H\ttr*itg fff 
 
 Name of Book and Volume, 
 
 Division 
 
 Range 
 
 Shelf.... 
 
 Received. &.&b&.. 187^ 
 
 University of California. 
 
 G-IFX OF 
 

 
 
 E TMENTS' OF MECHANICS. 
 
THE 
 
 ELEMENTS OF MECHANICS 
 
 JAMES RENWICK, LL.D., 
 
 PROFESSOR OF NATURAL EXPERIMENTAL' PHILOSOPHY 
 AND CHEMISTRY 
 
 C O L.-U M B I A COLLEGE, 
 
 NEW-YORK. 
 
 CAREY & LEA, C H E S N U T-S T R E E T. 
 
 1832. 
 
Entered according to the Act of Congress, in the year one thousand eight hun- 
 dred and thirty-twa, by James Renwick, in the Clerk's Office of the District Court 
 of the United States for the Southern District of New- York. 
 
 G.iMlOPKlNS & SUM, Pnnters. 
 
4 
 
 TO 
 
 CLEMENT C. MOORE, LL. D., 
 
 SENIOR TRUSTEE OF COLUMBIA COLLEGE, 
 THIS WORK IS RESPECTFULLY INSCRIBED, 
 AS A TOKEN OF GRATITUDE FOR MANY 
 AND SIGNAL FAVOURS, 
 
 V 
 
 * ANI) 
 
 OF HIGH RESPECT 
 
 FOR HIS WORTH AND VIRTUES 
 

 
 Jj : 
 
 
PREFACE. 
 
 THE work which is now submitted to the public, comprises a/portion 
 of the materials collected for the courses, it is my duty annually to deliver 
 in Columbia College. It was originally intended that the subject of 
 Practical Mechanics, should -have accompanied the Elements ; thus 
 forming a full treatise on the theoretic and practical parts of that useful 
 and interesting branch of Science. It was however found that in this 
 way the work would have assumed too bulky a form/ The appli- 
 cations of the elementary principles of the present work to the construc- 
 tion of Machines, have therefore been withheld, untiVme sense of the 
 public be declared as to its merits. Should the verdiet be favourable, 
 the author may be encouraged to proceed with the publication, not only 
 of the sequel to these elements, but with that of som of the other sub- 
 jects to which a labour of twelve years have been devyted, namely : Pure 
 Physics, Chemistry applied to the Arts, and Practical Astronomy. 
 
 In the discussion of subjects so extensive and Various, the author is 
 aware, that he has been denied the advantages tharare to be derived from 
 the division of labour, and has been unable to demote to any one object, 
 that steady attention, that can alone ensure entiri success. In spite of 
 these disadvantages, he ventures to submit the/present work to the pub- 
 lic, in the belief that it cannot fail to be useful to the student of Mechani- 
 cal Science. To those who have already nude progress, this volume 
 may present little novelty ; all that it contain/ of most value will be rea- 
 dily traced to obvious, if not familiar sources.) But to the learner, he can- 
 not but hope that it will offer in a condensed, md generally in a simple and 
 almost popular form, facts, principles, and Methods of investigation, that 
 he will find in no single work in any language, and which must be sought 
 for in various treatises, most .of them inaccessible to those who read no 
 ot her language but the English. 
 
V1U PREFACE. ^ , 
 
 The work presents the mixture of strict mechanical principles, with 
 the physical inferences from experiment and observation, that is de- 
 manded by the plan of teaching Natural Philosophy, which is generally 
 adopted in this country, and which is habitual in most of the English 
 Treatises on that general subject. It is therefore a combination of the 
 subject, styled by the French JVLecanique, with so much of the department 
 called by them Physique, as is necessary for its illustration, and for pre- 
 paring the way for > practical applications. Nor have the latter been wholly 
 omitted. They have, in the scope of the treatise, frequently come into 
 view, and have in all such cases been treated of in a concise manner. 
 
 In the use of the term " Mechanics", it has been employed as includ- 
 ing the whole science of Equilibrium and Motion, and therefore as com- 
 prising the departments of Hydrostaticks and Hydrodynamicks. 
 
 One object has been kept steadily in view, namely, that the student 
 shall not be compelled, after having mastered this treatise, to renew his 
 elementary studies, in case he should wish to rise to the higher applica- 
 tions of mechanical science. Should the author not have failed in this, 
 the present work, however humble, in the limitation of its direct applica- 
 tions to the mere works of human art, may serve as an introduction to the 
 science that investigates the mechanism of the universe. 
 
 In order to obviaU the necessity of continually quoting the authorities 
 for the various facts md investigations employed in the work, by which 
 the text would have encumbered, or the margin loaded with notes, a list 
 of the works that have been most frequently made use of is subjoined. In 
 some few cases, where foe investigations are copied without alteration, or 
 where they are not compUte, the passage of the author quoted, is expressly 
 referred to in a note. This list of authorities is far from complete ; to have 
 made it so would have appeared rather as a parade of research, than as a 
 security from the charge of quoting without proper acknowledgment ; it has 
 therefore been principally confined to those writers whose labours have 
 not become in some measure the common property of all who follow 
 them in this department of science. 
 
 COLOMBIA COLLEGE, New- York. February Is7, 1832. 
 
LIST 
 
 Of the Authors principally consulted in the compilation of this work. 
 
 BARLOW, . . . * . On Timber. 
 
 Du BUAT, .... Hydrodynamique. 
 
 BIOT, ..,.-. Physique. 
 
 BIOT and ARAGO, . . Observations Geodesiques. 
 
 BOSSUT, .... Hydrodynamique. 
 
 BOWDITCH, . . . Translation of Laplace. 
 
 COULOMB, .... Machines Simples. 
 
 DANIELL, .... Atmospheric Phenomena. 
 
 DELAMBRE, . . . Base du Systeme Metrique. 
 
 GAUTHEY, .... Construction des Ponts. 
 
 Do Canaux de Navigation. 
 
 GENIEYS, .... Distribution des Eaux. 
 
 HUTTON, .... Tracts on Gunnery. 
 
 KATER, Report on Weights and Measures. 
 
 LAPLACE, .... Mecanique Celeste. 
 
 NAVIER, .... Ponts Suspendus. 
 
 PECLET, .... Traite de Clialeur. 
 
 PLAYFAIR, .... Outlines of Natural Philosophy. 
 
 POISSON, . . . .' Mecanique. 
 
 PONTECOULANT, . . Systeme du Monde. 
 
 POUILLET, .... Physique. 
 
 PRONY, . . . . . Jaugage des Eaux. 
 
 RICHARD, .... Manuel d' Applications Mathema- 
 
 tiques. 
 
 SABINE, Experiments on the Pendulum. 
 
 VENTURI, .... Motion of Fluids. 
 
 VENTUROLI, . . . Mecanica. 
 
 YOUNG (Thomas), . Natural Philosophy. 
 


 TABLE OF CONTENTS. 
 
 BOOK I. 
 
 OF EQUILIBRIUM. 
 
 CHAP. I. 
 
 General Principles. PAGE 1 
 
 1. Definition of Mechanics. 
 
 2. Mode of determining when bodies are at rest or in mo- 
 
 tion. 
 
 3. Co-ordinates and axes to. which the position of points is 
 
 referred. 
 
 4. Change in the length and position of the co-ordinates in 
 
 case of motion. 
 
 5. Definition of Force. 
 
 6. Mode of measuring forces. 
 
 7. Determination of the direction of a force. 
 
 8. Definition of Equilibrium. 
 
 CHAP. II. 
 
 Equilibrium of Forces acting in the same line. 4 
 
 9. Conditions of equilibrium of forces acting in the same 
 line. 
 
 10. Definition of the Resultant of Forces and its Compo- 
 
 nents: 
 
 11. Cases of the relations between the resultant and its 
 
 components. 
 
 CHAP. III. 
 
 Equilibrium of Forces converging to a point. Composition and 
 
 Resolution of Forces. 6 
 
 12. Determination of the resultant of two forces converg- 
 ing to a point. 
 
 13. Composition and resolution of forces. 
 
 14. Conditions of equilibrium among three forces. 
 
 15. Polygon of forces. 
 
 16. Determination of the resultant of three rectangular for- 
 
 ces converging to a point. 
 
;i TABLE OF CONTENTS. 
 
 17. Resolution of any number offerees into three rectangular 
 
 forces. 
 
 18. Conditions of equilibrium in a system of forces, each 
 
 resolved into three, at right angles to each other. 
 
 19. Conditions of equilibrium of a system of forces acting 
 
 upon a surface. 
 
 CHAP. IV. 
 
 Equilibrium of Parallel Forces. PAGE 16 
 
 20. Method of proceeding in the consideration of the action 
 of parallel forces. 
 
 21. Case in which the point of application of a force may 
 
 be conceived to be changed. 
 
 22. Value and position of the resultant of two parallel 
 
 forces. 
 
 23. Conditions of equilibrium in a system of parallel forces. 
 
 24. Definition of the Centre of Parallel Forces. 
 
 25. Investigation of formulae for finding the centre of pa- 
 
 rallel forces. 
 
 26. Applications of the foregoing formula?. 
 
 27. Effect of the connexion of points into rigid systems. 
 
 28. Conditions of equilibrium of a system of parallel forces 
 
 acting upon a system of inflexible and inextensible 
 lines. 
 
 29. Conditions of equilibrium of the Funicular Polygon and 
 
 Catenaria. 
 
 CHAP. v. - 
 
 Equilibrium of Forces in the same plane, but neither parallel, nor 
 
 converging to a single point. 33 
 
 30. General determination of the resultant of forces acting 
 
 in the same plane. 
 31 and 32. Examination and definition of the Moment of 
 
 Rotation. 
 
 33. Investigation of the value of the moment of rotation of 
 the resultant of two forces in terms of the moments 
 of rotation of its components. 
 
 34 and 35. Conditions of equilibrium of a rigid system 
 having one fixed point. 
 
 36. Extension of the subject to forces not situated in one 
 
 plane. 
 
 37. Definition and determination of the value of the projec- 
 
 tion of a force. 
 
 38. Extension of the conditions of equilibrium of a rigid sys- 
 
 tem to points connected in any manner whatsoever. 
 
TABLE OF CONTENTS. XI11 
 
 BOOK II. 
 
 OF MOTION^ 
 
 CHAP. I. 
 
 Of motion in general. Uniform motion. General principles 
 
 of variable motion. PAGE 39 
 
 39. Principle of Inertia. 
 
 40. Definition of Velocity ; its value in terras of the space 
 
 and time. 
 
 41. Comparison of uniform motions in the same line. 
 
 42. Composition and resolution of motion. 
 
 43. Variable motions and the forces that cause them. 
 
 44. Accelerating forces. 
 
 45. Relation between the force and velocity in uniform mo- 
 
 tion. 
 
 46. Investigation of the circumstances of variable motion. 
 
 CHAP. II. 
 
 Of Rectilineal motion uniformly accelerated, or uniformly re- 
 tarded. 46 
 47. Definition of uniformly accelerated or retarded motion. 
 
 48. Definition of final velocity. 
 
 49. Investigation of the circumstances of uniformly accele- 
 
 rated motion. 
 
 50. Investigation of the circumstances of uniformly retarded 
 
 motion. 
 
 51. Relations of the forces that cause curvilinear motion. N 
 
 CHAP. III. 
 
 Of Curvilinear Motion. 50 
 
 52. Investigation of the general circumstances of curvilinear 
 motion. 
 
 CHAP. IV. 
 
 Of Parabolic Motion. 52 
 
 53. Investigation of the circumstances of the motion of a 
 body acted upon by two forces inclined to each other ; 
 of which one would produce uniform motion, and 
 the other is a constant force, always acting parallel 
 to itself. 
 
 CHAP. V. 
 
 Of the Motion of Points compelled to move on surfaces by the 
 
 action of accelerating forces. 56 
 
 54. Method of considering the action of a surface on bodies 
 compelled to move upon it. 
 
XJV TABLE OF CONTENTS. 
 
 55. Case of motion on a surface inclined at a constant 
 
 angle to the direction of an accelerating force, always 
 acting parallel to itself. 
 
 56. Investigation of the circumstances of motion on a plane 
 
 surface under the action of an accelerating force 
 always parallel to itself. 
 
 57. Circumstances of motion on a system of planes. 
 
 58. Circumstances of motion on a curve. 
 
 59. Circumstances of motion on a cycloid. 
 
 60. Circumstances of motion in a circular arc. 
 
 61. Definitions of Central Forces. 
 
 62. Cause of motion in circular orbits. 
 
 63. Uniformity of the velocity in circular orbits. 
 
 64. Circumstances of motion in circular orbits. 
 
 65. Relation between the periodic times and distances in 
 
 circular orbits. 
 
 66. General principles of motion in curvilinear orbits. 
 67 and 68. Principle of areas. 
 
 CHAP. VI. 
 
 Principle of D'Alembert. PAGE 71 
 
 69. Enunciation and illustration of the principle of D' Alem- 
 bert. 
 
 CHAP. VII. 
 
 Principle of Virtual Velocities. 72 
 
 70. Definition of Virtual Velocities. 
 71. Illustrations of the principle of virtual velocities. 
 
 BOOK III. 
 
 OF THE EQUILIBRIUM OF SOLID BODIES. 
 CHAP. I. 
 
 General properties of Matter. Division of Natural Bodies. 
 
 Measure of the Moving Force oj Bodies. 77 
 
 72. Essential properties of matter. 
 
 73. Extension of matter. 
 
 74. Divisibility of matter. 
 
 75. Definition of Body. Indestructibility of matter. 
 
 76. Atomic theory stated. 
 
 77. Impenetrability of matter. 
 
 78. Mobility of matter ; relations of the Density, Volume, 
 
 and Mass of bodies. 
 
 79 and 80. Division of bodies according to their mechani- 
 cal states. 
 
TABLE OF CONTENTS. XV 
 
 81. Forces that determine the mechanical state of bodies. 
 
 82. Division of the subject growing out of the different me- 
 
 chanical states of bodies. 
 
 83. Quantity of motion in bodies. 
 
 CHAP. II. 
 
 Attraction of Gravitation. PAGE 83 
 
 84. Fall of heavy bodies near the earth. 
 
 85. Direction of the force of gravity. 
 
 86. Convergence of these directions. 
 
 87. Relation of the attraction of gravitation to the quantity of 
 
 matter and velocity. 
 
 88. Question in relation, to the mutual attraction of bodies. 
 
 89. Experiment of Schehallien. 
 
 90. Principles of the experiment of Cavendish. 
 
 9 1 . Description of the apparatus of Cavendish. 
 
 92. Principles on which the mass of the planets is deter- 
 
 mined. 
 
 93. Consideration of gravity as an accelerating force. 
 
 94. Experiments of Galileo. , 
 
 95. Experiments and machine of Atwood. 
 
 96 and 97. Proofs of the mutual action of the earth and 
 moon. Law according to which the intensity of gra- 
 vity decreases. 
 
 98. Proofs of the universal influence of gravitation* 
 
 99. Force of gravity cannot be considered constant, even 
 
 at small distances from the surface of the earth. 
 
 100. Effect of the earth's rotation on the force of gravity at 
 
 the earth's surface. 
 
 101. Investigation of the relation between the centrifugal 
 
 force at the equator, and the whole force of gravity. 
 
 102. Effect of the combined action of gravitation and the cen- 
 
 trifugal force, on the figure of the earth. 
 
 103. Recapitulation of the laws of universal gravitation. 
 
 CHAP. III. 
 
 Of the Centres of Gravity and Inertia. 100 
 
 104. Direction of the force of gravity. Definition of Weight. 
 
 105. Definition of the Centre of Gravity. Application of the 
 
 conditions of equilibrium of parallel forces, to the 
 determination of the position of the centre of gravity. 
 
 106. Conditions of equilibrium of a body supported by a point. 
 
 107. Conditions of equilibrium of a body resting on an edge. 
 108 and 109. Conditions of equilibrium of a body resting on 
 
 a surface. 
 
 110. Effect of the figure of bodies on their stability. 
 
 111. Stability of bodies resting on inclined surfaces. 
 
 11?. Conditions of equilibrium in bodies resting on points. 
 
XVI TABLE OF CONTENTS, 
 
 113. Phenomena of rolling and sliding bodies. 
 
 114. Experimental methods of determining the position of 
 
 the centre of gravity. 
 
 115. Definition of Centre of Inertia. 
 
 . 
 
 CHAP. IV. 
 
 Of Friction. PAGE 112 
 
 116. Resistances to the motion of bodies. 
 
 117. Reference to the authors who treat of friction. 
 
 118. Varieties of friction. 
 
 119. Methods of determining the friction that prevents bodies 
 
 being set in motion. 
 
 120. Experiments and inferences in relation to the friction of 
 
 moving bodies. 
 
 121. Friction of rolling bodies. 
 
 122. Friction at the axles of wheels. 
 
 123. Theory of Coulomb. 
 
 124. Formula that expresses the results of the experiments on 
 
 friction. 
 
 125. Methods of lessening and overcoming friction. 
 
 126. Effects of friction in bringing bodies, moving near the 
 
 surface of the earth, to rest. 
 
 CHAP. v. 
 
 Of the Stiffness of Ropes. 124 
 
 j 127. Circumstances on which the stiffness of ropes depends. 
 
 128. Results of the experiments of Coulomb. 
 
 129. Effect of increasing the number of turns made by a rope 
 
 around a cylinder. 
 
 130. Data for calculating the stiffness of ropes obtained from 
 
 the experiments of Coulomb. 
 
 CHAP. VI. 
 
 Of the Mechanic Powers. 126 
 
 131. Definition of Machines. General principle of their 
 action. * 
 
 132. Definition of the mechanic powers. 
 
 133. Number and classification of the mechanic powers. 
 
 Of the Lever. 127 
 
 134. Definition of the Lever. 
 
 135. General condition of equilibrium in the lever. 
 
 136. Definition of the Power and the Weight. Classification 
 
 of the different kinds of lever. 
 
 137. Condition of equilibrium in a straight lever, on which 
 
 the power and weight act parallel to each other. 
 
 138. Condition of equilibrium of a lever, whether straight or 
 
 crooked in any direction of the power and weight. 
 
TABLE OF CONTENTS. XV11 
 
 139. Examples of the use of the different kinds of lever. 
 
 140. Principles on which the properties of the balance depend. 
 
 141. Error arising from inequality of the arms of a balance. 
 
 Mode of counteracting and correcting that error. 
 
 142. Properties of a good balance. 
 
 143. Condition of equilibrium in a compound system of levers. 
 
 144. Description and theory of the platform balance. 
 
 145. Description and theory of the steelyard. 
 
 146. Investigation of the effect of friction upon the lever. 
 
 Of the Wheel and Axle. PAGE 138 
 
 147. Condition of equilibrium in the Wheel and Axle. Various 
 forms of the wheel and axle. 
 
 148. Combinations of wheels and axles, by means of bands. 
 
 149. Wheels and Pinions. 
 
 1 50. Examination of the proper form to be given to the teeth of 
 
 wheels and pinions. 
 
 1 51. Change of intensity of force, and of velocity produced in 
 
 the action of wheels upon pinions, and of pinions upon 
 wheels. 
 
 152. Change in the direction of motion, produced by different 
 
 forms of the wheel and pinion. 
 
 153. Effect of friction upon the equilibrium of the wheel and 
 
 axle. 
 
 154. Allowance for friction in the action of wheels and pin- 
 
 ions. 
 
 Of the Pulley. 146 
 
 155. Conditions of equilibrium in the fixed and moveable 
 Pulley. 
 
 156. Conditions of equilibrium in combinations of pullies. 
 
 157. Effect of a want of parallelism in the ropes on the equi- 
 
 librium of pullies. 
 
 158. Friction of pullies, and modes of lessening it. 
 
 Of the Wedge. 151 
 
 159. Definition of the Wedge. 
 
 160. Condition of equilibrium in the isosceles wedge. 
 
 161. Practical action of the wedge. 
 
 162. Valuable applications of the wedge. 
 
 163. Extension of the principle of the wedge to bodies of 
 '-& other figures. 
 
 164 and 165. Effect of friction on the action of the wedge. 
 
 Of the Inclined Plane. 1 55 
 
 166. Definition of the Inclined Plane, and mode in which 
 the forces act upon it. 
 
 167. Conditions of equilibrium in the inclined plane. 
 
 168. Applications of the inclined plane. 
 
 169. Effect of friction on the equilibrium of the inclined plane, 
 
 3* 
 
TABLE OF CONTENTS. 
 
 Of the Screw. PAGE 157 
 
 170. Mode of forming the Screw, and conditions of its equi- 
 librium. 
 
 171. Effects of friction on the action of the screw. 
 
 172. Endless screw. 
 
 173. Valuable applications of the screw. 
 
 174. Hunter's improvement hi the screw. 
 
 175. Effect of friction on the endless screw, when combined 
 
 with the wheel and axle. 
 
 CHAP. VII. 
 
 Of the Strength of Materials. 165 
 
 176. Method of proceeding in considering the strength of 
 materials. 
 
 177. Hypothesis of Galileo. 
 
 178. Different modes in which forces act to destroy the aggre- 
 
 gation of bodies. 
 
 Of the Absolute Strength of Materials. 166 
 
 179. Analytic expressions for the absolute strength of mate- 
 rials. 
 
 180. Inferences from experiments, and tables of the absolute 
 strength of different substances. 
 
 Of the Respective Strength of Materials. 169 
 
 181. Inferences from the hypothesis of Galileo, in respect to 
 beams supported at one end. 
 
 182. Inferences from the same hypothesis in respect to beams 
 
 resting on two props. 
 
 183. Inferences in respect to beams firmly fixed to each end. 
 
 184. Examination of the effect of the inclination of a beam to 
 
 the horizon. 
 
 185. Consequences of an uniform distribution of the weight 
 
 that acts upon a beam. 
 
 186. Action of the beam's own weight. 
 
 187. Comparison between the inferences from the hypothesis 
 
 and the results of experiment, in the cases where they 
 nearly agree. 
 
 188. Discrepancies between the inferences from the hypothe- 
 
 sis, and the results of experiment. 
 
 189. Tables of the respective strength of various substances. 
 
 Of the Resistance of Bodies to a Force exerted to crush them. 181 
 
 190. Expressions for the resistance of a rectangular pillar to a 
 
 crushing force. 
 
 191. Tables of the resisting force of various bodies to a 
 crushing force. 
 
TABLE OF CONTENTS. XIX 
 
 Of the Strength of Torsion. PAGE 183 
 
 1 92. Laws which the resistance to Torsion follows. 
 1 93. Relative resistances to torsion of different substances. 
 
 CHAP. VIII. 
 
 Of the Equilibrium of Artificial Structures. 184 
 
 194. Principle on which the equilibrium of artificial struc- 
 tures is investigated. 
 195. Definition of the Strength, and of the Stress or Thrust. 
 
 Of the Equilibrium of Walls. 184 
 
 196. Conditions of equilibrium of a prismatic wall. 
 
 197. Effect of friction on the conditions of equilibrium. 
 
 198. Conditions of equilibrium of walls with sloping faces, 
 
 and with buttresses. 
 
 199. Applications of the theoretic principles. 
 
 Equilibrium of Columns. 187 
 
 200. Investigation of the conditions of equilibrium in columns. 
 
 201. Inferences from the investigation of 200. Figure of 
 
 columns loaded with weights. 
 
 202. Effect of a lateral thrust in the weight that acts on a col- 
 
 umn. 
 
 203. Figure of a column intended to resist the action of a 
 
 fluid. 
 
 Equilibrium of Terraces. 196 
 
 204. Mode in which earth acts on a terrace wall. 
 205. Investigation of the conditions of equilibrium. 
 
 Equilibrium oj Arches. 198 
 
 206. Reasons for the use of Arches. Different materials of 
 which they may be constructed. 
 
 207. Definitions and names of the parts of an arch. 
 
 208. Former mode of proceeding in the investigation of the 
 
 theory of arches. Its defects. 
 
 209. Manner in which arches are affected by the mutual 
 
 action of their parts. 
 
 210. Investigation of the position of the points of rupture in 
 
 an arch. 
 
 211. Inferences from the investigation in 210. 
 
 212. Relative thicknesses of piers and abutments. 
 
 213. List of the most remarkable stone arches. 
 
 Equilibrium of Dom es. 207 
 
 214. Comparison between domes and cylindric vaults. 
 
 215. Geometric properties of domes. 
 
 216. Description of some remarkable domes. 
 
XX TABLE OF CONTENTS. 
 
 Of Wooden Arches. PAGE 210 
 
 217. Comparison between the actions of wood and stone, 
 when employed in spanning openings. 
 
 218. Strength of trussed beams. 
 
 219. Strength of a combination of two straight beams. 
 
 220. Strength of a wooden arc of small curvature. 
 
 221. Strength of a wooden arc of greater curvature. 
 
 222. Applications to the construction of bridges. 
 
 223. Principle applied by Grubenman, in the construction of 
 
 wooden bridges. 
 
 Of Cast Iron Arches. 218 
 
 224. Comparison between the actions of cast iron, and those 
 
 of wood and stone in the construction of arches. 
 225. Instances of cast iron arches. 
 
 Of Chain Bridges. 219 
 
 226. Progress of the art of erecting chain bridges. 
 227. Theory of chain bridges. 
 228 to 234. Inferences made by Navier, from the theory. 
 
 238. Different cases in which the principle may be applied. 
 
 239. Notice of wire bridges. 
 
 BOOK IV. 
 OF THE MOTION OF SOLID BODIES. 
 
 CHAP. I. 
 
 Of Falling Bodies. 227 
 
 237. Action offerees whose direction passes through the cen- 
 tre of gravity of a solid body. 
 
 238. Investigation of the circumstances of the motion of a body 
 
 falling from rest near the surface of the earth, the 
 resistance of the air being left out of view. 
 
 239. Circumstances of the motion of a body projected verti- 
 
 cally upwards. 
 
 240. Investigation of the effect produced by the resistance of 
 
 the air, on the hypothesis that it varies with the square 
 of the velocity. 
 
 241. Discrepancies between the hypothesis used in 240, and 
 
 the actual law of the resistance of the air. 
 
 CHAP. II. 
 
 Of the Rotary Motion of Bodies. 232 
 
 242. Theory of the Moment of Inertia. 
 

 TABLE OP CONTENTS. XXI 
 
 243. Properties of the Centre of Gyration, and modes of 
 
 finding it. 
 
 244. Properties of the Centre of Percussion. 
 
 245. Consideration of the double motion of translation and 
 
 rotation produced by the action of a force whose direc- 
 tion does jiot pass through the centre of gravity of a 
 solid body. 
 
 246. Proposition in relation to the three principal axes of rota- 
 
 tion. 
 
 247. Variation in the velocity of the points of a body endued 
 
 with a double motion of rotation and translation, when 
 this velocity is estimated in the direction in which the 
 centre of gravity moves. 
 
 248. Centre of spontaneous rotation. Effects of the varying 
 
 velocity, spoken of in 247, on bodies moving in 
 resisting media. 
 
 249. Applications of the theory of the moment of inertia. 
 
 250. Mode of finding the moment of inertia in respect to a 
 
 given axis, when this moment is given in respect to 
 another line considered as an axis. 
 
 251. Applications of 250. 
 
 252. 
 
 CHAP. III. 
 
 Motion of Projectiles. PAGE 242 
 
 Circumstances of the motion of projectiles, abstracting 
 
 the resistance of the air. 
 
 253. Investigation of the effect of a resisting medium ; appli- 
 
 cation to the resistance of the air, upon the hypothesis 
 that it varies with the square of the velocity. 
 
 254. Application to military projectiles. Mode of action of 
 
 inflamed gunpowder. Inferences from the experi- 
 ments of Hutton, in respect to the expansive force of 
 gunpowder. 
 
 255. Effect of confined gunpowder according to the experi- 
 
 ments of Rumford. Consequences that follow in 
 certain cases. 
 
 256. Citation of the best experiments on military projectiles. 
 
 257. Original mode of determining the initial velocity of a pro- 
 
 jectile. 
 "258. Ballistic pendulum of Robins. 
 
 259. Effect of the length of the piece on the initial velocity. 
 
 260. Relation of the initial velocities to the charges of gun- 
 
 powder, and the weight of the balls. 
 
 261. Results ^f Hutton' s experiments on the resistance of the 
 
 air. Beduction of formulae for practical purposes. 
 
 262. Applications of the formulas of 261. 
 
 263. Inferences from the theory in respect to the maximum 
 
 charge of gunpowder. 
 
 264. Inferences in respect to the windage. 
 
TABLE OP CONTENTS. 
 
 265. Deviation of projectiles from the vertical plane, and its 
 
 cause. 
 
 266. Advantages of rifling the bores of guns. 
 
 267. Improvements in military ordnance resulting from the 
 
 experiments of Hutton. 
 
 268. Inquiry into the best figure for cannon. 
 
 269. Penetration of military projectiles into solid bodies. 
 
 Comparison of ancient and modern military engines. 
 
 270. Principles of the Ricochet and its applications. 
 
 CHAP. iv. 
 Theory of the Pendulum. PAGE 267 
 
 271. Circumstances of the motion of a Pendulum. 
 
 272. Theory of the motion of a simple pendulum in a circular 
 
 arc. 
 
 273. Isochronism of a simple pendulum moving in cycloidal 
 
 arcs. Method proposed to cause pendulums to move 
 in a cycloid. 
 
 274. Effect of difference of distance from the surface of the 
 
 earth on the lengths of isochronous pendulums. 
 
 275. Effect of the variation of the force of gravity on the 
 
 lengths of isochronous pendulums in different lati- 
 tudes. 
 
 276. Definition of the compound pendulum ; properties of the 
 
 Centre of Oscillation. 
 
 277. The centres of suspension and oscillation are convertible 
 
 points. 
 
 278. Investigation of the position of the centre of oscillation, 
 
 in homogeneous bodies of certain figures. 
 
 279. Effects of the resistance of the air on the motions of pen- 
 
 dulums. 
 
 CHAP. v. 
 
 Applications of the Pendulum. 277 
 
 280. Various purposes to which the pendulum is applicable. 
 
 281. Principles of the application of the pendulum as a mea- 
 
 sure of time. 
 
 282. Description and theory of compensation pendulums. 
 
 283. Methods in use for measuring the length of the pendu- 
 
 lum. 
 
 284. Method of Borda. 
 
 285. Reduction of the pendulum's vibrations to a cycloidaJ 
 
 arc ; corrections for the air's resistance ; determina- 
 tion of the length of the seconds pendulum from tint 
 of the experimental ; reduction to thp level of th* 
 sea ; effect of the figure and density n the grounJ "' 
 which the experiment is made. 
 
 286. Method of Kater. 
 
 287. Deduction of the measure of the force of grav/ry fro 
 
 length of the second's pendulum. 
 
 

 TABLE OF CONTENTS. XX111 
 
 288 and 289. Comparison of the methods of Borda and 
 Kater. 
 
 290. Results of the measures of the pendulum in different 
 
 places. 
 
 291. Principle on which the pendulum is applied as a stand- 
 
 ard of measure. 
 
 292. British system of weights and measures. 
 
 293. System of the state of New- York. 
 
 294. Comparison of the two foregoing systems. 
 
 295. French system of weights and measures. 
 
 296. Examination of the advantages and defects of the French 
 
 system. 
 
 297. Application of the principle of the pendulum in Caven- 
 
 dish's experiment. 
 
 CHAP. VI. 
 
 Of Collision. PAGE 300 
 
 298. Division of bodies into elastic and non-elastic. 
 
 299. Investigation of the consequences of the collision of non- 
 
 elastic bodies of spherical figure moving in the same 
 straight line. 
 
 300. Oblique impact of a non-elastic body against an immovea- 
 
 ble plane. 
 
 301. Oblique impact of non-elastic bodies of spherical figures. 
 
 302. Impact of non-elastic bodies moving in parallel lines. 
 
 303. Investigation of the consequences of the collision of 
 
 elastic bodies of spherical figure moving in the same 
 straight line. 
 
 304. Recapitulation of the inferences from the foregoing in- 
 
 vestigation. 
 
 305. Oblique impact of an elastic body against an immovea- 
 
 ble surface. 
 
 306. Constancy of the sum of the vires viva in the collision 
 
 of perfectly elastic bodies. 
 
 307. Gain of motion, estimated in a given direction, in the col- 
 
 lision of a small elastic body against a larger one. 
 
 308. Effects of impact when the elastic bodies do not move 
 
 in the same straight line. 
 
 309. The state of the common centre of gravity in respect to 
 
 motion or rest is not affected by the collision of bo- 
 dies. 
 
 310. Investigation of the effect of imperfect elasticity. 
 
 BOOK V. 
 
 OF THE EQUILIBRIUM OF FLUIDS. 
 CHAP. I. 
 
 General Characters of Fluid Bodies. 309 
 
 311. Distinctive property, and classification of fluids. 
 
XXIV TABLE OF CONTENTS. 
 
 312. Characters of liquids. 
 
 313 and 314. Investigation of the general conditions of the 
 
 equilibrium of fluids. 
 315. Difference in the action of solid and fluid masses. 
 
 CHAP. II. 
 
 Equilibrium of Gravitating Liquids. PAGE 314 
 
 316. Condition of equilibrium in gravitating liquids. 
 
 317. Figure of the surface of gravitating liquids. 
 
 318. Definition of the art of Levelling. 
 
 319. Description of the Water Level. 
 
 320. Description of the Spirit Level. 
 
 321. Mode of using the spirit level. 
 
 322. Description of the Mason's Level. 
 
 323. Correction of level for the sphericity of the earth. 
 
 324. Effect of atmospheric refraction on levelling. 
 
 325. Equilibrium of a homogeneous liquid in a bent tube. 
 
 326. Equilibrium of heterogeneous liquids. 
 
 327. Effects produced on the weight of solids immersed in 
 
 fluids. 
 
 328. Equilibrium of a s'olid floating at the surface of a liquid. 
 
 329. Comparison of the immersion of the same body floating 
 
 on different liquids. 
 
 CHAP. in. 
 
 Of the Pressure of Gravitating Liquids. 325 
 
 330. Investigation of the expressions for the pressure of a 
 liquid on a surface. 
 
 331. Inferences from the foregoing investigation. 
 
 332. Action of a liquid to change the direction and intensity 
 
 of the forces that act upon it. 
 
 333. Hydrostatic paradox. 
 
 334. Comparison of the theory with the results of experiment. 
 
 335. Properties of the centre of fluid pressure. 
 
 336. Investigation of the amount of pressures exerted by a 
 
 liquid on a body immersed in it. 
 
 CHAP. IV. 
 
 Of Specific Gravity. 333 
 
 337. Definition of Specific Gravity ; standard in which it is 
 estimated. 
 
 338. Changes produced in the density of water by variations 
 
 of temperature. 
 
 339. Principles of the method of specific gravities. 
 
 340. Modes of determining the specific gravity of solid bodies. 
 
 341. Modes of determining the specific gravity of liquids. 
 
 342. Method of ascertaining the weight of given bulks of 
 
 liquids. 
 
TABLE OF CONTENTS. XXV 
 
 348. Determination of the specific gravities ofliquids by glass 
 
 bubbles. 
 
 349. Table of Specific Gravities. 
 
 350. Application of the Method of specific gravities to deter- 
 
 mine the volume of bodies. 
 
 CHAP. v. 
 
 Of the nature and characters of Elastic Fluids, and of the 
 
 Pressure of the Jltmosphere. PAGE 347 
 
 351. Modes in which we become acquainted with the exist- 
 
 ence of Elastic Fluids. 
 
 352. Distinctive properties of permanently Elastic Fluids or 
 
 Gases. 
 
 353. Characters of condensable elastic fluids or Vapours. 
 
 354. Effect of heat on the tension of elastic fluids. 
 
 355. Law of Dalton. 
 
 356. Experiment of Torricelli. 
 
 357. Estimate of the amount of atmospheric pressure. 
 
 358. Principles of action, and description of the common pump 
 
 and syphon. 
 
 CHAP. VI. 
 
 Of the Air Pump. 359 
 
 359. Description of the Air Pump. 
 
 360. Gauges for the air pump. 
 
 361. Proofs afforded by the air pump of the pressure of the 
 
 atmosphere. / 
 
 362. Experimental proof that the pressure *< the air causes 
 
 the rise of liquids in the pump and in the Torricellian 
 apparatus. 
 
 363. Experimental proof that the air has weight. 
 
 364. Experimental proofs of the elasticity of air. 
 
 CHAP. vn. 
 Equilibrium of Permanently Elastic Fluids. 360 
 
 365. Experimental investigation of the law of Mariotte. 
 
 366. Conditions of the equilibrium of a gaseous atmosphere of 
 
 uniform temperature. 
 
 367. Law of Gay-Lussac. 
 
 368. Investigation of the relation between the densities and 
 
 volumes of gases under different circumstances of 
 temperature and pressure. 
 
 369. Properties common to liquids and elastic fluids ; buoy- 
 
 ancy of the atmosphere. 
 
 4* 
 
XX Vi TABLE Of CONTENTS. 
 
 CHAP. VIII. 
 
 Of the Equilibrium and Tension of Vapours. PAGE 368 
 370. Vapour is formed at other temperatures than that at which 
 ebullition occurs. 
 
 371. Relation of the weight and tension of vapour in a given 
 
 space, to the temperature and pressure. 
 
 372. Effect of unequal temperature, in the space occupied by 
 
 vapour, on its tension. 
 
 373. Expression for the relations between the temperatures, 
 
 tensions, and densities of vapours. 
 
 374. Experimental determinations of the tension of aqueous 
 
 vapour at different temperatures. 
 
 375. Dalton's Law of the tension of vapours of different sub- 
 
 stances. 
 
 376. Tension of mixtures of gases and vapours. 
 
 CHAP. IX. 
 
 Of the Specific Gravity of Elastic Fluids. 374 
 
 377. Method of determining the specific gravity of gases. 
 Table of the specific gravity of gases. 
 
 378. Method of determining the specific gravity of vapours. 
 
 Table of the specific gravity of vapours. 
 
 379. Table of the density and volume of steam at different 
 
 temperatures. 
 
 380. Of the Dew-Point, and method of finding it. 
 
 CHAP, X. 
 
 Of the Barometer and its Applications. 379 
 
 381. Various forms and modifications of the Barometer. 
 
 382. Theory of the Vernier. 
 
 383. Modes of rendering the barometer portable. 
 
 384. Method of filling and graduating barometers. 
 
 385. Accidental and periodic variations in atmospheric pres- 
 
 sure, and the height of the barometer. 
 
 386. Influence of the accidental variations on the weather. 
 
 387. Principle on which the barometer is applied in the 
 
 mensuration of heights. Effect of variations of 
 temperature on the results, and correction for differ- 
 ence of temperature. 
 
 388. Rules for observation, and for calculating heights by the 
 
 barometer. 
 
 389. Correction for moisture, and for the latitude of the place. 
 
 Example of the form of calculation. 
 
TABLE OP CONTENTS, XXVH 
 
 CHAP. XI. 
 
 Of the Attraction of Cohesion. PAGE 390 
 
 390. Experiment showing the existence of the Attraction of 
 
 Cohesion. % 
 
 391. Phenomena arising from the attraction of cohesion. 
 
 392. Theory of Capillary Attraction. 
 
 393. Relation between the forces of cohesion and aggrega- 
 
 tion. 
 
 394. Relation between the heights to which a liquid rises in 
 
 tubes of different diameters, in concentric tubes, and 
 between plates. 
 
 395. Effect of capillary action on aeriform fluids. 
 
 BOOK VI. 
 
 OF THE MOTION OF FLUIDS. . 
 CHAP. I. 
 
 Theory of the Motion of Liquids. 396 
 
 396. Hypothesis used in examining the motion of liquids. 
 397. Examination of the manner in which 'liquids actually 
 move. 
 
 CHAP. II. 
 
 Of the Motion of Liquids through orifices pierced in thin 
 
 plates. 400 
 
 398. Theory of this motion. 
 
 399. Shape of the jet determined by experiment. InAv-nce 
 
 of the contraction of the vein on the theoretic- esu ^ g ' 
 
 400. Phenomena of spouting liquids. 
 
 401. Formulae for the height and amplitude of a i*< of liquid. 
 
 402. Theory of the vortex formed in the disch^'ge of bquids 
 
 from vessels. 
 
 CHAP. III. 
 
 Of the Discharge of Liquids thr^gh short r*pes or adjut- 
 age- 
 
 403. General effect produced >/ applying * pipe to an orifice 
 through which a lia- ld issues. 
 
 404. Effects of adjutage* of differed forms. 
 
 405. Table of the ejects of adjudges. 
 
XXVlll TABLE OP CONTENTS. 
 
 CHAP. IV. 
 
 Of the Motion of Water in Pipes. PAGE 4 1 1 
 
 406. Theory of the motion of water in pipes of uniform bore. 
 
 407. Effect of variations in the diameter of the pipes. 
 
 408. Obstructions to which pipes are liable, and modes of ob- 
 
 viating them. 
 
 409. Effect of bends or elbows in the pipes. 
 
 410. Theory of the motion of water -in pipes diverging from a 
 
 main-pipe. 
 
 CHAP. v. 
 Of the Motion of Liquids in Open Channels. 421 
 
 411. Theory of the motion of liquids in open channels of uni- 
 form section. 
 
 412. Relations between the velocity at the surface, and the 
 
 mean velocity of a stream. 
 
 413. Effect of variations in the section of a stream, and of 
 
 bends in its course. Principle of the lateral commu- 
 nication of motion in fluids. 
 
 CHAP. VI. 
 
 Of Rivers. 42J 
 
 414. Circulation of aqueous matter between the land and 
 ocean. 
 
 415. Relation between the beds of rivers, and the force of 
 
 their currents. 
 
 416. Effects of the overflow of rivers. 
 
 417. Obstructions to which the navigation of rivers is liable. 
 x 4 IS. Modes of removing such obstructions. 
 
 419\Different methods of gauging streams. 
 
 CHAP. VII. 
 
 Of Canals. 43' 
 
 420. Company between the circumstances of canals, and of 
 
 421. Examination tf the proper form for the bed of a canal 
 
 at its union v ]th the source whence its waters are 
 drawn. \ 
 
 422. Investigation o f the i^ ?er figure o f banks of earth used 
 
 lor conning water. 
 
 423. Description of^ e form ou he bed and banks of a canal. 
 
 424. Different cases n^hich nav^ble canals are used. 
 
 425. Relation between h, area o?K Cana] , and the section o f 
 
 the vessels that nav^ a te it. 
 
 426. Description of a lock. 
 
 427. Mode of passing vessels thrbigh a 
 
TABLE OP CONTENTS. XX1X 
 
 428. Investigation of the strength of the walls of a lock. 
 
 429. Description of the gates of locks. 
 
 430. Investigation of the proper angle for the leaves of the 
 
 gates of locks. 
 
 431 . Principles on which the proper height of locks is deter- 
 
 mined. 
 
 432. Principles that determine the size of locks. 
 
 433. Defects of locks placed in juxta-position. 
 
 434. Mode of diminishing the waste of water in deep locks. 
 
 435. Estimate of the waste of water in canals, arising from 
 
 leakage and evaporation. 
 
 436. Feeders for canals. 
 
 437. Reservoirs for the supply of canals. 
 
 438. Use of reservoirs in clarifying the water of canals. 
 
 439. Culverts. 
 
 440. Aqueducts. 
 
 441. Waste-gates. 
 
 442. Inclined planes. 
 
 443. Description of the lock of Betancourt. 
 
 CHAP. VIII. 
 
 Of the Percussion and Resistance of Fluids. 459 
 
 444. Definition of the Resistance of a Fluid. 
 
 445. Theory of the resistance of fluids. 
 
 446. Experimental inferences of Bossut. Analysis of the 
 
 circumstances of the percussion of fluids. 
 
 447. Circumstances of the resistance to bodies immersed in 
 
 fluids. 
 
 448. Circumstances of the resistance to bodies floating at the 
 
 surface of liquids. 
 
 449. Limit to the velocity with which bodies can move at the 
 
 surface of liquids. 
 
 450. Formulae for the resistance of fluids by Bossut, and 
 
 Romme. 
 
 CHAP. IX. 
 
 Of the Motion of Waves. 467 
 
 & 451. Mode in which Waves may be formed at the surface of a 
 liquid. 
 
 452. Comparison between the motion of waves and the oscil- 
 
 lation of a liquid in a bent tube. 
 
 453. Theory of the oscillations of a liquid in a bent tube. 
 
 454. Application of the foregoing theory to the case of waves. 
 
 455. Reflection of waves. 
 
 456. Interference of systems of waves. 
 
 457. Propagation of waves through vertical orifices. 
 
 458. Examination of the phenomena of waves raised by the 
 
 winds. 
 
 459. Effect of inclined obstacles upon the motion of waves. 
 
 1 
 
TABLE OF CONTENTS. 
 
 CHAP. X. 
 
 Of the Motion of Elastic Fluids. 474 
 
 460. Theory of the Motion of Elastic Fluids. 
 
 461. Determination of the velocity with which atmospheric 
 
 air rushes into a vacuum. 
 
 462. Velocity of air of different densities entering spaces con- 
 
 taining air. 
 
 463. Velocities of gases other than atmospheric air. 
 
 464. Effect of the air's elasticity on its motion. 
 
 465. Influence of adjutages on the motion of air. 
 
 466. Effect of the lateral communication of motion in gases. 
 
 467. Motions of air produced by variations of temperature. 
 
 468. Tables of the velocity of steam. 
 
 CHAP. XI. 
 
 Of the Motion of Gases in Pipes. 479 
 
 469. Theory of the motion of air in Pipes. 
 470. Applications of the theory. 
 
 CHAP. xn. 
 
 Of the Motion of Air in Chimnies. 483 
 
 471. Theory of the motion of air in chimnies, abstracting the 
 resistance. 
 
 472. Effects of friction in general, and in flues of different sub- 
 
 stances. 
 
 473. Reason of the difference in the friction of air against 
 
 different substances. 
 
 CHAP. XII. 
 
 Of the Winds. 486 
 
 474. Definition and general cause of the Winds ; causes of 
 the diversities of temperature at different points on the 
 surface of the earth. 
 
 475. Effect of elevation on the temperature of places. 
 
 Formula that represents the circumstances that influ- 
 ence climate. 
 
 476. Influence of the nature of the surface on climate. 
 
 477. Temperature of lakes- and morasses ; effects of cultiva- 
 
 tion. 
 
 478. Recapitulation of the circumstances that influence the 
 
 temperature. 
 
 479. Manner in which the temperature and motion of the air 
 
 is affected by the temperature of the surface, and the 
 rotary motion of the earth. 
 
 480. Classification of the winds. 
 
 481. Phenomena of the Trade Winds. 
 
 482. Seat and character of the Variables. 
 
 483. Phenomena of the Monsoons. 
 
 484. Local modifications of the trade winds and monsoons. 
 
TABLE OF CONTENTS. XXXI 
 
 485. Westerly winds between the parallels of 30 and 40. 
 
 486. Variable winds of the temperate and frigid zones. 
 
 487. Land and Sea Breezes. 
 
 CHAP. XIV. 
 
 Of the Motion of Vapour in the Atmosphere. 502 
 
 488. Cause of the existence of Vapour in the atmosphere. 
 
 489. Phenomena of an atmosphere of vapour over a sphere of 
 
 uniform temperature and surface. 
 
 490. Phenomena in the case of an uniform decrease of tem- 
 
 perature from the poles to the equator. 
 
 491. Influence of the aerial part of the atmosphere. 
 
 492. Effects of diminution of temperature in rising in the aerial 
 
 atmosphere. 
 
 493. Motions in the aqueous atmosphere, growing out of the 
 
 unequal distribution of land and water. 
 
 494. Influence of the winds on the distribution of vapour. 
 
 495. Origin of springs and rivers. 
 
CORRIGENDA. 
 
 
 On PAGE 3 line 22, for " cos. 2 a+cos. 2 fc+eos. 2 c=0," read 
 
 " cos. 2 a+cos. 2 fc+ cos. 2 c = l." 
 
 14 line 12, for " cos. 2 a+cos. 2 6+cos. 2 c=0," read 
 "cos. 2 a+cos. 2 6+cos. 2 c = l." 
 
 42 line 24, for " *= +-7 ~77 ," read 
 
 <U - u 7J - V 
 
 <ff If 
 
 u/ -- x - 
 
 *"- 
 
 54 line 31, /or "g-" read "i/." 
 112 line 1, for "CHAP, v" read "CHAP, iv." 
 
 Wd Wd 
 
 160 line 7, /or "P= ," read " P = -Q- 
 
 Wdb 
 line 12, /or "P=-^- , read "P=-^- 
 
 270 (291) for "1 : 1+jfc," read "l+i h : 1." 
 320 (355) for " fc=j^ ," read "/i=^-." 
 
 327 line 10, for "substances," read "surfaces." 
 
 333 line 13, Jill in the blank, " ( )," with " (299)." 
 
 422 line 31, for "100," read "av." 
 
 449 line 4, for "friction," read "function." 
 
 437 line 53, for "level," read "lever." 
 
us.. A remains, < 
 
 
 BOOK I. 
 
 OF EQUILIBRIUM. 
 
 CHAPTER I. 
 
 GENERAL PRINCIPLES. 
 
 1. MECHANICS is the department of Physical Science which 
 treats of Motion, and of the construction of Machines. 
 
 2. Bodies appear to us to be in motion, when they change their 
 position in respect to other bodies that we conceive to be at rest ; 
 but even bodies that appear to our earlier investigation to be ab- 
 solutely at rest, may be in a state of rapid motion, as, for instance, 
 the body of the earth itself; which, to the uneducated and igno- 
 rant, appears solid and immovable, although it can be shown by 
 scientific proofs to be a state of rapid motion, both of rotation 
 and translation. Hence we refer motion, in the abstract, to un- 
 bounded and immovable space. As space is extended in three 
 dimensions, a body may move in the direction of any one of them, 
 or may have a motion intermediate between two or more of them : 
 it may rise or fall, approach or recede, pass to the right or to the 
 left, or its motion may be combined of two, or all three of those 
 varieties. 
 
 In order to render these circumstances of motion definite, we 
 refer the position of a point to three planes, supposed to be im- 
 movable in absolute space, and which cut each other at right 
 angles. 
 
 3. The perpendicular distances from the point to these three 
 planes, are called its Co-ordinates; the mutual intersection of any 
 two of the planes is called an Axis, and the common, intersection of 
 three planes is called the Origin of the co-ordinates. Each of 
 the axes is parallel to one of the co-ordinates, for the common 
 intersection of two of the planes is perpendicular to the third, 
 
 1 
 
2 OP EQUILIBRIUM. \JBook L 
 
 and the co-ordinates are by definition each drawn perpendicular 
 to one of the planes. 
 
 4. If the point be in motion, it will be shown by the change in 
 the length of one or more of the co-ordinates, and of the positions 
 of the points in which the co-ordinates cut the planes. 
 
 5. The cause by which a body is set in motion, whatever be 
 its nature, is called a Force. Forces may be of various descrip- 
 tions, but as they all produce motion, they may be compared 
 with each other, and made commensurable by means of the mo- 
 tion they produce ; and we judge of the intensity of a force by 
 the quantity of motion it is capable of causing. As we know 
 nothing of forces, except by their effects, we may hence assume 
 that the force is always measured by the quantity of motion it 
 impresses upon a point, and the latter is always proportioned to 
 the velocity of this point. 
 
 6. For the more convenient comparison of forces, we measure 
 them in terms of some conventional force, taken as the unit. Of 
 this we have a practical illustration in the manner in which the 
 forces of steam-engines and water-wheels are compared, in terms 
 of the conventional force called a horse-power. The intensity 
 of forces estimated in terms of some conventional unit, may then 
 be denoted by numbers, expressed algebraically by letters, or 
 represented by lines of definite magnitude. 
 
 7. The circumstances which must be known in respect to a 
 force, besides its intensity, are the place where it nets, or, its 
 point of application, and its direction. The point of application 
 is defined in the mode we have already explained, by referring 
 it by means of co-ordinates, to three rectangular planes. The 
 direction of a force is that in which it tends to cause a point to 
 move : it is usually represented by a straight line drawn in that 
 direction from the point of application ; and if upon this line be 
 set off the number of units from a scale, which corresponds to 
 the measure in terms of the conventional force, used as the 
 means of comparison, the force will be represented by it, both 
 in magnitude and direction. The direction will be defined in 
 respect to the three co-ordinates of the point of application, by 
 means of the three angles which it makes with these three lines. 
 
 In order to give the method all the extension of which it is 
 capable, these angles must be estimated of all magnitudes, from 
 to 180. The co-ordinates, by this method, need not be con- 
 ceived to be produced beyond the point of application ; and when in 
 calculation we employ the angular functions, those which have 
 different algebraic signs in the first and second quadrants, will 
 
Book /.] OP EQUILIBRIUM. 3 
 
 be best suited to express the position in which the line of direc- 
 tion lies. Thus when the cosine is used, and its algebraic sign 
 is positive, the line will lie on the side of the point of applica- 
 tion towards the plane to which the co-ordinate is perpendicular ; 
 and when it is negative, it will be turned towards the opposite 
 direction. 
 
 Between the cosines of the three angles the direction of a force 
 makes with the co-ordinates of its point of application, there is a 
 constant relation which may be thus expressed : 
 cos. 2 a+cos. 2 6+cos. 2 c = l; 
 
 for the line of direction will be the diagonal of a right-angled par- 
 all el opipedon, whose sides are in the direction of the three co- 
 ordinates, and the parts cut off from the latter will respectively 
 represent the cosines of the angles they make with the line of di- 
 rection, the latter being the radius ; now as the square of the 
 diagonal of a right angled parallelepiped is equal to the sum of 
 the squares of its three sides, the square of radius, or unity, is 
 equal to the sum of the squares of the three cosines. 
 
 If the line of direction lies in the same plane with the two co- 
 ordinates, with which it makes the angles a and 6, the angle c be- 
 comes a right angle, and its cosine =0, hence, in this case, 
 cos. 2 o-j- cos. 2 6=0. 
 
 When all the forces that are under consideration are parallel 
 to each other, one of the axes may be so taken as to be parallel 
 to their direction ; two of the angles in this case become right an- 
 gles, and the equation becomes 
 
 cos. 2 a=l. 
 
 8. When more than a single force acts upon a body, it is ob- 
 vious that it will riot move in a direction, or with an intensity 
 that is due to one of the forces alone, but will be influenced by 
 all the forces collectively. Hence, when a number of forces act 
 upon the same body, they respectively modify each other, and 
 may under certain circumstances completely neutralize each other. 
 When forces thus destroy each other's action, equilibrium is said 
 to exist among them, and the body on which they act is said 
 to be in equilibrio, under their joint action. It has been found 
 most easy to deduce the expressions which denote the motion of 
 a body, from those which denote the conditions of equilibrium; 
 hence it becomes necessary that the conditions, under which 
 forces produce equilibrium, should be first investigated. 
 
4 OP EQUILIBRIUM. [Book I. 
 
 CHAPTER II. 
 
 EQUILIBRIUM OP FORCES ACTING IN THE SAME LINE. 
 
 9. THE simplest case of equilibrium is when two forces act in the 
 same line, with equal intensities, but in contrary directions. We 
 represent this contrariety of direction by means of the Alge- 
 braic signs + and . When more than two forces act in the 
 same line, it is obvious that equilibrium can only exist when the 
 joint intensities of those that act in one direction, are exactly 
 equal to the joint intensities of those which act in opposition to 
 them. Expressing this difference of direction, by considering 
 one set of forces as negative in respect to the other, we obtain the 
 algebraic expression 
 
 A+B+C+&c.=0, 
 or in words. 
 
 Equilibrium exists among a number of forces acting in the 
 same straight line when their sum is equal to 0. 
 
 10. When a number of forces acting upon a body are not inequi- 
 librio, we may, without altering the circumstances under which 
 the body is placed, conceive their united action to be replaced by 
 that of a single force, under which the body would move exactly, 
 as if the whole continued to act. A force which thus produces 
 the same effect as a number of others, and may therefore identi- 
 cally replace them, is called their Resultant; the several forces 
 whose action it thus identically replaces are called its Compo- 
 nents. 
 
 11. If a force equal in magnitude to the Resultant, but contrary 
 in direction, be applied to the point at which the latter would 
 act if its components were removed, it will be obviously in ex- 
 act equilibrium with the components; for the case becomes that 
 of two equal for^s acting in the same straight line, but in con- 
 trary directions. Hence if any number of forces be in equilibrio, 
 any one of them must be equal in magnitude, and contrary in di- 
 rection to the resultant of all the rest. If, therefore, we have the 
 relations that exist between the Resultant and its components, 
 we can thence deduce the conditions of equilibrium of any forces 
 whatsoever. 
 
 These relations may for convenience of investigation be divided 
 into three cases, those of : 
 
Book /.] or EQUILIBRIUM. 5 
 
 1. Forces converging to a point ; 
 
 2. Parallel forces ; 
 
 3. Forces acting in one plane, but neither parallel nor con- 
 verging to a single point. 
 
 In considering the latter case, we shall have occasion to speak 
 of the conditions of equilibrium of forces acting in any direction 
 whatsoever, but all our applications can be referred to the case 
 of their being situated in one plane. 
 
OF EQUILIBRIUM. \Book I. 
 
 CHAPTER III. 
 
 EQUILIBRIUM OF FORCES CONVERGING TO A POINT. COMPO- 
 SITION AND RESOLUTION OF FORCES. 
 
 12. The Resultant of the two forces converging to a point, is 
 represented both in magnitude and direction, by the diagonal of 
 a parallelogram constructed on the two forces as sides. 
 
 First : Let the directions of the two forces be at right angles 
 to each other, and call the one X, and the other Y. Let R be 
 the unknown magnitude of the resultant, and a the angle which 
 it makes with the direction of X. If we suppose the two forces 
 to be extremely small and represented by their differentials rfX 
 and dY, and that they vary according to the same law, so that 
 when rfX becomes successively 2dX, 3dX, &c. dY becomes 
 2dY, 3dY, &c. ; it will be evident that the angle a will not vary, 
 and that the resultant will be constant in its direction ; its increase 
 will also follow the same law with its components, and if repre- 
 sented at first by dR, it will become in similar succession 2dR, 
 3dR, &c. Thus in the successive increments of the three forces, 
 
 the ratios between the resultant R and the two components X and 
 
 X 
 
 Y will remain constant. The relation -= being constant, may 
 
 1%- 
 be represented by a function of the constant angle a. This fact 
 may be expressed by the notation 
 
 g=9()- (1) 
 
 But as the angle comprised between Y and R is also constant, 
 the relation between these two quantities may be represented by 
 some function of it ; and this function will obviously be of the 
 same form with that which represents the former relation, or to 
 express it algebraically, the angle being the complement of a 
 
 !=<p(90'_ a). (2) 
 
Book I.] 
 
 OF EQUILIBRIUM. 
 
 In order to render the rest of the investigation more obvious, 
 we must have recourse to the anriexed figure. 
 
 In this, the rectangular forces X and Y are represented in magni- 
 tude and direction with the undetermined resultant R lying be- 
 tween them, for its direction must of necessity be intermediate, 
 and in the same plane with them. Let us now consider the force 
 X as the resultant of two others, the first of which x is in the di- 
 rection of the force R, and the second x' is at right angles to it ; 
 the angle comprised between the directions of X and x is the 
 same with that contained between X and R, or is equal to a, and 
 the angle contained between X and x' is 90 a. The same re- 
 lation will then exist between these three forces, taken by pairs, 
 that exists between X, Y, and R, or 
 
 (3) 
 
 -= 9 (90 a) 
 
 But from the equations (1) and (2) we have, multiplying by R, 
 X=R.<p(), Y=R.9(90 a), (4) 
 
 and from the equations (3) we obtain in like manner 
 
 ar=X.<p(), #'=X.9(90 a), (5) 
 
 substituting the values of 9 (a), and 9(90 a) from equations (1) 
 
 and (2), we obtain 
 
 _ _ 
 
 *~R ' x ~ R ' 
 
 Resolving Y in a similar manner into the two rectangular com- 
 ponents y and ?/', one of which y is in the direction of R, we ob- 
 tain by a similar operation 
 
 XY 
 
OF EQUILIBRIUM. [Book I. 
 
 The force R being the resultant of the two, X and Y, is also the 
 resultant of their four components x, y, x', y', whose values are 
 X 2 Y 2 XY XY 
 
 IT' IT' IT' IT' 
 
 but the two last x' and y', are equal in magnitude, and because 
 they respectively make right angles with R on its opposite sides ; 
 they act in the same line in contrary directions ; hence they mu- 
 tually destroy each other's actions, and the resultant R is made 
 up of the two remaining forces, which act in the same direction 
 with it, or 
 
 whence 
 
 R=v/(X 2 +Y 2 ), (6) 
 
 which is the expression for the magnitude of the diagonal of a 
 right angled parallelogram whose sides are X and Y ; therefore 
 the resultant of two rectangular forces is represented in magni- 
 tude by the diagonal of the parallelogram constructed on the two 
 forces as sides. 
 
 That it is also represented by it in direction will be obvious 
 from a few simple considerations. 
 
 The value of R being thus determined, call the angle which 
 
 the diagonal of the parallelogram makes with the side X, 6, then 
 
 X=R. cos. 6; (7) 
 
 substituting the value of X from the equation (4) and dividing 
 
 by R 
 
 cos. b=cp.(a). 
 
 The unknown function of the angle a may therefore be always 
 represented by the cosine of the known angle 6 ; and if there be 
 any case in which a = 6, the equality must hold good in all others. 
 Now if the forces X and Y be equal, the resultant R must be equi- 
 distant in direction from the directions of the two forces, and the 
 angle a will become the angle which the diagonal of the paral- 
 lelogram makes with the side X, or a=b ; therefore the resultant 
 of two rectangular forces is not only represented in magnitude, 
 but in direction, by the diagonal of the parallelogram constructed 
 upon the two forces as sides. 
 
 Next suppose that the two forces, X and Y, are not rectangu- 
 lar, but make with each other any other angle a. Resolve X in 
 two other forces, one of which, #, is in the direction of Y, the 
 other x perpendicular thereto ; the resultant will therefore be the 
 resultant of the three forces Y, x, and x', but as Y and x are in 
 the same direction, they have a resultant which is equal to their 
 sum, and R becomes the resultant of two rectangular forces, 
 whose magnitudes are respectively Y+#, and #', and from (6) 
 
Book /.] 
 
 OF EQUILIBRIUM. 
 
 or 
 
 but 
 
 (8) 
 
 (9) 
 
 JL 2 =3?+x' 2 , and a?=X cos. a, 
 substituting these values in the equation, (8), 
 
 R 2 =X 2 +2XY cos. a+Y 2 , 
 which shows that the resultant R is represented in magnitude by 
 the diagonal of the parallelogram constructed upon the forces X 
 and Y as sides. Also is it represented in direction, for the diago- 
 nal of the parallelogram constructed on Y-f-# and x' as sides, is 
 identical not only in magnitude, but in position with the diagonal 
 of the parallelogram constructed on X and Y. 
 
 The angle, </, may vary between and 180 ; when it ex- 
 ceeds a right angle, the quantity, x, becomes negative in respect 
 to Y ; the quantity, cos. a, also becomes negative, and the second 
 term of the equation is negative. 
 
 In order to render this part of the investigation more clear, we 
 annex a construction, in the two cases of obtuse and acute angles, 
 contained between the two forces. 
 
 13. The problems that have relation to finding the resultant of 
 two forces, when the components are given, and to finding the 
 components when the resultant is given, go by the name of 
 " The Composition and Resolution of Forces." All the questions 
 in which forces are to be composed or resolved, may be solved by 
 means of plane trigonometry ; and in general, of two forces, their 
 resultant, and the three angles they respectively make with each 
 other, any two being given, the remainder may be found. 
 
 (1). When the two components and the angle they contain are 
 given, we have from the equation, (9), 
 
 R= ^ (X a +2XY cos. a+Y 3 ) ; (10) 
 
 2 
 
10 Or EQUILIBRIUM. [Book L 
 
 when the angle, a, is a right angle, 
 
 X=R cos. a, Y=R sin. a. 
 (2). In the figure beneath, the force X, the resultant R, and a 
 line equal and parallel to Y, make up a triangle, of which the 
 angles are : the supplement of a ; the angle 6, equal to the angle 
 contained by the force Y and the resultant R ; and the angle c, 
 contained by the resultant and the force X ; hence as the sides of 
 triangles are proportioned to the sines of the opposite angles, 
 
 R : X : Y : : sin. a : sin. 6 : sin. c, (12) 
 
 and if the directions of X and Y are rectangular, 
 
 R=-^ =-Z-r, (is) 
 
 cos. c cos. b 
 X=R cos. c, Y=R cos. 6. (14) 
 
 14. Three forces converging to a point are in equilibrio when 
 each is proportioned to the sine of the angle contained by the di- 
 rections of the other two ; they are also in equilibrio when rep- 
 resented by the three sides of a plane triangle ; and hence the di- 
 rections of the three forces must lie in the same plane, and the 
 sum of any two must be greater than the third. 
 
Book /.] OF EQUILIBRIUM. 11 
 
 Let the three forces be X, Y and Z, (by 10), the force Z 
 
 will be equal to the resultant of X and Y and contrary in direc- 
 tion, hence the relation between their magnitudes may be ex- 
 pressed, by substituting Z forR in analogy (12), as follows: 
 
 Z : X : Y : : sin. a : sin. 6 : sin. c ; (15) 
 
 but the angle a, is the angle contained by the forces X and Y ; the 
 angle 6, is the supplement of the angle b' contained by the direc- 
 tions of Z and Y ; and the angle c is the supplement of the angle 
 c contained between the directions of Z and X ; and as the sines 
 of angles and their supplements have the same magnitude, 
 
 Z : X : Y : : sin. a : sin. 6' : sin. c ; (16) 
 
 or the forces are each proportioned to the sines of the angles con- 
 tained by the directions of the other* two. As the forces X and 
 Y, with their resultant R, are represented in magnitude by the 
 three sides of a triangle, so, as Z=R, 
 the three forces X, Y and Z are also rep- 
 resented in magnitude by the sides of a 
 triangle. This triangle may be formed by 
 drawing, through the extremity of the line 
 representing one of the forces X, a line 
 equal and parallel to Y, and joining the 
 ends of the last line to the point at which 
 the forces act. This last line is obvious- 
 ly equal to Z, and in the same direction 
 produced, for it is the diagonal of the par- 
 \_. allelogram, of which X and Y are sides. 
 
12 
 
 OP EQUILIBRIUM. 
 
 [Book I. 
 
 A triangle whose sides are proportioned to the forces, and per- 
 pendicular to their direction, may be formed as in the figure be- 
 neath, by drawing perpendiculars from points taken at will in the 
 
 Z 
 
 directions of the forces. It will be manifest that the three angles 
 a', &', c', of this triangle are respectively the supplements of the 
 three angles a, 6, c, that the direction, of the forces make with 
 each other ; hence the triangle thus constructed will be similar to 
 that formed in the former construction, whose sides represent 
 the magnitude of the three forces. 
 
 Three oblique forces cannot be in equilibrio, unless their di- 
 rections converge to a single point, for the resultant of any two 
 of them must be equal and opposite to the direction of the third ; 
 and hence its direction must pass through the point to which the 
 directions of the others converge. 
 
 As the resultant of any two of the forces lies in the same plane 
 with them, being the diagonal of a parallelogram of which they 
 are the sides, the third force, which is in the direction of this re- 
 sultant produced, must also lie in the same plane. The three 
 forces that are in equilibrio being represented in magnitude by 
 the three sides of a triangle, the sum of any two of which must be 
 greater than the third, the same must be true of the sum of the 
 magnitudes of any two of the forces. 
 
 15. When we have it in our power to find the resultant of any 
 two forces, we may proceed to find the resultant of three or 
 more ; for as the resultant identically replaces its components, 
 we may, after finding the resultant of any two of the forces, con- 
 ceive them to be removed, and the resultant substituted ; the re- 
 sultant of this and the third force, will be the resultant of the three 
 forces ; and this last resultant may be again combined with a 
 fourth force, and so on. 
 
 This problem may be illustrated by a remarkable geometric 
 construction. 
 
Book /.] 
 
 OP EQUILIBRIUM. 
 
 13 
 
 Let AB, AC, AD, AF, AG, represent a number offerees con- 
 curring to a point ; through the point B, draw the line BH, equal, 
 and parallel to AC ; through the point H, draw HI, equal and 
 parallel to AD ; through the point I, draw IK, equal and parallel 
 to AF ; and through the point J(, draw KL, equal and parallel to 
 AG ; then the line which joins L to A ; will be the common re- 
 sultant. 
 
 It is obvious that the line AH is the resultant of AB and AC ; 
 AI the resultant of AH and AD, or of AB, AC, and AD ; the 
 line AK, of AI and AF, or of AB, AC, AD and AF ; and the line 
 AL, of AI and AG, or of all the forces AB, AC, AD, AF, and 
 AG. 
 
 This construction is called the polygon offerees. If the poly- 
 gon close, or the last line, drawn parallel to the last force, end at 
 the point A, the forces are in equilibrio, and the resultant is equal 
 toO. 
 
 16. The resultant of three rectangular forces, converging to a 
 point, is represented in magnitude and in direction, by the diago- 
 nal of the rectangular parallelepiped, whose sides represent the 
 three forces in magnitude and direction. 
 
 The resultant R', of two of the forces X and Y, will be the diago- 
 nal of the rectangle which forms one of the faces of the paral- 
 lelepiped, of which these forces are sides, and 
 
14 OP EQUILIBRIUM. Book /.] 
 
 R'=x/(X 2 +Y 2 );' 
 
 the resultant R, of R', and Z will be the diagonal of the rectangle 
 of which these two forces are sides, and which will cut the paral- 
 lelepiped into two equal and similar prisms ; it will therefore be 
 the diagonal of the parallelepiped, and will be represented by the 
 formula 
 
 R= V (R' 2 +Z 2 ) = v/ (X 3 +Y 2 +Z 3 ) . (17) 
 
 If we call the angle the direction of R makes with X, a ; the 
 angle the direction of R makes with Y, b ; the angle the direc- 
 tion of R makes with Z, c ; these angles are, as has been 
 shown, connected by the following relation, 
 
 cos. 2 o-f-cos. 2 6-f-cos. 2 c=0. 
 
 We may find the values of X, Y and Z, when the force R, and 
 the angles a, 6, and c, are given, by the resolution of three plane 
 triangles, of each of which the hypothenuse and an angle are 
 given, thus : 
 
 X=Rcos. a, Y=Rcos. 6, Z=Rcos. c. (18) 
 
 17. In those investigations in mechanics where a number of 
 forces are concerned, it is usual to resolve them all into three forces 
 parallel to the three co-ordinates ; and the resultant of these three 
 sets of rectangular forces will obviously be the common resultant 
 of all the forces. The formulae given above (18) furnish a con- 
 venient mode of effecting this. 
 
 Call the several forces F, F', F", &c. the angles they respec- 
 tively make with the co-ordinates of their points of application a, 
 6, c, &c., a', b', c', &c., a", 6", c", &c. the values of X, Y and 
 Z, in (18) become 
 
 F cos. a, F cos. 6, F cos. c, &c. 
 
 and calling the three rectangular forces, which are the sum of the 
 components of all the original forces X, Y and Z, 
 
 X=F cos. a-fF' cos. 6+F" cos. c-h&c. ' '*"' 
 
 Y=F cos. a'+F' cos. &'+F" cos. c'-f &c. \ (19) 
 
 Z=F cos. a"+F' cos. 6"+F" cos. c"+&c. j 
 
 In these expressions, X, Y, and Z, are the components of the 
 resultant of all the forces resolved into three, mutually at right an- 
 gles to each other. The formulae show that the resultant of any 
 number of forces, resolved into a component in any given direc- 
 tion, is equal to the sum of the components of all the forces, re- 
 solved into directions parallel to the general resultant. 
 
 IS. The three forces, X, Y, and Z, are not situated in the same 
 plane, and hence can never be in equilibrio, so long as any one 
 of them has any magnitude ; for three oblique forces, in order to 
 be in equilibrio, must have their directions in one plane ; hence 
 the condition of equilibrium among any number of forces, each 
 
Book /.] O* EQUILIBRIUM. 15 
 
 resolved into three at right angles to each other, and parallel to 
 the axes of the co-ordinates, becomes 
 
 F cos. a+F' cos. 6-fF" cos. c+&c.=0 ~'<\\ 
 
 F cos. a'+F' cos. 6'+F" cos. c'+&c.=0 V (20) 
 
 F cos. a"-fF' cos. 6"+F" cos. c"+&c.-=0 j V ' 
 
 19. There is another case in which forces may be in equilibrio, 
 when they converge to a point ; this happens when they press 
 the point against a surface, which opposes a resistance to the mo- 
 tion of the point sufficient to prevent its penetrating under the 
 action of the forces. This resistance will be exerted in a direc- 
 tion which is perpendicular, or a normal, to the surface; for were 
 it exerted in an inclined direction, it might be resolved into two 
 components, one of which is parallel, the other perpendicular to 
 the surface ; the former would cause the point to move along the 
 surface, while the latter alone would oppose the progress of the 
 point ; now as mere resistance can never generate motion, how- 
 ever it may in other respects affect it, the resistance of the sur- 
 face must be exerted in the direction of the normal alone. 
 
 In order then that the point be in equilibrio, it is no longer ne- 
 cessary that the resultant of the forces shall be equal to 0, but 
 merely that its direction shall be a normal to the surface ; this it 
 must be, otherwise it would not be opposite in direction to the 
 resistance of the surface, and therefore would not be in equilibrio 
 with it. The action of the surface, being just sufficient to keep 
 the point in equilibrio, may be represented by a force equal to 
 the resultant of all the other forces, but acting in a contrary direc- 
 tion, or by R ; and the condition of equilibrium, they being re- 
 solved into three, parallel to their co-ordinates, will be (17) 
 
 V(X 2 +Y 2 +Z 2 ) R=0, (21) 
 
 while the pressure they exert upon the surface will be 
 
 (22) 
 
16 OF EQUILIBRIUM. [Book L 
 
 CHAPTER IV. 
 
 IV. EQUILIBRIUM op PARALLEL FORCES. CENTRE OF PAR- 
 ALLEL FORCES. 
 
 20. The conditions of equilibrium among parallel forces, may 
 be reached by steps similar to those employed in determining the 
 condition of equilibrium among converging forces. The method 
 of finding the resultant of two of them must be first investigated ; a 
 force equal and opposite to this, will be in equilibrio with the two 
 first. We may then proceed to the resultant of three, four, or 
 any number of forces; and a force equal, and opposite to it, will 
 cause equilibrium in the system, and the relations that exist among 
 them will show the condition of equilibrium. 
 
 Parallel forces cannot, act upon a single point; we therefore 
 suppose the points on which they act, to be connected in some 
 manner, and first, by an inflexible and inextensible line, which is 
 called their line of application. 
 
 There is one case in which parallel forces have no resultant. 
 This happens when there are but two that act on opposite sides 
 of the line of application, and are equal in magnitude; or when 
 the resultant of all those that act on one side of that line, is ex- 
 actly equal, but not directly opposite to the resultant of all those 
 that act on the other side. Two forces thus constituted, are called 
 a Couple. It will be obvious that their effect would be to cause 
 the line of application to revolve, and were it free, to move 
 around; the two forces would finally act in the same line, and 
 in opposite directions; they would then cease to be parallel. Two 
 such forces then, so long as they continue parallel, have no result* 
 ant, neither can they be in equilibrio. 
 
 21. In investigating the method of finding the resultant of par- 
 allel forces, it will be seen that we shall consider the point of ap- 
 plication of a force to be changed. A force will obviously 
 produce the same effect, whether it act at one or another point of 
 the same inflexible straight line. Of this we have a physical illus- 
 tration in the fact, that when a simple pressure is exerted through 
 the intervention of a rigid bar, it is wholly unimportant whether 
 the bar be long or short, provided its weight have no influence. 
 We may therefore conceive the point of application of a force to 
 be transferred to any other point in its direction, provided we 
 imagine the two points to be connected by an inflexible straight 
 line. 
 
 22. The resultant of two parallel forces is equal to their sum, 
 
Book!.'] OF EQUILIBRIUM. 17 
 
 parallel to their directions, and divides their line of application 
 into parts, reciprocally proportioned to the magnitude of the two 
 forces. 
 
 Suppose first, that, as in the figure, two converging forces act 
 upon the same side of the line, call them A and B ; their action 
 
 will not be changed by supposing their respective points of appli- 
 cation to be transferred to the point C, in which their directions 
 meet; and the direction of their resultant must therefore pass 
 through this point. From the point of convergence, then, its 
 point of application may be transferred to the point d, in which its 
 direction cuts the line of application, and this will therefore be the 
 point of application of the resultant. Call the angle at which the 
 forces are inclined to each other p, the value of the resultant from 
 (9) is 
 
 R= </ A 2 + AB cos. g +B 2 ) ; (23) 
 
 the relation of the forces A and B is from (12) 
 
 A : B : : sin. /3 : sin. a, (24) 
 
 /3 being the angle the direction of B makes with that of R, and 
 a the angle that the direction of A makes with that of R ; but the 
 ratio of the sines will be the same as that of the perpendiculars a 
 and 6, let fall upon the respective directions of the forces, from 
 the point of application of the resultant : hence 
 
 A : B : : b : a. (25) 
 
 We shall have occasion hereafter to recur to this step. 
 
 Now the equation (23), and the analogy (25) being true, what- 
 ever be the magnitude of the several angles the forces and t eir 
 resultant make with each other, will be true when the lines are 
 parallel. But when the lines are parallel, the angle =180 or 
 = and 
 
 cos. =1 ; 
 3 
 
IS 
 
 OF EQUILIBBIUM. 
 
 [Book I. 
 
 (26) 
 
 hence the equation (28) becomes 
 
 R=V(A 2 +AB+B 2 )and 
 R=A-r-B 
 
 or the resultant is equal to the sum of the forces. 
 
 The point of convergence being removed to an infinite distance, 
 the direction of the resultant becomes parallel to the directions 
 of the two forces ; and 
 
 the two perpendiculars a and b become proportioned to the parts 
 c and d into which the point of application of the resultant divides 
 the line of application of the forces, which is hence divided into 
 parts inversely proportioned to the two forces. 
 
 If the two parallel forces act on opposite sides of their line of 
 application, the same is also tnie ; but in this case, the opposition 
 of their direction is pointed out, by one of the forces being consi- 
 dered negative in respect to the other ; hence their algebraic sum 
 becomes their arithmetic difference. The point of application of 
 the resultant, will fall in the prolongation of the line of application, 
 beyond the point to which the greater force is applied ; and the 
 parts into which the line of application is divided, will be mea- 
 sured from the point of application of the resultant to those of the 
 two components. 
 
 To make this evident, suppose that to a line on which the two 
 forces, A and B are applied, a third force, C, is also applied equal 
 to the resultant, and opposite in direction; this force will be 
 negative in respect to the others, and will keep the system in equi- 
 librio ; the conditions of which will be thus expressed : 
 
 A+B+C=0, 
 
 (27) 
 
 any one of these forces then, is in equilibrio with the other two ; 
 and the resultant of these two would be equal to it, and opposite 
 
/.] 05 EQUILIBRIUM. 19 
 
 in direction. Let the two forces, whose resultant is required, be 
 B and C ; the resultant being equal and opposite to A, will have 
 a value, which, considering that one of the two forces B and C, 
 is negative in respect to the other, may be thus expressed : 
 
 and as the point of application of C divides the line to which A 
 and B are applied, into parts inversely proportioned to the two 
 forces, we have 
 
 A : B : : c : d, , 
 and 
 
 A+B :B: : c+d : d ; 
 but 
 
 A+B=C, 
 
 and c-\-d is the whole length of the line of application, measured 
 from the point to which A, or its equal and opposite force, R is 
 applied ; hence this line is again divided into parts inversely pro- 
 portioned to the magnitudes of the two forces. 
 
 23. The resultant of any number of parallel forces, acting; upon 
 points invariably connected with each other, may be found upon 
 the same principle by which ibe resultant of any number of 
 converging forces is found. Any two of them may be first re- 
 solved into a single one ; this may be combined with a third, and 
 the resultant found ; the second resultant may be combined with 
 a fourth, and so on. In this way it will be found, that the result- 
 ant of any number of parallel forces is equal to their sum. 
 
 Call the forces A, B, C, D, &c. 
 the first resultant R' being equal to the sum of A and B, 
 R'=A+B, 
 
 the value of the second resultant is 
 
 R'=R'-fC=A+B-fC, 
 
 and of the last resultant 
 
 R=A+B+C+D+&c. (28) 
 
 In this expression, as in the others, the forces that act in oppo- 
 
 site directions, must be considered as positive and negative in re- 
 
 spect to each other. The condition of equilibrium is obviously 
 
 A-{-B + C + &c.=:0. (29) 
 
 24. A change in the direction of two parallel forces, provided 
 they tlo not cease to be parallel, produces no change in the posi- 
 tion of the point to which their resultant is applied ; but the re- 
 sultant itself will have its direction changed, continuing always 
 parallel to its component. And in the several steps by which the 
 resultant of a number of parallel forces is found, it is obvious that 
 
20 
 
 OP EQUILIBRIUM. 
 
 [Book I. 
 
 the position of the several successive, and finally, of the last result- 
 ant, remains constant, however the forces vary in direction, pro- 
 vided they do not cease to be parallel. Hence, in a system of 
 parallel forces, if the/several forces revolve around their respec- 
 tive points of application without ceasing to be parallel, the re- 
 sultant will also revolve around its point of application, always 
 retaining its parallelism to its components. From this property, 
 the point of application of the resultant of a system of parallel for- 
 ces, is called their Centre. 
 
 25. It is important in many practical cases, to be able to deter- 
 mine the position of the centre of a given system of parallel forces, 
 applied to points forming an invariable, or rigid system. This is 
 effected by finding the value of its co-ordinates, in terms of the 
 several forces, and their respective co-ordinates. 
 
 Call the several forces A, B, C, &c., their respective co-ordi- 
 nates a, a', a'', 6, 6', 6'', c, c', c" &c., the resultant R, and its co- 
 ordinates X, Y, and Z. 
 
 In the following figure, let A and B represent the points of 
 
 application of two of the forces A and B, and R' the point of 
 application of their resultant R. Let O be the origin of the 
 co-ordinates O P, O N, O S, and a, 6, and x the co-ordinates 
 of the two forces, and their resultant, in respect to the plane of 
 P and O N. The line that unites the points in which the co-or- 
 dinates cut this plane, is a straight line, for the three forces are in 
 the same plane. Then as the line A B is divided by the result- 
 ant into parts inversely proportioned to A and B 
 
 A : B : : R'A : R'B ; 
 
 and 
 
 A-f B : B : : AB : R'B, 
 
 or 
 
 R': B : : AB : R'B. 
 
 (30) 
 
Book I.'] OP EQUILIBRIUM. 21 
 
 Draw a straight line through the point B, parallel to the line that 
 joins the points in which a, 6, and x cut the plane of O P and O N. 
 The respective distances of the points A and R from this line 
 will be a 6, and x 6. 
 
 From the similarity of triangles 
 
 AB : R'B : : a b : x b ; 
 comparing this with the analogy (30) 
 
 R': B : : a 6 : x b ; 
 whence 
 
 R'(# 6)=B(a 6) ; 
 
 now the resultant R' is equal to the sum of the two forces A and 
 B, and multiplying these equals by b we have 
 
 R=A6+B6 ; 
 adding the two last equations 
 
 R'x=Aa-\-Eb. (31) 
 
 A similar chain of reasoning, in respect to each of the other two 
 sets of co-ordinates, gives 
 
 (32) 
 K^=Aa'+i56" f : 
 
 by division 
 
 * = AoJBZ 
 
 (33) 
 
 K' 
 
 Let now C be the point of application of the third force C ; R" 
 the point of application of the resultant of the two forces C and R 7 , 
 of which the latter is the resultant of A and B. A similar inves- 
 tigation gives for the values of the co-ordinates x y' z of the 
 force R" 
 
 Rz+Cc" 
 R+C ; 
 
 substituting the values of R'.r, ~R'y and R'z, 
 
 ,_Aa+B6+Cc 
 
 = A+ B-f- C 
 
 ' A +B + C 
 Aa"+B6"+Cc' 
 A -f B -f C 
 
OF EQUILIBRIUM. 
 
 [Book I. 
 
 It is now obvious that the same method may be extended to 
 any number of forces whatsoever, and that we should finally ob- 
 tain for the values of the co-ordinates of any number of forces 
 the following equations : 
 
 X ~ 
 
 A -I- B+C+&C. 
 Ao'+By+C / +&c. 
 A + B+C + &C. 
 Aa"+Bfe"+C c "+&c. 
 A +B+C +&c. 
 In the case of equilibrium these equations become 
 
 =0 
 
 (34) 
 
 A H-B +C H-&C. 
 Aa'+B6'+Cc+&c. 
 A + B + C +&c. 
 
 =0 
 
 Aa"+Bfe"-r-Cc"+&c 
 A + B + C +& c . 
 
 The expressions (34) may be made to assume the following 
 form, which is more convenient in its application to the cases that 
 may occur in practice : 
 
 ~ 
 
 Y=: 
 
 F 
 
 Fy 
 
 2. F 
 
 (35) 
 
 The numerator of the three fractions being the sum of the pro- 
 ducts of the respective forces into their distances from the several 
 planes, and 2 . F being the sum of the forces themselves. 
 
 26. The most useful applications of these formulae, are to the 
 determination of the centres of parallel forces in lines, surfaces, 
 and solids. In these cases, the several magnitudes are sup- 
 posed to be divided into an infinite number of small parts or ele- 
 ments, each of which is acted upon by an equal parallel force. 
 The manner in which the foregoing formulae are transformed 
 into others adapted to this research, together with a few of their 
 applications will now be given. It will however be obvious, that 
 these formulae, having reference to the small elements of which 
 the magnitudes are made up, must be differential ; and that the 
 discovery of the position of the centre of parallel forces, can only 
 be effected by means of the integral calculus. 
 
Book /.] OP EQUILIBRIUM. 23 
 
 In the case of irregular solids and lines of double curvature,, it 
 is necessary to refer the position of their elements to three rec- 
 tangular planes ; and hence the three equations for the co-ordinates 
 of the centre of parallel forces, are necessary. In the case of sur- 
 faces and plane curves, no more than two of these equations are 
 necessary ; for the surface itself, or the plane in which the line lies, 
 may be taken as the plane passing through two of the axes, and it 
 will only be necessary to define the position in respect to these 
 two axes. 
 
 In lines that are symmetric on each side of one of their points, in 
 surfaces that are symmetric on eachsideof aline thattraverses them, 
 and in solids of revolution, but one of the equations is necessary ; 
 for one of the axes may in the first instance be supposed to pass 
 through the point on each side of which the line is symmetric, and to 
 be perpendicular to the line at that point ; in the second it may la 
 assumed as coinciding with the line that divides the surface into 
 two equal parts ; and in the third, it maybe taken as coinciding with 
 the axis, by a revolution around which the solid is described. In 
 all these cases, the centre of parallel forces will be situated some- 
 where in the common intersection of the two planes, its distance 
 from which is therefore =0, or if these be the planes on which Y 
 and Z fall, 
 
 Y=0, Z=0. 
 
 Restricting ourselves to cases that admit of the use of no more 
 than one equation (1). In symmetric curves, the equation (35) 
 
 2 .Fx 
 X ~2TF 
 becomes 
 
 in which I is the length of the line, and x the variable ordinate of 
 the element d/, in respect to the assumed plane. 
 
 (2) In plane surfaces the equation becomes 
 
 fxydx (37) 
 
 X -~T~ , 
 
 where s is the area of the surface, whose element is ydx. 
 
 (3) In solids of revolution, the formula for the position of their 
 centre of parallel forces becomes, if being the ratio of the circum- 
 ference of a circle to its diameter, 
 
 _J xfdx (38) 
 
24 oy EQUILIBRIUM. [Book I. 
 
 EXAMPLES. 
 
 (1) To find the centre of Parallel Forces of a circular arc. 
 Let B A D be the circular arc, whose length =1. 
 
 A. 
 
 Place the origin of the co-ordinates at the centre ; make the ra- 
 dius C A=r, CP=#, mP=7/, Am=s ; 
 the differential equation of the arc s is 
 
 ds 
 and 
 
 whence 
 
 x=r. cos.- 
 
 fxds=fxV(dx*+dy 2 }=r. 2 sin. + C. 
 
 When the arc Am becomes equal to AD 
 
 =-, and the arbitrary constant C =0 ; 
 
 I 
 
 and from the equation (36) 
 therefore 
 
 /X=2r 2 sin. ^;, 
 Let c represent the chord of the arc, then 
 
 c=2r. sin. 7^ ; 
 whence 
 
 and 
 
 ZX=rc, 
 
 wherefore the distance of the centre of gravity of a circular arc, 
 from the centre of the circle, is a fourth proportional to the length 
 of the arc, the chord, and the radius. 
 
 By analogous processes, the centres of parallel forces may be 
 found in other curves, that are symmetric on each side of the point 
 in which the axis cuts them, thus : 
 
Book /.] OP EQUILIBRIUM. 25 
 
 The centre of gravity of an arc of a cycloid, that is divided into 
 two equal parts by the diameter of the generating circle, is at one 
 third of the perpendicular height of the arc, from the vertex. 
 
 In a semicircle c=2r, and /=<TTT hence 
 err : 2r : : r : X ; 
 
 (2) To find the centre of parallel forces in a segment of a circle. 
 
 In the same figure let the radius CA^r and let the part of 
 AC intercepted between the centre and the chord BD a; let the 
 centre of the circle again be the origin of the co-ordinates ; the 
 equation of the circle will be 
 
 whence y= \/ (r 2 x 2 } ; 
 
 the expression (37) gives 
 
 X=2/V(a* y z }xdx. 
 Integrating between the values x=a, and v # 
 
 and the chord c = 2 V (r 2 a 2 ) 
 
 whence 
 
 Applying analogous methods, we obtain the following results : 
 A triangle has its centre of parallel forces, in the line drawn from 
 
 its vertex to the point, that bisects its base, at two thirds of its 
 
 length from the vertex. 
 
 In a trapezium, two of whose sides are parallel, call these 
 
 two sides c and d, and the straight line which bisects both, a, and 
 
 let the origin of the co-ordinates be in the point where this line 
 
 cuts the side c ; 
 
 a c+3d 
 
 A sector of a circle, has its centre of parallel forces in the radius 
 that divides it into two equal parts ; and its distance from the centre 
 of the circle is a fourth proportional to the arc, the chord, and two 
 thirds of the radius. 
 
 The distance of the centre of parallel forces of a parabola, from 
 its vertex, is equal to three fifths of its axis. 
 
 The centre of parallel forces in a cycloid, is in the diameter of 
 the generating circle, at the distance of one fourth of that line from 
 the vertex. 
 
 In the surfaces that bound solids of revolution, if being 
 the ratio of the circumference of a circle to its diameter, the 
 element of the surface is 2ydx, hence the equation (38) becomes 
 
26 OP EQUILIBRIUM. \BoOk I. 
 
 ~ f2<rfydx s 
 
 which being identical with (37,) shows, that the distance of the 
 centre of parallel forces, in the surface formed by the revolution 
 of a plane curve around an axis, is as far from the origin of the 
 co-ordinates, as the centre of parallel forces of the curve by whose 
 revolution it is generated. 
 
 (3) To find the centre of parallel forces, of a solid generated 
 by the revolution of an arc of an ellipse, or of the segment of a 
 spheroid. 
 
 The formula (38) is 
 
 fsnfdx 
 * 
 the equation of the curve is 
 
 a being the fixed axis, and c the revolving axis of the spheroid ; 
 whence 
 
 f(ax x 2 }xdx 
 K ~ f(ax x*)dx 
 
 Integrating, and taking the vertex for the origin of the co-ordi- 
 nates 
 
 i aa? i x 4 _4o 3# 
 X = 2 x* = 60 4x X ' 
 
 When the segment is a hemispheroid tf=a, and 2# a, which 
 being substituted for a, 
 
 5 
 
 X= 8* ; 
 
 its centre of parallel forces is therefore in the fixed axis, at a dis- 
 tance of |ths of its length from the vertex. 
 
 These expressions being independent of the value of c, are true 
 also of spheric segments. 
 
 In a hyperboloid of revolution 
 
 In a solid paraboloid, the centre of parallel forces is at the dis- 
 tance of two-thirds of the length of the axis, from the vertex. 
 
 Geometric methods also may be applied to the discovery of the 
 position of the centre of parallel forces. Thus in the case of a tri- 
 angle, it may be shown geometrically, as it has been analytically, 
 to be in the line that joins the vertex to the point which bisects 
 the base, at the distance of two thirds of its length from the vertex. 
 
Book /.] OP EQUILIBRIUM. 27 
 
 In the triangle ABC, bisect the base BC in E, the side AB in 
 D, draw AE, CD, and join DE. 
 
 The surface being divided into two equal parts by the line AE, 
 the centre of parallel forces will lie in this line ; for the same rea- 
 son it lies in the line CD, and must therefore be in the point g 
 where they intersect each other. 
 By the similarity of triangles, 
 
 gD : gC : : g-E : gA, 
 g-E : gA. : : DE : AC, 
 DE : AC : : AB : DB, 
 g-E : gA : : AB . DB ; 
 but because the line AB is bisected in D 
 AB : DB : : 1 : 2, 
 g-E : gA : : 1 : 2, 
 AE : gA : : 3 : 2, 
 therefore 
 
 g-A=|AE. 
 
 In a triangular pyramid the centre of parallel forces is in the 
 line that joins the vertex to the centre of parallel forces of the base, 
 at the distance of three fourths of that line from the vertex. 
 
OP EQUILIBRIUM. {Book L 
 
 Let ABCD be a triangular pyramid, to the point H that bisects 
 
 .A. 
 
 the line DC, draw AH, BH; make HE=i AH, HF=iBH; 
 
 E and F will be respectively the centres of parallel forces of the 
 surfaces ADC, DBC ; join EF, AF, BE. If the pyramid be re- 
 solved into elements, by means of planes parallel to DCB, it is 
 manifest that the line AF must pass through the centres of paral- 
 lel forces of all the elements ; their common centre of parallel 
 forces must therefore be in that line ; for the same reason it is the 
 line BF|: it is therefore at their common intersection, in the point g. 
 By similar triangles 
 
 Ag : gF : : AB : FE : : AH : EH, 
 AE : EH : : 3 : 1, 
 AH : EH : : 4 : 1, 
 
 Ag : gF : : 3 : 1, 
 
 Ag : AF: :3:4; 
 whence Ag-=f AF. 
 
 The first of these propositions may be applied geometrically to 
 find the centre of parallel forces of any polygon whatsoever ; for 
 it may be divided into a number of triangles, equal to the number 
 of sides less two. Let the centres of parallel forces be found in 
 two of the triangles, join them by a straight line, and divide it into 
 parts inversely proportioned to the areas of the two triangles, the 
 point of division is evidently their common centre of parallel 
 forces. If this point be joined to the centre of parallel forces of 
 the third triangle, and divided in a similar manner, the point of di- 
 vision becomes the common centre of parallel forces of the three 
 triangles. This method being continued, until all the triangles 
 into which the polygon has been divided have been used, the point 
 last obtained is the centre of parallel forces of the whole surface. 
 A solid bounded by plane surfaces may be divided by planes, 
 into a number of triangular pyramids. The second of these pro- 
 positions therefore gives us a geometric method, founded on the 
 
Book /.] 
 
 OP EQUILIBRIUM, 
 
 same principles, of finding the centre of parallel forces in any so- 
 lid whose boundaries are plane surfaces. 
 
 27. When the system of points on which a set of forces, whe- 
 ther parallel or not, acts, are not so connected as to form a rigid 
 and invariable system, the conditions of equilibrium are differ- 
 ent. They may be investigated in a general manner, but the 
 applications being limited in practice to a few cases, and the in- 
 vestigation difficult, we confine ourselves to the cases : of poly- 
 gons formed of invariable lines, whose angles are capable of 
 changing their magnitudes under the action of the forces that are 
 applied, until the system reaches a state of equilibrium, and when 
 the forces are parallel among themselves ; and of curves, formed 
 by the action of an infinite number of forces, upon the points of a 
 flexible but inextensible line. 
 
 28. When a system of parallel forces acts upon a system of in- 
 flexible and inextensible lines, forced to move around their con- 
 necting points, the sum of the forces will be equal to ; the forces 
 will be all in one plane, in which the lines at whose angles they 
 act are situated ; and the forces will be to each other respectively, 
 as the sum of the cotangents of the angles their directions make 
 with the lines, at whose point of concourse they themselves act. 
 
 Let a force F act at the angle A made by two inflexible lines 
 
 AT, AT', which angle is variable under the action of the forces ; 
 the system will be in equilibrio, when the lines are drawn, each in 
 its respective direction, by forces T and T' that are such as 
 would be in equilibrio with F when acting at the point A. This 
 is evident, because we can conceive the point of application of a 
 force removed to any point in its direction, without changing its 
 action, provided the points be, as in the case before us, connected 
 by inflexible and inextensible lines. 
 
 Let a and a' be the angles the direction of the force F makes 
 with T and T'; their sum is the angle these two forces make with 
 each other, hence by (16) 
 
30 
 
 OF EQUILIBRIUM. 
 
 [Book I. 
 
 F : T : T' : : sin. a+a' : sin. a! : sin. a. 
 
 (39) 
 
 As three forces in equilibrio around a point must be in one plane, 
 the direction of the lines AT, AT', will be in the same plane with 
 the direction of F. 
 
 It will be obvious that the more obtuse the angle, made by the 
 directions of the lines becomes, the greater will be the tension they 
 have to support under the action of the force F. If the direction 
 of the force F bisect the angle made by the two lines, each of them 
 will bear an equal tension. 
 
 Let us now suppose that, as in the figure beneath, a number of 
 
 parallel forces, F, F', F", F"', act upon a system of rigid and in- 
 flexible lines in such a manner as to cause equilibrium ; resolve 
 the force F into two others, T and X, in the direction of the two 
 lines at whose angles it acts ; the second force into two others, X' 
 and Y, in the direction of the two lines at whose angle it acts ; 
 resolve in like manner, F" into two Y' and Z ; and F'" into Z' and 
 T'. These forces will have from (16) the following relations : 
 
 F : T : X : : sin. (+') : sin. a' : sin. a "1 
 F' : X' : Y : : sin. (6+6') : sin. 6' : sin. b I . , m 
 F" : Y' : Z : : sin. (c+c') : sin. c' : sin. c ( 
 F'" :Z':T':: sin. (d+d') : sin. d' : sin. d J . 
 But in order that the polygon, formed by the lines at whose angles 
 the forces act, shall be in equilibrio, the forces by which each of 
 the lines is drawn at its opposite ends, must also be in equilibrio, 
 or 
 
 X=X', Y=Y', Z=Z'. 
 
 The several values of these forces, obtained from the analogies, 
 are therefore equal by pairs, or 
 
 F sin. a F' sin. 6' 
 
 sin. (aa') 
 F' sin. 6 
 
 sin. (6-RO : 
 F'"sin. c 
 
 sin. c+c 
 
 "sin. (6+6') 
 F" sin, c' 
 
 = sin. (c-K)' 
 F sin, d' 
 
 : sin. (d+d f ) 
 
 (41) 
 
Book /.] OF EQUILIBRIUM. 31 
 
 but when the forces are parallel sin. a'= sin. 6, sin. 6'= sin. c, 
 sin. c' sin. d. c 
 
 Multiplying the equations just given (41), by one or other o 
 these equals, we have 
 
 F sin. a . sin. '__F' sin. b sin. 6' 
 sin. (a+o')"~ sin. (6+6')" . , 
 
 F" sin, c sin. c'_F'" sin, d sin, ' ' 
 : sin. (c+c') sin. (d-\-d'} 
 
 but 
 
 sin. a sin. a' 1 
 
 sin. (+') ~~~cot. a+cot. a' ; 
 
 and so of the rest. The expressions may therefore take another 
 form, and become 
 
 F = F' ] 
 
 cot. a-j-cot. a'~~cot. 6+ cot. 6' I 
 F" F"' [ 
 
 ~cot. c+cot. c'~cot. d-|-cot. d' J ; 
 
 hence each of the forces is proportioned^ to the sum of the cotan- 
 gents of the two angles its direction makes with the two lines, at 
 whose junction it acts. 
 
 The lines are also in one plane, in which the forces F, F', F", 
 &c. likewise act. 
 
 The forces F, T and X, being in equilibrio around a point, are 
 in the same plane ; in which X' lies also ; and F' being parallel to 
 F, and drawn from a point in the direction of X is also in the 
 same plane ; Y lies in this plane also, because it is in equilibrio 
 with X' and F' ; and for the same reason that F' was in the same 
 plane with F, F" lies in the same plane with F' ; thus all the forces 
 lie in a single plane, and the polygon is a plane figure. 
 
 The forces which act to keep the system in equilibrio, being 
 parallel, their sum is equal to ; and if the system be attached 
 at the two extremities, to two fixed points, on which the tensions 
 resolved into two parallel directions, are T and T', 
 
 F+F'+F"+&c.+T+"T / =0. 
 
 The'sum of the parts of the tensions which act in directions, paral- 
 lel to those of the forces, will therefore be equal to the resultant 
 of all the other forces. 
 
 29. When the polygon has its two extremities fixed, it is called 
 the Funicular Polygon, because it is, as will be hereafter shown, 
 the figure a rope would assume when loaded by weights attached 
 to different points of its length. When the points at which the 
 forces act become infinitely near, the polygon becomes a curve 
 that may be called the funicular curve ; and when the weights are 
 equal, the curve is called the Catenaria. The research of the equa- 
 tions of the Catenaria is not strictly an object of elementary me- 
 
34 or EQUILIBRIUM. [Book I. 
 
 R=V(X a +Y 2 ); 
 
 call the angles its direction makes with the two axes, at the point, 
 in which if produced, they will meet, a and /3. Then by (18), 
 
 X a Y 
 
 cos. a= ,cos./3= 
 
 All that remains is to determine the co-ordinates of the point 
 of application. These may be determined, when the position 
 of the points of application of the several forces are determined, 
 and their co-ordinates given, by means of the principle 
 contained in formulae (34, and 35). In these, the values of 
 the several forces, with those of their co-ordinates, are to be sub- 
 stituted in a manner too obvious to heed description. 
 
 : 'MOflOO 
 
 31. The value of the resultant .of forces acting in one plane, 
 may also be determined by means of what are called their Mo- 
 ments of Rotalion. 
 
 To understand the meaning of this term, we shall recur to the 
 investigation of the value of the resultant of two parallel forces, 
 ( 22.) In the course of that, it was found that the perpendicu- 
 lar distances of the directions of two converging forces, acting 
 upon an inflexible line, from the point of application of their result- 
 ants, and the forces themselves, were in inverse proportion, or as 
 represented by, (25), 
 
 A : B : : b : a, 
 hence, 
 
 With two forces, having this relation, a third applied to the point of 
 application of the resultant, equal to it, and opposite in direction, 
 would cause the system to be in cquilihrio. Let us now suppose 
 that instead of applying a force to this point, it becomes fixed, but 
 in such a manner that the line of application of A and B, may be 
 free to revolve around it as a centre of motion. The two forces 
 although unequal in magnitude, are still in equilibrio, and each 
 will tend to make the line revolve with equal energy. This energy, 
 then, may be expressed by the two equal products A, and B6 ; 
 and in general, if we suppose the point of application of any force 
 to be united to a fixed point, by an inflexible line, the force will 
 act to cause the line to revolve around that poinf, with an energy 
 determined by the product of its intensity, into the perpendicular 
 distance of the fixed point, from the direction of the force ; 
 hence : 
 
 32. The moment of rotation of a force, in respect to a point, 
 is the product of the intensity of the force into the perpendicu- 
 lar let fall upon the direction of the force. 
 
 33. The moment of rotation of two forces, in respect to any 
 point situated in their plane, is equal to the sum or difference of 
 
/.] 
 
 OP EQUILIBRIUM. 
 
 35 
 
 the moments of rotation of its components, in respect to the same 
 point; to the difference, when the point falls within the angle, 
 formed by the directions of the two components ; to the sum 
 when the point falls without this angle. 
 
 Let F and F' be the two forces, R their resultant, converging 
 to the point d ; C the point whence the perpendiculars are let fall 
 
 let/,/, and r, be the three perpendiculars ; let the distance C d, 
 =c. Let each of the forces F, F', and R', be decomposed in two 
 others, one in the direction of c, the other perpendicular to it, 
 or in the direction B d E. 
 
 The value of the component of R, in the direction BdE, 
 will be 
 
 Rcos. ^RcZB; 
 but 
 
 'iO o: 
 , cos. 
 
 and the component of R, in the direction of B d E, becomes 
 
 :-;fcem| ' 
 
 In the same manner, the components of F and F', in the same 
 direction B d E, may be shown to be 
 
 , 
 
 these will be in the same direction, when the point C falls without 
 the angle, and in opposite directions when it falls within. Now, aa 
 these three forces would be in equilibrio, if R were applied in a 
 reversed direction ; their components in relation to two rectan- 
 
36 OP EQUILIBRIUM. [Book I. 
 
 gular axes, would be equal also ; the components of F and F', are 
 therefore equal to the component of R, and 
 
 R.:= F /+F/; 
 c c* c 
 
 multiplying by c 
 
 R,=F/+F/, 
 
 which expresses our proposition. 
 
 Extending the investigation in the usual manner to any num- 
 ber of forces, we have 
 
 , (45) 
 
 In this expression, it is obvious that the signs + and , express 
 
 the tendency of the force to turn the system, in one or the other 
 
 direction, around the centre of the Moments. 
 In case of equilibrium, the expression becomes 
 
 F/+F'/+F"/' HF&c. -0, (46) 
 
 or : 
 
 34. A system of forces, acting; in one plane upon a system of 
 points invariably connected, will be in equilibrio, when the sum of 
 the moments of all the forces that tend to make the system turn in 
 one direction, is equal to the sum of the moments of all the forces 
 that tend to make the system turn in an opposite direction. The 
 same proposition is also true in the case of the system being 
 firmly attached to the point C, or to the centre of the moments ; for 
 were the two sets of moments unequal, one or the other would 
 preponderate, and would make the system revolve. In this case 
 it is not necessary that the resultant of the forces should be equal 
 to 0, but merely, that, as the moment of the resultant is equal to the 
 sum or difference of the moments of all the forces, 
 
 Rr=0. 
 But this can only happen, if R have any magnitude, when 
 
 r=0; 
 hence the resultant must pass through the fixed point; and, 
 
 35. When a system of forces is applied to a system of points 
 invariably connected together in one plane, and having one fixed 
 point, the direction of their resultant must pass through that point, 
 or the system will not be in equilibrio. 
 
 36. If we suppose a straight line to be drawn through the 
 centre of the moments, and perpendicular to the plane, this line 
 will become an axis, on which the forces would tend to make the 
 system revolve ; and it will be no longer necessary that the forces 
 should act in one plane, but merely that they act in planes parallel 
 to each other ; for their moments, determined by lines drawn per- 
 
/.] OP EQUILIBRIUM. 37 
 
 pendicular both to their own direction and the axis, would remain 
 constant. 
 
 37. If the forces do not act in planes perpendicular to the fixed 
 axis, each of them may be resolved into two, one parallel to the 
 axis, the other lying in a plane perpendicular to it. It will then 
 be obvious, that the former produces no effect to make the system 
 revolve ; and no more of the force is exerted, for that purpose, 
 than is represented by the latter. The line that represents that 
 component of a given force, which acts in a plane different from 
 that in which it is itself situated, corresponds with its geometric 
 projection in that plane ; and calling the force A ; the projection 
 P ; and the angle the two planes make with each other i ; this 
 force may be found (14) by the formula. 
 
 P=A cos. i. (47) 
 
 38. The conditions of equilibrium in forces acting upon points 
 invariably connected, also hold good, when the points are con- 
 nected in any manner whatsoever; for it is evident that if the 
 system be in equilibrio, the state of equilibrium will not be 
 changed, by uniting the points of which it is composed in an inva- 
 riable manner. But in addition to the conditions, that are alone 
 necessary in points connected in an invariable manner, and are 
 common to systems connected in any manner whatsoever, there 
 will be others, that will depend upon the manner in which the 
 points are connected. 
 

 
 
>fi<te ciaad 91 < 
 
 BOOH. II. 
 
 OF MOTION. 
 
 ta 
 
 CHAPTER I. 
 
 Op MOTION IN GENERAL. UNIFORM MOTION. GENERAL 
 
 PRINCIPLES OP VARIABLE MOTION. 
 
 39. When a material point is acted upon by forces under whose 
 action it is not in equilibrio, it is set in motion, as it also would 
 be by the action of a single force. If it be not acted upon by any 
 force, as there is no reason that it should move in one direction 
 rather than another, it will remain at rest ; so also when once set 
 in motion, and no force act, or if the forces that do act are in 
 equilibrio, it must continue to move uniformly forwards in a 
 straight line ; for there is now no reason why it should change 
 either the rate, or the direction of its motion. Hence, all bodies 
 will continue in the same state, either of rest, or of motion uni- 
 formly forwards in a straight line; unless they be compelled to 
 change their state of rest or of motion, by the action of some force 
 impressed upon them. The truth of this principle is not obtained 
 from abstract reasoning alone, but is the uniform result of obser- 
 vation and experience. Although from what is observed to occur 
 in moving bodies near the surface of the earth, we might at first 
 sight infer that they had a natural tendency to come to rest ; still, 
 when we remark, that the more we lessen the resistances, the 
 longer is the continuance of the motion; and that we can in al- 
 most all cases, ascribe the diminution of the motion, or its change 
 of direction, to forces that we know from other circumstances to 
 be acting; we infer, that were these resistances not to act, the 
 body would go on uniformly in a straight line. This principle 
 is sometimes ranked as a property of matter, and is called its 
 Inertia. A similar inference may be deduced from the motions 
 of the heavenly bodies. In these, since the earliest record of au- 
 thentic observation, no change has been detected, that is not pe- 
 
40 OF MOTION. [Book II. 
 
 riodic ; their mean rates of motion have therefore been constant, 
 although no force has been applied to maintain their motions. 
 
 40. In order to represent the circumstances of motion, which 
 consist in the passage of a body through a portion of space in some 
 definite time, we make use of the term Velocity. Velocity is the 
 relation, or ratio, of the spaces described to the times employed 
 in describing them. It Will be obvious (hat in different motions, 
 such a relation does actually exist ; the more rapid the motion, 
 the less being the time occupied in describing a given space, and 
 the greater the space described in a given time. But space and 
 time are essentially heterogeneous quantities, and incapable of 
 any direct comparison. We are therefore compelled to resort to a 
 means of comparison, by adopting a method, in which the mea- 
 sure of each of these quantities may be considered as an abstract 
 number. Between such numbers, a ratio capable of being ex- 
 pressed does exist. In order to effect this object, we assume 
 some conventional unit for the space, as the foot, for instance : in 
 like manner we assume a conventional unit, for the measure of the 
 time. In terms of these, a ratio, or relation may be expressed. 
 As the velocity, in uniform motion, increases with the space de- 
 scribed, while the time diminishes, the relation between them 
 may be thus expressed : 
 
 .=4 ' (48) 
 
 where v is the velocity, s the space, and / the time. The first 
 of these will be denoted in the number of the conventional units 
 of space, described by the body in the unit of time. 
 
 From this equation we obtain expressions for the value of the 
 
 space s, and the time /. as follows : 
 
 k 8 
 
 s = v /, f=-. (49) 
 
 oil 
 
 41. When two uniform motions are to-be compared together ; 
 as for instance, when we wish to ascertain the time in which 
 bodies moving, or appearing to move, in the same line with dif- 
 ferent velocities, shall be at the same point, or shall appear to meet; 
 we may estimate the spaces, from some given point in the line, 
 actually, or apparently described, by the two bodies. Let s re- 
 present the distance from the fixed point at the time / ; b the dis- 
 tance from the same point, at the instant from which the time / is 
 estimated ; then the space becomes s b j and the first equation 
 (48) becoir.es , )n 'l 
 
Book //.] OF MOTION. 41 
 
 whence, 
 
 s=vt+b. (51) 
 
 The time /, may be either positive or negative ; when positive, it 
 denotes intervals of time subsequent to the passage of the body 
 through the fixed point ; when negative, it denotes intervals prior 
 to the body's reaching that point. So also, may s have positive 
 or negative values, which represent its position in respect to the 
 fixed point. In this manner the equation (51) will point out the 
 position of the body for every possible instant of time, in the line 
 that marks out the direction of its motion. 
 
 If there be another body moving, or appearing to move, uni- 
 formly in the same line with the first, whose velocity is v' ; whose 
 distance at the same time denoted by t, from the fixed point is s' ; 
 and the distance from the same point at the instant whence / is 
 estimated, is b', the equation of its motion will be 
 
 s'=v't+b.' (52) 
 
 In this equation, besides the same relations of positive and ne- 
 gative, among the quantities, that we have pointed out as appli- 
 cable to the former, the quantity v' will be negative, when the mo- 
 tion is in a direction contrary to that of the first body. 
 
 A comparison between the values of these two equations, will 
 point out the relative position of the two bodies in the direction 
 of their motion. When both are at the same instant, at the same 
 distance from the fixed point, s=s', whence, 
 
 which gives for the value of / 
 
 6' 6 
 
 If this value should be found negative, it denotes, that the bodies 
 meet before the instant whence the time is computed. 
 
 To give an instance of the application of these formulae : sup- 
 pose two bodies to be moving in the same line, and in the same 
 direction, with velocities in the relation of 1000 : 1. 
 
 Then, v = 1000, ' = 1. 
 
 Let A, represent the body whose velocity is greatest ; B, that 
 whose velocity is least. 
 
 Let A be situated, at the instant whence the time is estimated, 
 at the fixed point ; then, 
 
 6=0; 
 let B be situated at the same instant, at a distance represented by 
 
 6' = 1000, 
 then, 
 
 V 6=1000 
 v 1/= 999 
 
 6 
 
 < 
 
42 or MOTION. [Book 11' 
 
 b'b 1000 1 
 
 99 =1 999 = 1 - 001 ' 
 
 vb' _ 1000000 
 _ _ 
 
 = 1001555 = 1001,001; 
 
 hence the body A, will overtake the body B, at the end of one of 
 the units of time, and the e^th part of that unit ; and will have 
 described a space equal to 1001 g -$ units of space. But, esti- 
 mated decimally 1 the fractions become the infinite converging 
 series . 00 1 00 1 . The ancients, unaware of the fact, that the sum 
 of such an infinite series, was a finite quantity, reasoned from it, 
 to show that the one body would never overtake the other. The 
 argument employed by them, was called the Achilles, and the 
 conditions were the same as those we have chosen for our 
 example. 
 
 The same propositions may be applied to uniform motions, in 
 any lines whatsoever ; as for instance to motion in re-entering 
 curves, of which the circle would furnish the most simple instance ; 
 and the hour, minute, and second hand of a watch, supposed to 
 be fixed upon the same arbor, would afford an apt illustration. 
 
 Supposing them all to set off from the same point, 6 6' and 
 6' b" become equal to the circumference of the circle or *r, and 
 for two hands, 
 
 for all three hands 
 
 the application of this to calculation, is too obvious to require an 
 example. 
 
 42. If a point be impressed at the same time with two uni- 
 form motions, it moves in a line which is determined by their 
 joint effect ; and as each of these motions is due to a force, the 
 point will move as if it were actuated by a single force, which is 
 the resultant of the two forces ; hence, from the principles of 12, 
 it must move in the diagonal of the parallelogram, constructed 
 upon the two forces as sides. If actuated by any number of motions 
 whatsoever, it will move in the direction of the resultant of the 
 forces that cause these several motions. 
 
 Instances of bodies that are actuated at the same time, by more 
 than one motion, are innumerable in practice. All bodies re- 
 tained upon the surface of others, by means of attraction, friction, 
 or any other force, acquire the motion of the bodies on which 
 they rest. 
 
//.] OF MOTION. 43 
 
 Thus a body in a carriage, or mounted upon a horse, a person in 
 a vessel, and finally all bodies upon the surface of the earth, have 
 a motion due to that of the body on which they rest. This be- 
 comes evident upon a sudden cessation of the motion, and is not 
 instantly communicated, as may be perceived immediately after 
 the beginning of the motion. Thus, when a steam-boat is sud- 
 denly set in motion, we feel a tendency to move in a direction 
 apparently opposite ; this is due to the inertia, which would 
 leave us in our original position in respect to fixed objects on the 
 shore, were not the motion of the vessel communicated to us ; and 
 when the same vessel has her progress suddenly checked, we ex- 
 perience a tendency to move forwards, that remains until again 
 counteracted, in the mode in which the motion was first com- 
 municated. 
 
 A body thrown from a carriage In rapid motion, by a force act- 
 ins; perpendicular to the direction of the motion, does not fall to 
 the ground opposite to the point where the carriage was, when it 
 was projected, but opposite to the point the carriage has reached, 
 at the time it falls to the ground. 
 
 In feats of horsemanship, balls are thrown up vertically, at least 
 so far as the action of the rider influences them, but being im- 
 pressed by the motion of the horse, they fall again into the hand 
 of the rider ; the rider may spring directly upwards from the 
 saddle, and fall again upon it, although the horse be at full speed. 
 The directions in" which bodies move, being influenced by all the 
 motions with which they are impressed, the position of bodies 
 moving upon surfaces that are themselves in motion, is the same 
 in relation to points in the moving surface, at given instants of 
 time, as it would have been, had the surface remained at rest. 
 Thus we move from place to place, upon the deck of a vessel, 
 whose motion is not disturbed, with precisely the same effort, 
 that we would perform the same distance upon the land ; and were 
 we not aware of the vessel's motion from other circumstances, 
 would, as is done by children, ascribe the motion to the surround- 
 ing objects. 
 
 The same happens to us, from our situation on the surface of 
 the earth. This is impressed with a rapid motion, not only of 
 revolution, but of translation ; yet these being to all intents uni- 
 form, we ascribe the change of apparent position that our motion 
 causes in the heavenly bodies, to a proper motion existing in them ; 
 and it was ages after these apparent motions had been carefully ob- 
 served by astronomers, before the true cause of the phenomena 
 was detected. 
 
 43. Although it is the tendency of matter, if once set in motion, 
 to move forwards forever with uniform velocity, in the same direc- 
 
44 OP MOTION. [Book II. 
 
 tion, this is by no means the most frequent case of motion that 
 occurs in nature. There are in fact two distinct species offerees. 
 
 (1) Those which having acted for a time upon a body, abandon it 
 and leave it to go forward, so far as they are concerned, in a right 
 line with uniform velocity; these are called projectile forces. 
 
 (2) Those which act during the whole continuance of the body's 
 motion. Such forces will cause changes in the direction and ve- 
 locity of the body, and produce what in general terms, are called 
 Variable Motions. 
 
 44. As, when a point that has at one instant of time been at rest, 
 is afterwards found in motion, we infer the action of some force 
 to cause that motion ; so, when a point in motion, whose velocity 
 and direction have been determined, at some instant of time, is 
 afterwards found moving in a different direction, or with differ- 
 ent velocity, we also infer the action of some force to produce 
 this change. This force may either have acted for a greater or 
 less time, and then abandoned the point to itself, or it may have 
 been continually acting. A force of the latter description is called 
 an Accelerating Force, because had it acted upon a point origi- 
 nally at rest, it would have given the point an accelerated 
 velocity. 
 
 We judge of the fact of the motion of a point being accelerated, 
 by comparing the spaces described in equal times ; if after having 
 described a certain space in the unit of time, it shall be found 
 describing a greater space in an equal time, its motion has evi- 
 dently been accelerated. So, when after having described a given 
 space in the unit of time, it is afterwards found describing a less 
 space in the same time, its motion is retarded. An accelerating 
 force may produce a retarded motion ; for it may act in a direction 
 contrary to the motion originally impressed by some other force 
 upon the body ; and indeed most retarded motions are due to the 
 action of forces that may be considered under the general head of 
 Accelerating. 
 
 45. The time which a point takes to describe a given space, 
 will obviously depend upon the intensity of the force that causes 
 its motion. Thus a force of double the intensity, will cause a point 
 to describe twice the space in an equal time, and so on. As we 
 know absolutely nothing of the nature of forces, but find their 
 action to be proportioned to the spaces described, we might take 
 the spaces described in equal times as the measure of the forces ; 
 but as the velocities are proportioned to the spaces, they are also 
 proportioned to the forces ; and therefore in uniform motions, 
 where mere material points are concerned, the velocity and force 
 may mutually serve as each other's measures. All the princi- 
 ples and formulae that have been applied in the previous book to 
 
Booh //.] OP MOTION. 45 
 
 the composition offerees, are, therefore, applicable to the compo- 
 sition of velocities. 
 
 46. In motions produced by a force that acts continually, the 
 velocities are, as we have seen, variable. It is impossible for us 
 to determine from experience alone, whether such a force acts with- 
 out interruption, or whether it produces its effect by a succession of 
 impulses, separated by inappreciable intervals of time. Which- 
 ever of these modes of action be the true one, the results in both 
 cases will be the same ; for if we suppose the velocities to be re- 
 presented by the ordinates of a curve, whose abscissas represent 
 the times; the motion produced by a succession of impulses, sepa- 
 rated by infinitely small intervals of time, would correspond to a 
 polygon of an infinite number of sides; and this would be identical 
 with the curve. We therefore consider all variable motions, as 
 made up of a succession of uniform motions, each continued fora 
 very short space of time ; and the results obtained, from investi- 
 gations founded on this principle, are indentical with those that 
 would occur, were the velocity to be continually varying. 
 
 In the equation (48) the time and space s and tf, becoming in- 
 finitely small, will be represented by their differentials, and 
 
 =*,*=.*, * = , (53 ) 
 
 The intensity of an accelerating force, cannot, like that of a 
 force that produces uniform motion, be measured by the velocity, 
 for that is always varying, even although the force may remain 
 constant ; but it may be measured by the momentary variations 
 in the velocity. Let dv be the increase of the velocity during the 
 time dt ; then, since the increase in the velocity will be the same 
 as it would have been, had the action of the accelerating force not 
 been interrupted, dv will be equal to the product of the force into 
 its time of action dt, or calling the force/, 
 
 dv=fdt, 
 and 
 
 then since by (53) 
 we have 
 
 <> 
 
 Such are the general equations of variable motion ; and they 
 are applicable either to the case of its being accelerated or retarded ; 
 but in the latter case, dv is a negative quantity. 
 
OF MOTION. [Book II. 
 
 CHAPTER II. 
 
 OF RECTILINEAL MOTION UNIFORMLY ACCELERATED, OR UNIFORMLT 
 
 RETARDED. 
 
 47. When the spaces that a point describes in equal times, 
 increase or decrease by equal increments or decrements, the mo- 
 tion is said to be uniformly accelerated or retarded. In such a 
 case the quantities dv, and dt, in equation (54), are obviously con- 
 staat; hence the accelerating force/", is a constant force. 
 
 48. We call the velocity that a body would have, were the ac- 
 celerating force removed, its Final Velocity ; or, as we consider the 
 body to move uniformly, for infinitely small intervals of time, it 
 may be considered at its velocity as the end of the given time, 
 and used without the addition of the word final. The space de- 
 scribed from rest, in acquiring that velocity under the action of 
 the accelerating force, iscalled theSpace due to that Velocity ; and 
 the velocity acquired, is called the Velocity due to the Space. 
 
 49. In the motion of a point uniformly accelerated from a state 
 of rest, the velocities are proportioned to the times; the whole 
 spaces described, are proportioned to the squares of the times ; 
 and if the times be represented by the series of natural numbers, 
 the acquired velocities will be represented by the series of even 
 numbers ; the whole spaces, by the series of square numbers ; and 
 the spaces described in the successive units of time, by the series 
 of odd numbers. The measure of the accelerating force is equal 
 to twice the space described under its action from rest, in the 
 first unit of time ; and if the accelerating force be removed, the 
 velocity acquired in a given time by its action, is such as would 
 carry the point in an equal time, with uniform motion, through 
 twice the space it has passed through, in acquiring that velocity. 
 
 Suppose the point to have, at the instant the accelerating force 
 begins to act, and in the same direction, a velocity = a. It will ac- 
 quire during each successive unit of time, an additional velocity, 
 which as the force is, in the case under consideration, a constant 
 one, will be a constant increment. This increment, which will be 
 the measure of the accelerating force, we shall call g. The velo- 
 city being originally a, will become, at the end of the first unit of 
 time, a+g ; 
 
 at the end of the second unit, -f2g-; 
 
 and at the end of the time /, a ~^~gt 5 
 
 or calling the final velocity v, 
 
 v^a+gt. (57) 
 
jBook //.] OP MOTION. 47 
 
 Ift vary, and become t+dt, the space described , will vary 
 also, and become s+ds : ds then, is the space described in the time 
 dt. 
 
 If we suppose that for this small interval of time, the velocity is 
 constant, and equal to v, we have by (53), 
 
 ds=vdt, 
 substituting the value of v from (57) 
 
 ds=adt+gtdt. 
 Integrating 
 
 b being the distance from some fixed point in the direction of the 
 motion ; but when no more than a single point is concerned, 6 
 may be taken equal to 0, and the equation becomes 
 
 fif 
 
 s=at+~ (58) 
 
 eliminating t by means of equation (57). 
 v 2 a 2 
 
 If the body be at rest when the accelerating force begins to act, 
 a=0, and the equations (57), (58), (59), become 
 
 from these equations we readily obtain others, which with them, 
 give the value of the several quantities, each in terms of two of 
 the others, as follows, viz. 
 
 2s 2s 
 
 lit 
 
 (61) 
 
 - 
 
 If the motion continue only for the unit of time 
 
 v=2s. 
 
 The expression v=gt, in which g is a constant quantity, shows 
 us that the velocities acquired are proportioned to the times. 
 
 gf g 
 
 The expression *=~2~ in which ^ is constant, shows u 
 
 
48 or MOTION. [Book It. 
 
 that the whole spaces are proportioned to the squares of the 
 times. 
 
 2s 
 Applying the formula = to times taken as* the series of 
 
 natural numbers, and calling the space described in the first unit 
 of time, unity ; we have, for the successive values of t>, 
 
 2, 4, 6, 8, &c. 
 On the same hypothesis we have for the values of 5, 
 
 1,4,9,16, &c. 
 
 The successive differences of this series represent the spaces 
 described, during each successive unit of time, and are, 
 
 1, 3, 5, 7, &c. 
 
 The expression g=2s, when the motion has continued from 
 rest, for the unit of time, shows that the measure of the accele- 
 rating force is equal to twice the space described, from rest, in the 
 first unit of time. 
 
 2s 
 And the expression v=~^~ compared with the equation (48) of 
 
 s 
 
 uniform motion, v= j shows us, that a point, with the velocity ac- 
 quired by moving from rest under the action of a constant force, 
 would pass through twice the space in an equal time. 
 
 50. In the motion of a point that is equably retarded by the 
 action of a constant force, the velocities are as the times that 
 remain until the cessation of the motion ; and the spaces that 
 remain to be described, are as the squares of the times estimated 
 in the same manner, and as the squares of the velocities. The 
 point will go, before it loses its motion, through half the spacs 
 that it would have described, in the remaining time, with uniform 
 
 velocity. 
 
 t 
 
 In our previous investigation if applied to this case, g becomes a 
 negative quantity ; for the accelerating force acts in opposition to 
 the original velocity. Hence in retarded motion, 
 
 (61a) 
 
Book IL] OF MOTION. 48 
 
 when v=0, at which time the constant force will have completely 
 destroyed the initial velocity, 
 
 a a 3 a* 
 
 from which equations, the principles we have stated can be rea- 
 dily determined. 
 
 5t. If the original direction of the motion be not in the same 
 straight line, in which the accelerating force acts, the point must 
 describe a curve. For it will in the first instant of time describe 
 the diagonal of a parallelogram, whose sides represent the uniform 
 velocity, and the measure of the accelerating force; in this di- 
 rection it would tend to go forward, were it not again acted upon 
 by the accelerating force ; this action would produce a second 
 deflection, and so on ; and thus the point would describe a poly- 
 gon of an infinite number of sides, or a curve. Before then we 
 can determine the nature of the line the body describes, it be- 
 comes necessary to investigate the general properties of curvi- 
 linear motion. 
 
IK) OP MOTION. [Book II. 
 
 CHAPTER III. 
 OF CURVILINEAR MOTION. 
 
 52. The forces which concur to produce curvilinear motion 
 are, in general, resolved into three, parallel to three fixed rectan- 
 gular axes. This may be done, as explained in 16, in respect 
 to any forces whatsoever. The moving point will describe the 
 resultant of these three forces, which will identically replace all 
 the others that act. In the small elements of the time, it will 
 describe the diagonal of a parallelepiped, of which the three rec- 
 tangular forces, considered as producing uniform motion for the 
 small interval of time, are the sides. Now if we suppose perpen- 
 diculars to be let fall, from the successive positions of the moving 
 point, upon the three axes, the points in which these perpendicu- 
 lars cut theaxes, will move along with them; and the motion of each 
 of them will be such as would be due to a rectangular force, that 
 acts parallel to that axis. If any of the forces cease to act, the 
 motion in the directions parallel to the other axes will not be 
 changed. Each of the three rectangular forces, into which all 
 that act are resolved, is therefore independent of the others, and 
 its action may be determined upon the principles laid down in 
 46. 
 
