relatively to A. Then, by means of the rules of Article 
 70, the position is to be determined of an instantaneous axis 
 traversing A; when the motion of the whole body will be com- 
 pounded of a rotation about that axis and a translation of the 
 point A, carrying the instantaneous axis and the whole body along 
 with it. 
 
 72. Instantaneous Axis of a Rolling Body. The following prO- 
 
 position has many useful applications in mechanism : The instan- 
 taneous axis of a rigid body, which rolls without 
 slipping upon the surface of another rigid body, 
 passes through all the points in which those bodies ' 
 touch each other; for the particles in the rolling 
 body which at any given instant touch the fixed 
 'body without slipping must have, at that instant, 
 no velocity, and must therefore be, at that instant, 
 in the instantaneous axis. ~ . 
 
 Moreover, because an instantaneous axis is a 
 straight line, it follows that, in order that two surfaces which 
 touch each other at more than one point may roll on each other 
 without slipping, their points of contact must all lie in one straight 
 line : a property of plane, cylindrical, and conical surfaces only, 
 the terms " cylindrical " and " conical " being used in the general
 
 52 GEOMETRY OP MACHINERY. 
 
 sense, which comprehends cylinders and cones with bases of any 
 figure, as well as those with circular bases. 
 
 In fig. 41, let the plane of the paper represent a plane of 
 projection perpendicular to the straight line in which the fixed 
 and the rolling surfaces touch each other; let T be the projection, 
 and trace of that straight line, which is the instantaneous axis of 
 the rolling body. Let A be the projection at a given instant of a 
 point in the rolling body; then at that instant A is moving with 
 a velocity proportional to A T, and in a direction perpendicular to 
 the plane traversing A and the instantaneous axis, of which plane 
 A T is the trace. 
 
 It follows that the path traced by a point such as A in a rolling 
 body is a curve w/tose normal, A T, at dny given point, A, passes 
 through t/te corresponding position, T, of the instantaneous axis. 
 Curves of this class are called rolled curves; and some of them are 
 useful in mechanism, as will be explained farther on. 
 
 73. Composition of Rotation with Translation. From Article 52, 
 
 page 30, it appears that the single rotation of a body about a 
 fixed axis (such as O, fig. 19, page 26) may be regarded as com- 
 pounded of a rotation with equal angular velocity about a moving 
 axis parallel to the fixed axis (such as that whose trace is A, fig. 
 19), and a translation of that moving axis carrying the body along 
 with it in a circle round the fixed axis of the radius O A. A 
 similar resolution of motions may be applied to rotation about 
 an instantaneous axis. For example, the rotation of the rolling 
 body in fig. 41 about the instantaneous axis, T, may be conceived 
 to be made up of a rotation about another axis, C, parallel to the 
 instantaneous axis, and a translation of that axis. 
 
 The present Article relates to the converse process, in which 
 there are given a rotation of a secondary piece about an axis 
 occupying a fixed position in. the piece, and a translation of that 
 axis relatively to the frame in a direction perpendicular to itself 
 that is, parallel to the plane of rotation ; and it is required to find, 
 at any instant, the instantaneous axis so situated that a rotation 
 about it with the same angular velocity shall express the resultant 
 motion of the piece. 
 
 In fig. 41, let the plane of the paper be the plane of motion, 
 and let C be the projection and trace of the moving axis moving 
 relatively to the frame, but fixed as to its position in the secondary 
 piece. Let C U be the direction of the translation of that axis, 
 carrying the moving piece with it ; and let the velocity of transla- 
 tion be so related to the angular velocity of rotation as to be equal 
 to the velocity of revolution about the axis C, of a particle whose 
 
 ~ m velocity of translation -p. 
 
 distance from that axis is C 1 = *-s ; rr . Draw 
 
 angular velocity 
 
 C T of the length so determined, in a direction perpendicular
 
 ROLLING CYLINDER AND PLANE. 53 
 
 to C TJ, and pointing towards the right or towards the left of C U, 
 according as the rotation is right-handed or left-handed. Then T 
 will be the projection and trace qftJte required instantaneous axis; 
 so that if A is the projection of any point in the moving piece, 
 the direction of motion of that point is perpendicular to the plane 
 whose trace is A T; and the velocity-ratio of A and C is 
 
 AV _ TA 
 OU ~ T C' 
 
 The proof that T is the instantaneous axis is, that any particle 
 whose projection, at a given instant, coincides with T is carried 
 backward relatively to C by the rotation with a speed equal and 
 opposite to that with which C is carried forward by the translation; 
 so that the resultant velocity of every particle at the instant when 
 its projection coincides with T is nothing. 
 
 74. Rolling of a Cylinder on a Plane Trorlioid Cycloid. (A. 
 
 M., 386.) Every straight line parallel to the moving axis C, in a 
 cylindrical surface described about C with the radius C T, becomes 
 in turn the instantaneous axis. Hence the motion of the body is 
 the same with that produced by the rolling of such a cylindrical 
 surface on a plane, FTP, parallel to C and to U, at the distance 
 CT. 
 
 The path described by any point in the body, such as A, which 
 is not in the moving axis C, is a curve well known by the name of 
 trochoid. The particular form of trochoid, called the cycloid, is 
 described by each of the points in the rolling cylindrical surface. 
 
 75. Rolling of a Plane on a Cylinder; Involute Spirals. (A. 
 M., 387.) Another mode of representing 
 
 the combination of rotation with transla- 
 tion in the same plane is as follows : In 
 fig. 42, let O be the projection and trace 
 on the plane of motion of a fixed axis, 
 about which let the plane whose trace and 
 projection is O C (containing the axis O) 
 rotate (righthandedly, in the figure) with 
 a. given angular velocity. Let a secondary 
 piece have relatively to the rotating plane, 
 nnd in a direction perpendicular to it, a 
 translation with a given velocity. In the o 
 
 plane O C, and at right angles to the axis Fig. 42. 
 
 ^ . i f\m velocity of translation . . ,. . ,,,,-, 
 
 O, take O T = . , : , in such a direction that the 
 
 angular velocity 
 
 velocity 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 which the secondary piece has relatively to the rotating
 
 GEOMETRY OF MACHINERY. 
 
 plane. Then each point in the secondary piece which arrives at 
 the position T, or at any position in a line traversing T parallel to 
 the fixed axis O, is at rest at the instant of its occupying that 
 position; therefore the line traversing T parallel to the fixed axis 
 O is the instantaneous axis; the motion at a given instant of any 
 point in the secondary piece, such as A, is at right angles to the 
 plane whose trace is A T, perpendicular to the instantaneous axis ; 
 and the velocity of that motion bears to the velocity of the trans- 
 lation the ratio of T A to T O. 
 
 All the lines in the secondary piece which successively occupy 
 the position of instantaneous axis are situated in a plane of that 
 body, FTP, perpendicular to O Cj and all the positions of the 
 instantaneous axis are situated in a cylinder described about O 
 with the radius O T ; so that the motion of the secondary piece is 
 such as is produced by the rolling of the plane whose trace is P P 
 ow the cylinder whose radius is O T. Each point in the secondary 
 piece, such as A, describes a plane spiral about the fixed axis 0. 
 For each point in the rolling plane, P P, that spiral is the involute of 
 the circle whose radius is O T, being identical with the curve traced 
 by a pencil tied to a thread while the thread is being unwrapped 
 from the cylinder. For each point whose path of motion traverses 
 the fixed axis O that is, for each point in a plane of the secondary 
 piece traversing O and parallel to P P the spiral is Archimedean, 
 having a radius- vector increasing by a length eqxial to the circum- 
 ference of the cylinder at each revolution. 
 
 76. Composition of Rotations about Parallel Axes. (^1. M. y 388.) 
 
 In figs. 43, 44, and 45, let O be the projection and trace on the 
 
 Fig. 43. 
 
 Fig. 44. 
 
 Fig. 45. 
 
 plane of rotation of a fixed axis, and C the trace of a plane 
 traversing that axis, and rotating about it with the angular velocity 
 a. Let C be the projection and trace of an axis in that plane, 
 parallel to the fixed axis O; and about the moving axis C let a 
 secondary piece rotate with the angular velocity b relatively to the
 
 COMPOSITION OP ROTATIONS ABOUT PARALLEL AXES. 55 
 
 plane O C ; and let the directions of the rotations a and b be distin- 
 guished by positive and negative signs. The body is said to have the 
 rotations about the parallel axes O and C combined or compounded, 
 and it is required to find the result of that combination of parallel 
 rotations. 
 
 Fig. 43 represents the case in which a and b are similar in 
 direction : fig. 44, that in which a and b are in opposite directions, 
 and b is the greater; and fig. 45, that in which a and b are in 
 opposite directions, and a is the greater. 
 
 Let the trace C of the rotating plane be intersected in T by a 
 straight line parallel to both axes, in such a manner that the 
 distances of T (the trace of that line) 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: b : :GT :OT (1.) 
 
 When a and b are similar in direction, let T fall between O and C, 
 as in fig. 43 ; when they are contrary, beyond, as in figs. 44 and 45. 
 Then the velocity of the line T of the plane O C is a O T; and 
 the velocity of the line T of the secondary piece, relatively to the 
 plane O C, is b ' C T, equal in amount and contrary in direction to 
 the former; therefore each line of the secondary piece 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 b to represent arithmetical magnitudes, 
 and affix explicit signs to denote their directions, the direction of c 
 will be the same with that of the greater ; the case of fig. 43 will 
 be represented by the equation 2, already given; and those of figs. 
 44 and 45 respectively by 
 
 c = b a' } c = a b (2 A.) 
 
 The relative proportions of a, b, and c, and of the distances 
 between the fixed, moving, and instantaneous axes, are given by 
 the equation 
 
 a :b :c ::CT :OT :OC (3.) 
 
 The motion of any point, such as A, in the secondary piece, 
 according to the principle of Article 72, is at each instant at right 
 angles to the plane whose trace is A T, drawn from the point A
 
 56 GEOMETRY OF MACHINERY. 
 
 perpendicular to the instantaneous axis ; and the velocity of that 
 motion is 
 
 v = c -AT (4.) 
 
 77. Rolling of a Cylinder on a Cylinder Epitrochoids Epicycloids. 
 
 (A. M., 389.) All the lines in the secondary piece which succes- 
 sively occupy the position of instantaneous axis are situated in a 
 cylindrical surface described about C with the radius C T ; 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 secondary piece is that which is produced 
 by rolling the former cylinder on the latter cylinder. 
 
 In fig. 43 a convex cylinder rolls on a ^convex cylinder ; in fig. 
 44 a smaller convex cylinder rolls in a larger concave cylinder; 
 in fig. 45 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 C traces a circle. 
 
 When a cylinder rolls, as in fig. 44, within a concave cylinder 
 of double its radius, each point in the surface of the rolling cylinder 
 moves backwards and forwards in a straight line, being a diameter 
 of the fixed cylinder; each point in the axis of the rolling cylinder 
 traces a circle of the same radius with that cylinder; and each other 
 point in or attached to the rolling cylinder traces an ellipse of 
 greater or less eccentricity, having its centre in the fixed axis O. 
 This principle has been made available in instruments for drawing 
 and turning ellipses. 
 
 There is one case of the composition of rotations about parallel 
 .axes in which there is no instantaneous axis; and that is when the 
 two component rotations are of equal speed and in contrary direc- 
 tions; for then the resultant is simply a translation of the secondary 
 piece along with the moving axis. This may be illustrated by 
 referring to fig. 32, page 44, where the translation of the coupling- 
 rod C D may be looked upon as the resultant of the combination 
 of the rotation of the crank A C about A, with an equal and 
 contrary rotation, relatively to tlie crank, of C D about C. 
 
 78. Curvature of Involutes of Circles, Epicycloids, niitl Cycloids. 
 
 It is often useful to determine the radius of curvature of a rolled 
 curve at a given point, especially where the fixed curve and rolling 
 curve are circles, and where the tracing point is in the circum- 
 ference of the rolling curve. 
 
 In the case of the involute of a circle, the radius of curvature at
 
 CURVATURE OF ROLLED CURVES. 
 
 57 
 
 a given point is simply the length of a normal to the curve at that 
 point measured to the point where that normal touches the circle; 
 that is to say, it is the length of the straight part of the thread 
 used in drawing the involute. 
 
 In the case of a cycloid (traced by a point in the circumference 
 of a cylinder which rolls on a plane) the radius of curvature at a 
 given point is twice the length of the normal measured from that 
 point to the corresponding instantaneous axis. 
 
 In the case of an epicycloid the construction for finding the 
 radius of curvature is shown in fig. 46 ; the right-hand division of 
 the figure giving the construction for an external epicycloid, I A, 
 traced by a point, A, in the surface of a cylinder, the trace of 
 
 Fig. 46. 
 
 whose axis is C, rolling outside a fixed cylinder, the trace of whose 
 axis is O ; and the left-hand division giving the construction for 
 an internal epicycloid, I A', traced by a point, A', in the surface of 
 a cylinder, the trace of whose axis is C', rolling inside the same 
 fixed cylinder. The following description applies to both divisions 
 of the figure: it being observed that at the left-hand side the 
 letters are accented :
 
 58 GEOMETRY OF MACHINERY. 
 
 Let T be the trace of the instantaneous axis, or line of contact 
 of the cylinders, at the instant when the tracing point is at A; so 
 that A T is the normal to the epicycloid at A, and O T and C T 
 the radii of the fixed and rolling cylinders, being two parts of one 
 straight line. Through draw O E parallel to A C. Bisect T 
 in D, and draw the straight line A D E, cutting O E in E. Through 
 E draw E F parallel to T, and cutting A T (produced as far as 
 required) in F. Then A F will be the radius of curvature of the 
 epicycloid at the point A. 
 
 The following formula serves to find A F by calculation; 
 
 AT-OC* 
 AF = -CD x (L) 
 
 It is sometimes more convenient to calculate the distance, T F, of 
 the centre of curvature, F, from the instantaneous axis, T, and that 
 is done by the following formula : 
 
 AT-OD AT-OT. 
 CD 2CD 
 
 the use of which, in designing the teeth of wheels by Mr. Willis's 
 method, will appear farther on. 
 
 79. T JDraw Rolled Curves. A rolled curve may be drawn by 
 actually rolling a disc of the form of the rolling curve, carrying a 
 suitable tracing point, upon the edge of a disc of the form of the 
 fixed curve. But it needs much care to perform that operation with 
 accuracy, except with the aid of machinery specially contrived for 
 the purpose, such as is to be found in certain kinds of turning lathes. 
 
 For ordinary purposes in designing machinery, approximate 
 methods of drawing rolled curves are used, such as the following : 
 
 I. To draw approximately a rolled curve by the Jielp of tangent 
 circles. In fig. 47, let A B be the fixed curve, and A D the 
 rolling curve, touching the fixed curve at A, which is also the 
 position of the tracing point at starting. The curve A D rolls 
 from A towards B; and it is required to draw approximately the 
 curve traced by the point A. By Rule III. of Article 51, page 
 29, lay off on. each of the two curves A B and A D a series of 
 equal arcs, A 1, 12, 23, 34, <kc. Measure the straight chord from 
 1 to A on the curve A D, and with 11 = 1A as a radius, and the 
 point 1 on the curve A B as a centre, draw so much of a circle as 
 lies near the probable position of the rolled curve ; measure the 
 straight chord from 2 to A on A D, and with 22 = 2 A as a radius, 
 and the point 2 on the curve A B as a centre, draw in like manner 
 part of a circle; and go on, in the same way, drawing a series of 
 
 * The proof of this is as follows : Let the radius of the rolling cylinder, 
 C A = C T = r; let tliat of the fixed cylinder, O T = K, which is to be
 
 TO DRAW EOLLED CURVES. 
 
 circular arcs with the points 1, 2, 3, 4, &c., in the fixed curve A B, 
 for their centres, and for their radii the lines 11, 22, 33, 44, &c., 
 respectively equal to the distances 1 A, 2 A, 3 A, 4 A, &c., as 
 measured between points on the rolling curve. Then, with the 
 
 Kg. 47. 
 
 regarded as positive or negative according as the rolling cylinder rolls on the 
 outer or inner surface of the fixed cylinder; let the instantaneous radius or
 
 CO GEOMETRY OF MACHINERY. 
 
 free hand, or with the help of a bent spring, draw a curve, A E, 
 so as to touch all those circular arcs j this will be very nearly the 
 rolled curve required. 
 
 The curve A E is called the "Envelope" of the series of area 
 that it touches. 
 
 II. To find a series of points in a, rolled curve. Draw a series of 
 tangent circular arcs as in the preceding rule; then draw the 
 several normals, 11, 22, 33, 44, <fcc., as radii of those arcs; the 
 direction of each normal being determined by the principle, that at 
 the point where it meets the fixed curve A B, it makes an angle 
 with a tangent to that curve equal to the angle which the corre- 
 
 normal of the epicycloid, T A = p; and let the required radius of curvature, 
 AF = /o. 
 
 Let the angular velocity of the rolling cylinder, relatively to the rotating 
 plane O C, be denoted by b, and that of the plane C by a, so that the 
 resultant angular velocity of the rolling cylinder is a + b. Then, because 
 the angle C T A is the complement of one-half of the angle T C A, it is 
 
 evident that the angular velocity of T A is a + 5. But according to Article 
 
 2 
 76, a R = 6 r; therefore 
 
 In any indefinitely short time, d t, the normal sweeps through an angle 
 whose value in circular measure is 
 
 and the point A traces an arc of the length 
 
 ds=(a + b)pdt = b[l + 
 therefore the radius of curvature of the epicycloid at the point A is 
 
 + *j AT- PC 
 
 This formula is made to comprehend the case of a cycloid by making 
 R = oo, when it becomes p = 2 p; and that of the involute of a circle by 
 making r = oo, when we have p = p. When the epicycloid is internal, and 
 R and r denote arithmetical values of those radii, the sign is to be sub- 
 stituted for + both in the numerator and in the denominator of the formula. 
 The symbolical expression for equation 2 of the text is 
 
 with the same understanding as to the sign in the denominator. In the case 
 already referred to at the end of Article 77, when a cylinder rolls inside a 
 cylinder of twice its diameter, we have R. = 2 r, and the denominator of 
 the expression for p becomes = 0; showing that the radius of curvature is 
 infinite ; or, in other words, that the epicycloid traced is a straight line, as 
 stated in the text When the rolling cylinder is concave, r is negative.
 
 APPROXIMATE DRAWING OF EPICYCLOIDS. 61 
 
 spending chord on the rolling curve A D, makes with a tangent to 
 that curve at the corresponding point. Thus are found a series of 
 points, 1, 2, 3, 4, &c., on the rolled curve A E, at the ends of the 
 normals from the corresponding points on the fixed curve A B. 
 
 The two preceding Rules are applicable to fixed and rolling 
 curves of all figures whatsoever. When both curves are circles, 
 the finding of a series of points is facilitated by drawing the circle 
 C C', which contains the successive positions of the centre of the 
 rolling circle ; then marking those successive positions, 1', 2', 3', 4', 
 fec., on the circle C C', by drawing radii through the corresponding 
 points 1, 2, 3, 4, &c., on the circle A B ; then drawing the rolling 
 circle in its several successive positions (marked with dots in the 
 figure), and laying off the chords 11, 22, 33, 44, &c., of their proper 
 lengths upon those positions of the rolling circle, which chords will 
 be a series of normals to the rolled curve A E. 
 
 III. To approximate to the figure of an epicydoidal arc by means 
 of one circular arc. By the method of the preceding Rule draw the 
 normal to the epicycloidal arc in question at a point near its middle. 
 For example, if A 3 is the arc of the epicycloid A E, whose figure 
 is to be approximated to by means of one circular arc, draw the nor- 
 mal 22 by Rule II. Then conceive that normal to be represented 
 by A T in fig. 46, page 57 ; and by the method of Article 78 
 find the corresponding radius of curvature A F and centre of 
 curvature F. A circular arc described about F, with the radius 
 F A (fig. 46), will be an approximation to the epicycloidal arc. 
 
 This is the approximation used in Mr. Willis's method of 
 designing teeth for wheels, to be described farther on. It ensures 
 that the circular arc shall have, at or about the middle of its length, 
 the same position, direction, and curvature with the epicycloidal 
 arc for which it is substituted. Towards the ends of the arcs they 
 gradually deviate from each other. 
 
 IV. To approximate to the figure of an epicycloidal arc by means 
 of two circular arcs. This method of approximation is closer than 
 the preceding, but more laborious. It substitutes for an epicycloidal 
 arc a curve made up of two circular arcs; and the approximate 
 curve coincides exactly with the true curve at the two ends and at 
 one intermediate point, and has also the same tangents at its two ends 
 
 Suppose that A and B (fig. 48) are the two ends of the epicy- 
 cloidal arc to which an approximation is required, and that A C and 
 B C are normals to the arc at those points : the positions of the 
 ends of the arc and directions of its normals having been determined 
 by Rule II. of this Article. Let C be the point of intersection of 
 the normals. Draw the tangents A D perpendicular to A C, and 
 B D perpendicular to B C, meeting each other in D. Draw the 
 straight line DC, and bisect it in E. About E, with the radius 
 ED = EC, describe a circle, which will pass through the four
 
 2 
 
 GEOMETRY OF MACHINERY. 
 
 points, A, D, B, C. Draw the diameter F E G, bisecting the arc 
 A B in F and the arc B C A in G. 
 
 Draw the straight line G D, in which take GH = GA = GB. 
 
 Through H, parallel 
 to F E G, draw the 
 straight line H K L, 
 cutting A C in K 
 and B C in L. Then 
 about K, with the 
 radius K A = K H, 
 draw the circular arc 
 AH; and about L, 
 with the radius L H 
 = L B, draw the 
 circular arc H B : the 
 curve made up of 
 those two circular 
 arcs will be a close 
 approximation to the 
 epicycloidal arc, hav- 
 ing the same position 
 and tangents at its 
 two ends, and being 
 very near to the true 
 arc at all intermediate 
 points. 
 
 It maybe remarked 
 that G H = G A = 
 GB = /V /(HK-HL) 
 approximates very 
 closely to the mean 
 radius of curvature 
 of the epicycloidal arc 
 A B; also that the 
 process described is 
 applicable to the ap- 
 proximate drawing of 
 
 Pig. 48. many curves besides 
 
 epicycloids; and that 
 
 the ratio of the two radii, H L : H K, deviates less from equality 
 than that of the radii of any other pair of circular arcs which 
 can be drawn so as to touch A D in A and B D in B, and also to 
 touch each other at an intermediate point.* 
 
 (TT T 2 TT TT2 v 2 
 TT TT~TT T ~ ) * 3 * 
 
 ./, H L\2 . 
 ; or that I log. TT^S- I IS a minimum.
 
 RESOLUTION OP ROTATION. 63 
 
 80. Resolution of Rotation in Oeneral. The following proposi- 
 tions show how the rotation of a rigid body about a given axis, 
 fixed or instantaneous, may be resolved into two component 
 rotations about any two axes in the same plane with the actual 
 axis. 
 
 I. PARALLEL AXES. The rotation of a rigid body about a given 
 axis is equivalent to tJie resultant of two component rotations about 
 two axes parallel to the given axis and in the same plane, the angular 
 velocity of each of the three rotations being proportional to the distance 
 between the axes of the oilier two rotations. 
 
 In fig. 49, let the plane of the paper be perpendicular to the 
 three parallel axes, and let C be the trace of the axis of the 
 resultant rotation, and A and B the traces 
 of the axes of the component rotations ; all 
 three axes being in the same plane, whose 
 trace is A C B. Let the angular velocities 
 about 
 
 A, B, C, 
 
 be respectively proportional to 
 
 BO, CA, AB. 
 
 As the figure is drawn, all three angular 
 velocities are of the same sign, because A B 
 = B C + C A. If C lay beyond A and B, 
 49. instead of between them, A B would be the 
 
 difference of B C and C A, instead of their 
 sum ; and the lesser of these two distances and of the correspond- 
 ing angular velocities would have to be considered as negative. 
 
 Let H be the projection of a particle in the rigid body, which 
 particle is moving in a direction perpendicular to H C, with a 
 velocity proportional to C H A B. Then, first, from H let fall 
 H D perpendicular to A B; then, by the principles of Article 55, 
 page 33, the component velocity of H in the direction H D, 
 whether due to rotation about A, B, or 0, is the same with that 
 of a particle at D. Now the velocities of a particle at D due to 
 the rotations about 
 
 A, B, 
 
 are proportional respectively to 
 
 + AD-BC; -BD-CA; +CD-AB; 
 
 andCD-AB = AD-BC-BD-CA; therefore this com- 
 ponent of the velocity of the particle H due to the rotation about 
 C is the resultant of the corresponding components due to the 
 rotations about A and B respectively. 
 
 Secondly. Through H draw E G H F parallel to A C B, and 
 on it let fall the perpendiculars A E, B F, C G. Then, by the
 
 64 GEOMETRY OF MACHINERY. 
 
 principles of Article 55, page 33, the component velocities of H 
 along E F due to the rotations about the axes A, B, and C are 
 respectively equal to the velocities of E due to rotation about A, 
 of F due to rotation about B, and of G due to rotation about C ; 
 and because A E = B F = C G, these velocities are respectively 
 proportional to 
 
 BC, CA, AB; 
 
 But AB = BC + CA; therefore the component along E F of 
 the velocity of the particle H due to the rotation about C is the 
 resultant of the corresponding component velocities due to the rota- 
 tions about A and B respectively. Therefore the whole velocity of 
 the particle H due to rotation about C, x with an angular velocity 
 proportional to A B, is the resultant of the velocities of the same 
 particle due respectively to rotations about A, with an angular 
 velocity proportional to B C, and about B, with an angular velocity 
 proportional to C A. And this being true for every particle of the 
 rotating body, is true for the whole body : Q. E. D. 
 
 II. INTERSECTING AXES. Tlie rotation of a rigid body about a 
 given axis is equivalent to tfie resultant of two component rotations 
 about two axes in the same plane with the first axis, and cutting it in 
 one point; the angular velocities of the component and resultant rota- 
 tions being proportional respectively to the sides and diagonal of a, 
 parallelogram, which are parallel respectively to the three axes of 
 rotation. 
 
 In fig. 50 the upper right-hand part of the figure represents a 
 plane perpendicular to the resultant axis of rotation, O". F" is 
 the projection of any particle on that plane; and the direction of 
 motion of any particle whose projection is F" is perpendicular to 
 O" F". 
 
 O" Y" and 0" Z" are the traces of two planes perpendicular 
 to the first plane of projection and to each other; and D" and E" 
 are the projections of F" on those planes respectively. According 
 to the principle of Article 55, page 33, the component velocity 
 parallel to O" Y" of the particle whose projection is F' is the same 
 with the velocity of a particle at D" ; and its component velocity 
 parallel to 0" Z" is the same with that of a particle at E". 
 
 The upper left-hand part of the figure represents the plane 
 whose trace on the first plane of projection is O" Z'; 0' X', on this 
 second plane, is the axis of rotation; O' Z' is the trace of the first 
 plane of projection ; and D' is the projection of F", and is the same 
 point that is marked D" on the first plane. The lower part of the 
 figure represents the plane whose trace on the first plane of pro- 
 jection is 0" Y", and on the second plane, O' X'. On this third 
 plane O X is the axis of rotation, and also the trace of the second 
 plane; O Y is the trace of the first plane; E is the projection of
 
 RESOLUTION OP KOTATIOtf. 
 
 F, and is the same point that is marked E" on the first plane; O 
 is the projection of D" 
 
 The positions of the second and third planes are arbitrary so long; 
 as they both traverse the axis of rotation, and are at right angles 
 to each other. 
 
 In O X take O proportional to the angular velocity; and 
 make it point so that to an observer looking from O towards O 
 the rotation shall seem right-handed. 
 
 From O draw any two lines, O A and B, in the third plane; 
 from C draw C B parallel to A O, and C A parallel to B O, so as: 
 to complete the parallelogram O B C A. Then the proposition! 
 states, that a right-handed rotation about O A, with an angular 
 velocity proportional to O A, and a right-handed rotation about 
 O B, with an angular velocity proportional to O B, being combined, 
 are equivalent to the actual rotation. 
 
 To prove this for a particle at E, it is to be considered that the 
 motions impressed on E by the three rotations sepai*ately are each 
 of them perpendicular to the third plane; also, that the velocity 
 of E due to any one of the three rotations is proportional to the 
 angular velocity of that rotation multiplied by the perpendicular 
 distance of E from the axis of that rotation, and is therefore pro-
 
 66 GEOMETRY OF MACHINERY. 
 
 portional to the area of .a triangle having for its base the length 
 marked on that axis, to represent that angular velocity, and for its 
 summit the point Ej so that the velocities of a particle at E due 
 respectively to the rotations about 
 
 O A, OB, O C 
 
 are proportional respectively to the areas of the triangles 
 O A E, QBE, O C E. 
 
 Through A and B draw A G and B H perpendicular to O C, and 
 join E G and E H. Then, by plane geometry, 
 
 OAE=OGE;andOBE = O^H. E = G C E; 
 
 therefore 
 
 So that the velocity of E due to the actual rotation about O G is 
 the resultant of the velocities due to the rotations about O A and 
 OB; the angular velocities being proportional to the lengths laid 
 off on the axes respectively. 
 
 To prove the same proposition for a particle at D", whose 
 projection on the third plane is O, it is to be considered that the 
 perpendicular distance of this point from the three axes, O A, O B, 
 and O C, is identical, being the line marked O" D" and O' D' on 
 the first and second planes ; so that the velocities of D due to 
 the three rotations are simply proportional to the three angular 
 velocities. To represent, then, those three velocities as projected 
 on the third plane, draw O a, Ob, and O c perpendicular and 
 proportional respectively to O A, O B, and O C. It is evident 
 that O a, O b, and O c are the sides and diagonal of a paral- 
 lelogram similar to B C A ; and therefore that the velocity 
 of D" due to the actual rotation about O C is the resultant of the 
 velocities due to the rotations about O A and O B, the angular 
 velocities being proportional to the lengths laid off on the axes 
 respectively. 
 
 The proposition, therefore, is proved for both components of the 
 velocity of a particle at F"; and it holds for any particle whose 
 projection on a plane perpendicular to the axis O C is F"; that is, 
 for every particle of the body, and therefore for the whole body: 
 Q. KD. 
 
 It appeal's, then, that rotations, when represented by lengths 
 laid off on their axes proportional to their angular velocities, can 
 be compounded and resolved, like linear velocities, by constructing 
 parallelograms. 
 
 81. Rotations about Intersecting Axes Compounded. (A. J/., 392),
 
 COMPOSITION OP ROTATIONS. 67 
 
 In fig. 51, let O A be a fixed axis, and about it let the plane 
 A O C rotate with the angular 
 velocity a. Let the plane of 
 projection be that of those two 
 axes at a given instant. Let 
 be an axis in the rotating plane; 
 and about that axis let a second- 
 ary piece rotate with the angular* 
 velocity b relatively to the rotat- 
 ting plane; and let it be required 
 to find the instantaneous axis and the resultant angular velocity 
 of the secondary piece. From the principles of Article 80, 
 Proposition II., page 64, the following rule is deduced: 
 
 On O A take O a proportional to a; and on O C take O b 
 proportional to b. Let those lines be taken in such directions 
 that to an observer looking from O towards their extremities the 
 component rotations shall seem both right-handed. Complete the 
 parallelogram Obca; the diagonal c will be the instantaneous 
 axis; and its length will represent the resultant angular velocity. 
 Another mode of viewing the question is as follows : 
 
 Because the point O in the secondary piece is fixed, the instan- 
 taneous axis must traverse that point. The direction of that axis 
 is determined by considering that each point which arrives at that 
 line must have, in virtue of the rotation about O C, a velocity 
 relatively to the rotating plane, equal and directly opposed to that 
 which the coincident point of the rotating plane has. Hence it 
 follows that the ratio of the perpendicular distances of each point 
 in the instantaneous axis from the fixed and moving axes respect- 
 ively 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 if, in symbols, O T be the instantaneous axis, 
 
 sin AOT : sin COT : : b : a (1.) 
 
 The resultant angular velocity about this instantaneous axis is 
 found by considering that if 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 A, or from the resultant angular velocity, c, of the rigid 
 body about the instantaneous axis. That is to say, let C D, C E 
 be perpendiculars from C upon O A, T, respectively ; then 
 
 a CD - c-CE; 
 
 but C D : C E : : sin A O C : sin C O T; and therefore 
 sin C O T :,sin A Q C i : a : c; ,
 
 C8 GEOMETRY OP MACHINERY. 
 
 and, combining this proportion with that given in equation 1, we 
 obtain the following proportional equation : 
 
 sin C O T : sin A O T : sin A O C \ 
 
 : : a : b : c 
 
 : : Oa : Ob : O c. 
 
 That is to say, the angular velocities of the component and resultant 
 rotations are each proportional to tlie sine of the angle between the axes 
 of the other two; and the diagonal of the parallelogram O b c a repre- 
 sents both tlie direction of the instantaneous axis and the angular 
 velocity about tJiat axis. 
 
 82. Roiling Cones. (A. M., 393.) All \he lines which succes- 
 sively 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 secondary piece is such as would be 
 produced by the rolling of the former of those cones upon the 
 latter. Circular sections of the two cones are sketched in perspec- 
 tive in fig. 51. 
 
 It is to be understood that either of the cones may become a 
 flat disc, or may be hollow, and touched internally by the other. 
 For example, should ^ A T become a right angle, the fixed cone 
 would become a flat disc; and should ^AOT become obtuse, 
 that cone would be hollow, and would be touched internally by 
 the rolling cone; and similar changes maybe made in the rolling 
 cone. 
 
 The path described by a point in or attached to the rolling cone 
 is a spherical epitrochoid; and if that point is in the surface of the 
 rolling cone, that curve becomes a sp/terical epicycloid. It will be 
 shown in the next chapter how to draw such curves not exactly, 
 but with a degree of accuracy sufficient for practical purposes. 
 
 83. ReaoUuion of Helical Motion. The resolution of helical or 
 screw-like motion into rotation about an axis and translation along 
 that axis has already been treated of in the last section of the 
 preceding chapter. It remains to be shown how a helical motion, 
 may be regarded as compounded of two rotations about two axes 
 which are in different planes. 
 
 In fig. 52, let the lower part of the figure represent a plane of 
 projection, and O A and O B the projections upon that plane of 
 two axes which are both parallel to it, but not in the same plane. 
 Let the upper part of the figure represent a second plane of pro- 
 jection perpendicular to the first plane; and let F 1 G' be the 
 projection on that second plane of the common jierpend'cular of 
 those two axes (Article 36, page 14). Let a rigid body have a
 
 1VESOI.UTION 1 OP HELICAL MOTI01T. 
 
 69 
 
 motion compounded of two rotations about the two axes respect- 
 ively, with angular velocities represented by O A and O B, these 
 lines being drawn, as before, so that to an observer at O each 
 rotation shall appear right-handed. 
 
 Complete the parallelogram O A C B, and draw its diagonal 
 O C. Then, if the axes O A and O B were in the same plane, the 
 rotations about them, being combined, woiild be equivalent simply 
 to a rotation represented by O C, as in Articles 80 and 81, pages 
 C4 to 66. 
 
 Let the second plane of projection be now supposed parallel to 
 O C; and let F A', O' C', and G' B' be the respective projections 
 of O A, O C, and O B upon it. Draw A D, B E, and F O G 
 perpendicular to O C, and A F and B G parallel to O C. It is 
 obvious that D = F A', O E = G' B', and F O = O G. 
 
 According to Article 80, Proposition II., page 64, the rotation 
 represented by O A may now be regarded as compounded of a 
 rotation represented by O D, about an axis of which O D and 
 F' A' are the projections, and a rotation represented by O F, about 
 an axis of which F and the point F' are the projections; also, 
 the rotation represented by O B may be regarded as compounded 
 of a rotation represented by O E, about an axis of which O E and 
 G' B' are the projections, and a rotation represented by O G, about 
 an axis of which O G and the point G' are the projections. 
 
 Then, according to Article 76, page 54, the rotations about the 
 parallel axes F' A' and G' B', being combined, are equivalent to a
 
 70 .~ GEOMETRY OF MACHINERY. 
 
 rotation about an intermediate axis, O' C', in the same plane, with, 
 an angular velocity represented by 
 
 O C = O' C' = F' A' + G' B' ; 
 
 and that axis of resultant rotation divides the distance F' G' in. 
 the following proportion : 
 
 0' G' : F A' : G' B' 
 : : F G : O' G' : O' F'. 
 
 To find the point O' by graphic construction, draw F' H', 
 parallel to A O and G' H' parallel to B Q, cutting each other in 
 H'; then through H' draw H' O' C' parallel to O C. 
 
 Moreover, the component rotations represented by O F and G, 
 about the axes F and G', are of equal and opposite angular velocities ; 
 and therefore, according to Article 76, page 54, they are equivalent 
 to a translation in the direction O C, with a velocitv represented by 
 the product O F'FG. 
 
 That translation being compounded with the resultant rotation 
 represented by O C, gives finally, for the resultant motion of the 
 body, a Jtelical motion about the axis whose projections are O C and 
 O C'. 
 
 The pitch of that helical motion, or advance per turn, is found 
 by multiplying the rate of advance, O F, F G, by the time of one 
 
 /? t ^ Q O O 
 
 revolution, vy-p ; and is therefore equal to the circumference of 
 
 O F F C 1 ' 
 
 a circle whose radius is ^~ . Draw F' K' perpendicular to 
 u u 
 
 O B, and G' K' perpendicular to O A, cutting each other in K' 
 (which will be in the straight line H' O C'). Then it is evident 
 that F K' G' and C A O are similar triangles; and because 
 D A = F, we have the following proportion : 
 
 O F F' G' 
 C : F : : F G' : 0' K' = * 
 
 Therefore the pitch of the resultant helical motion is equal to the 
 circumference of a circle whose radius is O' K'; and the rate of 
 advance may be represented by the product O C 0' K'. 
 
 84. Roiling Myperboioids. Conceive the straight line O C to- 
 represent an indefinitely long straight edge, rigidly fastened to the 
 arm 0' F', and sweeping along with that arm round the axis O A; 
 then conceive the same straight line to be rigidly fastened to the 
 arm O' G, and to sweep along with this arm round the axis O B. 
 Thus are generated a pair of surfaces called tolling Hyperboloids,
 
 ROLLING HYPERBOLOIDS. 71 
 
 which touch each other all along the straight line O C. Fig. 53 
 shows the general appearance of a pair of rollers of that form; and 
 in fig. 54 the projections of their figures 
 are given with greater precision. If one 
 of those bodies is fixed, and the other made 
 to roll upon it, they continue to touch each 
 other in a sti'aight line, which is the in- 
 stantaneous axis of the rolling body; and 
 the rotation about that instantaneous axis 
 is accompanied by a sliding motion along Fig. 53. 
 
 the same axis, so as to give, as the resultant 
 
 compound motion, a helical motion about the instantaneous axis, 
 as described in the preceding Article. The following problem 
 sometimes occurs in mechanism : 
 
 Given, the angle between the directions of two axes, and the length 
 of their common perpendicular, to draw the projections of a pair of 
 rolling hyperbolo'ids of which these shall be the axes, and of which 
 one shall roll on the other, so as to have component angular velocities 
 bearing to each other a given ratio. 
 
 Let the lower part of fig. 54 (see next page) represent a plane to 
 which the two axes are parallel; and let O a and O 6 be their pro- 
 jections on that plane, with lengths laid off upon them proportional 
 to the intended component angular velocities. Draw b c parallel to 
 O a, and a c parallel to b, cutting each other in c; O c will be 
 the projection of the line of contact, or instantaneous axis; and the 
 length O c will represent the resultant angular velocity (as in the 
 preceding Article). 
 
 Through O, perpendicular to O c, draw O G' F', and lay off upon 
 it G' F' equal to the given common perpendicular; and let the 
 second plane of projection be perpendicular to the first plane, and 
 parallel to O c and G' F'. To find the projection of the line of 
 contact upon this second plane, proceed as in the preceding Article; 
 that is, draw F' H' and G' H' parallel respectively to O a and O b, 
 and H' O' parallel to c; H' 0' will be the required projection. 
 This projection may also be found, if convenient, by either of the 
 following methods : Draw G' K' perpendicular to O a, and F' ~K 
 perpendicular to O b, cutting each other in K' ; and then dra-w 
 H' K' O' parallel to A c; or otherwise : Draw g cf perpendicular 
 to O c, and divide F' G' in the following proportion : 
 
 fg : cf : eg 
 : : F' G' : O' F' : O' G'. 
 
 Draw TJ Y perpendicular to a, making U = Y = F 0'; 
 also draw Y O Z perpendicular to O b, making O Y = Z = 
 GO'; then U O Y and Y O Z will be the projections on the first
 
 72 GEOMETRY OP MACHINERY. 
 
 plane of the smallest transverse sections, or what may be called the 
 "throats" of the two hyperboloids; which transverse sections are 
 
 Fig. 54. 
 
 circles of the respective radii F' O' and G' O'. The projections of 
 those circles on the second plane of projection are the ellipses 
 IP O' Y' and V' O' Z', drawn according to the principles of Article 
 37, page 15. 
 
 To find the projections of a pair of circular transverse sections 
 of the two hyperboloi'ds, which shall cross each other in any 
 given point of, the line of contact, let T and T be the projections 
 of that point. Then draw T L perpendicular to a O, and T M 
 perpendicular to 6O; L and M will be the projections of the 
 centres of those circular sections on the first plane. Draw F' L' 
 and G' M' parallel, and L L' and M' M' perpendicular to O c ; 
 L' and M'will be the projections of those centres on the second 
 plane. In L O take LN = FO', and join NT; then in L T 
 produced, take L Q = N T; ; this will be the radius of the required 
 section of one hyperboloid; and Q will be a point in the hyperbola
 
 ROLLING HYPEKBOLOIDS OBLIQUE-ROLLING CYLINDERS. 73 
 
 TJ Q, which is the longitudinal section or trace of that surface on a 
 plane traversing the axis F 1 !/, and parallel to the first plane of 
 projection. Also, in M O take M P = G 1 0', and join P T; then 
 in M T produced take M R = P T ; this will be the radius of the 
 required section of the other hyperbolo'id ; and R will be a point 
 in the hyperbola Z R, which is the longitudinal section or trace of 
 this surface on a plane traversing the axis G M' and parallel to 
 the first plane of projection. 
 
 The projection, O T, of the line of contact is an asymptote to 
 Tx>th hyperbolas, UQ and ZR; and their other asymptotes are O W, 
 making L O W = L O T, and O X, making M O X = M O T. 
 
 The projections on the second plane of projection of the two 
 circular transverse sections which cross each other at the point 
 whose projections are T and T' are two ellipses, drawn according to 
 the principles of Article 37, page 15. 
 
 By the same process may be found the projections of any 
 required number of transverse sections of the two rolling hyperbo- 
 lo'ids, and of any required number of points, such as Q and R, in 
 their longitudinal sections. 
 
 Additional rules relating to the construction of such figures 
 will be given in the next chapter, in the articles which treat of 
 their application to skew-bevel wheels. 
 
 80. Cylinder Roiling Obliquely. The same kind of resultant 
 motion will take place, if for the rolling hyperbolo'ids there be 
 substituted a pair of cylinders described about the axes whose 
 projections are O A and O B, 
 tig. 52, page 69, with the 
 respective radii O' F' and 
 
 O' G'; provided the axis ofc . 
 
 the rolling cylinder is guided 
 so that the point where 
 it is met by the common 
 perpendicular F' G' shall 
 revolve in a circle of the 
 radius F ' G' round the axis 
 
 of the fixed cylinder, and so 
 
 that .the inclination of those -p- 55 
 
 two axes to each other shall 
 
 remain constant. The general appearance of such a pair of cylinders 
 is shown in fig. 55. They touch each other in a point only, and 
 not along a straight line, as the hyperboloids do. The uniform 
 transverse sections of such a pair of cylinders are identical with 
 those at the throats of the corresponding pair of hyperbolo'ids. 
 Further explanations as to obliquely-rolling cylinders will be 
 given in the next chapter, under the head of screw-gearing. 
 
 86. Cones Roiling obliquely. The same kind of resultant
 
 GEOJIETRY OF MACHINERY. 
 
 motion may be effected also by substituting for the pair of hyper- 
 boloids of fig. 54, page 72, the pair of cones which touch those 
 hyperboloiids in the pair of circles that cross each other in any given 
 point, T, of the instantaneous axis. To draw the projections of 
 
 those tangent cones, let O' T" 
 in fig. 56 (as in fig. 54) be 
 the instantaneous axis, O' the 
 point where it cuts the com- 
 mon perpendicular, and T' the 
 intended point of contact of 
 the cones. From O', pei-pen- 
 diculaj to O' T', draw O' h = 
 ' O' H' in fig. 56; join T h; and 
 
 -^_ / ^^_ G?K-^/L/ perpendicular to T' h draw hp", 
 
 cutting the instantaneous axis 
 in p'. Then a plane normal to 
 the instantaneous axis at p' 
 passes through the summits of 
 Fig. 56. both the required tangent 
 
 cones. Therefore, in fig. 54, 
 
 on O c lay off O p = O' p' of fig. 56, and draw qp r perpendicular to 
 O c, cutting O a in q and O b in r; q and r will be the projections of 
 the summits of the tangent cones on the first plane of projection. The 
 projections on the same plane of the longitudinal sections or traces 
 of these cones, upon planes traversing their axes parallel to the plane 
 of projection, are q Q and r R. Also, let the plane of fig. 56 be 
 the second plane of projection, and let F' L' and G' M', as in fig. 
 54, be the projections of the axes of the hyperboloiids, and F' O' G' 
 that of the common perpendicular. Draw q'p'r', cutting those 
 axes in q' and r ; these points will be the projections of the summit? 
 of the tangent cones on tlie second plane of projection. The projec- 
 tions of the bases of these cones on the same plane are the pair of 
 ellipses, with L' and M' for their centres, which cross each other at 
 the point T, as in fig. 54. The cones touch each other in the 
 point T" only, and not along a straight line, as the hyperbolo'ids do. 
 Further explanations as to obliquely-rolling cones will be given 
 in the next chapter, under the head of skew-bevel wheels. 
 
 87. Bauds or Flexible Secondary Pieces Cords Belts Chains. 
 
 (A: M., 400, 401.) The flexible pieces used in machinery may be 
 classed under three heads : Cords, which approximate to a round 
 form in section; Belts, which are flat; and C/iains, which consist of 
 a series of rigid links so connected together that the chain as a 
 whole is flexible. Mr. Willis gives them all the common name 
 of wrapping connectors; and for the sake of brevity in stating 
 principles that apply to them all, they may conveniently be called 
 bands.
 
 BANDS FLUID SECONDARY PIECES. 75 
 
 In treating of questions of pure mechanism, the centre line of a 
 band is treated as being of invariable length; for although no 
 substance is absohitely inextensible, and although when a band 
 passes over a curved surface the concave side is shortened and the 
 convex side lengthened, still the variations of length of the centre 
 line of the band are, or ought to be, practically inappreciable. 
 
 In order that the figure and motion of a band may be deter- 
 mined from geometrical principles alone, independently of the 
 magnitude and distribution of forces acting on it, its weight must 
 be insensible compared with the tension on it, and it must every- 
 where be tight; and when that is the case, each part of the band 
 which is not straight is maintained in a curved figure by passing 
 over a convex surface. When a band is guided by a given actual 
 surface, the centre line of that band may be regarded as guided by 
 an imaginary surface parallel to the actual surface, and at a 
 distance from it equal to half the thickness of the band. The line 
 in which the centre line of a band lies on such guiding surface is 
 the shortest line which it is possible to draw on that surface 
 between each pair of points in the course of the band. (It is a 
 well-known principle of the geometry of curved surfaces that the 
 osculating plane at each point of such a line is perpendicular to the 
 curved surface.) 
 
 Hence it appears that the motions of a tight flexible band, of 
 invariable length along its centre line and insensible weight, are 
 regulated by the following principles: 
 
 I. The length between each pair of points in the centre line of the 
 band is constant. 
 
 II. That length is the shortest line which can be drawn between its 
 extremities over the surface by which the centre line of the band 
 is guided. 
 
 The motions of a band are of two kinds 
 
 I. Travelling of a band along a track of invariable form; in 
 which case the velocities of all points of the centre line are equal. 
 
 II. Alteration of the figure of the track by the motion of the 
 guiding surfaces. 
 
 Those two kinds of motion may be combined. 
 
 The most usual problems in practice respecting the motions of 
 bands are those in which bands are the means of transmitting 
 motion between two pieces in a train of mechanism. Such problems 
 will be considered in the next chapter. 
 
 88. Fluid Secondary pieces. A mass of fluid may act as a 
 secondary piece in a machine; and in order that the motion of 
 such a mass may be a subject of pure mechanism, the volume 
 occupied by the mass must be constant ; and that not only for the 
 whole mass, but for every part of it, how small soever. In other 
 words, the fluid mass must in every part be of constant bulkiness;
 
 76 GEOMETRY OF MACHINERY. 
 
 this word being used to denote the volume filled by an unit of 
 mass; for example, the number of cubic feet filled by a pound, or 
 the number of cubic metres filled by a kilogramme. Every fluid, 
 whether liquid or gaseous, undergoes variations of bulkiness 
 through variations of pressure and of temperature ; but in mechan- 
 ism such variations of bulkiness may be either so small that they 
 may be disregarded for the practical purpose under consideration, 
 (as in the case of most liquids), or, if the fluid employed be gaseous, 
 they may be prevented by keeping the pressure and temperature 
 constant. 
 
 Under such conditions the motions of the particles of a fluid 
 mass are regulated by the following principle : 
 
 At a given series of sections of a stream of fluid of constant 
 bulkiness, the mean velocities at each instant of the particles in direc- 
 tions normal to those sections respectively) are inversely proportional 
 to tl\A areas of the sections.
 
 77 
 
 CHAPTER IV. 
 
 OF ELEMENTA11Y COMBINATIONS IN MECHANISM. 
 
 SECTION I. Definitions, General Principles, and Classification. 
 
 89. Elementary Combination* Defined. (A. M., 431.) An 
 
 "Elementary Combination" in Mechanism (a term introduced bv 
 Mr. Willis) consists of a pair of primary moving pieces, so con- 
 nected that one transmits motion to the other. In other words, 
 (to quote the Article Mechanics (Applied), in the eighth edition 
 of the Encyc. Urit.) 
 
 "An elementary combination in mechanism consists of two 
 pieces whose kinds of motion are determined by their connection 
 with the frame, and their comparative motion by their connection 
 with each other; that connection being effected either by direcfc 
 contact of the pieces, or by a connecting" (secondary) "piece" 
 (such as a band, or a link, or a mass of fluid), "which is not con- 
 nected with the frame, and whose motion depends entirely on the 
 motions of the pieces which it connects." 
 
 " The piece whose motion is the cause is called the driver; the 
 piece whose motion is the effect, the follower." 
 
 " The connection of each of those two pieces with the frame is 
 in general such as to determine the path of every moving point. In 
 the investigation, therefore, of the comparative motion of the 
 driver and follower, in an elementary combination, it is unneces- 
 sary to consider relations of angular direction, which are already 
 fixed by the connection of each piece with the frame; so that the 
 inquiry is confined to the determination of the velocity-ratio, and 
 of the directional-relation so far only as it expresses the connection 
 between forward and backward movements of the driver and 
 follower. When a continuous motion of the driver produces a 
 continuous motion of the follower, forward or backward, and a 
 reciprocating motion a motion reciprocating at the same instant, 
 the directional-relation is said to be constant. When a continuous 
 motion produces a reciprocating motion, or vice versa; or when a 
 reciprocating motion produces a motion not reciprocating at the 
 same instant, the directional-relation is said to be variable." 
 
 90. Line of Connection. In every class of elementary combina- 
 tions, except those in which the connection is made by reduplication 
 of cords, or by an intervening fluid, there is at least one straight
 
 78 
 
 GEOMETRY OP MACHINERY. 
 
 line called the line of connection of the driver and follower; being 
 a line traversing a pair of points in the driver and follower 
 respectively, which points are so connected that the component of 
 their velocity relatively to each other, resolved along the line of 
 connection, is null. 
 
 91. Comparative '.notions of Connected Points and Pieces. From 
 
 the definition of a line of connection it follows, that the components 
 of the velocities of a pair of connected points along their line of 
 connection are equal. And from this, and from the propei'ty of a 
 rigid body already stated in Article 54, page 32, it follows, that the 
 components, along a line of connection, of all t/ie points traversed by 
 that line, whether in the driver or in the follower, are equal. 
 
 The general principle which has just been stated serves to solve 
 every problem in which the mode of connection of a pair of pieces 
 being given it is required to find their comparative motion at a 
 given instant, or vice versa. 
 
 The following are the rules for applying that principle to the 
 three classes of problems which most frequently occur with refer- 
 ence to elementary combinations : 
 
 I. Pair of Points; or Pair of Sliding Pieces. In fig. 57, let A B 
 be a line of connection; and let it be taken as the axis of projec- 
 tion. Let A be a point in the driver, 
 and B a point in the follower, both in 
 the line of connection. Let A a', 
 -,!> A a" be the two projections of the 
 6 direction of motion of A at a given 
 instant; and let B b', B b" be the two 
 projections of the direction of motion 
 tT of B at the same instant. Lay off, 
 along the line of connection and in 
 the same direction, the equal distances 
 A a = B b ; draw a" a a', b" b b' perpendicular to the line of con- 
 nection; then A a' and A a", B b' and B b" will be the projections 
 
 of a pair of lines propor- 
 tional respectively to the 
 velocities of A and B at 
 that instant. The lengths 
 of those lines may be found 
 by the Rule of Article 19, 
 page 7. 
 
 II. Pair of Turning 
 Pieces. In fig. 58, let A B 
 be the line of connection of 
 a pair of turning primary 
 pieces. Let A and B be 
 the points where that line 
 
 Fig. 57. 
 
 Fig. 58.
 
 CONNECTED POINTS AND PIECES. 79 
 
 is met by the common perpendiculars from the axes of rotation of 
 the two pieces. (As to finding such common perpendiculars, see 
 Article 36, page 15.) Let A A' and B B' be the r abutments of 
 those two perpendiculars, drawn in opposite directions. Draw the 
 straight line A' B' (called the line of centres), cutting the line of 
 connection in I. 
 
 Then, because the component velocities of A and B along A B 
 are equal, the angular velocities (or the component angular 
 velocities) of the driver and follower about axes perpendicular to 
 A B must be to each other in the inverse ratio of the perpendiculars 
 A A' and B B'; or, what is the same thing, in the inverse ratio of 
 the segments I A' and I B' into which tlie line of centres is cut by 
 the line of connection. 
 
 Hence the following construction : In A B take A K = B I 
 (or B K = A I) ; and through K draw an oblique straight line in 
 any convenient direction, so as to cut A' A produced in c and B' B 
 produced in d; then the component angular velocities of the pieces 
 about two axes, A c and B d, perpendicular to the line of connection, 
 will be to each other in the direct ratio of A c to B c?. Also lay off, 
 in opposite directions, the angles B A a and./ B b, equal to the angles 
 which the two axes of rotation respectively make with the line of 
 connection, and draw c a and d b parallel to A B, cutting A a and 
 B b in a and b respectively. Then the ratio of A. a to B b will be 
 that of the resultant angular velocities of tJie two pieces. 
 
 Through A' and B' draw A' a' and B' b' parallel to A B; and 
 through a and b draw a e a and bf b' perpendicular to A B. Then 
 the proportion borne by c a = A e = A' a to d b = By = B' b' is 
 that of the component angular velocities of the two pieces about 
 axes parallel to the line of connection A B. Also A a and A' a' 
 represent the projections of the axis of rotation of the first piece 
 upon a pair of planes which cut each other in A e, one perpendicular 
 and the other parallel to the common perpendicular whose rabat- 
 ment is A A'; and B b and B r b' represent the projections of the 
 axis of rotation of the second piece upon a pair of planes which cut 
 each other in By* one perpendicular and the other parallel to the 
 common perpendicular whose rabatment is B B' 
 
 III. Turning Piece and Sliding Piece. In fig. 58, let A L be 
 the line of connection of a turning piece and a sliding piece, 
 and let it be taken for the axis of projection; and let one 
 of the planes of projection be parallel to the axis of the turning 
 piece. Let A a and A a be the projections of that axis; so 
 that A A' perpendicular to A L is the common perpendicular 
 of the axis and the line of connection. Take A a to represent the 
 angular velocity of the turning piece, and from a draw a c parallel 
 to L A, cutting A' A (produced if necessary) in c. Then A c will 
 represent the component angular velocity of the turning piece
 
 80 GEOMETRY OF MACHINERY. 
 
 about an axis, A c, perpendicular to A L ; and the product A A" A c 
 will represent the component velocity of any point in A L along that 
 line. 
 
 Let L be a point in the line of connection and in the sliding 
 piece; and let Li and Lm' be the projections of the direction of 
 motion of that piece. Lay off any convenient length, L I, to 
 represent the component velocity of the sliding piece along the line 
 of connection, and draw m I m perpendicular to that line ; then L m 
 and L m' will represent the two projections of the velocity of tlie 
 sliding piece. 
 
 Another construction is as follows : Having determined the angle 
 which the direction of motion of the sliding piece makes with the 
 line of connection A L, draw A' I, making the angle A A' I equal 
 to that angle; then the velocity of the sliding piece will be equal 
 to that of a point revolving at the end of the arm A' I, with the 
 angular velocity represented by A c. 
 
 92. Adjustment* of Speed The velocity-ratio of a driver and its 
 follower is sometimes made capable ofxbeing 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. Various such contrivances will be described 
 in a later chapter. 
 
 93. 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. In the case of a train of elementary combinations 
 the comparative motion of the last follower and first driver is found 
 by multiplying together all the velocity-ratios of the several element- 
 ary combinations of which the train consists, each ratio having the 
 directional-relation with which it is connected denoted by means of 
 the positive or negative algebraical sign, as the case may be. The 
 product is the velocity-ratio of the last follower and first driver; 
 and their directional-relation is indicated by the algebraical sign of 
 that product, found by the rules, that any number of positive 
 factors, and any even number of negative factors, give a positive 
 product; and that any odd number of negative factors give a 
 negative product. 
 
 94. Elementary Combinations Claused. The Only classification of 
 
 elementary combinations that is founded, as it ought to be, on 
 comparative motion, as expressed by velocity-ratio and directional- 
 relation, is that first given by Mr. Willis in his Treatise on Pure 
 Mechanism. Its general plan is as follows : 
 
 Class A : Directional-relation constant ; velocity-ratio constant. 
 
 Class B : Directional-relation constant ; velocity-ratio varying. 
 
 Class C: Directional -relation changing periodically; velocity- 
 ratio constant or varying.
 
 ELEMENTARY COMBINATIONS ROLLING CONTACT. 81 
 
 Each of those classes is subdivided by Mr. Willis into five 
 divisions, of which the characters are as follows : 
 
 Division I. Connection by rolling contact of surfaces, as iu 
 
 toothless wheels. 
 
 II. Connection by sliding contact of surfaces, as in 
 toothed wheels, cams, &c. 
 
 III. Connection by wrapping connectors or bands, 
 
 as in pulleys connected by belts, cords, or 
 chains. 
 
 IV. Connection by link-work, as in levers and cranks 
 
 connected by means of rods, &c. 
 
 V. Connection by reduplication of cords, as in blocks 
 
 and tackle used on board ship; 
 
 and to those five divisions may be added 
 
 Division VI. Connection by an intervening fluid, as in the 
 hydraulic press. 
 
 In the present treatise the principle of the classification of Mr. 
 Willis is followed ; but the arrangement (as in a Manual of Applied 
 Mechanics, already referred to) is modified by taking the mode of 
 connection as the basis of the primary classification. 
 
 With reference to classes B and C, in which the velocity-ratio is 
 or may be varying, it is to be observed that two kinds of problems 
 arise respecting velocity-ratio : the determination of the instan- 
 taneous velocity-ratio at the instant when the pieces are in one 
 given position; and the determination of the mean velocity-ratio 
 during the interval between two such instants : the latter quantity 
 is the ratio of the entire motions of the pieces during the interval. 
 
 SECTION II. Of Rolling Contact and Pitch Surfaces. 
 
 95. Pitch Surfaces* are those surfaces of a pair of moving pieces 
 which touch each other when motion is communicated by rolling 
 contact. The LINE OF CONTACT is that line which at each instant 
 traverses all the pairs of points of the pair of pitch surfaces which 
 are in contact. 
 
 The motion, relatively to the line of contact of their surfaces, of 
 a pair of primary pieces which move in rolling contact, is the same 
 with that of a secondary piece and a fixed piece, of which the former 
 rolls upon the latter, as already described in Article 72, page 51; 
 Articles 74 and 75, pages 53, 54; Article 77, page 56; Article 
 82, page 68, and Articles 84, 85, 86, pages 70 to 74 ; and therefore 
 the proper figures for the pitch surfaces of such primary pieces are 
 the same; that is to say, cylinders, cones, and hyperbolo'ids. 
 
 96. Toothless Wheels, Boilers, Toothless Backs. Of a pair of 
 
 O
 
 82 GEOMETRY OP MACHINERY. 
 
 primary moving pieces in rolling contact, both may rotate, or 
 one may rotate and the other have a motion of straight sliding. A 
 rotating piece, in rolling contact, is called a toothless wheel, and 
 sometimes a roller; a sliding piece may be called a toothless rack. 
 
 97. ideal Pitch Surfaces. The designing of pitch surfaces is 
 used not only with a view to the making of toothless wheels and 
 toothless racks (which are seldom employed), but much oftener as 
 a step towards determining the proper figures for wheels and racks 
 provided with teeth. 
 
 The pitch surface of a toothed wheel or of a toothed rack is an 
 ideal smooth surface, intermediate between the crests of the teeth 
 and the bottoms of the spaces between them, which, by rolling 
 contact with the pitch surface of another wheel, would com- 
 municate the same velocity -ratio that the teeth communicate by 
 their sliding contact. In designing toothed wheels and racks the 
 forms of the ideal pitch surfaces are first determined, and from 
 them are deduced the forms of the teeth. x 
 
 Wheels with cylindrical pitch surfaces are called spur wheels; 
 those with conical pitch surfaces, bevel w/ieels; and those with 
 hyperboloidal pitch surfaces, skew-bevel wheels. 
 
 98. The Pitch Line of a wheel, or, in circular wheels, the PITCH 
 CIRCLE, is the trace of the pitch surface upon a surface perpen- 
 dicular to it and to the axis ; that is, in spur wheels, upon a plane 
 perpendicular to the axis ; in bevel wheels, upon a sphere described 
 about the apex of the conical pitch surface; and in skew-bevel 
 wheels, upon an 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 line might be otherwise 
 defined, in most cases which occur in practice, simply as the trace 
 of the pitch surface upon a plane perpendicular to the axis of 
 rotation. 
 
 The PITCH POINT of a pair of wheels is the point of contact of 
 their pitch lines ; that is, the trace of the line of contact upon the 
 surface or surfaces on which the pitch lines are traced. 
 
 The pitch line of a rack is the trace of its pitch surface on a plane 
 parallel to its direction of motion, and containing its line of con- 
 nection with the wheel with which it works. 
 
 A SECTOR is a name given to a wheel whose pitch-line forms only 
 part of a circumference : sectors are used where the motion required 
 is reciprocating or "rocking," and does not extend to a complete 
 revolution. Everything stated respecting the figures of complete 
 wheels applies also to the figures of sectors. 
 
 99. General Conditions of Perfect Rolling Contact. (.[. J/., 439.) 
 
 The whole of the principles which regulate the motions of a pair 
 of primary pieces in perfect rolling contact follow from the single 
 principle, that each pair of points in the pitch surfaces which are in
 
 ROLLING CONTACT WHEELS WITH PARALLEL AXES. 83 
 
 contact at a given instant imist at that instant be moving in the same 
 direction with ttia same velocity. 
 
 The direction of motion of a point in a rotating body being per- 
 pendicular to a plane passing through its axis, the condition, that 
 each pair of points in contact with each other must move in the 
 same direction, leads to the following consequences : 
 
 I. That when both pieces rotate, their axes, and all their points 
 of contact, lie in the same plane. 
 
 II. That when one piece rotates and the other slides, the axis of 
 the rotating piece, and all the points of contact, lie in a plane per- 
 pendicular to the direction of motion of the sliding piece. 
 
 The condition, that the velocities of each pair of points of con- 
 tact must be equal, leads to the following consequences : 
 
 III. That the angular velocities of a pair of wheels, in rolling 
 contact, must be inversely as the perpendicular distances of any 
 pair of points of contact from the respective axes. 
 
 IV. That the linear velocity of a rack in rolling contact with a 
 wheel is equal to the product of the angular velocity of the wheel 
 by the perpendicular distance from its axis to a pair of points of 
 contact. 
 
 Respecting the line of contact, the above principles III. and IV. 
 lead to the following conclusions : 
 
 V. That for a pair of wheels with parallel axes, and for a wheel 
 and rack, the line of contact is straight, and parallel to the axes or 
 axis; and hence that the pitch surfaces are either cylindrical or 
 plane (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). 
 
 There is a sort of imperfect rolling contact which takes place 
 between hyperboloidal pitch surfaces; that is to say, there is a 
 sliding motion, but along the line of contact of the surfaces only; 
 so that the component motions of points in directions perpen- 
 dicular to the line of contact are the same as in perfect rolling 
 contact. This kind of motion will be considered in treating 
 specially of skew-bevel wheels. 
 
 100. Wheels with Parallel Axes. Given, the positions of the 
 parallel axes of a pair of wheels, and their velocity-ratio at a given 
 instant, to find the pitch-point. Fig. 59 represents the case in 
 which the directions of the rotations are contrary; fig. 60 that in 
 which they are the same. Let the plane of projection be perpendicu- 
 lar to the two axes, and let A and B be their traces; so that A B 
 is the line of centres. Perpendicular to A B draw A a and B 6
 
 GEOMETRY OF MACHINERY. 
 
 proportional to the intended angular velocities. Draw the straight 
 line a b, cutting A B (produced if necessary) in K. Lay off B I. = 
 
 2 
 
 Fig. 59. 
 
 A K (or A I = B K); I will be the required pitch-point or trace 
 of the line of contact. 
 
 The line of connection may be any straight line which traverses 
 I, or whose projection traverses I ; as C D. Let A C and B D be 
 perpendicular to the line of connection; then the velocities of the 
 points C and D are identical ; and the perpendiculars A C and B D 
 are inversely as the angular velocities of the pieces. 
 
 101. Wheel and Back. Given, at a given instant, the angular 
 velocity of a wheel and the linear velocity of a rack, to find their 
 pitch-point. In fig. 61, let the plane of projection be perpen- 
 j^ dicular to the axis of the wheel, and let A be 
 
 the trace of that axis. Draw A I perpen- 
 dicular to the direction of motion of the rack, 
 and make its length such that a point in the 
 wheel at I shall revolve with a velocity equal 
 to that of the rack; that is to say, make 
 
 . , linear velocity of rack ,, T ... 
 
 A I = : ,./..,,; then I will 
 
 angular velocity 01 wheel 
 
 be the required pitch-point. 
 
 The line of connection may be any straight 
 
 V 
 
 Fig. 61. 
 
 line which traverses I, or whose projection traverses I ; as C U. 
 Let fall the perpendicular A C; then the velocity of the point C in 
 the wheel is equal to the component velocity of the rack along 
 C U. Draw I V perpendicular to A I, to represent the whole 
 velocity of the rack, and from Y draw V U perpendicular to C U; 
 it is evident that I U is the component velocity along the line of 
 connection; and that AI:AC::IY:ITJ. 
 
 102. Circular wheel* in General. In order that, in an elemen- 
 tary combination of wheels, or of a wheel and rack, the velocity-
 
 CIRCULAR WHEELS STRAIGHT RACK. 
 
 85 
 
 ratio may be constant (so that the combination shall belong to Mr. 
 Willis's class A), it is obviously necessary that the pitch-point 
 during the entire revolution of each wheel should occupy an 
 invariable position in the line of centres; in other words, the pitch- 
 line of each wheel must be a circle, and that of a rack a straight 
 line. The corresponding forms of pitch-surface are : for a spur- 
 wheel, a circular cylinder; for a bevel-wheel, a cone with a circular 
 base, and sometimes a plane circular disc; for a rack, a plane; for 
 a skew-bevel wheel, a hyperbolo'id of revolution. Circular wheels 
 are by far the most common, the cases in which non-circular 
 wheels are used being comparatively rare. 
 
 103. Circular Spur-wheels. Given, a pair of parallel axes and 
 the constant velocity-ratio of a pair of wheels which are to turn 
 about them, to draw the pitch-circles of those wheels. Fig. 62 
 represents the case in which the directions of rotation are contrary; 
 fig. 63 that in which they are the same. Let A and B, as before, 
 be the traces of the axes on a plane perpendicular to them. Find 
 
 Fig. 62. 
 
 the pitch-point, I, as in Article 100, page 83. Then, about A and 
 B, with the radii A I and B I respectively, draw two circles; these 
 will be the pitch-circles required. 
 
 In fig. 62, where the rotations are contrary, and the pitch-point 
 between the axes, the pitch-sm-faces are both convex, and are said 
 to be in " outside gearing." In fig. 63, where the rotations are in 
 the same direction, and the pitch-point beyond the axis of most 
 rapid rotation, the smaller pitch-surface is convex and the larger 
 concave; and these are said to be in "inside gearing" 
 
 104. Circular Wheel and Straight Rack. Given, the axis of a 
 
 wheel, the direction of motion of a rack perpendicular to that 
 axis, and the distance from the axis of a point in the wheel 
 whose velocity is to be equal to that of the rack, to draw the 
 pitch-lines of the wheel and rack In fig. 64, let A be the trace
 
 GEOMETRY OF MACHINERY. 
 
 of the axis on a plane perpendicular to it. Draw A I perpen- 
 dicular to the direction in which the rack is to move, and of a 
 length equal to the given distance; then, 
 about A, with the radius A I, draw a circle, 
 and through I draw a straight line, M N, 
 touching that circle; these will be the re- 
 quired pitch-lines. 
 
 105. Circular Bevel Wheels. Given, a pair 
 
 of axes which intersect each other in a point, 
 and the constant velocity -ratio of two wheels 
 
 I 
 
 Fig. 64. 
 
 which are to turn about those axes, to draw projections of the 
 pitch-surfaces of those wheels. Let the plane of fig. 65 represent 
 the plane of the two axes; let O A and O B be their positions, and 
 
 O their point of intersection. Lay 
 off. on any convenient scale, along 
 those axes, the distances a and 
 O b respectively proportional to the 
 intended angular velocities (which, 
 in the example shown, are contrary). 
 Draw a c parallel to O b, and b c 
 parallel to O a, cutting each other 
 in c; draw the diagonal O c C; 
 this will be the line of contact; and 
 the required pitch-surfaces will be 
 parts of two cones described by 
 making C sweep round O A and 
 O B respectively, and having their 
 common summit at O. O C will be 
 one of the traces of both these cones ; 
 and their other traces will be O G, 
 making the angle A O G = A O C ; 
 and H, making the angle BOH 
 = BOC. 
 
 In any convenient position on the line of contact, mark a con- 
 venient breadth, C F, for the rims of both wheels, so that C F 
 shall be their actual line of contact. Draw GAG and F D K 
 perpendicular to O A, and C B H and F E L perpendicular to 
 O B; then CGKFand CHLF will be the projections of the 
 'two wheels on the plane of their axes. 
 
 To draw the projection of one-half of each of those wheels on a 
 plane perpendicular to its axis, about A, with the radius A C, 
 draw the semicircle C M G, and with the radius A R = D F draw 
 the semicircle R N" S; these will be parts of the pitch-circles of 
 which C A G and F D K are projections, and will form the re- 
 quired projection of one-half of the rim of the wheel whose axis is 
 O A; then, about B, with the radius B C, draw the semicircle
 
 BEVEL WHEELS SKEW-BEVEL WHEELS. 
 
 87 
 
 C P H, and with the radius B T = E F draw the semicircle 
 T Q U; these will be parts of the pitch-circles of which C B H 
 and F E L are projections, and will form the required projection 
 of one-half of the rim of the wheel 
 whose axis is O B. 
 
 The proper widths for the rims 
 of wheels will be considered 
 farther on. 
 
 The perspective sketch, fig. 66, 
 illustrates the case in which one 
 of the pitch-surfaces becomes a j-j~ gg 
 
 flat disc or ring. 
 
 106. Circular Skew-Bevel Wheels are used when a constant 
 velocity-ratio is to be communicated between two pieces which 
 turn about axes that are neither parallel nor intersecting. Their 
 pitch-surfaces are rolling hyperbolo'ids ; and the figures and principal 
 dimensions of such hyperbolo'ids are determined by the method 
 already described in Article 84, page 70, and shown in fig. 54, 
 page 72; it being understood that, in that figure, O a and b 
 represent the intended angular velocities in contrary directions of 
 the two wheels. 
 
 For the actual wheels, narrow zones or frusta only of the hyper- 
 boloi'ds are used, as shown in fig. 67. Where approximate accuracy 
 of form is sufficient, frusta of a pair of 
 tangent cones (or of tangent cylinders, if 
 the pitch-circles are the throats of the hyper- 
 boloi'ds) may be used; the figures of such 
 cones and cylinders being determined as 
 described in Articles 85 and 86, and shown 
 in figs. 55 and 56, pages 73, 74. 
 
 In all skew-bevel wheels the rolling 
 motion is combined with a relative sliding 
 motion along the line of contact, at a rate equal to the rate of 
 advance described in Article 83, page 70. 
 
 The present Article contains some additional rules, which may 
 have to be used in the designing and execution of skew-bevel wheels. 
 
 In fig. 68, let the vertical line through O represent the axis of a 
 skew-bevel wheel, O A = O a the radius of its throat, and O C' 
 a generating line, or line of contact, in that position in which it is 
 parallel to the plane of projection, which plane is supposed to pass 
 through the axis. 
 
 Draw the semicircle A B a; this will be the projection of half 
 the throat of the hyperboloi'd on a second plane of projection, 
 perpendicular to the axis of the wheel. 
 
 Let C' G' be the projection and trace of a plane perpendicular to 
 the axis, and chosen as a convenient plane for the pitch-circle in 
 
 Fig. 67.
 
 88 
 
 GEOMETRY OF MACHINERY. 
 
 the middle of the breadth of the rim of the intended wheel, and 
 
 let that projection cut O C' in C'. 
 
 I. To find the radius of the middle pitch- circle, and to draw its 
 
 projections. Through 
 B draw B C parallel 
 to O A; through C' 
 draw C' C parallel to 
 the axis, cutting B C 
 in C. Join O C; this 
 will be the required 
 radius, and the circle 
 D C G will be the 
 projection of the pitch- 
 circle on the second 
 plane; in G' C' pro- 
 duced take G' D' = 
 'O D = O G =0 C; 
 G' D' will be the pro- 
 jection of the pitch- 
 circle on the first 
 plane. 
 
 D' is a point in the 
 hyperbolic trace of the 
 hyperbolo'id on the first 
 plane; and by the same 
 process any number of 
 points in that trace 
 may be found. 
 
 II. To draw a nor- 
 mat to the pitch-surface 
 in the first plane of 
 projection. Perpendi- 
 cular to O C' draw 
 C' H, cutting the axis 
 of the wheel in H. 
 This line and O C will 
 be the projections of a 
 normal to the pitch- 
 surface at the point 
 whose projections are 
 C'andC. JoinHD'; 
 this line and O D 
 will be the projections 
 Fig. 68. of a normal to the 
 
 pitch - surface at the 
 point whose projections are D' and D.
 
 SKEW-BEVEL WHEELS. 89 
 
 III. To draw the traces of a tangent plane to the pitch surface at 
 the point D', D. The line D' D is the trace on the second plane of 
 projection; and the trace on the first plane isD'g* perpendicular 
 to D' H. 
 
 Another process for finding the trace D' q is as follows : About 
 D, with a radius, D E, equal to C B, draw a circular arc, cutting 
 the circle A B a in E. Through E draw E E' parallel to B O, 
 cutting O A in E'. The straight line D E' q will be the trace 
 required. 
 
 D E and D' E' are also the projections of a generating line of 
 the hyperboloi'd. 
 
 IV. Tangent cone. The siimmit of the tangent cone at the 
 pitch-circle D' G' is at the point q, and D' q is the trace of that 
 cone on the first plane of projection. When extreme accuracy of 
 form is not required, a portion of that cone, having the pitch-circle 
 D' G' at the middle of its breadth, may be used instead of the 
 exact hyperboloidal surface (Article 86, p. 73). 
 
 V. Normal spiral. The normal spiral is a curve on the 
 hyperboloidal siirface which cuts all the generating straight lines, 
 such as those whose projections are E' D', O C', &c., at right angles. 
 Its general form is indicated by the winding dotted curve which 
 traverses O and T in fig. 68. It has an uniform normal pitch, 
 O T, which is found as follows : From A let fall A S perpen- 
 dicular to O C' ; then the normal pitch of the normal spiral is 
 equal to the circumference of a circle whose radius is O S; that is 
 to say, 
 
 It is not necessary to draw precisely the projections of the normal 
 spiral ; but for purposes connected with the designing of teeth for 
 skew-bevel wheels it is useful to know its radius of curvature at 
 the pitch-circle chosen for the wheel. That is done as follows : 
 
 About G, with the radius GF = G/= BC, describe a circle. 
 cutting the circle A B a in F and/; from which two points draw 
 F F' and// ' parallel to B O. (Or otherwise, lay off O F' = of = E K. 
 F 1 G' and F G will be the two projections of a generating line.) 
 In H take Og = E' D'; join Y g,f g, and produce both these 
 lines as far as may be necessary. O F' g will be the rabatment of 
 the triangle whose projection is O F' G'. In O H produced, take 
 g h = H D' ; through h draw k h I perpendicular to F' G' k, and 
 cutting/' g I in 1: through I draw I m parallel to O A, cutting O H 
 produced in m; then g m will be the radius of curvature of every 
 normal spiral at the point where it crosses the pitch-circle G' D'. 
 
 (The object of making this construction above instead of below 
 the point g is merely to avoid confusion in the figure.)
 
 90 GEOMETRY OP MACHINERY. 
 
 Through g draw u gv parallel to k h I] this will be the rabatment 
 of a tangent to the normal spiral at the point G'. 
 
 To find the radius of curvature of a normal spiral at the throat 
 of the hyperbolo'id, in O H take O w - O A; draw xwy perpen- 
 dicular to C', and y z parallel to O A; O z will be the required 
 radius of curvature. 
 
 The lower part of the figure shows the projection on a plane 
 through the axis, of a pitch-circle equal to the pitch-circle G' D', 
 and at the same distance from the throat along the axis in the 
 opposite direction. D E R and D' E' R' are the two projections 
 of one generating line extending from one of those pitch-circles to 
 the other. G' F' R' is the projection of another such generating 
 line. The drawing of a pair of equal pitch-circles may sometimes 
 be useful in the making of patterns for the wheel and for its teeth. 
 
 P, P' and N", N' are the projections of points in the two edges 
 of the rim of the wheel. When the exact hyperboloidal pitch' 
 surface is to be used, and not merely a tangent cone, those points 
 are to be found by a process similar to that by which the projec- 
 tions D, D' are found. When a tangent cone is used as an 
 approximation, they are simply the intersections of two planes 
 perpendicular to the axis, with a tangent in the plane of the axis. 
 
 VI. Radius of curvature of hyperbolic trace. In constructing 
 the pitch-surface of a skew-bevel wheel, it is sometimes useful to 
 determine the radius of curvature of the hyperbolic trace of that 
 surface on a plane traversing the axis, at the point where that trace 
 cuts the pitch-circle, in order that a circular arc of that radius 
 may, if required, be used as an approximation to a small arc of 
 the hyperbolic curve. 
 
 In fig. 68 A, let O X be the axis of the hyperboloid, O A the 
 radius of its throat, O D an asymptote (being, as before, the pro- 
 jection of a line of contact that is parallel to the plane of projection), 
 and X Y the trace of the plane of the intended pitch-circle. Part 
 of the following process has already been described, but it will be 
 described again here, to make the explanation complete : Let D be 
 the point where X Y cuts the asymptote. Lay off X E = O A ; 
 join D E; and make X Y = D E; then X Y will be the radius of 
 the pitch-circle, and Y a point in the hyperbola. Perpendicular to 
 O D draw D F, cutting the axis in F; join F Y; this will be a 
 normal to the hyperbola at the point Y. Thus far the process has 
 already been described. 
 
 Through A draw A B parallel to the axis, cutting the asymptote 
 in B. From B, perpendicular to B, draw B C, cutting O A 
 produced in C. Then G will be the centre of curvature, and A C 
 the radius of curvature of the hyperbola at A ; that is, at the throat 
 of the hyperboloul. 
 
 In X Y, produced both ways as far as may be required, take
 
 SKEW-BEVEL WHEELS. 
 
 91 
 
 Y H = A O, Y L = A C, and Y G = Y F. In Y F take 
 Y K = A : join H F and K G. Through L, parallel to J 1 H, 
 
 Fig. 68 A. 
 
 draw L M, cutting F Y produced in M ; through M, parallel to 
 G K, draw M N, cutting X Y L produced in N j and through N
 
 92 GEOMETRY OP MACHINERY. 
 
 parallel to FH, draw N P, cutting F Y M produced in P; then 
 P will be the centre of curvature, and Y P the radius of curvature of 
 the hyperbola at Y.* 
 
 VII. Foci of hyperbolic trace. To find, if required, the foci of 
 the hyperbolic trace of the pitch-surface ; produce, in fig. 68 A, the 
 straight line O A, in both directions, as far as may be required, 
 and lay off in it O Q = O Q' = O B. Then Q and Q' will be the 
 two foci. The well-known property of a hyperbola, by means of 
 which it can be drawn when one point in it and the two foci are 
 given, is, that the difference of the distances from any point in the 
 curve to the foci is a constant quantity; for example, Y' Q' Y' Q 
 = A Q' AQ = 2AO. Instruments founded on this principle 
 are used for drawing hyperbolas. 
 
 107. Non-Circular Wheel* in General. (.1. M., 443.) Xon- 
 
 circular wheels are used to transmit a variable velocity-ratio 
 between a pair of parallel axes. In fig. 69, let C lf C 2 represent the 
 axes of such a pair of wheels; T r T 2 , a pair of points 
 which at a given instant touch each other in the line of 
 J contact (which line is parallel to the axes and in the 
 same plane with them); and Uj, U 2 , another pair of 
 ri points which touch each other at another instant of 
 the motion ; and let the four points, T 1? T 2 , U lf U 2 , be 
 in one plane perpendicular to the two axes and to 
 the line of contact. Then, for every such set of four 
 points, the two following equations must be ful- 
 filled: 
 
 0^ + C a F 2 = 0^ + 0,^ = 
 arc T x U, = arc T 2 U 2 ; 
 
 and those equations show the geometrical relations which must 
 exist between a pair of rotating surfaces in order that they may 
 move in rolling contact round fixed axes. 
 
 If one of the wheels be fixed and the other be rolled upon it, a 
 point in the axis of the rolling wheel describes a circle of the radius 
 GJ C 2 round the axis of the fixed wheel. 
 
 The equations are made applicable to inside gearing, by putting 
 instead of + and + instead of . 
 
 * The algebraical expressions of these operations are as follows : 
 
 Let A = 6; A B = a; O X = a;; X Y = y; 
 X F = m; Y F = n; Y P = P ; A C = po ; then 
 
 6; 
 
 OT= <
 
 NON-CIRCULAR WHEELS. 
 
 93 
 
 The angular velocity-ratio at a given instant has the value 
 
 CiT i: C 2 T 2 (2.) 
 
 Non-circular wheels, when without teeth, may be called Rolling 
 Cams; and in order that motion may be communicated by means 
 of a pair of rolling cams, and of a suitably shaped smooth rack or 
 sliding bar, it is necessary that the forces exerted by the two pieces 
 on each other should be such as to press their pitch-surfaces together. 
 
 The following are the general problems to be solved in designing 
 non-circular wheels : 
 
 I. Given, the axis and pitch-line of a non-circular wheel; to find 
 approximately the axis of another non-circular wheel which shall turn 
 in rolling contact with the first wheel, and of which an arc of a given 
 length on the pitch-line shall subtend a given angle. 
 
 In fig. 70, let the plane of projection be a plane perpendicular to 
 the axes of the wheels. Let A be the axis of the given wheel, 
 B C its pitch-line, and 
 B its pitch-point at a 
 given instant; and let 
 A B be part of the line 
 of centres. Also, let 
 B D be a straight tan- 
 gent to B C at B ; and 
 let the length of B D be 
 the length of the arc on 
 the pitch-line of the 
 second wheel which is 
 to subtend a given 
 angle. 
 
 In B D take B G = J B D, and about G, with the radius G D 
 = | B D, draw a circle, CDF, cutting the first pitch-line in C. 
 Then, according to Rule IV., Article 51, page 29, the arc B C 
 will be approximately equal in length to B D. Draw and measure 
 the straight line A C ; and in the line of centres take A E = A C. 
 Then draw the straight line E F, making, with the line of centres, 
 the angle H E F = the complement of half the angle that the 
 arc of a length equal to B D is to subtend, and cutting the circle 
 C D F in F. F will be approximately a point in the required pitch- 
 line of the second wheel; and B and F will be the two ends of an 
 arc approximately equal in length to B D and B C. To find the 
 axis of that wheel, find, by plane geometry, in the line of centres, 
 H B E, the centre, H, of a circle which shall traverse F and E; 
 H will be approximately the trace of the required axis. 
 
 II. To find a point in the second pitch-line whose distance from B, 
 as measured on that pitch-line, shall be approximately equal to any 
 given straight tangent, B L. Take B N = ^ B L; and about N ?
 
 94 GEOMETRY OF MACHINERY. 
 
 with a radius N L = f B L, draw a circular arc, cutting the first 
 pitch-line in K. Then B K will be approximately equal in length 
 to B L. Join and measure A K, and in the line of centres 
 take A M = A K. About H, with the radius H M, draw a 
 circular arc, M P, cutting the arc K L P in P ; P will be approxi- 
 mately the required point in the second pitch-line. 
 
 By repeating the same process, any number of points in the re- 
 quired pitch-line of the second wheel may be found approximately. 
 The error of the two preceding rules, in what may be considered 
 an extreme case viz., where the pitch-line of the first wheel 
 coincides with the straight tangent B D, and the angles B H F 
 and BAG are each half a right angle (as in designing a roller to 
 roll with a square roller) is about 0-003 of the length B D to be 
 laid off, and is in excess; the arc B F being too long by that 
 fraction of its length; and the error, in fractions of the arc, varies 
 nearly as the fourth power of the angle subtended by the arc. To 
 ascertain whether the error is sensible, and to correct it by a second 
 approximation, proceed as follows : 
 
 III. To obtain a closer approximation to the required axis 
 and pitch-line. Having drawn the pitch-line B F by Rule II., 
 measure its length in subdivisions by Rule I. of Article 51, page 
 28, and compare that length with B D, so as to ascertain the error. 
 Divide that error by B D, so as to express it in fractions of the 
 required length. Multiply the half-sum of the greatest and least 
 radii by the fraction expressing the ratio of the error to the required 
 length; the product will be a correction, which is to be applied to 
 the lengths of the line of centres, A H, and of each of the radii 
 H B, H F, HP, &c., of the second pitch-line ; that is to say, if 
 the pitch-line, as at first drawn, is too long, each of those straight 
 lines is to be shortened by having the correction subtracted 
 from it. 
 
 For example, in the extreme case already cited, where the first 
 pitch-line is a straight line, B D, perpendicular to A B, and sub- 
 tending half a right angle at A, and the second pitch-line is to 
 subtend also half a right angle at its axis H, let A B be taken 
 as unity; then we have (to three places of decimals) 
 
 BC = BD = AB = 1-000; 
 
 A C (coinciding with a straight line from A to D) = 1414; and 
 the application of Rule I. of this Article gives the following results 
 as first approximations: 
 
 A H = 2-267; H B = 1-267; H F = 0-853. 
 
 Upon drawing the second pitch-line, B F, by Rule II. of this 
 Article, and measuring it in subdivisions, it is found to be too
 
 NON-CIRCULAR WHEELS ELLIPTIC WHEELS. 95 
 
 long by 0-003 of its own length; which being nmltiplied by 
 
 TT T> I TT Tf 2-120 
 
 s = = 1-060, gives 0-003 for the correction to be 
 
 '-i a 
 
 subtracted from the line of centres and from each of the radii of the 
 second pitch-line. Thus are obtained the second approximations, 
 
 A H = 2-264; H B = 1-264; H F = 0-850. 
 
 As examples of non-circular wheels, the following may be 
 mentioned : 
 
 I. An ellipse rotating about one focus rolls completely round in 
 outside gearing with an equal and similar ellipse also rotating about 
 one focus, the distance between the axes of rotation being equal to 
 the major axis of the ellipses, and the velocity-ratio varying from 
 
 1 eccentricity , 1 + eccentricity , 
 
 r-r-i to , -r/ (see Article 108). 
 
 1 + eccentricity 1 eccentricity x 
 
 II. Lobed wheels, of forms derived from the ellipse, roll com- 
 pletely round in outside gearing with each other (see Article 109). 
 
 III. A hyperbola rotating about its farther focus rolls in inside 
 gearing, through a limited arc, with an equal and similar hyperbola 
 rotating about its nearer focus, the distance between the axes of 
 rotation being equal to the axis of the hyperbolas, and the velocity- 
 ratio varying between 
 
 eccentricity + 1 . 
 
 . . .- T and unity. 
 
 eccentricity 1 
 
 IV. Two logarithmic spiral sectors of equal obliquity rotate in 
 rolling contact with each other; or one logarithmic spiral sector 
 rotates in rolling contact with the oblique plane surface of a sliding 
 piece (see Article 110). 
 
 108. Elliptic wheels. The following rules are applicable to the 
 drawing of the pitch-lines of elliptic wheels, and the determination 
 of their comparative motions : 
 
 I. Given, the angle by which each wheel is alternately to overtake 
 and to fall behind the other, and the length of the line of centres, to 
 draw the ellipse which is the figure of both pitch-lines. 
 
 From a point, B, draw two straight lines, B F = B F' = half the 
 line of centres, making with each other the given angle F B F'. 
 Join F F 1 , bisect it in O, produce it both ways, and make O A = 
 O A' = half the line of centres. Draw B B' perpendicular to A A, 
 making O B' = O B. Then A A' is the major axis, B B' the 
 minor axis, O the centre, and F, F', the two foci of the required 
 ellipse, which may be drawn by means of a suitable instrument or 
 machine, or by the well-known process of putting an endless thread, 
 of a length = 2 A F' = 2 F A', round jtwo pins at the foci, and a
 
 OS 
 
 GEOMETRY OF MACHINERY. 
 
 pencil equal in diameter to those pins, and moving the pencil 
 round so as to keep the thread tight. In the workshop ellipses of 
 given dimensions can be drawn with great precision by means of 
 the turning lathe, fitted with apparatus to be afterwards referred to. 
 The wheels are to be centred, as shown in fig. 72, each upon one 
 of its foci. The fixed foci, which are thus placed in the axes of 
 the wheels, are marked A, B, in this figure, and the revolving foci, 
 C, D. The ellipses in fig. 72 are similar to that in fig. 71, but 
 drawn on one-half of the scale. 
 
 Fig. 71. 
 
 Fig. 72. 
 
 II. To find the angular motions and the angular velocity-ratio 
 corresponding to a given position of Hie pitch-point. Suppose both 
 wheels to have started from a position in which A, fig. 71, is the 
 pitch-point, being at the distance A F from the axis of one wheel, 
 and A F' from that of the other, so that the angular velocity-ratio 
 of the second wheel to the first is A F -f- A F. Let C be a new 
 position of the pitch-point. Draw C F, C F'. Then the angular 
 inotion of the first wheel is A F C ; that of the second wheel A F' C ; 
 the first wheel has overtaken the second wheel to the extent repre- 
 sented by the angle FCF = AFC-AF'C; and the velocity- 
 ratio of the second wheel to the first is C F -=- C F'. 
 
 AF 
 The angular velocity -ratio ranges between the limits . -, and 
 
 A F 
 
 - = ; and its mean value in each half-revolution is unity; becaiise 
 
 each half-revolution is made in the same time by both wheels. 
 The instantaneous velocity-ratio is unity when the pitch-point is at 
 B or B' because B F = B F'. 
 
 III. Given, at any instant, the position of one of the revolving foci, 
 to find the position of the otlwr revolving focus, and of the pitch-point. 
 In fig. 72, let A and B be the fixed foci. With a radius equal to 
 the distance between the foci, or double eccentricity (F F' in fig.
 
 ELLIPTIC WHEELS LOBED WHEELS. 97 
 
 71), draw circles about A and B. Let C be the given position of 
 one of the revolving foci. Then, with a radius C D A B (the 
 line of centres), draw a circular arc about C, cutting the circle round 
 B in D; this will be the other revolving focus. Join C D, cutting 
 A B in I; this will be the pitch-point. 
 
 If the wheels and their axles overhang the bearings, the revolving 
 foci, being at a constant distance apart, may be connected by means 
 of a link, C D, as shown in fig. 72. This is useful in elliptic 
 toothed wheels of great eccentricity, because of the teeth in certain 
 positions of the wheel being apt to lose hold of each other. 
 
 109. Lobed wheels * are wheels such as those shown in figs. 74 and 
 75, having two, three, or any 
 greater number of equal great- 
 est radii (such as those marked 
 F A" in fig. 74, and F A"' in 
 fig. 75), and also of least radii 
 (such as those marked F a" in 
 tig. 74, and F a"' in fig. 75). 
 Fig. 74 represents a two-lobed m 
 wheel, and fig. 75 a three-lobed a 
 wheel. An elliptic wheel may 
 be regarded as a one-lobed wheel. Let the difference between the 
 
 Fig. 74. 
 
 Fig. 75. 
 
 greatest and least radii of a lobed wheel be called the inequality ; 
 so that in an elliptic wheel (fig. 71) the inequality is the distance 
 
 * The properties of these wheels were discovered by the Reverend W. 
 
 Holditch.
 
 98 GEOMETRY OF MACHINERY. 
 
 between the foci, F F'. Then any pair of lobed wheels in which 
 the inequality is the same will, if properly shaped, work together 
 in rolling contact, and that whether their numbers of lobes are many 
 or few, the same or different; and this statement includes one- 
 lobed or elliptic wheels. 
 
 The advantage of wheels with two or more lobes is their being 
 self-balanced, which elliptic wheels are not. 
 
 The following are the rules for designing lobed wheels : 
 
 I. Given, in a pair of equal and similar lobed wheels, tlie angle by 
 which each wlieel is alternately to overtake and to fall behind the other 
 wlieel, tJie number of lobes, and t/ie mean radius, to find the inequality, 
 and Uience the greatest and least radii. Multiply the given angle by 
 the number of lobes; then, from a point B"', h'g. 73, draw two lines, 
 B"' F, B*' F', making with each other an angle equal to the product, 
 and make the length of each of them equal to the given mean 
 radius. Draw the straight line F F'; this will be the required 
 inequality. Bisect F F' in O, and produce it both ways; then lay 
 off O A"' = O a'" = B 1 " F = B"' F, the mean radius; then F A"' = 
 F' a"' will be the greatest radius, and F' A'" = F a!" the least 
 radius. 
 
 II. To find any number of points in the pitch-line. In fig. 73, 
 with the major axis A" a", and the foci F and F', draw a semi- 
 ellipse A'" B'" a". Then, in fig. 75, draw from the centre, F, the 
 straight lines marked F A'", dividing a complete revolution into as 
 many equal parts as there are to be lobes (in the present case, 
 three). Make each of these lines equal to the greatest radius 
 (F A"', fig. 73). Bisect the angles between them with the straight 
 lines marked F a"', fig. 75, and make each of the latter set of lines 
 equal to the least radius (F a"' , fig. 73). Divide the half-revolution 
 in fig. 73 into any convenient number of equal angles by the radi- 
 ating lines F 1, F 2, &c., meeting the ellipse at 1, 2, &c. Divide 
 each of the angles marked A'" F a'" in fig. 75 into the same number 
 of equal parts by radiating lines, and lay off upon them lengths, 
 F 1, F 2, &c., equal to those of the corresponding lines in fig. 73; 
 the points 1, 2, &c., in fig. 75, thus found, will be points in the 
 required pitch-line. 
 
 III. To find the positions of the mean radii of the required pitch- 
 line. Divide the angle A'" F B"', in fig. 73 by the number of 
 lobes, and lay off the quotient for each of the angles marked 
 A"' F B'" in fig. 75; then make each of the radii F B"' in fig. 75 
 equal to F B", in fig. 73; these will be the required mean 
 radii. 
 
 REMARK. The example in fig. 75 is a three-lobed wheel. The 
 two-lobed wheel of fig. 74 is drawn by a similar process; the ellipse 
 used for finding the radii being A" B' a" in fig. 73; the inequality 
 F'; and the angle by which. eacJU wheel alternately overtakes and
 
 LOBED WHEELS LOGARITHMIC SPIRAL SECTORS. 
 
 99 
 
 falls behind another equal and similar wheel being one-half of 
 F B" F', fig. 73. 
 
 IV. To draw the pitch-lines of a set of wheels of different numbers 
 of lobes, all of which shall work with each other in rolling con- 
 tact. The inequality must be the same in each wheel. Let F F', 
 fig. 73, be that inequality; and let O be the centre, A"' a"' the 
 major axis, and O B" the semi-minor axis of the ellipse which 
 serves for finding the radii of one of the set of wheels, which one 
 wheel is given. Divide B'" into as many equal parts as there 
 are lobes in the given wheel ; say, for example, three. To find the 
 figui'e of a wheel having any other number of lobes (say two), take 
 the point B" at that number of divisions from O ; join F B", F' B"; 
 lay off O A" = O a' = F B" = F' B"; draw the ellipse A" B" a" with 
 A" a" for its major axis, and F and F' for its foci ; this will be the 
 ellipse for determining the lengths of the radii of the new (two- 
 lobed) wheel. 
 
 The ellipse A' B' a' with the same foci, F F', whose minor semi- 
 axis, O B', is one division of O B"', is itself the pitch-line of the 
 one-lobed wheel, which will work in rolling contact with any wheel 
 of the set. 
 
 110. Logarithmic Spiral Sectors or Rolling Cams. A pair of 
 
 logarithmic spiral sectors may be used as rolling cams, to com- 
 municate by rolling contact an angular motion of limited extent, 
 in the course of which it is desired that the velocity-ratio shall 
 range between certain limits. The general nature of the figure 
 and position of such a pair of sectors may be represented by fig. 69, 
 page 92. 
 
 The only cases in which the dimensions and figures of such 
 sectors can be determined by plane geometry alone, without the 
 aid of calculation, are two, viz. : when the two sectors are equal 
 and similar, so that the sum of the greatest and least radii of each 
 of the two sectors is equal to the 
 line of centres; and when the 
 combination consists of one sector, 
 working with a sliding bar or 
 smooth rack. The following are 
 the rules applicable to such 
 cases : 
 
 I. Given, in fig. 76, the least and 
 greatest radii, O A and O B, of a 
 logarithmic spiral sector, and the 
 angle A O B between them, to find 
 intermediate points in tJie pitch-line 
 of such a sector, and to draw that 
 pitch-line approximately by means 
 of one or more circular arcs. 
 
 Fig. 76.
 
 100 GEOMETRY OF MACHINERY. 
 
 Describe a circle about the triangle O A B ; that is to say, bisect 
 any two of the sides of that triangle (C, E, and D being the three 
 points of bisection), and from the points of bisection draw perpen- 
 diculars to the sides, meeting in F, which will be the centre of the 
 circle through O, A, and B. Draw the diameter G F C H, bisect- 
 ing the arc A H B in H and the arc A O G B in G. Join O H 
 (which will be pei'pendicular to O G, and will bisect the angle 
 A O B); and about G, with the radius G A = G B, draw the 
 circular arc A K B, cutting O H in K. Then K will be a point 
 in the required spiral; and A K B will be the nearest approxi- 
 mation to the spiral arc traversing the three points, A, K 
 and B, that it is possible to make by means of one circular arc 
 only. 
 
 To find two additional points, and a closer approximation to the 
 curve, treat each of the triangles OAK and O K B as the 
 triangle O A B was treated; the result will be the finding of two 
 more points in the spiral, situated respectively in the radii which 
 bisect the angles A O K and K O B; and the drawing of two 
 circular arcs, one extending from A to K, and the other from K 
 to B, which will make a closer approximation to the spiral arc 
 than a single circular arc does. 
 
 The next repetition of the process will give four additional points 
 and four circular arcs; the next, eight additional points and eight 
 circular arcs; and so on to any degree of precision that may be 
 required. 
 
 The radius O K is a mean proportional between O A and O B ; 
 and this property enables its length to be found by calculation, if 
 required. 
 
 The obliquity of a logarithmic spiral, being the angle which a 
 tangent to the spiral at a given point makes with a tangent to a 
 circle described about the axis through that point, or the equal 
 angle which a normal to the spiral at the same point makes with a 
 radius drawn from that point to the axis, is uniform in a given 
 spiral. In fig. 76 the equal angles, O A G, O H G, and O B G, 
 are each of them less than the true obliquity of the spiral, 
 and the angle O K G is greater than the true obliquity. To 
 obtain the closest approximation to the true obliquity possible 
 without further subdividing the angle A O B, proceed as 
 follows : 
 
 II. To find the approximate obliquity. In H K take H L = ^ 
 H K; join L G; then L G will be the obliquity, very nearly. 
 In other words, L G will be very nearly parallel to a normal, 
 and perpendicular to a tangent, to the true spiral at the 
 point K. 
 
 II A. To find tJie approximate obliquity (Another method). By 
 Rule I. or Rule II. of Article 51, page 28. measure the approxi-
 
 LOGARITHMIC SPIRAL SECTORS. 101 
 
 mate length of the arc A B in fig. 76. Then, in fig. 77, draw the 
 straight line MN = OB OA; draw M P perpendicular to 
 
 M N ; and about N, with a radius 
 equal to the approximate length of 
 the arc A B, draw a circular arc, 
 cutting M P in P; join N P; 
 then the angle M P N will be 
 
 approximately the required obli- 
 quity. 
 
 III. Given (in fig. 76), one radius, 
 K, in a logarithmic spiral of a> 
 given obliquity, to draw approxi- 
 mately a short arc of that spiral 
 through K. Draw O G perpen- 
 dicular to O K ; draw K G, making 
 O K G = the angle of obliquity, 
 
 and cutting O K G in G; then, with the radius G K, draw a short 
 
 circular arc through K. 
 
 IV. To draw the pitch-line of a sliding bar which sliall work in 
 rolling contact with a given logarithmic spiral sector, A O B (fig. 77). 
 From the trace of the axis O draw O Q E, perpendicular to the 
 direction in which the bar is to slide, making O Q = O A, and 
 O K, = O B. Find the obliquity of the sector by means of one or 
 other of the preceding rules. Let I be any given position of the 
 pitch-point, and let T I S, traversing I perpendicularly to O Q R, be 
 parallel to the direction in which the bar is to slide. Draw the 
 straight line NIP, making the angle SIN = TIP = the 
 obliquity ; then draw Q N and R P parallel to T I S, and cutting 
 N I P in N and P respectively. The straight line NIP will be 
 the required pitch-line; and N and P will be the points in it 
 corresponding to A and B respectively in the pitch-line of the 
 sector. 
 
 At the instant when the pitch-point is at I, the velocity of the 
 sliding bar is equal to that of the point I in the sector; that is to 
 say, angular velocity x O I; agreeably to the general principle of 
 Article 101, page 84. 
 
 The following rules relate to the solution of questions respecting 
 logarithmic spiral sectors by calculation. 
 
 V. Given, two radii of a logarithmic spiral sector (as A 
 and O B, fig. 76), and the angle between them (A O B), to 
 find the obliquity of the spiral. Take the hyperbolic logar- 
 
 O T? 
 ithm * of the ratio ->c-v-; divide it by the angle A O B in 
 
 \J J\. 
 
 * Hyperbolic logarithm of a ratio = common logarithm x 2 '3026 
 nearly.
 
 102 GEOMETRY OP MACHINERY. 
 
 circular measure;* the quotient will be the tangent of the 
 obliquity. 
 
 VI. Given, the least and greatest radii of a logarithmic spiral 
 sector, and the angle between them, to find the lengths of a series of 
 intermediate radii, which shall divide that angle into a given number 
 of equal smaller angles. Take the difference between the logar- 
 ithms of the greatest and least radii; divide it by the given 
 number; then, commencing with the logarithm of the least radius, 
 compute by successive additions of the quotient a series of logar- 
 ithms, increasing by uniform differences up to the logarithm of the 
 greatest radius; these will be the logarithms of the required 
 intermediate radii. 
 
 VII. Given, one radius and tJte obliquity of a logarithmic spiral, 
 to find the length of a radius making a given angle with the given 
 radius. Multiply the given angle in circular measure (see first 
 footnote below) by the tangent of the obliquity; to the product 
 add the hyperbolic logarithm of the given radius ; the sum will be 
 the hyperbolic logarithm of the required radius; or otherwise, 
 multiply the product by 0-4343, and to the new product add the 
 common logarithm of the given radius; the sum will be the 
 common logarithm of the required radius. 
 
 VIII. Given, tlie difference between the greatest and least radii 
 of a logarithmic spiral sector, and the obliquity of its pitch-line, 
 to find the length of its pitch-line. Multiply the difference of the 
 radii by the cosecant of the obliquity, t 
 
 111. Frictional Cearing. To increase that friction or adhesion 
 between a pair of wheels which is the means of transmitting force 
 and motion from one to the other, their surfaces of contact are 
 sometimes formed into alternate ridges and grooves parallel to the 
 
 * Reduction of angles to circular measure 
 
 1 degree = -0174533 radius length, nearly. 
 30 degrees = 0-5236 ,, 
 
 60 degrees = 1 -0472 ,, 
 
 90 degrees = 1-5708 ,, 
 
 + In symbols, the equations of a logarithmic spiral are as follows : Let a 
 be the radius from whose directions angles are reckoned ; r, any other radius ; 
 6, the angle which r makes with a, in circular measure ; <$>, the obliquity of 
 the spiral ; , the length of the arc from a to r ; p, the radius of curvature at 
 the end of the radius r. Then 
 
 r = a e n ; tan <f> = s hyp. log. - 
 
 6 = cotan <j> hyp. log. - ; 
 
 = (r a) cosec <t> a cosec <p ( an _ j J ; 
 
 p r tan (p.
 
 103 
 
 plane of rotation, constituting what is called frictional gearing. 
 Fig. 78 is a cross-section of the rim of a wheel, illustrating the 
 kind of frictional gearing invented by Mr. Eobertson. Thecom- 
 parative motion of a pair of wheels thus ridged and grooved is nearly 
 the same with that of a pair of smooth wheels 
 in rolling contact, having cylindrical or conical 
 pitch-surfaces lying midway between the tops of 
 the ridges and bottoms of the grooves. 
 
 The relative motion of the surfaces of contact of 
 the ridges and grooves is a rotatory sliding or 
 grinding motion about the line of contact of the Fig. 78. 
 ideal pitch-surfaces as an instantaneous axis; and 
 the angular velocity of that relative grinding motion is equal to 
 the angular velocity of one wheel considered as rolling upon the 
 other as a fixed wheel; which may be found by the principles of 
 Article 77, page 56, and Article 82, page 68. 
 
 The angle between the sides of each groove is about 40; and it 
 is stated that the mutual friction of the wheels is about once and 
 a- half the force with which their axes are pressed towards each 
 other. 
 
 SECTION III. Of tlw Pitch and Number of the Teeth 
 of Wheels. 
 
 112. Relation between Teetb and Pitch-Surfaces Nature of the 
 
 Subject. (A. M., 446.) The most usual method of communicating 
 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 pre- 
 servation of a given velocity-ratio exactly, is by means of a series 
 of alternate ridges and hollows parallel, or nearly parallel, to the 
 successive lines of contact of the ideal toothless wheels or pitch- 
 surfaces, whose velocity- ratio would be the same with that of the 
 toothed wheels. The ridges are called teeth; the hollows, spaces. 
 The teeth of the driver push those of the follower before them, 
 and in so doing sliding takes place between them in a direction 
 across their lines of contact. 
 
 The properties of pitch-surfaces and pitch-lines, and the art of 
 designing them, have been explained in the preceding section. 
 The figures of teeth depend on the principles of sliding contact, 
 which belong to the ensuing section. The present section relates 
 to questions connected with the manner in which the pitch-line of 
 a wheel is divided by the acting surfaces of its teeth, without 
 reference to the figures of those surfaces; for such questions do not 
 require the principles of sliding contact for their solution. 
 
 113. Pitch Defined. (A. M., 447.) The distance, measured
 
 104 GEOMETRY OF MACHINERY. 
 
 along the pitch-line, from the front of one tooth to the front of the 
 next, is called the PITCH. 
 
 114. General Principle*. (A. M., 447.) The pitch, and the 
 number of teeth in wheels, are regulated by the following 
 principles : 
 
 I. In wheels which rotate continuously for one revolution or 
 more, it ia obviously necessary that the pitch should be an aliquot 
 part of the circumference of the pitch-line. 
 
 In racks, and in wheels which reciprocate without performing a 
 complete revolution, this condition is not necessary. Such wheels 
 are called sectors, as has been already stated. 
 
 II. In order that a pair of wheels, or a wheel and a rack, may 
 work correctly together, it is in all cases essential that the pitch 
 should be the same in each. 
 
 III. Hence, in any pair of wheels which work together, tlie 
 numbers of teeth in a complete circumference are inversely as the 
 numbers of whole revolutions in a, given Jime; or, in other words, 
 directly as the times of revolution. 
 
 IV. Hence, also, in any pair of wheels which rotate continuously 
 for one revolution or more, the ratio of the times of revolution 
 (being equal to that of the numbers of teeth), and its reciprocal, the 
 ratio of the numbers of revolutions in a given time, must both 
 be expressible in whole numbers. 
 
 V. In circular wheels, everything stated in the preceding 
 principles regarding the ratio of the numbers of revolutions in a 
 given time (in other words, of the mean angular velocity-ratio) is 
 true also of the angular velocity-ratio at every instant. 
 
 115. Frequency of Contact Hunting-Cog. Let n, N, be the re- 
 spective numbers of teeth in a pair of wheels, N being the greater. 
 Let t, T, be a pair of teeth in the smaller and lai-ger wheel respec- 
 tively, which at a particular instant work together. It is required 
 to find, first, how many pairs of teeth must pass the pitch-point 
 before t and T work together again (let this number be called a) ; 
 secondly, with how many different teeth of the larger wheel the 
 tooth t will work at different times (let this number be called b) ; 
 and thirdly, with how many different teeth of the smaller wheel 
 the tooth T will work at different times (let this be called c). 
 
 CASE 1. If n is a divisor of N, 
 
 
 (1.) 
 
 CASE 2. If the greatest common divisor of N and n be d, a 
 number less than n, so that n = m d, N = M d, then 
 
 a = mN = Mw = Mmc?;6 = M; c = m ....... (2.) 
 
 CASE 3. If N and n be prime to each other, 
 
 a = Nw; 6 = N; c-n ................... (3.)
 
 HUNTING-COG SMALLEST PINION ARITHMETICAL RULES. 105 
 
 It is considered desirable by millwrights, with a view to the 
 preservation of the uniformity of shape of the teeth of a pair of 
 wheels, that each given tooth in one wheel should work with as 
 many different teeth in the other wheel as possible. They there- 
 fore study to make the numbers of teeth in each pair of wheels 
 which work together such as to be either prime to each other, or to 
 have their greatest common divisor as small as is possible con- 
 sistently with the purposes of the machine. 
 
 When the ratio of the angular velocities of two wheels, being 
 reduced to its least terms, is expressed by numbers less than 
 those which can be given to wheels in practice, and it becomes 
 necessary to employ multiples of those numbers by a common 
 multiplier, which becomes a common divisor of the numbers of 
 teeth in the wheels, millwrights and engine-makers avoid the evil 
 of frequent contact between the same pairs of teeth, by giving one 
 additional tooth, called a hunting-cog, to the larger of the two 
 wheels. This expedient causes the velocity-ratio to be not exactly 
 but only approximately equal to that which was at first contem- 
 plated; and therefore it cannot be used where the exactness of 
 certain velocity-ratios amongst the wheels is of importance, as in 
 clockwork. 
 
 116. Smallest Pinion. The smallest number of teeth which it is 
 practicable to give to a pinion (that is, a small wheel), is regulated 
 by principles which will appear when the forms of teeth are con- 
 sidered. The following are the least numbers of teeth which can be 
 usually employed in pinions having teeth of the three classes of 
 figures named below, whose properties will be explained in the 
 sequel : 
 
 I. Involute teeth, 25 
 
 II. Epicycloidal teeth, 12 
 
 III. Round teeth, or staves, 6 
 
 117. Arithmetical Rules. For convenience sake the following 
 arithmetical rules are here given, as being useful in the designing of 
 toothed gearing. 
 
 I. 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 prime number, so as to obtain a 
 quotient not divisible again by the same prime number; 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
 
 106 GEOMETRY OF MACHINERY. 
 
 all the prime factors together; the product should be the given 
 number. 
 
 II. 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. 
 
 Divisor, 1420) 1808 (1, Quotient. 
 
 1420 
 Remainder, 388) 1420 (3, Quotient 
 
 1164 
 Remainder, 256) 388 (1, Quotient. 
 
 256 
 Remainder, 132) 256 (1, Quotient. 
 
 132 
 Remainder, 124) 132 (1, Quotient. 
 
 124 
 Remainder, 8) 124 (15, Quotient. 
 
 120 
 Remainder, 4) 8 (2, Quotient. 
 
 The last remainder, 4, is the required greatest common 
 measure. 
 
 III. To reduce the ratio of two numbers to its least terms, 
 divide both numbers by their greatest common measure. 
 
 . 1808 - 4 452 
 For example, j^-jj - 355- 
 
 IV. To express the ratio of two numbers in the form of a continued 
 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
 
 CONTINUED FRACTIONS. 107 
 
 c+1 
 
 d -f <fec., 
 
 the right-hand side is the continued fraction required. 
 
 To save space in printing, a continued fraction is often arranged 
 as follows : 
 
 The ratio of two incommensurable quantities is expressed by an 
 endless continued fraction. For example, the ratio of the diagonal 
 
 to the side of a square is expressed by 1 + - ^ : ~ r 5 - &c., 
 
 Zi T~ a ~r A T T 
 
 without end. 
 
 V. To form a series of approximations to a given ratio. Express 
 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 denominator 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, b, c, d t <fec. 
 
 1 n ri n" 
 Fractions, T > ^ ,>; 
 
 1 m m m 
 
 n _ + a _a n' _ I + b n ri^ n + c n' . 
 m = I+Q~i' m'~ Q + bm' 'm r '~m + cm'' } C ' 
 
 452 
 To take a particular case; let the given ratio be as before, -.' 
 
 then we have the following series :
 
 108 GEOMETRY OF MACHINERY. 
 
 Quotients, ..................... 1 3 1 1 1 15 2 
 
 1 1 4 5 9 H 219 452 
 Fractl0ns > ................ 1013 4 7 ll m 355 
 
 Less or greater than ) L G L G L G L G 
 given ratio, ......... J 
 
 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 difference 
 between any pair of consecutive converging fractions is equal to 
 unity divided by the product of their denominators; for example, 
 9 5 36 - 35 19 14 99-98 1 . . 77 A1 
 7 - 4 -TTT- = 28 J 7 - ll =: TTTT = 7T thirdl *> they 
 
 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, fourthly, the difference between any one of them 
 and the given ratio is less than the difference between that one and 
 the next fraction of the series. 
 
 Fractions intermediate between the converging fractions may be 
 
 -,, ,.- . hn + kri , n n' 
 
 found bv means of the formula -= - j - ,: where -and -. are 
 
 nm + Kin, m 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. 
 
 118. A Train of wheeiwork (A. M., 449,) consists of a series of 
 axes, each having upon it two wheels, one of which is driven by a 
 wheel on the preceding axis, while the other drives a wheel on the 
 following axis. If the wheels are all in outside gearing, the direction 
 of rotation of each axis is contrary to that of the adjoining axes. 
 In some cases a single wheel upon one axis answers the purpose. 
 both of receiving motion from a wheel on the preceding axis and 
 giving motion to a wheel on the following axis. Such a wheel is 
 called an idle wheel: it affects the direction of rotation only, and 
 not the velocity-ratio. 
 
 Let the series of axes be distinguished by numbers 1, 2, 3, 
 <fec ..... 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, N 2 , &c ..... N m _j ; and let the numbers of teeth in the driven 
 or following wheels be denoted by ris, each with the number of its 
 axis affixed; thus, n%, n s , &c ..... n m . Then the ratio of the 
 angular velocity a m of the m th axis to the angular velocity a^ of the 
 first axis is the prodxict of the m 1 velocity-ratios of the succes- 
 sive elementary combinations, viz. : 
 
 n m
 
 TRAINS OP WHEELS. 109 
 
 that is to say, the velocity-ratio of the last and first axes is the 
 ratio of the product of the numbers of teeth in the drivers to the 
 product of the numbers of teeth in the followers ; and it is obvious, 
 that so long as the same drivers and followers constitute the train, 
 the order in which they succeed each other does not affect the 
 resultant velocity-ratio. 
 
 Supposing all the wheels to be in outside gearing, then, as each 
 elementary combination reverses the direction of rotation, and as 
 the number of elementary combinations, m 1, is one less than 
 the number of axes, m, it is evident that if m is odd, the direction 
 of rotation is preserved, and if even, reversed. 
 
 It is often a question of importance to determine the numbers of 
 teeth in a train of wheels best suited for giving a determinate 
 velocity-ratio to two axes. It was shown by Young, that to do 
 this with the least total number of teeth, the velocity-ratio of each 
 elementary combination should approximate as nearly as possible to 
 3 - 5 9. This would in some cases give too many axes; and as a 
 convenient practical rule it may be laid down, that from 3 to 6 
 ought to be the range of the velocity-ratio of an elementary com- 
 bination in wheelwork. * 
 
 T> 
 
 Let rr, be the velocity-ratio required, reduced to its least terms, 
 u 
 
 and let B be greater than C. 
 
 T> 
 
 If ^ is not greater than 6, and C lies between the prescribed 
 \j 
 
 minimum number of teeth (which may be called t), and its double 
 2 t, then one pair of wheels will answer the purpose, and B and C 
 will themselves be the numbers required. Should B and C be 
 inconveniently large, they are if possible to be resolved into factors, 
 and those factors, or, if they are too small, multiples of them, used 
 for the numbers of teeth. Should B or C, or both, be at once 
 inconveniently large, and prime, or should they contain incon- 
 
 T> 
 
 veniently large prime factors, then, instead of the exact ratio ^, 
 
 * The following are some examples of the results of Young's rule, the first 
 line containing velocity-ratios, and the second, the numbers of elementary 
 combinations of wheels suited to give velocity-ratios intermediate between 
 the numbers in the first line : 
 
 1 7 24 88 315 1132 4064 14596 
 12345 6 7 
 
 The following are examples of the results of the modified rule, that the 
 lowest of the velocity -ratios for each elementary combination should range 
 from 3 to 6: 
 
 1 6 36 216 1296 7776
 
 110 GEOMETRY OF MACHINERY. 
 
 some ratio approximating to that ratio, and capable of resolution 
 into convenient factors, is to be found by the method of continued 
 fractions (see Article 117, page 106); also Willis On Mecltanism, 
 pages 223 to 238). 
 
 TO 
 
 Should ^ be greater than 6, the best number of elementary 
 O 
 
 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, m 1. 
 
 Then, if possible, B and C themselves are to be resolved each 
 into m 1 factors, which factors, or multiples of them, shall be not 
 less than t, nor greater than 6 t ; or if B and C contain incon- 
 veniently large prime factors, an approximate velocity-ratio, found 
 
 T> 
 
 by the method of continued fractions, is to be substituted for ^, as 
 
 O 
 
 before. When the prime factors of either B or C are fewer in 
 number than m I, the 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 order of 
 the drivers N, and of the followers n, is immaterial ; but to secure 
 equable wear of the teeth, as explained in Article 115, page 104, 
 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 ensured 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 multipliers : Let the required velocity-ratio, in its least 
 terms, be 
 
 B 360 
 C = 7 ' 
 
 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
 
 TRAINS DIAMETRAL AND RADIAL PITCH. Ill 
 
 to that which makes these factors least unequal, viz., 5 8 9. 
 Hence, resolving numerator and denominator into three factors 
 each, we have 
 
 B _ 5-8-9 
 
 C ~ 1-1-7* 
 
 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 ai-e 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 
 
 13 x 1 = 13; 17 x 1 = 17; 2 x 7 - 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 
 
 17 x 5 = 85; 13 x 8 = 104; 2 x 9 = 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 
 
 119. Diametral and Radial Pitch. The diametral pitch of a 
 circular wheel is a length bearing the same proportion to the pitch 
 proper, or circular pitch, that the diameter of a circle bears to its 
 circumference; and the radial pitch is half the diametral pitch. 
 In other words, the diametral pitch is to be found by dividing the 
 diameter of the pitch-circle by the number of teeth in the whole 
 circumference, and the radial pitch by dividing the radius by the 
 same number. In symbols, let p be the pitch, properly so called, 
 or circular pitch, as measured on the pitch circle, r the radius of 
 the pitch circle, or geometrical radius, and n the number of teeth; 
 
 q the diametral pitch, and f the radial pitch; then 
 
 1 
 
 113 2 r 355 
 
 q _ 113 _ r _nq _ 710 q 
 2 ~ 710^ ~ n' r = ~2 j p ~ 113 '2'
 
 112 GEOMETRY OF MACHINERY. 
 
 Wheels are sometimes described by stating how many teeth they 
 have for each inch of diameter; that is to say, by stating the recipro- 
 cal of the diametral pitch in inclines I - = ); and the phrases used 
 
 in so describing them are such as the following : A three-pitch wheel 
 is a wheel having three teeth for each inch of diameter ; so that 
 
 1 355 
 
 q = -z inch, and p = y^o o inch = 1 '04:72 inch; a ten-pitch 
 
 o 1 1 o x o 
 
 wlied is a wheel having ten teeth for each inch of diameter; so that 
 
 35 *5 
 
 q = O'l inch, and p = -y r inch = 0-31416 inch; and so on. 
 1 lo 
 
 The following are rules for solving questions regarding radial 
 and circular pitch by graphic construction. 
 
 I. Given, the circular pitch of a wheel, to find the radial pitch. 
 Draw a straight line equal to one-sixth part of the given circular 
 pitch, and then, by Rule IV. of Article X 51, page 29, find the two 
 ends of a circular arc approximately equal in length to that straight 
 line, and subtending 60. The chord of that arc will be the re- 
 quired radial pitch very nearly, being too long by about one-900th 
 part only. 
 
 This may be expressed in other words, as follows (see fig. 79) : 
 Let A B be a straight line equal to one-sixth of the given circular 
 pitch. Draw the equilateral triangle 
 ABC, bisect B C in D, and join AD; 
 in A B, take A E = ^ A B, and about 
 E, with the radius E B = | A B, draw 
 the circular ai'c B F, cutting A D pro- 
 duced in F; A F will be the required 
 approximate radial pitch. 
 
 If greater accuracy is reqxiired, make 
 the straight line equal to one-twelfth of 
 the circular pitch, and let the angle sub- 
 Fig. 79. tended by the arc be 30; the radius of 
 that arc will be the required radial pitch, 
 correct to one-1 4,400th part. 
 
 II. Gfoen, the radial pitch of a wheel, to find tlie circular pitch. 
 With a radius equal to six times the given radial pitch describe a 
 circle : mark upon the circumference of that circle a chord equal to 
 the radius, so as to lay off an arc equal to one-sixth part of the cir- 
 cumference; then, by Rule I. or II. of Article 51, page 28, draw a 
 straight line approximately equal in length to that arc; the length 
 of that straight line will be the required circular pitch, very nearly. 
 
 If Rule I. is used, the straight line will be too short by about 
 one-900th part; if Rule II. is used, it will be too long by about 
 one-3,600th part. If a closer approximation is required, measure
 
 POSITIONS OF PARALLEL AXES LAYING-OFF PITCH. 
 
 113 
 
 the circular pitch by both rules; then to the length, as measured 
 by Rule I., add four times the length as measured by Rule II., and 
 divide the sum by Jive; the quotient will be the required circular 
 pitch, correct to about one-40,000th part. 
 
 120. Relative Positions of Parallel Axes in Wheclvrork. I. Given, 
 the radial pitch and the numbers of teeth of a pair of wheels with 
 parallel axes, to find the length of the line of centres, or distance 
 between the axes. Multiply the radial pitch by the sum or by the 
 difference of the numbers of teeth, according as the wheels are in 
 outside or inside gearing. 
 
 II. Given, the length of the line of centres, and the numbers of 
 teeth, to find the radial pitch. Divide the given length by the sum 
 or the difference of the numbers, according as the wheels are in 
 outside or inside gearing. 
 
 III. Given, in fig 80, the perpendicular distance A A" between 
 the first and last axes of a train of wheels, which are to turn about 
 parallel axes all in one 
 
 plane, and the numbers of 
 teeth of the wheels; re- 
 quired, the positions of 
 the several pitch - points 
 and intermediate axes. 
 From one end, A, of the 
 straight line A A"', draw, 
 in any convenient different 
 direction, another straight 
 line A a'", on which lay 
 off, on any convenient 
 scale, a series of lengths 
 proportional to the num- 
 
 Fig. 80. 
 
 bers of teeth, viz. : A i for the first driver, i a' for the first follower; 
 a' i' for the second driver, i' a' for the second follower; and so on. 
 Let a" be the end of that series of lengths. Draw the straight 
 line a" A", and parallel to that line draw a series of straight 
 lines, i I, a' A, &c., through the points of division of A a", cutting 
 A A" in a corresponding series of points of division. Then A, A",, 
 <fec., will represent the intermediate axes, and I, I', &c., the pitch- 
 points. 
 
 121. Laying -off Pitch, and Subdivision of Pitch-Lines. The 
 
 laying-off of the pitch, or of any multiple of the pitch, on the pitch- 
 line of a wheel, is to be performed by means of Rule III. of Article 
 5 1 , page 29. The laying-off of the same length upon several different 
 pitch-lines, so as to find corresponding pitch-points upon them, may 
 be performed at one operation, as follows : Let the straight line 
 A G represent the given length. In A G take A C = \ AG; 
 and about C, with the radius C G = | A G, draw a circular arc.
 
 114 GEOMETRY OF MACHINERY. 
 
 D G D.'"" Let A D, A D', &c., be arcs of different pitch-lines, 
 touching A G in A, and cut off by the dotted circular arc ; each 
 
 of these arcs will be approxi- 
 mately equal in length to A G. 
 
 As to the operation of "pitch- 
 ing" that is. division of a pitch- 
 
 , y , ., ' -.IT f 
 
 circle or other pitch-line, or 01 
 any part of a pitch -line, into any 
 required number of parts, each 
 equal to the pitch see Article 
 51, Rules V. and VI., pages 29 
 and 30. 
 
 Circular and straight pitch- 
 Fig. 81. lines maybe subdivided by means 
 of "Dividing Engines." In a 
 
 dividing engine the piece upon which divisions are to be marked 
 is fixed upon a suitable support, capable^ of turning about an axis 
 or of sliding in a straight line, as the case may be, and moved by 
 means of a screw. By turning the screw a motion of any required 
 extent can be given to the piece, and repeated as often as may be 
 necessary; and after each such movement, a mark is made on the 
 surface to be divided by means of a sharp point or edge, having 
 a movement transverse to that of the piece to be divided.* 
 
 Machines are used by mechanical engineers, with movements on 
 the principle of dividing engines, which serve both to pitch wheels 
 or divide their pitch-circles, and to cut their teeth to the proper 
 shape. Such machines will be again mentioned further on. 
 
 SECTION IV. Sliding Contact Teeth, Screw-Gearing, and Cams. 
 
 122. General Principle of Sliding Contact. The line of connec- 
 tion, in the case of sliding contact of two moving pieces, is the 
 common normal to their surfaces at the point where they touch; 
 and the principle of their comparative motion is, that the com- 
 ponents, along that normal, of the velocities of any two points 
 traversed by it, are equal. This being borne in mind, all questions 
 of the comparative motion of a pair of primary pieces in sliding 
 contact may be solved by means of the Rules of Article 91, pages 
 78 to 80. 
 
 The acting surfaces of a pair of pieces in sliding contact may be 
 both plane or both convex, or one convex and one plane; but one 
 
 For descriptions of dividing engines for purposes of great precision, see 
 Ramsden's Description of an Engine for Dividing Mathematical Instruments, 
 1777 ; Ramsden's Description of an Engine for Dividing Straight Lines, 1779 ; 
 Holtzapffel On Turning and Mechanical Manipulation, voL ii., pages 639 
 to 654.
 
 SLIDING CONTACT FIGURES OF TEETH. 
 
 115 
 
 of them only can be concave; and in that case the other must be 
 convex, and of a curvature not flatter than that of the concave 
 surface. 
 
 123. Teeth of Wheels and Racks. General Principle. (A. M. 
 
 451.) The figures of the teeth of wheels and racks are regulated 
 by the principle, that the teeth sJiall give the same velocity-ratio by 
 their sliding contact which the ideal toothless pitch-surfaces would give 
 by tJieir rolling contact. 
 
 Let Bj, B 2 , in fig. 82, be parts of the pitch-lines of a pair of 
 wheels, I the pitch-point, and G v C 2 the traces of the axes. 
 According to Article 91, pages 78 to 80, the comparative velocity 
 of two connected pieces depends on the position of the point where 
 the line of connection cuts the line of centres. For a pair of smooth 
 pitch-surfaces, that point is the pitch-point I ; and for a pair of 
 surfaces in sliding contact, it is the point where the line of connection 
 of these surfaces (being, as stated in the preceding Article, their 
 common normal at the point where they touch) cuts the plane of 
 the axes. Hence the condition of the correct working of the teeth 
 of wheels and racks is the following : 
 
 The line of connection of the teeth should always traverse the pitch- 
 point. , 
 
 For example, in fig. 82, A x T 1 and A 2 T 2 may represent the 
 traces of parts of the acting surfaces of a pair 
 of teeth belonging to the driver and follower 
 respectively, Tj and T 2 a pair of particles in 
 these surfaces, which at a given instant touch 
 each other in one point, and P x T x T 2 P 2 the 
 common normal at that point; then that 
 normal ought always to traverse the pitch- 
 point I. 
 
 At the instant of passing the line of centres 
 the point of contact of a pair of teeth coincides 
 with the pitch-point. 
 
 124. Teeth Definitions of their Parts. 
 
 That part of the FRONT or acting surface 
 of a tooth which projects beyond the pitch- 
 surface is called the FACE; that part which lies within the 
 pitch-surface, the FLANK. The flanks of the teeth of the driver 
 drive the faces of the teeth of the follower, and the faces of the 
 teeth of the driver drive the flanks of the teeth of the follower. 
 The corresponding divisions of the BACK of a tooth may be called 
 the BACK-FACE and BACK-FLANK. The face of a tooth in outside gear- 
 ing is always convex ; the flank may be convex, plane, or concave. 
 
 When the motion of a pair of wheels is reversed, the backs of 
 the teeth become the acting surfaces. 
 
 By the PITCH-POINT OF A TOOTH is meant the point where the 
 
 Fig. 82.
 
 J16 
 
 GEOMETRY OF MACHINERY. 
 
 pitch-line of the wheel cuts the front of the tooth. At the instant 
 of passing the line of centres, the pitch-points of a pair of teeth 
 coincide with each other, with the point of contact, and with the 
 pitch-point of the pitch-lines. 
 
 The DEPTH of a tooth is the distance in the direction of a radius 
 from root to crest ; the extent to which the crest of a tooth projects 
 beyond the pitch-surface is called the ADDENDUM ; and a line parallel 
 to the pitch-line, and touching the crests of all the teeth of a wheel 
 or rack, is called the ADDENDUM-LINE, or, in a circular wheel, the 
 ADDENDUM-CIRCLE. The radius of the addendum-circle of a circular 
 wheel is called the REAL RADIUS, to distinguish it from the radius 
 of the pitch-circle, which is called the GEOMETRICAL RADIUS. 
 
 CLEARANCE or FREEDOM is the excess of the total depth above the 
 working depth, or, in other words, the least distance between the 
 crest of a tooth of one wheel and the bottom of the hollow between 
 two teeth of another wheel, with which the first wheel gears. 
 
 The pitch of a pitch-line is divided \by the fronts and backs of 
 the teeth into THICKNESS and SPACE. The excess of the space 
 between the teeth of one wheel above the thickness of the teeth of 
 another wheel with which the first wheel gears is called PLAY or 
 BACK-LASH; because it is the distance through which the pitch-line 
 of the driver moves after having its motion reversed before the 
 backs of the teeth begin to act. 
 
 125. ciiNtoiuarr i>imniioii of Teeth. The following are cus- 
 tomary dimensions for teeth, taken from a table which Mr. 
 Fairbairn gives in his treatise On Millwork (see fig. 83). 
 
 It is to be iinderstood that these customary dimensions may be 
 departed from when there is any sufficient reason for doing so. 
 Examples of this will appear in the sequel. 
 
 Fig. 83.
 
 DIMENSIONS OP TEETH INSIDE GEARING. 117 
 
 Let the pitch C D = p; then 
 Depth, total, A B = 075^; 
 
 Clearance or freedom, E B, ............ ) ,, n nc A A . . , 
 
 Also, play or back-lash, F D - C F } = / = 0>06 P + 0>04 inch ^ 
 
 Depth, working, A E = 075 p f' t 
 Addendum, A G = A A E; 
 
 Thickness, C F = P -^ ; 
 
 Space, F D = -/ 
 2> 
 
 Thickness of ring which carries the teeth (in a cast-iron wheel) 
 = thickness of tooth at root. 
 
 The least thickness sufficient for the teeth of a given pair of 
 wheels is a question of strength, depending on the force to be 
 exerted; and although such questions properly belong to a later 
 division of this treatise, it may be convenient to state here the 
 rule generally relied on : Divide the greatest pressure to be exerted 
 between a pair of teeth in pounds by 1,500; tlie square root of the 
 quotient will be the least proper thickness in inches. 
 
 For pressures expressed in kilogrammes, and thicknesses in 
 millimetres, the divisor becomes 1-055; the rule being in other 
 respects the same. 
 
 The least breadth sufficient for the fronts of teeth is a quantity 
 depending on dynamical principles, and belonging properly to the 
 next division; but for convenience it may here be stated that an 
 ordinary rule is as follows : Divide the greatest pressure to be exerted 
 in pounds by the pitch in inches, and by 160; the quotient will be the 
 breadth in inches. 
 
 For pressures in kilogrammes and dimensions in millimetres, 
 instead of dividing by 160, multiply by 9. 
 
 126. Teeth for inside Gearing The figures of the acting surfaces 
 of teeth for a pitch-circle in inside gearing are exactly the same 
 with those suited for the same pitch-circle in outside gearing; but 
 the relative positions of teeth and spaces, and those of faces and 
 flanks, are reversed; and the addendum-circle is of less radius than 
 the pitch-circle. All the rules in the ensuing Articles, with these 
 modifications, may be applied to inside gearing. 
 
 127. Common Velocity and Relative Velocity of Teeth Approach 
 
 And Recess Path of Contact. The common velocity of a pair of 
 * 0'04 inch = 1 millimbtre, nearly.
 
 118 GEOMETRY OF MACHINERY. 
 
 teeth is that component velocity along the line of connection which 
 is common to the pair of particles that touch each other at a given 
 instant. In fig. 82, page 115, let G l Pj, and C 2 P 2 be the two 
 common perpendiculars of the line of connection and the two axes 
 respectively, and let a^ and 2 denote the angular velocities about 
 those axes ; then the common component in question has the value 
 
 1 * C l P l = 2 ' 2 P 2 ..................... (!') 
 
 The relative velocity of a pair of teeth is the velocity with which 
 their acting surfaces slide over each other; and it is found as 
 follows : Conceive one of the pitch-surfaces to be fixed, and the 
 other to roll upon it, so that the line of contact (I, fig. 82, page 
 115) becomes an instantaneous axis; find the resultant angular 
 velocity (see Articles 73 to 77, pages 52 to 56, and Articles 81 and 
 82, pages 66 to 68), and multiply it by the perpendicular distance of 
 the point of contact of the teeth (T, fig. ^2) from the instantaneous 
 axis; the product will be the relative velocity required. That is 
 to say, let c denote the resultant angular velocity about the 
 instantaneous axis of the pitch-surface which is supposed to roll; 
 and in fig. 82 let I T be the perpendicular distance of the point of 
 contact from the instantaneous axis; then the relative velocity of 
 sliding is 
 
 c-IT ............................... (2.) 
 
 The values of the resultant angular velocity c (as has been 
 shown in the previous Articles, already referred to) are, for parallel 
 axes in outside gearing, c = a x + 2 ; for parallel axes in inside 
 gearing, c = a x a 2 ; and for intersecting axes, the diagonal of a 
 parallelogram, of which a x and er 2 are the sides. 
 
 While the point of contact, T, is advancing towards the pitch- 
 point I, the roots of the teeth are sliding towards each other; and 
 this relative motion is called the APPROACH. 
 
 The relative velocity gradually diminishes as the approach goes 
 on, and vanishes at the instant when I T = 0; that is, when the 
 point of contact coincides with the pitch-point; so that at that 
 precise instant the pair of teeth are in rolling contact. 
 
 After the point of contact has passed the pitch-point, the roots of 
 the teeth are sliding away from each other with a gradually 
 increasing relative velocity; and this relative motion is called the 
 KECESS. 
 
 During the approach the flank of the driver drives the face of 
 the follower; during the recess the face of the driver drives the 
 flank of the follower. 
 
 The extent of the sliding motion of a pair of teeth is equal, during 
 the approach, to the excess of the length of the face of the driven
 
 PATH AND ARC OF CONTACT OBLIQUITY OP ACTION. 119 
 
 tooth above the length of the flank of the driving tooth ; and during 
 the recess, to the excess of the length of the face of the driving 
 tooth above the length of the flank of the driven tooth. 
 
 The PATH OF CONTACT is the line traversing the various positions 
 of the point of contact, T (fig. 82, page 115). If the line of con- 
 nection preserves always the same position, the path of contact 
 coincides with it, and is straight; in other cases the path of contact 
 is curved. 
 
 It is divided by the pitch-point I into two parts : the path oj 
 approach, described by T in approaching the pitch-point; and 
 the path of recess, described by T after having passed the pitch- 
 point. 
 
 The path of contact is bounded where the approach commences 
 by the addendum-line of the follower; and where the recess ter- 
 minates, by the addendum-line of the driver. The length of the 
 path of contact must be such that there shall always be at least 
 one pair of teeth in contact; and it is better still, when practicable, 
 to make it so long that there shall always be at least two pairs of 
 teeth in contact; but this is not always possible. 
 
 128. Arc of Contact. (A. M., 454.) The arc of contact on a 
 pitch-line is that part of the pitch-line which passes the pitch-point 
 during the action of one given tooth with the corresponding tooth 
 of the other wheel. 
 
 In order that one pair of teeth at least may be in action at 
 each instant, the length of the arc of contact must be greater than 
 the pitch; and when practicable, it should be double the pitch, in 
 order that two pairs of teeth, at least, may be in action at each 
 instant; but this is not always practicable; and the most common 
 values are from 1 -4 to 1 -8 times the pitch. It is divided by the 
 front of the tooth to which it belongs into two parts : the arc of 
 approach, lying in advance of the front of the tooth ; and the arc of 
 recess, lying behind the front of the tooth. It is usual to make 
 the arcs of approach and of recess of equal length ; and in that 
 case each of them must be greater than half the pitch, and should, 
 if practicable, be made equal to the pitch. For a given pitch-line, 
 and a given pitch and figure of tooth, the length of those arcs 
 depends on the addendum, in a manner to be afterwards described. 
 
 129. Obliquity of Action. The obliquity of action of a pair of 
 teeth is the angle which the line of connection makes at any 
 instant with a tangent plane to the two pitch-surfaces; for ex- 
 ample, in fig. 82, page 115, the complement of the angle at I. 
 When the path of contact is a straight line, coinciding at every 
 instant with the line of connection, the obliquity is constant; in 
 other cases it is variable; and its mode of variation is usually such 
 that it diminishes during the approach, and increases again during 
 recess. In a dynamical point of view, it is advantageous to make
 
 120 GEOMETRY OF MACHINERY. 
 
 the obliquity as small as possible ; and, on the other hand, there is 
 & connection between the obliquity of action and the number of 
 teeth which makes it impracticable to use pinions of fewer than a 
 certain number of teeth with less than a certain maximum obliquity 
 of action. Mr. Willis, from an examination of the results of 
 ordinary practice, concludes that the best value on the whole for 
 the mean obliquity of action in toothed gearing is between 14 and 
 15. Such an angle may be easily constructed by drawing a right- 
 angled triangle whose three sides bear to each other the proportion 
 of the numbers 
 
 65 : 63 : 16; 
 
 when the required angle will lie opposite to the shortest side of the 
 triangle. The values of its chief trigonometrical functions are 
 
 sine, 16 ^ 65 = 0-2461538, nearly. 
 
 cosine, 63 + 65 = 0-9^92308, nearly. 
 
 tangent, 16 -=- 63 = 0-2539683, nearly. 
 
 cosecant, 65 + 16 = 4-0625. 
 
 cotangent, 63 -s- 16 = 3-9375. 
 
 The corresponding angle is 14 15'; being a little less than one-25th 
 part of a revolution. 
 
 130. The Teeth of Spur-wheels and Racks have acting surfaces 
 of the class called cylindrical surfaces, in the comprehensive sense of 
 that term; and their figures are designed by drawing the traces of 
 their surfaces on a plane perpendicular to the axes of the wheels 
 (or, in the case of a rack, to the axis of the wheel that is to gear 
 with the rack); which plane contains the pitch-lines and the line of 
 connection, and may be represented by the plane of the paper in iig. 
 82, page 115. The path of contact, also, is situated in the same 
 plane; and the angle of obliquity of action is at each instant equal 
 to the angle I G P, which the common perpendicular, C P, of the line 
 of connection and one of the axes makes with the line of centres, 
 Cj I C 2 . Because of the comparative simplicity of the rules for 
 drawing the figures of the teeth of spur-wheels, those rules are used, 
 with the aid of certain devices to be afterwards described, for 
 drawing the figures of the teeth of bevel wheels and skew-bevel 
 wheels also. 
 
 131. Involute Teeth for Circular Wheels. (A. J/., 457.) The 
 
 simplest of all forms for the teeth of circular wheels is that in 
 which the path of contact is a straight line always coinciding with 
 the line of connection, which makes a constant angle with the line 
 of centres, and is inclined at a constant angle of obliquity to the 
 common tangent of the pitch-lines. 
 
 In fig. 84, let C 1? C 2 , be the centres of two circular wheels, whose
 
 INVOLUTE TEETH RULES FOR DRAWING. 
 
 121 
 
 pitch-circles are marked B I} B 2 . Through the pitch-point I draw 
 the intended line of connection, P l P 2 , making, with the line of 
 
 the complement 
 P and C P 
 
 Fig. 84. 
 
 centres, the angle I P = 
 of the intended obliquity. 
 
 From Ci and C 2 draw 
 
 perpendicular to Pj P 2 , with which two per- 
 pendiculars as radii describe circles (called 
 base-circles) marked D x , D 2 . T 
 
 Suppose the base-circles to be a pair of 
 circular pulleys, connected by means of a ' 
 cord whose course from pulley to pulley is 
 P! I Pg. As the line of connection of those 
 pulleys is the same with that of the proposed Bjt 
 teeth, they will rotate with the required 
 velocity-ratio. Now, suppose a tracing point, 
 T, to be fixed to the cord, so as to be carried 
 along the path of contact, P x I P 2 . That 
 point will trace, on a plane rotating along 
 with the wheel 1, part of the involute of the base-circle ~D V and 
 on a plane rotating along with the wheel 2, part of the involute 
 of the base-circle D 2 , arid the two curves so traced will always cut 
 the line of connection at right angles, and touch each other in the 
 required point of contact T, and will therefore fu!61 the condition 
 required by Article 122, page 114. The teeth thus traced are 
 called Involute Teeth. 
 
 All involute teeth of the same pitch work smoothly together. 
 
 The following is the process by which the figures of involute 
 teeth are to be drawn in practice : 
 
 In fig. 85, let C represent the centre of the wheel, I the pitch- 
 point, C I the geometrical radius, BIB the pitch-circle, and let 
 the intended angle of obliquity of action be given, and also the 
 pitch. (In the example represented by the figure, the obliquity is 
 supposed to be 14^, as stated in Ai'ticle 129, page 120; and the 
 wheel has 30 teeth.) Then proceed by the following rules : 
 
 I. To draw tJte base-circle and the line of connection. About C, 
 with the radius C P = C I x cosine of obliquity (that is to say, in 
 
 /*O 
 
 the present example, C I), draw a circle, D P D ; this is the 
 by 
 
 base-circle. Then about I, with a radius I P = C I x sine of 
 
 = C I 
 
 J C 
 
 obliquity (that is to say, in the present example, ^ C I), draw a 
 
 bo 
 
 short circular arc, cutting the base-circle in P. Draw the straight 
 line P F I E ; this will be the line of connection ; and it will touch 
 the base-circle at P. 
 
 II. To find the normal pitch, the addendum, and the real
 
 GEOMETRY OF MACHINERY. 
 
 radius, and to draw the addendum-circle and the flank-circle. At 
 the pitch-point, I, draw the straight line I A, touching the pitch- 
 
 Fig. 85. 
 
 circle, and lay off upon it the length I A equal to the pitch. 
 From A let fall A E perpendicular to I R Then I E will be 
 what may be called the NORMAL PITCH, being the distance, as mea- 
 sured along the line of connection, from the front of one tooth to 
 the front of the next tooth. 
 
 The normal pitch is also the pitch on the base-circle; that is, the 
 distance, as measured on the base-circle, between the front of one 
 tooth and the front of the next. 
 
 The ratio of the normal pitch of involute teeth to the circular 
 pitch is equal to the ratio of the radius of the base-circle to that of 
 the pitch-circle ; that is to say, 
 
 IE 
 I A 
 
 C P . , , .. .. / 63 
 
 -7T-=- = cosine of obliquity I = -r-_ 
 C I \ oo 
 
 in the present example).
 
 INVOLUTE TEETH RULES FOR DRAWING. 123 
 
 In order that two pairs of teeth at least may always be in action, 
 the arc of contact is to consist of two halves, each equal to the pitch 
 (see Article 128, page 119). Lay off on the line of connection, 
 E P, the distance I F = I E. Then E F will be the path of con- 
 tact (Article 127, page 119), consisting of two halves, each equal to 
 the normal pitch. 
 
 Draw the straight line C E; this will be the real radius, and 
 the circle E G GT, drawn with that radius, will be the addendum- 
 circle, which all the crests of the teeth are to touch. Then, with 
 the radius C F, draw the circle P H (marked with dots in the 
 figure) ; this may be called the PLANK-CIRCLE, for it marks the inner 
 ends of the flanks of all the teeth. 
 
 The addendum is C E - C I. 
 
 III. To draw the ROOT CIRCLE; that is, the circle which the 
 bottoms of all the hollows between the teeth (or CLEARING CURVES, as 
 they are called) are to touch. First find, by drawing or by calcula- 
 tion, the greatest addendum of any wheel with which the given wheel 
 may have to gear; that is, the addendum of the smallest practicable 
 pinion of the same pitch and obliquity ; that is, the addendum of a 
 pinion in which the pitch subtends at the centre an angle approxi- 
 mately equal to the obliquity. With the obliquity already stated, 
 such a pinion has 25 teeth. To find the addendum of such a 
 pinion by drawing: Through F, parallel to P C, draw F L, as 
 cutting I C in L. Join L E ; then L E L I will be the required 
 greatest addendum. To find the greatest addendum by calculation, 
 let <p denote the obliquity, and p the pitch ; then 
 
 L E - L I = p cotan <p j J (3 sin 2 ? + 1) -1 1 . 
 
 With the angle of obliquity already stated, this gives 
 L E - L I = 0-343 p, very nearly j 
 
 and this is the origin of the value 0-35 p, which is very commonly 
 used for the addendum of teeth. 
 
 To the greatest addendum, thus found, add a suitable allowance 
 for clearance (Article 125, page 116), and lay off the sum I K 
 inwards from the pitch-circle along the radius. Then C K will 
 be the radius of the required root-circle. 
 
 IY. To draw the traces of tJte teeth. Mark the pitch-points of 
 the fronts of the teeth (I, I', &c.), according to the principles of 
 Article 121, page 113, and those of their backs, by laying off a 
 suitable thickness on the pitch-circle (see Article 125, page 116). 
 Obtain a "templet," or thin flat disc of wood or metal, having its 
 edge accurately shaped to the figure of the base-circle. Such a 
 templet is represented in plan by C D D, fig. 86, and in elevation
 
 124 GEOMETRY OF MACHINERY. 
 
 by D' D'. A piece of watch-spring, marked P M in plan, and P M' 
 in elevation, is to have its edges tiled so as to leave a pair of sharp 
 
 , p' V ?>' 
 
 Fig. 86. 
 
 projecting tracing-points, marked T", t', in Elevation, and T in plan. 
 One end of that spring, P, P', is to have a round hole drilled in it, 
 and to be fixed to the middle of the edge of the templet by means 
 of a screw, about which the spring is to be free to turn ; and the 
 other end, M, M', is to be fitted with a knob to hold it by. Place 
 the templet on the drawing (or pattern, as the case may be), so 
 that C shall coincide with the centre of the wheel, and D D with 
 the base-circle ; and also so that the lower of the two tracing-points, 
 when the spring is moved to and fro, shall pass through the pitch- 
 point of a tooth; then that tracing- point will draw the trace of the 
 front of the tooth; and by turning the templet about C, and repeat- 
 ing the process, the traces of the fronts of any required number of 
 teeth may be drawn. 
 
 To draw the traces of the backs of the teeth, the position of the 
 spring relatively to the templet is to be reversed, by turning it 
 about the screw at P, so as to use the tracing-point that was 
 previously uppermost 
 
 The distance, P T, from the screw to the tracing-points should 
 not be less than twice t/ie normal pitch. 
 
 V. The Clearing Curves are the traces of the hollows which lie 
 inside the flank-circle, F H, fig. 85. Their side parts ought to be 
 tangents to the inner ends of the flanks of the teeth (at F and H, 
 for example), and their bottom parts ought to coincide with the 
 root-circle through K. Those different parts may be joined 
 to each other by means of small circular arcs. In connection 
 with the figures of the side parts of those clearing curves, it may 
 be observed, that F L is a tangent to the inner end of the flank 
 I' F, and therefore to the clearing curve at that point; and that 
 tangents to the inner ends of other flanks may be drawn by re-
 
 INVOLUTE TEETH THEIU PECULIAR PROPERTIES. 125 
 
 peating the process by which F L is drawn, or by the following 
 process : About C, with the radius C N = P F, draw a circle* 
 F L N will be a straight tangent to that circle; and so also will 
 all the tangents to the flanks at their inner ends. Therefore, front 
 the inner ends of all the flanks, both front and back, draw straight 
 lines touching the circle C 1ST, and so placed that the straight lines 
 from the front and back flanks of the same tooth shall not cross 
 each other; these lines will show the proper positions for the side 
 parts of the clearing curves. When the flank-circle coincides with 
 the base-circle (as in the smallest pinion of a given pitch), the side 
 parts of the clearing curves coincide with the radii drawn from 
 the centre C to the inner ends of the flanks. 
 
 132. involute Teeth for Rack*. The following is the process of 
 designing the teeth of a straight rack which is to gear with an 
 involute-toothed wheel of a given pitch and a given obliquity : la 
 fig. 87, let A B be the pitch-line of the rack, and let A I = 1 1' be 
 the pitch. Lay off A I E 
 = the given angle of ob- 
 liquity, and from A let - r 7li--^ \ i\r 
 
 fall A E perpendicular to / \ ___H/ i -V- -- L 
 
 I E; then I E will be 1 -- -* '-^-- tllL.. 
 
 t\ie normal pitch ; further, j-j,, 87 
 
 if the path of contact is 
 
 to consist of two halves, each equal to half the normal pitch, 
 I E will be one of those halves; then in E I produced make 
 I F = I E, and I F will be the other half of the path of con- 
 tact. Through E, parallel to A B, draw EGG'; this will be 
 the addendum, line; through F, parallel to B A, draw F H; 
 this will be the flank-line, marking the inner ends of the acting 
 surfaces of the teeth. Perpendicular to A B draw I K, equal to 
 the greatest addendum in the set of wheels of the given pitch and 
 obliquity with an allowance for clearance added, as in Rule III. 
 of Article 131, page 123; through K, parallel to A B, draw a 
 straight line; this will be the root-line, with which the bottoms of 
 all the hollows between the teeth are to coincide. 
 
 The traces of the fronts of the teeth are straight lines perpen- 
 dicular to E F, and the fronts themselves are planes perpendicular 
 to E F. The backs of the teeth are planes inclined at the same 
 angle to A B in the contrary direction. 
 
 133. Peculiar Properties of Involute Teeth. Involute teeth have 
 
 some peculiar properties not possessed by teeth of other figures. 
 
 I. Sets of involute teeth have a definite and constant normal 
 pitch; being, as already explained, the distance between the fronts 
 of successive teeth, measured on the path of contact, or on the 
 circumference of the base-circle; and all w/ieels and racks vntl* 
 involute teeth of the same normal pitch gear correctly with each other-
 
 126 
 
 GEOMETKY OF MACHINERY. 
 
 II. The length of the line of centres, or perpendicular distance 
 between, the axes, of a pair of wheels with involute teeth of the 
 same normal pitch, or the perpendicular distance from the axis of 
 a wheel with involute teeth to the addendum-line of a rack with 
 which it gears, may be altered; and so long as the wheels, or wheel 
 and rack, are sufficiently near together to make the path of contact 
 longer than the normal pitch, and sufficiently far asunder for the 
 crests of each set of teeth to clear the hollows between the teeth 
 of the other set, the wheels, or the wheel and rack, will continue 
 to work correctly together, and to preserve their velocity-ratio; 
 although, in the case of a pair of wheels, the pitch-lines, the pitch 
 as measured on the pitch-lines, and the obliquity, will all be altered 
 when the length of the line of centres is altered. In other words, 
 the velocity-ratio of a pair of wheels with involute teeth of the 
 same normal pitch is the reciprocal of the ratio of the radii of their 
 
 base-circles, and depends on this ratio alone; and the velocity-ratio 
 of a wheel and rack with involute teeth of the same normal pitch
 
 INVOLUTE TEETH THEIR PECULIAR PROPERTIES. 
 
 127 
 
 depends solely on the radius of the base-circle of the wheel and 
 on the angle of obliquity of the line of connection. 
 
 Another way of stating this property of involute teeth is, that 
 the pitch-lines of wheels and racks with such teeth are arbitrary to 
 an extent limited only by the necessity of having a path of contact 
 of a certain length. 
 
 One practical result of this is (as Mr. "Willis first pointed out), 
 that the back-lash of involute teeth is variable at will, being capable 
 of being increased or diminished by moving the wheels, or the wheel 
 and rack, farther from or nearer to each other, and may thus be 
 adjusted so as to be no greater than is absolutely necessary in order 
 to prevent jamming of the teeth a property not possessed by teeth 
 of any other figure. 
 
 III. Given (in fig. 88), the centres, C, C, the base-circles, D D, 
 D' D', and the addendum-circles, A A, A' A', of a pair of spur- 
 wheels with involute teeth of a, given normal pitch, to find the line of 
 connection, the pitch-point, the pitch-circles, the pitch on the pitch- 
 circles, and the path of contact. 
 
 Draw a common tangent, P P', to the two base-circles in such a 
 position as to run from the driver to the follower in the direction 
 of motion. That common tangent will be the line of connection : 
 the point I, where it cuts the line of centres, will be the pitch-point : 
 two circles, B B and B' B', described about C and C' respectively, 
 and touching each other in I, will be the pitch-circles: the pitch 
 on the pitch-circles will be greater than the normal pitch in the 
 
 C I C' I 
 ratio 7^-^ = ,; and the part E E' of the line of connection which 
 
 Ox \j S. 
 
 lies between the two addendum-circles will be the path of contact. 
 IY. Given (in fig. 89), the centre, C, the base-circle, D T>, and the 
 addendum-circle, A. A, of a spur-wheel with involute teeth of a given 
 
 ri 
 
 Fig. 89. 
 
 normal pitch; also the pitch-line, B' B', and the addendum-line) A' A', 
 of a rack which is to have involute teeth of the same, normal pitch; to
 
 128 GEOMETRY OP MACHINERY. 
 
 find the pitch-point, the pitch-circle of the wheel, the line of connection, 
 the pitch, as measured on the pitch-lines, the path of contact, and the 
 position of the fronts of the teeth of the rack. 
 
 From C let fall C I perpendicular to B' B'; then I will be the 
 pitch-point ; and a circle, B B, of the radius C I, will be the pitch- 
 circle of the wheel. From I draw I P, touching the base-circle 
 D D; I P will be the line of connection. The pitch, as measured 
 on the pitch-lines, will be greater than the normal pitch, in the 
 
 C I 
 
 ratio ppg of the radius of the pitch-circle to that of the base-circle. 
 
 \J S. 
 
 The path of contact will be the part E E' of the line of connection, 
 which is contained between the addendum-line of the rack, A' A', 
 and the addendum-circle of the wheel, A A. The fronts of the 
 teeth of the rack are to be planes perpendicular to I P, or, in other 
 words, parallel to P C. 
 
 V. By the application of the preceding principles, two or more 
 wheels of different numbers of teeth, turning about one-axis, can. 
 be made to gear correctly with one wheel or with one rack; or two 
 or more parallel racks, with different obliquities of action, may be 
 made to gear correctly with one wheel, the normal pitches in each 
 case being the same ; and thus differential movements of various sorts 
 may be obtained. This is not possible with teeth of any other form. 
 
 The obliquity of the action of involute teeth is by many con- 
 sidered an objection to their use; and that is the reason why, 
 notwithstanding their simplicity and their other advantages, they 
 are not so often used as other forms. In anticipation of the subject 
 of the dynamics of machinery it may be stated, that the principal 
 effect of the obliquity of the action of involute teeth is to increase 
 the pressure exerted between the acting surfaces of the teeth, and 
 also the pressure exerted between the axles of the wheels and their 
 bearings, nearly in the ratio in which the radius of the pitch-circle 
 of each wheel is greater than the radius of the base-circle, and that 
 a corresponding increase of friction is produced by that increase of 
 pressure. In the example of Article 131, that ratio is 65 : 63. 
 
 134. Teeth for a Giren Path of Contact. In the three pre- 
 ceding Articles the forms of the teeth are found by assuming a 
 figure for the path of contact viz., the straight line. Any other 
 convenient figure may be assumed for the path of contact, and the 
 corresponding forms of the teeth found, by determining what 
 curves a point moving along the assumed path of contact will trace 
 on two discs, rotating round the centres of the wheels with angular 
 velocities, which bear that relation to the component velocity of the 
 tracing-point along the line of connection which is given by the 
 principles of Article 127, page 118. This method of finding the 
 forms of the teeth of wheels is the subject of an interesting treatise 
 by Mr. Edward Sang.
 
 TEETH TRACED BY ROLLING CURVES. 129 
 
 All wheels having teeth of the same pitch, traced from the same 
 path of contact, work correctly together, and are said to belong to 
 the same set. 
 
 135. Teeth Traced by Rolling Carres. (A M., 452.) From the 
 
 principles of Articles 122 and 123, pages 114, 115, it appears that at 
 every instant the position of the point of contact, T, of the acting 
 sm-faces of a pair of teeth (fig. 82, page 115), and the corresponding 
 position of the pitch-point I in the pitch-lines of the wheels to 
 which those teeth belong, are so related, that the line, I T, which 
 joins thorn, is normal to the surface of each of the teeth at the point 
 T. Now this is the relation which exists between the tracing- 
 point T, and the instantaneous aods or line of contact I, in a rolling 
 curve of such a figure, that, being rolled upon the pitch-line, its 
 tracing-point T . traces the outline of a tooth. (As to rolling 
 curves and rolled curves, see Ai-ticles 72, 74, 75, 77, 78, 79, pages 
 51 to 62.) 
 
 In order that a pair of teeth may work correctly together, it is 
 necessary" and sufficient that the instantaneous normals from the 
 pitch-point to the acting surfaces of the two teeth should coincide 
 at each instant; and this condition is fulfilled if the outlines of the 
 two teeth be traced by the motion of the same tracing-pomt, in rolling 
 the same rolling curve on the same side of the pitch-lines of the respec^ 
 tive wheels. 
 
 The flank of a tooth is traced while the rolling curve rolls inside 
 of the pitch-line; the face, while it rolls outside. 
 
 To illustrate this more fully, the following explanation is quoted 
 from the Article "MecJtanics (Applied)," in the Encyclopaedia JSrit- 
 annica(see fig. 90): "If any 
 curve, R, be rolled on the in- 
 side of the pitch-line, B B, of 
 a wheel, the instantaneous 
 axis of the rolling curve at 
 any instant will be at the 
 point I, where it touches the 
 pitch-line for the moment; 
 and consequently the line 
 A T, traced by a tracing- Fig. 90. 
 
 point T, fixed to the rolling 
 
 curve, will be everywhere perpendicular to the straight line T I ; so 
 that the traced curve A T will be suitable for the flank of a tooth, 
 in which T is the point of contact corresponding to the position I 
 of the pitch-point. If the same rolling curve R, with the same 
 tracing-point T, be rolled on the outside of any other pitch-line, it 
 will trace the face of a tooth suitable to work with the flank 
 AT. 
 
 "In like manner, if either the same or any other rolling curve 
 
 K
 
 130 GEOMETRY OF MACHINERY. 
 
 R' be rolled the opposite way, on the outside of the pitch-line B B, 
 so that the tracing-point T' shall start from A, it will trace the 
 face A T" of a tooth suitable to work with a flank traced by rolling 
 the same curve R' with the same tracing-point T' inside any other 
 pitch-line. 
 
 " The figure of the path of contact is that traced on a fixed plane 
 by the tracing-point, when the rolling curve is rotated in such a 
 manner as always to touch a fixed straight line E I E (or E' I' E', 
 as the case may be) at a fixed point I (or I'). 
 
 "If the same rolling curve and tracing-point be used to trace 
 both the faces and the flanks of the teeth of a number of wheels of 
 different sizes, but of the same pitch, all those wheels will work 
 correctly together, and will form a set. The teeth of a rack of the 
 same set are traced by rolling the rolling curve on both sides of a 
 straight line. 
 
 "The teeth of wheels of any figure, as well as of circular wheels, 
 may be traced by rolling curves on their nitch-lines ; and all teeth 
 of the same pitch, traced by the same rolling curve with the same 
 tracing-point, will work together correctly if the pitch-surfaces are 
 in rolling contact." 
 
 Involute teeth themselves might be traced by rolling a logarith- 
 mic spiral on the pitch-circle ; but it is unnecessary to explain this 
 in detail, as the ordinary method of tracing them is much more 
 simple. 
 
 136. Epicycloidal Teeth in General. For tracing the figures of 
 
 teeth, the most convenient rolling curve is the circle. The path of 
 contact which a point in its circumference traces is identical with 
 the circle itself; the flanks of the teeth for circular wheels are inter- 
 nal epicycloids, and their faces external epicycloids, and both flanks 
 and faces are cycloids for a straight rack. (See Article 74, page 
 53, and Article 77, page 56.) 
 
 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 
 tlie same set. 
 
 For a pitch-circle of twice the radius of the rolling or describing 
 circle (as it is called), the internal epicycloid is a straight line, 
 being a diameter of the pitch-circle ; so that the flanks of the teeth 
 for such a pitch-circle are planes radiating from the axis. For a 
 smaller pitch-circle, the flanks would be convex, and incurved or 
 under-cut, which would be inconvenient; therefore the smallest 
 wheel of a set should have its pitch-circle of twice the radius of the 
 describing circle, so that the flanks may be either straight or 
 concave. 
 
 In fig. 91, let B B be the pitch-circle of a wheel, C C the line of 
 centres, I the pitch-point, II the internal describing circle, and E.'
 
 EPICYCLOIDAi TEETH. 
 
 131 
 
 Fig. 91. 
 
 the external describing circle, so placed as to touch the pitch-circle 
 and each other at I; let E E be a straight tangent to the pitch- 
 circle at the pitch-point ; and let T I T be the path of contact, 
 consisting of the path of approach, T I, 
 and the path of recess, I T'. Each 
 of those arcs should be equal to the 
 pitch when practicable, in order 
 that there may be always at least two 
 pairs of teeth in action; but this is 
 not always possible; and the length 3- 
 of each of them in many cases is only 
 from 07 to 0-9 of the pitch, being 
 regulated by the customary practice 
 of making the addendum from 0'3 to 
 0-35 of the pitch. 
 
 The real radius of the wheel is 
 the distance from its centre to the 
 point T', at the outer end of the face 
 of a tooth; the dotted circle travers- 
 ing T' is the addendum-circle, and the perpendicular distance 
 from T to the pitch-circle B B is the addendum. 
 
 The Jlank-circle is a circle described about the centre of the 
 wheel, and traversing the point T; and the clearing curves (as in 
 the case of involute teeth, Article 131, Rule V., page 124) must 
 have a depth sufficient to clear the greatest addendum given to the 
 teeth, of any one of the set of wheels that are capable of gearing 
 with the wheel under consideration. 
 
 In passing the line of centres, the line of connection coincides with 
 the tangent E E, and the obliquity is nothing. The greatest angle of 
 obliquity of action is, during the approach, E I T, and during the 
 recess E I T'; and the mean angles of obliquity during the approach 
 and recess are the halves of those greatest angles respectively. From 
 the results of practical experience, Mr. Willis deduces the rule that 
 the mean obliquity should not exceed 15 3 , or one-twenty-fourth of 
 a revolution ; therefore the maximum obliquity should not exceed 
 30, or one-twelfth of a revolution; therefore the arcs I T and I T 
 should neither of them in any case exceed one-sixth of the circum- 
 ference of the describing circles to which they respectively belong; 
 from which it follows, that if either of those arcs is to be equal to 
 the pitch, the circumference of the describing circle ought not to be 
 less than six times the pitch; therefore the smallest pinion of a set 
 should have twelve teeth. 
 
 137. Tracing Epicycloidal Troth by Templets. The face of an 
 
 picycloidal tooth may be traced by rolling a templet of the form 
 of the describing circle upon a convex templet of the form of the 
 pitch-circle; and the flank, by rolling a templet of the form of the
 
 132 
 
 GEOMETRY OF MACHINERY. 
 
 describing circle upon a concave templet of tlie form of the 
 pitch-circle. 
 
 When the fixed templet is either convex (as when the face of 
 the tooth of a wheel is to be traced) or straight (as when either the 
 face or the flank of the tooth of a rack is to be traced), the rolling 
 templet may be prevented from slipping on the fixed templet by 
 connecting them together by means of a slender piece of watch- 
 spring, as follows : In fig. 92, C B B represents the fixed templet 
 
 of the form of the pitch-circle, and R the rolling templet of the 
 form of the describing circle. Q I P is a slender piece of watch- 
 spring, fastened by a screw at P to the edge of the fixed templet, 
 and by a screw at Q to the edge of the rolling templet. The spring 
 may have a sharp tracing-point formed at T on one of its edges, as 
 already described in Article 131, Rule IV., and shown in fig. 86, 
 page 124. A T, in fig. 92, represents part of the epicycloid traced 
 by the point T, and I the point of contact of the pitch-circle and 
 describing circle. The radius of each of the templets ought to be
 
 STRAIGHT-PLANKED TEETH. 
 
 133 
 
 made less than the radius of the circle which it represents, by 
 half the thickness of the spring P Q. 
 
 When the fixed templet is concave (for tracing the flanks of 
 teeth) this method of preventing the rolling templet from slipping 
 is not available. 
 
 138. Straight-Flanked Epicycloidnl Teeth. In the oldest form of 
 
 epicycloidal teeth, the traces of the flanks are straight lines radiat- 
 ing from the centre of the wheel, being the lines which would be 
 traced by a describing circle, of half the radius of the pitch-circle, 
 rolling inside the pitch-circle. Hence, in order that a pair of 
 wheels with teeth described according to this principle may gear 
 correctly together, the faces of the teeth of each wheel must be 
 traced by rolling upon the outside of its pitch-circle a describing 
 circle of half the radius of the other pitch-circle. 
 
 For example, in fig. 93, let C and C' be the centres of a pair of 
 
 A R 
 
 K 
 
 spur-wheels, and B B and B' B' their pitch-circles, touching each 
 other in the pitch-point I. Lay off the pitch-points of the fronts 
 and backs of the teeth on each of the pitch-circles, and draw 
 straight lines from the centres of the wheels to the points of division 
 of their respective pitch- circles; these lines will be the traces of the 
 flanks of the teeth. Bisect C I in B, and C' I in B', and about 
 B, and B,' respectively describe circles traversing I; these will be 
 the two describing circles for the faces of the teeth. Lay off, on
 
 134 GEOMETRY OF MACHINERY. 
 
 those two circles, sufficient lengths, I E and I E', for the two 
 divisions of the path of contact; that is to say, each of these lengths 
 must be greater than half the pitch, acd should be made as nearly 
 equal to the pitch as practicable. Then a circle, A A, described 
 about C through E', will be the addendum-circle of the first 
 wheel, and a circle, A' A', described about C', through E, will 
 be the addendum-circle of the second wheel. The two root- 
 circles, K K and K' K', are to be drawn so as to leave a 
 sufficient clearance between each of them and the opposite 
 addendum-circle. 
 
 To trace the front and back faces of the teeth of the first wheel, 
 roll the describing circle R' on the pitch-circle B B; to trace the 
 front and back faces of the teeth of the second wheel, roll the 
 describing circle E, on the pitch-circle B' B'. This may be done 
 with the aid of templets connected together by means of a spring, 
 as described in Article 137, page 131. 
 
 The traces of the flanks of the teeth of -a, rack, according to this 
 system, are straight lines perpendicular to the pitch-line, and those 
 of the faces are cycloids. The traces of the faces of a wheel that is 
 to gear with a rack are involutes of the pitch-circle. 
 
 Epicycloidal teeth described by this method are very smooth and 
 accurate in their action ; but they labour under the disadvantage 
 that the faces of the teeth of any given wheel are not suited to 
 work accurately with the flanks of the teeth of any wheel whose 
 radius differs from double the radius of the describing circle with 
 which they were traced. 
 
 139. Epicytloidnl Teeth Traced by an Uniform Describing 
 
 Circle. The property of working accurately with all teeth of the 
 same pitch, whatsoever the radius of the pitch-circle, is given to 
 epicycloidal teeth by tracing both the faces and the flanks of all 
 teeth of the same pitch, by rolling the same describing circle upon 
 the outside and the inside of the pitch-circle a system first intro- 
 duced by Mr. Willis. This method is illustrated by fig. 91, page 
 131, already described in Article 136. In order that the mean 
 obliquity of action may not in any case exceed 15, nor the 
 maximum obliquity 30, the circumference of the describing circle 
 employed is six times the pitch; so that its radius is six^imes the 
 radial pitch (see Article 119, page 111). According to tnis system, 
 the traces of both the flanks and the faces of the teeth of a rack 
 are cycloids. 
 
 140. Approximate Drawing of Epicycloidal Teeth. Various 
 
 approximate methods of drawing epicycloids have already been 
 described in Article 79, pages 59 to 62. The following are the 
 additional explanations required in order to show the application 
 of those methods to epicycloidal teeth : 
 
 I. By two pairs of Circular Arcs. In fig. 94, let I A be part
 
 APPROXIMATE EPICYCLOIBAL TEETH. 
 
 135 
 
 of the pitch-circle of a wheel. Draw the describing' circle touching 
 the pitch-circle at any convenient point, I, and outside or inside, 
 
 Fig. 94. 
 
 according as the face or the flank of a tooth is to be traced. 
 (Letters without an accent refer to the face; letters with an accent, 
 to the flank.) 
 
 Draw the straight tangent I P, equal in length to one of the two 
 divisions of the arc of contact, and in it take I D = I P. Then, 
 with the radius D P f I P, draw the circular arc A B; A and 
 B will be the two ends of the required epicycloidal arc. Join B I ; 
 and from A draw the straight tangent A C, cutting B I in G. 
 Then A C and B C will be the normals at the two ends of the 
 epicycloidal arc. Then proceed, according to Rule IV. of Article 
 79, pages 61 and 62, fig. 48, to draw two circular arcs approxi- 
 mating to the required curve; and perform the same operation 
 both for the face and for the flank of the tooth. 
 
 According to this method, the traces of the face and flank of a 
 tooth consist each of a pair of circular arcs, and the two arcs which 
 join each other at the pitch-point, A, of a tooth have a common 
 tangent there; because their centres are in the straight line 
 CTAC. 
 
 II. By one pair of Circular Arcs Mr. Willis's Method. Mr. 
 Willis first showed how to approximate to the figures of epicy- 
 cloidal teeth by means of two circular arcs one concave, for the 
 flank, the other convex, for the face; and each having for its radius 
 the mean radius of curvature of the epicycloidal arc. Mr. Willis's 
 rules may be deduced from the formula for finding the centre of 
 curvature of an epicycloid, which is given in Article 78, equation 2, 
 page 59; that formula being applied to the point in the epicy- 
 cloid whose normal meets the pitch-circle at a distance from the 
 pitch-point of the tooth to be traced equal to one-half of the 
 pitch, and the obliquity of that normal to the pitch-circle being 
 15.
 
 136 
 
 GEOMETRY OP MACHINERY. 
 
 In fig. 95, let B C be part of the pitch-circle, and A the 
 
 pitch-point of a tooth whose 
 front is to be traced. Lay 
 off, in opposite directions 
 from the point A, the arcs 
 A B and A C, each equal to 
 one-half of the pitch . Draw 
 the radii of the pitch-circle, 
 B D and C E, and through 
 the points B and C draw 
 the straight lines B F and C G, making angles of 75 with the 
 radii respectively; these lines are normals to the face and to the 
 flank of the tooth respectively. Let n denote the number of teeth 
 in the wheel. Lay off along the two normals the distances B F 
 and C G, as calculated by the following formulae : 
 ^ -.., pitch n _, p pitch t n 
 
 = 2 ' nTf ~12 ' = ^ ' n - 12 ' 
 
 Fig. 95. 
 
 then F will be the centre of curvature for the face, and G the 
 centre of curvature for the flank. 
 
 About F, with the radius F A, draw the circular arc A H ; this 
 will be the trace of the face of the tooth. About G, with the 
 radius G A, draw the circular arc A K ; this will be the trace of 
 the flank of the tooth. 
 
 To facilitate the application of this rule, Mr. Willis has published 
 tables of the values of B F and C G, and invented an instrument 
 called the " Odontograph." That instrument is an oblong piece 
 of card-board, F G K H, fig. 96, measuring about 13 inches by 7^ 
 
 inches. The oblique 
 edge, L H, makes an 
 angle of 75 with the 
 edge G F; so that when 
 the edge L H is laid 
 along a radius, O I, of 
 a pitch-circle, B B, 
 the edge GIF shows 
 the positions of normals 
 to acting surfaces of 
 teeth whose pitch- 
 points are at a distance 
 from I equal to half 
 the pitch. Along the 
 edge GIF two scales of equal parts are laid off in opposite 
 directions from the point I, where the straight line coinciding 
 with H L meets G F; the scale I F serving to mark the centres 
 for faces, and the scale I G the centres for flanks, at distances
 
 ODONTOGRAPH TEETH GEARING WITH STAVES. 137 
 
 from I computed by the formulae. Values of those distances for 
 different pitches and numbers of teeth, and other useful dimen- 
 sions, are given in tables which are printed on the sides of the 
 card-board. 
 
 141. Teeth Clearing with Round Slaves Trundles and Pin-wheels. 
 
 When two wheels gear together, and one of them has cylindrical 
 pins (called staves) for teeth, that one is called, if it is the larger of 
 the two, a pin-wheel, and if the smaller, a trundle. The traces of 
 the teeth of the other wheel are drawn in the following manner : 
 In fig. 97, let B 2 be the pitch-circle and C 2 the centre of the 
 trundle or pin- wheel, and let Bj B 1 be the pitch-circle of the other 
 wheel. Divide the pitch-circle, B x B p into arcs equal to the pitch, 
 and through the points of division trace a set of external epicy- 
 cloids by rolling the pitch-circle B 2 on the pitch-circle B v with the 
 centre of a stave for a tracing-point, as shown by the dotted lines; 
 then draw curves parallel to and within the epicycloids, at a dis- 
 tance from them equal to the radius of a stave. These will be the 
 fronts and backs of the required teeth. The clearing curves are 
 circular arcs of a radius equal to that of the staves. 
 
 Fig. 97. Fig. 98. 
 
 When the teeth drive the staves, the whole path of contact 
 consists of recess, and there is no approach ; for the teeth begin to 
 act on the staves at the instant of passing the line of centres. 
 When the staves drive the teeth, the whole path of contact consists 
 of. approach, and there is no recess; for the staves cease to act on 
 the teeth at the instant of passing the line of centres. The latter 
 mode of action is avoided where economy of power is studied, 
 because it tends to produce increased friction, for reasons to be 
 stated under the head of the Dynamics of Machines. 
 
 To drive a trundle in inside gearing, the outlines of the teeth of 
 the wheel should be curves parallel to internal epicycloids. A 
 peculiar case of this is represented in fig. 98, 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 equidistant staves; 
 and the internal epicycloids described by their centres, while the
 
 138 
 
 GEOMETRY OF MACHINERY. 
 
 pitch-circle of the trundle is rolling within that of the wheel, are- 
 three straight lines, diameters of the wheel, making angles of 60* 
 with each other. Hence the surfaces of the teeth of the wheel 
 form three straight grooves intersecting each other at the centre, 
 each being of a width equal to the diameter of a stave of the 
 trundle, with a sufficient addition for back-lash. 
 
 The following is the construction given by Mr. Willis for finding 
 in pin-wheels and trundles what is the greatest radius of stave 
 consistent with having an arc of contact not less than the pitch (see 
 fig. 99): 
 
 Let C be the centre of the wheel with teeth, and C' that of the 
 
 wheel with staves. On their 
 two respective pitch-circles lay 
 off the arcs I D and I B, each 
 equal to the pitch. Draw the- 
 straight line I B; draw also the 
 straight x line C E, bisecting the 
 angle I C D, and cutting I B in 
 E; then B E will be the greatest 
 radius that can be given to the 
 staves consistently with having 
 an arc of contact not less than 
 the pitch. 
 
 The proof is as follows: Be- 
 cause the fronts and backs of the 
 teeth are similar, the crest of 
 the tooth that acts on a stave at 
 B must be in the straight line 
 C E, that bisects the angle I C D. 
 When the centime of a stave is at 
 B, the point of contact of the 
 stave and tooth must be in the 
 line of connection I B. When 
 the staves have the greatest 
 radius consistent with the con- 
 tinuance of action, while the 
 centre of a stave moves from I to 
 B, the point of contact and the crest of the tooth coincide, and 
 are therefore at the point E, where I B and C E intersect. 
 
 Should C E pass beyond B, the proposed pair of wheels will not 
 work, and the design must be altered; and such is also the case 
 when C E either traverses the point B or cuts I B so near to B as 
 to give a radius too small for strength. 
 
 In practice, the radius B E ought to be made a little less than 
 that given by the Rule, in order that there may be no risk of 
 imperfect working through the effects of tear and wear. 
 
 Kg. 99.
 
 TRUNDLE PIN-RACK INTERMITTENT GEARING. 
 
 139 
 
 The smallest number of staves commonly met with in a trundle 
 is five. 
 
 A straight rack may have staves instead of teeth; it is then called 
 a pin-rack; and it is evident that the fronts and backs of the teeth, 
 of a wheel to gear with it should be parallel to involutes of the pitch- 
 circle of that wheel. On the other hand, a toothed straight rack 
 may gear with a trundle, and then the teeth of the rack are to be 
 traced by first rolling the pitch-circle of the trundle on the pitch- 
 line of the rack, so as to draw cycloids, and then drawing curves 
 parallel to and inside those cycloids, at a distance equal to the 
 radius of the staves. / 
 
 142. Intermittent Gear- 
 ing. The action of a pair 
 of wheels is said to be 
 intermittent when there 
 are certain parts of the 
 revolution of the driver 
 during which the follower 
 stands still. This is 
 effected by having a dead 
 arc, or portion without 
 teeth, such as A E, fig. 
 100, in the circumference 
 of the driver, to which 
 there corresponds a suit- 
 able gap in the series of 
 teeth of the follower, as 
 between C and D ; and in 
 most cases there are also 
 required a guide-plate, G- 
 H, fixed to one side of the 
 follower, which, when the 
 connection of the wheels 
 is renewed, is acted upon 
 by a pin, F, in the driver. 
 Supposing the radii and 
 
 the pitch of a pair of 
 
 wheels to be given, and 
 
 also the arc of repose by 
 
 which term is meant the 
 
 length upon the pitch- 
 circle of the driver of 
 
 that part which is to 
 
 pass during the pause in the movement of the follower the method 
 
 of designing those wheels so as to work with smoothness and 
 
 precision is as follows ; 
 
 Fig. 100.
 
 140 GKOSIETRY OP MACHINERY. 
 
 Draw the pitch-circles, and divide them as usual; draw also 
 the addendum-circles and root-circles according to the ordinary 
 rules. Mark the point, K, where the addendum-circles cut each 
 other at the receding side; this will be the point at which the 
 action of the teeth will terminate, at the instant when the pause 
 begins. Through K draw a curve suited for the front of a tooth of 
 the follower, and let C be the pitch-point of that tooth. Then, 
 starting from C, lay off the pitch-points of the fronts and backs of 
 the teeth of the follower, and draw those fronts and backs. Then, 
 looking towards the approaching side, mark the furthest tooth from 
 the line of centres, which is cut by the addendum-circle of the 
 driver; let D be the pitch-point of the face of that tooth. The 
 crest of the tooth D is to be cut away so as exactly to fit the 
 addendum-circle of the driver, and the teeth between it and the 
 tooth C are to be omitted, leaving a smooth part of the root-circle 
 between the front of C and the back of D ; this is the required 
 gap. 
 
 Measure the arc C D on the pitch-circle of the follower between 
 the fronts of the teeth C and D, and to its length add the length 
 of the intended arc of repose: from the sum subtract the space, B E, 
 that is to be left between each pair of teeth on the pitch-circle of 
 the driver; the remainder will be the dead arc, A. E, which is to be 
 laid off on the pitch-circle of the driver. The two ends of that arc 
 are to be bounded by curves like the front and back of a tooth of 
 the driver respectively : a front at A, a back at E ; and the inter- 
 vening part of the rim of the driver is to have a smooth edge 
 coinciding with its addendum-circle. 
 
 For the purpose of renewing the connection between the driver 
 and follower, the cylindrical pin F is to be fixed with its centre 
 in the pitch-circle of the driver, and the guide-plate G H is to be 
 fixed to the corresponding side of the follower. The acting edge 
 of the guide-plate is to be shaped like the front of a tooth for 
 working with the pin F (as in Article 141, page 137); and the 
 distance on the pitch-circle of the follower from the front of that 
 edge to the front of the tooth D is to be equal to the distance on 
 the pitch-circle of the driver from the front of the pin F to the 
 front of the tooth B ; so that when B is driving D, F shall at the 
 same time be driving G H. The end G of the guide-plate in the 
 position of repose should project just far enough inside the pitch- 
 circle of the driver to insure that the pin F shall meet it. 
 
 The action in working is as follows : Just before the pause, the 
 front, A, of the dead-arc drives the front of the tooth, C, in the 
 usual way throughout the ordinary path of contact; and then, as 
 there is a gap following C, the crest of the front A continues to 
 drive C until the crest of C reaches the position K, and clears the 
 addendum-circle of the driver. At that instant the driver loses
 
 TEETH FOR INTERMITTENT AND NON-CIRCULAR WHEELS. 141 
 
 hold of the follower, and at the same instant the top of the tooth D 
 comes in contact with the rim of the dead arc, which it is shaped 
 to fit; and this prevents the follower from moving until the dead 
 arc has passed clear of the tooth D. At this instant the pin F 
 begins to drive the guide-plate G H, and continues to do so until 
 the tooth B has begun to drive the tooth D, and the connection 
 is renewed. 
 
 If the pressure to be exerted is considerable, there may be a pair 
 of pins at F, one at each side of the driver, and a pair of guide- 
 plates at G H, one at each side of the follower. 
 
 The shortest arc through which the follower can be driven in 
 the interval between two pauses is C D; and such is the case when 
 B is the front of a second dead arc, and the tooth D is imme- 
 diately followed by a second gap. In this case it may be necessary 
 to cut away part of the outer side of the pin F, in order to insure 
 its clearing the tip, G, of the guide-plate when the next pause 
 begins. 
 
 It is easy to see how the same principles may be applied to the 
 designing of a wheel and rack with intermittent action. "When the 
 rack is the follower, a pair of similar and parallel racks, rigidly framed 
 together, may be made to gear with opposite edges of a spur-wheel, 
 having a toothed arc and a dead arc so arranged as to drive the two 
 racks alternately in opposite directions, and thus produce a 
 reciprocating motion of the piece of which they are parts. This 
 combination belongs to Class C of Mr. Willis's aiTangement. As. 
 to the form which it takes when one tooth only acts at a time, see 
 Article 164, further on. (See also Addendum, page 286.) 
 
 143. The Teeth of ivon-Circninr wheels may be traced by 
 rolling circles or other curves on the pitch-lines ; and when those 
 teeth are small, compared with the wheels to which they belong, 
 each tooth is nearly similar to the tooth of a circular wheel whose 
 pitch-circle has a radius equal to the radius of curvature of the 
 pitch-line of the actual wheel at the point where the tooth is 
 situated ; the tooth being traced by means of the same describing 
 circle which is xased for the circular wheel. 
 
 It is obvious that the use of an uniform describing circle for 
 teeth of a given pitch (as explained in Article 139, page 134) is 
 the most easily practicable method of tracing teeth for a non- 
 circular wheel. It may be carried out by means of templets, as 
 in Article 137, page 131. 
 
 The operation is necessarily much more laborious than the 
 corresponding operation for a circular wheel; because in a non- 
 circular wheel the teeth have figures varying with the curvature 
 of the pitch-line. 
 
 If the pitch-line of a non-circular wheel is one whose radii 
 of curvature at a series of points can be easily found, a series of
 
 142 GEOMETRY OF MACHINERY. 
 
 figures may be used for the teeth similar to the figures suited for 
 circular wheels of those radii; and in drawing those figures the 
 approximate methods of Article 140, pages 134 to 136, may be 
 employed.* 
 
 * The following relation between the radii of curvature at a pair of corre- 
 sponding points of a pair of pitch-lines that roll together, may be useful to 
 determine one of those radii of curvature when the other is known. Let 
 r and r' be the two segments into which the pitch-point divides the line of 
 centres at the instant when the pair of corresponding points in question are 
 in contact ; let p and p' be the two radii of curvature at these points, and 
 let be the angle which those radii make with the line of centres at the 
 instant before mentioned ; then 
 
 v> 
 
 (i.) 
 
 When the pitch-lines are in inside gearing, the greater of the two segments, 
 r, /, is to be made negative, and each radius of curvature is to be considered 
 as positive for a convex and negative for a concave pitch-line. 
 
 For a pair of equal elliptic pitch-lines, as in Article 108, page 93, the radii 
 of curvature at a pair of corresponding points are equal, and are therefore 
 both given by the following formulae : 
 
 or, 
 
 " = p ' = (r+)cos0 ; 
 
 and the same formulae apply to any pair of equal and similar lobed pitch-lines 
 of the class described in Article 109, page 97. 
 
 For a logarithmic spiral pitch-line (Article 110, page 99) the radius of 
 curvature at any point is given by the formula 
 
 and may be found approximately by construction, as already described in 
 the article referred to. 
 
 If one of the pitch-lines is straight (a case already used as an example hi 
 Article 107, page 92), the reciprocal of the radius of curvature of that line 
 is at every point equal to nothing ; so that equation 1 of this note becomes 
 (for the other pitch-line) 
 
 1 (4.) 
 
 Let c denote the length of the line of centres, and a the shortest distance 
 
 
 of the straight pitch-line from its own axis of motion ; then r' - - ; and 
 sequently equati 
 
 i c cos * e 
 
 r = c r 1 c -- 7. ; consequently equation 4 becomes 
 
 COS v 
 
 _ 
 
 p ~~ a c cos a 2 '" 
 
 ,_ a <* 
 
 P ~COS 2 CCOS 3
 
 TEETH OF NON-CIRCULAR WHEELS OP BEVEL WHEELS. 143 
 
 When a pair of non-circular wheels are connected by means of 
 teeth alone, care must be taken that the obliquity of the action of the 
 teeth does not become too great in certain positions of the wheels. 
 That obliquity is greatest at the instant when the obliquity of the 
 common tangent of the two pitch-lines at their pitch-point to a 
 perpendicular to the line of centres at that point is greatest, such 
 obliquity being in the direction of rotation of the follower; for, as 
 that is also the direction of the obliquity of the line of connection 
 of the teeth to the pitch-lines, those two obliquities are added 
 together at the instant in question; their sum being the total 
 obliquity of the line of connection to a perpendicular to the line of 
 centres. Excessive obliquity of action tends to produce great fric- 
 tion, and involves also the risk of the teeth either getting jammed 
 or losing hold of each other. In practice, the total obliquity of 
 action of the teeth of non-circular wheels is seldom allowed to exceed 
 about 50; or say, about 15 for the obliquity of the line of connec- 
 tion of the teeth to the pitch- lines, and 35 for the greatest obliquity 
 of the pitch-lines to a line perpendicular to the line of centres. 
 
 There is one case, however, in which it is not necessary to con- 
 fine the obliquity within such narrow limits; and that is when the 
 wheels have a pair of equal and similar elliptic pitch-lines centred 
 on two of their foci, and it is practicable to link the revolving foci 
 together, as shown in Article 108, fig. 72, page 96; for the link 
 preserves the connection accurately at the time when the obliquity 
 of the pitch-lines is greatest. In this case, indeed, the teeth may 
 be omitted throughout a pair of arcs at the two sides of each 
 elliptic pitch-line, each such toothless arc having a smooth rim of 
 the form of the pitch-line, and extending both ways from the end 
 of the minor axis to a pair of points perpendicularly opposite the 
 foci, or nearly so. (See page 292.) 
 
 144. Teeth of Bcvd-Wheeis. (A. M., 467.) The teeth of a bevel- 
 wheel have acting surfaces of the conical kind, generated by the 
 motion of a line travelling the apex of the conical pitch-surface, 
 while a point in it is carried round the traces of the teeth upon a 
 spherical surface described about that apex. 
 
 The operations of drawing the traces of the teeth of bevel- 
 wheels exactly, whether by involutes or by rolling curves, are in 
 every respect analogous to those for drawing the traces 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 de- 
 scribed 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 ;
 
 GEOMETRY OF MACHINERY. 
 
 I. Development of Teeth. Let O, fig. 101, be the common apex 
 of the pitch-cones, O B I, O B' I, of a pair of bevel-wheels ; O G, 
 O C', the axes of those cones ; I their line of contact. Perpen- 
 dicular to O I draw A I A', cutting the axes in A, A'; make the 
 
 outer rims of the patterns and of 
 the wheels portions of the cones 
 A B I, A B' I, of which the nar- 
 row zones occupied by the teeth 
 will be sufficiently near for practical 
 purposes to a spherical surface 
 described about O. As the cones, 
 A B I, A' B' I, cut the pitch-cones 
 at right angles in the outer pitch- 
 circles, I B, I B', they may be 
 called the normal cones. To find 
 the traces of the teeth xipon the 
 normal cones, draw on a flat surface 
 
 Fig. 101, 
 
 circular arcs, ID, I D', with the radii A I, A I ; those arcs will be 
 the developments of arcs of the pitch -circles, I B, I B', when the 
 conical surfaces, A B I, A' B' I, are spread out flat. Describe the 
 traces of teeth for the developed arcs as for a pair of spur-wheels, 
 then wrap the developed arcs on the normal cones, so as to make 
 them coincide with the pitch-circles, and trace the teeth on the 
 conical surfaces. 
 
 II. Traces and Projections of Teeth. Fig. 102 illustrates the 
 process of drawing the projection of a tooth of a bevel-wheel on a 
 plane perpendicular to the axis. In the first place, let A C 
 represent the common axis of the pitch-cone and normal cone ; A 
 being the apex of the normal cone. Let A I be the trace of the 
 normal cone on a plane traversing the axis ; and let 1 1', perpen- 
 dicular to I A, be part of the trace of the pitch-cone on the same 
 plane, of a length equal to the intended breadth of the toothed 
 rim of the wheel. C I perpendicular to A C is the radius of the 
 pitch-circle in which the pitch-cone and normal cone intersect each 
 other. About A, with the radius A I, draw the circular arc 
 DID, making D I = I D = half the pitch ; D I D will be the 
 development of an arc of the pitch-circle of a length equal to the 
 pitch. On the arc D I D lay off I G = I G = half the thickness 
 of a tooth on the outer pitch-circle. Then, by the rules for spur- 
 wheels, draw the trace, H G E G- H, of one tooth and a pair of 
 half-spaces, with a suitable addendum-circle through E, and a 
 suitable root- circle, H F H. 
 
 The straight line FIE will be the trace, upon a plane travers- 
 ing the axis, of the outer side of a tooth ; and E and F will be 
 the traces, on that plane, of the outer addendum-circle and root- 
 circle respectively. From E and F draw straight lines, E E' and
 
 TEETH OF BEVEL-WHEELS. 145 
 
 F F, converging towards the apex of the pitch-cone ; these will be 
 the traces of the addendum-cone and root-cone respectively. (For 
 want of space, the apex of the pitch-cone is not shown in fig. 102.) 
 Through I', parallel to F I E, draw F 7 1' E'j this wili be the trace. 
 
 xr, 
 
 \ 
 
 * 
 
 Fig. 102. 
 
 on a plane traversing the axis, of the inner side of a tooth ; an d 
 the points E', I', and F' will be respectively the traces of the 
 inner addendum-circle, inner pitch-circle, and inner root-circle. 
 
 Through A, parallel to C I, draw the straight line A i e, and 
 conceive this line to be traversed by a plane perpendicular to the 
 axis, as a new plane of projection. Through the points F, I, E> 
 F', I', E', draw straight lines parallel to C A, cutting A e in 
 /, i, e, f,' i', e ; these points, marked with small letters, will be 
 the projections, on the new plane, of the points marked with tho 
 corresponding capital letters.
 
 14:6 GEOMETRY OP MACHINERY. 
 
 Divide the depth, F E, of the tooth at its outer side into any 
 convenient number of intervals. Through the points of division 
 draw straight lines parallel to C A; these will cntfe in a series of 
 points, which will be the projections of the points of division of 
 F E. Through the points of division of F E, and also through 
 the projections of those points, draw circular arcs about A as a 
 centre. Measure a series of thicknesses of the tooth on the arcs 
 which cross F E, and lay off the same series of thicknesses on the 
 corresponding arcs which cross fe; a curve, h y e g h, drawn through 
 the points thus found, will be the required pi'ojection, on a plane 
 parallel to the axis, of the outer side of a tooth. 
 
 The projection, It! g 1 e g h', of the inner side of a tooth is found 
 by a similar process, except that the measuring and laying-off the 
 thicknesses is rendered unnecessaiy by the fact that each pair of 
 corresponding points in the projections of the outer and inner sides 
 lie in one straight line with A. For example, having drawn about 
 A a circular arc through i', di'aw the two^straight lines A g, A g ; 
 these will cut that arc in the points g', g', being the points in the 
 projection of the inner side corresponding to g, g in the projection 
 of the outer side ; and thus it is unnecessary to lay off the thick- 
 ness g' g. 
 
 145. Teeth of Skew-berel Wheels General Conditions. The 
 
 surfaces of the teeth of a skew -bevel wheel belong, like its pitch- 
 surface, to the hyperbolo'idal class, and may be conceived to be 
 generated by the motion of a straight line which, in each of its 
 successive positions, coincides with the line of contact of a tooth 
 with the corresponding tooth of another wheel. Those surfaces 
 may also be conceived to be traced by the rolling of a hyper- 
 boloidal roller upon the hyperbolo'idal pitch-surface, in the manner 
 described in Article 84, pages 70 to 73. 
 
 The conditions to be fulfilled by the traces of the fronts and 
 backs of the teeth on the hyperbolo'idal pitch-surface are : A. That 
 each of those traces shall be one of the generating straight lines of 
 the hyperbolo'id (Article 106, page 89); B. That the normal pitch, 
 measured from front to front of the teeth along the normal spiral 
 (Article 106, page 89), shall be the same in two wheels that gear 
 together (this second condition is always fulfilled if the two 
 pitch-surfaces are correctly designed, and the numbers of teeth 
 made inversely proportional to the angular velocities) ; and C. That 
 the teeth, if in outside gearing, shall be right-handed on both 
 wheels, or left-handed on both wheels ; and if in inside gearing, 
 contrary-handed on the two wheels. 
 
 Skew-bevel teeth may be said to be RIGHT-HANDED or LEFT- 
 HANDED, according to the direction in which the generating lines 
 of the teeth appear to deviate from the axis when looked at with 
 the axis upright, as in fig. 103, page 147. For example, the wheel
 
 SKEW-BEVEL TEETH. 
 
 in that figure 
 has left - handed 
 teeth ; for the 
 generating line 
 I' I deviates to the 
 left of the axis 
 A' A. The same 
 rule applies to the 
 direction in which 
 the crests of the 
 teeth appear to 
 deviate from the 
 radii of the wheel, 
 when looked at as 
 in the upper part 
 of fig. 105, page 
 150. 
 
 Right - handed 
 teeth have left- 
 handed normal 
 spirals, and left- 
 handed teeth 
 right-handed nor- 
 mal spirals. 
 
 146. Skew-bevel 
 Teeth Rules. I. 
 
 Normal Section of 
 a Tooth. In fig. 
 103, let A a A' be 
 the axis of a 
 skew-bevel wheel : 
 let a be the centre 
 of the throat of 
 its hyperbolo'idal 
 pitch-surface ; let 
 the dotted curve 
 through I be the 
 trace of that sur- 
 face on a plane 
 traversing the 
 axis ; and let C I 
 = ai be the radius 
 of the pitch-circle 
 at the middle of 
 the breadth of the 
 rim of theintended 
 
 , -03 
 
 103.
 
 148 
 
 GEOMETRY OF MACHINERY. 
 
 wheel, as found by Rule I. of Article 106, page 88. Draw by 
 Rules IL and III. of that Article, pages 88, 89, the normal I A 
 
 R 
 
 and tangent I I" T, to the trace of the pitch-surface at I. Then 
 find, by Rule V. of that Article, page 89, the radius of curvature 
 of the normal spiral at the point I, and lay off that radius of 
 curvature, I S, along the normal. 
 
 In fig. 104 (which is on a larger scale than fig. 103, for the sake 
 of distinctness), let A 0, as before, be the axis of the wheel, C I 
 the radius of the middle pitch-circle, I A the normal, and I S the 
 radius of curvature of the normal spiral ; draw I N" perpendicular 
 to I S. Then, by Rule V. of Article 106, page 89, find the angle 
 ( = O g F iu fig. 68, page 88) which a tangent to the normal spiral 
 makes with a tangent to the pitch-circle, and draw I P, making 
 that angle with I N. Lay off I P eqiial to tlie pitch as measured 
 on the middle pitch-circle ; let fall P N perpendicular to I N ; then
 
 SKEW- BEVEL TEETH. 149 
 
 I N will be the normal pitch at the middle pitch-circle. About S, 
 with the radius S I, draw a circular arc, and lay off on that arc 
 the distance, D D, equal to the normal pitch, one-half to each side 
 of I. Lay off the intended middle thickness, G G, of a tooth, one- 
 half to each side of I. Then draw, by the rules for spur-wheels, 
 the normal section, H G E G H, of a tooth, being its trace upon a 
 surface which cuts it normally at the middle of the breadth of 
 the rim of the wheel. 
 
 II. Trace of a Tooth on the Normal Cone. Through A in fig. 104 
 draw A i parallel and equal to C I, and through I draw li parallel 
 and equal to C A. About A, with the radius A i, draw the circular 
 arc d d, equal in length to I P, the pitch on the pitch-circle, and 
 having the middle of its length at the point i. This will be the 
 arc on the pitch-circle corresponding to the arc D D on the normal 
 spiral. 
 
 Divide E F, the middle depth of the tooth, into any convenient 
 number of intervals ; and through E and F and the points of 
 division draw straight lines parallel to I i, cutting A i e in a series 
 of corresponding points. Through the points in E F draw circular 
 arcs about S. Through the corresponding points of ef draw cir- 
 cular arcs about A. From the points where the arcs cut the trace 
 E G H measure oblique half-thicknesses to the centre line, E F, of 
 the tooth, along oblique lines drawn parallel to P I ; and lay off 
 those half-thicknesses at both sides of ef, along the arcs which cross 
 it. Through the points thus found draw the curve h g e g h ; this 
 will be the projection, on a plane perpendicular to the axis, of the 
 trace of a tooth upon the normal cone of the pitch-surface at the 
 middle of its breadth; that is, upon the cone whose trace is A I in 
 fig. 103. (If it be desired to draw the development of that trace, 
 lay off the oblique lialf-thicknesses along arcs drawn about A, 
 through the points of division of the radius A F I E. The result 
 is the drawing of an outline outside of, and nearly parallel to, 
 H G E G H. To prevent confusion, it is not shown in the figure.) 
 
 If the pitch-circle chosen is at the throat of the hyperboloi'd, the 
 normal cone becomes simply the plane of that circle ; and in fig. 
 104, A/ i e coincides with A F I E. 
 
 III. Projections of the Middle Lines of a Tooth. In fig. 103, let 
 FIE and fie, as before, represent the projections of the central 
 depth of a tooth, being part of a normal (E I F A, e if a) to the 
 pitch-surface at a point, I i, in the middle pitch-circle, whose radius 
 is C I ==a i; so that F,/, I, i, and E, e are the projections of the 
 middle points of the tooth at the root, at the pitch-surface, and at 
 the crest respectively; and let it be required to find the projections 
 of the middle lines of that tooth at the root, pitch-surface, and 
 crest respectively. 
 
 About a draw the circles//', i i', and e e ; being the projections,
 
 150 GEOMETRY OF MACHINERY. 
 
 on a plane perpendicular to the axis, of the root-circle through F,. 
 the pitch-circle through I, and the addendum-circle through E. 
 Draw also about a the pitch-circle at the throat of the hyperboloid, 
 and let a i' be its radius. Through i draw a straight line, i i' i', so 
 as to touch this throat pitch-circle, and let that straight line cut the 
 circle i i' in i and i'. Draw the straight lines//' /' and e e" e' 
 parallel to i i" i'. Then these three parallel lines will be the pro- 
 jections of the three middle lines before mentioned, on a plane 
 perpendicular to the axis. 
 
 Describe about a two circles touching // "/' and e e" e' respec- 
 tively. These will be respectively the root-circle and the addendum- 
 circle at the throat of the hyperboloid. The roots and crests of all 
 the teeth lie in a pair of hyperboloidal surfaces traversing this 
 pair of circles, and traversing also the pair of circles through F 
 and E. 
 
 The projection, on a plane traversing the axis, of the middle line 
 of the tooth on the pitch-surface is the tajngent I I" already found, 
 the points I" and i' being in one straight line parallel to a A. To 
 find the corresponding projections of the other two middle lines, 
 there are two methods. 
 
 First Method. From the points of contact/" and e", parallel to 
 a A, draw /" F" and e" E", cutting a i in F" and E' respectively. 
 Join F F" and E E". These will be the required projections. 
 
 Second Method. Lay off on the axis, a C' = a C, and a A' = 
 a A, and draw C' I' parallel to C I : then C' I' will be part of the 
 projection of a pitch-circle equal to C I. From i', parallel to 
 A a A', draw i' T, cutting C' I' in I'. Then i' and I' will be the 
 two projections of one pitch-point, and II" I' will be one straight 
 line. Join A' I'. This will be the projection of a normal to the 
 pitch-suiface at I'. Through /' and e' (which lie in one radius, 
 af i' e) draw /' F 7 and e' E' parallel to A a A', cutting A' I' in 
 F' and E' respectively. Join F F' and E E'. These will be the 
 required projections of the middle lines at the root and crest of the 
 tooth respectively. 
 
 IV. Complete Projection of a Tooth on a Plane Normal to the Axis. 
 Let the plane of projection in fig. 105 be normal to the axis of 
 the wheel, and (as in fig. 103) let a be the axis ; let the circles e e, 
 i i', and //', be the projections of the middle addendum-circle, 
 middle pitch-circle, and middle root-circle of the intended wheel ; 
 let the circles through e", i', and/" be the corresponding circles at 
 the throat of the pitch-suiiace ; and let the parallel straight lines 
 e e" e, i i' i", //'/', be the projections of the middle lines of a 
 tooth at the crest, pitch-surface, and root, drawn according to the 
 preceding rules. 
 
 At the end of the radius afi e construct, by the rules already given, 
 the projection of the trace of the tooth upon the middle normal
 
 SKEW-BEVEL TEETH. 
 
 151 
 
 cone, being the curve marked hgeghin fig. 104; and at the end 
 of the radius af *' ef construct a similar and equal figure. From 
 a series of points in the figure at/i e draw straight lines to the 
 corresponding points in the figure at/' i' e ; each of those straight 
 lines will be the projection of a 
 generating line of the surface of the 
 tooth. For example, the straight 
 lines from the corners of the crest 
 at e to the corresponding corners 
 of the crest at e! (both of which 
 lines touch the circle through e") 
 will be the projections of the 
 two edges of the crest; the 
 straight lines from the pair of 
 points where the curve at f i e 
 cuts the pitch-circle to the cor- 
 responding pair of points near 
 /' i' d (both of which lines touch 
 the circle through i') will be the 
 projections of the lines in which 
 the front and back of the tooth 
 respectively cut the pitch-surface ; 
 and the straight lines from the 
 bottoms of the clearing curves 
 near f to the corresponding points 
 near/' (both of which lines touch 
 the circle through/') will be the 
 projections of the lines marking the 
 bottoms of the hollows of which 
 these curves are the traces. 
 
 The projections of the outer and 
 inner sides of the tooth (being 
 portions of the outer and inner sides of the rim of the wheel) are 
 figures similar to the curve at / i e, and constructed by the same 
 method; the dimensions of the former being larger and those of 
 the latter smaller than the dimensions of that middle figure, in the 
 proportion in which the radii of the outer and inner pitch-circles 
 are respectively greater and smaller than those of the middle pitch- 
 circle. (As to those two circles, see Article 106, page 90.) The 
 tooth in fig. 105 is drawn of an exaggerated breadth, in order to 
 show more clearly the construction of the figure. 
 
 V. Modelling Skew-bevel Teeth. Construct a frame of rods to re- 
 present the axis A A' in fig. 103, the equal radii I and C' I', the 
 equal normals A F I E and A' F' I' E', and the generating line II'. 
 Make a pair of equal and similar templets, each of the shape and 
 dimensions of the normal section of a tooth, H G E G- H, fig. 104. 
 
 Fig. 105.
 
 152 GEOMETRY OF MACHINERY. 
 
 Fix those two templets to the frame at F I E and P I' E', fig. 103, 
 with their flat surfaces parallel to each other and normal to the 
 rod IP. Then a straight edge or a stretched wire, made to touch 
 the edges of the templets at a pair of corresponding points, will 
 mark one of the generating lines of a tooth ; and by the help of 
 this apparatus, teeth may be modelled suitable either for the pitch- 
 circle through C, or for that through C', or for the pitch-circle at 
 thf throat of the hyperboloi'd, or for any other pitch-circle on the 
 eame hyperboloid. 
 
 147. The Transverse Obliquity of Skew-bevel Teeth is the angle, 
 
 P T N, fig. 104, page 148 (equal to O g F' in fig. 68, page 88), which 
 the normal spiral makes with the pitch-circle; or it may be other- 
 wise defined as the angle which the generating line of a tooth on 
 the pitch-surface makes with the generating line of a tangent cone 
 at a given point, I. From the rule for finding that angle (Article 
 106, page 89), it is evident that, with a given hyperboloi'dal pitch- 
 surface, the transverse obliquity of the ^eeth is greatest at the 
 throat, and is the less the farther the middle pitch-circle of the 
 wheel is removed from the throat. Hence, it is generally advisable, 
 in designing skew-bevel wheels, to place the pitch-circles as far as 
 practicable from the throats of the hyperboloi'ds, because obliquity 
 of action tends to increase friction. 
 
 148. Skew-bevel Wheels In Double Pairs. Skew-bevel wheels 
 
 possess a property which ordinary bevel wheels do not viz., that 
 of being capable of combination by double pairs, as in fig. 106. The 
 upper part of the figure represents a projection on a plane parallel 
 to the line of contact, I I', and to the common perpendicular of the 
 axes F G. The lower part of the figure represents a projection on 
 a plane normal to the common perpendicular. Small letters in the 
 second projection correspond to capital letters in the first projection. 
 
 B and B' are two equal and similar wheels fixed on the shaft 
 A A', with pitch-surfaces forming parts of the same hyperboloi'd, 
 and at equal distances from its throat. They have equal and similar 
 teeth, with equal obliquities in the same direction ; and, in short, 
 both wheels may be cast from the same pattern. In the example 
 given, the teeth of both wheels are right-handed. In like manner, 
 I) and D' are two equal and similar wheels fixed on the shaft C C'; 
 B gears with D, and B' with D'. 
 
 This arrangement may be useful where it is desired, for the sake 
 of strength or of steadiness of motion, to divide the force exerted in 
 transmitting the motion between two pairs of wheels. 
 
 149. Teeth with Sloping Back*. The teeth described in the pre- 
 ceding Articles of this Section have their backs similar to their 
 fronts, so that the motion of the wheels may be reversed, the backs 
 then acting as the fronts did during the forward motion. There 
 are many cases in mechanism in which it is not necessary that the
 
 TEETH WITH SLOPING BACKS. 
 
 153 
 
 motion of the wheels should ever be reversed; and in such cases 
 the backs of the teeth of a pair of wheels are required simply to be 
 
 Fig. 106. 
 
 -of such shapes as to clear each other, without refei'ence to the trans- 
 mission of motion. The consequence of this is, that although the 
 traces of the backs must still belong to the same class of curves 
 with the traces of the fronts, their obliquity may be considerably 
 increased, the effect being to strengthen the teeth at their 
 roots.* 
 
 The most convenient curves for the traces of the backs of teeth 
 under those circumstances are involutes of a circle, for which there 
 may be substituted in practice circular arcs approximating to them; 
 
 This was first pointed out by Professor Willis.
 
 154 
 
 GEOMETRY OF MACHINERY. 
 
 and the method of drawing those arcs is as follows: Let fig. 107 
 represent the trace of part of a wheel with its teeth, that wheel 
 being the smallest wheel of a set that are to be capable of gearing 
 together; because the smallest wheel of such a set requires the 
 greatest addendum : let C be the centre, A A the addendum-circle, 
 
 Ac" 
 
 \ 
 
 Fig. 107. 
 
 B B the pitch-circle, D D the root-circle, and let E" I" J", ~EIJ, 
 E' I' 3', be the fronts of teeth designed according to the proper 
 rules, and F", F, F', the pitch-points of the backs of those teeth. 
 To any one of those 6ac& pitch-points, as F, draw the radius OF; 
 bisect F in G, and about G draw the semicircle F K C. Draw a 
 straight line, H K, perpendicular to and bisecting the distance, E F, 
 between the crest E and back pitch-point F; and let that straight 
 line cut the semicircle in K. About the centre C, with the radius 
 C K, draw the circle K" K K'; this will be the base-circle of the 
 required involutes (see Article 131, page 121). 
 
 To draw the circular arcs approximating to those involutes, 
 lay off, from the back pitch-points to the base-circle, the equal 
 distances F" K" = F K' = F K, &c; and about the respective
 
 SLOPING-BACKED TEETH STEPPED TEETH. 
 
 155 
 
 centres, K, K', K", &c.. draw the circular arcs E F L, E' F' L',. 
 E" F" L", &c. 
 
 In each of the larger wheels of the set, the radius of the base- 
 circle for the backs is to bear to the radius of the pitch-circle the 
 
 C K 
 constant proportion p- ^ n or ier that the backs of the teeth of all 
 
 the wheels of the set may have the same obliquity viz., the angle 
 KOF. 
 
 In a straight rack capable of gearing with any wheel of the set, 
 the traces of the backs of the teeth are to be straight lines, making 
 with the pitch-line an angle equal to C F K. 
 
 150. stepped Teeth. In order to increase the smoothness of the 
 action of toothed wheels, Dr. Hooke invented the making of the 
 fronts of teeth in a series of steps, as shown in fig. 108, where the 
 
 Fig. 108 A.
 
 15G GEOMETRY OP MACHINERY. 
 
 upper part of the figure is a projection of the rim of a wheel with 
 stepped teeth on a plane parallel to the axis, and the lower part 
 is a projection on a plane perpendicular to the axis. A wheel 
 thus formed resembles in shape a series of equal and similar toothed 
 discs placed side by side, with the teeth of each a little behind 
 those of the preceding disc. In such a wheel, let p be the circular 
 pitch, and n the number of steps. Then the path of contact, the 
 addendum, and the extent of sliding, are those due to the divided 
 
 q} 
 
 pitch -, while the strength of the teeth is that due to the thickness 
 
 corresponding to the total pitch p; so that the smooth action of 
 small teeth and the strength of large teeth are combined. The 
 action of small teeth is smoother and steadier than that of large 
 teeth, because they can be made to approximate more closely to 
 the exact theoretical figure; and also because the sliding motion of 
 one tooth upon another is of less extent. ,Ju the example shown 
 in fig. 108 there are four steps, so that the divided pitch is one- 
 fourth of the total pitch ; and the path of contact (E I F, in the 
 lower part of the figure) is of the length suited to the divided 
 pitch, being only one-fourth of the length which would have been 
 required had the fronts of the teeth not been stepped. 
 
 151. Helical Teeth, also invented by Dr. Hooke with the same 
 object, are teeth whose fronts, instead of being parallel to the line 
 of contact, of the pitch-cylinders of a pair of spur-wheels, cross that 
 line obliquely, so as to be of a screw-like or helical form : in other 
 words, they are teeth of the figure of short portions of screw-threads 
 (Article 58, page 36); the trace of each thread on a plane perpen- 
 dicular to the axis being similar to that of a stepped tooth, as 
 shown in the lower part of fig. 108. Fig. 108 A shows a projection 
 of the rim of a wheel with helical teeth on a plane parallel to 
 the axis. 
 
 In order that a pair of wheels with parallel axes and helical 
 teeth may gear correctly together, the teeth, besides being of the 
 same circular pitch, must have the same transverse obliquity; and 
 if in outside gearing, they must be right-handed on one wheel and 
 left-handed on the other. If in inside gearing, they must be either 
 right-handed or left-handed on both wheels. In fig. 108 A the 
 teeth are left-handed. In wheel- work of this kind the contact of 
 ach pair of teeth commences at the foremost 'end of the helical 
 fronts, and terminates at the aftermost end ; and the rims of the 
 wheels are to be made of such a breadth that the contact of one 
 pair of teeth shall not terminate until that of the next pair has 
 commenced. 
 
 Helical teeth ai'e open to the objection that they exert a laterally 
 oblique pressure, which tends to increase friction. 
 
 When, in designing a skew-bevel wheel, a portion of the tangent 
 cylinder at the throat of the hyperbolo'id (Article 106, pagfe 87;
 
 SCREW AND NUT SCREW GEARING. 157 
 
 find Article 85, page 73) is used as an approximation to the true 
 pitch-surface, the teeth of that wheel become screw-thi'eads, having 
 a transverse obliquity determined by the principles of Article 147, 
 page 152; and, as has been already stated in the article referred to, 
 they are either right-handed or left-handed in both -wheels. 
 
 152. Screw ami iVnt. The figure of a true screw, external or 
 internal, and the motion of a screw working in a corresponding 
 screw-shaped bearing, have been described in Articles 57 to 66, 
 pages 36 to 42. In the elementary combination of an external and 
 internal screiv, more commonly called a screw and nut, the two 
 pieces have threads, one external and the other internal, of similar 
 figures and eqiial dimensions, so as to tit each other truly; and one 
 of them turns about their common axis without translation, while 
 the other slides parallel to that axis without rotation. The best 
 form of section for the threads is rectangular. The comparative 
 motion is, that the sliding piece advances through a distance equal 
 to the pitch (viz., the " total axial pitch") during each revolution of 
 
 the turning piece. If the threads are < "fft.^^^ ' f tne sliding 
 piece is made to move towards an observer at one end of the axis 
 
 rotation, of the turning piece. The combination belongs 
 
 to Mr. Willis's Class A, because the velocity-ratio is constant ; and 
 the extent of the motion is limited by the length of the screw. 
 
 153. Screw Wheel-Work in General. Screw wheel-work consists 
 of wheels with cylindrical pitch-surfaces, having screw-threads or 
 helical teeth instead of ordinary teett One case of screw-gearing 
 has been described in Article 151, page 156 viz., that in which 
 the axes are parallel. The cases to which this and the following 
 articles relate are those in which the axes are not parallel; so 
 that the pitch-surfaces in an elementary combination are a pair 
 cylinders touching each other in one pitch-point, like those repre- 
 sented in Article 85, fig. 55, page 73. The pitch-point (O', fig. 55) 
 is obviously in the common perpendicular of the two axes (F' G', 
 fig. 55); and there is one straight line traversing the pitch-point 
 (OO, fig. 55), which is a tangent at once to the two pitch- 
 cylinders and to the acting surfaces or fronts of each pair of 
 threads at the instant when those surfaces touch each other at the 
 pitch-point : that straight line may be called the LINE OF CONTACT. 
 The angles of inclination of the screw-threads to the two axes (see 
 Article 63, page 40) are equal respectively to the angles made by 
 the line of contact with those axes. The PITCH-CIRCLES of the two 
 screws are the two circular sections of the pitch-cylinders which 
 traverse the pitch-point. The PLANE OF CONNECTION, or PLANE o?
 
 158 GEOMETRY OF MACHINERY. 
 
 ACTION, is a plane traversing the pitch-point normal to the line of 
 contact : that plane, of course, traverses the common perpendicular 
 of the axes. 
 
 When the line of contact is found by the rule given in Article 
 84, page 71, the cylindrical pitch-surfaces represent the tangent- 
 cylinders at the throats of a pair of hyperbolo'ids; and the screw- 
 threads are approximations to the skew-bevel teeth suited for that 
 combination, as already stated in Article 151, page 156. But in 
 many cases the line of contact has positions greatly differing from 
 this; and then the comparative motion becomes different from that 
 of a pair of skew-bevel wheels; the object of screw-gearing in such 
 cases being to obtain, with a given pair of cylindrical pitch- 
 surfaces, a velocity-ratio of rotation independent of the radii of 
 those surfaces; and such is the difference between approximate 
 skew-bevel gearing and screw gearing in general. 
 
 In every elementary combination in screw wheel-work, each of 
 the two pieces is at once a screw and a wheel; but it is customary, 
 when their diameters are very different, to call that which has the 
 smaller diameter the ENDLESS SCREW, or WORM, and that which has 
 the greater diameter the WORM-WHEEL. For example, in fig. Ill 
 (farther on) a! is the worm, or endless screw, and A' the worm- 
 wheel. The word " endless " is used because of the extent of the 
 motion being unlimited. 
 
 Screw wheel- work belongs to Mr. Willis's Class A, the velocity- 
 ratio being constant. 
 
 The following are the general principles of elementary combina- 
 tions in screw wheel- work : 
 
 I. The angular velocities of the two screws are inversely, and 
 their times of revolution directly, as the numbers of threads; 
 whence it follows that the angular velocity-ratio must be expressible 
 in whole numbers, as in the case of ordinary toothed wheels. 
 
 II. The divided normal pitch (see Article 66, page 42), as 
 measured on the pitch-cylinders, must be the same in two screws 
 that gear together. 
 
 III. The common component of the velocities of a pair of points 
 in the two screws at the instant when those two points touch 
 each other and pass the pitch-point, is perpendicular to the line of 
 contact and to the common perpendicular of the axes; in other 
 words, it coincides with the intersection of the plane of connection 
 and the common tangent-plane of the two pitch-cylinders. 
 
 IV. The circular or circumferential pitches of the two screws 
 (Article 42, page 66), as measured on their pitch-cylinders, are 
 proportional to the total velocities of points (called the surface 
 velocities) in those cylinders; and they bear the same proportion to 
 the divided normal pitch that those total velocities bear to their 
 common component.
 
 SCREW WHEEL- WORK. 159 
 
 Y. The relative transverse sliding of a pair of threads that are in 
 action takes place along the line of contact. 
 
 It will be shown in the next article that for a given pair of 
 axes and a given angular velocity-ratio the relative transverse 
 sliding is least when the pitch-cylinders are the tangent-cylinders 
 at the throats of a pair of skew-bevel hyperboloids. 
 
 154. Screw Wheel-Work Kales for Drawing. In figs. 109 and 
 
 110 the plane of projection is supposed to be the common tangent- 
 plane of the two pitch-cylinders; and I represents the pitch-point; 
 which is also the trace and projection of the common perpendicular 
 of the two axes. 
 
 I. Given, the projections of the two axes, the angular velocity- 
 ratio, and the radii of the two pitch-cylinders, to find the propor- 
 tionate values of their surface-velocities, and the proportionate 
 
 value and direction of the velocity of transverse sliding. The 
 two cylinders may be called respectively A and a. 
 
 In fig. 109, let I A and I a represent the projections of o 
 the two axes. Along those projections lay off lengths I A, 
 I a, proportional to the two angular velocities of rotation, 
 and pointing in the direction in which an observer must 
 look from I in order to make both rotations seem right- 
 handed. Draw the straight line A a, and divide it at K 
 into two parts inversely proportional to the radii of the 
 two pitch -cylinders ; in other words, let B and b denote 
 the radii of the cylinders A and a respectively, so that 
 B + b is the length of the common perpendicular, or line 
 of centres; and let K divide A a in the following proper- 
 tion: 
 
 B + b : B : 6 
 :: Aa :Ka : KA 
 
 Complete the parallelogram I Y K v; then IV, Iv, and 
 the diagonal I K, will be respectively proportional and 
 perpendicular to the surface velocity of the cylinder A, 
 the surface velocity of the cylinder a, and the velocity of 
 relative transverse sliding at the pitch-point I. 
 
 Or otherwise, by calculation; let -v- be the ratio of the 
 
 X&. 
 
 angular velocities, and _ that of the radii; then -^ is 
 
 ^r. 
 
 obviously the ratio of the surface velocities. 
 
 It is obvious that for a given pair of axes and a given 
 pair of angular velocities the velocity of transverse sliding 
 is least when I K is perpendicular to A a. But A a is _ 
 parallel to the line of contact of a pair of hyperbolo'idal Fig. 109. 
 pitch-surfaces for skew-bevel wheels having the given
 
 160 
 
 GEOMETRY OF MACHINERY. 
 
 velocity-ratio; and this is the demonstration of the statement in 
 the preceding article, that screws which coincide approximately 
 with skew-bevel wheels give the least possible transverse sliding 
 of the threads for a given pair of axes aud a given, velocity-ratio 
 (see page 159). 
 
 The proportionate value of the common component of tlie surface 
 velocities may be represented by the length of a perpendicular let 
 fall from either "V or v upon I K ; but the next rule gives a more 
 convenient way of representing both it and the transverse sliding. 
 
 II. To draw the line of contact, and to find the proportions borne 
 to tlie surface velocities by their common component, and by the trans- 
 verse sliding ; also the proportions borne to each other by the circular 
 pitches, tJie divided axial pitches, and the divided normal pitch. 
 
 In fig. 110 (as in fig. 109), let I represent the pitch-point, and 
 I A and I a the projections of the two axes. Perpendicular to I A 
 and I a respectively, draw I C and I c, of the proper lengths, and 
 in the proper directions, to represent the surface velocities of the 
 two pitch-cylinders at the point I; draw the straight line C c, 
 cutting the projections of the two axes in P and p respectively;
 
 SCREW GEARING. 161 
 
 and upon C c let fall the perpendicular I N (which will obviously 
 be parallel to I K in fig. 109). Through I draw TIT parallel to 
 Cc. 
 
 Then TIT will be the line of contact /IN will represent t he 
 common component of the surface velocities (and will also be the 
 trace of the plane of connection); Cc will represent the velocity of 
 transverse sliding; and the proportions of the several divided 
 pitches will be as follows : 
 
 I C : circular pitch of A. 
 
 : : I c : circular pitch of a. 
 : : I P : divided axial pitch of A. 
 
 ::!/>: divided axial pitch of a. 
 : : I N : divided normal pitch of both screws. 
 
 The figure may be regarded as part of the development of both 
 screws upon the common tangent plane of their pitch-cylinders. 
 (See Article 63, page 40. As to RACKS, see Addendum, page 289.) 
 
 The absolute lengths of the circular pitches are found by dividing 
 the pitch-circles into suitable numbers of equal parts, precisely as in 
 the case of spur-wheels (see Articles 112 to 121, pages 103 to 114); 
 and from them, by the aid of the proportions given by fig. 110, the 
 absolute lengths of the divided axial pitches and of the divided 
 normal pitch are easily found. For the total axial pitch of 
 either screw, multiply the divided axial pitch by the number of 
 threads. 
 
 III. To find the radii of curvature of the normal screw-lines. The 
 normal helix, or normal screw-line (see Article 65, page 41), of each 
 of the two screws touches I N at the pitch- point I ; and the plane 
 of connection of which I N is the trace is the common osculating 
 plane of the two normal screw-lines at I. Their radii of curvature 
 at that point both coincide with the common perpendicular of the 
 axes. The rule for finding such radii (Articles 64 and 65, page 41) r 
 when applied to this case, takes the following form : On I C lay 
 
 perpendicular 
 
 be the radius of curvature of the normal helix of the screw A. 
 A similar construction, substituting small for capital letters, serves 
 to find I r, the radius of curvature of the normal helix of the 
 screw a. 
 
 Fig. Ill represents two projections of the pitch-cylinders of a 
 pair of screws designed by the rules which have just been given, 
 and shows also the helical lines in which the fronts of the threads 
 cut those pitch-cylinders. The upper part of the figure is a pro- 
 jection on the plane of action, whose trace, in fig. 110, is I N. 
 A' a' is the common perpendicular of the two axes, and I the
 
 162 
 
 GEOMETRY OF MACHINERY. 
 
 pitch-point ; N'N' is the trace of the common tangent plane of the two 
 pitch-cylinders; and the arrow shows the direction of the common 
 
 component of their 
 
 surface velocities 
 
 at the point I'. R 
 and r are the centres 
 of curvature of the 
 two normal screw- 
 lines at the point 
 I'; and S S and s s, 
 described about R 
 and r respectively, 
 are their two oscu- 
 lating circles, whose 
 radii, I' R and I' r, 
 are found by Rule 
 III. 
 
 The lower part 
 of the figure is a 
 projection on the 
 r common tangent 
 plane of the pitch- 
 cylinders. A A and 
 a a are the pro- 
 jections of their two 
 axes; T I T is the 
 line of contact ; 
 N I N is the trace 
 of the plane of 
 action; and the 
 arrow marks the 
 direction of the 
 common component 
 of the surface 
 velocities at the 
 pitch-point I. 
 
 In the particular 
 example repre- 
 sented by figs. 109, 
 110, and 111, the 
 following are the 
 Fig. 111. principal data and 
 
 proportions : 
 
 Velocity-ratio = 20; 
 Number of threads of A, 40; of a, 2;
 
 THREADS FOR GEARING SCREWS. 1G3 
 
 . Ratios of radii and line of centres, 
 
 B + 6 :B :6 
 : : 11 :10 :1 
 
 Both screws right-handed. 
 
 155. Figures of Threads found by Cleans of Normal Scrcw-LineB. 
 
 By the following process threads may be designed for any gearing 
 screw, so that they shall gear correctly with threads designed on 
 the same principle for any other screw of the same normal pitch. 
 
 Let the screw to be provided with threads be, for example, the 
 screw A of fig. 111. Draw, by Rule III. of Article 154, page 161, 
 the osculating circle, S I' S, of its normal screw-line. Lay off' the 
 normal pitch upon that osculating circle, aud design the figure of a 
 tooth and two half-spaces of that pitch, with the proper addendum 
 and depth, as if the osculating circle were the pitch-circle of a spur- 
 wheel ; the figure so drawn will be the normal section of a thread, 
 being the trace of the thread upon a surface which cuts it at right 
 angles; and by the help of that section the threads may be made 
 of the correct figure. 
 
 The normal sections of the acting surfaces of a thread may be 
 either involutes of circles (Articles 131, 133, pages 120 to 128), or 
 epicycloids (Articles 136 to 140, pages 130 to 137). All screws 
 with involute threads of the same divided normal pitch gear correctly 
 together, and may be said to belong to one set; and they have the 
 same property with involute toothed wheels, of admitting of some 
 alteration of the distance between the axes. All screws of the 
 same divided normal pitch having epicycloi'dal teeth described by 
 the same rolling circle gear correctly together, and may be said to 
 belong to one set. 
 
 This method of designing the threads of gearing screws is believed 
 to be now published for the first time. 
 
 156. Figures of Threads designed on a Plane Normal to one Axi-. 
 
 In many cases which occur in practice the axes of the two 
 screws are perpendicular to each other; so that, in fig. 110, page 160, 
 A I P and a I p are at right angles, I C coincides with I p, and 
 I c coincides with I P; and therefore the divided axial pitch of 
 either screw is equal to the circular pitch of the other. In such cases, 
 and especially where the diameters of the pitch-cylinders are very 
 unequal, so that the larger screw is called a worm-wheel, and the 
 smaller an endless screw, it is often convenient to design the traces 
 of the threads on a plane normal to the axis of the worm-wheel, 
 and traversing the axis of the endless screw; and then it is evident 
 (as Mr. Willis appears to have been the first to show) that if the 
 traces of the threads of the worm-wheel be made like those of a 
 spur-wheel of the same radius and pitch, and those of the threads 
 of the screw like the traces of the teeth of a rack suited to gear
 
 164 
 
 GEOMETRY OF MACHINERY. 
 
 with that spur-wheel, the worm- wheel and screw will gear correctly 
 together. 
 
 Fig. 112 represents a worm-wheel and endless screw. 
 
 The lower part of the figure is a diagram drawn on the common 
 tangent plane of the pitch-cylinders. I is the pitch-point ; I C is 
 
 Fig. 112. 
 
 the divided axial pitch of the endless screw, being also the develop- 
 ment of the circular pitch of the worm-wheel; I c is the divided 
 axial pitch of the worm-wheel, being also the development of the 
 circular pitch of the endless screw. I N, perpendicular to C c, is 
 the development of the divided normal pitch of both screws; and 
 C c is the extent of transverse sliding which takes place while an 
 arc equal to the pitch passes the pitch-point
 
 GEARING SCREWS. IQfr 
 
 In the left-hand division of the upper part of the figure 
 the plane of projection is normal to the axis, A', of the worm- 
 wheel, and traverses the axis, a' a', of the endless screw. The 
 circle B B is the trace of the pitch -cylinder of the wheel the 
 straight line b b is the trace of the upper side of the pitch- 
 cylinder of the screw ; and those traces touch each other in the 
 pitch-point I'. The threads of the wheel, and those at the upper 
 side of the screw, are shown in section ; the traces of the threads of 
 the wheel are like those of the teeth of a spui'-wheel having the same 
 circular pitch, and B B for a pitch-circle; the traces of the threads 
 of the screw are like those of the teeth of a rack suited to gear with 
 that spur-wheel, and having b b for its pitch-line. The addendum- 
 circle, E E, of the worm-wheel, and the addendum-line, e e, of the 
 endless screw, are drawn as for a spur-wheel and rack. The lower 
 parts of the threads of the endless screw are shown in projection. 
 In the example given, both wheel and screw have right-handed 
 threads; the number of threads of the screw is two; of the wheel, 
 40 ; and the screw is represented as driving the wheel. The right- 
 hand division of the upper part of the figure shows the wheel in 
 section and the screw in projection; and the plane of projection 
 traverses the axis, A", of the wheel, and is normal to the axis, a", of 
 the screw; I" is the pitch-point. 
 
 The traces of the threads of the wheel in the left-hand division 
 of the upper part of the figure are involutes of a circle, and those 
 of the threads of the screw are straight lines. That shape, as in 
 the case of spur-wheels, enables the distance between the axes to 
 be varied to a certain extent without affecting the accuracy of the 
 action. But any shapes suited for the teeth of wheels and racks 
 may be employed. 
 
 If a set of worm-wheels be made of the same circular pitch and 
 obliquity of thread, and having the traces of the threads all 
 involutes or all epicycloids, traced by the same rolling circle; and 
 if a set of endless screws be made, all of the same divided axial 
 pitch, equal to the circular pitch of the wheels, and of an obliquity 
 of thread equal to the complement of the obliquity of the threads 
 of the wheels, and having the traces of the teeth, as the case may 
 be, all straight lines of the proper obliquity, or all epicycloids traced 
 by the same rolling circle that is used to trace the threads of the 
 wheels, then any one of the wheels will gear correctly with any 
 one of the screws. 
 
 157. ciose-Fitting Tangent Screws. In many cases the object of 
 screw-gearing is not the economical transmission of motive power, 
 but the production of small angular motions with great accuracy : 
 as, for example, when the principal wheel of a dividing engine, or 
 that of a machine for pitching and cutting the teeth of wheels, or 
 the wheel or sector which adjusts the direction of stroke of a
 
 166 GEOMETRY OF MACHINERY. 
 
 cutting tool in a shaping machine, is driven by a "tangent-screw" 1 
 situated relatively to the wheel in the manner already shown in 
 fig. 112. In such cases the screw has not only to move the wheel 
 into any required position, but to hold it there; and therefore it is 
 essential that there should be no back-lash. In order to ensure 
 this, together with the requisite precision of action, an exact copy 
 of the tangent-screw is made of steel, the edges of its thread are 
 notched, and it is hardened, so that it becomes a cutting tool : it 
 is then mounted in a suitable frame, so as to gear with the roughly 
 formed teeth or threads of the wheel, and turned so as to drive them; 
 in the course of which operation it cuts them to the proper figure. 
 The axis of the cutting screw is placed at first at a distance from 
 the axis of the wheel somewhat greater than the intended per- 
 manent distance ; and after each complete revolution of the wheel 
 the axes are brought a little nearer together, until the permanent 
 distance is attained; and by turning the screw in this last position, 
 the shaping of the teeth or wheel-threads is finished. From the 
 property of threads with traces similar to those of involute teeth, 
 which has already been mentioned in Article 156, page 165, it is 
 evident that this class of figures is peculiarly well suited to cases 
 in which the tangent-screw is made to cut the wheel, because of 
 the gradual diminution of the distance between the axes which 
 takes place during the process of cutting. 
 
 158. Oldkam'a 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. Various sorts of couplings will 
 be described and compared together in, 
 a later division of this treatise. The 
 present Article relates to a coupling for 
 parallel shafts, invented by Oldham, 
 which acts by sliding contact. It is 
 represented in fig. 113. C 1? C 2 are the 
 axes of the two parallel shafts; D 15 D 2 , 
 two discs facing each other, fixed on 
 Fig. 113. the ends of the two shafts respectively ; 
 
 Ej E 1? a bar sliding in a diametral 
 
 groove in the face of D A ; E 2 E 2 , a bar sliding in a diametral 
 groove in the face of D 2 : those bars are fixed together at A, at 
 right angles to each other, so as to form a rigid cross. The angular 
 velocities of the two discs and of the cross are all equal at every 
 instant; the middle point of the cross, at A, revolves in the dotted
 
 OLDHAM'S COUPLIXG PIN AND SLOT. 167 
 
 circle described upon the line of centres, Cj. C 2 , as a diameter, twice 
 for each turn of the discs and cross; the instantaneous axis of 
 rotation of the cross at any instant is at I, the point in the circle 
 Cj C 2 diametrically opposite to A ; and each arm of the cross slides 
 in its groove through a distance equal to twice the line of centres 
 during each half revolution, or twice the line of centres and back 
 again that is, four times the line of centres during each revolution. 
 Oldham's coupling belongs to Mr. Willis's Class A. The cross 
 may be strengthened by making its two bars take the form of 
 projecting diametral ridges on opposite sides of a third circular 
 disc Or the cross may consist of two grooves in the opposite sides 
 of such a disc, and instead of grooved discs, the two shafts may 
 cany cross bars fitting the grooves of the cross. 
 
 159. Pin and Straight Slot. The communication of a uniform 
 velocity-ratio by the sliding contact of a round pin with the sides 
 of a slot or groove has already been described in Article 141, page 
 137. A velocity-ratio varying in any manner may be communicated 
 by making the slot of a suitable figure, the principle of the com- 
 bination being, that the line of connection is a normal to the centre 
 line of the slot, traversing the centre line of the pin. The present 
 Article relates to cases in which the slot is straight and the 
 velocity-ratio variable. Three such cases are illustrated by figs. 
 114, 115, and 116, fui-ther on. Fig. 114 represents a coupling, 
 belonging to Mr. Willis's Class B, where two shafts turn about 
 the parallel axes A and B with equal mean angular velocities, 
 though the angular velocity-ratio at each instant is variable. Fig. 
 115 shows a crank turning continuously about the axis A, and 
 carrying a pin, C, which, by means of the slot F G, drives a lever 
 which rocks or oscillates about the axis B. Fig. 116 shows a 
 crank turning continuously about the axis A, and carrying a pin, 
 C, which, by means of the slot F G in the cross-head of the rod B, 
 gives a reciprocating sliding motion to that rod. The last two 
 combinations belong to Mr. Willis's Class C. 
 
 In practice, for the purpose of diminishing friction and pre- 
 venting back-lash, it is usual to make the pin turn in a bush which 
 slides in the slot; but that bush is not shown in the figures. 
 
 The following are the principles of the action of those three 
 combinations : 
 
 I. Coupling (fig. 114). In order that the directional relation of 
 the rotations may be constant, the crank-arm, A C, must be greater 
 than the line of centres, A B. 
 
 With a given crank-arm, A C, to find the position of the axis B 
 of the slot-lever, so that the crank and slot-lever shall alternately 
 overtake and fall behind each other by a given angle : With the 
 radius A C describe the circle D C E, and draw the diameter 
 DAE, with which the line of centres is to coincide. Lay off
 
 168 
 
 GEOMETRY OP MACHINERY. 
 
 EAH = EA7t = the complement of the given angle, and draw 
 H B h perpendicular to D A E. B will be the trace of the re- 
 quired axis. 
 
 At the instant when the centre of the pin is at H or h, the 
 
 angular velocities are equal; 
 and A H B = A h B is the 
 given angle beforementioned. 
 
 With a given position, C, of 
 the centre of the pin, to find 
 the angular velocity - ratio : 
 
 //jF/&W J From C, perpendicular to the 
 jvS/y%S centre line, B C, of the slot, 
 draw the line of connection, 
 C I, cutting the line of centres 
 in I ; then 
 
 Angular velocity of B A I 
 Angular velocity of A B I ' 
 or otherwise : draw A P par- 
 allel to B C and perpendicular 
 to C I; then 
 
 Angular velocity of B A P 
 Angular velocity of A ~~ B C* 
 
 The < ? > values of this ratio occur when the pin is at 
 
 ( E 1 
 
 < -p. > respectively; and they are as follows: 
 
 AE AC 
 
 Greatest, 
 
 BE AC - AB' 
 
 1 Least, 
 
 AD 
 
 AC 
 
 B D A C + A B* 
 The travel or length of sliding of the pin in ike slot is 
 FG = BF-BG = BD-BE; 
 
 and this takes place twice in each revolution. 
 
 II. Crank and Slotted Lever (fig. 115). As the crank-arm, A C, 
 in fig. 115, is shorter than the line of centres, A B, the slotted 
 lever, B G F, has a reciprocating or rocking motion. 
 
 With a given line of centres, A B, and a given semi-amplitude 
 or angular half-stroke of the rocking motion of the lever, A B K 
 = A B k, to find the length of crank-arm: From A let fall A K. 
 perpendicular to B K, or A k perpendicular to B & ; A K = A & 
 will be the required crank-arm. 
 
 K and k will be the two dead points; that is to say, the positions
 
 PIN AND SLOT. 
 
 169 
 
 of the centre of the pin at the two instants when the lever has no 
 velocity, having just ceased to move in one direction, and 
 just about to begin to move in the 
 opposite direction. 
 
 To find the angidar velocity-ratio at 
 the instant when the centre of the pin is 
 in a given position, C : Draw the corre- 
 sponding position, B C F, of the centre 
 line of the slot, and perpendicular to it 
 draw C I, cutting the line of centres in 
 I; then 
 
 Angular velocity of lever 
 
 AI 
 BI* 
 
 Angular velocity of crank 
 
 To find the travel of the pin in the 
 slot, lay off B G - B E, and B F = B D; 
 G and F will be the two ends of the 
 travel of the centre of the pin; and 
 FG=DE=2AC will be the length 
 of travel. 
 
 III. Crank and Slot-headed Sliding 
 Rod (fig. 116). The crank-arm, A C, in 
 this case is to be made equal to one-half Fig. 115. 
 
 of the intended length of stroke of the 
 
 sliding rod, B. Draw the circle described by C, the centre of the 
 pin, and let k A K be the diameter of that circle which is parallel 
 to the direction of motion of ^_^ 
 
 the rod; then K and k will 
 be the dead points, or posi- 
 tions of the centre of the pin 
 at the two instants when the 
 rod has no velocity. To find 
 the velocity-ratio of the rod 
 
 / 
 
 3K 
 
 V 
 
 -7 
 
 \ 
 
 E5t- 
 
 ^^^^^ "~ X 
 1 
 
 
 
 $ 
 
 Fig. 116 
 
 
 
 IK 
 
 and crank-pin when the 
 centre of the crank-pin is in 
 a given position, C : perpen- 
 dicular to the direction of 
 motion of the rod draw the 
 diameter DAE; this line 
 will correspond to the line of centres in the preceding problems; 
 then through C, and perpendicular to the centre-line, F G, of the 
 slot, draw the line of connection, C I, cutting DAE in I; the 
 following will be the required velocity-ratio : 
 
 Telocity of rod, B _ A I 
 Velocity of centre of pin, C AC*
 
 170 GEOMETRY OF MACHINERY. 
 
 The extent of travel of the, pin in the slot isFCf=DE = 
 
 160. Cams and Wipers iu General. Cams and wipers are those 
 primary pieces, with curved acting surfaces, which work in sliding 
 contact without being related to imaginary pitch-surfaces, as the 
 teeth of wheels and threads of screws are. The distinction between 
 a cam and a wiper is, that a cam in most cases is continuous in its 
 action, and a wiper is always intermittent; but a wiper is some- 
 times called a cam notwithstanding. A cam is often like a non- 
 circular sector or wheel in appearance; a wiper is often like a 
 solitary tooth. (As to "rolling cams," see Article 110, page 99.) 
 
 The solutions of all problems respecting the velocity-ratio and 
 directional relation in the action of cams and wipers are obtained 
 by properly applying the general principle of Article 122, page 114. 
 
 In most cases which occur in pi-actice, the condition to be- 
 fulfilled in designing a cam or a wiper does not directly involve the 
 velocity-ratio, but assigns a certain series of definite positions 
 which the follower is to assume when the driver is in a correspond- 
 ing series of definite positions. Examples of such problems will 
 be given in the following Articles. 
 
 161. Cam with Qroovc and Pin. Throughout the present Article 
 it will be supposed that the acting surface of the follower, which is 
 to be driven by the cam, is the cylindrical surface of a pin. It is 
 easy to see that without in any respect altering the action, a 
 cylindrical roller turning about a smaller pin may be substituted 
 for a pin in order to diminish friction. If the pin is to be driven 
 by the cam in one direction only, being made to return at the 
 proper time by the force of gravity or by the elasticity of a spring, 
 the cam may have only one acting edge; but if the pin is to be 
 driven back as well as forward by the cam, the cam must have two 
 acting edges, with the pin between them, so as to form a groove 
 or a slot of a uniform width equal to the diameter of the pin,, 
 with clearance just sufficient to prevent jamming or undue friction. 
 The centre of the pin may be treated as practically coinciding at 
 all times with the centre-line of such a groove, which centre-line 
 may be called the pitch-line of the cam. The most convenient way 
 to design a cam is usually to draw, in the first place, its pitch-line, 
 and then to lay off the half-breadth of the groove on both sides of 
 the pitch-line. When one acting edge only is required, it is to be 
 laid off on one side of a groove, the other side being omitted. 
 
 The line of connection at any instant is a straight line normal to 
 the pitch-line at the centre of the pin. 
 
 The surface in which the groove is made may be either a plane 
 or a surface of revolution ; a plane for a cam-plate which either 
 turns about an axis normal to its own plane or slides in a straight 
 line, and acts upon a pin whose centre moves in a plane parallel to- 
 that of the cam-plate; a solid of revolution, being either a cylinder,
 
 CAMS. 
 
 171 
 
 a cone, or a hyperboloi'd, for a cam whicli turns about an axis, and 
 acts on a pin whose centre has a reciprocating motion in a straight 
 line coinciding with a generating line of the surface of revolution. 
 
 The following example is a case of a rotating plane cam, giving 
 motion through a pin and lever to a rocking shaft whose axis is 
 parallel to the axis of rotation of the cam. 
 
 In fig. 117 the plane of projection is that of the cam-plate, and 
 is normal to the axes of the cam and of the level-. In the lower 
 part of the figure, A' represents the trace of the axis of the rocking
 
 172 GEOMETRY OP MACHINERY. 
 
 shaft, and C' the trace of the axis of the cam, so that A' C' is the 
 line of centres. The direction of rotation of the cam is shown by an 
 arrow. In the example, the direction is left-handed. The circular 
 arc, 6, described about A' with the radius A' 0, is the path to be 
 described by the centre of the pin; and the twelve points in that 
 arc, marked with numbers from to 11, are twelve positions which 
 the centre of the pin is to occupy at the end of twelve equal divisions 
 of a revolution of the cam. It is required to find the form of the 
 cam which will produce that motion in the pin. 
 
 In the upper part of the figure, let C represent the axis of the 
 cam; suppose that the cam is fixed, and that the line of centres, 
 C A, rotates about C, carrying the axis, A, of the rocking shaft along 
 with it, with an angular velocity equal and contrary to the actual 
 angular velocity of the cam. That supposition will not alter the 
 relative motions of the working pieces. With the radius C A 
 describe a circle to represent the supposed path of A relatively to 
 C; divide its circumference into twelve equal parts, and to the 
 points of division draw radii, C A , C A 1? C A,, <fec., to represent 
 twelve successive positions of the line of centres relatively to the 
 cam, as supposed to be fixed. Lay off the angles C A 0, C A l 1, 
 C A 2 2, &c., in the upper part of the figure respectively, equal to 
 the angles C' A' 0, C' A' 1, C' A' 2, &c., in the lower part of the 
 figure; and make each of the straight lines A 0, Aj 1, A 2 2, &c., 
 equal to the lever arm A' 0. The points thus found, 0, 1, 2, &c., 
 will be points in the pitch-line of the cam, and a curve drawn 
 through them will be the required pitch-line. 
 
 About each of the points 0, 1, 2, &c., draw a circle of a radius 
 equal to that of the pin : a pair of curves touching those circles 
 so as to be parallel to the pitch-line will mark the two sides of the 
 groove, without allowance for clearance. Clearance may be pro- 
 vided either by slightly diminishing the diameter of the pin or by 
 slightly increasing the width of the groove. If the lever is to be 
 raised by the cam, but brought down again by gravity, the outer side 
 of the groove may be omitted, and the cam will become a disc bounded 
 by the innermost of the three parallel curves shown in the figure. 
 
 The number of parts into which the revolution of the cam is 
 divided may be made more or less numerous according to the 
 degree of precision required. 
 
 It is easy to see how a similar method may be applied to the 
 designing of a cam-disc which shall produce a given motion in a 
 follower whose acting surface is of any given form. A figure is to 
 be constructed like the upper part of fig. 117, on the supposition 
 that the cam is fixed, and that the frame of the machine rotates 
 about the axis of the cam with an angular velocity equal and 
 contrary to the actual angular velocity of the cam. Then, just as 
 the pin in the upper part of fig. 117 is drawn in its several positions,
 
 CAMS. 
 
 173 
 
 0, 1, 2, &c., the trace of the acting surface of the follower is to be 
 drawn in its several successive positions ; and a line touching that 
 trace in all its positions will be the trace of the required cam-disc. 
 
 The dead points of a cam are the points in its pitch-line whicli 
 are at the greatest and least distances from its axis. In the 
 example shown in fig. 117 the dead points are and 6. "When 
 the centre of the pin is at those points it has no velocity. Any 
 part of the pitch-line which is an arc of a circle about C 
 corresponds to a pause in the motion of the pin. 
 
 162. Drawing a Cam by Circular Arcs. In many cases in which 
 cams have to be designed, the dead points alone are given by the condi- 
 tions of the problem, leaving the parts of the pitch-line between those 
 points to be. drawn according to convenience. .For example, in fig. 1 1 8j
 
 174 GEOMETRY OF MACHINERY. 
 
 C is the axis of the cam, and A and B are dead points; so that 
 C B and C A are respectively the least and greatest radii drawn 
 from the axis to the pitch-line; and the pitch -line at A and B 
 is normal to those radii respectively. The intermediate arcs of 
 the pitch-line are to be drawn of any convenient form, so as to 
 traverse A and B, and be normal to C A and C B. 
 
 The easiest way to draw such curves is by means of arcs of 
 circles. 
 
 The simplest case is when C A and C B are parts of one straight 
 line. The required pitch-line is then an eccentric circle, described 
 upon the straight line A C B as a diameter. 
 
 When C A and C B, as in the figure, are not parts of one 
 straight line, the following method may be used, being an extension 
 of Rule IV. of Article 79, page 61, and having the effect of giving 
 a pitch-line made up of four circular arcs, whose radii deviate less 
 from equality than those of any other combination of four circular 
 arcs which would answer the same purpose. 
 
 From A and B, perpendicular to A C and B C respectively, 
 draw A D and B D, cutting each other in D. These will be 
 tangents to the required pitch-line. Join C D ; bisect it in E ; and 
 about E, with the radius E G = E D, describe a circle which will 
 traverse the four points A, 0, B, D. Bisect the arc A C B in G. 
 About G, with the radius G A = G B, describe a circle ; and draw 
 the straight line D H G I, cutting that circle in H and I. Through 
 the points H and I, and parallel to D C, draw the straight lines 
 H Q and I P, cutting the circle A I B H in P and Q (the ends 
 of one diameter), and cutting also the straight line C B in M and 
 L, and the straight line A C produced in N and K. Then draw 
 four circular arcs, as follows : 
 
 The arc A P, described about the point K, 
 
 P T> T 
 
 >y * - ) )> }) **t 
 
 BQ, M, 
 
 Q A, JN ; 
 
 and those arcs will make up a pitch-line having C B and C A for 
 its greatest and least distances from the axis C, as required; and 
 also having its radii of curvature less unequal than is possible with 
 any other combination of four circular arcs, and no more, fulfilling 
 the required conditions. 
 
 When a cam is to have more than two dead points, each pair of 
 adjacent dead points are to be connected with each other by means 
 of two circular arcs, drawn according to Rule IV. of Article 79, 
 pages 61 and 62, fig. 48. 
 
 163. many-coiled Cams; Spiral and Conoidal Cams. When the 
 
 complete series of movements of a piece that is to be driven by a 
 cam extends over more than one revolution of the cam, there are
 
 CAMS WIPERS AND PALLETS ESCAPEMENTS. 
 
 175 
 
 cases in which the required result may be effected by means of a 
 groove in a cam-plate having a pitch-line of more than one coil ; 
 but difficulties in working may arise from the fact that the coils of 
 the groove must intersect each other. There are other cases in 
 which the motion required in the follower is of a kind that may 
 be produced by means of a spiral cam, 
 such as that shown in fig. 119. The 
 upper part of the figure is a projection 
 on a plane normal to the axis; the 
 lower part, a projection on a plane 
 parallel to the axis. A A' is the 
 spiral cam; B, a screw of an axial 
 pitch exactly equal to the axial pitch 
 of the cam. This screw, resting in a 
 fixed nut, forms one of the bearings 
 of the cam-shaft, and causes the 
 shaft and cam together to advance 
 along the axis at each revolution 
 through a distance equal to the pitch, 
 thus bringing a new coil of the 
 cam into action. The cam, A, may 
 also be made with a continuous con- 
 oidal surface, of which different parts 
 are brought into action at each revolu- 
 tion by the advance caused by the 
 screw B. It is evident that in spiral 
 and conoidal cams the extent of the 
 motion is limited. 
 
 164. Wipers and Pallets Escape- 
 ments. In fig. 120 a shaft rotating 
 about the axis A is provided with one 
 or more solitary teeth called wipers, 
 
 such as E. The action of the wipers upon the projecting parts of 
 the piece that they drive (which, for the sake of a general _ term, 
 may be called pallets) may be either intermittent or reciprocating. 
 
 I. As an example of intermittent action, one of the wipers repre- 
 sented in fig. 120, in moving from the position H to the position 
 E, is supposed to have driven before it a pallet from the position 
 G to the position F. The pallet projects from a vertical sliding 
 bar, or stamper, C. 
 
 B B is the addendum -circle of the wipers, and D D the addendum- 
 line of the pallets. Those lines cut each other at the point of 
 escape, E; and just at that point the pallet escapes from the wiper, 
 and the stamper, with its pallet, falls back to its original position, 
 and is ready to be lifted again by the next wiper. 
 
 The stamper and pallet referred to in this case are shaded. 
 
 Fig. 119.
 
 176 
 
 GEOMETRY OF MACHINERY. 
 
 IL As an example of reciprocating action, the sliding bar, C, of 
 
 the preceding example is supposed to have attached to it a frame, 
 
 d 
 
 Fig. 120. 
 
 e c c, at the opposite side of which is another pallet, g ; and this 
 pallet is so placed that immediately after the escape of the former 
 pallet, F, from the wiper at E, another wiper at h begins to act 
 upon the pallet g, and so to produce the return stroke of the frame, 
 C c c c. The point, e, where the addendum-line, d d, of the pallet 
 g cuts the addendum-circle, B B, of the wipers, is the point of 
 escape of the second pallet (whose position at the instant of escape 
 is marked/); and immediately afterwards a third wiper, arriving 
 at the position H, begins to produce a new forward stroke. 
 
 The length of stroke is represented in the figure by F G = f g* 
 It is evident that the number of wipers must be odd.
 
 ESCAPEMENTS. 
 
 177 
 
 This is the combination already referred to in Article 142, page 
 141. It belongs to a class of contrivances called escapements, 
 because of the escape of the follower from the action of the driver 
 at certain instants. There are many escapements which do not 
 belong to the subject of pure mechanism ; and amongst these are 
 found most of the escapements that are used in clocks and watches, 
 a-s being well suited to the regulation of those machines ; for in such 
 escapements the driver and follower are disconnected from each 
 other during the greater part of the movement. Only two more 
 escapements, therefore, will be described here. 
 
 III. Anchor Recoil Escapement. This escapement, though, not 
 well suited to the exact keeping of time, is used in old clockwork. 
 It is also used in vertical roasting jacks. The driver is a wheel 
 called the scape wheel, and the trace of its axis is represented by 
 the point A, fig. 121. E I F is its pitch-circle, cutting the line 
 
 v; 
 
 Fig. 121. 
 
 of centres, A B, in I. Y V is its addendum-circle. In the figure 
 the teeth are represented as cylindrical pins; in any case their 
 acting surfaces may be regarded as parts of cylinders, which, if the 
 teeth are sharp-pointed, are of insensible diameter. The arrow 
 near I shows the direction of rotation of the wheel. The point B 
 is the trace of the axis of the verge, or rocking shaft, to which a 
 reciprocating movement is to be given through the alternate action 
 of the teeth on the pallets, E, S and T U, which are the acting 
 
 H
 
 178 
 
 GEOMETRY OP MACHINERY. 
 
 surfaces of the crutch, S R T U. At the instant represented in 
 the figure, the crutch is at the middle of its swing, and in the act 
 of moving towards the left, through the action of the tooth E on 
 the pallet E, S. The swing of the crutch takes place while the 
 wheel moves through half the pitch ; at the end of which interval 
 the tooth E and pallet R S escape from each other, and another 
 tooth begins to act on the pallet T U, so as to make the crutch 
 swing towards the right, and so on alternately. The dotted circle 
 at F represents a tooth in the act of driving the pallet T U, at the 
 middle of the swing, towards the right. 
 
 To design the figures of the pallets, a method is to be employed 
 analogous to that described in Article 161, page 172; that is to say, 
 the crutch is to be supposed fixed, and the line of centres, B A, is 
 to be supposed to swing to and fro about the axis B, carrying 
 with it the axis A, through an angle equal to the angle through 
 which the crutch is actually to swing. 
 
 Lay off the angles A B = A B D =: x the semi- amplitude, or 
 half angle of swing; and make BC = BD = BA. Then C and 
 D are the two extreme positions of the axis A in its supposed 
 swinging motion. With a radius equal to that of the pitch-circle, 
 draw the arcs M N about C, and P Q about D; and with a 
 radius equal to that of the addendum-circle, draw the arcs m n 
 about C, and p q about D. From the point I lay off upon the 
 pitch-circle the arcs I E = I F = an odd number of times the 
 quarter-pitch; so that E I F shall be an odd number of half 
 pitches. The points E and F should be as near as practicable to 
 the points where two straight lines from B touch the pitch-circle. 
 About E and F draw circles to represent the traces of the acting 
 surfaces of the pins or teeth. Lay off, on the pitch-circle, the arcs 
 E G = F K = the quarter-pitch, with the radius of the acting 
 surface of a tooth deducted : this deduction is to ensure that between 
 the escape of a tooth from one pallet and the commencement of the 
 action of another tooth on the opposite pallet there shall be an 
 interval sufficient to enable the tooth that has just escaped to move 
 clear of the pallet which it has quitted. 
 
 About the centre B, through the point G, draw the circular arc 
 M G N, cutting the arc M N, ah-eady described about C, in 
 the points M and N. About the centre B, through the point 
 K, draw the circular arc P K Q, cutting the arc P Q, already 
 described about D, in the points P and Q. Through M, E, and 
 Q draw a continuous curve; this will be the pitch-line of the 
 pallet R S. Through N, F and P draw a continuous curve: 
 this will be the pitch-line of the pallet T U. Then, parallel to 
 those pitch-lines respectively, and at a distance from them equal 
 to the radius of the acting surface of a tooth, draw the traces, R S 
 and T U, of the acting surfaces of the pallets.
 
 ESCAPEMENTS CONNECTION BY BANDS. 179 
 
 The points of the pallets, at S and U, are to be cut off, so as not 
 to project within the circles q p and n m respectively. The traces 
 of the backs of the pallets, S W and U X, are to be circular arcs 
 described about B. 
 
 IV. Dead-beat Escapement. In the dead-beat escapement the 
 crutch swings each way through an arc of indefinite extent, in 
 addition to that through which it is driven by the action of the 
 teeth of the scape wheel ; and the scape wheel is made to pause in 
 its motion during each such additional swing, by its teeth bearing 
 against parts of the pallets whose surfaces are cylinders described 
 about the axis of the verge. The traces of these may be called 
 the dead arcs of the pallets. The recoil escapement shown in fig. 
 121, may be converted into a dead-beat escapement, as follows: 
 About B, with a radius equal to B M added to the radius of the 
 acting surface of a tooth, draw the circular arc B, Y; and also 
 about B, with a radius equal to B P, deducting the radius of the 
 acting surface of a tooth, draw the circular arc T Z : those two 
 arcs will be the required dead arcs of the pallets. 
 
 In order that a dead-beat escapement may go on working, there 
 must be a force, such as gravity or the elasticity of a spring, 
 continually tending to bring the crutch to its middle position, at 
 and near which the pallets are driven by the teeth ; hence its 
 principles are to a certain extent beyond the province of pure 
 mechanism. 
 
 In the dead-beat escapements of accurate clocks, the angle 
 through which the crutch swings is very small, and the angle 
 through which the teeth act on the pallets is still smaller; so that 
 in fig. 121 those angles may be looked upon as greatly exaggerated, 
 for the sake of distinctly showing the geometrical principles of 
 the combination. 
 
 SECTION V. Connection l>y Bands. 
 
 165. Bands and Pulleys Classed. (A. M., 478.) The word 
 lands may be used as a general term to denote all kinds of 
 flexible connecting pieces; and the word pulleys, when not other- 
 wise qualified, to denote all kinds of rotating pieces which are 
 connected with each other by means of bands. Bands may be 
 classed in the following manner ; which also involves a classification 
 of the pulleys to which the bands are suited : 
 
 I. Belts, which are made of leather, gutta percha, woven fabrics, 
 &c., are flat and thin, and require nearly cylindrical pulleys with 
 smooth surfaces. 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, and are in general without
 
 180 GEOMETRY OF MACHINERY. 
 
 ledges. A belt when in motion is shifted off a pulley, or from one 
 pulley on to another of equal size alongside of it, by pressing against 
 the "advancing side" of the belt; that is, that part of the belt 
 which is moving towards the pulley. Amongst belts may be 
 classed flat ropes. 
 
 II. Cords, made of catgut, leather, hempen or other fibres, or 
 wire, are nearly cylindrical in section, and require either drums 
 with ledges, or grooved pulleys. 
 
 III. Chains, which are composed of links or bars jointed together, 
 require wheels or drums, grooved, notched, and toothed, so as to 
 fit the links of the chains. Chains suited for this purpose are 
 called gearing chains. 
 
 Bands for communicating motion of indefinite extent are endless. 
 
 Bands for communicating reciprocating motion have usually their 
 ends made fast to the pulleys or drums w^ich they connect, and 
 which, when the extent of motion is less than a revolution, may be 
 sectors. 
 
 166. Principles of Connection by Bands. The line of connection 
 
 of a pair of pulleys connected by means of a band is the central 
 line or axis of that part of the band whose tension transmits the 
 motion. 
 
 The jjitch-surface of a pulley over which a band passes is the 
 surface to which the line of connection is always a tangent ; that is 
 to say, an imaginary surface whose distance from all parts of the 
 acting surface of the pulley that the band touches is equal to the 
 distance from the acting surface of the band to its centre line. 
 The pitch-surface of a pulley cannot be anywhere concave ; for 
 where the acting surface is concave, the band stretches in a 
 straight line across the hollow, and the pitch-surface is plane. 
 In ordinary pulleys for communicating a constant velocity-ratio 
 the pitch-surface is a circular cylinder; and its radius (called the 
 effective radius) is equal to the real radius of the pulley added to 
 half the thickness of the band. 
 
 The pitch-line of a pulley is the line on its pitch-surface in which 
 the centre-line lies of that part of the band which touches the 
 pulley. The line of connection is a tangent to the pitch-line. 
 When the line of connection is in a plane perpendicular to the axis 
 of the pulley, the pitch-line is the trace of the pitch-surface on 
 that plane: for example, the circular section of a cylindrical 
 pulley. When the line of connection is oblique to the axis, the 
 pitch-line is helical, or screw-like. 
 
 Problems respecting the comparative motion of pieces connected 
 by bands are solved by applying the principles of Article 91, page 
 78, taking A B in fig. 58 of that Article to represent the centre 
 line of that part of the band whose tension transmits the motion, 
 and A A' and B B' to represent the common perpendiculars from
 
 CONNECTION BY BANDS. 181 
 
 that line to the axes of the pulleys. When the pitch-surfaces of 
 the pulleys are circular cylinders, A A' and B B' represent their 
 effective radii. Rule II. of Article 91 shows how to find the 
 angular velocity-ratio of two pulleys whose proportionate dimen- 
 sions are given. The following is the converse rule for finding the 
 proportionate radii of two pulleys which are to transmit a given 
 angular velocity-ratio. In tig. 58, page 78, draw A a to represent 
 the projection of the axis of one pulley upon a plane parallel to 
 that axis traversing the line of connection, A B ; and draw B b to 
 represent a similar projection of the axis of the other pulley. Lay 
 off the distances A a and B & to opposite sides of A B, to represent 
 the intended angular velocities of the two pulleys. Draw A c and 
 B d perpendicular to A B ; and draw a c and b d parallel to A B, 
 cutting A c and B c? in c and d respectively. Then the lengths 
 A c and B d will represent the component angular velocities of the 
 pulleys about axes perpendicular to the line of connection, A B. 
 (In most cases which occur in practice, both the axes lie in planes 
 perpendicular to the line of connection; and then A o& and B6 
 coincide with A c and B d respectively.) 
 
 Draw the straight line c d, cutting the line of connection, A B, in 
 K. Then we have the proportion 
 
 BK : AK 
 
 : : effective radius of A : effective radius of B ; 
 
 and if one of those radii for example, that of A is given, the 
 other is found as follows : From A lay off A I = B K (or other- 
 wise, from B lay off B I = A K). Perpendicular to A B draw A A' 
 and B B' ; lay off A A' = the given radius of the pulley A, and draw 
 the straight line A I B', cutting B B' in B'; B B' will be the 
 required radius of B. 
 
 In the ordinary case, in which both axes lie in planes perpen- 
 dicular to the line of connection, it is evident that the velocities of 
 a pair of circular pulleys are inversely as their effective radii. 
 
 It is to be borne in mind that, especially as regards cases in which, 
 the 'axes do not both lie in planes perpendicular to the line of 
 connection, everything stated in the present Article is based on 
 the supposition that the band is perfectly flexible in all directions, 
 In the case of flat belts connecting pulleys whose axes are not both 
 in planes perpendicular to the line of connection, there are certain, 
 effects of the lateral stiffness of the belt which will be considered 
 farther on. 
 
 The velocity of the land is equal to that of a point revolving at 
 the end of the radius A A, fig. 58, page 78, with the angular 
 velocity represented by A c, and also to that of a point revolving 
 at the end of the radius B B', with the angular velocity represented
 
 182 GEOMETRY OF MACHINERY. 
 
 by B d. When a band connects a pulley with a sliding piece, the 
 comparative motion is given by Rule III. of Article 91, page 79. 
 
 Smooth bands, such as belts and cords, are not suited to com- 
 municate a velocity-ratio with precision, as teeth are, because of 
 their being free to slip on the pulleys; but the freedom to slip is 
 advantageous in swift and powerful machinery, because of its 
 preventing the shocks which take place when mechanism which is 
 at rest is suddenly thrown into gear, or put in connection with the 
 prime mover. A band at a certain tension is not capable of 
 exerting more than a certain definite force upon a pulley over 
 which it passes; and therefore occupies, in communicating its own 
 speed to the rim of that pulley, a certain definite time, depending 
 on the masses that are set in motion along with the pulley and the 
 speed to be impressed upon them; and until that time has elapsed 
 the band has a slip ping motion on the pulley; thus avoiding shocks, 
 which consist in the too rapid communication of changes of speed. 
 This will be further considered under the head of the Dynamics of 
 Machines. 
 
 167. Pulleys with Equal Angular Velocities When a pair of 
 
 pulleys turn about parallel axes in the same direction, with equal 
 angular velocities, their pitch-lines may be of any figure whatsoever, 
 curved or polygonal, provided they are equal and similar, and not 
 concave. Each of the two straight parts of the band is equal and 
 parallel to the line of centres; and those parts, if the pulleys are 
 circular and not eccentric, remain at a constant distance from the 
 line of centres; but have a reciprocating motion towards and from 
 that line if the pulleys are either eccentric or non-circular. A reel 
 is virtually a pulley whose pitch-line is a polygon with rounded 
 angles; and such is also the case with the expanding pulley, con- 
 sisting of four quadrants of a circle, which can be separated to a 
 greater or less distance from each other by means of screws. 
 
 168. Bands and Pulleys for n Constant Telocity- Ratio. In Order 
 
 that the velocity-ratio of a pair of pulleys may be constant, their 
 pitch-lines must be circular (except in the particular case specified 
 in the preceding Article, when the figure is not restricted to the 
 circle alone). 
 
 The band may be open or uncrossed, as in fig. 122; or it may be 
 crossed, as in fig. 123. With an open band the directions of rotation 
 
 Fig. 122. Fig. 123.
 
 BANDS AND PULLEYS. 183 
 
 are the same; with, a crossed band, contrary. In each of these 
 figures, 1 denotes the driving pulley, and 2 the following pulley; 
 Cj C 2 is the line of centres, and T x T 2 the line of connection; and 
 the angular velocity-ratio is expressed by 
 
 169. The Length of an landless Band, such as those shown in 
 figs. 122 and 123, consists of two straight parts, each equal to the 
 line of connection, and two circular arcs. When the band is 
 crossed, as in fig. 123, the circular arcs are of equal angular extent; 
 when the band is open, as in fig. 122, the angles subtended by the 
 two arcs make up one revolution. When the length of a band is 
 to be measured on a drawing, the circular parts may be rectified 
 graphically by Rule I. or Rule II. of Article 51, page 28. 
 
 To find the length of an endless band by calculation, let the line 
 of centres, Cj C 2 = c, and the effective radii of the pulleys, C x T 1 =r 1 ; 
 C 2 T 2 = r 2 ; r being the greater. Then each of the two equal 
 straight parts of the band is evidently of the length 
 
 Tj T 2 = /s/ c 2 (i\ + r 2 ) 2 for a crossed band; } 
 T 1 T 2 = J c 2 (r x r 2 ) 2 for an open band. J 
 
 Let ij be the arc to radius unity of the greater pulley, and i z that 
 of the less pulley, with which the band is in contact; then for a 
 crossed band 
 
 o . r, + 
 
 = i = K + 2 arc sin 
 
 and for an open band 
 
 (2.) 
 
 and the addition of the lengths of the straight and curved parts 
 gives the following total length : 
 For a crossed band, 
 
 L = 2 \/c 2 (r, + r 9 ) 2 + (r. + r 9 ) [ * + 2 arc sin -i 
 
 \ c 
 
 and for an open band, 
 
 * A* 
 
 L = 2 \/c 2 (r. r )^ + v(r+ + r 9 ) -f 2(r, r 9 ) arc sin * 
 
 \ 1 ,/ \ X Z/ \ X ^/ 
 
 (3.) 
 
 As the last of these equations would be troublesome to use 
 in a practical application to be mentioned in Article 171, an.
 
 184 
 
 GEOMETRY OF MACHINERY. 
 
 approximation to it, sufficiently close for practical purposes, is 
 obtained by considering, that if r 1 r 2 is small compared with c, 
 
 ^_ M \2 n* _ ff f __ nr 
 
 nearl, and arc sin -- - = - 1 
 
 C 
 
 _ ^_ M n* _ ff f 
 
 c 2 - (r-. r 9 ) 2 = c *-*= nearly, and arc sin -- - = - 1 
 
 2i G C 
 
 nearly; whence, for an open band, 
 
 L nearly = 2 c + * (r x + r 2 ) + &&-, ..... (3 A.) 
 