 The general equation of variable motion (56) is therefore ap- 
 plicable to this case ; and calling the three rectangular forces X, 
 Y and Z, and the spaces in these directions, #, y and z, it will be- 
 come 
 
 These equations contain all that it is necessary to know in respect 
 to the motion. For 1st, by an integration we obtain the values 
 of the three velocities in these directions ; these from (55) are 
 dx dy dz. , . 
 
 di' dt 1 dt' (63) 
 
 from the composition of which, the velocity of the moving body is 
 determined; 
 
 2. Another integration gives us the co-ordinates in terms of 
 the time : 
 
 3. By eliminating , we obtain the equations of the curve that 
 the body describes. 
 
 In the same case, of the resolution of all the forces into three 
 rectangular forces, we shall have 
 
 (64) 
 
Book II.} OF MOTION. 51 
 
 for, from the above equations we .obtain, by multiplying them 
 respectively by dx, dy, dz, and adding them together, 
 
 dxtfx+dytfy+dztfz^ 
 
 but as the axes are rectangular, 
 
 whence we obtain 
 
 dxd 3 x + dyd 2 y + dzd 2 z = dsd?s ; 
 therefore 
 
 dsd 2 s ds ds 
 z=~-= d. -~ 
 
 If the forces act in one plane, we have need only of the two 
 equations, 
 
 If but two forces, acting in the same plane, are concerned, it 
 may be simpler to employ them, without resolving them each into 
 three parallel to three rectangular axes ; the same principles ap- 
 ply to them as to three rectangular forces, and the equation (56) 
 becomes, calling the forces F and F', 
 
 When no more than two forces act, as the successive diagonals 
 are all in one plane, the curve is a plane curve, 
 
58 OF MOTION. [Eouh It 
 
 CHAPTER IV. 
 OF PARABOLIC MOTION. 
 
 53. If a point be acted upon by two forces not in the same straight 
 line, by virtue of one of which, it would describe a straight line 
 with uniform velocity, and of the other, a straight line with 
 uniformly accelerated velocity; and if the second force act paral- 
 lel to itself, during the continuance of the motion; the point will 
 describe a parabola under their joint action. The diameters of 
 this parabola are parallel to the directions of the accelerating 
 force, and its parameter is four times the space due to the velo- 
 city communicated by the first of the forces. 
 
 Let F' be the projectile force by whose action the body would 
 go on with uniform rectilineal velocity, and F the accelerating 
 force which begins as soon as the force F' ceases to act, then 
 from equation (66) 
 
 As F represents the accelerating force, call 
 
 V=g; 
 
 then, F' representing the force that produces the constant velocity, 
 
 F'=0; 
 hence 
 
 df df 
 
 piMtfi =*-JT 
 
 Integrating 
 
 A and B representing the initial velocities in the two directions ; 
 but as the accelerating force begins to act at the instant whence 
 the motion is to be computed, 
 
 A=0. 
 
 And we may measure B in terms of g-, and the space through 
 which the body should pass under its action, in order to acquire 
 the initial velocity ; by (61) 
 
 substituting these values, we have 
 df df 
 
Book II] 
 
 Integrating anew 
 
 OP MOTION. 
 
 53 
 
 In which expressions the constant quantities are omitted, for <=0, 
 /=0, f=Q, at the same time ; 
 eliminating t 
 
 f'*=4sf (67) 
 
 which is the equation of a parabola, whose ordinate and abscissa 
 are/' and /, and whose parameter is 4s. / being the abscissa, 
 shows that the diameters of the parabola, and of course its axis, 
 are parallel to the direction of the accelerating force. 
 
 To investigate the path by means of two rectangular co-ordi- 
 nates, X and Y, the directions of X being parallel to the direction 
 of the accelerating force. 
 
 In the figure beneath, let AP represent the action of the accele- 
 
 rating force, AT the direction in which the projectile force would 
 cause the body to move with uniform velocity. 
 
 Draw AQ to represent the direction of the axis X ; TR parallel 
 to AP ; and PR parallel to AQ ; join PM, then 
 AQ=:r, QM=7/, MR=i/-r-/; 
 call the L TAQ= L MPR , t, 
 then !/+//' sin. t, x=f cos. i ; 
 
 whence 
 
 x sin. t 
 
 but from (67) 
 therefore 
 
 <=. - 
 
 45 4* cos. 
 
 tan. i 
 
 4a cos. s i ' 
 
 (68) 
 
1t OF MOTIMW. [Bouh II 
 
 CHAPTER IV. 
 Or PARABOLIC Morion. 
 
 53. If point bo acted upon by two forces not in tbe same straight 
 line, by virtue of one of which, it would describe a straight line 
 with uniform velocity, and of the other, a straight line with 
 uniformly accelerated velocity; and if the second force act paral- 
 lel to iUelf, during the continuance of the motion; MM point will 
 describe a parabola under i ! ><i i joint action. The diameters of 
 this parabola are parallel to the directions of the accelerating 
 force, and its parameter is four times the space due to the vH., 
 city communicated hy the first of the forces. 
 
 Let F' be tbo projectile force by whose action the body w.ull 
 
 go on with UIHI'MIHI roctilinftal velocity, and F the acccl< i .MI.,^ 
 
 - In. l> U'giriM us noon an the force F' ceases to act, then 
 
 from .,|,i. ,(,.,,, 'Mi; 
 
 <ir ''* -. 
 
 As F represent* th aceoleroting force, call 
 
 V K; 
 ii> n, F' representing the force that produces the constant velocity, 
 
 I 0; 
 
 Integra ting 
 
 A and B representing the initial velocities in il. iu,, din -. -HMHI , 
 but as the accelerating force begins to act at the mitnm uh. n, 
 the motion is to be computed, 
 
 A0. 
 A n< I we may measure B in terms of #, and the space through 
 
 \vlnrl, )!,. |.,..lv slioul.l |,;,SM ,..!., ,| , ,,,-1 ...... u. oidrr to ar|uiru 
 
 the initial velocity t by (61) 
 
 substituting these flfaet, we have 
 
 If df 
 
 .1, **< ,// 
 
JBtwk 11] 
 
 OF 
 
 53 
 
 111 \\hirh expressions (In- rmislnill (]iiimlilii>s lire oiniltfd, for f 0, 
 
 /"O, /=0, at the iam<< limr ; 
 
 climmiilin;', / 
 
 /"=4</ (67) 
 
 Wllicll is llir .|!i.i!:..n . -I ;i pimiholll, \\hoc nidin.ili- .Mill llllHCINHn 
 
 arc/' aiul ./, mid \vli<..^- |;II,HIU id is is. / liriu^r iho iibrtciaia f 
 
 N|,(,\\;I thill Ihr ili.'inirlrr.s ol'lh,- |i;ir:il>iiln, ;m.l o| roin.si- ll -, u\i . 
 n n- [>!irjlll< l l lo lh<' iliHM-lion ofllic nrrcli-cililiH loH'f. 
 
 To in\- li".ilr III.- |.:ilh hv iiirinr; ul'luu n-clni|i>-iiliir ro-ordi- 
 lialrii, \ ..11,1 \ , Ihr (liiiTlmiis ni' \ linii>> |>lllllll<-l In llir dilTrlloil 
 of !hn iicci-lrinliut' I'OK . 
 
 In tlio liguro bcnuuth, lot AP roprenont the action of thes accele- 
 
 rating force, AT the direction in which the projectile force would 
 cause the body to move with uniform velocity. 
 
 Dl.'.u \<J lo irpiv.srill ihrdiMTlion of llir ;IM \ , I K ,,.,,. ,11. i 
 1<> AC; mid Pit pimdlnl (<> \o , |<>IM I'M, thru 
 
 AQ=j?, QM=y, MR=t/4-/; 
 call the /lTAQ=^MPR f t f 
 then !/+/=/' "in. i, =/' coi. i \ 
 
 \\h-nco 
 
 x in. i 
 
 I.NI IVi.m (Ii7) 
 (hnrloro 
 
 /-- 
 
 4f ooi. i ' 
 
 V" ; 
 
54 op MOTION. [Book II. 
 
 This last equation may be obtained directly, from the resolu- 
 tion of the two forces into two rectangular forces, one of which 
 is parallel to the direction of the accelerating force. 
 
 Call the measure of the accelerating force g ; as it acts in a 
 direction opposite to the co-ordinate Y, it will be negative, or g. 
 
 The alteration in the direction of X, being wholly due to the 
 force that produces a constant velocity, is =0. 
 
 The two equations (62) become 
 
 d 2 x_ d 2 y_ 
 dT 2 " ' ~dT~S' 
 Integrating, 
 
 y= ^-+ C f+c', x=bt+b'. (69) 
 
 Taking the origin of the co-ordinates at the point A, when /=0, 
 *=0, i/ 0, therefore 6'=0, c'=0. 
 
 Let v be the velocity due to the projectile force, or the initial 
 velocity, call the angle TAQ, as before, i. The components of 
 the initial velocity are v cos. t, acting in the direction of X ; and 
 v sin. f, acting in the direction ofy ; and from (63) 
 
 dx dy 
 
 = v C os. t, ji=v sin. ; 
 
 and as the constant quantities, 6 and c, represent these velocities, 
 
 b=v cos. ', cv cos. ; 
 therefore by (69) 
 
 gt 2 
 y= -g \-vt sin. t, x=vt cos. i ; 
 
 eliminating /, 
 
 u=x tan. t r~r - r~- ? 
 2v 2 cos. 2 t ' 
 
 but as v\/2gs (61), t? 2 =2g-5, by the substitution of which we 
 obtain, 
 
 A comparison of this with (67), by means of the preceding in- 
 vestigation, shows that this is the equation of a parabola, whose 
 axis is parallel to the direction of the accelerating force. 
 
 The maximum value of g is VS, the length of the axis of the 
 parabola. In this case cfa/=0, and the differential of (68) is 
 
 xdx 
 
 tan. t. cos. t==sin. t, 
 
 x=2a sin. t cos. ; 
 

 Book II. ] OP MOTION. 55 
 
 and as, 2 sin. i cos. =sin. 2 t, 
 
 x=s sin. 2t; (70) 
 
 substituting this value of x in (68), we obtain 
 
 y= SS m.*i. (71) 
 
 The value of x (70) is equal to AS ; this is half of AB, which 
 we shall call A ; therefore 
 
 A=2s sin. 2. (72) 
 
56 or MOTION. [Book 11. 
 
 CHAPTER V. 
 
 OF THE MOTION OF POINTS COMPELLED TO MOVE UPON SURFACES, 
 UNDER THE ACTION OF ACCELERATING FORCES. 
 
 54. When a point rests upon a given surface, this surface may 
 be considered as exerting a force to resist the pressure, which 
 force acts in the direction of a normal to the surface. In the case 
 of equilibrium, this is equal to the resultant of all the other forces, 
 19. But in the case of motion, the action of the surface must 
 be found, by decomposing the resultant of all the forces that act 
 into two components, one of which is parallel, the other a normal 
 to the surface. The last of these will represent the intensity with 
 which the point is pressed against the surface, by the forces, and 
 will be in equilibrio with the resistance of the surface ; the for- 
 mer therefore, will alone remain to cause the motion of the body. 
 
 55. The simplest case that can occur is where the surface is 
 plane, and is inclined at a given angle to the direction of an ac- 
 celerating force, which is always parallel to itself. In this case, the 
 point that is impelled by the force will describe a straight line, 
 which is the common intersection of the surface and the plane in 
 which the accelerating force acts. So also if a similar force act 
 upon a point placed on a curve surface, the point will describe a 
 plane curve, which is the common intersection of the surface and 
 the plane in which the force acts. In these instances the surfaces 
 may be considered as straight lines or plane curves. So also when 
 the force is constantly directed to a point, and its directions all 
 lie in one plane, the curve described is a plane curve. In these 
 several instances, the circumstances of the motion may be inves- 
 tigated in a more elementary manner than can be done by pur- 
 suing the general method of curvilinear motion, and referring the 
 forces to their rectangular co-ordinates. When the point moves 
 on a plane surface, we shall continue to use the name of the sur- 
 face ; but in all other instances, we shall merely name the curve 
 that is described. In our subsequent applications of the theory, 
 the modes, in which the path of a body may be made to coincide 
 with the several curves, will be pointed out. 
 
 56. When a point moves upon a plane surface, under the action 
 of an accelerating force whose intensity is constant, and whose 
 direction is always parallel to itself, that part of the accelerating 
 force which remains to cause the motion of the point, is also a 
 constant accelerating force, and is to the whole accelerating force 
 
Book II :\ 
 
 OF MOTION- 
 
 in the ratio of the cosine of the inclination of the surface to the 
 direction in which the point, if free, would move under the action 
 of that force. 
 
 Let A be the point, moving upon a plane surface, CD, under 
 the action of a constant force g-, whose direction makes with that 
 of the surface the angle i'. Draw BC perpendicular to BD, which 
 is parallel to the directions of the accelerating force, the length 
 of the plane being CD, we shall call BD its height. 
 
 Decompose the force g into two, one of which, p, is a normal 
 to the surface, the other,/ is parallel to it ; their values will be (14) 
 
 p=g. sin. t', / g cos. i'. 
 
 The action of the former is destroyed by the action of the sur- 
 face, the latter remains to move the point along it. 
 
 The motion of the point along the surface will be uniformly ac- 
 celerated, for /bears a constant relation to g-, which is a constant 
 accelerating force. 
 
 All the formulae in 49, that have reference to uniformly ac- 
 celerated motions, are applicable to this case, by substituting/, 
 or its value g cos. t , for g ; hence 
 
 v'=gt cos. e", 
 
 i) 
 8'=7r- 
 
 ~2g cos. t' 
 gt 2 cos. i' 
 
 t' 
 
 (73) 
 
 g cos. * 
 
 Comparing these with the quantities of the same kind in free mo- 
 tion, under the action of the same accelerating force, we have for 
 the ratio of the velocities acquired in equal times, 
 
 =cos. 
 v 
 
 (74) 
 
53 or MOTION. [Book //. 
 
 for the ratio of the spaces described in equal times, 
 
 =cos. t'; (75) 
 
 s 
 
 for the ratio of the spaces described in attaining equal velocities, 
 -r=cos. i" ; (76) 
 
 for the ratio of the times in which equal velocities are attained, 
 y=cos. t; (77) 
 
 the velocity, acquired in passing freely through the height of the 
 plane, is (61) 
 
 in moving along the plane surface, 
 
 t/=V(2gcos. i'-^ s 77') 
 
 these velocities are therefore equal, or 
 
 v=v'. (78) 
 
 In plane surfaces of unequal inclinations, but equal heights, 
 the velocities attained are therefore equal. 
 
 If the surface coincide in direction with the chord of a circle, 
 that terminates at either end of the diameter that corresponds 
 with the direction of the force, the time of describing the surface, 
 and passing freely through the diameter, will be equal. 
 
 Call the diameter c, 
 
 2<z 
 
 . '=^1P 
 
 the length of the chord is a cos. t , and the time of describing it, 
 2a cos. t' 2a 
 
 '-'T^r^T- (79) 
 
 And the same will be true of all chords, terminating at either ex- 
 tremity of the diameter which is parallel to the accelerating force. 
 The time of describing planes of equal inclination, is as the 
 square roots of their lengths. Call the lengths I and I' ; the times 
 /' and t", 
 
 21 21' 
 
 t'=V -- , t"=V -- ., 
 
 g COS. I g COS. I 
 
 21 21' 
 
 " t f ' 4 ff / / 
 
 ". r . i ' v ~i ' v ' 
 
 g-cos.1 gees, t 
 
 /' : t" : ; Jl : VI', (80) 
 
Book //.] 
 
 OF MOTION. 
 
 59 
 
 57. If a point move from rest, upon a system of plane surfaces,, 
 under the action of a constant accelerating force, it will acquire 
 the same velocity as it would have acquired, in moving under the 
 action of the same force, upon a single plane surface of a height 
 equal to that of the system ; or to that it would have acquired, in 
 moving freely through the height of the system. 
 
 Suppose firs t, that there are two planes, AB, BC ; let AE be 
 
 D S 
 
 a line perpendicular to the direction of 'the accelerating force ; 
 produce the plane BC, until it intersect AE in D. A point moving 
 from a state of rest under the action of the force, along AB, will 
 acquire the same velocity as in moving freely through DF, or in 
 moving along a surface in the direction BD ; it will therefore go 
 on in BC as if it had come from D, and will reach C with the ve- 
 locity it would have acquired in describing the plane DC, or in 
 passing freely through DG. 
 
 Had there been three planes in the system, the same mode of 
 reasoning would have led to the same result, and so for any num- 
 ber, or for a curved surface. 
 
 58. In this proposition, it is obvious that any resistance that 
 might take place at the angles of the planes, is left out of account. 
 If the planes in the system become infinitely small, the surface 
 becomes curved, and this proposition, therefore, holds true in res- 
 pect to the arc of a curve, in which the acquired velocity is the 
 same as that acquired in moving freely through its height. 
 
 59. A point moving in a cycloid, under the action of a con- 
 stant force describes every different arc that is terminated at the 
 extremity of the diameter of the generating circle, in equal times ; 
 and the time of describing each, is to the time of free motion, 
 through the diameter of the generating circle, as half the circum- 
 ference of a circle is to its diameter. 
 
60 
 
 OF MOTION. 
 
 [Book II. 
 
 If the constant force g-, that acts at the point C of the curve, 
 be resolved into two components, one of which is parallel, the 
 
 JL _*' 
 
 other perpendicular to the tangent at that place ; as the tangent 
 will coincide with the element of the curve ds, that part which re- 
 mains to cause the motion in the curve, will be to the whole accele- 
 rating force, as CB to DB. But of these, DB is a constant quan- 
 tity, and BC is half of the arc CE. Hence the force that acts 
 at any part of the curve, is proportioned to its distance from the 
 lowest point. Now if two different arcs of the curve, both ter- 
 minating at the point E, be supposed to be divided into an equal 
 number of elements, in each of which the velocity is constant ; 
 the forces will be proportioned to these elements, and they will 
 in consequence be described in each case in equal times ; and as 
 the number of elements are equal, the sums of these times will 
 be equal also, and both arcs will be described in equal times. 
 The whole semicycloid, FE, will therefore be described in the 
 same time with any of its smallest arcs. 
 
 The time of describing the semicycloid is easily found from 
 the general expression of variable motion, (53), 
 
 ds 
 
 hence 
 
 ds 
 
 (81) 
 
 but for the semicycloid, the diameter of whose generating circle 
 is a, 
 
 ive therefore obtain for the integral of this expression (81) 
 
 t=l<S~ g (82) 
 
 * being the ratio of the circumference of a circle to its diameter. 
 
 The time of the fall through the radius of the generating circle is 
 
 a tjf 
 
 V~ ; to which the last value of t has the ratio of ~ : 1. 
 
 o ^ 
 
J&00k //] OF MOTIOlf. 61 
 
 60. If the body describe a circular arc of very small size, the 
 time of describing it is the same as that in which an arc of a cy- 
 cloid is described, if both terminate at the point where the direc- 
 tion of the force is perpendicular to the curve ; but if the arcs have 
 any amplitude, the time of describing the circular arc will be 
 greater than that of describing the arc of a cycloid, by the quan- 
 tity ; in which h is the versed sine of the arc described, and 
 a the radius of the circle. 
 
 Suppose the body began to move from P, under the action of 
 an accelerating force, acting parallel to the radius C A, and to have 
 reached the point M. 
 
 Let AR=# ; RM=t/ ; AM=s ; the radius CA a, and AS=/s ; 
 we have for the point M, from the nature of the circle, 
 
 adx 
 
 and the velocity at M being due to the height SR, 
 
 g being the measure of the constant accelerating force ; hence 
 ds a dx 
 
 ' V(hx).(2ax x 2 ) ' 
 which can only be integrated by means of a series. For this pur- 
 pose the equation is resolved into the following form,* 
 
 dt=: 
 
 2a 
 
 2V g' V(hx-x 2 ) 
 which developed, and integrated from x=h to #=0, gives, 
 
 See Venturoli, Porsson, and Laplaae. 
 
OP MOTION. [Book II. 
 
 z h 1.3 2 h? 
 
 + &c -- < 83 > 
 
 In small arcs' the first term of the series is alone necessary, and 
 the expression becomes 
 
 '=5^ x 0+15)5 
 
 in very small arcs the series vanishes altogether, and 
 
 <=|V|. (84 a) 
 
 61. When the accelerating force, instead of acting parallel to 
 itself, is directed to a fixed point, it is called a central force. 
 
 It will be obvious from what has already been said, thbt a 
 point acted upon by a projectile force, and afterwards drawn to- 
 wards a centre by an accelerating force, must describe a curve. 
 
 The curve thus described is called a Trajectory, or Orbit. 
 
 If the point after moving completely around the centre of force, 
 again describe the same path, the orbit is said to be re-entering. 
 
 The time in which a re-entering orbit is described, counting 
 from the instant in which the moving point sets out from a given 
 position in the curve, until it return to it again, is called the 
 Periodic Time. 
 
 A line drawn from any point in the orbit is called a Radius 
 Vector. 
 
 62. The simplest case of central force is, where a body, con- 
 nected to a fixed point by an inflexible straight line, is impelled 
 by a projectile force at right angles to that line. The latter force 
 would have impressed upon the body, a motion with an uniform 
 velocity. The body will then, in consequence of its connexion 
 with the fixed point, describe a circle, of which that point is the 
 centre. If the connexion were to cease at any point in the curve, 
 the deflecting force would cease to act, and the body would go in 
 a straight line, whose direction would be a tangent to the curve. 
 The force acting at any point in the curve, must therefore be decom- 
 posed into two, one of which is in the direction of the curve, the 
 other in that of its radius. The last of these is called the centri- 
 fugal force, and is equal and directly opposite to the force that 
 draws the body towards the centre, and which, for distinction 
 sake, is called the centripetal force. 
 
 In elementary treatises, the consideration of the action of cen- 
 tral forces, is usually limited to the case of motions in circular 
 orbits, described under circumstances analogous to those we have 
 recited. If instead of an inextensible straight line, an attractive 
 force equal and directly opposed to the centrifugal force that has 
 
Book If.] OP MOTION. 63 
 
 just been defined, be substituted, the circumstances will be the 
 same as if the connexion were made by theinextensible line, and 
 the body will in like manner describe a circle. The relations 
 among the forces, velocities, times, and spaces, that are found in 
 respect to circles, may be applied to the case of other curves; for 
 a motion in a curve may, for a short space of time, be considered 
 as corresponding with that in a circle, whose radius is the radius 
 of curvature of that part of the curve. 
 
 63. In describing a circular orbit, under the action of two 
 forces, one of which is projectile, the other constant, and directed 
 to the centre of the circle, a point will move with uniform ve- 
 locity. 
 
 Let the point be situated in the curve at A, and suppose a pro- 
 
 G 
 
 jectile force to act, which in an infinitely small time, would carry 
 it forward in the direction AB, to the point B ; the central force 
 would in the same time cause a deflection to D. If the central 
 force were not to cease to act, the point would go on in the direc- 
 tion AD produced, and describe in an equal portion of time a 
 space DE equal to AD ; but the central force continuing to act, 
 brings it, at the end of an equal interval, to the point F ; and in 
 this way it would be found at the end of the third interval at H. 
 Now it is obvious that the- several triangular spaces CAB, CAD, 
 CDE, CDF, CEG, CFH, are all equal, and hence the radius 
 vector describes equal areas in equal times ; these areas are how- 
 ever in fact, sectors of the curve ; and in a circle, whose radii are 
 all equal, if the centre be also the centre of force, the arcs which 
 are described by the moving point, are also equal in equal times ; 
 in uneqiial times, proportioned to the times ; and the circle is de- 
 scribed with uniform velocity. 
 
 64. In circular orbits, the force is represented by the square 
 of the velocity divided by the radius. 
 
 The projectile velocity is such as would have been acquired 
 by falling freely, under the action of the central force, through 
 half the radius of the circle. 
 
64 
 
 OP MOTION. 
 
 [Book II. 
 
 If points revolve in different circles, the central forces are di- 
 rectly as the radii of the orbits, and inversely, as the squares of 
 the periodic times ; if the times be equal, they are therefore as 
 the radii ; if the radii be equal, they are inversely as the squares 
 of the periodic times. 
 
 Let a point describe a very small arc, AB, in the element of 
 C the time dt. During this small period, the versed 
 sine, AD, will be the space described under 
 the action of the central force ; this force may 
 therefore be measured by the velocity that would 
 have been acquired in moving through AD with 
 accelerated motion; the formula (61) gives 
 
 2s 2s 
 
 t?=-^-, whence f~p* 
 
 In this expression, 2* is twice the versed sine of 
 AB ; the very small arc, AB, may be consider- 
 ed as coinciding with its chord, and the chord 
 
 is a mean proportional between the radius and the versed sine ; 
 
 call the arc AB, o, and we have 
 
 2 a 3 a 2 
 2s=- 
 
 2r 
 
 and 
 
 now as the velocity in the circle is uniform, 
 
 whence 
 
 =-j and v*=(-.) ; 
 
 (85a) 
 
 /=v . ( 85 ) 
 
 To compare the intensity of this force with that of the central 
 force g, supposed to be constant ; from (61) 
 
 v 2 =gs, 
 whence 
 
 (86) 
 
 and when /is equal to g, 
 
 the space then to which the velocity is due, is equal to half the 
 radius. 
 
Book II.] or MOTION. 65 
 
 Let T be the periodic time, the circumference of the circle is 
 r, and (85a) 
 
 == ~T~ ' (87) 
 
 substituting this value of v in that of/, we have 
 
 (88) 
 
 whence the forces are directly as the radii of the orbits, and in- 
 versely, as the squares of the periodic times. 
 
 65. If the central force be such as varies in the inverse ratio of 
 the squares of the distances, the squares of the periodic times are 
 proportioned to the cubes of the distances. 
 
 If two forces, F and /, act in different circles, with intensi- 
 ties inversely proportioned to the squares of the distances, by (85) 
 V 2 v 2 
 
 F== RT' / = T' and 
 
 F:/: : V 2 : v 2 ; 
 but by hypothesis, 
 
 F:/::^:^; 
 
 therefore 
 
 V 2 u 3 
 
 . . . 2 . R2 . 
 
 R ' r ' 
 whence 
 
 V 2 : v 2 : :r 3 :R 3 ; 
 and as the times are inversely as the velocities, 
 
 T 2 : i 3 : : R 3 : r 3 , (89) 
 
 or the squares of the periodic times, are as the cubes of the radii. 
 
 66. Having thus investigated the circumstances of motion, in 
 those lines and curves that will be of most frequent application 
 in practice, by more elementary processes ; we shall next exhi- 
 bit the general principles that are applicable to any case what- 
 soever, by means of the resolution of all the forces into three, 
 parallel to three rectangular co-ordinates. 
 
 Call the resistance of the surface N; the angles that the rectan- 
 gular axes make with the normal to the surface a, 6, c ; the com- 
 ponents of the resistance will be (18) 
 
 N cos. a, N cos. 6, N cos. c ; (90) 
 
 the values of the three rectangular components of the active 
 forces are by (62) 
 
 v_^.' Y _^ 7 -^ 
 *-~dP ' *~ dP ' *~ df" 
 
65 or MOTION. [Book II. 
 
 to each of which must be added the component of N that acts in 
 its direction ; whence we have, for the equations of motion upon 
 a curve, 
 
 X-f-N cos. a ~~p ; 
 
 Y+N 
 
 cos. = 
 
 Z-f N cos. c- 
 
 (91) 
 
 the values of the velocities from (63), are 
 
 dx dy dz 
 
 dt' dt' df> 
 
 these may be reduced to the direction of the normal, by multi- 
 plying each by the cosine of the angle it makes with that line ; 
 and as the direction of the motion is at right angles with the nor- 
 mal, the sum of these three components must be 0, or 
 
 cos. + 37 cos. 6-f cos. c=0. (92) 
 
 dt dt dt 
 
 We may eliminate the unknown quantities in the three equa- 
 tions, (91) by adding them together, after multiplying the first by 
 dx, the second by dy, and the third by dz ; the sum of this addi- 
 tion becomes, if we take into account the equation (92), 
 
 "^y^^MH-TdH-Zfe (93) 
 
 If the quantity Xcta+Xcfy-t-Xcte, is the exact differential of the 
 three variable quantities, x, y, z, we obtain by integrating 
 
 *tbJ+*?= 1/( ,,,^ + C, (95) 
 
 dx dii dz 
 now as ~ f , are the components of the velocity in the 
 
 direction of the three axes ; the first member of this expression 
 represents the square of the velocity, and 
 
 v*=2f(x , y , z}. (96) 
 
 It may be remarked that there are several cases in practice, 
 where the expression Xdx-\-Ydy -\-Zdz, is not an exacf differential 
 of three variable quantities ; such for instance is the case, where 
 a body is acted upon by a fluid resistance, or by friction. 
 
 Where the three rectangular forces, X, Y, and Z, are each 
 equal to 0, 
 
 />,?/, z) = 0, 
 and 
 
 t> 3 =C, 
 or the square of the velocity, and consequently the velocity itself, 
 
Baok II.] oy MOTION. ($ 
 
 is constant. Thus, then, when the accelerating force ceases to 
 act, the body will continue to move with uniform velocity. 
 
 .If the forces, X, Y, and Z, retain a determinate magnitude, the 
 velocity is no longer constant, but it is independent of the curve 
 the body is compelled to describe ; for if we call A the velocity 
 that corresponds to another point of the curve, whose co-ordinates 
 are, a, 6, c; we can infer the same in respect to this, as to the ori- 
 ginal case, and 
 
 A 2 =C+2/(a , 6 , c), 
 which substracted from the former equation, gives 
 
 v*A 2 =2f(x ,y,z, ) 2/(a , 6 , c) ; (97) 
 
 it is obvious therefore, that the change in the squares of the velo- 
 cities, has no relation to the form of the curve described between 
 the two points. So also, when several bodies set off from a given 
 point, under the action of the same accelerating force, they will all, 
 on reaching another point, have the same velocities, however vari- 
 ous may be the lines described in the mean time, and different the 
 times of describing them. This is an obvious generalization of 
 what was found, in 58, to happen, when the motion took place in 
 plane curves, under the action of a constant force. 
 
 The further pursuit of this subject, by higher methods of analy- 
 sis, leads to a remarkable law which is called The Principle of 
 the Least Action. It may be thus expressed : 
 
 When a body is acted upon by forces whose relation is expressed 
 by the formula, (95) and the curve is not determined, it will choose 
 for the direction of its motion the shortest line that can be drawn 
 on the surface, or that in which the integral, fvds, is a minimum. 
 
 When the accelerating force ceases to act, the velocity is, as 
 we have seen, constant, and 
 
 fvds=vs, 
 
 and as s is the shortest space, the body will pass from one point 
 to another, in the shortest possible time. 
 
 For the further illustration of this subject, we refer to Bowditch's 
 translation of Laplace. 
 
 67. When a point is acted upon by but one accelerating force, 
 constantly directed towards a fixed point, the path will be a plane 
 curve; and the areas, described around this point by the radius 
 vector, are proportioned to the times employed in describing 
 them. 
 
 By (62), 
 
 ^_ Y <^_ #Z 
 
 W ' df~ ' ~d?~* 
 
 these are readily transformed into the following 
 
 7/X) . dt* 
 xZ} . d? 
 zY . df. 
 
68 OP MOTION. [Book II. 
 
 Let us suppose the origin of the co-ordinates to be at the centre 
 
 B 
 
 of force C, and let the line, CA, which joins the point in which 
 the body is at any given time, to the centre, represent the central 
 force. It will be obvious that the three sides of the parallelepiped 
 which represent the three rectangular components of CA, will be 
 the co-ordinates of the point A, and for any other magnitude of the 
 central force, they will be proportioned to these components ; 
 hence, 
 
 X : T : Z : : x : y : z, 
 and 
 
 *Y=7/X , z X=*Z , y Z=zY 
 xY yX=0 , zX a:Z=0 , yZ-zY=0, 
 
 and in the former equations the second members become =0 ; 
 integrating these, we have 
 
 xdy ydx=cdt 
 
 zdx xdx=c'dt 
 
 ydz zdy=c"dt. 
 
 If these expressions be multiplied, the first by 2, the second by 
 t/, the third by ar, they may be added, and we have 
 
 cz+c'y+c"z=0, 
 
 which is the equation of a plane surface ; whence we may infer 
 that the orbit is a plane curve. Take now the projection of the 
 point A, on the plane of #, y ; let A' be this projection ; the radius 
 vector, C A'=r ; and i?, the angle that determines its direction, in 
 respect to the axis parallel to x ; by (18) 
 
 x=r cos. v, yr sin. 0, 
 whence 
 
 xdy ydx=r 2 dv ; 
 
 and as the first number of this equation is, as we have seen, a con- 
 stant quantity, so also will be the second. But we may suppose 
 
Book //.] OF MOTION. 69 
 
 the arc of the curve, described by the body in passing between 
 the two successive positions, marked out by the small angle du, 
 to be a circle ; and the area described by the radius vector as a cir- 
 
 o _j 
 
 cular sector whose area is - , and this being the half of a con- 
 stant quantity, is itself a constant quantity ; ij; will also be equal to 
 ' and consequently, the area described in t is , wherefore 
 
 the area described during a given time, is proportioned to the times. 
 The areas then, that are described by the radius vector of the pro- 
 jection of a moving point, upon a plane passing through the centre 
 offeree, are proportioned to the times. Now as the orbit of the 
 body is itself a plane curve, and as the direction of the plane of 
 projection is arbitrary, this proposition is equally true in respect 
 to the areas described by the radius vector of the body itself, upon 
 the surface of the plane curve which forms the orbit. 
 
 This proposition is the same which has been investigated in 
 63. And is not only true of circular orbits, but of any orbits 
 whatsoever. 
 
 68. This same proposition may be deduced from more simple 
 considerations. In the first place, the curve described under the 
 action of a central force, must be a plane ; for, the direction of the 
 motion, in the first element of the time, will be in the plane formed 
 by its two components, the projectile, and the central force ; the 
 direction in which the moving point would tend to go on, were 
 the central force to cease to act, will likewise be in the same plane, 
 and the line, which joins the extremity of this tangential force 
 to the centre, will be in this plane also ; this line represents the 
 direction of the central force, during the second element of the 
 time, and the resultant is therefore still found in the original 
 plane ; and so of all the successive resultants, which make up the 
 curve, which lies therefore in one plane. 
 
 We have seen from 33, that the sum of the Moments of Ro- 
 tation of any number offerees, is equal to the Moment of Rota- 
 tion of their resultant. Now if we suppose the curve to be a 
 polygon of an infinite number of sides, and the velocities in each 
 of these to be uniform, the spaces described will be the measure 
 of the several forces ; the moment of rotation of the projectile 
 force will be the tangential line 4fy which represents that force, 
 
Of MOTION. 
 
 [Book H. 
 
 multiplied by the radius vector CA, or twice the triangle ABC ; the 
 moment of rotation of the force that acts in the 
 curve, is the rectangle under AC, and AD, or 
 twice the sectoral space, ADC. And as the di- 
 rection of the central force is in the radius 
 vector, its moment of rotation =0 ; hence the 
 twomoments of rotation, of the tangential force, 
 and of the force in the curve, are equal, as are al- 
 so their respective halves. At the point D, the 
 tangential force is that with which the curve 
 has been described ; its moment of rotation is 
 therefore equal to twice the space DCE, and 
 for the same reason, it is equal to twice the 
 second sectoral space, ADC, which is therefore equal to the sec- 
 toral space, ADF ; and these are the spaces described by the ra- 
 dius vector, in the equal elements of the time; hence the greater 
 spaces which these make up, will be proportioned to the times. 
 
Book //.] o MOTION. tl 
 
 CHAPTER VI. 
 
 PRINCIPLE OF D'ALEMBERT. 
 
 69. The formulae and conditions of the motion and equilibrium 
 of points, may be applied to systems of material points, or to such 
 bodies as actually exist in nature, by means of a self-evident prin- 
 ciple, first announced by D'Alembert, and which has been em- 
 ployed by all succeeding writers on Mechanics of any well 
 founded reputation. It may be expressed as follows : 
 
 If there be a body, or system of points materially connected in 
 any manner with each other, and which are acted upon by forces 
 given in magnitude and direction ; the action of these several 
 forces is modified by the connexion among the several points, 
 and they neither move in the direction, nor with the velocity 
 they would have, were they not connected. Still the forces that 
 must be compounded with those that cause the motion, in order 
 to make up the forces with which the points actually move, must 
 be such as are in equilibrio with each other, or that if they acted 
 upon the system alone, would produce no motion. The last 
 mentioned forces obviously represent the mutual action of the 
 points upon each other ; these could not of themselves cause mo- 
 tion, and are therefore in equilibrio. 
 
 To express this analytically, let the forces applied to the points 
 be represented by the velocities a, 6, c, &c. ; the velocities the 
 points actually have, by a b\ c ; the velocities which must be 
 combined with the first of these, in order to produce the last, a'', 
 6", c", &c., then 
 
 a"-f-fc"+c"-f&c.=:0; 
 
 for it is obvious that these velocities have no effect upon the act- 
 ual moving force of the system, which is due to the velocities a, 
 6, c, &c., alone ; and hence the forces which they represent mu- 
 tually destroy each other. 
 
 To apply this to systems of bodies, we must consider that the 
 moving force of each will depend not only on its velocity, but on 
 the number of equal material points it contains ; therefore, the 
 force, by which each body is impressed, is due to the product of 
 its number of points into its velocity; calling the former, A, B, 
 C, &c., we have 
 
 Aa"-r-B&"4-Cc"-|-&c.=0. (98) 
 
72 OF MOTION. [Book II. 
 
 CHAPTER VII. 
 
 PRINCIPLE OF VIRTUAL VELOCITIES. 
 
 70. If we suppose the equilibrium of any system of forces 
 whatsoever to be disturbed for an instant, and that each point to 
 which a force is applied, has a velocity, such as would have been 
 given to it, in the direction of the disturbance, by the force which 
 acts upon it. The velocities that the several points would ac- 
 quire, are called their Virtual Velocities. 
 
 71. In any system of forces whatsoever, the sum of the pro- 
 ducts of the several forces, into their respective virtual veloci- 
 ties, is equal to 0, or calling the forces, F, F', F", the virtual ve- 
 locities, v, v', v", 
 
 Ft>+FV+FV+&c.=0. (99 ) 
 
 Forces can only act, under certain definite circumstances, upon 
 their points of application. 
 
 1. All the forces may he applied to a single free point. 
 
 2. The point of application of the forces may be compelled to 
 rest upon a given surface. 
 
 3. It may be compelled, when it does move, to move along a 
 given surface. 
 
 4. The forces may act upon a system of points, united toge- 
 ther in any possible manner. 
 
 5. The system may have a fixed point, around which the others 
 must move, or a point that is compelled to describe a given sur- 
 face. 
 
 First. Let the forces act upon a single point, C, in the figure 
 beneath; let FC, FC', FC", &c., be their several directions 
 and RC, the .direction of their resultant R. 
 
 Draw a line, Cd, to represent the direction of the disturbance, or 
 that in which the point is supposed to move, and the distance to 
 
Book IL] OF MOTION. 73 
 
 which it is removed. This line will therefore represent the vir- 
 tual velocity of this point. Decompose each of the forces into 
 two others, one parallel to the line Cd, the other perpendicular to 
 it; the respective values of the components vill be, calling the 
 angles that they respectively make with Cd, a' and 6, 6', 6", &c., 
 
 R cos. #, F cos. 6, F cos. 6', F cos. 6", &c. 
 
 Now if perpendiculars be let fall from the point d, upon the several 
 directions of these forces, the angles these will respectively make 
 with the line Cd, will be equal to the angles the directions of the 
 forces make with it at the point C. The value of the projections, 
 r, /,/,/", &c., will therefore be symmetric with the values of 
 the components of the forces parallel to Cd, or callingthe lineCd, c, 
 
 r=c cos. a, f = c cos. 6, f'c cos. 6', &c. 
 
 Now, as the sum of the components of any force, estimated in a 
 given direction, is by (19) equal to the component of the force 
 estimated in the same direction, 
 
 R cos. a=F cos. 6-f-F' cos. &'-j-F" cos. 6"+&c. 
 
 multiplying this by the value of c, obtained from the foregoing ex- 
 pressions, we obtain 
 
 Rr=F/+F'/+F'/"+&c. 
 
 but r is the virtual velocity of the point C, and/,/',/", &c., are 
 obviously the components of that velocity, in the several directions 
 of the component forces, and are therefore so much of the vir- 
 tual velocity as is due to the action of the respective forces ; sub- 
 stituting then the letters u, i/, u", &c. for/,/,/', &c., and sup- 
 posing the system to be in equilibrio, we obtain 
 
 F+FV+FV'+&c.=0. 
 
 Second. In the case of the system of forces being such as 
 causes a point to rest upon a given surface, the action of this sur- 
 face may, 19, be represented, by introducing a force normal to 
 the surface in its direction, and equal to the resultant of the other 
 forces, which is also a normal to the surface. This force may 
 therefore be introduced among the forces in the above equation ; 
 calling it P, and the virtual velocity it causes, j>, we have 
 
 Pp-hF/fF/-r-F'/''-f&c.==0; 
 , and 
 
 F/-f-F/-j-F"/'+&c.=0 ; 
 
 the proposition is therefore true in respect to points compelled to 
 rest upon a surface by the action of the forces. 
 
 Third. The same reasoning applies to the case where the point 
 is compelled, if it move, to move along a given curve ; the mo- 
 tion being very small, may be considered as taking place in the 
 direction of the tangent to the curve, or perpendicular to the di- 
 10 
 
74 09 MOTION. [Book If. 
 
 rection of P ; the virtual velocity of p is therefore equal to 0, for 
 the angle of inclination becomes 90' ; and 
 
 P/>=0; 
 hence again, 
 
 F/+F'/-fF"/"+&c.=0. 
 
 ^- 
 
 Fourth. If the forces act upon points composing a system, in 
 which they are united in any manner whatsoever ; call the forces, 
 which we shall for the present restrict to three in number, F, F', 
 F". Each point is held in its position, in the case of equilibrium, 
 by actions exerted in consequence of its connexion with the other 
 points in the system ; these actions may be represented by forces, 
 coinciding, in their several directions, with the lines that join each 
 pointto the others which compose the system, and these forces will 
 have a determinate magnitude. Call the forces, that represent 
 these mutual actions, on the three several points, A, A', B, B', 
 C, C'. Now as each point will be in equilibrio under the action 
 of three forces, that which is applied to it, and the two which are 
 exerted upon it by the others, the principle of virtual velocities 
 holds good in respect to each of these three sets offerees, or 
 
 F'/"+Cc-f CV=0 ; 
 but the points being mutually connected, 
 
 A'=B, B'=C, C'=A ; 
 hence 
 
 F/+Aa+Ba'=0, 
 
 "/'+ Cc+Ac'=0 ; 
 adding these equations, 
 
 But the equilibrium of the system would subsist, if the forces F, 
 F', F", were suppressed, and the system supported by forces 
 equal and opposite to A, A', B, B', C, C', applied to the several 
 points ; for each of these pairs of forces has for its resultant a 
 force equal and opposite to F, F', and F" respectively. The 
 principle of virtual velocities is therefore applicable to this new 
 set of forces also ; or 
 
 A(a+c') B 
 subtracting this equation from the former, we have 
 
 And although our reasoning has, in order to prevent complexity, 
 been restricted to three points and three forces ; it is obvious 
 that it might have been extended to any number whatsoever. 
 
//.] op MOTION. 75 
 
 As we have assumed the points to be connected in any manner 
 whatsoever, it is obvious that the principle is applicable to all cases 
 of free systems of material points, whether they be united by in- 
 variable and inflexible lines, or by the loose aggregation that takes 
 place in the particles of fluids, or by any intermediate connexion 
 between that which is fixed and invariable, and that in which the 
 connexion is about to cease altogether. 
 
 Fifth. In the same manner, precisely, in which the case of 
 forces, acting upon a single free point, has been applied to those 
 of forces acting upon a point compelled to rest or to move upon a 
 given surface, may the above case be extended to those of systems 
 that have in them a fixed point, or a point compelled to move 
 upon a given surface, and thus the principle of Virtual Velocities 
 may be shown to be true in all possible cases. 
 
BOOK III. 
 
 OF THE EQUILIBRIUM OF SOL.ID BODIES, 
 
 CHAPTER I. 
 
 GENERAL PROPERTIES OF MATTER. DIVISION OF NATURAL BODIES. 
 MEASURE OF THE MOVING FORCES OF BODIES. 
 
 72. In order to apply the preceding principles to the circum- 
 stances which occur in nature, it becomes necessary, that we 
 should investigate not only the general abstract properties of mat- 
 ter, but also those which we find inherent in the greater part, if 
 not the whole, of these material substances that we can make the 
 objects either of experiment or observation. 
 
 These essential properties of matter, without which we cannot 
 conceive it to exist, are, Extension, Mobility, and Impenetrability. 
 
 73. Matter must occupy a certain portion of space, and thus be 
 extended in three dimensions, or is of that class of Geometric 
 magnitude, which is called a Solid. 
 
 74. Beingextended, it is of course capable of division, and were 
 we to reason in respect to this secondary property, as if the ma- 
 terial substance had the same geometric properties with the space 
 it occupies, it might be demonstrated that matter is infinitely di- 
 visible ; and thus, in many works on Mechanical Philosophy, it 
 has been inferred, from strict mathematical reasoning, that this is 
 the fact. It is however, obvious, that the reasoning which is ap- 
 plicable to mere geometric magnitude, has reference only to the 
 space occupied by matter, and not to matter itself. We may, there- 
 fore, reasonably doubt whether matter be infinitely divisible. 
 
 Still, the divisibility of matter may be carried to an extent, that 
 exceeds the limits that our senses can reach : thus by mere me- 
 chanical division, as in the hammering and drawing of metals, 
 the divisibility of the substances is very remarkable ; gold, when 
 manufactured into leaf, is beaten out, until the thickness of the 
 sheet does not exceed anVurth P art f an inch. 
 
7S Of THE EQUILIBRIUM [Book III. 
 
 In the mixed mechanical and chemical arts, divisibility is car- 
 ried still farther ; in making gold lace, for instance, where a rod 
 of silver is gilt by means of an amalgam, and then drawn into wire, 
 a single grain of gold covers 9600 square inches ; and when the 
 wire is flattened, the surface covered is nearly doubled in extent. 
 
 Chemistry furnishes still more marked instances of divisibility ; 
 particularly in the manner in which the colours produced by che- 
 mical tests and re-agents, are diffused throughout spaces of con- 
 siderable magnitude, by quantities of substances that are hardly 
 appreciable : 
 
 There are also substances, that we know by our senses to exist, 
 or can distinguish by the effects they are capable of producing, 
 which escape the nicest methods of chemical research ; thus, the 
 matter of which many of the most powerful odours is composed, 
 has not been detected by any analysis of the air which is its ve- 
 hicle ; and the pestilential miasmata which cause disease, and 
 even certain death, have baffled the most powerful, as well as 
 the most delicate methods of inquiry : 
 
 Another proof of the great divisibility of matter, may be drawn 
 from the minuteness of the living animals that can be distinguished 
 by the aid of the microscope ; Leuwenhoeck saw animals not ex- 
 ceeding the ten thousandth part of an inch in length, and each of 
 these is furnished with organs, has vessels in which fluids circu- 
 late, whose particles therefore must be small beyond all con- 
 ception. 
 
 Mere physical action is also capable of exhibiting a very exten- 
 sive divisibility, in the matter that is subject to it ; thus water, 
 when heated beyond a certain temperature is converted into steam, 
 which occupies about seventeen hundred times as much space as 
 the water whence it is generated, and yet no perceptible portion 
 of the space is devoid of aqueous matter. 
 
 75. The term Body, is employed to denote a determinate 
 quantity of matter, contained under some known figure, or exist- 
 ing in some peculiar mode. By mechanical, physical, or chemi- 
 cal action, influencing the divisibility of matter, a body may be 
 disintegrated ; its component parts wilt assume new forms and 
 characters, and constitute new bodies. By such forces constantly 
 acting in nature, the surface of the globe, and all the bodies that 
 we find upon it, are undergoing continual changes ; and are con- 
 stantly assuming new forms, or entering into new combinations. 
 In all these changes, however, no portion of matter is annihilated, 
 but its quantity continues invariable. We know in fact of no 
 agent in nature, that is capable of increasing or diminishing the 
 amount of the matter that exists in the Universe. Matter, there- 
 fore, might be inferred to be eternal ; but as we find that it is 
 
Book III.} OF SOLID BODIES* 79 
 
 not only a metaphysical truth, but one that the process of induc- 
 tion enables us to deduce, from a full investigation of all the phe- 
 nomena of nature : that, nothing can exist in any state whatever, 
 without some sufficient reason ; we have a right to infer, that the 
 permanence of matter, is the result of the action of a great and 
 all powerful first cause. 
 
 76. The indestructibility of matter might be urged as a proof 
 that it cannot be infinitely divisible, for that property could only 
 result in its annihilation. But we are enabled to draw more con- 
 clusive evidence^of the fact, that the divisibility of matter is finite, 
 from the modern discoveries of chemistry. In this science, we 
 are enabled to discover, and explain many important facts, by 
 means of the hypothesis, called the Atomic Theory ; this holds, 
 that all bodies are finally resolvable into particles, incapable of 
 farther division. From what has been said, it will however ap- 
 pear, that these atoms must be so small as to escape the mdst 
 acute of our senses, even when furnished with the most pow^r- 
 ful aids that the high improvement of the arts at present affords. 
 The proof of the existence of atoms, therefore, is not, and proba- 
 bly cannot be, complete ; for to constitute a theory that can be 
 received as absolutely true in philosophy, it is not only necessary 
 that it should fully explain all the phenomena, but that it shoild 
 be founded on evidence entirely independent of them. Stilljas 
 in the most minute state of division, to which bodies can be De- 
 duced in practice, we find them retaining their peculiar and infli- 
 vidual properties, we have a right to infer, that their divisibility 
 is in no case carried to an infinite extent ; we may, therefore, 
 assume, that they are made up of portions infinitely small ; th^se 
 are usually called Particles. 
 
 77. The term Impenetrability, as applied to denote a property 
 of matter, merely implies, that no two particles can occupy ihe 
 same space, at the same time. In this point of view, the hardest 
 and the softest substances are equally impenetrable. Impenetra- 
 bility is absolutely essential to the existence of matter, and has 
 an experimental proof in its indestructibility; for, were it not 
 impenetrable, two portions of matter might be reduced to a 
 single one, and one of them annihilated, which we have seen to 
 be impossible. It is to this properly, of excluding all other por- 
 tions of matter from the space that themselves occupy, that we 
 have recourse, in ascertaining the material existence of certain 
 substances, concerning which, inaccurate notions long prevailed. 
 
 78. The property of mobility is also deduced from experiment 
 and observation, for we find no body that cannot be set in mo- 
 tion, by an adequate force, nor any whose motion cannot in like 
 
80 OP THE EQUILIBRIUM [Book III. 
 
 manner, be arrested. It might also be shown to be a conse- 
 quence of the Impenetrability of matter, for as a body resists the 
 entrance of any other, into the space itself occupies, motion must 
 arise whenever a body, impelled by any force whatever, tends 
 to enter the space another previously occupies. 
 
 The sum of all the particles in a body, constitutes its mass, or 
 makes up its quantity of matter ; and in homogeneous bodies, the 
 quantity of matter is obviously proportioned to their respective 
 bulks. 
 
 In heterogeneous bodies, the quantity of matter has respect 
 not only to the bulk, but to the Density; the latter is the rela- 
 tion that the quantity of matter, in a given body, bears to its bulk. 
 
 This definition may be thus expressed : 
 
 D=^; (100) 
 
 whence we have 
 
 M 
 M=DB, andB=jj; (101) 
 
 The Densities then, are directly as the quantities of matter, 
 and inversely as the Bulks : 
 
 The Quantities of matter, are in the compound ratio of the 
 Densities and Bulks; and 
 
 The Bulks, are directly as the quantities of matter, and inversely 
 as ihe Densities. 
 
 79. Bodies differ from each other, in the greater or less diffi- 
 culty with which their particles may be separated. When the 
 separation of the particles requires the application of a determinate 
 force, the body is said to be solid ; when they may be divided, by 
 a force so small as to be inappreciable, they are said to be fluid. Of 
 flu ds, some have so small a capability of having their bulk changed 
 by pressure, as to have been considered as absolutely incompres- 
 sible ; such bodies are called Liquids ; of them, water at ordinary 
 temperatures is a familiar instance. Others again, are capable of 
 being compressed, and of occupy ing larger spaces, when the com- 
 pressing force is removed ; these are styled Gases, or Elastic 
 Fluids ; of these atmospheric air, may be cited as the type. 
 
 80. Many bodies are familiarly known to be capable, under 
 different physical circumstances, of existing in all the three dif- 
 ferent states : thus, water, when cooled below a certain tempera- 
 ture, passes into the solid form, and becomes Ice ; while if heat- 
 ed, beyond a certain limit, it assumes the gaseous state, and is 
 called Steam. As a general rule, deduced from various chemical 
 and purely physical facts, every body in nature is capable of assu- 
 
///.] OF SOLID BODIES. 81 
 
 ming, under proper circumstances, all of these three mechanical 
 states. 
 
 81. Heat is the great natural agent that is concerned in these 
 mechanical changes, and it is a general rule, that heat, existing in 
 that modification in which it is said to be latent, determines the 
 mechanical state that bodies assume. 
 
 When the latent heat is withdrawn, the body returns to its 
 original state ; thus steam, on parting with its latent heat, be- 
 comes water ; and water, on parting with its latent heat, becomes 
 ice : hence, we infer the action of another force in opposition to 
 that of heat; to this force, whose cause is, like that of heat, un- 
 known to us, we give the name of the Attraction of Aggregation. 
 These two great natural antagonist forces then, determine the 
 mechanical states in which bodies exist. 
 
 When the attraction of aggregation predominates, the body is 
 a Solid. 
 
 When they are equally balanced, the body is a perfect Liquid ; 
 and when the force of heat, exceeds that of the attraction of ag- 
 gregation, the body becomes an Elastic Fluid. 
 
 It will be hereafter seen, that no body has a constitution that 
 will entitle it to the name of a perfect liquid. In all known bo- 
 dies of this class, the force of attraction still preponderates over 
 that of heat, as will be made manifest when we examine the forms 
 in which small masses of liquids arrange themselves. 
 
 82. The attraction of aggregation is only known to us as act- 
 ing at insensible distances ; it is therefore unnecessary to enter 
 into any investigation of the ratio, in which its intensity dimi- 
 nishes, as the distance increases. Its absolute measure differs in 
 every different body, and must hence be determined experimen- 
 tally ; it constitutes the strength of the materials that are em- 
 ployed in practical mechanics, and will, under this name, become 
 the object of future consideration. Although the general princi- 
 ples of the two first books, apply equally to all bodies, whatever be 
 their state of aggregation ; still, they are modified in their action 
 by the peculiar mechanical properties of the different classes into 
 which we have found them to be divided. It is hence necessary 
 to consider the Mechanics of Solid, and of Fluid Bodies, sepa^ 
 rately. 
 
 83. In our previous inquiries, forces have been considered in 
 the abstract, and as acting upon material points of indeterminate 
 magnitude, although we have occasionally been compelled to use 
 the term Body. It ROW becomes necessary that we should be 
 able to estimate the force by which a body or aggregate of matter 
 is actuated. This, which is called the Quantity of Motion, may 
 
 11 
 
82 OF THE EQUILIBRIUM [Book III. 
 
 be ascertained from the following considerations. We may ob- 
 viously consider a moving body as made up of its particles, or of 
 a number of material points, and supposing its motion to be recti- 
 lineal, every particle will be actuated by a force, equal and paral- 
 lel to those which actuate the rest ; the whole force then will be 
 equal to the mass of the body, multiplied by the forces that actu- 
 ate its particles. If the motion be uniform, the velocity of each 
 of these particles is constant, and will represent the force by 
 which it is produced ; hence, the moving force of a body will be 
 represented by its mass, multiplied by its velocity. The same 
 will be obvious from other considerations ; for it is clear, that 
 when several bodies have equal velocities, that which has twice 
 the quantity of matter that another has, must have twice the quan- 
 tity of motion, and so on. In like manner, if the masses be equal, 
 the body which has the greatest velocity will have a quantity of 
 motion exactly proportioned to it. 
 
 This may be illustrated thus : 
 
 Let M be the mass of a body made up of the several particles 
 p, p'i p", &c., and v the common velocity, the sum of the several 
 forces by which they are actuated will be 
 
 ^v + &c. , 
 
 or pppc. Xv ; 
 
 and as the sum of the particles is equal to the mass, the expres- 
 sion for the quantity of motion,/, wiU become 
 
 /=Mr; (102) 
 
 whence 
 
 =^- (103) 
 
 From these equations the following consequences immediately 
 follow : 
 
 (1). The forces of moving bodies, or their^quantities of motion, 
 are represented by the products of their masses and velocities. 
 
 (2). In equal masses, the forces are proportioned to the velo- 
 cities. 
 
 (3). When bodies have equal quantities of motion, the velo- 
 cities are inversely as their masses. 
 
 (4). With equal velocities, the quantities of motion are pro- 
 portioned to the masses. 
 
Book ///.] OF SOLID BODIES, 83 
 
 CHAPTER II. 
 
 ATTRACTION OF GRAVITATION. 
 
 84. It is a fact demonstrated by universal experience, that all 
 heavy bodies, if unsupported, fall towards the surface of the earth. 
 This can only take place in consequence of the action of some 
 specific force, which has been supposed to reside in the earth it- 
 self, and which has been called the Attraction of Gravitation, or 
 more simply, Gravity. 
 
 85. The direction of this force may be ascertained by suspend- 
 ing a heavy body, from a fixed point, by a flexible string ; the 
 direction of the string will, as is obvious, mark the line in which a 
 body would tend to move, under the action of the force. 
 
 Such an apparatus is called the Plumb-Line. When adapted 
 to a ruler with parallel sides, it becomes a familiar and simple 
 instrument, of great practical utility. The plumb-line hanging 
 freely, being made to coincide with a line drawn along the ruler 
 parallel to its sides, the latter also point out the direction of gravity. 
 
 If a second ruler be adapted to the lower extremity of the first, 
 and make with it angles that are exactly 90, the face of the 
 second ruler will D*e found to adapt itself to the surface of stag- 
 nant waters; hence we infer that the direction of gravity-is per- 
 pendicular to the surface of standing water. We say however, 
 in general terms, that the direction of gravity is perpendicular 
 to the surface of the earth. 
 
 We are, in truth, compelled to refer the direction of gravity to 
 some more extended surface than that of the largest mass of 
 tranquil water : for it is a question that requires examination, 
 whether the direction of gravity, at a given place, be constant. 
 To ascertain this, it would be necessary to have some points 
 that we could consider absolutely fixed. Even the most stable 
 edifice has not sufficient permanence, to afford points that can be 
 relied upon as such. If, for instance, we were to find, that the 
 relative direction of the plumb-line and the wall of a building 
 had changed, we might at first sight doubt which had altered its 
 position ; for we know, that natural convulsions, or even gra- 
 dual decay, may alter the direction of the firmest walls. Even 
 the sides of a mountain could not be relied upon, for we know 
 of mountains being affected by earthquakes. 
 
 The sea, however, apparently unstable as it at first sight appears, 
 has, in its mean level, and in the limits within which its ebb and 
 flow are bounded, an instance of the greatest stability that we 
 
84 OF THE EQUILIBRIUM [Book III. 
 
 find on the face of our globe. Were the mean height of its sur- 
 face to change, or the limits of its rise to be much extended, it 
 would be attended with the most disastrous consequences, caus- 
 ing great inundations, or even another deluge. Were the sea to 
 become still, and to be no longer agitated, either by the waves 
 that are raised by the winds, or those which constitute its tides, 
 it would come to rest, in a position whose surface would be a 
 mean, between the limits within which its oscillations now take 
 place. This surface it will hereafter be shown, must be every 
 where perpendicular to the direction of gravity ; for the present, 
 we must receive this fact as true, from the experimental illustra- 
 tion that has been cited. It is this mean surface of the ocean, 
 supposed to be produced through the body of the continents, and 
 other portions of dry land, that we understand by the Surface of 
 the Earth. 
 
 86. The earth being a body nearly spherical, as can be shown 
 by a variety of astronomic facts, as well as from actual observa- 
 tion and measurement, all the directions of gravity converge 
 towards its centre. They would actually meet there, were the 
 earth a perfect sphere. 
 
 87. Ordinary observation might lead us at first sight to sup- 
 pose, that bodies are very unequally influenced by the attraction 
 of gravitation ; that its action had relation not only to their masses, 
 but was influenced also by their densities. Thus the metals and 
 stones fall with great velocity ; wood and other vegetable sub- 
 stances less rapidly ; while feathers, down, and other similar sub- 
 stances, seem hardly to be affected by the earth's attraction. 
 Others again actually rise from the surface, apparently in oppo- 
 sition to the direction of the attractive force, as vapour, smoke, 
 and clouds. When, however, we consider, that some bodies which 
 we may under ordinary circumstances see to fall, will, under 
 others, rise upwards, we are tempted to examine whether analo- 
 gous causes may not exist, to determine the floatation, or actual 
 elevation of others. Thus, for instance, a piece of wood that falls 
 in the open air, rises when placed in water, and floats at the sur- 
 face; and even iron will do the same, when plunged in a mass of 
 mercury. The air then, it is possible, may, by its buoyancy, pre- 
 vent altogether the fall of some bodies, and even support them 
 in the atmosphere ; and may, by its resistance, retard the descent 
 of others. Whether this be the fact or not, may be tested by 
 means of the air pump. This apparatus, as will hereafter be 
 seen, is capable of exhausting the greatest part of the air, from a 
 proper vessel, called a receiver ; and in the exhausted receiver of 
 an air-pump, solid bodies of every diversity of density fall in 
 times that are absolutely equal ; while the lightest vapours, and 
 
///.] OF SOLID BODIES. 85 
 
 smoke, descend and occupy the bottom of the vessel. Thus all 
 bodies are influenced by gravity, and the velocities of falling 
 bodies being thus shown to be equal, when not resisted by the 
 ir, whatever be their densities, it follows, that the forces which 
 actuate them, are proportioned to the masses, or quantities of 
 matter. 
 
 88. As the earth is situated in free and open space, it follows 
 from what has been said in relation to the inertia of matter, that 
 it cannot impress motion on a body, without parting with an equal 
 quantity of its own motion ; or, in simpler terms, that it must 
 move towards the falling body, with a quantity of motion, equal 
 to that with which the body moves towards it. But the mass of 
 the earth is so vast in respect to the bodies which we can perceive 
 to fall, that the velocity of the earth towards the falling body 
 will be wholly inappreciable. We cannot therefore test by ob- 
 servation, the fact, whether the fall of a heavy body is caused by 
 the mutual action of the earth and the body, or whether the body 
 alone moves. Neither can we admit mere reasoning to decide 
 whether the attraction be mutual, or is exerted solely by the 
 greater mass, which is, in this case, the earth. 
 
 89. If the attraction between a falling body and the whole 
 mass of the earth be mutual, it will follow, that it must also be 
 mutual between the parts that make up the mass ; and the at- 
 tracting force of the earth will be due to the sum of the separate 
 attracting forces with which its particles are endowed. A force 
 thus made up, will be influenced by irregularities on the sur- 
 face of the earth ; and large projecting masses, such as mountains, 
 would cause a deflection in the plumb-line, from a perpendicular 
 to the general surface of the earth. 
 
 Whether such a deviation do occur, can only be determined 
 by the aid of astronomic observation. The deviation must be at 
 most extremely small, for the size of the largest mountains bears 
 but a very small proportion to the whole mass of the earth : it is 
 therefore wholly imperceptible, except to the most accurate modes 
 of observation, and it is only in mountains of considerable size, 
 that it can be detected, even by them. But if such deviation of 
 the plumb-line, from the true vertical shall be detected, it furnishes 
 full and complete evidence, that the attraction of gravitation is 
 mutual between the bodies that are affected by it. 
 
 The deflection of the plumb-line by the attraction of a moun- 
 tain, was first suspected by Bouguer, one of the French acade- 
 micians who were sent to Peru for the purpose of measuring an 
 arc of the Meridian. In carrying on a series of triangles, in the 
 neighbourhood of the great mountain of Chimborazo, latitudes 
 were determined by means of observations of the stars, both on 
 
86 OP THE EQUILIBRIUM [Book HI. 
 
 the north and south sides of the mountain ; the itinerary measure 
 between two of these stations, was not found to correspond to the 
 difference of latitude, and as the apparent latitude depends on 
 the position of the zenith, pointed out by the plumb-line, this 
 discrepancy showed a deflection in the latter. The deflection ob- 
 served by Bouguer, did not appear to exceed 7" or 8". 
 
 In 1772, Maskelyne, the British Astronomer Royal, performed 
 a series of observations of the same character, at the base of the 
 mountain Schehallien, in Scotland. The result of these was, to 
 show conclusively, the fact of the deviation of the plumb-line ; 
 and from his having this sole object in view, and from the accu- 
 racy of his observations, we can place the most implicit confi- 
 dence in his inferences. The deviation was found to amount 
 to 54". 
 
 It will be at once seen, that, if the density and bulk of the 
 mountain be known, and the bulk of the earth, this experi- 
 ment affords a ready mode of determining the mass, and conse- 
 quently the density of the earth ; for the plumb-line will point 
 out the direction of the resultant of two forces, one of which is 
 the attractive force of the earth, the other that of the mountain ; 
 and the angle of deviation will enable us to calculate one of these, 
 when the other is given, upon the principles of the composition 
 and resolution of forces in 12. 
 
 Professor Playfair performed the geological examination that 
 was necessary to determine the nature of the materials com- 
 posing the mountain. Professor Button, upon these data, pro- 
 ceeded to calculate the mean density of the earth, which he 
 found, by his first calculation, to be 4. 56, the density of water 
 being taken as the unit ; but which,, on a careful revision of the 
 process, he has increased to 5. 
 
 Now as the surface of a large part of the globe is covered with 
 water, and the density of the earthy matter, that forms by far the 
 gratest portion of the residue, is little more than twice as great as 
 water, it becomes evident, that the earth is denser within than at 
 the surface ; and as we can detect no sudden increase of density, 
 it is probable that the variation in this respect is regular. 
 
 Observations on the attraction of mountains, made in the neigh- 
 bourhood of Marseilles, by the Baron de Zach, give analogous 
 results ; and we shall hereafter cite experiments, made with the 
 pendulum, that corroborate their accuracy. 
 
 90. The mutual attraction of bodies nenr the surface of the 
 earth, has been detected by Cavendish, and his observations have 
 furnished another determination of the mean density of the earth. 
 The apparatus employed by him is admirably suited to the pur- 
 pose for which it was intended. It was originally planned by 
 
Book ///.] OF SOLID BODIES. 87 
 
 Mitchell, a member of the Rbyal Society of London. This ex- 
 perimenter being prevented by illness, which proved fatal, from 
 completing his researches, left by will his apparatus to Francis 
 I. H. Wollaston, and from him it passed into the hands of Caven- 
 dish. The principle of this apparatus may be explained as fol- 
 lows : 
 
 If a bar of an inflexible substance be accurately poised by its 
 middle, in a horizontal position, by means of a thread or wire, the 
 nature of the thread or wire is such as to bring it to rest in one 
 particular position. A small force will be sufficient to withdraw 
 the bar from this position, but the twisting or torsion which this 
 deflection will cause in the wire, will gradually oppose an in- 
 creasing resistance, until this latter exceed the deflecting force ; 
 the torsion will then cause the bar to return to its original posi- 
 tion, whence the deflecting force will again compel it to move. 
 Hence the bar will oscillate between two points, determined by 
 the intensity of the deflecting force, and that of the torsion of 
 the wire. The rapidity of the oscillations will, upon principles 
 that we shall hereafter explain, furnish a measure of the intensity 
 of the deflecting force. 
 
 Now if bodies mutually attract each other, a considerable mass 
 of a heavy substance, of a metal for instance, placed in the same 
 horizontal plane with the bar, ought to cause a deflection, and 
 consequent oscillation ; for, in this position, the attractive force of 
 the Earth is completely counteracted by the exact poising of the 
 bar ; hence that influence, which would in most other cases cloak, 
 and render imperceptible, a force so much smaller as that exerted 
 by the heaviest masses we have it in our power to move, would 
 become neutralized. 
 
 91. Such being the principle, we shall proceed to describe the 
 apparatus more in detail. It is represented in the figures an- 
 nexed ; S and S' are the two spheres of metal, weighing each 350 Ibs ; 
 
 s- 
 
88 0* THE EQUILIBRIUM [Book III. 
 
 a box is represented in which the bar is shut up, in order to pre- 
 serve it from the action of currents of air ; s and s' are two small 
 balls suspended from the extremities of the moveable bar, and by 
 whose weight it is kept exactly balanced and in equilibrio. 
 
 The folio wing figure is a horizontal section of the same apparatus. 
 
 
 In the previous figure, will be seen the manner in which the two 
 small balls are suspended from the bar by a silver wire ; this wire 
 passes through the bar, and is attached to the vertical wire, f\ the 
 tenacity of the latter is just sufficient to bear, without risk of break- 
 ing, the weight of the bar and the two balls ss' 9 and its torsion is 
 the only force that opposes their oscillation. The two masses S 
 and S', are themselves supported by iron rods, attached to a hori- 
 zontal arm in such a manner as to have each a free motion in a se- 
 micircular arc, one on each side of the box : they can be placed 
 in either of the positions represented in the second figure, by an 
 operation that is performed on the outside of the chamber in 
 which the apparatus is enclosed. This chamber had neither doors 
 nor windows, and was illuminated by means of a lamp, which 
 was placed without the chamber, in order that the interior might 
 not be heated. An aperture opposite to the place of the lamp, 
 admitted the light to fall upon one end of the bar, and the oscil- 
 lations were noted by means of a small telescope, passed through 
 the wall, just beneath this opening. 
 
 The whole apparatus being at rest, and the masses SS' in a po- 
 sition at right angles to the bar, they were turned around until 
 they reached the position represented in the second figure. As soon 
 as this was done, the bar began to move, and the balls ss' to oscil- 
 late. 
 
 The amount of these oscillations furnishes a measure of the 
 attraction of the masses SS', at a given distance, and this being 
 known, and compared with the effects of the attraction of gravi- 
 tation, exerted by the earth, a simple statement in proportion 
 will give the mass of the latter. Its density was hence calculated 
 by Cavendish to be 5.48. 
 
 Hutton has detected an inacuracy in the calculations of Cavendish, 
 by correting which the result has been lessened, and brought more 
 nsar to the inference drawn from the experiments of Schehallien. 
 
Book III.] OP SOLID BODIES. 89 
 
 Still, the difference is greater, between the results of the two 
 methods, than it ought to be. The same distinguished mathema- 
 tician has. therefore, proposed that an experiment similar to that 
 of Schehallien, be repeated on the opposite faces of the great 
 Egyptian pyramid, at the height of one fourth from the base. The 
 regularity of the figure of this great artificial mass, and its proba- 
 ble identity of composition throughout, render this method so 
 accurate, as in all probability to settle this disputed question. 
 Until this be done, we may without much risk of error, assume that 
 the mean density of the earth is five times as great as that of water. 
 
 92. Knowing the density of the earth when compared with 
 water as the unit, the weight of a given bulk of water, and the 
 volume of the earth, we may proceed to calculate its weight, in 
 the units of a conventional system of measures. Astronomic 
 observations, and the calculations founded upon them, enable us 
 to compare the mass of the earth with that of the sun and moon; 
 knowing thus the mass of the sun, we calculate readily the mass 
 of any planet accompanied by a satellite; and the determination 
 of the mass of those planets that have no satellites, although less 
 easy, is also practicable ; thus, as has been well remarked, the 
 apparatus of Cavendish is in fact a balance, in which we can 
 weigh the vast bodies that compose the solar system. 
 
 93. As bodies fall to the earth when left without support, at 
 all places within our reach, it may be inferred that the attraction 
 of gravitation acts during the whole time of their descent, and is 
 therefore an accelerating force. This is also conclusively shown, 
 by the well known fact of the acceleration that takes place in the 
 motion of falling bodies ; these strike the earth with greater velo- 
 city, in proportion as the distance through which they have fallen 
 increases. We cannot however, have recourse to experiments 
 of this kind, in order to discover whether this accelerating force 
 be constant, or variable in its intensity ; for, as the air causes an 
 unequal velocity in bodies of unequal densities, and prevents the 
 fall of some altogether, and as we cannot obtain a vacuum of 
 sufficient extent for the experiment, we cannot receive the indi- 
 cations obtained by the fall, even of the densest bodies, as abso- 
 lutely accurate. 
 
 The resistance of the air, as will be hereafter seen, varies with 
 the square of the velocity ; hence, if the velocity of a falling body 
 be diminished in an arithmetic progression, the resistance of the 
 air decreases in geometric ratio. Could we therefore observe 
 the motion of a body actuated by gravity, under circumstances 
 where its motion would be diminished, without any alteration ta- 
 king place in the law of the accelerating force, we might obtain 
 
 12 
 
90 OF THE EQUILIBRIUM [Book III 
 
 that law, almost without its being affected by the resistance of 
 the air. 
 
 94. We may consider the force of gravity as acting parallel to 
 itself within small horizontal distances; a body therefore, moving 
 upon a plane slightly inclined to the horizon, will suit our pur- 
 pose ; for, by 56, the force by which a body is actuated, when 
 it moves on a plane, inclined to the direction of an accelerating 
 force at a constant angle, follows the same law with the accelera- 
 ting force itself. That is to say, that although the absolute velo- 
 city is lessened, the relation between the spaces described, during 
 different portions of the time of the motion's continuance, is con- 
 stant. Hence, should we find that the velocity with which a 
 body descends on an inclined plane is uniformly accelerated, we 
 may infer, that the force of gravity is constant, within the limits 
 of the plane's elevation. 
 
 Experiments were made by Galileo upon this principle. He 
 formed his inclined plane by stretching a cord twenty or thirty 
 feet in length, between the two fixed points, at different levels. The 
 difference of height between the two points must be small, and 
 the inclination of the plane is therefore small also ; the velo- 
 cities were diminished, according to the principle in 56, in the 
 ratio of the sine of the plane's inclination to the horizon. In or- 
 der to lessen the friction, the bodies were mounted upon wheels, 
 and the smallness of the velocities rendered the resistance of the 
 air insensible. Experiments made with this apparatus, showed 
 an uniform acceleration in the velocity of the bodies ; and hence, 
 that gravity was, within the limits occupied by the plane, a con- 
 stant accelerating force. 
 
 95. The machine of Atwood affords a more convenient and 
 elegant method of obtaining the same results. 
 
 Two bodies, of unequal weights, are united by a cord passing 
 over a pulley. It is therefore obvious that the preponderance 
 of the one will cause it to descend, and raise the other, through 
 an equal space ; but it is also obvious, that this motion cannot be 
 as rapid, as that which either body would have, if free. The re- 
 lation which the accelerating force, in such a system, bears to 
 the whole force of gravity may be easily investigated. 
 
 Let A and B be the two bodies, and g the force of gravity, 
 which is the accelerating force that acts upon them both, but 
 whose effects are modified by their mutual action. 
 
 Let t be the time elapsed since the motion began ; v the velo- 
 city of A ; v' the velocity of B. 
 
 These velocities are obviously equal, but in contrary directions. 
 
 During the time dt, the velocities will increase under the action 
 of the accelerating force, by quantities which will be represented 
 by dv and dv' ; during the same time the bodies, if falling freely, 
 
Book III.] or SOLID BODIES. 1 
 
 would each acquire (49) the velocity gdt. Hence the body A 
 will lose, in consequence of its connexion with B, a velocity rep- 
 resented by gdt dv ; and B will lose by its connexion with A, a 
 velocity represented by gdt dv. These are respectively 
 equal to a" and 6" in the formula (98) , Aa"+B&"+&c.=0, that 
 expresses the principle of D'Alembert, while A and B are res- 
 pectively equal to the quantities of the same name ; and the mo- 
 tion of the bodies being contrary, the second term becomes nega- 
 tive ; therefore, by substitution, we have 
 
 A(gdt dv} Bfecfc cfo')=0 ; 
 and as the velocities are equal, but with contrary signs, 
 
 dv=dv ; 
 therefore, 
 
 A(gdt dt>) B(gefc+dv)=0 ; 
 performing the multiplications, we obtain 
 
 AgdtBgdt Adv Bdv=0 ; 
 by transposition', 
 
 (A 
 dividing by A-J-B, 
 
 integrating, 
 
 A B 
 
 in this expression the arbitrary constant, a, represents the initial ve- 
 locity, which when the bodies move from rest, becomes equal to 
 ; in which case 
 
 The forces which impel equal moving bodies, or systems of 
 bodies, being proportioned to the velocities, we obtain the fol- 
 lowing law. 
 
 The force which remains to cause the descent of the heavier 
 body, in Atwood's machine, is to the whole force of gravity, as 
 the difference of the weights of the two bodies is to their sum. 
 
 Experiments with this apparatus, show a motion uniformly 
 accelerated ; hence we have a farther proof that the attraction of 
 gravitation is a constant force, within the space occupied by the 
 machine. 
 
 As the formula we have investigated, gives the relation between 
 the velocity the body A has, when united to B, and that which 
 it would havs when falling freely, this machine gives us a ready 
 mode of obtaining the velocity of falling bodies. Thus if the 
 body A have a weight of 101, and B of 99, their sum will be 
 200, their difference 2, and the proportion of the actual velocity, 
 to that obtained by falling freely T fo. Now a velocity no more 
 
93 O* THE EQUILIBRIUM [Book HI. 
 
 than T ioth part of that acquired by a falling body, is so slow, that 
 it will be perfectly easy to note the marks upon a vertical scale 
 attached to the instrument with which the descending weight 
 corresponds, at each beat of the clock that is also attached. By 
 such experiments, carefully conducted, it is found that the actual 
 descent, in the instance we have stated, is 1.93 inches in the first 
 second ; whence the fall of a heavy body from rest may be infer- 
 red to be 193 in. in the first second of time, or 16 feet and an inch. 
 
 The machine of Atwood is represented on the opposite page. 
 
 In this will be seen, 
 
 1. The pulley, , the extremities of whose axles each rest upon 
 two other wheels, bb, the axles of which rest on agate planes; in 
 this manner, as will hereafter be shown, the friction is reduced 
 to the smallest possible. 
 
 2. A divided scale along which the body A descends without 
 touching it. On thisscale are placed two moveable plates/and g, 
 one pierced in the form of a ring, the other a plane surface. The 
 weights are at first equal, and the preponderance of A is effected, 
 by laying upon it a body of the shape of a bar, which cannot pass 
 through the ring; thus the accelerating force may be removed 
 at any point, and the velocity will become uniform ; whence not 
 only the spaces described by the accelerated motion, but the final 
 velocities also, may be made the subjects of experiment. 
 
 3. In order to count the time, during which the motion takes 
 place, a clock beating seconds is attached to the side of the ma- 
 chine, and a spring is adapted, in such a way, that the weight A, 
 and the pendulum P, of the clock E, may be released at the same 
 instant of time. 
 
 Applied to ascertain the final velocities, this machine gives the 
 results obtained from theory in 56, namely, that the velocities 
 acquired by moving from rest, under the action of an accelerating 
 force, are such as to carry the body with uniform motion through 
 twice the space, in a time equal to that employed in acquiring 
 those velocities. 
 
 96. The inferences obtained from the experiments made with 
 the inclined plane of Galileo, and the machine of Atwood, have 
 been purposely restricted to the limits occupied by the respective 
 apparatus ; for it will, even at first sight, appear probable, that 
 the force of gravity, residing in the mass of the earth, although 
 apparently constant within small limits, must decrease as we 
 recede from the earth, according to some law dependent upon, or 
 in mathematical terms, according to some function of the dis- 
 tance. 
 
 Galileo, who first ascribed the fall of heavy bodies to a me- 
 chanical force exerted upon them by the earth, did not attempt to 
 
Book ///.] 
 
 OP SOLID BODIES. 
 
94 OP THE EQUILIBRIUM [ &00k III. 
 
 extend the action of that force to bodies not in the immediate 
 vicinity of the earth. Kepler, with more extended views, saw 
 that it could not be thus limited in its action, but failed in disco- 
 vering the law, according to which its intensity varies, in terms 
 of the distance. His views were in consequence neglected, and 
 almost completely forgotten. 
 
 Newton, like Kepler, saw that the action of gravity, which does 
 not diminish in a degree that it had been possible previous to his 
 time to detect, even at the top of the highest mountains, could not 
 cease suddenly. He hence inferred that it might extend to the 
 moon, and be the cause by which that body was compelled to 
 describe a re-entering orbit. But although it might extend to 
 the moon, it was improbable that it could act there, with an inten- 
 sity equal to that it exerts, at the surface of the earth. The direc- 
 tions of gravity, supposing the earth to be spherical, are directed 
 to the centre : hence the influence, whatever be its cause, is a radi- 
 ating one, and would probably follow the same law in its decrease, 
 as the radiating actions of light and heat. The force of these 
 natural agents, is well known to vary, in the inverse ratio of the 
 squares of the distances from the centre. Newton inferred, that 
 the decrease in the intensity of gravity, must follow the same 
 law; and that at the moon, the attractive force must be as much 
 less than it is at the surface of the earth, as the sqliare of the 
 radius of the earth is less than the square of the moon's distance 
 from the earth's centre. 
 
 The moon, if not acted upon by a force drawing it to the earth, 
 would, according to the principles of motion, 62, go off in the di- 
 rection of a tangent. Taking the motion of the moon for a short 
 space of time, and consequently in a small arc, the tangent of the 
 arc will represent the force with which the moon would proceed 
 in a rectilineal direction, the space between the tangent and the 
 extremity of the arc, will represent the measure of the deflecting 
 force. If the arc be taken which corresponds to the mean motion 
 of the moon for a minute of time, this space may be calculated to 
 be about sixteen feet. Now the measure of the force of gravity, 
 g-, at the surface of the earth is twice the space a body falls from 
 a state of rest in a second of time, or is in the nearest round num- 
 bers 32 feet. The mean distance of the moon is about 60 semi- 
 diameters of the earth, and the respective distances of the two 
 bodies have the ratio of 60 to 1. assuming with Newton, that the 
 force varies, in the inverse ratio of the squares of the distances, 
 the value of g f a the force of gravity at the moon, is 
 
 32 
 
 the space described in a given time, under the action of a constant 
 accelerating force, and within this small space the force may be 
 considered constant, is by the formula (61) 
 
///.] OP SOLID BODIES. 95 
 
 taking the value of g' given above as the measure of g the accele- 
 rating force in this formula, and making 
 
 *=6G", 
 or a minute of time, we have 
 
 s= 32+W 
 2-f-(60)' J 
 
 The deflection of the moon from a tangent to her orbit in a 
 minute of time, is therefore the same with that which it should be, 
 were she acted upon by an attracting force residing in the centre 
 of the earth, and varying in its intensity, in the inverse ratio of the 
 squares of the distances. 
 
 By an investigation analogous to this, Newton inferred that 
 the attractive force of the earth did not cease suddenly, but ex- 
 tended as far as the orbit of the moon, and acted there according 
 to the law observed in other forces, or influences, emanating 
 from a centre. 
 
 97. The experiments that have been cited 91, showing a mu- 
 tual attraction between gravitating bodies, it may readily be infer- 
 red, that the moon has an action upon the earth proportioned to its 
 mass. This attraction of the moon may be shown by its influ- 
 ence upon the liquid mass of the ocean, in which it forms waves, 
 that constitute an important part of the tides : it is also obvious 
 in various astronomic phenomena that are foreign to our subject. 
 
 9S. It being thus found that the attraction is mutual between 
 the earth and moon, analogy leads us to infer that the earth and 
 sun mutually attract each other, with forces proportioned to their 
 respective masses, and that the same mutual action takes place 
 between all the bodies that compose the solar system. The as- 
 tronomy of observation fully confirms this important truth ; anoT 
 physical astronomy, thus established on a secure basis, applies 
 the principles of mechanics to investigate the minute changes 
 and variations that this multitude offerees, exerted by moving 
 bodies, are constantly producing in each other's motions. Thus 
 the simple mechanical laws, which we are compelled to investigate 
 in order to explain the action of the most familiar machines,, are 
 the same that direct the vast mechanism of the heavens. Nor does 
 the research that these laws give rise to, stop at the bounds of 
 our own system. The fixed stars also obey the same universal 
 force; and the astronomers of Europe are now on the threshold 
 of discoveries in respect to them, more wonderful than even those 
 with which Laplace has closed the labour began by Newton, leav- 
 ing no motion in our planetary system, that is not reducible to 
 mechanical principles. 
 

 06 OF THE EQUILIBRIUM [Book III. 
 
 99. The law according to which gravity decreases, shows us, 
 that although to all appearance constant within the limits of our 
 experiments, it is not absolutely so ; and we have at present 
 modes of observation that will hereafter be noticed, by which 
 even the small decrease that occurs in distances within our reach, 
 may be rendered evident. 
 
 100. Were the earth at rest in space, and perfectly spherical 
 in its form, the force of gravity would be constant at every point 
 of its surface ; but if it be in motion this cannot be the case. 
 
 When a body revolves around a fixed axis, its several points 
 describe circles whose planes are perpendicular to, and their cen- 
 tres in, the axis. From t>4, it would appear that these points, 
 revolving in equal times, are influenced by centrifugal forces 
 that are proportioned to the radius of the circles they respect- 
 ively describe ; and the directions of the forces, are in the planes 
 of these circles. Now these forces must lessen the action of the 
 force of gravity, which is due to the attraction of the whole 
 mass of the earth, but they will affect it differently in different 
 latitudes. At the poles, the centrifugal force is equal to 0, for 
 the circle becomes a mere point; while it is greatest in points 
 situated in the equator, as that is the greatest circle of diurnal 
 rotation. At the equator, too, the centrifugal force acts in direct 
 opposition to the force of gravity, while in all other places they 
 are inclined to each other. 
 
 The law which the decrease of gravity, influenced by these 
 two circumstances, follows, upon the surface of the earth, between 
 the poles and the equator, may be thus investigated : 
 
 The general expression for the centrifugal force is (83) 
 
 call the diminution of gravity at any latitude/ 7 ; 
 the radius of the circle, described by any point on the surface, is 
 equal to the cosine of the latitude ; the time T is a sidereal day, 
 and is constant ; substituting the former value, and calling the 
 latitude L, we have 
 
 _4^cos. L 
 /= Tl . 
 
 but this force does not act in direct opposition to the force of 
 gravity, the one being parallel to the equator, while the other is 
 directed to the centre. We must therefore resolve the centrifugal 
 force f into two others, one of which is parallel, and the other 
 perpendicular to the surface of the earth, at the given latitude ; 
 the latter is that which acts to diminish the force of gravity, and 
 may be found by the formula of the Resolution offerees (11) 
 X=R cos. a, 
 

 Book fflj OP SOLID BODIES. 97 
 
 hence the expression for the value of the centrifugal force must be 
 multiplied by the cosine of the angle its direction makes with the 
 plumb-line, or by the cosine of the latitude, and the value of the 
 diminution of gravity, at any latitude, becomes 
 
 If then the earth were a perfect sphere, the centrifugal force 
 at the equator would bear to the force that lessens the attraction 
 of gravity at any latitude, the ratio of the radius of the earth 
 to the square of the cosine of the latitude. 
 
 101. The ratio of the centrifugal force at the equator, to the 
 whole force of gravity, may be ascertained in the following 
 manner : 
 
 Let G be the whole attractive force of the Earth at the sur- 
 face, g the apparent force of gravity at the Equator, then 
 
 ( 106 ) 
 
 In this expression, G is twice the space a body descends from 
 rest in a second of time, or 
 
 G=32.16feet. 
 
 and, flr=3.1416; 
 
 r is the Earth's radius, which calculated in feet, gives 
 
 r =41, 836420 feet; 
 the sidereal day reduced to seconds gives 
 
 T=86164"; 
 substituting these values, 
 
 #=32.160.1118; (107) 
 
 and as 
 
 32.16 
 
 0.1118" 288 ' 
 
 therefore 
 
 The ratio of the centrifugal force at the equator is to the ap- 
 parent force of gravity as 1 to 288 ; or the diminution of gra- 
 vity, between the poles and the equator, is aigth part of the 
 whole. 
 
 102. These relations have been investigated upon the hypo- 
 thesis of the earth's being a sphere, but this could only occur in 
 a moving body, in the case of its being a solid mass devoid of 
 all elasticity. Every point in a moving spherical body, being 
 influenced by a centrifugal force, greatest at the Equator and least 
 at the Poles, these points would tend to assume a state of equi- 
 
 13 
 
08 O* THE EQUILIBRIUM [JBook llL 
 
 librium, under the joint action of the centrifugal and gravitating 
 forces. If then the mass were originally spherical, the equato- 
 rial parts would tend to recede from, the polar to approach the 
 centre ; and were they free to move, the equatorial diameter would 
 be increased, and the polar would be diminished, until a state 
 of equilibrium were obtained. Now although we know nothing 
 certain, in respect to the state in which the interior of the earth 
 exists, and find its outer crust a solid body, yet in large basins or 
 cavities of that crust, a fluid, (the ocean,) exists, covering nearly 
 three-fourths of the surface. The level of this mass of fluid points 
 out, as has been seen, the mean surface of the earth. Now this 
 fluid mass could only be in a permanent state, if the general 
 shape of the crust had the form the fluid would itself assume. 
 Hence in whatever state the mass of the Earth may have been 
 originally created, its shape is that of a fluid body retained in 
 equilibrio, by the joint action of the centrifugal force and that of 
 gravity. This form has been investigated under a variety of hy- 
 potheses, and by different persons. It is not our province to enter 
 into these investigations; we shall therefore content ourselves 
 with simply stating a few of the results. 
 
 Newton, considering the earth as a homogeneous fluid mass, 
 endued with a rotary motion, and composed of particles, attract- 
 ing each other, according to his law of universal gravitation, took 
 it for granted, that it would assume the figure of an oblate sphe- 
 roid. With these data, he inferred that the flattening of such a 
 spheroid would be fths of the relation of the centrifugal force at 
 the equator, to the whole force of gravity. The latter has been 
 shown to be ^ 9 and hence the ratio between the equatorial and 
 polar axes would be, upon his hypothesis, 230 : 229. 
 
 Huygens, assuming the whole force to reside in the centre, 
 determined the ratio to be 578 : 577. 
 
 The last result has been shown since, to be consistent with the 
 theory of mutual attraction, under the hypothesis, that the earth is 
 infinitely dense at the centre, and infinitely rare at the surface. 
 Now as we have shown, ( 91) that the mean density of the earth 
 is greater than that of its surface, and hence inferred, that the den- 
 sity increases towards the centre, the figure of the earth must 
 vary between these limits, and the oblateness of the terrestrial 
 spheroid cannot be greater than ^^ nor less than 3 ^ . 
 
 Such being the theory, it will be obvious, that if a flattening at 
 the poles can in any manner be detected, such flattening would 
 furnish conclusive evidence of the diurnal motion of the earth. 
 The same would be shown by a decrease in the apparent action 
 of gravity, between the poles and the equator. 
 
Book III.} OF SOLID BODIES. 09 
 
 Both of these may be ascertained to exist, by methods that re- 
 main to be described, but the oblateness may and has been de- 
 tected by actual measurement. 
 
 103. To recapitulate the laws of Universal Gravitation. 
 (1.) It is common to all bodies, and mutual between them. 
 (2.) It is proportioned to the quantity of matter in the body. 
 (3.) Its intensity decreases, as the square of the distance from 
 the centre of attraction increases. 
 
100 OF THE CENTRE [Book 111. 
 
 CHAPTER III. 
 OF THE CENTRES OF GRAVITY AND INERTIA. 
 
 104. From what has been said in the last chapter, on the sub- 
 ject of the Attraction of Gravitation, it appears, that bodies near 
 the surface of the earth, being, like all others, influenced by it, 
 may be considered as acted upon by a number of gravitating 
 forces, tending to draw each particle, or material point, in their 
 mass, towards the centre. These forces, within the space occu- 
 pied by even the largest bodies, may be considered as equal ; 
 and although their directions actually converge, yet in any given 
 body the convergence is insensible ; no error can therefore arise 
 from considering them as absolutely parallel. 
 
 A heavy body, near the surface of the earth, may therefore 
 be considered as acted upon by a number of equal and parallel 
 forces. These forces have a resultant, which is equal to their 
 sum, and is identical with the weight of the body. This weight 
 will depend upon the volume of the body, its density, and the 
 intensity of gravity at the place in which it is situated. If we 
 call the volume or bulk B, the density D, and the measure of 
 the force of gravity g, the value of the weight W will be 
 
 W=B D g, 
 
 and as by (101) the mass, M=B D, 
 
 W=M^. 
 
 105. The point of application of this resultant, which, in the 
 abstract theory of parallel forces, has been called their Centre, is 
 in the case of gravitating bodies called the Centre of Gravity. 
 
 The formulae then of 16, by means of which the centre of 
 parallel forces is found, and the several inferences obtained from 
 those formulae in particular cases, are applicable to the subject 
 before us. So also are the inferences from the geometric inves- 
 tigations, in the same section. That this is true in respect to ho- 
 mogeneous solid bodies is evident, for they may be considered 
 as made up of a number of equal particles, uniformly distributed 
 throughout the mass. In practice, however, it frequently be- 
 comes necessary to determine the centres of gravity of lines and 
 surfaces, and even of abstract figures of three dimensions. For 
 this purpose, they are supposed to be divided into an infinite num- 
 ber of small and equal parts, each of which is influenced by an 
 equal gravitating force. 
 
Book ///.] OP GRAVITY. 1 01 
 
 The inferences before obtained, may be now recapitulated, in 
 reference to this individual case of parallel forces. 
 
 (1.) The centre of gravity of a straight line bisects it. 
 
 (2.) The centre of gravity of two straight lines is found by 
 joining the two points that bisect them, and dividing the line 
 that joins them, into parts reciprocally proportional to the mag- 
 nitudes of the two lines. Of three lines, the centre of gravity 
 may be found, by first finding the centre of gravity of two of 
 them ; this is then to be joined by a straight line to the point 
 that bisects the third, and this last line divided into parts recipro- 
 cally proportioned to the joint magnitude of the two first, and 
 the third line. The centre of gravity of four lines may be found, 
 by first finding the centre of three of them, and uniting it to the 
 point that bisects the fourth, which is then to be divided as in the 
 former case. In this manner the centre of gravity of the peri- 
 meter of a triangle, or other figure, bounded by straight lines, 
 may be determined. 
 
 (3.) The centre of gravity of the surface of a triangle, is in 
 the line that joins the vertex to the point that bisects the base, 
 at the distance of two-thirds of that line from the vertex. 
 
 (4.) The centre of gravity of a quadrilateral figure may be 
 found, by dividing it into two triangles, joining their respective 
 centres of gravity, and dividing the line that unites them into 
 parts reciprocally proportional to the area of the two triangles. 
 And in general the Centre of gravity of any polygon whatever 
 may be found by dividing it into triangles, the centre of gravity 
 of two of which is first found, and united to the centre of gra- 
 vity of the third by a line, that is to be divided in a similar ratio ; 
 this point is then to be united by a straight line, to the centre of 
 gravity of the fourth triangle, and so on. Any polygon, it is 
 well known, can be divided into as many triangles, less two, as 
 it has sides. 
 
 (5.) The centre of gravity of a circle is in its centre. This 
 point is also the centre of gravity of a ring contained between 
 two concentric circles. 
 
 The centre of gravity of a circular arc, is at a distance from 
 the centre of the circle, which is a fourth proportional to the 
 lengths of the arc, the chord, and the radius of the circle. 
 
 In a semicircle this distance is 
 
 =0.63662 r. 
 
 1.5708 
 
 (6.) The centre of gravity of a parallelogram is in the point 
 where its two diagonals intersect each other. 
 
 (7.) The centre of gravity of a parabola is in its axis, at the 
 distance of three-fifths of that line from the vertex. 
 
103 OP THE CENTRE [Book HI. 
 
 (8.) The centre of gravity of the surface of a solid, formed 
 by the revolution of a plane surface around a line, in respect to 
 which, all its parts are symmetrically situated, is the same as 
 the centre of gravity of the generating surface. 
 
 Thus : the centre of gravity of a hollow cylinder is in the 
 point that bisects its axis ; the centre of gravity of a hollow cone 
 is in its axis at a distance of two-thirds the length of that axis 
 from the vertex. The centres of gravity of hollow spheres, and 
 ellipsoids of revolution, are in their centres of magnitude. 
 
 (9.) The centre of gravity of the surface of a spheric segment 
 bisects its versed sine. 
 
 (10.) The centre of gravity of a triangular pyramid is in the 
 line that joins its vertex to the centre of gravity of the base, 
 and at the distance of three-fourths of that line from the vertex. 
 
 (11.) The centre of gravity of any solid figure, bounded by 
 plane surfaces may be found by dividing it into a number of tri- 
 angular pyramids, and proceeding in them as has been directed to 
 be done in regard to the triangles, into which plane surfaces are 
 divided for a similar purpose. 
 
 (12.) The centre of gravity of a solid cone, is in its axis at 
 the Distance of three-fourths of that line from the vertex. 
 
 (13.) The centre of gravity of a sphere is in its centre of 
 magnitude, as is the centre of gravity of a shell contained be- 
 tween two concentric spheres. 
 
 (14.) The centre of a gravity of a solid paraboloid is at a 
 distance from the vertex, equal to two-thirds of the axis. 
 
 (15.) The centre of gravity of a spheric segment is in its fixed 
 axis, at the distance of fths of its length from the vertex. 
 
 (16.) The centre of gravity of a cycloid, that is bisected by 
 the vertex, is in the diameter of the generating circle, at a dis- 
 tance of one-third of the perpendicular height from the vertical arc. 
 
 106. In order that a heavy body shall be supported, it is ne- 
 cessary that a force shall be applied to it, equal in magnitude, and 
 contrary in direction, to the force of gravity that acts upon it. 
 The direction of gravity is a vertical line, and its point of appli- 
 cation is the centre of gravity ; hence the supporting force must 
 act perpendicularly upwards, and must be applied either to the 
 centre of gravity itself, or somewhere in the vertical line pass- 
 ing through it, which is called its Line of Direction. 
 
 If the supporting force be applied to a single point in the body, 
 there are three cases that may occur : 
 
 (1.) In the first place, the point of support may be above the 
 centre of gravity ; in this case, the centre of gravity will be in the 
 vertical line passing through, or will be directly beneath, the point 
 of support ; this case occurs in all instances of suspension, where 
 
Book III] ofr GRAVil-t. 103 
 
 the line of support is vertical^ano! Trie centre of gravity is in the 
 line of support produced ; when this occurs, the centre of gra- 
 vity is in the lowest possible point. If the equilibrium of such 
 a system be in any manner disturbed, the body will oscillate on 
 each side of the vertical line, and if, as always happens in nature, 
 its motion be opposed by resistances, the body speedily returns 
 to rest in its original position; hence the equilibrium is said to 
 be stable. 
 
 (2.) The supporting force may be applied exactly to the cen- 
 tre of gravity. In this case, if the body be moved from its ori- 
 ginal position, the forces have their direction changed in respect 
 to any line taken in the body, and supposed to be at rest ; it is 
 therefore an instance of that case in parallel forces, where the forces 
 revolve around their points of application, without ceasing to be 
 parallel; for the result will be the same, whether the forces 
 themselves move, or the points of application turn around the 
 centre of force. Hence, a body supported by its centre of gra- 
 vity, will remain at rest in any position in which it is placed ; 
 the equilibrium is now said to be indifferent, as the body has no 
 greater tendency to remain in any one position than in another. 
 
 (3.) The point of support may be below the centre of gravity ; 
 in this case, as the opposite directions of the supporting and 
 gravitating forces must coincide in the same vertical line, the 
 centre of gravity will be immediately above the point of sup- 
 port. If the equilibrium be disturbed, the centre of gravity 
 must describe a circle around the point of support, hence the 
 centre of gravity is in the highest possible point; and as this 
 motion is in a curve concave to the horizon, the motion will 
 continue around the point in the same direction as at first, 
 until the centre of gravity come immediately beneath the point 
 of support, or until it meet some new point of support, by 
 means of which the centre of gravity may be sustained in a state 
 of stable equilibrium. As the body can never return of itself to 
 its original position, the equilibrium, when it is supported from 
 beneath, is said to be tottering, or unstable. 
 
 All feats of balancing depend upon these properties of the cen- 
 tre of gravity. They, generally speaking, consist in a skilful 
 application of a small force to retain the body in its position of 
 tottering equilibrium, even after the conditions are partially dis- 
 turbed. Sometimes the point of support is fixed ; the art then 
 consists in changing the distribution of the weight, in such a man- 
 ner as to bring back the line of direction of the centre of gra- 
 vity to the point of support. Sometimes the point of support is 
 moveable, and the skill is then shown, by changing its position 
 in such a manner, as to make the line of direction of the centre 
 
104 'OP THE CENTRE \BooklIl. 
 
 of gravity constantly move its position, so as to meet, and pass 
 through the point of support. 
 
 107. A body may be supported upon a sharp edge. In this case, 
 the centre of gravity will be in the vertical plane passing through 
 the edge: and here again the equilibrium may be stable, indiffer- 
 ent, or tottering, according as the centre of gravity lies below, 
 in, or above the line, which marks the meeting of the two sur- 
 faces that form the edge. 
 
 108. The body may rest upon a surface. In this case, equili- 
 brium can only occur when the line of direction of the centre of 
 gravity falls within the base on which the body stands. Al- 
 though the supporting force is a normal to the surface, and in 
 order that equilibrium may exist theoretically, the surface of sup- 
 port ought to be parallel to the horizon ; still there are certain 
 forces that act to prevent a body from sliding, even upon an in- 
 clined surface. Such forces will be hereafter examined and de- 
 scribed. It is sufficient for the present to state, that a body may 
 be in equilibrio upon a base of small inclination. This however 
 can only be the case, when, as in the former instance, the line of 
 direction of the centre of gravity falls within the base. 
 
 109. If the body be of such a form as to touch the horizontal 
 plane, on which it rests, only at a single point, the three several 
 species of equilibrium may exist, according to the form of the 
 body, and its position in respect to the plane. 
 
 If from the centre of gravity of the body, lines be supposed 
 to be drawn to every point of its surface, some of the lines will 
 be always normals to the surface, while others will be oblique. 
 If the body rest upon any of the points through which one of 
 these normals passes, equilibrium will take place ; if this normal 
 be the shortest line that can be drawn from the centre of gra- 
 vity to the surface, the equilibrium is stable, for the centre of 
 gravity will be directly above the point of support, and will also 
 be in the lowest possible position; hence any disturbing force 
 will only cause an oscillation in the body, which will finally re- 
 turn to rest in its original position. The same is the case if the 
 normal, although not absolutely the shortest line drawn from the 
 centre of gravity to the surface, is relatively shorter than those 
 contiguous to it. If the body rest upon the point where the 
 longest normal intersects the surface, the body rs in a state of tot- 
 tering equilibrium, and will, if disturbed, turn around until it 
 rest upon the shortest normal ; the same will occur, even if the 
 normal be not absolutely the longest line, but if it be longer than 
 the other lines, drawn from the centre of gravity to the surface, 
 which are in its immediate vicinity. If all the lines drawn from 
 
Book IfJ.] 
 
 OF GRAVlfY. 
 
 105 
 
 the centre to the surface be normals, the equilibrium will be in- 
 different, or if the normal on which it rests be situated in the 
 vicinity of other lines that are also normals. 
 
 As instances : 
 
 A portion of a homogeneous sphere, or of a spherical surface 
 equal to, or less than, a hemisphere, will have stable equilibrium; 
 the same will take place in an ellipsoid formed by the revolution 
 of an ellipse around its shorter axis, which will come to rest with 
 that axis in a vertical position. 
 
 A homogeneous sphere will remain in any position in which it 
 is placed upon a plane, and is hence in a state of indifference. 
 
 An egg, or an oblong ellipsoid of revolution, will be in a state 
 of tottering equilibrium, if poised upon its longer axis ; while if 
 laid on one side, as all the radii of the circular section, in which 
 the points of contact are situated, pass through the centre of gra- 
 vity, the equilibrium is indifferent. 
 
 A portion of a cylinder not greater than the half, cut off by a 
 plane parallel to its axis, and laid on the curved surface, comes to 
 restupon the line, in which a plane perpendicular to that by which 
 it is cut from the cylinder intersects the surface, and is there- 
 fore in a state of stable equilibrium. 
 
 These circumstances may be illustrated by the following 
 figures. 
 
 FIG. 1 FIG. 2. 
 
 Let the above figures represent bodies whose sections are se- 
 micircular, Fig. 1 being solid, Fig. 2 being hollow. In either 
 case, the normal #D will be the shortest line that can be drawn 
 from the centre of gravity, g-, to the curved surface. If any dis- 
 turbing force act, that is not sufficient to bring the body into the 
 position, in which lines passing through #, and A or B, are verti- 
 cal, the body will finally return to rest in the position in which 
 g-D is vertical : when the disturbing force is removed, it will os- 
 cillate in returning to rest, until the resistances overcome its mo- 
 tion. In the case of a spherical surface, the oscillations may take 
 place in any direction whatsoever, but in the case of a portion of 
 a cylinder, only in planes parallel to the circular section. In a 
 solid spheric segment, the distance Cg is only fths of CD ; while 
 in a hollow spheric segment, the centre of gravity will coincide 
 
 14 
 
106 
 
 OP THE CENTRE 
 
 [So ok HI 
 
 with that of the curve, and its distance from C will be nearly 
 fds of CD. A similar difference in position will take place, in 
 relation to portions of solid, and of hollow cylinders. Hollow 
 bodies of these classes, are, therefore, more stable than solid ones ; 
 and with equal weights, will more powerfully resist any effort to 
 disturb their equilibrium. 
 
 Jn the homogeneous sphere, one of whose great circles is re- 
 presented by the circle ADBE, Fig. 1, the centre of gravity, 
 FIG. 1. FIG, 2. 
 
 corresponds with the centre of magnitude, and the lines of direc- 
 tion will be all normals, and of equal length, upon whatever 
 point it rest; hence its equilibrium is that of indifference. If the 
 figure represent the circular section of a cylinder, a similar state 
 of indifference will exist in one direction. If in Fig. 2, the 
 elliptical curve ADBE, represent the section of an ellipsoid, 
 formed by the revolution of the curve upon its shorter axis, and 
 the solid thus formed be homogeneous, the centre of gravity will 
 be in the centre of magnitude g. The lines of direction #D and 
 #E, which are normals to the surface, are the shortest that can be 
 drawn within the body, it will therefore have, when resting upon 
 ci'.her of the points D or E, a stale of stable equilibrium. All 
 lines in the plane of #A, and#B, are also normals to the surface, 
 but are the longest lines of direction that can be drawn within 
 the body : hence, if resting upon the points A or B, it will be in a 
 state of tottering equilibrium, in case iho disturbing force act in 
 the plane of AB. If the same curve represent the section of an 
 ellipsoid, formed by revolution around its longer axis AB, the 
 lines g\ and #B, are normals, and the two longest lines of direc- 
 tion that can be drawn within the body, resting on the points A 
 and B ; therefore, its equilibrium is tottering. All the lines of di- 
 rection that can be drawn in the plane of DE, are equal among 
 themselves, and shorter than any other lines that can be drawn 
 within the body from the point g ; hence in respect to forces act- 
 ing in the plane of DE, the equilibrium is indifferent. 
 
 110. If the surface of a body resting on a plane, be also a 
 plane, and the line of direction of the centre of gravity fall 
 within it, the equilibrium is of course stable. If extrinsic forces 
 act to disturb the position of the body, the more extensive the 
 
Book ///.] 
 
 OF GRAVITY. 
 
 107 
 
 plane surface on which it rests, the nearer to the centre of the 
 surface the line of direction falls, and the lower the position of 
 the centre of gravity, the more the body will resist a force ap- 
 plied to overturn it. When the force that acts to overturn it is 
 sufficient for the purpose, the body, if not broken by its action, 
 will turn around one of its solid angles as a centre, or round one 
 of its edges as an axis. The centre of gravity must of course 
 rise in a circular arc, and with it the weight of the body ; the 
 resistance of the body to the effort to overturn it, will therefore 
 depend, not only upon its own weight, but upon the position and 
 curvature of the arc described by the centre of gravity. Some 
 of the cases that may occur in practice, are represented below. 
 
 The triangle ABC, Fig. 1, is the section of a pyramid or 
 cone whose centre of gravity, g, is at the distance of fths of its 
 
 FIG. 1. FIG. 2. 
 
 height from the vertex. In overturning, its centre of gravity 
 
 would describe a circular arc around the corner C. In Fig. 2 t 
 
 the triangle represents the section of a triangular prism, whose 
 
 centre of gravity is in the point g, at a distance of fds of its 
 
 height from the vertex. The weight in the former case will act 
 
 more directly to preserve the stability, while the disturbing force 
 
 will act more obliquely. The former is therefore the most stable. 
 
 In the figures beneath, Figs. 1 and 2, respectively repre- 
 
 FIG. 1. FIG 2. 
 
 B C 
 
 sent sections of prisms, the first of which has for its section a 
 trapezium, with parallel bases; the second is rectangular. The 
 centre of gravity of the second is at half its height, of the first, 
 at a distance from CD, represented by the formula 
 
 _o_ c 2d 
 A ~ 3 ' c d 
 

 108 
 
 OF THE CENTRE 
 
 [Book III. 
 
 c being the greater and d the lesser base ; it is obvious then that 
 its centre of gravity will be lower than in Fig. 2, and it will in 
 consequence be more stable. 
 
 Had Fig. 1 been the section of a truncated pyramid, the centre 
 of gravity would have been still nearer the base, and the stability 
 greater. Did Fig. 1 rest on the base AB, the results would be 
 directly opposite. 
 
 In two rectangular prisms, whose sections are Fig. I and 2, 
 the stability of the lowest, Fig. 2, is the greater of the two, 
 FIG. 1. FIG. 2. 
 
 their bases being equal ; and in two prisms of equal height, but 
 of different bases, that with the greatest base will have the great- 
 est stability. 
 
 If however the prism, Fig. 1, should cease to be homogeneous, 
 and be loaded with a weight towards the base CF, h}^ means of 
 which its centre of gravity is lowered to #', whose distance from 
 the plane of support is equal to g in Fig. 2, the two bodies will 
 have equal degrees of stability. 
 
 It may hence be inferred, that pyramids and cones, of small al- 
 titude, compared with theextent of their bases, are among the most 
 stable of all geometric figures. That walls with a broad base, and 
 whose faces incline inwards, are more stable than those whose 
 surfaces are parallel; that with equal bases, walls of the least 
 heights, and with equal heights, those with the greatest bases are 
 the most stable; that stability may be given to bodies, by con- 
 structing them in such a manner that their centre of gravity 
 may fall below the point in which it would be if they were ho- 
 mogeneous. 
 
 111. A body, whose sides are inclined in such a way that it 
 overhangs on one side of the base, may, notwithstanding;, be sta- 
 ble, if the centre of gravity fall within the base. And even if 
 the vertical line that passes through its centre of magnitude fall 
 without the base, the actual centre of gravity may be so lowered, 
 by a proper distribution of the weight, that its line of direction 
 shall fall within the base, and stability ensue. Thus in the city 
 of Pisa, in Italy, there is a tower that leans so much to one side, 
 

 Book III.} OF GRAVITY. 109 
 
 that it not only appears unsafe, but would be actually so, if ho- 
 mogeneous. Rut by an ingenious arrangement of the materials, 
 it is rendered stable. The lower parts are built of a heavy vol- 
 canic rock ; the middle of brick ; and the top of a light porous 
 stone, that will float on water : hence, the centre of gravity is 
 so low, that its line of direction falls within the base. 
 
 1 12. It is not necessary that the base shall be actually a plane 
 surface; but it is sufficient that the body rest upon points. If it 
 rest upon no more than two points, it is in the condition of a body 
 resting upon an edge, and the line of direction of the centre of 
 gravity must fall in the line that joins these points, otherwise the 
 body will not be in equilibrio. If itrest on more than two points, 
 the base is the surface formed by joining the points by straight 
 lines. It may in like manner rest on two edges, and the base 
 will be defined by supposing their extremities to be joined. If 
 the surface on which the body rests be irregular, it is best sup- 
 ported upon three points ; for these lie always in one plane, and 
 the stale of the body is precisely the same as if this plane were 
 applied to another. This principle is applied in practice to a 
 variety of surveying and astronomic instruments, to which sta- 
 bility is given by placing them upon three feet; if they had more 
 than three feet, they would rest firmly upon no surface but one 
 perfectly plane, or at least having in it an equal number of points, 
 on which to place the feet, that lie in one plane : while with three 
 feet, they can be placed steadily on the most irregular surfaces. 
 
 A three-legged table or chair stands firmly on the most un- 
 equal floor, while one with four legs is unsteady, except upon a 
 floor that is perfectly level. 
 
 1 13. We have stated that there are cases in nature, in which a 
 body will not descend an inclined plane. In such a case, if the 
 line of direction fall within the base, the body will be stable in 
 spite of the inclination of the plane ; and if it descend upon the 
 plane, it will slide down it. 
 
 Thus in the body beneath, if the centre of gravity be at g, 
 its line of direction falls within the base ; if the plane oppose a 
 resistance to its descent, it is stable ; but if the plane oppose no 
 resistance, it will slide down. 
 
 If a body beplaced onan inclined plane, 
 and the line of direction of the centre of 
 gravity, fall without the base, it will 
 turn around, until it apply itself to the 
 plane by a surface, within which its line 
 of direction will fall. 
 
110 
 
 OF THE CENTRE 
 
 [Eook III. 
 
 Thus the body whose section is represented beneath at A, will 
 be overturned, and come into the position B ; it will there re- 
 main at rest, if supported by a resistance in the plane, or will 
 slide down it if not supported. 
 
 If the body have no surface 
 wilhin which its line of direction 
 can fall, it will roll down the 
 inclined plane ; thus the body 
 whose section, represented be- 
 neath, is a regular polygon, will 
 roll down the inclined plane on 
 _ which it rests. 
 
 A cylinder or sphere, having a circular sec- 
 tion, will not rest, if homogeneous, on an in- 
 clined plane, at any of its points. But it may, if 
 loaded by a weight which will cause the line of 
 direction to fall upon, or above the place where 
 it rests upon the plane, either remain at rest, 
 or actually move up the inclined plane. 
 
 Thus in the body whose circular section is 
 represented beneath, an eccentric weight at 
 W will change the position of the centre of gravity from c to g, 
 and the body will roll up the plane, until the line of direction 
 fall upon the point at which it meets the plane, where it will 
 come to rest. 
 
 
 If this weight be moved, as may be done by a spring, in such 
 a manner as to be raised as much as it tends to fall by the rolling 
 motion of the body, the latter will exhibit the curious pheno- 
 menon, of a body apparently mounting in opposition to gravity. 
 
 A double cone may be made to appear to roll upwards, by 
 placing it between two edges, inclined to the horizon, and to each 
 other, and meeting at an acute angle at their lowest points. If 
 the double cone be laid at this angle, it rests upon its greatest 
 section ; and if the inclination of the planes to the horizon be 
 such, that the cone, when laid at other points of the plane, shall 
 
Book HI} OP INERTIA. Ill 
 
 rest upon two of its sections whose radius has lessened more than 
 the height of the plane has increased, the cone if laid at the 
 place where the planes meet, will roll along them, appearing to 
 ascend them, when in fact its centre of gravity is constantly de- 
 scending. 
 
 The distinguishing property of the centre of gravity in a 
 solid body, is, that if it be supported, the body is supported ; but 
 if it be not supported, the body will fall, and continue to fall, un- 
 til it meet a resistance of such a nature as to support this point. 
 
 114. In irregular bodies, whether the irregularity arise from 
 mere figure, or from an unequal distribution of matter throughout 
 their bulk, the mathematical methods of finding the position of the 
 centre of gravity are inapplicable. We may in such cases have 
 recourse to experimental methods, whose principles are founded 
 on the properties of the centre of gravity. 
 
 (1.) The body may be suspended alternately, from two differ- 
 ent points in its surface. The centre of gravity will, in either 
 case, lie immediately beneath the point of suspension ; it will there- 
 fore lie at the common intersection of the two lines that join the 
 two points of suspension to points in the body situated directly 
 benea,th each of them, when it is suspended from it. 
 
 (2 ) The body may be made to rest in equilibrio, in three dif- 
 ferent positions, upon a sharp edge; the vertical plane passing 
 through the edge, in each of the three several positions of the body, 
 will also pass through the centre of gravity, and the common 
 intersection of the three planes determines the situation of this 
 point. 
 
 115. If a body move in a straight line, under the action of any 
 other force than that of gravity, each of its particles may be sup- 
 posed to be actuated by an equal and parallel force; hence it will 
 act as if its whole mass were collected in the centre of these pa- 
 rallel forces. This point, which in gravitating bodies, as has just 
 been seen, is called the centre of gravity, is, in this case, called 
 the Centre of Inertia. Its position may therefore be found by 
 the same processes, whether analytic, geometric, or experimental, 
 by which the centre of gravity can be found. 
 
11)8 OF FKICTION. [Book HI. 
 
 CHAPTER V. 
 
 OF FRICTION. 
 
 
 
 116. So far as our investigations have hitherto proceeded, it 
 might appear, that so soon as equilibrium ceases to exist, among 
 the forces that act upon a body, it must be set in motion. This, 
 however, does not take place in practice ; for there are resist- 
 ances that are incapable themselves of causing motion, and which 
 therefore do not come within our original definition of the term 
 force ; these are yet effectual in retaining bodies at rest, after the 
 theoretic conditions of equilibrium are at end; they are, also, 
 capable of bringing bodies to rest, when they have been pre- 
 viously set in motion. Thus then, although they do not fall 
 within the strict definition of forces, still we cannot determine 
 the circumstances under which bodies move, without taking them 
 into account. The retardation they produce in motions arising 
 from other forces, is capable of being estimated in terms of a 
 conventional unit, precisely as if they were accelerating forces, 
 acting in directions opposed to those of the motion due to other 
 forces. Hence we may consider the action of these resistances, 
 precisely as if they were forces, always acting in directions op- 
 posite to those of a previously communicated motion, or to lhat 
 in which a body would tend to move, when its equilibrium is dis- 
 turbed. They are, in fact, passive or resisting forces, that are 
 only called into action under certain circumstances, but which 
 have, like active forces, a determinate measure, a definite inten- 
 sity, and a known point of application. 
 
 Of such resisting or retarding forces, the more important 
 are : 
 
 The resistance that the surfaces of solid bodies oppose to each 
 other's motions, or that one opposes to the motion of the other. 
 This is called Friction. 
 
 The resistance that certain bodies oppose to flexure; 
 
 The resistance of fluid media to bodies moving in them, and 
 which solids oppose to the motion of fluids. 
 
 The two first of these are of direct importance to our present 
 subject, and may be examined by the aid of principles that have 
 already been laid down. The consideration of the third must ne- 
 cessarily be postponed, until we treat of the mechanics of fluid 
 bodies. 
 
 117. The precise nature of friction is unknown to us, although 
 there is a well-founded hypothesis on the subject that shall here- 
 
III.} OF FRICTION. 113 
 
 after beeited. We are therefore compelled to have recourse to 
 experiment, in order to ascertain the laws its action follows. The 
 more important of these experiments, so far as they have been 
 recorded, are those of Coulomb, Vince, and Ximenes. To these 
 we shall recur, describing the manner in which they were per- 
 formed. 
 
 US. Friction, although always arising from the same general 
 cause, may be classed into three distinct varieties : 
 
 (1.) That which occurs when one body slides upon the surface 
 of another ; 
 
 (2.) The friction of bodies rolling; and 
 
 (3.) The friction at the axles of wheels. 
 
 These have all been separately and fully examined, and we 
 shall now proceed to describe the manner in which the various 
 experiments were made, and to indicate the result that have 
 been obtained from them. 
 
 119. If a body be placed upon a horizontal plane, and the line 
 of direction of its centre of gravity fall within its base, it will be 
 at rest, under two countervailing forces, its weight, and the 
 resistance of the plane. If the plane be gradually inclined, al- 
 though the equilibrium of these two forces is disturbed, because 
 they no longer act in direct opposition to each other, the body 
 will not at first move, but will remain at rest until the plane ac- 
 quire an inclination ; this inclination will be different, according 
 to the nature of the surface and figure of the body. At this in- 
 clination the body will be set in motion, and will slide or roll 
 down the inclined plane, according to the manner the line of di- 
 rection falls. Up to the beginning of its motion, it is supported 
 by its friction upon the plane, and this friction will be represented 
 in direction, by a force parallel to the plane on which it moves. 
 At the instant before motion begins, the three forces, namely, 
 the weight, the pressure, and the friction, are exactly in equili- 
 brio; their respective intensities may therefore be represented 
 upon the principles in 15. 
 
 Supposing the weight, W, to be known, and the angle * of the 
 plane's inclination to the horizon to be determined, the value of 
 the friction, F, and pressure, P, will be 
 
 F=W sin. , P=W cos. i ; 
 
 and the friction will be given in terms of the pressure, by the ex- 
 pression 
 
 F P tan. t. 
 
 Experiments conducted upon this principle, have given the 
 following results : 
 
 15 
 
1 14 OF FRICTION. [Book ///. 
 
 (1.) The friction is greatest between rough surfaces, and di- 
 minishes with the degree of polish that is given to them. 
 
 (2.) Friction is greater, all other things being equal, between 
 the surfaces of bodies that are of the same material, or homo- 
 geneous, than it is between bodies of different materials. 
 
 (3.) The rubbing surfaces remaining the same, the friction is 
 directly proportioned to the pressure. 
 
 (4.) The friction does not increase or diminish with the area 
 of the rubbing surface, the weight and the nature of the surface 
 remaining the same. 
 
 These experiments are limited, by their very nature, to the 
 determination of the resistance that prevents a body from being 
 set in motion ; and it might, at first sight, appear more than pro- 
 bable, that this force is more intense than the friction which re- 
 tards the velocity of a moving body. Neither do they give any 
 information whether the intensity of friction bears any relation 
 to the velocity. Another defect arises from the small num- 
 ber of the experiments that have been performed in this manner, 
 and we are hence uncertain, whether the results are applicable to 
 all cases whatsoever, or limited to a few particular instances. 
 
 120. The experiments of Coulomb, Vince, and Ximenes, were 
 performed in another manner. A body was drawn along a hori- 
 zontal table, by means of a weight attached to it by a cord, and 
 this cord passed over a pulley. The weight that produces a con- 
 stant velocity is obviously the measure of the friction, which is, 
 in this case, the resistance that opposes the motion of bodies ; 
 the weight necessary to set them in motion, is the measure of the 
 resistance that opposes their passage from a state of rest. 
 
 The different circumstances to be examined are : 
 
 (1.) The relation of the friction to the pressure ; 
 
 (2.) The effect of the nature of the surfaces, and of the manner 
 of their preparation, upon the friction; 
 
 (3.) The variation in the friction produced by a longer or 
 shorter continuance of the contact, previous to the application 
 of a force to overcome it ; 
 
 (4.) The influence of the extent of the surface ; 
 
 (5.) The change, if any, at different velocities. 
 
 (I.) As respects the relation of the friction to the pressure. 
 
 When at a maximum, all other circumstances remaining the 
 same, the friction was found to have a constant relation to tho 
 pressure. This maximum, as will hereafter be seen, is that which 
 opposes the force that acts to set a body in motion. 
 
 The friction of surfaces, that had been at rest until the maxi- 
 mum was attained, was found to be as follows, taking the mean 
 of the experiments : 
 
Book III.] OF FRICTION. 115 
 
 ^ 
 
 3 
 
 Of Oak resting upon Oak, the fibres being parallel, - - - 
 
 Of Oak resting upon Oak, the fibres being at right angles to 1 1 
 
 each other, - ) 3.75 
 
 Of Oak resting upon Fir, ------------ 
 
 Of Fir resting upon Fir, 
 
 1.78 
 
 Of Iron resting upon Oak, ----------- 
 
 5.5 
 
 Of Iron resting upon Iron, fi.- *s is. - - - ... - - 
 
 3*5 
 
 Of Iron resting upon Brass, --- -- 
 
 3.8 
 
 and if well polished, and the surfaces small, ------ _ 
 
 6 
 
 The friction of bodies in motion was found considerably less, 
 being at a mean as follows : 
 
 Of Oak moving on Oak, ----- 
 
 9.5 
 
 Oak moving on Fir, _-_._-------. 
 
 6.3 
 
 Fir moving on Fir, --------- 
 
 6 
 
 Elm moving on Elm, >---- 
 
 Iron or Copper moving on wood, --_--._._ 
 
 13 
 
 Iron on Iron, -_.---------_-_ 
 
 3.55 
 
 Iron on Copper, after long attrition, - _ 
 
 6 
 
 (2.) It was found that the metals could be polished separately, 
 until the minimum of friction was attained, but that in soft sub- 
 stances, it was necessary that they should be compressed, either 
 in the course of the experiments, or by previous pressure. Thus 
 i-n newly planed wood, the maximum friction was not far from 
 equal to the pressure, while after being some time in use, it fell 
 to the ratio at which it has been slated, of i. 
 
 The polishing of the surfaces of iron and copper, rubbing against 
 each other, reduced the friction nearly one half. 
 
 Unctuous substances, interposed between the rubbing surfaces, 
 diminish the friction ; and the experiments of Coulomb showed, 
 that those which are hardest have the greatest effect, when the 
 
116 OF FRICTION. [Book Iff. 
 
 weight is great; thus, although in light and delicate machinery, 
 the most limpid oils are hest, they are wholly ineffectual in great 
 pressures, when tallow must be employed. At still higher pres- 
 sures than those which became the subject of experiment, it ap- 
 pears from practical observations, that tcllow loses its power of 
 diminishing friction. So also is it less effective when the velo- 
 locity becomes sufficient to melt it. In this case it acts as if it 
 were oil, being well suited to diminish friction when the weight 
 is light, but of little effect in great pressures. When oleaginous 
 substances cease to have effect, as tallow, when the pressure be- 
 comes too intense, or the heat produced by the velocity is effi- 
 cient te render it fluid, solids of an unctuous texture have been 
 found to answer the purpose. Thus in carriages moving rapidly, 
 and in large machinery, plumbago has been mixed with tallow: 
 it has also been used separately ; and recently, soap-stone 
 (steatite) has been found efficacious in lessening friction when 
 the weight was so great that all other means failed. 
 
 Within the limits of the experiments, the mean friction, when 
 oleaginous matters were interposed, was found to be as follows : 
 
 Tallow interposed between two surfaces of oak, 
 
 If wiped off after application, ...._ 
 
 In very small surfaces, -----_----. 
 
 Tallow interposed between wood and oak, moving slowly on ) 1 
 
 each other, -.- . . . . _ j 35 
 
 Between brass and oak, under similar circumstances, - - - 
 
 47 
 After being sometime in use the friction was increased to ) 1 ,1 
 
 thrice these amounts, or to --.--.. j 12 & 16 
 The same result took place at increased velocities. 
 In the case of tallow, interposed between metallic surfaces, the 
 
 frictions were the same at all velocities. 
 
 Tallow, interposed between two surfaces of iron, gave - - - _L 
 
 10 
 
 Between iron and copper, - - - - - - - - .^i ;. JL 
 
 While 
 
 Oil interposed between the same surfaces, gave for the friction, 
 
 8 
 
 (3.) It was found, that in the case of the metals resting upon 
 each other, the maximum-of friction was reached instantly; in 
 wood resting upon wood, the maximum was reached in a few 
 in the contact of metals with wood, the maximum was 
 
Book HI.] OF FRICTION. l ^ 
 
 not attained until some days had elapsed ; and when grease was 
 interposed between any substances whatsoever, the time during 
 which the friction continued to increase, was still longer than in 
 the latter case. 
 
 The relations that have been stated in the two first instances, 
 were not found to be wholly independent of the amount of pres- 
 sure. That is to say, the ratio of the surface to the pressure was 
 not constant, but appeared to diminish at the higher pressures, 
 that were the objects of experiment. This diminution was even 
 more obvious, in the experiments ofVince and Ximenes, than in 
 those of Coulomb. 
 
 (4.) The magnitude of the surface was found to have an appre- 
 ciable effect upon the friction. In the case of oak moving upon 
 oak, in pressures from 100 to 4000 Ibs. per square foot, an adhe- 
 sion or additional resistance was found amounting to about If Ibs. 
 per square foot ; and in all other cases, an analogous increase was 
 remarked. 
 
 (5.) Under similar circumstances, the weight, that overcame 
 the friction, was found to be nearly the same at all velocities. 
 The slight variations that were observed, seemed rather to lead 
 to the conclusion, that at small velocities the friction increases 
 with the velocity, but that at great velocities, it diminishes in 
 the ratio of some small function of the velocity. 
 
 It thus appears that the variation in pressure, in magnitude of 
 surface, and in velocity, have but little effect, unless they become 
 very great. 
 
 We may therefore, in all usual cases, assume the following laws 
 as sufficiently accurate for practical purposes. 
 
 (1.) Between similar substances, under similar circumstances, 
 Friction is a constant retarding force. 
 
 (2. ) Friction is greatest between bodies whose surfaces are 
 rough, and is lessened by polishing them. 
 
 (3.) It is greater between surfaces composed of the same ma- 
 terial, than between bodies composed of different materials. 
 
 (4.) If the rubbing surfaces remain the same, the friction in- 
 creases directly as the pressure. 
 
 (5.) If the pressure continue the same, the friction has no re- 
 lation to the magnitude of the surface. 
 
 (6.) The relation between the pressure and the friction, can- 
 not be safely taken at less than 1, in calculating the effect of prime 
 movers. 
 
 On the other hand, when friction is to be substituted for a fixed 
 resistance, it cannot safely be estimated at more than ^. 
 
 Although the weight, which measures the intensity'of friction, 
 be constant, whatever be the velocity, still as the measure of a 
 force is not merely the weight that it is capable of raising, but de- 
 
118 or FRICTION. [Book HI. 
 
 pends also upon the rate at which the weight is lifted, the power 
 that is applied to overcome the constant friction, must increase 
 with the velocity at which that resistance is overcome. 
 
 The several laws that have just been laid down, are, as is ob- 
 vious from the result of the experiments, not absolutely, but only 
 nearly true, and cases occur occasionally in practice, in which 
 the minute effects that are due to the increase of pressure of the 
 surface, and of the velocity, become important. Thus : in launch- 
 ing a ship, the vast weight which it has, appears to become a force 
 capable of very much lessening the friction, and the vessel de- 
 scends down a plane of less inclination than a lighter body would. 
 At the Shoot of Alpnach, in Switzerland, where trees of great 
 size are conveyed along a trough of but small inclination, for 7 
 or 8 miles, in consequence of a velocity previously acquired in 
 falling through a curved spout of great inclination, the phenome- 
 na, as noted by Playfair, are such as can only be explained by as- 
 suming, that the friction of great masses, moving with great ve- 
 locities, is considerably less than that of smaller bodies moving 
 slowly. 
 
 121. The friction of rolling bodies has also been investigated 
 experimentally by Coulomb. The following are deductions from 
 his experiments : 
 
 (1.) Like the friction of sliding bodies, it is a constant force; 
 
 (2.) It is affected by the nature of the surface, so far as polish 
 is concerned, but is not lessened by the interposition of oleagi- 
 nous and unctuous substances; 
 
 (3 ) It is less between heterogeneous than between homogen- 
 eous substances ; 
 
 (4.) It is directly proportioned to the pressure ; 
 
 (5.) It has no relation to the magnitude of the surface ; 
 
 (6.) Its measure is much less than in the case of sliding'sur- 
 faces, and varies in the inverse ratio of the diameter of the rolling 
 body. 
 
 A cylinder of lignumvitse rolling on rulers of different kinds 
 of wood, and having a diameter of 32 inches, was not resisted 
 by a friction, at a mean, of more than T j . 
 
 122. The friction of the axles of wheels is still of another des- 
 cription ; it is less than that of sliding, and more than that of 
 rolling bodies. It follows, in all respects, the general laws of 
 sliding bodies. An axle of iron, turning in a box of oak, had a 
 friction of j ; when both were of wood, the friction was T V. 
 
 In metallic axles, resting in boxes of another metal, and well 
 coated with grease, the friction has been found to be no more 
 than J, . 
 
Book III.} OF FRICTION. 119 
 
 Of wooden axles in wooden boxes, when coated in a similar 
 manner, from T 1 ^ to ^ T . 
 
 Of iron in wood, also coated in grease, oV- 
 
 123. To adopt the theory of Coulomb, friction appears to arise 
 from the porosity of bodies. All, even the most dense, have large 
 spaces between the particles of which they are composed. When 
 they rest upon each other, the prominent parts of the one fall 
 into the cavities of the other, and are in a manner locked. In 
 hard substances, the maximum of this effect will be produced in 
 a short time ; but where they are soft, the maximum will not be 
 reached until the utmost compression the pressure is capable of 
 producing, is attained. 
 
 The polishing of bodies consists merely in rubbing down the 
 asperities, and multiplying the cavities in number, but diminish- 
 ing their depth ; hence they still interlock ; but the weight must be 
 raised but a small distance, in order to disengage the prominences 
 from the cavities, and the more perfect the polish the less will be 
 the height to which the weight must be lifted. The friction of bo- 
 dies in motion must obviously be less than that of bodies at rest, 
 because the projections of one substance require a certain defi- 
 nite time to adapt themselves to the cavities of the other, and the 
 difference will be greatest in those bodies that require the greatest 
 time to adapt themselves to each other. When unctuous matters 
 are interposed, they fill up the cavities and prevent the penetra- 
 tion of the prominent parts ; hence the resistance becomes almost 
 solely that which is due to the attraction of their own particles. 
 As the weight increases, liquids being more readily forced out of 
 vessels that contain them, and thus most easily disengaged from 
 the cavities, oppose less resistance than solids to penetration, and 
 are less efficacious in diminishing friction ; but on the other hand, 
 the more perfect the fluidity, the more easily are the particles of 
 liquid moved among each other ; and hence, so long as the pres- 
 sure is not sufficient to force them out of the cavities, they will 
 be better suited to diminish friction than thicker oils, or solid 
 grease. 
 
 The particles of homogeneous bodies are arranged in a similar 
 manner, whether by the action of crystallization, or their organi- 
 zation ; hence the cavities and asperities will fit better, and ap- 
 ply themselves more closely, than in heterogeneous bodies. 
 
 When a body slides upon another, one of two things must occur, 
 either the weight must be partially lifted, or the asperities must 
 be broken down before the sliding body ; either of these will re- 
 quire a force of some intensity, but when a body rolls, the very 
 act of rolling disengages the prominences from the cavities. The 
 action of axles, resting in sockets, is intermediate between that of 
 
120 OF FBICTION. [BOO/C HI. 
 
 rolling and that of sliding bodies, and hence has an intermediate 
 degree of advantage. It will be seen hereafter, that an adhesion 
 takes place between bodies which depends upon the extent of the 
 surfaces, and although this be extremely small, it is sufficient to 
 account for the small increase that follows the law of the surface. 
 
 The friction being in a certain degree removed from the bodies 
 themselves, and taking place among the particles of the oleaginous 
 substances, when the latter are used as coatings ; and as an in- 
 creased pressure will have an effect in overcoming the latter re- 
 sistance, it appears probable, that in this case, an increased pres- 
 sure may act in opposition to the other part of the friction, of 
 which it is the effective cause. 
 
 The property that bodies have of moving forward in the straight 
 line in which the force applied has been directed, would prevent 
 the prominent parts of bodies, moving with greater velocities, 
 from entering as deep into the cavities of those on which they 
 move,' as they would if moving with less velocities, and hence, at 
 great velocities, there ought to be an ascertainable diminution of 
 the friction. 
 
 124. The whole of the circumstances that are involved in fric- 
 tion, may be represented by .a formula, which is as follows: 
 
 F=/P <pP4ys-yv. (108) 
 
 In this, F represents the friction, P the pressure, S the area of 
 the surface, and V the velocity ; f is the ratio of the pressure to 
 the friction, under ordinary circumstances, say the fraction given 
 for the particular cases in 120; 9 is a very small function of the 
 pressure, whose amount has not been fully established from ex- 
 periment ; <p' is the cohesion of the unit of the surface ; and <p" a 
 very small coefficient of the velocity, whose magnitude is also 
 unknown. 
 
 125. The above theory, and the inferences from the experi- 
 ments whose results have been cited, point out the modes that 
 may be employed in practice to lessen or overcome the friction. 
 
 (1.) The line in which the power is applied, instead of being 
 parallel to the surface on which the body moves, may be slightly 
 inclined upwards. If the power be resolved into two components, 
 one of which is parallel, and the other perpendicular to the sur- 
 face, the first alone will be applied to the draught, and the other 
 will act to raise the weight of the body. By this latter action, 
 the parts of the surfaces that have been interlocked may be dis- 
 engaged. 
 
 (2.) The highest practicable degree of polish must be given ; 
 and the surfaces except in the case of rolling, coated with unctuous 
 matters. When the pressure is small, the most limpid oils should 
 be used ; as the pressure increases, those of more viscidity ; on a 
 
Book ///.] OP FRICTIOBT. 121 
 
 still further increase, tallow must be^em ployed ; when the pres- 
 sure becomes very great, or the velocity is such as to melt the 
 tallow, plumbago may be wfixed with that substance ; and finally, 
 in the greatest pressures that are to be found in practical mecha- 
 nics, plumbago alone, or soapstone, both in the state of fine pow- 
 der, may be employed. 
 
 (3.) The motion of rolling may be substituted for that of sli- 
 ding, when the body has a figure that will admit of it. 
 
 Thus tobacco in Virginia was formerly drawn to market, by 
 making the hogshead roll along the ground. This change of the 
 mode of motion, is not only advantageous in bodies of a circular 
 section, but may be made useful in others, although in them it 
 will become necessary to lift the weight at each turn the body 
 makes. If the force necessary to lift the weight, in such cases, be 
 not greater than the difference between rolling and sliding fric- 
 tions, an advantage will be gained. 
 
 (4.) If the body be of such a figure that it cannot advantage- 
 ously be made to roll, it may be set upon 'rollers, and the ad- 
 vantage which is due to their diameters and mode of motion, 
 will be attained. The body, in moving forward, will leave the 
 rollers behind it, which must therefore be lifted and carried for- 
 ward to receive it again ; hence this method is, generally speak- 
 ing, confined to short distances. When, however, the weight is 
 very great, it may be more advantageous than any other practica- 
 ble means : thus, for instance, the great rock, that forms the pe- 
 destal of the statue of Peter the Great, at St. Petersburgh, was 
 moved a distance of several versts, from the place where it was 
 found imbedded, to the banks of the Neva, upon small spheres of 
 metal, laid in a trough ; and after it was transported to the city, 
 was carried by the same method to the place where it was to be 
 set up. The removal of this vast mass would have been imprac- 
 ticable, by any of the ordinary means of transportation. 
 
 (5.) When rollers are inapplicable, the weight to be moved 
 may be laid upon a wheel carriage : here, besides the diminution 
 of friction which is obtained, a mechanical advantage is gained, 
 equivalent to the ratio of the diameter of the wheel to that of its 
 axle. 
 
 (8.) The use of wheels being attended with this mechanical 
 advantage, the extremities of their axles may be made to rest 
 upon the circumferences of other wheels, by means of which a 
 similar advantage may be gained, or may be made to press against 
 revolving bodies, instead of resting in cylindrical sockets. Such 
 applications of this principle are called friction wheels, and fric- 
 tion rollers. 
 
 16 
 
 
122 o* FRICTIOW. [Book III. 
 
 The most beautiful application of this kind, is that which was 
 adapted by Atvvood to his machine, which has already been spo- 
 ken of in 95. In this apparatus, the extremities of the axles 
 of the wheel, over which the cord, that connects the weights, 
 passes, are made each to rest upon the circumference of two 
 other wheels, and a mechanical advantage is gained in the ratio 
 that has just been mentioned. By this arrangement, the friction 
 is rendered wholly insensible, and interferes in no appreciable 
 degree with the results of the experiment. 
 
 In the patent blocks of Garnett, the axles, instead of moving 
 in cylindrical sockets, rest each upon six friction rollers, arranged 
 in a box, and by this means a similar advantage is gained. 
 
 An attempt, founded upon similar principles, is now making 
 by Wynans of New-Jersey, and applied to the wheels of car- 
 riages. 
 
 It will be obvious, that the advantage of wheels ceases, when 
 their effective friction becomes greater than that of their circum- 
 ferences upon the surfaces upon which they move. This is the 
 case upon hard smooth substances, such as ice, in which case 
 sledges are to be preferred to wheel carriages. 
 
 (7.) A knowledge of the fact, that a very great diminution of 
 the rubbing surface is attended with a sensible diminution of the 
 friction, may be applied with advantage in a limited number of 
 cases. Thus : where the surfaces are so hard as to admit of no 
 penetration, even where progressive motions are employed, the 
 moving body may have its surface diminished almost to an edge. 
 Of an application of this sort, we have an instance in the case of 
 skates. In rotary motions, the application of this principle is 
 more easy. Thus : although in machinery, generally, the axles 
 must be of a certain size, in order to bear the weight of the 
 wheels, and often the action of other pressures to which they are 
 subjected, these weights and pressures become so small in the 
 case of part of the train of wheels in the common watch, that 
 the bearing of the axles may be reduced to the smallest points 
 that can be made on hardened steel. These points, instead of 
 resting in sockets, are supported in small shallow cups of a hard 
 material, agate, ruby, or diamond. In motions of oscillation upon 
 axes, the axis may take the form of an edge of steel, and may be 
 made to rest upon a cylindrical surface, or even upon a polished 
 plane of some hard material ; of this we shall have instances in 
 the Balance and the Pendulum. 
 
 126. Friction is by far the most influential of the causes, by 
 which bodies moving near the surface of the earth are brought 
 to rest. If supported, they experience a friction from the body 
 that supports them ; if unsupported, they fall to the earth, either 
 
///.] OP FRICTION. 123 
 
 in a vertical or inclined direction ; if in a vertical direction, the 
 friction they meet in penetrating, rapidly destroys their motion, 
 even if the earth be soft where they fall : if in an inclined direc- 
 tion, of the two components of their metion, one of which is 
 perpendicular, the other parallel to the surface of the earth, the 
 former is at once destroyed by the resistance to penetration, the 
 other remains to carry the body along the surface, and this again 
 would be finally destroyed, by friction against the surface, even 
 did no other retarding force act. 
 
124 OT THE STIFFNESS [Book III. 
 
 CHAPTER T. 
 OF THE STIFFNESS OF ROPES. 
 
 127. It frequently becomes necessary, in practical mechanics, 
 to bend ropes over cylinders and rollers. This is the case even 
 in some of the elementary machines. When ropes are thus bent, 
 they always oppose a resistance, to overcome which it becomes 
 necessary to apply a part of the force that actuates the machine. 
 This resistance, it is obvious, may vary : 
 
 (1.) With the tension of the rope, or the weight by which it 
 is stretched ; 
 
 (2.) With the quality of the rope, depending upon the nature 
 of its materials, and the manner of its manufacture 5 
 
 (3.) With the size of the rope ; 
 
 (4.) With the diameter of the cylinder, over which it is bent. 
 
 128. The best experiments on this subject are also by Cou- 
 lomb. The first of the results obtained by him is : that like fric- 
 tion, the resistance of ropes to forces applied to bend them, is a 
 constant retarding force. The deductions, in respect to the cir- 
 cumstances that have been stated, are as follows, viz. : 
 
 (1.) The resistances of ropes are directly as the tensions to which 
 they are subjected ; 
 
 (2.) The resistance is greatest in ropes that have been strongly 
 twisted, in ropes coated with tar, and in new ropes. The ratio of 
 these is in some measure included in the next circumstance. 
 
 (3.) The resistance increases with some determinate power of 
 the diameter of the rope, which we shall call n : 
 In new tarred ropes, n=2, 
 In new white ropes, n=1.7, 
 In old ropes, n=1.5. 
 
 (4.) The resistances are inversely as the diameters of the cylin- 
 ders, around which the ropes are bent. 
 
 129. When a rope is wound more than once around a cylinder, 
 it is found that the resistances increase in geometric progression. 
 
 This principle is frequently applied in practice, when it is 
 wished to oppose rapidly increasing resistances to moving bodies : 
 thus, in arresting the progress of a vessel, a rope is turned again 
 and again around a post ; and a small number of turns will be- 
 come efficient to overcome any force not of sufficient intensity 
 to break the rope. 
 
 The resistance, in any particular case, is represented by the 
 formula, 
 
Book III.} o RO?US. 125 
 
 rn+pw, (109) 
 
 when m is the absolute resistance of the rope, and p the propor- 
 tion of the weight w, that is necessary to be added,in order to over- 
 come the increased resistance due to the addition of a weight. 
 
 1 30. To furnish data for the application, itis sufficient to quote a 
 single instance calculated from the experiments of Coulomb, 
 whence, by the principles we have laid cown, the resistance of 
 any other rope may be calculated. A roje not tarred, of an inch 
 in diameter, may be bent around a cylinder of 4 inches in diame- 
 ter, by a weight of T Vth of a pound, adced to T W of the weight 
 b)' which it is strained ; a tarred rope, o: % the same size, requires 
 a weight of about ^th of a pound, added .o J-yth of the weight by 
 which it is strained. From these two iistances, the resistance of 
 any other rope may be calculated by mtans of the principles that 
 have been laid down. 
 
 The sizes of ropes are usually estinated by the measures of 
 their circumference, as 1, 2, 3 inch, &c. Hence the instance* 
 given may serve as units. 
 
 The resistance, in all cases whatsoevtr, will be 
 
 , 
 
 In which expression,/?, w, and m are 35 in the former equation, 
 and have the values given in our instance ; n is the power from 
 128 ; cthe circumference of the rope; ind d the diameter of the 
 .cylinder, both expressed in inches, 
 
126 OP THE [Book III. 
 
 CHAPTER VI. 
 
 E MECHANIC POWERS. 
 
 131. A machine isaninstrument,by means of which we change 
 either the direction onthe intensity of a force, or both its di- 
 rection and intensity. The general principle of the equilibrium 
 of all machines whatsoever, is to be found in that of virtual veloci- 
 ties, By this, if the sevral points of the machine, on which the 
 forces act, were each to \e supposed to move under the action of 
 
 the force that is applied, 
 the products of all the i 
 latter being distinguishe 
 their directions, is equal I 
 have a fixed point, the pr 
 5th, becomes : Equilibri 
 sum of the products of all 
 into the respective virtua 
 
 jquilibrium will exist when the sum of 
 rces into their several velocities, the 
 as positive, or negative, according to 
 0. As machines, generally speaking, 
 position, in conformity with 71, case 
 m will exist in any machine, when the 
 c forces, on each side of the fixed point, 
 velocities of their points of application. 
 
 is exactly equal to the sum of the similar products on the oppo- 
 site side of the fixed poir,t. The value of this principle, as ap- 
 plied to the useful propeiiies of machines, will be discussed here- 
 after. For the present, Ive shall leave it, and proceed by more 
 direct methods to investigate the conditions of equilibrium in 
 the more simple forms of machines. 
 
 132. Machines are either simple or compound. The former 
 are the elementary parts, of which all compound machines are 
 made up, by combinations of various descriptions ; they are also 
 capable of being used singly. These simple machines are called 
 the Mechanic Powers. 
 
 133. The Mechanic Powers are six in number, viz. : The Le- 
 ver, the Wheel and Axle, the Pulley, the Wedge, the Inclined 
 Plane, and the Screw. 
 
 It has already been stated, that their properties may be re- 
 duced to a single principle ; but in the mode that has been chosen 
 for examining their conditions of equilibrium, it will be seen, 
 that the properties of four of them may be included in those of 
 two others. Hence the Mechanic Powers are arranged in two 
 divisions : to the first belong the Lever, the Wheel and Axle, 
 and the Pulley ; to the second, the Wedge, the Inclined Plane, 
 and the Screw. 
 
MECHANIC POWERS. 127 
 
 Oftfe Lever. 
 
 134. A lever is an inflexible bar, or rod, resting upon a fixed 
 axis or prop, that is called the Fulcrum, around which it is free 
 to move, under the action of the impressed forces. The general 
 condition of equilibrium, in the case of any number of forces 
 whatsoever, is an immediate deduction from the theory of the 
 moments of rotation in 34, and is as follows : 
 
 135. In any lever, whatsoever, equilibrium will exist, when 
 the sum of the products of the forces applied to it, on one side of 
 the fulcrum, into the perpendiculars let fall from that poi-nt upon 
 their respective directions, is exactly equal to the sum of similar 
 products, on the opposite side of the fulcrum. 
 
 Among the forces, it is obvious, that the weight of the lever 
 itself may be included, its direction being a vertical line, and its 
 point of application the centre of gravity of the bar. 
 
 136. The properties of the lever, and indeed of all the simple me- 
 chanic powers, are most usually limited to the case of the action 
 of no more than two forces. One of these is called the Power, the 
 other, the Weight. By the Weight, we understand the resistance 
 to he overcome ; by the Power, the force, whatever be its nature, 
 that is applied to overcome the resistance. In the usual mode of 
 treating the theory, the bar is supposed to be devoid of weight. 
 The points of application of these two forces, and the fixed point, 
 or fulcrum, may have three possible positions in respect to each 
 other, and we hence distinguish three different kinds of lever. 
 
 (1.) W T hen the fulcrum is between the power and the weight. 
 (2.) When the weight is between the power and the fulcrum. 
 (3.) When the power is between the weight and the fulcrum. 
 
 137. If the lever be straight, and the power and weight act 
 parallel to each other, equilibrium will exist, when the power 
 is to the weight in the inverse ratio of their respective distances 
 from the fulcrum; 
 
 or when Pa=W6. (Ill) 
 
 This case becomes the simple one in 22, of finding the point 
 of application of the resultant of two parallel forces ; for the fixed 
 point will be in the same state, as if it were acted upon by a force 
 equal and opposite to the resultant of the other two ; and it was 
 there demonstrated, that the point of application of the resultant 
 of two parallel forces, divides the line of application into parts 
 inversely proportioned to the intensities of the two forces. 
 
 138. In the case of a bent lever, or when upon a lever, whether 
 straight or crooked, the directions of the power and weight are 
 not parallel, equilibrium will exist when the two forces are to 
 
128 OP THE [Book IIL 
 
 each other inversely as the perpendiculars let fall from the ful- 
 crum upon their respective directions. 
 
 This may be deduced directly from (25),it,beinfi; again obvious, 
 as in the former case, that the fulcrum must be the point of the 
 application of the resultant of the two forces that keep the lever 
 in equilibrio ; and by that equation it is shown, that this point is 
 situated in such a manner, that the perpendiculars let fall from it, 
 upon the directions of the two forces, are in the inverse ratio of 
 the intensities of the two forces. 
 
 139. The lever is perhaps the most ancient, as it is still the 
 most familiar in its use of all the mechanic powers. The instances 
 of its practical application are almost too numerous to admit of 
 their being named. It will be sufficient to give a few, merely as 
 illustrations of its properties. 
 
 (1.) Of the first kind of lever, where, as in the figure beneath, 
 the fulcrum is between the power and the weight, we have in- 
 
 stances in the common crow-bar and handspike, in scissors, po- 
 kers, pincers, snuffers; and of a bent lever of this description, in 
 a hammer, when used for drawing nails. 
 
 (2.) In oars, the fulcrum is where the blade strikes the water ; 
 the weight is applied where the oar rests against the side, and the 
 power is applied by taking hold of the opposite extremity ; hence 
 
 they are levers of the second kind, in which, as in the figure, the 
 power is between the fulcrum and the weight. 
 
 The rudders of ships act upon similar principles. Of the same 
 kind of lever, cutting knives fixed at one end, doors moving upon 
 their hinges, the manner in which a weight is borne upon a wheel- 
 barrow, nut-crackers, &c., may be cited as instances. In these 
 two first kinds of lever, the distance of the power from the ful- 
 crum is greater than that of the weight; hence the weight has a 
 greater intensity than the power, when the two are in equilibrio, 
 and thus the power is capable of overcoming a resistance greater 
 than its own measure. 
 
Book ///.] MECHANIC POWERS. 129 
 
 (3.) Of the third kind of lever, in which, as represented on 
 the figure, the power is between the weight and the fulcrum, we 
 
 have examples in the common tongs, in sheep shears, in the man- 
 ner a ladder is raised against a wall. In this kind of lever, the 
 power being nearer to the fulcrum than the weight, the former 
 must have, in the case of equilibrium, a greater intensity than 
 the latter. In the two first kinds of lever, then, a given power 
 will raise a greater weight, or overcome a greater resistance than 
 it can when it acts directly; while, in the third kind of lever, 
 the weight raised, or the resistance overcome, is less than the 
 power is capable of doing, if it act without the intervention of 
 the lever. If the several figures be referred to, it will be seen 
 that these advantages in the two first cases, and this disadvantage 
 in the last, are compensated by the difference in the velocities of 
 the points of application, in case motion should take place. These 
 velocities will be represented by the arcs described. These are to 
 each other as the radii of the respective circles, of which they 
 are similar arcs, and as their radii are the arms of the levers, that 
 are to each other in the inverse ratio of the forces applied at their 
 extremities, the product of the weight into its velocity, will be 
 equal to that of the power into its velocity. This, it will be at 
 once seen, is no more than a case of the general principle of vir- 
 tual velocities. Hence, whenever intensity of force is gained by 
 means of the lever, it is always at the expense of an equal loss of 
 velocity; and where velocity is gained, it is gained at the ex- 
 pense of power. A similar inquiry into the velocities, with which 
 the points of application of the power and weight would move, in 
 the case of the equilibrium being disturbed, would show that 
 the same relation exists between the intensity offeree, and the 
 velocities lost or gained in actual motions. 
 
 140. The Balance is one of the most useful applications of the 
 lever. It is no more than a lever of the first kind, with arms of 
 equal lengths, resting upon a fulcrum. To the two extremities 
 are attached pans, or scales, in which heavy bodies may be placed. 
 It will be obvious, that when the two weights are equal, the ba- 
 lance will be in equilibrio under their joint action. If then a 
 certain number of units and fractions of any conventional system 
 
 17 
 
13* 0* TSX [BOO* W 
 
 of standard weights, be in one of the scales, it will just counter- 
 poise a substance placed in the other, whose weight must have an 
 equal value in that system. Hence, by means of a balance, the 
 unknown weight of any articles whatever, may be determined 
 by the aid of a set of properly graduated weights. 
 
 The balance, having this property, is of the most extensive 
 utility, not only in philosophical inquiries, but also in every va- 
 riety of trade, in which the articles cannot have their quantities 
 determined by measures of length, of surface, or of capacity, 
 either on account of their absolute nature, or in compliance with 
 custom. 
 
 141. It is difficult, in practice, to make the distances between 
 the two points whence the scales are suspended, and the fulcrum, 
 exactly equal. Still a sufficient degree of accuracy may be at- 
 tained for all the purposes of trade. But it sometimes happens 
 that balances, either by accident or design, have arms of unequal 
 lengths. In this case, the error may be detected by changing 
 the position of the two counterpoising weights from one scale to 
 the other. If, when thus transferred, they are still in equilibrio, 
 the balance is true, if they are not, it is false. 
 
 The actual weight may be obtained by means of a false balance, 
 by weighing the substance whose quantity is to be determined, 
 successively, in the two scales. Its true weight will be the geo- 
 metric mean between the known weights that counterpoise it, in 
 the two different positions. 
 
 of the nfcotoce vim m**r 
 
 scales; and o the tiro aims : then 
 nie property of uw lever, 
 
 P=W4, (111) 
 
 and W : P : W. (112} 
 
 When a balance is used for delicate investigations, any error 
 that might arise from an inequality in the arms, is readily ob- 
 viated by a simple process. The body, whose weight is to be 
 determined, b placed in one of the scales, and is counterpoised 
 by a substance capable of minute division, such as fine sand, placed 
 in the other. When the balance has been brought to rest in a 
 truly horizontal position, the body to be weighed Is removed, and 
 weights placed in the same scale, until they counterpoise the 
 substance that remains in the other scale. It will be obvious, 
 that the body whose weight is required, and the standard weights 
 
Book in.] XXCKAXK rOWXML 131 
 
 being both in equilibrio with the same pihiiiiirr, and having 
 acted upon the same arm of the lever, must he exactly equal to 
 each other. 
 
 142. The properties of a good balance are 
 
 (1.) That it should rest io a horizontal position when Inajij 
 with equal weights, and in an '"fljpf^ pfreitti?" when the weights 
 are not equal ; 
 
 (2.) That it should have great sensibility, so that a small pro- 
 portion of the weight with which it is loaded, added to either 
 scale, shall disturb the equilibrium ; 
 
 (3.) That it should be stable, or soon return to rest, after bar- 
 ing been put in motion by a change of the weijfes. 
 
 These properties depend in part upon a proper choice of the 
 point of suspension, in respect to the positions of the line that 
 joins the points whence the scales are suspended, and of the cen- 
 tre of gravity ; and in part upon accurate mechanical construc- 
 tion. 
 
 In the figure beneath, let A and B be the points whence the 
 
 scales are suspended; G, the centre of gravity; O, the centre of 
 suspension ; C, the point where the lines that join AJ3, and O,G, 
 intersect each other. 
 
 If the points O,C,G, were to correspond, the balance would 
 be, 1O7, in a state of indifference, it would be the most JIHIJ 
 ble to variations of weight, but would have no tendency to come 
 to rest in a horizontal position. The higher the point O, the 
 more stable will be the equilibrium, but. the less, all things else 
 being equal, will be the sensibility. The longer the arms, the 
 greater will be the moments of rotation of the weights, and con- 
 sequently the greater the sensibility. The centre of gravity will 
 be lowered by additional weights in the scales, and in this way 
 also the sensibility will be diminished. 
 
 If the point O should Call below C, the balance will be unsteady; 
 
133 OF THE [Book HI. 
 
 but if below both C and G, the balance will be in a state of tot- 
 tering equilibrium, and will be liable to be overturned. 
 
 In order that the balance shall come to rest in a horizontal po- 
 sition, not only must its arms be of equal lengths, but they, with 
 the scales attached, must be of equal weights. 
 
 In the actual construction of the balance it is important 
 
 (1.) That the motion shall be attended with as little friction as 
 possible : this is effected by making the axis of suspension of the 
 form that is called a knife-edge. A prismatic bar of hard steel is 
 passed through the beam of the balance, and is formed into an 
 edge beneath, by the intersection of two convex curves. In the 
 common balance, this is made to rest at each end on the surface 
 of a hollow ringj^r cylinder. 
 
 In some of the more accurate balances, the knife-edges rest 
 upon planes of polished agate. This is one of the cases that were 
 quoted in 125, in which the friction is reduced to the minimum, 
 by diminishing the rubbing surface to a mere edge. 
 
 As the friction is in the balance, as in other practical cases, pro- 
 portioned to the pressure, the greater the weight with which the ba- 
 lance is loaded, the less will be the sensibility ; and the latter is, as 
 has just been shown, also diminished, by the lowering of the centre 
 of gravity. To obviate this defect, some balances have been made 
 with a sliding weight beneath the point of suspension, by chang- 
 ing the position of which, the balance may be made more or less 
 sensible. 
 
 (2.) The distance hp.tween the centre of suspension, and the 
 points whence the scales hang, ought 10 remain exactly the same 
 during all the oscillations of the balance. This is sometimes ef- 
 fected by suspending the scales from knife-edges also. These are 
 passed through the ends of the beam with the edges uppermost ; 
 and the scales are hung from hooks or rings that rest upon them. 
 Sometimes, to make the touching surfaces the least possible, these 
 rings are ground on the inside to a sharp edge. 
 
Book III.] MECHANIC POWERS. 13S 
 
 A balance of a good construction is represented beneath. 
 
 In some of the best balances by Ramsden and Troughton, the 
 beam, instead of being a bar, is made of the form of two similar 
 and equal hollow cones, joined together at their greater bases. 
 
 Such a figure possesses far more strength than a solid bar. It 
 may, therefore, be made much lighter than almost any other 
 form. It is said that these instruments weigh so well, as to not$ 
 
13* op THE {Book III 
 
 differences of one-millionth part of the weight with which the 
 scales are loaded. 
 
 It is, however, an excellent balance that will weigh to the 
 TffTFoinrth of the weight with which it is loaded, and generally 
 speaking, balances do not weigh more nearly than from ^j^ to 
 smooth of the weight. 
 
 Balances of different sizes, strength, and materials, have differ- 
 ent degrees of accuracy. For weighing weights of different mag- 
 nitudes, then, several balances will be necessary, from those which 
 turn with weights of a small fraction of a grain, to those which 
 will bear several tons. 
 
 143. Levers may be combined together in such a manner as to 
 
 forma compound machine, as is the case in the figure. 
 
 Here the power P would be in equilibrio with a force acting 
 at W, when their relation was inversely as their distances, or 
 
 but, if instead of a weight, the end W be made to act upon the ex- 
 tremity of another lever, whose arms are p' and w\ then 
 
 Wj/= WV ; 
 
 and the third lever will have the following condition of equili- 
 brium : 
 
 whence 
 
 Ppp'p"=Www'w": (113) 
 
 therefore, 
 
 In a combination of levers, the power will be in equilibrio 
 with the weight, when the former is to the latter, as the con- 
 tinued product of all the arms of the lever, on which the weights 
 act, is to the continued product of all the arms on which the 
 power acts. 
 
 144. Combinations of levers may, upon this principle, be used 
 as weighing machines: for if the relation between the lengths of 
 their several arms be known, the relation between a known 
 weight, acting at one extremity of the first lever, to the unknown 
 mass which is in equilibrio with it at the farthest point of the 
 system, is also known ; and the weight of the latter will be deter- 
 minable. The most useful application of such a combination of 
 levers is in the platform balance, of which the following is a des- 
 cription : 
 
Book III.] 
 
 MECHANIC POWERS. 
 
 135 
 
 AB is a section of a platform of wood, of a rectangular shape, 
 resting upon a frame represented at H and I. CP", C'P", are levers 
 of the second class, having their fulcrums at C and CV Of these 
 there are four, diverging from the centre of the platform, in the 
 direction of i-ts semi-diagonals, to the four corners. At W and 
 \V, upon the two that are represented in the section, are pins, 
 which, by a small motion, may be brought in contact with the 
 platform, and thus may be made to raise it from the frame, and 
 bear its weight with that of the articles to be weighed. The ex- 
 tremities of these four levers rest upon a bar at P", which is sup- 
 ported at the point W", by another lever of the second class DP'. 
 The last is connected by a wire reaching from the extremity 
 P', of its longer arm, to the shorter arm of a lever of the first class ; 
 at the opposite end of which a scale, S, is suspended. A weight 
 placed in the scale S, will raise the point P', and with it the bar 
 P"; and the rise of the latter will cause the weight of the plat- 
 form, and the articles with which it is loaded, to press upon the 
 pins W, W. Thus a small weight in the scale, S, will be in 
 equilibrio with a large one on the platform AB, and their rela- 
 tion will be given by the formula (113). The lever, P'D, is 
 generally placed in a position at right angles to that in which it 
 is represented, projecting beyond one of the longer sides of the 
 rectangle : and it is evident, from mere inspection, that the four 
 levers, P"C, &c. act, so far as the conditions of equilibrium are 
 concerned, as a single one. Four are used, in order to bear the 
 platform at a sufficient number of points, and cause it to rise and 
 fall parallel to itself. 
 
 145. A Steelyard is another modification of the first kind of le- 
 ver, which is also used as a weighing machine. The lever in this 
 case has, as represented in the figure on the next page, unequal arms. 
 To the shorter of these, the substance to be weighed is attached, and 
 
136 OJ THE [Book III. 
 
 its weight is determined by means 
 of a constant known weight, that is 
 moved to different distances from 
 the fulcrum, until it be in equili- 
 brio with the substance to be weigh- 
 ed. If their distances from the ful- 
 crum be equal, the two weights are 
 equal ; if the constant weight be twice as far from the ful- 
 crum as the fixed point, whence the substance to be weighed 
 is suspended, it will be obvious that the latter weighs twice as 
 much as the former : and thus, as the distance of the constant 
 weight varies in arithmetical progression, the unknown weight 
 wilf vary as those distances. The longer arm of the steelyard is 
 therefore cut into equal divisions, and the unknown weight 
 is determined by the distance at which the constant moveable 
 weight is from the fulcrum, at the time equilibrium takes place. 
 
 The constant weight is suspended from the lever, by means of 
 a hook or ring that is cut beneath into a sharp edge, to enable it 
 to adapt itself to the notches that form the divisions of the longer 
 arm. Hence there is danger of these divisions being cut and 
 widened, until the instrument ceases to give true indications. So, 
 also, when the lever is inclined, the distance of the constant weight 
 varies. In few cases, indeed, can the steelyard be depended upon 
 for giving as true a measure of weight as the balance. If a scale 
 be suspended by knife-edges, from the longer arm of an unequal 
 lever, weights placed in it will have a value in determining the 
 weight of a substance suspended from the shorter arm, as much 
 greater than their true value, as the shorter arm is less than the 
 longer. Upon this principle they may be graduated ; and a weigh- 
 ing machine, thus constructed, will have an advantage over a 
 balance when very great weights are to be determined ; for the 
 short arm will be less liable to break under their action than the 
 arm of a balance, and the load upon the knife-edges will be much 
 less. 
 
 146. Were there no resistance to the motion of the lever, it 
 could be set in motion by the smallest addition either to the pow- 
 er or the weight; but in this case, as in all others, friction will 
 interpose ; and if the weight be set in motion by the power, the 
 latter must first receive an addition to its intensity, when merely 
 in equilibrio, which is equal to the moment of rotation of the 
 friction ; and in case it is desired that the weight shall set the 
 power in motion, the former must first receive a similar addition, 
 in order to be ready to cause motion by the smallest new accession 
 of force. 
 
Book ///.] MECHANIC POWERS'. 
 
 In the lever, the general equation of equilibrium is 
 
 137 
 
 if we let the resistance, whatever be its nature, =R, and let Rr be 
 its moment of rotation, then in the case in which the system is 
 ready to be set in motion, 
 
 Pp=Ww^Rr, . (114) 
 
 which will be a general condition in all machines whatsoever. 
 To apply this to the case of the lever : 
 
 Let AB be the lever acted upon by the parallel forces P and W, 
 and turning upon tjie cylinder C, as an axle ; call the arms p and 
 ic. Let the power, P, be on the point of setting the weight, W, in 
 motion. The pressure on the axle is equal to the resultant of 
 the two forces, P and W ; resolve this resultant into two forces, 
 one of which is a tangent to the axle, the other a normal. Let 
 m be the angle the tangential force makes with the direction of 
 the resultant, the two components are 
 
 (P+W) cos. m, 
 
 (P+W) sin. m . 
 
 that part of the friction which is directly opposed to the motion, 
 will have the same ratio to the whole friction,/, as the latter of these 
 components to the whole pressure, or will be 
 /(P+W) sin m, 
 
 1 
 and sin. m ~ ^/ (\-l-f *\ 
 
 whence the friction becomes 
 
 /(P+W) 
 
 and as r is its distance from the centre of motion, Rr in the gene- 
 ral formula (114) becomes 
 
 . . 
 
 and the condition of the state of the machine, in which motion is 
 about to begin, is 
 
1S8 or THE [Book III 
 
 The co-efficient,/, is in most cases, so small a fraction, that its 
 square,/ 2 , may be neglected, in which case the formula becomes 
 
 (118) 
 
 Of the Whetl and Axle. 
 
 147. The Wheel and Axle, as its name imports, is a wheel 
 firmly connected to an axle, and moving with it upon a common 
 axis. The power is applied to the circumference of the wheel ; 
 the weight to the circumference of the axle. It may, there- 
 fore, be considered as a lever having its fulcrum in the axis, and 
 the power, in a state of equilibrium, will be to the weight, as 
 the radius of the axle to the radius of the wheel, or 
 
 Pa=W6. (119 a) 
 
 But as the circumferences of wheels are proportioned to their 
 radii, the latter part of the proportion may be changed ; and 
 when the wheel and axle is in equilibrio, the power is to the 
 weight as the circumference of the axle to the circumference of 
 the wheel, or 
 
 P.2ira=W.2*6. (1196) 
 
 The simplest form of the wheel and axle is represented be- 
 neath. In it the power is applied by an endless rope passing over 
 
 
 the circumference of the wheel, and the weight to a rope coiled 
 or wound around the axle. Such is the form which is habitually 
 used in our warehouses. 
 
 It is not necessary that the wheel should be continuous: a 
 single spoke, or several, projecting from the axle, will be suffi- 
 cient. The axis may be either horizontal or vertical ; in the for- 
 mer case, an axle with bars or spokes, is called a Windlass ; in 
 the latter, a Capstan. 
 
 The windlass used in ships has a number of holes cut in the 
 direction of its length, upon four different parts of its circumfe- 
 rence ; handspikes or bars are placed, when the engine is to be 
 
Book ///.] 
 
 MECHANIC POWERS. 
 
 139 
 
 used, in the row of holes that is uppermost, and men springing 
 to these bars, act upon them partly by their muscular force, ?nd 
 partly by their weight. It is therefore an application of human 
 force that requires great exertion, and produces corresponding 
 effects for a short time. 
 
 The windlass which is used by well-diggers, has a bar at each 
 end ; this bar is bent at right angles, in order to furnish a con- 
 venient handle to the persons that work it. A handle thus formed 
 is called a Winch, and is of frequent application in many useful 
 cases. 
 
 A capstan with a single bar, to which a horse is harnessed, is 
 often used on the shores of our bays and rivers for the purpose 
 of drawing up logs and balks. In a ship's capstan, a number of 
 bars are inserted into the head, in the manner of the spokes of a 
 
 wheel. It is manoeuvred by men placed between these bars and 
 walking around. The exertion is therefore moderate, and can be 
 long continued. In large ships, the capstan passes through the 
 decks, and thus a gang of men may be applied to it on each deck. 
 In all these cases the weight is applied to a rope wound around 
 the axle. 
 
 148. A series of wheels and axles may be combined together, 
 as in the figure, by means of endless ropes or bands ; one of these 
 
140 
 
 O7 THE 
 
 [Book ///. 
 
 is passed over the circumference of a wheel, and the circumfe- 
 rence of the axle of the adjacent wheel ; the number of bands in 
 the system is always one less than the number of wheels. The 
 action is identical with that of a system of levers. The power, 
 therefore, will be in equilibrio with the weight, when the former 
 is to the latter, as the continued product of the radii of all the axles 
 is to the continued product of the radii of all the wheels. 
 
 149. Wheels and axles may also be combined, by making them 
 turn each other by the friction of their surfaces, and giving these 
 such a form as to exert a direct pressure upon each other. For 
 this purpose 
 
 (I.) Projecting pieces or cogs, as in the figure, may be adapted 
 
 to the circumference of the wheel, and the axle of the next may 
 be formed of two parallel circular plates, united by round staves, 
 arranged in the circumference of a circle : such an arrangement 
 is called a Cog-Wheel and Trundle : or, 
 
 (2.) The circumferences of both the wheels and axles may be 
 cut into teeth, as in the figure beneath. Such a modification is 
 called the Wheel and Pinion. 
 
Book 1IL] MECHANIC POWERS. 141 
 
 In conformity with the principles of a combination of levers, 
 of which this is an obvious application, equilibrium will exist in 
 a series of wheels and pinions, where the power is to the weight, 
 as the continued product of the radii of all the pinions, to the 
 continued product of the radii of all the wheels. 
 
 Were the radii of the pinions, and their number of teeth, exact 
 aliquot parts of the radii of the wheels, and of their number of 
 teeth, at each revolution of the wheel, the same tooth, upon its 
 circumference, would fall between the same two teeth of the 
 pinion. From this woufd arise an unequal wear. To prevent 
 this, the number of teeth on the wheel ought to be such as is 
 prime to the number of teeth qpon the pinion. This is usually 
 effected by adding one additional tooth to the wheels. A num- 
 ber of revolutions, then, equal to the number of teeth in the wheel, 
 must take place before the same two teeth of the adjacent wheel 
 and 'pinion can again come into contact. Such an additional 
 tooth is called the Hunting Cog. 
 
 To express the conditions of equilibrium in terms of the num- 
 ber of teeth : The power must be to the weight as the con- 
 tinued product of the number of teeth on all the pinions is to the 
 continued product of the teeth of all the wheels. 
 
 150. It is of great importance in systems of wheels and pi- 
 nions, that the teeth should have a proper curvature, in order 
 that the action of the power may be communicated to the weight 
 as directly, and with as little friction as is possible. It would 
 occupy too much space to enter fully into the detail of the con- 
 struction of the teeth of wheels, but it is necessary that the gene- 
 ral principles be explained. 
 
 We shall take the case of two wheels situated in the same 
 plane. 
 
 Let two circles touch each other, each being moveable around 
 the centre. A constant force in the direction of the common 
 tangent of the two circles, would make the circumference of 
 each revolve with equal velocities. In order that one of these 
 circles should transmit its motion to the other, simple friction 
 might at first be sufficient, but this would speedily wear them 
 away and thus disunite them. In practice, then, it is necessary, 
 that for the two circles that touch each other, two others described 
 around the same centres, and with diameters having the same 
 ratio to each other, should be substituted, and that on their cir- 
 cumferences should be placed teeth. These teeth, in order to keep 
 up an equable communication of motion, must satisfy this con- 
 dition : that in the action of the teeth, which will be in the di- 
 rection of the common normal of the surfaces in contact, the two 
 primitive circles shall be moved as if they were propelled by a 
 force in the direction of their common tangent. 
 
149 
 
 OF TUB 
 
 [Book III 
 
 Let there be two circles whose radii are AB, BD, that touch 
 each other at the point B : let there 
 be fixed upon one of the circles a 
 tooth terminated by the curve BM, 
 and on the other a radius AB, the 
 tooth, in turning, will move the radius 
 of the other circle ; the condition to 
 be fulfilled is that the curve BM shall, 
 in all its positions, touch the radius, 
 and the perpendicular to the radius 
 BA, at the point of contact, shall al- 
 ways pass through the fixed point B. 
 The curve which will satisfy the 
 conditions, is an epicycloid, /brmed 
 upon each of the circles by a point in 
 the other, supposing the latter to move 
 upon the circumference of the former, 
 in the same manner that a circle moves 
 upon a line when the common cycloid 
 is generated. 
 
 Thus in the figure, if t^o circle, whose radius is AB, be sup- 
 
Book III.] MECHANIC POWERS. 143 
 
 posed to move upon the circumference of the circle DBF, the 
 point F will describe an epicycloid DFE, and some portion of 
 this curve will be the proper figure to give to the projection of 
 the teeth. 
 
 The spaces between the teeth are formed by supposing the ra- 
 dius of either of the circles to be produced ; and that the point 
 on its extremity, shall, while the epicycloid is described, describe 
 a curve such as GHI ; this will give a form proper for the inter- 
 val of the teeth. The two curves cannot be made to unite except 
 at an angle ; hence it is necessary to join them by a straight line, 
 which is a common tangent. 
 
 If the teeth of one of the wheels have their forms determined, 
 the teeth of the other will no longer have the same form, but must 
 be modified so as to adapt themselves to the teeth of the first, in 
 conformity with the condition we have stated. 
 
 Thus in the case of the wheel and trundle, the staves of the 
 latter may be considered as teeth, whose sections are circular. 
 In this case, the cavities between the teeth will be portions of a 
 circle whose radius is the same as that of the staves of the trun- 
 dle ; the projections will be curves parallel to the epicycloid de- 
 scribed upon the circle whose radius is BD, by the circle whose 
 radius is AB. 
 
 However regular the curves may be, and however completely 
 they may satisfy the prescribed condition, there will be an in- 
 equality in their pressure on each other. This inequality may 
 be lessened by increasing the number, and lessening the size of 
 the teeth. In cases where intensity of force is gained at the ex- 
 pense of velocity, this inequality becomes less and less percepti- 
 ble at each addition of a wheel and pinion to the system ; but in 
 the case where velocity is gained, the apparent inequality is 
 multiplied in exact proportion to the increase of velocity. 
 
 In the former case, the successive variations will be repeated 
 several times within the time in which two teeth are in contact; 
 in the latter, they will be found to affect several teelh, each of 
 which will be unequally impelled. 
 
 151. The circumferences of the wheel and pinion that are in 
 contact will move, as has been seen, with equal velocities; their 
 moments of rotation are therefore proportioned to their respective 
 diameters. Hence, when wheels drive pinions, the intensity of 
 the force is diminished, and when pinions drive wheels, increased. 
 In the former case velocity is gained, in the latter it is lost. 
 
 152. When a motion is to be changed so that its direction 
 shall lie in a different plane from that in which the forces had before 
 acted, the wheel and pinion offers various modes of effecting the 
 change. Thus : the teeth may be cut upon the surface of a hol- 
 low cylinder, and stand at right angles to the plane in which the 
 
144 
 
 OF THE 
 
 [Book 111. 
 
 wheel revolves ; the axis of the pinion must then be perpendicular 
 to the axis of the wheel, and the motion will be taken off at right 
 angles ; such a wheel is called a Contrate Wheel. The same may 
 be effected by forming the teeth upon the surfaces of two conic 
 frusta, whose axes meet, and make with each other an angle, 
 which is the supplement of the required change of motion. The 
 figure of the teeth, in this case, is derived from a curve called the 
 spheric epicycloid. Such wheels are called Bevelled, or Mitre 
 geering, according as the angle their planes make with each other, 
 is right or oblique. 
 
 Let VAB, and VAD be sections of right cones ; wheels and 
 pinions constructed upon their frusta, will have the same me- 
 
 FIG. 1. 
 
 chanical properties as if they were in one plane. They will be- 
 sides change the direction of motion to a plane, making an angle 
 with that in which the original motion is performed, equal to the 
 complement of the angle made by their respective axes, VPJ, and 
 VC. In Fig. 1st, the motion is taken off at a right, in Fig. 2d, 
 at an obtuse angle. 
 
Book ///.] 
 
 MECHANIC POWERS. 
 
 145 
 
 In the figure beneath, a pair of mitre wheels, with their teeth 
 entering into each other, is represented. 
 
 Such wheels are sometimes equal in size, and answer no other 
 purpose than to change the direction of the motion. 
 
 153. In the original and simpler forms of the wheel and axle, the 
 friction of the pivots is estimated exactly as we have done it in the 
 case of the lever : but before the machine can be ready to be set in 
 motion, an additional force must be applied to overcome the resist- 
 ance of the rope or ropes. In the capstan and windlass there is 
 but one rope, and the equation of final equilibrium, when the 
 smallest force added to the power, will cause motion to begin, is 
 
 Po=W6+(P+W) i /r+6 (ro+p W) ; (119c) 
 
 in which expression, a is the radius of the wheel, b that of the 
 axle, and r of the gudgeon on which the motions are performed. 
 Coulomb, in order to give an instance of the application of his 
 theory, and the results of his experiments to practice, calculates 
 the joint resistances of the friction, and the rigidity of ropes, in 
 the case of raising 8000 Ibs. by means of a capstan, and finds that 
 one tenth part of the moving power must be expended upon these 
 resistances. 
 
 154. In the case of a wheel acting upon a pinion, or upon a 
 trundle, we shall not enter into a full investigation of the friction : 
 the old practical rule of allowing one eighteenth part of the poxver, 
 applied to its own arm of the lever into which the wheel may be 
 resolved, having been found to correspond with the results of the- 
 ory. A similar allowance must be made for every increase in the 
 
 19 
 
OJ THE 
 
 [Book 111 
 
 number of wheels and pinions in the system ; the additional fric- 
 tion being, in each, one eighteenth part of the force the wheel 
 exerts upon the pinion it drives. 
 
 Of the Pulley. 
 
 155. A Pulley is awheel, moveable upon an axis, and having 
 a groove cut upon its circumference, over which a cord passes. 
 It is enclosed in a box or case that supports the axle, which is 
 called its Block. The block may be either fixed to a firm sup- 
 port, or moveable. In both cases, the power is applied to one 
 end of the rope ; in the case of the fixed pulley No. 1, the weight 
 
 No. 1. No. 2. 
 
 is applied to the other extremity of the rope, in the moveable pul- 
 ley No. 2, the weight is suspended from the box or block. 
 
 In the fixed pulley, No. 1, the direction of the motion is 
 alone changed, for the power and weight have equal moments 
 of rotation, and hence may be considered as acting upon the 
 equal arms DC, CE, of a lever of the first kind, hence 
 
 P=W. (120) 
 
 In the moveable pulley No. 2, the rope is fastened at one end 
 to the fixed support, F ; this may also be considered as a lever, 
 but the fulcrum is not at the centre, but at the point D, hence 
 
 P : W : : DC : DE ; 
 
 but one of these lines being the radius, the other the diameter of 
 the same circle, 
 
 P=*W. (121) 
 
 In the moveable pulley, the direction of the force P is not 
 changed. But it is frequently desirable that the intensity of 
 the power shall not only be doubled, but that its direction shall 
 
Book ///.] 
 
 MECHANIC POWERS. 
 
 147 
 
 be changed. In order to effect this change, a fixed pulley may 
 be combined with a moveable pulley, as in the figure. 
 
 | s Here the condition of equilibrium is siill 
 
 ^ the same, or 
 
 156. Pullies, whether fixed or moveable, 
 may be combined together' in various man- 
 ners ; thus, as in the system beneath ; for 
 the weight that acts upon the moveable pulley 
 - 2 A, may be substituted a rope that 
 e is wound around a second pulley, 
 B ; to this, in like manner, a rope 
 passing over a third pulley, C, may 
 be applied, and the weight may 
 be attached to the box of C ; the 
 pulley, A. doubles the intensity of 
 the power, or 
 
 P=iW; 
 
 the pulley B does the same to the 
 force which acts upon it, which is 
 W and hence 
 
 The pulley, C produces the same 
 change in the force W", therefore, 
 
 W"=4W; 
 whence we obtain 
 
 P=| W. 
 
 It will be obvious, that in such 
 a system, the intensity of the pow- 
 er is increased in a geometric pro- 
 gression, whose common ratio is 
 
14S 
 
 OF THB 
 
 [Book Ilf. 
 
 2, and whose number of terms is the number of moveable pullies. 
 Hence, in a system whose number of moveable pullies is n, 
 
 2"(P)=W. (122) 
 
 This mode of combining pullies, is not convenient in practice, 
 and hence, in spite of its great power, it is not often used. A 
 system in which the number of fixed and moveable pullies is 
 equal, and all the moveable and all the fixed pullies are combined, 
 each kind in a single block, although it causes a less increase in 
 the intensity of the power, is, on account of its great convenience, 
 in much more extensive use. Such a system, composed of three 
 fixed and three moveable pullies, is represented below. It will 
 
 
Book ///.] MECHANIC POWERS. 149 
 
 be easily seen that as the rope is now continuous, the intensity of 
 the force cannot go on increasing in a geometric ratio. To deter- 
 mine the condition of equilibrium : the weight is supported, in 
 this case, by six ropes, or rather six separate parts of the. single 
 rope; each of them undergoes a tension due to the force P; they, 
 by their, united effort, support the weight, which is, therefore, 
 the resultant of these tensions, or of six equal and parallel forces; 
 hence 
 
 P=*W, 
 
 and for any number (ri) of moveable pullies, 
 
 2n P=W, 
 
 W 
 
 and P=^ (123) 
 
 A similar system may be formed by placing all the pullies 
 of each kind, in a box in such a manner, that their axis may be 
 the same. Their axles, however, must be separate ; for they will 
 not all revolve in equal times. The pair, composed of one fixed 
 and one moveable pulley, nearest to the power, having, at its cir- 
 cumference, the same virtual velocity with the power; while the 
 pair nearest to the weight has a virtual velocity no more than 
 twice that of the weight. The velocities of the several pairs of 
 pullies will, therefore, be as the series of natural nunbers,^ the 
 last term of which is the number of moveable pullies. 
 
 Out of this varying velocity grows an unequal wear upon the 
 different axles, and the evil would not be diminished by making 
 the pullies revolve at the same rate ; for this would be in fact 
 impracticable, without a vast increase of friction growing out of 
 the dragging of the rope over circumferences having naturally 
 different velocities, but which would be thus constrained to move 
 at the same rate by their connexion. 
 
 These defects are obviated in the blocks of White. In this, 
 each set of pullies is turned out of a single piece : the concentric 
 circles in the figure on the following page, are the projections of 
 the pullies, and are in an order of size corresponding lo the se- 
 ries of natural numbers. Hence a common rotation upon the 
 same fixed axis may be given to each set of pullies, which will 
 have the same velocity with the ropes that pass over them, and 
 a single axle to each block will be sufficient. 
 
150 
 
 OP THE 
 
 [Book III. 
 
 The modes in which pullies 
 may be combined, may be varied 
 almost infinitely : in them all, 
 however, the same principles are 
 applicable. It is* not necessary 
 then that we should pursue their 
 modifications to any greater ex- 
 tent. 
 
 157. When the ropes are not 
 parallel to each other, the direc- 
 tion of the forces becomes oblique, 
 and thus the effect of a pulley or 
 system of pullies, will be chang- 
 ed. The action of the power 
 and weight may, however, be 
 determined in all cases where the 
 angles the directions of the ropes 
 make with each other is known, 
 by means of the theorems of the 
 Composition and Resolution of 
 Forces 12. 
 
 158. The effect of friction up- 
 on pullies, and of the resistance of 
 ropes, may be calculated upon 
 the principles that have already 
 been laid down, in the case of the 
 wheel and axle. In a single pul- 
 ley, the equation of the state in 
 which the smallest additional 
 force will cause motion, is the 
 same as in the wheel and axle, 
 except that the radius of the 
 gudgeon r in (119c) becomes the 
 same as &, and the formula is 
 
 4-^W). (124) 
 
 In a complex system of pullies, this becomes difficult of ap- 
 plication. If, however, we assume, 1. That all the ropes are 
 parallel; 2. That all the wheels and all their axles are equal in 
 diameter; 3. That the function/, which represents the propor- 
 tion of the friction to the pressure, is very small ; 4. That the 
 resistance of the rope is proportioned to its tension, and that 
 hence, in formula (124) 77 = 0. Upon these assumptions, mak- 
 ing 7i the number of moveable pullies. and 
 
Book ///.] 
 
 MECHANIC POWERS. 
 
 161 
 
 the expression for the state in which the smallest addition to the 
 power will cause motion, may be found to be 
 
 (125 
 
 " 1. 
 
 The friction of pullies has been considerably lessened by an 
 application of friction wheels, (see 125,) made by the late Mr. 
 Garnett, of Brunswick, New-Jersey. Their plan is represented 
 beneath. 
 
 The axle of the pulley, it will be seen, rests upon six wheels, 
 enclosed in a box, and the friction is diminished in the ratio of 
 the diameters of these wheels to the diameters of their axles. 
 
 Of the Wedge. 
 
 159. The Wedge is a triangular prism, of some hard material, 
 whose section is,generally speaking, isosceles. The power is applied 
 perpendicularly to the surface on which it acts; the weight is a 
 resistance which is resolved into two parts, each of which acts 
 perpendicularly upon the other two faces of the wedge. In or- 
 der that equilibrium shall exist, these forces must, 14, converge 
 to a point within the wedge, and must be to each other in the 
 ratio of the sides of the triangular section on which they act. 
 
 It will be at once seen from 14, that three oblique forces 
 can only be in equilibrio when they converge to a point, and, 
 three such forces are proportioned in magnitude to the three 
 sides of a triangle formed by lines drawn perpendicular to the 
 direction of the forces. 
 
1 52 OF THE [Book III. 
 
 160. In an isosceles wedge, the weight is applied to the two 
 equal sides, the third side may be called the head of the wedge; 
 and equilibrium will exist when the power is to the weight as 
 the thickness of the head of the wedge is to twice the length of 
 either of its sides. This is an obvious inference from the gene- 
 ral proposition, and has the form of the following equation, a 
 being the thickness of the wedge, and b the length of either of 
 its sides. 
 
 Wa 
 P =-2b' . 
 
 161. The theory of the equilibrium of the wedge is of little 
 use in practical mechanics, for it is generally used in splitting or 
 cleaving, and is rarely, or never, acted upon by the constant ap- 
 plication of a force, but is impelled by a moving body striking 
 against it at intervals, or as it is styled, by percussion. The ef- 
 fects, in this case, are proportioned to the weight of the moving 
 body multiplied by the square of its velocity. For the aggre- 
 gation of the particles of the body into which it is thus driven, 
 may be considered as a constant retarding force, and from (616) 
 
 a 2 
 
 * = 2g : 
 
 it therefore appears, that the distance to which a body, retarded 
 by a constant force, will go, before it loses its whole velocity, is 
 proportioned to the square of its velocity ; hence the striking body 
 will continue to impel the wedge, until the latter has produced 
 an action proportioned to the weight of the former, multiplied by 
 the square of its velocity. 
 
 The effect of the wedge is still farther increased by the agita- 
 tion produced by collision, among the particles of the body into 
 which it isdriven; they are by thisaction more easily penetrated, 
 as is obvious from the fact that repeated blows will destroy the 
 aggregation of the strongest substances. This total destruction 
 of aggregation is, however, only finally effected by numerous 
 shocks, unless they be of great intensity ; and, generally speak- 
 ing, the force of aggregation becomes, so soon as the earlier 
 blows cease to be felt, as strong, to all appearance, as it was at 
 first. Hence, in the act of splitting or cleaving bodies by a 
 wedge, the body closes forcibly upon the wedge at the instant the 
 striking body ceases to act, and by its pressure produces a fric- 
 tion, sufficiently great to retain the wedge in the position to 
 .which it has been driven. This great friction adds another most 
 important property to the wedge, namely : that, although im- 
 pelled by an intermitting force, it does not return back to its 
 original position, when that force ceases to act, but retains all 
 
- 
 
 MECHANIC POWERS. 
 
 153 
 
 the advantage derived from previous impulses, to the whole of 
 which any new impulse is superadded. 
 
 162. The valuable applications of the wedge are : 
 
 (1.) To all splitting and cleaving instruments, and to every 
 variety of edge tools. 
 
 (2.) To obtain great pressures by means of small forces, wedges 
 being driven between a firm obstacle and the body to be com- 
 pressed. 
 
 A printing press by Rust, of New- York, is also an applica- 
 tion of the wedge. The two rollers, A and B, fall into cavities 
 
 F 
 
 of the wedge CDEF, and are fixed, one to the upper part of the 
 frame, the other to the platten. When the press is to be set in 
 action, the small end, C.D, of the wedge is drawn out by a com- 
 bination of levers, the rollers then slide along the plane faces of 
 the wedge, and a great pressure is produced. 
 
 (3.) To raise great weights to asmall height : each successive 
 impulse applied to the wedge raises the weight a small distance, 
 whence it does not again fall ; for the friction retains the wedge 
 in its place, until a new blow be struck. 
 
 One of the most valuable and beautiful applications of the 
 wedge, is to the lifting blocks of Seppings, by means of which 
 the vast weights of the largest ships can be raised and supported, 
 when placed in dock for repair. A section of these blocks is re- 
 presented beneath. 
 
 A modification of these blocks, which is said to be even more 
 convenient in use, has been invented by Thomas, an engineer in 
 the employ of the American Navy Department. 
 
 (4.) There is an application of the wedge that is called the 
 Lewis, which is employed in speedily attaching great weights to 
 
 20 
 
 . 
 
1.54 
 
 OP THS 
 
 [Book 1IL 
 
 the engines that are employed to move them. In a large stone, 
 for instance, a hole of the figure of a truncated cone, that does not 
 differ much from a cylinder, is cut, the larger base forming its 
 bottom ; into this is dropped an instrument of the following form. 
 A cylinder is cut by two planes equally inclined to a plane pass- 
 ing through its axis, and when the three pieces are laid togeth.er, 
 the whole has the exact cylindrical form. A section of this in- 
 strument is represented beneath; the side AB is placed lowest, 
 and a force applied to the ring C ; the effect of this upon the 
 
 C 
 
 wedge-formed piece, will be, to force the outer pieces against the 
 aides of the cavity, and the pressure thus produced, will cause so 
 much friction as to prevent the apparatus from being withdrawn 
 by any force not sufficient to break the substance in which the 
 hole is cut. By this simple and ingenious instrument, stones of 
 several tons in weight m,ay be firmly and suddenly attached to 
 engines employed to raise them, and as instantly detached, when 
 their weight is supported, and no longer acts upon the ring. 
 
 163. The principle of the wedge may be applied to the case 
 of pressures upon its triangular bases as well as upon its inclined 
 faces, and to bodies of pyramidal and conical figures : in all of 
 these, the same effects are produced by a succession of blows, as 
 in the simple triangular prism ; and friction acts in the same 
 manner to prevent their being withdrawn. Of this form, we find 
 applications in all piercing tools, and in nails, spikes, treenails, 
 and other similar instruments; they are employed for uniting 
 the parts of such bodies as permit them to penetrate without any 
 
HI.] MECHANIC POWEHS. 
 
 great difficulty, when driven by successive blows, yet which 
 close and retain them in their place, by means of great friction. 
 
 164. It will be obvious from what has been stated in speaking 
 t)f the applications of this mechanic power, that friction, although 
 it always resists the action of any power vvhatever,upon machines 
 formed of such materials as nature furnishes, is not, on that ac- 
 count, an absolute loss. Indeed, few of the mechanic powers 
 could act, were there no friction ; the friction of the ropes, in 
 the wheel and axle, and in the pulley, causes the cylinders on 
 which they rest, to turn ; and all the most valuable applications 
 of the wedge, derive the principal part of their usefulness from 
 the action of friction. So also it will be seen, that one of the two 
 remaining powers would be of little use, were it not for its friction. 
 
 The parts of machines, of buildings, and other mechanical 
 structures, could not be held together were it not for friction ; 
 and we shall find instances in which systems, that would other- 
 wise be in a state of tottering equilibrium, are rendered stable by 
 friction. 
 
 165. The friction which attends the use of the most valuable 
 applications of the wedge, is not such as can be reduced to cal- 
 culation, nor indeed is it important that it should. 
 
 Of the Inclined Plane. 
 
 166. The Inclined Plane is an instrument formed of a plane 
 surface, in any position whatsoever, except parallel or perpendi- 
 cular to the horizon. A body, placed upon such a surface, is 
 actuated by its own weight, exerted in a direction perpendicular 
 to the horizon, and which, being a constant force, would cause it 
 to descend, as shown in 47, with uniformly accelerated velo- 
 city ; its descent is retarded by a constant force consisting in the 
 resistance of the plane. These two forces, as may be deduced 
 from 56, have the ratio of the length of the plane to the length 
 of its base, or of the cosine of the plane's inclination to the hori- 
 zon, to unity, or 
 
 W 
 
 R= cos - 
 
 The force with which the body tends to descend the plane, 
 may be represented by VV sin. i ; and as the power, if applied in 
 a direction parallel to the line in the surface of the plane, in 
 which the body would tend to descend, must, in order to cause 
 equilibrium, be equal to the force with which the body tends to 
 descend 
 
 P=W sin. i. (127) 
 
156 or THE \Book III. 
 
 The sine of the angle of inclination is the ratio between the length 
 of the plane and its height, hence : 
 
 167. In art inclined plane, the power, when inequilibrio with 
 the weight, must have to it the ratio of the height of the plane 
 to its length. The inclined plane, in this case, may be considered 
 as a wedge in equilibrio, under the action of three forces, the re- 
 lation between two of which, will determine the conditions of 
 equilibrium. 
 
 But a weight may be supported upon an inclined plane, when 
 the direction o-f the power is not parallel to the plane. In this 
 case, the power must be resolved into two forces, one parallel to 
 the surface of the plane, the other perpendicular to it, the latter 
 has no effect, and the support \vill be wholly due to the former 
 of the two components. By 13, the value of this component 
 will be, calling the angle the direction the power makes with 
 the surface of the plane, a, 
 
 P cos. o; 
 
 and the condition of equilibrium will be represented by the ana- 
 logy 
 
 P cos. a : W : : h : /, (128) 
 
 in which h is the height, and / the length of the plane; and there- 
 fore, 
 
 P cos. o=W sin. t. 
 If the former act parallel to the base 
 
 a=, 
 and 
 
 P=W tan. t; (129) 
 
 if we call the length of the base b, we have 
 
 h 
 T =tan.i; 
 
 therefore, 
 
 P: W : : h : b -, or (130) 
 
 when the power acts parallel to the horizontal base of the plane, 
 equilibrium will exist when the power is to the weight as the 
 height of the plane to the length of the base. 
 
 168. The inclined plane is a mechanic power of a frequent, 
 nay, of almost constant application, in cases almost Loo numerous 
 to be recited. Thus: we take advantage of natural slopes to 
 raise bodies upon them, with powers of less intensity than they 
 would require if raised vertically. We lift weights into wheel 
 carriages by temporary inclined planes, formed of parts, either 
 fixed or moveable, with which they are furnished. In raising 
 

 
 Book HI.] MECHANIC POWERS. 157 
 
 weights from cellars, or from one story of a building to another, 
 we convert the stairs into inclined planes, by laying skids upon 
 them. In all these cases, upon the principles in 125, we save 
 friction by making the body roll instead of sliding. We may 
 also combine this mechanic power, in the case of rolling bodies, 
 with a modification of the moveable pulley : thus, when a barrel 
 is to be raised from a cellar, after laying skids upon the steps, 
 and thus converting them into an inclined plane, we take a cou- 
 ple of ropes, and making them fast at the top of the plane, pass 
 them around tho barrel, and bring them back again to the top of 
 the plane : a force applied to these ropes causes the barrel to re- 
 volve like a roller, and the intensity of the power, in addition to 
 the gain it acquires from the plane, is doubled by the action of 
 this temporary pulley. Inclined planes are also frequently con- 
 structed expressly for the purpose of enabling a power to coun- 
 terbalance a weight of greater intensity : thus in saw-mills, the 
 logs are drawn to the place where they are to be sawn, upon a fix- 
 ed inclined plane. This principle has also been applied in the 
 cases of roads, railways, and canals. In Morton's Marine Rail- 
 way, vessels of great size are drawn from the water upon an in- 
 clined plane, by a power exerted through thejntervention of mo- 
 difications and combinations of the wheel and axle. 
 
 169. The friction of the inclined plane is easily determined, 
 for it is always a function of the pressure ; and the state in which 
 the smallest addition to the power will cause motion, may be 
 represented by the formula, 
 
 P=W sin. t+/cos. i. (131) 
 
 Of the Screw. 
 
 170. The Screw is a mechanic power that may be considered 
 to be formed, as in the figure, by wrapping an inclined plane 
 
 around a cylinder : it is composed of a spiral ridge or thread up- 
 on the surface of a cylinder, which cuts every line that can be 
 drawn upon its surface, and parallel to its axis, at an equal angle. 
 The weight is applied to the extremity of the cylinder, and the 
 power acts to turn the screw around ; thus tending to propel the 
 weight in the direction of the axis of the screw; it therefore be- 
 comes necessary that the screw shall move in a cavity to which 
 
153 OP THK [Book III, 
 
 its thread adapts itself, or in a screw formed upon the surface of a 
 hollow cylinder, which the solid screw exactly fills. The 
 weight may be applied to the circumference of the cylinder, in 
 the direction of a tangent to this circle. The screw, in this case, 
 is an inclined plane, whose height is the distance between the 
 convolutions of the thread, or, as we usually style it, the distance 
 between the thread ; for although there be but a single thread 
 wound around the cylinder, yet, as when we view it, we see nume- 
 rous convolutions, we call each of them a thread, as if they were 
 actually separate. The base of Ihis inclined plane is the circumfe- 
 rence of the screw, hence from 1 67, the power is to the weight, 
 when in equilibrio, as the distance between the threads is to the 
 whole circumference of the screw. It is far more usual to apply 
 
 the power, as in the figure, to a lever at right angles to the axis 
 of the screw, or to a circular head, to whose plane the axis of the 
 screw is a normal. In this case the condition of equilibrium is, 
 that the power be to the weight, as the distance between the 
 threads of the screw is to the whole circumference described by 
 the point to which the power is applied. 
 
 Call the distance between the threads d, the circumference of 
 the screw c ; call the intensity of the force exerted by the power 
 at the circumference of the screw, W. 
 
 By the principle of the wheel and axle, 
 P : W : : c : C ; 
 whence 
 
 By the principle just laid down for the screw, 
 W : W : : d : c ; 
 
 Wd 
 and W' = -- ; 
 
 whence 
 
 Wd=PC, \ 
 
 and P : \V : : d : C ; } 
 
Book III.] MECHANIC POWERS. 153 
 
 171. Friction affects the screw in the same manner that it does 
 the wedge ; the friction of the solid upon the hollow screw being 
 sufficient to prevent it from returning after the power ceases to 
 act. Hence the screw may be applied to many of the purposes 
 for which the wedge is used. The difference in their use is, that in 
 the wedge the power most frequently acts by a succession of blows j 
 in the screw it acts in the manner of a constant force. Hence 
 the latter may be used for raising great weights to a small height, 
 and is the most frequent instrument used for accumulating force, 
 in order to apply it to pressure. It therefore forms an essential 
 part of the coining engine, the notaries' and printing, as well as 
 various other presses. It is, also, like the modifications of the 
 wedge, employed for fastening, and holding bodies together. 
 This it does the more effectually, inasmuch as the screw must 
 either be turned around and withdrawn, by reversing the motion 
 by which it entered, or forced out, by breaking its own threads, 
 or those of the hollow screw formed to receive it in the bodies to 
 be united. It will therefore unite most bodies more effectually 
 than nails or spikes, and is alone applicable to the union of 
 such hard bodies as cannot be penetrated by arty form of the 
 wedge ; such are, for instance, the metals, in the mass of which 
 hollow screws, fitted to receive the solid screw, may be cut. 
 
 172. A screw may be applied to the teeth of a wheel, and will, 
 by its simple revolution, cause the wheel to revolve. In this case, 
 the screw need have no progressive motion, and is therefore form- 
 ed upon a rod that is free to turn, but does not advance forwards. 
 The hollow screw is no part of such a system, and the machine 
 is a combination of the screw with the wheel axle. 
 
160 
 
 OF THE 
 
 [Book in. 
 
 In such a combination, the screw is called endless. It is rep- 
 resented beneath : 
 
 
 The power is applied to the head of the screw, the weight to 
 the circumference of the axle : this power will be in equilibrio 
 with a force W, acting where the threads of the screw tend to 
 turn the wheel, at whose circumference, by (132), 
 
 PC W'd 
 
 TV^-g-, orP= ; 
 
 and the force W will be in equilibrio, with the weight acting 
 upon the circumference of the axle, where, by (11 9 a), 
 
 hence 
 
 Wdb 
 
 ca 
 
 (133) 
 
 Equilibrium, therefore, will exist in the case of the perpetual 
 screw acting upon a wheel and axle, when the power is to the 
 weight as the distance between the threads of the screw multi- 
 plied by the radius of the axle, is to the circumference of the head 
 of the screw multiplied by the radius of the wheel. 
 
 173. If the solid screw, and the hollow one to which it is adapt- 
 ed, be of a hard material, the motion cannot be free, unless the 
 
Book ///.] MECHANIC POWERS. 161 
 
 intervals between their respective threads be as equal as the na- 
 ture of materials will admit. As screws may be cut with many 
 threads, within a small space, they may be applied to divide that 
 space into as many parts as there are threads of the screw within 
 it, and each of these parts will be determined by a complete re- 
 volution of the screw around its axis. But as the screw may 
 have a circular head adapted to it, and the circumference ot this 
 head may be divided into many equal parts, the division may be 
 carried further, to the determination of as many parts of the inter- 
 val between the threads, as there are divisions upon the head of 
 the screw ; for as a complete revolution of the head of the screw 
 corresponds to a progressive motion of the axis of the cylinder 
 on which it is cut, through a distance equal to the interval be- 
 tween the adjacent threads ; so a partial revolution will corres- 
 pond to a progressive motion in the screw, that bears the same 
 ratio to the distance between the threads, that this partial revo- 
 lution bears to the whole circumference. In like manner, if a 
 perpetual screw be applied to the teeth of a wheel, and both be 
 of a hard material, the teeth, catching in succession in the thread 
 of the same screw, must be equal among themselves. A complete 
 revolution of the screw will move a point upon the circumference 
 of the wheel, through a space equal to the distance between the 
 threads, or to the breadth of the tooth upon the wheel ; if a head be 
 adapted to the screw, and its circumference divided into equal 
 parts, portions of the revolution of the screw may be estimated 
 by means of them, and these will be the measure of corresponding 
 parts of the progressive motion of the wheel, through the inter- 
 val between the threads of the screw. 
 
 This property of the screw is applied to several important 
 purposes. 
 
 (1.) Many of the micrometers which are used in telescopes, 
 for the measurement of such angles as are included within their 
 field, are moved by means of screws; the numbers of whose re- 
 volutions, and the parts of a revolution, determined by the divi- 
 sions of the head of the screw, furnish the measure of the angle. 
 The whole breadth of the field is first determined by observing 
 the time it takes an equatorial star to traverse the diameter, and 
 this time is reduced to degrees, minutes, and seconds. To take an 
 instance: let the circle ABCD represent the field of a telescope, 
 in which are seen the horizontal and vertical wires AC, BD ; 
 the wire EF, is moved by the micrometer screw ; the breadth of 
 
 21 
 
162 
 
 OF THE 
 
 [Book 111- 
 
 the field having been determined by observation, it is next as- 
 certained by experiment how many revolutions and parts of a 
 revolution are required to move the wire across the field ; and 
 the value of each revolution and part is calculated by a simple, 
 proportion. The moveable wire is then placed so as to appear to 
 coincide with the vertical one, and one of the objects being made 
 to coincide with the latter, the screw is turned until the moveable 
 wire coincides with the other object; then counting the revolutions 
 of the screw, and observing the portions in excess upon the head 
 of the screw, it will be at once seen that a measure of the angle 
 is obtained in terms nf tha known divisions of the screw. 
 
 (2.) A screw moving a straight bar forward, affords the means 
 of dividing it equally, and to great minuteness, by means of di- 
 visions on the head of the screw. Such is the principle of the 
 Straight-line Dividing-Engine of Ramsden. 
 
 (3.) A circular plate may be made to revolve by means of an 
 endless screw. A circular limb laid upon the plate, and concen- 
 tric with it, may be divided, by ascertaining the proportion each 
 thread of the screw bears to the whole circumference of the cir- 
 cle. More minute divisions may be obtained by dividing the 
 head of the screw. Such is the principle of the circular-di- 
 viding engine of Ramsden. In a very beautiful application of 
 the same principle, which is now used by Mr. Patten, an instru- 
 ment maker, in New-York, the circular plate has 360 teeth on 
 its circumference : each revolution of the screw, therefore, cor- 
 responds to a single degree, and the more minute divisions are 
 obtained by divisions upon its head. For divisions of instruments 
 that do not require much care, the screw is turned by a treadle, 
 at each motion of which, a cutting tool makes its mark upon the 
 
Book III.} 
 
 MECHANIC LOWERS. 
 
 163 
 
 limb to be divided. The same engine has been applied to straight 
 lines, by placing upon the circular plate, a second plate accurately 
 centered, whose circumference is exactly a yard; the rod (o be 
 divided is moved forward by friction against this plate, and con- 
 sequently, by a complete revolution of the plate, through a straight 
 line a yard in length : ten revolutions of the screw, therefore, 
 correspond to an inch, each revolution to the tenth of an inch, 
 and lesser divisions are obtained by means of the divisions upon 
 the head of the screw. 
 
 (4.) The same principle was applied in the astronomical in- 
 struments of the last century, to subdivide the smallest divisions 
 cut upon their limbs. As this method has now been superseded 
 by others, it requires no explanation in the present treatise. 
 
 174. It will appear from reference to 172, that the increase 
 in the intensity of the power, will, in the screw, if the diameter 
 of its head, or the length of the bar applied to turn it, remain 
 constant, be in the inverse ratio of the distance between the 
 threads; hence, in order that a small power shall counterba- 
 lance a weight of great intensity, the thread must be very much 
 diminished. In this event, the strength of the, thread may be so 
 far lessened as to be incapable of bearing, without breaking, the 
 intensity of the resistance. A common screw, then, has a limit to 
 its capacity of changing the intensity of the force, in the strength 
 of the material of which it is constructed. "Screws, with fine 
 threads, have the additional disadvantage, that, after their work is 
 perfoi med, it requires a considerable time to turn them back again 
 to the point at which their action commenced, in order to apply 
 them to overcome a new resistance. Both of these defects have 
 been remedied by a compound screw, invented by Dr. Hunter. 
 In the figure, AB is a screw of large thread, and consequently of 
 
 w 
 
14 OF TBS [Book ///. 
 
 little power ; but it is hollow within, and cut into threads, to re- 
 ceive the second solid screw, BC. The threads of the latter are 
 a lillle less than those of the former. Were they exactly equal, 
 when the screw AB is turned, the screw BC would enter into 
 its cavity just as far as the former is pressed forward, and the 
 point C would remain at rest. But as the threads of BC are 
 smaller than those of AB, the former recedes during a single re- 
 volution of the latter, through a space equal only to its own 
 thread ; and hence the point C is pressed forward, through the 
 difference between the respective threads. The velocity of the 
 weight then is no more than the difference in the velocities of the 
 screws, and by the principle of virtual velocities, which is most 
 convenient for our purpose, the condition of equilibrium is: the 
 power shall be to the weight, as the circumference of the circle 
 described by the power, is to the difference between the distances 
 of the threads in the two screws. If now there be ten threads of 
 the outer screw in the space of an inch, and ten of the inner in 
 the space of nine-tenths of an inch, the difference will be T 4 7 th 
 of an inch, and the intensity of the power is as much increased 
 as it would be in acting upon a screw having an hundred threads 
 within an inch; while the strength of the threads of the least 
 screw would be more than nine limesas great, and capable, there- 
 fore, of bearing more than nine times the strain. It is easy in 
 practice to make the greater screw carry the lesser back with it 
 to its primitive position, and hence it can be placed in the posi- 
 tion in which it is again t,o begin to act, in a tenth part of the 
 time, in the case we have assumed, and in proportion for any 
 other difference between the distances of the threads. A third 
 screw may be placed within the second, and the intensity of the 
 power again increased upon the same principle. 
 
 175. The friction of a simple screw depends not merely upon 
 the nature of the materials, and their polish, but also upon the 
 greater or less tightness with which the solid and hollow screws 
 apply themselves to each other. It would in consequence be 
 impossible to reduce the action of the friction to any fixed law. 
 In the endless screw, when combined with the wheel and axle, 
 ii rmv be investigated; and the equation ofilhe st.ite in which an 
 ad lition to the p wer, however small, would cause motion, will 
 be found to be, using ihe notation of (133) 
 
 p=w - - ' (134) 
 
Book ///.] STREHGTH, &C. 165 
 
 CHAPTER VII. 
 
 OF THE STRENGTH OF MATERIALS. 
 
 176. The force with which the particles of a solid body resist 
 the forces that tend to separate them, and which constitutes their 
 strength, may be found, in particular cases, by experiment. If 
 the experiments be multiplied, both in respect to the species of 
 substances, and to the size and circumstances under which por- 
 tions of a given substance arc employed, a general law might be 
 finally obtained, whence algebraic expressions of the relation be- 
 tween the strength, the dimensions, and the material, might be 
 deduced. Or we may, by assuming an hypothesis to represent 
 the probable manner in which bodies are made up, deduce from 
 it general formulae, which, applied to cases that occur in practice, 
 will be sufficient, in almost every instance, to represent the phe- 
 nomena. The latter of these methods, although not the most 
 accurate, shall be employed by us; but we shall carefully note 
 the differences that exist between the inferences from the hypo- 
 thesis, and the results of experiments. 
 
 177. Galileo was the author of the hypothesis that is most 
 generally employed by writers on mechanics, and which will 
 suffice in all usual cases. He assumes that all solid bodies are 
 composed of a great number of parallel and equal fibres, perfectly 
 inflexible and inextensible ; that when they break, the several 
 fibres give way in succession, and the body turns upon the fibre 
 or fibres that are the last to give way, as upon a hinge. 
 
 Leibnitz observing the flexure that takes place in bodies be- 
 fore they break, assumed as the basis of his hypothesis, the fact, 
 that every body admits before it breaks, of a certain degree of 
 extension : the fibres, therefore, are both extensible and flexible; 
 and he inferred that the strength of each fibre, instead of being 
 equal, varied with its quantity of extension, or was proportioned 
 to its distance from the fixed point around which the beam is sup- 
 posed to turn. 
 
 The hypothesis of Galileo has been recently extended by Bar- 
 low, who has, by introducing the circumstance of the occurrence 
 of a flexure before the beam breaks, been enabled to explain all 
 the cases which appeared, by a comparison of the hypothesis 
 with experiment, to be anomalous. 
 
 178. The force which tends to destroy the aggregation of the 
 particles of a solid body, may act in four different manners. 
 

 16(5 STRENGTH OF [Book HI. 
 
 (I.) It may tend to tear the body asunder by exerting an ac- 
 tion in the direction of its fibres. 
 
 (2.) It may tend to break the body across, by a transverse 
 strain, exerted either perpendicularly, or obliquely to the direc- 
 tion of its fibres. 
 
 (3.) It may tend to crush the body. 
 
 (4.) It may tend to separate the particles by means of torsion, 
 twisting or wrenching the body, by an action in a plane perpen- 
 dicular to its axis. 
 
 The resistance which bodies oppose to a force acting in the 
 first of these modes, is called the Absolute Strength. In fibrous 
 bodies, it is different, according as the force is applied in the di- 
 rection of the fibre, or at right angles to it : for although by hy- 
 pothesis we should conceive fibres to exist, even in the transverse 
 direction, yet, in nature, none such are found ; and the aggre- 
 gation is far more easily destroyed in the latter case than in the 
 former. 
 
 The resistance to a fracture across the body, is called its Re- 
 spective Strength ; and we may call the resistance to a twisting 
 force, the Strength of Torsion. 
 
 It needs no reasoning to show that the measure of the strength, 
 even in the same body, will be different in each of these different 
 cases; and such is the difference in the manner in which the 
 particles of bodies of different natures are united, that there can 
 be no general law that will represent the relations of these four 
 species of resistance. But in conformity with our hypothesis, it 
 will be obvious that, in all the several cases, the resistance to 
 Jracture will vary with the number of fibres, which, in homo- 
 geneous bodies will depend upon the area of their sections. It 
 must also depend upon the manner in which the force acts to 
 break the body. 
 
 Of the Absolute Strength of Materials. 
 
 179. When the strain is exerted in the direction of the fibres, 
 the force that tends to break a body will be direciiy opposed to 
 its force of aggregation, and the resistance must depend upon the 
 cohesive force of each fibre, and upon their number, but upon no 
 other circumstance. 
 
 To enable us to express this analytically : 
 
Book ///.] MATERIALS. 167 
 
 Let. ABCD, represent the area of a prismatic beam or rod ; 
 x, and y, the ordinate and abscissa 
 B of any point of its surface ; 8 the ab- 
 solute strength of one of its differential 
 elements, abed. The force of the ag- 
 gregation, may be represented by the 
 weight which is just sufficient to destroy 
 
 d 
 
 it, which we shall call U. 
 
 The area of the section will be 
 2jydx. 
 
 The expression for the absolute strength will therefore be 
 
 V=2afydx, (135) 
 
 for the area of each of the elements will be 2ydx, its strength 
 2sydx, and the strength of the whole will be found by integrating 
 this expression, in which 2s is constant. 
 
 In a rectangular bar, whose lineal dimensions are a and 6 
 ab=2fydx, 
 
 and 
 
 U=sab, (136) 
 
 whence 
 
 ab 
 
 =0 
 
 .. 
 In a circle whose radius is r, 
 
 2fydx=xr 2 , 
 and 
 
 U^rfr 2 *, (138) 
 
 whence 
 
 .=-;pr . (139) 
 
 180. When experiments are made upon the resistance of rods, 
 of dimensions given in some conventional unit of lineal measure, 
 to a direct strain, exerted by means of loads estimated in some 
 conventional unit of weight ; the value of s may be found in 
 terms of the latter, and in reference to an element of the surface, 
 whose magnitude is the square of the unit of length. The most 
 valuable experiments that we have of this sort, are those of Bar- 
 low, upon wood. These were made upon cylindrical rods, and 
 the strength, s of the formulae, deduced for an element of the area 
 of a square inch. The results are contained in the second of the 
 following tables. The first table has been compiled from various 
 other sources. 
 
168 STRENGTH OF [Book III. 
 
 TABLE I. 
 
 ABSOLUTE STRENGTH OF THE METALS. 
 
 Cast Steel, .... 140000 Ibs. 
 
 Gold, (according to Morveau) 80000 Ibs. 
 
 Wrought Iron, (Swedish,) 72000 Ibs. 
 
 do (English,) - 56000 Ibs. 
 
 do do in the form of chains, 48000 Ibs. 
 
 Bronze, (Gun Metal,) \ .-' . - 36000 Ibs. 
 
 Wrought Copper, - 33000 Ibs. 
 
 Cast do - 19000 Ibs. 
 
 Brass, - - 17000 Ibs. 
 
 Tin, 4700 Ibs. 
 
 Lead, - - 1800 Ibs. 
 
 TABLE II. 
 
 ABSOLUTE STRENGTH OF DIFFERENT KINDS OF WOOD DRAWN IN 
 THE DIRECTION OF THEIR FIBRES. 
 
 Boxwood, - "" ; ' - 20000 Ibs. 
 
 Ash, - - - 17000 Ibs. 
 
 Teak, - - 15000 Ibs. 
 
 Norway Fir, - - - 12000 Ibs. 
 
 Beech, - - - 11 500 Ibs. 
 
 Canada Oak, - - 11400 Ibs. 
 
 Russia Fir, - - - 10700 Ibs. 
 
 Pitch Pine, - 10400 Ibs. 
 
 English Oak, - 10000 Ibs. 
 
 American White Pine, 9900 Ibs. 
 
 Pear Tree, - - - 9800 Ibs. 
 
 Mahogany, - - 8000 Ibs. 
 
 Elm, - - - 5800 Ibs. 
 
 TABLE III. 
 
 ABSOLUTE COHESIVE STRENGTH OF WOOD DRAWN IN A DIRECTION i-T 
 RIGHT ANGLES TO THE FIBRES. 
 
 Teak, .... 818 Ibs. 
 
 American White Pine, - - 757 Ibs. 
 
 Norway Fir, - - 648 Ibs. 
 
 Beech, - - -. 615 Ibs. 
 
 English Oak, - 598 Ibs. 
 
 Canada Oak, - - 588 Ibs. 
 
 Pitch Pine, - 588 Ibs. 
 
 Elm, - H/>"* 509 Ibs. 
 
 Ash, - >-. - - - 359 Ibs. 
 
Book ///.] MATERIALS. 169 
 
 Of the Respective Strength of Materials. 
 
 181. To apply the hypothesis of Galileo to the case of a trans- 
 verse strain, we shall suppose in the first place, the substance to 
 have the form of a prismatic beam, that it is firmly inserted at one 
 end into a fixed support, lies in a horizontal position, and is acted 
 upon by a weight that presses at the end that is not fixed ; that 
 the fracture takes place at the point of support, beginning at the 
 upper side, on which the weight presses, and terminating at the 
 other. The beam then, its fibres being by hypothesis inflexible 
 and inextensible, will turn around the latter point in a vertical 
 plane. At the instant of fracture, the two forces that act are in 
 equilibrio with each other, their respective moments of rotation 
 must be therefore equal. 
 
 In the prismatic beam, whose section is ABCD, the strength 
 _ n will be represented by the expression 
 (135), 2s ydx ; but as the effort of the 
 weight will cause the beam to turn 
 around a horizontal axis passing through 
 the lowest point, the moment of rotation 
 of the strength of each element, will be 
 found by multiplying 2s f ydx into the per- 
 
 pendicular distance of its centre of gra- 
 vity from this axis. Now the mean of all these distances will be 
 the distance of the centre of gravity G, of the whole, from the 
 point where the fracture terminates. The moment of rotation of the 
 whole resistance will therefore be, calling this distance c, 
 
 2csfydx. 
 Let EFHG represent the longitudinal section of the beam, 
 
 fixed at EG to a firm support, and pressed by a weight acting at 
 F. Let the length EF=/; the moment of rotation of the 
 weight will be I W, and at the instant of breaking, 
 lW=2csfydx ^ 
 
 or w= ^fy^ x [ (140) 
 
 22 
 
170 STRENGTH OF [Book III. 
 
 In rectangular beams, whose two dimensions are a and 6, 
 2/ ydx 
 
 c = 
 and the equation (140) becomes 
 
 2/ ydx=ab, 
 b 
 
 (141) 
 In square beams, whose side is a, 
 
 W=g. (1416) 
 
 In cylindric beams, whose radius is r, 
 
 or 3 * 
 W=- - [ . (142) 
 
 In a hollow cylindric beam, whose cavity is a cylinder having the 
 same axis as the outer surface, the radii of the two cylinders 
 being R and ', 
 
 If the area of the eylindric ring be equal to n'r 2 of (142) 
 
 W=^ ; (144) 
 
 andthe strength will be to the strength of the solid cylinder of (142) 
 as R : r, that is to say : In different cylindric beams, having the 
 same quantity of the same material, and equal lengths, but having 
 different diameters, in consequence of cavities of greater or less 
 size within them, the strengths are in the direct ratio of their di- 
 ameters. 
 
 We may obtain the comparative dimensions of solid and hollow 
 cylinders, that will bear equal weights, by a comparison of the 
 formulae (142) and (143), whence we obtain 
 
 r 3 = R 3__ R /3 
 
 r = V [R(R'-0].l 
 
 If the cavity of the hollow cylinder be not concentric, the strength 
 should increase, according to the hypothesis, as the cavity ap- 
 proaches the lower side, when it is fixed at one end only ; for in 
 this case, the centre of gravity of the section will be further re- 
 moved from the point where the fracture terminates. 
 
 In a beam of the shape of an isoceles triangle, whose base is 
 e, and altitude ?, 
 
 if the edge be placed uppermost, 
 
IIL] MATERIALS. 171 
 
 and 
 
 W=-^; (146) 
 
 if the base be placed uppermost, 
 _2t 
 
 c= / ; 
 
 and 
 
 W~*, ,(147) 
 
 In a square beam, whose diagonal is vertical, 
 
 and 
 
 In a rectangular beam, when / and * are constant, the strength 
 varies with a6 2 , (141). This proposition may be applied to a case 
 thafmay sometimes be useful in practice, namely : To cut the 
 strongest possible beam out of a given cylinder. Thus a tree, 
 although a cone or conoid, may, for all useful purposes, be con- 
 sidered as a cylinder ; for the size of the rectangular beam that 
 can be cut from it, will be determined by the area of its smaller 
 end. 
 
 Let r be the given diameter of the cylinder, * and y the lineal 
 dimensions of the required beam. When this is the strongest 
 possible, xy 2 will be a maximum ; 
 
 x e :y z :r-: : 1 : 2 : 3. (149) 
 
 The cylinder must therefore be so cut, that the squares of the 
 breadth of the beam, of its depth, and of the diameter of the cylin- 
 der, shall be in the proportion of 1 : 2 : 3. 
 
 
172 
 
 STRENGTH OF 
 
 [Book IIL 
 
 This admits of an easy geometric construction. In the circu- 
 
 A 
 
 lar section of the cylinder draw the diameter AB, divide the di- 
 ameter into three equal parts by the points F and G ; from the 
 points F and G, draw the perpendiculars FC, GD, towards op- 
 posite sides, cutting the circle in the points C and D ; join AC, 
 CB, BD, AD ; the parallelogram ADBC, will have the required 
 property, and will he the section of the strongest beam that can 
 be cut from the cylinder, whose diameter is AB. 
 It will be at once seen that 
 
 AB a : BC 2 : AC 3 : : 1 : 2 : 3. 
 
 (150) 
 
 1S2. When abeam, lying in a horizontal position, rests upon 
 two props, and is broken by a weight placed at an equal distance 
 from the two props, we may consider the laws of its strength as 
 included in the general case of a beam supported at one end on- 
 ly ; for if, according to the hypothesis, it break without bending, 
 we may conceive it to be formed of two beams, each inserted in 
 a firm support at the place of fracture, and acted upon at each end 
 by a force equal to half the weight that just breaks it, but which 
 is directed upwards instead of downwards. This force, which 
 is equal to half the weight, will act at a distance which is equal 
 to half the length ; its effort is therefore equal to no more than 
 a fourth part of the effort of the same weight, applied to the same 
 beam, if supported at one end only; and as this effort must be 
 just equal, at the instant of breaking, to the transverse strength 
 of the beam, the latter will be four times as strong as when sup- 
 ported at one end only. 
 
MATERIALS. 173 
 
 In the beam ABCD, supported at C and D, and to which a 
 
 755- 
 
 weight, W, is applied at the point E, which bisects AB. If we 
 suppose the half AEFD, to be firmly fixed at EF, it will be bro- 
 ken by a force applied at A, in the vertical direction, which 
 is equal to \ W ; the moment of rotation of this force will be equal 
 to the respective strength of the beam, and as the distance EA 
 is J/ f (140) ^ 
 
 =2scfydx, 
 
 and 
 
 8scfydx 
 W=^ L - ; (151) 
 
 In rectangular beams, one of whose faces is horizontal, we obtain 
 in the same manner as (141), 
 
 2sab 2 
 W= . (152) 
 
 In square beams, (141 6), 
 
 2sa? 
 W= . (153) 
 
 In cylindric beams, (142), 
 
 W= ; (154) 
 
 with a similar inference for the case of a concentric hollow cylin- 
 der, as in (143) and (144). 
 
 But if the hollow be not concentric, the strength will, in the 
 present case, increase with the approach of the cavity to the up- 
 per surface. And so in triangular beams, the proportions of (146) 
 and (147) will still hold good, but they will be stronger with the 
 edge uppermost. 
 
 In a square beam, whose diagonal is vertical, (148), 
 
 2sa' { v/2 
 w= __ . (155) 
 
 183. In the case of a beam lying horizontally, and firmly fixed 
 at each end, the resistance will be equal to that of a single beam 
 supported by two props, added to those of two beams fixed at one 
 end only ; for it is obvious, that three fractures must be produ- 
 ced, one in the middle, and one at each end ; at the first, the re- 
 
174 STRENGTH OF [Book III. 
 
 sistance will be equal to that of the supported beam, or four times 
 as great as that of the same beam, if supported at one end only. 
 The resistance in each of the latter cases, will be that of a beam 
 of half the length fixed at one end only, or one fourth of the 
 last resistance ; the whole resistance will therefore be six times 
 as great as that of the same beam when fixed at one end only, or 
 
 (156) 
 
 In rectangular beams, (141) and (152), 
 
 In square beams, (141 6) and (153), 
 
 3sa' 
 W= . (158) 
 
 In cylindric beams, (142) and (154), 
 
 6*1*3 
 
 W= . (159) 
 
 If experiments be made with rectangular beams, in either of the 
 three several positions, the absolute strength, s, is determinable 
 by means of them, for calling the weights that just break them, 
 W, W, and W". 
 
 In beams firmly fastened at one end, from (141), 
 2/W 
 
 In beams, supported at each end on props, from (152) 
 IW 
 
 In beams firmly fastened at each end, from (157) 
 JW" 
 
 184. When the beam, instead of lying in a horizontal position, 
 is inclined, an increase of its strength takes place, which we shall 
 proceed to investigate. 
 

 Book ///.] MATERIALS. 175 
 
 Let either of the beams, whose longitudinal section is ABCD, 
 
 lie in an inclined position, and let the angle of inclination be t, the 
 moment of rotation of the weight will become 
 
 WJ cos. i. (163) 
 
 The effort the weight exerts to break the beam, is no longer 
 exerted directly, but it is unnecessary to take this obliquity into 
 account, if bodies be constituted, as represented by our hypothe- 
 sis ; for the number of fibres that act to resist fracture, is still the 
 same. In bodies that are not fibrous, as, for instance, in those that 
 are crystallized, the inference would probably be different ; but, on 
 this point, we are unable to refer to the results of any experiments. 
 It will be obvious that the same reasoning will be true, whether 
 the beam be inclined upwards or downwards, and is applicable to 
 the cases of its being supported, or firmly fixed at both ends, as 
 well as to that of its being fixed at one end only. 
 
 185. When the weight that tends to break a beam is not accu- 
 mulated at a single point, but is uniformly distributed over its 
 whole length ; its effort is diminished to the half of what it ex- 
 erts, when, in the case of a beam fixed at one end, it acts at the 
 opposite extremity ; or when, in the case of a beam supported or 
 fixed at both ends, it acts in the middle. This will be obvious 
 in the first case, from the consideration that the point, where the 
 weight acts, will be in the middle of the beam, instead of being 
 at its end; hence its moment of rotation becomes | /W; and as 
 the other two cases are deduced immediately from the first, the 
 same principle applies to them also. 
 
 186. We have hitherto omitted the action of the weight of the 
 beam itself. 
 
 In small beams indeed, their own weights are of little impor- 
 tance, and need hardly be taken into account in the experiments ; 
 but in large beams this is not the case, as may easily be seen. 
 
176 STRENGTH OF [Book HI. 
 
 The weight which breaks a beam is made up of its own weight, 
 and the weight which is applied for the purpose : the former acts 
 at the centre of gravity of the beam, which in prismatic beams is 
 in the middle of their length. Its moment of rotation, therefore, 
 will be V/, calling the weight of the beam V ; the joint effort 
 of the two will therefore be, in the case of a beam supported at 
 one end, and placed horizontally, and if the additional weight act 
 at the extremity, 
 
 (W+fV)/; 
 and the formula (140) for the strength will become 
 
 (164) 
 
 In beams that are similar, we may substitute, for 2cf ydx, the cube 
 of any one of their homologous dimensions multiplied by a con- 
 stant co-efficient ; let then 
 
 fP=2cfydx: 
 and the above equation becomes 
 
 W+$V=sfl*. (165) 
 
 In similar beams of the same homogeneous material, the weight 
 is a function of the cube of their homologous dimension, as will 
 be half the weight, or 
 
 d>P=iV, 
 and 
 
 W+9/ 3 =s// 2 . (166) 
 
 It will therefore be evident, that while the strength of similar 
 beams increases only as the square of one of their homologous 
 dimensions, the effort of their own weight to break them increases 
 with the cube ; and thus a Hmit will be reached, when 
 
 The same principle applies equally to the other two cases, in 
 which beams are supported, or fixed at both ends. 
 
 187. We shall now recapitulate the results of our hypothesis, 
 and state what discrepancies have been observed between them, 
 and the inferences from actual experiment. 
 
 (1.) In any prismatic beam whatsoever, thestrength is directly 
 proportioned, to the area of its section, and to the distance of its 
 centre of gravity from the point where the fracture terminates ; 
 and inversely, to the length of the beam. 
 
 (2.) The strengths of beams lying in a horizontal position, 
 when fixed at one end only ; when supported by a prop at each 
 end ; and when firmly fixed at both ends, are as the numbers 
 1:4:6. That is to say : a beam firmly fixed at both ends, is 
 six times, a beam merely supported at both ends, four times as 
 strong as when it is fixed at one end only. 
 
Book ///.] MATERIALS, 177 
 
 These several inferences from the hypothesis, agree within all 
 usual limits, with the results of experiments. The discrepancies 
 are : that the strengths increase in a ratio a little greater than the 
 square of the depth, in rectangular heams ; and decrease rather 
 more rapidly than the inverse ratio of the length. 
 
 The second of the above propositions admits of the following 
 cases : 
 
 (a) In beams of the same material, with equal and similar sec- 
 tions, and unequal lengths, the strengths are inversely propor- 
 tioned to the lengths. 
 
 The lengths being equal : 
 
 (6) In rectangular beams of the same materials, the strengths 
 are proportioned to the product of the breadth by the square of 
 the depth ; 
 
 (c) In square beams, the strengths are proportioned to the cubes 
 of the sides of the square sections ; 
 
 (fl] In solid cylindric beams, the strengths are proportioned to 
 the cubes of the radii ; 
 
 (e) In hollow cylindric beams, having the same quantities of 
 material distributed around cylindric cavities of different diame- 
 ters, the strengths are directly as the diameters. 
 
 (3.) Large beams are weaker in proportion than small ones, 
 for their own weight constitutes a part of the force that tends to 
 break them ; and in similar solid bodies, the stress growing out 
 of their own weight increases as the cubes of their homologous 
 dimensions, while the strength only increases with the squares. 
 
 We see from ihN, that models may be strong, and capable of 
 bearing a stress far beyond any that can be applied to them ; yet 
 that machines constructed exactly similar to them in proportion, 
 and of like materials, but of increased dimensions, may become 
 too weak to bear even their own weight ; that there must be a 
 limit to the size and extent of any structures that can be erected 
 by the hand of man ; and that a similar limit exists even in the 
 works of nature. Thus in organic bodies, mountains, hills, trees, 
 the size they can attain, without risk of disintegration, isrestricted 
 within certain bounds. In the animal creation, the same princi- 
 ple applies, and the limit is sooner reached. 
 
 Our theory would show that when a body becomes weak in 
 consequence of an increase of length, strength may at first 
 be added by increasing its breadth, and still more by increasing 
 its thickness, the length remaining constant. Here, how- 
 ever, the weight is increased in a greater ratio than the length, 
 and finally becomes excessive. The same quantity of material may 
 assume a stronger form by being fashioned into a hollow tube ; 
 yet here again a limit is reached when, the circumference of the 
 
 23 
 
178 STRENGTH OF [Book III 
 
 tube becomes so thin as to be liable to be crushed by the forces 
 that act upon it. 
 
 In animals, we find that the smaller classes have bones far 
 more slender in proportion than those of the larger kinds. Their 
 muscles are far less thick in proportion to their length ; and their 
 masses are diminished in a proportion much more rapid than that 
 of the cubes of their similar dimensions. We find small animals 
 capable of lifting weights greater in proportion to those of their 
 own bodies, than larger animals can ; and in spiteof this additional 
 effort, they are enabled to continue their exertions for a longer 
 period without fatigue. To obtain the greatest possible strength 
 with the least possible weight, the bones of animals have the 
 form of hollow tubes, as have the quills and feathers of birds. 
 In the vegetable kingdom, we find trees and plants made up of 
 bundles of hollow tubes ; and in those where great strength and 
 comparative lightness are necessary, these are again arranged so 
 as to form a hollow cylinder; as, for instance, in the whole family 
 of the gramina. 
 
 (4.) When beams are in an inclined position, their strength, 
 which we shall call F, becomes 
 
 F=Wcos. t. (168) 
 
 This deduction from the hypothesis, is true in practice, so long 
 as the beam does not bend under the effort of the weight applied 
 to break it. 
 
 (5.) When the pressure, instead of acting upon a single point, 
 at the extremity of a beam fixed at one end, or in the middle of a 
 beam supported or fixed at both ends, is equally distributed through- 
 out the whole beam, twice the weight will be required to break it. 
 
 188. As far as we have recited the results of the hypothesis, 
 they agree in all useful cases with the deductions from experi- 
 ment. But somp of ih^ rules, deduced from the hypothesis, do 
 not coincide with what occurs in practice. 
 
 Thus : 
 
 It has been shown that there is a difference in the strength of 
 triangular beams, according to their position, with an edge or a 
 face uppermost; and that this difference follows different laws, 
 according to the manner in which the beam is supported. Ex- 
 periment absolutely contradicts this; for in neither of the three 
 different positions in which beams have been tried, has any im- 
 portant difference been found in the strength of a triangular 
 beam, when placed with an edge, or with a face uppermost. 
 
 So also, hollow cylindric beams have not been found stronger, 
 when the cavity has been nearer to the side where the fracture 
 terminates, as they ought to be in conformity with hypothesis; 
 neither is a square beam stronger when its diagonal is vertical. 
 
Book HI.} MATERIALS. 179 
 
 These remarkable discrepancies, together with the less impor- 
 tant ones that have been noted in the preceding section, arise 
 from an omission in the hypothesis; this does not take into ac- 
 count the elasticity and consequent flexure which materials un- 
 dergo before they actually break. These circumstances have 
 been introduced into the investigation by Barlow, in his treatise 
 on "the Strength and Stress of Timber;" to this work, then, 
 we refer for the mode in which the following formulae, that 
 coincide almost exactly with the results of experiment, have been 
 obtained. Using the same notation as before, and calling the 
 angle of deflection d 9 
 
 In a beam fixed at one end and loaded at the other, 
 
 In a beam supported at both ends, and loaded at the middle, 
 
 ~ 4a6 2 cos. 3 d' 
 In a beam fixed at each end, and loaded in the middle, 
 IW 
 
 ~ ' 
 
 In these formulae, s' differs froms in our formulae (160), (161), 
 and (162), inasmuch as it is the strength of the element ydx in 
 (140), (151), and (156), instead of being the strength of 2ydx. 
 We have preferred our mode of estimating the respective strength, 
 inasmuch as it retains the connexion with the formula (139), 
 which gives the value of the absolute strength. 
 
 189. In order to make our formulae applicable to practice, we 
 subjoin a table of the respective strength of various bodies, re- 
 duced to an element of the size of a cubic inch. 
 
 TABLE 
 
 OF THE RESPECTIVE STRENGTH OF VARIOUS SUBSTANCES, 
 
 Metals. 
 
 Wrought Iron, Swedish, - 22000 Ibs. 
 
 do English, 18000 Ibs. 
 
 Cast Iron, - 16000 Ibs. 
 
 Wood. 
 
 Teak, .... 4900 Ibs. 
 
 Ash, - - 4050 Ibs. 
 
 Canada Oak, 3500 Ibs. 
 
 English Oak, 3350 Ibs. 
 
 Pitch Pine, - - - 3250 Ibs. 
 
180 STRENGTH OF [Book III. 
 
 Beech, - 3100 Ibs. 
 
 Norway Fir, - - 2950 Ibs. 
 
 American White Pine, - - 2200 Ibs. 
 
 Elm, .... 1013 Ibs. 
 
 We arc without any good experiments on the respective 
 strength of stone. It would, however, appear from some expe- 
 riments of Gauthey, that, in soft freestone, 
 
 s=68.7 Ibs. 
 In hard freestone, 
 
 5=72.75 Ibs. 
 And from some experiments recorded by Barlow, that in brick 
 
 5=64 Ibs. 
 
 This would make the respective strength of stone and brick, far 
 beneath that of wood, or iron. 
 
 The preceding table may be applied to the calculation of the 
 strength of horizontal beams, of any figure whatever, by the three 
 formulae, (HO), (151), and (15ti). These may, however, assume 
 a more convenient form for practice, by calling the area of the 
 beam's section, A. 
 
 Let then 
 
 the area of the transverse section of the beam in sq. 
 
 inches ; 
 c=the distance of the centre of gravity of A, from the point where 
 
 the fracture terminates, in inches, 
 f=the length of the beam in inches, 
 s=the number from the preceding table. 
 W=the measure of the beam's strength, being the weight which 
 
 will just break it in pounds. 
 
 In beams fixed at one end, and loaded at the other, 
 
 W=^. (172) 
 
 In beams supported at each end, and loaded in the middle, 
 
 4c5A 
 W=- T -. (173) 
 
 In beams fixed at each end, and loaded in the middle, 
 
 W= -. (174) 
 
 The particular formulae, for rectangular square and cylindric 
 beams, will be found at (141), (1416), (142), (152), (153), (154), 
 (157), (158), and (159). 
 
Book ///.] 
 
 MATERIALS. 
 
 181 
 
 Of the Resistance of Bodies to a Force exerted locvush them. 
 
 190. The resistance of bodies to forces that act to crush them, 
 would, at first sight, appear to follow the same law with the ab- 
 solute strength ; that is to say : it must be proportioned to 
 the area of the body, and the force of aggregation of its particles. 
 Kxperiment, however, shows, that the thickness of the substance 
 has an important influence on the pressure it is capable of bear- 
 ing, without being crushed. In the first place, it is found that 
 very thin plates are readily crushed ; their resistance next in- 
 creases with the thickness, until it reach a maximum ; and finally, 
 decreases slowly, with a farther increase of thickness. It has 
 been attempted to frame a mathematical theory, that should re- 
 present these circumstances; assuming, that the body was com- 
 posed of flexible fibres, and that the crushing took place in conse- 
 quence of a bending in the fibres. 
 
 The respective strength, F, of the pillar, ABCD, supposed to 
 be rectangular, is represented by the formula (141). 
 
 A 
 
 satf 
 
 the effort of the weight W, exerted to break it, may be 
 represented by a force, bearing a constant relation to the 
 weight and applied to the middle of the column, say, at the 
 
 WJ 
 
 point E ; its moment of rotation will be -r- , and, 
 
 .=*() 
 
 \ 21 ) 5 
 
 whence, 
 
 and in a square beam, 
 
 (173) 
 
 (174) 
 
 191. In bodies whose sections are similar, it may be inferred 
 that the resistance to a force exerted to crush them, ft propor- 
 tioned, directly to the cubes of the homologous dimensions of the 
 sections, and inversely to the squares of their lengths, The most 
 complete set of experiments that we have upon the variation in 
 the strength of columns of different lengths, are those of Mus- 
 chenbrook, and they correspond closely with the above theory. The 
 subject, however, has not been so fully tested as to authorize us 
 to receive any theory with implicit confidence. We shall, in 
 
182 STRENGTH OF [Book III. 
 
 consequence, give the absolute results of experiments, in the 
 terms of the weights and dimensions of the original record, 
 without attempting to reduce them to a conventional unit. 
 
 TABLE. 
 
 EXPERIMENTS MADE BY RONDELET, ON THE RESISTANCE OF DIFFERENT 
 SPECIES OF STONE IN CUBIC BLOCKS, OF THE SIZE OF 5 CENTIMETRES 
 IN EACH DIMENSION. 
 
 Spec. Grav. Crushing Weight. 
 
 Kilogramme*. 
 
 Swedish Basalt, - - - 3.065 47.809 
 
 Basalt of Auvergne, No. 1, 3.014 44.250 
 
 do No. 2, 2.884 51.945 
 
 do No. 3, 2.756 28.858 
 
 Porphyry, - 2.798 50.021 
 
 Green Granite, (Vosges,) No. 1, 2.854 15.487 
 
 Grey do (Brittany,) 2.737 16.353 
 
 Granite, (Vosges,) No. 2, 2.664 20.482 
 
 Granite, (Normandy,) - 2.662 17.555 
 
 Granite, (Champ du Boul,) - 2.643 20.441 
 
 Granite, (Oriental Rose,) - 2.662 22.004 
 
 Black Marble, (Flanders,) - 2.721 19.719 
 
 White Veined Marble, - 2.701 7.455 
 
 White Statuary Marble, 2.695 8.176 
 
 Experiments made at the same time upon cubes of stone, whose 
 sides varied from 3 to 6 centimetres, showed that the strengths 
 varied nearly in proportion to the areas of their bases, and were 
 not influenced by the thickness. This corresponds to the law of 
 absolute strength in 179, and differs from that we have given 
 for the resistance to crushing. On the other hand, when cubes 
 of the same size were placed one upon another, a diminution in 
 the resistance was found that does not differ much from the lat- 
 ter law. 
 
 By experiments made by Rennie, on the resistance of cast 
 iron to pressure, it was found that the maximum strength was 
 reached at thicknesses of from ;|ths to .', an inch, and that a prism 
 a quarter of an inch square, and half an inch in depth, was crushed 
 by 10,000 Ibs. 
 
 A cube of cast copper, i inch each way, was crushed by 
 731S Ibs. 
 
 A cube of wrought copper, of the same size, by 6440 Ibs. 
 
 Of brass, by 10304 Ibs. 
 
 Of cast tin, by 966 Ibs. 
 
 Of lead, by 4S3 Ibs. 
 
 An inch cube of elm, is crushed by 12S4 Ibs. 
 
 Of American pine, by - 16C6 Ibs. 
 
Book ///.] MATEHIALS. 183 
 
 Of Norway fir, by ,-v - 1928 Ibs. 
 
 Of English oak, by 3860 Ibs. 
 
 Of the Strength of Torsion. 
 
 192. By the experiments of Coulomb, the resistance of wires 
 of the same material, to a force exerted to twist them, appears 
 to increase with the fourth power of their diameters. A similar 
 result follows nearly, from experiments by Renniei on square 
 bars of cast iron. It would also appear, from a simple course of 
 reasoning, that the resistance must diminish with the distance of 
 the point in the rod to which the twisting force is applied, from 
 the place where it is fixed ; for the rod tends, under the action 
 of the force, to form a screw, the distance between whose 
 threads is the same as the distance between the two points. And 
 in the inclined plane which the screw forms when developed, 
 the twisting force will act as if it tended to raise a weight along 
 the surface of the plane, whose altitude is the constant lineal dimen- 
 sion of the base of the rod. 
 
 193. We have no experiments on the absolute resistance to 
 torsion : the following are the relative resistances of different 
 materials deduced from the experiments of Rennie. 
 
 Lead, - ... 1000 
 
 Tin, 1438 
 
 Copper, - 4312 
 
 Brass, - 4688 
 
 Gun Metal, 5000 
 
 Swedish Iron, - 9500 
 
 English Iron, 10125 
 
 Cast Iron, 10600 
 
 Blister Steel, 16688 
 
 Sheer Steel, 17063 
 
 Cast Steel, ... 19562 
 
 
 * 
 
184 EQUILIBRIUM OF [Book III. 
 
 CHAPTER VIII. 
 
 OF THE EQUILIBRIUM OF ARTIFICIAL STRUCTURES. 
 
 194. Every building, machine, or other nrtificial structure, 
 may be considered as made up of a system of forces, and in order 
 that it shall be stable, it is necessary, that in this system, the 
 forces which tend to overthrow it shall not exert an effort greater 
 than those which tend to sustain. If equilibrium exist among all 
 the forces that act, the structure will be stable, until some new 
 force be applied to disturb this state of equilibrium ; but in order 
 that it shall be permanent, the sustaining forces must have so great 
 a preponderance over those which tend to destroy, that no acci- 
 dental application of extrinsic force shall be able to overcome the 
 equilibrium. Hence it becomes a matter of great importance in 
 mechanics, that we should be able to state the conditions of equi-> 
 librium that exist, among the forces that are to be found in action 
 in those structures which are of most frequent occurrence. 
 
 195. When by the action of a disturbing force, any part of a 
 structure is removed from its place, it can only move in one of 
 two ways; it may be pushed directly forwards upon the base-on 
 which the structure rests, or upon the adjacent parts of the struc- 
 ture itself; or it may revolve about some fixed point or line. If 
 we call the resultant of the forces, that tend to support a structure, 
 its Strength, and that of the forces that tend to destroy it, the 
 Stress, or Thrust ; in the former case, equilibrium must exist be- 
 tween the strength and the stress ; and in the latter, between the 
 moments of rotation of these two resultants. 
 
 Of the Equilibrium of Walls. 
 
 196. A wall may be considered as a prismatic structure ; and, 
 in its most simple case, as symmetric on each side of the vertical 
 plane in which the stress acts, and in which its own centre of gra- 
 vity falls. In this case we may leave the mass of the solid itself 
 out of question, and examine no more than the conditions of equi- 
 librium of its vertical section. 
 
Book ///.] ARTIFICIAL STRUCTURES. 185 
 
 Let ABCD be the vertical section of the wall. Let the stress 
 
 S be resolved into two forces, R and T, of which R acts in a ho- 
 rizontal, and T in a vertical direction, it will be obvious that the 
 wall cannot be moved from its place, except by a progressive mo- 
 tion from B towards A, or by a rotary motion around the point 
 A. 
 
 The resistance to the former of these, will be the friction. The 
 pressure on the base, AB, will be the sum of the weight of the 
 wall, M, and the force, T, or M+T ; the friction then will be 
 
 /(M+T) ; 
 and the condition of equilibrium will be 
 
 R=/(M+T). (175) 
 
 To estimate the moments of rotation of the forces: from the 
 centre of gravity, g-, let fall a perpendicular, g-X, on AD ; from 
 S, the point of application of the stress, let fall the perpendicular, 
 SE, on the same line ; let AX w, AE=/, ES=r ; the moment 
 of rotation of the weight will be Mm, of the force T, T/, and of 
 the force R, Rr ; the two former will concur to preserve the sta- 
 bility, the latter to destroy it, and the condition of equilibrium will 
 be 
 
 Rr=Mm+Tf. (176) 
 
 If the wall have a rectangular section, and be of homogeneous 
 materials ; let its height =a, its thickness =6, its density = G ; 
 then, as the mass is the product of the bulk by the density, 
 
 M=a6 G. (177) 
 
 The distance of the line of direction of the centre of gravity 
 from the point A, on which the wall would tend to turn, will be 
 
 ** 
 
 The resistance to a horizontal strain will be 
 
 6/G; (178) 
 
 The moment of the resistance to a rotary motion will be 
 
 i6 2 G; (179) 
 
 Therefore, in a rectangular wall, the resistance to a horizontal 
 thrust increases with its thickness ; and the resistance to an ef- 
 fort to overturn it, with the square of the thickness. 
 24 
 
ISO EQUILIBRIUM O? [Book HI. 
 
 197. In addition to the action of the stress, to move the wall 
 horizontally, or to overturn it, that part of the stress which is 
 exerted in a horizontal direction, tends to break the wall ; that 
 part which acts in the vertical direction, tends to crush it. The 
 manner of the action of a force of the latter description has been 
 explained in 190. Did ihe wall consist of one piece of a homo- 
 geneous substance, the resistance to the former of these forces 
 may be considered as a case of a beam fixed at one end, which 
 has been examined in 181. But walls are composed for the 
 most part of separate portions of heavy substances, held in their 
 place, partly by the friction of their surfaces, and partly by the 
 tenar.it \- of cements ; hence, for the quantity .5, in (140), \ve must 
 sub*ti i:te the value of these resistances. In respect to both of 
 thes rii ciniistanres. we have excellent experiments by Boistard, 
 which are to be found in the Treatise of Gauthey 4 '//e la Con- 
 strtic'inn des Punlx. " By these it appears, that the friction of 
 chiselled stores upon each other is constantly four-fifths of the 
 presMire ; that the resistance both of mortar and water cement, is 
 proportioned to the surface, and is equal, in the former, to 
 1426j Ibs. per square foot, in the latter, to 7561 |bs. When 
 plunged in water, however, the strength of the latter is rather 
 increased than lessened, while the former retains little or no te- 
 nacity. The friction, it is obvious, will be greatest in the low- 
 est joints, and in a rectangular wall, will decrease uniformly from 
 the base to the top. 
 
 19S. When a wall is to resist a horizontal strain, it may be 
 strengthened by making one of the faces sloping, or by building 
 buttresses or counterforts, projecting from it at right angles. The 
 advantages derived from these different methods, may be thus 
 investigated. 
 
 (1.) Let/) be the base of the sloping part of the wall, being 
 the addition to 6 of (177) at the base, and on the side opposite to 
 that where the strain acts; the equation (175) will become 
 
 R=o/G(6+ip); (180) 
 
 and the equation, (176), 
 
 Rr=aG(^ 2 +6p-fip 2 ) ; (181) 
 
 If the slope were on the same side with that on which the strain 
 acts, the moment of resistance would be 
 
 Rr=G(ifc 2 +ify>+ip 2 ). (182) 
 
 This latter method would therefore be less advantageous than the 
 former. Were the wall to be increased uniformly in thickness, 
 by the quantity p, the moment of resistance would be 
 
 and this would be the least advantageous method of the three. 
 
III.] ARTIFICIAL STRUCTURES. 187 
 
 (2.) Let the figure represent the horizontal plan of the wall 
 
 i- 
 
 Cr 
 
 
 iv 
 
 with three of its buttresses, the intervals between which are each 
 divided into two equal parts in the points G, H ; if the horizontal 
 strain act uniformly along the whole wall, its resultant will fall in 
 the vertical plane that divides it into two equal parts ; and in this 
 section, its centre of gravity will also fall ; let this vertical plane 
 be AB ; let the height of the wall =a, its breadth =6 ; ihe projec- 
 tion of the buttress, AD=c ; its thickness MN=p ; and the in- 
 terval GD=d ; the equation for the resistance will become (175) 
 R=afG(bd+cp) ; (183) 
 
 for the moment of the resistance, (176), 
 
 Kr*aG(|6d+6e<i+j4>). (134) 
 
 By an investigation similar to that made in the former case, it 
 will be found, that buttresses applied to the side of the wall on 
 which the strain acts, are less advantageous than those on the op- 
 posite side, and that either are preferable to an addition of their 
 quantity of material to a wall of uniform thickness. 
 
 199. These principles are applied to a very great extent in 
 architecture. Thus the walls of the temples of ancient Ejrypt, 
 which are the most stable buildings containing chambers, that 
 have been erected by the hand of man, and are pressed by heavy 
 stone roofs, have a considerable external slope; the same form 
 is given to the terrace walls of fortifications, which are besides 
 strengtfiened within by buttresses. In the wonderful buildings 
 of the middle ages, known by the name of Gothic, the walls have 
 such large and numerous apertures as almost to disappear en- 
 tirely ; but in them, the stress of massive vaults of stone is well 
 sustained, by means of buttresses of great projection, and which, 
 by being built with an external slope, unite the advantages of 
 both methods. As an instance of this, may be cited the buttresses 
 of King's College Chapel at Cambridge ; these project at their 
 base more than twenty feet from the body of the building, and 
 the whole space between them is almost completely occupied by 
 windows. 
 
 Equilibrium of Columns. 
 
 200. The reasoning in 1 90 and 191, might be considered 
 as directly applicable to the case of columns having a weight to 
 
. 
 
 188 EQUILIBRIUM OF [Book III. 
 
 support upon their summits. It is not, however, sufficiently strict 
 to be admitted, when the length is great in proportion to the 
 area of the section. A column, as then stated, if its materials 
 be either wholly incompressible, or after they have been compress- 
 ed as far as their elasticity will admit, can only give way, by a 
 flexure taking place either in the. whole mass, or in the fibres of 
 which it is composed. It hence becomes necessary to take into 
 account, the mode in which the stress acts to produce this flex- 
 ure, and the manner in which resistance of the material opposes 
 it. 
 
 Let us take the case of an elastic plate Mwin, infinitely thin, 
 
 and fixed at the point M in such a manner, that however it may 
 be bent, the direction of the .tangent, MT, shall remain constant. 
 Let us suppose, that this^pUte is submitted to the action of a 
 number of forces acting in the same plane, which will be that of 
 the curvature. Call the resultant of the components of these 
 forces that are parallel to the axis AY, P, and that of components 
 parallel to AX, Q ; the co-ordinates, x and y,'of the curve being 
 referred to the same two axes. It is required to determine the 
 conditions of equilibrium of the forces P and Q, and the elasticity 
 of the plate which will tend to restore it to the direction of the 
 tangent MT. 
 
 If at any point of the plate in, the part Mn becomes fixed, and 
 the part mn perfectly rigid, a case that will not affect the condi- 
 tions of equilibrium : the effect of the forces, P and Q, will be to 
 tend to turn the part mn around the point m, and the action of the 
 elasticity of the plate will be exerted to turn the same part of the 
 plate in a contrary direction : hence the elasticity may be con- 
 sidered as a force acting perpendicularly to the line m/, and whose 
 intensity, E, in the case of equilibrium, must be equal to the sum 
 of the moments of rotation of P and Q, in respect to the point m. 
 If p and q, be the distances of these forces from the axes Ax 
 and At/, their distances from wi, will be p #, and 9 x. The 
 equation of equilibrium will therefore be 
 
 P(p *) + Q(9 *)=E. (185) 
 
 At any other point than m, the elasticity of the plate will act with 
 a different intensity. The law usually assumed to represent the 
 
Book ILL] 
 
 ARTIFICIAL STRUCTURES. 
 
 169 
 
 elasticity is, that the force is proportioned to the tension, or is at 
 any given point, inversely proportioned to the radius of curvature. 
 If, then, E be the value of the elasticity at a point whose radius 
 of curvature is equal to unity, at any other point the elasticity will 
 be 
 
 *>=-- (186) 
 
 i+;6r 
 
 The well known value of p is 
 
 9=' 
 
 dx* 
 
 whence we obtain 
 
 E 
 
 (187) 
 
 We may consider this equation with greater ease, by confining 
 ourselves to the cases that may occur in practice. 
 
 (1). If we suppose that the plate is so situated that the tangent 
 MT, is parallel to the axis AX, and that the force P is the sole 
 force, and acts at the end N of the plate ; 
 
 then Q=0, and if we call the length of the plate /, 
 
 (188) 
 
 dx 
 
 If the curvature of the plate be very small, -7- will also become 
 
 dx 2 
 very small, and its square - - may be neglected in the second 
 
 member of the equation ; we may take therefore 
 
 E &. ; (189) 
 
190 EQUILIBRIUM OF [Book Hi. 
 
 Integrating we obtain 
 
 *(Hr>= E ' < 190 > 
 
 and there is no need of an arbitrary constant ; for when *=0 
 
 S= 0i 
 
 Integrating again, we obtain 
 
 (191) 
 
 in which 'k represents the distance AM. 
 
 If we make the versed sine of the curvature MB=;/*, when x=e, 
 y=k /; and we obtain for the value of/, neglecting its algebraic 
 sign, 
 
 in which we have a relation between the length of an elastic plate, 
 and the force that is capable of producing a given curvature. 
 