 C 
 
 in which it is sufficiently accurate for practical purposes to make 
 *=3f 
 
 170. Pulleys with Flat Belt*. It has already been stated in Article 
 165, page 179, that a flat belt tends to move towards that part of 
 the pulley whose radius is greatest, or to "climb," as the phrase is; 
 and that pulleys for such belts are therefore made without ledges, 
 and with a slight swell or convexity at the middle of the rim, in 
 order that the belt may tend to remain there. The swell usually 
 allowed in the rims of pulleys is one twenty-fourth part of the. 
 breadth. 
 
 The tendency to climb is produced by the lateral stiffness of 
 the belt, in the following manner: When the part of the belt 
 which touches the pulley deviates towards one side, as in fig. 124, 
 the part which is approaching the pulley 
 is made to deviate towards the opposite 
 side; and thus, after the pulley has 
 turned through a small angle, the devia- 
 tion of the belt is corrected. 
 
 A crossed belt is twisted half round 
 in passing from one pulley to another, 
 as shown in fig. 123, so as to bring the 
 same side of the belt into contact with 
 both pulleys. The principal object of 
 this is, that the two straight parts of the 
 belt may pass each other flatwise where 
 they cross, so as not to resist each other's 
 motion. Another object, in the case of 
 leather belts, is to bring the rougher side 
 of the leather into contact with both 
 pulleys. 
 
 It has already been stated that the 
 position which a belt assumes upon a pulley is determined by the 
 position of its advancing side; that is, of the part of the belt 
 which is approaching the pulley. In the contrivance called the 
 "fast and loose pulley" for engaging and disengaging machinery, 
 a belt driven by a suitable driving pulley is provided with two 
 similar and equal following pulleys, mounted side by side upon 
 
 Fig. 124.
 
 BELTS AND PULLEYS SPEED CONES. 
 
 185 
 
 one axis; one of these pulleys is made fast to the shaft; the other 
 turns loosely upon it. The belt, when in motion, can be shifted by 
 means of a fork, that guides its advancing side to the fast pulley 
 or to the .loose pulley at will, so as to engage or disengage the 
 shaft on which those pulleys are fitted. The driving pulley is 
 made of a breadth equal to the breadths of the fast and loose 
 pulleys together. 
 
 The lateral stiffness of a belt is also made available for the 
 pui'pose of keeping it in its place on the pulleys when their axes 
 are not parallel, as in fig. 125, which is sketched in isometrical 
 perspective. C l C x and C 2 C 2 are the axes; E x E 2 , their common 
 
 C2 
 
 Fig. 125. 
 
 perpendicular. In order that the belt may remain on the pulleys, 
 the central plane of each pulley must pass through tlie point of 
 delivery of the other pulley that is, the point where the belt 
 leaves the other pulley ; or, in other words, the centred planes of the 
 two pulleys sliould intersect in the straight line which connects the 
 two points of delivery. In fig. 125, Dj arid D<, are the two points of 
 delivery; and the pulleys are so placed that D i D 2 is the line of 
 intersection of their central planes. It is easy to see that this 
 arrangement does not admit of the motion being reversed; for 
 when that takes place, D l and D 2 cease to be the points of delivery, 
 and become the points where the belt is received; and it is at once 
 thrown off the pulleys. 
 
 171. Speed Cones (A. M., 483) are a contrivance for varying 
 and adjusting the velocity-ratio communicated between a pair of 
 parallel shafts by means of a belt, and may be either continuous 
 cones or conoids, as in fig. 126, A, B, whose velocity-ratio can be
 
 186 
 
 GEOMETRY OF MACHINERY. 
 
 varied gradually while they are in motion by shifting the belt; or 
 sets of pulleys whose radii vary by steps, as in fig. 126, C, D in 
 which case the velocity-ratio can be changed by shifting the belt 
 from one pair of pulleys to another while the machine is at rest. 
 
 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, pages 183, 184, must be constant. 
 
 For a crossed belt, as at A and C, L depends solely on the line of 
 centres, c, and on the sum of the radii, r^ + r z . 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 
 pairof continuous cones, generated 
 by the revolution of two straight 
 lines inclined opposite ways to 
 their respective axes at equal 
 angles, and by a set of pairs of 
 pulleys in which the sum of the 
 radii is the same for each pair. 
 
 For an open belt the following 
 practical rule is deduced from the 
 approximate equation (3 A.) of 
 Article 169, page 184 : 
 
 Let the speed-cones be equal 
 and similar conoids, as in fig. 126, 
 B, but with their large and small 
 ends turned opposite ways. Let 
 
 Fig. 126. 
 
 r x be the radius of the large end of each, r z that of the small end, 
 r that of the middle ; and let y be the swell or convexity, measured 
 perpendicular to the axis, of the arc by whose revolution each of 
 the conoids is generated; then 
 
 y = ( \~J?> ( L ) 
 
 and 
 
 'i 
 
 + y;- 
 
 .(2.) 
 
 x = 3^ nearly enough for the present purpose. 
 
 To find the swell, y, by graphic construction : in fig. 126 E, draw 
 
 A -Q A B = 3^ times the line of 
 
 centres; from B, perpendicular 
 to A B, draw B C = the difference 
 between the greatest and least 
 Fig. 126 E. radii; join A C, and cut off from 
 
 it A D = A B; D C will be the 
 required swell
 
 SPEED CONES PULLEYS FOR CORDS. 187 
 
 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. 126 D, let a series of 
 differences of the radii, or values of r l r 2 , be assumed; then, for 
 each pair of pulleys, the half-sum of the radii is to be computed 
 from the difference by the formula 
 
 (3.) 
 
 r being the value of that half-sum when the radii are equal; 
 and finally, the radii are to be computed from their half-sum and 
 half-difference, as follows : 
 
 .(4.) 
 
 172. Pulleys for Ropes and Cords require ledges to prevent the 
 band from slipping off; for even fiat ropes have not sufficient 
 lateral stiffness to make them remain, of themselves, on the convexity 
 of a pulley. A cord, in passing round a pulley, lies in a groove, 
 sometimes called the gorge of the pulley ; if the object of the pulley 
 is merely to support, guide, or strain the cord, the gorge may be 
 considerably wider than the cord ; if the pulley is to drive or to be 
 driven by the cord, so as to transmit motive power, the gorge must 
 in general fit the cord closely, or even be of a triangular shape, so 
 as to hold it tight. Sometimes the gorge of a pulley which is to be 
 driven by a cord at a low speed has radial ribs on its sides, in order 
 to give it a firmer hold of the cord. 
 
 The groove of a pulley for a wire rope should not grasp it tightly, 
 lest the rope be injured; and the motion must be communicated 
 by means of the ordinary friction alone. M. C. F. Hirn has 
 introduced, with good success, the practice of filling the bottoms of 
 the grooves of iron pulleys for wire ropes moving at a high speed 
 with gutta percha, jammed in tight. This will be again referred to 
 in treating of the dynamics of machinery, and of its construction. 
 
 When a cord does not merely pass over a pulley, but is made 
 fast to it at one end, and wound upon it, the pulley usually 
 becomes what is called a drum or a barrel. A drum for a round 
 rope is cylindrical, and the rope is wound upon it in helical coils. 
 Each layer of coils increases the effective radius of the drum by an 
 amount equal to the diameter of the rope. A drum for a flat rope 
 is of a breadth simply equal to the breadth of the rope, which is 
 wound upon it in single coils, each of which increases the effective
 
 188 GEOMETRY OF MACHINERY. 
 
 radius by an amount equal to the thickness of the rope; and 
 instead of ledges it often has pairs of arms, forming as it were 
 skeleton ledges. 
 
 173. Guide Pulleys. A guide pulley merely changes the direction 
 of a band on the way from the pulley which drives the band to the 
 pulley which is driven by it. Guide pulleys ai'e useful chiefly 
 to change the direction of a round cord which communicates 
 motion between two other pulleys whose pitch-circles are not in 
 the same plane. In a case of that kind the following is the rule 
 for finding a proper position for a guide pulley : By the Rule of 
 Article 27, page 10, find the line of intersection of the planes of 
 the pitch-circles of the driving and following pulley respectively. 
 From any convenient point in that line draw tangents to the 
 proper sides of the two pitch-circles, to represent the centre-lines 
 of two straight parts of the band; then, by the rule of Article 22, 
 page 8, draw the rabatment of the angle wiiich these straight lines 
 make with each other. Let A C Bin fig. 127 represent that 
 rabatted angle; draw a straight line, C D, 
 bisecting it ; and about any convenient point, 
 D, in that straight line describe a circle 
 touching the two straight lines, C A, C B: 
 this will be the pitch-circle of a suitable 
 guide pulley. 
 
 174. Straining Pulley*. A straining pulley 
 is used to bring a band to the degree of tension 
 which is necessary in order to enable it to 
 transmit motion from a driving pulley to a 
 following pulley. A straining pulley, as ap- 
 Fig. 127. plied to a flat belt, is usually pressed, by means 
 
 of a lever, against one of the parts of the belt 
 
 which extends between the driving and following pulleys, so as to 
 push that part of the belt towards the line of centres. The effect 
 of this is to tighten the belt and increase the friction exerted 
 between it and the pulleys which it connects. This is one of the 
 contrivances used for engaging and disengaging machinery. The 
 straining or tightening pulley is usually applied to the returning 
 part of the belt ; that is, the part which moves from the driving 
 pulley towards the following pulley. 
 
 Sometimes a straining pulley hangs in a loop or bight of a cord, 
 and is loaded with a weight, as in fig. 128, farther on. 
 
 175. Eccentric and Non-Circular Pulleys are used for transmitting 
 
 a varying velocity-ratio. For example, in fig. 128 the pitch-line 
 of the pulley A is an eccentric circle, and might be a curve of any 
 figure presenting no concavity ; the pitch-line of B is circular and 
 centred on its axis in the figure; but it, too, might be eccentric or 
 non-circular. D E is the line of connection, being the centre-line
 
 ECCENTRIC AND NON-CIRCULAR PULLEYS. 
 
 189 
 
 of the driving part of the cord, and a tangent to both pitch-lines; 
 
 and the cord is kept tight by a loaded straining pulley at C. The 
 
 angular velocities of the pulleys 
 
 A and B at any given instant are 
 
 inversely as the perpendicular 
 
 distances A D and B E of their 
 
 axes from the line of connection; 
 
 or in symbols, let a and b be those 
 
 angular velocities ; then 
 
 6 _ A_D 
 
 a ~ BE' 
 
 There is one instance in which 
 no straining pulley is required; 
 and that is when the pitch-lines 
 of the driving pulley and of the 
 following pulley are a pair of 
 equal and similar ellipses, centred 
 on two of their foci, A, A', as Fig. 128. 
 
 shown in fig. 129, and connected 
 
 by means of a crossed cord. The mean angular velocities are equal 
 and opposite, each entire revolution being performed in the same 
 
 time by both pulleys; and the velocity- ratios at different instants 
 are the same as in a combination of a pair of elliptic wheels having 
 the same foci and the same line of centres. In the figure, E I E 
 and E' I K represent the pitch-lines of such a pair of elliptic 
 wheels : the pitch point being always at the intersection, I, of the 
 two straight parts of the cord. 
 
 To design such a pair of elliptic pulleys, the data required are 
 the line of centres, A A', and the angle by which each pulley is 
 alternately to overtake and to fall behind the other pulley. Then,
 
 190 GEOMETRY OF MACHINERY. 
 
 by Rule I. of Article 108, page 95, find the foci; and about those 
 foci draw any ellipse that is not larger than the ellipse suited, 
 according to the same rule, for the pitch-line of a wheel to work in 
 rolling contact; the ellipse so drawn will be suitable for the pitch- 
 lines of both pulleys, C D and C' D'. The pulleys, like the wheels 
 described in Article 108, will rotate in the same manner as if 
 the revolving foci were connected with each other by a straight 
 link, B B', equal to the line of centres, A A'; and their corre- 
 sponding positions and velocity-ratio at any given instant may be 
 found by Rules II. and III. of Article 108, pages 96, 97. 
 
 Amongst non-circular pulleys are fusees, used in watch- work ; 
 in which the pitch-line is a spiral described on a conoidal 
 surface. 
 
 Non-circular pulleys may be indefinitely varied in figure with- 
 out difficulty; for the possibility of keeping the band tight by 
 means of a straining pulley removes the necessity of preserving 
 certain relations between the pitch-lines, as in rolling contact. 
 
 176. Chain Pulleys and Gearing Chains. A chain pulley in some 
 
 cases is merely a circular grooved pulley for guiding a chain : or a 
 cylindrical barrel on which a chain is wound, being made fast at 
 one end to the barrel, as in cranes ; and those need no special 
 description. But when a chain is to drive or to be driven by a 
 pulley to which it is not made fast, the acting surface of the pulley 
 must be adapted to the figure of the chain, so as to insure a 
 sufficient hold between them. Amongst chain pulleys of this kind 
 are included capstans and windlasses. 
 
 The pitch-line of a true chain pulley is a polygon, as exemplified 
 in figs. 130 and 131, in each of which figures the angles of the 
 pitch polygon are marked by black spots, and its sides by dotted 
 lines. Each side of the pitch polygon is equal to what may be 
 called the pitch, or effective length, of a link of the chain. When 
 the links consist of flat bars of equal length, connected by means 
 of cylindrical pins, as in fig. 130, the pitch of each link is the same, 
 being the distance between the centres of two pins; and the pitch- 
 line accordingly is an equilateral polygon (in the figure a regular 
 hexagon). When the chain consists of oval links, like those of a 
 chain-cable, as in fig. 131, the pitch of a link which lies flatwise on 
 the rim of the pulley is equal to its longer internal diameter plus 
 the diameter of the iron, and the pitch of a link which stands 
 edgewise on the rim of the pulley is equal to its longer interval 
 diameter minus the diameter of the iron; so that the pitch polygon 
 has long and short sides alternately (in the figure there are 
 twelve sides six long and six short; and the length of a long side 
 is to that of a short side as 5 to 3). In fig. 130 the pulley is 
 simply a polygonal prism; in fig. 131 it has hollows to fit those 
 links which stand edgewise.
 
 GEARING CHAINS AND PULLETS SUSPENDED PULLEYS. 191 
 
 Each of the pulleys shown in these figures has teeth; and the 
 traces of the acting surfaces of the teeth are circular arcs, described 
 about the adjacent angles of the pitch polygons. In fig. 130 the 
 
 Fig. 130. 
 
 Fig. 131. 
 
 chain consists of double and single links alternately; and the sides 
 of the pulley are provided alternately with single teeth and with 
 pairs a single tooth where each double link lies, and a pair of 
 teeth for each single link to lie between. Sometimes the pulley is 
 provided with single teeth only one in the middle of each side on 
 which a double link lies. Chains of the shape shown in fig. 130 
 are made with various numbers of parallel and similar bars in each 
 link, according to the strength required. Of course, the number of 
 bars in a link is even and odd alternately. Such chains are also 
 sometimes made with links of leather, connected together by brass 
 pins, and are used to communicate motion between cylindrical 
 drums. The object of this is to have greater flexibility than is 
 possessed by a flat leather belt. In fig. 131 each short side only 
 of the polygon is provided with a pair of teeth, which receive a link 
 standing edgewise between them, and press against the end of a 
 link that lies flatwise. 
 
 Sometimes a chain pulley consists of a number of radiating 
 forks, forming as it were a reel; this is called a sprocket-wheel. 
 Sometimes it has a triangular gorge, with radiating ribs on the 
 inner sxirface of each of the ledges. 
 
 177. Suspended Pulley*. When rotation is transmitted, by means 
 of two pairs of pulleys connected by cords, from one shaft through
 
 GEOMETRY OF MACHINERY. 
 
 C 
 
 an intermediate shaft to a third shaft, having its axis in one 
 straight line with the first shaft, the waste of work in over- 
 coming friction may be diminished by supporting the intermediate 
 shaft without bearings : its weight being simply hung by means of 
 the cords from the pulleys on the other two shafts ; and care being 
 taken to load the intermediate shaft so as to produce the tension 
 on the cords which is required in order to transmit the motion. 
 
 "What that tension ought to be 
 is a question belonging to the 
 dynamics of machinery. This 
 contrivance appears to have 
 - A been first introduced by Sir 
 William Thomson. In fig. 132 
 A is the first and C the third 
 shaft, and B is the intermediate 
 shaft, suspended by means of 
 the cords that pass round its 
 pulleys; D, D are heavy round 
 discs, of the weights required in 
 order to give sufficient tension 
 Fig. 132. to the cords. The shaft B, 
 
 and all the pieces which it 
 carries, should be very accurately balanced. 
 
 SECTION VI. Connection by Linkwork. 
 
 178. Definitions. (A. M., 484.) The pieces which are connected 
 by linkwork, if they rotate or oscillate, are visually called cranks, 
 beams, and levers. The link by which they are connected is a 
 rigid bar, which may be straight or of any other figure : the straight 
 figure, being the most favourable to strength, is used when there 
 is no special reason to the contrary. The link is known by 
 various names under various circumstances, such as coupling-rod, 
 connecting-rod, crank-rod, eccentric-rod, &c. It is attached to the 
 pieces which it connects by two pins, about which it is free to 
 turn. The effect of the link is to maintain the distance between 
 the centres of those pins 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 perpen- 
 dicular let fall from its connected point upon its axis of rotation is 
 the arm or crank-arm. If the motions of the pieces are performed 
 parallel to one plane, or about one central point, the pins are almost 
 always cylindrical, with their axes perpendicular to the plane, 
 or traversing the point, as the case may be. In all other cases the 
 acting surfaces of the pins must be portions of spheres described 
 about the connected points making what are called ball-and'
 
 LINKWORK DEAD POINTS. 193 
 
 socket joints; unless universal joints are used, which will be 
 described further on. 
 
 179. Principles of Connection. (A. AT. 485.) All questions as 
 to the comparative motions of a pair of pieces connected by a link 
 may be solved by means of the general principles and rules given 
 in Article 91, pages 78 to 80, and illustrated by figs. 57 and 58. 
 The axes of rotation of a pair of turning pieces connected by a 
 link are almost always parallel to each other, and perpendicular to 
 the line of connection; in which case the angular velocity-ratio at 
 any instant is the reciprocal of the ratio of the common per- 
 pendiculars let fall from the line of connection upon the axes of 
 rotation. 
 
 Another method of treating questions of linkwork is to find, by 
 the principles of Article 69, pages 46 to 50, the instantaneous 
 axis of the link; for the two connected points move in the same 
 manner with two points in the link, considered as a rigid 
 body. 
 
 If a connected point belongs to a turning piece, the direction of 
 its motion at a given instant is perpendicular to the plane contain- 
 ing the axis and crank-arm of the piece. If a connected point 
 belongs to a shifting piece, the direction of its motion at any 
 instant is given, and a plane can be drawn perpendicular to that 
 direction. 
 
 The line of intersection of the planes perpendicular to the paths 
 of the two connected points at a given instant is the instantaneous 
 axis of the link at that instant; and the velocities of the connected 
 points are directly as their distances from that axis. 
 
 In drawing on a plane surface, the two planes perpendicular to 
 the paths of the connected points are represented by two lines 
 (being their traces on a plane normal to them), and the instanta- 
 neous axis by a point ; and should the length of the two lines 
 render it impracticable to produce them until they actually inter- 
 sect, the velocity-ratio of the connected points may be found by 
 the principle, that it is equal to the ratio of the segments which a 
 line parallel to the line of connection cuts off from any two lines 
 drawn from a given point perpendicular respectively to the 
 paths of the connected points. Examples will be given further 
 on. 
 
 180. Dead Pointa. (A. M,, 486.) If at any instant the plane 
 traversing one of the crank-arms and its axis of rotation 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 indeter- 
 minate. 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 that instant is null; unless it 
 
 o
 
 194 
 
 GEOMETRY OF MACHINERY. 
 
 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. 
 
 181. Conpied Parallel shafts. (A . M., 487.) There are only two 
 cases in which an. uniform angular velocity-ratio (being that of 
 equality) is communicated by linkwork. One of them 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 connec- 
 tion 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. (See fig. 32, page 44.) 
 
 This elementary combination belongs to Willis's Class A. 
 
 182. Drag-Link. The term drag-Link is applied to a link, as 
 
 C D, fig. 133, which connects 
 together two cranks, A C and 
 B D, so as to make them 
 perform a complete revolu- 
 tion in the same time and 
 in the same direction. The 
 cranks may be equal or un- 
 equal. If the two axes (whose 
 traces in the figure are A and 
 B) are parts of one straight 
 line (that is, if the points A 
 
 and B coincide), the angular velocities of the cranks are equal at 
 
 every instant, and the combination belongs to Willis's Class A; 
 
 and such is the action of the drag-link when used as a coupling. 
 
 If the axes are not parts of one straight line (so that A and B are 
 
 different points), the velocity-ratio varies, and the combination 
 
 belongs to Class B. 
 
 In most cases the crank which is the driver goes foremost, and 
 
 pulls the follower after it; and hence the name of " drag-link." 
 The following are rules to be observed in determining the 
 
 dimensions of the parts. 
 
 I. In order that the directional relation -may be constant, 
 each of the crank-arms, A C, B D, should be longer than the line of 
 centres, A B. 
 
 II. For the same reason, and also in order that there may be 
 no dead-points, tfie length of the line of connection, C D, should be 
 greater than the lesser segment, E F, and less than tJie greater 
 
 Fig. 133.
 
 DRAG- LINK. 19$ 
 
 segment, F G, into which the diameter, E G, of the greater of the two 
 circles described by the connected points is divided by the other circle. 
 This principle holds also when those circles are equal, and is then 
 applicable to the diameter of either of them. In other words 
 C D is to be made 
 
 AC-BD, 
 BD-AB. 
 
 The comparative motions may be found by either of the follow- 
 ing rules : 
 
 III. To find tlie angular velocity-ratio in a given position of the 
 cranks : on the line of connection, C D, let fall from the axes the 
 perpendiculars, A L, B M ; then 
 
 Angular velocity of B D _ A L 
 Angular velocity of A C B M' 
 
 Or otherwise : produce the line of connection, C D, till it cuts the 
 line of centres in I ; then 
 
 Angular velocity of B D _ I A 
 Angular velocity of A C IB' 
 
 When C D is parallel to A B the angular velocities are equal. 
 
 IV. To find the linear velocity -ratio of the connected points : 
 in a given position of the cranks produce the crank-arms until they 
 intersect ; their point of intersection, K, will be the trace of the 
 instantaneous axis of the link ; then 
 
 Telocity of D _ K D 
 Velocity of C "KIT 
 
 The limits between which that velocity-ratio fluctuates are 
 
 , when B D traverses A, and ^ -r-=, when A 
 
 A. \j .A. O .A. Jt> 
 
 traverses B. 
 
 The two shafts, in their rotation, may be regarded as alternately 
 overtaking and falling behind each other by an angle which we 
 may call the angular displacement. The complete angular dis- 
 placement is attained in two opposite directions alternately, at the 
 two instants when the angular velocities of the shafts are equal : 
 that is, when the line of connection is parallel to the line of 
 centres. The following is a rule for designing a drag-link motion 
 with equal cranks, which shall produce a given angular displace- 
 ment; and although not the only rule by which that problem, 
 might be solved, it appears to be the simplest in its application. 
 
 V. In fig. 134 draw two straight lines, C O c, D O d, cutting
 
 196 
 
 GEOMETRY OF MACHINERY. 
 
 Fig. 134. 
 
 each other at right angles in the point ; lay off along those lines 
 the equal lengths C = O D. From C and D draw the straight 
 lines C A, D B, making the angles OCA=ODB = half tlw 
 given angular displacement, and cutting O d and O c respectively 
 
 ,-- , in A and B. Join A B 
 
 /''" "\ and C D. Then A B will 
 
 represent the line of centres; 
 AC and BD the two crank- 
 arms; and C D the line of 
 connection. 
 
 The position of the parts 
 represented will be that in 
 which the angle between 
 the crank-arms is least. To 
 show, if required, the posi- 
 tion -of the parts when that 
 angle is greatest, lay off 
 O c and O d equal to O C 
 and O D, and join A c, 
 B d, and c d. 
 
 183. I.iiilv for Contrary 
 
 Rotation*. The only other 
 elementary combination by liukwork which belongs to Willis's 
 Class B is that in which two equal cranks, rotating about parallel 
 axes in contrary directions, are connected by means of a link equal 
 in length to the line of centres. This has been already described 
 in Article 108, page 97, and represented in fig. 72, page 96, as a 
 contrivance to aid the action of elliptic wheels. There are two 
 dead points in each revolution which the pins pass at the instant 
 when the line of connection coincides with the line of centres; 
 consequently the link is not well adapted to act alone, and requires 
 a pair of elliptic wheels, or of elliptic pulleys (Article 175, page 
 189), to ensure the accurate transmission of the motion. 
 
 184. Kiinkwork with Reciprocating motion Crank and Beam 
 
 Crank and Piston-Rod. (A. M., 488.) The following are examples 
 of the most frequent cases in practice of linkwork belonging to 
 Willis's Class C, in which the directional relation is recipro- 
 cating; and in determining the comparative motion, they are 
 treated by the method of instantaneous axes, already referred to in 
 Article 179, page 193: 
 
 Example I. Two Turning Pieces with Parallel A xes, such as a 
 beam and crank (fig. 135). Let C v C. 2 , be the parallel axes of the 
 pieces; Tj, T 9 , their connected points; C i T x , C 2 T 2 , their crank 
 arms; T l T 2 , the link. At a given instant let v l be the velocity 
 of T, ; v a that of T . 
 
 j 
 
 To find the ratio of those velocities, produce C 1 T v C 2 T 2 , till
 
 LINKWOBK ECCENTRIC LENGTH OF STROKE. 
 
 197 
 
 they intersect in K; K is the instantaneous axis of the link or 
 connecting-rod, and the velocity-ratio is 
 
 Vl :^ 2 ::KT 1 :KT 2 (1.) 
 
 Should K be inconveniently far off, draw any triangle with its sides 
 respectively parallel to C i Tj, C 2 T 2 , and T 3 T 2 ; the ratio of the 
 two sides first mentioned will be the velocity-ratio required. For 
 example, draw C 2 A parallel to C L T L , cutting Tj T 2 in A; then. 
 
 C 2 A 
 
 C 2 T 2 . 
 
 .(2.) 
 
 . 136. 
 
 Example II. Rotating Piece and Sliding Piece, such as a piston- 
 rod and crank (fig. 136). Let C 2 be the axis of a rotating piece, and 
 T, E, the straight line along which a sliding piece moves. Let T 1? 
 T 2 , be the connected points; C 2 T 2 , the crank arm of the rotating 
 piece; and T l T 2 , the link or connecting rod. The points T I} T 2 , 
 and the line T t R, are supposed to be in one plane, perpendicular 
 to the axis C 2 . Draw T l K perpendicular to T 1 R, intersecting 
 C 2 T 2 in K ; K is the instantaneous axis of the link ; and the rest 
 of the solution is the same as in Example I. 
 
 185. (A. M., 489.) An Eccentric (fig. 137) being a circular disc 
 keyed on a shaft, with whose axis its centre does 
 
 dot coincide, and used to give a reciprocating 
 motion to a rod, is equivalent to a crank whose 
 connected point is T, the centre of the eccentric 
 disc, and whose crank arm is C T, the distance 
 of that point from the axis of the shaft, called 
 the eccentricity. 
 
 An eccentric may be made capable of having its eccentricity 
 altei-ed by means of an adjusting screw, so as to vary the extent of 
 the reciprocating motion which it communicates, and which is 
 called the throw, or travel, or length of stroke. 
 
 186. (A. M., 490.) The length of stroke of a point in a recipro- 
 cating piece is the distance between the two ends of the path in 
 
 Fig. 137.
 
 198 
 
 GEOMETRY OF MACHINERY. 
 
 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 con- 
 tinuously rotating piece (Article 180, page 193). 
 
 I. When the crank-arm and the path of the connected point in the 
 reciprocating piece are given, to find the stroke and the dead points. 
 If the connected point in the reciprocating piece moves in a straight 
 line traversing and perpendicular to the axis of the turning piece, 
 the length of stroke is obviously twice the crank -arm. If that 
 connected point moves in any other path, let P P, in fig. 138, 
 represent that path, A the trace of the crank- 
 axis, and A D A E the crank-arm. From 
 the point A to the path F F lay off the dis- 
 tances A B = the line of connection the 
 crank-arm, and A C = the line of connection 
 + the crank-arm; thensB C will be the stroke 
 of the connected point in the reciprocating 
 piece. Draw the straight lines C E A and 
 BAD, cutting the circular path of the crank- 
 pin in the points E and D : these will be the 
 dead points. 
 
 II. When the crank-arm, A D = A E, the 
 length of the line of connection, and the dead 
 points, D and E, are given, to find the two ends 
 of the stroke of the connected point in the 
 reciprocating piece. In D A and A E pro- 
 duced, make D B and E C each equal to the 
 length of the line of connection ; B and C will 
 be the required ends of the stroke. 
 
 When the path of the connected point in 
 the reciprocating piece is a straight line, the 
 preceding principles may be thus expressed in 
 algebraical symbols : 
 Fig. 138. Let S be the length of stroke, L the length 
 
 of the line of connection, and R the crank- 
 arm. Then, if the two ends of the stroke are in one straight line 
 with the axis of the crank, 
 
 S = 2R; ..(1.) 
 
 and if their ends are not in one straight line with that axis, then 
 S, L R, and L + R, are the three sides of a triangle, having the 
 angle opposite S at that axis; so that if 6 be the supplement of the 
 arc between the dead points, 
 
 S 2 = 2 (L2 + R2) _ 2 (L2 - R2) cos 
 
 2 L 2 + 2 R2 - S2 \ (2.) 
 
 008 6 - 2 (L 2 - R 2 )
 
 MEAN AND EXTREME VELOCITY-RATIOS. 199 
 
 187. Mean Velocity Ratio. In dynamical questions respecting 
 machines, especially when the mode of connection is by linkwork, 
 it is often requisite to determine the mean ratio of the linear 
 velocities of a pair of connected points during some definite period ; 
 which mean ratio is simply the ratio of the distances moved 
 through by those points in that period. Three cases may be dis- 
 tinguished, according as the combination of linkwork belongs to 
 Willis's Class A, Class B, or Class C. 
 
 In Class A the mean velocity-ratio is identical with the velocity- 
 ratio at each instant. For examples, see Article 181, page 194, 
 and Article 182, page 194. 
 
 In Class B the mean velocity-ratio of the connected points during 
 each complete revolution is that of the circumferences of the 
 circles in which they move. For examples, see Article 182, page 
 194, and Article 183, page 196. 
 
 In Class C the mean velocity-ratio of the connected points may 
 be taken either for a whole revolution of the revolving point and 
 double stroke of the reciprocating point, or it may be taken 
 separately for the forward stroke and return stroke of the recipro- 
 cating point, where it has different values for these two parts of 
 the motion. In the former case it is expressed by the ratio of 
 twice the length of stroke of the reciprocating point to the 
 circumference of the circle described by the revolving point j that 
 is to say, for example, in fig. 138, page 198, by the ratio 
 
 2BC 
 
 Circumference D G E H ' 
 
 In the latter case, the two mean velocity-ratios are expressed by 
 the proportions borne by the length of stroke of the reciprocating 
 point, to the two arcs into which the dead points divide the path 
 of the revolving point. For example, in fig. 138, those two ratios 
 are respectively 
 
 BO , B C 
 
 and 
 
 Arc D G E' Arc E H D' 
 
 The most frequent case in practice is that represented in fig. 
 136, page 197, where the reciprocating point moves in a straight 
 line traversing the axis about which the revolving point moves ; 
 and in that case the mean velocity- ratio for each single stroke and 
 for a whole revolution is 
 o 
 - = 0-63662 nearly. 
 
 188. Extreme Telocity-Ratios. In those cases in which one of 
 the points connected by a link revolves continuously, while the 
 other has a reciprocating motion, it is often desirable to determine 
 the greatest value of the ratio borne by the velocity of the recipro-
 
 200 
 
 GEOMETRY OF MACHINERY. 
 
 eating point to that of the revolving point. The general principle 
 upon which that greatest ratio depends is shown in fig. 139. 
 in which T represents the reciprocating point, and T the revolving 
 
 point; T T', the line of 
 connection; and C T, the 
 crank-arm. Let C A be 
 perpendicular to the direc- 
 tion of motion of the 
 reciprocating point T', and 
 let A be the point where 
 the line of connection cuts 
 C A; then, as has been, 
 
 Fig. 139. 
 
 already shown in Article 184, page, 196, 
 
 Velocity of T' _ C A 
 Velocity of T ~ CT' 
 
 and at the instant when that ratio is greatest, A is at its greatest 
 distance from C ; therefore, at that instant the direction of motion 
 of the point A in the line of connection is along that line itself. 
 Draw T' K parallel to C A, produce C T till it cuts T' K in K, the 
 instantaneous centre of motion of the link, and join K A; then 
 the direction of motion of the point A in the line of connection at 
 any instant is perpendicular to A K ; and therefore, at the instant 
 when C A is greatest, A K is perpendicular to A T'. Upon this 
 proposition depends the determination of the greatest value of the 
 
 C A 
 
 ratio n ; but that determination cannot be completed by 
 (_/ J. 
 
 geometry alone ; for it requires the solution of a cubic equation, 
 as stated in the footnote.* 
 
 * In fig. 139, let the crank-arm C T = a ; let the line of connection 
 T T* = b ; these two quantities being given ; and when the ratio of the 
 velocity of T to that of T is a maximum, let the angle C T' T = 6, and the 
 angle A C T = <j>. 
 
 Solve the following cubic equation : 
 
 sin 6 sin 4 sin 
 
 = 0, 
 
 (1.) 
 
 so as to determine the value of sin 2 0, which is the only root of that 
 equation that is positive and less than 1. Next, calculate the value of 
 the angle <, or those of its trigonometrical functions, by the help of one or 
 more of the following equations (each of which implies the others) : 
 
 tan <f> = cos sin = _ ; 
 
 m 
 
 , ( sin 2 sin 4 i 
 
 ""^^in-sinse-sin'o;/
 
 EXTREME VELOCITY-RATIO DOUBLING OSCILLATIONS. 201 
 
 An approximate solution of this question may, however, be 
 obtained by plane geometry, when the line of connection, T T', is 
 not less than about twice the crank-arm, C T. It consists in 
 treating the angle at T as if it were a right angle (from, which it 
 differs by the angle A K T); and thus we obtain 
 
 C A .Or . / (0 T2 + T T' 2 ) 
 
 Q-Tp = (nearly) ^-^ = - T T/ . 
 
 When T T' is great as compared with C T, the error of this 
 solution is inappreciable, or nearly so; when T T' = 2 C T, the 
 approximate solution is too small by about one per cent., and is 
 therefore near enough for practical purposes; when T T' becomes 
 less than 2 C T, the error rapidly increases, so as to make the 
 approximate solution inapplicable; but cases of this last kind are 
 very uncommon in practice. 
 
 189. Doubling of Oscillations by IJnkwork. When two recipro- 
 
 cating pieces are connected by means of a link, the follower may 
 be made to perform two oscillations or strokes for one of the 
 driver, in the following manner: In fig. 140, let the driver be an 
 arm or lever, A B ; A its axis of motion, and B its connected point. 
 
 Fig. 140. 
 
 Let C be the connected point of the follower, and B C the link. 
 Then the parts of the combination are to be so arranged that the 
 straight line C c, which traverses the two ends of the stroke of the 
 point C, shall traverse also the axis A, and shall bisect the arc of 
 
 and finally, calculate the required greatest velocity-ratio by the following 
 formula : 
 
 C A cos (9 - <j>) . 
 
 CT = cost* 
 
 In the two extreme cases the values of that ratio are as follows : When 5 
 is immeasurably longer than a, C A ~- C T sensibly = 1 ; when 6=0,
 
 202 
 
 GEOMETRY OF MACHINERY. 
 
 motion, b B b', of the connected point B. The result will be, that 
 while the point B performs a single stroke, from b to b', the point 
 C will perform a double stroke, from c to C and back again. 
 
 If C is a point in a second lever, that second lever may, by means 
 of a similar arrangement, be made to drive a third lever, so as again 
 to double the frequency of the strokes ; and thus, by a train of 
 linkwork, the last follower may have the frequency of its strokes 
 increased, as compared with those of the first driver, in a ratio 
 expressed by any required power of 2. 
 
 190. Slow motion by Link work. As has been already explained 
 in Article 180, page 193, when the connected point in the driver 
 of an elementary combination by linkwork is at a dead point, the 
 
 velocity of the connected 
 point of the follower is 
 nothing; and when the 
 connected point of the 
 driver is near a dead 
 point, the motion of the 
 connected point of the 
 follower is comparatively 
 very slow, and gradually 
 increases as the connected 
 point of the driver moves 
 away from the dead point. 
 "When, therefore, it is 
 Fig. 141. desired that the motion of 
 
 a follower shall, at and 
 
 near a particular position of the combination, be very slow as 
 compared with that of the driver, or as compared with that of 
 the follower itself when in other positions, arrangements may be 
 used of the class which is exemplified in fig. 141 and fig. 141 A. 
 
 In fig. 141 the lever A B, turning about an axis at A, drives, 
 by means of the link B D, the lever C D, which turns about an 
 axis at C. When the driving lever is in the position marked A B, 
 it is in one straight line with the link B D; so that B is a dead 
 point, and the velocity of the follower is null. As the connected 
 point of the driver advances from B towards b, the connected point 
 of the follower advances from D towards d, with a comparative 
 velocity which is at first very small, and goes on increasing by 
 degrees. When the motion is reversed, the comparative velocity 
 of the latter point gradually diminishes as it returns from d 
 towards D, 'and finally vanishes at the last-named point. Motions 
 ef this kind are useful in the opening and closing of steam-valves, 
 in order to prevent shocks. 
 
 Fig. 141 A shows a train of two elementary combinations of the 
 same kind with that just described; the effect being to make the
 
 LINKWORK UNIVERSAL JOINT. 
 
 203 
 
 motion of a third connected point, E, quite insensible during a 
 certain part of the motion of the first connected point, B. 
 
 Fig. 141 A. 
 191. (A. M., 491.) Hoohe's Coupling, or Universal Joint (fig. 142), 
 
 is a contrivance for coupling shafts whose axes intersect each other 
 in a point. 
 
 Let be the point of intersection of the axes C^, 2 , and 
 * their angle of inclination to each 
 other. The pair of shafts C 1} C 2 , 
 terminate in a pair of forks, F 1? F 2 , 
 in bearings at the extremities of 
 which turn the pivots at the ends of 
 the arms of a rectangular cross 
 having its centre at O. This cross 
 is the link; the connected points are 
 the centres of the bearings F 1? F 2 . 
 At each instant each of those points 
 moves at right angles to the central 
 plane of its shaft and fork ; therefore the line of intersection of the 
 central planes of the two forks, at any instant, is the instantaneous 
 axis of the cross; and the velocity-ratio of the points F 1? F 2 (which, 
 as the forks are equal, is also the angular velocity-ratio of the shafts), 
 is equal to the ratio of the distances of those points from that 
 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 instantaneous value fluctuates 
 between the limits, 
 
 Fig. 142.
 
 204 
 
 GEOMETRY OP MACHINERY. 
 
 . when F, is in the plane of the axes; 
 
 etj cos i 
 
 = cos i when F 2 is in that plane. 
 
 ...(1.) 
 
 
 The following is the geometrical construction for finding the 
 position of one of the shafts which corresponds to any given position 
 of the other; also the velocity-ratio corresponding to that position : 
 Let the shaft whose position is given be called the first shaft, and 
 the other the second shaft; and let the corresponding arms of the 
 cross be called the first and second arms respectively. 
 
 In fig. 143, let O be the point of intersection of the axes of the 
 two shafts, and let the plane of projection be a plane traversing O, 
 
 and normal to the axis of the second shaft. Let A a be the trace 
 of the plane of the two axes, and COG, perpendicular to A O a, 
 a normal to that plane. With any convenient radius, O A, describe 
 a circle about O. Lay off the angle A O D equal to the angle *, 
 which the axes of the shafts make with each other. Through D, 
 
 O B 
 
 parallel to C C, draw D B, cutting O A in B; then ~ = cos i is 
 
 vJ J\. 
 
 the velocity-ratio of the second to the first axis, when the first arm 
 
 O A 1 
 
 coincides with O C and the second with O A ; and =-^r = : = 
 
 O B cost 
 
 sec i is the velocity-ratio, when the first arm coincides with A, 
 and the second with C.
 
 UNIVERSAL JOINTS. 205 
 
 About O, with the radius O B, describe a circle. Draw the 
 radius O E/, cutting the two circles in E and f respectively, and 
 making the angle A . O f = the given angle which the first arm 
 makes with the plane of the axes: in other words, let O/be the 
 rabatment of the first arm, made by rabatting a plane normal to 
 the first axis upon the plane of projection. Through E, parallel to 
 O C, draw E F l ; and through f, parallel to O A, draw/ F l ; the 
 point F x will be the projection of the point whose rabatment is/! 
 Draw the straight line O F l ; this will be the projection of the first 
 arm on a plane normal to the second axis. Then perpendicular to 
 O F L draw O F 2 ; this will be the required position of the second 
 arm. 
 
 The projection of the path of the point F l is the ellipse C B C. 
 
 To find the angular velocity-ratio corresponding to the given 
 position of the arms; about any convenient point, G, in A O a, 
 describe a circle through O, cutting F x O and f O (produced if 
 required) in H and h respectively; from which points draw H K 
 and h k parallel to O C, and cutting A O a in K and k respectively. 
 Then we have 
 
 kh 
 
 
 The particular form of universal joint shown in fig. 142 is chosen 
 in order to exhibit all the parts distinctly. In practice, the joint 
 is often made much more compact, the forks not having more 
 space between them and the cross than is necessary in order to 
 admit of the required extent of motion of the cross-arms, and the 
 cross being sometimes made in the form of a circular disc, or of a 
 ring, or of a ball, with four pivots projecting from its circumference. 
 "Where the angle of obliquity of the two shafts (i) is small, each of 
 the forks is often made in the form of a round disc on the end of 
 the shaft, having a pair of projecting horns or lugs to carry the 
 bearings of the pivots. 
 
 The universal joint belongs to Willis's Class B. When used as 
 a coupling, it is liable to the objection, that although the mean, 
 velocity-ratio is uniform, being that of equality, the velocity-ratio 
 at each instant fluctuates, and thus gives rise to vibratory and 
 unsteady motion. 
 
 192. (A. M., 492.) The Double nooke' Joint (fig. 144) is used to 
 obviate the vibratory and unsteady motion caused by the fluctuation 
 
 * In algebraical symbols, let </> t = A Of, and < a = A F a , be the angles 
 made by the first and second arm respectively at a given instant with the 
 plane of the axes of the shafts ; then 
 
 tan <f> 1 tan < a = cos i; and 
 a a _ d <f>v _ sin 2 (f> 3 _ tan </>, + cotan <p t 
 a x (J, #! ~~ sin 2 <p 1 ~ tan < 2 + cotan <j> a
 
 206 GEOMETRY OF MACHINERY. 
 
 of the velocity-ratio which has already been mentioned. Between 
 the two shafts to be connected, Cj, C 3 , there is introduced a short 
 
 intermediate shaft, C 2 , making equal 
 angles with Cj and C 3 , connected 
 with each of them by a Hooke's 
 joint, and having both its own forks 
 in the same plane. The effect of 
 this combination is, that the angular 
 Fig. 144. velocities of the first and third shafts 
 
 are equal to each other at every 
 
 instant ; and that the planes of the first and third forks make, at 
 every instant, equal angles with the plane of the three axes. Hence, 
 as regards the comparative motion of the first and third shafts, the 
 double Hooke's joint belongs to Class A; but as regards the motion 
 of the second or intermediate shaft, it belongs to Class B.* 
 
 The double Hooke's joint works correctly when the third shaft 
 is parallel to the first, as well as in the position shown in the 
 figure. 
 
 193. iiookc-nnti-oidimm Coupling. This name may be given to 
 an universal joint in which the pivots of the cross are capable of 
 sliding lengthwise as well as of turning in their bearings in the 
 horns of the forks. It combines the properties of Hooke's coupling 
 with that of Oldham's coupling, formerly described (Article 158, 
 page 166); that is to say, it is capable of transmitting motion 
 between shafts whose axes are neither parallel nor intersecting. It 
 acts by sliding contact and linkwork combined: when single, it 
 belongs to Class B; and when double, with the axes of the three 
 shafts in parallel planes, and the first and third making equal 
 angles with the intermediate axis, to Class A. 
 
 194. Intermittent Linkwork Click niul Ratchet. A click OF catch, 
 
 being a reciprocating bar (such as B C in figs. 145 and 146) acting 
 upon a ratchet wheel or rack, which it pushes or pulls through a 
 certain arc at each forward stroke, and leaves at rest at each back- 
 ward 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 
 fixed catch, or pall, or detent (such as b c in fig. 145), turning on a 
 fixed axis, prevents the ratchet wheel or rack from reversing its 
 motion. 
 
 * Let i be the angle of inclination of C t and C 2 , and also that of C 2 and 
 C s . Let #!, < 2 , </> 3 , be the angles made at a given instant by the planes of 
 the forks of the three shafts with the plane of their axes, and let a lt a 2 , a s , 
 be their angular velocities. Then 
 
 tan < 2 tan < s cos i tan < t tan $ 2 ; 
 whence tan < 3 = tan </>j ; and a a = a a .
 
 CLICK AND RATCHET. 
 
 207 
 
 The effective stroke, being the space through which the ratchet is 
 driven by each forward stroke of the click, is necessarily once, or a 
 
 Fig. 145. 
 
 whole number of times, the pitch of the teeth of the ratchet ; and 
 it is obvious that the length of the total stroke of the click must 
 
 Fig. 146.
 
 203 GEOMETRY OF MACHINERY. 
 
 be greater than the effective stroke, and less than the next greater 
 whole number of times the pitch. It is advisable, when practicable, 
 to make the excess of the total above the effective stroke no greater 
 than is just sufficient to ensure that the click shall clear each 
 successive tooth of the ratchet. In figs. 145 and 146 the effective 
 stroke is once the pitch of the ratchet; in fig. 147, twice the 
 pitch. 
 
 A- catch may be made to drop into its place in front of each 
 successive tooth either by gravity or by the pressure of a spring, 
 according to the circumstances of the case. 
 
 Some clicks act by thrusting, as B C in fig. 145, and B C in fig. 
 146 ; others by pulling, as b c in fig. 145. 
 
 The direction of the pressure between a click and a tooth is 
 nearly a normal to the acting surfaces of the click and tooth at 
 the centre of their area of contact; for example, in fig. 145, the 
 dotted lines marked C D, c d, and in fig. 146, the dotted line 
 marked C D. In order that a click may bs. certain not to lose its 
 hold of the tooth, that normal ougM to pass inside the axis of motion 
 of a thrusting click, and outside the axis of motion of a pulling click. 
 For example, in fig. 145, C D passes inside the axis B, and c d 
 passes outside the axis 6; the words "inside" and "outside" 
 being used to denote respectively nearer to and further from the 
 ratchet. 
 
 It is convenient, though not essential, that a click for driving a 
 wheel should be carried by an arm concentric with the wheel ; such 
 as the arms A B in fig. 145, and A B in fig. 146. In such cases 
 the total angular stroke of the click-arm (marked B A B' in fig. 
 145, and B' A B" in fig. 146) must be a little greater than the 
 effective angular stroke, which is once, or a whole number of times, 
 the pitch-angle of the teeth of the wheel. The axis of motion of 
 the click-arm may, however, be placed elsewhere if necessary, pro- 
 vided care is taken that in all positions of the arm the line of 
 pressure passes to the proper side of the axis of motion of the click. 
 (See figs. 148, 149, further on.) 
 
 Fig. 146 represents a tumbling or reversible click, shaped so as to 
 act upon the teeth of an ordinary toothed wheel. In its present 
 position it drives the wheel in the direction pointed out by the 
 arrow : by throwing it over into the position marked with dotted 
 lines, it is made, when required, to drive the wheel the contrary 
 way. 
 
 It is easy to see that the acting surfaces of clicks, and the teeth 
 of ratchets on which they act, may be shaped in a variety of ways 
 besides those exemplified in the figures. 
 
 195. Silent click. This is a contrivance for avoiding the noise 
 and the tear and wear which arise from the sudden dropping of 
 the common click into the space between the teeth of the ratchet-
 
 SILENT CLICK DOUBLE-ACTING CLICK. 
 
 JJU!) 
 
 wheel. The wheel is like an ordinary toothed wheel B C is the 
 click, which, in the example, is made to push the teeth. It is 
 carried by one branch, A B, of a bell-crank lever, which has a 
 rocking motion about the same axis with the wheel. The other 
 
 Fig. 147. 
 
 branch of the bell-crank lever has two studs or pins in it, E and E*. 
 Between these pins is the driving arm, A F, which has a recipro- 
 cating motion about the same axis, and is connected by a link, 
 G H, with the click. 
 
 B A B' is the total angular stroke of the bell-crank lever; 
 D B D' is the angle through which the click must be moved in 
 order to lift it clear of the teeth. The sum of these angles, 
 B A B' + D B D', is = F A F", the angular stroke of the driving 
 arm. The positions of the studs, E and E', are so adjusted, that 
 the driving arm in passing from the one to the other moves through 
 the angle F A F'= D B D'; being the angular motion that lifts 
 the click clear of the teeth before the return stroke, or makes it 
 take hold before the forward stroke. During those parts of the 
 motion of the driving arm and click, the bell-crank lever stands 
 still : its forward and return strokes are made by the driving arm 
 pressing against the studs E and E' respectively. 
 
 196. Double-Acting Click. This is the contrivance sometimes 
 called, from its inventor, " the lever of La Garousse." It consists 
 of two clicks making alternate strokes, so as to produce a nearly 
 continuous motion of the ratchet which they drive ; that motion 
 being intermitted for an instant only at each reversal of the 
 direction of movement of the clicks. In fig. 148 the clicks act by 
 pushing; in fig. 149, by pulling. The former arrangement is on 
 the whole the best adapted to cases in which the mechanism
 
 210 
 
 GEOMETRY OF MACHINERY. 
 
 requires considerable strength ; such as windlasses on board ship. 
 Each single stroke of the click-arms advances the ratchet through 
 one-half of its pitch. 
 
 Corresponding points in the two figures are marked with the 
 same letters; and as fig. 148 contains some parts which do not 
 
 Fig. 148. 
 
 occur in fig. 149, the former will, in the first place, be referred to 
 in explaining the principles to be followed in designing such 
 combinations. 
 
 Let the figure and dimensions of the ratchet-wheel be given, 
 and let A be its axis, and B B its pitch-circle; that is, a circle 
 midway between the points and roots of the teeth. 
 
 Having fixed the mean obliquity of the action of the clicks 
 that is, the angle which their lines of action, at mid-stroke, are to 
 make with tangents to the pitch-circle draw any convenient radius 
 of the pitch-circle, as L A, and from it lay off the angle LAD, 
 equal to that obliquity. On A D let fall the perpendicular L D, 
 and with the radius A D describe the circle CO; this will be the 
 base-circle, to which the lines of action of the clicks are to be 
 tangents. (As to base-circles, see also Article 131, page 121.) Lay 
 off the angle DAE equal to an odd number of times half the pitch- 
 angle; then through the points D and E in the base-circle draw two 
 tangents, cutting each other in F. Draw F G, bisecting the angle
 
 DOUBLE-ACTING CLICKS FRICTIONAL CATCH. 
 
 211 
 
 Fig. 149. 
 
 at F, and take any convenient point in it, G, for the trace of the 
 axis of motion of the rocking-shaft which carries the click-arms. 
 From G let fall G H 
 and G K perpendicu- 
 lar to the tangents 
 F D H and E F K; 
 then H and K will be 
 the positions of the 
 centres of motion of 
 the two clicks at mid- 
 stroke; and G H and 
 F K will represent the 
 dick-arms. Let L and 
 M be the points where 
 D H and E K respec- 
 tively cut the pitch- 
 circle; then H L and 
 K M will be the 
 lengths of the two clicks. The effective stroke of each click will be 
 equal to half the pitch, as measured on the base-circle C C; and the 
 total stroke must be as much greater as is necessary in order to 
 make the clicks clear the teeth. 
 
 In fig. 149, where the clicks pull instead of pushing, the obliquity 
 is nothing; and the consequence is that the base-circle, C C, coin- 
 cides with the pitch-circle, B B, and that the points L and M coin- 
 cide respectively with D and E. 
 
 197. Fricrtonal Catch. The frictional catch (called sometimes 
 the "silent feed-motion") is a sort of intermittent linkwork, founded 
 on the dynamical principle, that two surfaces will not slide on each 
 other so long as the angle which the direction of the pressure 
 exerted between them makes with their common normal at the 
 place where they touch each other is less than a certain angle called 
 the angle of repose, which depends on the nature of the surfaces, 
 and their state of roughness or smoothness, and of lubrication. 
 The smoother and the better lubricated the surfaces, the smaller is 
 the angle of repose. 
 
 In trigonometrical language, the angle of repose is the angle 
 whose tangent is equal to the co-efficient of friction: that is, to the 
 ratio which the friction between two surfaces, being the force which 
 resists sliding, bears to the normal pressure; or, what is the same 
 thing, it is the angle whose sine is equal to tlie ratio that the friction, 
 bears to the resultant pressure when sliding takes place. The 
 subject of friction, and of the angle of repose, properly belong to 
 the dynamical part of this treatise, and will be mentioned in greater 
 detail further on. For the present purpose it is sufficient to state 
 that the sine of the angle of repose for metallic surfaces in a
 
 212 
 
 GEOMETRY OF MACHINERY. 
 
 K." 
 
 moderately smooth state, and not lubricated, as deduced from the 
 experiments of Morin, ranges from 0'15 to O2, or thereabouts; so 
 
 that an angle whose 
 sine is one-seventh of 
 radius may be con- 
 sidered to be less than 
 the angle of repose of 
 any pair of metallic 
 surfaces which are in 
 the above-mentioned 
 condition. 
 
 The frictional catch, 
 though always depend- 
 ing on the principle 
 just stated, is capable 
 of great variety in 
 xletail. The arrange- 
 ment represented in 
 fig. 150 is constructed 
 in the following man- 
 ner : 
 
 The shaft and rim 
 of the wheel to be 
 acted upon are shown 
 in section. A K is 
 the catch-arm, having 
 a rocking motion 
 about the axis A of 
 the wheel; the link 
 by which it is driven 
 is supposed to be 
 jointed to it at K; 
 and K' K" represents 
 Fig. 150. the stroke, or arc of 
 
 motion, of the point 
 
 K ; so that K' A K" is the angular stroke of the catch-arm. L is 
 a socket, capable of sliding longitudinally on the catch-arm to a 
 small extent; a shoulder for limiting the extent of that sliding 
 motion is marked by dotted lines. The socket arid the part of 
 the arm on which it slides should be square, and not round, to 
 prevent the socket from turning. From the side of the socket 
 there projects a pin at D, from which the catch D G H hangs. 
 M is a spring, pressing against the forward side of the catch. 
 G and H are two studs on the catch, which grip and carry forward 
 the rim, B B C C, of the wheel during the forward stroke, by means 
 of friction, but let it go during the return stroke.
 
 FRICTIONAL CATCH SLOTTED LINK BAND LINKS. 213 
 
 A similar frictional catch, not shown in the figure, hanging 
 from a socket on a fixed instead of a moveable arm, at any 
 convenient part of the rim of the wheel, serves for a detent, 
 to hold the wheel still during the return stroke of the moveable 
 catch-arm. 
 
 The following is the graphic construction for determining the 
 proper position of the studs G and H : Multiply the radii of the 
 outer and inner surfaces, B B and C C, of the rirn of the wheel by 
 a co-efficient a little less than the sine of the angle of repose say 
 ^ and with the lengths so found as radii describe two circular 
 arcs about A ; the greater (marked E) lying in the direction of for- 
 ward motion, and the less (marked F) in the contrary direction. 
 Prom D, the centre of the pin, draw D E and D F, touching those 
 two arcs. Then G, where D E cuts B B, and H, where D F cuts 
 C C, will be the proper positions for the points of contact of the 
 two studs with the rim of the wheel. For the force by which the 
 catch is driven during the forward stroke acts through D ; that 
 force is resolved into two components, acting along the lines D G E 
 and F H D respectively; and those lines make with the normals to 
 the rim of the wheel, at G and H respectively, angles less than the 
 angle of repose of a pair of metallic surfaces that are not lubricated. 
 Should it be thought desirable, the positions of the holding studs, 
 or of one of them, may be made adjustable by means of screws or 
 otherwise. 
 
 The stiffness of the spring M ought to be sufficient to bring the 
 catch quickly into the holding position at the end of each return 
 stroke. 
 
 The length of stroke of a frictional catch is arbitrary, and 
 may, by suitable contrivances, be altered during the motion. Con- 
 trivances for that purpose will be described further on. 
 
 A pair of frictional catches may be made double-acting, like the 
 double-acting clicks of the preceding Article. 
 
 198. Slotted i Jnk. A slotted link is connected with a pin at one 
 of its ends, not by a round hole fitting the pin closely, but by an 
 oblong opening or slot with semicircular ends. This is an example 
 of intermittent linkwork; the intermission in its action taking 
 place during the middle part of each stroke, while the pin is 
 shifting its position relatively to the link from the one end of the 
 slot to the other. That intermission takes effect by producing a 
 pause in the motion of that piece which is the follower, and which 
 may be either the jink or the pin ; and the stroke of the follower 
 is shorter than that of the driver by an extent corresponding to 
 the length of the slot, as measured from centre to centre of its two 
 
 O ' 
 
 semicircular ends. 
 
 199. Band r,ink. "Where tension alone, and not thrust, is 
 to act along a link, it may be flexible, and may consist either
 
 214 
 
 GEOMETRY OP MACHINERY. 
 
 of a single band, or of an endless band passing round a pair 
 of pulleys which turn round axes traversing and moving with 
 the connected points. For example, in fig. 151, A is the axis 
 
 __ of a rotating shaft, B that of a crank-pin, 
 
 x''" "X, C the other connected point, and B 
 
 /' \ the line of connection; and the connection 
 
 is effected by means of an endless band, 
 passing round a pulley which is centred 
 /y\ u P on Q) an d round the crank-pin itself, 
 \ /y^^^l which acts as another pulley. The pulleys 
 
 are of course secondary pieces; and the 
 motion of each of them belongs to the 
 subject of aggregate combinations, being 
 compounded of the motion which they have 
 along with the line of connection, B C, and 
 of their respective Dotations relatively to 
 that line as their line of centres; but the 
 motion of the points B and C is the same as 
 if B C were a rigid link, provided that forces 
 act which keep the band always in a state 
 of tension. 
 
 This combination is used in order to lessen 
 the friction, as compared with that which 
 takes place between a rigid link and a pair of 
 pins; and the band employed is often a leather chain, of the kind 
 already mentioned in Article 176, page 191, because of its 
 flexibility. 
 
 SECTION VII. Connection % Plies of Cord, or 6y 
 Reduplication. 
 
 200. General Explanations. (A. M., 494.) The combination of 
 pieces connected by the several plies of a cord, rope, or chain, con- 
 sists of a pair of cases or frames called blocks, each containing one 
 or more pulleys called sheaves. One of the blocks (A, figs. 152, 153), 
 called the fixed block, ex fall-block, is fixed; the other, called the 
 fly-block, or running block, B, is moveable to or from the fall- 
 block, with which it is connected by means of a rope, or fall, of 
 which one end is fastened either to a fixed point or to the running 
 block, while the other end, C, called the hauling part, is free ; and 
 the intermediate portion of the rope passes alternately round the 
 pulleys in the fixed block and running block. The several plies of 
 the rope are called by seamen parts; and the part which has its 
 end fastened is called the standing part. The whole combination 
 is called a tackle or purchase. When the hauling part is the driver, 
 and the running block the follower, the two blocks are being drawft
 
 CONNECTION BY PLIES OF CORD. 
 
 215 
 
 together; when the running block is the driver, and the hauling 
 part the follower, the two blocks are being pulled apart. 
 
 201. Velocity-Ratios. (-4. M. t 495, 496.) The velocity-ratio chiefly 
 considered in a purchase is that between the velocities of the 
 running block, B, and of the hauling part, C. That ratio is 
 expressed by the number of plies of rope by which the running 
 
 Fig. 152. 
 
 Fig. 153. 
 
 block 
 
 is connected with the fall-block. Thus, in fig. 152, 
 and in fig. 153, C + B = 6. A tackle is called a 
 
 C-r B 
 
 twofold
 
 216 GEOMETRY OF MACHINERY. 
 
 purchase, a threefold purchase, and so on, according to the value of 
 the velocity-ratio C -^- B. For example, fig. 152 is a sevenfold 
 purchase, and fig. 153 a sixfold purchase. 
 
 The velocity of any ply or part of the rope is found in the follow- 
 ing manner : For a ply on the side of the fall-block, A, next the 
 Jiauling-part, C, it is to be considered what would be the velocity of 
 that ply if it were itself the hauling part : that is to say, the ratio 
 of its velocity to that of the running block is expressed by the 
 number of plies between the ply in question and the point of attach- 
 ment of the standing part. For a ply on the side of the fall-block 
 furthest from the hauling part, 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. If the standing part is 
 attached to a fixed point, as in fig. 153, its velocity is nothing; if to 
 the .running block, as in fig. 152, its velocity is equal to that of 
 the block. The comparative velocities of tlie several parts of the 
 ropes are expressed by the upper row of figures. The lower row 
 of figures express the velocities of the several parts relatively to 
 the running block. 
 
 202. Ordinary Form of Pulley-Blocks. A block, as XlSed On board 
 
 ship, consists of an oval sltell, usually of elm or metal, containing 
 one or more pulleys, called sheaves, of lignum-vitse or metal, turn- 
 ing about a cylindrical wrought-iron pin. The round 
 hole in the centre of a wooden sheave is lined with 
 a gun-metal tube called the bushing. The part of the 
 sheave-hole through which the rope or chain reeves 
 is called the swallow. In the bottom and sides of a 
 A block is a groove called the score, into which fits the 
 strop or strapping of rope or iron by which the block is 
 hung or secured to its place. Ordinary blocks con- 
 taining one pin are called single, double, treble, &c., 
 according to the number of sheaves that turn about 
 that pin side by side. Each sheave turns in a separate 
 hole in the shell. Fig. 154 shows examples of the 
 forms of iron pulley-blocks commonly used in machinery 
 on land. A is a treble block ; B, a double block. The 
 block B has an eye for the attachment of the standing 
 part of the rope. 
 
 203. White's Pulleys. When the sheaves of a block, 
 as in the ordinary form, are all of the same diameter, they 
 all turn with different angular velocities, because of the 
 different velocities of the plies of rope that pass over 
 Fig. 154. them. But by making the effective radius of each 
 sheave proportional to the velocity, relatively to the 
 ~blocJc, of the ply of rope which it is to carry, the angular velocities 
 of the sheaves in one block may be rendered equal; so that the
 
 WHITE'S PULLEYS COMPOUND PURCHASES. 
 
 217 
 
 sheaves may be made all in one piece, having two journals which 
 turn in fixed bearings. 
 
 These are called "White's Pulleys," from the inventor; and they 
 are represented in figs. 152 and 153, page 215: having been chosen 
 to illustrate the general principles of the action of blocks and tackle, 
 because of the clearness with which they show the positions of all 
 the parts of the rope. 
 They are not, however, 
 much used in practice, 
 because the unequal 
 stretching of different 
 parts of the cord pre- 
 vents the combination 
 from working with that 
 degree of accuracy which 
 is necessary in order that 
 any advantage may be 
 obtained by means of it 
 over the common con- 
 struction. 
 
 204. Compound Pur- 
 chases. A compound 
 purchase consists of a 
 train of simple pur- 
 chases; that is to say, 
 the hauling part of one 
 tackle is secured to the 
 running block of another, 
 and so on, for any num- 
 ber of tackles. In prac- 
 tice, however, the number 
 of tackles in a compound 
 purchase is almost always 
 two; and then the rope 
 that has the running 
 block secured to it is 
 usually called the pen- 
 dant, and the rope that 
 is directly hauled upon 
 by hand, the fall. 
 
 The velocity-ratio is, 
 as in other trains of 
 
 elementary combinations, Fi ffg 155, 
 
 the product of the velo- 
 city-ratios belonging to the elementary or simple tackles of which 
 the compound purchase consists.
 
 218 
 
 GEOMETRY OF MACHINERY. 
 
 For example, in fig. 155, A B C is a twofold purchase; and 
 at D, its pendant is secured to the fly-block of a threefold pur- 
 chase, D E F, whose hauling part is F G. The velocity-ratio of 
 D to B is 2, and that of G to D is 3 ; so that the velocity-ratio 
 of G to B is 2 x 3 = 6 ; and the compound purchase is sixfold. 
 
 205. Rope and Space Required for a Purchase. An elementary 
 
 or simple purchase requires no more space to work in than the 
 greatest distance from outside to outside of the fixed and running 
 blocks. The least length of rope sufficient for it may be found 
 as follows : To the greatest distance between the centres of the 
 blocks add half the effective circumference of a sheave (see Article 
 166, page 180); multiply the sum by the number of plies of rope 
 which connect the blocks with each other; and to the product add 
 the least length of the hauling part required under the circum- 
 stances of the particular case. 
 
 A compound purchase requires a length of space to work in 
 equal to the whole distance traversed by the fly-block of the last 
 purchase in the train (viz., that whose hauling part is free), with 
 a sufficient additional length added for the blocks and their 
 fastenings. 
 
 206. Obliquely-acting Tackle. The parts of the rope of a tackle, 
 instead of being parallel to each other and to the direction of 
 motion of the running block, may make various angles with that 
 direction. For example, in fig. 156, B is the running block, and B b 
 
 Fig. 156. 
 
 its line of motion; and in the case represented, that block hangs 
 from two parts of a rope the standing part, B A, and another 
 part, B C. To find the velocity-ratio of the hauling part, D, to the 
 running block, B : from the centre, B, of that block, draw straight 
 lines, B a, B c, parallel to the parts of the rope by which it hangs; 
 at any convenient distance from B, draw the straight line a b c
 
 OBLIQUE TACKLE TILLER-ROPES. 
 
 219 
 
 perpendicular to B b, and cutting all the straight lines which 
 diverge from B; then, 
 
 asB6:istoBa+ Be, 
 
 : : so is the velocity of B 
 : to the velocity of D ; 
 
 and the same rule may be extended to any number of parts, thus : 
 velocity of D sum of lengths cut off on lines diverging from B 
 velocity of B ~ B 6 
 
 The combination belongs to Class B; because, owing to the con- 
 tinual variation of the obliquity of the parts of the rope, the velocity- 
 ratio is continually changing. 
 
 206 A. Tiller -Ropes. The tiller of a ship is a horizontal lever 
 projecting from the rudder-head, by means of which the position of
 
 220 GEOMETRY OF MACHINERY. 
 
 the rudder is adjusted. It usually points forward ; that is, in the 
 contrary direction to the rudder itself. In ships of war the tiller is 
 usually put over, or moved to one side or to the other, by means of a 
 pair of obliquely-acting twofold tackles, made of raw hide ropes, 
 which haul it respectively to starboard (that is, towards the right) 
 and to port (that is, towards the left), when required. The hauling 
 parts of both tackles are guided by fixed pulleys so as to be wound 
 in opposite directions round one barrel, which is turned by means 
 of the steering-wheel. * 
 
 Fig. 156 A is a plan of this combination. A is the rudder-head ; 
 A B, the tiller, shown as amidships, or pointing right ahead; 
 D B F G is the starboard tiller-rope; D' B F' G', the port tiller- 
 rope. These ropes are made fast to eye-bolts at D and D' ; at B 
 they are rove through blocks that are secured to the tiller ; at F 
 and F' they are led round fixed pulleys; and G and G' are their 
 hauling parts, which are led, by means of pulleys which it is unne- 
 cessary to show in the figure, to the barrel of the steering-wheel. 
 
 A b is the position of the tiller when put over about 40 to star- 
 board ; and the corresponding positions of the tiller-ropes are 
 D b F G and D' b F G'. 
 
 In order that the tiller-ropes may never become too slack, it is 
 necessary that the sum of the lengths of their several parts should 
 be nearly constant in all positions of the tiller; that is to say, that 
 we should have, in all positions, 
 
 T>b + bY + T>'b + bF' nearly = 2 (D B + B F). 
 
 That object is attained, with a rough approximation sufficient for 
 practical purposes, by adjusting the positions of the points D, D', 
 and F, F', according to the following rule : 
 
 RULE. About A, with the radius A B, describe a circle. Make 
 
 2 
 A C = 5 A B ; and through C, perpendicular to A B, draw a 
 
 9 
 
 straight line cutting that circle in D and D'. These will be the 
 points at which the standing parts of the ropes are to be made fast. 
 
 Then produce A B to E, making B E = A B ; and through E, 
 
 
 perpendicular to A B E, draw F' E F, making E F = E F' = - 
 
 C D ; F and F' will be the stations for the fixed blocks. 
 
 When the angle B A 6 is about 40, the sum of the lengths of 
 the parts of the ropes is a little greater than when the tiller is 
 amidships; but the difference (which is about one-oOth part of the 
 length expressed in the preceding equation) is not so great as to 
 
 " See Peake's Rudimentary Treatise on Shipbuilding, second volume, pp. 
 66, 162 ; also Watts, Rankine, Napier, and Barnes On Shipbuilding, p. 202.
 
 TILLER-ROPES HYDRAULIC CONNECTION. 221 
 
 cause any inconvenient increase of tightness. For angles not 
 exceeding 30 the approximation to uniformity of tightness is 
 extremely close. 
 
 SECTION VIII. Hydraulic Connection. 
 
 207. General Nature of the Combinations. The kind of Com- 
 binations to which the present section relates are those in which 
 two cylinders fitted with moveable pistons are connected with each 
 other by a passage, and the space between the pistons is entirely 
 filled with a mass of fluid of invariable volume. 
 
 Any liquid mass may be treated, in most practical questions 
 respecting the transmission of motion, as if its volume were in- 
 variable, because of the smallness of the change of volume produced 
 in a liquid by any possible change of pressure. For example, in 
 the case of water, the compression produced by an increase in the 
 intensity of the pressure to the extent of one atmosphere (or 14' 7 
 Ibs. on the square inch), is only one-20,000th part of the whole 
 volume. (See Article 88, page 75.) 
 
 The volume, then, of the mass of fluid enclosed in the space 
 between two pistons being invariable, it follows that if one piston 
 (the driver) moves inwards, sweeping through a given volume, the 
 other piston (the follower) must move outwards, sweeping through 
 an exactly equal volume ; otherwise the volume of the space con- 
 tained between the pistons would change; and this is the principle 
 upon which the comparative motion in hydraulic connection depends. 
 
 208. Cylinders, Pistons, and Plungers. A piston is a primary 
 piece, sliding in a vessel called a cylinder. The motion of the piston 
 is most commonly straight; and then the bearing surfaces of the 
 piston and cylinder are actually cylindrical, in the mathematical 
 sense of that word. 
 
 When the motion of a piston is circular, the bearing surfaces 
 of the piston, and of the vessel in which it slides, are surfaces of 
 revolution described about the axis of rotation of the piston ; but 
 that vessel, in common language, is still called a cylinder, although 
 its figure may not be cylindrical. 
 
 A plunger is distinguished from an ordinary piston in the follow- 
 ing way : The bearing surface of a cylinder for a plunger consists 
 merely of a collar, of a depth sufficient to prevent the fluid from 
 escaping; and the plunger slides through that collar, and has a 
 bearing surface of a length equal to the depth of the collar added 
 to the length of stroke; so that during the motion different parts 
 of the surface of the plunger come successively into contact with the 
 same surface of the collar. On the other hand, an ordinary piston 
 has a bearing surface of a depth merely sufficient to prevent the 
 fluid from escaping; and the cylinder has a bearing surface of a
 
 GEOMETRY OF MACHINERY. 
 
 length equal to the depth of that of the piston added to the length 
 of stroke; so that during the motion the same surface of the piston 
 comes into contact successively with different parts of the surface 
 of the cylinder. For example, in fig. 157, A is a plunger, working 
 through the collar B in the cylinder C; and in fig. 158, A is an 
 
 Fig. 157. 
 
 Fig. 158. 
 
 ordinary piston, working in the cylinder B. The action of plungers 
 and of ordinary pistons in transmitting motion is exactly the same ; 
 and in stating the general principles of that action, the word 
 piston is used to include plungers as well as ordinary pistons. 
 
 The volume swept by a piston in a given time is the product of 
 two factors transverse area and length. The transverse area is 
 that of a plane bounded by the bearing surface of the piston 
 and cylinder, and normal to the direction of motion of the piston, 
 so that it cuts that surface everywhere at right angles. In a 
 straight-sliding piston that plane is normal to the axis of the 
 cylinder; in a piston moving circularly, it traverses the axis of 
 rotation of the piston: in other words, the area is that of a 
 projection of the piston on a plane normal to its direction of 
 motion. 
 
 "When the motion of the piston is straight, the length of the 
 volume swept through is simply the distance moved by each point 
 of the piston. When the motion is circular, that length is
 
 HYDRAULIC CONNECTION COMPARATIVE VELOCITIES. 223 
 
 the distance moved through by the centre of the area of the 
 piston.* 
 
 So long as the transverse area and length of the space swept by 
 a piston are the same, it is obvious that the form of the ends of 
 that piston does not affect the volume of that space. 
 
 When the space in the cylinder which contains the fluid acted 
 on by a piston is traversed by a piston-rod, the effective transverse 
 area is equal to the tranverse area of the piston, with that of the 
 rod subtracted. For example, in fig. 158, the upper division of 
 the cylinder is traversed by the piston-rod C, working through the 
 stuffing-box D ; hence the effective transverse area in that division 
 of the cylinder is the difference between the transverse areas of the 
 piston A and rod C. In the lower division of the cylinder, where 
 there is no rod, the whole transverse area of the piston is effective. 
 A trunk acts in this respect like a piston-rod of large diameter. 
 
 209. Comparative Velocities of Pistons. From the equality of the 
 volumes swept through by a pair of pistons that are connected with 
 each other by means of an intervening fluid mass of invariable 
 volume, it obviously follows that the velocities of the pistons are 
 inversely as their transverse areas. 
 
 The transverse areas are to be measured, as stated in the pre- 
 ceding Article, on planes normal to the directions of motion of the 
 pistons; and when the motion of a piston is circular, the velocity 
 referred to in the rule is that of the centre of its transverse area. 
 
 Let A and A' denote the transverse areas of the two pistons 
 marked with those letters in fig. 159, page 224, and v and v' their 
 
 v' A. 
 
 velocities; then their velocity-ratio is - = ,. 
 
 v j\. 
 
 As the velocity-ratio of a given pair of connected pistons is con- 
 stant, the combination belongs to Willis's Class A. 
 
 210. Comparative Velocities of Fluid Particles. It may Sometimes 
 
 be required to find the comparative mean velocities with which 
 
 * To find the distance of the centre of a plane area from an axis in the 
 plane of that area : divide the area, by lines parallel to that axis, into a 
 number of narrow bands ; let d x be the breadth of one of those bands, and 
 
 y its length ; then y d x is the area of that band ; and / y d x is the 
 whole area. Let x be the distance from the axis to the centre of the band 
 y d x; then x y d x is the geometrical moment of ^that band, and I x y d x 
 
 is the geometrical moment of the whole area relatively to the axis ; which 
 moment, being divided by the area, gives the required distance of the centre 
 of the area from, the axis, viz., 
 
 /xyd x 
 , = . (See Article 293, page 334.) 
 
 ydx
 
 224 
 
 GEOMETRY OP MACHINERY. 
 
 the fluid particles flow through a given section of the passage which 
 connects a pair of pistons; it being understood that the mean 
 velocity of flow through a given section of the passage denotes the 
 mean value of the component velocities, in a direction normal to 
 that section, of all the particles that pass through it. From the 
 fact that in a given time equal volumes of fluid flow through all 
 sectional surfaces that extend completely across the passage, it 
 follows that the mean velocity of flow through any such section is 
 inversely as its area (a principle already stated in Article 88, page 
 76); and this principle applies to all possible sections, transverse 
 and oblique, plane and curved. 
 
 For example, in fig. 159, let B denote the area of a transverse 
 section, B B, of the passage which connects the two cylinders, and 
 
 c 
 
 B \ 
 
 A 
 
 B! 
 
 Fig. 159. 
 
 u the mean velocity with which the particles of fluid flow through 
 that section; then v, as before, being the velocity of the piston 
 whose transverse area is A, we have 
 
 A 
 
 B' 
 
 Also, let C denote the ai'ea of an oblique section, C C, of the passage, 
 and w the mean component velocity of the fluid particles in a 
 direction normal to that section; then 
 
 w A . w B 
 - = 7^; and - = -. 
 v C u n 
 
 C 
 
 211. Use of Valrcs Intermittent Hydraulic Connection. Valves 
 
 are used to regulate the communication of motion through a fluid
 
 VALVES PUMP HYDRAULIC PRESS. 225 
 
 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 drives 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 of 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 con- 
 stitutes a pump. 
 
 II. When the fluid drives the piston, the valves must be opened 
 and shut by mechanism, or by hand. In this case the cylinder is 
 a working cylinder. 
 
 It is by the aid of valves that intermittent "hydraulic connection 
 between two pistons is effected; and the action produced is 
 analogous to that of the click, ratchet, and detent, in intermittent 
 link-work. 
 
 For example, in the Hydraulic Press, the rapid motion of a small 
 plunger in a pump causes the slow motion of a large plunger in 
 a working cylinder; and the connection of the pistons is made 
 intermittent by means of the discharge valve of the pump; being 
 a valve which opens outwards from the pump and inwards as 
 regards the working cylinder. The pump draws water from a 
 reservoir, and forces it into the working cylinder: during the 
 inward stroke of the pump plunger, the plunger of the working 
 cylinder moves outward with a velocity as nmch less than that of 
 the pump plunger as its area is greater. At the end of the inward 
 stroke of the pump plunger, the valve between the pump and the 
 working cylinder closes, and prevents any water from returning 
 from the working cylinder into the pump; and it thus answers the 
 purpose of the detent in ratchet-work (see page 206). During the 
 outward stroke of the pump plunger that valve remains shut, and 
 the plunger of the working cylinder stands still, while the pump is 
 again filling itself with water through a valve opening inwards. 
 When the piston of the working cylinder has finished its outward 
 stroke, which may be of any length, and may occupy the time of 
 any number of strokes of the pump, it is permitted to be moved 
 inwards again by opening a valve by hand and allowing the water 
 to escape. 
 
 A hydraulic press is often furnished with two, three, or more 
 pumps, making their inward strokes in succession, and so producing 
 
 Q
 
 226 GEOMETRY OF MACHINERY. 
 
 a continuous motion of the working plunger. This is analogous to 
 the double-acting click (page 209). 
 
 211 A. Flexible Cylinders and Pistons. By an extension of the 
 
 use of the word "cylinder," it maybe made to include vessels made 
 wholly or partly of a flexible material, which answer the purpose 
 of a cylinder with its piston, by altering their shape and internal 
 capacity ; such as bellows. Questions as to this class of vessels 
 may be approximately solved according to purely geometrical 
 principles, by assuming the flexible material of which they are 
 made to be inextensible. 
 
 In bellows, and pumps constructed on the principle of bellows, 
 the vessel must have at least a pair of rigid ends, which, being 
 moved alternately from and towards each other, answer the purpose 
 of a piston. If those ends are equal and similar, and connected 
 together by sides that may be assumed to be inextensible and 
 perfectly flexible, the volume of fluid alternately drawn in and 
 forced out may be taken as nearly equal ^to the area of one end 
 multiplied by the distance through which the centre of area of one 
 end moves alternately towards and from the other end. 
 
 Another example is furnished by a Irind of pump, in which a 
 circular orifice in one of the sides of a box is closed by a rigid flat 
 disc of smaller diameter, and a bag in the form of a conical frustum 
 of leather, or some other suitable material the inner edge of the 
 leather being made fast to the disc, and the outer edge to the cir- 
 cumference of the orifice. In working, the disc is moved alternately 
 inwards and outwards, so as to draw the conical bag tight in 
 opposite directions alternately. To find the virtual area of piston, 
 add together the area of the disc, the area of the orifice, and four 
 times the area of a circle whose diameter is the half-sum, of the 
 diameters of the disc and orifice, and divide the sum by six. 
 That virtual area, multiplied by the length of stroke, gives nearly 
 the volume of fluid moved per stroke. 
 
 In Bourdoris pumps and engines an elastic metal tube, of a 
 flattened form of transverse section, is bent so as to present the 
 figure of a circular arc. The internal capacity of the tube is 
 varied by alternately admitting and expelling fluid ; the effect of 
 which is to flatten the curvature of the tube when its capacity is 
 increased, and to sharpen that curvature when that capacity is 
 diminished; so that if one end of the tube is fixed in position and 
 direction, the other end has an oscillating motion. 
 
 In fig. 81, page 114, the arcs A D, A D', A D* may be taken 
 to represent successive positions of the tube; A being its fixed end, 
 and I) its moveable end. The path of the moveable end, D D' D", 
 is nearly an arc of a circle of the radius C G = f of the length of 
 the tube. The capacities of the tube in its several different posi- 
 tions, A D, A D', A D", &c., vary nearly in the inverse ratio ofti&
 
 FLEXIBLE PUMPS - TRAINS. 227 
 
 arcs G D, G D', G D", &c. ; so that if the capacity of the tube, when 
 in a given position, is known, we can calculate its capacity in any 
 other position, and the volume of fluid admitted or expelled in 
 passing from any given position to any other.* 
 
 SECTION IX. Miscellaneous Principles respecting Trains. 
 
 212. Converging Trains. The essential principles of a train of 
 mechanism have been stated in Article 93, page 80. Two or 
 more trains may converge into one; that is to say, two or more 
 primary pieces, which are followers in different trains, may all act 
 as drivers to one primary piece. In such cases the comparative 
 motion in each of the elementary combinations formed by the one 
 follower with its several drivers is fixed by the nature of the 
 connection; and thus the comparative motions of all the pieces are 
 determined. As an example of converging trains, we may take a 
 compound steam engine, in which two or more pistons drive one 
 shaft, each by its own connecting-rod and crank. 
 
 213. Diverging Trains. One train of mechanism may diverge 
 into two or more; that is to say, one primary piece may act as 
 driver to two or more primary pieces, each of which may be the 
 commencement of a distinct train. In this case, as well as in that 
 of converging trains, the comparative motions of all the pieces are 
 determined. 
 
 Examples of diverging trains might be multiplied to any extent. 
 One of the most common cases is that in which a number of differ- 
 ent machines in a factory are driven by one prime mover: all 
 those machines are so many diverging trains. In many instances 
 there are diverging trains in one machine ; thus in almost every 
 
 * Let A D' be the position for which the capacity of the tube is known, 
 and let V be that capacity. Let A D and AD" be the positions of the tube 
 at the two ends of its stroke ; let V arid V" be the corresponding capacities ; 
 and let the lengths of the arcs G D, G D', G D" be denoted by s, s', s" 
 respectively. Then we have 
 
 V s = V s' = V" s"; and - : * : - : : V : V : V" 
 
 S S S 
 
 (1.) 
 
 The volume of fluid admitted or expelled at each stroke is as follows : 
 
 V" - V = V s' ( \ - - 1 ) = V(, '). ....... (2.) 
 
 \ s s i s s 
 
 The length of stroke of the point D is s s" ; hence the apparatus may 
 be regarded as equivalent to a cylinder and piston of that length of stroke, 
 and of the following transverse area : 
 
 V" _ V V' s' 
 
 - - =1 = 1-4 .................................. (3.) 
 
 "
 
 228 GEOMETRY OF MACHINERY. 
 
 machine tool there are at least two diverging trains one to pro- 
 duce the cutting motion, and the other the feed motion. 
 
 214. Train for diminishing Mm (nation-, of Speed. The fluctua- 
 
 tions in the velocity-ratio, when a revolving and a reciprocating 
 point are connected by means of a link, have been stated in Article 
 184, pages 196, 197, and in Article 188, pages 199 to 201. In 
 some cases it is desirable that the velocity-ratio of a reciprocating 
 point to a revolving point should be more nearly uniform. For 
 this piirpose a train of two combinations may be used, the first 
 primary piece being a rotating shaft, which may be called A; the 
 second, another rotating shaft, which may be called B; and the 
 third, the reciprocating piece, C. The connection of A with B is 
 by means of a pair of equal and similar two-lobed wheels (see 
 Article 109, page 97); and a crank on B, by means of a connecting- 
 rod, drives C. The two-lobed wheels are to be so placed that the 
 shortest radius of the wheel on B shall be^ in gearing with the 
 longest radius of the wheel on A at the instants when the crank is 
 passing its dead-points. The result to be aimed at in the arrange- 
 ment is, that each quarter-stroke of C shall be made as nearly as 
 possible in the time of one-eighth of a revolution of A ; and in order 
 that this may be the case, the following should be the angles moved 
 through by the two shafts respectively in given times : 
 
 Shaft A, .................. o 45 90 135 180 
 
 Shaft B, commencing at j 
 
 a dead-point of the ~o 60 90 120 180 
 
 crank, .................. ) 
 
 Hence it appears that B is alternately to overtake and to fall 
 behind A by 15. This angle, then, being given, the rules of 
 Article 109, page 98, are to be applied to the designing of the pitch- 
 lines of the wheels. The greatest and least radii of those wheels are 
 approximately 0'634 and 0'366 of the line of centres respectively. 
 
 The following are the comparative velocities, at different instants, 
 of a revolving point in A at a given distance from its axis, of a 
 revolving point in B at the same distance from its axis, and of a 
 point in C connected by a very long link with the point in B * : 
 
 * Mr. Willis, in his Treatise on Mechanism, investigates the figures of a pair 
 of wheels on A and B for giving exact uniformity to the ratio C -4- A. The 
 equations are as follows : Let c be the line of centres ; r, a radius of the 
 wheel on B, making the angle with the shortest radius ; r', the corresponding 
 radius of the wheel on A, making the angle 6' with the longest radius of this 
 wheel ; then we have 
 
 IT sin i Q, IT . 
 
 r = ' ~ ; ^ = c - r ; and fl' = versm 0. 
 
 Mr. Willis points out that the forms of the pitch-lines given by the equations 
 must in practice be slightly modified at the points which gear together when 
 the crank is at its dead-points.
 
 TRAINS COMBINATIONS IN CLASSES. 229 
 
 1 80 
 
 Angles moved through \ 
 by A,.. -- < 
 
 
 
 1-732 
 o 
 o 
 
 45 
 
 0-866 
 0-866 
 0-750 
 
 90 
 Q'577 
 
 I'OOO 
 
 0-577 
 
 135 
 
 0-866 
 0866 
 0-750 
 
 J 
 
 Velocity-ratio B 
 Velocity-ratio C 
 Velocity-ratio C 
 
 -A, 
 *B, 
 
 -A, 
 
 O 
 
 o 
 Mean value of each of the velocity-ratios C -4- B and C -f- A, 0-637. 
 
 A similar adjustment may be made by connecting the shafts A 
 and B by means of an universal joint (Article 191, page 203); the 
 fork on the shaft B being so placed as to have its plane perpen- 
 dicular to the plane of the axes when the crank is at its dead-points ; 
 the angle made by those axes with each other should be that whose 
 cosine is 0-577, viz., 54|. 
 
 The Double Hooke's Joint (Article 192, page 205) is an example 
 of a train in which the fluctuation of the velocity-ratio is corrected 
 exactly. 
 
 SECTION X. References to Combinations arranged in Classes. 
 
 215. Object of this Section. In the preceding sections the various 
 elementary combinations in mechanism have been arranged accord- 
 ing to the mode of connection. The object of the present section 
 is to give a list of such combinations, arranged according to Mr. 
 Willis's system that is, according to the comparative motion 
 with references to the previous Articles and pages of this treatise, 
 where the several combinations are described. Two deviations 
 from or modifications of Mr. Willis's system are used; first, the 
 addition, at the commencement of each Class, of references to places 
 where the comparative motions of two points in one primary piece 
 are treated of; and secondly, the placing of combinations in which 
 the connection is intermittent, iti a class by themselves, entitled 
 Class D. 
 
 216. CLASS A. Directional -Relation Constant Velocity-Ratio 
 Constant. 
 
 COMBINATIONS. 
 
 Velocity-Ratio that of Equality alone. 
 
 ARTICLES. PAGES. 
 
 Pair of Points in one straight-sliding Primary Piece, 43 22 
 
 Sliding Contact, Oldham's Coupling, 158 1 66 
 
 Bands, equal and similar Non-circular Pulleys, 167 182 
 
 Linkwork, Coupled Parallel Shafts, 181 194 
 
 Drag-link: Shafts in one straight line, 182 194 
 
 Double Hooke's Joint, 192 205 
 
 Double Hooke-and-Oldham Coupling,.... 193 206
 
 230 GEOMETRY OF MACHINERY. 
 
 Any Constant Velocity -Ratio. 
 
 
 
 
 ARTICLES. 
 
 PAGES. 
 
 Pair of Points in one Rotating Primary Piece 
 
 53 
 
 31 
 
 Pair of Points in one Screw, 
 
 60 
 
 37 
 
 Rolling Contact: Circular Toothless Wheels and 
 
 C 'to* 
 
 
 Sectors, and Straight Racks, 
 
 V tO 
 
 (106 
 
 
 92 
 
 Rolling Contact: Frictional Gearing, 
 
 in 
 
 IO2 
 
 
 ' 112 
 
 103 
 
 
 to 
 
 to 
 
 Sliding Contact; Circular Toothed Wheels and 
 
 141 
 
 J 39 
 
 Sectors, and Straight Racks, * 
 
 144 
 
 MS 
 
 
 to 
 
 to 
 
 
 151 
 
 157 
 
 
 1 J 52 
 
 157 
 
 Sliding Contact: Screw Gearing, v < 
 
 to 
 
 to 
 
 
 [157 
 
 166- 
 
 
 165 
 
 179 
 
 Bands and Pulleys, < 
 
 i77 
 
 to 
 
 
 200 
 
 214 
 
 Blocks and Tackle, 
 
 to 
 
 to 
 
 
 205 
 
 218 
 
 
 207 
 
 221 
 
 Hydraulic Connection : Pistons and Cylinders, 
 
 to 
 
 to 
 
 
 2IO 
 
 224 
 
 217. CLASS B. Directional-Relation Constant; Telocity-Ratio 
 Variable. 
 
 Mean Velocity-Ratio that of Equality alone. 
 
 j 108 95 
 
 Rolling Contact : Smooth Elliptic and Lobed Wheels, J to 
 
 I X 9 99 
 
 Sliding Contact : Toothed Elliptic and Lobed Wheels, 143 141 
 
 Pin and Slot Coupling, 159 167 
 
 Crossed Cord and Elliptic Pulleys, 175 189 
 
 Linkwork: Drag-Link, 182 194 
 
 Link for Contrary Rotations, 183 196 
 
 Single Hooke's Joint, 191 203 
 
 Single Hooke-and-Oldham Coupling, .... 193 206 
 
 Any Mean Velocity-Ratio. 
 
 | 107 92 
 
 Rolling Contact : Non-Circular Wheels and Sectors, < to to 
 
 (no 102
 
 COMBINATION IN CLASSES MECHANICAL POWERS. 231 
 
 Any Mean Velocity- Ratio Continued. 
 
 ABTICI^S. PACKS. 
 
 Sliding Contact : Teeth of Non-Circular Wheels and \ 
 
 Sectors, / T 3 MI 
 
 Bands with Non-Circular Pulleys, 175 188 
 
 Linkwork with Rocking Cranks and Levers... .. ! 4 *9 
 
 ( 190 202 
 
 tf 218 
 
 Blocks and Tackle, obliquely acting, 1 , to 
 
 ! 
 
 2O6 A 
 
 221 
 
 218. CLASS C. Directional-Relation Variable. 
 
 Sliding Contact : Pin and Slot, 159 168 
 
 ( 160 170 
 Cams, J to to 
 
 1 16 3 175 
 
 i 184 196 
 
 Linkwork : Rotating Cranks and Eccentrics, <J to to 
 
 ( 188 201 
 
 Levers for Multiplying Oscillations, 189 201 
 
 Band-links, 198 213 
 
 219. CLASS D. Intermittent Connection. 
 
 Sliding Contact: Intermittent Wheel- work, 142 139 
 
 Wipers and Pallets; Escapements, 164 175 
 
 I" 194 206 
 
 Linkwork: Clicks and Ratchets, < to to 
 
 I 196 211 
 
 Frictional Catches, 196 211 
 
 Slotted Link, 199 213 
 
 Hydraulic Connection: Yalves, Pumps, Hydraulic J 211 224 
 
 Press, Bellows, )2iiA 
 
 I 227 
 
 SECTION XI. Comparative Motion in the "Mechanical Poivers." 
 
 220. Clarification of the mechanical Powers. "Mechanical 
 
 Powers" is the name given to certain simple or elementary ma- 
 chines, all of which, with the single exception of the pulley, are 
 more simple than even an elementary combination of a driver and 
 follower; 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
 
 GEOMETRY OF MACHINERY. 
 
 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 shows both these classifications : 
 
 NEWER CLASSIFICATION. 
 
 THE LEVER, 
 
 OLDER CLASSIFICATION. 
 
 / The Lever. 
 
 ""( The Wheel and Axle. 
 f The Inclined Plane. 
 
 THE INCLINED PLANE, < The Wedge. 
 
 ( The Screw. 
 
 THE PULLEY, 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 dynamical 
 division of this Treatise. 
 
 221. I^erer Wheel nnd Axle. In the lever and the wheel and 
 axle of the older classification, which are both comprehended under 
 the lever of the newer classification, 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 stated in Article 53, page 31 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 
 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. 161 to 164; 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. 160. 
 
 In each of these figures the plane 
 of projection is normal to the axis, 
 and A is the trace of the axis. In 
 fig. 160, D and F are the traces of 
 two cylindrical pitch-surfaces. In 
 Fig. 160. && s - 161 * ^64, D and F are the 
 
 projections of the driving and 
 following points respectively.
 
 MECHANICAL POWERS. 
 
 233 
 
 The axis of a lever is often called tine 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. 161, 1G2, and 163. In other cases the lever is said to be bent, 
 as in fi". 164. 
 
 Fig. 161. 
 
 Fig 162. 
 
 D 
 
 Fig. 163. 
 
 The straight lever is said to be of one or other of three kinds, 
 according to the following classification : 
 
 In a lever of the first kind, fig. 161, the 
 driving and following points are at oppo- 
 site sides of the fulcrum A. 
 
 In a lever of the second kind, fig. 162, the 
 driving and following points are at the same 
 side of the fulcrum, and the driving point is 
 the further from the fulcrum. 
 
 In a lever of the third kind, fig. 163, the 
 driving and following points are at the same side of the fulcrum, 
 and the following point is the further from the fulcrum. 
 
 222. inclined riane Wedge. 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 43, pages 22, 23. 
 
 In the inclined plane, fig. 165, A A is the trace of a fixed plane; 
 B, a block sliding on that 
 plane in the direction B C; 
 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, C, in B C, let fall C D 
 perpendicular to B D; then 
 B D -:- B C is the ratio of 
 the component velocity in 
 
 Fig. 165. 
 
 the direction B D to the entire velocity of B. 
 
 In fig. 166, 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-
 
 GEOMETRY OF MACHINERY. 
 
 tion C D to the position c d; and if G e be drawn normal to the 
 plane C D, the ratio borne by the component velocity of the wedge 
 
 Fig. 166. 
 
 in a direction normal to its oblique face to its entire velocity will 
 be expressed by C e : C c. 
 
 223. Screw. 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 
 63, page 40) into a straight line : being the hypothenuse 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. 1 65 be taken to represent the pitch of the screw, and 
 D C, perpendicular to B D, the circumference of the circle described 
 by the point in question about the axis, B C will 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 -f- 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 C. This shows why the screw is comprehended under the 
 general head of the inclined plane, in the newer classification of 
 the mechanical powers. 
 
 224. Pulley. The term pullet/, in treating of the mechanical 
 powers, means any purchase or tackle of the class already described 
 in Section VII. of this Chapter, pages 214 to 221.
 
 235 
 
 CHAPTER Y. 
 
 OF AGGREGATE COMBINATIONS IN MECHANISM. 
 
 SECTION I. General Explanations. 
 
 225. Aggregate Combination Defined. "Aggregate Combinations" 
 
 is a term introduced by Professor Willis, to denote those assem- 
 blages of pieces in mechanism in which the motion of one follower 
 is the resultant of motions impressed upon it b} more than one 
 driver. The number of independently-acting drivers which impress 
 directly a compound motion on one follower cannot be greater than 
 three; because each driver determines the motion of at least one 
 point in the follower; and the determination of the motion of three 
 points in a body determines the motion of the whole body. In 
 most cases which occur in practice, the number of independent 
 drivers which act directly on one follower is two. 
 
 226. General Principle of tbeir Action. The follower which has 
 
 such a compound motion directly communicated to it by more 
 than one primary piece must necessarily be a secondary piece, 
 as denned in Article 37, page 17; its motion at any instant is 
 the resultant of the motions impressed upon it separately by the 
 pieces which act as its drivers; and the determination of that 
 resultant motion depends upon the principles already explained in 
 Chapter III. of this Division, pages 43 to 75. Several examples of 
 the motion of secondary pieces have been given in the preceding 
 Chapter, in treating of those secondary pieces, such as links and 
 bands, and the sheaves of running blocks, which act as connectors 
 in elementary combinations. 
 
 227. Aggregate Combination* terminating in a Primary Piece. 
 
 Very often an aggregate combination is of the nature of a train; 
 and although a secondary piece receives in the first instance a 
 compound motion from two or from three primary pieces, that 
 secondary piece communicates motion in the end to a primary 
 piece. In such cases the motion of that last primary follower may 
 be determined, by finding the motions which would be communi- 
 cated to it through the intermediate secondary piece or pieces by 
 the several primary drivers acting separately, and taking the 
 resultant of those motions. 
 
 228. Shifting Trains A secondary piece in an aggregate com- 
 bination has very often a form like that of a primary piece, and
 
 23G 
 
 GEOMETRY OF MACHINERY. 
 
 is distinguished from a primary piece only by the fact that its 
 bearings, instead of being carried by the fixed frame, are carried by 
 
 a moving frame; that moving 
 frame being one of the primary 
 pieces from which the second- 
 ary piece receives its motion. 
 For example, a wheel may turn, 
 about an axis which is carried 
 by an arm that turns about 
 another axis. The compound 
 motions of which such second- 
 ary pieces are capable have 
 been treated of in Articles 72 
 to 79, pages 51 to 62, and 
 Articles 81 to 86, pages 66 to 
 74. When such a secondary piece is to drive or to be driven by a 
 primary piece, or another secondary piece "hot carried by the same 
 
 Fig. 1G8. 
 
 moving frame, special contrivances, which may be called shifting 
 trains, have to be used in order to keep up the connection between 
 the two pieces during their various changes of relative position. 
 The following are examples : 
 
 I. When two pieces turning about parallel axes are connected 
 by toothed gearing, and one of them is free to shift its position, 
 along its axis relatively to the other, the LONG or BROAD PINION 
 may be used. In fig. 167 A A and B B are a pair of parallel 
 axes; C, a spur-wheel on A A; D, a pinion on B B; and the 
 breadth of the pitch-surface of D is made greater than that of C by 
 a length equal to the distance through which D is capable of being 
 shifted longitudinally. 
 
 II. When a toothed wheel, C C, fig. 168, gears with a rack,
 
 SHIFTING TRAINS. 237 
 
 D D, and either the rack is to be capable of turning about an axis, 
 B B, parallel to its pitch-line, or the axis A of the wheel is to be 
 capable of being moved round the axis B B at the end of an arm, 
 F A, the CIRCULAR RACK is to be used, being, as represented in 
 the figure, a solid of revolution generated by the rotation of the 
 trace of the rack-teeth about the axis B B. The pitch-line D D 
 becomes the trace of an imaginary pitch-cylinder generated by its 
 revolution about the axis B B; and the pitch-point E is the point 
 of contact of that cylinder with the pitch-cylinder of the wheel. 
 
 It is easy to see that by fixing a broad pinion on one part of a 
 shaft, and a circular rack on another, that shaft may receive at 
 the same time two independent motions of rotation about its axis 
 and translation along its axis respectively, from two different spur- 
 wheels; the result being a helical motion; and this is one of the 
 simplest of aggregate combinations. 
 
 III. TRAIN-ARM. When rotation is to be transmitted from a 
 fixed axis to a shifting axis, or from one shifting axis to another, 
 and the relative motion of the two axes is such that their distance 
 apart, and the angle which their dii-ections make with each other, 
 do not change, in other words, when one of the two axes revolves 
 round the other as if it were carried by a rotating arm, the con- 
 nection between those axes may be kept up by means of one rigid 
 frame, which carries any combination or train of mechanism suitable 
 for transmitting rotation from the one axis to the other: such a 
 frame is called a train-arm. 
 
 The general principles of the velocity-ratios which are communi- 
 cated by means of train-arms will be stated further on; but at 
 present one particular case requires special mention, it is that in 
 which the train carried by the arm is such that the two axes con- 
 nected by it are parallel, and the angular velocities of the pieces 
 which turn about them equal and in the same direction. In fig. 
 169 the plane of projection is supposed to be normal to the two 
 axes to be connected. A and B the traces of 
 those two axes, and A B their common per- 
 pendicular. A moveable frame or train-arm 
 connects the bearings of the axes with each 
 other, so that the distance A B is invariable ; 
 and that frame carries a train of mechanism ***-, 
 
 such as to transmit the angular velocity of 
 the piece which turns about A unchanged in pj g 1C9 
 
 velocity and direction to the piece which 
 turns about B. For example, those pieces may hare pairs of 
 parallel and equal cranks linked together by coupling-rods; or 
 they may be equal and similar pulleys connected by a band; or 
 equal and similar toothed wheels, with an intermediate wheel 
 gearing with both. The result is, that while the train-arm turns
 
 238 GEOMETRY OF MACHINERY. 
 
 into any other position, such as A b, the angular velocities of the 
 pieces which rotate about the axes A and B respectively continue 
 to be equal in magnitude and identical in direction. 
 
 IV. When rotation is to be transmitted between a pair of axes 
 whose common perpendicular alters in length as well as in direc- 
 tion, a COMPOUND TRAIN-ARM may be used, consisting of two or 
 more train-arms jointed together at intermediate axes. For 
 example, in fig. 170, A and C are the traces of two such axes. B 
 
 is the trace of an intermediate axis, 
 
 ,- ---..^ connected by means of two train-arms 
 
 \ with A and with C respectively, so 
 
 that the distances A B and B C are 
 y -* c invariable; while A B can be turned 
 ,-'' into any angular position about A, 
 
 S such as A b, and B C into any angular 
 
 position about B, such as b c. Then 
 the relative position of A and C can 
 be altered either in direction or in 
 pj J7Q distance, so long as their distance apart 
 
 does not exceed A B + BO; and the 
 transmission of motion will still be kept up by means of the trains 
 that are carried by the train-arms. 
 
 V. When motion is transmitted between two axes by means of 
 a band, the connection may be maintained during changes of the 
 relative position of those axes by means of STRAINING PULLIES and 
 GUIDING PULLIES so arranged as to keep the band tight. 
 
 229. methods of Treating Problems respecting Aggregate Com- 
 binations. The methods by which problems respecting aggre- 
 gate combinations are solved may be distinguished into two 
 classes. 
 
 I. In one class a piece which may be regarded as a train-arm, 
 or moving frame (and which may be designated by B), has a given 
 motion relatively to the fixed frame, A, of the machine; and at 
 the same time a secondary moving piece, C, has a given motion 
 relatively to B. The resultant of those two given motions is the 
 motion of C relatively to A; and the general rules for finding it in 
 various cases have been stated in Articles 73 to 77, pages 52 to 5 G, 
 and Articles 81 to 86, pages 66 to 74. 
 
 II. In the other class of methods the motions of three points in 
 a secondary piece that is free to move in all directions, or, more 
 frequently, the motions of two points in a secondary piece that is 
 guided so as to move in one plane, or about one fixed point, are 
 given; and the motion of the piece as a whole is to be deduced 
 from them. The general rules for doing this have been given 
 in Articles 69 to 71, pages 45 to 51. 
 
 There ia no difference in principle between the kinds of problems
 
 AGGREGATE VELOCITIES AND AGGREGATE PATHS. 239 
 
 that are treated by those two classes of methods respectively; the 
 choice of methods is a matter of convenience only. 
 
 230. Aggregate Combinations classed according to their Purposes 
 
 Aggregate Velocities Aggregate Paths. The classification of aggre- 
 gate combinations which will be adopted throughout the rest of 
 this Chapter is that of Mr. Willis, and is founded on the purposes 
 which the combinations are designed to effect. Those purposes 
 are distinguished into (I.) aggregate velocities, and (II.) aggregate 
 paths. 
 
 I. When an aggregate velocity is the object aimed at, the final 
 piece of the train is usually a primary piece, whose comparative 
 velocity, by the help of an aggregate combination, is made either 
 to have a certain constant value or to vary according to a law 
 which it might be difficult or impossible to realize by means of a 
 train of elementary combinations only. 
 
 II. When an aggregate path is the object aimed at, a point in a 
 secondary piece is made, by means of an aggregate combination, to 
 move in a path of a figure which may be different from that which 
 a point in a primary piece would describe. 
 
 The only paths which points in primary pieces can describe are 
 straight lines, circles, and screw lines;* and paths of all other 
 figures must be described by the help of aggregate combinations. 
 Sometimes, indeed, it is found convenient to use aggregate com- 
 binations for describing, either exactly or approximately, even 
 those elementary paths themselves the straight line, the circle, 
 ani the screw-line. For example, there is a numerous class of 
 aggregate combinations called parallel motions, whose object is to 
 make a point move sensibly in a straight line. 
 
 * La other words, paths in which both the curvature and the tortuosity 
 are either none or uniform. The curvature of a path is the reciprocal of the 
 radius of curvature. The tortuosity is the reciprocal of the length, measured 
 along the path, in the course of which the radius of curvature rotates round 
 a tangent to the path as an axis, through the angle which subtends an arc 
 equal to radius. In the case of a helix, or screw-line, let r be the radius of 
 the cylinder on which the screw-line is described, and p the pitch of that 
 
 line ; let q = be the radius of a circle whose circumference is equal to 
 
 ^ *7T 
 
 the pitch ; let be the obliquity of the screw -line to a plane normal to its 
 axis ; let p be its radius of curvature ; and let a be the reciprocal of the 
 tortuosity. Then q = r tan 0; and according to Article 64, page 41, the 
 radius of curvature is 
 
 p = r + = r sec 2 0. 
 r 
 
 Also, it can be shown that the reciprocal of the tortuosity is 
 
 <r = q -i = q cosec 2 = 2 r cosec 2 = p cotan 0. 
 2
 
 240 
 
 GEOMETRY OF MACHINERY. 
 
 A further subdivision of the purposes of aggregate combinations 
 leads to the following classification : 
 
 AGGREGATE VELOCITIES. 
 
 Production of Uniform Velocity-Ratios (as in Willis's Class A). 
 Production of Varying Velocity-Ratios (as in Willis's Classes 
 B and C). 
 
 AGGREGATE PATHS. 
 
 Description of Curved Paths, (Ellipses, Epicycloids, &c.) 
 Description of Sensibly Straight Paths (Parallel Motions). 
 
 231. ConTerging Aggregate Combinations. This term may be 
 
 applied to denote those trains in which the drivers in an aggregate 
 combination are themselves the followers in 
 aggregate combinations. By means of trains 
 of that kind, any number of component 
 motions may be combined. Suppose, for 
 example, that a piece, A, is driven jointly 
 by B and C, and that B is driven jointly 
 by D and E, and C by F and G ; then the 
 motion of A is the resultant of four com- 
 ponent motions, due respectively to the 
 actions of D, E, F, and G. 
 
 SECTION II. Production of Uniform 
 Aggregate Velocity- Ratios. 
 
 232. Differential Pulley and Windlass. 
 
 In this combination, two pulleys, B and C 
 (fig. 171), of different radii, rotate as one 
 piece about a fixed axis, A. An endless 
 chain, B D E C L K H, passes over both 
 pulleys. The rims of the pulleys are shaped 
 so as to hold the chain, and prevent it from 
 slipping. The lines in the figure represent 
 the pitch- lines of the pulleys and the centre 
 line of the chain respectively. As to the 
 relation between those lines and the actual 
 figures of the pieces, see Articles 166, 176, 
 pages 180, 190. One of the bights or loops 
 in which the chain hangs, D E, passes under 
 and supports the running block F. The 
 other loop or bight, H K L, hangs freely; 
 and very often the combination is driven by 
 hauling upon the part H K; which therefore 
 may be called the hauling part. It is evident 
 
 Kg. 171.
 
 DIFFERENTIAL PULLEY. 241 
 
 that the velocity of the hauling part is equal to that of the 
 pitch-circle B. Sometimes the compound pulley is driven by 
 other means; as by a second endless chain acting on a sprocket- 
 wheel. 
 
 In order that the velocity-ratio may be exactly uniform, the 
 radius of the sheave F should be an exact mean between the radii 
 of B and C : but it is not necessary to follow this rule strictly in 
 practice. In stating the velocity- ratio, however, it will be assumed 
 that the rule has been observed. 
 
 Let the velocities of the pitch-circles of B and be denoted by 
 B and C respectively. Then the proportion of those velocities to 
 each other is 
 
 C AC 
 B ~ AB 
 
 Let F denote the velocity of the running block. Then, if C were 
 a fixed point, and consequently C E a "standing part" of the chain, 
 the value of F would be ^ B, and the direction of its motion would 
 be upward (agreeably to the principles of Article 201, page 215). 
 Also, if B were a fixed point, and B D a standing part, the value 
 of F would be ^ C ; the negative sign being used to denote down- 
 ward motion. The actual value of F is the resultant of those two 
 components; that is to say, 
 
 whence we have the comparative motion of the larger pitch-circle 
 B, and the running block F, expressed by the following velocity- 
 ratio : 
 
 F_l f CH AB - AC 
 B~ 2 \ ' BJ 2 AB 
 
 The velocity of the running block is the same with that of the 
 pitch-circle of a pulley of the radius A G = - ~ , turning 
 
 with the same angular velocity with the actual differential or 
 compound pulley. 
 
 To calculate the length of chain required for a differential 
 pulley, take the following sum: half the circumference of A + 
 half the circumference of B + half the circumference of F + 
 twice the greatest distance of F from A + the least length of 
 the loop H K L. This last quantity is fixed according to 
 convenience.
 
 242 
 
 GEOMETRY OF MACHINERY. 
 
 The differential windlass or differential barrel (fig. 172) is identi- 
 cal in principle with the differential pulley; the difference in con- 
 struction being, that in the differential 
 windlass the running block hangs in the 
 bight of a rope whose two parts are wound 
 round, and have their ends respectively 
 made fast to, two barrels of different radii, 
 which rotate as one piece about the axis 
 A. The differential windlass is little used 
 in practice, because of the great length of 
 rope which it requires. That length is 
 expressed by the following sum : Twice 
 the least distance of the running block 
 from A + half circumference of running 
 
 T> 
 
 block + ^ x total distance through which 
 
 Fig. 172. F i s lifted ; and the last term is often an 
 
 inconveniently great quantity. 
 
 233. Compound Screw*. (A. M., 505.) A compound screw con- 
 sists of two screws cut upon the same spindle, and each having 
 a nut fitted upon it. The screw turns : one of the nuts is usually 
 fixed, so that the screw in turning in that nut is made to advance ; 
 the other nut slides, but does not turn ; and the sliding motion of 
 the second nut relatively to the first nut is the resultant of the 
 advance of the screw relatively to the first nut, and of a motion 
 equal and opposite to the advance of the screw relatively to the 
 second nut; that is to say, the second nut moves relatively to the 
 first nut as if it were acted upon by a single screw of a pitch equal 
 to the difference between the pitches of the two screw-threads that 
 are cut on the spindle; supposing those threads to wind the same 
 way. But if the threads are contrary-handed, for the difference 
 of their pitches is to be substituted the sum. 
 
 Fig. 173 represents a differential screw : that is, a compound screw 
 
 Fig. 113. 
 
 in which the threads wind the same way. N^ and N 2 are the two 
 nuts; Sj Sj, the longer-pitched thread ; S 2 S 2 , the shorter- pitched 
 thread : in the figure both those threads are left-handed. At each 
 turn of the screw the nut N 2 advances relatively to Nj through a 
 distance equal to the difference of the pitches. The use of the
 
 COMPOUND SCREW EPICYCLIC TRAINS. 243 
 
 differential screw is to combine the slowness of advance due to a fine 
 pitch with the strength of thread which can be obtained by means 
 of a coarse pitch only. 
 
 Fig. 174 represents a compound screw in which the two threads 
 are contrary-handed; and the effect of each turn of the screw is to 
 alter the distance between the nuts N x and N 2 by an amount 
 equal to the sum of the pitches of the threads, which are usually 
 equal to each other. This combination is used to tighten the 
 couplings of railway carriages. 
 
 234. Epityclic Trains with Uniform Action. An epicyclic train 
 
 for producing an uniform aggregate velocity-ratio consists essen- 
 tially of four parts, whose general arrangement may be held to be 
 represented by the diagram in Fig. 175 viz., the primary wheels 
 B and C, turning about the same 
 axis, O, with different uniform velo- 
 cities ; the train-arm A, being a 
 moveable frame, turning with an 
 uniform velocity about the same 
 axis; and the shifting train of 
 secondary pieces, carried by the 
 train-arm A, and transmitting an 
 uniform velocity-ratio from B to 
 C, in the manner of an ordinary 
 train. The shifting train may Fig. 175. 
 
 consist of any kind of mechanism 
 
 belonging to Class A; such as circular toothed wheels, whether 
 spur, bevel, or skew-bevel; screw-gearing; circular pulleys and 
 bands; links with equal parallel cranks; and double universal joints. 
 The comparative motions of the three primary pieces, A, B, and 
 C, are determined in the following manner : Let a, b, and c re- 
 present numbers proportional to the respective angular velocities of 
 those pieces; it being understood that rotations in one direction 
 are to be considered as positive, and those in the contrary direction 
 as negative. 
 
 First, suppose that B is fixed relatively to A ; that is to say, 
 that it simply turns along with A, having the same angular 
 velocity; or, in symbols, that b = a; then it is evident that C mus* 
 turn along with A also, with the same angular velocity; that is to 
 say, on this supposition, we have c a. 
 
 Next, let B have a different angular velocity from A; then 
 b a will represent the angular velocity of B relatively to A. 
 
 Determine, from the construction of the shifting train, the ratio 
 of the velocity of to that of B, as if the train-arm A were fixed ; 
 and denote that ratio by n; taking care to mark the value of n as 
 positive or negative, according as the rotations of B and C are in 
 similar or contrary directions. That ratio will also be the ratio
 
 GEOMETRY OF MACHINERY. 
 
 which the angular velocity of C relatively to A bears to the angii- 
 lar velocity of B relatively to A, when the train-arm A is in 
 motion ; that is to say, in symbols, 
 
 and this is the general equation of the action of an epicyclic 
 tarain. 
 
 Two particular cases may be distinguished, according as the 
 wheel C or the train-arm A is the follower in the combina- 
 tion. 
 
 CASE I. The wheel B and the train-arm A are driven by means 
 of diverging trains, with angular velocities proportional to given 
 numbers, b and a; then the proportionate angular velocity of C is 
 given by the following formula : 
 
 c = n (b a) + a = n b + (1 - n] a ............ (2.) 
 
 CASE II. The primary wheels B and C are driven by means of 
 diverging trains with angular velocities proportional to given 
 numbers, 6 and c; then the proportionate angular velocity of the 
 train-arm a is given by the following formula : 
 
 c n b b c 
 
 a = 
 
 1 -n ~ _ 1 1 -n (3.) 
 
 n 
 
 In some examples of both cases one of the primary wheels is fixed. 
 Let B be that wheel ; then 6 = 0; and we have 
 
 .(4.) 
 
 One of the uses of epicyclic trains is to obtain with precision 
 velocity-ratios in toothed wheel-work which are expressed by 
 numbers whose factors are too large to be suitable for the teeth 
 
 ft 
 
 of wheels. For example, r may be such a ratio; and it may be 
 
 
 
 possible to divide ^ into two parts, as expressed by the follow- 
 ing formula : 
 
 -= -- 
 
 such that each of those parts is expressed by numbers whose factors 
 are not too large ; and then, by using a train-arm with the velocity-
 
 EPICYCLIC TRAINS. 245 
 
 ratio -, and a shifting train with the velocity-ratio n, the required 
 
 velocity-ratio may be obtained with precision by means of wheels 
 of moderate size.* 
 
 Another use of epicyclic trains is to make the train-arm move, 
 for purposes of regulation (as in certain governors), with a 
 velocity proportional to the difference between the velocities of 
 the primary wheels B and C. This is best effected by causing the 
 primary wheels B and C to rotate in contrary directions, and to 
 connect them by means of a shifting train such that, when the 
 train-arm is at rest, the angular velocities of those wheels are 
 equal and opposite. This amounts to making n 1 in equation 
 3, and c = a negative quantity, say k; and then the expression 
 for the angular velocity of the train-arm becomes 
 
 b - k 
 
 For example, in Fig. 176, O is a vertical spindle, about which the 
 equal and similar bevel wheels B 
 aud C turn in opposite directions. 
 A is the train-arm, being a hori- 
 zontal spindle carried by a collar 
 which turns about the vertical 
 spindle. The shifting train consists 
 of a bevel wheel turning about the 
 spindle A, and gearing with the 
 wheels B and C. In order to pro- 
 duce a balance of forces, two, and Fig. 176. 
 
 sometimes three or four, equal and 
 
 similar horizontal spindles like A project from the collar, and carry 
 equal and similar bevel wheels. In the figure two are shown. The 
 result is, that when the wheels B and C turn in opposite directions 
 with equal speed, the train-arm stands still; but when the velocities 
 of those wheels become unequal, the train-arm turns in the direction 
 
 * The solution of this problem is to be obtained in any particular case by 
 a series of trials conducted generally in the following manner : Let n be an 
 
 approximation to the ratio -,, not containing factors exceeding what is con- 
 sidered a convenient limit (values of n may be found by the method of 
 continued fractions, Article 117, page 107). Then make a = l _ n J and 
 
 try whether the ratio ^ contains inconveniently large factors. The trial is 
 
 to be repeated with the various different values of n, until a satisfactory 
 result is arrived at. This method cannot fail, provided it is c only, and not 
 b, which contains inconveniently large factors.
 
 246 GEOMETRY OF MACHINERY. 
 
 of the greater of the two velocities, with a speed equal to half their 
 difference. Other applications of epicyclic trains, where the last 
 follower is a secondary piece, will be mentioned under the head of 
 aggregate paths. 
 
 SECTION III. Production of varying Aggregate Velocity-Ratios. 
 
 235. The Reciprocating Endless Screw may be used where it is 
 desired that there shall be periodic fluctuations in the ratio of the 
 speed of the follower to that of the driver. In this combination a 
 wheel is driven by a rotating screw, as in fig. 112, page 164, which 
 screw has at the same time a reciprocating motion along its axis. 
 
 236. Epicyclic Trains with Periodic Action are used for the Same 
 
 purpose. This is effected by communicating, by means of suitable 
 mechanism, such as a cam, or a crank and link, the required 
 reciprocating motion to the train-arm A, fig. 175, page 243. The 
 angular velocity of the follower, C, is expressed, as in Article 234, 
 
 by 
 
 c = n b + (1 n) a; 
 
 in which n b is a constant term, and (1 n) a a periodically 
 varying term; the factor 1 n being constant, and the factor a 
 periodic. 
 
 236 A. The Snn-and-Planet Motion is a SOft of epicyclic train 
 
 with periodic action. In fig. 177, C is a shaft which overhangs 
 
 Fig. 177. 
 Its bearing, and carries on its overhanging end a toothed wheel, C E,
 
 SUN-AND-PLANET MOTION ECCENTRIC GEARING. 247 
 
 called the sun-wheel. This gears with another toothed wheel, D E, 
 called the planet-wheel, which is made fast to the connecting-rod 
 D B, which hangs from one end of the lever or walking-beam, 
 A B. At the centre, D, of the wheel D E is a pin which is con- 
 nected with the shaft C by a link or bridle, C D (shown by 
 dotted lines); so that it revolves round the axis of C like a 
 crank-pin, making one revolution for each double-stroke of the 
 beam A B. 
 
 In the first place, to determine the mean ratio of the linear 
 velocity of the pin D to that of the pitch-circle of the sun-wheel, 
 C E, it is to be observed that the latter velocity is at every instant 
 equal to that of the pitch-point E in the planet- wheel. Now, the 
 motion of the planet-wheel is one of translation in a circle along 
 with the pin D, compounded with an angular oscillation to and fro 
 along with the rod D B. Hence the mean linear velocity of D is 
 equal to that of the pitch-circle of C E. 
 
 Secondly, as to the mean ratio of the angular velocity of the bridle 
 C D to that of the sun- wheel C E, it is obvious that as the mean 
 linear velocities of D, and of the pitch-circle of C E, are equal, their 
 mean angular velocities are inversely as the radii C E and D ; or 
 in symbols 
 
 mean angular velocity of C D C E 
 mean angular velocity of C E CD' 
 
 In the sun-and-planet motion, as originally contrived and con- 
 structed by Watt, the sun-wheel and planet- wheel were made of 
 equal radii; so that C D was = 2 C E; and the sun- wheel made 
 two turns for each revolution of the planet- wheel round it. 
 
 Thirdly, as to the ratio of the linear velocities of the points D 
 and E at any instant ; this is to be found by producing D C till it 
 cuts A B in I, which will be the instantaneous axis of the planet- 
 wheel j and then taking the proportion, 
 
 velocity of D ID. 
 velocity of E ~ I E 
 
 The mean value of this ratio is unity, as already stated. It attains 
 its greatest and least values in the two positions of the combination 
 when B D and C D are in one straight line, so that I coincides 
 with Bj and then its values are respectively 
 
 BD BD 
 
 and 
 
 BD-DEBD+DE 
 
 237. Eccentric Gearing. This is a combination for producing a 
 periodically varying velocity-ratio by means of a train of circular 
 wheels, one of which turns eccentrically about an axis. It is
 
 248 GEOMETRY OP MACHINERY. 
 
 nearly, but not exactly, equivalent in its action to a pair of elliptic 
 toothed wheels (Article 100, page 95). In fig. 178, A is the axis 
 of a shaft, which carries an eccentric circular toothed wheel. This 
 
 Fig. 178. 
 
 gears with a second toothed wheel, centred on a moveable axis, C, 
 which gears with a third toothed wheel, centred on a fixed axis, D. 
 The centres of the three wheels are linked together by the two 
 train-arms B C, C D; so that the wheels are kept always in 
 gearing, while the centre pin B revolves round the axis A. Sup- 
 pose the wheels on B and D to be of equal size. Then, if the train- 
 arms were fixed, the rotation of the first wheel about B would 
 produce a rotation of the third wheel about D, with equal speed 
 and in the same direction. The effect of the revolving of B about 
 A is to combine that rotation of D with an alternate increase and 
 diminution of speed, corresponding to the alternate diminution and 
 increase of the angle BCD. The greatest and least values of the 
 velocity-ratio take place when the line of connection, C B, touches 
 the two sides of the circle described by B about A; that is to say, 
 when that line is in the two positions marked OBI and C J B' 
 respectively. Let I and J be the points where C B cuts the line 
 of centres, D A, when in those positions; then the two correspond- 
 ing values of the velocity-ratio of D to A are respectively 
 
 AI A J 
 
 - 
 
 238. Aggregate Unkwork in General. A combination in aggre- 
 gate linkwork is usually of the following kind : A bar, or other 
 rigid body, capable of moving parallel to a given plane, has two of 
 its points connected by means of rods with two drivers: A third 
 point is connected by means of a rod with a follower. The motions
 
 AGGREGATE LINKWORK. 249 
 
 of the first two points, as compared with those of their drivers, are 
 determined by the principles of elementary combinations in link- 
 work, and so also is the motion of the follower, as compared with 
 that of the third point; but the determination of the motion of the 
 third point from that of the first two is a problem to be solved by 
 the principles of the motion of secondary pieces, Article 69, pages 
 45 to 48 ; that is, by the process of finding the instantaneous axis, 
 or by some equivalent process. 
 
 In most of the particular cases of aggregate velocities obtained 
 by linkwork which occur in practice, the three points in the bar 
 are either situated in one straight line to which their motions are 
 perpendicular, or are so nearly in that position that their com- 
 parative motions, as determined on the supposition of their being 
 in it exactly, are sufficiently near to the truth for practical 
 purposes. In such cases let A and B, figs. 179 and 180, be the 
 
 Fig. 179. Fig 180. 
 
 two points whose velocities at a given instant are given, and 
 the third point. Draw A a and B b perpendicular to A B, and of 
 lengths proportional to the given velocities, and in the proper 
 direction; join a 6; draw C c also perpendicular to A B, cutting 
 a b in c; C c will represent the velocity of C. The following 
 formula is the symbolical expression of the same rule, in which 
 a, b, and c denote the velocities of A, B, and C respectively : 
 
 a-BC + 6-0 A 
 
 TBT 
 
 The formula, as it stands, is applicable to the case in which C lies 
 between A and B, the velocities a, b, and c being treated as posi- 
 tive or negative according to their directions. When C lies 
 beyond B, B C is to be treated as negative, and C A as positive ; 
 when beyond A, C A is to be treated as negative, and B C as 
 positive. 
 
 The velocity of C may be regarded as the resultant of two com- 
 ponents, - , which would be its velocity if B were fixed ; and 
 
 A. xi 
 
 -, which would be its velocity if A were fixed. 
 
 AB 
 
 Trains of aggregate linkwork may be used to combine any
 
 250 GEOMETRY OF MACHINERY. 
 
 number of component motions. For example, in fig. 181, A, B, D, 
 and E receive motion from four different drivers : C has a motion 
 
 whose components depend 
 
 kl on the motions of A and B, 
 P 1 P and F a motion whose com- 
 ponents depend on the mo- 
 tions of D and E ; and G has 
 a motion whose components 
 depend on the motions of 
 C and F, and therefore on 
 the motions of A, B, D, and E, jointly. 
 
 239. Harmonic motion in Aggregate liinkwork. By harmonic 
 
 motion is to be understood the motion of a point which moves to 
 and fro in a straight line in such a manner that its velocity at every 
 instant is equal to the component, parallel to that straight line, of 
 another point which revolves uniformly in a circle. The length of 
 the straight line is called the travel of the reciprocating point, and is 
 equal to the diameter of the circle. (As to the component velocities 
 of a revolving point, see Article 55, pages 34, 35.) 
 
 Harmonic motion is exactly realized by any point in a slot- 
 headed sliding rod, driven by an uniformly rotating crank, as 
 explained in Article 159, page 169. The angle which the crank 
 makes with its dead points is called, in mathematical language, the 
 phase of the motion. The velocity of the reciprocating point varies 
 proportionally to the sine of the phase; and the distance of that 
 point from its middle position varies as the cosine of the phase. 
 
 Harmonic motion is approximately realized by any point in a 
 piece, such as a piston, which is driven by means of a connecting- 
 rod and an uniformly rotating crank. The extent of error in that 
 approximation may be expressed either in the form of greatest error 
 in position or of greatest error in velocity. The greatest error in 
 position is the distance of the reciprocating point from the middle 
 of its travel, when the crank is midway between its dead points ; 
 and when the line of stroke passes through the axis of the crank, 
 its value may be found either by constructing a figure, or by the 
 following formula : 
 
 in which I denotes the length of the line of connection, and c that 
 of the crank-arm. The comparative error in position is the ratio of 
 this error to the half-travel c; that is to say, 
 
 which, when I is many times greater than c, is nearly equal to
 
 HARMONIC MOTION IN AGGREGATE LINKWORK. 251 
 
 
 
 Q-J. The greatest error in velocity is the proportionate excess of 
 
 the greatest velocity of the reciprocating piece above that of the 
 crank-pin, as found by the rules of Article 188, pages 199 to 201. 
 When I is not less than 2 c, the value of the error in velocity is 
 given approximately by the expression 
 
 V 
 
 or 2j 2 , nearly. 
 
 When the line of stroke does not pass through the axis of rota- 
 tion of the crank, there are other errors arising from the two dead 
 points not being diametrically opposite. Those errors may be found 
 by applying the rules of Article 196, page 198. 
 
 The present and the following Article relate to cases in which 
 two points in a bar receive given transverse movements, which are 
 either exactly harmonic, or so nearly so that they may be treated 
 as harmonic for practical purposes, and are also of equal period, 
 and have a given constant difference of phase; and it is required to 
 find the extent of travel and the relative phase of the motion of a 
 third point, situated either exactly or nearly in one straight line 
 with the first two. 
 
 The following is the general rule for the solution of all 
 such cases. Some of its applications will be given in the next 
 Article: 
 
 RULE. In fig. 182 draw the straight line A B to represent the 
 bar in question, and let A and B represent the points whose motions 
 are given, and C the point whose motion is to be found. Perpen- 
 dicxalar to A B, draw A a to represent the half-travel of A, and B b 
 to represent the half-travel of B. These distances may be laid off in 
 both directions, so that a a shall represent the whole travel of A, 
 and b b that of B. The difference of phase of A and B is supposed 
 to be given ; that is to say, A moves as if driven by a crank A' A" 
 ( A a), and B as if driven by a crank B' B" ( = B b), which cranks 
 rotate with the same angular velocity, and make a given constant 
 angle with each other. 
 
 At A and B lay off the angles B A D = A B D, each equal to 
 half the difference of phase ; and about the triangle A D B describe 
 a circle. Join a b, a b, and through the point of intersection, E, 
 draw the straight line D E, cutting the circle in F. Join F A, 
 F B; then the angle A F B will be equal to the given difference of 
 phase. Lay off F a' = A a, and F b' = B b ; then F of and F b' will 
 represent the two cranks which actually or virtually drive A and 
 B, in their angular position relatively to each other. Join a b' ; 
 this will be parallel to A B (because it can be shown by plane
 
 252 
 
 GEOMETRY OF MACHINERY. 
 
 geometry that FAB and F a' b' are similar triangles). Finally, 
 draw the straight line F C, cutting a b' in d ; then the point C will 
 
 Fig. 182. 
 
 move almost exactly as if it were driven by a crank-arm, C' C', 
 equal in length to F c', and having the angular position relatively 
 to the cranks that drive A and B which F c has relatively to F a' 
 and F b '; that is to say, being in advance of the crank which drives 
 A by the angle a! F c', and behind the crank which drives B by 
 the angle 6' F c'.
 
 LINK-MOTIONS. 253 
 
 The travel of C may be represented in the figure by drawing 
 perpendicular to A B, the straight line cc = 2Cc = 2Fc'. 
 
 When the extent of travel of A and B is the same, part of the 
 trouble of the construction is saved; for the point F is found 
 simply by laying off the angles B A F = A B F, each equal to 
 half the supplement of the difference of phase. 
 
 The construction which has been described solves the problem 
 by drawing alone. Sometimes it may be convenient to use calcu- 
 lation combined with drawing; and then the whole process consists 
 in drawing the triangle F a b' in any convenient position, with its 
 legs, F a' and F I', equal to the half-travel of the points A and B 
 respectively, and its angle, a F &', equal to the difference of phase 
 of their motions, and dividing, by calculation, the base a' b' at c in 
 the same proportion in which A B is divided at C. 
 
 240. Link- motions for Slide-Valves belong to the kind of com- 
 binations mentioned in the preceding Article. The bar which 
 receives harmonic motion is called the link; it is in general slightly 
 curved, and only sometimes straight. Two points in it, marked 
 A and B in figs. 183 to 186, receive approximately-harmonic 
 motions from two eccentrics, E and F, on the engine-shaft, O, called 
 respectively the forward and the backward eccentrics. The link 
 carries a slider, 0. That slider is attached to the head of the slide- 
 valve spindle either directly (as shown at C in figs. 183, 184, and 
 185), or by means of an intermediate rod, C X (as in figs. 186, 
 187). The slider is capable of being adjusted to different positions in 
 the link, either by shifting the link (as in figs. 183, 184, and 
 185, which represent Stephenson's link-motion) or by shifting the 
 slider (as in fig. 186, which represents Gooch's link-motion), or by 
 shifting the link and the slider at the same time in opposite 
 directions (as in Allan's link-motion, represented in fig. 187). In 
 Stephenson's link-motion the form of the link is an arc of a circle, 
 concave towards the shaft, and of a radius equal to the length of 
 the eccentric rods E A, F B. In Gooch's link-motion the figure 
 of the link is an arc of a circle described about the head, X, of the 
 valve-spindle. In Allan's link-motion the link is straight, and the 
 adjustment of the proportions of the mechanism for shifting it will 
 be described presently. In each case the object is, that the shifting 
 of the position of the slider, C, relatively to the link, A B, shall 
 not cause any sensible alteration of the middle position of the 
 slide-valve. In each of the figures, O D represents the crank of 
 the engine to which the link-motion belongs, the positions of the 
 parts being those which they take when that crank is at a dead- 
 point. 
 
 In each of the figures, also, the eccentrics are represented simply 
 by points, E, F, which mark the centres of the eccentric discs. 
 It has already been explained, in Article 195, page 197, that an
 
 254 
 
 GEOMETRY OP MACHINERY. 
 
 eccentric is equivalent to a crank whose arm coincides with the 
 eccentricity; that is, the distance from the axis of rotation to the 
 centre of figure of the disc. 
 
 Fig. 183. 
 
 Fig. 184. 
 
 The general problem in questions as to the action of link-motions 
 is this : the dimensions of the parts being given, and the angles
 
 LINK-MOTIONS. 255 
 
 made by the eccentric-arms E, OF, with the crank O D ; also 
 the position of the slider C in the link A B; to find the length 
 and positions of what may be called the single virtual eccentric-arm 
 O g; that is to say, the arm of a single eccentric, which, if con- 
 nected with the valve-spindle by a rod marked g C in figs. 183 
 and 184, and g c in figs. 186 and 187, would produce, approxi- 
 mately, the same motion of the valve-spindle that the actual 
 mechanism produces. 
 
 The solution of that problem consists generally of two steps; 
 the first being to find the two virtual eccentric arms, O e, Of, 
 which, on the supposition of the eccentric rods, E A, F B, being 
 indefinitely long, or of slotted cross-heads being used instead of 
 eccentric rods, would be equivalent in their action to the actual 
 eccentrics with their oblique rods; and the second step being to 
 find the single virtual eccentric arm, O g, whose action is equivalent 
 to the combined action of those two, on the same supposition of an 
 indefinite length of rod or a slotted cross-head being used. 
 
 There are two different arrangements of the eccentric rods, which 
 are said to be crossed or open according as they cross each other or 
 not when the crank O D is pointing away from the cylinder. In 
 figs. 183 and 186 the rods are open; in figs. 184 and 187 they are 
 crossed. The two following rules apply to either arrangement : 
 
 I. To find the virtual forward and backward eccentric arms. 
 Through A and B draw straight lines parallel to the line of stroke, 
 O X, of the valve-spindle, and mark on those straight lines the 
 ends of the travel of the points A and B respectively, a', a', and 
 b'y b'. These points are to be found by Rule 1. of Article 186, 
 page 198. Bisect a' a' in a, and b' b' in b. Join O a and O 6; 
 these straight lines will pass nearly, though not exactly, through 
 the dead-points of the eccentrics E and F respectively. Lay off 
 the angles X O e = a O E and X O / - 6 O F, and make 
 O e = a a' and Of = b b f . Then O e and O / will be the required 
 virtual forward and backward eccentric arms. 
 