 (2). If we suppose that the force P in its turn becomes equal to 
 0, and that the force Q is applied to the extremity N of the plate, in 
 
 such a manner that its prolongation passes through the point M, 
 at which point the plate rests upon a plane AY, the plate will 
 
 then be curved, as in M m N. If we again neglect -j-j the equa- 
 
 "2T 
 tion of equilibrium will become 
 
 Q(9 !/)= E ^-- ( 193 ) 
 
 Multiplying by cfy, and integrating we have 
 
 J-.2 
 
 In order to determine the constant quantity A, let / be the 
 greatest value of i/, and we have at the same time 
 
 !/=9 /? 
 and 
 
 -^=0; therefore 
 ay 
 
Book ///.] ARTIFICIAL STRUCTURES. 101 
 
 and 
 
 A=/ 3 -?', 
 substituting this value of A in (193a) it becomes 
 
 E-, (184) 
 
 or 
 
 dy 
 
 dx= 
 
 whose integral is 
 
 qy=fsm. xV^jr ; (195) 
 
 in which there is no need of an arbitrary constant, for when 
 #=0, y=q. 
 
 If we suppose that the axis AX is identical with the line MN, 
 and that the ys are positive, when reckoned downwards from this 
 line, then the equation becomes 
 
 y=f S m.xV*jr. (196) 
 
 In this expression y becomes =0, when j?=0, or #=/; and 
 when the curvature is very small, we may consider / as coinci- 
 ding with MN ; upon this hypothesis then, 
 
 sin. JV~ = 0; 
 
 and this can only be true when />/ vr is an exact multiple of half 
 
 the circumference of a circle ; let m be any whole number, and #, 
 the relation of the circumference of a circle to its diameter ; we 
 can express this fact as under 
 
 /A- 
 V E~ 2 ' 
 
 from which we obtain for the value of Q, 
 
 and the equation (196) takes the form 
 
 Wl*flf 
 
 y=/sin. t x. (198) 
 
 When m=l which is the smallest possible value, 
 
 \ ^2 
 
 Q= 4^ , (199) 
 
 and 
 
 y=/flin- ~ x ; (200) 
 

 
 EQUILIBRIUM OF [Book If I. 
 
 from these we may deduce that when the value of Q is less than 
 in (199), it will not produce any effect on the elastic plate. 
 
 This last case will represent the action of a weight W, upon an 
 elastic plate, standing vertically upon a horizontal plane. We 
 have for the value of the weight that will produce no flexure, 
 
 W=^ (201) 
 
 If instead of a single elastic plate of infinite thinness, we con- 
 sider the forces to act upon a body of definite magnitude ; let the 
 forces be as before, P and Q, acting in the plane XY, 
 
 parallel to the axes AX AY, let'Mmn be the curvature which the 
 action of the forces has given to one of the generating lines of the 
 surface of the body, and suppose that through some point m, of this 
 curve a plane passes, to which the curve at that point is a nor- 
 mal, and that its intersection with the body be represented by a 
 curve such as mm'. Refer this curve to two rectangular axes aw, 
 and at", that lie in its plane, of which aw is perpendicular to the 
 plane in which the forces P and Q act, and touches the curve mm 
 on the side where the action of these forces tends to bend the 
 body ; call the co-ordinates of this curve w and v. 
 
 The body may be considered as made up of an infinite num- 
 ber of elastic fibres ; and if we suppose, as before, that the part 
 M becomes fixed, and the part mn inflexible, we shall have again 
 for the condition of equilibrium, an equality between the moments 
 of rotation of P and Q, and the force of elasticity. The latter 
 
 E 
 for any one fibre, whose base is dw x dv is by (186), dw dv ; 
 
 E 
 
 its moment in respect to the axis aw is dw vdv. The moment of 
 
 F 
 
 rotation of the whole base mm' will be ffdw vdv, and the con- 
 dition of equilibrium (186), 
 
 p(p-. x ) + Q(q y ) = -ffdw vdv. (202) 
 
 If the section of the body were a rectangle, one of whose sides 
 
HI.] ARTIFICIAL STRUCTURES. 193 
 
 corresponds to av, 
 
 ffdw vdv=^ . (203) 
 
 If the section were a circle, whose radius is r, 
 
 ffwdv dv=r* . (204) 
 
 If the body were a cylindric tube, 
 
 ffdw fldr=,r(R 3 d' 2 ), (205) 
 
 and the relation between the diameters of a solid cylinder, and a 
 cylindric tube, of equal strengths, would be 
 
 It will be therefore obvious, by comparing the four last equations 
 with (141), (142), (143), (144), and (145), that the resistance 
 to flexure follows the same laws as the resistance to fracture, so 
 far as the strength has reference to the area of the section ; but 
 as respects the length, it will be seen by reference to (197), and 
 (199), that the strength to resist flexure, diminishes with the in- 
 crease of the square of that dimension. 
 
 To apply this to the case of a column. Suppose a solid to be 
 
 placed vertically that it is charged with a weight W at its up- 
 per extremity, and that all its horizontal sections are circular. In 
 this case, W = Q, and if we take for the origiri of the co-ordinates 
 
 the point that bisects the height, f=q ; the first term in (202) 
 may be neglected. Combining (202) and (204), we obtain for 
 the equation of equilibrium, 
 
 E 
 
 : 1JT 3 . (207) 
 
 The smallest quantity of material, or the greatest strength with 
 a given quantity will be obtained, when the resistance to flexure 
 is the same in every section. In this case the radius of curvature 
 of the flexure the beam undergoes, is constant, and the curve an 
 arc of a circle, whose radius is . 
 
 In a solid of equal resistance, the curve assumed in bending will 
 have a constant radius of curvature, or be a circle whose radius is 
 p. The value of y will become 
 
 substituting this value of?/ in the preceding equation, we obtain 
 
 This will be a maximum when x=Q, or the column must be 
 thickest in the middle of its length. 
 
 Were this analysis carried farther, it might be inferred that 
 the column should diminish towards each end, to a point, and 
 should therefore have the figure of a spindle. 
 
 In the foregoing investigation, however, it has not been taken 
 
 25 
 
194 EQUILIBRIUM OF [Book III. 
 
 into view, that the resistance to a crushing force depends upon 
 the number of fibres, and consequently in a homogeneous body, 
 on the area of the surface pressed. The above inference will, 
 therefore, only hold good in the impossible case, that all the fibres 
 should be curved, and meet at the two extremities in a point. 
 
 Let us next take the case of a column resting on a horizontal 
 base, and of a circular section throughout, formed by the revolution 
 of a curve around the axis ; and that a force acts, which is distri- 
 buted in the ratio of the external surfacesof the horizontal sections, 
 and whose directions are all parallel to each other, and perpendi- 
 cular to the axis. 
 
 Call the resultant of these forces R. It is obviously an instance 
 of the case No. (1), of our preliminary analysis. 
 
 The equation of equilibrium obtained from (202) and (203) is 
 
 Rx=r* . 
 
 P 
 
 If n be the pressure upon the unit of the surface, we have 
 
 E 
 
 whence we obtain 
 
 Sh" (209) 
 
 the generating curve is, therefore, one convex towards its axis, 
 and in which the abscissas are proportioned to the squares of the 
 ordinates. 
 
 201. From the preceding course of reasoning it may be in- 
 ferred : 
 
 That the strength of a column is proportioned directly to its 
 area, and inversely to the square of its length; that, were its fibres 
 capable of being collected at each end into a point, it ou^ht to 
 have the form of a spindle. As this, however, is not the case, 
 it may be inferred, that, were reference to be had only to the 
 weight the column is to support, it ought to be somewhat thicker 
 in the middle of its length. As it. has also its own weight to sup- 
 port, the swell ought to lie lower than half the height, for in this 
 way the centre of gravity will be lowered, and the base enlarged ; 
 and thus the resistance to a force exerted to overturn it, or to one 
 tending to thrust it horizontally from its place, will be also increa- 
 sed. 
 
 The practice, then, of architects is founded on reason, and is as 
 follows : In the Grecian Doric, the columns are truncated cones, 
 whose least base is uppermost. In the Roman Doric, in the Ionic, 
 the Corinthian, and Composite orders, the columns are sometimes 
 made cylindrical for one third of their height from the base, and 
 
 ' 
 
Book HI.] ARTIFICIAL STRUCTURES. 195 
 
 for the remaining two thirds of their height, are frusta of cones ; 
 at other times they swell at one third of Iheir height, the whole 
 external surface being one of revolution, formed by a curve con- 
 cave towards the axis. It is most usual to make this curve a 
 conchoid. 
 
 202. When the weight exerts a lateral thrust, the upper and 
 lower surfaces of the column remaining constant, it may be infer- 
 red from 1 98, that the axis of the column ought not to be vertical, 
 but should be slightly inclined, towards the side against which 
 the weight acts. By a recent examination of the columns of the 
 Parthenon, it hns been found that this principle is there applied, 
 and to great advantage. The columns of the external perystyle 
 in this perfect specimen of Grecian architecture, all have their 
 axes slightly inclined inwards. 
 
 203. When a column has no weight other than its own to sup- 
 port, but is acted upon by a stress exerted perpendicular to the 
 axis, the form should be that of a conoid, formed by the revolu- 
 tion of a. curve convex towards the axis. And which, when the 
 stress is proportioned to the surface, as is the case in the action of 
 a fluid, should be a curve whose abscissas are in arithmetic, 
 and ordinates in geometric progression. We' have instances of 
 the application of this principle in the works of nature, in the 
 manner in which the trunks of tree rises from their roots, and 
 their branches from their insertion into the trunk, or into larger 
 branches. In the arts, we find a beautiful application of it, by 
 Smeaton, in the construction of the Edystone Lighthouse. In this 
 remarkable edifice, obstacles of the most appalling character were 
 to be overcome in its erection ; and it is frequently exposed to 
 violent swells of the sea. The rock on which it was built is in- 
 clined : advantage was taken of this circumstance to cut it into 
 steps, to each of which one of the lower courses of stones is adapted. 
 These steps are so formed, that one at least of the stones of each 
 course is dovetailed to the rock ; the remaining stones are so 
 cut, that none of them can be removed, without being lifted, unless 
 the dovetail should be disunited ; and throughout all the courses, 
 no one of the outer stones can be removed horizontally without 
 moving all those in the same course. In addition to the tenacity of 
 the cement, the courses were connected by dowels of the form of 
 cubes, of a hard stone, one half of each of which is inserted into 
 each course. This structure has now stood in its perilous situation 
 for seventy years, and has borne without injury, the great stress to 
 which it is exposed. A structure of similar character has more re- 
 cently been erected on the Bell Rock, at the mouth of the Firth 
 of Forth, in Scotland. 
 
198 EQUILIBRIUM or [BooAf ///. 
 
 Equilibrium of Terraces. 
 
 204. When earth is piled up into mounds or terraces, we may 
 conceive that the faces of these were at first vertical ; the earth 
 composing the upper part of the face, being loose, will separate 
 itself and fall; and thus the base will be gradually enlarged, and 
 the face become more and more inclined, until the friction on 
 the surface of the inclined plane, thus formed, becomes equal to 
 the force with which the earth would tend to descend it. At this 
 point, any farther wear of the mound would cease, and it would 
 become stable, each particle on its surface being in equilibrio, un- 
 der the action of its own weight, the resistance of the surface of 
 which it forms a part, and the retarding force of friction. 
 
 In looie soils, the earth will not be supported until the surface 
 make an angle of 60 with the vertical plane ; in tenacious soils, 
 the support may take place when the angle reaches 54. 
 
 205. If it be required to support a terrace by a vertical wall, it 
 must be so constructed that it shall be able to bear the horizontal 
 thrustof the prismatic massofearth, which lies above the plane, that 
 would form the surface of a bank that would be itself supported. 
 But this prism is itself partially supported by friction, and we must, 
 before we can ascertain its horizontal thrust, ascertain how much 
 of the force it would exert in a horizontal direction, is counterac- 
 ted by the friction. 
 
 Let us suppose a weight, W, to lie upon a plane inclined to the 
 horizon at an angle p ; and that a force, A, is applied in a horizon- 
 tal direction, which, with the friction, just supports the weight. If 
 we resolve each of the forces R and A, into two others ; one of 
 which is parallel, the other perpendicular to the plane, they 
 become 
 
 R cos. fx, R sin. /m ; 
 
 and 
 
 A sin. fj., R cos. p. 
 
 The two which are parallel to the plane, act in contrary directions ; 
 their sum is, therefore, 
 
 R cos. j& A sin. fx. 
 The other two concur in direction ; their sum, therefore, is 
 
 R sin. j^-f-A cos. p, 
 
 and represents the whole pressure on the plane ; the friction 
 will, therefore, be 
 
 /(Rsin. fx)H-/(Acos. <x) ; 
 
 and as the friction is equal to a force that will just support the 
 weight upon the plane, this value of the friction will be equal to 
 that of the forces that act parallel to the plane. Forming an equa- 
 
Book ILL] 
 
 ARTIFICIAL STRUCTURES. 
 
 197 
 
 tion of these two expressions, we obtain from it for the value of 
 
 . (210) 
 
 tan. 
 
 If, than, the earth be supported by a wall, the second member of 
 this equation will represent the horizontal thrust of the earth. 
 
 Let us now apply this to the investigation of the horizontal thrust 
 of the prism, BCE. 
 
 Let the angle CBE=fji ; 
 
 the line BC^a; 
 
 and the variable ordinate in the direction of BC=x. 
 The element of the surface will be 
 xdx tan. /x ; 
 and if g be the density of the earth, its weight will be 
 
 gxdx tan. p ; 
 the horizontal thrust of the element will be 
 
 or 
 
 tan. 
 
 If we make 
 
 1 /tan. p \ 
 
 ') 
 
 cot. jx / 
 
 1 /tan. p _ 
 
 the thrust of the element is 
 
 WLgxdx. 
 
 Integrating, and making #=a, we obtain for the value of the whole 
 thrust 
 
 (211) 
 
 The moment of the thrust of the element will be 
 M.gxdx(a x). 
 
 Integrating, and making x=a, we have for the moment of the 
 whole thrust 
 
198 EQUILIBRIUM OF [Book ///. 
 
 The expression, (210), will become =0, when tan. jx=0, or 
 when tan. f/,= between these two limits there will be a value 
 
 when the thrust will be a maximum, determined by making 
 dM=0 ; from this we obtain 
 
 if we substitute this value in (210), we obtain for the thrust, 
 
 ia-tan. a f*; 
 and for the moment of the thrust, 
 
 a 3 -tan. V 
 
 The angle whose tangent is /-f V (I / 2 ), is just double of 
 that whose tangent is _, and the latter angle is that at which the 
 
 earth would just be supported by its own friction ; in loose earth, 
 then, 
 
 fx=30, and tan. fx==>/i ; 
 in tenacious earth, 
 
 (A =27, and tan. ^= V ^ nearly. 
 Hence, in the first, the thrust will be 
 
 *a ff> (212) 
 
 and its momentum, 
 
 rV^- (213) 
 
 In the second, the thrust will be 
 
 and its momentum, 
 
 j f a>g. (215) 
 
 It has been already seen, (179), that the moment of the resist- 
 ance of a wall, whose altitude is a, to a horizontal thrust, is 
 
 and this, in the case of equilibrium, must be exactly equal to the 
 moment of the thrust, or in the case of loose earth, 
 
 (216) 
 whence we obtain for the thickness of a rectangular wall, 
 
 fe=j ^(aVG). (217) 
 
 The value of the thickness of a wall with any given slope, may 
 be in like manner obtained from (181), and that of a wall with 
 buttresses from (184). 
 
 Of the Equilibrium of Arches. 
 
 206. When an aperture of considerable extent is to be covered 
 by a mass of any material whatsoever, it will appear at once, 
 from what has been said in 186, that there is a limit to the use 
 of beams or horizontal lintels, growing out of the difference be- 
 
Book ///.] ARTIFICIAL STRUCTURES, 199 
 
 tween the ratios, in which the respective strength, and the action 
 of their own weight to break them, increase. This limit will be 
 reached earlier in stone than in any other material of which we have 
 treated, inasmuch as its respective strength is but small, while 
 its weight is great. So also, in this material, it is frequently dif- 
 ficult to obtain pieces sufficiently large to form lintels, even of a 
 size within the limit at which they would become too weak. In 
 all these cases, we have recourse to what is called an arch. 
 
 An arch differs from a lintel, inasmuch as it is composed of a 
 number of pieces of the material, arranged in such a u.anuer as 
 mutually to sustain each other; and as each piece has but small 
 dimensions, measured in a horizontal direction, each will sustain 
 a considerable vertical pressure, while the greater part of the 
 force that is applied to them is borne by their mutual action up- 
 on each other's surfaces. Arches, generally speaking, have 
 curved or polygonal surfaces, forming the lower part of their 
 mass, which are concave towards the horizon. They are support- 
 ed at the extremities upon walls ; and in this case, the resistance 
 they oppose to the forces that tend to destroy them, is principally 
 that with which their materials resist a force that tends to crush 
 them. But there are also cases in which the arch is a curve or 
 polygon, convex towards the horizon, in which case their princi- 
 pal resistance is due to the absolute strength of the material. 
 
 As both the absolute strength, and the resistance to compres- 
 sion are more intense in all materials, than their respective 
 strength ; and as, in addition, the forces that tend to destroy an 
 arch, act in most cases obliquely, it is at once obvious, that an 
 aperture of far greater length can be covered by an arch, than 
 can be done by any other application of the same material. 
 
 The circumstances that affect arches will differ according to 
 the materials of which they are constructed. These are princi- 
 pally, stone or brick, cast iron, wood, wrought iron in the shape 
 of chains, and ropes ; arches of. the three former substances re- 
 quire no distinctive appellation, we shall call arches of the last 
 two kinds, Arches of Suspension. 
 
 207. A stone or brick arch is composed of a number of prisms, 
 whose section is a trapezium ; these may be considered as trun- 
 cated wedges, and are called Voussoirs. They are generally of 
 an uneven number ; the odd one, which occupies the vertex of 
 the arch, is called the Keystone. The vertical walls on which 
 they rest, are called Abutments, and when there are two con- 
 tiguous arches, the intermediate wall is called a Pier. The point 
 where the vertical wall meets the curve of the arch, is called the 
 Spring of the arch. The distance between the piers or abutments 
 that support a single arch, its Span. The lower or inner curve 
 
EQUILIBRIUM OP F Book ///. 
 
 of an arch is called its Intrados; the upper or outer curve, its 
 Extrados. The intervals between the voussoirs are called 
 Joints. 
 
 208. In considering the theory of arches, it has been usual to 
 proceed, either by assuming a given intrados, and investigating 
 the relative size of the voussoirs ; or assuming the magnitude of 
 the voussoirs, to investigate the curve of the intrados. In both 
 cases, the voussoirs have been considered as wedges, perfectly free 
 to move upon each other, or resisted neither by friction nor the 
 tenacity of cement. 
 
 By the former method, it may be shown, that the magnitude 
 of the voussoirs, in order that equilibrium should exist, should 
 be to each other as the portions cut from a horizontal line passing 
 through the vertex of the arch, by the prolongations of the joints. 
 In the case of an arch, whose intrados is a portion of a circle, the 
 relative weights of the voussoirs would be the differences of the 
 tangents; and if the arch were semicircular, the lower voussoirs 
 must be infinite. 
 
 By the latter method, it may be shown, that the arch of equili- 
 brntion, if the voussoirs be equal, must be a homogeneous cate- 
 naria ; and, in the case of unequal voussoirs, a catenarian curve 
 loaded with weights proportioned to those of the voussoirs. And 
 in the case of weights, varying in the relation of the differences of 
 the series of tangents, the curve would become a portion of a circle. 
 
 Both methods being founded upon the same hypothesis, give, 
 not only in the last quoted instance, but in all others, similar re- 
 sults. 
 
 An arch of equilibration, determined in either mode, would 
 have its centre of gravity in the highest possible position, and 
 would therefore be in a state of tottering equilibrium. Hence, any 
 action, however small, exerted upon it, would shake the voussoirs 
 from their position, and the arch would be destroyed. So far is this 
 from being the case in practice, that arches may be considered 
 among the most permanent of structures ; hence, it is obvious, 
 that so far from the friction and the adhesion of the cements 
 being quantities that may be safely neglected, or for which a 
 mere correction may be applied 1o an hypothesis from which 
 they are first abstracted, they constitute, in truth, forces as essen- 
 tial to the conditions of equilibrium, as the mutual pressure of the 
 voussoirs themselves. 
 
 - it would also appear from these hypotheses, that an arch would 
 exert no horizontal thrust upon its abutments, unless the face of 
 the abutment were not a tangent to the arch at its spring; and 
 hence, that a thrust would not be created by flattening the arch, 
 provided that the radius of curvature at the spring coincided with 
 
ARTIFICIAL 8TRUCTUKES. 
 
 201 
 
 EookllL] 
 
 the horizontal line passing through it. All these inferences are, 
 in like manner, contradicted by experiment. 
 
 In order, then, to the establishment of a true theory of arches, 
 it is necessary that experiment and observation should be pre- 
 viously called in, to show the exact circumstances under which 
 arches change their form, or actually give way. Such experi- 
 ments were first made on very small models, by Davisy, at JVlont- 
 pelier, about the year 1732 ; and were repeated on a larger scale 
 by Boistard, in 1800. Similar experiments were also made by 
 Cauthey ; and the same author has recorded his observations up- 
 on a number of broken bridges, and others in such a state of de- 
 cay as to require their being taken down. Perronet likewise, at a 
 previous date, had made accurate observations upon the change 
 of figure undergone by new arches, at the moment of removing 
 their centres. 
 
 209. Arches are built by laying their voussoirs upon a tem- 
 porary arch or frame of wood, called a Centre, whose upper sur- 
 face has the form it is intended to give to the arch. In laying 
 the voussoirs, it is found that the lower ones retain their position 
 simply by virtue of the friction upon the faces on which they 
 rest, and that they may be laid wholly independently of the 
 centre. They do not begin to slide, until the inclination of the 
 faces becomes equal to about 40. At this time they begin to 
 press upon the centre, which would have its form changed in con- 
 sequence, were it not loaded at the summit, for the purpose of 
 counteracting this change of figure. 
 
 As the number of voussoirs increases, a pressure begins to take 
 place on the upper part of the centre, which tends to press it 
 against the spring of the arch ; the upper voussoirs tend to turn 
 around their lower angle, and the joints open at the extrados. 
 When the keystone is placed, and the centre removed, the open 
 joints close again, and new openings and motions appear in the 
 arch, to represent which, we must have recourse to a figure: 
 
 The upper parts of the arch, from D to d, are no longer sup- 
 ported, except by their mutual pressure ; this tends to close their 
 
 26 
 
202 EQUILIBRIUM OV [Book III. 
 
 joints at the extrados, and press them from each side towards the 
 point E : this point becomes a point of support for both halves of 
 the arch. The joints of the upper part of the intrados will open. 
 The pressure on the point E, necessarily reacts towards the abut- 
 ments and the lower parts of the arch, which it tends to over- 
 throw by causing them to turn around their outer angles K and Ar. 
 In consequence of this pressure and reaction, each half of the arch 
 separates into parts at some intermediate points, D or d. These 
 serve as points of support for the higher parts, and to transmit 
 their action towards the abutments. If the latter do not possess 
 sufficient stability to resist the pressure of the arch, it separates 
 into four parts, which, in breaking, turn around the points K,D,E, 
 d and AJ, as upon hinges. If the abutments are capable of meet- 
 ing the pressure, it still manifests itself by closing the joints of 
 the extrados near the point E, and of the intrados near the points 
 d and D ; causing those of the intrados to open near the point E, 
 and of the extrados near the points D and d. 
 
 The position of the points d and D, which are called Points of 
 Rupture, depends upon the figure of the vault, and the distribu- 
 tion of the weight it supports. In determining the strength of 
 an arch, it is important to know their position. It is always at 
 the weakest part of the arch ; but this is not necessarily that where 
 it is thinnest, as we shall see hereafter. 
 
 In the experiments it was found, that : 
 
 In semicircular arches, that did not rest on abutments, the 
 points of rupture were at an angle of 30 from the spring. 
 
 In oval arches, formed of three circular arcs, at 50, of the 
 small arc rising from the spring. 
 
 In flat arches, the points of rupture were at the spring, as they 
 were in all circular arches whose versed sine was less than a 
 fourth part of the chord. 
 
 In all cases the whole mass, say the arch and its abutments, 
 tended to divide into four parts, turning upon points in the intra- 
 dos, at the spring or base of the abutment, and the vertex, and 
 separating at two intervening points. 
 
 210. To investigate the action of the forces that tend to produce 
 these motions. 
 
 Suppose the arch to be divided into two symmetric parts, by 
 the vertical plane passing through the line EC, the relations of 
 the forces will be identical on each side of EC. Let K be the 
 origin of the co-ordinates ; let x and y be the horizontal and ver- 
 tical co-ordinates of the point D ; x' and y' the co-ordinates of the 
 point E. 
 
 Resolve the forces that act at D into two, Xand Y, parallel to the 
 jtwo co-ordinates ; those that act at E into two also, X' and Y'. 
 
///.] ARTIFICIAL STRUCTURES. 203 
 
 By the principle of vertical velocities, 70, the equation of equi- 
 librium will be 
 
 '=0. (218) 
 
 The forces that act at D, act upon a lever KD, which we shall 
 caliyj those at E upon another DE, which we shall call g-, their 
 values in terms of the co-ordinates are 
 
 g=^ [(*'-*)"+ ('-)']; 
 
 the differentials of which are respectively 
 
 and 
 
 (a?' ar). (dx f dx) + (y'y)-(dy'dy) =0. 
 Multiplying the first members of these equations respectively by 
 the constant co-efficients / and /', we have 
 
 Ixdx -\-lydy ; 
 and 
 
 l'(x'x')dx t l(x > x}dx+ &c. 
 
 If these be added to the equation (218), we shall find from it 
 that the sum of the terms, that involve the differential of any one 
 of the co-ordinates =0, or 
 
 TLdx+lxdx l'(x' x)dx=Q ; 
 T[dy+lydyl'(y'y]dy=0 ; 
 r' x)dx'=0 ; 
 
 dividing by dx, dx', dy and cfy', we obtain 
 
 and eliminating / and /', 
 
 ')jr (Y+T>=0. I ( 219 ) 
 
 The forces which act are the weights of the two portions of the 
 arch, applied to their respective centres of gravity, M and N ; call 
 these weights m and n. 
 
 The arch being in equilibrio, 
 
 X=0. 
 
 If for m we substitute its two parallel components acting at D and 
 E, their magnitudes will be 
 
 FQ 
 
 at E, m ; 
 
204 EQUILIBRIUM OF [Book III. 
 
 EF 
 and at D, wi = Y'. 
 
 The force m resolved in like manner., gives for its action, at D, 
 
 KS 
 
 hence the sum of the forces that act at D, is 
 _ EF KS 
 
 Y ~ m EQ"'" n KR ; 
 and 
 
 DQ 
 
 FQ 
 
 because it is only the component of m JTQ in the direction of E 
 
 that acts. Substituting these values in the two equations, (219) 
 we obtain identical values for both terms of the first member of 
 the first ; in the second, we have 
 
 S KR=0. (220) 
 
 In order, then, that equilibrium shall exist, the first term must be 
 equal to the sum of the two last ; and for stability it ought to be 
 greater. 
 
 The deductions from this theory are abundantly simple. 
 
 From the value of the horizontal thrust, X', it appears that it 
 increases with the length of the line FQ, or with the approach of 
 the centres of gravity of the upper parts of the arch to their sum- 
 mit. 
 
 We may, therefore, see that it would be possible to investigate 
 a curve for the extrados, when the intrados is given, that would 
 form an arch without any horizontal thrust whatsoever. When 
 the centre of gravity of the higher part of the half arch falls in the 
 middle, which it will very nearly do, when the arch is very low 
 and the vault of uniform thickness, 
 
 and the formula becomes 
 
 m DO 
 
 . ^ . KU n.KS i.KR=0. (221) 
 
 If the arch be flat, EQ becomes equal to the thickness of the 
 voussoirs. If we call this thickness e, and the span of the arch 
 
 - , the expression will become 
 
 m / 
 
 -- . - . KU n.KS w.KR=0. (222) 
 
 6 
 
Book III.] ARTIFICIAL STRUCTURES. 205 
 
 From this it follows, that the horizontal thrust of a flat arch, is 
 equal to the fourth part of its weight, multiplied by the ratio of 
 the length to the thickness. Not only does the horizontal thrust 
 diminish with the approach of the centre of gravity of the upper 
 part of the half arch to its vertex, but the same causes a diminu- 
 tion in the pressure on the keystone. 
 
 In very low arches, this pressure becomes equal to the eighth 
 part of the weight of the whole arch, multiplied by the relation of 
 the span to the whole height of the arch, measured from the top 
 of the keystone. 
 
 In flat arches, the pressure on the keystone is just half the hori- 
 zontal thrust. 
 
 211. It will, therefore, be at once seen, that it is advantageous 
 to diminish the depth of the keystone as much as possible ; for 
 this will diminish both the pressure upon itself, and the horizon- 
 tal thrust. No more depth, therefore, should be given to the key- 
 stone, than is just sufficient to insure safety from the joint action 
 of the pressure upon it, and the agitation produced by extrinsic 
 causes. The value of the resistance to pressure in various kinds 
 of stone, has been given in 191. It will be proper to allow for 
 its diminution in the ratio of the squares of the depth, which in 
 this case will be the horizontal dimension of the keystone, and to 
 proportion the surface so as to bear three times the weight that 
 is capable of crushing it. 
 
 In arches of the form of a circular arc of but few degrees, the 
 horizontal thrust is very great; and even when a sufficient thick- 
 ness is given to the abutment, absolute safety is not obtained : for 
 the effort of the thrust may be sufficient to overcome the tenacity 
 of the cement, and separate the haunches of the arch from the abut- 
 ment. This risk is even greater in arches of wood or iron, rest- 
 ing upon stone abutments, and there are several instances on 
 record, of bridges of these materials having fallen as soon as the 
 centering was removed. 
 
 The first application of the formula (220), is to the determina- 
 tion of the position of the points of rupture. When the figure of 
 the arch is given, the resolution of this problem presents no great 
 difficulty in its principles. These are as follow : An arch ne- 
 cessarily breaks in its weakest part, which is that in which its re- 
 sistance has the least ratio to the forces that act upon it, and which 
 is not in consequence' necessarily the thinnest part of the vault. 
 The point of rupture will therefore be, where the moment of the 
 force that tends to overturn the lower part, is the greatest possi- 
 ble, when compared with the forces that tend to sustain it. 
 
206 EQUILIBRIUM OF [Book III. 
 
 This ratio is 
 
 FQ DQ 
 
 roKR-f-nKS ; (223) 
 
 when this is a maximum, its differential=0. 
 
 In circular arches, however, this simple principle is attended 
 with practical difficulties, in consequence of the transcendental 
 quantities which the nature of the circle introduces. In arches 
 formed of several circular arcs, the calculation becomes wholly 
 impossible. In place, then, of a direct mathematical investigation, 
 the position of the points of rupture, determined by the experi- 
 ments of which we have spoken, is assumed as the basis of the 
 calculation. The resulting thicknesses of the abutments, is con- 
 siderably less than is usually given in practice, and requires to 
 be increased, in order to allow for want of firmness in the founda- 
 tion ; and to augment the pressure on the plane, where the arch 
 meets the foundation, so much as by the friction to prevent sli- 
 ding, should the adhesion of the mortar be insufficient. A pow- 
 erful aid might be obtained in resisting the latter action, by uni- 
 ting the courses of stone by means of dowels. We give the thick- 
 ness of abutments, calculated by Gauthey, whose work we have 
 followed in our analysis, for arches of 60 French metres in span; 
 this will serve as a guide in similar inquiries. 
 
 Species of Arch. 
 
 Thickness of Abut- 
 ments. 
 
 Position of Points of 
 Rupture. 
 
 METKES. 
 
 Semicircular, 1. 32 
 Oval flattened id, 1. 62 
 do do ': th 2. 24 
 Circular arc of 60, 3.09 
 
 13 30' 
 
 31 30' 
 40 so' 
 00' 
 
 212. It has, generally speaking, been usual to give to the piers of 
 bridges the same thickness as to their abutments. But this is by 
 no means necessary ; for when two arches rest on the same pier, 
 their horizontal thrusts mutually counterbalance each other ; and 
 the pier has no other stress to bear than that of the superincum- 
 bent weight. It is frequently advantageous in practice to make 
 the piers as thin as possible ; as for instance, when a bridge cros- 
 ses a rapid stream, or one subject to sudden floods. The ancient 
 bridges, and those of England, generally speaking, have their 
 abutments equal to their piers; and this is absolutely necessary 
 when the arches are built in succession. But when all the arches 
 are carried up simultaneously, and their centres struck at the same 
 time, it becomes practicable to give dimensions to the piers suited 
 merely to the stress they are afterwards to sustain. Such is the 
 present practice of the French engineers. 
 
Book III.} ARTIFICIAL STRUCTURES. 207 
 
 213. Arches belong as a principal feature to no architecture 
 more early than that of the Romans. Belzoni indeed states, that 
 he found at Thebes, the relics of arches of brick that seemed to- 
 be of a date prior to the Persian conquest. But even were the 
 arch then known, it was but little used. The oldest arch in ex- 
 istence is that of the Cloaca Maxima at Rome, the architects of 
 which were Etrurians; and in the works of the Romans, we find 
 arches superior in magnitude to any that have hitherto been con- 
 structed, with but one exception, that we shall presently state. 
 The arches of the bridge of Trajan, over the Danube, were 
 semicircular, raised on lofty piers and abutments, and had a span 
 of ISO feet. 
 
 The bridge of Vieillebrioude, in France, approaches more nearly 
 to the bridge of Trajan in dimensions, than any other modern 
 bridge; it was built in 1454, and has a span of 178 feet. And as 
 the bridge of Trajan has long since fallen, it was until within a 
 year, the largest arch in existence. 
 
 The bridges of Gignac and Lavaur, in France, have each arches 
 of 160 feet. 
 
 * The arches of the bridge of Neuilly, and the centre arch of the 
 bridge of Mantes, have each 127 feet span. 
 
 The great arch of the bridge of Verona, in Italy, has 160 feet. 
 The marble bridge at Florence, built by Michael Angelo, has 
 138 feet. 
 
 The span of the centre arch of Waterloo bridge, in London, is 
 120 feet, and of that of the new London bridge, 140 feet. 
 
 At the present moment a bridge is constructing over the Dee, 
 at Chester, in England, whose span is 200 feet. Should this stand 
 after the centre is removed, it will be the greatest stone arch of 
 ancient or modern times. 
 
 In consequence of the abundance and excellence of other mate- 
 rials in the United States, our stone bridges are neither numerous 
 nor important. The most beautiful specimen of this species of 
 architecture that we possess, is the acqueduct bridge at the Little 
 Falls of the Mohawk. 
 
 Equilibrium of Domes. 
 
 214. The same principles are applicable to domes or spherical 
 vaults ; and if the curve, in the figure on p. 201, that represents 
 a section of an arch, be now assumed for the section of a dome, the 
 circumstances that take place in its fracture will also be represented, 
 it having points ot fracture, and dividing into four parts. But in 
 arches which are cylindrical, or vaults that are cylindroidal, the 
 rupture takes place in the horizontal plane passing through the 
 
203 EQUILIBRIUM or [Hook III. 
 
 points D and d t while in a dome, the line of rupture is a circle. In 
 domes, the weight of the upper portions will be less than in arches, 
 the points of rupture will lie higher, the horizontal thrust, and the 
 pressure on the keystone will also be less. Domes, therefore, will 
 be borne by less abutments than arches ; but as domes are, gene- 
 rally speaking, raised upon a lofty wall, instead of resting upon an 
 abutment, it is inconvenient to give this wall a great thickness; 
 and hence artificial means, that will be presently mentioned, are 
 resorted to, in order to give the requisite resistance. 
 
 215. Domes are of more easy construction than arches, for each 
 course keys itself, and they may even be erected without any 
 permanent centre ; while an arch must be finished, and the key- 
 stone placed, before it supports itself; each course in a dome serves 
 as a keystone to those that are beneath, and apertures of large di- 
 mensions may therefore be left in the middle of domes. These 
 serve for the admission of light, or may be made the base of other 
 more elevated parts of the structure. 
 
 Domes derive much of their beauty from a geometrical property 
 they possess, which is as follows: the common intersection ofa 
 sphere, and a cylinder whose axis is directed to the centre of the 
 sphere, is a circle, and of course a plane curve. Hence, if any num- 
 ber of arches be arranged on the sides of a regular polygon, a dome 
 may be built resting upon them ; and a cylindrical tower may be 
 built upon the opening in the centre of a dome, and may in its 
 turn become the support of a second dome. 
 
 216. Domes had their origin among the Etruscans, whose tem- 
 ples were circular in plan, and covered with a simple hemispheric 
 vault. The Romans borrowed this species of structure from that 
 neighbouringnation,and brought it to great perfection. The finest 
 antique specimen that remains of this species of building, is that 
 which goes by the name of the Pantheon. This building has a 
 circular ground plan, on which is raised a lofty cylindric wall that 
 bears a hemispheric dome, 144 feel in diameter. As the walls have 
 not of themselves sufficient stability to support with certainty the 
 lateral thrust, they are loaded at ihe spring of the arch by ma- 
 sonry, accumulated in the following mode: The generating curve 
 of the inner surface, or intrados of the dome, is a semicircle, the 
 
Book IIL] ARTIFICIAL STRUCTURES. 209 
 
 extrados is a less portion of a greater circle, as represented in the 
 
 n 
 
 B 
 
 figure ; hence it becomes possible to raise the external vertical 
 faces AB, of the wall, much higher than the internal CD; and a 
 great weight rests upon the spring of the arch, while the lower 
 portions of the arch itself are also strengthened. 
 
 The domes of a number of Christian churches were built at 
 the intersection of the aisles that form a cross ; they were hence 
 borne upon four arches, to which they were applied upon the 
 principle stated in the last section. 
 
 To give greater elevation, the first dome was, in the progress 
 of architecture, terminated immediately above the keystones of 
 the arches, and a second dome raised upon the circular ring that 
 constituted the opening. A bolder architect proposed to raise a 
 cylindric wall upon this ring, and support upon it the second 
 dome. Thus was gradually reached the sublime conception of 
 the dome of St. Peter's. In realizing this conception, various 
 practical difficulties presented themselves. The principle ap- 
 plied in the Pantheon was inapplicable, in consequence of the 
 great mass of material it would require, that might have increased 
 the pressure beyond that which the abutments of the supporting 
 arches could bear. A flattened external dome would have been 
 invisible, except from a distance, and wholly deficient in beauty. 
 For these reasons, the dome that was seen from within, was made 
 of the smallest practicable, but of uniform thickness, and the part 
 seen from without, the half of an oblong spheroid. In the dome of 
 St. Peter's, these two domes are both of masonry, and spring from 
 a common base, as in the figure, diverging as soon as the outer 
 
 and innner curve of each, intersect 
 each other. In the case of St. Paul's, 
 in London, the inner dome is of brick, 
 the outer a wooden frame, bearing a co- 
 vering of sheet lead. To support the 
 frame that bears the outer dome, a trun- 
 cated cone of brick rises from the inner 
 dome, and bears the smaller cupola or 
 
 27 
 
210 EQUILIBRIUM 0? \BuokUL 
 
 lantern, which in St. Peter's is borne 
 on the ring that terminates the outer 
 dome. A section of the dome of St. 
 Paul's is represented in the annexed 
 figure. 
 
 Although in these different modes, 
 lightness and beauty were both 
 gained, the resistance to the hori- 
 zontal thrust is so much diminished, 
 that the domes could not have been supported by the balancing 
 of their parts, or by the cohesion of cement; to remedy this de- 
 fect, the lower courses of the dome of St. Peter's are bound by 
 strong hoops of wrought iron ; and at St. Paul's, chains are laid 
 in a groove, cut in the stone ring whence the dome springs, and 
 secured by melted lead. 
 
 Domes and groined arches are formed in Gothic architecture, 
 at the junction of vaults formed by the intersection of two 
 circular arcs. The lateral thrust is, in these cases, met by but- 
 tresses formed on the principles of 198, and the resistance to it 
 is further increased, by loading the buttresses with heavy masses 
 of stone, assuming the form of pinnacles. These, which form 
 one of the chief embellishments of Gothic architecture, are beau- 
 tiful, not only from their graceful figures, but from their evident 
 adaptation to an important purpose. 
 
 Of Wooden Arches. 
 
 217. Wood may be applied to the construction of arches also, 
 or to the formation of frames that may answer as a substitute for 
 arches. The application of the principles employed in the pre- 
 ceding chapter, to this material, is, however, different from that 
 which is adapted to the theory of stone arches. While in stone 
 arches the mass is made up of separate parts, which divide in par- 
 ticular points, and move around others as if they had no cohesion, 
 so that the respective strength may be considered as nothing, 
 and the whole available resistances are, that of friction, and that 
 which the material opposes to a crushing force ; the use of wood 
 brings into efficient action its respective strength, and the resist- 
 ance to separation may, in some cases, become that furnished by 
 the absolute strength. The length that can be safely given to a 
 single beam, supported or fixed at each end, and lying in a hori- 
 zontal position, is limited by its own weight, as has been seen in 
 186. 
 
 218. If the force that acts be greater than can be borne by a 
 single beam, two may be united in such a manner as to act like 
 
Book Hl t ] ARTIFICIAL STRUCTURES. 211 
 
 a single piece. Thus, if a beam be merely laid upon another, as 
 
 T> 
 
 AB upon CD, each will resist the effort to bend it, which must 
 precede fracture, merely with its own force; the system is there- 
 fore only twice as strong, in respect to the sum of the forces that act, 
 as the single beam. If then, thesingle beam have reached thelimitat 
 which it breaks by its own weight, they will both be broken. 
 If now the two beams be united, as may be simply done by dow- 
 els or pins, in such a way that one cannot bend without, causing 
 an equal bending in the other, the two will act as a single beam, 
 and the strength will be four times as great (141), as that of the 
 single beam, or twice as great as the united strength of the two 
 acting separately ; hence, the two thus united, will now not only 
 bear their own weight, but require an additional weight equal to 
 their own, to break them. Two beams may be still more ad- 
 vantageously united by cutting their adjacent surfaces into the 
 form of the teeth of a saw, turned in opposite directions, as in the 
 figure. If these be united by screw bolts, or .by straps of metal, 
 
 both must bend together, and hence act to resist fracture like a 
 single beam. This method is called Trussing. 
 
 219. When the limit of strength is reached, either in a single 
 beam or in this arrangement, two beams may next be placed in 
 an inclined position, pressing against each other, as in the figure. 
 
 The action of the weight being now oblique to the direction of 
 the beam, the effort of the weight will decrease with the cosine 
 of the angle of inclination, 184. 
 
 Let F be the strength of the compound beam, and W the 
 weight just sufficient to break it. 
 
 F=Wcos.z. 
 
 A lateral thrust will also take place at the points A and B, which 
 may be represented by 
 
 Wcot. . (224) 
 
212 EQUILIBRIUM 07 [Book III 
 
 220. If instead of two beams inclined at an angle, a piece of 
 
 -A. B 
 
 small curvature be substituted, the effort of the weight will be 
 diminished, 134, in the ratio of the cosine of the angle CAB j 
 and the resistance of the wood to flexure, as well as its strength, 
 will be enhanced even in a higher ratio, in consequence of the 
 difficulty of bending a curved piece in the direction opposed to 
 its curvature. The latter advantage is of course gained only in 
 the case of the fibres of the wood being also curved ; for, if the 
 fibres be cut across, the strength will be diminished ; because the 
 lateral cohesion of the wood, which (see 180) is far less than 
 its respective strength, is now the only resistance that remains. 
 
 Whenever the line CA does not cut the intrados, we may with- 
 out error consider the half of the arch as a straight piece of 
 equal dimensions, loaded at one end by a weight acting vertically, 
 and standing itself upright. This weight will be equal to so 
 much of the force as acts in the direction AC., and the condition 
 of equilibrium, between this component of the whole of the dis- 
 turbing forces that act, and the strength of the arc will be given 
 by the formula (199). 
 
 in which I is the length of the arc CA. 
 
 We have in the preceding chapter omitted the question of the 
 elasticity of wood, and may, in this case, substitute for that pro- 
 perty the respective strength. Using this, it will be obvious that 
 there are three different methods of valuing E, according as the 
 pieces that form an arch, when there are more than one, are com- 
 bined : when they all act distinctly ; when they are merely com- 
 bined by the vertical posts of the arch, or by other pieces crossing 
 them ; and when they are so united as to form a body, no part of 
 which can move without affecting all the rest. 
 
 In the first case, supposing each of the pieces to have a rectan- 
 gular section, whose breadth is o, and depth 6 ; let 10 be the num- 
 ber from the table of relative strength, n the number of pieces. 
 
 E=na6 2 M>; (225) 
 
 In the last case, 
 
 E=n 2 a6 2 tp; (226) 
 
 and in the second case it will be intermediate. 
 
 It is, however, hardly possible to unite beams in such a way as 
 
Book III.] ARTIFICIAL STRUCTURES. 213 
 
 to give them the entire strength determined by the last formula. 
 And in any case whatever, a large allowance ought to be made 
 after calculating the value of E. 
 
 If W be a weight resting upon the vertex of the arch, 
 
 Q=W cos. i. 
 
 But if the weight be uniformly distributed over the arch, 
 Q=iW cos. i. 
 
 221. If, however, the curvature be considerable, we can no 
 longer consider the wooden arc as formed of two straight pieces, 
 but must have recourse to principles in some respects similar to 
 those of stone arches. In the arc ACB, a force acting at C, will 
 
 not break the beam at that point, 
 but at two points, D and d, inter- 
 mediate between it and the two 
 points of support; and the lower 
 parts, AD and Bc?,will turn around 
 A and B, as in the case of a stone 
 arch, (see 209). The point of 
 rupture is easily determined in 
 this case ; for, the respective 
 strength of a -beam of equal thick- 
 ness, being uniform throughout, if the momentum of the stress 
 be abstracted, whether the beam be straight or crooked, the rup- 
 ture must occur where the effortof the weight acts most directly 
 upon the beam. This point will be determined by letting fall a 
 perpendicular from the centre of the arc, c, upon the line CA. 
 
 Having thus determined the point of rupture, we may proceed 
 to determine the conditions of equilibrium between the force P 
 that acts in the direction CA, and the resistance at the point D. 
 This may be done by conceiving that the part DA is immovea- 
 ble, and that the part DC being firmly fixed at D, is acted 
 upon, by a force that tends to bend it, in the direction C A. This 
 will not affect the condition of equilibrium, and it becomes an 
 application of the formula, (192). 
 
 , .. 
 
 In which / is the length of the arc CA, and /the versed sine of 
 the curvature which the force P is capable of producing. From 
 this we obtain for the value of P 
 3E/ 
 
 '"~T' (227) 
 
 But / is a function of P, and the value of the latter is still in- 
 volved in the expression, it is, therefore, necessary to have the 
 means of determining /. This can always be safely done by 
 
EQUILIBRIUM OF [Book II L 
 
 taking as the value of/, the maximum flexure, or that which pre- 
 cedes rupture. This deduced from experiment, is 
 
 222. In applying these principles to the construction of bridges, 
 two different methods have been pursued. 
 
 (1.) A continuous arc has been formed by bending plank, ar- 
 ranged so that none of their joints should be opposite, and united 
 by bolts and iron straps, in such a manner as to act as a single 
 piece. Such is the principle of the bridges erected in Europe 
 under the direction of Wiebeking. Of these the bridge of Bam- 
 berg is the most remarkable. Its span is 221 feet. 
 
 In this bridge, the road passes over the summit of the arch, 
 which is therefore flat, and has a great lateral thrust. If this be 
 not carefully opposed by a proper connexion with the abutments, 
 and by giving them a sufficient weight, the bridge may be de- 
 stroyed by it. This was the case in two bridges erected on a 
 similar principle at Paterson, New-Jersey. 
 
 It is, however, by no means necessary, except where it is de- 
 sired to leave room for the passage of vessels, to make the road 
 pass over the vertex of the arch. The nature of the material, 
 which is both light and strong, admits of the arch being formed 
 of several separate ribs. Carriages and passengers may pass be- 
 tween these upon a horizontal road, resting on' timbers, support- 
 ed by-thearch from above. If the timbers, supported by each 
 rib, be made to form a continuous chord, and be connected with 
 the rib at the spring of the arch, they will, in addition, oppose 
 their absolute strength to the horizontal thrust, which may thus 
 be entirely done away. The ribs, too, may be made with a much 
 greater curvature, and will be both stronger in consequence, and 
 have less horizontal thrust. 
 
 Such is the principle of the very beautiful bridge erected by 
 Burr, over the Delaware at Trenton, N. J., the larger arches of 
 which have 194 feet span. 
 
 (2.) An arch of timber may be formed of pieces arranged in the 
 form of a polygon; such an arch would be in equilibrio, had it 
 the form of the funicular polygon, 29 ; but as the equilibrium 
 would be tottering, it is better to make the system rigid, in which 
 case, it is unnecessary to seek or observe the law of equilibrium. 
 The system rnay be made rigid by extending some of the timbers 
 beyond the points where they intersect the others, framing them 
 together, and connecting them again with others, forming trian- 
 gular frames, which cannot alter their shape without breaking. 
 Such is the principle of the wooden bridges of Hampton, and 
 Cambridge, in England. The largest of these, however, has less 
 
///.] ARTIFICIAL STRUCTURES. 215 
 
 than 50 feel span. This method is extremely faulty, and no 
 bridge of any great span constructed after it, has been of long du- 
 ration. A better mode of making the system rigid, is, to make 
 it double, and interpose, between the arch pieces, queen-posts, 
 AB, A'B' ? &e. forming by means of mortices and tenants, a series 
 
 of open voussoirs, or quadrilateral frames. To prevent these 
 from changing their figure, a frame of the figure of a St. Andrew's 
 cross is placed in each. This principle was adopted in the bridge 
 erected over the Piscataqua, near Portsmouth, New-Hampshire, 
 whose span is 256 feet. 
 
 (3.) The two methods may be combined; of this we have an 
 instance in the great arches constructed by Wernwag, one over 
 the Schuylkill, near Philadelphia, the other over the Delaware 
 at Easton. The former has 340 feet span, and is the largest 
 wooden arch now in existence. A project for an arch upon this 
 principle, of 400 feet span, is given by Tredgold, in his work on 
 the principles of carpentry. 
 
 223. In all the methods of extending wooden structure across 
 openings, of which we have hitherto spoken, the principle of the 
 arch has been taken as the leading and prominent feature. Far 
 more simple considerations have led to the construction of wooden 
 bridges, of greater span than any we have hitherto cited. 
 
 The simplest mode of spanning an opening is a beam ; but we 
 have seen that this has an early limit in the size of timber, which 
 cannot be obtained of sufficient depth to enable a long beam to 
 bear its own weight. It might, however, occur, that as it is pos- 
 sible by trussing, as explained in 218, to obtain beams of great- 
 er strength than single trees will afford, so it would be practicable 
 to build a structure which should act upon the principle of a beam. 
 The first thing that ought to be determined, for this purpose, is 
 the figure that would have, under equal size, the greatest degree 
 of strength; such is one that would have the moment of its 
 strength equal in every part of its length. 
 
 The beam having a rectangular section, whose constant breadth 
 is a, and variable depth v, the strength of any section, if supported 
 at both ends, will be (152) 
 
EQUILIBRIUM OF [Book III. 
 
 the resistance to flexure will be (203) 
 E 
 
 P 
 
 If we call the force that acts to bend it, P, we shall have for 
 the condition of equilibrium 
 
 Px E 
 
 -=-= 2ot? 3 . (228) 
 
 2 p 
 
 If the weight be uniformly distributed, we shall have for the 
 value of P, in terms of the weight w, borne by the unit of the 
 beams' length, 
 
 and the equation (228) becomes 
 E 
 
 whence 
 
 x ow 
 
 r<S (229) 
 
 which is the equation of a straight line, making, with the horizon, 
 
 pw 
 an angle, whose tangent is -/(p ) And as the circumstances 
 
 are similar on each side of the middle of the beam, ^the solid 
 of greatest strength is an isosceles triangle. We cannot, how- 
 ever, diminish the thickness of the beam at its points of support 
 to 0, and hence the figure becomes a pentagon, as represented 
 beneath, two of whose sides are parallel and equal, and two of 
 whose angles are right angles. 
 
 If we examine the action of the weight" to^cause this beam to 
 bend, we shall see that the fibres nearest the upper part would 
 be compressed, and those nearest the lower would be lengthened, 
 and thus a line acb might be drawn, which would separate the 
 extended from the contracted fibres. Such a line is called the 
 Neutral Axis. In arranging the pieces of wood of which the 
 bridge is composed, the best method will obviously be, that which 
 shall bring their absolute strength most nearly into direct oppo- 
 sition, to the contractions and expansions, which the beam, if of 
 a single piece, would undergo in bending, but shall which give them 
 
Book ///.] 
 
 ARTIFICIAL STRUCTURES. 
 
 217 
 
 a firm bearing on the abutments. These pieces may then be 
 united by vertical posts, which will form the whole into a series 
 of open triangular frame work ; to these posts may be attached 
 a horizontal trussed beam, which will answer to bear the road. 
 Such a plan of frame work is represented beneath, and it will 
 be obvious, that the horizontal beam will destroy the lateral 
 thrust. 
 
 This is the principle that was adopted by a Swiss carpenter of 
 the name of Grubenman, in the construction of the bridges of 
 Schaff hausen and Wettingen. The first of these had a span of 
 365 feet, and appeared to be divided into two spans, resting on 
 an intermediate pier. But the use of this support was in 
 opposition to the desires of the architect, and he had the skill to 
 leave it questionable whether the bridge derived any strength 
 from it or not. 
 
 The bridge of Wettingen had a span of 384 feet. Both of 
 these remarkable bridges were of sufficient strength to bear any 
 probable load. Both were unfortunately destroyed during the 
 campaign of 1799, and neither have been replaced. 
 
 When from the position of the bridge, the road must pass over 
 its summit, the beam beneath may be suppressed, and the system 
 takes the form represented in the figure, which is that of the 
 
 bridge of Kandel, in the canton of Berne, constructed by Rit- 
 ter. In this there is a horizontal thrust, which must be counter- 
 acted by the resistance of the piers. 
 
 A modification of the same form, proposed by Gauthey, is re- 
 presented beneath. 
 
 It may be objected to the bridges of Grubenman, that on many 
 
 28 
 
218 
 
 EQUILIJ1KIUAI OF 
 
 [Book III. 
 
 parts of them the moments of the forces that tend to change the 
 figure become very greatj in consequence of the great length of 
 the arms on which they act. On this account, a frame of this 
 description has, in many cases, been combined with a bent arc 
 of plank, like that of the bridges of Wiebeking and Burr. 
 When this is introduced, the resistance it opposes to bending 
 seems sufficient, and no other stress need be guarded against ex- 
 cept that which tends to absolute fracture. In some of the 
 bridges proposed by Gauthey, therefore, the long pieces that are 
 extended to prevent the former action, are suppressed, and the 
 arch takes the following form: 
 
 A still better arrangement, founded on the same principle, is 
 to be found in the bridge of Wernwag, that crosses the Delaware, 
 at New Hope. The principle of this is exhibited beneath. 
 
 M 
 
 
 _ 
 
 An ingenious application of the principle of the beam, has 
 been made by an American engi-neer, ElhicI Town. His bridge 
 is composed of a rectangular frame of timber, connected by di- 
 agonal braces. 
 
 This bridge will have great strength to resist fracture, but little 
 to resist flexure, in consequence of the length of the horizontal 
 beams, and the mobility at the angles of the braces. It is, how- 
 ever, lighter than any other plan that has ever been proposed, 
 and might, by a few obvious improvements, be made capable of 
 spanning great openings. 
 
 Of Cast Iron Arches. 
 
 224. Cast iron bridges may be erected either of continuous 
 ribs and bands, or by forming the material into skeleton vous- 
 soirs. In the first case, their theory is the same as that of wood- 
 en arches, substituting the resistance of cast iron for that of wood ; 
 in the second case, their theory is similar to that of stone arches, 
 
ARTIFICIAL STRUCTURES. 219 
 
 substituting the value of the strength of the one material for the 
 other. Stone, a material of small respective strength, and more 
 than double the weight of wood, requires that the intrados should 
 have a continuous surface, and the space between it and the ex- 
 trados is often of necessity filled up. Wood, a material of great- 
 er respective strength, of less resistance to a crushing force, and 
 less weight, may be arranged in separate and similar ribs, and a 
 lightness in the arch, far more than proportioned to the different 
 densities of the substances, being thus attained, spans of far great- 
 er extent may be compassed than by stone. Cast iron holds an 
 intermediate rank between these two materials : being possessed 
 of more strength than either, it has three times the density of 
 stone; but it may, like wood, even if formed into voussoirs, bear- 
 ranged in separate ribs ; the voussoirs, too, need no more ma- 
 terial than will form a flaunch around their periphery, and furnish 
 supporting braces within. Still, iron will form an arch much 
 heavier than wood ; and the limit at which it is crushed by its 
 own weight is, therefore, sooner reached. 
 
 Of the two methods which have been mentioned, that of con- 
 tinuous ribs is objectionable, because cast iron, froru its crystalline 
 structure, has no great tenacity ; hence voussoirs are better, 
 by means of which the resistance to crushing is substituted for 
 the respective strength. 
 
 225. After the full manner in which the theories of stone and 
 wooden arches have been discussed, it is unnecessary to enter in- 
 to a repetition of the principles, even for the purpose of applying 
 them to another material. We shall, therefore, be content with 
 giving a list of the principal arches of cast iron that have hither- 
 to been constructed. 
 
 The first instance of an iron bridge is that of Coalbrookdale 
 in England. Its span is 100 feet. The main rib is composed of 
 no more than two pieces, meeting at the vertex of the arch. It 
 is, therefore, an instance of the first kind of cast iron arch. It 
 was erected about the year 1776. The second cast iron bridge 
 is at Build was, near Coalbrookdale. Its span is 131 feet. The 
 date of its erection is 1795. 
 
 One at Sunderland, in England, over the Wear, erected in 
 1796, has a span of 209 feet. It is the first instance of the for- 
 mation of cast iron into voussoirs. 
 
 The bridge of Austerlitz, at Paris, has five arches of 106 feet 
 each, and is composed of skeleton voussoirs, very scientifically 
 arranged. 
 
 Of Chain Bridges. 
 
 226. Bridges may be supported by means of chains, or ropes. 
 
220 EQUILIBRIUM OP [Book HI. 
 
 In the earlier forms, planks were laid directly upon the the flexi- 
 ble material, which was stretched between two firm supports ; the 
 chain or rope was, for the convenience of passage, brought into a 
 position as nearly horizontal as possible. This method is, however, 
 inconvenient in practice, and diminishes the resistance of the ma- 
 terial ; for it will be seen by reference to 27, that, considering 
 the bridge as a funicular polygon of an infinite number of sides, 
 the effort of a weight acting in a vertical direction to break it, will 
 be much increased by diminishing the curvature ; and, after all, 
 the road could never cease to have an inconvenient slope. 
 
 A far better application of the principle was carried into effect 
 about the year 1796, by Mr. James Findley of Pennsylvania. In- 
 stead of attempting to render the road nearly level by the tension 
 of the chains, he made them of such length that the versed sine 
 should be not less than one seventh of the span. The road was 
 suspended from the chain, and might therefore be rendered per- 
 fectly horizontal. Chains, iron rods, or beams of wood may be 
 used to effect the suspension. Forty bridges of this description 
 were erected iu the United States previous to the year 1808. As 
 wrought iron is the material that has the greatest absolute 
 strength; as the chains by which the road is suspended may be 
 multiplied, and the longitudinal beams having thus many points of 
 support, need not be very thick; and as wood is the lightest ma- 
 terial of which a road can be constructed, it is susceptible of de- 
 monstration, that an arch of greater span may be constructed upon 
 the principle of Mr. Findley, than in any of the modes that we 
 have yet spoken of, or indeed in any other manner that has yet 
 been proposed. It is only wonderful that chain bridges have not 
 yet come into more general use, for there are innumerable cases 
 in which they possess greater advantages than those of any other 
 material. 
 
 It was not until 1814, that Findley's method attracted 
 the attention of European engineers. At that date it was 
 in contemplation to shorten the post road from London to Liver- 
 pool, by effecting a passage across the Mersey at Runcorn, a 
 position in which no other material would have been applicable ; 
 for the locality required an arch with a span of 1000 feet. Tel- 
 ford, therefore, proposed a bridge composed of a timber road 
 suspended by chains, identical in principle and form with those 
 of Findley; and having instituted a series of interesting and useful 
 experiments on the strength of wrought iron, he proved beyond 
 all question the practicability of the project. It has not however 
 been carried into effect. 
 
 In 1S18, a bridge formed of iron wire, bearing a path for foot 
 passengers, which had been erected the year before by the Earl 
 of Buchan, across the Tweed at Dry burgh, was carried away, 
 
///.] ARTIFICIAL STRUCTURES. 221 
 
 and replaced immediately by a chain bridge, which was the first 
 erected in Europe on.Findley's principle. About the same period, 
 Telford presented a project for one of similar character, across 
 the straits of the Menai, which has since been executed. 
 
 The latter is, in point of extent of span, the most remarkable 
 bridge in existence. The distance between the centre of the piers 
 is 560 feet, arid the road is elevated 100 feet above the level of the 
 highest tides. The height of the supports above the level of the 
 road, which height corresponds with the versed sine of the arc, 
 was intended at first to have been no more than 35 feet, but has 
 been increased to 50 feet. 
 
 Before the completion of the bridge over the Menai, Captain 
 Brown, so well known as the introducer of chain cables for ships, 
 completed the construction of a chain bridge over the Tweed, 
 near Berwick : this was opened to the public in 1820, and was 
 the first executed in Europe, which was adapted for the convey- 
 ance of loaded carriages. Since that period, numerous other 
 bridges, and several piers for the reception of vessels have been 
 constructed in Great Britain, and the method has been successfully 
 introduced into France. 
 
 227. A bridge formed of chains with a road suspended from 
 it, is in the condition of the funicular polygon, 29. Its theory 
 may, however, be more easily reduced to calculation Tor practi- 
 cal purposes, by considering it as a catenaria, and the inequality 
 in the distribution of the weight is too small to cause any error 
 in practice, from assuming it to be loaded at every point with an 
 equal weight. 
 
 If we take one of the points of suspension of the chain, for the 
 origin of the co-ordinates, x and y ; and if a be the angle the 
 curve makes with its tangent at that point ; w the weight borne 
 by each unit of the length of the chain, assumed to be constant ; 
 and T the tension of the chain at the point of suspension, we have 
 by the theory of the catenaria, 28, for the equation of the curve 
 
 A cos. a ( A wy \/[(A wy} 2 A 2 cos. 2 a] ) 
 
 ~^~ g * \ ~ A(l sin. a) ~~ ; I 
 
 for the versed sine/, equal to the height of the point of suspension 
 above the level of the horizontal tangent of the curve, 
 
 for the half span, = \ s, 
 
 A cos. a i cos. a \ 
 
 i s = - log. (- - . -- ) , (446 
 
 w \l sin. a / 
 
 for half the length of chain, = * /, 
 
EQUILIBRIUM OP [Book 111. 
 
 A sin. a 
 
 * '= i- ; (<) 
 
 and for the relation between /and /, 
 
 ,1 cos. a 
 
 f=l - - 
 sin. a 
 
 from (44a) and (44c), we obtain for the values of A, 
 
 id wf 
 
 A = - - = - - ^ -- . (230) 
 
 2 sin. a 1 cos. a 
 
 The direct determination of the value of the angle a, from the 
 properties of the catenaria, is not easy ; but when the versed sine 
 does not exceed T ^ th of the span, the curve does not differ sensi- 
 bly from a circle, that would have the same lines for its tangents 
 at the points of suspension : from the properties of the circle, the 
 value of the tangent of the angle a, is found in terms of its radius 
 R, and chord d, to be as follows, v:z : 
 
 d 
 
 ~ 
 
 and the value of the radius may, from the properties of the same 
 curve, be found by the subsidiary formulae, 
 
 ' 
 
 These formulae and principles, will, generally speaking, suffice 
 for the calculation of any of the cases that can occur in practice. 
 But fora more full discussion of the theory in which all the cir- 
 cumstances are taken into account, we refer to the work of 
 Navier, Rapport et Memoire sur les Ponts suspendus, Paris, 
 1823. 
 
 228. From this we quote the following practical rules that are 
 immediate inferences from his investigations. In increasing the 
 span of chain bridges, there is no reason to fear that the changes 
 of figure, growing out of the action of passing loads, will be aug- 
 mented. These changes may even he rendered less sensible, in 
 bridges of Widepan, by making the versed sine of the curve bear 
 a less proportion to the extpnt of span. In fact, as the ratio of 
 the versed sine to the span diminishes, the curve of the bridge 
 will approach more and more near to a straight line, and at this 
 limit the figure of the chains is invariable, whatever be the man- 
 ner in which the weight is distributed, provided they be, as the 
 hypothesis assumes, incxtensible. On the other hand, by dimi- 
 nishing the versed sine, the constant tension, that the weight of 
 the road exercises on the chains, is increased also, as well as the 
 variable tension growing out of accidental loads ; and both of these 
 tensions would become infinitely great, if the chain were stretched 
 in a straight line. 
 
Book III.'] ARTIFICIAL STRUCTURES. 223 
 
 229. In calculating the dimensions of the iron chains which 
 compose the inverted arch, as well as those by which the road is 
 suspended from them, the absolute strength, 179, should be 
 made at least twice as great as the greatest probable tension ; for 
 an iron rod will, at a limit of strain, a little greater than the 
 half of its absolute strength begin to stretch; and its elasticity will 
 be so much impaired by the strain, that it will not restore itself 
 to its original dimensions. 
 
 230. Iron is subject to fracture on sudden changes cf tempera- 
 ture. This appears to arise from unequal expansion ; the outer parts 
 being sooner affected, expand or contract earlier than those within. 
 To prevent any danger from this cause, heavy loads should not 
 be permitted to pass, for some hours after any great and sudden 
 change in the temperature of the air shall have occurred. As 
 changes sufficient to cause danger are by no means frequent, such 
 precaution cannot be productive of any important inconvenience. 
 
 231. Bridges formed of chains are liable to two species of os- 
 cillations : the one vertical, growing out of the passage of weights ; 
 the other horizontal, arising from the action of the wind, or other 
 external disturbing forces. In respect to the' former, they are 
 more likely to produce injury in small than in large arches, for 
 both the extent and velocity of the vibrations decrease with the 
 increase of the span ; the extent of this kind of oscillation follows 
 the inverse ratio of the squares of the chord of the curve ; while 
 the velocity decreases with the length itself. 
 
 232. Had the chains no weight to support besides their own, 
 they would be readily caused to oscillate in a horizontal direction, 
 and would follow in their vibrations the law of pendulums, the 
 time of performing them being independent of the intensity of 
 the disturbing force. The mathematical investigation shows, 
 that the amplitude of these oscillations would diminish in a ratio 
 more rapid than that in which the length of the chains increases. 
 
 The chains, however, are connected with the timber road in 
 such a manner, that they cannot oscillate in a horizontal direction 
 without causing it to change its figure, either horizontally .or 
 vertically. As it is easy, by a proper system of framing, to ren- 
 der the road almost inflexible in a horizontal direction, these os- 
 cillations can be at most but small, unless the disturbing force be- 
 come of sufficient intensity to cause the rupture of the wood 
 work. So long, then, as the tendency to oscillate is but small, 
 it may be performed without meeting with much resistance ; but 
 so soon as it begins to increase, it is opposed by the whole ri- 
 gidity and weight of the timber road. 
 
224 EQUILIBRIUM OF [ Hook III. 
 
 233. The strength of wood, when drawn in the direction of 
 its fibres, being very great, amounting, ISO, to about one-eighth 
 part of the strength of wrought iron, the former material may be 
 used to suspend the road, from the principal chains, to great ad- 
 vantage, particularly as regards economy of construction. 
 
 234. The expense of the chains is a minimum, when the versed 
 sine of the catenaria has to its chord the relation of 1 : 2\/2 ; but 
 this is a proportion that is rarely admissible in practice. 
 
 235. The establishment of chain bridges offers a great variety 
 of different cases, that may comport with arrangements more or 
 less simple in their structure, thus : 
 
 (1.) The bridge may cross a ravine situated in a passage en- 
 closed between rocks, that are high enough to afford firm and 
 fixed points for fastening the chains ; in such a case, and particu- 
 larly when the breadth of the gorge exceeds 4 or 500 feet, a chain 
 bridge may not only be the most economical, but often the only 
 practicable method of passage. 
 
 (2.) The space to be traversed by the arch, may offer firm 
 supports for the chains, at a sufficient height on one side only. 
 In this case the curve may advantageously have the form of a 
 half catenaria, being attached at a proper height to the loft}' bank, 
 and having for its tangent at the opposite bank, a horizontal line. 
 
 (3.) When both banks are low, the chains must be attached to 
 artificial supports. These are sometimes masses of masonry ; at 
 others, frames of cast iron; and if they are not themselves suffi- 
 ciently solid to sustain the tension, they must be reinforced by 
 chains extending from them, in directions opposite to that in which 
 those which support the bridge are suspended, and which must be 
 firmly fastened to the ground. 
 
 (4.) This last mode may, in some cases, be modified to advan- 
 tage, by advancing the piers, or artificial supports, into the space 
 that is to be traversed. In this case, the chains that strengthen 
 the piers are made to bear a portion of the bridge, and the whole 
 may assume the arrangement figured beneath, of one whole and 
 
 two half arches. This was the plan proposed by Telford, for 
 the bridge at Runcorn. 
 
 (5.) A single pier may, in some cases, be erected in the mid- 
 dle of the space, and the bridge take the form of two half cate- 
 

 Book III.'] ARTIFICIAL STRUCTURES. 225 
 
 nariae, as represented in the figure. A bridge of this shape was 
 
 constructed in England, under the direction of Brunei, to be 
 erected in the Island of Bourbon. 
 
 When the extent of the proposed bridge is great, various com- 
 binations of whole and half arches may be formed, according to 
 the local circumstances. 
 
 The theory would show that, while the height of the points 
 that support the chain may be increased indefinitely, there is 
 no practical limit to the extent of the span, except that at which 
 the chains would, if suspended vertically, break by their own 
 weight; for additional strength may be gained by increasing the 
 proportionate magnitude of the versed sine of the curve. This 
 investigation is however of little value in practice, for besides the 
 immense expense of artificial supports of great height, the oscil- 
 lations are rendered, as has been seen, more intense by such an 
 increase. For this last reason, the proportion originally em- 
 ployed by Findley of |, has been reduced to ^ in most instances. 
 
 If the proportion between the span and the versed sine is con- 
 stant, the length of the former has a theoretic limit. This has 
 been calculated by Navier, under the assumption that the relation 
 is y'j, and found to be 2900 feet. As no bridge has yet been 
 erected of a span as great as 600 feet, we are probably still be- 
 neath the practical limit. It cannot, however, be advised to at- 
 tempt the construction of chain bridges much exceeding the last 
 mentioned extent, and the increase of the span will probably be 
 made by gradual steps, as has been the case in other instances. 
 
 236. Bridges of wire, and round iron of different sizes, have 
 also been used; in these the road rests directly upon the wires. 
 More recently, in France, Seguin has proposed to use iron in 
 this form, and to suspend the road from it, as in the chain bridges 
 of Findley. He has urged in favour of his proposition several 
 plausible arguments, among which are, the greater strength ob- 
 tained by an equal quantity of metal in this form, and the entire 
 removal of the risk that arises in chains from an imperfect weld- 
 ing of the links. 
 
 29 
 
 

 
 
 

 BOOK IV. 
 
 OF THE MOTION OF SOLID BODIES. 
 
 CHAPTER I. 
 
 OP FALLING BODIES. 
 
 237. From what has been said in the preceding book, it will 
 be seen, that, when a force whose direction passes through the 
 centre of gravity, acts for an instant of time upon a body, and 
 then abandons it to itself, the motion is exactly such as it would 
 have were all tfre matter of the body collected in that centre, 
 and the force were to act there with an intensity equal to that 
 which it actually has. 
 
 We, in truth, know of no such forces in nature. No body can 
 instantaneously acquire the velocity due to the force impressed, 
 but must pass through inferior degrees of motion, requiring a cer- 
 tain time for the purpose. But there are innumerable cases, 
 where this time is so small as to be absolutely inappreciable, and 
 we may therefore, without committing any error, assume that 
 the action is instantaneous. 
 
 So, also, as every body is composed of a number of separate par- 
 ticles, which in solids are united by the attraction of aggregation, 
 (a force that however intense, does not render the system abso- 
 lutely rigid or invariable,) a greater or less time will be required 
 to convey the impulse from the point to which it is originally ap- 
 plied, and to distribute it throughout the system. This time is, 
 like the other, so small as to be inappreciable ; still there are many 
 cases where we see sure indications of the motion having been 
 communicated by degrees ; and there are even some, where we 
 take advantage of this circumstance in our practical applications. 
 
 The centre of gravity of any body acted upon by a force, the 
 duration of whose action is so small as to be insensible, and whose 
 direction passes through that point, moves uniformly forwards 
 in the straight line which marks the direction of the force. If 
 two such forces act, the centre of gravity moves uniformly in 
 
228 OF FALLING BODIES. [Book IV. 
 
 the diagonal of a parallelogram, whose sides are parallel to the 
 direction, and represent in magnitude the intensity of the forces. 
 And thus of any number of forces, according to the principles of 
 42. 
 
 238. When a body falls near the surface of the earth, it is acted 
 upon by accelerating forces, whose directions are perpendicular 
 to that surface. Within the small space that any body, whose 
 motion can usually become the object of investigation, occupies, 
 these forces may be considered as parallel to each other. Their 
 resultant, then, will pass through the centre of gravity, to which 
 we may therefore conceive, that a single accelerating force is ap- 
 plied. 
 
 The attraction of gravitation to which this action is due, varies 
 ( 100) at different points of the earth's surface, and decreases as 
 we rise from the surface, ( 9G), according to a determinate law. 
 Both of these circumstances may aho be neglected without 
 causing any sensible error; and hence, a body left without sup- 
 port, near the earth's surface, may be considered as a body moving 
 from rest, under the action of a constant accelerating force. It 
 will therefore move in a straight line, in the direction of the force, 
 or perpendicular to the surface of the earth, and with an uni- 
 formly accelerated velocity. All the formulae of 49, are there- 
 fore applicable to this case, provided a value be assigned to g> 
 the measure of the accelerating force. 
 
 By the methods described in Book III., Chap. I., and others 
 to which we shall hereafter refer, it has been inferred, that a bo- 
 dy moves from rest in a second of time, under the action of the 
 force of gravity, through a distance of 16 r V feet. Hence we have 
 for the value of g, (60), 321 feet. 
 
 In most of the calculations in which the formulae of 49 are 
 employed, it is sufficient to take the approximation of 
 
 g=32 feet. 
 Substituting this value we obtain 
 
 V (231) 
 
 2s 
 
 T'~^' J 
 
 239. A heavy body projected upwards from the surface of the 
 earth, will be retarded by a constant force, which will finally 
 destroy the upward motion. It will then begin to fall ; the mo- 
 tion upwards w^ill be equably retarded, the motion downwards 
 equably accelerated. 
 
Book IV. OF FALLING BODIES. 229 
 
 For the time of flight, and the height to which it rises, we have, 
 by the substitution of the same value of g, from (61 a) and 
 (61 6), 
 
 Hence the body will rise to the height whence it must have 
 fallen, in order to acquire the initial velocity ; and the times of 
 ascent and descent will be, exactly equal. The whole time of 
 
 i} 
 flight will be . 
 
 240. When the motion of bodies falling near the surface of the 
 earth is actually observed, it is found to differ materially from 
 what would be obtained from the calculation of the preceding 
 formulae. Thus Dr. Desaguliers found that a ball of lead of two 
 inches in diameter, took 41 seconds to fall from the dome of St. 
 Paul's, in London, to the pavement, a distance of 272 feet. Now 
 by the formula (231), 
 
 s=l6l 2 . 
 
 It ought therefore, in the same time, to have fallen through 
 324 feet. In a body of less diameter and similar density, and in 
 one of equal diameter and less density, the difference between 
 the formula and the experiment would have been still greater. 
 The cause of this difference is the resistance of the air, a retarding 
 force analogous to friction, but which follows a different law. 
 
 In a subsequent part of the work, we shall examine the nature 
 and character of this resistance. For the present, it is sufficient 
 to state, that it is usually assumed to increase in the ratio of the 
 square of the velocity. This is nearly true when the distance 
 through which a body falls does not exceed a few hundred feet: 
 and it therefore may not be wholly useless to investigate the mo- 
 tion of falling bodies upon this hypothesis. 
 
 Taking the notation of 237, the resistance of the air may be 
 expressed upon the above hypothesis, by multiplying v 3 by a con- 
 stant co-efficient.* Call this co-efficient g-Ar 2 , the retarding force 
 will be, 
 
 the accelerating force of gravity being g, the actual force,/, which 
 accelerates the body's motion, will be 
 
 f=g- ff W; 
 
 /=g.(l&V). 
 
 From (53) and (54), we may deduce the equations 
 /<& dr, and fds=vdv, 
 
 * Venturoli, vol. I. p. 89 
 
230 OF FALLING BODIES. [Book IV. 
 
 which become 
 
 dv 1 
 
 ^' =I =r (233) 
 
 f<fo 
 
 z d *=lkv- J 
 
 Integrating, and remarking for the determination of the constant 
 quantity, that when /=0, we have at the same time s=0 and 
 ?=0, we obtain the two equations, 
 
 I 
 
 and eliminating r, obtain a third, 
 
 1 / gkt gkt 
 
 ~ 
 
 From equation (234) it will be seen, that the greatest possible 
 value of v cannot exceed T . Hence, if the body fall from a 
 
 considerable height, the velocity may finally become uniform. 
 
 The motion of a rising body might be investigated, by taking 
 for the retarding force/, 
 
 /=*+* 
 
 The constant co-efficient, g-fc 2 , is -found to be proportional to 
 the density of. the air, and in bodies of similar figures, to be in the 
 inverse ratio of their homologous dimensions, and of the density of 
 the body. Call the density of the air D', let m be a constant co- 
 efficient, D the density of the body, and r the homologous di- 
 mension, which, in a spherical body is the radius, we have 
 
 whence we have for the value of k 
 
 and for the constant velocity which cannot be exceeded, 
 
 (flT D 
 B> 
 
 All things else being equal, the maximum velocity will be pro- 
 portioned to the square root of the density of the falling body ; and 
 hence, the denser the body, the longer it will continue to be acce- 
 lerated, the greater will be the constant velocity acquired, and the 
 shorter the time of its descent through a given distance. 
 
Book IF. OF FALLING BODIES. 231 
 
 So also, all things else being equal, the constant velocity will 
 be proportioned to the square root of the radii of spherical bodies, 
 and hence the larger the body of the same material, the greater 
 will be its constant final velocity, and the less the time of its fall 
 through a given distance. 
 
 241. The law thus ascertained in respect to bodies falling 
 through the air, namely, that their acquired velocity can never 
 exceed a certain limit, and finally becomes uniform, is true in 
 all eases, where a body impelled by an accelerating force is re- 
 tarded by another force, whose intensity increases in a higher 
 ratio than the simple velocity. It is also true, as will be obvious, 
 when the accelerating force decreases with the increase of the 
 velocity, and the retarding force is constant. 
 
 It will be seen from (237) that if the density of the air vary, 
 as is actually the case, the resistance must vary also ; and that 
 the co-efficient of the square of the velocity cannot be constant, 
 as we have assumed in our hypothesis, but will be less in rare 
 than in dense air. 
 
 To investigate the motion of a falling body, in such a manner 
 as to include all the circumstances, it would be necessary then to 
 take into account its figure and density ; the variation in the in- 
 tensity of gravity at different distances from the surface of the 
 earth ; the variation in the density of the atmosphere under dif- 
 ferent pressures and at different temperatures. The problem 
 would therefore become extremely complex, even were the re- 
 sistance of air of a given density to increase exactly, as assumed 
 in our hypothesis, with the square of the velocity. It fortu- 
 nately happens, however, that there are few or no cases in prac- 
 tical mechanics, in which a greater degree of accuracy in the de- 
 termination of the motion of falling bodies is required, than is to 
 be obtained from the original formulae (231): and hence, even 
 the investigation we have copied from Venturoli, and which may 
 be found under another form in Poisson, is almost wholly a mat- 
 ter of mere curiosity. 
 
 When a body falls from rest under the action of the attraction 
 of gravitation, as the direction of the force passes through its 
 centre of gravity, it acquires no rotary motion. This is not ne- 
 cessarily the case when it is projected upwards, for the projectile 
 force may be applied to a point other than its centre of gravity ; 
 the body will, in consequence, assume a rotary motion as well as 
 one in a vertical direction. It therefore becomes necessary that 
 we should consider the circumstances of motions of this character. 
 
232 ROTARY MOTION [Book 1 V. 
 
 CHAPTER II. 
 
 OF THE ROTARY MOTION OF BODIES. 
 
 242. It has been shown, 82, that the measure of the moving 
 force of a body is the product of its mass into its velocity. This 
 is also called its quantity of motion, and is the measure of a force, 
 that, if acting for an instant, and then abandoning the body to 
 itself, would communicate to it the given velocity. 
 
 A solid body may be considered as a number of material points 
 or particles of matter connected with each other in such a way 
 as to form an invariable system. If each of these points be acted 
 upon by an equal and parallel force, or if a single force, or the 
 resultant of several, be applied to the centre of gravity, or in a 
 line whose direction passes through the centre of gravity, the 
 body will move in a straight line ; and as the points will be in 
 equilibrio around the centre of gravity, they will each proceed 
 also in a straight lino. But if the forces that act upon each point 
 be not equal and parallel, or the resultant do not pass through 
 the centre of gravity, each point will have a tendency to niove 
 in the direction and with the intensity of the force impressed. 
 This tendency will be modified by the mutual connexion of the 
 points ; and hence, although each forms a part of a mass whose 
 general velocity, if the forces cease to act, is constant, yet as 
 each will have a different velocity, a rotary motion must ensue, 
 around some point comprised within the system. It therefore 
 becomes necessary as a preliminary to the general investigation 
 of the conditions of the motion of solid bodies, to determine the 
 laws of rotary motion. 
 
 Let us first assume that but a single force acts, and that the 
 body must revolve around a fixed axis. 
 
 Let ABCD be a system of points lying in one plane, and in- 
 variably connected with each other, and with an axis passing 
 through the point S, whence the distances to the points respect- 
 ively are a, 6, c , d ; let A be the point to which the force is 
 
Book /F.J o BODIES. 233 
 
 applied, and v the velocity it would have 
 were it not connected with the other 
 points, u the velocity it has in conse- 
 quence of its forming a part of the sys- 
 tem. The quantity of motion lost by 
 A, on this account, will be 
 
 A (v-u) ; 
 
 and the moment of rotation of this force, 
 in respect to S, 
 
 As the whole system is invariably connected, the time of revo- 
 lution of all the points will be the same, and their respective ve- 
 locities will be to that of A, as their distances from S ; hence 
 their respective velocities will be 
 
 bu cu du 
 
 a a a 
 
 The quantities of motion acquired by B, C, & D, will there- 
 fore be 
 
 B bu C cu Ddu 
 
 and their respective moments of rotation 
 
 u Cc'u Dd 2 ^ 
 
 T~ ' a ' T~ ' 
 
 By the principle of D'Alembert, these several moments of ro- 
 tation must be in equilibrio with each other, or 
 
 . B6' W + 
 Aa (t? w) 
 
 whence 
 
 A a, v 
 
 The moment of the system in respect to the point S, will be 
 
 (A a 2 +B 6 2 +C c'-r-D d 2 ) ~; (239) 
 
 each point in the system will therefore exert a force determined 
 by multiplying it by the square of its distance from the axis, and 
 by the velocity at A. 
 
 The sum of these products, extended to any number of points, 
 Aa 2 +B6 2 +CcH-DdM-&e. (240) 
 
 is called the Moment of Inertia of the system, in respect to the 
 fixed point. 
 
 The angular velocity is -, which is the value of the angle de- 
 30 
 
234 ROTARY MOTION [Book IV. 
 
 scribed around the point S, in a second of time, in parts of the 
 radius ; and to determine it in portions of a circle, the quantity 
 
 
 - must be multiplied by 57. 29578 ; the value of the arc that is 
 
 equal to the radius. 
 
 For the value of the angular velocity we have from (238) 
 
 These propositions are obviously true of a system composed 
 of any number of points whatever, situated in the same place. 
 They are also true of a system lying in different planes, the dis- 
 tances a, 6, c, &c., being in this case the perpendiculars let fall 
 from the several points upon the fixed axis. 
 
 243. In any system of points, or bodies, that are compelled to 
 revolve around a fixed axis, there may be found a point in which 
 if they were all collected, a given force applied at any distance 
 from the axis will communicate the same angular velocity as if 
 it were applied at the same distance from the axis to the system, 
 in its original state. This point is called the Centre of Gyration. 
 
 To find the centre of gyration, in the same system that we 
 have just considered. 
 
 Let x be the distance of the centre of gyration from the axis : 
 the moment of inertia of the system, if united in this point, will be 
 (A-f-B+C+D)* 2 ; 
 
 u 
 and as the angular velocity - is to be the same in both cases, this 
 
 must be equal to the moment of inertia of the system in the ori- 
 ginal state, or 
 
 (A+B-fC+D) ^=Aa 2 -r-B6 a -f Cc 2 +Dd a , 
 whence 
 
 = ^ " A+B+C+D ( 242 ) 
 
 In order to apply this to the case of a solid body, the number of 
 points must be supposed infinite : call each of them dm, and the 
 variable distance from the axis, r, the formula will become 
 
 (f9*d 
 JdZ 
 
 244. There will be a point in the radius SA, to which if an 
 obstacle be applied sufficient to stop the rotary motion of the 
 system, there will be no motion communicated to the axis S. 
 And if a resistance be there applied, the whole of the force of the 
 system will be exerted to overcome it. This point is called the 
 Centre of Percussion. In this point then, if all the matter in the 
 
Book IV '.1 OP BODIES. 235 
 
 system were collected, the moment of rotation will be equal to 
 that of the system in its original state. 
 
 To find the position of this point : let x be its distance from S : 
 the moment of rotation of the system if collected in that point 
 will be 
 
 (A+B+C+D-r&c.)*--; 
 a 
 
 which must by hypothesis be equal to 
 
 Aa 2 +B6 3 +Cc 2 +DdM-&c.- . 
 a 
 
 whence we obtain for the value of x in all cases 
 Aa 2 +B6 2 +C C 2 +&c. . 
 
 Aa+B6-fCc+&c. ' ^ } 
 
 and for the integral equation, 
 
 *=7^- < 245 > 
 
 245. We have seen that a force whose direction passes through 
 the centre of gravity of a body, would give it a rectilineal direc- 
 tion only ; while if the force do not pass through the centre of 
 gravity, it must cause it to revolve. The case' of its being com- 
 pelled to revolve on a fixed axis has been examined. If, however, 
 the body have no fixed point, it will not only acquire a rotary, 
 but a progressive motion, or one of translation. In order to ex- 
 amine the manner in which these two different species of motions 
 will take place : 
 
 Let ABCD be a section of the body, passing through its centre 
 of gravity, G, and the point F to which the force that produces the 
 motion is applied ; let FH represent this force in magnitude and 
 direction ; from the centre of gravity G, draw GI perpendicular to 
 FH; and in GI produced on the other side of G, take GK equal to 
 
 M ^ 
 
 GI. It will be evident that the condition of equilibrium of the 
 
236 ROTARY MOTION [Book IV. 
 
 system will not be changed by applying two forces, KL & KM, 
 to the point K, each of which is equal to the half of FH, provided 
 they be parallel to FH, and act in opposite directions. The force 
 FH may then be considered as the resultant of four forces, say its 
 two halves FN, and HN, and the two assumed forces, KL and 
 KM. Of these four forces, two, FN and KL, concur to produce 
 rectilineal motion, in the direction, and with the intensity of their 
 resultant ; the direction of their resultant passes through the point 
 G, is parallel to their direction, and equal in magnitude to the ori- 
 ginal force FH. The other two forces, HN and KM, will concur 
 to produce a rotary motion, around an axis passing through the same 
 point G ; and this axis will be a normal to the plane ABCD : 
 hence 
 
 If a force be applied in any direction to a body which is free to 
 move, it will cause its centre of gravity to describe a straight line, 
 parallel to the direction of the force ; and will communicate to the 
 body a quantity of motion equal to its own intensity. The velo- 
 city may of course be found by dividing the force by the mass 
 of the body. 
 
 If the force do not pass through the centre of gravity, it will, 
 besides, communicate to the body a rotary motion around an axis 
 passing through its centre of gravity ; and this axis will be a normal 
 to the plane, passing through the centre of gravity and the di- 
 rection of the force. The angular velocity, and other circum- 
 stances of the rotary motion, may be computed as if the axis of 
 rotation were fixed. For it will be at once seen, that if we were 
 to apply to the body a force in a direction passing through the 
 centre of gravity, equal in magnitude, and contrary in direction, to 
 its motion of translation, we should destroy this motion altogether, 
 but should not in any manner affect its motion of rotation. 
 
 When a body revolves upon an axis, every point will acquire 
 a centrifugal force proportioned to its distance from the axis. 
 Hence the axis of rotation will have a constant position, only when 
 these centrifugal forces are in equilibrio around it. In all other 
 cases, the axis of rotation must undergo a change. A homoge- 
 neous sphere may revolve permanently upon any one of its diame- 
 ters. An ellipsoid of revolution may revolve permanently around 
 the axis of the generating curve, or upon any one of its equato- 
 rial diameters; but upon no other line, for the several points that 
 compose this solid will not be symmetrically situated in respect to 
 any other. A homogeneous cylinder may revolve permanently 
 upon its geometric axis, or upon any diameter of the circle that 
 bisects the axis. 
 
 246. A further examination of the properties of revolving bo- 
 dies, leads to a remarkable proposition ; the investigation of 
 which exceeds the limits within which our subject is restricted. 
 It is as follows : 
 
Book I P.] OJT BODIES. 237 
 
 In any body whatever, however irregular, there are three axes 
 of permanent rotation at right angles to each other, upon any one 
 of which, if the body revolve, the centrifugal forces of its several 
 points will be in equilibrio. These three axes, have also this re- 
 markable property, that the moment of inertia in respect to them, 
 is either a maximum or a minimum ; that is to say, is greater or 
 less than if the body revolved around any other line as an axis. 
 
 247. When a body has a double motion, of rotation and trans- 
 lation impressed upon it, the centre of gravity will, as has been 
 seen, move forwards with uniform velocity ; the other points in 
 the body will move with velocities that are continually varying. 
 Those on which the rotary and direct motions concur for an in- 
 stant, and which are most distant from the axis of rotation, will 
 move with the greatest velocity ; and those in which these motions 
 are opposed, will move with the least ; and some of the points, 
 most distant from the axis of rojtation, will actually have a motion 
 in a direction contrary to that of the centre of gravity. 
 
 248. There will also be a point in the system, in which at any 
 instant, the progressive and rotary motions will exactly balance 
 each other. This point is called the Centreof Spontaneous Rotation. 
 The motion of the body far any instant of time, may be considered 
 asa simple rotary motion around this centre. This variety in the 
 rate at which the different points of a body move, when endued 
 both with a motion of rotation and translation, has no effect when 
 the body moves forward, without meeting with resisting forces ; 
 but when these act, it produces marked changes in the direction 
 and circumstances of their motion. We shall have occasion to 
 recur to these circumstances hereafter, in treating of motions in 
 resisting media. For the present we shall confine ourselves to 
 what happens when a body is rising or falling in the air, near the 
 surface of the earth. 
 
 The resistance it meets with from that medium, being a function 
 of the velocity, will act unequally upon opposite sides of the body, 
 unless the rotation be around a vertical line; and this unequal re- 
 sistance will produce a deviation from the original direction of 
 the motion. Hence, a heavy body, although projected vertically 
 upwards, rarely or never falls back upon the exact point whence 
 it was projected. 
 
 There are various instances of the same kind to be met with, when 
 the air or other resistances act upon a revolving body, and the 
 deviation may become so considerable, as to bring a body projected 
 horizontally, back to the point where the motion began. Thus, if 
 a disk of metal, such as a coin, placed in a vertical position, upon 
 a plane surface, be impelled by a force applied to either extremity 
 of its horizontal diameter, it will acquire a rotary and a progres- 
 
238 ROTARY MOTION [Book IV. 
 
 sive motion, the former being around its vertical diameter; the un- 
 equal action of the air upon the opposite sides of the vertical axis, 
 concurring with the friction of the plane on which it rests, will 
 cause it to describe a series of re-entering curves. A billiard ball 
 placed on a plane surface, and impelled by a force which gives its 
 lower side a motion of rotation contrary to the direction of its cen- 
 tre of gravity, will have its progressive motion destroyed by the 
 friction, and will, afterwards, by virtue of the rotary motions it re- 
 tains, roll back towards the place whence it originally set out. Cases 
 of a similar nature are too frequent and familiar to need enumera- 
 tion. 
 
 249. The general expression, (240), for the value of the mo- 
 ment of inertia, may be applied to particular cases by means of 
 the calculus. 
 
 Call the moment of inertia S ; let dm be an element of the 
 figure whose moment is sought, and x the distance from the axis 
 of rotation ; then 
 
 8=fj*dm. (246) 
 
 To adapt this to individual instances : 
 
 (1). To find the moment of inertia of a straight line, revolving 
 around an axis perpendicular to itself, and passing through one of 
 its extremities : 
 then dm becomes dx, and 
 
 S=, 
 integrating 
 
 8 4< 
 
 and when x=a, 
 
 S=. (247) 
 
 (2). To find the moment of inertia of the circumference of a 
 circle, in respect to an axis passing through the centre, and perpen- 
 dicular to the plane of the circle : 
 
 The element of the curve being ds, and its distance from the 
 axis or radius, a, 
 
 integrating 
 
 and when s becomes the whole circumference, or 2tfa, 
 
 S=2*a 3 . (248) 
 
 (3). To find the moment of inertia of the circumference of a 
 circle in respect to a diameter ; let x and y be the ordinate and 
 
 abscissa; the element ds~ ; its distance from the diameter = i/, 
 
 and 
 
Book IV.} oy BODIES. 239 
 
 The integral of ydx, when * becomes the whole circumference, is 
 equal to the area or to ra 2 , therefore, 
 
 S=rf. (249) 
 
 (4). To find the moment of inertia of the area of a circle whose 
 radius is a, in respect to an axis passing through its centre, and 
 perpendicular to its plane : 
 
 Take an elementary ring whose radii are z, and z-Mz, the area 
 of this ring will be, 2zdz, and its moment of inertia, considering 
 it as the circumference of a circle, will be 2*a?dz, by case (2) ; 
 hence, 
 
 S 
 Integrating 
 
 a^fffl; 
 and when z a, 
 
 S=i,r^a 4 . (250) 
 
 (5). In a similar manner it may be concluded, that the moment 
 of inertia of the area of a circle in respect to one of its diameters, 
 
 S=i*a 4 . (251) 
 
 (6). To find the moment of inertia of a solid, formed by the 
 revolution of a curve, in respect to the axis of rotation : 
 
 The figure being symmetric may be referred to no more than two 
 co-ordinates, x and y ; take for the element the solid contained 
 between two circles, whose distances from the origin of the co-or- 
 dinates are respectively #, and x-\-dx t the moment of rotation of 
 the element will be, by case (3),,, ajjhj 
 
 and that of the whole solid, 
 
 S=.i tf/ydar. (252) 
 
 To apply this formula to the case of a cylinder, whose length is 6, 
 and the radius of whose base is a, 
 
 S=itfa 4 &. (253) 
 
 In a cone, the radius of whose base is o, and whose altitude is 6, 
 
 S= T V<'&. (254) 
 
 In a hemisphere, 
 
 S T *j*a 5 . (255) 
 
 In a sphere, 
 
 S=JLtfa 5 . (256) 
 
 250. When the moment of inertia of a body in respect to any 
 axis is known, it is easy to find its moment of inertia in respect 
 to any other axis, provided it be parallel to the first. 
 
240 
 
 
 ROTARY MOTION 
 
 First let the given axis pass through the centre of gravity : then 
 let GG' be this axis, and CC' another parallel to it, in respect to 
 
 which the moment of inertia is sought ; let an element of the sys- 
 tem be situated at M, and let its mass be dm : suppose a plane to 
 pass through M, perpendicular to the two axes GG', and CC', and 
 in this plane draw the lines MG', MC, and join CG' ; let fall MP 
 from the point M, perpendicular on CG'. Let MG=r, MC=y, 
 then 
 
 (sea) 
 
 (3SS) 
 
 in the right angled triangles MCP, and MG'P, ' 
 MC 3 =MP a +CP 2 
 
 MP a =MG' 2 G'P 3 , 
 
 and 
 
 CP a =(G'P+CG') a , 
 
 whence 
 
 MC 2 =MG /3 +2(CG'+G'P)+CG' 2 , 
 
 and S' will be equal to/dn?, multiplied by the second member of 
 the above equation ; but dm multiplied by the line GP, is the mo- 
 ment of inertia of the element dm, in respect to a plane passing 
 through the centre of gravity G; and, therefore, fdm multiplied 
 by the varying length of this line =0, by the property of the cen- 
 tre of parallel forces, 25. The term of which it forms a part is, 
 therefore, also =0, and disappears ; hence, if we call the distance 
 between the centre of gravity, and the point C, through which the 
 second axis passes, &; 
 
 S'^fdm^+fdmk* ; (257) 
 
 and by substituting S, for its value, and integrating the second 
 term, we obtain 
 
 (258) 
 
Book IT.} or BODIES. 241 
 
 The moment of inertia S", in respect to any other parallel axis, 
 whose distance from the centre of gravity is /b', is 
 
 S"=S-|-mfc /2 ; (259) 
 
 and by substituting the value of S, obtained from the last equation, 
 S"=S'-r-m(&' 2 #). (260) 
 
 251. This principle may be applied to the discovery of the 
 moment of inertia, in cases different from those that have already 
 been investigated. Thus : 
 
 (6). To find the moment of inertia of a circle, in respect to an 
 axis parallel to a diameter : we have for its moment of inertia in 
 respect to that diameter, (251), i<m* : 
 
 Call the distance between the diameter and axis, k ; m becomes 
 the area of the circle, or tfa 2 , and 
 
 S=itfa 4 +*a 3 &. (261) 
 
 (7). To find the moment of inertia of a solid of revolution, in 
 respect to an axis passing through its vertex, and perpendicular to 
 the axis of rotation x : 
 
 We may take for the element of the solid in this case, a circle 
 whose radius is ?/, and whose distance from the axis is x. Its 
 moment of inertia, in respect to the axis, will be, 247, 
 
 and to the vertex, 
 and its mass is 
 
 whence 
 
 $=ify*dx+fx*tfdx. (262) 
 
 If the axis pass through any other point, whose distance from 
 the centre of gravity is r, 
 
 S = A fy*dx+r*fy 2 dx . (263) 
 
 31 
 
242 MOTION OF [Book IF'. 
 
 CHAPTER III. 
 
 Or THB MOTION OF PROJECTILES. 
 
 252. It his been shown, 23S, that a body projected upwards 
 from any point on the earth's surface, is retarded both by the ac- 
 tion of the force of gravity, and the resistance of the air. If pro- 
 jected downwards from a point near the earth, the former would 
 accelerate its motion with its whole intensity ; the latter would 
 still act to retard it. But if it be projected in any other direction 
 than a vertical line, the resistance of the air continues to act ac- 
 cording to the same law, while the force of gravity is now exert- 
 ed in a direction inclined to that in which it would tend to move, 
 under the action of the projectile force. Let us in the first place 
 conceive the resistance of the air to be removed. To whatever 
 point of the body the projectile force is applied, its centre of 
 gravity, 244, will acquire a motion, in a direction parallel to 
 that of the force, with uniform velocity. The force of gravity, 
 within the limits in which projectiles move, may be considered 
 as constant, and although its directions are all converging, no im- 
 portant error can arise from considering it. as acting always par- 
 allel to itself. The case, therefore, becomes the same as that we 
 have considered in Book II. Chap. IV. Hence we may infer : 
 
 That if a body were projected from a point near the surface of the 
 earth in a direction that is not a normal to that surface, and no 
 other accelerating force were to act but that of gravity, it would 
 describe a parabola, whose diameters are normals to the horizon- 
 tal plane : 
 
 That it would have the greatest horizontal range, when pro- 
 jected at an angle of 45 degrees to the horizon : 
 
 That it would rise to the greatest height, the more nearly its 
 original direction approaches to the vertical ; and that the time 
 of flight will also increase with the increase of the angle of eleva- 
 tion. 
 
 If we call h the height whence the body would have fallen to 
 acquire the initial velocity v ; d the horizontal distance to which 
 the projectile is carried ; i the angle of inclination ; and y the force 
 of gravity, we obtain from (72) and (61), 
 
 , (264) 
 
 which is a maximum when 
 
 * t=45; 
 
 in which case the equation becomes 
 
Book If r .} PKOJKCTILES. 243 
 
 &=*.= *- (265) 
 
 g 32 
 If we suppose the initial velocity to be 2000 feet per second, 
 
 d= 125000 feet, 
 or nearly twenty four miles. 
 
 253. When projectiles move with but small velocity, the dis- 
 crepancy between the parabolic theory, and what is found to oc- 
 cur in practice, is but small; but with increasing velocities, as 
 the air's resistance increases in a higher ratio than the velocity, 
 this discrepancy becomes very great. The general effect of a re- 
 sisting medium may be thus investigated.* 
 
 Let g-R represent the resistance of the medium, which will be 
 exercised in a direction contrary to that of ds, the element of the 
 curve at any given point. Refer the curve to two axes, one ver- 
 tical, t/, the other horizontal, #, both passing through the point of 
 projection. 
 
 If we decompose the resistance, g-R, into two forces parallel to 
 the two axes, that parallel to x will be 
 
 **' 
 
 that parallel to t/, 
 
 and these being retarding forces, will have the negative sign. The 
 first will express the whole disturbing force in the direction paral- 
 lel to x ; but in the direction parallel to ?/, the force of gravity acts 
 also : hence the whole forces parallel to the two co-ordinates be- 
 come 
 
 and 
 
 ^ ds ' 
 
 From the general equations of curvilinear motion, 62, we 
 have 
 
 and 
 
 whence we have 
 
 dx _d*x 
 
 *Venturoli, Vol. 1. p. 95. 
 
244 MOTION OP [Book IV. 
 
 dy_d*y 
 
 gg* Sfjr- 
 
 The velocity in the direction of a: is constant by (236), there- 
 fore, by taking the differential of both, and substituting in the se- 
 cond the value of d 2 /, we obtain 
 
 and taking the differential of the second, we have 
 
 eliminating dt and d z t from this equation, we have for the equation 
 of the curve, 
 
 Whence, if the law of the resistance were known, we might pro- 
 ceed to determine the nature of the curve. 
 
 Assuming that the resistance of the air varies with the square 
 of the velocities, it has been attempted to ascertain the curve 
 which a projectile would describe. Its differential equation 
 has not, however, yielded to the integral calculus ; and were the 
 resistance to bear, as. we hereafter shall see that it does, a more 
 complex relation to the velocity, even the differential equation 
 would become difficult to obtain. Although this curve does not 
 satisfy the actual conditions of the resistance, it will still be a 
 matter of interest to state what is known in respect to it, from its 
 differential equation, and from a method of determining points 
 through which it passes, that has been thence deduced. While 
 the two branches of the parabola, measured from the horizontal 
 plane, are similar and equal to each other, and have equal bases ; 
 the two branches of the curve that would be described under the 
 action of a projectile force, the attraction of gravitation, and a 
 resistance varying with the squares of the velocity, are unequal ; 
 the base of its ascending branch being longer than that of the de- 
 scending branch. 
 
 In the parabola, the times of describing the two branches are 
 equal ; and the velocity of the projectile, when it again reaches 
 the horizontal plane passing through the point of projection, is 
 equal to the initial velocity. In the curve under consideration, 
 the time of describing the ascending branch is less than that of 
 describing the descending branch, and the final velocity is less 
 than the initial velocity, and may become constant. 
 
 254. The most important application of our subject is to what 
 are styled military projectiles. These are impelled, by the ex- 
 plosion of gunpowder from instruments having cylindrical cavi- 
 ties, and which are, in general terms, called Pieces of Ordnance. 
 
 When gunpowder is inflamed, the whole is converted into an 
 
Book IF'.] PROJECTILES. 245 
 
 elastic fluid, whose bulk is many times greater than that of the 
 solid whence it is derived, and whose elastic force is enhanced by 
 the high temperature at which it is generated. 
 
 By the experiments of Hutton, to which we shall presently 
 refer, gunpowder would appear to expand itself with a velocity 
 of 5000 feet per second, and to exert a force at least 2000 times 
 as great as the pressure of the atmosphere. This deduction was 
 obtained from its action upon balls, fired from pieces of ordnance. 
 The weight of these is small compared with the force exerted by 
 the gunpowder, and their motion is at first but little resisted ; 
 hence they are set in motion before the whole of the gunpowder 
 is fired, and do not sustain its entire action. It is even well 
 established by experience, that no inconsiderable quantity of the 
 gunpowder isjoften blown, uninflamed, out of the gun, by the ex- 
 plosion of the rest. 
 
 255. Count Rumford made experiments in another manner. 
 The gunpowder was in very small quantities, and was placed in 
 a small vessel of wrought iron, without a vent. The ignition 
 was accomplished, by heating this vessel red hot from without. 
 In this way the escape of elastic fluid, that takes place in pieces 
 of ordnance from the vent, was avoided. T^he resistance to be 
 overcome was a weight of great amount, when compared with 
 the quantity of gunpowder, being a brass battering cannon of the 
 size called a twenty-four pounder ; the cascable, or spherical knob 
 that terminated the breech of this gun, was fitted to the vessel 
 containing the gunpowder, by grinding, in such a manner as to 
 make an air-tight joint. Experimenting with this apparatus, 
 Rumford inferred that the force exerted by confined gunpowder, 
 was equivalent to the pressure of 100000 atmospheres. 
 
 This would, at first sight, appear to exceed the limits of credi- 
 bility ; but there are cases in which gunpowder does actually ex- 
 ert a force far greater than it could, were its energy no more 
 than was inferred by Hutton. As instances of this kind, may be 
 cited what happens in the blasting of rocks, and in military 
 mines, particularly in the effect produced by the latter, that is 
 called the globe of compression. In this, besides throwing up a 
 large conoid of earth, the gunpowder shakes the ground to a 
 considerable distance around it, acting with sufficient energy to 
 overturn and destroy walls cf solid masonry. There being this 
 great difference in the action of gunpowder, when it is exerted 
 against a body that is easPy set in motion, and when it is closely 
 confined ; it will be at ence seen that great dangers may arise, 
 when, from accident or intention, the projectile to be launched 
 from a piece of ordnance is resisted in its motion. Thus, if the 
 muzzle of a gun be inserted in water, if a portion of air be left 
 
246 MOTION OF [Book IV. 
 
 between a wad and the rest of the charge ; if the projectile be 
 of a hard material, and of such a shape that it may strike before 
 it issues from the piece : in all these cases, the strength of the 
 material of which the piece is formed may not be sufficient to 
 resist the accumulation of force, and bursting may be the conse- 
 quence. So, also, if the wad be of a cohesive material, such as 
 tarred yarn ; and particularly when it is so large as to enter the 
 piece with difficulty, similar consequences may ensue. To the 
 latter cause we may with certainty attribute the bursting of guns 
 in the navy of the United States, and to the frequent loss of them 
 in the proof. We have ourselves witnessed a case in the proof 
 of guns, where the balls made their way through the sides of the 
 piece, and large portions of the wad remained sticking to the 
 bore in front of them. 
 
 256. The theory of projectiles, as has been seen, cannot be 
 completed by the aid of mathematics, and it hence becomes ne- 
 cessary, in order that the practice of gunnery may become sure, 
 that experiment be made upon a considerable variety of pieces of 
 ordnance, with projectiles of various descriptions, at different an- 
 gles of elevation, and with varying charges of gunpowder. The 
 best general experiments of this nature have been made by Hut- 
 ton and Robins. In addition, the artillery services of several Eu- 
 ropean nations, and particularly that of France, are in possession of 
 manuscript tables of the effects of their several descriptions of 
 ordnance, derived from the experiments that are annually making 
 at their schools of practice. The experiments of Hutton, Ro- 
 bins, and some of less extent made by Count Rumford, being 
 alone accessible, we shall be compelled to confine ourselves to 
 them : they are, however, sufficient for the foundation of a cor- 
 rect theory, on which a sound practice may be established. 
 
 257. The first point to be ascertained in experiments on mili- 
 tary projectiles, is the initial velocity, which serves as the basis 
 of all the other investigations. In the older experiments, this 
 was done by pointing the piece vertically upwards, and observing 
 the time that elapsed between the discharge and the return of 
 the ball to the ground ; half of this was assumed as the time of 
 descent, and the velocity calculated by the formula, (231) 
 
 v=32t. 
 
 This method was found to give a velocity, that, if small, 
 agreed tolerably well, in the ranges calculated from it, with the 
 parabolic theory. But at the usual velocities of military pro- 
 jectiles, it was found to give results that varied very much from 
 that theory. 
 
 
Book IV. PROJECTILES. 247 
 
 258. Robins next invented an apparatus styled by him the Bal- 
 listic Pendulum. This is composed of a large mass of wood, sus- 
 pended by a rod, from centres on which it is free to vibrate. 
 The ball is fired against this, and the velocity communicated to 
 it being measured, that of the ball may be easily determined. 
 The whole of the ball's motion, provided it did not pass through 
 the wooden mass, would be communicated to the latter ; or rather 
 the ball and it would go on with a common velocity, whence the 
 quantity of motion could be determined, by multiplying the ve- 
 locity by the joint mass of the ball and the pendulum. This pro- 
 duct, divided by the mass of the ball, would give the velocity of 
 the latter. 
 
 So far, the principle is simple : a difficulty, however, arises in 
 the determination of the velocity of the pendulum ; this does 
 not go on with uniform motion, but rises in a circular arc, until 
 the force of gravity checks its motion, and causes it again to re- 
 turn to the point whence it set out. But if the arc be known, the 
 doctrine of the motion of pendulums, which will be explained 
 hereafter, enables us to calculate the velocity that is destroyed in 
 describing it. 
 
 Another difficulty arises from the fact that the pendulum will 
 describe different arcs, according to the greater or less distance of 
 the point the ball strikes, from the centre of motion ; and it will 
 also be obvious from 243, that if the ball be not fired against 
 the centre of percussion, a part of the force will be expended 
 upon the axis on which the pendulum hangs. This difficulty was 
 removed by Hutton, who investigated a theorem whence the ve- 
 locity could be calculated, when the position of the centres of gra- 
 vity and of percussion, and that of the point where the ball strikes, 
 are known j his formula is as follows : viz. 
 
 v=5.6727 g c 
 
 in which 6 is the weight of the ball, 
 
 p the weight of the pendulum, 
 
 g the distance between the centre of gravity and the 
 point of suspension, 
 
 the distance between the centre of percussion and 
 the point of suspension, 
 
 1 the distance from the point struck by the ball to the 
 point of suspension, 
 
 c the chord of the arc described by the pendulum, and 
 r its radius. 
 
 The position of the centre of gravity was determined experi- 
 mentally, by balancing the pendulum upon an edge, according to 
 the principles of 114. 
 The quantity o, was ascertained by suspending the pendulum 
 
248 MOTION or [Book IV. 
 
 and finding the time of its vibration in very small arcs, whence 
 the length can be calculated, as will be hereafter explained in 
 treating of pendulums. 
 
 The chord c, was measured, by attaching to the pendulum a 
 graduated tape, which was drawn out by the motion of the pen- 
 dulum, from between two steel edges. 
 
 Hutton also determined the initial velocity of the ball from the 
 recoil of the gun, by means of the principle of inertia 39. By 
 this it is apparent that the gunpowder must communicate to the 
 gun as much motion as it is capable of giving to the ball, but in a 
 contrary direction. The gun rendered heavier by additional 
 weights, was suspended in the same manner as the pendulum, and 
 its velocity of recoil ascertained in the manner that has been de- 
 scribed in the last section. 
 
 Experimenting in these ways, it was found that the greatest 
 initial velocity of a military projectile, does not much exceed two 
 thousand feet per second. 
 
 259. As the elastic fluid generated by the firing of gunpowder, 
 acts upon the ball during the whole time of its continuance in 
 the piece, tending to expand with a velocity of 5000 feet per 
 second, while the ball does not acquire a velocity much greater 
 than 2900 feet : the latter must be accelerated during the whole 
 of its continuance in the piece, provided its velocity does pot be- 
 come so great, that the resistance of the air, and the friction to 
 which it is subjected, are sufficient to render its velocity constant. 
 This acceleration is not, however, uniform, but becomes less and 
 less as the ball moves forward, for the following reasons: 
 
 (1.) The elasticity of the fluid proceeding from the inflamed 
 gunpowder, decreases with the increase of the space it occupies. 
 
 (2.) The elasticity depends in part upon the temperature, and 
 this will decrease as the gas expands, and also by the conducting 
 power of the piece. 
 
 ($.) Forces of this nature act with more intensity upon bodies 
 at rest than upon bodies in motion. 
 
 (4.) The ball is resisted by the air, a retarding force that in- 
 creases in a higher ratio than does the velocity of the ball, and 
 which may finally render the latter constant. See 239. 
 
 Thus, although an increase in the length of the bore does, in 
 all cases that can occur in practice, increase the initial velocity, 
 still it does not do so in the same ratio in which the length of the 
 bore is increased. 
 
 The experiments of Hutton showed that this increase was in 
 a ratio not as great as that of the square root of the length of the 
 piece, but in a ratio greater than that of the cube root ; and 
 that if/ be the length of the piece, the ratio is almost exactly 
 
 /. (266) 
 
PROJECTILES. 249 
 
 260. The velocities communicated to balls of equal weights, 
 in pieces of the same length, by unequal quantities of powder, 
 were as the square roots of the quantities of powder. 
 
 The velocities communicated to balls of different weights, in 
 pieces of equal lengths, by unequal quantities of pov/der, were 
 inversely as the square roots of the weights of the balls. 
 
 If p be the quantity of gunpowder, 6 the weight of the ball, 
 and a co-efficient, constant for all pieces of similar form, 
 
 .. (267) 
 
 This co-efficient, m, may be safely taken at 1600 feet in most of 
 the cases that occur in practice. 
 
 A remarkable consequence follows from the above formula : 
 If the weight of the ball be increased, by using solid instead of 
 hollow balls, or making them of denser substances, the velocity, 
 all other things being equal, does not decrease more rapidly than 
 in the inverse ratio of the square root of the weight; and hence 
 a heavy ball will have a greater quantity of motion than a lighter 
 one, projected from the same piece, with equal quantities of 
 powder. 
 
 261. The resistance of the air was also carefully examined by 
 Hutton ; we have not space to enter into the detail of his expe- 
 riments : it is sufficient to state, that at all velocities, from 300 
 to 1100 feet per second, he found that the resistances might be 
 thus expressed : 
 
 r=mv 2 +nv. (268) 
 
 In this formula, v is the velocity, and m and n constant co-effi- 
 cients, which for balls of the diameter of 2 inches, are, 
 m= .00002665, 
 n= -.00388. 
 
 The resistance to balls, is found to vary with the squares of 
 their diameters. The formula for any other diameter (a) will there- 
 fore be 
 
 r =( mv 3__ W7) ) a * t (269) 
 
 in which 
 
 m = . 000007657, 
 w=.000175. 
 
 Calculations founded upon this formula, are found difficult in 
 practice ; and hence a co-efficient is chosen from the experiments, 
 which will affect only the square of the velocity, at such velocities 
 as are most frequently employed in practice. Call this co-effi- 
 cient c, the whole resistance to the body expressed in Ibs. will be 
 ct> 3 a 3 , 
 32 
 
MOTION OF [Book 
 
 which will be equal to the value of the part of the retarding force 
 in 239, represented by Ar, multiplied by the weight of the 
 body, or 
 
 whence 
 
 *=^, (270) 
 
 and gk is the measure of the retarding force. 
 
 Let t be the initial velocity, u the velocity remaining after 
 moving through a distance, x ; in order to find the value of x : 
 
 By the general formula of variable motion, (53) 
 . do dv 
 
 f=sV=di=-fo ; 
 
 and substituting x for *, 
 t dv 
 
 f=dx~i 
 whence 
 
 v dv=fdx ; 
 
 substituting the value of/ or gk 2 , and considering that the force 
 is a retarding one, we obtain 
 
 A J* <"'"' 
 
 v dv=g <&"* ; 
 
 and 
 
 to dv 
 
 *"*Z3 
 
 Integrating, and considering that when ar=o, 0=u, we obtain 
 
 To give this a more convenient form, and to adapt it to the case 
 of balls of cast iron : 
 
 The weight of a cubic inch of cast iron is 4.3 oz., hence 
 
 tc=.5236a 3 X 4.3=2.25a 3 
 
 in ounces avoirdupois, g 1 the measure of the force of gravity, is 
 32 feet, and the co-efficient, c, suited to velocities of from 1200 to 
 1400 feet per second is .000007657 ; hence we have 
 
 w 
 
 = 58Ha; 
 gca* 
 
 multiplying this co-efficient by 2.30258, in order that we may 
 substitute common for hyperbolic logarithms, we obtain 
 
 *= 13380. log. ; (272) 
 
Book /F".] PROJECTILES. 251 
 
 from this we obtain for the values of u and 0, 
 
 A still more convenient form may be given to this, by consi- 
 dering that if D be the density of the ball, 
 
 w=.5236a 3 D; 
 
 which, taking atmospheric air as the unit, gives us 
 20 Da u 
 
 9 log.-; (274) 
 
 whence 
 
 log. v log. u Q |. 
 
 The calculation of the two last formulae could be made more 
 easy, if they had the form 
 
 log. t,=log. r-flog. m ) 
 
 log. v=log. u ^log. m. ) ' v / 
 
 If we make 
 
 20 Da 
 
 this condition may be satisfied by finding a constant number, by 
 
 which if - be multiplied, it will give a logarithm, which, within 
 c 
 
 the usual limits of the range of military projectiles, roes not 
 differ much from 
 
 the fraction 0.43429448, is one that is best suited for this pur- 
 pose ; hence, in the preceding formulse, 
 
 log. n=-X 0.43429448. . (277) 
 
 To find the time employed in describing the space *, we have 
 
 (53) 
 
 dx 
 
 
 from which may be obtained, by substitution and integration, em- 
 ploying afterwards the quantities c and MI, according to the prin- 
 ciples just laid down, 
 
 =; (m-l); (278) 
 
253 MOTION OP [Book IV. 
 
 if the initial velocity, and the time of flight be given, we have for 
 the value of the space x from (277) 
 
 and for m, from the previous equation, 
 
 ,n= +1. (279) 
 
 In order to determine the angle of projection at which a projec- 
 tile with a given velocity will strike the horizontal plane ai a given 
 distance ; we must consider, that the projectile, as soon as it 
 leaves the piece, is acted upon by the force of gravity, by which, 
 if we abstract from the air's resistance, it will in a time, /,fall through 
 a space, s 1 whose value is (231) 
 s=16 t 3 . 
 
 If we consider the space, ar, which the ball passes through, as 
 coinciding with the horizontal distance, (and this would be the 
 case if the ball described a straight line) we have, for the tangent 
 of the angle of elevation f , 
 
 16/ 3 . 
 Tan. i= (280) 
 
 and substituting the value of t from (278) 
 
 16 
 Tan.t = ^ c 2 (m I) 2 ; (281) 
 
 from which may be obtained, by substituting the value of w, 
 
 16 / a? \ 
 Tan. i=- y (-+*) 5 (282) 
 
 whence we have 
 
 tan. 
 
 262. To enable us to apply these formulae to practice: Cast 
 iron has a density, in terms of water, as the unit of 7.4, as deter- 
 mined by Hutton, who also assumes that the relation of the den- 
 sities of water and air are as 1000: If ; hence we have for the 
 density of cast iron in terms of air, 6054. 
 
 We have next to ascertain the diameters of the more usual pro- 
 jectiles, in fractions of feet. These, as will he hereafter more 
 particularly described, are, in the case of cannon, distinguished 
 by their weights. Their denominations and dimensions, are as 
 follows : 
 
Book 
 
 PROJECTILES. 
 
 253 
 
 .Ball of 
 42 Ibs. 
 32 Ibs. 
 24 Ibs. 
 18 Ibs. 
 12 Ibs. 
 
 9 Ibs. 
 
 6 Ibs. 
 
 4 Ibs. 
 
 Diameter in inches. 
 7.018 
 6.410 
 5.823 
 5.292 
 4.623 
 4.200 
 3.668 
 3.204 
 
 Diameter in feet. 
 0.5848 
 0.5342 
 0.4852 
 0.4410 
 0.3852 
 0.3500 
 0.3057 
 0.2670 
 
 fed 
 
 $ 
 
 
 
 
 
 
 *ji 
 
 H 
 
 LOGARITHMS OF 
 
 LOGARITHMS OF 
 
 CO 
 
 s 
 
 H 
 
 
 
 H-l 
 
 
 to 
 
 20 Da 
 
 1 
 
 J 
 
 <t 
 
 5 
 
 
 -X 0.43439449 
 
 PO 
 
 Q ~ 
 
 Q 
 
 9 
 
 c 
 
 42 Ibs. 
 
 0.5848^ 
 
 
 3.8958373 
 
 5.7420470 
 
 32 Ibs. 
 
 0.5342 
 
 
 3.8565338 
 
 5.7813505 
 
 24 Ibs. 
 
 0.4852 
 
 
 3.8147507 
 
 5.8231336 
 
 18 Ibs. 
 
 0.4410 
 
 
 3.7732685 
 
 5.8646158 
 
 12 Ibs. 
 
 0.3852 
 
 " 6054 ' 
 
 3.7145162 
 
 5.9233681 
 
 9 Ibs. 
 
 0.3500 
 
 
 3.6728979 
 
 , 5.9649864 
 
 6 Ibs. 
 
 0.3057 
 
 
 3.6151253 
 
 6.0127590 
 
 4 Ibs. 
 
 0.2670 .. 
 
 
 3.5553412 
 
 6.0825431 
 
 The pieces of ordnance most frequently employed in the prac- 
 tice of artillery are, Cannon, Mortars, Howitzers, and Carron- 
 ades. The diameters of the bore of all these different pieces 
 are called their Calibres. 
 
 Cannon vary from twelve to twenty-four, or even more calibres 
 in length ; they are moveable upon two axles that are prolongations 
 of the same cylinder, whose axis is a normal to the vertical plane 
 passing through the axis of the piece, and which are called Trun- 
 nions ; by these they are attached to a carriage, on which they may 
 be moved from place to place. The form of the carriage admits them 
 to be elevated or depressed, a few degrees above or below the 
 horizontal plane. They are distinguished, according to their use, 
 into Navy, Battering, Garrison, and Field Guns or Pieces. 
 
 The axis of the trunnions is usually in the vertical plane pas- 
 sing through their centre of gravity, but intersects it below that 
 point. 
 
 Mortars are short pieces of ordnance whose trunnions are at 
 their breech. By these they are adapted to beds. In the 
 English service, they are permanently fixed to these beds, at an 
 angle of 45 with the horizon. In the French, and in our land 
 service, they are moveable upon their trunnions, so that they may 
 be fired at any angle of elevation. 
 
 Howitzers have a form similar to that of mortars, but have 
 their trunnions situated like those of cannon. They are mounted 
 
254 MOTION OF [Book 
 
 on the same kind of carriages as cannon, but as they are shorter, 
 are capable of greater angles of elevation. 
 
 Carronades are short cannon, used almost wholly in the naval 
 service. They have no trunnions, but are connected with their 
 carriages or slides by iron straps, passed through a cylindrical 
 opening, made in a piece cast on their lower sides for the purpose. 
 All these pieces of ordnance are now cast solid, and the cylindri- 
 cal cavity that contains the charge is cut out by a rotary motion ; 
 whence it is called the Bore. 
 
 The bore of mortars, howitzers, and carronades, is made of 
 smaller diameter towards the breech ; thus assuming the shape of 
 two cylinders united by a portion of a spherical surface. The 
 smaller part of the bore is of such length as to receive the maxi- 
 mum service charge of gunpowder, and is called the Chamber. 
 Some cannon also have chambers, as have the better description of 
 small arms. The formulae that have been given above, are appli- 
 cable to cannon, howitzers and carronades, but not to mortars, 
 unless fired at small angles of elevation ; a case that rarely occurs 
 in practice ; for it is only at small angles of elevation that the 
 actual path of the projectiles approaches nearly to the distance, 
 measured in a straight line. 
 
 The distance called Point Blank, in the English service, is 
 estimated from the position of the piece to the point at which 
 the curved path of the projectile intersects the horizontal plane 
 on which the carriage is placed ; the axis of the piece being also 
 horizontal. The term, de But en Blanc, of the French, fre- 
 quently translated point blank, supposes the line of sight to be 
 directed in a horizontal line over the highest points of the breech 
 and muzzle. As these lie on the surface of a cone, the line of sight 
 is inclined to the axis of the piece, the Litter of which is, in con- 
 s.equcnce, inclined upwards from the horizon ; the path of the 
 projectile being a curve, will cut the line of sight in rising, at no 
 great distance from the mouth of the piece, and again descending, 
 will cut it a second time, at a distance that will vary with the ve- 
 locity, arvd the angle of inclination. The distance from the piece, 
 to the point where the path of the projectile cuts the line of sight 
 the second time, is the distance de But en Blanc. To strike a 
 point at this distance, the piece is aimed directly at the object, by 
 means of sights, cut into the breech, and the swell of the muzzle; 
 and the method will be as accurate as any method in gunnery can 
 well be, whether the point aimed at lies in the same horizontal 
 plane with the piece, or be elevated or depressed in respect to 
 that plane, within the limits of elevation and depression, that the 
 carriage of the piece will admit. 
 
 In order to extend the advantages of the method of direct aim, 
 
Book IV.] PROJECTILES. 255 
 
 to objects more remote than the distance de But en Blanc, the 
 French artillerists adapt a moveable sight, called the Hausse, to 
 the breech of the piece ; when this is raised, and. the line of 
 sight again directed to the object, the axis of the piece will be 
 more elevated than before, and the horizontal range in consequence 
 greater. When the abject is nearer than the distance de But 
 en Blanc, the original or natural line of sight must be directed to 
 a point below the object. 
 
 The calculation of the angles of elevation corresponding to the 
 natural line of sight, and to that given by known elevations of the 
 Hausse, may be ascertained as follows : 
 
 The angle of elevation will be equal to the least angle of a 
 right-angled plane triangle, of which one side is the difference be- 
 tween the radii at the breech, and at the muzzle, and the other 
 the length of the piece : hence, if R and r, be the respective 
 radii, and / the length, 
 
 . R r 
 
 tan. i= : ' 
 
 if now the Hausse be used, let M be the height to which it is ele- 
 vated, 
 
 . R+H r 
 tan. i= - - -- . 
 
 The application of these principles and formulae to practice, 
 may be illustrated by the following 
 
 EXAMPLES. 
 
 (1). A 24 pound ball is projected with a velocity of \2QQfeet 
 per second, required the velocity it will have, after passing 
 through a distance of 900 feet ? 
 
 The formulae are, 
 
 log. v=log. u log. m, (276) 
 
 log. m X 0.4343 ; (277) 
 
 and 
 
 w=1200 feet ; 
 
 x 900 feet ; 
 log. (w=1200,) . . 3.0791812 
 
 log. ix 0.4343 5.8231336 
 
 3 c 
 log. (#=900,) . . 2.9542425 
 
 log. (*X 0.4343- log. w) 8.7773761 No. 0.0598930 
 
 log. (i>=1045) -.. ;' , . . 3.0192882 
 
256 MOTION or [Book IV. 
 
 (2). A 24-pound ballis fired at an object distant 9QQ feet, and 
 will require there a velocity of 1000 feet to produce the proper 
 effect, required the initial velocity with which it should be pro- 
 jected ? 
 
 The formula is 
 
 log. M=log. 0-Rog. m ; (276) 
 
 The calculation of the preceding example gives 
 
 log. ro= . . . . ., . . V" 0.0598930 
 log. (v=1000) . ... 3.0000000 
 
 log. ( w =1147 feet) ..... 3.0598930 
 
 (3). A 24-pound ball being projected with an initial velocity of 
 1200 feet per second, required the time it will take to pass 
 through the first 900 feet ? 
 
 The formula is 
 
 ; (278) 
 
 we have, as before, 
 
 log. m=0.0598930, 
 and 
 
 ro=1.147, 
 m 1=0.147; 
 
 log. m 1, ...... - 9.1673173 
 
 log. c, (from subsidiary table), . . . 3.8147507 
 
 ar. co. log. (w=1200 feet, ) . . 6.9208188 
 
 log. (/=0."799,) ...... 9.9028868 
 
 or nearly 8-tenths of a second. 
 
 4. The elevation of the axis of a 24-pound gun, when pointed 
 de But en Blanc, is 1.20' ; with what initial velocity, should the 
 ball be projected to strike an object, distant 2000 feet ? 
 
 The formula is, 
 
 log. (ar 2000,) ..... 3.3130300 
 
 2 
 
 log. ar 1 , ... 6.6160600 
 
 ar. co. log. c, . 6.1852493 
 
 2.8013093 
 
Book IV^ PROJECTILES. 357 
 
 PROJECTILES. 
 
 X 2 
 
 = 632.86 Brought forward, 
 
 #=2000.00 
 
 +z =2632.86 log. . . .;> . 3.4204278 
 
 log. 16, ....... 1.2041200 
 
 ar. co. tan. ', . 1.6331055 
 
 2)6.2576533 
 
 log. (tt=1345.32) 3.1288266 
 
 (5). At what angle of elevation should a 24-pound gun be fired, 
 in order to strike amark at a distance of26S2jeet, the initial velo- 
 city being 1500 feet per second ? 
 The formula is, 
 
 y+* ) (282) 
 
 log. (#=2682 feet,) . . . . ' . 3.4284588 
 
 2 
 
 log. x* '.'V^ ..... > . 6.8569176 
 
 ar. co. log. c, ...... 6.1852493 
 
 a? 
 
 log. (=1102) V. V .... 3.0421679 
 
 #=2682 
 
 \- x = 3784 log ...... 3.5779511 
 
 c 
 
 log. 16, - . 1.2041200 
 
 log. u 2 =3. 1760913 X 2 ar. co. 3.6478174 
 
 tan. (t=l.32'.29 / ',) ..... 8.4298885 
 
 (6) What is the range de But en Blanc, of a 24-pound gun, 
 whose natural elevation is 1.20'., and initial velocity 1400 feet per 
 second 1 
 The formula is, 
 
 tan. i 
 
 log. (w=1400) ...... 3.1461280 
 
 2 
 
 log. w 3 Carried over, 6.2922560 
 
 33 
 

 MOTIOXOF [Book IV. 
 
 log. w a Brought over, 6.2922560 
 
 tan (i = l.20) 8.3668945 
 
 ar. co. log. 16 8.7958890 
 
 ar. co. log. c 6.1852493 
 
 log. 0.436797 . . 9.6402798 
 
 add 0.25 
 
 log. 0.686797 ' " ~ * .-...?-' 9.8368285 
 
 which divided by 2, gives .... 9.9684142 
 
 whose number is 0.931996 
 deduct 0.500000 
 
 log. . . 0.431996 9.6354798 
 
 log. c j .v. (-*- . ="{-.tf(^'.'V-"'.. V 3.8147507 
 log. 2819.88 \. ' .' .' . . ^ : *' 3.4502305 
 
 The distance de but en Wane is therefore nearly 2820 feet. 
 
 The foregoing tables are not applicable to the case of projectiles, 
 fired at angles exceeding four or five degrees, in consequence of 
 the error which arises from taking the horizontal distance as equal 
 to the actual path, and from considering the tangent of the angle 
 
 i ft/** 
 of elevation to be equal to . Both of which assumptions are 
 
 obviously far different from the truth, when the path acquires 
 any sensible curvature. The approximations that have been 
 made to the determination of the problem of the motion of 
 shells, and other projectiles, fired at great angles of elevation, 
 may be seen in Hutton's tracts. In actual service, tables are 
 used, calculated for every particular species of ordnance; and 
 hence, it is unnecessary to attempt giving any general rules. 
 
 263. The value which we have taken for the co-efficient of v 3 , 
 in the formulae, is that which corresponds to initial velocities of 
 12 to 1500 per second. Greater velocities may be given, as 
 has been stated, to military projectiles ; say upwards of 2000 feet 
 per second. To give these velocities, is, however, attended 
 with a much increased expenditure of gunpowder ; for the resist- 
 ances increase, afterwards, in much higher ratio. The cause of 
 this is, that the projectile, in passing through the atmosphere, 
 forms a wave of condensed air in its front, while the air is rare- 
 fied behind it ; it is hence resisted by a medium of greater den- 
 sity, while it derives little or no support from behind. At a ve- 
 
Book IP.] PROJECTILES. 
 
 locity of 1342 feet, the air will no longer follow it, and a 
 vacu^wn will be left behind it : hence, any initial velocity, however 
 gprat, will be speedily reduced to that limit, and will require to 
 j>roduce it a greater charge of gunpowder than would be con- 
 sistent with the formula (267) 
 
 A velocity of 1300 feet per second is given by a charge of one 
 third of the weight of the ball j and this is the greatest charge 
 that ought to be admitted in service, except in one particular 
 case, to which we shall hereafter refer. This is the regulation 
 charge for cannon in the British and American navies ; and al- 
 though the velocity it gives is often greater than is advisable, for 
 reasons that will be presently stated, still it would be inexpe- 
 dient to reduce it farther, in consequence of the injury that 
 gunpowder sustains from the moist air, in contact with the 
 ocean. In the land service, on the other hand, where this de- 
 terioration does not take place to such an extent, the service 
 charge, except in the case that has been referred to, need never 
 exceed |th or th of the weight of the ball. 
 
 264. As balls for the larger species of ordnance are made of a 
 hard material, cast-iron, they cannot fit the bore of the piece ex- 
 actly, without endangering its bursting. Hence there is a differ- 
 ence between the calibre of the piece, and the diameter of the 
 ball, which is called the windage. The more perfect the work- 
 manship of the bore, and the more accurately the balls are cast, 
 the less may be the windage. It has, therefore, been considerably 
 diminished with the improvement of the mechanic arts. 'In dif- 
 ferent services, the practice, in this respect, has been different. 
 In some, the windage has been made to bear a certain proportion 
 to the diameter of the ball ; in others, it is a constant quantity in 
 all cavities. Were the principal dangers to be apprehended, a want 
 of sphericity in the balls, or of regularity in the bore, the former 
 would be the correct method ; but in the present state of the arts 
 in our country, the chief risk will arise from an increase in the 
 diameter of the ball by oxidation, and this may be met by a wind- 
 age constant in all calibres. Sir Howard Douglas has proposed 
 that it be reduced in the British naval service to little more than 
 ith of an inch, from 0.33 in. in the 42 pound gun, and 0.22 in. in 
 the 12. With this reduced windage, he infers that an equal ve- 
 locity may be given with fths of the powder now used. That 
 an increased velocity will be a consequence of the diminution of 
 the windage will be obvious, when it is considered that the ball 
 moves from rest to a velocity that never exceeds fths of that with 
 which the elastic fluid tends to expand itself. Hence a portion 
 of the gas will escape without acting ; and this will bo deter- 
 
260 MOTION OP [Book IV. 
 
 mined by the size of the eccentric ring, formed hy the circular 
 sections of the bore and the ball. 
 
 265. In consequence of the windage, the ball will rest on tha 
 bottom of the bore, or will strike against it as it passes out, even, 
 when supported by a wooden seat or bottom, in such manner 
 that its axis coincides with the axis of the piece. In either of 
 these cases, and one or other must occur, the ball will acquire a 
 rotary motion around an axis that is not parallel to the axis of 
 the piece. It will thus happen, as will be easily seen from 241, 
 that the ball will deviate, not only from a parabolic path, but 
 from a plane curve; that it will, according to the direction of the 
 axis around which the rotary motion takes place, be deflected to 
 the right or to the left of the vertical plane, passing through the 
 axis of the piece ; and that it may rise above, or fall below the curve 
 that would be described by a body not revolving. In striking 
 against the bore, it may be reflected and thrown to the opposite 
 side ; and this may occur more than once before it leaves the 
 mouth of the piece; this will cause a change in the initial di- 
 rection, and concur in giving a rotary motion, by the combina- 
 tion of which, very considerable deviations may occur. These 
 deviations may also be affected by irregularities, the ball passing 
 from one side to the other of the same vertical plane. Such de- 
 viations may arise from cavities on the surface of the balls, or 
 from their not being homogeneous within. 
 
 Inequalities in the bore may also contribute to cause deviations ; 
 and hence, particular guns will always throw their shot to the 
 right, and others to the left of the line of sight. 
 
 The same weight even of the same parcel of gunpowder, will 
 often produce different initial velocities ; and different parcels 
 often differ in strength. From all these circumstances, the prac- 
 tice of gunnery is attended with a great deal of uncertainty ; and 
 even the best theory is no more than a guide, and does not give 
 results that are to be implicitly relied upon. 
 
 The deviations from the vertical plane, of which we have just 
 spoken, will be enhanced by increasing the velocity ; and hence 
 it is, that charges less than those generally employed, will be 
 most advantageous in many cases. For if a ball, after passing 
 through a given space, retains sufficient velocity to do the injury 
 it is intended to effect, it will strike the mark with more cer- 
 tainty, if discharged with a less initial velocity; and the required 
 range will be better attained by an increased elevation, than by 
 an increased charge. In all cases, greater uncertainty in the at- 
 tainment of the object aimed at, is produced by the use of too 
 great a charge of gunpowder. 
 
 266. The deviation of projectiles from the vertical plane, might 
 be obviated by giving them a rotary motion around an axis co- 
 
Book IV.] PROJECTILES. 261 
 
 inciding with the axis of the piecp. This has, however, been 
 found impracticable in the larger species of ordnance. In small 
 arms, it may be effected by what is called rifling the barrel. This 
 consists in cutting in the metal that surrounds the bore, shallow 
 grooves, extending from the muzzle to the lodgment of the ball; 
 these are spiral, making about one revolution, more or less, ac- 
 cording to the length of the barrel, around the bore. The ball 
 being of a soft metal, lead, is cast a little larger than the bore, 
 and is, in loading, forced in ; it is, therefore, cut by the hard 
 sides of the bore into a form adapting itself to the spiral grooves 
 or rifles ; and when the piece is discharged, it derives from them 
 a rotary motion around the axis of the piece. As there will be 
 no windage, a less proportionate Charge will give an appropriate 
 velocity than in any other species of ordnance, and the aim is far 
 more sure and certain. 
 
 267. It has been seen that the initial velocity, although in- 
 creased by increasing the length of the piecs^ increases in a much 
 less ratio, or with a power of the length be^ veen \^ s square and 
 cube root, which may be represented by f . 
 
 It has also been seen that great velocities are n^ attended with 
 proportionate ranges, and cause uncertainty in the-,i m> \^ ma y 
 hence be concluded that neither very great lengths >j the bore, 
 nor large charges of gunpowder, are ever necessary. Vith small 
 charges, the metal of the piece is less strained than w% large 
 and thus not only may the length, but the- thickness of th* piece 
 be reduced. The results of the experiments of Robins ana. Hut- 
 ton, have led to the lessening of the size and weight of mo^t of 
 the pieces of ordnance. A great and sudden improvement w\s, 
 in consequence, made in the artillery services of Europe, about 
 the commencement of the wars of the French revolution. No 
 field-piece has now a bore of more than 18 calibres in length, 
 which is, or was, lately, the regulation in the French service. In 
 the English service, the regulation length is fourteen calibres, 
 while in the American, during the late war, it was reduced to 
 twelve, and the pieces weighed no more than Icwt. to each pound 
 of ball. These were found to be sufficient for all purposes of the 
 service. An unwise policy has lately led to the alteration of the 
 model, by giving the bore the proportions of the French pieces, 
 yet without increasing the weight; it has, however, been found, 
 that pieces of the new model, even after standing proof, have 
 burst in the schools of practice. 
 
 In cannon other than field pieces, a reduction of the length is 
 not always practicable : thus in battering guns, a certain length 
 in front of the trunnions is absolutely necessary, for they are, 
 generally speaking, fired from embrasures of earth, which would 
 be injured by the gas expanding in every direction from the 
 
MOTION or [Book IV, 
 
 mouth of a short gun. The battering guns of the French service 
 are, therefore, made of the form of two frusta of cones, united at 
 the trunnions; that nearest the breech diminishing more rapidly 
 than that towards the muzzle. We shall see that this form, al- 
 though well adapted for this object, is not as strong as one formed 
 on another principle. The three calibres of French battering 
 guns have all, for the same reason, equal lengths. Navy guns 
 should also project to a certain distance beyond the side of the 
 vessel ; and the same reasons apply to garrison as to battering 
 guns. It is, besides, convenient to have them of the same model, 
 that the same pieces may be used for either purpose. In the 
 American service, it may be here stated, instead of the three ca- 
 libres used by the French fo r battering guns, the 24, 16, and 12 
 pounders, there is but one. tn e 18 pounder. 
 
 In field pieces, there no objection to shortening the smaller 
 pieces, and hence this' 1335 of guns, of each different nation, have 
 bores of a constant D m er of calibres in length. 
 
 Cannon were fo mer ly made of an alloy of copper and tin, 
 which is generaiy> but improperly, called brass. It had the ad- 
 vantage of a g at degree of tenacity, but was objectionable from 
 its great den^Yt an d high cost. It was also liable, in rapid ser- 
 vice, to sof tin a d bend. Now that the charge has been reduced, 
 cast-iron although less tenacious than the brass, has been sub- 
 stituted f r it* m tne ship, garrison, and battering guns of 
 Europ^m nations. But it has not been found practicable, 
 gene-aMy speaking, to use cast-iron in the lighter species of 
 ordnance, (field pieces) ; for although it is a general rule that 
 snail vessels, of similar dimensions and material, will bear a 
 greater strain than large ones, still there is a circumstance in the 
 casting of iron that more than counteracts this. It is found that 
 articles made from the same cast-iron, and drawn from the same 
 charge of a furnace, will be w.eaker in proportion as they are more 
 rapidly cooled ; and, therefore, the small masses of field pieces, cool- 
 ing most rapidly, are weaker than the guns of other descriptions. 
 But the iron made from the ores of Sweden and the United States, 
 with charcoal, is of such excellent quality, that field pieces may 
 be safely made of it, weighing even less than brass guns of equal 
 calibre and length. 
 
 268 In respect to the shape of cannon, it will be at once seen 
 that the part in which the charge of powder is situated, must bear 
 the greatest strain ; and it seems probable, although it would be 
 difficult to reduce it to the test of mathematical analysis, that the 
 greatest effort is exerted by the expanding gas, at the point where 
 the ball is lodged. In all the cases of burst guns that we have 
 examined, th^breech was entire, a fact that seems to corroborate 
 
Book IV.] PROJECTILES. 263 
 
 this inference. Rumford has, therefore, proposed to make the 
 thickness of metal greatest at this point, and has planned a gun 
 of beautiful proportions, swelling in a curve from the breech to 
 the point assumed for the lodgment of the ball, and again con- 
 tracting in a curve to the projection of the muzzle. It is, how- 
 ever, impossible to assign the exact point at which the ball will 
 ba lodged, in consequence of the difference in the space occupied 
 by different charges of gunpowder. Hence the form given to 
 the American navy 32-pounder, and battering 18, is to be prefer- 
 red ; this is cylindric from the base ring to the trunnions, and 
 conical thence to the swell of the muzzle. Our navy 42-pounder, 
 which is formed on the principles of the French battering gun, 
 having a great weight in the breech, and being comparatively 
 thin at the lodgment of the ball, is much weaker under an equal 
 weight, than if it were formed upon the same principles as the 32- 
 pounder. 
 
 269. The next circumstance to be considered, is the manner 
 in which military projectiles penetrate the obstacles they encoun- 
 ter. The resistance of any homogeneous body may be considered 
 as a uniform retarding force : hence, if we call this force/, the ve- 
 locity with which the ball strikes the obstacle v, and the space to 
 which it penetrates before it loses its whole motion s, we have 
 from 
 
 The penetration of bodies of the same size, figure, and weight, 
 will, therefore, be proportioned directly to the squares of their 
 velocities, and inversely to the strength of the substance into 
 which they enter. 
 
 If the bodies be balls of different diameters and densities, their 
 own moving force, (their velocities being equal,) will be propor- 
 tioned to their densities, and the cubes of their diameters ; for it 
 is as their masses ; the body into which they enter, will resist 
 with a force proportioned to the section of the ball, or to the square 
 of its diameter ; and, therefore, the depth penetrated, will be as 
 the quotient of these quantities, or directly as the density and the 
 diameter of the ball. 
 
 Introducing the cases of different velocities and different kinds 
 of obstacles, we have for the general rule : that balls penetrate 
 into obstacles to depths that are proportioned directly to their 
 densities, their diameters, and the squares of their velocities; 
 and inversely, to the resisting force of the obstacle. 
 
 If the same ball strike against the same obstacle, with different 
 velocities, the depths are proportioned, as has been shown, to the 
 squares of the velocities. The same rule will apply to the depth 
 
MOTION OF [Book ir. 
 
 that remains to be penetrated after a portion of the velocity has 
 been lost, and the remaining velocities will be proportioned to the 
 square roots of the remaining depths. Let us suppose the whole 
 depth to be divided into units of length ; the ratio between the 
 original velocity and that remaining after penetrating any number 
 of units, n will be 
 
 >/(-*) 
 
 V/5 ' 
 
 and the velocity los.t will be 
 
 Vs V(s n) 
 
 -77- '* 
 
 The velocities lost in passing through the several units of space, 
 will form a series of the following terms : 
 
 that this series is one that rapidly in- 
 
 And it will be at once seen 
 
 - obstacle, is much greate 
 
 It will therefore happen that balls passing with great veloci- 
 ties through thin obstacles, may do so without communicating 
 any perceptible motion. When balls pass with great velocities 
 through a substance that is elastic, the hole is smooth, and is often 
 ,of less diameter than the ball ; but if the velocity be small, the 
 obstacle will be torn and rent. 
 
 The prodigious effect of modern military engines in destroy- 
 ing the walls of fortresses, depends upon the principle that the 
 penetration is proportioned to the squares of the velocities. The 
 actual quantity of motion in the battering ram of the ancients, 
 might be made equal to that of a 24-pound ball ; but the former 
 produced no other effect than that of agitation, by which, however, 
 a wall might be destroyed, after a long series of blows. When 
 cannon are used for destroying walls, or as it is technically called, 
 battering in breach, the shot of all the guns in the battery are first 
 directed so as to cut, by their penetration, a horizontal groove in 
 the lowest part of the wall that can be seen from their position. 
 They are next directed so as to cut two vertical grooves at each 
 end of the horizontal one ; and thus a quadrangular portion is se- 
 parated from the rest of the wall. The cannon are then fired in 
 vollies against this portion, until the agitation it receives from 
 the shock is sufficient to cause it to fall. In the latter part of the 
 operation, cannon have no advantage over the battering ram, but 
 
PROJECTILES. 265 
 
 the former part of it, by which it is in fact rendered most efficient, 
 could not be performed by the ram. 
 
 The effect being proportioned to the square of the velocity, 
 this is the case to which reference was made in 261, where the 
 charge may exceed with advantage d of the weight of the ball. 
 It is usually made fds. The batteries in breach are also estab- 
 lished, as near as possible, to the wall to be destroyed, in order 
 that the velocity may be but little diminished by the air's resist- 
 ance. 
 
 In other cases, great velocities may be injurious : thus, if a ball 
 pass through the obstacle, it will communicate less motion than 
 if it lodge ; and the greater the velocity, the less will be the in- 
 jury done. For this reason, in close naval engagements, great 
 velocities are less destructive of the enemy's vessel than smaller 
 ones. 
 
 Upon this principle is founded the introduction of the car- 
 ronade into the naval service. This species of ordnance is short ; 
 and being loaded with no more gunpowder than |th of the weight 
 of the ball, may ha^e but little thickness of metal. It may, 
 therefore, be used in the place of long guns of much smaller ca- 
 libres, while the effect of its projectiles, in close action, is greater 
 than that of a ball of equal weight from a long gun. 
 
 270. When projectiles strike against a hard substance, in an 
 oblique direction, they are reflected according to laws that will 
 be hereafter examined. So, also, on the surface of water, balls 
 impinging at small angles, rise again, and perform a second 
 curved path ; and this may be repeated several times. The resist- 
 ance of the water will be proportioned to that component of the 
 moving force of the ball, whose direction i* a normal to the sur- 
 face of the water : when this becomes less than the gravity of 
 the ball, it will no longer rise. This motion, in successive bounds 
 over the surface of ground or water, is called the ricochet. It has 
 been applied to great advantage in the attack of fortified places ; 
 and gives to guns placed upon the shore, in proper positions, great 
 advantages over those opposed to them in ships. In the attack 
 of fortified places, the first or more distant batteries are no longer 
 placed ia^front of the part to be attacked, but in the prolongation 
 of its faces, and opposite to the returning sides of the fortress. 
 The guns in these batteries are fired at elevations of 4 or 5, 
 with charges of gunpowder that enable the balls in the descend- 
 ing part of their path, just to raze the opposing parapet: they, 
 therefore, bound along, parallel to the direction of the front to be 
 attacked, dismount the guns, and destroy the defenders. 
 
 Under the fire of these ricochet batteries, approaches are made 
 to points sufficiently near for the erection of batteries in breach ; 
 
 34 
 
' 
 
 266' MOTION OF, &LC. [Book I y. 
 
 by these the walls are destroyed. It is in consequence of this me- 
 thod, which was invented by Vauban, that the means of the at- 
 tack of fortresses have become superior to those of defence, and 
 that the time of the resistance of a fortress can be calculated with 
 almost mathematical precision. 
 
 When balls are fired from shipping, and strike the water, they 
 never rise in any of their bounds as high as the point whence 
 they are projected. Hence, in firing from a ship at an object on 
 the land that is higher than the deck on which the gun is placed, 
 there is no other chance of striking it but by a direct aim ; while, 
 if a ball be fired from the land, it will strike the vessel, if it be 
 within the limit of its recochets ; and if*the height of the battery 
 be not so great as to permit the balls to rise above the vessel. It 
 thus happens that two or three heavy guns, placed upon the land 
 in a proper position, are more than a match for the heaviest ship, 
 while towers, and walls with embrasures, may be destroyed by 
 the superiority in the quantity of guns that can be arranged, in a 
 given space, on shipboard. 
 
Book ]VJ\ THEORY OF THE PENDULUM. 267 
 
 CHAPTER IV. 
 
 THEORY or THE PENDULUM. 
 
 271. If a gravitating body be suspended by a string or rod from 
 a fixed point, it will hang in a vertical position ; but if it be raised 
 from that position laterally, the string or rod remaining inflex- 
 ible, and then permitted to escape, it will, under the joint action 
 of gravity, and the tension of the string or rod, descend in an 
 arc of a circle of which the point of suspension will be the centre. 
 When it reaches the vertical position, it will, 58 have acquired 
 a velocity equal to that acquired by falling vertically through the 
 versed sine of the arc, and which would tend to carry it forwards 
 with uniform velocity in a horizontal direction. The tension of 
 the string will cause it to continue to move in the circle of which 
 the point of suspension is the centre ; it will, therefore, after 
 passing the vertical line, rise in a circular arc, until its whole 
 velocity be destroyed, which, if no other force but gravity act, 
 will be, when it reaches a height on the opposite side of the ver- 
 tical, equal to that whence it at first fell. On reaching this point, 
 it will again descend, and passing the vertical, rise to the point 
 whencelt originally set out. From this it will a second time 
 descend, and would thus continue to vibrate for ever. But it 
 will be seen that two forces act to retard the motion, namely, the 
 resistance of the atmosphere, and friction around the point of 
 suspension. By virtue of these, the circular arcs described are 
 gradually lessened, until a state of rest be reached; the suspend- 
 ing string or rod becoming stationary in a vertical position. 
 
 A body thus suspended and caused to vibrate, is called a Pen- 
 dulum. 
 
 The nassage from its highest position on one side of the ver- 
 tical, until it reach the greatest height on the opposite i ide, u 
 called an Oscillation. 
 
 272. In order to study the theory of this species of motion, 
 we imagine to ourselves'a gravitating point, suspended by meai 
 of an inflexible line, devoid of gravity. The case then becomes 
 that of & 60, a gravitating point, compelled to move upon a cir 
 cular surface, by an accelerating force whose direction is always 
 parallel to itself. 
 
268 THEORY OF [JBook IV. 
 
 The time t, of descent in a small circular arc, of which a is the 
 radius, and h the versed sine, is by (84) 
 <jf a h 
 
 hence the whole time of an oscillation, T, becomes, substituting 
 / the length of the pendulum for a, 
 
 T=W 1 X(1+ -| ; (285) 
 
 O 
 
 and finally, in evanescent arcs, 
 
 T=W- ; 
 
 o 
 
 whence we obtain for the values of / and g 4 , 
 
 and when T = l", 
 
 g=W. (288) 
 
 To compare the time of the oscillation of a pendulum in a very 
 
 small arc, with that of the descent of a heavy body : Let the dis- 
 
 tance the body falls be half the length of the pendulum, and we 
 
 have by (61) 
 
 '=4 
 
 which compared with (286) gives 
 
 T :::*: 1. (289) 
 
 When the respective times in which two pendulums vibrate, or 
 
 their numbers of oscillations, in equal times, are known, and the 
 
 length of one of them has been determined, that of the other may 
 
 be calculated 
 
 Let / and I' be their respective lengths ; 
 T and T' their times of oscillating ; 
 N and N' the numbers of vibrations in equal times, then 
 T : T' : : N' : N ; 
 
 for the number of oscillations is inversely as the times. 
 We have for the value of the times (286) 
 
 whence, T : T' ::>//: ^/', 
 
 and N: N': : Jl' : v'/; 
 
 therefore, when the respective times are known, and I is given, 
 
 
Book IV.} THE PENDULUM. *C9 
 
 and when N and N' are known, and I' given, 
 
 N"J' 
 l=-jj . (290) 
 
 From these formulae it may be concluded : 
 
 (1.) That, the force of gravity remaining constant, the times 
 of the vibrations of pendulums of unequal lengths are respectively 
 as the square roots of their lengths, and inversely as the square 
 roots of the numbers of their vibrations in a given time. 
 
 (2.) That, the force of gravity still remaining constant, the 
 lengths of different pendulums are directly as the squares of the 
 times of their oscillations, and inversely as the squares of the 
 numbers of their oscillations in a given time. 
 
 (3.) That the time of the oscillation of a pendulum, is to the 
 time that a heavy body would fall freely by the force of gravity, 
 through half its length, as the circumference of a circle is to its 
 diameter. 
 
 (4.) In different positions, the intensity of the force of gravity 
 may be measured by the length of the pendulum that vibrates in 
 equal times at the different places ; and the intensity of gravi- 
 tation is always directly proportioned to the length of the pendu- 
 lum that beats seconds at the place. 
 
 273. If the pendulum vibrate in a cycloid, the formula 
 
 becomes true, 59, whatever be the amplitude of the arc : all 
 the above propositions are, therefore, absolutely true of pendu- 
 lums vibrating in cycloidal arcs. 
 
 It has been attempted to make pendulums vibrate in cycloidal 
 arcs, upon the following principles, which are theoretically true ; 
 although, as we shall hereafter see, they are not susceptible of 
 being applied to any practical purpose. This attempt has been 
 founded on the following principles. 
 
 rt- 
 
270 
 
 THEORY OF 
 
 [Book IV. 
 
 It is a property of the cycloid, that its evolute is an equal cy- 
 cloid, placed in an inverted position. Hence, if the length of 
 the pendulum SP be bisected, and on the line ADB, drawn 
 
 through the point of bisection, perpendicular to SP, a cycloid be 
 constructed, the diameter of whose generating circle is DP; and 
 if two half cycloids be constructed tangents to the lines SD, and 
 ADB, the latter will be the evolutes of the corresponding semicy- 
 cloids. If then a pendulum be suspended from S, by a flexible rgd, 
 of the length SP, it will, when moved from its vertical position and 
 permitted to oscillate, apply itself to the two curves $A, SB ; and 
 it is evident, that during these oscillations, it will describe the 
 curve APB, or a part of it. 
 
 When the circular arc does not exceed 4 or 5 degrees, the re- 
 lation of the time of oscillation in it to that in a cycloidal arc is 
 given by the formulae (2S5) and (2S6), or is as 
 
 l:l+i&: (291) 
 
 when the arc exceeds that amount, it becomes necessary to intro- 
 duce more terms of the series in (84). 
 
 From this series, the excess of the times of oscillation in circular 
 arcs over those in a cycloid, has been calculated, as follows, viz. 
 In an arc of 30 on each 
 
 side of the vertical . . . 0.01675 
 of 15 .... 0.00426 
 
 of 10 .... 0.00190 
 
 of 5 . . . 0.00012 
 
 of 21 .... 0.00003 
 
 So thifc when the circular arc. in which a pendulum vibrates, 
 does not exceed 2^ on each side of the vertical, or 5 in all, the 
 
 
Book /KJ TIIJ: PENDULUM. 271 
 
 excess in the length of the time of its oscillations over those in a 
 cycloid does not exceed 9" per day. 
 
 274. It has been seen, 96, that the intensity of gravity va- 
 ries as we proceed from the surface of the earth, in the inverse 
 ratio of the squares of the distances. 
 
 If we call the radius of the earth R, the distance above the 
 mean surface A, the force of the gravity at the surface g, and at 
 the height ft, g', we have 
 
 whence we obtain 
 
 2ft 
 
 (2ft ft 2 \ 
 l + tt~+B2l 
 It R 2 / 5 
 
 or, neglecting the last term, 
 
 [~ 2ft~l 2g' h 
 
 and 
 
 In the same manner we obtain for the values of the lengths of 
 the pendulums, at the surface, and at the height ft, 
 
 These relations are, however, only true in the case impossi- 
 ble in practice, that the pendulum is raised through the air with- 
 out resting, as it must, upon some projecting part of the solid 
 crust of the earth. In this event, a local attraction of the mass of 
 ground on which it rests will interfere with the law. 
 
 275. The length of the pendulum that vibrates seconds at any 
 place, is proportioned, as has been stated, to the force of gravity 
 at that part of the earth's surface. That is to say, to the differ- 
 ence between the whole force of gravity and the centrifugal force. 
 We may hence obtain the relation between the lengths of pendu- 
 lums in different latitudes. 
 
 If the centrifugal force at the equator be called/, the latitude 
 L, the force of gravity at the pole, or the absolute measure of that 
 force, G, the relative force at the equator g-, 
 
 g-=G-/; andG= ff +/5 
 
 and the centrifugal force at the latitude L, will be, 100, 
 /cos. 3 L. 
 
THEORY OF 
 
 Then if g' be the apparent force of gravity at the latitude, ] 
 
 g-'=G-/cos. a L; 
 
 substituting the value of G from the above equation, 
 g / =g > +/-/co 8 . 2 L=g+/(l-cos. 2 L), 
 
 '=+/ sin. 2 L. ( 2 
 
 As the pendulums of the different places have the same r 
 as the gravitating forces, we have, if E be the length of the [ 
 dulum at the equator, and d the difference between the length 
 the polar and equatorial pendulums, for the difference betw 
 the length m of the pendulum at the equator, and at the latil 
 
 w=E+dsin. 2 L; (2 
 
 and for the length n, at another latitude L', 
 
 n=E + d sin. a L'. (2 
 
 If m and n be given, the lengths at the pole, and the equ 
 may be found ; for by subtraction of the foregoing express 
 we obtain 
 
 m n= (E+d sin. 2 L) (E-fd sin. 2 L') =<2(sin. 2 L sin. 2 
 whence we obtain for the value of d, 
 
 m n 
 
 sin. (L+L') . sin.(L L') ; 
 
 and d being given, we have the value of E from either of the 
 muhE, (295) and (296), 
 
 E=m dsin. 2 L I ff 
 
 E^TI dsin. 2 L' ( 
 
 The ratio- will be the same as^, which expresses the din: 
 E # 
 
 tion of gravity from the pole to the equator. 
 
 When this ratio is given, the ellipticity of the terrestrial sphe 
 may be determined ; for it has been shown, by writers on j 
 sical astronomy, that if this ratio be subtracted from 5 of the 
 portion of the centrifugal force at the equator, to the appa 
 force of gravity there, the remainder expresses the oblatene* 
 flattening at the poles. 
 
 When more than two observations are to be combined, it 
 be best done by the method of least squares, as may be sec 
 the Mecanique Celeste, in a paper of Dr. Adrain in the An 
 can Philosophical Transactions, and in Sabine's Experiment 
 the Pendulum. 
 
 276. The Simple Pendulum, such as we have assumed it 
 the purpose of investigating the theory, cannot exist in prac 
 for we can neither make use of a body so small as to be consid 
 as a single gravitating point, nor abstract from the weight oi 
 
Book IV. THE PETfDULUM. 273 
 
 rod by which it is suspended. Such pendulums as can be actually 
 constructed, are called Compound Pendulums. 
 
 in every compound pendulum, there is necessarily a point, in 
 which if all the matter were collected, a simple pendulum will 
 be formed, whose oscillations will be performed in the same time 
 as those of the compound pendulum. This point is called the 
 Centre of Oscillation. Its essential property is, obviously, that 
 the sum of the moments of rotation of the several points of which 
 it is made up, will be equal to the moment of rotation of the whole 
 mass, if situated at the centre of osciUation. 
 
 Calling the distance from the centre of oscillation ar, the seve- 
 ral points of which the body is made up, A, B, C, &c., their res- 
 pective distances from the centre of suspension, a, 6, c, &c., we 
 have for the sum of their moments of rotation, by 241, 
 
 (Aa 2 -l-B6 2 4-Cc 2 +&c.) - ; 
 
 and for the moment of rotation of the whole mass, if collected in 
 the centre of* oscillation, 
 
 (Aa+B6+Cc+&c.) x - ; 
 a 
 
 whence we obtain 
 
 Hence, the centre of oscillation in a vibrating body is identical 
 with the centre of percussion in a revolving body ; and the rule 
 for finding its position may be thus expressed in words : Divide 
 the moment of inertia of the body by its moment of rotation; the 
 quotient is the distance of the centre of oscillation, from the cen- 
 tre of suspension. 
 
 If the points, instead of being united by a line devoid of weight, 
 compose a solid body, the same proposition will be true, provided 
 the distances be measured from the several points, perpendicularly 
 to the horizontal line passing through the point of suspension ; and 
 if the body, instead of being suspended by a point, is supported on 
 the last mentioned line as an axis, the propositions are still true in 
 respect to it. So, also, if a horizontal line be drawn through the 
 centre of oscillation, every point in it will be at an equal distance 
 from the axis of suspension ; and hence any point in the former 
 line, which may be called the axis of oscillation, will have the 
 properties of a centre of oscillation. 
 
 From the properties of the centre of gravity, the quantity 
 
 Aa+B6-fCc-h&c. 
 
 is equal to the mass multiplied by the distance of the centre of 
 gravity of the body, from the centre of suspension, or calling the 
 
 35 
 
274 THEORY OF [Book IV. 
 
 former, M, the latter, -, to Mg. And if we call the moment of 
 inertia, S, the formula, (299), becomes 
 
 S 
 
 -S' 
 
 and 
 
 S=Mgx. (300) 
 
 If the pendulum be suspended by its axis of oscillation, that 
 being parallel to the axis of suspension, the moment of rotation, 
 S', now becomes, (259), 
 
 S' = S m(k' 2 #) , 
 = S m(k'+k) (k'k) ; 
 
 but k'+k is the distance between the axes of suspension and os- 
 cillation, or =x, therefore, 
 
 S'=S mkx+mk'x; 
 but, by (246), 
 
 hence, 
 
 S'=mkx ; 
 
 and the moment of rotation is mk ; hence, the distance, #', from 
 the axis of oscillation, when the pendulum is suspended by it, to 
 the line that becomes the axis of oscillation, is 
 
 Ha?-- ( 301 > 
 
 277. Thus it appears, that if a compound pendulum be sus- 
 pended by its axis of oscillation, the axis of suspension becomes 
 the axis of oscillation, or that the centres of oscillation and suspen- 
 sion are convertible points. Therefore, if a pendulum be suspended 
 upon an axis passing through its centre of oscillation, it will vi- 
 brate in the same time as when suspended from its ordinary axis 
 of suspension ; and conversely, if a pendulum be found to vibrate 
 in equal times when suspended from two different axes, and if one 
 of these be called the axis of suspension, the other becomes the axis 
 of oscillation ; and the distance between these axes is the length of 
 a simple pendulum, whose oscillations would be isochronous, with 
 those of the compound pendulum. 
 
 278. When the length of a pendulum is spoken of, we do j 
 understand its physical length, or distance between its two ex- 
 treme ends ; but, by this term we understand the distance between 
 its centres of oscillation and suspension, or the length of the iden- 
 tical simple pendulum. The determination of the position of the 
 centre of oscillation of a pendulum, therefore, frequently becomes 
 an important subject of inquiry in practice. This may be effected 
 in the case of all solids formed by the revolution of symmetric 
 curves, by means of the principle in 276, that the distance be- 
 
Book IF.] THE PENDULUM. 275 
 
 tween the centres of suspension and oscillation, may be found by 
 dividing the moment of inertia by the moment of rotation, both 
 in reference to the axis of suspension. 
 
 (1). Thus in the case of a straight line, suspended by one of 
 its extremities, the moment of inertia is ~, (247), the moment 
 
 of rotation ~ ; hence, we have for the distance between the cen- 
 tres of oscillation and suspension, 
 
 '=y. (802) 
 
 In the same manner we may obtain, using the formulae, (253) and 
 (254) : 
 
 (2). In a cylinder suspended by the middle of one of its bases, 
 a being the length, and, 6, the radius of the base, 
 
 2a b 
 
 ~3~ + 2^' ( 303 ) 
 
 (3). In a cone suspended by the vertex, the radius of whose 
 base is 6, 
 
 4a b a 
 
 /= T+5^ < 304 ) 
 
 When the cone is a right cone, 
 
 a=b, 
 and 
 
 l=a. 
 
 In which case the centre of oscillation is in the middle of the base. 
 (4). In a sphere whose radius is r, and which is suspended from 
 a point on its surface, from (256) 
 
 l==~ . (305) 
 
 (5). In a sphere whose radius is r, and which is suspended by 
 a line devoid of weight, that unites a point in its surface to the 
 centre of suspension : if c be the length of this line, we have, by 
 using the principles of 250, 
 
 . 2r* 
 
 279. It has already been mentioned that pendulumsin all the ca- 
 ses in which they can be employed, are resisted by the friction upon 
 the axis of suspension, and by the resistance of the medium in which 
 they move, namely, the atmosphere. The arcs described, therefore, 
 gradually become less and less, until the motion ceases altogether, 
 and the pendulum reaches a state of rest. The resistance at the 
 axis of suspension is usually diminished to the lowest practicable 
 
276 THEORY OF, &c. [Book 
 
 limit, and hence it becomes unnecessary to apply any correction 
 for it, except so far as its effect becomes apparent in the gradual 
 lessening of the are of oscillation. 
 
 In respect to the resistance of the air, as the velocity of the 
 pendulum is at most but small, that part of it which varies with 
 the square of the velocity may be neglected, and the resistance 
 may be considered as directly proportional to the velocity. Were 
 this absolutely true, although the motion would be retarded, and 
 each oscillation increased in duration, the time of the performance 
 of each would be equally affected, and the isochronism in arcs of 
 a cycloid would not be altered, while in circular arcs the variation 
 in isochronism would be allowed for, in the correction for the ex- 
 tent of the arc. The whole retardation would, if this were true, 
 be allowed for by applying a correction for the buoyancy of the 
 air. The gravity of the pendulum is diminished by its falling 
 through that fluid; and hence, if m represent the absolute density 
 of the pendulum, m', the density of the air, the correction 
 would be 
 
 m m 
 
 Such, at least, is the only correction that has hitherto been em- 
 ployed to determine, from the actual oscillations of a pendulum, 
 what would occur were it to move in a vacuum or space void 
 of air. Recent experiments by Bessel, and by Sabine, have shown 
 that this correction is insufficient. Bessel's experiments consisted 
 in making pendulums of similar figures, but of unequal densities, 
 vibrate in air ; and in making the same pendulums vibrate in two 
 different media, say air and water. From these, he inferred the 
 difference that would ensue, were the pendulum to move in a space 
 void of air. Sabine, on the other hand, instituted a direct com- 
 parison between the oscillations of a pendulum in air, and in such 
 a vacuum as may be obtained by the air-pump. These different 
 modes of experimenting lead to similar conclusions, namely, that 
 the correction necessary to be applied to reduce the motion 
 in air, to that in a vacuum, is in all cases greater than is re- 
 presented by the formula that has just Keen stated ; and that it 
 depends upon the figure of the pendulum. It must, therefore, 
 be ascertained for every different shape that is given to the pendu- 
 lum, by actual experiment, for it can hardly be susceptible of re- 
 duction to any precise formula. The consequence of this disco- 
 very, upon some experiments with the pendulum, will hereafter 
 be cited. 
 
Book IF'.] APPLICATIONS OF THE PENDULUM. 877 
 
 CHAPTER V. 
 
 APPLICATIONS OP THE PENDULUM. 
 
 280. The pendulum may be applied to three several impor- 
 tant purposes. 
 
 (1.) To measure portions of time, or to subdivide the units 
 we derive from astronomic phenomena, into smaller and equal 
 portions. 
 
 (2.) To determine the measure of the force of gravity, at 
 different places, and under different circumstances; and thus to 
 enable us to infer the variation in the apparent intensity that is 
 due to the centrifugal force; and the variation in the actual in- 
 tensity at the surface, that is due to the figure of the earth. Hence 
 the figure of the earth may be inferred. 
 
 (3.) The pendulum has been employed as a standard of mea- 
 sure. 
 
 281. The pendulum is employed as a measure of time, upon 
 the principle that its oscillations, in very small circular arcs, are 
 isochronous, provided the length of the pendulum do not vary, 
 and the intensity of gravity,- at a given place, remains constant. 
 The latter, we have every reason to believe, is unchangeable in 
 the same part of the earth's surface, and it will be seen that there 
 are methods by which the former may be rendered nearly inva- 
 riable also. 
 
 When pendulums are employed as a measure of time, they are 
 adapted to instruments called Clocks. These instruments an- 
 swer the two-fold purpose : 
 
 (1.) Of restoring to the pendulum, at each oscillation, as much 
 force as it loses, in consequence of the resistance of the air, and 
 the friction at the axis of suspension ; and 
 
 (2.) Of registering the number of oscillations performed by 
 the pendulum. To accomplish the latter object, they are so con- 
 structed, that by simple inspection, the interval of time, or num- 
 ber of oscillations that have elapsed since the clock was last ob- 
 served, can be at once determined. The rate of the oscillations 
 of the pendulum can be changed by altering its length ; and 
 thus by comparing the clock with astronomic observations, the 
 day may be divided into the usual number of conventional parts. 
 The greater number of clocks have pendulums that oscillate once 
 in each second of time ; and when we speak of the length of a 
 pendulum at a given place, we mean one whose beats are an ex- 
 
APPLICATIONS OF [Book IV. 
 
 act second, or jsho part of a mean solar day. Some clocks have 
 pendulums that beat half seconds, and others again oscillate but 
 once in two seconds. According to the principles in 271, the 
 pendulums of the former must have no more than one fourth of 
 the length, and of the latter, must be four times as long as the 
 second's pendulum. 
 
 It was at one time attempted to make the pendulums, used as 
 measures of time, vibrate in arcs of a cycloid. This has been 
 abandoned, in consequence of the mechanical difficulties of mak- 
 ing the cycloidal cheeks, to which it should adapt itself: of the 
 impossibility of obtaining a flexible string of invariable length ; 
 and of the doubt that must exist in pendulums of any convenient 
 form, ifl respect to the position of tho point, the centre of oscilla- 
 tion, that ought to describe the curve. 
 
 282. The pendulum of a clock is usually composed of a weight, 
 or bulb, of a lenticular form, adapted to an axis of suspension by a 
 metallic rod. As both the rod and the bulb are liable to vary in 
 magnitude, by variations in temperature, the position of the cen- 
 tre of oscillation, and consequently the length of the pendulum, 
 must be undergoing perpetual alterations. Thus the law of iso- 
 chronism will be no longer true, and the clock will not move at 
 an uniform rate, nor mark equal divisions of time. To obviate 
 this defect, various modifications of the original simple form have 
 been contrived, which are called Compensation Pendulums. 
 Their general principle is identical, and consists in making the 
 pendulum of two substances that expand in opposite directions, 
 in such a manner as to keep the centre of oscillation at a constant 
 distance from the axis of suspension. 
 
 Various methods have been proposed and employed for this 
 purpose : 
 
 (1.) The rod of a clock pendulum is often attached to a short 
 piece of flexible metal or spring ; this may be griped by two plane 
 surfaces, pressed against it by screws ; and its effectual length 
 will be measured from the lower edge of these surfaces, provided 
 the pressure be sufficient to support its whole weight. Now if a 
 bar of the same metal with that which forms the rod of the pen- 
 dulum, and of an equal length, be adapted to the back of the 
 clock-case ; and if it be so applied to a horizontal arm, attached 
 to the spring that bears the pendulum, that it shall in its contrac- 
 tions and expansions cause the spring to slide between the surfaces 
 on which it rests ; every change in the length of the pendulum 
 rod, under the influence of temperature, would be exactly coun- 
 teracted. The objection that applies to this method is, that if 
 the friction of the surfaces against the spring be sufficient to bear 
 the whole weight of the pendulum, it will interfere with the ac- 
 
Book If^J THE PENDULUM. 279 
 
 tion of the compensation; while if it be not, the effective length 
 of the pendulum is no longer determined by them. Such is the 
 plan of compensation proposed by Dr. Fordyce, which has the 
 advantage of- great simplicity, although in consequence of the 
 defects that have been stated, it is not perfect. 
 
 (2.) The rod and bulb of a pendulum being separate bodies, 
 and the former generally passing through the latter, which only 
 rests upon a projecting part, it occurred to Graham, that were 
 they to be made of two different substances, the expansion of the 
 rod downwards, might be compensated by the expansion of the 
 bulb upwards. There is not, however, a sufficient difference in 
 the expansibilities of the solid metals, to allow this principle to 
 be carried into effect by means of them. But mercury expands 
 about 16 times as much as steel; and hence, could the rod be 
 made of the latter metal, and the bulb of the former, in a ratio of 
 dimension of about 16:1, a compensation might be effected. 
 To form the bulb, the mercury is placed in a cylindrical jar of 
 glass; and in order to support the jar from beneath, the rod is 
 divided into two branches, connected at the lower end by a plate, 
 having thus a shape analogous to a stirrup. This pendulum, 
 called the Mercurial Pendulum of Graham, is perhaps the most 
 perfect of all compensations. Its original adjustments are, how- 
 ever, more difficult than those of some others we shall describe ; 
 and if it should be removed from the place where it was origi- 
 nally adjusted, the experiments must be again repeated. It is, 
 therefore, only used infixed observatories. 
 
 To investigate the relation between the length of the column of 
 mercury in the vase of the mercurial pendulum, and the whole 
 length of the rod : 
 
 Let / be the whole length of the steel rod ; s the lineal expan- 
 sion of steel ; y the unknown length of the column of mercury ; 
 m the cubic expansion of mercury ; g the lineal expansion of 
 glass. Suppose that the conditions of the problem require the 
 centre of gravity of the mercury to remain in a constant position, 
 which would be true, provided the position of the centre of gravity 
 of the rod also remained constant. The distance of the centre 
 of gravity of the mercury in the vase from the bottom, or extre- 
 
 y 
 
 mity of the pendulum, is ^, and the distance, L, of its centre of 
 gravity from the axis of suspension, is, 
 
 T-/ y . 
 2 ' 
 
 and for a perfect compensation, this expression must be a con- 
 stant quantity, however / and g vary under the influence of tem- 
 perature. 
 
880 APPLICATIONS or [Book IV. 
 
 If r be the radius of the vase that contains the mercury at the 
 standard temperature ; V, the volume of mercury, will be 
 
 V=*r 2 i/. (308) 
 
 At t degrees, the radius of the vase will become 
 
 y will also vary and become y' ; and, at the same time, the vol- 
 ume of mercury will become 
 
 hence 
 
 V(l+mO=irf*(l+ gtfT ; (309; 
 
 dividing this by the equation (308), we obtain 
 
 (1+ffOY 
 l+m<= y ; (310; 
 
 whence we have 
 
 which is the value of the height of the column of mercury at the 
 new temperature, t. But as g is but a small quantity, its seconc 
 power may be neglected without any sensible error ; and the ex 
 pression will become 
 
 y'=y+y(m2g)t. (312 
 
 The constant distance, L, will at the same temperature be 
 
 L=J_ f+Jd y; (313 
 
 and substituting the value oft/', 
 
 ; (314 
 
 and as this must be equal to the first value, the variable term af 
 fected by t must be =0, or 
 
 Is |(w 2o-) = 0; (315 
 
 whence we have for the value of?/, 
 
 y=l 2 * ; (316 
 
 m 2g ' 
 
 taking the exact expansions of the three substances between th 
 boiling and freezing points, we have 
 
 a=0.00124, 
 w=0.01848, 
 g=0.00088 ; 
 and substituting these values, we obtain 
 
 I 
 ^-6/75' 
 
Book IV.] 
 
 THE PENDULUM. 
 
 281 
 
 s b s b 
 
 1) 5 5s 
 
 (3.) Harison, finding that Graham had failed in applying the 
 solid metals as a compensation to the two separate parts of the 
 pendulum, abandoned that method, and sought for an application 
 of them to the rod alone. Inferring from experiments on the 
 expansion of brass and steel, that their relative expansion was 
 nearly in the ratio of 3 : 5, he planned a frame of which the fol- 
 lowing figure will give an idea. It was composed of nine paral- 
 lel bars, and from a fancied resemblance, was called the Gridiron 
 Pendulum. The five bars marked s, are of steel, the four 
 marked 6, are of brass ; the centre rod of steel is fixed at top to 
 the cross-bar, connecting the two contiguous brass rods, but slides 
 freely through the two lower bars that cross its direction. This 
 centre rod bears the bulb. The remaining 
 rods are fastened to the cross pieces at both 
 ends, and the outer and upper cross-piece 
 is attached to the axis of suspension. It 
 will be apparent, from inspection, that the 
 steel rods will, in their expansion, tend to 
 lengthen the pendulum, while the brass 
 rods will, in theirs, tend to shorten it. If 
 these two expansions exactly counterba- 
 lance each other, the length of the pendu- 
 lum will remain invariable. The four 
 brass rods act by pairs, and, therefore, as if 
 there were but two ; and no more than 
 three of the five steel rods are to be con- 
 sidered, for four of them also are connected 
 in pairs. Hence, if the expansion of the 
 sum of three of the steel rods added to the 
 expansion of the rod that connects the grid- 
 iron frame to the axis of suspension be 
 equal to the expansion of two of the brass 
 rods, the condition of compensation is ful- 
 filled. 
 
 To investigate the lengths that should be given to the brass 
 rods in the gridiron pendulum of Harison : 
 
 Let L be the distance from the axis of suspension to the ex- 
 treme end of the pendulum ; and let ihe condition of compensa- 
 tion be that this length shall remain constant. Let / be the com- 
 mon length of the outer pair of steel rods ; /' that of the inner pair; 
 a the distance from the gridiron frame to the axis of suspension 
 36 
 
APPLICATIONS OF [Book IV. 
 
 occupied by a steel rod ; and b the length of the steel rod that 
 bears the bulb, occupying the middle of the frame. Let X and 
 / be the several lengths of the two pairs of brass rods. We have 
 from the construction of the apparatus, 
 
 L=o+ W+& X X' ; (317) 
 
 if we call the expansions of brass and steel, B and S, we have for 
 the constant value of L, after a change of temperature of/ de- 
 grees, 
 
 L=o+/+r+& x V 
 
 + [(a-f W+&) . S (X+X') . B] ; (818) 
 and in this equation, the variable term affected by S and B =0, 
 
 (a+/-H'+6)S (X+X')B=0; 
 but we have from our first equation, 
 
 and substituting this value of the foregoing, we obtain 
 
 (L+X+X) S (X+X') B=0 ; (319) 
 
 whence 
 
 (X+X'). (B S) = 
 
 If we take the approximate ratio of expansion used by Hari- 
 son, we have 
 
 3L 
 
 or for the joint length of the two pairs of brass bars, one and a 
 half times the whole distance, from the axis of suspension to the 
 extremity of the pendulum. 
 
 These investigations may serve as a guide in the construction 
 of these two species of pendulum, but they are obviously inex- 
 act. Nor is it necessary that they should be more accurate ; for 
 the adjustment of each must finally be made by experiment ; and 
 reference must be had, not only to the pendulum itself, but to 
 the clock ; for the action of the clock on the pendulum will be 
 affected by changes of temperature, and the pendulum must meet 
 this, as well as its own variations. 
 
 In the gridiron pendulum, as has been seen, five rods, three 
 of steel and two of brass, are sufficient for the purpose of com- 
 pensation ; the other four are added for the purpose of making 
 it symmetric, and causing the line of direction of its centre of 
 gravity to pass, when the pendulum is at rest, through the centre 
 of magnitude of the bulb. 
 
 (4.) The expansion of an alloy of S pts of zinc and one of tin, 
 is to that of steel nearly as 2 : 1. Hence, a smaller number of 
 bars of these two substances will furnish a compensation. Thus 
 
Book IV.\ THE PENDULUM. 
 
 the pendulum applied by Breguet to his clocks, is composed of 
 no more than five rods, three of steel and two of zinc; two of 
 the steel rods and the two of zinc being combined in pairs, so 
 that it may be considered as composed, so far as the principle of 
 compensation is concerned, of no more than three rods. 
 
 The pendulum of Harison has been improved by Troughton, 
 who has substituted for the two pairs of brass rods, two cylinders 
 of the same metal sliding one within the other, to which the iron 
 rods are attached. The principle is obviously the same, but it 
 has some advantages over the original form, inasmuch as it is 
 less liable to external injury, and the brass tubes will not bend 
 under the upward pressure of their expansion, to which rods of 
 the same metal are liable. 
 
 Such are the more important of the forms of compensation 
 pendulums. The number and variety of those that have been 
 proposed is very great, and it would occupy too much space to 
 enter into a description of them. Those who feel a curiosity to 
 examine their different structures, will find an admirable paper 
 on the subject by Kater, in the work on Mechanics, which bears 
 the joint names of himself and Dr. Lardner. 
 
 283. In order to determine the intensity of gravity, by means 
 of the pendulum, it becomes necessary to measure its length; 
 that is to say, to determine the distance between its axis of sus- 
 pension and centre of oscillation. Two principal methods are 
 now employed for this purpose, those of Borda and Kater. 
 
 284. In Borda's method, the experimental pendulum, from 
 the measure of which the length of the second's pendulum is to 
 be inferred, is composed of a sphere of platinum, suspended by 
 a slender wire of iron, from a knife-edge of steel resting on plane 
 surfaces of polished agate. This form, employing the densest of 
 known substances, and the slenderest wire that is sufficient to 
 bear it with safety, approaches as nearly as possible to the hypo- 
 thetical simple pendulum. 
 
 Its length, considered as a pendulum, or the distance between 
 its centres of suspension and oscillation, is determined by calcu- 
 lation from its total physical length, obtained by actual measure- 
 ment. To effect this measurement, the pendulum is rendered 
 stable, by screwing up from beneath, a cup-shaped vessel, that 
 just catches the ball of the pendulum, without bearing any part 
 of its weight. A scale of iron is then applied to it, on which the 
 physical length is marked. An improved method consists in 
 screwing up from beneath a plane of polished steel, until it just 
 touches the platinum sphere ; the pendulum is then removed, 
 and to its place is adapted a scale, by means of knife-edges simi- 
 
284 APPLICATIONS OP [Book IV. 
 
 lar to those of the pendulum. This scale is composed of two parts, 
 one of which is firmly fastened to the knife-edge, and is shorter 
 than the pendulum ; the other slides upon this, and is moved by 
 a screw. The scale being thus placed, the moveable part is de- 
 pressed by means of the screw, until it just touches the steel plate; 
 the length of the two portions united, that is to say, of the part 
 fixed to the knife-edge, added to that of the projection of the 
 moveable part, is of course just equal to the physical length of the* 
 experimental pendulum. 
 
 The theoretic length, or the distance between the axes of sus- 
 pension and oscillation, is next deduced, upon the principle of its 
 being a sphere suspended by a line void of weight, by the for- 
 mula (306), 
 
 Such at least would be the principle, were the wire and sphere 
 the only parts of the pendulum, and the former devoid of weight. 
 As, however, neither of these is true, particularly as parts must 
 be adapted to attach the wire to the knife-edge and to the sphere, 
 a much more complex formula must be used in practice. We re- 
 fer for this to the " Base du Systeme Metrique," and to Delam- 
 bre's Treatise on Astronomy. 
 
 In the original apparatus of Borda, the length of the experimen- 
 tal pendulum was four times the length of the second's pendulum. 
 The time of its oscillation was determined by a method called 
 that of Coincidences. For this purpose, the pendulum was sus- 
 pended upon knife edges, resting on planes of agate, in front of a 
 well-regulated astronomical clock, having a compensation pendu- 
 lum. The knife edge was moved upon the agate planes, until the 
 wire of the experimental pendulum, as viewed through a small 
 telescope, placed at the distance of 12 or 15 feet in front of Iho 
 clock, exactly coincided with the centre of the circle, bounding 
 the lenticular bulb of this clock pendulum. This point was mark- 
 ed by drawing two black lirfcs on a white ground, making each 
 an angle of 45 with the horizon ; a black screen was so placed 
 as to hide just half the wire of the experimental pendulum. The 
 two pendulums being set in motion, an observer placed at the tele- 
 scope, would see the wire, and the point marked upon the clock 
 pendulum disappear behind the edge of the screen, at each of 
 their alternate oscillations. If, when first observed, they did not 
 pass the edge of the screen at the same instant of time, they would, 
 provided the one were not exactly four times as long as the other, 
 gradually approach, until both would disappear at the same in- 
 stant. The time marked by the clock is then noted, as the instant 
 of the first coincidence. 
 
Book IV.] THE PENDULUM. 285 
 
 It is usual to make the experimental pendulum a little longer 
 than four times that of the clock ; hence the former makes 
 a little less than one oscillation for every two of the latter. Af- 
 ter the interval of four seconds, the wire and the cross will be 
 again in the field of the telescope at the same time, but the cross 
 will precede the wire. At each successive interval of four seconds, 
 the distance at which they pass each other, will increase until the 
 interval ofthe times of their respective disappearances amounts to 
 1 ". After this they will approach, until they again pass the eye, 
 and disappear behind the screen at the same instant, which is no- 
 ted as that of a second coincidence. During this interval, the 
 clock pendulum will have gained two oscillations upon the experi- 
 mental pendulum ; that is to say, the number of the oscillations of 
 the experimental pendulum, will have been one less than half the 
 number of seconds marked by the clock ; the latter number is 
 obtained by simple inspection ofthe dial ofthe clock. 
 
 The observation of the coincidences is continued, until the ex- 
 perimental pendulum has lost too much of its mQtion to render 
 them easily distinguishable, and the record of the times is col- 
 lected in a set. 
 
 2S5. In order to reduce the oscillations performed by the ex- 
 perimental pendulum, to a cycloid, or an infinitely small circular 
 arc ; the extent of the arcs of vibration on each side of the verti- , 
 cal, are observed at each coincidence. The correction is applied 
 upon the principle of the formula, (2S5), in whieli the last terms 
 of the series of (S3), are neglected. This, in very small arcs, 
 becomes very nearly 
 
 sin. 2 a 
 
 -nr 
 
 It is more convenient, however, to apply the correction to the 
 whole set, in which case the mean of the first and last arcs of 
 vibration may be taken. But as the arcs decrease in fact in a geo- 
 metric progression, a more correct formula has been calculated 
 by Borda, which is as follows : 
 
 j'\ 
 ra v ~ i ~ j ~*~- v ~ ^ ; ^ '321} 
 
 32 M (log. sin. a log. sin. a' ' 
 in which a and ' are the greatest and least amplitudes ofthe 
 oscillations; and M, the modulus of the tables of logarithms, 
 =2.30255509. 
 
 The pendulum vibrating in air, the number of oscillations it is 
 observed to make, must be corrected for the buoyajicy of the air, 
 which is done by the formula (307). No correction has hith- 
 erto been applied for the variation in the arc's resistance, growing 
 
286 APPLICATIONS Off 
 
 out of the figure of the pendulum, detected by Bessel and Sabine, 
 as stated in 279. 
 
 As the temperature of the apparatus may vary during the series 
 of coincidences, the length obtained by measurement will proba- 
 bly not be the same as that at which anj' one of the coincidences 
 has occurred, and the latter will differ from each other. The 
 wire being. capable of expansion and contraction, by changes of 
 temperature, it will be necessary to reduce the whole to some 
 standard temperature. So also must the length of the rod, by 
 which it is measured, be corrected for the temperature at which 
 that part of the operation is performed. 
 
 The corrections for the expansion by temperature, may be ob- 
 tained from the tables of the expansion of the metals that are to 
 be found in authors on physical subjects : or, they may be ob- 
 tained from experiments on the varying rate of the pendulum's 
 own oscillations, at different degrees of heat. The last method 
 was used by Sabine, and is the best, inasmuch as the actual ex- 
 pansion of the apparatus is obtained, instead of the mean expansion 
 of the species of substance employed. 
 
 When the length of the experimental pendulum is known, and 
 the number of oscillations it performs in a given time, say in a 
 mean solar day, is determined and reduced to a cycloidal arc and 
 a vacuum ; the length of the pendulum that would vibrate seconds, 
 may be at once obtained ; for it has been shown, 272, that the 
 respective lengths of pendulums, are inversely as the squares of 
 their numbers of vibration in equal times; hence, as there are 
 6400 seconds in a mean solar day, if L be the lenyth of the expe- 
 r mental pendulum, corrected for temperature ; N Ihe number of 
 vibrations it performs in a mean solar day, corrected for the arc 
 of vibration, and for the buoyancy of the air : the length / of the 
 pendulum that vibratos seconds, will be 
 
 (86400) 2 ' 
 
 It still remains that the length thus obtained should be reduced 
 to the mean surface of the earth, or to the level of the sea. The 
 altitu/le of the place of observation above this surface, must, 
 therefore, be determined, and the reduction performed by the 
 formula, (293), 
 
 The form and nature of the ground will influence this result ; and 
 hence, when the elevation is considerable, the length thus obtained 
 will not be that which would be found at the level of the sea. It 
 has been stated by Dr. Young, in the Transactions of the Royal 
 
Book IT.} TUB PENDULUM. 287 
 
 Society of London, that if the station be upon a table land of a 
 density equal to fds of the mean density of the earth, the diminu- 
 tion of the force of gravity will be no more than one-half of what 
 is due to the height above the level of the sea; and that in the 
 most unequal country, there will be at least lh to be deducted 
 from the correction obtained by the above formula, (293). In 
 Sabine's investigation, this correction has been multiplied by the 
 constant coefficient, 0.6. 
 
 It has also been discovered by Sabine, whose remark has been 
 confirmed by Biot, that the nature of the ground on which the 
 experiment is performed, affects the length of the pendulum in 
 all places. The attraction of gravitation being greater upon dense 
 earthy substances than it is upon rare. 
 
 As an instance of the effect of elevated ground, we shall cite 
 the observations of Carlini, at Mount Cenis. The pendulum 
 measured at the top of the mountain, and reduced to the level of 
 the sea by the usual method, had a length in French measure of 
 0.^993708 ; while a pendulum, for the same latitude deduced from 
 the pendulum of Bourdeaux, would not have been longer than 
 0.^993498. The difference of 0.^000210 is, therefore, due to 
 the local attraction. This observation may be cited as being 
 among those whence the density of the earth has been inferred. 
 
 The method of Borda has been improved by Biot, and the ap- 
 paratus rendered more convenient. The length of the experi- 
 mental pendulum has been reduced to one that differs but little 
 from that of the pendulum of the clock : a copper wire has been 
 substituted for one of steel, as less liable to rust; and the whole 
 apparatus enclosed in a glass case, to render it less exposed to the 
 action of currents of air, and to sudden changes of temperature. 
 With this diminished length, the pendulum that moves fastest 
 still gains two oscillations upon the other, between each coinci- 
 dence. 
 
 2S6. The method of Kater is founded upon the principle, 
 276, that the centres of suspension and pscillation are con- 
 vertible points; and conversely, that if a pendulum vibrate in 
 equal times, upon two different parallel axes, one of these has the 
 relation to the other of the axis of suspension to that of oscilla- 
 tion. If then a pendulum be taken, into which two knife edges, 
 turned in opposite directions, are inserted ; if the distance be- 
 tween these knife edges is very nearly that which can be esti- 
 mated to exist between the centres of oscillation and suspension ; 
 and if a moveable weight be adapted to the rod, this weight may 
 be so adjusted by trials, that the pendulum shall oscillate in equal 
 times, when suspended by either axis. 
 
288 
 
 APPLICATIONS OF 
 
 [Book 
 
 Cls 
 
 The form of the pendulum of Kater is such as is represented 
 beneath. 
 
 A The rod AEFA', is a bar of hammered brass, to 
 
 which is adapted the bulb B, of cast brass, and of 
 a form no more different from a cylinder than is 
 necessary for the convenience of casting. The 
 knife edges made of wootz, are represented at a 
 and a' ; they are formed of two planes, meeting at 
 an angle of about 60. The moveable weight is 
 in two parts, and slides on the bar AA'. The 
 part c, is capable of being firmly fixed to the bar 
 by a screw, whose head is represented ; the part 
 d, is attached to the part c, by a screw of a. fine 
 thread, by means of which a slow motion may be 
 given to d after c is made fast. The pendulum is 
 brought nearly to its adjustment, by sliding the 
 whole weight along the rod ; c is then firmly fast- 
 ened by its screw, and the adjustment is completed 
 by the slow motion of d. 
 
 The portions E and F, of the rod A A', were 
 in the original apparatus of Kater, made of black- 
 ened wood. 
 
 The observations of the coincidences, by means 
 of which the pendulum is adjusted, and whence 
 the value of its oscillations is determined for cal- 
 culation, when adjusted, are made as follows : 
 The clock pendulum being at rest, a small tele- 
 scope is placed directly in front of it, and the ex- 
 perimental pendulum is suspended in such a man- 
 ner that one of its blackened terminations, just 
 hides a white circular spot, drawn upon the lens 
 of the clock pendulum, and concentric with it. 
 The diaphragm of the telescope has two plates 
 with vertical edges ; these are pressed forward by 
 screws, until they appear in optical contact with 
 the blackened bar. When the pendulums are set 
 in motion, during the greater part of the oscilla- 
 ^^^^^ tions, the bar and the white spot will both be seen ; 
 
 but, from time to time, for a few contiguous oscil- 
 lations, the bar will wholly hide the white spot. Kater took, 
 as the instant of coincidence, the beat of the pendulum that fol- 
 lowed the first passage in which no part of the white spot was 
 seen. This method is objectionable, inasmuch as the same ob- 
 server will, under different circumstances of light, continue to 
 
Book IV ,] THE PENDULUM. 289 
 
 see the spot a longer or shorter time before it is wholly ob- 
 scured. The tact of observation, even in the same observer, will 
 differ at different times, and will improve by practice; while, 
 when the experiments of different observers are compared, a 
 marked difference will be found to exist in their powers of vi- 
 sion. 
 
 For these reasons, Sabine, in his experiments, adopted the plan 
 of observing, not only the disappearance, but also the reappear- 
 ance of the white disk ; noting the second succeeding the former, 
 and preceding the latter : the mean of the two was taken as the 
 instant of coincidence. The author had a good opportunity of 
 testing the value of these two different methods, when associated 
 with Sabine, in the experiments to determine the length of the 
 pendulum at New- York. Had the experiments made by the two 
 observers been compared upon the plan of Kater, a considerable 
 discrepancy would have ensued ; but when compared by the me- 
 thod of Sabine, the results were nearly identical. 
 
 The length of the experimental pendulum, in Kater's method, 
 is determined directly, without reference to calculation,, by mea- 
 suring the distance between the. knife edges. This is effected 
 by means of a scale furnished with powerful microscopes ; to one 
 of these a micrometer is adapted. With this apparatus, the 
 10,000th part of an inch becomes a measurable 1 quantity. The 
 method of Kater requires the same corrections and reductions as 
 that of Borda; thus the time of oscillation must be corrected for 
 the arc of vibration, and for the buoyancy of the air ; the length 
 must be corrected for the temperature, and the second's pendu- 
 lum, calculated from the observations, reduced to the level of the 
 sea. 
 
 The method of Kater is liable to one objection, namely, that 
 in conformity with the views of Bessel, it will be unequally re- 
 sisted by the air, when suspended by its different knife-edges. 
 A modification of the apparatus proposed by Bailey, is less liable 
 to this objection : this pendulum is a simple bar, without a bulb, 
 and the adjustment is effected by filing away those portions 
 whose weight is in excess. 
 
 287. In order to determine the measure of the force of gravity 
 at any given place, when the length of the second's pendulum is 
 known, we have the formula (289) 
 
 g=l*. 
 
 At New-York, the length of the pendulum, or /, is 39.1 in, 
 nearly, or 3 ft. 2583, hence 
 
 g=32 ft. 1576, 
 or nearly 32| feet. 
 37 
 
290 APPLICATIONS OF [Book IV. 
 
 A heavy body will, in consequence, fall in vacuo during the 
 first second of time, 271, through a space equal to 16^ feet. 
 
 288. It has been stated that the apparent intensity of gravity, 
 or the difference between its absolute force, and the diminution 
 growing out of the earth's rotation, may be immediately deduced 
 from a measure of the second's pendulum. In the method of 
 Borda, the experimental pendulum is made to vibrate in the se- 
 veral places in which it is desired to ascertain this quantity ; but 
 as the length of the suspending wire may vary, it becomes ne- 
 cessary to go through the whole process at each station. 
 
 In the method of Kater, as the distance between the knife 
 edges is invariable, except by changes of temperature, the influ- 
 ence of which is known, one careful measurement will suffice for 
 any number of stations. /The original pendulum may, therefore, 
 be carried from station to station, and its coincidences observed. 
 A direct comparison between those observed at different places, 
 givesan immediate determination of the length of the pendulum 
 that would oscillate in a second, at the several different stations. 
 
 The method has been still farther improved by its author. A 
 pendulum, having but one knife edge at the usual point of sus- 
 pension, is suspended in front of the clock, in the place where 
 the original experiment was made, with the pendulum with con- 
 vertible axes. The rate of its oscillations being determined, its 
 length can be calculated by the formula (290), from that of the 
 original pendulum. It may then be carried to other places, and 
 the length of the pendulum of the place determined from the rate 
 of its oscillations, in the same manner. In this way, Kater him- 
 self determined the length of the second's pendulum at the more 
 important stations of the British Trigonometrical Survey. Sabine 
 has since employed the same method, at a variety of stations, 
 from Ascension, in lat. 7, 55', 48", S. ; to Spitsbergen, in lat. 
 79. 49'. 58". N. 
 
 This part of Kater's method, as applicable to observations, at 
 different places, is much more convenient than that of Borda, 
 even as improved by Biot. It may also be performed in a much 
 less time ; thus, for instance, Biot was engaged for three months 
 at Unst, in completing his measure of the pendulum, while Kater 
 effected his in three weeks. 
 
 289. The relation between the lengths of the pendulum, at 
 different places, may also be determined by means of a clock fur- 
 nished with a pendulum whose rod is not liable to have its di- 
 mensions changed by transportation, except in consequence of 
 variations of temperature. Such a clock was used in the first 
 
Book 
 
 THE PENDULUM. 
 
 291 
 
 expeditions of Parry and Ross, and the absolute length was ascer- 
 tained, by comparison with the original experiment of Kater. 
 
 The method of Kater is still imperfect, inasmuch as the length 
 determined at the original station, and therefore at all others, 
 still rests upon his own single experiment ; and it has not yet 
 been ascertained, how far it is possible for the same experimenter, 
 or another equally well qualified, to reproduce an identical re- 
 sult. Until this question be settled, i^ must remain questionable 
 whether the differences in the measure of the pendulum at the 
 same place, by the two different methods of Borda and Kater, 
 arise from the methods themselves, or are involved in the origi- 
 nal determination on which the results of the latter method are 
 founded. 
 
 290. By the use of these two metho Is, the pendulum has been 
 measured in various places in both hemispheres, by Kater, Sa- 
 bine, Biot, Bouvard, Matthieu, Arago, Chaix, Freycinet, and 
 Duperrey. Some of these results are to be found in the following 
 
 TABLE. 
 
 STATION. 
 
 NORTH 
 LATITUDE. 
 
 ill 
 
 .00 
 
 3 e : 
 
 PENDULUM. 
 
 At fhe 
 
 Station. 
 
 Reduced to 
 the level of 
 the Sea. 
 
 St. Thomas, . . . 
 Maranham, ... 
 Sierra Leone, 
 Trinidad, .... 
 Jamaica, .... 
 Formontera, . . . 
 New- York, 
 Bourdeaux, . . . 
 Paris .... 
 
 0".24'.41" 
 2.31 .43 
 8 .29 .28 
 10.38.56 
 17.56.07 
 38.39.56 
 40 .42 .43 
 44 .50 .26 
 48.50 .14 
 51 .31 .08 
 55 .58 .39 
 60.45.26 
 70 .04 .05 
 79 .49 .58 
 
 21ft 
 77 
 190 
 21 
 9 
 606 
 67 
 56 
 230 
 921 
 69 
 30 
 29 
 21 
 
 39.02069 
 39.01197 
 39.01954 
 39.0187^ 
 39.03508 
 39.09176 
 39.10153 
 39.11282 
 39.12843 
 39.13908 
 39.15540 
 39.17145 
 39.19512 
 39.21464 
 
 39.02074 
 39.01214 
 39.01997 
 39.01884 
 39.03510 
 39.09325 
 39.10168 
 39.11295 
 39.12894 
 39.13929 
 39.15546 
 39.17151 
 39.19519 
 39.21469 
 
 London, .... 
 Leith, 
 
 TTust 
 
 Harnmerfest, . . . 
 Spitsbergen, . . . 
 
292 
 
 APPLICATIONS OP 
 
 A part of the observations that have been made in the south- 
 ern hemisphere will be found in the following 
 
 TABLE. 
 
 STATION. 
 
 SOUTH 
 LATITUDE. 
 
 PENDULUM REDUCED 
 TO THE LEVEL OF 
 THE SEA. 
 
 Ascension, 
 
 7 l> .55'.48" 
 
 39.02410 
 
 Bahia, .... 
 Isle of France, . . 
 Port Jackson, . . 
 Malouine Islands 
 
 10.38.56 
 20.09.19 
 33.51.39 
 51 .31 .44 
 
 39.02435 
 39.04793 
 39.08049 
 39.13695 
 
 The general inference of Sabine, from a combination of his 
 own experiments with those of Kater^ and those made at the 
 stations of the French trigonometric survey, is that the length of 
 the equatorial pendulum =39 in. 01569 ; the increase of its length 
 from the equator to the pole =O in. 20227. The formula, (296), 
 for the length /', at any intermediate latitude L, becomes (296), 
 
 /'=39.01569-|-0.20227 sin. 2 L. 
 
 A more recent French deduction, into which the observations 
 of Duperrey and Freycinet enter, gives 
 
 J'=39.01741 +.23505 sin.= L. 
 
 The result obtained by Sabine gives, for the oblateness of the 
 terrestrial spheroid, oj^; and for the centrifugal force at the 
 equator, T -J- T th part of the whole force of gravity. The last de- 
 duction gives for the centrifugal force, T J 7 . The centrifugal 
 force is usually stated at T 4 F , which lies between the above in- 
 ferences ; and this is the value that we shall employ. 
 
 291. It had, until the discovery of the influence of local at- 
 traction by Sabine, been generally concluded, that the pendulum 
 vibrating seconds in a given latitude, at the level of the sea, was 
 a constant and invariable quantity. Its length is also capable of 
 comparatively easy determination ; as all the observations con- 
 nected with its measure may be made within the space of a few 
 weeks. Hence it has been proposed as a standard of measure. 
 Were the first inference true, and were the reduction to the level 
 of the sea independent of local influence, no better method could 
 well be devised for the purpose of re-establishing standards of 
 lineal measure that have been lost, or are suspected of being al- 
 tered by age. It has also been proposed to use the length of the 
 pendulum not only as the standard, but as the unit of lineal mea- 
 sure. This, however, is objectionable, except in the case where 
 
:i ' -, 
 
 Book IV.] THE PENDULUM. 293 
 
 the customary unit of a country differs but little from the length 
 o;" the pendulum. Such is the case with the measure of Denmark ; 
 aid hence, underthe auspices of Schumacher, a system of weights 
 aad measures has been formed in that country, of which the pen- 
 dilum forms both the unit and the standard. 
 
 In cases where the difference between the unit in actual use 
 and the length of the pendulum is considerable, it is better to 
 retain the ancient unit, and define its relation to the pendulum. 
 For this purpose, it will be evident from what has been said in 
 relation to the influence of local circumstances, that it will not 
 be safe to use the pendulum of a given latitude; but that the only 
 admissible method is to take the pendulum measured in a par- 
 t cular place as the standard. 
 
 When a unit of lineal measure has been defined, in relation 
 t some standard existing in nature, its square will serve as a 
 uiit of measures of surface ; its cube, or some aliquot part, 
 as the unit of measures of capacity ; and the weight of its cube, 
 filed with pure water at some given temperature, will furnish a 
 Uiit of measures of weight. It has, however, been found more 
 eisy to determine the weight of water that a measure of capacity 
 wll hold, than to ascertain its cubic contents; and hence, in 
 sone systems, the unit of capacity has been defined by declaring 
 what number of the units of weight it shall contain. 
 
 292. -This being premised, we shall proceed to describe the 
 principles upon which a reform 'has been effected in the standards 
 of England, and of the State of New- York, in both of which the 
 pendulum has been assumed as the basis. 
 
 The standard of measure in Great Britain is the pendulum, vi- 
 brating seconds in a cycloidal arc, in a vacuum, and at the level 
 of the sea, in the latitude of London, 51 31' OS" N. 
 
 The unit of measures of length is the Yard of such magnitude 
 that the pendulum shall measure, 39 in. 13929. The yard is di- 
 vided into three feet ; the foot into twelve inches ; and for cloth 
 measure the binary subdivision is permitted. Greater measures 
 of length are multiples of the yard, derived as in the ancient 
 system. 
 
 The square of the yard, or of any other unit of length, may be 
 used as a unit of superficial measure. 
 
 The standard temperature, to which measures of length or sur- 
 face are to be reduced, is 62 of Fahrenheit's thermometer, and 
 the material of which the standard yard is made is brass. 
 
 The unit of weight is the Troy Pound, of such magnitude that 
 a cubic inch of water, at 62, weighs 252 grs. 458, there being 
 5760 grs. in this pound. 
 
294 APPLICATIONS OF [Book 1 1 ' . 
 
 \ 
 
 The avoirdupois pound is also used, and isdefinedas beingequpl 
 to 7000 grs..Troy. 
 
 The unit of measures of capacity is the Gallon, which is a vefr- 
 sel thai holds exactly ten avoirdupois pounds of water, at tBe 
 temperature of 62. 
 
 The bushel holds eighty pounds of water at the same tempe- 
 rature. 
 
 293. The standard of the state of New-York is the pendulum 
 vibrating seconds in a cycloidal arc, and in a vacuum in Colum- 
 bia College in the city of New-York. 
 
 The unit of lineal measure is the Yard, which is of such mag- 
 nitude as to bear to the pendulum the proportion of 1,000000 to 
 1,086158. 
 
 Its usual subdivisions are allowed to be employed ; and is 
 standard temperature is that of melting ice. It is identical will 
 the present British standard yard, which has been restored in is 
 magnitude to that used previous to the revolution, and whi<p 
 had continued in use in the State of New-York ; but in the coir 
 parison, each is to be taken at its own standard temperatur . 
 The standard temperature of the English system is 62 Fahre - 
 heit ; of that of the State of New-York, 32. 
 
 The unit of measures of weight, is the Avoirdupois Poutd, 
 of such magnitude that a cubic foot of pure water, at its maxi- 
 mum density, shall weigh 1000 oz. or G2lbs. 
 
 The unit of dry measures of capacity, is the Gallon, a vessel 
 of such magnitude as to hold exactly lOlbs. of pure water, at its 
 maximum density. The bushel, therefore, holds 80lbs. 
 
 The unit of liquid measure is also a gallon, containing eight 
 pounds of distilled water, at its maximum of density. The 
 adoption of this unit, was a deviation from the original plan, 
 which contemplated but one set of measures of capacity for so- 
 lids and fluids, and it has impaired the symmetry of the system. 
 
 294. To the English system, it is to be objected : that it as- 
 sumes for its standard the pendulum of a particular latitude, which 
 will not be constant, in consequence of local circumstances ; that 
 the determination on which the length of this standard has been 
 performed in a private building, the house of Mr. Brown ; that 
 it retains two units of weight, of the same denomination, but of 
 different magnitudes; and that its standard temperature is wholly 
 arbitrary, founded on no natural phenomenon, and dependent 
 upon a conventional thermometric scale. The mode of defining 
 its unit of weight, moreover, involves a fractional quantity, ami 
 the bulk of water employed in the determination, namely, a cu- 
 bic inch, is too small. 
 
Book IT.] THE PENDULUM. 295 
 
 To the system of the State of New- York none of these objec- 
 tions apply, except so far as relates to the double system of mea- 
 sures of capacity. The standard is the pendulum of a particular 
 place; and that, so far as is known, is invariable; that place is 
 a public building, readily accessible ; the standard temperature* 
 are defined by physical states of water, in respect to which there 
 can be no error, and which are independent of thermometric 
 scales. The unit of weight is determinable from a bulk of water 
 of sufficient magnitude. 
 
 295. The French system of weights and measures has for its 
 standard a quadrant of the meridian. The unit of measures of 
 length is the Metre, which is a ten millionth part of the* quad- 
 rant. 
 
 The unit of superficial measure, is the Are, a surface 10 metres 
 each way, or 100 square metres. 
 
 The unit of measures of capacity, is the Litre, a vessel con- 
 taining the cube of a tenth part of the metre. 
 
 The unit of weight is the Gramme which is equal to the weight 
 of the cube of the hundredth part of the metre, filled with distil- 
 led water, at its maximum of density, or to the 1000th part of 
 the weight of a litre of Water. 
 
 The standard temperature of the measures of length is that of 
 melting ice. 
 
 The whole of the divisions and multiples of the units were de- 
 cimal, and the principal of nomenclature adopted, was to prefix 
 the Greek numerals to the decimal multiples, and the Romaa 
 numerals to the decimal subdivisions of the units. 
 
 Thus the measures of length are, 
 
 Myriametre = 10000 metres. 
 Kilometre = 1000 metres. 
 . Hectometre = 100 metres. 
 Decametre = 10 metres. 
 Metre = 1 metre. 
 
 Decimetre = *-$ metre. 
 Centimetre = ,1^ metre. 
 Millimetre = , ^ metre. 
 The measures of Surface are 
 
 Hectare = 10.000 sq. metres. 
 
 Are = 100 sq. metres. 
 
 Centiare = 1 sq. metre. 
 
 The measures of Capacity are 
 
 Kilolitre = 1000 litres. 
 
 Hectolitre 100 litres. 
 
 Decalitre = 10 litres. 
 
 Litre = 1 litre. 
 
 
296 APPLICATIONS OF [Book IV. 
 
 Decilitre = T V litre. 
 Centilitre = r litre. 
 The weights are 
 
 Myriogramme = 10000 grammes. 
 
 Kilogramme = 1000 grammes. 
 
 Hectogramme = 100 grammes. 
 
 Decagramme = 10 grammes. 
 
 Gramme = 1 gramme. 
 
 Decigramme = T V gramme. 
 
 Centigramme T ^ gramme. 
 
 Milligramme = , 7 V F gramme. 
 
 296. No system can be imagined more perfect and beautiful, in 
 a scientific point of view, than this system of the French nation. 
 It is founded upon a standard existing in nature, and invariable, 
 and which is susceptible of determination to such a degree of ex- 
 actitude, that no probable error that can, in the present state of 
 science, be committed in the measure of degrees, will affect the 
 small fraction of the standard that forms the unit of length. 
 From its decimal division, it is exactly consistent with our usual 
 system of arithmetic; and its nomenclature is systematic, and of 
 easy recollection. Still it is not without/ault, even in a scientific 
 point of view. The measures of length are incapable, for instance, 
 of application to astronomic purposes, in which we use the semi- 
 diameter of the earth, and not its quadrant as the unit ; and these 
 two magnitudes are incommensurable. Neither are we aware 
 that a measure of the meridian in other countries, particularly 
 in our own hemisphere, would reproduce the same magnitude 
 for the quadrant that was obtained in France. The measure of 
 a sufficient arc, whence to determine the length of a quadrant, is 
 an operation of grest expense, and would occupy a long time. 
 Hence, in presenting the types of the units to the National Assem- 
 bly, the commission propose to verify them, if suspected of alter- 
 ation, and reproduce them, if lost, by reference to the pendulum 
 of the Observatory of Paris; thus recurring to the same natural 
 standard that had been rejected by them in the outset. The 
 metre is, therefore, after all the labour that was expended in its 
 determination, no more than a conventional length, whose rela- 
 tion to the second's pendulum of a particular place is well deter- 
 mined. It has also been found impracticable to introduce the 
 decimal division into the measure of angles; and after strenuous 
 attempts for that purpose, and the laborious construction of new 
 tables, even the astronomers of France have returned to the an- 
 cient division of the circle. 
 
 The objections, in a practical point of view, are more nume- 
 rous, and have been found insuperable. Thus, however well-cal- 
 
Book IV.~\ THE PENDULUM. 297 
 
 culated for scientific purposes, and even for those of commerce, 
 the decimal multiples of the unit may be, decimal subdivisions 
 have been found unsuited for the purpose of retail traffic ; for this 
 object no other than a binary system can, with convenience, be 
 used. In fact,' in the subdivisions of the unit, no other method 
 appears to be consistent with nature ; and those systems which are 
 founded on division by two, appear to defy any attempts to alter 
 them. Thus the system of money in the United States, which is 
 strictly decimal, is only used in written calculations; while the 
 old binary division of the Spanish dollar is retained in all retail 
 operations, in spite of the barbarous nomenclature that is applied 
 to it in some of the States. Several of the units of the French 
 system, or their decimal divisions and multiples, are unsuited to 
 ordinary transactions ; subdivisions suitable to these were, there- 
 fore, first introduced clandestinely, and afterwards sanctioned by 
 law. Thus a measure of the length of about a foot, is the most 
 convenient for many mechanical uses ; and for this purpose a 
 measure of the third part of a metre was formed, called the Me- 
 trical Foot, to which the ancient duodecimal subdivision was ap- 
 plied. The kilogramme differing but little from two ancient 
 pounds, its half has become the unit of weight in actual use, and 
 is called the Metrical Pound ; to this, also, a binary division has 
 been applied, and the decimal system in the desending scale has 
 not only failed in being introduced into commerce, but has been 
 abolished by authority. 
 
 Thus there are at present in France, in fact, three diverse sys- 
 tems ; the ancient, which is not wholly abandoned ; the decimal 
 system of the commission ; and a system derived from the latter, 
 to which the ancient names, and many of the ancient subdivisions 
 are applied. 
 
 The system proposed, and partially introduced by the French 
 philosophers, may, therefore, be considered as a splendid failure, 
 worthy however of a better fate, from the scientific skill with 
 which the operations connected with it were executed. It is also 
 memorable for the light it has thrown on all analogous processes, 
 and the actual benefit the researches, in respect to it, have confer- 
 rld upon physical science. Warned by the example of the French, 
 the British, Danish, and American governments, have wisely re- 
 stricted themselves to the verification of the measures in actual 
 use, and their restoration to their true dimensions. The two for- 
 mer, and the state government of New-York, have referred them 
 to the pendulum, a standard existing in nature, determinate, and 
 easily determinable. 
 
 The determination of units of measures of capacity, and of 
 weight, from standards existing in nature, involves certain nice- 
 
 38 
 
293 APPLICATIONS OF [Book IV. 
 
 ties founded on the mechanics of fluid bodies. These will be fully 
 explained in a subsequent part of this work. 
 
 297. Among the applications of the theory of the t pendulum, 
 may also be classed the principle of the calculations by means of 
 which the density of the earth is ascertained from the experiment 
 of Cavendish, 91. The balls attached to the balance being set 
 in motion, and caused to oscillate by the attraction of the masses 
 of lead presented to them, may evidently be considered as a hori- 
 zontal pendulum actuated by that attractive force. By comparing 
 the length of this pendulum with that of a common pendulum, 
 that would oscillate in the same time, we may obtain the relation, 
 between the attractive force of the spheres of lead, and of the earth, 
 considered as a sphere. Trie equation that expresses this relation 
 may be thus investigated : 
 
 Let us consider the bodies attached to the extremities of the 
 horizontal bar of Cavendish's apparatus, as if their masses were 
 collected in a single point, and abstract the mass of the bar itself, 
 so that each arm of the balance may be considered as a simple 
 pendulum. 
 
 Let the length of the arm =a; the distance of the centre of 
 gravity of the attracting mass, from the point of suspension of the 
 Dar? = c ; the angular distance between the end of the bar, and 
 the centre of gravity of the attracting mass, at the time motion 
 begins, =a ; their mean angular distance in the oscillations =/3 ; 
 the measure of the attractive force exerted at the distance of the 
 unit of lineal measure in which the distances are estimated, by a 
 mass whose magnitude is equal to the unit of weight in which the 
 masses are estimated, =/; the distance between the attracting 
 mass, and the end of the bar at the time motion begins, =c. 
 
 Call the measure of the attractive force of the earth, g-, the ra- 
 dius of the earth, R, and its mass m, we shall have for the value 
 of g, in trms of m,/, and R, 
 
 '',,.; <"& M 
 
 We should in like manner have for the value of the attractive force 
 of the mass employed in the experiment, provided its distance 
 from the end of the pendulum were constant, in terras of its mass 
 wi', of the force/, and the distance c, 
 
 But this force does not act directly : it must, therefore, be decom- 
 posed into two, one of which is perpendicular, the other parallel 
 to the bar that oscillates. The former alone acts, and its value 
 will be from 13, 
 
 w'/o sin. a 
 
Book 77^.] THE PENDULUM. 299 
 
 We are next to consider that this is not a constant force, but that 
 it acts with the least intensity at the time motion begins, and in- 
 creases until the bar approaches nearest to the attracting mass. 
 At this point, the torsion of the wire of suspension overcomes the 
 motion, and causes the bar to return. When the deflection =/3, 
 the two forces balance each other, and at this time we have the 
 mean value of the attractive force which will therefore be a func- 
 tion of f3. And we may without ^investigation assume, what might 
 be shown by a rigorous analysis, that it is inversely proportioned 
 to (3, or that to find the value of the attractive force that acts on 
 the balance at its mean intensity, the foregoing expression must 
 be multiplied by 1. Hence we have for the value of the attractive 
 
 force ', that acts to cause the oscillations, 
 
 The general expression, (186), gives for the value of the time of 
 the oscillations of any pendulum under the action of an attractive 
 force, g-, 
 
 T=W . 
 
 S 
 
 If we call the length of the common pendulum, whose oscillations 
 are synchronous with those of the bar /, we have by substituting 
 the value of g, from (a), and squaring 
 
 and by substituting the value of g-', from (6), and also squaring 
 T2 _ *V/36 
 
 m'fa sin. a ' 
 hence 
 
 ^R 2 /_ V/36 
 
 mf ~m'fa sin. a ' 
 
 
 m m'a sin. a 
 
 whence we obtain 
 
 m I R 2 a sin. a 
 
 and all the quantities in the second number of the expression may 
 be obtained from experiment ; and thus the ratio between the 
 mass of the earth, and that used in the experiment, will admit .of 
 calculation. 
 
 * See Poisson,vol. ii. p. 42. 
 
300 or COLLISION. [Book IV. 
 
 CHAPTER VI. 
 OF COLLISION. 
 
 298. The simplest mode in which motion can be communicated 
 from one body to another is by collision. It is unnecessary for 
 us to inquire whether actual contact takes place in this case be- 
 tween the particles of which bodies are composed. It is sufficient 
 for us to know that the result of all experiment is precisely such 
 as would happen, were the contact actually to occur. 
 
 All bodies in nature are more or less compressible, and when 
 they have been compressed, tend in agreateror less degree, tore- 
 cover their original figure. In some, this tendency is extremely 
 small ; in others it is considerable ; it is styled their Elasticity. 
 When bodies restore themselves to their original figure, after be- 
 ing compressed, and their particles in restoring the figure, return 
 with a force equal to that by which the compression was effected, 
 they are said to be perfectly elastic. If they did not yield to com- 
 
 Sression, they would be hard, and wholly devoid of elasticity, 
 f the latter class, there are probably no bodies in nature ; but 
 there are some that retain any figure that may be impressed upon 
 them, having little or no tendency to restore themselves to their 
 original shape. 
 
 Gases and vapours are, within certain limits, perfectly elastic; 
 bodies of other classes differ materially in this respect. 
 
 299. In investigating the laws of collision, we consider bodies ei- 
 ther as perfectly elastic, or as wholly devoid of elasticity ; and may 
 thence finally conclude what would occur in the case of imperfect 
 elasticity. The simplest law is that which governs the collision 
 of non elastic bodies ; from this, too, the circumstances of the col- 
 lision of perfectly elastic bodies, may be directly deduced. The 
 former of these must, in consequence, be first investigated : 
 
 Let A and B be two non-elastic bodies, homogeneous, and of a 
 spherical figure ; and let their centres move in the same straight 
 line, in such a manner, that, when they strike, the point of impact 
 shall be in the line that joins their centres. Let a and 6, be their 
 respective velocities, which if in the same direction, will have the 
 same, if in contrary directions, opposite algebraic signs. When 
 they strike against each other, each will yield to a greater or less 
 extent, until the whole action have taken place, and the time in 
 which this occurs may be considered as inappreciably small ; in 
 all cases it is in fact insensible. So soon as the whole action has 
 taken place, the two bodies will move forward with equal velocity, 
 
Book IV.} OF COLLISION. 301 
 
 for their is no force to act, provided they be non-elastic, to separate 
 them. Call this common velocity, v, the quantity of velocity lost 
 or gained by A, will be a u ; and that lost or gained by B, will 
 be 6 v ; and these will be positive, or negative quantities, accord- 
 ing to the conditions of the problem. 
 
 The quantities of motion the bodies respectively lose or gain, 
 will be 
 
 Ao Ar, and B6 BTJ ; 
 hence by the principle of D' Alembert, 69, 
 
 (Ao Aw) + (B6 Bu)=0 ; 
 whence we obtain for the value of r, 
 
 The whole quantity of motion will be, 
 
 Ar-r-Bu ; (322 
 
 and the respective quantities are, Au, and Bt>. 
 
 If one of the bodies, A, be alone in motion, and B at rest, 6=0 
 and 
 
 If the bodies move in opposite directions, a and 6 will have con- 
 trary signs ; and if a be positive, 
 
 Aa B6 
 
 If the bodies be equal, and the velocities equal, but in contrary 
 directions, 
 
 v=0. (325) 
 
 If the bodies have equal quantities of motion in opposite directions 
 Aa=B6, (326) 
 
 and 
 
 A: B:: 6:0; 
 
 in which case we again have 
 
 t=0. (327) 
 
 To express these formulae in words : 
 
 When two non-elastic homogeneous bodies of a spherical form 
 impinge against each other, at a point situated in the line that 
 joins their centres, if the sum of their quantities of motion before 
 impact, be divided by the sum of their masses, the quotient is 
 the common velocity after impact. The quantities of motion 
 must be considered as having the same or contrary signs, accord- 
 ing as the motions are in the same or contrary directions. 
 
 If one of the bodies is at rest, its quantity of motion is to be di- 
 vided by the sum of the masses. 
 
308 or COLLISION. [Book /F. 
 
 If th~ bodies be equal, and move in opposite directions with 
 equal velocities, they destroy each other's motions, and both come 
 to rest. The same is the consequence if unequal bodies meet, 
 with equal quantities of motion, or which is the same thing, when 
 their velocities in opposite directions are inversely as their 
 masses. 
 
 It will be also seen from inspection of the formula (321), that 
 the change of motion that takes place in each of the bodies is 
 equal, and that the change in their velocities is inversely as their 
 masses. 
 
 The sum of their motions, estimated in the same direction, is 
 the same before and after impact, and the state of their centre of 
 gravity, whether it be at rest or in motion, is not changed by 
 their mutual action. 
 
 If one of the bodies be infinitely large in respect to the other, 
 and at rest, then the common velocity becomes infinitely small, 
 and no error can arise in taking 
 
 t-=0. 
 
 Such is the case in all obstacles that are considered as immovea- 
 ble, which are so only in consequence of their great magnitude, 
 or from their being firmly connected with the surface of the earth. 
 If their surfaces be plane, they may be considered as portions of 
 spheres of infinite magnitude. Hence, when a non-slastic body 
 strikes against a plane surface in the direction of a normal to it, 
 it will be brought to rest upon it ; and so, as we may consider 
 curved surfaces, when the impact takes place at a single point, as 
 made up of planes, rest will again take place whenever a spheri- 
 cal and homogeneous non-elastic body strikes in the direction of a 
 normal to the surface, against any immoveable obstacle. 
 
 300. If a spherical body strike against an immoveable plane 
 surface in any other direction than t.hat of a normal, resolve its 
 moving force into two components, one of which is in the direc- 
 tion of a normal to the surface, and the other coincides with the 
 surface. That part which is in the direction of the normal, will be 
 destroyed by the resistance of the surface ; the other part will 
 remain acting upon the body, which will, therefore, move along 
 the surface under its influence ; and the new direction will be 
 defined by its being in the plane, passing through the original di- 
 rection of the moving body. 
 
 301. If the two spherical bodies d/> not move in the same 
 straight line, but have the directions of their motions inclined to 
 each other, the bodies will go on together in the direction of the 
 resultant of their respective quantities of motion, and the sum of 
 their new motions will be represented in magnitude by this re- 
 sultant. 
 
Book 
 
 OF COLLISION. 
 
 SOS 
 
 The determination of the direction and quantity of motion in this 
 case may be effected by a simple geometric construction, which is 
 as follows: Draw from the pointof concourse lines to represent the 
 magnitudes and directions of the forces whose intensities are the 
 respective products of the masses of the two bodies into their ve- 
 locities ; call these lines A and B. Resolve one of these, B, by 
 the construction of a right angled parallelogram into two others, 
 one of which, a, is parallel, the other, 6, perpendicular to A : pro- 
 duce A on the other side of the point of concourse, until the 
 length of the produced part=A-fa; produce B in like man- 
 ner until the part produced b : on the two lines produced, A-fa 
 and 6, construct a parallelogram which will be rightangled, and 
 draw the diagonal. The sum of the motions of the two bodies 
 after impact, will be represented in magnitude and direction by 
 this diagonal ; and if it be divided into two parts, proportioned to 
 the masses of the two bodies, these will represent their respective 
 quantities of motion. 
 
 302. If two bodies, moving in parallel lines, strike against 
 each other, the circumstances of their motion, after impact, will 
 differ from the case we have considered, in which their centres of 
 gravity move in the same straight line. 
 
 Suppose, first, that the bodies strike in such a manner that the 
 direction of one of them shall be a common normal to their sur- 
 faces. 
 
 Let KLM, NOP, be sections of the two bodies, in a plane 
 
304 or COLLISION. [Book IV. 
 
 passing through their respective centres of gravity, and the line 
 AC, which represents the direction of A ; this direction will have 
 for its distance, ar, from the centre of gravity of B, the line BC. 
 Let a be the velocity that remains to the body A, after impact. 
 The mass B being acted upon by a force whose direction does 
 not pass through its centre of gravity, will derive from it two mo- 
 tions, one of which is progressive, the other rotary, 245, around 
 an axis passing through its centre of gravity, and perpendicular 
 to the plane assumed for the section of the body. Let u be the 
 velocity of the progressive motion, t<>, the angular velocity of the 
 rotary motion. Let a and 6 represent the respective motions of 
 A and B before impact. 
 
 The body A will gain or lose a velocity represented by a a' 
 the body B, a velocity represented by 6 M, and by the principle 
 ofD'Alembert, 
 
 A. (a a') -f- B (6 w) =0. (328) 
 
 The body, B, will revolve with a force equal to that which A 
 loses by the collision. If then the moment of inertia of the 
 body, B, in respect to the axis passing through its centre of gra- 
 vity, = S, we shall have (241) for the value of the angular velo- 
 city, 
 
 Ax (a a') 
 u= - ^j -- -. (329) 
 
 For the same reason that spherical bodies, 297, do not sepa- 
 
 rate after impact, the body, A, will go on with a common velocity 
 
 with the point C, of the body B. This velocity is u xu, hence, 
 
 a'=u xw. (330) 
 
 If the bodies do not strike in a point that is in the common nor- 
 mal to their two surfaces, both will be caused to revolve, and the 
 rotary motion of A, may be estimated upon the same principles. 
 
 303. When the bodies that impinge against each other are 
 elastic, they will no longer go on together with a common velo- 
 city, but will be separated by the action of their elasticities. If 
 the elasticity be perfect, each body will tend to restore itself to 
 its original Torm, with a force equal to that by which the change 
 of figure has been produced ; this force will be equal to the 
 change of motion it would undergo were it non-elastic. What- 
 ever quantity of motion, then, a body loses or gains, when non- 
 elastic, will be exactly doubled if it be perfectly elastic. On this 
 principle, we may proceed to investigate the laws of the collision 
 of perfectly elastic bodies. 
 
 Supposing the bodies to be spherical, and to move in the same 
 straight line, using the same notation as in 297 ; their common 
 velocity, if non-elastic would be (321) 
 
 A+B 
 
Book 1V.~\ OF COLLISION. 305 
 
 the body A, would, in this case, lose or gain a velocity repre- 
 sented by 
 
 v a 
 
 and it will lose or gain as much more on account of its perfect 
 elasticity ; hence its actual velocity will be 
 
 a'=2 a; (331) 
 
 substituting the value of v, we have 
 / Ao B6\ 
 
 = ~ 
 
 whence 
 
 Aa B&+2B6 
 
 a== ~A+B - ( 332 ) 
 
 or 
 
 (A B)a-f2B6 
 
 a '= ~A+B~ ~ ; < 333 > 
 
 pursuing the same course of reasoning in respect to B, we obtain 
 B6 A6+2 Ao 
 
 "A+B - ' (334) 
 
 or 
 
 (B A) 6+2 Aa 
 
 -A+S- ( 33 *) 
 
 when the bodies are equal, 
 
 a' =6, and b'a. (336) 
 
 The relative velocity before impact, is a 6, after impact, 
 a ' 6' 5 by subtraction of the two last, we obtain 
 
 a 6 = a'+6', (337) 
 
 and 
 
 a-r-a'=6+&', (333) 
 
 whence we have for the sums of their motions before and after 
 impact, 
 
 Aa-|-B&=Aa'+B&'. (339) 
 
 Hence we may infer, that 
 
 304. When two perfectly elastic and homogeneous spherical 
 bodies, moving in the same straight line, impinge against each 
 other, they separate after impact, and move each with a different 
 velocity ; if we estimate the velocities that would be l8st or 
 gained by each, if non-elastic, and add or subtract this, to or from, 
 the common velocity they would have upon the same hypothesis, 
 we obtain their actual velocities after impact. 
 
 If the masses of the bodies be equal, their velocities are inter- 
 changed. Thus, if one only of the bodies be in motion, it will 
 come to reit after impact, and the other will move forward with 
 the original velocity of the first ; if they move in opposite direc- 
 
 39 
 
306 OF COLLISION. [Book IV. 
 
 tions, each will return in the direction whence it came, with the 
 velocity the other had : if they move in the same direction with 
 unequal velocities, the more swift overtaking the more slow, that 
 which moved most rapidly will move with the less velocity the 
 other had before impact, and vice versa. Whether the bodies 
 be equal or unequal, the difference in their velocities, or what 
 is styled their Relative Velocity, will be the same in amount both 
 before and after impact; but they will have contrary signs. 
 
 When an elastic body strikes against another that is at rest, 
 and larger than itself, the former will return in the direction in 
 which it was proceeding before impact; and if the latter be infi- 
 nitely large when compared with the former, the motion of re- 
 turn or reflection will have the same velocity with that of impact. 
 For the same reason, a body perfectly elastic will be reflected 
 back by a plain surface, in a direction opposite to that in which 
 it came, and with a velocity equal to that it originally had. 
 
 305. If the impact be oblique, the reflection will not take place 
 in the same straight line, but will still occur ; and it may be in- 
 ferred, from an application of the principle of "elasticity to 298, 
 that the directions of reflection and incidence will both be situa- 
 ted in the same plane perpendicular to the surface of impact, and 
 that the angles of incidence and reflection will be equal to each 
 other. 
 
 306. There are several cases where, when we have occasion to 
 estimate mechanical effect, it becomes necessary to consider the 
 product of the mass by the square of the velocity. This product is 
 called the Vis Viva of the body, in reference to which it is esti- 
 mated. When non-elastic bodies inpinge against each other, the 
 sum of these products of the two bodies does not remain constant ; 
 but it is otherwise with bodies that are perfectly elastic. 
 
 If we compare the sums of the forces estimated in the above 
 manner in the case of non elastic bodies, it will be found that they 
 have diminished ; in the case of perfectly elastic bodies, these 
 products remain constant, as may be readily shown. 
 
 Taking the formula, (339), 
 
 Aa+B6=Aa'+B6', 
 and 
 
 A(_a')=B(&-6'); 
 then by (338), 
 
 ". (340) 
 
 multiplying these two equations, we have 
 whence we obtain 
 
Book IV. 1 OF COLLISION. 307 
 
 307. When a small elastic body in motion strikes against a 
 greater elastic body at rest, it has been stated that the fortnet 
 would be reflected ; the larger will be set in motion, and will 
 proceed forward in the direction in whfch the smaller was pro- 
 ceeding before impact. This will be readily seen by consider- 
 ing, that were the bodies not elastic, the whole motion would be 
 distributed among them in proportion to their masses ; hence, 
 when we add as much more motion to that the larger body would 
 receive under this hypothesis, the sum will he more than the origi- 
 nal quantity of motion in the smaller body. The whole quan- 
 tity of motion in the two bodies is not on this account increased, 
 for the smaller body is reflected with a quantity of motion ex- 
 actly equal to the increased amount communicated to the greater 
 body, and the algebraic sum of the motions estimated in one di- 
 rection is constant. 
 
 When several elastic bodies, increasing in magnitude, are ar- 
 ranged in the same line, touching each other, and one smaller 
 than the least of them strikes against it, each will in turn be re- 
 flected, and communicate to the succeeding one a greater quantity 
 of motion than itself has. This increase of motion is greatest 
 when the masses of the bodies increase in geometric progression. 
 
 Let there be three elastic homogeneous bodies of spherical 
 figure, A, B and C, of which A is in motion, and strikes against B, 
 in a line passing through the centres of the three bodies ; the ve- 
 locity communicated to B will be, (334), 
 
 A+5 
 
 and to C, calculated also from (334), 
 
 4ABa 4Art 
 
 ( A+B)X(B+C)- A I)(B+C) 
 
 4Aa (342) 
 
 AC 
 A+B-f-g i-C ; 
 
 if the magnitudes of A and C are given, the last expression will 
 obviously be a maximum, when 
 
 B 3 -AC, 
 
 or when A, B and C, are in geometric progression, and the same 
 may be proved of any number of elastic bodies whatsoever. 
 308. When the bodies before collision move in parallel lines, 
 or when they meet in lines making an angle with each other, 
 the principles of 299 may be made use of to estimate the quan- 
 tities of motion that would be lost or gained by each, if non elas- 
 
308 OF COLLISION. [Book IV. 
 
 tic, and the change that is produced in the direction of the mo- 
 tion. Knowing, then, that both the change in the direction, and 
 in the quantity of motion, is doubled by perfect elasticity, the 
 manner in which the bodies will move in the first case, and the 
 direction and intensity in the second, subsequent to collision, 
 may be investigated. In these cases, the general formulae of the 
 composition and resolution of forces will meet every possible in- 
 stance. 
 
 309. If we examine into the state of the centre of gravity, we 
 shall find that whether it be in rest or in motion before the colli- 
 sion of the bodies, that state of rest or motion will not be affected 
 by their impact ; the same has been found to be the case in the 
 collision of non-elastic bodies ; and this is a constant law in what- 
 ever manner the bodies act upon each other, and whatever be 
 their respective natures. 
 
 310. When the elasticity of bodies that impinge against each 
 other is not perfect, the quantity of motion lost or gained by 
 each, in consequence of their elasticities, will be less than in the 
 case of perfect elasticity. 
 
 Let p be the relation of the force of elasticity to the compress- 
 ing force, the formulae, (333) and (335), will become 
 
 B(o 6) 
 a < =a _ ( l+p) 
 and 
 
 If we estimate the loss of vis inta, which takes place in conse- 
 quence of collision, we shall find it to be 
 AB(o 6) a 
 
 A+B * (343) 
 
 when the bodies are perfectly elastic, p=l, and this expression 
 =0 ; and when non-elastic, />=0, and the expression is a maxi- 
 mum. 
 
BOOK V. 
 
 OF THE EQUILIBRIUM OF FLUIDS. 
 
 CHAPTER I. 
 
 GENERAL CHARACTERS 01- FLUID BODIES. 
 
 311. Fluids are distinguished from solid bodies, 79, by the 
 greater degree of ease with which their particles may be separa- 
 ted. It is the distinctive and characteristic property of fluids, 
 that these particles may be moved among each other by the 
 smallest possible force. The physical agent that causes fluidity is 
 heat, existing in a latent state. 
 
 When the repulsive force, exerted by heat, exactly balances the 
 attractioi\ of aggregation that exists among the particles, the body 
 is a liquid ; when the force of heat preponderates, the body has 
 a tendency to expand itself; this tendency is opposed, and may 
 be overcome by pressure, and hence the body will occupy spaces 
 of different extents, that will depend upon the intensity of the 
 compressing force : such a fluid is said to be elastic. 
 
 We know of no perfect liquids in nature ; in them all the force 
 of the attraction of aggregation slightly exceeds the repulsive 
 force of heat, as is manifested by small portions of liquids ar- 
 ranging themselves in the form of spherical drops, and by their 
 particles resisting forces that tend to separate them, although with 
 a feeble intensity ; this resistance is called Viscidity. 
 
 312. Liquids, in examining the theory of their equilibrium 
 and motion, are considered as incompressible, and consequently 
 wholly devoid of elasticity. A want of this property was, for a 
 longtime, considered to be essential to liquids. It might, how- 
 ever, have been inferred, that as liquids are capable of contracting 
 and expanding with changes of temperature, they were not ab- 
 solutely incompressible; and that a mechanical agent of intensi- 
 ty equal to heat, might cause them to change their bulk. 
 
310 GENERAL CHARACTERS [Book V. 
 
 This inference has been confirmed by the experiments of Can- 
 ton, and more recently by those of Perkins and Oersted, who 
 have shown, that the bulk of water may be diminished by pres- 
 sure. This diminution in bulk is about .000048 for a pressure 
 equivalent to 15 Ibs. upon a square inch of surface ; an amount 
 that, as will be shown hereafter, constitutes that unit of pressure 
 which is called an Atmosphere. 
 
 313. To determine the nature and intensity of the forces that 
 act upon any given particle of a fluid mass, we may, in conse- 
 quence of the very small size of the particles, and the apparent 
 continuity of the mass, ascribe to these particles any figure that 
 we think proper, or which is most convenient foe the purposes of 
 our investigation. 
 
 If we take then a given particle of the fluid, and refer its posi- 
 tion to three rectangular co-ordinates, X, Y and Z, we may con- 
 sider the particle as occupying a space extended in three dimen- 
 sions, and having the figure of a parallelepiped, whose dimensions 
 are dr, dy and dz ; and may assume for its volume the product 
 of its three dimensions. If we call its density s, its mass will be 
 adxdydz. (344) 
 
 The forces that act upon this particle, must be either inherent 
 in the particle itself, and intrinsic ; or must be exercised m the 
 form of pressure by the surrounding particles, and therefore ex- 
 trinsic. 
 
 The intrinsic forces may, 17, be resolved into thtfe, acting 
 at right angles to each other, in the directions of rfie Aree axes, 
 X, Y, Z ; if we call these components P, Q and K, the forces 
 that solicit the particle will be found by multiplying /hese compo- 
 nents respectively by its mass, or they will be 
 P* dx dy dz, > 
 
 Qsdxdydz, } (345) 
 
 .>/ j d (:Kj ye >,{) * R* dx dy dz. I 
 
 The extrinsic forces are pressures, awl may be represented in 
 each case by taking the surface dh on which the pressure is ex- 
 erted, and multiplying it by the keight of a column of homoge- 
 neous matter, whose weight is equal to this pressure, and by the 
 density of this column. If we /suppose the density of the column 
 to be 1, and call its height p, the pressure will be 
 
 pdh (346) 
 
 This pressure must be exercised in the direction of a normal to 
 the surface dk : for were the force oblique ; of its two rectangular 
 components, one in the direction of the surface itself, the other in 
 that of its normal ; the latter alone could produce any effect. And 
 it is therefore obvious, that all the forces that can act upon a given 
 surface of a fluid particle, must be finally reducible to two, act- 
 
Book V.~\ OF FLUID BODIES. 311 
 
 ing in opposition to each other, in the direction of its normal ; 
 these forces may be called 
 
 pdk, a.ndp'dk. (347) 
 
 And in the case of equilibrium, these pressures must be equal, 
 and 
 
 P=P'- (348) 
 
 If the surface pressed change its direction, without having its 
 position in the mass of fluid changed, the value of p will remain 
 constant. For : suppose the surface to turn around one of its 
 edges, until it make with its original position the angle t : The 
 magnitude of the surface that is pressed by the column, the latter 
 remaining unchanged in area and volume, will be 
 
 dk 
 
 cos. i ' 
 
 the length of the perpendicular height of the column in its new po- 
 sition, from the surface, will be 
 
 p cos. i ; 
 
 the product of these two quantities is the amount of the pressure, 
 or 
 
 dk ' n 
 
 { . p cos. t=dkp. 
 
 But in different parts of the same mass, the quantity p will vary, 
 and will depend upon, or be a function of X, Y and Z. 
 
 If then we take another surface contiguous to the first, the 
 column that measures the pressure upon it will have for its alti- 
 
 tude, 
 
 and the pressure will be, if the surface have the same area as the 
 first, 
 
 To apply these principles to the case of the elementary parallele- 
 piped : p is, as has been stated, a function of the three co-ordi- 
 nates ; we may, therefore, assume for the value of its differential, 
 
 and the three co-efficients, L, M and N, will be the differentials 
 of jp, in the respective directions of X, Y and Z. 
 
 The face whose surface is dij dz will be pressed in the direction 
 of a normal, which direction will be the same as that of #, by a 
 force represented by 
 
 p dy dz ; 
 
 the opposite and equal face of the parallelepiped will be pressed 
 by a force in the opposite direction, whose measure is a column 
 or prism, whose base is dy dz, and altitude p+L dx ; this force 
 will therefore be 
 
 (p-f L dx]dy dz ; 
 
312 GENERAL CHARACTERS [Book V. 
 
 and upon the remaining four surfaces, the pressures will be 
 
 p dy dx, 
 (p+Kz)dy dx, 
 
 p dz dx, 
 
 (p+M dy)dz dx. 
 
 These six forces, acting in opposite directions by pairs* may 
 be resolved into three, which are 
 
 L dx dy dz, \ 
 
 M dx dy dz, } (349) 
 
 N dx dy dz. ) 
 
 These three rectangular forces will be the resultants of all the ex- 
 trinsic forces that act upon the particle. If with these be com- 
 bined the rectangular resultants of the intrinsic forces, (345), we 
 have, for the rectangular resultants of all the forces that act, 
 (Pa L)dxdydz, } 
 (Q M)dx dy dz, } (350) 
 
 314. The nature and intensity of the forces that act upon any 
 given particle of the fluid, being obtained in the preceding ex- 
 pressions, we may proceed to investigate the conditions of equi- 
 librium that exist among them. 
 
 As every particle of a fluid mass has, by the definition of that 
 class of bodies, perfect freedom of motion, except such resistances 
 as may arise from the pressure of the surrounding particles; every 
 particle must in consequence, when the mass is in equilibrio, be 
 also in equilibrio under the forces that act upon it. The three 
 rectangular resultants of the forces, (350), must therefore be res- 
 pectively =0. Hence, in the expressions, (350), we have 
 
 L=P, M=Q, N=R; 
 
 if we multiply these equations respectively by dx, dy, dz, and add 
 the products together, we obtain 
 
 dp=s(P rf#+Q <%+R dz), (351) 
 
 which is the equation that gives the conditions of equilibrium. 
 
 315. The resistance to the motion of the particles of fluids, is 
 restricted to the pressure of the surrounding mass ; while in solids, 
 not only does this resistance act, but one of far greater intensity, 
 namely, the attraction of aggregation that exists among their par- 
 ticles. The above condition of equilibrium therefore exists in 
 solids as well as in fluids; that is to say, that if it obtain, the 
 mass will be in equilibrio, but we cannot infer, that if it do not 
 obtain, the mass will therefore cease to be in equilibrio. 
 
 It results from the nature of fluid bodies, whose particles are 
 free to move, that if any one of them be set in motion, all the 
 
Book V.~\ OF FLUID BODIES. 313 
 
 rest must be set in motion also ; and that the application of any 
 force to any one of the particles, in addition to those whose re- 
 sultants are given in the equation (351), will affect the equilibri- 
 um of all the rest; therefore a pressure applied to any point of a 
 fluid mass, will be propagated in all directions, and influence 
 every particle of which the mass is made up. A pressure ap- 
 plied to any point of a solid body, would, in like manner, be 
 propagated in all directions, were it not that it is counteracted by 
 the attraction of aggregation. Hence, the property usually as- 
 sumed as the basis of the mechanics of fluid bodies, namely, that 
 of propagating pressure equally in all directions, is not distinc- 
 tive ; and solids, in which the attraction of aggregation is weak, 
 or in which it has been overcome by mechanical action, will press 
 to a certain extent in conformity with the laws that govern the 
 pressure of fluids. This is found to be the case in masses of sand 
 and loose earth, that often produce mechanical effects similar, al- 
 though not absolutely identical with those produced by fluids. 
 In consequence of this mode of action, the investigation in 196, 
 of the strength of a wall that will support the pressure of earth, 
 does not always give sure results. 
 
 40 
 
314 EQUILIBRIUM OF [Book V. 
 
 % 
 
 CHAPTER II. 
 
 OF THE EQUILIBRIUM OF GRAVITATING LIQUIDS. 
 
 316. All fluids upon or near the surface of the earth, are in- 
 fluenced by the attraction of gravitation ; in liquids thus situated, 
 no other cause exists for the mutual action of their particles upon 
 each other. This force is always exerted in the direction of a 
 line tending to the centre of the earth, at which point all the di- 
 rections of the force of gravity at the surface would meet, were 
 the earth a perfect sphere. The direction then, at any place, is 
 a vertical line. 
 
 If we suppose that this direction coincides with the axis *, the 
 quantities P, and Q, in the direction of the other axes, become 
 in liquids =0. R, alone remains, and if we consider this as 
 equal to unity, the equation (351) becomes 
 
 dp=*d2; (352) 
 
 and integrating, 
 
 p=(A+Z); * 
 
 the force p, which measures the pressure upon any point of a gra- 
 vitating liquid, will, therefore, be a function of the variable quan- 
 tity 2, or will depend upon its vertical depth beneath the surface 
 of the fluid. Hence, every point situated at the same distance 
 below the surface of a mass of fluid, will undergo an equal pres- 
 sure. 
 
 At the surface of a gravitating fluid, the quantity dp, which de- 
 pends upon the depth of the fluid, is = ; hence, the equation 
 (351) of the surface, when in equilibrio, becomes 
 
 (Pdx+Qdi/+Rd*)=0; 
 if we take the density *=1, 
 
 P dx+ Q efy+R dz=0 ; (353) 
 
 and in the case of a liquid, 
 
 dz=0. (354) 
 
 317. When a gravitating liquid is placed in an open vessel, its 
 particles move in consequence of the fundamental property of 
 fluids, until they reach a state in which the conditions of equili- 
 brium are satisfied. It will then come to rest, and assume an 
 uniform and constant surface. The last equation (354) shows 
 that every point in this surface will be at an equal distance from 
 the plane to which the axis z is perpendicular ; the surface will, 
 therefore, be plane, and at right angles to the direction of the 
 force of gravity. The last part of the proposition is also true, 
 
Book T 7 ".] GRAVITATING LIQUIDS. 315 
 
 when the extent of the surface becomes so great that the direc- 
 tions of gravity can no longer be considered as parallel ; but in 
 this case, thesurface itself becomes curved, and beingevery where 
 at right angles to the force of gravity, is parallel to the mean sur- 
 face of the terrestrial spheroid. Such a surface is said to be 
 level, and is spontaneously assumed by all stagnant fluids upon 
 the surface of the earth ; and in masses that are agitated by ex- 
 trinsic forces, as the ocean, which is in constant motion under 
 the action of the winds, and the causes that produce the tides, 
 the mean surface is level. 
 
 That the surface of a liquid, when in a state of equilibrium un- 
 der the action of any force whatsoever, must be perpendicular to 
 the direction of that force, may be shown in another manner. 
 For if we assume that the surface has not this direction, the force 
 that acts on any particle, may be resolved into two, one perpen- 
 dicular, the other parallel to the surface ; the latter then will not 
 be counteracted by fluid pressure, and the particle will move un- 
 der its influence, which is contrary to the hypothesis of equili- 
 brium. This mode of considering the subject shows in addition, 
 what does not appear from the equation of equilibrium, namely, 
 that the force that causes a fluid to assume a definite surface, must 
 be directed from without the mass towards its interior. 
 
 318. The art of levelling consists in drawing a line, which 
 shall every where intersect the direction of gravity at right an- 
 gles. Such a line is in fact a curve, although, it may be con- 
 sidered as straight, within narrow limits. Our instruments do 
 not furnish the means of drawing the level curve, and we are, 
 therefore, compelled to content ourselves with obtaining tangents 
 to it, at the several points of observation. If these tangents be of 
 no great length, and if they be equally produced each way from 
 the point of contact with the curve, they form a polygon, that 
 may, without sensible error, be considered as identical with the 
 curve itself. 
 
 The levelling instruments, in most frequent use, are: 
 
 The Water Level, the Spirit Level, and the Mason's Level. 
 
 319. The water level is formed of a tube of glass, bent twice 
 at right angles. If a fluid be placed in this tube, its surfaces will 
 rise in each branch to the same height above the mean surface, 
 as will be shown in a succeeding section. If the sight be directed 
 over these two surfaces, the line of sight will be of course a tan- 
 gent to the level curve. 
 
 320. The essential part of a spirit level consists of a cylin- 
 drical tube of glass, containing a portion of spirits of wine 
 that nearly fills It ; this tube is hermetically sealed at each end. 
 
316 EQUILIBRIUM OF [Book V, 
 
 The space in the tube that is not occupied by the spirit, contains 
 air, which appears in the form of a bubble. The air being much 
 lighter than the spirit, the bubble will always occupy the highest 
 part of the tube ; and when the tube is truly horizontal, the bub- 
 ble will come to rest at a distance exactly equal from each end. 
 Any deviation from a horizontal position, will be marked by an 
 approach of the bubble to the end of the tube that is most ele- 
 vated. 
 
 In order to render the indications of the level more precise, it 
 has been improved by grinding the interior of the tube into the 
 form of a portion of a circular ring of large radius ; and the axis 
 of the tube is in consequence no longer a straight line, but an arc 
 of a circle. If the radius of this circle be known in some con- 
 ventional unit of measure, and divisions of the same unit be 
 marked upon the tube, the deviation of the bubble from the ex- 
 act middle of the tube may be measured by these divisions, and 
 hence the inclination of the chord or line that joins the extreme 
 points of the axis, estimated. 
 
 This level is usually mounted upon a tripod stand, that bears 
 two plates that are ground to fit each other ; one of these is at 
 rest, the other moves upon it, around a vertical axis, and carries 
 with it the tube. The conditions on which accuracy depends in 
 this part of the arrangement, are, that the axis of the plates' motion 
 shall be truly vertical ; and that the axis of the tube shall be pa- 
 rallel to the surfaces of the plates, and of course perpendicular to 
 the vertical axis of motion. 
 
 Whether both of these conditions be fulfilled, may be ascer- 
 tained by one and the same operation, and the instrument ad- 
 justed, if necessary. The direction of the axis of the motion of 
 the plate, is capable of change in respect to the stand, by means 
 of four screws, called the levelling screws of the instrument. 
 Such, at least, is the more usual number, and to assume that num- 
 ber as adopted, will best suit the purpose of explanation ; al- 
 though three screws would probably afford a more convenient 
 adjustment, as may be inferred from 112. The tube being brought 
 directly over two of the screws, they are moved in opposite di- 
 rections until the bubble exactly occupies the middle of the tube ; 
 the axis of the tube is, therefore, level. In order to determine whe- 
 ther it be parallel to the plates, and whether the axis of the latter be 
 vertical, the moveable plate is turned around its axis through the 
 half of a complete revolution ; the position of the levellingtubeis, 
 therefore, reversed. If in this new position, the bubble still oc- 
 cupy the middle of the tube, the adjustment is complete. If, 
 however, the bubble have approached to either end, one half of 
 its change of position is obviously owing to a want of parallelism 
 in the motions of the plate and the tube ; the other half, to the 
 
Book J 7 ".] GRAVITATING LIQUIDS. 317 , 
 
 axis not being vertical. In order to correct the latter error, the 
 levelling screws are again moved, until the bubble have passed 
 back through about half the space thit marks its change of posi- 
 tion. To enable us to remedy the former error, the glass tube 
 that contains the spirits of wine is enclosed in another tube of 
 brass, open at its upper surface, to permit the bubble to be seen, 
 and the divisions of the scale to be read. This outer tube is at- 
 tached to the bar that supports it, at one end, by a hinge, whose 
 axis is horizontal, and at the other by a vertical screw. The 
 bubble having been moved back by the means just described, 
 through half the distance it before passed through, in conse- 
 quence of the error in adjustment, is brought exactly to the mid- 
 dle of the tube by the last named screw. The instrument being 
 then turned around until it reach its original position, the bubble 
 is again inspected; if it now come to rest exactly in the middle 
 of the tube, the adjustment is complete ; if not, the preceding 
 operations must be repeated, until the bubble stand exactly in the 
 middle, in both positions of the tube. 
 
 In order to direct the vision to the signals placed at the sta- 
 tions whose relative level is to be determined, sights must be 
 adapted to the tube ; that which has superseded all others, is the 
 telescope ; and the level, instead of being mounted as we have 
 assumed, for the sake of more easy illustration, upon a simple 
 bar, is attached by the hinge and vertical screw to the tube of 
 the telescope. The telescope itself is mounted upon the horizon- 
 tal bar that turns around with the moveable plate. For this pur- 
 pose, the latter is furnished with two vertical supports; these 
 have the form of rods, divided at their upper end into two in- 
 clined branches, and thus having the form of the letter Y, whence 
 this part of the apparatus derives its name. One of these Ys has 
 a motion in a vertical direction, by means of a screw placed be- 
 neath it. The telescope has upon its tube two collars, at an ap- 
 propriate distance from each other, that are accurately turned, 
 and rest upon the Ys. It is obvious, that the axis of the tele- 
 scope, or more properly, the line that joins the centres of the two 
 collars, must be rendered parallel to the axis of the spirit level. 
 To ascertain if this be the case, and to correct it if it be not : the 
 previous adjustment having been completed, and the bubble 
 brought exactly to the middle of the tube, the telescope is lifted 
 from the Ys, and again replaced in an inverted position ; that is 
 to say, each of the collars is made to rest in the Y that had before 
 borne the other. If in this inverted position the bubble still 
 occupy the middle of the tube, this adjustment is complete ; but 
 if it do not, one half of the error is due to the position of the pa- 
 rallel plates, the other half to the position of the Ys. In order, 
 therefore, to effect this adjustment, the bubble is brought back to 
 
318 EQUILIBRIUM OF [Book J r . 
 
 the middle of the tube, partly by the levelling screws of the in- 
 strument, and partly by the screw that has been described as giv- 
 ing a vertical motion to one of the Ys. 
 
 A telescope having a field of view of a definite extent, it is 
 necessary that some line should be defined within it, to point the 
 direction of the vision to the signals. This is effected in the 
 levelling instrument, by means of the intersection of two wires, 
 the one vertical, and the other horizontal. These are placed in 
 the common focus of the two lenses of the telescope, and are, in 
 consequence of one of the optical properties of that instrument, 
 as distinctly visible as the signal. The line passing through the 
 eye, and the intersection of these wires, is called the Line of 
 Collimation. In order to adjust this ; the telescope, after the 
 instrument has received the previous adjustment, is directed 
 towards a moveable vane, on which a horizontal line is drawn ; 
 the vane is raised or lowered, until this line apparently coincides 
 with the horizontal wire, in the focus of the telescope. The tele- 
 scope is next turned half round, in the Ys, on its own horizontal 
 axis; if the line on the vane correspond exactly with the hori- 
 zontal wire, the instrument requires no farther adjustment; but 
 if these lines do not coincide, half the error is evidently due to 
 the positron of the vane ; the other half to that of the horizontal 
 wire in the focus of the telescope. The correction is again double ; 
 the vane is moved through half the space that intervenes between 
 the upper and lower visible position of the telescope ; and the 
 horizontal wire of the telescope, is moved through the other half, 
 by means of screws, adapted for the purpose, to the tube of the 
 telescope. 
 