 II. To find the single virtual eccentric arm. Draw the straight 
 line ef, and in it take the point g, dividing e/'in the same pro- 
 portion in which the slider C divides A B. O g will be the 
 required virtual single eccentric arm. 
 
 The preceding rules are applicable to all the three constructions 
 of link-motion. But for each particular construction there are 
 special rules by which the process may be simplified, to the extent 
 of dispensing with the whole or part of the detailed process, except 
 for certain principal positions of the slider in the link. 
 
 III. In GoocKs Link-Motion (fig. 186) the link is hung or attached 
 to a fixed pin by means of the rod L J, and the alteration of the 
 position of the slider C in it is effected by shifting a lever (not 
 shown), one end of which is connected, by means of the rod N M,
 
 256 
 
 GEOMETRY OF MACHINERY.
 
 LINK-MOTIONS. 257 
 
 with the valve-rod C X. The result is that the virtual forward 
 and backward eccentric arms O e and O f are the same for all 
 positions of the slider in the link, and have only to be found once 
 for all. The figure shows this motion with open eccentric- rods ; 
 and the angle e Of = EOF - aOb; but it is also made with 
 crossed eccentric-rods, and then the angle e Of = EOF + aO b. 
 
 IV. In Stephensoris Link-Motion (figs. 183, 184) the link is 
 attached by the rod J L to the end of a lever (not shown) ; and by 
 shifting that lever the position of the link relatively to the slider 
 is changed when required. The consequence is that the virtual 
 eccentric arms O e and O / are different in length and position for 
 every different position of the slider in the link; and the application 
 of the general Rules I. and II. to a variety of such positions becomes 
 a tedious process. The time and labour, however, required for 
 that process are to a great extent saved by using the following 
 approximate method, which is sufficiently accurate for practical 
 purposes : When the link-motion is in full forward gear that is, 
 when C coincides with A the actual forward eccentric arm O E is 
 itself the virtual eccentric arm. When the link-motion is in full 
 backward gear that is, when C coincides with B the actual 
 backward eccentric arm O F is itself the virtual eccentric arm. 
 For intermediate positions proceed as follows (see fig. 185) : Draw 
 the link A B in that position in which the straight line A B is 
 parallel to the straight line E F, and let H be the centre of 
 curvature of the link when in that position ; its radius, H A = H B, 
 being equal to the length of each of the eccentric-rods E A, F B. 
 Then, iftJie eccentric-rods are open, draw E K parallel to A H, and 
 F K parallel to B H, cutting each other in K ; and through the 
 three points E, F, K describe a circle ; the arc E g F of that circle will 
 be a very close approximation to the curve that contains the ends 
 of all the virtual single eccentric radii. For a given position, C, of 
 the slider, take the point g, dividing the arc E F in the same pro- 
 portion in which C divides A B; and O g will be the required 
 virtual eccentric radius. If the eccentric-rods are crossed, draw E K' 
 parallel to A' H, and F K' parallel to B' H, cutting each other in 
 K'; and through the three points E, F, K' describe a circle. The arc 
 E g 1 F of that circle will be a very close approximation to the curve 
 containing the ends of all the virtual single eccentric radii. In this 
 arc take the point g, dividing it in the same proportion in which C 
 divides the link; and O g will be the required virtual eccentric radius. 
 
 V. Allan's Link-Motion Centre of the Shifting-Lever. In Allan's 
 link-motion (fig. 187) the middle point, J, of the link, and any 
 convenient point, M, in the valve-rod C X, are attached by rods 
 to the two ends of a lever, N L, which turns about an axis at P; 
 and the position of the slider C in the link A B is adjusted by 
 moving the handle P Q. In order that the motion may work 
 
 s
 
 258 
 
 GEOMETRY OF MACHINERY. 
 
 correctly, the centre of the shifting-lever must be properly placed ; 
 and the following is the rule for finding its position: In fig. 188, 
 
 Fig. 187.
 
 LINK-MOTIONS. 
 
 259 
 
 draw the isosceles triangle O A B, in which A B is the length of 
 
 the link, and A and O B are each equal to the length of an 
 
 eccentric - rod. About O 
 
 draw a circular arc through 
 
 A and B. Bisect the 
 
 straight line A B in J; 
 
 join O J, and produce it, 
 
 making J X equal to the 
 
 length of the valve-rod; 
 
 and mark, in J X, the 
 
 position of the point M to 
 
 which the rod M N is to 
 
 be attached. About X, 
 
 with the radius X J, 
 
 draw a circular arc, J B', 
 
 cutting the arc through 
 
 A and B in B'. Make 
 
 B' A' = B A, and O A = 
 
 O A; join B' A', and bisect 
 
 it in J'. Also join X B'; 
 
 and in it take X M' = X M. 
 
 Then M M' and J J' will 
 
 be the distances through 
 
 which M and J are respec- 
 
 tively to be shifted in order 
 
 to shift the slider C from 
 
 mid-gear to full gear that 
 
 is, from J to B, or to A, as 
 
 the case may be. Draw 
 
 the straight line M' J', cut- Fig. 188. 
 
 ting M J in R, and through 
 
 R draw R P perpendicular to X; then the axis of the shifting 
 
 lever is to be placed at a convenient point in the line R, P. 
 
 VI. Allan's Link-Motion Virtual Eccentric Arm. By the ap- 
 plication of the general Rules I. and II., find the 
 virtual eccentric arms in full forward gear, O e; in 
 full backward gear, Of; and in mid-gear, g ; and 
 draw them in a diagram, as in fig. 189. Then 
 through the points e, g, and /draw a circular arc; 
 this will be approximately the curve containing the 
 ends of all the virtual eccentric arms for different positions of 
 the slider; and it is to be used like the corresponding curve for 
 Stephenson's link-motion. If the rods are open, this arc will be 
 concave towards O, asegf; if crossed, convex, as e g f. * 
 
 * On the subject of Link-Motions, see Zeuner's Schiebersteuerungen, and 
 M'Farlane Gray's Geometry of the Slide- Valve. 
 
 
 
 Fig. 189.
 
 260 
 
 GEOMETBY OF MACHINERY. 
 
 It is to be remarked that in order to show distinctly the 
 principles of the construction of the figures illustrating this article, 
 all those dimensions which give rise to errors that is, to deviations 
 from the exact law of harmonic motion are exaggerated ; such as 
 the lengths of the link A B, and of the eccentric-arms O E and O F, 
 as compared with that of the eccentric-rods E A and F B. In 
 ordinary practice the link is from one-third to one-fifth, and each 
 of the eccentric-arms about one-twentyfifth of the length of an 
 ; eccentric-rod; and the effect of these propor- 
 
 tions is to make the deviation of the resultant 
 motion of the slide-valve from true harmonic 
 motion practically inappreciable. 
 
 241. Differential Harmonic motions. Two 
 
 primary pieces, having different motions, may 
 be regarded as constituting an aggregate com- 
 bination with respect to the motion of one of 
 them relatively to t^e other; because that 
 motion is the resultant of two components : 
 for example, if A be taken to denote the frame, 
 and B and C the two primary pieces, the motion 
 of C relatively to B is the resultant of the 
 motion of B relatively to A, and of a motion 
 equal and contrary to the motion of B relatively 
 to A. This principle has been already stated 
 in Article 42, page 21 The following is one 
 of its most frequent applications: In fig. 190, 
 let b and c be two pieces which have approxi- 
 mately harmonic motions in parallel directions 
 and of equal periodic time, but diffei'ing in 
 phase, given to them respectively by two cranks 
 or eccentric-arms, A B and A C, which turn as 
 one piece with the same angular velocity about 
 axis A, the angle BAG being the difference 
 of phase. Then the motion of the slide c re- 
 latively to the slide b is approximately the same 
 with that which would be produced by a crank 
 or eccentric-arm, B C, turning with the same 
 \ angular velocity; that is to say, it is an ap- 
 | proximately harmonic motion of the same 
 
 ', V A /' ; periodic time with the two elementary motions 
 
 \ \^ / : of B and C ; its half-travel is equal to B C, and 
 
 \ / its phase at any instant is that corresponding to 
 
 ^- -'' the direction of B C at that instant. This is what 
 
 Fig. 190. may be called a differential harmonic motion; 
 
 and upon such motions depends the action of 
 
 double slide-valves and moveable slide-valve seats in steam engines.
 
 CIRCULAR AGGREGATE PATHS. 
 
 261 
 
 SECTION IV. Production of Curved Aggregate Paths. 
 
 242. circular Aggregate Paths. Some circular aggregate paths 
 are traced by means of mechanical combinations, which are capable 
 also of tracing ellipses, if required; and these will be described 
 further on. The present Article relates to combinations in which 
 circular paths alone are traced. 
 
 Amongst such combinations may be classed the coupling-i-od 
 shown in fig. 32, Article 68, page 44; for every point in or rigidly 
 attached to that rod traces a circle of a radius equal in length to 
 the crank-arms by which the rod is carried; and the same takes 
 place in every case in which a secondary piece has a motion of 
 circular translation without rotation. For example, in fig. 191, A 
 is a centre pin, carrying a fixed spur-wheel in other words, a spur- 
 wheel without rotation. About the axis of that wheel there turns 
 a disc, carrying a set of diverg- 
 ing epicyclic trains. Each 
 epicyclic train consists of a 
 spur-wheel, B, gearing with 
 the fixed wheel A, and an- 
 other spur-wheel, C, gearing 
 with B. The last spur-wheel, 
 C, is exactly equal in radius 
 and in number of teeth to 
 the fixed wheel A; and the 
 consequence is, that each of 
 the wheels marked C has an 
 angular velocity equal to that 
 of A that is to say, equal to 
 nothing : in other words, when 
 the disc rotates, the wheels 
 marked C have a motion of 
 circular translation without 
 
 rotation. Let E be any point in one of the wheels C ; and draw A D 
 equal and parallel to C E; then E traces a circle round D, exactly 
 equal to the circle which the centre C of the wheel to which E 
 belongs traces round A at the same time. This combination is used 
 in spinning wire ropes. Each of the wheels C carries a bobbin from 
 which a wire or a strand is paid out as the spinning goes on; and 
 the effect of the absence of rotation in the wheels C is, that the 
 wires or strands are spun together without being twisted, which 
 would overstrain the material. 
 
 The combination shown in fig. 192 serves to guide a point C 
 along an arc, B C A, of a circle of a radius so great that it would 
 be inconvenient to guide the point C by connecting it directly with 
 
 Fig. 191.
 
 262 
 
 GEOMETRY OF MACHINERY. 
 
 the centre of the circle. It is based upon this well-known 
 geometrical principle : Let A and B be any two fixed points in the 
 circular arc to be traced; then the two chords C A, C B make 
 with each other a constant angle at C viz., 
 the supplement of one-half of the angle which 
 the arc A B subtends at the centre of the 
 circle. Two rods are fastened together at C, 
 so as to make with each other the proper con- 
 stant angle; and they are guided by passing 
 through sockets at A and B, which sockets are 
 free to turn about A and B respectively, but 
 not to move otherwise. Then, when the rods 
 are made to slide through the sockets, the 
 point C traces the required circular arc. The 
 angle made by the rods with each other may 
 be made adjustable by means of a screw or 
 otherwise, so as to vary the curvature of the 
 arc when required. 
 
 243. Epitrochoidai Paths. An epitrochoid is 
 the curve traced by a point in or rigidly at- 
 tached to a circle which rolls either inside or 
 outside of another circle (called the base-circle) ; 
 also, if two circles (as the pitch-circles of two 
 spur-wheels) turn in rolling contact with each 
 other about fixed axes, a point rigidly attached 
 to one of those circles traces an epitrochoid 
 upon a disc rigidly attached to the other. 
 
 When the tracing-point is in the circum- 
 ference of the rolling-circle, the curve traced 
 becomes that particular kind of epitrochoid 
 that is called an epicycloid. The properties 
 of this curve have already been explained in 
 Article 78, page 56, with a view to its adapta- 
 tion to the figures of the teeth of wheels. 
 
 If the circumferences of the rolling-circle and 
 of the base-circle are commensurable with each 
 other, the epitrochoid returns into itself, and 
 has a finite number of lobes or coils viz., the 
 Fig. 192. denominator of the fraction which, being in its 
 
 least terms, expresses the ratio borne by the 
 circumference of the rolling-circle to that of the base-circle. If 
 those circumferences are incommensurable, the epitrochoid does 
 not return into itself, so that the number of its lobes or coils is 
 indefinite. 
 
 When the rolling-circle rolls outside a base-circle of equal-radius, 
 the epitrochoid is one-lobed, and is called a cardioid.
 
 ITROCHOIDAL PATHS. 
 
 263 
 
 In the examples shown in figs. 193, 194, and 195 the ratio of 
 
 c 
 
 Kg. 194.
 
 364 
 
 GEOMETRY OF MACHINERY. 
 
 the rolling-circle to the base-circle is -, so that the epitrochoids are 
 
 o 
 
 three-lobed. Each figure shows 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, b; and the tracing-points of the external and in- 
 ternal rolling- circles are marked C and c respectively. 
 
 In fig. 193 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. 194 the tracing-points are inside the 
 rolling-circles; and the curves traced are prolate epitrochoids, 
 distinguished by their wave-like form. In fig. 195 the tracing- 
 
 Fig. 195. 
 
 points are outside the rolling-circles; and the curves traced are 
 curtate epitrochoids, distinguished by their looped form. 
 
 An important property of curves traced by rolling has already 
 been mentioned viz., that at every instant the straight line 
 joining the tracing-point and the pitch-point, or point of contact of
 
 ROLLED PATHS IN GENERAL. 
 
 265 
 
 the rolling-curve and base-curve, is normal to the traced curve at 
 the tracing-point. 
 
 The distance B C or b c 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. (See page 290.) 
 
 244. Rolled Paths in General. An infinite variety of curves 
 may be traced by rolling different pairs of non-circular curves upon 
 each other. In practice it is most convenient to limit this process 
 to pairs of non-circular curves which are capable of turning in 
 rolling contact about fixed parallel axes ; that is to say, which are 
 suitable for the pitch-lines of wheels, according to the principles 
 explained in Article 107, page 92. Suppose any such pair of 
 pitch-lines to turn in rolling contact with each other; then a point 
 rigidly attached to one of them (the rolling-line) will trace upon a 
 disc rigidly attached to the other (the base-line) a third curve ; at 
 every point in that third or traced curve the normal will be the 
 straight line traversing the tracing-point and the pitch-point, or 
 point of contact of the rolling-line and base-line.* 
 
 Fig. 195 A. 
 
 * The following rule gives an approximate method of determining the 
 figures of a rolling-curve and base-curve suited for tracing one lobe of a given 
 curve, B C, about a given pole, A (see fig. 195 A) ; the term pole being used to
 
 266 
 
 GEOMETRY OF MACHINERY. 
 
 24o. Elliptic Path* traced by Rolling form a particular case of 
 internal epitrochoids. In fig. 196 is represented a rolling-circle, 
 which rolls inside a base-circle of exactly twice its radius. Then 
 
 Fig. 196. 
 
 denote the trace of the axis of the hase -wheel upon the plane of the given 
 curve : 
 
 Let A B and A C be the greatest and least distances of the given curve 
 
 from the pole. Make the line of centres A D = A E = ; and 
 
 the tracing-arm D B = C E = 
 
 AB AC 
 
 With the line of centres A D 
 
 as radius, draw a circle, D E ; this will be the path of the axis of the rolling- 
 curve. Take a series of points, F 1} F 2 , F 3 , &c., in the curve to be traced, 
 and from each of them lay off, to the circumference of the circle, D E, a 
 distance, F G, equal to the tracing-arm; thus finding a corresponding
 
 ELLIPTIC PATHS TRAMMEL. 2G7 
 
 (considering a quarter of a revolution at a time), while the centre 
 of the rolling- circle traces a quadrant, B b, 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 c, of an ellipse whose semiaxes are A C = 
 AB + BC, andAc=CD = AB-BC; also a point C' outside 
 the rolling-circle, but rigidly attached 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 internal, and the 
 latter an external, ellipse. The proportions of the axes of either of 
 them may be indefinitely varied by adjusting the position of the 
 tracing-point; 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 that is, to the radius of the base-circle. 
 
 This is the principle of the mechanism commonly used for 
 turning ellipses. (See Addendum, page 290.) 
 
 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. 
 
 246. A Trammel is a substitute for a pair of rolling- circles 
 suited for tracing ellipses; but it is less used in mechanism, than in 
 drawing instruments. It depends on the following principles : 
 That every point in the circumference of the rolling-circle in fig. 
 196 traces a straight line through A; and that consequently, if 
 two points in a rigid body, so chosen as to be in the circumference 
 of one circle through A, be so guided as to move in straight lines 
 traversing A, the whole body will move as if it were carried by 
 that circle, rolling inside a circle of twice the radius. 
 
 In fig. 197, let X X and Y Y be the centre lines of two straight 
 grooves, cutting each other in A ; and let B and C be the centres 
 of two pins which slide along those grooves respectively at an 
 invariable distance, B C,from each other. Through B, perpendicular 
 to X X, and through C, perpendicular to Y Y, draw the straight 
 lines B G, C G, cutting each other in G; this point will evidently 
 be the trace of the instantaneous axis of a rigid body attached to B 
 and C. Join A G, bisect it in D, and about D draw a circle 
 
 series of points, G^ &c., in that circle. (Two only of these points are 
 lettered, to prevent confusion. ) 
 
 Draw the radii AG,, &c. Then, through the points F,, &c., draw a 
 series of normals, Fj H,, &c., cutting their corresponding radii in a series of 
 points, HJ, &c. These will be points in the base-curve. 
 
 To construct the rolling-curve, draw, in a separate diagram, the tracing- 
 arm gf = G F; draw the radii g h, &c., equal to the corresponding lines 
 G x Hj, &c., in the original diagram, and making the angles f g h lt &c., 
 equal to the corresponding angles 'F 1 G a H t , &c., in the original diagram; 
 then h lt &c., will be points in the required rolling-curve; g being the trace 
 of its axis, which is at the invariable distance A D from the axis of the 
 base-curve ; and/ the tracing-point.
 
 268 
 
 GEOMETRY OF MACHINERY. 
 
 through A and G. That circle will also traverse B and C, because 
 the angles A B G and A C G are right angles. Now, the length of 
 
 // 
 
 Fig. 197. 
 
 the chord B C in that cii-cle is constant, and it subtends at the 
 circumference the constant angles Y A X and B G C; therefore 
 the diameter A G of that circle is constant in all positions of the 
 pins B and C as they slide along the grooves ; therefore the several 
 positions of the instantaneous centre, G, are all in one circle de- 
 scribed about A ; therefore the motion of a rigid body attached to 
 the pins B and is the same as if it were carried by the circle 
 A B G C rolling inside a circle of twice the radius described about 
 A. Hence the point D in such a rigid body traces a circle about 
 A ; and any other point, such as E, traces an ellipse whose semi- 
 axes are respectively equal to A D + D E and A I) D E. The 
 straight line G E is at every instant a normal to the ellipse traced 
 by the point E. 
 
 In the ordinary form of the trammel, represented in fig. 198, the
 
 TRAMMEL. 
 
 2G9 
 
 grooves X X and Y Y are at right angles to each other; and the 
 moving rigid body is a straight rod, B C F, carried by two pins at 
 B aiid C, which are fixed in blocks that slide along the grooves. 
 
 Fig. 198. 
 
 The tracing-point is moveable, and can be adjusted to any required 
 position along the rod. When midway between B and C, at D, it 
 
 B C 
 traces a circle of the radius ; when at any other intermediate 
 
 a 
 
 point, such as B, it traces an ellipse whose semiaxes are equal re- 
 spectively to E B and E C; and if the rod be prolonged, a point F 
 in the prolongation traces an ellipse whose semiaxes are equal re- 
 spectively to F B and F C. In each case the axes of the ellipse 
 coincide with the centre lines of the grooves X X and Y Y. 
 
 When the trammel is oblique-angled at A, the positions of the 
 axes of the ellipse described by a given tracing-point, E (fig. 197), 
 are found as follows : Produce E D till it cuts the virtual rolling-
 
 270 
 
 GEOMETRY OF MACHINERY. 
 
 circle in a; join B a, C a; then E a B and E a C will be equal to 
 the angles made by the longer axis with X X and Y Y respectively; 
 and the shorter axis will of course be perpendicular to the longer. * 
 247. Feathering Paddle-Wheels exemplify a class of aggregate 
 combinations in which linkwork is the means of producing the 
 aggregate motion. Each of the paddles is supported by a pair of 
 journals, so as to be capable of turning about a moving axis parallel 
 to the axis of the paddle-wheel, while its position relatively to 
 that moving axis is regulated by means of a lever and rod con- 
 necting it with another fixed axis. Thus, in fig. 199, A is the 
 axis of the paddle-wheel ; K the other fixed axis, or eccentric-axis ; 
 B, E, N", C, P, M, D the axis of a paddle at various points of its 
 
 Fig. 199. 
 
 revolution round the axis A of the wheel ; B F, E H, 1ST Q, C R, 
 P S, M L, D G, the stem-lever of the paddle in various positions; 
 K F, K H, K Q, K R, K S, K L, K G, various positions of the 
 guide-rod which connects the stem-lever with the eccentric-axis. 
 When the end of the paddle-shaft overhangs, and has no outside 
 
 * The oblique-angled trammel is believed to be the invention of Mr. 
 Edmund Hunt.
 
 FEATHERING PADDLES. 271 
 
 bearing, the eccentric-axis may be occupied by a pin fixed to the 
 paddle-box framing; but if the paddle-shaft has an outside as well 
 as an inside bearing, the inner ends of the guide-rods are attached 
 to an eccentric collar, large enough to contain the paddle-shaft and 
 its bearing within it, and represented by the small dotted circle that 
 is described about K. One of the rods, called the driving-rod, is 
 rigidly fixed to the collai-, in order to make it rotate about the axis 
 K ; the remainder of the rods are jointed to the collar with pins. 
 
 The object of the combination is to make the paddles, so long as 
 they are immersed, move as nearly as possible edgewise relatively 
 to the water in the paddle-race. The paddle-race is assumed to be 
 a uniform current moving horizontally, relatively to the axis A, 
 with a velocity equal to that with which the axes B, &c., of the 
 paddle-journals revolve round A. Let E be the position of a 
 paddle-journal axis at any given instant; conceive the velocity of 
 the point E in its revolution round A to be resolved into two com- 
 ponents, a normal component perpendicular, and a tangential com- 
 ponent parallel, to the face of the paddle. Conceive the velocity of 
 the particles of water in the paddle-race to be resolved in the same way. 
 Then, in order that the paddle may move as nearly as possible edge- 
 wise relatively to the water, the normal components of the velocities 
 of the journal E and of the particles of water should be identical. 
 
 Let B be the lowest point of the circle described by the paddle- 
 journal axes; that is, let A B be vertical. Draw the chord E B. 
 Then it is evident that the component velocities of the points B 
 and E along E B are identical. But the velocity of B is identical 
 in amount and direction with that of the water in the paddle-race. 
 Therefore the face of a paddle at E should be normal to the chord 
 E B, or as nearly so as possible. Another way of stating the same 
 principle is to say that a tangent, E C, to the face of the paddle 
 should pass through the highest point, G, of the circle described by 
 the paddle-journal axes. CAB being tie vertical diameter of that 
 circle. 
 
 It is impossible to fulfil this condition exactly by means of the com- 
 bination shown in the figure ; but it is fulfilled with an approximation 
 siifficient for practical purposes, so long as the paddles are in the water, 
 by means of the following construction : Let D and E be the two 
 points where the circle described by the paddle-journals cuts the sur- 
 face of the water. From the uppermost point, C, of that circle draw 
 the straight lines C E, C D, to represent tangents to the face of a 
 paddle at the instant when its journals are entering and leaving 
 the water. Draw also the vertical diameter C A B, to represent a 
 tangent to the face of a paddle at the instant when it is most 
 deeply immersed. Then draw the stem-lever projecting from the 
 paddle in its three positions, D G, B F, E H. In the figure, that 
 lever is drawn at right angles to the face of the paddle; but the
 
 272 GEOMETRY OP MACHINERY. 
 
 angle at which it is placed is to a certain extent arbitrary, though 
 
 it seldom deviates much from a right angle. The length of the 
 
 3 
 stem-lever is a matter of convenience : it is usually about ^ of the 
 
 u 
 
 depth of the face of a paddle. Then, by plane geometry, find the 
 centre, K, of the circle traversing the three points, G, F, and H; 
 K will mark the proper position for the eccentric-axis; and a circle 
 described about K, with the radius K F, will traverse all the 
 positions of the joints of the stem-levers. 
 
 From the time of entering to the time of leaving the water, 
 paddles fitted with this feathering gear move almost exactly as 
 required by the theory ; but their motion when above the surface 
 of the water is very different, as the figure indicates. 
 
 To find whether, and to what extent, it may be necessary to notch 
 the edges of the paddles, in order to prevent them from touching 
 the guide-rods, produce A K till it cuts the circle G F H in L; 
 from the point L lay off the length, L M, of the stem-lever to the 
 circle D B E , and draw a transvei-se section of a paddle with the 
 axis of its journals at M, its stem-lever in the position M L, and 
 its guide-rod in the position L K. This will show the position 
 of the parts when the guide-rod approaches most closely to the 
 paddle. 
 
 Some engineers prefer to treat the paddle-race as undergoing a 
 gradual acceleration from the point where the paddle enters the 
 water to the point of deepest immersion. The following is the con- 
 sequent modification in the process of designing the gear : Let the 
 final velocity of the paddle-race be, as before, equal to that of the 
 point B in the wheel, and let the initial velocity be equal to that 
 of the point b, at the end of a shorter vertical radius, A b. Let D 
 be the axis of a paddle-journal in the act of entering the water, 
 and E the same axis in the act of leaving the water. Join b D 
 and B E; draw the face of the paddle at D normal to D b, the 
 face of the paddle at B vertical, as before, and the face of the 
 paddle at E normal to E B. Then draw the stem-lever in its three 
 positions, making a convenient constant angle with the paddle- 
 face ; and find the centre of a circle traversing the three positions 
 of the end of the stem-lever; that centre will, as before, mark the 
 proper position for the eccentric-axis. 
 
 248. Spherical Epitrochoidal Paths Z-Crank. A point rigidly 
 
 attached to a cone which rolls on another cone describes a spherical 
 epitrochoid, situated in a spherical surface whose centre is at the 
 common apex of the two cones. This sort of aggregate motion is 
 illustrated by Mr. Edmund Hunt's Z-crank. 
 
 In fig. 200, A A is a rotating shaft, carrying at B, B, two crank- 
 arms, which project in opposite directions, and are connected with 
 ach other by means of a cylindrical crank-pin, B B. The shaft,
 
 Z-CRANK. 
 
 273 
 
 crank-arms, and crank-pin, are all rigidly fastened together; and 
 they rotate as one piece about the axis A A. 
 
 The crank-pin is contained within a hollow cylindrical sleeve 01 
 
 tiibe, fitting it accurately, but not tightly, and free to rotate, 
 relatively to the pin, about the axis B B, but not to shift longi- 
 tudinally. From that tube, and in a plane normal to B B, and 
 traversing C, the intersection of B B and A A, there project any 
 required number of arms, such as C D, C D. Those arms move as 
 one piece with the tube; and each of them, at its end D, is con- 
 nected, by means of a ball-and-socket joint, with a link, and through 
 the link with a piston-rod, which has a reciprocating sliding move- 
 ment parallel to A A. 
 
 The lower part of the figure shows the position of the mechanism 
 after a quarter of a revolution has been made from the position 
 represented in the upper part of the figure. 
 
 The motion of the sleeve, with its arms, is compounded of a. 
 rotation about the fixed axis A A, and of a rotation with equal 
 speed in the contrary direction about the revolving axis B B. 
 
 Therefore, in the plane of those axes at any instant draw C E, 
 bisecting the obtuse angle B C A, and C E will be the instantaneous 
 axis of the sleeve and arms. 
 
 T
 
 274 GEOMETRY OF MACHINERY. 
 
 Draw C F, making the angle A C F = ACE; and C G, making 
 the angle B C G = B C E. Then EOF will be the trace of a 
 fixed cone, having A C A for its axis; and EGG will be the trace 
 of a rolling cone, having B C B for its axis; and the motion of the 
 sleeve with its arms will be the same as if it were rigidly attached 
 to the rolling cone. 
 
 Each of the points marked D describes a spherical epitrochoid, 
 shaped like a long and slender figure of 8, and situated in the surface 
 of a sphere of the radius C D, whose trace in the figure is marked 
 by a dotted circle. The trace of the fixed cone on that sphere is 
 projected in the figure by the straight line E F; that of the rolling 
 cone, in the upper part of the figure, by the straight line E G, and 
 in the lower part by a dotted ellipse. The centre of the base of the 
 rolling cone is marked H and h in the two parts of the figure 
 respectively. 
 
 SECTION V. Parallel Motions. 
 \ 
 
 249. Parallel motions in General. A parallel motion is a com- 
 bination of turning pieces in mechanism, usually links and levers, 
 designed to guide the motion of a reciprocating piece either exactly 
 or approximately in a straight line, so as to avoid the friction which 
 arises from the use of straight guides for that purpose. Its most 
 common application is to the heads of piston-rods. 
 
 Some parallel motions are exact; that is, they guide the piston- 
 rod head or other reciprocating piece in an exact straight line; but 
 these parallel motions cannot always be conveniently made use of. 
 Other parallel motions are only approximate; that is, the path of 
 the piece which they guide is near enough to a straight line for the 
 practical object in view; and these are the most frequent. They 
 are usually designed upon the principle, that the two extreme 
 posit.ions and the middle position of the guided point shall be 
 exactly in one straight line; care being taken, at the same time, 
 that the deviations of the intermediate parts of the path of that 
 point from that straight line shall be as small as possible. 
 
 There are purposes for which no merely approximate parallel 
 motion is sufficiently accurate ; such as the guiding of the tool in 
 a planing machine, whose motion ought to be absolutely straight. 
 
 25 U. Exact Parallel Motions. When a wheel- rolls round inside 
 a ring of exactly twice its radius, any point in the pitch-circle of 
 the wheel traces a straight line, being a diameter of the pitch- 
 circle of the ring (Article 245, page 266). This combination, then, 
 has sometimes, though seldom, been used as an exact parallel motion 
 for a piston-rod; the head of the piston-rod being jointed to a pin 
 at the pitch- circle of the rolling wheel, and the crank to another 
 pin at the centre of that wheel.
 
 275 
 
 d- 
 
 201. 
 
 Fig. 201 represents another exact parallel motion, first proposed, 
 it is believed, by Mr. Scott Russell. 
 
 The arm C D turns on the axis 0, and is jointed at D to the 
 middle of the bar A D B, whose 
 length is double of that of C D, and 
 .one of whose ends, B, is jointed to a 
 slider, sliding in straight guides 
 along the line C B. Draw B E 
 perpendicular to C B, cutting G D 
 produced in E; then E is the in- 
 stantaneous axis of the bar A D B; 
 and the direction of motion of A is 
 at every instant perpendicular to E 
 A; that is, A moves along the 
 straight line A C a. While the 
 stroke of A is A C a, extending to 
 
 equal distances on either side of C, and equal to twice the chord of 
 the arc D d, the stroke of B is only equal to twice the deflection 
 of that arc ; and thus A is guided through a comparatively long 
 stroke by the sliding of B through a comparatively short stroke, 
 and by rotatory motions at the joints C, D. B. This may be called 
 the grasshopper parallel motion. 
 
 251. Approximate Grasshopper Parallel motion. The point B, 
 
 instead of sliding between straight guides, may be carried by the 
 end of a lever which, in its middle position, is parallel to a A, 
 and which is so long that the deviation of the arc described by B 
 from a straight line is insensible. (See also page 292.) 
 
 252. Watt's Parallel motion General Description. The general 
 
 construction of the ordinary form of Watt's approximate parallel 
 motion is shown in fig. 202. A c is 
 one arm of the walking-beam of the 
 engine, turning about an axis at the 
 main-centre c. A B is the main- 
 link, connecting the end A of the 
 beam with the piston-rod B D. T t 
 is the back-link, equal and parallel c 
 to the main-link; and B T is the 
 parallel-bar, equal and parallel to 
 the part A t of the walking-beam, 
 and completing the parallelogram 
 A t T B. The piston-rod head, B, is 
 to be guided so that its highest, middle, and lowest positions shall be 
 in one straight line; and this is effected by guiding in the same 
 manner the point P, where the straight line B c cuts the back-link. 
 The guiding of the point P is effected by means of the radius-bar, or 
 bridle, C T, which is a lever that turns about an axis at C, and is 
 
 Fig. 202.
 
 276 
 
 GEOMETRY OF MACHINERY. 
 
 jointed at T to the back-link. From the point P a pump-rod is 
 often hung. Some variations in detail will be explained further 
 on. The links, bridle, and parallel-bar usually consist each of a 
 pair of equal and similar pieces, connected respectively with the 
 two sides of the piston-rod head. 
 
 Fig. 203 shows on a larger scale the principle according to 
 which the point P is guided. S s is the line of stroke of that 
 point; P I} P 9 , and P 3 , its upper, middle, and lower positions. 
 T P t (shown in three positions, numbered 1, 2, and 3 respectively) 
 
 Fig. 803. 
 
 is the link; CT and ct, the two levers which guide that link, 
 each shown in three positions. In designing a parallel motion 
 the principal problem is to adjust the relative positions and pro- 
 portions of the levers and link, so that the three points P x , P 2 , and 
 P 8 shall lie in the straight line S s. There are also subordinate 
 problems, of which the first has for its object to make the devia- 
 tions of the point P from the straight line of stroke, in positions 
 intermediate between P 1; P 2 , and P 3 , as small as possible. It will 
 be shown further on that those deviations arise from the varying 
 obliquity of the link T t to the line of stroke S s ; therefore each 
 lever should be so placed relatively to the link that the greatest 
 obliquity of the link shall be as small as possible ; and for that 
 purpose the link, in its positions of greatest obliquity to the right 
 and left of the line of stroke respectively, should make equal angles 
 at opposite sides of that line ; which condition is to be fulfilled by 
 making the middle position of each lever perpendicular to the 
 line of stroke, and making that line bisect, in the points M and in,
 
 PARALLEL MOTIONS. 
 
 "211 
 
 the deflections T 2 V and t<,v of the arcs described by the points 
 1 and t respectively. This adjustment is to be made for the beam 
 and Jink before proceeding to design the parallel motion strictly 
 so called ; that is, the position and dimensions of the bridle rela- 
 tively to the beam and link. 
 
 Another subordinate problem is to design the parallelogram, 
 when a second point, such as B in fig. 202, is to be guided. 
 
 253. Walt's Parallel Motion Rules for Designing. I. Given (in 
 
 fig. 204) the line of stroke, G D, of a piston-rod, the middle position 
 
 of its head, B, and the centre, A, of a walking-beam or lever, which. 
 
 in its middle position, A D, is 
 
 perpendicular to G D : to find 
 
 the radius of the lever, so that 
 
 the link connecting it with B 
 
 shall deviate equally to the 
 
 two sides of G D during the 
 
 motion; also, the length of 
 
 the link. 
 
 Make D E = - stroke; join 
 
 A E; and perpendicular to it 
 draw E F, cutting A D pro- 
 duced in F; A F will be the 
 required radius. Join F B; 
 this will be the link. 
 
 II. Given the data and 
 results of Rule I. ; also the 
 point, G, where the middle 
 position of a bridle or second 
 lever, connected with the same 
 
 Fig. 204. 
 
 link, cuts G D : to find the second lever, so that the two extreme 
 positions of B shall lie in the same straight line, G B D, with the 
 middle position. 
 
 Through G draw a straight line, L G K, perpendicular to G D ; 
 produce F B till it cuts that line in L; this point will be one end 
 of the required second lever at mid-stroke, and F L will be the 
 entire link. Then, in D G lay off D H = G B; join A H, and 
 produce it till it cuts L G K in K ; this will be the centre for tho 
 second lever. 
 
 III. Given (in fig. 205) the middle positions, A C and B E, 
 (parallel to each other) and the extreme positions, A D, A D', B F, 
 B F 7 , of two levers centred at A and B respectively : required the 
 radii of those levers, so that a link connecting them shall deviate 
 in direction by equal angles alternately to the two sides of a per- 
 pendicular to A C and B K 
 
 From B let fall B G perpendicular to A C (produced if necessary).
 
 278 
 
 GEOMETRY OF MACHINERY. 
 
 Draw A H bisecting the angle CAD. Also, at the point G, lay off 
 \F the angle AGH = - E B F; 
 
 and let G H cut A H in H. 
 Draw H K perpendicular to 
 H A, and H L perpendicular to 
 H G, cutting A G in K and L 
 respectively. In B E lay off 
 B M = G L. Join K M. Then 
 will A K and B M be the re- 
 quired radii of the levers, and 
 K. M the link connecting 
 them. 
 
 IY. In the link K M, found 
 by the preceding rule, to find 
 the point whose middle and two- 
 extreme positions lie in one 
 straight line; also the length 
 of stroke of that point. 
 
 Through H draw P H N per- 
 pendicular to A K and B M. 
 The required length of stroke 
 is 4 P H. The required point 
 "' is R, where P H N cuts K M; 
 but as the acuteness of the 
 angle of intersection causes this 
 method of finding B, to be want- 
 ing in precision, it is better to 
 proceed as follows : Draw the 
 straight line A B, cutting N P 
 in Q ; then lay off P R = N Q : 
 R will be the required point. 
 
 Y. Given (in fig. 206 or in fig. 
 206 A) the main-centre, A, the 
 middle position of the main- 
 lever, A F, the piston-rod head, 
 B, and its length of stroke; 
 the radius, A F, of the lever, 
 and the main-link, F B, having 
 been found by Rule I. Let the 
 figure represent those parts at 
 mid-stroke; and let it be re- 
 quired to construct a parallel 
 motion consisting of a parallelo- 
 gram, C E D F (in which C E = 
 F D is called the parallel-bar, 
 
 fig. 205. 
 
 D 
 
 Fig. 206.
 
 PARALLEL MOTIONS. 
 
 279 
 
 or bridk ' H 
 
 
 i B, cutting 
 
 lever H E, such that the 
 middle and extreme posi- 
 tions of G shall lie in one 
 straight line. 
 
 (The point G shows 
 where a pump-rod may, if 
 convenient, be jointed to - 
 the back-link). 
 
 Fig. 206 B represents 
 that case of the application 
 of this rule in which the 
 points B and C coincide 
 with each other; and it 
 requires no separate de- 
 scription except the sub- 
 stitution of B for C in the 
 
 Rule. This construction Fi S- 206 * 
 
 is the same with that which 
 
 is roughly represented in r B 
 
 fig. 202. 
 
 VI. Given (in fig. 207) 
 the main-centre, A; also 
 the main-lever or walking- 
 beam, A F, and main-link, 
 F B, found by Rule I. Let 
 B be the middle position, 
 and C and C' the two ex- 
 treme positions, of the 
 point where the parallel- 
 bar is jointed to the main- 
 link, whether at the piston- Fig. 206 B. 
 rod head or at some other 
 
 point. Let A G and A G' be the two extreme positions of the 
 main-lever. Let H be a convenient point in the straight line, B H, 
 parallel to F A, chosen as the centre for the bridle ; and let H J 
 be a convenient length chosen for the radius of that lever; J being 
 the middle position of the point where it is jointed to the parallel- 
 bar. 
 
 Required, the points where the back-link ought to be jointed to 
 the main-lever, A F, and the bridle, H J, respectively. 
 
 About H, with the radius H J, describe a circular arc; then take 
 the length, B J, of the parallel-bar in the compasses, and lay off the
 
 280 
 
 GEOMETRY OF MACHINERY. 
 
 same length from C aud C' respectively to K and K' in that arc : 
 C K aud C' K.' will be the extreme positions of the mrallel-bar. 
 
 Fig. 207 
 
 Join H K, H K': these will be the extreme positions of the bridle. 
 Then, by Rule III., find the ends, M and N, of the back-link, M N, 
 which is to connect the bridle with the main-lever. 
 
 The two extreme positions of the back-link are marked P Q, 
 FQ'. 
 
 If it be desired to joint a pump-rod to the back-link, the proper 
 point for that purpose may be found by Rule IV.
 
 PARALLEL MOTIONS. 
 
 281 
 
 254. Watt's Parallel .notion Extent of Deviation. The follow- 
 ing rule serves to determine 
 the extent of the greatest 
 deviation of the guided point 
 in Watt's parallel motion from 
 an exactly straight path. In 
 fig. 208, let D G be the line of 
 stroke ; A F and K L, the two 
 levers ; F L, the link ; and B, 
 the guided point : the moving 
 parts being in their middle 
 positions. About A and K 
 respectively, draw the circular 
 arcs F / and L Z, being the 
 paths of the points F and L, 
 cutting the line of stroke in/ 
 and I Take the length F L 
 of the link in the compasses; 
 lay it of from one of those 
 arcs to the other, by trial, or 
 by the eye, so as to be parallel 
 to D G; and mark its ends,/' 
 arid I. 
 
 Draw the straight line/ I', 
 cutting K L in G'; this will be 
 the position of the link when its 
 deviation towards K is greatest; 
 and G G' will be that deviation. 
 The greatest deviation towards 
 A will be of equal extent, and 
 will take place when the levers 
 make nearly equal angles with 
 their middle position in the 
 contrary direction. * 
 
 Fig. 208. 
 
 * To calculate the greatest deviation approximately, let the radii A F and 
 K L be denoted respectively by a and a'; the distances, A D and K G, of 
 the centres from the line of stroke, by b and 6' ; the distance I) G by c; the 
 length of the link F L by 1; and the greatest deviation G G' by d. Also 
 make 
 
 -tf^ and 
 
 Then 
 
 Df= 
 
 d = 
 
 f;, very nearly. 
 
 (i.) 
 
 \fiott continued on nextpayt.
 
 282 GEOMETRY OF MACHINERY. 
 
 As is shown in the note, the fraction expressing the propor- 
 tion borne by the deviation to the length of stroke varies, to a 
 rough approximation, inversely as the length of the link, inversely 
 as the cube of the mean of the lengths of the levers, and directly 
 as the fourth power of the stroke ; and when the angular stroke is 
 42, and the link about equal in length to the levers, that fraction 
 is about ToVtrth. In order to make the deviation visible in the 
 figure, the dimensions which tend to increase it are much ex- 
 aggerated beyond those which are admissible in practice. 
 
 The rules of Article 253 give the following values for some of the data 
 in the preceding formula in terms of the stroke s :- 
 
 o' = b' + 
 
 16 b" 
 a + a'-b-b'^l 
 
 \/i'*<ri: 
 
 V 
 
 When the stroke is not too great, compared with the lengths of the levers, 
 we may also put 
 
 /nearly = /' nearly = ~- 
 o 
 
 whence we obtain the following approximate formula, as being near enough 
 to the truth in practice, for the proportion borne by the greatest deviation 
 to the length of stroke : 
 
 d , s* (b + fr) 
 
 7 near] y = iWFFs* ........................... < 2 -> 
 
 For example, let 6 = 6' = c; and let s = b; which gives an angular stroke 
 of about 56^; then 
 
 = 0-00065. 
 
 3 
 
 Next, let b = b' = c; and let s = -gb; which gives an angular stroke of 
 
 about 42 ; then 
 
 The Radius of Mean Curvature of the path of the guided point is found by 
 the following formula : 
 
 32 d
 
 PARALLEL MOTIONS. 
 
 283 
 
 255. Tracing Approximate Circular Arcs by the Parallel lUotion. 
 
 An intermediate point in the link of a parallel motion, not 
 coinciding with the point whose path approximates to a straight 
 line, moves nearly in an arc of a circle ; and this fact may be made 
 useful in the approximate tracing of arcs of long radius. The 
 approximation is not sufficient for purposes which require the 
 utmost precision, such as cutting the slots in curved links for slide- 
 valve gear; but where less precision is necessary, as in shaping 
 barrel staves, it might prove useful. The following are rules 
 relating to this process : 
 
 I. To find the radius of a circular arc traversing the extreme and 
 middle positions of a given point in the link of a parallel motion. 
 In fig. 209, let A Fand K L be the levers, in their middle position; 
 F L, the link; B, the point which is guided nearly in a straight 
 
 Fig. 209. 
 
 line; D G, its line of stroke; f, another point in the link. Per- 
 pendicular to G D, draw yd a, cutting it in d. In D G- produced, 
 lay off G h = B d; join h K, and produce it till it cutsfd a in a' r 
 this will be the centre of the circular arc traversing the middle and 
 extreme positions of the pointy"; and af will be the radius of that 
 arc. 
 
 II. To design a parallel motion which shall guide a point ap- 
 proximately along a given circular arc, B "D C, fig. 210, of which D 
 is the middle point and A the centre. 
 
 Draw the straight chord B C, cutting AD in E; bisect the 
 deflection D E in F; through F draw F G parallel to B C (and 
 thei-efore perpendicular to A D). Choose any convenient position, 
 I, for the centre of motion of one of the levers, and draw I G
 
 284 
 
 GEOMETRY OF MACHINERY. 
 
 and 
 
 parallel to D A. Lay off G H = E B - B C; join I H; 
 
 perpendicular to this line draw H K. cutting I G produced in K ; 
 I K will be one of the levers. Join K D ; this will be part of the 
 link; and M, where K D cuts F G, will lie the point in the link 
 which moves nearly in a straight line ; but that point is found with 
 more precision as follows : Join I A, cutting G D in L, and lay 
 off F M = G L. Produce K M D; and make M X = M K ; then 
 
 Fig. 210. 
 
 draw IS" P parallel and equal to I K; K N will be the link, and 
 K" P the other lever, centred at P. 
 
 III. To find approximately the greatest deviation of the point 
 which is guided nearly in a circular arc. Find by the rule of the 
 preceding Article the greatest deviation of the point which moves 
 nearly in a straight line. The greatest deviation of any other
 
 PARALLEL MOTIONS. 
 
 285 
 
 point in the link, from its approximate circular path, is proportional 
 very nearly to the product of the lengths of the segments into which 
 it divides the link. For example, in fig. 209 we have 
 
 greatest deviation of f ~Lf'f~F 
 greatest deviation of B = L B B F D< 
 
 r greatest deviation of D K D D N 
 
 and in ng. 210, ^ ^-. -^r^ = T , -., , nearly. 
 
 greatest deviation of M K M M N J 
 
 256. Roberta's Parallel Motion. This might also be called the 
 W parallel motion, from its shape. It is considered to be a con- 
 venient form for horizontal engines. Two cases of this combination 
 are shown in the figures: a simpler case in fig. 211 ; a more com- 
 plex case in fig. 212. 
 
 I. In Fig. 211, let D E be the straight line of stroke which the 
 guided point is to follow approximately, and A the middle of 
 that line. Draw two D 
 
 isosceles triangles, 
 DBA, ACE, with 
 equal legs, D B A B 
 = A C = C E. The 
 length of each of these 
 legs should not be less 
 thanD Ax 0-843; but 
 it may be as much 
 greater as the avail- 
 able space will permit ; 
 and the greater it is 
 the more accurate will 
 be the motion. Join Fig. 211. 
 
 B C, which of course 
 
 is = D A = A E. Then ABC represents a rigid triangular frame, 
 of which the apex A is the guided point; while the angles of the 
 base, B and C, are jointed to two levers or bridles, B D, C 
 which turn about axes at D and E respectively. The two extreme 
 positions of the triangle ABC are marked respectively D B C' and 
 E B" C", the points B" and C' coinciding. 
 
 The reason for prescribing that the length of each leg of the 
 triangles shall not be less than the base x 0-843 is, that this pro- 
 portion between these lengths makes the points B', CT, and E lie in 
 one straight line, and the points C", B", and D in another straight 
 line in the two extreme positions of the combination respectively. 
 
 II. The second arrangement, fig. 212, is to be used in those 
 cases in which it may be inconvenient to have the axes of motion 
 D and E of the bridle-levers traversing the line of stroke. Let 
 A' A A" be the line of stroke, and A its middle point Draw an
 
 286 
 
 GEOMETRY OF MACHINERY. 
 
 isosceles triangle, A B C, of a convenient size and figure, with its 
 apex, A, at the middle point of the stroke, and its base, B C, parallel 
 to the line of stroke. Draw the same triangle in its two extreme 
 positions, A' B' C' and A" B" C', leaning over equally in opposite 
 
 Fig. 212. 
 
 directions ; the sides A' B r and A" C" may conveniently be made 
 vertical; but this is not essential. Find, by plane geometry, the 
 centre D of a circular arc traversing the three points B', B, B" ; also 
 the centre E of a circular arc traversing the three points C', C, C'. 
 D B and E C will be the two bridle-levers, and D and E the bases 
 of their axes of motion. 
 
 ADDENDUM TO ARTICLE 142, PAGE 141. 
 
 Intermittent Oearing Counter- Wheels Oencva Stop. In the inter- 
 mittent toothed wheelwork described at pages 139 to 141, the wheels and their 
 teeth are so designed that, during the transmission of motion, the velocity- 
 ratio has a constant value. In some cases, however, it is not necessary that 
 the velocity-ratio should be constant, provided only that the follower per- 
 forms a certain part of a revolution for each revolution of the driver, as in 
 mechanism for counting revolutions. The simplest mechanism of that sort 
 consists of a toothed wheel of the ratchet form (Article 194, page 207, fig. 
 145), driven by a wiper or single tooth (Article 164, page 175), which pro- 
 jects from a rotating cylinder, and has its length adjusted so that the arc of 
 contact is equal to the pitch of the ratchet wheel. This requires no special 
 explanation. But there are cases in which the abruptness of the action of 
 the wiper would be disadvantageous, and in which it is desirable, in order 
 to prevent shocks, that the follower should be set in motion and stopped by 
 insensible degrees. 
 
 The following is the most precise method of designing a pair of wheels 
 turning about parallel axes, in which the follower is to count the revolu- 
 tions of the driver by turning through a certain aliquot part of a revolution 
 for each revolution of the driver ; the action being absolutely without shock, 
 and capable of taking place in either direction. 
 
 In fig. 100 A, let A B be the line of centres, A the trace of the axis of the 
 driver, and B that of the follower. In the example shown in the figure, the
 
 INTERMITTENT GEARING COUNTER-WHEELS. 
 
 287 
 
 follower is to make om-fiftli of a revolution for each revolution of the driver ; 
 but the same rules are applicable to any given number of aliquot parts. 
 
 Draw straight lines radiating from B (marked by dots in the figure), so as 
 to divide the angular space round B into twice the given number of equal 
 
 Fig. 100 A. 
 
 aliquot parts ; the line of centres, B A, being one of these radiating lines. 
 Then from A let fall perpendiculars A C and A D on the two radiating lines 
 which lie nearest to the line of centres, and complete the regular polygon of 
 which B is the centre, and A C and A D are two of the half-sides. (In the
 
 288 GEOMETRY OF MACHINERY. 
 
 figure this is a regular pentagon. ) On the line of centres lay off A E = A C 
 = A D; and about B, with the radius B E, describe a circle ; this circle will 
 cut the alternate radiating lines in a set of points, such as F, which will be 
 the centres of the semicircular bottoms of a set of notches that will gear in 
 succession with a pin carried by the driver. Having assumed a convenient 
 radius for that pin, draw a circle to represent it about the point C as a 
 centre. The pin will be carried by a plate or arm projecting from the axis 
 A, in a different plane from that of the driven wheel. For the bottom of the 
 notch C F draw about F, as a centre, a semicircle of a radius equal to that 
 of the pin C, increased by an allowance sufficient for clearance ; and for the 
 two sides of that notch draw two straight lines touching that semicircle, 
 and parallel and equal to F C. Draw all the other notches of the driven 
 wheel of the same figure and dimensions. 
 
 In the intervals between the notches the rim of the driven wheel is to 
 consist of a series of equal hollow circular arcs, described respectively about 
 the angles of the polygon with a radius such as to leave a thickness of 
 material sufficient for strength and durability at each side of each of the 
 notches ; for example, the arc G H K is described about the centre A, so as 
 to leave a sufficient thickness of material at G and K. 
 
 The periphery of the driver is to consist of a cylindrical surface extending 
 round the dead arc, H K M, and fitting smoothly, but not tightly, into eacli 
 of the hollows, such as G H K, in the rim of the follower; and of a hollow, 
 H L M, of a depth and figure such as to clear the horns, such as G and K, 
 of the notches in the follower. The angular extent of that hollow, HAM, 
 is to be equal to the angle CAD, that is, to the supplement of C B D, and 
 is to lie so that the radius A C shall bisect it. 
 
 The effect of this construction is as follows : While the pin moves through. 
 the arc C E D, the follower is driven through the angle C B D ; and as the 
 pitch-point evidently moves from A to E, and then back to A again, the 
 angular velocity of the follower gradually increases from nothing to a 
 maximum, and then gradually diminishes to nothing again. At the point D 
 the pin leaves the notch, and while it moves through the arc DIG, the 
 follower remains at rest, and is kept steady by the dead arc, M K H, fitting 
 into one of the hollows in its periphery. When the pin arrives again at C, 
 it enters and drives a second notch, and so on. The combination evidently 
 works with rotation in either direction. 
 
 The Geneva Stop is the name given to the form of this combination that is 
 employed when the object is that the follower shall stop the driver after it 
 has turned through a certain number of revolutions and fractions of a revo- 
 lution. For that purpose one of the notches is to be filled up, as shown by 
 the dotted semicircle at N, so as to leave only a recess fitting the pin in the 
 position C or D. The extent of rotation to which the driver will then be 
 limited is expressed by as many revolutions as there are intervals in the 
 circumference of the follower, less the angle CAD; and as the angle CAD 
 is the supplement of C B D, this is expressed in algebraical symbols as 
 follows : Let n be the number of intervals in the circumference of the 
 followei ; then the driver is limited to the following number of turns : 
 
 For example, in the figure, we have n 5 ; therefore, if one of the notches 
 is stopped, the rotation of the driver will be limited to 
 
 5 5 + -r= 47 turns. 
 2 5 
 
 This contrivance is used in watches, to prevent their being overwound. 
 Very often a hammer-headed tooth is used instead of the cylindrical pin C.
 
 GENEVA STOP RACKS IN SCREW GEARING. 289 
 
 This enables the horns of the notches to be shortened ; but it gives more 
 backlash, and less precision and smoothness of action. 
 
 ADDENDUM TO ARTICLE 154, PAGE 161. 
 
 Racks in Screw Centring. When a straight rack gears with a screw, 
 the pitch-surface of the rack is a plane, touching the pitch-cylinder of the 
 screw. The traces of the teeth of the rack on its pitch-plane are parallel 
 straight lines, and are the development on that plane of the traces of the 
 threads of the screw upon its pitch-cylinder. The principal rule to be used 
 in designing a rack and screw is a modification of Rule II. of Article 154, 
 and is as follows : 
 
 In fig. 110 A, let the plane of projection be the pitch-plane of the rack; 
 and let A I P be the projection of the axis 
 of the screw : being also the straight line in 
 which the pitch-surfaces touch each other. 
 Let I c represent the direction in which 
 the rack is to slide ; and let the length I c 
 represent the surface-velocity of the rack. 
 Having assumed a proper transverse ob- 
 liquity for the teeth of the rack, draw the 
 straight line c P, to represent the trace of 
 a tooth.* 
 
 Draw I C perpendicular to I A, cutting 
 P c in C ; then 1 will represent the surface- 
 velocity of the screw; and c C will represent 
 the velocity of transverse sliding of the threads 
 of the screw on the teeth of the rack. On 
 C c P let fall the perpendicular I N; this 1^ \ A 
 
 will represent the common component of the c \ 
 
 surface velocities. Also, the proportions 
 borne to each other by the divided pitches F 'S uo * 
 
 are as follows : as 
 
 I c : longitudinal pitch of rack 
 : : I N : divided normal pitch of rack and screw 
 : : I C : circular pitch of screw 
 : : I P : divided axial pitch of screw. 
 
 Having assumed a convenient absolute value for the longitudinal pitch ol 
 the rack, find, by the help 
 of the diagram, the circular . V 
 
 pitch of the screw ; multiply 
 that circular pitch by a con- 
 venient number of threads, 
 for the circumference of the 
 pitch-cylinder, and find its 
 radius by construction or 
 by calculation. 
 
 Fig. HOB is a projection 
 of a rack, C C', and screw, 
 A A', showing the traces of 
 the teeth and threads oil 
 the pitch-surfaces. 
 
 During the action of a 
 tooth and a thread, the 
 
 * The proper obliquity for the rack-teeth depends on friction^ a. ^^ffi 
 ynamics of Machinery ; but it may here be stated, that when the screw is to dme 
 
 Dynam
 
 290 
 
 GEOMETRY OF MACHINERY. 
 
 pitch-point travels along the whole length of the line of contact of the 
 pitch-surfaces, whose projection is A A'. In the figure the breadths of those 
 surfaces are such, that at any given instant there are at least three pitch- 
 points. 
 
 The teeth and threads are to be designed by drawing their normal sections, 
 as described in Article 155, page 163. The normal section of a tooth of the 
 rack will be that adapted to its normal pitch. The normal section of a 
 thread of the screw is to be found by the help of the osculating circle of the 
 normal helix, as found by Rule III. of Article 154, page 161, which is re- 
 peated here for convenience. On I C in fig. 110 A lay off I B equal to the 
 radius of the pitch-cylinder ; then, perpendicular to I C, draw B D, cutting 
 I N in D ; then, perpendicular to I N, draw D R, cutting I C in R ; I Ji 
 will be the radius of curvature of the normal helix. 
 
 A screw and rack are used in one form of Sellers' s planing machine. 
 
 ADDENDUM TO ARTICLE 243, PAGE 265, AND ARTICLE 245, 
 PAGE 267. 
 
 Epitrochoidnl Paths. In tracing epitrochoidal paths (including ellipses) 
 it is obviously not essential that there should actually be two wheels having 
 the fixed and rolling circles respectively for their pitch-circles, provided 
 only that the wheel which carries the tracing-point is carried by a train-arm 
 and driven by an epicyclic train, so as to have the same motion with the 
 
 imaginary rolling circle. As 
 
 to the principles which regulate 
 
 \ the motion of the rolling circle, 
 
 \D see Articles 76, 77, pages 54 to 
 
 ? __ \ 56. As to the action of the 
 
 .'*" *~ % v \ epicyclic train, see Article 234, 
 
 X v \ pages 243, 244. The wheel- 
 \ \ work of the epicyclic train may 
 be varied in detail according to 
 convenience, so long as it gives 
 the required velocity -ratio. 
 
 To exemplify this, let it be 
 required to design an instru- 
 ment for tracing ellipses by 
 rolling motion; and in Fig. 
 196 A, let A be the centre of 
 \ // the fixed circle D, and B the 
 
 \ v / / centre of the rolling circle C, 
 
 X^ ,/' ; whose diameter, A I (as already 
 
 ~^~..__ ,.''' stated in Article 245), is the 
 
 radius of the fixed circle. The 
 /' simplest gearing obviously con- 
 
 / sists of an internally toothed 
 
 / fixed ring whose pitch-circle is 
 
 Fig. 196 A. D, and a spur-wheel whose 
 
 pitch-circle is C ; but this com- 
 bination may be inconvenient in practice, and it may be desirable to use an 
 epicyclic train in outside gearing. The train-arm will of course be repre- 
 sented by A B; and the absolute angular velocity of the wheel centred on B, 
 which is to carry the tracing-arm, must be made equal and contrary to that 
 
 (as is almost always the case in practice), the angle I c P should be acute, and equal to the 
 complement of the angle of repose of the teeth, say about 85, if the teeth are smooth and 
 greasy.
 
 AGGREGATE PATHS. 291 
 
 of the train-arm. To effect this, let d be a fixed spur-wheel centred on A ; 
 let c be a spur-wheel centred on B, and of half the radius of the fixed wheel ; 
 and let e be an idle wheel of any convenient radius, gearing with both the 
 wheels before mentioned. Then, if a be taken to denote the'angular velocity 
 of the train-arm, the angular velocity of the fixed wheel d relatively to the 
 train-arm will be = a ; and the angular velocity of the wheel c, and of the 
 tracing-arm which it carries, relatively to the train-arm will be = 2 a ; 
 therefore the absolute angular velocity of the tracing-arm will be a 2 a 
 = a ; that is to say, equal and contrary to that of the train-arm, as 
 required. 
 
 As represented in the figure, this instrument is capable of tracing various 
 ellipses in which either the half-sum or the half-difference of the semiaxes 
 is = A B ; and it is easy to see that by jointing the train-arm at E, the 
 centre of the idle wheel, and providing the means of fixing its two divisions, 
 A E and E B, so as to make different angles with each other, the distance 
 A B may be varied within certain limits. 
 
 ADDENDUM TO ARTICLE 244, PAGE 265. 
 
 Aggregate Paths traced by Cam-Motions. An endless variety of 
 aggregate paths may be traced by means of combinations in which rolling 
 action is combined with the action of a cam. 
 
 Suppose two axes, not intersecting, to be connected by means of a pair, 
 or of a train, of suitable wheels, so as to have any required velocity -ratio ; 
 and suppose that along with one axis there rotates a disc on which an 
 aggregate path is to be traced ; and that along with the other axis there 
 rotates a cam of any required figure, which cam, by acting on a sliding bar, 
 causes the tracing-point to move along the line of centres, or common per- 
 pendicular of the axes, towards and from the axis that carries the disc, 
 according to a law determined by the figure of the cam ; then the tracing- 
 point will trace on the disc a curve whose figure will depend on that of the 
 cam, and on the velocity -ratio of the cam and disc. 
 
 (As to the action of cams, see Article 161, pages 170 to 173.) 
 
 Combinations of this sort are used in lathes for ornamental turning. The 
 tracing-point is the point of a suitable cutting tool ; and the cam which 
 regulates its motion is commonly called the copy-plate, or shaper-plate. (See 
 Northcott On Turning, 1868, Part III.)* 
 
 * To express by algebraical symbols the action of this combination, let the polar equation 
 of the pitch-line of the cam be 
 
 r=f(8) a-) 
 
 <6 being the angle made by the radius r with a fixed radius. Also, let a be the length of the 
 line of centres, and b the constant distance along that line from the pitch-line of the cam to 
 the tracing-point 
 Let the polar equation of the curve traced by that point on the disc be 
 
 r> if (6'); (2.) 
 
 and let n be the ratio borne by the angular velocity of the cam to that of the disc. Then the 
 quantities in the equations 1 and 2 are connected with each other by the following pair of 
 equations : 
 
 r + r> = a 6; (o.) 
 
 B = nff (4.) 
 
 Hence, if the figure of the cam is given that is, if the function/ in equation 1 is given the 
 figure of the traced curve is determined by the following equation: 
 
 I, ^ / Ot\ /H \ 
 
 T a u j \n u ) , * v'-.i 
 
 and if the figure of the traced curve is given that is, if the function <? in equation 2 is given 
 the figure of the cam is determined by the following equation: 
 
 r = <*-!>-<? ( 9 -\ (6.)
 
 GEOMETRY OF MACHINERY. 
 
 ADDENDUM TO ARTICLE 251, PAGE 275. 
 
 Approximate Grasshopper Parallel motion. Another form of approxi- 
 mate parallel motion of the grasshopper kind is designed as shown in fig. 
 r 201 A. Let A, A', A" be the 
 
 extreme and middle positions 
 of the guided point, lying in 
 one straight line. Draw the 
 straight line A' B B' perpen- 
 dicular to A A' A"; and lay 
 off the intended length of the 
 guiding-bar, A B = A' B' = 
 A" B, so as to find the extreme 
 positions, B and B', of its 
 further end. This end may 
 be guided either by straight 
 guides, or by a lever centred 
 at a point, C, equidistant from 
 B and B' ; that lever being so 
 long as to make the point B 
 describe a very flat circular 
 arc, deviating very little from 
 a straight line. 
 
 Choose a convenient point, D, for the attachment of the bridle to the bar 
 A B, and lay off A" D" = A' IX = A D, so as to find the extreme and middle 
 positions of that point. Then, by plane geometry, find the centre E of a 
 circular arc traversing the points D, D', D" ; E will be the trace of the axis 
 of motion of the bridle E D. The error of this parallel motion is the less 
 the nearer D is to the middle of A B. 
 
 Fig. 201 A. 
 
 ADDENDUM TO ARTICLE 143, PAGE 143. 
 
 Involute Teeth for Elliptic Wheels are designed by drawing an ellipse 
 confocal with the elliptic pitch-line, and having its major axis smaller in a 
 fixed proportion, and then drawing, for the traces of the fronts and backs of 
 the teeth, involutes of the smaller ellipse. The proportion in which this 
 ellipse is smaller than the pitch-ellipse should be such, that every tangent 
 to the smaller ellipse shall cut the fronts of two teeth at least between that 
 ellipse and the pitch -line. The pair of smaller ellipses in a pair of elliptic- 
 toothed wheels are analogous in their motion to a pair of elliptic pulleys ; as 
 to which, see Article 175. page 189.
 
 293 
 
 CHAPTER VI. 
 
 OF ADJUSTMENTS. 
 
 257. Adjustments Defined and Classed. The Word "adjust- 
 
 tneuts" was introduced by Professor Willis, in. order to compre- 
 hend tinder one general term all contrivances for varying at will 
 the comparative motions in a machine. Every adjustment may 
 be regarded as an aggregate combination in which the action is 
 temporary or intermittent; and the various kinds of adjustments 
 might have been classed under the head of "Aggregate Combina- 
 tions," in the preceding chapter; but it is more convenient to treat 
 of them by themselves. Various contrivances which belong to the 
 class of adjustments have already been described under the head of 
 "Elementary Combinations," as well as of aggregate combinations: 
 these will be specified in their order further on. Other contriv- 
 ances belonging to the class of adjustments involve the application 
 of the principles of dynamics and of the strength of materials, to 
 such an extent that their description, at all events in detail, must 
 be reserved for later divisions of this book. 
 
 When adjustments are classed according to the purposes to 
 which they are applied, they may be arranged as follows : 
 
 Traversing-Gear and Feed-Motions; 
 Engaging, Disengaging, and Reversing-Gear; 
 Gear for varying Speed or Stroke. 
 
 258. TraTersing-tear and Feed-motions in Ocneral. By travers- 
 
 ing-gear is meant the mechanism by means of which a machine, 
 consisting of framework and moving pieces, is shifted from place 
 to place without being thrown out of connection with the driver 
 from which it receives its motion; such, for example, as the 
 mechanism by which the truck in a travelling crane, that carries 
 the hoisting machinery, is made to move to different positions on 
 a travelling platform, which itself is capable of being moved to 
 different positions on a fixed framework; or the mechanism by 
 which the arm in a drilling machine is made to move to various 
 positions, carrying with it the boring-tool and the machinery by 
 which that tool is driven ; or that by which the tool-holder in a 
 shaping machine is turned into various positions, according to the 
 varying directions in which the strokes of the tool are to be made. 
 By a feed-motion is meant the mechanism in a machine-tool by
 
 294 GEOMETRY OF MACHINERY. 
 
 means of which, after a stroke has been made, either the cutting- 
 tool or the work (that is, the piece of material operated upon) is- 
 shifted into a new position, preparatoiy to making the next cut ; 
 for example, in a lathe for turning axles, the feed-motion causes 
 the tool to shift, at each revolution of the axle that is being turned, 
 through a certain distance in a direction parallel to the axis of 
 rotation; and in a sawing machine, the feed-motion causes the log 
 of wood that is being sawn to advance through a certain distance 
 either during or after each cut of the saw. Some feed-motions 
 are continuous in their action ; others are intermittent. 
 
 It is obvious that the general principles of traversing-gear, and 
 of those feed-motions in which the tool is shifted, are those of shifting- 
 trains, already stated in Article 228, pages 235 to 238. The con- 
 sideration of traversing-gear and feed-motions in detail belongs to 
 the subject of the construction of machinery, and nuist therefore 
 be deferred. 
 
 SECTION I. Of Engaging, Disengaging, and Reversing-Gear. 
 
 259. General Explanations. Engaging and Disengaging-Gear, or 
 sometimes Disengaging and Ke-engaging-Gear, is the name given to 
 those contrivances by means of which the connection between a 
 follower and its driver can be begun and stopped at will; in other 
 words, by means of which the combination can be thrown into gear 
 and out of gear when required. For brevity's sake, such con- 
 trivances may be called simply Disengagements. Disengagements 
 may be classed in different ways. According to one mode of 
 classification, they are distinguished into those which, in the com- 
 munication of motion, act by pressure, and those which act by 
 friction. Disengagements which act by pressure are precise and 
 definite in their action; that is, the connection between the pieces 
 that are thrown into gear at a given instant is established at once, 
 in a certain definite position of the pieces, and with a certain 
 definite velocity-ratio. Disengagements which act by friction 
 are to a certain extent indefinite in their action; that is, the 
 velocity-ratio corresponding to the complete establishment of the 
 connection is produced by degrees; and the relative position of the 
 pieces when the connection is completely established is uncertain. 
 In certain cases the definite action of the former class of disen- 
 gagements is necessary : in other cases it is unnecessary ; and in 
 these the frictional class of disengagements have a great advantage, 
 because of their avoiding the shocks and straining actions which 
 accompany sudden changes of velocity. The principles upon 
 which such straining actions depend belong to the dynamics of 
 machinery. 
 
 By another mode of classification, disengagements are arranged
 
 DISENGAGEMENTS REVERSING-GEAK CLUTCH. 295 
 
 according to the kind of mechanism of which they consist, as 
 follows : 
 
 I. Disengagements by means of Couplings; where two pieces 
 that turn about one axis are coupled or uncoupled at pleasure; so 
 that when coupled, they turn as one piece. These may transmit 
 motion either by pressure or by friction. 
 
 II. Disengagements with Rolling Contact. These always transmit 
 motion by friction. 
 
 III. Disengagements with Sliding Contact. These transmit motion 
 by pressure; and in most cases they act by throwing toothed 
 wheels or screws into and out of gear. 
 
 IV. Disengagements by Bands transmit motion by friction. 
 
 V. Disengagements by Linkwork transmit motion by pressure. 
 
 VI. Disengagements with Hydraulic Connection transmit motion, 
 by the pressure of a fluid; and they are made to act by the opening 
 and shutting of valves. 
 
 Reversing-Gear usually consists simply of a double set of 
 engaging and disengaging-gear ; that is to say, an arrangement of 
 mechanism by means of which the follower can, when required, be 
 thrown into gearing with one or other of two drivers that drive it 
 in opposite directions, or may be disengaged from both. 
 
 It is obvious that all the combinations in which the connection 
 is intermittent (enumerated in Article 219, page 231) are 
 examples of self-acting disengagements; and that some of them 
 (such as the escapements described in Article 164, pages 176 to 
 179) are examples of self-acting reversing-gear. 
 
 260. cimch. A clutch is a sort of coupling, in which one 
 rotating piece drives another piece that turns about the same axis, 
 by means of two or more projecting claws or horns, that fit into 
 corresponding recesses, or lay hold of corresponding horns, on the 
 second piece. In a disengaging clutch the driving piece is a 
 cylindrical box or collar with suitable horns, which is capable of 
 easily sliding lengthwise upon a rotating shaft, and is made to 
 rotate constantly along with the shaft, by having in its internal 
 cylindrical surface a slot or longitudinal groove, fitting a 
 longitudinal key or feather that projects from the shaft. In the 
 outer cylindrical surface of the clutch is a circular groove, into 
 which there fit easily the rounded ends of the prongs of a forked 
 hand-lever, by means of which the clutch can be shifted lengthwise 
 on the shaft through a distance sufficient to engage its horns with 
 or disengage them from those of the following piece. The following 
 piece may be another length of shaft, turning about the same axis; 
 or it may be a wheel or a pulley, loose upon the same shaft with 
 the clutch. 
 
 Sometimes the acting faces of the clutch, instead of being planes 
 traversing the axis of rotation, are inclined backwards as regards the
 
 296 GEOMETRY OF MACHINERY. 
 
 direction of motion, at an angle of 15, or thereabouts. The 
 effects of this are, that a certain forward pressure must be continu- 
 ally exerted by the lever on the clutch when in gear, in order to 
 make it keep its hold; and that any sudden acceleration of one of 
 the parts of the coupling causes the clutch to lose its hold, and 
 thus prevents the transmission of a shock to the machinery which 
 is driven by means of it. 
 
 261. Friction-Clutch Friction-Cones Friction-Sector* Friction- 
 
 Discs. In the friction-clutch the following piece is a circular disc, 
 having a hoop which grasps it, and which can be tightened or 
 slackened by means of screws until the friction between the hoop 
 and the disc is just sufficient to transmit the required power. 
 The hoop has two projecting horns, corresponding to those of the 
 clutch. When this combination is thrown into gear, the clutch 
 instantly communicates its own velocity to the hoop ; but the hoop 
 at first slips on the disc, which is set in motion by degrees; and 
 thus dangerous shocks are avoided. 
 
 In the friction-cones the driver, as in the case of the clutch, is a 
 cylindrical box, turning along with the shaft, and capable of being 
 shifted lengthwise by means of a hand-lever; but instead of horns, 
 it has a disc with a rim turned to a very accurate and smooth 
 convex conical surface. The follower is a disc whose rim is turned 
 to a concave conical surface, exactly fitting that of the driver. 
 When the driver is pushed forward by means of the lever, so as to 
 press the two conical surfaces together, it gradually imparts its rota- 
 tion to the follower by means of the friction of those surfaces. On 
 drawing back the driver by means of the lever, the connection 
 immediately ceases. 
 
 The angle of obliquity of the conical surfaces should be just 
 great enough to prevent any risk of their becoming jammed against 
 each other, so as to prevent disengagement; and for that purpose 
 an angle of 10 or thereabouts is sufficient. 
 
 In the fractional sector coupling (invented by Mr. Bodmer) 
 the follower is a cylindrical box, loose on the shaft, and carrying a 
 circular disc-plate with a hoop-shaped rim. The inner cylindrical 
 surface of that rim is turned true and smooth. The driver 
 consists of a boss fixed on and turning with the shaft, and carrying 
 an expanding disc composed of two sectors, with true and smooth 
 cylindrical rims, fitting the inner surface of the rim of the followers. 
 Those sectors can be simultaneously moved from or towards 
 the shaft by means of right and left-handed screws, turned by 
 levers and links; the links lie parallel to the shaft, and are jointed 
 to a collar which is shifted by means of a forked lever, as in the 
 ordinary clutch. When the sectors are moved outwards, they fit 
 tightly to the inside of the hoop-shaped rim of the follower, and 
 by their friction communicate to it the rotation of the shaft.
 
 DISENGAGEMENTS. 
 
 207 
 
 When moved inwards, they cease to touch that rim, and tho 
 connection ceases. (See Fairbairn On Millwork, Part II, Edition of 
 1863, pp, 91, 92.) 
 
 In Mr. R. D. Napier's disengaging-gear the pushing forward 
 in the usual way of a cylindrical clutch-box causes two segmental 
 pieces to grasp between them a drum that rotates with the shaft, and 
 so to communicate rotation to a disc to which they are attached. 
 
 In the friction-disc disengagement (Mr. Weston's invention) a 
 set of flat discs are made to turn along with the shaft by means of 
 grooves and feathers, and are capable of shifting longitudinally to 
 a small extent. Between each pair of that first set of discs is 
 placed a disc belonging to a second set, which are loose on the 
 shaft, but are made to turn along with it when required, by press- 
 ing them between the discs of the first set. The second set of discs, 
 by means of grooves and feathers at their outer edges, carry along 
 with them in their rotation a wheel or a pulley concentric with 
 the shaft. 
 
 262. Disengagements acting by Rolling Contact. A pair of 
 
 wheels acting on each other by rolling contact may be engaged and 
 disengaged when required by pressing them together and drawing 
 them asunder, the axis of one of them being made moveable for 
 that purpose; and this is practised in grooved frictional gearing of 
 the kind described in Article 111, page 150. 
 
 The principle of another method of effecting engagement and 
 disengagement by wheels in rolling contact is shown in fig. 213. 
 
 213. 
 
 A and B are the traces of the fixed axes of a pair of smooth wheels, 
 whose surfaces do not touch each other : A being the driver and B
 
 298 GEOMETRY OF MACHINERY. 
 
 the follower. C is the trace of the moveable axis of an intermediate 
 idle wheel, which drives B, and is driven by A ; D being the pitch- 
 point of A and C, and E the pitch-point of C and B. The straight 
 line D E is the common line of connection of the three wheels; 
 and as pressure only, and not tension, can be transmitted along 
 that line from the first wheel to the third wheel, the connection 
 ceases if the motion is reversed. To disengage the wheels while 
 in motion forwards, the axis C is shifted so as to put an end to 
 the contact at D or at E, or at both those points. 
 
 The angles of obliquity, C D E = C E D, which the line of 
 connection D E makes with the two lines of centres, A C and 
 B C, ought to be a little greater than the " angle of repose " of the 
 surfaces of the wheels, in order that the wheel C may not become 
 jammed between the wheels A and B; but it ought not to be 
 greater than is just sufficient to prevent the risk of jamming; in 
 order that the force with which C must be pressed towards A and 
 B may not become unnecessarily great. The value of that force 
 and of the angles of obliquity will be considered under the head 
 of the "Dynamics of Machinery ;" meanwhile, in anticipation of that 
 division of this treatise, the following values are given of the angle 
 of repose for different surfaces : 
 
 Cylindrical surfaces without grooves. 
 
 Metal on Metal; dry, 10; slimy, 8; greasy, 4. 
 
 Metal on Oak ; dry, 28 ; wet, 14. 
 
 Metal on Elm; dry, 13. 
 
 Leather on Metal ; dry, 29i; wet, 20; greasy, 13. 
 
 Leather on Oak, 17. 
 
 Grooved metal surfaces, as in frictional gearing; about 28. 
 
 The construction, therefore, for designing this disengagement is as 
 follows : Construct an isosceles triangle C D E, with the angles at 
 D and E each a little greater than the angle of repose; produce 
 C D and C E, laying off upon them D A and E B proportional to 
 the radii of the wheels to be connected ; join A B. Then the pro- 
 portions borne respectively by A D, B E, and C D, to A B, will 
 be those which the radii of the wheels are to bear to the line of 
 centres. 
 
 263. Disengagements and Reversing-Oear acting by Sliding Con- 
 tact. A pair of toothed wheels, whether spur, bevel, or skew- 
 bevel, may be thrown into or out of gear by shifting one of them 
 along its axis. This sort of disengagement belongs to the class in 
 which motion is transmitted by pressure; so that the velocity-ratio' 
 and the relative position of the pieces are definite, and the com- 
 munication of motion abrupt. Another way of making it act is 
 to have the wheels always in gearing with each other, and to 
 effect the engagement and disengagement of one of them by
 
 DISENGAGEMENTS. 
 
 299- 
 
 means of a clutch upon the shaft that carries it: as in Article 260 
 page 295. 
 
 The most common kind of reversing-gear which acts by means 
 of toothed wheels is shown in fig. 214; A is the driving-shaft 
 carrying a bevel- wheel which drives in contrary directions a pair of 
 bevel-wheels, B, C, that turn loose 
 on the driven shaft, D D. A 
 double clutch, E, sliding along 
 a feather on the latter shaft, is 
 made, by means of a collar and 
 lever, to lay hold of the one or 
 the other of the bevel-wheels 
 B, C, according to the direction in 
 which the shaft D D is to 
 rotate. 
 
 264. Disengagements ami Re- 
 rersing-Gear by Bands. When 
 
 rotation is transmitted from one 
 
 shaft to another by means of a 
 
 belt and a pair of pulleys, the 
 
 form of engaging and disengaging- 
 
 gear employed is the "fast and 
 
 loose pulley " already described in 
 
 Article 170, pages 184 and 185. 
 
 The fork mentioned there is called 
 
 a belt-guide, or belt-shifter. It is 
 
 evident that the contrivance of the fast and loose pulley is applicable 
 
 to belts alone, and not to cords and chains. 
 
 Reversing -gear by means of belts with fast and loose pulleys is 
 arranged in the following way : on the driven shaft is one fast 
 pulley, between two loose pulleys, one for each of the two belts, 
 which run in opposite directions. In the act of reversing the 
 motion, care should be taken that the belt which has been driving 
 the fast pulley is shifted completely on to its own loose pulley 
 before any part of the other belt is shifted on to the fast pulley. 
 
 A method of engaging and disengaging connection by bands, 
 applicable to cords as well as to belts, is to tighten and slacken the 
 band when required, by means of a straining pulley, as already 
 described in Article 174, page 188. 
 
 265. Disengagements and Reversing-Gear acting by Iiinkwork. 
 
 Amongst disengagements acting by linkwork are all the examples 
 of intermittent linkwork described in Article 194 to 197, pages 
 206 to 213; and in most of those examples, besides the periodical 
 disengagement which takes place at each return stroke, there- 
 exists also the means of making a permanent disengagement, by 
 fixing the click or catch so as to prevent it from taking hold of the 
 
 Fig. 214.
 
 300 
 
 GEOMETRY OF MACHINERY. 
 
 teeth. In fig. 146, page 207 (described at page 208), is an 
 example of reversing-gear in linkwork. 
 
 In ordinary linkwork (as distinguished from click-and-ratchet- 
 work) the means of disengagement consist in connecting the link 
 with the pin at one of the connected points by means of a gab 
 (as at A, tig. 215); that is, a deep notch wi^h plane sides and a 
 
 semi-cylindrical bottom, fit- 
 ting the pin accurately but 
 easily. The link is thrown 
 out of gear when required, by 
 
 Fig. 215. moving the gab clear of the 
 
 pin, either by hand or by 
 
 suitable mechanism. Sometimes the gab is provided with spread- 
 ing jaws, to enable it the more easily to lay hold of the pin when 
 the connection is to be re-engaged. 
 
 Another case of disengagement by linkwork is that of the liooks 
 in a Jacquard loom. At each shot or stroke of the loom there are 
 certain threads of the warp that have to be raised and lowered 
 again, while other threads remain at rest ; the order and arrange- 
 ment of threads so treated being varied at each shot, in a manner 
 depending on the pattern to be woven. In fig. 216, B C is a hook, 
 of which, the lower end is connected with a thread : the hook is 
 
 kept in a nearly vertical posi- 
 tion by passing easily through 
 a hole at D in a horizontal 
 sliding - bar, D E, called a 
 needle; and the hooked upper 
 end, at B, overhangs a hori- 
 zontal bar or rail, A, which is 
 carried by a frame having a 
 vertical reciprocating motion, 
 of the extent represented by 
 A A'. In the position shown 
 by full lines and capital letters 
 the hook stands ready to be 
 lifted by the rail A; but when 
 the needle is drawn back to 
 the position d e, the hook is 
 made to assume the position 
 
 Fig. 216. C d b, shown in dotted lines, 
 
 in which it stands disengaged 
 
 from the rail, and remains at rest during the next stroke. The 
 needles are usually drawn back by means of springs, and pushed 
 forward by the forward stroke of a drum, F, which turns about a 
 horizontal axis, and has also, along with that axis, a reciprocating 
 motion in the direction of the length of the needles. The drum is 
 
 s e 
 
 o 
 
 C
 
 DISENGAGEMENTS JACQUARD HOOKS VALVES. 301 
 
 of the form of a polygonal prism, usually square, as in the figure; 
 its acting face is covered with an oblong card (of pasteboard or 
 sheet metal), having holes in it opposite the ends of those needles 
 which are not to be pushed forward. The drum does not rotate 
 during its forward stroke, when it is pushing the needles ; but 
 during the return stroke a catch pulls it round so as to bring a 
 new face opposite the needles, with a new card upon it, having a 
 proper arrangement of holes for the next stroke. The cards, in 
 sufficient number to produce the entire pattern, are linked together 
 at their longer edges, so as to form, as it were, a flat chain, which 
 hangs over the drum, by whose rotation they are brought round 
 one by one to act on the ends, E, of the needles. 
 
 The Jacquard Apparatus, of drum, cards, needles, and hooks, may 
 be applied to many purposes besides that of lifting the threads of a 
 warp. 
 
 2(36. Disengagements acting by Hydraulic Connection Valreo. 
 When the driver and follower are two pistons, and the former trans- 
 mits motion to the latter by means of an intervening mass of fluid 
 (as in Articles 207 to 211 A, pages 221 to 227), the engagement 
 and disengagement are effected by opening and closing a valve in 
 the passage through which the displaced fluid flows : as has 
 been already stated in Article 211, page 224. If the forward 
 motion of the driving piston is to go on while the valve is closed, 
 some other outlet must be opened for the fluid which it displaces. 
 
 A reversing action takes place in hydraulic connection, when a 
 stream of fluid is admitted, by means of suitable valves, so as to act 
 alternately on the two sides of a piston, as in a double-acting water- 
 pressure engine. 
 
 In all cases in which the motion of a piston driven by a fluid is 
 reversed, an outlet, with a suitable valve, must be provided for the 
 escape from the cylinder, during the return stroke, of the mass of 
 fluid by which the previous forward stroke was produced. 
 
 267. Principle* of the Action of Valves. It would be out of 
 
 place here to describe in detail the various kinds of valves used in. 
 machinery; and therefore a summary only of the general principles 
 of their construction and action, so far as those principles can be 
 considered as forming part of the Geometry of Machines, will now 
 be given, chiefly abridged from A Manual oftlie Steam Engine and 
 other Prime Movers. 
 
 Valves in general, considered with reference to the means by 
 which they are moved, may be divided into three principal classes : 
 Valves, sometimes called clacks, which are opened and shut by 
 the pressure of the fluid that traverses their openings, and are 
 usually intended for the purpose of permitting the passage of the 
 fluid in one direction only, and stopping its return ; valves moved 
 by hand; and valves moved by mechanism. When a piston,
 
 302 GEOMETRY OP MACHINERY. 
 
 drives a fluid, as in ordinary pumps, the valves are usually moved 
 by the fluid: when the fluid drives the piston, it is in general 
 necessary that the valves should be moved by hand or by mechan- 
 ism. In water-pressure engines that work occasionally and at 
 irregular intervals, such as hydraulic hoists and cranes, the valves 
 are usually opened and shut by hand; in those which work periodi- 
 cally and continuously, they are moved by mechanism connected 
 with the engine. Valves, when considered with reference to the 
 kind of motion by which they open and shut the ports, or orifices 
 to which they are fitted, may be distinguished into Drop-valves, 
 which are opened and shut by being lifted up and set down ; Flap- 
 valves, which turn on a hinge ; and Slide-valves. 
 
 The seat of a valve is the fixed surface on which it rests, or 
 against which it presses. 
 
 The face of a valve is that part of its surface which comes in 
 contact with the seat. 
 
 When a valve occurs in the course of a^>ipe or passage, the valve- 
 box or chamber, being that part of the passage in which the valve 
 works, should always be of such a shape as to allow a free passage 
 for the fluid when the valve is open, so that the fluid may pass the 
 valve with as little change of area of the stream as possible ; and if 
 necessary for that purpose, the valve-chamber may be made of 
 larger diameter than the rest of the passage. 
 
 A valve moved by mechanism has almost always a periodical 
 reciprocating motion, by which it is alternately opened and shut. 
 The simplest mode by which that motion can be given is by a 
 crank, or an eccentric, carried by some continuously-rotating piece, 
 and acting through a rod ; as in Articles 1 84 to 1 86, pages 1 9 6 to 1 98 ; 
 and such is the ordinary way of moving slide-valves. Drop-valves 
 are sometimes worked by the same kind of mechanism, with the 
 addition of a contrivance for setting them down veiy gently, of the 
 kind described in Article 190, pages 202, 203; or by means of 
 cams or wipers (Articles 160 to 164, pages 170 to 175). 
 
 The principal forms of valves are the following : 
 
 I. The Bonnet-Valve or Conical Valve is the simplest form of 
 drop-valve, and is a flat or slightly arched circular plate whose 
 face, being formed by its rim, is sometimes a frustum of a cone, 
 and sometimes a zone of a sphere, the latter figure being the best. 
 Its seat, being the rim of the circular orifice which the valve closes, 
 is of the same figure with the face or rim of the valve, and the 
 valve-face and its seat are turned and ground to fit each other 
 exactly, so that when the valve is closed no fluid can pass. The 
 thickness of a valve of this form is usually from a fifth to a tenth 
 of its diameter, and the mean inclination of its rim about 45. 
 
 To ensure that the valve shall rise and fall vertically, and always 
 return to its seat in closing, it is sometimes provided with a spindle,
 
 VALVES. 303 
 
 moving through a ring or cylindrical socket. A knob on the end 
 of the spindle prevents the valve from rising too high. When the 
 valve is to be moved by hand or by mechanism, the spindle may 
 be continued through a stuffing-box, and connected with a handle 
 or a lever, so as to be the means of transmitting motion to the 
 valve. 
 
 II. The Ball Clack is a drop-valve of the form of an accurately- 
 turned sphere. When of large size, it is in general hollow, in order 
 to reduce its weight. Its face is its entire surface : its seat is a 
 spherical zone. 
 
 III. The Divided Conical Valve is composed of a series of con- 
 centric rings. The largest ring may be considered as a bonnet- 
 valve, in which there is a circular orifice, forming a seat for a 
 smaller bonnet- valve, in which there is a smaller circular orifice, 
 forming a seat for a still smaller bonnet- valve, and so on. This 
 arrangement enables a large opening for the passage of fluid to be 
 formed with a moderate upward motion of each division of the valve. 
 
 IV. The Double-beat Valve is a drop-valve so contrived as to 
 enable a large passage for a fluid to be opened and shut easily under 
 & high pressure. Fig. 217 represents a section of the valve, with 
 its seats and chamber, and fig. 218 a plan of the valve alone. 
 
 The valve shown in the figure is for the purpose of opening and 
 .shutting the communication between the pipes A and B, 
 
 Fig. 217. Fig 218. 
 
 The pipe B is vertical, and its upper rim carries one of the two 
 valve-seats, which are of the form of the frustum of a cone, and 
 each marked a.
 
 304 GEOMETRY OF MACHINERY. 
 
 A frame C, composed of radiating partitions, fixed to and resting 
 on the upper end of the pipe B, carries a fixed circular disc, whose 
 rim forms the other conical valve-seat. 
 
 The valve D is of the form of a turban, and has two annular 
 conical faces, which, when it is shut, rest at once on and fit equally 
 close to the two seats, a, a. When the valve is raised, the fluid 
 passes at once through the cylindrical opening between the lower 
 edge of the valve and the upper edge of the pipe B, and through 
 the similar opening between the upper edge of the valve and the 
 rim of the circular disc. 
 
 The greatest possible opening of the valve is when its lower edge 
 is midway between the disc and the rim of the pipe B, and is given, 
 by the following formula : 
 
 Let 
 
 d^ be the diameter of the pipe B j 
 
 d z , that of the disc ; 
 
 h, the clear height from the pipe to the disc, less the thickness of 
 the valve ; 
 
 A, the greatest area of opening of the valve; then 
 
 A = 3-1416 ^^? . h . (L) 
 
 and in order that this may be at least equal to the area of the pipe 
 B, viz., -7854 dl, we should have 
 
 h at least = , * (2.) 
 
 which, if, as is usual, d i = d^ gives 
 
 h ut least = -I; ...(2 A.) 
 
 4 ' 
 
 but h is in general considerably greater than the limit fixed by this 
 rule. 
 
 If the upper and lower seats are of equal diameter, the valve is 
 called an equilibrium-valve; and this is the kind of double-beat 
 valve most commonly used in steam engines. In water-pressure 
 engines, pumps, and hydraulic appai'atus generally, the lower valve- 
 seat is generally made a little larger than the upper. 
 
 V. A common Flap- Valve is a lid which opens and shuts by 
 turning on a hinge. The face and seat are planes. 
 
 A pair of flap- valves placed hinge to hinge constitute a "butterfly 
 dock." The chamber of a flap- valve should be of considerably 
 greater diameter than the valve. 
 
 VI. A Flexible Flap- Valve consists of a piece of some flexible 
 material, such as waterproof canvas or India rubber. It may be 
 rectangular, so as to have one edge fixed to the seat, and the
 
 VALVES. 
 
 opposite edge attached to a bar, by moving which it is opened and 
 shut; or it may be circular, and fixed to the seat at the centre; 
 and this is the form usually adopted for self-acting flexible flap- 
 valves in pumps. The seat of the flap consists of a flat horizontal 
 grating, or a plate perforated with holes. To prevent a circular 
 flap-valve from rising too high, it is usually provided with a guard, 
 which is a thin metal cup formed like a segment of a sphere, grated 
 or perforated like the valve-seat, to which it is bolted at the centre. 
 When the valve is raised by a current from below, it applies itself 
 to the bottom of the cup. When the current is reversed, the fluid 
 from above, pressing on the valve through the holes in the cup, 
 drives it down to its seat again. 
 
 VII. The Disc-and-Pivot Valve, or Throttle-Valve, consists of a 
 thin flat metal plate or disc, which, when shut, fits closely the 
 opening of a pipe or passage, generally circular in section, but 
 sometimes rectangular. The valve turns upon two pivots OP 
 journals, placed at the extremities of a diameter traversing its 
 centre. 
 
 When the valve is turned so as to lie edgeways along the pas- 
 sage, the current of fluid passes with very little obstruction : when 
 it is turned transversely, the current is stopped, or nearly stopped. 
 By placing the valve at various angles, various openings can be 
 made. If the valve, when shut, is perpendicular to the axis of the 
 pipe, the opening for any given inclination of the valve to that 
 axis is proportional to the coversed-sine of the inclination. If the 
 valve is oblique when shut, the opening at a given inclination is 
 proportional to the difference between the sine of that inclination 
 and the sine of the inclination when shut. 
 
 The face of this valve is its rim; its seat is that part of the 
 internal surface of the passage which the rim touches when the 
 valve is shut; and those surfaces ought to be made to fit very 
 accurately, without being so tight as to cause any difficulty in 
 opening the valve. 
 
 One of the journals of the valve usually passes through a bush 
 or a stuffing-box in the pipe, so as to afford the means of commu- 
 nicating motion to the valve from the outside. 
 
 VIII. Slide- Valves. The seat of a slide-valve consists of a plane 
 metal surface, very accurately formed, part of which is a rim sur- 
 rounding the orifice or port, which the valve is to close, and from 
 
 - to of the breadth of that orifice, while the remainder extends 
 
 T: &\) 
 
 to a distance from the orifice equal to the diameter of the valve, in 
 order that the valve, when in such a position as to leave the port 
 completely open, shall still have every part of its face in contact 
 with the seat. 
 
 The valve is of such dimensions as to cover the port together
 
 306 GEOMETRY OF MACHINERY. 
 
 with that portion of the seat which forms a rim surrounding the 
 port. The face of the valve must be a true plane, so as to slide 
 smoothly on the seat. As to the periodical motion of slide- 
 valves, see the next Article. 
 
 Rotating slide-valves are sometimes used, in which the valve and 
 its seat are a pair of circular plates, having one or more equal and 
 similar orifices in them. The passage is opened by turning the 
 valve about its centre until its openings are opposite to those of 
 the seat, and shut by turning it so that its openings are opposite 
 solid portions of the seat. (See page 314.) 
 
 IX. A Piston-Valve is a piston moving to and fro in a cylinder, 
 whose internal surface is the valve-seat. The port is formed by a 
 ring or zone of openings in the cylinder, communicating with a 
 passage which surrounds it; and by moving the piston to either 
 side of those openings, that passage is put in communication with 
 the opposite end of the valve-cylinder. 
 
 X, Cocks. This term is sometimes applied to all valves which 
 are opened and shut by hand; but its proper application is to those 
 valves which are of the form of a frustum of a cone, or conoid, 
 turning in a seat of the same figure. 
 
 In the most common form of cock, the seat is a hollow cone of 
 slight taper, having its axis at right angles to the pipe in whose 
 course it occurs. The valve is a cone fitting the seat accurately, 
 and having a transverse passage through it of the same figure and 
 size with the bore of the pipe, so that, in one position it forms 
 simply a continuation of the pipe, and offers no obstruction to the 
 current, while, by turning it into different angular positions, the 
 opening may be closed either partially or wholly. 
 
 268. Periodical motion of slide-Vaires. The motion of a slide- 
 valve driven by a crank or an eccentric is a case of approximate 
 harmonic motion, as already described in Article 239, page 250; 
 and in most cases which occur in practice, it may be treated, with- 
 out material error, as if it were exact harmonic motion : that is 
 to say, the travel or length of stroke of the slide is twice the 
 eccentric-arm; the slide is in its middle position when the eccentric- 
 arm is sensibly at right angles to the line of its dead-points; 
 in other words, when the phase of its revolution is sensibly 90; 
 and the displacement of the slide from its middle position at 
 any given instant is sensibly equal to the eccentric-arm multiplied 
 by the cosine of the phase. For example, in fig. 220 (page 308), 
 the straight line F A L, bisected in A, represents twice the eccen- 
 tric-arm ; so that A F and A L respectively represent the displace- 
 ments of the slide at the two dead-points of the revolution of the 
 eccentric, when the phase is respectively and 180. On those 
 two lines as diameters describe two equal circles, A H F G A, 
 and ANLPA; then, when the phase is = ^ FAD, the dis-
 
 MOTION OF SLIDE-VALYE. 307 
 
 placement is = AG; and when the phase is = .^F AM, the 
 displacement is = AN, in the contrary direction to that of the 
 displacement A G. 
 
 tinder the geometry of machinery are comprehended the rules 
 by which the movement of the slide-valve of an engine is made to 
 bear certain relations to that of the crank with which the piston 
 is connected. The following are terms used in those rules : 
 
 The two opposite sides of the port, or oblong opening in the 
 seat of a slide-valve, are distinguished as the induction-side and the 
 eduction-side; the former being the side at which the fluid enters 
 the port; the latter, the side at which it is discharged. 
 
 The lap, or cover, of a slide-valve at one of its edges is the extent 
 to which that edge overlaps the adjoining edge of 
 the port which it covers when the slide-valve is 
 in its middle position. In fig. 219 is a section 
 of part of a vertical slide-valve and its port; 
 W is the lower port of a cylinder; X, the 
 lower half of the slide-valve, in its middle 
 position; U is the induction-side, and V the 
 eduction-side, of the port; C is the induction- Fig. 219. 
 edge, and P the eduction-edge of the valve; U C 
 is the lap on the induction-side, and Y P the lap on the eduction- 
 side : the hollow part of the valve opposite X is called the exhaust- 
 cavity. 
 
 It is evident that the opening and closing of the port at either 
 side take place at the instants when the displacement of the slide 
 in a direction away from that side is equal to the lap at that side ; 
 and that the port remains open at that side so long as the displace- 
 ment in the proper direction is greater than the lap. Thus, the 
 port W remains open at the side U, so long as the displacement of 
 the slide towards P is greater than U C ; and at the side V, so 
 long as the displacement of the slide towards C is greater than 
 V P. If the lap at either side is nothing, the opening and closing 
 at that side take place in the middle position of the slide ; and the 
 port remains open at that side during half a revolution of the 
 eccentric. 
 
 The instant at which the port is first opened at the induction- 
 side is called the instant of admission; that at which it is closed, 
 of suppression, or cut-off; that at which it is first opened at the 
 eduction-side, the instant of release; that at which it is closed at 
 the eduction-side, the instant of compression, 
 
 By the angular advance of the eccentric is to be understood the 
 angle at which the eccentric-arm stands in advance of that position, 
 which would bring the slide-valve to mid-stroke when the crank is 
 at its dead-points : in other words, the excess above 90 of the 
 phase of the eccentric when the phase of the crafk is ; or in.
 
 308 
 
 GEOMETRY OF MACHINERY. 
 
 symbols, phase of eccentric phase of crank 90. When the 
 slide is at its middle position at the same instant at which the 
 crank is at a dead-point, the angular advance is nothing. 
 
 RULE I. Given, the positions of the crank at the instants of 
 admission and cut-off; to find the proper angular advance of the 
 eccentric, and the proportion of the lap on the induction-side to 
 the half-travel of the slide.* 
 
 In fig. 220, let A B and A C be the positions of the crank at 
 the beginning and end of the forward stroke ; let the arrow show 
 the direction of rotation j let X a; be perpendicular to B C ; let 
 A D be the position of the crank at the instant of cut-off, and 
 
 Fig. 220. 
 
 A E its position at the instant of admission. Draw A F, 
 bisecting the angle E A D; A F will represent the position of the 
 
 * The method used in this and the following rules is that of Professor Dr. 
 Zeuner, of the Swiss Federal Polytechnic School at Zurich, published in his 
 treatise on Slide-valve Gearing, entitled, Die.Schiebersteuerungen.
 
 MOTION OP SLTDE-VALVE. 309 
 
 crank at the instant when the slide is at the forward end of its 
 stroke ; and FAX will be the angular advance of the eccentric. 
 
 Lay off the distance A F to represent the half-travel; and on 
 A F as a diameter describe the circle A H F G, cutting A D in 
 
 G and A E in H ; then -^-^ = will be the required ratio of 
 
 lap at the induction-side to half-travel; and A G = A H will 
 represent that lap, on the same scale on which A F represents the 
 half-travel 
 
 On the same scale, I K represents the width of opening of the 
 valve at the beginning of the stroke, sometimes called the " lead of 
 the slide." Strictly speaking, this is the lead of the induction-edge 
 of the slide only; the lead of the centre of the slide being A K; 
 that is, its distance from its middle position at the beginning of the 
 forward stroke. 
 
 RULE II. Given, the data and results of the preceding rule, and 
 the position, A M, of the crank at the instant of release; to find 
 the ratio of lap on the eduction-side to half-travel, and the position 
 of the crank at the instant of compression. Produce FA to L, 
 making A L = A F ; on ALas a diameter draw a circle cutting 
 
 A N 
 
 A M in N ; then -r-y will be the required ratio of lap at eduction- 
 side to half-travel. 
 
 About A draw the circular arc N" P, cutting the circle A L 
 again in P; join A P: then A P will be the required position of 
 the crank at t/te instant of compression. 
 
 RULE III. Given, the data and results of Rule I., and the 
 position, A Q, of the crank at the instant of compression; to find 
 the ratio of lap at the eduction-side to half-travel, and the position 
 of the crank at the instant of release. Produce F A as before ; 
 onAL = FAasa diameter draw a circle cutting A Q in P ; 
 
 AP 
 
 -r-= will be the required ratio of lap at tfie eduction-side to half- 
 
 -V -Lj 
 travel. 
 
 About A draw the circular arc P N, cutting the circle A L 
 again in N; join A N; A N will be the position of the crank at 
 the instant of release. 
 
 RULE IV. Given, the angular advance of the eccentric, the 
 half-travel of the slide, and the lap at both sides; to find the 
 positions of the crank at the instants of admission, cut-off, release, 
 and compression. Draw the straight lines BAG and X A a; per- 
 pendicular to each other; and take B and C to represent the dead- 
 points. Let the arrow denote the direction of rotation. Draw 
 F A L, making the angle F A X = the angular advance of the 
 eccentric; and make A F = A L = half-travel. On A F and
 
 310 GEOMETRY OF MACHINERY. 
 
 A L as diameters, draw circles. About A, with a radius equal to 
 the lap at the induction-side, draw an arc cutting the circle on 
 A F in H and G; also, with a radius equal to the lap at the 
 eduction-side, draw an arc cutting the circle on A L in N and P. 
 Draw the straight lines A H E, A G D, A N M, A P Q. These 
 will represent respectively the positions of the crank at the instants 
 of admission, cut-off", release, and compression. 
 
 The eccentric may act on the slide, not directly, but through a 
 train of levers and linkwork. The effect of this on the application 
 of the mles is merely to substitute for the actual eccentric-arm a 
 virtual eccentric-arm equal to the half-travel of the slide. 
 
 The effects of the link-motion, of double slides, and of move- 
 able slide-valve seats, in modifying the length and position of the 
 virtual eccentric-arm, have been already described in Articles 239 
 to 241, pages 250 to 260. 
 
 SECTION II. Of Adjustments for Changing Speed and Stroke. 
 
 269. General Explanations. All methods of changing the velo- 
 city-ratio of an elementary combination in a machine operate by 
 changing the position of their line of connection ; for on the posi- 
 tion of that line the velocity-ratio depends, according to the principle 
 already explained in Article 91, page 78. In some cases the 
 combination contains two or more pairs of acting surfaces (such as 
 wheels or pulleys), one or other of which can be thrown into gear 
 according to the velocity-ratio required; and then it is in general 
 necessary to stop the motion in order to change the velocity-ratio. 
 In other cases there are contrivances for changing the velocity -ratio 
 by degrees while the machine is in motion. 
 
 In the case of linkwork the change of velocity-ratio is often 
 connected with a change of length of stroke. 
 
 Many of the most ordinary and useful adjustments for changing 
 speed have already been described under the head of elementary 
 or of aggregate combinations; and in such cases it will be suffi- 
 cient in the present section to refer to the place where the detailed 
 description is to be found. 
 
 Adjustments for changing speed, like engaging and disengaging- 
 gear, may in most cases be distinguished into two classes, according 
 as the connection is made by pressure or by friction. In the former 
 case the change of velocity-ratio is definite, and in most instances 
 sudden; in the latter case, gradual, and to a certain extent indefinite. 
 
 270. Changing Speed by Friction-Wheels. To obtain changes of 
 
 speed by means of friction- wheels, a pair of parallel shafts are to be 
 provided with as many pairs of wheels as there are to be different 
 velocity-ratios; each pair of wheels being connected with each 
 other, not directly, but by means of an intermediate idle wheel.
 
 CHANGING SPEED. 311 
 
 which can be thrown into or out of gear at pleasure, as in the 
 second method of disengagement described in Article 262, page 
 297; the only difference being that whereas in that Article the 
 two principal wheels of the pair are described as being equal, in 
 the present case they will in general be unequal. The rule as to 
 the obliquity of the line of connection is the same. (See page 298.) 
 
 A combination of friction-wheels in which the velocity-ratio 
 is changed by degrees during the 
 motion, is shown in fig. 221. (It 
 forms part of Morin's Integrating 
 Dynamometer.) A is a plane cir- 
 cular disc, turning about an axis 
 perpendicular to its own plane. B 
 is a wheel driven by the friction of Fig. 221. 
 
 the disc against its edge ; and it turns 
 
 about an axis that cuts the axis of A at right angles. The angu- 
 lar velocity of B varies proportionally to its distance from the 
 centre of A, and is varied by altering that distance. 
 
 271. changing Speed by Toothed Wheels. The ordinary method 
 of producing precise and definite changes of the angular velocity- 
 ratio of two rotating shafts is by means of change-wheels : that is 
 to say, there are several pairs or trains of wheels, suited to a cer- 
 tain series of velocity-ratios; and one or other of those pairs or 
 trains of wheels is thrown into gear according to the comparative 
 speed that is wanted at the time. 
 
 Sometimes the wheels are made so as to be put on the shafts 
 and taken off at pleasure. If an intermediate idle wheel is not 
 used, between two shafts connected by pairs of change -wheels, there 
 must be as many pairs of change- wheels as there are different 
 velocity-ratios; because the sum of the geometrical radii of each 
 pair must be equal to the line of centres; but by the help of an 
 intermediate idle wheel, any two wheels which are not so large as 
 to touch each other may be put into connection; so that by a 
 proper choice of numbers of teeth, the number of different ratios 
 may be made equal to the product of the number of different 
 wheels that can be fitted on one shaft into the number that can 
 be fitted on the other after the first has been fitted. 
 
 Change-wheels ai-e frequently arranged so as to be thrown into 
 or out of gear by shifting the whole series longitudinally along 
 with the shaft that carries them. For example, in fig. 222 A A 
 and B B are a pair of parallel axes; and the transverse lines 
 marked 1, 2, 3, &c., represent the radii of two series of change- 
 wheels carried by shafts turning about those axes respectively. 
 To each wheel of one series there corresponds a wheel in the other 
 series, marked with the same figure ; and any such pair can be 
 thrown into gear when required, by shifting the shaft A longitudi-
 
 312 
 
 GEOMETRY OF MACHINERY. 
 
 nally. To place the wheels on the shafts so as to occupy the least 
 possible space, the following rules are to be observed :- Let b denote 
 the breadth of the rim of a wheel, plus a small allowance for clear- 
 ance. Range the radii of the wheels on A in such a manner that 
 the greatest shall be in the middle, with a diminishing series on 
 
 ff 7 
 
 
 
 
 
 
 
 
 ! 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 B J 
 
 L 
 
 2 
 
 
 3 
 
 F 
 
 L 
 K. 2 
 
 22. 
 
 > 6 
 
 \ 
 
 t 
 
 3 
 
 each side of it. Then, commencing at the two ends of the double 
 series, make the two endmost intervals between the middle planes 
 of the wheels on the axis A (1 2 and 7 6 in the figure), each = b ; 
 the pair of intervals next them (2 3 and 6 5 in the figure), each 
 = 26; the next pair (3 4 and 54), each = 36; and so for any 
 number of intervals that may be required. Then make the in- 
 terval between the middle planes of each pair of wheels on the axis 
 B greater by one breadth, b, than the corresponding interval on the 
 axis A. 
 
 272. Changing Speed by Bands nnd Pulleys. The most convenient 
 
 way of changing the velocity-ratio of rotation of a pair of shafts, 
 where absolute precision in the ratio is not required, is by means 
 of "speed-cones," which have already been described in Article 175, 
 page 185. When a series of pulleys is used with radii changing 
 step by step, the motion must be stopped in order to shift the band 
 from one pair of pulleys to another ; and this is applicable to cords 
 as well as to belts. When tapering conoidal pulleys are used, the 
 belt can be shifted, and the velocity-ratio gradually changed, while 
 the machineiy is in motion; and this is applicable to belts only. 
 
 273. Changing Stroke in Linktrork. The principles upon which 
 the length of stroke in linkwork depends have been explained in 
 Article 186, page 197. When a piece receives a reciprocating 
 motion from a lever, a crank, or an eccentric, the simplest way of 
 changing the length of stroke is to change the distance of the con- 
 nected point in the lever, crank, or eccentric, from its axis of 
 motion. In the case of a continuously rotating crank or eccentric, 
 this can be done by means of an adjusting screw, the motion being
 
 CHANGING SPEED. 
 
 313 
 
 stopped when an alteration is to be made; but in the case of a 
 reciprocating lever, the pin to which the connecting-rod is jointed 
 may be carried by a stud, capable of sliding in a slot in the lever, 
 and having its position in that slot adjusted by means of a rod and 
 a handle which can be shifted while the machinery is in motion. 
 Sufficient examples of the latter kind of action have already been 
 given under the head of link-motions, in Article 240, pages 253 to 
 260. 
 
 Fig. 223 represents a train of linkwork proposed by "Willis, for 
 adjustingthe velocity-ratio 
 and comparative length of 
 stroke of two reciprocating 
 points. The points to be 
 connected are marked D 
 and E; and D A and E A 
 are their lines of stroke, 
 intersecting each other in 
 A. A B is a train-arm 
 centred at A, and capable 
 of being adjusted to any 
 required angular position. 
 At B, the other end of the 
 train-arm, is centred the 
 reciprocating lever B C, 
 
 equal in length to B A, and e 
 
 connected with the points 
 D and E by the links D Fig. 223. 
 
 and C E. 
 
 While the lever B C oscillates through a small angle to either 
 side of B A, the motions of D and E are very nearly equal to the 
 component motions of C along A D and A E respectively ; that is 
 to say, we have, at any given instant, the following proportion 
 very nearly exact : 
 
 Velocity of C : velocity of D : velocity of E 
 :: BA : BF : BG; 
 
 in which B F and B G denote the lengths of perpendiculars let 
 fall from B on A D and A E respectively ; and the same propor- 
 tions hold very nearly for the lengths of stroke of those three 
 points; hence those proportions can be made to assume any re- 
 quired value while the mechanism is in motion, by adjusting the 
 position of the train-arm A B. 
 
 274. Changing Speed with Hydraulic Connection. The Com- 
 parative speed of a piston driven by a fluid may be altered by 
 altering the number of driving-pistons which force the fluid into 
 the cylinder of the driven piston at the same time. For example,
 
 314 GEOMETRY OF MACHINERY. 
 
 in some hydraulic presses it is desirable to diminish step by step 
 the ratio which the velocity of the press-plunger bears to that of 
 the pump-plungers; and that is done by forcing water into the 
 press-cylinder at first by means of several pumps at once, and 
 diminishing their number as the process goes on, until at last only 
 one is kept at work. 
 
 ADDENDUM TO ARTICLE 267, PAGE 306. 
 
 Slide-Valve*. Another class of rotating slide-valves is that in 
 which the seat of the valve forms part of a cylindrical surface, 
 usually concave; the face of the valve forms an arc of a correspond- 
 ing cylindrical surface, convex when the seat is concave ; and the 
 reciprocating motion of the valve takes place by rocking, or oscillat- 
 ing rotation, about the axis of the cylindrical surfaces. The " Cor- 
 liss " valves are an example of this. 
 
 A straight-sliding slide-valve and its seat are also sometimes of 
 a cylindrical form, the reciprocating motion taking place parallel 
 to the axis of the cylinder. 
 
 There are instances of plane-faced slide-valves which have motions 
 of curvilinear translation, produced by aggregate combinations of 
 linkwork : for example, Hunt's slide-valves.
 
 PART II 
 
 DYNAMICS OF MACHINERY. 
 
 CHAPTER I. 
 
 SUMMARY OF GENERAL PRINCIPLES. 
 
 275. Natnre 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 mech- 
 anism; and therefore much of the present part of this treatise 
 will consist of references back to Part I.* 
 
 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 detennination 
 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. 
 
 The present Chapter contains a summary of the principles of 
 dynamics that word being taken in the comprehensive sense in 
 which it is used in Thomson and Tait's Natural Philosophy, to 
 
 * A large portion of the present Part, and especially of the second Chapter, 
 although originally written for this work, has already appeared as an Intro- 
 duction to A Manual of the Steam Engine and other Prime Movers; for that 
 book would have been incomplete without an explanation of the dynamical 
 principles of the action of machines in general.
 
 31 G DYNAMICS OF MACHINERY. 
 
 denote the science of forces, whether employed in balancing each 
 other or in producing motion. The ensuing Chapters will contain 
 the special application of those principles to machines. 
 
 276. Force* Action and Kcanioii. Every force is an action 
 exerted between a pair of bodies, tending to alter their condition 
 as to relative rest and motion; and it is exerted equally, and in 
 contrary directions, upon each body of the pair. That is to say, 
 if A 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 analo- 
 gous to that of relative motion, explained in Article 42, page 21. 
 
 The forces chiefly to be considered in machines are the follow- 
 ing : 
 
 I. Gravity, exerted between the parts of the machine, fixed and 
 moving, and the whole mass of the earth. The action of the earth 
 on the machine alone requires to be considered in practice; for 
 although the re-action of the machine on the earth is equal and 
 opposite, the enormous mass of the earth, as compared with the 
 machine, causes the effects of that re-action to be inappreciable. 
 This is the only case in which re-action may be disregarded. 
 
 II. Forces exerted between parts of the machine and contiguous 
 external bodies, solid or fluid. Sometimes those bodies support 
 the foundations of the machine : sometimes they drive the 
 machinery ; as when the impulse or the pressure of a fluid drives 
 an engine: sometimes they are moved by it; as in the lifting of 
 loads, the overcoming of friction against external bodies, the 
 working of machine tools, <tc. 
 
 III. Forces exerted between a moving piece and tlie frame, at 
 their bearing surfaces. These forces may be distinguished into 
 pressure and friction. By the pressures exerted by the bearings 
 the moving piece is kept in its proper place and path; by friction 
 its motion is resisted. The equal and opposite re-actions of the 
 moving piece on the frame tend to strain the frame; and the 
 making of the frame so as to be capable of bearing them involves 
 questions of strength, belonging to the Third Part of this treatise. 
 
 IV. Forces exerted between connected moving pieces. These too 
 may be distinguished into pressure and friction. 
 
 When exerted along the line of connection, they serve to trans- 
 mit motion and motive power; when exerted transversely to it, 
 they produce either a straining effect, or a waste of mechanical 
 work, or both. Here the equality of action and re-action is of 
 great importance. The force which is exerted between a driver 
 and a follower along their line of connection is a driving force, 
 otherwise called an effort, as regards the motion of the follower, 
 and a resistance as regards the motion of the driver.
 
 FORCES. 317 
 
 V. Forces exerted between the different parts of one piece, 
 whether fixed or moving. These constitute the stress, by which 
 the piece resists the tendency of the forces applied to it externally 
 to overstrain it or to break it: and they belong to the subiect of 
 the Third Part. 
 
 277. Forces, how Determined and Expressed. A force, as re- 
 
 spects one of the two bodies between which it acts, is determined, 
 or made known, when the following three things are known 
 respecting it: first, the place, or part of the body to which it 
 is applied ; secondly, the direction of its action ; thirdly, its 
 magnitude. 
 
 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 & machine, 
 fixed or moving, that force may be treated as if it were con- 
 centrated at a point or points, determined by suitable pro- 
 cesses; and such is the use of those numerous propositions in statics 
 which relate to forces concentrated at points; or single forces, as 
 they are called. 
 
 The DIRECTION of a force is that of the motion which it tends 
 to pi'oduce. A straight line drawn through the point of applica- 
 tion of a single force, and along its direction, is the LINE OF 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 foi-ce is expressed arithmetically by stating 
 in numbers its ratio to a certain unit or standard 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 
 standard pound avoirdupois; which is the weight, in the latitude 
 of London, of a certain piece of platinum kept in a public office. 
 (See the Act 18 and 19 Viet., cap. 72; also a paper by Professor 
 "W. H. Miller, in the Philosophical Transactions for 1856.) 
 
 For the sake of convenience, or of compliance with custom, other 
 units of weight are occasionally employed in Britain, bearing certain 
 ratios to the standard pound ; such as
 
 318 DYNAMICS OF MACHINERY. 
 
 Tlie grain = roVor of a pound avoirdupois. 
 
 The troy pound = 5,760 grains = 0-82285714 pound avoirdupois. 
 
 The hundredweight = 112 pounds avoirdupois. 
 
 The ton = 2,240 pounds avoirdupois. 
 
 The French standard of -weight is the kilogramme, 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 centigrade, or 
 39 0> 4 Fahrenheit, and under the pressure which supports a baro- 
 metric column of 760 millimetres of mercury ; but it is in reality a 
 little greater. 
 
 A comparison of French and British measures of weight and of 
 size is given in a table at the end of this volume. 
 
 A kilogramme is 2-20462125 Ibs. avoirdupois. 
 A pound avoirdupois is 0-4535926525 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 unit of velocity is in Britain one foot per second; in 
 
 France one metre per second. 
 
 The unit of mass is the mass of so much matter as weighs one 
 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 co-efficients is denoted by the 
 letter g. 
 
 A single force may be represented in a drawing by a straight 
 line ; the position of the line showing the line of action of the force, 
 and an arrow-head its direction ; a point in the line marking the 
 point of application of the force ; and the length of the line repre- 
 senting the magnitude of the force. 
 
 277 A. measures of Force and mass. If by the unit of force 
 
 is understood the weight of a certain standard, such as the 
 avoirdupois pound, then the mass of that standard is 1 + g; and
 
 MASS RESULTANT AND COMPONENT FORCES. 319 
 
 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; for g is the velocity 
 which a body's own weight, acting unbalanced, impresses on it 
 in a second. This is the system employed in many scientific 
 writings, and in particular, in Thomson and Tait's Natural Philo- 
 sophy. It has great advantages 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. 
 
 278. 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 combined 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 conv 
 ponents are related to each other exactly in the same manner with 
 the velocities and directions of resultant and component motions; 
 and all the rules of Article 41, pages 18 to 21, are applicable to 
 forces as well as to motions, and need not be repeated here. 
 
 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 
 resultant is to be found by methods which will be stated further 
 on. 
 
 The following are additional rules as to resultant and component 
 forces not explicitly given in Article 41 : 
 
 I. If the component forces act along one line, all in the same 
 direction, their resultant is equal to their sum; if some act in one 
 direction and some in the contrary direction, the resultant is their 
 algebraical sum; that is to say, add together separately the forces 
 which act in the two contrary directions respectively; the difference 
 of the two sums will be the amount of the resultant, and its 
 direction will be the same with that of the forces whose sum is 
 the greater. This principle applies also to the magnitudes and 
 directions of parallel forces not acting in one line. 
 
 II. Triangle of Forces. Given, the directions of three forces 
 which balance each other, acting in one plane and through one
 
 320 DYNAMICS OF MACHINERY. 
 
 point ; construct a triangle whose sides make the same angles with 
 each other that the directions of the forces do ; the proportions of 
 the forces to each other will be the same with those of the cor- 
 responding sides of that triangle. Unless three forces act in one 
 plane and through one point, they cannot balance each other. 
 
 To solve the same question by calculation ; let A, B, C, stand 
 for the magnitudes of the three forces ; A O B, B O C, C A, for 
 the angles between their directions; then 
 
 sin B O C : sin C O A : sin A B : : A : B : C. 
 
 Each of those three forces is equal and opposite to the resultant 
 of the other two. 
 
 III. Polygon of Forces. To find the resultant of any number 
 (Fj, F 2 , F 3 , <fec., fig. 224), of forces in different directions, acting 
 through one point, O. Commence at the point of application, and 
 construct a chain of lines representing the forces in magnitude, 
 and parallel to them in direction, (0 A = and || F p A B = and 
 || F^ B G = and || F 3 , &c.) Let D be the end of that chain; join 
 ^ ^ O D ; this will represent the required 
 
 resultant; and a force (F 5 ) equal and 
 opposite to O D will balance the given 
 forces. This rule applies to the pro- 
 jections of the forces on any given 
 plane. 
 
 To solve the same question by cal- 
 Fig. 224. culation instead of by construction : 
 
 IV. (When the forces act in one 
 
 plane.) Assume any two directions at right angles to each other 
 as axes; resolve each force into two components (X, Y) along those 
 axes; take the resultants of those components along the two axes 
 separately (2 X, 2 Y); these will be the rectangular components 
 oft/ie resultant R of all the forces; that is to say, 
 
 and if * be the angle which B, makes with X, 
 
 2X 2Y 
 
 
 V. (When the forces act in different planes.) Assume any 
 three directions at right angles to each other as axes; resolve each 
 force into three components (X, Y, Z) along those axes; take the- 
 resultants of the components along the three axes separately (2 X, 
 2 Y, 2 Z); these will be the rectangular components oftlie resultant
 
 FORCES COUPLES. 321 
 
 of all tlie forces; and its magnitude and direction will be given by 
 the following equations: 
 
 K = A/ | (2 X)2 + (2 Y) 2 -r (2 Z) 2 1 
 
 2X 2Y 2Z 
 
 cos = -jj-; cos /3 = -^-- cos y = ^-. 
 
 279. Couples. In fig. 225, let F, F represent a couple of equal, 
 parallel, and opposite forces, applied to a rigid body, and not acting 
 in the same line; L, the perpendicular 
 distance between their lines of action; 
 then F is the force of the couple, L the j 
 arm, span, or leverage; and the product 
 force x leverage = F L is the statical 
 moment of the couple, which is right or 
 left-handed according as the couple tends j,. 225 
 
 to impress right-handed or left-handed 
 
 rotation on the body (Article 48, page 25). All the forces which 
 produce and resist the motion of rotating pieces in a machine 
 act in couples. Couples of equal moment acting in the same 
 direction and in the same plane, or in parallel planes, are equivalent 
 to each other. 
 
 Comparison of Measures of Statical Moment. 
 
 Kilogrammfetres, 
 
 Inch-lb. = 0-01152 1 
 
 12 = i Ft.-lb. = 0-138254 
 
 112= 9s" = i Inch-cwt. = 1-29037 
 
 1,344= 112 = 12 = i Foot-cwt. = 15-4844 
 
 2,240 = i86| = 20 = if = i Inch-ton = 25-8074 
 
 26,880 = 2,240 = 240 = 20 =12 = i Foot-ton =309-689 
 
 I. To find the resultant moment of any number of couples 
 acting on a rigid body in the same plane, or in parallel planes. 
 Take the sums of the right-handed and left-handed moments 
 separately; the difference between those sums will be the result- 
 ant moment, whicli will be right-handed or left-handed according 
 to the direction of the moments whose sum is the greater. 
 
 II. To represent the moment of a couple by a single line. Upon 
 any line perpendicular to the plane of the couple, set off a length 
 proportional to the moment (0 M, fig. 225), in such a direction 
 that to a spectator looking from O towards M the couple shall 
 seem right-handed. The line O M is called the axis of the couple. 
 
 Couples as represented by their axes are compounded and re- 
 solved like velocities, and like single forces, by the Rules o$
 
 322 
 
 DYNAMICS OF MACHINERY. 
 
 Article 41, pages 18 to 21, and by those of Article 278, page 
 319. 
 
 III. To find the resultant of a single force, F, applied to a rigid 
 body at 0, and a couple, M, acting on the same body in the same 
 
 or in a parallel plane. Conceive 
 the force F to be shifted in 
 that plane, parallel to itself, to 
 the left if the couple is right- 
 handed, to the right if the couple 
 is left-handed, through a dis- 
 tance, O A, found by dividing 
 M by F. The shifted single 
 force, F acting through A, will 
 Fig. 226. be the resultant required. 
 
 (The combination of a single 
 
 force with a couple acting in a plane perpendicular to the line of 
 action of the force cannot be further simplified.) 
 
 IY. To resolve a single force into a single force acting in a dif- 
 ferent but parallel line, and a couple. In fig. 227, let F be the 
 given force acting in the line E D, and B a given point not in E D. 
 Through B conceive a pair of equal and contrary forces to act in 
 a line parallel to E D; viz., + F* equal to F and in the same 
 direction; and F* equal to F and in the con- 
 trary direction; also, let fall B A perpendicular 
 to E D. Then the original force F acting through 
 A is resolved into the equal and parallel force F 7 
 acting through B, and the couple of forces F and 
 F 7 , with the arm A B and moment F x A B; 
 which couple is right or left-handed according as 
 B lies to the right or left of F, relatively to a 
 -^ spectator looking in the direction towards which 
 F acts. 
 
 F x A B is called the moment of the force F 
 relatively to the point B; or relatively to the 
 axis O X traversing B in a direction perpen- 
 dicular to the plane of F and A B ; or relatively 
 to a plane traversing B perpendicularly to A B. 
 280. Parallel Forces. I. To find the resultant of two parallel 
 forces. The resultant is in the same plane with, and parallel to, 
 the components. It is their sum or difference, according as they 
 act in the same or contrary directions ; and in the latter case its 
 direction is that of the greater component. To find its line of 
 action by construction, proceed as follows : Fig. 228 representing 
 the case in which the components act in the same direction, fig. 
 229 that in which they act in contraiy directions. Let A D and 
 B E be the components. Join A E and B D, cutting each other
 
 PARALLEL FORCES. 
 
 323 
 
 in F. In B D (produced in fig. 229) take B G = D F. Through 
 O draw a line parallel to the components; this will be the line of 
 
 H 
 
 Fig. 228. 
 
 action of the resultant. To find its magnitude by construction: 
 parallel to A E, draw B C and D H, cutting the line of action 
 of the resultant in C and H; C H will represent the resultant 
 required; and a force equal and opposite to C H will balance A D 
 and B E. 
 
 To find the line of action of the resultant by calculation; make 
 either 
 
 AD -DB 
 OH ; 
 
 BE-DB 
 OH 
 
 When the two given parallel forces are opposite and equal, they 
 form a couple, and have no single resultant. 
 
 II. 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 /^ \ *s 
 line A C B, fig. 230, and measure the dis- 
 tances between the points where it cuts the 
 lines of action. Then each force will be pro- \ \ 
 portional to the distance between the lines \ \ 
 of action of the other two. The direction of \ >A/ 
 the middle force, C, is contrary to that of the 
 other two forces, A and B. Fig. 230. 
 
 In symbols, let A, B, and C be the forces; then,
 
 324 DYNAMICS OF MACHINERY. 
 
 A-fB + 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. 
 
 III. 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 
 Fig. 231. action in any convenient position ; and in 
 
 fig. 231 or fig. 232, let A, B, C, D represent 
 
 the points where the four lines of action cut the plane. Draw the 
 K 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 
 >B formed by the other three points. 
 
 In fig. 231 the directions of the forces at A, B, 
 and C are the same, and are contrary to that of 
 Fig. 232. the force at D. In fig. 232 the forces at A and D 
 act in one direction, and those at B and C in the 
 contrary direction. 
 In symbols, 
 
 BCD:CDA:DAB:ABC 
 : : A : B : C : 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 
 opposite to the components of the fourth. Hence the rule serves 
 to resolve a force into three parallel components not acting in one 
 plane. 
 
 IV. To find the Resultant of any number of Parallel forces. For 
 the magnitude and direction of the resultant take the algebraical 
 sum of the components as if they acted along one line (Article 
 278, page 319). This may be denoted by R = 2 F. For the 
 position of the resultant proceed as follows : In any plane per- 
 pendicular to the lines of action of the parallel forces take an axis 
 of moments in any convenient position. Multiply each component 
 force (F) by its perpendicular distance (x) from that axis, so as to 
 obtain its moment (F x) relatively to the axis. Mark those 
 moments as positive or negative according to the direction in which 
 they tend to turn the body to which they are applied about the 
 axis; and take their algebraical sum, which will be the resultant
 
 PARALLEL FORCES SPECIFIC GRAVITY HEAVINESS DENSITY. 325 
 
 moment (M = 2 F a;). Divide the resultant moment by the result- 
 ant force ; the quotient will be the perpendicular distance of the 
 line of action of the resultant from the axis of moments; viz: 
 
 _M_2-Fa; 
 *-R- ; S~F ' 
 
 The algebraical sign of this distance will indicate its direction. 
 
 Take another axis of moments in the same plane, and per- 
 pendicular to the first axis; and by a similar operation find the 
 perpendicular distance of the resultant from the second axis. 
 The position of the resultant will then be completely deter- 
 mined. 
 
 If R = 0, the resultant is a couple. If M = 0, the line of 
 action of the resultant traverses the axis of moments. 
 
 281. Specific Gravity Heaviness Density IJulkiiicss. I. Specific 
 
 Gravity is the ratio of the weight of a given bulk of a given sub- 
 stance 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 U Fahr., pure water expands in 
 the ratio of 1-001118 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 Ibs. avoirdupois to the cubic foot; in 
 French measures, in kilogrammes to the cubic decimetre (or to the litre). 
 The values of the heaviness of water at 39 0> 1 Fahr., and at 62 
 Fahr., are respectively 62-425 and 62-355 Ibs. 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 
 to 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 
 u Density " is often used to denote the i-atio of the heaviness of a 
 given gas to that of air, at the same temperature and pressure. 
 
 IV. The Budkiness of a substance is the number of units of 
 volume which an unit of weight fills; and is the reciprocal of the 
 heaviness. In British measures bulkiness is most conveniently 
 expressed in cubic feet to the Ib. avoirdupois ; in French measures, 
 in cubic decimetres (or in litres) to tlie kilogramme. 
 
 Rise of temperature produces (with certain exceptions) increase of 
 bulkiness. The following are examples of rates of expansion in bulk, 
 in rising from the freezing to the boiling point of water: that is 
 from Cent, or 32 Fahr., to 100 Cent, or 212 Fahr. The lineat 
 expansion of a solid body is one-third of its expansion in bulk.
 
 326 
 
 DYNAMICS OP MACHINERY. 
 
 Perfect gases, 0-365 ; air at ordinary pressures, 0'366 ; water, 
 0-04775; spirit of wine, 0'1112; mercury, 0-018153; oil, linseed 
 and olive, 0'08 ; wrought iron and steel, 0-0036 ; cast iron, 0-0033 ; 
 copper, 0-0055; bronze, 0-0054; brass, 0-0065: brick, common, 
 0-0106 ; fire-brick, 0-0015 ; glass, 0-0027. 
 
 TABLE OF HEAVINESS AND SPECIFIC GRAVITY. 
 
 Weight of a cubic 
 foot in 
 
 GASES, at 32 Fahr., and under one atmosphere : lb - avoirdupois. 
 
 Air, 0-080728 
 
 Carbonic acid, 0-12344 
 
 Hydrogen, 0-005592 
 
 Oxygen, 0-089256 
 
 Nitrogen, 0-078596 
 
 Steam (ideal), 0-05022 
 
 LIQUIDS, at 32 Fahr. (except Water, ^ 6igh f0 o t f & cubic l$5r 
 
 which is taken at 39'! Fahr.) : Ibs. avoirdupois. pure water = 1. 
 
 Water, pure, at 39'!, 62-425 i-ooo 
 
 sea, ordinary, 64-05 1-026 
 
 Alcohol, pure, 49'3& 0-791 
 
 proof spirit, 57'i8 0-916 
 
 44'7o 0-716 
 
 Mercury, 848-75 13-596 
 
 Naphtha, 52*94 0-848. 
 
 Oil, linseed, 58-68 0-940. 
 
 olive, 57-12 0-915 
 
 whale, 57'^2 0*923; 
 
 of turpentine, 54'3i 0-870 
 
 Petroleum, 54'8t 0-878 
 
 SOLID MINERAL SUBSTANCES, non-metallic : 
 
 Brick, 125 to 135 2 to 2-167 
 
 Brickwork, 112 1-8 
 
 Coal, anthracite, 100 1-602 
 
 bituminous, 77-4 to 89-9 1-24 to 1-44 
 
 Coke, 62'43to 103-6 roo to 1-66 
 
 Glass, crown, average, 156 2-5 
 
 flint, 187 3-0 
 
 green, 169 2-7 
 
 plate, 169 2-7 
 
 Granite, 164^0172 2-63 to 2-76 
 
 Limestone (including marble), 169 to 175 2-7 to 2 -g 
 
 magnesian, 178 2-85 
 
 Masonry, 116 to 144 1-85 to 2-3
 
 HEAVINESS AND SPECIFIC GRAVITY. 327 
 
 SOLID MINERAL SUBSTANCES, non- Weigh / 00 > t f fn Cnbic g?S 
 
 metallic continued, Ibs. avoirdupois. purewater' = l. 
 
 Mortar, 109 175 
 
 Sand (damp), 118 1-9 
 
 (dry), 88-6 1-42 
 
 Sandstone, average, 144 2-3 
 
 various kinds, 130 to 157 2-08 to 2-52 
 
 METALS, solid : 
 
 Brass, cast, 487 to 524-4 7-8 to 8-4 
 
 wir e, 533 8 '54 
 
 Bronze, 524 8-4 
 
 Copper, cast, 537 8-6 
 
 sheet, 549 8-8 
 
 hammered, 556 8-9 
 
 Gold, 1186 to 1224 19 to 19-6 
 
 Iron, cast, various, 434 to 456 6-95 to 7 '3 
 
 average, 444 7-11 
 
 Iron, wrought, various, 47 4 to 487 7 -6 to 7 -8 
 
 avei-age, 480 7-69 
 
 Lead, 712 11-4 
 
 Platinum, 1311 to 1373 2I * 22 
 
 Silver, 655 10-5 
 
 Steel, 487 to 493 7-8 to 7-9 
 
 Tin, 456 to 468 7'3 to 7'5 
 
 Zinc, 424 to 449 6-8 to 7-2 
 
 TIMBER : * 
 
 Ash, 47 0-753 
 
 Bamboo, 25 0-4 
 
 Beech, 43 0-69 
 
 Box, 60 0-96 
 
 Elm, 34 0-544 
 
 Fir:RedPine, 30 to 44 0-48 to 0-7 
 
 Spruce, 30 to 44 0-48 to 07 
 
 American Yellow Pine, 29 0-46 
 
 Larch, 31 to 35 0-5 to 0-56 
 
 Mahogany, Honduras, 35 0-56 
 
 Spanish, 53 0-85 
 
 Oak, European, 43 to 62 0-69 to 0-99 
 
 American, Red, 54 0>8 7 
 
 Teak, 41 to 55 0-66 to 0-88 
 
 "Willow........ 25 0-4 
 
 Yew, 5 ' 8 
 
 * The Timber in every case is supposed to be dry.
 