 We determine in these adjustments, and in the subsequent use 
 of the instrument, the proper position of the bubble, by observing 
 the place occupied by its two extremities ; for the bubble has no 
 direct mode of showing the position of its exact middle. The 
 length of the bubble is constantly varying with changes of tempe- 
 rature, for the spirits of wine expand and contract, when heated 
 or cooled, and the air being elastic, accommodates itself to this 
 change of bulk. Exposure to the sun affects the temperature, 
 and his rays sometimes fall obliquely and partially. In the latter 
 case, the indications of the spirit level, are uncertain, and the 
 bubble will move from its position, when the instrument itself is 
 perfectly at rest. 
 
 321. In using the level, it may be placed at one of the poinfs, 
 whose difference of elevation is to be determined ; when the in- 
 strument has been levelled by bringing the bubble to the middle 
 of the tube, by the motion of the levelling screws, and it has been 
 ascertained that the bubble will continue in that position while it 
 
Book VJ\ GRAVITATING LIQUIDS. 319 
 
 moves around the vertical axis, the telescope is directed to the other 
 point ; at this a staff is set up in a vertical position, on which a vane 
 slides. The vane has a horizontal line marked upon it, and an as- 
 sistant slides the vane upon the staff, until this horizontal line ap- 
 pears to coincide with the horizontal wire of the telescope. The 
 difference between the height of the eye-piece of the instrument, 
 and that of the horizontal line upon the vane measured from the 
 points on which they respectively stand, is the difference of level. 
 This determination wiM require corrections, unless the dis- 
 tance between the points be small. It is also liable to error, from 
 a want of accurate previous adjustment in the instrument, and 
 from the bubble not being exactly in the middle of the tube. The 
 latter may be allowed for, by noting the divisions that point out 
 the position of the level, and thence calculating the effect of the 
 inclination of the axis of the telescope. 
 
 All these sources of error may be in a great degree compen- 
 sated by the second, and more usual method. In this, the instru- 
 ment is placed as nearly as possible at an equal distance from each 
 of the points, whose difference of level is to be ascertained. A 
 staff and sliding vane is set up at each of these points, and the tele- 
 scope directed to them in turn. The lines upon the vanes having 
 been made to coincide with the horizontal wire of the telescope, 
 the difference in the altitudes of these lines, above the points on 
 which the staves rest, is the difference of the respective levels. 
 If the place of observation is exactly equidistant from the 
 two points, the sphericity of the earth will equally affect both 
 vanes, and the apparent difference will be the same as the true. So 
 also if the axis of the telescope be from any cause inclined, or the 
 line of collimation do not coincide with it, the errors that arise 
 will be equal in both directions. 
 
 When a line of considerable length is to be levelled, it should 
 be divided into parts by stations at equal distances; from 180 to 
 200 feet is a convenient space for this purpose. Staves being set 
 up at the two first stations, the levelling instrument is placed at an 
 equal distance from each of them, and the observations made, as 
 has been directed : the staff from the first station is then moved 
 forward to the third ; that at the second remaining stationary. The 
 instrument is next moved for ward to a point equidistant from the 
 second and third station, and a new pair of observations made. In 
 recording the observations, the heights of the vanes, as seen from 
 each position of the instrument, are arranged in two columns ; the 
 heights observed by looking towards the first station, or back- 
 wards, being set in one column ; those obtained by looking for- 
 wards in the other. The difference between the sums of the num- 
 bers in the columns, gives the difference of level of the extreme 
 stations. 
 
320 EQUILIBRIUM OP [Book V. 
 
 322. The mason's level is composed of a ruler, to which a plumb- 
 line is adapted, a line being drawn to mark the proper position of 
 the cord, and an opening made to receive the plummet. A second 
 ruler is adapted to the lower part of the first, whose lower edge 
 is at right angles to the line with which the cord of the plumb-line 
 is made to coincide ; when the latter is brought to its proper 
 position, the lower edge of the ruler is of course level, being at 
 right angles to the plumb-line, and consequently to the direction 
 of gravity. 
 
 A modification of this instrument, that is useful in many cases, 
 may be constructed by suspending the plumb-line from a point at 
 the intersection of two bars of equal lengths, which, therefore, 
 form sides of an isosceles triangle ; the position of the plumb- 
 line is determined, by making it coincide with the middle of a 
 bar, that is parallel to the line that would constitute the base of 
 the triangle. Levels of this last form are sometimes constructed 
 of considerable size, the bars being six or seven feet in length. 
 They may be used for laying out ditches and trenches, for the 
 distribution of water for agricultural purposes, but are not suffi- 
 ciently accurate for the purposes of engineers. 
 
 323. The correction for the sphericity of the earth, applied to 
 the first method of observation described in the preceding section, 
 is obtained by considering the horizontal distance as a mean pro- 
 portional between the earth's diameter, and the distance of the 
 point at which the tangent cuts the diameter that passes through 
 the point, whose level is to be determined, from the surface of the 
 earth. This hypothesis is no more than an approximation, but is 
 sufficiently near the truth, to be employed on most occasions. 
 
 Upon this hypothesis, if a be the horizontal distance between 
 the two points ; D the earth's diameter ; and /i the height of the 
 horizontal vane above the level of the point at which the instru- 
 ment is placed, all estimated in the same unit of lineal measure, 
 
 *=. (355) 
 
 324. Observations, made with a levelling instrument at one of 
 the points, whose relative heights are to be determined, may 
 also be affected by atmospheric refraction. This is not the case 
 when the instrument is placed half way between them. The cor- 
 rection to be used in the former case is not properly an object of 
 investigation in mechanics, as it is derived from the principles of 
 experimental physics. 
 
 325. When a homogeneous gravitating liquid, instead of being 
 contained in a single vessel, or collected in a great mass in hol- 
 low basins on the surface of the earth, is placed in bent tubes, or 
 
 
V.~\ GRAVITATING LIQUIDS. 321 
 
 in vessels communicating with each other at bottom ; the sur- 
 faces of the liquid in the several vessels or branches, will all form 
 portionsof the same general surface oflevel ; or will, within a small 
 space, be equally elevated or depressed below the same horizon- 
 tal plane. For : 
 
 If in the first place we suppose the liquid to be contained in a 
 tube or vessel, with no more than two branches, we may consider 
 it as divided into two columns, by an imaginary surface, where 
 the branches communicate. In order that equilibrium shall exist, 
 the pressures upon this surface, exerted by the two columns, must 
 be equal. Now, the fluid pressure upon a surface is a function of 
 the vertical co-ordinate z, 314, or will depend upon the vertical 
 depth of the surface pressed, beneath the horizontal surface of the 
 liquid. This quantity, z, must therefore be equal, in the case of 
 equilibrium, in both branches, and the respective surfaces must be 
 on the same level. The same mode of investigation will apply to 
 the case of three, or any number of branches or vessels, commu- 
 nicating beneath the surface of the fluid. 
 
 If two heterogeneous fluids, incapable of being mixed, be placed 
 
 in the two branches of a bent tube, the pressure exerted by one of 
 
 them on the surface of contact, will be by the integration of (352), 
 
 p=fsdz>, (356) 
 
 and if s' be the density of the second fluid, and z' its depth, in the 
 case of equilibrium, we have in the same manner, 
 
 p=fs'dz f . (357) 
 
 In the case of liquidity, when each of these fluids may be con- 
 sidered of uniform density throughout, we obtain by the integra- 
 tion of the second members of the above equations, both of 
 which are equal top of (346), 
 
 AZ=*V ; 
 and 
 
 z : z' : : s' : * ; (358) 
 
 hence : 
 
 326. The heights to which columns of heterogeneous liquids 
 rise above the common surface of contact, are inversely propor- 
 tioned to their respective densities. 
 
 If two immiscible liquids be introduced into the same vase, 
 the denser of the two will occupy the lower part of the vase ; for 
 the freedom of motion which the particles of each possess, will 
 allow it to descend, by its superior gravity, through the rarer. 
 Hence, if the above proposition be submitted to the test of experi- 
 ment, the denser liquid must be first introduced into the tube ; 
 for if the rarer be first introduced, the denser will descend 
 through it, separate it into two columns, and occupy the bend of 
 the tube. When the denser liquid has been introduced into the 
 bent tube, the rarer, being poured into one of the branches, will, 
 
 41 
 
322 EQU1LIBK1UM 07 [Book V* 
 
 by its pressure, force down the level of the denser in that branch, 
 while that in the other will be elevated ; but the depression will 
 never carry the surface of contact beyond the bend in the tube, 
 which will, therefore, continue to be occupied by the denser fluid. 
 The surface of contact must be level, for the depths, z, andz', 
 are constant quantities. 
 
 327. When a solid body is wholly immersed beneath the sur- 
 face of a fluid, it occupies a space identical with what it did be- 
 fore immersion ; for mere immersion does not alter its volume. 
 The whole volume of the fluid mass, when it contains the solid, 
 will, therefore, be increased as much as is equal to the volume of 
 the solid. 
 
 As the volumes of bodies of equal weights are inversely as their 
 densities, the weights of fluid, displaced by bodies of different 
 densities wholly immersed, will also be inversely as their respec- 
 tive densities. 
 
 When a body is thus immersed, its own weight tends to cause 
 it to descend in the direction of the force of gravity ; but this 
 tendency will be counteracted, in a greater or less degree, by the 
 action of the fluid. If instead of a solid body, we consider the 
 case of one of the elementary particles of the fluid, and ascribe to 
 it, upon the principle of 313, the figure there assumed for one 
 of the particles of the fluid : the pressures of the fluid upon the 
 four vertical faces of the parallelepiped, will exactly counter- 
 balance each other ; the pressure on the upper surface, will be 
 measured by a column of the fluid, (346), whose area is that of 
 this surface of the particle, and whose altitude, is the vertical depth 
 of the same surface, beneath the level of the fluid (considered as ho- 
 mogeneous); this depth in the case of a liquid, or other homoge- 
 neous fluid, will be the actual depth. The downward pressure on 
 the lower base will be the sum of the weight of the superincum- 
 bent column, and the weight of the solid particle itself, while the 
 upward pressure on the same base will be measured by a column 
 of the fluid, whose altitude is the vertical depth of this lower sur- 
 face. As the solid particle identically replaces in bulk an equal vol- 
 ume of the liquid, the difference of these two pressures will be the 
 difference of the weights of the two particles ; this, if the densities 
 of the two be equal, will be =0. The same will also be true, as 
 will appear from the same section, 31 3, whatever be the direction 
 of the surfaces of the particle. Now, as the solid body is made 
 up of its particles, what is true in respect to one of them in this 
 instance, will be true of its whole mass, and the tendency of the 
 solid body to descend, under the action of gravity, will be dimi- 
 nished by a force, whose measure is the weight <5f a volume of 
 the fluid, equal to the bulk of the solid body. Hence, if the solid 
 
Book V.] GRAVJTATIN* LIQUIDS. 323 
 
 have the same density as the fluid, it will remain at rest in any 
 part of the latter in which it may be placed : if its density be 
 greater, it will descend ; and if its density be less than that of 
 the liquid, the solid will rise ; but the force with which it moves, 
 in either case, will have for its measure the difference in the 
 weights of equal volumes of the two substances. 
 
 It will therefore be seen, that when a solid body is immersed 
 in a fluid, it appears to lose as much of its weight as is equal to 
 the weight of a mass of fluid whose volume is the same with its 
 own. This loss of weight was by some considered as actual, and 
 not merely apparent. This, however, is not the case, for the 
 tendency to descend under the action of the attraction of gravi- 
 tation is not destroyed, but merely opposed ; the weight of the 
 body, which is the sum of the gravitating forces exerted upon 
 its several particles, 105, still remains ; but these forces 
 are opposed by others acting in a contrary direction, and their 
 joint resultant is of course less than the weight of the body. To 
 render this more clear, if a vase be taken that contains a liquid, 
 and if a solid body be immersed in it; although the latter will 
 appear to lose a portion of its weight, the joint weight of the 
 vase, the liquid and the solid, will still be the same, as if the lat- 
 ter were weighed separately, and its weight added to the joint 
 weight of the other two substances. 
 
 328. When the solid has less density than the liquid, the re- 
 sultant of the two sets of forces will be negative, and the appa- 
 rent loss of weight of the solid will be greater than its own 
 weight : it will in consequence rise to the surface of the li- 
 quid. Having reached the surface, it will continue to rise, and 
 elevate a portion of its volume above the surface, until equili- 
 brium between the two sets of forces be restored. Thedownward 
 pressure on the lower surface becomes in this case no more than 
 the weight of the solid itself; the upward pressure, in order to 
 be equal to this, must have for its measure columns of fluid, 
 whose joint weight is equal to that of the solid ; and the joint 
 volume of these columns will therefore be equal to the volume of 
 the portion of the solid immersed. Hence, when a solid body 
 of less density than that of the liquid, has risen to, or is placed 
 upon, the surface of the latter, it will float there, displacing as 
 much of the liquid as is equal in weight to the whole weight of 
 the solid body. 
 
 The force which acts downwards, being the weight of 
 the solid body, has for its point of application the centre of gra- 
 vity of the solid ; the force that acts upwards is the resultant of 
 a number of parallel forces exerted by an infinite number of ver- 
 
324 EQUILIBRIUM, &c. [Book V. 
 
 tical columns of the liquid, which together make up a volume 
 equal to that of the part of the solid that is immersed. The re- 
 sultant of such a system of forces will have for its point of appli- 
 cation the centre of parallel force ; this being determined upon, 
 the principles of 105, will be the same with the centre of gra- 
 vity of the part of the solid immersed. And as it is necessary, 
 not only that two opposing forces in equilibrio should be equal 
 in intensity, but that they should act in the same straight line, 
 this condition is also necessary for the the existence in equili- 
 brium ; and, 
 
 A solid will float in equilibrio upon the surface of a liquid, 
 when it has displaced a volume of the liquid whose weight is 
 equal to its own, and when the centres of gravity of the whole 
 solid and of the part immersed are situated in the same vertical 
 line. 
 
 329. If the same solid body be placed in succession in two dif- 
 ferent liquids, whose densities are both greater than its own, it 
 will float at the surface of both. The part of the solid immersed 
 will displace equal weights of the two liquids, but the volumes 
 displaced will be different, if their densities be not the same. 
 Now, as the volumes of equal weights of different bodies are in- 
 versely as their respective densities, the parts of the same solid 
 that are immersed in two different liquids, on whose surface it 
 successively floats, are inversely as their respective densities. 
 
 If the floating body have the form of a prism, and float, in con- 
 formity with the second of the above conditions of equilibrium, 
 with its axis in a vertical position ; as the horizontal area of the 
 part immersed will be constant, the body will be immersed to 
 depths measured vertically along its axis, that are inversely pro- 
 portioned to the densities of the liquid on which it floats. 
 
Book V.} PRESSURE, &c. 325 
 
 CHAPTER III. 
 
 y 
 
 OF THE PRESSURE OF GRAVITATING LIQUIDS. 
 
 330. The pressure of gravitating liquids upon surfaces im- 
 mersed in them, or upon the sides of the vessels that contain 
 them, may be easily determined from the principles of the pre- 
 ceding chapter. 
 
 The equation (352), gives us for the value of p, 
 
 and if the origin of the co-ordinate, z, be at the surface of a homo- 
 geneous liquid, 
 
 psz ; 
 
 substituting this value in the expression (346), for the pressure, 
 P, we have 
 
 dP=dksz. 
 
 The measure of the pressure upon an infinitely small surface, 
 therefore, is the weight of a column of the liquid, whose base is 
 equal to the surface pressed, and altitude equal to its depth be- 
 neath the surface of the fluid. 
 
 Upon a plane surface of determinate magnitude, lying in a ho- 
 rizontal position, the whole pressure is the sum of the partial pres- 
 sures upon all its elementary portions, and will therefore be 
 
 P=te, (359) 
 
 3 f*v or will be measured by the weight of a prism of the liquid, whose 
 base is equal to the surface pressed, and whose altitude is equal 
 to the distance of the plane from the horizontal surface of the li- 
 quid. If the plane be inclined to the horizon, call the distances 
 of its respective elements from the surface of the liquid, z 7 , z", 
 z", &c., the sum of the partial pressures will be 
 
 But if we call the distance of the centre of gravity of the plane 
 beneath the level surface of the liquid, Z, 
 
 and 
 
 P=skZ ; (359a) 
 
 and this will be true of any surface, whether plane or curved. 
 
 These several expressions, applied to the case of a liquid, con- 
 tained in a vessel, are wholly independent of the bulk of the li- 
 quid itself; for they include no other quantities than the area of 
 the surface pressed, and the depth of the liquid. Hence : 
 
TREISFRK OP [Book V. 
 
 33 J . The pressure of a liquid, upon a horizontal plane, is equal 
 to the weight of a prism of the liquid whose section is equal to 
 the area of the plane ; and whose altitude is equal to the depth of 
 the plane beneath the surface of the liquid. 
 
 The pressure of a liquid upon a surface that is not horizontal, 
 is equal to the weight of a prism of the liquid, whose section is 
 equal to the area of the surface pressed, and whose altitude is 
 equal to the depth of the centre of gravity of that surface beneath 
 the level surface of the liquid. 
 
 Upon equal and similarly situated surfaces, the pressures are as 
 the depths of the liquid ; and at equal depths, the pressures are 
 as the surfaces. 
 
 Upon a given surface, the pressure depends wholly upon the 
 depth of the liquid above its centre of gravity, and has no refe- 
 rence to the quantity of liquid ; and thus, in vessels having equal 
 bases, and in which the liquid stands at equal heights, the pres- 
 sures on the bases are equal, however different may be the res- 
 pective capacities of the vessels. 
 
 If the vessel be prismatic, and the base horizontal, the pressure 
 on the base is exactly equal to the weight of the liquid it contains ; 
 but if the vessel be wider at top than at bottom, the pressure on the 
 base is less than the weight of the liquid it contains ; while if the 
 vessel be narrower at top, the pressure on the base exceeds the 
 whole weight of the liquid. 
 
 332. When a liquid is in equilibrio in a vase, its surface is, as 
 has been shown, 315, horizontal : on the sides of the vessel, 
 equilibrium exists in consequence of the fluid pressure being ex- 
 actly counteracted by the resistance of the solid boundary. This 
 resistance may be represented by a force, whose direction is a 
 normal to the surface : hence the liquid pressure must also act in 
 the normal, and will always have a direction perpendicular to 
 the point of the surface on which it acts. In addition to the pres- 
 sure on the bottom of the vessel, the sides are also pressed by 
 forces, estimated as has been stated above ; and thus in a pris- 
 matic vessel, not only will the base sustain a pressure equal to 
 the whole Weight of the fluid, but the sides will also sustain pres- 
 sures perpendicular to their several surfaces : the measure of these 
 forces is a weight of a prism of the liquid, whose horizontal section 
 is etfual to the area pressed, and whose altitude is equal to the depth 
 at which the centre of gravity of the sides lies beneath the horizon- 
 tal surface of the liquid. Thus, in a cube, filled with liquid, the 
 pressure on the base is equal to the whole weight of the liquid, 
 and acts vertically in the same direction with the force of gravity. 
 On each of the four vertical faces, the pressure is equal to 
 half the weight of the liquid mass, and acts in a horizontal di- 
 
Book V.~\ GRAVITATING LIQUIDS. 327 
 
 rection. A liquid, therefore, acting on the sides and base of a 
 vessel, both changes the intensity and the direction of the force 
 which acts, namely, the weight of the mass itself impelled by the 
 force of gravity in a vertical direction. 
 
 A liquid then, is, in fact, by the definition of 131, a machine ; 
 and masses of liquid may, and frequently are, used to produce ef- 
 fects analogous to those produced by machines. 
 
 333. In a close vessel, to which a lateral tube is adapted, if both 
 be filled with a liquid, the pressure on the sides, the top and the 
 base, will depend upon the areas of these substances, and on the 
 height of the liquid in the lateral tube. This will be true, what- 
 ever be the respective dimensions of the vessel and the tube adapt- 
 ed to it ; and thus any quantity of liquid, however small, may be 
 made to counterbalance any other quantity however great. This 
 principle is usually styled the Hydrostatic Paradox. 
 
 When a liquid is contained in a cjose vessel, and a pressure is 
 exerted upon it by means of a piston fitted tightly to an opening 
 made in one of its sides, this pressure may be represented by the 
 weight of a mass of fluid resting upon the piston as a base ; the 
 action of the piston will therefore produce the same effect upon 
 the liquid mass as a column of liquid, whose area is equal to that 
 of the piston, and whose altitude is such, that the weight of the 
 prism of liquid that has this area and altitude, shall be equal to 
 the pressure upon the piston ; the pressure exerted by the piston 
 will therefore be equally felt upon all the sides of the vessel, and 
 will upon an area equal to that of the piston, be equal to the 
 whole pressure the latter exerts. 
 
 334. These principles may be subjected to the test of experi- 
 ment, which fully confirms them ; and experiments are of import- 
 ance in this department of mechanics, inasmuch as we know no- 
 thing of the nature of the particles of which fluids are composed, 
 nor of the manner of their action upon each other; the hypothe- 
 sis on which we have proceeded in the mathematical investiga- 
 tion of the conditions of equilibrium, and of the measure of pres- 
 sure, is obviously inaccurate, in omitting the viscidity, and the 
 compressibility of liquids. When, however, we find that the 
 deductions from the hypothesis are exactly consistent with ex- 
 periment; we infer that the neglect of these circumstances pro- 
 duces no sensible error. The experimental illustrations are as 
 follow : 
 
 (1.) If water or any other liquid be contained in an open vessel, 
 it assumes a horizontal surface, except so far as it is influenced near 
 the sides, by a force that will be hereafter the object of investi- 
 gation. 
 
328 PRESSURE OF [Book V. 
 
 (2.) If two or more immiscible liquids be poured into the same 
 vessel, they arrange themselves in the order of their respective 
 densities, the denser liquids occupying the lowest places. 
 
 (3.) If a liquid be poured into a bent tube, or into vessels com- 
 municating at bottom, it rises in both branches of the tube, and in 
 all the vessels, to a common level, whatever be the areas of the 
 branches of the tube, or the cubic contents of the several vessels. 
 
 (4.) Two heterogeneous liquids, placed in opposite branches of 
 a bent tube, rise to heights above their surface of contact inverse- 
 ly proportioned to their respective densities; the denser liquid 
 will occupy the bend of the tube, and the surface of contact will 
 be in the branch occupied by the rarer liquid. 
 
 (5.) The phenomena of bodies denser than liquids, in which 
 they are placed, sinking; and those which are rarer, rising and 
 floating at the surface, are too familiar to need illustration. 
 
 (6.) The quantity of weight lost by a body immersed in a li- 
 quid, may be shown to be equal to the weight of an equal volume 
 of the liquid, by means of a cylindric bucket which a solid of the 
 same form exactly fills. If the bucket and solid be placed in one 
 of the scales of a balance and counterpoised, and the solid be af- 
 terwards suspended from the scale, in a vessel containing a liquid, 
 the former counterpoise will now preponderate : but if the bucket 
 be filled with a portion of the liquid, .equilibrium is restored. 
 
 (7.) The quantity of liquid displaced by a floating body, may be 
 measured by performing the experiment in a graduated glass jar ; 
 and it will be found, that the increase in the space the liquid oc- 
 cupies, is exactly equal to a volume of the liquid whose weight 
 is identical with that of the solid body. 
 
 (8.) The same experiment performed with two different liquids, 
 and the same solid body, shows that the quantities the latter dis- 
 places of each respectively, are inversely as their densities. 
 
 (9.) Vases of different figures and areas, and therefore containing 
 different bulks of a liquid, may be adapted by screws to one and 
 the same base, held in its place by a counterpoising force ; and it 
 will be found that this opposing force is overcome in each of 
 them, so soon as the liquid acquires the same depth. The same 
 will be found true, whether the base be horizontal or inclined. 
 The force that keeps the base in its place, may be a weight act- 
 ing through the intervention of a lever of the first kind ; this 
 weight therefore furnishes a measure of the liquid's constant pres- 
 sure in the vases of different figures ; and this weight will be 
 found, if the lever have equal arms, to be equal to the weight of 
 a prism of water, having a section equal to the base on which 
 the liquid presses, and an altitude equal to the depth of the liquid 
 in the vessel. 
 
f* 
 
 Book V.~\ GRAVITATING LIQUIDS. 329 
 
 (10.) An apparatus, called the hydrostatic bellows, is formed 
 by connecting two circular or elliptical planks, by a flexible 
 band of leather, in such a manner as to be water tight; to this 
 a tall tube is adapted'. If a liquid be poured into this apparatus 
 through the tube, it will be found that it is capable of counterpoi- 
 sing a weight resting upon the upper plank, equal to the weight 
 of a prismatic mass of water, whose section is equal to the area of 
 the bellows, and altitude equal to that of the liquid in the tube. 
 
 (11.) The effect of pressures, exerted by means of pistons, may 
 be best illustrated by the machine called Bramah's press, or by 
 the forcing pump. 
 
 335. When a gravitating liquid is placed in a vessel, the pres- 
 sure it exerts upon equal surfaces of its sides, will increase with 
 the depth, as is obvious from the investigations of 314 ; or as 
 may be easily shown, by supposing the liquid to be divided into 
 an infinite number of horizontal strata. 
 
 If the respective distances of these, from the surface, be z', z", 
 z'", &c., the respective pressures on equal surfaces, will be 
 
 sz'dk, sz"dk, sz'"dk, &c. 
 
 quantities continually increasing with the variable co-ordinate, z. 
 Hence : 
 
 Although the pressure upon a given base, horizontal, inclined, 
 or vertical, depends upon the depth to which its centre of gravity 
 is immersed below the level of the liquid, that point will not be, 
 in the case of an inclined or vertical surface, the point of applica- 
 tion of the resultant of the fluid pressures. This point of appli- 
 cation is called the Centre of Pressure, and must be lower than 
 the centre of gravity of the surface pressed. The true position 
 of the centre of pressure, may be thus investigated in the case 
 of a plane surface, in which it will always fall in the plane itself. 
 
 Let k represent the area of the surface, dk will be one of its 
 small elements ; let z represent the distance of the centre of gra- 
 vity of the whole surface, from the upper level of the liquid, and 
 z\ z", z'", &c., the distances of its several elements. 
 
 The several partial pressures will be 
 
 z dk, z' dk, z" dk, &c. ; 
 and the sum of their moments of rotation, , 
 
 The whole pressure will be 
 
 Zk;, 
 
 and if we call the distance of the centre of pressure, c, the mo*- 
 ment of rotation will be 
 
 Z&c; 
 
 42 
 
330 PRESSUKE OF [Booh V. 
 
 and this will be equal to the sum of the moments of the partial 
 pressures, or 
 
 zkc=dk(.z 2 +z'-+z" a +&tc.) ; 
 whence 
 
 - : (360) 
 
 JLtK 
 
 but we have 
 
 Zk^dJ^z+z'+z"), 
 and, therefore, 
 
 
 or if we call the several elements, A, B, C, &c., their respective 
 distances from the surface, o, 6, c, &c., 
 
 A+B6+Cc+&c. ' 
 
 a formula identical with ( ), which we have found adapted to 
 ?&e investigation of the position of the centre of oscillation. 
 This may be given in a more convenient form. 
 Let x and y, be the vertical and horizontal co-ordinates of the 
 surface, kthe-depth of its upper edge beneath the level of the li- 
 quid, the area will be 
 
 the several distances, z, z', z", &c. ; 
 
 h+dx, h+2dx, h+3dx, Sic. : 
 the expression (361) will, therefore, become 
 
 and when the plane that is pressed by the liquid, reaches to the 
 level surface of the latter, 
 
 (86* 
 From these formulae we obtain the following useful results : 
 
 In a parallelogram, whose upper side is on a level with the 
 surface of the liquid, the centre of pressure is at a distance of two 
 thirds of its'height beneath that surface. 
 
 In a triangle, whose base is on a level' with the same surface, 
 this distance is one half its altitude ; but if the vertex be in the 
 surface of the liquid, the distance is three-fourths. 
 
 336. When a solid body is wholly immersed in a liquid, its 
 surfaces undergo pressures ; the measures of these will be the 
 weight of columns of the liquid, having the surface pressed for a 
 base, and the depth of its centre of gravity for the altitude. The 
 whole amount of the pressures will, therefore, depend upon the 
 magnitude of the surface of the solid, and have no reference to its 
 
 
 
 
Book 
 
 GRAVITATING LIQUIDS. 
 
 volume or density. But when the directions of these pressures 
 are taken into account, they have a resultant which depends upon 
 2 volume solely, and is, as might indeed be inferred without 
 investigation, the same with the loss of weight the solid experien- 
 ces, or is equivalent to (he weight of a mass of the fluid, equal in 
 volume to the immersed solid. 
 
 A more strict investigation is, however, necessary : 
 Let the body immersed be, ABQ, and suppose its whole mass 
 to be divided into an infinite number of small cylindrical columns, 
 
 lying in a horizontal position. Let 
 % be one of these, terminated at 
 the surface in the bases, B6, Q^, 
 and distant from the level of the fluid 
 by the depth LBz. 
 
 Let BN be a normal to the sur- 
 face, B6, which may be considered 
 as plane. Let 6H be the projection 
 of this surface upon a plane passing 
 through 6, and perpendicular to BQ. 
 6H will be the section of the cylin- 
 der, perpendicular to it3 axis, and 6H~ -B6, cos. NBQ. 
 
 The liquid exercises upon the base, B6, a pressure whose 
 measure, taking the density of the liquid=l, is 
 
 and whose direction is the normal BN. If this pressure be de- 
 composed into three rectangular forces, that which acts horizon- 
 tally in the direction BQ, will be 
 
 z.Bfc cos. NBQ=*./jH. 
 
 In the same manner we find the horizontal pressure on the oppo- 
 site base, Q</, to be 
 
 *-#Q ; 
 
 but the surfaces, 6H and gQ, are equal to each other, and the 
 horizontal pressures are, therefore, equal ; they are also opposite 
 to each other, and hence their resultant =0. 
 
 And in like manner it may be shown, that all the horizontal 
 pressures parallel to BQ, mutually destroy each other. 
 
 If we next suppose the body to be divided into an infinite num- 
 ber of columns also horizontal, but at right angles to BQ, the same 
 result will follow. And these two sets of forces at right angles to 
 each other, make up the whole horizontal pressures ; for each 
 partial pressure may be resolved into two forces at right angles to 
 each other, and all of these are included in the two sets of which 
 we have spoken. The horizontal pressures upon the surface of 
 a solid, immersed in a liquid, are, therefore, in equilibrio. 
 
 Next let us suppose the volume of the solid to be divided into 
 an infinite number of vertical columns, and let one of these be 
 A6, terminated by the bases, Aa, B6 ; the lower base being im- 
 mersed to the depth, LB=z ; the upper to the depth LA =z'. 
 
333 PRESSURE OP, &c. [Book V. 
 
 The normal pressure upon the upper surface, Aa, resolved into 
 three rectangular forces, gives for the vertical pressure, 
 
 z'.AI. 
 The vertical pressure on the lower surface is in like manner, 
 
 z.VH, 
 and 
 
 AI=BH. 
 The resultant of these two opposite vertical pressures is therefor 
 
 0-*') ; BH, 
 
 and is in the direction BA. It is, therefore, equal and opposite 
 to the weight of the column of fluid that would occupy the space 
 of the column A6. And the resultant of all the vertical pressures 
 would be equal and opposite to the weight of a mass of the liquid, 
 equal in volume to the solid body immersed. 
 
 In the same manner it may be shown that the resultant of all 
 the pressures, exerted by a liquid upon the vessel that contains 
 it, is equal to the weight of the liquid itself. 
 
 This, however, does not contradict the law, that the amount 
 of the pressures exerted outwards have for their measure the 
 weight of a column of the liquid whose base is equal to the whole 
 surface of the vessel, and whose altitude is the depth of its centre 
 of gravity beneath the surface. 
 
00k V.} SPECIFIC GRAVITIES. 333 
 
 CHAPTER IV. 
 
 OF SPECIFIC GRAVITIES. 
 
 337. The specific gravity of a body is the ratio of its weight 
 the weight of an equal volume of some other body. In this 
 
 jneral sense, it is equivalent to density, which is the relation 
 jtween the weights of equal volumes of different bodies. But 
 hile density is an abstract term, and is determined by the di- 
 ict comparison of the bodies in question, specific gravity is re- 
 tive, and is a numerical value of the density obtained by com- 
 irison with some conventional standard. The standard, in 
 ;neral use for this purpose, is water. As this substance may 
 mtain gaseous, earthy, and saline impurities, it will only answer 
 le purposes of a standard when freed from them. This may 
 ; effected by distillation. The heat of boiling drives off all the 
 iseous matter, and, in distillation, the solid substances are left 
 ;hind in the still. Newly fallen rain water, at a distance from 
 *bitations, or that obtained by the melting of clean snow, is also 
 ifficiently pure for the purpose. 
 
 338. Water, like all other substances, is liable to changes of 
 jlume by changes in its temperature ; hence, it can only be em- 
 loyed as a standard, at some conventional temperature, to which 
 le results of the experiments for determining specific gravities 
 iust be reduced. The English usually take, for this purpose, a 
 lean atmospheric temperature, say about 60 ; the French, the 
 smperature at which water has its maximum of density. The 
 tter is by far the most convenient and scientific method, for it 
 
 hardly possible to perform experiments at the exact tempera- 
 ire chosen by the English, in consequence of the practical diffi- 
 alty of counteracting, by additions of colder or warmer water, 
 ic constant variations in the temperature of the air; and this 
 icthod is dependent for its accuracy upon the indications of a 
 lennometer, the divisions upon whose scale are arbitrary. On the 
 Lher hand, it is always possible to obtain water at its maximum 
 snsity, and easy to maintain it in that state. Thus a vessel of 
 ater, on which a small portion of ice floats, will have always in 
 s lower parts, water of the maximum density ; for so long as this 
 ate has not been attained by the water, the cooler portions will 
 nk to the lower part of the vessel , in consequence of their greater 
 eights under equal bulks ; but so soon as the maximum of den- 
 ty is attained by the lowest portions of the liquid, this descent 
 
SPECIFIC ORATITIES. [Book T~. 
 
 ceases. Thus the French standard temperature is best suited to 
 all cases where great accuracy is required, as it is better to obtain 
 a result from direct experiments, that require no correction, than 
 to correct those obtained under other circumstances. 
 
 In the cases that most frequently occur in practice, such nicety 
 is unnecessary, and the experiment may be performed with water 
 of any temperature ; but the temperature must be noted, and a 
 correction applied for it. 
 
 This correction rests upon the density of water, at the experi- 
 mental temperature. We therefore subjoin a table of the den- 
 sities of water, at different sensible heats, the maximum of den- 
 sity being employed as the unit. 
 
 TABLE. 
 
 Of the Densities of Water at different Temperature*. 
 
 32. 
 
 0.9998918 
 
 50. 
 
 0.9997825 
 
 34. 
 
 0.9999428 
 
 55. 
 
 0.9995324 
 
 36. 
 
 0.9999761 
 
 60. 
 
 0.99918S6 
 
 38. 
 
 0.9999944 
 
 65. 
 
 0.99S7549 
 
 39.4 
 
 1.0000000 
 
 70". 
 
 0.9982239 
 
 41. 
 
 0.9999950 
 
 75. 
 
 0.9976255 
 
 43". 
 
 0.9999739 
 
 80. 
 
 0.9969316 
 
 45. 
 
 0.9999378 
 
 85. 
 
 0.9961497 
 
 339. The densities of bodies might be compared, by immersing 
 them, in succession, in a prismatic vessel, containing a liquid of 
 less density than either. The liquid in which a solid is immersed, 
 occupies a space as much greater than it did before, as is equal to 
 the volume of the solid. Hence, in a vessel of constant and 
 known area, the relation between the changes of level caused by 
 the immersion of different solid bodies, gives the relation between 
 their densities; and when the weight of a given bulk of water is 
 known, the same method would give their respective specific 
 gravities in terms of water as the unit. 
 
 This was the original method proposed by Archimedes, in the 
 famous problem of the crown of Hiero. 
 
 The method now employed, depends upon the principle that 
 a body, when immersed in a fluid, loses as much of its weight as 
 is equal to the weight of an equal volume of the fluid. 327. 
 
 Let S be the specific gravity of the solid body ; *', that of the 
 liquid ; IP, the absolute weight of the solid ; IP', its weight in the 
 liquid ; ID u?', will be its loss of weight in the liquid, and, there- 
 fore, the weight of a mass of the liquid, equal in volume to the 
 
Book /^.] SPECIFIC GRAVITIES. 335 
 
 solid. If B be this common volume, the definition of specific 
 gravity gives us 
 
 w X M? w' 
 
 8=, and ''= 
 
 whence 
 
 w ww' 
 
 *_*^* f ( 364 ) 
 
 and 
 
 8=," " 
 
 rature, 
 
 w w'" 
 If the liquid employed be pure water, at its standard tempe- 
 
 and 
 
 c 
 
 w w' 
 
 To obtain then the specific gravity of a solid body, its weight 
 must be divided by the loss of weight in pure water, of the stand- 
 ard temperature. If the water be at any other temperature, we 
 obtain the approximate specific gravity, by dividing the weight 
 of the body by its loss of weight in the water; this may be re- 
 duced to the true specific gravity, as will be seen from the final for- 
 mula. (364), by multiplying the proximate specific gravity by the 
 density of water, at the temperature at which the experiment is 
 made ; the density of water at the standard temperature being 
 taken as the unit. For this purpose, the table in the preceding 
 section may be used. 
 
 340. In determining specific gravities, we use a common ba- 
 lance, to which apparatus intended to facilitate the process of 
 weighing the body in water, is adapted. When the solids are in 
 the form of masses united by the attraction of aggregation of 
 their particles, and are denser than water, the process is ex- 
 tremely simple. The body is weighed in air, and then in water, 
 and the first weight is to be divided by the difference of the two 
 weights ; a correction is then applied^ as has been just stated, for 
 the temperature of the water, if it be not at its maximum density. 
 
 When the solid is in a state of fine^powder, it must be placed in 
 a vessel whose weight in air and in water, have been previously 
 determined. When the vessel containing the powder is weighed 
 in air and in water, the difference between these, and the weights 
 of the vessel alone, in air and in water, are the weights of the 
 powder under the two different circumstances. The calculation 
 may then be performed as before. 
 
33C SPECIFIC GRAVITIES. [Book V. 
 
 When the solid is soluble in water, it often happens that a li- 
 quid may be found in which it is not soluble. The density of 
 this liquid being known, by methods hereafter to be explained, 
 the body may be weighed first in air and then in this liquid ; we 
 obtain, by the same method of calculation, its specific gravity, in 
 terms of the liquid as the unit. This, as will be seen from (364) 
 must be multiplied by the specific gravity of the liquid, in order 
 to obtain that of the solid in terms of water ; for it is obviously 
 unimportant whether we use water at a temperature different from 
 the standard, or a liquid whose density in respect to water is 
 known. 
 
 When the solid is less dense than water, its weight in that 
 liquid cannot be obtained directly ; but it may be attached to 
 another, sufficiently dense to cause both to sink. The weight of 
 the light body in water will obviously be the difference in the 
 weights of the two bodies when united, and of the heavy body 
 alone. 
 
 The principle and method of calculation ma}? be best illustrated 
 by symbols : 
 
 Let / be the weight of the light body ; fe, that of the heavy body, 
 air ; h', of the same body in water ; c, the joint weight of the two 
 bodies in air ; c', their joint weight in water. Then c c', and 
 h h', will be the respective losses of weight, and the loss of 
 weight of the light body will be 
 
 (o-O-(A-V); 
 hence, 
 
 The divisory in this formula, will always exceed the dividend, 
 and the resulting specific gravity is a fraction, which is usually 
 obtained in the decimal form. 
 
 When no convenient fluid can be found in which the solid 
 is insoluble, its specific gravity cannot be obtained by means of 
 the hydrostatic balance. 
 
 341. The specific gravity of a liquid may be determined with 
 the hydrostatic balance, in various ways. 
 
 (I.) A solid of convenient size and form, usually a bulb of 
 glass of known weight, may be weighed in water, and in the 
 liquid whose specific gravity is sought. The respective losses 
 of weight, are the weights of equal bulks of the two fluids; and 
 water being the unit, we divide the loss of weight in the liquid 
 by the loss of weight in water. 
 
 For if we call the loss of weight in \vater, w" ; the loss in the 
 other liquid, /, tke formula (365) becomes 
 
 S=~ (367) 
 
 ! ' ,-;p * || 
 
 * 
 
 , 
 
Book f.] SPECIFIC GRAVITIES. 337 
 
 (2.) A phial of known weight may be taken, filled with water 
 and weighed j the increase of weight is the weight of the con- 
 tained water; it is then emptied, and filled with the liquid, 
 whose weight is obtained in the same manner: we have again 
 weights of equal bulks of the substances, whence the specific 
 gravity may be obtained, as in the preceding instance. 
 
 (3.) We know, by accurate experiments, the weight of a given 
 bulk of water ; hence, if a phial of known weight and internal 
 capacity be taken, the weight of the water it contains is known. 
 It is, therefore, only necessary to fill the phial with the liquid 
 whose specific gravity is sought, and weigh it. The mode of 
 calculation is obvious. 
 
 342. In determining the weight of a given volume of water, or 
 of any other liquid, it is found much more easy in practice to 
 weigh a solid of known dimensions and weight in the liquid, 
 than to make a vessel of a given capacity, and fill it with the 
 liquid. This arises from the great ease and certainty with which 
 the external dimensions of a solid can be measured, compared 
 with those with which the internal capacity of a vessel can be 
 guaged. 
 
 Sir George Shuckburgh, in order to ascertain the weight of a 
 given bulk of water, (a cubic inch,) in Troy grains, made use of 
 three solids of the same material, but of different forms. One 
 was a sphere, another a cylinder, the third a cube : the material 
 was brass. These having been constructed with' every possible 
 care, were afterwards measured by a scale furnished with pow- 
 erful microscopes. In this way the lineal dimensions were as- 
 certained, and the inequalities, inseparable from the best mate- 
 rials, and most accurate workmanship, detected : from these data, 
 their volumes were computed. 
 
 Experiments, with the same three solids, were also made by 
 Kater, in the researches on which the British standard of weights 
 and measures is founded. 
 
 In the French investigations for establishing the basis of their 
 metrical system, a similar method was used, with a body of cylin- 
 drical form, and of larger dimensions than those used by Shuck- 
 burgh. 
 
 The experiments of Kater make the weight of a cubic inch of 
 pure water at the temperature of 62, 252 grs. 458, Troy. 
 
 The State of New-York has, as stated in Book IV., Chapter 
 V., taken for the basis of its standard of weight the avoirdupois 
 pound, of such magnitude that the cubic foot of pure water, at its 
 maximum density, weighs 1000 oz. or 62lbs. 
 
 The same difficulty that we have stated, exists in determining 
 standards of cubic measure, by means of the internal capacity ; 
 
 43 
 
333 srjECinc GRAVITIES. [Book V. 
 
 hence, the British Government, and the Legislature of the State 
 of New-York, have defined their units of capacity, by prescribing 
 the number of pounds of distilled water, of the standard tempe- 
 rature that the vessel shall contain. This method is preferable 
 to that of defining it by the number of cubes of the measure of 
 length that it shall comprise. 
 
 343- In the instructions for determining specific gravities of 
 33S, it will have been noted that the weight in air is spoken of, 
 instead of the absolute weight referred to in the theory of 337. 
 These are not identical, for the air being a fluid, presses upon the 
 solid, and produces a loss of weight analogous to that caused by 
 liquids, as demonstrated in 325. Indeed, as the air has an uni- 
 form density within the space the solid under experiment occu- 
 pies, the effect is in fact identical, and the solid loses as much of 
 its absolute weight as is equal to the weight of its volume of air. 
 In different bodies, of equal weight, this loss will be in the in- 
 verse ratio of their densities, and thus the rarer bodies will be 
 the most affected. In very accurate investigations, it becomes 
 necessary to take the buoyancy of the air into account. The 
 principle on which this correction is founded, may be inferred 
 from the theory that has already been laid down. The exact 
 amount, and the variations to which it is subject, must be defer- 
 red until we have investigated the properties of air. 
 
 When the weights and the substance to be weighed are homo- 
 geneous, both are equally affected by the buoyancy of the air. 
 Hence, there area few cases in which no correction is necessary, 
 even in the most accurate investigations : thus, in prescribing the 
 weight of a given measure of distilled water, for the purpose of 
 defining the unit of weight, the British and New-York statutes 
 declare that the experiments shall be made with brass weights, 
 under which denomination the experimental solid is included. 
 
 344. In many practical cases, methods more speedy than are 
 furnished by the hydrostatic balance, are necessary. This neces- 
 sity occurs more frequently in the determination of the specific 
 gravities of liquids, and particularly in ascertaining the value of 
 spirituous liquors, both for the convenience of commerce, and the 
 collection of revenue. In these cases, an instrument called the 
 Hydrometer, is employed. 
 
 A hydrometer consists essentially of three parts : a stem, on 
 which divisions are drawn; a bulb or hollow float, to render it 
 buoyant in a liquid ; and a weight by which the stem may be 
 made to float in a vertical position. These parts are exhibited in 
 the following figure, in which A is the stem, B the bulb, and C 
 the weight 
 
Book 
 
 PBCIFIC GRAVITIES. 
 
 339 
 
 The principle on which the hydrometer is 
 used, is, that a body net to float on two different 
 fluids, of te.s density than itself, will displace 
 volumes of them, that ore inveiveJy as their re- 
 spective densities. 330. 
 
 It is, therefore, obvious, that the hydrometer 
 must he less dense than the rarest liquid in 
 which it is intended to be used ; and that in the 
 most dense, it must sink at least so far as to 
 cover its bulb; otherwise, it would in the one 
 case sink to the bottom, and in the other, the 
 liquid would not reach the divisions of ihe stem. 
 Delicacy of indication will be promoted in 
 the hydrometer by making the stem slender, 
 in which case the differences of the depths to 
 which it sinks will be considerable for small 
 changes of specific gravity. On the other hand, 
 the extent of differing specific gravities, or what 
 is styled the scale, will be limited, when the 
 stem is slender, unless it be made of inconve- 
 nient length. When great accuracy is required, 
 the stem must be necessarily slender; and in 
 order to obtain measures of specific gravities, 
 not included within the scale of a single instrument, two or more 
 hydrometers must be used. 
 
 This is generally the case with the hydrometers used by che- 
 mists, of which the material is glass. Glass is well adapted to 
 the purpose, from its cleanliness; the case with it may be fa- 
 shioned by the blowpipe; and from its being acted upon by but 
 few chemical liquids. On the other hand, its fragility is an ob- 
 jection to its use by manufacturers, and in the hands oif fiscalofii- 
 cers. 
 
 The necessity of having more than a single instrument, may 
 be obviated by adapting moveable weights to the same hydro- 
 meter, in such manner that they may be removed, or added, ac- 
 cording to the density of the liquid. 
 
 For these reasons the hydrometers in more frequent use, are 
 made of metallic substances, and have metallic weights. 
 
 Dicas' hydrometer, which is prescribed by law in the customs 
 of the United States, has 36 moveable weights, and the stem has 
 10 divisions: the weights are so adjusted that in a given liquid, 
 if the hydrometer sink with one of the weights to the on the 
 stem, the next weight shall sink it 10. Hence the divisions 
 amount to 360. This instrument is extremely delicate, and well 
 
340 SPECIFIC CHAYITIES. [Book V. 
 
 suited for nice investigations, but is too complex and trouble- 
 some in its use for ordinary purposes. 
 
 A hydrometer upon similar principles, but of greater simpli- 
 city, has also been constructed in Boston. 
 
 A still more simple instrument has been introduced by South- 
 worth, of New York. The material is also silver, and there is 
 but one moveable weight. All these instruments measure spe- 
 cific gravities from that of pure water, or 1, to that of alcohol or 
 0.825. 
 
 Silver is objectionable as a material for hydrometers, unless it 
 be gilt, in consequence of its liability to tarnish, and the necessity 
 of cleaning it, which will wear away the metal, and alter the in- 
 dications of the instrument. 
 
 345. The hydrometer may be used for determiningthe weights 
 and specific gravities of solids. For this purpose the instrument 
 is made so much lighter than water, as to require the addition of 
 a considerable weight to sink it to a mark on the middle of its 
 stem. This weight being known, in some conventional unit or 
 standard, the body to be weighed is placed upon a cup prepared 
 for the purpose at the top of the instrument; weights 
 are added until the hydrometer sink to the mark at 
 which it before stood ; the joint weight of the body, 
 and the additional weights, is, therefore, the same as 
 that which was found by experiment to bring the in- 
 strument to this position. It only remains to subtract 
 the weight added when the body is placed in the cup, 
 from that which when used alone sinks it to the proper 
 level ; the difference is the weight required. 
 
 If the specific gravity is to be ascertained, the bo- 
 dy is next placed in a second cup, adapted for the pur- 
 pose, to the top of the weight that steadies the instru- 
 ment, where it is of course immersed in the water. 
 Its loss of weight will be apparent by a rise in the 
 stem of the instrument ; fresh weights are added to 
 bring it again down to the original mark, and these are 
 the measure of this loss. 
 
 Such is the hydrometer of Nicholson, the form and arrange- 
 ment of which may be understood from the inspection of the an- 
 nexed figure. 
 
 An instrument for weighing, founded upon similar principles, 
 but far superior in accuracy, has recently been constructed by 
 Mr. Hassler, to be employed in the rectification of the weights 
 and measures used in the customs of the United States. A hoi- 
 
BoOOk V.} SPECIFIC GRAVITIES. 341 
 
 low bulb of glass, of the figure of a flask, is placed in a vase that 
 is set upon a bracket ; three steel rods are sealed to the neck of 
 the flask, and bear a metallic plate, cut in such a manner as to 
 form three horizontal arms that project beyond the sides of the 
 vase; from these arms three rods proceed downwards, two of 
 which embrace the bracket, and the third descends in front of 
 it ; these rods bear a shelf or scale on which weights and the 
 substance to be weighed are placed. By the use of the bracket, 
 the position of the weight is thus brought beneath the vase, 
 and the equilibrium of the apparatus is rendered stable, while in 
 Nicholson's hydrometer the equilibrium is tottering. The opera- 
 tion of weighing may, therefore, be performed with greater ease. 
 The size of the instrument is also much greater than is ever given 
 to Nicholson's ; and as the whole exposed to the liquid is of glass, 
 except the rods, mercury may be used instead of water, by which 
 liquid the capacity of the instrument to bear heavy weights, is in- 
 creased more than thirteen fold. 
 
 346. The specific gravity of bodies, is one of their most im- 
 portant distinctive characters, and must hence be included among 
 the properties by which they are described and defined. By 
 means of it, we may frequently determine the nature of bodies, 
 without having recourse to any other means ; and in all cases it 
 is an aid in the discovery of the class to which substances hitherto 
 unexamined are to be referred. We therefore use- the method of 
 specific gravities in many branches of physical science, in several 
 of the arts ; and although in commerce it is as yet applied to but 
 few purposes, its use is capable of much farther extension. 
 
 The seven ancient metals, which are still in most frequent use, 
 have such marked differences in density, that they may frequently 
 be distinguished from one another by their specific gravities ; and 
 if alloyed or adulterated, the determination of the specific gravi- 
 ties of the compounds will, in many cases, detect the mixture. 
 
 Thus in gold, whether in the shape of bullion, or of coin, great 
 specific gravity is often a test of its value. Until recently it was 
 a sure one ; for gold, before the discovery of platinum, was the 
 densest of all known substances, and any admixture is sure to pro- 
 duce a diminution in the specific gravity. Recently an alloy of 
 platinum has been discovered, that possesses many of the external 
 characteristics of gold. Platinum being the densest of substances, 
 we now know that even the appropriate density of alloys of gold 
 may be given by it, to the alloys of which the former metal forms 
 a part. These, however, want the ductility and malleability of 
 gold, and when they have the same colour, have less density than 
 its alloys. Among the other ancient metals, tin is the least dense, 
 and hence when impure, its specific gravity is increased. 
 
342 SPECIFIC GRAVITIES. [000k V. 
 
 The adulterations of drugs, and pharmaceutical preparations, 
 may be frequently detected by the change of specific gravity they 
 produce. Acids, in particular, can have their puriiy tested in al- 
 most all cases. 
 
 In the operations of nractical chemistry, the strength of solu- 
 tions, the proper state of concentration suited for crystallization, 
 for fermentation, and JistiJJation, are known by the use of the 
 hydrometer. This instrument is therefore of value in the aits of 
 brewing, distilling, sugar refining, as well as in the processes that 
 are strictly chemical. .-* 
 
 The value and character of metallic ores, may also be in many 
 cases inferred from their specific gravity ; and in general it forms a 
 marked distinction of minerals, by means of which they may be 
 classed, and their constituent parts inferred. 
 
 Such are a few of the more important purposes to which the 
 method of specific gravities, is applicable. 
 
 347. When two substances, whose specific gravitiesare known, 
 ore mixed mechanically, it might at first sight be inferred that the 
 proportions in which they exist in the compound, would be rea- 
 dily inferred from its specific gravity. This will be apparent 
 from the following investigation : 
 
 Let w and iv'be the weights of the two bodies ; 
 # and g' their respective specific gravities ; 
 G the specific gravity of the compound. 
 
 The weigh* of the compound is w+iv'. The weight of its vol- 
 ume of water (364) is 
 
 w+w' 
 
 ~~G 
 
 The respective weights in water, of the volumes of the two com- 
 ponents, are 
 
 w . to' 
 
 7 and 7 ! 
 
 hence 
 
 =-4-; (368) 
 
 * S S 
 whence we obtain 
 
 w = ^ (ic+i*Q ; 369) 
 
 and 
 
 ,_(r -G)^, , ,, (370^ 
 
 * "^~ _ j f u -+" x(, ) * v / 
 
 These formula? are, however, of little or no value in practice ; 
 
SPECIFIC 
 
 343 
 
 for when two bodies are mixed, their joint volume is rarely or 
 never the same as the sum of their respective volumes. In gene- 
 ral, the joint volume is less than the sum of the separate volumes. 
 This diminution of volume, that occurs in most cases of mixture, 
 is called Concentration. The most remarkable instance of con- 
 centration that has been noted, is that stated by professor Robin- 
 son, who says, that when 40 pts. of platinum are alloyed with 5 
 pts. of iron, the bulk of the mass is but 39. This alloy, then, of 
 the densest simple substance with one of little more than a third 
 of its density, gives a compound even more dense than the first. In 
 the case of the mixtures of alcohol and water, that form the spi- 
 rituous liquors of commerce, the concentration produces marked 
 effects, and must be taken into account in all the estimates of their 
 value that are drawn from their specific gravities. The densities 
 of such mixtures have been made the subject of accurate experi- 
 ments by Gilpin, the results of a part of which are comprised in 
 the following 
 
 TABLE 
 
 Of the densities of mixtures of alcohol and water, at the temperature of 
 600 of Fahrenheit. 
 
 PTS. OF 
 
 WATER. 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 
 10 
 9 
 8 
 7 
 6 
 5 
 4 
 3 
 2 
 1 
 
 
 PTS. OF 
 
 ALCOHOL. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 10 
 10 
 10 
 10 
 10 
 10 
 10 
 10 
 10 
 10 
 10 
 
 SPECIFIC GRAVITIES. 
 
 1.00000 
 
 0.98654 
 
 0.97771 
 
 0.97074 
 
 0.96437 * 
 
 0.95804 
 
 0.95181 
 
 0.94579 
 
 0.94018 
 
 0.93493 
 
 0.93002 
 
 0.92449 
 
 0.91933 
 
 0.91287 
 
 0.90549 
 
 0.89707 
 
 0.88720 
 
 0.87569 
 
 0.88208 
 
 0.8456S 
 
 0.82500 
 
344 SPECIFIC GRAVITIES. [Book V. 
 
 We have not reduced this table to the standard of the maximum 
 density of water; for the habitual custom, not only in England, but 
 in France, where the spirits of commerce are concerned, is to re- 
 duce the densities at the observed temperatures, to that of 60. 
 Upon this principle the hydrometer or alcoometer, planned by 
 Gay Lussac, and used in the French excise, is graduated. 
 
 In ascertaining the values of spirituous liquors, for the purpose 
 of the collection of duties in the United States, four different 
 stages of proof are established by law ; each of these pays a specific 
 duty. The spirit that lies between any two of these in density, 
 is counted as belonging to the lower proof. This method is un- 
 fair in practice, and has been made the source of actual, if not of 
 technical frauds upon the revenue. The true and equitable 
 principle is that adopted by the French government ; by this 
 the duty is estimated upon the quantity of alcohol of the specific 
 gravity of 0.825, at the temperature of 60, that is contained in 
 the mixture. 
 
 348. One other method of determining the specific gravities 
 of liquors remains to be mentioned. A number of bubbles may 
 be blown from a glass tube, having different densities. Their 
 specific gravities, having been determined by experiment, in li- 
 quids of known densities, are marked upon them. 
 
 When the specific gravity of a liquid is to be determined, they 
 are thrown into it, until one is found that remains quiescent in 
 any part of the liquid in which it is placed, and with the specific 
 gravity of this bubble, that of the liquid is identical. 
 
 349. We subjoin a table of specific gravities compiled from the 
 most accurate authorities. 
 
 TABLE 
 
 Of the specific gravities of bodies, in terms of Mater, at Us maximum of 
 
 density. 
 
 f Rolled, .... 22.6667 
 
 I Wire, .... 21.0396 
 
 Platinum, <> Hammered , . . 2 0.3356 
 
 I Purified, . . . 19.4981 
 
 i Hammered, . . . 19.3600 
 
 Pure gold. J Cast< .... 19.2562 
 
 Gold 22 carats fine cast, . . 17.4846 
 
 do 20, do " cast, . 15.7075 
 
 Tungsten, ... . 17.6000 
 
 Mercury, 13.5967 
 
 Lead, 11.3512 
 
 Palladium 11.3000 
 
 Rhodium, 1 
 
r.j SPECIFIC aRAVITIES. 
 
 Silver cast, . . .->. (,,;/ . . 10.4743 
 
 Bismuth, . . - {r-:j..v* 9.8220 
 
 Copper, '*,.;;*> = ;" i*-. 8.7880 
 
 Molybdenum, i.^ ..-^ '',- t .\.. . , .; 8.6110 
 
 Arsenic, . . ,. -., -. f . 8.3080 
 
 Nickel, . . . ' . . . 8.2790 
 
 Uranium, 8.1000 
 
 Soft Steel, | Hammered, 
 
 (Cast, . . . ..:-;. r- 7.8324 
 Hard Steel, I Hammered, 
 
 ( Cast, .... 7.8156 
 
 Cobalt, . .... 7.8119 
 
 Bar Iron, 7.7873 
 
 Tin, 7.2907 
 
 Cast Iron, . * -\ '.,-. . . 7.2083 
 
 Zinc, .... ...... "/. 6.8604 
 
 Antimony, ...... 6.7120 
 
 Tellurium, 6.1150 
 
 Chromium, ...... 5.9000 
 
 Iodine, . . -. J .... 4.9480 
 
 Sulphate of Baryta, . '^;V -,''.', . . 4.4300 
 
 Zircon, .... *'V ' . 4.4157 
 
 Ruby and Sapphire, (oriental) . . v V 4.2830 
 
 Diamond, from ..... 3.5307 
 
 to 3.5007 
 
 Flint Glass, ' ' 3.3290 
 
 Fluor Spar, 3.1908 
 
 Tourmaline, 3.1552 
 
 Native Sulphur, 2.0329 
 
 Sodium, . . . 0.9726 
 
 Potassium, ... . 0.8651 
 
 Sulphuric Acid, . . . . . 1,8408 
 
 Nitric Acid, , 1.5500 
 
 Water from the Dead Sea, . . 1.2402 
 
 Nitric Acid, 1.2176 
 
 Sea Water, 1.2062 
 
 Milk, 1.0300 
 
 Distilled Water, . . 1.0000 
 
 Bourdeaux Wine, ... 0.9938 
 
 Burgundy Wine, ... 0.9214 
 
 Olive Oil, . . . . . 0.9152 
 
 Muriatic Ether, . "... 0.8740 
 
 Spirits of Turpentine, . . 0.8697 
 
 Naptha, . . ' 0.8475 
 Standard Alcohol of Gilpin, . 
 Alcohol perfectly pure, 
 Sulphuric Ether, . 
 
 44 
 
 345 
 
346 SPECIFIC GRAVITIES. [Book V. 
 
 350. If the weight of a given bulk of water be known, the 
 Hydrostatic balance may be applied when geometric methods fail, 
 to determine the volume or cubic contents of bodies. For this pur- 
 pose they must be weighed in air, and in water, and as their 
 loss of weight is the weight of their volume of water, their cubic 
 contents may be calculated. 
 
 Their volume B in cubic inches, when the weights are taken 
 in troy grs., will be 
 
 w w' 
 
 Their volume in cubic feet, when the weights are taken in avoirdu- 
 pois ounces will be 
 
 When the specific gravity of a body is known, we may calcu- 
 late its volume when its weight is given, or its weight when its 
 volume is given upon similar principles. 
 
 If B be the volume, and W the weight of the body, g its speci- 
 fic gravity, w the weight of the cubic unit of water in the unit of 
 measure, we have 
 
 W=BW>, (373) 
 
 and 
 
 B= . (374) 
 
 gto 
 
Book V.] ELASTIC FLUIDS. 347 
 
 CHAPTER V. 
 
 Oj- THE NATURE AND CHARACTERS or ELASTIC FLUIDS, AND OF THE 
 PRESSURE OF THE ATMOSPHERE. 
 
 351. We become acquainted with the material existence of 
 the air that composes the atmosphere of our earth, and which is 
 not at first as obvious to our senses as that of solids and liquids, 
 in various ways : 
 
 (1.) We observe it moving in currents of wind, capable of 
 producing powerful mechanical effects ; 
 
 (2.) We find it resisting the entrance of other substances into 
 the space it occupies, thus : when a glass jar is inverted and press- 
 ed into a mass of water, the water enters it at first to but a small 
 distance ; and whatever be the depth to which it is forced down, 
 the air will still occupy a part of the jar ; 
 
 (3.) We may weigh it, and thus show that, like other mate- 
 rial substances, it is affected by the attraction of gravitation. 
 
 The fluid nature of air may be shown by the freedom with 
 which bodies at the surface of the earth move in it ; and by its 
 exerting pressures in all directions, and therefore upwards as 
 well as downwards : thus, if a glass vessel be filled with water, 
 and a piece of paper be laid upon the surface of the water, pro- 
 jecting over the edges of the vessel ; the vessel may be carefully 
 inverted without spilling the water ; and the liquid will after- 
 wards remain in the vessel, in which it is supported by the pres- 
 sure of the air. 
 
 The experiment (2) may be extended to show another es- 
 sential property of air, namely, its elasticity ; for, as the ves- 
 sel descends in the mass of water, the water rises, although, as 
 has been stated, it never wholly fills it. The space occupied by 
 the air, is therefore lessened by the pressure of the liquid ; if the 
 vessel be permitted to rise, the water descends, until the mouth 
 of the vessel reach the surface, when the air again occupies the 
 same space it did at first. It is, therefore, capable of having its 
 bulk diminished by pressure, and of restoring itself to its origi- 
 nal volume when the pressure is removed, or is elastic. 
 
 352. This elasticity is considered as permanent, for, no mere 
 change of temperature, or in its relations to latent heat, will con- 
 vert atmospheric air into a liquid or solid form. It is indeed said 
 that intense pressure will reduce it to a liquid state, and we know 
 that its chemical constituents do, when in combination, become 
 
343 CHARACTERS OJ [JBoO/C V. 
 
 both liquid and solid. Still the difference in this respect, is so mark- 
 ed between the air of our atmosphere and another class of elastic 
 fluids, that we are not warranted, in treating of its mechanics, to 
 refuse to receive the permanence of its elasticity, as one of its 
 principal characteristic properties. 
 
 The researches of chemistry have shown us that the air of our 
 atmosphere is not homogeneous in its chemical constitution, but 
 that it is a mixture of several fluids of the same mechanical na- 
 ture, differing in their chemical properties. So, also, have a num- 
 ber of other fluids permanently elastic in character, that form no 
 perceptible portion of the atmosphere, been discovered ; to the 
 whole of these we give the common epithet of Gas. 
 
 A gas, then, is a material substance, belonging to the class of 
 fluids, gravitating or possessed of weight, and permanently elas- 
 tic. 
 
 353. When water is heated in a close vessel, an elastic fluid is 
 generated ; this finally acquires such expansive force as to break 
 the vessel : if boiled in a glass vessel, with an open neck, the 
 space, not occupied by water, will appear perfectly transparent 
 and colourless ; but a cloud will appear to issue from the neck. 
 This cloud, if examined, will be found to be composed of water 
 in a state of minute division, and we infer that the space within 
 the vessel is filled with an elastic fluid. To render this more evi- 
 dent, adapt a syringe to the neck of the vessel, containing a pis- 
 ton fitted air tight, and depressed to the bottom of the syringe ; 
 when the water begins to boil, the piston will begin to rise, and 
 will speedily reach the top of the syringe; if cold water be now 
 applied to the exterior of the syringe, the piston will be caused sud- 
 denly to descend ; we infer, therefore, that the elastic fluid, that 
 had been generated is now condensed. The visible cloud that is- 
 sues from a vessel, was formerly called vapour or steam ; we now 
 apply those terms to the invisible elastic fluid ; and we distin- 
 guish it from gases, as being condensible by physicial means, 
 which they are not. 
 
 Numerous experiments show, that water is not the only sub- 
 stance, that is, when heated, converted into an elastic fluid ; few 
 or no substances indeed exist, that do not become volatile at a 
 greater or less degree of temperature, and all these volatilized 
 matters are condensed when the heat is withdrawn. The name 
 of vapour is hence applied to them all. 
 
 Vapour, then, is a material substance, existing in the fluid form, 
 elastic, but not permanently so, being capable of condensation by 
 cold. 
 
 We therefore subdivide elastic fluids into two classes : Gases, 
 or permanently clastic fluids ; and Vapours, or condensible elas- 
 
Book 
 
 ELASTIC FLUIDS. 349 
 
 tic fluids. Physically speaking, the line that separates these two 
 classes, is not distinctly marked. There are some substances 
 classed as gases, that are condensible with no great difficulty; 
 thus ammonia becomes liquid at very low temperatures ; in- 
 deed, few or none of the gases are capable of retaining their elas- 
 tic form under intense pressure ; for even atmospheric air has, 
 as stated by Perkins, been reduced to the liquid form. On the 
 other hand, there are some vapours that, under physical or me- 
 chanical circumstances, differing in no great degree from those 
 we find existing at the surface of the earth, would appear perma- 
 nently elastic : thus, the vapour of ether is formed under ordi- 
 nary circumstances at 98, and would, in a climate of that tem- 
 perature, be gaseous. 
 
 The elastic force with which gases or vapours tend to expand 
 themselves, is called their Tension. 
 
 354. The tension of elastic fluids is increased by heat, and di- 
 minished by cold ; thus, under a given pressure, a given weight 
 of atmospheric air, and in general, of any elastic fluid, is found, 
 when heated, to occupy a greater space. This may be simply 
 shown^by enclosing a portion of air in a bladder, and exposing it 
 to heat; the bladder will be distended and finally burst. If the 
 bladder be completely filled, and then cooled, it will cease to be 
 distended, and the distention may be restored by raising it again 
 to its original temperature. 
 
 355. The mass that composes the atmosphere of our earth 
 being a fluid, the general properties of that class of bodies would 
 lead us to infer, that it will tend, from the freedom of motion 
 among its particles, to distribute itself uniformly over the whole 
 surface of the earth. This is confirmed by experience, for we 
 find the atmosphere exerting a pressure, that taken at a mean, is 
 constant at every point upon the mean surface of the earth. 
 
 This property is not confined to the mass, but is inherent in 
 each of its several chemical constituents. We learri this from 
 the researches of Dalton, who has shown conclusively that every 
 different elastic fluid tends to distribute itself uniformly over the 
 surface of the earth, and thus to form a separate atmosphere of 
 itself. This tendency is exerted precisely as if no other elastic 
 fluid were present ; but the rapidity with which the distribution 
 takes place, is affected by the fluid resistance of the other elastic 
 fluids : and we shall see, hereafter, that in the present physical 
 state of our planet, the several elastic fluids that make up our 
 atmosphere, are constantly tending to a state of equilibrium, that 
 they never completely attain This discovery of Dalton, we 
 shall find of great value ; for the present, we shall merely state, 
 that it has fully explained the fact, observed by chemists, but not 
 
350 PRESSURE OF [Book V. 
 
 to be accounted for upon the principles of that science alone, of 
 the constant proportion of the two gases that compose the prin- 
 cipal part of atmospheric air. 
 
 356. The fluid pressure of the atmosphere was first exhibited 
 by Torricelli. It having been found that the height to which 
 water rises in a common pump, was limited ; and it being infer- 
 red by him that the cause was mechanical, he saw that this, 
 whatever were its nature, must on the principle of 326, raise 
 a mass of denser fluid to a less height. He, therefore, inferred 
 that the experiment of the pump might be performed with an ap- 
 paratus of less^size, provided he used mercury instead of water. 
 Taking, therefore, a glass tube of about three feet in length, he 
 adapted a piston to it, and plunged the end in a basin of mer- 
 cury. On drawing up the piston, the mercury followed until it 
 reached the height of SO inches, at which the fluid ceased to 
 rise ; on drawing up the piston farther, a space was left between 
 it and the surface of the mercury. Now as the limit to which 
 water rises, in a well-constructed common pump, is 34 feet; and 
 as this height is to 30 inches in the inverse ratio of their respect- 
 ive densities, (See Table of Specific Gravities, 346), the pres- 
 sures of these two columns of the liquids are equal, and require 
 an equal force for their support, if the force be not identical. 
 Torricelli at once, and as was considered at the time, with great 
 boldness, inferred that the forces which supported the water in 
 the pump, and the mercury in the tube, were identical, and that 
 it was to be sought in the fluid pressure of the atmosphere. As 
 the weight of the atmosphere was then unknown, and the fact of 
 its compressibility not understood, the boldness of his inference 
 staggered even some of the most enlightened of his cotempora- 
 ries. 
 
 Toricelli also planned a more convenient mode of performing 
 his experiment. Taking a tube of the same length as before, he 
 closed it at one end, and filled it with mercury ; placing the 
 finger upon the open end of the tube, he inverted it in a basin 
 of the same liquid; on removing the finger, the mercury sub- 
 sided from the sealed end of the tube, until it reached the height 
 of about 30 inches above the level of the mercury in the basin, 
 at which it remained stationary. 
 
 That the cause which supports a column of water, is identical 
 with that which sustains the column of mercury, in these expe- 
 riments, may be shown, by pouring water upon the surface of 
 the mercury in the basin. If the tube be now raised until its 
 open end be above the level of the mercury, but remain still im- 
 mersed in the water, the mercury in the tube will, from its su- 
 perior weight, fall through the water beneath it; the water will 
 
Book J 7 ".] THE ATMOSPHERE. 351 
 
 enter the tube to supply its place, but so far from ceasing to rise 
 at a height of 30 inches, rushes to the top of the tube, and would 
 fill it, even were its length as great as 34 feet. 
 
 Pascal, who did not at first admit the conclusions of Torricelli, 
 planned an experiment v/hich furnished complete evidence of 
 the accuracy of the views of that philosopher. It has been shown, 
 316, that that the pressure of a fluid, upon a given base, is a 
 function of the depth. If then the cause of the rise of the mer- 
 cury be the pressure of the atmosphere, it is obvious that on car- 
 rying the Torricellian apparatus to a greater height above the 
 level of the sea, the pressure must diminish, and the column of 
 mercury that it will support will be lessened in height. This ex- 
 periment being performed at the base, and on the summit of the 
 mountain Puy dt Dome, in Auvergne in France, gave this anti- 
 cipated result; a diminution in the height of the column of mer- 
 cury being found amounting to about 3 inches, for a change of 
 level of 3<500 feet. 
 
 357. The apparatus of Torricelli, then, demonstrates conclu- 
 sively the existence of atmospheric pressure. It also enables us 
 to measure its amount. For the pressure of the column of mer- 
 cury, with which that of the atmosphere must be in equilibrio, is, 
 331, equal to the weight of a prism of the fluid whose height is 
 30 inches. Upon a square inch, the bulk of this prism is 30 
 cubic inches ; and this quantity of mercury has a weight that dif- 
 fers but little from 15 Ibs. The air of the atmosphere, therefore, 
 presses, at the level of the sea, with a force equivalent to 15 Ibs. 
 upon each square inch of surface. This quantity of 15 Ibs. per 
 square inch, constitutes that unit in which the pressures and ten- 
 sions of elastic fluids are measured, and which is called an Atmos- 
 phere. 
 
 Were atmospheric air a fluid of uniform density throughout, 
 its height above the mean surface of the earth might be calculated 
 from the Torricellian experiment ; for if we call the density of 
 water 1000, that of mercury is 13600 ; and that of air at a mean, 
 at the level of the sea is usually estimated at If ; the mean height 
 of the column of the mercury in the barometer, at the same 
 level, is thirty inches, or 2i feet. Calculated from these data, the 
 height of an atmosphere of uniform density is 27600 feet. 
 
 The Torricellian apparatus, with additions and under modifi- 
 cations that will be hereafter described, goes at present by the 
 name of the Barometer. 
 
 358. The common pump, to an observation on the action of 
 which the Torricellian experiment was due, is an apparatus whose 
 
359 
 
 PRESSUHE OF 
 
 [Book V. 
 
 A' 
 
 form and mode of action may be understood from the annexed 
 figure. AA', is a pipe of any convenient length, 
 less than the limit to which water may be raised by 
 atmospheric pressure ; BB', a barrel generally of 
 greater diameter than the pipe ; c, a valve open- 
 ing upwards, at the junction of the pipe and barrel ; 
 d, a piston, moveable by means of the rod e ; in 
 this piston there is also a valve opening up- 
 wards. When the piston is raised, the air in the 
 barrel between the two valves is expanded, and 
 its tension diminished ; the air in the barrel, there- 
 fore, opens the valve c, and the whole mass of air 
 tends to become less dense ; but as this pipe com- 
 municates with water in a basin, that is pressed at 
 its surface by the air of the atmosphere, this pres- 
 sure causes the water to rise in the tube, until the 
 tension of this confined air becomes equal to the 
 pressure of the atmosphere. On depressing the 
 piston d y the valve in it opens, and air passes upwards from the 
 barrel as the former descends ; but the valve c[is closed by the 
 downward pressure, and the volume of water that has entered the 
 pipe, remains. On again raising the piston, the same action takes 
 place as at first, and an additional quantity of water enters the 
 pipe : a second depression of the piston causes the same effects as 
 the first. Thus a column of water will be raised by the alter- 
 nating motion of the piston, until that fluid reaches the piston in 
 its lower position. On raising the piston when the water has 
 reached it, that fluid will be compelled to follow it by the pres- 
 sure of the atmosphere ; when the piston is again depressed, the 
 water flows through the valve situated in it ; and on its being 
 again raised, the water will be lifted upon its surface until it 
 reaches and flows out of the spout F. 
 
 Although in theory the limit of the height to which water may 
 be raised by the common pump, measured from the surface in 
 the basin beneath, to the highest position of the moveable piston, 
 is 34 feet ; it is not found practicable, with pumps of ordinary 
 structure, to raise that liquid more than about 28 feet. This dif- 
 ference arises from the difficulty of making the apparatus abso- 
 lutely air-tight. 
 
 Liquids are supported in a syphon, upon the same principle 
 that they are raised in the common pump. If a bent tube be filled 
 with a liquid, and inverted, without permitting the liquid to 
 
 escape. 
 
 and one of its branches be immersed in a vessel contain- 
 
 ing a mass of the same liquid, so much of the liquid in the tube as 
 is above the level plane, of which the surface of that in the vessel 
 
Book VJ\ ELASTIC FLUIDS. 353 
 
 forms a portion, will be supported by the pressure of the atmos- 
 phere ; and the columns, in the two branches of the tube, lying 
 above this level, exactly balance each other ; but the remainder 
 of the column, in the branch that is not immersed, will cause the 
 column of which it is a part, to preponderate and descend ; the 
 pressure of the atmosphere will cause an equal quantity of liquid 
 to rise in the opposite branch to supply its place, and thus a con- 
 tinual stream will flow from the open end of the tube, until the 
 level of the liquid in the vessel descends as low as the orifice on 
 the branch of the tube that is not immersed. 
 
 The apparatus called the Syphon, is a bent tube, and has usu- 
 ally branches of unequal length, the shorter of which is immersed 
 in the vessel. Hence the flow of liquid will continue until its 
 level descends as low as the opening in the immersed branch ; 
 the air will then enter the syphon, and the whole of the liquid 
 will be discharged from it. 
 
 Instead of filling the syphon, and inverting it, the air may be 
 extracted by a pump, or by the action of the lungs ; the mouth 
 is, in the latter case, applied to a pipe adapted for the purpose, 
 and the end of the syphon that is not immersed, closed by the 
 finger, or by a stopcock. The apparatus, in this form, is called 
 a Crane, and is used for the purpose of decanting liquids. The 
 flow of liquid from a syphon, will have a retarded velocity, pro- 
 vided the vessel be permitted to empty itself. But if it be kept 
 full, by any appropriate means, the velocity remains constant. 
 
 45 
 
354 OF THE AIK PUMP. [Book V. 
 
 CHAPTER VI. 
 
 OP THE AIR PUMP. 
 
 359. The mechanical properties of the air may be best inves- 
 tigated experimentally, by means of the apparatus called the Air 
 Pump. 
 
 If an instrument similar in construction to a common pump, but 
 of more accurate workmanship, be adapted to a tight vessel, the 
 action, which in the common pump diminishes the tension of the 
 air in such a manner as to allow water to be forced up by the pro- 
 sure of the atmosphere, will now act merely to rarefy the air con- 
 tained in the vessel. When the piston is raised, the air in the 
 vessel expands in consequence of its elasticity, opens the lower 
 valve, and fills the whole space beneath the piston with air of an 
 uniform tension, but of diminished density. When the piston is 
 depressed, the air contained between it, and the lower valve, is 
 compressed until it reach the same density as the external atmos- 
 phere ; it then opens the upper valve and passes out. On raising 
 the piston again, a farther rarefaction takes place ; and thus, at 
 each alternate motion of the piston a new expansion takes place, 
 and a portion of the air originally contained in the vessel, passes 
 out. 
 
 We call the space within the receiver, from which the air has 
 been extracted, the Vacuum of the Air Pump. This is less per- 
 fect than that at the top of the barometer, which is called the Tor- 
 ricellian Vacuum. 
 
 The first air pump was invented by Otto Guericke, of the city 
 of Magdeburg, and had the form we have used for our illustration. 
 The joint between the pump and the close vessel, was rendered 
 air tight by a collar of leather, moistened with water. We now 
 use in the several joints of the pump, leather dipped in a liquid 
 oleaginous matter. 
 
 The air pump has received many modifications in form, and 
 improvements in structure, since the time of its invention, of 
 which the following are among the most important. 
 
 A close vessel is ill suited for the purpose of introducing sub- 
 stances, or apparatus for experiment, into the vacuum of the air 
 pump ; but if an open receiver be taken, and placed upon a plate, 
 with an orifice in the middle of which the pump communicates 
 by a pipe ; when the pump is set in action, it will tend to rarefy 
 the air in the receiver, and the pressure of the atmosphere will 
 fix the latter to the plate of the pump. The receiver will then. 
 
Book F.] OP THE AIR PUMP. 355 
 
 if the joints between it and the plate be airtight, be to all intents 
 and puposes a close vessel. To render this joint air tight, the 
 plate was at first covered with a piece of oiled leather. It was 
 however found, that the evaporation of the oil from the leather, 
 lessened the extent of rarefaction, by continually supplying a fresh 
 elastic fluid. For this reason, then, the plate of the pump (usu- 
 ally made of brass) has been ground; the rim of the receiver is 
 also ground, in order to adapt itself to the plate. In this way the 
 joint will frequently be air tight, without any other precaution, 
 but more generally it is necessary to touch the rim of the receiver 
 with a little oil. 
 
 As oil is acted upon by brass, and corrodes it, glass plates have 
 been substituted for brass, and are ground in the same manner. 
 
 The working of the piston of the pump, is opposed by the pres- 
 sure of the atmosphere ; hence another pump barrel has been add- 
 ed, and the two pistons made to act alternately, the one rising 
 as the other falls. By this method, the pressures on the pistons 
 are made to oppose each other. To work the pistons with greater 
 facility, their rods are cut into teeth, forming racks : between 
 these, and interlocking with both, is placed a pinion, or toothed 
 wheel. The latter is turned by a winch, and sometimes by a 
 double handle. 
 
 The resistance of the atmosphere may be in a great degree re- 
 moved, by making the cylinder of the pump air tight ; the piston 
 rod must, in this case, work through a collar of leathers ; a late- 
 ral spout is therefore placed to permit the escape of the air, and this 
 contains a third valve, opening upwards. A pump of this struc- 
 ture works with greater ease, as the exhaustion of the receiver 
 increases, while in those with open barrels, the resistance aug- 
 ments with the exhaustion. 
 
 In the earlier pumps, both the valves were opened by the elas- 
 ticity of the air ; and thus a limit to the exhaustion was attained, 
 in the resistance of the valve to an effort to open it. In order to 
 increase the power of the pump in exhaustion, the lower valve is 
 now made to open by the rise of the piston, and is shut by its de- 
 scent. 
 
 Such an arrangement is represented on the succeeding page, 
 being a section of the air pump of Demoutier. 
 
 P P' is the piston in which is seen the upper valve s ; the lower 
 valve is represented at c; it has a conical form, and is attached to 
 
356 
 
 01 THE AIR PUMP. 
 
 [Book V. 
 
 a rod ct y that passes through the piston, and its leather packing, 
 with so much friction as to be air tight. As soon as the piston 
 begins to rise, the rod ct will, in consequence of the friction, 
 move with the piston, and the valve c opens ; but the rod moves 
 through no more space than is just necessary to open the valve, 
 being checked by a shoulder at r, that strikes against a plate that 
 forms a lid to the barrel of the pump. 
 
 The valves of the pumps that first succeeded that of Guericke, 
 were made of strips of bladder, or of oiled silk : in the pump just 
 represented, the upper valve is of steel, resting on an oiled leather 
 seat; the lower valve is a leather conic frustum, applying itself to 
 a hollow frustum formed in a steel plate. In the best English 
 pumps, the valves and seats are both of metal, and the form that 
 of a conic frustum. 
 
 We have stated that oil and brass mutually act upon each other; 
 hence, the barrels and pistons that were originally made of brass, 
 and rendered air tight, by packing of leather soaked in oil, were 
 rapidly worn. To remedy this defect, the pistons are now made 
 of steel, which is effectually preserved from rust by the oil used 
 in the packing, and the barrels of the best pumps are made of 
 glass. 
 
 360. The power of air pumps, to exhaust the receivers, placed 
 upon their plates, is measured by means of instruments called 
 Gauges. These are of several kinds. 
 
 (1). The Barometer Gauge. This consists of a glass tube about 
 30 inches in length, and open at both ends ; it is placed in a ver- 
 tical position attached to the stand of the pump ; the lower end 
 is plunged in a basin of mercury ; the upper is cemented air tight 
 
Book K] OP THE AIR PUMP. 357 
 
 to a pipe that leads to the opening in the plate of the pump, by 
 which the air passes from the receiver ; or it may communicate 
 in some other manner with the valves of the pump. As the air 
 in the receiver is exhausted, the pressure of the external atmos- 
 phere, forces the mercury in the basin to rise in the glass tube. 
 The difference between the height to which the mercury is thus 
 raised, and that at which it is supported at the same time in the 
 sealed Torricellian tube, or Barometer, is obviously a measure of 
 tension of the air yet remaining in the receiver. We shall here- 
 after see that this tension is exactly proportioned to the quantity 
 of air that remains. 
 
 (2). A tube 6 or S inches in length, sealed at one end, and 
 filled with mercury, is inverted in a basin of the same liquid. 
 The mercury is of course supported by the pressure of the atmos- 
 phere, and continues to fill the tube. This apparatus is placed 
 beneath a receiver, communicating with the valves of the pump. 
 During the first stages of the exhaustion, the mercury still remains 
 supported, but so soon as the tension oT the contained air becomes 
 less than is sufficient to support such a z J.um-; of mercury, that 
 liquid begins to fall in the tube ; the height at which it stands 
 above the level of that in the basin, is a rne'.'-ure of the tension 
 of the remaining air. This is called the Short A-aromete.* Gauge. 
 
 (3). The Syphon Gauge acts upon the same >nncipl3 as the 
 last; but instead of plunging the open end of : .3 tube in a basin 
 of mercury, it is bent upwards ; the whole is their fare composed 
 of two parallel branches, one open, and the other closed. The 
 latter is filled, before the action of the pump begir s, with mer- 
 cury. 
 
 These two gauges are placed beneath separate receivers, ex- 
 pressly adapted for them to an additional part of the pump, in 
 order that they may be used at the same time that other receivers 
 are placed upon the principal plate. 
 
 (4). The Pear-Gauge a vessel of glass of the shape denoted 
 by its name, having a small opening at the larger end, is sunk 
 by means of weights into a basin of mercury, with its opening 
 downwards. This apparatus being placed beneath a receiver, and 
 the pump set in action, the air contained in the pear-shaped vessel 
 will expand, and make its way through the mercury : thus, the 
 exhaustion withki it will correspond with that effected in the 
 receiver. On re-admitting the air to the receiver, the pressure 
 upon the mercury in the basin will force it into the pear-shaped 
 vessel, until the portion of air left within it be restored to about 
 its original density. Hence, th- relation between the part of the 
 vessel not filled by the mercury, and its whole capacity, will be a 
 measure of the exhaustion of the pump. 
 
358 OP THE AIR PUMP. [Book V. 
 
 When this gauge is used with a pump, to whose plate the re- 
 ceiver is adapted by means of a sheet of leather soaked in oil, it 
 will exhibit a far higher degree of exhaustion than any of the three 
 preceding gauges. . This grows out of the fact that the elastic va- 
 pour of the oil, which affects them, cannot enter it. This obser- 
 vation led to the improvement we have mentioned, of fitting the 
 receivers to the plate of the pump by grinding. 
 
 361. The air pump affords a great variety of proofs of the pres- 
 sure of the atmosphere, of which a few may be cited : 
 
 (1). In the manner in which the receiver is fixed to the plate 
 of the pump, in such a way as to act as a close vessel ; in its re- 
 maining firmly attached to the plate after the air is exhausted. 
 
 (2). The hand placed upon an open receiver, will in like man- 
 ner be firmly fastened to it ; and the pressure which was before 
 equal on both sides of the hand, and therefore imperceptible, will 
 become painful. 
 
 (3). A square flask of glass, screwed to the opening in the 
 plate of the pump, will be broken, as will a glass plate, ground 
 to fit an open receiver. 
 
 (4). A small receiver, suspended by means of a rod passing 
 through a collar of leathers, within a larger receiver, may be de- 
 pressed by means of the rod, till it rests on the plate of the pump. 
 If this small receiver do not cover the orifice by which the plate 
 communicates with the valves, the re-admission of air into the 
 larger receiver, fixes the smaller to the plate of the pump; nor can 
 it be removed until the air be again exhausted. 
 
 (5). Two hemispheres of brass ground to fit each other, may 
 be affixed to the plate of the pump by means of a pipe proceeding 
 from one of them, and which is screwed to the orifice. If the 
 air be exhausted, they will be found to be pressed powerfully 
 together. If there be a stop-cock on the pipe of communication, 
 it may be closed, and the apparatus detached. The spheres held 
 together by the pressure of the atmosphere may be then separated 
 by means of weights ; one of them being suspended, and the 
 weights attached to the other. The quantity of weight required to 
 separate them, will be a measure of the pressure they sustain : as 
 the surface of a sphere of known diameter can be calculated, this 
 was used as a measure of the atmosphere's pressure upon each unit 
 of square measure. It is, however, affected by the air which 
 remains in the apparatus, and which the pump cannot wholly ex- 
 haust : it is, therefore, inferior in accuracy to the tube of Torricelli. 
 
 Experiments for illustrating the pressure of the atmosphere by 
 means of the air pump, may be multiplied to a great extent, but 
 the above are sufficient for our purpose. 
 
Book P.] OF THE AIR PUMP. 359 
 
 362. That the pressure of the atmosphere is the cause of the 
 rise of fluids in pumps, and of mercury in the Torricellian ex- 
 periment, may also be shown by means of the air-pump. Thus, 
 if a model of a pump be passed through a collar of leathers, adapt- 
 ed to the top of a receiver, it will be found to be without effect, 
 when the reservoir in which it is plunged is not pressed by air. 
 So if a barometer be in like manner inserted into the top of a re- 
 ceiver, the mercury will fall in the tube as the air is exhausted 
 from the receiver. 
 
 363. To show the weight of air, a flask furnished with a stop- 
 cock is weighed ; it is then screwed to the plate of the pump, 
 and the air exhausted ; the stopcock being closed, it may be re- 
 moved without admitting air, and being again weighed, it will be 
 found lighter than before. To obtain the exact weight of a given 
 bulk of air, demands certain precautions which will be hereafter 
 stated. 
 
 364. The elasticity of the air is manifest from the manner in 
 which the air-pump itself works ; it may also be exhibited in va- 
 rious other manners, thus : 
 
 (1.) If a bladder, partially inflated with air, be placed under 
 a receiver, and the air withdrawn from the latter, the bladder 
 swells, and is distended; on re-admitting the air to the receiver, 
 the bladder collapses to its original dimensions. 
 
 (2.) If the bladder, when partially filled, be loaded with 
 weights, and placed beneath a receiver, the contained air will, in 
 expanding itself as the receiver is exhausted, lift the weights. 
 
 (3.) If a glass mattrass be inverted in a vessel containing a 
 liquid, and placed beneath the receiver of an air-pump ; as the 
 air is exhausted from the receiver, that confined in the mattrass 
 by the liquid will expand, and escape in visible bubbles through 
 the fluid mass. On re-admitting the air, the liquid will be forced 
 up the neck of the mattrass, and into the bulb, which it will, if 
 the exhaustion have been sufficient, nearly fill. On a second ex- 
 haustion, the air will expand, as will be manifested by the de- 
 scent of the liquid confined in the mattrass; and the latter will be 
 forced back a second time on the re-admission of air. 
 
360 EQUILIBRIUM OF [Book V. 
 
 CHAPTER VII. 
 
 EQUILIBRIUM OF PERMANENTLY ELASTIC FLUIDS. 
 
 365. Although the elasticity of the air is demonstrated by the 
 experiments cited at the close of the last chapter, they do not 
 exhibit the law of that elasticity ; nor do they point out what re- 
 lation the force, with which it expands itself, bears to the force 
 by which it is compressed. This is a question whose importance 
 demands a particular investigation, which may be performed by 
 the following experiments. 
 
 (1.) Take a glass vessel with a narrow neck, (one nearly of a 
 spherical figure is best adapted for the purpose), and pour into it 
 a small portion of mercury ; if we then screw, air-tight, to its 
 neck a slender tube of glass open at both ends, of the length of 
 at least thirty inches, whose lower end is wholly immersed in the 
 mercury, the air that fills the upper part of the vessel will be 
 wholly separated from the general mass of the atmosphere. If 
 this apparatus be placed beneath a tall receiver, upon the plate of 
 the air pump, and the air exhausted ; the pressure on the air, 
 confined in the vessel, will be lessened in proportion as the ten- 
 sion of the air in the receiver is diminished ; the elastic force of 
 the confined air will, in consequence, cause a column of mercury 
 to rise in the tube, and the height of this column will be a mea- 
 sure of the difference between the tension of the air in the re- 
 ceiver, and that of the air of its original density confined in the 
 vessel. If the pump have a long barometer gauge (No. 1 of 
 357), the mercury that rises in it furnishes a similar measure of 
 the difference between the pressure of the external atmosphere 
 and the tension of the air remaining in the receiver. 
 
 The relation between the two columns of mercury furnishes a 
 direct comparison between the pressure of the atmosphere, and the 
 elasticity of the air that composes the part on which that pressure 
 takes place; for the confined air has obviously been enclosed un- 
 der that very pressure. On comparing the heights of these two 
 columns, during all the periods of the exhaustion, they will be 
 found to all appearance exactly equal, the mercury rising in the 
 two iubespaHpa*9U. The small change in the bulk of the con- 
 fined air, growing out of the mercury it forces out of the vessel, 
 is insensible. Hence, it is obvious that the elasticity of air, 
 at the ordinary density of the atmosphere, is exactly equal to the 
 pressure. 
 
Book VJ\ PERMANENTLY ELASTIC FLUIDS. 361 
 
 (2.) To examine the effect of increased pressure, we take a cy- 
 lindrical glass tube bent into two branches ; one of these is closed, 
 and the other open. Mercury is then poured in until it fill the 
 bend of the tube ; a portion of the air that would be thus confined 
 in the closed branch is permitted to escape, by inclining the ap- 
 paratus, until the mercury stands at the same level in bolh branch- 
 es of the tube. The confined air has then no other pressure to 
 sustain than that of the atmosphere, for the columns of mercury 
 in the two branches counterbalance each other. The air, there- 
 fore, that is confined, has, as may be inferred from the preceding 
 experiment, the same density as that of the adjacent stratum of 
 the atmosphere. In order to increase the pressure, mercury is 
 poured into the open branch of the tube ; the confined air is then 
 affected by a pressure equal to the sum of that of the atmosphere, 
 and that of the column of mercury measured from the level at 
 which it stands in the close branch of the tube. 
 
 In this experiment, the level of the mercury in the closed 
 branch of the tube will be found to rise as the pressure caused by 
 the column in the open branch increases, thus marking a conden- 
 sation in the enclosed air. By the uniform result of all experi- 
 ments, after employing the precautions that will be presently 
 stated, it is found that the space occupied by the confined air, is 
 inversely as the pressure to which it is subjected. 
 
 In order to express this fact, let B be the original volume of 
 the air, under the pressure, /?, namely that of the atmosphere ; B' 
 a volume under the increased pressure, p' ; 
 
 whence we obtain 
 
 B'=^; (376) 
 
 and for the volume B", under any other pressure p", 
 
 Ep 
 
 "O/> __ * . 
 
 'p'" 
 
 dividing bv B' we obtain 
 
 whence 
 
 B"=-^l. (376) 
 
 P 
 
 By which we may calculate the relation of the volumes of the 
 same mass of air under any given pressures whatsoever. 
 The precautions to which we have referred, grow out of the 
 
 following circumstances : 
 
 46 
 
362 EQUILIBRIUM OF [Book V. 
 
 (a) Air in being com pressed has its temperature raised ; hence 
 the apparatus must be permitted to cool down to the temperature 
 of the surrounding air before the observation is made. 
 
 (b) All air contains a greater or less proportion of aqueous va- 
 pour, and this, as we shall presently see, is not affected by pres- 
 sure in exactly the same manner as air, under similar circum- 
 stances. In order to make the experiment perfectly satisfactory, 
 the tube must be well dried, and the moisture withdrawn from 
 the air it contains, by exposing the latter to the contact of hygro- 
 metric substances, for some hours before the experiment is made. 
 
 The space being found, after these precautions have been taken, 
 to be inversely proportioned to the pressure, it may be inferred 
 that in the case of the condensation of air, such as is usually found 
 at the surface of the earth, the density is always proportioned to 
 the pressure ; and to this, the tension, or elastic force, is equal. 
 
 (3.) It remains that the law of elasticity should be determined 
 for air of diminished density. For this purpose, take the Torri- 
 cellian apparatus and fill the tube partly with mercury ; the re- 
 maining part will continue to contain air. On placing the fin- 
 ger upon the open end, and inverting the tube, the air will 
 rise through it to the close end, and will, so long as the finger is 
 tightly pressed on the opening, occupy the same space it did at 
 first. But on immersing the open end of the tube in a basin of 
 mercury, the confined air is no longer compressed by the whole 
 force of the atmosphere, for the latter must also support the co- 
 lumn of mercury. The confined air then, being less compressed, 
 expands itself, and causes the mercury to descend ; and it finally 
 comes to rest in such a position that the sum of the pressure of 
 the column of mercury, and of the tension of the confined air, is 
 equal to the pressure of the atmosphere. 
 
 To ascertain the law from this experiment, let /be the tension 
 of the air after its expansion ; />, the pressure of the atmosphere ; 
 6, the original volume of the air, and x, that to which it expands 
 itself; it is found that 
 
 whence we obtain 
 
 l=- r (377) 
 
 which is identical with the formula (375) ; hence the law is the 
 same, in the case of rarefied as in that of condensed air. 
 
 The same experiments may be performed with dry gases, and 
 the results are found the same as when they are performed with 
 atmospheric air. Hence, the air of the atmosphere and all other 
 
Book 
 
 PERMANENTLY CLASTIC FLUIDS. 363 
 
 dry gases, at constant temperatures, occupy spaces that are in- 
 versely as the pressures to which they are subjected. This law 
 was originally discovered by Mariotte, and goes by his name. It 
 is true at all mean and usual pressures, but ceases to be so at cer- 
 tain limits. Were it absolutely true, the smallest possible quan- 
 tity of air would, on the pressure being wholly removed, occupy 
 a space infinitely great ; while there could be no space so small 
 into which the largest mass of the air could not be compressed by a 
 sufficient force. The limit is found, on the one hand, in the fact 
 that nearly all the gases have been condensed into the liquid 
 form ; and even atmospheric air, as is stated by Perkins, has been 
 reduced to that state. The limit, on the other hand, appears ra- 
 ther to arise from the relation of expanding air to heat ; for in 
 the higher regions of the atmosphere, the expansion produces a 
 cold so intense, that the diminution of temperature will finally 
 produce an effect equal and contrary to that caused by the re- 
 moval of pressure. Laplace, in stating this limit, has expressed 
 the opinion : that although the existence of the limit, and conse- 
 quently the finite extent of the atmosphere, is capable of demon- 
 stration ; the exact height at which it ceases to expand is not 
 within the reach of calculation. We are therefore compelled to 
 have recourse to a physical fact to estimate the probable extent 
 of the atmosphere of the earth ; this is the phenomenon of twi- 
 light, whence it is inferred, that the air of the atmosphere still re- 
 tains a sufficient density to reflect the rays of light, at a distance 
 of forty miles from the surface of the earth. 
 
 Were atmospheric air capable of indefinite expansion, the mass 
 that surrounds the earth would have distributed itself around the 
 earth and moon, in the ratio of their respective masses, by the 
 influence of the attraction of gravitation ; and generally, had any 
 one planet, at the time of the creation, been surrounded by an at- 
 mosphere of such a character, all would have derived "from it 
 atmospheres distributed inasimilar ratio. Now the moon has no 
 perceptible atmosphere, and we have no reason to conclude that 
 the other bodies of the solar system have atmospheres, of the 
 density and mass that might have been inferred from the law of 
 Mariotte : hence, again, we find a corroboration of the opinion, 
 that the atmosphere of the earth has a finite extent. The same 
 was inferred by Wollaston, from astronomic observations, into 
 the detail of which it is not our province to enter. 
 
 366. We may now proceed to investigate the conditions of 
 equilibrium in the air that composes our atmosphere, under the 
 supposition that it is of uniform temperature throughout. 
 
EQUILIBRIUM OF [Book V. 
 
 If we use the notation of 316, the density s being, by the law 
 of Mariotte, directly proportioned to the pressure, p, 
 
 s=mp ; (378) 
 
 in which equation, m is a constant co-efficient to be determined 
 by experiment. 
 
 The equation (352), 
 
 dp=3 dz, 
 will give 
 
 mz=log. p ; 
 and if c be the modulus of the tables, 
 
 ,="". 
 
 If, as is most convenient, we conceive the origin of the co-or- 
 dinate 2, to be at the mean surface of the earth, or at the level 
 of the sea, l 
 
 dp=sdz\ (379) 
 
 and substituting the value of 5, 
 
 dp= mp dz. (380) 
 
 If P be the pressure of the atmosphere at the level of the sea, 
 when 2=0, p=P ; and integrating the foregoing equation, we 
 obtain 
 
 roz=log. P log. p, (381 ) 
 
 and 
 
 p=Pc~ --. (382) 
 
 Hence : 
 
 The altitude of any point in the atmosphere, above the level 
 of the sea, is proportional to the difference between the logarithms 
 of the respective pressures-, and the difference of level between 
 any two points in the atmosphere, is in a similar manner propor- 
 tioned to the difference of the logarithms of the pressures at these 
 two points. The column of mei-ury in the Torricellian tube is 
 the measure of the pressure at the po iot where it is placed, and 
 therefore the difference in the logarithms of the height of the 
 two columns, at the two points, is in like manner proportioned 
 to their difference of level. 
 
 In the equation (380), if p=0, z becomes infinite ; hence we 
 reach the conclusion already stated, of the infinite extent of 
 the atmosphere, when the relations of air of different densities 
 to temperature is not taken into account. 
 
 367. If the temperature be not constant, as it is not in practice, 
 it becomes necessary to take into view the dilatation and contrac- 
 tion of the air by changes of temperature. We owe the original 
 determination of the law of the dilatation of air to Gay Lussac. 
 Similar inferences were obtained separately, at about the same 
 
Book V.] PERMANENTLY ELASTIC FLUIDS. 365 
 
 period, by Dalton. Still more accurate experiments have since 
 been performed by Petit and Dulong. It is not our province to 
 enter into the detail of this inquiry, but it is sufficient to express 
 the general law, which is : 
 
 All permanently elastic fluids expand an equal proportion of 
 their volume, estimated at some given temperature, for equal in- 
 crements of temperature, and this proportion is the same in them 
 all. Within the usual limits at which experiments or observa- 
 tions are made, the expansion of mercury in a glass tube, or its 
 rise in the thermometer, is also uniform : hence, every perma- 
 nently elastic fluid expands an equal and constant quantity for 
 every degree of thermometric heat, and this is usually estimated 
 in the form of a fraction of its bulk at the temperature of melting 
 ice. The experiments of Petit and Dulong make this fraction 
 0.002083 for every degree of Fahrenheit's thermometer. 
 
 368. The two laws of Mariotte and Gay Lussac combined, 
 will enable us to determine the density of atmospheric air, under 
 any pressure and at any temperature, its density at some given 
 pressure and temperature being first known. 
 
 If a be the constant co-efficient of dilatation, 0.00208 ; i the 
 number of degrees of the thermometer above or below 32' ; any 
 given unit of bulk of air, under the mean pressure of the atmos- 
 phere, and at the temperature of 32, will become at the tempera- 
 ture t, 
 
 and if P be the mean pressure of the atmosphere, and P' any 
 other pressure, this bulk will, by the law of Mariotte, become 
 (at)P 
 -p7 (383) 
 
 As the volumes are inversely as the densities, if D be the den- 
 sity under the pressure P, and temperature 32; and D' the density 
 at the pressure P', and temperature 
 
 whence 
 
 ( 384 > 
 
 And so in respect to any other gas, whose density at the mean 
 pressure P, and temperature 32, is D", we obtain for its density 
 D'", at the temperature <, and pressure P', 
 
366 EQUILIBRIUM OF [Book V. 
 
 and by comparison with the last formula, (384), 
 DP' 
 j)" )'" (386) 
 
 Thus the densities of gases preserve the same relation to each 
 other at every temperature and every pressure. 
 
 369. By reference to the investigations of Chapter III., and 
 comparing them with the characters of elastic fluids, it will be 
 seen that they possess, in some respects, properties in common 
 with liquids. Thus : 
 
 (1.) The pressure of an elastic fluid will be proportioned to 
 the surface on which it presses, and is measured by the weight 
 of a prism of the fluid, of uniform density throughout, whose base 
 is equal to the surface pressed, and whose altitude is such, that if 
 the fluid were of uniform density, the pressure on the unit of 
 surface would be the same as it is found to be under the law of 
 variable density. 
 
 (2.) A solid body, immersed in an elastic fluid, loses as much 
 weight as is equal to the weight of the fluid it displaces, or of a 
 volume of the fluid equal to its own. Hence the intense pres- 
 sure of 15 pounds on each square inch of the surface of our bo- 
 dies, and which on the body of a middle-sized man, amounts, if 
 the mere intensity of the pressures be estimated, without refer- 
 ence to their counteracting directions, to 11 tons, has for its re- 
 sultant, 336, only the weight of a mass of air equal to our bo- 
 dies in volume; and as the direction of this resultant is upwards, 
 so far from being crushed by this pressure, we are actually sup- 
 ported by it. If, however, the pressure be removed from any 
 part of our bodies, it becomes manifest on the rest, and presses 
 the part to which the exhaustion is applied against the apparatus 
 that operates the exhaustion. Such is the case of the experiment, 
 361, No. 2. 
 
 This buoyancy of the air will render the determination of the 
 specific gravity of bodies, by the methods of Chap. IV., inaccu- 
 rate ; for instead of their absolute weight, we obtain the weight 
 less that of an equal bulk of air. A correction may, however, he 
 applied to the approximate specific gravity, which rests on the 
 following principles. 
 
 Let to be the absolute weight, 
 
 \V the weight in air, 
 
 W the weight in water, 
 
 S the specific gravity as usually obtained, 
 
 ' the true specific gravity, 
 
 m the specific gravity of air. 
 
 The absolute weight will be greater than the weight in air by the 
 weight of a mass of air equal in bulk to the body. As it should 
 
Book V.] PERMANENTLY ELASTIC FLUIDS. 367 
 
 lose in water w W, it will lose in air m (w W), we have for 
 the value of the absolute weight, 
 
 '). 
 
 We may, without sensible error, substitute W for u>, in the 
 quantity w W', and the equation becomes 
 w>=W-t-m(W W'). 
 
 We may then apply the correction m (W W') to the weight 
 in air, before dividing by the loss of weight ; and the latter may 
 be taken in reference to this corrected weight instead of the 
 weight in air. 
 
 The value of m will be determined hereafter. 
 
 The exact value of the absolute specific gravity will be 
 
 5=S+Wl(l S). 
 
368 EQUILIBRIUM AND [Book V. 
 
 CHAPTER VIII. 
 
 OF THE EQUILIBRIUM AND TENSION OF VAPOURS. 
 
 370. Water, which boils under the mean pressure of the atmo- 
 sphere at 212, boils under diminished pressures at a lower tem- 
 perature, and may, by increased pressures, be prevented from 
 boiling until it attain a greater degree of sensible' heat. The 
 same is the case with other liquids; each has a determinate de- 
 gree of temperature at which it boils, under the mean pressure 
 of the atmosphere ; and the boiling point is raised or lowered, 
 with the increase or diminution of the pressure to which it is 
 subjected. 
 
 In the ebullition, an elastic fluid is generated, and it is only 
 when the tension of this elastic fluid becomes equal to the pres- 
 sure, that the action called boiling, takes place. These elastic 
 fluids may, by the abstraction of their latent heat, be restored to the 
 liquid form, and hence belong to the subdivision we have styled 
 vapours. 
 
 Although that rapid formation of vapour, which is attended 
 with ebullition, takes place under a given pressure, at a con- 
 stant temperature in each different liquid ; still the liquid is ca- 
 pable of rising in the form of vapour at lower temperatures. 
 Thus, although water, under ordinary circumstances, does not 
 boil below 212, yet if left exposed in an open vessel, a gradual 
 diminution of its bulk will take place at lower temperatures, and 
 it will finally be wholly dissipated. This has been conclusively 
 shown by Dalton, to be owing to the formation of vapour of a 
 temperature identical with that of the water whence it proceeds; 
 and of an elastic force, identical with that which would proceed 
 from water, boiling at that temperature, under diminished pres- 
 sure. Even ice is capable of furnishing vapour of its own tempe- 
 rature, and an appropriate tension. The same is the case with 
 other liquids; although all boil under given pressures, at some 
 temperature constant in each ; all give out vapour insensibly at 
 less elevated temperatures. 
 
 371. To measure the tension of aqueous vapour, a portion of 
 water may be introduced into the Torricellian apparatus, where it 
 will float upon the surface of the mercury. Vapour of the tem- 
 perature of the water, will immediately form in the vacuum. Its 
 pressure will be indicated by its causing the mercury to descend 
 from its original height ; the tension of the vapour will be mea- 
 
Book V.~\ TENSION OF VAPOURS. 369 
 
 sured by the space through which the mercury falls ; or by the 
 difference in the height of the mercury in the tube in which the 
 experiment is performed, and that at which it stands in the baro- 
 meter. 
 
 If the tube be of considerable length, and if, instead of a basin, 
 another tube of larger diameter than the former be used to hold 
 the mercury in which it is inverted, the pressure upon the vapour 
 contained in the Torricellian vacuum, may be varied. Thus : By 
 raising the inverted tube the pressure may be diminished ; in this 
 case it is found that so long as any water remains, new vapour 
 of the same tension will be generated, and the altitude of the mer- 
 cury in the tube will remain constant. If the tube be depressed, 
 the increase of pressure causes the condensation of a portion 
 of the contained vapour, and the mercury will almost instantly 
 rise to its original height above the surface of the mercury in 
 the outer tube ; or if the depression be performed slowly, the 
 mercury will move up the inner tube exactly in proportion as 
 the latter is pressed down, and will stand at a constant altitude. 
 
 If the quantity of water in the tube be so small that in raising 
 the tube the whole will be converted into vapour, it will be found 
 that after that time the tension of the confined vapour diminishes, 
 as does that of air, with the increase of the space it occupies. 
 
 It may hence be inferred that the law of Mariotte is inapplica- 
 ble to aqueous vapour, so long as it is in contact with the liquid 
 whence it is generated ; but this law becomes true at diminished 
 pressures, when no water is present. The same is found to be 
 true in the case of all the vapours, on which experiment has been 
 made. All vapours then, have, at a given temperature, a max- 
 imum of density and tension that cannot be exceeded ; and on 
 attaining which, they are condensed into the liquid form by any 
 attempt to compress them. 
 
 The essential characteristic of a vapour, therefore, is, that at a 
 given temperature, no more than a limited quantity by weight 
 can exist in a given space ; so that by diminishing the space, the 
 whole excess is reduced by pressure to the liquid form, without 
 any increase taking place in the tension. While, when the 
 space that gases occupy is diminished by pressure, their tension, 
 at a constant temperature, is increased exactly as the space they 
 occupy is lessened. 
 
 372. It may be easily ascertained in the course of these expe- 
 riments, that the tension of the vapour is increased by an increase 
 of temperature, and vice versa. We shall hereafter inquire into 
 the law that this increase follows. If then vapour be formed, or 
 admitted into a space unequally heated, it might at first sight ap- 
 pear that the vapour would have unequal tensions. This, how- 
 
 47 
 
370 EQUILIBRIUM AND [Book V. 
 
 ever, is not the fact, for in consequence of its elasticity and fluid 
 nature, it would tend to an equilibrium of tension ; the warmer 
 portions passing to the parts of the space that are less heated, 
 would have their temperature lowered, and tension diminished ; 
 and thus, in a space unequally heated, the maximum of tension, 
 when equilibrium is established, is no more than what is due to 
 the minimum temperature. 
 
 373. When a vapour, after being formed, is heated out of con- 
 tact with any of the liquid whence it is generated, it tends to ex- 
 pand, and its elastic force is increased exactly as if it were a gas. 
 
 Hence, at a constant density, its pressure becomes, 368, 
 
 F=P(l=FoO; (387) 
 
 and under a constant pressure, its density is, 368, 
 
 D 
 
 (388) 
 
 But when a vapour is heated in contact with the liquid whence 
 it is generated, the latter acquires the power of furnishing vapour 
 until the maximum of tension due to that temperature is attained. 
 Thus not only does the expansive force increase with the tempe- 
 rature, according to the law of Gay Lussac, but with the density 
 caused by the new evaporation, according to the law of Mariotte. 
 
 The relation between the temperatures, densities, and tensions 
 of vapours might, therefore, seem susceptible of determination. 
 
 In effect, in the case of water which boils at 180 above the 
 freezing point, an investigation analogous to that of 367, 
 gives us, 
 
 ^ ^ F (1 + 180 a) 
 
 D=D P (1+oQ ( 389 ) 
 
 We are however unable by this, to determine the tensions inde- 
 pendently of the densities, and are, therefore, compelled to have 
 recourse to experiment, in order to determine the elastic force of 
 steam and other vapours at given temperatures. From these ex- 
 periments empirical formulae may be deduced. 
 
 374. The best experiments on the tension of aqueous vapour 
 at low temperatures, are those of Dalton, which are comprised in 
 the following table. 
 
Book V.] 
 
 TENSION OP VAPOURS. 
 
 371 
 
 TABLE 
 
 Aqueous Vapour, at temperatures below 
 of the column of Mercury they are capa- 
 
 Te.perature. 
 
 117 
 122 
 127 
 132 
 137 
 142 
 147 
 152 
 157 
 162 
 167 
 172 
 177 
 182 
 187 
 192 
 197 
 202 
 207 
 212 
 
 Of the Maximum Tension of 
 212, estimated in the height 
 ble of supporting. 
 
 Te m peratur, j **% 
 
 2 0.068 
 
 12 0.096 
 
 0.115 
 0.168 
 0.200 
 0.237 
 
 42 0.283 
 
 47 0.339 
 
 52 0.401 
 
 57 0.474 
 
 62 0.560 
 
 67" 0.655 
 
 72 0.770 
 
 77 0.910 
 
 82 LOT 
 
 87 1.24 
 
 92 1.44 
 
 97 1.68 
 
 102 1.98 
 
 107 2.32 
 
 112 2.68 
 
 From the numbers in this table the following formula for the 
 elastic force P, has been deduced, in terms of degrees of the tem- 
 perature /, reckoned from 32, 
 
 P = .1781(l-f-.0060 7 . (390) 
 
 A formula better adapted to atmospheric temperatures, is 
 
 P=.18+.007 1 +.00019 P (391) 
 
 The tension of aqueous vapour or steam, at temperatures above 
 the boiling point, is usually estimated in atmospheres. The la- 
 test experiments are those of Dulong and Arago, which give the 
 results of the following table, as far as 24 atmospheres from ob- 
 servation, and as far as 50 from calculation. 
 
 The formula used in the calculations, and which is deduced 
 from the observations, is 
 
 P = (l + . 00397 O 5 ; (392) 
 
 t being estimated from 32. 
 
 3.09 
 
 3.50 
 
 4.00 
 
 4.60 
 
 5.29 
 
 6.05 
 
 6.87 
 
 7.81 
 
 8.81 
 
 9.91 
 
 11.25 
 
 12.72 
 
 14.22 
 
 15.86 
 
 17.80 
 
 19.86 
 
 22.13 
 
 24.61 
 
 27.20 
 
 30.00 
 
372 EQUILIBRIUM AND [Book V. 
 
 TABLE 
 
 Of the Elastic force of Steam at corresponding temperatures, from 1 to 
 
 50 Atmospheres. 
 
 Tension in I Corresponding I Tension in I Corresponding 
 Atmospheres. | Temperature. | Atmospheres. | Temperature. 
 
 1 212 9 350.8 
 1 229.2 10 358.9 
 
 2 250.5 11 366.8 
 2 263.8 12 374. 
 
 3 275.2 13 380.6 
 3i 285. 1 14 386.9 
 
 4 294 16 398.5 
 4i 301.3 18 408.9 
 
 5 308.8 20 418.5 
 54 314. 22 427.3 
 
 6 320.4 24 435.6 
 6 326.5 30 457.2 
 
 7 331.7 40 486.6 
 
 8 341.8 50 508.6 
 
 375. In experimenting on the tensions of the vapours of sub- 
 stances other than water, Dalton discovered that a remarkable law 
 held good within the limits at which his experiments were made. 
 This law goes by his name, and is as follows, viz: 
 
 Every different liquid has a determinate temperature at which 
 it boils under the mean pressure of the atmosphere. At this tem- 
 perature the elastic force of its vapour is just equal to the pres- 
 sure of the atmosphere. At other temperatures equidistant from 
 the boiling points of the different liquids, their respective tensions 
 are still equal. 
 
 Thus : water boils at 212, and ether at 96, the tension of the 
 vapours being in both cases the same ; when water is heated un- 
 der pressure to 250. 5, its tension is doubled, for an increase in 
 temperature of 38. 5 ; and so, the vapour of ether at the tempera- 
 ture of 96-r-38, 5=134. 5, has an elastic force equivalent to 
 two atmospheres. 
 
 376. By an apparatus similar in principle to the tube of Mari- 
 otte, 365, (2) air may be enclosed, and subjected to any given 
 pressure. If this air be perfectly dry, and if water be passed up 
 into the space occupied by the air, it will be found that it evapo- 
 rates at all temperatures whatsoever, furnishing a vapour whose 
 tension may be determined by the change it causes in the column 
 of mercury that confines and compresses the air. The rapidity 
 with which this vapour is given out, will be affected by the pres- 
 sure ; being instantaneous, as has been shown, in vacuo, and be- 
 coming less rapid for every increase in the pressure. 
 
Book F.] TENSION or VAPOURS. 373 
 
 After a time, however, the pressure of the vapour adds, to the 
 original tension of the confined air, an elastic force which is ex- 
 actly equivalent to the maximum tension of vapour of the tem- 
 perature at which the experiment is made. The evaporation then 
 ceases, and the joint tension of the confined air and vapour re- 
 mains constant. 
 
 Thus the presence of air does not effect the maximum tension 
 of vapour of a given temperature, but merely retards its forma- 
 tion; and the quantity of aqueous matter in the elastic form that 
 can exist in a given space is the same, whether that space be void 
 of any other substance, or filled with air of any density whatsoever. 
 Vapour then, like a gas, tends to distribute itself around the earth 
 in an atmosphere, and the formation of such an atmosphere is not 
 prevented, but merely retarded, by the presence of an aeriform at- 
 mosphere. 
 
 Vapour when mingled with a permanently elastic fluid, may 
 be condensed by the same two causes that induce its precipitation 
 in vacuo : namely, by an increased pressure, and by a diminished 
 temperature. 
 
 The laws that are applicable to the mixture of two gases, or of 
 a gas with a vapour, are true of the mixture of any number of elas- 
 tic fluids, whether they be permanently so or not. And thus in 
 a given space, any number of elastic fluids whatever may be en- 
 closed, without these substances interfering with each other. It 
 is only necessary that they be subjected to a pressure equal to the 
 sum of their several tensions. 
 
374 SPECIFIC GRAVITY [Book V. 
 
 CHAPTER IX. 
 
 OF THE SPECIFIC GRAVITY OF ELASTIC FLUIDS. 
 
 377. We may, as has been stated in 363, obtain the weightof 
 a mass of atmospheric air, by taking a flask furnished with a stop- 
 cock, weighing it, and then adapting it to the plate of the air- 
 pump to exhaust the air. The stop-cock being closed, the flask 
 may be removed, and again weighed ; the difference between its 
 weight under the two different circumstances, is the weight of the 
 air that has been withdrawn. 
 
 As the air of the atmosphere always contains moisture, a more 
 correct measure of the weight of the air the flask is capable of 
 containing, may be obtained, by taking the last weight as the ab- 
 solute weight of the flask; air that has been carefully dried by 
 hygrometric substances is then introduced, and the flask again 
 weighed. We thus obtain its weight, free of the influence of the 
 vapour it contains under ordinary circumstances. 
 
 The weight of any other gas may be ascertained in a similar 
 manner, by introducing it, after being well dried, into a flask, 
 whence the air has been previously exhausted. 
 
 In performing these experiments, it will be obvious that the 
 quantity of gas that will enter the vessel, depends upon the pres- 
 sure at which it is introduced, and upon its own temperature ; for 
 under different pressures it will have different densities, the tem- 
 perature remaining constant ; and at different temperatures the 
 density will also vary, the pressure remaining constant. Hence, 
 both the temperature and pressure, at which the experiments are 
 made, must be noted. 
 
 The flask in which the experiment is made, will also vary in 
 size, under the influence of temperature ; the effect thus produced, 
 in consequence of the dilatability of the glasss, must therefore be 
 taken into account. 
 
 If the gas cannot be introduced perfectly dry, its hygrometric 
 state must be observed. 
 
 The operations of weighing are performed in the air of the at- 
 mosphere ; hence, the apparent weight of the flask, whether full 
 or empty, will be less than the true by the weight of an equal vol- 
 ume of air. This loss of weight will be affected by the pressure 
 and temperature of the air, and by the quantity of aqueous vapour 
 it contains. 
 
 The air-pump does not exhaust the whole of the air from the 
 flask, and hence the proportion that remains, which will be indi- 
 cated by the guage of the pump, must be noted. 
 
Book P.] or ELASTIC FLUIDS. 375 
 
 Such are the principles on which the determination of the 
 
 weight of atmospheric air, and different gases depend. The detail 
 
 | of the operations, and the manner of applying the corrections may 
 
 be seen by reference to Biot : Traite Complet de Physique, 
 
 The capacity of the flask being known, the weight thus ob- 
 tained may be compared with that of an equal bulk of water, and 
 the specific gravity determined upon the principles of Chap. IV, 
 in terms of that liquid as a unit. It has been thus found that the 
 specific gravity of atmospheric air, at the temperature of 32, is 
 0.001299055. Its density at any other temperature and pressure, 
 may be obtained by means of the formula (384). This fraction is 
 the value of m, in the formula for the absolute specific gravity of 
 bodies, 369. 
 
 This comparison is necessary, in order to connect the tables of 
 the specific gravities of gases with those of solid and liquid bodies, 
 in the latter of which water is employed as the unit of density. 
 
 When the specific gravities of gases are sought, it is usual to 
 determine them in terms of some body of that class, taken as the 
 unit. Atmospheric air, which, when freed from moisture, has 
 an uniform constitution in all parts of the globe, is most conve- 
 nient for this purpose. It is, therefore, most frequently employed 
 in mere mechanical investigations. But for many purposes in 
 chemistry, hydrogen possesses superior advantages, particularly 
 from the fact of the numbers that represent the den-sities being 
 whole, in consequence of the great levity of that gas. Some 
 chemical writers employ oxygen as the unit. 
 
 We subjoin a table of the specific gravities of some of the gases, 
 in terms of atmospheric air. 
 
 TABLE 
 
 Of the Specific Gravities of Gases. 
 
 Atmospheric Air . ' ' 1.0000 
 
 HydriodicGas . 'V ' 4.4288 
 
 Silico-Fluoric Gas . * 3.5735 
 
 Chlorine - V' 1 '- . . > < 2.4216 
 
 Sulphurous Acid Gas . 2.1930 
 
 Cyanogen ~^MT. ' i 1,8064 
 
 Nitrous Oxide . i >.n. ; -i> .5269 
 
 Carbonic Acid Gas ; .v v .5245 
 
 Muriatic Acid Gas ..;. ;-; .2474 
 
 Sulphuretted Hydrogen '. iu .1912 
 
 Oxygen >,!.-" . . .1026 
 
 Nitrogen '".'.. . . 0.9757 
 
SPECIFIC GRAvrrr [Book V. 
 
 Carbonic Oxide . . 0.9569 
 
 Ammonia . . . 0.5967 
 
 Carburetted Hydrogen . 0.5596 
 
 Hydrogen . . . 0.0688 
 
 378. The density of vapour may be determined by introducing 
 i known weight of the substance that yields it, into a receiver 
 ntaining mercury, supported by the pressure of the atmosphere, 
 d inverting the receiver in a vessel also containing mercury, 
 upon the principle of the Torricellian experiment. The outer 
 vessel is tall enough to contain a mass of some transparent fluid 
 sufficient depth to cover the receiver. The whole apparatus 
 is then heated ; and when the whole of the substance, whose va- 
 pour is under experiment, has been evaporated, the space the va- 
 iour occupies and its temperature are noted. We then have a 
 Jlk of the vapour, whose weight is the same as that of the sub- 
 stance whence it was generated ; the temperature is known, and 
 >e pressure can be determined by means of the columns of mer- 
 cury and of the surrounding liquid compared with the indication 
 the barometer at the time. In this manner, it is found that the 
 density of steam at the temperature of 212 is ' part of the 
 ensity of water at 32. The densities of vapours compared 
 with atmospheric air as the unit, are as follows: 
 
 TABLE 
 
 Of the Specific Gravities of the Vapours of Different Substances, at their 
 boiling temperatures, under the mean pressure of the atmosphere. Jit- 
 motphtric air at the temperature o/32, and under the same pressure, 
 being taken as the unit. 
 
 Vapour of Iodine , . . 8.6111 
 
 of Oil of Turpentine . 5.0130 
 
 of Nitric Acid . . 3.1805 
 
 of Sulphuret of Carbon . 2.6447 
 
 of Sulphuric Ether . 2.5860 
 
 of Pure Alcohol . . 1.6133 
 
 of Water . . . 0.6235 
 
 379. The density of a vapour, under a given pressure, and at 
 a given temperature being known, that under any other pressure, 
 and at its corresponding temperature, can be calculated by the 
 formula (389). In this manner the density of steam, in terms 
 of water, and the relative volumes occupied by given weights of 
 aqueous vapour, have been calculated. The pressures, and the 
 temperatures, above the boiling point, have been taken from older 
 experiments than those of Arago and Dulong. They have been 
 purposely retained in order to show the view of this subject that 
 is still most generally received. 
 
Book F.] 
 
 ELASTIC FLUIDS. 
 
 377 
 
 TABLE 
 
 Of the Density and Volume of Aqueous Vapour, at its Maximum Ten- 
 sion. The Unit of Density being water at the temperature o/32, and 
 that of volume, the volume oj an equal weight of water, also at 32. 
 
 32 
 
 41 
 50 
 59 
 
 68 
 770 
 
 86 
 95 
 104 
 113 
 122 
 131 
 140 
 149 
 158 
 167 
 176 
 185 
 194 
 203 
 512 
 251.6 
 291.2 
 330.8 
 370.4 
 
 a 
 it 
 
 ti 
 U 
 t( 
 it 
 
 U 
 U 
 
 1 
 
 2 
 
 4 
 
 8 
 
 16 
 
 0,0000053 
 
 0,0000073 
 
 0,0000097 
 
 0.0000131 
 
 0.0000173 
 
 0.0000227 
 
 0.0000297 
 
 0.0000390 
 
 0.0000499 
 
 0.0000637 
 
 0.0000710 
 
 0.0001022 
 
 0.0001261 
 
 0.0001592 
 
 0.0001964 
 
 0.0002388 
 
 0.0002936 
 
 0.0003557 
 
 0.0004261 
 
 0.0005074 
 
 0.0005900 
 
 0.00110 
 
 0.00210 
 
 0.00399 
 
 0.00760 
 
 188600 
 137000 
 103000 
 76330 
 57800 
 44050 
 33670 
 25640 
 20030 
 15690 
 14080 
 9784 
 7930 
 6281 
 5091 
 4187 
 ' 3406 
 2811 
 2346 
 1971 
 1696 
 909 
 476 
 250 
 131 
 
 traeted from that of Darnell. 
 
 48 
 
SPECIFIC GRAVITY, &C. [Book V. 
 
 TABLE 
 
 Of the Force, Weight, and Expansion of Aqueous Vapour, at different 
 degrees of temperature, from to 92. 
 
 Temperature. 
 
 Elastic force in 
 inches of mercury. 
 
 Weig-ht of a cubic 1 ., 
 foot in grains. | Evasion. 
 
 
 
 0.064 
 
 0.789 
 
 .000 
 
 12 
 
 0.096 
 
 1.156 
 
 .024 
 
 22 
 
 0.139 
 
 1.642 
 
 .045 
 
 32 
 
 0.200 
 
 2.317 
 
 .066 
 
 42 
 
 0.283 
 
 3.214 
 
 .087 
 
 52 
 
 0.401 
 
 4.468 
 
 .108 
 
 62 
 
 0.560 
 
 6.126 
 
 .129 
 
 72 
 
 0.770 
 
 8.270 
 
 .150 
 
 82 
 
 1.070 
 
 11.293 
 
 .170 
 
 92 
 
 1.440 
 
 14.931 
 
 .191 
 
 212 
 
 30.000 
 
 257.191 1.441 
 
 380. The presence of aqueous vapour in atmospheric air, may be 
 shown by gradually cooling down a polished surface. When such 
 a surface reaches the temperature that corresponds to the max- 
 imum tension of the vapour, a thin film of dew will be deposited, 
 and cloud the surface. If the temperature of the surface be 
 ascertained, we can by means of the above table determine the 
 quantity of vapour contained in each cubic foot. Were the ta- 
 ble complete to every degree of the thermometer, simple in- 
 spection would give us the weight in grs. opposite to the observed 
 temperature of the surface, provided the air were also of that 
 temperature. But as the temperature of the air is always higher, 
 the vapour will be expanded, and hence, the weight given in the 
 third column, requires to be corrected, by multiplying it by the 
 fractions, representing the ratio of the two expansions, at the 
 temperatures of precipitation, and of the air. 
 
 The temperature of precipitation, or that to which the surface 
 is reduced at the moment it begins to be clouded, is called the 
 Dew-Point. 
 
 The best instrument that has yet been contrived for observing 
 it, is the Hygrometer of Daniell. 
 