 328 DYNAMICS OF MACHINERY. 
 
 WEIGHT OP CUBES, RODS, PLATES, BARS, AND SPHERES. 
 
 
 A. 
 
 B. 
 
 c. 
 
 D. 
 
 E. 
 
 F. 
 
 
 Cubic 
 
 Bound 
 Rod, 
 
 Square 
 Bar, 
 
 Plate, 
 1 ft x 1 ft. 
 
 Cubic 
 
 Sphere, 
 
 
 Inch. 
 
 1 ft long 
 
 1ft x 1 
 
 
 foot 
 
 diam 
 
 
 
 1 in. diam. 
 
 in. x 1 in. 
 
 x in. 
 
 
 ' 
 
 
 Ibs. 
 
 Ibs. 
 
 Ibs. 
 
 Ibs. 
 
 Ibs. 
 
 
 Brass, cast, average, ... 
 
 0-298 
 
 2'8l 
 
 3-58 
 
 43 "0 
 
 516 
 
 0-I56 
 
 
 O'^oS 
 
 2'pl 
 
 3'7 
 
 44'4 
 
 533 
 
 0-l62 
 
 
 J 
 
 O'^O^ 
 
 2-86 
 
 3-64 
 
 437 
 
 524 
 
 0-I59 
 
 
 O J 
 
 0-318 
 
 2-99 
 
 3-81 
 
 ~<J 1 
 
 4575 
 
 549 
 
 0-166 
 
 hammered,.... 
 
 J 
 
 0-322 
 
 3-3 
 
 3-86 
 
 46-3 
 
 556 
 
 0-168 
 
 Iron, cast, average,.... 
 
 0-257 
 
 2-42 
 
 3-08 
 
 37-0 
 
 444 
 
 0-134 
 
 Iron, wrought, average, 0-278 
 
 2-62 
 
 3'33 
 
 40-0 
 
 480 
 
 0-146 
 
 
 0-412 
 
 
 
 3-88 
 
 4-94 
 
 59'3 
 
 712 
 
 0-216 
 
 
 0-283 
 
 2-67 
 
 * y 
 
 3-40 
 
 40-8 
 
 ~ 
 
 4 
 
 490 
 
 0-148 
 
 
 **^o 
 
 0*267 
 
 1 
 
 2-52 
 
 J i 
 
 3-21 
 
 38-5 
 
 9 S 
 
 462 
 
 0-140 
 
 Zinc, averaere. .. 
 
 fl 
 
 o-2S2 
 
 2-38 
 
 X 
 
 TO 3 
 
 ' ' ' 
 
 36-3 
 
 436 
 
 0-132 
 
 282. Centre of CSrayltj Moment of Weight. RULE I. The centre 
 
 of gravity of a body of uniform heaviness is its centre of magnitude. 
 (See supplement to this Chapter, page 334.) 
 
 RULE II. To find the moment of a body's weight relatively to 
 a given plane of moments ; multiply the weight by the perpendicular 
 distance of the body's centre of gravity from the given, plane. 
 
 NOTE. In comparing together or combining the moments of 
 weights which lie some at one side and some at the other side of a 
 plane of moments, those moments are to be distinguished into 
 positive and negative, according to the sides of the plane at which 
 the weights lie. 
 
 RULE III. To find the common centre of gravity of a set of 
 detached bodies; find their several moments relatively to a con- 
 venient fixed plane ; find the resultant of those moments by adding 
 together, separately, the positive and negative moments, and taking 
 the difference between the two sums, which will be positive or 
 negative according as the positive or negative sum is the greater. 
 Divide that resultant moment by the total weight ; the quotient 
 will be the perpendicular distance of the common centre of gravity 
 from the fixed plane ; and its positive or negative sign will show at 
 which side of the plane that centre lies. If necessary, repeat the 
 same process for a second and a third fixed plane, so as to deter- 
 mine the position of the required centre completely. The two or 
 three planes (as the case may be) are usually taken perpendicular 
 to each other. 
 
 RULE IV. To find the centre of gravity of a body consisting of 
 parts of unequal heaviness; find separately the centres of those 
 parts, and treat them as detached weights by Rule III.
 
 DISTRIBUTED FORCES VARIED MOTION. 329 
 
 283. The Centre of Pressure in a plane surface is the point 
 traversed by the resultant of a pressure that is exerted at that 
 surface. 
 
 RULE, Conceive that upon the pressed surface as a base, there 
 stands a prismatic solid of a height at each point of that surface 
 proportional to the intensity of the pressure; the point in the 
 pressed surface at the foot of a perpendicular from the centre of 
 magnitude of the solid (see supplement to this Chapter) will be the 
 centre of pressure. 
 
 When the intensity is uniform, the centre of pressure is at the 
 centre of magnitude of the pressed surface. 
 
 284. The entre of Buoyancy of a solid wholly or partly immersed 
 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. 
 
 285. The Resultant of a Distributed Force 1. To find the 
 
 resultant of a body's weight; find the centre of gravity of the body; 
 the resultant will be a single force equal to the weight, acting 
 vertically downwards through the centre of gravity. 
 
 II. To find the resultant of a pressure; find the centre of 
 pressure (as in Article 283); the resultant will be a single force 
 equal in amount to the pressure, and acting in the same direction. 
 and through the centre of pressure. The amount of the pressure 
 is equal to the area of the pressed surface, nmltiplied by the mean 
 intensity of the pressure, and is also equal to the weight of the 
 imaginary prismatic solid mentioned in Article 283. 
 
 286. 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 metre ; 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. (See Article 
 302.) 
 
 287. Principles Relating to Varied motion. An unbalanced force 
 applied to a body produces change of momentum equal in amount 
 to and coincident in direction with the impulse exerted by the 
 force. Impulse is the product of the force into the time during 
 which it acts in seconds. Momentum is the product of the mass of 
 a body into its velocity in units of distance per second. As to the 
 units of force and of mass, see Article 277A, page 318. A body 
 receiving an impulse re-acts against the body giving the impulse, 
 with an equal and opposite impulse. The following are rules based 
 on the equality of impulse and momentum : 
 
 I. To find what impulse is required to produce a given change 
 in the velocity of a given mass; multiply the mass by the change 
 in its velocity, in units of distance per second.
 
 330 DYNAMICS OF MACHINERY. 
 
 (If the change consists in acceleration, the impulse must be 
 forward; if in retardation, backward.) 
 
 II. To calculate what unbalanced effort or unbalanced resist- 
 ance, as the case may be, is required to produce a given increase or 
 diminution of a body's speed in a given time or in a given distance. 
 
 Case I. If the time is given ; multiply the weight of the mass 
 by its change of velocity; divide by g, and by the time in seconds. 
 
 Case II. If the distance is given ; multiply the Aveight of the 
 mass by the change in the half-square of its velocity, and divide by 
 g, and by the distance. (For values of g, see page 318.) 
 
 III. To find the re-action of an accelerated or retarded body; 
 find the force required to produce the change of velocity; the 
 re-action will be equal and opposite. 
 
 The momentum and re-action of a body of any figure undergoing 
 translation are the same as if its whole mass were concentrated at 
 its centre of gravity. 
 
 The principles of this and the following Article will be further 
 explained and exemplified in the next Chapter. 
 
 288. Deviated Motion and Centrifugal Force. To make a body 
 
 move in a curve, some other body must guide it by exerting on it 
 a deviating force directed towards the centre of curvature. The 
 revolving body re-acts on the guiding body with an equal and 
 opposite centrifugal force. 
 
 To find the deviating and centrifugal force of a given mass- 
 revolving with a given velocity in a circle of a given radius : 
 multiply the weight of the mass by the square of its linear velocity, 
 and divide by the radius; or otherwise: multiply the mass by the 
 square of its angular velocity of revolution, and multiply by the 
 radius : the result will be the value of the deviating and centri- 
 fugal forces in absolute units, which may be converted into units 
 of weight by dividing by g. 
 
 The resultant centrifugal force of a rigid body of any shape is the 
 same in amount and direction (though not the same in distribution) 
 as if the whole mass were collected at its centre of gravity. 
 
 288 A. Falling Bodies. The following rules apply to a body fall- 
 ing without sensible resistance from the air : 
 
 I. To find the velocity acquired at the end of a given time; 
 multiply the time in seconds by g (see page 318). 
 
 II. To find the height of fall in a given time; multiply the 
 square of the time in seconds by ^ g = 16-1 feet = 4 - 904 metres. 
 
 III. To find the height of fall corresponding (or " due ") to a 
 given velocity ; divide the half- square of the velocity by g. 
 
 IV. To find the velocity due to a given height; multiply the 
 height by 2 g, and extract the square root. 
 
 Zi = 8-025 feet = 4429 metres.
 
 GRAVITY MENSURATION OP AREAS. 
 
 331 
 
 TABLE OP HEIGHTS DUE TO VELOCITIES. 
 
 v = "Velocity in feet per second. 
 h = Height in feet = v 2 -f- 6 4 -4. 
 
 This table is exact for latitude 54|, and near enough to exact- 
 ness for practical purposes in all parts of the earth's surface. 
 
 V 
 
 h 
 
 I 
 
 01553 
 
 2 
 
 06211 
 
 3 
 
 13975 
 
 4 
 
 24845 
 
 5 
 
 38820 
 
 6 
 
 55901 
 
 7 
 
 76087 
 
 8 
 
 '99379 
 
 9 
 
 1-2578 
 
 10 
 
 1-5528 
 
 ii 
 
 1-8789 
 
 12 
 
 2-2360 
 
 13 
 
 2-6242 
 
 14 
 
 3-0435 
 
 15 
 
 3*4938 
 
 16 
 
 3'975 2 
 
 17 
 
 4-4876 
 
 18 
 
 5-0311 
 
 19 
 
 5-6056 
 
 20 
 
 6'2II2 
 
 21 
 
 6-8478 
 
 22 
 
 7-5I55 
 
 23 
 
 8-2143 
 
 2 4 
 
 8-9441 
 
 25 
 
 9-7050 
 
 26 
 
 10-497 
 
 V 
 
 7* 
 
 27 
 
 1 1 -320 
 
 28 
 
 12-174 
 
 29 
 
 13-059 
 
 30 
 
 J 3"975 
 
 3i 
 
 14-922 
 
 32 
 
 15-901 
 
 32-2 
 
 16-100 
 
 33 
 
 16-910 
 
 34 
 
 17-950 
 
 35 
 
 19-022 
 
 36 
 
 20-124 
 
 37 
 
 21-257 
 
 38 
 
 22-422 
 
 39 
 
 23-618 
 
 40 
 
 24-845 
 
 4i 
 
 26-102 
 
 42 
 
 27-391 
 
 43 
 
 28-711 
 
 44 
 
 30-062 
 
 45 
 
 3 I- 444 
 
 46 
 
 32-857 
 
 47 
 
 34'3 01 
 
 48 
 
 35776 
 
 49 
 
 37-283 
 
 5o 
 
 38-820 
 
 52 
 
 41-987 
 
 54 
 
 45-280 
 
 56 
 
 48-695 
 
 58 
 
 52-235 
 
 60 
 
 55'9 01 
 
 62 
 
 59-688 
 
 64 
 
 63-602 
 
 64-4 
 
 64-400 
 
 66 
 
 67-640 
 
 68 
 
 71-800 
 
 70 
 
 76-087 
 
 72 
 
 80-496 
 
 74 
 
 85-029 
 
 76 
 
 89-688 
 
 78 
 
 94-472 
 
 80 
 
 99*379 
 
 82 
 
 104-41 
 
 84 
 
 109-56 
 
 86 
 
 114-84 
 
 88 
 
 120-25 
 
 90 
 
 125-78 
 
 92 
 
 131-43 
 
 94 
 
 137-20 
 
 96 
 
 143-10 
 
 100 
 
 155-28 
 
 SUPPLEMENT TO CHAPTER I. 
 
 Rules for the Mensuration of Figures and finding of Centres of 
 Magnitude. 
 
 289. To Measure any Plane Area. Draw an axis or base-line, 
 A X, 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,
 
 332 DYNAMICS OF MACHINERY. 
 
 which breadths will, of course, be one more in number than the 
 intervals. (For example, in fig. 233, the length of the figure is 
 divided into ten equal intervals, and eleven breadths are measured 
 at b , b v &c.) Then the following rules are exact, if the sides 
 of the figure are bounded by straight lines, and by 
 "| parabolic curves not exceeding the third degree, and 
 
 are approximate for boundaries of any other figures. 
 
 RULE A. ("Simpsons First Rule" to be used 
 when the number of intervals is even.) Add together 
 the two endmost breadths, twice every second inter- 
 mediate breadth, and four times each of the remain- 
 ing intermediate breadths; multiply the sum by the 
 common interval between the breadths, and divide by 
 3; the result will be the area required. 
 
 For two intervals the multipliers for the breadths 
 are 1, 4, 1; 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." 
 Fig. 233. RULE B. ("Simpsons Second Rule," to be used 
 when the number of intervals is a multiple of 3.) 
 Add together the two endmost breadths, twice every third inter- 
 mediate 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. 
 
 RULE C. (" Mcrrifidd'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 opposite 
 the intervals between the breadths. Then take the differences 
 of those differences, or second differences, and write them opposite 
 the intervals between the first differences, distinguishing them into 
 positive and negative according to the following principles: 
 
 First Differences. Second Difference. 
 
 Positive increasing, or ) -r, ... 
 
 -V-T- ,. ,. . . , .' > Positive. 
 
 Negative diminishing, J 
 
 Negative increasing, or ) ivr f 
 
 Positive diminishing, / a e ^ atl 
 
 In the column of second differences there will now be two blanks 
 opposite the two endmost breadths; those blanks are to be filled up
 
 MENSURATION OF AREAS OP VOLUMES. 333 
 
 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 between them. 
 
 The area of the whole figure may be found 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 common 
 interval. 
 
 In symbols, let y be an actual breadth, and y' the corresponding 
 
 corrected breadth ; then y' = y =-^ A 2 y- 
 
 RULE D. ("Common Trapezoidal Rule" 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. 
 
 290. 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 289, page 332 ; and the result 
 of the calculation will be the volume required. If Rule C is used, 
 the volume will be obtained in separate layers. 
 
 METHOD II. By Prisms or Columns (" Woolley's Rule"}. 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. 
 
 291. To measure the Length of nnr Curve. Divide it into short 
 
 arcs, and measure each of them by Rule I of Article 51, page 28.
 
 334 DYNAMICS OF MACHINERY. 
 
 292. Centre of Magnitude Oencral 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 
 fi<nire 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 plana 
 
 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, 
 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, sepai-ately, 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 shows at which side of the plane the required centre 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 
 finding 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 sufficient. 
 
 293. 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. 233 (of 
 Article 289, page 332). Draw an axis, A X, in a convenient 
 position, divide it into equal intervals, measure breadths at th$
 
 CENTRE OP A PLANE AREA. 335 
 
 ends and at the points of division, and calculate the area, as in 
 Article 289. 
 
 Then multiply each breadth by its distance from one end of the 
 axis (as A) ; consider the products as if they were the breadths of 
 a new figure, and proceed by the rules of Article 289 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 
 position being in general perpendicular to A X), and by a similar 
 process 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. 
 
 RULE B. If convenient, the distance of the required centre 
 from a plane cutting an axis at one of the intermediate points of 
 division, 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 resultant moment of tha 
 whole figure, which being divided by the whole area, the quotient 
 will be the distance of the required centre from the plane of 
 division. 
 
 REMARK. When the resultant moment is = 0, the centre is in 
 the plane of division. 
 
 RULE C. 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 28% page 331 ; distinguish the 
 products into right-handed and left-handed, according 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 f 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.* 
 
 * 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 
 planes perpendicular to the area and to each other, and x and y the per-
 
 336 DYNAMICS OF MACHINERY. 
 
 294. 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 290) ; 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. 
 
 295. Centre of Magnitude of n Curred I^lne. RULE A. Toflnd 
 
 approximately the Centre of Magni- 
 tude of a very Flat Curved Line. 
 In fig. 234, let A D B be the arc. 
 Draw the straight chord A B, which 
 Fig. 234. bisect in C ; draw C D (the deflec- 
 
 tion of the arc) perpendicular to 
 
 A B; from D lay off D E = C D; E will be very nearly the 
 centre required. 
 
 This process is exact for a cycloidal arc whose 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 I) E too 
 small ; the error, for an arc subtending 60, being about -3^7 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 flat, divide it into 
 very flat arcs ; find their several centres of magnitude by Rule A, 
 and measure their lengths ; then treat the whole curve as a com- 
 pound figure, agreeably to Rule II. of Article 292, page 334. 
 
 296. Special Figures. I. TRIANGLE (fig. 235). From any t wo 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 
 Fig. 235. side, and cut off one- third part from that line, 
 
 commencing at the side. 
 
 II. QUADRILATERAL (fig. 236). Draw the two diagonals A C 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 
 
 pendicular distances of the centre of magnitude of the area from the same 
 planes; then 
 
 _JJ'xdxdy _ ffy d x d y 
 
 ~ -
 
 CENTRES OF MAGNITUDE. 
 
 Fig. 236. 
 
 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 
 pi-eceding rule, will be the centre required 
 
 III. PLANE POLYGON. Divide it into 
 triangles; find their centres, and measure 
 their areas; then treat the polygon as a 
 compound figure made up of the triangles, 
 by Rule II. of Article 292, page 334. 
 
 IV. PRISM OR CYLINDER WITH PLANE 
 PARALLEL ENDS. Find the centres of the 
 ends; 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. 
 
 V. TETRAHEDRON, OR TRIANGULAR PYRAMID (fig. 237). Bisect 
 any two opposite edges, as A D and ^ 
 
 B C, in E and F; join E F, and bisect 
 it in G ; 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; this 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 Ptule II. of Article 
 290. 
 
 VIII. CIRCULAR ARC. In fig. 238, let A B be the arc, and C 
 the centre of the circle of which it is part. 
 
 Bisect the arc in D, and join C 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. CIRCULAR SECTOR, C A D B, fig. 238. 
 Find C E as in the preceding rule, and 
 make C F = f C E ; F will be the centre re- 
 quired. 
 
 X. SECTOR OF A FLAT RING. Let r be the external and r' the 
 internal radius of the ring. Draw a circular arc of the same 
 
 \ Q '5 
 
 ' rtO ,___ A *> 
 
 angular extent with the sector, and of the radius -^ g ,- 2 -, 
 
 and find its centre of magnitude by Rule VIII. 
 
 z 
 
 Fig. 23;
 
 338 
 
 CHAPTER II. 
 
 OP THE PERFORMANCE OP WORK BY MACHINES. 
 
 SECTION I. Of Resistance and Work. 
 
 297. The Action of a Machine is to produce Motion against Resist- 
 ance. For example, 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 resistance 
 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 offers to having part of its surface removed ; and so of 
 other machines. 
 
 298. Work. (A. M., 513.) The action of a machine is measured, 
 or expressed as a definite quantity, by multiplying the motion which 
 it produces 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 
 many 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 overcome 
 in kilogrammes, and quantities of work in what are called kilo- 
 grammetres, one kilogrammetre being the work performed in lifting 
 a weight of one kilogramme through a height of one metre. 
 
 The following are the proportions amongst those units of distance, 
 resistance, and work, with their logarithms :
 
 WORK POWER VELOCITY. 339 
 
 Logarithms. 
 
 One metre = 3-2808693 feet, 0-515989 
 
 One foot = 0-30479721 metres, 1-484011 
 
 One kilogramme = 2-20462 Ibs. avoirdupois, 0-343334 
 
 One Ib. avoirdupois = '453593 kilogramme, 1-656666 
 
 One kilogrammetre = 7-23308 foot-pounds, 0-859323 
 
 One foot-pound = 0-138254 kilogrammetres, 1-140^77 
 
 299. 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 (A. M., 661). It may be expressed in units 
 of work (such as foot-pounds) per second, per minute, or per hour, 
 as the case may be; but there is a peculiar unit of power appro- 
 priated to its expression, called a HORSE-POWER, which is, in Britain, 
 
 550 foot-pounds per second, 
 or 33,000 foot-pounds per minute, 
 or 1,980,000 foot-pounds per hour. 
 
 This is also called an actual or real horse-power, to distinguish it 
 from a nominal horse-power, the meaning of which will afterwards 
 be explained. It is greater than the performance of any ordinary 
 horse, its name having a conventional value attached to it. 
 
 In France, the term FORCE DE CHEVAL, or CHEVAL-VAPEUR, is 
 applied to the following rate of work : 
 
 Foot-lbs. 
 
 75 kilogramme'tres per second = 54 2 ^ 
 
 or 4,500 kilogrammetres per minute = 32,549 
 or 270,000 kilogrammetres per hour = 1,952,932 
 
 being about one-seventieth part less than the British horse-power. 
 
 300. Velocity. If the velocity of the motion which a machine causes 
 to be performed against a given resistance be given, then the pro- 
 duct 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 
 tmits of time. 
 
 It is usually most convenient, for purposes of calculation, to 
 express the velocities of the parts of machines either in feet per 
 second or in feet per minute. For certain dynamical calculations 
 to be afterwards referred to, 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.
 
 340 DYNAMICS OP MACHINERY. 
 
 Comparison of Different Measures of Velocity. 
 
 
 
 Miles 
 
 Feet 
 
 Feet 
 
 Feet 
 
 
 
 per hour. 
 
 per second. 
 
 per minute. 
 
 per hour. 
 
 
 
 I 
 
 = 1-46 
 
 = 88 
 
 5280' 
 
 
 
 0-6818 
 
 = I 
 
 = 60 
 
 3600 
 
 
 
 0-01136 
 
 = 0-016 
 
 i 
 
 60 
 
 1 nautical mile 
 per hour, or 
 "knot," 
 
 I 
 
 0*0001893 
 = 1-1508 
 
 = 0-00027 
 = 1-688 
 
 = 0-016 = 
 = 101-27 
 
 I 
 
 6076 
 
 The units of time being the same in all civilized countries, the pro- 
 portions amongst their units of velocity are the same with those 
 amongst their linear measures. 
 
 301. Work in Terms of Angular Motion.^ (A. M., 593.) "When a 
 
 resisting force opposes the motion of a part of a machine which 
 moves round a fixed axis, such as a wheel, an axle, or a crank, the 
 product of the amount of that resistance into its leverage (that is, 
 the perpendicular distance of the line along which it acts from the 
 fixed axis) is called the moment, or statical moment, of the resist- 
 ance. 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 a foot-pound, but which, nevertheless, is a quantity 
 of an entirely different kind from a foot-pound of work. (Seep. 321.) 
 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 
 
 2 * = 6-2832 
 
 denote, as is customary, the ratio of the circumference of a circle to 
 its radius. Then the distance through which the given resistance 
 is overcome is expressed by 
 
 the leverage x 2 X T ; 
 
 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 
 = tlw resistance x tfo leverage x 2 * x T.
 
 WORK. 341 
 
 But the product of the resistance into the leverage is what is called 
 the moment of the resistance, and the product 2 it T is called the 
 angular motion of the rotating body ; consequently, 
 
 The work performed 
 = the moment of the resistance x the angular motion. 
 
 The mode of computing the work indicated by this last equation 
 is often more convenient than the direct mode already explained in 
 Article 298. 
 
 The angular motion 2 v T of a body during some definite unit of 
 time, as a second or a mimite, is called its angular velocity; that is 
 to say, angular velocity is the product oftJie turns and fractions of a 
 turn made in an unit of time into the ratio (2 K = 6-2832) oftlie 
 circumference of a circle to its radius. Hence it appears that 
 
 The rate of work 
 = the moment of the resistance x the angular velocity. 
 
 302. Work in Terms of Pressure ansl Volume. (A. M., 517.) 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 surface at which the pressure acts, if that area is per- 
 pendicular 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 area 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 
 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 area 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
 
 342 
 
 DYNAMICS OP MACHINERY. 
 
 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 7-7 of a cubic foot 
 
 in volume. 
 
 The following table gives a comparison of various units in which 
 the intensities of pressures are commonly expressed. (A. M., 86.) 
 
 Pounds on the 
 square foot. 
 144 
 
 I 
 
 One pound on the square inch, 
 
 One pound on the square foot, 
 
 One inch of mercury (that is, weight 
 of a column of mercury at 32 
 Fahr., one inch high), 7'73 
 
 One foot of water (at 3y-l Fahr.), 62-425 
 
 One inch of water, 5-2021 
 
 One atmosphere, of 29-922 inches 
 
 of mercury, or 760 millimetres, 2116-3 
 
 One foot of air, at 32 Fahr., and 
 under the pressure of one atmo- 
 sphere, 0*080728 
 
 One kilogramme on the square 
 
 metre, 0-204813 
 
 One kilogramme on the square 
 
 millimetre, 204813 
 
 One millimetre of mercury, 2 -7 847 
 
 Pounds on the 
 square inch. 
 
 0-4912 
 
 o-4335 
 0-036125 
 
 0-0005606 
 0-00142231 
 
 1422-31 
 0-01934 
 
 303. Algebraical Expressions for Work. (A. M., 515, 517, 593.) 
 
 To express the results of the preceding articles in algebraical sym- 
 bols, 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 = 2vT = 6-2832 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 H I is the statical moment of the resistance. Sup- 
 posing the resistance to be a pressure, exerted between a piston and 
 a fluid, let A be the area or projected area of the piston, and p the 
 intensity of the pressure in pounds per unit of area.
 
 WORK. 343 
 
 Then the following expressions all give quantities of work in the 
 given time in foot-pounds : 
 
 B s ; i B ? ; p As ; ip A. I. 
 
 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. 
 
 304. Work against an Oblique Force. (A. M., 511.) The resist- 
 ance 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 resistance required. That resolution is effected by means of the 
 well known principle of the parallelogram of forces as follows : 
 
 In fig. 239, let A represent the point at which a resistance is over- 
 
 come, A B the direction in which 
 
 B 
 
 ' that point is moving, and let A F 
 be a line whose direction and 
 length represent the direction and 
 pi 239 magnitude of a force obliquely 
 
 opposed to the motion of A. 
 From F upon B A produced, let fall the perpendicular F B ; the 
 length of that perpendicular will represent the magnitude of the 
 lateral component of the oblique force, and the length A B. will 
 represent the direct component or resistance. 
 
 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. 
 
 305. Summation of Quantities of Work. In every machine, resist- 
 ances are overcome during the same interval of time, by differ- 
 ent 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 
 
 2 -Us; (1.) 
 
 in which 2 denotes the operation of taking the sum of a set of
 
 344 DYNAMICS OP MACHINERY. 
 
 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.) 
 
 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 translation only, the velocities of every point at a given in- 
 stant are equal and parallel ; hence, in a given interval of time, the 
 motions of all the points are equal; and the work performed is to 
 be found by multiplying the sum of the resistances into the motion 
 as a common factor ; an operation expressed algebraically thus 
 
 (3.) 
 
 II. 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 oft/ie moments of the resist- 
 ances relatively to the axis into the angular motion as a common 
 factor an operation expressed algebraically thus 
 
 i2 -Rl- ............................. (4.) 
 
 The sum denoted by 2 R I is the total moment of resistance of the 
 piece in question. 
 
 III. In every train 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 science called by Mr. Willis, Pure 
 Mechanism. 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 E, 
 is to be overcome, moves with a velocity bearing the ratio n : 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
 
 WORK. 345 
 
 overcome, and the results added together, as the following symbol 
 denotes : 
 
 2-wR, (5.) 
 
 that sum is the equivalent resistance at the driving point; 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 
 
 s 2 -R (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 : 1 to that of the driving axle. Then the 
 equivalent moment of resistance at the driving axle is n R I ; 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 
 
 a-rolU (7.) 
 
 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 -nlRl (8.) 
 
 IV. Centre of Gravity. The work performed in lifting a body 
 is the product of the weight of the body into the height through which 
 its centre of gravity is lifted. (See Article 282, page 328.) 
 
 If a machine 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 of all the 
 weights into the height through 
 which their common centre of 
 gravity is lifted. 
 
 306. Representation of Work 
 
 by an Area As a quantity of 
 work is the product of two 
 " quantities, a force and a motion, 
 Fig.24'>. it may be represented by the 
 
 area of a plane figure, which is the product of two dimensions. 
 
 s
 
 34G 
 
 DYNAMICS OF MACHINERY. 
 
 7^ 
 
 p 
 ~7 
 
 ^ 
 
 .- < 
 
 ^ 
 
 c 
 
 \ 
 
 13 
 ^^ 
 
 a 
 
 :^- 
 
 
 
 S -A. B J' " 
 Fig. 241. 
 
 Let the base of the rectangle A, fig. 240, represent one foot of motion, 
 and its height one pound of resistance; then will its area represent 
 one foot-pound of work. 
 
 In the larger rectangle, let the base OS represent a certain 
 motion s, on the same scale with the base of the unit-area A ; and 
 let the height OR 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 O S O 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. 
 
 807. Work against Varying Resistance. (A. M., 515.) In fig. 241, 
 
 let distances as before, be re- 
 presented by lengths measured 
 along the base line O X of the 
 figure; and let the magnitudes 
 of the resistance overcome at 
 each instant be represented by 
 the lengths of ordinates drawn 
 perpendicular to O X, and paral- 
 lel to O Y : For example, 
 when the working body has moved through the distance repre- 
 sented by O S, let the resistance be represented by the ordinate S R. 
 
 If the resistance were constant, the summits of those ordinates 
 would lie in a straight line parallel to O X, like R B in fig. 240 ; 
 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 ERG, fig. 241, which is not parallel to O 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 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 OX 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 re-
 
 WORK. 347 
 
 presented by A B, let the resistance be represented by A C, and so 
 for other divisions. 
 
 Then the work performed during the division of the motion re- 
 presented by AB, on the supposition of alternate constancy and 
 abrupt variation of the resistance, is represented by the rectangle 
 AB -AC ; 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 ta 
 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 differ from the area O E G F by an amount less 
 than any given difference. 
 
 Therefore the area OEGF, bounded by tlie straight line OF, which 
 represents tJie motion, by the line E G, whose ordinates represent the 
 values of the resistance, and by the two ordinates O E and F G, repre- 
 sents exactly the work performed. (See Article 289, page 331.) 
 
 The MEAN RESISTANCE during the motion is found by dividing, 
 the area O E G F by the motion OF. 
 
 308. Useful Work and i.ost 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 work. 
 
 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 
 over-coming 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.
 
 348 DYNAMICS OP MACHINERY. 
 
 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. 
 
 309. Friction. (Partly extracted and abridged from A. J/., 189, 
 190, 191, 204, and 669 to 685). The most frequent cause of loss 
 of work in machines is friction being that force which acts be- 
 tween two bodies at their surface of contact so as to resist their 
 sliding on each other, and which depends 011 the force with which 
 the bodies are pressed together. The following law respecting the 
 friction of solid bodies has been ascertained by experiment : 
 
 The friction which a given pair of solid bodies, with (fair surfaces 
 in a given condition, are capable of exerting, is simply proportional 
 to the force with which t/iey are pressed together. 
 
 There is a limit to the exactness of the above law, when the 
 pressure becomes so intense as to crush or grind the parts of the 
 bodies at and near their surface of contact. At and beyond that 
 limit the friction increases more rapidly than the pressure; but 
 that limit ought never to be attained at the bearings of any 
 machine. 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, however, of the friction of rest over the friction of 
 motion, is instantly destroyed by a slight vibration; so that the 
 friction of motion is alone to be taken into account as causing con- 
 tinuous loss of work. 
 
 As to 'materials for bearings, see pages 462, 463, 464. 
 
 The friction between a pair of bearing surfaces is calculated by 
 multiplying the force with which they are directly pressed together, 
 by a factor called the co-efficient of friction, which has a special 
 value depending on the nature of the materials and the state of the 
 surfaces as to smoothness and lubrication. Thus, let R denote the 
 friction between a pair of surfaces ; Q, the force, in a direction per- 
 pendicular to the surfaces, with which they are pressed together ; 
 and /the co-efficient of friction; then
 
 FEICTI03T. 
 
 349 
 
 (1.) 
 
 The co-efficient of friction of a given pair of surfaces is the tan- 
 gent 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 each other. 
 
 The following is a table of the angle of repose <P, the co-efficient 
 of friction f= tan <P, and its reciprocal 1 :f, for the materials of 
 mechanism condensed from the tables of General Morin, and 
 other sources, and arranged in a few comprehensive classes. The 
 values of those constants which are given in the table have re- 
 ference to the friction of motion* (See pp. 399 and 476.) 
 
 No. 
 
 SURFACES. 
 
 9 
 
 / 
 
 I:/ 
 
 1 
 
 Wood on wood, dry, 
 
 14 to 26V 
 
 25 to '5 
 
 4 to 2 
 
 I 
 
 ,, soaped, 
 
 1H to 2* 
 
 2 to -04 
 
 5 to 25 
 
 3 
 
 Metals on oak, dry, 
 
 264 to 31 
 
 5 to '6 
 
 2 to 1-C7 
 
 4 
 
 wet. .. 
 
 13Jtol4i 
 
 24 to ' 6 
 
 4-17 to 3-85 
 
 5 
 
 
 1H 
 
 2 
 
 5 
 
 ft 
 
 Metals on elm, dry, 
 
 n y To 14 
 
 2 to '25 
 
 5 to 4 
 
 7 
 
 Hemp on oak, drv, 
 
 28 
 
 53 
 
 1-89 
 
 8 
 
 wet ... 
 
 18V 
 
 33 
 
 3 
 
 9 
 
 Leather on oak 
 
 15 to~19j 
 
 27 to -38 
 
 3-7 to 2-86 
 
 in 
 
 Leather on metals, drv, 
 
 29i 
 
 56 
 
 1-79 
 
 n 
 
 
 20 
 
 36 
 
 2-78 
 
 12 
 
 STTeasV. .., 
 
 13 
 
 23 
 
 4-35 
 
 13 
 
 oilv. .. 
 
 8i 
 
 15 
 
 C-G7 
 
 14 
 
 Metals on metals, drv, 
 
 8} to ll^ c 
 
 15 to -2 
 
 6'G7 to 5 
 
 15 
 
 n wet 
 
 16i 
 
 3 
 
 3-33 
 
 16 
 17 
 
 18 
 
 Smooth surfaces, occasionally greased. 
 continually greased. 
 OT ' best results, 
 
 4 to 4i 
 3 " 
 I 30 to 2 
 
 07 to -08 
 05 
 03 to '036 
 
 14 3 to 12-5 
 20 
 33-3 to 27-G 
 
 19 
 
 Bronze on lignum vitas, constanth- wet, 
 
 3? 
 
 05? 
 
 20? 
 
 * In a paper, of which an abstract appeared in the Comptes Eendus of the 
 French Academy of Sciences for the 26th of April, 1858, M. H. Bochet 
 describes a series of experiments which have led him to the conclusion, that 
 the friction between a pair of surfaces of iron is not, as had formerly been 
 believed, absolutely independent of the velocity of sliding, but that it 
 diminishes slowly as that velocity increases, according to a law expressed by 
 the following formula. Let 
 
 R denote the friction ; 
 
 Q, the pressure ; 
 
 v, the velocity of sliding, in metres per second = velocity in feet per second 
 X 0-3048; 
 
 /, a, y, constant co-efficients ; then 
 
 R _/+ y a v 
 Q~~ 1 + a v 
 
 The following are the values of the co-efficients deduced by M. Bochet from
 
 350 DYNAMICS OF MACHINERY. 
 
 310. Unguent*. Three results in the preceding table, Nos. 16, 17. 
 and 18, have reference to smooth firm surfaces of any kind, greased 
 or lubricated to such an extent that the friction depends chiefly on 
 the continual supply of unguent, and not sensibly on the nature of 
 the solid surfaces; and this ought almost always to be the case in 
 machinery. Unguents should be thick for heavy pressures, that 
 they may resist being forced out, and thin for light pressures, that 
 their viscidity may not add to the resistance. 
 
 Unguents may be divided into four classes, as follows : 
 
 I. Water, which acts as an unguent on surfaces of wood and 
 leather. It is not, however, an unguent for a pair of metallic 
 surfaces; for when applied to them, it increases their friction. 
 
 II. Oily unguents, consisting of animal and vegetable fixed oils, 
 as tallow, lard, lard oil, seal oil, whale oil, olive oil. The vegetable 
 drying oils, such as linseed oil, are unfit for unguents, as they 
 absorb oxygen, and become hard. The animal oils are on the 
 whole better than the vegetable oils. x 
 
 III. Soapy unguents, composed of oil, alkali, and water. For a 
 temporary purpose, such as lubricating the ways for the launch of 
 a ship, one of the best unguents of this class is soft soap, made from 
 whale oil and potash, and used either alone or mixed with tallow. 
 But for a permanent purpose, such as lubricating railway carriage 
 axles, it is necessary that the unguent should contain less water 
 and more oil or fatty matter than soft soap does, otherwise it 
 woxild dry and become stiff by the evaporation of the water. The 
 best grease for such purposes does not contain more than from 25 
 to 30 per cent, of water; that which contains 40 or 50 per cent, 
 is bad. 
 
 IV. Bituminous' unguents, in which liquid mineral hydrocarbons 
 are used to dissolve and dilute oily and fatty substances. 
 
 The intensity of the pressure between a pair of greased surfaces 
 ought not to be so great as to force out the unguent. The follow- 
 ing formula agrees very fairly with the results of practice : 
 
 Let v be the velocity of sliding, in feet per second; p, the greatest 
 proper intensity of pressure, in Ibs. on the square inch; then 
 
 44800 
 
 p ~ 60 v + 20* 
 p ought not in any case to exceed 1200. 
 
 his experiments, for iron surfaces of wheels and skids rubbing longitudinally 
 on iron rails : 
 
 /, for dry surfaces, 0'3, 0'25, 0'2; for damp surfaces, 0'14. 
 
 a, for wheels sliding on rails, 0'03; for skids sliding on rails, 0'07. 
 
 <y, not yet determined, but treated meanwhile as inappreciably small.
 
 FRICTION OF A BAND. 351 
 
 As to the measurement of friction, see Article 348, page 395. 
 
 310 A. Friction of a Band. A flexible band may be used either 
 to exert an effort or a resistance upon a drum or pulley. In either 
 case, the tangential force, whether effort or resistance, exerted 
 between the band and the pulley, is their mutual friction, caused 
 by and proportional to the normal pressure between them. 
 
 In fig. 242, let C be the axis of a pulley A B, round an arc of 
 which there is wrapped a band, T l A B T 2 ; let the outer arrow 
 represent the direction in which the band slides, or tends to slide, 
 relatively to the pulley, and the inner arrow the direction in which 
 the pulley slides, or tends to slide, relatively to 
 the band. 
 
 Let TJ be the tension of the free part of the 
 band at that side towards which it tends to draw 
 the pulley, or from which the pulley tends to 
 draw it; T 2 , the tension of the free part at the 
 other side; T, the tension of the band at any 
 intermediate point of its arc of contact with the 
 pulley ; 6, the ratio of the length of that arc to 
 the radius of the pulley; d 8, the ratio of an 
 indefinitely small element of that aix; to the 
 radius; R = T x T 2 , the total friction between 
 the band and the pulley; d R, the elementary Fig. 242. 
 
 portion of that friction due to the elementary 
 arc d 6; f, the co-efficient of friction between the materials of the 
 band and pulley. 
 
 Then it is known that the normal pressure at the elementary arc 
 d 6 is T d 6 ; T being the mean tension of the band at that elemen- 
 tary arc ; consequently, the friction on that arc is 
 
 d ~R=fTd 6. 
 
 Now, that friction is also the difference between the tensions of the 
 band at the two ends of the elementary arc; or 
 
 which equation being integrated throughout the entire arc of con- 
 tact, gives the following formulae : 
 
 R =1?! - T 2 = T x (1 - -/) = T 2 (e/ - 1). 
 
 "When a belt connecting a pair of pulleys has the tensions of its 
 two sides originally equal, the pulleys being at rest ; and when the 
 pulleys are set in motion, so that one of them drives the other by
 
 352 DYNAMICS OF MACHINERY. 
 
 means of the belt; it is found that the advancing side of the belt 
 is exactly as much tightened as the returning side is slackened, so 
 that the mean tension remains unchanged. The ratio which it 
 bears to the force, R, to be transmitted, is given by this formula : 
 
 T a + T 2 _ ef* + 1 
 
 If the arc of contact between the band and pulley, expressed in 
 turns and fractions of a turn, be denoted by ?i, 
 
 6 = 2 T n; &<> = lO 2 ' 7288 /"; .................. (3.) 
 
 that is to say, e? 6 is the antilogarithm, or natural number, corre- 
 sponding to the common logarithm 2 7288 fn. 
 
 The value of the co-efficient of friction, f, depends on the state 
 and material of the rubbing surfaces. For leather belts on iron 
 pulleys, the table of Article 309, page 349, shows that it ranges 
 from *56 to '15. In calculating, by equation 2 of this Article, the 
 proper mean tension for a belt, the smallest value, f -15, is to 
 be taken, if there is a probability of the belt becoming wet with oiL 
 The experiments of Messrs. Henry R. Towne and Robert Briggs, 
 however (published in the Journal of the Franklin Institute for 
 1868), show that such a state of lubrication is not of ordinary 
 occurrence; and that in designing machinery, we may in most 
 cases safely take f = 0-42. Professor Reuleaux (Constructionslehre 
 fur Maschinenbau) takes f = 0-25. The following table shows 
 the values of the co-efficient 2-7288 / by which n is multiplied iu, 
 equation 3, corresponding to different values of f', also the corre- 
 sponding values of various ratios amongst the forces, when the arc 
 of contact is half a circumference : 
 
 /= 0-I 5 ' 2 5 0*42 0-56 
 
 2'7288/ = 0-41 0-68 1-15 1-53 
 
 Let 6 = -, and n = ^; then 
 
 T! -s- T 2 = 1-603 2-188 3758 5-821 
 
 Tj -7- R = 2-66 1-84 1-36 i-2i 
 
 T x + T 2 -:- 2 R = 2-16 1-34 0-86 071 
 
 In ordinary practice, it is usual to assume T 2 = R; Tj = 2 R; 
 Tj + T 2 -s- 2 R = 1-5. This corresponds to/ = 0-22 nearly. 
 
 For a wire rope on cast-iron, f maybe taken as= 0-15 nearly; 
 and if the groove of the pulley is bottomed with gutta-percha, 
 0-25. 
 
 When an endless band runs at a very high velocity, its centri- 
 fugal force has an indirect effect on the friction, which will be 
 considered further on. (See page 441.)
 
 FRICTION 353 
 
 311. The Work Performed againat 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 is 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 ?' be the outer and inner radii; then the 
 leverage of the friction is 
 
 2 i* _ r ' s 
 
 For " Schiele's anti-friction pivot," whose longitudinal section is 
 the curve called the "tractrix," the moment of friction is/ x the 
 load x the external radius. This is greater than the moment for 
 an equally smooth flat pivot of the same radius; but the anti-fric- 
 tion pivot has the advantage, inasmuch as the wear of the surfaces 
 is uniform at every point, so that they always fit each other accu- 
 rately, and the pressure is always uniformly distributed, and never 
 becomes, as is the case in other pivots, so intense at certain points 
 as to force out the unguent and grind the surfaces. 
 
 In the cup and ball pivot, the end 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 
 somewhat less radius than the concave surfaces of the cups. The 
 moment of friction of this pivot is at first almost inappreciable, 
 from the extreme 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 angular velocity of the rolling surfaces relatively to each 
 
 2A
 
 354 DYNAMICS OF MACHINERY. 
 
 other. The following are approximate values of the arm in decimala 
 of afoot : 
 
 Oak upon oak, '6 (Coulomb). 
 
 Lignum-vitse on oak, 0-004 
 
 Cast-iron on cast-iron, 0-002 (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 one degree of Fahrenheit, for every 772 foot- 
 pounds of lost work. 
 
 The heat produced by friction, when moderate in amount, ig 
 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 isprevented 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 Fahrenheit, with 
 good mineral unguents, about 30. The effect of friction upon 
 the efficiency of machines will be considered further on, in 
 Section IV. 
 
 312. work or Acceleration. (A. Jlf.,12, 521-33, 536, 547, 549, 554, 
 689, 591, 593, 595-7.>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 
 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. (See Article 287, page 329.) 
 
 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 ; 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, acts 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.
 
 ACCELERATION GRAVITY MASS MOMENTUM. 355 
 
 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 
 is, 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 velocity '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 v l 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 
 
 (1.) 
 
 If the acceleration be at any different rate per second, the force 
 necessary to produce that acceleration, being the resistance on the 
 driving body due to the acceleration of the driven body, bears tJie same 
 proportion to the driven body's weight which the actual rate of accele- 
 ration bears to tJie rate of acceleration produced by gravity acting 
 freely. (In metres per second, g = 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 v t feet per 
 second ; and let that velocity increase at an uniform rate, so that 
 at an instant t seconds later, it is v 2 feet per second. 
 
 Let /denote the rate of acceleration; then 
 
 and the force R necessary to produce it will be given by the pro- 
 portion, 
 
 <7:/::W:R; 
 
 that is to say, 
 
 R= /W = W(. 2 -. l) ........ 
 
 9 ff* 
 
 w 
 
 The factor , in the above expression, is called the MASS of the 
 
 g 
 
 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 27 7 A, page 318.) 
 
 \V v 
 
 The product - of the mass of a body into its velocity at any 
 9
 
 356 DYNAMICS OF MACHINERY. 
 
 instant, is called its MOMENTUM; so that the resistance due to a 
 given acceleration is equal to the increase of momentum divided by 
 t/ie time which that increase occupies. 
 
 If the product of the 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 the impulse by which it is caused. 
 
 If the rate of acceleration is not constant, but variable, the force 
 B> varies along with it. In this case, the value, at a given instant 
 
 d 77 
 
 of the rate of acceleration, is represented \)jf = -> , and the cor- 
 
 Ci t 
 
 responding value of the force is 
 
 /W_W dv 
 "~i -~j"dt~ 
 
 The WORK 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 equal 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 ot 
 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 oj 
 the body, multiplied by the increase of the half-square of its velocity. 
 
 To express this by symbols, in the case of an uniform rate of 
 acceleration, let s denote the distance moved through by the driven 
 Dody during the acceleration; then 
 
 which being multiplied by equation 3, gives for the work of accel- 
 eration, 
 
 W 
 
 = ' 
 
 g ' 
 
 In the case of a variable rate of acceleration, let v denote the mean 
 Telocity, and d s the distance moved through, in an interval of time 
 d t so short that the increase of velocity d v is indefinitely small 
 compared with the mean velocity. Then 
 
 ds = vdt' f ........................... (7.)
 
 ACCELERATION. 357 
 
 which being multiplied by equation 4, gives for the work of accel- 
 eration during the interval d t, 
 
 Rds = . ^ . v dt 
 g dt 
 
 r= . vdv; (8.) 
 
 */ 
 
 and the integration of this expression (see Article 11 A) gives for 
 the work of acceleration during a finite interval, 
 
 f T, , W f , W v*. -v> 
 ]Kd S = -Jvdv=;-- . JLZ_J (9.) 
 
 being the same with the result already arrived at in equation 6. 
 
 From equation 9 it appears that tlie work performed in producing 
 a given acceleration depends on the initial and final velocities, Vj and 
 V;,, and not on tlie 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 h of the weight of the body into the 
 height of fall. Comparing this with equation 6, we find 
 
 * = T, < la > 
 
 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 difference of the Jieights due to its 
 initial andjmal 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 lost; as when 
 a marine engine performs work of acceleration on the water that is 
 struck by the propeller. 
 
 Work of acceleration performed on the moving pieces of tho 
 prime mover itself, or of the machinery driven by it, is not neces- 
 sarily lost, as will afterwards appear. 
 
 313. Summation of Work of Acceleration moment of Inertia. 
 
 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 
 "~T v\-
 
 358 DYNAMICS OP MACHINERY. 
 
 The process of finding that sum is facilitated and abridged iu 
 certain cases by special methods. 
 
 I. Accelerated Rotation Moment of Inertia. Let a denote the 
 angular velocity of a solid body rotating about a fixed axis; _ that 
 is, as explained in Article 46, 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 
 
 v = ar; .............................. (2.) 
 
 and if in a given interval of time the angular velocity is accelerated 
 from the value a, to the value a 2 , the increase of the velocity of the 
 particle in question is 
 
 v 3 - v 1 = r(a a - a,) .......................... (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 angular 
 velocity; that is to say, in symbols, 
 
 w v\ - v\ wr * al - aj 
 
 ~~ "" 
 
 g 2 g 2 "V" 
 
 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 pro- 
 ducts of the weight of each particle (w) into tJte square of its radius- 
 vector (r 8 ), and to multiply the sum so obtained (2 w r 2 ) by the in- 
 
 crease of the half-square of tlie angular velocity ( - (aj a,)], and 
 
 divide by the rate of acceleration due to gravity (g). The result, 
 viz.: 
 
 (5.) 
 g 2 ) 2g 
 
 is the work of acceleration sought. In fact, the sum 2 w r* is tfie 
 vxight of a body, which, if concentrated at the distance unity from
 
 MOMENT OF INERTIA. 359 
 
 the aods of rotation, would require tlie same work to produce a given 
 increase of angular velocity which the actual body requires. 
 
 The term MOMENT OF INERTIA is applied in some writings to the 
 sum 2 w r 3 , and in others to the corresponding mass 2 w r* -f- g. 
 For purposes of mechanical engineering, the sum 2 w r' is, on the 
 whole, the most convenient, bearing as it does the same relation to 
 angular acceleration which weight does to acceleration of linear 
 velocity. 
 
 The Radius of Gyration, or Mean Radius of a rotating body, is a 
 line whose square is the mean of the squares of the distances of its 
 particles from the axis; and its value is given by the following 
 equation : 
 
 = 
 
 so that if we put "W = 2 w for the weight of the whole body, the 
 moment of inertia may be represented by 
 
 (7.) 
 
 The following are additional rules relating to moments of inertia 
 and radii of gyration : 
 
 RULE II. Given, the moment of inertia of a body about an 
 axis traversing its centre of gravity in a given direction ; to find 
 its moment of inertia about another axis parallel to the first; 
 multiply the mass (or weight) of the body by the square of the per- 
 pendicular distance between the two axes, and to the product add 
 the given moment of inertia. 
 
 RULE III. Given, the separate moments of inertia of a set of 
 bodies about parallel axes traversing their several centres of gravity ; 
 required, the combined moment of inertia of those bodies about a 
 common axis parallel to their separate axes ; multiply the mass (or 
 weight) of each body by the square of the perpendicular distance of 
 its centre of gravity from the common axis ; add together all the 
 products, and all the separate moments of inertia; the sum will be 
 the combined moment of inertia. 
 
 RULE IY. To find the square of the radius of gyration of a 
 body about a given axis ; divide the moment of inertia of the body 
 about the given axis by the mass (or weight) of the body. 
 
 RULE Y. Given, the square of the radius of gyration of a 
 body about an axis traversing its centre of gravity in a given 
 direction ; to find the square of the radius of gyration of the same 
 body about another axis parallel to the first.-; to the given square 
 add he square of the perpendicular distance between the two 
 axes.
 
 360 DYNAMICS OF MACHINERY. 
 
 TABLE OP SQUARES OF RADII OF GYRATION. 
 
 BODT. 
 
 Axis. 
 
 RADIUS.* 
 
 I. Sphere of radius r 
 
 Diameter 
 
 Polar axis 
 Axis, 2a 
 
 Diameter 
 Diameter 
 
 Longitudinal 
 axis, '2a 
 
 Longitudinal 
 axis, 2a 
 
 Longitudinal 
 axis, 2a 
 
 Longitudinal 
 axis, 2a 
 
 Transverse 
 diameter 
 
 Transverse 
 axis, 26 
 
 Transverse 
 diameter 
 
 Transverse 
 diameter 
 
 Axis, 2a 
 
 Axis, 2a 
 Diagonal, 26 
 
 2r 
 
 IL Spheroid of revolution polar semi- 
 axis a equatorial radius T 
 
 5 
 
 2r* 
 5 
 
 6 a + c 
 
 III. Ellipsoid semi-axes , b, c, 
 
 IV. Spherical shell external radius r, 
 internal /, 
 
 5 
 
 2(r 8 -">) 
 
 V. Spherical shell, insensibly thin ra- 
 dius r, thickness dr, 
 
 5 ( r a _ rs) 
 2r 
 
 VI. Circular cylinder length 2a, radius 
 
 3 
 r 
 
 VII. Elliptic cylinder length 2a, trans- 
 verse semi-axes b, c, 
 
 2 
 
 fc a + c* 
 
 VIII. Hollow circular cylinder length 2a, 
 external radius r, internal r', 
 
 4 
 r + r' 2 
 
 IX. Hollow circular cylinder, insensibly 
 thin length 2o, radius r, thick- 
 
 2 
 
 r 
 
 r 2 a s 
 4~ + T 
 
 c' a* 
 4 + 3 
 
 r* + / a * 
 
 X. Circular cylinder length 2a, radius 
 
 XL Elliptic cylinder length 2o, trans- 
 verse semi-axes b, c, 
 
 XII. Hollow circular cylinder length 2a, 
 external radius r, internal r', 
 
 XIII. Hollow circular cylinder, insensibly 
 thin radius r thickness dr 
 
 4 h 3 
 
 r s a 
 "2 + T 
 6 s + c 
 
 XIV. Rectangular prism dimensions 2a, 
 26. 2c, . . 
 
 XV. Rhombic prism length 2a, diagonals 
 26, 2c. .. 
 
 3 
 
 b 9 +c 
 
 XVT. Rhombic prism, as above, 
 
 6 
 
 c ^a 
 
 
 6 + 3
 
 CENTRE OP PERCUSSION. 361 
 
 314. Centre of Percussion Equivalent Simple Pendnlnm Equiva- 
 lent Fir-wheel In calculations respecting the rotation of a rigid 
 body about a given axis, it is often convenient to conceive that for 
 the actual body there is substituted its equivalent simple pendulum; 
 that is, a body having the same total 
 mass, but concentrated at two points, of 
 which one is in the axis : also the same 
 statical moment, and the same moment 
 of inertia. 
 
 The following are the rules for doing 
 this : 
 
 I. To find the centre of percussion of 
 a given body turning about a given axis. 
 
 In fig. 243, let X X be the given axis, 
 and G the centre of gravity of the body. 
 From G let fall G C perpendicular to 
 X X. Through G draw G D parallel Fig. 243. 
 
 to X X, and equal to the radius of 
 
 gyration of the body about the axis G D. Join C D. Then will 
 C E = C D = ^ G D 2 + C G 2 = the radius of gyration of the 
 body about X X. From D draw D B perpendicular to C D, 
 cutting C G produced in B. Then will B be the centre of percus- 
 sion of the body for the axis X X. 
 
 G D 2 
 
 To find B by calculation; make G B = ~ 
 
 Vjr (j 
 
 C is the centre of percussion for an axis traversing B parallel to 
 XX. 
 
 II. To convert the body into an " equivalent simple pendulum " 
 for the axis X X, or for an axis through B parallel to X X; divide 
 the mass of the body into two parts inversely proportional to G G 
 and G B, and conceive those parts to be concentrated at C and B 
 respectively, and rigidly connected together. 
 
 (Let W be the whole mass, and C and B the two parts; then 
 W-GB ^ W-GC\ 
 CB; ; CB 7 
 
 (The " equivalent simple pendulum " has the same weight with 
 the given body, and also the same moment of weight, and the same 
 moment of inertia, with the given body, relatively to an axis in the 
 given direction X X, traversing either C or B.) 
 
 III. Equivalent Ring, or Equivalent Fly-wheel. "When the given 
 axis traverses the centre of gravity, G, there is no centre of per- 
 cussion. The moment of the body's weight is nothing, and its 
 moment of inertia is the same as if its whole mass were concentrated 
 in a ring of a radius equal to the radius of gyration of the body. 
 That ring may be called the " equivalent ring," or " equivalent 
 
 BY* W iJvvl*
 
 362 DYNAMICS OF MACHINERY. 
 
 315. Reduced inertia. If in a certain m:\cliine, 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 
 performed in producing the corresponding acceleration in the piece 
 in question is the same with that which would have been required 
 if a weight n 3 W had been concentrated at the driving point. 
 : If a similar calculation be performed for each moving piece in the 
 machine, and the results added together, the sum 
 
 2-w 9 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 v 2 the final velocity of the driving point, the work of accelera- 
 tion of the whole machine is 
 
 (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 "co-efficient of 
 steadiness" for reasons which will afterwards appear. 
 
 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 ; 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 
 
 (3.) 
 
 in performing the calculation expressed by the formula (1.) 
 
 316. Summary of Variona Kinds of Work. In order to present at 
 
 one view the symbolical expression of the various modes of perform- 
 ing 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 resistances, any 
 one of which is represented by R, are overcome at points, the re- 
 spective ratios of whose velocities to that of the driving point are 
 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 velocity of the end of its radius of gyration) to the velocity of 
 the driving point; and that the driving point, whose mean velocity
 
 SUMMABT OP WORK-i-DEVIATING FORCE. 
 
 is v = - , undergoes the acceleration d v. Then the whole work 
 a t 
 
 performed during the interval in question is 
 
 9 
 
 The mean total resistance, reduced to the driving point, may be 
 computed by dividing the above expression by the motion of the 
 driving point ds = vdt, giving the following result: 
 
 ~ . 2 W+ SR+ 4? S n' 2 W ..... (2.) 
 
 as gdt 
 
 SECTION II. Of Deviating and Centrifugal Force. 
 
 317. Deviating Force of a Single Body. (A. M., 537.) 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. (A. M., 510, 512.) 
 
 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 Height due Body's Deviating 
 
 curvature. to velocity. weight. force. 
 
 In the case of projectiles 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 rela- 
 tive motion. In machines, deviating force is supplied by the 
 strength or rigidity of some body, which guides the revolving mass, 
 making it move in a curve. 
 
 A pair of free bodies attracting each other have both deviated 
 motions, the attraction of each guiding the other; and their devia- 
 tions of motion relatively to their common centre of gravity 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.
 
 3C4 DYNAMICS OF MACHINERY. 
 
 318. Centrifugal Force (A, M., 538) 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 already stated, every force is an action 
 between two bodies ; and deviating force and centrifugal force are 
 but two different names for the same force, applied to it according 
 as the condition of the revolving body or that of the guiding body 
 is under consideration at the time. 
 
 319. A Revolving Pendulum is one of the simplest practical appli- 
 cations of the principles of deviating force, and is described here 
 because its use in regulating the speed of prime movers will after- 
 wards have to be referred to. It consists of a 
 ball A, suspended from a point C by a rod C A 
 of small weight as compared with the ball, and 
 revolving in a circle about a vertical axis C B. 
 The tension of the rod is the resultant of the 
 weight of the ball A, acting vertically, and of its 
 centrifugal force, acting^horizontally ; and there- 
 fore the rod will assume such an inclination that 
 
 height B C weight or ,, , 
 
 TJJ t\Aj _ _ __ O __ - ^* /I \ 
 
 radius A B centrifugal force ~ v 2 '"^ '' 
 
 where r = A B. Let T be the number of turns per second of the 
 pendulum, and a its angular velocity; then 
 
 and therefore, making B C = h, 
 
 _JL_ 
 
 4 * z T 2 
 
 / *t. T * , j rr j N 0-8154 foot 9-7848 inches ... 
 = (in the latitude of London) - =^ - = - - (*) 
 
 320. Deviating Force in Terms of Angular Velocity. (.1. J/., 540.) 
 When a body revolves in a circular path round a fixed axis, as 
 IH almost always the case with the revolving parts of machines, the 
 i-adius of curvature of its path, being the perpendicular distance of 
 the body from the axis, is constant ; and the velocity v of the body 
 is the product of that radius into the angular velocity; or symboli- 
 cally, 
 
 If these values of the velocity be substituted for v in equation 1 of 
 Article 317, it becomes
 
 CENTRIFUGAL FORCE AND COUPLE. 303 
 
 _W-4*s T2 r 
 
 (!' 
 
 9 9 
 
 321. Resultant Centrifugal Force. (A. M,, 603.) The whole cen- 
 trifugal force of a body of any figure, or of a system of connected 
 bodies, rotating about an axis, is the same in amount and direction 
 as if the whole mass were concentrated at the centre of gravity of 
 the system. That is to say, in the formula of Article 320,W is to 
 be held to represent the weight of the whole body or system, and r 
 the perpendicular distance of its centre of gravity from the axis ; 
 and the line of action of the resultant centrifugal force Q is always 
 parallel to r, although it does not in eveiy case coincide with r. 
 
 When the axis of rotation traverses the centre of gravity of the 
 body or system, the amount of the centrifugal force is nothing; 
 that is to say, the rotating body does not tend to pull its axis as a 
 whole out of its place. 
 
 The centrifugal forces exerted by the various rotating pieces of 
 a machine against the bearings of their axles are to be taken into 
 account in determining the lateral pressures which cause friction, 
 and the strength of the axles and framework. 
 
 As those centrifugal forces cause increased friction and stress, 
 and sometimes, also, by reason of their continual change of direc- 
 tion, produce detrimental or dangerous vibration, it is desirable to 
 reduce them to the smallest possible amount ; and for that purpose, 
 unless there is some special reason to the contrary, the axis of ro- 
 tation of every piece which rotates rapidly ought to traverse its 
 centre of gravity, that the resultant centrifugal force may be no- 
 thing. 
 
 322. Centrifugal Couple Permanent Axis. It is not, however, 
 
 sufficient to annul the effect of centrifugal force, that there should 
 be no tendency to shift the axis as a whole; there should also be 
 no tendency to turn it into a new angular position. 
 
 To show, by the simplest possible example, that the latter ten- 
 dency may exist without 
 the former, let the axis of 
 rotation of the system 
 shown in fig. 245 be the 
 centre line of an axle rest- 
 ing in bearings at E and F. 
 At B and D let two arms 
 project perpendicularly to 
 that axle, in opposite direc- 
 tions in the same plane, 
 carrying at their extremi- Fl S- 
 
 ties two heavy bodies A and C. Let the weights of the arms be
 
 366 DYNAMICS OF MACHINERY. 
 
 insensible as compared with the weights of those bodies; and let 
 the weights of the bodies be inversely as their distances from the 
 axis ; that is, let 
 
 A AB = C C D. 
 
 Let AC be a straight line joining the centres of gravity of A 
 and C, and cutting the axis in G; then G is the common centre of 
 gravity of A and C, and being in the axis, the resultant centrifugal 
 force is nothing. 
 
 In other words, let a be the angular velocity of the rotation j 
 then 
 
 The centrifugal force exerted on the axis by A 
 
 a- A-AB 
 
 ~7~ 
 
 The centrifugal force exerted on the axis by G 
 cfC-CT) 
 
 9 \ 
 
 and those forces are equal in magnitude and opposite in direction; 
 so that there is no tendency to remove the point G in any direc- 
 tion. 
 
 There is, however, a tendency to turn the axis about the point G, 
 "being the product of the common magnitude of the couple of cen- 
 trifugal forces above stated, into their leverage ; that is, the perpen- 
 dicular distance B D between their lines of action. That product 
 is called the moment of the centrifugal couple; and is represented by 
 
 Q'BD; ..... ......................... (1.) 
 
 <3 being the common magnitude of the equal and opposite centri- 
 fugal forces. 
 
 That couple causes a couple of equal and opposite pressures of 
 the journals of the axle against their bearings at E and F, in the 
 directions represented by the arrows, and of the magnitude given 
 by the formula 
 
 these pressures continually change their directions as the bodies 
 A and C revolve; and they are resisted by the strength and 
 rigidity of the bearings and frame. It is desirable, when practi- 
 cable, to reduce them to nothing ; and for that purpose, the points 
 B, G, and D should coincide; in which case the centre line of the 
 axle E F is said to be a permanent axis. 
 
 When there are more than two bodies in the rotating system, 
 the centrifugal couple is found as follows :
 
 CENTRIFUGAL COUPLE. 
 
 367 
 
 Let X X, fig. 246, represent the axis of rotation; G, the centre 
 of gravity of the rotating body or system, situated in that axis ; so 
 that the resultant centrifugal force is nothing. 
 
 Let W be any one of the parts of 
 -which the body or system is com- 
 posed, so that, the weight of that 
 part being denoted by W, the 
 weight of the whole body or sys- 
 tem may be denoted by 2 W. 
 
 Let r denote the perpendicular 
 distance of the centre of "VV from 
 the axis ; then 
 
 W a?r 
 
 Fig. 246. 
 
 is the centrifugal force of "W", pull- 
 ing the axis in the direction x W. 
 
 Assume a pair of axes of co-ordinates, G Z, G Y, 
 perpendicular to X X' and to each other, and fixed 
 relatively to the rotating body or system that is, 
 rotating along with it. 
 
 From W let fall W y perpendicular to the plane 
 of G X and G Y, and parallel to G Z ; also W z, 
 perpendicular to the plane of GX and GZ, and " Y 
 
 parallel to G Y ; and make Fig. 247. 
 
 sc y ^~ji w / ^ y i sc % w *?/ ^ & \ \JT ss oc. 
 
 Then the centrifugal force which "W exerts on the axis, and which 
 is proportional to r, may be resolved into two components, in the 
 direction of, and proportional to, y and z respectively, viz. : 
 
 W a 2 v 
 
 parallel to G Y, and 
 
 9 
 
 parallel to G Z; 
 
 9 
 
 and those two component forces, being both applied at the end of 
 the lever G x = x, exert moments, or tendencies to turn the axis 
 X X' about the point Z, expressed as follows : 
 
 W a * y X , tending to turn G X about G Z towards G Y; 
 9 
 
 W q * z x } tending to turn G X about G Y towards G Z. 
 
 ff 
 In the same manner are to be found the several moments of the
 
 3C8 DYNAMICS OF MACHINERY. 
 
 centrifugal forces of all the other parts of which the body or system 
 consists; and care is to be taken to distinguish moments which 
 tend to turn the axis towards G Y or G Z from those which tend to 
 turn it from, those positions, by treating one of these classes of 
 quantities as positive, and the other as negative. 
 
 Then by adding together the positive moments and subtracting 
 the negative moments for all the parts of the body or system, are 
 to be found the two sums, 
 
 -' . S'Wya:; - . S-Wzaj; ................. (3.) 
 
 9 9 
 
 which represent the total tendencies of all the centrifugal forces to 
 turn the axis in the planes of G Y and G Z respectively. 
 
 In fig. 247 lay down G Y to represent the former moment, and 
 G Z, perpendicular to G Y, to represent the latter. Then the dia- 
 gonal G M of the rectangle G Z M Y will represent the resultant 
 moment of what is called the CENTRIFUGAL COUPLE, and the direc- 
 tion of that line will indicate the direction in which that couple 
 tends to turn the axis G X about the point G. Its value, and its 
 angular position, are given by the equations, 
 
 
 The condition which it is desirable to fulfil in all rapidly rotating 
 pieces of machines, that the axis of rotation shall be a permanent 
 axis, is fulfilled when each of the sums in the formula 3 is nothing; 
 that is, when 
 
 S ' W y a; = 2-W zx = 0, ............. (5.) 
 
 The question, whether the axis of a rotating piece is a permanent 
 axis or not, is tested experimentally by making the piece spin round 
 rapidly with its shaft resting in bearings which are suspended by 
 chains or cords, so as to be at liberty to swing. If the axis is not 
 a permanent axis, it oscillates; if it is, it remains steady. 
 
 When the axis of rotation traverses the centre of gravity of the 
 piece, there is said to be a STANDING BALANCE; when it is also a 
 permanent axis, there is said to be a RUNNING BALANCE. 
 
 SECTION III. Of Effort, Energy, Powkr, and Efficiency. 
 
 323. Effort is a name applied to a force which acts on a body in 
 the direction of its motion (A. M., 511). 
 
 If a force is applied to a body in a direction making an acute
 
 EFFORT - UNIFORM SPEED. 369 
 
 angle with the direction of the body's motion, the component 
 of that oblique force along the direction of the body's motion is an 
 effort. That is to say, in fig. 248, let A B represent the direction 
 in which A is moving; let A F repre- 
 sent a force applied to A, obliquely to 
 that direction; from F draw F P per- 
 pendicular to A B; then A P is the 
 
 effort due to the force A F. The trans- F 
 
 Fig 248 
 verse component P F is a lateral force, 
 
 like the transverse component of the oblique resisting force in 
 Article 304. 
 
 To express this algebraically, let the entire force A F = F, the 
 effort ATP = P, the lateral force FT? = Q, and the angle of obli- 
 quity P A F = t. Then 
 
 a.) 
 
 Q = F sin 6 
 324. Condition of Uniform Speed. (A. M., 510, 512, 537.) Ac- 
 
 cording to the first law of motion, in order that a body may move 
 uniformly, the forces applied to it, if any, must balance each other ; 
 and the same principle holds for A. machine consisting of any num- 
 ber 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 Section 2 ; 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 can be shown from the principles of statics (that is, the science 
 of balanced forces), that in any body, system, or machine, that con- 
 dition 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 
 products of the resistances into the velocities of the points where they 
 a/re overcome. 
 
 Thus, let v be the velocity of a driving point, 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 P V = 2 B, V' ........................... (1.) 
 
 If there be only one driving point, or if the velocities of all the 
 2u
 
 370 DYNAMICS OF MACHINERY. 
 
 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 v; reducing 
 the above equation to 
 
 (2.) 
 
 Referring now to Article 305, let the machine be one in which the 
 comparative 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; 
 
 then the condition of uniform speed may be thus expressed : 
 P = 2-wK; (3.) 
 
 that is, the total effort is equal to the sum of the resistances reduced to 
 the driving point. 
 
 325. Energy Potential Energy. (A. J/., 514, 517, 593, 660.) 
 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 energy to be afterwards referred 
 to), 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 
 100 x 20 = 2,000 foot-pounds; which means, that in descending 
 from its actual elevation to the lowest point of its course, the 
 weight is capable of performing work 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 Ibs., the weight of 
 the store of water is 100,000,000 Ibs., which being multiplied by 
 the height through which 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. 
 
 326. Equality of Energy Exerted and Work Performed- 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,
 
 WORK AND ENERGY ACCELERATION. 371 
 
 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, 
 ' the distance through which it is driven; then 
 
 s = v t-, s'=v t; 
 
 and multiplying equation 1 of Article 324 by the time t, we obtaia 
 the following equation : 
 
 2-Ptf = 2'R?/ = 2-Ps = 2 -Rs'; (1.) 
 
 which expresses the equality of energy exerted, and work per- 
 formed, for constant efforts and resistances. 
 
 When the efforts and resistances vary, it is sufficient to refer to 
 Article 307, to show that the same principle is expressed as 
 follows : 
 
 2 J Yds = 2 I Rds'; (2.) 
 
 where the symbol / 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. 
 
 327. Various Factors of Kiicrgy. 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 301 and 302, pages 340 to 341. 
 
 328. The Energy Exerted in Producing Acceleration (A. M., 549) 
 
 is equal to the work of acceleration, whose amount has been inves- 
 tigated in Articles 312 and 313, pages 354 to 357. 
 
 329. The Accelerating Effort (A. M., 554) by which a given in- 
 crease 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 312 and 313. Referring, therefore, to equations 4 and 8 
 of Article 312, we find the two following expressions, the first of 
 which gives the accelerating effort required to produce a given
 
 372 DYNAMICS OF MACHINERY. 
 
 acceleration dv in a body whose weight is W, when the time dt m 
 which that acceleration is to be produced is given, and the second, 
 the same accelerating effort, when the distance d s = v d t in which 
 the acceleration is to be produced is given : 
 
 P- W - 
 ' 
 
 _W vdv_W d (a 2 ) 
 g d s ~ g ' 2 d s'" 
 
 Referring nex.t to Article 313, page 357, we find from equations 
 5, 6, and 7, that the work of acceleration corresponding to an increase 
 d a in the angular velocity of a rotating body whose moment of 
 inertia is I, is 
 
 I d (a 2 ) I a d a 
 
 ~ ~~~ 
 
 Let d t be the time, and d i = a d t the a/hgular 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 : 
 
 (3.) 
 
 I a d a Id (a 2 ) 
 
 s . - - = . - . 
 
 g d i g 2 d i 
 
 Lastly, referring to Article 315, page 362, 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 of 
 
 v' 
 
 the driving point v the ratio = n y then the accelerating effort 
 
 v 
 
 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 d 8 = v d t, that point may undergo the acceleration d v, 
 and each weight W the corresponding acceleration n d v, is given 
 by one or other of the two formulae 
 
 zn*W dv 
 
 ~' ........................ (5 ' 
 
 W vdv _ zn* W djv*) 
 ' ds ~~ g ' 2ds
 
 STOKED AND BESTORED ENERGY ACTUAL ENERGY. 373 
 
 330. Work During Ketardation Energy Stored and Restored. 
 
 {A. M., 528, 549, 550.) In order to cause >a given retardation, or 
 diminution of the velocity of a given body, in a given time, or 
 while it traverses a given distance, 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 equal to the 
 retardation. 
 
 It is for this reason that it has been stated, in Article 312, 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 329 simply by putting R, the 
 symbol for a resistance, instead of P, the symbol for an effort, and 
 d v, the symbol for a retardation, instead of d v, the symbol for 
 an acceleration. 
 
 331. The Actual Energy (A. M., 547, 589) 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 
 ike weight of tlie body into the height from which it must fall to 
 acquire its actual velocity ; that is to say, 
 
 ................................ w 
 
 The total actual energy of a system of bodies, each moving with 
 its own velocity, is denoted by 
 
 and when those bodies are the pieces of a machine, whose velocities
 
 374 DYNAMICS OP MACHINERY. 
 
 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 
 
 being the product of the reduced inertia (or co-efficient 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 t 
 and moment of inertia 2 W r 2 = I, is 
 
 that is, the product of the moment of inertia into tJte height due to the 
 velocity, a, of a point, whose distance from the axis of rotation is 
 unity. 
 
 When a given amount of energy is alternately stored and re- 
 stored by alternate increase and diminution in the speed of a ma- 
 chine, 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 engine, and it is inquired how high it will 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. 
 
 332. A Reciprocating Force (A. M., 556) is a force which acts 
 alternately as an effort and as an equal and opposite resistance, 
 according to the direction of motion of the body. Such a force is 
 the weight of a moving piece whose centre of gravity alternately 
 rises and falls j and such is the elasticity of a perfectly elastic body.
 
 RECIPROCATING FORCE PERIODICAL MOTION. 375 
 
 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; so that by the motion of a body alternately 
 against and with a reciprocating force, energy is stored and re- 
 stored, 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 h^W 
 
 is stored while the centre of gravity is rising, and restored while 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. , 
 
 333. Periodical motion. (A. M., 553.) 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 and the resistances, exactly as if the speed 
 were uniform and the reciprocating forces null; and all the equa- 
 tions of Articles 324 and 326 are applicable to periodic motion, pro- 
 vided that in the equations of Article 324, and equation 1 of Article 
 326, P, R, and v are held to denote the mean values of the efforts, 
 resistances, and velocities, that s and s' are held to denote spaces 
 moved through in one or more entire periods, and that in equa-
 
 376 DYNAMICS OF MACHINERY. 
 
 tion 2 of Article 326, the integrations denoted by / be held to 
 
 extend to one or more entire periods. 
 
 These principles are illustrated by the steam engine. The velo- 
 cities of its moving parts are continually varying, 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 and work, 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 pis- 
 ton 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 work- 
 ing. 
 
 " I. In the first place is considered the action of the machine 
 during one or more whole periods, with a view to the determina- 
 tion 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 thaii 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." . 
 
 334. Starting and stopping. (A. M., 691.) 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 331, page 373. 
 
 If, in order to stop a machine, the effort of the prime mover is 
 simply suspended, the machine will continue to go until work has 
 been performed in overcoming 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, 
 
 335. The Efficiency of a machine is a fraction expressing the ratio
 
 EFFICIENCY AND COUNTER-EFFICIENCY. 377 
 
 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 machines is 
 to bring their efficiency and counter-efficiency as near to unity as 
 possible, 
 
 As to useful and lost work, see Article 308. The algebraical ex- 
 pression of the efficiency of a machine having uniform or periodical 
 motion, is obtained by introducing the distinction between useful 
 and lost work into the equations of the conservation of energy. 
 Thus, let P denote the mean effort at the driving point; s, the space 
 described by it in a given interval of time, being a whole number 
 of periods of revolutions; R I} the mean useful resistance; s v the 
 space through which it is overcome in the same interval ; R 2 , any 
 one of the wasteful resistances; s 2 , the space through which it is 
 overcome; then 
 
 ? = !*! + ? - R 2 s 2 ; .................. (1.) 
 
 and the efficiency of the machine is expressed by 
 
 R, s t R| 81 (9 . 
 
 Ps R^ + s-Rs*/" 
 
 In many cases the lost work of a machine, R 2 s 2 , 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 de- 
 pends 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 
 
 R,*, 1 
 
 .(3.) 
 
 1+A + fT- 
 "i *i 
 
 and the first of these is the mathematical expression of what Mr. 
 Moseley calls the " modulus " of a machine. 
 
 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 overcom- 
 ing its own friction. Hence the efficiency of such an intermediate 
 piece is the ratio of the work performed by it in driving the follow- 
 ing 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 effi- 
 ciencies of the series of moving pieces which transmit energy from the 
 driving point to the working point. The same principle applies to a
 
 378 DYNAMICS OP MACHINERY. 
 
 train of successive machines, each driving that which follows it; and 
 to counter-efficiency as well as to efficiency. 
 
 336. 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, 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 299, page 339. It is equal to 
 0-9863 of a British horse-power; and a British horse-power is 
 1-0139 force de cheval. 
 
 337. General Equation. The following general equation pre- 
 sents at one view the principles of the action of machines, whether 
 moving uniformly, periodically, or otherwise : 
 
 ^ g 
 
 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 l the 
 velocity at the beginning, and v 2 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. 
 
 R d ', the work performed in overcoming any resistance during 
 the interval in question; 
 
 P d s, 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. 
 
 338. The Principle of Virtual Velocities, when applied to the 
 
 uniform motion of a machine, is expressed by equation 3 of Article 
 324, already given in page 369 ; or in words as follows : The effort 
 is equal to the sum of the resistances reduced to the driving point;
 
 VIRTUAL VELOCITIES MECHANICAL POWERS. 379 
 
 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 passive resistances, is expressed by 
 means of a modified form of the general equation of Article 337, 
 obtained in the following manner : Let n denote either the ratio 
 borne at a given instant by the velocity of a given working point, 
 where the resistance E. 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 weight is W; let n' denote the corresponding ratio 
 for the mean velocity of a mass whose weight is W, undergoing 
 
 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 effort 
 at the instant, or the mean effort during the given interval, as the 
 case may be, is given by the following equation : 
 
 - n ' Wdv ', 
 gd t 
 
 If the ratio ri, which the velocity of the mass W bears to that of 
 
 ,, , . dV ridv , dv 
 
 the driving point, is constant, we may put -j = -77-, where -, 
 
 tli t (It O/t 
 
 denotes the rate of acceleration of the driving point; and then the 
 
 third term of the foregoing expression becomes -=-- 2 n' 2 W, as in 
 
 gat 
 
 formula 2 of Article 316, page 363. 
 
 339. Forces in the Mechanical Powers, neglecting Friction Pur- 
 chase. The mechanical powers, considered as means of modifying 
 motion only, have been considered in Articles 221 to 224, pages 
 231 to 234. When friction is neglected, any one of the mechan- 
 ical 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 of 
 "resistance" for "weight" is made in order to express the fact.
 
 380 DYNAMICS OF MACHINERY. 
 
 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 
 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 principles of statics; 
 viz. : In the case of the LEVER (including the wheel and axle), from 
 the balance of couples of equal and opposite moments; in the 
 case of the INCLINED PLANE (including the wedge and the screw), 
 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 mechanical power is the resultant of the 
 effort, the resistance, the latei*al 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 explained in connec- 
 tion with the pulley, in Article 201, pages 215, 216. 
 
 The following are the results of the principle of virtual velocities, 
 as applied to determine the purchase in the several mechanical 
 powers : 
 
 I. LEVER. 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
 
 MECHANICAL POWERS PURCHASE. 381 
 
 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 IV. WEDGE. Here the 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. 165, page 
 233, and Cc in fig. 166, page 234) to that component of the 
 velocity (represented by B D in fig. 165, page 233, and C e in fig. 
 166, page 234) which is directly opposed to the resistance: it 
 being understood that the effort 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 link; 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. SCREW. 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 effort (P) which drives the screw be applied to a point 
 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 mkes 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 - 2 r 2 +> 2 ); therefore the purchase of 
 the screw, neglecting friction, is expressed as follows : 
 
 R s/T^r 2 + 
 
 P p 
 
 length of one coil of path of driving point 
 pitch 
 
 VI. PULLET. (See Articles 200 and 201, pages 214 to 216.) 
 In the pulley without friction, the purchase is the ratio borne by 
 the resistance which opposes the advance of the running block to 
 the effort exerted on the hauling part of the rope ; and it is ex- 
 pressed by the number of plies of rope by which the running block 
 is connected with the fixed block. 
 
 VII. The HYDRAULIC PRESS, 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, A, fig. 159, page 224; and by the 
 effort, the force which drives inward the pump-plunger, A'. The 
 intensity of the pressure exerted between each of the two plungers
 
 382 DYNAMICS OF MACHINERY. 
 
 and the fluid is the same; therefore the amount of the pressure 
 exerted betwf jn 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 : 
 
 E, A transverse area of press-plunger 
 P A' transverse area of pump-plunger ' 
 
 and this is the reciprocal of the ratio of the velocities of those 
 plungers, as already shown in Article 209, page 223. 
 
 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 effort being limited in amount. 
 
 \ 
 SECTION IY. Of Dynamometers. 
 
 340. Dynamometers are instruments for measuring and recording 
 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 distance through 
 which that force is exerted to be observed independently. The 
 following are examples of this class : 
 
 a. The weight of a solid body may be so suspended as to balance 
 the resistance, as in Scott Russell's experiments on the resistance 
 of boats. (Edin. Trans., xiv.) 
 
 b. The weight of a column of liquid may be employed to balance 
 and measure the effort required to drag a carriage or other body, 
 as in Milne's mercurial dynamometer. 
 
 c. The available energy of a prime mover may be wholly ex- 
 pended in overcoming friction, which is measured by a weight, as 
 in Prony's dynamometer (described further on). 
 
 d. A spring balance may be interposed between a prime mover 
 and a body whose resistance it overcomes. 
 
 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 abscissae represent the distances moved through, 
 its ordinates the resistances overcome, and its area the work per- 
 formed (as in fig. 241, page 346). 
 
 A dynamometer of this class consists essentially of two principal 
 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
 
 DYNAMOMETEBS. 383 
 
 with a velocity bearing a known constant proportion to the velo- 
 city with which the resistance is overcome. The spring carries a 
 pen or pencil, which marks on the paper or card the required line. 
 The following are examples of this class of instruments: 
 
 a. Morin's Traction Dynamometer. 
 
 b. Morin's and Hirn's Rotatory Dynamometers. 
 
 c. The Steam Engine Indicator. 
 
 III. Instruments called Integrating Dynamometers, which re- 
 cord the work performed, but not the resistance and motion 
 separately. 
 
 341 . Prony's Friction Dynamometer , < * a^ 
 
 measures the useful work performed 1 
 by a prime mover, by causing the 
 whole of that work to be expended 
 in overcoming the friction of a 
 brake. In fig. 249, A represents a Fig. 249. 
 
 cylindrical drum, driven by the 
 prime mover. The block D, attached to the lever B C, and the 
 smaller blocks with which the chain E is shod, form a brake which 
 embraces the drum, and which is tightened by means of the screws 
 F, F, until its friction is sufficient to cause the drum to rotate at 
 an uniform speed. The end C of the lever carries a scale G, in 
 which weights are placed to an amount just sufficient to balance 
 the friction, and keep the lever horizontal. The lever ought to be 
 so loaded at B that when there are no weights in the scale, it shall 
 be balanced upon the axis. The lever is prevented from deviating 
 to any inconvenient extent from a horizontal position by means of 
 safety-stops or guards, H, K. 
 
 The weight of the load in the scale which balances the friction 
 being multiplied into the horizontal distance of the point of suspen- 
 sion C from the axis, gives the moment of friction, which being 
 multiplied into the angular velocity of the drum, gives the rate of 
 useful work or effective power of the prime mover. 
 
 As the whole of that power is expended in overcoming the fric- 
 tion between the drum and the brake, the heat produced is in 
 general considerable; and a stream of water must be directed on 
 the rubbing surfaces to abstract that heat. 
 
 The friction dynamometer is simple and easily made; but it is 
 ill adapted to measure a variable effort; and it requires that when 
 the power of a prime mover is measured, its ordinary work should 
 be interrupted, which is inconvenient and sometimes impracticable. 
 
 342. ITIoriH'w Traction Dynamometer. The descriptions of this 
 
 and some other dynamometers invented by General Morin are 
 abridged from his works, entitled Sur quelques Appareils dynamo- 
 metriques and Notions fondamentales de Mecanique. 
 
 Fig. 250 is a plan and fig. 250 a an elevation of a dynamometer
 
 384 
 
 DYNAMICS OF MACHINERY. 
 
 for recording by a diagram the work of dragging a load horizontally. 
 a a, b b are a pair of steel springs, through which the tractive 
 force is transmitted, and which serve by their deflection to measure 
 
 Fig. 250. 
 
 Fig. 250 A. 
 
 that force. They are connected together at the ends by the steel 
 links f,f. The effort of the prime mover is applied, through the 
 link r, to the gland d, which is fixed on the middle of the fore- 
 most spring; the equal and opposite resistance of the vehicle is 
 applied to the gland c, which is fixed on the middle of the after- 
 most spring. When no tractive force is exerted, the inward faces 
 of the springs are straight and parallel; when a force is exerted, 
 the springs are bent, and are drawn apart, through a distance pro- 
 portional to the force. The springs are protected against being 
 bent so far as to injure them by means of the safety bridles i, i, 
 with their bolts e, e. Those bridles are carried by the after-gland, 
 and their bolts serve to stop the foremost spring when it is drawn 
 forward as far as is consistent with the preservation of elasticity 
 and strength. 
 
 The frame of the apparatus for giving motion to the paper band 
 is carried by the after-gland. The principal parts of that apparatus 
 are the following : 
 
 I, store drum on which the paper band is rolled, before the com- 
 mencement of the experiment, and off which it is drawn as the 
 experiment proceeds;
 
 DYNAMOMETERS. 385 
 
 gr, taking-up drum, to which one end of the paper band is glued, 
 and which draws along and rolls up the paper band with a velocity 
 proportional to that of the vehicle. Fixed on the axis of this drum 
 is a fusee having a spiral groove round it, whose radius gradually 
 increases at the same rate as that at which the effective radius of 
 the drum g is increased during its motion by the rolling of succes- 
 sive coils of paper upon it. The object of this is to prevent that 
 increase of the effective radius of the drum from accelerating the 
 speed of the paper band ; 
 
 TO is a drum which receives through a train of wheelwork and 
 endless screws a velocity proportional to that of the wheels of the 
 vehicle, and which, by means of a cord, drives the fusee. The 
 mechanism is usually so designed that the paper moves at one- 
 fiftieth of the speed of the vehicle. 
 
 Between the drums I and g there are three small rollers to sup- 
 port the paper band and keep it steady. 
 
 One of the safety bridles carries a pencil, k, which, being at rest 
 relatively to the frame of the recording apparatus, traces a 
 straight line on the band of paper as the latter travels below 
 the pencil. That line is called the zero line, and corresponds to 
 O X in fig. 241, page 346. 
 
 An arm fixed to the forward gland carries another pencil, whose 
 position is adjusted before the experiment, so that when there is 
 no tractive force its point rests on the zero line. During the ex- 
 periment, this pencil traces on the paper band a line such as 
 ERG, fig. 241, whose ordinate or distance from any given point 
 in the zero line is the deflection of the pair of springs, and pro- 
 portional to the tractive force, at the corresponding point in the 
 journey of the vehicle. 
 
 The areas of the diagrams drawn by this apparatus, representing 
 quantities of work, may be found either by the method described 
 in Article 289, page 331, or by an instrument for measuring the 
 areas of plane figures, called the Planimeter, or Platometer, of which 
 various forms have been invented by Ernst, Sang, Clerk Maxwell, 
 Amstler, and others. 
 
 A third pencil, actuated by a clock, is sometimes caused to mark 
 a series of dots on the paper band at equal intervals of time, and 
 so to record the changes of velocity. 
 
 When one vehicle (such as a locomotive engine) drags one or 
 more others, the apparatus may, if convenient, be turned hind side 
 before, and carried by the foremost vehicle. In such a case the 
 motion of the band of paper ought to be derived, not from a driving- 
 wheel, which is liable to slip, but from a bearing-wheel. 
 
 When the apparatus is used to record the tractive force and 
 work performed in towing a vessel, the apparatus for moving the 
 paper band may be driven by means of a wheel or fan, acted upon 
 
 2 c
 
 386 DYNAMICS OF MACHINERY. 
 
 by the water; in which case the ratio of the velocity of the band 
 to that of the vessel should be determined by experiment. 
 
 Owing to the varieties which exist in the elasticity of steel, the 
 relation between the deflections of the springs and the tractive 
 forces can only be roughly calculated beforehand, and should be 
 determined exactly by direct experiment that is, by hanging 
 known weights to the springs, and noting the deflections. 
 
 The best form of longitudinal section for each spring is that 
 which gives the greatest flexibility for a given strength, and con- 
 sists of two parabolas, having their vertices at the two ends of the 
 spring, and meeting base to base in the middle ; that is to say, the 
 thickness of the spring at any given point of its length should be 
 proportional to the square root of the distance of that point from 
 the nearest end of the spring. To express this by a formula, let 
 
 c be the half-length of the spring ; 
 
 h, the thickness in the middle; 
 
 x, the distance of any point in the spring from the end nearest 
 to it; 
 
 h', the thickness at that point; then 
 
 The breadth of each spring should be uniform, and, according to 
 General Morin, should not exceed from 1 to 2 inches. Let it be 
 denoted by b. 
 
 The following is the formula for calculating beforehand the 
 probable joint deflection of a given pair of springs under a given 
 tractive force : 
 
 Let the dimensions c, h, b be stated in inches, and the force P 
 in pounds. 
 
 Let y denote the deflection in inches. 
 
 Let E denote the modulus of elasticity of steel, in pounds on the 
 square inch. Its value, for different specimens of steel, varies from 
 29,000,000 to 42,000,000, the smaller values being the most com- 
 mon. Then 
 
 The deflection should not be permitted to exceed about one- 
 tenth part of the length of the springs. 
 
 343. Morin'B Rotatory Dynamometer is represented in figs. 251, 
 
 251 a, and is designed to record the work performed by a prime 
 mover in transmitting rotatory motion to any machine. A is a fast 
 pulley, and C a loose pulley, on the same shaft. A belt transmits 
 motion from the prime mover to one or other of those pulleys 
 according as it is desired to transmit motion to the shaft or not.
 
 DYNAMOMETERS. 
 
 387 
 
 A third pulley, B, on the same shaft, carries the belt which 
 transmits motion to the machine to be driven. This pulley is also 
 loose on the shaft to a certain extent, so that it is capable of mov- 
 
 Fig. 251. 
 
 . 251 A. 
 
 ing, relatively to the shaft, backwards and forwards through a small 
 arc, sufficient to admit of the deflection of a steel spring by which 
 motion is transmitted from the shaft to the pulley. 
 
 One end of that spring is fixed to the shaft, so that the spring 
 projects from the shaft like an arm, and rotates along with it. 
 The other end of the spring is connected with the pulley B near its 
 circumference, and is the means of driving that pulley ; so that the 
 spring undergoes deflection proportional to the effort exerted by 
 the shaft on the pulley. 
 
 A frame projecting radially like an arm from the shaft, and 
 rotating along with it, carries an apparatus, similar to that used in 
 the traction dynamometer, for making a band of paper move radi- 
 ally with respect to the shaft with a velocity proportional to the 
 speed with which the shaft rotates. A pencil carried by this frame 
 traces a zero line on the paper band; and another pencil carried 
 by the end of the spring, traces a line whose ordinates represent 
 the forces exerted, just as in the traction dynamometer. 
 
 The mechanism for moving the paper band is driven by a toothed 
 ring surrounding the shaft, and kept at rest while the shaft rotates 
 by means of a catch. When that catch is drawn back, the toothed 
 ring is set free, rotates along with the shaft, and ceases to drive 
 the mechanism; and thus the motion of the paper band can be 
 stopped if necessary. (See page 446.) 
 
 344. In the Torsion Dynamometer (otherwise called " Paiidy- 
 namometer") of M. G-. A. Him, the torsion of the rotating shaft 
 which transmits power is made the means of measuring and record-
 
 388 DYNAMICS OF MACHINERY. 
 
 ing, by a self-acting apparatus, the moment of the couple by which 
 the shaft is driven, and the work done by that couple. Two trains of 
 wheels, driven from the shaft at two different points, communicate 
 rotations of equal speed in opposite directions about one axis to 
 two bevel-wheels which gear with an intermediate bevel-wheel at 
 opposite sides of its rim, forming a combination like that shown in 
 fig. 176, page 245. The axis of the third wheel, corresponding to 
 the train-arm A in fig. 176, indicates by its position one-half of the 
 angle through which the shaft is twisted between the spur-wheels, 
 and communicates its motion to the pencil of the recording appara- 
 tus; which pencil, as in other recording dynamometers, draws a 
 line on a strip of paper that is moved at a speed proportional to 
 the speed of the shaft that transmits the power. (See Annales des 
 Mines, 1867, vol. xi.) 
 
 The only perfectly accurate way of determining the relation 
 between the displacement of the pencil and the moment transmitted 
 by the shaft, is to ascertain by direct experiment the twisting effect 
 of a known couple when applied to the shaft. But a probable 
 approximate value of that relation may be calculated as follows : 
 Let M be the twisting moment ; x, the length of that part of the 
 shaft whose angular torsion is to be determined ; h, its diameter; 
 C, the co-efficient of transverse elasticity of the material; 6, the 
 angle of torsion, in circular measure; then,* 
 
 32 
 
 = 0.098 ................. (1.) 
 
 Let n be the ratio which each of the contrary angular velocities 
 of the bevel-wheels corresponding to B and C in fig. 176 bears to 
 the angular velocity of the shaft, and y the length of an index cor- 
 responding to the train-arm, A, in that figure; then the angular 
 
 n d 
 displacement of that index is ^-; and the linear displacement of 
 
 a 
 
 its end (which may be denoted by z) is 
 
 therefore the following formula expresses the relation between the 
 moment M, and the displacement z; 
 
 = 0-196 t ............ 
 
 z nyd lo nxy nxy 
 
 Should the shaft be hollow, let h' be its internal diameter; then 
 in each of the preceding formulae for h 4 substitute h* - /if*. 
 The following are values of the co-efficient C : 
 
 * Manual of Applied Mechanics, Article 322, page 357.
 
 PANDYNAMOMETER SPIRAL SPRINGS. 389 
 
 Dimensionsin Inches,.... Millimetres. 
 
 Forcesin Lbs., Kilogrammes. 
 
 Cast Iron, about 2,850,000 2,000 
 
 Wrought Iron, from 8,500,000 6,000 
 
 to 10,000,000 7,000 
 
 Steel, from 10,000,000 7) 
 
 to. 12,000,000 8,400 
 
 Calculation may be used preliminary to the designing of the 
 apparatus, in order to find approximately the extent of the dis- 
 placement of the recording pencil; but the exact relation of that 
 displacement to the twisting moment exerted through the shaft 
 ought always to be determined by experiment. 
 
 345. Elasticity of Spiral Springs. As spiral or helical springs 
 are much used in dynamometric apparatus, it is convenient here to 
 state the laws of their resistance to extension and compression. 
 
 In order that such a spring may be an accurate instrument for 
 measuring forces that is, in order that the proportion borne by 
 the load acting on the spring to the extension or compression 
 which it produces may be constant the figure of the spring should 
 be a true helix, as described in Article 58, page 36. 
 
 It is more favourable to accuracy to measure a force by the ex- 
 tension than by the compression of a spiral spring ; because during 
 extension it preserves almost exactly a truly helical form, and the 
 coils remain in a cylindrical surface; whereas during compression 
 the middle coils are apt to swerve sideways, so as to make the 
 spring lose the proper figure. There are cases, however, in which 
 the use of the compression of the spring is unavoidable ; and then 
 it is kept approximately in its proper figure by being enclosed 
 in a cylindrical casing, which ought to be so large as not to 
 impede the longitudinal motion of the spring. 
 
 The pair of equal and opposite forces by which a spiral spring is 
 stretched should act exactly along the axis of the helix; for which 
 purpose the ends of the spring should be made fast to a pair of 
 strong and stiff arms, each of which should be perpendicular to the 
 helix, and should lie along a radius of the cylinder on which the 
 helix is described, so that the inner ends of the arms may be in 
 the axis of the helix; and at those inner ends the forces to be 
 measured should be applied. The best form of section for the 
 wire of which the spring is made is circular; because the extension 
 of the spring depends on the torsion of the wire ; and the laws of 
 torsion are known with greater precision for a circular form of 
 section than for any other. 
 
 The following formulse show the relations between the load and 
 the extension or compression of the spring : 
 
 Let r be the radius of the cylinder containing the helical centre 
 line of the spiral spring, as measured from the axis to the centre of
 
 390 DYNAMICS OF MACHINERY. 
 
 the wire; n, the number of coils of which the spring consists; d, 
 the diameter of the wire; C, the co-efficient of rigidity or transverse 
 elasticity of the material; /, the greatest safe shearing stress upon 
 it; W, any load not exceeding the greatest safe load; v, the 
 corresponding extension or compression ; W 1? the greatest safe 
 steady load ; v v the greatest safe extension or compression ; then. 
 
 W_ C^ 4 _0-196/<P 12-566 nfr* 
 
 ~*' l ~ ' v * ~ 
 
 Gd 
 
 The greatest safe sudden load is ~ . 
 
 If the wire of which the spring is made is square, and of the 
 dimensions d x d, the load for a given deflection is greater than 
 for a round wire of the diameter d, in the ratio of 281 to 196, or 
 of 1-43 to 1, or of 10 to 7, nearly. 
 
 The values of the co-efficient, C, of transverse elasticity of steel 
 and charcoal iron wire, in Ibs. on the square inch, range between 
 10,500,000 and 12,000,000; and in kilogrammes on the square 
 millimetre, from, 7,400 to 8,400, nearly. 
 
 By the greatest safe stress is to be understood the greatest stress 
 which is certain not to impair the elasticity of the spring by fre- 
 quent repetition; say 30,000 Ibs. on the square inch. 
 
 W 
 
 The value of the ratio borne by the load to the extension 
 
 v 
 
 ought to be ascertained by direct experiment for every spring that 
 is used in dynamometers or indicators. 
 
 346. steam 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 afford the means of computing, according to the principles of 
 Articles 302 and 307, 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 difference of those quantities, 
 which is the available or effective energy exerted 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. 252. A B 
 is a cylindrical case. Its lower end, A, contains a small 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
 
 INDICATOR. 
 
 391 
 
 Fig. 252. 
 
 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 casing. 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 shows 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. 
 
 F 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 pulled 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 
 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. 
 253 it is represented by A A.
 
 392 
 
 DYNAMICS OF MACHINERY. 
 
 Fig. 253. 
 
 Then 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 C D E B. 
 The ordinates drawn to that curve 
 from any point in the atmospheric 
 line, such as H K and H G, indi- 
 cate the differences between the 
 r pressure of the steam and the at- 
 mospheric pressure at the corre- 
 sponding 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 E B 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 gen- 
 erally omitted; in which case the atmospheric pressure may be 
 assumed at its mean value, being 14 '7 Ibs. on the square inch, or 
 2116-3 Ibs. on the square foot, at and near the level of the sea. 
 
 Let A O = H F be ordinates representing the pressure of the 
 atmosphere. Then O F Y 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 the steam. Let O B and L E be ordi- 
 nates touching the ends of the diagram; then 
 
 O L 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 O 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, 
 represents the effective work of the steam on the piston, during the 
 complete stroke. 
 
 Those areas can be found by the Rules of Article 289, page 
 331 ; and the common trapezoidal rule, D, page 333, is in general 
 sufficiently accurate. The number of intervals is usually ten, and 
 of ordinates eleven. 
 
 The mean forward pressure, the mean back pressure, 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.
 
 INDICATOR. 393 
 
 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 Allowing 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 & be the first, b n 
 the last, and 6 l5 b z , &c., the intermediate breadths; then let n be 
 the number of intervals, and b m the mean breadth ; then 
 
 and this represents the mean forward pressure, mean back pressure, 
 or mean effective pressure, as the case may be. Let p e be the 
 mean effective pressure; then the effective energy exerted by the 
 steam on the piston during each double stroke is the product of 
 the mean effective pressure, the area of the piston, and the length 
 of stroke, or 
 
 2> e A S ; .............................. (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 
 
 p e A.^s; ............................ (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 effective 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 mean 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 
 number of double strokes or revolutions in a minute, gives the 
 indicated power per minute ; that is to say, if p" denotes the gen- 
 eral mean effective pressure, the indicated power per minute is 
 
 p"A'2Ns; ..................... . ..... (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.
 
 394 DYNAMICS OF MACHINERY. 
 
 The re-actions of the moving parts of the indicator, combined with 
 the elasticity of the spring, 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 showing the pressures on a visible scale. In 
 Kichards's indicator this is effected by the help of a train of very 
 light linkwork, which causes the pencil to show the movements 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. 
 
 347. integrating Dynamometer* record simply the work per- 
 formed in dragging a vehicle or driving a machine, without 
 recording separately the force and the motion. In that of Morin 
 this is effected by means of a combination which has already been 
 described in Article 270, page 311, and illustrated in fig. 221. In 
 that figure (which see) A represents a plane circular disc, made to 
 rotate with an angular velocity proportional to the speed of the 
 motion of the vehicle or machine, and B a small wheel driven by 
 the friction of the disc against its edge, and having its axis parallel 
 to a radius of the disc. The wheel B, and some mechanism which 
 it drives, are carried by a frame which is carried by a dynamometer 
 spring, and so adjusted that the distance of the edge of B from the 
 centre of A is equal to the deflection of the spring, and propor- 
 tional to the effort. 
 
 The velocity of the edge of B at any instant being the product 
 of its distance from the centre of A into the angular velocity of A r 
 is proportional to the product of the effort into the velocity of the 
 vehicle or machine that is, to the rate at which work is performed; 
 therefore the motion of the wheel B, in any interval of time, is 
 proportional to the work performed in that time; and that work can 
 be recorded by means of dial-plates, with indexes moved by a train 
 of wheelwork driven by the wheel B. 
 
 In Moison's integrating dynamometer a ratchet-wheel is driven 
 by the strokes of a click. (See Articles 194 to 196, pages 206 to 
 211.) The number of these strokes in a given time is proportional
 
 DYNAMOMETERS MEASUREMENT OP FRICTION. 395 
 
 to the speed of the machine whose work is to be measured ; and by 
 means of a dynamometer-spring the length of each stroke of the 
 click is adjusted so as to be proportional to the effort exerted at 
 the time. The result is that the total extent of motion of the 
 ratchet-wheel in a given time is proportional to the work performed. 
 It is obvious that the frictional catch might be applied to this 
 apparatus (Article 197, page 211). 
 
 348. measurement of Friction. Under the head of Dynamo- 
 meters may be classed apparatus for the experimental measurement 
 of friction. 
 
 If by means of any kind of dynamometer whose use does not 
 involve the interruption of the performance of the ordinary work 
 of a train of mechanism, we measure the power transmitted at two 
 parts of that train, the difference will be the power expended in 
 overcoming the friction of the intermediate parts. Hirn's Pan- 
 dynamometer (Article 344, page 387) seems well adapted for ex- 
 periments of this class. The power of a steam engine, as exerted 
 in the cylinder, may be measured by means of the indicator, and 
 the power transmitted to machinery which that engine drives, by 
 a suitable dynamometer; and the difference will be power ex- 
 pended chiefly in overcoming the friction of the intermediate 
 mechanism. 
 
 Special apparatus for measuring the friction of axles is used, not 
 only for purposes of scientific investigation as to the co-efficients 
 of friction of different pairs of surfaces in different states, but for 
 practically testing the lubricating properties of oil and grease. 
 Two forms of apparatus may be described. 
 
 I. Statical Apparatus. A short cylindrical axle, of a convenient 
 diameter (say 2, 3, or 4 inches), is supported at its ends by bearings 
 on the top of a pair of strong fixed standards. The ends of the 
 axle overhang their bearings, and carry a pair of equal and similar 
 pulleys, by means of which it is- driven at a speed equal, or nearly 
 equal, to the greatest intended working speed of the axles with 
 which the unguents to be tested are to be used in practice. The 
 object of driving the axle at both ends is to ensure great steadiness 
 of motion. The driving-gear ought to be capable of reversing the 
 direction of rotation. At the middle of its length the axle is 
 turned so as to form a very accurate and smooth journal, of a 
 length equal to from 1^ to 2^ times its diameter. Upon that 
 journal there hangs a plumber-block or axle-box, fitted with a 
 suitable bush or bearing. That plumber-block is rigidly connected 
 with a heavy mass of suitable material, such as cast iron, so as to 
 form as it were a pendulum hanging from the journal in the 
 middle of the axle, and of a weight suited to produce a pressure 
 on the journal equal to the greatest pressure to which the unguent 
 is to be exposed in practice (see Article 310, page 353). The
 
 396 DYNAMICS OF MACHINERY. 
 
 pendulum is furnished with an index and graduated arc, to show 
 its deviation from a vertical position. 
 
 The hanging plumber-block having been supplied with the 
 unguent to be tested, the axle is to be driven at full speed, first in 
 one direction, and then in the contrary direction, and the two 
 contrary deviations of the pendulum observed. Let 6 denote the 
 half-sum of those deviations, expressed in circular measure to 
 radius unity; c, the distance from the axis of rotation to the centre 
 of gravity of the pendulum; r, the radius of the journal; let W be 
 the weight of the pendulum : then the mean statical moment of 
 the pendulum is 
 
 W c sin 6 = W c 6 nearly; 
 
 and that moment balances the moment of friction (Article 311, 
 page 356), whose value is f W r nearly, and will be afterwards 
 shown to be exactly 
 
 W r sin tp, 
 
 I being the angle of repose. Equating, therefore, those two equal 
 moments, we find 
 
 r sin <D = c sin 6- } and 
 
 C C & 
 
 sin <p=y nearly = - sin 6 = nearly. (1.) 
 
 The distance, c, of the centre of gravity of the pendulum from the 
 axis may be found experimentally, by applying a known weight 
 at a known horizontal distance from the axis, so as to make the 
 pendulum deviate, and observing the deviation. Let P be the 
 weight so applied, x its leverage, the deviation which it produces; 
 then, if there were no friction, we should have 
 
 Pa; 
 
 W sin 
 
 In order to eliminate the effects of friction from the determination 
 of c, the load P with the leverage x should be applied at contrary 
 sides, so as to increase the deviation of the pendulum, while the 
 axle is rotating in the two contrary directions. 
 
 Let sin 6 be the mean of the sines of the deviations produced by 
 friction alone, and sin the mean of the sines of the deviations 
 produced by the friction and the load P together; then we shall 
 have 
 
 /ox 
 
 ' " 
 
 W (sin - 
 
 II. Dynamic or Kinetic Apparatus. To measure the friction of 
 an axle by means of its retarding effect upon a rotating mass, the axle
 
 MEASUREMENT OF FRICTION. 397 
 
 is supported on suitable bearings at its ends, as in the Statical 
 Apparatus just described ; and at the middle of its length it has 
 fitted on it, and accurately balanced, a round disc acting as a fly- 
 wheel, of weight sufficient to produce the required pressure on the 
 bearings. (See Article 310, page 353.) The numbers of turns 
 made by the axle are counted and indicated by means of a light and 
 easily-driven train of small wheels, with dial-plates and indexes. 
 
 The axle is provided with driving-gear of a kind which can be 
 instantly disengaged when required; for example, a fast pulley on 
 one overhanging end, with a loose pulley alongside of it, the loose 
 pulley being carried, not by the fly-axle itself, but by a separate 
 axle in the same straight line with the fly-axle. 
 
 After the axle with its fly-disc has been set in motion at a speed 
 greater than the working speed of the axles to which the unguent 
 to be tested is to be applied in practice, the driving-gear is to be 
 disengaged; when the speed of rotation will undergo a gradual 
 retardation through the friction of the journals. The numbers of 
 turns made in a series of equal intervals of time (for example, 
 intervals of thirty seconds, or of sixty seconds, or of a hundred 
 seconds) are to be observed on the counting dials, and noted down. 
 
 Let W denote the weight of the whole rotating mass, con- 
 sisting of the axle with its fly-disc; f, the radius of gyration 
 of that mass. (See Article 313, page 357). Let t be the uni- 
 form length in seconds of the intervals of time during which 
 the numbers of revolutions are recorded ; and in one of those 
 intervals let the disc make n revolutions, and in the next 
 interval n' revolutions. Then the mean angular velocity is, 
 
 during the first interval, - - , and during the second interval, 
 
 o 
 
 ; and treating the rate of retardation as sensibly uniform, the 
 I 
 
 retardation which takes place during the t seconds which elapse 
 from the middle of the first interval to the middle of the second 
 interval is 
 
 2 * (n - n') 
 ~~ 
 
 and to produce that retardation in the course of t seconds in a body 
 whose moment of inertia is Wg 2 , there is required a retarding 
 moment of the following value : 
 
 x = ,(n-fWf ...... 
 
 z 
 
 Part of the retarding moment is due to the resistance of the air; 
 but if the fly is a smooth round disc without arms, this may be
 
 398 DYNAMICS OP MACHINERY. 
 
 neglected for the purpose of the experiments, and the whole moment 
 treated as due to axle-friction. Let r be the radius of the journals, 
 and f the co-efficient of friction : then, as before, the moment of 
 friction is very nearly/" W r; and by equating this to the retard- 
 ing moment, and dividing both sides of the equation by W r, we 
 obtain the following formula for the co-efficient of friction : 
 
 2 > (n - tiQ e 2 m 
 
 * 
 
 When the numbers of revolutions have been observed during a 
 series of more than two equal intervals of time, the formula 2 for 
 the co-efficient of friction is to be applied to each consecutive pair 
 of intervals, and a mean of the results taken. 
 
 The radius of gyration e and the radius of the journals r should 
 of course be expressed in the same units of measure. In British 
 measures, feet are the most convenient for the present purpose. 
 The constant factor has the following values : 
 
 -- - 
 
 g ' 5-125 feet 1-56 metre'" 
 
 Similar experiments may be made with a disc rotating about a 
 vertical axis, and supported by a pivot; regard being had to the 
 value of the moment of friction of a pivot, as stated in Article 311, 
 page 356. 
 
 To find the square e 2 of the radius of gyration by experiment, 
 fix a pair of slender pins in the two faces of the disc at two points 
 opposite each other, and near its circumference; hang up the disc 
 with its axle by these pins, and make it swing like a pendulum in 
 a plane perpendicular to its axis; count the number of single 
 swings in some convenient interval of time ; calculate their number 
 per second, and let N denote that number. Then calculate the 
 length L of the equivalent simple pendulum, by the following 
 formula : 
 
 The constant factor of this expression, being the length of the 
 seconds pendulum, has approximately the following values : 
 
 J3 = 3-26 feet = 0-992 metre ................ (5.) 
 
 Let C be the distance from the point of suspension to the axis 
 of figure of the disc and axle; then the square of the radius of 
 gyration is calculated as follows : 
 
 e 2 ^ C(L-C) ......................... (6.)
 
 MEASUREMENT OF FRICTION. 399 
 
 When the object of the experiments is not to obtain absolute 
 values of the co-efficient of friction, but merely to compare one 
 specimen of unguent with another, it is sufficient to compare 
 together the rates of retardation with the two unguents in equal 
 intervals of time. 
 
 III. Comparison of Heating Effects. For the same purpose of 
 comparing unguents with each other, without measuring the friction 
 absolutely, the heating effects of the friction with different unguents 
 are sometimes compared together. The apparatus used is similar 
 to that described under the head of (I.) Static Apparatus ; except that 
 there is no reversing-gear,and that the pendulum, or loaded plumber- 
 block, has no index nor graduated arc, and is provided with a ther- 
 mometer, having its bulb immersed in the passage through which 
 the unguent flows from the grease-box to the journal. Another 
 thermometer, hung on the wall of the room, shows the temperature 
 of the air. The axle is driven at its proper speed, until the 
 temperature shown by the first-mentioned thermometer ceases to 
 rise; and then the elevation of that temperature above the 
 temperature of the air is noted. (See Article 310, page 353.) 
 
 In all experiments for the purpose of comparing unguents with 
 each othei-, care should be taken to remove one sort of unguent 
 completely from the rubbing surfaces, grease-box, and passages, 
 before beginning to test the effect of another sort, lest the mixture 
 of different sorts of unguents should make the experiments incon- 
 clusive. 
 
 ADDENDUM TO ARTICLE 309, PAGE 348. 
 
 Friction of Pistons and Plungers. From experiments made by 
 Mr. William More and others, it appears that the friction of 
 ordinary pistons and plungers may be estimated at about one- 
 tenth of the amount of the effective pressure exerted by the fluid 
 on the piston. 
 
 The friction of a plunger working through a cupped leather 
 collar is equal to the pressure of the fluid upon a ring equal in 
 circumference to the collar, and of a breadth which, according to 
 Mr. More's experiments, is about 04 of the depth of bearing-sur- 
 face of the collar; and according to the experiments of Messrs. 
 Hick and Luthy, from -01 to -015 inch (= from -25 to -375 
 millimetres), according to the state of lubrication of the collar.
 
 400 
 
 CHAPTER III 
 
 OF REGULATING APPARATUS. 
 
 349. 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 &fly; or it may act by adjusting the power of the 
 prime mover to the work to be done, when it is called a governor. 
 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 I. Of Brakes. 
 
 350. 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 con- 
 venience 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 (as in Prony's Dynamometer, Article 341, page 383). 
 
 III. Pump-brakes, in which the resistance employed is the 
 friction amongst the particles of a fluid forced through a narrow 
 passage. 
 
 IV. Fan-brakes, in which the resistance employed is that of a 
 fluid to a fan rotating in it. 
 
 351. 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. 
 
 To stop a machine, the brake must employ work to the amount 
 of the whole actual energy of the machine, as already stated in
 
 BRAKES. 401 
 
 Article 334. To retard a machine, the brake must employ work 
 to an amount equal to the difference between the actual enei-giea 
 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 
 
 R = - ............................... (1.) 
 
 n ^ ' 
 
 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 311, page 354. 
 
 352. 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 9 ; see page 349). The component of that oblique pressure in 
 the direction of a tangent to the rim of the drum is the friction. 
 (R); 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 
 
 f being the co-efficient of friction, and the proper value of R being 
 determined by the principles stated in Article 351. 
 
 It is in general desirable that the brake should be capable of 
 effecting its purpose when pressed against the drum by means of
 
 402 DYNAMICS OP MACHINERY. 
 
 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 Ibs., or 150 Ibs., 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 Ibs. or 25 Ibs. 
 
 353. 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 will run 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. 
 
 354. Flexible Brakes. (A. M., 678.) A flexible brake embraces 
 a greater or less arc of the rim of a drum or pulley whose motion 
 it resists. In some cases it consists of an iron strap, of a radius 
 naturally a little greater than that of the drum ; so that when left 
 free, the strap remains out of contact with the drum, and does not 
 resist its motion; but when tension is applied to the ends of the 
 strap, it clasps the drum, and produces the required friction. The 
 rim of the drum may be either of iron or of wood. In other cases 
 the brake consists of a chain, or jointed series of iron bars, usually 
 faced with wooden blocks on the side next the drum. When ten- 
 sion is applied to the ends of the chain, the blocks clasp the drum 
 and produce friction ; when that tension is removed, the blocks are
 
 FLEXIBLE BRAKES. 
 
 403 
 
 drawn back from the drum by springs to which they are attached, 
 and the friction ceases. 
 
 The following formulae are exact for perfectly flexible continuous 
 bands, and approximate for elastic straps and for chains of blocks. 
 Their demonstration has already been siven in Article 310 A, pa^e 
 351. 
 
 In fig. 254, let A B be the drum, and C its axis, and let the 
 direction of rotation of the drum be indicated 
 by the arrow. Let T t and T 2 represent the 
 tensions at the two ends of the strap, which 
 embraces the rim of the drum throughout the 
 arc A B. The tension T L exceeds the tension 
 T 2 by an amount equal to the friction between 
 the strap and drum, R ; that is, 
 
 R = T i - T 2 . 
 
 Let c denote the ratio which the arc of contact, 
 A B, bears to the circumference of the drum ; 
 f, the co-efficient of friction between the strap 
 and drum ; then the ratio T x : T 2 is the number 
 whose common logarithm is 2 -7288 fc, or 
 
 T 
 
 Fig. 254. 
 
 1 = 10 27288/c = N . (1.) 
 
 which number having been found, is to be used in the following 
 formulae for finding the tensions, T 1? T 2 , required in order to pro- 
 duce a given resistance, R : 
 
 N 
 Backward or greatest teuton, T x = R -^ =; (2.) 
 
 Forward or least tension, 
 
 T- = R 
 
 i 
 
 N - 1 
 
 .(3.) 
 
 The following cases occur in practice : 
 
 I. When it is desired to produce a great resistance compared 
 with the force applied to the brake, the backward end of the brake, 
 where the tension is T v is to be fixed to the framework of the 
 machinery, and the forward end moved by means of a lever or 
 other suitable mechanism; when the force to be applied by means 
 of that mechanism will be T 2 , which, by making N sufficiently 
 great, may be made small as compared with R. 
 
 II. When it is desired that the resistance shall always be less than 
 a certain given force, the forward end of the brake is to be fixed, 
 and the backward end pulled with a force not exceeding the given 
 force. This will be T^ and, as the equation 2 shows, how great
 
 404 DYNAMICS OF MACHINERY. 
 
 soever N may be, R will always be less than T r This is the 
 principle of the brake applied by Sir William Thomson to 
 apparatus for paying out submarine telegraph cables, with a view 
 to limiting the resistance within the amount which the cable can 
 safely bear. 
 
 In any case in which it is desired to give a great value to the 
 ratio N", the flexible brake may be coiled spirally round the drum, 
 so as to make the arc of contact greater than one circumference. 
 
 355. Pump-Brakes. .The resistance of a fluid, forced by a pump 
 through a narrow orifice, may be used to dispose of superfluous 
 energy; as in the "cataract," or "dash-pot." 
 
 The energy which is expended in forcing a given weight of fluid 
 through an orifice is found by multiplying that weight into the 
 height due to the greatest velocity which its particles acquire in 
 that process, and into a factor greater than unity, which for each 
 kind of orifice is determined experimentally, and whose excess 
 above unity expresses the proportion which the energy expended in 
 overcoming the friction between the fluid and the oi'ifice bears to 
 the energy expended in giving velocity to the fluid. 
 
 The following are some of the values of that factor, which will 
 be denoted by 1 + F : 
 
 For an orifice in a thin plate, 1 + F = 1-054 ................ (1.) 
 
 For a straight uniform pipe of the length I, and whose hydraulic 
 mean depth, that is, the area divided by the circumference of its 
 cross-section, is m, 
 
 1 + F = 1-505 + ^. . ...(2.) 
 
 m 
 
 For cylindrical pipes, m is one-fourth of the diameter. 
 
 The factor /in the last formula is called the co-efficient of friction 
 of the fluid. For water in iron pipes, the diameter d being ex- 
 pressed in feet, its value, according to Darcy, is 
 
 /= 0-005 (1+J-.L) ...................... (3.) 
 
 For air, /= 0-006 nearly. ......................... .'(4.) 
 
 The greatest velocity of the fluid particles is found by dividing 
 the volume of fluid discharged in a second by the area of the outlet 
 at its most contracted part. When the outlet is a cylindrical 
 pipe, the sectional area of that pipe may be employed in this 
 calculation ; but when it is an orifice in a thin plate, there is a 
 contracted vein of the issuing stream after passing the orifice, whose 
 area is on an average about 0-62 of the area of the orifice itself; 
 and that contracted area is to be employed in computing the
 
 PUMP-BRAKES. 405 
 
 velocity. Its ratio to the area of the orifice in the plate is called 
 the co-efficient of contraction. 
 
 The computation of the energy expended in forcing a given 
 quantity of a given fluid in a given time through a given outlet, is 
 expressed symbolically as follows : 
 
 Let V be the volume of fluid forced through, in units of volume 
 per second. 
 D, the heaviness of the fluid (see page 326). 
 
 A, the area of the orifice. 
 
 c, the co-efficient of contraction. 
 
 v, the velocity of outflow. 
 
 R, the resistance overcome by the piston of the pump in driving 
 the water. 
 
 , the velocity of that piston. 
 
 Then 
 
 = .T[' ............................. < 5 '> 
 
 and 
 
 (6.) 
 
 the factor 1 + F being computed by means of the formulae 1, 2, 3, 4. 
 To find the intensity of the pressure (p) within the pump, it is 
 to be observed, as in Article 302, that if A' denotes the area of the 
 piston, 
 
 V = A' u; B. = p A'; ..................... (7.) 
 
 consequently, 
 
 (8.) 
 
 that is, the intensity of the pressure is that due to the weight of a 
 vertical column of the fluid, whose height is greater than that due to 
 the velocity of outflow in the ratio 1 + F : 1. 
 
 To allow for the friction of the piston, about one-tenth may in 
 general be added to the result given by equation 6. (Seo page 399.) 
 
 The piston and pump have been spoken of as single ; and such 
 may be the case when the velocity of the piston is uniform. When 
 a piston, however, is driven by a crank on a shaft rotating at an 
 uniform speed, its velocity varies ; and when a pump-brake is to 
 be applied to such a shaft, it is necessary, in order to give a 
 sufficiently near approximation to an uniform, velocity of outflow, 
 that there should be at least either three single acting pumps, 
 driven by three cranks making with each other angles of 120, or 
 a pair of double-acting pumps, driven by a pair of cranks at right 
 angles to each other ; and the result will be better if the pumps
 
 406 DYNAMICS OF MACHINERY. 
 
 force the fluid into one common air vessel before it arrives at the 
 resisting orifice. 
 
 That orifice may be provided with a valve, by means of which its 
 area can be adjusted so as to cause any required resistance. 
 
 A pump-brake of a simple kind is exemplified in the apparatus 
 called the "cataract" for regulating the opening of the steam valve 
 in single-acting steam engines. It is fully described in most 
 special treatises on those engines.* 
 
 356. Fan-Brakes- A fan, or wheel with vanes, revolving in 
 water, oil, or air, may be used to dispose of surplus energy; and 
 the resistance which it causes may be rendered to a certain extent 
 adjustable at will, by making the vanes so as to be capable of being 
 set at different angles with their direction of motion, or at different 
 distances from their axis. 
 
 Fan-brakes are applied to various machines, and are usually 
 adjusted so as to produce the requisite resistance by trial. It is, 
 indeed, by trial only that a final and exact adjustment can be 
 effected; but trouble and expense may be saved by making, in the 
 first place, an approximate adaptation of the fan to its purpose by 
 calculation. 
 
 The following formulae are the results of the experiments of 
 Duchemin, and are approved of by Poncelet in his Mecanique 
 Inditstrielle : 
 
 For a thin flat vane, whose plane traverses its axis of rotation, let 
 
 A denote the area of the vane ; 
 
 I, the distance of its centre of area from the axis of rotation; 
 
 s, the distance from the centre of area of the entire vane to 
 the centre of area of that half of it which lies nearest the axis of 
 rotation ; 
 
 v, the velocity of the centre of area of the vane ( = a I, if a is 
 the angular velocity of rotation) ; 
 
 D, the heaviness of the fluid in which it moves; 
 
 II I, the moment of resistance ; 
 
 k, a co-efficient whose value is given by the formula 
 
 k = 1-254 + 1-6244-^; (1.) 
 
 v S 
 
 then 
 
 HI = IkVA-f-. (2.) 
 
 
 When the vane is oblique to its direction of motion, let i denote 
 
 * Pump-brakes have been applied to railway carriages by Mr. Laurence 
 Hill. Hydraulic buffers, -which act on the same principle, have been applied 
 to railway carriages by Colonel Clark, R.A.
 
 FAN-BRAKES FLUCTUATIONS OF SPEED. 407 
 
 the acute angle which its surface makes with that direction ; then 
 the result of equation 2 is to be multiplied by 
 
 1 + 
 
 .(3.) 
 
 It appears that the resistance of a fan with several vanes increases 
 nearly in proportion to the number of vanes, so long as their dis- 
 tances apart are not less at any point than their lengths. Beyond 
 that limit the law is uncertain. 
 
 SECTION II. Of Fly- Wheels. 
 
 357. Periodical Fluctuation* of Speed in a machine (A. M., 689) 
 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 330, page 373. 
 
 To determine the greatest fluctuation of speed in a machine 
 moving periodically, take ABC, in fig. 255, 
 to represent the motion of the driving point 
 during one period; let the effort P of the prime 
 mover at each instant be represented by the 
 ordinate of the curve D G E I F; and let the 
 sum of the resistances, reduced to the driving 
 point as in Article 305, at each instant, be Fig. 255. 
 
 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, C F. Then the integral, 
 
 (P - R) d s, 
 
 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, represented by the equal 
 area E K F I. Let those equal quantities be each represented by 
 A E. Then the actual energy of the machine attains a maximum 
 value at B, and a minimum value at A and C, and A E is the 
 difference of those values. 
 
 Now let V Q be the mean velocity, v l the greatest velocity, v z the 
 least velocity of the driving point, and 2 n 2 W the reduced inertia 
 of the machine (see Article 315, page 362); then
 
 408 DYNAMICS OF MACHINERY. 
 
 which, being divided by the mean actual energy, 
 
 gives 
 
 v\ - P| _ A E . 
 
 ~ ................. 
 
 and observing that v = (t?j + v 2 ) -f- 2, we find 
 t?j - t> 2 _ A E _ g A E 
 
 .(3.) 
 
 a ratio which may be called the co- efficient of fluctuation of speed 
 or of unsteadiness. 
 
 The ratio of the periodical excess and deficiency of energy A E 
 
 to the whole energy exerted in one period or revolution, / P d s, 
 lias been determined by General Morin for steam engines under 
 various circumstances, and found to be from ^ to -^ for single- 
 cylinder engines. For a pair of engines driving the same shaft, 
 with cranks at right angles to each other, the value of this ratio 
 is about one-fourth, 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 -r- / 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 fi , 
 
 Length of crank 
 
 = -105 -118 -125 -132 
 
 EXPANSIVE CONDENSING ENGINES. 
 Connecting rod crank x 5. 
 
 Fraction of stroke at7.1_ _L _ 2. * 
 which steam is cut off, J 3" ~ ~ 6 ~? 8 
 
 = .163 -173 -178 -184 -189 -191
 
 PLY-WHEELS. 409 
 
 EXPANSIVE NON-CONDENSING ENGINES. 
 
 Steam cut off at - - 
 
 234o 
 
 A E -^ Tp d s = -160 -186 -209 -232 
 
 For double-cylinder expansive engines, the value of the ratio 
 A E -f- jPds 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. 
 
 358. Piy-ivhccia. (A. M., 690.) A fly-wheel is a wheel with a 
 heavy rim, whose great moment of inertia being comprehended in 
 the reduced moment of inertia of a machine, reduces the co-efficient 
 
 of fluctuation of speed to a certain fixed amount, being about Q ^ for 
 
 Oa 
 
 ordinary machinery, and ^ or ^ for machinery for fine purposes. 
 
 Let be the intended value of the co-efficient of fluctuation of 
 m 
 
 speed, and A E, as before, the fluctuation of energy. If this is to 
 be provided for by the moment of inertia, I, of the fly-wheel alone, 
 let a Q be its mean angular velocity; then equation 3 of Article 
 357 is equivalent to the following : 
 
 . 
 ( ' 
 
 j. _ m g A E t 
 
 ft 2 ,.. -v ; 
 
 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 
 effort 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 connection 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
 
 410 DYNAMICS OF MACHINERY. 
 
 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 357; 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 and 
 inside radii a calculation which may be expressed thus : 
 
 ............................ (3.) 
 
 whence the weight of the rim is given by the formula 
 _7ngrAE_mgrAE 
 : ~a*r2~ t/2 
 
 if V be the velocity of the rim of the fly-wheel. 
 
 In millwork the ordinary values of the product m g, 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 metres. 
 
 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 III. Of Governors. 
 
 359. 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 GOVERNOR. 
 
 The special construction of the different kinds of regulators is a 
 subject for a treatise on prime movers. In the present treatise it
 
 KEGITLATORS AND GOVERNORS. 411 
 
 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 distinguish 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 
 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 differential-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 difference 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 governors, 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 ar& 
 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. 
 
 360. 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
 
 412 
 
 DYNAMICS OF MACHINERY. 
 
 opposite sides, a pair of revolving pendulums, which, by the positions 
 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 319; 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. 256), and T the number of turns per 
 second ; then 
 
 g _ -815 foot _ 9-78 inches _ 0-248 metre 
 
 '** " ' ~ n rr\ct " TTV> f 1 1 m.l V * / 
 
 T 2 
 
 T 2 
 
 If the rods of the revolving pendulums are jointed, as in fig. 
 257, not to a point in the vertical axis, but to a pair of points, 
 
 Fig. 257 
 
 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 balls 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 
 neglecting the effects both of the weight and of the centrifugal 
 force of the rods. These effects may, if required, be taken into 
 account approximately, as follows : Let B be the weight, and b the 
 radius, of a ball ; let R be the weight of a rod, and r the length 
 from O to the centre of B; let /* be the height from the centre of 
 B to O, and /*' the corrected height ; then 
 
 and the number of revolutions per second will correspond nearly 
 to this corrected height.
 
 GOVERNORS. 413 
 
 The ordinary steam engine governor invented by Watt, which 
 is represented in fig. 256, is a position-governor, and acts on the 
 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 shown in the figure, the lever O C forms 
 one piece with the ball-rod O B, and the lever O c with the ball- 
 rod O b ; 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 spindle, 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 357, let the utmost extent of fluctuation of the 
 speed of the engine between its highest and lowest limits be the 
 
 fi-action of the mean speed; let h be the altitude of the governor 
 
 m 
 
 corresponding to the mean speed; and let k 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 : 
 
 _Ly_ (l _ _LY : :7 i: ; 
 2m/ \ 2m/ 
 
 and consequently 
 
 361. loaded Pcndiiliim-Governbr. 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 
 
 2 B 
 1 + a given absolute variation of altitude in moving the 
 
 .A. 
 
 regulator produces a proportionate variation of speed smaller than
 
 414 DYNAMICS OF MACHINERY. 
 
 in the common governor, in the ratio r _ ,-, ; and the governor 
 
 is said to be more sensitive than a common governor, in the ratio of 
 A : A + 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 q B is to be substituted for 2 B. 
 
 In the one case 2 B, and in the other 2 q B, is the wqight, 
 applied directly at A. which would be statically equivalent to the 
 load B, applied where it is. 
 
 362. Parabolic Pendulum-Governors. In fig. 258, let B X be 
 the axis of the spindle^ and E the centre of one 
 of the balls, which, as it moves towards or 
 from the spindle, is guided so as to describe a 
 parabolic arc, K E, with the vertex at K. 
 Let E F be a normal to the parabola, cutting 
 the axis in F, The vertical height of F above 
 E is constant, being equal to twice the focal 
 distance of the parabola ; hence this governor 
 is absolutely isochronous. That is to say, the 
 balls cannot remain steady in any position 
 except at one particular speed of rotation; 
 Y being that corresponding to an altitude equal 
 to twice the focal distance of the parabola; 
 and any deviation of the speed above or below 
 that value causes the balls to move continuously outwards and 
 upwards, or inwards and downwards, as the case may be, until their 
 action on the regulator restores the proper speed. The force with 
 which the balls tend to shift their position vertically, when a de- 
 viation of speed occurs, is expressed very nearly by - -; in 
 
 which A is the collective weight of the balls, n the proper number 
 of revolutions in a given time, and A n the deviation from that 
 number. The balls may be guided in various ways, viz : 
 
 I. By hanging each of them by means of a flexible spring from 
 a cheek, L H, of the form of the evolute of the parabola. To find 
 a series of points in the parabola and its evolute, let h be the 
 altitude; then from the vertex K lay off K A = K B = \h- A 
 will be the focus, and the horizontal line B Y the directrix. Draw 
 A C parallel to an intended position of the ball-rod; bisect it in D;
 
 GOVERNORS. 415 
 
 draw D E perpendicular to A C, and C E parallel to B X ; the 
 intersection E will be a point in the parabola, and E D a tangent. 
 Then parallel to C A, draw E F ; this will be a normal, and a posi- 
 tion of the ball-rod. From F, parallel to D E, draw F G, cutting 
 C E produced in G; and from G, parallel to B Y, draw G, cutting 
 EF produced in H; this will be a point in the evolute. To ex- 
 press this algebraically, let B C = y and C E = z be the co-ordinates 
 of the parabola; and let B M = d and M H = y' be those of its 
 evolute. Then we have 
 
 II. Another method of guiding the balls is to support them by 
 means of a pair of properly curved arms, on which they slide or roll. 
 On the top of the balls there rests a horizontal plate or bar, which 
 communicates their vertical movements to the regulator. 
 
 III. Approximate Parabolic Governor. In Farcot's governor, 
 the rod E H, in its middle position, is hung from a joint, H, at the 
 end of an arm, M H ; this gives approximate isochronism. The 
 co-ordinates of the point H are found by the rules already given. 
 
 36 2 A. Loaded Parabolic Governor. When the balls of a para- 
 
 bolic governor are guided in the second manner described in the 
 preceding article, and support above them a plate or bai*, to which 
 their vertical movements are communicated, an additional load 
 may be applied to them by means of that plate. Let A be the 
 collective weight of 'the balls; B, the additional load; then the 
 altitude corresponding to a given speed is greater than in the 
 unloaded governor, in the ratio of A + B : A ; and the speed 
 corresponding to a given altitude is greater, in the ratio of 
 ,J (A + B) : vA ; and by varying the load, the speed of the 
 governor may be varied at will. 
 
 363. Isochronous Gravity-Governor (JRankind s). In this form 
 
 of governor (see fig. 259) the four* centrifugal balls marked B are 
 balanced, as regards gravity, about the joint A, on the spindle 
 AM. D, D are sliders on the ball-rods; D C, D C, levers jointed 
 to the sliders, and 'centred on a point in the spindle at C, and of 
 a length DC CA; G G, a loaded circular platform hung from 
 the levers C D, C D, by links E F, E F ; H, an easy-fitting collar, 
 jointed to the steelyard lever H K, whose fulcrum is at K; L, a 
 weight adjustable on this lever. This governor is truly isoch- 
 ronous ; the altitude h of a revolving pendulum of equal speed is 
 given by the equation 
 
 B A B 2 
 -2D-CD' 
 in which B is the collective weight of the centrifugal masses, and
 
 416 
 
 DYNAMICS OP MACHINERY. 
 