 Another instrument, also well fitted for the purpose, has more 
 recently been invented by Pouillet. It is foreign to our pur- 
 pose to enter into the detail of the structure of these beautiful 
 and ingenious instruments. 
 
Book V.] BAROMETER AND ITS APPLICATIONS. 319 
 
 CHAPTER X. 
 
 OF THE BAROMETER AND ITS APPLICATIONS. 
 
 381. The Barometer is constructed by giving to the Torricel- 
 lian apparatus a support that unites its tube and basin ; its form 
 may also be changed in such a manner as to substitute a more 
 convenient receptacle for the mercury than the latter. A scale 
 is then adapted, by means of which the altitude of the column of 
 mercury may be measured in some conventional unit of length. 
 
 The French use in their barometers, the metre as the unit, and 
 the scale exhibits its decimal Divisions. The English divide their 
 scale into inches, and these are subdivided decimally. The former 
 assume for the mean height of the mercury in the barometer, at 
 the level of the sea, 0.760 metres ; the latter, 30 English inches. 
 These heights, however, even when the pressure remains inva- 
 riable, are affected by the change that takes place in the density 
 of the mercury, under changes of temperature. 
 
 When the barometer is intended to remain stationary, in a 
 single place, the original form of a wide basin, in which the end 
 of the tube is immersed, is still used ; and as a considerable change 
 in the height of the mercury in the tube will not produce any 
 sensible difference of level in the basin, it has the advantage of 
 needing no correction. 
 
 If the open end of the tube be bent upwards, it is called the 
 syphon barometer, and the pressure of the air upon the mercury 
 in the open branch of the tube, will produce the same effect as it 
 does when acting upon that in the basin, in the original apparatus 
 of Torricelli. A scale of the same measure of length must be 
 adapted to each branch of the tube, and the position of the mer- 
 cury noted upon both, in order to determine the difference of 
 level. 
 
 To increase the length of the divisions that correspond to a 
 given change of level in the mercury, various plans have been 
 proposed ; all, however, except two, have gone wholly out of 
 use, and therefore require no description. These two are the 
 wheel barometer of Hooke, and the conical barometer. 
 
 The form of the wheel barometer, is as follows : adapt a float 
 of iron to the open branch of the syphon barometer, and counter- 
 poise it by a weight attached to a cord passing over a pulley ; 
 the weight must be of such magnitude that when the mercury 
 subsides in the tube, the iron float shall preponderate and follow 
 
380 BAROMETER AND \Eook V. 
 
 the mercury in its descent ; but when the mercury rises, the float 
 being buoyant upon it, is drawn up by the counterpoise. In this 
 motion the pulley will be turned around, and if an index be affixed 
 to its axis, the latter will traverse around a dial concentric with 
 the pulley. On this dial the divisions may be marked. 
 
 The conical barometer is a slender tube with a conical bore, 
 the open end having the largest diameter. It is found that a 
 column of mercury will remain suspended, in such a tube, by the 
 pressure of the atmosphere, if it be carefully inverted, and be not 
 agitated. This column will assume a length which corresponds 
 to the pressure of the atmosphere. If this pressure be lessened, 
 the mercury will fall until the column of mercury, which, as 
 it descends into a wider part of the tube, must decrease in alti- 
 tude, is again in equilibrio with atmospheric pressure. 
 
 If the pressure, on the other hand, increase, the mercury will 
 be forced up; but the length of its column will increase, in con- 
 sequence of its entering a portion of the tube of less diameter, 
 and the rise will cease when this length becomes the measure of 
 the increased pressure. 
 
 It will be obvious that in both these cases, the change in the 
 position of the mercury must be considerably greater than that 
 which will take place in a tube of uniform bore. 
 
 3S2. The invention of the Vernier has, in a great degree, re- 
 moved the necessity of seeking for a form of the barometer that 
 shall have a scale of greater length than that which corresponds 
 to the change of level in a tube of uniform bore, placed in a ver- 
 tical position. The scale of the barometer being fixed, the ver- 
 nier consists in a moveable scale that slides along it, and whose 
 lower or upper extremity carries the index that is made to cor- 
 respond with the surface of the mercury in the tube. The length 
 of this sliding scale is made exactly equal to some given number 
 of divisions upon the fixed scale ; and it is subdivided into a 
 number of equal parts, exceeding, or falling short, by one, the 
 number of divisions upon the corresponding length of the scale. 
 
 The theory of this instrument is as follows : 
 
 Let a be the length of the fixed scale, which the length as- 
 sumed for the vernier exceeds or falls short of by 1 division ; 
 let n be the number of divisions in a, and which is the same with 
 the number of divisions in the vernier. 
 
 The length of the vernier will be, 
 
 a 
 The length of each division on the fixed scale is - ; 
 
Book F.] ITS APPLICATIONS. 381 
 
 The length of each division of the vernier will be 
 1 a a a 
 
 And as the length of each division of the fixed scale is -, the dif- 
 
 a 
 ference in the length of the respective divisions is ^f -^ . 
 
 To take the case of the common barometer ; the inch is di- 
 vided into ten parts, and eleven of these are taken for the length 
 of the vernier, which is divided into ten equal parts ; a then is 
 equal to 1 inch, w=10, and 
 
 _ 1 . , 
 2 -10om 
 
 which will have the negative sign. Hence the changes in its po- 
 sition will be indicated by looking down its scale, and counting 
 the number of that division, reckoned from the upper end of the 
 vernier, that corresponds with a division of the fixed scale. The 
 index is therefore placed at the top of the vernier ; the inch and 
 tenth next below the index, give the height to the first place of 
 decimals, and the second place is given by the indication of the 
 vernier. 
 
 In barometers where a greater degree of accuracy is required, 
 the inch is divided into 20 equal parts ; the vernier is made equal 
 in length to 24 of these, and is divided into 25 equal parts. In 
 this case, 
 
 a=1.25 inch, 
 
 n= 25 
 a 1.25 
 
 The difference in the length of the respective divisions then is 
 _i_ part of an inch, and its sign is positive ; hence, the index is 
 placed at the bottom of the vernier, and its indications sought by 
 counting upwards from the index, until that division be reached, 
 which exactly corresponds to a division of the fixed scale. 
 
 383. In some of the applications of the barometer, it is neces- 
 sary that it should be safely portable from place to place. This 
 may be effected in various ways. 
 
 ( 1 . ) The mercury may be enclosed in a leathern bag, adapted, in- 
 stead of a basin, to the bottom of the tube. If a screw be applied 
 beneath the bag, the mercury may, by its pressure, be forced up 
 until it strike against the top of the tube ; the instrument may 
 then be carried, in an inverted, or in a horizontal position, with- 
 out risk or danger from the striking of the mercury against the 
 top of the tube. 
 
 (2.) The open end of a syphon barometer, may be made of 
 two pieces, united by a stopcock of a material that is not acted 
 
382 BAROMETER AND [Book V. 
 
 upon by mercury. On inclining the tube, the mercury strikes 
 against its top and fills it. In this position, if the stopcock be 
 closed, the mercury will not be able to escape when the vertical 
 position of the instrument is restored ; neither can it oscillate, for 
 it will completely occupy the whole of the tube. 
 
 (3.) In the barometer of Gay Lussac, the necessity of a stop- 
 cock on the open branch of the syphon, is obviated by contract- 
 ing the tube at the bend to very small dimensions, and continu- 
 ing this contraction for some distance up the closed branch of the 
 tube. The external air cannot enter through the tube thus con- 
 tracted, and it may, therefore, be safely inverted, and carried 
 from place to place. 
 
 (4.) In the very perfect and complete portable barometer of 
 Sir George Shuckburgh, the mercury is enclosed in a wooden 
 cistern, closed at the bottom by a flexible diaphragm of leather. 
 This is moved by a screw until the mercury fills the tube. Over 
 the mercury an ivory float is placed, th*t is brought by the action 
 of the screw to a mark on the stem, that shows when the mercury 
 is at the level whence the divisions on the scale have been mea- 
 sured. This float has a ring between it and the mercury that is 
 pressed up when the mercury rises, and closes the opening 
 through which the float passes. 
 
 (5.) In the portable barometer of Englefield, a cistern of wood 
 is cemented to the glass tube, and the whole is packed in a rod 
 of wood, of the size of a common walking cane. A piston works 
 in the cistern, by means of a screw, and can be raised until it 
 force the mercury to the top of the tube. The construction of 
 this instrument has been much improved by Daniell ; partly in 
 some particulars that will be stated in the next section, and partly 
 by applying a cistern of cast-iron, and adapting a table to the 
 instrument, by which the correction for the expansion of the glass 
 and mercury, by heat, can be obtained by inspection. The com- 
 parative diameters of the tube and cistern are also given, and the 
 height at which the scale has been adapted to the tube. At all 
 other heights, a correction must be applied for the change of level 
 in the cistern, for which this relation between these diameters is 
 the element. 
 
 384. In filling the barometer with mercury, and applying the 
 scale, a variety of precautions are necessary. The mercury must 
 be purified by chemical means from all extrinsic substances, for 
 they will alter its density ; but when properly purified, the den- 
 sity and character of the mercury are always identical. 
 
 The mercury must be completely purged of air, and air must 
 be carefully excluded from the tube ; for even a small quantity 
 of air, rising to the top of the tube, will produce a considerable 
 
Book P.] ITS APPLICATIONS. 383 
 
 depression in the column of mercury. To separate both the air 
 that is contained in the mass of mercury, and that which it can- 
 not fail to imbibe in the act of being poured into the tube, the 
 mercury must be boiled in the tube itself. This is done by fill- 
 ing the tube at first only to a third part of jts length, and boiling ; 
 mercury, that has been heated, is then added in several distinct 
 portions, and each successive portion is heated until it boils. 
 After the tube is nearly filled, as it would endanger it to complete 
 the boiling, the residue is added from a parcel of mercury that 
 has been boiled separately. The reason of adding mercury that 
 has been previously heated, is to prevent the glass from breaking 
 by being suddenly cooled. The effectual exclusion of the air 
 may be ascertained after the tube has been inverted in the cistern, 
 by inclining the tube until the mercury rises to the top ; if it 
 strike hard, and with a sharp sound, the air has been completely 
 driven off. 
 
 In affixing the scale, it is usual to set it by comparison with 
 another barometer; but Daniell has, in all the barometers made 
 under his direction, applied scales divided by actual measure- 
 ment from the surface of the mercury in the basin ; and the po- 
 sition of the mercury at the time that this division is performed, 
 is noted upon the outside of the case of the instrument, as the 
 neutral point, as has already been stated in the preceding section. 
 
 The introduction of air, when the barometer is inverted for 
 carriage, and again restored to its proper position for use, is diffi- 
 cult to be avoided, in barometers of the usual construction ; for 
 there is no adhesion between glass and mercury, by which the 
 passage of an extrinsic substance can be prevented. To obviate 
 this defect, Daniell has welded a ring of platinum to the bottom 
 of the tube ; between this and the mercury there is an attraction, 
 that will prevent the entrance of air. 
 
 Besides the correction for temperature, a correction is required 
 for a depression caused in the mercury by capillary action. 
 
 385. Although the mean height of the barometer at the level 
 of the sea is about 30 inches, yet it is far from standing constantly 
 at that height. It is found on the contrary to vary in a greater 
 or less degree at every point on the surface of the earth, and va- 
 riations of the same character are found to take place at all alti- 
 tudes that have been reached. These variations are either peri- 
 odical or accidental. In equatorial regions the former are the 
 more important ; but in temperate climates, the periodic variations 
 appear, on a first inspection, to be completely masked by those 
 which are accidental. The whole amount of variation too, appears 
 to increase as we recede from the poles to the equator, although it 
 is influenced in a very great degree by local circumstances. Thus 
 
384 BAROMETER AND 
 
 at New- York the variation does not much exceed 1J inches, 
 while in Great Britain it is as great as 3 inches. 
 
 To separate the accidental from the periodic variations in the 
 barometer, a long series of observations must be made, at hours 
 of the day chosen for their adaptation to the purpose. If the 
 mean height be alone sought, the hour of noon is well suited, and 
 a series of observations, continued for some years, will show 
 whether there be any change due to the season of the year. For 
 the horary oscillations, more frequent observations must be made. 
 Intemperate climates, Ramond has proposed as best .adapted 
 to the purpose, the hours of 9 A. M., noon, 3, and 9 P. M. 
 Daniell has directed the use of the hours 3, and 9 A. M., and 3 
 and 9 P. M. If no more than three observations are to be made, 
 the latter author has chosen 8 A. M., 4 P. M., and midnight. 
 Under the equator, Humboldt has shown that the maximum of 
 height takes place at the hours of 9 A. M., and 11 P. M., the 
 minimum at 4 A. M., and 4 P. M. In Paris, Ramond has shown 
 that the times of maximum and minimum, vary with the season ; 
 in winter the hours of the maximum are 9 A. M., and 9 P. M., 
 of the minimum at 3 A. M ; in summer the maxima occur at 8 
 A. M., and IIP. M., the minimum at 4 P. M. These horary 
 variations appear to be less in high latitudes than at the equator, 
 while the accidental variations follow, as has been stated, a dif- 
 ferent law. 
 
 386. These variations are the consequence and indications of 
 changes in the pressure of the atmosphere. Those which we 
 have styled accidental, are from longexperience found to produce 
 changes in the state of the weather. The nature of the change 
 portended, is however different in different countries, and there 
 is but one general rule, namely, that a sudden fall of the mer- 
 cury always portends a high wind. 
 
 The barometer may therefore be used to prognosticate the state 
 of the weather, and will be effectual for this purpose, wherever a 
 number of observations has been made sufficient to detect the law, 
 that the variations of the one follow in respect to the other. It 
 is also used to great advantage on ship-board, to foretell gales of 
 wind. 
 
 387. As the barometer furnishes a measure of the pressure of 
 the atmosphere, and as this pressure varies, 366, with a change 
 in the distance from the mean surface of the earth, or with the 
 altitude of the place of observation above the level of the sea, the 
 barometer is used for the purpose of measuring differences in al- 
 titude, and ascertaining the absolute height of places above the 
 ocean. The principle on which this may be performed, has al- 
 ready been stated in 366. 
 
Book V.\ IT APPLICATIONS. 385 
 
 If the atmosphere were of uniform temperature throughout, we 
 have from (381), for the value of the difference of level z, 
 
 z=m.(log. P logp) ; 
 
 or, as the columns of mercury in the barometer are the measures of 
 the pressures, P and p, we may consider those letters as repre- 
 senting the number of inches and decimals in those columns re- 
 spectively. 
 
 The value of the constant co-efficient m, is determined by expe- 
 riment, and is, by the observations of Gen. Roy, at the tempera- 
 ture of melting ice, and estimated in English measure, 10000 fa- 
 thoms. Hence, 
 
 z= 10000 (log. P log.p), 
 or in feet, 
 
 2=60000 (log. P log. p) . (393) 
 
 The height of the columns of mercury i? affected by tempera- 
 ture ; hence, a correction must be Applied to the observed 
 columns of mercury to reduce them to a common temperature. 
 
 Mercury expands itself 0.0001016 of its bulk at 32, for every 
 degree of heat : hence, if the temperature of the columns of mer- 
 cury be known, the reduction of each to that standard temperature 
 is easy, and the mode obvious. IVo sensible error can arise, 
 however, in ordinary atmospheric temperatures, from considering 
 the above fraction as the rate of expansion between the tempera- 
 tures which the mercury in the barometer has at the two stations. 
 Hence, the correction may be applied to but one of the columns, 
 and it is most convenient to do so, to that which has the lowest 
 temperature, and which will most commonly be that observed 
 at the highest station. If T/ and t ', be the two temperatures, the 
 co-efficient denoting this correction will be 
 
 1 +0.0001016 (T' t'} . (394) 
 
 Difference of temperature will also affect the density of the air ; 
 and hence the difference of the height of the mercurial columns 
 will not be the same at other temperatures, at each or either of the 
 places, as it would be, had both the temperature of 32. A cor- 
 rection is therefore needed for this cause, which will affect the co- 
 efficient m. Air dilates, as has been stated, 367, 0.002083 of 
 its bulk, for every degree of temperature reckoned from 32. This 
 fraction, therefore, is not constant at all temperatures. It is, 
 however, usual to consider it as such, and to apply it, by means of 
 the mean temperature of the air at the two stations, above the free- 
 zing point : hence, the correction consists in multiplying m by 
 
 .002083 
 
 or which is the same, by 
 
 1 + 0.0010415 (T-H 64) . (395) 
 
 49 
 
386 BAIIOMETBR AND [Book V. 
 
 The formula (393) therefore becomes, when these corrections 
 are taken into account, 
 
 z= 10000 [1 +0. 001041 (T-H 64)] , 
 
 j* P (396) 
 
 p [1+0.0001016 (T f )] . 
 
 This formula is sufficiently near the truth for most of the cases 
 that can occur in practice. When much accuracy is required, and 
 especially where the difference of level is great, there are other 
 circumstances to be taken into account. 
 
 (1). The mean height of the mercury in the barometer is af- 
 fected by the difference in the apparent force of gravity at dif- 
 ferent latitudes, according to the law in 100. A correction, 
 therefore, may be needed on this account ; the element of which 
 is the latitude of the place. This, however, is at most small, 
 and is generally neglected. 
 
 (2). A more important cause of error exists in the mixture of 
 atmospheric air with aqueous vapour, which is subject to different 
 laws of pressure and equilibrium. It has been considered by an 
 authority of no less weight than that of Laplace, to be impracti- 
 cable in the present state of our knowledge, to apply a correction 
 for tins circumstance. This difficulty has, however, been over- 
 come by Daniell, who has shown that his hygrometer can be ap- 
 plied to the purpose ; and has published tables that accompany it, 
 by means of which this correction can be applied. These tables, 
 and the principles on which they are founded, may be seen on re- 
 ference to the "Quarterly Journal of Science, edited at the 
 British Institution", No 25. 
 
 3SS. In applying the barometer as a measure of differences of 
 level, several precautions are necessary. In order to prevent the 
 observations being affected by the periodic, and still more by 
 the accidental variations in the mean pressure of the atmosphere, 
 they should be made simultaneously at the two places whose dif- 
 ference of level is sought. Hence, two observers, and two instru- 
 ments, are necessary. 
 
 The barometers should each, have a thermometer inclosed in 
 their case, in order to mark the temperature of the mercury they 
 contain. These are called the Attached Thermometers. 
 
 Each observer should be furnished with a separate thermome- 
 ter, to note the temperature of the air. 
 
 One of the observers remaining stationary, the other may move 
 with his instruments from place to place, and thus observe at a 
 number of stations. In order to a comparison of observations, 
 the observer who remains at the same place should take observa- 
 tions at prescribed intervals of time, say from 10' to 15' ; the one 
 who proceeds from place to place, should note the time of each 
 
" : 
 
 Book V.~\ ITS APPLICATIONS. 367 
 
 of his observations. Those nearest in point of time, must of course 
 betaken as the objects of comparison. 
 
 The calculation is performed as follows : The column of mer- 
 cury at the station whose temperature is lowest, is to be corrected 
 for that element. This correction consists in adding to the 
 height of that column, its continual product by the constant frac- 
 tion 0.0001016, and by the difference of the indications of the two 
 attached thermometers. 
 
 The difference between the logarithms of the other column of 
 mercury, and this corrected column is then to be taken. 
 
 This difference multiplied by 10000, is the approximate differ- 
 ence of altitude between the two stations in fathoms, and is ob- 
 tained at once, by moving the decimal point four places to the 
 right. 
 
 The approximate difference of level is lastly to be corrected, 
 for the difference of the temperature of the air at the two stations 
 from 32. This correction is obtained by multiplying the excess 
 of the mean of the two detached thermometers above 32, or their 
 defect below 32, by the constant fraction 0.002083, and the pro- 
 duct by the approximate altitude. This correction is added when 
 this mean temperature exceeds 32, and subtracted when it 
 is less. 
 
 389. If the correction for the moisture of -the atmosphere, 
 whose element is obtained from the indications of the hygrometeV 
 of Daniell, is not employed, it will be better to substitute for 
 the constant fraction 0.002083, which represents the expansion 
 of dry atmospheric air for each degree of Fahrenheit's thermome- 
 ter, the fraction 0.00244, which by the experiments of Gen. Roy, 
 is consistent with a mean state of moisture. 
 
 The co-efficient m, which we have stated at 10000 fathoms, 
 has been inferred by Raymond, from a great number of observa- 
 tions, to be 18336 metres, at the latitude of 45. This is equivalent 
 to 10025 fathoms, and makes the number given in our formula 
 in error, about ? ^th part. The whole change in the intensity of 
 gravity from the pole to the equator is, as has been shown, 100, 
 aigth part of the force of gravity, on the hypothesis of the earth's 
 having a spherical figure ; but in consequence of the spheroidal 
 figure of the earth, the apparent intensity is still farther lessened 
 at the equator, and the ratio of the two forces is, 290, actually 
 as great as T | , or more exactly 0.005674. 
 
 As the value of m is determined for the latitude of 45, it be- 
 comes necessary to reduce observations at otherlatitudes, to the lati- 
 tude of 45. The value of the correction may be thus found : 
 
 Let g be the force of gravity at the equator; /the centrifugal 
 force there, will be =0.005674 g. 
 
388 BAROMETER AND [Book V. 
 
 From (294), we have for the value of y, the force of gravity in 
 lat. 45, 
 
 7 
 whence 
 
 for the force g', at any other latitude L, we have 
 
 g-'=g+/sin. a L ; 
 substituting the value of g, from the foregoing equation, 
 
 or 
 
 but as 
 
 cos. 2 L= 12 sin. 3 L; 
 we obtain 
 
 and substituting the value of/, 
 
 g'= y -_0.002837 cos. 2 L . 
 
 Whence a correction may be deduced to be applied to the quan- 
 tity z, in the formula, when the height is considerable, and the 
 latitude distant from 45. 
 
 The formula for the complete calculation then becomes, 
 2=10000 (10.002837 cos. 2 L) . 
 
 r / \ H 
 
 I 1+0.002083 (-2~ 32) J . 
 
 p 
 
 log * 
 
 KI+O.OOOIOGI (T 0] ' 
 
 We annex a form of the calculation 
 
 EXAMPLE. 
 
 Calculation of the difference in level of two stations, A and B, at which 
 the following observations were made cotemporaneously. 
 
 A 28.691 T=84 T'=76.5 
 
 B 28.791 *=78 l'=82 
 
 =49 T' f= 5.5 2L=91.32' 
 

 Book F.] ITS APPLICATIONS. 389 
 
 log. 0. 002837= 7.45286 log. 28.691 = 1.45775 
 log. cos. 910.32'= 8.42746 log. 27.791 = 1.44390 
 log. 0.000076= 5.88032 
 
 log. [H-(0.0001061 X 5.5)=0.99946]= 9,99975 
 
 1.44365 
 141 = 10000X0.01410 
 
 log. 141=2.14922 
 
 log. [1 + (0.002083X49) = 1.10207]=0.04221 
 
 log. [(10.000076) =0.999924] =9.99999 
 
 log. - - - 155.39 fathoms =2.19142 
 
 6 
 
 932.34 feet. 
 
 If we take m=60l50 feet, the last part of the calculation will 
 be, 
 
 log. 0.0140 =8.14922 
 
 log. 60150 =4.77924 
 
 log. 1.10207 =0.04221 
 
 log. 0.999924=9.99999 
 
 log. 934.68ft. =2.97066 
 
39 ATTRACTION OP [Book !> r . 
 
 CHAPTER XI. 
 
 OF THE ATTRACTION OF COHESION. 
 
 390. The conditions of the equilibrium of fluids that have been 
 investigated in the preceding chapters, are occasionally affected 
 by an action that takes place between their particles, and those of 
 solid bodies in contact with them. This action is called the At- 
 traction of Cohesion, or from the most remarkable class of the 
 phenomena to which it gives rise, Capillary Attraction. 
 
 The existence of this attraction, and its capability of exerting 
 a determinate force, may be shown by a very simple experiment. 
 
 If a disk of any substance that is capable of being moistened by 
 a liquid, be suspended from the arm of a balance, and counter- 
 poised by weights ; and if a vessel containing the liquid be raised 
 from beneath, until its surface just touch the disk, they will be 
 found to adhere. This adhesion may be overcome by adding 
 weights to the opposite arm of the balance ; or rather the disk 
 may be drawn away from the mass of liquid, for it will still carry 
 with it a film of the liquid ; and the force exerted by the addi- 
 tional weight does not overcome the cohesive force, but only the 
 attraction of aggregation that exists between the particles of the 
 liquid. 
 
 391. Phenomena due to the same cause are observed in a va- 
 riety of cases. Thus : the surface of water or alcohol, in a glass 
 vessel, is slightly raised around the edges; if the glass be di- 
 minished in size, the elevation of the fluid at the edge will in- 
 crease ; and in a tube of glass of small diameter, a column will 
 be supported within it, when plunged in a liquid, above the level 
 of the general surface. In tubes of very small diameter, called, 
 from their size, Capillary Tubes, this column may amount to some 
 inches. 
 
 In other cases, a depression exists within the tube ; thus, when 
 one of glass is immersed in mercury, that liquid will be obviously 
 lower within the tube than it is without. 
 
 In cases where the liquid is raised in a tube, its upper surface 
 assumes a concave form, which, in small tubes, differs but little 
 from a portion of a sphere ; and in cases where a liquid is de- 
 pressed, the upper surface is convex. 
 
 Similar phenomena occur between two tubes of different di- 
 ameters, placed one within the other, and between two parallel 
 plates. If two plates are inclined to each other, and meet at one 
 
Book V.] COHESION. 391 
 
 of their edges, the liquid will rise between them, its surface as- 
 suming the form of a curve. 
 
 Two solid bodies that would not otherwise adhere, may be 
 made to stick together with considerable force, by this action. 
 Thus, if two plates of glass be moistened with water, aiyi then 
 pressed together, they require a considerable effort to separate 
 them ; and two plates of polished brass may be in the Same man- 
 ner united by oil or melted tallow. In the latter cas^, the in- 
 terposed substance becomes solid on cooling, and the force with 
 which it resists an effort to separate the plates, becomes very 
 great. In general terms, when two solid bodies are matte to co- 
 here by the intervention of a substance that can be applied in a 
 "liquid state, and which afterwards becomes solid, the cohesion is 
 rendered more intense. This principle is applied in the process 
 of soldering the metals. 
 
 392. In order to examine the phenomena of the rise of liquids 
 in capillary tubes : 
 
 Let us suppose a prismatic tube, standing in a vertical posi- 
 tion ; that its sides are perpendicular to its base ; and that it is 
 supported in a vessel, in such a manner that it plunges at its base 
 into a liquid of such a nature as to rise in the tube above its na- 
 tural level. 
 
 The attraction of the tube has a very limited sphere of action, 
 for the height to which fluids rise in tubes of different thickness, 
 provided their interior diameters be the same, is constant. Any 
 small column of the liquid, situated in or near the axis of the tube, 
 will not be affected by this attraction, but must be supported by 
 the action of the adjacent columns of fluid. It is therefore clear, 
 that the action of the tube upon the column, immediately in contact 
 with it, is the final cause of the elevation of the whole mass. A 
 ring of the fluid is first raised ; this raises a st^pnd ; the second 
 a third, and so on, until the weight of the fluid exactly balances 
 the attractive forces that are exerted by the sides of the tube. In 
 order to determine the conditions of equilibrium, let us conceive 
 the tube to be produced in the form of a syphon, by a part of no 
 thickness, and which, therefore, does not by its attraction inter- 
 fere with the conditions of equilibrium, and does not prevent the 
 re-action of the fluid partieles contained within it upon those in 
 the tube. It is obvious that the column contained within this 
 imaginary tube, will exactly replace, in its fluid action, the whole 
 mass contained in the vessel ; and in the case of equilibrium, the 
 pressure in the two branches of the tube must be identical. But 
 the columns of fluid in the two branches are of unequal heights ; 
 the difference of pressure that results from this inequality, must, 
 therefore, be counteracted by the attractions of the prism and the 
 fluid ; these are exerted in a vertical direction in the original 
 branch of the tube. 
 
392 ATTRACTION OF [J3ook V. 
 
 As the prism is assumed to be vertical, its base is horizontal. 
 The fluid contained in the additional tube, is attracted vertically 
 downwards: (l,) by its own particles ; (2,) by the fluid that sur- 
 rounds it. But these two attractions are counteracted by the like 
 attractions that the fluid contained in the second vertical branch 
 sustains. The fluid of the vertical branch of the second tube is 
 besides attracted vertically upwards by the fluid in the first tube. 
 Bui this attraction is destroyed by the attraction the former exerts 
 upon the latter column of fluid. These reciprocal attractions 
 may therefore be disregarded. 
 
 Finally the fluid in the second tube is attracted vertically upwards 
 by the first tube ; hence there results a vertical force that contri- 
 butes to destroy the excess of pressure due to the elevation of the 
 fluid in the first tube. This force tends to destroy the excess of 
 pressure exerted in opposition to it, by the column of fluid raised 
 in the original tube above the general level. 
 
 This force we shall call P. 
 
 The forces that act upon the fluid contained in the original 
 tube, are as follows : 
 
 (1.) It is attracted by itself; but as the reciprocal attractions of 
 the particles of a solid body do not cause any motion, we may ab- 
 strKct this attraction, for we may, in a tube of uniform bore, stand- 
 ing in a vertical position, consider the vertical pressure as if it 
 were produced by a solid body filling the tube. 
 
 (2.) The elevated fluid is attracted downwards by the liquid co- 
 lumn contained in the part beneath the level of the external fluid ; 
 but it attracts in its turn with an equal force, and these attractions 
 mutually destroy each other. 
 
 (3.) The fluid is also attracted downwards by the fluid that sur- 
 rounds the ideal prolongation of the tube ; hence there results 
 a vertical force directed downwards, that we shall call P', its 
 sign being negate, in order to represent that its direction is op- 
 posite to thf< of the force P. The forces would be exactly equal 
 if the tpi>e were composed of the same material with the fluid. 
 Thuir inequality is therefore due to the difference in the intensity 
 of the attractive forces exerted by the particles of the fluid upon 
 each other, and by the particles of the tube upon those of the fluid. 
 If we take r and r' to represent these respective intensities, we 
 have 
 
 P : P' : : r : r'. (398) 
 
 (4. ) The fluid in the first tube is attracted vertically upwards by 
 the matter of the tube, and this force is obviously equal to P. 
 
 The whole of the attractive forces that act, are, therefore, 
 
 2 P P', (399) 
 
 and they arc equal to the weight of the column that they raise. 
 
 If a- represent the force of gravity, D the density of the fluid, 
 and V the volume of the elevated column, 
 
 g DV=2P P'. (400) 
 
Book J 7 ".] COHESION. 393 
 
 It is obvious from this equation, that the quantity V, will always 
 have the same sign with the quantity 2 P P'. 
 Hence : 
 
 393. When the attraction of the particles of the fluid for each 
 other is half that which the particles of the tube have for those 
 of the fluid, the level within and without the tube will be 
 the same ; when the former is less than half the latter, the fluid 
 will be raised ; but when it is greater, the surface of the fluid will 
 be depressed. 
 
 The action being exerted, at imperceptible distances, will only 
 be felt by the columns in immediate contact with the tube ; we 
 may, in consequence, neglect the curvature of the tube, and con- 
 sider it as developed into a plane surface. 
 
 The attractive force, P, will therefore be proportioned to the 
 size of this plane, and may be represented by r C, C being the 
 contour of the surface. And for a like reason, 
 
 P'=r'C; 
 whence 
 
 g-DV = (2r r')C. (401) 
 
 This formula is applicable to all the cases that have been ob- 
 served in practice, and the results that flow from it are consistent 
 with observation. 
 
 In cylindrical tubes : 
 
 Let a be the radius of the interior of the tube, h the height of 
 the column, measured from the level of the fluid without, to the 
 curved surface it supports or depresses. The volume of this 
 column is 
 
 *a?h. (402) 
 
 To this, must be added the volume of the meniscus in which 
 it terminates, which will be the difference between the volume of 
 a cylinder, whose height and base are both equal to a, and the 
 hemisphere whose radius is a, or to 
 
 2*a? tea 3 
 
 to? g , or simply to -g~. 
 
 The whole volume of fluid raised, will therefore be 
 
 and the contour of the base C, is 
 
 substituting these values in the foregoing equation, we have 
 
 g-D (a 2 h+*a 3 ) = (2r r') 2a ; (403) 
 
 and dividing by ta, 
 
 (-* a\ /2r r'\ 
 ft + 3)= 2 (-Tcr) < 404 > 
 
 50 
 
394 ATTBAOTION OF [Book V- 
 
 If the tubes be of the same material, and be plunged in the 
 same fluid, whose temperature is constant, r, ^^ sf and D will be 
 the same in every case, and the second member of the equation 
 will be a constant quantity, which we may call A, and 
 
 a /*+ =A. (405) 
 
 whence 
 
 a A 
 M-3--. (406) 
 
 When the tube is small, the height, ft, is great, compared with 
 
 a 
 the radius a, and the quantity ~, may be neglected, or will be 
 
 masked by the errors of observation ; we may therefore assume, 
 
 h=~ (407) 
 
 Hence : 
 
 394. In capillary tubes, liquids will rise or be depressed, ac- 
 cording to the relation between the attractive forces stated in the 
 preceding section, to heights inversely proportioned to the di- 
 ameters of the tubes. 
 
 Applying the formula (402), by a similar method, to the case 
 of two parallel solid surfaces, at a small distance from each other, 
 which we shall call J, we would obtain 
 
 A=-j. (408) 
 
 Hence : 
 
 A liquid will be raised or depressed between two parallel 
 plates, at small distances from each other, to heights that are 
 inversely proportioned to the distances between the plates ; 
 and these heights will be the same as those to which the same 
 liquid would rise in a tube of the same material with the plates, 
 whose radius is equal to the distance between the plates. 
 
 Two concentric tubes, or a tube surrounding a solid cylinder, 
 may be considered as a case of this kind ; the liquid will rise be- 
 tween them to half the height it would in a tube whose diameter 
 is equal to their distance. 
 
 Such is the theory of Laplace, which, neglecting certain small 
 quantities, coincides with the observations of former experi- 
 menters, but which by the more accurate experiments of Gay Lus- 
 sac, has been found to be true, even in its full extent ; the refine- 
 ments introduced by tbe latter, having enabled him to detect the 
 small variations in level that had escaped those who preceded 
 him. 
 
 Laplace has also investigated the nature of the surface that 
 
 
Book V.*\ COHESION. 395 
 
 would be assumed by a liquid. We have not spa^ce, nor would 
 it be consistent with an elementary treatise, to enter into his 
 beautiful and complete analysis; we shall, therefore, content 
 ourselves with stating the results of observation which are con- 
 firmed by the theory. 
 
 In tubes of small diameter, the surface of the liquid, whether 
 elevated or depressed, is always spherical ; but in the former case, 
 it is a concave portion, in the latter, a convex one, of a sphere. 
 
 395. Aeriform fluids are also affected by capillary attraction ; 
 and this is, in some cases so intense, as to condense them into a 
 volume considerably less than that which they occupy under or- 
 dinary pressures. The most remarkable instance of this sort is, 
 the absorption of the gases by charcoal. This substance, when 
 recently burnt, takes up within twenty-four hours, according to 
 the experiments of Saussure, of 
 
 Ammonia . . 90 times its volume. 
 Muriatic Acid Gas 85 " 
 
 Sulphurous Acid 65 " 
 
 Sulphuretted Hydrogen 55 " 
 Nitrous Oxide 40 " 
 
 Carbonic Acid . 35 " 
 
 OlefiantGas . 35 " 
 Oxygen . 9.25 
 
 Nitrogen . . 7.5 " 
 
 Hydrogen . . 1.75 " 
 
 Solutions also, generally ranked among chemical phenomena, 
 are unquestionably due to the mechanical action of the particles 
 of the solvent upon those of the solid ; and this attraction is op- 
 posed by the attraction of aggregation that the particles of the 
 latter have for each other. The mixture of liquids, particularly 
 when attended with the diminution of bulk called Concentration, 
 may be also. included among the cases of cohesive attraction. 
 

 
 
 
; rAi, 
 
 BOOK VI. 
 
 OF THE MOTION OF FLUIDS. 
 
 AS'p&Wno G;J/ ' 
 
 CHAPTER I. 
 
 THEORY OF THE MOTION OF LIQUIDS. 
 
 396. We may, by the application of the principle of D'Alem- 
 bert to the equations of the equilibrium of fluids, deduce the gene- 
 ral equations of their motion. These equations are, however, ex- 
 tremely complicated, and are incapable of complete integration. 
 It has, therefore, been more usual to proceed by means of an hy- 
 pothesis, that appears at first sight to approximate to the truth. 
 This hypothesis considers the fluid to be divided into a number 
 of horizontal layers or strata, and that the particles in each of 
 these separate layers have a common velocity. Each of the 
 layers then would continue parallel to itself, and is composed of 
 the self-same particles throughout the whole duration of its mo- 
 tion ; and any given layer will descend and occupy the place of 
 that which is immediately beneath it, and so in succession. 
 
 If this hypothesis be applied to the case of a liquid contained 
 in a vessel of irregular figure, it might be demonstrated, that in 
 moving through it, each different layer will have a velocity in- 
 versely proportioned to the area of the section of the vessel at 
 which this layer is situated. 
 
 This hypothesis is, however, in many cases, at utter va- 
 riance with what is observed in practice. When, for instance, 
 a liquid is placed in a prismatic vessel that is permitted to dis- 
 charge itself through an orifice in the bottom, the surface ceases 
 at once to be level, being depressed immediately above the ori- 
 fice. The surface, therefore, becomes concave, and the particles 
 that compose it, instead of tending to descend in a horizontal layer, 
 appear to move, as it were, upon an inclined plane, to the point 
 that is most depressed, and thence to descend vertically to the 
 orifice. Not only does this tendency to the vertical line appear 
 
398 THEORY OF THE [Book V{. 
 
 at the surface, but it is also manifested in the inferior strata : thus. 
 if a liquid of less density be poured upon the surface of the liquid 
 that is first introduced into the vessel, it speedily joins the de- 
 scending current and passes out; but it does not pass out unmix- 
 ed, for the two liquids are intimately mingled, until the whole of 
 the lighter be discharged. This tendency in the lower strata, 
 towards the column that is immediately above the orifice, is also 
 manifested by placing in the vessel powders of a density equal to 
 that of the liquid ; these will be seen to move towards the orifice, 
 in curves of various degrees of convergence, and will unite them- 
 selves with the effluent stream. 
 
 397. These phenomena may be accounted for, by supposing 
 that each particle of the liquid descends towards Ihe orifice ex- 
 actly as if it were unconnected with the surrounding mass. It 
 would therefore acquire an uniformly accelerated velocity ; the co- 
 lumn would be broken, and spaces left between the particles, to- 
 wards which the pressure of the adjacent liquid would impel other 
 particles, that would thus join in the current, and occupy the 
 void spaces of the column. 
 
 If the orifice be pierced in the side of the vessel, the particles 
 of liquid will still move towards it like falling bodies, but will 
 describe a curve instead of a vertical line. If the orifice be 
 made in a part of the vessel that will permit it to be directed up- 
 wards, the particles will again reach it, under circumstances 
 similar to those which are found in bodies moving, under the ac- 
 tion of gravity, upon curved surfaces whose tangents make with 
 a horizontal line, the same angle that the direction of the orifice 
 makes with the horizontal plane. 
 
 When the motion begins, the particles immediately in contact 
 with the orrfice, move from a state of rest ; and those that lie be- 
 tween them and the surface cannot be accelerated without accele- 
 rating those beneath them ; thus a resistance will be opposed to 
 the descent that will for a time prevent the particles that proceed 
 from the surface of the liquid, from acquiring the velocity a falling 
 solid body would attain in passing through the same space. The 
 time for which this resistance will produce an appreciable effect is 
 but short, for so soon as the first particle that issues shall have 
 fallen through a space equal to the altitude of the fluid above the 
 orifice, its velocity will become equal to that of the particles pro- 
 ceeding from the surface, would acquire at the orifice ; and it will 
 no longer retard the column that follows it. 
 
 If the fluid move in a vessel of variable section, its velocity does 
 not vary with the area, but there is a column or vein that moves 
 in it, precisely as if it were in a, prismatic vessel, while in the 
 parts whose areas are greatest, eddies are formed. The nature 
 
- 
 
 Book J 7 /.] MOTION OF LIQUIDS. 399 
 
 and character of these eddies will be more particularly considered 
 hereafter. 
 
 The vein that moves according to the law of gravity, will be 
 resisted by the viscidity of the neighbouring particles of the li- 
 quid ; it will also be retarded, in consequence of the particles 
 that join it in its course having a less velocity than those which 
 proceed from a higher level. The former will receive a part of 
 the motion of the latter, and the whole will move forward with a 
 common velocity. If the vein nearly fill up the vessel in which 
 it is moving, it will meet with a resistance analagous to friction 
 from the sides, and will be influenced by the attraction of cohe- 
 sion. 
 
 If these circumstances be left out of account, we may consider 
 the velocity of any part of a vein of a gravitating liquid in mo- 
 tion, to be such as would be due to the height of the surface of 
 the liquid, above the point whose motion is considered. In some 
 cases, the circumstances to which we have referred are of no mo- 
 ment, and may safely be neglected ; in others, they affect the ve- 
 locity in a high degree, and even render constant that motion 
 which would otherwise be uniformly accelerated. 
 
 It therefore becomes necessary to distinguish different cases 
 of the motion of liquids. The more important of these cases are 
 five in number : viz., 
 
 (1.) The motion of liquids that issue from orifices pierced in 
 thin plates. 
 
 (2.) The motion of liquids through orifices cut in thick plates, 
 or through short tubes adapted to orifices. 
 
 (3.) The motion of liquids in long tubes or pipes. 
 
 (4.) The motion of liquids in open channels. 
 
 (5.) The motion of liquids over the edges of the lower portion 
 of the irregular sides of the vessel or basin that contains them. 
 
 We shall take up the consideration of these several cases in the 
 order in which they have been mentioned. 
 
400 . MOTION OF [Book VI. 
 
 CHAPTER II. 
 
 OF THE MOTION OF LIQUIDS THROUGH ORIFICES PIERCED IN THIN 
 
 PLATES. 
 
 398. If we abstract the circumstances of which we have spo- 
 ken in the close of the last chapter, a liquid, on reaching an orifice, 
 will have a velocity due to the height of the level of the liquid 
 above the orifice. To represent this in a formula : 
 
 Let be the velocity with which the liquid issues ; h the verti- 
 cal height of the surface of the liquid above the orifice ; g the 
 measure of the force of gravity ; then by (61), 
 
 (409) 
 
 From this formula a variety of consequences immediately fol- 
 low. 
 
 (1.) If the vessel be kept constantly full, the velocities of the 
 effluent fluids, from any orifice given in position, is constant. 
 
 (2.) From orifices pierced at different heights in the side of a 
 vessel kept constantly full, the velocities are as the square roots 
 of the depths of the orifices beneath the level surface of the 
 liquid. 
 
 (3.) If the vessel be permitted to empty itself, the velocity 
 with which the liquid will issue from a given aperture is equally 
 retarded. 
 
 (4.) As none of the circumstances of which we have spoken 
 will affect the area of the column of liquid, discharged from a 
 given orifice, the quantities discharged in the elements of the 
 time will be directly proportioned to the velocities. 
 
 Therefore, as in a vessel permitted to empty itself, the velocities 
 are uniformly retarded, it is obvious from 49, that twice as 
 much liquid should flow from a given orifice, in the unit of time, 
 when the vessel is kept constantly full, as should flow from the 
 same orifice in the same time, when the vessel is permitted to 
 empty itself; and so, when the vessel is permitted to empty it- 
 self, it should occupy twice as much time to discharge a given 
 quantity, as would suffice for an equal discharge from a vessel 
 kept constantly full. 
 
 (5.) In emptying a vessel through an orifice in its bottom, the 
 quantities discharged in equal times should decrease as the series 
 of odd numbers. 
 

 Book VI ^ 
 
 LIQUIDS. 
 
 401 
 
 (6.) When a fluid issues from a vessel in a vertical direction, 
 it will rise or fall in a vertical line ; and if it be directed upwards, 
 as it has an initial velocity due to the height of the surface of the 
 liquid above the orifice, it should rise, if we abstract from the re- 
 sistances it meets, to the level of that surface. The velocity in 
 the jet will be uniformly retarded. 
 
 (7.) If a liquid spout from an orifice in any other than a verti- 
 cal direction, the joint action of its effluent motion, with a velo- 
 city due to the depth, and of the force of gravity, would, if no 
 other forces acted, cause it to describe a parabola, whose directrix 
 will lie in the horizontal plane, coinciding with the surface of 
 the liquid, ( 53). 
 
 (8.) If different orifices be pierced in a horizontal direction, in 
 the vertical sides of a vessel, which is kept constantly filled with 
 a liquid, the effluent velocities of each vein of liquid will vary, 
 and each will have a different distance to descend before it reaches 
 the horizontal plane that coincides with the bottom of the ves- 
 sel. Hence the curves described in each case will be different. 
 
 The comparative distances to which the several jets will pass 
 over this horizontal plane may be thus investigated : 
 
 Let AB be the side of a prismatic vessel, at the point D, in 
 which an orifice is pierced, whence a fluid kept constantly at 
 the level of the point A issues in a horizontal direction. Bisect 
 AB in C, and around C describe a semicircle ; draw the ordinate 
 DE through the point D. 
 
 51 
 
 ! 
 
402 MOTION OF [Book VI. 
 
 Let 
 
 DB=a, 
 DE=*. 
 
 The liquid, in issuing from the point d, will have a velocity due 
 to the height /i, and equal to 
 
 this velocity would carry it with uniform velocity through twice 
 the distance /i, in the same space of time that it has taken to ac- 
 quire that velocity. But so soon as it leaves the orifice, it has its 
 direction changed and describes a parabola. Under the action 
 of this deflecting force, it will reach the horizontal plane BF, in 
 the same time that it would have fallen through the height DB, 
 if it were influenced by the force of gravity alone. The times of 
 describing AD, and DB, are respectively 
 
 2h 2a 
 
 V~, and V-> 
 
 in the first of these times, the projectile force would carry it with 
 uniform motion through the space 2ft, and in the second, through 
 the distance at which the jet of fluid strikes the horizontal plane, 
 which we shall call d. As the spaces are as the times, we have 
 the analogy 
 
 2h 2a 
 V : V ::2h:d, 
 
 o 6 
 
 whence we obtain for the value o 
 
 The distance then from the perpendicular side of the vessel at 
 which the parabolic jet strikes the horizontal plane on which the 
 vessel stands, is equal to twice the line ordinately applied to a se- 
 micircle, of which the vertical depth of the fluid is the diameter, 
 From the centre of the circle, or half the height of the liquid, 
 the horizontal range will be the greatest ; and at equal distances 
 above or below this point, the horizontal ranges are equal. 
 
 399. When a liquid issues from an orifice with a given velo- 
 city, whether it be such as is determined from the foregoing ab- 
 stract theory, or modified by physical circumstances, it might be 
 at first sight concluded that the quantity discharged in the linit 
 of time, might be determined by multiplying the area of the ori- 
 fice by the velocity. This, however, is not the case. The liquid 
 does not issue in a prismatic shape, and hence its quantity is not 
 measured by the contents of a prism of which the orifice is the 
 base. On the contrary, the vein of liquid is obviously contracted 
 
Book VL] ' LIQUIBS. 403 
 
 soon after it issues forth, and again spreads out to dimensions 
 larger than those of the orifice. That this ought to be the case, 
 will be understood from reference to the foregoing theory. For, 
 as the particles move from the lower layers of the liquid to join 
 the vein directed towards the orifice, they have a motion oblique 
 to that of the general current ; but from the particles that compose 
 it, they receive a change of direction, and at the same time re-act 
 upon them. Thus the particles that issue from the edges of the 
 orifice, will have a direction inclined to the axis of the jet, and 
 the stream, of which they form a part, must contract in dimen- 
 sions. As these directions will cross each other, some of the par- 
 ticles will, below the point to which they converge, be forced 
 outward from the axis. Thus the shape of the jet will be one 
 formed of two truncated conoidal frusta; one of these will have 
 the orifice for its base, and a definite altitude ; while the other 
 will have the smaller base of the former for its lesser base, and 
 an unlimited altitude. The investigation by analytic means of 
 the exact figure of these conoidal frusta, were it practicable, 
 would yet be attended with no valuable results ; for the contrac- 
 tion in the vein is connected with the change in velocity grow- 
 ing out of the viscidity of the liquid, and the mutual action of its 
 particles ; therefore, the separate effects of these tw>o different 
 actions cannot be distinguished in experiment. We, in conse- 
 quence, consider that the true velocity is given by the principles 
 of the preceding section, and that the whole diminution in the 
 quantity discharged, is due to the contraction of the vein. 
 
 The contraction that occurs in the form of a jet of fluid, issuing 
 from an orifice, is too apparent to have escaped notice, even at 
 an early period. Newton, however, was the first who attempted 
 to ascertain the amount of this contraction. In this he was not 
 perfectly successful, but conceived that he had found it to be in 
 the ratio 5 : 7, or of \/l : \/2. 
 
 If a be the area of the orifice, q the quantity discharged, and 
 we use the same notation as before, 
 
 and upon the hypothesis that the quantity is found by multiplying 
 the area of the orifice by the velocity, 
 
 q=aV2gh. (410) 
 
 This is called the Theoretic Discharge. If it be reduced in the 
 ratio given by Newton, we have 
 
 q=aVgh. (411) 
 
 As the change in this formula is made in the second part of it 
 which represents the velocity, a false inference has been drawn 
 
404 MOTION OP [Book VI. 
 
 by some writers, who, forgetting the circumstance of the con- 
 traction of the vein, have stated that the velocity itself is dimin- 
 ished, and becomes no more than is due to half the height of the 
 fluid above the orifice. 'This, however, is an obvious error; the 
 velocity is but little affected when the liquid issues from an ori- 
 fice pierced in a thin plate, and the diminution in the actual dis- 
 charge compared with the theoretic, is principally due to the 
 contraction of the vein. 
 
 The best experiments on the phenomena of liquids issuing 
 from orifices pierced in thin plates, are those of Bossut. From 
 these it appears, 
 
 ( 1 ). That, except when a vessel is nearly exhausted, no sensible 
 error can arise from considering the velocities as due to the height 
 of the level surface of the liquid above the orifice. 
 
 The narrowest part of a jet of liquid, issuing from an orifice, 
 is called the Vena C on tract a ; this is situated at a distance from 
 the orifice, when of a circular figure, that is equal to its radius. 
 
 (2). The figure of a vertical jet, lying between a circular ori- 
 fice and the vena contracta, is nearly a conic frustum, whose two 
 bases have to each other the ratios of 62 : 100, or nearly as 5 : 8, 
 instead of 5 : 7, as stated by Newton. 
 
 The figure of the jet, beyond the vena contracta, is also sensi- 
 bly a conic frustum, the angle of whose vertex is 32. 
 
 In orifices of any other figure, the same ratio is nearly true be- 
 tween the dimensions of the orifice and the section of the liquid 
 at the vena contracta ; but the figure of the jet is not pyramid- 
 ical ; at a small distance from the orifice, if of a rectangular shape, 
 it assumes the form of a cross, whose arms He in the direction of 
 the diagonals of the orifice; beyond this, the section again be- 
 comes rectangular, with diagonals parallel to the sides of the ori- 
 fice, and this figure is retained in the subsequent enlargement of 
 the vein. 
 
 Analogous changes of figure take place in the section of the 
 vein of liquid, when the orifice has the figure of a triangle or a 
 polygon. 
 
 (3). So long as the vessel continues to hold a column of liquid at 
 a considerable height above the orifice, the actual discharges from 
 orifices of any figure whatsoever, are to the theoretic nearly in 
 the ratio ffa ; and the velocities and quantities are nearly pro- 
 portioned to the square roots of the depth of the liquid. 
 
 (4). Small orifices discharge rather less than the reduced quan- 
 tity, large orifices rather more ; and of orifices of equal area and 
 unequal circumferences, those with the smallest circumference 
 discharge the greatest quantity. 
 
Book VL] LIQUIDS. 405 
 
 400. When the liquid spouts vertically upwards, there is a de- 
 viation from the theoretic height that can be easily perceived. 
 That this should be the case, will be obvious when we consider 
 that the particles of liquids are retarded by the column that has 
 preceded them, the particles of which are moving with a con- 
 tinually diminishing velocity ; and that the particles, after reach- 
 ing; their utmost height, tend to return in a vertical direction: a 
 part of the force of the ascending column must therefore be ap- 
 plied to force them to one side. In consequence of the latter 
 circumstance, it has been found that a jet of liquid, when slightly 
 inclined, rises higher than if pointed vertically upwards. When 
 the original velocity is due to a great height, and is, in conse- 
 quence, large, the resistance of the air becomes a powerful re- 
 tarding force, and hence creates a limit beyond which no head of 
 water, however great, can cause a vertical jet of liquid to rise. 
 The height to which a liquid rises vertically upwards, and which 
 is, therefore, always lower than the level of the liquid in the 
 vessel whence it issues, is* called the Effective Head. 
 
 It has been ascertained by the experiments of Mariotte, that 
 a head of five French feet and an inch, produces a vertical jet of 
 five feet. 
 
 If H, and H', be the actual heads of two masses of water that 
 form vertical jets ; fe, and A', the effective hea'ds, the experiments 
 of Mariotte, give the following relation between them. 
 H h h? 
 
 and 
 
 H-(H' /O^+fc, (413) 
 or taking the above data, where 
 
 , . ' H '= 5 T2- , -" ! ;aS >, 
 
 h'=5 
 
 A; ' -(414) 
 
 whence the real head that will produce a jet of any given height, 
 can be estimated. 
 
 401. If the velocity due to the effective head be employed in 
 the parabolic theory, instead of the actual velocity, the results 
 will be nearly identical with those that actually take place. 
 Hence, from the theory of projectiles, 250, we have for the 
 height to which an inclined jet will rise, 
 
 hsin. 2i; (415) 
 
406 MOTION or [Book VI. 
 
 and for the horizontal distance to which it will reach, 
 
 ZJisin. 2i. (416) 
 
 402. The rules of 399, are only found to hold good in prac- 
 tice when the height of the column of liquid in the vessel is large, 
 compared with the area of the orifice ; as the height lessens and 
 the vessel becomes nearly empty, the velocity of discharge is di- 
 minished below that due to the height ; and the contraction of 
 the vein increases. This grows partly out of a rotary motion 
 that often takes place in the fluid in the vessel, and causes a cen- 
 trifugal force that lessens the action of gravitation. The parti- 
 cles, therefore, that reach the orifice in a vertical direction, have 
 a less velocity than they would otherwise acquire; and those 
 whose direction is oblique, are less powerfully acted upon, and 
 continue their oblique course longer. 
 
 The formation of the vortex, whose rotary motion produces 
 these effects, may be thus explained : 
 
 The particles that enter the vein directed towards the orifice, 
 have a motion that may be resolved into two, one in the vertical, 
 the other in a horizontal direction. The latter is obviously a 
 centripetal force. If a third force act upon any one of the par- 
 ticles, in any other direction than that of these two components, 
 it will cause the particle to deviate in a horizontal direction ; 
 for one of its component* will be horizontal, and thus the motion 
 in the direction of the radius will become circular, if the disturb- 
 ing force be of sufficient intensity, or spiral, if it be less intense. 
 In the latter case, it may be considered as taking place in circles, 
 successively decreasing in magnitude, and the laws of circular 
 motion of 64, will be applicable. This disturbing force may 
 proceed from extrinsic causes, but it may arise also from irregu- 
 larity in the figure of the vessel, or from the position of the ori- 
 fice being in any other point than the centre of magnitude of a 
 base of regular figure; for in either of these cases, the particles 
 that join the vein at a given level, will reach it with different 
 inclinations and velocities, and will therefore effect each other's 
 motions. 
 
 If we call the velocity of rotation of any particle is its distance 
 from the axis of the vein, r, and the time of a revolution, /, the 
 velocity of rotation will be constant ; and will, therefore, in the 
 successively decreasing circles, be inversely proportioned to the 
 radii ; and the areas of the circles will be as the times of de- 
 scribing them, or the times will be directly as the squares of the 
 radii. The centrifugal forcp will therefore be, 64, inversely as 
 the cubes of the radii, or distances from the axis ; it may, there- 
 fore, in approaching the vein, give the particle a force that will 
 enable it to resist the action of the descending particles of the 
 
Book Vl.~\ LIQVIDS. 407 
 
 vein that it tends to join. The greater the height whence the 
 latter have descended, the greater will be their action to destroy 
 the centrifugal force, which will be constant ; and hence, the 
 cavity that will be formed by the latter, on the surface of the liquid, 
 will increase as the depth of liquid in the vessel diminishes. The 
 figure of the section of the cavity has been investigated on these 
 principles by Venturi, and has been shown to be a curve convex 
 towards the axis of the vein. The curve is one of the third 
 order. 
 
408 THE DISCHARGE [Book VI. 
 
 r - CHAPTER HI. 
 
 OF THE DISCHARGE OF LIQUIDS THROUGH SHORT PIPES OR 
 ADJUTAGES. 
 
 403. When a liquid, instead of flowing through an orifice pla- 
 ced in a thin plate, issues by a short pipe ; and if the pipe and 
 the liquid be of such materials as will act mutually upon each 
 other by the attraction of cohesion, the edges of the orifice will 
 exert a force upon the filaments of liquid in contact with them ; 
 the consequence of this should be their deviation from their ori- 
 ginal direction towards the vertical, and a consequent increase in 
 the dimensions of the vena contracta. 
 
 This theory is fully confirmed by experiment, whence it appears, 
 that the adaptation of pipes, to the orifices whence liquids issue, 
 increases the quantities discharged. Such additional tubes are 
 called Adjutages. 
 
 404. Adjutages of different forms, have different degrees of ad- 
 vantage in this respect, that can only be determined by experi- 
 ment. 
 
 When a cylindrical tube is adapted to a circular orifice, the dis- 
 charge is increased, until the length amount to four times its di- 
 ameter; after this limit, it again decreases in consequence of fric- 
 tion in the tube. 
 
 The increase in the discharge is in the ratio of 82 : 62. 
 
 The same increase still takes place, if the tube be contracted in 
 the form of a frustum of a cone, whose altitude is at the distance 
 of half the radius from the orifice, and whose lesser base has an area 
 of T 6 ^, of the area of the orifice ; and again spread out to its ori- 
 ginal size, by the adaptation of another conic frustum. 
 
 If the first cone be merely inserted in a cylindric tube, the in- 
 crease is only in the ratio of 77 : 62. 
 
 If the tube have the form just described of two truncated cones, 
 adapted to each other at their lesser bases ; the greater bases ha- 
 ving the same area with the orifice, the lesser that of the vena con- 
 tracta, or T 6 T 2 ; the cone next the orifice an altitude equal to its 
 radius, the other cone an angle at the vertex of 36, the discharge 
 is increased in the ratio of 92 : 62. If the latter cone be length- 
 ened until the area of its greater base becomes one half more than 
 that of the orifice, the discharge is increased in the ratio of 
 94 : 62. 
 
Book VI.] OF LIQUIDS. 409 
 
 Thus it appears that in an adjutage, a part of which is contracted 
 to the area of the vena contracta, or to no more than .62 of the 
 orifice, the actual discharge may approach within .06 of the theo- 
 retic. 
 
 If then there exists a right to draw water through a pipe of 
 given dimensions, the quantity determined by theory may be 
 increased in the ratio of 132 : 100, by merely uniting the pipe 
 to the receiver by a truncated cone, the area of whose lesser base 
 is that of the pipe; whose larger base has an area that bears to 
 theless, the ratio of 100 : 62 ; and whose altitude is half the diame- 
 ter of the greater base. A near approach is obtained to this form, 
 by making the diameters of the base as 10 : 8, and the height of 
 the cone fths of the diameter of the tube. 
 
 A still fartner increase in the ratio of 150 : 100 may be ob- 
 tained by making the tube spread out at its place of discharge, in 
 a conical form, at an angle of 1-6 with its axis ; and this increase 
 will be obtained, even if a cylindrical tube of considerable length 
 intervene between the two cones. 
 
 It has been stated by some writers that this last increase wirl-. 
 take place, whatever be the length of the intervening cylindrical 
 tube. But this is not the case beyond that limit at which the 
 velocity of the water in the pipe becomes constant, or when the 
 retarding and accelerating forces counteract each other. 
 
 Similar results take place in channels of forms other than cyl- 
 indric; in them all, an increase of the liquid they will carry, may 
 be effected by giving the tubes the forms the liquid would assume 
 under the mutual action of its particles. 
 
 405. These results, in the case of tubes of circular section, are 
 very remarkable, and are worthy of exhibition in a tabular form. 
 
 TABLE 
 
 Of the quantities of a liquid discharged in equal times from adjutages of 
 different figures. 
 
 From an orifice in a thin plate, . . . 0.62 
 
 Through a short cylindrical tube, whose area is 
 
 the same as that of the orifice, . . 0.82 
 
 Through a tube contracted at the distance of half 
 
 its diameter from the. orifice to an area of .62, 0.82 
 
 Through a tube of the figure of two truncated 
 cones, whose least base is .62, and whose 
 two greater bases are equal to the orifice, . 0.92 
 
 Through a tube formed of similar cones, whose 
 length is increased until the area of its place 
 of discharge, is one half greater than that of 
 
 the orifice, 0.94 
 
 52 ^ N 
 
 \ 
 
410 THE DISCHARGE. [Book VL 
 
 Theoretic discharge, . . . . . 1.00 
 Discharge through a given aperture, connected 
 with the reservoir by a truncated cone of 
 which the aperture is the lesser base, . . 1.32 
 Discharge through the same aperture, connected 
 with the reservoir in the same manner, and 
 which has another truncated tube adapted to 
 it, the angle of whose vertex is 32, . . V _ 1-50 
 
/ 
 
 . 
 
 Book VI] OF LIQUIDS. 41 I 
 
 ^A ten Cf':- rro i - 
 
 CHAPTER IV. 
 
 OF THE MOTION OP WATER IN PIPES. 
 
 406. When the tube that is adapted to an orifice, by which 
 water flows from a reservoir, is of a length greater than four times 
 the diameter of the orifice, the velocity is retarded ; this retarda- 
 tion is caused by a resistance, arising from the friction of the 
 liquid against the sides of the tube. Under the action of this re- 
 sistance, the velocity of the liquid, which at first varies with the 
 square root of its depth, will finally become constant. 
 
 The law which this resistance follows, has been determined by 
 experiment. It has been thus found to be a function of the velo- 
 city, and of such a nature that it may be conveniently divided 
 into two parts ; the first of which is directly as the velocity ; the 
 second directly as its square. The resistance also varies with the 
 surface, by which the liquid is in contact with the channel in 
 which it runs. 
 
 To express this law analytically : 
 
 Let v be the velocity ; 
 
 a, and /?, constant co-efficients, determine)! by experiment ; 
 
 s the surface ; 
 
 The friction,/, will be 
 
 /=(o*+j8) (417) 
 
 When a liquid moves in a tube of uniform diameter with a con- 
 stant velocity, the current fills the whole of the tube ; the parti- 
 cles of the liquid in immediate contact with the tube, will be most 
 resisted by friction ; but in consequence of the viscidity common 
 to all liquids, they will receive motion from the neighbouring par- 
 ticles, and will in turn retard them ; hence, although the velocity 
 of all the particles situated in a given transverse section of the 
 tube is not constant, it may, without any sensible error, be con- 
 sidered as such. 
 
 Let us then suppose that the space occupied by a liquid, that 
 has acquired an uniform velocity in a tube, is divided into a great 
 number of layers, infinitely thin, by means of planes perpendicu- 
 lar to the axis of the tube. Let A, B, C, D, be one of the layers 
 
 into which the fluid is divided. The 
 motion being uniform, the resultant 
 of all the forces that act upon it is 
 =0, or, 39, they are in equilibrio. 
 Among the forces that accelerate 
 are the fluid pressures ; if that upon 
 the unit of surface of the face AB, 
 
MOTION OF [Book VI. 
 
 be p, that on the unit of the face, C, D, will be 
 
 p+dp. 
 
 The forces which oppose the motion of the liquid will be : 
 (1). The difference of the fluid pressures on the opposite sur- 
 faces of the layer. If a be the area of these surfaces, this resis- 
 tance will be 
 
 a dp. 
 
 (2). The friction. This as has been seen, (417) will be rep- 
 resented by 
 
 but in a thin layer, we may substitute the circumference, c, of the 
 pipe, multiplied by the differential of the length /, for s, and this 
 resistance will become 
 
 c(a.v+(3v 2 )dl. (418) 
 
 On the other hand, the force which tends to move the liquid in 
 the tube, is that component of its weight which lies in the direction 
 of the axis of the tube. 
 
 Let i be the inclination of this axis to the vertical ; the whole 
 weight of the layer will be 
 
 a dig ; - 
 and its component in the direction of the axis of the tube, 
 
 a dig cos. . 
 If the difference of level of the points, A and C, be dz, we have 
 
 dz=dicos.i, (419) 
 
 therefore 
 
 a dig cos. i=agdz . (420) 
 
 When the motion is constant, 39, this force must be in equi- 
 librio with the two first, or 
 
 a gdz=a dp+c (ar+/3r 2 ) dl . (421) 
 
 Integrating, and introducing for the constant quantity, the initial 
 pressure at the origin of the tube, P, we have 
 
 a gz=a(pP)+c(av+(3v 2 ) ; (422) 
 
 which, when / becomes equal to the length of the tube, becomes, 
 calling the pressure at its place of discharge, P', 
 
 agz=a(P f P)+c(a.v+(3v 2 )l ; (423) 
 
 whence we obtain 
 
 . (424) 
 
 If the diameter of the tube be D, 
 
 and 
 
 P--/P' p\ 
 
 ar+/3v a =| D -_ ' . (425) 
 
-*V.'v 
 
 Book VI.} WATER IN PIPES. 413 
 
 If H be the column of liquid that presses on the origin of the 
 tube, and H', that which presses on its place of discharge, we 
 have 
 
 P=gH, P'=H'; 
 
 and substituting we obtain 
 
 _ T37 I-TT 
 
 . (426) 
 
 If the origin of the ordinate, z, be taken at the surface of the col- 
 umn, H, 
 
 z H'-fH=H H'; (427) 
 
 and if the discharge take place in the open air, 
 
 z H'+H=H ; 
 in which case 
 
 at>+/3i> 2 =I)g3-. (428) 
 
 By the researches of Prony, who compared fifty different ex- 
 periments made on tubes, for conveying water, 
 
 -=0.00017, 
 g 
 
 fi * 
 
 / -=0.003416, 
 
 S . , * :$k 
 
 therefore, the quantities being estimated in irietres, 
 
 1 TT 
 
 0.00017 v-f 0.003416 v 2 =-Vg.-j.- (429) 
 
 Whence we obtain, by taking a value for v, deduced in a particu- 
 lar case from experiment, 
 
 / H\ 
 
 0=_0.024S829-r- V (0.000619159+717.885 D 2 -yj (430) 
 
 which becomes, when the quantities are estimated in English feet, 
 
 DHx 
 
 0.02375+8201.6 ) . (431) 
 
 The co-efficients a and /3, become, if applied to the English 
 foot as the unit, 
 
 a=0.00017, 
 /3=0.000104; 
 and neglecting the term that involves the velocity simply ; which, 
 
414 MOTION op [Book VI. 
 
 as the co-efficient is small, may be done, if the velocity be not 
 great, without any sensible error, we have 
 
 1 TT 
 
 0.000104 7> 3 =- D p (432) 
 
 and 
 
 H 
 
 D 9 =2404 Dy ; (433) 
 
 =46.82 V (D y) . (434) 
 
 If the diameter of the tube be estimated in English inches, as 
 is most usually the case, and the other quantities in feet, 
 
 TT 
 
 u 3 =200 D y ; (435) 
 
 and 
 
 t> = 14.142x/(Dy). (436) 
 
 If Q represent the quantity discharged by the tube in the* unit 
 of time 
 
 Q=* -4-. (437) 
 
 and 
 
 4Q 
 t>=^p; (438) 
 
 substituting this in the equation (428), and neglecting the first 
 power of v, we have 
 
 16 Q 3 1 H 
 
 and if we make 
 
 t _64/3 
 
 6 ~ * 
 we have 
 
 I 
 and 
 
 calculating the numeric value, we obtain 
 
 6=0.0000688, 
 and 
 
 Q=38.13>/(D 5 T ), (441) 
 
 in which all the lineal dimensions are in English feet, and the 
 quantity in cubic feet. 
 
 

 Book VL] WATER IN PIPES. 415 
 
 For the quantity in cubic feet, when the diameter D of the 
 tube is given in inches, we have , 
 
 / Wx 
 
 (442) 
 
 For the quantity in English statute gallons, D being in inches 
 as before, 
 
 Q= . 48635 N/ (D 5 y . (443) 
 
 For the quantity in standard liquid gallons of the Slate of New- 
 York. 
 
 TTv 
 
 (444) 
 
 The formulas for the value of D, mfien Q, H, and /, are given, 
 can be easily obtained from the foregoing, the fundamental ex- 
 pression being, 
 
 i J\. 
 
 (445) 
 
 The formula (434), is similar to that of Prony, for metres, 
 which is 
 
 H\ 
 
 T) < 446 ) 
 
 but which reduced to English measure, would be 
 
 Hv 
 T)' (447) 
 
 the co-efficient being 48.5, instead of 46.82, as we have made it. 
 
 Prony's formula, however, appears to be in excess, except 
 when the velocity is considerable ; that of (434) is probably in 
 defect, except at small velocities. 
 
 The formula of Du Buat, who led the way in these investiga- 
 tions, is 
 
 n n //I /% * \ (448.) 
 
 In which, 
 
 V is the velocity in English inches ; 
 d half the radius of the pipe ; 
 
 H 
 
 * the mean slope of the pipe which is equivalent to -y of Pro- 
 ny's formula. 
 
 log. The hyperbolic logarithm of the quantity to which it is 
 prefixed. 
 
 The computation by this formula may be facilitated by means 
 of subsidiary tables, the best set of which are to be found in the 
 Edinburgh Cyclopedia, article, Hydrodynamics. 
 
 :{.jr!' m&* ;' W^^#$^ 
 
 j ^-M'^-^8'1 
 
 4 '. .4. , 
 
. 
 
 416 MOTION OP [Book VI. 
 
 The^ formula of Eytelwein, is in English feet. 
 DH 
 50 
 
 407. The foregoing investigation is only applicable to the case 
 of a pipe of uniform slope, lying in the same vertical plane, and 
 of a constant section. If the diameter be not constant, the dis- 
 charge will obviously be due to the area of its least section ; but 
 will be affected by the same causes that influence the discharge 
 of fluids through orifices and adjutages. The velocity may be 
 calculated as above, and be multiplied by the smallest area 
 of the pipe. The amount thus obtained, must then be increased 
 or diminished by the use of the co-efficient, which may be ob- 
 tained from the table in & 405, according to the nature and form 
 of the contraction. ^* 
 
 The quantity D, used in the calculation of the velocity, must 
 be the general diameter of the pipe, for the friction will obviously 
 be principally due to it, or nearly so, and not to those portions 
 that are contracted. 
 
 The necessity of continuing a pipe of uniform bore, from the 
 place where it receives the liquid it is to carry, to the place where 
 it is to discharge, is therefore manifest; but at its two extremi- 
 ties it should have conical adjutages. 
 
 408. Pipes that convey water, are liable to two species of ob- 
 struction, that tend to diminish the effective areas of their section, 
 and thus lessen the quantity they would otherwise discharge. 
 All spring or river water contains gaseous matter : this will often 
 escape and separate itself in consequence of its expansive force; 
 hence lodgments of air may take place in^the higher parts of the 
 pipe, and where the pipe, after having risen, is bent, and again 
 descends. A self-acting apparatus has been planned to permit 
 the escape of such lodgments of air. It consists of a valve of the 
 form of a sphere, that is placed in a vertical cylinder, adapted to 
 the upper bends of the pipe ; in the cap that closes this cylinder, 
 a hole is cut for the seat of the valve, and is carefully ground to 
 the figure of a hollow zone, of a sphere of the same radius as the 
 spherical valve. The valve is made of metal, and-is hollow, in 
 order that it may be light enough to be buoyant in water. When 
 the pipe runs full of water, the pressure of the liquid keeps the 
 sphere closely applied to its seat ; but when a lodgment of air 
 takes place, the sphere falls ; the valve is therefore opened, and 
 the air escapes; the water which follows lifts the sphere, and ap- 
 plies it to the seat that has just been described. The only pre- 
 caution in using this is, to take care that the valve seat shall be 
 lower than the level of the water in the reservoir whence the 
 pipe is supplied. 
 
Book VL~\ WATER IN PIPES. 417 
 
 A simple stopcock, that is occasionally opened, will answer 
 the same purpose, but is not self-acting. 
 
 The pipe may be interrupted at the places where the air is 
 likely to lodge and the water discharged into a basin, whence it 
 is again drawn by the prolongation of the pipe. In this case, all 
 the advantage derived from the superior height of the water in 
 the original reservoir is lost. This may be obviated by raising 
 the pipe in this place, by artificial means, to the height due to 
 the velocity of the liquid, which will be as much less than the 
 original head, as is due to the friction. 
 
 Such an arrangement is called a Souterazi. 
 
 It possesses, when applied to very long lines of pipes, an im- 
 portant advantage ; for the water conveyed in them becomes vapid 
 and disagreeable, but will, by exposure to air, recover its quali- 
 ties. 
 
 Deposits of earthy matter often take place in the lower angles 
 of a pipe. These arise from substances that are either mechani- 
 cally mixed, or held in solution in the water. These deposits 
 may be removed by throwing a cork, to which a string is attached, 
 into the pipe. If the space left in the pipe be sufficient to ad- 
 mit its passage, it will carry one end of the string to the place of 
 discharge, and an instrument for cleansing the pipe adapted to 
 the other end, may be drawn through the pipe by means of it. 
 
 Stopcocks may be adapted to the lower angles of the pipe, and 
 opened at proper intervals ; the current they cause will carry 
 with it any earthy matter that has not become indurated. 
 
 Short tubes may be placed beneath the lower angles, communi- 
 cating with the pipe, by means of a smaller vertical pipe. The de- 
 posit will take place in them, instead of the main pipe, and they 
 may be removed as often as necessary, and cleansed. 
 
 409. If the pipe be not of uniform slope, or do not lie wholly 
 in the same vertical plane, the water moving in it will experience 
 a resistance at the angles. The amount of the resistance has been 
 ascertained by the experiments of Du Buat. He found it to be 
 proportioned to the square of the velocity, and to the square of 
 the sine of the deviation of the tube from its original direction, 
 to the number of bends or elbows in the tube. 
 
 For a single angle, therefore, this resistance may be thus ex- 
 pressed : 
 
 R=mv 2 sin. H i, 
 
 and for any number of elbows, 
 
 R=-mu 2 2, sin. s t. (450) 
 
 53 
 
.418 MOTION OF [Book VI. 
 
 The co-efficient, m, as determined by Du Buat, is in French 
 inches, :.-.-5.*i 
 
 =0.000336, 
 
 2998.5 
 in metres, iV ! *- 
 
 m=0.0123 ; 
 and in English feet, 
 
 m=0.039. 
 In the latter case the formula (450) becomes 
 
 R=0.0039t; 2 2.(sin. 2 *) . (451) 
 
 This formula ceases to be true when the angles of deviation 
 exceed 30. 
 
 410. The general formulae, (428) and (440), for the velocity 
 and quantity discharged by a pipe, are only applicable to the case 
 of a single pipe of uniform bore throughout, or where there are 
 a few definite contractions in the course of such a pipe. They 
 are not adapted to the circumstances of lateral tubes that diverge 
 from a main pipe, for the purpose of distributing a liquid to dif- 
 ferent points, as is frequently necessary in the supply of cities 
 with water. Into these the water will enter with a velocity, that 
 is due to its pressure upon the part of the main pipe, to which the 
 lateral tube is adapted. 
 
 Water, as may be inferred from the preceding investigations, 
 moves in a tube in consequence of the pressure upon its origin, 
 and the weight of the particles in the descending branches, and 
 is resisted by the friction against the pipe, and the weight of the 
 particles in the ascending branches. One part of the moving 
 power is, therefore, employed in generating the velocity of the 
 liquid ; another in overcoming friction ; while the third is ex- 
 pended upon the resistance of the columns that act in a direction 
 contrary to the motion. The last is the principal element that 
 determines the pressure on the tube. The main tube may there- 
 fore be considered as a reservoir whence the lateral pipe is with- 
 drawn, and all the circumstances determined upon the principles 
 that we have applied to the main tube, and its reservoir. The 
 pressure on any given point will, theoretically speaking, be due to 
 the difference between the actual height, and that due to the velo- 
 city of the liquid. This pressure cannot be always practically 
 determined with an accuracy sufficient for the purpose. It will 
 be seen, however, in the following investigation, that the dis- 
 charge of the lateral pipes may be determined by means of ex- 
 pressions, into which the pressure does not enter. 
 
 Let Q, be the quantity of water the main pipe would deliver 
 at the point whence the first lateral pipe diverges ; 
 
Book Vl.~\ 
 
 WATER IN PIPES. 
 
 419 
 
 D, the diameter of this pipe ; 
 
 L, its whole length ; 
 
 X X' X" ... X"- 1 , the partial lengths of the main pipe, whose 
 sum is equal to L ; 
 
 Z, the difference of level between the water in the reservoir, 
 and the opening of the first lateral tube ; 
 
 Z' Z" . . . . Z n ~S the difference of level between each two con- 
 secutive branches. 
 
 H' H" H'" .... H", the heights due to the pressures on the 
 points whence the successive branches diverge ; 
 
 q' d' I' z' ) 
 
 q" d" I" z" \ similar elements for each separate branch ; 
 
 qn fa l n z n j 
 
 cy the constant quantity in the formulae (440), to (444), accor- 
 ding to the measure employed ; we have for the several parts of 
 the main pipe, and its branches from (440), 
 
 w 
 
 (6) 
 
 (452) 
 
 r 7" _ W" 
 
 The equations being in all to the number of 2w, or twice as 
 many as there are lateral pipes, 
 
 To eliminate H', H", &c., we first combine the equations (a] 
 
 and (c), (a), (c), and (e), &c. ; we thus obtain 
 
 Q 2 X+(Q q'} 2 X'=c 2 DXZ+Z' H") , (g) 
 
 Q 2 X+(Q f/) 2 X' +(Q q' .... g-7X-' = 
 
 c 2 D 5 (Z+Z'- Z"- 1 H n ) . (h) 
 
 Then by combining the equations (a] and (6), (d} and (g*), &c. 
 we have 
 
 Z z'} , () 
 
 (Jfc) 
 
 [Q 2 X+(Q q'} 2 ^ ____ +(Q q' - - g n 
 c 2 D 5 d n5 (Z+Z' ..... +Z"-' 
 
420 LIQUIDS IN [Book VI. 
 
 From the last equation we may obtain the values d' d", to 
 which are 
 
 5 -^ 
 
 v 
 
 (453) 
 
 cD 5 (Z-r-Z" a") (Q 2 +(Q q'W 
 These two equations are sufficient to determine the law ; and in 
 order that they shall give rational values for the diameters of the 
 lateral pipes, it is necessary that the denominators of the fractions 
 should be positive*. 
 
 "See, "Essai sur les Moytns de conduire deleter et dc distribuer les Eauz, par M. 
 Genieys. 
 
Book VL\ OPEN CHANNELS. 421 
 
 V 
 
 
 
 CHAPTER V. 
 
 OF THE MOTION OF LIQUIDS IN OPEN CHANNELS. 
 
 411. When a liquid issuing from a reservoir enters into an 
 open channel, the general direction of the bed must be inclined 
 downwards, otherwise it would not continue to flow ; and the 
 surface will have a slope from the reservoir towards the place of 
 discharge. Being acted upon by the force of gravity, the liquid 
 will have a tendency to assume, an accelerated velocity. It rarely 
 however happens, and only when the slope is very great, or the 
 length of the channel small, that this acceleration does occur; in 
 some cases the velocity, so far from increasing with the distance 
 from the reservoir, or source, diminishes. In most instances the 
 mean velocity is found for long distances, to be uniform, and to 
 change only with changes in the nature and character of the bed. 
 
 When a stream flows in a channel with uniform mean velocity, 
 it is said to be in train ; this can only occur when no accelerating 
 force acts, or when the sum of the accelerating and retarding 
 causes is =0. 
 
 The circumstances, then, of the uniform motion of a liquid in 
 an open channel of uniform section, may be made the basis of the 
 theory of the motion of liquids in open channels of any figure or 
 variety of dimension whatsoever; and the variations from the 
 simple theory which the change of dimension may produce, can, 
 if necessary, be applied as corrections to the inferences. 
 
 The principal retarding force, to which the motion of liquids 
 in open channels is liable, is the friction upon their beds. This, 
 according to the experiments of Coulomb, will be a function of the 
 velocity and of the surface directly, and be inversely as the area 
 of the section of the stream ; and as in the case of tubes, that part of 
 the resistance that is a function of the velocity, has two terms ; in 
 one of which the first, and in the other, the second power of the 
 velocity are involved. This observation forms the basis of the 
 theory. 
 
 This retarding force may be thus expressed : 
 
 f=g-(av+(3v*). (454a) 
 
 w ' *', - 
 
 In which formula a and /3 are constant co-efficients, determined 
 in some particular case by experiment ; 
 /, the friction ; 
 of, the measure of the gravitating force ; 
 
422 LIQUIDS IN [Book VI. 
 
 a, the length of the perimeter of that surface, which is in contact 
 with the liquid ; 
 
 to, the area of a transverse section. 
 
 Now let 
 
 1= the length of the axis of the channel ; 
 
 1 H= the difference of level between the places where the chan- 
 nel receives and delivers the liquid ; we may obtain by a course 
 of reasoning similar to that in 402. 
 
 to H 
 
 The quantity - , is that which is called the Hydraulic mean 
 s 
 
 depth, or Radius ; this is usually represented by R. The quan- 
 tity , is the slope or inclination of the tube, which is represented 
 
 by I. Using these symbols, we have 
 
 av+f3v 2 =gRI. (454) 
 
 According to the investigations of Prony, the constant numbers 
 applicable to this equation, are, after division by g, when the mea- 
 sures are taken in metres, 
 
 a=0.0004445, 
 /3=0.0003093; 
 or in English feet, 
 
 a=0.0004445, 
 /3=0.0000912; 
 and neglecting at% we have 
 
 .0000912t> 2 =RI ; (455) 
 
 whence 
 
 t> 2 =12000 RI, 
 t?=109.53v/RI. 
 
 Eytelwein makes the numbers for English feet, 
 a=0.000243, 
 ^=0.00001113 ; 
 whence we obtain, again neglecting m, 
 
 v=94.87VRI . (456) 
 
 If we employ the same constant number for /3, that we have 
 made use of in tubes, 402, we have 
 
 v = 93.64v/RI; (457) 
 
 or taking the simpler formula of Prony, as in (447), 
 
 VZ-97VRI . (458) 
 
 By the application of experiment to determine the value of u, 
 
 in a particnlar case, Prony obtains a general formula, applicable 
 
Book F/.] OPEN CHANNELS. 423 
 
 both to tubes and open channels ; this is as follows, the measures 
 being in English feet : 
 
 -y=_0.154+N/ (0.0238-1-32806 G) . (459) 
 
 In this formula, in the case of pipes, 
 
 G^ID^-; (460) 
 
 in the case of open channels, 
 
 G=RI=^.^. '.-,- (461) 
 
 In the former case, therefore, it is identical with (431). 
 
 412. The velocity, v, in the foregoing investigations is the 
 mean velocity, and will not be that of all parts of the stream. 
 Those portions which are nearest to the sides and bottom of the 
 channel, will be directly resisted by them ; and although in con- 
 sequence of the x viscidity, they will derive motion from those 
 more distant, and will retard them, the latter will have the 
 greater velocity. On the other hand, the liquid pressure will 
 tend to accelerate the threads of fluids which lie deepest beneath 
 the surface of the stream. By the combination of these two ac- 
 tions, the greatest velocity in a symmetric open channel will be 
 in its middle, and at a small distance below the surface. Between 
 the greatest velocity at or near the surface, and the mean velocity, 
 there is a constant relation, as has been shown by the experimen- 
 tal researches of Du Buat. The formula derived by him from his 
 experiments is however faulty, and has been converted by Prony, 
 into the following form : 
 
 Let Vj be the mean velocity ; 
 V, the greatest velocity ; 
 
 -*& 
 
 From this, may be derived the following mean value of v, 
 
 v =0.816458 V; (463) 
 
 or, what is in most cases sufficiently accurate, 
 
 0=4 V ' ( 464 ) 
 
 o 
 
 413. When the channels in which streams run, are neither of 
 uniform section throughout, nor directed in a straight line, varia- 
 tions from the above theoretic inferences take place. 
 
 Bends and elbows in the stream obstruct its course, and dimi- 
 nish the discharge ; the current will in such cases be directed from 
 one of the points towards the next, which it will tend to wear 
 away, and in the bays that intervene, eddies take place. 
 
 The formation of eddies may be thus explained. In conse- 
 
424 MOTION OP LIQUIDS. [Book VI. 
 
 quence of the viscidity of liquids, a current that is in motion, 
 tends to carry with it the neighbouring masses of a similar fluid 
 nature ; thus, if a current be passing through a space greater than 
 it is capable of occupying, without changing its velocity, the 
 lateral fluid will join the stream, and tend to increase its quan- 
 tity ; by this flow the level will be lowered, and the pressure of 
 the adjacent masses will impel a current towards the lowest point, 
 which will be that whence the fluid is first drawn. In a wide 
 channel, the main stream will be increased by this action, and on 
 one or both sides of it, a current will be formed in an opposite 
 direction, into which, when the space is again contracted, or 
 another point reached, the excess of fluid that has united itself 
 with the main stream will flow, and thus keep up the circulation. 
 If, on the other hand, the increase in the section of the channel be 
 permanent, eddies will at first be formed, but the velocity of the 
 main stream will gradually diminish, until it become capable of 
 filling its new bed. 
 
Book VI. ] or RIVERS. 425 
 
 CHAPTER VI. 
 OF RIVERS. 
 
 414. The vapour which is raised from the whole surface of the 
 earth, and particularly from the ocean, tends to distribute itself 
 uniformly according to the mechanical law, mentioned in 355, 
 and known by the name of its discoverer, Dalton. Thus, the ex- 
 cess of moisture furnished to the atmosphere, from the surface of 
 the sea, is borne towards the land ; upon the latter, as will here- 
 after be shown, the causes that produce precipitation are, in gene- 
 ral, more frequent than on the ocean ; hence, more moisture falls 
 on the surface of the land, than is evaporated from it. The loss 
 arising from this excess of precipitation is again supplied from the 
 ocean, in conformity with the law we have cited. It thus hap- 
 pens that in most parts of the continents and islands, the quantity 
 of moisture that falls to the surface, in the form of rain, hail, 
 snow and dew, exceeds that which is evaporated from their sur- 
 face. This excess partly runs over the surface of the ground, and 
 partly penetrates into it. In the latter case it frequently meets 
 impervious strata ; along these the water is carried by its gravity, 
 until they break out to the day. The water will there form springs. 
 These unite with that running upon the surface, and descend to 
 the lowest points of vallies, where they form lakes, or, if the 
 slope be sufficient, streams of various magnitudes. Lakes when 
 they increase to such a height as to overtop the lower parts of 
 their barriers, or acquire a sufficient pressure to force their way 
 through them, also give rise to streams. Such streams running 
 in the lower parts of vallies, and upon the lines of the greatest slope, 
 unite at the junction of two or more vallies, and mix their waters 
 in a channel of greater magnitude ; and thus, by successive junc- 
 tions, if the extent of country be sufficient, rivers of great mag- 
 nitude are formed. 
 
 415. The formation of the beds and channels of rivers appears, 
 even on the most cursory observation, to have been effected by 
 the action of their own waters : from the continued effort of their 
 streams, a species of equilibrium has taken place between the mo- 
 tion of their currents, and the resistance of the soil over which they 
 run. Thus : when the velocity is very great, the beds are com- 
 posed of solid rock; in the next stage of diminishing velocity, 
 the beds are composed of large rolled stones ; when the velocity 
 decreases still farther, the bottom is composed of gravel ; then of 
 
 54 
 
426 or RITERS. [Book VL 
 
 sharp sand ; and finally, when the water becomes stagnant, of 
 fine argillaceous particles, or mud. 
 
 Still the equilibrium between the active force of the stream 
 and the resistance of its bed, is not absolutely perfect ; a slow 
 and gradual action takes place on all parts of its bed, impercepti- 
 ble, except in the larger class of streams, or under extraordinary 
 circumstances ; but this action is sure, and after the lapse of 
 years will be found to have produced most important effects. 
 
 It has been determined by the observations of Du Buat, that 
 fine clay will not resist the action of a current, whose velocity, 
 at bottom, exceeds three inches per second : fine sand begins to 
 resist 'when the velocity falls below six inches ; coarse sand, at 
 a velocity of eight inches; gravel, at velocities from seven to 
 tw.elve inches ; pebbles of an inch in diameter, at two feet ; and 
 angular fragments, of the size of an egg, at three feet per second. 
 
 Rivers cannot be considered as forming, throughout their whole 
 course, channels of uniform slope ; on the contrary, they are, as 
 a general rule, most inclined near their sources, and become less 
 and less so as they approach the sea. They may, however, in 
 most cases, be divided into portions; each of these may be con- 
 sidered as composed of a current flowing with a constant area, 
 and having an uniform slope at its surface. These portions are 
 frequently separated by marked physical barriers. 
 
 Under the long conflict of the active forces of running water, 
 and the passive resistance of their beds, the latter have, in almost 
 all cases, assumed a state of apparent permanence ; the changes 
 that take place in them are slow, and only perceptible by the 
 comparison of their states at long distant periods. 
 
 416. Rivers are subject to an increase and diminution in the 
 quantity of their waters. This variation sometimes bears but a 
 small proportion to their mean magnitude, at others is of great 
 extent. It is in some cases periodic, varying with the season, 
 being produced by the melting of the snows, in other cases, is 
 subject to no fixed law. 
 
 When the variation in the bulk of a stream is small, its bed 
 has usually for its section a concave curve, whose versed sine 
 bears a greater ratio to its chord, when formed in earth that op- 
 poses a great resistance, and of course, when the stream is rapid, 
 than it does in soils of less tenacity. The resistance of solid rock 
 may render this rule untrue when the bed is formed in that sub- 
 stance. ;.: 
 
 When the variation in the bulk of the stream is great, and its 
 slope small, there is usually a bed of the same form- as in the other 
 case, suited to convey the stream when it does not much exceed 
 its mean magnitude ; this is enclosed on one or both sides by the 
 
Book VI.] OF RIVERS. 427 
 
 alluvial deposits of the river. These usually assume the shape 
 of an inclined plane, sloping from the bank of the main stream 
 towards the adjacent country. The manner in which this shape 
 is produced, may still be witnessed in the streams of many parts 
 of our own country. The river, as its volume increases, flows 
 with greater rapidity, and carries with it an increased quantity of 
 earthy matter ; when it rises so far as to overtop the banks which 
 bound it when not swollen, if trees and bushes grow upon them, 
 they will catch and retain the larger and heavier parts, while the 
 lighter alone will remain suspended. Thus the greater propor- 
 tion of the deposit takes place on the edge of the ordinary bed. 
 This process may still be annually witnessed on the Mississippi ; 
 and the height of the natural dyke that is thus formed, is in- 
 creased by the quantity of drift wood that is retained by the 
 nearest obstacles. This natural barrier, so long as the periodic 
 overflow is unimpeded, more than counteracts any rise that may 
 take place in the bed, from deposits at those times when the 
 stream has less than its mean velocity. 
 
 It frequently however happens in cultivated countries, that it 
 becomes necessary to restrain the periodic overflow. For this 
 purpose dykes are erected on the borders of the usual channel, 
 their erection being facilitated by the natural form of the adja- 
 cent ground. In this case, the substances carried by the stream 
 are deposited in its bed, instead of being spread over the whole 
 surface, to which the inundation had before extended. The level 
 of the surface of the stream will in consequence rise, until the 
 slope may finally become too small to convey it, and it may seek 
 an outlet in other directions. It is thus rendered indispensable 
 to raise the dykes to correspond to the increased height of the 
 surface of the stream. By a long continued process of this kind, 
 the bottom of the bed of the Po has been raised higher than the 
 level of the adjacent country, and the surface of its stream over- 
 tops the houses of Ravenna ; so also the ancient channel of the 
 Rhine has been filled up, until it will no longer carry its waters, 
 and they have sought outlets in other directions. 
 
 A similar action is going on upon the main outlet of the Mis- 
 sissippi, where the superior magnitude of the stream makes a 
 change in its bed, and in the height of its waters, when full, more 
 marked in a few^ years than it is in the Po and Rhine in cen- 
 turies. 
 
 A river that requires to be confined by dykes, is often crooked 
 in its course. The danger of overflow may, in this case, He les- 
 sened upon principles derived from our investigations. It will 
 be seen by reference to the formula (456), that the velocity varies 
 with the square root of the slope ; if then the course be rendered 
 
4*8 OF RIVERS. [Book VI. 
 
 straight, the fall between two given points remaining constant, 
 while the distance in the direction of the stream is lessened, the 
 slope is augmented, and a bed of given dimensions will carry a 
 greater quantity of water, in a given time. 
 
 417. Among the many important purposes that rivers subserve, 
 that of navigation is worthy of particular notice. 
 
 Rivers, according to their size and importance, are sometrmes 
 navigable for vessels of various descriptions and sizes ; bqt are, 
 at others, frequently liable to obstructions and impediments, that 
 either interrupt or prevent their navigation altogether. 
 
 These obstructions and impediments may be arranged in seve- 
 ral distinct classes. 
 
 (1.) Rivers maybe obstructed by Falls or Cataracts, consisting 
 in a sudden change of level. 
 
 (2.) Rocky barriers may cross the bed of a river ; these will 
 prevent its forming a channel sufficiently large to discharge its 
 waters with its usual mean velocity. In this case, the flow of 
 the higher part being impeded, the water accumulates above the 
 barrier, until the slope becomes so great immediately over it, as 
 to increase the velocity there to an extent adapted to the discharge 
 of the stream, through the contracted space. Such obstructions 
 are called Rapids. 
 
 In large streams the rocky barrier sometimes lies so deep that 
 navigation is not obstructed, but merely embarrassed, by a cur- 
 rent of increased velocity. Such is the rapid at the outlet of 
 Lake Erie, and a more magnificent instance is to be seen in the 
 Race in Long Island Sound. 
 
 In small streams they interrupt the navigation altogether. 
 
 (3.) When rivers, after running in a mountainous country, 
 reach one of less slope, their velocity is diminished ; the earthy 
 particles they had before been able to carry with them, are in 
 consequence deposited, 'the bed is filled up, and the river seeks a 
 discharge by spreading itself. Thus the breadth of the channel 
 is increased, and its depth diminished. A similar consequence 
 may follow when the velocity of a river is lessened by its meet- 
 ing the tide within its own channel. By a combination of these 
 two causes, the obstructions that exist in the Hudson River, near 
 Albany, have been formed. 
 
 (4.) Where a river that carries large quantities of earthy mat- 
 ter enters the ocean, the conflicting action of the two masses of 
 water, causes a cessation of motion that influences a deposit ; bars 
 are thus created in the sea, in advance of the mouth of the river. 
 In the case of large rivers, these may become the basis of islands, 
 which are gradually connected with the main land, while the force 
 of the current sweeps out the intervening channels, and new bars 
 
Book VI.} OF RIVERS. 429 
 
 are formed beyond them. In this manner the Deltas of rivers 
 were originally deposited, and, in some instances, still continue 
 to protrude themselves into the sea. 
 
 (5.) A river may carry a sufficient quantity of water to admit 
 of navigation, and may be unobstructed by falls, rapids, or shal- 
 lows, but may be so rapid as either to prevent an ascending trade, 
 or to diminish the area of the stream so far as not to admit of a 
 vessel floating in it. 
 
 418. Each of these obstructions has its appropriate remedy, 
 the application of which may however be, in some cases, imprac- 
 ticable, either from physical circumstances, or the great expen- 
 diture they involve. 
 
 (1.) When a river is obstructed by falls, we make a lateral 
 channel, fed from the upper level, and apply locks to it, upon 
 principles that will be explained under the head of canals. 
 
 (2.) If rapids have upon them a sufficient depth of water to float 
 the vessels, artificial mechanical means may be brought in aid 
 thus : the power of men or of animals may be applied from the 
 shore ; powerful steamboats may be used in towing ; or the ves- 
 sel may have wheels adapted to it, the area of whose paddles is 
 greater than its own section. In this last case, if a barrel be 
 adapted to the wheels, and a rope passed around it and fastened to 
 the shore above the rapid, the wheels being more powerfully acted 
 upon by the current than the vessel is, will turn around, coil up 
 the rope, and drag the vessel forward. It however generally 
 happens that the impediment of a rapid must be overcome by a 
 lateral channel, as described in the preceding instance. 
 
 (3.) As the formation of shallows arises from a diminution in 
 the velocity of a stream, it may be prevented, and the obstacle 
 may even be removed, by increasing the velocity. This may be 
 done by contracting the horizontal dimensions of the bed. The 
 upper portions of the stream being thus retarded, the level rises, 
 until the slope, as in the natural formation of a rapid, becomes 
 sufficient for the discharge. With this increased velocity, the 
 stream will have sufficient force to carry away the earthy matter 
 it before deposited, and a permanent improvement in its depth 
 will take place. \ 
 
 As a stream tends to continue of uniform section, even in an 
 increased channel, forming eddies in the bays and hollows, it is 
 not necessary that this contraction should be effected by conti- 
 nuous and parallel lateral dykes. It is sufficient that piers be 
 built out, alternately from each bank, at distances from each other 
 equal to about the breadth it is proposed to give the stream. The 
 heads of these piers should be arranged, if possible, in two pa- 
 rallel straight lines, in order that the stream may assume a straight 
 
430 OF RIVERS. [Book VI. 
 
 course, in which, as has been already explained, it will have the 
 greatest velocity. The deposit, which was before uniformly dis- 
 tributed over the bed, will now take place between the piers. 
 In all such cases, a straightening of the channel is advantageous. 
 
 If islands be formed in the shallow parts, as often occurs, all 
 the branches of the river that surround them, except one, should 
 be closed, by weirs that rise to the ordinary level of the stream, 
 and which will therefore permit a discharge through these lateral 
 channels when it is swollen. If there be several islands on either 
 side of the main channel, it may be often advantageous to unite 
 them by a longitudinal dyke, in order to prevent the stream from 
 spreading between them. 
 
 (4.) A similar principle will direct the operations intended for 
 the removal of bars at the mouths of rivers. It becomes neces- 
 sary to give the stream such a velocity as will make its force 
 preponderate over the resistance of the ocean. In large rivers, 
 the extent of the bars puts them beyond the reach of improve- 
 ment. In small streams, piers may be built out from the main 
 land, from each side of the mouth of the river, gradually inclin- 
 ing to each other. The waters being all confined between them, 
 and acquiring from the contraction an increased velocity, will 
 not only cease to make farther deposits upon the bar, but will 
 carry. away so much of it as lies between the piers. The deposit 
 will now take place behind the piers on either hand. Where 
 the direction of the river makes a small angle with the line of 
 the coast, a single pier or jetty may be sufficient. Of this me- 
 thod we have fine instances in the ancient port of Dunkirk, and 
 in the present port of Havre : and similar principles have been 
 successfully applied at Buffalo, on Lake Erie. 
 
 (5.) A rapid stream may be rendered navigable by building 
 dams or weirs across it from place to place : between each two 
 of these, the depth of the water will be increased and its velocity 
 Diminished; by both of these changes navigation may be ren- 
 dered more easy. The passage of these dams by vessels, was 
 originally effected in one of two different methods ; these require 
 illustration, in consequence of their being the germs of more 
 perfect means, that are still in use. The first of these methods 
 was the Sluice; the second, the Inclined Plane. 
 
 To form a sluice, an aperture is left in the mass of each of the 
 dams that divide the river into successive ponds, of different 
 levels. This is for the greater part of the time closed by a gate ; 
 and as this gate will have an unequal pressure on its opposite 
 sides, in consequence of the difference of level, it can only be 
 opened by a vertical motion. 
 
 In the canals of China, the gate is formed of heavy beams, 
 dropped into vertical grooves, made in the masonry of the dam. 
 
Book VL~\ or RIVERS. 431 
 
 These being piled upon each other, the lower ones are thus loaded 
 with a weight that sinks them to the bottom of the passage, and 
 they oppose an effectual barrier to the flow of the water through 
 them ; for the small quantity that may penetrate between their 
 joints, will be wholly unimportant. 
 
 When the gate is opened, the water will discharge iteslf with 
 great velocity through the passage. The vessels of the descending 
 trade are committed to this current: and as they would run the 
 risk of being swept into the eddies it will form, men are stationed 
 on each bank, furnished with poles, to keep them in the direc- 
 tion of the effluent stream. By this discharge, the level of the 
 water in the upper pond, is lowered ; the velocity of the current 
 that flows through the passage is diminished in consequence. 
 Natural agents, such as the force of men or animals, are then ap- 
 plied to draw up the vessels that carry the ascending trade. 
 
 If the change of level be considerable, the danger in passing 
 down, and the force required to draw the vessels up, are both too 
 great. This method, therefore, becomes impracticable. In the 
 infancy of inland navigation, the inclined plane of a rude and 
 imperfect form, was introduced in cases of this sort. In a part 
 of the weir was introduced a mound of masonry, of the form of 
 a triangular prism. Of the two inclined planes that composed 
 its upper surface, one descended to the bottom of the lower, and 
 one to that of the upper level of the navigation, as in the figure 
 beneath. 
 
 The vessel, in passing from one level to the other, was first 
 drawn up on one of the inclined planes, and passing the ridge, 
 was resisted in its descent on the other. For this purpose ropes 
 were passed around the vessel, and applied to capstans situated 
 on each side. 
 
 The sluice, in its original form, is still used in China, as is the 
 inclined plane. The sluice is also still used in the Low Countries, 
 and in the valley of the Po, in cases where the change of level 
 is not more than two or three feet. In almost all other instances, 
 locks have been substituted for sluices, and several locks, in the 
 cases to which the inclined plane was formerly applied. 
 
 These methods of improving the navigation of rivers, have 
 the advantage of requiring a comparatively small original cost ; 
 
432 OF RIVERS. Book VL~\ 
 
 they may, therefore, be used in countries of thin population, and 
 small wealth. They have the disadvantage on the other hand, 
 that they are liable to injury from floods, while the navigation 
 is itself subject to vicissitudes from variation in the supply of the 
 stream. The passage of vessels through sluices, and over the 
 original form of the inclined plane, is difficult, and requires the 
 application of force of an expensive character. For all these 
 reasons, it has been the uniform result of experience, that in 
 spite of the great excess of the first cost, it is, wherever the ca- 
 pital can be obtained, better to make a canal parallel to the river, 
 than to attempt to improve the navigation in its own bed. The 
 principle on which the construction of canals rests, are contained 
 in the following chapter. 
 
 419. When a river is to have its navigation improved, or when 
 water is to be drawn from it for supplying pipes, for irrigation, 
 or any other useful purpose, it is, generally speaking, necessary 
 to ascertain the quantity of water that it furnishes. For some 
 physical investigations, the whole quantity that flows in its bed, 
 may be the object of research. But for mechanical purposes, it is, 
 generally speaking, best to limit the inquiry to the minimum 
 supply, for on this will depend the certainty with which the stream 
 can be depended upon for most practical uses. 
 
 In a stream of so small a size that barriers can, without diffi- 
 culty, be erected across it, a dam that interrupts its course, is 
 constructed at some convenient point. In this a gate is placed, 
 formed of a rectangular frame, and a shuttle that can be raised 
 or lowered at pleasure. The shuttle being closed, the passage of 
 the water is interrupted, and it would rise until it found a dis- 
 charge over the dam : before it reaches this height, the shuttle 
 is drawn up, and the water passes out. When this discharge is 
 just sufficient to prevent the level of the water from rising far- 
 ther, but not sufficient to cause it to fall, it will be obvious that 
 the gate discharges the exact quantity of water that the stream 
 furnishes. 
 
 This quantity may be calculated on the following principles. 
 The area of the open part of the gate may be considered as an 
 aperture in a thin plate, and must, therefore, receive a correction 
 for the vena contracta. The velocity, which will vary in every 
 horizontal element of the orifice, will have for its mean, that 
 which is due to the depth of the centre of pressure of the opening 
 beneath the surface of the fluid. This velocity being found by 
 the principles of 401, and the formula (61), 
 
 is to be multiplied by the area of the aperture and the constant 
 quantity 0.62, 408. 
 
Book 
 
 OF RIVERS. 
 
 433 
 
 In streams whose section has an area of more than two square 
 feet, this method would become expensive and troublesome. To 
 gauge streams of an area of from two to ten or twelve square feet, 
 we have recourse to another method. 
 
 The area of the stream is carefully measured. Its velocity 
 may next be determined by the aid of an apparatus, called from 
 its inventor, the Tube of Pitot. A tube ABC is taken and bent 
 until one of its branches is at right an- 
 gles to the direction of the other. This 
 tube is open at both ends, and is con- 
 tracted at the opening C of its shorter 
 branch. It is placed in the stream, the 
 branch AB in a vertical position, and 
 the horizontal branch is turned around 
 until the stream enters freely and di- 
 rectly into the orifice E. The water 
 entering the tube with a determinate 
 velocity, has such a force as would cause 
 it to rise above the level of the surface 
 of the stream, to the height whence it 
 must have fallen to acquire that velocity. 
 In the tube, it will be resisted by fric- 
 tion, and the height will be lessened. 
 The height to which it rises may be de- 
 termined by applying a graduated rod 
 to the outside of the tube, if of glass. 
 If, however, the tube be not transparent, 
 and it is usually metallic, a rod, be, is 
 
 placed in the vertical branch of such specific gravity as to float on 
 water. This rod being graduated, will mark the rise of the li- 
 quid. The corresponding velocity may then be obtained by the 
 usual formula (61.) 
 
 ' 
 
 which is sufficiently accurate, except when the velocity is small. 
 
 Experiments may be made in different parts of the section of 
 the stream, and the mean of the results taken as the mean velo- 
 city. It is, however, generally speaking, sufficient to measure 
 the velocity in this manner, near the middle of the surface, and 
 thence to deduce the mean velocity by the formula (462) or 
 (463). . 
 
 This velocity, multiplied by the area, will give the discharge 
 in the unit of time. 
 
 To use the tube of Pitot, it is convenient to attach it to a 
 wooden stand, by which it can be placed in the bed of the stream 
 and afterwards levelled. 
 
 55 
 
434 
 
 OF RIVERS. 
 
 [Book VI. 
 
 This instrument is, therefore, limited to streams of not more 
 than four or five feet in depth. 
 
 In streams of greater size, the velocity of the surface may be 
 measured by a float. For this purpose, a part of the stream must 
 be chosen, where it flows between its banks without eddies. The 
 area of its section may then be determined by measuring its 
 breadth, and its depth, at equal intervals, from bank to bank, as 
 in the following figure, where AB represents the breadth, and 
 
 i> 
 
 BE, FG, HI, depths measured at three points, dividing the 
 breadth into four equal parts. The area may then be considered 
 as made up of the two triangles, ADE, BHI, and the two trape- 
 zoids, DFGE, and FHIG, or of any other number of the latter 
 that the circumstances may require. A float is next thrown into 
 the stream, and the time in which it is carried through a given 
 distance, that is accurately measured upon Its banks, noted by a 
 time-keeper. The velocity in the unit of time thence deduced, 
 is reduced, as before, to the mean velocity, and multiplied by 
 the area. The product is of course the discharge. The floats 
 used in this operation, may be constructed in various modes. 
 Thus : 
 
 A piece of cork may have a weight attached to it by a string 
 that shall render their united mass but little less dense than water. 
 No more of the cork will then float than is sufficient to render it 
 visible, and it will not be liable to disturbance from the action of 
 the wind. In a very short time it will acquire the whole velo- 
 city of the part of the current in which it is placed. It will also, 
 after a greater or less time, be carried to that part of the stream 
 that has the greatest velocity. If the float, and the weight at- 
 tached to it, have similar figures, and are of equal size, it will be 
 obvious that the velocity assumed by it will be the mean of that 
 of the two horizontal layers of the stream in which they lie. As 
 this condition may be always conveniently adopted in practice, 
 there is no need of investigating the correction that would be due 
 to any difference in these respects. If the distance between the 
 two bodies be small, their velocity may, without sensible error, 
 be considered as that of the surface of the stream. 
 
 The float may have the form of a hollow cylinder, as for in- 
 stance, a piece of a reed, closed at one end. This may be ballasted 
 
Book Vl.~\ OF RIVERS. 435 
 
 by shot or other small weights, until no more than a small por- 
 tion of it be elevated above the surface. 
 
 If the length of the tube bear a considerable proportion to the 
 depth of the stream, it will float in a position considerably in- 
 clined to the vertical, in consequence of the difference of the 
 velocity with which the fluid moves at different depths; and in 
 all cases, there will be a greater or less inclination. An investi- 
 gation of the nature and direction of the forces that act, may en- 
 able us to determine the velocity at the surface when the velocity 
 of the reed and its inclination to the horizon are given. Such an 
 investigation may be found in Venturoli, Vol. II. p. 218. It is, 
 however, better not to use this instrument, except when it floats 
 nearly vertically, in which case the error is unimportant, and 
 therefore needs no correction. 
 
 In streams of still larger size, a part is chosen in which the 
 water flows for a distance of 1 or 200 yards, without eddies. 
 The area is measured as above, and the mean of several measure- 
 ments is to be preferred to a single one. The sum of the length^ 
 of the lower sides of the triangles and quadrilateral figures, is 
 taken as the length of the curved section of the bed. The area 
 divided by this, gives the quantity R, in (454), or the hydraulic 
 mean depth. The slope of the surface, or I, of the same formula, 
 is next obtained by levelling, taking the difference in the altitude 
 of the surface of the stream at two points, whose distance is as- 
 certained by measurement. These two quantities being given, 
 the mean velocity is deduced by means of the formulae, (462) or 
 (463), and this multiplied by the area, gives the discharge in the 
 unit of time. 
 
 An instrument used for a variety of other purposes, and called 
 a Dynamometer, may also be applied to measure the velocity of 
 a stream. The essential part of this instrument is a spring, the 
 quantity of whose contraction, under pressures of known inten- 
 sities, has been determined by experiment. If a plane surface 
 be attached to such a spring, by a cord or other convenient 
 method, the action of the stream will compress the spring, and 
 the amount of compression will measure the action of the water, 
 which will be given by the index of the instrument, in some con- 
 ventional unit of weight. This action may also be considered 
 as represented by the weight of a prism of the fluid, whose area is 
 equal to the area of the surface acted upon, and whose altitude is 
 that through which a heavy body must have fallen to acquire the 
 velocity. If then the number of units of some given cubic mea- 
 sure of water that is equivalent to the weight to be calculated, 
 and divided by the area of the plane, the quotient is that height ; 
 from this the velocity may be calculated by the usual formula. 
 
436 or RIVERS. [Book VL 
 
 This principle is not absolutely true, as will be seen when we 
 treat of the percussion and resistance of liquids. It is, however, 
 sufficiently near the truth for all practical purposes. 
 
 One other method of gauging a stream, remains to be men- 
 tioned. It is applicable to the case where it is traversed by a 
 dam whose upper surface is horizontal, and over which the water 
 discharges itself in a sheet, forming a water-fall. The velocity 
 with which a fluid passing through such an aperture as the upper 
 surface of the dam would represent, is discharged, is, according 
 to the reasonings and experiments of Du Buat, 
 
 in which expression b is the breadth of the sheet" of water, and 
 h its depth, both expressed in English feet. The formula for 
 the French metre, is 
 
 v=0.1895bVh\ (464) 
 
 These formulae comprise the last case of the motion of liquids 
 recapitulated in 397. 
 
Book VI] . OF CANALS. 437 
 
 CHAPTER VII. 
 
 Or CANALS. 
 
 420. Canals are open artificial channels formed for the con- 
 veyance of water. They may be used for the purposes of navi- 
 gation ; for the supply of those intended for navigation, in which 
 case they are called Feeders; for the supply of cities ; in the 
 draining of morasses, and for irrigating land for agricultural pur- 
 poses. Canals differ in character from rivers ; the latter have, 
 by the long action of antagonist forces, formed beds, that gene- 
 rally speaking occupy the lowest levels in vallies, and follow the 
 lines of greatest slope. The former have some conventional slcrpe 
 suited to their object, and which is usually uniform, or are ab- 
 solutely level ; they follow this prescribed line along the sides of 
 hills, or through their masses, and are often carried at a considera- 
 ble height over streams, ravines, and even broad vallies. Rivers 
 rise from humble origin, but uniting with others as they pro- 
 ceed, and receiving the discharge from lateral vallies, swell 
 in bulk in spite of the causes of waste that affect them equally 
 with canals. Canals, on the other hand, maintain an uniform 
 section throughout their course, and their volume diminishes in 
 consequence of those causes ; while if collateral supplies are 
 broyght in, they need be no more than equal to the restoration 
 of what has been -wasted : if these collateral supplies exceed this 
 amount, means must be provided to get rid of the excess. 
 
 421. Canals take their rise in reservoirs, either in the form of 
 natural streams and lakes, or of artificial basins, that collect the 
 surface water. The fluid contained in these has usually a velocity 
 less than that required in the canal ; and even if it be a rapid 
 stream, whence the canal proceeds, there will be a difference in 
 the directions of the two currents, Hence the water in the 
 canal will not at once assume the required velocity; and if the 
 bed have a constant section until it unite with the reservoir, the 
 area of the stream will diminish as it recedes from the reser- 
 voir, until the constant velocity adapted to the slope and cir- 
 cumstances of the channel be attained. This diminution of 
 area can only take place by a diminution in the depth of 'the 
 water in the canal. Hence at the origin of canals, whose area 
 does not vary, a fall will be formed, by which the area of the 
 stream will be diminished. If then it be important that the 
 
438 OF CANALS. [Book VI. 
 
 canal shall be filled, it must spread out as it approaches the reser- 
 voir. If we suppose it to be influenced by the same causes that 
 affect the vena contracta, and that the channel has a rectangular 
 section of uniform depth, the plan of the channel would be form- 
 ed on each side by a logarithmic curve, whose axis and greater 
 and less ordinates have the proportion of 5 : 6.25 : 4 ; the second 
 number being half the breadth of the channel at the reservoir ; 
 and the third half its uniform breadth. If the figure were investi- 
 gated on the principle that the velocity of water moving in a 
 channel varies inversely as its area, it would be found that the 
 proper figure of the banks would be a parabola, whose vertex is 
 at the point where the breadth of the channel becomes constant. 
 Neither of these methods of investigation is free from objection ; 
 but it is evident from experiment and observation, that in order 
 that water shajl enter into a channel without forming a fall, or 
 that it shall completely fill it, the channel must, at the reservoir, 
 have a width greater than the breadth of the uniform section that 
 it has at other points ; and this increase of breadth should take 
 place in the form of a curve convex towards the axis of the 
 canal. 
 
 Such a form is to be found in nature when streams take their 
 rise in lakes, or other similar reservoirs. If a canal be formed 
 in soft earth, it will gradually wear away the earth until it as- 
 sume such a form ; but in solid rock such a shape cannot be spon- 
 taneously assumed. Even in canals made in soft earth, it is bet- 
 ter to give the requisite shape artificially, than to wait for the 
 slow action of the water. 
 
 422. The shape of the section of a canal is usually a trapezium, 
 two of whose sides are parallel and horizontal ; the other two 
 equally inclined to the horizon. This inclination will depend 
 upon the nature of the soil in which the canal is formed, being 
 least in tenacious earth, and greatest in loose soils. No soil will 
 maintain itself when the base of the slope is less than one and a 
 third times its height, or in the proportion of four to three. While 
 in loose soils the slope must be at least as great as in the propor- 
 tion of two to dne ; or the base twice as great as the height. The 
 force that acts upon the bank is the pressure of the water, and 
 this is partly exerted to push the bank aside, and partly to over- 
 turn it by a rotary motion. In banks of earth, where the mate- 
 rial has little cohesive strength, the former effort is most likely 
 to be injurious ; while in masses of masonry, the latter is the more 
 important 
 
 To investigate the proper thickness of a bank, whose height 
 and slope and the material of which it is composed are given : 
 
Book ft.] OF CANALS. 439 
 
 Let ABCD be the section of a prismatic mound of earth, 
 pressed by water on the side AB ; let the slopes on each side be 
 
 A B equal, and suppose the bank 
 
 itself to be composed of mate- 
 terials of infinite cohesive force, 
 but capable of being moved 
 horizontally in the direction 
 AD, by a force equal to the 
 friction among its particles. 
 B B e. c 
 
 Let AB, the length of the face = a ; 
 
 BAE, the base of the slope, = b ; 
 
 ABE, the vertical height, = h ; 
 
 the angle of the slope, ABC, = i ; 
 
 AD, the thickness of the top of the bank = x ; 
 
 the density of the material, that of water being unity, D ; 
 the co-efficient of the friction = /; 
 
 the thickness of the bank at bottom will be = x + 2 b ; 
 
 The pressure on the line AB will be, 331, 
 
 la/t; 
 
 this force acts at the centre of pressure in a direction per- 
 pendicular to the face of the bank. If then it be resolved into 
 two components, one in a horizontal, the other in the vertical 
 direction, the former only will tend to thrust the bank from its 
 plane ; the latter will add to the weight that presses on the base, 
 will increase the friction, and consequently add to the stability. 
 The first of these components will be, 13, 
 
 ^ ah sin. i ; 
 the second, 
 
 \ ah cos. i ; 
 but as 
 
 h b 
 
 sin. i~, and cos* i , 
 4 . ct> a 
 
 these forces become respectively 
 
 i/i 3 , and %hb\ 
 
 of which that represented by /i 2 , tends to thrust the bank from 
 its place. 
 
 This force is resisted by the friction on the base, which is a 
 function of the whole pressure, and this is made up of the weight 
 of the bank, and the force \ h b derived from the liquid pressure 
 acting downwards. 
 
 The weight of the bank is found by multiplying its area by its 
 density, and is 
 
OP CANALS. [Book 
 
 we therefore have, as the condition of equilibrium, 
 
 (465) 
 whence we obtain for the value of 2-, 
 
 - 
 
 slope 
 
 (467) 
 
 When the bank is triangular, #=0, and the base of the slope 
 becomes 
 
 to find the co-efficient/, we have 
 
 < 468 > 
 
 The mean density of earthy substances being about 2, we have 
 in the case where 26=3/i, 
 
 f= T 2_ = 0.133; 
 and when 6 =2 A, 
 
 /=A; 
 
 and, as the pressure at the surface is evanescent, and a triangular 
 mound would of course resist it, these values may be applied in 
 other investigations. 
 
 In practice, however, the water will enter into the pores of 
 the earth, and thus the surface exposed to pressure will be much 
 increased. 
 
 To allow for this, let the co-efficients /be reduced to one half, 
 or in firm earth to 0.06, and in loose earth, to 0.05. We shall 
 then have, in the latter case, in a bank of triangular section, 
 
 and the whole base of the bank, 
 
 26=8/i. (469) 
 
 When the mass of the bank is given, as that part of the liquid 
 pressure which tends to thrust the bank aside, diminishes as the 
 inclination of the face on which it acts increases ; while on the 
 opposite side, loose earth supports itself at a slope whose base is 
 to the height as 2 : 1 : we may infer, that the maximum of strength 
 will be attained when the slope on which the water presses has 
 for its base 
 
 6=6 /i; 
 
 and when the base, 6', of the opposite side, is 
 
 In loose earth, when the bank has a trapezium for its section, 
 and 6=2/?, we obtain for the value of #, from (466), 
 
 (470) 
 
Book J7.] OP CANALS. 441 
 
 in firm earth, where 6=1 i h, 
 
 x=2,3(K). (471) 
 
 These give the thickness of the bank at the surface of the water ; 
 but as it is usual to make the bank of a canal one foot higher 
 than the level of the water it contains, we would have for the 
 thickness at its upper surface, in the case of loose earth, in feet, 
 
 x=2.5(h) 4, (472) 
 
 and in firm earth, 
 
 o?=2.3(/i) 3. (473) 
 
 423. A canal is usually confined between a bank on one side, 
 whose dimensions may be determined on the foregoing princi- 
 ples, and a towing path, the breadth of whose upper surface must 
 be sufficient for a road, on which the animals employed in draugfit 
 may easily pass each other. If, then, the dimensions deduced 
 above be not sufficient for this last object, the breadth of the up- 
 per surface of the towing path must be increased to at least nine 
 feet. For the other bank, the .usual rule is, to make its breadth 
 at top equal to the height measured from the bottom of the canal ; 
 but in this case there should be abermoffrom 1 to 1 feet at the le- 
 vel of the water, by which the thickness of the bank at the. water's 
 edge will, in usual cases, nearly coincide with our formulae, (470) 
 or (471, and which will have the advantage .of preventing the 
 wash of the banks from falling into thelcanal. 
 
 To prevent the entrance of rain water, a ditch called the coun- 
 ter ditch is formed on the outside of each of the banks. This is 
 particularly necessary in side lying-ground, where the rain may 
 produce injurious effects. 
 
 The profile of a well-constructed canal will therefore present 
 the following figure : 
 
 And when the breadth and depth are given, the relations between 
 the depth of excavation and the height of embankment, that will 
 just suffice to give the proper form, is a simple geometric problem. 
 
 424. Canals are applied to the purposes of inland navigation 
 in several cases : 
 
 (1.) As has been already stated, they may be employed to pass 
 obstacles upon rivers, that are in other respects navigable, or may 
 be constructed parallel to a stream whence they derive their sup 
 ply of water. 
 
 56 
 
442 OF CANALS. [Book VI. 
 
 (2). They may be made to communicate between two naviga- 
 tions of eqaal levels, drawing their supply from both, or between 
 two of different heights, drawing their supply from the higher. 
 
 (3). They may form a communication between two naviga- 
 tions, passing over ground higher than either. This is at present 
 the most usual case of canal navigation. Such canals are said to 
 have a summit level, or to be a point de partage. 
 
 425. The dimensions of navigable canals will depend upon the 
 section of the vessels intended to navigate them. They must in 
 the first place be wide enough to permit two vessels to pass each 
 other with freedom ; for this purpose the breadth at bottom is 
 usually made twice as great as the breadth of beam of the vessels j 
 in the second place, the depth is usually made one foot more than 
 their draught of*water. This is done for two reasons : first, be- 
 cause it is found that vessels are much more impeded, when the 
 channel in which they move is shallow, than when it is deeper ; 
 and secondly, in order to pro vide for any accidental defect of wa- 
 ter, or for deposits of earth thatmay lodgein the canal. In eitherof 
 these cases, the vessels would be exposed to take the ground, 
 were there not an extra depth of water. 
 
 426. The bed of a canal must be absolutely level, or have no 
 more slope than is sufficient to convey water to replace that 
 which may be wasted. Hence, when the navigations it is to con- 
 nect are of different levels ; when constructed on the bank of a 
 stream whose surface must of course slope ; or when it surmounts 
 a summit, it must be made in a series of channels at different 
 levels, and means must be provided to pass vessels from one level 
 to another. That which is most usually employed is called a 
 Lock. 
 
 A lock is a four-sided chamber, contained between two parallel 
 walls in the direction of its length, and by two gates or sluices, 
 situated at the two extremities. 
 
 The bottom of a lock is a floor of wood, or masonry, on the 
 same level with the bottom of the lower of the two reaches of 
 the canal that it serves to connect. The gates rise to the height 
 of the water in the upper reach, and the walls to the height of the 
 banks of that reach. The lower gate reaches to the floor of the 
 lock, the upper gate is usually established upon a transverse wall, 
 the top of which is on a level with the bottom of the upper reach, 
 and which is called the Breast Wall. 
 
 When the water in the lock is upon a level with the surface 
 of the water in the higher reach, it is said to be full ; when on 
 a level with that in the lower reach of the canal, it is said to be 
 empty. The gates are suspended from the walls in such a man- 
 
Book F7.] 
 
 CANALS. 
 
 443 
 
 ner. as to close, when they undergo a pressure from above ; they 
 may be opened when the pressure on each side is equal. 
 A plan and section of a lock are represented beneath. 
 FIG. 2. FIG. 1. 
 
 CO 
 
 
 Fig. 1st, is a longitudinal section ; fig. 2d, a horizontal plan 
 of a lock. 
 
 A, B, C, D, are the walls of which E and F are the parts that 
 enclose the chamber. 
 
 M, G, N, D, the breast wall on which the upper gate is sup- 
 ported. 
 
444 OF CANALS. [Book VI 
 
 HH, recesses in which the upper gate is received when open. 
 
 1 1, similar recesses to receive the lower gate. 
 
 K, sill of the upper gate. 
 
 L, sill of the lower gate. 
 
 O, upper opening of the culverts that form a communication 
 between the waters of the upper and lower levels of the canal, in 
 order to fill the lock. 
 
 V, vault in the breast wall into which these culverts enter. 
 
 WW, level of the water in the upper level of the canal. 
 
 WW, do " ^ lower do. 
 
 The operations of filling the lock from the upper level of the 
 canal, and of emtying it, by a discharge into the lower level, 
 may be effected by means of small gates, or wickets, sliding in 
 grooves in the timbers of the gates. 
 
 \T<* this method no objections apply in the lower gate ; but in 
 the apper, the water spouting from the top of the breast wall may 
 injure the lock, or enter into the vessel contained in it, which 
 may thus be destroyed, or sunk. Hence, passages called Culverts, 
 have been formed in the masonry of the walls of the lock, to 
 which wickets are adapted ; these passes opening from the water in 
 the upper level, are inclined iri such a manner as to discharge 
 themselves below the surface of the water in the lock, even when 
 as much as possible is drawn off. In the Canal du Centre, in 
 France, the culverts descend vertically, and discharge themselves 
 into an open vault, formed by the thickness of the breast wall. 
 A lock has recently been invented in this country by Mr. Can- 
 vass White, in which the breast wall is omitted, and the upper 
 gate is as tall as the lower one. In this form, the culverts in the 
 walls become unnecessary, and the breast wall which is for many 
 reasons, the weakest part of a lock, is suppressed. The bottom of 
 the upper and lower levels, are united by a slope, rising from 
 the sill of the upper gate. 
 
 427. When a vessel is to rise from the lower to the upper level of 
 the canal, it may either find the lock full, or empty ; in the for- 
 mer case, the surplus water must be discharged through the 
 wickets, until it reach a common level on each side of the lower 
 gate ; in the latter, the water in the lock, and in the lower level 
 are already at an equal height. The pressure being therefore 
 equal on each side of the lower gate, it may be opened, and the 
 boat drawn forward into the lock. The lower gates are next shut 
 behind it, and the wickets or culverts that communicate with the 
 water above opened. This will therefore pass into the lock, and 
 as it finds no exit through the lower gate, will fill the lock ; as the 
 lock fills, the vessel, buoyant on the surface, rises; the filling of 
 the lock, and rise of the vessel continue, until the water stand on 
 
Book F/.] op CANALS. 445 
 
 each side of the upper gate, at a common level. The pressure on 
 each side of the upper gate being then equal, it may be opened, 
 and the vessel drawn forward into the upper level of the canal. 
 
 If the canal be empty when a boat is to descend, it must be fill- 
 ed, or if full, kept so until the upper gate be opened, and the ves- 
 sel admitted. The upper gate is then closed behind the vessel, 
 and the water discharged from the lock, through the wickets of 
 the lower gate, until the water within, and in the level below, 
 reach the same height, when the lower gate may be opened, and 
 the vessel drawn out. 
 
 The lock appears to have taken its origin from the accidental 
 juxta-position of two sluices, in the action of which its important 
 and valuable properties were discovered. It seems to have been 
 first intentionally used in the Canal of Martizana, in Italy, about 
 the eleventh century. '**;; 
 
 428. To determine the thickness of the longitudinal walls that 
 confine a lock, when the depth of water, and the nature of the 
 material is given, we must in the first place consider, that being 
 built of masonry, the resistance to lateral thrust, is that of the 
 friction of stone upon stone, at the joints, and of the cohesive force 
 of the stone at other points ; the former is aided by the cohesive 
 force of the mortar, and these resistances being both great, the 
 water will exert a more powerful influence to overturn the wall, 
 than to move it laterally. As the pressure at the surface of the 
 water is 0, and as the wall may be built in a vertical position, we 
 may assume it, for the purpose of investigation, to have a section 
 of the figur^ of a right angled triangle. 
 
 Let ABC, be a section of the wall ; let the height AB, =p ; 
 the thickness BC, =#; 
 
 v* the liquid pressure on the face AB, will 
 
 be, 417, ' 
 
 and it will act to overturn the wall at the 
 point P, the centre of pressure, with a 
 moment of rotation represented by 
 
 
 The resisting force will be the weight 
 
 \ ** * 
 
 ;B "~ C and to find its moment of rotation it 
 
 must be multiplied by the perpendicu- 
 lar distance of its direction from the point C, which is | x. This 
 moment of rotation, then, will be 
 
 and in the case of equilibrium between the two forces, 
 
446 or CANALS. [Book VI. 
 
 \#=\V*\ (474) 
 
 whence we obtain 
 
 if we take the density of the materials to be 2 f we have 
 
 x=p , (476) 
 
 or the thickness of the wall at bottom, should be equal to half its 
 height. 
 
 In the construction of locks, the thickness of the walls at bottom is 
 made equal to half the depth of the water they contain when full. 
 But the top of the wall is higher than the top of the gates, which 
 determines the highest level of the water, in order to prevent any 
 accidental overflow ; and the section of the wall is not triangular, 
 but quadrilateral. The top must have a sufficient thickness to 
 admit of the masonry being carried up in two faces of ashler, 
 which are filled up within by irregular pieces, laid in water proof 
 cement, and grouted, in order to prevent any filtration. This 
 thickness cannot be less than from 3 to 4 feet. When the wall is 
 of brick, it may be of less thickness at top, but must be covered 
 with a coping strongly clamped. When the depth of water to be 
 supported is less than 7 or 8 feet, the wall will have two parallel 
 faces ; at greater depths the outer face slopes, until it has a thick- 
 ness at bottom of half the depth of water. 
 
 The walls of the lock are continued at its two ends, until they 
 meet the banks of the canal ; and as the latter is wider than the lock, 
 these portions diverge, and are called' the Wing Walls; those at 
 the lower end of the canal decrease in height, until they meet the 
 earthen bank. These walls are represented in the profile, and 
 section at ADandBC. 
 
 . These walls having less stress to undergo than those of the 
 chamber of the lock, need not be thicker at bottom than one third 
 of their own height. 
 
 429. The gates of locks are frames of timber, covered with 
 plank ; the lower gate reaches from 4he bottom of the lock ; the 
 upper from the top of the breast wall, to the level of the water in 
 the upper reach of the canal. 
 
 As frames of a quadrilateral figure are liable to change their 
 shape when suspended at one end, the frame must have diagonal 
 braces, or the plank may be put on in a diagonal direction. The 
 gate is suspended, by adapting a gudgeon to the bottom of one of 
 its uprights, and passing an iron collar built into the wall over its 
 upper end. This post must therefore project below the bottom 
 of the gate, and a circular cavity is made in the bottom of the 
 lock, to receive its gudgeon. This post, as well as that on the 
 
Book VL~\ 
 
 OF CANALS. 
 
 , 
 
 447 
 
 opposite side of the gate, is sufficiently tall to rise above the top 
 of the wall, in order to be framed into the lever that serves to 
 open and shut the gate. This lever extends, when the gate is 
 shut, over the top of the wall, and serves as a counterpoise to the 
 gate. The form of the gates will be understood by reference to 
 the figure. 
 
 
 
 // 
 
 
 .f; 
 
 
 
 'AWJ\VA 
 
 
 \YAWA\\\\, 
 
 
 'A\\YA\\\V/ 
 
 
 \YA\WA\l 
 
 
 mrmr/ 
 
 
 The bottom of the lock is lower, within the space where the 
 gates swing, than at any other place, and the gates when shut, 
 rest against a sill. The walls also, are built in such a form as to 
 admit the gate into their thickness, when open, in such manner 
 that its face, and that of the wall, shall then be in the same plane. 
 The outer side of the post on which the gate hangs is rounded, and 
 the stones are cut, where it touches the wall, into a curved hollow, 
 that permits its motion, and to which the retired part of the wall 
 is a tangent. 
 
 When the lock has a small breadth, say not exceeding four 
 feet, both of the gates may be formed of a single leaf, and are, 
 when shut, at Bright angles to the faces of the walls. When the 
 breadth does not exceed six or seven feet, the upper gate may 
 still be in a single leaf; but in this case the lower gate, and in 
 all locks whose breadth exceeds seven feet, both gates must be 
 formed of two equal, and similar leaves, suspended from the op- 
 posite sides of the lock. The reason of this is, that the pressure 
 would then become too great to be borne by a single length of 
 timber, supported at one end. 
 
 430. When lock gates are made of two leaves, they must be 
 so placed as to afford each other mutual support. For this pur- 
 pose, instead of shutting in such a manner as to lie in one plane, 
 
443 OF CANALS. [Book VI. 
 
 the two leaves make an angle with each other, and the sill has the 
 figure of an isosceles triangle, whose vertex is turned towards the 
 upper level of the canal. This arrangement will be seen in the 
 draught, on page 443. 
 
 It will be obvious, that if this angle be a right angle, the tim- 
 bers of each leaf will receive the pressure of the other in the di- 
 rection of their length, and will therefore afford the greatest 
 mutual support ; while if they lie in one plane, they will not give 
 each other any support. But in the former case, the timbers will 
 be longer, and have less strength to bear the liquid pressure that 
 acts upon the leaf of which they are a part, than in the latter. 
 Upon these principles, the proper angle at which the gates should 
 meet, in order to possess a maximum of strength, may be investi- 
 gated. 
 
 The pressure of water upon the leaf AC, when the depth is 
 constant, will be proportioned to the length of AC, which we shall 
 
 call / ; and the strength of the timbers which form the gate will 
 be inversely proportioned to I ; call the pressure on the unit of 
 surface, P ; and S, the respective strength of the material. We 
 have in the case of equilibrium, 
 
 and 
 
 S=P/ 2 . 
 
 The strength, then, in order to resist the pressure, must be a 
 constant function of the square of the horizontal dimension of the 
 leaf; or if we call the breadth of the lock 2a, and the projection 
 CD of the sill #, a constant function of (a 2 +^) 
 
 The leaf AC, also derives strength from the other leaf BC : if 
 we decompose this force into two parts, one of which is perpen- 
 dicular, the other parallel to AC, and which may be represented 
 by CE, EB ; that represented by EB, will alone act to support 
 the leaf AE. But this force will vary with the sine of the angle 
 A, while x varies with the tangent ; it will therefore be a constant 
 
V \ 
 
 Book rf.] OF CANALS. 449 
 
 ' . * 
 
 function of ax. The whole resistance which a gate of a given 
 breadth will oppose to the pressure of the water, will decrease 
 with the former of these quantities, and increase with the latter, 
 of both of which it is a constant friction, or will be greatest when 
 ax (a 2 +# 2 ) is a maximun, or when 
 
 x=-a=a s ia. 30. 
 
 Hence, the greatest strength will be attained when the pro- 
 jection CD of the sill is one fourth of the breadth of the lock, 
 and the angle that the gate makes with the wall of the lock is 
 60. 
 
 When the length of the horizontal timbers that form the leaves, 
 of a gate, is determined by means of the breadth, and the most 
 advantageous projection thus investigated, their dimensions may 
 be calculated by considering them as beams supported at each 
 end, upon the principles of 189. 
 
 431. The height which is given to locks, is the sum of two 
 quantities : the depth of the water in the lower level of the canal, 
 and the difference between the level of the two surfaces. The 
 latter is called the Fall of the Lock. The fall that is most advan- 
 tageous, will depend upon a variety of circumstances. 
 
 The nature of the ground, in some cases, prescribes ,the 
 proper fall ; but in the hands of a skilful engineer, the quantity 
 of fall may, generally speaking, be settled upon other principles. 
 
 The fall of the locks of a given canal should be constant, so 
 that the water discharged from one may just suffice to sup- 
 ply those below it. In this case, the successive levels should 
 be each capable of containing one lock full of water, in addi- 
 tion to that which is necessary for the navigation, without over- 
 flow. The service of the canal would then be performed without 
 interruption, and without waste. This rule, however, neglects 
 the loss of water by evaporation arid leakage : a better principle 
 would be, that the fall of the several locks should decrease from 
 the point at which the canal receives water, until a new supply 
 be admitted, when they are again to be restored to their original 
 height. 
 
 It is next to be remarked that locks of great fall expend a great 
 quantity of water, and can therefore be used only when that is 
 abundant. 
 
 In locks of small fall, boats require nearly as much time to pass 
 each of them, as to pass one of greater height. Hence, when a 
 given fall is distributed among several locks, instead of a single 
 one, the delays are much increased. 
 
 The thickness of the walls increasing with the depth of water, 
 
 57 
 
450 or CANALS. [Book VI. 
 
 with which their height increases also, the cost of masonry wll 
 increase with the square of the depth. This depth is not the 
 fall of the lock, but is the sum of that quantity, and the constant 
 depth of the canal. Thus the whole cost of the masonry of a set 
 of locks, to overcome a given fall, will decrease, with the increase 
 of their fall, and diminution of their number, to a certain limit, 
 easily determined in each particular case, and afterwards increase. 
 In this investigation, other circumstances will come into view, 
 such as the cost of wood and iron work ; the expense of making 
 and securing the foundations. 
 
 Locks are liable to danger' from the filtration of water through 
 the earth in which they are placed: this may, in some cases, be 
 so great, that the bottom of the lock may be pressed upwards by 
 a liquid pressure, due to the whole surface of the lock and the 
 head of water in the upper level. To counteract this pressure, we 
 have the weight of the lock itself, and that of the water it contains. 
 In a high lock, when empty, the former force may preponderate, 
 and the lock be forced upwards. 
 
 Taking all these circumstances into account, it has been laid 
 down as a rule, that the fall of locks should not exceed ten, nor 
 fall short of eight feet. 
 
 432. The length of a lock will depend upon that of the largest 
 vessels that are to pass it, and will be such that a vessel can lie, 
 without risk of touching, between the angle of the Isaves of the 
 lower gate and the breast wall. 
 
 The breadth of locks must, in like manner, be adapted to that 
 of the vessels, and is usually made one foot greater than their 
 breadth of beam. So also, as has been seen, the dimensions of 
 the canal itself will depend upon that of the vessels that are to 
 navigate it. 
 
 In determining the appropriate size of vessels, the nature of the 
 adjacent navigable communications must be taken into account, 
 as well as the expense of construction, and the character of the 
 moving power employed upon them. As a general rule, the ex- 
 pense of construction increases even in a higher ratio than the 
 dimensions of the canal ; hence, when it is intended to connect 
 two navigations already in existence, it should be calculate^ for 
 the smallest vessels that usually ply upon them. 
 
 The moving power employed, is usually that of horses, walk- 
 ing upon the towing path of the canal ; and a loss of power would 
 ensue if the boats were smaller than is calculated for the draught 
 of a single horse. It has been found, by long experience, that a 
 horse readily drags twenty-five tons, in a boat weighing five tons, 
 upon a canal, or thirty tons in all. Boats of this tonnage are, 
 therefore, well-adapted to canal navigation, and may have a 
 
Book VI.'] OF CANALS. 
 
 length of about 60 feet, a breadth of beam of 8 feet, and a draught 
 of water of 4 feet. On the other hand, it is to be considered, 
 that the resistance to boats of similar figure, increases with the 
 area of their midship frame, or in a ratio no higher than the square 
 of their homologous dimensions ; while the tonnage they carry 
 increases with the cube : and that large vessels, on a canal, do 
 not require a greater number of hands to navigate them than 
 small ones. Two are, in all cases, sufficient ; the one to steer 
 the boat, the other to drive the horse. 
 
 433. In the earlier canals, it was a frequent custom to build 
 locks in juxta-position, like steps of stairs, the upper gate of the 
 one answering as the lower gate of the next, and so on. This 
 disposition is faulty. (1.) Inasmuch as it causes a greater use of 
 water. And (2.) Because it delays the navigation. 
 
 When the locks have a space between them, sufficient for two 
 boats to pass each other, and all the locks are of the same height, 
 a single lock full of water will suffice for the passage of a boat 
 from the summit to the lowest level'of the canal ; and if the trade 
 exactly alternate, the ascending boats will find the locks empty; 
 and they will be filled, in readiness for the descending boats, by 
 the operation of raising those that are ascending. When the locks 
 are in juxta-position, the first descending boat passes through the 
 system with but one lock full of water, but when another is to rise, 
 it finds all the locks empty ; and hence, there must be drawn from 
 the higher level as many locks full of water as there are locks in 
 the system. The second descending boat finds all the locks full, 
 and the whole of them must be discharged in order to permit it 
 to pass. It is therefore evident, that when the locks are combined 
 in a system, they use a quantity of water as many times greater 
 than when they are separate, as there are locks in the system. 
 This waste will not be obviated wholly, by merely placing a basin 
 between the locks, in which two boats may pass each other ; but 
 this basin must be large enough to receive and retain a lock full 
 of water, and to permit that same quantity to be drawn from it 
 without rendering the water too shallow. If the basin have not 
 sufficient capacity, a part of the descending water will run to 
 waste, and the boats may take the ground. If the circumstances 
 of the ground bring the locks very near to each other, the breadth 
 of the basin between them must be increased, until it comply with 
 the above condition. In general, this condition may be attained 
 by increasing the space between the locks ; and it may be shown 
 by calculation, that in a canal, the locks of which have eight feet 
 fall, there should not be-a less space than 200 yards between two 
 contiguous locks. 
 
 In respect to the loss of time arising from the juxta-position of 
 
452 or CANALS. [Book VI 
 
 locks: In an alternating trade, but one vessel can be in the sys- 
 tem at a time, while if there be a space between the locks, there 
 may be an ascending vessel in each lock, and a descending one 
 in each intermediate basin. It is therefore evident that if n be 
 
 the number of locks combined in a system, no more than -th 
 
 part of the number of boats that would pass, were the locks iso- 
 lated, can pass the system in a given time, when they are in juxta- 
 position.. 
 
 There are, however, local cases, in which a system of locks, 
 in juxta-position, must be made use of, instead of an equal num- 
 ber of single ones with basins between them. In this case, the 
 defects we have spoken of, may be obviated by making two com- 
 plete systems by the side of each other. The expense of con- 
 struction is thus increased, but it may be more than compensated 
 by saving in other parts of the work. Thus, at Lockport, on the 
 great Western Canal of the State of New-York, where the upper 
 level of the canal lies on the top of a ridge of compact limestone, 
 that descends rapidly to the lower level, it was found much 
 cheaper to make two collateral systems of locks, than to exca- 
 vate the space necessary for intermediate basins. In such a com- 
 bination, one of the system's is used for the ascending, the other 
 for the descending trade. 
 
 434. Where circumstances compel the use of deep locks, the 
 waste of water that they occasion may be lessened, by making 
 lateral basins to receive a part of the discharge, when a boat is 
 descending, and to restore it to the lock when a boat is to ascend. 
 One on this plan, is described by Belidor, but the necessity of 
 such a form can rarely occur, and the cost of construction would 
 be very great. 
 
 435. The supply of water that is needed for a canal, depends: 
 (l.j Upon the lockage, which will follow the law of the num- 
 ber of vessels that will probably pass. It is usual to assume, that 
 a vessel will expend one lock full of water on each side of the 
 summit. This will certainly be sufficient when the locks are not 
 in juxta-position. 
 
 (2.) Upon the evaporation from the surface, from which must 
 be deducted the quantity of rain that falls upon it. It has been 
 found by experiments made upon the canal of Languedoc, that 
 the annual quantity of evaporation is 32 inches. That is to say, 
 that the allowance for this cause of waste must be equal to a pa- 
 rallelepiped of water, whose base is the whole surface of the water 
 in the canal, and whose altitude is 32 inches. In most calcula- 
 tions, it has been customary to take this altitude at 36 inches. 
 
Book VI.] OF CANALS, 453" 
 
 (3.) Upon the leakage. This may take place through the 
 banks of the canal, or through the gates of its locks. In the 
 former case, it will depend upon the nature of the soil and the 
 extent of the canal ; in the latter, it will be a constant quantity, 
 in canals whose' gates have equal dimensions ; for the leakage 
 through the gate nearest the summit, will supply that which takes 
 place in all the gate on the same side of the summit. 
 
 The leakage through the banks is greatest in new canals; for 
 the banks, if undisturbed, are gradually tightened by their own 
 pressure, and by the particles of earth which the water deposits 
 in filtering through them. When the soil is porous, the canal 
 may be lined with an earth retentive of water, or a portion of the 
 middle of each bank may be built up with a soil of this character. 
 If placed in the middle of the bank, a tough tenacious^clay may 
 answer the purpose. But when upon the inner surface, it is ne- 
 cessary that it should as well resist the action of running water, 
 as the entrance of stagnant. For this latter purpose, a loamy 
 earth, mingled with small pebbles, has been found to be the best. 
 The operation of lining a bank with earth retentive of moisture, 
 is called Puddling. 
 
 When such precautions have been taken, and the banks have 
 become consolidated, it is the estimate of European engineers, 
 that the leakage is twice as much as the evaporation, or amounts 
 to six feet upon the surface of the canal, annually. In our coun- 
 try, this estimate has been found insufficient : it is, however, 
 rather to be ascribed to a defect in the mechanical construction, 
 than to any difference in the physical circumstances. 
 
 436. When a canal unites two navigable streams, and can de- 
 rive its waters from one or both of them, or when it is parallel to 
 a considerable stream, it may be considered that the supply of 
 water cannot fail to be sufficient. But when it passes over a 
 ridge or summit that divides waters falling in two directions, 
 to obtain a sufficient supply will require considerable pains and 
 precautions. As the canal will seek to traverse the ridge at its 
 lowest points or gaps, a channel cut from the summit level, and 
 continued along the side of the higher parts of t}ie ridge, will in- 
 tercept all the waters that flow upon the surface of these higher 
 parts ; and if it have a slope towards the canal, will convey them 
 to its summit level. Such a channel is called a Feeder, and the 
 supply it will furnish will be more certain should it intersect the 
 course of streams. The quantity of water it will certainly fur- 
 nish, may be ascertained by gauging the streams : the slope that 
 must be given to it, will probably be determined by local circum- 
 stances ; and the proper dimensions to convey the given quantity 
 upon the given slope, may be ascertained upon the principles of 
 411. 
 
454 OF CANALS. [Book VI. 
 
 Even when no stream of magnitude is traversed by the feeder, 
 it may, if the extent of ground higher than it be considerable, re- 
 ceive a sufficient quantity of surface water to feed a canal. This 
 may be judged of, by knowing the extent of surface that the feeder 
 will drain, and the quantity of rain that falls upon it. A par- 
 ticular investigation will be necessary in relation to the latter cir- 
 cumstance, for the quantity of rain is far more influenced by local 
 causes, and particularly by difference of altitude, than the quan- 
 tity evaporated. 
 
 437. In almost all climates, the quantity of water furnished by 
 streams, or directly by rain, is various at different seasons. At 
 one time in the year the supply will be in excess, at another in 
 defect. To guard against the consequences of this inequality, 
 reservoirs must be constructed. These are made by closing up the 
 entrances of vallies, into which the feeders are conducted, and 
 must lie at so high a level that the water will run from their bot- 
 tom into the canal ; for, when this is not the case, no more water 
 can be drawn from them, than lies above the level, whence water 
 will run to it. As reservoirs are liable to evaporation from their 
 surface, they ought to be constructed in places that will admit of 
 their containing a large quantity of water, with the smallest pos- 
 sible surface ; their banks ought therefore to be steep, and capa- 
 city obtained by increasing their'depth, rather than their breadth. 
 The water should be drawn from their surface, in order that it 
 may be free from earthy matter, which the liquid remaining at 
 rest in the reservoir, will deposit. 
 
 The strength of the walls and banks, by which water is re- 
 tained in reservoirs, follows the same law as that of the banks of 
 canals, 422, or the walls of locks, 428, according to the mate- 
 rial of which they are constructed. 
 
 438. It is in all cases of extreme importance, that none but 
 clear water should be admitted into a canal. If this precaution be 
 not observed, the canal will fill up, and lodgments will take place 
 in the locks, that will prevent the working of the gates. Hence, 
 reservoirs may fulfil an important purpose, by clarifying the 
 water, even when unnecessary to equalize the supply. In the 
 original execution of the canal of Languedoc, no precautions were 
 taken to admit none but clear water ; on the contrary, efforts were 
 made to introduce into the canal, every stream whose course it 
 intersected. The consequence was, that after a few years, it was 
 under contemplation to abandon it, rather than incur the great 
 expenditure demanded for maintaining it of a depth sufficient for 
 navigation. Vauban, however, who was deputed for that especial 
 purpose, pointed out the means of preventing this difficulty from 
 
Book VI.] OF CANALS. 455 
 
 occurring again. For the same reason, it is necessary in the 
 New- York canals to discharge the water from them, and excavate 
 annually several inches of sediment. The banks are in conse- 
 quence exposed to the action of the frost, and are rendered liable 
 to give way on the re-admission of the water. These inconve- 
 niences may be obviated by certain simple precautions : 
 
 (1.) No water should ever be admitted into a canal, until it 
 has remained in a reservoir long enough to discharge its impuri- 
 ties ; and the water should, as far as possible, be admitted at but 
 a few points in the higher levels, whence the rest are to be sup- 
 plied. 
 
 (2.) When a stream intersects the course of a canal, it is not to 
 be admitted, but passed under it by a culvert, or over it by an 
 aqueduct. < '-V 
 
 (3.) The rain, and surface waters of the country, that lie 
 higher than the canal, must be intercepted by the counter ditch, 
 and passed at proper places beneath the canal, to the lower 
 country. 
 
 439. When vtrater is to be passed beneath a eanal, a Culvert 
 must be constructed. This is an arched channel of masonry that 
 is built beneath the bed of the canal, and reaches from one side of 
 its embankment to the other. If the water to be conveyed, lie at 
 about the same level with that in the canal, the culvert has the form 
 of an inverted syphon, and acts upon the principle of a water pipe. 
 The water may then be raised to nearly the same level at the 
 end of the culvert, opposite to that at which it enters, and may 
 resume its former channel. When used to convey water whose 
 level is considerably lower than that in the canal, it may be con- 
 sidered as an open channel, the arch performing no other office 
 than to support the embankment of the canal. The nrcre surface 
 waters should be conveyed beneath the canal, at regular intervals, 
 and by small culverts; but streams will require culverts adapted 
 to their capacity ; and U*is generally better to unite a number of 
 small culverts, than to make the arched passage of too large a 
 size. 
 
 440. When the stream is large, or its valley wide, it becomes ne- 
 cessary to convey the canal across it by an aqueduct. An aque- 
 duct is a bridge, that instead of carrying a road, contains the chan- 
 nel of a canal between its parapets. If built of masonry, it also 
 contains the towing path and bank of the canal, and a sufficient 
 mass of earth beneath its channel, to prevent filtration. The 
 breadth of the channel is usually but half as great as in other 
 parts of the canal, and therefore but one boat can pass the aque- 
 duct at a time. 
 
456 
 
 OF CANALS. 
 
 [Book VI. 
 
 The following figure represents the section of an aqueduct of 
 masonry. 
 
 In our country, aqueducts are frequently constructed, by form- 
 ing a channel for the canal, of plank confined by frames of tim- 
 ber, as in the figure beneath. The towing path is also formed of 
 wood. 
 
 In England, cast iron formed 
 into plates, and united by screw 
 bolts and nuts, is often used for 
 aqueducts; they may be supported 
 on pillars of stone or iron, and 
 are perhaps the very best kind of 
 aqueducts, except where the cost 
 of the material much exceeds that 
 of slone or wood. 
 
 The most remarkable aqueduct 
 of this description, is upon the Ellesmere Canal, across the valley 
 of Llangollen, in Wales. 
 
 The principles upon which the dimensions of the several parts 
 of aqueducts, may be calculated, are the same with those of 
 bridges, and of the banks of canals, or the walls of locks, and 
 need not to be repeated. 
 
 441. Feeders or other sources of supply, may bring into a ca- 
 nal more water than can be used in lockage, or wasted by evapo- 
 ration and leakage. This, running over the gates of the locks, 
 would cause a current that must injure the banks, and if the levels 
 be long, might swell and overflow them. To prevent the water 
 accumulating, Waste Gates are constructed at intervals. These are 
 built of masory, and have the shape of a triangular prism, the 
 edge of which is at the level beyond which the water is not to 
 be permitted to rise. If made in the towing path, a bridge is 
 provided for the passage of the horses employed in draught. 
 
 These waste gates are placed where a natural channel has existed, 
 by the continuation of which the surplus waters may be carried 
 

 . t 
 Book F/.] OP CANALS. 457 
 
 off; or where such channels do not occur, an artificial channel is 
 formed for the purpose. 
 
 442. When a country is mountainous, and the construction of 
 canals would be attended with great and sudden changes of leve'l, 
 locks become too expensive to permit their application; and 
 it thus happens that in many districts, where, canals might be 
 supplied with abundance of water, and the wants of commerce 
 demand them, they are notwithstanding considered impracticable. 
 In such cases modifications of the inclined plane, rendered self-act- 
 ing, might no doubt be effected. But although often proposed, 
 and in forms to which no reasonable objection can be applied, 
 they have not yet been brought into successful action. 
 
 443. When water is scarce, canals are impracticable, or at least 
 susceptible of conveying only a limited trade. Some of the modi- 
 fications of the inclined plane use less water than locks, and might 
 be advantageously employed in such cases. 
 
 Betancourt has proposed a lock, which vessels may pass with- 
 out any expenditure of water whatever. A basin of area equal to 
 the lock is built beside it, and communicates with it at bottom. In 
 this basin is placed a water tight case that has nearly the same area 
 with it, and is of the same density with water. A small force will 
 therefore be sufficient to sink or raise it at pleasure. If the gates 
 of the lock be closed, and this case be depressed, it forces water 
 from the basin to the lock, until it reach the level of the upper 
 reach of the canal, and a boat may be thus raised up. If a boat is 
 to descend, the gates are again closed , and the case drawn upwards, 
 the water in the lock then flows back into the lateral basin, and 
 the boat floating upon it -falls to the lower level. As the depth 
 of water in the basin varies, a consequent variation will be de- 
 manded in the force that elevates or depresses the case. This is 
 effected by a counterpoise, moving at the extremity of the arm of 
 a bent level, in an arc of a circle whose plane is vertical. The 
 whole arrangement may be understood by the inspection of the 
 figure on the succeeding page. 
 
 5S 
 
453 
 
 OF CANALS. 
 
 [Book VI 
 
 ABODE, are the walls enclosing the chamber of the lock H, 
 and the lateral chamber I. F, is a wall separating the chamber of 
 the lock, from the lateral basin. In this wall is the arched opening 
 G, forming a communication between the chamber, H, and basin, 
 I. L, represents the upper gate of the lock resting on the breast 
 wall. K,a hollow plunger, by the elevation or depression of which 
 the common level of the water in the basin and chamber, is raised 
 or lowered, m m, a chain passing over the pully, N, and connect- 
 ing the plunger, K, with the bent-lever, m P, at the extremity, 
 P, of which the counterpoise is situated. 
 
 This counterpoise moves in the quadrantal arc, r P s ; and 
 when the plunger is depressed to its greatest depth, presses verti- 
 cally on the fulcrum, 0, and has no action on the plunger ; in other 
 positions in the quadrant, its action, as has been demonstrated by 
 Betancourt, increases in precisely the same ratio as that part of the 
 weight of the plunger, which is not supported by the fluid pres- 
 sure of the water in the basin. 
 
T^ook PL] PERCUSSION, &c. 459 
 
 '^KT 1 '."*; 
 
 CHAPTER VIII. 
 
 OP THE PERCUSSION AND RESISTANCE OF FLUIDS. 
 
 s 
 
 444. When a surface is exposed to the action of a fluid, in mo- 
 tion, or when a surface in motion impinges against a fluid, if we 
 have no regard to what occurs after impact, the circumstances 
 may be considered as identical ; that is to say, the action will, in 
 the one case, depend upon the velocity with which the fluid strikes 
 the surface; in the other, on the velocity with which the solid 
 strikes the fluid. When both are in motion, the effect will ob- 
 viously depend upon the difference between the two velocities, 
 or, what is called the relative velocity, and either may be con- 
 sidered as being in motion with this velocity. Whichever of the 
 two is in motion, or if both be in motion, we may consider the 
 action as identical with a resistance, opposed by a plane at rest to 
 a fluid moving with the relative velocity ; and the term Resist- 
 ance may be, in all cases, employed to denote the action. 
 
 The theoretic investigation of this problem is attended with 
 great difficulty, inasmuch as the particles of the fluid interfere with 
 each other's action, even before impact, and continue to act after 
 impact, according to laws that it is impossible to ascertain. 
 
 445. If we suppose that the fluid particles strike in guccession 
 against the surface exposed to them, losing by their action so 
 much of their velocity as is in the direction of a normal to the 
 surface, and that they then cease to act, either upon the surface, 
 or on the remaining particles, we have the hypothesis that is most 
 commonly employed in this investigation, and which we shall 
 now make use of. 
 
 Let a fluid, whose density is s, strike with a velocity r, at right 
 angles against a plane surface whose area is A. Let h be the 
 height due to the velocity v, dx the space through which the fluid 
 passes in the time dt. 
 
 We have therefore from, (53) 
 dx 
 
 The quantity of fluid that strikes against the surface in the time 
 dt, will be Adx, and its mass 
 
 Aadar; (478) 
 
 and as it moves with the velocity u, its quantity of motion in dt 
 
 will be 
 
 Asvdx-, (479) 
 
460 PERCUSSION AND [Book VL 
 
 and in the unit of time 
 
 A sv dx 
 
 -^r-' 
 
 dx 
 
 substituting the value of -77 from (477), we have for the resist- 
 ance R, 
 
 R=A.sv*=2gA.sh. (481) 
 
 If the fluid be water, s=l, and 
 
 R=At> 2 =2gA&. (482) 
 
 In the case of direct impact then, the action of a liquid in mo- 
 tion against a surface, should be proportioned to the area of the 
 surface, and the square of the velocity. 
 
 If the fluid strike against the plane at an angle of incidence t, 
 the velocity must be resolved into two components, one of which 
 is parallel, the other perpendicular to the plane ; the latter will be 
 represented by v sin. t, and the formula (482) becomes 
 
 R=A (v sin. i) 2 =A v* sin. 2 i. (483) 
 
 In oblique incidences then, the action of the fluid should be 
 proportioned to the area of the surface, the square of the velocity, 
 and the square of the sine of the angle of incidence. 
 
 To apply this theory to cases that may occur in practice : 
 Let the body be a prism whose section is an isosceles triangle ; 
 let the area^of the rectangular ^se be A ; the angle of the vertex 
 of t^e triangle 2 : The resistance on each of the sides is 
 
 ~ A v 2 sin. 2 t, 
 
 
 
 and the whole resistance, 
 
 A 2 sin. 2 . (484) 
 
 If the triangle be right-angled, the resistance on its faces 
 will bo 
 
 AT) 2 
 
 -g-. (485) 
 
 It may also be shown, by the application of the calculus to the 
 same theory, that the resistance to a half cylinder, is two thirds 
 of that to the rectangle which forms its base ; and that the resist 
 ance to a hemisphere is one half of that to a plane surface of the 
 size of its great circle. 
 
 446. These investigations being of rio practical value, we 
 shall omit them, and proceed to the results of experiment, in the 
 case to which the hypothesis bears the closest analogy, namely, 
 that of a jet of fluid striking against a plane surface. 
 
 From the bestiexperiments that have been made in this case, 
 namely, those of Bossut, it has been concluded : 
 
Book VI. RESISTANCE OF FLUIDS. 461 
 
 (1.) That under similar circumstances, the action of the fluid 
 is nearly proportioned to the area of its section. 
 
 (2.) That all other circumstances being equal, the force is nearly 
 proportioned to the square of the velocity. 
 
 (3.) That the force of the fluid, when the plane is oblicme to 
 the direction of the current, does not follow the law ol the 
 squares of the sines of the angles of inclination. At least at the 
 angle of <JO, the resistance is always less than would be due to 
 that ratio. x ( 
 
 (4.) The absolute measure of the force with which the fluid 
 strikes the plane directly, is not constant, or as given in (482) 
 
 2A gh, 
 
 but varies according to the ratio that the surface bears to the 
 section of the vein. If the surface be much greater than the 
 section of the vein of fluid that strikes it, the above formula 
 is true; but as they approach more nearly to equality, the force 
 of impact becomes less, until, when they are about equal, the 
 force may be represented by 
 
 -Agfc; (486) 
 
 or is no more than three eighths of that derived from the hypo- 
 thesis. 
 
 When the phenomena with which these experiments are 
 attended are carefully observed, they are found to be as follows: 
 
 The vein of fluid is enlarged as it approaches the surface, form- 
 ing a conoid whose curvature is convex towards the axis, and 
 varies with the greater or less size of the surface. The particles, 
 after striking the surface, move off, if the surface be small, at a 
 great angle of obliquity ; and this angle lessens as the surface 
 increases, qntil they move parallel to the surface, or as it were, 
 slide along it. When this takes place, the resistance becomes 
 equal to that deduced from the theory, and is no farther increased 
 by an increase of the extent of the surface. But if the surface 
 be surrounded by a ledge or rim, the resistance is increased be- 
 yond that deduced from the hypothesis ; and in a concave sur- 
 face, the resistance which ought, according to the hypothesis, be 
 less than if it were plane, is also increased beyond that given by 
 the formula (482). 
 
462 PERCUSSION 'AND [ Book J 7 /. 
 
 These circumstances may be reduced to the test of analysis. 
 
 Let PQ be a surface exposed to the action of a jet of liquid, 
 and let the curved lines, ALX, BY represent the boundaries of a 
 section of the vein made by a plane passing through the axis 
 MN, and let the vein be a solid, generated by the revolution of 
 either of these similar curves, an*d assume the surface acted upon 
 to have a circular section. Let M be the point where the vein com- 
 mences to spread. We may conceive that the particles in motion 
 divide themselves at the point M, leaving within them a conoid of 
 stagnant fluid PMQ, and that the action upon the plane is trans- 
 mitted to it by this conoid. We may also consider that the velocity 
 in the current that surrounds PMQ, is constant, as there is no ob- 
 vious reason for its change, and that it is equal to that the vein 
 had before it began to spread. 
 
 This being premised, let the area of the vein at A MB = a ; its 
 constant velocity = v; and, referring the curve to the axis MN, let 
 MG = x; GH = g- the arc MH =s, and HL, the breadth of the 
 current at H =z ; the radius of the osculating circle, at H =R ; 
 the angle NPT of the obliquity with which the particles leave the 
 surface = <p. As the velocity is conceived to be constant, the 
 area of the current contained between the corresponding circular 
 sections of the outer and inner cone will be constant also, and 
 this area is formed at H, by the revolution of the line HL, 
 which is a normal to both the curves. If HL be bisected in O, 
 and OF = y' be drawn perpendicular to the axis, this area will be 
 
 2 ir y'z. 
 
 If we take an elementary ring of the moving fluid, whose sec- 
 tion is H /, its mass will be 
 
 2 if g'z ds ; 
 and its inner boundary, generated by the revolution of the line H h, 
 
Book VI.] RESISTANCE OF FLUIDS. 463 
 
 Now by the principles of 65, every element of the fluid con- 
 tained in such a ring presses upon its inner boundary with a cen- 
 tral force, represented by (85), 
 
 and this force multiplied by the mass, 2 < y'z ds, will give the- 
 moving force with which the fluid in the elementary ring presses 
 upon its inner boundary, 2 if yds. This product is 
 
 2 
 
 T7 . 2iry f ds. (487) 
 
 This pressure may be represented by the weight of a cylinder of 
 the fluid that has for the area of its base 2 if y cfo, and for its alti- 
 tude P, or by 
 
 2* g P y ds ; 
 and 
 
 whence we obtain for the value of P, 
 v*z y' 
 
 ; (488) 
 
 and from the general distinctive property of fluids, every point 
 of the surface, PQ, will be pressed by a column whose alti- 
 tude is P. 
 
 If now in the preceding equation we substitute, for z and R, 
 their respective values, 
 
 d ' d~s 
 we obtain 
 
 dx 
 2gPy dy= A M . j- g ; (489) 
 
 and integrating 
 
 Av 2 . (490) 
 
 dx dx 
 
 When 2/=0, we have ^ = 1, and when ?/=PN, we have JJ=\/ 
 
 sin. cp ; we therefore have for the integral, 
 
 g . P X PN 2 =At> 2 (1 sin. 9) ; (491) 
 
 but the first member of this equation is obviously the value of the 
 resistance, for it is the weight of a cylinder of the fluid whose base 
 is the circle PN, and whose altitude is P ; the resistance, there- 
 fore, is 
 
 Av 2 (1 sin. 9). (492) 
 
464 PERCUSSION AND [Book VI. 
 
 It is evident from this, that as the angle <p decreases, the resist- 
 ance should increase until <p=90, or when the fluid moves along 
 the plate ; in this case it becomes 
 
 Av 2 , 
 
 according to the hypothesis, and these results are in conformity 
 with the experiments. 
 
 If a ledge be formed around the surface PQ, the fluid cannot 
 escape on'the side opposite to that on which it impinges, or move 
 along the surface ; but must be thrown off in a direction making 
 the angle 9 on the same side of PQ with the axis MN ; in this 
 
 dx dx 
 
 case -T- becomes negative, and when /=PN, -7-= sin. <p ; 
 
 the expression for the resistance therefore becomes 
 
 Afl 2 (l-fsin. <p), (493) 
 
 and when <p=90, 
 
 2Ar 2 . (494) 
 
 We may thus explain the very great difference in the resistance 
 which a body of the form of a portion of a hollow sphere meets, 
 when moving in a fluid, with its different surfaces opposed to it. This 
 difference, it will appear from the theory, may be in a relation as 
 great as four to one ; the concave surface being resisted four times 
 as much as the convex ; and in practice, the difference is even 
 greater. Upon this principle the parachute is applied to balloons. 
 
 447. When a surface, instead of being struck by a vein of fluid, 
 is immersed in a mass of that description, and has a relative velo- 
 city in respect to it, growing out either of its own motion, or that 
 of the fluid, or a combination of both ; a similar conoid of stagnant 
 water will be formed in front of it, and a similar diminution of the 
 resistance will take place. We can only ascertain the quantity 
 of this diminution by experiment, which shows that when the 
 area of the stream is great in proportion to that of the surface, 
 the resistance may be represented by 
 
 . 
 
 2 : (495) 
 
 or is no more than the half of that pointed out by the hypothe- 
 sis. 
 
 If, however, the channel be narrow, and the surface fills up a 
 large portion of it, the resistance augments. The experiments 
 of Du Buathave given the following results; 
 
 Let the resistance in a channel of indefinitely large size be 
 10000, and M the ratio between the area of the channel and that 
 of the obstacle, the resistance in the channel will be 
 84600 
 
Book P7.] RESISTANCE OF LIQUIDS. 465 
 
 If, then, the obstacle fill up half the channel, the resistance is 
 more than double what it is in an indefinitely large channel, 
 and we may consider the channel as large enough to get nd of 
 this cause of increased resistance* when its area exceeds 6| times 
 that of the obstacle. 
 
 448. When the surface acted upon, is under the circumstances 
 of a floating body, one part being immersed in the fluid, the other 
 floating above it, the level of the water is raised on the side on 
 which the fluid acts, and a depression takes place on the opposite 
 side. 
 
 The fluid striking against a plane surface with the velocity, , 
 will tend to rise in front of it, to the height due to this velocity, or 
 to (60) 
 
 In like manner, a depression will take place behind a solid body, 
 and if the surface be a plane parallel to the anterior surface, the 
 whole difference of level will be, 
 
 v* 
 
 ff* 
 
 c 
 
 Upon an elementary rectangle, whose constant breadth = &, and 
 depth = dx, the action will be due in part .to the velocity with 
 which it moves through the fluid, and partly to a pressure due to 
 the depth, #, and may therefore be represented (495) by 
 
 bdx(v V2gx) 2 . (497) 
 
 Integrating, we have, for the additional pressure growing out of 
 the rise of the fluid, 
 
 An equal resistance will grow out of the depression behind, so 
 that the whole increase growing out of this cause, will be 
 
 bv* 
 TV-"- (499) 
 
 O 
 
 It will be obvious, however, that the fluid cannot rise to the height 
 due to the whole velocity, and therefore although the resistance 
 growing out of this cause, probably varies with the fourth power of 
 the velocity, it has a co-efficient considerably less than is deter- 
 mined by the above investigation. 
 
 449. We may therefore infer that the resistance of water to a 
 body floating upon its surface, is composed of two separate forces, 
 the one due to the impulse of the fluid ; the other to the wave 
 raised before, and the depression that takes place behind it. The 
 first varies with the square, the second with the fourth power of 
 the velocities. The first, however, is alone sensible at moderate 
 
 59 
 
466 PERCUSSION, &c. [Book VL 
 
 velocities, but the second becomes the most intense at great velo- 
 cities, and must finally cause a limit beyond which the speed of a 
 body moving at the surface of a liquid cannot be carried. No 
 such resistance affects bodies wholly immersed in a liquid, and 
 hence, as the limit of speed depends upon the square of the velo- 
 city, it is not as soon attained as in the former case. 
 
 450. Our investigations appear to show that the resistance to 
 surfaces inclined to the direction at which the fluid strikes them, 
 varies with the square of the sine of the angle of incidence. This 
 is found to be far from true, in the case of bodies moving in mas- 
 ses of fluids, when submitted to the test of experiment. The 
 theory of Juan, makes this resistance to vary with the sine sim- 
 ply ; this is however equally, or even more faulty, except when 
 the angle is great ; and even in this case it gives a result below 
 the truth. 
 
 If the inclination of the surface to the direction of the fluid, do 
 not exceed 30, or the angle of incidence is between 60 and 90, 
 the formula (483) corresponds nearly with the truth; at inclina- 
 tions from 30, to 60, the following formula deduced by Bossut, 
 from experiment, will give results nearly accurate. 
 
 R=f. 2 s in. fM-0.003 (90 ) 3 - 25 .. (500) 
 
 At still greater obliquities, the following formula of Romme, 
 gives a nearer approximation. 
 
 ou 3 2+ sin. 2 
 
 * < R=-r- 30 . 
 
* ''? 
 
 Book VI.] fUE MOTION OF WAVES. , 467 
 
 CHAPTER IX. 
 
 Or THE MOTION OF WAVES. 
 
 451. When a pressure is applied to any portion of the surface 
 of a liquid, the column pressed is shortened or diminished in 
 depth; the excess will enter into the surrounding columns, and 
 the pressure will, from the nature of fluids, be propagated to them. 
 They will in consequence rise above their original level. But 
 not being sustained by a hydraulic pressure, they will again 
 fall, and thus acquiring a velocity due to their height, will de- 
 scend below the level, until that velocity is overcome by the ac- 
 tion of the adjacent columns. These will, in consequence, receive 
 a similar motion, which they will in turn propagate. Thus a se- 
 ries of ridges and intermediate cavities, will be formed upon the 
 surface, and appear to propagate themselves in circles, from the 
 column to which the original impulse was given. This motion 
 that appears to take place on the surface, is not a progressive mo- 
 tion of the mass, for the particles of liquid that pass from one col- 
 umn to another, return again to that to which they originally be-' 
 longed ; and even this takes place below the surface. The pro- 
 pagation of motion, therefore, that takes place at the surface, is 
 the communication of a tendency to oscillate, and cannot, so long 
 as these oscillations are unimpeded, give a progressive motion to 
 the particles at the surface. 
 
 452. The phenomena of the motions of waves have been com- 
 pared to those of a fluid oscillating in a bent tube, and although 
 the analogy is not complete, yet there are so many points of co- 
 incidence, as to authorize us to adopt this phenomenon as the foun- 
 dation of our theory. 
 
468 
 
 THE MOTION 
 
 [Book VI. 
 
 Let AB, CD, represent two branches of a bent tube, equally in- 
 clined to the horizon. Let the section of the branch AB w, the 
 
 section of the branch CD=n. Let the vertical height EF of the 
 tube =p, and the height of the level of the liquid in the branch 
 AB, above the common level when at rest = a. 
 
 The branches being equally inclined, the spaces, in the direc- 
 tion of the axes of the respective branches, through which the li- 
 quid oscillates, will be proportioned to the vertical altitude of these 
 spaces. Call the vertical height through which the liquid oscillates 
 in the branch AB, z. Then as the liquid that is depressed in the 
 one branch, enters into the other, the quantity that rises above the 
 original level in each will be equal ; the altitudes in the respective 
 tubes, will be inversely as their respective areas ; and the vertical 
 height of the oscillation in CD, will be 
 
 mz 
 
 453. To apply this to the case of waves diverging in circles 
 from a point. They may be considered as formed by the vibra- 
 tions of liquids, in a series of concentric cylindric tubes; the in- 
 tervals between which are infinitely small. Their respective 
 areas, are therefore proportioned to the circumferences of the cir- 
 cles that form the sections of the cylindric tubes, and as these are 
 proportioned to their diameters, the heights to which a circular 
 system of waves rises, will decrease in the inverse ratio of their 
 distances from the point of original action, or in an arithmetic 
 progressiou. 
 
 The force by which the fluid is caused to oscillate, will be the 
 difference of the pressures of the columns in the two branches ; 
 this will be measured, 331, by multiplying the area, m, by the 
 
Book V[.~\ OF WAVES. 469 
 
 % ' .-'.i" ' .'&> 
 
 height a, supposing the density of the liquid =1, or will be ma. 
 The column of fluid, whose pressure in the opposite branch is 
 equal, will be measured by the product of the area, w, into a height 
 which is to a, inversely as the respective areas, or by 
 
 amn 
 
 n 
 
 and the motion acquired by the column in AB, will continue until 
 the pressure in CD become equal to the original force in AB, or 
 until the column in CD becomes 
 
 2am; 
 
 hence, the space through which the oscillation in CD is per- 
 formed, becomes equal to 
 
 2m 
 
 n 
 and 
 
 z=2 a . 
 The column in CD will now preponderate, and by a similar 
 
 course of reasoning may be shown to descend through , and 
 
 to elevate the column in AB through 2 a. Thus, if we abstract 
 from friction, the force exerted in each oscillation will remain 
 constant ; the oscillations will therefore be constant in extent, and 
 the spaces described will be proportioned to the forces ; whence 
 we may at once infer that the oscillations are isochronous. 
 
 The times of these isochronous oscillations will depend upon 
 the space through which the motion is performed, and the inten- 
 sity of the moving force ; and as the-latter is equal in both direc- 
 tions of the oscillation, the circumstances will be the same as if 
 both branches of the tube had equal areas. 
 
 Let the branches be of equal diameters, and let m, be the com- 
 mon area of the two branches, and a, the original elevation or de- 
 pression from the common level. The difference between the 
 masses contained in the two branches of the tube, will be 2 am, 
 and their sum will be Im ; /, being the whole length of the column 
 in both branches, or of the axis of the tube. Applying the principle 
 of D'Alembert, it may be shown as in the investigation of the prin- 
 ciple of Atwood's machine, 95, that the accelerating force at the 
 beginning of an oscillation is represented by the difference of the 
 masses divided by their sum, or by 
 
 2am _ 
 
 2a 
 which is equal to -y . But this force will be variable, 
 
 because the difference of level in the two branches of the tube is 
 continually changing. When the liquid in one branch has de- 
 scended through the distance #, it will have risen an equal quan- 
 
THE MOTION [Book VL 
 
 tity in the other branch, and the accelerating force/, will have be- 
 come 
 
 _ 2 a 2x 
 
 f= ~T- ; 
 
 and, as in the formula, (104a), 
 
 di=^^ gdt ; (502) 
 
 and from the general formula, (53), 
 
 . . . ., 
 
 substituting this value of dt, in (502), we have 
 vdv=^ (2adx2xdx} . 
 Integrating, and rejecting the constant quantity, because when 
 
 whence 
 
 v =^/ JL# (2 ax a?) ] ; (503) 
 
 substituting this value of 0, in the general expression, (53), we 
 have 
 
 dt=- 
 
 whence 
 
 integrating, and rejecting the arbitrary constant, because when 
 x=0, t=0, we have when #=a, 
 
 (505) 
 
 i . ,1^ . O 
 
 which is the time of half an oscillation. The time of a complete 
 I - oscillation is therefore 
 
 t=v(j>-). , (506) 
 
 O 
 
 This is by (286) the time in which a cycloidal pendulum, whose 
 length is half //would perform its vibrations. 
 
 454. To apply this to the case of waves. A wave is contin- 
 ually oscillating between its highest and lowest points, and the 
 motion being considered analogous to that of a fluid in a bent 
 tube, the time of an. undulation will be that of the oscillation of a 
 
/.] OF WAVES. 471 
 
 pendulum, whose length is half the distance between the highest 
 and lowest points. The wave will return to its original height, 
 or will appear to pass over the distance between two contiguous 
 elevations or depressions in twice this time, or in the time of a 
 single vibration of a pendulum, whose length is four times as great 
 as that whose oscillations correspond with those of the waves. 
 But if we abstract from the elevation of the wave, the length of 
 such a pendulum is the distance between two contiguous elevations, 
 or which is called the Breadth of the wave. 
 
 If we call this breadth 6, the time T, of a wave running its 
 breadth, may be represented by 
 
 T= V - = * V . (507a) 
 
 S S 
 
 Fnom this we obtain for the value of the mean velocity with which 
 the wave is propagated, by substituting in the general formula, 
 
 the value of *=2 _-/, and the foregoing value of T, 
 
 455. When a series of waves are proceeding from a centre, and 
 meet a vertical obstacle, they are reflected ; for the effect of the 
 obstacle will be the same as if a new impulse were given to the col- 
 umn, causing an elevation or depression, equal to that they ac- 
 tually have ; and this will be propagated like the original impulse, 
 but in a contrary direction. The waves moving in circular arcs, 
 the reflection takes place in similar circular, arcs ; and thus, the 
 series of reflected waves will proceed exactly as if it came 
 from a centre as far distant behind the vertical obstacle, as the 
 original centre is in front of it. If the series of waves flow par- 
 allel, and of equal height, which, as we shall presently see, may oc- 
 cur, the reflected waves will diminish in height, as if they pro- 
 ceeded from behind the obstacle, and the joint elevation of the 
 wave in immediate contact with the obstacle will be higher than 
 any other. 
 
 456. The original, and reflected waves, being neither of them 
 attended with a progressive motion, will not interfere with each 
 other's progress ; but where the elevations of a wave of each series 
 correspond, the elevation is the sum of the two ; and where the 
 depressions correspond, the resulting depression is also the sum 
 of the two depressions. Where a depression of one series corre- 
 sponds with an equal elevation in another, the surface of the liquid 
 will not change its level. And in two series of circular waves, 
 there will be certain points symmetrically arranged in curves, 
 
472 THE MOTION [Book VI. 
 
 as might be readily shown by a diagram, in which these effects 
 will counteract each other ; and thus, although different series of 
 waves do not interfere with each other's propagation, they do 
 still neutralize each other's effects, at particular points. This ac- 
 tion is styled Interference. 
 
 457. If the obstacle against which waves strike, have a vertical 
 opening in it, of small horizontal breadth, the oscillating col- 
 umns that reach it, will act there, as an impulse originally ex- 
 erted at that point would have done ; and^ hence, a new series of 
 waves will appear to proceed from the orifice. If the breadth of 
 the orifice be increased, new series of waves will appear to pro- 
 ceed from it, but they will no longer have the figure of circles, 
 for the motion of oscillation will be propagated through the ori- 
 fice, and act most powerfully in the direction of a sector, whose 
 centre is at the point whence the original impulse proceeded. 
 
 458. When the wind acts to raise waves, they do not diverge 
 from a centre, but usually proceed in parallel lines, straight, or 
 nearly so. If the impulse were momentary, the waves would 
 decrease in height, in consequence of the viscidity of the fluid, 
 and the friction among its particles. But as winds blow for a 
 space of time of some duration, the original impulse is increased 
 rather than diminished, and thus waves continue to rise, and their 
 propagation may take place with increased, rather than diminished 
 altitude. The increase in height will continue, until the sum of 
 the columns elevated above the general level, and the friction 
 become equal to the disturbing cause. The limit to which a single 
 series of waves can be raised by the wind, has been inferred to be 
 no more than 6 feet. As wind blowing over the surface of smooth 
 water, moves parallel to it, the original cause of waves being 
 raised by the wind, is friction ; but after the waves are raised, the 
 wind acts upon that surface which is inclined to it; and its force 
 may be resolved into two components, one of which tends to in- 
 crease the elevation. The whole force of the wind also tends to 
 give a progressive motion, to the mass of water included in the 
 elevation of the wave ; and thus the shape of the waves ceases to 
 be a figure, with two surfaces equally inclined to the horizon ; and 
 the surface on the side opposite to the wind, has its inclination 
 increased. This increase may become so great as to make the 
 wave project beyond its base ; in which case, the force of gravity 
 will cause the summit to break, and roll over the surface of the 
 wave beneath. 
 
 If a wind, after having raised a series of waves, shall cease to 
 blow, and another arise from the opposite point of the compass, 
 the latter will act against surfaces more inclined to the horizon, 
 than the other did, and will thus produce a greater effect. It there* 
 
Book F/] or WATER. 473 
 
 fore happens, that at sea, the highest waves are raised by sudden 
 changes of wind, or when a wind blows in a direction contrary to 
 that in which the motion of oscillation is propagated. Any change 
 in the direction of the wind, will create a new series of waves 
 crossing the first ; and thus the elevations and depressions, or the 
 total height of the waves may be increased. It is in this man- 
 ner that the very great excess of the height of waves beyond 
 the limit stated for a wind blowing in a constant direction, is 
 caused. And in conformity, we find the ocean comparatively 
 smooth in those fegions of the earth that are the seat of constant 
 winds, and that the height of waves is greatest in those regions 
 where changes of wind are most frequent. 
 
 Wind acting by its friction to raise waves, it may be inferred 
 that if any substance capable of lessening friction be interposed, 
 the elevation that would otherwise take place is lessened. And 
 in consequence, it has been found that when oil is poured 
 upon the surface of water, waves are rarely formed except by the 
 most intense winds ; and if poured upon waves already formed, 
 it permits the viscidity and friction of the water to act to bring 
 them to rest ; thus, oil may be used to lessen the dangers to 
 which vessels are exposed, by the violence of the oscillation of 
 waves, which is in some cases very great. 
 
 459. When waves meet ah inclined obstacle, the columns in 
 which we have conceived them to vibrate, are lessened in depth, 
 and thus their fluid pressure is diminished. The waves no longer 
 meeting with the same resistance as before, the liquid acquires a 
 progressive motion, which will carry it up the inclined surface, 
 until its moving force is counteracted by the weight of the quan- 
 tity thus elevated above its original level. For this reason, 
 breakers or surf, form upon shelving coasts, whatever be the di- 
 rection of the wind. 
 
 When waves are raised by the wind, the influence being ex- 
 erted wholly upon the surface cannot penetrate to any great depth. 
 From 30 to 40 feet, is inferred to be about the greatest distance 
 from the surface, to which the agitation reaches. It is otherwise 
 with those waves that are formed by the 'attraction of the sun and 
 moon, and which constitute the tides. 
 
 60 
 
474 MOTION OF [Book VL 
 
 CHAPTER X. 
 
 OF THE MOTION OF ELASTIC FLUIDS. 
 
 460. When an elastic fluid moves through an aperture into a 
 vacuum, it is usually considered as contained in an open vessel, 
 through an orifice in which it passes with a velocity due to a col-, 
 umn of the fluid, of sufficient height to give it by its pressure, the 
 density at which it is found. This hypothesis is correct, so far 
 as regards the equality of velocity between an elastic fluid con- 
 tained in a close, and in an open vessel ; provided their densities 
 be identical, for the elastic force is by the law of Mariotte, 365, 
 exactly equal to the pressure by which any given density is pro- 
 duced. 
 
 In consequence of this same law, the density of an elastic fluid, 
 contained in an open vessel, will decrease, as we rise from the 
 surface of the earth ; and in order to produce the usual existing 
 pressure, the open vessel must be considered as extending to the 
 utmost limits of the atmosphere. Instead, however, of investiga- 
 ting the circumstances that would actually take place, we con- 
 sider the elastic fluid as reduced to the liquid state, and as being 
 of uniform density throughout ; the height of the column in the 
 vessel therefore becomes that which was, 357, styled the height 
 of a homogeneous atmosphere. 
 
 If we call this height, ft, we have from 407, 
 
 u=\/2 gh ; 
 and taking A=27600 feet, as determined in 357, we have 
 
 u= 1328 feet. (508) 
 
 To take a more exact determination, and which will be applicar 
 ble to our succeeding researches. . 
 
 At the temperature of 32, and under a pressure of 30 inches 
 of mercury, the density of that metal, in terms of air as the unit, 
 is 10467; hence the height of a homogeneous atmosphere, of 
 the temperature of 32, is 25268 feet, and 
 
 u= 1295 feet. (508a) 
 
 461. Air, therefore, of the temperature of 32, will rush into 
 a vacuum with a velocity of 1295 feet per second. If the tem- 
 perature be about 60, the velocity becomes 132S feet, at which 
 it is usually stated in English books. 
 
 It may be at once inferred from this investigation, that when 
 the temperature of air varies, its velocity in entering a vacuum 
 will vary also. 
 
Book VI.~\ ELASTIC FLUIDS. 475 
 
 In fact, if m be the expansion of air for each degree of Fahren- 
 heit's thermometer ; <, the number of degrees of the thermome- 
 ter, reckoned from the freezing point, h becomes h (l-p-wif), and 
 
 t>'=</[2&(l+mO]; ( 509 ) 
 
 whence, taking the above value of 1295 feet for v, at 32, we have 
 for ', at the temperature oH-f-32 , 
 
 i/=1295v/(l4-w*). (510) 
 
 4<52. When the density of the air varies, and we abstract the 
 variation of temperature with which such variations are usually 
 attended, the height of a homogeneous atmosphere, of the new 
 density, will be the same as before, and the velocity will not vary. 
 
 The velocity of an elastic fluid in entering a vacuum will, by 
 this reasoning, be always the same with that which a liquid of 
 similar density, and capable ofexertingan equal pressure with it, 
 would flow from a vessel. And in this form the rule may be ex- 
 tended to the case of air, or other elastic fluids, rushing from a 
 vessel into a space containing an elastic"fluid of a different density. 
 The velocity will be in all cases the same as that with which a liquid 
 of similar density, and capable of exerting a pressure equal to the 
 difference of the two tensions, would flow. Thus air, of a tension 
 of two atmospheres having, if its temperature^be the same, double 
 the density of that of the atmosphere, will flow out of a vessel into 
 the open a*ir, with half the constant velocity at which air would 
 enter a vacuum. 
 
 When air rushes into a vessel in which a vacuum has been 
 previously formed, its velocity is diminished as the vessel fills 
 with air, and should, according to the hypothesis become =0, 
 when- the air in the vessel acquires a density equal to that of the 
 air in the space whence it flows. The velocity being considered 
 as due to the altitude of a homogeneous atmosphere, the motion 
 in this case is considered as retarded by a motion growing out of 
 a fall, through an atmosphere of equal and uniform density, by 
 whose pressure the density acquired at the moment in the ves- 
 sel, would be produced. 
 
 In this view of the subject, the velocity with which air rushes 
 into a close vessel, which it finally fills with a mass of density 
 equal to its own, is equably retarded. 
 
 463. In gases other than atmospheric air, the velocities with 
 which thcry enter a vacuum, are in the inverseratio of the square 
 roots of their densities, for : 
 
 If /t, be the height of a homogeneous atmosphere of atmos- 
 pheric air ; /i', the height of a homogeneous atmosphere of another 
 gas ; D, the density of the gas, that of atmospheric air being 
 
476 MOTION OF [Book VI. 
 
 unity ; the heights, in order to produce the same pressure, must 
 be inversely as their densities, and 
 
 ' 
 
 .; 
 
 Hence, ', the velocity of a gas, whose density is D, will be 
 
 *=^- (511) 
 
 464. This theory is far from being perfectly satisfactory, par- 
 ticularly as it is, obvious that the whole of the effects that may be 
 due to the elasticity of the air, are o.mitted. The most import- 
 ant of these, perhaps, is that which takes place when air rushes 
 into a vessel, in which a vacuum has previously been formed. In 
 this case our theory would appear to show that the velocity is 
 uniformly retarded, until it becomes ; and the density of the 
 air that has entered the vessel the same as that without. This 
 does not occur in practice, for the motion will continue after the 
 densities become equal, and the air in the vessel will be con- 
 densed ; it will then re-act and expand, and the state of rest will 
 be aquired by a series of oscillations. 
 
 465. It has been ascertained by the experiments of D'Aubuisson, 
 that air, in passing through an orifice pierced in a thin plate, is 
 affected like a liquid, and'forms a vena contracta, whose area is, 
 as in the case of a liquid, 0.62 of the area of the orifice. The 
 application of cylindric adjutages, increases the quantity that is- 
 sues to 0.93, and a conical tube to 0.95. The adjutage may be 
 twenty or thirty times the diameter of the orifice in length, before 
 the discharge begins to diminish in consequence of the friction. 
 
 466. The principle 1 , 413, of the lateral communication of mo- 
 tion, holds good in gases, as well as in liquids. Thus, liquids in 
 motion carry with them a current of the air that is in contact with 
 them ; and gases, or vapours in motion, carry with them the 
 neighbouring air. 
 
 The latter fact may be conclusively established by the phe- 
 nomena of the Eolipyle. This instrument is a boiler in which 
 steam is generated, and permitted to escape from a narrow aper- 
 ture. It has for ages been employed to excite combustion. Now 
 steam alone, unmixed with atmospheric air, wpuld extinguish 
 flame, instead of increasing its intensity ; and the fact of its being 
 increased, proves that a current of atmospheric air joins the efflu- 
 ent steam, and is carried with it through the burning fuel. 
 
 We may upon this principle explain a curious fact, observed in 
 the efflux of air from a bellows, or other machine, in which it is 
 compressed ; it has also been observed in the escape of steam from 
 
; - '- v 
 
 Book F/.] ELASTIC FLUIDS. 477 
 
 the safety valves of boilers. If a circular disk of four or five 
 times the diameter of the orifice, be placed close to it, not only 
 will it not be forced away by the current of elastic fluid, but will 
 be retained near the orifice by a force of considerable intensity, 
 in so much, that if the orifice be directed downwards, the disk 
 will be supported in spite of its gravity, even when formed of a 
 dense metallic substance. 
 
 That this ought to be the case, may be readily understood 
 when we consider, that an elastic fluid issuing from the orifice, 
 A B, and having its course interrup- 
 ted by the plate, G E will assume 
 the form of a conoid, D A B C, con- 
 taining the cavity, G F E ; this cavity 
 will at the beginning of the action be 
 filled with a conoid of ,the elastic 
 
 fluid. But if a lateral communication of motion take place, the 
 fluid contained in this conoid will join itself to the stream that es- 
 capes at the edges of the plate, and a vacuum will be formed in 
 the conoidal space, GE F ; the pressure of the atmosphere acting 
 upon the surface of the plate, G F, will therefore press it towards 
 the orifice. As it approaches the orifice, the action of the fluid 
 will become more intense ; for it will strike against the disk ; the 
 surface beneath which the vacuum exists, 'will diminish, and thus 
 the force that acts o repel the disk from the 'orifice may prepon- 
 derate, and th% disk be forced back ; but this force diminishes 
 as the disk recedes, while the surface, to which the atmospheric 
 pressure is due, increases : thus the forces that tend to move the 
 disk in opposite directions, will be continually varying, and under 
 this variation the disk will assume an oscillating motion. 
 
 467. Air may not only be set in motion by the difference in 
 pressure, arising from mechanical expansion, or condensation, but* 
 also by the physical action of heat, which changes its density. 
 
 If by any cause whatsoever, the equilibrium of temperature of a 
 mass of air be disturbed, the parts which are most heated become 
 less dense than those which surround them, and therefore .tend 
 to rise ; the space that they before occupied, will be supplied by 
 the adjacent air, and thus a circulating motion will take place. 
 
 The force with which a portion of a mass of air that has been 
 heated, will tend to rise, is by the principle of 334, equal to the 
 difference between its own weight, and the weight of an equal 
 mass of the same air, before it was heated. 
 
 If the air that is heated be free, it will, both in consequence of 
 the resistance it meets with in rising, and the tendency of elastic 
 fluids to distribute themselves over a given surface, in such man- 
 ner that the pressure shall become uniform, mix with the air 
 
478 MOTION OF [Book VI. 
 
 through which it rises ; it will also assume a common tempera- 
 ture with the latter, in consequence of the radiation of its heat. 
 
 If the air be heated in a straight tube, or close channel, having 
 an aperture at both ends ; and if the two ends are not at the same 
 level, it will rise towards the upper end, and will not mix with 
 other air, or give out much of its heat, until it reach the higher 
 opening. ^Here it will again tend to mix and distribute itself 
 through the adjacent air. In this manner the motion of air in 
 chimnies takes place. 
 
 If air be disseminated through a space unequally heated, and 
 its parts acquire the temperature of the portions of that space 
 which they occupy, a motion of circulation must also take place; 
 and this will continue, so long as the unequal distribution of tem- 
 perature continues. 
 
 In this manner, the currents in the atmosphere, called Winds, 
 are generated, as will hereafter be more fully explained. 
 
 468. The density of steam does not vary exactly as its pressure, 
 but follows the specific law stated in 373. Taking the relative 
 densities and temperatures of steam, as given in the table of 374, 
 the following results have been obtained. 
 
 TABLE 
 
 Of the velocity with which steam of different^ensions enters a vacuum. 
 
 Tension in I Velocity in I Tension in I Velocity in 
 Atmospheres. feet. Atmospheres. | feet. 
 
 1 1910 5 2040 
 
 2 1978 10 2080 
 
 3 2007 15 2122 
 ^ '' 4 2023 20 2142 
 
 TABLE 
 
 Of the velocity with which steam of different tensions enters a space con- 
 laining atmospheric air. 
 
 T. in Af. | V. in Ft. | T. in At. | V. in Ft. 
 
 H 874 5 1834 
 
 1J- 1154 8 1952 
 
 2 1400 12 2027 
 
 3 1647 16 2070 
 
 4 1761 20 "2095 
 
Book PL] GASES. 479. 
 
 CHAPTER XI. 
 
 OF THE MOTION OF GASES IN PIPES. 
 
 469. When the tube in which gases are in motion is long, they 
 are retarded by friction, in a manner analogous to that observed 
 in water and other liquids. Hence, although the velocity of an 
 elastic fluid cannot finally become constant in a tube, many of the 
 circumstances are in other respects similar to those stated in chap- 
 ter IV. It is however important, that we should investigate 
 them more closely in the case of air. 
 
 Let H be the height of a column of mercury that measures the 
 pressure of the air on entering the tube ; h, a similar quantity at 
 the place of discharge ; 6, the altitude of the barometer at the 
 time ; tf, the height of the thermometer above or below 32 ; m^, 
 the expansion of air for each degree of the thermometer ; d, the 
 diameter of the pipe ; V, the velocity ; and let 
 
 Suppose the mean height of the barometer to be 30 inches, or 
 2.5 feet. 
 
 Taking, as in 459, the density of mercury, in terms of air, at 
 32, to be 10467, the relation of the density of mercury to that 
 of the air that is issuing from the pipe, will be 
 
 /2.5T \ 
 
 10467^) . (512) 
 
 The velocity t*, at the place of discharge, will be (509) 
 
 2.5 T\ 
 
 - 
 
 h. 10467-^; (513) 
 
 and extracting the square root of the numbers under the radical 
 
 frj 
 
 0=266 V (h. fc) . (514) 
 
 The velocities at different points in the tube will be inversely 
 as the densities of the air at those points ; for as the motion is 
 continuous, the same quantity of air must pass through every dif- 
 ferent section in an equal time. The densities at the ends being 
 proportioned to the pressures, will be proportioned to 6+H, and 
 6+/i; the mean density will be proportioned to 
 
 H+fc 
 H2-. (515) 
 
 And it will be obvious that the mean velocity may be obtained by 
 an analogy, of which the mean density is the first term ; the 
 
430 MOTION OF [Book VI. 
 
 pressure at the place of discharge, the second ; and the velocity 
 v, the third. Hence, we have for the mean velocity, V, 
 
 V=266. > 
 
 H+/I (516) 
 
 The experiments of D'Aubuisson have shown that the resist- 
 ance, of which H h is the measure, is proportioned to the square 
 of the velocity. Other experiments show that it is directly pro- 
 portioned to the length of the tube L, and inversely to its diameter. 
 Hence, if N be the constant co-efficient of the resistance, 
 
 H fc=N.--; (517) 
 
 and substituting in this expression the value of t> from (514) we 
 obtain 
 
 whence 
 
 ' ' (519) 
 
 or in a more convenient form, 
 
 H 
 
 NLT - (520) 
 
 d (6+/0 ? 
 
 h being still involved in the second member of this equation, it 
 can only be resolved by an approximation. This may be ob- 
 tained from knowing that in the cases that most usually occur in 
 
 T 
 practice, , . is a quantity that varies but little, and which may, 
 
 therefore, without sensible error, be considered as constant. If 
 then we make 
 
 T 
 
 we have 
 
 ^ H 
 
 (521) 
 
 when h r H Snd L, are estimated in English feet, and d in inches, 
 
 c=0.002, 
 arid the formula becomes 
 
 h= ~17~ (522) 
 
 : .*.,.) 0.002 --r+1 . 
 
Book VI.] GASES IN PIPES. 481 
 
 The pressure under which the, air issues, being thus obtained, 
 the effluent velocity is given by the formula (514) 
 
 "= 266 
 and here again we may make use of a constant quantity for 
 
 i . , , without any sensible error. 
 
 To find the quantity in cubic feet, the diameter of the pipe be- 
 ing given in feet, the above formula must be multiplied by 
 ifd 2 
 
 or if the diameter be given in inches, by 
 
 4 X 144 576 ' 
 and calling the quantity discharged Q, 
 
 (523) 
 
 T 
 
 If we take, instead of ^ , ^, a constant quantity, determined 
 
 from experiment in the case of the blowing machines of furnaces, 
 we have, h being estimated in feet, d in inches, and Q in cubic 
 feet, 
 
 Q=79.44d 2 x//i. (524) 
 
 If the area of the aperture, through which the air is discharged, 
 should differ from the mean area of the tube, as is frequently the 
 case in blowing engines, the velocity may be determined upon 
 the principle, that in different sections of the same tube, the velo- 
 cities are inversely as the areas, or, if d be the diameter of the 
 orifice of discharge, D that of the rest of the tube, as 
 d 2 
 
 If the orifice of discharge be in , thin plate, this co-efficient would 
 become 
 
 d 2 
 0.62 jj^-, 
 
 and if formed by a cylindrical tube, 
 
 d 2 
 0.93 ^r. 
 
 This being the case most usual in practice, we have for the 
 quantity discharged, from (524) 
 
 d 4 
 Q=74.34 jp-v/A. (525) 
 
 61 
 
 

 482 MOTION OF [Book VL 
 
 470. It will be obvious, from what has been stated, that the 
 most important application of this subject is to the tubes, by 
 which air is conveyed from blowing machines to excite the com- 
 bustion of blast furnaces ; in these, a knowledge of the quantity 
 of air they convey, is often of great importance. 
 
 Another useful practical case is, that of the conveyance of in- 
 flammable gas in pipes, for the purpose of 'illumination. The 
 above investigations, and the formulae thence deduced, are ap- 
 plicable in this instance also ; for it has been found that the re- 
 sistance to the motion of carburetted hydrogen in tubes, not only 
 follows the same laws, but is equal in quantity to that which re- 
 tards the motion of atmospheric air. 
 
 The same principles might be applied to determine the velocity 
 of effluence, in terms of the pressure upon the entrance of the 
 tube. The pressure of the issuing air may be easily determined 
 from (522) ; we have not considered it necessary to enter into 
 the investigation of formulae for this purpose. 
 
Book VL] AIR IN CHIMNIES. 483 
 
 "if. ' 
 
 CHAPTER XII. 
 
 OP THE MOTION OP AIR IN CHIMNIES. 
 
 471. A chimney may be considered as a tube, in any position 
 except horizontal, at the lower opening of which air is heated by 
 an extrinsic cause. This air will, in conformity with what has 
 been stated in Chapter X, rise and pass out at the upper opening ; 
 its place will be supplied by air pressing from the space adjacent 
 to the lower opening. If this be heated in its turn, as it enters 
 the chimney, to the same temperature, it will rise with a force 
 equal to that possessed by what preceded it ; and so soon as the 
 whole tube is filled with the heated air, the velocity will become 
 uniform. 
 
 The circumstances will obviously be the same as if a tube were 
 adapted to the bottom of the chimney and filled with air of the 
 original temperature, while the chimney is filled with the heated 
 air ; and thus the air will move as in an inverted syphon, in 
 which two columns of fluid, of different densities but of equal 
 altitudes, press against each other. For as the action of the ex- 
 ternal air may be considered equal on both openings of the chim- 
 ney, the acceleration it produces in the one column, and the re- 
 tardation in the other, may be neglected. 
 
 Let A, be the height of the column of air in the chimney ; t the 
 external temperature ; ? that of the air in the chimney ; m the 
 dilatation of the air for each degree of the thermometer. The 
 length of the column of cold air reduced to 32, will be 
 
 1 A 
 
 l+W 
 The same column at the temperature t' 9 will be (509), 
 
 >' -" '. v>. (527) 
 
 and this will be the height to which the velocity would be due, 
 with which the air would enter the chimney, if that were void of 
 air ; the actual velocity will be due to the difference between this 
 height and ft, or to 
 
 " 
 
 and we have for the value of v from (509) 
 
 =avF:iffi(S^-l)], (529) 
 

 484 MOTION OF [Book VI. 
 
 the quantity 1-f tm, is, generally speaking, so small that it may 
 be neglected ; and we, therefore, have 
 
 v = 7 [Zghm (**')] , (531) 
 
 and we may take for the height to which the velocity is due, 
 hm(tt'). . 
 
 472. Such would be the theory were the air to meet with no re- 
 sistance in its passage through the chimney. But it is obvious that 
 it will meet with friction ; that its temperature and consequent 
 ascensive force will diminish. Thus the velocity with which 
 the heated air would otherwise begin to ascend will be lessened ; 
 and its excess of elastic force will continually decrease, from the 
 origin to the summit. At the extremity of the chimney, the 
 elastic force will obviously be proportioned to the^xcess of temper- 
 ature that it retains at that point. And as we may consider that 
 the same quantity of air passes through every different section 
 of the chimney in the same time, it follows that at every point 
 the velocity is inversely as the densit}^ and therefore that the 
 velocity decreases from the origin to the summit. 
 
 If P be the pressure under which the air enters the chimney ; 
 p the pressure it retains at the summit. The loss of motion 
 growing out of resistances in the chimney, may be represented 
 byP p. 
 
 The resistance appears from experiment to be directly pro- 
 portioned to the square of the velocity, and the length of the chim- 
 ney ; and inversely to its diameter. 
 
 If p be estimated as the height of a column of the heated air, 
 under the pressure of the atmosphere, 
 
 p=hm(tt'}', (532) 
 
 and from (509) and (531), 
 
 v= v/2gft, (533) 
 
 calling the velocity V ; the diameter of the tube D ; its length 
 L ; and the co-efficient of the resistance K ; the law of the resist- 
 ance just stated, gives us 
 
 V 2 L 
 
 whence 
 
 KV 2 L 
 
 substituting this value in the foregoing equation, (533), we have 
 
 K V 2 L 
 
 (534) 
 

 Book VI.'] AIR IN CHIMNIES. 485 
 
 whence we obtain for the value of V, 
 
 / PD \ 
 
 V =^^(]5+2iIx)' (535) 
 
 and for that of K, 
 
 _ V 2 D 
 
 ' 
 
 By means of which last formula, the co-efficient of the friction 
 can be obtained by experiment. This has been done by Peclet, 
 who has found when the values of P, D, L, and V, are given in 
 French metres : 
 
 (1.) That in brick chimnies, 
 
 K=0.0127. (537) 
 
 (2.) In wrought iron flues, 
 
 K=0.0050. (538) 
 
 (3.) In chimnies or flues of cast iron, > : ," . 
 
 K=0.0025. (539) 
 
 473. The great difference between the values of the co-effi- 
 cient of the friction, in tubes of different substances, is a remark- 
 able fact; particularly as it differs essentially from what is ob- 
 served in the motion of water in pipes, where the substance of 
 which they are composed has no essential influence on the ve- 
 locity. 
 
 This discrepancy may be explained by the difference in the 
 attraction of the two fluids for the substances of which the tubes 
 are composed. Water adheres, by its attraction of cohesion, to 
 the pipes, and moistens them. It thus in fact, in running in 
 tubes, rubs against the water that adheres to them, and the fric- 
 tion is a constant quantity ; for it always takes place between 
 surfaces, both of which are composed of the same substance. 
 While in the case of the motion of air, the friction takes place 
 between it and the material of which the tube is composed, and 
 its co-efficient should therefore be different in tubes of different 
 materials. 
 
 The dimensions of chimnies depend upon the quantity of air 
 which the combustible requires for /its perfect ignition, and upon 
 the velocity which it will assume in them. The former i*an 
 object of chemical and physical investigation, and would be out 
 of place in a work on Mechanic^. The latter is determined by 
 the foregoing equations. 
 
486 THE WINDS. [nook 
 
 CHAPTER VIII. 
 
 OF THE WINDS. 
 
 474. Winds are currents in the atmosphere, that are generally 
 if not always caused by a disturbance in its equilibrium, owing 
 to the unequal and variable distribution of temperature upon 
 the surface. 
 
 The temperature of the surface of the earth, is due to two an- 
 tagonist causes : 
 
 (1). The constitution of the mass of the earth and its atmos- 
 phere : 
 
 (2). The reception of the radiant heat of the sun. 
 
 The heat arising from the first of these causes radiates, and in 
 consequence of this radiation, the earth would continually grow 
 cooler. This waste is supplied by the heat that radiates from 
 the sun. It has been proved incontestibly by Laplace, that the 
 mean temperature of the earth is now constant, and has been so 
 for 2000 years ; hence, the quantity of heat that radiates from the 
 earth, and the quantity received from the sun, exactly balance 
 each other: that is to say, for any Aeries of years, the sum of the 
 quantities received by the whole earth from the sun, is just equal 
 to the quantity that radiates. 
 
 This equality does not exist for short periods of time. The 
 radiation from the surface goes on continually, although not 
 uniformly, being greatest from the portions that are most 
 heated ; while the reception of heat at aoy given place, only takes 
 place while the sun is above the horizon. 
 
 The quantity of heat received from. the sun, upon a given sur- 
 face, varies in a given natural day with the altitude of the sun 
 above the horizon, in consequence of the greater or less obliquity 
 of his rays. It also differs from day to day, in consequence of 
 the variation in the length of the natural day, and in the meri- 
 di^i altitude of the sun. 
 
 The first of these variations grows out. of the rotation of the 
 earth upon its axes, in the space of a day ; the second, arises from 
 the revolution of the earth around the sun, in an annual orbit 
 inclined to the equator. 
 
 This orbit being an ellipse, of which the sun occupies one of 
 the foci, and the arcs of the ellipse described not being propor- 
 tioned to the times, the sun's daily apparent motion is not equa- 
 ble. 
 
Book VL~\ THE WINDS. 487 
 
 The orbit being inclined to the axis of rotation, causes a vari- 
 ation in the length of the natural day, and in the sun's meridian 
 altitude; hence flow the vicissitudes of the seasons. The diame- 
 ter of the orbit that passes through the sun, does not pass through 
 the equinoctial points, but is nearer to the solsticial points than 
 to the equinoctial ; hence the equinoxes do not divide the year 
 into two equal parts, but the summer of the northern hemisphere 
 is about 7| days longer than the summer of the southern. It fol- 
 lows that the northern hemisphere receives more heat from the 
 sun, than the southern; and a balance taking place in the quantity 
 of heat received and radiated, not only in the whole earth, but in 
 the two separate hemispheres, the northern hemisphere is 
 warmer than the southern ; therefore, even the mean place of the 
 equator of temperature does not coincide with the astronomic 
 equator, but lies north of it. -^ . 
 
 Did the plane of the earth's orbit coincide with the terrestrial 
 equator, the days and nights would be constantly equal in every part 
 of the globe. The sun's rays at noon would fall vertically upon 
 points in the equator, and would be tangents to the earth at the 
 poles ; and were there no lateral communication by radiation or 
 the conducting property of the materials of the earth, and its at- 
 mosphere, the quantity of heat received at noon, from the sun, 
 at every different place, would obviously be proportioned to the 
 cosine of the latitude. The mean temperature of the days in 
 every latitude would be constant; and this mean diurnal tem- 
 perature would follow a regular law of decrease, from the equator 
 to the poles. Taken for a whole year, the mean temperature, if 
 disturbing causes did not act, should follow a similar law. The 
 variations at the surface are so sudden as to cloak this law, except 
 when studied by the comparison of a long series of thermometric 
 observations ; but when we penetrate beneath the surface of the 
 earth, to a depth sufficient to be removed from the influence of 
 the changes at the surface, we find this law to hold good : > fV 
 
 The temperature at the depth of from 60, to SO feet beneath 
 the surface of the earth, is constant in every different climate, and 
 corresponds closely with the mean temperature at the surface. 
 
 At the equator, the days and nights are of equal lengths through- 
 out the year ; and the meridian zenith distance of the sun, never 
 amounts to more than 231. At the polar circle, the natural days 
 vary in length from 24 hrs. to hrs., and the change in the me- 
 ridian altitude is from to 47. Thus, at the equator, the tem- 
 perature varies but little on each side of the mean ; while, were 
 the earth of uniform surface, the extent of the variation on each 
 side of the mean rate should increase regularly from the equator, 
 to the polar circles. Within the polar circles, the sun does not 
 
488 THE WINDS. [Book VI. 
 
 rise for several days on each side of the winter solstice, and is 
 above the horizon for several days near the summer solstice; 
 while at the poles, the natural day and night have each a length 
 of six months. Hence it might be inferred, that even greater 
 variations on each side of the mean temperature should occur 
 within the polar circles, and that the variation should be a maxi- 
 mum at the pole. 
 
 The quantity of heat derived from the sun on the clay of the 
 summer solstice, has been calculated to be nearly equal in the 
 lats. of 45 and 60 ; and when the sun's declination exceeds 18, 
 the quantity of heat received in 24 hours at the pole is not less 
 than it is at the equator, during the twelve hours the sun is above 
 the horizon. 
 
 It thus happens that were there no disturbing cause, nor any 
 means by which the excess of heat might be conveyed from one re- 
 gion to another,' the distribution of heat at the surface would be 
 continually varying. The distribution according to a regular law 
 of decrease, from the equator to the poles, would only take place 
 near the time of the equinoxes ; while at the solstices the parallels 
 receiving the greatest quantity of heat would be without the 
 tropics ; and parallels in the frigid and torrid zones might re- 
 ceive equal quantities of heat on the same day. 
 
 As however the earth has at no great depth, the mean tem- 
 perature of the climate, this will tend in a high latitude to prevent 
 the surface from acquiring a heat as great as that actually com- 
 municated on the hottest days ; and thus the heat of the surface 
 in such latitudes will never rise as high on the warmest days, as 
 is consistent with the quantity actually received. In the same 
 manner the surface of high latitudes never cools as low as is con- 
 sistent with the quantity of radiation from the surface, in the ab- 
 sence of any supply from the sun. 
 
 Local circumstances that will hereafter be stated, affect the 
 range of sensible heat ; and thus, places in the same latitude, many 
 have very different maxima and minima of temperature; and the 
 amount of variation may be much greater in a given place, than 
 it is in another of the same latitude. 
 
 Certain physical causes interfere in high latitudes, to prevent 
 the extent of the changes of temperature being as great as they 
 would be, in consequence of the great difference in the altitude of 
 the sun, and of its continuance above the horizon at different sea- 
 sons ; these will be stated in their proper place; and thus the 
 greatest alternations seem to take place, in thelat. of from 35 to 
 50. In New-York, the annual range of the thermometer, from 
 its summer maximum to its winter minimum, sometimes exceeds 
 "*"* ; and the difference between the mean temperature of the 
 
Booh VI.] THE WINDS. 489 
 
 hottest, and that of the coldest month, amounts to 56 ; at Pekin, 
 the latter difference is 60, while atFunchal, it is no more than 10. 
 
 475. Difference of elevation above the surface of the earth, has 
 a great effect upon the temperature of places. The air of the at- 
 mosphere is from its elastic nature, denser at the mean surface of 
 the earth, than in higher regions ; and air has an increased capa- 
 city for heat when it becomes rarer ; hence, in the higher parts 
 of the atmosphere an intense cold prevails, and the temperature 
 of the land decreases with its elevation above the level of the ocean. 
 So intense is the action of this cause, and so speedily is it sensible 
 in rising from the earth, that even in the heart of the torrid zone 
 mountains exist whose tops are covered with perpetual snow. On 
 these, it is therefore evident, that the thermometer never rises 
 much above 32. 
 
 It has been inferred, but without sufficient reason, that the 
 mean temperature of the limit of perpetual snow is 32, but ob- 
 servation shows that it is in all cases lower, and the limit appears 
 to arise rather from the mean temperature of the warmest month, 
 than from that of the entire year. 
 
 It has been estimated that the temperature decreases as we re- 
 cede from the surface of the earth, at the rate of about 1 for 
 every 270 feet. 
 
 In consequence of the great variation that takes place in the 
 quantity of heat received from the sun, in temperate and frigid 
 climates, while the radiation has a much less range, an accumula- 
 tion takes place, at those seasons when the sun is highest at 
 noon, and remains longest above the horizon, by which the tem- 
 perature increases for some time after the solstice ; a correspond- 
 ing diminution in temperature goes on after the shortest day. Thus 
 it happens that the greatest heat in middle latitudes occurs about 
 a month after the summer solstice, and the greatest cold about an 
 equal time after the winter solstice. 
 
 An empirical formula, that very nearly corresponds with ob- 
 servation, has been framed to represent the temperatures at dif- 
 ferent seasons, and at altitudes above the level of the sea, in all 
 latitudes. 
 
 Let M be the mean temperature in lat. 45 ; 
 M+E, the mean temperature at the equator ; 
 L, the latitude of the place ; 
 F, a co-efficient determined by observation ; 
 H, the altitude of the place above the level of the sea ; 
 /, the sun's longitude. 
 
 Then we have for the mean diurnal temperature, on the day 
 for which the longitude / is given, 
 
 . 2 L+F sin. (I 30) ^ . (540) 
 62 
 
490 THE WINDS. [Book VI. 
 
 If F 15, the formula gives results that are on the average 
 true, in the western part of Europe, and in the North Atlantic. 
 
 476. The nature of the surface has a great effect upon the dis- 
 tribution of temperature, and upon the distance that exists between 
 the extremes of heat and cold in different parts of the globe. 
 
 The surface of the earth is partly of solid land, and partly wa- 
 ter. Within the former, the communication of heat is extremely 
 slow, and hence the surface of the land adapts its temperature more 
 closely to the quantity of heat received daily, than the surface of 
 the ocean. The latter, when exposed to heat that varies from place 
 to place, is set in motion ; for so long as the temperature of the 
 surface does not fall below 40, the water expands, and the columns 
 in the warmer parts increasing in altitude, while they diminish in 
 density, a current is caused from the parts most heated, to those 
 which are colder; a counter current is formed in the water be- 
 neath, in which the colder portions flow towards the zone of 
 greatest heat. In this manner, so much of the heat derived from 
 the sun, as exceeds the radiation, is conveyed at the surface, from 
 the heated regions, to those which are colder. 
 
 This motion ceases, however, when the temperature of the 
 surface falls below 40, beneath which degree any diminution of 
 the temperature of the water will render it lighter than that which 
 is beneath, and the heated portion sinks, instead of rising. 
 
 When the sun shines upon the land, its calorific rays penetrate 
 to but a small depth, say no more than a few inches ; its surface 
 is in consequence rapidly heated, when the heat received exceeds 
 that which is radiated : when the latter is in excess, the loss of 
 heat is principally confined to the surface, which is therefore 
 rapidly cooled. 
 
 In water, when the reception of heat exceeds the radiation, the 
 calorific rays penetrate to a considerable depth, say 20 to 30 feet ; 
 the heat being thus distributed through a large mass, the superfi- 
 cial temperature is but slowly altered. When on the other hand, 
 the radiation is in excess, the upper portions, on parting with their 
 heat, contract, and becoming heavier than the water which is be- 
 neath, descend until they reach the bottom, or a stratum of the 
 fluid of equal temperature with themselves; a circulation is thus 
 kept up, and the heat lost, although equal, or even superior in 
 quantity to that withdrawn from the land, is again derived from 
 a large mass ; the diminution of the superficial temperature is 
 therefore slow. When however the surface is qpoled below 40, 
 this motion ceases. 
 
 From the combination of these circumstances, it happens that 
 the temperature of the surface of the ocean is more constant than 
 that of the land ; that it can be reduced to certain laws easily dis- 
 
Book VL] THE WINDS. 491 
 
 covered from observation ; and that it follows much more closely 
 than the land, the law of a regular diminution of temperature, 
 from the equator to the poles. The variations on each side of the 
 mean temperature, are also less on the ocean than they are upon 
 the land. These rules likewise hold good in islands, and to a less 
 extent in countries adjacent to the ocean ; these portions of the 
 land have climates of less vicissitude than the interior of conti- 
 nents. 
 
 477. Great and deep lakes have a similar, although less impor- 
 tant influence on climate ; for although the extent of their sur- 
 face be not sufficiently great to cause any distribution of heat by 
 currents, the difference between the quantities of heat received 
 and radiated, affect not their surface alone, but their whole mass. 
 Their surface, therefore, like that of the ocean, preserves a more 
 uniform temperature than that of the land. 
 
 When a lake cools, the motion that we have described in 
 speaking of the ocean, in which the cooler parts descend, and by 
 which the heat is withdrawn from the whole mass, goes on until 
 the temperature throughout becomes 40. Water at this tem- 
 perature reaches its maximum of density, the motion of descent 
 ceases, and the surface will be speedily cooled to the temperature/ 
 of congelation. Deep lakes, however, descend to such depths/^ 
 to come into contact with those strata of the earth's mass 
 tain the mean temperature of the climate ; from these tj 
 will derive heat; and thus it may happen that a deep>^ e > f no 
 great superficial extent, is never frozen. Such phep^" ena occ "r 
 in the small lakes of the western part of the stat^" New- York, 
 the surface of which never freezes. 
 
 Shallow lakes and morasses tend to ma keX climate colder ; for 
 the cold produced at their surfaces not onty by evaporation, but 
 by radiation, cannot long be compensatory an internal motion. 
 
 The draining of morasses, renders A climate warmer, as does 
 the cutting of forests, and the extern of cultivation. The ef- 
 fects of the latter causes appear to/xtend beyond the region where 
 they operate directly. Thus, t*e cultivation of France and Ger- 
 many, has changed the climate of Italy ; and thus, the clearing of 
 the forests of the interior of the United States, has raised the 
 mean annual temperature of the seacoast. 
 478. To recapitulate our general inferences : 
 
 (1). Upon the land, the zone of greatest sensible heat will be 
 a little north of the equator on the days of the equinox; but 
 will on other days of the year vary in position ; and will be found 
 in the interior of continents about a month after the solstices, 
 in latitudes as high as from 40 to 50. In the ocean, on the 
 
492 THE WINDS. [Book VI. 
 
 contrary, the zone of maximum temperature does not vary in its 
 position more than 8, and is always to the north of the equator. 
 From this zone, the heat of the surface of the ocean decreases 
 uniformly to a latitude of from 28 to 30. Beyond this limit, on 
 either side of the equator to the latitude of 50, the heat of the 
 surface is alternately greater or less than would be consistent 
 with a regular decrease, according to the law of the cosine of the 
 latitude; but after the equinoxes, it appears to coincide for a 
 short time with the results of that law. 
 
 (2). Elevated countries are colder than those more near the 
 level of the sea. 
 
 To these, it is to be added, that the western sides of the two 
 great continents are sensibly warmer, or have a higher mean 
 temperature than the eastern. 
 
 (3). The change in the density, caused by change of tem- 
 perature, produces currents in the ocean ; the surface of water 
 also becomes more slowly heated, and parts with its heat less 
 rapidly than the surface of land exposed to an equal action of the 
 sutx's rays. The ocean therefore enjoys a more equable tempera- 
 ture than the land, and influences in a similar manner the climate 
 of islands and seacoasts. 
 
 (4), Cultivation appears to raise the mean temperature, and 
 CGl *ainly ameliorates the climate. In the United States, this effect 
 appe^s to k e we n jnarke^ but is attended with an anomaly. The 
 duratioi O f intense cold has been sensibly lessened, but the dimi- 
 nution ot he i en gth o f the winter is wholly in its earlier part ; 
 on the othe.. hand, frosts are experienced at later dates in the 
 spring than foi^ er iy. uch are the more important circumstan- 
 ces that influence Climate, and on them a theory of the winds may 
 be founded. 
 
 479. The air of our a^osphere receives heat from, and com- 
 municates it to, the parts ^f the earth on which it presses. Those 
 parts of it in immediate cou. ac t, acquiring or parting with heat 
 readily; their volumes and to ns i ons are therefore changed, a 
 disturbance of equilibrium takes place, and motion ensues. Thus 
 fresh portions of air are brought into contact with the surface of 
 the earth, and the influence of its changes of temperature extended. 
 These motions in the atmosphere, concur therefore with those of 
 the ocean, of which we have already spoken, to moderate the vi- 
 cissitudes of heat, to which the surface would otherwise be sub- 
 jected. 
 
 The lower stratum of the air of the atmosphere, tends in con- 
 sequence, to an equilibrium of temperature with the surface be- 
 neath it ; this state it however never reaches, or never retains 
 for more than a short space of time ; besides, in its own tendency 
 
Book F/.] THB WINDS. 493 
 
 to move, until a state of equilibrium of temperature be attained, it 
 is set into a continual, and frequently violent motion. 
 
 This state of equilibrium, it may be stated, is not that of uni- 
 form temperature throughout ; but would be one of uniform tem- 
 perature at the mean surface of the earth, and of a temperature 
 regularly decreasing from that surface upwards, in conformity 
 with the relations of the air's diminishing density to specific heat. 
 
 Had the air no motion growing out of such disturbances of tem- 
 perature, its inertia, and the friction that takes place between it 
 and the earth, and among its own particles, would cause it to as- 
 sume precisely the same angular velocity with the part of the sur- 
 face immediately beneath it. In its motions it must therefore be 
 considered as acted upon by two forces ; the one arising from the 
 disturbance of the equilibrium of temperature ; the other, from 
 the rotary motion of the parallel whence it begins to move over 
 the surface. 
 
 From the foregoing considerations it will be seen that the 
 earth's atmosphere must be in a state of almost constant motion, 
 forming the currents that are styled Winds. 
 
 Upon the greater part of the surface of the ocean, these are re- 
 ducible to fixed and determinate laws. Upon continents, and in 
 high latitudes upon the ocean, although we may assign the gene- 
 ral causes of the winds, yet the order and periods of their recur- 
 rences are irregular. 
 
 480. The winds may be divided into classes, which we shall 
 enumerate before proceed ing to explain their causes. They are 
 
 1. The Trade Winds; 
 
 2. Monsoons ; 
 
 3. The local variations of the Trades and Monsoons ; 
 
 4. The regular Westerly Winds ; 
 
 5. The variable winds of continents, and of temperate and po- 
 lar climates ; 
 
 6. The land and sea breezes. 
 
 The theory of the winds has derived most important accessions 
 from the researches of Daniell, whose labours we shall make use 
 of in the explanation of these phenomena. As it is unnecessary 
 to enter into any strict calculations in relation to them, we shall, 
 in this discussion, dispense with the use of algebraic notation. 
 
 481. Were the earth a sphere of uniform temperature, and at 
 rest in space ; its atmosphere a perfectly dry and permanently elas- 
 tic fluid ; the height of the latter would be constant over every point 
 of the earth's surface, and its density and elasticity, at equal ele- 
 vations, every where the same. The column of mercury that 
 it would support in the barometer, would therefore be the same at 
 
494 TitE WINDS. [Book VI. 
 
 every point on the surface of the sphere ; and equal at equal 
 heights above the surface. The atmosphere would be absolutely 
 at rest ; and as its elasticity is proportioned to the pressure, the 
 density would decrease in geometrical progression, while the dis- 
 tance from the surface of the sphere increases in arithmetical. 
 When air is rarefied, its capacity for heat is increased, and vice 
 versa; the sensible heat of the atmosphere must therefore decrease 
 as the altitude increases ; and as this changes the volume of elastic 
 fluids, even under equal pressures, the barometer alone will no 
 longer be the exact measure of the progressive density, but must 
 be associated with the thermometer. Any change of temperature 
 that affects every part of the sphere, would cause an increase in 
 the elasticity of the atmosphere, and in its consequent height, 
 without producing any motion in the lateral direction, or any 
 change in the pressure upon the surface; but the pressure will be 
 changed at all other altitudes. 
 
 If the temperature of the sphere, instead of being equal at every 
 point, were greatest at the equator, and decreased towards the 
 poles, the pressure on every point of the surface would still con- 
 tinue the same ; but the altitude of the atmospheric column would 
 become greatest at the equator, and its specific gravity at the sur- 
 face less there than at the poles. The heavier fluid at the poles 
 must, by its greater weight pass beneath, and displace the lighter, 
 and a current will be established in the lower part of the atmos- 
 phere, from the poles towards the equator. The difference in the 
 specific gravity of the polar and equatorial columns becomes less 
 as \ve ascend into the atmosphere ; while the elasticity, which is 
 constant at the surface, varies with the height, and the barometer 
 stands higher at equal elevations in the equatorial, than in the 
 polar column. It will hence happen, that, at some definite height, 
 the unequal density of the lower strata will be compensated ; and 
 a counter-current will take place in the higher regions from the 
 equator towards the poles. 
 
 The heights at which this would happen, under certain circum- 
 stances may be calculated, and the velocity of each current deter- 
 mined. This has been done by Daniell, to whose work the reader 
 is referred, for the process and inferences. From his investi- 
 gations it appears, that the lower current directed from the poles 
 towards the equator, extends to the height of two miles and a half, 
 gradually diminishing in velocity from the surface upwards. At 
 the last mentioned height the counter current begins, and its ve- 
 locities gradually increase from that altitude upwards. 
 
 The velocity and direction of these currents may be affected by 
 the partial rarefaction or condensation of any of the columns ; and 
 such change of density will naturally take place, in consequence 
 
Book VL] TUB WINDS. 495 
 
 of the vicissitudes of the seasons, and the alternations of day and 
 night. 
 
 If the sphere revolve around its polar diameter, as an axis, an 
 apparent modification will take place in the direction of the cur- 
 rents. The lower current, coming from a point whose velocity 
 of rotation is less than that at which it arrives, will appear to be 
 affected with a motion, in a direction contrary to that of the revo- 
 lution of the sphere ; while the upper current, being under op- 
 posite circumstances, will be apparently affected in an opposite 
 manner. 
 
 The earth revolves around its axis in a direction from west to 
 east ; and hence the great equatorial currents, that are in fact di- 
 rected, on the north side of the equator, from north to south, and 
 on the south side from south to north, appear in both cases de- 
 fleeted towards the east. 
 
 In the months of April and October, such a state of things does 
 actually take place upon the earth ; hence N. E. winds prevail 
 at those periods throughout the whole northern, and S. E. winds 
 throughout the whole southern hemisphere ; the hemispheres 
 being divided by the equator of temperature, and not by that of 
 latitude. 
 
 At other seasons, the regular law of decreasing temperature is 
 interrupted even upon the surface of the ocean, at latitudes of from 
 28 to 32 ; and is not to be recognised upon the land ; hence 
 these winds are constant only within these limits, and in the 
 open ocean. These constant currents are called the Trade Winds, 
 and from their directions, the N. E. and S. E. Trades. 
 
 The velocity of rotation changes more for a given difference of 
 latitude in high than in low latitudes; hence the apparent devia- 
 tion, from a true northern or southern direction, will be greatest 
 near the outer verge of the trade winds, and least near their cen- 
 tral zone. 
 
 4S2. This central zone that divides the trade winds, has a 
 breadth, varying at different seasons, from 2\ to 9 degrees. It 
 corresponds with the equator of temperature, and hence varies 
 in position, 474, a few degrees, but is always on the northern 
 side of the equator. Within this narrow zone the winds are 
 subject to no regular law, and hence it is said to be the seat of 
 the Variables. In this space the velocities of the currents pro- 
 ceeding in opposite directions destroy each other, and an accu- 
 mulation, as has been stated, would take place, did not the air 
 rise and join the counter-current, that continually flows in the 
 higher regions. 
 
 At the outer limits of the regular trades, it might be inferred 
 that the descent of the counter-current would form a narrow zone. 
 
496 THE WINDS. [Book VI. 
 
 of winds uncertain in direction, and generally light ; such a zone 
 is distinctly marked, and well known in the North Atlantic 
 Ocean. Whether it be found in the Pacific and South Atlantic 
 Oceans cannot be stated, for the want of careful and sufficient 
 recorded observations. 
 
 The trade winds, as may be inferred from this theory, prevail 
 in the open ocean, in the Atlantic and Pacific, between the lati- 
 tudes of 30 N. and 27 S. On entering them from either side, 
 the deviation, growing out of the rotation of the earth, towards 
 the east, is greatest ; and this deviation becomes less and less as 
 the equatorial zone is approached ; in the immediate vicinity of 
 this zone, the wind is nearly due N. on the northern side of the 
 equator, and nearly due S. on the southern. 
 
 483. In the Indian Ocean, winds changing their direction half 
 yearly, and blowing regularly in each direction for nearly six 
 months, are experienced. These periodic winds are called the 
 Monsoons. The cause of them is to be found in the position of 
 this ocean in respect to the adjacent continents. 
 
 To the north of the Indian Ocean extends the whole mass of 
 the old continent, with the exception of the southern extremity 
 of Africa. The ocean and the land, thus placed, are acted upon 
 by the sun, in his annual course, with different degrees of inten- 
 sity at different seasons. In the summer of northern latitudes, 
 the sun is vertical over large portions of the continent, and ac- 
 cording to the principles of 476, the superficial heat of the 
 land being more speedily raised, even by an equal exposure to 
 the sun, becomes greater than that of the ocean. The denser air 
 at the surface of the ocean therefore presses towards the land, 
 causing a current whose absolute motion is from south to north. 
 On the northern side of the equator, coming from a point whose 
 velocity of rotation is greater than that of the points it meets in 
 its course, it has an apparent deflection towards the west and 
 forms a S. W. wind. Hence in the Indian Ocean, the south- 
 western monsoon blows between the months of April and No- 
 vember, on the north side of the equator. 
 
 The causes that produce the S. W. monsoon, also operate on 
 the southern side of the equator, as far as 11 S. The current 
 they cause, pressing N. towards the Equator, from a parallel 
 that has a less velocity of rotation, appears as a S. E. wind. 
 
 When the sun is on the southern side of the equator, the old 
 continent, losing by radiation more heat than it receives, becomes 
 colder than the Indian Ocean. The air above it, therefore, 
 presses to the south, and the influence extends as far as 11 S. 
 For reasons the converse of the preceding, the apparent direction 
 becomes N. E. on the northern side of the equator, and N. W 
 
Book PL] THE WINDS. 4i>7 
 
 on the southern. To the south of this parallel the regular S. E. 
 trade wind blows continually, in the Indian as well as in the other 
 Oceans. 
 
 484. The most important modifications of the trade winds, 
 growing out of local circumstances, are as follows : 
 
 The continent of Africa, over which the sun is continually 
 vertical, is always heated at the surface, for reasons already as- 
 signed, 476, to a temperature higher than the adjacent ocean. 
 Hence, in the Gulf of Guinea, a wind sets almost constantly to- 
 wards the land, and is modified in its direction by the tending of 
 the shore. Between the region in which this sea breeze blows, 
 and that in which the trade winds begin again to prevail, these 
 two winds, diverging from the same space, cause an exhaustion, 
 which is supplied by a counter-current in higher parts of the at- 
 mosphere. Within this interval there is a portion of the surface 
 of the ocean that is the seat of almost perpetual calms. 
 
 The course of the trade winds is interrupted by the continent 
 of America, hence their influence is not felt until at some dis- 
 tance from the coast of the Pacific Ocean. 
 
 Upon the eastern coast of North America, in the summer sea- 
 son of the respective hemispheres, the greater heat of the land 
 draws a current from the ocean ; by this the extent of the trade 
 wind is increased. The course of this part is, however, E. or 
 even S. E. on the coast of Florida, Georgia, &c. 
 
 The monsoons in the neighbourhood of the land, have their 
 courses deflected also, and sometimes their influence merges alto- 
 gether in the land and sea breeze. 
 
 485. Between the parallels of 30 and 40, in both the North 
 and South Pacifies, a westerly wind blows almost constantly ; 
 intermitting only for a short space of time, after each equinox, 
 when a regular distribution of temperature, over the whole earth, 
 gives rise to the N. E. and S. E. trade winds. 
 
 In the Northern Atlantic this wind is not constant, in conse- 
 quence of the comparative want of breadth of that ocean, by 
 which it is subjected to the influence of the contiguous continents. 
 A westerly wind is, however, the prevailing wind in this ocean, 
 except in the months of April and October, when a N. E. wind 
 is more frequent. 
 
 The cause of the existence of such westerly currents may be 
 thus explained. 
 
 Within the tropics, and to a short distance beyond them, the 
 variations of temperature, from a law of regular decrease from 
 the equator towards the poles, are so small as to be insensible, 
 and hence, as has been stated, the trade winds are constant within 
 
 63 
 
498 THE WINDS. [Book VI. 
 
 certain limits. Without these limits, the parallels are alternately 
 warmer and colder, according to the reason, than would be con- 
 sistent with the law of regular decrease. In some one parallel, 
 the deviation from this law will be the greatest. This may be 
 taken as about the parallel of 40, in which, as may be seen from 
 the examples of New-York and Pekin, the vicissitudes of tem- 
 perature are excessive. When this parallel is more heated than 
 is consistent with the law of the mean temperatures, the course 
 of the great current from the poles towards the equator is inter- 
 rupted, the atmosphere in contact with the surface of the earth 
 will be accelerated on the side of this parallel nearest to the pole ; 
 on the side nearest the equator, the air increases in density, and 
 hence moves in a direction contrary to that which it would have 
 if the temperature decreased regularly towards the poles. To 
 counteract the condensation that would hence arise in the paral- 
 lel of 40, a counter-current takes place. The lower current 
 coming from a parallel whose velocity of rotation is greater than 
 that which it reaches, is apparently impressed with a motion 
 from W. to E. and becomes a S. VV. wind on the north side of 
 the equator, and N. W. on its southern side. 
 
 When this parallel has a lower temperature than is consistent 
 with the law of regular decrease, it has been demonstrated by 
 Daniell, that the atmospheric pressure would be diminished ; 
 for this reason a current would set toward it on both sides, in or- 
 der to restore the equilibrium, and thus the two causes so different 
 in themselves, will produce similar effects, and winds deflected 
 towards the west will again take place. 
 
 The slow changes that take place in the temperature of the sur- 
 face of the ocean, growing out of the causes stated in 476, make 
 the parallel in which these opposite influences operate to produce 
 this effect, nearly constant in position. And it is for a similar 
 reason, that the monsoons do not vary gradually in intensity and 
 direction with the declination of the sun, but intermit wholly 
 for a time, and then assume the new direction. 
 
 The interval of the monsoons is attended with great oscillations 
 in the atmosphere; great accumulations tnke place in some places, 
 attended with corresponding rarefactions in others ; these mu- 
 tually re-act upon each other ; thus violent storms, the Typhoons 
 of the Indian seas, occur in the interval of the monsoons. 
 
 486. Upon the continents, thechangesoftemperature from day 
 to day, and the alternations of heat from day to night, are rapid and 
 frequent; hence there is no constancy in the direction or inten- 
 sity of the winds. In high latitudes, even in the open sea, simi- 
 lar inequalities occur. Hence, the land and ocean, in lati- 
 tudes higher than 40, are the seat of winds that can be reduced 
 
Rook VL~\ THE WINDS. 499 
 
 to no fixed laws, and the frequency of whose changes increases 
 with the latitude. 
 
 In the blowing of these variable winds, the inertia of the air 
 tends to cause accumulations in the parts towards which they 
 blow, and expansions in those whence they come ; the elastic 
 nature of the air allows these to increase, until the moving force 
 is destroyed, when a returning current is formed which will again 
 cause similar condensations and exhaustions. Thus the varia- 
 tions in the height of the barometer, which have been noted in 
 385, become of greater extent in high, than in low latitudes; 
 and when winds have gradually expended their force, a wind in 
 a direction exactly contrary often succeeds. 
 
 Although the variable winds of temperate climates are subject 
 to no fixed laws, still we may often find in the local circum- 
 stances of countries, reasons why certain winds should blow more 
 frequently than others. Such winds are called the Prevailing 
 Winds, of the particular climate. 
 
 In the northern and middle portions of the seaboard of the Uni- 
 ted States, the great prevailing winds are the N. W., the N. E ,the 
 S. W., and the S. E. By an attentive examination of the cir- 
 cumstances of the country, we may easily show why these should 
 be of frequent occurrence, and probably prevail to the exclusion of 
 all others. 
 
 A great current of the ocean called the Gulf Stream, proceeds 
 from the Gulf of Mexico, and runs nearly parallel to the coast of 
 North America, as far as the banks of Newfoundland. This 
 current during the winter months, is much warmer than the 
 neighbouring continent ; hence a current of air frequently sets 
 from the land towards the ocean, which forms the N. W. wind 
 of the United States. 
 
 In summer, although the land becomes warmer than the Gulf 
 Stream, the great difference of temperature between the sea- 
 board and the interior, in which at. the lat. of 60, and at the 
 depth of six feet beneath the surface, the ground is entirely frozen, 
 is sufficient to account for the N. W. being a frequent wind. 
 
 The trade winds are interrupted in their course by the great 
 chain of mountains that traverse nearly the whole continent of 
 America. This interruption causes an accumulation of air 
 against their sides. It cannot be lessened by a return in the di- 
 rection whence it came, in consequence of the perpetual current 
 of the trade winds. It will, therefore, when no other cause for 
 a prevailing wind exists, press over the whole continent; and, if 
 the accumulation have been going on fora long time, will exert a 
 force that no other wind of our climate does. 
 
 TheN. E. wind of the United States, as well as that of Europe, 
 
500 THE WINDS. [Book VI. 
 
 maybe considered as the great current, directed towards the equa- 
 tor, and exerting its influence when no other cause is in action. 
 That it should be frequent, may be explained from the fact, that 
 the cleared and cultivated portions of the United States, will often 
 be under the circumstances of a regular increase of temperature, 
 from the N. E. to the S. W. This wind was formerly confined 
 to a strip extending but a few miles from the coast. As the coun- 
 try has been cleared, its influence has been more extended, and it 
 is annually becoming more prevalent. 
 
 If a N. W. wind have prevailed for some time, as a current of 
 air must, according to the law of lateral communication of mo- 
 tion, follow the direction of the Gulf Stream, an accumulation 
 must take place over the latter, which, seeking to restore the equi- 
 librium of pressure, presses towards the continent, and causes a 
 S. E. Wind. 
 
 487. The land and sea breezes are winds whose period is a 
 single day, the former prevailing when the sun is below, the latter 
 when he is above the horizon. The causes are to be sought in the 
 different manner, in which land and water are affected by radia- 
 tion, and the direct heat of the sun. During the day, the surface 
 of the land, as will be seen by reference to 170, becomes more 
 heated than that of the adjacent ocean. Hence a current begin- 
 ning at some hour in thannorning, and continuing until the sun 
 is near setting, will flow from the water towards the land. At 
 night the water remains warm, while the surface of the land cools 
 rapidly, and hence the current sets from the land towards the 
 water. 
 
 Of all the winds in the climate of New- York, a north wind is 
 perhaps the most rare. It however sometimes blows for two or 
 three days together, and is remarkable for the extreme heat with 
 which it is attended. The cause of this high temperature seems 
 to be, that a north wind prevents the action of the sea breeze, that 
 would otherwise act to temper the climate, and moderate its vicis- 
 situdes. 
 
 The variable winds of temperate climates, as maybe seen from 
 what has been said, arises either from a condensation in some part 
 of the atmosphere, or a rarefaction in others, or both may concur. 
 
 When the former is the cause, the wind proceeds forward, ex- 
 tending itself in the direction towards which it appears to blow. 
 Such is the case with the N. W. wind of our climate. 
 
 Within the tropics, the islands and countries situated upon 
 the sea-coast, have their climates tempered by winds styled the 
 Land and Sea Breezes. The sea breeze begins to blow three or 
 four hours after sunrise, and continues to blow until a little after 
 sunset. The land breeze commences three or four hours after 
 
Book VI.} THE WINDS. 501 
 
 sunset, and continues until about sunrise. The cause of these 
 alternating winds may be thus explained : The land and sea, 
 being both equally exposed to the action of the sun during the 
 day, the former as explained in 476, becomes more heated at 
 the surface than the latter. The air in contact with the land, in 
 consequence expands and rises, while that over the sea presses in 
 to supply the place. At night, the surface of the land parts with 
 its heat most rapidly, and the course of the current is reversed. 
 
 These winds are not confined to the tropics, but may be ob- 
 served on the sea-coast of countries situated in latitudes as high 
 as 45, during the summer of the hemisphere in which they are 
 situated. Thus, at New- York, a sea breeze is experienced almost 
 daily during the months of June, July, and August ; and a land 
 breeze may be occasionally observed during the same months. 
 
 When the latter cause occurs, the wind will appear to blow first 
 in the quarter towards which it is directed. Such is the N. E. 
 wind of our climate, which begins to blow in Florida many hours 
 before it is felt at Boston. 
 
 The manner in which a wind of this character may arise, and 
 thus extend itself to witidward, may be illustrated by referring to 
 what occurs on opening the gate of a sluice, in which case the 
 current sets towards the opening ; but the motion begins in im- 
 mediate contact with the sluice, and is propagated in a direction 
 contrary to the current. 
 
502 MOTION OF VAPOUR IN [Book VI. 
 
 CHAPTER XIV. 
 
 OF THE MOTION OP VAPOUR IN THE ATMOSPHERE. 
 
 488. It will be seen by reference to 370, that water forms 
 vapour at all temperatures whatsoever, of a tension and density 
 having relation to the temperature, according to the tables of 374 
 and 379. Now, as a great portion of the surface of the earth is 
 covered with that liquid, it follows, that vapour will rise from 
 it, and by the general property of elastic fluids to form atmos- 
 pheres independent of each other, will tend to distribute itself 
 over the surface of the earth ; and it would assume the tension and 
 elasticity due to the temperature of the space it occupies, did no 
 opposing force act to prevent it. 
 
 489. Were the earth a sphere, wholly covered with water, of 
 uniform temperature throughout its surface, and if we suppose 
 the aerial atmosphere not to exist, the water would form an atmos- 
 phere of vapour, whose pressure would be equal to the elastic 
 force of vapour at the constant temperature of the surface. The 
 temperature of this aqueous atmosphere would not be uniform 
 throughout, but would be so only at the surface, for the higher por- 
 tions undergoing less pressure would expand, and their tempera- 
 ture would diminish to that corresponding to the tension and den- 
 sity of the vapour at the given point. To take an instance : Were 
 a sphere whose surface is wholly covered with water, and which 
 has no aerial atmosphere, to have at every point on its surface the 
 temperature of 32 ; an atmosphere of vapour would be formed 
 around it, whose tension would be 0.2 inches, and whose tem- 
 perature at the surface would be 32. At the altitude of 30,000 
 feet it may be calculated that the tension would be diminished 
 one-half, or to 0. 1 inch, and the temperature of the vapour to 
 13. 
 
 The atmosphere of vapour would be in perfect equilibrium, and 
 at rest, over the whole surface of the sphere ; and would, by its 
 pressure, prevent the formation of any more vapour. No preci- 
 pitation would occur in any part, and the whole mass would be 
 clear and transparent. 
 
 An uniform increase of the temperature of the surface, would 
 cause the formation of new vapour ; the tension of the whole 
 would become uniform ; the temperature of its lower parts would 
 be the same as that of the surface of the sphere, and an analogous 
 decrease of temperature and tension would take place at increas- 
 ing elevations. 
 
-- ; " ***'"' * : 
 
 ; 
 
 Book K/.] ,THE ATMOSPHERE. 503 
 
 " i-' 
 
 490. But if the temperature of the sphere were to become 
 unequal, the circumstances would be different. If we assume it 
 to follow the law of its mean temperature, being warmest at the 
 equator, and to decrease in heat according to some definite law, 
 from the equator to the poles ; the tension of the vapour over the 
 whole surface, would be due to the minimum temperature, 372, 
 or to that of the poles; but the. evaporation being due to the heat 
 of the different points on the surface, would be determined by 
 that heat, and go on continually ; while at the poles, an equal and 
 rapid condensation would take place. The vapour would in this 
 case, flow in mass, from the equator towards the poles, and the 
 precipitation at the latter points would raise the level of the ocean, 
 until currents were formed, by which all the condensed water 
 would flow back to the equator. 
 
 491. If some retarding force were to act, by which the flow of 
 vapour is resisted in its course from the equator towards the 
 poles ; the precipitation would be distributed throughout the 
 whole sphere, except at the zone of greatest temperature. Con- 
 tinual evaporation would go on at the equator; continual pre- 
 cipitation at the poles, and both evaporation and precipitation at 
 all intermediate latitudes. 
 
 Such a retarding force is to be found in the aerial atmosphere. 
 The vapour, although it constantly tends to form an atmosphere, 
 according to its own mechanical laws, is resisted in its motions 
 by the aerial atmosphere through which it is compelled as it 
 were to filter ; and thus, were the circumstances of which we 
 have already spoken, to be all that affect the mixed mass of air 
 and vapour, condensation would be taking place in the higher re- 
 gions at all latitudes, attended at the same time with evaporation 
 from beneath. 
 
 492. The relations of air under different pressures to heat, are 
 different from that of vapour, and the temperature of an aerial at- 
 mosphere diminishes much more rapidly with increasing eleva- 
 tions, than that of an aqueous atmosphere; here then, we find 
 upon the principles of 368, a new cause of precipitation, which 
 would in the higher regions be attended with a corresponding in- 
 crease in the evaporation from beneath. 
 
 If then the earth were wholly covered with water, continual 
 rains, or at least perpetual clouds, would be experienced every 
 where but at the equator. 
 
 493. As, however, rather more than one-fourth part of the 
 earth's surface is dry land ; this produces a very marked change 
 in the circumstances of the atmosphere. The land furnishes a 
 comparatively small quantity of vapour. Hence, as vapour tends 
 
504 MOTION OF VAPOUR IN [Book VI. 
 
 \ 
 
 to fojrra an atmosphere of itself, distributed according to the rela- 
 tions of temperature and tension, over the whole surface, the va- 
 pour formed over the surface of the ocean would continually press 
 towards the land, until a state of equilibrium could be attained. 
 
 If the land be warmer than the ocean, the vapour would be 
 heated above its original temperature, and a greater quantity by 
 weight, could exist without deposition in a gvven space; hence, 
 the vapour that might otherwise be precipitated over the ocean, 
 ' would be diverted towards the land, and even there no deposit 
 might ensue. 
 
 If the land be colder than the adjacent ocean, vapour will still 
 flow towards it, but it will now be condensed upon it, and a part 
 at least of the condensation that would otherwise take place upon 
 the ocean, will take place upon the land. 
 
 The ocean has a temperature far less variable than that of the 
 land, and thus both will be affected with alternations of rain and 
 sunshine, according to the relations between their temperatures. 
 In the temperate and frigid zones, these will be subject to no 
 fixed laws ; but in the torrid zone, seasons of considerable length 
 will be wet or dry, according to the latitude of the place and the 
 declination of the sun. 
 
 494. The flow of the vapour, in conformity with its own me- 
 chanical laws, is not only retarded by the mere resistance of the 
 atmosphere, but is affected by the winds. When the course of 
 the wind coincides with the direction the vapour would assume 
 under its own pressure, the flow of the vapour is accelerated ; 
 when the. contrary is the case, it is retarded ; and it may thus 
 happen, that some districts of the continents are wholly deprived 
 of moisture, while others receive a superabundant proportion. 
 
 Winds also agitate and mix together masses of air of different 
 temperatures, and when these contain moisture, of a tension ap- 
 proaching to the maximum due to their respective temperatures, 
 precipitation must almost always ensue. That this must be the 
 case, will appear from the consideration, that the quantity of va- 
 pour that can exist in a given space, varies in geometric progres- 
 sion, while the temperature varies in arithmetic ; and the tempera- 
 ture that results from the mixture of equal masses of air, is the 
 arithmetic mean of their respective temperatures. As the latter 
 is always greater than the geometric mean, the quantity of vapour, 
 if both masses of air approached to saturation, will be greater than 
 is consistent with the resulting temperature, and the excess must 
 be precipitated. 
 
 495. It will be obvious, from what has been stated, that vapour 
 is in almost all cases pressing from the ocean towards the land ; 
 
Book F7.J -THE ATMOSPHERS. f-,* 405 
 
 while upon the latter a precipitation must ensue, often greats* 
 than upon an equal surface of the ocean. In this excess of pre- 
 cipitation we are to seek the origin of springs and rivers ; by ihe 
 latter this excess is restored to the sea, to be again evaporated, 
 and thus keep up the continual circulation. 
 
 496. So long as aqueous matter remains in the state of vapour, 
 it is transparent. On its first condensation a cloud appears. The 
 manner of the formation of .clouds is as follows : Water on its 
 first condensation tends to unite in the form of hollow globules, 
 or vesicles containing air; as it parts at the same time with its 
 latent heat, the air, as well within the vesicles as between them, 
 is rarefied, and the united mass of water and rarefied air may re- 
 main as light as an equal bulk of atmospheric air, or even lighter. 
 Clouds may therefore remain floating in the atmosphere, or even 
 rise. As this heat is dissipated, the clouds grow heavier and fall, 
 while the air in the vesicles losing its elasticity, permits them 
 to be broken by the internal pressure. The water then runs into 
 drops, which, being many times heavier than atmospheric air, 
 descend, forming Rain. 
 
 Clouds may be formed in all cases where the temperature of 
 the ground is lower than that at which the vapour, mixed with 
 atmospheric air, can remain permanent. Thus: whenever a 
 warm wind flows over a cold surface, mists and fogs take place ; 
 and if the difference of temperature be considerable, they may 
 break into rain. For an equal difference of temperature between 
 the ground and air, it may be shown by calculations formed on 
 the table of 308, that the greatest quantity of precipitation will 
 take place, when the two unequal temperatures are both high. 
 Thus the causes that would produce heavy rains in warm climates, 
 may produce no more than fogs, or dense mists, in those that are 
 colder. 
 
 Clouds may also be formed on sudden changes of wind, upon 
 the principle explained in 491, when two masses of air are 
 mixed that are both nearly saturated with moisture. It is to this 
 cause that nearly all the rains of temperate climates are due. The 
 passing of warm winds over cold surfaces, rarely produces more 
 than mists or fogs, except in warm climates. 
 
 When clouds, after being formed, begin to descend, in conse- 
 quence of the dissipation of the heat, by the rarefaction arising 
 from which they are supported, they often reach strata of the 
 atmosphere comparatively dry, and of higher temperature than 
 they themselves possess. In such a case, the vapour may be again 
 taken up, and the cloud dissipated. Thus clouds are frequently 
 seen to roll down the sides of the mountains, and to disappear at 
 a certain level ; this is a proof of a dry state of the air beneath, 
 
 64 
 
506 MOTION OP VAPOUR IN [Book VI. 
 
 and is therefore considered by the inhabitants of mountainous 
 eountries, as a prognostic of fair weather. 
 
 When a cloud, on the other hand, formed in high and cold re- 
 gions, passes in its descent through strata saturated with moisture, 
 or nearly so, it may cool them until precipitation ensue; the pre- 
 cipitated moisture, uniting itself to the descending cloud, will 
 augment the intensity of the rain it causes. Thus the same rain 
 will be more copious in vallies, than upon the neighbouring moun- 
 tains ; and the difference is so sensible^n this respect, that it has 
 been detected by means of the rain-gauges at the observatory of 
 Paris, one of which is upon the ground, the other upon the ter- 
 raced roof of the building. 
 
 497. When the precipitation of vapour ensues at temperatures 
 below the freezing point, Snow is formed ; the particles of the con- 
 densed aqueous matter being free to move in any direction, ar- 
 range themselves under the action of their mutual attraction, in 
 the manner of crystals. These crystals have usually the figure of 
 six-pointed stars ; and the aggregation of broken crystals of this 
 shape forms flakes of snow. 
 
 498. Hail is a phenomenon that is not completely explained; 
 the best theory on the subject, although not absolutely satisfactory, 
 is as follows : 
 
 It is known that when water is frozen in a torricellian vacuum, 
 it granulates and assumes the form of hail ; hail also reaches the 
 ground with a very great velocity : hence we may conclude, that 
 it is formed in very rare air, and in a high region of the atmos- 
 phere. The decomposition of organic substances, is constantly 
 giving out hydrogen gas, and this, from its specific levity, rises 
 to the higher regions of the atmosphere ; hence, as no gas can re- 
 main long over another unmixed, it mingles with atmospheric air, 
 and becomes susceptible of hoing inflamed by electricity. Should 
 it be thus acted upon, it forms water, will be condensed into a 
 space much less than it formerly occupied, and would leave a va- 
 cuum, did not the adjacent portions of air rush in to fill the void. 
 The sudden rarefaction of this air will produce an intense cold ; 
 the newly formed water will be frozen, and under circumstances 
 that will cause it to granulate ; descending from a lofty region, it 
 will have great velocity ; formed from hydrogen gas, and by the 
 electric discharge, it will occur most frequently during the sum- 
 mer months, and accompany lightning. 
 
 499. It was shown in the preceding chapter, that the earth retains 
 a constant mean temperature, under the joint action of solar and 
 terrestrial radiation ; but that the rate of these is unequal, not only 
 at different seasons, but from hour to hour ; the former ceases alto- 
 
Book VL~\ THE ATMOSPHERE. 507 
 
 gether at the setting of the sun, while the latter continues for a 
 time undiminished. Hence the surface of the earth cools rapidly 
 after sunset, and may speedily reach the dew-point of the air in 
 contact with it. So soon as this is the case, moisture begins to 
 be precipitated, and a cloud is formed, the descent of the water of 
 which this is composed, forms the deposit, that we call Dew. The 
 cooling will be propagated slowly upwards, and the cloud will ap- 
 pear to rise ; notwithstanding which, the moisture of which it is 
 composed, actually falls. After some hours, the earth and air 
 will assume the same temperature, and the cloud will disappear. 
 
 The first morning rays of the sun, passing horizontally through 
 the air, will heat it, long before their influence can be felt upon 
 the ground. The air will therefore acquire a greater capacity for 
 moisture : if there be any water in the vicinity, vapour will rise, 
 and propagate itself through the mass ; but as the ground still re- 
 mains colder, a new precipitation will ensue ; thus dew will again 
 be formed, and moisture occur in the morning. 
 
 500. When the surface of the ground, or of any other substance, 
 is cooled by radiation to the temperature of 32, the dew is frozen, 
 and takes the form of white or hoarfrost; this may often be de- 
 posited, when the temperature both of the air and of the ground 
 at a very small depth, is above that of freezing. 
 
 501. When clouds exist in the atmosphere, the radiation is im- 
 peded, and dew will not be formed. Thus a want of dew is 
 usually a prognostic of rain. 
 
 When the air is still, dew is most copious, and thus it falls in 
 greatest abundance in sheltered situations, and frosts will continue 
 later in the spring, and begin earlier in autumn, in vallies than on 
 the open hills in the vicinity. 
 
 The motion of air mixes the portion cooled by contact with the 
 earth, with that which is not, and brings new masses into contact ; 
 hence, although the loss of heat by radiation, may be as great or 
 even greater, the ground will receive heat from the air, and the 
 change of temperature will be less. In conformity, heavy dews 
 do not fall during the prevalence of high winds, and hoar frosts 
 rarely occur while they blow. 
 
 Surfaces that radiate well, will be most cooled, and will in con- 
 sequence receive the greatest quantity of dew ; and thus of land 
 frequently tilled, and that which is left undisturbed, the latter will 
 derive most moisture from the atmosphere in this form. 
 
 502. In the prosecution of this subject, we have in some degree 
 trespassed upon the limits of pure physical science : this was how- 
 ever necessary, in order to give a correct view of the phenomena, 
 but the discussion is incomplete from the propriety of confining 
 ourselves as closely as possible to what is strictly mechanical. 
 
508 CONCLUSION. [Book VI. 
 
 So also in a variety of other cases, it has been judged expe- 
 dient to extend our investigations into collateral branches of 
 knowledge, while subjects of no small extent, and more imme- 
 diately connected with Mechanics, have been passed over. This 
 extension on the one hand, and omission on the other, have been 
 determined by a view to the practical applications of our subject. 
 These applications were originally intended to have formed a 
 part of the work, and by means of them its true scope and ob- 
 jects would have become apparent. In its present form, how- 
 ever, the author trusts it will be found an useful introduction to 
 the study of a most important and interesting branch of science, 
 whether it be considered in its immediate connexion with the 
 arts, or in its bearing upon knowledge of a more elevated char- 
 acter. 
 
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