 D the load, suspended directly at D, to which the actual load is 
 statically equivalent. The load D, and consequently the altitude 
 
 Fig. 259. 
 
 and the speed, can be varied at will, by shifting the weight L; 
 which can be done either by hand or by the engine itself. The 
 regulator may be acted on by the other end of the lever H K. 
 The levers C D, C D should be horizontal when in their middle 
 position ; and then the ball-rods will slope at angles of 45. Two
 
 GOVERNORS. 417 
 
 positions of the parts of the governor when the rods deviate from 
 their middle position, are shown by dotted lines and accented 
 letters. If convenient, the links E F, E F may be hung directly 
 from the slides D, D. 
 
 The theory of this governor is illustrated by fig. 2 GO. In 
 any position of the parts, let A G be the axis of rotation ; 
 A B, a ball-rod carrying a ball at B ; C, the point at which the 
 lever C D = C A is jointed to the spindle; D, the central point 
 of the slider at the end of that lever. About C draw the circle 
 A D Q, cutting the axis of rotation in Q; join D Q; and draw 
 D R, and B P perpendicular to A Q. 
 
 Then when the position of the parts varies, and the speed is 
 constant, the moment of the centrifugal force of the balls relatively 
 to A varies proportionally to B P P A, and therefore propor- 
 tionally to the area of the right-angled triangle A P B ; and the 
 moment relatively to A of the load which acts on the point D 
 varies proportionally to D E, and therefore to the area of the 
 right-angled triangle A D Q ; but the areas of the triangles A B P 
 and A I) Q bear a constant ratio to each other viz., that of 
 A B 2 to A Q 2 ; therefore the moment of the centrifugal force at a 
 constant speed, and the moment of load, bear a constant ratio to 
 each other in all positions of the parts of the governor ; and if 
 they are equal in one position, they are equal in every position ; 
 and if unequal in one position, they are unequal in every position. 
 Therefore the governor is truly isochronous. 
 
 To express algebraically the relations between the dimensions, 
 the revolving mass, the load, and the speed ; let B be the collec- 
 tive weight of the four balls; D, the total load which is actually 
 or virtually applied at the points D, D; let the length of each 
 ball-rod A B b ; and let the length of each of the levers C D = c. 
 In any position of the governor, let the angle Q A B = 9. Then, 
 because A C D is an isosceles triangle, we have the angle Q C D 
 = 26. It is also evident that B P = b sin 6; A P = b cos 0; 
 DR = c < sin20=2c- cos 6 sin 6. 
 
 Let n, as before, be the number of revolutions per second. 
 Then the centrifugal moment of the balls relatively to A is 
 
 g 
 
 and the statical moment of the load relatively to A is 
 
 2 D - D E, = 4 D c cos 6 sin 6; 
 
 which two moments, being equated to each other, and common 
 factors struck out, give the following equation : 
 
 B& 2 
 
 -^ = 4Dc; 
 
 2E
 
 418 DYNAMICS OF MACHINERY. 
 
 and therefore 
 
 B6 2 E A B 2 
 
 as has already been stated.* 
 
 364. Fluctuations of Isochronous GoTeriiors. When a truly 
 
 isochronous governor is rapid in its action on the regulator, and 
 meets with little resistance from friction, it may sometimes happen 
 that the momentum of the moving parts carries them beyond the 
 position suited for producing the proper speed ; so that a deviation 
 from the proper speed takes place in the contrary direction to the 
 previous deviation, followed by a change, in the contrary direction, 
 in the position of the governor, which again is carried too far by 
 momentum; and so on; the result being a series of periodical 
 fluctuations in the speed of the engine. When this is found to 
 occur, it may be prevented by the use of a piston working in an 
 oil-cylinder or dash-pot; which will take away the momentum 
 of the moving parts, and cause the regulating action of the governor 
 to take place more slowly, without impairing its accuracy. 
 
 365. Balanced, or Spring Governors, (Silver's, Weirs, Hunt's, 
 
 .Sir W. Thomson's, &c.) In this class of governors, often called 
 Marine Governors, as being specially suited for use on board ship, 
 the action of gravity on the balls is either self-balanced, or made, by 
 rapid rotation, so small compared with the centrifugal force as to 
 be unimportant. The centrifugal force is opposed by springs. To 
 make such a governor isochronous, the springs ought to be so 
 arranged as to make the elastic force exerted by them vary in the 
 simple ratio of the distance from the centres of the balls to the axis. 
 In order that the action of gravity on the balls may be self- 
 balanced, if there are two balls only, they must move in opposite 
 directions, in a plane perpendicular to the axis of rotation : which 
 axis may have any position, but is usually horizontal. They might 
 be guided by sliding on rods perpendicular to the spindle; but 
 they are more frequently guided by combinations of linkwork, dif- 
 ferent forms of which are exemplified in Weir's governor and in 
 Hunt's governor. If there are four balls, they are carried by a pair 
 of arms like the letter X, as in fig. 259 (but with the spindle 
 usually horizontal instead of vertical), and such is the arrangement 
 in Silver's Marine Governor. The springs in balanced governors 
 are seldom fitted tip with a view to perfect isochronism; but for 
 marine engines this is unimportant, as the principal object of 
 applying governors to them is to prevent changes of speed so great 
 
 * It has been pointed out by Mr. Edmund Hunt that this form of governor 
 is virtually a parabolic governor; for the common centre of gravity of the 
 balls and of the load moves in a parabola, of a focal distance equal to half 
 the altitude given by the formula.
 
 GOVERNORS. 
 
 and sudden as to be dangerous ; such as those which tend to occur 
 when the screw-propeller of a vessel pitching in a heavy sea is 
 alternately lifted out of and plunged into the water. 
 
 Rules showing the relation between the deflection of a straight 
 spring, or the extension of a spiral spring, and the elastic force 
 exerted by the spring, have already been given in Article 342, page 
 386, and Article 345, page 389. 
 
 366. Disengagement-Governors. The most complete example of 
 a disengagement-governor is that commonly used for water-wheels, 
 and sometimes also for 
 the steam engine. The 
 peculiar parts of this 
 governor are represent- 
 ed in fig. 261. A A is 
 part of the spindle of a 
 pair of revolving pen- 
 dulums similar to those 
 in an ordinaiy governor ; 
 B, a cylindrical slider, 
 hung from the ball-rods 
 by links whose lower 
 ends are shown at C, C. 
 D is a tooth or cam 
 projecting from the 
 slider, and sweeping 
 round as the spindle and 
 
 pendulums rotate. To make the slider rotate truly with the spindle, 
 the part of the spindle on which it slides may either be made 
 square, or may have a projecting longitudinal feather fitting easily 
 a groove in the inside of the slider. 
 
 E is one end of a lever capable of turning about a vertical axis 
 (not shown), and provided with a fork of four prongs, F, F, G, H. 
 The prongs F, F are just far enough apart to clear the tooth D, 
 as it sweeps round, when the spindle is turning at its proper speed, 
 and the ball-rods and slider in their middle position; and the lever 
 E is then in its middle position also. The prong G is below, and 
 the prong H above, the level of the prongs F, F; and when the 
 lever is in its middle position, the clear distance of G and H from 
 the cylindrical surface of the slider B is one-half of the distance of 
 F, F from that surface. When the spindle begins to fall below its 
 proper speed, the slider moves downwards until the tooth D strikes 
 the prong G, and drives the lever E to one side. Should the 
 spindle begin to turn faster than the proper speed, the slider rises 
 until the tooth D strikes the prong H, and drives the lever E to 
 the contrary side. The lever E acts through any convenient train 
 of mechanism upon the clutch of a set of reversing-gear, like the 
 
 Fig. 261.
 
 420 
 
 DYNAMICS OF MACHINERY. 
 
 combination shown in fig. 214, Article 2G3, page 299. The driving- 
 shaft of that combination is continually driven by the engine. 
 When the lever E is in its middle position, the following shaft is 
 disengaged from the driving-shaft, and remains at rest. "When 
 the lever E is shifted to one side or to the other, the reversing-gear 
 drives that following shaft in one direction or in the other; and its 
 motion, being transmitted by a suitable train to the regulator, 
 corrects the deviation of spee'd. So soon as the spindle resumes 
 its proper speed, the tooth D, by striking one or other of the prongs 
 F, F, replaces the lever E in its middle position, and disengages 
 the regulating train. 
 
 367. In Differential GoTcniors the regulation of the prime 
 mover is effected by means of the difference between the velocity 
 of a wheel driven- by it and that of a wheel regulated by a revolving 
 pendulum. This class of governors is exemplified by fig. 262, 
 
 representing Siemens's differential 
 governor as applied to prime 
 movers. X A is a vertical dead- 
 centre or fixed spindle about 
 which the after-mentioned pieces 
 turn ; G is a pulley driven by 
 the prime mover, and fixed to a 
 bevel-wheel, which is seen below 
 it ; E is a bevel- wheel similar to 
 the first, and having the same 
 apex to its pitch-cone. To this 
 wheel are hung the revolving 
 masses B, of which there are 
 usually four, although two only 
 are shown. Those masses form 
 
 sectors of a ring, and are surrounded by a cylindrical casing, 
 F. When the masses revolve with their proper velocity, they 
 are adjusted so as nearly to touch this casing ; should they 
 exceed that velocity, they fly outwards and touch the casing, 
 and are retarded by the friction. Their centrifugal force may 
 be opposed either by gravity or by springs. For practical 
 purposes, their angular velocity of revolution about the vertical axis 
 may be considered constant. G, G are horizontal arms projecting 
 from a socket which is capable of rotation about A, and carrying 
 vertical bevel-wheels, which rest on E and support C, and transmit 
 motion from C to E. There are usually four of the arms G, G, 
 with their wheels, though two only are shown. H is one of those 
 arms which projects, and has a rod attached to its extremity to act 
 on the regulator of the prime mover, of what sort soever it may be. 
 When C rotates with an angular velocity equal and contrary to 
 that of E with its revolving pendulums, the arms G, G remain 
 
 // 
 
 
 \ 
 
 B 
 
 1 
 
 "f 
 
 'A 
 
 
 Fig. 262.
 
 GOVERNORS. 421 
 
 at rest; but should C deviate from that velocity, those arms rotate 
 in one direction or the other, as the case may be, with au angular 
 velocity equal to one-half of the difference between the angular 
 velocity of C and that of E (see Article 234, page 245), and con- 
 tinue in motion until the regulator is adjusted so that the prime 
 mover imparts to C an angular velocity exactly equal to that of 
 the revolving masses B, B. 
 
 There are various modifications of the differential governor, but 
 they all act on the same principle. 
 
 368. In Pump-Governors each stroke of the prime mover to 
 be regulated forces, by means of a small piimp, a certain volume of 
 oil into a cylinder fitted with a plunger, like a hydraulic press. 
 The oil is discharged at an uniform rate through an adjustable 
 opening, back into the reservoir which supplies the pump. When 
 the prime mover moves faster or slower than it > proper speed, the 
 oil is forced into the cylinder faster or slower, as the case may be, 
 than it is discharged, so as to raise or to lower the plunger ; and 
 the plunger communicates its movements to the regulator, so as to 
 correct the deviation of speed. 
 
 The Bellows-Governor acts on the same principle, using air in- 
 stead of oil, and a double bellows instead of a pump and a plunger- 
 cylinder 
 
 369. In Fan-Governors the greater or less resistance of air or of 
 some liquid to the motion of a fan driven by the prime mover, causes 
 the adjustment of the opening of the regulator.
 
 422 
 
 CHAPTER IV. 
 
 OP THE EFFICIENCY AND COUNTER-EFFICIENCY OF PIECES, 
 COMBINATIONS, AND TRAINS, IN MECHANISM. 
 
 370. Nature and Division of the Subject. The terms Efficiency 
 
 and Counter-efficiency have already been explained in Article 335, 
 page 377; 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 309 to 311, pages 
 348 to 354. In the present Chapter are to be set forth the effects 
 of wasteful resistance, and especially of friction, on the efficiency 
 and counter-efficiency of single pieces, and oiscorubinations 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 that 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 U be the useful work to be 
 done per second; c, c', c", &c., the counter-efficiencies of the several 
 parts of the train; T, the total energy to be expended per second; 
 then 
 
 T = c-c'-c"-&c....U (1.) 
 
 When the mean effort 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 Article 
 324, page 309, and Article 338, page 379), 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 P be the effort required, in 
 the absence of wasteful resistance; P, the actual effort required; 
 then 
 
 P = c-c' 'c"-&c....P Q ; (2.) 
 
 and in determining the efficiency or the counter-efficiency of a 
 single piece, the most convenient method of proceeding often con-
 
 EFFICIENCY OF PKIMABY PIECES. 423 
 
 gists in comparing together the efforts required to drive that piece, 
 with and without friction, and thus finding the ratios 
 
 P P 
 
 -^ = efficiency; ~- = counter-efficiency. (3.) 
 
 ^o 
 
 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 connection employed in elementary combinations. 
 
 SECTION I. Efficiency and Counter-efficiency of Primary Pieces. 
 
 371. Efficiency of Primary Pieces in Ociieral. 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 Resistance, and denoted by R; 
 
 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 (<?), whose tangent (f = tan $>) is the co-efficient of friction 
 (see Article 309, page 349) ; and the amount of the friction is 
 expressed by Q sin 0, or very nearly by _/Q, when the co-efficient 
 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 R is vertical, by Rule I. of Article 280, page 322 ; and if 
 inclined, by the Rules given or referred to in Article 278, page 
 319. 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
 
 424 DYNAMICS OF MACHINERY. 
 
 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 dii'ection 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. N 
 
 371 A. Conditions Assumed to be Fulfilled. In all the problems 
 
 treated of in this section, the following conditions are assumed to 
 be fulfilled: First, that except when otherwise specified, the 
 forces other than bearing-pressures which are applied to the piece 
 under consideration that is, the useful resistance, the weight, and 
 the effort act either in parallel directions, or exactly or nearly in 
 one plane, parallel to the planes of motion of the particles of the 
 piece; secondly, that the acting parts of the piece do not overhang 
 the bearings; and thirdly, that the bearing-surfaces fit each other 
 easily without any grasping or pinching. As to the object of the 
 fulfilment of such conditions, and the effects of departure from, 
 them, the following explanations have to be made : 
 
 I. The bearing-surface of many primary pieces, and especially of 
 rotating pieces, is in general divided into two parts; for example, 
 an axle is very often supported by two journals. If the forces 
 other than bearing-pressures which are applied to the moving piece, 
 are parallel to each other, the parts of the bearing-pressure will also 
 be parallel to them and to each other; and the sum of the frictional 
 resistances due to the two parts of the bearing-pressure will be 
 simply equal to the frictional resistance due to the whole bearing- 
 pressure treated as one force. The same will be the case when 
 the forces other than bearing-pressures act in one plane, parallel 
 to the planes of motion of the particles of the piece; and will be 
 nearly the case when, although those forces act in different planes, 
 the transverse distance between their planes of action is small 
 compared with the distance between the planes of action of the 
 two components into which the bearing-pressure is divided. 
 
 But when that condition is not fulfilled, the friction at the 
 bearings, being proportional to the sum of the two components
 
 EFFICIENCY OF MECHANISM. 425 
 
 into which the bearing-pressure is divided, will be greater than 
 the friction due simply to the resultant bearing-pressure considered 
 as one force ; and the efficiency of the piece will be diminished. 
 
 II. The effect upon the friction, and upon the work lost in 
 overcoming it, produced when the acting parts of a moving piece 
 overhang its bearings, may be approximately calculated and allowed 
 for in the following manner: 
 
 Suppose that the bearing- surface of a primary piece, whether 
 sliding or turning, is divided into two parts ; and that the trans- 
 verse distance between the centres of those two parts that is, the 
 distance in a direction perpendicular to the planes of motion of 
 the particles of the piece is denoted by c. Let the plane of action 
 of the forces other than bearing-pressures be situated outside the 
 space between the two parts of the bearing-surface, and at the 
 transverse distance z from the centre of the nearer of those parts ; 
 and consequently at the distance z + c from the centre of the 
 further of them. Let Q be the resultant bearing-pressure. The 
 two components of that resultant pressure, exerted at the two 
 parts of the bearing-surface, will be contrary to each other in 
 direction ; and their values will be respectively, 
 
 Q (z + c) 
 at the nearer part, ; 
 
 and at the further part, 
 
 v 
 
 The total friction will be the sum of two components exerted at 
 the two parts of the bearing-surface respectively, and will be pro- 
 portional to the arithmetical sum of the two components of the 
 bearing-pressure ; that is, to the force 
 
 Q(c + 2z), 
 
 whereas, if the plane of action of the resultant of the given force 
 and the effort had not overhung the bearings, the friction would 
 have been simply proportional to Q. Hence the effect of that 
 plane's overhanging the bearings by the distance z, is to increase 
 the friction approximately in the ratio of 
 
 III. As to the condition that the bearing-surfaces should fit 
 each other easily, it is necessary in order that the bearing-pressure 
 may not contain, to any appreciable extent, pairs of components 
 which balance each other, being transverse to the direction of the
 
 426 
 
 DYNAMICS OF MACHINERY. 
 
 resultant bearing-pressure; for such components cause an un- 
 necessary addition to the friction. The ratio in which the friction 
 of a tight-fitting bearing exceeds that of an easy -fitting bearing of 
 the same dimensions and figure, is very nearly equal to that in 
 which the whole area of the bearing-surface exceeds the area of 
 the projection of that surface on a plane normal to the direction of 
 the resultant bearing-pressure. 
 
 When the use of bearing-surfaces in pairs, oblique to the plane 
 of the pressure and motion, is unavoidable (as, for example, in the 
 case of the Y-shaped bearings of a planing machine), their effect 
 may be allowed for by increasing the co-efficient of friction in the 
 ratio above-mentioned; which is expressed by the secant of the 
 equal angles which the normals to the bearing-surfaces make with 
 that plane. 
 
 372. Efficiency of a Straight-sliding Piece In fig. 263, let A A 
 
 be a straight guiding-surface, upon which there slides, in the direc- 
 
 Fig. 263. 
 
 tion marked by the feathered arrow, the moving piece B. Let 
 CD represent the given force, being the resultant of the useful 
 resistance and of the weight of the piece B. (The figure shows 
 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. 
 
 Draw CN perpendicular to A A; and CF making the angle 
 N C F = the angle of repose. Through D, parallel to C J, draw 
 the straight line D H Q, cutting C N in H, and C F in Q ; and 
 through H and Q, and parallel to D C, draw H K and Q Kj, 
 cutting C J in K and Kj respectively. Produce H C to H', and 
 Q C to g, making C H 1 = H C, and C Q' = Q C.
 
 EFFICIENCY OF SLIDING-PIECE OF AXLE. 427 
 
 Then, in the absence of friction, C H' will represent the resultant 
 bearing-pressure exerted upon B by A A; and CK = DHwill 
 represent the force in the given direction C 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 x the actual 
 driving force required; and we shall have 
 
 C K C K 
 
 the efficiency = /-v^ ; and the counter-efficiency = l . 
 {-> iVj U .K.Q 
 
 If from D, K , and K x there be let fall upon A A the perpen- 
 diculars D R, K P , and K x P a , C R will represent the direct 
 resistance to the advance of B; C P , the direct effort in the 
 absence of friction; and C P 1? the direct effort taking friction into 
 account; so that the distance P P : 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 = Pi/ (the given. 
 force) ; let the acute angle A C I) be denoted by , and the acute 
 angle A C J by /3 ; and let <J> denote the angle of repose N C Q. 
 
 Then, in the triangle C D H, we have ^.T) C H = , and 
 
 a 
 
 C H D = J /3; and in the triangle C Q D, we have ^ D C Q 
 
 A 
 
 i? *7r 
 
 = 77 + <p, and ^L CQD = ft <P; consequently 
 
 A 2i 
 
 P H = R- H^ ; p Q = K cos ( ' - *X 
 
 COS ft ' COS (ft + <p) ' 
 
 whence it follows that the efficiency and counter-efficiency are 
 given by the following equations : 
 
 P D H cos * cos (0+9) 1 /tan ft 
 Efficiency = ^, = ^-^ = - ) - \ = ^ J-^T- - (1.) 
 J P x D Q cos /3 cos ( <p) l+/tan x ' 
 
 PI 1 +/tan * , . 
 
 Counter-efficiency = = - ................... (2.) 
 
 It is to be remarked, that the efficiency diminishes to nothing 
 when cotan ft = f; that is to say, when /3 is the complement of the- 
 angle of repose, (p. In other words, if the oblique effort is applied 
 in the direction C Q, no force, how great soever, will be sufficient* 
 to keep the piece B in motion. 
 
 373. Efficiency of an Axle. In fig. 264, let the circle A A- A 
 represent the trace of the bearing-surface of an axle on a plane- 
 perpendicular to its axis of rotation, O in other words, the trans- 
 verse section of that surface. Let the arrow near the letter N 
 represent the direction of rotation. Let C D be the given: force;
 
 DYNAMICS OF MACHINERY. 
 
 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. 264. 
 
 In toothed wheel -work the lines of action of the useful resist- 
 ance and of the effort may be taken as coinciding with the lines of 
 connection of the rotating piece with its follower and with its 
 driver respectively. In pulleys connected with each other by 
 bands, special principles have to be attended to, which will be 
 explained in the ensuing Article. 
 
 Let r denote the radius of the bearing-surface. 
 
 About describe the small circle B B, with a radius = f 
 sin =fr, very nearly. Draw the line of action, C 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 D and C J intersect each other in a point, C, as 
 shown in the figure, CTQ 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, CTQ will be parallel to both. 
 The centre of bearing-pressure is at Q ; and Q T = <p, the angle 
 of repose. 
 
 In the former case the efficiency may be found by parallelo-
 
 EFFICIENCY OF AXLE. 429 
 
 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 CON in H, 
 and C T Q in E. Through H and E, parallel to D C, draw H P 
 and E P r Then, in the absence of friction, H C would represent 
 the bearing-pressure, and C P = D H the effort; the actual bear- 
 ing-pressure is represented by E C, and the actual effort by C P l = 
 D E. Hence the efficiency and counter-efficiency are as follows : 
 
 P!~ DE' P ~DH"' 
 
 Another method, applicable whether the forces are inclined or 
 parallel, is as follows : From the axis of rotation O, let fall O L 
 and O M 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 absence of friction is 
 
 From 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 L x and T Mj perpendicular respec- 
 tively to the same pair of lines of action; then the actual effort 
 will be 
 
 Pi = R' . 
 
 1 T 
 
 Hence the efficiency and the counter-efficiency have the following 
 value : 
 
 P _0 
 ~ 
 
 P x ~ O M 6 T JV 
 ?i _ M o ' T L i 
 
 P O - o L O T M; 
 
 J 
 
 The same results are expressed, to a degree of approximation 
 sufficient for practical purposes, by the following trigonometrical 
 formula : Let O L = I ; O M = m ; ^ C O L = ; ^ C O M 
 = /3. Then we have, very nearly, 
 
 fr 
 **o m -fr sin ft _ m ,^ . 
 
 I -\-fr sin a ~~ .fr 
 1 + -- '
 
 430 DYNAMICS OP MACHINERY. 
 
 In making use of the preceding formula, it is to be observed 
 that the contrary algebraical signs of sin and sin /3 apply to those 
 cases in which the two angles and ft lie at contrary sides of O C. 
 In the cases in which those angles lie at the same side of C, their 
 algebraical signs are the same; and in the formula they are to be 
 made both positive or both negative, according as /3 is less or greater 
 than ; so that the efficiency may be always expressed by a frac- 
 tion less than unity. That is to say, 
 
 1 - J-- sin ft 
 If ^ ;> = - _. ............. (3 A.) 
 
 fr 
 p 1 + sin ft 
 
 If ^ ;>=- -_ - ............... (3 a) 
 
 1 1 + ^ sin 
 
 When the lines of action intersect, let C be denoted by c ; 
 then I = c cos , and m = c cos ft ; and consequently the three 
 preceding equations take the following form : 
 
 p 
 A and et of contTfvry si^ns * 
 
 f T 
 1 - tan ft 
 c 
 
 (4\ 
 
 *1 
 
 .ft and a, of the same sign ; 
 p 
 
 f r > 
 
 1 + tan a, 
 c 
 
 1 - ^ tan ft 
 c 
 
 \*-) 
 
 ft ^ P 
 
 1 tan* 
 c 
 
 c 
 
 (A.T>\ 
 
 i- -^ p ~ 
 
 1 4. fj. tan * 
 
 \* a -J 
 
 c 
 
 When the lines of action of the forces are parallel, we have sin ft 
 and sin = + 1 or 1, as the case may be; and the formulae 
 take the following shape : 
 
 When I and m lie at contrary sides of O, the piece is a " lever 
 of the first kind ; " and
 
 EFFICIENCY OF AXLE. 431 
 
 "When I and m lie at the same side of O ; 
 
 If m ~^ I, the piece is a " lever of the second kind ;" and 
 
 "T 
 
 If m ^. I, the piece is a " lever of the third kind ;" and 
 
 (As to levers of the first, second, and third kinds, see Article 
 221, page 233.) 
 
 The following method is applicable whether the forces are inclined 
 or parallel; in the former case it is approximate, in the latter 
 exact. Through O, perpendicular to O 0, draw TJ O V, cutting 
 the lines of action of the given force and of the effort in II 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 
 
 
 P! ~~ O V T IT ' 
 
 It is evident that with a given radius and a given co-efficient of 
 friction, the efficiency of an axle is the greater the more nearly 
 the effort 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. 
 
 374. Axles of Pulleys connected by Bands. When the rotating 
 piece which turns with an axle consists of a pair of pulleys, one 
 receiving motion from a driving pulley, and the other communi- 
 cating motion to a following pulley, regard must be had to the fact 
 that the useful resistance and the driving effort are each of them 
 the difference of a pair of tensions; and that it is upon the resultant 
 of each of those pairs of tensions (being their sum, if they act parallel 
 ; to each other) that the axle-friction depends. 
 
 The principles according to which the tensions required at the 
 two sides of a band for transmitting a given effort are determined, 
 have been stated in Article 310 A, pages 351, 352. 
 
 The belt which drives the first pulley may be called the driving 
 J>elt; that which is driven by the second pulley, the following belt.
 
 432 DYNAMICS OF MACHINERY. 
 
 The tensions on the two sides of the following belt are given ; and 
 the moment of the useful resistance is that of their difference, 
 acting with a leverage equal to the effective radius of the second 
 pulley. Let p be that radius; T l and T 2 the two tensions; then 
 the moment of the useful resistance is 
 
 For the actual useful resistance there is to be substituted a force 
 equal to the resultant of T x and T 2 , and exerting the same moment. 
 That is to say, let y denote the angle which the two sides of the 
 band make with each other ; then for the actual useful resistance 
 is to be substituted a force, 
 
 R" = J {Tl + T- + 2 Tj T 2 cos y }, ............ (1.) 
 
 acting at the following perpendicular distance from the axis of 
 rotation : 
 
 p(Tl -T 2 ) 
 R"~ 
 
 And this is to be compounded with the weight of the rotating 
 piece, to find the given force R' of the rules in the preceding 
 Article, whose perpendicular distance from the axis will be 
 
 R' 
 
 (3.) 
 
 The value of k may be expressed in terms of the ratio of the 
 tensions to each other, and independently of their absolute values, 
 
 T 
 
 as follows : Let N = ~* be the ratio of the two tensions found 
 
 *-2 
 
 by the rules of Article 310 A, page 351. Then 
 
 _ 
 
 " N /{N 2 + 1 + 2 N cos -/} ' 
 
 In like manner, for the actual line of action of the effort by 
 which the first pulley is driven is to be substituted the line of 
 action of a force exerting the same moment, and equal to the 
 resultant of the tensions of the two sides of the driving-band. 
 The perpendicular distance m of this line of action from the axis 
 of rotation is given by the following formula: Let p be the 
 effective radius of the pulley; N', the ratio of the greater to the 
 lesser tension ; y, the angle which the two sides of the band make 
 with each other ; then 
 
 p'(N' 1) 
 
 "^{N' 2 + 1 + 2 N' COSy'}' 
 
 There are many cases in practice in which the two sides of each
 
 EFFICIENCY OF AXLES OF A SCREW OF SHAFTING. 433 
 
 of the bands may be treated as sensibly parallel; and then we 
 have simply, 
 
 = 
 
 N - 1 
 
 (6.) 
 
 J: - 
 
 N'+ 1 
 
 And if, moreover, as frequently happens, the weight of the pulleys 
 and axle is small compared with the tensions, we may neglect it, 
 and make E,' = R" and l = k, preparatory to applying the rules 
 of the preceding Article to the determination of the efficiency. 
 
 375. Efficiency of a Screw. The efficiency of a screw acting as a 
 primaiy 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 63, page 40 ; and as to the 
 efficiency of a piece sliding along a straight guide, see Article 372, 
 page 426. 
 
 376. Efficiency of Long Lines of Horizontal Shafting. In a line 
 
 of horizontal shafting for transmitting motive power to long 
 distances in a mill, a great part of the wasted work is spent in 
 overcoming the friction produced simply by the weight of the 
 shaft resting on its bearings; and the efficiency and counter- 
 efficiency as affected by this cause of loss of power can be con- 
 sidered and calculated separately. 
 
 For reasons connected with the principles of the strength of 
 materials, to be explained further on, the cube of the diameter of a 
 shaft of uniform diameter must be made to bear a certain propor- 
 tion to the driving moment exerted upon it to keep up its rotation. 
 That is to say, let M x denote that moment; h, the diameter of the 
 shaft; then 
 
 ........................... (1.) 
 
 A being a co-efficient whose values in practice range, according 
 to cii-cumstances to be explained in the Third Part of this treatise, 
 
 for forces in Ibs. and dimensions in inches, from 300 to 1,800; 
 and for forces in kilogrammes and dimensions in millimetres, 
 from 0-21 to 1*26. 
 
 Let w denote the heaviness of iron; /, the co-efficient of friction j 
 then the weight of an unit of length of the shaft is
 
 434 DYNAMICS OF MACHINERY. 
 
 the friction per unit of length is, very nearly, 
 
 and the moment of friction per unit of length is 
 
 ^-fw h* = -3927 fw W> nearly. ............... (2.) 
 
 o 
 
 Let L be the length of a shaft of uniform diameter, such that the 
 whole driving moment is exhausted in overcoming its own friction. 
 This may be called the exhaustive length. Then we must have 
 
 M! = A h s = -3927 fw h* L; and therefore 
 
 L = 
 
 -3927/uf " ...................... > 
 
 For lengths in feet, and diameter in inches, we have MJ = ; being 
 
 o 
 
 the weight in pounds of a rod of iron a foot long and an inch 
 square. For lengths in metres, and diameters in millimetres, we 
 have w = -0077 nearly; being the weight of a rod of iron one 
 metre long and one millimetre square. Let/= O'OSl; then the 
 following are the values of the exhaustive length L corresponding 
 to different values of A : 
 
 A, British measures, 300 600 1,200 1,800 
 
 French, 0-21 0-42 0^84 1*26 
 
 L, feet 4>5o 9,000 18,000 27,000 
 
 metres 1,365 2,730 5,460 8,190 
 
 It is obvious that the efficiency of a length, I, of shafting of 
 uniform diameter is given by the expression 
 
 M being the driving moment in the absence of friction ; M I} the 
 
 actual driving moment; and =r, the fraction of that moment ex- 
 
 JLi 
 
 pended on friction; also, that the counter-efficiency is 
 M, L 
 
 .(5.) 
 
 M ~L -t" 
 
 "When, besides its own weight, the shaft is loaded with the weights 
 of pulleys and tensions of belts, the effect of such additional load
 
 EFFICIENCY OF LINES OF SHAFTING. 435 
 
 may be allowed for, with a degree of accuracy sufficient for practical 
 purposes, in the following manner: Find the magnitude of the 
 resultant of the weight of the shaft and additional load ; and let m 
 be the ratio which it bears to the weight of the shaft. Then the 
 modified value of the exhaustive length is to be found by putting 
 m w instead of w in the denominator of the expression (3.) : that is 
 to say 
 
 L = -3927 fmw ........................ ( 6<) 
 
 The waste of work in a long line of shafting may be diminished, 
 and the efficiency increased, by causing it to taper, so that the cube 
 of the diameter shall at each cross-section be proportional to the 
 moment exerted there. The most perfect way of fulfilling that 
 condition is to make the diameter diminish continuously in geo- 
 metrical progression; the generating line or longitudinal section 
 of the shaft being a logarithmic curve. Let h be the diameter at 
 the driving end, x the distance of a given cross-section from that 
 end, and y the diameter at that cross-section; then 
 
 (7.) 
 
 in which e S L is the reciprocal of the natural number, or anti- 
 
 sc 
 logarithm, corresponding to the hyperbolic logarithm ^-= , and to 
 
 the common logarithm ^-= . Let I be the total length of such 
 
 a tapering shaft, and M the useful working moment exerted at 
 its smaller end; then we have 
 
 Efficiency, 
 
 .(8.) 
 
 M -J 
 Counter-efficiency, ~- = e lj . 
 
 This cannot be perfectly realized in practice; but it can be 
 approximated to by making the shaft consist of a series of lengths, 
 or divisions, each of uniform diameter, and increasing in diameter 
 step by step. 
 
 Let - now denote the length of one of those divisions ; the 
 
 n 
 
 number of divisions being n. The counter-efficiency of each 
 division is expressed by 
 
 J_i 
 
 n
 
 436 DYNAMICS OP MACHINERY. 
 
 and consequently, the counter-efficiency of the whole shaft is, 
 
 L 
 
 .(10.) 
 
 M A ~ ^ L -- 
 
 n 
 
 The diameters of the lengths of shafting, beginning at the driving- 
 end, form a diminishing geometrical progression, of which the 
 common ratio is 
 
 SECTION II. Efficiency and Counter-efficiency of Modes of 
 Connection in Mechanism. 
 
 377 Efficiency of modes of Connection In General. In an ele- 
 mentary combination consisting of two pieces, a driver and a 
 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 efficiency of the connection. In 
 calculating the efficiency or the counter-efficiency of a train of 
 mechanism, therefore, the factors to be multiplied together 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. 
 
 378. Efficiency of Rolling Contact. The work lost when one 
 primary piece drives another by rolling contact is expended in 
 overcoming the rolling resistance of the pitch-surfaces, a kind of 
 resistance whose mode of action has been explained in Article 311, 
 page 353; and the value of that work in units of work per second 
 is given by the expression a b N"; in which N is the normal pressure 
 exerted by the pitch-surfaces on each other; b, 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 354); 
 and a, the relative angular velocity of the surfaces. 
 
 The useful work per second is expressed by u fN, in which./ is 
 the co-efficient of friction of the surfaces, and u the common velocity 
 of the pitch-lines. Hence the counter-efficiency is 
 
 i a b /i \ 
 
 = 1 + (1.) 
 
 uf
 
 EFFICIENCY OF MODES OF CONNECTION. 
 
 437 
 
 Let jp 1 and p z be the lengths of two perpendicxilars let fall 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-, vJ ' 
 
 which value being substituted in equation (1), gives for the counter- 
 efficiency the following value : 
 
 .(2.) 
 
 It is assumed that the normal pressure is not greater than is 
 necessary in order to give sufficient friction to communicate the 
 motion. 
 
 It is evident, from the smallness of b, that the lost work in this 
 case must be almost always a very small fraction of the whole. 
 
 379. Efficiency of Sliding Contact in General. In fig. 205, 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 ; whence V W will represent 
 the velocity of sliding, and T U, per- 
 pendicular to V W, the common com- 
 ponent of the velocities of the two 
 particles along their line of connection 
 RTF. 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; 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 surfaces at T, lay 
 off T P to represent the effort exerted by the driver on the 
 
 Fig. 2j65.
 
 438 DYNAMICS OF MACHINERY. 
 
 follower, and T R ( = 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 309, page 349), and inclined in 
 the proper direction to represent forces opposing the sliding motion ; 
 draw P Q and R S parallel to C 0. 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 driver, and P Q = R S will represent 
 the friction which is overcome, through the distance V "W, in each 
 second; while the useful resistance, T R, is overcome through the 
 distance T U. Hence the useful work per second is T U T R ; 
 the lost work is V W R S ; and the counter-efficiency is 
 
 YW-RS 
 C=1+ ...................... L 
 
 Let the angle U T Y = *, the angle U T W = ft, and let /be 
 the co-efficient of friction. Then we have 
 
 YW RS 
 
 T- = tan * + tan ft; 
 
 and consequently 
 
 c = 1 + /(tan* + tan/3) .................... (2.) 
 
 380. Efficiency of Teeth. It has already been shown, in Article 
 127, page 118, that the relative velocity of sliding of a pair of 
 teeth in outside gearing is expressed at a given instant by 
 
 (! + oj) 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 pass- 
 ing the pitch-point. Its mean value may be assumed, with suf- 
 ficient 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 co- 
 efficient of friction; then the work lost per second through the 
 friction of the teeth is 
 
 (a l + og) tfP. 
 Let u be the common velocity of the two pitch-circles ; 6, the
 
 EFFICIENCY OF TEETH. 439 
 
 mean 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 ex- 
 pressed by 
 
 P u cos 6 
 
 so that the counter-efficiency is 
 
 c - 1 + K + ag) */ . n\ 
 
 u cos d 
 
 Let r x and r t be the radii of the two pitch-circles ; then we have 
 
 u u 
 
 a i = ~> a z = '> 
 r i r z 
 
 and consequently 
 
 c= 
 
 - + -\ ................ (2.) 
 
 r i r 2 ) 
 
 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), 
 
 fp 
 and if the pitch be denoted by p, we have t sec ^; and there- 
 
 2i 
 
 fore 
 
 ._ 1 + A/l + 
 
 2 ( r t 
 
 where W A and n z 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 -r of that 
 
 assumed in equation (3); and the work lost in friction is less in the 
 same proportion. 
 
 The preceding rules have been stated in the form applicable to 
 spur-wheels. In order to make them applicable to bevel- wheels, 
 all that is necessary is to understand that the measurements of 
 radii, distances, and obliquity, are to be made, not on the actual 
 pitch-circles, but on the pitch-circles as shown on the development of 
 the normal cones; as to which, see Article 144, page 144. 
 
 When there is a transverse component in the relative velocity of 
 sliding (as in gearing- screws, Article 154, page 160), the fractional 
 value of the work lost in friction is to be first computed as if for a 
 pair of spur-wheels whose pitch-circles are the osculating circles of 
 the normal screw-lines (see Article 154, pages 161, 162; and Article 
 155, page 163). Then find in what ratio the velocity of sliding is
 
 440 DYNAMICS OP MACHINERY. 
 
 increased by compounding the transverse component with the 
 direct component (a x 4- a 2 ) t; and increase the fraction of work lost 
 through friction in the same proportion. 
 
 381. Efficiency 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 
 Reuleaux (Construclionslehre fur Maschinenbau, 307). 
 
 Let T be the mean tension of the belt ; S, its sectional area ; 
 r, the radius of the pulley; b, a constant divisor determined by ex- 
 periment; R', the resistance due to stiffness; then 
 
 6 (for leather) = 3-4 inch = 87 millimetres. 
 
 To apply this to an endless belt connecting a pair of pulleys of 
 the respective radii r x and r z , let T x and T 2 be the tensions of the 
 two sides of the belt, as determined by the rule of Article 310 A, 
 page 351. Then the useful resistance is T, T 2 ; the mean ten- 
 
 T "f- T 
 sion is i ? ; an( j the additional resistance due to stiffness is 
 
 2 
 consequently the counter-efficiency is 
 
 T t + T 2 S f 1 
 
 1 -J_ 1 _ 
 
 _ N + l 81 
 
 r 
 
 2(N 
 
 Article 
 S, of a leather belt is given by the formula 
 
 .(2.) 
 
 T 
 
 denoting =p, as in Article 374, page 432. The sectional area, 
 
 where p denotes the safe working tension of leather belts, in units 
 of weight per unit of area; its value being, according to Morin,
 
 EFFICIENCY OF BANDS. 441 
 
 0-2 kilogramme on the square millimetre, or 
 285 Ibs. 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 : 
 
 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; Tj^ being 
 the tension at the driving-side of the belt, as calculated by the rules 
 of Article 310 A, page 351, exclusive of centrifugal tension: 
 
 and the following formula for the counter-efficiency : 
 T 2 w v 2 
 
 -4 + s ri 1 
 
 
 2( Tl -T 2 ) 
 
 The questions of areas of bands and centrifugal tension will be 
 further considered in the part of this treatise relating to the strength 
 of machinery. 
 
 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., 
 
 where D is the diameter of the rope, and b' = 54 millimetres = 
 2-125 inches.
 
 442 
 
 DYNAMICS OF MACHINERY. 
 
 D 2 S 
 
 In all the formulae, -TT is to be substituted for . . The proper 
 o o 
 
 value of D 2 is given by the formula 
 
 T i 
 " P" 
 
 .(8.) 
 
 where p' = 1000 for measures in inches and Ibs. ; and 
 
 p' = 07 for measures in millimetres and kilogrammes. 
 382. Efficiency of Linkwork. In fig. 266, let G I T V C 2 T 2 be two 
 
 levers, turning about parallel axes at C t and C 2 , and connected with 
 each other by the link T x T 2 ; T 1 and T 2 being the connected points. 
 
 Fig. 266. 
 
 The pins, which are connected with each other by means of the 
 link, are exaggerated in diameter, for the sake of distinctness. Let 
 G! T x be the driver, and C 2 T 2 the follower, the motion being as 
 shown by the arrows. From the axes let fall the perpendiculars 
 Cj PJ, C 2 P 2 , 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 T x T 2 . Let M 2 be the working moment ; and 
 let M be the driving moment in the absence of friction; then we 
 have 
 
 _M 2 j_C L P 1 
 - 
 
 To allow for the friction of the pins, multiply the radius of each 
 pin by the sine of the angle of repose; that is, very nearly by the
 
 EFFICIENCY OF LHTKWOBK OF TACKLE. 443 
 
 co-efficient of friction; and with the small radii thus computed, 
 Tj Aj and T 2 K y draw small circles about the connected points. 
 Then draw a straight line, Q x A x B x Q 2 B 2 A^ 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 
 Cj Qj, C 2 Q 2 ; 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 
 
 M 2 -C 1 Q 1 
 Mi - 0,0, 
 
 Comparing this with the value of the driving moment without 
 friction, we find for the counter-efficiency 
 
 M 1 _C L Q Ll C A P 2 . 
 
 -M -C 2 Q 2 -C 1 P 1 - - (L) 
 
 and for the efficiency 
 
 1 _ MO _ C 2 Q 2 C : Pj 
 c ~ M x ~ Cj Q! C 2 P 2 - 
 
 (See page 449.) 
 
 383. Efficiency of Blocks and Tackle. (See Articles 200, 201, 
 pages 214 to 216.) 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 427. Let cf denote the counter-efficiency of the rope, 
 when passing over a single sheave, determined by the principles 
 of Article 381, the tension being taken as nearly equal to 
 
 T> 
 
 : where E, is the useful load, or resistance opposed to the motion 
 n 
 
 of the running block. R -=- n is also the effort 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 
 
 so that the actual or effective purchase, instead of being expressed 
 by n, will be expressed by
 
 444 DYNAMICS OP MACHINERY. 
 
 384. Efficiency of Connection by means of a Fluid. When motion 
 
 is communicated from one piston to another by means of an inter- 
 vening mass of fluid, as described in Articles 207 to 210, pages 221 
 to 224, the efficiencies and counter-efficiencies of the two pistons 
 have in the first place to be taken into account; which quantities 
 are to be determined by means of the principles stated at page 
 399; that is to say, with ordinary workmanship and packing, the 
 efficiency of each piston may be taken at 0-9 nearly; while with a 
 carefully made cupped leather collar the counter-efficiency of a 
 plunger may be taken at the following value : 
 
 i-i 6 ; ............................. GO 
 
 in which d is the diameter of the plunger; and b a constant, 
 whose value is from O'Ol to 0-015 of an inch, or from 0'25 to 
 0'38 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 pressure on the plunger is *- and the pressure 
 
 required to overcome the friction is p c b. 
 
 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 resis- 
 tance 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, 
 
 .(2.) 
 
 is the velocity of the stream of fluid. Let b denote the wetted 
 border or circumference of the passage; then, 
 
 (3.) 
 
 = , 
 
 is what is called the hydraulic mean depth of the passage. In a 
 cylindrical pipe, m = diameter. Let I be the length of the
 
 EFFICIENCY OF CONNECTION BY FLUID. 445 
 
 passage, and w the heaviness of the liquid. Then the loss of 
 pressure in overcoming the friction of the passage is 
 
 ; fl w v 2 
 
 -27' 
 
 in which g denotes gravity, and / a co-efficient of friction whose 
 value, for water in cylindrical cast-iron pipes, according to the 
 experiments of Darcy, is 
 
 /= 0-0051+;. .................. <ft> 
 
 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 of the fluid is 
 
 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 work 
 through friction of the passage, depending on principles similar to 
 those of the friction of liquids; and there is a further loss through 
 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, aboiit Jive 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 
 
 1 25*4 
 
 * When the diameter is expressed in millimetres, for r -5 substitute . -.
 
 446 DYNAMICS OF MACHINERY. 
 
 its friction may be estimated at one per cent, of the total or absolute 
 pressure of the air, for 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 T^th of the capacity of the compressing pumps has been 
 found to answer in practice. The air may be thus 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 ma^ be calculated approxi- 
 mately by taking two-sevenths of the logarithm of the absolute 
 working pressure of the compressed air in atmospheres, and finding 
 the corresponding natural number. That is to say, let p Q denote 
 one atmosphere (= at the level of the sea 14-7 Ibs. on the square 
 inch, or 10333 kilogrammes on the square metre); let p be the 
 absolute working pressure of the air, so that p^ p is the effective 
 pressure; then the counter-efficiency due to the escape of heat is, 
 
 -' 
 
 From examples of the practical working of compressed air, 
 when 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. 
 
 ADDENDUM TO ARTICLE 343, PAGE 386. 
 
 Rotatory Dynamometers Epicyclic Train Dynamometer. The 
 
 term of " epicyclic- train dynamometers" may be applied to those 
 instruments in which the power to be measured is transmitted 
 through an epicyclic train, and the effort exerted is measured by 
 means of the force required to hold the train-arm at rest. In 
 King's dynamometer, for example, there is a train of wheel- work
 
 DYNAMOMETERS PULLEYS TELODYNAMIC TRANSMISSION. 447 
 
 of which the principle (though not the details) is sufficiently well 
 represented by fig. 176, page 245. The bevel- wheel B is driven 
 by the prime mover; and through the bevel- wheels (or bevel- wheel, 
 there being usually only one) carried by the arm A, it drives the 
 bevel-wheel C, which drives the working machinery. The train- 
 arm A is kept steady by a weight, or by a spring; and it is obvious 
 that the moment of that force relatively to the common axis of 
 rotation of B, C, and A, must be double of the moment trans- 
 mitted from B to C; which latter moment that is, half the 
 moment of the weight or spring that holds A steady, being 
 multiplied by 2 x the number of turns in a given time, gives 
 the work done in that time. This apparatus may be made to 
 record its results on a travelling strip of paper, like other kinds of 
 dynamometers. 
 
 ADDENDA TO ARTICLE 381, PAGE 440. 
 
 I. Use of Believing Rollers between Pulley*. When a pair of 
 
 pulleys connected with each other by means of a band are near 
 together, the bearings of their shafts may be relieved from the 
 pressure due to the tension of the band by placing between the 
 pulleys a smooth idle wheel or roller, turning in rolling contact 
 with them both. The axis of rotation of the roller should be in 
 the same plane with those of the pulleys; and two out of the three 
 shafts should have their bearings so fitted up as to be capable of a 
 small extent of motion in a direction perpendicular to the axes of 
 rotation, in order that the distances of those axes from each other 
 may adjust themselves when the band is tightened, and that the 
 tension of the band and the pressure transmitted through the roller 
 may balance each other without the aid of pressures at the 
 bearings. 
 
 II. Efficiency of Telodynamic Transmission. The phrase " Telo- 
 
 dynamic Transmission" is used to denote Mr. C. F. Hirn's method 
 of transmitting motive power to long distances by means of an 
 endless wire rope, connecting a pair of large pulleys, and moving 
 at a high speed. The pulleys are made of cast iron; and each of 
 them has at the bottom of its groove a dovetail-shaped recess filled 
 with gutta-percha, which is driven in and rammed tight by means 
 of a mallet; the wire rope bears against the gutta-percha bottom 
 of the groove ; and this is found both to transmit an effort better, 
 and to ensure greater durability of the rope and pulleys, than 
 when the rope bears against a cast-iron surface. 
 
 The ordinary speed of the rope is from 50 to 80 feet per second; 
 and with wrought-iron pulleys, it is considered that it might be 
 increased to 100 feet per second. The effort to be transmitted is 
 calculated from the power to be transmitted, by expressing that
 
 448 DYNAMICS OF MACHINERY. 
 
 power in units of work per second, and dividing by the speed. 
 The available tensions at the driving and returning sides of the 
 rope are calculated by the rules of Article 310 A, page 351; in 
 practice it is considered sufficiently accurate to make the former 
 twice, and the latter once, the effort to be transmitted. To each of 
 those tensions is to be added the centrifugal tension (see Article 
 381, page 441) in order to obtain the total tensions. The trans- 
 verse dimensions of the rope are adapted to the total tension at the 
 driving side of the rope, by the application of rules to be given in 
 the Part of this Treatise relating to strength. 
 
 In order that the rope may not be overstrained by the bending 
 of the wires of which it consists, in passing round the driving and 
 following pulleys, the diameter of each of those pulleys should not 
 be less than 140 times the diameter of the rope, and is sometimes 
 as much as 260 times. 
 
 The distance between the driving and following pulleys is not 
 made less than about 100 feet; for at less distances shafting is 
 more efficient ; nor is it made more than x 500 feet in one span, 
 because of the great depth of the catenary curves in which the 
 rope hangs. When the distance between the driving and follow- 
 ing pulleys exceeds 500 feet, the rope is supported at intermediate 
 points by pairs of bearing pulleys, so as to divide the whole dis- 
 tance into intervals of 500 feet or less. 
 
 The bearing pulleys are constructed in the same way with the 
 driving and following pulleys, and of about half the diameter. 
 
 The loss of work due to the stiffness of the rope may be regarded 
 as insensible; because when the diameters of the pulleys are 
 sufficient, the wires of which the rope is made straighten them- 
 selves by their own elasticity after having been bent. 
 
 It has been found by practical experience that the losses of 
 power in this apparatus are nearly as follows, in fractions of the 
 whole power transmitted : 
 
 Overcoming the axle-friction of the driving and 
 
 following pulleys, about ^r, or ..................... 0-0250 
 
 Overcoming the axle-friction of each pair of bear- 
 
 ing pulleys, about ^, or ........................... o-ooii 
 
 Hence the efficiency of telodynamic transmission may be 
 estimated at 
 
 N being the number of pairs of intermediate bearing pulleys.* 
 
 For detailed information on the subject of Telodynamic Transmission, 
 see the following authorities: Notice sur la Transmission Telodynamique,
 
 FRICTION CAUSED BY OBLIQUE CONNECTING-ROD* 449 
 
 ADDENDUM TO ARTICLE 382, PAGE 442. 
 
 Effect of Obliquity of a Connecting-Rod on Friction. The alter- 
 
 nate thrust and tension along the connecting-rod is almost always 
 an important component, and sometimes the most important com- 
 ponent, of the force which is balanced by the pressure of the bear- 
 ings of a crank-shaft ; and the lateral component of that alternate 
 thrust and tension is the cause of the friction of the guides by 
 which the head of the piston-rod is made to move in a straight 
 line, when there is no parallel motion. 
 
 The direction of the connecting-rod is continually changing 
 between certain limits; and this causes a continual change in the 
 ratio borne by the whole force exerted along that rod, and by its 
 lateral component, to its direct component. 
 
 Let r be the crank-arm, c the length of the connecting-rod; then 
 the mean value of the ratio which the lateral component bears to 
 the direct component is very nearly as follows : 
 
 Q _ 0-7854 r 
 
 p - ^(c 2 - 0-617 r 2 )' ' 
 
 and if /be the co-efficient of friction of the guides, the counter- 
 efficiency of the piston-rod head will be nearly 
 
 The mean ratio borne by the total force (T) exerted along the 
 connecting-rod to its direct component (P) is nearly as follows : 
 
 C 2_ 0-617 
 
 and the axle-friction of the crank-shaft is increased nearly in that 
 ratio, beyond what it would be if the obliquity of the connecting- 
 rod were insensible.* 
 
 par C. F. Him (Colmar, 1862). Reuleaux, Constructionslehre fur Mas- 
 chinenbau (Braunschweig, 1854 to 1862), 324 to 342. 
 
 * The exact solution of these questions is given by the aid of elliptic 
 functions ; but for practical purposes the approximate solution in the text 
 is sufficient. 
 
 2o
 
 PAET III. 
 
 MATERIALS, CONSTRUCTION, AND STRENGTH OF 
 MACHINERY. 
 
 CHAPTER I. 
 
 OP MATERIALS USED IN MACHINERY. 
 
 385. General Explanations. The materials used in machinery 
 are of two principal kinds inorganic and organic. 
 
 The inorganic materials consist almost wholly of metals; for 
 although stony and earthy materials occur in the foundations of 
 fixed machines, and in houses which contain machinery, they 
 are little used in machinery itself. 
 
 The organic materials consist chiefly of wood, of vegetable and 
 animal fibre in the form of ropes and bands, and of indian rubber 
 and gutta percha, and a few miscellaneous substances. 
 
 The present chapter gives a summary of those properties of 
 materials upon which their use in machinery and millwork 
 depends; and it is necessarily to a great extent identical with 
 those parts of A Manual of Civil Engineering which treat of the 
 same materials. 
 
 SECTION I. Of Iron and Steel. 
 
 386. Kinds of iron and Steel. The metallic products of the iron 
 manufacture are of three principal kinds malleable iron, cast iron, 
 and steel. Malleable iron is pure or nearly pure iron. Cast iron is 
 a granular and crystalline compound of iron and carbon, more or 
 less mixed with uncombined carbon in the form of plumbago. It 
 is harder then pare iron, more brittle, and less tough. Steel is a 
 compound of iron with less carbon than there is in cast iron ; it is 
 harder than cast iron, and tougher than wrought iron, though less 
 ductile; and it is the strongest of all known substances for its 
 dimensions. It is also the strongest of all metals for its weight; 
 but in the comparison of tenacity with weight, steel and all metals 
 are exceeded by many kinds of organic fibre. There are many 
 intermediate gradations between pure iron and the hardest steel, 
 some of which are known by such names as " steely iron " and 
 " semi-steel."
 
 IMPURITIES OP IRON CAST IRON. 451 
 
 387. impurities of iron. The strength and other good qualities 
 of iron and steel depend mainly on the absence of impurities, and 
 especially of sulphur, phosphorus, silicon, calcium, and magnesium. 
 
 Sulphur and calcium, and probably also magnesium, make iron 
 " red-short" that is, brittle at a red heat ; phosphorus and silicon 
 make it "cold-short" that is, brittle at low temperatures. These 
 are both serious defects; but the latter is the worse. 
 
 Sulphur comes in general from coal or coke used as fuel. Its 
 pernicious effects can be avoided altogether by using fuel which 
 contains no sulphur; and hence the strongest and toughest of all 
 iron is that which is smelted, reduced, and puddled either with 
 charcoal, or with coke that is free from sulphur. 
 
 Phosphorus comes in most cases from phosphate of iron in the 
 ore, or from phosphate of lime in the ore, the fuel, or the flux. The 
 ores which contain most phosphorus are those found in strata 
 where animal remains abound. 
 
 Calcium and Silicon are derived respectively from the decom- 
 position of lime and of silica by the chemical affinity of carbon 
 for their oxygen. The only iron which is entirely free from those 
 impurities is that which is made by the reduction of ores that 
 contain neither silica nor lime : such as pure magnetic iron ore, 
 pure haematite, and pure sparry iron ore. 
 
 388. Cast iron is the product of the process of smelting iron 
 ores. In that process the ore in fragments, mixed with fuel and 
 with flux (that is to say, with a substance such as lime, which 
 tends to combine with the earthy constituents of the ore), is 
 subjected to an intense heat in a blast-furnace, and the products 
 are slag, or glassy matter formed by the combination of the flux 
 with the earthy ingredients of the ore, and pig iron, which is a 
 compound of iron and carbon, either unmixed, or mixed with a 
 small quantity of uncombined carbon in a state of plumbago. 
 
 The ore is often roasted or calcined before being smelted, in 
 order to expel carbonic acid and water. 
 
 The total quantity of carbon in pig iron ranges from two to five 
 per cent, of its weight. 
 
 Different kinds of pig iron are produced from the same ore in 
 the same furnace under different circumstances as to temperature 
 and quantity of fuel. A high temperature and a large quantity of 
 fuel produce gray cast iron, which is further distinguished into 
 No. 1, No. 2, No. 3, and so on; No. 1 being that produced at the 
 highest temperature. A low temperature and a deficiency of fuel 
 produce white cast iron. Gray cast iron is of different shades of 
 bluish-gray in colour, granular in texture, softer and more easily 
 fusible than white cast iron. White cast iron is silvery white, 
 either granular or crystalline, comparatively difficult to melt, 
 brittle, and excessively hard.
 
 452 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 It appears that the differences between those kinds of iron 
 depend not so much on the total quantities of carbon which they 
 contain as on the proportions of that carbon which are respec- 
 tively in the conditions of mixture and of chemical combination 
 with the iron. Thus, gray cast iron contains one per cent., and 
 sometimes less, of carbon in chemical combination with the iron, 
 and from one to three or four per cent, of carbon in the state of 
 plumbago in mechanical mixture ; while white cast iron is a homo- 
 geneous chemical compound of iron with from two to four per cent, 
 of carbon. Oi the different kinds of gray cast iron, No. 1 contains 
 the greatest proportion of plumbago, No. 2 the next, and so on. 
 
 There are two kinds of white cast iron, the granular and the 
 crystalline. The granular kind can be converted into gray cast 
 iron by fusion and slow cooling ; and gray cast iron can be con- 
 verted into granular white cast iron by fusion and sudden cooling. 
 This takes place most readily in the best iron. Crystalline white 
 cast iron is harder and more brittle than granular, and is not 
 capable of conversion into gray cast iron by fusion and slow 
 cooling. Gray cast iron, No. 1, is the most easily fusible, and 
 produces the finest and most accurate castings; but it is deficient 
 in hardness and strength; and therefore, although it is the best 
 for castings of moderate size, in which accuracy is of more impor- 
 tance than strength and stiffness, it is inferior to the harder and 
 stronger kinds, No. 2 and No. 3, for pieces requiring great strength 
 and stiffness. 
 
 The presence of plumbago renders cast iron comparatively weak 
 and pliable, so that the order of strength and stiffness among 
 different kinds of cast iron from the same ore and fuel is as fol- 
 lows : 
 
 Granular white cast iron. 
 
 Gray cast iron, No. 3. 
 
 No. 2. 
 
 No. 1. 
 
 Crystalline white cast iron is not introduced into this classifica- 
 tion because its extreme brittleness makes it unfit for use in 
 machinery. 
 
 Granular white cast iron, also, although stronger and harder 
 than gray cast iron, is too brittle to be a safe material for the 
 entire mass of any piece in a machine that is exposed to shocks; 
 but it is used to form a hard and impenetrable skin to a piece of 
 gray cast iron by the process called chilling. This consists in 
 lining the portion of the mould, where a hardened surface is re- 
 quired, with suitably-shaped pieces of iron. The melted metal, on 
 being run in, is cooled and solidified suddenly where it touches the 
 cold iron; and for a certain depth from the chilled surface, varying
 
 CAST IRON. 453 
 
 from |th to ^-incli in different kinds of iron, it takes the white 
 granular condition, while the remainder of the casting takes the 
 gray condition. 
 
 Even in castings which are not chilled by an iron lining to the 
 mould, .the outermost layer, being cooled more rapidly than the 
 interior, approaches more nearly to the white condition, and forms 
 a skin, harder and stronger than the rest of the casting. 
 
 A strong kind of cast iron called toughened cast iron, is pro- 
 duced by the process invented by Mr. Morries Stirling, of adding 
 to the cast iron, and melting amongst it, from one-fourth to one- 
 seventh of its weight of wrought-iron scrap. 
 
 Malleable Cast Iron is made by the following process: The 
 castings to be made malleable are imbedded in the powder of red 
 haematite (which consists almost wholly of peroxide of iron) ; they 
 are then raised to a bright red heat (which occupies about 24 
 hours), maintained at that heat for a period varying from three to 
 five days, according to the size of the casting, and allowed to cool 
 (which occupies about 24 hours more). The oxygen of the haema- 
 tite extracts part of the carbon from the cast iron, which is thus 
 converted into a sort of soft steel ; and its tenacity (according to 
 experiments by Messrs. A. More & Son) becomes about three times 
 that of the original cast iron. 
 
 389. The strength of east irou of every kind, like that of 
 granular substances in general, is marked by two properties : the 
 smallness of the tenacity (which is on an average about 16,000 or 
 18,000 Ibs. on the square inch) as compared with the resistance to 
 crushing (which ranges from 80,000 to 110,000), and the different 
 values of the stress immediately before rupture of the same kind of 
 iron in bars torn directly asunder, and in beams of different forms 
 when broken across. 
 
 For the results of experiments on the strength of various kinds 
 of cast iron, see the tables of the following chapter. 
 
 The strength of cast iron to resist cross breaking was found 
 by Mr. Fairbairn to be increased by repeated meltings up to the 
 twelfth, when it was greater than at first in the ratio of 7 to 5 nearly. 
 After the twelfth melting that sort of strength rapidly fell off. 
 
 The resistance to crushing went on increasing after each succes- 
 sive melting; and after the eighteenth melting it was double of its 
 original amount, the iron becoming silvery white and intensely 
 hard. 
 
 The transverse strength of No. 3 cast iron was found by Mr. 
 Fairbairn not to be diminished by raising its temperature to 600 
 Fahr. (being about the temperature of melting lead). At a red 
 heat its strength fell to two-thirds. 
 
 390. Castings for machinery. The best course for an engineer 
 to take, in order to obtain cast iron of a certain strength, is not to
 
 454 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 specify to the founder any particular kind or mixture of pig iron, 
 but to specify a certain minimum strength which the iron should 
 show when tested by experiment. 
 
 As to the appearance of good iron for castings, it should have 
 on the outer surface a smooth, clear, and continuous skin, with 
 regular faces and sharp angles. When broken, the surface of 
 fracture should be of a light bluish-gray colour and close-grained 
 texture, with considerable metallic lustre ; both colour and texture 
 should be uniform, except that near the skin the colour may be 
 somewhat lighter and the grain closer; if the fractured surface is 
 mottled, either with patches of darker or lighter iron, or with crystal- 
 line spots, the casting will be unsafe; and it will be still more unsafe 
 if it contains air-bubbles. The iron should be soft enough to be 
 slightly indented by a blow of a hammer on an edge of the casting. 
 When cut by tools of different kinds, the iron should show a 
 smooth, compact, and bright surface, free from bubbles and other 
 irregularities, of an uniform colour, and capable of taking a good 
 polish. 
 
 Castings are tested for air-bubbles by ringing them with a 
 hammer all over the surface. 
 
 Cast iron, like many other substances, when at or near the 
 temperature of fusion, is a little more bulky for the same weight 
 in the solid than in the liquid state, as is shown by the solid iron 
 floating on the melted iron. This causes the iron as it solidifies to 
 fill all parts of the mould completely, and to take a sharp and 
 accurate figure. The solid iron contracts in cooling from the 
 melting point down to the temperature of the atmosphere, by 
 about one per cent, in each of its linear dimensions, or one-eightk 
 of an inch in a foot nearly; and therefore patterns for castings 
 are made larger in that proportion than the intended pieces of 
 cast iron which they represent. 
 
 The rate of linear expansion of cast iron between the freezing 
 and boiling points of water is about 'ooi 1 1. 
 
 A convenient instrument in making patterns for castings is a 
 contraction-rule / that is, a rule on which each division is longer 
 in the proportion already mentioned than the true length to which 
 it corresponds. 
 
 In designing patterns for castings, care must be taken to avoid 
 all abrupt variations in the thickness of metal, lest parts of 
 the casting near each other should be caused to cool and contract 
 with unequal rapidity, and so to split asunder or overstrain the 
 iron. It is advantageous also that castings, especially those for 
 moving pieces in machinery, such as wheels, should be of sym- 
 metrical figures, or as nearly so as is consistent with their purposes, 
 in order that they may have no tendency to become distorted while 
 cooling.
 
 CAST IRON MALLEABLE IRON. 455 
 
 Iron becomes more compact and sound by being cast under 
 pressure; and hence cast-iron cylinders, pipes, columns, shafts, 
 and the like, are stronger when cast in a vertical than in a 
 horizontal position, and stronger still when provided with a Jiead, 
 or additional column of iron, whose weight serves to compress the 
 mass of iron in the mould below it. The air-bubbles ascend and 
 collect in the head, which is broken off when the casting is cool. 
 
 Care should be taken not to cut or remove the skin of a piece 
 of cast iron more than is absolutely necessary, at those points 
 where the stress is intense. In order that this rule may be carried 
 out in pieces (such as toothed wheels) which are shaped to an 
 accurate figure by cutting or abrading tools, care should be taken to 
 make them as nearly as practicable of the true figure by casting 
 alone, so that the depth of skin to be cut away may be as small 
 as possible. 
 
 391. Wrought or malleable iron in its perfect condition is sim- 
 ply pure iron. It falls short of that perfect condition to a greater 
 or less extent owing to the presence of impurities, of which the 
 most common and injurious have been mentioned, and their effects 
 stated, in Article 387, page 451 ; and its strength is in general 
 greater or less according to the greater or less purity of the ore 
 and fuel employed in its manufacture. 
 
 Malleable iron may be made either by direct reduction of the 
 ore or by the abstraction of the carbon and various impurities 
 from pig iron, mainly by means of oxygen. The latter is the 
 more common process ; and the ordinary method of carrying it on, 
 by stirring the iron in a reverberatory furnace, is called puddling. 
 The oxygen which carries off the carbon in the process of puddling 
 comes partly from the air and partly from a bed of cinder and 
 oxide of iron, called the fettling, with which the bottom of the 
 furnace is covered. The bloom, or lump of iron drawn from the 
 puddling furnace, is hammered, to drive out the cinder with which 
 it is mixed a compound of silica and protoxide of iron ; it is then 
 rolled into bars, which are cut into lengths, fagotted into bundles, 
 re-heated, and re-rolled, until bars are obtained of the required 
 dimensions. The jibrous structure of bar iron is owing to the 
 process of fagotting and rolling, by which it is made. In Mr. 
 Bessemer' s process a blast of air is blown through the molten 
 iron, in a large vessel or retort, until the carbon and silicon are 
 oxidized and removed. 
 
 Strength and toughness in bar iron are indicated by a fine, 
 close, and uniform fibrous structure, free from all appearance of 
 crystallization, with a clear, bluish-gray colour and silky lustre 
 on a torn surface where the fibres are shown. 
 
 Plate iron of the best kind consists of alternate layers of fibres 
 crossing each other. It should have a hard, smooth skin, some-
 
 456 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 what glossy, and when broken, should show perfect uniformity of 
 structure, and be free from all tendency to split into layers. 
 
 To examine the internal structure of iron, whether in bars or 
 in plates, a short piece may be notched on one side, near the 
 middle, and bent double. During this process the uncut part 
 should not break ; and if the iron gives way at all, it should do 
 so by splitting along the fibres near the bottom of the notch. 
 The fitness of bar iron for structures, machines, and srnithwork 
 of different kinds, is tested by bending and punching it cold, and 
 by punching and forging it hot, so as to ascertain whether it shows 
 any signs of brittleness either when cold or when hot (called 
 " cold-short" and " red-short").* 
 
 Malleable iron is distinguished by the property of welding : two 
 pieces, if raised nearly to a white heat, and pressed or hammered 
 firmly together, adhering so as to form one piece. In all opera- 
 tions of which welding forms a part, such as rolling and forging, 
 it is essential that the surfaces to be welded should be brought 
 into close contact, and should be perfectly clean and free from 
 oxide of iron, cinder, and all foreign matter. 
 
 In all cases in which several bars are to be fagotted and ham- 
 mered, or rolled into one, attention should be paid to the manner 
 in which they are " piled" or built together, so that the pressure 
 exerted by the hammer or the rollers may be transmitted through 
 the whole mass. If this be neglected, the finished bar, plate, or 
 other piece, may show flaws, marking the divisions between the 
 bars of the pile. 
 
 Wrought iron, although it is at first made more compact and 
 strong by relieating and hammering, or otherwise working it, soon 
 reaches a state of maximum strength; after which all reheating 
 and working rapidly make it weaker. Good bar iron has in gen- 
 eral attained its maximum strength; and, therefore, in all opera- 
 tions of forging it, whether on a great or small scale, by the 
 steam-hammer or by that in the hand of the blacksmith, the 
 desired size and figure ought to be given with the least possible 
 amount of reheating and working. 
 
 It is of great importance to the strength of all pieces of forged 
 iron that the continuity of the fibres near the surface should be as 
 little interrupted as possible; in other words, that the fibres near 
 the surface should lie in layers parallel to the surface, t 
 
 * For full information as to the tests to which iron and steel are subjected 
 by the Admiralty regulations, reference may be made to Chapter xviii of the 
 Treatise on Shipbuilding, by E. J. Reed, Esq., C.B., Chief Constructor of the 
 Royal Navy. 
 
 t On this subject, see a Paper by the Author of this work, in the Proceed- 
 ings of the Institution of Civil Engineers for 1843. See also the Transactions 
 Of the Institution of Engineers in Scotland for 1862-63, pages 37, 41, 43.
 
 MALLEABLE IRON STEEL AND STEELY IRON. 457 
 
 Another important pi'inciple in designing pieces of forged iron 
 which are to sustain shocks and vibrations, is to avoid as much as 
 possible abrupt variations of dimensions, and angular figures, 
 especially those with re-entering angles; for at the points where 
 such abrupt variations and angles occur, fractures are apt to 
 commence. If two parts of a shaft, for example, or of a beam 
 exposed to shocks and vibrations, are to be of different thicknesses, 
 they should be connected by means of curved surfaces, so that the 
 change of thickness may take place gradually, and without re- 
 entering angles. 
 
 392. Steel and steely iron. Steel is a compound of iron with 
 from - 5 to 1-5 per cent, of its weight of carbon. These, according 
 to most authorities, are the only essential constituents of steel. 
 
 The term " steely iron" or " semi-steel " may be applied to 
 compounds of iron with less than 0-5 per cent, of carbon. They 
 are intermediate in hardness and other properties between steel 
 and malleable iron. 
 
 In general such compounds are the harder and the stronger, and 
 also the more easily fusible, the more carbon they contain. Those 
 kinds which contain less carbon, though weaker, are more easily 
 welded and forged, and from their greater pliability, are the fitter 
 for pieces that are exposed to shocks. 
 
 Impurities of different kinds affect steel injuriously in the same 
 way with iron. 
 
 There are certain foreign substances which have a beneficial 
 effect on steel. One 2,000th part of its weight of silicon causes 
 molten steel to cool and solidify without bubbling or agitation; 
 but a larger proportion is not to be used, as it would make the 
 steel brittle. The presence of manganese in the iron, or its intro- 
 duction into the crucible or vessel in which steel is made, improves 
 the steel by increasing its toughness and making it easier to weld 
 and forge. 
 
 Steel is distinguished by the property of tempering ; that is to 
 say, it can be hardened by sudden cooling from a high temperature, 
 and softened by gradual cooling; and its degree of hardness or 
 softness can be regulated with precision by suitably fixing that 
 temperature. The ordinary practice is, to bring all articles of 
 steel to a high degree of hardness by sudden cooling, and then to 
 soften them more or less by raising them to a temperature which 
 is the higher the softer the articles are to be made, and letting 
 them cool very gradually. The elevation of temperature previous 
 to the annealing or gradual cooling is produced by plunging the 
 articles into a bath of a fusible metallic alloy. The temperaure 
 of the bath ranges from 430 to 560 Fahr. 
 
 According to the experiments of Mr. Kirkaldy, a great increase 
 of strength is produced by hardening steel in oil.
 
 458 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 Steel is made by various processes, which have of late become 
 very numerous. They may all be classed under two heads viz., 
 adding carbon to malleable iron, and abstracting carbon from cast 
 iron. The former class of processes, though the more complex, 
 laborious, and expensive, is preferred for making steel for cutting 
 tools and other fine purposes, because of its being easier to obtain 
 malleable iron than cast iron in a high state of purity. The latter 
 class of processes is well adapted for making great masses of steel 
 and steely iron rapidly and at moderate expense. The following 
 are some of the different kinds of steel, and the processes by which 
 they are made : 
 
 Blister Steel is made by a process called " cementation" which 
 consists in imbedding bars of the purest wrought iron (such as 
 that manufactured by charcoal from magnetic iron ore) in a layer 
 of charcoal, and subjecting them for several days to a high tem- 
 perature. Each bar absorbs carbon, and its surface becomes 
 converted into steel, while the interior is in a condition interme- 
 diate between steel and iron. Cementation x may also be performed 
 by exposing the surface of the iron to a current of carburetted 
 hydrogen gas at a high temperature. Cementation is sometimes 
 applied to the surfaces of articles of malleable iron, in order 
 to give them a skin or coating of steel, and is called " case- 
 hardening." 
 
 Shear Steel is made by breaking bars of blister steel into lengths, 
 making them into bundles or fagots, and rolling them out at a 
 welding heat, and repeating the process until a near approach to 
 uniformity of composition and texture has been obtained. It is 
 used for various tools and cutting implements. 
 
 Cast Steel is made by melting bars of blister steel in a crucible, 
 along with a small additional quantity of carbon (usually in the 
 form of coal-tar) and some manganese. It is the purest, most 
 uniform, and strongest steel, and is used for the finest cutting 
 implements. 
 
 Another process for making cast steel, but one requiring a 
 higher temperature than the preceding, is to melt bars of the 
 purest malleable iron with manganese, and with the whole 
 quantity of carbon required in order to form steel. The quality 
 of the steel, as to hardness, is regulated by the proportion of 
 carbon. A sort of semi-steel, or steely iron, made by this process, 
 and containing a small proportion of carbon only, is known as 
 homogeneous metal. 
 
 The making of large masses of steel by adding the proper 
 ingredients to liquid malleable iron has been much facilitated by 
 the use of Siemens's regenerative furnace, which enables a very 
 high temperature to be kept up, with an ease and economy 
 unknown before.
 
 459 
 
 Steel macfe by the air-blast is produced from molten pig iron by 
 Mr. Bessemer's process. In the first place, the carbon is removed 
 by the air-blast, so that the vessel is full of pure malleable iron in 
 the melted state; and then carbon is added in the proper propor- 
 tion, along with manganese and silicon. The usual way of adding 
 the carbon is by running into the vessel a sufficient quantity of a 
 compound called " spiegeleisen," consisting of highly carbonized 
 cast iron and manganese. The steel thus produced is run into 
 large ingots, which are hammered and rolled like blooms of 
 wrought iron. 
 
 Puddled Steel is made by puddling pig iron, and stopping the 
 process at the instant when the proper quantity of carbon remains. 
 The bloom is shingled and rolled like bar iron. 
 
 The broken surface of a piece of steel shows a mass of veiy 
 small crystalline grains, finer than those of cast iron. Uniformity 
 in the size and colour of the grains is a mark of good steel ; and 
 the smaller they are, the finer and the harder is its quality. In 
 fine cast steel the grains are so small as not to be separately distin- 
 guishable by the naked eye ; and the fracture presents a smooth 
 but dull surface, of an uniform slate-gray colour. 
 
 As to expansion by heat, see page 326. 
 
 393. Strength of Wrought Iron and Steel. The numerical results 
 
 of experiments on the strength of wrought iron and steel will be 
 found in the tables between this chapter and the next. 
 
 Wrought iron, like fibrous substances in general, is more tena- 
 cious along than across the fibres ; and its tenacity, or resistance to 
 tearing asunder, is greater than its resistance to crushing, except 
 when in the form of blocks whose lengths are less than, or but 
 little greater than, their diameters. 
 
 The ductility of wrought iron often causes it to yield by degrees 
 to a load, so that it is difficult to determine its strength with pre- 
 cision. 
 
 Wrought iron has its longitudinal tenacity considerably in- 
 creased by rolling and wire-drawing; so that the smaller sizes of 
 bars are on the whole more tenacious than the larger; and iron 
 wire is more tenacious still, as is shown in the Tables. 
 
 Wrought iron is weakened by too frequent reheating and 
 forging; so that, even in the best of large forgings, the tenacity is 
 only about three-fourths of that of the bars from which the forgings 
 were made, and sometimes even less. 
 
 The strength both of iron and steel is injured by the action of 
 tools which overstrain the particles in the neighbourhood of the 
 portion of material which they remove, and especially by punch- 
 ing. In the case of steel, the strength lost through punching is 
 partially, but not wholly, restored by annealing. The drilling of 
 holes has no such weakening effect.
 
 460 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 Plate iron is somewhat less tenacious crosswise than lengthwise; 
 but the difference ought not to exceed about one-tenth. 
 
 For details as to co-efficients of strength in iron and steel, refer- 
 ence must be made to the tables of the next chapter ; but the 
 following short table gives a condensed view of the values of the 
 ultimate tenacity which ought to be shown by really good bars and 
 plates of iron and steel, fit to be used as materials in making 
 machinery : 
 
 _ .. Kilogrammes 
 
 rp Tnrt? On the S 4 Uare 
 
 Square Inch. imHl S 
 
 Iron, large forgings, from 40,000 28 
 
 to 50,000 35 
 
 Iron Plates, lengthwise, from 50,000 35 
 
 to 60,000 42 
 Do. crosswise, at least 90 per 
 
 cent, of tenacity lengthwise. 
 
 Iron Bars and Rods, from\ 55,000 39 
 
 to 65,000 46 
 
 Do., rivet iron, at least, 60,000 42 
 
 IronWire, 90,000 63 
 
 Mild Steel, from 70,000 49 
 
 to 90,000 63 
 
 Hard Steel, from 90,000 63 
 
 to 110,000 77 
 
 Hardest Cast Steel, 130,000 91 
 
 It is higlily important also that the iron and steel of which 
 pieces exposed to shocks and vibrations are to be made should 
 possess toughness; and this may be tested by observing in what 
 proportion the length of the piece is increased at the instant before 
 breaking. The ultimate elongation of really good and tough 
 specimens of iron and steel, as ascertained in Mr. Kirkaldy's 
 experiments, was nearly as follows, in fractions of the original 
 length : 
 
 Bar Iron, from 0-15 to 0-30 
 
 Plate Iron, lengthwise, from 0-04 to 0-17 
 
 Do. crosswise, from 0-015 to o - n 
 
 Steel Bars, from ox>5 to 0-19 
 
 Steel Plates, from 0-03 to 0-19 
 
 394. Preset-ration of iron. Continual motion, especially of a 
 vibratory kind, tends to prevent the rusting of iron and steel;* 
 hence most of the moving pieces in machinery have little or no 
 
 * See Mallet, " On the Corrosion of Iron," in the Reports of the British 
 -Association for 1843 and 1849.
 
 IRON STEEL VARIOUS METALS. 461 
 
 need of any special means of protection, except shelter from the 
 weather and proper care in keeping them clean. But the frame- 
 work of machines may often require some protection against cor- 
 rosion. The corrosion of iron is a sort of slow combustion, during 
 which the iron combines with oxygen, and produces rust. The 
 ordinary methods of preserving iron consist principally in prevent- 
 ing the access of oxygen to the metal. 
 
 Cast iron will often last for a long time without rusting, if care 
 be taken not to injure its skin, which is usually coated with a film 
 of silicate of the protoxide of iron, produced by the action of the 
 sand of the mould on the iron. Chilled surfaces of castings are 
 without that protection, and therefore rust more rapidly. 
 
 The corrosion of iron is more rapid when partly wet and partly 
 dry, than when wholly immersed in water or wholly exposed to 
 the air. It is accelerated by impurities in water, and especially 
 by the presence of decomposing organic matter or of free acids. It 
 is also accelerated by the contact of iron with any metal which is 
 electro-negative relatively to the iron, or, in other words, has less 
 affinity for oxygen (such as copper), or with the rust of the iron 
 itself. If two portions of a mass of iron are in different condi- 
 tions, so that one has less affinity for oxygen than the other, the 
 contact of the former makes the latter oxidate more rapidly. In 
 general, hard and crystalline iron is less rapidly oxidable than 
 ductile and fibrous iron. Cast iron and steel decompose rapidly in 
 warm or impure sea-water. 
 
 The following are amongst the ordinary methods of preserving 
 iron : 
 
 I. Boiling in coal-tar, especially if the pieces of iron have first 
 been heated to the temperature of melting lead. 
 
 II. Heating the pieces of iron to the temperature of melting 
 lead, and smearing their surfaces, while hot, with cold linseed oil, 
 which dries and forms a sort of varnish. 
 
 III. Painting with oil paint, which must be renewed from time 
 to time. The linseed oil process is a good preparation for paint- 
 ing. 
 
 IV. Coating with zinc, commonly called " galvanizing." This 
 is efficient, provided it is not exposed to acids capable of dissolving 
 the zinc ; but it is destroyed by sulphuric acid in the atmosphere 
 of places where much coal is burned. It lasts well at sea. 
 
 V. Coating with tin, applied to thin sheet iron. 
 
 SECTION II. Of Various Metals and Alloys. 
 
 395. zinc Tin Lead Copper. These are the metals which, 
 next to iron, occur most frequently in machinery ; but, owing to 
 their softness, none of them are suited, in a pure state, for frame-
 
 462 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 work or for moving pieces. It is by compounding them, so as to 
 form alloys, that sufficient hardness is obtained. Zinc, lead, and 
 copper are used for vessels to hold liquids which would corrode 
 iron, and for flexible tubes; zinc and tin, as already mentioned, 
 are used for coating iron, to preserve it ; copper, having great 
 tenacity when rolled and hammered, is used for making boilers 
 into which substances are to be introduced which would be in- 
 jurious to iron, or be injured by it; also for making rivets for 
 leather driving-belts. 
 
 As to the heaviness of those metals, see pages 327, 328; as to 
 their expansion by heat, see pages 326, 327 ; as to their strength, 
 see the tables of the next chapter. Their melting points are as 
 follows : 
 
 Fahrenheit. Centigrade. 
 
 Tin, 426 219 
 
 Lead, 630 332 
 
 Zinc, about 700 about 370 
 
 Copper, about 2550, about 1400 
 
 396. Bronze and Brass are the names given to alloys of copper 
 with tin and with zinc. The name brass, in common language, is 
 applied to such alloys indiscriminately; but, strictly speaking, 
 bronze is the proper name of the alloys of copper with tin ; brass, 
 that of the alloys of copper with zinc. 
 
 Bronze is at least equal to copper in tenacity, and is consider- 
 ably superior in hardness and resistance to crushing. Brass is 
 inferior to copper in strength. Both bronze and brass make good 
 castings, which quality is not possessed by copper. 
 
 These properties render bronze and brass (and especially bronze, 
 where strength is required) suitable both for framework and for 
 moving pieces in machinery. 
 
 Bronze is used, in particular, for the bushes or bearings of rotat- 
 ing shafts, because it has the hardness requisite for durability, and 
 at the same time is not so hard and durable as iron. This latter 
 quality ensures that the shaft shall not be worn by the bearing, 
 but the bearing by the shaft. 
 
 As zinc is cheaper than tin, alloys of copper with zinc are pre- 
 ferable to those of copper with tin in those cases in which strength 
 and durability are of secondary importance. 
 
 The following general principle should be observed in the manu- 
 facture of all alloys whatsoever, as being essential to the soundness, 
 strength, and durability of the compound metal : The quantities 
 of the constituents should bear definite atomic proportions to each 
 other. 
 
 For example, the chemical equivalents of copper, tin, zinc, and 
 lead bear to each other the following proportions :
 
 ALLOYS. 463 
 
 Copper. Tin. Zinc. Lead. 
 
 63.5 118 65.2 207 
 
 and the proportions in which they are combined in any alloy should 
 be expressed by multiples of those numbers. 
 
 When this rule is not observed, the metal produced is not a 
 homogeneous compound, but a mixture of two or more different 
 compounds in irregular masses, shown by a mottled appearance 
 when broken; and those masses being different in expansibility 
 and elasticity, tend to separate from each other ; and being different 
 in chemical composition, they produce electric circuits and promote 
 corrosion. 
 
 The following is a list of the most useful alloys of copper with 
 tin and zinc : 
 
 COMPOSITION. 
 BY EQUIVALENTS. BY WEIGHT. 
 
 ' * ' ' * ALLOYS OF COPPER WITH TIN. 
 
 Copper. Tin. Copper. Tin. 
 
 12 i 762 118 Very hard bronze. 
 
 14 i 889 118 Hard bronze for machinery bearings. 
 
 ,- e a ( Bronze, or gun-metal : contracts in cool- 
 
 io i 1016 118 < f & ., ,,. -j.1 
 
 I ing from its melting point, TTO. 
 
 18 i 1143 118 Bronze somewhat softer. 
 
 20 i 1270 118 Soft bronze for toothed wheels, &c. 
 
 COMPOSITION. 
 Br EQUIVALENTS. BY WEIGHT. 
 
 ' * > ' * > ALLOYS OF COPPEK WITH ZINC. 
 
 Copper. Zine. Copper. Zinc. 
 
 4 i 254 65.2 Malleable brass. 
 
 ,. ( Ordinary brass : melting point, 1869 
 ' \ Fahr. : contracts in cooling, ^V 
 
 ( Yellow metal for sheathing and fasten- 
 
 3 2 190.5 no.4 < c i . 
 
 * { ings of ships. 
 
 -c ( Spelter-solder, for brazing copper and 
 
 4 Q 254 19^.6 { ^. 
 
 ^ J \ iron. 
 
 Various alloys of copper, tin, and zinc are used in machinery, 
 and may be regarded as modifications of true bronze, produced by 
 substituting one or two equivalents of zinc for tin. They are less 
 expensive than true bronze, but not so tough. 
 
 397. other Alloys. The strongest of all alloys yet known is 
 Aluminium Bronze, as a reference to the tables of the strength of 
 metals will show. Different sorts contain from 5 to 10 per cent, 
 of aluminium, and from 95 to 90 per cent, of copper; and if 31 '5 
 be taken as the equivalent of copper, and 137 as that of aluminium, 
 their atomic constitution is probably from 8 to 4 equivalents of 
 copper to 1 equivalent of aluminium.
 
 464 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 Alloys of copper with lead, called pot-metal, are used for cocks 
 and valves where strength is unimportant ; but they are weak and 
 brittle ; and in bronze for bearings, lead is an adulteration. 
 
 Soft Metal, or Babbitt's Metal, consists of 50 parts of tin, 1 of 
 copper, and 5 of antimony. It may be considered as a sort of 
 metallic grease. It is used to make bearings for heavily loaded 
 shafts, in the following way : A bronze or cast-iron bush is pre- 
 pared, with a recess about a quarter of an inch deep in its bearing- 
 surface, bounded at the ends by ledges, to prevent the soft metal 
 from escaping; the soft metal in a melted state is run into that 
 recess, either round a core of the shape and size of the journal or 
 round the journal itself. 
 
 Soft Solder, used for soldering tin-plate, when of the best 
 quality, is a compound of 4 equivalents of tin to 1 of lead; or by 
 weight, very nearly 2 parts of tin to 1 of lead. It melts at 360 
 Fahr. Its ultimate tenacity is about 7,500 Ibs. on the square inch. 
 
 SECTION III. Of some Stony ^Materials. 
 
 398. Stone Bearings for Shafts have occasionally been used. The 
 natural stones fit for this purpose are those which are wholly free 
 from grittiness, and are somewhat inferior in hardness to iron; 
 such as gypsum, pure clay slate, pure compact limestone and 
 marble, and silicate of magnesia, or soapstone, the last being the 
 best. Stones containing crystals of quartz, such as sandstone, 
 sandy limestones and slates, &c., are not suitable. A material 
 called adamas is sometimes used for bearings : it consists of silicate 
 of magnesia ground, calcined, moulded by hydraulic pressure into 
 blocks of suitable figures, and baked. The advantage of silicate of 
 magnesia consists in its combining a certain greasiness of surface 
 with a degree of hardness sufficient for durability. 
 
 SECTION IV. Of Wood and other Organic Materials. 
 
 399. structure of Wood. Wood is the material of trees belong- 
 ing almost exclusively to that class of the vegetable kingdom in 
 which the stem grows by the formation of successive layers of 
 wood all over its external surface, and is therefore said by botanists 
 to be exogenous. 
 
 The tissues of which wood consists are distinguished into two 
 kinds cellular tissue, consisting of clusters of minute cells; and 
 vascular tissue, or woody fibre, consisting of bundles of slender 
 tubes, the latter being distinguished from the former by its fibrous 
 appearance. The difference, however, between those two kinds of 
 tissue, although very distinct both to the eye and to the touch, is 
 really one of degree rather than of kind ; for the fibres or tubes of
 
 STRUCTURE OI WOOD. 465 
 
 vascular tissue are simply very much elongated cells, tapering to 
 points at the ends, and breaking joint with each other. 
 
 The tenacity of wood when strained along the grain depends on 
 the tenacity of the walls of those tubes or fibres ; the tenacity of 
 wood when strained across the grain depends on the adhesion of 
 the sides of the tubes and cells to each other. Examples of the 
 difference of strength in those different directions are given in the 
 tables. 
 
 When a woody stem is cut across, the cellular and vascular 
 tissues are seen to be arranged in the following manner : 
 
 In the middle of the stem is the pith, composed of cellular tissue, 
 inclosed in the medullary sJteath, which consists of vascular tissue 
 of a particular kind. From the pith there extend, radiating 
 outwards to the bark, thin partitions of cellular tissue, called 
 medullary rays; between these, additional medullary rays extend 
 inwards from the bark, to a greater or less distance, but without 
 penetrating to the pith. 
 
 When the medullary rays are large and distinct, as in oak, they 
 are called " silver grain." 
 
 Between the medullary rays lie bundles of vascular tissue, forming 
 the woody fibre, arranged in nearly concentric rings or layers round 
 the pith. In most cases each ring is the result of a year's growth 
 of the tree. These rings are traversed radially by the medullary 
 rays. The boundary between two successive rings is marked more 
 or less distinctly by a greater degree of porosity, and by a difference 
 of hardness and colour. 
 
 The rings are usually thicker at that side of the tree which has had 
 most air and sunshine, so that the pith is not exactly in the centre. 
 
 The wood of the entire stem may be distinguished into two 
 parts the outer and younger portion, called " sap-wood" being 
 softer, weaker, and less compact, and sometimes lighter in colour 
 than the inner and older portion, called "heart-wood." The heart- 
 wood is alone to be employed in those structures and machines in 
 which strength and durability are required. 
 
 The number of rings of sap-wood ranges from five to forty and 
 upwards in different sorts of wood, and is greatest in trees of the 
 pine and fir kind. 
 
 The structure of a branch is similar to that of the trunk from 
 which it springs, except as regards the difference in the number of 
 annual rings, corresponding to the difference of age. A branch 
 becomes partially imbedded in those layers of the trunk which are 
 formed after the time of its first sprouting; it causes a perforation 
 in those layers, accompanied by distortion of the fibres, and consti- 
 tutes what is called a knot. (On various matters mentioned in 
 this Article, see Balfour's Manual of Botany, Part I., chapters 
 i. and ii.) 
 
 2n
 
 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 400. Clarification of Wood. For mechanical purposes, trees may 
 be classed according to the structure of the wood ; and upon a com- 
 parison of that structure in different kinds of trees, a division into 
 two great classes at once suggests itself, which exactly corresponds 
 with a botanical division, viz. : 
 
 PINE- WOOD, comprising all timber trees belonging to the coni- 
 ferous order; and 
 
 LEAF- WOOD, comprising all other timber trees. 
 
 Beyond this primary division, the place of a tree in the botanical 
 system has little or no connection with the structure of its timber. 
 
 In the following table those two great classes are subdivided 
 according to a system proposed by Tredgold, founded, in the first 
 place, on the greater or less distinctness of the medullary rays : 
 
 CLASS I. Pine-Wood. (Natural order Coniferce.) 
 
 Examples: Pine, Fir, Larch, Cowrie, Yew, Cedar, 
 Juniper, Cypress, &c. 
 
 CLASS II. Leaf- Wood. (Non-coniferous trees.) 
 
 Division 1. With distinct large medullary rays. 
 
 (The trees in this division form part of the 
 natural order Amentacece.) 
 
 Subdivision 1. Annual rings distinct. 
 Example : Oak. 
 
 Subdivision 2. Annual rings indistinct. 
 
 Examples : Beech, Plane, Sycamore, &c. 
 
 Division 2. Without distinct large medullary rays. 
 
 Subdivision 1. Annual rings distinct. 
 Examples : Chestnut, Ash, Elm, <fec. 
 
 Subdivision 2. Annual rings indistinct. 
 
 Examples : Mahogany, Walnut, Box, Teak, Green- 
 heart, Mora, Lignum-vitae, <fcc. 
 
 The chief practical bearings of the foregoing classification are 
 as follows : 
 
 Pine-wood, or coniferous timber, in most cases contains turpen- 
 tine. It is distinguished by straightness in the fibre and regularity 
 in the figure of the trees; qualities favourable to its use for long 
 pieces in framework. At the same time, the lateral adhesion of 
 the fibres is small, so that it is much more easily shorn and split 
 along the grain, or torn asunder across the grain, than leaf-wood; 
 and is therefore less fitted to resist thrust or shearing stress, or 
 any kind of stress that does not act along the fibres. Even the 
 toughest kinds of pine-wood are easily wrought ; and this quality,
 
 TIMBER. 467 
 
 combined with lightness and stiffness, makes certain kinds, such 
 as deal, specially well suited for making patterns for large castings. 
 A peculiar characteristic of pine-wood (but one which requires 
 the microscope to make it visible) is that of having the vascular 
 tissue "punctated;" that is to say, there are small lenticular 
 hollows in the sides of the tubular fibres. This structure is prob- 
 ably connected with the smallness of the lateral adhesion of 
 those fibres to each other. Pine-wood is, on the whole, inferior to 
 leaf-wood for works of carpentry and machinery in exposed 
 situations; because the strong kinds (as pine and fir) are deficient 
 in durability; and the durable kinds (as cedar and cypress) are 
 deficient in strength. 
 
 In Leaf-wood, or non-coniferous timber, there is no turpentine. 
 The degree of distinctness with which the structure is seen, whether 
 as regards medullary rays or annual rings, depends on the degree 
 of difference of texture of different parts of the wood. Such 
 difference tends to produce unequal shrinking in drying ; and 
 consequently those kinds of wood in which the medullary rays and 
 the annual rings are distinctly marked, are more liable to warp 
 than those in which the texture is more uniform. At the same 
 time, the former kinds of wood are, on the whole, the more 
 flexible, and in many cases are very tough and strong, which 
 qualities make them suitable for pieces that have to bear shocks. 
 
 401. Appearance of Good Timber. There are certain appear- 
 ances which are characteristic of strong and durable wood, to what 
 class soever it belongs. In the same species of wood, that specimen 
 will in general be the strongest and the most durable which has 
 grown the slowest, as shown by the narrowness of the annual 
 rings. 
 
 The cellular tissue, as seen In the medullary rays (when visible), 
 should be hard and compact. 
 
 The vascular or fibrous tissue should adhere firmly together, and 
 should show no woolliness at a freshly-cut surface; nor should it 
 clog the teeth of the saw with loose fibres. 
 
 If the wood is coloured, darkness of colour is in general a sign 
 of strength and durability. 
 
 The freshly-cut surface of the wood should be firm and shining, 
 and should have somewhat of a translucent appearance. A dull, 
 chalky appearance is a sign of bad timber. 
 
 In wood of a given species, the heavier specimens are in general 
 the stronger and the more lasting. 
 
 Amongst resinous woods, those which have least resin in their 
 pores, and, amongst non-resinous woods, those which have least 
 sap or gum in them, are in general the strongest and most last- 
 ing. 
 
 Timber should be free from such blemishes as " clefts," or cracks
 
 468 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 radiating from the centre ; " cup-shakes," or cracks which partially 
 separate one annual layer from another; " upsets," where the fibres 
 have been crippled by compression ; "rind-galls," or wounds in a 
 layer of the wood, which have been covered and concealed by 
 the growth of subsequent layers over them; and hollows, or spongy 
 places, in the centre or elsewhei-e, indicating the commencement of 
 decay. 
 
 402. Example* of pim-Wood. The following are a few ex- 
 amples of timber of this class : 
 
 I. PINE timber is the wood of various species of the genus 
 Pinus, the best being that of the Red Pine, or Scottish Fir (Pinus 
 sylvestris), grown in the north of Europe. This wood is stiff, 
 strong, and straight-grained, and well suited for large framing. 
 
 Pine timber is also obtained from various other species, chiefly 
 North. American, of which, the best are the Yellow Pine (Pinus 
 variabilis) and White Pine (Pinus Strobus). It is softer and less 
 durable than the Red Pine of the north of Europe, but lighter, and 
 can be had in larger logs. 
 
 Timber similar in its properties to the best kinds of pine is 
 produced by the Kauri or Cowrie of New Zealand (Dammara 
 Australis). 
 
 II. WHITE FIR, or DEAL timber of the best kind, is the wood of 
 the Spruce Fir (Abies excelsa), grown in the north of Europe. 
 
 This is an excellent kind of timber for light framing and joiners' 
 work, and is specially well suited for making patterns of machinery. 
 
 Amongst other kinds of spruce fir applied to the same purposes 
 are the North American White Spruce (Abies alba), and Black 
 Spruce (Abies nigra). 
 
 403. Examples of I,enf-Wood with Large Rays. I. OAK timber 
 
 belongs to the first subdivision of Tredgold's system. It is the 
 strongest, toughest, and most lasting of those grown in temperate 
 climates, and is well suited for framing in which strength, tough- 
 ness, and durability are required ; but it has in general the defect, 
 which is a serious one as regards machinery, of being subject to 
 warp. It is obtained from various species or varieties of the 
 botanical genus Quercus. 
 
 The wood of the oak contains gallic acid, which contributes to 
 the durability of the timber, but corrodes iron. Metal fastenings 
 for oak should therefore be of copper, or its alloys; or, if of iron, 
 they should be well coated with zinc. 
 
 The following are examples of trees belonging to Tredgold'a 
 second subdivision : 
 
 II. BEECH (Fagus sylvatica), common in Europe. 
 
 III. American PLANE (Plalanus occidentalis), common in North 
 America. 
 
 IV. SYCAMORE (Acer pseudo-platanus), also called Great Mapble,
 
 EXAMPLES OP WOOD. 4G9 
 
 and in Scotland and the north of England, Plane; common in 
 Western Europe. 
 
 All these afford compact wood of uniform texture. They are 
 valuable for blocks which have to resist a crushing force. They last 
 well when constantly wet (especially beech), but when alternately 
 wet and dry they decay rapidly. 
 
 404. Examples of Leaf- Wood without Large Rays. The examples 
 
 of timber in this Article belong to the first subdivision of the 
 second division according to Tredgold's system, having no large 
 distinct medullary rays, and having the divisions between the 
 annual rings distinctly marked by a more porous structure. They 
 are in general strong, but flexible; and therefore, in machinery, 
 they are suitable for pieces in which the power of bearing shocks 
 is of more importance than rigidity. 
 
 I. The ASH (Fraxlnus excelsior] furnishes timber whose tough- 
 ness and flexibility render it superior to that of all other European 
 trees for making handles of tools, shafts of carriages, spokes of 
 wooden wheels, and the like; but which is not sufficiently stiff and 
 durable to be used in framing. 
 
 II. The common ELM (Ulmus campestris) and smooth-leaved 
 ELM (Ulmus glabra) yield timber which is very durable when 
 constantly wet, but not when alternately wet and dry. Its strength 
 across the grain, and its resistance to crushing, are comparatively 
 great; and these properties render it useful for some parts of 
 mechanism, such as cogs of wheels and shells of ships' blocks. 
 There are other European species of elm, such as the Wych Elm 
 ( Ulmus montana) ; but their timber is inferior to that of the two 
 species named. 
 
 A North American species, the Rock Elm, is said to be not only 
 durable under water, but straight-grained and tough, so as to be 
 well suited for framing. 
 
 405. Examples of Leaf-Wood without Large Rays continued. 
 
 The kinds of timber mentioned in this Article are examples of the 
 second subdivision of Tredgold's second division, having no large 
 distinct medullary rays, and no distinct difference of compactness 
 in the rings. This uniformity of structure is accompanied by 
 comparative freedom from warping; and hence this subdivision 
 contains various sorts of wood which are specially well adapted 
 both for framing and for moving pieces in machinery, where ac- 
 curacy and constancy of form are required. 
 
 I. MAHOGANY (Swietenia, Mahagoni) is produced in Central 
 America and the West India Islands that of the former region 
 being commonly known as " Bay Mahogany ;" that of the latter, as 
 " Spanish Mahogany." When of good quality, it is very straight- 
 grained, very strong in all directions (though easily split along the 
 grain), very durable, and preserves its shape under varying circum-
 
 470 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 stances as to heat and moisture, better than any other kind of 
 timber which can be procured in equal abundance. Mahogany 
 varies much in quality ; bay mahogany being in general superior to- 
 Spanish mahogany in strength, stiffness, and durability, and in the 
 size of the logs, which are from 24 to 48 inches square. Bay 
 mahogany of good quality is probably the best of all timber for the 
 framing of machinery. Spanish mahogany is the more highly 
 valued for ornamental purposes. Spanish mahogany is distin- 
 guished by having a white chalky substance in its pores, those of 
 bay mahogany being empty. 
 
 II. LIGNUM- VITJE (Guaiacum officinale) is produced in the West 
 India Islands. It is remarkable for heaviness, compactness, tough- 
 ness, and hardness, and for the property of resisting a crushing 
 force with nearly equal strength across and along the grain a 
 pi'operty which makes it specially useful for rollers, sheaves, and 
 other moving pieces in mechanism. In converting logs into- 
 sheaves, the direction of the fibre of the timber is parallel to the 
 axis of the sheave. The heart-wood is yellowish-green, the sap- 
 wood greenish-yellow ; and it is considered advisable, in cutting it 
 into pieces suitable for sheaves, to leave a ring of sap-wood all 
 round the heart-wood, which is thus protected against too rapid 
 drying, and prevented from splitting. 
 
 Properties similar to those of Lignum-vitse are possessed by Box- 
 wood (JSuxus sempervirens), Ebony (Brya ebenus, and other genera 
 and species), Ironwood (Mesua Nagaha), and various other woods, 
 chiefly tropical 
 
 The same subdivision embraces various kinds of timber grown 
 in tropical climates, which are highly valued for shipbuilding pur- 
 poses, and which would be suitable also for the framing of machines 
 such as the Teak (Tectona grandis) and Saul (Shorea robusta) of 
 India, and the Greenheart (Nectandra Hodicei), Mora (Mora ex- 
 cdsa), and Sabicu (Acacia proximo] of South America and the West 
 Indies. 
 
 406. Seasoning. Seasoning timber consists in expelling, as far 
 as possible, the moisture which is contained in its pores. 
 
 Natural Seasoning is performed simply by exposing the timber 
 freely to the air in a dry place, sheltered, if possible, from sun- 
 shine and high winds. The seasoning yard should be paved and 
 well drained, and the timber supported on stone or cast-iron 
 bearers, and piled so as to admit of the free circulation of air over 
 all the surfaces of the pieces. 
 
 Natural seasoning to fit timber for cai'penters' work, usually 
 occupies about two years; for joiners' work and machinery, about 
 four years ; but much longer periods are sometimes employed. 
 
 To steep timber in water for a fortnight after felling it, extracts 
 part of the sap, and makes the drying process more rapid.
 
 SEASONING AND PRESERVATION OP WOOD. 471 
 
 Artificial Seasoning consists in diying the timber in an oven by 
 means of a current of hot air. It occupies from seven to nine 
 days for each inch of the thickness of the piece of timber. 
 
 In the course of drying, timber loses weight and shrinks in its 
 transverse dimensions. The loss of weight ranges in different 
 examples from 6 per cent, to 40 per cent.; and the transverse 
 shrinking from 2 per cent, to 8 per cent., the most common rate 
 being 3 per cent. The sorts of wood which shrink most in 
 drying are the most subject to warp. 
 
 407. Durability, Decay, anil Preset-ration of Wood. All kinds 
 
 of timber are more lasting when kept constantly dry, and at the; 
 same time freely ventilated. 
 
 Timber kept constantly wet is softened and weakened; but it 
 does not necessarily decay. Various kinds of timber, some of 
 which have been already mentioned, such as greenheart, elm and 
 beech, possess great durability in that condition. 
 
 The situation which is least favourable to the duration of timber 
 is that of alternate wetness and dryness, or of a slight degree of 
 moisture, especially if accompanied by heat and confined air. 
 
 Timber exposed to confined air alone, without the presence of 
 any considerable quantity of moisture, decays by " dry rot," which 
 is accompanied by the growth of a fungus, and finally converts the 
 wood into a fine powder. 
 
 Amongst the most efficient means of preserving wood, are good 
 seasoning and the free circulation of air. 
 
 Protection against moisture is afforded by oil paint, provided 
 that the timber is perfectly dry when first painted, and that the 
 paint is renewed from time to time. A coating of pitch or tar 
 may be used for the same purpose. 
 
 Protection against the dry rot may be obtained by saturating 
 the timber with solutions of metallic salts, such as sulphate of iron, 
 sulphate of copper, bichloride of mercury, and chloride of zinc. 
 
 Timber is protected against wet rot, dry rot, and white ants, by 
 saturation with the liquid called commercially " creosote" which is 
 a kind of pitch oil. 
 
 408. strength of Timber. Amongst different specimens of 
 timber of the same species, those which are most dense in the dry 
 state are in general also the strongest. 
 
 Tables of the results of experiments on the strength of different 
 kinds of timber, strained in various ways, are given in the next 
 chapter. 
 
 The following are some general remarks as to the different ways 
 in which the strength of timber is exerted : 
 
 I. The Tenacity along the grain, depending, as it does, on 
 the tenacity of the fibres of the vascular tissue, is on the whole 
 greatest in those kinds and pieces of wood in which those fibres
 
 472 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 are straightest and most distinctly marked. It is not materially 
 aftected by temporary wetness of the timber, but is diminished 
 by long-continued saturation with water, and by steaming and 
 boiling. 
 
 The Tenacity across the grain, depending chiefly on the lateral 
 adhesion of the fibres, is always considerably less than the tenacity 
 along the grain, and is diminished by wetness and increased by 
 dryness. Very few exact experiments have been made upon it. 
 Its smalluess in pine-wood as compared with leaf-wood forms a 
 marked distinction between those two classes of timber, the pro- 
 portion which it bears to the tenacity along the grain having been 
 found to be, by some experiments 
 
 In pine-wood, from l-20th to l-10th. 
 
 In leaf-wood, from l-6th to l-4th and upwards. 
 
 II. The Resistance to Shearing, by sliding of the fibres on each 
 other, is the same, or nearly the same, with x the tenacity across the 
 grain. 
 
 III. The Resistance to Crushing along the grain, depending, as 
 it does, on the resistance of the fibres to being crippled, or " upset," 
 and split asunder, is greatest when their lateral adhesion is greatest, 
 and was found by Mr. Hodgkinson to be nearly twice as great for 
 dry timber as for the same timber in the green state. In most 
 kinds of timber, when dry, it ranges from one-half to two-thirds of 
 the tenacity. 
 
 Experiments have been made on the crushing of timber across 
 the grain, which takes place by a sort of shearing; but they have 
 not led to any precise result, except that timber in general is both 
 more compressible and weaker against a transverse than against a 
 longitudinal pressure; and consequently, that intense transverse 
 compression of pieces of timber ought to be avoided. Certain 
 special kinds of timber are valued for the property of resisting 
 compression across the grain well. Of these the most generally 
 used is lignum-vitse, already mentioned in Article 405, page 470. 
 
 IV. The Modulus of Rupture of timber, which expresses its 
 resistance to cross-breaking, is usually somewhat less than its tenacity; 
 but seldom much less. 
 
 409. ue of Wood in machinery. The following tabular ar- 
 rangement of the more ordinary kinds of wood, according to the 
 purposes in machinery to which they are applicable, is principally 
 based on a similar table given by Holtzapffel in his treatise on 
 Mechanical Manipulation. 
 
 FRAMEWORK. 
 
 Strong, stiff, durable, and free from warping. 
 Mahogany.
 
 tTSE OP WOOD IN MACHINERY. 473 
 
 Strong longitudinally, stiff, and straight-grained. 
 Pine, Deal. 
 
 Strong, tough, and durable. 
 Oak, Teak, Saul. 
 
 Tough and pliable. 
 Ash. 
 
 Strong against pressure. 
 
 Elm (durable when wet), Beecn. 
 
 LEVERS AND CONNECTING-RODS. 
 Strong and stiff. 
 
 Pine, Deal, Mahogany. 
 
 Strong and tough. 
 Oak, Teak. 
 
 Tough and pliable. 
 
 Ash, Hazel, Hickory, Lancewood. 
 
 PULLEYS, SHEAVES, ROLLERS. 
 
 Lignum-vitse, Box, Mahogany. 
 BEARINGS FOR SHAFTS. 
 
 Box, Beech, Holly, Lignum-vitse, Elm. 
 
 When wood is used for bearings, the ends of the fibres should be 
 exposed to the pressure. 
 
 COGS. 
 
 Crabtree, Hornbeam, Locust, Beech. 
 PATTERNS. 
 
 Deal, Mahogany, Pine, Alder. 
 
 In machinery whose speed is liable to be suddenly changed or 
 checked, it is often useful to make some of the parts which trans- 
 mit the motion of wood, although the whole of the remainder mav 
 be of iron; the object being that the wood, by yielding to a shock, 
 may prevent it from damaging the iron ; and also that in the event 
 of breakage occurring, it may take place in the wooden parts, which 
 can be replaced more easily and at less cost than the iron parts. 
 
 For example, the great spur fly-wheel by means of which a 
 steam engine or a water-wheel drives the machinery of a mill is 
 very generally a mortise-wheel; that is to say, a cast-iron wheel 
 with rectangular sockets called mortises in its rim, into which are 
 fitted wooden teeth called cogs. The pinion which those teeth 
 drive is wholly of cast iron. Wooden cogs are made double the 
 thickness of cast-iron teeth that have to bear the same pressure. 
 
 Another instance of the application of the same principle is when,
 
 474 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 in a steam engine that drives an iron rolling mill, the middle part 
 of the thickness of the connecting-rod, which transmits thrust, is 
 made of wood, the tension being transmitted by means of a 
 wrought-iron strap. 
 
 409 A. Pasteboard, composed of layers of paper perpendicular to 
 the pressure, is sometimes used for bearings of shafts. 
 
 410. Organic Materials for Bands, Leather, Gutta-Pcrchn, Indian 
 
 Rubber, Cotton, Flax, and Hemp. I. Leather Belts. The ordinary 
 material for driving-belts in machinery is ox-leather from the 
 back of the animal. It is of a nearly uniform thickness, ranging 
 from ^ to of an inch (from 4 to 6 millimetres). It is to be had 
 in pieces up to 4| or 5 feet long, and about 8 inches broad. The 
 several lengths of leather of which a belt is made are spliced and 
 cemented together, and fastened to each other by means of pins or 
 rivets of copper or of soft brass. The two ends of the belt are 
 connected with each other by a lacing of thongs, or by copper or 
 brass pins. 
 
 A belt is said to be single or double according as it is made of one 
 or of two thicknesses of leather. 
 
 The inside of the leather is rougher than the outside, and is placed 
 next the pulleys ; crossed belts being twisted so as to bring the 
 same side of the leather in contact with both pulleys (fig. 123, page 
 182). 
 
 Leather belts, when new, are not quite of the heaviness of water 
 say about 60 Ibs. per cubic foot; but after having been for some 
 time in use, they become thinner and denser by compression, and 
 are then about as heavy as water. The weight of single belting 
 may be approximately estimated as follows : 
 
 Per foot length and inch breadth, 0-068 Ib. 
 
 Per square metre of surface, 4 kilogrammes. 
 
 The following table shows the results of experiments by Mr. 
 Henry R. Towne on the ultimate tenacity of belts, compared with 
 the practical rule of General Morin as to their safe working tension. 
 The tensions in lengths of belt are calculated from the above 
 estimate of the heaviness. 
 
 TTTTTMITK TVvinrv Lbs. per Kilos, per Feet of Metres of 
 
 In. wide. Mm.w&e. Belt. Belt. 
 
 The solid leather, 675 12 10,000 3,000 
 
 At the rivet-holes of the splices, 382 6-8 5,600 1,700 
 
 At the lacing, 210 375 3,100 940 
 
 SAFE WORKING TENSION (see ) ,, 
 
 page 441), ..... j 45 
 
 II. Raw Hide Belts. The process of tanning, which makes 
 leather durable, impairs its strength ; the tenacity of raw hide
 
 ORGANIC MATERIALS FOR BANDS. 475 
 
 being about once and a half that of tanned leather. When raw- 
 hide is used for belts or for ropes, it is soaked with grease to keep 
 it pliable and protect it against the action of air and moisture. 
 
 III. Gutta-Percha is sometimes used for flat belts. They are 
 made of the same dimensions with leather belts for transmitting the 
 same force, and are nearly of the same weight. 
 
 IV. Woven Belts are made of a flaxen or cotton fabric ; a suffi- 
 cient number of plies being used to give a thickness equal to that 
 of leather belts, and cemented together with indian rubber. When. 
 made of flax, they are said to be about three times more tenacious 
 than tanned leather belts of the same transverse dimensions. 
 
 V. Ropes and Cords, when of organic materials, are made of 
 leather, raw hide, and catgut, and of flax, hemp, and other vege- 
 table fibre. Round cords of leather and of hide are made by 
 twisting strips of those materials into round strands, and spinning 
 or plaiting those strands into ropes. The ultimate tenacity, when 
 the material is of the best kind, may be taken, for leather, as given 
 by the table in the first division of this Article and for raw hide, 
 as one and a half that of leather. Assuming the heaviness, when 
 well twisted, to be equal to that of water, this will give the follow- 
 ing results : 
 
 ULTIMATE TKUArrTT Feet of Mfetres of Lbs. on the K. on the 
 
 ULTIMATE J.BNACITT. Eope Kope _ circular circular 
 
 Inch. Mm. 
 
 Leather, 10,000 3,000 3,360 2-36 
 
 Eaw Hide, 15^ 4>5 00 5>4Q 3'54 
 
 Working Tension 
 factor of safety, 6. 
 
 Leather, I ,^7 5 5^o 0-39 
 
 Raw Hide, 2,500 750 840 0-59 
 
 Hemp, as used in ropes, is spun, or "laid up," with a right- 
 handed twist into yarns. Yarns are laid up left-handed into 
 strands. Three strands laid up right-handed make a hawser; three 
 hawsers laid up left-handed make a cable. Hempen ropes are 
 classed according to the number and arrangement of their strands. 
 The following are the commonest kinds : 
 
 Hawser-laid rope, 3 strands. 
 
 Cable-laid rope = 3 hawsers twisted together, 9 strands. 
 
 Shroud-laid rope = core or heart surrounded by. 4 strands. 
 
 The girth squared is the dimension commonly employed in 
 calculating the weight and strength of hempen ropes. The proof, 
 or testing load, is given by multiplying the girth squared by one of 
 the factors in the following table ; the breaking load is from two to- 
 three times the proof load; the working load is about one-fourth of 
 the proof load : that is, about one-tenth of the breaking load.
 
 7G MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 Multiplier 
 for Proof Load. 
 Lbs. 
 
 Multiplier Proof Strength 
 for Weight. in Length of Kopa. 
 Lb. per Feet. Metre*. 
 100 Fathoms. 
 
 42O 
 336 
 269 
 
 23-1 
 
 21-5 
 
 10,920 
 
 9,000 
 7>5o 
 
 2,740 
 2,290 
 
 Hawser-laid rope, 
 Shroud-laid 
 Cable-laid 
 
 Tarred Hopes have about three-fourths of the strength of white 
 ropes of the same size. 
 
 Wire ropes contain a certain proportion of organic material in 
 the hempen cores round which the wires are spun, but it does not 
 sensibly contribute to their strength, which will be more fully 
 considered further on. 
 
 ADDENDUM TO FOOT-NOTE TO ARTICLE 309, PAGES 349 AND 350. 
 
 From some experiments made by Mr. R. D. Napieiy forming the subject of s paper 
 on friction and unguents, read before the Philosophical Society of Glasgow, on the 16th 
 December, 1874, by that gentleman, and experiments made by an eminent foreigner, 
 but believed to be not yet published , it may be safely deduced that the friction between 
 two bodies is a function of the force with which they are pressed together and of their 
 relative velocity of motion. It is further probable that for substances without unguents, 
 the friction increases with the velocity to a certain maximum, and then diminishes. 
 Mr. Napier believes his experiments shew "that with mineral oils the co-efficient of 
 friction is less at higher than at lower velocities, and that with animal and vegetable 
 oils the reverse is the case ;" and further, with the employment of unguents the friction 
 has been found to increase with the velocity and vice versd. and also to diminish with 
 the velocity and vice versd. A very small co-efficient of friction was found to obtain 
 when a small " quantity of water was allowed to run on the top of oil." As Mr. Napier 
 points out, there is necessity for further experiment.
 
 477 
 
 GENERAL TABLES OF THE STRENGTH OF 
 MATERIALS. 
 
 TABLE OP THE RESISTANCE OF MATERIALS TO STRETCHING AND 
 TEARING BY A DIRECT PULL, in pounds avoirdupois per square 
 incli. 
 
 MATERIALS. 
 
 Tenacity, 
 
 or Resistance to 
 
 Tearing. 
 
 Modulus of 
 
 Elasticity, 
 
 or Resistance to 
 
 Stretching. 
 
 STONES, NATURAL AND ARTIFICIAL : 
 Brick, ) 
 
 Cement, } 28 to 3' 
 
 Glass, 9,400 8,000,000 
 
 glate | 9,600 13,000,000 
 
 ( to 12,800 to 16,000,000 
 
 Mortar, ordinary, 50 
 
 METALS: 
 
 Brass, cast, 18,000 9,170,000 
 
 wire, 49,000 14,230,000 
 
 Bronze (Copper 8, Tin i) r 36,000 9,900.000 
 
 Aluminium, 73,000 
 
 Copper, cast, 19,000 
 
 sheet, 30,000 
 
 bolts, 36,000 
 
 wire, 60,000 17,000,000 
 
 Iron, cast, various qualities, I, I3 ' 4 I4,ooo,ooo 
 
 ( to 29,000 to 22,900,000 
 
 average, 16,500 17,000,000 
 
 Iron, wrought, plates, 51,000 
 
 joints, double ri vetted, 35,7 
 
 single rivetted, 28,600 
 
 , , , ,, f 60,000 ) 
 
 bars and bolts, j to 70^000 / 29,000,000 
 
 hoop, best-best, 64,000 
 
 ( 70,000 ) 
 
 wire, < , > 25,300,000 
 
 ( to 100,000 J 
 
 wire-ropes, 90,000 15,000,000 
 
 Lead, sheet, 3,3o 720,000 
 
 Steel bars, j . IO0 ' 000 . 2 9,ooo,ooo 
 
 ( to 130,000 to 42,000,000 
 
 Steel plates, average, 80,000 
 
 Tin, cast, 4,600 
 
 Zinc,. 7,000 to 8,000
 
 478 
 
 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 MATERIALS. 
 
 Tenacity, 
 
 or Resistance to 
 
 Tearing. 
 
 Modulus of 
 
 Elasticity, 
 
 or Resistance to 
 
 Stretching. 
 
 TIMBER AND OTHER ORGANIC FIBRE: 
 
 Acacia, false. See " Locust." 
 
 Ash (Fraxinus excelsior), 17,000 1,600,000 
 
 Bamboo (Bamhusa arundinacea), 6,300 
 
 Beech (Fagus sylvatica), 11,500 1,350,000 
 
 Birch (Betula alba), 15,000 1,645,000 
 
 Box (Buxus sempervirens), 20,000 
 
 Cedar of lieba.non(CedrusLibani), 11,400 486,000 
 
 Chestnut (Castanea Vesca), { 10,000) IjI4OjOOO 
 
 I TJO 1 2>OOO I 
 
 Elm (Ulmus campestris),.. I 7,ooo 
 
 14,000 ( to 1,340,000 
 
 Fir: Red Pine (Pinus sylvestris), (. I2 ' 1,460,000 
 
 ( to 14,000 to 1,900,000 
 
 Spruce (Abies excelsa),.. L r 4o,ooo 
 
 12,400 ( to 1,800,000 
 
 Larch (LarixEuropcea),... L 9 ' 900,000 
 
 ( to 10,000 to 1,360,000 
 
 Flaxen Yarn, about 25,000 
 
 Hazel (Corylus Avellana), 18,000 
 
 Hempen Ropes, from 12,000 to 16,000 
 
 Hide, Ox, undressed, 6,300 
 
 Hornbeam (Garpinus Betulus),. 20,000 
 
 Lancewood (Guatteria virgata), 23,400 
 
 Leather, Ox, 4,200 24,300 
 
 Lignum- Yitae (Guaiacum qffici- \ g 
 
 Locust (Robinia Pseudo-A coda), 1 6,000 
 
 f 8 ooo ) 
 -tyLanogany(oWi6f^7izctirz aiiagoni), s Q / 1,255,000 
 
 Maple (Acer campestris), 10,600 
 
 Oak, European (Quercus sessili- ( 10,000 1,200,000 
 
 flora&ud Quercus pedunculata), (to 19,800 to 1,750,000 
 
 American Red (Quercus \ 
 
 ru-bra), } IO > 2 5 ^0,000 
 
 Silk fibre, 52,000 1,300,000 
 
 Sycamore(AcerPseudo-Platanus). 13,000 1,040,000 
 
 Teak, Indian (Tectona grandis), 15,000 2,400,000 
 
 African, (?) 21,000 2,300,000 
 
 Whalebone, 7, 700 
 
 Yew (Taxus baccata), 8,000
 
 TABLES OF STRENGTH. 479 
 
 II. 
 
 TABLE OF THE RESISTANCE OF MATERIALS TO SHEARING AND 
 DISTORTION, in pounds avoirdupois per square inch. 
 
 . , Transverse 
 
 MA Resistance Elasticity, 
 
 ,, MATERIALS. to. O r Resistance to 
 
 METALS : Shearing. Distortion. 
 
 Brass, wire-drawn, 5j33>oo 
 
 Copper, 6,200,000 
 
 Iron, cast, 27,700 2,850,000 
 
 wrought, 5o,oool 8 >Soo,ooo 
 
 [to 10,000,000 
 
 TIMBER : 
 
 Fir: Red Pine, 5 ooto 800 I, 6 *' oo 
 
 (to 116,000 
 
 Spruce, 600 
 
 Lai-ch, 97otoi,7oo 
 
 Oak, 2,300 82,000 
 
 Ash and Elm, 1,400 76,000 
 
 III. 
 
 TABLE OF THE RESISTANCE OF MATERIALS TO CRUSHING BY A 
 DIRECT THRUST, in pounds avoirdupois per square inch. 
 
 Resistance 
 MATERIALS. to 
 
 Crushing. 
 
 STONES, NATURAL AND ARTIFICIAL: 
 
 Brick, weak red, 5 go to 800 
 
 strong red, 1,100 
 
 ,, fire, 1,700 
 
 Chalk, 330 
 
 Granite, 6,500 to 11,000 
 
 Limestone, marble, 5;5o 
 
 granular, 4,000 to 4,500 
 
 Sandstone, strong, 5?5 
 
 ordinary, 3,300 to 4,400 
 
 Rubble masonry, about four-tenths of cut stone. 
 
 METALS : 
 
 Brass, cast, 10,300 
 
 Bronze, Aluminium, 132,000 
 
 Iron, cast, various qualities, 82,000 to 145,000 
 
 average, 112,000 
 
 wrought, about 36,000 to 40,000
 
 480 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 Resistance 
 
 MATERIALS. to 
 
 Crushing. 
 TIMBER,* Dry, crushed along the grain : 
 
 Ash, .................................................. 9,000 
 
 Beech, ................................................ 9,360 
 
 Birch, ................................................ 6,400 
 
 Blue-Gum (Eucalyptus Globulus), .............. 8,800 
 
 Box, .................................................. 10,300 
 
 Bullet-tree (Achras Sideroxylon), ............... 14,000 
 
 Cabacalli, ........................................... 9>9oo 
 
 Cedar of Lebanon, ................................. 5 , 860 
 
 Ebony, "West Indian (Brya Ebenus), .......... 19,000 
 
 Elm, .................................................. 10,300 
 
 Fir: Red Pine, .................................... 5,375 to 6,200 
 
 Larch, ................................ . ....... 5,570 
 
 Hornbeam, ......................................... 7,3o 
 
 Lignum- Vitse, ...................................... 9*900 
 
 Mahogany, .......................................... 8,200 
 
 Mora (Mora excelsa), .............................. 9j9 
 
 Oak, British, ....................................... 10,000 
 
 Dantzic, ....................................... 7,7 
 
 American Red, .............................. 6,000 
 
 Teak, Indian, ....................................... 12,000 
 
 Water-Gum (Tristania nerifolia), ............. 11,000 
 
 IV. 
 
 TABLE OF THE RESISTANCE OP MATERIALS TO BREAKING ACROSS, 
 in pounds avoirdupois per square inch. 
 
 Resistance to Breaking, 
 MATERIALS. or 
 
 Modulus of Rupture.f 
 
 STONES: 
 
 Sandstone, 1,100 to 2,360 
 
 Slate, 5,ooo 
 
 The resistances stated are for dry timber. Green timber is much weaker, having 
 sometimes only half the strength of dry timber against crushing. 
 
 f The modulus of rupture is eighteen times the load which is required to break a bar 
 of one inch square, supported at two points one foot apart, and loaded in the middle 
 between the points of support.
 
 TABLES OF STRENGTH. 481 
 
 Resistance to Breaking, 
 MATERIALS. or 
 
 Modulus of Rupture. 
 
 METALS: 
 
 Iron, cast, open- work beams, average, 17,000 
 
 solid rectangular bars, var. qualities, 33,000 to 43,500 
 
 average, 40,000 
 
 wrought, 40,000 to 54,000 
 
 TIMBER: 
 
 Ash, 12,000 to 14,000 
 
 Beech, 9,000 to 12,000 
 
 Birch, 11,700 
 
 Blue-Gum, 16,000 to 20,000 
 
 Bullet-tree, 15,900 to 22,000 
 
 Cabacalli, 15,000 to 16,000 
 
 Cedar of Lebanon, 7>4QO 
 
 Chestnut, 10,660 
 
 Cowrie (Dammara austral'is), 11,000 
 
 Ebony, West Indian, 27,000 
 
 Elm, 6,000 to 9,700 
 
 Fir: Red Pine, 7,100 to 9,540 
 
 Spruce, 9,900 to 12,300 
 
 Larch, 5,000 to 10,000 
 
 Greenheart (Nedandra Eodicei), 16,500 to 27,500 
 
 Lancewood, 17,350 
 
 Lignum- Vitse, 12,000 
 
 LoCUSt, II,2OO> 
 
 Mahogany, Honduras, 11,500* 
 
 Spanish, 7,600^ 
 
 Mora, 22,000* 
 
 Oak, British and Russian, 10,000 to 13,600 
 
 Dantzic, 8,700 
 
 American Red, 10,600- 
 
 Poon, i3>3 
 
 Saul, 16,300 to 20,700 
 
 Sycamore, 9,600 
 
 Teak, Indian, 12,000 to 19,000 
 
 African, 14,980 
 
 Tonka (Dipteryx odorata), 22,000 
 
 "Water-Gum, I7>46o 
 
 "Willow (Salix, various species),..... 6,6co 
 
 "1 i
 
 482 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 V. MISCELLANEOUS SUPPLEMENTARY TABLE, 
 
 Material. 
 
 Dimensions. 
 
 Tearing 
 
 Load, 
 Ibs. 
 
 Length of 
 1 Ib. weight, 
 in feet. 
 
 Tenacity in 
 feet of the 
 Material. 
 
 Cast steel bar 
 
 I in. X I in. 
 
 130,000 
 
 O"2Q7 
 
 38 610 
 
 Charcoal iron wire, 
 
 area i sq. in. 
 
 IOO,OOO 
 
 O"? 
 
 30,000 
 
 Iron wire rope, 
 
 girth 1*27 in. 
 
 4..4.SO 
 
 6-0 
 
 -^6880 
 
 Iron bar strong 
 
 I in X I in 
 
 60.000 
 
 O"? 
 
 18 ooo 
 
 Boiler plate, strong, 
 Teak wood, 
 
 area i sq. in. 
 I in X I in 
 
 5O,OOO 
 I5,OOO 
 
 0-3 
 vo 
 
 15,000 
 
 CA OOO 
 
 Deal, 
 
 I in. X i in. 
 
 I2,OOO 
 
 A'O 
 
 48,OOO 
 
 Hempen hawser, 
 
 girth i in. 
 
 1,050 
 
 26*0 
 
 27,3OO 
 
 Hempenrope,cable-laid, 
 
 girth 10 in. 
 
 67,200 
 
 0-279 
 
 18,750 
 
 VI. SUPPLEMENTARY TABLE FOR WROUGHT IRON AND STEEL. 
 
 Description of Material. 
 
 MALLEABLE IRON. 
 
 Tenacity in Ibs. per Square Inch. 
 Lengthwise. Crosswise. 
 
 Ultimata 
 Extension. 
 
 Wire average, 
 
 86,000 
 
 T. 
 
 
 Wire weak, 
 
 71,000 
 
 Mo. 
 
 
 Yorkshire (Lowmoor), . . . 
 
 64,200 
 
 F. 52,490 
 
 F. 
 
 from 
 
 66,390 ) 
 
 
 f 0-20 
 
 to 
 
 60,075 J 
 
 
 
 ( 0-26 
 
 Yorkshire (Lowmoor) } 
 
 
 
 
 and Staffordshire > 
 
 59,740 
 
 F. 
 
 O'2 to O'25 
 
 rivetiron, j 
 
 
 
 
 Charcoal bar, 
 
 63,620 
 
 F. 
 
 O'2 
 
 Staffordshire bar, . . .from 
 to 
 
 62,231 ) 
 56,7i5j 
 
 N. 
 
 f -302 
 
 \ -186 
 
 Yorkshire bridge iron,... 
 
 49,93 
 
 F. 43,940 
 
 F. -04; -029 
 
 Staffordshire bridge iron, 
 
 47,600 
 
 F. 44,385 
 
 04; -036 
 
 Lanarkshire bar,. . . from 
 
 64,795 \ 
 
 N 
 
 f -158 
 
 to 
 
 5i,327 f 
 
 XI 
 
 V2 3 8 
 
 Lancashire bar, from 
 
 60,110 ) 
 
 
 f -169 
 
 to 
 
 53,775 J 
 
 
 
 \'2l6 
 
 Swedish bar, from 
 
 48,933 1 
 
 
 (-264 
 
 to 
 
 41,25! J 
 
 * 
 
 1-278 
 
 Russian bar, from 
 
 59,096 ) 
 
 "N" 
 
 
 to 
 
 49,564 J 
 
 XI 
 
 1 4I 33 
 
 Bushelled iron from ) 
 turnings, J 
 
 55,878 
 
 N. 
 
 166 
 
 Hammered scrap, 
 
 53,420 
 
 N. 
 
 248 
 
 Angle-iron from ) from 
 
 61,260 ) 
 
 "\r 
 
 
 various districts, J to 
 
 50,056 j 
 
 IS. 

 
 TABLES OF STRENGTH. 
 
 483 
 
 TABLE continued, 
 
 Description of Material. 
 
 Straps from van- ) from 
 ous districts, ... J to 
 Bessemer's iron, cast ) 
 ingot,.. .. f 
 
 Tenacity in Ibs. per Square Inch. Ultimate 
 Lengthwise. Crosswise. Extension. 
 
 55,937 1 xr f -108 
 41,386 / \-o 4 8 
 
 41,242 YV 
 
 Bessemer's iron, ham- ) 
 mered or rolled, J 
 Bessemer's iron, boiler ) 
 
 72,643 
 
 w. 
 
 
 t)late. . . .1 
 
 00,319 
 
 w . 
 
 
 Yorkshire plates,... from 
 to 
 
 58,487 \ 
 52,000 j 
 
 N 55,033) 
 46,221 / 
 
 jq- J ' I0 9; '59 
 \ -170; -113 
 
 Staffordshire plates, from 
 
 56,996 ) 
 
 N 51,251 \ 
 
 jj r4; -034 
 
 to 
 
 46,404 / 
 
 ' 44,764 J 
 
 (13; -059 
 
 Staffordshire plates, ) 
 best-best, charcoal, J 
 
 45,0X0 
 
 F. 41,420 
 
 F. -05; -045 
 
 Staffordshire ) from 
 
 59,820 
 
 F. 54,820 
 
 F. -05; -038 
 
 plates, best-best, j to 
 
 49,945 
 
 F. 46,470 
 
 F. -067; -04 
 
 Staffordshire plates, best, 
 
 61,280 
 
 F. 53,820 
 
 F. -077; -045 
 
 Staffordshire plates, ) 
 common, J 
 
 50,820 
 
 F. 52,825 
 
 F. -05; -043 
 
 Lancashire plates, 
 
 48,86* 
 
 F. 4fj,oiii 
 
 F. '043: '028 
 
 Lanarkshire plates, from 
 
 T-^J^ V O 
 
 53,849 \ 
 
 Tjr**j 
 
 N 48,848) 
 
 T*J, V*.VJ 
 
 N / <0 33; '014 
 
 to 
 
 43,433 j 
 
 ' 39,544 J 
 
 1 *093; "046 
 
 Durham plates, 
 
 ST.24S 
 
 N. 46,712 
 
 *o8o; '064. 
 
 Effects of Reheating and Rolling. 
 
 . 4f V, I ** 
 
 ^^y> wv "r 
 
 Puddled bar, 
 
 4.3.QO4 " 
 
 
 
 The same iron five } 
 
 T^o,y T 
 
 
 
 times piled, reheated > 
 
 61,824 
 
 
 
 and rolled .... 1 
 
 
 OL 
 
 
 The same iron eleven \ 
 
 
 
 
 times piled, reheated V 
 
 43,94 
 
 
 
 and rolled... ,. 1 
 
 
 
 
 Strength of Large Forgings. 
 
 Bars cut out of \ from 47,582 ) 
 large forgings, j to 43,759 J 
 
 Bars cut out of large ) 6 
 forgings, .............. ( ** 
 
 44,5^8 
 36,824 
 
 -231; -168 
 -205; -064
 
 484 
 
 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 TABLE continued. 
 
 Ultimata 
 
 Description of Material. 
 
 Lengthwise. Crosswise. Extension. 
 
 STEEL AND STEELY IRON. 
 
 
 
 
 
 Caststeelbars,rol- ) from 
 
 132,909 ) 
 
 K 
 
 f -052 
 
 
 led and forged, / to 
 
 92,015 j 
 
 
 \-i53 
 
 
 Cast steel bars, rolled ) 
 
 
 
 
 
 and forged,.... J 
 
 130,000 
 
 . 
 
 
 
 Blistered steel bars, ) 
 rolled and forged . . . , J 
 
 104,298 
 
 N. 
 
 097 
 
 
 Shear steel bars, rolled I 
 and forged. . . . . 1 
 
 118,468 
 
 K 
 
 135 
 
 
 Bessemer's steel bars, 1 
 rolled and forged,... ) 
 
 111,460 
 
 N". 
 
 055 
 
 
 Bessemer's steel bars, ) 
 cast ingots J 
 
 63,024 
 
 W. 
 
 N 
 
 
 
 
 Bessemer's steel bars, ) 
 hammered or rolled, J 
 
 152,912 
 
 W. 
 
 
 
 Spring steel bars, ham- ) 
 
 
 N 
 
 180 
 
 
 mered or rolled, J 
 
 7 ,5 9 
 
 * 
 
 
 
 Homogeneous metal { 
 bars, rolled, j 
 
 90,647 
 
 N. 
 
 137 
 
 
 Homogeneous metal ) 
 
 
 
 
 
 v 
 bars, rolled, f 
 
 93,000 
 
 . 
 
 
 
 Homogeneous metal ) 
 
 80 'T2A. 
 
 N. 
 
 it 
 
 
 bars, forsred... .. I 
 
 y>i ^4 
 
 
 9 
 
 
 Puddled steel ) f 
 bars, rolled and V lro 
 forsred. . . 
 
 71,484 ) 
 62,768 / 
 
 N. 
 
 (191 
 ( '091 
 
 
 Puddled steel bars, ) 
 rolled and forged,... j 
 
 90,000 
 
 F. 
 
 
 
 Puddled steel bars, ) 
 rolled and forged,... J 
 
 94,752 
 
 il 
 
 
 
 Mushet's gun-metal, 
 
 103,400 
 
 F. 
 
 0-034 
 
 
 Cast steel plates,.... from 
 to 
 
 96,289 ) 
 75,594 J 
 
 TJ- 97,38 I 
 ' 69,082 j 
 
 N i' 57 ' 
 t -198; 
 
 096 
 196 
 
 Cast steel plates,... hard, 
 
 102,900 \ 
 
 F. 
 
 r-o3i 
 
 
 soft, 
 
 85,400 J 
 
 
 l -031 
 
 
 Homogeneous metal ) 
 plates, first quality, j 
 
 96,280 \ 
 
 N ' I 
 
 ( -086; 
 
 144 
 
 Homogeneous metal ) 
 plates, second quality, j 
 
 72,408] 
 
 1 
 
 I -059; 
 
 032 
 
 Puddled steel ) from 
 plates, j to 
 
 102,593 ) 
 
 XT 85,365) 
 ' 67,686 1 
 
 XT (-028; 
 \ -082: 
 
 013 
 OT? 
 
 Puddled steel plates,.... 
 
 ^600 
 
 F. J 
 
 \ 
 
 0-125 
 
 tJ
 
 CABLES OP STRENGTH 1 . 485 
 
 TABLE continued. 
 
 n,, ,.- ^ xr^a,.!,,! Tenacity in Ibs. per Square Inch. TJltimata 
 
 Lengthwise. Crosswise. Extension. 
 
 Coleford Gun-metal. 
 
 Weakest, 108,970 \ -190 
 
 Strongest, '.. 160,540 > F. '030 
 
 Mean of ten sorts, 137,340) -072 
 
 In the preceding table the following abbreviations are used for 
 the names of authorities : 
 
 C., Clay; F., Fairbairn; H., Hodgkinson; M., Mallet; Mo., 
 Morin; N.,* Napier & Sons; R, Rennie; T., Telford; W., 
 Wilmot. 
 
 The column headed "Ultimate Extension" gives the ratio of the 
 elongation of the piece, at the instant of breaking, to its original 
 length. It furnishes an index (but a somewhat vague one) to the 
 ductility of the metal, and its consequent safety as a material for 
 resisting shocks. 
 
 When two numbers separated by a semicolon appear in the 
 column of ultimate extension (thus -082; '057), the first denotes 
 the ultimate extension lengthwise, and the second crosswise. 
 
 VII. RESILIENCE OF IRON AND STEEL. 
 
 Metal under Tension. 
 
 Cast iron "Weak, 
 
 Ultimate 
 Tenacity. 
 
 I3.4OO 
 
 "Working 
 Tenacity. 
 
 4,4.67 
 
 Modulus 
 of 
 Elasticity. 
 
 14,000 ooo 
 
 Modulus 
 of 
 Resilience. 
 
 I'42'l 
 
 .Average, .. 
 
 i6,c;oo 
 
 fJ.^oo 
 
 I7,OOO,OOO 
 
 I-78 
 
 Stronsf. . . 
 
 20,000 
 
 0,667 
 
 32,900 ooo 
 
 4'08 
 
 Bar iron Good average, .. 
 Plate iron Good average, 
 Iron wire Good average, 
 Steel Soft, 
 
 60,000 
 50,000 
 90,000 
 
 QO.OOO 
 
 2O,OOO 
 16,667 
 3O,OOO 
 3O,OOO 
 
 29,OOO,OOO 
 24,OOO,OOO? 
 25,300,000 
 
 29,000 ooo 
 
 I 379 
 
 11-57? 
 
 35*57 
 
 3l'O3 
 
 Hard. . . 
 
 iq2,OOO 
 
 44.OOO 
 
 42,000.000 
 
 46-10 
 
 In the above Table of Resilience the working tenacity is for a 
 "dead" or steady load. The modulus of resilience is calculated 
 by dividing the square of that working tenacity by the modulus of 
 elasticity. 
 
 * The experiments whose extreme results are marked N. were conducted 
 for Messrs. R. Napier & Sons by Mr. Kirkaldy. For details, see Transac- 
 tions of the Institution of Engineers in Scotland, 1858-59; also Kirkaldy On 
 the Strength of Iron and Steel.
 
 486 
 
 MATERIALS, ' CONSTRUCTION, AND ' STRENGTH. 
 
 VIII. SUPPLEMENTARY TABLE FOR CAST IRON. 
 
 Kinds of Iron. 
 
 Direct 
 Tenacity. 
 
 Resistance 
 to Direct 
 Crushing. 
 
 Modulus of 
 Rupture 
 of Square 
 Bars. 
 
 Modulus 
 of 
 Elasticity. 
 
 No. 1. Cold blast, jj m 
 No. 1. Hot blast, |[ rom 
 
 12,694 
 17,466 
 13,434 
 
 56,455 
 80,561 
 
 72,193 
 
 36,693 
 
 39-771 
 29,889 
 
 I4,OOO,OOO 
 15,380,000 
 
 11,539,000 
 
 No. 2. Cold blast, fi orn 
 
 16,125 
 I3>34S 
 
 88,741 
 68,532 
 
 35,316 
 
 33,453 
 
 15,510,000 
 12,586,000 
 
 /to 
 No. 2. Hot blast, fi ora 
 
 18,855 
 13,505 
 
 102,408 
 82,734 
 
 39,609 
 28,917 
 
 17,036,000 
 12,259,000 
 
 No. 3. Cold blast, -j[ rom 
 
 17,807 
 14,200 
 
 IO2,03O 
 76,900 
 
 38,394 
 35,88i 
 
 16,301,000 
 14,281,000 
 
 No. 3. Hot blast, It 
 
 15,508 
 15,278 
 
 115,400 
 101,831 
 
 47,061 
 35,640 
 
 22,908,000 
 15,852,000 
 
 No. 4. Smelted by coke) 
 without sulphur......... C 
 
 23,468 
 
 104,881 
 
 43,497 
 4i,7i5 
 
 22,733,000 
 
 Toughened cast iron, -|, r 
 No. 3. Hot blast after first) 
 
 23,461 
 25,764 
 
 129,876 
 119,457 
 98,560 
 
 39,690 
 
 
 
 No. 3. Hot blast after) 
 twelfth melting, C 
 
 
 
 163,744 
 
 56,060 
 
 
 
 No. 3. Hot blast after) 
 eighteenth melting, ... 
 Malleable cast iron............. 
 
 48 ooo 
 
 197,120 
 
 25,350 
 
 
 
 
 
 
 
 
 It is to be understood that the numbers in one line of the pre- 
 ceding table do not necessarily belong to the same specimen of iron, 
 each number being an extreme result for the kiud of iron, specified 
 in the first column.
 
 487 
 
 CHAPTER IL 
 
 PRINCIPLES AND RULES RELATING TO STRENGTH AND STIFFNESS. 
 
 411. 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. It is to a certain extent identical 
 with a similar summary which appeared in A Manual of Civil 
 Engineering, but modified to adapt its principles to the problems 
 which occur in machinery. Various special problems relating to 
 machinery will be considered in the third Chapter. 
 
 SECTION I. Of Strength and Stiffness in General. 
 
 412. fjoad, sires*, strain, strength. The load, or combination 
 of external forces, which is applied to any piece, moving or fixed, 
 in a 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, ac- 
 companied by strain, 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 
 fracture 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 different kinds of strength as there are 
 different ways in which it can be strained or broken, as shown ia 
 the following classification ;
 
 488 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 Strain. Fracture. 
 
 -,-,, (Extension Tearing. 
 
 Elementary j Compression...... Crushing. 
 
 ( Distortion Shearing. 
 
 Compound < Twisting Wrenching. 
 
 ( Bending Breaking across. 
 
 413. C'o-cfflciento 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 avoirdupois on the square inch. 
 As to other units, see Article 302, page 342. 
 
 Co-efficients 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. Tables of their values arq given at the end of 
 the volume. 
 
 Co-efficients 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. 
 
 414. 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 
 as put on by imperceptible degrees, and which remains steady ; such 
 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 shown that in most cases which occur in practice a 
 live load produces, or is liable to produce, twice, or very nearly 
 iwice, 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
 
 FACTORS OF SAFETY. 489 
 
 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 apparent 
 factor of safety has to be made greater than the real factor of 
 safety in those cases in which 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 appai'ent 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 shows the ordinary values of real factors of 
 safety : 
 
 RIIAL FACTORS OP SAFKTT. 
 Dead Load. Live 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 
 
 Effort, nearly. 
 
 Steam engines acting against a constant 
 resistance 
 
 Single engine, ................................. 1*6 9*6 
 
 I I 
 
 Pair of engines driving cranks at right ) 
 
 i / 
 
 angles,.... ......... ... .......... . ......... J 
 
 Three engines driving equiangular ) ,-. 
 
 cranks, .................................... J x 5 3
 
 490 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 Ordinary cases of varying effort and) z 
 
 resistance, j 
 
 Linesof shafting in millwork; apparent ] 
 
 factor of safety for twisting stress j 
 
 due to motive power, to cover allow- \- from 18 to 36 
 
 ances for bending actions, accidental | 
 
 extra load, &c., j 
 
 Almost all the experiments hitherto made on the strength of 
 materials give co-efficients or moduli of ultimate strength; that is r 
 co-efficients 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 co-efficient 
 of working stress, which is to be used in the calculation, along 
 with the working 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. 
 
 415. The Proof or Testing by experiment of the strength of a 
 piece of material is conducted in two different ways, according ix> 
 the object in view. 
 
 I. If the piece is to be afterwards tised, 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. 
 
 IL If the piece is to be 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
 
 PROOF OR TESTING STIFFNESS PLIABILITY. 491 
 
 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. 
 
 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 
 showed that supposition to be erroneous, by proving that in most 
 materials a set is produced by almost any load, how small so- 
 ever. 
 
 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 ap- 
 pears 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. (See page 444.) 
 
 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. 
 
 416. Stiffness or Rigidity, Pliability, their moduli or Co-efficients* 
 
 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 co-efficient 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 which are used in machinery, 
 the moduli of stiffness, though not exactly constant, are nearly 
 constant for stresses not exceeding the proof strength.
 
 4D2 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 The reciprocal of a modulus of stiffness may be called a " modidus 
 of pliability ;" that is to say, 
 
 f a*-*r Intensity of Stress 
 
 Modulus of Stiffness - ^- = ; 
 
 Strain 
 
 Modulus of Pliability = =r r . 
 
 Intensity ot Stress 
 
 The use of specific moduli of stiffness will be explained in the 
 sequel. Values of them are given in the tables prefixed to this 
 chapter. 
 
 417. The Elasticity of n Soiiti consists of stiffness, or resistance 
 to change of figure, combined with the power of recovering the 
 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 Co-efficients 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 
 
 418. Resilience or Spring is the quantity of mecJianical work re- 
 quired 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. 
 
 419. Height* or Length* of Moduli of Stiffness and Strength. 
 
 The term height or length, as applied to a modulus or co-efficient 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 Ibs. on the square foot 
 
 Heaviness of material in Ibs. to the cubic foot 
 __ Modulus in Ibs. on the square inch 
 
 ~ Weight of 12 cubic inches of the material' 
 Height of a modulus in inches 
 
 Modulus in Ibs. on the square inch 
 " Heaviness of material in Ibs. to the cubic inch '
 
 RESISTANCE TO DIRECT TENSION. 493 
 
 Height of a modulus in metres 
 
 __ Modulus in kilogrammes on the square metre _ 
 Heaviness of material in kilogrammes to the cubic mStre' 
 
 Several examples of this mode of stating the intensity of stress 
 have already been given; as at pages 474, 475; and in the Tables, 
 page 482. 
 
 SECTION II. Of Resistance to Direct Tension. 
 
 420. Strength, Stiffiic**, and Resilience of a Tie. The Word lie 
 
 is here used to denote any piece in framing or in mechanism, such 
 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 co-efficients 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 
 
 P=pS .............................. (1.) 
 
 If the sectional area varies at different points, the least 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 X 
 
 and * = 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, for examples of which, see the Tables. Then 
 
 (2.) 
 
 Let/' denote the proof tension of the material, so that/' S is the 
 proof load of the tie; then the proof extension is/' -?- K 
 
 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
 
 494 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 load is/' x -T- E. The force which acts through this space has for 
 its least value 0, for its greatest value P f S, and for its mean 
 value f S -T- 2; so that the work done in stretching the tie to the 
 proof strain is 
 
 '"^ ' 
 
 _ 
 E " E ' 2 ' 
 
 The co-efficient y 2 -f- E, by which one-half of the volume of the 
 tie is multiplied in the above formula, is called the MODULUS OF 
 RESILIENCE. For examples of its value, see the Tables, page 485. 
 
 A sudden pull of f S -H 2, or one-half of the proof load, being 
 applied to the bar, will produce the entire proof strain ofy -f- E, 
 which is produced by the gradual application of the proof load 
 itself; for the work performed by the action of the constant force 
 /' S -=- 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 toy 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. 
 
 421. 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 ; 
 
 f, the ultimate tenacity of the material, in pounds per square 
 inch; 
 
 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 147 Ibs. on the square 
 inch or thereabouts. 
 
 Then 
 
 and the proper proportion of thickness to radius is given by the 
 formula, 
 
 U. .............................. (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 Iba.
 
 CYLINDRICAL AOT> SPHERICAL SHELLS. 495 
 
 per square inch. That of a double-rivetted joint, per square inch 
 of the iron left between the rivet holes (if drilled, and not punched), 
 is the same; that of a single-rivetted joint somewhat less, as the 
 tension is not uniformly distributed. It is convenient in practice 
 to state the tenacity of ri vetted joints in Ibs. per square inch of 
 the entire plate; and it is so stated in the annexed table, in which 
 the results for ri vetted joints are from the experiments of Mr. 
 Fairbairn, and that for a welded joint from an experiment by Mr. 
 Dunn. The joints of plate-iron boilers are single ri vetted ; but 
 from the manner in which the plates break joint, the tiltimate 
 tenacity of such boilers is considered to approach more nearly to 
 that of a double-rivetted joint than to that of a single-rivetted 
 joint. 
 
 Wrought-iron plate joints, double-rivetted, the dia- 
 meter of each hole being A of the pitch, or dis- 
 tance from centre to centre of holes, 35,000 
 
 Wrought-iron plate joints, sing]', rivetted, 28,000 
 
 Wrought-iron boiler shells, with single-rivetted joints 
 
 properly crossed, 34,000 
 
 Wrought-iron retort, with a welded joint, 3>75 
 
 Cast-iron boilers, cylinders, and pipe? (average), 16,500 
 
 Malleable cast-iron cylinders, 48,000 
 
 422. Thick Hollow Cylinders and Spheres. The assumption that 
 
 the tension in a hollow cylinder or sphere is uniformly distributed 
 throughout the thickness of the shell is approximately true only 
 when the thickness is small as compared with the radius. 
 
 Let R represent the external and r the internal radius of a thick 
 hollow cylinder, such as a hydraulic press, the tenacity of whose 
 material is f, and whose bursting pressure is p. Then we must 
 have 
 
 R 2 -r* P. , n 
 
 R2 + ^ ~f> 
 
 and, vjonseqnently, 
 
 by means of which formula, when r, f, and p are given, R may be 
 computed. 
 
 In the case of a hollow sphere the following formulae give the 
 ratios of the bursting pressure to the tenacity, and of the external 
 to the internal radius : 
 
 P. 2R3-2r3. 
 
 /" R 3 + 2 r 8 '"' W
 
 496 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 f+2p } (4) 
 
 2f- P J" "W 
 
 SECTION III. Of Resistance to Distortion and Shearing. 
 
 423. 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 
 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 
 prismatic particle, so as to distort it; and let be the distortion, 
 expressed by the tangent oftlte difference between each of the distorted 
 angles of the prism and a right angle; then 
 
 (1.) 
 
 is the modulus of transverse elasticity, or resistance to distortion; of 
 which examples are given in the tables, page 479. 
 
 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 
 
 as 2* 
 
 Q 
 
 The ratio ^ of the modulus of transverse elasticity to the modulus 
 of direct elasticity defined in Article 420, page 493, has dif- 
 
 ferent values for different materials, ranging from to -. For 
 
 J 
 
 wrought iron and steel it is about _. 
 
 o
 
 EESISTANCE TO SHEARING. 497 
 
 The ultimate shearing strength, or modulus of resistance to 
 shearing, in other words, the intensity of the greatest shearing 
 stress when the body is on the point of giving way, is, in wrought 
 iron and steel, and most other metals, equal, or nearly equal, to the 
 tenacity : in cast iron it is about once and a half greater than the 
 tenacity; in timber it is nearly equal to the tenacity across the 
 grain. (See the Tables, page 479.) 
 
 424. Strength of Fastenings and Joint-Pins. The connecting 
 
 pieces already referred to as being exposed to the action of a shear- 
 ing force may be distinguished into fastenings, such as rivets, keys, 
 wedges, gibs and cottars, and screws, by which two pieces are 
 secured together so as to act as one piece; and joint-pins, by which 
 two pieces are so connected as to be free to turn about the joint. 
 It is obvious that the figure of a joint-pin, as well as that of the 
 hole or socket in which it works, must be that of a surface of 
 revolution, such as a circular cylinder; and that the fit, though 
 accurate, must be easy, like that of an axle in its bearings. Most 
 fastenings and joint-pins are exposed to a bending as well as to a 
 shearing action, and in some cases the most severe stress is that 
 arising from the bending action; but in other cases the most severe 
 stress is that produced by the shearing load. These latter cases 
 are as follows : All rivets, keys, and other fastenings which are 
 tightly jammed in their holes; all cylindrical joint-pins, fixed at 
 one end, in which the length of the loaded part is less than one- 
 third of the diameter; and all cylindrical joint-pins, fixed at both 
 ends, in which the length of the loaded part is less than two- 
 thirds of the diameter. 
 
 In order that the shearing stress on a connecting piece may be 
 uniformly distributed over the cross-section, it is necessary that 
 the fastening should be held so tight in its hole or socket that the 
 friction at its surface may be at least of equal intensity to the 
 shearing stress; and then the intensity of that stress is represented 1 
 simply by P -=- A; P being the shearing load, and A the area 
 which resists it. 
 
 But when the connecting piece fits easily, as must always be the 
 case with joint-pins, the greatest intensity of the stress, to which the 
 strength of the connecting piece must be adapted, exceeds the 
 mean intensity P -r A, in a ratio which depends on the figure of 
 the cross- section ; and whose values, for the ordinary figures, are 
 
 3 
 
 for rectangular cross-sections, --. ; 
 
 a 
 
 4 
 for circular and elliptic cross-sections, ^ ; 
 
 and the sectional area must accordingly be made greater in that 
 
 2K
 
 498 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 ratio than the area which would have been sufficient had the 
 connecting piece fitted tightly. 
 
 The chief kinds of connecting pieces, to which these principles 
 have to be applied, will now be considered separately. 
 
 425. Rivets are made of the most tough and ductile metal. (See, 
 for example, " Rivet Iron," in pages 460 and 482.) 
 
 The ordinary dimensions of rivets in practice are as follows : 
 
 Diameter of a rivet for plates less than half an inch thick, about 
 double the thickness of the plate. 
 
 For plates of half an inch thick and upwards, about once and a- 
 half the thickness of the plate. 
 
 Length of a rivet before clenching, measuring from the head = 
 sum of the thicknesses of the plates to be connected + 2| x 
 diameter of the rivet. 
 
 The longitudinal compression to which a rivet is subjected 
 during the operation of clenching, whether by hand or by machinery, 
 tends to make it fit its hole tightly, andHhus to produce uniform 
 distribution of the stress; but as such uniformity cannot be ex- 
 pected to be always realized, it is usual to assume, in practice, that 
 there is a deviation from uniformity of shearing stress sufficient to 
 neutralize the greater toughness of the metal in the rivets than in 
 the plates which they connect; and, therefore, the distance apart 
 of the rivets used to connect two pieces of metal plate together is 
 regulated by the rule, that the joint sectional area of the rivets shall 
 be equal to the sectional area of plate left after punching the rivet 
 holes. This rule leads to the following algebraical formula : 
 
 Let t denote the thickness of the plates ; 
 d, the diameter of a rivet ; 
 n, the number of ranks of rivets ; 
 
 it being understood that the rivets which form a rank stand in a 
 line perpendicular to the direction of the tension which tends to 
 pull the plates asunder. 
 
 c, the pitch, or distance from centre to entre of the adjoin- 
 ing rivets in one rank; then 
 
 7854 n & 
 c = d + (1.) 
 
 Each plate is weakened by the rivet holes in the ratio 
 c - d -7854 n d 
 
 t + -7854 n d 3 
 
 .(2.) 
 
 In " single-rivetted " joints, n = 1; in " double-rivetted " joints, 
 n = 2, and the two ranks of rivets form a zig-zag; in " chain-
 
 RESISTANCE TO SHEARING. 499 
 
 rivetted" joints, n may have any value greater than 1. A single- 
 rivetted joint is weakened by unequal distribution of the tension 
 on the plate in the ratio of 4 : 5. 
 
 Suppose that in a chain-rivetted joint the pitch c from centre 
 to centre of the rivets is fixed, so as not to weaken the plates below 
 a given limit; then in order to find how many ranks of rivets there 
 should be, in other words, how many rivets there should be in 
 each file, the following formula may be used : 
 
 426. Pins, Keys, Wedges, Gibs, and Cottars. These fastenings 
 
 are, like rivets, themselves exposed to a shearing load, while they 
 serve to transmit a pull or thrust from one piece in framework or 
 mechanism to another; and the rule for determining their proper 
 sectional area is the same, with this modification only, that it is 
 safest in most, if not in all cases, to allow for the possibility of an 
 easy fit, according to the rule stated at the end of Article 424, page 
 497. 
 
 In order that a wedge, key, or cottar may be safe against slipping 
 out of its seat, its angle of obliquity ought not to exceed the angle 
 of repose of metal upon metal, which, to provide for the contin- 
 gency of the surfaces being greasy, may be taken at about 4. 
 (Article 309, page 349.) 
 
 427. Bolts and Screws. If a bolt has to withstand a shearing 
 stress, its diameter is to be determined like that of a cylindrical 
 pin. If it has to withstand tension, its diameter is to be determined 
 by having regard to its tenacity. In either case the effective 
 diameter of the bolt is its least diameter; that is, if it has a screw 
 on it, the diameter of the spindle inside the thread. It is to be 
 observed, however, that in order to provide for possible irregularities 
 in the distribution of the stress, it is customary to use for screws a 
 very large factor of safety, ranging from 12 to 15; the mean in- 
 tensity of the working stress on wrought-iron screws being only 
 about 4,000 Ibs. on the square inch, or 2'8 kilogrammes on the 
 square millimetre. 
 
 The ordinary form of section of the thread of a fastening screw 
 is an isosceles triangle with the angles rounded; and according to 
 the proportions recommended by Mr. Whitworth, the angle at the 
 summit is 55, making the height of the triangle = 0-96 of its base. 
 One-sixth of that height is taken away by the rounding of the edge 
 of the thread, and another sixth by the rounding of the bottom of 
 the groove, leaving two-thirds, or 0' 64 of the base; and as the base 
 of the triangle is the pitch of the screw, the projection of the thread 
 is 0*64 of the pitch.
 
 500 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 The pitch should not in general be greater than one-fifth of the 
 effective diameter, and may be considerably less : for example, one- 
 tenth and one-twelfth are ordinary proportions. 
 
 In order that the resistance of a screw or screw-bolt to rupture 
 by stripping a triangular thread may be at least equal to its resist- 
 ance to direct tearing asunder, the length of the nut should be at 
 least one-half of the effective diameter of the screw; and it is 
 often in practice considerably greater; for example, once and a 
 half that diameter. 
 
 The head of a bolt is usually about twice the diameter of the 
 spindle, and of a thickness which is usually greater than five- 
 eighths of that diameter. 
 
 SECTION IV. Of Resistance to Twisting and Wrenching. 
 
 428. 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 twisting 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. 
 
 429. Strength of a Cylindrical Shaft. (A. M., 321.) A Cylindrical 
 
 shaft, A B, fig. 267, being subjected to the twisting moment of a 
 
 pair of equal and opposite couples ap- 
 plied to the cross-sections A and B, it 
 is required to find the condition of 
 stress and strain at any intermediate 
 cross-section, such as S, and also the 
 angular displacement of any cross- 
 Fj 'g- 267. section relatively to any other. 
 
 From the uniformity of the figure of 
 
 the bar, and the uniformity of the twisting moment, it is evident 
 that the condition of stress and strain of all cross-sections is the 
 same ; also, because of the 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
 
 TORSION. 501 
 
 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 
 
 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 
 
 ax 
 
 (2.) 
 
 The STRENGTH of the shaft is determined in the following man- 
 ner: Let q i 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 woi-king moment of torsion is to 
 be determined. Let r { be the external radius of the axle. Then. 
 g-j is the value of q at the distance r from the axis; and at any 
 other distance, r, the intensity of the shearing stress is 
 
 Conceive the cross-section 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 2 vrdr; 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 those three quantities, is 
 
 which being integrated for all the rings from the centre to the cir- 
 cumference of the cross-section, gives for the moment of torsion, 
 and of resistance to torsion,
 
 502 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 if h = 2 TI be the diameter of the shaft, 
 
 (^ = 1-5708; ^ = 0-196 nearly). 
 
 If the axle is liollow, h being the diameter of the hollow, the 
 moment of torsion becomes 
 
 16 
 
 The following formulae serve to calculate the diameters of shafts 
 when the twisting moment and stress are given ; solid shafts : 
 
 *,.(?! 
 
 1 \ Q- 
 
 *i /' 
 
 hollow shafts 
 
 .(6.) 
 
 which last formula serves to compute the diameter of a hollow 
 axle, when the ratio h : h^ of its internal and external diameter 
 has been fixed. 
 
 Values of the ultimate shearing strength of various substances 
 are given in the Tables. As for the working stress, a long 
 series of practical trials has shown that wrought-iron axles bear a 
 stress of 9,000 Ibs. per square inch, or 6-3 kilogrammes on the 
 square millimetre, for any length of time, if well manufactured of 
 good material, the factor of safety being about 6. If the ultimate 
 shearing stress of cast iron, 27,000 Ibs. on the square inch, is 
 divided by the same factor, the modulus of working stress is found 
 to be 4,500 Ibs. on the square inch, or nearly 3-2 kilogrammes on 
 the square millimetre. 
 
 It is chiefly in the shafting of mills that those large apparent 
 factors of safety are met with, referred to in Article 414, page 490. 
 
 430. Angle of Torsion. Suppose a pair of diameters, originally 
 parallel, to be drawn across the two circular ends, A and B, fig. 
 267, page 500, of a cylindi-ical shaft, solid or hollow; it is proposed 
 to find the angle which the directions of those lines make with 
 each other when the shaft is twisted, either by the working moment 
 of torsion, or by any other moment. 
 
 This question is solved by means of equation (2) of Article 429, 
 page 501, which gives for the angle of torsion per unit of length, 
 
 dO q 
 dx~ Gr
 
 TORSION. 503 
 
 The condition of the shaft being uniform at all points of its length, 
 the above quantity is constant; and if x be the length of the shaft, 
 and 6 the angle of torsion sought, expressed in length of arc to 
 
 radius 1, we have -= - and therefore, 
 x dx 
 
 I. Let the moment of torsion be the working moment, for which. 
 
 Q = 1 = 2 ?i. 
 r r x A. ' 
 
 the value taken for the modulus, q v being the safe working stress. 
 Then the angle of working torsion is 
 
 and is the same whether the shaft is solid or hollow. This formula 
 gives the angle 6 in circular 'measure; that is, in arc to radius 
 unity; so that if at each end of the shaft there is an arm of the 
 length y, the displacement of the end of one of those arms relatively 
 to the other will be y 6, 
 
 Values of C, the co-efficient of transverse elasticity, are given in 
 the tables. In calculating the working torsion of wrought-iron 
 shafts, we may make 
 
 o =from ioo to ioo ................. (3>) 
 
 II. The proof torsion, to which a shaft may be twisted by a 
 gradually applied load when testing it, may be made double the 
 working torsion. 
 
 III. Let the moment of torsion have any amount, M, consistent 
 
 with safety. Then for we have to put its value in terms of M 
 and Aj j and the results are as follows : 
 
 a 2 M 
 
 For solid shafts, - = T', and 
 
 v <7tr\' 
 
 32 M x 1r . Mo; 
 
 6 = = 10 ' 2 
 
 For hollow shafts, - = 7^ -r x ; and 
 
 r K r - r
 
 504: MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 32 M x 
 
 = 10>2 nTXI - J^ nearlv 
 
 ~ * G (hi - fcj) (A* - 
 
 An example of the application of equation (4) lias already been 
 given in Article 344. 
 
 431. The Resilience of a. Cylindrical Shaft is the product of one- 
 
 half of the moment of proof torsion into the corresponding angle 
 of torsion; and it is given by the following equation : 
 
 M & ir q\ hi x f .. , . f , "I 
 
 =- = a for a solid shaft; or 
 
 Z 10 \j 
 
 (1.) 
 
 M 6 , q\ (% - Kg) x. ... ... 
 
 ~7T = Tz ' ri /a for a hollow snaft - 
 
 -i lo \j ri^ 
 
 432. Shafts not circular in Section. When the cross-section of 
 a shaft is not circular, it is certain that the x ratio - of the shearing 
 
 stress at a given point to the distance of that point from the axis 
 of the shaft is not a constant quantity at different points of the 
 cross-section, and that in many cases it is not even approximately 
 constant; so that formulae founded on the assumption of its being 
 constant are erroneous. The mathematical investigations of M. de 
 St. Venant have shown how the intensity of the shearing stress is 
 distributed in certain cases. 
 
 The most important case in practice to which M. de St. Tenant's 
 method has been applied is that of a square shaft; and it appears 
 that its moment of torsion is given by the formula, 
 
 M = 0-281 g l h s nearly; 
 in which h is one side of the square cross-section. 
 
 SECTION V. Of Resistance to Sending and Cross- Breaking. 
 
 433. Resistance to Rending 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 
 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 transverse to 
 the axis of the piece. 
 
 Conceive a beam which is acted upon 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
 
 RESISTANCE TO BENDING. 505 
 
 extemal transverse forces which act 011 that part, and constitute 
 the load on it, do not necessarily balance each other. Their result- 
 ant may be found by Rule IV. of Article 280, page 324. That 
 resultant is called the Shearing Load at the cross-section under con- 
 sideration, and it is balanced by the Shearing 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; the resultant 
 thrust and the resultant tension are equal and opposite, and com- 
 pose a couple, whose moment is the moment of stress, equal and 
 opposite to the bending moment. 
 
 In problems respecting the transverse strength and stiffness of 
 beams there are four processes : first, 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 severely; secondly, to determine 
 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; thirdly, to determine the relations between the dimen- 
 sions and figure of the beam and the deflection produced by the 
 bending moments, in order that the beam may be so designed as 
 to have sufficient stiffness or sufficient flexibility, according to its 
 purpose. 
 
 434. Calculation of Shearing Xioada and Bending Moments. 
 
 In the formulae 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 F and M at any given 
 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. 268 represents a beam supported at both ends, and loaded 
 between them. Fig. 269 represents a bracket; that is, a beam,
 
 606 
 
 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 supported and fixed at one end, and loaded on a projecting portion. 
 P, Q, represent in each case the supporting forces; in fig. 268, W v 
 
 "W^ W 3 , <fcc., represent portions of the load; in fig. 269, "W re- 
 presents the endmost portion of the load K and W 1} W^ W 3 , other 
 portions; in both figures, Aa^, Aa; 2 , Ax 3 , <fcc., 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 in- 
 definitely short portion of the beam may be denoted by d x, 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 following 
 steps : 
 
 STEP I. To find the Supporting Forces or Bearing Pressures, P 
 and Q. Assume any convenient point in the longitudinal axis 
 as origin of co-ordinates, and find the distance X Q of the resultant 
 of the load from it, by Rule IY. of Article 280, pages 324, 325 ; 
 that is to say, 
 
 1 
 
 .(2.) 
 
 Then, by Rule II. of Article 280, page 323, find the two sup- 
 porting forces or bearing pressures, P and Q ; that is to say, let R 
 be the resultant load, and P R and R Q its distances from the 
 points of support; and make
 
 SHEARING LOADS AND BENDING MOMENTS. 507 
 
 PQ:PR:QPv) 
 
 : :R : Q : P. f W 
 
 STEP II. To find tJte Shearing 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. 268), and 
 in a bracket to be at the loaded point farthest from the fixed end 
 (as A, fig. 269). 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. 268. 
 
 - 01 JJ_0 ) 
 
 p "W" "W 
 
 ' " 1 VV 2> 
 
 -F=W 
 
 In Fig. 269. 
 
 &C. j 
 
 and generally, and generally, 
 
 F = P 2 Wj...(4.) 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, 
 
 /3f 
 w d x; (6.) 
 
 In a bracket, F = / w dx; (7.) 
 
 in which 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 
 ends, 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 outer point of support, and its 
 value 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, 
 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. To find the Bending Moments at a Series of Sections.
 
 508 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 At the origin A there is no bending moment. Multiply the 
 length of each of the intervals A x 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 successively the bending 
 moments at the inner ends of the other intervals.* 
 That is to say, bending moment 
 
 at the origin A ; M = ; 
 
 at the line of action of "W 1 ; M l = F 01 A #1 ; 
 
 W 2 ; M 2 = F 01 A x l + F 13 &x; 
 
 &c. fcc. 
 
 and generally, M = 2 -F Ax (8.) 
 
 If the divisions A x are of equal lengths, this becomes 
 
 M = A^'SY; (9.) 
 
 and for a continuously distributed load, 
 
 M 
 
 = ( X ~Fdx (10.) 
 
 J o 
 
 Substituting for F, equation (10), its values as given by equa- 
 tions (6) and (7) respectively, we obtain the following results : 
 For a beam supported at both ends, 
 
 /if rx 
 I wdx* 
 J o 
 
 /*' 
 (x -x) wdx-, (11.) 
 o 
 
 For a beam fixed at one end, 
 
 -M= (" [* wda?= I"' (x'-x)wdx; (12.) 
 
 J J -' 
 
 in the latter of which equations the symbol M denotes that the 
 bending moment acts downwards. 
 
 The Greatest Bending itiomont 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. 
 
 STEP IV. To deduce the Shearing Load and Sending Moment in 
 one Seam from those in another Seam similarly supported and 
 loaded. This is done by the aid of the following principles : 
 
 When beams differing in length and in the amounts of tlie loads 
 upon them are similarly supported, and have their loads similarly dis- 
 tributed, the shearing loads at corresponding sections in them vary as 
 
 See Mr. Herbert Latham's work On Iron Bridges.
 
 SHEARING LOADS AND BENDING MOMENTS. 509 
 
 the total loads, and the bending moments as the products of the loads 
 and lengths. 
 
 The length between the points of support of a beam supported at 
 the ends, as in fig. 268, is often called the span. 
 
 435. 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; 
 
 v), 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. 
 
 I. Bracket fixed at one end and loaded ) -,,- -r-r T /i \ 
 
 > M = c \V ...... (1.) 
 
 at the other, ............................ j v ' 
 
 II. Bracket fixed at one end and uni- ) , .. _ c "W w c 2 /g 
 formly loaded, ......................... / == 2 = 2 ' 
 
 III. Beam supported at both ends and 1 
 
 loaded at an intermediate point, I _ (c 2 a; 2 )"W , 
 whose distance from the middle of f = ~~ ' ~2~c ' ' 
 the span is x, ........................... j 
 
 IY. Beam supported at both ends and) , ... __ 
 
 loaded in the middle, ................ j (* =0); M = -g-. 
 
 V. Beam supported at both ends and ) 1VT cW we 2 . 
 
 uniformly loaded, ..................... / M = ~^~ : = ~2~ ^ 
 
 VI. If a beam has equal and opposite couples applied to its two 
 ends; for example, if the beam in fig. 270 has the couple of equal 
 and opposite forces P x applied at A and B, and the couple of 
 equal and opposite forces P 2 at C and D, 
 
 and if the opposite moments P x A B 
 
 = P 2 C D = M are equal, then each 
 
 of the endmost divisions, A B and 
 
 C D, is in the condition of a bracket 
 
 fixed at one end and loaded at the 
 
 other (Example I.); and the middle Fig. 270. 
 
 division, B 0, is acted upon by the 
 
 uniform bending moment M, and by no shearing load. 
 
 VII. Let a beam of the half-span c be loaded with an uni- 
 formly 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
 
 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 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 
 
 then the solutions are as follows : 
 
 CASE 1. When - = or ^ -= ; ; x = m (c a); and 
 c 1 + m 
 
 2 
 
 ~ ~ __.. a in , 
 
 CASE 2. When = or ^ = = - : x = a : and 
 c I + m 
 
 In the following case both sets of formulae give the same result; 
 
 v a m i \ ^ 
 
 when = = i - : x = a = m (c a) : and 
 c 1 +m 
 
 436. Bending moments produced by Iiongitndinal 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 (see Article 293, page 334), that 
 force exerts a moment upon that cross-section equal to the product 
 of the force into the perpendicular 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 uniform 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. 
 
 437. moment of Stress Transverse Strength. The bending mo- 
 ment 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
 
 TRANSVERSE STRENGTH. 511 
 
 direction. Thus, fig. 271 represents a side view of a short portion 
 
 of a bent beam ; C C' is a layer, originally 
 
 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', length- Fig. 271. 
 
 ened, by the bending action of the load. 
 There is one intermediate surface, O O', 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, in 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. 
 272, which represents a vertical longitudinal section of a rectangular 
 beam, supported at the ends, and loaded at intermediate points. 
 It is covered with a network 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. 272. 
 
 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 the crushing of the compressed side, A A', or 
 by the tearing of the stretched side, B B'. 
 
 In fig. 273, A represents a 
 
 beam of a granular material, like ^ A - 
 
 cast iron, giving way by the 
 
 crushing of the compressed side, 
 
 out of which a sort of wedge is Fig. 273. 
 
 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
 
 512 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 in practice, the longitudinal stress of the layers of a beam may, 
 without material error, be assumed to be uniformly varying, its 
 intensity being simply proportional to the distance of the layer 
 from the neutral surface. 
 
 Let fig. 274 represent a cross-section of a beam (such as that 
 represented in fig. 271), A the compressed side, B 
 the extended side, C any layer, and O the neutral 
 axis of the section, being the line in which it is cut 
 - by the neutral surface. Let p denote 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 -f- y is a constant 
 Fig. 274. quantity. Let that constant be denoted for the 
 
 present by a. 
 
 Let z be the breadth of the layer C, and d y its thickness; 
 Then the amount of stress alon it is 
 
 the amount of the stress along all the layers at the given cross- 
 section is 
 
 a I yzdy; 
 
 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 = Q, ......................... (1.) 
 
 and the neutral axis traverses the centre of magnitude of the cross- 
 section. This principle enables the neutral axis to be found by the 
 aid of the methods explained in Article 293, page 334. 
 
 To find the greatest value of the constant p -r- y consistent with 
 the strength of the beam at the given cross-section, let y a be the 
 distance of the compressed side, and y b that of the extended side 
 from the neutral axis ; f a the greatest thrust, and f b the greatest 
 tension, which the material can bear in the form of a beam; com- 
 pute^ -f- ?/ a , andy s -f- y b , and adopt the less of those two quantities 
 as the value of p + y, which may now be denoted by f -f- y ; 
 f being f a or f b , and y^ being y a or y 6 , according as the beam is 
 liable to give way by crushing or by tearing. 
 
 For the best economy of material, the two quotients ought to be 
 equal ; that is to say, 
 
 J fa fb fa. + Jb. /I A \ 
 
 = = = - ? i ................ \ A A v 
 
 2/1 2/ 2/6 * 
 
 and this gives what is called a cross-section of equal strength.
 
 MOMENT OF RESISTANCE. 513 
 
 The moment relatively to the neutral axis, of the stress exerted 
 along any given layer of the cross-section, is 
 
 y pzd y = y z zdy; 
 2/1 
 
 and the 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=pyzdy = <- y z zdy; (2.) 
 
 J y\J 
 
 or making I y 2 z d y = I, 
 
 M - - (2 A.^ 
 
 " * ' 
 
 When the breaking load is in question, the co-efficient y is what 
 is called the MODULUS OF RUPTURE of the material. 
 
 When the proof load or working load is in question, the co- 
 efficient f 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 modulus /is usually 9,000 Ibs. 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^ 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 
 
 I - n>bh*, (3.) 
 
 (4.) 
 
 n' and m' being numerical factors depending on the form of section; 
 and making ri -f- m' = n, the moment of resistance may be thus 
 expressed, 
 
 M = nfbh?. ......................... (5.) 
 
 Hence it appears that the resistances of similar cross-sections to 
 cross-breaking are as their breadths and as the squares of their depths. 
 
 2L
 
 514 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 Another way of expressing the moment of resistance is as 
 follows : Let S be the sectional area of the beam, then we have 
 
 I=:&'/i 2 S; ........................ (3 A.) 
 
 in which Iff h 2 is the radius of gyration of the cross-section, k' being 
 a numerical factor depending on the form of section. Then making 
 % -T- TO' = k, the moment of resistance may be thus expressed : 
 
 S (5 A.) 
 
 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 Articles 434, 
 435, 436, pages 505 to 510, to the value of the moment of resistance 
 of the beam, at the cross-section where that greatest bending 
 moment acts, as given in equation (5) or equation (5 A) of this 
 Article. 
 
 The depth h is usually fixed by considerations of stiffness, to be 
 explained further on; and then the unknown quantity is either 
 the breadth, b, or sectional area, S, according as equation (5) or equa- 
 tion (5 A) is made use of. Sometimes, as when the cross-section is 
 circular or square, we have b = h; and then we have h 3 , instead of 
 b /i 2 in equation (5), which is solved so as to give h by extraction of 
 the cube root. The following are the formulae for these calcula- 
 tions : 
 
 M 
 
 and when h = b, 
 
 /Msi 
 
 (7 A.) 
 
 M 
 
 In finding the value of the geometrical moment of inertia I of 
 cross-sections of complex figure, the following rules are useful : 
 
 If a complex cross-section is made up of a number of simple 
 figures, conceive the centre of magnitude of each of those figures 
 to be traversed by a neutral axis parallel to the neutral axis of the 
 whole section. Find the moment of inertia of each of the com- 
 ponent figures relatively to its own neutral axis; multiply its area 
 by the square of the distance between its own neutral axis and 
 the neutral axis of the whole section; and add together all the 
 results so found, for the moment of inertia of the whole section. 
 To express this in symbols, let S' be the area of any one of the 
 component figures, ^' the distance of its neutral axis from the
 
 MOMENT OF RESISTANCE. 
 
 515 
 
 neutral axis of the whole section, I' its moment of inertia relatively 
 to its own neutral axis; then the moment of inertia of the whole 
 section is 
 
 I = 2 I' + 2 
 
 A'. 
 
 .(8.) 
 
 When the figure of the cross-section can be made by taking 
 away one simpler figure from another, both the area and the 
 moment of inertia of the subtracted figure are to be considered 
 as negative, and so treated, in making use of equation (8). 
 
 EXAMPLES OF THE NUMERICAL FACTORS IN EQUATIONS (3), (4), 
 
 (5), AND (7). 
 
 
 I 
 
 rf-ft 
 
 M 
 
 
 bh y 
 
 h 
 
 /&*" 
 
 I. llectan "le b li 
 
 1 
 
 l 
 
 1 
 
 (including square) 
 
 12 
 
 2 
 
 6 
 
 IL Ellipse- 
 Vertical axis h, } 
 
 IT 1 
 
 ft j. on A. 
 
 1 
 
 J7T _ 1 
 QO ]<)? 
 
 Horizontal axis b > 
 
 
 2 
 
 
 (including circle) ) 
 
 = 0-0491 
 
 
 = 0-0982 
 
 IIL Hollow rectangle, b h V h'; 
 also I - formed section, 
 
 1 / b'h'\ 
 
 1 
 
 I/ b'h'^ 
 
 where b' is the sum of the 
 breadths of the lateral 
 hollows ... 
 
 12V bh 3 ) 
 
 2 
 
 6V bh 3 ) 
 
 
 
 
 
 IV. Hollow square ) 
 W A" ... ( 
 
 - (l--\ 
 12 V />*/ 
 
 1 
 2 
 
 -d-^} 
 
 6 V A 4 / 
 
 
 
 
 
 
 * (\ b ' h ' 3 \ 
 
 1 
 
 -A v*!\ 
 
 
 64\ bh 3 } 
 
 2 
 
 32 V b h 3 ) 
 
 
 -(i-n) 
 
 1 
 
 -(l ^ 
 
 
 64 V h'/ 
 
 2 
 
 32 V A 4 / 
 
 VIL Isosceles triangle ; base b, } 
 height h ; y t measured > 
 from summit ) 
 
 1 
 36 
 
 2 
 3 
 
 1 
 24 
 
 
 

 
 516 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 EXAMPLES OF THE NUMERICAL FACTOR k IN EQUATIONS (5 A) 
 AND (7 B). 
 
 FORM or CKOSS-SECTION. i = ;. 
 
 / /* o 
 
 I. Rectangle, ......................... ^. 
 
 II. Ellipse and circle, ................. . 
 
 III. Hollow rectangle, 
 
 S = b h b' h'; also I-shaped 
 
 section, b' being the sum of 7 . ? ,3 
 
 the depths of the lateral 1 -- j-i-r 
 hollows, ........................ bh , 
 
 IV. Hollow square, S = 7i 2 - A' 2 ,... -l + 
 
 V. Do., very thin (approx.), -. 
 
 . V 
 l - 
 
 I/, Vh'3\ 
 VI. Hollow ellipse, ..................... g (l - j^) + 
 
 VII. Hollow circle, ..................... ~ (l + ^). 
 
 o \ ftv 
 
 VIII. Do., very thin (approx.), -.. 
 
 IX. T-shaped section; flange A, _, , 
 
 web C; S - A + C (approx.), v + 4 A ) 
 
 6 (C + A) (C + 2 A)' 
 
 X. I-shaped section; flanges A, B; 
 webC; S = A + B + C; the 
 beam supposed to give Wa 7 at r , 
 the flange A (approx.), ........ O (Q-^4 A + 4 B) + 12 A B 
 
 6(C + 2B)(A + B + C) 
 X. A. Do., do., the beam sup- 
 
 C + 2A)(A + B + C) * 
 XI. I-shaped section, with equal 
 
 flangesA - B;S - C + 2A -,/ . A 
 
 (approx.), ........................ i(l + 4A _ ) 
 
 V n ; 6 \ C + 3 A/
 
 BEAMS DEFLECTION. 517 
 
 438. Longitudinal Section* of Uniform Strength are those ill 
 
 which the dimensions of the cross-section are varied in such a 
 manner that its safe working moment of resistance is equal to the 
 working bending moment at each section of the beam, and not 
 merely at the section where the bending moment is greatest. That 
 moment of resistance, for figures of the same kind, being propor- 
 tional to the breadth and to the square of the depth, can be varied 
 either by varying the breadth, the depth, or both. The law of 
 variation depends upon the mode of variation of the moment of 
 flexure of the beam from point to point, and this depends on the 
 distribution of the load and of the supporting forces, in a way 
 which has been stated in previous Articles. When the depth of 
 the beam is made uniform, and the breadth varied, the vertical 
 longitudinal section is rectangular, and the horizontal longitudinal 
 section is of a figure depending on the mode of variation of the 
 breadth. When the breadth of the beam is made uniform, and the 
 depth varied, the horizontal longitudinal section is rectangular, 
 and the vertical longitudinal section is of a figure depending on 
 the mode of variation of the depth. When the beam, or the body 
 which acts as a beam, is of circular cross-section, so that the breadth 
 and depth are equal, each being a diameter of the cross-section, the 
 diameter varies as the cube root of the bending moment. This 
 case occurs in axles which are exposed to a bending moment, and 
 not to a twisting moment. The following are examples of the results 
 of those principles : 
 
 I. Fixed at one end, loaded at the other; b k z varies as the 
 distance from the loaded end. 
 
 II. Fixed at one end, uniformly loaded; b h z varies as the square 
 of distance from the free end. 
 
 III. Supported at ends, loaded at an intermediate point; b A 2 
 varies as the distance from the adjacent point of support. 
 
 IV. Supported at ends, uniformly loaded; b A 2 varies as the 
 product of the distances from the points of support. 
 
 In applying the principles of this Article, it is to be borne in 
 mind that they do not take the shearing load into account; and 
 that, consequently, the figures described in the above examples 
 may require, at and near the points where the shearing load is 
 greatest, some additional sectional area, to enable them to with- 
 stand that load, especially in examples III. and IV.; for in these 
 cases the shearing load is greatest at the points of support, where 
 there is no bending moment. 
 
 439. Deflection of Beam*. Four sorts of problems occur in con- 
 nection with the deflection and the stiffness of beams; first, to find 
 the proof, or greatest safe deflection, being the deflection under the 
 proof load; secondly, the deflection under any given load, not ex- 
 ceeding the proof load; thirdly, to find the dimensions of a beam
 
 518 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 which shall have a given deflection under its proof load, or under 
 some other given load; fourthly, from the observed deflection it may 
 be required to deduce the intensity of the most severe stress. 
 The following are the rules : 
 
 To find the curvature (that is, the reciprocal of the radius of 
 curvature) of an originally straight beam at a given cross- 
 section. 
 
 I. The cross-section under its proof stress. Divide the proof 
 stress (/i) by the distance of the most severely-strained particles 
 from the neutral axis, and by the modulus of elasticity; the 
 quotient will be the proof curvature; 
 
 II. The bending moment given. Divide the bending moment 
 by the moment of inertia of the given cross-section (see Article 
 437, page 513), and by the modulus of elasticity of the material. 
 In symbols, let r be the radius of curvature ; then 
 
 III. To find the inclination of the longitudinal axis of the beam 
 to its original direction at a given point. Divide the length of 
 the beam into small intervals (d x) ; multiply the length of each 
 
 interval by the curvature at its centre ( giving the product - J ; 
 
 add together the products for the intervals from a point where the 
 beam continues horizontal to the point where the inclination is 
 required j the sum will be the required inclination ; that is, 
 
 IV. To find the deflection. Multiply the length of each small 
 interval by its inclination (obtaining the product i d x) ; add to- 
 gether those products for the intervals extending between the 
 highest and lowest points of the beam ; the sum will be the required 
 deflection ; that is, 
 
 v= I i 
 
 dx. (4.) 
 
 The preceding is the general method. The following are special 
 rules: 
 
 Let c be the half-span of a beam supported at both ends, or the 
 length of a bracket fixed at one end ; h, the extreme depth : and 6, the 
 extreme breadth of the beam ; W, any given load ; f t , the proof
 
 DEFLECTION. 519 
 
 stress ; m! h, the distance of the most severely-strained layer from 
 the neutral axis ; I, the moment of inertia of the greatest cross- 
 section ; m", n", m'", n" 1 , numerical factors (see Table below). 
 
 V. Greatest inclination under proof load ; 
 
 ^J&Tf (5 '> 
 
 VI. Proof deflection ; 
 
 v,=^A^- (6.) 
 
 VII. Greatest inclination under a given load, W; 
 
 m!" We 2 
 
 El " '"^ l ' ) 
 VIII. Deflection under a given load, W; 
 
 of /ft \ 
 
 1 El 
 
 Proof Load. 
 EXAMPLES. Factors for 
 
 Any Load. 
 Factors for 
 
 A. UNIFORM CROSS-SECTION. 
 
 Slope. 
 m" 
 
 Deflection. 
 n" 
 
 Slope. Deflection. 
 m" n'" 
 
 I. Constant Moment of Flex- ) 
 
 1 
 
 1 
 
 
 
 II. Fixed at one end. loaded ) 
 
 1 
 
 I 
 
 I 
 
 1 
 
 at other, J 
 
 2 
 
 3 "' 
 
 " 2 '" 
 
 ... 3 
 
 III. Fixed at one end, uni- ) 
 
 1 
 
 1 
 
 I 
 
 1 
 
 formly loaded, J 
 
 3 
 
 .... 4 .... 
 
 " 6 '" 
 
 ... g 
 
 IV. Supported at both ends, ) 
 
 1 
 
 1 
 
 1 
 
 1 
 
 loaded in middle, J 
 
 2 " 
 
 3 " 
 
 .. 4 ... 
 
 ... 6 
 
 V. Supported at both ends, ) 
 
 2 
 
 5 
 
 1 
 
 5 
 
 uniformly loaded, J 
 
 3 
 
 12"" 
 
 .. 6 ... 
 
 "'48 
 
 B. UNIFORM STRENGTH AND UNI- 
 
 
 
 
 
 FORM DEPTH. 
 
 
 
 
 
 (The curvature of these is uniform.) 
 
 VI. Fixed at one end, loaded ) 
 
 
 1 
 
 
 1 
 
 at other, J 
 
 .. 
 
 .... 2 .... 
 
 ... 
 
 ... ^ 
 
 VII. Fixed at one end, uni- ) 
 
 
 1 
 
 1 
 
 1 
 
 formly loaded, j 
 
 .. 
 
 .... 2 .... 
 
 .. 2 ... 
 
 "' 4 
 
 VIII. Supported at both ends, ) 
 
 i 
 
 1 
 
 1 
 
 1 
 
 loaded in middle, J 
 
 
 .... 2 .... 
 
 .. 2 ... 
 
 " 4 
 
 IX. Supported at both ends, ) 
 uniformly loaded, J 
 
 1 
 
 1 
 .... 2 .... 
 
 1 
 .. 4 ... 
 
 1
 
 520 
 
 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 EXAMPLES. 
 
 C. UNIFORM STRENGTH AND UNI- 
 FORM BREADTH. 
 
 X. Fixed at one end, loaded ) 
 atother, .................... J 
 
 Proof Load. Any Load. 
 
 Factors for Factors for 
 
 Slope. Deflection. Slope. Deflection. 
 
 XII. Supported at both ends, ) 9 
 loaded in middle, ....... J w 
 
 XL Fixed at one end, uni- ) 1 
 
 formly loaded, ........... / infinite. infinite ^ 
 
 1-08 0-5708 0-3927 0-H27 
 
 IX. Given, the half-span, c, and the intended proof deflection, V L , 
 of a proposed beam; to find the proper value of the greatest depth, 
 Ji make 
 
 Q 
 
 _ 
 
 l - 
 
 
 (taking n" from the preceding table, and making m! h , as before, 
 denote the distance from the layer in which the stress is f^ to the 
 neutral axis). 
 
 X. To deduce the greatest stress in a given layer of a beam from 
 the deflection found by experiment. 
 
 Let h be the depth of the beam at the section of greatest stress, 
 and y the distance from the neutral axis of that section to that 
 layer of the beam at which the greatest stress is required: 
 
 c, the half-span of a beam supported at both ends, or the length 
 
 of the loaded part of a beam supported at one end ; 
 n", the factor for proof deflection, already explained; 
 E, the modulus of elasticity of the material ; 
 v, the observed deflection; 
 then the intensity of the required stress is 
 
 <"> 
 
 XI. To find the deflection of an uniform beam produced by its 
 own weight, or by an uniform load bearing a given proportion, 
 1 + m, to the weight of the beam. Let w be the heaviness of the 
 material of which the beam consists; r 2 = I -H S, the square of the 
 radius of gyration of its cross-section; ri", as before, the factor for
 
 BEAMS DEFLECTION RESILIENCE. 521 
 
 deflection under a given load; then, for a beam supported at both 
 ends, 
 
 and for a bracket fixed at one end, 
 
 v' = 
 
 A table of values of r 2 will be given at p. 525. The application 
 of this problem to shafts for transmitting power will be explained 
 in the next Chapter. 
 
 440. Beam fixed at the Ends. When a beam is not merely 
 supported, but fixed in direction at its two ends, it bends into the 
 form of a curve which has two points of inflection ; being convex 
 upwards at the points of support, and concave upwards in the 
 middle. The following are the two most important cases; the 
 cross-section of the beam being supposed uniform in both : 
 
 I. Load concentrated at middle of span. The bending moments 
 at the points of support and at the middle of the span are equal 
 and contrary, and each equal to half of the bending moment upon 
 an equal and similarly loaded beam with ends merely supported; 
 
 "W c 
 
 that is, M = . 
 
 Factor for proof deflection, n" = ^. 
 
 Factor for deflection under a given load, n"' = -^. 
 
 II. Load uniformly distributed. The bending moment at the 
 middle of the span is one-third, and the contrary bending moment 
 at each point of support two-thirds, of what the bending moment 
 in the middle of the span woxild be if the ends were merely sup- 
 
 w c 2 
 ported. That is, the most severe bending moment is M = ^-. 
 
 Factor for proof deflection, n" = -^ 
 
 Factor for deflection under a given load, n" = j^. 
 
 441. The Resilience of a Beam (A, M., 305) is the work per- 
 formed in bending it to the proof deflection ; in other words, the 
 energy of the greatest shock which the beam can bear without 
 injury; such energy being expressed by the product of a weight 
 into the height from which it must fall to produce the shock in
 
 522 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 question. This, if the load is concentrated at or near one point, 
 is the product of half the proof load into the proof deflection ; that 
 is to say, let P be the proof load; then the resilience is 
 
 Let W be the weight of a mass which is let fall upon the beam 
 from the height z. Then the whole height through which that 
 mass falls, before the beam reaches its proof deflection, is z + v; 
 and the whole energy of the blow which it gives to the beam is 
 W (z + Vj); which being equated to the resilience, gives the 
 following equation : 
 
 W(* + Ul ) = ; ...................... (2.) 
 
 an equation which enables any one of the four quantities, W, z, P, 
 v v to be calculated when the other three ore given. 
 
 If the load is distributed, the length of the beam is to be divided 
 into a number of small elements, and half the proof load on each 
 element multiplied by the distance through which that element is 
 depressed. The integral of the products will be the resilience. 
 
 SECTION VI. Of Resistance to Thrust or Pressure. 
 
 442. Resistance to Compression and Direct Crushing. Resistance 
 
 to longitudinal compression, when the proof stress is not exceeded, 
 is sensibly equal to the resistance to stretching, and is expressed by 
 the same modulus of elasticity, denoted by E (page 493). When 
 that limit is exceeded, it becomes irregular. (See Article 420, 
 page 493.) 
 
 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 (page 493) are 
 approximately applicable, substituting thrust for tension, and using 
 the proper modulus of resistance to direct crushing instead of the 
 tenacity.
 
 RESISTANCE TO THRUST. 
 
 523 
 
 Blocks whose lengths are less than about once-and-a-half their 
 diameters offer greater resistance to crushing than that given by the 
 Rules ; but in what proportion is uncertain. 
 
 The modulus of resistance to direct crushing, as the Tables show, 
 often differs considerably 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 by splitting (fig. 275) into a number of nearly 
 prismatic 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. 
 
 Fig. 275. Fig. 276. 
 
 Fig. 278. 
 
 II. Crushing by shearing or sliding of portions of the block along 
 oblique surfaces of separation is characteristic of substances of a 
 granular texture, like cast iron, and most kinds of stone and brick. 
 Sometimes the sliding takes place at a single plane surface, like 
 A B in fig. 276 ; sometimes two cones or pyramids are formed, like 
 c, c, in fig. 277, which are forced towards each other, and split or 
 drive outwards a number of wedges surrounding them, like w, w, 
 in the same figxire. In substances which are crushed by shearing, 
 the resistance to crushing is always much greater than the tenacity; 
 Sometimes the block splits into four wedges, as in fig. 278. 
 
 III. Crushing by bulging, or lateral swelling and spreading of 
 the block which is crushed, is characteristic of ductile and tough 
 materials, such as wrought iron. Owing to the gradual manner in 
 which materials of this nature give way to a crushing 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 u 
 
 IV. Crushing by buckling or crippling is characteristic of fibrous 
 substances, 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. The
 
 524 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 resistance of such substances to crushing is in genei-al considerably 
 less than their tenacity, especially where the lateral adhesion of the 
 fibres to each other is weak compared with their tenacity. The 
 resistance of most kinds of timber to crushing, when dry, is from 
 
 1 2 
 
 -j to of the tenacity. Moisture in the timber weakens the lateral 
 
 '2i o 
 
 adhesion of the fibres, and reduces the resistance to crushing to 
 about one-half of its amount in the dry state. 
 
 443. Crashing br Cross-Breaking. Long struts and pillars in 
 framework, and rods, bars, or links in machinery, which transmit 
 thrust, give way by bending sideways and breaking across. Let P 
 be the breaking load of such a piece ; S, its sectional area ; I, its 
 length ; r, the least geometrical radiiis of gyration of its cross-section ; 
 f and c, two co-efficients depending on the material ; then 
 
 I. For a piece fixed in direction at both ends; 
 
 - ......................... 
 
 c 1* 
 
 II. For a piece jointed at both ends (such as a link or connect- 
 ing-rod in machinery) ; 
 
 p f 
 
 s = : 4T 2 ............................ (2>) 
 
 1 -I- - 2 
 
 c r 
 
 III. For a piece jointed at one end and fixed in direction at the 
 other (such as a piston-rod); 
 
 P _ / 
 
 S ~ ~ IFF ......................... ' 
 
 + 9 c r* 
 
 The square of the radius of gyration referred to is given by the 
 expression, 
 
 where S is the area of cross-section of the piece, and I the geome- 
 trical moment of inertia of that cross-section about a neutral axis 
 perpendicular to the direction in which the piece is most flexible. 
 (See Articles 437, 439, pages 513, 519.) 
 
 VALUES OF THE CONSTANTS FOB THE BREAKING LOAD. 
 
 / c 
 
 Lbs. on the Square Inch. 
 
 Malleable iron, ..................... 36,000 36,000 
 
 Cast iron, ............................ 80,000 6,400 
 
 Dry timber, strong kinds, ........ 7,200 3,000
 
 CRUSHING BY CROSS-BREAKING COLLAPSING. 
 
 TABLE OF VALUES OF r 2 FOR DIFFERENT FORMS OF 
 CROSS-SECTION. 
 
 Solid rectangle; least dimen- ) 72 , 
 
 sion - h; / 
 
 Hollow square tube; dimen-) ,, 2 /'2\ . 19 
 
 sions, outside, h ; inside, h' ; J "'' 
 
 Thin square cell; side h; 7t 2 -f- 6. 
 
 Thin rectangular cell; breadth, ) 7t 2 h + 3 b 
 
 b; depth, h; J "B'TTT"' 
 
 Solid cylinder; diameter = h;... h z -f- 16. 
 
 Hollow cylinder; diameter,) ,, ,,, . 
 
 outside, h; inside, A'; J 
 
 Thin hollow cylinder; dia- ) 72 
 
 meter = A; J 
 
 Angle iron of equal ribs; ) 72-91 
 
 breadth of each = b; j ~ " 
 
 Angle iron of unequal ribs; ) , 2 , 1 /J9 ,. 
 
 greater, b ; less, h; / + /4 "> 
 
 Cross of equal arms ; 7i 2 -^ 24. 
 
 H-iron ; breadth of flanges, b ; } 7>2 A 
 
 their joint area, A; area of - ' 
 
 web, B; j 12 A+ B' 
 
 Channel iron; depth of flanges) r A A T> \ 
 
 + % thickness of web, h; \h* \ + uP i 
 
 area of web, B; of flanges, A; j I 12 (A + B) 4 (A + B) 2 J 
 
 All the dimensions being in the same units of measure. 
 
 444. Collapsing of Tubes. When a thin hollow cylinder, such 
 as an internal boiler flue, is pressed from without, it gives way by 
 collapsing, under a pressure whose intensity was found by Mr. 
 Fairbairn (Pkilos, Trans., 1858) to vary nearly according to the 
 following laws : 
 
 Inversely as the length ; 
 
 Inversely as the diameter ; 
 
 Directly as a function of the thickness, which is very nearly 
 the power whose index is 2-19 ; but which for ordinary practical 
 purposes may be treated as sensibly equal to the square of the 
 thickness. 
 
 The following formula gives approximately the collapsing pressure, 
 p, in Ibs. on the square inch, of a plate-iron flue, whose length, I, 
 diameter, d, and thickness, t, are all expressed in the same units of 
 measure: 
 
 ^ = 9,672,000^ (1.)
 
 526 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 For kilogrammes on the square millimetre, the constant co- 
 efficient becomes 6,800. 
 
 Mr. Fairbairn having strengthened tubes by rivetting round 
 them rings of T-iron, or angle iron, at equal distances apart, found 
 that their strength is that corresponding to the length from ring to 
 ring. 
 
 He also found that the collapsing pressure of a tube of an elliptic 
 form of cross-section is found approximately by substituting for d, 
 in the preceding formula, the diameter of the osculating circle at 
 the flattest part of the ellipse; that is, let a be the greater, and b 
 the lesser semi-axis of the ellipse ; then we are to make
 
 527 
 
 CHAPTER III. 
 
 OF SPECIAL PRINCIPLES RELATING TO STRENGTH AND STIFFNESS 
 IN MACHINES. 
 
 445. Subjects of this Chapter. In the designing of machines 
 with a view to sufficient strength and stiffness, certain special 
 principles must be kept in view besides those general principles 
 which are applicable to machines in common with structures. The 
 first section of this Chapter gives a summary of those principles', 
 the remaining sections relate to the strength and stiffness of certain 
 special parts of machines. 
 
 SECTION I. Summary of Principles. 
 
 446. lioad in machines. In most examples of machinery the 
 whole load must be treated as a live load, because of its action 
 being accompanied with vibration ; and also in many cases because 
 the straining action of the load operates upon different sets of 
 particles in succession, and comes with more or less suddenness upon 
 such sets of particles. In some of these latter cases the straining 
 action of the load upon a given particle is periodically reversed ; 
 for example, the bending moment exerted on a rotating shaft causes 
 alternate tension and thrust to be exerted upon the same particle, 
 as it passes alternately to the stretched and to the compressed side 
 of the axle. 
 
 Hence the real factor of safety in machinery is seldom less 
 than 6. 
 
 There are exceptional cases in which, owing to the smoothness 
 of the motion and the steadiness of the straining action, the load 
 may be considered as intermediate between a dead load and a live 
 load, so that a smaller factor of safety is sufficient; such, for 
 example, as the transmission of power through bands of such length 
 as to hang in a sensibly curved form. 
 
 447. Straining Actions computed from Power. The straining 
 
 actions on moving pieces can be in some cases wholly, and in others 
 partly, determined from the power transmitted, and from the speed, 
 by methods of calculation which will be described and exemplified 
 further on. The cases in which the straining action can be wholly 
 determined from the power transmitted are those which fulfil the 
 following conditions : uniformity of effort, absence of lateral com- 
 pon&nts in the straining forces, and smallness of the straining
 
 528 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 actions due to the weight and to the re-action of the piece itself, 
 and of pieces carried by it, so that those parts of the straining 
 action may be treated as insensible. 
 
 The rules for computing straining actions from power trans- 
 mitted are the following : 
 
 I. To compute the effort exerted along a given line of con- 
 nection; divide the power transmitted, in units of work per second, 
 by the common component along the line of connection of the 
 velocities of the connected points. 
 
 If the power is given in horses-power, reduce it in the first place 
 to units of work per second, by multiplying by 550 for foot-lbs., or 
 by 75 for kilogrammetres. 
 
 II. To compute the straining moment exerted through a given 
 rotating piece; divide the power transmitted, in units of work in 
 a given time, by the angular motion in the same time : that is, by 
 2 if times the number of turns in that time. 
 
 In symbols, let U be the power, in units of work per minute ; 
 N, the number of revolutions per minute; M, the straining mo- 
 ment; then 
 
 U 0-159155 U 
 M = - ^ = - ^ (1.) 
 
 2 "f N N 
 
 This formula gives the moment in the same denomination with 
 the work. If the work is given in foot-lbs. per minute, and the 
 moment is required in inch-lbs., the above expression must be 
 multiplied by 12 ; that is, 
 
 M = H ^ = ' XT (2.) 
 
 2 -N N v ' 
 
 Let H P denote the number of horses-power transmitted 3 so 
 that 
 
 U in foot-lbs. per minute = 33000 H P ; and 
 U in kilogramnostres per minute = 4500 HP; 
 
 then we have 
 
 63000 H P 
 
 M in inch-lbs. = 
 M in foot-lbs. = 
 
 N 
 
 5250 H P 
 N j 
 
 TIT- i-i :u 716-2 HP ' 
 
 Jtt in kilogrammetres = r^ (y.) 
 
 The_ formula for kilogrammetres is adapted to the French horse- 
 power, which is about one-seventieth part less than the British.
 
 STBAINING ACTIONS IN MACHINES. 52!> 
 
 In the cases in which part only of the straining action can be 
 determined from the power transmitted, the causes of additional 
 straining action are the following: Excess of maximum effort 
 above mean effort; lateral components in straining forces; weight 
 of the piece itself and of pieces carried by it ; re-actions of the piece 
 itself and of pieces carried by it, when undergoing acceleration or 
 retardation. It has already been stated in Article 414, page 488, 
 that such additional straining actions are sometimes calculated 
 expressly, and sometimes allowed for by using an apparent factor 
 of safety greater than the mean factor of safety in a suitable pro- 
 portion. 
 
 There are cases in which the best method of calculating the 
 straining action is to determine directly the greatest load, without 
 reference to the power transmitted. 
 
 448. Alternate Strains. Pieces are often met with in machinery 
 which are strained alternately in opposite directions, such being 
 especially the case when the motion is reciprocating : for example, 
 the piston-rod and connecting-rod of a steam engine, which are 
 subjected alternately to tension and to thrust; and the beam of a 
 steam engine, which is exposed alternately to bending actions ia 
 opposite directions. Such pieces must be adapted to resist effi- 
 ciently the straining action in either direction, and especially that 
 which is most severe. This principle is applicable to framing as 
 well as to moving pieces. 
 
 449. straining Effects of Re-action. When the particles of a 
 piece undergo changes of speed and direction, their re-actions pro- 
 duce straining effects resembling those produced by their weights; 
 due regard being had to the directions of those re-actions, and t<> 
 the ratios which they bear to the weights of the particles. For 
 example, if a particle of the weight w undergoes the acceleration 
 d v } in, the time d t, the re-action of that particle is 
 
 _ 
 
 9 dt"~ "' 
 
 and is exerted in a direction opposite to that of the acceleration 
 (Article 287, page 330) ; and if a particle of the weight w revolves 
 with the angular velocity a, in a circle of the radius r, its re-action 
 (or centrifugal force) is 
 
 and is exerted in a direction away from the centre of the circle 
 (Article 288, page 330). 
 
 In many cases of reciprocating motion in machinery, the motion 
 of the reciprocating mass is harmonic (as to the meaning of which, 
 see Article 239, page 250); and then its greatest re-action is equal 
 
 2 M
 
 530 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 to what its centrifugal force would be if it revolved in a period 
 equal to the time of a double stroke, in a circle of a radius equal 
 to the half-stroke. Let T be the period, or time of a double stroke 
 in seconds; x, the half-stroke; w, the weight of the reciprocating 
 mass; then its greatest re-action is 
 
 w x /Q x 
 
 * ............................. (3 
 
 4 *2 q 
 
 The co-efficient - is the reciprocal of -~z, which is, as already 
 g 4 * z 
 
 stated in Article 319, page 364, the altitude of a revolving 
 pendulum whose period is one second; that is, nearly, 0-815 foot, 
 or 9-78 inches, or 248 millimetres. 
 
 The moment of re-action of a mass which undergoes an accelera- 
 tion of angular velocity, d a, in the interval of time d t, is given by 
 the expression 
 
 Ida ... 
 
 in which I denotes the moment of inertia of the rotating mass 
 (Article 313, page 358). If the mass has a rocking or oscillating 
 motion, following the harmonic law, about its axis, the greatest 
 moment of re-action is as follows : 
 
 4*2 16 
 
 in which T is the periodic time of a complete or double oscilla- 
 tion, and 6 the semi-amplitude; that is, the angle in circular 
 measure through which the stroke, or oscillation, extends to each 
 
 side of the middle position of the rocking body. Values of j - 2 
 
 have already been given. 
 
 The moments of re-action given by the formulae (4) and (5) may 
 constitute twisting moments upon shafts, or bending moments 
 upon levers. 
 
 450. Framework. The load which strains the framework of a 
 machine consists partly of the weight of that framework itself; but 
 principally of the bearing-pressures exerted by the moving pieces. 
 How those bearing-pressures are to be determined has already been 
 shown in the course of Part II., Chapter IV., Section I. The 
 framework ought to be so designed as to make the bearing- 
 pressures at different points, or the components of those bearing- 
 pressures, as far as possible balance each other. "When this 
 principle is perfectly carried out, the pressure exerted by the 
 machine on its foundation will consist simply of its weight; all
 
 FRAMEWORK STIFFNESS AND PLIABILITY. 531 
 
 the horizontal components of the bearing-pressures, and all the 
 bearing-pressures which act in couples, being mutually balanced. 
 This, however, is possible only when the prime mover, the 
 working machinery, and the material operated upon, are all 
 carried by one connected assemblage of framework. In other 
 cases, all that can be attained is an approximation to the balance of 
 horizontal pressures and of couples. When two bearings occur 
 near each other that are exposed to opposite pressures, or to 
 pressures containing opposite components, it is in general ad- 
 visable, in designing the frame, to connect those bearings with 
 each other as directly as possible, by means of a strut or of a tie. 
 
 451. stiffness and Pliability. In all cases in which precision of 
 movement is required, stiffness is essential both to the moving 
 pieces and to the framework of a machine. It is ensured, first, 
 by causing the pieces exposed to strain to resist it as far as 
 practicable by direct tension and direct thrust, rather than by 
 twisting or bending stress (Article 420, page 493; and Article 
 442, page 522); and secondly, where indirect modes of exerting 
 stress are unavoidable, to give the piece such transverse dimensions 
 as are necessary in order to prevent the extent to which it yields 
 from exceeding a certain limit (Article 430, page 502; Article 
 439, page 517). According to the first of those principles, the 
 framework and the moving pieces of a machine, where rigidity is 
 required, should consist, as far as practicable, of struts and ties; 
 according to the second principle, where beams have to be used, 
 the depth and span, and where shafts have to be used, the diameter 
 and span, are to be so proportioned to each other as to prevent 
 the ratio of the deflection to the span from exceeding a certain 
 
 limit (usually from to ftnn ). The special rule applicable 
 
 \ 1,^00 ^,UUU/ 
 
 to shafts will be given further on. As to beams, see page 520. 
 
 On the other hand, there are cases in which absolute precision 
 of movement is unnecessary, and in which pliability is an advan- 
 tage, as giving the power of withstanding shocks. This advantage 
 is possessed by leathern belts, and by raw hide and hempen ropes, 
 because of the great extensibility of the materials. Wire ropes, 
 when stretched tight, possess it to a less degree ; but when of a 
 span sufficient to- hang visibly in curves, the power of alteration 
 of curvature constitutes a kind of pliability, which enables shocks 
 to be borne ; and the same remark applies to chains when hanging 
 slack. Pliability in the shape of compressibility, where thrust has 
 to be resisted, as in connecting-rods, is obtained by using timber, 
 as has already been stated in Article 409, page 474. Beams, and 
 pieces acting as beams, are made flexible to any extent required, by 
 making the depth sufficiently small in comparison with the span, 
 the breadth being at the same time made sufficiently great to give
 
 532 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 the requisite strength; or by using tough and pliable kinds of 
 timber, such as those mentioned in Article 409, page 473, as 
 possessing those qualities. 
 
 452. Compound stress. Both in moving pieces and in framework, 
 but especially in moving pieces, straining actions of different kinds 
 are sometimes compounded : as direct tension or direct thrust with 
 bending, or bending with twisting. In such cases the resultant 
 stress arising from the combination must be taken into account. 
 The rules applicable to the cases of this sort which commonly occur 
 in practice will be given in the course of the ensuing sections of 
 this Chapter. 
 
 SECTION II. Special Eules as to Bands, Rods, and Links. 
 
 453. Belts and Cords at moderate Speeds. The effective working 
 
 tension required at the driving side of a band is to be found by the 
 rules already given in Article 310 A, pages 351, 352. When the 
 speed at which the band runs is such that the centrifugal tension 
 may be disregarded, and when the band is a belt or cord of 
 organic material, such as leather, raw hide, gutta percha, or hemp, 
 the working tension is to be divided by a suitable co-efficient of 
 working strength, so as to give, according to the nature of the 
 co-efficient employed, either the weight per unit of length, or the 
 sectional area ; or, in the case of flat belts of a given thickness, the 
 breadth; or, in the case of cords, the square of the diameter, or 
 the square of the girth. Co-efficients adapted to those different 
 methods of calculation, and to different materials, have already 
 been given in Article 410, pages 474, 475, 476. 
 
 454. Allowance for Centrifugal Tension. When the speed is SO 
 
 great that it becomes necessary to allow for centrifugal tension, 
 the co-efficient of working strength to be used is that which is 
 expressed in the form of an equivalent length of the band itself. 
 Let that length be denoted by b. Let v be the velocity with which 
 the band is to run; then the centrifugal tension, expressed in 
 
 v z 
 length of band, is ; and this is exerted at every point of the 
 
 band, in addition to the effective tension required for the trans- 
 mission of power; so that after deducting the centrifugal tension, 
 the strength which remains available to resist the effective tension is 
 
 (I-) 
 
 when expressed in length of band. Therefore, let Tj be the 
 effective working tension required at the driving side of the band; 
 w S, the weight of an unit of length of the band required ; then
 
 SPECIAL RULES AS TO BANDS. 533 
 
 M>S = -^-g- ........................... (2.) 
 
 , V* 
 -- 
 
 9 
 
 The weight per unit of length is expressed in the form of a 
 product, to S ; in which S denotes the sectional area, and w the 
 heaviness of the material. 
 
 455. Wire Ropes present a case in which direct tension is 
 combined with an additional stress produced by the bending of the 
 wires round the pulleys. Let D be the diameter of a pulley; d, 
 that of a single wire; E, the modulus of elasticity of the wire; then. 
 the bending produces a stress which is tensile at one side of the 
 wire, and compressive at the other, and whose intensity, in units 
 
 p* 7 
 
 of weight on the unit of area, is -~- ; and in length of the rope, 
 
 HI 7 
 
 =: : w being the heaviness of the material. 
 w D 
 
 Let b, as before, be the safe working strength expressed in length 
 of rope ; v, the velocity at which the rope runs ; then the strength 
 in length of rope, available to resist the effective working tension 
 at the driving side, is 
 
 .pr 
 wD 
 
 n\ 
 (1.) 
 
 and the weight per unit of length, w S, of a rope suited to bear the 
 effective working tension T 1? is given by the following equation : 
 
 g w 
 
 The most convenient way of using this formula is to fix a mini- 
 mum value for the ratio D -4- d, in which the diameter of the pulley 
 is to exceed that of a single wire, and thence to deduce the value of 
 the stress E d + w D, produced by bending. Then, having calcu- 
 lated the weight, w S, per unit of length, the diameter, d, of a single 
 wire is to be deduced from that weight, and the least prope* 
 diameter for a driving pulley, D, by multiplying d by the previously 
 fixed ratio. 
 
 An ordinary value of D -f- d is 2000. 
 
 A wire rope of the ordinary construction consists of six strands 
 spun round a hempen core; and each of the strands consists of six 
 wires spun round a smaller hempen core, so that there are thirty- 
 six wires in all. The diameter of a single wire is given with suf- 
 ficient accuracy for the present purpose by the formula,
 
 534 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 . . / (w S in Ibs. per foot\ , . 
 
 a (in fractions of an inch) = A/ I * J.(p.) 
 
 or 
 
 (in millimetres) = ^(4^ w S in kilogrammes per metre). (3A.) 
 
 The following are values of the moduli of elasticity and strength 
 for ropes made of the best charcoal iron wire. Steel wire ropes 
 may be taken as having about the same modulus of elasticity, and 
 as being stronger than iron in the proportion of 4 to 3 nearly. 
 
 Feet of Metres of 
 
 Rope. Rope. 
 
 E 
 
 Modulus of Elasticity, -, ....... 7,500,000 2,286,000 
 
 Ultimate Tenacity, ............... 26,880 8,193 
 
 Proof Tension, .................... ^,44 4>96 
 
 Working Tension with steady 
 
 action (factor of safety, 3|),* 7,680 2 >34 
 Working Tension with unsteady 
 
 action (factor of safety, 6),... 4,480 T )3^5 
 
 456. Deflection and Length of Bands. The form in which a band 
 hangs between two pulleys which it connects, is that of a catenary. 
 In cases which occur in practice, the parabola may be used as 
 an approximation to the catenary, without sensible error. This 
 gives the following approximate formula for the deflection of the 
 band at the middle of its span, below a straight line joining its 
 two points of suspension : 
 
 in which c is the half-span, measured along the before-mentioned 
 straight line, whether horizontal or sloping; b is the length of 
 rope equivalent to the available tension (exclusive of centrifugal 
 tension), and y is the deflection. 
 
 Let i be the angle of inclination of the span of the band to the 
 horizon; and s the length of the part of the band which hangs in 
 a curve between the two points of suspension; then 
 
 4 ?/ 2 
 s nearly = 2 c + -rr. cos 2 i ................ (2.) 
 
 3 c 
 
 When the span is horizontal, cos 2 i = 1. 
 
 The driving and returning parts of the band have different 
 
 * This value of the working tension is calculated from the co-efficient of 
 stress given by Reuleaux, as applicable to Hirn's telodynamic transmission. 
 (Constructionslehre fur Maschinenbau, 329.)
 
 BANDS CHAINS TENSION-RODS. 535 
 
 tensions (see Article 310A, page 352), and therefore different de- 
 flections. Their lengths are to be calculated separately, and added 
 together, along with the lengths of the circular parts of the band 
 which pass round the pulleys. 
 
 457. Chains. Chains consisting of oval links, when the tendency 
 of each link to collapse is resisted by means of a cross-bar called a 
 stay or stud, as in fig. 279, have a strength equal to 
 
 that due to the collective sectional area of the two 
 
 sides of the link. The tenacity of the iron in the 
 
 link is reduced by the processes of forging and welding 
 
 so as to be from f to f of that of the cable-iron 
 
 bolt from which it is made; so that, taking the 
 
 ultimate tenacity of cable-iron bolts at 60,000 
 
 Ibs. on the square inch, that of a stud chain is 
 
 from 52,500 to 45,000 Ibs. on the square inch; 
 
 and about 7,500 Ibs. on the square inch may be 
 
 taken as a safe working modulus of tension with a 
 
 live load : the smaller of the preceding co-efficients 
 
 being divided by 6 as a factor of safety. The test load is about half 
 
 the breaking load, or three times the working load. An unstudded 
 
 chain has about two-thirds of the strength of a studded chain of 
 
 the same dimensions. 
 
 The following are the ordinary proportions of the links of a 
 stud-chain, as used for ships' cables and rigging, in terms of the 
 diameter of the bolts from which they are made. 
 
 Length: outside, 6 diameters; inside, 4 diameters. 
 
 Breadth: outside, 3^ diameters; inside, 1^ diameter. 
 
 Thickness of stay : at ends, 1 diameter ; at middle, T*V diameter. 
 
 The weight of a stud- chain, of these proportions, in Ibs. per 
 foot, is found by multiplying the square of the diameter of the 
 cable-iron in inches by 9, very nearly; and its weight in kilo- 
 grammes per metre, by multiplying the square of that diameter 
 in millimetres by 0-0208. 
 
 In designing chains made of flat links connected by pins, regard 
 must be paid to the principles of Article 424, page 497, so as to 
 give the dimensions of the pins their due proportions to those of 
 the links. 
 
 458. Rods or Links for Tension. The following are the rules 
 applicable to the ordinary cases of rods or links for transmitting 
 tension ; such as piston-rods in single-acting steam engines. 
 
 I. When the resultant tension acts along the longitudinal axis 
 of the rod, that is, along a straight line traversing the centres of all 
 the cross-sections, the area of cross-section is to be proportioned to 
 the load according to the rules of Article 420, page 494; the 
 modulus of greatest working stress being taken at 9,000 Ibs. 
 on the square inch for wrought iron; 2,500 for cast iron (which,
 
 536 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 if shocks are to be borne, is not a suitable material for this purpose); 
 and 1,000 for timber of straight-grained and tough kinds. 
 
 II. Should the resultant tension act, not along the axis of the 
 rod, but at a distance from it, whose greatest value may be denoted 
 by x, let P denote the load ; then the tensile action is combined 
 with a bending moment, P x. 
 
 Let S be the area of cross- section ; h, the depth of the rod that is, 
 its diameter in the plane of the bending moment ; k, the numerical 
 factor in equation (5 A) of Article 437, page 514, and in the Table 
 of page 516; the greatest additional intensity of tension produced 
 by the bending moment is 
 
 Pa; 
 
 and the total intensity of the greatest tension is 
 P 
 
 and consequently, if ./be the modulus of working stress, the proper 
 sectional area is given by the formula, 
 
 s = ^ 
 
 n) 
 
 III. In a tension rod which is horizontal or inclined, the 
 additional stress produced by the bending action of its own weight 
 may require to be taken into consideration. Let w be the heavi- 
 ness of the material; c, the half -span between the points of 
 support measured along the axis of the rod; i, the angle of 
 inclination of that axis to the horizon; then the bending moment is 
 
 ,.. w S c 2 cos i 
 
 n/\ _ _ ___ 
 
 " 2~ 
 and the greatest stress produced by that moment is 
 
 M we 2 cost ._. 
 
 2kh 
 
 The easiest way to make use of this formula in practice is to 
 assume in the first place a convenient value for h; calculate p by 
 equation (2) ; and then make 
 
 --. 
 
 f-p 
 
 (3.) 
 
 IV. If the rod has a transverse reciprocating motion, the re-action 
 due to that motion will produce alternate bending actions in.
 
 STRENGTH OF TENSION-RODS AND LINKS. 5C7 
 
 opposite directions. Let z be the semi-amplitude of the transverse 
 motion that is, half its total extent; let n be the number of 
 double swings in a second; then make 
 
 (in which j^ = 0-815 foot = 0-248 me-tre nearly); then the 
 
 additional stress produced by this motion is to be found by putting 
 m w instead of iv cos i iu eq\iation (2), and is to be provided for in 
 the same way with the stress p of that equation. If one end of 
 the rod has a transverse reciprocating motion, while the other has 
 no such motion, or if the two ends have motions of different 
 amplitudes, make z equal to the semi-amplitude of the transverse 
 motion of the centre of the rod ; the result will be near enough 
 to the truth for practical purposes. 
 
 V. If weight and re-action both take effect in the same vertical 
 plane, make 
 
 we 2 ... 
 
 and xise this value of p in equation (3). 
 
 VI. In designing the fastenings for transmitting the tension at 
 the ends of rods or links, regard is to be had to the principles of 
 Article 424, page 497. 
 
 VII. The sides of an eye at the end of a tension-rod are usually 
 made so as to have a collective sectional area equal to once-and-a- 
 halfthat of the rod; because the uneven distribution of the stress 
 in them diminishes their strength to about two-thirds of what it 
 would be if the stress were uniform ; and the same rule is appli- 
 cable to a strap answering the purpose of an eye. 
 
 459. Rods or i>iui(H for Reciprocating stress. When a rod trans- 
 mits alternately tension and thrust of equal amount, the most 
 severe straining action is usually that produced by the thrust; and 
 the proper dimensions are to be found by the rules of Article 443, 
 page 524. 
 
 The piston-rods and connecting-rods of double-acting steam 
 engines are examples of this. Piston-rods are to be treated as 
 struts fixed at one end and jointed at the other; connecting-rods 
 as struts jointed at both ends. 
 
 The ways of using the rules referred to are as follows : 
 
 I. In ordinary cases of the piston-rods and connecting-rods of 
 steam engines, an approximate sectional area may be calculated 
 in the first place, by multiplying the area of the piston by the 
 greatest intensity of the effective pressure of the steam, so as to
 
 538 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 find the greatest working load, and dividing it by a modulus of 
 working stress, which may be 2,500 Ibs. on the square inch for a 
 piston-rod, and 1,750 Ibs. on the squai-e inch for a connecting-rod 
 at the middle of its length. It is usual to give the connecting-rod 
 a swell in the middle; so that at the ends it is of the same area 
 with the piston-rod. The ratio of the greatest to the least 
 diameter is about that of 6 to 5. 
 
 This rule may be considered as giving a safe value for the trans- 
 verse section, when the length of the rod does not exceed about 
 thirty-six times its diameter. When the proportion of length to 
 diameter is greater, the following rule may be applied to wrought- 
 iron rods. 
 
 II. Let f be the intensity of the safe working thrust along a 
 short rod say 6,000 Ibs. on the square inch, or 4 - 2 kilogrammes 
 on the square millimetre, for wrought iron. Let S be the sectional 
 area of the rod; I, its length; h, its diameter; P, the amount of 
 the greatest working thrust; then we have 
 
 P _ / S _ . n \ 
 
 2 > ........................ \ / 
 
 in which the values of c 1 are as follows : 
 
 For a round rod, jointed at both ends, .............. 562 
 
 fixed at one end, ................... I ? 2 7o 
 
 fixed at both ends, ............... 2,250 
 
 For a square rod, j ointed at both ends, .............. 7 50 
 
 fixed at one end, ................... 1,690 
 
 fixed at both ends, ................. 3,ooo 
 
 But in a round rod, we have h 2 - 1-273 S nearly; and in a square 
 rod, h 2 = S; consequently we may make 
 
 c 1 ;* 2 ^ S; ......................... (2.) 
 
 in which a has the following values : 
 
 a. 
 For a round rod, jointed at both ends, .............. 7*6 
 
 fixed at one end, ................... 1,611 
 
 fixed at both ends, ................. 2,864 
 
 For a square rod, j ointed at both ends, .............. 75 
 
 fixed at one end, ................... 1,690 
 
 fixed at both ends, ................. 3,ooo 
 
 Thus equation (1) may be made to take the following form : 
 
 P (a S + Z 2 ) = a/S 2 ; .................... (3.) 
 
 which quadratic equation, being solved, gives the following formula 
 for the sectional area :
 
 HODS FOR RECIPROCATING STRESS. 530 
 
 III. Sometimes, as has already been stated in page 474, a 
 connecting-rod is made of a square bar of wood to transmit the 
 thrust, bound with a wrought-iron strap to transmit the tension. 
 In this case the transverse area of the strap is to be determined by 
 the principles of Article 458, with a working modulus of tension 
 of about 6,000 Ibs. on the square inch (because the stress may be 
 unequally distributed near the ends of the strap, where it bends 
 round the bushes that hold the pins); and that of the square 
 timber bar is to be found by equation (4) of the preceding rule, 
 with the following values for the constants : 
 
 f = 720 Ibs. on the square inch; 
 
 = 0-5 kilogramme on the square millimetre; 
 
 a = 62-5. 
 af = 45,000 Ibs. on the square inch; 
 
 = 32 kilogrammes on the square millimetre. 
 
 In a compound rod of this kind there is a tendency to slacken 
 the hold of the pins which it connects, through the shortening of 
 the compressed bar; and means must therefore be provided of 
 tightening it from time to time by wedges or screws. 
 
 IV. When the form of cross-section chosen is such that the 
 ratio I 2 -f- r 2 , which the length of the rod is to bear to the radius 
 of gyration of its cross-section, can be approximately determined 
 independently of the sectional area, that area is to be found simply 
 by the following formula : 
 
 ~Tt r T> \ 
 
 (5-) 
 
 For values of r 2 and of c, see Article 443, pages 524, 525. 
 
 The commonest example of the class of figures to which this 
 method is applicable is the cross-shaped section, which is well 
 adapted for transmitting thrust; and for which we have 
 r 2 = h 2 -j- 24; and consequently 
 
 cr 2 = 1500 A 2 (6.) 
 
 "V. The braced connecting-rod may be looked upon as a modifi- 
 cation of the compound rod mentioned in Rule III. of this Article. 
 The thrust-bar occupies the middle of the combination, and may 
 be of timber or of cast iron. From the middle of its length diverge 
 four transverse arms, in the shape of a cross; and four tie-rods 
 extend from end to end of the bar, and at the middle of their 
 length are made to spread asunder by the arms of the cross. The
 
 540 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 length of those arms is, in a great measure, a matter of practical 
 convenience ; about one-twelfth of the length of the compound rod 
 is usual. The effect on the thrust-bar is to increase its strength, 
 so as to be equal to that of a bar of the same section and half 
 the length ; that is to say, in equations (4) and (5), for a and c, 
 substitute 4 a and 4 c. The amount of stress on the tension-rods 
 is increased in the same ratio with their length. 
 
 SECTION III. Special Rules relating to Axles and Shafts. 
 
 460. Oeucral Explanation* a* to Shafts, Axles, and Journals. 
 
 The words axle and shaft are, to a certain extent, used indiscri- 
 minately ; but it may be held that in most cases the term shaft 
 implies the transmission of motive power along the rotating piece 
 denoted by that term, and consequently the exertion of a twisting 
 moment at each cross-section, to be found by the principles of 
 Article 447, page 528; while an axle fh general is subjected to a 
 bending moment only. 
 
 The parts of a shaft which rest on the bearings ai-e called pivots, 
 collars, gudgeons, and journals. 
 
 Pivots and collars are for bearing end-thrust (see Article 311, 
 page 353). Gudgeons and joui'nals are for bearing transverse 
 pressure. It was proposed by Buchanan, in his Treatise on Mill- 
 work, to apply the word gudgeon only to the bearing part at the 
 end of a shaft or axle, which is exposed to bending action alone, 
 and not to twisting action; and journal to an intermediate bearing 
 part through which a twisting moment is or may be exerted ; but 
 the custom of using the word journal in both senses indiscrimi- 
 nately is so prevalent, that it is impracticable to carry out Buchanan's 
 suggestion. The terms end-journal and neck-journal, or simply 
 neck, may serve to distinguish them. 
 
 Cast iron may be used for shafts where no shocks are to be 
 borne. In other cases the proper materials are wrought iron and 
 mild steel. The greatest proper values of the real modulus of 
 working stress are the following, or nearly so : 
 
 Lbs. on the Kilos, on the 
 
 square inch. square millimetre. 
 
 Cast Iron, 4>5 3'i6 
 
 Wrought Iron,* 9,000 6-33 
 
 Steel, i3>5 9'5 
 
 As to the use and treatment of cast iron for this purpose, as 
 well as for machinery in general, see Article 390, pages 453-455. 
 In the case of wrought-iron shafts, it is important that the 
 
 * The working modulus for wrought iron is the result of a series cf 
 practical trials of railway carriage axles, extending over many years, by 
 Lieutenant David Rankine aud the Author.
 
 JOURNALS GUDGEONS. 541 
 
 continuity of the fibres at and near the surface should be as little 
 broken as possible, and that where the stress is severe there should 
 be no re-entering angles in the outline ; for at places where the 
 fibres are interrupted, and at re-entering angles, cracks are apt to 
 commence, which gradually extend inwards, and at length reduce 
 the sound part of the axle to so small a diameter that it snaps in 
 two. This process has been known to occupy two or three years, 
 and sometimes more. (See Article 391, pages 456, 457; and the 
 authorities referred to in. the second note to page 456.) Hence, 
 when it is necessary that a journal should be of a diameter 
 materially smaller than the main body of the axle, the parts of 
 different diameters should be connected by a curved and not by an 
 angular shoulder; and the reduction of diameter should as far as 
 possible be produced by forging, rather than by cutting or turning ; 
 the process of turning being used only to give precision to the 
 shape. 
 
 461. Gudgeons or End-Journals. A gudgeon, or journal at the 
 end of a shaft, exposed to the transverse bending action of the 
 bearing pressure only, has to fulfil two conditions : to have a trans- 
 verse section sufficient to bear the bending moment safely; and to 
 have a longitudinal section sufficient to prevent the unguent from 
 being forced out by the pressure. 
 
 Let "W denote the greatest working load, or bearing pressure; x, 
 the length of the gudgeon ; h, its diameter. Let p be the proper 
 value of the pressure per unit of area of longitudinal section (as 
 to which, see Article 310, page 350); and let f be the modulus of 
 working stress. Then the condition as to proper lubrication is 
 expressed by the following equation : 
 
 pxh = W; ........................... (1.) 
 
 and the condition as to strength by 
 
 'fh3 = Wx; ........................ (2.) 
 
 in the latter of which equations provision is made for the con- 
 tingency of the whole load being concentrated on the outer end 
 of the gudgeon. The fact is, that in well-made and well-fitted 
 machinery, the resultant load acts through the middle of the 
 length of the gudgeon, or very nearly so; and that the bending 
 
 *\V *c 
 moment at the shoulder is only about ^ , instead of W as; but 
 
 a 
 
 the error is on the safe side. By elimination from those two 
 equations are obtained the following formulse : 
 
 /32 W 2 \l 
 diameter, h = ( yj 4 ; (3.)
 
 542 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 length, * = h- 
 
 The following is a convenient form of equation (3) for practical 
 use : Let 
 
 The following are examples of the values of the co-efficients. In 
 each case, f 9,000 Ibs. on the square inch, or 6-33 kilogrammes 
 on the square millimetre. 
 
 p. Ibs. on the square ) 
 
 1 inch, ............ . ..... \ 4 5o 300 225 150 
 
 p. kilogrammes on the 1 /- f 
 
 et .11. v. > O'^IO O'2II O'lSS O'IO6 
 
 square millimetre,... J 
 
 a, Ibs. per square inch, 630 ^514 446 364 
 
 a, kilogrammes per ) , 
 
 squarl millimetre,...} '^ '3 6 * 0-314 0-256 
 
 o 
 
 p .......................... i '40 1-71 1-98 2-42 
 
 In the case of cast iron, for which the working modulus of 
 stress is one-half of that for wrought iron, each of the values 
 of a and of x -f- h in the above table is diminished in the ratio of 
 1: J2 = 0-707 : 1. 
 
 Another method of calculation is to fix an arbitrary value for 
 
 sc 
 
 the ratio of the length to the diameter; say -= = m; then we have, 
 
 'i 
 
 by equation (2), 
 
 and consequently 
 
 if a' be taken to denote 
 
 32m' 
 
 The values of the ratio of length to diameter, approved by 
 Fairbairn in his Treatise on MiUwork, are, for cast iron, 1-5, 
 and for wrought iron, 1-75. The following table shows the corre- 
 sponding values of a'.
 
 GUDGEONS CRANK- PINS BEARING AXLES. 543 
 
 Cast Iron. "Wrought Iron. 
 
 y, Ibs. on the square inch, ................... 4,500 9,000 
 
 f, kilogrammes on the square millimetre, 3'i6 6-33 
 
 a', Ibs. on the square inch, .................. 294 504 
 
 a, kilogrammes on the square millimetre, 0-207 '345 
 
 All the rules for end-journals apply also to crank-pins. 
 
 462. Bearing Axles is a term which may be used to distinguish 
 those axles which have to bear a bending action, but not a twisting 
 action. 
 
 An axle in this condition is to be treated as a beam; the 
 bending moments at a series of cross-sections being calculated by 
 the rules of Articles 434 to 436, pages 505 to 510. In making 
 those calculations, it is xisual to assume, as in the preceding Article, 
 that the bearing-pressures act through the extreme ends of the 
 gudgeons. 
 
 The cross-section adopted must be one adapted to resist bending 
 actions in all directions; such as a circle, solid or hollow, a solid 
 octagon, or a cross. In wrought iron and steel, the solid and 
 hollow circular sections are suitable; in cast iron, the hollow 
 circular and the cross-shaped sections; in timber, the solid 
 octagon. 
 
 In the case of circular sections, the diameter is to be calculated 
 from the bending moment by equation (7 A) of Article 437, page 
 514: that is to say, let M be the greatest working bending 
 moment ; f, the modulus of working stress ; m, the ratio of the 
 diameter of the hollow (if any) to the outside diameter; then 
 
 / 32 M \\ M 
 
 = 
 
 \\ f 
 
 / = \ (1 - 
 
 if a" be taken to denote f -f- 32. The following are examples of 
 the co-efficients : 
 
 Cast Iron. Wrought Iron. Steel. 
 
 /, Ibs. on the square inch, ................... 4,5oo 9,000 13,500 
 
 f, kilogrammes on the square millimetre, 3-16 6-33 9-5 
 
 ", Ibs. on the square inch, .................. 441 882 1,323 
 
 ", kilogrammes on the square millimetre, 0*31 0-62 0-93 
 
 It is of course to be understood that, when British measures are 
 used, M is to be expressed in inch-lbs. 
 
 The solid octagonal section may be considered as practically 
 equivalent to its inscribed circle. 
 
 For the cross-shaped section, the outside diameter, h, is in the first 
 place to be fixed; and then the requisite sectional area is to be
 
 544 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 calculated by equation (7 B) of Article 437, page 514, making 
 k = ^- in that equation; that is to say, 
 
 3 - 12 M 
 
 " 
 
 fh '" 
 
 Bearing shafts are frequently tapered from the place of greatest 
 bending moment towards the points of support, so as to give a 
 longitudinal section of equal strength (as to which see Article 438, 
 page 517), or as near an approximation to such a section as is con- 
 sistent with practical convenience. 
 
 463. Neck-jTonrnaia. A neck-journal (often called simply a neck, 
 and by Buchanan, a journal) is an intermediate part of a shaft or 
 axle, turned to a smooth and truly cylindrical surface, so as to fit 
 its bearing easily, as stated in Ai'ticle 371 A, page 424. If it is 
 exposed to bending action only, its diameter is to be determined by 
 the rules of the preceding Article ; if to twisting action, or to 
 twisting and bending combined, by rule which will be given in 
 the ensuing Articles. 
 
 The lengths of neck-journals are to be calculated so as to give 
 the requisite area for bearing the pressure, according to the prin- 
 ciples of Article 310, page 350. 
 
 464. Shafts under Torsion. A shaft which transmits motive 
 power is exposed to a twisting moment throughout its whole 
 length. The first step towards determining the proper diameter 
 for the shaft is to calculate that twisting moment from the power 
 to be transmitted and the speed of rotation, according to the 
 principles of Article 447, page 528. From the power and speed, 
 the result obtained in the first place is the mean twisting moment; 
 and the greatest twisting moment may either be deduced from the 
 mean twisting moment directly, or may be provided for, together 
 with various causes of additional stress, by using a sufficiently 
 large apparent factor of safety, as already stated in Article 414, 
 page 489. 
 
 The greatest values of the real modulus of working stress in 
 shafts under torsion correspond to about 6, as a real factor of 
 safety. The apparent modulus may be considerably less when a 
 greatly increased apparent factor of safety is used. The apparent 
 factor of safety is sometimes as high as 36. 
 
 The formulae to be used in calculating the diameter of a shaft 
 from the twisting moment, or from the power and speed, as the- 
 case may be, are as follows. Let M be the moment; H P, the real 
 horses-power to be transmitted ; N, the number of revolutions per 
 minute. Then, 
 
 3//MN 
 
 A)= 
 
 3// 
 
 " V(
 
 STRENGTH OF SHAFTS. 545 
 
 The values of the co-efficients A and B being as follows : 
 
 where q 1 is the modulus of working stress, real or apparent, ac- 
 cording to the way in which the moment M is calculated : 
 
 C 8G 
 
 where C denotes the number of units of work per minute in a 
 horse-power. 
 
 The following Table shows a series of examples of the constants 
 which occur in these formulae : 
 
 FACTORS OF SAFETY. 
 
 For Cast Iron, ............................................. 6* 9 12 18 
 
 Wrought Iron, .................. 6* 9 12 18 24 36 
 
 Steel, .................... . 6* 9 13^ 18 27 36 
 
 MODULUS OF STRESS. 
 on the square) 
 
 9'6 6-4 4-3 3-2 2'i i'6 1-07 
 ,. M 
 
 V ALUES OF A = ,-5-. 
 
 h 6 
 
 ^inct^'^l 2 '7o '>So 1,200 900 600 4S o 300 
 
 M 2t5IEj 22 5 i5o loo 75 50 37-5 *S 
 
 "ARSSSSi*! '9 - 6 ' 8 4 ' 6 3 0-42 0-315 o,r 
 
 N/t 3 
 
 VALUES OF B = 
 
 HF 
 
 h in inches, 23*3 35 52*5 70 105 140 210 
 
 h in centimetres, 380 570 855 1,140 1,710 2,280 3,420 
 
 Apparent factors of safety, ranging from 6'6 to 10, are found in. 
 the paddle-shafts and propeller-shafts of steam-vessels, the real 
 factor of safety being seldom above 6. Apparent factors of safety 
 ranging from 18 to 36 are met with in the shafting of mills; and 
 they correspond to what are called "light shafting" and "heavy 
 shafting " respectively. 
 
 465. Span between Bearings of Shafts. The ordinary rules for 
 
 * Real factor of safety. 
 
 f- A grammetre is one-1, 000th part of a kilogrammotre. 
 2N
 
 546 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 fixing the greatest span between the bearings of a line of shafting 
 are based on the principle that the deflection produced by 
 the weight of the shaft itself, and by any additional transverse 
 load which may be applied to it, should not exceed a certain 
 fraction of the span. Different authors give different values for 
 that fraction. In the following formulae the value adopted is one- 
 2,000th; and the rule obtained agrees with that given by Mr. 
 Molesworth in his Pocket Book. 
 
 A. general formula for the deflection in such cases is given in 
 Article 439, page 521, and is as follows: 
 
 m being the proportion which the additional load bears to the 
 weight of the shaft; c, the half-span; w, the heaviness of the 
 material ; E, its modulus of elasticity; ri^, a numerical co-efficient; 
 and r, the radius of gyration of the cross-section of the shaft. In 
 most cases, the load may be treated as uniformly distributed; and 
 as there may be a coupling near each bearing, the shaft is to be 
 
 treated as simply supported, and not fixed, at each bearing; so that 
 
 g 
 n"' = JQ. The values of the square of the radius of gyration, for 
 
 the figures of cross-section which occur in practice, are : 
 
 h z 
 For a solid circular section of the diameter h, r 2 = ^ ; 
 
 For a hollow circular section; diameter outside, ) 2 _ h z + h' z f 
 h; inside, h'; ) r ~ \Q > 
 
 Cross-shaped section; breadth over arms, h; 
 
 r 2 = =- approximately. 
 
 E 
 The value of - may be taken at 7,500,000 feet, or 90,000,000 
 
 inches, or 2,286,000 metres. 
 
 Putting for n'" the value already stated, we have 
 
 , ,, , 5 w c 4 , ,. 
 
 v = ( L + m) . ., 2 ) an d consequently 
 
 24 E ^'r\ 
 
 "5(1 +m)w c : "^ ' 
 
 giving for the span between the bearings, 
 
 192 E v' r 2
 
 STIFFNESS AND STRENGTH OF SHAFTS. 547 
 
 "' 1 E 
 
 Let = ; let have the value stated above: and let the 
 
 c 1,000 w 
 
 section be a solid circle ; then 
 
 ' 
 in which 
 
 a = 216,000 inches = 5,500 metres nearly. 
 When the shaft is loaded with pulleys and with the tensions of 
 
 their belts, it is usual to assume m, ^ 
 
 o 
 
 The result of equation (3) is a span which is not to be exceeded 
 in the construction of a line of shafting of a given diameter. 
 
 466. Shafts under Combined Bending and Twisting Actions. 
 
 When a shaft is strained by bending and twisting actions com- 
 bined, two cases may be distinguished : first, where the bending 
 moment and twisting moment are both given, and the diameter of 
 the shaft is to be found; for example, where the pressure exerted 
 on a crank-pin produces combined bending and twisting actions on 
 a journal; and, secondly, where the bending moment is produced 
 by the weight and re-action of the shaft itself, and therefore 
 depends on the diameter. 
 
 I. When a given twisting moment, . M, is combined with a 
 given bending moment, M^ make 
 
 ^(M 2 + M' 2 ) + M' = M"; (1.) 
 
 and find the diameter required in order to bear safely a twisting 
 
 moment equal to M", by means of equation (1) of Article 464, page 
 
 543. An example of this problem, solved graphically, is shown in 
 
 fig. 280, which represents a shaft having a crank 
 
 at one end. At the centre of the crank-pin, P, is 
 
 applied the pressure of the connecting-rod; and 
 
 at the centre of the bearing, S, acts the equal and 
 
 opposite resistance of that bearing. Representing 
 
 the common magnitude of those forces by P, 
 
 they form a couple whose moment is 
 
 C 
 
 P-SP. 
 
 Draw 8 Q bisecting the angle P S M. On S Q 
 
 let faL the perpendicular P Q. From Q let fall Fig. 280. 
 
 <5 M perpendicular to S M. 
 
 Calculate the diameter of the shaft, as if to resist a twisting 
 moment, M" = 2 P S M, and it will be strong enough to resist
 
 548 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 the combined bending and twisting action of P applied at the 
 point marked P. 
 
 The greatest longitudinal stress on the particles of a shaft, 
 produced by the bending action of its own weight, and of any 
 additional load or re-action bearing a given proportion, m, to its 
 own weight, is found by applying equation (2) or equation (5) of 
 Article 458, pages 536 and 537, to the case of a solid cylindrical 
 body ; that is to say, when the proper substitutions are made, we 
 have the following value for the intensity of that stress : 
 
 . . 4 w c 2 . 
 
 ;; = (cos \ + m) - ; ..................... (2.) 
 
 in which c is the half-span between the bearings; h, the diameter; 
 w, the heaviness of the material (being, for iron, 0-278 Ibs. per 
 cubic inch, or 7,690 kilogrammes per cubic mtre); i, the inclina- 
 tion of the shaft to the horizon; and m the proportion borne by 
 the additional load and re-action to the weight of the shaft. 
 
 Let q be the real intensity of tlie greatest stress due to twisting. 
 It is given by the formula 
 
 16 M 
 
 where M is the real value of the greatest working twisting 
 
 (1 / 
 - = 5-1 nearly. J That stress is a shearing stress; 
 
 and is equivalent to tension and thrust combined, and of the same 
 intensity, q, exerted in two directions perpendicular to each other, 
 tangential to the cylindrical surface of the shaft, and making 
 angles of 45 with the axis. 
 
 The resultant stress due to the twisting stress, q, and the bending 
 stress, p, combined, is found by the following formula, which has 
 been demonstrated by writers on the internal equilibrium of 
 elastic solids : 
 
 s = 
 
 where s denotes the intensity of the resultant stress. 
 
 The conditions which the diameter of a shaft ought to fulfil are 
 expressed by the following equation, derived from equation (4) : 
 
 S 2 _ sp _ ? 2 = o ; (4 A.) 
 
 in which, for s, is to be put a safe working value of the resultant 
 stress (say 8,000 Ibs. on the square inch, or 5*6 kilogrammes on the 
 square millimetre), and for p and q, their values in terms of M, c, 
 w, and h, as given by equations (2) and (3) respectively. The
 
 STRENGTH AND STIFFNESS OF SHAFTS. 549 
 
 equation then becomes of the sixth order; and it is to be solved so 
 as to find h. This can be done by approximation only; and a 
 convenient method of approximation is as follows : Assume for q 
 an approximate value, q', somewhat less than that of s (say 
 q' = O9 s). Then calculate an approximate value, h', of the 
 diameter, from equation (3), viz. : 
 
 5-1 MU 
 
 Then calculate, for p, an approximate value, p', from equation (2), 
 viz. : 
 
 , , 4 w c 2 ... 
 
 p = (cos i + m)^; .. ................ (6.) 
 
 and from the approximate value of p' calculate a second approxi- 
 mate value of q, as follows : 
 
 Should this agree with the first approximate value, q, the 
 approximate diameter, h', will answer; arid should there be a 
 difference, a second approximation, h", to the required diameter is 
 to be computed, as follows : 
 
 < & > 
 
 When, as is usually the case, the difference, q' q", is small com- 
 pared with q 1 , the following formula for the second approximation 
 is sufficiently near the truth : 
 
 A third approximation might be found by repeating the process; 
 but the second approximation will, in general, be found accurate 
 enough for practical purposes. 
 
 467. Centrifugal whirling of shaft*.* Any small deflection of the 
 centre line of a shaft from the straight axis of rotation gives rise, on 
 the one hand, to centrifugal force, tending to make the deflection 
 become greater; and, on the other hand, to elastic stress, resisting 
 the deflection, and tending to straighten the centre line again. The 
 resistance to deflection may be shortly called the stiffness. In. 
 very small deflections, the centrifugal force and the stiffness both 
 increase according to the same law, being both sensibly propor- 
 
 * The substance of this Article first appeared in the Engineer of the 
 9th April, 1869.
 
 550 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 tional to the deflection simply; hence, whichever of them is the 
 greater for an indefinitely small deflection, continues to be the 
 greater until some deflection is reached which causes a sufficient 
 difference between their laws of variation. The consequence is, 
 that if, for an indefinitely small deflection, the centrifugal force is 
 equal to or greater than the stiffness, the shaft must go on per- 
 manently whirling round in a bent form, to the injury of itself 
 and the adjoining machinery and framing : a kind of motion which 
 may be called centrifugal whirling. On the other hand, if for an 
 indefinitely small deflection the stiffness is greater than the 
 centrifugal force, centrifugal whirling is impossible. 
 
 For a shaft of a given length, diameter, and material, there is a 
 limit of speed ; and for a shaft of a given diameter and material, 
 turning at a given speed, there is a limit of length, below which 
 centrifugal whirling is impossible. 
 
 The mathematical expression of the conditions of the problem 
 leads to a linear differential equation of thq fourth order, integrable 
 by means of circular and exponential functions. The integrals 
 are (as might have been expected) identical in form with those 
 obtained by Poisson in his investigation of the transverse vibrations 
 of elastic rods (Traite de Mecanique, Vol. II., 528) ; and some of 
 the numerical results calculated by Poisson are applicable to the 
 present problem. The relation between the limits of length and of 
 speed depends on the way in which the shaft is supported. The 
 only two cases which will here be given are those respectively of a 
 shaft supported on two bearings at its ends, and of an overhanging 
 shaft with one end fixed in direction. 
 
 E 
 
 Let be the modulus of elasticity of the material, expressed in 
 
 units of height of itself (say, for wrought iron, about 7,500,000 
 feet, or 2,286,000 metres); r 2 , the square of the radius of gyration 
 of the cross-section of the shaft (= h z -f- 16 for a cylindrical 
 shaft of the diameter h) ; and a, the angular velocity of rotation, 
 ( = 2 x number of turns per second). Calculate a certain length, 
 &, as follows : 
 
 5 = (*fl*. (1.) 
 
 \ W Q/ / 
 
 Then the limit of span, below which centrifugal whirling is 
 impossible, bears a ratio to b depending on the manner in which, 
 the shaft is supported; for example, 
 
 Shaft supported at the ends; span, 2 c, < ~~_ Q.iiig j f j (2-) 
 
 Shaft overhanging: direction of one end f = 0-595 * b ) ,o\ 
 fixed; length, c, ( = 1-87 b | \ /
 
 (E A r 2 \ J- 
 =-=f) 4 (4.) 
 w w 2 / 
 
 SHAFTS CENTRIFUGAL WHIRLING. 551 
 
 In practical calculations it may be convenient to put, instead 
 ci A 
 of -j i , -g ; where n is the number of revolutions per second, and 
 
 A = j^2 ( = 0-815 foot, or 0-248 metre, nearly) is the altitude of 
 
 a revolving pendulum which makes one revolution in a second. 
 This gives, for the value of b, 
 
 b 
 
 It is obvious that r should be expressed in the same units of 
 
 E 
 
 measure with and A; for example, in feet, if they are expressed 
 
 in feet. 
 
 The inverse formulae for the limit of speed below which centri- 
 fugal whirling is impossible in a shaft of a given length, I, are of 
 course as follows : 
 
 Make b = 2 c x 0-3 183 for a shaft supported at the two 
 ends, and of the half span, c; (5.) 
 
 Or b = 0-5347 c for an overhanging shaft; (6.) 
 
 Then the limit of speed, in revolutions per second, is 
 
 n 
 
 --JrVnr < 7 ') 
 
 E A 
 
 The following are approximate values of - and its square 
 
 and fourth roots, for British and French measures : 
 
 W \ W 
 
 Feet, ......................... 6,100,000 2,470 49-7 
 
 Metres, ....................... 566,000 752-4 27-4 
 
 An additional mass turning along with the shaft, such as a 
 pulley, has little effect on the centrifugal force when it is in the 
 usual position ; that is, close to or near to a bearing. 
 
 The effect of an additional rotating load distributed uniformly 
 
 E 
 along the shaft may be allowed for by diminishing the height
 
 552 MATERIALS, CONSTRUCTION; AND STRENGTH. 
 
 of the modulus of elasticity in the same proportion in which the 
 weight of the shaft itself is less than the gross load. 
 
 The effect of an additional rotating load at a point not near a 
 bearing has not yet been investigated. The problem is capable of 
 solution by means of the general integrals already known ; but it is 
 not of much practical importance; for when a shaft is so long and 
 so rapid in its rotation as to require precautions against centri- 
 fugal whirling, the first precaution is to avoid loading it with 
 rotating masses which are not very near the bearings. 
 
 468. Dimension* of Couplings. Couplings have already been, 
 referred to in connection with disengaging and re-engaging gear, 
 in Article 260, page 295. An ordinary shaft-coupling may be 
 described as consisting of two discs at the ends of the two lengths 
 of shafting to be coupled, which two discs, when put together, form 
 one cylinder; each of the two discs being cylindrical, and having, 
 on the side which faces the other disc, alternate projecting and 
 receding sectors : the projecting sectors forming claws, which fit 
 into the hollows and between the claws of the opposite disc. When 
 there is only one projecting sector or claw on each disc, of a semi- 
 cylindrical figure, the coupling is the circular half-lap coupling, 
 introduced by Mr. Fairbairn. It is described by him in the follow- 
 ing words (Treatise on Mills and Mittwork, Part II., page 81): 
 " It is perfectly round, and consists of two laps, turned to a gauge, 
 and, when put together by a cutting machine, it forms a complete 
 
 cylinder. A cylindrical box is fitted over these, 
 
 and fixed by a key, grooved half into the box and half into the 
 shaft. The whole is then turned in the lathe to the same centres 
 as the bearings of the shaft " 
 
 The following are the proportions given by Mr. Fairbairn for 
 this coupling : 
 
 Area of the coupling, =2 x area of the shaft. 
 
 Or, in other words, diameter of) , , 1/40 ,. run. 
 
 coupling, } = 1>4142 x ^ameter of shaft. 
 
 Length of lap, = diameter of shaft. 
 
 Length of box, = 2 x diameter of shaft. 
 
 To which may be added, outside ) O i j- * t, CL 
 
 diameter of box, } = 2 * x diameter of shaft. 
 
 469. Rushes ami Plumber-Blocks. The following rules for the 
 dimensions of the brasses or bushes of bearings for journals, and of 
 the plumber-blocks which carry those bushes, are given on the 
 authority of Mr. Molesworth : 
 
 BUSH: thickness of metal at bottom = 0-15 inch + from 0-09 to 
 
 0'12 diameter of journal. 
 thickness of metal at sides = f thickness at bottom.
 
 PLUMBER-BLOCKS STRENGTH OF TEETH. 553 
 
 PLUMBER- BLOCK : thickness of sole-plate = 0'3 diameter of journal. 
 
 thickness of cover = 04 diameter of journal. 
 
 diameter of bolts, if two in number = 0'25 dia- 
 
 meter of journal. 
 
 if four in number 0'18 dia- 
 
 meter of journal. 
 
 SECTION IV. Special Rules relating to Pulleys, Wheels, Teeth, 
 and Levers. 
 
 470. Teeth and Rims of Wheels, and Dimensions depending on 
 
 them. The teeth of wheels are made strong enough to resist the 
 bending moment which may arise from the whole force transmitted 
 by a pair of wheels happening to act on one corner of one tooth, 
 such as D in fig. 281. 
 
 In fig. 281 A, let the shaded part represent a portion of a cross- 
 section of the rim of the 
 wheel A of fig. 281, and let 
 E H K P be the face of a 
 tooth, on one corner of which, 
 P, acts the force represented 
 by that letter. Conceive 
 
 any sectional plane, E F, to ^ ^ *"' 281 * 
 
 intersect the tooth from the 
 side EP to the crest PK, 
 and let P G be perpendicular "" 
 
 to that plane. Let h be the 
 thickness of the tooth, and let E F = b, P G = I. 
 
 Then the bending moment at the section E F is P I, and the 
 greatest stress produced by that moment at that section is 
 
 6 PZ 
 
 which is a maximum when ^ P E F = 45, and b = 2 I, having 
 then the value, 
 
 3P 
 
 Consequently, the proper thickness for the tooth is given by the 
 equation 
 
 .(i.) 
 
 This formula is Tredgold's; according to whom the proper value 
 for the greatest working stress,^ is 4,500 Ibs. on the square inch,
 
 554 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 when the teeth are of cast iron; and about 1,125 Ibs. on the square 
 inch, when they are wooden cogs ; being equivalent respectively to 
 3 -2 and to - 8 kilogrammes on the square millimetre; so that a 
 wooden cog has twice the thickness of a cast-iron tooth fitted to bear 
 the same pressure. 
 
 The thickness having been thus determined, the pitch is to be 
 deduced from it by the following formula : 
 
 h + 0-02 inch h + 0'5 millimetre 
 
 and all the other dimensions of the teeth, and of the ring which 
 carries them, are to be calculated from the pitch, by means of rules 
 which have already been given in Article 125, page 117. 
 
 It has already been stated in the rules referred to, that the 
 depth of the ring, when cast along with the teeth, is equal to the 
 thickness of a tooth at its root. In a ftiortise-wheel, with a cast- 
 iron ring, and wooden cogs fixed into mortises in the ring, the depth 
 of the ring is about equal to the pitch. The thickness of metal in 
 the ring regulates the thickness of metal in other parts of the 
 wheel, which should be the same, or nearly the same. 
 
 The hoop-shaped ring which carries the teeth of a wheel is often 
 strengthened by means of a rib or feather in a plane normal to the- 
 axis, and sometimes by means of two such ribs, so as to make the 
 cross-section of the rim T-shaped, or trough-shaped, as the case may 
 be. The sectional area of the rim in such cases is made nearly 
 equal to that of the arms, which is determined by means of 
 principles to be stated further on. 
 
 The breadth of a wheel is regulated by that of the teeth, which 
 is found by one of the rules already referred to. The breadth of a 
 pulley is regulated by that of the belt which is to run upon it ; and 
 the swell (as already stated in Article 170, page 184), is one-24th 
 of the breadth. 
 
 471. Boss and Arms of a Wheel or Puller. When a wheel or 
 
 pulley is cast in one piece, the nave or boss is usually about twice 
 the diameter of the shaft on which it is fixed, and sometimes a 
 little less. When the arms are cast separate from the boss, and 
 inserted into sockets in it, where they are fixed by means of bolts 
 or of wedges, the boss is from three to four times the diameter of 
 the shaft. 
 
 A small wheel or pulley often has instead of arms, a thin flat 
 disc, whose thickness may be made equal to the thickness of metal 
 round the eye of the boss. In rapidly revolving pulleys, the arms 
 are often made of a flat oval section, so as to cleave the air 
 edgewise. The arms of wheels are usually of a T-shaped, cross- 
 shaped, or tubular form of section.
 
 STRENGTH OP WHEELS AND PULLEYS. 555 
 
 To calculate the bending moment which each of the arms of a, 
 wheel has to resist, let M be the greatest moment of the effort 
 transmitted by the wheel; n, the number of arms; r, the geomet- 
 rical radius of the wheel, from the axis to the pitch-line; x, the 
 length of an arm, from the boss to the rim ; M', the bending 
 moment on each arm ; then two cases may be distinguished. 
 
 I. If the arms and rim are made in one piece, either by casting 
 or by welding; 
 
 <>> 
 
 and this formula is applicable also to wheels like the paddle wheels 
 of a steamer, in which wrought-iron arms are rigidly bolted or 
 rivetted both to the boss and to the rim. 
 
 II. If the arms are cast along with segments of the rim, and 
 fastened into sockets in the boss; 
 
 ...(2.) 
 nr ^ ' 
 
 This second formula is based on the supposition that the joint 
 where an arm is inserted into the boss cannot safely be trusted to 
 bear any part of the bending moment. This is not strictly correct, 
 but it is an error on the safe side. 
 
 The transverse section of the arms is to be adapted to bear 
 safely the working moment thus found, by the aid of the rules of 
 Article 437, pages 514 to 516. 
 
 In Case I, the greatest bending moment is exerted on each arm 
 at two points, close to the rim and close to the boss respectively; 
 in Case II, the greatest bending moment is exerted close to the 
 rim. 
 
 Another way of adjusting the strength of the arms to the 
 moment exerted through them is as follows : Having fixed the 
 figure and dimensions of an arm according to convenience, calculate 
 the working moment to which it is adapted ; let this be denoted 
 by M' ; then the number of arms required is given by the following 
 formulae : 
 
 In Case I; n = g-j ................... ( 3 -) 
 
 In Case II; n = ~ ; .................... (4.) 
 
 The real working modulus of stress for cast iron in these cal- 
 culations should not exceed 4,500 Ibs. on the square inch, or 3 '2 
 kilogrammes on the square millimetre; and for wrought iron,
 
 556 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 9,000 Ibs. ou the square inch, or 6 -3 kilogrammes on the square 
 millimetre. 
 
 472. Centrifugal Tension in Wheels and Pnlleys. The rim of a 
 
 wheel, moving with the velocity v, is subjected to a centrifugal 
 
 v 2 
 tension whose amount is equal to the weight of a length of that 
 
 i/ 
 
 rim (including teeth, if it is a toothed wheel). This is resisted 
 by the tenacity of the rim at its smallest cross-section (or by 
 the fastenings of the rim, if it is made in segments), partly 
 assisted by the tenacity of the arms. Each of the arms has to 
 bear its own centrifugal tension, which, at a point close to the boss, 
 is equal to the weight of a length of the arm itself expressed 
 
 (r*\ v 2 
 I * )tr ', v being the radius of the wheel, and 7 J that of the 
 ir 1 / Zg 
 
 boss ; and on the whole, it is an error on the safe side to make the 
 rim strong enough to bear its own centrifugal tension without aid 
 from the arms. This fixes a limit of safety as to speed, for a rim 
 of a given, material and construction. Let m be the ratio in which 
 the mean sectional area is greater than the effective sectional area, 
 f the greatest working tensile stress, and let w be the heaviness of 
 the material ; then the greatest proper velocity, being that which 
 produces the stress /J is given by the formula : 
 
 The modulus. , for cast iron may be taken at 800 feet, or 244 
 w 
 
 metres; so that when m = 1, as in a pulley, or a fly-wheel without 
 teeth, we have v = 160 feet, or 49 metres, per second nearly. Let 
 a cast-iron spur fly-wheel be so designed that m = '2; then v = 113 
 feet, or 34 metres, per second nearly. 
 
 f 
 The modulus, -, for wrought-iron wheel-tyres that are not 
 
 welded, but rolled out of perforated discs, may be taken at 2,400 
 feet, or 730 metres. 
 
 473. Tension-Arm* of Vertical Water-Wheels. The weight of 
 
 a great vertical water-wheel, of the construction introduced by 
 Hewes, is hung from a cast-iron boss by means of wrought-iron 
 tension-rods. The load is distributed amongst the rods which, at 
 a given instant, point obliquely or vertically downwards from the 
 boss; and the amount of the tension on each rod is proportional 
 nearly to the square of the cosine of its inclination to the vertical. 
 
 The mean value of that square is nearly ; and, at any instant, 
 
 2 
 
 half the total number of rods point downwards. Hence, let f be
 
 STRENGTH OF WHEELS AND LEVERS. 557 
 
 the intensity of the tension on the rods which point vertically 
 downwards; S, the sectional area of a rod; n, the number of rods; 
 W, the load; then 
 
 and 
 
 g _ /2 \ 
 
 For the working value off, we may take from 9,000 to 10,000 Ibs. 
 on the square inch; or from 6 - 3 to 7 kilogrammes on the square 
 millimetre. (See A Manual of the Steam-Engine and other Prime 
 Movers, Article 155, pages 181, 182.) 
 
 474. Braced wheels. Instead of transmitting power between 
 the boss and the rim of a wheel by means of the resistance of the 
 arms to bending, the arms may be so placed as to transmit power 
 by their direct tension and thrust; and for that purpose they must 
 not be radial, but must lie in the direction of tangents to a circle of 
 a radius somewhat smaller than that of the boss. Let r" denote the 
 radius of this circle; n, the number of arms; M, the greatest 
 moment transmitted; then the amount of the greatest stress along 
 an arm is given by the following expression : 
 
 M 
 
 This is tension for one half of the arms, and thrust for the other 
 half; and their dimensions are to be determined by the rules of 
 Article 459, pages 537 to 539. 
 
 475. r.evei-8, jtruuiN, and Cranks have usually one or two arms, 
 as the case may be ; and each arm is in the condition of a bracket ; 
 the greatest bending moment being exerted at that cross-section 
 which traverses the fulcrum, or axis of motion. In the crank of a 
 steam engine, the greatest bending moment is identical with the 
 greatest twisting moment exerted on the shaft to which the crank 
 is fixed. 
 
 In ordinary cases it is unnecessary to add anything to the rules 
 which have already been given in Article 434 to 438, pages 504 
 to 517, for determining bending moments, and the transverse 
 dimensions required in order to resist those moments. Cranks are 
 usually rectangular in section; levers and walking beams are 
 sometimes rectangular and sometimes I-shaped. The bending 
 moment is in most cases exerted in contrary directions alternately, 
 so that the cross-section must be made symmetrical about the 
 neutral axis; and for the modulus of stress must betaken a safe 
 working value of that kind of stress against which the material is
 
 558 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 weakest; tension for cast iron, thrust for wrought iron; for 
 example : 
 
 Cast Iron. Wrought Iron. 
 
 Lbs. on the square inch, 3000 6000 
 
 Kilogrammes on the square millimetre, ... 2-1 4-2 
 
 Holes made in a lever for the purpose of inserting pins should 
 be as near as possible to the neutral layer; that being the position 
 in which the removal of a given area of material weakens the 
 lever least. 
 
 The following rules for the proportionate dimensions of a steam- 
 engine crank, made to be keyed on the end of a shaft, are those 
 deduced by Mr. Bourne from the practice of Messrs. Boulton and 
 Watt: 
 
 Diameter of 
 Shaft-Journal 
 multiplied by 
 
 Crank- web; thickness produced to centre of shaft,... 075 
 
 breadth produced to centre of shaft, 1-50 
 
 Large Eye; breadth, 1-75 
 
 thickness, 0-45 
 
 Diameter of 
 
 Piston-Rod 
 
 multiplied by 
 
 Crank- web; thickness produced to centre of pin, ITO 
 
 breadth produced to centre of pin, i'6o 
 
 Small Eye; breadth, 1-87 
 
 thickness, 0-63 
 
 Crank-pin Journal; diameter, 1-40 
 
 length, 1-60 
 
 As to Crank-pins, see pages 541 to 543. 
 
 Trussed or framed levers are sometimes used; as in the walking 
 'beams of American river steamers. A beam of that sort consists 
 mainly of a cast-iron cross, having the ends of its arms tied 
 together by four wrought-iron tie-rods, forming a lozenge-shaped 
 figure. The long arms are from twice to three times the length of 
 the short arms. The long arms are always in a state of thrust ; the 
 tipper and lower tie-rods alternately are subjected to tension; and 
 the upper and lower short arms of the cross alternately are sub- 
 jected to a thrust equal to twice the load. The load, the thrust 
 along a long arm, and the tension on a tie-rod, are to each other 
 nearly in the proportions of the length of a short arm, the length 
 of a long arm, and the length of a tie-rod.
 
 559 
 
 CHAPTER IT. 
 
 ON THE PRINCIPLES OF THE ACTION OF CUTTING TOOLS. 
 
 476. General Explanation*. In making the bearing and work- 
 ing surfaces of the parts of a machine, it is only a rough approxi- 
 mation to the required figure that can be obtained by casting, by 
 forging, or by pressure. The precision of form which is essential 
 to smooth motion and efficient working is given by means of 
 cutting tools. The object of the present chapter is to give a brief 
 statement of the principles upon which the action of such tools 
 depends. For detailed information respecting them, reference 
 may be made to the second volume of Holtzapffel's Treatise on 
 Mechanical Manipulation, extending from page 457 to page 1025, 
 and to Mr. Northcott's Treatise on Lathes and Turning ; and for 
 a very clear summaiy account of their nature and use, to an 
 Essay by Mr. James Nasmyth, published at the end of the later 
 editions of Buchanan's Treatise on Millwork. 
 
 The appendix to Holtzapffel's volume contains two essays of 
 much value on the general principles of cutting tools one by Mr. 
 Babbage, and the other by Professor Willis. 
 
 477. Characteristics of Catting Tools in General. The usual 
 
 material for cutting tools is steel, of a degree of hardness suited 
 to that of the material to be cut. Every cutting tool has at least 
 one cutting edge; and sometimes three or more edges meet and 
 form a point, two or more of those edges being cutting edges; so 
 that the form of the cutting part of a tool is that of a wedge, or 
 of a pyramid, as the case may be. A cutting edge is formed by 
 the meeting of two surfaces, generally plane, and sometimes curved. 
 When a surface forming a cutting edge is oblique to the original 
 surfaces of the bar out of which the tool is made, that surface is 
 called a chamfer or bevel. The angle at which those surfaces 
 meet may be called the cutting angle. It ranges from about 
 15 to 135, according to the nature of the material to be cut, and 
 the way in which the tool is to act upon it. Examples of cutting 
 angles of tools for different purposes will be mentioned further on. 
 A narrow cutting edge at the end of a bar-shaped tool is often 
 called the point of the tool ; the body of the tool is called the shaft 
 or the blade; the term shaft being usually applied to tools with a 
 cutting point or narrow edge at one end, and blade to those which, 
 have a longitudinal cutting edge; but blade is often applied to
 
 560 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 both kinds of tools. A bar-shaped shaft is sometimes called a stem. 
 The blade or shaft of a tool is sometimes made of iron, and edged 
 or pointed with steel. A larger piece, to which the blade is fixed, 
 is called the stock; and in the case of hand-tools, that part of the 
 stock which is grasped by the hand is called the handle. The 
 stocks and handles of hand-tools are xisually of wood of some strong 
 and tough kind. Where steady pressure is to be exerted, stiff 
 woods are to be chosen, such as beech and mahogany ; where blows 
 are to be given, flexible woods, such as ash. Machine-tools are 
 held in tool-holders of different sorts, made of cast and wrought 
 iron. A rest is a fixed or moveable support for a cutting tool , in 
 machine tools, the rest carries the tool-holder. 
 
 The term machine-tool is often applied, not merely to the cutting 
 implement itself, but to the whole machine of which it forms 
 part. 
 
 The piece of material which is being cut by a tool is commonly 
 called the work. A given relative motion of the work and cutting 
 tool may be obtained either by keeping the work fixed and 
 moving the tool, or by keeping the tool fixed and moving the work, 
 or by a combination of both those motions. 
 
 478. Classification of Cutting Tools. The following classification 
 is that of Holtzapffel. Cutting tools, according to their mode of 
 action on the bodies to which they are applied, are divided into 
 Shearing, Paring, and Scraping tools ; the following being the 
 characters by which those tools are distinguished from each 
 other. 
 
 I. A Shearing Tool acts by dividing a plate or bar of the 
 material operated on into two parts by shearing ; that is to say, 
 by making these two parts separate from each other by sliding at 
 the surface of separation. 
 
 II. A Paring Tool cuts a thin layer or strip, called a shaving, 
 from the surface of the work; thus producing a new surface. 
 
 III. A Scraping Tool scrapes away small pai'ticles from the 
 surface of the work; thus correcting the small irregularities 
 which may have been left by a paring tool. 
 
 479. Shearing and Punching Tools. A pair of shears for 
 
 cutting iron usually consists of two blades; the lower fixed, 
 and the upper inoveable in a vertical direction. The inner 
 faces of the blades are plane, and are so fitted as to slide past 
 each other very closely, but without appreciable friction. The 
 ordinary angle for the cutting edges is from 75 to 80. In. 
 shears for cutting plates, the edge of the lower blade is horizontal ; 
 that of the upper blade has an inclination of from 3 to 6, in 
 order that the shearing may begin at one edge of the plate, and 
 go on gradually towards the other edge. Fig. 282 represents a 
 cross-section of the blades of a pair of shears, with their cutting
 
 SHEARING AND PUNCHING TOOLS. 
 
 561 
 
 fig. 282. 
 
 edges at A and B, and having between them a plate or bar, C C, 
 which is to be shorn through at the plane whose trace is the 
 dotted line A B. 
 
 The blades of shears for cutting angle-iron bars have V-shaped 
 edges, to suit the form of cross-section of the bars. 
 
 The class of shearing tools comprehends also tools for punching ; 
 the only difference being in the 
 form of the surface at which 
 shearing takes place; which form 
 determines that of the cutting 
 edges of the tools. The general 
 principle of the action of those 
 tools is shown in fig. 283; in 
 which C C is a plate, lying upon 
 the die, represented in vertical 
 section by B B; that is, a flat 
 disc or block of hard steel, having 
 in it a hole a little larger than 
 the hole that is to be punched 
 in the plate. A is the punch, 
 of a cylindrical figure, its base 
 being of the size and figure of 
 the hole to be made. The descent 
 of the punch causes the material 
 of the plate C C to separate by 
 shearing at the surface whose 
 traces are marked by dotted lines, 
 and drives out a piece called a 
 wad, which drops through the 
 hole in the die. The wad is 
 slightly conical; its upper end 
 being of the size of the punch, 
 and its lower end nearly of the 
 size of the hole in the die. 
 
 The ordinary difference of dia- 
 meter between the punch and 
 the die ranges from 0-1 to O3 
 of the thickness of the plate. 
 Sometimes the die is made pro- 
 portionately larger, in order that 
 the holes may be more conical. 
 
 The smaller end only of a punched hole is accurate in size 
 and figure, and smooth. The larger end is more or less irregular 
 and ragged. When it is necessary that a punched hole should 
 be made regular and smooth, that is done by scraping with a 
 pyramidal tool called a rimer or broach. 
 
 2o 
 
 Fig. 283.
 
 562 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 The pressure required for shearing or for punching may be 
 calculated by means of the following formula : 
 
 P=/c; .............................. (1.) 
 
 in which t is the thickness of the plate to be shorn or punched ; c, 
 the length of the cut; that is, the breadth of the plate, for shearing, 
 on the circumference of the hole, for punching; so that ct is the 
 area of shorn surface ; and f, the co-efficient of ultimate resistance 
 to shearing; which may be estimated, for malleable iron, at 
 
 50,000 Ibs. on the square inch, or 
 
 35 kilogrammes on the square millimetre. 
 
 The work of shearing or punching is estimated by Weisbach 
 (Ingenieur, 129) as equal to that of overcoming the resistance P 
 through one-sixth of the thickness of tlie plate; that is to say, it is 
 expressed by 
 
 and this is the net, or useful work. The same author estimates 
 the counter-efficiency of shearing and punching machines at from 
 1^ to 1^; so that taking the higher ot those estimates, the total 
 work of such a machine at one stroke is 
 
 (3l) 
 
 and this formula is to be tised in calculating the power required to 
 drive such machines, and the dimensions of their fly-wheels (see 
 page 410). 
 
 480. Paring and Scraping Tools in Oeneral. The general nature 
 of the modes of action of paring and of scraping tools is illustrated 
 by figs. 284 and 285; paring being represented by fig. 284; scrap- 
 ing by fig. 285. In each figure, the tool is supposed to act upon a 
 cylindrical piece of work in a turning lathe. The arrow in each 
 figure represents the direction of motion of the woi-k relatively to 
 the tool. The plane of each figure is parallel to the direction of 
 the cutting motion, and perpendicular to the face of the work. 
 The point A is the trace of the cutting edge of the tooL FAB 
 is a normal to the face of the work, and is also the trace of a plane 
 normal to the direction of the cutting motion. A C is the trace of 
 a plane tangent to the face of the work. Had the process repre- 
 sented been planing instead of turning, A C would have been the 
 trace of the face of the work itself. A D is the trace of the 
 face-plane or front-plane of the cutting edge. The angle A C D 
 is called by Babbage the angle of relief. !
 
 PARING AND SCRAPING TOOLS. 
 
 563 
 
 A E is the trace of the upper plane of the cutting edge : so called 
 because of its position in the process of turning, when the plane 
 
 Fig. 284. 
 
 A C is vertical, and the tool cuts upwards. In other opera- 
 tions, such as planing and slotting, the plane A E may be turned 
 
 Fig. 285. 
 forwards or downwards; the fact being, that in every case it is th
 
 564 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 real front plane as regards the cutting motion ; but the custom is 
 to call it the upper plane. 
 
 D A E is the angle of the tool, or cutting angle. In paring tools 
 the cutting angle is always acute, as in fig. 284; having values 
 depending on the nature of the material to be cut, of which ex- 
 amples will be given further on. In scraping tools, the cutting 
 angle may be acute, right, or obtuse; in fig. 2S5, it is nearly a 
 right angle. 
 
 The angle C A E, made by the upper plane A E with the tangent 
 plane A C, may, for the sake of convenience, be called the walking 
 angle; so that the working angle is equal to the sum of the cutting 
 angle and the angle of relief; or in symbols, C A E = DAE + 
 CAD.* 
 
 It is upon the working angle that the distinction between 
 properly shaped paring and scraping tools depends. That distinc- 
 tion may be stated as follows : .Every ^paring tool has an acute 
 working angle; every right or obtuse working angle makes a scraping 
 tool. 
 
 An acute working angle, if it is too great to suit the material on 
 which it acts, may give a scraping instead of a paring tool. 
 
 As regards the effect of the cutting angle upon the action of a 
 tool, it is obvious, that as the working angle is greater than the 
 cutting angle, an obtuse or even a right cutting angle gives a 
 scraping tool; and that an acute cutting angle too may give a 
 scraping tool, if the tool is held so as to make the working angle 
 right or obtuse; but that an acute cutting angle is essential to a 
 paring tool. 
 
 In well-made paring tools, the difference between the working 
 angle and the cutting angle, or angle of relief, C A D, is not made 
 greater than experience has proved to be necessary in order to 
 prevent excessive friction between the face-plane of the tool and 
 the face of the work; that is, about 3. Any increase of that 
 angle is to be avoided; because it tends to make the tool dig into 
 the work, and to produce "chattering;" and because, by diminish- 
 ing the cutting angle, it weakens the tool. 
 
 The position of the shaft of the tool as held by the tool-holder 
 should in all cases be such that the back and forward motion of 
 adjustment of the cutting edge, A, by which the depth of the cut 
 is regulated, shall take place exactly in the plane FAB, perpendi- 
 cular to the face of the work; that is to say, in the case of a 
 turning lathe, in a plane traversing the axis of rotation. 
 
 The tools shown in the figures have the front ends of their shafts 
 bent, hooked, or cranked, as it is termed, in such a manner as to 
 
 " Mr. Babbage calls E A F the angle of escape; so that the angle of escape 
 is the complement of the working angle. In carpenters' hand-planes, the 
 working angle is called the pitch.
 
 PARING AND SCRAPING TOOLS. 
 
 5G5 
 
 ensure that the cutting edge, A, shall not le in advance of the axis 
 of the straight part of the shaft of the tool. The object of this 
 arrangement is, that any deflection which the tool-shaft may 
 undergo through excessive resistance to the cut. may tend to 
 move the cutting edge, A, away from the work, and not to make it 
 dig into it. 
 
 In tools for rough work, and having very stiff shafts, the cranked 
 shape is unnecessary ; and then the upper side of the shaft is in 
 the plane B AF; the proper position of the vipper plane of the 
 cutting edge being given by means of a hollow or flute in the upper 
 side of the tool. 
 
 Fig. 286 represents a paring tool designed on the same principles 
 with that shown in fig. 284, but with two cutting edges, and 
 a three-sided pyramidal 
 point. The upper part of 
 the figure, marked with 
 capital letters, is an eleva- 
 tion ; the lower part, mark- 
 ed with italic letters, is a 
 plan, or horizontal pro- 
 jection ; and corresponding 
 letters mark correspond- 
 ing points. The cutting 
 edges in the plan are 
 marked a b and a c ; abed 
 and a c f d are the pro- 
 jections of the two face- 
 planes; ad is the projec- 
 tion of the front edge; 
 and a b g h c that of the 
 upper plane. In the ele- 
 vation are shown one of the 
 cutting edges, A B; one 
 of the face-planes, ABED; 
 and the front edge, A D. 
 Sometimes the front edge 
 is rounded, together with 
 
 the angles cab andfd e; thus connecting the two straight cutting 
 edges by means of a short curved cutting edge. Sometimes the 
 whole cutting edge is a curve ; and then, instead of two face-planes, 
 there is a cylindrical front surface. 
 
 The relations amongst the angles made by the planes and edges 
 of such a tool as that shown in fig. 286 may be determined either 
 by spherical trigonometry, or by the rules in descriptive geometry 
 given in this book, Articles 24 to 30, pages 9 to 12. They are 
 treated of also in an essay by Mr. Willis, already mentioned as 
 
 Fig. 280.
 
 566 
 
 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 having been published in the appendix to the second volume of 
 Holtzapffel On HecJianical Manipulation. 
 
 A tool with one curved or two straight cutting edges may be 
 used to cut a groove, or to pare a layer by successive shavings off 
 the surface of a piece of work. In the latter case the shaft of the 
 tool is to be so formed and held, that one of the straight cutting 
 edges (for example, a b}, or one side of the curved edge, touches the 
 pared face of the work, and the other (for example, a c) cuts into 
 the side of the unpared part of the layer that is being removed; 
 and according as the tool is shaped and placed so as to lie to the- 
 right or to the left of the unpared part of the layer, it is called a 
 right-side or a left-side tool. Thus, in a right-side tool, a b touches 
 the pared face ; a c, the side of the unpared layer ; and in a left- 
 side tool, a c touches the pared face, and a b the side of the 
 unpared layer. A tool with one cutting edge which is parallel to the 
 face of the work, as in fig. 284, or a tool with two cutting edges, as 
 in fig. 286, making equal angles with the face, is called a, face-tool. 
 Fig. 287 shows how the principle of having a small angle of relief, 
 
 with a sufficiently acute cutting 
 angle, is applied to drills or 
 boring bits. The figure shows 
 a front view, lettered A, B, C, 
 &c. ; an edge view, lettered 
 A 1 , B', C', <fcc. ; and an end 
 view, lettered in italics. A A 
 is the axis of rotation; B C, 
 B C, the cutting edges ; and 
 the requisite acuteness is given 
 to the cutting angle (marked 
 D' C' E' in the edge view) by 
 means of a flute or curved 
 hollow parallel to each of the 
 cutting edges. 
 
 481. Catting Angles of Tools. 
 
 The best cutting angles for 
 paring tools suited to different 
 materials have been ascertain- 
 ed by practical experience. 
 The following are the principal 
 results : 
 
 MATERIALS. Cutting Angles. 
 
 Wood, from 20 to 45 
 
 Iron and Steel, from 60 to 70 
 
 (The smaller angles for the softer kinds; 
 
 the greater for the harder.) 
 Brass and Bronze, 80 and upwards.
 
 CCTTIXG TOOLS. 567 
 
 In the case of scraping tools, the size of the cutting angle is a 
 question mainly of convenience and strength; for the same tool 
 which is a paring tool, when the working angle is only a little 
 greater than the cutting angle, becomes a scraping tool when the 
 working angle is sufficiently increased. 
 
 It may be considered, however, that in order to give sufficient 
 strength to the tool, the least cutting angle for a scraping tool 
 should not be less than the cutting angle of a paring tool suited 
 to the same material. 
 
 The cutting angle of scraping tools for iron ranges from 60 
 to nearly 135; the former value being met with in triangular 
 scrapers for finishing plane surfaces; the latter, in octagonal broaches 
 for enlarging and correcting conical holes. 
 
 482. Speed of Cutting Tools. The speed of the cutting motion 
 of tools is limited by the heat produced by their action : it must 
 not be so great as to cause that heat to affect the temper of the 
 steel. Hence it is less, the harder the material of the work. The 
 following are examples. 
 
 Speed of Cutting Motion. 
 MATERIAL Feet per Minute. 
 
 White Cast Iron, .................. about 5 
 
 Steel, and Gray Cast Iron, ....... from 10 to 20 
 
 Wrought Iron, ..................... from 15 to 25 
 
 Brass and Bronze, .................. from 40 to 100 
 
 Wood, ................................. from 300 to 2000 
 
 Higher speeds may be used for planing, and for ordinary turn- 
 ing, where the tool and the cut surface are freely exposed to the 
 air, than for drilling and boring. 
 
 483. Resistance and Work of Paring Tools. The following esti- 
 
 mate of the resistance to the motion of a tool paring iron, and of 
 the work done, is based on that given by Weisbach : 
 
 Let R denote the resistance to the cutting motion of the tool; 
 b the breadth, and t the thickness of the shaving which it pares off; 
 BO that b t is the sectional area of the shaving; then 
 
 in which f is a co-efficient depending on the hardness and tough- 
 ness of the material. The value of /for iron is estimated by Weis- 
 bach at 50,000 Ibs. on the square inch, or 35 kilogrammes on the 
 square millimetre. For steel, it is probably from once-and-half to 
 twice as great. Let I be a given length of shaving; then the work 
 done in paring that length off is 
 
 PI =flbt .......................... (2.) 
 
 Let w be the heaviness of the material; then w I b t is the
 
 568 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 weight of material pared off; and the work done in that process is 
 
 f 
 evidently equal to tli& work of lifting that weight to the height ; 
 
 whose value for iron is 
 
 15,000 feet, or 
 4,570 metres. 
 
 The counter-efficiency of the machines in which paring tools are 
 used may be estimated at from 1 '3 to 1*5; or say, on an average, 
 1'4; so that the total work of a machine in paring away a given 
 weight of iron may be estimated as being equal to that of lifting 
 the same weight to a height of 
 
 21,000 feet, or 
 
 6,400 metres. 
 
 \ 
 
 The work done by cutting tools produces heat, which, unless 
 abstracted, tends to injure the tools by raising their temperature. 
 In order to abstract the heat and keep the temperature moderate, 
 the point of the tool, in cutting wrought iron and steel, is moistened 
 with a suitable liquid, such as oil, or a solution of carbonate of soda 
 in water. Pure water should not be xised, as it causes iron and 
 steel to rust. Cast iron, brass, and bronze are cut with dry tools. 
 
 484. Combinations of Cutting Tools. The boring bit, already 
 mentioned in Article 480, page 566, is in fact a combination of two 
 cutters or paring tools. A combination of several paring tools, 
 working side by side, is seen in the tool sometimes called a comb, 
 used in screw-cutting lathes, for cutting several turns of the thread 
 at the same time. Cutters following each other in succession 
 occur in taps and dies, for cutting internal and external screws by 
 hand. A circular cutter, or rose-cutter, used in shaping the 
 teeth of wheels, is itself a toothed wheel, each of its teeth being a 
 paring tool. The teeth of a saw form a series of small paring tools 
 or scraping tools, according to their working angles; and by the 
 process of setting them that is, bending them alternately to the 
 right and left they are made alternately into left-side and right- 
 side tools, so as to cut the two sides of the saw-kerf. 
 
 485. Motions of Machine-Tools in Genera!. In most examples 
 
 of machine-tools, the relative motion of the tool and the work is 
 the resultant of three component motions, usually at right angles 
 to each other, or of two out of those three : the cutting motion, the 
 traversing motion, or transverse feed motion, and the advancing 
 feed motion; the first two taking place parallel to the face of the 
 work, and the third in a direction normal to it. 
 
 The cutting motion is the most rapid of the three, being that by 
 which the tool acts on the face of the work, leaving a narrow strip 
 or band from which a portion of the material has been pared or
 
 CUTTING TOOLS RULED SURFACES. 569 
 
 scraped away. In many instances, the cutting motion is effected 
 by a motion of the work, the tool remaining fixed, and such is the 
 case especially in turning and screw-cutting lathes, and in almost 
 all planing machines. There are other operations in which the 
 cut is made by a motion of the tool; such as drilling, boring, 
 slotting, and shaping. The speed of the cutting motion has been 
 already mentioned in Article 482, page 567. 
 
 The transverse feed motion takes place parallel to the face of the 
 work, and at right angles to the cutting motion : it is that motion 
 by which the tool is made to shift its position relatively to the 
 work, so as to make a series of parallel cuts side by side, leaving a 
 series of strips or bands which compose a surface of any required 
 extent. It is sometimes continuous, and sometimes intermittent. 
 The general nature of transvei-se feed motions has already been 
 described in Article 258, page 293. 
 
 The rate at which the traversing motion takes place in paring a 
 continuous surface depends on the breadth of the cut; which, in 
 iron, ranges from O'Ol to 0-05 inch ( = from 0-25 to 1-25 millimetre). 
 In screw-cutting, the traverse at each revolution is equal to the 
 pitch of the screw. 
 
 The advancing feed-motion is that by which, after a certain depth 
 of material has been cut away from the face of the work, the tool is 
 advanced so as to cut away an additional depth. This is very 
 often an intermittent motion; and in turning and planing ma- 
 chines it is Tisually an adjustment, made from time to time by 
 hand. Its extent, at each adjustment, is equal to the depth of the 
 cut; which, in iron, ranges from the smallest appreciable quantity 
 up to 0-75 inch (19 millimetres) in ordinary cases. 
 
 486. Staking Ruled Surfaces Planing Slotting Shaping. A 
 
 ruled surface is one in which every point is traversed by a straight 
 line lying wholly in the surface. Such a straight line is called a 
 generating line of the surface ; and the surface may be regarded as 
 generated by the motion of the straight line. A ruled surface may 
 be cut to any required degree of precision, by the successive strokes 
 of a tool, each stroke being made along a straight generating line 
 of the surface. 
 
 The class of ruled surfaces comprehends the following different 
 kinds, amongst others : 
 
 I. All straight surfaces: that is, surfaces in which the straight 
 generating lines are all parallel to each other. Such surfaces may 
 be either plane or cylindrical; the term cylindrical surfaces em- 
 bracing not only those whose transverse sections are circular, but 
 those whose transverse sections are curves of any figure, such as 
 the acting surfaces of the teeth of spur-wheels (Article 130, page 
 120). It has already been stated, in Article 38, page 17, that the 
 bearing surfaces of sliding primary pieces must all be straight.
 
 570 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 II. All conical surfaces: that is, surfaces in which the straight 
 generating lines all pass through one point. The transvei-se 
 sections of such a surface may be circular, as in the pitch-cone of a 
 circular bevel wheel (Article 105, page 86); or they may be non- 
 circular curves, as in the acting surfaces of the teeth of such a 
 wheel (Article 144, page 143). 
 
 III. Hyperboloidal or skew-bevel surfaces: in which the straight 
 generating lines, not intersecting in one point, traverse a series of 
 similar transverse sections of the surface. Such transverse sections 
 may be circular, as in the pitch-surface of a circular skew-bevel 
 wheel (Article 106, page 87); or non-circular, as in the acting 
 surfaces of skew-bevel teeth (Article 145, page 146). 
 
 The PLANING MACHINE cuts straight surfaces of all kinds; and 
 is used especially for the cutting of plane surfaces to a certain 
 degree of approximation ; that is to say, the longitudinal straigJit- 
 ness of the surface is perfect; but the transverse straightness is 
 approximate. In a planing machine of ordinary construction, 
 there is a fixed horizontal bed, very strongly and stiffly supported : 
 its essential parts being a pair of most accurately made straight, 
 parallel, horizontal guides, of a V-shape in cross-section. In those 
 guides there slides a pair of straight, parallel, horizontal, triangular 
 bars, forming the supports of a stiff and strong horizontal table, 
 having in it several slots or oblong openings, for the purpose of 
 enabling the work to be secured to it. From the two sides of the 
 bed rise a pair of standards, carrying a straight horizontal saddle 
 that bridges across the table. The saddle, by means of straight 
 horizontal guides, supports the tool-holder, which has a transverse 
 traversing motion. The sliding surfaces of the saddle and tool- 
 holder are usually of a dove-tail shape in cross-section. 
 
 The cutting action is effected by a longitudinal motion of the 
 table carrying the work: the gearing which communicates that 
 motion ought to be extremely smooth and accurate in its action ; 
 such as the rack and helical pinion described at page 289 ; and the 
 pitch-point of the rack and pinion ought to be as directly as possible 
 below the cutting tooL As to the speed, see Article 482, page 567. 
 
 During the return stroke, the tool is lifted clear of the work, 
 and the motion of the rack and pinion is reversed by means of self- 
 acting reversing gear; such, for example, as that mentioned in 
 Article 264, page 299; and the train of wheelwork which produces 
 the return stroke is so proportioned as to give an increased speed 
 of motion to the table, usually about double that of the cutting 
 stroke. 
 
 The transverse traversing motion of the tool-holder is inter- 
 mittent, being made during the return stroke of the table. The 
 combination which directly produces it is usually a transverse 
 horizontal screw driving a nut that is fixed to the tool-holder.
 
 PLANING SLOTTING SHAPING SCRAPING. 571 
 
 The screw is driven by a suitable train of wheelwork, the first wheel 
 of which is driven by a click, usually of the reversible kind 
 (Article 194, page 207). The extent of traverse after each cutting 
 stroke can be regulated by the help of adjustments of length of 
 stroke (Article 273, page 312), or of change-wheels (Article 271, 
 page 311). 
 
 In some large planing machines for very heavy work, the 
 cutting stroke is effected by a longitudinal motion of a strong 
 frame carrying the saddle and the tool-holder; the table with the 
 work being at rest. 
 
 Eor cutting straight surfaces of more or less complex cross-section, 
 and especially for cutting straight grooves and straight rectangular 
 holes, such as key-ways and slots, the SLOTTING-MACHINE is used. 
 In this machine the tool-holder or cutter-bar usually slides verti- 
 cally in a guiding groove in the slide-head, which is carried by a 
 strong overhanging frame. Below the slide-head is a table to 
 which the work is secured, capable of being turned about a vertical 
 axis, and traversed horizontally in two rectangular directions, so 
 as to bring the work into any required position relatively to the 
 cutting-tool. 
 
 A SHAPING MACHINE differs from a slotting machine mainly in 
 having a slide-head that is capable of being turned into different 
 positions, so as to cause the tool to make strokes in different direc- 
 tions when required. It is used for cutting ruled surfaces of 
 various kinds. Circular cutters (page 568), driven by suitable 
 shifting trains (Article 228, page 235), are sometimes used in 
 shaping machines. 
 
 487. Scraping Plane Surfaces. When the highest practicable 
 degree of accuracy is required in a plane surface, its form may 
 in the first place be given approximately by the planing machine, 
 but it must be finished by the hand-scraper. Scrapers for iron are 
 usually made of very hard steel, with edges of 60. 
 
 When an existing standard plane surface (or planometer, as it 
 is sometimes called) is available, it is smeared with a very thin 
 coating of a mixture of red chalk and oil. The new plane, in its 
 approximate condition, is laid face to face on the standard plane, 
 and gently rubbed on it. The prominent places on the new plane 
 pick up the colouring matter, and are marked by it; and thus the 
 workman is guided to the parts that require scraping down. The 
 process is repeated again and again until the new plane fits the 
 standard plane with the required degree of precision. 
 
 In the absence of a standard plane, three approximately plane cast- 
 iron plates are made, stiffened at the back by ribs. One pair of 
 those are taken in the first place ; and one of them being smeared 
 with a suitable mixture, they are repeatedly rubbed together, so 
 as to mark the prominent places, and both are scraped, until
 
 572 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 they fit each other with a certain degree of accuracy. At this 
 stage of the process, they may be both plane; or both spherical, and 
 of the same radius, one being convex and the other concave. Then 
 the two plates first taken are compared in succession with the third 
 plate, and the operations of rubbing and scraping repeated, with 
 the plates combined by pairs in every possible way, until all three 
 plates accurately fit each other in every combination and position ; 
 when they must necessarily be truly plane. This is the process by 
 which standard planes are made; and when a set of three have been 
 made, it is usual to reserve one of them very carefully for testing 
 from time to time the accuracy of the other two, which are em- 
 ployed as standards of planeness and straightness for ordinary use. 
 
 488. making Surfaces of Revolution Turning* Drilling, and 
 
 Boring. A turning-lathe usually contains the following principal 
 parts. The bed, truly plane and horizontal. The head-stocks, or 
 supports for the axis of rotation of the work; one fixed, and 
 the other capable of being shifted longitudinally on the bed 
 to a greater or less distance from the fixed headstock, so as to 
 suit the size of the work. The saddle, which slides longitudinally 
 on the bed, carrying the rest, which carries the tool-holder. The 
 rest has longitudinal and transverse traversing motions, usually 
 produced by means of screws and nuts, acting on slides with dove- 
 tail-shaped straight bearing surfaces ; the position of the tool-holder 
 is adjustable vertically and horizontally. 
 
 The longitudinal traversing motion of the saddle is sometimes 
 produced by a pinion driving a rack, like the motion of the table 
 in a planing machine, and sometimes by a strong and very 
 accurately made screw, extending the whole length of the bed ; the 
 latter method is used in screw-cutting lathes. Many lathes are 
 provided both with a guide-screw and with a rack-and-pinion 
 motion for traversing, either of which can be used at pleasure. 
 The guide-screw is commonly reserved for screw-cutting, and the 
 rack and pinion used for ordinary purposes. 
 
 The moveable headstock carries the screw-spindle, which does 
 not rotate, but can be slid back and forward by means of a screw, 
 in order to adjust the position of its point, which forms one of the 
 supports of the work. The fixed or fast headstock carries the 
 lathe-spindle, which is a rotating horizontal shaft, driven at a 
 proper speed by means of a suitable belt and pulleys ; the speed is 
 capable of adjustment by means of speed-cones, usually of the 
 stepped kind described in Article 171, page 185. 
 
 The journals of the lathe-spindle are in most cases made slightly 
 conical, and are tightened in their bearings, when required, by 
 means of screws acting endwise. 
 
 The ends of both spindles project inwards from the headstocks: 
 they are capable of being fitted with various contrivances for
 
 TURNING. 573 
 
 supporting and holding the work. The screw-spindle usually, and 
 the lathe-spindle sometimes, is fitted with a conical point of steel 
 called a centre, the angle at the point ranging from 60 for wood, 
 to 80 or 90 for metal ; such points support the work and keep it 
 truly centred on the axis of rotation. The lathe-spindle can also 
 be fitted with chucks of different sorts ; being discs provided with 
 holes, pins, and other means of holding the work, and causing 
 it to rotate along with the lathe-spindle; or with a mandril or 
 cylindrical continuation of the spindle, on which wheels and 
 pulleys, and other pieces of work having eyes in their centres, can 
 be keyed for the purpose of being turned. 
 
 A chuck in the form of a large circular disc is called a, faceplate. 
 Some lathes have face-plates on both spindles ; and then the two 
 spindles are driven at the same speed, by means of two pinions on 
 one shaft, gearing with teeth on the rims of the face-plates. 
 
 The greatest radius of the work which can be turned in a given 
 lathe is limited by the height of the axis of rotation above the bed ; 
 and the lathe is described as a " twelve-inch lathe," a " twenty-four- 
 inch lathe," &c., according to that height. 
 
 The tool-holder is adjusted so that the point or cutting part of 
 the tool is exactly in a horizontal plane traversing the axis of rotation. 
 The direction of rotation is such that the surface of the work moves 
 downwards at the point of the tool, which accordingly cuts upwards. 
 
 The screws and nuts, or the pinions and racks, by which the 
 traversing motions of the tool-holder are produced, are driven 
 from the lathe-spindle through trains containing change-wheels 
 (Article 271, page 311); and by means of these the velocity-ratio 
 and directional-relation of the cutting motion and of the traversing 
 motion can be adjusted so as to produce the required resultant or 
 aggregate relative motion. As to the rate of traverse per revolu- 
 tion, see Article 485, page 569. 
 
 When the word traversing is used without qualification, it gener- 
 ally means that the tool traverses in a direction parallel to the axis 
 of the lathe, so as to turn a cylindrical surface. When the tool is 
 made to move in the direction of a radius perpendicular to the 
 axis, it turns a plane surface ; and the process is called surfacing. 
 This is very often the means used of making a plane approximately, 
 previous to correcting it by scraping. By combining those two 
 motions, so as to make the tool traverse in a straight line cutting 
 the axis obliquely, a conical surface is turned. When the point of 
 the tool is made to traverse in a circle, one diameter of which 
 coincides with the axis, a spherical surface is turned. A hyper- 
 boloidal surface might be turned by making the point of the tool 
 traverse along one of its straight generating lines (see Article 84, 
 page 70 ; Article 106, page 87). 
 
 All the preceding operations are examples of circular turning,
 
 .574 MATERIALS, CONSTRUCTION, AND STRENGTH. 
 
 in which the point of the tool describes, relatively to the work, a 
 circle about the axis, if the traversing motion be neglected, or a 
 helix or spiral of a pitch equal to the traverse per revolution, if 
 this component of the motion be taken into account. In eccentric 
 turning, the point is made to describe, relatively to the work, 
 paths of various other kinds, such as eccentric circles, ellipses, epi- 
 cycloids, and arbitrary curves of various sorts. Such aggregate 
 paths are produced, sometimes by epicyclic trains carried by the 
 chuck which holds the work, as in the eccentric chuck, elliptic chuck, 
 and geometric chuck; sometimes by the action of cams or shaper- 
 plates on the tool-holder. The actions of such combinations have 
 been treated of in Part I., Chapter V., Section IV., pages 261 to 
 267; and in the Addenda, pages 290, 291. 
 
 Drilling and Boring may be looked upon as modifications of 
 turning, applied to the making of concave surfaces of revolution, 
 and especially of hollow cylinders. The word boring is often, 
 applied to both processes; but when drilling and boring are dis- 
 tinguished from each other, drilling means the making of a 
 cylindrical hole by a tool which advances endways, cutting 
 shavings from the flat or conical bottom of the hole (see fig. 287) ; 
 and boring, the enlarging and correcting of a hollow cylindrical sur- 
 face already made; such as that of a cast-iron steam-engine cylinder. 
 
 In drilling, the tool or drill usually rotates about a vertical 
 axis; and it is very often driven by a shifting train, carried by a 
 jib or train-arm. (As to shifting trains, see Article 228, pages 235 
 to 238.) This is in order that the position of the drill may be 
 shifted to various parts of the work. The train-arm or jib projects 
 horizontally from a strong hollow standard, containing the vertical 
 shaft that drives the shifting train. The work is supported by a 
 table, which is often made so as to be capable of being turned 
 about a vertical axis, and shifted horizontally on slides in two 
 rectangular directions, in order to bring different points in the 
 work below the drill. 
 
 The feed-motion is given sometimes by gradually lowering the 
 drill, sometimes by gradually raising the table. 
 
 In a multiple drilling machine (used for making rows of holes in 
 iron plates) a set of drills are driven from one shaft by means of 
 skew-bevel pinions or other suitable mechanism. The feed motion is 
 given by raising the table. The forms of drilling tools are very various. 
 
 In a boring machine, the inner surface of a hollow cylinder is 
 pared by means of one or more tools carried by a cutter-bar or 
 cutter-head; being a cylinder a little smaller than the hollow 
 cylinder to be bored. When the work is a very large cylinder, it 
 is usually fixed; and the rotation and traversing motion are both 
 given to the cutter head. 
 
 489. Screw-Cutting. The operation of cutting screws is per-
 
 SCREW-CUTTING WHEEL-CUTTING. 575 
 
 formed in a lathe; the work rotates, and the tool-holder is made 
 to traverse longitudinally by means of tho guide-screw, already 
 mentioned at page 572. The nut by means of which the guide- 
 screw drives the saddle is a clasp-nut (see Addenda, page 576): 
 which can be thrown into or out of gear with the guide-screw 
 when required. The guide-screw is made with great care and 
 precision. An ordinary value of its pitch is half an inch. The 
 velocity-ratio and directional-relation of the motions of the guide- 
 screw and of the lathe-spindle are adjusted by means of change- 
 wheels to the pitch and direction of the screw to be cut, according 
 to the following principles. 
 
 T Speed of Rotation of Guide-Screw Pitch of New Screw 
 Speed of Rotation of Lathe-Spindle ~~ Pitch of Guide-Screw 
 
 II. The direction, right or left-handed, of the new screw, is 
 similar or contrary to that of the guide-screw, according as the 
 directions of rotation of the guide-screw and of the lathe-spindle 
 are similar or contrary. 
 
 490. Wheel-Cutting. A wheel-cutting machine, for shaping the 
 teeth of wheels, may be regarded as a special form of the shaping 
 machine, in some cases combined with the turning lathe. The 
 wheel to be cut is fixed on mandrils carried at the end of a rotating 
 spindle, mounted on a head-stock. Sometimes that spindle acts 
 as a lathe-spindle, while the wheel is being turned. When the 
 pitching and tooth-cutting are to be begun, a large worm-wheel, 
 permanently fixed on the spindle, is made to gear with a tangent 
 screw, by means of which it is successively turned through a series 
 of angles, each equal to the pitch-angle; first, for the purpose of 
 pitching the wheel, or marking the pitch-points of the teeth on the 
 pitch-circle, and then for the purpose of changing the position of 
 the wheel after each tooth has been cut, preparatory to cutting the 
 next tooth. The figures of the teeth are given approximately by 
 casting, and finished by cutting. 
 
 Each stroke of the cutter is guided so as to take place along a 
 straight line. In spur-wheels that straight line is parallel to the 
 axis; in bevel wheels, it traverses the apex of the pitch-cone; in 
 skew-bevel wheels, it is a generating line of the hyperboloidal 
 surfaces of the teeth. When a single cutter is used, the slide in 
 which it works is guided into the proper positions for the suc- 
 cessive strokes by a templet shaped like a tooth or like the space 
 between two teeth. In cutting the teeth of spur-wheels, a rotating 
 circular cutter is used; and the form of the cutting edges of its 
 teeth is the counterpart of that of the space between two teeth. 
 
 The cutting of worm-wheels by means of a rotating cutter which 
 is a copy of the screw that is to gear with the wheel, has already 
 been mentioned in Article 157, page 166.
 
 576 APPENDIX. 
 
 ADDENDUM TO ARTICLE 263, PAGE 298. 
 
 C'lnp-IVut. In certain machine-tools, the traversing motion of the tool- 
 holder is produced by a screw which rotates about a fixed axis, and drives 
 longitudinally (as described in Article 152, page 157) a nut that fits it truly, 
 without pinching it. That nut is made in two halves, capable of being 
 clasped round the screw or unclasped, according as the combination is to be 
 thrown into or out of gear. When the combination is about to be thrown 
 into gear, either the nut or the screw must, if required, be shifted longi- 
 tudinally, so as to bring the threads of the one opposite to the grooves of 
 the other. 
 
 ADDENDUM TO TABLE V., PAGE 482. 
 
 Strength and Elasticity of Silk and Flax. 
 
 Silken Flaxen Flaxen 
 
 Thread. Thread. Yarn in Canvas. 
 
 Ultimate tenacity 
 
 In lineal feet of material, 120,000 95,000 from 52,000 to 59,000 
 
 In metres, 36,600 29,000 from 15,900 to 18,000 
 
 Modulus of elasticity 
 
 In lineal feet of material, 3,000,000 
 In metres, 914,400 
 
 ADDENDUM TO ARTICLE 465, PAGE 547; AND ARTICLE 466, 
 PAGE 549. 
 
 Braced Shaft. The description given in Article 459, page 539, of a 
 braced connecting-rod, applies also to the general construction of a braced 
 shaft. The object of that construction is to diminish the deflection and the 
 bending stress of a shaft of long span, through the support given to the shaft 
 at the middle of the span by the bracing. The following is the easiest way 
 of computing the dimensions : The diameter of the shaft, h, is to be calcu- 
 lated so as to bear safely the twisting moment, combined with a bending 
 moment due to half the actual span. Let c be that half-span ; and y the 
 length of one of the arms of the bracing cross, which is to be fixed according 
 
 to convenience. Let be the safe working tension of the bracing rods, in 
 
 length of themselves ; say, 3,000 feet, or 900 metres. Let d be the proper 
 diameter for a tension-rod. Then the proper ratio for the diameter of each 
 of the four tension-rods to the diameter of the shaft is given by the following 
 formula : 
 
 2wc- 
 
 Vfy 
 j ** 
 ^ - 
 
 I . 2 w c 
 m 1 
 
 fy 
 
 The tension-rods should be tightened by means of screws just sufficiently to 
 prevent any perceptible slackness of the rods which successively come upper- 
 most as the shaft rotates. 
 
 ADDITIONAL AUTHORITIES. 
 
 Drauncs OF MACHINES: ~!&ose\efs Mechanics of Engineering. 
 
 CONSTRUCTION OP MACHISJBS: D. JL Clark On' the txi.ibiled Machinery of 18C2 (especially 
 Fart III., pages 128 to 210).
 
 APPENDIX. 577 
 
 ADDENDA TO SECTION VIIL, PAGE 221. 
 
 The Accumulator. The application of hydraulic power has been much 
 extended of late years, and especially with reference to machine tools. Machines for 
 rivetting, punching, and shearing iron plates, are now being worked by water- 
 pressure. Heavy loads, such as swing bridges, cranes, hoists, &c., are also readily 
 moved by hydraulic machinery. 
 
 The pressures used are very high, and, to obtain these, recourse is had to an 
 "accumulator." The accumulator is on the principle of the hydraulic press, viz. 
 a movable weighted plunger, raised by the action of force-pumps. The movable 
 part can be loaded with what weights are necessary, and raised by the pressure of 
 the water forced into the cylinder in which the plunger works : the pressure of the 
 water will then correspond to the load which it supports. By means of a separate 
 pipe leading to another plunger or piston connected with, say, the die of a rivetting 
 machine, hydraulic connection is maintained with the cylinder, and thus, when the 
 stroke of the die is made, a corresponding fall of the weighted part also takes place, 
 whereby a sudden expenditure of the work previously accumulated is brought about. 
 Variations in pressure and lengths of stroke are necessary for different operations ; 
 in rivetting machines, however, a load of 5 tons, having a fall of about 2 feet, 
 under a pump pressure of 1,500 Ibs. per square inch, is sufficient with a travel of 
 die of 24 inches to close up a rivet, the final pressure on the die being about 40 tons. 
 
 Hydraulic Itnm. By means of the hydraulic ram, water at a low head may 
 be made available for raising a portion of the same water to a higher level than the 
 supply. 
 
 The action of this machine depends upon utilising part of the energy of the supply- 
 water in closing a heavy clack-valve, and in compressing air in air-vessels connected 
 by clack-valves. During this process a loss of water takes place, as the closing of 
 the heavy clack-valve is dependent upon the increased velocity of flow due to this 
 waste. When the valve shuts, the outflow is stopped, and the energy of the water 
 is now spent in compressing the air in the air-vessels, by which means part of the 
 water is forced upwards through the delivery-pipe. As the velocity of the water 
 has now been reduced, the heavy clack-valve falls, and the process is repeated. 
 
 As the efficiency of such a machine will be the useful work performed in raising; 
 the water above the reservoir, divided by the work due to the fall of the water run 
 
 to waste, the ratio -' . - will represent the efficiency, where H = fall from reservoir 
 
 VB*~ 
 
 to ram, h = height of delivery above reservoir, Q quantity run to waste, 
 q = quantity delivered. The results as to efficiency differ considerably, depend- 
 ing on mechanism, and proportion of height of fall and height of delivery; the best 
 results appearing to be obtained when the proportion is as 1 ; 10 ; the efficiency so> 
 
 fi Q H 
 obtained will be about '6, and consequently q = -y . 
 
 ADDENDUM TO ARTICLE 262, PAGE 297, 
 
 Disengagements acting by Rolling Contact. In fig. 213, the radii A D 
 and B E of the two smooth wheels to be connected are equal; but those radii may, if 
 required, be made unequal, the essential condition of the proper working of the combi- 
 nation being that the angle, A C B, made at the centre of the intermediate wheel by the 
 two lines of centres, should be a little less than twice the complement of the angle of 
 repose. (See page 298.) 
 
 2p
 
 678 APPENDIX. 
 
 ADDENDUM TO ARTICLE 353, PAGE 402. 
 
 C'oniinuoiiN Brakes. Continuous brakes are now largely in use on railway 
 lines, and greatly facilitate the rapid stoppage of the trains through the application 
 of the brake resistance to the wheels of the engine and to the wheels of the carriages 
 composing the train. Various forms of such brakes have been tried ; the method 
 most in favour being that in which, through air pressure acting on suitable 
 mechanism under each carriage, tie brake blocks may be applied with rapidity to 
 the wheels. The air is subjected to pressure by a pump on the engine, and conveyed 
 by a flexible and easily jointed pipe to the points of application. From some recent 
 experiments it appears that the greatest retarding force is obtained when the wheela 
 are not skidded, but just on the point of being skidded. 
 
 ADDENDUM TO ARTICLE 392, PAGE 457. 
 
 Steel. Steel is now being largely used for engineering purposes, such as rails, 
 ship and boiler plates, &c. This steel, as at present manufactured, has a tensile 
 strength of about 28 tons per square mch, with an elongation of from 20 to 25 per 
 ~xnt. on a length of 8 inches ; strips cut from plates, heated to a cherry-red, and 
 cooled in water at 80 Fahr., can be bent double without breaking. Steel wire 
 appears to have a tensile strength of from 80 to 100 tons per square inch. 
 
 Since the safe working strength of plate steel is now recognised as equal to 6 4 
 tons per square inch, it may be expected to enter largely into the construction of 
 bridges of large span, as a considerable saving of weight is obtained. 
 
 In the case of boilers subjected to high pressures, steel may be used with much 
 advantage, as a reduction in thickness of the plates may be effected, allowing of a 
 more uniform distribution of the strain due to internal pressure ; the thinner plate 
 will also more readily transmit heat 
 
 ADDENDUM TO ARTICLE 397, PAGE 463. 
 
 Phosphor Bronze. This metal, an ailoy of copper, tin, and phosphorus, is 
 found to possess a high tensile strength and great endurance as regards wear, and is 
 being used principally for bearings and machinery. 
 
 The various strengths of this metal appear to vary greatly according to the pro- 
 portions of materials used, hence different alloys are employed for different classes of 
 work; the range of ultimate tensile strength being from 22,000 Ibs. to 57,000 Ibs. 
 per square inch, whilst the ultimate compressive strength varies from about 21,500 
 Ibs. to 45,000 Ibs. per square inch. 
 
 It also appears that wire made of this metal possesses the quality of high tensile 
 strength. 
 
 manganese Bronze. The strength and freedom from corrosion of manganese 
 bronze constitute it an excellent material for screw propellers, as blades of lighter 
 section can be used than are required for steel. 
 
 The transverse strength appears to be about double that of gun-metal and steel, 
 and its tensile strength varies from 24 to 40 tons per square inch. 
 
 ARTICLE 309, PAGE 348. 
 
 From experiments made by Capt. Douglas Galton, C.B., F.R.S., on the effect of 
 brakes upon railway trains, it appears that 
 
 (1.) The retarding effect of a wheel sliding upon a rail is not much less than when
 
 APPENDIX. 
 
 579 
 
 braked with such a force as would just allow it to continue to revolve, the distance 
 due to friction of the wheel on the rail being only about of the friction between tha 
 wheel and the brake blocks. 
 
 (2.) The coefficient of friction between the brake blocks and the wheels varies 
 inversely according to the speed of the train ; thus, with cast-iron brake blocks on 
 steel tires, the coefficient of friction when just moving was '330, 
 
 At 10 miles per hour '242 
 20 -192 
 
 30 
 40 
 50 
 60 
 
 164 
 140 
 116 
 074 
 
 Further, it was found that this coefficient was affected by time ; thus, starting at 
 27 miles per hour, the coefficient was -171, after 5 sees. -130, after 10 sees. -119, 
 after 15 sees. '081, and after 20 sees. '072; at 47 miles per hour the coefficient at 
 starting was -132, falling after 10 sees, to '070; and at 60 miles per hour falling 
 from -072 to -058. 
 
 These coefficients are further influenced by material and weather. It was found 
 that the distance run by a train on the level at 50 miles per hour varied with the 
 percentage of the total weight of train used for retardation, as follows with 5 per 
 cent. 555 yds., 10 per cent 277 yds., 20 per cent 139 yds., 30 per cent. 92 yds. 
 
 The author points out, among other conditions, that a perfect continuous brake 
 should be fitted to act npon all the wheels of engine and carriages; that it should 
 exert upon the brakes of each pair of wheels within two seconds a force of about 
 twice the load on these wheels ; that the brake-block pressure should be such that the 
 friction between the block and wheel may not be greater than the adhesion between 
 the wheel and the rail ; and that the action should be automatic, in the event of a 
 separation of the train or failure of connections. 
 
 ARTICLE 359, PAGE 410. 
 
 Governors. Amongst various forms of marine engine governors, Dunlop's 
 pneumatic governor has been found very serviceable for compound marine engines. 
 In this governor the controlling power lies in the level of the water outside the 
 ship, as this varies from time to time with the pitching of the vessel. The apparatus 
 consists of an air vessel, which is placed in communication with the sea by a cock. 
 The air becoming compressed, the action is communicated to the governor, which 
 consists of a casing containing an air-tight space in connection with a pipe, and 
 fitted with an india-subber diaphragm. This diaphragm as it rises and falls (due to 
 the pressure of the air, as acted upon by the varying level of the waves) acts upon 
 a spring, which communicates with the throttle valve by means of a link. 
 
 Feed-water heater* are being introduced into boilers, with good results. The 
 principle upon which these appliances depend is the delivery of the feed-water to the 
 boiler at a uniformly high temperature, such as 200 F. and higher, and the freeing 
 of the water from impurities. Exhaust steam is used, and in some cases live steam 
 from the boiler. The exhaust steam is passed into tubes surrounded by the feed- 
 water contained in a strong casing, and, after circulating, passes off. The water 
 receives so much heat from the surfaces of the tubes, and is afterwards further heated 
 by a coil of pipes containing live steam. Any matter precipitated as a deposit is 
 received in a filter chamber containing charcoal, and the water thus purified and 
 heated passes to the boiler. 
 
 Fuel Feeder*. Of late years many improvements have been made in connection 
 with steam boilers, with the view of increasing the economy of the fuel, and thereby 
 rendering the boiler a more efficient steam generator. The loss due to emission of smoke
 
 580 APPENDIX. 
 
 may in many cases be met by careful firing. Appliances known as " mechanical 
 stokers" have been introduced, whereby the fuel is gradually fed to the furnace through. 
 a hopper arrangement in front, means being adopted to work the coal to the back of the 
 furnace. Another method of fuel feeding has been recently tried, in which the coal 
 is charged upon a movable truck, which is pushed inside the grate, and below the 
 fire-bars; the fresh coal is then lifted by gearing and pushed up below the burning 
 fuel, and thus partially cokes before being consumed. Forced draught is used on 
 board some steam-vessels, such as torpedo boats. 
 
 ARTICLE 407, PAGE 471. 
 
 Preservation of timber, for railway sleepers, &c., from decay is attempted by 
 various processes. Kyanising consists in the immersion of the timber in a solution 
 of corrosive sublimate, in the proportion of 1 part of sublimate to 100 parts of water, 
 the wood being left in the solution for two days and upwards, according to thick- 
 ness. In another process chloride of zinc is used. Creosoting consists in first placing 
 the timber in a cylinder and applying superheated *team, and thereafter establishing 
 a vacuum in the cylinder ; by these operations the sap is withdrawn ; thereafter a 
 "dead oil," containing creosote heated to about 160, is forced at a pressure of about 
 20 Ibs. into the cylinder, and fills up the space from which the sap has been removed. 
 This process, however, varies with the condition of the wood when used.* 
 
 Fans. Ventilation of the workings in mines is now usually accomplished by 
 fans. Of these there are various forms, some of the pump order, and some depending 
 on centrifugal action. The Giubal fan, belonging to the latter class, appears to be 
 now largely used, and has been described as a " centrifugal ventilator, with eight or 
 ten flat blades set backwards from the radius, encased with mason work, and pro- 
 vided with a regulating shutter at the discharge orifice." Much additional advan- 
 tage is obtained from the addition of a short chimney or vertical outlet for the air, 
 having a form enlarging outwards to the top. An efficiency of about 60 per cent, is 
 claimed for these fans. 
 
 The Giubal fans, running about 60 revolutions per minute, with diameter of 
 30 feet, give about 100,000 cubic feet of air per minute; whilst fans of 37 feet 
 diameter at the same speed give as much as 200,000 cubic feet. 
 
 Where the pits are very deep, furnace-ventilation is thought to be fully more 
 economical than fans, as much as from 150,000 to 200,000 cubic feet of air per 
 minute having been obtained by a single large furnace. 
 
 ARTICLE 207, PAGE 221. 
 
 Hydraulic machinery. Hydraulic machinery has now been applied success- 
 fully to the raising of water from deep mines. In mines of ordinary depth, a steam- 
 engine at the surface, by means of rods passing down the shaft, works a set of ram 
 placed at different points in the shaft. When the depth is great, the weight of the 
 pump rods becomes excessive, and other methods have to be adopted. 
 
 Recently, extensive hydraulic arrangements have been carried out in Nevada, where- 
 1600 galls, per minute had to be raised from a mine about 2400 feet deep, and 
 delivered at a point 800 feet above the bottom of the shaft, f 
 
 The arrangement consists of a steam-engine, with pumps and accumulator at the 
 surface, which forces water down to a hydraulic engine placed at the foot of the 
 shaft ; the latter acts upon pumps by which the water is elevated to a height of over 
 800 feet. 
 
 * For further information on the preservation of timber, see Trans. American Society of Civil 
 Engineers. Vol. XL 
 
 t Se Trant. Intt. Engineers and Stiff builders in Scotland, VoL XXV.
 
 APPENDIX. 581 
 
 The steam-engine on the surface is compound, with a 10-feet stroke, the steam 
 pressure being 80 Ibs. The high-pressure cylinder, 35 inches in diameter, is placed 
 before and in line with the low-pressure cylinder, of 70 inches diameter. The pumps 
 are four in number, and have 84-inch plungers, worked direct from the cross-head of 
 the piston rod. The water is forced through an 8-inch pipe to an air vessel 25 inches 
 in diameter and 70 feet high, and thereafter down the shaft to the hydraulic engine at 
 bottom. The exhaust water returns to the surface through 10-inch pipes, to be again 
 used. 
 
 The pressure in the air vessel is 960 Ibs. per square inch, whilst the pressure 
 on the pipes at the bottom of the shaft is about 2000 Ibs. per square inch. 
 
 The underground machinery consists essentially of two hydraulic engines fitted with 
 four plungers, each 64- inches in diameter, and having a 10-feet stroke. Each pair 
 of these plungers carries a second 14-inch plunger, which raises the water to a height 
 of about 800 feet to the Sutro tunnel. The inlet and outlet valves are so arranged 
 that the one shuts the other. Cast-iron air vessels are used and rubber valves. 
 
 Hydraulic machinery is now largely used for engineering work,- such as rivetting, 
 flanging, punching, shearing, &c., and for moving swing bridges and working 
 cranes. The pressures used vary from 700 to 1500 Ibs. per square inch, an accumu- 
 lator being employed. 
 
 Hydraulic machinery is now also used on board war ships for loading the heavy 
 guns, and in some cases the ordinary gun-carriage is done away with, the guns simply 
 resting on blocks, which are free to slide backwards or forwards with the motion of 
 the gun. Hydraulic cylinders are used for checking the sliding motion due to the 
 recoil, and also serve for placing the gun again in position. 
 
 ARTICLE 233, PAGE 242. 
 
 Hastie's Differential Screw-Nut consists of a nut having two screw threads of 
 different pitch. Thus, for fixing a piston to its rod, a nut is used with a screw cut 
 on the inside to fit the thread on the piston rod. On the outside of the nut another 
 screw is cut of a finer pitch than the internal thread (the difference in practice is 
 about one-tenth). A corresponding thread to that upon the outside of the nut is cut 
 upon the piston. 
 
 The nut being first entered on the piston rod, its outer thread is then entered into 
 the piston block, and, being screwed home, draws the piston rod into its place at one- 
 tenth part of the speed of the nut itself. A corresponding power is in this manner 
 obtained to make a tight fit. 
 
 Any tendency to slip back is also overcome, the angle of the thread being very fine, 
 or equivalent to a screw having forty threads to the inch. 
 
 ARTICLE 376, PAGE 433. 
 
 Hollow shafts of Whitworth compressed steel are now frequently 
 propeller shafts of our large ocean-going steamships. In some cases the cc. these 
 are also made hollow. Considerable saving of weight is obtained by^ gr ea t e ** 
 shafts, as the outermost parts of a shaft or axle are those which gi.crank snatt 
 resistance to the torsional action to which they are subjected. Whemre shrunk on 
 is solid, it is usually built up of several pieces that is to say, the crra fire to induce 
 to the crank pin and straight part of the shaft, after being heatedly be proper ly i 
 the necessary expansion. Great care has to be taken so that aj reach to 26 incnw 
 line. Solid crank shafts for engines of 8000 or 10,000 horse-p ' 
 diameter.
 
 582 APPENDIX. 
 
 The diameter of a crank shaft, in terms of the indicated horse-power, may be 
 expressed as follows : 
 
 * / I.H.P. x 100 
 
 d = 
 
 N 
 Where d= diameter of shaft in inches, and N = number of revolutions per minute. 
 
 ARTICLE 240, PAGE 253. 
 
 Valve Gear motions. Various forms of link motion have been devised with 
 the view of obtaining the necessary reciprocating motion of the valve without the 
 intervention of eccentrics. One of these, known as Joy's valve gear, appears to 
 carry this out with great precision. 
 
 In this arrangement the eccentric is done away with, and the necessary movement 
 obtained from the motion of the connecting rod itself. 
 
 Fig. 1 shows the application of this arrangement to locomotive engines, and 
 Fig. 2 to marine engines (see page 583). 
 
 A link, L (see Fig. 1), whose extremities are secured to the connecting rod, C r 
 and to a radius bar, B, has a lever, E, attached to it, whose centre or fulcrum works 
 in a slot, S, curved to the radius of a link, K, which connects the other end of the 
 lever with the valve spindle, V. 
 
 The slot can be turned or rotated so as to incline on either side of the vertical at 
 will. By giving the slot these inclined positions, forward or backward motions of 
 the engine are obtained. 
 
 In some cases the slot is dispensed with, and the fulcrum of the lever is carried by 
 a radius rod. The path of the centre, or fulcrum, in this case can be shifted by means 
 of a weigh lever and hand wheel (see Fig. 2). 
 
 One great advantage of this form of valve gear, when applied to locomotives, is 
 that the slide-valve can be placed on the top of the cylinder, and thus larger 
 cylinders and greater bearing surfaces for the driving axles obtained. Another 
 advantage appears to be that it gives four points of acceleration and four points of 
 retardation in the stroke, these points of acceleration corresponding to the points of 
 "admission" and of "cut-off ;" and by this peculiarity not only is the admission 
 and the cut-off executed very much more smartly than by link, but the "release" 
 occurs somewhat later and the compression very much later, and, as a result of this, 
 an independent expansion valve is unnecessary. 
 
 The dotted lines in Fig. 1 show the paths of the various centres. 
 
 ARTICLE 381, PAGES 440-447. 
 
 able Tramways. Cable tramways have now been used with successful 
 -s in some of the large cities in the United States. 
 
 Nn Francisco an endless steel wire rope, 3 inches in circumference and 11,000 
 * * is supported on sheaves 11 inches in diameter, placed 39 feet apart, the 
 rJT > * n an * ron t uoe placed between the car rails and beneath the roadway, 
 Throujppeniug of about one inch wide is left. 
 
 anD > whiciig opening the connection between the car is made, and consists of an 
 onstantly g-yj the rope when the car is to be set in motion. The rope is kept 
 "^Dged at chjby means of a powerful steam-engine. Additional sheaves are 
 f he rope passel of direction of the rope. 
 
 *he carg are saiipd grip pulleys at the engine-house, 8 feet in diameter. 
 y horses. The 8to$ e more eas 'ly started and brought to rest than when drawn 
 me rope, and putting' o* e k effected by allowing the " grip " to slacken its hold of 
 Chicago abou f 4 " brakes. 
 
 a of these tramways are laid down, some of the ropes- 
 
 \
 
 APPENDIX. 
 
 583
 
 584 APPENDIX. 
 
 being over 27,000 feet in length. These ropes are made of steel 1| inches in 
 diameter, and have a breaking strength of 39 tons. 
 
 Wire Ropes. From a comparison of the strength of iron and steel wire ropes, 
 it appears that the steel rope is nearly double the strength of the iron rope. 
 
 Let L = breaking load in tons, and C = circumference of rope in inches, 
 then 
 
 (C\ 2 
 ) for Iron wire ropes, 
 
 /C\ 2 
 
 L = 25 X I TT I for Steel wire ropes, 
 and 
 
 (C\ 2 
 I for Hemp ropes. 
 
 In the East Eiver Bridge, New York, there are four cables, each of which 
 contains 5296 galvanised steel oil-coated wires. Thejse wires are laid parallel and 
 wrapped together, making a cable of 15| inches diameter. 
 
 The ultimate strength of such a cable is estimated at 12,200 tons. 
 
 Corrugated Flues. From recent experiments on steel corrugated flues by Mr. 
 Parker, of Lloyds', the following formulas are proposed for strength : 
 
 rr,.. _.*.!. u 60,000 x t 
 Ultimate crushing strength in Ibs. per square mch = , 
 
 where t is the thickness of plate, and d the mean diameter of furnace. 
 Working strength in Ibs. per square inch ^ H ', 
 
 where T is the thickness of plate in sixteenths of an inch, and D the greatest 
 diameter of the furnace in inches. With the latter rule the margin of safety appears 
 to be fully 5. 
 
 The experiments were carried out with a flue having the corrugation 14 inches deep 
 and 6 inches apart. The steel plate showed a tenacity of 22*7 tons per square inch, 
 with an elongation of 35 per cent, in a length of 10 inches. 
 
 ARTICLE 376, PAGES 433 and 582. 
 
 Strength of Shaft* for Screw Steamers. 
 
 Where d = diameter of crank shaft in inches. 
 
 D 2 = square of diameter of low-pressure cylinder in inches, or the gum of the 
 
 squares, if there are more than one. 
 P = absolute pressure, or boiler pressure + 15. 
 S stroke in feet. 
 
 y, has the following values: 170 for two cranks set at right angles, and 180 for 
 three cranks set at 120 apart. 
 The diameter of the intermediate shafting = *95 d.
 
 APPENDIX. 585 
 
 ARTICLE 472, PAGE 556. 
 
 Fly- Wheels. The stress on the rim of a fly-wheel due to centrifugal fores may 
 be found as follows : 
 
 The outward radial tension due to centrifugal force on a unit, say, of one cubic inch, 
 
 of the material in the rim is (see p. 364), and the total bursting tension to be 
 
 resisted by the tenacity of the material in a ring of one square inch section in the rim 
 
 will be - x r x 12 (see p. 494). 
 gr 
 
 11} 1""* 
 
 Let / = tensile strength of the material, then x 12 = / or v 
 
 Let the material of the rim be cast-iron where w '26 lb., and let/= 1,500 Ibs. 
 
 as safe working strength; then v 124 feet per second, as a safe 
 
 A/ 12 x -26 
 
 velocity for the rim. 
 
 This value will of course vary with the assigned value of/; 100 feet per second being 
 sometimes given as the extreme safe limit of speed. 
 
 Strength of Leather Bells. Ultimate strength about 3200 Ibs. per sq. inch; 
 working strength about one-eighth of this. 
 
 2-inch cotton rope, working strength COO Ibs. 
 
 Strength of Wires. 
 
 Pure Copper wire, .... about 18 tons per sq. inch. 
 Phosphor bronze, . . . ,, 45 ,, ,, 
 
 Steel wire, . . . . . 90 to 140 ,, 
 
 Strength of Lead Pipes. 
 
 4,400 x t 
 
 where p bursting pressure in Ibs. per square inch. 
 t = thickness in inches. 
 d = internal diameter in inches.
 
 586 
 
 APPENDIX. 
 
 
 Gramme in a grain. 
 Kilog. in a Ib. avoirdupois. 
 Tonnes in a ton. 
 Metres in a foot. 
 Millimetres in an inch. 
 Kilometre in a mile. 
 Square metre in a square foot 
 
 Square inillim. in a square inc 
 
 Cubic metre in a cubic foot. 
 Kilogrammetre in a foot-poun 
 ( Kilogrammes-to-the-metre 
 ( a pound-to-the-foot. 
 ( Kilogrammes -to -the -squai 
 metre in a pound-to-tli 
 ( square-foot. 
 ( Kilog. -to -the - square -mil 
 J metre in a pound-to-tli 
 (^ square-inch. 
 ( Kilogrammes - to - the - cub: 
 
 6 P 1 -! "^ 
 
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 f*
 
 APPENDIX. 
 
 587 
 
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 INDEX. 
 
 ABSOLUTE unit offeree, 318. 
 Acceleration, 330. 
 
 Work of, 354. 
 
 Accumulator, 577. 
 Action and re-action, 31G. 
 Actual energy, 373. 
 Addendum of a tooth, 116. 
 Adjustments in mechanism, 293. 
 
 of speed, 80. 185, 310-313. 
 
 of stroke, 310-312. 
 
 Aggregate combinations, 235. 
 
 paths, 239, 261, 291. 
 Air Friction of in pipes. 404. 
 Transmission of motive power by, 
 
 445. 
 
 Allan's link motion, 257. 
 Alloys, 461. 
 
 Aluminium bronze, 463, 477. 
 Angle of repose, 211, 298, 349. 
 Angles of tools Cutting, 566, 
 Angular velocity, 24. 
 Approach of tee'th, 118. 
 Arcs Measurement of, 27. 
 Axis Fixed, 24. 
 ,, Instantaneous, 46. 
 
 of a rolling body, 51. 
 Neutral, 512. 
 ,, Temporary, 45. 
 Axles and shafts Efficiency of, 427, 431, 
 
 433, 449. 
 
 Friction of, 396. 
 Strength, of, 540, 543. 
 
 BABBITT'S metal, 464. 
 Back-lash, 116. 
 Back of a tooth, 115, 152. 
 Balance of a machine, 365. 
 
 ,. of effort and resistance, 370. 
 
 375. 
 
 Running, 368. 
 Standing, 368. 
 Ball-and-socket joint, 192. 
 Ball clack, 303. 
 Band-link, 213. 
 Bands Classed, 179. 
 Connection by, 180. 
 Deflection of, 534. 
 Efficiency of, 440, 447. 
 .. Friction of, 351. 
 
 Bands Length of, 183. 
 ,, Materials for, 474. 
 ,, Motion of, 74, 184. 
 Strength of, 532, 585 
 ,, with circular pulleys, 182. 
 ,, with polygonal pulleys, 182. 
 Barrel pulley, 187. 
 Beams Bending action on, 504. 
 ,, Deflection of, 517. 
 ,, fixed at the ends, 521. 
 ,, in Hnkwork, 192. 
 ., Longitudinal sections of uniform 
 
 strength for, 517. 
 ,, Resilience of, 521. 
 Strength of, 513. 
 j ,, Strength of walking, 557. 
 Bearing-pressure, 423. 
 
 ,, Intensity of, 350. 
 
 Bearings Dimensions of bushes and plum- 
 ber blocks for, 552. 
 Forms of, 17, 353. 
 ,, Friction of, 353. 
 Lubrication of, 350. 
 Materials for, 462. 
 Bellows Motion of, 226. 
 Belt Flat driving, 184. 
 Motion of, 74, 184. 
 Strength of, 474, 532, 585. 
 with fast and loose pulleys, 184. 
 with speed cones, 18.5. 
 Bending moments Calculation of, 505. 
 Bending Moment of resistance to, 510, 
 
 513. 
 
 ,, Resistance to, 504. 
 Bevel-wheels, 82, 86. 
 Blocks and tackle, 214. 
 Block-brake, 401. 
 Boiler flues Strength of, 525, 584. 
 
 shells Strength of, 494, 578. 
 Bolts Dimensions and strength of, 499. 
 Bonnet valve, 302. 
 Boring, 574. 
 Braced shaft, 576. 
 Bracket, 521. 
 Brakes. 400. 
 
 Block, 401. 
 Continuous, 578. 
 Fan, 406. 
 Flexible 402.
 
 590 
 
 INDEX. 
 
 Brakes Pnmp or hydraulic, 404. 
 
 Brass, 462, 477. 
 
 Brasses, 462, 552. 
 
 British and French, measures, 586. 
 
 Broach, 561. 
 
 Bronze, 462, 477, 578. 
 
 Bulkiness, 325. 
 
 Bushes, 18, 552. 
 
 CABLE TRAMWAYS. 582. 
 Cam-motions in turning, 291. 
 Cams, 170. 
 
 Rolling, 99. 
 Spiral and conoidal, 174. 
 ,, To draw by circular arcs, 173. 
 Capstan, 190. 
 
 Castings Iron for machinery, 453. 
 Cast-iron, 451. 
 
 Malleable, 453. 
 
 Resilience of, 485. 
 
 ., Strength of, 453, 477, 479, 
 
 481, 486. 
 
 Tools for cutting, 567. 
 Cataract, 404. 
 Catch, 206. 
 
 Frictional, 211. 
 Centre of a curved line, 336. 
 of a plane area, 334. 
 of a volume, 336. 
 of buoyancy, 329. 
 of gravity, 328, 345. 
 of magnitude, 334. 
 of mass, 328, 345. 
 of percussion, 361. 
 of pressure, 329. 
 of special figures, 336. 
 Centrifugal couples, 365, 368. 
 force, 330, 364. 
 force Balance of, 368. 
 force Resultant, 365. 
 tension, 441, 532. 
 whirling of shafts, 549. 
 Chains Gearing, 190. 
 Motion of, 74. 
 Strength of, 535. 
 Change wheels, 311. 
 Changing speed, 80, 185, 310-313. 
 
 stroke, 310-312. 
 Chilling cast-iron, 452. 
 Chuck, 574. 
 
 Circle Involute of, 53, 56. 
 Projections of, 15. 
 Circular aggregate paths, 261. 
 ,, arcs Measurement of, 27. 
 ., bevel wheels, 86. 
 ,, pulleys, 182. 
 skew bevel wheels, 87. 
 spur wheels, 85. 
 wheels, 84. 
 Clasp-nut, 576. 
 
 Clearance of teeth, 116. 
 Clearing curves of teeth, 123. 
 Click, 206. 
 
 DouMe-acting, 209. 
 Frictional, 211. 
 Silent, 208. 
 Clutch, 295. 
 Cocks, 306. 
 
 Co-efficient of elasticity, 492. 
 of friction,' 3 19, 579. 
 of steadiness, 362. 
 ,, of strength, 488. 
 Cog Hunting, 1U4. 
 Cogs, 104, 473. 
 
 Strength of, 553. 
 Wooden, 473. 
 Collapsing Resistance to, 525. 
 Collar for plunger, 221. 
 
 for shaft, 353. 
 Comb for screw-cutting, 5C8. 
 Combinations Aggregate, 235. 
 
 Elementary, 77, 80, 436. 
 
 Common measure Greatest, 106. 
 Comparative motion, 22. 
 
 in elementary combi- 
 
 nations, 78. 
 in rotating pieces, 31, 
 
 35. 
 ,, of rigidly connected 
 
 points, 32. 
 Component, 319. 
 
 motions, 18. 
 
 velocities in a rotating piece, 
 
 33. 
 
 Composition of forces, 319. 
 of motions, 18. 
 of rotation with translation, 
 
 52, 
 
 of rotations, 54. 
 Compound screw, 242. 
 Compression Longitudinal, 522. 
 Cones Pitch, 86. 
 Rolling, 68, 73. 
 Speed, 185. 
 Conical valve, 302. 
 Connected points Motion of, 32. 
 Connecting-rod, 192, 197, 449. 
 
 Effect of obliquity o^ 
 
 449. 
 
 Strength of, 524, 537. 
 
 Connection Line of, 32, 77. 
 Continuous brakes, 578. 
 Contraction of cast iron, 454. 
 Copper, 461. 
 Copy-plate, 291. 
 Cords, 180. 475. 
 
 Pulleys for, 187. 
 Corrosion of iron, 461. 
 Corrugated flues, 584. 
 Cottars Strength of, 499.
 
 INDEX. 
 
 591 
 
 Counter-efficiency, 377, 422. 
 Counter-wheels, 286. 
 Coupled parallel shafts, 44, 194. 
 Couples, 321. 
 
 Centrifugal, 365, 368. 
 Statical, 321. 
 Coupling Circular half-lap, 552. 
 Dimensions of, 552. 
 Double Hooke's, 205. 
 Drag link, 194. 
 Hooke-and-Oldham, 206. 
 Hooke's, 203. 
 Oldham's, 166. 
 Pin and slot, 167. 
 rod, 192. 
 
 Crank and beam Motion of, 196. 
 and piston-rod Motion of, 196. 
 and slot, 167. 
 in linkwork, 192. 
 rod, 192. 
 Strength of, 557. 
 Z, 272. 
 Crushing by bending, 524. 
 
 ,, Direct Resistance to, 522. 
 Curvature, 518. 
 
 Curved lines Measurement of, 28. 
 Cutter Circular, 568. 
 Cutting tools, 559. 
 Cycloid, 53. 
 Cylinder Flexible, 226. 
 
 Hydraulic, 221. 
 Cylinders Hollow, 494. 
 
 ,, ,, Resistance to burst- 
 
 ing of, 494, 495. 
 
 ,, Resistance to col- 
 
 lapsing of, 525. 
 Cylinders^Pitch, 83, 85. 
 
 Rolling, 53, 56, 73. 
 
 DASH-POT, 404. 
 Dead-beat escapement, 179. 
 Dead points in linkwork, 193. 
 ,, of cam, 173. 
 
 of pin and slot, 168, 169. 
 
 Deflection of bands, 534. 
 of beams, 517. 
 ,, of shafts, 545. 
 of steel springs, 386, 389. 
 Density, 325. 
 Deviating force, 330, 363. 
 Diametral pitch, 111. 
 Differential pulley, 240; screw-nut, 581. 
 
 harmonic motions, 260. 
 Disengagements, 294. 
 Disengagement by a clasp-nut, 576. 
 by a clutch, 295. 
 by bands, 184, 185, 188, 
 
 299. ' 
 
 - by fast and loose pulleys, 
 184. 
 
 Disengagement by friction-cones, sectors, 
 and discs, 296. 
 
 by linkwork, 299. 
 
 by smooth wheels, 297, 
 
 577 
 
 by teeth, 298. 
 
 by valves, 301. 
 
 Double beat valve, 303. 
 Drag-link, 194. 
 Drawings of a machine. 5. 
 Drill, 566. 
 Drilling, 574. 
 Driver, 77. 
 Driving point, 23. 
 Drum, 187. 
 
 Dynamics of machinery, 315. 
 Dynamometer, 382. 
 
 Friction, 383. 
 
 Integrating, 394. 
 
 Rotatory, 386, 446. 
 
 ,, Torsion, 387. 
 
 ,, Traction, 383. 
 
 ECCENTRIC, 197. 
 
 gearing, 247. 
 
 pulleys, 188. 
 ,, rod, 192, 197. 
 Effect and power, 378. 
 Efficiency and counter-efficiency, 315, 376, 
 
 422. 
 Efficiency of bands, 440, 447. 
 
 of block and tackle, 443. 
 of fluid connection, 444. 
 of linkwork, 442. 
 of a machine, 315. 
 of primary pieces, 423. 
 of rolling contact, 436. 
 of a screw, 433. 
 
 of a shaft or axle, 427, 433,449. 
 of a sliding piece, 426, 437. 
 of teeth, 438. 
 Effort, 316, 368. 
 
 Accelerating, 371. 
 
 ., and resistance Balance of, 370, 
 
 375. 
 Elasticity, 492. 
 
 of spiral springs, 389. 
 Elementary combinations, 77. 
 
 classed, 80, 229. 
 
 ,, Efficiency of, 
 
 436. 
 illipses traced by the trammel, 267. 
 
 ,, by rolling, 266. 
 
 lliptic pulleys, 189. 
 wheels, 95. 
 
 Involute teeth for, 292. 
 
 nergy, 370. 
 
 Actual (or kinetic), 373. 
 and work General equation of 
 378.
 
 592 
 
 INDEX. 
 
 Energy exerted and work done Equality 
 
 of, 370, 375. 
 Potential, 370. 
 stored and restored, 373, 375, 407. ! 
 Engaging and disengaging gear, 294. 
 Epfcyclic train, 243, 246. 
 Epicycloid, 56. 
 Epicycloidal teeth, 130. 
 Epitrochoid, 56, 262, 290. 
 
 traced by rolling, 262, 290. 
 Escapements, 175. 
 
 Anchor recoil, 177. 
 
 Dead-beat, 179. 
 
 Expansion of iron, 325, 454. 
 
 FACE of a tooth, 115. 
 
 Factor Prime of a number, 105. 
 
 Factors of safety, 488, 545. 
 
 Falling bodies, 330. 
 
 Fans tor mines, 580. 
 
 Fan-brake, 406 ; governor, 421. 
 
 Fastenings Strength of, 497. 
 
 Feed-motions in machine tools, 293, 569. 
 
 Flank of a tooth, 115. 
 
 circle of teeth, 123. 
 Flaxen fibre Strength of, 576. 
 Flexible secondary pieces, 74. 
 Floating metal, 454. 
 Fluid secondary pieces, 75, 221. 
 Fly-wheels, 361, 407, 409, 556, 585. 
 Follower, 77. 
 Foot pounds, 338. 
 Force Absolute unit of, 318. 
 
 Accelerating and retarding, 329. 
 and mass Measures of, 318. 
 Centrifugal, 330, 364, 368. 
 de cheval, 338. 
 Deviating, 330, 363. 
 Ordinary units of, 317. 
 Reciprocating, 374. 
 Forces, 316. 
 
 Composition and resolution of, 
 
 319. 
 
 -Parallel, 322. 
 Fractions Continued, 10R. 
 
 Converging, 108. 
 Frame, 17- 
 
 Framework Straining actions on, 530. 
 Freedom of teeth, 116. 
 French and British measures, 586. 
 Friction, 348. 
 
 clutch, 296. 
 
 Co-efficient of, 349, 579. 
 
 cones, 296. 
 
 couplings and disengagements, 
 
 296. 
 
 discs, 296. 
 dynamometer, 383. 
 Heat produced by, 354, 399. 
 ,, in machines, 351. 
 
 Friction Measurement of, 395. 
 
 ,, of air in pipes, 404. 
 
 ,, of a band, 351. 
 
 of axles, 396. 
 
 ,, of bearings, 353. 
 
 ,, of pistons and plungers, 399. 
 
 of pivots, 353. 
 
 ,, of water in pipes, 404. 
 
 ,, Resistance of, 348. 
 
 ,, sectors, 296. 
 
 Table of co-efficients of, 349. 
 
 . , Work done against, 353. 
 Friciional catch, 211. 
 
 gearing, 102. 
 
 GEAR Disengaging and re-engaging, 294. 
 
 Reversing, 295. 
 Gearing chains, 190. 
 
 Frictional, 102. 
 --.Intermittent,' 139, 286. 
 Screw, 157, 163, 289, 439. 
 Slide valve, 253. 
 - Toothed, 114. 
 Geneva stop, 286. 
 Geometry Descriptive Elementary rules 
 
 in, 3. 
 
 of machinery, 3. 
 ' Rules in relating to planes, 9. 
 Rules in relating to straight 
 
 lines, 6. 
 
 Gibs and cottars Strength of, 499. 
 Gooch's link motion, 255. 
 Governors, 410, 579. 
 
 Balanced, or spring, 418. 
 Bellows, 421. 
 Differential, 420. 
 Disengagement, 419. 
 Fan, 421. 
 Fluctuations of, 418. 
 Isochronous gravity, 415. 
 Loaded, 413. 
 M Loaded parabolic, 415. 
 Parabolic, 414. 
 Pendulum, 411. 
 Pump, 421. 
 Grasshopper parallel motion, 275, 292. 
 Gravity, 316. 
 
 Centre of, 328, 345. 
 Motion produced by, 330, 357, 
 Specific, 325. 
 Table of specific, 326. 
 Grease, 350. 
 Gudgeons, 18, 540. 
 
 Strength of, 541. 
 Guide pulleys, 188. 
 Gyration Radius of, 359, 514. 
 table, 360. 
 
 HARMONIC motion, 250, 529. 
 Heaters for feed-water, 579.
 
 INDEX. 
 
 593 
 
 Heat of friction, 354, 399. 
 Heaviness, 325. 
 
 Table of, 326. 
 Heights due to velocities Table of, 331. 
 Helical motion, 36. 
 
 Resolution of, 68. 
 teeth, 156. 
 Helix, 38. 
 
 ,, Normal, 41. 
 Hide Raw belts, 474. 
 Hooke's coupling, 203, 205. 
 Hooke-and-Oldham coupling, 206. 
 Horse-power, 339, 378, 393. 
 Hunting-cog, 104. 
 Hydraulic connection, 221, 580. 
 
 Comparative velo- 
 
 cities in, 223. 
 
 Efficiency of, 444. 
 
 Intermittent, 224. 
 
 Hydraulic cylinder, 221. 
 Hydraulic press, 225, 381. 
 
 >5 >t Strength of, 495. 
 
 Hydraulic Ram, 577. 
 Hyperboloids Pitch of, 87. 
 
 Rolling, 70, 87. 
 Hypocycloid, or internal epicycloid, 56. 
 
 IMPULSE, 356. 
 Inclined plane, 232, 381. 
 Indicated power, 390. 
 Indicator, 390. 
 
 diagram, 392. 
 Inertia Moment or, 359, 513. 
 
 Reduced, 362. 
 Inside gearing, 85, 117. 
 Instantaneous axis, 46, 51. 
 Intermittent gearing, 139, 286. 
 Involute of circle, 53, 56. 
 teeth, 120, 292. 
 Iron, 450. 
 Bar, 455. 
 Cast, 450, 451. 
 Corrosion of, 461. 
 Expansion of, 325, 454. 
 forgings, 456. 
 
 Impurities of, 451. 
 I, Kinds of, 450. 
 Malleable, or wrought, 450, 455. 
 
 Malleable cast, 453. 
 -Pig, 451. 
 -Plate, 455. 
 w Preservation of, 460. 
 
 Resilience of, 485. 
 Steely, 457, 484. 
 
 -Strength of, 453, 456, 459, 477, 
 
 479, 481, 482. 
 Tools for cutting, 559. 
 Welding of, 456. 
 
 JACQUAKD hooks, 300. 
 Joint Ball and socket, 192. 
 
 Joint Double universal, 205. 
 
 Universal, 203. 
 Joint-pins and fastenings Strength of, 
 
 497. 
 Joints Strength of rivetted, 495. 
 
 Strength of welded, 495. 
 Journals, 18. 
 Journals Friction of, 353. 
 
 Strength and dimensions of, 541, 
 544. 
 
 KEYS Strength of, 497, 499. 
 Kilogramme, 318. 
 Kilogrammetre, 321, 338. 
 Kinetic energy, 373, 
 Knot, or nautical mile, 340. 
 
 LAP of slide valve, 307. 
 
 Lathes, 572. 
 
 Lead, 461. 585. 
 
 Lead of slide valve, 309. 
 
 Leather Strength of, 474, 585. 
 
 Lever, 192, 232, 380. 
 
 Strength of, 557. 
 Line of connection, 32, 77. 
 Line of contact, 157. 
 Link, 192. 
 
 Band, 213. 
 , Drag, 194. 
 
 for contrary rotations, 196. 
 Slotted, 213. 
 Strength of, 535, 537. 
 Link-motions for slide valves, 253, 582. 
 lankwork Aggregate, 248. 
 Connection by, 192. 
 
 Doubling of oscillations by, 
 
 201. 
 
 Efficiencj of, 442, 449. 
 Harmonic motion in, 250. 
 Intermittent, 206. 
 Length of stroke in, 197. 
 Slow motion by, 202. 
 Velocity -ratios in, 199. 
 with reciprocating motion, 196. 
 Lobed wheels, 97. 
 Lubrication, 348, 350, 395. 
 
 MACHINES Efficiency of, 315. 
 
 Frame of, 17. 
 
 General equation of the action 
 of, 378. 
 
 Moving pieces in, 17. 
 
 Straining actions in, 527. 
 Machinery Dynamics of, 315. 
 
 Geometry of, 3. 
 
 Use and parts of, 1. 
 Magnitude Centre of, 334. 
 Malleable cast-iron, 453. 
 
 iron, 450, 455. 
 Mass, 318, 355.
 
 594 
 
 INDEX. 
 
 Mass Centre of, 328, 345. 
 Materials used in machinery, 450. 
 Measure Greatest common, 106. 
 Measures Comparative table of French 
 
 and British, 586. 
 Mechanical powers Comparative motion 
 
 in, 231. 
 
 Forces in the, 379. 
 
 Mechanism Aggregate combinations in, 
 
 235. 
 Elementary combinations in, 
 
 77, 80. 
 ,, Elementary combinations in 
 
 classed in detail, 229. 
 Primary pieces in, 17. 
 Pure, 3. 
 Secondary pieces in, 43. 
 Mensuration of areas, 331. 
 
 of curved lines, 27, 333. 
 of geometrical moments, 334. 
 of volumes, 333. 
 Modulus of a machine, 377. 
 of elasticity, 492. 
 of pliability, 492. 
 of resilience, 494. 
 of rupture, 513. 
 ,, of stiffness, 491. 
 Moment of a plane area, 223, 335. 
 of inertia, 359, 513. 
 of resistance, 513. 
 Moments Geometrical, 334. 
 
 Statical, 321. 
 Momentum, 329, 355, 356. 
 Mortise-wheel, 473, 554. 
 Motion Comparative, 22. 
 
 Comparative in rotating pieces, 
 
 31, 35. 
 
 Helical, 36. 
 
 of a rigid body Unrestricted, 50. 
 of connected points, 32. 
 Periodic, 33, 196, 246, 375, 407. 
 Relative, 21, 30. 
 Resolution and composition of, 18. 
 
 NECKS of shafts, 544. 
 Neutral axis, 512. 
 
 surface, 511. 
 Nominal horse-power, 339. 
 Non-circular wheels, 92. 
 Normal helix, 41. 
 
 Normal pitch of gearing-screws, 158, 160, 
 163. 
 
 of screw-line, 41. 
 
 of teeth, 122. 
 Nut Clasp, 576. 
 ., or internal screw, 36. 
 Nuts for bolts, 499, 500. 
 
 ODONTOGRAPH, 136. 
 Oil, 350. 
 
 Oldham coupling, 166. 
 Outside gearing, 85. 
 
 PADDLE-WHEELS Feathering, 270, 
 Pallets, 175. 
 Pandynamometer, 387. 
 Parallel forces, 322. 
 motions, 274. 
 
 Extent of deviation of, 
 
 281. 
 
 Grasshopper, 275, 292. 
 Robert's, 285. 
 Rules for designing, 
 
 277. 
 Tracing approximate 
 
 circular arcs by, 283. 
 Watt's, 275. 
 
 Paring tools, 562. 
 Pasteboard for bearings, 474. 
 Paths Aggregate, 239, 261. 
 Pendulum, 361. 
 
 governor, 411. 
 Revolving, 364. 
 Percussion Centre of, 361. 
 Periodic motion, 33, 196, 246, 375, 407. 
 Phosphor Bronze, 578. 
 Pig-iron, 451. 
 Pillars, 524. 
 
 Pin and slot Connection by, 167. 
 Pinions, 105. 
 
 Long or broad, 236. 
 Pin-rack, 139. 
 
 Pins-Strength of, 497, 499. 
 Pin-wheel, 137. 
 Piston, 221. 
 
 Friction of, 399. 
 Work performed by a, 341. 
 Piston-rod, 223. 
 
 Efficiency of, 449. 
 Strength of, 524. 
 Pitch of a screw Axial, 37, 42. 
 Circumferential, 42. 
 
 Diametral and radial, 
 
 , Divided, 42. 
 Normal, 41, 158, 160, 163. 
 Pitch-circles, 82, 157. 
 Pitch-cones, 86. 
 Pitch-cylinders, 83, 85, 
 Pitch-lines, 82. 
 Pitch-point, 82, 115. 
 Pitch-surfaces, 81, 180. 
 Pitch of teeth, 103. 
 Pitching, 113, 575, 
 Pivots, 18. 
 
 Pivot Friction of, 353. 
 Plane surfaces Scraping of, 571. 
 Planing, 569. 
 
 ,, machine, 570. 
 Planometer, 571.
 
 INDEX. 
 
 595 
 
 Plate-iron, 455. 
 Plate-joints, 495, 498. 
 Pliability 491, 492, 531. 
 Plumber-blocks, 652. 
 Plunger, 221. 
 
 Friction of, 399. 
 Pneumatic connection, 445. 
 Point Driving and working, 23. 
 Polygon of forces, 320. 
 Potential energy, 370. 
 Pot-metal, 464. 
 Power, 339. 
 
 and effect, 378. 
 Horse, 339, 378, 393. 
 Indicated, 390. 
 Powers Mechanical, 231, 379. 
 Press Hydraulic, 225, 381, 495. 
 Pressure Centre of, 329. 
 
 Intensity of, 329, 342. 
 
 on bearings, 350. 
 Primary moving pieces Efficiency of, 423. 
 Motions of, 17,24. 
 Work of, 344. 
 
 Prime factors, 105. 
 
 mover, 1. 
 
 Projection of points and lines, 3, 6. 
 Proof strength, 487, 490. 
 Puddling iron, 455. 
 Pulley, 214, 234, 381. 
 Pulley-blocks, 214, 443. 
 Pulleys, 179. 
 
 Circular, 182. 
 
 Differential, 240. 
 
 Eccentric, 188. 
 
 Elliptic, 189. 
 
 Fast and loose, 184. 
 
 for chains, 190. 
 
 for flat belts, 184. 
 
 for ropes and cords, 187. 
 
 Guide, 188. 
 
 Non-circular, 188. 
 Polygonal, 182. 
 
 Speed,_185. 
 
 Straining, 188. 
 
 Suspended, 191. 
 Pump brake, 404. 
 Punching tools, 560. 
 Purchase, 214, 217. 
 
 RABATMENT, 4. 
 Rack Circular, 237. 
 
 gearing with screw, 289. 
 Racks Teeth of, 120. 
 
 n Toothless, 81. 
 Radial pitch, 111. 
 Radius of gyration, 359, 360, 514, 
 Ratchet and click, 206. 
 Rate of work, 339. 
 
 Ratio Approximations to a given, 107 
 Reaction, 316. 
 
 Reaction of accelerated and retarded 
 
 bodies, 330, 529. 
 ,, of a revolving body, 330. 
 ,, Straining effects of, 529. 
 Reciprocating force, 374. 
 Reduplication, 214, 443. 
 Regulating apparatus for machinery, 400. 
 Relative motion, 21. 
 
 ,, in a rotating piece, 30. 
 
 Repose Angle of, 211, 298, 349. 
 Resilience, 485, 492, 504, 521. 
 Resistance, 316. 
 
 ,, due to acceleration, 354. 
 ,, Mean, 347. 
 of friction, 348. 
 Reduction of to the driv- 
 ing point, 344. 
 Resolution of forces, 319. 
 ,, of motions, 18. 
 ,, of rotation, 63. 
 Resultant force, 319, 329. 
 
 motion, 18. 
 Reversing-gear, 295. 
 
 ,, by belts, 299. 
 
 by linkwork, 299. 
 
 by teeth, 299. 
 
 by valves, 301. 
 
 Revolution Motion of, 26. 
 Rigidity, 491. 
 Rimer, 561. 
 
 Rivets Strength of, 498. 
 Rivetted joints Strength of, 495. 
 Rod Connecting, 192, 197, 449, 524, 537. 
 Eccentric, 192, 197. 
 Piston, 223, 449, 524. 
 Rolled curves, 52. 
 
 To draw, 58. 
 Tracing of by mechan- 
 ism, 265 r 
 Rollers, 81. 
 
 Rolling, 51, 56, 68, 70. 
 cams, 93, 99. 
 ,, cones, 68, 73. 
 ,, contact Connection by, 81. 
 Efficiency of, 436. 
 , , General conditions of, 82. 
 resistance, 353. 
 Root-circle of teeth, 123. 
 Ropes Strength of, 475, 582, 584. 
 
 Wire, 447, 533, 534, 584. 
 Rotation about a fixed point, 48. 
 
 Composition and resolution of, 
 
 54, 63. 
 
 ,, compounded with translation, 52. 
 ,, of a primary piece, 24. 
 ,, of a secondary piece, 43, 45. 
 Ruled surfaces Cutting of, 569. 
 
 SAFETY Factors of, 488. 
 Saw, 568.
 
 596 
 
 INDEX. 
 
 Scraping, 562, 571. 
 Screw, 18. 
 
 and nut, 157, 576. 
 Comparative motion in, 37. 
 Compound, 242. 
 Differential, 242. 
 Efficiency of, 433. 
 Endless, 163. 
 
 as a mechanical power, 234, 381. 
 Motion and figure of, 36. 
 Pitch of, 36. 
 Reciprocating endless, 246. 
 Strength of, 499. 
 Tangent, 165. 
 wheel work, 157. 
 with clasp-nut, 576. 
 Screw-cutting by lathe, 575, 
 
 by taps and dies, 568. 
 
 Screw-gearing, 157. 
 
 Efficiency of, 439. 
 
 Figures of threads in, 163. 
 with rack, 289. 
 
 Screw-line, or helix, 38. 
 
 Axial pitch of, 38, 42. 
 Circumferential pitch of, 42. 
 Curvature of, 41. 
 Development of, 40. 
 Divided pitch of, 42. 
 Normal pitch of, 41. 
 Screws Right and left-handed, 37. 
 Secondary moving pieces, 43. 
 
 Flexible, 74. 
 -Fluid, 75. 
 
 . Rotation of, 45 
 
 Translation of, 
 
 44. 
 
 Sectors Logarithmic spiral, 99. 
 Shaft Braced 576. 
 
 ,, Centrifugal whirling of, 549. 
 Shafting Efficiency of long lines of, 433. 
 Span between bearings of, 545. 
 Shafts and axles Efficiency of, 427, 433, 
 
 449. 
 
 Shafts Resilience of, 504. 
 Resistance of to twisting, 600. 
 Stifihess of, 545, 576. 
 Strength of, 501, 540, 644, 617, 
 
 576, 581, 584. 
 Shaper-plate, 291. 
 Shaping machine, 571. 
 Shearing Resistance to, 496. 
 
 tools, 560. 
 Sheaves, 214, 216. 
 Shells Boiler, 494, 578. 
 Silk Strength of, 576. 
 Simpson's rules, 332. 
 Skew-bevel wheels, 82, 87, 146, 152. 
 Slide-valves, 305, 314. 
 
 Motion of, 260. 306. 
 Link motions for, 253. 
 
 Slide-valves Movable-seated, 260. 
 Sliding contact Connection by, 114. 
 
 ,, Efficiency of, 437. 
 
 Sliding piece Efficiency of, 426, 449. 
 Slot and pin Connection by, 167. 
 Slotted link, 213. 
 Slotting, 569. 
 
 ,, machine, 571. 
 Soft metal, 464. 
 Solder Hard, or spelter, 463. 
 
 Soft, 464. 
 Specific gravity, 325. 
 
 table of, 326. 
 
 Speed Adjustments of, 80. 
 ,, ,, by bands and pul- 
 
 leys, 185, 312. 
 
 by friction- 
 
 wheels, 310. 
 by toothed 
 
 wheels, 311. 
 
 \ by valves, 313. 
 
 Speed of cutting tools, 567. 
 ,, of rotation, 24. 
 ,, Periodic fluctuations of, 35, 247, 
 
 375, 407. 
 
 ,, Uniform Condition of, 369. 
 Speed cones, 185. 
 Spirals, 53, 99. 
 Springs Deflection of straight steel, 
 
 386. 
 
 Elasticity of spiral, 389. 
 Sprocket-wheel, 191. 
 Spur wheels, 82, 120. 
 Starting a machine, 376. 
 Staves or pins for wheels, trundles, and 
 
 racks, 137. 
 
 Steadiness Co-efficient of, 362. 
 Steel, 450, 457, 578. 
 Annealing of, 457. 
 Kinds of, 458. 
 Resilience of, 485. 
 springs Elasticity of, 386, 389. 
 Strength of, 459, 477, 484, 578. 
 Tempering, 457. 
 Stephenson's link motion, 257. 
 Stepped teeth, 155. 
 Stiffness Moduli of, 491, 492. 
 
 ,, and pliability in machines, 531. 
 of beams and shafts, 517, 547. 
 Stone bearings for shafts, 464. 
 Stopping a machine, 376, 400. 
 Strain, 487. 
 Straining actions in machines, 527. 
 
 ,, ,, Alternate, 529. 
 
 effects of re-action, 529. 
 Strength and stiffness Principles and 
 
 rules, 487-558. 
 Co-efficients or moduli of, 488 
 
 492. 
 of fastenings, 497.
 
 INDEX. 
 
 597 
 
 Strength of materials Tables of, 477-486, 
 
 576, 584, 585. 
 Testing of, 490. 
 
 Ultimate, proof, and working, 
 
 487. 
 Stress, 317, 487. 
 
 Moment of, 510. 
 Stroke Adjustment of, 310, 312. 
 
 Length of in linkwork, 197. 
 Struts Strength of long, 524. 
 short, 522. 
 
 Sun-and-planet motion, 246. 
 Surfaces Making of ruled, 569. 
 Scraping of plane, 571. 
 
 TABLES of alloys of copper, tin, and zinc, 
 
 463. 
 British and French weights and 
 
 measures, 586. 
 classification of woods, 466. 
 
 data for calculating fly-wheels, 
 
 408. 
 
 elementary combinations in me- 
 chanism, in classes, 229-231. 
 ,, expansion by heat, 326. 
 ,, factors for deflection, 519, 520. 
 for dimensions of axles, 
 
 543. 
 
 for gudgeons, 542, 543. 
 
 for shafts, 545. 
 
 for strength of struts, 
 
 524, 538. 
 for transverse strength, 
 
 515, 516. 
 
 factors of safety, 489, 545. 
 friction, 349. 
 ,, heaviness, density, specific 
 
 gravity, 326-328. 
 
 heights due to velocities, 331. 
 ,, measures, British and French, 
 
 586. 
 
 of intensity of pres- 
 
 sure, 342. 
 of resistance and 
 
 work, 339. 
 of statical moment, 
 
 321. 
 of velocity, 340. 
 
 squares of radii of gyration, 
 
 360, 525. 
 strength of belts, 474. 
 
 of iron and steel, 460, 
 
 477, 479, 481-486. 
 
 ofmaterialsgenerally, 
 
 477-486. 
 
 of ropes, 475, 476. 
 
 of wire ropes, 534. 
 
 uses of wood in machinery, 
 
 472. 
 weights, British and Frencb.,586. 
 
 Tackle, 214. 
 
 Efficiency of, 443. 
 Tangent screw, 165. 
 Teeth Arc of contact of, 119. 
 Common and relative velocity oS, 
 
 117. 
 
 Dimensions of, 116. 
 Efficiency of, 438. 
 Epicycloidal, 130. 
 Approximate, 134. 
 
 Figures of, 115. 
 for a given path of contact, 128. 
 for inside gearing, 117. 
 for intermittent gearing, 139, 286. 
 gearing with round staves, 137. 
 -Helical, 156. 
 
 Involute for circular wheels, 1 20. 
 for elliptic wheels, 292. 
 
 for racks, 125. 
 
 normal pitch of, 122. 
 
 peculiar properties of, 
 
 125. 
 
 Machine for cutting, 575. 
 Obliquity of action of, 119. 
 of mitre or bevel wheels, 143. 
 of non-circular wheels, 141. 
 of skew-bevel wheels, 146. 
 of spur-wheels and racks, 120. 
 Parts of, 115. 
 Pitch and number of, 103. 
 Pitching, or laying off pitch of, 
 
 113, 575. 
 
 ,, Stepped, 155. 
 Strength of, 553. 
 traced by rolling curves, 129. 
 with sloping backs, 152. 
 Wooden, or cog, 473. 
 Telodynamic transmission, 447. 
 Tempering of steel, 457. 
 Tension Resistance to, 493. 
 Tension rods Strength of, 535. 
 Testing of strength, 490. 
 Thread of screw, 36. 
 Thrust Bending action of, 524. 
 ,, Resistance to, 522. 
 )? Strength of rods under, 537. 
 Tie Strength, stiffness, and resilience of, 
 
 493. 
 
 Tiller, 219. 
 Tiller-ropes, 219. 
 Timber, 464-482, 580. 
 Tin, 461. 
 Tools, 559. 
 M Combinations of, 568. 
 
 Counter-efficiency of machine, 
 
 568. 
 tt Cutting angles of, 566. 
 
 Cutting in general, 559. 
 it Motions of machine, 568. 
 , Paring, 562.
 
 598 
 
 INDEX. 
 
 Tools Punching, 561. 
 
 ,, Resistance and work of, 567. 
 
 ,, Scraping, 562. 
 
 Shearing, 560. 
 
 ,, Speed of cutting, 567. 
 Torsion, 500. 
 
 Angle of, 388, 502. 
 ,, dynamometer, 387. 
 ,, Resistance to, 500. 
 Traces of lines and surfaces, 5. 
 Traction dynamometer, 383. 
 Train-arm, 237. 
 Trains Epicyclic, 243, 246. 
 
 ,, of mechanism, 80, 227. 
 
 ,, of wheel work, 108. 
 
 Shifting, 235. 
 Trammel, 267. 
 Translation of a secondary piece, 44. 
 
 straight, 18. 
 Transverse strength, 504. 
 Traversing Rate of, 569. 
 Traversing-gear, 293. 
 Triangle of forces, 319. 
 Trochoid, 53. 
 Trundle, 137. 
 Trunk for piston, 223. 
 Turning Aggregate combinations of 
 
 mechanism used in, 243, 266, 290, 
 
 291. 
 
 Turning lathes Action of, 572. 
 Twisting, 500. 
 
 ULTIMATE strength, 487. 
 Unguents, 350. 
 
 Testing friction with, 395. 
 Universal joint, 203. 
 Unit of force, 317. 
 Unit of mass, 318. 
 
 VALVES Action of, 301. 
 
 Principal lands of, 302. 
 Slide, 305. 
 
 Use of, 224. 
 Velocities Component, 33. 
 
 Virtual, 378. 
 Velocity, 339. 
 
 Aggregate, 239. 
 
 Angular, 24, 341. 
 
 Mean and extreme compara- 
 tive, 199. 
 
 ,, Measures of, 340. 
 
 ratio, 22, 199. 
 Virtual velocities, 378. 
 
 WATER Friction of in pipes, 404. 
 
 wheels Tension arms of, 556. 
 Watt's parallel motion, 275. 
 Wedge, 232, 381. 
 
 Strength of, 499. 
 Weight, 317, 325. 
 
 Weights and measures Comparative table 
 
 of British and French, 586. 
 Welding, 456. 
 Wheel and axle, 232, 380. 
 and rack, 84, 85. 
 cutting, 575. 
 Wheels Bevel, 82, 86. 
 Braced, 557. 
 Circular, 85. 
 Elliptic, 95. 
 J} Fly, 361,407, 556, 585. 
 Lobed, 97. 
 
 Non-circular, 92. 
 Pitch-surfaces, pitch-lines, pitch- 
 points of, 82. 
 
 Skew-bevel, 82, 87, 152. 
 Sprocket, 191. 
 Spur, 82. 
 
 Strength of arms of, 554, 556. 
 x of rims of, 553, 556. 
 
 of teeth of, 553. 
 Teeth of, 115. 
 Toothless, 81. 
 with parallel axes, 83. 
 Worm, 163. 
 White's pulley, 216. 
 Windlass, 190. 
 
 Differential, 242. 
 Wipers, 170, 175. 
 
 Wire-ropes Deflection and length of, 534. 
 Strength of, 533, 584, 585. 
 Transmission of power bv, 
 
 447, 582. 
 Wood, 464. 
 
 Appearance of good, 467. 
 
 Classification of, 466. 
 
 Examples of, 468. 
 
 Preservation of, 471, 580. 
 
 ,, Seasoning of, 470. 
 
 Strength of, 471, 478, 479, 480, 
 
 481, 482. 
 
 Structure of, 464. 
 Use of in machinery, 472. 
 Wooden cogs, 473. 
 Work, 338. 
 
 ,, against an oblique force, 343. 
 ,, against varying resistance, 346. 
 ,, Algebraical expressions for, 342. 
 and energy General equation of, 
 
 378. 
 done, and energy exerted Equality 
 
 of, 370, 375. 
 
 done during retardation, 373. 
 in terms of angular motion, 340. 
 in terms of pressure and volume, 
 
 341. 
 
 Measures of, 339. 
 of acceleration, 354. 
 ,, of machines, 338. 
 Rate of, 339.
 
 INDEX. 
 
 599 
 
 Work represented by an area, 345. 
 Summary of various kinds of, 
 
 362. 
 
 Summation of, 343. 
 Useful and lost, 347. 
 Working load, 487. 
 point, 23. 
 
 Worm-wheel, 163. 
 Wrenching, 500. 
 Wrought iron, 455. 
 
 Z-CBANK, 272. 
 Zinc, 461. 
 
 BBU. AND BAIN, PRINTERS, MITCHELL STREET, GLASGOW.
 
 A 
 
 CATALOGUE 
 
 OF 
 
 STANDARD WORKS 
 
 PUBLISHED BY 
 
 CHARLES GRIFFIN & COMPANY. 
 
 I. Religious Works, I 
 
 II. Works on Medicine and the Allied Sciences, 5 
 III. General Scientific and Technical Works, . 14 
 
 IV. Educational Works, 25 
 
 V. Works in General Literature, . . .31 
 
 LONDON: 
 12 EXETER STREET, STRAND.
 
 INDEX. 
 
 AITKEN(Sir W., M.D.), Science and Practice 
 
 of Medicine, 5 
 
 Outlines, 5 
 
 ANDERSON (M'Call) on Skin Diseases, . 6 
 BELL (R.), Golden Leaves, . . . .31 
 BERINGER.J. J. & C., Assaying, . . 14 
 BLVTH (A. W.), Hygiene and Public Health, 7 
 
 Foods and Poisons, 7 
 
 BROUGH (B. H.), Mine Surveying, . . 14 
 
 BROWNE (W. R.), Student's Mechanics, . 14 
 
 Foundations of Mechanics, . . .14 
 
 Fuel and Water, 14 
 
 BRYCE (A. H.), Works of Virgil, ... 25 
 BUNYAN'S Pilgrim's Progress (Mason), . i 
 CAIRO AND CATHCABT, Surgical Ward-Work, 5 
 CHEEVER'S (Dr.), Religious Anecdotes, . i 
 COBBETT (Wm.), Advice to Young Men, . 31 
 
 -^ Cottage Economy, 31 
 
 ^ English Grammar 25 
 
 ^ French do 25 
 
 Legacy to Labourers, .... 32 
 
 Do. Parsons, ..... 32 
 COBBIN'S Mangnall's Questions, . .25 
 
 COLERIDGE on Method 26 
 
 CRAIK (G.), History of English Literature, . 26 
 
 Manual of do., . . . 26 
 
 CRIMP (W. S.), Sewage Disposal Works, . 14 
 CRUTTWELL'S History of Roman Literature, 27 
 
 Specimens of 
 
 Early Christian Literature, 
 
 CRUDEN'S CONCORDANCE, by Eadie, 
 CURRIE (J.), Works of Horace, . 
 
 Caesar's Commentaries, 
 
 DALGAIRNS' Cookery 
 
 DAVIS (J. R. A.), An Introduction to Biology, 
 The Flowering Plant, .... 
 DICK (Dr.), Celestial Scenery, 
 Christian Philosopher, .... 
 DOERING and GRAEME'S Hellas, . 
 D'ORSEY (A. J.), Spelling by Dictation, 
 DUCKWORTH (Sir D., M.D.), Gout, . 
 DUPRE & HAKE, Manual of Chemistry, . 
 EADIE (Rev. Dr.), Biblical Cyclopaedia, 
 Cruden's Concordance, .... 
 
 Classified Bible, 
 
 Ecclesiastical Cyclopaedia, 
 
 Dictionary of Bible, .... 3 
 ELBORNE (W.), Pharmacy and Materia Medica, 8 
 EMERALD SERIES OF STANDARD 
 
 AUTHORS, 33 
 
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 THE MANAGEMENT OF LABOUR AND OF THE LYING-IN 
 PERIOD. In 8vo, with Illustrations. Cloth, 7/6. 
 
 " Fully accomplishes the object kept in view by its author. . . . Will be found 
 of GREAT VALUE by the young practitioner." Glasgow Medical Journal. 
 
 LEWIS (W. Bevan, L.R.C.P., M.R.C.S., Medical 
 
 Director of the West- Riding Asylum, Wakefield) : 
 
 MENTAL DISEASES (A Text-book of) : With Special Reference to 
 the Pathological Aspects of Insanity. For the Use of Students and Practi- 
 tioners. With 1 8 Li tho- Plates, Charts, and Illustrations. Royal 8vo, 28/. 
 "A splendid addition to the literature of mental diseases. . . . Every page 
 bristles with important facts. As a standard work on the pathology of mental diseases 
 the volume should occupy a prominent place." Dublin Medical Journal.
 
 ro CHARLES GRIFFIN <L- CO.'S PUBLICATIONS. 
 
 By Professors LANDOIS and STIRLING. 
 
 HUMAN PHYSIOLOGY 
 
 (A TEXT-BOOK OF): 
 
 Including Histology and Microscopical Anatomy. 
 WITH SPECIAL REFERENCE TO PRACTICAL MEDICINE, 
 
 By DR. L. LANDOIS, 
 
 PROF. OF PHYSIOLOGY, UNIVERSITY OF GREIFSWALD. 
 
 Translated from the Sixth German Edition, with Annotations and Additions, 
 By WM. STIRLING, M.D., Sc.D., 
 
 BRACKKNBURY PROFESSOR OP PHYSIOLOGY IN OWENS COLLEGE, AND VICTORIA UNIVERSITY, 
 MANCHESTER ; EXAMINER IN THE UNIVERSITIES OF OXFORD AND CAMBRIDGE. 
 
 Royal 8vo, Handsome Cloth. 34s. 
 
 Bit|j facrg Dimerous Illustrations. 
 THIRD ENGLISH EDITION. 
 
 GENERAL CONTENTS. 
 
 PART I. Physiology of the Blood, Circulation, Respiration, Digestion, Absorption, 
 Animal Heat, Metabolic Phenomena of the Body. 
 
 PART II. Secretion of Urine; Structure of the Skin; Physiology of the Motor 
 Apparatus ; the Voice and Speech ; General Physiology of the Nerves ; Electro-Physiology ; 
 the Brain ; Organs of Vision, Hearing, Smell, Taste, Touch ; Physiology of Development. 
 
 * # * Since its first appearance in 1880, Prof. LANDOIS' TEXT- 
 BOOK OF PHYSIOLOGY has been translated into three Foreign 
 languages, and passed through five large editions. 
 
 To meet the wishes of Students, the THIRD ENGLISH EDITION 
 has been issued in ONE VOLUME, printed on specially prepared 
 paper. Numerous Additions have been made throughout, bringing 
 the work abreast in all respects of the latest researches in Physiology 
 and their bearing on Practical Medicine ; and the number of 
 Illustrations has also been largely increased from 494 in the 
 First to 692 in the present Edition. 
 
 "So great are the advantages offered by Prof. LANDO'S* TEXT-BOOK, from the 
 EXHAUSTIVE and EMINENTLY PRACTICAL manner in which the subject is treated, that 
 it has passed through FOUR large editions in the same number of years. . . . Dr. 
 STIRLING'S annotations have materially added to the value of the work. Admirably adapted 
 for the PRACTITIONER. . . . With this Text-book at command, NO STUDENT COULD FAIL 
 IN HIS EXAMINATION." The Lancet. 
 
 "One of the MOST PRACTICAL WORKS on Physiology ever written, forming a 'bridge* 
 between Physiology and Practical Medicine. ... Its chief merits are its completeness 
 and conciseness. . . . The additions by the Editor are able and judicious. . . . 
 EXCELLENTLY CLEAR, ATTRACTIVE, AND SUCCINCT." Brit. Med. Journal. 
 
 "The great subjects dealt with are treated in an admirably clear, terse, and happily- 
 illustrated manner. At every turn the doctrines laid down are illuminated by reference to 
 facts of Clinical Medicine or Pathology." Piactitioner. 
 
 " We have no hesitation in saying that THIS is THE WORK to which the PRACTITIONER 
 will turn whenever he desires light thrown upon, or information as to how he can best 
 investigate, the phenomena of a COMPLICATED OR IMPORTANT CASE. To the STUDENT it 
 will be EQUALLY VALUABLE." Edinburgh Medical Journal. 
 
 "LANDOIS and STIRLING'S work cannot fail to establish itself as one of the most useful 
 and popular works known to English readers." Manchester Medical Chronicle. 
 
 As a work of reference, LANDOIS and STIRLING'S Treatise OUGHT TO TAKE, THB 
 FORKMOST PLACE among the text-books in the English language. The woodcuts are 
 noticeable for their number and beauty." Glasgow Medical Journal. 
 
 " Unquestionably the most admirable exposition of the relations of Human Physiology 
 to Practical Medicine that has ever been laid before English readers." Students' Journal.
 
 MEDICINE AND THE ALLIED SCIENCES. 
 
 By Drs. MEYER and FERGUS. 
 
 Now Beady, with Three Coloured Plates and numerous Illustrations. 
 Royal 8vo, Handsome Cloth, 25s. 
 
 DISEASES OF THE EYE 
 
 (A PRACTICAL TREATISE ON), 
 
 BY EDOUARD MEYER, 
 
 Prof, a VEcole Pratique, de la Faculty de Medecine de Paris, 
 Chev. of the Leg. of Honour, &c. 
 
 Translated from the Third French Edition, with Additions as 
 
 contained in the Fourth German Edition, 
 By F. FERGUS, M.B., Ophthalmic Surgeon, Glasgow Infirmary. 
 
 The particular features that will most commend Dr. Meyer's work 
 to English readers are its CONCISENESS, its HELPFULNESS in explana- 
 tion, and the PRACTICALITY of its directions. The best proof of its 
 worth may, perhaps, be seen in the fact that it has now gone through 
 three French and four German editions, and has been translated into 
 most European languages Italian, Spanish, Russian, and Polish and 
 even into Japanese. 
 
 Opinions of the Press. 
 
 " A GOOD TRANSLATION OP A GOOD BOOK. . . . A SOUND GUIDE in the diagnosis and treatment of 
 the various diseases of the eye that are likely to fall under the notice of the general Practitioner. 
 The Paper, Type, and Chromo-Lithographs are all that could be desired. . . . We know of no work 
 in which the DISEASES and DEFORMITIES of the LIDS are more fully treated. Numerous figures illus- 
 trate almost every defect remediable by operation." Practitioner. 
 
 " A VBRYTRUSTWOKTHT GUIDE in all respects. . . . THOROUGHLY PRACTICAL. Excellently trans- 
 lated, and very well got up. Type, Woodcuts, and Chromo-Lithographs are alike excellent." 
 Lancet. 
 
 " Any Student will find this work of GREAT VAI.UK. . . . The chapter on Cataract is excellent. 
 . . . The Illustrations describing the various plastic operations are specially helpful.' 1 Brit. 
 Mtd. Journal. 
 
 "An EXCELLENT TRANSLATION of a standard French Text-Book. . . . We can cordially recom- 
 mend Dr. Meyer's work. It is essentially a PRACTICAL ;WORK. The Publishers have done their part 
 in the tasteful and substantial manner characteristic of their medical publications." Opti:/iatmic 
 Rtviwi. 
 
 A 3
 
 12 CHARLES ORIFFIN & CO:S PUBLICA TIONS. 
 
 LINN (S.H., M.D., D.D.S., Dentist to the Imperial 
 
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 THE TEETH : How to preserve them and prevent their Decay. A 
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 Plates and Diagrams. Crown 8vo. Cloth, 2/6. 
 
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 THE SANITARY CONTRASTS OF THE CRIMEAN WAR. 
 Demy 8vo. Cloth limp, 1/6. 
 
 "A most valuable contribution to Military Medicine." British Medical JournaL 
 "A most concise and interesting Review." Lancet. 
 
 MACALISTER (A., M.D., F.R.S., Professor of 
 
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 HUMAN ANATOMY : Systematic and Topographical (A Text-book 
 of), including the Embryology, Histology, and Morphology of Man, with 
 Special Reference to the Requirements of Practical Surgery and Medicine. 
 With 816 Illustrations. Medium 8vo. 365. 
 
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 THE CENTRAL NERVOUS ORGANS. A Guide to the Study 
 of their Structure in Health and Disease. Translated, with annotations 
 and additions, by ALEX. HILL, M.A., M.D., Master of Downing College, 
 Cambridge. With all the Original Illustrations. Medium 8vo, 253. 
 
 PARKER (Prof. W. Kitchen, F.R.S., Hunterian 
 
 Professor, Royal College of Surgeons): 
 
 MAMMALIAN DESCENT: being the Hunterian Lectures for 1884. 
 Adapted for General Readers. With Illustrations. In 8vo, cloth, 10/6. 
 
 "A very striking book ... as readable as a book of travels. Prof. PARKER 
 is no Materialist." Leicester Post. 
 
 PORTER (Surgeon-Major J. H., Late Assistant 
 
 Professor of Military Surgery in the Army Medical School): 
 
 THE SURGEON'S POCKET-BOOK : an Essay on the Best Treat- 
 ment of the Wounded in War ; for which a Prize was awarded by Her 
 Majesty the Empress of Germany. Specially adapted to the PUBLIC 
 MEDICAL SERVICES. With 152 Illustrations and folding-plate, fcp. 
 8vo, roan, 7/6. Third Edition, Revised and Enlarged. By Brigade- 
 Surgeon C. H. Y. GODWIN, of the Army Medical School. 
 
 " Every Medical Officer is recommended to have the ' Surgeon's Pocket-Book ' by 
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 Precis of Field-Service Medical Arrangements for Afghan War. 
 
 "A complete vade mecum to guide the military surgeon in the field." British 
 Medical Journal. 
 
 " A capital little book of the greatest practical value. ... A surgeon with this 
 Manual in his pocket becomes a man of resource at once." Westminster Review.
 
 MEDICINE AND THE ALLIED SCIENCES. 13 
 
 SEXTON (A. Humboldt, F.C.S., Professor of 
 
 Metallurgy, Glasgow and West of Scotland Technical College) : 
 
 1. QUANTITATIVE ANALYSIS (Outlines of). For the Use of 
 Students. With Numerous Illustrations. Third Edition, Crown 8vo. 
 Cloth, 3 /. 
 
 " A good and useful book. . . . Really supplies a want." The Lancet. 
 
 " This is an admirable little volume, and well fulfils its purpose." Schoolmaster. 
 
 2. QUALITATIVE ANALYSIS (Outlines of). For the use of 
 Students. With numerous Illustrations. Second Edition. Crown 8vo. 
 Cloth, 3/6. 
 
 " The Work of a thorough practical Chemist . . . and one which may be 
 unhesitatingly recommended." Brit. Medical Journal. 
 
 STIRLING (William, M.D., D.Sc., Owens College, 
 
 Manchester) : 
 
 A TEXT-BOOK OF HUMAN PHYSIOLOGY (See tinder Landois 
 and Stirling, page 10). 71iird Edition. 
 
 OUTLINES OF PRACTICAL PHYSIOLOGY : including Experi- 
 mental and Chemical Physiology. With special reference to Clinical 
 Medicine. A Laboratory Handbook for the use of Students. Second 
 Revistd Edition. With 234 Illustrations. In large Crown 8vo, 9/. 
 
 " An excellent treatise, which we can thoroughly recommend." Lancet. 
 " May be confidently recommended as a guide to the Student of Physiology." 
 Glasgow Medical Journal. 
 
 OUTLINES OF PRACTICAL HISTOLOGY. With 344 Illustra- 
 tions. Large Crown 8vo. Cloth, 12/6. 
 
 THORBURN (John, M.D., F.R.C.P., Late 
 
 Professor of Obstetric Medicine, Owens College and Victoria University, 
 Manchester ; Obstetric Physician to the Manchester Royal Infirmary) : 
 
 THE DISEASES OF WOMEN (A Practical Treatise on). Prepared 
 with Special Reference to the Wants of the General Practitioner and 
 Advanced Student. With Chromo-lithograph, and over 200 Illustrations. 
 Royal 8vo, handsome cloth, 2i/. 
 
 The ENTIRE WORK is IMPARTIAL and INSTRUCTIVE, and in every way worthy of its 
 author." British Medical Joni nal. 
 
 THORBURN (Wm., B.S., B.Sc., M.D., F.R.C.S., 
 
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 THE SURGERY OF THE SPINAL CORD (A Contribution to). 
 With Diagrams, Tables, and Illustrations. Medium 8vo. Handsome 
 cloth, I2s. 6d. 
 
 THORNTON (J. Knowsley, M.B., M.C., Surgeon 
 
 to the Samaritan Free Hospital for Women, &c., &c.) : 
 
 THE SURGERY OF THE KIDNEYS. Being the Harveian 
 Lectures for 1889. With Illustrations and Tables. Demy8vo. Cloth, $s.
 
 14 CHARLES GRIFFIN & CO:S PUBLIC A TIONS. 
 
 GENERAL SCIENTIFIC AND TECHNICAL WORKS, 
 
 BERINGER (C, F.I.C., F.C.S., late Chief Assayer 
 
 to the Rio Tinto Company); and BERIXGER (J. J., F.I.C , F.C.S., 
 Public Analyst for, and Lecturer to the Mining Association of, Cornwall) : 
 
 ASSAYING (A Text-Book of): for the Use of Students, Mine 
 Managers, Assayers, &c. With numerous Tables and Illustrations. 
 Crown 8vo. Cloth, 10/6. 
 
 "A REALLY MERITORIOUS WORK, that may be safely depended upon either for systematic 
 instruction or for reference." Nature. 
 
 BROWNE (Walter R., M.A., M. Inst. C.E., 
 
 F.G.S., late Fellow of Trinity College, Cambridge) : 
 
 THE STUDENT'S MECHANICS : An Introduction to the Study 
 of Force and Motion. With Diagrams. Crown 8vo. Cloth, 4/6. 
 
 " Clear in style and practical in method, 'THE STUDENT'S MECHANICS,' is cordially 
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 "The merits of the work are especially conspicuous in its clearness and brevity . . . 
 deserves the attention of all who have to teach or learn the elements of Mechanics. 
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 FOUNDATIONS OF MECHANICS. 
 
 Papers reprinted from the Engineer. In crown 8vo, I/. 
 
 FUEL AND WATER: A Manual for 
 
 Users of Steam and Water. By Prof. SCHWACKHOFER and W. R. 
 BROWNE, M.A. (See p. 23.) 
 
 BROUGH (B.H., F.G.S., Instructor of Mine 
 
 Surveying, Royal School of Mines) : 
 
 MINE SURVEYING (A Text-Book of) : for the Use of Managers 
 of Mines and Collieries, Students at the Royal School of Mines, City 
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 " Supplies a long-felt want." Iron. 
 
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 SEWAGE DISPOSAL WORKS. A Guide to the Construction of 
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 33 Lithographic Plates, Tables, and Illustrations in the Text. Medium 
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 SCIENTIFIC AND TECHNICAL WORKS, 15 
 
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 BRIDGE-CONSTRUCTION 
 
 (A PRACTICAL TREATISE ON): 
 
 Being a Text-Book on the Design and Construction of 
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 FOR THE USE OF STUDENTS, DRAUGHTSMEN, AND ENGINEERS. 
 
 BY 
 
 T. CLAXTON FIDLER, M. INST. C.E. 
 TKHftb IFiumerotts Woodcuts anb 17 Xitbograpbtc plates. 
 
 THE object of this book is to describe the modern practice of 
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 " As an exposition of the LATEST advances of the Science, we are glad to welcome this 
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 Review. 
 
 " Mr. Fidler's book is one which every Student of Mechanics ought to possess, and 
 which merits, as it will receive, the appreciative attention of all practical men." Scotsman. 
 
 A full Prospectus of the above important work may be 
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 16 CHARLES GRIFFIN & CO.'S PUBLICATIONS. 
 
 GURDEN (Richard Lloyd, Authorised Surveyor 
 
 for the Governments of New South Wales and Victoria) : 
 
 TRAVERSE TABLES : computed to Four Places Decimals for every 
 Minute of Angle up to 100 of Distance. For the use of Surveyors and 
 Engineers. Second Edition. Folio, strongly half-bound, 2 1/. 
 
 * < * Published with Concurrence of the Surveyors- General for New South 
 Wales and Victoria. 
 
 "Those who have experience in exact SURVEY-WORK will best know how to appreciate 
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 enable the user to ascertain the sines and cosines for a distance of twelve miles to within 
 half an inch, and this BY REFERENCE TO BUT ONE TABLE, in place of the usual Fifteen 
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 ensure to every user, and as every Surveyor in active practice has felt the want of such 
 assistance, few knowing of their publication will remain without them." Engineer. 
 
 " We cannot sufficiently admire the heroic patience of ttie author, who, in order to 
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 the proofs." Engineering. 
 
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 FROM SOURCE TO SEA : or, Gleanings about Rivers from many 
 Fields. A Chapter in Physical Geography. Cloth elegant, 3/6. 
 
 " Excellent reading . . a book of popular science which deserves an extensive 
 circulation." Saturday Review. 
 
 JAMIESON (Andrew, C.E., F.R.S.E., Professor 
 
 of Engineering, Glasgow and West of Scotland Technical College) : 
 
 STEAM AND THE STEAM ENGINE (A Text-Book on) : Specially 
 arranged for the use of Science and Art, City and Guilds' of London 
 Institute, and other Engineering Students. With 200 Illustrations and 
 Four Folding- Plates. Fifth Edition. Crown 8vo. Cloth, 7/6. 
 
 " The BEST BOOK yet published for the use of Students." Engineer. 
 
 " This is undoubtedly the MOST VALUABLE and MOST COMPLETE hand-book of reference 
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 1. STEAM AND THE STEAM ENGINE (An Elementary Manual 
 on), forming an introduction to the larger Work by the same Author. 
 With numerous Illustrations and Examination Questions at the end of 
 each Lecture. Second Edition. Crown 8vo. Cloth, 3/6. 
 
 2. MAGNETISM AND ELECTRICITY (An Elementary Manual on). 
 With numerous Illustrations and Examination Questions. Crown 8vo. 
 Part I. Magnetism, is. Part II. Voltaic Electricity, is. 6d. Part III. 
 Frictional Electricity, is. 6d. Or complete in Cloth, 33. 6d. 
 
 3. APPLIED MECHANICS (An Elementary Manual on). With 
 Diagrams and Examination Questions. Crown 8vo.
 
 SCIENTIFIC AND TECHNICAL WORKS. 17 
 
 M'MILLAN (W. G., F.I.C., F.C.S.), Chemist and 
 
 Metallurgist to the Cossipore Foundry and Shell- Factory, Calcutta. 
 
 ELECTRO-METALLURGY (A Treatise on): Embracing the Appli- 
 cation of Electrolysis to the Plating, Depositing, Smelting, and Refining 
 of various Metals, and to the Reproduction of Printing Surfaces and Art- 
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 MUNRO (John, C. E.) and JAMIESON 
 
 (Andrew, C.E V F.R.S.E.): 
 
 A POCKET-BOOK OF ELECTRICAL RULES AND TABLES, 
 for the use of Electricians and Engineers. Pocket Size. Leather, 8/6. 
 Sixth Edition, revised and enlarged. With numerous Diagrams. 
 
 ** The SIXTH EDITION has been thoroughly Revised and Enlarged 
 by about 120 pages and 60 new Figures. 
 
 /'WONDERFULLY PERFECT. . . . Worthy of the highest commendation we can 
 give it." Electrician. 
 
 "The STERLING VALUE of Messrs. MUNRO and JAMIESON'S POCKET-BOOK." 
 Electrical Revieiv. 
 
 MUNRO (R. D.), STEAM BOILERS: Their 
 
 Defects, Management, and Construction. A Manual for all concerned 
 in the care of Steam Boilers, but written with a special view to the 
 wants of Boiler-Attendants, Mill-Mechanics, and other Artisans. With 
 Numerous Illustrations. Crown 8vo, Cloth 3/6. 
 
 " The volume is a valuable companion for workmen and engineers engaged about 
 Steam Boilers, and ought to be carefully studied, and always at hand. ' Colliery 
 Guardian. 
 
 "The subjects referred to are handled in a trustworthy, clear, and practical manner. 
 . . . The book is VEKY USEFUL, especially to steam users, artisans, and young 
 engineers." Engineer. 
 
 PHILLIPS (J. Arthur, F.R.S.,M. Inst. C.E.,F.C.S., 
 
 F.G.S., Ancien Eleve de 1'Ecole des Mines, Paris): 
 
 ELEMENTS OF METALLURGY : a Practical Treatise on the Art 
 of Extracting Metals from their Ores. With over zoo Illustrations, many 
 of which have been reduced from Working Drawings, and two Folding- 
 Plates. Royal 8vo, 848 pages, cloth, 367. NEW EDITION by the 
 Author and Mr. ft. JSauerman, F.G.S. 
 
 General Contents. 
 
 I. A TREATISE on FUELS and REFRACTORY MATERIALS. 
 II. A Description of the principal MINERALS, with their DISTRIBUTION. 
 III. STATISTICS of the amount of each METAL annually produced throughout the 
 
 World. 
 IV. The METHODS of ASSAYING the different ORES, together with the PROCESSES 
 
 of METALLURGICAL TREATMENT. 
 
 " ' Elements of Metallurgy ' possesses intrinsic merits of the highest degree. Such a 
 work is precisely wanted by the great majority of students and practical workers, and its 
 very compactness is in itself a first-rate recommendation. ._ . . In our opinion, the 
 BEST WORK EVER WRITTEN ON THE SUBJECT with a view to its practical treatment.'* 
 Westminster Revievv. 
 
 " The VALUE OF THIS WORK is ALMOST INESTIMABLE. There can be no question that 
 the amount of time and labour bestowed upon it is enormous. . . . There is certainly 
 no Metallurgical Treatise in the language calculated to prove of such general utility. * 
 Miningjoumal. 
 
 MANY NOTABLE ADDITIONS 
 
 WILL BE FOUND IN THE SECTIONS DEVOTED TO 
 
 IRON, LEAD, COPPER, SILVER, AND GOLD, 
 
 Dealing luitft New Processes and Development.
 
 iS CHARLES GRIFFIN & CO.'S PUBLICATIONS. 
 
 Demy 8vo, Handsome cloth, 18s. 
 
 PHYSICAL GEOLOGY AND 
 PALEONTOLOGY, 
 
 ON THE BASIS OF PHILLIPS. 
 
 BY 
 
 HARRY GOVIER SEELEY, F. R. S., 
 
 PROFESSOR OF GEOGRAPHY IN KING'S COLLEGE, LONDON. 
 
 TKHttb frontispiece in Cbromo^Lftbograpbg, an& Jllustratfons, 
 
 " It is impossible to praise too highly the research which PROFESSOR SEELEY'S 
 ' PHYSICAL GEOLOGY ' evidences. IT is FAR MORE THAN A TEXT-BOOK it is 
 a DIRECTORY to the Student in prosecuting his researches." Extract Jrom the 
 Presidential Address 10 the Geological Society, 1885, by Rev. Professor Bonney, 
 >.Sf. t LL.D., F.R.S. 
 
 " PROFESSOR SEELEY maintains in his ' PHYSICAL GEOLOGY ' the high 
 reputation he already deservedly bears as a Teacher. . . . It is difficult, 
 in the space at our command, to do fitting justice to so large a work. . . . 
 The final chapters, which are replete with interest, deal with the Biological 
 aspect of Palaeontology. Here we find discussed the origin, the extinction, 
 succession, migration, persistence, distribution, relation, and variation of species 
 with other considerations, such as the Identification of Strata by Fossils, 
 Homotaxis, Local Faunas, Natural History Provinces, and the relation of 
 Living to Extinct forms." Dr. Henry Wood-ward, F.R.S. , in the " Geological 
 Magazine" 
 
 ' ' A deeply interesting volume, dealing with Physical Geology as a whole, 
 and also presenting us with an animated summary of the leading doctrines and 
 facts of Palaeontology, as looked at from a modern standpoint." Scotsman. 
 
 " PROFESSOR SEELEY'S work includes one of the most satisfactory Treatises 
 on Lithology in the English language. ... So much that is not accessible 
 in other works is presented in this volume, that no Student of Geology can 
 afford to be without it." American Journal of Engineering. 
 
 " Geology from the point of view of Evolution." Westminster Review. 
 
 *' PROFESSOR SEELEY'S PHYSICAL GEOLOGY is full of instructive matter, 
 wfcilst the philosophical spirit which it displays will charm many a reader. 
 From early days the author gave evidence of a powerful and eminently original 
 genius. No one has shown more convincingly than the author that, in all 
 ways, the past contains within itself the interpretation of the existing world. " 
 Annals of Natural History.
 
 SCIENTIFIC AND TECHNICAL WORKS. 19 
 
 Demy 8vo, Handsome cloth, 34s. 
 
 STRATIGRAPHICAL GEOLOGY 
 AND PALEONTOLOGY, 
 
 ON 
 
 THE BASIS OF PHILLIPS. 
 
 BY 
 
 ROBERT ETHERIDGE, F. R. S., 
 
 THE NATURAL HIST. DEPARTMENT, BRITISH MUSEUM, LATE PAL/EONTOLOGIST TO 1 
 
 GEOLOGICAL SURVEY OF GREAT BRITAIN, PAST PRESIDENT OF THE 
 
 GEOLOGICAL SOCIETY, ETC. 
 
 TWlftb dfcap, Numerous aable0, an& abfrt*sf flMates. 
 
 "In 1854 Prof. JOHN MORRIS published the Second Edition of his 'Catalogue 
 of British Fossils,' then numbering 1,280 genera and 4.000 species. Since 
 that date 3,000 genera and nearly 12,000 new species have been described, 
 thus bringing up the muster-roll of extinct life in the British Islands alone to 
 3,680 genera and 16,000 known and described species. 
 
 "Numerous TABLES of ORGANIC REMAINS have been prepared and 
 brought down to 1884, embracing the accumulated wealth of the labours of 
 past and present investigators during the last thirty years. Eleven of these 
 Tables contain every known British genus, zoologically or systematically placed, 
 with the number of species in each, showing their broad distribution through 
 time. The remaining 105 Tables are devoted to the analysis, relation, 
 historical value, and distribution of specific life through each group of strata. 
 These tabular deductions, as well as the Palseontological Analyses through the 
 text, are, for the first time, fully prepared for English students." Extract from 
 Author's Preface. 
 
 \* PROSPECTUS of the above important -work perhaps the MOST ELABORATE of 
 its kind ever written, and one calculated to give a new strength to the study 
 of Geology in Biitain may be had on application to the Publishers. 
 
 It is not too much to say that the work will be found to occupy a place 
 entirely its own, and will become an indispensable guide to every British 
 Geologist. 
 
 " No such compendium of geological knowledge has ever been brought together before." 
 Westminster Review. 
 
 "If PROF. SEELEY'S volume was remarkable for its originality and the breadth of its views, 
 Mr. ETHERIDGE fully justifies the assertion made in his preface that his book differs in con- 
 struction and detail from any known manual. . . . Must take HIGH RANK AMONG WORKS 
 
 OF REFERENCE." Atke>Uf1tm,
 
 to CHARLES GRIFFIN Jk CO.'S PUBLICATIONS. 
 
 SCIENTIFIC MANUALS 
 
 BY 
 
 W. /. MACQUORN RANKINE, C.E., LLD., F.R.S,, 
 
 Late Regius Professor of Civil Engineering in the University of Glasgow. 
 
 THOROUGHLY REVISED BY W. J. MILLAR, C.E., 
 
 Secretary to the Institute of Engineers and Shipbuilders in Scotland. 
 In Crown 8vo. Cloth. 
 
 f. RANKINE (Prof.): APPLIED MECHANICS: 
 
 comprising the Principles of Statics and Cinematics, and Theory of Struc- 
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 Edition, 12/6. 
 
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 V. RANKINE (Prof.): USEFUL RULES and 
 
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 ** THE MECHANICAL TEXT- BOOK forms a simple introduction to PROFESSOR RANKINK'S 
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 SCIENTIFIC AND TECHNICAL WORKS. 21 
 
 PROF. RANKINK'S WORKS (Continued). 
 
 VII. RANKINE (Prof.): MISCELLANEOUS 
 
 SCIENTIFIC PAPERS. Royal 8vo. Cloth, 31/6. 
 
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 With Memoir by Professor TAIT, M. A. Edited by W. J. MILLAR, C.E. 
 With fine Portrait on Steel, Plates, and Diagrams. 
 
 _ " No more enduring Memorial of Professor Rankine could be devised than the publica- 
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 By SIR EDWARD REED. 
 
 Royal 8uo, Handsome Cloth, 25s. 
 
 THE STABILITY OF SHIPS. 
 
 BY 
 
 SIR EDWARD J. REED, K.C.B., F.R.S., M.P., 
 
 KNIGHT OF THE IMPERIAL ORDERS OF ST. STANILAUS OF RUSSIA ; FRANCIS JOSEPH OF 
 AUSTRIA; MEDJIDIE OF TURKEY; AND RISING SUN OF JAPAN; VICE- 
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 With numerous Illustrations and Tables. 
 
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 22 
 
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 SCIENTIFIC AND TECHNICAL WORKS. 23 
 
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 CRAIK'S ENGLISH LITERATURE. 
 A COMPENDIOUS HISTORY OF 
 
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 By GEORGE LILLIE CRAIK, LL.D., late Professor of History and 
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 GENERAL CONTENTS. 
 
 INTRODUCTORY. 
 
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 EDUCATIONAL WORKS. 27 
 
 WORKS BY REV. C. T. CRUTTWELL, M.A., 
 
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 EDUCATIONAL WORKS. 29 
 
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